Programming the Dynamic Analysis of Structures

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ISBN: 9780419156109 | 0419156100
Cover: Hardcover
Copyright: 12/6/2002

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This book presents a series of integrated computer programs in Fortran-90 for the dynamic analysis of structures, using the finite element method. Two dimensional continuum structures such as walls are covered along with skeletal structures such as rigid jointed frames and plane grids. Response to general dynamic loading of single degree freedom systems is calculated, and the author also examines multi degree of freedom systems (including earthquake analysis). Each chapter covers a different aspect of analytic theory and the corresponding program segments. It will be an essential tool for practising structural and civil engineers, whilst also being of interest to academics and postgraduate students.

Preface

xvi

Single Degree of Freedom System - I

(19)

Introduction

(1)

Inertial force

(1)

Damping force

(1)

Single degree of freedom system

(2)

Mathematical study of the SDOF system

(1)

Influence of gravitational forces

(1)

Solution of the differential equation

(2)

Solution to the free vibration problem

(1)

Undamped free vibration

(1)

Units used in dynamic analysis

(1)

Example

(1)

Damped free vibration

(2)

Underdamped system

(1)

Critically damped case

(1)

Critical damping

(1)

Damping in structures

(1)

Overdamped system

(1)

Measurement of damping

(2)

Summary of free vibration analysis

(1)

Systems subjected to harmonic excitation

(1)

Dynamic magnification factor

(2)

Response near resonance

(1)

Response to base excitation

(2)

Single Degree of Freedom System - II

(30)

Introduction

(1)

Laplace transform method

(1)

Laplace transform: examples

(1)

Inverse Laplace transform

(1)

Inverse Laplace transform: examples

(1)

Laplace transform of derivatives

(1)

Solution of differential equations

(1)

Some useful results

(4)

Unit step function

(1)

Shift theorems

(2)

Convolution theorem

(1)

Summary of some results

(1)

General solution of Duhamel integral

(2)

Solution of Duhamel integral: specific examples

(7)

Suddenly applied load

(1)

Rectangular impulse loading

(2)

Triangular impulse loading

(3)

Exponentially decaying force

(1)

Shock spectra

(3)

Rectangular impulse

(1)

Triangular impulse

(2)

Numerical evaluation of Duhamel integral

(3)

Program: Duhamel

(3)

Use of Duhamel: example

(1)

Earthquake response spectra

(4)

Introduction to Multiple Degree of Freedom Systems

(34)

Introduction

(2)

Eigen-value problem

(13)

Determinants

(2)

Cramer's rule

(1)

Positive definiteness

(1)

Eigen-values of [K-ω2 M]: examples

(3)

Eigen-vectors of [K-ω2 M]

(6)

General and special eigen-value problems

(2)

Conversion from general to special eigen-value problem

(1)

Properties of eigen-values and eigen-vectors

(5)

Characteristic equation

(1)

Orthogonal property of eigen-vectors

(2)

Arbitrary vector in terms of eigen-vectors

(2)

Mode superposition method

(8)

Undamped forced response

(1)

Undamped forced response: Example

(1)

Square root of the sum of the squares approximation

(2)

Damped forced response

(1)

Damped forced response: Example

(3)

Response to base acceleration

(4)

Earthquake participation factors: Example

(3)

Matrix Routines

(23)

Introduction

(1)

Matrix manipulations

(1)

Cholesky factorisation

(5)

Solution of simultaneous equations by Cholesky decomposition

(2)

Converting a general to a special eigen-value problem

(2)

Crout factorisation

(3)

Solution of simultaneous equations by Crout decomposition

(1)

Gaussian elimination method

(4)

Gaussian elimination method: example

(2)

Determinant of a matrix

(1)

Determinant from Gaussian elimination

(1)

Determinant from Cholesky factorisation

(1)

Efficient storage schemes

(3)

Rectangular format

(1)

Vector or Skyline format

(3)

Fortran-90 programs

101

(6)

Program Cholfact

101

(1)

Program Crout

102

(2)

Program Gauss

104

(3)

Solution Methods for the General Eigen-Value Problem

107

(67)

Introduction

107

(1)

Forward iteration

107

(5)

Forward iteration: example

109

(3)

Inverse iteration and higher frequencies

112

(2)

Gram-Schmidt method of orthogonalisation

112

(2)

Calculation of higher frequencies

114

(3)

Calculation of higher frequencies: example

114

(3)

Simultaneous iteration

117

(1)

Shift technique

117

(5)

Shift technique: example

118

(2)

Higher frequencies and shift technique

120

(2)

Transformation methods

122

(1)

Jacobi diagonalisation of the general eigen-value problem

123

(9)

Calculation procedure for the Jacobi method

126

(1)

General Jacobi method: example

126

(4)

Determination of eigen-vectors

130

(2)

Lanczos method of tri-diagonalisation

132

(10)

Lanczos method of tri-diagonalisation: example

135

(3)

Comments on convergence of Lanczos method

138

(3)

Selective orthogonalisation

141

(1)

Sturm sequence

142

(10)

Interleaving of eigen-values: example

146

(1)

Sturm sequence property and sign counts

147

(3)

Sign counts: example

150

(1)

Isolating a specific eigen-value by root bisection

151

(1)

QR factorisation

152

(5)

QR factorisation: example

154

(3)

Gerschgorin limits on eigen-values

157

(1)

Gerschgorin limits: example

157

(1)

Fortran-90 Programs

158

(16)

Program Lanczgen

159

(6)

Program Genjacobi

165

(4)

Program Qrlancz

169

(5)

Solution of Large-Scale Eigen-Value Problems

174

(39)

Introduction

174

(1)

Simultaneous iteration method

174

(12)

Theoretical basis

174

(3)

Implementation of the simultaneous iteration method

177

(2)

Example of the simultaneous iteration method

179

(7)

Subspace iteration method

186

(9)

Rayleigh quotient

186

(3)

Implementation of the subspace iteration method

189

(2)

Subspace iteration method: example

191

(4)

Ritz vectors

195

(3)

Superposition of Ritz vectors

195

(1)

Generation of Ritz vectors

196

(1)

Generation of Ritz vectors: example

196

(2)

Lanczos vectors

198

(3)

Generation of Lanczos vectors: example

199

(1)

Use of Lanczos vectors

200

(1)

Fortran-90 Programs

201

(12)

Program Simult

201

(6)

Program Subspace

207

(6)

Line Elements

213

(43)

Introduction

213

(1)

Bar element: differential equation

213

(3)

Natural frequency and mode shapes of bar element

216

(2)

Beam element: flexural deformations only

218

(3)

Natural frequency and mode shapes of beam element

221

(4)

Simply supported beam: mode shapes

221

(2)

Cantilever beam: mode shapes

223

(2)

Standards results for beam vibration

225

(1)

Beam element with shear deformation included

225

(3)

Effect of axial load on natural frequency of beams

228

(3)

Shaft element

231

(3)

Approximate shape functions and stiffness coefficients

234

(1)

Approximate dynamic stiffness matrix for a bar element

234

(2)

Derivation of approximate dynamic stiffness matrix

236

(1)

Method of minimum total potential

237

(14)

Approximate bar element dynamic stiffness matrix

237

(2)

Approximate beam element dynamic stiffness matrix including flexural deformation only

239

(3)

Approximate beam element dynamic stiffness matrix including flexural and shear deformations

242

(7)

Approximate beam-column element dynamic stiffness matrix

249

(2)

Approximate dynamic stiffness matrix of a shaft element

251

(1)

Galerkin's weighted residual method

251

(5)

Undamped free longitudinal vibrations

252

(1)

Undamped free flexural vibrations including bending deformations only

253

(3)

Finite Elements

256

(42)

Introduction

256

(1)

Summary of results from the Theory of Elasticity

256

(3)

Equilibrium

257

(1)

Strain-stress relationship

257

(1)

Stress-strain relationship

258

(1)

Strain energy

258

(1)

Strain-displacement relationship

258

(1)

Equilibrium equations in terms of displacements

258

(1)

A plane stress rectangular element: shape functions

259

(2)

Galerkin's weighted residual method

261

(4)

Integrals of shape functions

264

(1)

Explicit K and M matrices

265

(1)

Derivation of K and M matrices by minimisation of total potential

265

(2)

Strain energy

265

(1)

Loss of potential by external loads

265

(1)

Loss of potential by inertial forces

266

(1)

Matrices K and M

267

(1)

Concept of mapping and isoparametric elements

267

(7)

Change of variables and implicit functions

271

(2)

Cartesian derivatives of shape functions

273

(1)

Numerical integration by Gaussian quadrature

274

(2)

Evaluation of K and M matrices of isoparametric elements

276

(2)

In-plane quadratic isoparametric element

278

(1)

Summary of results from the Theory of Thin Plates

278

(3)

Equilibrium

279

(1)

Moment-curvature relationship

279

(1)

Strain energy

280

(1)

Equations of equilibrium in terms of displacements

280

(1)

Frequencies of vibration of a simply supported plate

280

(1)

A Kirchoff rectangular plate-bending element: shape functions

281

(4)

Conditions that displacement functions must satisfy

285

(1)

Galerkin's weighted residual method

285

(5)

Derivation of K and M matrices by minimisation of total potential

290

(2)

Strain energy

290

(1)

Loss of potential by external loads

290

(1)

Loss of potential by inertial forces

291

(1)

Matrices K and M

291

(1)

A conforming rectangular thin plate element

292

(1)

Mindlin-Reissner isoparametric quadratic thick plate-bending element

292

(3)

Isoparametric quadratic flat shell element

295

(1)

Reduced integration

296

(2)

Finite Strip Method

298

(23)

Introduction

298

(1)

Bending finite strip with simply supported ends

298

(4)

In-plane finite strip with simply supported ends

302

(3)

Auxiliary nodal line technique

305

(5)

Thin plate-bending strip with an auxiliary nodal line

306

(2)

In-plane strip with an auxiliary nodal line

308

(2)

Mindlin-Reissner plate-bending finite strip

310

(3)

Cubic spline strip method

313

(6)

Cubic splines

313

(2)

Example

315

(1)

Incorporation of boundary conditions

316

(3)

K and M matrices for cubic spline finite strip

319

(2)

In-plane cubic spline finite strip

319

(1)

Cubic spline plate-bending finite strip

319

(2)

Direct Stiffness Method

321

(16)

Introduction

321

(1)

Member axis stiffness matrix of line elements

321

(2)

Member axis stiffness matrix of finite elements

323

(1)

In-plane finite elements

323

(1)

Plate-bending finite elements

323

(1)

Flat shell finite elements

323

(1)

Member axis stiffness matrix of finite strip elements

324

(1)

Member axis to global axis stiffness matrix

324

(6)

Two-dimensional pin-jointed member

324

(1)

Two-dimensional rigid-jointed member

325

(2)

Plane grid member

327

(1)

Flat shell finite strip

328

(2)

Structural stiffness matrix

330

(4)

Automatic assembly of structural stiffness matrix

334

(2)

Mass matrices

336

(1)

Direct Integration

337

(23)

Introduction

337

(1)

Direct numerical integration

337

(9)

Linear acceleration method

338

(1)

The Wilson-θ method

339

(5)

The Newmark method

344

(1)

The central difference method

345

(1)

Explicit and implicit integration schemes

346

(1)

Stability of integration schemes

346

(1)

Accuracy of integration schemes

347

(1)

Non-linear analysis

347

(2)

Non-linear analysis: the Wilson-θ method

347

(1)

Non-linear analysis: Newmark method

348

(1)

Equilibrium iterations

349

(1)

Programs

349

(11)

Program Wilson

349

(5)

Program Newmark

354

(3)

Program Central

357

(3)

Subroutines and Programs

360

(74)

Introduction

360

(1)

Terminology used in the programs

360

(1)

Subroutines for data input

361

(3)

Subroutine Cordgen

361

(1)

Subroutine Intgen

362

(1)

Subroutine Nfgen

363

(1)

Subroutines to determine the size of stiffness matrix

364

(3)

Subroutine Maxeqn

364

(1)

Subroutine Bandwd

365

(1)

Subroutine Nskyline

365

(2)

Subroutines for Cholesky factorisation

367

(2)

Subroutine Cholesky

367

(1)

Subroutine Backsub (A,B,C)

368

(1)

Subroutine Forwsub (A,B,C)

369

(1)

Stiffness matrices of line elements

369

(4)

Subroutine Stifrj

370

(1)

Subroutine Stifgr

371

(2)

Consistent mass matrices of line elements

373

(6)

Subroutine Readmasframe

373

(1)

Subroutine Smasrj

374

(2)

Subroutine Smasgr

376

(3)

Simultaneous iteration

379

(5)

Subroutine Simit

379

(5)

Subroutines for Sturm sequence check

384

(3)

Subroutine Sturm

384

(1)

Subroutine Gausred

385

(2)

Program Frasim.F90

387

(3)

Finite element analysis: plate bending

390

(7)

Subroutine Dstifpb

390

(7)

Program Dplsim.F90

397

(3)

Subroutines for direct integration

400

(7)

Subroutine Mult6(A,B,C)

400

(1)

Subroutine Backs1(A,B,C)

401

(1)

Subroutine Forscal

402

(1)

Subroutine Acelcal

402

(1)

Subroutine Step

403

(3)

Program Frastep.F90

406

(1)

Program Dplstep.F90

407

(1)

Finite strip method: plate bending

407

(5)

Subroutine Dstifstp

407

(2)

Program Stripsim.F90

409

(3)

Spline finite strip

412

(12)

Subroutine Bandwdspln

412

(1)

Subroutine Nskylinespln

413

(1)

Subroutine Dsplnpb

414

(6)

Subroutine Spline

420

(4)

Program Splnsim.F90

424

(4)

Data preparation for programs

428

(6)

Data input to program Frasim.F90

428

(3)

Data input to program Dplsim.F90

431

(1)

Data input to program Splnsim.F90

431

(2)

Data input to program Frastep.F90

433

(1)

Data input to program Dplstep.F90

433

(1)

Selected References

434

(3)

Index

437

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Programming the Dynamic Analysis of Structures

Summary

Author Biography

Table of Contents

Supplemental Materials

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Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Programming the Dynamic Analysis of Structures

Summary

Author Biography

Table of Contents

Supplemental Materials

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