This point is evident when there is an error in the assumed failure mode and order. The error can be caused by a correlation among the members. For example, two identical components composed of the same material and same manufacturer can fail at the same time under the same environmental condition, regardless of the distribution of structural stress load. Therefore, assuming the failure of important components to be statistically independent or fully correlated, the upper and lower bound of the probability of combined failure modes can be predicted by combining such component’s failures. Due to the assumption of  correlation  among  failure  modes,  the  suggested  method

Fig. 4. Side view and observation points for the target bridge.

provides upper and lower bounds for the probabilities of failures.

The time–cost saving effect increases significantly with the increase in correlated components (Fig. 2). For example, when 20 failure cases of components are considered, the permutation method yields 2.433 × 1018 possible failure modes, the same number of possible modes for the structural analyses. That    is

2.32 × 1012 times larger than the number of possible modes to be considered by the proposed combination method.

3. Reliability evaluation based on response surface method

3.1. Target bridge

An arch bridge in Korea is considered for the analysis at the preliminary design stage before the construction. The  bridge is designed to resist ultimate limit states of flexure, shear, and buckling conditions. The finite element model for the bridge system is shown in Fig. 3, which shows 1492 nodes and 1570 frame elements. The bridge has 2 lanes and is 14.3 m in width and 360 m in span length.

Fig. 4 shows the critical components, which are determined based on the results of finite element analysis for the evaluation of component reliability. The observation points in Fig. 4 are for each risk event due to the failure of critical components.

1564 A.S. Nowak, T. Cho / Journal of Constructional Steel Research 63 (2007) 1561–1569

Table 2

Probabilistic properties of random variables

Random variables Bias

fac- tora

C.O.V.bReference

Fig. 5. The section considered to check the stress under tensile axial force with flexural moment.

Arch ribs

Axial strength f PU 1.05 0.10 Assumed

External axial force f E 1.05 0.10 Nowak [8]

Table 1

Girders Elasticity of steel ES 1.00 0.06 Tabsh and Nowak [7]

External force acting on the section of tie-girder (kN)

Flexural resistant moment

MR 1.12 0.135    Tabsh and Nowak [7]

Resistant shear force SR 1.08 0.12 Nowak [8]

External flexural moment

ME 1.05 0.10 Nowak [8]

Maximum 540.310    152.830    161.440    622.410     13 163.770