Problem $1$:

Consider the simply supported beam and its superimposed loading in Figure $1.$

ACTUALLY SOLVE ALL PARTS

Figure $1.$ Beam loading model for problem $\backslash$#$1.$

The beam is laterally braced at the support points, at the concentrated loads and at midspan. The dead loads do not include beam weight. A$992$ steel is used. The loads shown in the diagram are service loads.

Provide the following:

a$)$ Determine the factored reactions and draw the factored shear and moment diagrams. Locate and label the points of maximum factored shear and moment for the worst$-$case load combination.

b$)$ Calculate the lateral$-$torsional buckling modification factor, $\backslash ($ C$\_\{$b$\}\backslash ),$ for the midspan segment of the beam.

c$)$ Select an efficient W$24$ beam to handle the factored moment assuming the midspan range governs design. Page $6$ shows the properties for all W$24\text{'}$s$.$ Clearly show that the resistance $\backslash (\backslash$left$(\backslash$phi $\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$n$\}\}\backslash$right$)\backslash )$ exceeds demand $(\backslash (\backslash$left$.\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$u$\}\}.\backslash$right$)\backslash )$ "Efficient" means that the ratio of $\backslash (\backslash$mathrm$\{$M$\}\_\{$u$\}/\backslash$phi $\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$n$\}\}\backslash )$ should be within the range of $\backslash (0\mathrm{.}90<mtext></mtext><mi>\backslash </mi><mi>)</mi>$ ratio

d$)$ Reconsider the endspans $-$ using the beam you selected, do they need to be considered? Why or why not? Briefly explain your reasoning.

e$)$ Determine if the beam is sufficient to resist the factored shear.

f$)$ Determine the worst$-$case live load and total load deflections. Compare to the limitations presented in the lecture notes assuming that a finished ceiling is present below the beam. Remember, we use service loads to calculate deflections!