Problem $3$:

Two cantilever beams AB and CD are connected to an axial bar BD$.$ A rigid block is located a small distance $s$ above point $B.$ Beam AB has flexural rigidity $EI$ and length $L.$ Beam CD has flexural rigidity $2EI$ and length $L.$ Bar BD has axial rigidity $EA,$ coefficient of thermal expansion $,$ and length $h.$ Note the following:

The gap $s$ is much smaller than $L(sL).$

For simplicity, assume $EA=\frac{1EI}{2}\frac{L}{L^2}$ and $h=\frac{L}{4}.$

A uniform temperature change $T$ is then applied to bar BD$.$ Do the following:

a$.$ Determine the required temperature change $T_c$ so that beam $AB$ just contacts the rigid block. Your answer should be in terms of $,s,$ and $L.$

b$.$ Determine the axial force in bar $BD,F,$ and the deflection of point $D,{}_D,$ if the applied temperature change exceeds the critical value found in part a$.,$ i$.$e$.,T>T_c.$ Your answers should be in terms of $EI,L,s,,$ and $T.$