Problem $2,$ Selected Distribution Models.

Reference: Prof. Daniele Veneziano, MIT, Civil and Environmental Engineering

A plain concrete column is subjected to a random axial load W with lognormal distribution, mean value Kilo Newton, a unit of force$]$ and coefficient of variation $V_W=0\mathrm{.}2.$ The resulting compressive stress $$ is given by $=\frac{W}{A},$ where A is the cross$-$sectional area of the column. The crushing strength of concrete $f_c$ also has lognormal distribution, with mean value meter$)$ and coefficient of variation $V_{f_c}=0\mathrm{.}2.$

$($a$)$ Obtain the probability density function $($PDF$)$ of the applied stress $($which depends on A $).$

$($b$)$ If the column has a $0\mathrm{.}40m0\mathrm{.}40m$ square cross$-$section, what is the probability $P_F$ that it fails i$.$e$.$