Problem 1. Consider a steel cantilever beam with length of l = 18 ft under a distributed random dead load. Furthermore, assume that yield stress and section modulus S are also random variables. All the random variables are uncorrelated with the following information.

1. Estimate reliability in terms of bending stress with 500, 1000, and 10000 samples. Summarize the results in a table.
2. Plot the empirical pdf for all of your generated random numbers. Evaluate the mean and standard deviation from your generated data, and compare to the requested values in a table.

Comment on the pdf 's, do they look reasonable to you?
3. Plot the dead load data on a normal probability paper using our previous functions.

4. Compare your results with FOSM method (bonus:5 pts ).

Note. The demand is expressed as M S , where M is the moment due to the load at the xed end.

Problem 2. We would like to write a MATLAB function to generate random numbers from Extreme type I (Gumbel) distribution. The CDF is given as FX(x) = e-^e-(z-1)/s, with parameters of l (location) and s (scale). The quantile function (inverse cumulative distribution function), Q(p), of a Gumbel distribution is given by

Q(p) = 1 - sln(-ln(p))

the variate Q(U) has a Gumbel distribution with parameters l (location)and s (scale) when the random variate U is drawn from the uniform distribution on the interval (0; 1). Therefore, this simply means that your function should rst call the rand function of MATLAB to generate U. It will then use the above formula to transform into Gumbel (Recall the equation
on the last page of handout 4). The function should get three things as input: number of data n, l and s the parameters of the requested Gumbel distribution. It should then, as an output, return a vector of numbers.

Problem 3. Consider the following performance function for bending of a rectangular reinforced concrete beam under moments due to dead and live loads.

where the deterministic dimensions and rebar area are given as follows: b = 15 in, d = 24 in and a = 5 in2. The random variables are assumed to be uncorrelated with the following information.

Estimate probability of failure with Monte Carlo simulation using 500; 1000; 10000 samples. Summarize the results in a table. What would your nal answer be?