$[1]$ Problem $1$:

The figure below shows a beam under distributed loads with a rectangular cross$-$section. The nodes and ele$-$

ments have been labelled and numbered on the beam as shown in the figure.

Figure $1$: Beam configuration I

Material Properties, load values and dimensions:

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

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? $(12pts)$

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 $2.(8pts)$

d$)$ Find the unknown reaction forces and moments at nodes $1$ and $3(12pts)$

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

pts$)$

f$)$ Sketch the Bending moment and shear force force diagram for each element in the beam $(8tts)$