VE 9972 | Punching Design of Interior Column in Flat Slab with Unsymmetrical Cross-Section According to ACI 421

Description

Analytical Solution

1. Calculation of the Effective Slab Depth

The effective slab depth is calculated from the slab thickness, concrete cover, and nominal diameter of the flexural reinforcement:

\(
\mathsf{
d
=
h-c-d_b
}
\)

\(
\mathsf{
=
7.00\,in.
-
0.75\,in.
-
\frac{5}{8}\,in.
=
5.62\,in.
}
\)

2. Calculation of the Critical Section Geometry

The critical section is located at a distance of \(\mathsf{d/2}\) from the column face.

The dimensions of the critical section are:

\(
\mathsf{
b_1
=
c_x+d
=
12.00\,in.
+
5.62\,in.
=
17.62\,in.
}
\)

\(
\mathsf{
b_2
=
c_y+d
=
20.00\,in.
+
5.62\,in.
=
25.62\,in.
}
\)

The length of the critical perimeter is:

\(
\mathsf{
b_o
=
2\left(b_1+b_2\right)
=
2\left(17.62+25.62\right)
=
86.5\,in.
}
\)

According to the reference solution, the effective area and the polar moment of inertia of the critical section are:

\(
\mathsf{
A_c
=
486\,\mathrm{in.}^{2}
}
\)

\(
\mathsf{
J_c
=
28.0
\times
10^{3}\,\mathrm{in.}^{4}
}
\)

The maximum distance from the centroid of the critical section is:

\(
\mathsf{
x
=
\frac{b_1}{2}
=
\frac{17.62}{2}
=
8.81\,in.
}
\)

Column Cross-Section with Critical Perimeter

3. Calculation of the Fraction of Moment Transferred by Shear

The fraction of the unbalanced moment transferred by eccentric shear is calculated according to ACI 421.1R Eq. (4.2b):

\(
\mathsf{
\gamma_v
=
1
-
\frac{1}
{
1+
\frac{2}{3}
\sqrt{\frac{b_1}{b_2}}
}
}
\)

\(
\mathsf{
=
1
-
\frac{1}
{
1+
\frac{2}{3}
\sqrt{\frac{17.62}{25.62}}
}
=
0.36
}
\)

4. Calculation of the Applied Punching Shear Stress

The maximum punching shear stress is calculated from the direct shear force and the unbalanced moment transferred by eccentric shear:

\(
\mathsf{
v_u
=
\frac{V_u}{A_c}
+
\frac{\gamma_v M_{uy}x}{J_c}
}
\)

\(
\mathsf{
=
\frac{110\times10^{3}\,lb}
{486\,\mathrm{in.}^{2}}
+
\frac{
0.36
\cdot
600\times10^{3}\,lb\,in.
\cdot
8.81\,in.
}
{
28.0\times10^{3}\,\mathrm{in.}^{4}
}
=
294\,psi
}
\)

Using the strength-reduction factor

\(
\mathsf{
\phi
=
0.75
}
\)

the design shear stress is:

\(
\mathsf{
\frac{v_u}{\phi}
=
\frac{294\,psi}{0.75}
=
392\,psi
}
\)

5. Calculation of the Concrete Shear Resistance

The ratio of the long side to the short side of the column is:

\(
\mathsf{
\beta_c
=
\frac{c_y}{c_x}
=
\frac{20}{12}
=
1.67
}
\)

The nominal concrete shear resistance without shear reinforcement is determined from the minimum of the following expressions:

• \(\mathsf{\left(2+\frac{4}{\beta_c}\right)\sqrt{f'_c}}\)
• \(\mathsf{\left(2+\frac{40d}{b_o}\right)\sqrt{f'_c}}\)
• \(\mathsf{4\sqrt{f'_c}}\)

For the present geometry:

\(
\mathsf{
v_n
=
\min
\left[
4.4\sqrt{f'_c},
4.6\sqrt{f'_c},
4.0\sqrt{f'_c}
\right]
=
253\,psi
}
\)

Thus:

\(
\mathsf{
v_n
=
4\sqrt{f'_c}
=
253\,psi
}
\)

Since

\(
\mathsf{
\frac{v_u}{\phi}
=
392\,psi
>
253\,psi
=
v_n
}
\)

headed shear reinforcement is required.

6. Calculation of the Required Shear Reinforcement

In the presence of headed shear reinforcement, the shear stress resisted by the concrete is:

\(
\mathsf{
v_c
=
3\sqrt{f'_c}
=
190\,psi
}
\)

The shear stress to be resisted by the headed studs is therefore:

\(
\mathsf{
v_s
\geq
\frac{v_u}{\phi}
-
v_c
=
392\,psi
-
190\,psi
=
202\,psi
}
\)

The required stud area per spacing is:

\(
\mathsf{
\frac{A_v}{s}
\geq
\frac{v_s b_o}{f_{yt}}
}
\)

\(
\mathsf{
\geq
\frac{
202\,psi
\cdot
86.5\,in.
}
{
51{,}000\,psi
}
=
0.34\,in.
}
\)

7. Selection of the Headed Stud Arrangement

The first peripheral line of studs and the spacing between adjacent peripheral lines must satisfy:

\(
\mathsf{
s_1
\leq
0.5d
}
\)

and

\(
\mathsf{
s
\leq
0.5d
}
\)

With

\(
\mathsf{
0.5d
=
0.5
\cdot
5.62\,in.
=
2.81\,in.
}
\)

the selected values

\(
\mathsf{
s_1
=
2.25\,in.
}
\)

and

\(
\mathsf{
s
=
2.75\,in.
}
\)

satisfy the spacing requirements.

Using ten studs with a cross-sectional area of approximately
\(\mathsf{0.11\,in.^2}\) per stud, the provided stud area per peripheral line is:

\(
\mathsf{
A_v
=
10
\cdot
0.11\,\mathrm{in.}^{2}
=
1.10\,\mathrm{in.}^{2}
}
\)

The provided stud area per spacing is:

\(
\mathsf{
\frac{A_v}{s}
=
\frac{1.10\,\mathrm{in.}^{2}}
{2.75\,in.}
=
0.40\,in.
}
\)

Since

\(
\mathsf{
0.40\,in.
>
0.34\,in.
}
\)

the selected headed shear reinforcement is adequate.

Side View of Punching Shear Reinforcement

8. Verification of the Outer Critical Section

The outer critical section is located at a distance of \(\mathsf{d/2}\) beyond the outermost peripheral line of studs.

For ten peripheral lines, the distance between the column face and the outermost stud line is:

\(
\mathsf{
s_1
+
9s
=
2.25\,in.
+
9
\cdot
2.75\,in.
=
27.0\,in.
}
\)

Therefore, the distance of the outer critical section from the column face is:

\(
\mathsf{
\alpha d
=
27.0\,in.
+
\frac{5.62\,in.}{2}
=
29.8\,in.
}
\)

The corresponding factor is:

\(
\mathsf{
\alpha
=
\frac{\alpha d}{d}
=
\frac{29.8}{5.62}
=
5.3
}
\)

At this critical section, the design shear stress is:

\(
\mathsf{
\frac{v_u}{\phi}
=
125\,psi
}
\)

The nominal shear resistance outside the shear-reinforced zone is:

\(
\mathsf{
v_n
=
2\sqrt{f'_c}
=
126\,psi
}
\)

Since

\(
\mathsf{
125\,psi
<
126\,psi
}
\)

the extent of the shear-reinforced zone is adequate.
\(
\)

Results

The results from RFEM 6 are presented below.

The RFEM 6 results are compared with the analytical reference solution in the following table.

Evaluation

The comparison shows the following:

• The effective slab depth and the critical perimeter are in very good agreement.
• The polar moment of inertia differs by approximately 1.7%. This minor deviation places the RFEM result on the conservative side.
• The largest deviation occurs in the applied punching shear stress and results from the different treatment of moment transfer.
• The concrete shear resistance is reproduced almost exactly.
• The selected headed shear reinforcement satisfies the design requirements.

The selected headed shear reinforcement and the extent of the shear-reinforced zone satisfy the corresponding design requirements.