Problem $1$

Part A 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 roller support D$.$

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

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

$($d$)$ What is the definition of a general stiffness term $K_{ij},$ where i and $j$ correspond to the row i and column $j$ of the stiffness matrix, respectively?

$($e$)$ 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$-f{t}^2.$

$($f$)$ Solve for the displacements $D_1$ and $D_2.$

Part B

The analysis of the beam structure shown above was performed considering two unknown displacements and all external loads and support settlements applied. The following displacements were obtained $($note that these may not be the correct values obtained for $1.f)$:

$D=\{[D_1],[D_2]\}=\{[+0\mathrm{.}0002155$rad$],[-0\mathrm{.}001401$rad$]\}$

Using these results, use force recovery to find the following reactions:

$C_y=$

$M_A=$