FIGURE 3.30 Spectral radii for Newmark method and Wilson-0 method. (Reprinted from Ref. [1, p. 357] by courtesy of Marcel Dekker, Inc.)

For reasonable accuracy, Dt/T = 0.1 is recommended, where T is the natural period of highest mode considered in the analysis. Properly selecting the number of modes depends on an individual problem. For the direct integration approach, frequency analysis is not needed. Therefore, the periods of desired methods are not available. A suggested approach is to follow the building code that stipulates the number of modes, and the lower bound of period should be considered. Generally, we can use one Dt and another slightly smaller Dt to find solutions continuously, until two successive solutions are reasonably close. We can also inspect the exciting function to select approximate Dt (for instance, Dt should not be greater than the intervals of the earthquake records).

3.6.5 Nondeterministic Analysis Introduction to Stochastic Seismic Response Analysis

As discussed in Section 3.6.1, when a force cannot be specified as a definite function by time, the response analysis should be determined through a stochastic approach. Earthquake force is a typical example that has inherent uncertainty in magnitude and in time variation. For brief presentation, the seismic response is chosen for discussion in this section. For a structure subjected to an earthquake, the motion equation is similar to Equation 3.94 and is expressed as

where xg is the ground motion acceleration and {In} is the influence coefficient factor. By the modal analysis method, the motion equation is decomposed to N (d.o.f. of the system) independent second-order differential equations

€'+ 2pipix'i+ p2 x'= r;€g i = 1,2,..., N (3.113)

where x' is the modal displacement for the i'th modes and has the relation with the floor displacement as {x} = [F']{x'} and [F'] is the normalized modal matrix. pi and pi are damping coefficient and angle frequency for the ith mode. ^ — {F'}T{in} is called a participation factor, which implies how much the ¿th mode participates in synthesizing the structural total load. In random analysis, the weakly stationary input to a linear system leads to a weakly stationary output. For the weakly stationary stochastic process X, the autocorrelation function RX(t) and the spectral density function SX(o) are a Fourier transform pair and with the following relationship:


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