A simply supported steel beam $($compact section$)$ is subjected to a point load $$ in the middle. The

length of the beam is $18$ ft$.$ The plastic modulus $$ of the beam about the axis of bending is $803$

$.$

The load $$ is a random variable with a mean value of $10\mathrm{.}2$ k and a coefficient of variation of $11$

percent. The yield stress of the steel, $,$ is also a random variable with a mean value of $40\mathrm{.}3$ ksi

and a coefficient of $11\mathrm{.}5$ percent $($Hint: the moment demand for a simply supported beam is the

maximum moment at midspan as $$

$4$

$.$ Thus, the performance function is as $=$

$$

$4$

$).$

$($a$)$ Formulate a performance function $$ that can be used to describe the failure condition.

Consider bending only $($Hint: consider $=$ capacity $$ demand; failure corresponds to the

case when $<0).$

$($b$)$ Formulate an alternative performance function Y involving the product and quotient of

random variables. Consider bending only.

$($c$)$ If $$ and $$ are both normal random variables, calculate the probability of failure using the

results of section $($a$).$

$($d$)$ If $$ and $$ are both lognormal random variables, calculate the probability of failure using

the results of section $($b$).$