Critical Section Properties and Equations for Actual Stress

This section discusses the calculation of punching resistance for an unreinforced section.

Notation

A = area of one side of the critical section, in²

b_o = total length of the critical section, in.

b₁ = width of the critical section measured in the direction of the span for which moments are determined, in.

b₂ = width of the critical section measured in the direction perpendicular to b1, in.

d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, as outlined in ACI 318, in.

I_xx = moment of inertia for bending about the x-axis for the entire critical section, in⁴

${\bar{I}}_{x x}$ = moment of inertia contribution about the x-axis for an individual side of the critical section, calculated with respect to the centroid of the critical section, in⁴

I_yy = moment of inertia for bending about the y-axis for the entire critical section, in⁴

${\bar{I}}_{y y}$ = moment of inertia contribution about the y-axis for an individual side of the critical section, calculated with respect to the centroid of the critical section, in⁴

I_xy = product of inertia for the entire critical section, in⁴

${\bar{I}}_{x y}$ = product of inertia contribution for an individual side of the critical section, calculated with respect to the centroid of the critical section, in⁴

L = length of one side of the critical section, in.

M_ox = joint reaction (moments from columns above and below) about the x-axis at the centroid of the column utilizing a "right-hand rule" for sign convention, kip-in

M_oy = Joint reaction (moments from columns above and below) about the y-axis at the centroid of the column utilizing a "right-hand rule" for sign convention, kip-in

M_ux = column reaction, moment about the x-axis at the centroid of the critical section, kip-in

M_uy = column reaction, moment about the y-axis at the centroid of the critical section, kip-in

v_u = shear stress located at some point on the critical section, ksi

V_u = axial column reaction, located at the centroid of the column with an upward column reaction being positive, kips

x = x-coordinate of the centroid of the entire critical section, in.

${\bar{x}}_{s i d e}$ = x-coordinate of the centroid of a side of the critical section, in.

x_col = x-coordinate of the centroid of the column, in.

x_point = x-coordinate of the point at which you are calculating stresses, in.

y = y-coordinate of the centroid of the entire critical section, in.

${\bar{y}}_{s i d e}$ = y-coordinate of the centroid of a side of the critical section, in.

y_col = y-coordinate of the centroid of the column, in.

y_point = y-coordinate of the point at which you are calculating stresses, in.

γ_vx = fraction of unbalanced moment about the x-axis transferred by eccentricity of shear, in accordance with ACI 318

γ_vy = fraction of unbalanced moment about the y-axis transferred by eccentricity of shear, in accordance with ACI 318

θ = angle between a side of the critical section and the positive x-axis

Equations for Calculation of Shear Stress

The equations presented are derived from basic mechanics of materials. A similar formulation can be found in the article "Design of Stud Shear Reinforcement for Slabs" by Ghali & Elgabry, ACI Structural Journal, May-June 1990. The values of γ_vx and γ_vy are always calculated about the principal axes of the critical section.

$ν_{u} = \frac{V_{u}}{b_{o} d} + \frac{(y_{p o int} - \bar{y}) [γ_{v x} M_{u x} I_{y y} + γ_{v y} M_{u y} I_{x y}]}{I_{x x} I_{y y} - I_{x y}^{2}} - \frac{(x_{p o i n t} - \bar{x}) [γ_{v y} M_{u y} I_{x x} + γ_{v x} M_{u x} I_{x y}]}{I_{x x} I_{y y} - I_{x y}^{2}}$
$M_{u x} = M_{o x} + V_{u} (y_{c o l} - \bar{y})$
$M_{u y} = M_{o y} - V_{u} (x_{c o l} - \bar{x})$
$I_{x x} = \sum_{s i d e s = 1}^{n} {\bar{I}}_{x x}$
$I_{y y} = \sum_{s i d e s = 1}^{n} {\bar{I}}_{y y}$
$I_{x y} = \sum_{s i d e s = 1}^{n} {\bar{I}}_{x y}$
${\bar{I}}_{x x} = \frac{d L^{3}}{12} \sin^{2} (θ) + L d {(\bar{y} - {\bar{y}}_{s i d e})}^{2}$
${\bar{I}}_{y y} = \frac{d L^{3}}{12} \cos^{2} (θ) + L d {(\bar{x} - {\bar{x}}_{s i d e})}^{2}$
${\bar{I}}_{x y} = \frac{d L^{3}}{12} \sin (θ) \cos (θ) + L d (\bar{x} - {\bar{x}}_{s i d e}) (\bar{y} - {\bar{y}}_{s i d e})$
$γ_{v} = 1 - \frac{1}{1 + \frac{2}{3} \sqrt{\frac{b_{1}}{b_{2}}}}$

Note: Equation a) is based upon standard strength of materials equations for bending in an asymmetric section. If the moments are applied about one or more axis of symmetry, then I_xy = 0 and equation a) reduces to the more familiar:

v_{u} = \frac{V_{u}}{b_{o} d} + \frac{γ_{v x} M_{u x} (y_{p o int} - \bar{y})}{I_{x}} - \frac{γ_{v y} M_{u y} (x_{p o int} - \bar{x})}{I_{y}}