$[1]$ Problem $1$:

The figure below shows a beam under distributed loads with a rectangular cross$-$section. The nodes and elements have been labelled and numbered on the beam as shown in the figure.

Figure $1$: Beam configuration I

Material Properties, load values and dimensions:

$\backslash [$

E$=105\backslash$mathrm$\{$GPa$\},$ b$=70\backslash$mathrm$\{$~mm$\},$ d$=40\backslash$mathrm$\{$~mm$\},$ L$=100\backslash$mathrm$\{$~mm$\},$ P$=80\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}$

$\backslash ]$

Note: Assume that node $1$ is the origin of the system

a$)$ Construct the System Free Body Diagram for the beam, showing nodal degrees of freedom and system level forces and moments. Which of these are known and unknown? $(12$ pts$)$

b$)$ Find the equivalent forces and moments due to distributed loads at nodes $1,2$ and $3.(12$ pts$)$

c$)$ Find the value of deflection and the slope of the deflected beam at node $2.(\backslash (\backslash$mathbf$\{8\}\backslash$mathbf$\{$~ p t s$\}\backslash ))$

d$)$ Find the unknown reaction forces and moments at nodes $1$ and $3(\backslash (\backslash$mathbf$\{12\}\backslash$mathbf$\{$~ p t s$\}\backslash ))$

e$)$ Find the stress in x$-$direction at the beam cross$-$section at node $2,\backslash (\backslash$sigma$\_\{$x$\}\backslash )$ as a function of y $-$coordinate. $(8$ pts$)$

f$)$ Sketch the Bending moment and shear force force diagram for each element in the beam $(\backslash (\backslash$mathbf$\{8\}\backslash$mathbf$\{$~ p t s$\}\backslash ))$