Pile group

The loads on each pile of a group under a rigid cap: vertical and battered piles in any layout, axial load, moments about both axes and horizontal loads, checked against the pile capacities the engineer enters. Two analyses are offered (USACE EM 1110-2-2906, 4-5): the classic rigid-cap formula with axial piles, and the stiffness method of Hrennikoff and Saul (the method of the Corps program CPGA) with lateral pile springs. The cap itself is designed with the pile cap type; the piles' capacities come from the geotechnical and structural pile design (the laterally loaded pile type gives a lateral capacity).

Inputs of the pile group #

• Pile layout: a rectangular grid (piles along x and y, spacings sx and sy, centred on the load point and numbered row by row from −y, each row from −x), or entered coordinates from the load point. Entered piles may be battered: the batter n (n vertical to 1 horizontal; 0 = vertical) and the plan direction the pile leans toward going down (0° = +x, 90° = +y). The pile diameter or width and shape (drawing, spacing and edge checks).
• Analysis:
  • Rigid cap, axial piles: all piles alike and acting axially. With vertical piles the reactions follow R = P/n ± Mx·y/Σy² ± My·x/Σx² about the group centroid (solved in general form, so any layout works) and the horizontal load is shared equally. With battered piles the axial-only stiffness solution resolves the horizontal load through the batter (an elastic-center type solution); a load the batter cannot resolve is NOT FEASIBLE.
  • Rigid cap, elastic piles (stiffness method): each pile is an axial spring ka = C33·A·E/L and the lateral springs of a long pile in a soil of constant modulus Es (Hetenyi's semi-infinite beam on springs, as in CPGA: 4EIβ³, 2EIβ², 2EIβ for a head fixed into the cap, 2EIβ³ for a pinned head, β = (Es/4EI)^¼). The 6 × 6 stiffness of the cap gives its displacements and every pile's axial force, shear and head moment. Inputs: pile length L, area A, modulus E, moment of inertia I, the axial stiffness factor C33 (about 1.0 for end bearing to 2.0 for friction piles, EM 1110-2-2906 4-5; a load test is best) and the soil modulus Es (horizontal subgrade modulus times the pile width, constant with depth, reduced by the engineer for close spacing).
• Loads: per load type (D, L, Lr, S, R, W, E), unfactored, at the load point: P (down), Mx (adds compression on the +y piles), My (adds compression on the +x piles), Hx, Hy. Applied at the top of the cap (H then adds H·h at the pile heads) or at the pile-head level.
• Cap: the edge distance (pile center to cap edge; the cap is the pile group plus the edge distance), thickness h, and optionally the cap and soil-cover weight as dead load.
• Capacities (per pile): compression, tension (0 = no tension allowed), lateral (0 = not checked) and, for fixed heads in the stiffness method, the head moment (0 = not checked); the group capacity (block failure, efficiency; 0 = not checked). With ASD (ASCE 7-22 2.4.1) these are allowable loads; with LRFD (2.3.1) factored resistances.

Auto-design (editor toolbar) varies the piles along x and y and their spacing (grid layout; the y spacing keeps its ratio to the x spacing): fewest piles, then the smallest cap. The pile type, capacities, loads and analysis are kept; piles at entered coordinates are kept as entered. See Auto-design.

Checks performed #

The drawings show the cap plan (piles dashed with their marks, arrows toward the toe of battered piles, pile rows, edge and overall dimensions, the load point) and section A along x with the piles below the cap at their batter. The sheet notes list each pile's largest and smallest axial force by mark. Groups of more than 20 piles mark the first pile of each row.

Limitations #

• The cap is rigid; flexible caps and mats on piles are not covered.
• No torsion about the vertical axis is applied, and the piles carry no torsion.
• The lateral springs assume a long pile in a soil of constant modulus; p-y (nonlinear) springs, group reduction factors and pile-soil-pile interaction are not modelled (enter a reduced Es).
• The pile capacities, block failure and group efficiency are the engineer's inputs; nothing is assumed.
• Rigid method with battered piles: piles act axially only (the horizontal load goes to the batter).

Verification examples #

Verification status: 18 values compared in 5 examples (5 published, 0 hand calculations from the cited equations): 18 within tolerance, 0 recorded deviations. Code basis: the provisions it uses were read in the target edition.

Published worked examples (backend/tests/test_verify_pile_group.py):

In CPGA's sign convention a positive MY adds compression on the −x piles; the example is entered with My = −173.4 kip-ft.