Problem $1$

For the beam shown below:

$($a$)$ Determine the degree of kinematic indeterminacy, DKI. Show your work for credit. Neglect axial deformations and take advantage of the zero moment condition $($free rotation$)$ at the hinge location.

$($b$)$ Determine the joint force vector, hat$(Q).$

$($c$)$ Determine the fixed$-$end force vector, $Q^0.$

$($d$)$ Determine the coefficients for ALL columns of the stiffness matrix for the structure shown below. Assume that all members have an $EI=100,000$ kip$-$ft ${?}^2$

$($e$)$ Solve for the unknown displacements.

$($f$)$ Using these results from $(1.$e$),$ use force recovery to find the following: $($f$\mathrm{.}1)$ reaction $B_y$; $($f$\mathrm{.}2)$ reaction $M_A$; $($f$\mathrm{.}3)$ bending moment at the right end of the beam segment $AB,M_{BA}$; $($f$\mathrm{.}4)$ bending moment at the left end of the beam segment $BC,M_{BC}.$

$($g$)$ Confirm your previous results using SAP $2000.$ Based on your SAP $2000$ results, provide plots showing: $($g$\mathrm{.}1)$ the deformed shape; $($g$\mathrm{.}2)$ the values of displacements D$1$ and D$2$; $($g$\mathrm{.}3)$ the reactions; and $($g$\mathrm{.}4)$ the bending moment diagram.