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

load $($ wsDL $)$ of $10k\frac{N}{m}$ and a live load $($ wLL $)$ of $5k\frac{N}{m}.$ A concentrated live load $(P{L}_L)$ 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.

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 self weight of the beam in $k\frac{N}{m}$

b$)$ Find the modulus of rupture, and the modulus of elasticity of the concrete.

c$)$ The gross moment of inertia $(Ig{r}_r)$ of the Uncracked section $($ignoring steel$).$

d$)$ Compute the moment $M_{cr}$ at which the beam will crack and find the stress in the concrete

and the steel just prior to cracking. Draw the strain diagram and the stress diagram.

e$)$ Find the service dead and live loads and factored dead and live loads.

f$)$ Draw the shear and the moment diagram due to factored dead and live loads.

g$)$ What is the moment in the beam due to service dead and live loads $(Ms)$ and factored dead

and live loads $(M_f).$ The self$-$weight of the beam has to be included.