The reliability of a E-metre long simply supported uniform beam under a uniformly distributed load, w, is to be investigated. The beam can either fail in bending, shear or combined bending and shear. The random variables are w, the bending capacity M and the shear capacity V of the beam. The statistics of the random variables and the performance functions (failure means value of function is less than zero) of the three failure modes are as follows: _MM 91(X)= 910w w)= 1140125sz 600 kNm 0.1 Normal orlognormal (10:01 )_ V 05 L w w V 400 kN 0.15 Normal or Iognormal '92 '92 2 wt. WM w 100 kN/m 0.25 Normal or Iognormal .93 (X) = .93 (M, V, W) = 1 - 8M W Perform Direct Monte Carlo simulation assuming that M, V and w are independent normal random variables, for N of (i) 100 and (ii) 1000. Remember to freeze the generated data. Estimate (a) the failure probability of each mode; (b) the failure probability of the beam; and (c) the Chi-square Goodness of Fit test at 95% confidence level for the generated data of the random variables. Repeat for the case where M, V and w are independent Iognormal random variates