Problem $3$ The elastic beam in the figure is loaded with uniform load $(W)$ and has critical moment capacity

$(M).$ Assume that $W,L,$ and $M$ are uncorrelated random variables with

${}_W=2k\frac{N}{m},{}_W=0\mathrm{.}4k\frac{N}{m}$

${}_L=4m,{}_L=0\mathrm{.}4m$

${}_M=5kN*m,{}_M=0\mathrm{.}4kN*m$

Calculate the reliability index for the limit state $M-M_{\text{m}\text{a}\text{x}}=0$ by the first$-$order reliability analysis. Note that

the limit state is nonlinear in the basic variables. $[$Hint: use Lagrange multiplier method$].$