Page - 864 - in Book of Full Papers - Symposium Hydro Engineering

and variance of performance function; but they do not provide any information concerning the shape of its PDF. Therefore, the probability of events can be determined merely based on the assumed PDF for performance function by usually using normal distribution function. Recently, most of the reliability analysis, especially in the study of liquefiable soils, have been done using these methods [11-21]. Finally, it is worth noting that simulation methods are somewhat more accurate comparing with the previous ones. By the use of these methods, it is possible to predict the probability of event by simulating stochastic input parameters and implementing in repetitive calculations. The striking feature of these methods is that they can be used in complex mathematical problems, where the closed-form solution of which is not possible [22]. Thanks to the development of computer technology and available personal computer, application of these methods has been increased significantly in engineering problems. Monte Carlo simulation method is one of the most applicable methods, which will be discussed in the next section [1]. 4. MONTE CARLO SIMULATION METHOD In Monte Carlo Simulation (MCS), a mathematical or empirical operator 𝐹 (𝑋 ) with the variable 𝑋 ranging from 𝑥 1 to 𝑥 𝑛 is continuously calculated in which the operator (s) are random or have uncertainty with prescribed probability distributions [22-23]. MCS is actually an accurate reliability analysis method, applicable for collapse problems including slope stability, retaining walls and foundations [24-25]. In the analysis, stochastic values for input parameters are chosen; afterwards, the probability density function of stochastic parameters – which include any shape but normal, log-normal and beta distribution functions are generally implemented based on the characteristics of the stochastic variables, used to obtain the performance function. It is worth mentioning that the current procedure is repeated so far as proper statistical distribution for performance function is obtained enabling the user to determine the mean and standard deviation of performance function and, finally, prediction of the probability of events. Generally, this method consists of four steps as follows [25]: 1. Generating stochastic values for each of stochastic variables according to assigned probability density function. 2. Computing performance function using a proper deterministic method based on generated values in previous step. 3. Repeating steps 1 and 2 for as many times as required. 4. Determining probability distribution function of performance function and calculating the probability of events. The number of required Monte Carlo trials is dependent on the level of confidence in the solution and the amount of stochastic variables. Based on the statistical theory, Eq. (8) has been recommended for the number of iterations [25]: [8] N = ( d2 4(1 − ε)2 ) m 864