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Normally, For Earthquake Loading Direct Numerical Integration Is Very Slow

The most general approach for the solution of the dynamic response of structural systems is the direct numerical integration of the dynamic equilibrium equations. This involves, after the solution is defined at time zero, the attempt to satisfy dynamicequilibrium at discrete points in time. Most methods use equal time intervals at ∆t , 2 ∆t , 3∆t ........ N∆t . Many different numerical techniques have previously been presented; however, all approaches can fundamentally be classified as either explicit or implicit integration methods. Explicit methods do not involve the solution of a set of linear equations at each step. Basically, these methods usethe differential equation at time “ t ” to predict a solution at time “ t + ∆t ”. For most real structures, which contain stiff elements, a very small time step is required in order to obtain a stable solution. Therefore, all explicit methods are conditionally stable with respect to the size of the time step. Implicit methods attempt to satisfy the differential equation at time “ t ” after thesolution at time “ t − ∆t ” is found. These methods require the solution of a set of linear equations at each time step; however, larger time steps may be used. Implicit methods can be conditionally or unconditionally stable.



There exist a large number of accurate, higher-order, multi-step methods that have been developed for the numerical solution ofdifferential equations. These multistep methods assume that the solution is a smooth function in which the higher derivatives are continuous. The exact solution of many nonlinear structures requires that the accelerations, the second derivative of the displacements, are not smooth functions. This discontinuity of the acceleration is caused by the nonlinear hysteresis of most structural materials, contactbetween parts of the structure, and buckling of elements. Therefore, only single-step methods will be presented in this chapter. Based on a significant amount of experience, it is the conclusion of the author that only single-step, implicit, unconditional stable methods be used for the step-by-step seismic analysis of practical structures.

In 1959 Newmark [1]presented a family of single-step integration methods for the solution of structural dynamic problems for both blast and seismic loading. During the past 40 years Newmark’s method has been applied to the dynamic analysis of many practical engineering structures. In addition, it has been modified and improved by many other researchers. In order to illustrate the use of this family of numericalintegration methods consider the solution of the linear dynamic equilibrium equations written in the following form:
& & & Mu t + Cu t + Ku t = Ft


The direct use of Taylor’s series provides a rigorous approach to obtain the following two additional equations:
& u t = u t-∆t + ∆t u t-∆t +

∆t 2 ∆t 3 & & & &t- ∆t +...... & u t-∆t + u 2 6


& & & & u t = u t-∆t + ∆t u t-∆t +∆t 2 & &t- ∆t + ...... & u 2




Newmark truncated these equations and expressed them in the following form:
& u t = u t-∆t + ∆t u t- ∆t +

∆t 2 & & u t-∆t + β ∆t 3&&& u 2

(20.2a) (20.2b)

& & & & u t = u t-∆t + ∆t u t-∆t + γ ∆t 2&&& u

If the acceleration is assumed to be linear within the time step, the following equation can be written:&& = & u & & & & ( u t − u t − ∆t ) ∆t


The substitution of Equation (20.3) into Equations (20.2a and b) produces Newmark’s equations in standard form

1 & & & & & u t = u t-∆t + ∆t u t-∆t + ( − β ) ∆t 2 u t- ∆t + β ∆t 2 u t 2
& & & & & & u t = u t-∆t + (1 − γ ) ∆t u t-∆t + γ ∆t u t

(20.4a) (20.4b)

Newmark used Equations (20.4a, 20.4b and 20.1) iteratively, for each time...
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