Situation:

$6.$ A $9-$meter long simply$-$supported beam is made up of $W12106$ section. It is subjected to a $20k\frac{N}{m}$ uniformly distributed load $($self$-$weight included$),$ and two concentrated loads, $150kN$ and $100kN,$ located at the third points. The allowable bending stress is $148\mathrm{.}8$MPa.

Properties $ofW867$ :

$I_x=388\mathrm{.}344{10}^{6}m{m}^4$

$A=20130m{m}^2$

$d=330mm$

$b_f=310mm$

$t_f=25mm$

$t_w=15mm$

Fig. STD$-04$

a$.$ Determine the maximum moment in the beam $(kN*m).$

b$.$ If the beam is to be reinforced by $20mm-$thick plates at the top and bottom, what is the required width $(mm)?$

c$.$ What is the required length $(m)$ of the reinforcing plates?