This book describes the theory of all the methods of static and dynamic analyses, be they linear or nonlinear, both materially and geometrically. The book is intended to give the reader a sound appreciation of the basic and advanced methods of structural analyses. Specific emphasis is given to the advanced analysis codes of MSC.NASTRAN and LS/DYNA.
Computational Finite Elements and Methods of Static and Dynamic Analyses
PDF document. 505 pages.
ACKNOWLEDGEMENTS. 6
LIST OF SYMBOLS AND NOTATIONS. 7
1 INTRODUCTION.. 10
2 CONCEPTS OF COMPUTATIONAL FINITE ELEMENT STRUCTURAL ANALYSIS. 11
2.1 Overview of the Finite Element Method. 11
2.1.1 GL, ML Static Finite Element Elemental Formulation. 12
2.1.1.1 The Principle of Virtual Displacements of The Principle of Virtual Work (Equivalent to the Variational Method or The Principle of Minimum Potential Energy) 12
2.1.1.2 The Method of Weighted Residuals. 13
2.1.2 GL, ML Static Finite Element Global Formulation. 15
2.1.3 GNL, MNL Static Finite Element Elemental Formulation. 18
2.1.4 GNL, MNL Static Finite Element Global Formulation. 23
2.1.5 How Nonlinear Analysis Varies From Linear Analysis. 26
2.1.5.1 Geometric Nonlinearity. 26
2.1.5.2 Material Nonlinearity. 28
2.1.5.3 Contact and Boundary Conditions Nonlinearity. 28
2.1.6 GL, ML and GNL, MNL Dynamic Finite Element Elemental and Global Formulation. 28
2.2 Static Analysis Concepts. 29
2.2.1 Overview of Methods of Structural Static Analyses. 29
2.3 Dynamic Analysis Concepts. 30
2.3.1 Overview of Methods of Structural Dynamic Analyses. 30
2.3.1.1 Modal Analyses. 30
2.3.1.2 Forced Response Analyses. 30
2.3.2 Nature of Trial Solution in Harmonic Frequency Vibrations. 32
2.4 Static and Dynamic Analyses Sequences. 33
3 METHODS OF STATIC ANALYSES. 36
3.1 GL, ML Static Analysis by The Implicit Direct Stiffness Method. 36
3.1.1 Mathematical Formulation of Analysis. 36
3.1.2 Concepts of Geometric Stiffness or P-D (KGA From KEA) Analysis 37
3.1.3 Performing P-D (KGA From KEA) Linear Static Analysis – Direct Approach. 42
3.1.4 Performing P-D (KGA From KEA) Linear Static Analysis – Modal Approach; And Hence P-D Based Buckling 44
3.1.4.1 Objectives. 44
3.1.4.2 Mathematical Formulation. 44
3.1.4.3 Critical Load Case Considerations. 46
3.1.4.4 Modelling (Modal) Imperfections. 47
3.1.4.5 Design. 51
3.1.4.6 Limitations. 52
3.1.4.7 Methodology. 53
3.1.5 MSC.NASTRAN Decks. 56
3.1.5.1 GL, ML Static Analysis. 56
3.1.5.2 GL, ML P-D (KGA From KEA) Static Analysis. 65
3.1.5.3 GL, ML P-D (KGA From Exact or Approximate KTA) Static Analysis 66
3.1.6 Hand Methods Verification. 67
3.1.6.1 Static Displacements by the Unit Load Method of the Virtual Work Principle. 67
3.1.6.2 BMD and SFD of Structures of Low Static Indeterminacies by Flexibility Analysis. 70
3.1.6.3 Bending Moments, Shear Force, and Displacements of Structural Elements by the Stiffness Method. 92
3.1.6.4 Bending Moments and Shear Force of Structures of Low Kinematic Indeterminacies by Moment Distribution. 93
3.1.6.5 Summary of Deflections and Effects. 117
3.1.6.6 Member Buckling Check Using The P-D Method To Incorporate Imperfections, Residual Stresses. 158
3.1.6.7 Overall Portal Frame and Multi-Storey Building P-D Analysis Using The Amplified Sway Method. 171
3.2 GL, ML Buckling Analysis by The Implicit Linearized Eigenvalue Analysis. 172
3.2.1 Linearization of Tangent Stiffness Matrix and Formulating The Linear Eigenvalue Problem.. 172
3.2.2 Problem Reduction and Trial Modes in Linear Buckling Analysis. 178
3.2.3 Concepts of Linearized (KGA From KEA) Buckling Analysis. 179
3.2.4 MSC.NASTRAN Decks. 180
3.2.4.1 GL, ML (KGA From KEA) Buckling Analysis. 180
3.2.4.2 GL, ML (KGA From Exact or Approximate KTA) Buckling Analysis 186
3.2.5 Hand Methods Verification. 189
3.2.5.1 Member or Local (KGA From KEA) Buckling. 189
3.2.5.2 Overall System (KGA From KEA) Buckling. 197
3.3 GL, MNL Plastic Collapse Analysis by the Implicit Linear Simplex Programming.. 208
3.3.1 Mathematical Formulation of Analysis. 208
3.3.2 Mathematical Proof 211
3.3.3 Displacements and Rotations at Collapse. 215
3.3.4 Plastic Limit Analysis of Simple Framed Structures. 216
3.3.4.1 Portal Frame. 216
3.3.4.2 Parabolic Arch. 222
3.3.4.3 Displacements and Rotations at Collapse. 223
3.3.5 Plastic Limit Analysis of Rectangular Multi-Storey Multi-Bay Frames with Improved Data Generation. 225
3.3.5.1 Two Storey Frame With Concentrated Loading. 226
3.3.5.2 Three Storey Frame With Distributed Load. 228
3.3.5.3 Multi-Bay Portal Frame. 231
3.3.5.4 Vierendeel Truss. 233
3.3.6 Hand Methods Verification. 234
3.3.6.1 Plastic Collapse Analysis by Solving Equilibrium Equations. 234
3.3.6.2 Plastic Collapse Analysis by Virtual Work (Hobbs) 238
3.3.6.3 Plastic Collapse Analysis by Virtual Work (Chryssanthopoulos) 240
3.4 GNL, MNL, Contact Nonlinear Static and Buckling Analysis by The Implicit Tracing The Equilibrium Path Method. 256
3.4.1 Mathematical Formulation of Tracing the Equilibrium Path. 256
3.4.2 Newton-Raphson Load Control, Displacement Control or Arc-Length Control Algorithm.. 259
3.4.3 Equilibrium Paths, Stability of Equilibrium Paths, Critical Points, Stability of Critical Points. 260
3.4.4 MSC.NASTRAN Decks. 263
3.4.4.1 GNL, MNL Load Control, Displacement Control or Arc-Length Control Static Analysis. 263
3.4.4.2 Nonlinear Static and Linearized Eigenvalue Buckling Analysis. 274
3.4.4.3 Nonlinear Static and Linear Eigenvalue Modal Dynamic Analysis 275
3.4.4.4 Restart From Nonlinear Static Analysis SOL 106 Into Nonlinear Static Analysis SOL 106. 276
3.4.4.5 Restart From Nonlinear Static SOL 106 Into Linear Solution Schemes SOL 107 to SOL 112. 276
3.4.4.6 Implicit Nonlinear Static (and Dynamic Analysis) SOL 400. 276
3.4.4.7 Implicit Nonlinear Static (and Dynamic Analysis) SOL 600. 276
3.5 GNL, MNL Static and Buckling Analysis by Dynamic Relaxation. 277
3.5.1 Dynamic Relaxation of the Explicit Finite Difference Scheme Solving Newton’s Dynamic Equilibrium ODE (LS-DYNA) 277
3.5.2 LS-DYNA (GNL, MNL Explicit Transient) Dynamic Relaxation Cards. 280
4 METHODS OF DYNAMIC ANALYSES. 282
4.1 GL, ML Implicit Real Modal (Eigenvalue) Analysis. 282
4.1.1 Mathematical Formulation of Analysis. 282
4.1.2 MSC.NASTRAN Decks. 286
4.1.2.1 GL, ML Real Modal Analysis. 286
4.1.2.2 GL, ML P-D (KGA From KEA) Real Modal Analysis. 287
4.1.2.3 GL, ML P-D (KGA From Exact or Approximate KTA) Real Modal Analysis. 289
4.1.3 Hand Methods Verification. 290
4.1.3.1 Natural Frequency and Free Vibration Response of SDOF Systems 290
4.1.3.2 Natural Frequencies of MDOF Systems. 293
4.1.3.3 Natural Frequencies of Distributed Systems. 305
4.1.3.4 Natural Frequencies By Exactly Solving the Partial Differential Equilibrium Equation. 306
4.1.3.5 Approximate Formulae. 311
4.2 GL, ML Implicit Direct Complex Modal (Eigenvalue) Analysis. 313
4.2.1 Mathematical Formulation of Analysis. 313
4.2.2 Complexity of Modes of Vibration. 314
4.2.3 Complex Modal Analysis To Determine Modal Damping Values. 314
4.2.4 MSC.NASTRAN Decks. 315
4.2.4.1 GL, ML Complex Modal Analysis. 315
4.2.4.2 GL, ML P-D (KGA From KEA) Complex Modal Analysis. 317
4.2.4.3 GL, ML P-D (KGA From Exact or Approximate KTA) Complex Modal Analysis. 319
4.2.5 Hand Methods Verification. 320
4.2.5.1 Determination of Damped Natural Frequency and the Maximum Dynamic Displacement, umax for Free Damped Vibration Due to Initial Displacement and/or Initial Velocity by Classically Solving the SDOF Linear ODE and Maximizing the Solution. 320
4.3 GL, ML Implicit (Real) Modal Frequency Response Analysis. 323
4.3.1 Nature of the Dynamic Loading Function. 323
4.3.2 Mathematical Formulation of Analysis. 323
4.3.3 Capability of A Finite Number of Modes To Model The Static and Dynamic Response of Structure. 328
4.3.4 Complex Response F(w) and Amplification D(w) With Elemental and/or Modal Structural Damping. 331
4.3.5 Representation of A MDOF System As A SDOF system 332
4.3.6 MSC.NASTRAN Decks. 334
4.3.6.1 GL, ML Modal Forced Frequency Response Analysis. 334
4.3.6.2 GL, ML P-D (KGA From KEA) Modal Forced Frequency Response Analysis. 337
4.3.6.3 GL, ML P-D (KGA From Exact or Approximate KTA) Modal Forced Frequency Response Analysis. 339
4.3.7 Hand Methods Verification. 340
4.3.7.1 Determination of Maximum Dynamic Displacement for Deterministic Frequency Domain Loading by Transforming the Coupled MDOF Linear Damped ODEs To a Set of Independent (Uncoupled) SDOF ODEs and Solving the Independent Equations in a Manner Similar to Solving a SDOF ODE.. 340
4.4 GL, ML Implicit (Complex) Modal Frequency Response Analysis. 349
4.4.1 Mathematical Formulation of Analysis. 349
4.5 GL, ML Implicit Direct Frequency Response Analysis. 352
4.5.1 Nature of the Dynamic Loading Function. 352
4.5.2 Mathematical Formulation of Analysis. 352
4.5.3 MSC.NASTRAN Decks. 355
4.5.3.1 GL, ML Direct Forced Frequency Response Analysis. 355
4.5.3.2 GL, ML P-D (KGA From KEA) Direct Forced Frequency Response Analysis. 365
4.5.3.3 GL, ML P-D (KGA From Exact or Approximate KTA) Direct Forced Frequency Response Analysis. 367
4.5.4 Hand Methods Verification. 368
4.5.4.1 The Theory of the Dynamic Magnification Factor for Undamped Motion and Hence the Determination of Maximum Dynamic Displacement, umax for Deterministic Harmonic Loading by Classically Solving the SDOF Linear Undamped ODE and Maximizing the Solution. 368
4.5.4.2 The Theory of the Dynamic Magnification Factor for Damped Motion and Hence Determination of Maximum Dynamic Displacement, umax for Deterministic Harmonic Loading by Classically Solving the SDOF Linear Damped ODE and Maximizing the Solution. 370
4.5.4.3 The Theory of Vibration (Base) Isolation and the Force Transmitted Into Rigid Foundation by the Damped Structure Subjected to Deterministic Harmonic Loading. 375
4.5.4.4 The Theory of Vibration (Base) Isolation and the Determination of Maximum Dynamic Displacement, umax for Deterministic Harmonic Support Motion (Displacement, Velocity or Acceleration) by Classically Solving the SDOF Linear Damped ODE (in Absolute and Relative Terms) and Maximizing the Solution. 377
4.6 GL, ML Frequency Domain Analysis – Deterministic and Random Dynamic Response Analysis. 380
4.6.1 Mathematical Preliminaries of Representing Dynamic Characteristics in the Frequency Domain. 380
4.6.2 GL, ML Vibration Testing. 381
4.6.2.1 Vibration Testing for Model Correlation. 381
4.6.2.2 Vibrating Testing For Analysis Procedure Verification. 384
4.6.3 GL, ML Steady-State Response of Deterministic Periodic (Not Necessarily Harmonic) Long Duration Excitation Utilizing Fourier Series (or Generally Utilizing Fast Fourier Transforms FFT) 385
4.6.3.1 Fourier Series. 385
4.6.3.2 Discrete Fourier Series. 386
4.6.3.3 Discrete Fourier Series in Complex Notation. 387
4.6.3.4 Double Sided Discrete Fourier Series in Complex Notation. 388
4.6.3.5 Normalized Double Sided Discrete Fourier Series in Complex Notation 389
4.6.3.6 Symmetrical Normalized Double Sided Discrete Fourier Series in Complex Notation. 390
4.6.3.7 Symmetrical Normalized Single Sided Discrete Fourier Series in Complex Notation. 391
4.6.3.8 Practicalities of the Specification of the Fast Fourier Transform (FFT) Representation. 392
4.6.3.9 MSC.NASTRAN Fast Fourier Transform Analysis Methodology. 394
4.6.4 GL, ML Steady-State Response of Random, Gaussian, and Stationary (and Ergodic) Excitations Utilizing Power Spectral Density (PSD) Functions. 397
4.6.4.1 Statistic of Time Domain Function. 397
4.6.4.2 Definition of the Power Spectral Density (PSD) 401
4.6.4.3 Validity of the PSD Representation. 404
4.6.4.4 Generation and Specification of the PSD.. 405
4.6.4.5 Statistical Information Provided by the PSD.. 406
4.6.4.6 MSC.NASTRAN Random Analysis Methodology. 409
4.7 GL, ML Implicit (Real) Modal Transient Response Analysis. 413
4.7.1 Nature of the Dynamic Loading Function. 413
4.7.2 Mathematical Formulation of Analysis. 413
4.7.3 Capability of A Finite Number of Modes To Model The Static and Dynamic Response of Structure. 418
4.7.4 Concepts of Equivalent Static Force. 427
4.7.5 Structural Damping in Time Domain Analyses. 429
4.7.6 MSC.NASTRAN Decks. 431
4.7.6.1 GL, ML Modal Forced Transient Response Analysis. 431
4.7.6.2 GL, ML P-D (KGA From KEA) Modal Forced Transient Response Analysis. 434
4.7.6.3 GL, ML P-D (KGA From Exact or Approximate KTA) Modal Forced Transient Response Analysis. 436
4.7.7 Hand Methods Verification. 437
4.7.7.1 Determination of Maximum Dynamic Displacement for Deterministic Time Domain Loading by Transforming the Coupled MDOF Linear Undamped ODEs To a Set of Independent (Uncoupled) SDOF ODEs and Solving the Independent Equations in a Manner Similar to Solving a SDOF ODE.. 437
4.7.7.2 Determination of Max Dynamic Displacement for Deterministic Time Domain Support Motion (Displacement, Velocity or Acceleration) by Transforming the Coupled MDOF Linear Undamped ODEs (In Relative Terms) To a Set of Independent (Uncoupled) SDOF ODEs (In Relative Terms) and Solving the Independent Equations in a Manner Similar to Solving a SDOF ODE.. 446
4.7.7.3 Determination of Maximum Dynamic Displacement for Deterministic Time Domain Loading by Transforming the Coupled Distributed System Linear Damped ODEs (from the governing PDE) To A Set of Independent (Uncoupled) SDOF ODEs and Solving the Independent Equations in a Manner Similar to Solving a SDOF ODE.. 450
4.8 GL, ML Implicit Direct Transient Response Analysis. 451
4.8.1 Nature of the Dynamic Loading Function. 451
4.8.2 Mathematical Formulation of Analysis. 451
4.8.3 MSC.NASTRAN Decks. 454
4.8.3.1 GL, ML Direct Forced Transient Response Analysis. 454
4.8.3.2 GL, ML P-D (KGA From KEA) Direct Forced Transient Response Analysis. 464
4.8.3.3 GL, ML P-D (KGA From Exact or Approximate KTA) Direct Forced Transient Response Analysis. 466
4.8.4 Hand Methods Verification. 467
4.8.4.1 Determination of Maximum Dynamic Displacement, umax by Solving the SDOF Undamped/Damped Linear Equation of Motion ODE for Deterministic Time Domain Loading With/Without Initial Conditions Using the Convolution Integral (Duhamel’s integral) 467
4.8.4.2 Determination of Maximum Dynamic Displacement, umax by Solving the SDOF Undamped/Damped Linear Equation of Motion ODE (In Relative Terms) for Deterministic Time Domain Support Motion (Displacement, Velocity or Acceleration) With/Without Initial Conditions Using the Convolution Integral (Duhamel’s integral) 474
4.9 GNL, MNL Implicit and Explicit Direct Transient Response Analysis. 475
4.9.1 Nature of the Dynamic Loading Function. 475
4.9.2 Deterministic Non-Periodic Short Duration Impulse (a.k.a. Blast) Loading Functions With Subsequent Wave Propagation. 476
4.9.3 Projectile Crash (a.k.a. Impact) (and Impulsive Blast) Analysis With Subsequent Wave Propagation. 477
4.9.4 Brittle Snap or Redundancy Check Excitation. 480
4.9.5 GNL, MNL Explicit Central FD Scheme for Newton’s Dynamic Equilibrium ODE (LS-DYNA) 485
4.9.5.1 Solution of Partial or Ordinary Differential Equations Using Finite Difference (FD) Schemes. 485
4.9.5.2 Mathematical Formulation of Analysis – Explicit Central Finite Difference Scheme. 486
4.9.5.3 Stability. 487
4.9.5.4 Accuracy. 488
4.9.6 GNL, MNL Implicit Newmark Scheme for Newton’s Dynamic Equilibrium ODE (MSC.NASTRAN) 489
4.9.6.1 Mathematical Formulation of Analysis – Implicit Newmark Scheme 489
4.9.6.2 Stability. 491
4.9.6.3 Accuracy. 491
4.9.7 Comparison Between Implicit and Explicit Time Integration Schemes. 492
4.9.8 MSC.NASTRAN Decks. 494
4.9.8.1 GNL, MNL Direct Forced (Implicit) Transient Response Analysis 494
4.9.8.2 Nonlinear Static Analysis and Nonlinear Transient Analysis. 498
4.9.8.3 Restart From Nonlinear Static Analysis SOL 106 Into Nonlinear Transient Analysis SOL 129. 498
4.9.8.4 Restart From Nonlinear Transient Analysis SOL 129 Into Nonlinear Transient Analysis SOL 129. 498
4.9.8.5 Implicit Nonlinear (Static and) Dynamic Analysis SOL 400. 499
4.9.8.6 Implicit Nonlinear (Static and) Dynamic Analysis SOL 600. 499
4.9.8.7 Explicit Nonlinear Dynamic Analysis SOL 700. 499
4.9.9 LS-DYNA Decks. 500
4.9.9.1 GNL, MNL Direct Forced (Explicit) Transient Response Analysis 500
4.9.10 Hand Methods Verification. 502
4.9.10.1 Determination of Displacement Response Time History by Solving the SDOF Nonlinear (in Stiffness, Damping and Displacement) Equation of Motion ODE for Deterministic Time Domain Loading With/Without Initial Conditions by Implicit Newmark-b Time Integration Schemes. 502
BIBLIOGRAPHY.. 504
