1 Foundations of Dynamics
1.1 Kinematics of particles
1.2 Kinetics of particles
1.3 Power, work, and energy
1.4 Conservation of energy
1.5 Dynamics of rigid bodies
1.6 Example
1.7 The Euler{Lagrange equations
1.8 Summary
2 Numerical Solution of Ordinary Di_erential Equations
2.1 Why numerical methods?
2.2 Practical implementation
2.3 Analysis of a first order equation
2.4 Analysis of second order di_erential equations
2.4.1 The central di_erence method
2.4.2 The generalized trapezoidal rule
2.4.3 Newmark's method
2.5 Performance of the methods
2.6 Summary
3 Single-Degree-of-Freedom Systems
3.1 The SDOF oscillator
3.2 Undamped free vibration
3.3 Damped free vibration
3.4 Forced vibration
3.4.1 Suddenly applied constant load
3.4.2 Sinusoidal load
3.4.3 General periodic loading
3.5 Earthquake ground motion
3.6 Nonlinear response
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xiv Contents
3.7 Integrating the equation of motion
3.8 Example
4 Systems with Multiple Degrees of Freedom
4.1 The 2{DOF system as a warm-up problem
4.2 The shear building
4.3 Free vibration of the NDOF system
4.3.1 Orthogonality of the eigenvectors
4.3.2 Initial conditions
4.4 Structural damping
4.4.1 Modal damping
4.4.2 Rayleigh damping
4.4.3 Caughey damping
4.4.4 Non-classical damping
4.5 Damped forced vibration of the NDOF system
4.6 Resonance in NDOF systems
4.7 Numerical integration of the NDOF equations
5 Nonlinear Response of NDOF Systems
5.1 A point of departure
5.2 The shear building, revisited
5.3 The principle of virtual work
5.4 Nonlinear dynamic computations
5.5 Assembly of equations
5.6 Adding damping to the equations of motion
5.7 The structure of the NDOF code
5.8 Implementation
6 Earthquake Response of NDOF Systems
6.1 Special case of the elastic system
6.2 Modal recombination
6.3 Response spectrum methods
6.4 Implementation
6.5 Example
7 Special Methods for Large Systems
7.1 Ritz projection onto a smaller subspace
7.2 Static correction method
7.3 Summary
8 Dynamic Analysis of Truss Structures
8.1 What is a truss?
8.2 Element kinematics
8.3 Element and nodal static equilibrium
8.4 The principle of virtual work
8.5 Constitutive models for axial force
Contents xv
8.6 Solving the static equations of equilibrium
8.7 Dynamic analysis of truss structures
8.8 Distributed element mass
8.9 Earthquake response of truss structures
8.10 Implementation
8.11 Example
9 Axial Wave Propagation
9.1 The axial bar problem
9.2 Motion without applied loading
9.3 Classical solution by separation of variables
9.4 Modal analysis with applied loads
9.5 The Ritz method and _nite element analysis
9.5.1 Dynamic principle of virtual work
9.5.2 Finite element functions
9.5.3 A slightly di_erent formulation
9.5.4 Boundary conditions
9.5.5 Higher order interpolation
9.5.6 Initial conditions
9.6 Axial bar dynamics code
10 Dynamics of Planar Beams: Theory
10.1 Beam kinematics
10.1.1 Motion of a beam cross section
10.1.2 Strain{displacement relationships
10.1.3 Normal and shear strain
10.2 Beam kinetics
10.3 Constitutive equations
10.4 Equations of motion
10.4.1 Balance of linear momentum
10.4.2 Balance of angular momentum
10.5 Summary of beam equations
10.6 Linear beam theory
10.6.1 Linearized kinematics
10.6.2 Linearized kinetics
10.6.3 Linear equations of motion
10.6.4 Boundary conditions
10.6.5 Initial conditions
11 Wave Propagation in Beams
11.1 Propagation of a train of sinusoidal waves
11.1.1 Bernoulli{Euler beam
11.1.2 Rayleigh beam
11.1.3 Timoshenko beam
11.2 Solution by separation of variables
xvi Contents
11.3 The Bernoulli{Euler beam
11.3.1 Implementing boundary conditions
11.3.2 Natural frequencies
11.3.3 Orthogonality of the eigenfunctions
11.3.4 Implementing the initial conditions
11.3.5 Modal vibration
11.3.6 Other boundary conditions
11.3.7 Wave propagation
11.3.8 Example: Simple{simple beam
11.4 The Rayleigh beam
11.4.1 Simple{simple Rayleigh beam
11.4.2 Orthogonality relationships
11.4.3 Wave propagation: Simple{simple beam
11.4.4 Other boundary conditions
11.4.5 Implementation
11.5 The Timoshenko beam
11.5.1 Simple{simple beam
11.5.2 Wave propagation
11.5.3 Numerical example
11.6 Summary
12 Finite Element Analysis of Linear Beams
12.1 The dynamic principle of virtual work
12.1.1 The Ritz approximation
12.1.2 Initial conditions
12.1.3 Selection of Ritz functions
12.1.4 Beam _nite element functions
12.1.5 Ritz functions and degrees of freedom
12.1.6 Local to global mapping
12.1.7 Element matrices and assembly
12.2 The Rayleigh beam
12.2.1 Virtual work for the Rayleigh beam
12.2.2 Finite element discretization
12.2.3 Initial conditions for wave propagation
12.2.4 The Rayleigh beam code
12.2.5 Example
12.3 The Timoshenko beam
12.3.1 Virtual work for the Timoshenko beam
12.3.2 Finite element discretization
12.3.3 The Timoshenko beam code
12.3.4 Veri_cation of element performance
12.3.5 Wave propagation in the Timoshenko beam
Contents xvii
13 Nonlinear Dynamic Analysis of Planar Beams
13.1 Equations of motion
13.2 The principle of virtual work
13.3 Tangent functional
13.4 Finite element discretization
13.5 Static analysis of nonlinear planar beams
13.5.1 Solution by Newton's method
13.5.2 Static implementation
13.5.3 Veri_cation of static code
13.6 Dynamic analysis of nonlinear planar beams
13.6.1 Solution of the nonlinear di_erential equations
13.6.2 Dynamic implementation
13.6.3 Example
13.7 Summary
14 Dynamic Analysis of Planar Frames
14.1 What is a frame?
14.2 Equations of motion
14.3 Inelasticity
14.3.1 Numerical integration of the rate equations
14.3.2 Material tangent
14.3.3 Internal variables
14.3.4 Speci_c model for implementation
14.4 Element matrices
14.4.1 Finite element discretization
14.4.2 Local to global transformation
14.5 Static verification
14.6 Dynamics of frames
14.6.1 Earthquake ground motion
14.6.2 Implementation
14.6.3 Examples
14.6.4 Sample input function
A Newton's Method
A.1 Linearization
A.2 Systems of equations
B The Directional Derivative
B.1 Ordinary functions
B.2 Functionals
C The Eigenvalue Problem
C.1 The algebraic eigenvalue problem
C.2 The QR algorithm
C.3 Eigenvalue problems for large systems
C.4 Subspace iteration
xviii Contents
D Finite Element Interpolation
D.1 Polynomial interpolation
D.2 Lagrangian interpolation
D.3 Ritz functions with hp interpolation
D.4 Lagrangian shape functions
D.5 C0 Bubble functions
D.6 C1 Bubble functions
E Data Structures for Finite Element Codes
E.1 Structure geometry and topology
E.2 Structures with only nodal DOF
E.3 Structures with non-nodal DOF
F Numerical Quadrature
F.1 Trapezoidal rule
F.2 Simpson's rule
F.3 Gaussian quadrature
F.4 Implementation
F.5 Examples
Index