A key property of a steel beam is its modulus of elasticity E, which measures the beam’s stiffness. Stiffer beams (larger E values) bend less when loaded and thus can typically bear larger loads in a structure. The modulus of elasticity of a beam is uncertain due to variations in the manufacturing process. A startup company can produce very low cost beams with a mean E of 30,000 kips/in2 and a coefficient of variation of 0.4.
(a) On a single graph, plot the PDFs if E is assumed to be normally, lognormally, and gamma distributed. Plot the PDFs for E values between 10,000 and 50,000 kips/in2. Make sure to label the axes and use a legend to label the curves.
(b) On a second graph, plot the CDFs if E is assumed to be normally, lognormally, and gamma distributed. Again, plot the curves for E values between 10,000 and 50,000 kips/in2. Make sure to label the axes and use a legend to label the curves.
(c) Let’s say that standards require the beam to be rated using the F = 0.01 value (99% of all manufactured beams will have an E value above this rating). What value of E has F = 0.01 for each of the three distributions? How much does the choice of the distribution affect this value?
(d) The company has been trying to improve their manufacturing so that the beams can be rated with E = 25,000 kips/in2? What is the F associated with E = 25,000 for each of the distributions? Have they achieved their goal?