Problem $\backslash$#$2[40$ points total$]$: Consider the two$-$span continuous beam supported on expansive soil. Over time, the fixed support at $\backslash ($ C $\backslash )$ has displaced vertically up an unknown magnitude $(4)$ due to heaving. Span $\backslash ($ A B $\backslash )$ is also subjected to an upward concentrated force $(10$ kips $)$ at midspan. Note that both spans have equal flexural rigidity $\backslash ($ E I$=5,000\backslash )$ kip$-$ft $\backslash (\{\}^\{2\}\backslash ).$

Upward displacement, $\backslash (\backslash$Delta $\backslash )$

Assuming that the internal moment at the pinned support at $\backslash ($ B $\backslash )$ is not to exceed $30$ k $-$ ft $($i$.$e$.,\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$B$\}\backslash$boldsymbol$\{$C$\}\}=+\backslash$mathbf$\{30\}\backslash$mathbf$\{$k $-$ f t$\}\backslash )$ clockwise$),$ determine the largest magnitude of $\backslash (\backslash$Delta $\backslash )$ that the beam can withstand. Use the slope$-$deflection method.

Additional note$/$hint: To save time, not every slope$-$deflection equation needs to be determined to fully solve the problem!