pSc P3 Nq,

2 p4 Dq, Nq ( = 4p| + Pi (1 + 4p2)s + p^s2 D® = s[pgPi + 4p2p2s + 2pgpi(2p2 + 2p2 - 1)s2 + 4p2p2s3 + pgPiS4]

After the RMS values of the response are obtained, one can calculate the probability of safety in duration t by

where D is the safety range assigned for the response X and P{X(0) e D} is the probability of the initial X value in the range D. vD is called the D-outcrossing rate, expressing the rate of response X going out of range D

which is the sum of up-crossing rate v+ as upper limit, a and down-crossing rate v- as bottom limit, b, for the safety range D = [a b]. For the stationary Gaussian process with mean m and variance a2X, the up-crossing and down-crossing rate for ±a are


If the probability of the initial value X in range D is 1, Equation 3.126 becomes

The probability of safety for the range D = [a -a] can be obtained by substituting Equations 3.127 and 3.128 into Equation 3.129a as


In summary, one can determine the RMS value of the response Xi from Equations 3.122 and 3.125 with a given RMS value of the ground acceleration in Equation 3.120, in which the soil layer property is

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