Deep Foundations, Bridge Pier Pile Caps & Soil Stress Overlap
Converse-Labarre Friction Pile Group Efficiency Formula
Calculate Stress-Shadow Angle theta (deg), Converse-Labarre Group Efficiency eta_g (%), Unfactored Sum of Single Piles (kN), and Interaction-Reduced Group Ultimate Capacity (kN).
Twelve 1,200 kN friction piles DO NOT carry 14,400 kN at 3d spacing—overlapping shear bulbs cut efficiency to 71%!
When an m-by-n grid of friction piles of diameter d (m) is driven under a rigid bridge-pier pile cap at center-to-center spacing s (m, typically 2.5d to 4.0d), skin-friction stress bulbs from adjacent shafts overlap in soft-to-medium clay, lowering capacity per pile below an isolated load-test pile Q_single! Under the AASHTO/classical Converse-Labarre Group Efficiency Equation, Stress-Overlap Shadow Angle is theta_deg = DEGREES(ATAN(d / s)) and Group Efficiency is eta_g = 1 - (theta_deg / 90) * ((n * (m - 1) + m * (n - 1)) / (m * n)), giving Group Ultimate Capacity Q_group = m * n * Q_single * eta_g! Multiplying m * n * Q_single naively ignores bulb overlap and overstates foundation capacity by nearly 30%!
Verify on a 3 x 4 (12-Pile) Bridge Pier Cap—Pile_Rows m = 3 (B2), Piles_Per_Row n = 4 (B3), Pile_Shaft_Diameter d = 0.50 m (B4), Center_to_Center_Spacing s = 1.50 m (3.0d spacing, B5), and Single_Pile_Ultimate_Capacity Q_1 = 1,200.0 kN (B6): Shadow Angle theta equals ATAN(0.50 / 1.50) = 18.43 deg (E2), Converse-Labarre Group Efficiency hits 70.98% (E3), Unreduced 12-Pile Sum is 14,400.00 kN (E4), and True Group Ultimate Capacity equals 10,221.41 kN (E5)!
| Converse-Labarre Pile Cap KPI (3x4 Grid, d=0.50 m, s=1.50 m, Q1=1,200 kN) | Formula | Verified Output |
|---|---|---|
| Shadow Overlap Angle theta (deg) | ATAN(Pile_Dia_d / Spacing_s)*180/PI() | 18.43° (11.31° @ s = 2.50 m) |
| Group Efficiency eta_g (%) | (1 - (th/90)*((n*(m-1)+m*(n-1))/(m*n)))*100 | 70.98% (82.20% @ s = 2.50 m) |
| Naive Unreduced Sum m*n*Q1 (kN) | Rows_m * Cols_n * Single_Pile_Q1 | 14,400.00 kN |
| Group Ultimate Capacity (kN) | m * n * Q1 * eta_g | 10,221.41 kN (11,836.42 @ 2.50 m) |
Converse-Labarre Rectangular Grid Pile Group Efficiency Formula
Validate integer m >= 1, integer n >= 1, m * n >= 2 (a group has at least 2 piles!), Pile_Dia d > 0, Spacing s >= 2 * d (minimum constructible clear spacing!), and Q_1 > 0!
=LET(mr_w, B2, nc_l, B3, dp_m, B4, sp_m, B5, q1_k, B6, IF(OR(mr_w < 1, INT(mr_w) <> mr_w, nc_l < 1, INT(nc_l) <> nc_l, mr_w * nc_l < 2, dp_m <= 0, sp_m < 2 * dp_m, q1_k <= 0), "Invalid Pile Group Input", LET(th_d, ATAN(dp_m / sp_m) * 180 / PI(), eg_f, 1 - (th_d / 90) * ((nc_l * (mr_w - 1) + mr_w * (nc_l - 1)) / (mr_w * nc_l)), ROUND(eg_f * 100, 2))))Wider 5d Pile Spacing Overlap Relief Check (Set Spacing s = 2.50 m)
Widen Center_Spacing B5 from 1.50 m (3d) to 2.50 m (5d) under an enlarged pile cap: Shadow Angle E2 drops to 11.31 deg, lifting Group Efficiency E3 from 70.98% to 82.20% and adding +1,615.01 kN of Group Capacity E5 (up to 11,836.42 kN)!
=IF(ISTEXT(E3), E3, LET(th_d, ATAN(B4 / B5) * 180 / PI(), eg_f, 1 - (th_d / 90) * ((B3 * (B2 - 1) + B2 * (B3 - 1)) / (B2 * B3)), ROUND(B2 * B3 * B6 * eg_f, 2)))Try the example
Use the original values or change them to test your understanding.