Page - 863 - in Book of Full Papers - Symposium Hydro Engineering

0.413 0.415 0.419 10 0.966 1.001 1.027 15 1.962 2.107 2.225 20 3.646 4.156 4.597 25 6.385 7.899 9.353 30 10.642 14.667 19.134 35 16.958 26.834 39.937 40 25.886 48.385 86.045 45 37.876 86.4 192.66 50 As shown in Tables 1 and 2, non-associativity has a negligible effect on 𝑁 ᵧ for internal friction angles below 25 degrees but it becomes more significant for larger friction angle. The numerical solutions can approximate 𝑁 ᵧ factor for each dilation angle in smooth and rough foundations. Results are given for both rough and smooth footings. for rough footings: [6] Nᵧ = e0.66+5.11tanφ ∗ tanφ∗ For smooth footings: [7] Nᵧ = e 5.11tanφ ∗ tanφ∗ 3. RELIABILITY ANALYSIS METHODS In reliability analysis, there are three specific categories including analytical, approximate and simulation methods which can be used in different conditions based on their drawbacks and advantages. The analytical methods consider probability density functions of input parameters mathematically defined and the performance function (e.g. safety factor), integrated over input variables; accordingly, probability density function of the performance function can be determined. The drawback of these methods is that they are limited to problems with few numbers of stochastic variables. However, it is possible to assess the probability feature of the problems using ideal mathematical viewpoint. It is worth mentioning that few studies have been done using analytical methods of reliability analysis in view of their mathematical complexity. Jointly Distribution Random Variables (JDRV) approach is the one utilizes mathematical techniques [4-7] by numerical integration in order to find probability distribution function (PDF) of performance function, but it is not able to consider correlation coefficient between stochastic parameters. However, approximate solutions compute the probability of events with some indicators such as mean value and variance of input stochastic parameters. It should be mentioned that conventional approximate solutions are based on three methods: First Order Reliability Method (FORM) [8], First Order Second Moment (FOSM) [9] and Point Estimation Method (PEM) [10]. Each of them uses several simplifying assumptions for prediction of failure probability that somewhat reduces the accuracy. Furthermore, approximate solutions estimate the mean 863