Question $1(11$ marks$)$

Consider the following scenario. You need to construct a bridge to cross a river as seen in the schematic below. The bridge will rest at a height above the water level at a gap length of $\backslash ($ L$=\backslash )10$ m $.$

You choose the following arrangement of truss members to create the bridge.

a$)$ Define the force equilibrium for the structure to obtain matrix $\backslash (\backslash$boldsymbol$\{$A$\}\backslash ).$

b$)$ Define the force deformation relationship and the continuity of deformation for the structure. Assume that the members are homogeneous with a Young's modulus of $\backslash ($ E $\backslash )$ and are of constant cross$-$sectional area $\backslash ($ A $\backslash ).$

c$)$ Assuming a Young's modulus of $\backslash ($ E$=1\mathrm{.}2\backslash$mathrm$\{$GPa$\}\backslash )$ and a rectangular cross$-$sectional area of the planks with $\backslash ($ w$=1\backslash$mathrm$\{$~m$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$h$\}=0\mathrm{.}25\backslash$mathrm$\{$~m$\}\backslash )($indicated below, where $\backslash ($ w $\backslash )$ points out of the $\backslash ($ x $\backslash )-\backslash ($ y $\backslash )-$plane$)$ calculate the vertical displacement along the $\backslash ($ y $\backslash )-$axis for a vertical load of $1$ kN at node $2.$ Ignore the mass of the structure.