Dynamic Analysis of Structures

Front Matter
Copyright
Dedication
Preface
General concepts and principles of structural dynamics
	Chapter outline
	Introduction
	Types of dynamic loads
	Dynamic degrees of freedom
	Dynamic model and formulation of the equation of motion of SDOF systems
	Derivation of the equations of motion using dAlemberts principle
	Principle of virtual displacements
	Hamiltons principle
	Lagranges equations
		Derivation of Lagranges equations
		Lagrange multipliers
		Small displacements
			Potential energy and stiffness matrix
			Kinetic energy and mass matrix
		Raleighs dissipation function
	Influence of the gravity loads
	Problems
	References and further reading
Single-degree-of-freedom systems: Free vibrations
	Chapter outline
	Introduction
	Free undamped vibrations
	Free damped vibrations
		Critically damped system
		Underdamped system
		Overdamped system
	Conservation of energy in an undamped system
	Problems
	References and further reading
Chapter 3: Single-degree-of-freedom systems: Forced vibrations
	3.1. Introduction
	3.2. Response to harmonic loading
		3.2.1. Response of undamped systems to harmonic loading
		3.2.2. Response of damped systems to harmonic loading
	3.3. Response to arbitrary dynamic loading-Duhamels integral
		3.3.1. Undamped vibrations
		3.3.2. Damped vibrations
	3.4. Analytical evaluation of the Duhamel integral-applications
		3.4.1. Response to step function load
		3.4.2. Response to ramp function load
		3.4.3. Response to step function load with finite rise time. Static load
	3.5. Response to impulsive loads
		3.5.1. Rectangular pulse load
		3.5.2. Triangular pulse load
		3.5.3. Asymmetrical triangular pulse load
		3.5.4. Response to piecewise linear loading
	3.6. Response to a periodic loading
		3.6.1. Periodic loads
		3.6.2. Fourier series
		3.6.3. Response of the SDOF system to periodic excitation
	3.7. Response to unit impulse
		3.7.1. The delta function or Diracs delta function
		3.7.2. Response to unit impulse
		3.7.3. Response to arbitrary loading
		3.7.4. The reciprocal theorem in dynamics
	3.8. Problems
	References and further reading
Numerical integration of the equation of motion
	Chapter outline
	Introduction
	The central difference method
	The average acceleration method
	The analog equation method
	Stability of the numerical integration methods
		Errors in the numerical integration
		Difference equations
		Difference equations and stability of the numerical integration methods
		Stability of the central difference method
		Stability of the average acceleration method
		Stability of the analog equation method
	Accuracy of the numerical integration
	Problems
	References and further reading
Chapter 5: Nonlinear response: Single-degree-of-freedom systems
	Chapter outline
	5.1. Introduction
	5.2. The central difference method
	5.3. The average acceleration method
	5.4. The analog equation method
	5.5. Problems
	References and further reading
Response to ground motion and vibration isolation
	Chapter outline
	Introduction
	Equation of motion: Relative displacement
		Response spectra
	Equation of motion in terms of the total displacement
	Vibration isolation
		Transmission of force
		Transmission of motion
	Problems
	References and further reading
Damping in structures
	Chapter outline
	Introduction
	Loss of energy due to damping
	Equivalent viscous damping
	Hysteretic damping
	Coulomb damping
		Free vibrations with Coulomb damping
		Forced vibrations with Coulomb damping
	Damping modeling via fractional derivatives
		Introduction
		The fractional derivative
	Measurement of damping
		Free vibration decay method
		Resonance amplitude method
		Width of response curve method
	Problems
	References and further reading
Generalized single-degree-of-freedom systems-Continuous systems
	Chapter outline
	Introduction
	Generalized single-degree-of-freedom systems
	Continuous systems
		Introduction
		Solution of the beam equation of motion
		Free vibrations of beams
			The simply supported beam
			The cantilever beam
		Orthogonality of the free-vibration modes
		Forced vibrations of beams
	Problems
	References and further reading
Analysis in the frequency domain
	Chapter outline
	Introduction
	Complex form of the Fourier series
	Complex dynamic response to periodic load
	Fourier integral representation of a nonperiodic load
	Response to a nonperiodic load
	Discrete Fourier transform
	Application of the discrete Fourier transform to dynamic analysis
	Fast Fourier transform
		The Sande-Tukey algorithm
	Problems
	References and further reading
Multi-degree-of-freedom systems: Models and equations of motion
	Introduction
	Systems with localized mass and localized stiffness
	Systems with distributed mass and localized stiffness
	Systems with localized mass and distributed stiffness
		The method of influence coefficients
			Elastic forces
			Damping forces
			Inertial forces
	Systems with distributed mass and distributed stiffness
		The method of global shape functions
	Mixed systems
	Transformations of the equations of motion
	Problems
	References and further reading
The finite element method
	Introduction
	The finite element method for the plane truss
		Properties of the plane truss element
			The method of the Lagrange equations
			The method of virtual work
		Transformation of the nodal coordinates of the truss element
		Equation of motion of the plane truss
		Steps to formulate the equations of motion for a plane truss by the finite element method
		Modification of the equations of motion due to the supports of the structure
	The finite element method for the plane frame
		Properties of the plane frame element
			The method of the Lagrange equations
			The method of virtual work
		Transformation of the nodal coordinates of the plane frame element
	Static condensation: Guyans reduction
	Flexural vibrations of a plane frame
	Reduction of the degrees of freedom due to constraints
	Axial constraints in the plane frame
	The finite element method for the plane grid
		Properties of the plane grid element
		Transformation of the nodal coordinates of the plane grid element
	The finite element method for the space frame
		Properties of the space frame element
		Transformation of the nodal coordinates of the space frame element
	The finite element method for the space truss
		Properties of the space truss element
		Transformation of the nodal coordinates of the space truss element
	Rigid bodies within flexible skeletal structures
		Rigid bodies in spaces frames
		Rigid bodies in spaces trusses, plane grids, plane frames, and plane trusses
	Problems
	References and further reading
Multi-degree-of-freedom systems: Free vibrations
	Chapter outline
	Introduction
	Free vibrations without damping
	Orthogonality of eigenmodes
	Eigenmodes of systems with multiple eigenfrequencies
	The linear eigenvalue problem
		The standard eigenvalue problem of linear algebra
		Properties of the eigenvalues and eigenvectors
		The generalized eigenvalue problem
	The Rayleigh quotient
	Properties of eigenfrequencies and modes of MDOF systems without damping: A summary
	Solution of the vibration problem without damping
	The method of mode superposition
	Solution of the vibration problem with damping
		Direct solution of the differential equation
		Linearization of the quadratic eigenvalue problem
		The use of a proportional viscous damping matrix
	Construction of a proportional damping matrix
		Rayleigh damping
		Additional orthogonality conditions: Caughey damping matrix
		Construction of the proportional damping matrix using the modal matrix
	Problems
	References and further reading
Numerical evaluation of the eigenfrequencies and eigenmodes
	Chapter outline
	Introduction
	The vector iteration method
		The inverse vector iteration method
		Convergence of the inverse vector iteration method
	Computation of higher-order eigenpairs
		The vector purification method
		The inverse vector iteration method with shifts
	Free or partially supported structure
	Problems
	References and further reading
Multi-degree-of-freedom systems: Forced vibrations
	Introduction
	The mode superposition method
	Modal contribution in the mode superposition method
		Modal participation
		Static correction method
		Error in mode superposition method due to truncation of higher modes
	Reduction of the dynamic degrees of freedom
		Static condensation
		Kinematic constraints
	Rayleigh-Ritz method
		Ritz transformation
		Approximation using Ritz vectors
	Selection of Ritz vectors
		Method of natural mode shapes
		The method of derived Ritz vectors
	Support excitation
		Multiple support excitation
		Uniform support excitation
	The response spectrum method
	Comparison of mode superposition method and Rayleigh-Ritz method
	Numerical integration of the equations of motions-Linear MDOF systems
		The central difference method (CDM)-Linear equations
		The average acceleration method (AAM)-Linear equations
		The analog equation method (AEM)-Linear equations
	Numerical integration of the equations of motions-Nonlinear MDOF systems
		The average acceleration method (AAM)-Nonlinear equations
		The analog equation method (AEM)-Nonlinear equations
	Problems
	References and further reading
Dynamic analysis of multistory buildings
	Chapter outline
	Introduction
	The multistory building
		The concept of the multistory element
		Nodal displacement matrix, nodal force matrix, transformation matrix, and stiffness matrix of the MSE
		Mass matrix of the MSE and multistory building
		Equation of motion of the multistory building
	Dynamic response of multistory buildings due to ground motion
	Problems
	References and further reading
Base isolation
	Chapter outline
	Introduction
	Analysis of the one-story building with base isolation
		Linear response of the isolation systems
		Modeling of nonlinear response of isolation systems
			Linear springs or laminated rubber bearings with flat sliders
			Linear springs or rubber bearings and nonlinear dampers
			Friction pendulum bearing
			High damping rubber bearing or lead rubber bearing-Bilinear model
			Hysteretic isolators-Bouc-Wen model
	The multistory building with base isolation
		The equation of motion of the multistory building with base isolation
		Reduction of the DOF of the superstructure using mode shapes
		Reduction of the superstructure DOF using Ritz vectors
		Linear response of the isolation system
		Nonlinear response of the isolation system
	Problems
	References and further reading