Punching Shear Check with Column Moment — Two-Way Shear, Moment Transfer and the Depth Required

Design checks

Calculation

1. 1Effective depth
   d = h - c_c - d_b
   d = 10.00 - 0.75 - 0.625 = 8.625
   d=8.625 in

   The average of the two layers' depths.

2. 2Critical section, interior columnACI 318-19, document reference
   b_0 = \textstyle\sum l_i, \quad b_1 = c_1 + d \ \text{(or run to an edge)}, \ b_2 = c_2 + d
   b_0 = 28.63 + 28.63 + 28.63 + 28.63 = 114.50, \quad b_{1} = 28.63, \ b_{2} = 28.63
   b_0=114.50 in

   d/2 from the column's faces; sides lost to a slab edge: none. b1 is the section's extent along local 2, b2 along local 3; for M2 they swap.

3. 3Polar moments of the critical sectionACI 318-19, document reference
   J_c = \textstyle\sum \left( \frac{d\,l^3}{12} + \frac{l\,d^3}{12} \right)_{\parallel} + \sum \left( d\,l\,r^2 \right)_{\perp} \ \text{(about the centroid)}
   J_{c3} = 137928, \quad J_{c2} = 137928
   J_{c3}=137928 in⁴

   Jc3 for M3 (the stress varies along local 2), Jc2 for M2 (along local 3). Sides along the stress's variation by their own second moment; sides across it by d·l·r².

4. 4Fraction of the moment carried by eccentric shearACI 318-19, document reference
   \gamma_v = 1 - \frac{1}{1 + \frac{2}{3}\sqrt{b_1/b_2}}
   \gamma_{v3} = 1 - \frac{1}{1 + \frac{2}{3}\sqrt{28.63/28.63}} = 0.400, \quad \gamma_{v2} = 0.400
   \gamma_{v3}=0.400
5. 5Size effectACI 318-19, document reference
   \lambda_s = \sqrt{\frac{2}{1 + d/10}} \le 1
   \lambda_s = \sqrt{\frac{2}{1 + 8.625/10}} = 1.000
   \lambda_s=1.000

   Two-way slab without shear reinforcement, ACI 318-19 (d in inches).

6. 6Concrete two-way shear strengthACI 318-19, document reference
   v_c = \min\!\left(4,\ 2 + \frac{4}{\beta},\ 2 + \frac{\alpha_s d}{b_0}\right) \lambda_s \lambda \sqrt{f'_c}
   v_c = \min(253.0,\ 379.5,\ 317.1) = 253.0, \quad \beta = 1.00,\ \alpha_s = 40,\ \sqrt{f'_c} = 63.25
   v_c=253.0 psi

   4 governs. αs = 40, 30, 20 for 4, 3, 2 sides; √f'c ≤ 100 psi.

7. 7Design shear strengthACI 318-19, document reference
   \phi v_c, \quad \phi = 0.75
   \phi v_c = 0.75 \times 253.0 = 189.7
   \phi v_c=189.7 psi
8. 8Shear on the section (Entered by hand)
   V_u = |P|, \quad v_{ug} = \frac{V_u}{b_0 d}
   V_u = 150.0, \quad v_{ug} = \frac{150.0 \times 1000}{114.50 \times 8.625} = 151.9
   V_u=150.0 kip
9. 9Unbalanced moments about the section's centroidACI 318-19, document reference
   M_{u3} = \pm M_3 + V\,e_2, \quad M_{u2} = \pm M_2 + V\,e_3
   M_{u3} = 40.00 + 0.00 = 40.00, \quad M_{u2} = 25.00 + 0.00 = 25.00
   M_{u3}=40.00 kip·ft

   Interior column: the centroid is at the column's centre, so no V·e term. The senses of M3 and M2 shown are those giving the worst stress.

10. 10Largest shear stress on the sectionACI 318-19, document reference
    v_u = \frac{V_u}{b_0 d} + \frac{\gamma_{v3} M_{u3}\, x}{J_{c3}} + \frac{\gamma_{v2} M_{u2}\, y}{J_{c2}}
    v_u = 151.9 + \frac{0.400 \times 40.00 \times 12000 \times 14.313}{137928} + \frac{0.400 \times 25.00 \times 12000 \times 14.313}{137928} = 184.3
    v_u=184.3 psi

    At the section's corner (+14.31, +14.31) in from the column's centre along local 2 and 3; x and y measured from the section's centroid.

11. 11Effective depth requiredACI 318-19, document reference
    d_{req} = \min\{d : v_u(d) \le \phi v_c(d)\ \text{for every combination}\}, \quad h_{req} = d_{req} + c_c + d_b
    d_{req} = 8.44, \quad h_{req} = 8.44 + 0.75 + 0.625 = 9.82 \quad (d = 8.625 \text{ provided})
    d_{req}=8.44 in

    Found by search: the section, Jc, γv, vc (with αs·d/b0 and λs) and any deducted load are recomputed at each depth.

12. 12Moment transferred by flexure, M3 (local 2)ACI 318-19, document reference
    \gamma_f = 1 - \gamma_v, \quad b_{slab} = c + 1.5h \times 2, \quad A_s = \frac{0.85 f'_c b d}{f_y}\left(1 - \sqrt{1 - \frac{2 \gamma_f M_{sc}}{\phi\, 0.85 f'_c b d^2}}\right)
    \gamma_f M_{sc} = 0.600 \times 40.00 = 24.00, \quad b_{slab} = 50.0, \quad A_s = 0.63
    A_s=0.63 in²

    Top steel within 50.0 in centred on the column (cut at a free edge), φ = 0.9; Msc the largest over the combinations (Entered by hand), about the section d/2 out.

13. 13Moment transferred by flexure, M2 (local 3)ACI 318-19, document reference
    \gamma_f = 1 - \gamma_v, \quad b_{slab} = c + 1.5h \times 2, \quad A_s = \frac{0.85 f'_c b d}{f_y}\left(1 - \sqrt{1 - \frac{2 \gamma_f M_{sc}}{\phi\, 0.85 f'_c b d^2}}\right)
    \gamma_f M_{sc} = 0.600 \times 25.00 = 15.00, \quad b_{slab} = 50.0, \quad A_s = 0.39
    A_s=0.39 in²

    Top steel within 50.0 in centred on the column (cut at a free edge), φ = 0.9; Msc the largest over the combinations (Entered by hand), about the section d/2 out.

Assumptions

• The column is a rectangle c1 × c2: c1 along its local 2 axis (ETABS t3, across which M3 bends) and c2 along local 3 (t2, across which M2 bends). A circular column is entered as the square of the same area.
• d = h − cover − db, the average depth of the two layers of bars.
• The column's force P punches the slab whatever its sign; the moments are taken about the critical section's centroid, adding the column load's eccentricity from it at an edge or corner.
• Each moment is taken in both senses and the worst combination of them governs: a column's moment sign does not say which way it bends the slab.
• γv of the moment about each axis is carried by shear varying linearly about the section's centroid; γf = 1 − γv by flexure within c + 1.5h each side. γf is not raised for low shear (the code's optional adjustment is not taken).
• Nothing is deducted for load inside the critical section, so Vu = P (conservative).