Question $3$

$[25$ Marks$]$

A pin$-$jointed truss is shown in Figure Q$3.$ Assuming that both elements have the same Young's modulus E but that element BC has cross section area $1\mathrm{.}5$ times the cross section area of the element AC$.$ Use the stiffness matrix method to:

i$.$ Establish all element stiffness matrices in global coordinates

ii$.$ Assemble the global stiffness matrix for the truss

$[7$ marks$]$

iii. Find the displacements in node C

$[7$ marks$]$

iv$.$ Calculate the internal force for element A$-$C

$[7$ marks$]$

$[4$ marks$]$

In expanded form for a pin$-$jointed truss the element stiffness matrix in global co$-$ordinates $(x,Y)$ is:

$=cos{}_{ij}=sin{}_{ij}$

$[K_{ij}]=\frac{AE}{L}\left[\begin{array}{ccccccccccccc}{}^2& & -{}^2& -& {}^2& -& -{}^2& -& {}^2& & -{}^2& & {}^2\end{array}\right]$

$-$

$-$

$-{}^2$

$-$

$-{}^2$

$$