Page - 865 - in Book of Full Papers - Symposium Hydro Engineering

where N is the number of Monte Carlo trials, d is the normal standard deviation corresponding to the level of confidence, 𝜀 is the desired level of confidence and m is the amount of stochastic variables. Table 3 represents some confidence level (𝜀 ) with corresponding standard deviation (Std) with regard to the number of stochastic variables. Table 3 Standard deviations according to confidence levels Standard deviation(std) confidence level (𝜀 ) 1.282 80% 1.645 90% 1.96 95% 2.576 99% To generate random numbers, often probability distribution functions that generate numbers betwen zero and one are used such as uniform probability distribution function. Then using Eq. 9 these numbers are converted to numbers with standard normal probability distribution. [9] N1 = √−2lnR1 cos(2πR2) 𝑅 1 and 𝑅 2 are random numbers generated using uniform distribution and 𝑁 1 is an independent random number generated using standard normal distribution. This process will be continued until enough random numbers are generated. The generated random numbers have mean of zero and standard deviation of one. To convert numbers with standard normal distribution (𝑁 𝑖 ) to numbers with the normal distribution having the mean of 𝜇 𝑥 and the stress standard deviation of 𝜎 𝑥 (𝑋 𝑖 ), Eq. 10 is applied. [10] N1 = √−2lnR1 cos(2πR2) [11] Xi = μx + Niσx In Eq. 10, 𝑋 𝑖 is a generated random number with normal distribution that is used in the calculation of the performance function and 𝑁 𝑖 is a random number with standard normal distribution. The presented equations are provided for the case that random numbers are generated independently. 5. CALCULATION OF THE BEARING CAPACITY OF DAM FOUNDATION USING MONTE CARLO SIMULATION METHOD In this method, the foundation bearing capacity can be analyzed for various performance levels and soil bearing capacity is calculated depending on the desired level of confidence. Probabilistic analysis for bearing capacity of dam 865