Problem $1.$ Beam Analysis

A simply supported $700$ WB $150$ beam $($bending about its strong axis$)$ is made of Grade $300$ steel and heavily welded. The beam length is $10$ m $.$ The beam self$-$weight is ignored.

a$)$ Calculate the maximum factored uniformly distributed load, $w^{**},$ that can be carried if the beam is fully braced over its total length by a floor diaphragm.

b$)$ Calculate the maximum factored moment, $M^{**},$ that can be carried if full lateral$-$torsional bracing $($without warping restraint$)$ is provided at the beam ends. The moment demand is uniform over the beam length. Use $k_t=k_l=k_r=1\mathrm{.}0.$

c$)$ Calculate the maximum factored moment, $M^{**},$ that can be carried if full lateral$-$torsional bracing $($without warping restraint$)$ is provided at the beam ends. The beam bends in double curvature with a linear moment diagram such that the moment demand at one end of the member is $0\mathrm{.}75$ times that at the other end.

d$)$ Calculate the maximum factored uniformly distributed load, $w^{**},$ that can be carried if full lateraltorsional bracing $($without warping restraint$)$ is provided at the beam ends and partial restraint is also provided at the mid$-$span. The load is applied at the top flange.

e$)$ If the deflection limit due to live load is $\frac{L}{300},$ what is the maximum factored uniformly distributed load, $w^{**},$ that can be carried if the ratio of dead load $(G)$ : live load $(Q)$ is $1$:$5?$

Hint: Deflection is a serviceability limit state, therefore the live load used for calculating deflection is $w_Q,$ whereas the maximum factored uniformly distributed load, $w^{**},$ is based on the load combination: $1\mathrm{.}2w_G+1\mathrm{.}5w_Q.$

f$)$ Calculate the dependable shear capacity, $V_v,$ and the corresponding maximum factored uniformly distributed load, $w^{**}.$

$($i$)$ By comparing the maximum factored load, $w^{**},$ based on shear capacity $($Part $($f$)),$ deflection limitation $($Part $($e$))$ and flexural capacity $($from Part $($d$)),$ which one governs for the design?

$($ii$)$ If the beam is supported on a $450$ mm wide plate at either end, and if the beam continues $250$ mm beyond the centre of the support, is it satisfactory? Use the governing load from Question $($i$).$ Compare the web yielding strength, $R_{by},$ and web buckling strength, $R_{bb},$ to the reaction force at the support.

g$)$ Use computer software $($such as EXCEL$)$ to develop a $M_b$ vs $L_e$ curve with ${}_m=1\mathrm{.}0$ and ${}_m=1\mathrm{.}7.$ What is the maximum $L_e$ before the member strength starts to decrease due to FLT buckling when ${}_m=1\mathrm{.}7?$

Hint: You can compare your ${}_m=1\mathrm{.}0$ curve with the attached charts.