Probabilistic evaluation for the implicit limit-state function of stability of a highway tunnel in China

This paper presents a practical algorithm for the first-order reliability method (FORM) to deal with the implicit nature of a limit-state function (LSF) in reliability assessment of stability of a working highway tunnel, which is constructed in a mountainous area in China. First, a mechanical model to determine the LSF that is not explicitly known with complex non-linear behaviour is formulated for the primary support provided by the combination of shotcrete and rockbolts. After reviewing concisely the basic concepts relevant to the FORM, the central difference approximation is subsequently introduced to estimate the partial derivatives of the LSF. By consideration of Taylor’s formula, this LSF can be transformed into an equation involving a single unknown described as the reliability index, and then the resulting solution procedure for determination of the reliability index can be rendered based on the derivation rule of compound function. A flow for tunnel reliability problem posed by the implicit LSF in applying the FORM is further developed. Eventually, the proposed methodology for the LSF in the non-explicit circumstance is used to perform the probabilistic evaluation for such a working tunnel where the choices of values of step length coefficient affecting the calculation results are suggested. Comparisons are made with the Monte-Carlo simulation method (MCSM) to assess the computational accuracy and efficiency of the algorithm proposed in this paper. It is shown that the MCSM used to obtain the “exact” solutions entails the formidable computational effort, whereas the current algorithm that alleviates the computational labour dramatically can provide an efficient way of implementing reliability calculations for the implicit LSF and offer results which are found to be in satisfactory agreement with the “exact” ones when the step length coefficient is properly taken.