Consider a pile subject to two types of vertical loads, Q and QD. The capacity of the pile consists of a shaft resistance R, and a toe resistance RT. All the four components are considered random variables that follow the normal distribution. The pile will fail when the sum of resistances is smaller than the sum of loadings (see the performance function, G(x)) below. Q+Q R 5000 Rs Q-N(150 kN, 30 kN) Q-N(300 KN, 20 kN) RN (600 KN, 200 KN) R, N (200 kN, 80 KN) x =(Q, QD: R, R} G(x) = R +R-Q-Q 1. Assume all the four variables are statistically independent. a. Calculate the reliability index and failure probability ps of the pile using the Monte Carlo Simulation Method. b. Determine the minimum number of simulations to achieve a COV pf of 0.1. 2. Assume Rs and RT are correlated with a coefficient of correlation of p=0.5. Other variables remain statistically independent. Calculate the reliability index and failure probability py of the pile using the Monte Carlo Simulation Method. 3. Repeat Question (1) using the Mean-value First-Order Reliability method (MFORM). 4. Repeat Questions (1) and (2) using the Advanced First-Order Reliability Method (AFORM).