Rigid and structural pile cap

The group-cap tool couples two named calculations: the Saul/USACE CPGA rigid-body pile-load distribution and an edition-specific reinforced-concrete cap check. It never treats a plausible pile reaction as proof that the cap itself has adequate flexure or shear resistance.

Two explicit analysis scopes

• Distribution only preserves all six cap actions (Fx, Fy, Fz, Mx, My, Mz) and returns cap displacement plus every pile-head reaction. It makes no concrete resistance claim.
• Distribution + structural accepts explicit cap, column/pedestal, pile-head footprints, normal-weight concrete, four reinforcement mats, and already-factored load combinations. It checks flexure, one-way shear, column/pedestal punching, pile punching, required reinforcement, and the governing demand/capacity ratio.

Rigid-cap distribution

Each pile contributes its vertical, lateral, and optional rotational head springs at its declared plan coordinates. The full 6×6 stiffness K = Σ B_iᵀ k_i B_i retains coupling for asymmetric layouts. Solving for u = [ux, uy, uz, rx, ry, rz] gives, for example, the compression-positive pile reaction Q_i = kz(uz + rx·y_i − ry·x_i). This is the classic Saul (1968) formulation used by USACE CPGA and discussed in EM 1110-2-2906.

Structural cap method

The current structural method is deliberately frozen to the AASHTO LRFD Bridge Design Specifications, Second Edition with 2002 interims, matching the public FHWA NHI-04-041 pile-footing worked examples. A newer AASHTO edition will be a separate method ID; the engine will not substitute revised coefficients or resistance factors silently.

Flexure is cut at each loaded-area face. One-way shear uses the edition-specific critical section and effective shear depth. The demand per unit width follows the FHWA NHI-04-041 Step 8.11 outer-row estimate: the piles beyond each cut are split about the cap centreline, the heavier half is mirrored across the cap (for example R = [2(P1 + P2) + P3]/L), and the result is spread over the full cap width, so a biaxial moment that loads one end of a row more heavily is not averaged away. Column and pile punching perimeters are offset bydv/2, clipped at cap edges, and include fractional pile-head pressure when a rectangular footprint crosses a cut. βc is the long-to-short side ratio of the load (AASHTO 5.13.3.6.3; FHWA GEC 12 p. 476): the column for column punching and the pile footprint for pile punching. Where an edge-clipped pile perimeter on a narrow cap is more elongated than 2 and than the pile, its ratio (the FHWA NHI-04-041 Step 7.11 reading) is used instead, with a warning, because it gives the lower resistance. The four top/bottom X/Y mats carry physical bar area, diameter, spacing, cover, and layer order so d is auditable.

Loads and combinations

Every combination has a stable ID and contains already-factored loads. Cap self-weight is applied only through its explicit combination factor and the declared concrete unit weight; no hidden 1.0 or 1.25 default is inserted. In structural mode, loads act at the column/pedestal centroid and are transformed to the cap origin. The structural cap outline is always centered on the pile-coordinate origin (0, 0); it is not re-centered on the pile pattern. Measure pile and column coordinates from the cap center, not a corner. A pile pattern whose centroid is off the origin is analysed as an eccentric pattern and the response carries a warning. The sign convention is right-handed with z downward: positive Fz is downward and positive pile axial reaction is compression.

Checks and governing utilization

• Combination envelope — maximum pile compression, uplift, and lateral reaction across all combinations.
• Flexure — actual moment, minimum design moment, resistance, required/provided reinforcement, tension face, and DCR at four faces.
• One-way shear — cut coordinate, demand, resistance, effective shear depth, and DCR in ±X and ±Y.
• Punching — separate column/pedestal and applicable pile checks with demand, bo, dv, resistance, and DCR.
• Governing utilization — the largest DCR with its combination and location; over 1.0 is a failed structural check.

For supplied factored reactions from up to eight equal circular piles, the separate combined-pile punching assessment evaluates GEC 12 subset, edge and corner bounds. It complements this historical cap method; it does not replace the cap's one-way shear and flexure checks.

Engineering warnings

Some inputs are legal but need engineering judgment. The response then carries plain-language warnings, shown above the results in the tool, in the MCP result card, and in the PDF report's engineering-warnings table:

• Rotational pile-head stiffness: the rigid-cap solve omits the CPGA lateral–rotational coupling terms b15/b24 (USACE TR ITL-89-3 ¶60–61; Saul 1968), so the reported head moments are kθ times the cap rotation only. Obtain fixed-head pile moments from a coupled pile-head analysis.
• Reinforcement below 60 ksi: AASHTO LRFD 5.4.3.1 permits yield strengths below 60 ksi only with the Owner's approval.
• Edge distance and spacing: a pile closer than 9 in from its side to the cap edge, or pile pairs closer than the greater of 30 in or 2.5 pile widths, are below the AASHTO LRFD 10.7.1.2 minimums (FHWA GEC 12, FHWA-NHI-16-009, p. 470).
• Eccentric pile pattern: a pile-pattern centroid away from the cap centre usually means coordinates were measured from a corner; confirm the pattern is intended.

Public benchmark and status

Automated tests reproduce FHWA NHI-04-041 Design Step 8.11: its 20-pile geometry and biaxial load formula give Pile 1 ≈ 290.45 kip; the 14 reactions outside the column punching perimeter give ≈ 2509 kip; and the published bo ≈ 625.61 in, dv ≈ 36.40 in, and factored punching resistance ≈ 4082 kip are recovered within worked-example rounding. SI and US twins execute one inch-kip-ksi calculation path and agree to conversion precision.

Primary public sources: the FHWA abutment pile-footing example, pier pile-footing example, and USACE EM 1110-2-2906.

Applicability and fail-closed limits

Structural mode currently requires all of the following:

• a perfectly rigid rectangular cap on uncoupled linear springs for vertical piles;
• rectangular column/pedestal and pile-head bearing footprints wholly inside the cap;
• normal-weight concrete (λ = 1) with a unit weight of 135–160 pcf (21.21–25.13 kN/m³): the AASHTO LRFD 5.2 normal-weight range of 135–155 pcf plus the C3.5.1 allowance of 5 pcf for reinforcement;
• f′c 2.4–10 ksi and fy 40–75 ksi (below 60 ksi the response warns, per AASHTO 5.4.3.1);
• under-reinforced sections within c/d ≤ 0.42 and separated individual punching regions;
• no transverse shear reinforcement in the resistance equations;
• factored vertical load and biaxial overturning only; and
• compression pile reactions only—pile-head anchorage/pullout is not represented.

Overlapping footprints, impossible bar layers, out-of-range materials, horizontal or torsional structural load transfer, uplift reactions, overlapping individual punching regions, and substantial overlap with the loaded area all fail clearly. Flexible caps, battered piles, strut-and-tie regions, shear friction, development/anchorage, durability, seismic detailing, and newer-code checks remain outside this method ID.