Structural Concrete Design, Flat-Plate Floors & Column Footings
Two-Way Slab Punching Shear Critical Perimeter (b0) Formula
Calculate ACI 318 Critical d/2 Perimeter b_0 (mm) around an Interior Column, Nominal Two-Way Shear Strength V_c (kN), Factored Shear Resistance phi*V_c (kN), and Demand-to-Capacity Ratio.
Offset the punching shear cone by d/2 on ALL sides—adding d to EACH column dimension!
Sudden brittle punching-shear failure around interior building columns is the #1 collapse hazard in beamless flat-plate parking garages and mat footings! Under ACI 318, the truncated 45° shear pyramid is evaluated along Critical Perimeter b_0 located d / 2 outward from every face of a rectangular interior column (c_1 x c_2 mm): because d / 2 is added on BOTH left and right faces (and front/back faces), Critical Perimeter is b_0 = 2 * (c_1 + d) + 2 * (c_2 + d) (NOT 2 * c_1 + 2 * c_2 + 2 * d, a frequent exam error that misses half the corner offset)! For square/low-aspect columns without shear studs, concrete nominal two-way shear capacity is V_c = (0.33 * SQRT(f_c_prime) * b_0 * d) / 1,000 kN with shear reduction factor phi = 0.75!
Verify on an Interior Flat-Plate Column Connection—Column_Width c_1 = 450.0 mm (B2), Column_Depth c_2 = 450.0 mm (B3), Effective_Slab_Depth d = 200.0 mm (B4 -> Critical Perimeter b_0 = 4 * (450 + 200) = 2,600.00 mm in E2), Concrete_Strength f_c_prime = 30.0 MPa (B5), and Factored_Column_Shear_Transfer V_u = 650.0 kN (B6): Nominal Punching Capacity V_c hits (0.33 * SQRT(30) * 2,600 * 200) / 1,000 = 939.89 kN (E3), Factored Resistance phi * V_c equals 704.92 kN (E4), and Demand-to-Capacity Utilization V_u / (phi * V_c) equals 0.922 (<= 1.000 PASS, E5)!
| Two-Way Punching KPI (450x450 mm Col, d=200 mm, fc'=30 MPa, Vu=650 kN) | Formula | Verified Output |
|---|---|---|
| Critical Perimeter b0 at d/2 (mm) | 2 * (c1 + d) + 2 * (c2 + d) | 2,600.00 mm (2,800 @ d = 250 mm) |
| Nominal Concrete Shear Vc (kN) | (0.33 * SQRT(fc') * b0 * d) / 1000 | 939.89 kN (1,265.24 @ d = 250 mm) |
| Design Punching phi*Vc (kN) | 0.75 * Nominal_Vc_kN | 704.92 kN (948.93 @ d = 250 mm) |
| Punching DCR Ratio Vu / (phi*Vc) | Factored_Shear_Vu / (0.75 * Vc) | 0.922 (0.685 @ d = 250 mm) |
ACI 318 Interior-Column Two-Way Punching Capacity & DCR Formula
Validate c_1 >= 150, c_2 >= 150, column aspect max(c_1,c_2)/min(c_1,c_2) <= 2.0 (where the 0.33 coefficient governs!), Slab_Depth d >= 100, f_c_prime in [17, 70] MPa (SQRT(f_c_prime) <= 8.3 MPa per ACI), and V_u > 0!
=LET(c1_m, B2, c2_m, B3, dp_s, B4, fc_m, B5, vu_k, B6, IF(OR(c1_m < 150, c2_m < 150, MAX(c1_m, c2_m) / MIN(c1_m, c2_m) > 2, dp_s < 100, fc_m < 17, fc_m > 68, vu_k <= 0), "Invalid Slab Input", LET(b0_p, 2 * (c1_m + dp_s) + 2 * (c2_m + dp_s), vc_n, (0.33 * SQRT(fc_m) * b0_p * dp_s) / 1000, ROUND(0.75 * vc_n, 2))))Drop-Panel +50 mm Slab Thickening Quadratic Gain Check (Set Effective Depth d = 250 mm)
Add a 50 mm column drop panel raising Effective_Depth B4 from 200.0 mm to 250.0 mm (+25%): because d enlarges BOTH shear depth AND perimeter b_0 (2,800 mm in E2), Design Punching Capacity E4 jumps +34.6% to 948.93 kN, slashing DCR E5 to 0.685!
=IF(ISTEXT(E4), E4, LET(b0_p, 2 * (B2 + B4) + 2 * (B3 + B4), pv_c, 0.75 * ((0.33 * SQRT(B5) * b0_p * B4) / 1000), ROUND(B6 / pv_c, 3)))Try the example
Use the original values or change them to test your understanding.