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By setting time difference t — 0, Equation 3.114b becomes

where a\ is called variance of X or mean square. The square root of the mean square is aX, known as the root-mean-square (RMS). For a zero mean stochastic process E(X(t)) — 0, RMS represents the magnitude of the response. Let the design value be xi — m-Sx, (3.116)

where l represents ith d.o.f. and m is the coefficient related to the probability of the safety. By the sum of the modal contribution, the mean square value of the lth d.o.f. can be obtained as

where F'H is the (th element in the ¿th modal vector, ri is the participation factor of the ith mode, S^(o) is the spectral density function for the ground acceleration input, and Hi(m) is the frequency response function for the ith mode

Hi (o)= -r-;-; j p2 - o2 + 2j Pipi o j in which pi, pi are structural frequencies and damping coefficient of the ith mode, respectively. The spectral density function for a stationary ground acceleration is

which was proposed by Kanai and Tajimi [5,6] based on the study of the frequency content from ground motion records. In Equation 3.119, pg and pg are prevailing frequency and damping ratio describing soil layer characteristics, S0 is the intensity of the excitation and can be determined by the strength of the ground motion. By setting the ground acceleration RMS as on the left-hand side of Equation 3.115 and substituting SXf_(o) of Equation 3.119 into the right-hand side of that equation, S0 can be obtained after integration as

2Pg4

The integral in Equation 3.117 can be found after the substitution of Equation 3.119 by the theorem of residue [7], and the results are f+1 jHi (o)^ (o) do — § ^ (3.121a)

Nq, = (1 + 4r2)r, + 4Pg(r? + r2)s + 4r2r,s2 + rgs3; s = pg/p, Dqt = rgr, + 4r2r2s + (2rg + 2r2 - i)s2 + 4rgr2s3 + rr,s4

Substituting Equation 3.121a into Equation 3.117 yields

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