Question $3$

$($a$)$ Derive crippling$/$ Euler load for a pin$-$ ended column.

$[12$ marks$]$

$($b$)$ A timber column $15cm20cm$ and $6$ m long has both ends fixed. Determine:

$($i$)$ The critical buckling load; derive all expressions used.

$[9$ marks$]$

$($ii$)$ Safe load for the column if factor of safety is $3.$

$[4$ marks$]$

Question $4$

$($a$)$ A cantilever beam of length I carries a UDL of $wk\frac{N}{m}$ extending from the clamped end to the midpoint of the length of the beam. Show that the expression for the maximum deflection of the beam is given by: $\frac{W^4}{\text{R}\text{R}\text{A}\text{F}\text{t}}.$

$[15$ marks$]$

$($b$)$ Explain the application of the moment area method to determine the slope and deflection of cantilever beams.

$[10$ marks$]$