Problem $2($Full credit $=2$ pts$)$: Design a wood beam using $4"$ dimensional lumber $($the width of

the section is set as $b=4",$ and you need to select the depth $d$ of the section$)$ to carry a uniform

distributed load of $300$ plf over a $L=12\text{'}$ span. The lumber is No$.2$ southern yellow pine $($SYP$),$

where $F_b=1,100,F_v=80,$ and $E=1\mathrm{.}6{10}^6,$ adequately glued, screwed, and bolted. The

deflection requirement is $x{M}_{\text{m}\text{a}\text{x}}=5,400lb-ft,S_{\text{m}\text{i}\text{n}}=58\mathrm{.}91i{n}^{3}4124xdS>S_{\text{m}\text{i}\text{n}}w=309\mathrm{.}572l\frac{b}{f}t{M}_{\text{m}\text{a}\text{x}}=5,572\mathrm{.}3lb\frac{W}{?}{V}_{\text{m}\text{a}\text{x}}=1,857\mathrm{.}43lb\frac{w}{?}=0\mathrm{.}217f_v=70\mathrm{.}75$

Deflection: $=0\mathrm{.}217f_b=905\mathrm{.}57$

Shear stress: $V_{\text{m}\text{a}\text{x}}=1,857\mathrm{.}43lb-ft(\frac{w}{s}$elf$-$weight$),f_v=70\mathrm{.}75$

Deflection: $=0\mathrm{.}217D.$ You should consider the beam's self$-$weight.

Hints:

$(1)$ Use the textbook Appendix $6$ for the geometric properties and self$-$weights $of$ dimensional

lumber sections. Use the geometric properties about the $x$ centroidal axis.

$(2)$ First select a lumber section based $on$ the bending stress with the uniform distributed load

only and without the beam's self$-$weight, and then perform bending, shear, and deflection

check with the sum $of$ the uniform distributed load and the self$-$weight $(whichis$ uniform

distributed$).$

$M_{\text{m}\text{a}\text{x}}=5,400lb-ft,S_{\text{m}\text{i}\text{n}}=58\mathrm{.}91i{n}^3,$ Select $412(or$ other $4xd$ sections with $S>S_{\text{m}\text{i}\text{n}})$

Add the self$-$weight $to$ the external uniform distributed load: $w=309\mathrm{.}572l\frac{b}{f}t$

Bending stress: $M_{\text{m}\text{a}\text{x}}=5,572\mathrm{.}3lb-ft(\frac{W}{s}$elf$-$weight$),f_b=905\mathrm{.}57$

Shear stress: $V_{\text{m}\text{a}\text{x}}=1,857\mathrm{.}43lb-ft(\frac{w}{s}$elf$-$weight$),f_v=70\mathrm{.}75$

Deflection: $=0\mathrm{.}217$

answers are in red please show how to get them.