Beam flexure verification: 12×20 rectangular, 3#8

This page walks through the complete ACI 318-19 flexure check for test case W1: a 12×20 in rectangular beam with 3-#8 bars, f′c = 4,000 psi, Grade 60 steel. Every intermediate value is pinned by a regression test that runs on each code change. This is one of 30 cases in the beam verification report.

Case definition

Section	12 × 20 in (b = 12, h = 20)
Reinforcement	3-#8 (A_s = 2.37 in²), 1 layer
Effective depth	d = 17.625 in (1.5 in cover + #3 stirrup + half #8 bar)
Materials	f′c = 4,000 psi (β₁ = 0.85), Grade 60 (f_y = 60 ksi)
Applied moment	M_u = 100 k-ft = 1,200 kip-in

Step 1: Whitney stress block depth

The equivalent rectangular stress block (ACI 318-19 §22.2.2.4.1) replaces the nonlinear concrete stress distribution with a uniform 0.85 f′c acting over depth a:

a = \frac{A_s \, f_y}{0.85 \, f'_c \, b} = \frac{2.37 \times 60}{0.85 \times 4 \times 12} = \frac{142.2}{40.8} = 3.485\ \text{in}

Neutral axis depth from β₁ (Table 22.2.2.4.3; β₁ = 0.85 for f′c ≤ 4,000 psi):

c = \frac{a}{\beta_1} = \frac{3.485}{0.85} = 4.100\ \text{in}

Step 2: Net tensile strain and φ factor

Net tensile strain in the extreme tension steel at nominal strength (ACI 318-19 §9.3.3.1):

\varepsilon_t = 0.003 \times \frac{d - c}{c} = 0.003 \times \frac{17.625 - 4.100}{4.100} = 0.003 \times 3.299 = 0.00990

Yield strain ε_ty = f_y / E_s = 60 / 29,000 = 0.00207. The tension-controlled threshold is ε_ty + 0.003 = 0.00507. Since ε_t = 0.00990 > 0.00507, the section is tension-controlled and φ = 0.90 (Table 21.2.2):

\varepsilon_t = 0.00990 > \varepsilon_{ty} + 0.003 = 0.00507 \quad \Rightarrow \quad \phi = 0.90

Step 3: Design moment capacity

Nominal moment from the internal couple (ACI 318-19 §22.2):

\phi M_n = \phi \, A_s \, f_y \left(d - \frac{a}{2}\right) = 0.90 \times 2.37 \times 60 \times \left(17.625 - \frac{3.485}{2}\right) = 0.90 \times 2{,}258.5 = 2{,}032.6\ \text{kip-in} = 169.39\ \text{k-ft}

M_u = 100 k-ft < φM_n = 169.39 k-ft: PASS (DCR = 0.590).

Pinned values

These values are locked by regression tests. A drift beyond tolerance fails the build before it can deploy.

Quantity	Symbol	Value	ACI reference
Whitney block depth	a	3.485 in	Cl 22.2.2.4.1
Neutral axis depth	c	4.100 in	Cl 22.2.2.4.1
Net tensile strain	ε_t	0.00990	Cl 9.3.3.1
Strength reduction factor	φ	0.90	Table 21.2.2
Design moment capacity	φM_n	169.39 k-ft	Cl 22.2