Reliability analyses of underground openings with the point estimate method

Reliability analyses usually need to treat a large amount of data to obtain satisfactory results. The most common reliability tool is Monte Carlo simulation (MCS). This technique demands a large number of evaluations of the mechanical behavior of the problem in question. Thus, MCS are less suitable for complex geotechnical numerical models like underground excavation stability. To overcome this limitation, approximation techniques are used in the field of structural reliability. The probability of failure Pf has been frequently estimated by means of the point estimate method (PEM) introduced by Rosenblueth (1975). Recently, some researches have focused on the enhancement of the PEM by improved sampling techniques and applying higher-order moments to approximate the probabilities of failure. This paper presents a comparison of the accuracy of three different schemes of point estimate methods: Rosenblueth (1975), Hong (1998) and Zhao and Ono (2000) to estimate the probability of failure of wall convergence of a circular tunnel and the face stability of a shallow tunnel by using higher moment approximation of the reliability index. Monte Carlo simulation (with or without importance sampling) approximations were performed to serve as a benchmark for evaluating the accuracy of the PEM methods. Results show that the classical second moment approximations are not suitable for tunnel wall convergence response, presenting errors larger than 250% in the estimate of reliability indices. The best results were obtained with Zhao and Ono′s PEM with fourth moment approximation (FM-3) of the reliability index presenting errors below 20%. Regarding the face stability, all PEMs yielded accurate results with errors below 15%. Finally, the use of PEM is suggested only for preliminary analyses because of its general lack of accuracy.