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(2) By selecting the time or space-dependent input variables for the limit state function, the stochastic probability of occurrence can be analyzed.
(3) Using least square method to construct the response surface function, the probabilistic prediction of the failure of the target bridge system can be analyzed.
Using the fitted function of selected important variables, the limit state function is used for the safety evaluations with less numbers of simulation analyses. The choice of analysis points
where P is a probability; Pf is the probability of failure of the i th element; FR(q) is the cumulative distribution function (CDF); qi is the normalized base variables of load; and Pf i is the probability of failure of i th component; R represents the resistance [12].
The multiplication of probabilities shown in the fifth line of the derivation above is permissible because we assumed that the events, R1 > q1, R2 > q2, . . . are statistically independent of the series system.
A parallel system can consist of ductile or brittle elements. A parallel system with n perfectly ductile elements is in a state of failure when all of the elements fail (i.e., yield). Let Ri represent the strength of the i th element in such a system. The system strength will be the sum of all the strengths of the elements or:
n
R = . Ri . (2)
i =1
The probability of failure of the system can be determined as follows. Assume a deterministic load q is acting on the system, and let qi be the portion of the total load taken by the
A.S. Nowak, T. Cho / Journal of Constructional Steel Research 63 (2007) 1561–1569 1563
i th element. Failure of this element corresponds to the event Ri < qi, and the probability of failure of the element is given by Eq. (2). With this information, the probability of failure of the system can be determined as follows:
Pf = FR(q) = 1 − P[( R1 > q1) ∩ ( R2 > q2)
∩ · · · ( Rn > qn)]
= 1 − P( R1 > q1) P( R2 > q2) ∩ · · · P( Rn > qn)
= 1 − [1 − P( R1 ≤ q1)]
× [1 − P( R2 ≤ q2)] · · · [1 − P( Rn ≤ qn)]
Fig. 2. The required number of structural analyses by permutation method and combination method.
where R is the system resistance and Ri is the resistance of each element. If we assume that all of the element resistances follow the same CDF, then the mean and variance of the system resistance can be expressed in terms of the element parameters
µe and σ 2 as follows:
Fig. 3. Finite element model for the target bridge (deformed shape of the target bridge in moving load case).
where ρ is the coefficient of correlation.
To determine the reliability index for the entire system, we must first look at how the component reliability, βe , is related to the mean and standard deviation of the strength for each element. For the i th element, the limit state function is:
gi () = Ri − qi . (6)
2.3. System reliability as predicted by the combination of failure modes
The upper and lower bound of the probability of failures were predicted by the conventional system reliability assessment assuming a parallel connection between the main girder and arch ribs. However, the system failure can start from different modes. Furthermore, although the Pf for a system can be evaluated using the system reliability analysis of the assumed failure mode scenarios, the combination or the order of the failure mode is hard to be analyzed completely using the system reliability analysis.