고체역학 (Beer) — Pure Bending: Problem 4.53

4.53 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 \(\sigma_s\) and \(\sigma_c\). 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.