OpenSees Command Language Manual
Below you will find brief information for OpenSees Command Language. This manual outlines commands for earthquake engineering simulation, focusing on numerical simulation of nonlinear structural and geotechnical systems. Covers static, reversed-cyclic, and dynamic time-series analyses.
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Open System for Earthquake Engineering Simulation (OpenSees) OpenSees Command Language Manual Silvia Mazzoni, Frank McKenna, Michael H. Scott, Gregory L. Fenves, et al. Printed on 19 July, 2006 iii Contents Introduction 11 Notation ........................................................................................................................................11 Copyright ......................................................................................................................................13 Introduction to the Tcl command language ..................................................................................14 Tcl Commands Format.................................................................................................................15 Example Tcl Commands ..............................................................................................................16 Additional Tcl Resources .............................................................................................................18 OpenSees Interpreter...................................................................................................................19 OpenSees 20 OpenSees Features .....................................................................................................................22 WHY OPENSEES? ......................................................................................................................22 + The library of materials, elements and analysis commands makes OpenSees a powerful tool for numerical simulation of nonlinear structural and geotechnical systems................22 + The OpenSees library of components is ever-growing and at the leading edge of numericalsimulation models ..............................................................................................................22 + The OpenSees interface is based on a command-driven scripting language which enables the user to create more-versatile input files.......................................................................22 + OpenSees is not a black box, making it a useful educational tool for numerical modeling22 + You can create your own material, element or analysis tools and incorporate them into OpenSees ..........................................................................................................................22 + NEES is supporting integration of OpenSees as the simulation component of laboratory testing.................................................................................................................................22 MODELS: .....................................................................................................................................23 + linear & nonlinear structural and geotechnical models ...................................................23 SIMULATIONS: ............................................................................................................................24 + static push-over analyses ...............................................................................................24 + static reversed-cyclic analyses .......................................................................................24 + dynamic time-series analyses.........................................................................................24 + uniform-support excitation...............................................................................................24 + multi-support excitation ...................................................................................................24 MODELING FEATURES: .............................................................................................................24 MODEL-BUILDING FEATURES: .................................................................................................25 ANALYSIS FEATURES:...............................................................................................................28 ModelBuilder Object .....................................................................................................................31 Domain Object..............................................................................................................................32 Recorder Object ...........................................................................................................................33 Analysis Object.............................................................................................................................34 Model-Building Objects 36 model Command 37 Basic Model Builder......................................................................................................................37 build Command ............................................................................................................................38 node Command 39 mass Command 40 Contents constraints objects iv 41 Single-Point Constraints...............................................................................................................41 fix Command ................................................................................................................................41 fixX Command..............................................................................................................................42 fixY Command..............................................................................................................................43 fixZ Command ..............................................................................................................................43 Multi-Point Constraints .................................................................................................................44 equalDOF Command ...................................................................................................................44 rigidDiaphragm Command ...........................................................................................................45 rigidLink Command ......................................................................................................................46 uniaxialMaterial Command 47 -- Contributed Uniaxial Materials 48 BARSLIP Material ........................................................................................................................48 Concrete04 Material -- Popovics Concrete Material ....................................................................61 Concrete04 -- Material Behavior 65 FatigueMaterialExample.tcl 67 Limit State Material 77 Shear Limit Curve.........................................................................................................................78 Axial Limit Curve ..........................................................................................................................79 Shear Limit Curve.........................................................................................................................80 Example Files for Limit State Material..........................................................................................82 PINCHING4 Material ....................................................................................................................88 PINCHING4 Uniaxial Material Model Discussion.........................................................................95 PyTzQz Uniaxial Materials .........................................................................................................109 PySimple1 Material ....................................................................................................................109 TzSimple1 Material.....................................................................................................................110 QzSimple1 Material ....................................................................................................................111 PyLiq1 Material...........................................................................................................................112 TzLiq1 Material...........................................................................................................................113 PySimple1Gen Command..........................................................................................................115 TzSimple1Gen Command..........................................................................................................116 ReinforcingSteel -- Material Behavior 131 Elastic Material ...........................................................................................................................131 Elastic-Perfectly Plastic Material ................................................................................................132 Elastic-Perfectly Plastic Gap Material ........................................................................................134 Elastic-No Tension Material .......................................................................................................136 Parallel Material..........................................................................................................................137 Series Material ...........................................................................................................................138 Hardening Material .....................................................................................................................140 Concrete01 Material -- Zero Tensile Strength............................................................................141 Concrete01 -- Material Behavior 145 Concrete02 Material -- Linear Tension Softening ......................................................................145 Concrete02 -- Material Behavior 152 Concrete03 Material -- Nonlinear Tension Softening................................................................152 Concrete03 -- Material Behavior 157 Contents v Steel01 Material .........................................................................................................................157 Steel01 -- Material Behavior 164 Steel02 Material -- Giuffré-Menegotto-Pinto Model with Isotropic Strain Hardening .................165 Steel02 -- Material Behavior 170 Hysteretic Material......................................................................................................................170 Hysteretic -- Material Behavior 173 Viscous Material .........................................................................................................................173 nDMaterial Command 175 Elastic Isotropic Material ............................................................................................................175 J2 Plasticity Material...................................................................................................................176 Plane Stress Material .................................................................................................................176 Plate Fiber Material ....................................................................................................................177 Template Elasto-Plastic Material................................................................................................177 FluidSolidPorousMaterial Material .............................................................................................182 updateMaterialStage ..................................................................................................................184 section Command 185 Elastic Section............................................................................................................................187 Uniaxial Section..........................................................................................................................187 Fiber Section ..............................................................................................................................189 Fiber Command..........................................................................................................................191 Quadrilateral Patch Command...................................................................................................192 Circular Patch Command ...........................................................................................................194 Straight Layer Command ...........................................................................................................196 Circular Layer Command ...........................................................................................................197 Section Aggregator.....................................................................................................................199 Elastic Membrane Plate Section ................................................................................................202 Plate Fiber Section .....................................................................................................................202 Bidirectional Section...................................................................................................................203 Isolator2spring Section: Model to include buckling behavior of an elastomeric bearing ..........204 element Command 213 Contents vi Truss Element ............................................................................................................................213 Corotational Truss Element........................................................................................................214 Elastic Beam Column Element...................................................................................................215 NonLinear Beam-Column Elements...........................................................................................216 Nonlinear Beam Column Element..............................................................................................216 Beam With Hinges Element .......................................................................................................218 Displacement-Based Beam-Column Element............................................................................222 Zero-Length Elements................................................................................................................222 Zero-Length Element..................................................................................................................223 Zero-Length ND Element ...........................................................................................................224 Zero-Length Section Element ....................................................................................................225 Quadrilateral Elements...............................................................................................................226 Quad Element ............................................................................................................................226 Shell Element .............................................................................................................................227 Bbar Plane Strain Quadrilateral Element ...................................................................................227 Enhanced Strain Quadrilateral Element.....................................................................................228 Brick Elements ...........................................................................................................................229 Standard Brick Element..............................................................................................................229 Bbar Brick Element.....................................................................................................................231 Eight Node Brick Element ..........................................................................................................232 Twenty Node Brick Element.......................................................................................................234 u-p-U element.............................................................................................................................236 FourNodeQuadUP Element .......................................................................................................237 BeamColumnJoint Element........................................................................................................238 Beam-Column Joint Element Discussion ...................................................................................243 Beam-Column Joint Element Discussion 243 Soil Models and Solid-Fluid Fully Coupled Elements 264 PressureDependMultiYield ..............................................................................................265 PressureDependMultiYield02 ..........................................................................................272 PressureIndependMultiYield ............................................................................................277 updateMaterialStage ........................................................................................................283 updateParameter .............................................................................................................284 FluidSolidPorousMaterial .................................................................................................285 FourNodeQuadUP ...........................................................................................................286 Nine_Four_Node_QuadUP..............................................................................................288 BrickUP ............................................................................................................................290 Twenty_Eight_Node_BrickUP..........................................................................................292 block Command 295 block2D Command.....................................................................................................................296 block3D Command.....................................................................................................................298 region Command 300 Geometric Transformation Command 302 Linear Transformation ................................................................................................................302 P-Delta Transformation ..............................................................................................................308 Corotational Transformation.......................................................................................................309 Time Series 311 Contents vii Constant Time Series.................................................................................................................312 Linear Time Series .....................................................................................................................313 Rectangular Time Series............................................................................................................314 Sine Time Series ........................................................................................................................315 Path Time Series........................................................................................................................316 pattern Command 317 plain Pattern ...............................................................................................................................317 load Command ...........................................................................................................................318 sp Command ..............................................................................................................................319 eleLoad Command .....................................................................................................................320 UniformExcitation Pattern ..........................................................................................................321 MultipleSupport Pattern..............................................................................................................322 groundMotion Command............................................................................................................323 Plain GroundMotion....................................................................................................................323 Interpolated GroundMotion ........................................................................................................324 imposedMotion Command .........................................................................................................324 Recorder Objects 326 Node Recorder ...........................................................................................................................326 EnvelopeNode Recorder ............................................................................................................328 Drift Recorder .............................................................................................................................329 Element Recorder ......................................................................................................................330 EnvelopeElement Recorder .......................................................................................................332 Display Recorder........................................................................................................................333 Plot Recorder .............................................................................................................................334 playback Command....................................................................................................................335 Analysis Objects 336 constraints Command 339 Plain Constraints ........................................................................................................................341 Penalty Method ..........................................................................................................................341 Lagrange Multipliers ...................................................................................................................342 Transformation Method ..............................................................................................................343 numberer Command 344 Plain Numberer ..........................................................................................................................344 RCM Numberer ..........................................................................................................................345 system Command 346 BandGeneral SOE......................................................................................................................347 BandSPD SOE ...........................................................................................................................347 ProfileSPD SOE .........................................................................................................................347 SparseGeneral SOE...................................................................................................................347 UmfPack SOE ............................................................................................................................348 SparseSPD SOE ........................................................................................................................348 test Command 349 Norm Unbalance Test ................................................................................................................349 Norm Displacement Increment Test...........................................................................................350 Energy Increment Test...............................................................................................................351 algorithm Command 352 Contents viii Linear Algorithm .........................................................................................................................352 Newton Algorithm .......................................................................................................................352 Newton with Line Search Algorithm ...........................................................................................353 Modified Newton Algorithm ........................................................................................................354 Krylov-Newton Algorithm............................................................................................................354 BFGS Algorithm .........................................................................................................................354 Broyden Algorithm......................................................................................................................355 integrator Command 356 Load Control...............................................................................................................................358 Displacement Control .................................................................................................................358 Minimum Unbalanced Displacement Norm................................................................................359 Arc-Length Control .....................................................................................................................360 Newmark Method .......................................................................................................................361 Hilbert-Hughes-Taylor Method ...................................................................................................362 analysis Command 363 Static Analysis ............................................................................................................................363 Transient Analysis ......................................................................................................................364 VariableTransient Analysis.........................................................................................................365 rayleigh command 367 eigen Command 368 analyze Command 369 dataBase Commands 370 FileDatastore Command ............................................................................................................370 save Command ..........................................................................................................................371 restore Command.......................................................................................................................371 Miscellaneous Commands 372 print Command...........................................................................................................................372 reset Command..........................................................................................................................373 wipe Command ..........................................................................................................................373 wipeAnalysis Command .............................................................................................................374 loadConst Command..................................................................................................................374 getTime Command.....................................................................................................................375 nodeDisp Command...................................................................................................................375 video Command .........................................................................................................................375 play Command ...........................................................................................................................376 nodeBounds ...............................................................................................................................376 display Command.......................................................................................................................377 How To.... 378 Run OpenSees...........................................................................................................................379 ...Define Units & Constants ........................................................................................................382 ...Generate Matlab Commands ..................................................................................................383 ...Define Tcl Procedure...............................................................................................................384 ...Read External files ..................................................................................................................385 Building The Model.....................................................................................................................387 ...Define Variables and Parameters .................................................................................387 ...Build Model and Define Nodes .....................................................................................389 ...Build Model and Define Nodes using Variables............................................................390 ...Define Materials ............................................................................................................391 Contents ix ...Define Elements............................................................................................................392 Defining Output ..........................................................................................................................393 ...Define Analysis-Output Generation ..............................................................................393 ...Define Data-Plot During Analysis..................................................................................394 Gravity Loads .............................................................................................................................394 ...Define Gravity Loads ....................................................................................................394 ...Run Gravity Analysis.....................................................................................................395 Static Analysis ............................................................................................................................395 ...Define Static Pushover Analysis ...................................................................................395 ...Run Static Pushover Analysis.......................................................................................396 Dynamic Analysis .......................................................................................................................398 ...Define Dynamic Ground-Motion Analysis .....................................................................398 ...Run Dynamic Ground-Motion Analysis ...................................................................................398 ...Combine Input-File Components ............................................................................................399 ...Run Parameter Study..............................................................................................................400 ...Run Moment-Curvature Analysis on Section ..........................................................................401 ...Determine Natural Period & Frequency ..................................................................................403 Getting Started with OpenSees 404 Introduction.................................................................................................................................406 Download OpenSees .................................................................................................................407 Run OpenSees...........................................................................................................................409 Problem Definition ......................................................................................................................413 Model Builder .............................................................................................................................414 Nodes .........................................................................................................................................415 Elements ....................................................................................................................................417 Recorders ...................................................................................................................................418 Summary of Model-Building Input File .......................................................................................418 Loads and Analysis ....................................................................................................................421 1. Load definition ........................................................................................................................421 2. Analysis definition and features .............................................................................................422 3. Analysis execution..................................................................................................................423 Gravity Loads .............................................................................................................................423 Summary of Gravity Loads.........................................................................................................427 Lateral Loads -- Static Pushover................................................................................................427 Lateral Loads -- Cyclic Lateral Load ..........................................................................................428 Lateral Loads -- Dynamic ground motion ...................................................................................429 Getting Going with OpenSees (under development) 433 Problem Definition ......................................................................................................................434 Model Building............................................................................................................................435 Variables and Units ....................................................................................................................435 Model Builder .............................................................................................................................437 Nodal Coordinates & Masses, Boundary Conditions .................................................................438 Materials .....................................................................................................................................439 Element Cross Section...............................................................................................................441 Elements and Element Connectivity ..........................................................................................442 Gravity and other Constant Loads .............................................................................................443 Summary of Defining Structural Model ......................................................................................443 Error-Checking Tip for Model Building .......................................................................................449 Recorders for Output..................................................................................................................454 Analysis Components 455 Script Utilities Library 456 Contents x matTest.tcl..................................................................................................................................456 RCcircSection.tcl ........................................................................................................................458 RCcircSectionFEDEAS.tcl .........................................................................................................459 RCFrameDisplay.tcl ...................................................................................................................461 MomentCurvature.tcl ..................................................................................................................462 ReadSMDFile.tcl ........................................................................................................................463 RotSpring2D...............................................................................................................................465 StFramePZLdisplay.tcl ...............................................................................................................466 Wsection.tcl ................................................................................................................................467 RigidFrame3Ddisplay.tcl ............................................................................................................468 Units&Constants.tcl ....................................................................................................................469 MatlabOutput.tcl .........................................................................................................................469 genPlaneFrame.tcl .....................................................................................................................470 References 473 Index 477 11 CHAPTER 1 Introduction This document is intended to outline the basic commands currently available with the OpenSees interpreter. This interpreter is an extension of the Tcl/Tk language for use with OpenSees. OpenSees is an object-oriented framework for finite element analysis. OpenSees' intended users are in the research community. A key feature of OpenSees is the interchangeability of components and the ability to integrate existing libraries and new components into the framework (not just new element classes) without the need to change the existing code. Core components, that is the abstract base classes, define the minimal interface (minimal to make adding new component classes easier but large enough to ensure all that is required can be accommodated). In This Chapter Notation.................................................................... 11 Copyright.................................................................. 13 Introduction to the Tcl command language ............... 14 OpenSees Interpreter............................................... 19 Notation The notation presented in this chapter is used throughout this document. Input values are a string, unless the first character is a $, in which case an integer, floating point number or variable is to be provided. In the Tcl language, variable references start with the $ character. Tcl expressions can also be used as input to the commands where the input value is specified by the first character being a $. Optional values are identified in enclosing <> braces. When specifying a variable quantity of values, the command line contains (x $values). The number of values required, x, and the types of values, $values, are specified in the description of the command. An arbitrary number of input values is indicated with the dot-dot-dot notation, i.e. $value1 $value2 ... Chapter 1 Introduction 12 The OpenSees interpreter constructs objects in the order they are specified by the user. New objects are often based on previously-defined objects. When specified as an object parameter, a previously-defined object must have already been added to the Domain. This requirement is specified in the description of the command arguments. Also, and very important: Do not use greek fonts ANYWHERE in your documentation, not only do they not come out in Html, they are not allowed in tcl, which is where this is important. I recommend spelling out the letters: alpha, beta, gamma, eps (short for epsilon), etc. I recommend to not use them in your figures either, as you would be unable to refer to them. If you must use them you have to do it very carefully, as it may be okey for internal variables. Example command: node $nodeTag (ndm $coords) <-mass (ndf $MassValues)> This line executes the node command (page 39) assigns coordinates and masses to a specified node. The $nodeTag argument is an integer tag identifying the node.The coordinate arguments are specified with the parentheses () because the number of arguments is dependent on the definition of the model (ndm (page 37)): two arguments in 2D and three in 3D. The mass specification at the node definition is optional. Therefore, it is enclosed in <> braces. The number of mass arguments is also dependent on the model definition, depending on the number of degrees of freedom assigned to a node (ndf (page 37)). Chapter 1 Introduction 13 Copyright Copyright © 1999,2000 The Regents of the University of California. All Rights Reserved. Permission to use, copy, modify, and distribute this software and its documentation for educational, research and non-profit purposes, without fee, and without a written agreement is hereby granted, provided that the above copyright notice, this paragraph and the following three paragraphs appear in all copies. Permission to incorporate this software into commercial products may be obtained by contacting the University of California. [Bill Hoskins Office of Technology Licensing, 2150 Shattuck Avenue #150 Berkeley, CA 94720-1620, (510) 643-7201] This software program and documentation are copyrighted by The Regents of the University of California. The software program and documentation are supplied "as is", without any accompanying services from The Regents. The Regents does not warrant that the operation of the program will be uninterrupted or error-free. The end-user understands that the program was developed for research purposes and is advised not to rely exclusively on the program for any reason. IN NO EVENT SHALL THE UNIVERSITY OF CALIFORNIA BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, INCLUDING LOST PROFITS, ARISING OUT OF THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF THE UNIVERSITY OF CALIFORNIA HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. THE UNIVERSITY OF CALIFORNIA SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE SOFTWARE PROVIDED HEREUNDER IS ON AN "AS IS" BASIS, AND THE UNIVERSITY OF CALIFORNIA HAS NO OBLIGATIONS TO PROVIDE MAINTENANCE, SUPPORT, UPDATES, ENHANCEMENTS, OR MODIFICATIONS. Chapter 1 Introduction 14 Introduction to the Tcl command language The Tcl scripting language was chosen to support the OpenSees commands, which are used to define the problem geometry, loading, formulation and solution. These commands are one-line commands which have specific tasks, as described in this manual. The Tcl language provides useful programming tools, such as variables manipulation, mathematical-expression evaluation and control structures. Tcl is a string-based scripting language which allows the following: Variables and variable substitution Mathematical-expression evaluation Basic control structures (if , while, for, foreach) Procedures File manipulation More information on Tcl commands can be found at its web site: Tcl/Tk Primer (http://dev.scriptics.com/scripting/primer.html) (http://dev.scriptics.com/scripting/primer.html) Handy Tcl commands: incr - Increment the value of a variable: set a 1 incr a a book reference: Brent Welch <[email protected]>, Ken Jones, and Jeff Hobbs: Practical Programming in Tcl and Tk, (http://www.beedub.com/book/) 4th Edition ISBN: 0-13-038560-3, June, 2003 (http://www.beedub.com/book/) Chapter 1 Introduction 15 Tcl Commands Format Tcl scripts are made up of commands separated by new lines or semicolon (;). The basic syntax for a Tcl command is: command $arg1 $arg2 ... command name of the Tcl command or user-defined procedure $arg1 $arg2 ... arguments for the command Tcl allows any argument to be nested command: command [nested-command1] [nested-command2] where the [] are used to delimit the nested commands. The Tcl interpreter will first evaluate the nested commands and will then evaluate the outer command with the results to the nested commands. The most basic command in Tcl is the set command: set variable $value for example: set a 5 The Tcl interpreter regards a command starting with the pond sign (#) to be a comment statement, so it does not execute anything following the #. For example: # this command assigns the value 5 to the variable a set a 5 The pound sign and the semicolon can be used together to put comments on the same line as the command. For example: set a 5; # this command assigns the value 5 to the variable a Chapter 1 Introduction Example Tcl Commands arithmetic procedure for & foreach functions >set a 1 >proc sum {a b} { for {set i 1} {$i < 10} {incr i 1} { 1 return [expr $a + $b] puts “i equals $i” >set b a } } a >sum 2 3 >set b $a 5 1 >set c [sum 2 3] set sum 0 >expr 2 + 3 5 foreach value {1 2 3 4} { 5 > set sum [expr $sum + $value] >expr 2 + $a } 3 puts $sum >set b [expr 2 + $a] 10 3 > > file manipulation procedure & if statement >set fileId [open tmp w] >proc guess {value} { anumber global sum >puts $fileId “hello” if {$value < $sum} { puts “too low” >close $fileID >type tmp } else { if {$value > $sum} { hello > puts “too high” } else { puts “you got it!”} } >source Example1.tcl } > guess 9 too low > 16 Chapter 1 Introduction Additional Tcl Resources Here are additional resources for Tcl: http://www.freeprogrammingresources.com/tcl.html (http://www.freeprogrammingresources.com/tcl.html) (a large list of helpful resources) http://www.tcl.tk/man/ (http://www.tcl.tk/man/) (Tcl/Tk manual pages) http://www.mit.edu/afs/sipb/user/golem/doc/tcltk-iap2000/TclTk1.html (http://www.mit.edu/afs/sipb/user/golem/doc/tcltk-iap2000/TclTk1.html) (a tutorial describing many commands by describing their implementation in a short program) http://www.beedub.com/book/ (http://www.beedub.com/book/) (some sample chapters from Practical Programming in Tcl and Tk, by Welch and Jones) http://philip.greenspun.com/tcl/ (http://philip.greenspun.com/tcl/) (not the most readable tutorial IMHO, but it does have Tickle-me-Elmo ;) It can be accessed from the link below as well.) http://www.tcl.tk/scripting/ (http://www.tcl.tk/scripting/) 17 Chapter 1 Introduction 18 http://hegel.ittc.ukans.edu/topics/tcltk/tutorial-noplugin/index.html (http://hegel.ittc.ukans.edu/topics/tcltk/tutorial-noplugin/index.html) (a short tutorial on the essential Tcl commands, also includes a manual of Tcl/Tk commands at the website below) http://hegel.ittc.ukans.edu/topics/linux/man-pages/index/index-mann.html (http://hegel.ittc.ukans.edu/topics/linux/man-pages/index/index-mann.html) http://pages.cpsc.ucalgary.ca/~saul/personal/archives/Tcl-Tk_stuff/tcl_examples/ (http://pages.cpsc.ucalgary.ca/~saul/personal/archives/Tcl-Tk_stuff/tcl_examples/) (Tk widgets with screenshots) OpenSees Interpreter The main abstractions of OpenSees will be explained using the OpenSees interpreter. The interpreter is an extension of the Tcl (page 15) scripting language. The OpenSees interpreter adds commands to Tcl for finite element analysis. Each of these commands is associated (bound) with a C++ procedure that is provided. It is this procedure that is called upon by the interpreter to parse the command. In this document we outline only those commands which have been added to Tcl by OpenSees. For OpenSees we have added commands to Tcl for finite element analysis: Modeling – create nodes, elements, loads and constraints Analysis – specify the analysis procedure. Output specification – specify what it is you want to monitor during the analysis. HELP OpenSees Documentation Web Page (http://opensees.berkeley.edu/OpenSees/primer.html) http://opensees.berkeley.edu/cgi-bin/OpenSeesCommands.pl (http://opensees.berkeley.edu/cgibin/OpenSeesCommands.pl) 19 CHAPTER 2 OpenSees What is OpenSees? •ᾉAn object-oriented software framework for simulation applications in earthquake engineering using finite element methods. OpenSees is not a code. •ᾉA communication mechanism within PEER for exchanging and building upon research accomplishments. •ᾉAs open-source software, it has the potential for a community code for earthquake engineering. Chapter 2 In This Chapter OpenSees Features ................................................. 22 ModelBuilder Object ................................................. 31 Domain Object ......................................................... 32 Recorder Object ....................................................... 33 Analysis Object ........................................................ 34 OpenSees 20 Chapter 2 21 OpenSees Features + The library of materials, elements and analysis c powerful tool for numerical simulation of nonlinea systems + The OpenSees library of components is ever-gro numerical-simulation models + The OpenSees interface is based on a command enables the user to create more-versatile input files. + OpenSees is not a black box, making it a useful modeling + You can create your own material, element or an into OpenSees + NEES is supporting integration of OpenSees as laboratory testing WHY OPENSEES? Chapter 2 MODELS: + linear & nonlinear structural and geotechnical models 22 Chapter 2 SIMULATIONS: + static push-over analyses + static reversed-cyclic analyses + dynamic time-series analyses + uniform-support excitation + multi-support excitation MODELING FEATURES: MODEL-BUILDING CAPABILITIES: model command (page 37) node command (page 39) mass command (page 40) constraints objects uniaxialMaterial command (page 47) nDMaterial command (page 175) section command (page 185) element command (page 213) block command (page 295) region command Geometric Transformation command (page 302) 23 Chapter 2 Time Series command (page 311) pattern command (page 317) ANALYSIS CAPABILITIES constraints command (page 339) numberer command (page 344) system command (page 346) test command (page 349) algorithm command (page 352) integrator command (page 356) analysis command (page 363) rayleigh command (page 367) eigen command (page 368) dataBase command RECORDER/OUTPUT CAPABILITIES Node Recorder (page 326) EnvelopeNode Recorder Drift Recorder Element Recorder (page 330) EnvelopeElement Recorder Display Recorder (page 333) Plot Recorder playback Command MODEL-BUILDING FEATURES: UNIAXIAL MATERIALS (page 47): 24 Chapter 2 25 Elastic Material (page 131) Elastic-Perfectly Plastic Material Elastic-Perfectly Plastic Gap Material Elastic-No Tension Material Parallel Material Series Material Hardening Material Concrete01 Material -- Zero Tensile Strength (page 141) Concrete02 Material -- Linear Tension Softening (page 145) Concrete03 Material -- Nonlinear Tension Softening (page 152) Steel01 Material (page 157) Steel02 Material -- Giuffré-Menegotto-Pinto Model with Isotropic Strain Hardening (page 165) Hysteretic Material (page 170) Viscous Material BARSLIP Material Bond_SP01 - - Strain Penetration Model for Fully Anchored Steel Reinforcing Bars Concrete04 Material -- Popovics Concrete Material (page 61) Fatigue Material Limit State Material PINCHING4 Material (page 88) PyTzQz Uniaxial Materials Reinforcing Steel Material MULTIDIMENSIONAL/nD MATERIALS: Elastic Isotropic Material (page 175) J2 Plasticity Material Plane Stress Material Plate Fiber Material (page 177) Template Elasto-Plastic Material FluidSolidPorousMaterial Material (page 182) PressureIndependMultiYield Material PressureDependMultiYield Material ELEMENTS: Truss Element Chapter 2 Corotational Truss Element Elastic Beam Column Element (page 215) Nonlinear Beam Column Element (page 216) Beam With Hinges Element (page 218) Displacement-Based Beam-Column Element (page 222) Zero-Length Element Zero-Length ND Element Zero-Length Section Element (page 225) Quad Element Shell Element Bbar Plane Strain Quadrilateral Element (page 227) Enhanced Strain Quadrilateral Element Standard Brick Element Bbar Brick Element Eight Node Brick Element (page 232) Twenty Node Brick Element u-p-U element FourNodeQuadUP Element (page 237) BeamColumnJoint Element SECTIONS: Elastic Section (page 187) Uniaxial Section (page 187) Fiber Section (page 189) Section Aggregator Elastic Membrane Plate Section Plate Fiber Section (page 202) Bidirectional Section Isolator2spring Section: Model to include buckling behavior of an elastomeric bearing ANALYSIS FEATURES: 26 Chapter 2 27 Linear Equation Solvers (page 346) -- provide the solution of the linear system of equations Ku = P. Each solver is tailored to a specific matrix topology. Profile SPD (page 347) -- Direct profile solver for symmetric positive definite matrices Band General -- Direct solver for banded unsymmetric matrices Band SPD -- Direct solver for banded symmetric positive definite matrices Sparse General -- Direct solver for unsymmetric sparse matrices Sparse Symmetric -- Direct solver for symmetric sparse matrices UmfPack General -- Direct UmfPack solver for unsymmetric matrices Full General -- Direct solver for unsymmetric dense matrices Conjugate Gradient -- Iterative solver using the preconditioned conjugate gradient method Eigenvalue Solvers -- provide the solution of the generalized eigenvalue problem Kv = MvL Symmetric Arpack -- Arpack solver for symmetric matrices Band Arpack -- Arpack solver for banded matrices DOF Numberers (page 344) -- number the degrees of freedom in the domain Plain (page 344) -- Uses the numbering provided by the user RCM (page 345) -- Renumbers the DOF to minimize the matrix band-width using the Reverse Cuthill-McKee algorithm Static Integrators (page 356) -- determine the next time step for an analysis Load Control (page 358) -- Specifies the incremental load factor to be applied to the loads in the domain Displacement Control (page 358) -- Specifies the incremental displacement at a specified DOF in the domain Minimum Unbalanced Displacement Norm (page 359) -- Specifies the incremental load factor such that the residual displacement norm in minimized Arc Length (page 360) -- Specifies the incremental arc-length of the load-displacement path Transient Integrators (page 356) -- determine the next time step for an analysis including inertial effects Newmark (page 361) -- The two parameter time-stepping method developed by Newmark Hilbert-Hughes-Taylor (page 362) -- The three parameter Hilbert-Hughes-Taylor timestepping method Chapter 2 28 Central Difference -- Approximates velocity and acceleration by centered finite differences of displacement Solution Algorithms (page 352) -- Iterate from the last time step to the current Linear Algorithm -- Uses the solution at the first iteration and continues Newton Algorithm (page 352) -- Uses the tangent at the current iteration to iterate to convergence Modified Newton Algorithm -- Uses the tangent at the first iteration to iterate to convergence Newton with Line Search Algorithm Krylov-Newton Algorithm BFGS Algorithm Broyden Algorithm Convergence Tests (page 349) -- Accept the current state of the domain as being on the converged solution path Norm Unbalance -- Specifies a tolerance on the norm of the unbalanced load at the current iteration Norm Displacement Increment -- Specifies a tolerance on the norm of the displacement increments at the current iteration Energy Increment -- Specifies a tolerance on the inner product of the unbalanced load and displacement increments at the current iteration Relative Unbalance Relative Displacement Increment Relative Energy Increment Constraint Handlers (page 339) -- Handle the constraints defined on the domain Plain Constraints (page 341) -- Removes constrained degrees of freedom from the system of equations Lagrange Multipliers (page 342) -- Uses the method of Lagrange multipliers to enforce constraints Penalty Method (page 341) -- Uses penalty numbers to enforce constraints Transformation Method (page 343) -- Performs a condensation of constrained degrees of freedom Chapter 2 29 analysis Command (page 363) -- defines what type of analysis is to be performed Static Analysis (page 363) -- solves the KU=R problem, without the mass or damping matrices. Transient Analysis (page 364) -- solves the time-dependent analysis. The time step in this type of analysis is constant. The time step in the output is also constant. Variable Transient Analysis (page 365) -- performs the same analysis type as the Transient Analysis object. The time step, however, is variable. This method is used when there are convergence problems with the Transient Analysis object at a peak or when the time step is too small. The time step in the output is also variable. ModelBuilder Object The model builder constructs As in any finite element analysis, the analyst's first step is to subdivide the body being studied into elements and nodes, to define loads acting on the elements and nodes, and to define constraints acting on the nodes. The ModelBuilder is the object in the program responsible for building the following objects in the model and adding them to the domain: Node (page 39) Mass (page 40) Material (page 175, page 47) Section (page 185) Element (page 213) LoadPattern (page 317) TimeSeries (page 311) Transformation (page 302) Block (page 295) Constraint (page 339) Chapter 2 30 Domain Object The Domain object is responsible for storing the objects created by the ModelBuilder (page 31) object and for providing the Analysis (page 336) and Recorder (page 33) objects access to these objects. Figure 1: Domain Object Chapter 2 31 Recorder Object The recorder object monitors user-defined parameters in the model during the analysis. This, for example, could be the displacement history at a node in a transient analysis, or the entire state of the model at each step of the solution procedure. Several Recorder (page 326) objects are created by the analyst to monitor the analysis. What does a recorder do? Monitors the state of a domain component (node, element, etc.) during an analysis Writes this state to a file or to a database at selected intervals during the analysis There are also recorders for plotting and monitoring residuals Once in a file, the information can be easily post-processed. Analysis Object The Analysis objects are responsible for performing the analysis. The analysis moves the model along from state at time t to state at time t + dt. This may vary from a simple static (page 363) linear analysis to a transient (page 364, page 365) non-linear analysis. In OpenSees each Analysis object is composed of several component objects, which define the type of analysis how the analysis is performed. Chapter 2 Figure 2: Analysis Object 32 33 CHAPTER 3 Model-Building Objects These objects are used to create the physical model. In This Chapter model Command ...................................................... 37 node Command........................................................ 39 mass Command ....................................................... 40 constraints objects.................................................... 41 uniaxialMaterial Command ....................................... 47 nDMaterial Command .............................................. 175 section Command .................................................... 185 element Command ................................................... 213 Soil Models and Solid-Fluid Fully Coupled Elements 264 block Command ....................................................... 295 region Command...................................................... 300 Geometric Transformation Command....................... 302 Time Series .............................................................. 311 pattern Command..................................................... 317 34 CHAPTER 4 model Command This command is used to construct a ModelBuilder object. Currently there is only one type of ModelBuilder accepted. For an example of this command, refer to the Model Building Example (page 389) In This Chapter Basic Model Builder.................................................. 37 build Command ........................................................ 38 Basic Model Builder This command is used to construct the BasicBuilder object. model BasicBuilder -ndm $ndm <-ndf $ndf> $ndm dimension of problem (1,2 or 3) $ndf number of degrees of freedom at node (optional) (default value depends on value of ndm: ndm=1 -> ndf=1 ndm=2 -> ndf=3 ndm=3 -> ndf=6) These additional commands allow for the construction of Nodes (page 39), Masses (page 40), Materials (page 175, page 47), Sections (page 185), Elements (page 213), LoadPatterns (page 317), TimeSeries (page 311), Transformations (page 302), Blocks (page 295) and Constraints (page 339). These additional commands are described in the subsequent chapters. EXAMPLE: Chapter 4 model Command model basic -ndm 3 -ndf 6; # 3 spacial dimensions, 6 DOF's per node For an example of this command, refer to the Model Building Example (page 389) build Command This command is used to invoke build() (????) on the ModelBuilder (page 31) object. build This command has no effect a BasicBuilder (page 37) object, but will on other types of ModelBuilder (page 31) objects. 35 36 CHAPTER 5 node Command This command is used to construct a Node object. It assigns coordinates and masses to the Node object. node $nodeTag (ndm $coords) <-mass (ndf $MassValues)> $nodeTag integer tag identifying node $coords nodal coordinates (ndm (page 37) arguments) $MassValues nodal mass corresponding to each DOF (ndf (page 37) arguments) (optional) The optional -mass string allows analyst the option of associating nodal mass with the node EXAMPLE: node 1 0.0 0.0 node 2 0.0 120. 0.0; # x,y,z coordinates (0,0,0) of node 1 0.0; # x,y,z coordinates (0,120,0) of node 2 For an example of this command, refer to the Model Building Example (page 389) 37 CHAPTER 6 mass Command This command is used to set the mass at a node. mass $nodeTag (ndf $MassValues) $nodeTag integer tag identifying the node associated with the mass $MassValues mass values corresponding to each nodal degrees of freedom (ndf (page 37) values) EXAMPLE: mass 2 2.5 0.0 2.5 0.0 0.0 0.0; # define mass in x and z coordinates For an example of this command, refer to the Model Building Example (page 389) 38 CHAPTER 7 constraints objects From Cook: " A constraint either prescribes the value of a DOF (as in imposing a support condition) or prescribes a relationship among DOF. In common terminology, a single-point constraint sets a single DOF to a known value (often zero) and a multi-point constraint imposes a relationship between two or more DOF.... For example, support conditions on a three-bar truss invoke single-point constraints, while rigid links and rigid elements each invoke a multi-point constraint." In This Chapter Single-Point Constraints ........................................... 41 Multi-Point Constraints ............................................. 44 Single-Point Constraints The following commands construct homogeneous single-point boundary constraints. fix Command This command is used to construct homogeneous single-point boundary constraints. fix $nodeTag (ndf $ConstrValues) $nodeTag integer tag identifying the node to be constrained $ConstrValues constraint type (0 or 1). ndf (page 31) values are specified, corresponding to the ndf degrees-of-freedom. The two constraint types are: 0 unconstrained 1 constrained EXAMPLE: fix 1 1 1 1 1 1 1; # node 1: fully fixed Chapter 7 fix 2 0 1 0 0 1 0 -mass 2.5 0.0 2.5 0.0 0.0 0.0; torsion, translational masses in x-z plane only constraints objects 39 # node 2: restrain axial elongation and For an example of this command, refer to the Model Building Example (page 389) fixX Command This command is used to construct multiple homogeneous single-point boundary constraints for all nodes whose x-coordinate lies within a specified distance from a specified coordinate. fixX $xCoordinate (ndf $ConstrValues) <-tol $tol> For example, this command is used when specifying boundary conditions for a series of nodes lying in a plane parallel to the y-z plane in global coordinates. $xCoordinate x-coordinate of nodes to be constrained $ConstrValues constraint type (0 or 1). ndf (page 31) values are specified, corresponding to the ndf degrees-of-freedom. The two constraint types are: $tol 0 unconstrained 1 constrained user-defined tolerance (optional, default = 1e-10) EXAMPLE: fixX 0.0 1 1 1 1 1 1 -tol 0.1; # fully restrain all nodes in y-z plane at origin (x=0.0) fixY Command This command is used to construct multiple homogeneous single-point boundary constraints for all nodes whose y-coordinate lies within a specified distance from a specified coordinate. fixY $yCoordinate (ndf $ConstrValues) <-tol $tol> For example, this command is used when specifying boundary conditions for a series of nodes lying in a plane parallel to the x-z plane in global coordinates. Chapter 7 constraints objects $yCoordinate y-coordinate of nodes to be constrained $ConstrValues constraint type (0 or 1). ndf values are specified, corresponding to the ndf (page 37) degrees-of-freedom. 40 The two constraint types are: $tol 0 unconstrained 1 constrained user-defined tolerance (optional, default = 1e-10) EXAMPLE: fixY 0.0 1 1 1 1 1 1 -tol 0.1; # fully restrain all nodes in x-z plane at origin (y=0.0) fixZ Command This command is used to construct multiple homogeneous single-point boundary constraints for all nodes whose z-coordinate lies within a specified distance from a specified coordinate. fixZ $zCoordinate (ndf $ConstrValues) <-tol $tol> For example, this command is used when specifying boundary conditions for a series of nodes lying in a plane parallel to the x-y plane in global coordinates. $zCoordinate z-coordinate of nodes to be constrained $ConstrValues constraint type (0 or 1). ndf values are specified, corresponding to the ndf (page 37) degrees-of-freedom. The two constraint types are: $tol 0 unconstrained 1 constrained user-defined tolerance (optional, default = 1e-10) EXAMPLE: fixZ 0.0 1 1 1 1 1 1 -tol 0.1; # fully restrain all nodes in x-y plane at origin (z=0.0) Chapter 7 constraints objects 41 Multi-Point Constraints The following commands construct multi-point boundary constraints. equalDOF Command This command is used to construct a multi-point constraint between nodes. equalDOF $rNodeTag $cNodeTag $dof1 $dof2 ... $rNodeTag integer tag identifying the retained, or master node (rNode) $cNodeTag integer tag identifying the constrained, or slave node (cNode) $dof1 $dof2 ... nodal degrees-of-freedom that are constrained at the cNode to be the same as those at the rNode Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. EXAMPLE: equalDOF 2 3 1 3 5; # impose the traslational displacements in x and z directions, and rotation about the y-axis of node 3 to be the same as those of node 2. rigidDiaphragm Command This command is used to construct a number of Multi-Point Constraint (MP_Constraint) objects. These objects will constraint certain degrees-of-freedom at the listed slave nodes to move as if in a rigid plane with the master node. rigidDiaphragm $perpDirn $masterNodeTag $slaveNodeTag1 $slaveNodeTag2 ... Chapter 7 $perpDirn constraints objects 42 direction perpendicular to the rigid plane (i.e. direction 3 corresponds to the 1-2 plane) The rigid plane can be the 1-2, 1-3 or 2-3 plane $masterNodeTag integer tag identifying the master node $slaveNodeTag1 $slaveNodeTag2 ... nodes that are to be constrained to the master node NOTE: The constraint object is constructed assuming small rotations. NOTE: The rigidDiaphragm command works only for problems in three dimensions with sixdegrees-of-freedom at the nodes (ndf (page 37) = 6). EXAMPLE: rigidDiaphragm 2 2 4 5 6; 2. constrain nodes 4,5,6 to move as if in the same x-z plane as node rigidLink Command This command is used to construct a single MP_Constraint object. rigidLink $type $masterNodeTag $slaveNodeTag $type string-based argument for rigid-link type: rod only the translational degree-of-freedom will be constrained to be exactly the same as those at the master node beam both the translational and rotational degrees of freedom are constrained. $masterNodeTag integer tag identifying the master node $slaveNodeTag integer tag identifying the slave node to be constrained to master node NOTE: The constraint object constructed for the beam option assumes small rotations EXAMPLE: rigidLink beam 2 3; # connect node 3 to node 2 via a rigid link-beam. 43 CHAPTER 8 uniaxialMaterial Command This command is used to construct a UniaxialMaterial object which represents uniaxial stressstrain (or force-deformation) relationships. The valid queries to any uniaxial material when creating an ElementRecorder (page 330) are 'strain,' 'stress,' and 'tangent.' In This Chapter -- Contributed Uniaxial Materials .............................. 48 Elastic Material ......................................................... 131 Elastic-Perfectly Plastic Material............................... 132 Elastic-Perfectly Plastic Gap Material ....................... 134 Elastic-No Tension Material...................................... 136 Parallel Material........................................................ 137 Series Material ......................................................... 138 Hardening Material ................................................... 140 Concrete01 Material -- Zero Tensile Strength........... 141 Concrete02 Material -- Linear Tension Softening...... 145 Concrete03 Material -- Nonlinear Tension Softening152 Steel01 Material ....................................................... 157 Steel02 Material -- Giuffré-Menegotto-Pinto Model with Isotropic Strain Hardening ...................................................... 165 Hysteretic Material.................................................... 170 Viscous Material ....................................................... 173 44 CHAPTER 9 -- Contributed Uniaxial Materials These additional materials are contributions to OpenSees and should be available in the latest executable. If you have questions about these objects, please do contact the individual authors directly. In This Chapter BARSLIP Material .................................................... 48 Bond_SP01 - - Strain Penetration Model for Fully Anchored Steel Reinforcing Bars....................................................... 50 Concrete04 Material -- Popovics Concrete Material.. 61 Fatigue Material........................................................ 65 Limit State Material................................................... 77 PINCHING4 Material ................................................ 88 PyTzQz Uniaxial Materials ....................................... 109 Reinforcing Steel Material ........................................ 117 BARSLIP Material Contact Author: This command is used to construct a uniaxial material that simulates the bar force versus slip response of a reinforcing bar anchored in a beam-column joint. The model exhibits degradation under cyclic loading. Cyclic degradation of strength and stiffness occurs in three ways: unloading stiffness degradation, reloading stiffness degradation, strength degradation. uniaxialMaterial BarSlip $matTag $fc $fy $Es $fu $Eh $db $ld $nb $depth $height <$ancLratio> $bsFlag $type <$damage $unit> $matTag unique material object integer tag $fc positive floating point value defining the compressive strength of the concrete in which the reinforcing bar is anchored Chapter 9 -- Contributed Uniaxial Materials $fy positive floating point value defining the yield strength of the reinforcing steel $Es floating point value defining the modulus of elasticity of the reinforcing steel $fu positive floating point value defining the ultimate strength of the reinforcing steel $Eh floating point value defining the hardening modulus of the reinforcing steel $ld floating point value defining the development length of the reinforcing steel $db point value defining the diameter of reinforcing steel $nb an integer defining the number of anchored bars $depth floating point value defining the dimension of the member (beam or column) perpendicular to the dimension of the plane of the paper $height floating point value defining the height of the flexural member, perpendicular to direction in which the reinforcing steel is placed, but in the plane of the paper $ancLratio floating point value defining the ratio of anchorage length used for the reinforcing bar to the dimension of the joint in the direction of the reinforcing bar (optional, default: 1.0) $bsFlag string indicating relative bond strength for the anchored reinforcing bar (options: "Strong" or "Weak") $type string indicating where the reinforcing bar is placed. (options: "beamtop", “beambot” or "column") $damage string indicating whether there is full damage in the material or no damage (optional, options: “Damage”, “NoDamage” ; default: Damage) $unit string indicating the type of unit system used (optional, options: “psi”, “MPa”, “Pa”, “psf”, “ksi”, “ksf”) (default: “psi” / “MPa”)* NOTE: Model Characteristics: 45 Chapter 9 -- Contributed Uniaxial Materials 46 The uniaxial material model uses the Pinching4 material (page 88) model (Ref. Pinching4 material model). The response envelope for the bar-slip springs does not represent strength deterioration, but once the slip demand exceeds 3mm (0.12 in), strength deterioration due to cyclic loading initiates. As a result the bond-slip springs always exhibit positive stiffness, but strength deterioration upon reloading to a previously observed slip demand. Reloading and Unloading Stiffness deterioration are also simulated. The damage index for unloading and reloading stiffness degradation is evaluated the same say as the Pinching4 material (ref. Pinching4 material doc.) but the index for strength degradation δf i is specified as ( (~ ) ) δf i = gF1 ⋅ d max + gF 3 ≤ gFLim With degradation model parameters gF*. It should be noted in here that the deterioration parameters for unloading, reloading stiffness and strength degradation cannot be modified by the user and are defined to represent observed behavior. *NOTE: The model includes predefined bond strengths, so there is the necessity to include in Units in this material model. For default one can specify units in psi (i.e. pounds/inch2) or in MPa (i.e. N/mm2). The code detects units in psi if the compressive strength of concrete is greater than 1000 otherwise it takes it as MPa system. The optional variable $unit will help the user to specify other different types of unit systems according to one’s choice, but currently it is limited to the unit systems as specified above. The user should also take care to specify the units of length in the corresponding matching units. (note: Pa = N/m2; ksf = kilo-pound/ft2) Bond_SP01 - - Strain Penetration Model for Fully Anchored Steel Reinforcing Bars URL: http://www.uwm.edu/~jzhao/Bond_SP01_pages/Bond_index.html http://www.uwm.edu/~jzhao/Bond_SP01_pages/Bond_index.html Contact Authors: Jian Zhao, University of Wisconsin, Milwakee [[email protected]] Sri Sritharan, Iowa State University [[email protected]] Chapter 9 -- Contributed Uniaxial Materials 47 This command is used to construct a uniaxial material object for capturing strain penetration effects at the column-to-footing, column-to-bridge bent caps, and wall-to-footing intersections. In these cases, the bond slip associated with strain penetration typically occurs along a portion of the anchorage length. This model can also be applied to the beam end regions, where the strain penetration may include slippage of the bar along the entire anchorage length, but the model parameters should be chosen appropriately. uniaxialMaterial Bond_SP01 $matTag $Fy $Sy $Fu $Su $b $R This model is for fully anchored steel reinforcement bars that experience bond slip along a portion of the anchorage length due to strain penetration effects, which are usually the case for column and wall longitudinal bars anchored into footings or bridge joints $matTag Unique material object integer tag $Fy Yield strength of the reinforcement steel $Sy Rebar slip at member interface under yield stress and may be obtained from d (in ) $ Fy ( psi ) $ Sy (in ) = 0.1 b ( 2 * alpha + 1) 4000 f ' ( psi ) c 1 alpha + 0.013(in ) or 1 db(mm) $Fy(MPa alpha ) (2*alpha+1) +0.34(mm) $Sy(mm) =2.54 8437 f '(MPa ) c db: rebar diameter $Fy: yield strength of the reinforcement steel fc': concrete compressive strength of the adjoining connection member alpha: parameter used in the local bond-slip relation and can be taken as 0.4 in accordance with CEB-FIP Model Code 90 $Fu Ultimate strength of the reinforcement steel $Su Rebar slip at the loaded end at the bar fracture strength $Su = (30~ 40)*$Sy Chapter 9 -- Contributed Uniaxial Materials $b Initial hardening ratio in the monotonic slip vs. bar stress response (0.3~0.5) $R Pinching factor for the cyclic slip vs. bar response (0.5~1.0) stress NOTE: $Fu $bK $Fy K $Sy $Su slip 48 Chapter 9 -- Contributed Uniaxial Materials 49 stress Figure 1: Monotonic bar stress vs. slip response as modelled in Bond_SP01 K K slip K Figure 2: Cyclic bar stress vs. slip response as modelled in Bond_SP01 80 $R = 1.0 $R = 0.6 Bar stress (ksi) 40 0 -40 -80 -0.6 -0.4 -0.2 0 Loaded-end slip (in) 0.2 0.4 0.6 Chapter 9 -- Contributed Uniaxial Materials 50 Figure 3: Pinching effect represented by $R in Bond_SP01 Model background: Capturing the structural response and associated damage require accurate modeling of localized inelastic deformations occurring at the member end regions as identified by shaded areas in Figure 4. These member end deformations consist of two components: 1) the flexural deformation that causes inelastic strains in the longitudinal bars and concrete, and 2) the member end rotation, as indicated by arrows in Figure 4, due to reinforcement slip. The slip considered here is the result of strain penetration along a portion of the fully anchored bars into the adjoining concrete members (e.g., footings and joints) during the elastic and inelastic response of a structure. Ignoring the strain penetration component may appear to produce satisfactory force-displacement response of the structural system by compromising strain penetration effects with greater contribution of the flexural action at a given lateral load. However, this approach will appreciably overestimate the strains and section curvatures in the critical inelastic regions of the member, and thereby overestimate the structural damage. Building frame Bridge bent Structural wall Figure 4: Expected inelastic regions at the column and wall ends Chapter 9 -- Contributed Uniaxial Materials 51 The zero-length section element available in OpenSees may be used to accurately model the strain penetration effects (or the fixed end rotations shown in Figure 4). Zero-length section elements have been generally used for section analyses to calculate the moment corresponding to a given curvature. To model the fixed-end rotation, the zero-length section element should be placed at the intersection between the flexural member and an adjoining member representing a footing or joint as shown in Figure 5. A duplicate node is also required between a fiber-based beam-column element and the adjoining concrete element as shown in Figure 5. The translational degree-of-freedom of this new node (i.e., node j in Figure 5) should be constrained to the other node (i.e., node i in Figure 5) to prevent sliding of the beam-column element under lateral loads because the shear resistance is not included in the zero-length section element. N M Node k Sec. n beamcolumn element . . . Sec. 2 zero-length section element Sec. 1 Node j Node i 0 Figure 5: Adding a zero-length section element to a beam-column element The zero-length section element in OpenSees is assumed to have a unit length such that the element deformations (i.e., elongation and rotation) are equal to the section deformations (i.e., axial strain and curvature). The material model for the steel fibers in the zero-length section element represents the bar slip instead of strain for a given bar stress. The uniaxial material model Bond_SP01 is developed for steel fibers in the zero-length section elements. Note on Material Model for Concrete Fibers Chapter 9 -- Contributed Uniaxial Materials 52 Similar to the model proposed for the steel fibers, a material model describing the monotonic response and hysteretic rules is also required for the concrete fibers. The combination of using the zero-length section element and enforcing the plane section assumption at the end of a flexural member impose high deformations to the extreme concrete fibers in the zero-length element. These deformations would likely correspond to concrete compressive strains significantly greater than the strain capacity stipulated by typical confined concrete models. Such high compressive strains at the end of flexural members are possible because of additional confinement effects expected from the adjoining members and because of complex localized deformation at the member end. Without further proof, it is suggested that the concrete fibers in the zero-length section element follow a concrete model in OpenSees (e.g., Concrete02). To accommodate the large deformations expected to the extreme concrete fibers in the zero-length element, this concrete model may be assumed to follow a perfectly plastic behavior once the concrete strength reduces to 80% of the confined compressive strength. A parametric study has indicated that the simulation results would not be very sensitive to the compressive strain chosen to trigger the perfectly plastic behavior for the concrete fibers in the zero-length section element. Reference: Zhao, J., and S. Sritharan. Modeling of strain penetration effects in fiber-based analysis of reinforced concrete structures. ACI Structural Journal. (Accepted for publication in 2006). example ss_ic1.tcl # bridge T-joint # ----------------# Units: kips, in. # Pushover analysis multiple dispBeamColumn with strain penetration # modeling the circular RC column in a bridge T-joint tested by # Sri Sritharan # "Seismic response of column/cap beam tee connections # with cap beam prestressing" Ph.D. thesis, UCSD # Create ModelBuilder (with two dimensions and 2 DOF/node) model basic -ndm 2 -ndf 3 # Create nodes # tag X node 1 0.0 node 2 0.0 node 3 0.0 node 4 0.0 node 5 0.0 node 6 0.0 node 7 0.0 node 8 12.0 node 9 19.0 node 10 40.0 node 11 61.5 node 12 84.0 node 13 12.0 Y -48.0 -30.0 -12.0 0.0 12.0 30.0 48.0 0.0 0.0 0.0 0.0 0.0 0.0 Chapter 9 -- Contributed Uniaxial Materials 53 # Fix supports at base of column # Tag DX DY RZ fix 1 1 1 0 fix 7 1 0 0 #equalDOF $rNodeTag $cNodeTag $dof1 $dof2 equalDOF 8 13 2 # Define materials for nonlinear columns ##column Core CONCRETE uniaxialMaterial Concrete01 uniaxialMaterial Concrete01 ##column Cover CONCRETE uniaxialMaterial Concrete01 ##column stub CONCRETE uniaxialMaterial Elastic ##beam CONCRETE ecu uniaxialMaterial Concrete01 ## STEEL rebar ##STEEL02 $cR1 $cR2 uniaxialMaterial Steel02 0.04 1.0 0.04 uniaxialMaterial Steel02 0.00 1.0 0.00 uniaxialMaterial Elastic #uniaxialMaterial StrPen01 $Cd $db $fc uniaxialMaterial StrPen01 0.0 1.0 4.35 tag 1 200 tag 2 tag 3 f'c -6.38 -6.38 f'c -4.56 E 2280 ec0 -0.004 -0.004 ec0 -0.002 tag $a1 f'cu -5.11 -5.11 f'cu 0.0 f'c 4 -5.76 $a2 5 $a3 65.0 tag $a4 29000 6 62.8 29000 7 29000 $Tag $sy $fy $su 400 0.02 65 0.7 ecu -0.014 -0.014 ecu -0.006 ec0 f'cu -0.002 0.0 -0.006 $Fy $E $b $R0 0.02 18.5 0.925 0.15 0.02 18.5 0.925 0.15 $fu $Kz $R 97.5 0.50 0.7 1.0 1.0 $la 25.0 # Define cross-section for nonlinear column set colDia 24; # bending in this direction (local and global y) set cover 1.38; set bcent 1.81; # [expr $cover+0.197+0.5] set As 0.60; # Area of no. 7 bar set R [expr $colDia/2.0] set Rc [expr $colDia/2.0-$cover] set Rb [expr $colDia/2.0-$bcent] section Fiber 1 { # core concrete fibers patch circ # concrete cover fibers patch circ # reinforcing fibers layer circ } section Fiber 2 { # core concrete fibers patch circ # concrete cover fibers patch circ # reinforcing fibers layer circ } 1 70 22 0.0 0.0 0.0 $Rc 0.0 360.0 2 70 2 0.0 0.0 $Rc $R 0.0 360.0 5 14 $As 0.0 0.0 $Rb -90.0 244.3 3 70 22 0.0 0.0 0.0 $Rc 0.0 360.0 3 70 2 0.0 0.0 $Rc $R 0.0 360.0 5 14 $As 0.0 0.0 $Rb -90.0 244.3 Chapter 9 section Fiber 5 { # core concrete fibers patch circ # concrete cover fibers patch circ # reinforcing fibers layer circ } -- Contributed Uniaxial Materials 200 70 22 0.0 0.0 0.0 $Rc 0.0 360.0 2 70 2 0.0 0.0 $Rc $R 0.0 360.0 400 14 $As 0.0 0.0 $Rb -90.0 244.3 $aw -$bw $aw secID 1 1 1 1 1 1 1 1 1 1 1 transfTag # Define cross-section for nonlinear cap beam set bmw 27; # bending in this direction (local and global y) set bmh 24; # bending in this direction (local and global y) set cover 1.38; set bcent 1.81; # [expr $cover+0.197+0.5] set aw [expr $bmw/2.0] set ac [expr $bmw/2.0-$cover] set ab [expr $bmw/2.0-$bcent] set bw [expr $bmh/2.0] set bc [expr $bmh/2.0-$cover] set bb [expr $bmh/2.0-$bcent] section Fiber 3 { # concrete fibers patch quad 4 48 1 # reinforcing fibers layer straight 6 7 $As layer straight 6 7 $As } 54 -$bw -$aw $bw -$aw -$bb $bb $ab $ab -$bb $bb -$ab -$ab $bw # Define column elements ## Geometry of coloumn elements geomTransf Linear 1 # Number of integration points along length of element set np 5 # Create the columns using displacement controlled beam-column elements # tag ndI ndJ nsecs element dispBeamColumn 1 1 2 $np 3 element dispBeamColumn 2 2 3 $np 3 element dispBeamColumn 3 3 4 $np 3 element dispBeamColumn 4 4 5 $np 3 element dispBeamColumn 5 5 6 $np 3 element dispBeamColumn 6 6 7 $np 3 element dispBeamColumn 7 4 8 $np 2 element dispBeamColumn 8 13 9 $np 1 element dispBeamColumn 9 9 10 $np 1 element dispBeamColumn 10 10 11 $np 1 element dispBeamColumn 11 11 12 $np 1 # zeroLengthSection tag ndI ndJ secID element zeroLengthSection 12 8 13 5 # Set up and perform analysis ## Create recorders recorder Node -file topdispac.out -time -node 12 -dof 2 disp recorder Node -file rot4ac.out -time -node 4 -dof 3 disp recorder Node -file rot8ac.out -time -node 8 -dof 3 disp recorder Drift drift84ac.out 4 8 2 1 recorder Drift curvatureac.out 8 9 3 1 Chapter 9 -- Contributed Uniaxial Materials recorder Drift curvatureac1.out 13 9 3 1 recorder Element -file secstrspac.out -time -ele 8 section 5 fiber -$Rb 0 5 stressStrain recorder Element -file secstrsnac.out -time -ele 8 section 5 fiber $Rb 0 5 stressStrain recorder Element -file secstrcac.out -time -ele 8 section 5 fiber $Rc 0 1 stressStrain recorder plot topdispac.out Node12_Ydisp 10 10 300 300 -columns 2 1 set P -90.0 pattern Plain 1 "Constant" { load 12 $P 0.0 0.0 } # Define analysis parameters integrator LoadControl 0 system SparseGeneral -piv test NormDispIncr 1.0e-4 2000 numberer Plain constraints Plain algorithm KrylovNewton analysis Static # Do one analysis for constant axial load analyze 1 # Define reference force pattern Plain 2 "Linear" { load 12 0.0 1.0 0.0 } set dU1 -0.02 set dU2 0.02 # Perform the analysis integrator DisplacementControl 12 2 $dU1 analyze 33 integrator DisplacementControl 12 2 $dU2 analyze 65 integrator DisplacementControl 12 2 $dU1 analyze 32 integrator DisplacementControl 12 2 $dU1 analyze 67 integrator DisplacementControl 12 2 $dU2 analyze 134 integrator DisplacementControl 12 2 $dU1 analyze 67 integrator DisplacementControl 12 2 $dU1 analyze 101 integrator DisplacementControl 12 2 $dU2 analyze 202 integrator DisplacementControl 12 2 $dU1 analyze 101 integrator DisplacementControl 12 2 $dU1 analyze 135 integrator DisplacementControl 12 2 $dU2 analyze 270 integrator DisplacementControl 12 2 $dU1 analyze 135 55 Chapter 9 -- Contributed Uniaxial Materials 56 integrator DisplacementControl 12 2 $dU1 analyze 202 integrator DisplacementControl 12 2 $dU2 analyze 404 integrator DisplacementControl 12 2 $dU1 analyze 202 materialmodels.tcl # load pattern test # ----------------# Jian Zhao, Iowa State University # June 20, 2004 # ----------------#----------Create Model (2-D 3dof problem-plane truss) model basic -ndm 2 -ndf 3 # Create nodes # tag X Y node 1 0.0 node 2 1.0 node 3 1.0 0.0 0.0 1.0 # Fix supports at node 1 # Tag DX DY fix 1 1 1 fix 2 0 1 fix 3 0 0 RZ 1 1 1 # Define faked material # unconfined uniaxialMaterial Concrete01 # confined CONCRETE uniaxialMaterial Concrete01 # unconfined ratio ft et0 uniaxialMaterial Concrete03 0.1 2.5 0.0025 # STEEL02 $a1 $a2 $a3 uniaxialMaterial Steel02 0.04 1.0 0.04 #uniaxialMaterial StrPen01 $Cd $db $fc uniaxialMaterial StrPen01 0.0 1.0 4.35 set mat 5 # Define truss element # tag element truss element truss element truss #----------DONE Create Model CONCRETE 1 tag f'c 2 CONCRETE ft0 3 tag -4.5 ec0 -6.0 tag beta -6.0 f'c -0.002 -0.004 f'c etu -0.004 tag $Fy 4 68.1 tag $sy 5 0.022 jnode 1 1 2 A 2 3 3 ec0 0.0 f'cu -5.0 ec0 -0.006 ecu -0.014 f'cu ecu f'cu ecu -5.0 -0.014 0.5 0.45 0.0015 $E $b $R0 $cR1 $cR2 29000 0.010 22 0.925 0.15 $fy $su $fu $Kz $R 60 0.77 90 0.5 0.65 -node 2 -dof 1 $a4 1.0 $la 25.0 inode 1 2 3 #----------Create Analysis # Create displacement command recorders recorder Node -file Mattag 1 $mat 1 $mat 1 $mat commanddisp.out -time Chapter 9 -- Contributed Uniaxial Materials disp # Create reaction force recorders (for checking material models) recorder Element -file reaction.out -time -ele 1 57 axialForce pattern Plain 1 "Series -dt 0.01 -filePath cycload.txt" { sp 2 1 1 } integrator LoadControl .01 system SparseGeneral -piv test NormUnbalance 1.0e-6 100 numberer Plain constraints Transformation algorithm Newton analysis Static set numIncr 310 analyze $numIncr wipe Concrete04 Material -- Popovics Concrete Material Contact Authors: Laura Lowes: http://www.ce.washington.edu/~lowes/ Michael Berry: [email protected] This command is used to construct a uniaxial Popovics concrete material object with degraded linear unloading/reloading stiffness according to the work of Karsan-Jirsa and tensile strength with exponential decay. uniaxialMaterial Concrete04 $matTag $fc $ec $ecu $Ec <$ft $et> <$beta> $matTag unique material object integer tag $fc floating point values defining concrete compressive strength at 28 days (compression is negative)* $ec floating point values defining concrete strain at maximum strength* $ecu floating point values defining concrete strain at crushing strength* $Ec floating point values defining initial stiffness** $fct floating point value defining the maximum tensile strength of concrete Chapter 9 -- Contributed Uniaxial Materials $et floating point value defining ultimate tensile strain of concrete $beta floating point value defining the exponential curve parameter to define the residual stress (as a factor of $ft) at $etu 58 *NOTE: Compressive concrete parameters should be input as negative values. **NOTE: The envelope of the compressive stress-strain response is defined using the model 57000 fc proposed by Popovics (1973). If the user defines Ec = (in psi) then the envelope curve is identical to proposed by Mander et al. (1988). Model Characteristic: For loading in compression, the envelope to the stress-strain curve follows the model proposed by Popovics (1973) until the concrete crushing strength is achieved and also for strains beyond that corresponding to the crushing strength. ε f ci = f c′ ci εc Esec = f ci ε ci n= n ε n − 1 + ci εc n Ec Ec − Esec where subscript i refer to the stress/strain at any load step. For unloading and reloading in compression, the Karsan-Jirsa model (1969) is used to determine the slope of the curve. For tensile loading, an exponential curve is used to define the envelope to the stress-strain curve: ft Ec f t ε t − ft Ec εi − β σ i = ft ft where β multiplier refers to the stress at ultimate tensile strain, generally the value considered is 0.1. For unloading and reloading in tensile, the secant stiffness is used to define the path. Chapter 9 -- Contributed Uniaxial Materials 59 The following figures show example stress-strain histories generated using the model. -4 Tensile response stress 0.4 1.5 cyclic tension x 10 1 0.2 0.5 0 0 0.5 1 1.5 strain 0 0 500 Compressive response stress 1500 2000 x 10 cyclic compression 3 0.01 2 0.005 1 0 0 2 4 strain 6 8 0 0 200 400 600 800 1000 -3 x 10 -3 Tensile-Compressive response 3 10 2 stress 1000 -4 x 10 cyclic tension-compression 5 1 0 0 -1 -2 0 2 4 strain 6 8 -3 x 10 -5 0 200 400 600 800 1000 1200 Chapter 9 -- Contributed Uniaxial Materials 60 cyclic tension compression response envelope 3 2.5 stress (ksi) 2 1.5 1 0.5 0 -0.5 -1 0 1 2 3 strain 4 5 6 7 -3 x 10 Reference: Mander, J. B., Priestley, M. J. N., and Park, R. (1988). "Theoretical stress-strain model for confined concrete." Journal of Structural Engineering ASCE, 114(8), 1804-1825. Popovics, S. (1973). " A numerical approach to the complete stress strain curve for concrete." Cement and concrete research, 3(5), 583-599. Karsan, I. D., and Jirsa, J. O. (1969). "Behavior of concrete under compressive loading." Journal of Structural Division ASCE, 95(ST12). 61 CHAPTER 10 Concrete04 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Fatigue Material Author: Patxi Uriz Contact Information: Exponent - Failure Analysis Associates 149 Commonwealth Drive Menlo Park, CA 94025 [email protected] The fatigue material uses a modified rainflow cycle counting algorithm to accumulate damage in a material using Miner’s Rule. Element stress/strain relationships become zero when fatigue life is exhausted. uniaxialMaterial Fatigue $matTag $tag <-E0 $E0> <-m $m> <-min $min> <-max $max> This material model accounts for the effects of low cycle fatigue. A modified rainflow cycle counter has been implemented to track strain amplitudes. This cycle counter is used in concert with a linear strain accumulation model (i.e. Miner’s Rule), based on Coffin-Manson log-log relationships describing low cycle fatigue failure. This material “wraps” around another material and does not influence the stress-strain (or force-deformation) relationship of the parent material. Once the Fatigue material model reaches a damage level of 1.0, the force (or stress) of the parent material becomes zero (1.0x10-8 times the call to the material). If failure is triggered in compression, the material stress is dropped at the next zero-force crossing (i.e. compression force never “drops” to zero). The Fatigue material assumes that each point is the last point of the history, and tracks damage with this assumption. If failure is not triggered, this pseudo-peak is discarded. Chapter 10 Concrete04 -- Material Behavior 62 The material also has the ability to trigger failure based on a maximum or minimum strain (i.e. not related to fatigue). The default for these values is set to very large numbers. The default values are calibrated parameters from low cycle fatigue tests of European steel sections Ballio and Castiglioni (1995), for more information about how material was calibrated, the user is directed to Uriz (2005). $matTag unique material object integer tag $tag Unique material object integer tag for the material that is being “wrapped” $E0 Value of strain at which one cycle will cause failure (default 0.191) $m Slope of Coffin-Manson curve in log-log space (default -0.458) $min Global minimum value for strain or deformation (default -1e16) $max Global maximum value for strain or deformation (default 1e16) Example Command: Example command: uniaxialMaterial Fatigue 2 1 -E0 0.191 -m -0.458 where 2 is the Fatigue material tag, and 1 is the material that is being wrapped, the -E0 and -m values I have listed here are the default values. Valid recorder objects for the material are ‘stress’,’tangent’, ‘strain’, ‘stressStrain’, and ‘damage’. The stress, strain, and tangent recorder options must be available in the material that you are wrapping. Chapter 10 FatigueMaterialExample.tcl FatigueMaterialExample.tcl 63 Chapter 10 FatigueMaterialExample.tcl # Run an example script to test the fatigue material model ## Define model and loads model BasicBuilder -ndm 2 -ndf 2 node 1 0.0 0.0 node 2 0.0 0.0 fix 1 1 1 fix 2 0 1 uniaxialMaterial Steel01 1 60.0 29800.0 0.003 uniaxialMaterial Fatigue 2 1 element zeroLength 1 1 2 -mat 2 -dir 1 pattern Plain 1 "Linear" { # nd load 2 FX 1.0 0.0 0.0 } ## Recorders recorder Element -file "Damage.out" -time -ele 1 material 1 damage recorder Element -file "StressStrain.out" \ -time -ele 1 material 1 stressANDstrain ## Set analysis parameters test EnergyIncr 1.0e-8 200 0 algorithm Newton system UmfPack numberer RCM constraints Plain analysis Static ## Source the displacement history, and initialize analysis parameters #source RandomStrainHstory1.tcl source RandomStrainHstory2.tcl set LoopLength [array size disp] set h 1 set controlNode 2 set currentDisp [nodeDisp $controlNode 1 ] puts [format " \n STARTING DISPLACEMENT = %5.3f \n" $currentDisp] 64 Chapter 10 Figure 1. DamageExample1.jpg FatigueMaterialExample.tcl 65 Chapter 10 Figure 2. DamageExample2.jpg FatigueMaterialExample.tcl 66 Chapter 10 RandomStrainHistory1.tcl FatigueMaterialExample.tcl 67 Chapter 10 set disp(1) -0.0110 set disp(2) -0.0106 set disp(3) -0.0129 set disp(4) -0.0089 set disp(5) -0.0113 set disp(6) -0.0069 set disp(7) -0.0079 set disp(8) 0.0020 set disp(9) -0.0118 set disp(10) -0.0057 set disp(11) -0.0180 set disp(12) -0.0097 set disp(13) -0.0195 set disp(14) -0.0152 set disp(15) -0.0321 set disp(16) -0.0281 set disp(17) -0.0323 set disp(18) -0.0145 set disp(19) -0.0179 set disp(20) -0.0098 set disp(21) -0.0297 set disp(22) -0.0219 set disp(23) -0.0304 set disp(24) -0.0172 set disp(25) -0.0199 set disp(26) -0.0175 set disp(27) -0.0261 set disp(28) -0.0201 set disp(29) -0.0243 set disp(30) -0.0089 set disp(31) -0.0092 set disp(32) 0.0047 set disp(33) -0.0104 set disp(34) 0.0087 set disp(35) 0.0048 set disp(36) 0.0059 set disp(37) 0.0022 FatigueMaterialExample.tcl 68 Chapter 10 FatigueMaterialExample.tcl 69 Chapter 10 RandomStrainHistory2.tcl FatigueMaterialExample.tcl 70 Chapter 10 set disp(1) -0.0058 set disp(2) 0.0028 set disp(3) -0.0043 set disp(4) -0.0040 set disp(5) -0.0088 set disp(6) -0.0048 set disp(7) -0.0057 set disp(8) 0.0120 set disp(9) 0.0077 set disp(10) 0.0138 set disp(11) 0.0056 set disp(12) 0.0125 set disp(13) 0.0019 set disp(14) 0.0036 set disp(15) -0.0053 set disp(16) 0.0028 set disp(17) -0.0011 set disp(18) 0.0084 set disp(19) 0.0024 set disp(20) 0.0154 set disp(21) 0.0078 set disp(22) 0.0123 set disp(23) -0.0006 set disp(24) 0.0022 set disp(25) -0.0022 set disp(26) 0.0027 set disp(27) 0.0012 set disp(28) 0.0033 set disp(29) -0.0059 set disp(30) 0.0058 set disp(31) -0.0081 set disp(32) 0.0005 set disp(33) -0.0092 set disp(34) -0.0024 set disp(35) -0.0066 set disp(36) -0.0042 set disp(37) -0.0094 FatigueMaterialExample.tcl 71 Chapter 10 FatigueMaterialExample.tcl 72 References Uriz, Patxi (2005) “Towards Earthquake Resistant Design of Concentrically Braced Steel Structures,” Doctoral Dissertation, Structural Engineering, Mechanics, and Materials, Department of Civil and Environmental Engineering, University of California, Berkeley, December 2005 Ballio, G., and Castiglioni, C. A. (1995). "A Unified Approach for the Design of Steel Structures under Low and/or High Cycle Fatigue." Journal of Constructional Steel Research, 34, 75-101. 73 CHAPTER 11 Limit State Material Contact Author: Ken Elwood: http://www.civil.ubc.ca/faculty/KElwood/KElwood.html This command is used to construct a uniaxial hysteretic material object with pinching of force and deformation, damage due to ductility and energy, and degraded unloading stiffness based on ductility. Failure of the material is defined by the associated limit curve. uniaxialMaterial LimitState $matTag $s1p $e1p $s2p $e2p $s3p $e3p $s1n $e1n $s2n $e2n $s3n $e3n $pinchX $pinchY $damage1 $damage2 $beta $curveTag $curveType. $matTag unique material object integer tag $s1p $e1p stress and strain (or force & deformation) at first point of the envelope in the positive direction $s2p $e2p stress and strain (or force & deformation) at second point of the envelope in the positive direction $s3p $e3p stress and strain (or force & deformation) at third point of the envelope in the positive direction (optional) $s1n $e1n stress and strain (or force & deformation) at first point of the envelope in the negative direction* $s2n $e2n stress and strain (or force & deformation) at second point of the envelope in the negative direction* $s3n $e3n stress and strain (or force & deformation) at third point of the envelope in the negative direction (optional)* $pinchX pinching factor for strain (or deformation) during reloading $pinchY pinching factor for stress (or force) during reloading $damage1 damage due to ductility: D1(µ-1) $damage2 damage due to energy: D2(Ei/Eult) $beta power used to determine the degraded unloading stiffness based on ductility, µ-β (optional, default=0.0) $curveTag an integer tag for the LimitCurve defining the limit surface $curveType an integer defining the type of LimitCurve (0 = no curve, 1 = axial curve, all other curves can be any other integer) Chapter 11 Limit State Material 74 *NOTE: negative backbone points should be entered as negative numeric values In This Chapter Shear Limit Curve..................................................... 78 Axial Limit Curve ...................................................... 79 Shear Limit Curve..................................................... 80 Example Files for Limit State Material ...................... 82 Shear Limit Curve This command is used to construct a shear limit curve object that is used to define the point of shear failure for a LimitStateMaterial object. Point of shear failure based on empirical drift capacity model from Chapter 2. limitCurve Shear $curveTag $eleTag $rho $fc $b $h $d $Fsw $Kdeg $Fres $defType $forType <$ndI $ndJ $dof $perpDirn $delta>. $curveTag unique limit curve object integer tag $eleTag integer element tag for the associated beam-column element $rho transverse reinforcement ratio (Ast/bh) $fc concrete compressive strength (psi) $b column width (in.) $h full column depth (in.) $d effective column depth (in.) $Fsw floating point value describing the amount of transverse reinforcement (Fsw = Astfytdc/s) $Kdeg If positive: unloading stiffness of beam-column element (Kunload from Figure 4-8) if negative: slope of third branch of post-failure backbone (see Figure 4-6) $Fres floating point value for the residual force capacity of the post-failure backbone (see Figure 4-6) Chapter 11 Limit State Material $defType integer flag for type of deformation defining the abscissa of the limit curve 1 = maximum beam-column chord rotations 2 = drift based on displacment of nodes ndI and ndJ $forType integer flag for type of force defining the ordinate of the limit curve* 0 = force in associated limit state material 1 = shear in beam-column element $ndI integer node tag for the first associated node (normally node I of $eleTag beam-column element) $ndJ integer node tag for the second associated node (normally node J of $eleTag beam-column element) $dof nodal degree of freedom to monitor for drift** $perpDirn perpendicular global direction from which length is determined to compute drift** $delta drift (floating point value) used to shift shear limit curve 75 NOTE: * Option 1 assumes no member loads. ** 1 = X, 2 = Y, 3 = Z Axial Limit Curve This command is used to construct an axial limit curve object that is used to define the point of axial failure for a LimitStateMaterial object. Point of axial failure based on model from Chapter 3. After axial failure response of LimitStateMaterial is forced to follow axial limit curve. limitCurve Axial $curveTag $eleTag $Fsw $Kdeg $Fres $defType $forType <$ndI $ndJ $dof $perpDirn $delta>. $curveTag unique limit curve object integer tag $eleTag integer element tag for the associated beam-column element $Fsw floating point value describing the amount of transverse reinforcement (Fsw = Astfytdc/s) $Kdeg floating point value for the slope of the third branch in the post-failure backbone, assumed to be negative (see Figure 4-6) Chapter 11 Limit State Material $Fres floating point value for the residual force capacity of the post-failure backbone (see Figure 4-6) $defType integer flag for type of deformation defining the abscissa of the limit curve 1 = maximum beam-column chord rotations 2 = drift based on displacment of nodes ndI and ndJ $forType integer flag for type of force defining the ordinate of the limit curve* 0 = force in associated limit state material 1 = shear in beam-column element 2 = axial load in beam-column element $ndI integer node tag for the first associated node (normally node I of $eleTag beam-column element) $ndJ integer node tag for the second associated node (normally node J of $eleTag beam-column element) $dof nodal degree of freedom to monitor for drift** $perpDirn perpendicular global direction from which length is determined to compute drift** $delta drift (floating point value) used to shift axial limit curve 76 NOTE: * Options 1 and 2 assume no member loads. ** 1 = X, 2 = Y, 3 = Z Shear Limit Curve This command is used to construct a shear limit curve object that is used to define the point of shear failure for a LimitStateMaterial object. Point of shear failure based on empirical drift capacity model from Chapter 2. limitCurve Shear $curveTag $eleTag $rho $fc $b $h $d $Fsw $Kdeg $Fres $defType $forType <$ndI $ndJ $dof $perpDirn $delta>. $curveTag unique limit curve object integer tag $eleTag integer element tag for the associated beam-column element $rho transverse reinforcement ratio (Ast/bh) $fc concrete compressive strength (psi) Chapter 11 Limit State Material $b column width (in.) $h full column depth (in.) $d effective column depth (in.) $Fsw floating point value describing the amount of transverse reinforcement (Fsw = Astfytdc/s) $Kdeg If positive: unloading stiffness of beam-column element (Kunload from Figure 4-8) if negative: slope of third branch of post-failure backbone (see Figure 4-6) $Fres floating point value for the residual force capacity of the post-failure backbone (see Figure 4-6) $defType integer flag for type of deformation defining the abscissa of the limit curve 1 = maximum beam-column chord rotations 2 = drift based on displacment of nodes ndI and ndJ $forType integer flag for type of force defining the ordinate of the limit curve* 0 = force in associated limit state material 1 = shear in beam-column element $ndI integer node tag for the first associated node (normally node I of $eleTag beam-column element) $ndJ integer node tag for the second associated node (normally node J of $eleTag beam-column element) $dof nodal degree of freedom to monitor for drift** $perpDirn perpendicular global direction from which length is determined to compute drift** $delta drift (floating point value) used to shift shear limit curve NOTE: * Option 1 assumes no member loads. ** 1 = X, 2 = Y, 3 = Z 77 Chapter 11 Limit State Material Example Files for Limit State Material CenterCol_basicModel.tcl 78 Chapter 11 Limit State Material # Example analysis file for LimitStateMaterial # # Units: kip, in # KJE, Feb 2003 source tags.tcl # Number of analysis steps and step size #set nSteps 35000 set nSteps 6000 set dlamda 0.1 # Loading option set loading push #set loading cyclic # Analysis control option set control load #set control displ ########################### # Build model ########################### model BasicBuilder -ndm 2 -ndf 3 ################################ # Define nodal mesh and B.C.s ################################ set L 58.0 # tag X Y node 1 0.0 0.0 node 2 0.0 0.0 node 3 0.0 $L node 4 0.0 $L node 5 0.0 $L # fix fix fix tag DX DY RZ 1 1 1 1 4 0 0 1 5 1 1 1 ############################## # Create column section ############################## source CenterColSecFiber.tcl ################################# # Define the beam-column element ################################# geomTransf PDelta 1 set nint 5 element nonlinearBeamColumn $bcTag 2 3 $nint $flexSec 1 -iter 5 1e-15 #################################### # Define the zero-length end springs #################################### #rigid material uniaxialMaterial Elastic $rigidMatTag 9.9e9 #bottom of column slip spring 79 Chapter 11 Limit State Material CenterColShearSpring.tcl # Defines Shear Spring # Units: kip, in # KJE, Feb 2003 # slopes of shear spring backbone set rigidSlope 1700; #Values when using zero-length spring (G*Av/L) set negSlope -8; # residual shear capacity set Vr 3.0; # strengths for initial response set Vi1 25.0; set Vi2 30.0; set Vi3 45.0; # stiffness of unloading slope for flexural component set kf 24.7 ;# measured off hysteresis plot from analysis # define limit surface using shear drift model # tag eleTag # rho f'c b h d Fsw # Kdeg Fres defType forType nodeI nodeJ dof perpDirn limitCurve Shear $shearCurveTag $bcTag\ 0.0018 3517.0 9.0 9.0 7.75 11.87\ $kf $Vr 2 0 1 4 1 2 0.0 # define HystereticMaterial uniaxialMaterial LimitState $shearTag\ $Vi1 [expr $Vi1/$rigidSlope] $Vi2 [expr $Vi2/$rigidSlope] $Vi3 [expr $Vi3/$rigidSlope]\ [expr -$Vi1] [expr -$Vi1/$rigidSlope] [expr -$Vi2] [expr -$Vi2/$rigidSlope] [expr -$Vi3] [expr $Vi3/$rigidSlope]\ $pinchX $pinchY $damage1 $damage2 $beta $shearCurveTag 2 0 CenterColSecFiber.tcl # Defines Center Column Section # Units: kip, in # KJE, Feb 2003 set h 9.0 set b $h set beta1 100.0 set beta2 100.0 set spall nospalling set alphaS 0.015 # Set parameters for fiber section set nfCore 28 set nfCover 4 set c 1.0 set z [expr $h/2.0-$c] set y [expr $b/2.0-$c] 80 Chapter 11 Limit State Material 81 set nz [expr -$z] set ny [expr -$y] set zc [expr $z+$c] set yc [expr $y+$c] set nzc [expr -$zc] set nyc [expr -$yc] set fc -3.517 set fc [expr -3.517*$beta2/100.0] set fcu [expr $fc*$beta1/100.0] set fy 69.5 set Es 29000.0 set esy [expr $fy/$Es] set esu 0.15 set fu [expr $fy+$alphaS*$Es*($esu-$esy)] # Define parameters for elastic section set Ec 3400 set A [expr $h*$b] set Ig [expr $h*$h*$h*$b/12] set Mcr set Kcr 228.0 1.2e-4 set My set Ky 554.0 ;# for trilinear model (Mmax from UCFyber) 6.3e-4 ;# based on UCFyber results set Ry [expr $Ky*$L/6.0] ; # yield rotation set EI [expr $My/$Ky] set alpha 0.05 set Ku set Mu set Ru 0.05 [expr $My+$alpha*$EI*($Ku-$Ky)] 0.5 ;# this assumes alpha is very small so diff between M-curv alpha and M-rot alpha is negligible set pinchX 0.5 set pinchY 0.4 set damage1 0.0 set damage2 0.0 set beta set Ic 0.4; #only to be used with version 1.3 [expr $EI/$Ec] set slipStiff 91000 ;# see CenterColumnMomCurv.xls set Rslipy [expr $My/$slipStiff] set alphaSlip [expr $alpha/(1.0+(1.0-$alpha)*($slipStiff*$L/6/$EI))] set Rslipu [expr $Rslipy+($Mu-$My)/($alphaSlip*$slipStiff)] # Create axial failure spring source CenterColAxialSpring.tcl # Create shear failure spring source CenterColShearSpring.tcl section Aggregator $shearAxialOnlySec $shearTag Vy $axialFailTag P Chapter 11 # Create fiber section # Define uniaxialMaterials # tag f'c epsc f'cu epscu uniaxialMaterial Concrete01 $coreTag $fc -0.002 $fcu -0.0052 if {$spall == "spalling"} { uniaxialMaterial Concrete01 $coverTag $fc -0.002 0.0 } elseif {$spall == "nospalling"} { uniaxialMaterial Concrete01 $coverTag $fc -0.002 $fcu -0.0052 -0.0060 } else { puts stderr "Invalid spalling option: $spall" } # tag fy E hardening ratio #uniaxialMaterial Steel02 $steelTag 69.5 29000 $alphaS uniaxialMaterial Hysteretic $steelTag $fy $esy $fu $esu\ -$fy -$esy -$fu -$esu 1.0 1.0 0 0 # Define the fiber section section Fiber $flexSec { # Define the concrete patch with fibers for unidirectional bending patch quadr $coreTag 1 $nfCore $ny $z $ny $nz $y $nz $y $z # Define the four cover patches patch quadr $coverTag 1 $nfCover $nyc $zc $nyc $nzc $ny $nz $ny $z patch quadr $coverTag 1 $nfCover $y $z $y $nz $yc $nzc $yc $zc patch quadr $coverTag 1 $nfCore $ny $nz $nyc $nzc $yc $nzc $y $nz patch quadr $coverTag 1 $nfCore $nyc $zc $ny $z $y $z $yc $zc # Define the reinforcement explicitly using fiber command # yloc zloc area matTag fiber -3.250 3.250 0.2 $steelTag 0 fiber -3.250 -3.250 0.2 $steelTag 0 fiber 3.250 -3.250 0.2 $steelTag 0 fiber 3.250 3.250 0.2 $steelTag 0 fiber -3.187 0.0 0.31 $steelTag 0 fiber 3.187 0.0 0.31 $steelTag 0 fiber 0.0 -3.187 0.31 $steelTag 0 fiber 0.0 3.187 0.31 $steelTag 0 } # moment-rotation end springs for slip (assume elastic) uniaxialMaterial Elastic $centerSlipTag $slipStiff set Acenter $A set Eccenter $Ec set Iccenter $Ic CenterColAxialSpring.tcl # Defines Axial Spring # Units: kip, in # KJE, Feb 2003 set Fsw [expr 11.87/1.0] Limit State Material 82 Chapter 11 Limit State Material 83 # Axial elastic stiffness (only one axial spring used) set axialElasticSlope [expr 99.0*$Ec*$A/$L]; #99 times more rigid than column set axialNegSlope -90.0; #slope = axial load/axial displ} # residual capacity set Pr 5.0; # axial loads used for setting initial elastic slope set P1 65.0; set P2 75.0; set P3 85.0; # define limit surface # tag eleTag Fsw Kdeg Fres defType forType nodeI nodeJ dof perpDir delta eleRemove limitCurve Axial $axialCurveTag $bcTag $Fsw $axialNegSlope $Pr 2 2 1 4 1 2 0.0 0 # define LimitStateMaterial uniaxialMaterial LimitState $axialFailTag\ $P1 [expr $P1/$axialElasticSlope] $P2 [expr $P2/$axialElasticSlope] $P3 [expr $P3/$axialElasticSlope]\ [expr -$P1] [expr -$P1/$axialElasticSlope] [expr -$P2] [expr -$P2/$axialElasticSlope] [expr -$P3] [expr $P3/$axialElasticSlope]\ 0.5 0.5 0.0 0.0 0.0 $axialCurveTag 1 Tags.tcl # UniaxialMaterial tags set coreTag 1 set coverTag 2 set steelTag 3 ;# core concrete ;# cover concrete ;# steel set shearTag 4 ;# shear limit state material set momTag 5 ;# mom-curv hysteretic model set axialTag 6 ;# elastic axial force-strain model set momDegTag 7 ;# degrading moment-curv hysteretic model set axialFailTag 8 ;# axial limit state material set centerSlipTag 9 ;# elastic slip spring set rigidMatTag 10 set softMatTag 11 # Section tags set flexSec 1 set shearSec 2 set flexTopSec 3 set flexBotSec 4 set axialSec 5 set shearAxialSec 6 set axialOnlySec 7 set flexShearSec 8 set shearAxialOnlySec 9 # Limit Curve tags set shearCurveTag 1 set axialCurveTag 2 # Element tags Chapter 11 set bcTag 99 cyclic.txt 1.2557524028480316e-003 1.2557524028480316e-003 3.0608252487525078e-003 1.5641001656305775e-003 1.1015791633965932e-003 -1.8313432524451514e+000 PINCHING4 Material Contact Author: Limit State Material 84 Chapter 11 Limit State Material 85 This command is used to construct a uniaxial material that represents a ‘pinched’ loaddeformation response and exhibits degradation under cyclic loading. Cyclic degradation of strength and stiffness occurs in three ways: unloading stiffness degradation, reloading stiffness degradation, strength degradation. uniaxialMaterial Pinching4 $matTag $ePf1 $ePd1 $ePf2 $ePd2 $ePf3 $ePd3 $ePf4 $ePd4 <$eNf1 $eNd1 $eNf2 $eNd2 $eNf3 $eNd3 $eNf4 $eNd4> $rDispP $rForceP $uForceP <$rDispN $rForceN $uForceN > $gK1 $gK2 $gK3 $gK4 $gKLim $gD1 $gD2 $gD3 $gD4 $gDLim $gF1 $gF2 $gF3 $gF4 $gFLim $gE $dmgType $matTag unique material object integer tag $ePf1 $ePf2 $ePf3 $ePf4 floating point values defining force points on the positive response envelope $ePd1 $ePd2 $ePd3 $ePd4 floating point values defining deformation points on the positive response envelope $eNf1 $eNf2 $eNf3 $eNf4 floating point values defining force points on the negative response envelope (optional, default: negative of positive envelope values) $eNd1 $eNd2 $eNd3 $eNd4 floating point values defining deformations points on the negative response envelope (optional, default: negative of positive envelope values) $rDispP floating point value defining the ratio of the deformation at which reloading occurs to the maximum historic deformation demand $rForceP floating point value defining the ratio of the force at which reloading begins to force corresponding to the maximum historic deformation demand $uForceP floating point value defining the ratio of strength developed upon unloading from negative load to the maximum strength developed under monotonic loading $rDispN floating point value defining the ratio of the deformation at which reloading occurs to the minimum historic deformation demand (optional, default: $rDispP) Chapter 11 Limit State Material $rForceN floating point value defining the ratio of the force at which reloading begins to the force corresponding to the minimum historic deformation demand (optional, default: $rForceP) $uForceN floating point value defining the ratio of the strength developed upon unloading from a positive load to the minimum strength developed under monotonic loading (optional, default: $rForceP) $gK1 $gK2 $gK3 $gK4 $gKLim floating point values controlling cyclic degradation model for unloading stiffness degradation $gD1 $gD2 $gD3 $gD4 $gDLim floating point values controlling cyclic degradation model for reloading stiffness degradation $gF1 $gF2 $gF3 $gF4 $gFLim floating point values controlling cyclic degradation model for strength degradation $gE floating point value used to define maximum energy dissipation under cyclic loading. Total energy dissipation capacity is defined as this factor multiplied by the energy dissipated under monotonic loading. $dmgType string to indicate type of damage (option: “cycle”, “energy”) 86 Chapter 11 Limit State Material 87 NOTE: Figure 1: Definition of Pinching4 Uniaxial Material Model Damage Models: Stiffness and strength are assumed to deteriorate due to the imposed “load” history. The same basic equations are used to describe deterioration in strength, unloading stiffness and reloading stiffness: k i = k 0 ⋅ (1 − δk i ) Chapter 11 Limit State Material 88 where ki is the unloading stiffness at time ti, ko is the initial unloading stiffness (for the case of no damage), and δk i (defined below) is the value of the stiffness damage index at time t . i d max i = d max 0 ⋅ (1 + δd i ) where d max i is the deformation demand that defines the end of the reload cycle for increasing d deformation demand, max 0 is the maximum historic deformation demand (which would be the deformation demand defining the end of the reload cycle if degradation of reloading stiffness is ignored), and (fmax )i δd i (defined below) is the value of reloading stiffness damage index at time t . = (fmax )0 ⋅ (1 − δf i ) i Chapter 11 where (fmax )i is the current envelope maximum strength at time ti, Limit State Material (fmax )0 89 is the initial envelope δf maximum strength for the case of no damage, and i (defined below) is the value of strength value index at time ti. δk δd δf i and i , may be defined to be a function of displacement history The damage indices, i , only ($dmgType = “cycle”) or displacement history and energy accumulation ($dmgType = “energy”). For either case, all of the damage indices are computed using the same basic equation. If the damage indices are assumed to be a function of displacement history and energy accumulation, the unloading stiffness damage index, ~ δk i = gK1 ⋅ d max ( ) gK 3 Ei + gK 2 ⋅ E monotonic where d min i d max i ~ d max = max , def min def max Ei = ∫ dE load history E monotonic = gE ⋅ dE monotonic load history ∫ gK 4 ≤ gKLim δk i is computed as follows: Chapter 11 Limit State Material 90 with Emonotic equal to the energy required to achieve under monotonic loading the deformation that defines failure, defmax and defmin the positive and negative deformations that define failure. The δd δf i and i , are computed using the same equations with degradation other damage indices, model parameters gK* replaced by gF* and gD*, as is appropriate. The above expressions were meant for “Energy” type damage. The user specification of “Energy” type damage implements damage due to displacement as well as energy. Other type of damage can be activated: “Cycle” which implements damage due to displacement as well as damage accrued due to load cycle counting. The expressions for the “Cycle” damage are given below. If the damage indices are assumed to be a function only of the displacement history, the unloading stiffness damage index, (~ ) δk i = gK1 ⋅ d max gK 3 δk i is computed as follows: + gK 2 ⋅ (Cycle ) gK 4 ≤ gKLim where d min i d max i ~ d max = max , def min def max with Cycle equal to the number of cycles accrued in the loading history, defmax and defmin the δd δf i and i, positive and negative deformations that define failure. The other damage indices, are computed using the same equations with degradation model parameters gK* replaced by gF* and gD*, as is appropriate. EXAMPLE: main input file: RCyclicPinch.tcl (page 101) supporting files: procUniaxialPinching.tcl (page 106) procRCycDAns.tcl (page 108) Chapter 11 Limit State Material 91 PINCHING4 Uniaxial Material Model Discussion PINCHING4 Uniaxial Material Model Discussion The example files (RCyclicPinch.tcl (page 101), procUniaxialPinching.tcl (page 106), procRCycDAns.tc (page 108)l) create a one-dimensional structural model consisting of a single truss element of length 1.0 and area 1.0 (Figure 1). The Pinching4 uniaxial material model is used to simulate the stress-strain response of the truss material. The truss is subjected to a pseudo-static cyclic loading. Several files are provided that include different input parameters for the Pinching4 material model and result in different load-displacement histories for the truss structure. Refer to the documentation about the Pinching4 uniaxial material model for additional information. Input for the Pinching4 Uniaxial Material Model Refer to the documentation of the Pinching4 uniaxial material model for an explanation of the following notation. Response Envelopes: In these examples the pinching material model is demonstrated with two different loaddeformation response envelopes. Envelope 1 (Figure 2) defines a hardening-type response while Envelope 2 (Figure 2) defines a softening-type response. Envelope 1 (Figure 2): ePd1 ePf 1 0.0001 2 − eNd1 − eNf 1 ePd 2 ePf 2 0.0055 6 − eNd 2 − eNf 2 = = ePd 3 ePf 3 0.0188 7 − eNd 3 − eNf 3 ePd 4 ePf 4 0.0189 7.2 − eNd 4 − eNf 4 Chapter 11 Limit State Material 92 Envelope 2 (Figure 2): ePd1 ePf 1 0.0001 2 − eNd1 − eNf 1 ePd 2 ePf 2 0.0055 6 − eNd 2 − eNf 2 = = ePd 3 ePf 3 0.0188 7 − eNd 3 − eNf 3 ePd 4 ePf 4 0.0189 0.2 − eNd 4 − eNf 4 Load-Unload Response Parameters: The form of the load-unload response, and the extent of pinching in the response history, is defined by the following six parameters. In each of the examples, the following values are used. [rDispP rForceP uForceP] = [rDispN rForceN uForceN ] = [0.5 0.25 0.05] Chapter 11 Limit State Material 93 Strength and Stiffness Degradation Parameters: The Pinching4 uniaxial material model simulates degradation of stiffness and strength under cyclic loading. The example files demonstrate, individually and in combination, each of the two stiffness degradation options and the one strength degradation option. The following parameters are used to define strength and stiffness degradation, as needed, in the example files. gK1 gK 2 gK 3 gK 4 gKLim 1.0 0.2 0.3 0.2 0.9 gD1 gD 2 gD3 gD 4 gDLim 0.5 0.5 2.0 2.0 0.5 = gF1 gF 2 gF 3 gF 4 gFLim 1.0 0.0 1.0 1.0 0.9 gE 10.0 Tcl Scripts: The following tcl script files are used to run the examples: RCyclicPinch.tcl (page 101) procUniaxialPinching.tcl (page 106) procRCycDAns.tcl (page 108) Lines should be commented out as necessary within RCyclicPinch.tcl to exercise different degradation models of the Pinching4 uniaxial material model and different load histories. If the structure is subjected to a monotonic load history, the load-displacement history is shown in Figure 2 results with the actual response history depending on the envelope chosen for the material model. If the structure is subjected to a cyclic load history (Figure 3), one of the loaddisplacement histories shown in Figure 4 results, with the exact response depending on the strength and stiffness degradation model employed. In this case the damage type activated was “Energy”. The case in which the damage type activated was “Cycle” is shown in Figure 5. Figures: Chapter 11 Figure 3: Structural Model Figure 1: Structural Model Limit State Material 94 Chapter 11 Figure 4: Response Envelopes Figure 2: Response Envelopes Figure 5: Cyclic Displacement History Figure 3: Cyclic Displacement History Limit State Material 95 Chapter 11 Limit State Material 96 Chapter 11 Limit State Material 97 Figure 6: LoadDeformation Response Histories ("Energy type damage") Figure 4: Load-Deformation Response Histories (“Energy type damage”) Figure 7: Plot showing Both Stiffness and Strength Degradation for Damage Type "Energy" and "Cycle" Figure 5. Plot showing Both Stiffness and Strength Degradation for Damage Type "Energy" and "Cycle" RCyclicPinch.tcl ############################################################################################# ##################### # Test example for PINCHING MATERIAL # Chapter 11 Limit State Material 98 # Written: N.Mitra # # Description: uniaxial material with user defined envelope (softening type used here) and damage parameters # # Date: May 04 2002 # ## Model subjected to reverse Cyclic Loading # ## File Name: RCyclicPinch.tcl # # refer to Pinching-Type Material Model.doc for full explanation of the parameters # ############################################################################################# ##################### wipe #create the ModelBuilder object model BasicBuilder -ndm 2 -ndf 2 # add nodes - command: node nodeId xCrd yCrd node 1 0.0 0.0 node 2 1.0 0.0 ## please keep the follwoing procedures on the same path source procUniaxialPinching.tcl source procRCycDAns.tcl ##### Positive/Negative envelope Stress/Load Chapter 11 Limit State Material ### stress1 stress2 stress3 stress4 set pEnvelopeStress [list 2.0 6.0 7.0 0.2] set nEnvelopeStress [list -2.0 -6.0 -7.0 -0.2] ##### Positive/Negative envelope Strain/Deformation ### strain1 strain2 strain3 strain4 set pEnvelopeStrain [list 0.0001 0.0055 0.0188 0.0189] set nEnvelopeStrain [list -0.0001 -0.0055 -0.0188 -0.0189] ##### Ratio of maximum deformation at which reloading begins ### Pos_env. Neg_env. set rDisp [list 0.5 0.5] ##### Ratio of envelope force (corresponding to maximum deformation) at which reloading begins ### Pos_env. Neg_env. set rForce [list 0.25 0.25] ##### Ratio of monotonic strength developed upon unloading ### Pos_env. Neg_env. set uForce [list 0.05 0.05] 99 Chapter 11 ##### Coefficients for Unloading Stiffness degradation ### gammaK1 gammaK2 gammaK3 gammaK4 gammaKLimit set gammaK [list 1.0 0.2 0.3 0.2 0.9] #set gammaK [list 0.0 0.0 0.0 0.0 0.0] ##### Coefficients for Reloading Stiffness degradation ### gammaD1 gammaD2 gammaD3 gammaD4 gammaDLimit set gammaD [list 0.5 0.5 2.0 2.0 0.5] #set gammaD [list 0.0 0.0 0.0 0.0 0.0] ##### Coefficients for Strength degradation ### gammaF1 gammaF2 gammaF3 gammaF4 gammaFLimit set gammaF [list 1.0 0.0 1.0 1.0 0.9] #set gammaF [list 0.0 0.0 0.0 0.0 0.0] set gammaE 10 # material ID set matID 1 # damage type (option: "energy", "cycle") set dam "energy" Limit State Material 100 Chapter 11 Limit State Material 101 # add the material to domain through the use of a procedure procUniaxialPinching $matID $pEnvelopeStress $nEnvelopeStress $pEnvelopeStrain $nEnvelopeStrain $rDisp $rForce $uForce $gammaK $gammaD $gammaF $gammaE $dam # add truss elements - command: element truss trussID node1 node2 A matID element truss 1 1 2 1.0 1 # set the boundary conditions - command: fix nodeID xResrnt? yRestrnt? fix 1 1 1 fix 2 0 1 pattern Plain 1 Linear { load 2 1 0 } recorder Node -file RCyclicPinchR.out -node 2 -dof 1 disp # build the components for the analysis object system ProfileSPD constraints Plain test NormDispIncr 1.0e-8 20 Chapter 11 Limit State Material 102 algorithm Newton numberer RCM ## analysis type used in the procedure is Static set peakpts [list 0.0001 0.001 0.002 0.003 0.005 0.006 0.007 0.009 0.01 0.011 0.012 0.013 ] set increments 10 set nodeTag 2 set dofTag 1 ## start procedure for feeding in ## Reverse Cyclic loading to the model by Disp. control procRCycDAns $increments $nodeTag $dofTag $peakpts # print the results at nodes print node procUniaxialPinching.tcl ############################################################################################# ######## ## # procUniaxialPinching.tcl # # procedure for activating the pinching material given its parameters in the form of list # Chapter 11 Limit State Material 103 # created NM ([email protected]) dated : Feb 2002 # ############################################################################################# ######## proc procUniaxialPinching { materialTag pEnvelopeStress nEnvelopeStress pEnvelopeStrain nEnvelopeStrain rDisp rForce uForce gammaK gammaD gammaF gammaE damage} { # add material - command: uniaxialMaterial ...... paramaters as shown #uniaxialMaterial Pinching4 tag #### stress1P strain1P stress2P strain2P stress3P strain3P stress4P strain4P #### stress1N strain1N stress2N strain2N stress3N strain3N stress4N strain4N #### rDispP rForceP uForceP rDispN rForceN uForceN #### gammaK1 gammaK2 gammaK3 gammaK4 gammaKLimit #### gammaD1 gammaD2 gammaD3 gammaD4 gammaDLimit #### gammaF1 gammaF2 gammaF3 gammaF4 gammaFLimit gammaE $damage uniaxialMaterial Pinching4 $materialTag [lindex $pEnvelopeStress 0] [lindex $pEnvelopeStrain 0] [lindex $pEnvelopeStress 1] [lindex $pEnvelopeStrain 1] [lindex $pEnvelopeStress 2] [lindex $pEnvelopeStrain 2] [lindex $pEnvelopeStress 3] [lindex $pEnvelopeStrain 3] [lindex $nEnvelopeStress 0] [lindex $nEnvelopeStrain 0] [lindex $nEnvelopeStress 1] [lindex $nEnvelopeStrain 1] [lindex $nEnvelopeStress 2] [lindex $nEnvelopeStrain 2] [lindex $nEnvelopeStress 3] [lindex $nEnvelopeStrain 3] [lindex $rDisp 0] [lindex $rForce 0] [lindex $uForce 0] [lindex $rDisp 1] [lindex $rForce 1] [lindex $uForce 1] [lindex $gammaK 0] [lindex $gammaK 1] [lindex $gammaK 2] [lindex $gammaK 3] [lindex $gammaK 4] [lindex $gammaD 0] [lindex $gammaD 1] [lindex $gammaD 2] [lindex $gammaD 3] [lindex $gammaD 4] [lindex $gammaF 0] [lindex $gammaF 1] [lindex $gammaF 2] [lindex $gammaF 3] [lindex $gammaF 4] $gammaE $damage } Chapter 11 Limit State Material 104 procRCycDAns.tcl ############################################################################################# ######## # # # procRCycDAns.tcl # # procedure for reverse cyclic displacement control analysis given the peak pts. # analysis type used : STATIC # Written : N.Mitra # # # ############################################################################################# ######## proc procRCycDAns { incre nodeTag dofTag peakpts} { set x [lindex $peakpts 0] set fir [expr $x/$incre] integrator DisplacementControl $nodeTag $dofTag 0.0 1 $fir $fir # create the analysis object analysis Static # perform the analysis analyze $incre integrator DisplacementControl $nodeTag $dofTag 0.0 1 [expr -$fir] [expr -$fir] analyze [expr 2*$incre] integrator DisplacementControl $nodeTag $dofTag 0.0 1 $fir $fir analyze $incre for {set j 1} {$j < [llength $peakpts]} {incr j 1} { set tx [lindex $peakpts $j] set tinc [expr $tx/$fir] set rt [expr int($tinc)] integrator DisplacementControl $nodeTag $dofTag 0.0 1 $fir $fir analyze $rt integrator DisplacementControl $nodeTag $dofTag 0.0 1 [expr -$fir] [expr -$fir] analyze [expr 2*$rt] integrator DisplacementControl $nodeTag $dofTag 0.0 1 $fir $fir analyze $rt } Chapter 11 Limit State Material 105 ################################ end procRCycDAns.tcl ####################################### } PyTzQz Uniaxial Materials This section describes commands that are used to construct uniaxial materials for p-y, t-z and q-z elements for modeling soil-structure interaction developed at UC Davis. There is available documentation on UCD COmpGeomech work within OpenSees framework at: http://sokocalo.engr.ucdavis.edu/~jeremic/OpenSees/UCD_CG_OpenSees_Commands_2up.pd f (http://sokocalo.engr.ucdavis.edu/~jeremic/OpenSees/UCD_CG_OpenSees_Commands_2up.p df) Also, there is a more in depth writeup (lecture notes) at: http://sokocalo.engr.ucdavis.edu/~jeremic/CG/CompGeomechanicsLectureNotes.pdf (http://sokocalo.engr.ucdavis.edu/~jeremic/CG/CompGeomechanicsLectureNotes.pdf) PySimple1 Material This command is used to construct a PySimple1 uniaxial material object. uniaxialMaterial PySimple1 $matTag $soilType $pult $Y50 $Cd <$c>. $matTag Unique material object integer tag. $soilType soilType = 1 Backbone of p-y curve approximates Matlock (1970) soft clay relation. soilType = 2 Backbone of p-y curve approximates API (1993) sand relation. Chapter 11 Limit State Material $pult Ultimate capacity of the p-y material. Note that “p” or “pult” are distributed loads [force per length of pile] in common design equations, but are both loads for this uniaxialMaterial [i.e., distributed load times the tributary length of the pile]. $Y50 Displacement at which 50% of pult is mobilized in monotonic loading. $Cd Variable that sets the drag resistance within a fully-mobilized gap as Cd*pult. $c The viscous damping term (dashpot) on the far-field (elastic) component of the displacement rate (velocity). Default = 0.0. Nonzero c values are used to represent radiation damping effects. 106 NOTE: Full documentation of the PyLiq1 command is found in PySimple1_documentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/PySimple1_docum entation.PDF) TzSimple1 Material This command is used to construct a TzSimple1 uniaxial material object. uniaxialMaterial TzSimple1 $matTag $tzType $tult $z50 <$c>. $matTag Unique material object integer tag. $tzType tzType = 1 Backbone of t-z curve approximates Reese and O’Neill (1987) relation. tzType = 2 Backbone of t-z curve approximates Mosher (1984) relation. $tult Ultimate capacity of the t-z material. Note that “t” or “tult” are distributed loads [force per length of pile] in common design equations, but are both loads for this uniaxialMaterial [i.e., distributed load times the tributary length of the pile]. $z50 Displacement at which 50% of tult is mobilized in monotonic loading. $c The viscous damping term (dashpot) on the far-field (elastic) component of the displacement rate (velocity). Default = 0.0. Nonzero c values are used to represent radiation damping effects. Chapter 11 Limit State Material 107 NOTE: Full documentation of the TzLiq1 command is included in: TzSimple1_documentation.PDF (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/TzSimple1_docum entation.PDF) QzSimple1 Material This command is used to construct a QzSimple1 uniaxial material object. uniaxialMaterial PySimple1 $matTag $qzType $qult $Y50 <$suction $c>. $matTag Unique material object integer tag. $soilType qzType = 1 Backbone of q-z curve approximates Reese and O’Neill’s (1987) relation for drilled shafts in clay. qzType = 2 Backbone of q-z curve approximates Vijayvergiya’s (1977) relation for piles in sand. $qult Ultimate capacity of the q-z material. Note that “q” or “qult” are stresses [force per unit area of pile tip] in common design equations, but are both loads for this uniaxialMaterial [i.e., stress times tip area]. $z50 Displacement at which 50% of pult is mobilized in monotonic loading. Note that Vijayvergiya’s relation (qzType=2) refers to a “critical” displacement (zcrit) at which qult is fully mobilized, and that the corresponding z50 would be 0. 125zcrit. $suction Uplift resistance is equal to suction*qult. Default = 0.0. The value of suction must be 0.0 to 0.1.* $c The viscous damping term (dashpot) on the far-field (elastic) component of the displacement rate (velocity). Default = 0.0. Nonzero c values are used to represent radiation damping effects.* NOTE: Full documentation of the QzSimple1 command is found in: QzSimple1_Documentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/QzSimple1_docum entation.PDF) *NOTE: Optional variables suction and c must either both be omitted or both be included. Chapter 11 Limit State Material PyLiq1 Material This command is used to construct a PyLiq1 uniaxial material object. uniaxialMaterial PyLiq1 $matTag $soilType $pult $Y50 $Cd $c $pRes $solidElem1 $solidElem2. $matTag Unique material object integer tag. $soilType soilType = 1 Backbone of p-y curve approximates Matlock (1970) soft clay relation. soilType = 2 Backbone of p-y curve approximates API (1993) sand relation. $pult Ultimate capacity of the p-y material. Note that “p” or “pult” are distributed loads [force per length of pile] in common design equations, but are both loads for this uniaxialMaterial [i.e., distributed load times the tributary length of the pile]. $Y50 Displacement at which 50% of pult is mobilized in monotonic loading. $Cd Variable that sets the drag resistance within a fully-mobilized gap as Cd*pult. $c The viscous damping term (dashpot) on the far-field (elastic) component of the displacement rate (velocity). Default = 0.0. Nonzero c values are used to represent radiation damping effects. $pRes Minimum (or residual) p-y resistance that the material retains as the adjacent solid soil elements liquefy (i.e. at ru = 1.0). $solidElem1 Element object integer tag for a solid element from which PyLiq1 will obtain mean effective stresses and pore pressures. $solidElem2 Element object integer tag for a solid element from which PyLiq1 will obtain mean effective stresses and pore pressures. 108 Chapter 11 Limit State Material 109 NOTE: Full documentation of the PyLiq1 command is found in: PyLiq1_Documentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/PyLiq1_documenta tion.pdf) NOTE: The implementation of PyLiq1 requires that the specified soil elements consist of FluidSolidPorousMaterials in FourNodeQuad elements. To model the effects of liquefaction with PyLiq1, it is necessary to use the material stage updating command: updateMaterialStage –material $matNum –stage $sNum where the argument matNum is the material number (for PyLiq1) and the argument sNum is the desired stage (valid values are 0 & 1). With sNum=0, the PyLiq1 behavior will be independent of any pore pressure in the specified solidElem’s. When updateMaterialStage first sets sNum=1, PyLiq1 will obtain the average mean effective stress in the two solidElem’s and treat it as the initial consolidation stress prior to undrained loading. Thereafter, the behavior of PyLiq1 will depend on the mean effective stresses (and hence excess pore pressures) in the solidElem’s. The default value of sNum is 0 (i.e., sNum=0 if updateMaterialStage is not called). Note that the updateMaterialStage command is used with some soil material models, and that sNum=0 generally corresponds to the application of gravity loads (e.g., elastic behavior with no excess pore pressure development) and sNum=1 generally corresponds to undrained loading (e.g., plastic behavior with excess pore pressure development). The analysis for gravity loading cannot use the "algorithm Linear" command because the relevant soil materials do not currently work properly with this command. Instead, the "algorithm Newton" or some other option must be used. TzLiq1 Material This command is used to construct a PyLiq1 uniaxial material object. uniaxialMaterial TzLiq1 $matTag $soilType $tult $z50 $c $solidElem1 $solidElem2. $matTag Unique material object integer tag. $tzType tzType = 1 Backbone of t-z curve approximates Reese and O’Neill (1987) relation. tzType = 2 Backbone of t-z curve approximates Mosher (1984) relation. Chapter 11 Limit State Material $tult Ultimate capacity of the t-z material. Note that “t” or “tult” are distributed loads [force per length of pile] in common design equations, but are both loads for this uniaxialMaterial [i.e., distributed load times the tributary length of the pile]. $z50 Displacement at which 50% of tult is mobilized in monotonic loading. $c The viscous damping term (dashpot) on the far-field (elastic) component of the displacement rate (velocity). Default = 0.0. Nonzero c values are used to represent radiation damping effects. $solidElem1 Element object integer tag for a solid element from which TzLiq1 will obtain mean effective stresses and pore pressures. $solidElem2 Element object integer tag for a solid element from which TzLiq1 will obtain mean effective stresses and pore pressures. 110 NOTE: Full documentation of the TzLiq1 command is included in: TzLiq1_Documentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/TzLiq1_documentat ion.pdf) NOTE: The implementation of TzLiq1 requires that the specified soil elements consist of FluidSolidPorousMaterials in FourNodeQuad elements. To model the effects of liquefaction with TzLiq1, it is necessary to use the material stage updating command: updateMaterialStage –material $matNum –stage $sNum where the argument matNum is the material number (for TzLiq1) and the argument sNum is the desired stage (valid values are 0 & 1). With sNum=0, the TzLiq1 behavior will be independent of any pore pressure in the specified solidElem’s. When updateMaterialStage first sets sNum=1, TzLiq1 will obtain the average mean effective stress in the two solidElem’s and treat it as the initial consolidation stress prior to undrained loading. Thereafter, the behavior of TzLiq1 will depend on the mean effective stress (and hence excess pore pressures) in the solidElem’s. The default value of sNum is 0 (i.e., sNum=0 if updateMaterialStage is not called). Note that the updateMaterialStage command is used with some soil material models, and that sNum=0 generally corresponds to the application of gravity loads (e.g., elastic behavior with no excess pore pressure development) and sNum=1 generally corresponds to undrained loading (e.g., plastic behavior with excess pore pressures development). The analysis for gravity loading cannot use the "algorithm Linear" command because the relevant soil materials do not currently work properly with this command. Instead, the "algorithm Newton" or some other option must be used. Chapter 11 Limit State Material 111 PySimple1Gen Command This command is used to construct output files containing material properties for PySimple1 uniaxial materials. The PySimple1Gen command constructs PySimple1 materials (Boulanger, 2003) for pre-defined zeroLength elements. PySimple1Gen $file1 $file2 $file3 $file4 $file5 <$file6> $file1 The name of an input file containing soil and pile properties required to define the PySimple1 materials. $file2 The name of an input file containing information about nodes that define the mesh in the domain. $file3 The name of an input file containing information about the zeroLength elements to be assigned PySimple1 materials (hereafter called p-y elements). $file4 The name of an input file containing information about the beam column elements that are attached to p-y elements. $file5 The name of the output file to which the PySimple1 materials are written. $file6 The name of the output file to which the applied patterns are written (optional). The command has been structured such that $File2, $File3, $File4, $File5 and $File6 can be sourced directly by OpenSees from within a master tcl file. Hence $File2, $File3 and $File4 serve two purposes: 1 They provide information to PySimple1Gen to create the PySimple1 materials. 2 They can be sourced directly in a master tcl file to define the nodes, zeroLength elements for p-y materials, and pile elements, respectively. Furthermore, $File5 and $File 6 serve the following purpose: 1 They can be sourced by OpenSees from within a master tcl file to define the PySimple1 materials and the applied patterns, respectively. The intended use of the files is demonstrated in an example problem in the Appendix: PySimple1GenDocumentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/PySimple1GenDoc umentation.pdf) Chapter 11 Limit State Material 112 TzSimple1Gen Command The TzSimple1Gen command constructs TzSimple1 materials (Boulanger, 2003) for pre-defined zeroLength elements. TzSimple1Gen $file1 $file2 $file3 $file4 $file5 <$file6> $file1 The name of an input file containing soil and pile properties required to define the TzSimple1 materials. $file2 The name of an input file containing information about the nodes that define the mesh in the domain. $file3 The name of an input file containing information about the zeroLength elements that are to be assigned TzSimple1 materials (hereafter called tz elements). $file4 The name of an input file containing information about the beam column elements that are attached to tz elements. $file5 The name of the output file to which the TzSimple1 materials are written. $file6 The name of the output file to which the applied patterns are written (optional). The command has been structured such that $File2, $File3, $File4, $File5 and $File6 can be sourced directly from within a master tcl file. Hence $File2, $File3 and $File4 serve two purposes: 1 They provide information to TzSimple1Gen to create the TzSimple1 materials. 2 They can be sourced directly in a master tcl file to define the nodes, zeroLength elements for tz materials, and pile elements, respectively. Furthermore, $File5 and $File6 serve the following purpose: 1 They can be sourced directly in a master tcl file to define the TzSimple1 materials and the applied patterns. Chapter 11 Limit State Material 113 The dual use of the files is demonstrated in an example problem in the Appendix: TzSimple1GenDocumentation.pdf (http://peer.berkeley.edu/~silvia/OpenSees/manual/documents/PyTzQzMats/TzSimple1GenDoc umentation.pdf) Reinforcing Steel Material Contact Authors: Jon Mohle M.S., P.E. Sashi Kunnath: http://cee.engr.ucdavis.edu/faculty/kunnath/kunnath.htm This command is used to construct a ReinforcingSteel uniaxial material object. This object is intended to be used in a reinforced concrete fiber section as the steel reinforcing material. uniaxialMaterial ReinforcingSteel $matTag $fy $fu $Es $Esh $esh $eult < -GABuck $lsr $beta $r $gama > < -DMBuck $lsr < $alpha >> < -CMFatigue $Cf $alpha $Cd > < -IsoHard <$a1 <$limit> > > $matTag unique material object integer tag $fy Yield stress in tension (see Figure 1) $fu Ultimate stress in tension $Es Initial elastic tangent $Esh Tangent at initial strain hardening $esh Strain corresponding to initial strain hardening $eult Strain at peak stress -GABuck Buckling Model Based on Gomes and Appleton (1997) Chapter 11 Limit State Material 114 $lsr Slenderness Ratio (see Figure 2) $beta Amplification factor for the buckled stress strain curve. (see Figure 3) $r Buckling reduction factor r can be a real number between [0.0 and 1.0] r=1.0 full reduction (no buckling) r=0.0 no reduction 0.0<r<1.0 linear interpolation between buckled and unbuckled curves $gamma -DMBuck Buckling constant (see Figures 3 and 4) Buckling model based on Dhakal and Maekawa (2002) $lsr Slenderness Ratio (see Figure 2) $alpha Adjustment Constant usually between 0.75 and 1.0 Default: alpha=1.0, this parameter is optional. -CMFatigue Coffin-Manson Fatigue and Strength Reduction $Cf Coffin-Manson constant C (see Figure 5) $alpha Coffin-Manson constant α (see Figure 5) $Cd Cyclic strength reduction constant (see Figure 6 and Equation 3) -IsoHard Isotropic Hardening / Diminishing Yield Plateau $a1 Hardening constant (default = 4.3) $limit Limit for the reduction of the yield plateau. % of original plateau length to remain (0.01 < limit < 1.0 ) Limit =1.0, then no reduction takes place (default =0.01) -MPCurveParams Menegotto and Pinto Curve Parameters $R1 (default = 0.333) $R2 (default = 18) $R3 (default = 4) R = R2 (1.0 − R3ε t ) ( ε y' ) R1 see Fig 6b Chapter 11 Limit State Material 115 NOTE: This simulation is based on the Chang and Mander(1994) uniaxial steel model. The simulation has incorporated additional reversal memory locations to better control stress overshooting (default is 10 branches but this can be easily modified by changing the variable “LastRule_RS” within the header file “ReinforcingSteel.h”). The cycle counting method implemented in the routine achieves the same result as rainflow counting. Fatigue parameters are based on the Coffin-Manson equation for plastic strain amplitude as indicated in Figure 6a. The buckling simulations incorporated consist of a variation on Gomes and Appleton(1997) and Dhakal and Maekawa(2002). The buckling and fatigue portions of this simulation are still being further enhanced and refined. Additional buckling and fatigue options should be available in the near future. Figure 1: Material Constants BACKBONE CURVE: The backbone curve shown in Figure 1 is used as a bounding surface for the reinforcing bar simulation. This backbone curve is shifted as described by Chang and Mander (1994) to account for Isotropic hardening. This backbone can be obtained by utilizing simple tension test data. Within the material class, the backbone curve is transformed from engineering stress space to natural stress space (accounting for change in area as the bar is stressed.) This allows the single backbone to represent both tensile and compressive stressstrain relations. The tension and compression backbone curves are not the same in engineering stress space for this model! This transformation assumes small strain relations described by Dodd and Restrepo-Posada (1995) The softening region (strain greater than eult), shown in Figure 1, is a localization effect due to necking and is a function of the gage length used during measurement. This geometric effect is ignored in this simulation. In this simulation, it is assumed that there is no softening in natural stress space. Because the simulation always converts back to engineering stress space, you will observe some softening in the tension response due to the reduction in area, however this will be much smaller than that shown in the original backbone curve proposed by Chang and Mander. Chapter 11 Limit State Material 116 DIMINISHING YIELD PLATEAU: It has been observed that when a reinforcing bar is subjected to plastic strain reversals within the yield plateau, strain hardening will initiate at a lower strain that that of the same bar loaded monotonically. Additionally, isotropic hardening can result from repeated strain reversals and is commonly related to accumulated plastic strain. These two aspects of the stress-strain behavior of steel bars are somewhat related and that by shortening the yield plateau as a function of accumulated plastic strain, the model will have some capability to simulate both the diminishing yield plateau and isotropic hardening. The Chang and Mander model, on which this formulation is based, models only anisotropic hardening by shifting the backbone curves and by targeting previous reversal points on the backbone curves. By adding a component of isotropic hardening, the model has additional capabilities and is able to more accurately simulate test data. Accumulated plastic strain is tracked within the material model for each branch, plateau adjustments are made only in the outer branches for simplicity. The plastic strain due to the backbone curve is ignored so that a monotonically loaded sample can be calibrated to a monotonic test sample more easily. Figure 2: Slenderness Defined Chapter 11 Limit State Material 117 Figure 3: Buckling Parameters GOMES AND APPLETON BUCKLED CURVE: Figure 3 describes the use of the buckling parameters modified from Gomes and Appleton(1997). β is an amplification factor that allows the user to scale the buckling curve. This is useful to adjust the location of the bifurcation point. The r factor is used to adjust the curve between the buckled curve and the unbuckled curve. The variable r can only be a real number between 0.0 and 1.0. Chapter 11 Limit State Material 118 The γ factor is the positive stress location about which the buckling factor is initiated. This factor was introduced to avoid kinks in the reloading branch. The implementation of the γ factor is shown in Figure 3. The basic idea is that the stress strain curves are reduced toward the positive stress γfsu. γ should be between 0.0 and 1.0. A γ of 0.0 will factor to the zero stress axis. This will usually produce a kink in the reloading curve at the zero stress location. Good results have been obtained using the following values for the buckling constants. β r γ 1.0 0.4 0.5 β r or γ 2.0 0.0 0.5 Figure 4 displays the buckling behavior due to the variation of the different constants. The response shown on the upper left is the unbuckled case. In each of the other cases, buckling behavior is defined by the constants shown. Chapter 11 Limit State Material 119 Figure 4: Effect of Sample Parameters in the Gomes and Appleton Buckling Model DHAKAL AND MAEKAWA BUCKLED CURVE: The buckling model described in this section is based on Dhakal and Maekawa(2002). This model takes two terms, lsr and α. lsr is the slenderness ratio as described in Figure 2 and α is an amplification factor. Dhakal and Maekawa suggest a value of α =1.0 for linear strain hardening and α =0.75 for elastic perfectly plastic material behavior. The material model in this implementation is neither linear strain hardening nor elastic perfectly plastic. However, since the material model does include strain hardening α=1.0 has been assumed as the default value. Figure 5 shows the unbuckled vs buckled stress strain response curves. Chapter 11 Limit State Material 120 Figure 5: Effect of Suggested Parameters in the Dhakal and Maekawa Buckling Model CYCLIC DEGRADATION: Cf and α are factors used to relate the number of half cycles to fracture to the half cycle plastic strain amplitude (Figure 6a). Plastic strain half cycle amplitude ε is defined by Equation 1. The total half cycle strain amplitude, t , is shown in Figure 6b as the change in strain from reversal A to reversal B. Cf and α are used to define a cumulative damage factor, D, as described in Equation 2. Chapter 11 Figure 6a: Coffin-Manson Constants ε p = εt − Limit State Material 121 Figure 6b: Half Cycle Terms Defined σt Es ∆ε p D = ∑ C f (1) 1 α (2) The cumulative damage factor is zero at no damage and 1.0 at fracture. Once a bar has been determined to have fractured, the strength is rapidly degraded to zero. Chapter 11 Limit State Material 122 Figure 7: Strength Reduction A degrade constant, K1, is used to describe loss in strength due to damage or other phenomenon resulting in softening due to plastic reversals. The degradation is currently assumed to have a simple linear relationship with D. This is used to correlate strength degradation to the cumulative damage factor. This linear relationship is shown in Equation 3. φSR = K1D (3) Alternately this simple linear equation can be rewritten in a way that makes the strength degradation independent of the number of half cycles to failure. Keeping the failure and degradation terms independent is convenient for calibration. Equation 3 is rewritten below utilizing the strength degradation constant Cd. 1 ∆ε p α φSR = ∑ Cd (4) The constants K1 and Cd can be related as shown in Equation 5. Cd = Cf K1α (5) Suggested starting values have been obtained from data reported by Brown and Kunnath (2000) for bars with a slenderness of 6. Keep in mind that this experimental data is limited and additional calibration may be necessary to capture realistic behavior in a reinforcing bar embedded in concrete and influenced by other factors such as confinement. Chapter 11 α: 0.506 Cf: 0.26 Cd: 0.389 Limit State Material 123 Sample Simulations of Degradation behavior α is best obtained from calibration of test results. α is used to relate damage from one strain range to an equivalent damage at another strain range. This is usually constant for a material type. Cf is the ductility constant used to adjust the number of cycles to failure. A higher value for Cf will result in a lower damage for each cycle. A higher value Cf translates to a larger number of cycles to failure. Cd is the strength reduction constant. A larger value for Cd will result in a lower reduction of strength for each cycle. The four charts shown in Figure 8 demonstrate the effect that some of the variables have on the cyclic response. Chapter 11 Limit State Material 124 Chapter 11 Limit State Material 125 Figure 8: Fatigue and Degradation Parameter Examples In Figure 8, the upper left response contains no strength degradation by setting the Cd variable to 0.0. The upper right response shows strength degradation due to the suggested values of Cf, α, and Cd. The response shown on the lower left demonstrates the change in the response when the suggested values of Cf and α are used with Cd=0.6. Making the value of Cd larger results in less strength reduction due to damage. The response on the lower right once again returns to the suggested values but Cf is changed to 0.15. This results in a more rapid accumulation of damage causing the bar to fail sooner. Note however that the strength degradation is unaffected by the more rapid accumulation of damage. The strength reduction and failure are not interdependent making the model easier to calibrate. References 1 Chang, G. and Mander, J. (1994). “Seismic Energy Based Fatigue Damage Analysis of Bridge Columns: Part I – Evaluation of Seismic Capacity.” NCEER Technical Report 940006. 2 Dodd, L. and Restrepo-Posada, J. (1995). “Model for Predicting Cyclic Behavior of Reinforcing Steel” J. Struct. Eng., 121(3), 433-445. 3 Gomes, A., and Appleton, J. (1997). “Nonlinear Cyclic Stress-Strain Relationship of Reinforcing Bars Including Buckling.” Eng. Struct., 19(10), 822–826. 4 Brown, J. and Kunnath, S.K. (2000). “Low Cycle Fatigue Behavior of Longitudinal Reinforcement in Reinforced Concrete Bridge Columns.” NCEER Technical Report 00-0007. 5 Dhakal, R. and Maekawa, K. (2002). “Modeling for Postyield Buckled of Reinforcement” J. Struct. Eng., 128(9), 1139-1147. 126 CHAPTER 12 ReinforcingSteel -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Elastic Material This command is used to construct an elastic uniaxial material object. uniaxialMaterial Elastic $matTag $E <$eta> $matTag unique material object integer tag $E tangent $eta damping tangent (optional, default=0.0) Chapter 12 ReinforcingSteel -- Material Behavior Figure 8: Elastic Material Elastic-Perfectly Plastic Material This command is used to construct an elastic perfectly-plastic uniaxial material object. uniaxialMaterial ElasticPP $matTag $E $epsyP <$epsyN $eps0> $matTag unique material object integer tag $E tangent 127 Chapter 12 ReinforcingSteel -- Material Behavior $epsyP strain or deformation at which material reaches plastic state in tension $epsyN strain at which material reaches plastic state in compression (optional, default: tension value) $eps0 initial strain (optional, default: zero) 128 Chapter 12 ReinforcingSteel -- Material Behavior 129 Figure 9: ElasticPerfectly Plastic Material Elastic-Perfectly Plastic Gap Material This command is used to construct an elastic perfectly-plastic gap uniaxial material object. uniaxialMaterial ElasticPPGap $matTag $E $Fy $gap <$eta> <damage> $matTag unique material object integer tag $E tangent stiffness $Fy stress or force at which material reaches plastic state $gap initial gap (strain or deformation) $eta hardening ratio (=Eh/E), which can be negative damage a switch to accumulate damage in the material. If damage is omitted, default value, the gap material "re-centers" on load reversal. NOTE: To create a compression-only gap element, NEGATIVE values need to be specified for $Fy and $gap. Chapter 12 ReinforcingSteel -- Material Behavior Figure 10: ElasticPerfectly Plastic Gap Material Elastic-No Tension Material This command is used to construct a uniaxial elastic-no tension material object. uniaxialMaterial ENT $matTag $E 130 Chapter 12 ReinforcingSteel -- Material Behavior $matTag unique material object integer tag $E elastic model in compression In tension, there is zero stress. Figure 11: Elastic-No Tension Material 131 Chapter 12 ReinforcingSteel -- Material Behavior 132 Parallel Material This command is used to construct a parallel material object made up of an arbitrary number of previously-constructed UniaxialMaterial (page 47) objects. uniaxialMaterial Parallel $matTag $tag1 $tag2 ... $matTag unique material object integer tag $tag1 $tag2 ... identification of materials making up the material model The parallel material is represented graphically: Figure 12: Parallel Material Chapter 12 ReinforcingSteel -- Material Behavior 133 In a parallel model, strains are equal and stresses and stiffnesses are additive: Series Material This command is used to construct a series material object made up of an arbitrary number of previously-constructed UniaxialMaterial (page 47) objects. uniaxialMaterial Series $matTag $tag1 $tag2 ... $matTag unique material object integer tag $tag1 $tag2 ... identification of materials making up the material model Chapter 12 ReinforcingSteel -- Material Behavior The series material is represented graphically: Figure 13: Series Material In a series model, stresses are equal and strains and flexibilities are additive: 134 Chapter 12 ReinforcingSteel -- Material Behavior 135 Figure 14: Series Material Relationship Hardening Material This command is used to construct a uniaxial material object with combined linear kinematic and isotropic hardening. The model includes optional visco-plasticity using a Perzyna formulation (REF???) uniaxialMaterial Hardening $matTag $E $sigmaY $H_iso $H_kin <$eta> $matTag unique material object integer tag $E tangent stiffness $sigmaY yield stress or force $H_iso isotropic hardening Modulus $H_kin kinematic hardening Modulus $eta visco-plastic coefficient (optional, default=0.0) Figure 15: Hardening Material Chapter 12 ReinforcingSteel -- Material Behavior 136 Concrete01 Material -- Zero Tensile Strength This command is used to construct a uniaxial Kent-Scott-Park concrete material object with degraded linear unloading/reloading stiffness according to the work of Karsan-Jirsa and no tensile strength. (REF: Fedeas). uniaxialMaterial Concrete01 $matTag $fpc $epsc0 $fpcu $epsU $matTag unique material object integer tag $fpc concrete compressive strength at 28 days (compression is negative)* $epsc0 concrete strain at maximum strength* $fpcu concrete crushing strength * $epsU concrete strain at crushing strength* *NOTE: Compressive concrete parameters should be input as negative values. The initial slope for this model is (2*$fpc/$epsc0) Chapter 12 Figure 16: Concrete01 Material -- Material Parameters Concrete01 Material -- Material Parameters ReinforcingSteel -- Material Behavior 137 Chapter 12 ReinforcingSteel -- Material Behavior 138 Chapter 12 ReinforcingSteel -- Material Behavior Figure 17: Typical Hysteretic StressStrain Relation of Concrete_1 Model Typical Hysteretic Stress-Strain Relation of Concrete_1 Model 139 140 CHAPTER 13 Concrete01 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Concrete02 Material -- Linear Tension Softening This command is used to construct a uniaxial concrete material object with tensile strength and linear tension softening. This is a Fedeas Material and the source code may not be available, yet. uniaxialMaterial Concrete02 $matTag $fpc $epsc0 $fpcu $epscu $lambda $ft $Ets $matTag unique material object integer tag $fpc compressive strength* $epsc0 strain at compressive strength* $fpcu crushing strength* $epsU strain at crushing strength* $lambda ratio between unloading slope at $epscu and initial slope $ft tensile strength $Ets tension softening stiffness (absolute value) (slope of the linear tension softening branch) Chapter 13 Concrete01 -- Material Behavior *NOTE: Compressive concrete parameters should be input as negative values. 141 Chapter 13 The initial slope for this model is (2*$fpc/$epsc0) Figure 18: Concrete02 Material -- Material parameters Concrete02 Material -- Material parameters Concrete01 -- Material Behavior 142 Chapter 13 Concrete01 -- Material Behavior 143 Chapter 13 Concrete01 -- Material Behavior Figure 19: Typical Hysteretic StressStrain Relation of Concrete_2 Model Typical Hysteretic Stress-Strain Relation of Concrete_2 Model Figure 20: Hysteretic Stress-Strain Relation of Concrete_2 Model in Tension-Compression Hysteretic Stress-Strain Relation of Concrete_2 Model in Tension-Compression 144 Chapter 13 Concrete01 -- Material Behavior 145 Chapter 13 Concrete01 -- Material Behavior Figure 21: Hysteretic Stress-Strain Relation of Concrete_2 Model in Tension-Compression (Detail) Hysteretic Stress-Strain Relation of Concrete_2 Model in Tension-Compression (Detail) Figure 22: Comparison of Hysteretic Behavior of Concrete_1 and Concrete_2 model Comparison of Hysteretic Behavior of Concrete_1 and Concrete_2 model 146 147 CHAPTER 14 Concrete02 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Concrete03 Material -- Nonlinear Tension Softening This command is used to construct a uniaxial concrete material object with tensile strength and nonlinear tension softening. This is a Fedeas Material and the source code may not be available, yet. uniaxialMaterial Concrete03 $matTag $fpc $epsc0 $fpcu $epscu $lambda $ft $epst0 $ft0 $beta $epstu $matTag unique material object integer tag $fpc compressive strength* $epsc0 strain at compressive strength* $fpcu crushing strength* $epsU strain at crushing strength* $lambda ratio between unloading slope at $epscu and initial slope (=2*$fpc/$epsc0) $ft tensile strength $epst0 tensile strain at the transition from nonlinear to linear softening Chapter 14 Concrete02 -- Material Behavior $ft0 tensile stress at the transition from nonlinear to linear softening $beta exponent of the tension softening curve $epstu ultimate tensile strain 148 Chapter 14 Concrete02 -- Material Behavior *NOTE: Compressive concrete parameters should be input as negative values. Figure 23: Concrete03 Material -- Material Parameters Concrete03 Material -- Material Parameters 149 Chapter 14 Concrete02 -- Material Behavior 150 Chapter 14 Concrete02 -- Material Behavior Figure 24: Hysteretic stress-strain relation of Concrete_3 model in Tension-Compression Hysteretic stress-strain relation of Concrete_3 model in Tension-Compression Figure 25: Hysteretic stress-strain relation of Concrete_3 model in Tension-Compression (Detail) Hysteretic stress-strain relation of Concrete_3 model in Tension-Compression (Detail) 151 152 CHAPTER 15 Concrete03 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Steel01 Material This command is used to construct a uniaxial bilinear steel material object with kinematic hardening and optional isotropic hardening described by a non-linear evolution equation (REF: Fedeas). uniaxialMaterial Steel01 $matTag $Fy $E0 $b <$a1 $a2 $a3 $a4> $matTag unique material object integer tag $Fy yield strength $E0 initial elastic tangent $b strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent) $a1, $a2, $a3, $a4 isotropic hardening parameters: (optional, default: no isotropic hardening) $a1 isotropic hardening parameter, increase of compression yield envelope as proportion of yield strength after a plastic strain of $a2*($Fy/E0). $a2 isotropic hardening parameter (see explanation under $a1) Chapter 15 Concrete03 -- Material Behavior $a3 isotropic hardening parameter, increase of tension yield envelope as proportion of yield strength after a plastic strain of $a4*($Fy/E0) $a4 isotropic hardening parameter (see explanation under $a3) 153 Chapter 15 Concrete03 -- Material Behavior 154 Chapter 15 Concrete03 -- Material Behavior Figure 26: Steel01 Material -- Material Parameters of Monotonic Envelope Steel01 Material -- Material Parameters of Monotonic Envelope Figure 27: Steel01 Material -- Hysteretic Behavior of Model w/o Isotropic Hardening Steel01 Material -- Hysteretic Behavior of Model w/o Isotropic Hardening 155 Chapter 15 Concrete03 -- Material Behavior Figure 28: Hysteretic Behavior of Model with Isotropic Hardening in Compression Steel01 Material -- Hysteretic Behavior of Model with Isotropic Hardening in Compression 156 Chapter 15 Concrete03 -- Material Behavior 157 Figure 29: Steel01 Material -- Hysteretic Behavior of Steel_1 Model with Isotropic Hardening in Tension Steel01 Material -- Hysteretic Behavior of Steel_1 Model with Isotropic Hardening in Tension Chapter 15 Concrete03 -- Material Behavior 158 159 CHAPTER 16 Steel01 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Chapter 16 Steel01 -- Material Behavior 160 Steel02 Material -- Giuffré-Menegotto-Pinto Model with Isotropic Strain Hardening This command is used to construct a uniaxial Giuffre-Menegotto-Pinto steel material object with isotropic strain hardening. uniaxialMaterial Steel02 $matTag $Fy $E $b $R0 $cR1 $cR2 $a1 $a2 $a3 $a4 $matTag unique material object integer tag $Fy yield strength $E initial elastic tangent $b strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent) $R0, $cR1, $cR2 control the transition from elastic to plastic branches. Recommended values: $R0=between 10 and 20, $cR1=0.925, $cR2=0.15 $a1, $a2, $a3, $a4 isotropic hardening parameters: (optional, default: no isotropic hardening) $a1 isotropic hardening parameter, increase of compression yield envelope as proportion of yield strength after a plastic strain of $a2*($Fy/$E). $a2 isotropic hardening parameter (see explanation under $a1) $a3 isotropic hardening parameter, increase of tension yield envelope as proportion of yield strength after a plastic strain of $a4*($Fy/$E) $a4 isotropic hardening parameter (see explanation under $a3) Chapter 16 Steel01 -- Material Behavior 161 Chapter 16 Steel01 -- Material Behavior Figure 30: Steel02 Material -- Material Parameters of Monotonic Envelope Steel02 Material -- Material Parameters of Monotonic Envelope Figure 31: Steel02 Material -- Hysteretic Behavior of Model w/o Isotropic Hardening Steel02 Material -- Hysteretic Behavior of Model w/o Isotropic Hardening 162 Chapter 16 Steel01 -- Material Behavior 163 Chapter 16 Steel01 -- Material Behavior Figure 32: Steel02 Material -- Hysteretic Behavior of Model with Isotropic Hardening in Compression Steel02 Material -- Hysteretic Behavior of Model with Isotropic Hardening in Compression Figure 33: Steel02 Material -- Hysteretic Behavior of Model with Isotropic Hardening in Tension Steel02 Material -- Hysteretic Behavior of Model with Isotropic Hardening in Tension 164 165 CHAPTER 17 Steel02 -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Hysteretic Material This command is used to construct a uniaxial bilinear hysteretic material object with pinching of force and deformation, damage due to ductility and energy, and degraded unloading stiffness based on ductility. uniaxialMaterial Hysteretic $matTag $s1p $e1p $s2p $e2p <$s3p $e3p> $s1n $e1n $s2n $e2n <$s3n $e3n> $pinchX $pinchY $damage1 $damage2 <$beta> $matTag unique material object integer tag $s1p $e1p stress and strain (or force & deformation) at first point of the envelope in the positive direction $s2p $e2p stress and strain (or force & deformation) at second point of the envelope in the positive direction $s3p $e3p stress and strain (or force & deformation) at third point of the envelope in the positive direction (optional) Chapter 17 Steel02 -- Material Behavior $s1n $e1n stress and strain (or force & deformation) at first point of the envelope in the negative direction* $s2n $e2n stress and strain (or force & deformation) at second point of the envelope in the negative direction* $s3n $e3n stress and strain (or force & deformation) at third point of the envelope in the negative direction (optional)* $pinchX pinching factor for strain (or deformation) during reloading $pinchY pinching factor for stress (or force) during reloading $damage1 damage due to ductility: D1(mu-1) $damage2 damage due to energy: D2(Eii/Eult) $beta power used to determine the degraded unloading stiffness based on ductility, mu-beta (optional, default=0.0) *NOTE: negative backbone points should be entered as negative numeric values 166 Chapter 17 Figure 34: Hysteretic Material Steel02 -- Material Behavior 167 168 CHAPTER 18 Hysteretic -- Material Behavior Each material was subjected to a series of ten uniaxial tension and compression strain histories. The following is the response of this material to such strain excursions. The data shown are the normalized stresses versus strain. In the normalization, the steel stress was divided by the yield stress Fy and the concrete stress was divided by the absolute value of compressive strength fc - to maintain positive tension and negative compression. The first figure shows all 10 load patterns together, which are subsequently shown individually. Viscous Material This command is used to construct a uniaxial material object with a non-linear elastic stressstrain-rate relation given by: stress =C(strain-rate)alpha. uniaxialMaterial Viscous $matTag $C $alpha $matTag unique material object integer tag $C tangent $alpha damping tangent 169 CHAPTER 19 nDMaterial Command This command is used to construct an NDMaterial object which represents stress-strain relationships at the integration points of continuum and force-deformation elements. The valid queries to any ND material when creating an ElementRecorder (page 330) are 'strain,' 'stress,' and 'tangent.' In This Chapter Elastic Isotropic Material........................................... 175 J2 Plasticity Material................................................. 176 Plane Stress Material ............................................... 176 Plate Fiber Material .................................................. 177 Template Elasto-Plastic Material .............................. 177 FluidSolidPorousMaterial Material ............................ 182 Elastic Isotropic Material This command is used to construct an ElasticIsotropic material object. nDMaterial ElasticIsotropic $matTag $E $v $matTag unique material object integer tag $E elastic Modulus $v Poisson's ratio The material formulations for the ElasticIsotropic object are "ThreeDimensional," "PlaneStrain," "Plane Stress," "AxiSymmetric," and "PlateFiber." These are the valid strings that can be passed to the continuum elements (page 231, page 227, page 228, page 226, page 227, page 229) for the type parameter. Chapter 19 nDMaterial Command 170 J2 Plasticity Material This command is used to construct a J2Plasticity material object. nDmaterial J2Plasticity $matTag $K $G $sig0 $sigInf $delta $H $matTag unique material object integer tag $K bulk Modulus $G shear Modulus $sig0 initial yield stress $sigInf final saturation yield stress $delta exponential hardening parameter $H linear hardening parameter Plane Stress Material This command is used to construct a plane-stress material wrapper which converts any threedimensional material into a plane stress material via static condensation. nDMaterial PlaneStress $matTag $threeDtag $matTag unique material object integer tag $threeDTag material tag for a previously-defined three-dimensional material Chapter 19 nDMaterial Command 171 Plate Fiber Material This command is used to construct a plate-fiber material wrapper which converts any threedimensional material into a plate fiber material (by static condensation) appropriate for shell analysis. nDMaterial PlateFiber $matTag $threeDTag $matTag unique material object integer tag $threeDTag material tag for a previously-defined three-dimensional material Template Elasto-Plastic Material This command is used to construct the template elasto-plastic material object. nDMaterial Template3Dep $matTag $ElmatTag -YS $ys -PS $ps -EPS $eps <ELS1 $el> <-ELT1 $et> $matTag unique material object tag $ElmatTag previously defined elastic nDMaterial (such as ElasticIsotropic3D (page 175), PressureDependentElastic3D) tag $ys yield surface variable, previously defined in Yield Surface (page 178) object $ps potential surface variable, previously defined in Potential Surface (page 179) object $eps elasto-plastic state variable, previously defined in EPState (page 182) object $el scalar (isotropic) evolution law variable, previously defined in Evolution Law (page 181) object $et tensorial (kinematic) evolution law variable, previously defined in Evolution Law (page 181) object Chapter 19 nDMaterial Command 172 Yield Surface This command sets the yield surface variable ys to be the specified type. Currently these include: Drucker-Prager yield surface, Rounded Mohr-Coulomb (Willam-Warnke) yield surface, von Mises yield surface, Cam-Clay yield surface and Leon yield surface. set ys "-YieldSurfaceType <parameter list>" Valid strings for YieldSurfaceType are DP, VM and CC. For Drucker-Prager yield surface set ys "-DP" For von Mises yield surface set ys "-VM" For rounded Mohr-Coulomb (Willam-Warnke) yield surface set ys "-RMC01" For Cam-Clay yield surface set ys "-CC $M" $M Slope of the critical state line in p-q space For Leon yield surface set ys "-Leon $fc $ft $e $c" $fc compressive strength $ft tensile strength $e excentricity of yield surface (usually 0.6-0.7) $c cohesion Chapter 19 nDMaterial Command 173 Potential Surface This command is used to set the potential surface variable $ps to the specific surface (or directly to the flow directions). Currently included are: Drucker-Prager potential surface, Rounded MohrCoulomb (Willam-Warnke) potential surface, von Mises potential surface, Cam-Clay potential surface and Leon potential surface. set ps "-PotentialSurfaceType <parameter list>" Valid strings for PotentialSurfaceType are DP, VM and CC. For the Drucker-Prager potential surface set ps "-DP" For the von Mises potential surface set ps "-VM" For rounded Mohr-Coulomb (Willam-Warnke) potential surface set ps "-RMC01" For the Cam-Clay potential surface set ps "-CC $M" $M Slope of the critical state line in p-q space For Leon potential surface set ps "-Leon $fc $ft $e $c" $fc compressive strength Chapter 19 nDMaterial Command $ft tensile strength $e excentricity of yield surface (usually 0.6-0.7) $c cohesion 174 Evolution Law This command is used to set the evolution law variable el to the specified type. There are two types of evolutions laws implemented: scalar (isotropic) evolution and tensorial (kinemartic) evolution. For scalar evolution law, there are linear scalar evolution law and nonlinear scalar evolution law. For tensorial evolution law, there are linear tensorial evolution law and nonlinear tensorial evolution law. set el "-EvolutionLawType <parameter list>" Valid strings for EvolutionLawType are Leq, NLp, LEij, and NLEij For linear scalar evolution law set el "-Leq $a" For Cam-Clay type nonlinear scalar evolution law set el "-NLp $e_o $lambda $k For linear tensorial evolution law set et "-LEij $a1" For Armstrong-Frederick type nonlinear tensorial evolution law set et "-NLEij $h_a $C_r" $a linear hardening coefficient $e_o initial void ratio $lambda nonlinear evolution law constant (Cam-Clay type) Chapter 19 nDMaterial Command 175 $k nonlinear evolution law constant (Cam-Clay type) $a1 linear tensorial evolution law constant $h_a nonlinear tensorial evolution law constant (Armstrong-Frederick type) $C_r nonlinear tensorial evolution law constant (Armstrong-Frederick type) EPState This command is used to set the Elasto-Plastic State, which includes two states. To set the initial stress tensor to variable sts, if –stressp $sts is used in eps: set sts "$sigma_xx $sigma_xy $sigma_xz $sigma_yx $sigma_yy $sigma_yz $sigma_zx $sigma_zy $sigma_zz" To assign to the Elasto-Plastic state variable eps the specified state parameters set eps "<-NOD $nt> -NOS $ns $scalar1 $scalar2 ... <-stressp $sts>" $sigma_xx $sigma_xy $sigma_xz $sigma_yx $sigma_yy $sigma_yz $sigma_zx $sigma_zy $sigma_zz Initial stress tensor components (Default = 0.0), $nt number of tensorial internal variables $ns number of scalar internal variables $scalar1 $scalar2 ... corresponding initial values of scalar internal variables $sts initial stresses Chapter 19 nDMaterial Command 176 FluidSolidPorousMaterial Material FluidSolidPorousMaterial couples the responses of two phases: fluid and solid. The fluid phase response is only volumetric and linear elastic. The solid phase can be any NDMaterial (page 175). This material is developed to simulate the response of saturated porous media under fully undrained condition. nDMaterial FluidSolidPorousMaterial $tag $nd $soilMatTag $combinedBulkModul $tag unique material object integer tag $nd Number of dimensions, 2 for plane-strain, and 3 for general 3D analysis. $soilMatTag The material number for the solid phase material (previously defined). $combinedBulkMod ul Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water typically), and n the initial porosity. NOTE: 1. Buoyant unit weight (total unit weight - fluid unit weight) should be used in definition of the finite elements composed of a FluidSolidPorousMaterial. 2. During the application of gravity (elastic) load, the fluid phase does not contribute to the material response. OUTPUT INTERFACE: The following information may be extracted for this material at given integration point, using the OpenSees Element Recorder facility (McKenna and Fenves 2001): "stress", "strain", "tangent", or "pressure". The "pressure" option records excess pore pressure and excess pore pressure ratio at a given material integration point. Chapter 19 nDMaterial Command 177 updateMaterialStage This command is used to update a PressureDependMultiYield, a PressureIndependMultiYield, or a FluidSolidPorous (page 182) material. To conduct a seismic analysis, two stages should be followed. First, during the application of gravity load (and static loads if any), set material stage to 0, and material behavior is linear elastic (with Gr and Br as elastic moduli). A FluidSolidPorous (page 182) material does not contribute to the material response if its stage is set to 0. After the application of gravity load, set material stage to 1 or 2. In case of stage 2, all the elastic material properties are then internally determined at the current effective confinement, and remain constant thereafter. In the subsequent dynamic (fast) loading phase(s), the deviatoric stressstrain response is elastic-plastic (stage 1) or linear-elastic (stage 2), and the volumetric response remains linear-elastic. updateMaterialStage -material $tag -stage $sNum $tag previously-defined material object integer tag $sNum desired stage: 0 – linear elastic, 1 – plastic, 2 – Linear elastic, with elasticity constants (shear modulus and bulk modulus) as a function of initial effective confinement. 178 CHAPTER 20 section Command This command is used to construct a SectionForceDeformation object, hereto referred to as Section, which represents force-deformation (or resultant stress-strain) relationships at beamcolumn and plate sample points. What is a section? • A section defines the stress resultant force-deformation response at a cross section of a beam-column or plate element Types of sections: Elastic – defined by material and geometric constants • Resultant – general nonlinear description of force-deformation response, e.g. momentcurvature • Fiber – section is discretized into smaller regions for which the material stress-strain response is integrated to give resultant behavior, e.g. reinforced concrete The valid queries to any section when creating an ElementRecorder (page 330) are 'force' and 'deformation.' Chapter 20 section Command 179 Figure 35: Section Representation In This Chapter Elastic Section.......................................................... 187 Uniaxial Section........................................................ 187 Fiber Section ............................................................ 189 Section Aggregator................................................... 199 Elastic Membrane Plate Section............................... 202 Plate Fiber Section ................................................... 202 Bidirectional Section................................................. 203 Isolator2spring Section: Model to include buckling behavior of an elastomeric bearing .................................................. 204 Elastic Section This command is used to construct an ElasticSection object. section Elastic $secTag $E $A $Iz <$Iy $G $J> $secTag unique section object tag $E Young's Modulus $A cross-sectional area of section $Iz second moment of area about the local z-axis $Iy second moment of area about the local y-axis (optional, used for 3D analysis) $G Shear Modulus (optional, used for 3D analysis) $J torsional moment of inertia of section (optional, used for 3D analysis) This command is useful for patch tests of the nonlinear beam-column elements (page 216). It also allows nonlinear beam-column elements to be used for elastic analysis. EXAMPLE: section Elastic 1 29000 100 100000 80000 20000 100000; IDtag 1 # create elastic section with Chapter 20 section Command 180 Uniaxial Section This command is used to construct a UniaxialSection object which uses a previously-defined UniaxialMaterial (page 47) object to represent a single section force-deformation response quantity. (Formerly known as Generic1d section, which is still accepted by OpenSees) section Uniaxial $secTag $matTag $string $secTag unique section object tag $matTag previously-defined UniaxialMaterial (page 47) object $string the force-deformation quantity to be modeled by this section object. One of the following strings is used: P Axial force-deformation Mz Moment-curvature about section local z-axis Vy Shear force-deformation along section local y-axis My Moment-curvature about section local y-axis Vz Shear force-deformation along section local z-axis T Torsion Force-Deformation EXAMPLE: section Uniaxial 1 1 Mz; # create sectionID-tag 1 from UniaxialMaterialID-tag 1 for the moment-curvature about section local z-axis. Chapter 20 section Command 181 Fiber Section The FiberSection object is composed of Fiber objects. A fiber section has a general geometric configuration formed by subregions of simpler, regular shapes (e.g. quadrilateral, circular and triangular regions) called patches. In addition, layers of reinforcement bars can be specified. The subcommands patch (page 192) and layer (page 197, page 196) are used to define the discretization of the section into fibers. Individual fibers, however, can also be defined using the fiber (page 191) command (During generation, the Fiber objects are associated with uniaxialMaterial (page 47) objects, which enforce Bernoulli beam assumptions. The geometric parameters are defined with respect to a planar local coordinate system (y,z). See figures. section Fiber $secTag { fiber <fiber arguments> patch <patch arguments> layer <layer arguments> } An example fiber section is shown in the Figure. Chapter 20 Figure 36: Fiber Section section Command 182 Chapter 20 section Command 183 Fiber Command This command is used to construct a UniaxialFiber object and add it to the section. fiber $yLoc $zLoc $A $matTag $yLoc y coordinate of the fiber in the section (local coordinate system) $zLoc z coordinate of the fiber in the section (local coordinate system) $A area of fiber $matTag material tag of the pre-defined UniaxialMaterial (page 47) object used to represent the stress-strain for the area of the fiber NOTE: in 2D (page 37) bending is about the local z-axis EXAMPLE: fiber 0.0 0.0 1.0 1; materialIDtag 1 Figure 37: Fiber Command # create a single fiber of area 1.0 at the origin (0,0) of the section, using Chapter 20 section Command 184 Quadrilateral Patch Command This command is used to construct a Patch object with a quadrilateral shape. The geometry of the patch is defined by four vertices: I J K L, as illustrated in the Figure. The coordinates of each of the four vertices is specified in sequence -- counter-clockwise. patch quad $matTag $numSubdivIJ $numSubdivJK $yI $zI $yJ $zJ $yK $zK $yL $zL $matTag material integer tag of the previously-defined UniaxialMaterial (page 47) object used to represent the stress-strain for the area of the fiber $numSubdivIJ number of subdivisions (fibers) in the IJ direction. $numSubdivJK number of subdivisions (fibers) in the JK direction. $yI $zI y & z-coordinates of vertex I (local coordinate system) $yJ $zJ y & z-coordinates of vertex J (local coordinate system) $yK $zK y & z-coordinates of vertex K (local coordinate system) $yL $zL y & z-coordinates of vertex L (local coordinate system) Chapter 20 section Command NOTE: in 2D (page 37) bending is about the local z-axis EXAMPLE: patch quad $coreMatTag 8 8 -$b -$h $b -$h $b $h -$b $h; # define core patch with 8 subdivisions within a rectange of width 2b and depth 2h 185 Chapter 20 section Command Figure 38: Quadrilateral Patch Circular Patch Command This command is used to construct a Patch object with a circular shape. patch circ $matTag $numSubdivCirc $numSubdivRad $yCenter $zCenter $intRad $extRad <$startAng $endAng> $matTag material integer tag of the previously-defined UniaxialMaterial (page 47) object used to represent the stress-strain for the area of the fiber $numSubdivCirc number of subdivisions (fibers) in the circumferential direction. $numSubdivRad number of subdivisions (fibers) in the radial direction. $yCenter y & z-coordinates of the center of the circle $zCenter $intRad internal radius $extRad external radius $startAng starting angle (optional. default=0.0) $endAng ending angle (optional. default=360.0) NOTE: in 2D (page 37) bending is about the local z-axis 186 Chapter 20 section Command 187 EXAMPLE: patch circ $coreMatTag 8 8 0.0 0.0 0.0 $h; # define core patch with 8 subdivisions within a whole circle of diameter 2h Chapter 20 section Command 188 Figure 39: Circular Patch Straight Layer Command This command is used to construct a straight layer of reinforcing bars. layer straight $matTag $numBars $areaBar $yStart $zStart $yEnd $zEnd $matTag material integer tag of the previously-defined UniaxialMaterial (page 47) object used to represent the stress-strain for the area of the fiber $numBars number of reinforcing bars along layer $areaBar area of individual reinforcing bar $yStart $zStart y and z-coordinates of starting point of reinforcing layer (local coordinate system) $yEnd $zEnd y and z-coordinates of ending point of reinforcing layer (local coordinate system) NOTE: in 2D (page 37) bending is about the local z-axis EXAMPLE: layer straight $steelMatTag 10 0.11 -b -h b -h; # define steel layer of 10 bars with area 0.11 at bottom of section of width 2b by 2h Chapter 20 section Command 189 Figure 40: Straight Layer Circular Layer Command This command is used to construct a circular layer of reinforcing bars. layer circ $matTag $numBar $areaBar $yCenter $zCenter $radius <$startAng $endAng> $matTag material integer tag of the previously-defined UniaxialMaterial (page 47) object used to represent the stress-strain for the area of the fiber $numBar number of reinforcing bars along layer $areaBar area of individual reinforcing bar Chapter 20 $yCenter $zCenter $radius $startAng section Command 190 y and z-coordinates of center of reinforcing layer (local coordinate system) radius of reinforcing layer $endAng starting and ending angle of reinforcing layer, respectively. (Optional, Default: a full circle is assumed 0-360) NOTE: in 2D (page 37) bending is about the local z-axis EXAMPLE: layer circ $steelMatTag 10 0.11 0.0 0.0 $h 0 360; # define circular steel layer of 10 bars with area 0.11 uniformly distributed along circumference of circle of diameter 2h Chapter 20 section Command 191 Figure 41: Circular Reinforcing Layer Section Aggregator This command is used to construct a SectionAggregator object which groups previously-defined UniaxialMaterial (page 47) objects into a single section force-deformation model. section Aggregator $secTag $matTag1 $string1 $matTag2 $string2 ....... <section $sectionTag> $secTag unique section object integer tag $matTag1, $matTag2 ... previously-defined UniaxialMaterial (page 47) objects $string1, $string2 .... the force-deformation quantities corresponding to each section object. One of the following strings is used: <-section $sectionTag> P Axial force-deformation Mz Moment-curvature about section local z-axis Vy Shear force-deformation along section local y-axis My Moment-curvature about section local y-axis Vz Shear force-deformation along section local z-axis T Torsion Force-Deformation specifies a previously-defined Section (page 185) object (identified by the argument $sectionTag) to which these UniaxialMaterial (page 47) objects may be added to recursively define a new Section (page 185) object NOTE: The UniaxialMaterial (page 47) objects aggregated in this Section (page 185) object are uncoupled from each other as well as from the Section (page 185) object represented by $sectionTag, if present. There are two main tasks that can be performed using the Section Aggregator: Chapter 20 section Command 192 1. Group previously defined uniaxial materials to describe stress resultant section behavior Figure 42: Section Aggregator 1 EXAMPLE: section Aggregator 1 2 Vy 5 Mz; #create new section with IDtag 1, taking the existing material tag 2 to represent the shear and the existing material tag 5 to represent the moment. Chapter 20 section Command 193 2. Add to an existing section Figure 43: Section Aggregator 2 EXAMPLE: section Aggregator 2 2 Vy -section 4; # create new section with IDtag 2, taking the existing material tag 2 to represent the shear and adding it to the existing section tag 4, which may be a fiber section where the interaction betweeen axial force and flexure is already considered. Chapter 20 section Command 194 Elastic Membrane Plate Section This command is used to construct an ElasticMembranePlateSection object, which is an isotropic section appropriate for plate and shell analysis. section ElasticMembranePlateSection $secTag $E $nu $h $rho $secTag unique section object tag $E Elastic Modulus $nu Poisson's Ratio $h thickness of the plate section $rho mass density of the material (per unit volume) Plate Fiber Section The plate fiber section takes any plate fiber material (page 177) and, by thickness integration, creates a plate section appropriate for shell analysis. section PlateFiber $secTag $fiberTag $h $secTag unique section object tag for section being constructed $fiberTag material tag for a previously-defined plate fiber material (page 177) $h thickness of the plate section Chapter 20 section Command 195 Bidirectional Section This command is used to construct a Bidirectional section object which is the two-dimensional generalization of a one-dimensional elasto-plastic model with linear hardening. section Bidirectional $matTag $E $sigY $H_iso $H_kin $matTag unique section object integer tag $E Elastic Modulus $sigY yield stress $H_iso isotropic hardening Modulus $H_kin kinematic hardening Modulus then why do we have kinematic and isotripic hardening parameters??????*********** Chapter 20 section Command 196 Isolator2spring Section: Model to include buckling behavior of an elastomeric bearing Author: Keri L. Ryan Assistant Professor, Utah State University Contact: http://www.engineering.usu.edu/cee/faculty/kryan/ This command is used to construct an Isolator2spring section object, which represents the buckling behavior of an elastomeric bearing for two-dimensional analysis in the lateral and vertical plane. An Isolator2spring section represents the resultant force-deformation behavior of the bearing, and should be used with a zeroLengthSection element. The bearing should be constrained against rotation. section Iso2spring $matTag $tol $k1 $Fyo $k2o $kvo $hb $PE <$Po> $matTag unique section object integer tag $tol tolerance for convergence of the element state. Suggested value: E-12 to E-10. OpenSees will warn if convergence is not achieved, however this usually does not prevent global convergence. $k1 initial stiffness for lateral force-deformation $Fyo nominal yield strength for lateral force-deformation $k2o nominal postyield stiffness for lateral force-deformation $kvo nominal stiffness in the vertical direction $hb total height of elastomeric bearing $PE Euler Buckling load for the bearing $Po axial load at which nominal yield strength is achieved (optional, default = 0.0, i.e. no strength degradation) Chapter 20 section Command 197 Model Characteristic: This material model is based on a two-spring mechanical model of an elastomeric bearing, originally developed by Koh and Kelly [1987] (Figure 1). The model yields the approximate results as predicted by stability analysis of a multi-layer bearing. The axial flexibility of the bearing is modeled by an additional vertical spring in series (not shown in Figure 1). The original model included only linear material behavior. The rotational stiffness is given by where is the Euler buckling load as a function of bending stiffness EIs and bearing height hb. The nominal shear stiffness and the vertical stiffness , where G is the shear modulus, Ec is the compression modulus, A is the cross-sectional area, and tr is the total height of rubber. Ec and G can be related by the bearing shape factor S [Kelly 1997]. Chapter 20 section Command 198 Figure 1: Two-spring model of an isolation bearing in the undeformed and deformed configuration. In this implementation, the linear shear spring has been replaced by a bilinear spring to represent the nonlinear behavior observed in elastomeric and lead-rubber bearings [Ryan et. al 2005] (Figure 2). The nonlinear behavior is implemented by rate-independent plasticity with kinematic hardening [Simo and Hughes 1998]. The behavior of the nonlinear spring is controlled by the initial stiffness k1, yield strength Fyo, and postyield stiffness k2o. Figure 2: Bilinear shear spring and parameters. Also included is an optional variation of strength with axial load, to represent the inability of lead-plug bearings to achieve their full strength when lightly loaded, as has been experimentally observed. An empirical equation for the yield strength as a function of compressive load has been developed from experimental data: Chapter 20 section Command 199 where P is the compressive load on the bearing and Po is the axial load at which approximately 63% of the nominal strength is achieved (Figure 3). The bearing is assumed to have an acting yield strength of zero in tension. If not specified, Po = 0, which means that the strength equals the nominal yield strength and no strength degradation occurs. Figure 3: Empirical model for yield strength degradation. The equilibrium equations and kinematic constraints for the two-spring model form a system of five nonlinear equations (below) which are solved by an internal iterative Newton algorithm (this is iteration within the material object). The stiffness matrix is formed by taking differentials of the equilibrium and kinematic equations. The return mapping algorithm is implemented at each iteration to determine the state of the shear spring. Chapter 20 section Command 200 Not defined previously, v is the deformation due to the vertical spring and ubv is the total vertical deformation, including the geometric effect of tilting. Analysis of the linear two-spring model leads to the following approximate coupled lateral force-deformation and vertical force-deformation equations: where is the critical buckling load for the bearing, and That is, lateral stiffness decreases as the axial load on the bearing approaches the critical load, and vertical flexibility increases in the laterally deformed configuration. Example: The following example demonstrates a simple cyclic lateral load test, and was used to produce the lateral force deformation behavior shown in Figure 4 by variation of the parameters P/Pcr and P/Po in the script file (page 210). Chapter 20 section Command 201 Figure 4: Cyclic lateral force deformation behavior of a bearing as a function of axial load: (a) postyield stiffness degrades as axial load P approaches the critical load Pcr, (b) yield strength degrades as axial load decreases relative to Po. References: Ryan, Keri L., James M. Kelly and Anil K. Chopra (2005). “Nonlinear model for lead-rubber bearings including axial-load effects” Journal of Engineering Mechanics, ASCE, 131(12). Kelly, James M. (1997). Earthquake-Resistant Design with Rubber. Springer-Verlag. Koh, C.-G. and Kelly, J. M. (1987). “Effects of axial load on elastomeric isolation bearings”, Rep. No. UCB/EERC-86/12. Earthquake Engineering Research Center, University of California, Berkeley. Kelly, James M. (2003). “Tension buckling in multilayer elastomeric bearings”, Journal of Engineering Mechanics, ASCE, 129(12):1363-1368. Simo, J. C and T. J. R. Hughes (1998). Computational Inelasticity. New York, NY, Springer. CyclicLoading_Iso2spring.tcl # Test Example for ISOLATOR2SPRING material # Written: K. Ryan # Date: November 22, 2005 # # Description: A Single DOF isolator supporting a rigid mass. Generates lateral force deformation through Chapter 20 section Command # application of a pseudo-static cyclic (sinusoidal) lateral force. Force deformation behavior varies # as a function of the axial load P and normalized values of P/Pcr and P/Po # # Units: cm, dimensionless set g 982.0 set PI [expr 2.0 * asin(1.0)] # Define the model builder model BasicBuilder -ndm 2 -ndf 3 # Assumed parameters which define system # Period associated with second stiffness k2o set Tb 2.5 # Normalized axial load parameter P/Pcr. Vary to produce data for Figure 4a set PPcr 0.001 # Normalized axial load paramter P/Po. Vary to produce data for Figure 4b set PPo 0.25 # Unit mass set M 1.0 # Bearing shape factor set S 20.0 # Yield displacement set uy 1.0 # Strength coefficient (determines yield strength) set mu 0.1 set tol 1.0e-10 # Parameters computed from the assumed ones set wb [expr (2.0*$PI/$Tb)] set k2o [expr $wb*$wb*$M] set Fyo [expr $mu*$M*$g + $k2o*$uy] set k1 [expr $Fyo/$uy] set kvo [expr 6.0*$S*$S*$k2o] set Pcr [expr $M*$g/$PPcr] set Pe [expr 50.0*$Pcr] set Ps [expr 1.0/50.0*$Pcr] set hb [expr $Ps/$k2o] set Po [expr $M*$g/$PPo] # Define nodes node 1 0.0 0.0 node 2 0.0 0.0 # Define single point constraints (Constrain the isolator against rotation) fix 1 1 1 1 fix 2 0 0 1 # Create Iso2spring section section Iso2spring 1 $tol $k1 $Fyo $k2o $kvo $hb $Pe $Po # id ndI ndJ mattag element zeroLengthSection 1 1 2 1 # Apply gravity load on isolated mass set P [expr $M*$g] pattern Plain 1 "Linear" { load 2 0.0 -[expr $P] 0.0 } system BandSPD 202 Chapter 20 constraints Plain numberer Plain test EnergyIncr 1.0e-15 100 algorithm Newton integrator LoadControl 0.1 1 0.1 0.1 analysis Static initialize analyze 10 loadConst -time 0.0 # Estimated postyield stiffness degradation set k2oest [expr $k2o*(1.0-$PPcr*$PPcr)] # Estimated yield strength degradation set Fyoest [expr $Fyo*(1.0-exp(-$PPo))] # Define amplitude of cyclic force to induce approx. 20 cm deformation set Fu [expr $Fyoest + $k2oest*20.0] # Apply sinusoidal lateral load set SinePath "Trig 0.0 1.5 1.0" pattern Plain 2 $SinePath { load 2 $Fu 0.0 0.0 } wipeAnalysis recorder Node -file Node2.out -node 2 -dof 1 disp recorder Element 1 -file Elem1.out force system BandGeneral constraints Plain numberer Plain test EnergyIncr 1.0e-15 100 algorithm Newton integrator LoadControl 0.01 1 0.001 0.1 analysis Static analyze 1500 print ele 1 print node 2 section Command 203 204 CHAPTER 21 element Command This command is used to construct an Element object. In This Chapter Truss Element .......................................................... 213 Corotational Truss Element ...................................... 214 Elastic Beam Column Element ................................. 215 NonLinear Beam-Column Elements ......................... 216 Zero-Length Elements.............................................. 222 Quadrilateral Elements ............................................. 226 Brick Elements ......................................................... 229 FourNodeQuadUP Element...................................... 237 BeamColumnJoint Element ...................................... 238 Truss Element This command is used to construct a truss element object. There are two ways to construct a truss element object: One way is to specify an area and a UniaxialMaterial (page 47) identifier: element truss $eleTag $iNode $jNode $A $matTag the other is to specify a Section (page 185) identifier: element truss $eleTag $iNode $jNode $secTag $eleTag unique element object tag $iNode $jNode end nodes $A cross-sectional area of element $matTag tag associated with previously-defined UniaxialMaterial (page 47) $secTag tag associated with previously-defined Section (page 185) Chapter 21 element Command 205 When constructed with a UniaxialMaterial (page 47) object, the truss element considers strainrate effects, and is thus suitable for use as a damping element. The valid queries to a truss element when creating an ElementRecorder (page 330) object are 'axialForce,' 'stiff,' deformations,' 'material matArg1 matArg2...,' 'section sectArg1 sectArg2...' There will be more queries after the interface for the methods involved have been developed further. Corotational Truss Element This command is used to construct a Corotational Truss (CorotTruss) element object. A corotational formulation adopts a set of corotational axes which rotate with the element, thus taking into account an exact geometric transformation between local and global frames of reference. There are two ways to construct a Corotational Truss element object: One way is to specify an area and a UniaxialMaterial (page 47) identifier: element corotTruss $eleTag $iNode $jNode $A $matTag the other is to specify a Section (page 185) identifier: element corotTruss $eleTag $iNode $jNode $secTag $eleTag unique element object tag $iNode $jNode end nodes $A cross-sectional area of element $matTag tag associated with previously-defined UniaxialMaterial (page 47) object $secTag tag associated with previously-defined Section (page 185) object NOTE: When constructed with a UniaxialMaterial (page 47) object, the truss element considers strain-rate effects, and is thus suitable for use as a damping element. The valid queries to a corotational truss element when creating an ElementRecorder (page 330) object are 'axialForce,' 'stiff,' 'material $matNum matArg1 matArg2...,' 'section $secNum sectArg1 sectArg2...' Chapter 21 element Command 206 Elastic Beam Column Element This command is used to construct an elasticBeamColumn element object. The arguments for the construction of an elastic beam-column element depend on the dimension of the problem, ndm (page 31): For a two-dimensional problem: element elasticBeamColumn $eleTag $iNode $jNode $A $E $Iz $transfTag For a three-dimensional problem: element elasticBeamColumn $eleTag $iNode $jNode $A $E $G $J $Iy $Iz $transfTag $eleTag $iNode unique element object tag $jNode end nodes $A cross-sectional area of element $E Young's Modulus $G Shear Modulus $J torsional moment of inertia of cross section $Iz second moment of area about the local z-axis $Iy second moment of area about the local y-axis $transfTag identifier for previously-defined coordinate-transformation (page 302) (CrdTransf) object The valid queries to an elastic beam-column element when creating an ElementRecorder (page 330) object are 'stiffness' and 'force.' Chapter 21 element Command 207 NonLinear Beam-Column Elements There are basically two types of Nonlinear Beam-Column Elements Force based elements Distributed plasticity (nonlinearBeamColumn (page 216)) Concentrated plasticity with elastic interior (beamWithHinges (page 218)) Displacement based element Distributed plasticity with linear curvature distribution (dispBeamColumn (page 222)) Nonlinear Beam Column Element This command is used to construct a nonlinearBeamColumn element object, which is based on the non-iterative (or iterative) force formulation, and considers the spread of plasticity along the element. element nonlinearBeamColumn $eleTag $iNode $jNode $numIntgrPts $secTag $transfTag <-mass $massDens> <-iter $maxIters $tol> $eleTag $iNode unique element object tag $jNode end nodes $numIntgrPts number of integration points along the element. $secTag identifier for previously-defined section (page 185) object $transfTag identifier for previously-defined coordinate-transformation (page 302) (CrdTransf) object $massDens element mass density (per unit length), from which a lumpedmass matrix is formed (optional, default=0.0) $maxIters maximum number of iterations to undertake to satisfy element compatibility (optional, default=1) $tol tolerance for satisfaction of element compatibility (optional, default=10-16) The integration along the element is based on Gauss-Lobatto quadrature rule (two integration points at the element ends). Chapter 21 element Command 208 The element is prismatic, i.e. the beam is represented by the section (page 185) model identified by $secTag at each integration point. The -iter switch enables the iterative form of the flexibility formulation. Note that the iterative form can improve the rate of global convergence at the expense of more local element computation. The valid queries to a nonlinear beam-column element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' and 'section $secNum secArg1 secArg2...' Where $secNum refers to the integration point whose data is to be output. Useful references for this element are found in the References (page 473, http://www.) chapter. Beam With Hinges Element Contact Author: Michael H. Scott Oregon State University http://web.engr.oregonstate.edu/~mhscott/ This command is used to construct a beamWithHinges element object, which is based on the non-iterative (or iterative) flexibility formulation, and considers plasticity to be concentrated over specified hinge lengths at the element ends. Note, the beamWithHinges element localize plastic hinging at the element ends only. This type of element divides the element in three parts: two hinges at the ends, and a lineareleastic region in the middle. The hinges are defined by assigning to each a previously-defined section. The length of the each hinge is also specified by the user: Chapter 21 element Command 209 While the integration of distributed-plasticity force-base elements distributes the gauss points along the entire element length, the beamWithHinges element localizes the integration points in the hinge regions. Two integration points per hinge are used to be able represent the curvature distribution accurately -- Gauss-Radau Integration: Two integration points per hinge, however, require too much computational overhead. A Modified Gauss-Radau Integration, developed by Scott et al., would apply the Gauss-Radau hinge integration over 4Lp instead of Lp. Elastic properties are then applied to the interior integration points, where a closed-form solution is used: Chapter 21 element Command 210 There are many advantages to this formulation over the other types of beamWithHinges: Nonlinear behavior is confined to the integration points at the element ends. The user only needs to specify the length of each hinge. Captures largest bending moment at the ends. Represents linear curvature distributions exactly. Characteristic length is equal to Lp when deformations localize. The arguments for the construction of the element depend on the dimension of the problem, ndm (page 31). For a two-dimensional problem: element beamWithHinges $eleTag $iNode $jNode $secTagI $Lpi $secTagJ $Lpj $E $A $Iz $transfTag <-mass $massDens> <-iter $maxIters $tol> For a three-dimensional problem: element beamWithHinges $eleTag $iNode $jNode $secTagI $Lpi $secTagJ $Lpj $E $A $Iz $Iy $G $J $transfTag <-mass $massDens> <-iter $maxIters $tol> $eleTag $iNode unique element object tag $jNode end nodes $secTagI identifier for previously-defined section (page 185) object corresponding to node i $Lpi hinge length at node i $secTagJ identifier for previously-defined section (page 185) object corresponding to node j $Lpj hinge length at node j $E Young's Modulus $A area of element cross-section $Iz section moment of inertia about the section local z-axis $Iy section moment of inertia about the section local y-axis $G Shear Modulus $J torsional moment of inertia of cross section Chapter 21 element Command $transfTag identifier for previously-defined coordinate-transformation (page 302) (CrdTransf) object $massDens element mass density (per unit length), from which a lumpedmass matrix is formed (optional, default=0.0) $maxIters maximum number of iterations to undertake to satisfy element compatibility (optional, default=1) $tol tolerance for satisfaction of element compatibility (optional, default=10-16) 211 The -iter switch enables the iterative form of the flexibility formulation. Note that the iterative form can improve the rate of global convergence at the expense of more local element computation. NOTE: The elastic properties are integrated only over the beam interior, which is considered to be linear-elastic. The valid queries to a beamWithHinges element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' 'rotation' (hinge rotation), or 'section $secNum secArg1 secArg2...' Where $secNum refers to the integration point whose data is to be output. NOTE: The beamWithHinges element used here has four elastic sections and two fiber section at the end. Therefore for output in specifying $secNum: $secNum=1: hinge at node i $secNum=6: hinge at node j Reference: Scott, M.H. and G.L. Fenves. "Plastic Hinge Integration Methods for Force-Based Beam-Column Elements", Journal of Structural Engineering, ASCE, 132(2):244-252, February 2006. Displacement-Based Beam-Column Element This command is used to construct a dispBeamColumn element object, which is a distributedplasticity, displacement-based beam-column element. element dispBeamColumn $eleTag $iNode $jNode $numIntgrPts $secTag $transfTag <-mass $massDens> Chapter 21 $eleTag $iNode element Command 212 unique element object tag $jNode end nodes $numIntgrPts number of integration points along the element. $secTag identifier for previously-defined section (page 185) object $transfTag identifier for previously-defined coordinate-transformation (page 302) (CrdTransf) object $massDens element mass density (per unit length), from which a lumpedmass matrix is formed (optional, default=0.0) The integration along the element is based on the Gauss-Legendre quadrature rule (REF???). The element is prismatic, i.e. the beam is represented by the section model identified by $secTag at each integration point. The valid queries to a displacement-based beam-column element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' and 'section $secNum secArg1 secArg2...' Where $secNum refers to the integration point whose data is to be output. Zero-Length Elements Zero-length elements connect two points at the same coordinate. Zero-Length Element This command is used to construct a zeroLength element object, which is defined by two nodes at the same location. The nodes are connected by multiple UniaxialMaterial (page 47) objects to represent the force-deformation relationship for the element. element zeroLength $eleTag $iNode $jNode -mat $matTag1 $matTag2 ... -dir $dir1 $dir2 ... <-orient $x1 $x2 $x3 $yp1 $yp2 $yp3> $eleTag $iNode unique element object tag $jNode end nodes $matTag1 $matTag2 ... tags associated with previously-defined UniaxialMaterials (page 47) $dir1 $dir2 ... material directions: 1,2,3 translation along local x,y,z axes, respectively Chapter 21 4,5,6 element Command 213 rotation about local x,y,z axes, respectively the orientation vectors can be specified for the element (optional): $x1 $x2 $x3 vector components in global coordinates defining local x-axis (vector x) $yp1 $yp2 $yp3 vector components in global coordinates defining vector yp which lies in the local x-y plane for the element: the local z-axis is defined by the cross product between the vectors x and yp If the optional orientation vectors are not specified, the local element axes coincide with the global axes. The valid queries to a zero-length element when creating an ElementRecorder (page 330) object are 'force,' 'deformation,' 'stiff,' and 'material $matNum matArg1 matArg2 ...' Where $matNum is the tag associated with the material whose data is to be output. Zero-Length ND Element This command is used to construct a zeroLengthND element object, which is defined by two nodes at the same location. The nodes are connected by a single NDMaterial (page 175) object to represent the force-deformation relationship for the element. element zeroLengthND $eleTag $iNode $jNode $matTag <$uniTag> <-orient $x1 $x2 $x3 $yp1 $yp2 $yp3> $eleTag $iNode unique element object tag $jNode end nodes $matTag tag associated with previously-defined NDMaterial (page 175) object $uniTag tag associated with previously-defined UniaxialMaterial (page 47) object which may be used to represent uncoupled behavior orthogonal to the plane of the NDmaterial response. If the NDMaterial (page 175) object is of order two, the response lies in the element local x-y plane and the UniaxialMaterial object may be used to represent the uncoupled behavior orthogonal to this plane, i.e. along the local z-axis. If the NDMaterial (page 175) object is of order three, the response is along each of the element local exes. the orientation vectors can be specified for the element (optional): Chapter 21 element Command $x1 $x2 $x3 vector components in global coordinates defining local x-axis (vector x) $yp1 $yp2 $yp3 vector components in global coordinates defining vector yp which lies in the local x-y plane for the element: 214 the local z-axis is defined by the cross product between the vectors x and yp If the optional orientation vectors are not specified, the local element axes coincide with the global axes. NOTE: The zeroLengthND element only represents translational response between its nodes The valid queries to a zero-length element when creating an ElementRecorder (page 330) object are 'force,' 'deformation,' 'stiffness,' and 'material matArg1 matArg2 ...' Zero-Length Section Element This command is used to construct a zeroLengthSection element object, which is defined by two nodes at the same location. The nodes are connected by a single SectionForceDeformation (page 185) object to represent the force-deformation relationship for the element. element zeroLengthSection $eleTag $iNode $jNode $secTag <-orient $x1 $x2 $x3 $yp1 $yp2 $yp3> $eleTag $iNode unique element object tag $jNode $secTag end nodes tag associated with previously-defined Section (page 185) object the orientation vectors can be specified for the element (optional): $x1 $x2 $x3 vector components in global coordinates defining local x-axis (vector x) $yp1 $yp2 $yp3 vector components in global coordinates defining vector yp which lies in the local x-y plane for the element: the local z-axis is defined by the cross product between the vectors x and yp If the optional orientation vectors are not specified, the local element axes coincide with the global axes. The section (page 185) force-deformation response represented by section string P acts along the element local x-axis, and the response for code Vy along the local y-axis. The other modes of section response follow from this orientation. Chapter 21 element Command 215 The valid queries to a zero-length element when creating an ElementRecorder (page 330) object are 'force,' 'deformation,' 'stiffness,' and 'section secArg1 secArg2 ....' NOTE: the ZeroLength section element only has 1 section therefore you do not need to identify the section in the recorder command .. Example: recorder Element -file Element1.out -time -ele 1 section fiber 0.10 0.10 stressStrain Quadrilateral Elements Quad Element This command is used to construct a FourNodeQuad element object which uses a bilinear isoparametric formulation. element quad $eleTag $iNode $jNode $kNode $lNode $thick $type $matTag <$pressure $rho $b1 $b2> $eleTag unique element object tag $iNode $jNode $kNode $lNode four nodes defining element boundaries, input in counterclockwise order around the element. $thick element thickness (constant) $type string representing material behavior. Valid options depend on the NDMaterial (page 175) object and its available material formulations. The type parameter can be either "PlaneStrain" or "PlaneStress." $matTag tag associated with previously-defined NDMaterial (page 175) object $pressure surface pressure (???? sign convention????****) $rho element mass density (per unit volume) from which a lumped element mass matrix is computed (optional, default=0.0) $b1 $b2 constant body forces defined in the isoparametric domain (optional, default=0.0) Consistent nodal loads are computed from the pressure and body forces. Chapter 21 element Command 216 The valid queries to a Quad element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' and 'material $matNum matArg1 matArg2 ...' Where $matNum refers to the material object at the integration point corresponding to the node numbers in the isoparametric domain. Shell Element This command is used to construct a ShellMITC4 element object, which uses a bilinear isoparametric formulation in combination with a modified shear interpolation to improve thin-plate bending performance. element ShellMITC4 $eleTag $iNode $jNode $kNode $lNode $secTag $eleTag unique element object tag $iNode $jNode $kNode $lNode four nodes defining element boundaries, input in counterclockwise order around the element. $secTag tag associated with previously-defined SectionForceDeformation (page 185) object. Typically, corresponds to some PlateFiberSection (page 202), elastic or otherwise Should the element be required to compute a mass matrix, a consistent translational element mass matrix is computed. Rotational-inertia terms are ignored. The valid queries to a shell element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' and 'material matArg1 matArg2 ...' Chapter 21 element Command 217 Bbar Plane Strain Quadrilateral Element This command is used to construct a four-node quadrilateral element object, which uses a bilinear isoparametric formulation along with a mixed volume/pressure B-bar assumption. This element is for plane strain problems only. element bbarQuad $eleTag $iNode $jNode $kNode $lNode $matTag $eleTag unique element object tag $iNode $jNode $kNode $lNode four nodes defining element boundaries, input in counterclockwise order around the element $matTag tag associated with previously-defined NDMaterial (page 175) object Should the element be required to compute a mass matrix, a consistent translational element mass matrix is computed. Rotational-inertia terms are ignored. Enhanced Strain Quadrilateral Element This command is used to construct a four-node quadrilateral element, which uses a bilinear isoparametric formulation with enhanced strain modes. element enhancedQuad $eleTag $iNode $jNode $kNode $lNode type $matTag $eleTag unique element object tag $iNode $jNode $kNode $lNode four nodes defining element boundaries, input in counterclockwise order around the element. type string representing material behavior. Valid options depend on the NDMaterial (page 175) object and its available material formulations. The type parameter can be either "PlaneStrain" or "PlaneStress." $matTag tag associated with previously-defined NDMaterial (page 175) object Chapter 21 element Command 218 Should the element be required to compute a mass matrix, a consistent translational element mass matrix is computed. Rotational-inertia terms are ignored. The valid queries to a zero-length element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' and 'material matArg1 matArg2 ...' Brick Elements Standard Brick Element This element is used to construct an eight-node brick element object, which uses a trilinear isoparametric formulation. element stdBrick $eleTag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $matTag $eleTag unique element object tag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 eight nodes defining element boundaries, input order is shown in the figure $matTag tag associated with previously-defined NDMaterial (page 175) object Chapter 21 element Command 219 Should the element be required to compute a mass matrix, a consistent translational element mass matrix is computed. Rotational-inertia terms are ignored. Figure 44: NodeNumbering for Standard Brick Element Note: Node numbering for this element is different from that for the eight-node brick (page 232) (Brick8N) element. Chapter 21 element Command 220 Bbar Brick Element This command is used to construct an eight-node mixed volume/pressure brick element object, which uses a trilinear isoparametric formulation. element bbarBrick $eleTag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $matTag $eleTag unique element object tag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 eight nodes defining element boundaries, input order is shown in the figure $matTag tag associated with previously-defined NDMaterial (page 175) object Should the element be required to compute a mass matrix, a consistent translational element mass matrix is computed. Rotational-inertia terms are ignored Chapter 21 element Command 221 Figure 45: Node Numbering for EightNode ThreeDimensional Element Eight Node Brick Element The command is used to construct an eight-node three dimensional brick element object, which is based on tensor operation. element Brick8N $eletag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $matTag $bf1 $bf2 $bf3 $massDens $eletag unique element object tag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 eight node coordinates, input order is shown in the figure $matTag material tag associated with previsouly-defined NDMaterial object $bf1 $bf2 $bf3 body force in the direction of global coordinates x, y and z $massDens mass density (mass/volume) The valid queries to a Brick8N element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' stress', 'gausspoint' or 'plastic'. The output is given as follows: 'stress' the six stress components from each Gauss points are output by the order: sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_xz,sigma_yz 'gausspoint' the coordinates of all Gauss points are printed out 'plastic' the equivalent deviatoric plastic strain from each Gauss point is output in the same order as the coordinates are printed Chapter 21 Figure 46: Node Numbering for EightNode ThreeDimensional Element element Command 222 Chapter 21 element Command 223 Twenty Node Brick Element The element is used to construct a twenty-node three dimensional element object element Brick20N $eletag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $node9 $node10 $node11 $node12 $node13 $node14 $node15 $node16 $node17 $node18 $node19 $node20 $matTag $bf1 $bf2 $bf3 $massDen $eletag unique element object tag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $node9 $node10 $node11 $node12 $node13 $node14 $node15 $node16 $node17 $node18 $node19 $node20 twenty node coordinates, input order is shown in the figure $matTag material tag associated with previsouly-defined NDMaterial (page 175) object $bf1 $bf2 $bf3 body force in the direction of global coordinates x, y and z $massDen mass density (mass/volume) The valid queries to a Brick20N element when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' stress', 'gausspoint' or 'plastic'. The output is given as follows: 'stress' the six stress components from each Gauss points are output by the order: sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_xz,sigma_yz 'gausspoint' the coordinates of all Gauss points are printed out 'plastic' the equivalent deviatoric plastic strain from each Gauss point is output in the same order as the coordinates are printed Chapter 21 Figure 47: Node Numbering for TwentyNode ThreeDimensional Element element Command 224 Chapter 21 element Command 225 u-p-U element This command is used to construct a u-p-U element object, which include two types: eight node element and twenty node element. For eight-node element: element Brick8N_u_p_U $eleTag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $matTag $bf1 $bf2 $bf3 $n $alpha $soildDens $fluidDens $k1 $k2 $k3 $K_fluid $P For twenty-node element: element Brick20N_u_p_U $eleTag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $node9 $node10 $node11 $node12 $node13 $node14 $node15 $node16 $node17 $node18 $node19 $node20 $matTag $bf1 $bf2 $bf3 $n $alpha $soildDens $fluidDens $k1 $k2 $k3 $K_fluid $P $eleTag unique element object tag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 .... node coordinate (either eight or twenty), input order is shown in the figure $matTag material tag associated with previsouly-difined NDMaterial object $bf1 $bf2 $bf3 body force in the direction of global coordinates x, y and z $n porosity $alpha 1-Ks/Kt (ratio of void space =1 for soils, =0.6 for concrete...) $soildDens solid density $fluidDens fluid density $k1 $k1 $k3 coefficient of permeability in the direction of x, y and z $K_fluid fluid bulk modulus $P pressure... not used currently (set to 0.0) Chapter 21 element Command 226 The valid queries to a Brick8N_u_p_U and Brick20N_u_p_U elements when creating an ElementRecorder (page 330) object are 'force,' 'stiffness,' stress', 'gausspoint' or 'plastic'. The output is given as follows: 'stress' the six stress components from each Gauss points are output by the order: sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_xz,sigma_yz 'gausspoint' the coordinates of all Gauss points are printed out 'plastic' the equivalent deviatoric plastic strain from each Gauss point is output in the same order as the coordinates are printed FourNodeQuadUP Element FourNodeQuadUP is a four-node plane-strain element using bilinear isoparametric formulation. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot's theory of porous medium. Each element node has 3 degrees-offreedom (DOF): DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p). element quadUP $eleTag $iNode $jNode $kNode $lNode $thick $type $matTag $bulk $fmass $hPerm $vPerm <$b1 $b2 $t> $eleTag unique element object tag $iNode, $jNode, $kNode, $lNode Four element node (previously defined) numbers in counter-clockwise order around the element $thick Element thickness $type The string "PlaneStrain" $matTag Tag of an NDMaterial object (previously defined) of which the element is composed $bulk Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water), and n the initial porosity. $fmass Fluid mass density $hPerm Permeability coefficient in horizontal direction $vPerm Permeability coefficient in vertical direction Chapter 21 element Command 227 $b1, $b2 Optional body forces in horizontal and vertical directions respectively (defaults are 0.0) $t Optional uniform element normal traction, positive in tension (default is 0.0) TYPICAL RANGE OF PERMEABILITY COEFFICIENT (m/s) Gravel >1.0x10-3 Sand Silty Sand 1.0x10-5 ~ 1.0x10- 1.0x10-7 ~ 1.0x10- 3 5 Silt Clay 1.0x10-9 ~ 1.0x10- <1.0x10-9 7 OUTPUT INTERFACE: Pore pressure can be recorded at an element node using OpenSees Node Recorder (page 326): recorder Node <-file $fileName> <-time> <-node ($node1 $node2 ...)> -dof 3 vel Note: dof 3 is for pore pressure output. The valid queries to a quadUP element when creating an ElementRecorder (page 330) are 'force', 'stiffness', or 'material matNum matArg1 matArg2 ...', where matNum represents the material object at the corresponding integration point. BeamColumnJoint Element This command is used to construct a two-dimensional beam-column-joint element object. The element may be used with both two-dimensional and three-dimensional structures; however, load is transferred only in the plane of the element. element beamColumnJoint $eleTag $Nd1 $Nd2 $Nd3 $Nd4 $Mat1 $Mat2 $Mat3 $Mat4 $Mat5 $Mat6 $Mat7 $Mat8 $Mat9 $Mat10 $Mat11 $Mat12 $Mat13 <$eleHeightFac $eleWidthFac> $eleTag an integer identifying the element tag in the domain $Nd1,$Nd2,$Nd3,$ Nd4 tag associated with previously defined nodes $Mat1 uniaxial material tag for left bar-slip spring at node 1 $Mat2 uniaxial material tag for right bar-slip spring at node 1 Chapter 21 element Command $Mat3 uniaxial material tag for interface-shear spring at node 1 $Mat4 uniaxial material tag for lower bar-slip spring at node 2 $Mat5 uniaxial material tag for upper bar-slip spring at node 2 $Mat6 uniaxial material tag for interface-shear spring at node 2 $Mat7 uniaxial material tag for left bar-slip spring at node 3 $Mat8 uniaxial material tag for right bar-slip spring at node 3 $Mat9 uniaxial material tag for interface-shear spring at node 3 $Mat10 uniaxial material tag for lower bar-slip spring at node 4 $Mat11 uniaxial material tag for upper bar-slip spring at node 4 $Mat12 uniaxial material tag for interface-shear spring at node 4 $Mat13 uniaxial material tag for shear-panel $eleHeightFac floating point value (as a ratio to the total height of the element) to be considered for determination of the distance in between the tension-compression couples (optional, default: 1.0) $eleWidthFac floating point value (as a ratio to the total width of the element) to be considered for determination of the distance in between the tension-compression couples (optional, default: 1.0) 228 Chapter 21 element Command 229 NOTE: Node 3 external node Node 4 shear panel bar-slip spring (typ.) Node 2 internal node Node 1 interface-shear spring (typ.) Chapter 21 element Command 230 Figure 48: BeamColumnJoint Element The valid queries to a BeamColumnJoint element when creating an ElementRecorder (page 330) are as follows: ‘internalDisplacement’ - returns the displacements of the internal joint nodes. ‘externalDisplacement’ - returns the displacement of the external joint nodes. ‘deformation’ - generates a four-column matrix in which the first column is the contribution to the total joint shear deformation of all of the bar-slip components of the joint, the second is the deformation contribution of the interface shear springs, the third is the deformation contribution of the shear-panel and the fourth is the total shear deformation of the joint. ‘node1BarSlipL’ the Left at node 1. - returns the load-deformation response history of the Bar-Slip spring on ‘node1BarSlipR’ the Right at node 1. - returns the load-deformation response history of the Bar-Slip spring on ‘node1InterfaceShear’- returns the load-deformation response history of the Interface-Shear spring at node 1. ‘node2BarSlipB’ - returns the load-deformation response history of the Bar-Slip spring on the Bottom at node 2. ‘node2BarSlipT’ the Top at node 2. - returns the load-deformation response history of the Bar-Slip spring on ‘node2InterfaceShear’- returns the load-deformation response history of the Interface Shear spring at node 1. ’node3BarSlipL’ the Left at node 3. - returns the load-deformation response history of the Bar-Slip spring on ‘node3BarSlipR’ the Right at node 3. - returns the load-deformation response history of the Bar-Slip spring on ‘node3InterfaceShear’- returns the load-deformation response history of the Interface-Shear spring at node 3. ‘node4BarSlipB’ - returns the load-deformation response history of the Bar-Slip spring on the Bottom at node 4. ‘node4BarSlipT’ the Top at node 4. - returns the load-deformation response history of the Bar-Slip spring on Chapter 21 element Command 231 ‘node4InterfaceShear’- returns the load-deformation response history of the Interface Shear spring at node 4. ‘shearPanel’ - returns the load-deformation response history of the Shear-Panel spring. EXAMPLE: main input file: PR1.tcl (page 247) supporting files: procMKPC.tcl (page 257) procUniaxialPinching.tcl (page 106) procRC.tcl (page 259) References: Lowes, Laura N.; Mitra, Nilanjan; Altoontash, Arash A beam-column joint model for simulating the earthquake response of reinforced concrete frames PEER-2003/10 (http://nisee.berkeley.edu/elibrary/getdoc?id=1288878) Pacific Earthquake Engineering Research Center, University of California, Berkeley 2003 59 pages (400/P33/2003-10) Chapter 21 Beam-Column Joint Element Discussion 232 Beam-Column Joint Element Discussion Beam-Column Joint Element Discussion The example files (PR1.tcl (page 247), procMKPC.tcl (page 257), procUniaxialPinching.tcl (page 106), procRC.tcl (page 259)) create a model of a RC beam column sub-assemblage (Figure 1). The cruciform is subjected to constant gravity load at nodes 4 and 7 and pseudo-static cyclic lateral load under displacement control at node 10. The beam-column-joint region (element number 7) is represented using a beamColumnJoint element (Figure 2), and the beams and columns (element numbers 1 through 6) are modeled using the nonlinearBeamColumn element. The beam-column joint consists of 13 components that may have different material constitutive models; in this example 9 of the 13 components utilize the nonlinear material model – Pinching4. Figure 3 shows the displacement history for node 10. Chapter 21 Figure 49: Cruciform model Figure 1: Cruciform model Figure 50: Beam Column Joint Element Figure 2: Beam Column Joint Element Beam-Column Joint Element Discussion 233 Chapter 21 Figure 51: Displacement history for Node 10 Figure 3: Displacement history for Node 10 Beam-Column Joint Element Discussion 234 Chapter 21 Beam-Column Joint Element Discussion 235 Tcl Scripts: The following tcl script files are used to run the examples: PR1.tcl (page 247) procMKPC.tcl (page 257) procUniaxialPinching.tcl (page 106) procRC.tcl (page 259) The p-delta response of cruciform, along with the response of each of the nonlinear joint components is shown below. The shear panel response shows the moment-curvature relationships whereas the bar slips at the beam top and bottom are represented by the force-slip plots (Figure 4). Chapter 21 Beam-Column Joint Element Discussion Figure 52: Nonlinear Joint component response Figure 4: Nonlinear Joint component response Figure 53: P-delta response of the cruciform Figure 5. P-delta response of the cruciform PR1.tcl ############################################################################ # Test example for BEAM COLUMN ELEMENT JOINT ------- PARK RUITONG TEST SPECIMEN Unit 1 # Written: N.Mitra # Description: 4 noded 12 dof element having 12 springs and a shear panel # Date: Feb 16 2003 ## Model consisting of a crucifix with beams and columns and a joint 236 Chapter 21 Beam-Column Joint Element Discussion 237 ## File Name: PR1.tcl # refer to Beam-Column-Joint Element.doc for full explanation of the parameters ################################################################################ #create the ModelBuilder object model BasicBuilder -ndm 2 -ndf 3 ## unit name ---- PR1 set fName "PR1"; source procMKPC.tcl source procUniaxialPinching.tcl source procRC.tcl # all dimensions are in here as MPa (conversion factors are required in certain places) set Strfactor 145; set Lenfactor [expr 1/25.4]; ## Y taken as the inplane dim. against which the bending takes place set colY 406; set colZ 305; set bmY 457; set bmZ 229; # covers set colCov 43; set bmCov1 42; set bmCov2 33; set bmCov $bmCov1; # y,z,x dimension of the joint respectively set JointWidth [expr $colY]; set JointHeight [expr $bmY]; set JointDepth $colZ ; set BeamLengthIn 645; set BeamLengthOut 1271; set ColumnLengthClear 1008; set JointVolume [expr $JointWidth*$JointHeight*$JointDepth]; ############################################################ ################# material properties of column section ############################################################ set CUnconfFc -45.9; set CUnconfEc -0.002; set CTSspace 60; set CTSlength 1853.53; set CTSFy 282; set CTSarea 28.3; set CFy 498.0; set CEs 196600.0; set CsHratio 0.004216; set CAs 201.06; procMKPC $CUnconfFc $CUnconfEc $colY $colZ $colCov $CTSspace $CTSlength $CTSFy $CTSarea $Strfactor $Lenfactor Chapter 21 Beam-Column Joint Element Discussion 238 set CUnconfFcu [lindex $concreteProp 2]; set CUnconfEcu [lindex $concreteProp 3]; set CConfFc [lindex $concreteProp 4]; set CConfEc [lindex $concreteProp 5]; set CConfFcu [lindex $concreteProp 6]; set CConfEcu [lindex $concreteProp 7]; ######################################################### ########################### material properties of beam section ##################################################### set BUnconfFc -45.9; set BUnconfEc -0.002; set BTSspace 80; set BTSlength 1036; set BTSFy 282; set BTSarea 28.3; set BFy 294.0; set BEs 210400.0; set BAs 201.06; set BsHratio 0.002322; procMKPC $BUnconfFc $BUnconfEc $bmY $bmZ $bmCov $BTSspace $BTSlength $BTSFy $BTSarea $Strfactor $Lenfactor set BUnconfFcu [lindex $concreteProp 2]; set BUnconfEcu [lindex $concreteProp 3]; set BConfFc [lindex $concreteProp 4]; set BConfEc [lindex $concreteProp 5]; set BConfFcu [lindex $concreteProp 6]; set BConfEcu [lindex $concreteProp 7]; #################################################################### ######################### details for the material models of bar slip of the beam #################################### set bs_fc [expr -$BUnconfFc]; set bs_fs $BFy; set bs_es $BEs; set bs_fsu 434; set bs_dbar 16; set bs_esh [expr $BsHratio*$BEs]; set bs_wid $colZ; set bs_dep $bmY; set bsT_nbars 5; set bsB_nbars 2; set bs_ljoint $colY; ################################################################################ ########################## details for the material models of bar slip of the column ################################## set cs_fc [expr -$CUnconfFc]; set cs_fs $CFy; set cs_es $CEs; set cs_fsu 660; set cs_dbar 16; set cs_esh [expr $CsHratio*$CEs]; set cs_wid $colZ; set cs_dep $colY; set cs_nbars 3; set cs_ljoint $bmY; ##################################################################### ############### add nodes - command: node nodeId xCrd yCrd ####################################################### Chapter 21 Beam-Column Joint Element Discussion node 1 0.0 0.0 node 2 0.0 $ColumnLengthClear node 3 [expr -$BeamLengthOut-$BeamLengthIn-$JointWidth/2] node 4 [expr -$BeamLengthIn-$JointWidth/2] [expr $ColumnLengthClear+$JointHeight/2] [expr $ColumnLengthClear+$JointHeight/2] node 5 [expr -$JointWidth/2] [expr $ColumnLengthClear+$JointHeight/2] node 6 [expr $JointWidth/2] [expr $ColumnLengthClear+$JointHeight/2] node 7 [expr $BeamLengthIn+$JointWidth/2] [expr $ColumnLengthClear+$JointHeight/2] node 8 [expr $BeamLengthOut+$BeamLengthIn+$JointWidth/2] [expr $ColumnLengthClear+$JointHeight/2] node 9 0.0 [expr $ColumnLengthClear+$JointHeight] node 10 0.0 [expr 2*$ColumnLengthClear+$JointHeight] # add material Properties - command: uniaxialMaterial matType matTag ... #command: uniaxialMaterial Elastic tag? E? uniaxialMaterial Elastic 1 10000000000.0 ##################################################################### ############## inelastic beam column elements ########################################################## uniaxialMaterial Concrete01 10 $BUnconfFc $BUnconfEc $BUnconfFcu $BUnconfEcu uniaxialMaterial Concrete01 20 $BConfFc $BConfEc $BConfFcu $BConfEcu uniaxialMaterial Steel02 30 $BFy $BEs $BsHratio 18.5 0.925 0.15 0.0 0.4 0.0 0.5 uniaxialMaterial Concrete01 40 $CUnconfFc $CUnconfEc $CUnconfFcu $CUnconfEcu uniaxialMaterial Concrete01 50 $CConfFc $CConfEc $CConfFcu $CConfEcu uniaxialMaterial Steel02 60 $CFy $CEs $CsHratio 18.5 0.925 0.15 0.0 0.4 0.0 0.5 ########### for columns /////////////////////////////////////////////////////////////////////// set z [expr $colZ/2.0]; set y [expr $colY/2.0]; section Fiber 1 { patch rect 50 8 1 [expr $colCov-$y] patch rect 40 2 1 [expr -$y] patch rect 40 2 1 [expr $y-$colCov] 239 [expr $colCov-$z] [expr $y-$colCov] [expr $colCov-$z] [expr $colCov-$y] [expr $colCov-$z] [expr $y] [expr $z-$colCov] [expr $z-$colCov] [expr $z-$colCov] Chapter 21 Beam-Column Joint Element Discussion patch rect 40 8 1 [expr -$y] [expr -$z] patch rect 40 8 1 [expr -$y] [expr $z-$colCov] [expr $y] layer straight 60 3 $CAs [expr $y-$colCov] layer straight 60 2 $CAs 0.0 layer straight 60 3 $CAs [expr $colCov-$y] [expr $y] 240 [expr $colCov-$z] [expr $z] [expr $colCov-$z] [expr $y-$colCov] [expr $colCov-$z] 0.0 [expr $z-$colCov] [expr $z-$colCov] [expr $colCov-$z] [expr $colCov-$y] [expr $z-$colCov] } #################### for beams /////////////////////////////////////////////////////////////////// set z [expr $bmZ/2.0]; set y [expr $bmY/2.0]; section Fiber 2 { patch rect 20 8 1 [expr $bmCov1-$y] [expr $bmCov1-$z] patch rect 10 2 1 [expr -$y] patch rect 10 2 1 [expr $y-$bmCov1] patch rect 10 8 1 [expr -$y] [expr -$z] patch rect 10 8 1 [expr -$y] [expr $z-$bmCov1] [expr $bmCov1-$z] [expr $bmCov1-$y] [expr $bmCov1-$z] layer straight 30 3 $BAs [expr $y-$bmCov1] $bmCov1] [expr $y] [expr $y] [expr $y] [expr $bmCov1-$z] ##############///////////////////////////////////////////////////////////////////////////////////// ## add geometric transformation -command: geomTransf transfType ... ## geomTransf Linear tag? geomTransf Linear 1 geomTransf Linear 2 element nonlinearBeamColumn 1 1 2 5 1 2 element nonlinearBeamColumn 2 9 10 5 1 2 [expr $z-$bmCov1] [expr $z-$bmCov1] [expr $z] [expr $y-$bmCov1] [expr $bmCov1-$z] } [expr $z-$bmCov1] [expr $bmCov1-$z] [expr $bmCov1-$z] layer straight 30 2 $BAs [expr $y-$bmCov1-$bmCov2] [expr $z - $bmCov1] layer straight 30 2 $BAs [expr $bmCov1-$y] $bmCov1] [expr $y-$bmCov1] [expr $z - [expr $y-$bmCov1-$bmCov2] [expr $bmCov1-$y] [expr $z - Chapter 21 Beam-Column Joint Element Discussion 241 element nonlinearBeamColumn 3 3 4 3 2 1 element nonlinearBeamColumn 4 4 5 2 2 1 element nonlinearBeamColumn 5 6 7 2 2 1 element nonlinearBeamColumn 6 7 8 3 2 1 ##### end element formation as well as material defination for beams and columns ###################### ########################################################################################## # for beam bottom set matID1 21 set matID2 22 uniaxialMaterial BarSlip $matID1 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsB_nbars $bs_wid $bs_dep strong beamBot uniaxialMaterial BarSlip $matID2 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsB_nbars $bs_wid $bs_dep strong beamBot ## %%%%%%%%%%%%%%% equivalent statement can be made in other way #uniaxialMaterial BarSlip $matID1 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsB_nbars $bs_wid $bs_dep 1.0 strong beamBot damage #uniaxialMaterial BarSlip $matID2 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsB_nbars $bs_wid $bs_dep 1.0 strong beamBot damage # for beam top set matID3 31 set matID4 32 uniaxialMaterial BarSlip $matID3 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsT_nbars $bs_wid $bs_dep strong beamTop uniaxialMaterial BarSlip $matID4 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsT_nbars $bs_wid $bs_dep strong beamTop ## %%%%%%%%%%%%%%% equivalent statement can be made in other way #uniaxialMaterial BarSlip $matID3 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsT_nbars $bs_wid $bs_dep 1.0 strong beamTop damage #uniaxialMaterial BarSlip $matID4 $bs_fc $bs_fs $bs_es $bs_fsu $bs_esh $bs_dbar $bs_ljoint $bsT_nbars $bs_wid $bs_dep 1.0 strong beamTop damage # for columns set matID5 41 Chapter 21 Beam-Column Joint Element Discussion 242 set matID6 42 set matID7 43 set matID8 44 uniaxialMaterial BarSlip $matID5 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep strong column uniaxialMaterial BarSlip $matID6 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep strong column uniaxialMaterial BarSlip $matID7 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep strong column uniaxialMaterial BarSlip $matID8 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep strong column ## %%%%%%%%%%%%%%% equivalent statement can be made in other way #uniaxialMaterial BarSlip $matID5 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep 1.0 strong column damage #uniaxialMaterial BarSlip $matID6 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep 1.0 strong column damage #uniaxialMaterial BarSlip $matID7 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep 1.0 strong column damage #uniaxialMaterial BarSlip $matID8 $cs_fc $cs_fs $cs_es $cs_fsu $cs_esh $cs_dbar $cs_ljoint $cs_nbars $cs_wid $cs_dep 1.0 strong column damage #################### end material formation for bar slip ######################################################## ############################ material for shear panel ############################################################# ## Positive/Negative envelope Stress set p1 2.1932; set p2 4.0872; set p3 4.4862; set p4 [expr $p3*1e-3]; ## stress1 stress2 stress3 stress4 set pEnvStrsp [list [expr $p1*$JointVolume] [expr $p2*$JointVolume] [expr $p3*$JointVolume] [expr $p4*$JointVolume]] set nEnvStrsp [list [expr -$p1*$JointVolume] [expr -$p2*$JointVolume] [expr -$p3*$JointVolume] [expr $p4*$JointVolume]] ## Positive/Negative envelope Strain ## strain1 strain2 strain3 strain4 set pEnvStnsp [list 0.0002 0.004465 0.0131 0.0269] set nEnvStnsp [list -0.0002 -0.004465 -0.0131 -0.0269] Chapter 21 Beam-Column Joint Element Discussion ## Ratio of maximum deformation at which reloading begins ## Pos_env. Neg_env. set rDispsp [list 0.25 0.25] ## Ratio of envelope force (corresponding to maximum deformation) at which reloading begins ### Pos_env. Neg_env. set rForcesp [list 0.15 0.15] ## Ratio of monotonic strength developed upon unloading ### Pos_env. Neg_env. set uForcesp [list 0.0 0.0] ## Coefficients for Unloading Stiffness degradation ## gammaK1 gammaK2 gammaK3 gammaK4 gammaKLimit set gammaKsp [list 1.13364492409642 #set gammaKsp [list 0.0 0.0 0.0 0.0 0.0 0.10111033064469 0.0 0.0] #### Coefficients for Reloading Stiffness degradation ### gammaD1 gammaD2 gammaD3 gammaD4 gammaDLimit set gammaDsp [list 0.12 0.0 0.23 #set gammaDsp [list 0.0 0.0 0.0 0.0 0.0 0.95] 0.0] #### Coefficients for Strength degradation ### gammaF1 gammaF2 gammaF3 gammaF4 gammaFLimit set gammaFsp [list 1.11 0.0 0.319 #set gammaFsp [list 0.0 0.0 0.0 0.0 0.0 0.125] 0.0] set gammaEsp 10.0 uniaxialMaterial Pinching4 5 [lindex $pEnvStrsp 0] [lindex $pEnvStnsp 0] \ [lindex $pEnvStrsp 1] [lindex $pEnvStnsp 1] [lindex $pEnvStrsp 2] \ [lindex $pEnvStnsp 2] [lindex $pEnvStrsp 3] [lindex $pEnvStnsp 3] \ [lindex $nEnvStrsp 0] [lindex $nEnvStnsp 0] \ 0.91652498468618] 243 Chapter 21 Beam-Column Joint Element Discussion 244 [lindex $nEnvStrsp 1] [lindex $nEnvStnsp 1] [lindex $nEnvStrsp 2] \ [lindex $nEnvStnsp 2] [lindex $nEnvStrsp 3] [lindex $nEnvStnsp 3] \ [lindex $rDispsp 0] [lindex $rForcesp 0] [lindex $uForcesp 0] \ [lindex $rDispsp 1] [lindex $rForcesp 1] [lindex $uForcesp 1] \ [lindex $gammaKsp 0] [lindex $gammaKsp 1] [lindex $gammaKsp 2] [lindex $gammaKsp 3] [lindex $gammaKsp 4] \ [lindex $gammaDsp 0] [lindex $gammaDsp 1] [lindex $gammaDsp 2] [lindex $gammaDsp 3] [lindex $gammaDsp 4] \ [lindex $gammaFsp 0] [lindex $gammaFsp 1] [lindex $gammaFsp 2] [lindex $gammaFsp 3] [lindex $gammaFsp 4] \ $gammaEsp energy #################### end material formation for shear panel ########################################### ##element BeamColumnJoint tag? iNode? jNode? kNode? lNode? matTag1? matTag2? matTag3? matTag4? ## matTag5? matTag6? matTag7? matTag8? matTag9? matTag10? matTag11? matTag12? matTag13? ## <element Height factor?> <element Width factor?> ## please note: the four nodes are in anticlockwise direction around the element ## requires material tags for all 13 different components within the element. ## the first 12 being that of spring and the last of the shear panel element beamColumnJoint 7 2 6 9 5 41 42 1 21 31 1 43 44 1 22 32 1 5 #element beamColumnJoint 7 2 6 9 5 1 1 1 1 1 1 1 1 1 1 1 1 1 ## %%%%%%%%%%%%%%% equivalent statement can be made in other way #element beamColumnJoint 7 2 6 9 5 41 42 1 21 31 1 43 44 1 22 32 1 5 1.0 1.0 #element beamColumnJoint 7 2 6 9 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0 1.0 # set the boundary conditions - command: fix nodeID xResrnt? yRestrnt? fix 1 1 1 0 fix 2 0 0 0 fix 3 0 1 0 fix 4 0 0 0 fix 5 0 0 0 fix 6 0 0 0 fix 7 0 0 0 fix 8 0 1 0 fix 9 0 0 0 Chapter 21 Beam-Column Joint Element Discussion fix 10 0 0 0 pattern Plain 2 Linear { load 4 0 -55000 0 -const load 7 0 -55000 0 -const } system ProfileSPD constraints Plain integrator LoadControl 0 1 0 0 test NormDispIncr 1e-8 150 algorithm Newton numberer RCM analysis Static analyze 1 loadConst -time 0.0 pattern Plain 1 Linear { #load nd? Fx? Fy? Mz? load 10 1 0 0 } set rbbt "_RBbt"; set rbtp "_RBtp"; set dlcbr "_DLCbr"; set sp "_Sp"; set jdf "_Jdf"; set lbbt "_LBbt"; set lbtp "_LBtp"; set drcbr "_DRCbr"; set ulcbr "_ULCbr"; set urcbr "_URCbr"; set RBbt [concat $fName$rbbt]; set RBtp [concat $fName$rbtp]; set Sp [concat $fName$sp]; set LBbt [concat $fName$lbbt]; set LBtp [concat $fName$lbtp]; set DLCbr [concat $fName$dlcbr]; set DRCbr [concat $fName$drcbr]; set URCbr [concat $fName$urcbr]; set ULCbr [concat $fName$ulcbr]; set Jdf [concat $fName$jdf]; recorder Node $fName.out disp -load -node 10 -dof 1 recorder Element 7 -file $RBbt.out node2BarSlipB stressStrain recorder Element 7 -file $RBtp.out node2BarSlipT stressStrain recorder Element 7 -file $LBbt.out node4BarSlipB stressStrain recorder Element 7 -file $LBtp.out node4BarSlipT stressStrain recorder Element 7 -file $DLCbr.out node1BarSlipL stressStrain recorder Element 7 -file $DRCbr.out node1BarSlipR stressStrain recorder Element 7 -file $ULCbr.out node3BarSlipL stressStrain 245 Chapter 21 Beam-Column Joint Element Discussion 246 recorder Element 7 -file $URCbr.out node3BarSlipR stressStrain recorder Element 7 -file $Sp.out shearpanel stressStrain recorder Element 7 -file $Jdf.out deformation set peakpts [list 0.1 10 10 30 30 45 45 60 60 75 75 90 90 105 105] set increment 10 set nodeTag 10 set dofTag 1 procRC $increment $nodeTag $dofTag $peakpts # print the results at node and at all elements print node #print element procMKPC.tcl ############################################################################################# ########## # # procMKPC.tcl ## procedure for evaluating the confined concrete material envelope points based upon the modified ## kent park procedure. The procedure takes in the unconfined concrete and confining steel properties. ## created : NM ([email protected]) dated: Dec. 2002 ############################################################################################# ########## proc procMKPC { CUnconfFc CUnconfEc Y Z Cov TSspace TSlength TSFy TSarea Strfactor Lenfactor } { set CUnconfEcu -0.004; set SecWid [expr $Lenfactor*$Z]; set SecDep [expr $Lenfactor*$Y]; set cover [expr $Lenfactor*$Cov]; set UFc [expr -$Strfactor*$CUnconfFc]; set Ue0 [expr -$CUnconfEc]; set Uecu [expr -$CUnconfEcu]; set hoopSpc [expr $Lenfactor*$TSspace]; set hoopLngth [expr $Lenfactor*$TSlength]; set hoopFy [expr $Strfactor*$TSFy]; set hoopArea [expr $TSarea*$Lenfactor*$Lenfactor]; # ratio of volume of rectangular steel hoops to volumne of concrete core measured to outside of peripheral hoops set rhoS [expr ($hoopLngth*$hoopArea)/(($SecWid-2*$cover)*($SecDep-2*$cover)*$hoopSpc)]; # width of concrete core measured to outside of peripheral hoop set b [expr $SecWid - 2*$cover]; Chapter 21 Beam-Column Joint Element Discussion 247 set temp [expr $b/$hoopSpc] set e50u [expr (3+0.002*$UFc)/($UFc - 1000)]; set e50h [expr 3*$rhoS*pow($temp,0.5)/4]; set Zm [expr 0.5*($UFc-1000)/(3+0.002*$UFc)]; set Z [expr 0.5/($e50u + $e50h - $Ue0)]; set K [expr (1 + $rhoS*$hoopFy/$UFc)]; # unconfined ultimate compressive strength set UFcu [expr -$UFc*(1-$Zm*($Uecu-$Ue0))/$Strfactor]; #cracking strain in confined concrete set Ce0 [expr -$K*$Ue0]; # cracking stress in confined concrete set CFc [expr -$K*$UFc/$Strfactor]; # ultimate stress in confined concrete set CFcu [expr 0.2*$CFc]; # ultimate strain in confined concrete set Cecu [expr -(0.8/$Z - $Ce0)]; global concreteProp; set concreteProp [list $CUnconfFc $CUnconfEc $UFcu $CUnconfEcu $CFc $Ce0 $CFcu $Cecu]; #puts [lindex $concreteProp 0] return $concreteProp; } procUniaxialPinching.tcl ############################################################################################# ######## ## # procUniaxialPinching.tcl # # procedure for activating the pinching material given its parameters in the form of list # # created NM ([email protected]) dated : Feb 2002 # ############################################################################################# ######## Chapter 21 Beam-Column Joint Element Discussion 248 proc procUniaxialPinching { materialTag pEnvelopeStress nEnvelopeStress pEnvelopeStrain nEnvelopeStrain rDisp rForce uForce gammaK gammaD gammaF gammaE damage} { # add material - command: uniaxialMaterial ...... paramaters as shown #uniaxialMaterial Pinching4 tag #### stress1P strain1P stress2P strain2P stress3P strain3P stress4P strain4P #### stress1N strain1N stress2N strain2N stress3N strain3N stress4N strain4N #### rDispP rForceP uForceP rDispN rForceN uForceN #### gammaK1 gammaK2 gammaK3 gammaK4 gammaKLimit #### gammaD1 gammaD2 gammaD3 gammaD4 gammaDLimit #### gammaF1 gammaF2 gammaF3 gammaF4 gammaFLimit gammaE $damage uniaxialMaterial Pinching4 $materialTag [lindex $pEnvelopeStress 0] [lindex $pEnvelopeStrain 0] [lindex $pEnvelopeStress 1] [lindex $pEnvelopeStrain 1] [lindex $pEnvelopeStress 2] [lindex $pEnvelopeStrain 2] [lindex $pEnvelopeStress 3] [lindex $pEnvelopeStrain 3] [lindex $nEnvelopeStress 0] [lindex $nEnvelopeStrain 0] [lindex $nEnvelopeStress 1] [lindex $nEnvelopeStrain 1] [lindex $nEnvelopeStress 2] [lindex $nEnvelopeStrain 2] [lindex $nEnvelopeStress 3] [lindex $nEnvelopeStrain 3] [lindex $rDisp 0] [lindex $rForce 0] [lindex $uForce 0] [lindex $rDisp 1] [lindex $rForce 1] [lindex $uForce 1] [lindex $gammaK 0] [lindex $gammaK 1] [lindex $gammaK 2] [lindex $gammaK 3] [lindex $gammaK 4] [lindex $gammaD 0] [lindex $gammaD 1] [lindex $gammaD 2] [lindex $gammaD 3] [lindex $gammaD 4] [lindex $gammaF 0] [lindex $gammaF 1] [lindex $gammaF 2] [lindex $gammaF 3] [lindex $gammaF 4] $gammaE $damage } procRC.tcl ############################################################################################# ########### Chapter 21 Beam-Column Joint Element Discussion 249 # # procRC.tcl ## procedure for setting up a reversed cycle loading scheme. The input are mainly the ## peak points for the loading. ## The procedure primarily uses Displacement control for loading, if it fails uses ArcLength control ## created : NM ([email protected]) dated: Sep 2002 ############################################################################################# ########### proc procRC { incre nodeTag dofTag peakpts } { set displayTag 0; set numTimes 150; set x [lindex $peakpts 0]; set dU [expr $x/$incre]; #set dU0 [expr $dU/1000]; set dU0 [expr $dU/10000]; integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU $dU analysis Static analyze $incre integrator DisplacementControl $nodeTag $dofTag 0.0 1 [expr -$dU] [expr -$dU] analyze [expr 2*$incre] integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU $dU analyze $incre ## end the first peak pt start for others for {set j 1} {$j < [llength $peakpts]} {incr j 1} { set y [lindex $peakpts $j] set dSt [expr $y/$dU] set dS [expr int($dSt)] test NormDispIncr 1e-8 $numTimes $displayTag Chapter 21 Beam-Column Joint Element Discussion algorithm Newton ############# start loading cycle ################## set t 0; while {$t != $dS} { integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU $dU set ok [analyze 1] incr t 1; if {$ok != 0} { # if {$t == $dS} {break}; puts "Displacement control failed ..... trying Arc-Length control" set currentDisp [nodeDisp $nodeTag $dofTag] puts "Current Displacement is $currentDisp" # algorithm Linear test NormDispIncr 1e-6 $numTimes $displayTag #algorithm ModifiedNewton # integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU0 $dU0 # integrator DisplacementControl $nodeTag $dofTag 0.0 10 $dU0 $dU0 integrator ArcLength [expr $dU0] 1.0 # set ok [analyze 1] analyze 1 } # puts "that worked ..... back to regular Newton " test NormDispIncr 1e-8 $numTimes $displayTag # algorithm Newton } ################## end of loading cycle, start unloading cycle ######## set t 0; while {$t != [expr 2*$dS]} { integrator DisplacementControl $nodeTag $dofTag 0.0 1 [expr -$dU] [expr -$dU] set ok [analyze 1] incr t 1; 250 Chapter 21 Beam-Column Joint Element Discussion 251 if {$ok != 0} { # if {$t == [expr 2*$dS]} {break}; puts "Displacement control failed ..... trying Arc-Length control" set currentDisp [nodeDisp $nodeTag $dofTag] puts "Current Displacement is $currentDisp" # algorithm Linear test NormDispIncr 1e-6 $numTimes $displayTag #algorithm ModifiedNewton # $dU0] integrator DisplacementControl $nodeTag $dofTag 0.0 1 [expr -$dU0] [expr - # $dU0] integrator DisplacementControl $nodeTag $dofTag 0.0 10 [expr -$dU0] [expr integrator ArcLength [expr $dU0] 1.0 # set ok [analyze 1] analyze 1 } # puts "that worked .... back to regular Newton " test NormDispIncr 1e-8 $numTimes $displayTag # algorithm Newton } ############# end of unloading cycle, start reloading cycle ########### set t 0; while {$t != $dS} { integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU $dU set ok [analyze 1] incr t 1; if {$ok != 0} { # if {$t == $dS} {break}; puts "Displacement control failed ..... trying Arc-Length control" set currentDisp [nodeDisp $nodeTag $dofTag] puts "Current Displacement is $currentDisp" # algorithm Linear test NormDispIncr 1e-6 $numTimes $displayTag #algorithm ModifiedNewton Chapter 21 Beam-Column Joint Element Discussion # integrator DisplacementControl $nodeTag $dofTag 0.0 1 $dU0 $dU0 # integrator DisplacementControl $nodeTag $dofTag 0.0 10 $dU0 $dU0 integrator ArcLength [expr $dU0] 1.0 # set ok [analyze 1] analyze 1 } # puts "that worked .... back to regular Newton " test NormDispIncr 1e-8 $numTimes $displayTag # algorithm Newton } ######## reloading cycle completed ############################# if {$ok == 0} { puts "analysis succesful at $y mm displacement"; } else { puts "analysis could not proceed fine beyond $y mm displacement"; } } } 252 253 CHAPTER 22 Soil Models and Solid-Fluid Fully Coupled Elements This chapter describes the user interfaces for: 1) a number of NDMaterial models developed for simulating nonlinear, drained/undrained soil response under general 3D cyclic loading conditions, and 2) a number of 2D and 3D solid-fluid fully coupled elements for simulating pore water pressure dissipation/redistribution. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. This documentation is last updated at 8/5/2005 by Zhaohui Yang ([email protected] (mailto:[email protected])) and Ahmed Elgamal ([email protected] (mailto:[email protected])). In This Chapter PressureDependMultiYield ....................................... 265 PressureDependMultiYield02 ................................... 272 PressureIndependMultiYield..................................... 277 updateMaterialStage ................................................ 283 updateParameter...................................................... 284 FluidSolidPorousMaterial.......................................... 285 FourNodeQuadUP.................................................... 286 Nine_Four_Node_QuadUP ...................................... 288 BrickUP .................................................................... 290 Twenty_Eight_Node_BrickUP .................................. 292 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 254 PressureDependMultiYield PressureDependMultiYield material is an elastic-plastic material for simulating the essential response characteristics of pressure sensitive soil materials under general loading conditions. Such characteristics include dilatancy (shear-induced volume contraction or dilation) and nonflow liquefaction (cyclic mobility), typically exhibited in sands or silts during monotonic or cyclic loading. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. When this material is employed in regular solid elements (e.g., FourNodeQuad, Brick), it simulates drained soil response. To simulate soil response under fully undrained condition, this material may be either embedded in a FluidSolidPorousMaterial (see below), or used with one of the solid-fluid fully coupled elements (see below) with very low permeability. To simulate partially drained soil response, this material should be used with a solid-fluid fully coupled element with proper permeability values. During the application of gravity load (and static loads if any), material behavior is linear elastic. In the subsequent dynamic (fast) loading phase(s), the stress-strain response is elastic-plastic (see MATERIAL STAGE UPDATE below). Plasticity is formulated based on the multi-surface (nested surfaces) concept, with a non-associative flow rule to reproduce dilatancy effect. The yield surfaces are of the Drucker-Prager type. OUTPUT INTERFACE: The following information may be extracted for this material at a given integration point, using the OpenSees Element Recorder facility (McKenna and Fenves 2001): "stress", "strain", "backbone", or "tangent". For 2D problems, the stress output follows this order: σxx, σyy, σzz, σxy, ηr, where ηr is the ratio between the shear (deviatoric) stress and peak shear strength at the current confinement (0<= ηr<=1.0). The strain output follows this order: εxx, εyy, γxy. For 3D problems, the stress output follows this order: σxx, σyy, σzz, σxy, σyz, σzx, ηr, and the strain output follows this order: εxx, εyy, εzz, γxy, γyz, γzx. The "backbone" option records (secant) shear modulus reduction curves at one or more given confinements. The specific recorder command is as follows: recorder Element –ele $eleNum -file $fName -dT $deltaT material $GaussNum backbone $p1 <$p2 …> Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 255 where p1, p2, … are the confinements at which modulus reduction curves are recorded. In the output file, corresponding to each given confinement there are two columns: shear strain γ and secant modulus Gs. The number of rows equals the number of yield surfaces. nDMaterial PressureDependMultiYield $tag $nd $rho $refShearModul $refBulkModul $frictionAng $peakShearStra $refPress $pressDependCoe $PTAng $contrac $dilat1 $dilat2 $liquefac1 $liquefac2 $liquefac3 <$noYieldSurf=20 <$r1 $Gs1 …> $e=0.6 $cs1=0.9 $cs2=0.02 $cs3=0.7 $pa=101> Octahedral shear stress τ $τ f Shear stress-strain at p'r Octahedral shear strain γ $Gr $γmax $tag A positive integer uniquely identifying the material among all nDMaterials. $nd Number of dimensions, 2 for plane-strain, and 3 for 3D analysis. $rho Saturated soil mass density. $refShearModul (Gr) Reference low-strain shear modulus, specified at a reference mean effective confining pressure refPress of p’r (see below). $refBulkModul (Br) Reference bulk modulus, specified at a reference mean effective confining pressure refPress of p’r (see below). $frictionAng (f) Friction angle at peak shear strength, in degrees. Chapter 22 $peakShearStra (gmax) Soil Models and Solid-Fluid Fully Coupled Elements 256 An octahedral shear strain at which the maximum shear strength is reached, specified at a reference mean effective confining pressure refPress of p’r (see below). Octahedral shear strain is defined as: γ = [ 2 ( ε xx − ε yy )2 + (ε yy − ε zz )2 + (ε xx − ε zz )2 + 6ε xy2 + 6ε yz2 + 6ε xz2 3 ] 1/ 2 $refPress (p’r) Reference mean effective confining pressure at which Gr, Br, and γmax are defined. $pressDependCoe (d) A positive constant defining variations of G and B as a function of instantaneous effective confinement p’: G = Gr ( p′ d ) p ′r B = Br ( p′ d ) p ′r $PTAng (φPT) Phase transformation angle, in degrees. $contrac A non-negative constant defining the rate of shear-induced volume decrease (contraction) or pore pressure buildup. A larger value corresponds to faster contraction rate. $dilat1, $dilat2 Non-negative constants defining the rate of shear-induced volume increase (dilation). Larger values correspond to stronger dilation rate. $liquefac1, $liquefac2, $liquefac3 Parameters controlling the mechanism of liquefaction-induced perfectly plastic shear strain accumulation, i.e., cyclic mobility. Set liquefac1 = 0 to deactivate this mechanism altogether. liquefac1 defines the effective confining pressure (e.g., 10 kPa) below which the mechanism is in effect. Smaller values should be assigned to denser sands. liquefac2 defines the maximum amount of perfectly plastic shear strain developed at zero effective confinement during each loading phase. Smaller values should be assigned to denser sands. liquefac3 defines the maximum amount of biased perfectly plastic shear strain γb accumulated at each loading phase under biased shear loading conditions, as γb=liquefac2 x liquefac3. Typically, liquefac3 takes a value between 0.0 and 3.0. Smaller values should be assigned to denser sands. See the references listed at the end of this chapter for more information. $noYieldSurf Number of yield surfaces, optional (must be less than 40, default is 20). The surfaces are generated based on the hyperbolic relation defined in Note 2 below. Chapter 22 $r , $Gs Soil Models and Solid-Fluid Fully Coupled Elements 257 Instead of automatic surfaces generation (Note 2), you can define yield surfaces directly based on desired shear modulus reduction curve. To do so, add a minus sign in front of noYieldSurf, then provide noYieldSurf pairs of shear strain (γ ) and modulus ratio (Gs) values. For example, to define 10 surfaces: … -10 γ1 Gs1 … γ10 Gs10 … See Note 3 below for some important notes. $e Initial void ratio, optional (default is 0.6). $cs1, $cs2, $cs3, $pa Parameters defining a straight critical-state line ec in e-p’ space. If cs3=0, ec = cs1 − cs 2 log( p ′ / p a ) else (Li and Wang, JGGE, 124(12)), ec = cs1 − cs2( p ′ / p a ) cs3 where pa is atmospheric pressure for normalization (typically 101 kPa in SI units). All four constants are optional (default values: cs1=0.9, cs2=0.02, cs3=0.7, pa =101). NOTE: The friction angle φ defines the variation of peak (octahedral) shear strength τf as a function of current effective confinement p’: τf = 1 2 2 sin φ p′ 3 − sin φ Octahedral shear stress is defined as: τ = 1 (σ xx − σ yy )2 + (σ yy − σ zz )2 + (σ xx − σ zz )2 + 6σ xy2 + 6σ yz2 + 6σ xz2 3 [ ] 1/ 2 Chapter 22 1 Soil Models and Solid-Fluid Fully Coupled Elements 258 (Automatic surface generation) At a constant confinement p’, the shear stress τ (octahedral) - shear strain γ (octahedral) nonlinearity is defined by a hyperbolic curve (backbone curve): Gγ τ= 1+ γ p′r γ r p′ d where γr satisfies the following equation at p’r: τf = G r γ max 2 2 sin φ p r′ = 3 − sin φ 1 + γ max / γ r (User defined surfaces) The user specified friction angle f is ignored. Instead, f is defined as follows: sin φ = 3 3 σ m / p r′ 6 + 3 σ m / p r′ where sm is the product of the last modulus and strain pair in the modulus reduction curve. Therefore, it is important to adjust the backbone curve so as to render an appropriate f. If the resulting f is smaller than the phase transformation angle fPT, fPT is set equal to f. Also remember that improper modulus reduction curves can result in strain softening response (negative tangent shear modulus), which is not allowed in the current model formulation. Finally, note that the backbone curve varies with confinement, although the variations are small within commonly interested confinement ranges. Backbone curves at different confinements can be obtained using the OpenSees element recorder facility (see OUTPUT INTERFACE above). 1 The last five optional parameters are needed when critical-state response (flow liquefaction) is anticipated. Upon reaching the critical-state line, material dilatancy is set to zero. 1 SUGGESTED PARAMETER VALUES Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 259 For user convenience, a table is provided below as a quick reference for selecting parameter values. However, use of this table should be of great caution, and other information should be incorporated wherever possible. Loose Sand (15%-35%) Medium Sand (35%-65%) Medium-dense Sand (65%-85%) Dense Sand (85%-100%) Chapter 22 rho (ton/m3) 1.7 Soil Models and Solid-Fluid Fully Coupled Elements 1.9 2.0 2.1 refShearModul (kPa, 5.5x104 at p’r=80 kPa) 7.5x104 1.0x105 1.3x105 refBulkModu (kPa, at p’r=80 kPa) 1.5x105 2.0x105 3.0x105 3.9x105 frictionAng 29 33 37 40 peakShearStra (at p’r=80 kPa) 0.1 0.1 0.1 0.1 refPress (p’r, kPa) 80 80 80 80 pressDependCoe 0.5 0.5 0.5 0.5 PTAng 29 27 27 27 contrac 0.21 0.07 0.05 0.03 dilat1 0. 0.4 0.6 0.8 dilat2 0 2 3 5 liquefac1 (kPa) 10 10 5 0 liquefac2 0.02 0.01 0.003 0 liquefac3 1 1 1 0 e 0.85 0.7 0.55 0.45 260 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 261 PressureDependMultiYield02 PressureDependMultiYield02 material is modified from PressureDependMultiYield material, with: 1) additional parameters ($contrac3 and $dilat3) to account for Kσ effect, 2) a parameter to account for the influence of previous dilation history on subsequent contraction phase ($contrac2), and 3) modified logic related to permanent shear strain accumulation ($liquefac1 and $liquefac2). Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. nDMaterial PressureDependMultiYield02 $tag $nd $rho $refShearModul $refBulkModul $frictionAng $peakShearStra $refPress $pressDependCoe $PTAng $contrac1 $contrac3 $dilat1 $dilat3 <$noYieldSurf=20 <$r1 $Gs1 …> $contrac2=5. $dilat2=3. $liquefac1=1. $liquefac2=0. $e=0.6 $cs1=0.9 $cs2=0.02 $cs3=0.7 $pa=101> $contrac3 A non-negative constant reflecting Kσ effect. $dilat3 A non-negative constant reflecting Kσ effect. $contrac2 A non-negative constant reflecting dilation history on contraction tendency. $liquefac1 Damage parameter to define accumulated permanent shear strain as a function of dilation history. (Redefined and different from PressureDependMultiYield material). Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 262 $liquefac2 Damage parameter to define biased accumulation of permanent shear strain as a function of load reversal history. (Redefined and different from PressureDependMultiYield material). Others See PressureDependMultiYield material above. Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 263 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 264 NOTE: The following values are suggested for the model parameters. Dr=30% Dr=40% Dr=50% Dr=60% Dr=75% Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 265 rho (ton/m3) 1.7 1.8 1.9 2.0 2.1 refShearModul (kPa, at p’r=80 kPa) 6x104 9x104 10x104 11x104 13x104 16x104 22x104 23.3x104 24x104 26x104 (K0=0.5) (K0=0.47) (K0=0.45) (K0=0.43) (K0=0.4) frictionAng 31 32 33.5 35 36.5 PTAng 31 26 25.5 26 26 peakShearStra (at p’r=101 kPa) 0.1 refPress (p’r, kPa) 101 pressDependCoe 0.5 Contrac1 0.087 0.067 0.045 0.028 0.013 Contrac3 0.18 0.23 0.15 0.05 0.0 dilat1 0. 0.06 0.06 0.1 0.3 dilat3 0.0 0.27 0.15 0.05 0.0 e 0.85 0.77 0.7 0.65 0.55 refBulkModu (kPa, at p’r=80 kPa) Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 266 PressureIndependMultiYield PressureIndependMultiYield material is an elastic-plastic material in which plasticity exhibits only in the deviatoric stress-strain response. The volumetric stress-strain response is linearelastic and is independent of the deviatoric response. This material is implemented to simulate monotonic or cyclic response of materials whose shear behavior is insensitive to the confinement change. Such materials include, for example, organic soils or clay under fast (undrained) loading conditions. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. During the application of gravity load (and static loads if any), material behavior is linear elastic. In the subsequent dynamic (fast) loading phase(s), the stress-strain response is elastic-plastic (see MATERIAL STAGE UPDATE below). Plasticity is formulated based on the multi-surface (nested surfaces) concept, with an associative flow rule. The yield surfaces are of the Von Mises type. OUTPUT INTERFACE: The following information may be extracted for this material at a given integration point, using the OpenSees Element Recorder facility (McKenna and Fenves 2001): "stress", "strain", "backbone", or "tangent". For 2D problems, the stress output follows this order: σxx, σyy, σzz, σxy, ηr, where ηr is the ratio between the shear (deviatoric) stress and peak shear strength at the current confinement (0<= ηr<=1.0). The strain output follows this order: εxx, εyy, γxy. For 3D problems, the stress output follows this order: σxx, σyy, σzz, σxy, σyz, σzx, ηr, and the strain output follows this order: εxx, εyy, εzz, γxy, γyz, γzx. The "backbone" option records (secant) shear modulus reduction curves at one or more given confinements. The specific recorder command is as follows: recorder Element –ele $eleNum -file $fName -dT $deltaT material $GaussNum backbone $p1 <$p2 …> where p1, p2, … are the confinements at which modulus reduction curves are recorded. In the output file, corresponding to each given confinement there are two columns: shear strain γ and secant modulus Gs. The number of rows equals the number of yield surfaces. Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 267 nDmaterial PressureIndependMultiYield $tag $nd $rho $refShearModul $refBulkModul $cohesi $peakShearStra <$frictionAng=0. $refPress=100. $pressDependCoe=0. $noYieldSurf=20 <$r1 $Gs1 …> > Octahedral shear stress τ $τ f Shear stress-strain at p'r Octahedral shear strain γ $Gr $γmax $tag A positive integer uniquely identifying the material among all nDMaterials. $nd Number of dimensions, 2 for plane-strain, and 3 for 3D analysis $rho Saturated soil mass density. $refShearModul (Gr) Reference low-strain shear modulus, specified at a reference mean effective confining pressure refPress of p’r (see below). $refBulkModul (Br) Reference bulk modulus, specified at a reference mean effective confining pressure refPress of p’r (see below). $cohesi (c) Apparent cohesion at zero effective confinement. $peakShearStra (gmax) An octahedral shear strain at which the maximum shear strength is reached, specified at a reference mean effective confining pressure refPress of p’r (see below). $frictionAng (f) Friction angle at peak shear strength in degrees, optional (default is 0.0). $refPress (p’r) Reference mean effective confining pressure at which Gr, Br, and γmax are defined, optional (default is 100.). Chapter 22 $pressDependCoe (d) Soil Models and Solid-Fluid Fully Coupled Elements 268 An optional non-negative constant defining variations of G and B as a function of initial effective confinement p’i (default is 0.0): G = Gr ( p i′ d ) p r′ B = Br ( p i′ d ) p r′ If φ=0, d is reset to 0.0. $noYieldSurf Number of yield surfaces, optional (must be less than 40, default is 20). The surfaces are generated based on the hyperbolic relation defined in Note 2 below. $r , $Gs Instead of automatic surfaces generation (Note 2), you can define yield surfaces directly based on desired shear modulus reduction curve. To do so, add a minus sign in front of noYieldSurf, then provide noYieldSurf pairs of shear strain (γ ) and modulus ratio (Gs) values. For example, to define 10 surfaces: … -10 γ1 Gs1 … γ10 Gs10 … See Note 3 below for some important notes. NOTE: 1 The friction angle φ and cohesion c define the variation of peak (octahedral) shear strength τf as a function of initial effective confinement p’i : τf = 1 2 2 sin φ 2 2 p i′ + c 3 − sin φ 3 Automatic surface generation: at a constant confinement p’, the shear stress τ (octahedral) shear strain γ (octahedral) nonlinearity is defined by a hyperbolic curve (backbone curve): Gγ τ= 1+ γ p′r γ r p′ d where γr satisfies the following equation at p’r : τf = G r γ max 2 2 sin φ 2 2 p r′ + c= 3 − sin φ 3 1 + γ max / γ r Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 269 (User defined surfaces) If the user specifies f=0, cohesion c will be ignored. Instead, c is defined by c=sqrt(3)*sm/2, where sm is the product of the last modulus and strain pair in the modulus reduction curve. Therefore, it is important to adjust the backbone curve so as to render an appropriate c. If the user specifies f>0, this f will be ignored. Instead, f is defined as follows: sin φ = 3 ( 3 σ m − 2c) / p ′r 6 + ( 3 σ m − 2c) / p ′r If the resulting f<0, we set f=0 and c=sqrt(3)*sm/2. Also remember that improper modulus reduction curves can result in strain softening response (negative tangent shear modulus), which is not allowed in the current model formulation. Finally, note that the backbone curve varies with confinement, although the variation is small within commonly interested confinement ranges. Backbone curves at different confinements can be obtained using the OpenSees element recorder facility (see OUTPUT INTERFACE above). 1 SUGGESTED PARAMETER VALUES For user convenience, a table is provided below as a quick reference for selecting parameter values. However, use of this table should be of great caution, and other information should be incorporated wherever possible. Soft Clay Medium Clay Stiff Clay Chapter 22 rho (ton/m3) 1.3 Soil Models and Solid-Fluid Fully Coupled Elements 1.5 1.8 refShearModul (kPa) 1.3x104 6.0x104 1.5x105 refBulkModu (kPa) 6.5x104 3.0x105 7.5x105 cohesi (kPa) 18 37 75 peakShearStra 0.1 0.1 0.1 frictionAng 0 0 0 pressDependCoe 0 0 0 270 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 271 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 272 updateMaterialStage This command is used to update a PressureDependMultiYield, PressureDependMultiYield02, PressureIndependMultiYield, or FluidSolidPorous material. To conduct a seismic analysis, two stages should be followed. First, during the application of gravity load (and static loads if any), set material stage to 0, and material behavior is linear elastic (with Gr and Br as elastic moduli). A FluidSolidPorous material does not contribute to the material response if its stage is set to 0. After the application of gravity load, set material stage to 1 or 2. In case of stage 2, all the elastic material properties are then internally determined at the current effective confinement, and remain constant thereafter. In the subsequent dynamic (fast) loading phase(s), the deviatoric stress-strain response is elastic-plastic (stage 1) or linear-elastic (stage 2), and the volumetric response remains linear-elastic. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. updateMaterialStage -material $tag -stage $sNum $tag Material number. $sNum desired stage: 0 - linear elastic, 1 – plastic, 2 - Linear elastic, with elasticity constants (shear modulus and bulk modulus) as a function of initial effective confinement. Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 273 updateParameter This command is used to update: 1. Parameters of PressureDependMultiYield or PressureIndependMultiYield material. Currently, two material parameters, reference low-strain shear modulus Gr and reference bulk modulus Br, can be modified during an analysis. 2. Permeability parameters of solid-fluid fully coupled elements including FourNodeQuadUP, Nine_Four_Node_QuadUP, BrickUP, and Twenty_Eight_Node_BrickUP. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. updateParameter -material $tag -refG $newVal updateParameter -material $tag -refB $newVal updateParameter -element $tag -hPerm $newVal updateParameter -element $tag -vPerm $newVal $tag Material/element number. $newVal New parameter value. Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 274 FluidSolidPorousMaterial FluidSolidPorousMaterial couples the responses of two phases: fluid and solid. The fluid phase response is only volumetric and linear elastic. The solid phase can be any NDMaterial. This material is developed to simulate the response of saturated porous media under fully undrained condition. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. OUTPUT INTERFACE: The following information may be extracted for this material at given integration point, using the OpenSees Element Recorder facility (McKenna and Fenves 2001): "stress", "strain", "tangent", or "pressure". The "pressure" option records excess pore pressure and excess pore pressure ratio at a given material integration point. nDMaterial FluidSolidPorousMaterial $tag $nd $soilMatTag $combinedBulkModul $tag A positive integer uniquely identifying the material among all nDMaterials $nd Number of dimensions, 2 for plane-strain, and 3 for general 3D analysis. $soilMatTag The material number for the solid phase material (previously defined). $combinBulkModul Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water typically), and n the initial porosity. 1 NOTE: 1 Buoyant unit weight (total unit weight - fluid unit weight) should be used in definition of the finite elements composed of a FluidSolidPorousMaterial. 2 During the application of gravity (elastic) load, the fluid phase does not contribute to the material response. Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 275 FourNodeQuadUP FourNodeQuadUP is a four-node plane-strain element using bilinear isoparametric formulation. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot's theory of porous medium. Each element node has 3 degrees-offreedom (DOF): DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p). Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. OUTPUT INTERFACE: Pore pressure can be recorded at an element node using OpenSees Node Recorder: recorder Node <-file $fileName> <-time> <-node ($nod1 $nod2 …)> -dof 3 vel See OpenSees command manual (McKenna and Fenves 2001) for nodal displacement, velocity, or acceleration recorders. The valid queries to a quadUP element when creating an ElementRecorder are 'force', 'stiffness', or 'material matNum matArg1 matArg2 ...', where matNum represents the material object at the corresponding integration point. element quadUP $eleTag $iNode $jNode $kNode $lNode $thick $matTag $bulk $fmass $hPerm $vPerm <$b1=0 $b2=0 $t=0> $eleTag A positive integer uniquely identifying the element among all elements $iNode, $jNode, $kNode, $lNode Four element node (previously defined) numbers in counter-clockwise order around the element $thick Element thickness $matTag Tag of an NDMaterial object (previously defined) of which the element is composed Chapter 22 $bulk Soil Models and Solid-Fluid Fully Coupled Elements 276 Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water), and n the initial porosity. $fmass Fluid mass density $hPerm Permeability coefficient in horizontal direction $vPerm Permeability coefficient in vertical direction $b1, $b2 Optional gravity acceleration components in horizontal and vertical directions respectively (defaults are 0.0) $t Optional uniform element normal traction, positive in tension (default is 0.0) TYPICAL RANGE OF PERMEABILITY COEFFICIENT (cm/s) Gravel Sand Silty Sand Silt Clay >1.0x10-1 1.0x10-3 ~ 1.0x10-1 1.0x10-5 ~ 1.0x10-3 1.0x10-7 ~ 1.0x10-5 <1.0x10-7 Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 277 Nine_Four_Node_QuadUP Nine_Four_Node_QuadUP is a 9-node quadrilateral plane-strain element. The four corner nodes have 3 degrees-of-freedom (DOF) each: DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p). The other five nodes have 2 DOFs each for solid displacement. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot's theory of porous medium. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. OUTPUT INTERFACE: Pore pressure can be recorded at an element node using OpenSees Node Recorder: recorder Node <-file $fileName> <-time> <-node ($nod1 $nod2 …)> -dof 3 vel See OpenSees command manual (McKenna and Fenves 2001) for nodal displacement, velocity, or acceleration recorders. The valid queries to a Nine_Four_Node_QuadUP element when creating an ElementRecorder are 'force', 'stiffness', or 'material matNum matArg1 matArg2 ...', where matNum represents the material object at the corresponding integration point. element 9_4_QuadUP $eleTag $Node1 $Node2 $Node3 $Node4 $Node5 $Node6 $Node7 $Node8 $Node9 $thick $matTag $bulk $fmass $hPerm $vPerm <$b1=0 $b2=0> 4 7 3 8 9 6 1 5 2 $eleTag A positive integer uniquely identifying the element among all elements $Node1,… $Node9 Nine element node (previously defined) numbers (see figure above for order of numbering). Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 278 $thick Element thickness $matTag Tag of an NDMaterial object (previously defined) of which the element is composed $bulk Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water), and n the initial porosity. $fmass Fluid mass density $hPerm, $vPerm Permeability coefficient in horizontal and vertical directions respectively. $b1, $b2 Optional gravity acceleration components in horizontal and vertical directions respectively (defaults are 0.0) Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 279 BrickUP BrickUP is an 8-node hexahedral linear isoparametric element. Each node has 4 degrees-offreedom (DOF): DOFs 1 to 3 for solid displacement (u) and DOF 4 for fluid pressure (p). This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot's theory of porous medium. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. OUTPUT INTERFACE: Pore pressure can be recorded at an element node using OpenSees Node Recorder: recorder Node <-file $fileName> <-time> <-node ($nod1 $nod2 …)> -dof 3 vel See OpenSees command manual (McKenna and Fenves 2001) for nodal displacement, velocity, or acceleration recorders. The valid queries to a BrickUP element when creating an ElementRecorder are 'force', 'stiffness', or 'material matNum matArg1 matArg2 ...', where matNum represents the material object at the corresponding integration point. element brickUP $eleTag $Node1 $Node2 $Node3 $Node4 $Node5 $Node6 $Node7 $Node8 $matTag $bulk $fmass $PermX $PermY $PermZ <$bX=0 $bY=0 $bZ=0> 7 8 6 5 4 1 $eleTag 3 2 A positive integer uniquely identifying the element among all elements Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 280 $Node1,… $Node8 Eight element node (previously defined) numbers (see figure above for order of numbering). $matTag Tag of an NDMaterial object (previously defined) of which the element is composed $bulk Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water), and n the initial porosity. $fmass Fluid mass density $permX, $permY, $permZ Permeability coefficients in x, y, and z directions respectively. $bX, $bY, $bZ Optional gravity acceleration components in x, y, and z directions directions respectively (defaults are 0.0) Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 281 Twenty_Eight_Node_BrickUP Twenty_Eight_Node_BrickUP is a 20-node hexahedral isoparametric element. The eight corner nodes have 4 degrees-of-freedom (DOF) each: DOFs 1 to 3 for solid displacement (u) and DOF 4 for fluid pressure (p). The other nodes have 3 DOFs each for solid displacement. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot's theory of porous medium. Please visit http://cyclic.ucsd.edu/opensees (http://cyclic.ucsd.edu/opensees) for examples. OUTPUT INTERFACE: Pore pressure can be recorded at an element node using OpenSees Node Recorder: recorder Node <-file $fileName> <-time> <-node ($nod1 $nod2 …)> -dof 3 vel See OpenSees command manual (McKenna and Fenves 2001) for nodal displacement, velocity, or acceleration recorders. The valid queries to a Twenty_Eight_Node_BrickUP element when creating an ElementRecorder are 'force', 'stiffness', or 'material matNum matArg1 matArg2 ...', where matNum represents the material object at the corresponding integration point. element 20_8_BrickUP $eleTag $Node1 … $Node20 $matTag $bulk $fmass $PermX $PermY $PermZ <$bX=0 $bY=0 $bZ=0> 7 15 6 16 8 14 20 13 5 19 18 17 4 11 12 1 $eleTag 3 10 9 2 A positive integer uniquely identifying the element among all elements Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 282 $Node1,… $Node20 20 element node (previously defined) numbers (see figure above for order of numbering). $matTag Tag of an NDMaterial object (previously defined) of which the element is composed $bulk Combined undrained bulk modulus Bc relating changes in pore pressure and volumetric strain, may be approximated by: Bc ≈ B f / n where Bf is the bulk modulus of fluid phase (2.2x106 kPa for water), and n the initial porosity. $fmass Fluid mass density $permX, $permY, $permZ Permeability coefficients in x, y, and z directions respectively. $bX, $bY, $bZ Optional gravity acceleration components in x, y, and z directions directions respectively (defaults are 0.0) Chapter 22 Soil Models and Solid-Fluid Fully Coupled Elements 283 References Elgamal, A., Lai, T., Yang, Z. and He, L. (2001). "Dynamic Soil Properties, Seismic Downhole Arrays and Applications in Practice," State-of-the-art paper, Proc., 4th Intl. Conf. on Recent Advances in Geote. E.Q. Engrg. Soil Dyn. March 26-31, San Diego, CA, S. Prakash (Ed.). Elgamal, A., Yang, Z. and Parra, E. (2002). "Computational Modeling of Cyclic Mobility and Post-Liquefaction Site Response," Soil Dyn. Earthquake Engrg., 22(4), 259-271. Elgamal, A., Yang, Z., Parra, E. and Ragheb, A. (2003). "Modeling of Cyclic Mobility in Saturated Cohesionless Soils," Int. J. Plasticity, 19(6), 883-905. McKenna, F. and Fenves, G. (2001). "The OpenSees Command Language Manual: version 1.2," Pacific Earthquake Engineering Center, Univ. of Calif., Berkeley. (http://opensees.berkeley.edu). Parra, E. (1996). "Numerical Modeling of Liquefaction and Lateral Ground Deformation Including Cyclic Mobility and Dilation Response in Soil Systems," Ph.D. Thesis, Dept. of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY. Yang, Z. (2000). "Numerical Modeling of Earthquake Site Response Including Dilation and Liquefaction," Ph.D. Thesis, Dept. of Civil Engineering and Engineering Mechanics, Columbia University, NY, New York. Yang, Z. and Elgamal, A. (2002). "Influence of Permeability on Liquefaction-Induced Shear Deformation," J. Engrg. Mech., ASCE, 128(7), 720-729. Yang, Z., Elgamal, A. and Parra, E. (2003). "A Computational Model for Liquefaction and Associated Shear Deformation," J. Geotechnical and Geoenvironmental Engineering, ASCE, 129(12), 1119-1127. 284 CHAPTER 23 block Command The block command is used to generate meshes of quadrilateral or brick finite element. The block2D (page 296) command generates meshes of quadrilateral elements in two or three dimensions. In three dimensions, a two-dimensional surface appropriate for shell analysis is generated. The block3D (page 298) command generates three-dimensional meshes of eight-node brick solid element. In This Chapter block2D Command................................................... 296 block3D Command................................................... 298 Chapter 23 block Command 285 block2D Command The block2D command generates meshes of quadrilateral elements in two or three dimensions. In three dimensions, a two-dimensional surface appropriate for shell analysis is generated. block2d $nx $ny $e1 $n1 element (element arguments) { 1 $x1 $y1 <$z1> 2 $x2 $y2 <$z2> 3 $x3 $y3 <$z3> 4 $x4 $y4 <$z4> <5> <$x5> <$y5> <$z5> <6> <$x6> <$y6> <$z6> <7> <$x7> <$y7> <$z7> <8> <$x8> <$y8> <$z8> <9> <$x9> <$y9> <$z9> } $nx $ny number of elements in the local x and y directions of the block, respectively $e1 $n1 starting element and node number for generation, respectively element string defining which quadrilateral element (page 227, page 228, page 226, page 227) is being used (element arguments) list of data parameters for element being used. This list may include, but is not limited to, a $matTag number {$x1, .... $x9} {$y1, .... $y9} coordinates of the block elements in two dimensions {$z1, .... $z9} coordinate of the block elements in third dimension (optional, default=0.0) Only the first four nodes (1-4) are required. Nodes 5-9 are used to generate curved meshes. The user may specify any combination of nodes 5-9, omitting some of them if desired. Chapter 23 block Command NOTE: this command only recognizes variable substitutions when the command arguments are placed in quotes rather than braces EXAMPLE: block2d $nx $ny $e1 $n1 element (element arguments) { 1 $x1 $y1 <$z1> 2 $x2 $y2 <$z2> 3 $x3 $y3 <$z3> 4 $x4 $y4 <$z4> <5> <$x5> <$y5> <$z5> <6> <$x6> <$y6> <$z6> <7> <$x7> <$y7> <$z7> <8> <$x8> <$y8> <$z8> <9> <$x9> <$y9> <$z9> } Figure 54: Node Numbering for NineNode block2D 286 Chapter 23 block Command 287 block3D Command The block3D command generates three-dimensional meshes of eight-node brick solid element. block3d $nx $ny $nz $e1 $n1 element elementArgs { 1 $x1 $y1 $z1 2 $x2 $y2 $z2 3 $x3 $y3 $z3 4 $x4 $y4 $z4 5 $x5 $y5 $z5 6 $x6 $y6 $z6 7 $x7 $y7 $z7 8 $x8 $y8 $z8 <9> <$x9> <$y9> <$z9> ... <27> <$x27> <$y27> <$z27> } $nx $ny $nz number of elements in the local x,y and z-direction of the block $e1 starting element number for generation $n1 starting node number for generation element define which brick element (page 231, page 229) is being used elementArgs list of data parameters for element being used. This list may include, but is not limited to, a $matTag number {$x1, .... $x27} {$y1, .... $y27} {$z1, .... $z27} coordinates of the block elements in three dimensions. Chapter 23 block Command 288 NOTE: this command only recognizes variable substitutions when the command arguments are placed in quotes rather than braces Only the first eight nodes (1-8) are required. Nodes 9-27 are used to generate curved meshes. The user may specify any combination of nodes 9-27, omitting some of them if desired. 289 CHAPTER 24 region Command The region command is used to label a group of nodes and elements. This command is also used to assign rayleigh damping parameters to the nodes and elements in this region. region $regTag <-ele ($ele1 $ele2 ...)> <-eleRange $startEle $endEle> <-ele all> <-node ($node1 $node2 ...)> <-nodeRange $startNode $endNode> <node all> <-rayleigh $alphaM $betaK $betaKinit $betaKcomm> The region is specified by either elements or nodes, not both. If elements are defined, the region includes these elements and the all connected nodes. If nodes are specified, the region includes these nodes and all elements whose external nodes are prescribed. $regTag unique integer tag $ele1 $ele2 ... tags of elements -- selected elements in domain (optional, default: omitted) $startEle $endEle tag for start and end elements -- range of selected elements in domain (optional, default: all) all all elements in domain (optional & default) $alphaM $betaK $betaKinit $betaKcomm Arguments to define Rayleigh damping matrix (optional, default: zero) OR: $regTag unique integer tag $node1 $node2 ... node tags -- select nodes in domain (optional, default: all) $startNode $endNode tag for start and end nodes -- range of nodes in domain (optional, default: all) all all nodes in domain (optional & default) $alphaM $betaK $betaKinit $betaKcomm Arguments to define Rayleigh damping matrix (optional, default: zero) The damping matrix D is specified as a combination of stiffness and mass-proportional damping matrices: D = $alphaM * M + $betaK * Kcurrent +$betaKinit * Kinit + $betaKcomm * KlastCommit Chapter 24 region Command The mass and stiffness matrices are defined as: M mass matrix used to calculate Rayleigh Damping Kcurrent stiffness matrix at current state determination used to calculate Rayleigh Damping Kinit stiffness matrix at initial state determination used to calculate Rayleigh Damping KlastCommit stiffness matrix at last-committed state determination used to calculate Rayleigh Damping NOTE: a region is defined by either nodes or elements, not both. EXAMPLE: region 1 -ele 1 5 -eleRange 10 15 region 2 -node 2 4 6 -nodeRange 9 12 290 291 CHAPTER 25 Geometric Transformation Command The geometric-transformation command (geomTransf) is used to construct a coordinatetransformation (CrdTransf) object, which transforms beam element stiffness and resisting force from the basic system to the global-coordinate system. The command has at least one argument, the transformation type. Each type is outlined below. In This Chapter Linear Transformation .............................................. 302 P-Delta Transformation ............................................ 308 Corotational Transformation ..................................... 309 Linear Transformation This command is used to construct a linear coordinate transformation (LinearCrdTransf) object, which performs a linear geometric transformation of beam stiffness and resisting force from the basic system to the global-coordinate system. For a two-dimensional problem: geomTransf Linear $transfTag <-jntOffset $dXi $dYi $dXj $dYj> For a three-dimensional problem: geomTransf Linear $transfTag $vecxzX $vecxzY $vecxzZ <-jntOffset $dXi $dYi $dZi $dXj $dYj $dZj> $transfTag unique identifier for CrdTransf object Chapter 25 $vecxzX $vecxzY $vecxzZ Geometric Transformation Command X, Y, and Z components of vecxz, the vector used to define the local x-z plane of the local-coordinate system. The local y-axis is defined by taking the cross product of the x-axis and the vecxz vector. These components are specified in the global-coordinate system X,Y,Z and define a vector that is in a plane parallel to the x-z plane of the local-coordinate system. These items need to be specified for the three-dimensional problem. $dXi $dYi $dZi joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node i (the number of arguments depends on the dimensions of the current model) (optional) $dXj $dYj $dZj joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node j (the number of arguments depends on the dimensions of the current model) (optional) The element coordinate system is specified as follows: 292 Chapter 25 Geometric Transformation Command 293 The x-axis is the axis connecting the two element nodes; the y- and z-axes are then defined using a vector that lies on a plane parallel to the local x-z plane -- vecxz. The y-axis is defined by taking the cross product of the x-axis and the vecxz vector. The section is attached to the element such that the y-z coordinate system used to specify the section corresponds to the y-z axes of the element. Figure 55: Definition of the Local Coordinate System Chapter 25 Figure 56: Definition of Rigid Joint Offset (note: check sign of dXi,etc components) Geometric Transformation Command 294 Chapter 25 Geometric Transformation Command The following figures should aid in understanding the vector vecxz definition: 295 Chapter 25 Geometric Transformation Command 296 Chapter 25 Geometric Transformation Command 297 P-Delta Transformation This command is used to construct the P-Delta Coordinate Transformation (PDeltaCrdTransf) object, which performs a linear geometric transformation of beam stiffness and resisting force from the basic system to the global coordinate system, considering second-order P-Delta effects. For a two-dimensional problem: geomTransf PDelta $transfTag <-jntOffset $dXi $dYi $dXj $dYj> For a three-dimensional problem: geomTransf PDelta $transfTag $vecxzX $vecxzY $vecxzZ <-jntOffset $dXi $dYi $dZi $dXj $dYj $dZj> The element coordinate system and joint offset values are specified as in the Linear transformation (page 302). $transfTag unique identifier for CrdTransf object $vecxzX $vecxzY $vecxzZ X, Y, and Z components of vecxz, the vector used to define the local x-z plane of the local-coordinate system. (These items need to be specified for the three-dimensional problem.) These components are specified in the global-coordinate system X,Y,Z and define a vector that is in a plane parallel to the x-z plane of the local-coordinate system. These items need to be specified for the three-dimensional problem. $dXi $dYi $dZi joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node i (the number of arguments depends on the dimensions of the current model (page 37)) (optional) $dXj $dYj $dZj joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node j (the number of arguments depends on the dimensions of the current model (page 37)) (optional) Chapter 25 Geometric Transformation Command 298 Corotational Transformation This command is used to construct the Corotational Coordinate Transformation (CorotCrdTransf) object, which performs an exact geometric transformation of beam stiffness and resisting force from the basic system to the global coordinate system. For a two-dimensional problem: geomTransf Corotational $transfTag <-jntOffset $dXi $dYi $dXj $dYj> For a three-dimensional problem: geomTransf Corotational $transfTag $vecxzX $vecxzY $vecxzZ <-jntOffset $dXi $dYi $dZi $dXj $dYj $dZj> NOTE: The Corotational transformation is only available with the Win32 version of OpenSees (http://opensees.berkeley.edu/OpenSees/binaries.html). The element coordinate system and joint offset values are specified as in the Linear transformation (page 302). $transfTag unique identifier for CrdTransf object $vecxzX $vecxzY $vecxzZ X, Y, and Z components of vecxz, the vector used to define the local x-z plane of the local-coordinate system. (These items need to be specified for the three-dimensional problem.) These components are specified in the global-coordinate system X,Y,Z and define a vector that is in a plane parallel to the x-z plane of the local-coordinate system. $dXi $dYi $dZi joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node i (the number of arguments depends on the dimensions of the current model (page 37)) (optional) $dXj $dYj $dZj joint offset values -- absolute offsets specified with respect to the global coordinate system for element-end node j (the number of arguments depends on the dimensions of the current model (page 37)) (optional) 299 CHAPTER 26 Time Series While there is no timeSeries command in the language, a number of commands take as the argument a list of items which defines the TimeSeries object to be constructed as part of the command, such as the LoadPattern (page 317) and groundMotion (page 323) objects. Time series act differently depending on what type of object they are applied to: LoadPattern (page 317) object: Load factors are applied to the loads and constraints groundMotion (page 323) object: Load factors are applied at the DOF in a ground motion The type of TimeSeries objects available are presented in this chapter. NOTE: The TimeSeries objects are handled by the Tcl interpreter as lists. Therefore, they can be defined a-priori within quotes "" and given a variable name. EXAMPLE: set Gaccel "Series -dt $dt -filePath $outFile -factor $GMfatt"; # time series information pattern UniformExcitation 2 1 -accel $Gaccel; In This Chapter Constant Time Series............................................... 312 Linear Time Series ................................................... 313 Rectangular Time Series .......................................... 314 Sine Time Series ...................................................... 315 Path Time Series...................................................... 316 # create uniform excitation Chapter 26 Time Series 300 Constant Time Series This command creates a ConstantSeries TimeSeries (page 311) object and associates it to the LoadPattern (page 317) object being constructed. Constant <-factor $cFactor> $cFactor load-factor coefficient. (optional. default = 1.0) The load factor to be applied to the loads and constraints in the LoadPattern object is constant and equal to $cFactor. Figure 57: Constant Time Series Linear Time Series This command creates a LinearSeries TimeSeries (page 311) object and associates it to the LoadPattern (page 317) or groundMotion (page 323) object being constructed. Linear <-factor $cFactor> Chapter 26 $cFactor Time Series 301 load-factor coefficient. (optional. default = 1.0) The load factor to be applied to the loads and constraints in the LoadPattern or groundMotion object is equal to $cFactor* time Figure 58: Linear Time Series Rectangular Time Series This command creates a RectangularSeries TimeSeries (page 311) object and associates it to the LoadPattern (page 317) object being constructed. Rectangular $tStart $tFinish <-factor $cFactor> $tStart start time when the load factor is applied $tFinish end time when the load factor is applied $cFactor load-factor coefficient. (optional. default = 1.0) Chapter 26 Time Series 302 The load factor to be applied to the loads and constraints in the LoadPattern object is constant and equal to $cFactor during the domain time from $tStart to $tFinish Figure 59: Rectangular Time Series Sine Time Series This command creates a TrigSeries TimeSeries (page 311) object and associates it to the LoadPattern (page 317) object being constructed. Sine $tStart $tFinish $period <-shift $shift> <-factor $cFactor> $tStart start time when the load factor is applied $tFinish end time when the load factor is applied $period characteristic period of sine wave $shift phase shift (radians) (optional. default = 0.0) $cFactor load-factor coefficient. (optional. default = 1.0) The load factor applied to the loads and constraints in the LoadPattern object is equal to: Chapter 26 Time Series Figure 60: Sine Time Series Path Time Series This command associates a TimeSeries (page 311) object of the type PathSeries or PathTimeSeries (if the increment is not constant) to a LoadPattern (page 317) object. There are many ways to specify the load path. For a load path where the values are specified at constant time intervals: Series -dt $dt -values {list_of_values} <-factor $cFactor> where the values are specified in a list included in the command Series -dt $dt -filePath $fileName <-factor $cFactor> where the values are taken from a file specified by $fileName 303 Chapter 26 Time Series 304 For a load path where the values are specified at non-constant time intervals: Series -time {list_of_times} -values {list_of_values} <-factor $cFactor> where both time and values are specified in a list included in the command Series -fileTime $fileName1 -filePath $fileName2 <-factor $cFactor> where both time and values are taken from a file specified by $fileName1 (for the time data) and $fileName2 (for the values data) $cFactor load-factor coefficient. (optional. default = 1.0) The load factor to be applied to the loads and constraints in the LoadPattern object is equal to $cFactor*(user-defined series) 305 CHAPTER 27 pattern Command The pattern command is used to construct a LoadPattern object, its associated with the TimeSeries (page 311) object and the Load (page 318) and Constraint (page 339) objects for the pattern. In This Chapter plain Pattern ............................................................. 317 UniformExcitation Pattern......................................... 321 MultipleSupport Pattern ............................................ 322 plain Pattern This command is used to construct an ordinary LoadPattern (page 317) object in the Domain (page 32). pattern Plain $patternTag (TimeSeriesType arguments) { load (load-command arguments) sp (sp-command arguments) eleLoad (eleLoad-command arguments) } $patternTag unique pattern object tag TimeSeriesType arguments list which is parsed to construct the TimeSeries (page 311) object associated with the LoadPattern object. load ... list of commands to construct nodal loads -- the NodalLoad (page 318) object sp ... list of commands to construct single-point constraints -- the SP_Constraint (page 319) object eleLoad ... list of commands to construct element loads -- the eleLoad (page 320) object Chapter 27 pattern Command 306 NOTE: The TimeSeries object is handled by the Tcl interpreter as a list and can be defined apriori and given a variable name. EXAMPLE # define LoadPattern 1. impose load in a linear manner pattern Plain 1 Linear { ; load 3 100 0. 0. 0. 0. 20.; # apply force and moment at node 3 } load Command This command is used to construct a NodalLoad object. load $nodeTag (ndf $LoadValues) The nodal load is added to the LoadPattern being defined in the enclosing scope of the pattern command. $nodeTag node on which loads act $LoadValues load values that are to be applied to the node. Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. EXAMPLE load 3 100 0. 0. 0. 0. 20.; # apply force Fx=100 and moment Mz=20 at node 3 Chapter 27 pattern Command 307 sp Command This command is used to construct a single-point non-homogeneous constraint (SP_Constraint) object. sp $nodeTag $DOFtag $DOFvalue $nodeTag node on which the single-point constraint acts $DOFtag degree-of-freedom at the node being constrained. Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. $DOFvalue reference value of the constraint to be applied to the DOF at the node. EXAMPLE sp 3 1 0.1; # impose displacement Dx=0.1 at node 3 Chapter 27 pattern Command 308 eleLoad Command The elasticBeamColumn (page 215), forceBeamColumn (nonlinearBeamColumn (page 216)) and dispBeamColumn (page 222) elements all handle ele loads. The syntax for 2D is as follows: For uniformly-distributed load: eleLoad -ele $eleTag1 <$eleTag2 ....> -type -beamUniform $Wz <$Wx> For point load: eleLoad -ele $eleTag1 $eleTag2 -type -beamPoint $Pz $xL <$Px> The syntax for 3D is as follows: For uniformly-distributed load: eleLoad -ele $eleTag1 <$eleTag2 ....> -type -beamUniform $Wy $Wz <$Wx> For point load: eleLoad -ele $eleTag1 $eleTag2 -type -beamPoint $Py $Pz $xL <$Px> $eleTag1 previously-defined-element tags <$eleTag2 .....> additional previously-defined-element tags (optional) $Wy 3D: uniformly-distributed load in local y direction $Wz 2D: uniformly-distributed load in direction perpendicular to element longitudinal axis 3D: uniformly-distributed load in local z direction Chapter 27 pattern Command $Wx uniformly-distributed load in direction parallel to element longitudinal axis $Py 3D: point load in local y direction $Pz 2D: point load in direction perpendicular to element longitudinal axis 309 3D: point load in local z direction $xL location of point load, prescribed as fraction of element length $Px point load in direction longitudinal to element longitudinal axis (applied at the same location as $Pz) were x axis is along the member. NOTE: the forceBeamColumn (nonlinearBeamColumn (page 216)) element might have problems converging; the current code needs to be redone, 'MICHAEL' has promised to get around to it. UniformExcitation Pattern This command is used to construct a UniformExcitation load pattern object. pattern UniformExcitation $patternTag $dir -accel (TimeSeriesType arguments) <-vel0 $ver0> $patternTag unique pattern object tag $dir direction of excitation (1, 2, or 3) used in formulating the inertial loads for the transient analysis. TimeSeriesType arguments TimeSeries (page 311) object associated with the acceleration record used in determining the inertial loads. $vel0 initial velocity to be assigned to each node (optional, default: zero) NOTE: The TimeSeries object is handled by the Tcl interpreter as a list and can be defined apriori and given a variable name. Chapter 27 pattern Command 310 EXAMPLE: set Gaccel "Series -dt $dt -filePath $outFile -factor $GMfatt"; # time series information pattern UniformExcitation 2 1 -accel $Gaccel; with IDtag 2 in direction 1 # create uniform excitation MultipleSupport Pattern This command is used to construct a MultipleSupportExcitation load pattern object. pattern MultipleSupport $patternTag { groundMotion (groundMotion-command arguments) imposedMotion (imposedMotion-command arguments) } $patternTag unique pattern object tag groundMotion ... list of commands to construct the GroundMotions (page 323) object that is then added to the object to define the multiplesupport excitation that is being imposed on the model imposedMotion ... list of commands to construct the ImposedSupportSP (page 324) constraint object that is then added to the object to define the multiple-support excitation that is being imposed on the model groundMotion Command The groundMotion command is used to construct a GroundMotion object used by the ImposedMotionSP (page 324) constraints in a MultipleSupportExcitation (page 322) object. Chapter 27 pattern Command 311 Plain GroundMotion This command is used to construct a plain GroundMotion object. Each GroundMotion object is associated with a number of TimeSeries (page 311) objects, which define the acceleration, velocity and displacement records. groundMotion $gMotionTag Plain <-accel (accelSeriesType accelArgs)> <-vel (velSeriesType velArgs)> <-disp (dispSeriesType dispArgs)> <-int (IntegratorType intArgs)> $gMotionTag unique GroundMotion (page 323) object tag <-accel (accelSeriesType accelArgs)> TimeSeries (page 311) objects defining the acceleration record (optional). <-vel (velSeriesType velArgs)> TimeSeries (page 311) objects defining the velocity record (optional) <-disp (dispSeriesType dispArgs)> TimeSeries (page 311) objects defining the displacement record (optional) <-int (IntegratorType intArgs)> If only the acceleration record is specified, the user has the option of specifying the TimeSeriesIntegrator (page 356) that is to be used to integrate the acceleration record to determine the velocity and displacement record (optional, default: Trapezoidal) NOTE: The TimeSeries object is handled by the Tcl interpreter as a list and can be defined apriori and given a variable name. NOTE: Any combination of the acceleration, velocity and displacement time-series can be specified. Chapter 27 pattern Command 312 Interpolated GroundMotion This command is used to construct an InterpolatedGroundMotion object. groundMotion $gMotionTag Interpolated $gmTag1 $gmTag2 ... -fact $fact1 $fact2 ... $gMotionTag unique GroundMotion (page 323) object tag $gmTag1 $gmTag2 ... ground motions which have already been added to the MultipleSupportExcitation (page 322) object. $fact1 $fact2 ... factors that are used in the interpolation of the ground motions to determine the ground motion for this InterpolatedGroundMotion object. imposedMotion Command This command is used to construct an ImposedMotionSP constraint which is used to enforce the response of a dof at a node in the model. The response enforced at the node at any give time is obtained from the GroundMotion (page 323) object associated with the constraint. imposedMotion $nodeTag $dirn $gMotionTag $nodeTag node where response is enforced $dirn dof of enforced response Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. $gMotionTag pre-defined GroundMotion (page 323) object tag NOTE: The GroundMotion (page 323) object must be added to the MultipeSupportExcitation (page 322) pattern before the ImposedMotionSP constraint. Chapter 27 FMK: ADD TIME-SERIES INTEGRATORS pattern Command 313 314 CHAPTER 28 Recorder Objects The recorder commands are used to construct a Recorder object, which is used to monitor items of interest to the analyst at each commit(). In This Chapter Node Recorder......................................................... 326 EnvelopeNode Recorder .......................................... 328 Drift Recorder........................................................... 329 Element Recorder .................................................... 330 EnvelopeElement Recorder...................................... 332 Display Recorder...................................................... 333 Plot Recorder ........................................................... 334 playback Command.................................................. 335 Node Recorder The Node type records the displacement, velocity, acceleration and incremental displacement at the nodes (translational & rotational) recorder Node <-file $fileName> <-time> <-node ($node1 $node2 ...)> <nodeRange $startNode $endNode> <-region $RegionTag> <-node all> -dof ($dof1 $dof2 ...) $respType $fileName file where results are stored. Each line of the file contains the result for a committed state of the domain (optional, default: screen output) -time this argument will place the pseudo time of the as the first entry in the line. (optional, default: omitted) $node1 $node2 ... tags nodes where response is being recorded -- select nodes in domain (optional, default: all) $startNode $endNode tag for start and end nodes where response is being recorded -- range of nodes in domain (optional, default: all) Chapter 28 Recorder Objects $RegionTag tag for previously-defined selection of nodes defined using the Region command. (optional) all where response is being recorded -- all nodes in domain (optional & default) $dof1 $dof2 ... degrees of freedom of response being recorded. 315 Valid range is from 1 through ndf (page 37) , the number of nodal degrees-of-freedom. $respType defines response type to be recorded. The following response types are available: disp displacement* vel velocity* accel acceleration* incrDisp incremental displacement "eigen i" eigenvector for mode i reaction nodal reaction NOTE: $respType must be the last argument in this command. Do not forget the double quote in the eigen response type. *NOTE: For transient analysis, the response output recorded depends on the input: UniformExcitation (page 321) Pattern(input acceleration record): Relative (wrt supports) disp, vel and accel MultipleSupport (page 322)Pattern (input displacement record): Absolute disp, vel and accel Example: recorder Node -file nodeDisp.out –time –node 1 5 -nodeRange 10 25 -dof 2 disp recorder Node -file node34.eig -time -node 3 4 -dof 1 2 3 "eigen 2" recorder Node -file Rnode.out -time -node 1 3 5 -dof 1 2 reaction Chapter 28 Recorder Objects 316 EnvelopeNode Recorder The Node type records the envelope of displacement, velocity, acceleration and incremental displacement at the nodes (translational & rotational). The envelope consists of the following: minimum, maximum and maximum absolute value of specified response type. recorder EnvelopeNode <-file $fileName> <-node ($node1 $node2 ...)> <nodeRange $startNode $endNode> <-region $RegionTag> <-node all> -dof ($dof1 $dof2 ...) $respType $fileName file where results are stored. Each line of the file contains the result for a committed state of the domain (optional, default: screen output) $node1 $node2 ... where response is being recorded -- select nodes (optional, default: all) $startNode $endNode tag for start and end where response is being recorded (optional, default: all) $RegionTag tag for previously-defined selection of nodes defined using the Region command. (optional) all where response is being recorded -- all nodes in domain (optional & default) $dof1 $dof2 ... degrees of freedom of response being recorded. Valid range is from 1 through ndf (page 37) , the number of nodal degrees-of-freedom. $respType defines response type to be recorded. The following response types are available: disp displacement vel velocity accel acceleration incrDisp incremental displacement NOTE: $respType must be the last argument in this command. Example: recorder EnvelopeNode -file EnvelopeNode.out –time –node 1 5 -nodeRange 10 25 -dof 2 disp Chapter 28 Recorder Objects 317 Drift Recorder The Drift type records the displacement drift between two nodes. The drift is taken as the ratio between the prescribed relative displacement and the specified distance between the nodes. recorder Drift -file $fileName <-time> -iNode ($inode1 $inode2 ...) -jNode ($jnode1 $jnode2 ...) -dof ($dof1 $dof2 ...) -perpDirn ($perpDirn1 $perpDirn2 ...) $fileName file where results are stored. Each line of the file contains the result for a committed state of the domain -time this argument will place the pseudo time of the as the first entry in the line. (optional, default: omitted) ($inode1 $inode2 ...) set of i nodes for which drift is recorded ($jnode1 $jnode2 ...) set of j nodes for which drift is recorded ($dof1 $dof2 ...) set of nodal degrees of freedom to monitor for drift Valid range is from 1 through ndm (page 37), the number of nodal degrees-of-freedom. ($perpDirn1 $perpDirn2 ...) set of perpendicular global directions from which length is determined to compute drift (1 = X, 2 = Y, 3 = Z) Example: recorder Drift -file drift.out -time -iNode 1 2 -jNode 3 4 -dof 1 -perpDirn 2 Chapter 28 Recorder Objects 318 Element Recorder The Element type records the response of a number of elements. The response recorded is element-dependent and depends on the arguments which are passed to the setResponse() element method. recorder Element <-file $fileName> <-time> <-ele ($ele1 $ele2 ...)> <-eleRange $startEle $endEle> <-region $regTag> <-ele all> ($arg1 $arg2 ...) $fileName file where results are stored. Each line of the file contains the result for a committed state of the domain (optional, default: screen output) -time this argument will place the pseudo time of the as the first entry in the line. (optional, default: omitted) $ele1 $ele2 ... tags of elements whose response is being recorded -- selected elements in domain (optional, default: omitted) $startEle $endEle tag for start and end elements whose response is being recorded -- range of selected elements in domain (optional, default: all) $regTag previously-defined tag of region of elements whose response is being recorded -- region of elements in domain (optional) all elements whose response is being recorded -- all elements in domain (optional & default) $arg1 $arg2 ... arguments which are passed to the setResponse() element method The setResponse() element method is dependent on the element type, and is described with the element Command (page 213). Beam-Column Elements (page 218, page 222, page 215, page 216) : Common to all beam-column elements: globalForce – element resisting force in global coordinates (does not include inertial forces) example: recorder Element -file ele1global.out -time -ele 1 globalForce localForce – element resisting force in local coordinates (does not include inertial forces) example: recorder Element -file ele1local.out -time -ele 1 localForce Chapter 28 Recorder Objects 319 Sections: (page 185) section $secNum – request response quantities from a specific section along the element length, $secNum refers to the integration point whose data is to be output force – section forces example: recorder Element -file ele1sec1Force.out –time -ele 1 section 1 force deformation – section deformations example: recorder Element -file ele1sec1Force.out –time -ele 1 section 1 deformation stiffness – section stiffness example: recorder Element -file ele1sec1Force.out –time -ele 1 section 1 stiffness stressStrain – record stress-strain response. example: recorder Element -file ele1sec1Force.out –time -ele 1 section 1 fiber $y $z <$matID> stressStrain $y local y coordinate of fiber to be monitored* $z local z coordinate of fiber to be monitored* $matID previously-defined material tag (optional) *NOTE: The recorder object will search for the fiber closest to the location ($y,$z) on the section and record its stress-strain response NOTE: the ZeroLength section element (page 225) only has 1 section therefore you do not need to identify the section in the recorder command .. Example: recorder Element -file Element1.out -time -ele 1 section 1 fiber 0.10 0.10 stressStrain EnvelopeElement Recorder The Element type records the response of a number of elements. The response recorded is element-dependent and depends on the arguments which are passed to the setResponse() element method. The envelope consists of the following: minimum, maximum and maximum absolute value of specified response type. recorder EnvelopeElement <-file $fileName> <-ele ($ele1 $ele2 ...)> <-eleRange $startele $endele> <-ele all> <-region $regTag> ($arg1 $arg2 ...) Chapter 28 Recorder Objects $fileName file where results are stored. Each line of the file contains the result for a committed state of the domain (optional, default: screen output) $eleID1 $eleID2 ... tags of elements whose response is being recorded -- selected elements in domain (page 32) (optional, default: omitted) $startele $endele tag for start and end elements whose response is being recorded -- selected elements in domain (page 32) (optional, default: all) all elements whose response is being recorded -- all elements in domain (page 32) (optional & default) $regTag tag of region of elements whose response is being recorded -region of elements in domain (page 32) (optional) $arg1 $arg2 ... arguments which are passed to the setResponse() element method 320 The setResponse() element method is dependent on the element type, and is described with the element Command (page 213). Beam-Column Elements (page 218, page 222, page 215, page 216) : Common to all beam-column elements: globalForce – element resisting force in global coordinates (does not include inertial forces) example: recorder EnvelopeElement -file ele1global.out -time -ele 1 globalForce localForce – element resisting force in local coordinates (does not include inertial forces) example: recorder EnvelopeElement -file ele1local.out -time -ele 1 localForce Sections: (page 185) section $secNum – request response quantities from a specific section along the element length, $secNum refers to the integration point whose data is to be output force – section forces example: recorder EnvelopeElement -file ele1sec1Force.out –time -ele 1 section 1 force deformation – section deformations example: recorder EnvelopeElement -file ele1sec1Force.out –time -ele 1 section 1 deformation stiffness – section stiffness example: recorder EnvelopeElement -file ele1sec1Force.out –time -ele 1 section 1 stiffness Chapter 28 Recorder Objects 321 stressStrain – record stress-strain response. example: recorder EnvelopeElement -file ele1sec1Force.out –time -ele 1 section 1 fiber $y $z stressStrain $y local y coordinate of fiber to be monitored* $z local z coordinate of fiber to be monitored* *NOTE: The recorder object will search for the fiber closest to the location ($y,$z) on the section and record its stress-strain response NOTE: the ZeroLength section element (page 225) only has 1 section therefore you do not need to identify the section in the recorder command .. Example: recorder Element -file Element1.out -time -ele 1 section fiber 0.10 0.10 stressStrain Display Recorder This recorder opens a graphical window for displaying of graphical information. recorder display $windowTitle $xLoc $yLoc $xPixels $yPixels <-file $fileName> $windowTitle title of graphical window $xLoc $yLoc horizontal and vertical location of graphical window (upper leftmost corner) $xPixels $yPixels width and height of graphical window in pixels $fileName in addition to the window display, information is sent to a file to redisplay images at a later time. (optional) A TclFeViewer object is constructed. This constructor adds a number of additional commands to OpenSees, similar to the construction of the BasicBuilder (page 37). These additional commands are used to define the viewing system for the image that is place on the screen. These commands are currently under review and will be presented in a future version of this document. Chapter 28 Recorder Objects 322 Plot Recorder This recorder type opens a graphical window for the plotting of the contents of the prescribed file recorder plot $fileName $windowTitle $xLoc $yLoc $xPixels $yPixels -columns $xCol0 $yCol0 <-columns $xCol1 $yCol1><-columns $xCol2 $yCol2> ... $fileName source file of plotted data $windowTitle title of graphical window $xLoc $yLoc horizontal and vertical location of graphical window in pixels (upper left-most corner) $xPixels $yPixels width and height of graphical window in pixels $xCol0 $yCol0 Column number to be plotted in X-axis and Y-axis, respectively. One set of columns must be defined. $xCol1 $yCol1 Additional lines may be plotted on the same graph by repeating the -columns command. These data come from the same source file. (optional) $xCol2 $yCol2 playback Command This command is used to invoke playback on all Recorder objects constructed with the recorder command (page 326). playback $commitTag $commitTag integer used to invoke the record() method (????) 323 CHAPTER 29 Analysis Objects The Analysis objects are responsible for performing the analysis. The analysis moves the model along from state at time t to state at time t + dt. This may vary from a simple static (page 363) linear analysis to a transient (page 364, page 365) non-linear analysis. In OpenSees each Analysis object is composed of several component objects, which define the type of analysis how the analysis is performed. In general terms, the analysis objects are used to solve the following time-dependent equilibrium-equation problem for a transient analysis: -- transient equilibrium Where FI is the acceleration-dependent inertial force vector, FR is the velocity (damping) and displacement-dependent (stiffness) resisting-force vector. P(t) is the external applied-force vector. The acceleration, velocity and displacement vectors are all time-dependent. The component classes consist of the following: ConstraintHandler (page 339) -- determines how the constraint equations are enforced in the analysis -- how it handles the boundary conditions/imposed displacements DOF_Numberer (page 344) -- determines the mapping between equation numbers and degrees-of-freedom AnalysisModel (page 363) -- defines what time of analysis is to be performed Integrator -- determines the predictive step for time t+dt SolutionAlgorithm (page 352) -- determines the sequence of steps taken to solve the nonlinear equation at the current time step SystemOfEqn/Solver -- within the solution algorithm, it specifies how to store and solve the system of equations in the analysis Chapter 29 Figure 61: Analysis Object In This Chapter constraints Command .............................................. 339 numberer Command ................................................ 344 system Command .................................................... 346 test Command.......................................................... 349 algorithm Command ................................................. 352 integrator Command................................................. 356 analysis Command................................................... 363 rayleigh command .................................................... 367 eigen Command....................................................... 368 analyze Command ................................................... 369 dataBase Commands............................................... 370 Analysis Objects 324 Chapter 29 Analysis Objects 325 326 CHAPTER 30 constraints Command This command is used to construct the ConstraintHandler object. The ConstraintHandler object determines how the constraint equations are enforced in the analysis. Constraint equations enforce a specified value for a DOF, or a relationship between DOFs. The degrees of freedom can be broken down into UR, the retained DOF's, and UC, the condensed DOF's: Chapter 30 constraints Command 327 The Plain Constraints (page 341) command is used to enforce homogeneous single-point constraints, such as the case of homogeneous boundary conditions, where all boundary conditions are fixity, using single-point constraints (page 41, http://www.). For this case: UC=0.0 The other constraints commands are used for ALL other cases, such as the case of nonhomogeneous single-point constraints using the sp command (page 319), multi-point constraints (page 44), imposed motions (page 324) and multi-support excitation (page 322). For such cases, the relationship between DOF's can be written as:UC=CRCUR. Where CRC is a matrix of constants. The following constraints handlers are currently availble: Penalty Method (page 341) -- consists of adding large numbers to the stiffness matrix and the restoring-force vectors to impose a prescribed zero or nonzero DOF. This method applies very stiff elements (numerically) at the boundary conditions. These additional stiffnesses affect the eigenvalues/eigenvectors in a transient analysis. This is the recommended method for a static analysis. Lagrange Multipliers (page 342) -- apply the method of lagrange multipliers to the system of equations, thus enlarging the size of the materices. Once Labrange Multipliers have been applied, the resulting stiffness matrix is no longer positive definite. Therefore, this method should be used only if there are condition-number problems with the penalty method. Transformation Method (page 343) -- transforms the stiffness matrix by condensing out the constrained DOF's. This method reduces the size of the system for multi-point constraints. This is the recommended method for a transient analysis. However, this method should not be used when nodes are constrained in series. For example, U3 is constrained to U2, which is then constrained to U1. Of the different methods, "the Lagrange multiplier method is more attractive than the transformation method if there are few constraint equations that couple many DOF. However, Lagrange multipliers are active at the structure level, but transformation equations can be applied at either the structure level or element by element. The latter has the appeal of disposing of constraints at an early stage, when the matrices are small and manageable". (Cook) "In comparison with Lagrange multipliers, penalty functions have the advantage of introducing no new variables. However, the penalty matrix may significantly increase the bandwidth of the structural equations, depending on how DOF are numbered and what DOF are coupled by the constraint equations. Penalty functions have the disadvantage that penalty numbers must be chosen in an allowable range: large enough to be effective but not so large as to provoke numerical difficulties". (Cook) More information and examples on these methods are discussed in detail in the Cook book. Chapter 30 constraints Command 328 In This Chapter Plain Constraints ...................................................... 341 Penalty Method ........................................................ 341 Lagrange Multipliers ................................................. 342 Transformation Method ............................................ 343 Plain Constraints This command creates a PlainHandler which is only capable of enforcing homogeneous singlepoint constraints. If other types of constraints exist in the domain, a different constraint handler must be specified. constraints Plain Penalty Method This command is used to construct a PenaltyConstraintHandler which will cause the constraints to be enforced using a penalty method. The penalty method consists of adding large numbers to the stiffness matrix and the restoring-force vectors to impose a prescribed zero or nonzero DOF. constraints Penalty $alphaSP $alphaMP $alphaSP factor used when adding the single-point constraint into the system of equations $alphaMP factor used when adding the multi-point constraint into the system of equations In this method the potential-energy equation which makes up the system of equations is augmented by a penalty function {t}T[alpha]{t}/2 where [alpha] is a diagonal matrix of "penalty numbers". The resulting system of equations is of the form: [K + CTalphaC]U = [R+CTalphaQ] Chapter 30 constraints Command 329 Where CTalphaC can be called the penalty matrix. C and Q are matrices containing constants, K is the stiffness matrix, U represents the DOF and R the restoring forces. If alpha=0 the constraints are ignored. As alpha grows, U changes in such a way that the constraint equations are more nearly satisfied. In this case, however, the analysis becomes error prone, as the system represents a stiff region supported by a flexible region. NOTE: The Penalty Method affects the maximum eigenvalues of the system and may cause problems in a Transient analysis. The Penalty Method is discussed in detail in the Cook Book -- Concepts and Applications of Finite Element Analysis. "Guideline for choice of alpha: If computer words carry approximately p decimal digits, experience has shown that alpha should not exceed 10p/2". (Cook) Lagrange Multipliers This command is used to construct a LagrangeConstraintHandler which will cause the constraints to be enforced using the method of Lagrange multipliers. constraints Lagrange <$alphaSP> <$alphaMP> $alphaSP factor used when adding the single-point constraint into the system of equations (optional, default=1.0) $alphaMP factor used when adding the multi-point constraint into the system of equations (optional, default=1.0) NOTE: Values for $alphaSP and $alphaMP other than 1.0 are permitted to offset numerical roundoff problems. NOTE: The system of equations is not positive definite due to the introduction of zeroes on the diagonal by the constraint equations: From Cook: "Lagrange's method of undetermined multipliers is used to find the maximum or minimum of a function whose variables are not independent but have some prescribed relation. In structural mechanics the function is the potential energy and the variables are the DOF". Chapter 30 constraints Command 330 Transformation Method This command is used to construct a TransofrmationConstraintHandler which will cause the constraints to be enforced using the transformation method. constraints Transformation NOTE: With the current implementation, a retained node in an MP_Constraint cannot also be specified as being a constrained node in another MP_Constraint. The constraint equations takes the following form: 331 CHAPTER 31 numberer Command This command is used to construct the DOF_Numberer object. The DOF_Numberer object determines the mapping between equation numbers and degrees-of-freedom -- how degrees-offreedom are numbered. Plain (page 344) -- nodes are assigned degrees-of-freedom arbitrarily, based on the input file. This method is recommended for small problems or when sparse solvers are used, as they do their own internal DOF numbering. RCM (page 345) -- nodes are assigned degrees-of-freedom using the Reverse CuthillMcKee algorithm. This algorithm optimizes node numbering to reduce bandwidth using a numbering graph. This method will output a warning when the structure is disconnected. As certain system of equation and solver objects do their own mapping, i.e. SuperLU, UmfPack, Kincho's, specifying a numberer other than plain may not be needed. In This Chapter Plain Numberer ........................................................ 344 RCM Numberer ........................................................ 345 Plain Numberer This command is used to construct a PlainNumberer object. numberer Plain The Plain numberer assigns degrees-of-freedom to the nodes based on how the nodes are stored in the domain. Currently, the user has no control over how nodes are stored. Chapter 31 numberer Command 332 RCM Numberer This command is used to construct a RCMNumberer object. numberer RCM The RCM numberer uses the reverse Cuthill McKee (REF?) algorithm to number the degrees of freedom. 333 CHAPTER 32 system Command This command is used to construct the LinearSOE and LinearSolver objects to store and solve the system of equations in the analysis. Figure 62: system command In This Chapter BandGeneral SOE.................................................... 347 BandSPD SOE......................................................... 347 ProfileSPD SOE ....................................................... 347 SparseGeneral SOE................................................. 347 UmfPack SOE .......................................................... 348 SparseSPD SOE ...................................................... 348 Chapter 32 system Command 334 BandGeneral SOE This command is used to construct an un-symmetric banded system of equations object which will be factored and solved during the analysis using the Lapack band general solver. system BandGeneral BandSPD SOE This command is used to construct a symmetric positive definite banded system of equations object which will be factored and solved during the analysis using the Lapack band spd solver. system BandSPD ProfileSPD SOE This command is used to construct symmetric positive definite profile system of equations object which will be factored and solved during the analysis using a profile solver. system ProfileSPD SparseGeneral SOE This command is used to construct a general sparse system of equations object which will be factored and solved during the analysis using the SuperLU solver. system SparseGeneral <-piv> -piv perform partial-pivoting (optional, Default: no partial pivoting is performed) Chapter 32 system Command 335 UmfPack SOE This command is used to construct a general sparse system of equations object which will be factored and solved during the analysis using the UMFPACK solver. (REF?) system UmfPack SparseSPD SOE This command is used to construct a sparse symmetric positive definite system of equations object which will be factored and solved during the analysis using a sparse solver developed at Stanford University by Kincho Law. (REF?) system SparseSPD 336 CHAPTER 33 test Command This command is used to construct a ConvergenceTest object. Certain SolutionAlgorithm (page 352) objects require a ConvergenceTest object to determine if convergence has been achieved at the end of an iteration step. The convergence test is applied to the following equation: The test perform the following checks: Norm Unbalance Norm Displacement Increment Energy Increment In This Chapter Norm Unbalance Test .............................................. 349 Norm Displacement Increment Test ......................... 350 Energy Increment Test ............................................. 351 Norm Unbalance Test This command is used to construct a CTestNormUnbalance object which tests positive force convergence if the 2-norm of the b vector (the unbalance) in the LinearSOE (page 346) object is less than the specified tolerance. test NormUnbalance $tol $maxNumIter <$printFlag> $tol convergence tolerance $maxNumIter maximum number of iterations that will be performed before "failure to converge" is returned $printFlag flag used to print information on convergence (optional) Chapter 33 test Command 0 no print output (default) 1 print information on each step 2 print information when convergence has been achieved 4 print norm, dU and dR vectors 5 at convergence failure, carry on, print error message, but do not stop analysis 337 The test performs the following check: Norm Displacement Increment Test This command is used to construct a CTestNormDispIncr object which tests positive force convergence if the 2-norm of the x vector (the displacement increment) in the LinearSOE (page 346) object is less than the specified tolerance. test NormDispIncr $tol $maxNumIter <$printFlag> $tol convergence tolerance $maxNumIter maximum number of iterations that will be performed before "failure to converge" is returned $printFlag flag used to print information on convergence (optional) 0 no print output (default) 1 print information on each step 2 print information when convergence has been achieved 4 print norm, dU and dR vectors 5 at convergence failure, carry on, print error message, but do not stop analysis The test performs the following check: Chapter 33 test Command 338 Energy Increment Test This command is used to construct a CTestEnergyIncr object which tests positive force convergence if one half of the inner-product of the x and b vectors (displacement increment and unbalance) in the LinearSOE (page 346) object is less than the specified tolerance. test EnergyIncr $tol $maxNumIter <$printFlag> $tol convergence tolerance $maxNumIter maximum number of iterations that will be performed before "failure to converge" is returned $printFlag flag used to print information on convergence (optional) 0 no print output (default) 1 print information on each step 2 print information when convergence has been achieved 4 print norm, dU and dR vectors 5 at convergence failure, carry on, print error message, but do not stop analysis The test performs the following check: 339 CHAPTER 34 algorithm Command This command is used to construct a SolutionAlgorithm object, which determines the sequence of steps taken to solve the non-linear equation. In This Chapter Linear Algorithm ....................................................... 352 Newton Algorithm ..................................................... 352 Newton with Line Search Algorithm .......................... 353 Modified Newton Algorithm....................................... 354 Krylov-Newton Algorithm .......................................... 354 BFGS Algorithm ....................................................... 354 Broyden Algorithm.................................................... 355 Linear Algorithm This command is used to construct a Linear algorithm object which takes one iteration to solve the system of equations. algorithm Linear Newton Algorithm This command is used to construct a NewtonRaphson algorithm object which uses the NewtonRaphson method to advance to the next time step. algorithm Newton NOTE: The tangent is updated at each iteration. Chapter 34 algorithm Command Newton with Line Search Algorithm This command is used to construct a NewtonLineSearch algorithm object which uses the Newton-Raphson method with line search to advance to the next time step. algorithm NewtonLineSearch $ratio $ratio limiting ratio between the residuals before and after the incremental update (between 0.5 and 0.8) If the ratio between the residuals before and after the incremental update is greater than the specified limiting ratio the line search algorithm developed by Crissfield (REF?) is employed to try to improve the convergence. 340 Chapter 34 algorithm Command 341 Modified Newton Algorithm This command is used to construct a ModifiedNewton algorithm object which uses the Modified Newton-Raphson method to advance to the next time step. The difference between this method and the Newton-Raphson method is that the tangent stiffness is not updated at each step, thus avoiding expensive calculations needed in multi-DOF systems. However, more iterations may be needed to reach a prescribed accuracy. algorithm ModifiedNewton NOTE: The tangent at the first iteration of the current time step is used to iterate on the next time step. Krylov-Newton Algorithm This command is used to construct a KrylovNewton algorithm object which uses a modified Newton method with Krylov subspace acceleration to advance to the next time step. algorithm KrylovNewton The accelerator is described by Carlson and Miller in "Design and Application of a 1D GWMFE Code" from SIAM Journal of Scientific Computing (http://epubs.siam.org/sambin/dbq/toclist/SISC) (Vol. 19, No. 3, pp. 728-765, May 1998). BFGS Algorithm This command is used to construct a BFGS algorithm object for symmetric systems which performs successive rank-two updates of the tangent at the first iteration of the current time step. algorithm BFGS <$count> $count number of iterations within a time step until a new tangent is formed Chapter 34 algorithm Command 342 Broyden Algorithm This command is used to construct a Broyden algorithm object for general unsymmetric systems which performs successive rank-one updates of the tangent at the first iteration of the current time step. algorithm Broyden <$count> $count number of iterations within a time step until a new tangent is formed 343 CHAPTER 35 integrator Command This command is used to construct the Integrator object. The Integrator object determines the meaning of the terms in the system of equation object Ax=B. The Integrator object is used for the following: determine the predictive step for time t+dt specify the tangent matrix and residual vector at any iteration determine the corrective step based on the displacement increment dU The system of nonlinear equations is of the form: Static analysis: Transient analysis: Chapter 35 integrator Command 344 The type of integrator used in the analysis is dependent on whether it is a static analysis (page 363) or transient analysis (page 364): STATIC ANALYSIS* LoadControl (page 358) DisplacementControl (page 358) MinUnbalDispNorm (page 359) ArcLength (page 360) TRANSIENT ANALYSIS Newmark (page 361) Hilbert-Hughes-Taylor Method (HHT (page 362)) *NOTE: static integrators should only be used with a Linear TimeSeries (page 313) object with a factor of 1.0. In This Chapter Load Control............................................................. 358 Displacement Control ............................................... 358 Minimum Unbalanced Displacement Norm............... 359 Arc-Length Control ................................................... 360 Newmark Method ..................................................... 361 Hilbert-Hughes-Taylor Method.................................. 362 Load Control This command is used to construct a StaticIntegrator object of type LoadControl integrator LoadControl $dLambda1 <$Jd $minLambda $maxLambda> Chapter 35 integrator Command $dLambda1 first load increment (pseudo-time step) in the next invocation of the analysis (page 363) command. $Jd factor relating load increment at subsequent time steps. (optional, default: 1.0) $minLambda $maxLambda arguments used to bound the load increment (optional, default: $dLambda1 for both) 345 The load increment at iterations i, dLambda(i), is related to the load increment at (i-1), dLambda(i-1), and the number of iterations at (i-1), J(i-1), by the following: dLambda(i) = dLambda(i-1)*Jd/J(i-1) Displacement Control This command is used to construct a StaticIntegrator object of the type DisplacementControl. integrator DisplacementControl $nodeTag $dofTag $dU1 <$Jd $minDu $maxDu> $nodeTag node whose response controls the solution $dofTag degree-of-freedom whose response controls the solution. Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. $dU1 first displacement increment (pseudo-time step) in the next invocation of the analysis command $Jd factor relating displacement increment at subsequent time steps. (optional, default: 1.0) $minDu $maxDu arguments used to bound the displacement increment (optional, default: $dU1 for both) The displacement increment at iterations i, dU(i), is related to the displacement increment at (i1), dU(i-1), and the number of iterations at (i-1), J(i-1), by the following: dU(i) = dU(i-1)*Jd/J(i-1) Chapter 35 integrator Command 346 Minimum Unbalanced Displacement Norm This command is used to construct a StaticIntegrator object of type MinUnbalDispNorm. integrator MinUnbalDispNorm $dlambda11 <$Jd $minLambda $maxLambda> $dLambda11 first load increment (pseudo-time step) at the first iteration in the next invocation of the analysis (page 363) command. $Jd factor relating first load increment at subsequent time steps. (optional, default: 1.0) $minLambda $maxLambda arguments used to bound the load increment (optional, default: $dLambda11 for both) The load increment at iterations i, dLambda1(i), is related to the load increment at (i-1), dLambda1(i-1), and the number of iterations at (i-1), J(i-1), by the following: dLambda1(i) = dLambda1(i-1)*Jd/J(i-1) Arc-Length Control This command is used to construct a StaticIntegrator object of type ArcLength. Arc-length methods are used to enable the solution algorithm to pass limit points, such as maximum and minimum loads, and snap-through and snap-back responses. At these limit points, the stability of the numerical system is dependent on whether the analysis is performed under load or displacement control. In structural analysis, these limit points are characteristic of cracking of reinforced concrete and of buckling of shells. integrator ArcLength $arclength $alpha (??? is this the correct order, at the workshop Frank had it different) $arclength $alpha SEE FMK arclength value Chapter 35 integrator Command 347 The equilibrium equation can be written in the from: the arc-length method aims to find the intersection of the above equation with s=constant, where s is the arc-length, defined by: and the scaling parameter ψ is required because the load contribution depends on the adopted scaling between the load and displacement terms. for the arc-length methods, one should replace the differential form of the equation for ds with an incremental form: ??? where ∆l is the fixed 'radius of the desired intersection. In the arc-length method the load parameter λ becomes a variable, adding one to the n displacement variables and equations. Newmark Method This command is used to construct a TransientIntegrator object of type Newmark. integrator Newmark $gamma $beta <$alphaM $betaK $betaKinit $betaKcomm> $gamma Newmark parameter γ $beta Newmark parameter β $alphaM $betaK $betaKinit $betaKcomm Arguments to define Rayleigh damping matrix (optional, default: zero) The damping matrix D is specified as a combination of stiffness and mass-proportional damping matrices: Chapter 35 integrator Command 348 D = $alphaM * M + $betaK * Kcurrent +$betaKinit * Kinit + $betaKcomm * KlastCommit The mass and stiffness matrices are defined as: M mass matrix used to calculate Rayleigh Damping Kcurrent stiffness matrix at current state determination used to calculate Rayleigh Damping Kinit stiffness matrix at initial state determination used to calculate Rayleigh Damping KlastCommit stiffness matrix at last-committed state determination used to calculate Rayleigh Damping Hilbert-Hughes-Taylor Method This command is used to construct a TransientIntegrator object of type HHT or HHT1. integrator HHT $gamma <$alphaM $betaK $betaKinit $betaKcomm> $gamma Newmark parameter γ $alphaM $betaK $betaKinit $betaKcomm Arguments to define Rayleigh damping matrix (optional, default: zero) The damping matrix D is specified as a combination of stiffness and mass-proportional damping matrices: D = $alphaM * M + $betaK * Kcurrent +$betaKinit * Kinit + $betaKcomm * KlastCommit The mass and stiffness matrices are defined as: M mass matrix Kcurrent stiffness matrix at current state determination Kinit stiffness matrix at initial state determination KlastCommit stiffness matrix at last-committed state determination Chapter 35 integrator Command 349 350 CHAPTER 36 analysis Command This command is used to construct the Analysis object (page 336), which defines what type of analysis is to be performed. The following analysis types are available: Static Analysis (page 363) -- solves the KU=R problem, without the mass or damping matrices. Transient Analysis (page 364) -- solves the time-dependent analysis. The time step in this type of analysis is constant. The time step in the output is also constant. Variable Transient Analysis (page 365) -- performs the same analysis type as the Transient Analysis object. The time step, however, is variable. This method is used when there are convergence problems with the Transient Analysis object at a peak or when the time step is too small. The time step in the output is also variable. All currently-available analysis objects employ incremental solution strategies. In This Chapter Static Analysis.......................................................... 363 Transient Analysis .................................................... 364 VariableTransient Analysis ....................................... 365 Static Analysis This command is used to construct a StaticAnalysis object. analysis Static This analysis object is constructed with the component objects previously created by the analyst. If none has been created, default objects are constructed and used: Component Object Default object Chapter 36 analysis Command SolutionAlgorithm (page 352), StaticIntegrator (page 358, page 356, page 358, page 359) NewtonRaphson (page 352) EquiSolnAlgo with a CTestNormUnbalance (page 349) with a tolerance of 1e-6 and a maximum of 25 iterations ConstraintHandler (page 339) PlainHandler (page 341) ConstraintHandler DOF_Numberer (page 344) RCM (page 345) DOF_Numberer LinearSOE (page 346), LinearSolver (page 346) profiled symmetric positive definite (page 347) LinearSOE and Linear Solver Integrator LoadControl StaticIntegrator (page 358) with a constant load increment of 1.0 351 Transient Analysis This command is used to construct a DirectIntegrationAnalysis object. analysis Transient This analysis object is constructed with the component objects previously created by the analyst. If none has been created, default objects are constructed and used: Component Object Default object SolutionAlgorithm (page 352), TransientIntegrator (page 356) NewtonRaphson (page 352) EquiSolnAlgo with a CTestNormUnbalance (page 349) with a tolerance of 1e-6 and a maximum of 25 iterations ConstraintHandler (page 339) PlainHandler (page 341) ConstraintHandler DOF_Numberer (page 344) RCM (page 345) DOF_Numberer LinearSOE (page 346), LinearSolver (page 346) profiled symmetric positive definite (page 347) LinearSOE and Linear Solver Chapter 36 Integrator analysis Command 352 Newmark TransientIntegrator (page 361) with γ=0.5 and β=0.25 VariableTransient Analysis This command is used to construct a VariableTimeStepDirectIntegrationAnalysis object. analysis VariableTransient This analysis object is constructed with the component objects previously created by the analyst. If none has been created, default objects are constructed and used: Component Object Default object SolutionAlgorithm (page 352), TransientIntegrator (page 356) NewtonRaphson (page 352) EquiSolnAlgo with a CTestNormUnbalance (page 349) with a tolerance of 1e-6 and a maximum of 25 iterations ConstraintHandler (page 339) PlainHandler (page 341) ConstraintHandler DOF_Numberer (page 344) RCM (page 345) DOF_Numberer LinearSOE (page 346), LinearSolver (page 346) profiled symmetric positive definite (page 347) LinearSOE and Linear Solver Integrator Newmark TransientIntegrator (page 361) with γ=0.5 and β=0.25 353 CHAPTER 37 rayleigh command This command is used to assign damping to all previously-defined elements and nodes: rayleigh $alphaM $betaK $betaKinit $betaKcomm The damping matrix D is specified as a combination of stiffness and mass-proportional damping matrices: D = $alphaM * M + $betaK * Kcurrent +$betaKinit * Kinit + $betaKcomm * KlastCommit The mass and stiffness matrices are defined as: M mass matrix used to calculate Rayleigh Damping Kcurrent stiffness matrix at current state determination used to calculate Rayleigh Damping Kinit stiffness matrix at initial state determination used to calculate Rayleigh Damping KlastCommit stiffness matrix at last-committed state determination used to calculate Rayleigh Damping 354 CHAPTER 38 eigen Command This command is used to perform a generalized eigenvalue problem to determine a specified number of eigenvalues and eigenvectors. eigen <$type> $numEigenvalues $type $numEigenvalues eigen-value analysis type: frequency solve: K- λ M generalized solve: K- λ M (default) standard solve: K- λ I number of first eigenvalues (λ) to be determined The eigenvectors are stored at the nodes and can be printed out using Node Recorder (page 326) or the Print command (page 372). NOTE: The current eigensolver is able to solve only for N-1 eigenvalues, where N is the number of interial DOFs. When running into this limitation, the user should add negligible masses to additional unrestrained degrees of freedom. For example, for a 3-dof/node problem, instead of: mass 1 100 0 0 the user should input: mass 1 100 1e-9 0 where dof-2 at node 1 is unrestrained. 355 CHAPTER 39 analyze Command This command is invoked on the Analysis object constructed with the analysis command (page 363). analyze $numIncr <$dt> <$dtMin $dtMax $Jd> $numIncr number of load steps $dt time-step increment. Required if a transient analysis (page 364) or variable time step transient analysis (page 365) was specified. $dtMin $dtMax minimum and maximum time steps Required if a variable time step transient analysis (page 365) was specified. $Jd number of iterations performed at each step Required if a variable time step transient analysis (page 365) was specified. This command RETURNS: 0 successful <0 unsuccessful 356 CHAPTER 40 dataBase Commands This command is used to construct a FE_Datastore object. Currently there is only one type of Datastore object available. In This Chapter FileDatastore Command........................................... 370 FileDatastore Command This command is used to construct the FE_Datastore object. database $type $dbName $type $dbName database type: File outputs database into a file MySQL creates a SQL database BerkeleyDB creates a BerkeleyDB database database name. If the database type is File, the command will save the data into a number of files, e.g. $dbName.id11 for all ID objects of size 11 that sendSelf() is invoked upon. The invocation of this command will add the additional commands save and restore to the OpenSees interpreter to allow users to save and restore model states. Chapter 40 dataBase Commands save Command This command is used to save the state of the model in the database. save $commitTag $commitTag unique identifier that can be used to restore (page 371) the state at a later time restore Command This command is used to restore the state of the model from the information stored in the database. restore $commitTag $commitTag unique identifier used to restore the state at the model when the save (page 371) command was invoked 357 358 CHAPTER 41 Miscellaneous Commands These are a few additional miscellaneous command used in OpenSees In This Chapter print Command......................................................... 372 reset Command........................................................ 373 wipe Command ........................................................ 373 wipeAnalysis Command ........................................... 374 loadConst Command................................................ 374 getTime Command................................................... 375 nodeDisp Command................................................. 375 video Command ....................................................... 375 nodeBounds ............................................................. 376 display Command..................................................... 377 print Command This command is used to print output. To print all objects of the domain: print <$fileName> To print node information: print <$fileName> -node <-flag $flag> <$node1 $node2 ...> To print element information: print <$fileName> -ele <-flag $flag> <$ele1 $ele2 ...> $fileName fileName for printed output (optional, Default: stderr -- screen dump) Chapter 41 Miscellaneous Commands $flag integer flag to be sent to the print() method, depending on the node and element type (optional) $node1 $node2 ... node tag for selected-node output (optional) 359 Default: all $ele1 $ele2 ... element tag for selected-element output (optional) Default: all reset Command This command is used to set the state of the domain to its original state. reset The command invokes revertToStart() on the Domain (page 32) object. wipe Command This command is used to destroy all constructed objects. wipe This command is used to start over without having to exist and restart the interpreter (page 19). wipeAnalysis Command This command is used to destroy all objects constructed for the analysis. wipeAnalysis Chapter 41 Miscellaneous Commands 360 This command is used to start a new type of analysis. This command does not destroy the elements (page 213), nodes (page 39), materials (page 175, page 47), etc. It does destroy the solution strategies: the algorithm (page 352), analysis (page 363), equation solver (page 346), constraint handler (page 339), etc. loadConst Command This command is used to invoke setLoadConst() on all LoadPattern (page 317) objects which have been created up to this point. loadConst <-time $pseudoTime> -time $pseudoTime set pseudo time in the domain to $pseudoTime (optional, default: zero) getTime Command This command returns the time in the domain. getTime nodeDisp Command Returns the displacement or rotation at specified node. nodeDisp $nodeTag $dof $nodeTag node tag $dof degree-of-freedom tag Valid range is from 1 through ndf (page 37), the number of nodal degrees-of-freedom. (???) Chapter 41 Miscellaneous Commands 361 video Command This command is used to construct a TclVideoPlayer object for displaying the images in a file created by the recorder display (page 333) command. video -file $fileName -window $windowName $fileName fileName of images file created by the recorder display (page 333) command $windowName name of window to be created The images are displayed by invoking the play (page 376) command. Chapter 41 Miscellaneous Commands 362 play Command This command is used to play the TclVideoPlayer object created by the video (page 375) command. play nodeBounds This commands returns a 5-element list of the max and min coordinates of the nodes defined in the domain: nodeBounds The command returns: "Xmin Ymin Zmin Xmax Ymax Zmax", independently of the domain dimensions, 1D, 2D or 3D, zeros are returned for the extra dimensions. This command is useful when setting up a display. Example: 3D set Xmin [lindex [nodeBounds] 0]; set Ymin [lindex [nodeBounds] 1]; set Zmin [lindex [nodeBounds] 2]; set Xmax [lindex [nodeBounds] 3]; set Ymax [lindex [nodeBounds] 4]; set Zmax [lindex [nodeBounds] 5]; 2D set Xmin [lindex [nodeBounds] 0]; set Ymin [lindex [nodeBounds] 1]; set Xmax [lindex [nodeBounds] 3]; set Ymax [lindex [nodeBounds] 4]; 1D set Xmin [lindex [nodeBounds] 0]; set Xmax [lindex [nodeBounds] 3]; Chapter 41 Miscellaneous Commands 363 display Command This command is used to display the model defined in the domain, and is associated with the display recorder (page 333): display $arg1 $arg2 $amplification Example usage: set dAmp 10; # this amplification is dependent on the model size if {$ShapeType == "ModeShape" } { display -$nEigen 0 $dAmp; # display mode shape for mode $nEigen } elseif {$ShapeType == "NodeNumbers" } { display 1 -1 0 ; # display node numbers } elseif {$ShapeType == "DeformedShape" } { display 1 5 $dAmp; } # display deformed shape 364 CHAPTER 42 How To.... In this chapter, some examples on how to generate input for OpenSees for specific tasks will be presented. In This Chapter Run OpenSees......................................................... 379 ...Define Units & Constants ...................................... 382 ...Generate Matlab Commands................................. 383 ...Define Tcl Procedure............................................. 384 ...Read External files ................................................ 385 Building The Model................................................... 387 Defining Output ........................................................ 393 Gravity Loads ........................................................... 394 Static Analysis.......................................................... 395 Dynamic Analysis ..................................................... 398 ...Combine Input-File Components ........................... 399 ...Run Parameter Study ............................................ 400 ...Run Moment-Curvature Analysis on Section ......... 401 ...Determine Natural Period & Frequency ................. 403 Chapter 42 How To.... Run OpenSees There are three ways that OpenSees/Tcl commands can be executed: Interactive Commands can be input directly at the propt, as shown in the figure (Win32 version): Figure 63: Run OpenSees -Interactive 365 Chapter 42 How To.... 366 Execute Input File at OpenSees prompt This method is the most-commonly used one. An external file containing the input commands can be generated a-priori (inputFile.tcl) and be executed at the OpenSees prompt by using the source command. The generation of the input script files is presented in this chapter. The file execution is shown in the figure (Win32 version): Figure 64: Run OpenSees -- Source File Batch Mode The previously-created input file containing the Tcl script commands necessary to execute the analsis can also be executed at the MS-DOS/Unix prompt, as shown in the figure (Win32 version): Chapter 42 Figure 65: Run OpenSees -- Batch Mode How To.... 367 Chapter 42 How To.... 368 ...Define Units & Constants The OpenSees interpreter does not process units. Units, however, can be used when entering values if these units are defined previously. The unit definition consists of two parts: the basic units are defined first, all other units are subsequently defined. The basic units are assigned a value of one and all OpenSees output is in these units. It is very important that all basic units are independent of each other. The unit-definition file can contain both metric and Imperial units, as can the basic units. Hence, the input files may contain mixed units. Constants, such as π and g can also be defined apriori. An example of unit and constant definition is given in the following: Procedure to define units and constants # ----Units&Constants.tcl--------------------------------------------------------# define basic units set in 1.; set sec 1.; set kip 1.; set ksi [expr $kip/pow($in,2)]; # define dependent units set psi [expr $ksi/1000.]; set ft [expr 12.*$in]; set lb [expr $kip/1000]; set pcf [expr $lb/pow($ft,3)]; set ksi [expr $kip/pow($in,2)]; set psi [expr $ksi/1000.]; # define metric units set cm [expr $in/2.54]; set meter [expr 100.*$cm]; set MPa [expr 145*$psi]; # define constants set PI [expr 2*asin(1.0)]; set g [expr 32.2*$ft/pow($sec,2)]; set U 1.e10; # a really large number set u [expr 1/$U]; # a really small number Chapter How To.... 369 CHAPTER ...Generate Matlab Commands Matlab is a common tool for post-processing. Matlab command files can be generated using the Tcl scripting language. Using this technique ensures that the same analysis parameters are used. Here is an example. # script to generate .m file to be read by matlab # -------------MatlabOutput.tcl------------------------------------------------------------------set Xframe 1; # this parameter would be passed in set fDir "Data/"; file mkdir $fDir; # create directory set outFileID [open $fDir/DataFrame$Xframe.m w]; # Open output file for writing puts $outFileID "Xframe($Xframe) = $Xframe;"; # frame ID puts $outFileID "Hcol($Xframe) = $Hcol;"; # column diameter puts $outFileID "Lcol($Xframe) = $Lcol;"; # column length puts $outFileID "Lbeam($Xframe) = $Lbeam;"; # beam length puts $outFileID "Hbeam($Xframe) = $Hbeam;"; # beam depth puts $outFileID "Bbeam($Xframe) = $Bbeam;"; # beam width puts $outFileID "Weight($Xframe) = $Weight;" ; # superstructure weight close $outFileID ...Define Tcl Procedure The procedure is a useful tool available from the Tcl language. A procedure is a generalized funcion/subroutine using arguments. Whenever the Tcl procedure is invoked, the contents of body will be executed by the Tcl interpreter. An example of a Tcl procedure is found in RCcircSec.tcl. It defines a procedure which generates a circular reinforced concrete section with one layer of steel evenly distributed around the perimeter and a confined core: # ----RCcircSec.tcl-----------------------------------------------------------------------# by Michael H. Scott # Define a procedure which generates a circular reinforced concrete section Chapter How To.... 370 # with one layer of steel evenly distributed around the perimeter and a confined core. # Formal arguments # id - tag for the section that is generated by this procedure # ri - inner radius of the section # ro - overall (outer) radius of the section # cover - cover thickness # coreID - material tag for the core patch # coverID - material tag for the cover patches # steelID - material tag for the reinforcing steel # numBars - number of reinforcing bars around the section perimeter # barArea - cross-sectional area of each reinforcing bar # nfCoreR - number of radial divisions in the core (number of "rings") # nfCoreT - number of theta divisions in the core (number of "wedges") # nfCoverR - number of radial divisions in the cover # nfCoverT - number of theta divisions in the cover # Notes # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis proc RCcircSection {id ri ro cover coreID coverID steelID numBars barArea nfCoreR nfCoreT nfCoverR nfCoverT} { section fiberSec $id { set rc [expr $ro-$cover]; # Core radius patch circ $coreID $nfCoreT $nfCoreR 0 0 $ri $rc 0 360; # Core patch patch circ $coverID $nfCoverT $nfCoverR 0 0 $rc $ro 0 360; # Cover patch if {$numBars <= 0} { return } set theta [expr 360.0/$numBars]; # Angle between bars layer circ $steelID $numBars $barArea 0 0 $rc $theta 360; # Reinforcing layer } } This procedure is invoked by the following commands, assuming that all arguments have been defined previously in the input generation: source RCcircSection.tcl; RCcircSection $IDcolFlex $riCol $roCol $cover $IDcore $IDcover $IDsteel $NbCol $AbCol $nfCoreR Chapter How To.... 371 $nfCoreT $nfCoverR $nfCoverT NOTE: the file containing the definition of the procedure (RCcircSec.tcl) needs to be sourced before the procedure is invoked. ...Read External files External files may either contain Tcl commands or data. Common input file The external file may contain a series of commands that is common in most analyses. One set of Tcl commands that can be stored in a external file are ones which define units. An example of an external file that may want to be read within the input commands is the unitdefinition file presented earlier (units&constants.tcl (page 469)). This file is invoked with the following command: source units.tcl Repeated Calculations An external file may contain a series of calculations that are repeated. An example of this is a parameter study: set Hcolumn 66; source analysis.tcl set Hcolumn 78; source analysis.tcl The analysis.tcl file contains the commands that set up and execute the entire analysis. External Data File The following commands open a data file (filename=inFilename), read the file row by row and assign the value of each row to the a single variable (Xvalues). If there are more than one value in the row, $Xvalues is a list array, and the individual components may be extracted using the lindex command. The user may change the commands to be exectued once the data-line has been read to whatever is needed in the analysis. # ----ReadData.tcl-------------------------------------------------------------------if [catch {open $inFilename r} inFileID] {; # Open the input file and check for error puts stderr "Cannot open $inFilename for reading"; # output error statement } else { foreach line [split [read $inFileID] \n] { ; # Look at each line in the file if {[llength $line] == 0} {; # Blank line --> do nothing continue; Chapter How To.... 372 } else { set Xvalues $line; # execute operation on read data } } close $inFileID; ; # Close the input file } Building The Model ...Define Variables and Parameters In the Tcl scripting language variables may be used to represent numbers. Once defined, these variables can be use instead of numbers in Tcl and OpenSees commands. When they are being recalled, the variables are precedented by the symbol $. If this symbol is not used, the variable name is interpreted as a string command and an error may result. A few examples are given: MATERIAL PARAMETERS: # -----MaterialParameters.tcl------------------------------------------------------# nominal compressive strength of concrete set fc [expr -4.0*$ksi]; set Ec [expr 57*$ksi*sqrt(-$fc/$psi)]; # Concrete Elastic Modulus set fc1C [expr 1.26394*$fc]; # CONFINED concrete (mander model), max stress set eps1C [expr 2.*$fc1C/$Ec]; # strain at maximum stress set fc2C $fc; # ultimate stress set eps2C [expr 2.*$fc2C/$Ec]; # strain at ultimate stress set fc1U $fc; # UNCONFINED concrete (parabolic model), max stress set eps1U -0.003; # strain at maximum stress set fc2U [expr 0.1*$fc]; # ultimate stress set eps2U -0.006; # strain at ultimate stress set Fy [expr 70.*$ksi]; # STEEL yield stress set Es [expr 29000.*$ksi]; # elastic modulus of steel set epsY [expr $Fy/$Es]; # steel yield strain set Fy1 [expr 95.*$ksi]; # steel stress post-yield set epsY1 0.03; # steel strain post-yield set Fu [expr 112.*$ksi]; # ultimate stress of steel set epsU 0.08; # ultimate strain of steel Chapter How To.... 373 set Bs [expr ($Fu-$Fy)/($epsU-$epsY)/$Es]; # post-yield stiffness ratio of steel set pinchX 1.0; # pinching parameter for hysteretic model set pinchY 1.0; # pinching parameter for hysteretic model set damage1 0.0; # damage parameter for hysteretic model set damage2 0.0; # damage parameter for hysteretic model set betaMUsteel 0.0; # degraded unloading stiffness for hysteretic material based on MU^(-beta) set betaMUjoint 0.0; timoshenko value of 0.5 # degraded unloading stiffness for hysteretic material based on MU^(-beta) -- set betaMUph 0.0; timoshenko value of 0.5 # degraded unloading stiffness for hysteretic material based on MU^(-beta) -- set G $U; # Torsional stiffness Modulus set J 1.; # Torsional stiffness of section, place here just to keep with G set GJ [expr $G*$J]; # Torsional stiffness ELEMENT PARAMETERS: # -----ElementParameters.tcl---------------------------------------------------set Hcol [expr 5.*$ft]; # column diameter set Lcol [expr 36*$ft]; # column length set Hbeam [expr 8.*$ft]; # beam depth set Lbeam [expr 36.*$ft]; # beam length set GrhoCol 0.0125; # column longitudinal-steel ratio set Weight [expr 2000.*$kip]; set Bbeam $Hcol; # beam width set Rcol [expr $Hcol/2]; set Acol [expr $PI*pow($Rcol,2)]; # column cross-sectional area set cover [expr $Hcol/15]; # column cover width set IgCol [expr $PI*pow($Rcol,4)/4]; set IyCol $IgCol; # elastic-column properties set IzCol $IgCol; # elastic-column properties set IzBeam [expr $Bbeam*pow($Hbeam,3)/12]; # beam gross moment of inertia, about horizontal Z-axis set IyBeam [expr $Hbeam*pow($Bbeam,3)/12]; # beam gross moment of inertia, about the vertical Y-axis set Abeam [expr $Hbeam*$Bbeam*10000]; # beam cross-sectional area, make it very very stiff # superstructure weight # COLUMN radius # column gross moment of inertia, uncracked # define COLUMN section parameters # number of column longitudinal-reinforcement bars set NbCol 20; set AsCol [expr $GrhoCol*$Acol]; # total steel area in column section set AbCol [expr $AsCol/$NbCol]; set riCol 0.0; set roCol $Rcol; # outer radius of column section set IDcore 1; # bar area of column longitudinal reinforcement # inner radius of column section # ID tag for core concrete Chapter set IDcover 2; # ID tag for cover concrete set IDsteel 3; # ID tag for steel set nfCoreR 8; # number of radial fibers in core set nfCoreT 16; # number of tangential fibers in core set nfCoverR 2; # number of radial fibers in cover set nfCoverT 16; # number of tangential fibers in cover set IDcolFlex 2; # ID tag for column section in flexure, before aggragating torsion set IDcolTors 10; # ID tag for column section in torsion set IDcolSec 1; # ID tag for column section set IDcolTrans 1; # ID tag for column transformation, defining element normal set IDbeamTrans set np 5; 2; How To.... 374 # ID tag for beam transformation, defining element normal # Number of integration points GRAVITY PARAMETERS: # -----GravityParameters.tcl---------------------------------------------------# define GRAVITY paramters # gravity axial load per column set Pdl [expr $Weight/2]; set Wbeam # gravity dead load distributed along beam length [expr $Weight/$Lbeam]; set Mdl [expr $Wbeam*pow($Lbeam,2)/12]; set Mass [expr $Weight/$g]; set Mnode [expr $Mass/2]; # nodal moment due to distributed dl # mass of superstructure # nodal mass for each column joint ANALYSIS PARAMETERS: # -----AnalysisParameters.tcl---------------------------------------------------# Displacement increment for pushover analysis set DxPush [expr 0.1*$in]; set DmaxPush [expr 0.05*$Lcol]; set DtAnalysis [expr 0.005*$sec];# time-step Dt for lateral analysis set DtGround [expr 0.02*$sec]; # time-step Dt for input grond motion # maximum displamcement for pushover analysis set TmaxGround [expr 35 *$sec]; # maximum duration of ground-motion analysis set gamma 0.5; # gamma value for newmark integration set beta 0.25 # beta value for newmark integration ...Build Model and Define Nodes This example shows how to set up the geometry of the structure shown in the Figure (page 390). These commands are typically placed at the beginning of the input file, after the header remarks. # construct model builder using the model Command (page 37) wipe; # clear data from past analysis Chapter model basic -ndm 3 -ndf 6; How To.... 375 # modelbuilder: basic (page 37), ndm= no. dimensions, ndf= no. # Define nodes ------ frame is in X-Y plane (X-horizontal, Y-vertical) using the node Command (page 39) node 1 0. 0. 0.; # base of left column node 2 360. 0. 0.; # base of right columnn node 3 0. 120. 0.; # top of left column node 4 360. 120. 0.; # top of right columnn # Define Boundary Conditions and nodal mass using the fix Command (page 41) ! 1: restrained, 0: released fix 1 1 1 1 1 1 1; fix 2 1 1 1 1 1 1; fix 3 0 1 1 1 1 0 -mass [expr 2000/2/386.4] 0. 0. 0. 0. 0. fix 4 0 1 1 1 1 0 # define mass at node 4 using the mass Command (page 40): mass 4 [expr 2000/2/32.2/12] 0. 0. 0. 0. 0. # NOTE: The second command assigning the mass of a specific node will override any previous mass assignments to that node. ...Build Model and Define Nodes using Variables This example sets up the geometry of the cantilever column shown in the Figure (page 390) -using variables. # -------Build&Nodes.tcl------------------------------------------------------wipe; # clear data from past analysis model basic -ndm 3 -ndf 6; # define units and constants # source units.tcl; # if contained in external file # Define nodes ------ frame is in X-Y plane (X-horizontal, Y-vertical) using the node Command (page 39) Chapter node 1 0. 0. 0.; # base of left column node 2 $Lbeam 0. 0.; # base of right columnn node 3 0. $Lcol 0.; # top of left column node 4 $Lbeam $Lcol 0.; # top of right columnn How To.... 376 # Define Boundary conditions using the fix Command (page 41) ! 1: restrained, 0: released fix 1 1 1 1 1 1 1; # fully-fixed support fix 2 1 1 1 1 1 1; fix 3 0 1 1 1 1 0 -mass $Mass 0. 0. 0. 0. 0.; fix 4 0 1 1 1 1 0; # define the mass at node 4 using the mass Command (page 40): mass 4 $Mass 0. 0. 0. 0. 0.; # NOTE: The second command assigning the mass of a specific node will override any previous mass assignements to that node. ...Define Materials The following is an example on how to define materials for reinforced-concrete structures. The examples assume that the variables have been defined apriori. If these commands are placed into an external file they can be used in a number of analyses without significant modifications using the source command. # -------MaterialsRC.tcl---------------------------------------------------------set ConcreteMaterialType "inelastic"; # options: "elastic","inelastic" # options: "elastic","bilinear","hysteretic" set SteelMaterialType "hysteretic"; # CONCRETE if {$ConcreteMaterialType =="elastic"} { uniaxialMaterial Elastic $IDcore $Ec uniaxialMaterial Elastic $IDcover $Ec } if {$ConcreteMaterialType == "inelastic"} { uniaxialMaterial Concrete01 $IDcore $fc1C $eps1C $fc2C $eps2C; # Core concrete uniaxialMaterial Concrete01 $IDcover $fc1U $eps1U $fc2U $eps2U; # Cover concrete } # STEEL if {$SteelMaterialType == "elastic"} { uniaxialMaterial Elastic $IDsteel $Es } if {$SteelMaterialType == "bilinear"} { uniaxialMaterial Steel01 } $IDsteel $Fy $Es $Bs Chapter How To.... 377 if {$SteelMaterialType == "hysteretic"} { uniaxialMaterial Hysteretic $IDsteel $Fy $epsY $Fy1 $epsY1 $Fu $epsU -$Fy -$epsY -$Fy1 -$epsY1 $Fu -$epsU $pinchX $pinchY $damage1 $damage2 $betaMUsteel } ...Define Elements # -----ELEMENTS.tcl------------------------------------------------------# COLUMNS set ColumnType "inelastic"; # options: "rigid" "elastic" "inelastic" set np 5; # number of integration points # source RCcircSection.tcl; # proc to define circular fiber section for flexural characteristics RCcircSection $IDcolFlex $riCol $roCol $cover $IDcore $IDcover $IDsteel $NbCol $AbCol $nfCoreR $nfCoreT $nfCoverR $nfCoverT # Define torsional stiffness uniaxialMaterial Elastic $IDcolTors $GJ; section Aggregator $IDcolSec geomTransf Linear $IDcolTors $IDcolTrans T -section $IDcolFlex; # attach torsion and flexure # Linear: no second-order effects 0 0 1; if {$ColumnType == "rigid"} { set $IyCol [expr $IyCol*$IyCol]; set $IzCol [expr $IzCol*$IzCol]; element elasticBeamColumn 1 1 3 $Acol $Ec $G $J $IyCol $IzCol $IDcolTrans element elasticBeamColumn 2 2 4 $Acol $Ec $G $J $IyCol $IzCol $IDcolTrans } if {$ColumnType == "elastic"} { element elasticBeamColumn 1 1 3 $Acol $Ec $G $J $IyCol $IzCol $IDcolTrans element elasticBeamColumn 2 2 4 $Acol $Ec $G $J $IyCol $IzCol $IDcolTrans } if {$ColumnType == "inelastic"} { element nonlinearBeamColumn 1 1 3 $np $IDcolSec $IDcolTrans element nonlinearBeamColumn 2 2 4 $np $IDcolSec $IDcolTrans } # BEAM geomTransf Linear $IDbeamTrans 0 0 1; element elasticBeamColumn 3 3 4 $Abeam # define orientation of beam local axes $Ec $G $J $IyBeam $IzBeam $IDbeamTrans; Chapter How To.... 378 Defining Output ...Define Analysis-Output Generation Different output may be generated depending on whether the analysis is static or dynamic. Here is an example: # -------Output.tcl------------------------------------------------------------------------set fDir "Data/"; # create directory file mkdir $fDir; set ANALYSIS "Static"; # this variable would be passed in # this variable would be passed in set IDctrlNode 3; if {$ANALYSIS == "Static"} { # Record nodal displacements -NODAL DISPLACEMENTS set fDstatFrame DStatFrame[expr $Xframe] set iNode "$IDctrlNode"; # can add node numbers to this list foreach xNode $iNode { set fNode Node$xNode set Filename $fDir$fDstatFrame$fNode recorder Node $Filename.out disp -time -node $xNode -dof 1 6; }; # end of xNode # Record element forces and deformations - COLUMNS set iEL "1 2" set fFstatFrame FStatFrame[expr $Xframe] set fDstatFrame DStatFrame[expr $Xframe] foreach xEL $iEL { set fEl El[expr $xEL] set iSEC "1 3 5" set Ffilename $fDir$fFstatFrame$fEl recorder Element $xEL -time -file $Ffilename.out localForce foreach xSEC $iSEC { set fSec Sec[expr $xSEC] set Dfilename $fDir$fDstatFrame$fEl$fSec recorder Element $xEL -file $Dfilename.out -time section $xSEC deformations }; }; }; # end of xSEC # end of xEL # end of static analysis Chapter set ANALYSIS "Dynamic"; # this variable would be passed in set GroundFile "ElCentro"; # this variable would be passed in if {$ANALYSIS == "Dynamic"} { set fDDynaFrame DDynaFrame[expr $Xframe] set fGroundFile $GroundFile set Filename $fDir$fDDynaFrame$fGroundFile # Record nodal displacements recorder Node $Filename.out disp -time -node $IDctrlNode -dof 1 }; # end of dynamic analysis ...Define Data-Plot During Analysis # ---------RecorderPlot.tcl-----------------------------------------------------------------------------------set pfile "Data/node.out"; recorder Node $pfile disp -time -node $IDctrlNode -dof 1 set title PushFrame$Xframe; recorder plot $pfile $title 0 0 350 350 -columns 2 1 set pfile "Data/Elem1.out"; set title PushElem1; recorder Element 1 -time -file $pfile globalForce recorder plot $pfile $title 400 0 350 350 -columns 2 1 Gravity Loads ...Define Gravity Loads # ---------DefineGravity.tcl-------------------------------------------------------------------------------------set GravSteps 10 pattern Plain 1 Linear { load 3 0. -$Pdl 0. 0. 0. -$Mdl ; # Fx Fy Fz Mx My Mz load 4 0. -$Pdl 0. 0. 0. +$Mdl } system UmfPack; # solution procedure, how it solves system of equations How To.... 379 Chapter constraints Plain; test NormDispIncr 1.0e-5 How To.... 380 # how it handles boundary conditions, enforce constraints 10 0; algorithm Newton; # renumber dof's to minimize band-width numberer RCM; integrator LoadControl [expr 1./$GravSteps] 1 [expr 1./$GravSteps] [expr 1./$GravSteps] analysis Static initialize; # this command will not be necessary in new versions of OpenSees ...Run Gravity Analysis # ---------RunGravity.tcl-----------------------------------------------------------------------------------analyze $GravSteps # run gravity analysis loadConst -time 0.0; # keep gravity load and restart time -- lead to lateral-load analysis Static Analysis ...Define Static Pushover Analysis The following commands are executed once the gravity loads have been defined and applied # ---------DefinePushover.tcl-------------------------------------------------------------------------------------------------set analysis "STATIC"; # this variable would be passed in # the following settings do not need to be here if they have been defined in the gravity analysis system UmfPack; constraints Plain; test NormDispIncr 1.0e-5 10 0; algorithm Newton; numberer RCM; analysis Static; # -----------set PUSHOVER "DispControl"; # run displacement-controlled static pushover analysis pattern Plain 2 Linear { load $IDctrlNode 100.0 0.0 0.0 0.0 0.0 0.0 } if {$PUSHOVER == "LoadControl"} { integrator LoadControl 0.2 4 0.1 2.0 set Nsteps 20 Chapter How To.... 381 } elseif {$PUSHOVER == "DispControl"} { integrator DisplacementControl $IDctrlNode 1 $DxPush 1 $DxPush $DxPush set Nsteps [expr int($DmaxPush/$DxPush)] } else { puts stderr "Invalid PUSHOVER option" } ...Run Static Pushover Analysis While running a static pushover analysis may take a single command, convergence may not always be reached with a single analysis-parameter setting. A Tcl script which tries different solutions can be incorporated to improve the chances of convergence. No convergence issues The following command executes the static push-over analysis when convergence is not a problem. # ---------RunPushover.tcl-----------------------------------------------------------------------------------analyze $Nsteps Convergence attemps The following Tcl script should be incorporated in the input file to run a number of attempts at convergence: # ---------RunPushover2Converge.tcl---------------------------------------------------------------set ok [analyze $Nsteps] # if analysis fails, try the following, performance is slowed inside this loop if {$ok != 0} { set ok 0; set maxU $DmaxPush set controlDisp 0.0; test NormDispIncr 1.0e-8 20 0 while {$controlDisp < $maxU && $ok == 0} { set ok [analyze 1] set controlDisp [nodeDisp $IDctrlNode 1] if {$ok != 0} { puts "Trying Newton with Initial Tangent .." test NormDispIncr 1.0e-8 algorithm Newton -initial 1000 1 Chapter How To.... set ok [analyze 1] test NormDispIncr 1.0e-8 20 0 algorithm Newton } if {$ok != 0} { puts "Trying Broyden .." algorithm Broyden 8 set ok [analyze 1] algorithm Newton } if {$ok != 0} { puts "Trying NewtonWithLineSearch .." algorithm NewtonLineSearch .8 set ok [analyze 1] algorithm Newton } }; }; # end while loop # end original if $ok!=0 loop if {$ok != 0} { puts "DispControl Analysis FAILED" puts "Do you wish to continue y/n ?"; gets stdin ans; if {$ans == "n"} done; # include if want to pause at analysis failure # not recommended in parameter study # as it interrupts batch file } else { puts "DispControl Analysis SUCCESSFUL" } Dynamic Analysis ...Define Dynamic Ground-Motion Analysis # ---------DefineDynamic.tcl---------------------------------------------------------------wipeAnalysis system UmfPack 382 Chapter How To.... 383 constraints Plain test NormDispIncr 1.0e-8 20 0; algorithm Newton numberer RCM integrator Newmark $gamma $beta $alphaM $betaK $betaKcomm $betaKinit; analysis Transient set Nsteps [expr int($TmaxGround/$DtAnalysis)]; # read a PEER strong motion database file, extracts dt from the header and converts the file # to the format OpenSees expects for uniform ground motions source ReadSMDFile.tcl; set dir "GMfiles/" set outFile $dir$GroundFile.g3; # set variable holding new filename set inFile $dir$GroundFile.th ReadSMDFile $inFile $outFile dt; # convert the ground-motion file set GMfatt [expr $g*$GMfact]; # data in input file is factor of g set Gaccel "Series -dt $dt -filePath $outFile -factor $GMfatt"; pattern UniformExcitation 2 1 -accel $Gaccel; # time series information # create uniform excitation ...Run Dynamic Ground-Motion Analysis # ---------RunDynamicGM.tcl-----------------------------------------------------------analyze $Nsteps $DtAnalysis; ...Combine Input-File Components A series of Tcl-script input-file components have been presented in this section. These components can be combined to perform a static lateral-load analysis of the portal frame under consideration using the source command: # ----FullStaticAnalysis.tcl--------------------------------------------------------wipe model basic -ndm 3 -ndf 6 source Units&Constants.tcl source MaterialParameters.tcl source ElementParameters.tcl Chapter How To.... 384 source GravityParameters.tcl source AnalysisParameters.tcl source MatlabOutput.tcl source BuildModel&Nodesw/Variables--portal.tcl source materialsRC.tcl source RCcircSec.tcl source Elements.tcl source Output.tcl source DefineGravity.tcl source runGravity.tcl source DefinePushover.tcl source RunPushover2Converge.tcl This method of breaking the input file into components is convenient when the size of the problem does not permit manageability of a single input file. ...Run Parameter Study In a parameter study, the same analysis is performed on a number of models where only certain properties are varied, such as column height. There are two common types of parameter studies shown in this section: series and parallel parsing. The following diagram illustrates the difference between series and parallel parsing for two parameter lists [1 2 3] and [A B C]: Parsing in series In this type of study, one parameter is held constant, while the others are parsed in sequence: # -----ParameterStudySeries.tcl---------------------------------------------------source units.tcl set iHcol "[expr 5.*$ft] [expr 6.5*$ft]"; # column diameter Chapter set iLcol "[expr 36*$ft] [expr 42*$ft]"; How To.... 385 # column length # initialize Frame Counter, used in output set Xframe 0; foreach Hcol $iHcol { foreach Lcol $iLcol { set Xframe [expr $Xframe+1]; set ANALYSIS "Static"; source Analysis.tcl* }; # close iLcol loop }; # close iHcol loop *NOTE: The file Analysis.tcl contains all the model and analysis commands. Parsing in parallel In this study, the ith elements of each parameter list are considered together, resulting in fewer study models. # -----ParameterStudyParallel.tcl---------------------------------------------------source units.tcl set iHcol "[expr 5.*$ft] [expr 6.5*$ft]"; # column diameter set iLcol "[expr 36*$ft] [expr 42*$ft]"; # column length set Xframe 0; # initialize Frame Counter, used in output foreach Hcol $iHcol Lcol $iLcol{ set Xframe [expr $Xframe+1]; set ANALYSIS "Static"; source Analysis.tcl* }; # close iHcol & iLcol loop *NOTE: The file Analysis.tcl contains all the model and analysis commands. ...Run Moment-Curvature Analysis on Section A procedure for performing section analysis (only does moment-curvature, but can be easily modified to do any mode of section reponse): # ---------MPhiProc.tcl------------------------------------------------------------------------------- Chapter How To.... # Sets up a recorder which writes moment-curvature results to file # make sure the node and element numbers are not used elsewhere in the model # this procedure is set up for a 3-D problem: 3 dimensions/node, 6 dof/node # Arguments # secTag -- tag identifying section to be analyzed # axialLoad -- axial load applied to section (negative is compression) # maxK -- maximum curvature reached during analysis # numIncr -- number of increments used to reach maxK (default 100) proc MomentCurvature {secTag axialLoad maxK {numIncr 100} } { node 1001 0.0 0.0 0.0; # Define two nodes at (0,0) node 1002 0.0 0.0 0.0 fix 1001 1 1 1 1 1 1; # Fix all degrees of freedom except axial and bending fix 1002 0 1 1 1 1 0 element zeroLengthSection 2001 1001 1002 $secTag recorder Node Mphi.out disp -time -node 1002 -dof 6;# output moment & curvature integrator LoadControl 0 1 0 0; # Define analysis parameters system SparseGeneral -piv; # Overkill, but may need the pivoting! test NormUnbalance 1.0e-9 10 numberer Plain; constraints Plain; algorithm Newton; analysis Static; pattern Plain 3001 "Constant" { load 1002 $axialLoad 0.0 0.0 0.0 0.0 0.0 }; # Define constant axial load analyze 1; # Do one analysis for constant axial load pattern Plain 3002 "Linear" { load 1002 0.0 0.0 0.0 0.0 0.0 1.0 }; # Define reference moment set dK [expr $maxK/$numIncr]; # Compute curvature increment # Use displacement control at node 1002 for section analysis, dof 6 integrator DisplacementControl 1002 6 $dK 1 $dK $dK analyze $numIncr; } # Do the section analysis 386 Chapter How To.... 387 When including this procedure, ensure that the node and element numbers used by it are not used elsewhere in the OS model. The above procedure may be incorporated into the static pushover analysis file: # ----MomentCurvature.tcl--------------------------------------------------------wipe model basic -ndm 3 -ndf 6 source Units&Constants.tcl source MaterialParameters.tcl source ElementParameters.tcl source GravityParameters.tcl source materialsRC.tcl source RCcircSec.tcl RCcircSection $IDcolSec $riCol $roCol $cover $IDcore $IDcover $IDsteel $NbCol $AbCol $nfCoreR $nfCoreT $nfCoverR $nfCoverT source MPhiProc.tcl set phiYest [expr $epsY/(0.7*$Hcol)]; # estimate yield curvature set axialLoad -$Pdl; # define axial load -- +tension in Mom-curv analysis set maxK [expr 20*$phiYest]; # maximum curvature reached during analysis MomentCurvature $IDcolSec $axialLoad $maxK; ...Determine Natural Period & Frequency The natural period and frequency of the structure can be determined at any point during the analysis using the eigen (page 368) command. In turn, these quantities can be stored as variables and used in defining analysis parameters, such as rayleigh-damping parameters: # ------------PeriodFreq&Damping.tcl----------------------------------------------------------------# determine Natural Period, Frequency & damping parameters for SDOF set $xDamp 0.02; # damping ratio (0.02-0.05-typical) set lambda [eigen 1] set omega [expr pow($lambda,0.5)] set Tperiod [expr 2*$PI/$omega]; # period (sec.) puts $Tperiod set alphaM # stiffness-prop. RAYLEIGH damping parameter; D = alphaM*M 0; # stiffness proportional damping; set betaK 0; set betaKcomm [expr 2*$xDamp/$omega]; +beatK*KCurrent # mass-prop. RAYLEIGH damping parameter; Chapter How To.... +betaKcomm*KlastCommitt set betaKinit 0; # initial-stiffness proportional damping +beatKinit*Kini 388 389 CHAPTER 43 Getting Started with OpenSees Under the NEESgrid support, a Getting Started with OpenSees (http://peer.berkeley.edu/~silvia/OpenSees/gettingstarted/) document has been produced. Technical Report NEESgrid-TR 200421 www.neesgrid.org Getting Started with OpenSees Silvia Mazzoni, Frank McKenna, Gregory L. Fenves Pacific Earthquake Engineering Research Center University of California, Berkeley OpenSees version 1.6.0 Chapter 43 Getting Started with OpenSees 390 please send questions and comments about the manual to [email protected] The documentation and validation examples of the Getting Started with OpenSees Manual were supported by NEESgrid subcontract award 04-344 from the NEESgrid System Integrator (SI) under NSF grant CMS-0117853 In This Chapter Introduction .............................................................. 406 Download OpenSees ............................................... 407 Run OpenSees......................................................... 409 Problem Definition .................................................... 413 Model Builder ........................................................... 414 Nodes....................................................................... 415 Elements .................................................................. 417 Recorders................................................................. 418 Summary of Model-Building Input File ...................... 418 Loads and Analysis .................................................. 421 Gravity Loads ........................................................... 423 Summary of Gravity Loads ....................................... 427 Lateral Loads -- Static Pushover .............................. 427 Lateral Loads -- Cyclic Lateral Load ......................... 428 Lateral Loads -- Dynamic ground motion.................. 429 Chapter 43 Getting Started with OpenSees 391 Introduction Modern earthquake engineering utilizes modeling and simulation to understand the behavior and performance of systems during earthquakes. With the support of the National Science Foundation, the Pacific Earthquake Engineering Research Center (PEER) has developed the Open System for Earthquake Engineering Simulation, OpenSees for short, as a software platform for research and application of simulation for structural and geotechnical systems. The OpenSees software framework uses object-oriented methodologies to maximize modularity and extensibility for implementing models for behavior, solution methods, and data processing and communication procedures. The framework is a set of inter-related classes, such as domains (data structures), models, elements (which are hierarchical), solution algorithms, integrators, equation solvers, and databases. The classes are as independent as possible, which allows great flexibility in combining modules to solve simulation problems for buildings and bridges, including soil and soil-structure-foundation interaction, and most recently including reliability computational modules. The open source software is managed and made available to users and developers through the OpenSees website at http://opensees.berkeley.edu (http://opensees.berkeley.edu). The software architecture and open-source approach for OpenSees provide many benefits to users interested in advanced simulation of structural and geotechnical systems with realistic models of nonlinear behavior. First, the modeling approach is very flexible in that allows selection and various combinations of a number of different element formulations and material formulations, along with different approximations of kinematics to account for largedisplacements and P-D effects. As an open-source project, developers and researchers are using the extensible features of the software architecture to add additional capability. A second advantage is that there is a wide range of solution procedures and algorithms that the user can adapt to solve difficult nonlinear problems for static and dynamic loads. Another feature is that OpenSees has a fully programmable scripting language for defining models, solution procedures, and post-processing that can provide simple problem solving capability, as illustrated in this manual, or very sophisticated modeling and parameters studies of large, complex systems. Finally, OpenSees provides a flexible interface to computer resources, storage and databases, and network communication to take advantage of high-end computing systems. Structural and geotechnical models can be analyzed from desktop PC’s to parallel computers within OpenSees. As an advanced platform for computational simulation, OpenSees provides an important resource for the National Science Foundation-sponsored George E. Brown, Jr. Network for Earthquake Engineering Simulation (NEES (http://www.nees.org)), and it has been adopted by NEESgrid (http://www.neesgrid.org) System Integration project as the NEES simulation component. The NEESgrid decision to utilize OpenSees and adapt it to interface with other NEESgrid resources provides an important capability for NEES researchers and users. The modular design of OpenSees means that it can be customized for the integrating physical and computation simulation through data repositories, visualization, and hybrid control for advanced experimental methods, all of which meet important NEES objectives. Chapter 43 Getting Started with OpenSees 392 Open source software, such as OpenSees, requires community support and participation. The objective of this “Getting Started” manual is to provide an introduction so that users of OpenSees can obtain, install, and begin using the software to solve problems. Download OpenSees To download and install OpenSees the user is required to download both the OpenSees and Tcl/Tk packages. The OpenSees and Tcl/Tk packages can both be downloaded from the OpenSees binaries webpage http://opensees.berkeley.edu/binaries.html. This page can be found using the quick links pull down menu from any of the OpenSees web pages. The binaries download page will be similar to that shown below. Chapter 43 Getting Started with OpenSees 393 At this page the user is required to download two files OpenSees.X.X.X.exe and tcl/tk Y.Y.Y. This can be done by selecting (clicking) on the links located in the box labeled DOWNLOAD Windows Binaries. The file downloaded by clicking on the OpenSees.X.X.X.exe link is a zip file, from which the OpenSees.X.X.X.exe file can be extracted using your favorite extractor. The user can place this executable anywhere they wish. It is recommended to place it in a directory near where all your scripts will be stored. The file downloaded by clicking the tcl/tk Y.Y.Y link is an installer for Tcl/Tk. By clicking on the file the user is brought through a series of screens. The first screen shows the package information for Y.Y.Y, the second screen has the license agreement, you must accept the license to proceed with the installation. The third screen specifies the installation mode and location. Here the user must change the default installation location from C:/Tcl to C:/Program Files/Tcl (there is a space between Program and Files), as shown in the image below. Chapter 43 Getting Started with OpenSees 394 This is the only change the user must make. User now just keeps selecting Next until done. Run OpenSees There are three ways that OpenSees/Tcl commands can be executed: Interactive Commands can be input directly at the propt, as shown in the figure (Win32 version): Chapter 43 Figure 66: Run OpenSees -Interactive Getting Started with OpenSees 395 Chapter 43 Getting Started with OpenSees 396 Execute Input File at OpenSees prompt This method is the most-commonly used one. An external file containing the input commands can be generated a-priori (inputFile.tcl) and be executed at the OpenSees prompt by using the source command. The generation of the input script files is presented in this chapter. The file execution is shown in the figure (Win32 version): Figure 67: Run OpenSees -- Source File Batch Mode The previously-created input file containing the Tcl script commands necessary to execute the analsis can also be executed at the MS-DOS/Unix prompt, as shown in the figure (Win32 version): Chapter 43 Figure 68: Run OpenSees -- Batch Mode Getting Started with OpenSees 397 Chapter 43 Getting Started with OpenSees 398 Problem Definition A portal frame will be used to demonstrate the OpenSees commands. A structural model will be defined first. Subsequently, a number of static and dynamic analyses will be defined and implemented. The structural model consists of the planar portal frame shown in the figure below: Figure 69: Getting Started -- Problem Definition, Geometry The columns and beam will be modeled as elastic elements. At a more advanced level, these elements can be replaced by more refined element models. In the analysis phase, the frame will be subjected to three different load cases: 1 DISPLACEMENT-CONTROLLED LATERAL PUSHOVER; 2 DISPLACEMENT-CONTROLLED REVERSED CYCLIC LATERAL LOADING; 3 DYNAMIC GROUND-MOTION-INPUT TRANSIENT ANALYSIS. Chapter 43 Getting Started with OpenSees 399 In all cases, however, the frame will be subjected to constant static gravity loads: Model Builder Defining the model builder expands the Tcl command library to include OpenSees-specific commands, such as node and element definition, etc. Currently, there is only one model builder available, basic model builder (page 37), this is the model builder that includes all the commands presented in this library. The model builder also defines the number of dimensions (ndm) and degrees of freedom per node (ndf): model BasicBuilder -ndm $ndm <-ndf $ndf> For a 2-D problem, you really only need three degrees of freedom at each node, the two translations in the plane and the rotation about the plane's normal: model basic -ndm 2 -ndf 3 Chapter 43 Getting Started with OpenSees 400 Nodes At this point the user needs to decide which units will be used. In this demonstration, inches and kips will be used for length and force. Seconds will be used for time. The assignment of node and element numbers is defined in the figure below: Figure 70: Getting Started -- Nodes & Elements In a 2D problem only the x and y coordinates need to be defined, using the node (page 39) command: node $nodeTag (ndm $coords) <-mass (ndf $MassValues)> node 1 node 2 node 3 node 4 0 504 0 504 0 0 432 432 Chapter 43 Getting Started with OpenSees 401 The boundary conditions are defined using the fix (page 41) command: fix $nodeTag (ndf $ConstrValues) with three degrees of freedom per node are constrained: fix 1 1 fix 2 1 fix 3 0 fix 4 0 1 1 0 0 1 1 0 0 where a fixed constraint is defined with a 1, a free constraint is defined with a 0. Nodal masses are typically defined at the same time as the nodal coordinates. The nodal mass is used to calculate the eigenvalues and to perform the dynamic analysis. Only the nodal mass in the horizontal direction will be defined in this demonstration. Nodal masses can either be defined within the node (page 39) command, or the can be "appended" using the mass (page 40) command: mass $nodeTag (ndf $MassValues) mass 3 5.18 0. mass 4 5.18 0. 0. 0. The mass value was calculated by dividing the nodal weight (1/2 of the total super-structure weight) by the gravitational constant g (32 ft/sec): Chapter 43 Getting Started with OpenSees 402 Elements The elastic columns and beams are defined using the elastic beam column element (page 215). The characteristics of a 2-D elastic element depend on the material modulus and the section area and moment of inertia. Because the elements in this frame represent reinforced-concrete elements, the value of 4227 ksi for the elastic modulus of concrete will be used. The following values represent the area and moment of inertia of the columns and beams: COLUMNS BEAM Area Iz The transformation command defines how the element coordinates correlate to the global model coordinates. In a 2D problem, element orientation does not need to be considered, and can be the same for all elements. The linear transformation (page 302) will be used in this demonstration: geomTransf Linear $transfTag <-jntOffset $dXi $dYi $dXj $dYj> geomTransf Linear 1 The following commands define the two columns (element 1 and 2) and the beam (element 3): element elasticBeamColumn $eleTag $iNode $jNode $A $E $Iz $transfTag element elasticBeamColumn 1 element elasticBeamColumn 2 element elasticBeamColumn 3 1 2 3 3 4 4 3600 3600 5760 4227 4227 4227 1080000 1080000 4423680 1 1 1 Chapter 43 Getting Started with OpenSees 403 Recorders The recorder command is used to define the analysis output. The node recorder will be used to output the horizontal and vertical displacements at node 3 into a file named Node3.out: recorder Node <-file $fileName> <-time> <-node ($node1 $node2 ...)> <nodeRange $startNode $endNode> <-region $RegionTag> <-node all> -dof ($dof1 $dof2 ...) $respType recorder Node -file Node3.out -time -node 3 -dof 1 2 disp The element recorder will be used to output the element forces. Element forces for element 1 will be output into file Element1.out: recorder Element <-file $fileName> <-time> <-ele ($ele1 $ele2 ...)> <-eleRange $startEle $endEle> <-region $regTag> <-ele all> ($arg1 $arg2 ...) recorder Element -file Element1.out -time -ele 1 force Summary of Model-Building Input File The following is a compilation of all the commands necessary to build the model: model basic -ndm 2 -ndf 3 # nodal coordinates: node 1 0 0 node 2 504 0 node 3 0 432 node 4 504 432 # bondary conditions: fix 1 1 1 1 fix 2 1 1 1 fix 3 0 0 0 fix 4 0 0 0 # nodal masses: mass 3 5.18 0. 0. mass 4 5.18 0. 0. # transformation: Chapter 43 Getting Started with OpenSees geomTransf Linear 1 # connectivity: element elasticBeamColumn 1 1 3 3600 4227 1080000 1 element elasticBeamColumn 2 2 4 3600 4227 1080000 1 element elasticBeamColumn 3 3 4 5760 4227 4423680 1 # recorders recorder Node -file Node3.out -time -node 3 -dof 1 2 disp recorder Element -file Element1.out -time -ele 1 force The above commands can be input line by line at the OpenSees command prompt: 404 Chapter 43 Getting Started with OpenSees 405 Otherwise, they can be saved into an input file called example.tcl. This file can then be sourced in from the OpenSees command line: Chapter 43 Getting Started with OpenSees 406 Loads and Analysis In OpenSees applying loads is a three-step process: 1. You must first define the loads in a load pattern 2. You must then define the analysis and its features 3. The loads are then applied when you execute the analysis 1. Load definition Loads are defined using the pattern command. Three types of patterns are currently available: a. plain Pattern -- this pattern is used to define the following: i. nodal loads, such as gravity loads and lateral loads (or load-controlled nodal displacements) ii. single-point constraints, such as displacement control at a node (typically used for a constant displacement at a node) iii. element loads, such as distributed gravity loads along the element (this is a new option, which still needs documentation). b. UniformExcitation Pattern -- this type of pattern imposes a user-defined acceleration record to all fixed nodes, in a specified direction. c. MultipleSupport Pattern -- this type of pattern imposes a user-defined displacement record at specified nodes, in a specified direction, or a ground-motion record. Chapter 43 Getting Started with OpenSees 407 2. Analysis definition and features The analysis-definition part of OpenSees allows the user to define the different linear and nonlinear analysis tools available. For each analysis, the following items need to be defined, preferably in this order: constraints The constraints command is used to construct the ConstraintHandler object. Constraints enforce a relationship between degrees-offreedom. The ConstraintHandler object determines how the constraint equations are enforced in the analysis. numberer The numberer command is used to construct the DOF_Numberer object. The DOF_Numberer object determines the mapping between equation numbers and degrees-of-freedom -- how degrees-offreedom are numbered. system The system command is used to construct the LinearSOE and LinearSolver objects to store and solve the system of equations in the analysis. test The test command is used to construct a ConvergenceTest object. Certain SolutionAlgorithm objects require a ConvergenceTest object to determine if convergence has been achieved at the end of an iteration step. algorithm The algorithm command is used to construct a SolutionAlgorithm object, which determines the sequence of steps taken to solve the non-linear equation. integrator The integrator (page 356) command is used to construct the Integrator object. The Integrator object determines the meaning of the terms in the system of equation object. The Integrator object is used for the following: determine the predictive step for time t+dt specify the tangent matrix and residual vector at any iteration determine the corrective step based on the displacement increment dU Chapter 43 analysis Getting Started with OpenSees 408 The analysis (page 363) command is used to construct the Analysis object. This analysis object is constructed with the component objects previously created by the analyst. All currently-available analysis objects employ incremental solution strategies. There are three types of analysis currently available: Static Analysis (page 363) Transient Analysis (page 364) VariableTransient Analysis (page 365) 3. Analysis execution The analysis is executed using the analyze (page 369) command. This command moves the analysis forward by the specified number of steps. Gravity Loads Gravity loads are independent of the type of lateral loading and here they are considered part of the structural model. NODAL FORCES & MOMENTS Because the beam is an elastic element, the vertical load distributed along the horizontal member can be represented by nodal forces and moments. The nodal forces are distributed equally to the two end nodes. The nodal bending moments are equal and opposite: The nodal force is equal to one half of the superstructure weight: Chapter 43 Getting Started with OpenSees 409 the distributed load is calculated by dividing the total load by the beam length: The bending moment is then calculated from the distributed load: LOAD PATTERN DEFINITION Like all loads in OpenSees, gravity loads require two steps. The first step defines the load into a load pattern (page 317), the second applies the load pattern and the associated gravity load. The plain pattern (page 317) command with a linear time series (page 313) is used in the load definition: pattern Plain $patternTag (TimeSeriesType arguments) { load $nodeTag (ndf $LoadValues) } pattern Plain 1 Linear { load 3 0.0 -2000 -168074 load 4 0.0 -2000 168074 } CREATE ANALYSIS The constraints (page 339) command is used to construct the ConstraintHandler object. Constraints enforce a relationship between degrees-of-freedom. The ConstraintHandler object determines how the constraint equations are enforced in the analysis. constraints Transformation Chapter 43 Getting Started with OpenSees 410 The numberer (page 344) command is used to construct the DOF_Numberer object. The DOF_Numberer object determines the mapping between equation numbers and degrees-offreedom -- how degrees-of-freedom are numbered. With the RCM numberer nodes are assigned degrees-of-freedom using the Reverse Cuthill-McKee algorithm numberer RCM The system (page 346) command is used to construct the LinearSOE and LinearSolver objects to store and solve the system of equations in the analysis. The BandGeneral command is used to construct an un-symmetric banded system of equations object which will be factored and solved during the analysis using the Lapack band general solver system BandGeneral The test (page 349) command is used to construct a ConvergenceTest object. Certain SolutionAlgorithm objects require a ConvergenceTest object to determine if convergence has been achieved at the end of an iteration step. The convergence test is applied to the following equation: The NormDispIncr test performs the following check: Norm Displacement Increment test NormDispIncr $tol $maxNumIter <$printFlag> test NormDispIncr 1.0e-6 6 The algorithm (page 352) command is used to construct a SolutionAlgorithm object, which determines the sequence of steps taken to solve the non-linear equation. algorithm Newton The integrator (page 356) command is used to construct the Integrator object. The Integrator object determines the meaning of the terms in the system of equation object. The Integrator object is used for the following: determine the predictive step for time t+dt specify the tangent matrix and residual vector at any iteration determine the corrective step based on the displacement increment dU The system of nonlinear equations is of the form: static analysis: Chapter 43 Getting Started with OpenSees 411 The type of integrator used in the analysis is dependent on whether it is a static analysis or transient analysis: static analysis LoadControl (page 358) DisplacementControl (page 358) MinUnbalDispNorm (page 359) ArcLength (page 360) integrator LoadControl $dLambda1 <$Jd $minLambda $maxLambda> integrator LoadControl 0.1 The analysis (page 363) command is used to construct the Analysis object (page 336). This analysis object is constructed with the component objects previously created by the analyst. All currently-available analysis objects employ incremental solution strategies. analysis Static The analyze (page 369) command is used to apply the gravity load (in 10 steps) and all loads that have been defined at this point, and the loadConst (page 374) command maintains the gravity load constant for the remainder of the the analyses and resets the current time to zero. analyze $numIncr <$dt> <$dtMin $dtMax $Jd> loadConst <-time $pseudoTime> analyze 10 loadConst -time 0.0 Summary of Gravity Loads The gravity loads can be placed into a file, called GravityLoads.tcl with the following commands: pattern Plain 1 Linear { load 3 0.0 -2000 -168074 Chapter 43 Getting Started with OpenSees 412 load 4 0.0 -2000 168074 } constraints Transformation numberer RCM system BandGeneral test NormDispIncr 1.0e-6 6 algorithm Newton integrator LoadControl 0.1 analysis Static analyze 10 loadConst -time 0.0 This file can be sourced in after the example.tcl file: source example.tcl source GravityLoads.tcl Lateral Loads -- Static Pushover The following commands assume that the example.tcl and the gravityloads.tcl files have been run. The wipe command should be used at the beginning of the input to clear any previous OpenSees-objects definition: wipe source example.tcl source gravityloads.tcl In this analysis, a lateral displacement of increasing amplitude is imposed at the free nodes (3 and 4). The imposed displacements are applied using a displacement-control integrator, where the load factors are scaled to reach the desired displacement (compared to an imposeddisplacement analysis). This method is the most efficient when you have a non-strengthdegrading system. The first step is to define the load pattern. To do so, we create a new load pattern (ID tag 2) for the lateral loads pattern Plain 2 Linear { load 3 100.0 0.0 0.0 load 4 100.0 0.0 0.0 } Most of the analysis features that were defined in the gravity analysis are still valid since this type of analysis is also static. The loads, however, are applied differently. While gravity was applied as a load, using the LoadControl integrator, the DisplacementControl integrator is used in this pushover: integrator DisplacementControl $nodeTag $dofTag $dU1 <$Jd $minDu $maxDu> The load is applied to node 3, in the direction of DOF 1, with a displacement increment of 0.1 integrator DisplacementControl 3 1 0.1 Chapter 43 Getting Started with OpenSees 413 A total of a 1-inch displacement needs to be applied, hence 10 steps are needed: analyze 10 Lateral Loads -- Cyclic Lateral Load The following commands assume that the example.tcl and the gravityloads.tcl files have been run. The wipe command should be used at the beginning of the input to clear any previous OpenSees-objects definition: wipe source example.tcl source gravityloads.tcl In this analysis, a lateral displacement cycle (positive and negative) of a prescribed amplitude is imposed at the free nodes (3 and 4). The imposed displacements are applied using a displacement-control integrator, where the load factors are scaled to reach the desired displacement (compared to an imposed-displacement analysis). This method is the most efficient when you have a non-strength-degrading system. The first step is to define the load pattern. To do so, we create a new load pattern (ID tag 3) for the lateral loads pattern Plain 3 Linear { load 3 100.0 0.0 0.0 load 4 100.0 0.0 0.0 } Most of the analysis features that were defined in the gravity analysis are still valid since this type of analysis is also static. The loads, however, are applied differently. While gravity was applied as a load, using the LoadControl integrator, the DisplacementControl integrator is used in this pushover. Similarly, while in the pushover analysis a single load increment was used, variable load increments are used to reverse the loading from positive to negative, and back to positive. The load is applied to node 3, in the direction of DOF 1, with a displacement increment of 1 for the first rise to amplitude 1, -2 for the reversal to amplitude -1, and again positive 1 for the reversal back to amplitude zero: integrator DisplacementControl 3 1 0.1 analyze 10 integrator DisplacementControl 3 1 -0.2 analyze 10 integrator DisplacementControl 3 1 0.1 analyze 10 this can be put into a foreach loop: foreach Dincr {0.1 -0.2 0.1} { integrator DisplacementControl 3 1 $Dincr analyze 10 Chapter 43 Getting Started with OpenSees 414 } Lateral Loads -- Dynamic ground motion The following commands assume that the example.tcl and the gravityloads.tcl files have been run. The wipe command should be used at the beginning of the input to clear any previous OpenSees-objects definition: wipe source example.tcl source gravityloads.tcl The dynamic ground-motion analysis is a transient, rather than static, type of analysis. Therefore, most of the analysis components should be redefined. First of all, the load pattern needs to be defined. The load pattern here consists of defining an acceleration time-history applied at the support nodes. The time-history is defined in a file named BM68elc.th, taken from the PEER strong-motion database. The first lines of this file are: PACIFIC ENGINEERING AND ANALYSIS STRONG-MOTION DATA BORREGO MOUNTAIN 04/09/68 0230, EL CENTRO ARRAY #9, 270 ACCELERATION TIME HISTORY IN UNITS OF G NPTS= 4000, DT= .01000 SEC -.1368849E-02 -.1659410E-02 -.1466880E-02 -.6865326E-03 -.6491235E-03 -.6172128E-03 -.5942289E-03 -.5720329E-03 -.5517003E-03 -.5367939E-03 -.5300330E-03 -.5315104E-03 -.5389920E-03 -.5492582E-03 -.5592027E-03 -.5659958E-03 -.5672101E-03 -.5617805E-03 -.5502959E-03 -.5347288E-03 -.5176619E-03 -.5013709E-03 -.4873454E-03 -.4763228E-03 -.4683559E-03 -.4626830E-03 -.4579708E-03 -.4512405E-03 -.4376077E-03 -.4130071E-03 -.3772566E-03 -.3363394E-03 -.3030926E-03 -.2926074E-03 -.3144186E-03 -.3668375E-03 -.4373818E-03 -.5104884E-03 -.5745380E-03 -.6248976E-03 -.6621411E-03 -.6878470E-03 -.7014600E-03 -.6985488E-03 -.6737667E-03 -.6258232E-03 -.5616336E-03 -.4955459E-03 -.4432164E-03 -.4144737E-03 -.4103149E-03 -.4251781E-03 -.4493034E-03 -.4742715E-03 -.4942019E-03 -.5021476E-03 -.4907415E-03 -.4549906E-03 -.4008473E-03 -.3457893E-03 -.3076823E-03 -.2975411E-03 -.3148410E-03 -.3469618E-03 -.3746677E-03 -.3814194E-03 -.3598713E-03 -.3127679E-03 -.2503611E-03 -.1870004E-03 -.1381307E-03 -.1166398E-03 -.1296596E-03 -.1730662E-03 -.2365307E-03 -.3092055E-03 -.3795541E-03 -.4410602E-03 -.4886626E-03 -.5170590E-03 -.5252823E-03 -.5139311E-03 -.4858098E-03 -.4459504E-03 -.3992677E-03 -.3528575E-03 -.3119584E-03 -.2767747E-03 -.2493392E-03 -.2343354E-03 -.2328132E-03 -.2443271E-03 -.2710242E-03 -.3142711E-03 -.3756243E-03 -.4585760E-03 -.5641321E-03 -.6901264E-03 -.8277058E-03 -.9398496E-03 -.9604577E-03 -.8289034E-03 -.5332316E-03 -.1380318E-03 .2342108E-03 .4651973E-03 .5026886E-03 .3746881E-03 .1623773E-03 -.5159882E-04 -.2224880E-03 -.3475301E-03 -.4529523E-03 -.5485990E-03 -.5963545E-03 -.5423656E-03 -.3657646E-03 -.1048402E-03 .1563278E-03 .3291475E-03 .3576328E-03 .2409380E-03 .2377646E-04 -.2316319E-03 -.4694656E-03 -.6625028E-03 -.8119362E-03 -.9298628E-03 -.1028468E-02 -.1108383E-02 -.1131224E-02 -.1020689E-02 -.7165112E-03 -.2327270E-03 .3181869E-03 .7732246E-03 .1012043E-02 .1013671E-02 .8560355E-03 .6690806E-03 .5646968E-03 .5919845E-03 .7377345E-03 .9490117E-03 .1157766E-02 .1288701E-02 .1282487E-02 .1122759E-02 .8318915E-03 .4619941E-03 Chapter 43 .7578448E-04 -.5752344E-03 -.2406227E-03 .7517190E-03 -.4397852E-03 -.1227062E-02 -.2595554E-03 -.4995743E-03 -.1628483E-03 .9316287E-03 -.9508948E-03 -.6359309E-03 -.4939353E-03 -.4264229E-03 -.3998632E-04 .8846478E-03 -.1343749E-02 .1519731E-03 -.6127490E-03 -.3601404E-03 .1649808E-03 .5857181E-03 -.1561289E-02 .9751270E-03 Getting Started with OpenSees 415 -.6275750E-03 -.3004083E-03 .4539122E-03 .1074495E-03 -.1537678E-02 .1659683E-02 A number of tcl scripts are available to the user at the openSees web site which have been written for specific tasks. The file ReadSMDFile.tcl is a script procedure which parses a ground motion record from the PEER strong motion database by finding dt in the record header, then echoing data values to the output file. This file should be saved in the same directory as the OpenSees executable: # -----------------------------------------------------------------------------------------------------------------------------------------# READSMDFILE.tcl # Written: MHS # Date: July 2000 # A procedure which parses a ground motion record from the PEER strong motion database by finding dt in the record header, then # echoing data values to the output file. Formal arguments # inFilename -- file which contains PEER strong motion record # outFilename -- file to be written in format G3 can read # dt -- time step determined from file header # Assumptions # The header in the PEER record is, e.g., formatted as follows: # PACIFIC ENGINEERING AND ANALYSIS STRONG-MOTION DATA # IMPERIAL VALLEY 10/15/79 2319, EL CENTRO ARRAY 6, 230 # ACCELERATION TIME HISTORY IN UNITS OF G # NPTS= 3930, DT= .00500 SEC proc ReadSMDFile {inFilename outFilename dt} { # Pass dt by reference upvar $dt DT # Open the input file and catch the error if it can't be read if [catch {open $inFilename r} inFileID] { puts stderr "Cannot open $inFilename for reading" } else { # Open output file for writing set outFileID [open $outFilename w] # Flag indicating dt is found and that ground motion # values should be read -- ASSUMES dt is on last line # of header!!! set flag 0 # Look at each line in the file foreach line [split [read $inFileID] \n] { if {[llength $line] == 0} { # Blank line --> do nothing continue } elseif {$flag == 1} { # Echo ground motion values to output file puts $outFileID $line } else { # Search header lines for dt foreach word [split $line] { # Read in the time step if {$flag == 1} { set DT $word break } # Find the desired token and set the flag Chapter 43 Getting Started with OpenSees 416 if {[string match $word "DT="] == 1} { set flag 1 } } } } # Close the output file close $outFileID # Close the input file close $inFileID } } # ------------------------------------------------------------------------------------------------------------------------------------------ Once this file has been created, it can be read-in by the input file and used in the analysis procedure, where an acceleration time-series (page 311) is defined and used in a UniformExcitation (page 321) load pattern: Series -dt $dt -filePath $fileName <-factor $cFactor> pattern UniformExcitation $patternTag $dir -accel (TimeSeriesType arguments) <-vel0 $ver0> # create load pattern source ReadSMDFile.tcl ReadSMDFile BM68elc.th BM68elc.acc dt set accelSeries "Series -dt $dt -filePath BM68elc.acc -factor 1"; pattern UniformExcitation 2 1 -accel $accelSeries The stiffness and mass-proportional damping factors can then be defined using the rayleigh (page 367) command: rayleigh $alphaM $betaK $betaKinit $betaKcomm There are three different stiffnesses the user can use, the current value, the initial value, or the stiffness at the last committed state (which is what is used here): # set damping factors rayleigh 0. 0. 0. [expr 2*0.02/pow([eigen 1],0.5)] The various analysis components are defined as follows: # create the analysis wipeAnalysis constraints Plain numberer RCM system UmfPack test NormDispIncr 1.0e-8 10 algorithm Newton integrator Newmark 0.5 0.25 analysis Transient Chapter 43 Getting Started with OpenSees 417 In a transient analysis, the analyze command also requires a time step. This time step does not have to be the same as the input ground motion. The number of time steps is equal to the total duration of the analysis (10 seconds) divided by the time step (0.02): analyze [expr 10/0.02] 0.02 418 CHAPTER 44 Getting Going with OpenSees (under development) This document will walk you through the following: define gravity loads 1 Define the recorders for output 2 Define the analysis components. Three types of analyses will be set up: a static displacement-controlled pushover analysis a displacement-controlled reversed cyclic analysis a dynamic ground-motion-input transient analysis In This Chapter Problem Definition.................................................... 434 Model Building ......................................................... 435 Recorders for Output ............................................... 454 Analysis Components .............................................. 455 Chapter 44 Getting Going with OpenSees (under development) 419 Problem Definition The structural system that will be used in this document is the SDOF cantilever column shown in the figure below. The column cross section consist of a reinforced-concrete fiber section with different material properties for the confined core and the unconfined cover. Chapter 44 Getting Going with OpenSees (under development) 420 Figure 71: Problem Definition -- Geometry This structure will be subjected to gravity loads, as well as three types of lateral load, applied in three different analyses. These loads are shown in the figure: Figure 72: Problem Definition -- Loads Model Building This chapter describes the main steps necessary to define the physical model of a structure. Variables and Units The authors highly recommend that the user define variables and use these variables for the input commands. Variables can be defined using the Tcl "set" command: set Radius 5; set Diameter [expr $Radius/2.]; # define radius of section # define section diameter Chapter 44 Getting Going with OpenSees (under development) 421 Therefore, it is recommended that the user define all material and geometric properties, as variables: set fc -5000; set fy 60000; Because OpenSees does not use internal units, the user must keep track of the types of units being used. For example, the user must define all length units in either inches or meters, etc., consistently throughout. The Tcl feature of being able to handle variables enables the user to define units as variables, and hence use them in building the model. Here is an example: First of all, the basic units need to be defined. The OpenSees output will be in these units: set in 1.; set sec 1.; set kip 1.; # define basic unit -- length # define basic unit -- time # define basic unit -- weight (or define force, but not both) The basic units must be independent of each other. Once they have been defined, additional units that are made up of these basic units can be defined: set ksi set psi set ft [expr $kip/pow($in,2)]; [expr $ksi/1000.]; [expr 12.*$in]; # define engineering units It is a good idea to define constants at the same time that units are defined: set g set PI set U set u [expr 32.2*$ft/pow($sec,2)]; [expr 2*asin(1.0)]; 1.e10; [expr 1/$U]; # gravitational acceleration # define constants # a really large number # a really small number Once the units have been defined, model variables can be defined in terms of these units and, hence, they don't all have to be defined in the basic units: set fc [expr -5500*$psi]; # CONCRETE Compressive Strength, ksi (+Tension, -Compression) set Ec [expr 57000.*sqrt(-$fc/$psi)];# Concrete Elastic Modulus set Fy [expr 68.*$ksi]; # STEEL yield stress set Es [expr 29000.*$ksi]; # modulus of steel set epsY [expr $Fy/$Es]; # steel yield strain It is also a good idea to use variables for IDtags of materials, sections, elements, etc. This is done to ensure that the same ID tag is not used when defining the input. Also, it makes it easier in writing the input to use a variable name that makes sense. Here is an example: # set up parameters for column section and element definition set IDcore 1; # ID tag for core concrete set IDcover 2; # ID tag for cover concrete Chapter 44 set IDsteel 3; Getting Going with OpenSees (under development) 422 # ID tag for steel For the example structural model, the following variables need to be defined: # define GEOMETRY variables set Hcol [expr 6.*$ft]; # column diameter set Lcol [expr 36*$ft]; # column length set GrhoCol 0.015; # column longitudinal-steel ratio set Weight [expr 3000.*$kip]; # superstructure weight set Rcol [expr $Hcol/2]; # COLUMN radius set Acol [expr $PI*pow($Rcol,2)]; # column cross-sectional area set cover [expr 6*$in]; # column cover width set G $U; # Torsional stiffness Modulus set J 1.; # Torsional stiffness of section set GJ [expr $G*$J]; # Torsional stiffness # define COLUMN REINFORCEMENT variables set NbCol 20; # number of column longitudinal-reinforcement bars set AsCol [expr $GrhoCol*$Acol]; # total steel area in column section set AbCol [expr $AsCol/$NbCol]; # bar area of column longitudinal reinforcement # define GRAVITY variables set Mass [expr $Weight/$g]; # mass of superstructure set Mnode [expr $Mass]; # nodal mass for each column joint # define DAMPING variables from $xDamp --SDOF system, use stiffness proportional damping only set xDamp 0.02; # modal damping ratio # ------ set analysis variables set DxPush [expr 0.1*$in]; # Displacement increment for pushover analysis set DmaxPush [expr 0.05*$Lcol]; # maximum displamcement for pushover analysis set gamma 0.5; # gamma value for newmark integration set beta 0.25; # beta value for newmark integration set DtAnalysis [expr 0.005*$sec]; # time-step Dt for lateral analysis set DtGround [expr 0.02*$sec]; # time-step Dt for input grond motion set TmaxGround [expr 50.*$sec]; # maximum duration of ground-motion analysis Model Builder For a 2-D problem, you really only need three degrees of freedom at each node, the two translations in the plane and the rotation about the plane's normal: model basic -ndm 2 -ndf 3 Chapter 44 Getting Going with OpenSees (under development) 423 Nodal Coordinates & Masses, Boundary Conditions NODAL COORDINATES AND NODAL MASSES Once the dimensions of the problem is defined, it recommended that the user define the coordinates of the nodes, the mass associated with each node and DOF and the boundary conditions at the nodes. The nodal coordinates are defined using the node (page 39) command. The number of parameters associated with this command are referenced to the model command. Nodal masses can be defined at the same time as the coordinates. In the two-dimensional problem considered here, only the x and y coordinates of each node need to be defined, and three mass parameters (two translation and one rotation of the plane) need to be defined: node 1 0. 0.; node 2 0. $Lcol direction, only # column base is located at the origin of the plane -mass $Mnode 0. 0.; # the column end has one translational mass in the x the nodal mass (page 40) can also be defined using the mass command: mass 2 $Mnode 0. 0.; # this command supersedes any previous mass definition at this node. BOUNDARY CONDITIONS The boundary conditions are defined using the fix (page 41) command. The tag 0 represents an unconstrained (free) degree of freedom, the tag 1 represents a constrained (fixed) DOF. For the structure under consideration, the column base is completely fixed (1-1-1) and the end is free (00-0). Three DOF's need to be defined here, the two translations and the rotation in the x-y plane: fix 1 1 1 1; fix 2 0 0 0; # fixed base # free end Chapter 44 Getting Going with OpenSees (under development) 424 Materials Once the nodes have been defined, the next step towards defining elements is the material (page 175, page 47) definition. This step may not be necessary when using elastic element (page 213) or sections (page 185), as the materials are defined with the element or section. There are two types of materials currently available in OpenSees, uniaxial materials (page 47) and nDmaterials (page 175). The different types of concrete and steel materials are among the uniaxial materials. There are three types of concrete available: 1 Concrete01 (page 141): uniaxial Kent-Scott-Park concrete material object with degraded linear unloading/reloading stiffness according to the work of Karsan-Jirsa and no tensile strength 2 Concrete02 (page 145): uniaxial concrete material object with tensile strength and linear tension softening 3 Concrete03 (page 152): uniaxial concrete material object with tensile strength and nonlinear tension softening. Concrete02 will be used for the structure under consideration, as the tensile strength of the concrete is of interest in the elastic range, and modeling nonlinear tension softening is not considered necessary for the purpose of the example. The cover and core concrete will be modeled as different materials, using the same material type, but different stress and strain characteristics and different material tags. Steel01 (page 157) will be used for the reinforcing steel. Because some material characteristics are dependent on others, it is recommended that the user define the material properties using variables. # Confined concrete: set fc [expr -5.5*$ksi]; # CONCRETE Compressive Strength, ksi (+Tension, -Compression) set Ec [expr 57*$ksi*sqrt(-$fc/$psi)]; # Concrete Elastic Modulus set fc1C [expr 1.26394*$fc]; # CONFINED concrete (mander model), maximum stress set eps1C [expr 2.*$fc1C/$Ec]; # strain at maximum stress set fc2C $fc; # ultimate stress set eps2C [expr 5*$eps1C]; # strain at ultimate stress # Unconfined concrete: set fc1U $fc; # UNCONFINED concrete (todeschini parabolic model), maximum stress set eps1U -0.003; # strain at maximum stress set fc2U [expr 0.1*$fc]; # ultimate stress set eps2U -0.006; # strain at ultimate stress # Concrete02 variables: set lambda 0.1 ; # ratio between unloading slope at $epscu and initial slope set ftC [expr -$fc1C/10.]; # tensile strength +tension set ftU [expr -$fc1U/10.]; # tensile strength +tension set Ets [expr $Ec/10.]; # tension softening stiffness # reinforcing steel set Fy [expr 68.*$ksi]; # STEEL yield stress set Es [expr 29000.*$ksi]; # modulus of steel set epsY [expr $Fy/$Es]; # steel yield strain Chapter 44 Getting Going with OpenSees (under development) 425 set Fu [expr 95.2*$ksi]; # ultimate stress of steel set epsU 0.1; # ultimate strain of steel set E2 [expr ($Fu-$Fy)/($epsU-$epsY)]; # post-yield tangent stiffness set Bs [expr $E2/$Es]; # post-yield stiffness ratio of steel To facilitate referencing the different material types, the user should set up material tags as variables: # set up parameters for column section and element definition set IDcore 1; # ID tag for core concrete set IDcover 2; # ID tag for cover concrete set IDsteel 3; # ID tag for steel The materials are defined using the uniaxialMaterial (page 47) command: uniaxialMaterial Concrete02 $IDcore $fc1C CONCRETE uniaxialMaterial Concrete02 $IDcover $fc1U COVER CONCRETE uniaxialMaterial Steel01 $IDsteel $Fy $Es $Bs; $eps1C $fc2C $eps2C $lambda $ftC $Ets; # CORE $eps1U $fc2U $eps2U $lambda $ftU $Ets; # # REINFORCING STEEL Element Cross Section Some element types require that the element cross section be defined a-priori, this is done using the section (page 185) command. The section is used to represent force-deformation (or resultant stress-strain) relationships at beam-column and plate sample points. While there are many types of sections available, the fiber (page 191) section will be used to define the cross section of the column in the structure under consideration. A fiber section has a general geometric configuration formed by subregions of simpler, regular shapes (e.g. quadrilateral, circular and triangular regions) called patches. In addition, individual or layers of reinforcement bars can be specified.The fiber section can be defined as a combination of the following: fiber (page 191) -- a single fiber can be defined, such as a single reinforcing bar. The coordinates, associated area and material tag are prescribed with the fiber. (The coordinates are given with respect to the plane of the cross section, a coordinate transformation is later defined in the input using the transformation command) patch (page 192) -- a patch defines an area that has a regular shape: quadrilateral (page 192) or circular (page 194). A different material can be associated with each patch. layer (page 197, page 196) -- a layer defines a layer of reinforcement that has a regular shape: straight (page 196) or circular. (page 197) A different material can be associated with each layer. The circular cross section of reinforced concrete will be defined using the patch and layer commands. First of all, it is important to define the variables: # Notes # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size Chapter 44 Getting Going with OpenSees (under development) 426 # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis set IDcolFlex 2; # ID tag for column section in flexure, before aggregating torsion set riCol 0.0; # inner radius of column section set roCol $Rcol; # outer radius of column section set nfCoreR 8; # number of radial fibers in core (number of "rings") set nfCoreT 16; # number of tangential fibers in core (number of "wedges") set nfCoverR 2; # number of radial fibers in cover set nfCoverT 16; # number of tangential fibers in cover # cover - cover thickness, has been defined with the geometry # IDcore - material tag for the core patch, has been defined with the materials # IDcover - material tag for the cover patches, has been defined with the materials # IDsteel - material tag for the reinforcing steel, has been defined with the materials # NbCol # number of column longitudinal-reinforcement bars, has been defined with the geometry # AbCol # bar area of column longitudinal reinforcement, has been defined with the geometry The fiber cross section is defined as follows: section fiberSec $IDcolFlex { set rc [expr $roCol-$cover]; # Core radius patch circ $IDcore $nfCoreT $nfCoreR 0 0 $riCol $rc 0 360; # Define the core patch patch circ $IDcover $nfCoverT $nfCoverR 0 0 $rc $roCol 0 360; # Define the cover patch set theta [expr 360.0/$NbCol]; # Determine angle increment between bars layer circ $IDsteel $NbCol $AbCol 0 0 $rc $theta 360; # Define the reinforcing layer } Elements and Element Connectivity Once the element cross section has been defined, additional mechanical properties must be associated (aggregated) to it. Elastic torsion needs to be added to the column under consideration, using an elastic (page 131) uniaxial material: set IDcolTors 10; # ID tag for column section in torsion set IDcolSec 1; # ID tag for column section uniaxialMaterial Elastic $IDcolTors $GJ; # Define torsional stiffness section Aggregator $IDcolSec $IDcolTors T -section $IDcolFlex; # attach torsion to flexure and create a new section IDtag The geometric transformation (page 302) is used to relate the local element, and section, coordinates to the global system coordinates (simple here for a 2-D problem): set IDcolTrans geomTransf Linear $IDcolTrans; 1; # ID tag for column transformation, defining element normal # Linear: no second-order effects The element, a nonlinearBeamColulmn (page 216) element, and its connectivity, are defined as follows: set np 5; element nonlinearBeamColumn 1 1 2 # Number of integration points $np $IDcolSec $IDcolTrans Chapter 44 Getting Going with OpenSees (under development) 427 Gravity and other Constant Loads Gravity loads are independent of the type of lateral loading and are considered part of the structural model. These loads are first defined: # apply constant gravity load (and other constant loads) set Pdl [expr $Weight]; # gravity axial load per column pattern Plain 1 Linear { load 2 0.0 -$Pdl 0.0 } The above defines the gravity load (ID=1) as a load in the negative y-direction at node 2 with a magnitude Pdl And then applied: # set up solution procedure system UmfPack; constraints Plain; transformation # set up convergence criteria test NormDispIncr 1.0e-5 10 0; algorithm Newton; stiffness at every iteration numberer RCM; (optimization) # set up load steppring integrator LoadControl 0.1 1 0.1 0.1; to converge, Dmax, Dmin # set up type of analysis, static for gravity analysis Static initialize # RUN GRAVITY ANALYSIS analyze 10 loadConst -time 0.0 # solution procedure, Super-LU, how it solves system of equations # how it handles boundary conditions, enforce constraints through the # tolerance, max no. of iterations, and print code , 1: every iteration # use Newton's solution algorithm: updates tangent # renumber dof's to minimize band-width # variable load-stepping: Do initial incr., desred no. of iterations Summary of Defining Structural Model # DEFINE UNITS set in 1.; # define basic unit -- length set sec 1.; # define basic unit -- time set kip 1.; # define basic unit -- weight (or define force, but not both) set ksi [expr $kip/pow($in,2)]; # define engineering units set psi [expr $ksi/1000.]; set ft [expr 12.*$in]; set g [expr 32.2*$ft/pow($sec,2)]; # gravitational acceleration set PI [expr 2*asin(1.0)]; # define constants set U 1.e10; # a really large number set u [expr 1/$U]; # a really small number # define GEOMETRY variables Chapter 44 Getting Going with OpenSees (under development) 428 set Hcol [expr 6.*$ft]; # column diameter set Lcol [expr 36*$ft]; # column length set GrhoCol 0.015; # column longitudinal-steel ratio set Weight [expr 3000.*$kip]; # superstructure weight set Rcol [expr $Hcol/2]; # COLUMN radius set Acol [expr $PI*pow($Rcol,2)]; # column cross-sectional area set cover [expr $Hcol/15]; # column cover width set G $U; # Torsional stiffness Modulus set J 1.; # Torsional stiffness of section set GJ [expr $G*$J]; # Torsional stiffness # define COLUMN REINFORCEMENT variables set NbCol 20; # number of column longitudinal-reinforcement bars set AsCol [expr $GrhoCol*$Acol]; # total steel area in column section set AbCol [expr $AsCol/$NbCol]; # bar area of column longitudinal reinforcement # define GRAVITY variables set Mass [expr $Weight/$g]; # mass of superstructure set Mnode [expr $Mass]; # nodal mass for each column joint # define DAMPING variables from $xDamp --SDOF system, use stiffness proportional damping only set xDamp 0.02; # modal damping ratio # ------ set analysis variables set DxPush [expr 0.1*$in]; # Displacement increment for pushover analysis set DmaxPush [expr 0.05*$Lcol]; # maximum displamcement for pushover analysis set gamma 0.5; # gamma value for newmark integration set beta 0.25; # beta value for newmark integration set DtAnalysis [expr 0.005*$sec]; # time-step Dt for lateral analysis set DtGround [expr 0.02*$sec]; # time-step Dt for input grond motion set TmaxGround [expr 50.*$sec]; # maximum duration of ground-motion analysis # define ModelBuilder model basic -ndm 2 -ndf 3; # basic: modelbuilder type, ndm= number of dimensions, ndf= #dof/node # Nodal Coordinates and Nodal Masses node 1 0. 0.; # column base is located at the origin of the plane node 2 0. $Lcol -mass $Mnode 0. 0.; # the column end has one translational mass in the x direction, only # Boundary Conditions fix 1 1 1 1; # fixed base fix 2 0 0 0; # free end # Confined concrete: set fc [expr -5.5*$ksi]; # CONCRETE Compressive Strength, ksi (+Tension, -Compression) set Ec [expr 57*$ksi*sqrt(-$fc/$psi)]; # Concrete Elastic Modulus set fc1C [expr 1.26394*$fc]; # CONFINED concrete (mander model), maximum stress set eps1C [expr 2.*$fc1C/$Ec]; # strain at maximum stress set fc2C $fc; # ultimate stress set eps2C [expr 5*$eps1C]; # strain at ultimate stress # Unconfined concrete: set fc1U $fc; # UNCONFINED concrete (todeschini parabolic model), maximum stress set eps1U -0.003; # strain at maximum stress set fc2U [expr 0.1*$fc]; # ultimate stress set eps2U -0.006; # strain at ultimate stress # Concrete02 variables: set lambda 0.1 ; # ratio between unloading slope at $epscu and initial slope set ftC [expr -$fc1C/10.]; # tensile strength +tension set ftU [expr -$fc1U/10.]; # tensile strength +tension set Ets [expr $Ec/10.]; # tension softening stiffness # reinforcing steel set Fy [expr 68.*$ksi]; # STEEL yield stress set Es [expr 29000.*$ksi]; # modulus of steel set epsY [expr $Fy/$Es]; # steel yield strain set Fu [expr 95.2*$ksi]; # ultimate stress of steel set epsU 0.1; # ultimate strain of steel Chapter 44 Getting Going with OpenSees (under development) 429 set E2 [expr ($Fu-$Fy)/($epsU-$epsY)]; # post-yield tangent stiffness set Bs [expr $E2/$Es]; # post-yield stiffness ratio of steel # set up parameters for column section and element definition set IDcore 1; # ID tag for core concrete set IDcover 2; # ID tag for cover concrete set IDsteel 3; # ID tag for steel uniaxialMaterial Concrete02 $IDcore $fc1C $eps1C $fc2C $eps2C $lambda $ftC $Ets; # CORE CONCRETE uniaxialMaterial Concrete02 $IDcover $fc1U $eps1U $fc2U $eps2U $lambda $ftU $Ets; # COVER CONCRETE uniaxialMaterial Steel01 $IDsteel $Fy $Es $Bs; # REINFORCING STEEL # element cross-section # Notes # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis set IDcolFlex 2; # ID tag for column section in flexure, before aggregating torsion set riCol 0.0; # inner radius of column section set roCol $Rcol; # outer radius of column section set nfCoreR 8; # number of radial fibers in core (number of "rings") set nfCoreT 16; # number of tangential fibers in core (number of "wedges") set nfCoverR 2; # number of radial fibers in cover set nfCoverT 16; # number of tangential fibers in cover # cover - cover thickness, has been defined with the geometry # IDcore - material tag for the core patch, has been defined with the materials # IDcover - material tag for the cover patches, has been defined with the materials # IDsteel - material tag for the reinforcing steel, has been defined with the materials # NbCol # number of column longitudinal-reinforcement bars, has been defined with the geometry # AbCol # bar area of column longitudinal reinforcement, has been defined with the geometry section fiberSec $IDcolFlex { set rc [expr $roCol-$cover]; # Core radius patch circ $IDcore $nfCoreT $nfCoreR 0 0 $riCol $rc 0 360; # Define the core patch patch circ $IDcover $nfCoverT $nfCoverR 0 0 $rc $roCol 0 360; # Define the cover patch set theta [expr 360.0/$NbCol]; # Determine angle increment between bars layer circ $IDsteel $NbCol $AbCol 0 0 $rc $theta 360; # Define the reinforcing layer } # element connectivity set IDcolTors 10; # ID tag for column section in torsion set IDcolSec 1; # ID tag for column section uniaxialMaterial Elastic $IDcolTors $GJ; # Define torsional stiffness section Aggregator $IDcolSec $IDcolTors T -section $IDcolFlex; # attach torsion to flexure and create a new section IDtag set IDcolTrans 1; # ID tag for column transformation, defining element normal geomTransf Linear $IDcolTrans; # Linear: no second-order effects set np 5; # Number of integration points element nonlinearBeamColumn 1 1 2 $np $IDcolSec $IDcolTrans # apply constant gravity load (and other constant loads) set Pdl [expr $Weight]; # gravity axial load per column pattern Plain 1 Linear { load 2 0.0 -$Pdl 0.0 } # set up solution procedure system UmfPack; # solution procedure, Super-LU, how it solves system of equations constraints Plain; # how it handles boundary conditions, enforce constraints through the transformation # set up convergence criteria test NormDispIncr 1.0e-5 10 0; # tolerance, max no. of iterations, and print code , 1: every iteration Chapter 44 Getting Going with OpenSees (under development) algorithm Newton; stiffness at every iteration numberer RCM; (optimization) # set up load steppring integrator LoadControl 0.1 1 0.1 0.1; # set up type of analysis, static for gravity analysis Static initialize # RUN GRAVITY ANALYSIS analyze 10 loadConst -time 0.0 430 # use Newton's solution algorithm: updates tangent # renumber dof's to minimize band-width # variable load-stepping # print to screen that you are done with this step: puts "Done with Structural Model Generation" Please note the last command, it communicates to the user that all commands preceding it have been executed. The above commands can be submitted to OpenSees one-by-one, or they can be saved into a file, say input1.tcl. Once the input file has been saved, it can be executed at the OpenSees command prompt: Chapter 44 Getting Going with OpenSees (under development) 431 Of course, you will likely not get the clean response I got above. Say you make a mistake. Likely it is a simple mistake it and you go and fix it (most errors that we ALL commit are simple typing errors). To check it, you need to source the input file once more: Chapter 44 Getting Going with OpenSees (under development) 432 OpenSees does not allow you to define objects with the same IDtag more than once. To solve this problem, without exiting and re-entering OpenSees, you can use the wipe (page 373) command: Once you get confirmation that all the input commands have been executed correctly, you are ready to move on to the next step. Type exit, press enter and go on. Chapter 44 Getting Going with OpenSees (under development) 433 Error-Checking Tip for Model Building To enable error-checking, it is good practice to place markers within the input file. Markers are simple statements to be output to the screen using the puts (http://docsrv.sco.com:507/en/man/html.TCL/puts.TCL.html) tcl command. These markers tell the user what commands have been completed. This way, if an error occurs, the user is able to find the location of the error. Here is an example, where puts commands are placed here and there in the input1.tcl file: puts "Begin units defintion" # DEFINE UNITS set in 1.; # define basic unit -- length set sec 1.; # define basic unit -- time set kip 1.; # define basic unit -- weight (or define force, but not both) set ksi [expr $kip/pow($in,2)]; # define engineering units set psi [expr $ksi/1000.]; set ft [expr 12.*$in]; set g [expr 32.2*$ft/pow($sec,2)]; # gravitational acceleration set PI [expr 2*asin(1.0)]; # define constants set U 1.e10; # a really large number set u [expr 1/$U]; # a really small number puts "Units have been defined" # define GEOMETRY variables set Hcol [expr 6.*$ft]; # column diameter set Lcol [expr 36*$ft]; # column length set GrhoCol 0.015; # column longitudinal-steel ratio set Weight [expr 3000.*$kip]; # superstructure weight set Rcol [expr $Hcol/2]; # COLUMN radius set Acol [expr $PI*pow($Rcol,2)]; # column cross-sectional area set cover [expr $Hcol/15]; # column cover width set G $U; # Torsional stiffness Modulus set J 1.; # Torsional stiffness of section set GJ [expr $G*$J]; # Torsional stiffness # define COLUMN REINFORCEMENT variables set NbCol 20; # number of column longitudinal-reinforcement bars set AsCol [expr $GrhoCol*$Acol]; # total steel area in column section set AbCol [expr $AsCol/$NbCol]; # bar area of column longitudinal reinforcement # define GRAVITY variables set Mass [expr $Weight/$g]; # mass of superstructure set Mnode [expr $Mass]; # nodal mass for each column joint # define DAMPING variables from $xDamp --SDOF system, use stiffness proportional damping only set xDamp 0.02; # modal damping ratio # ------ set analysis variables set DxPush [expr 0.1*$in]; # Displacement increment for pushover analysis set DmaxPush [expr 0.05*$Lcol]; # maximum displamcement for pushover analysis set gamma 0.5; # gamma value for newmark integration set beta 0.25; # beta value for newmark integration set DtAnalysis [expr 0.005*$sec]; # time-step Dt for lateral analysis set DtGround [expr 0.02*$sec]; # time-step Dt for input grond motion set TmaxGround [expr 50.*$sec]; # maximum duration of ground-motion analysis puts "All inital variables have been defined" # define ModelBuilder model basic -ndm 2 -ndf 3; # basic: modelbuilder type, ndm= number of dimensions, ndf= #dof/node puts "the Model has been built" # Nodal Coordinates and Nodal Masses Chapter 44 Getting Going with OpenSees (under development) 434 node 1 0. 0.; # column base is located at the origin of the plane node 2 0. $Lcol -mass $Mnode 0. 0.; # the column end has one translational mass in the x direction, only # Boundary Conditions fix 1 1 1 1; # fixed base fix 2 0 0 0; # free end puts "Nodal Coordinates, Nodal Masses, and Boundary Conditions have been defined" puts "Begin material definition" # Confined concrete: set fc [expr -5.5*$ksi]; # CONCRETE Compressive Strength, ksi (+Tension, -Compression) set Ec [expr 57*$ksi*sqrt(-$fc/$psi)]; # Concrete Elastic Modulus set fc1C [expr 1.26394*$fc]; # CONFINED concrete (mander model), maximum stress set eps1C [expr 2.*$fc1C/$Ec]; # strain at maximum stress set fc2C $fc; # ultimate stress set eps2C [expr 5*$eps1C]; # strain at ultimate stress # Unconfined concrete: set fc1U $fc; # UNCONFINED concrete (todeschini parabolic model), maximum stress set eps1U -0.003; # strain at maximum stress set fc2U [expr 0.1*$fc]; # ultimate stress set eps2U -0.006; # strain at ultimate stress # Concrete02 variables: set lambda 0.1 ; # ratio between unloading slope at $epscu and initial slope set ftC [expr -$fc1C/10.]; # tensile strength +tension set ftU [expr -$fc1U/10.]; # tensile strength +tension set Ets [expr $Ec/10.]; # tension softening stiffness # reinforcing steel set Fy [expr 68.*$ksi]; # STEEL yield stress set Es [expr 29000.*$ksi]; # modulus of steel set epsY [expr $Fy/$Es]; # steel yield strain set Fu [expr 95.2*$ksi]; # ultimate stress of steel set epsU 0.1; # ultimate strain of steel set E2 [expr ($Fu-$Fy)/($epsU-$epsY)]; # post-yield tangent stiffness set Bs [expr $E2/$Es]; # post-yield stiffness ratio of steel # set up parameters for column section and element definition set IDcore 1; # ID tag for core concrete set IDcover 2; # ID tag for cover concrete set IDsteel 3; # ID tag for steel puts "All material variables have been defined" uniaxialMaterial Concrete02 $IDcore $fc1C $eps1C $fc2C $eps2C $lambda $ftC $Ets; # CORE CONCRETE uniaxialMaterial Concrete02 $IDcover $fc1U $eps1U $fc2U $eps2U $lambda $ftU $Ets; # COVER CONCRETE uniaxialMaterial Steel01 $IDsteel $Fy $Es $Bs; # REINFORCING STEEL puts "All materials have been defined" puts "Being element cross-section definition" # element cross-section # Notes # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis set IDcolFlex 2; # ID tag for column section in flexure, before aggregating torsion set riCol 0.0; # inner radius of column section set roCol $Rcol; # outer radius of column section set nfCoreR 8; # number of radial fibers in core (number of "rings") set nfCoreT 16; # number of tangential fibers in core (number of "wedges") set nfCoverR 2; # number of radial fibers in cover set nfCoverT 16; # number of tangential fibers in cover # cover - cover thickness, has been defined with the geometry # IDcore - material tag for the core patch, has been defined with the materials # IDcover - material tag for the cover patches, has been defined with the materials Chapter 44 Getting Going with OpenSees (under development) 435 # IDsteel - material tag for the reinforcing steel, has been defined with the materials # NbCol # number of column longitudinal-reinforcement bars, has been defined with the geometry # AbCol # bar area of column longitudinal reinforcement, has been defined with the geometry puts "All element-cross-section variables have been defined" puts "Begin section definition" section fiberSec $IDcolFlex { set rc [expr $roCol-$cover]; # Core radius patch circ $IDcore $nfCoreT $nfCoreR 0 0 $riCol $rc 0 360; # Define the core patch patch circ $IDcover $nfCoverT $nfCoverR 0 0 $rc $roCol 0 360; # Define the cover patch puts "All patches have been defined" set theta [expr 360.0/$NbCol]; # Determine angle increment between bars layer circ $IDsteel $NbCol $AbCol 0 0 $rc $theta 360; # Define the reinforcing layer puts "All layers have been defined" } puts "End of element-cross-section definition" puts "Begin Element Connectivity" # element connectivity set IDcolTors 10; # ID tag for column section in torsion set IDcolSec 1; # ID tag for column section uniaxialMaterial Elastic $IDcolTors $GJ; # Define torsional stiffness section Aggregator $IDcolSec $IDcolTors T -section $IDcolFlex; # attach torsion to flexure and create a new section IDtag puts "All has been aggregated" set IDcolTrans 1; # ID tag for column transformation, defining element normal geomTransf Linear $IDcolTrans; # Linear: no second-order effects set np 5; # Number of integration points element nonlinearBeamColumn 1 1 2 $np $IDcolSec $IDcolTrans puts "End Element defintion" puts "Begin Gravity loads" # apply constant gravity load (and other constant loads) set Pdl [expr $Weight]; # gravity axial load per column pattern Plain 1 Linear { load 2 0.0 -$Pdl 0.0 } puts "Gravity Load Pattern has been defined" # set up solution procedure system UmfPack; # solution procedure, Super-LU, how it solves system of equations constraints Plain; # how it handles boundary conditions, enforce constraints through the transformation # set up convergence criteria test NormDispIncr 1.0e-5 10 0; # tolerance, max no. of iterations, and print code , 1: every iteration algorithm Newton; # use Newton's solution algorithm: updates tangent stiffness at every iteration numberer RCM; # renumber dof's to minimize band-width (optimization) # set up load steppring integrator LoadControl 0.1 1 0.1 0.1; # variable load-stepping # set up type of analysis, static for gravity analysis Static puts "Static Analysis for Gravity and Constant loads has been defined" initialize # RUN GRAVITY ANALYSIS analyze 10 loadConst -time 0.0 puts "End of Gravity Analysis" # print to screen that you are done with this step: puts "Done with Structural Model Generation" Chapter 44 Getting Going with OpenSees (under development) The OpenSees control window will look like this: 436 Chapter 44 Getting Going with OpenSees (under development) 437 When there is an error, it can be localized: Based on the markers, the erroneous command is located within the section definition, between the two puts commands. puts "Begin section definition" section fiberSec $IDcolFlex { set rc [expr $roCol-$cover]; # Core radius patch circ $IDcore $nfCoreT $nfCoreR 0 0 $riCol $rc 0 360; # Define the core patch HELLO patch circ $IDcover $nfCoverT $nfCoverR 0 0 $rc $roCol 0 360; # Define the cover patch puts "All patches have been defined" set theta [expr 360.0/$NbCol]; # Determine angle increment between bars layer circ $IDsteel $NbCol $AbCol 0 0 $rc $theta 360; # Define the reinforcing layer puts "All layers have been defined" } Chapter 44 Getting Going with OpenSees (under development) 438 Recorders for Output The recorder commands are currently being modified and the documentation will be updated accordingly. So stay tuned..... The following, therefore, may not yet work well............. Recorders are used to monitor the state of the model at each analysis step. They can be placed at nodes, elements, sections, fibers.... First of all, it is convenient to place all output data in a subdirectory: file mkdir data set FileName0 "data/" The name $FileName0 will be appended to each filename so that the output be placed in the data subdirectory. The first recorder of interest on the cantilever column is a recorder to measure the horizontal and vertical displacements of the column end, node 2, as well as the rotation. The node (page 326) recorder is used: set FileName1 Node2DxDyQz; # it is very good practice to give the output file a descriptive name set $FileName $FileName0$FileName1 recorder Node $FileName.out disp -node 2 -dof 1 2 3; The next items of interest are the element forces and deformations, using the elemen (page 330)t recorder. For the nonlinearBeamColumn (page 216) element, you really only need the forces at the element ends: set FileName1 Fel1 set $FileName $FileName0$FileName1 recorder Element 1 -time -file $FileName.out localForce; # output local element forces for element 1 The element deformations at the cross-section level can be recorded at any integration point. For this example, the deformation at the first integration point (the base of the column) will be recorded: set FileName1 Del1sec1 set $FileName $FileName0$FileName1 recorder Element -file $FileName.out -ele 1 section 1 deformations Recorders can also be placed anywhere on a fiber section to measure fiber stresses and strains. When more than one material may occupy the location specified (such as a steel bar at the edge of the confined-concrete core), a preferred material can be specified. The location of the recorder is specified using the local coordinate system. If no fiber is located at that coordinate, a blank file will be output (very common error). set FileName1 SSel1sec1 set $FileName $FileName0$FileName1 recorder Element -file $FileName.out -ele 1 section 1fiber $Rcol 0. $IDcore stressStrain; 439 CHAPTER 45 Analysis Components 440 CHAPTER 46 Script Utilities Library A library of Tcl Procedures is included in this chapter. The user sould modify the procedures to meet the needs of the analysis. In This Chapter matTest.tcl................................................................ 456 RCcircSection.tcl ...................................................... 458 RCcircSectionFEDEAS.tcl........................................ 459 RCFrameDisplay.tcl ................................................. 461 MomentCurvature.tcl ................................................ 462 ReadSMDFile.tcl ...................................................... 463 RotSpring2D............................................................. 465 StFramePZLdisplay.tcl ............................................. 466 Wsection.tcl.............................................................. 467 RigidFrame3Ddisplay.tcl........................................... 468 Units&Constants.tcl .................................................. 469 MatlabOutput.tcl ....................................................... 469 genPlaneFrame.tcl ................................................... 470 matTest.tcl A script for testing uniaxial materials with a single DOF truss. This script requires an additional file (http://opensees.berkeley.edu/OpenSees/examples/pattern1.txt). # matTest.tcl: SDOF truss to test uniaxial material models # Units: kip, in # MHS, Sept 1999 # email: [email protected] model BasicBuilder -ndm 1 -ndf 1 # Define nodes node 1 0.0 node 2 1.0 # Fix node 1 fix 1 1 # Define uniaxialMaterial Chapter 46 # Script Utilities Library tag f'c epsc f'cu epscu uniaxialMaterial Concrete01 1 -5.0 -0.002 -1.0 -0.004 # Define truss element with unit area # tag ndI ndJ A matTag element truss 1 1 2 1.0 1 set dt 1.0 set filename pattern1.txt ;# Increment between data points ;# Filename containing data points set factor 0.006 ;# Factor applied to data values # Read displacement pattern from file # Note, any pattern type can be used here: Linear, Path, Sine, etc. pattern Plain 1 "Series -dt $dt -filePath $filename -factor $factor" { # Set reference displacement value # node dof value sp 2 1 1.0 } # Impose monotonic displacements #pattern Plain 2 "Linear -factor $factor" { # sp 2 1 1.0 #} # Record nodal displacements (same as strains since truss length is 1.0) recorder Node truss.out disp -load -node 2 -dof 1 # Record truss force (same as stress since truss area is 1.0) recorder Element 1 -time -file force.out force system UmfPack constraints Penalty 1.0e12 1.0e12 # Set increment in load factor used for integration # Does not have to be the same as dt used to read in displacement pattern set dl $dt integrator LoadControl $dl 1 $dl $dl test NormDispIncr 1.0e-6 10 algorithm Newton numberer RCM analysis Static analyze 10000 441 Chapter 46 Script Utilities Library 442 RCcircSection.tcl # Define a procedure which generates a circular reinforced concrete section # with one layer of steel evenly distributed around the perimeter and a confined core. # Formal arguments # id - tag for the section that is generated by this procedure # ri - inner radius of the section # ro - overall (outer) radius of the section # cover - cover thickness # coreID - material tag for the core patch # coverID - material tag for the cover patches # steelID - material tag for the reinforcing steel # numBars - number of reinforcing bars around the section perimeter # barArea - cross-sectional area of each reinforcing bar # nfCoreR - number of radial divisions in the core (number of "rings") # nfCoreT - number of theta divisions in the core (number of "wedges") # nfCoverR - number of radial divisions in the cover # nfCoverT - number of theta divisions in the cover # # Notes # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis # proc RCcircSection {id ri ro cover coreID coverID steelID numBars barArea nfCoreR nfCoreT nfCoverR nfCoverT} { # Define the fiber section section fiberSec $id { # Core radius set rc [expr $ro-$cover] # Define the core patch patch circ $coreID $nfCoreT $nfCoreR 0 0 $ri $rc 0 360 # Define the cover patch Chapter 46 Script Utilities Library 443 patch circ $coverID $nfCoverT $nfCoverR 0 0 $rc $ro 0 360 if {$numBars <= 0} { return } # Determine angle increment between bars set theta [expr 360.0/$numBars] # Define the reinforcing layer layer circ $steelID $numBars $barArea 0 0 $rc $theta 360 } } RCcircSectionFEDEAS.tcl # Define a procedure which generates a circular reinforced concrete section # with one layer of steel evenly distributed around the perimeter and a confined core. # Writes section information in FEDEAS format to the TCL file stream fedeas # # Formal arguments # id - tag for the section that is generated by this procedure # ri - inner radius of the section # ro - overall (outer) radius of the section # cover - cover thickness # coreID - material tag for the core patch # coverID - material tag for the cover patches # steelID - material tag for the reinforcing steel # numBars - number of reinforcing bars around the section perimeter # barArea - cross-sectional area of each reinforcing bar # nfCoreR - number of radial divisions in the core (number of "rings") # nfCoreT - number of theta divisions in the core (number of "wedges") # nfCoverR - number of radial divisions in the cover # nfCoverT - number of theta divisions in the cover # fedeas - file stream to which FEDEAS information is written # # # # Notes Calling procedure should define a TCL file stream, e.g. set fedeas [open fedeas.out w] Chapter 46 # The center of the reinforcing bars are placed at the inner radius # The core concrete ends at the inner radius (same as reinforcing bars) # The reinforcing bars are all the same size # The center of the section is at (0,0) in the local axis system # Zero degrees is along section y-axis # Assumes G3 material tags and FEDEAS material tags are consistent Script Utilities Library 444 # proc RCcircSectionFEDEAS {id ri ro cover coreID coverID steelID numBars barArea nfCoreR nfCoreT nfCoverR nfCoverT fedeas} { # Define the fiber section section fiberSec $id { puts $fedeas fsection; # FEDEAS fsection command puts $fedeas $id; # fsection id puts $fedeas 2,0; # 2 patches, reference axis is geometric centroid # Core radius set rc [expr $ro-$cover] # Define the core patch patch circ $coreID $nfCoreT $nfCoreR 0 0 $ri $rc 0 360 puts $fedeas $coreID,2,$nfCoreR,$nfCoreT,1,1; # matID,circular,nfRad,nfAng,propIJ,propJK puts $fedeas 0,0; # (y,z) center of patch puts $fedeas $ri,$rc; # R1,R2 puts $fedeas 0,360; # theta1,theta2 puts $fedeas 0,0; # NOT USED # Define the cover patch patch circ $coverID $nfCoverT $nfCoverR 0 0 $rc $ro 0 360 puts $fedeas $coverID,2,$nfCoverR,$nfCoverT,1,1 puts $fedeas 0,0 puts $fedeas $rc,$ro puts $fedeas 0,360 puts $fedeas 0,0 if {$numBars <= 0} { puts $fedeas 0 puts $fedeas "" return } # Determine angle increment between bars Chapter 46 Script Utilities Library set theta [expr 360.0/$numBars] puts $fedeas 1; # Number of layers # Define the reinforcing layer layer circ $steelID $numBars $barArea 0 0 $rc $theta 360 puts $fedeas $steelID,2,$numBars,,$barArea; # matID,circular,numBars,<barSize>,barArea puts $fedeas 0,0; # (y,z) center of arc puts $fedeas $rc; # RL puts $fedeas $theta,360; # theta1,theta2 puts $fedeas "" } } RCFrameDisplay.tcl # a window showing the displaced shape recorder display DispShape 10 10 300 300 -wipe # next three commmands define viewing system, all values in global coords vrp 288.0 150.0 0 # point on the view plane in global coord, center of local viewing system vup 0 1 0 # dirn defining up direction of view plane vpn 0 0 1 # direction of outward normal to view plane # next three commands define view, all values in local coord system prp 0 0 100 # eye location in local coord sys defined by viewing system viewWindow -400 400 -400 400 # view bounds uMin, uMax, vMin, vMax in local coords plane 0 150 # distance to front and back clipping planes from eye projection 0 # projection mode port -1 1 -1 1 # area of window that will be drawn into fill 1 display 1 0 10 # fill mode 445 Chapter 46 Script Utilities Library 446 MomentCurvature.tcl # A procedure for performing section analysis (only does moment-curvature, but can be easily modified to do any mode of section reponse. # MHS # October 2000 # # Arguments # secTag -- tag identifying section to be analyzed # axialLoad -- axial load applied to section (negative is compression) # maxK -- maximum curvature reached during analysis # numIncr -- number of increments used to reach maxK (default 100) # # Sets up a recorder which writes moment-curvature results to file # section$secTag.out ... the moment is in column 1, and curvature in column 2 proc MomentCurvature {secTag axialLoad maxK {numIncr 100} } { # Define two nodes at (0,0) node 1 0.0 0.0 node 2 0.0 0.0 # Fix all degrees of freedom except axial and bending fix 1 1 1 1 fix 2 0 1 0 # Define element # tag ndI ndJ secTag element zeroLengthSection 1 1 2 $secTag # Create recorder recorder Node section$secTag.out disp -time -node 2 -dof 3 # Define constant axial load pattern Plain 1 "Constant" { load 2 $axialLoad 0.0 0.0 } Chapter 46 Script Utilities Library # Define analysis parameters integrator LoadControl 0 1 0 0 system SparseGeneral -piv; # Overkill, but may need the pivoting! test NormUnbalance 1.0e-9 10 numberer Plain constraints Plain algorithm Newton analysis Static # Do one analysis for constant axial load analyze 1 # Define reference moment pattern Plain 2 "Linear" { load 2 0.0 0.0 1.0 } # Compute curvature increment set dK [expr $maxK/$numIncr] # Use displacement control at node 2 for section analysis integrator DisplacementControl 2 3 $dK 1 $dK $dK # Do the section analysis analyze $numIncr } ReadSMDFile.tcl A procedure which converts a PEER strong motion database (http://peer.berkeley.edu/smcat/) file to OpenSees format # # -------READSDMFILE.TCL------------------------------------------------read gm input format # # Written: MHS # Date: July 2000 # 447 Chapter 46 # A procedure which parses a ground motion record from the PEER # strong motion database by finding dt in the record header, then # echoing data values to the output file. # # Formal arguments # inFilename -- file which contains PEER strong motion record # outFilename -- file to be written in format G3 can read # dt -- time step determined from file header # # Assumptions # The header in the PEER record is, e.g., formatted as follows: # PACIFIC ENGINEERING AND ANALYSIS STRONG-MOTION DATA # IMPERIAL VALLEY 10/15/79 2319, EL CENTRO ARRAY 6, 230 # ACCELERATION TIME HISTORY IN UNITS OF G # NPTS= 3930, DT= .00500 SEC proc ReadSMDFile {inFilename outFilename dt} { # Pass dt by reference upvar $dt DT # Open the input file and catch the error if it can't be read if [catch {open $inFilename r} inFileID] { puts stderr "Cannot open $inFilename for reading" } else { # Open output file for writing set outFileID [open $outFilename w] # Flag indicating dt is found and that ground motion # values should be read -- ASSUMES dt is on last line # of header!!! set flag 0 # Look at each line in the file foreach line [split [read $inFileID] \n] { if {[llength $line] == 0} { # Blank line --> do nothing continue } elseif {$flag == 1} { Script Utilities Library 448 Chapter 46 Script Utilities Library # Echo ground motion values to output file puts $outFileID $line } else { # Search header lines for dt foreach word [split $line] { # Read in the time step if {$flag == 1} { set DT $word break } # Find the desired token and set the flag if {[string match $word "DT="] == 1} { set flag 1 } } } } # Close the output file close $outFileID # Close the input file close $inFileID } } RotSpring2D # Procedure which creates a rotational spring for a planar problem # rotSpring2D.tcl # SETS A MULTIPOINT CONSTRAINT ON THE TRANSLATIONAL DEGREES OF FREEDOM, # SO DO NOT USE THIS PROCEDURE IF THERE ARE TRANSLATIONAL ZEROLENGTH # ELEMENTS ALSO BEING USED BETWEEN THESE TWO NODES # # Written: MHS # Date: Jan 2000 # # Formal arguments # eleID - unique element ID for this zero length rotational spring 449 Chapter 46 Script Utilities Library # nodeR - node ID which will be retained by the multi-point constraint # nodeC - node ID which will be constrained by the multi-point constraint # matID - material ID which represents the moment-rotation relationship # for the spring proc rotSpring2D {eleID nodeR nodeC matID} { # Create the zero length element element zeroLength $eleID $nodeR $nodeC -mat $matID -dir 6 # Constrain the translational DOF with a multi-point constraint # retained constrained DOF_1 DOF_2 ... DOF_n equalDOF $nodeR $nodeC 1 2 } StFramePZLdisplay.tcl # a window to plot the nodal displacements versus load if {$displayMode == "displayON"} { recorder plot StFramePZL1.out Node4Xdisp 10 400 300 300 -columns 3 1 # a window showing the displaced shape recorder display g3 10 10 300 300 -wipe prp 288.0 150.0 100.0 vrp 288.0 150.0 0 vup 0 1 0 vpn 0 0 1 viewWindow -400 400 -400 400 plane 0 150 port -1 1 -1 1 projection 0 fill 1 display 2 0 10 } Wsection.tcl Procedure for creating a wide flange steel fiber section # Wsection.tcl: tcl procedure for creating a wide flange steel fiber section # written: Remo M. de Souza # date: 06/99 # modified: 08/99 (according to the new general modelbuilder) 450 Chapter 46 # input parameters # secID - section ID number # matID - material ID number # d = nominal depth # tw = web thickness # bf = flange width # tf = flange thickness # nfdw = number of fibers along web depth # nftw = number of fibers along web thickness # nfbf = number of fibers along flange width # nftf = number of fibers along flange thickness proc Wsection { secID matID d tw bf tf nfdw nftw nfbf nftf} { set dw [expr $d - 2 * $tf] set y1 [expr -$d/2] set y2 [expr -$dw/2] set y3 [expr $dw/2] set y4 [expr $d/2] set z1 [expr -$bf/2] set z2 [expr -$tw/2] set z3 [expr $tw/2] set z4 [expr $bf/2] # section fiberSec $secID { # nfIJ nfJK yI zI yJ zJ yK zK yL zL patch quadr $matID $nfbf $nftf $y1 $z4 $y1 $z1 $y2 $z1 $y2 $z4 patch quadr $matID $nftw $nfdw $y2 $z3 $y2 $z2 $y3 $z2 $y3 $z3 patch quadr $matID $nfbf $nftf $y3 $z4 $y3 $z1 $y4 $z1 $y4 $z4 } } RigidFrame3Ddisplay.tcl # a window showing the displaced shape set displayType "PLAN" #set displayType "PERSPECTIVE" # a window showing the displaced shape Script Utilities Library 451 Chapter 46 recorder display g3 10 10 300 300 -wipe if {$displayType == "PERSPECTIVE"} { prp -7500 -5000 50000 vrp 0 -500 250 vup 0 0 1 vpn 0 -1 0 viewWindow -200 400 -300 300 } if {$displayType == "PLAN"} { prp 0 0 1000 vrp 0 0 0 vup 0 -1 0 vpn 0 0 -1 viewWindow -200 200 -200 200 } plane 0 1e12 port -1 1 -1 1 projection 1 fill 0 display 1 0 10 Units&Constants.tcl Procedure to define units and constants # ----Units&Constants.tcl--------------------------------------------------------set in 1.; # define basic units set sec 1.; set kip 1.; set ksi [expr $kip/pow($in,2)]; set psi [expr $ksi/1000.]; set ft [expr 12.*$in]; set lb [expr $kip/1000]; set pcf [expr $lb/pow($ft,3)]; set ksi [expr $kip/pow($in,2)]; # define dependent units Script Utilities Library 452 Chapter 46 Script Utilities Library set psi [expr $ksi/1000.]; # define metric units set cm [expr $in/2.54]; set meter [expr 100.*$cm]; set MPa [expr 145*$psi]; # define constants set PI [expr 2*asin(1.0)]; set g [expr 32.2*$ft/pow($sec,2)]; set U 1.e10; # a really large number set u [expr 1/$U]; # a really small number MatlabOutput.tcl # script to generate .m file to be read by matlab # -------------MatlabOutput.tcl------------------------------------------------------------------# this parameter would be passed in set Xframe 1; set fDir "Data/"; file mkdir $fDir; # create directory set outFileID [open $fDir/DataFrame$Xframe.m w]; # Open output file for writing puts $outFileID "Xframe($Xframe) = $Xframe;"; # frame ID puts $outFileID "Hcol($Xframe) = $Hcol;"; # column diameter puts $outFileID "Lcol($Xframe) = $Lcol;"; # column length puts $outFileID "Lbeam($Xframe) = $Lbeam;"; # beam length puts $outFileID "Hbeam($Xframe) = $Hbeam;"; # beam depth puts $outFileID "Bbeam($Xframe) = $Bbeam;"; # beam width puts $outFileID "Weight($Xframe) = $Weight;" ; # superstructure weight close $outFileID genPlaneFrame.tcl # Define a procedure which will generate nodes and elements for a plane frame having absolute column line locations in the list #'columnLine', absolute girder line locations in the list 'girderLine', # section IDs for the columns and girders, 'columnID' and 'girderID', and # 'nIntPt' integration points along every member. # # Notes: automatically fixes the base nodes # only geneartes nonlinearBeamColumn elements 453 Chapter 46 Script Utilities Library # allows columns and girders to be spaced arbitrarily # does not add nodal masses or loads, but can be extended to do so # starts node numbering at 1 454 # # Formal arguments # columnLine - a list of column line locations # The actual argument would be defined as so, set columns {0 120 240 360} # girderLine - a list of grider line locations # The actual argument would be defined as so, set girders {180 300 420} # columnID - an integer representing the section ID (tag) for the columns in the frame # girderID - an integer representing the section ID (tag) for the girders in the frame # nIntPt - an integer representing the number of integration points along each member in the frame proc genPlaneFrame {columnLine girderLine columnID girderID nIntPt} { set n 1; # Node number counter geomTransf Linear 1; # Geometric transformation for all elements # For each column line foreach xLoc $columnLine { node $n $xLoc 0 # Fix the base node fix $n 1 1 1 incr n 1 # For each girder line foreach yLoc $girderLine { node $n $xLoc $yLoc incr n 1 } } # Useful variables set numCol [llength $columnLine] set numGir [llength $girderLine] set e 1; # Element number counter # For each column line for {set i 1} {$i <= $numCol} {incr i 1} { Chapter 46 Script Utilities Library # Node number at the base of this column line set bottom [expr ($i-1)*$numGir + $i] # Node number at the top of this column line set top [expr $i*$numGir + $i] # Travel up this column line creating elements for {set j $bottom} {$j <= [expr $top-1]} {incr j 1} { element nonlinearBeamColumn $e $j [expr $j+1] $nIntPt $columnID 1 incr e 1 } } # Difference in node numbers I and J for any girder in the frame set delta [expr $numGir+1] # For each girder line for {set j 1} {$j <= $numGir} {incr j 1} { # Node number at the left end of this girder line set left [expr $j+1] # Node number at the right end of this girder line set right [expr ($numCol-1)*$numGir + $numCol + $j] # Travel across this girder line creating elements for {set k $left} {$k < $right} {incr k $delta} { element nonlinearBeamColumn $e $k [expr $k+$delta] $nIntPt $columnID 1 incr e 1 } } }; #end proc 455 456 CHAPTER 47 References API(1993). Recommended Practice for Planning, Design, and Constructing Fixed Offshore Platforms. API RP 2A - WSD, 20th ed., American Petroleum Institute. Boulanger, R. W. (2003). The PySimple1 Material. http://opensees.berkeley.edu (http://opensees.berkeley.edu/). Cook, R. D., D. S. Malkus, M. E. Plesha. "Concepts and Applications of Finite Element Analysis." John Wiley & Sons, 1989. Crisfield, M. A. "Non-linear Finite Element Analysis of Solids and Structures." John Wiley & Sons, vol. 1, 1991 Elgamal, A., Lai, T., Yang, Z. and He, L. (2001). "Dynamic Soil Properties, Seismic Downhole Arrays and Applications in Practice," State-of-the-art paper, Proc., 4th Intl. Conf. on Recent Advances in Geote. E.Q. Engrg. Soil Dyn. March 26-31, San Diego, CA, S. Prakash (Ed.). Elgamal, A., Yang, Z. and Parra, E. (2002). "Computational Modeling of Cyclic Mobility and Post-Liquefaction Site Response," Soil Dyn. Earthquake Engrg., 22(4), 259-271. Elgamal, A., Yang, Z., Parra, E. and Ragheb, A. (2003). "Modeling of Cyclic Mobility in Saturated Cohesionless Soils," Int. J. Plasticity, 19(6), 883-905. Georgiadis, M. (1983). “Development of p-y curves for layered soils.” Proc., Geotechnical Practice in Offshore Engineering, ASCE, pp. 536-545. Lowes, Laura N.; Mitra, Nilanjan; Altoontash, Arash A beam-column joint model for simulating the earthquake response of reinforced concrete frames PEER-2003/10 (http://nisee.berkeley.edu/elibrary/getdoc?id=1288878) Pacific Earthquake Engineering Research Center, University of California, Berkeley 2003 59 pages (400/P33/2003-10) Matlock, H. (1970). "Correlations of design of laterally loaded piles in soft clay."Proc. Offshore Technology Conference, Houston, TX, Vol 1, No.1204, pp. 577-594. McKenna, F. and Fenves, G. (2001). "The OpenSees Command Language Manual: version 1.2," Pacific Earthquake Engineering Center, Univ. of Calif., Berkeley. (http://opensees.berkeley.edu (http://opensees.berkeley.edu/)). Mosher, R. L., (1984) "Load Transfer Criteria for Numerical Analysis of Axially Loaded Piles in Sand," US Army Engineering Waterways Experimental Station, Automatic Data Processing Center, Vicksburg, Mississippi, January. Chapter 47 References 457 Parra, E. (1996). "Numerical Modeling of Liquefaction and Lateral Ground Deformation Including Cyclic Mobility and Dilation Response in Soil Systems," Ph.D. Thesis, Dept. of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY. Reese, L.C., and O'Neill, M. W., (1987) "Drilled Shafts: Construction Procedures and Design Methods," Report No. FHWA-HI-88-042, U.S. Department of Transportation, Federal Highway Administration, Office of Implementation, McLean, Virginia. Vijayvergiya, V.N. (1977) "Load-Movement Characteristics of Piles," Proceedings, Ports 77 Conference, American Society of Civil Engineers, Long Beach, CA, March. Yang, Z. (2000). "Numerical Modeling of Earthquake Site Response Including Dilation and Liquefaction," Ph.D. Thesis, Dept. of Civil Engineering and Engineering Mechanics, Columbia University, NY, New York. Yang, Z. and Elgamal, A. (2002). "Influence of Permeability on Liquefaction-Induced Shear Deformation," J. Engrg. Mech., ASCE, 128(7), 720-729. Yang, Z., Elgamal, A. and Parra, E. (2003). "A Computational Model for Liquefaction and Associated Shear Deformation," J. Geotechnical and Geoenvironmental Engineering, ASCE, December (in press). For the Force-Based Beam-Column Element: Neuenhofer, Ansgar, FC Filippou. Geometrically Nonlinear Flexibility-Based Frame Finite Element (http://ascelibrary.aip.org/vsearch/servlet/VerityServlet?KEY=JSENDH&smode=strresults&sort=r el&maxdisp=25&origquery=%28flexibility%29+&threshold=0&pjournals=IJGNAI%2CJAEEEZ%2 CJAEIED%2CJBENF2%2CJCRGEI%2CJCCOF2%2CJCCEE5%2CJCEMD4%2CJLEED9%2CJ ENMDT%2CJOEEDU%2CJGGEFK%2CJHEND8%2CJHYEFF%2CJITSE4%2CJIDEDH%2CJM ENEA%2CJMCEE7%2CJPCFEV%2CJPEPE3%2CJSENDH%2CJSUED2%2CJTPEDI%2CJUP DDM%2CJWRMD5%2CJWPED5%2CLMEEAZ%2CNHREFO%2CPPHMF8%2CPPSCFX%2C& pyears=&possible1=flexibility&possible1zone=article&possible3=Neuenhofer&possible3zone=au thor&bool3=and&OUTLOG=NO&viewabs=JSENDH&key=DISPLAY&docID=1&page=1&chapter =0&aqs=). ASCE Journal of Structural Engineering, Vol. 124, No. 6, June, 1998. ISSN 07339445/98/0006-0704-0711. Paper 16537. pp. 704-711. Chapter 47 References 458 Neuenhofer, Ansgar, FC Filippou. Evaluation of Nonlinear Frame Finite-Element Models (http://ascelibrary.aip.org/vsearch/servlet/VerityServlet?KEY=JSENDH&smode=strresults&sort=r el&maxdisp=25&origquery=%28flexibility%29+&threshold=0&pjournals=IJGNAI%2CJAEEEZ%2 CJAEIED%2CJBENF2%2CJCRGEI%2CJCCOF2%2CJCCEE5%2CJCEMD4%2CJLEED9%2CJ ENMDT%2CJOEEDU%2CJGGEFK%2CJHEND8%2CJHYEFF%2CJITSE4%2CJIDEDH%2CJM ENEA%2CJMCEE7%2CJPCFEV%2CJPEPE3%2CJSENDH%2CJSUED2%2CJTPEDI%2CJUP DDM%2CJWRMD5%2CJWPED5%2CLMEEAZ%2CNHREFO%2CPPHMF8%2CPPSCFX%2C& pyears=&possible1=flexibility&possible1zone=article&possible3=Neuenhofer&possible3zone=au thor&bool3=and&OUTLOG=NO&viewabs=JSENDH&key=DISPLAY&docID=2&page=1&chapter =0&aqs=). ASCE Journal of Structural Engineering, Vol. 123, No. 7, July, 1997. ISSN 07339445/97/0007-0958-0966. Paper No. 14157. pp. 958-966. Neuenhofer, Ansgar, FC Filippou. ERRATA -- Geometrically Nonlinear Flexibility-Based Frame Finite Element. ASCE Journal of Structural Engineering, Vol. 124, No. 6, June, 1998. ISSN 0733-9445/98/0006-0704-0711. Paper 16537. pp. 704-711. Taucer, Fabio F, E Spacone, FC Filippou. A Fiber Beam-Column Element for Seismic Response Analysis of Reinforced Concrete Structures. Report No. UCB/EERC-91/17 (http://nisee.berkeley.edu/elibrary/getdoc?id=237794). Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley. December 1991. Spacone, Enrico, V Ciampi, FC Filippou. A Beam Element for Seismic Damage Analysis. Report No. UCB/EERC-92/07. (http://nisee.berkeley.edu/elibrary/getdoc?id=241936) Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley. August 1992. BeamWithHinges Element: Scott, M.H. and G.L. Fenves. "Plastic Hinge Integration Methods for Force-Based Beam-Column Elements", Journal of Structural Engineering, ASCE, 132(2):244-252, February 2006. ReinforcingSteel Material: Chang, G. and Mander, J. (1994). “Seismic Energy Based Fatigue Damage Analysis of Bridge Columns: Part I – Evaluation of Seismic Capacity.” NCEER Technical Report 94-0006. Dodd, L. and Restrepo-Posada, J. (1995). “Model for Predicting Cyclic Behavior of Reinforcing Steel” J. Struct. Eng., 121(3), 433-445. Gomes, A., and Appleton, J. (1997). “Nonlinear Cyclic Stress-Strain Relationship of Reinforcing Bars Including Buckling.” Eng. Struct., 19(10), 822–826. Brown, J. and Kunnath, S.K. (2000). “Low Cycle Fatigue Behavior of Longitudinal Reinforcement in Reinforced Concrete Bridge Columns.” NCEER Technical Report 00-0007. Chapter 47 References 459 Dhakal, R. and Maekawa, K. (2002). “Modeling for Postyield Buckled of Reinforcement” J. Struct. Eng., 128(9), 1139-1147. Concrete04 Element: Mander, J. B., Priestley, M. J. N., and Park, R. (1988). "Theoretical stress-strain model for confined concrete." Journal of Structural Engineering ASCE, 114(8), 1804-1825. Popovics, S. (1973). " A numerical approach to the complete stress strain curve for concrete." Cement and concrete research, 3(5), 583-599. Karsan, I. D., and Jirsa, J. O. (1969). "Behavior of concrete under compressive loading." Journal of Structural Division ASCE, 95(ST12). Useful Resources: National Information Service for Earthquake Engineering -- NISEE. The Earthquake Engineering Online Archive (http://nisee.berkeley.edu/elibrary/) (full text reports, images and data). http://nisee.berkeley.edu/elibrary/ National Information Service for Earthquake Engineering -- NISEE. Earthquake Engineering Abstracts Database (http://nisee.berkeley.edu/eea.html): http://nisee.berkeley.edu/eea.html American Society of Civil Engineer -- ASCE. Online Journals Program (http://www.pubs.asce.org/journals/jrns.html). http://www.pubs.asce.org/journals/jrns.html 461 Index Arc-Length Control • 27, 345, 347, 412 Axial Limit Curve • 75 -- Contributed Uniaxial Materials • 44 . ...Build Model and Define Nodes • 34, 35, 36, 37, 39, 375 ...Build Model and Define Nodes using Variables • 376 ...Combine Input-File Components • 384 ...Define Analysis-Output Generation • 379 ...Define Data-Plot During Analysis • 380 ...Define Dynamic Ground-Motion Analysis • 383 ...Define Elements • 378 ...Define Gravity Loads • 380 ...Define Materials • 377 ...Define Static Pushover Analysis • 381 ...Define Tcl Procedure • 370 ...Define Units & Constants • 369 ...Define Variables and Parameters • 373 ...Determine Natural Period & Frequency • 388 ...Generate Matlab Commands • 370 ...Read External files • 372 ...Run Dynamic Ground-Motion Analysis • 384 ...Run Gravity Analysis • 381 ...Run Moment-Curvature Analysis on Section • 387 ...Run Parameter Study • 385 ...Run Static Pushover Analysis • 382 A Additional Tcl Resources • 17 algorithm Command • 24, 28, 324, 337, 340, 352, 353, 361, 411 analysis Command • 24, 29, 324, 346, 347, 351, 356, 361, 409, 412 Analysis Components • 440 Analysis Object • 31 Analysis Objects • 30, 324, 351, 412 analyze Command • 356, 409, 412 B BandGeneral SOE • 27, 335 BandSPD SOE • 27, 335 BARSLIP Material • 25, 44 Basic Model Builder • 12, 34, 35, 36, 37, 184, 186, 187, 189, 191, 316, 317, 322, 376, 400 Bbar Brick Element • 26, 170, 221, 288 Bbar Plane Strain Quadrilateral Element • 26, 170, 218, 286 Beam With Hinges Element • 26, 208, 209, 319, 321 Beam-Column Joint Element Discussion • 233 BeamColumnJoint Element • 26, 228 BFGS Algorithm • 28, 342 Bidirectional Section • 26, 196 block Command • 23, 29, 34, 285 block2D Command • 285, 286 block3D Command • 285, 288 Bond_SP01 - - Strain Penetration Model for Fully Anchored Steel Reinforcing Bars • 25, 46 Brick Elements • 219 BrickUP • 280 Broyden Algorithm • 28, 343 build Command • 35 Building The Model • 373 C CenterCol_basicModel.tcl • 78 CenterColAxialSpring.tcl • 82 CenterColSecFiber.tcl • 80 CenterColShearSpring.tcl • 80 Circular Layer Command • 182, 190, 426 Circular Patch Command • 187, 426 Concrete01 -- Material Behavior • 141 Concrete01 Material -- Zero Tensile Strength • 25, 137, 425 Concrete02 -- Material Behavior • 148 462 Index Concrete02 Material -- Linear Tension Softening • 25, 141, 425 Concrete03 -- Material Behavior • 153 Concrete03 Material -- Nonlinear Tension Softening • 25, 148, 425 Concrete04 -- Material Behavior • 61 Concrete04 Material -- Popovics Concrete Material • 25, 57 Constant Time Series • 301 constraints Command • 24, 28, 29, 34, 306, 324, 327, 352, 353, 361, 410 constraints objects • 23, 38 Copyright • 13 Corotational Transformation • 299 Corotational Truss Element • 26, 206 cyclic.txt • 84 CyclicLoading_Iso2spring.tcl • 201, 202 Element Recorder • 24, 43, 170, 179, 206, 207, 209, 212, 213, 214, 215, 216, 217, 219, 222, 224, 227, 228, 231, 319, 439 Elements • 403 Elements and Element Connectivity • 427 Energy Increment Test • 28, 339 Enhanced Strain Quadrilateral Element • 26, 170, 218, 286 EnvelopeElement Recorder • 24, 320 EnvelopeNode Recorder • 24, 317 EPState • 172, 176 equalDOF Command • 41 Error-Checking Tip for Model Building • 434 Evolution Law • 172, 175 Example Files for Limit State Material • 78 example ss_ic1.tcl • 52 Example Tcl Commands • 16 D F dataBase Commands • 24, 357 Defining Output • 379 Displacement Control • 27, 345, 346, 352, 412 Displacement-Based Beam-Column Element • 26, 208, 212, 309, 319, 321 display Command • 364 Display Recorder • 24, 322, 362, 364 Domain Object • 30, 306, 321, 360 Download OpenSees • 393 Drift Recorder • 24, 318 Dynamic Analysis • 383 Fatigue Material • 25, 61 FatigueMaterialExample.tcl • 63 Fiber Command • 182, 184, 426 Fiber Section • 26, 182 FileDatastore Command • 357 fix Command • 38, 376, 377, 402, 424 fixX Command • 39 fixY Command • 39 fixZ Command • 40 FluidSolidPorousMaterial • 275 FluidSolidPorousMaterial Material • 25, 177, 178 FourNodeQuadUP • 276 FourNodeQuadUP Element • 26, 227 E eigen Command • 24, 355, 388 Eight Node Brick Element • 26, 220, 222 Elastic Beam Column Element • 26, 207, 309, 319, 321, 403 Elastic Isotropic Material • 25, 170, 172 Elastic Material • 25, 127, 427 Elastic Membrane Plate Section • 26, 195 Elastic Section • 26, 180 Elastic-No Tension Material • 25, 131 Elastic-Perfectly Plastic Gap Material • 25, 130 Elastic-Perfectly Plastic Material • 25, 128 eleLoad Command • 306, 309 element Command • 23, 29, 34, 205, 319, 321, 361, 425 Element Cross Section • 426 G genPlaneFrame.tcl • 454 Geometric Transformation Command • 23, 29, 34, 207, 208, 212, 213, 292, 427 getTime Command • 361 Getting Going with OpenSees (under development) • 419 Getting Started with OpenSees • 390 Gravity and other Constant Loads • 428 Gravity Loads • 380, 409 groundMotion Command • 300, 301, 311, 312, 313 H Hardening Material • 25, 136 Index Hilbert-Hughes-Taylor Method • 27, 345, 349 How To.... • 365 Hysteretic -- Material Behavior • 169 Hysteretic Material • 25, 166 I imposedMotion Command • 311, 313, 328 integrator Command • 24, 27, 312, 344, 352, 353, 408, 411 Interpolated GroundMotion • 313 Introduction • 11, 392 Introduction to the Tcl command language • 14 Isolator2spring Section Model to include buckling behavior of an elastomeric bearing • 26, 197 463 Model Building • 421 model Command • 23, 34, 40, 41, 42, 298, 299, 307, 308, 313, 318, 346, 361, 375 ModelBuilder Object • 29, 30, 35, 38, 39, 207, 211 Model-Building Objects • 33 Modified Newton Algorithm • 28, 342 MomentCurvature.tcl • 447 MultipleSupport Pattern • 311, 313, 316, 328 Multi-Point Constraints • 41, 328 N Lagrange Multipliers • 28, 328, 330 Lateral Loads -- Cyclic Lateral Load • 414 Lateral Loads -- Dynamic ground motion • 415 Lateral Loads -- Static Pushover • 413 Limit State Material • 25, 73 Linear Algorithm • 28, 340 Linear Time Series • 301, 345, 410 Linear Transformation • 292, 298, 299, 403 load Command • 306, 307 Load Control • 27, 345, 352, 412 loadConst Command • 361, 412 Loads and Analysis • 407 nDMaterial Command • 23, 29, 34, 170, 177, 214, 216, 218, 219, 221, 224, 361, 425 Newmark Method • 27, 345, 348, 353 Newton Algorithm • 28, 340, 352, 353 Newton with Line Search Algorithm • 28, 341 Nine_Four_Node_QuadUP • 278 Nodal Coordinates & Masses, Boundary Conditions • 424 node Command • 12, 23, 29, 34, 36, 361, 376, 401, 402, 424 Node Recorder • 24, 228, 315, 355, 439 nodeBounds • 363 nodeDisp Command • 361 Nodes • 401 Nonlinear Beam Column Element • 26, 208, 309, 310, 319, 321, 427, 439 NonLinear Beam-Column Elements • 180, 208 Norm Displacement Increment Test • 28, 338 Norm Unbalance Test • 28, 337, 352, 353 Notation • 11 numberer Command • 24, 27, 324, 332, 352, 353, 411 M O mass Command • 23, 29, 34, 37, 376, 377, 402, 424 materialmodels.tcl • 56 Materials • 425 MatlabOutput.tcl • 454 matTest.tcl • 441 Minimum Unbalanced Displacement Norm • 27, 345, 347, 352, 412 Miscellaneous Commands • 359 Model Builder • 400, 423 OpenSees • 19 OpenSees Features • 21 OpenSees Interpreter • 18, 360 J J2 Plasticity Material • 25, 171 K Krylov-Newton Algorithm • 28, 342 L P Parallel Material • 25, 133 Path Time Series • 304 pattern Command • 24, 29, 34, 300, 301, 302, 303, 304, 306, 361, 410 P-Delta Transformation • 298 464 Index Penalty Method • 28, 328, 329 PINCHING4 Material • 25, 46, 84 PINCHING4 Uniaxial Material Model Discussion • 91 Plain Constraints • 28, 328, 329, 352, 353 Plain GroundMotion • 312 Plain Numberer • 27, 332 plain Pattern • 306, 410 Plane Stress Material • 25, 171 Plate Fiber Material • 25, 172, 195 Plate Fiber Section • 26, 195, 217 play Command • 362, 363 playback Command • 24, 323 Plot Recorder • 24, 323 Potential Surface • 172, 174 PR1.tcl • 232, 233, 236, 237 PressureDependMultiYield • 255 PressureDependMultiYield02 • 262 PressureIndependMultiYield • 267 print Command • 355, 359 Problem Definition • 399, 420 procMKPC.tcl • 232, 233, 236, 247 procRC.tcl • 232, 233, 236, 249 procRCycDAns.tcl • 90, 91, 93, 104 procUniaxialPinching.tcl • 90, 91, 93, 102, 232, 233, 236, 248 ProfileSPD SOE • 27, 335, 352, 353 PyLiq1 Material • 108 PySimple1 Material • 105 PySimple1Gen Command • 111 PyTzQz Uniaxial Materials • 25, 105 Q Quad Element • 26, 170, 216, 286 Quadrilateral Elements • 216 Quadrilateral Patch Command • 182, 185, 426 QzSimple1 Material • 107 R RandomStrainHistory1.tcl • 67 RandomStrainHistory2.tcl • 70 rayleigh command • 24, 354, 417 RCcircSection.tcl • 443 RCcircSectionFEDEAS.tcl • 444 RCFrameDisplay.tcl • 446 RCM Numberer • 27, 332, 333, 352, 353 RCyclicPinch.tcl • 90, 91, 93, 97 ReadSMDFile.tcl • 448 Recorder Object • 30, 31 Recorder Objects • 31, 315, 323 Recorders • 404 Recorders for Output • 439 Rectangular Time Series • 302 References • 72, 209, 457 region Command • 23, 290 Reinforcing Steel Material • 25, 113 ReinforcingSteel -- Material Behavior • 127 reset Command • 360 restore Command • 358 rigidDiaphragm Command • 41 RigidFrame3Ddisplay.tcl • 452 rigidLink Command • 42 RotSpring2D • 450 Run OpenSees • 366, 395 S save Command • 358 Script Utilities Library • 441 Section Aggregator • 26, 192 section Command • 23, 29, 34, 179, 192, 205, 206, 208, 209, 211, 213, 215, 217, 320, 321, 425, 426 Series Material • 25, 134 Shear Limit Curve • 74, 76 Shell Element • 26, 170, 217, 286 Sine Time Series • 303 Single-Point Constraints • 38, 328 Soil Models and Solid-Fluid Fully Coupled Elements • 254 sp Command • 306, 308, 328 SparseGeneral SOE • 27, 335 SparseSPD SOE • 27, 336 Standard Brick Element • 26, 170, 219, 288 Static Analysis • 29, 31, 324, 345, 351, 381, 409 Steel01 -- Material Behavior • 160 Steel01 Material • 25, 153, 425 Steel02 -- Material Behavior • 166 Steel02 Material -- Giuffré-Menegotto-Pinto Model with Isotropic Strain Hardening • 25, 161 StFramePZLdisplay.tcl • 451 Straight Layer Command • 182, 189, 426 Summary of Defining Structural Model • 428 Summary of Gravity Loads • 412 Summary of Model-Building Input File • 404 system Command • 24, 27, 334, 337, 338, 339, 352, 353, 361, 411 Index T Tags.tcl • 83 Tcl Commands Format • 15, 18 Template Elasto-Plastic Material • 25, 172 test Command • 24, 28, 337, 411 Time Series • 24, 29, 34, 300, 301, 302, 303, 304, 306, 310, 312, 417 Transformation Method • 28, 328, 331 Transient Analysis • 29, 31, 324, 345, 351, 352, 356, 409 Truss Element • 25, 205 Twenty Node Brick Element • 26, 224 Twenty_Eight_Node_BrickUP • 282 TzLiq1 Material • 109 TzSimple1 Material • 106 TzSimple1Gen Command • 112 U UmfPack SOE • 27, 336 Uniaxial Section • 26, 181 uniaxialMaterial Command • 23, 24, 29, 34, 43, 133, 134, 181, 182, 184, 185, 187, 189, 190, 192, 205, 206, 213, 214, 361, 425, 426 UniformExcitation Pattern • 310, 316, 417 Units&Constants.tcl • 372, 453 updateMaterialStage • 178, 273 updateParameter • 274 u-p-U element • 26, 226 V Variables and Units • 421 VariableTransient Analysis • 29, 31, 324, 351, 353, 356, 409 video Command • 362, 363 Viscous Material • 25, 169 W wipe Command • 360, 433 wipeAnalysis Command • 360 Wsection.tcl • 451 Y Yield Surface • 172, 173 Z Zero-Length Element • 26, 213 Zero-Length Elements • 213 Zero-Length ND Element • 26, 214 465 Zero-Length Section Element • 26, 215, 320, 322 ">
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Key features
Nonlinear structural & geotechnical simulation
Growing component library
Command-driven scripting interface
Educational tool for numerical modeling
Integration with laboratory testing
Customizable materials & elements
Frequently asked questions
Static push-over, static reversed-cyclic, and dynamic time-series analyses are supported.
Includes model-building, analysis capabilities, and recorder objects.
Single-point and multi-point constraints can be modeled via various commands.
