Problem $2\mathrm{.}3$ A concrete beam $8$ m in span carries a superimposed uniformly distributed dead

load $(w_{SDL})$ of $10k\frac{N}{m}$ and a live load $(w_{LL})$ of $5k\frac{N}{m}.$ A concentrated live load $(P_L)$ of $50$ kN

also acts at the center of the span as shown below. The beam cross section is $300mm400mm$

and it is reinforced with $4-25mm$ dia. bars. The clear cover to the reinforcement is $50$ mm $.$

Assume concrete strength $f{c}^{\text{'}}=40$MPa and the yield stress of steel is fy$=420$ Mpa with an E$-$value

of steel as $200$ Gpa $.$ Compute the following:

a$)$ Find the cracked moment of inertia $(I_{cr})$ of the cracked section.

b$)$ The moment $M_{cr}$ at which the beam will crack, find the stress in the concrete and the steel just

after cracking. Draw the strain diagram and the stress diagram.

a$)$ If the magnitude of moment acting on the beam is $3M_{cr.}.$ Find stress in concrete and

steel.

c$)$ From the shear and the moment diagram due to factored dead and live loads, what is the

required ultimate shear capacity $V_u$ and the ultimate moment capacity $M_u.$