Given a Bernoulli beam with a fixed support in A and a simple support in $B.$ The loads and dimensions are shown in the figure. The beam has constant flexural rigidity $EI.$ It is assumed that for $F=0$ the beam is straight and stress free. The $M-($moment$-$curvature$)$ diagram is assumed to be bilinear for each crosssection.

Determine the limit load by performing the elastic$-$plastic analysis. $(8$p$)$

Estimate the limit load by applying the upper$-$bound theorem. $(2$p$)$