Programming the Dynamic Analysis of Structures

by Bhatt; Prab

• ISBN13: 9780419156109
• ISBN10: 0419156100
• Edition: 1st
• Format: Hardcover
• Copyright: 2002-12-06
• Publisher: CRC Press

Summary

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.

Author Biography

Dr Prabhakara Bhatt is Senior Lecturer in Civil Engineering at Glasgow University

Table of Contents

Preface	xvi
Single Degree of Freedom System - I	1	(19)
Introduction	1	(1)
Inertial force	2	(1)
Damping force	2	(1)
Single degree of freedom system	2	(2)
Mathematical study of the SDOF system	4	(1)
Influence of gravitational forces	5	(1)
Solution of the differential equation	5	(2)
Solution to the free vibration problem	7	(1)
Undamped free vibration	8	(1)
Units used in dynamic analysis	9	(1)
Example	9	(1)
Damped free vibration	9	(2)
Underdamped system	9	(1)
Critically damped case	10	(1)
Critical damping	11	(1)
Damping in structures	11	(1)
Overdamped system	12	(1)
Measurement of damping	13	(2)
Summary of free vibration analysis	15	(1)
Systems subjected to harmonic excitation	15	(1)
Dynamic magnification factor	16	(2)
Response near resonance	17	(1)
Response to base excitation	18	(2)
Single Degree of Freedom System - II	20	(30)
Introduction	20	(1)
Laplace transform method	20	(1)
Laplace transform: examples	21	(1)
Inverse Laplace transform	21	(1)
Inverse Laplace transform: examples	21	(1)
Laplace transform of derivatives	22	(1)
Solution of differential equations	23	(1)
Some useful results	23	(4)
Unit step function	24	(1)
Shift theorems	24	(2)
Convolution theorem	26	(1)
Summary of some results	27	(1)
General solution of Duhamel integral	28	(2)
Solution of Duhamel integral: specific examples	30	(7)
Suddenly applied load	30	(1)
Rectangular impulse loading	31	(2)
Triangular impulse loading	33	(3)
Exponentially decaying force	36	(1)
Shock spectra	37	(3)
Rectangular impulse	37	(1)
Triangular impulse	38	(2)
Numerical evaluation of Duhamel integral	40	(3)
Program: Duhamel	43	(3)
Use of Duhamel: example	46	(1)
Earthquake response spectra	46	(4)
Introduction to Multiple Degree of Freedom Systems	50	(34)
Introduction	50	(2)
Eigen-value problem	52	(13)
Determinants	53	(2)
Cramer's rule	55	(1)
Positive definiteness	55	(1)
Eigen-values of [K-ω2 M]: examples	56	(3)
Eigen-vectors of [K-ω2 M]	59	(6)
General and special eigen-value problems	65	(2)
Conversion from general to special eigen-value problem	66	(1)
Properties of eigen-values and eigen-vectors	67	(5)
Characteristic equation	67	(1)
Orthogonal property of eigen-vectors	68	(2)
Arbitrary vector in terms of eigen-vectors	70	(2)
Mode superposition method	72	(8)
Undamped forced response	72	(1)
Undamped forced response: Example	73	(1)
Square root of the sum of the squares approximation	74	(2)
Damped forced response	76	(1)
Damped forced response: Example	77	(3)
Response to base acceleration	80	(4)
Earthquake participation factors: Example	81	(3)
Matrix Routines	84	(23)
Introduction	84	(1)
Matrix manipulations	84	(1)
Cholesky factorisation	85	(5)
Solution of simultaneous equations by Cholesky decomposition	86	(2)
Converting a general to a special eigen-value problem	88	(2)
Crout factorisation	90	(3)
Solution of simultaneous equations by Crout decomposition	92	(1)
Gaussian elimination method	93	(4)
Gaussian elimination method: example	95	(2)
Determinant of a matrix	97	(1)
Determinant from Gaussian elimination	97	(1)
Determinant from Cholesky factorisation	97	(1)
Efficient storage schemes	98	(3)
Rectangular format	98	(1)
Vector or Skyline format	98	(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