Dynamics of Structure and Foundation - A Unified Approach: 1. Fundamentals

884 Pages·2008·6.645 MB·English

by  Chowdhury, Indrajit

Dynamics of Structure and
Foundation – A Unified Approach
1. Fundamentals

Dynamics of Structure and
Foundation – A Unified Approach
1. Fundamentals
Indrajit Chowdhury
Petrofac International Ltd
Sharjah, United Arab Emirates
Shambhu P. Dasgupta
Department of Civil Engineering
Indian Institute of Technology
Kharagpur, India
Alsoavailable:
DynamicsofStructureandFoundation–AUnifiedApproach
1.Fundamentals
IndrajitChowdhury&ShambhuP.Dasgupta2009,CRCPress/Balkema
ISBN:978-0-415-47145-9(Hbk)
ISBN:978-0-203-88527-7(eBook)
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Contents
Preface xv
1 Introduction 1
1.1 Whythisbook 1
1.2 Why the topic of dynamics? 2
1.3 The demography of the book 4
2 Theory of elasticity and numerical methods
in engineering 9
2.1 Mechanics of continua: Stress and strain 9
2.2 Concept of strain 9
2.2.1 Displacement field 9
2.2.2 Concept of small domain 10
2.2.3 Body undergoing small deformation 13
2.2.4 Strain tensor 15
2.2.5 Derivative of a vector fixed in a moving
reference 18
2.2.6 Physical interpretation of strain tensor 24
2.2.7 Cubical dilatation 27
2.2.8 Transformation of strains 29
2.2.9 Equations of compatibility 34
2.3 Stresses 37
2.3.1 Concept of stress 37
2.3.2 Principal stresses and strains, invariants 42
2.3.3 Cauchy’s stress quadric and Mohr diagram 44
2.3.4 Plane stress conditions 48
2.3.5 Plane strain conditions 49
2.3.6 Octahedral stresses and strains 50
2.3.7 Spherical and deviatoric stress components 51
2.4 Constitutive relations 52
vi Contents
2.5 Equations of equilibrium 59
2.5.1 Some useful expressions 59
2.5.2 Differential equations at a point (general) 60
2.5.3 Differential equations at a point (in terms
of stresses) 60
2.5.4 Differential equations at a point (in terms
of displacements) 61
2.5.5 General solution 62
2.5.6 Two-dimensional cases 62
2.6 Theorems of elasticity 65
2.6.1 Principles of superposition 65
2.6.2 Strain energy 66
2.6.3 Virtual work 70
2.7 Mechanics of homogeneous isotropic elastic bodies 79
2.7.1 Material derivative of volume integral 82
2.7.2 The equations of continuity 83
2.7.3 The equations of motion 83
2.7.4 Moment of momentum 85
2.7.5 Basic equation of motion of an elastic body 86
2.7.6 Various strain measures 87
2.7.7 Solution of the three-dimensional equation 88
2.7.8 Static solutions with no body forces 90
2.8 Some basics 92
2.8.1 Summary of governing equations/relations 92
2.8.2 Lame’s equations [combining all equations,
governing differential equation in terms
of u, v, w] 93
2.9 Some classical solutions of elastostatics 94
2.9.1 Kelvin (1848) problem – A single force
acting in the interior of an infinite solid.
(Malvern 1969, Fung 1965) 94
2.9.2 Boussinesq (1878) problem – A normal force
acting on the surface of a semi-infinite solid 95
2.9.3 Cerruti (1882) problem – A tangential force
acting on the surface of a semi-infinite solid
(Mindlin 1936, Love 1944), [sameas
Boussinesq’s problem, only the load acting
on the surface is horizontal] 96
2.9.4 Mindlin’s (1936) solution 97
2.9.5 Theories of Elastodynamics 102
2.10 Numerical methods in engineering: Basicsand
applications 102
2.10.1 Introduction 102
Contents vii
2.10.2 Approximate methods applied to boundary
value problems 106
2.11 The Finite Difference Method (FDM) 115
2.11.1 Application to ordinary differential
equations (ode) 119
2.11.2 Application to partial differential equations 130
2.11.3 Laplace and Biharmonic equations 147
2.11.4 Irregular meshes or grids 154
2.11.5 Laplace operator with irregular mesh 155
2.11.6 Bi-harmonic equations with irregular meshes 156
2.11.7 Refined finite difference analysis 157
2.11.8 Free edged plates with different boundary
conditions 159
2.11.9 Finite difference in polar co-ordinate 161
2.11.10 Finite difference solution for initial value
problem 163
2.11.11 Finite difference solution for initial-
boundary value problem 163
2.11.12 Finite difference application in dynamics 165
2.12 The finite element method 167
2.12.1 The finite element club and its members 167
2.12.2 Brief history on the development of finite
element method 168
2.12.3 The basic philosophy 171
2.12.4 Displacement based derivation of stiffness
matrix 172
2.12.5 Plane strain CST element 182
2.12.6 Why constant strain and how effective is
the element? 182
2.12.7 Why convergence improve with refined
meshes 183
2.12.8 The constitutional laws which bound the
developers 183
2.12.9 The rule of polynomial – the entry rule to
developers club 184
2.12.10 How do we select the polynomial function
correctly? 186
2.12.11 The law of convergence – the three
commandments 187
2.12.12 Non-conforming elements an exception to
the law 189
2.12.13 Natural coordinates: the gateway to
numerical analysis through computer 189
viii Contents
2.12.14 Numerical integration technique used for
FEM 191
2.12.15 Gauss quadrature scheme for numerical
integration 191
2.12.16 Stiffness matrix for 4-nodded rectangular
element under plane strain condition 194
2.12.17 Iso-parametric formulation for elements
with arbitrary shape 199
2.12.18 Other form of isoparametric elements 210
2.12.19 Iso-parametric formulation of CST element 211
2.12.20 Condensation – The Houdini trick of
vanishing nodes 214
2.12.21 Alternative method of deriving a
quadrilateral element 217
2.12.22 The Reverse Logic – How correct it is? 219
2.12.23 Incompatible or Non-conforming element –
Where two wrongs make one right 219
2.12.24 How tough is this lawbreaker? 225
2.12.25 Taylor’s improved incompatible
quadrilateral 225
2.12.26 Higher order finite elements – The second
generation members of the FEM family 234
2.12.27 Lagrange’s interpolation function – An
extension to school co-ordinate geometry 238
2.12.28 Elements of Serendipidity family – named
after Princes of Serendip 242
2.12.29 Other type of higher order elements 247
2.12.30 Plate element – the problem child of FEM
family 248
2.12.31 Triangular plate element in bending – the
Catch-22 element 249
2.12.32 DKT Plate element 252
2.12.33 Rectangular plate element in bending mode 254
2.12.34 Four-nodded quadrilateral plate element in
bending 258
2.12.35 Three Dimensional Hexahedral Element –
One last to bore you 263
2.12.36 Twenty-nodded hexahedral element 268
2.12.37 The patch and eigenvalue test – The
performance warranty certificates 269
2.12.38 A retrospection on what we presented so far 275
2.12.39 The assemblers – the tailors who stitches
the pieces to give final shape 275
Contents ix
2.12.40 Formulation of the global stiffness matrix 276
2.12.41 Transformation in space for 3D analysis 281
2.12.42 Members vertical in space – aspecialcase 283
2.12.43 Global stiffness matrix and transformation
of finite element continuum 291
2.12.44 Implementing the boundary condition 291
2.12.45 Formulating specified support displacement 293
2.12.46 Calculation of element stress and
displacements 295
2.12.47 Solution of equilibrium equation 302
2.12.48 Gaussian elimination – The technique of
back substitution 303
2.12.49 The LDLT decomposition technique 304
2.12.50 Frontal wave solution – Iron’s technique
reflecting present consumer market 307
2.12.51 The World of Boris Galerkin – A look at
finite element beyond stress analysis 310
2.12.52 Thermal analysis of composite wall in one
dimension 312
2.12.53 The user domain-rookies, fakes, control
freaks and clever Ivans 319
2.12.54 Finite element model of table top centrifugal
compressor with dynamic soil-structure
interaction 322
2.12.55 Static soil-structure interaction analysis of a
pedestrian subway below ground 327
3 Basics of lumped parameter vibration 335
3.1 Introduction 335
3.2 Single-degree-of freedom 336
3.2.1 Free vibration: Undamped case 336
3.2.2 Forced vibration 351
3.2.3 Steady-state analysis: Mechanical impedance
method 355
3.2.4 Q-values and their interpretation 356
3.2.5 Power absorption 360
3.2.6 Heavy damping 362
3.2.7 Frequency dependent loading 363
3.2.8 Dissipation of energy 371
3.2.9 Velocity squared damping 373
3.2.10 Solid damping 374
3.2.11 Analysis of friction forces
(Coulomb friction, dry friction) 374