2. A simply-supported steel beam has a diameter of 1.25 in, and design requirements demand that the deflection in any direction at x = 10 in should be less than δ = 0.00375 in. Find the probability of failure given that P_z ~ N(100,8²) lb and P_y ~ N(200,10²) lb using the First Order Second Moment Method. Note that P_z and P_y are independent.

Solution

Step 1: Define the limit-state function.

               Limit-state function:

, where .

               Therefore,

                Then,

where  is the distance to forces from origin,    is the length of the beam, ,  is the Young's modulus of steel, and  is the moment of inertia about y or z axis.

Step 2: Differentiate the limit-state function.

Step 3: Find the mean and standard deviation of the limit-state function.

Step 4: Evaluate the probability of failure.

Probability of failure: 0.0238525                                                                                                                    Ans.