Civil Engineering Association

Dynamic Analysis of Structures reflects the latest application of structural dynamics theory to produce more optimal and economical structural designs. Written by an author with over 37 years of researching, teaching and writing experience, this reference introduces complex structural dynamics concepts in a user-friendly manner. The author includes carefully worked-out examples which are solved utilizing more recent numerical methods. These examples pave the way to more accurately simulate the behavior of various types of structures. The essential topics covered include principles of structural dynamics applied to particles, rigid and deformable bodies, thus enabling the formulation of equations for the motion of any structure.


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