Does ACI 318-19 use the simplified 2*sqrt(f'c) formula for footing shear?

No. ACI 318-19 Table 22.5.5.1 replaced the old simplified formula with a detailed expression that depends on the longitudinal reinforcement ratio rho_w. For lightly reinforced footings, the detailed formula can give a substantially lower Vc than the old 2*sqrt(f'c). The engine uses Vc = min(8*rho_w^(1/3), 5) * lambda_s * lambda * sqrt(f'c) * bw * d.

One-way shear in footings per ACI 318-19

Q: Why is the size effect factor lambda_s equal to 1.0 for footings?

ACI 318-19 Section 13.2.6.2 exempts shallow foundations from the general size effect reduction. Footings are supported by distributed soil reaction rather than point loads, and their failure behavior is better constrained than flexural members like beams or slabs.

Q: When does one-way shear govern over punching shear in a footing?

One-way shear tends to govern for long rectangular footings, large columns relative to the footing width, or footings with low reinforcement ratios (which reduce Vc in the detailed formula). Wall footings have only a one-way shear check because the loaded area extends the full wall length and no closed punching perimeter forms.

One-way (beam) shear checks whether the footing slab can resist shear on a plane across its full width, at a distance d from the column or wall face. ACI 318-19 Section 13.2.7.2 defines the critical section, and Table 22.5.5.1 gives the concrete shear strength V_c using a reinforcement-ratio-dependent formula. This page covers one-way shear for both isolated footings and wall footings. Punching (two-way) shear is covered on its own spoke. Flexural reinforcement, which determines the ρ_w used in the V_c formula, is covered separately.

Why the critical section is at d from the column face

Loads applied between the column face and a distance d from it can transfer to the column through a diagonal compression strut (arching action), bypassing beam-action shear entirely. Only the upward soil pressure beyond d must be resisted by the shear strength of the concrete cross-section.

Contrast with punching shear, which uses a critical perimeter at d/2 from the column face. The punching failure mechanism is a truncated cone around the column; the steeper critical angle means the failure surface forms closer to the column.

ACI 318-19 Section 13.2.7.2 (cross-referenced by 7.4.3.2) codifies this: the critical section for one-way shear is at a distance d from the face of the column or wall. When the footing edge is less than d from the column face, no one-way shear check is needed (the entire cantilever is within the arching zone).

Applicable clauses

Clause	Requirement
§13.2.7.2	Critical section at d from the face of the column or wall.
§7.4.3.2	Beam shear cross-reference for one-way members.
Table 22.5.5.1	V_c without shear reinforcement (detailed ρ_w formula).
§22.5.3.1	√f'_c limited to 100 psi.
§13.2.6.2	λ_s = 1.0 for shallow foundations (size effect exemption).
Table 21.2.1	φ = 0.75 for shear.

Factored shear demand V_u

The factored shear at the critical section equals the upward soil pressure times the tributary area beyond d:

V_u = q_u \cdot b_w \cdot \left(\ell - d\right)

q_u = factored bearing pressure (P_u / (L × B) for isolated, or P_{u,per ft} / B for wall footings)
ℓ = cantilever from the column face to the footing edge = (footing dimension − column dimension) / 2
d = effective depth (d_bot for the direction with bottom bars, d_top for the perpendicular direction)
b_w = footing width perpendicular to the shear plane

If ℓ ≤ d, the critical section falls beyond the footing edge and V_u = 0. No one-way shear check is needed in that direction.

Concrete shear strength V_c (Table 22.5.5.1)

ACI 318-19 Table 22.5.5.1 provides two expressions for V_c of members without shear reinforcement. The smaller governs:

V_{c,1} = 8\,\rho_w^{1/3}\,\lambda_s\,\lambda\,\sqrt{f'_c}\;b_w\,d

V_{c,2} = 5\,\lambda_s\,\lambda\,\sqrt{f'_c}\;b_w\,d

V_c = \min\!\left(V_{c,1},\;V_{c,2}\right)

ρ_w = A_s,provided / (b_w × d), the longitudinal reinforcement ratio from the flexure check
λ_s = 1.0 for footings (Section 13.2.6.2)
λ = lightweight concrete factor (1.0 for normal weight)
√f'_c in psi, capped at 100 psi per Section 22.5.3.1
b_w and d in inches; result in lb, divide by 1,000 for kip

The first expression depends on ρ_w: a footing with lighter reinforcement has a lower V_c. This is a significant change from ACI 318-14, which used the simplified 2√f'_c formula that did not depend on reinforcement ratio. For lightly reinforced footings, the new formula can give substantially lower capacity.

Size effect exemption for footings

ACI 318-19 Section 13.2.6.2 sets λ_s = 1.0 for one-way shear in shallow foundations. The general formula λ_s = √(2 / (1 + d/10)) applies to beams and slabs, but footings are exempt because distributed soil reaction constrains their behavior differently from members loaded by concentrated forces.

Strength check

The strength check compares the factored demand against the reduced capacity:

\phi\,V_c \geq V_u

where φ = 0.75 (Table 21.2.1 for shear). The demand-capacity ratio:

\text{DCR} = \frac{V_u}{\phi\,V_c}

DCR ≤ 1.0 passes. If DCR > 1.0, increase the footing thickness (which increases d and reduces the shear arm simultaneously).

Two directions and effective depth

Isolated footings are checked in both directions:

L-direction: shear plane across the full width B. Uses d = d_bot (bottom reinforcement layer). b_w = B.
B-direction: shear plane across the full length L. Uses d = d_top (second reinforcement layer, one bar diameter above the bottom). b_w = L.

d_{bot} = h - \text{cover} - d_b / 2

d_{top} = h - \text{cover} - 3\,d_b / 2

The B-direction has a shorter effective depth and longer shear arm, so it typically produces a slightly higher DCR than the L-direction for a square footing.

Wall footings

Wall footings are checked in the transverse direction only, per a 1-ft strip (b_w = 12 in). The cantilever is ℓ = (B − wall thickness) / 2, and d = d_bot. There is no longitudinal one-way shear check for wall footings (the wall itself spans in that direction).

Worked example: isolated footing

Input	Value
f'_c	4,000 psi
λ	1.0 (normal-weight)
Column	16 × 16 in, interior
Footing	7 ft × 7 ft × 20 in thick
Reinforcement	#6 bars, 7 each way (A_s = 3.08 in²), 3 in cover
P_u	150 kips (concentric)

Step 1: Effective depths

d_bot = 20 − 3 − 0.75/2 = 16.625 in
d_top = 20 − 3 − 3(0.75)/2 = 15.875 in

Step 2: Factored soil pressure

q_u = 150 / (7 × 7) = 3.061 ksf

Step 3: L-direction shear demand

Cantilever: ℓ = (84 − 16) / 2 = 34 in = 2.833 ft
Shear arm: 2.833 − 16.625/12 = 2.833 − 1.385 = 1.448 ft
V_u = 3.061 × 7 × 1.448 = 31.0 kip

Step 4: L-direction V_c

ρ_w = 3.08 / (84 × 16.625) = 0.00221
√f'_c = 63.2 psi (< 100; no cap)
V_c,1 = 8 × 0.00221^1/3 × 1.0 × 1.0 × 63.2 × 84 × 16.625 / 1,000 = 92.0 kip
V_c,2 = 5 × 1.0 × 1.0 × 63.2 × 84 × 16.625 / 1,000 = 441.6 kip
V_c = min(92.0, 441.6) = 92.0 kip

Step 5: L-direction check

φV_c = 0.75 × 92.0 = 69.0 kip
DCR = 31.0 / 69.0 = 0.45 → OK

Step 6: B-direction shear demand

Shear arm: 2.833 − 15.875/12 = 2.833 − 1.323 = 1.510 ft
V_u = 3.061 × 7 × 1.510 = 32.4 kip

Step 7: B-direction V_c

ρ_w = 3.08 / (84 × 15.875) = 0.00231
V_c,1 = 8 × 0.00231^1/3 × 1.0 × 1.0 × 63.2 × 84 × 15.875 / 1,000 = 89.2 kip
V_c,2 = 5 × 1.0 × 1.0 × 63.2 × 84 × 15.875 / 1,000 = 421.7 kip
V_c = min(89.2, 421.7) = 89.2 kip

Step 8: B-direction check

φV_c = 0.75 × 89.2 = 66.9 kip
DCR = 32.4 / 66.9 = 0.48 → OK

The B-direction has a higher DCR (0.48 vs 0.45) due to the smaller effective depth d_top. In both directions, the ρ_w-based formula (V_c,1) governs by a wide margin over the cap (V_c,2), as is typical for footings with moderate reinforcement ratios.

Frequently asked questions

Why is the critical section for one-way shear at d from the column face, not at d/2?

Loads applied between the column face and a distance d from it can transfer directly to the column by diagonal compression (arching action). Only loads beyond d must be carried by beam-action shear. Punching shear uses d/2 because the failure is a truncated cone around the column with a steeper critical angle.

Does ACI 318-19 use the simplified 2√f'_c formula for footing shear?

No. ACI 318-19 Table 22.5.5.1 replaced the old simplified formula with a detailed expression that depends on the longitudinal reinforcement ratio ρ_w. For lightly reinforced footings, the detailed formula can give a substantially lower V_c than the old 2√f'_c. The formula is V_c = min(8ρ_w^1/3, 5) × λ_s × λ × √f'_c × b_w × d.

Why is the size effect factor λ_s equal to 1.0 for footings?