Dynamic Analysis Of Structures

Introduction to Dynamic Analysis Of Structures

All real physical structures, when subjected to loads or displacements, behave dynamically.  The additional inertia forces, from Newton’s second law, are equal to the mass times the acceleration.  If the loads or displacements are applied very slowly then the inertia forces can be neglected and a static load analysis can be justified. Hence, dynamic analysis is a simple extension of static analysis.

In addition, all real structures potentially have an infinite number of displacements.Therefore, the most critical phase of a structural analysis is to create a computer model, with a finite number of massless members and a finite number of node (joint) displacements, that will  simulate the behavior of the real structure.  The mass of a structural system, which can be accurately estimated, is lumped at the nodes.  Also, for linear elastic structures the stiffness properties of the members, with the aid of experimental data, can be approximated with a high degree of confidence.

However, the dynamic loading, energy dissipation properties and boundary (foundation) conditions for many structures are difficult to estimate.  This is always true for the cases of seismic input or wind loads.

STEP BY STEP SOLUTION METHOD

The most general solution method for dynamic analysis is an incremental method in which the equilibrium equations are solved at times  ∆ ∆ ∆ t   t   t , 2 , 3 ,   etc.  There are a large number of different incremental solution methods.

In general, they involve a solution of the complete set of equilibrium equations at each time increment.  In the case of nonlinear analysis, it may be necessary to reform the stiffness matrix for the complete structural system for each time step.  Also, iteration may be required within each time increment to satisfy equilibrium.  As a result of the large computational requirements it can take a significant amount of time to solve structural systems with just a few hundred degrees-of-freedom.

MODE SUPERPOSITION METHOD
The most common and effective approach for seismic analysis of linear structural systems is the mode superposition method.  This method, after a set of orthogonal vectors are evaluated, reduces the large set of global equilibrium equations to a relatively small number of uncoupled second order differential equations.  The numerical solution of these equations involves greatly reduced computational time.

RESPONSE SPECTRA ANALYSIS

The basic mode superposition method, which is restricted to linearly elastic analysis, produces the complete time history response of joint displacements and member forces due to a specific ground motion loading [1,2].  There are two major disadvantages of using this approach.  First, the method produces a large amount of output information that can require an enormous amount of computational effort to conduct all possible design checks as a function of time.  Second, the analysis must be repeated for several different earthquake motions in order to assure that all the significant modes are excited, since a response spectrum for one earthquake, in a specified direction, is not a smooth function.

SOLUTION IN THE FREQUENCY DOMAIN
The basic approach, used to solve the dynamic equilibrium equations in the frequency domain, is to expand the external loads  F(t) in terms of Fourier series or Fourier integrals.  The solution is in terms of complex numbers that cover the time span from  -∞ to  ∞.  Therefore, it is very effective for periodic types of loads such as mechanical vibrations, acoustics, sea-waves and wind [1].

SOLUTION OF LINEAR EQUATIONS
The step-by-step solution of the dynamic equilibrium equations, the solution in the frequency domain, and the evaluation of eigenvectors and Ritz vectors all require the solution of linear equations of the following form:

AX=  B

Where A is an ’N by N’ symmetric matrix which contains a large number of zero terms.  The ’N by M’ X displacement and B load matrices indicate that more than one load condition can be solved at the same time.

UNDAMPED HARMONIC RESPONSE
The most common and very simple type of dynamic loading is the application of steady-state harmonic loads of the following form:

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