was also compared with the conventional system reliability analysis, providing the upper and lower bound of probabilities of failure of the target arch bridge system.

2. Reliability evaluation and the combination of failure modes

2.1. Component reliability by response surface method

A bridge system can be modeled as parallel–series connections. If it is fully correlated, the series connection system fails with the failure of the weakest element, while the parallel connection system fails with the failure of the strongest element. The parallel connection system exhibits different failure modes between brittle and ductile materials. Each connection has been considered independently and also for the fully correlated case.

It is troublesome to evaluate the limit state function in an explicit way for complicated structures, composed of many elements. Probabilistic finite element method or response surface method can be used to calculate the probability of failure ( Pf ) based on implicit limit state function [4]. The whole calculation process for a nonlinear process or for multi- degree freedom structures becomes very expensive due to the complex formalization of the finite element method or the large degrees of freedom. Furthermore, it is almost impossible to differentiate the derivative terms for the reliability calculation by Monte Carlo Simulations (MCS) or by First Order Second Moment (FOSM) method [9].

Response Surface Method (RSM) was developed by Box and Wilson [5] as a method of statistical regression analysis for the analysis of chemical factory operation, and it is now widely

Fig. 1. Central and axial points in (a) central composite design and (b) Bucher–Bourgund method.

(shown as the center, axis points and factorial points in Fig. 1) is very important to obtain the precise fitting to the global behavior. In this study, the Bucher–Bourgund method, which does not use cross-coupled terms and factorial points [6], was used.

2.2. Conventional system reliability

There are two basic types of systems: series systems and parallel systems. A series system is referred to as a weakest link system because the failure of the system corresponds to the failure of the weakest element in the system.

Assuming that the strengths of the elements are all statistically independent, we can calculate the probability of failure as follows:

Pf = FR(q) = P( R < q) = 1 − P( R > 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)]

used  in  many  areas  such  as  physics,  engineering,  medical n n

science  and  sociology  for  the  probabilistic  evaluation  of a

= 1 − .[1 − FRi (qi )] = 1 − .[1 − Pf i ] (1)

system.  Approximating  the  structural  response  by  the input

random variables, the response surface is used to evaluate  the

reliability and the probability of failure.

The RSM method has the following advantages.

(1) The implicit limit state function consists of user-specified selected input variables, which might be proven to be important or sensitive. The sensitivities can be easily checked by using the limit state function. After defining the limit state function, it can be easily differentiated to obtain the probability of failure. Therefore, unlike Monte Carlo Simulation, it is possible to calculate the extremely smaller probability of occurrence even for a very complicated structure with large degrees of freedom.