Question $\backslash$# $1$:

A beam$-$bar system is subjected to the loading shown in Figure $1.$ Beam $1-2$ is fixed at node $1$ while at node $2$ the beam is connected to a rotational spring with stiffness $\backslash ($ k$\_\{\backslash$theta$\}=5000\backslash$mathrm$\{$kNm$\}/\backslash$mathrm$\{$rad$\}\backslash ).$ Also, the beam is suspended to a vertical bar connecting nodes $2$ and $4,$ as shown in figure. The bar is fixed at node $4.$ A linearly varying distributed axial load is applied to the bar with intensity $\backslash ($ P$\_\{2\}=90\backslash$mathrm$\{$kN$\}/\backslash )\backslash ($ m $\backslash )$ and $\backslash ($ P$\_\{4\}=50\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ at nodes $2$ and $4,$ respectively. Additionally, a moment $\backslash ($ M$\_\{2\}=50\backslash$mathrm$\{$kNm$\}\backslash )$ is applied at node $2.$ For this beam$-$bar system, determine the following using the finite element method $($hand calculations$)$:

a$)$ the vertical displacement and the slope at point $2$

b$)$ draw the stress distribution along the length of the vertical bar $2-4$

c$)$ Evaluate the moment and shear values at the fixed support at node $1.$

Considering the same beam$-$bar system, develop a SAP$2000$ model to answer parts $($a$),($b$),$ and $($c$)$ above. Compare the results obtained from your hand calculations to those resulting from the SAP$2000$ model