A singly reinforced beam will be provided with the reinforcing bars corresponding to $90\%$ of the maximum reinforcement ratio specified in the ACI Code for a tension$-$controlled $($i$.$e$.,$ underreinforced$)$ design.

The materials used have the following properties:

$f_c^{\text{'}}=4000,f_y=80000,{}_y=0\mathrm{.}002$

a$.$ Estimated the steel area $A_s.$ You do not need to choose rebars. $(10$pt$)$

b$.$ Sketch the stress and strain distributions in the beam cross$-$section. The diagrams should show the location of the neutral axis and the compressive$/$tensile forces acting on the cross$-$section. $(5$pt$)$

c$.$ Calculate the nominal moment strength $M_n$ and the design moment strength $M_n[$in kips$-$ft$].$ Briefly explain how the formula of moment calculation can be derived. $(25$pt$)$

d$.$ Assume that the design moment strength is less than the factored moment. Hence, compression rebars will be added. The effective depth is $d^{\text{'}}={1\mathrm{.}5}^{\text{'}\text{'}}.$ Determine the steel area of compression rebars $A_s^{\text{'}}$ so that the compression bars yield at the flexural failure. $(10$pt$)$

e$.$ Based on $A_s^{\text{'}}$ determined in part d calculate the nominal moment strength $M_n$ and the design moment strength $M_n[$in kips$-$ft$].$ Also briefly explain how the formula of moment calculation can be deriven. $(25$pt$)$

A singly reinforced beam will be provided with the reinforcing bars corresponding to $90\%$ of the maximum reinforcement ratio specified in the ACI Code for a tension$-$controlled $($i$.$e$.,$ underreinforced$)$ design.

The materials used have the following properties:

$f_c^{\text{'}}=4000,f_y=80000,{}_y=0\mathrm{.}002$

a$.$ Estimated the steel area $A_s.$ You do not need to choose rebars. $(10$pt$)$

b$.$ Sketch the stress and strain distributions in the beam cross$-$section. The diagrams should show the location of the neutral axis and the compressive$/$tensile forces acting on the cross$-$section. $(5$pt$)$