Calculation of Ultimate Section State

This section documents the procedure used by RAM Concrete Beam to calculate the neutral axis depth, c, and strain in the extreme tension steel at the ultimate state, ε_t. The neutral axis of bending is located by enforcing equilibrium on the section when the strain in the extreme compression (top) fiber is ε_cu = 0.003 (see Figure below).

Strains and stresses for section at ultimate state

Equating the force in the concrete to the sum of the forces in the reinforcing steel bars,

F_c = ∑F_s

F_c = 0.85f'_cβ₁cb_f

Σ F_{s} = E_{s} A_{s} ϵ_{s} = E_{s} A_{s} (\frac{d}{c} - 1)

0.85f'_cβ₁b_fc² - E_sA_sc - E_sA_sd = 0

This quadratic is then solved for c. With the values of c and ε _c known, the assumption of strain compatibility allows the steel strain to be determined using similar triangles.

ϵ_{t} = ϵ_{c u} (\frac{d}{c} - 1)