Question $2$

The singly symmetric column shown in Fig. $2a$ is loaded by a concentric compression force $(P).$ The crosssection dimensions are also shown in Fig. $2$a$.$ The flange width $($a$)$ and column length $($L$)$ are given in the table with respect to your last digit of SID. The column can be assumed to be simply supported against flexure and torsion at both ends, i$.$e$.u=u^{\text{'}\text{'}}=v=v^{\text{'}\text{'}}===0$ at the ends. The material yield stress is $320$MPa. The geometric parameters may be calculated by using THIN$-$WALL$-2.$

a$)$ Determine the critical elastic buckling load of the column. Check both results of the buckling load and the buckling mode against THIN$-$WALL$-2.$

b$)$ Determine the ultimate strength of the column according to AS$/$NZS$4600.$

c$)$ Assume now that the top flange is connected to sheeting along the full length of the column and that the sheeting provides an elastic lateral restraint to the top flange, as shown in Fig. $2$b$.$ The spring constant per unit length of column is denoted by $k.$ The distance from the shear centre $(S)$ to the centre line of the top flange is denoted by $d.$ Determine the displacement of the top flange in terms of the generalised buckling displacements $(u_b,v_b,{}_b).$

d$)$ Explain how you would modify the energy equation for this buckling problem to include the effect of the sheeting, $($see eqn. $(225)$ of the lecture notes on flexural$-$torsional buckling$).$

e$)$ Draw separate sketches of the column cross$-$section for different values of $k$ at mid$-$span showing the flexural and flexural$-$torsional buckling modes you would expect and suggest at least one longitudinal displacement field $(u_b,v_b,{}_b)$ for the buckling modes. Give reasons for your answer.

$\backslash$table$[[$Last digit of SID,$a(mm),L(mm)$