Dynamics of Structure and Foundation - A Unified Approach

1 Introduction *1.1 Why this book *1.2 Why the topic of dynamics? *1.3 The demography of the book 2 Theory of elasticity and numerical methods in engineering *2.1 Mechanics of continua: Stress and strain *2.2 Concept of strain *2.2.1 Displacement field *2.2.2 Concept of small domain *2.2.3 Body undergoing small deformation *2.2.4 Strain tensor *2.2.5 Derivative of a vector fixed in a moving reference *2.2.6 Physical interpretation of strain tensor *2.2.7 Cubical dilatation *2.2.8 Transformation of strains *2.2.9 Equations of compatibility *2.3 Stresses *2.3.1 Concept of stress *2.3.2 Principal stresses and strains, invariants *2.3.3 Cauchy's stress quadric and Mohr diagram *2.3.4 Plane stress conditions *2.3.5 Plane strain conditions *2.3.6 Octahedral stresses and strains *2.3.7 Spherical and deviatoric stress components *2.4 Constitutive relations *2.5 Equations of equilibrium *2.5.1 Some useful expressions *2.5.2 Differential equations at a point (general) *2.5.3 Differential equations at a point (in terms of stresses) *2.5.4 Differential equations at a point (in terms of displacements) *2.5.5 General solution *2.5.6 Two-dimensional cases *2.6 Theorems of elasticity *2.6.1 Principles of superposition *2.6.2 Strain energy *2.6.3 Virtual work *2.7 Mechanics of homogeneous isotropic elastic bodies *2.7.1 Material derivative of volume integral *2.7.2 The equations of continuity *2.7.3 The equations of motion *2.7.4 Moment of momentum *2.7.5 Basic equation of motion of an elastic body *2.7.6 Various strain measures *2.7.7 Solution of the three-dimensional equation *2.7.8 Static solutions with no body forces *2.8 Some basics *2.8.1 Summary of governing equations/relations *2.8.2 Lame's equations [combining all equations, governing differential equation in terms * of u, v, w] *2.9 Some classical solutions of elastostatics *2.9.1 Kelvin (1848) problem - A single force acting in the interior of an infinite solid. *(Malvern 1969, Fung 1965) *2.9.2 Boussinesq (1878) problem - A normal force acting on the surface of a semi-infinite solid *2.9.3 Cerruti (1882) problem - A tangential force acting on the surface of a semi-infinite solid *(Mindlin 1936, Love 1944), [same as Boussinesq's problem, only the load acting on the surface is horizontal] *2.9.4 Mindlin's (1936) solution *2.9.5 Theories of Elastodynamics *2.10 Numerical methods in engineering: Basics and applications *2.10.1 Introduction *2.10.2 Approximate methods applied to boundary value problems *2.11 The Finite Difference Method (FDM) *2.11.1 Application to ordinary differential equations (ode) *2.11.2 Application to partial differential equations *2.11.3 Laplace and Biharmonic equations *2.11.4 Irregular meshes or grids *2.11.5 Laplace operator with irregular mesh *2.11.6 Bi-harmonic equations with irregular meshes *2.11.7 Refined finite difference analysis *2.11.8 Free edged plates with different boundary conditions *2.11.9 Finite difference in polar co-ordinate *2.11.10 Finite difference solution for initial value problem *2.11.11 Finite difference solution for initialboundary value problem *2.11.12 Finite difference application in dynamics *2.12 The finite element method *2.12.1 The finite element club and its members *2.12.2 Brief history on the development of finite element method *2.12.3 The basic philosophy *2.12.4 Displacement based derivation of stiffness matrix *2.12.5 Plane strain CST element *2.12.6 Why constant strain and how effective is the element? *2.12.7 Why convergence improve with refined meshes *2.12.8 The constitutional laws which bound the developers *2.12.9 The rule of polynomial - the entry rule to developers club *2.12.10 How do we select the polynomial function correctly? *2.12.11 The law of convergence - the three commandments *2.12.12 Non-conforming elements an exception to the law *2.12.13 Natural coordinates: the gateway to numerical analysis through computer *2.12.14 Numerical integration technique used for FEM *2.12.15 Gauss quadrature scheme for numerical integration *2.12.16 Stiffness matrix for 4-nodded rectangular element under plane strain condition *2.12.17 Iso-parametric formulation for elements with arbitrary shape *2.12.18 Other form of isoparametric elements *2.12.19 Iso-parametric formulation of CST element *2.12.20 Condensation - The Houdini trick of vanishing nodes *2.12.21 Alternative method of deriving a quadrilateral element *2.12.22 The Reverse Logic - How correct it is? *2.12.23 Incompatible or Non-conforming element - Where two wrongs make one right *2.12.24 How tough is this lawbreaker? *2.12.25 Taylor's improved incompatible quadrilateral *2.12.26 Higher order finite elements - The second generation members of the FEM family *2.12.27 Lagrange's interpolation function - An extension to school co-ordinate geometry *2.12.28 Elements of Serendipidity family - named after Princes of Serendip *2.12.29 Other type of higher order elements *2.12.30 Plate element - the problem child of FEM family *2.12.31 Triangular plate element in bending - the Catch-22 element *2.12.32 DKT Plate element *2.12.33 Rectangular plate element in bending mode *2.12.34 Four-nodded quadrilateral plate element in bending *2.12.35 Three Dimensional Hexahedral Element - One last to bore you *2.12.36 Twenty-nodded hexahedral element *2.12.37 The patch and eigenvalue test - The performance warranty certificates *2.12.38 A retrospection on what we presented so far *2.12.39 The assemblers - the tailors who stitches the pieces to give final shape *2.12.40 Formulation of the global stiffness matrix *2.12.41 Transformation in space for 3D analysis *2.12.42 Members vertical in space - a special case *2.12.43 Global stiffness matrix and transformation of finite element continuum *2.12.44 Implementing the boundary condition *2.12.45 Formulating specified support displacement *2.12.46 Calculation of element stress and displacements *2.12.47 Solution of equilibrium equation *2.12.48 Gaussian elimination - The technique of back substitution *2.12.49 The LDLT decomposition technique *2.12.50 Frontal wave solution - Iron's technique reflecting present consumer market *2.12.51 The World of Boris Galerkin - A look at finite element beyond stress analysis *2.12.52 Thermal analysis of composite wall in one dimension *2.12.53 The user domain-rookies, fakes, control freaks and clever Ivans *2.12.54 Finite element model of table top centrifugal compressor with dynamic soil-structure interaction *2.12.55 Static soil-structure interaction analysis of a pedestrian subway below ground 3 Basics of lumped parameter vibration *3.1 Introduction *3.2 Single-degree-of freedom *3.2.1 Free vibration: Undamped case *3.2.2 Forced vibration *3.2.3 Steady-state analysis: Mechanical impedance method *3.2.4 Q-values and their interpretation *3.2.5 Power absorption *3.2.6 Heavy damping *3.2.7 Frequency dependent loading *3.2.8 Dissipation of energy *3.2.9 Velocity squared damping *3.2.10 Solid damping *3.2.11 Analysis of friction forces (Coulomb friction, dry friction) *3.2.12 Response under impulsive loading *3.2.13 General solution for any arbitrary forcing system *3.2.14 Response spectra *3.2.15 Earthquake type of excitation *3.3 Stability of dynamic solutions *3.3.1 Phase planes and stability of solution *3.3.2 Basics of differential equation *3.3.3 Homogeneous Systems with Constant Coefficients, Phase Plane, Critical Points *3.3.4 Phase plane method for SDOF system *3.3.5 Self-excited oscillations *3.3.6 Autonomous system *3.3.7 State space method *3.3.8 State speed *3.3.9 Stability of the solution *3.4 Multiple-degrees-of-freedom systems *3.4.1 Free vibration: Undamped system *3.4.2 Steady-state analysis: Mechanical impedance method *3.4.3 Coupled translation and rotation *3.4.4 Forced vibration *3.4.5 Semi-definite systems *3.5 Nonlinear systems *3.5.1 Free vibrations *3.5.2 Forced vibrations *3.5.3 Large amplitudes in response: Order and chaos 4 An introduction to soil-structure systems under statical condition *4.1 Introduction *4.1.1 What we did twenty years ago *4.1.2 The Present Scenario. *4.2 Soil-structure interaction *4.3 Static soil-structure interaction *4.4 Non uniform contact pressure *4.5 Various soil models-the tools in the toolkit *4.5.1 Winkler springs *4.5.2 Estimation of sub-grade modulus *4.6 Evaluation of nodal springs *4.6.1 So the ground rule is *4.7 Limitations/advantages of Winkler spring model *4.8 Finite element models *4.8.1 Plate element *4.9 Finite element analysis of plate with soil stiffness based on isotropic elastic half space theory *4.9.1 Displacement profile of soil under a foundation based on half space theory *4.10 Finite grid method/equivalent beam element, the unsung work horse *4.11 FEM application for problems of class 2D *4.12 Plane stress and plane strain condition *4.12.1 Plane stress condition *4.12.2 Plane strain condition *4.13 FEM model for the vertical cut problem *4.14 Infinite finite element a logical paradox *4.15 Basis of formulation of the infinite element *4.15.1 What does it really mean? *4.15.2 Why did we transform the co-ordinate and what did we gain out of it? *4.16 Material property affecting the model *4.17 Relation between sub-grade modulus and modulus of elasticity *4.18 Selection of Poisson's ratio *4.19 Limitation and advantages of finite element method in static soil structure interaction problem 5 Concepts in structural and soil dynamics *5.1 Introduction *5.2 A brief history of dynamic analysis of structure and foundation in civil engineering *5.2.1 Basic concepts *5.2.2 Orthogonal transformation or the transformation basis *5.2.3 Direct integration technique, the alternate approach *5.2.4 Wilson-Theta method *5.3 Eigen value analysis *5.3.1 Some techniques for eigen value analysis *5.3.2 Standard Jacobi's technique *5.3.3 Generalized Jacobi technique *5.3.4 Dynamic analysis based on finite element method *5.4 Introduction to soil and elasto-dynamics *5.4.1 Development of soil dynamics to the present state of art *5.4.2 One-dimensional propagation of wave through an elastic medium *5.4.3 Three-dimensional propagation of waves in an infinite elastic medium *5.4.4 Propagation of waves in polar co-ordinates *5.4.5 Reflection/Refraction *5.4.6 Where does this all lead to? *5.4.7 Some background on integral transforms and other mathematical theorems *5.5 Halfspace elastodynamic solution *5.5.1 Lamb's solution for two-dimensional problem *5.5.2 Pekeris' solution for surface pulse *5.5.3 Pekeris' solution for buried pulse *5.5.4 Interpretation of Pekeris' solution *5.5.5 Chang's Solution to dynamic response for horizontal surface loading *5.6 Geotechnical earthquake analysis *5.6.1 Soil dynamics and earthquake *5.6.2 Waves induced by underground blast *5.7 Geotechnical analysis of machine foundations *5.7.1 Soil dynamics and machine foundation *5.7.2 Reissner's method *5.7.3 Sung and Quinlan's method *5.7.4 Bycroft's solution for dynamic response of foundation *5.7.5 Reissner and Sagoci's method of torsional oscillation *5.7.6 Hseih's method for dynamic response of foundation *5.7.7 Lysmer and Richart's model for dynamic response of foundation *5.7.8 Hall's analog for sliding and rocking vibration *5.7.9 Vibration of rectangular footings resting on elastic half-space *5.7.10 Rigid strip footing *5.7.11 Luco and Westmann solution for rigid strip footing *5.7.12 Dynamic response of circular footings *5.7.13 Vibration of an elastic half space under rectangular loading *5.8 Vibration of embedded footings *5.8.1 Embedment effect on foundation *5.8.2 Research carried out in India *5.8.3 Energy transmitted from a circular area *5.9 Finite element solution for foundation dynamics *5.9.1 Soil dynamics and finite element analysis *5.9.2 Use of structural boundary conditions *5.9.3 Use of spring or boundary elements *5.9.4 Use of transmitting/silent boundaries with finite elements *5.9.5 Standard viscous and Rayleigh boundary elements *5.9.6 Paraxial boundaries *5.9.7 Infinite finite elements *5.9.8 Epilogue References Subject index