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
bo = 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.
Ixx = moment of inertia for bending about the x-axis for the entire critical section, in4
= 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
Iyy = moment of inertia for bending about the y-axis for the entire critical section, in4
= 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
Ixy = product of inertia for the entire critical section, in4
= 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.
Mox = 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
Moy = 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
Mux = column reaction, moment about the x-axis at the centroid of the critical section, kip-in
Muy = column reaction, moment about the y-axis at the centroid of the critical section, kip-in
vu = shear stress located at some point on the critical section, ksi
Vu = 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.
= x-coordinate of the centroid of a side of the critical section, in.
xcol = x-coordinate of the centroid of the column, in.
xpoint = x-coordinate of the point at which you are calculating stresses, in.
y = y-coordinate of the centroid of the entire critical section, in.
= y-coordinate of the centroid of a side of the critical section, in.
ycol = y-coordinate of the centroid of the column, in.
ypoint = y-coordinate of the point at which you are calculating stresses, in.
γvx = fraction of unbalanced moment about the x-axis transferred by eccentricity of shear, in accordance with ACI 318
γvy = 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 γvx and γvy are always calculated about the principal axes of the critical section.