Strand7 Theoretical Manual Contents

 

Theor ore etic tical Manual Theorr etica Theo eticall back backggr ound to the Stra Strand7 fi f i nite ni te elem element ent ana analysis lysi s system

Edition 1 January 2005 Strand7 Release 2.3 © 2004-2005 Strand7 Pty Limited  All rights reserved  reserved  

 

Contents

Preface Chapter Outline Conventions and Notation  Chapter 1 Introduction .............................. ................................................................. ..................................................................... ..................................................................... .............................................1 ..........1  1.1 Governing Equations and Finite Element Approximation.......................................................................2  1.2 Minimum Potential and Virtual Work Principles  ................................. .................................................................... ........................................................4 .....................4  1.3 Element Field Functions ............................... ................................................................. ..................................................................... ....................................................................6 .................................6  1.4 Element Stiffness Matrix .............................. ................................................................ ..................................................................... ....................................................................7 .................................7 1.5 Element Mass Matrix ............................... .................................................................. ..................................................................... ..................................................................... .......................................8 ....8  1.6 Element Damping Matrix.............................. ................................................................ ..................................................................... ....................................................................9 .................................9  1.7 Element Nodal Load Vector .............................. ................................................................ ..................................................................... ............................................................1 .........................10 0  1.8 Assembly of Global Equations .............................. ................................................................. ...................................................................... ......................................................1 ...................12 2  1.9 Solution of Global Equations ................................. .................................................................... ...................................................................... ......................................................1 ...................14 4  1.10 Calculation of Element Results ................................ ................................................................... ..................................................................... ................................................1 ..............15 5  1.11 Convergence of Finite Element Solutions ............................. ................................................................ ..................................................................1 ...............................15  1.12 Units in Finite Element Data................................ ................................................................... ...................................................................... ......................................................1 ...................16 6  Chapter 2 Structural Materials................................. ................................................................... ..................................................................... ............................................................17 .........................17  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Generalized Hooke’s Law .................................. .................................................................... ..................................................................... ............................................................1 .........................17 7  Isotropic Material ................................ ................................................................... ..................................................................... ..................................................................... ...........................................18 ........18  Orthotropic Material ................................. .................................................................... ..................................................................... ..................................................................... .....................................23 ..23   Anisotropic Material ................................. .................................................................... ..................................................................... ..................................................................... .....................................27 ..27  Laminate Material ............................... .................................................................. ..................................................................... ..................................................................... ...........................................28 ........28  User-defined Material .............................. ................................................................. ..................................................................... ..................................................................... .....................................41 ..41  Material Temperature Dependence ................................ .................................................................. ..................................................................... ...........................................41 ........41  Material Time Dependence ............................... ................................................................. ..................................................................... ............................................................4 .........................45 5 

Chapter 3 Line Elements (Beams)   ................................................................. .................................................................................................... ......................................................47 ...................47  3.1 3.2 3.3 3.4

Spring-Damper Element ............................... ................................................................. ..................................................................... ..................................................................4 ...............................47  Cable Element ................................ .................................................................. ..................................................................... ..................................................................... ................................................5 ..............50 0  Truss Element ................................. ................................................................... ..................................................................... ..................................................................... ................................................5 ..............52 2  Cutoff Bar Element ................................... ...................................................................... ..................................................................... ..................................................................... .....................................55 ..55 

3.5 Point Contact Element .................................. .................................................................... ..................................................................... ..................................................................5 ...............................57 3.6 Beam Element ................................ .................................................................. ..................................................................... ..................................................................... ................................................6 ..............63 3 

 

 

Strand7 Theoretical retical Manual

3.7 User-defined Beam Element ................................... ..................................................................... ..................................................................... ..................................................... .................. 71  3.8 Pipe Element ................................... ..................................................................... ..................................................................... ...................................................................... ............................................... ............ 74  3.9 Connection Element ................................. .................................................................... ..................................................................... ..................................................................... ..................................... 78  3.10 Principal Axis System ................................. .................................................................... ..................................................................... ................................................................ ..............................80 80  3.11 Local Axis System of Beam Cross-Section ................................. .................................................................... ........................................................... ........................82 82  3.12 Offset ............................. ................................................................ ...................................................................... ..................................................................... ................................................................ ..............................83 83  3.13 End Release.................................. .................................................................... ..................................................................... ...................................................................... ............................................... ............ 85  3.14 Support ................................ ................................................................... ..................................................................... ..................................................................... ........................................................... ........................87 87   3.15 Distributed Load................................. ................................................................... ..................................................................... ..................................................................... ......................................... ....... 89  3.16 Concentrated Load ................................ ................................................................... ..................................................................... ..................................................................... ..................................... 90  3.17 Pre Tension ................................... ..................................................................... ..................................................................... ...................................................................... ............................................... ............ 91  3.18 Temperature Gradient ................................ ................................................................... ..................................................................... ................................................................ ..............................92 92  3.19 Heat Transfer ................................ .................................................................. ..................................................................... ...................................................................... ............................................... ............ 93  3.20 Element Results ................................. ................................................................... ..................................................................... ..................................................................... ......................................... ....... 95  Chapter 4 Surface Elements (Plates)  ............................... .................................................................. ...................................................................... ............................................... ............ 99  4.1 Shape Functions for 2D Isoparametric Elements .............................. ................................................................. ..................................................... .................. 99  4.2 Local Axis System ............................... ................................................................. ..................................................................... ..................................................................... ....................................... ..... 107  4.3 Plane Stress, Plane Strain and Axisymmetric Elements .................................. .................................................................... ..................................11 110 

 

..................................................................... .................................. ..................................................................... .................................................................... ..................................11 117

4.4 Shear Plate/Shell 4.5 PanelElement Element ............................... .................................................................. ..................................................................... .................................................................... ..................................12 126  4.6 3D Membrane Element ................................. .................................................................... ..................................................................... .............................................................. ............................13 131  4.7 Offset ................................ ................................................................... ...................................................................... ..................................................................... .............................................................. ............................13 132  4.8 Support .................................. ..................................................................... ..................................................................... ..................................................................... ......................................................... ......................13 134  4.9 Pre Stress ................................... ...................................................................... ..................................................................... ..................................................................... ................................................... ................ 136  4.10 Edge Load ................................ ................................................................... ..................................................................... ..................................................................... ................................................... ................ 137  4.11 Face Load ................................ ................................................................... ..................................................................... ..................................................................... ................................................... ................ 137  4.12 Edge Release ............................... ................................................................. ..................................................................... ...................................................................... ............................................. .......... 139  4.13 Temperature Gradient ................................ ................................................................... ..................................................................... .............................................................. ............................13 139  4.14 Heat Transfer ................................ .................................................................. ..................................................................... ...................................................................... ............................................. .......... 139  4.15 Analysis and Material Type Combinations .................................. ..................................................................... ......................................................... ......................14 142  4.16 Element Results ................................. ................................................................... ..................................................................... ..................................................................... ....................................... ..... 143  Chapter 5 Solid Elements (Bricks) ............................... ................................................................. ..................................................................... ................................................... ................ 145  5.1 Shape Functions for Brick Elements ............................... .................................................................. ..................................................................... ....................................... ..... 146  5.2 Local Axis System ............................... ................................................................. ..................................................................... ..................................................................... ....................................... ..... 157  5.3 Element Formulation ................................ ................................................................... ..................................................................... .................................................................... ..................................15 158  5.4 Integration Scheme .................................. ..................................................................... ..................................................................... .................................................................... ..................................15 159  5.5 Support .................................. ..................................................................... ..................................................................... ..................................................................... ......................................................... ......................16 161  5.6 Pre Stress ................................... ...................................................................... ..................................................................... ..................................................................... ................................................... ................ 161  5.7 Face Load ................................... ...................................................................... ..................................................................... ..................................................................... ................................................... ................ 161  5.8 Heat Transfer .................................. .................................................................... ..................................................................... ...................................................................... ............................................. .......... 162  5.9 Element Results ................................... ..................................................................... ..................................................................... ..................................................................... ....................................... ..... 163 

 

Contents

Chapter 6 Links.................................. .................................................................... ..................................................................... ...................................................................... ....................................................165 .................165  6.1 Master-Slave Link ............................... .................................................................. ..................................................................... ..................................................................... .........................................16 ......165  6.2 Sector Symmetry Link ................................... ..................................................................... ..................................................................... ................................................................1 .............................166  6.3 Coupling Link .................................. .................................................................... ..................................................................... ..................................................................... ..............................................1 ............168 68  6.4 Pinned Link ................................ .................................................................. ..................................................................... ...................................................................... ....................................................1 .................169 69  6.5 Rigid Link .............................. ................................................................. ..................................................................... ..................................................................... ..........................................................1 .......................170 70  6.6 Shrink Link ................................. ................................................................... ..................................................................... ...................................................................... ....................................................1 .................172 72   6.7 2-Point Link ................................ .................................................................. ..................................................................... ...................................................................... ....................................................1 .................173 73  6.8 Implementation of Links ............................... ................................................................. ..................................................................... ................................................................1 .............................173  6.9 Rigid Link Clusters ................................... ...................................................................... ..................................................................... ..................................................................... ...................................17 175 5  6.10 Heat Transfer ................................ .................................................................. ..................................................................... ..................................................................... ..............................................1 ............176 76  Chapter 7 The Solvers ............................... .................................................................. ..................................................................... ..................................................................... .........................................177 ......177  7.1 7.2 7.3 7.4 7.5

Linear Static Solver ..................................................................... ....................................................................................................... ..................................................................... ...................................17 179 9  Linear Buckling Solver .................................................................... ....................................................................................................... ................................................................1 .............................186  Nonlinear Static Solver ................................. ................................................................... ..................................................................... ................................................................1 .............................189  Natural Frequency Solver .................................. .................................................................... ..................................................................... ..........................................................1 .......................194 94  Harmonic Response Solver ................................................................ ................................................................................................... ..........................................................1 .......................197 97 

 

.................................................................. ............................... ..................................................................... ..............................................2 ............202 02

7.6 Linear Transient Dynamic Solver  7.7 Nonlinear Transient Dynamic Solver .............................. ................................................................ ..................................................................... .........................................20 ......205  7.8 Spectral Response Solver ................................. ................................................................... ..................................................................... ..........................................................2 .......................206 06  7.9 Steady State Heat Solver  .................................. .................................................................... ..................................................................... ..........................................................2 .......................212 12  7.10 Transient Heat Solver ................................................................... ...................................................................................................... ................................................................2 .............................214  Chapter 8 Structural Dynamics ............................... ................................................................. ..................................................................... ..........................................................219 .......................219  8.1 8.2 8.3 8.4 8.5 8.6 8.7

Dynamic Loading ................................ ................................................................... ..................................................................... ..................................................................... .........................................21 ......219  Mass................................. .................................................................... ..................................................................... ..................................................................... ................................................................2 .............................220  Damping ............................... .................................................................. ..................................................................... ..................................................................... ..........................................................2 .......................223 23  Base Excitation............................... ................................................................. ..................................................................... ..................................................................... ..............................................2 ............227 27  Time Domain Numerical Integration ............................... ................................................................. ..................................................................... .........................................22 ......229  Mode Superposition ................................. .................................................................... ..................................................................... ..................................................................... ...................................23 233 3  Mass Participation .............................. ................................................................. ..................................................................... ..................................................................... .........................................23 ......237

 

Chapter 9 Heat Transfer ..................................................................... ....................................................................................................... ..................................................................... ...................................239 239  9.1 Basic Theory ................................... ..................................................................... ..................................................................... ..................................................................... ..............................................2 ............239 39  9.2 Finite Element Formulation ............................... ................................................................. ..................................................................... ..........................................................2 .......................242 42  9.3 Using Heat Transfer Results for Structural Analysis .............................. ................................................................ ..............................................2 ............244 44  9.4 Other Applications of the Steady Heat Solver .............................. ................................................................. ..........................................................2 .......................245 45  .................................................................... ...................................................................... ....................................................247 .................247  Chapter 10 Material Nonlinearity .................................

10.1 Material Model Classification and General Description ................................. .................................................................... ...................................24 248 8  10.2 Elasto-Plastic Material ................................ .................................................................. ..................................................................... ................................................................2 .............................249  10.3 Nonlinear Elastic Material ............................... ................................................................. ..................................................................... ..........................................................2 .......................256 56 

 

10.4 Rubber MaterialSoil ................................................................... ................................ ..................................................................... ..................................................................... .........................................25 ......258 10.5 Duncan-Chang Material ............................... .................................................................. ...................................................................... ....................................................2 .................269 69 

 

 

Strand7 Theoretical retical Manual

Chapter 11 Geometric Nonlinearity .................................. ..................................................................... ...................................................................... ............................................. .......... 275  11.1 11.2 11.3 11.4 11.5

Basic Concepts and Stiffness Matrix Formulation ................................ ................................................................... ............................................. .......... 276  Formulation of Linear Buckling Problems ............................... ................................................................. .............................................................. ............................28 282  Stress and Strain Output ................................. ................................................................... ..................................................................... ......................................................... ......................28 283  Geometric Nonlinear Analysis vs Linear Buckling Analysis ............................... ........................................................... ............................28 286  Typical Examples of Geometric Nonlinear Analysis................................... ..................................................................... ....................................... ..... 289 

Chapter 12 Numerical Procedures............................... ................................................................. ..................................................................... ................................................... ................ 291

  12.1 Matrix Decomposition ................................. .................................................................... ..................................................................... .............................................................. ............................29 291  12.2 Eigenvalue Extraction ................................. .................................................................... ..................................................................... .............................................................. ............................29 295  12.3 Solution of Nonlinear Equations ............................... .................................................................. ...................................................................... ............................................. .......... 299  12.4 Matrix Static Condensation ................................... ..................................................................... ..................................................................... ................................................... ................ 305  12.5 Numerical Integration .................................. ..................................................................... ..................................................................... .............................................................. ............................30 306  12.6 Fast Fourier Transform ............................... .................................................................. ..................................................................... .............................................................. ............................31 317  Chapter 13 Post Processing ............................... .................................................................. ..................................................................... .............................................................. ............................ 321  13.1 Element Results Processing ................................. ................................................................... ..................................................................... ................................................... ................ 321  13.2 Element Results Presentation .............................. ................................................................ ..................................................................... ................................................... ................ 326  13.3 Deformed Elements Display ................................. ................................................................... ..................................................................... ................................................... ................ 328  13.4 Sub-modelling ............................... ................................................................. ..................................................................... ...................................................................... ............................................. .......... 329   Chapter 14 Special Topics ................................... ...................................................................... ..................................................................... .............................................................. ............................ 331  14.1 Load on Projected Length/Projected Area .............................. ................................................................ .............................................................. ............................33 331  14.2 Structural and Non-Structural Mass .............................. ................................................................. ..................................................................... ....................................... ..... 332  14.3 Tables .................................. ..................................................................... ..................................................................... ..................................................................... ......................................................... ......................33 333  14.4 Bandwidth Minimization ................................... ..................................................................... ..................................................................... ......................................................... ......................33 338  14.5 Local Singularity and Drilling Degree of Freedom  ................................ ................................................................... ............................................. .......... 342  14.6 Suppression of Drilling Degree of Freedom ................................ ................................................................... ......................................................... ......................34 343  14.7 Torsion Constant and Shear Area Calculation ................................. .................................................................... ................................................... ................ 344  Appendix A Attribute Summary .............................. ................................................................ ..................................................................... ......................................................... ...................... 347   A.1  A.2  A.3  A.4

Node Attributes .............................. ................................................................ ..................................................................... ...................................................................... ............................................. .......... 347 Beam Attributes Attributes ................................... ..................................................................... ..................................................................... ..................................................................... ....................................... ..... 353  Plate Attributes ............................... ................................................................. ..................................................................... ...................................................................... ............................................. .......... 363  Brick Attributes ............................... ................................................................. ..................................................................... ...................................................................... ............................................. .......... 372 

Appendix B Guidelines to Finite Element Modelling................................... ...................................................................... ............................................. .......... 379  B.1 Understanding the Problem .............................. ................................................................ ..................................................................... ......................................................... ......................37 379  B.2 Familiarization with the FE Method and the Software ................................. ................................................................... ....................................... ..... 380  B.3 Using Quality Assurance Tools .................................. ..................................................................... ...................................................................... ............................................. .......... 380  B.4 Summary .............................. ................................................................. ..................................................................... ..................................................................... ......................................................... ......................38 384 

 

Contents

Appendix C Stress and Strain ................................. ................................................................... ..................................................................... ..........................................................385 .......................385  C.1 C.2 C.3 C.4 C.5 C.6

Conventions .............................. ................................................................ ..................................................................... ...................................................................... ....................................................3 .................385 85  Transformation ............................... ................................................................. ..................................................................... ..................................................................... ..............................................3 ............387 87  Principal Stresses .............................. ................................................................. ..................................................................... ..................................................................... .........................................38 ......388  Stress Invariants ................................ ................................................................... ..................................................................... ..................................................................... .........................................38 ......389  Deviatoric and Hydrostatic Stresses .............................. ................................................................ ..................................................................... .........................................38 ......389  Equivalent/Effective Stresses ............................... .................................................................. ...................................................................... ....................................................3 .................390 90

  Appendix D List of Abbreviations ............................... .................................................................. ...................................................................... ....................................................393 .................393  Bibliography.................................. ..................................................................... ..................................................................... ..................................................................... ..........................................................395 .......................395  Index ................................ .................................................................. ..................................................................... ..................................................................... ..................................................................... .........................................321 ......321 

 

 

Preface

 The finite finite ele lem ment me meth thod od (FEM ) is to tod day th the most pow powe erfu rful num numerica rical tool tool av available fo forr the the analys lysis of structure structures. T he method is is used extensive xtensivelly in virtuall virtually y all the engine nginee ering ri ng disci discipl pliines, nes, not only for the ana anallys ysiis of structures tructures and other other engineering ri ng systems tems, but increasingly singly to drive dri ve the design. design. Strand7 is Stra is a softwa software sys system tem base based on the fini finite te element metho method. d. I t provi provides des a visua visual environment envir onment for applying the method to real world engineering problems.  This manual is not not inte intended to be be a comp ompre reh hensive ive re review iew of the finite finite ele lem ment meth thod od.. There There are thousands of books, journals and other resources dedicated to the concepts, the theory and the appli pplications ti ons of the the method (e (e.g. see see the bibli bibliography). ography). I nstea nstead, this thi s manua manuall aims to provi provide de Stra Strand7 users with relevant theoretical information about the finite element method as it applies to Strand7.  This should ould giv give e users fu furt rth her insigh ight into into th the behaviour iour of the their Stra Strand7 mod mode els and th therefo fore re bett tte er interpretation nterpretation of analys analysiis resul results; ts; it it shoul should d also encoura ncourage sensible nsibl e use of Strand7 Strand7 by highli highl ighting ghti ng the embedde mbedded d assumptions umptions and limit li mita ations ti ons of both the me method thod and its implementation. ntation. T he manual manual attempts to present its material in a simple and straightforward manner, focusing on the physical meaning of the subject matter rather than the highly mathematical one.  The reader is assumed to have some ome know knowle led dge of statics tics, dy dynamic ics s, me mechanic ics s of materials rials,, tth he th the eor ory y of elasti elasticit city y and str stre ess analysi nalysis. F amili mil iarity ri ty with with numeri numerica call analys ysiis, ma matri trix computation computation and and the basics basics of the the fi fini nite te element method method woul would d be advantage dvantageous. A basic basic understanding of matr matriix notation notati on is is also assume assumed.  This ma manual should ould be use used in conju onjun nction tion with with ot oth her Stra trand7 doc ocu umentation tion such as the the Using

 Strand7  Strand 7 handbook, the Strand7 V er i fificcati tio on M anual nual, and the Strand7 O nline H elp.

 

 

Chapter Outl Outlin inee

Chapter 1 gives a brief summary of the finite element formulation covering the basic concepts of the theory. I t provides provi des basic informatio nformation n for a relative relatively ly new new use user of the fini finite te element method thod and introduces concepts referred to in the following chapters. Chapter hapter 2 descri scribe bes s the line linea ar materi materia al models ava vaiilable in Stra Strand7. T he models models range range from from gene neralized ralized H ooke’s ooke’s mate teri ria al models for for isotropic, otropic, orthotro orthotropi pic c and and anisotropic anisotropi c materials to lami lamina nate te materi ma teria al models and use user -defined -defi ned materi materia al models for pla pl ate/ te/she shell elements. lements. T hi his s chapter chapter concludes with a description of the way Strand7 treats temperature-dependent and time-dependent material properties. Chapter 3 gives a detailed description of all the available line elements, referred to as beams in Strand7. Strand7 provides a variety of different line elements for various applications, such as spring-damper elements for static and dynamic analysis, catenary elements for cable structures, contact elements for contact problems or changing boundary conditions, pipe elements for pipe systems, thin and thick beam elements, nts, and user user -defi -defined ned beam beam el element nts. s. C hapter hapter 4 gi give ves s a detai detailed desc descri ripti ption on of all the the surface elements, ments, referre referred to as pl pla ates in Stra Strand7, including 3-node and 6-node triangular elements, and 4-node, 8-node and 9-node quadrilateral elements. ments. T hese elements can be used for for plane plane stress stress and plane strain strain analysis, analysis, axisymmetr triic sol oliid analysis, plate pl ate and shell shell analysis, 3D 3D membrane membrane analysis nalysis and heat heat ttransfer ransfer analysis. C hapter 5 gives a de deta taiiled descri cripti ption on of all the three-dime -di mens nsiional (3D ) soli solid eleme elements in Strand7, includi ncluding ng 4-node 4-node and and 10-no 10-node de tetr tetrahe ahedral dral elements, 5-node -node and 13-nod 13-node e pyrami pyramid elements, 6-node -no de and 15-nod 15-node e wedge elements, elements, and 8-node, 8-node, 16-no 16- node de and 20-node 20-node hexahedral el elements. ements. T hese hese elements are use used for for 3D conti continuum nuum probl proble ems in structura structur al and heat ttrans ransfer fer anal analys ysiis. C hapter 6 discusses links li nks,, iincludi ncluding ng the physi physical function function of the differe different nt links li nks,, their corre corresponding ponding mathema themati tica cal expres xpressions and and imple impleme mentation wi withi thi n Strand7. T he avai vailable link link types types include master-slave, sector symmetry, coupling, pinned, rigid, shrink and 2-point links. C hapter hapter 7 introdu ntroduce ces the Strand7 Strand7 sol solve vers. rs. T hese solvers can be used to anal analyse a va vari rie ety of probl problems ems encountered ncountered in industr industry y and rese research. F or instance i nstance, they can can be used for static static analysis nalysis (linear and nonlinear), buckling analysis, dynamic analysis (eigenvalue problems, harmonic response, spectral response res ponse,, linea near and nonl nonliinea near tra transient) and heat heat ttrans ransfer fer anal nalys ysiis ((ste stea ady-sta dy-state te and and trans transiient). nt) .

 

 

Strand7 Theoretical retical Manual

C hapter hapter 8 introduce introduces s topics on structural structural dynamics, mics, including types types of dynamic dynamic loa loadi ding, ng, modell modelliing of structural mass and damping, time domain integration, the mode superposition method, and base exci xcitation. tation. T hese hese topics topi cs relate to a number number of the the Stra Strand7 sol solve vers rs including includi ng natura tural ffreque requency, harmonic response, spectral response and transient analysis. C hapt hapter er 9 prese presents the basic theor theory y of heat transfer, transfer, includi including ng conducti conduction, on, convecti convection on and and radi radia ati tion on (the three modes of heat transfer supported by Strand7), the finite element formulation of heat transfer and the application of the heat solver to other types of field problems. Cha hapte pterr 10 iintroduce ntroduces s ma mate teri ria al nonli nonline nea ari rity. ty. M ate teri ria al models models for elasto lasto-pl -pla asti tic, c, nonline nonli nea ar elas elasti tic c and rubbe rubb er materi materia als are descri scribe bed, d, toge together ther with with the D uncan uncan-Cha -C hang ng model model ffor or nonl nonliinear analysis ysis of soil soi ls. Chapter hapter 11 int intro roduce duces geome geometri tric c nonline nonli nea arity. ri ty. I t covers covers the bas basic conce concepts pts of la large displace displacement and finite (large) strain, the formulation of the geometric stiffness matrix, solution strategies, strain measures and the use of the geometric nonlinear solver for buckling and post-buckling analysis. Chapter 12 describes some of the numerical procedures that are employed in Strand7, most of which are essenti ntia al ffor or any sof software tware implementation ntation of the fini finite te element ment method. thod. T hi his s includes matr matriix decomposition, eigenvalue extraction, time-domain integration for transient dynamic problems, modifiied Ne modif N ewton-R wton -R aphson phson ite iteration ration for solving solving nonli nonline nea ar equa equati tions, ons, ma matrix tri x stati tic c condens condensa ati tion, on, numerica ri cal iintegrati ntegration on schem scheme es fo forr the the element ment matr matriix ca calculations culations and the F ast Four Fouriier Trans T ransfo form. rm. Chapter 13 covers the procedures used in Strand7 for the post-processing of the results, including result extrapolation and averaging, contouring, display of deformed elements and sub-modelling. Chapter 14 discusses various special topics including projected load, structural and non-structural mass, tables tab les and thei their applica applicati tion, on, bandwidth mini minimization mization of the global stiff ne nes ss ma matrix tri x and and node reorderi reordering ng methods, the treatment of drilling degrees of freedom and local singularity suppression, and the calculation of the torsion constant and shear area for beam cross-sections. A ppendix ppendi x A summa summari rizes zes the node and and element attri attr ibutes suppor supported ted by Stra Str and7 and the way they are used by the solvers. A ppendix ppendi x B is a guideline guideli ne to fi fini nite te element model modelling li ng includi including ng a very useful ch che ecklis ckli st for the fini fi nite te element modelling process in practice. A ppendi ppendix x C summari rizes zes stress stress and strain strain conventions, conventions, trans transfo formatio rmations, ns, pri principal stress stresses, stress stress invariants, deviatoric stresses and effective stresses. A ppendix ppendix D is i s a bi bibli bliography ography that lis lists some of the the important books and arti articles cles on the topic topi c of fini fi nite te element analysis.