Chapter 4.5, Problem 53P

The design of a reinforced concrete beam is said to be balanced if the maximum stresses in the steel and concrete are equal, respectively, to the allowable stresses σ_s and σ_c.. Show that to achieve a balanced design the distance x from the top of the beam to the neutral axis must be

$x=\frac{d}{1+\frac{{\sigma }_s{E}_c}{{\sigma }_c{E}_s}}$

Where E_c and E_s are the moduli of elasticity of concrete and steel, respectively, and d is the distance from the top of the beam to the reinforcing steel.

Fig. P4.53 and P4.54