What is gamma_v and why does it matter?

gamma_v is the fraction of the unbalanced moment that is transferred by eccentricity of shear stress on the critical perimeter; the rest (gamma_f) is transferred by flexure. Ignoring gamma_v underestimates the peak shear stress at the column face, because the moment adds stress on one side of the perimeter and relieves it on the other.

Eccentric punching shear per ACI 318-19

Q: What changed in ACI 318-19 for punching shear?

The three-equation approach in Table 22.6.5.2 with alpha_s by column position is the same as in ACI 318-14, and the eccentric shear stress expression is also unchanged. What calcnote adds is the full asymmetric Jc for edge and corner positions, rather than treating every column as an interior one.

When a column delivers a moment to the footing, or sits at the edge or corner where the critical perimeter is no longer symmetric about the column, the shear stress around the perimeter is not uniform. ACI 318-19 handles this by transferring part of the moment through eccentric shear: the fraction γ_v of the moment adds a linearly varying stress to the direct shear stress, computed from the polar moment of inertia J_c of the critical section. This page covers γ_v, J_c for interior, edge and corner columns, and the peak-stress check. For the concentric case and the three v_c equations, see punching shear in footings.

Applicable clauses

ACI 318-19 reference	Topic
Cl 8.4.4.2.3	Moment transfer by eccentricity of shear.
R8.4.4.2.3	Commentary: the γ_v fraction and the J_c expressions.
Cl 8.4.4.2	γ_v = 1 − γ_f (fraction transferred by eccentric shear).
Cl 22.6.5.3	v_u = V_u/(b_od) + γ_vM_uc/J_c.
Table 22.6.5.2	α_s by column position (40 interior, 30 edge, 20 corner).

When eccentric punching applies

Check the eccentric stress whenever either of these is true:

Moment transfer at the column. The column delivers an unbalanced moment M_u to the footing in addition to the axial load P_u, in one or both directions.
Column not centered on the critical perimeter. At an edge or corner column the perimeter is truncated by the footing edge, so even a pure axial load acts eccentrically with respect to the centroid of the perimeter.

A column at the center of the footing with no moment needs only the concentric check. As soon as a moment is present, the peak stress on the perimeter exceeds the average V_u/(b_od), and the design must be checked against that peak.

The γ_v fraction (Cl 8.4.4.2)

Part of the unbalanced moment is carried by flexure in a strip around the column; the remainder is carried by shear stress on the critical perimeter:

\gamma_f = \frac{1}{1 + \tfrac{2}{3}\sqrt{b_1 / b_2}} \qquad \gamma_v = 1 - \gamma_f

b₁ = dimension of the critical section measured in the direction of the moment.
b₂ = dimension of the critical section perpendicular to b₁.

For a square interior critical section (b₁ = b₂), γ_f = 0.60 and γ_v = 0.40: forty percent of the moment is transferred by eccentric shear. A critical section that is longer in the direction of the moment (large b₁/b₂) transfers more by shear, so γ_v rises; a shorter one transfers less.

Polar moment of inertia J_c

J_c is the polar moment of inertia of the critical section, treated as a thin-walled shape of thickness d about its own centroid. Its form depends on how many sides of the perimeter exist. Throughout this page, d is the average effective depth and the column face is taken flush with the footing edge at edge and corner positions.

Interior column (symmetric perimeter)

The perimeter is a closed rectangle of b₁ × b₂, with b₁ = c₁ + d and b₂ = c₂ + d. The centroid is at the center, so the distance to the extreme fiber is c = b₁/2:

J_c = \frac{d\,b_1^3}{6} + \frac{b_1\,d^3}{6} + \frac{d\,b_2\,b_1^2}{2}, \qquad c = \frac{b_1}{2}

Edge column (asymmetric perimeter)

The perimeter has three sides: a back face of length b_back = c_parallel + d, parallel to the free edge, and two legs of length ℓ = c_perp + d/2 running toward the free edge. For eccentricity perpendicular to the free edge (the direction of b₁ = ℓ) the centroid shifts toward the back face. The distance from the back face to the centroid is c_AB and the remainder of the leg is c_CD:

c_{AB} = \frac{\ell^2}{b_{back} + 2\ell}, \qquad c_{CD} = \ell - c_{AB}

J_c = b_{back}\,d\,c_{AB}^2 + 2\left[\frac{d\,\ell^3}{12} + \frac{\ell\,d^3}{12} + \ell\,d\left(\frac{\ell}{2} - c_{AB}\right)^2\right]

For eccentricity parallel to the free edge (the direction of b₁ = b_back) the section is symmetric about the load line, so the centroid does not shift:

J_c = \frac{d\,b_{back}^3}{12} + \frac{b_{back}\,d^3}{12} + \frac{\ell\,d\,b_{back}^2}{2}, \qquad c = \frac{b_{back}}{2}

Corner column (asymmetric in both directions)

The perimeter has two sides, each d/2 beyond the column face: the leg b_par = c_par + d/2 and the leg b_perp = c_perp + d/2, with b_perp lying in the direction of the eccentricity. The centroid is shifted in both directions, so eccentricity in either direction uses an asymmetric section. c_AB is measured from the b_par leg:

c_{AB} = \frac{b_{perp}^2}{2\,(b_{par} + b_{perp})}, \qquad c_{CD} = b_{perp} - c_{AB}

J_c = b_{par}\,d\,c_{AB}^2 + \frac{d\,b_{perp}^3}{12} + \frac{b_{perp}\,d^3}{12} + b_{perp}\,d\left(\frac{b_{perp}}{2} - c_{AB}\right)^2

Moments in the two directions each produce their own stress term, and the terms are superimposed on the direct shear stress.

Worked comparison: one column, three positions

A 16 × 16 in column on a footing with d = 16.0 in, evaluated at each position with the section properties above. The last column is the peak eccentric stress contributed by 100 kip-in of unbalanced moment.

Position and moment axis	b₁ × b₂ (in)	γ_v	c_AB / c_CD (in)	J_c (in⁴)	Stress per 100 kip-in (psi)
Interior	32 × 32	0.400	16.0 / 16.0	371,371	1.72
Edge, eccentricity parallel to free edge	32 × 24	0.435	16.0 / 16.0	251,221	2.77
Edge, eccentricity perpendicular to free edge	24 × 32	0.366	7.2 / 16.8	97,485	6.31
Corner (either axis, square column)	24 × 24	0.400	6.0 / 18.0	54,272	13.27

Stress = γ_v × M_u × c_max / J_c with c_max = max(c_AB, c_CD). The same moment produces about 3.7 times the interior stress at an edge column (eccentricity perpendicular to the edge) and about 7.7 times at a corner column, which is why edge and corner footings are the cases where ignoring J_c is most unconservative.

Stress calculation (Cl 22.6.5.3)

The factored shear stress on the critical perimeter is the direct shear stress plus the stress from the fraction of the moment transferred by shear:

v_u = \frac{V_u}{b_o\,d} \pm \frac{\gamma_v\,M_u\,c}{J_c}

The moment adds stress on one side of the centroid and relieves it on the other. The governing value is the larger one, found at the extreme fiber farthest from the centroid:

c_{max} = \max(c_{AB},\, c_{CD}), \qquad v_{u,max} = \frac{V_u}{b_o\,d} + \frac{\gamma_v\,M_u\,c_{max}}{J_c}

Moments in both directions each contribute a term. The demand-capacity ratio compares the peak stress with the design capacity from Table 22.6.5.2:

DCR = \frac{v_{u,max}}{\phi\,v_c}, \qquad \phi = 0.75

Here V_u is the net shear at the critical section (column load minus the soil reaction inside the perimeter), b_o is the perimeter of the actual three-sided or two-sided section at edge and corner columns, and v_c is the smallest of the three Table 22.6.5.2 equations.

α_s by column position (Table 22.6.5.2)

Column position	α_s	Critical section
Interior	40	4 sides, symmetric.
Edge	30	3 sides, centroid shifted perpendicular to the free edge.
Corner	20	2 sides, centroid shifted in both directions.

α_s enters equation (c) of Table 22.6.5.2, v_c,c = (2 + α_sd/b_o)λ_sλ√f'_c. Because edge and corner perimeters have a smaller b_o and a smaller α_s, this equation is more likely to govern there than at interior columns.

What calcnote computes

For every isolated footing, calcnote runs the concentric check and, when a moment is entered, the full eccentric check at the column position you select:

Interior, edge and corner columns. The perimeter, b_o, α_s, γ_v and J_c follow the position, including the shifted centroid (c_AB, c_CD) at edge and corner columns.
Moments about both axes. Each direction gets its own γ_v and J_c, and the stress terms are added to the direct shear stress.
Peak stress and DCR. v_u,max is reported next to φv_c, with γ_v, J_c, c_AB and c_CD shown so the result can be checked by hand.

Scope note: the edge and corner sections assume the column face is flush with the footing edge.

See the full eccentric check on a real footing: open the footing demo, or see an edge column with moment to inspect γ_v, J_c and the peak stress. Free, no signup.

Frequently asked questions

What is γ_v and why does it matter?

γ_v is the fraction of the unbalanced moment that is transferred by eccentricity of shear stress on the critical perimeter; the rest (γ_f) is transferred by flexure. Ignoring γ_v underestimates the peak shear stress at the column face, because the moment adds stress on one side of the perimeter and relieves it on the other.

How is J_c different for edge vs interior columns?

Interior columns have a symmetric critical perimeter, so the centroid sits at the middle and J_c has a single closed form. Edge and corner columns have an asymmetric perimeter whose centroid shifts toward the back face, so the distances to the two extreme fibers differ (c_AB and c_CD) and J_c must be computed about the shifted centroid. For the same moment, the peak stress at an edge or corner column is several times the interior value.

Why does calcnote use c_max instead of c at both faces?

It is the conservative choice. On an asymmetric perimeter the two faces carry different stresses, and the larger one occurs at the face farthest from the centroid. Using c_max = max(c_AB, c_CD) gives the governing stress for either direction of moment, so the check does not depend on the sign of the moment.

What changed in ACI 318-19 for punching shear?

The three-equation approach in Table 22.6.5.2 with α_s by column position is the same as in ACI 318-14, and the eccentric shear stress expression is also unchanged. What calcnote adds is the full asymmetric J_c for edge and corner positions, rather than treating every column as an interior one.

Sources

ACI 318-19: Cl 8.4.4.2 (γ_v = 1 − γ_f), Cl 8.4.4.2.3 and R8.4.4.2.3 (moment transfer by eccentricity of shear, J_c commentary), Cl 22.6.5.3 (combined shear stress), Table 22.6.5.2 (v_c equations and α_s).