Fundamentals of Structural Dynamics

1 Foundations of Dynamics1.1 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 xiii 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