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

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

Biography

Indrajit Chowdhury, Shambhu P. Dasgupta

‘The two-volume set will be useful as a reference for professionals involved in earthquake or dunamic analysis or the design of machine foundations in the oil, gas, and energy sector. The books can also be used in advanced courses in structural dynamics, soil dynamics, analysis and machined foundations, and earthquake engineering.’
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