A beam fixed at both ends spans $20m$ and is subjected to a dead load $13k\frac{N}{m}$ and a live load of $20k\frac{N}{m}.$ It also supports a point load carrying a live load $4kN,$ located at $10m$ from the left support. The beam is adequately supported against lateral movement of the compression flange at the beam ends and at one$-$fifth points. The maximum allowed dead load deflection is $\frac{L}{360}$ or $25mm,$ whichever governs. Select a suitable $W$ shaped beam, grade $345W$ steel to satisfy the limit states. $(25$ minutes$)$

$NTSDL=13k\frac{N}{m},LL=20R\frac{N}{m}-0$ istri.

Nowl$-LL=4$RNox$1cm$

Asrume There is $N_{0}OL$:$.DL=0$

$M_1=\frac{w{l}^2}{8}+\frac{pl}{4}=\frac{(46\mathrm{.}25)(20000{)}^2}{8}+\frac{(6)(10000)}{4}=23\mathrm{.}12{10}^8$

$($:$V_1=\frac{wl}{2}+\frac{P}{2}=\frac{(46\mathrm{.}25)(20000)}{2}+\frac{6}{2}=462503-l\}$

$M_M=\frac{5w{l}^4}{384EI}+\frac{P{l}^3}{48EI}=\frac{(5w{l}^4)(48E)+(P{l}^3)(384E)}{\text{I}}=>$

$=[[(5)(46\mathrm{.}25)(20{10}^3)][(48)(200)]+[(6)(10{10}^3)](384)(200)]$

$=(3\mathrm{.}552{10}^{23})+(46\mathrm{.}08{10}^8)$