Laterally loaded piles (p-y)
The COM624P p-y finite-difference method — pile inputs, every soil model and parameter, head conditions, and how to read the response.
A laterally loaded pile resists horizontal load by mobilizing the soil around it. The p-y method models that interaction as a beam — the pile — supported by a bed of independent, nonlinear springs — the soil. PileCalc solves it with the COM624P finite-difference formulation (Wang & Reese, FHWA-SA-91-048), the same engine that underlies LPILE and RSPile. This page explains the method and every input on the lateral tool. Use the free p-y curve generator to inspect one published soil model before running a full pile.
The p-y method
The pile is governed by the beam-column equation on a nonlinear (Winkler) foundation. At every depth x, the soil pushes back with a reaction p (force per unit length) that depends on the local deflection y through a p-y curve:
Here EI is the pile's flexural rigidity, Pₓ the axial load (the P-Δ term), and E_py the secant modulus of the p-y curve at the current deflection. Because the springs soften as they load, E_py changes with y — which is what makes the response nonlinear: doubling the load more than doubles the deflection.
Each soil model defines the shape of its p-y curve from the soil's strength and stiffness. A defining feature for clays is the deflection that mobilizes half the ultimate resistance:
where b is the pile diameter and ε₅₀ the strain at half the failure stress in a triaxial test. The smaller ε₅₀, the stiffer the soil's response near the surface.
The finite-difference solver
The pile is divided into a number of equal segments — the increments — and the governing differential equation is replaced by a finite-difference approximation at each node. Because the soil springs are nonlinear, the solver iterates: it assumes a stiffness, solves the linear system, reads new secant moduli off the p-y curves at the resulting deflections, and repeats until the deflections stop changing.
Two diagnostics tell you the solution is trustworthy. Convergence means the iteration settled. The equilibrium residual — the applied load minus the integrated soil reaction — should be essentially zero; a large residual means the model did not balance and the result should not be used.
Direct vs. Georgiadis layering
Pile inputs
The pile is modeled as an elastic beam. For a uniform pile you provide its geometry and bending stiffness directly.
The length of the pile below the head, in contact with soil.
Why it matters. Beyond a critical length the head response stops changing — the pile behaves as “long” and added length does nothing. Too short and the tip itself translates and rotates, changing the entire profile. Length is usually set by axial capacity and the depth to a competent bearing layer.
The pile width or diameter, b.
Why it matters. Diameter appears throughout the p-y curves — ultimate resistance scales with b, and y₅₀ = 2.5·ε₅₀·b — and in the section's bending stiffness. A larger pile attracts more soil resistance and is stiffer. Typical driven/bored piles are 0.3–2 m.
The bending stiffness EI = E · I — modulus times moment of inertia of the section.
Why it matters. EI governs how much the pile bends. A stiffer pile spreads load deeper and deflects less, but it can attract more bending moment. For sections that crack or yield, use moment–curvature to get an effective EI. A 0.6 m solid concrete pile is on the order of 1.9×10⁵ kN·m².
The length of pile standing above the ground surface, between the head and the soil — the depth of the ground surface below the pile head.
Why it matters. Models a pile that projects above grade — a pier column, or a pile exposed by scour. The free length acts as a cantilever and adds substantial deflection. Set it to 0 when the head is at the ground surface.
Sectioned & tapered piles
Not every pile is prismatic. Switching the pile editor to sectioned replaces the single diameter and EI with a stack of sections that tile the pile from head to tip — you enter each section's bottom depth, exactly like soil layers. A stepped pipe, a cased drilled shaft with a rock socket, or a belled base are all stacks of two or three sections. Each section is either prismatic (its own diameter and EI) or tapered. The solver uses the section properties at every node: the bending stiffness enters the finite-difference equations locally, and the p-y curves are built with the local width, so a larger lower section attracts more soil resistance where it actually is.
Tapered sections
A tapered section varies its diameter continuously from a top value to a bottom value — a tapered steel pole, a timber pile, or the transition cone of a belled shaft. You choose the cross-section it represents: a constant-wall pipe or a solid circle, with the material's Young's modulus E. At each node the solver interpolates the local diameter and computes EI from the actual geometry — I ∝ D³t for a thin pipe, I ∝ D⁴ for a solid circle — a geometric taper, not a linear blend between two EI values (which would understate the stiffness loss, since I falls much faster than D). The p-y curve width follows the local diameter too, so both the structural and the soil sides of the model see the true shape.
Soil models
Each soil layer is assigned a p-y model calibrated to its material. Picking the right family matters more than fine-tuning a parameter — the models produce genuinely different resistance shapes. Choose the model in each layer's editor; the parameters that appear adapt to your choice.
| Model | Use for | Key parameters |
|---|---|---|
| Soft clay — Matlock (1970) | Soft to medium clay, submerged | c, ε₅₀, J |
| Stiff clay above water — Welch & Reese (1972) | Stiff clay, no free water | c, ε₅₀, J (optional k) |
| Stiff clay below water — Reese | Stiff clay, submerged (cyclic-sensitive) | c, ε₅₀, k |
| Sand — Reese (1974) | Sand, wedge + flow model | φ, k |
| Sand — API / O'Neill–Murchison | Sand, the API tanh form | φ, k |
| Silt / cemented c-φ — Reese (1974) | Silts and cemented soils with both c and φ | c, φ, k, J |
| Weak rock — Reese (1997) | Weathered / weak rock | qu, Eir, RQD, k_rm |
| Piedmont residual — Simpson & Brown (2003) | Residual silty sands / saprolite (Piedmont province) | Eₛᵢ |
| ISO 19901-4 clay — Jeanjean et al. (2017) | Offshore clay, large-diameter piles | sᵤ (DSS), α |
| Liquefied sand — Rollins et al. (2005) | Fully liquefied sand (post-triggering) | depth only |
| Elastic (Winkler) subgrade | Linear springs, back-analysis | E_py |
| User-defined p-y curves | Load-test-derived or site-specific curves | depth-tagged (y, p) tables |
For clays, the soft-clay and stiff-clay families differ in how resistance is mobilized and how cyclic loading degrades it. For sands, the Reese model builds an explicit passive wedge while the API model uses a calibrated hyperbolic-tangent curve; the two agree closely for typical sands. The elastic model is a linear spring p = E_py·y for back-calculation or code checks. The liquefied-sand model (Rollins et al., 2005) is a concave-up curve fitted to full-scale blast-liquefaction tests — resistance is a function of depth alone, with no strength inputs. The remaining specialty models are described below.
Piedmont residual soil — Simpson & Brown (2003)
The partially weathered micaceous silty sands and silts of the US Southeast Piedmont geologic province — residual soil and saprolite — fit neither the clay nor the sand criteria well. Simpson & Brown (2003) derived a dedicated p-y model from five full-scale lateral load tests at the Auburn NGES. Unlike the hyperbolic or wedge-based curves, it is a secant-modulus-degradation law: the curve starts on a linear branch with initial slope b·Eₛᵢ, then the secant modulus decays logarithmically with normalized deflection y/b, reaching a plateau at y/b = 0.0375.
The single soil input is the initial modulus of subgrade reaction Eₛᵢ (force/length³), best estimated from an in-situ test: Eₛᵢ ≈ 22·N₆₀ (pci) from SPT, 0.118·qc (kPa) from CPT, 0.076·E_DMT (psi) from the dilatometer, or 0.235·E_PMT (psi) from the pressuremeter — the dilatometer and pressuremeter correlations are the ones Simpson & Brown recommend. Near the surface Eₛᵢ is reduced to half its value at grade, ramping linearly to full value one diameter down. Only a static (monotonic) curve is published.
ISO 19901-4 clay — Jeanjean et al. (2017)
The modern offshore alternative to Matlock: the monotonic clay framework of Jeanjean et al. (2017), adopted by ISO 19901-4 to replace the classic soft-clay criterion. The ultimate resistance is pu = Np·su·b with a depth-dependent bearing factor Np that runs from 3.22 at the surface to a deep flow-around cap of 9 + 3α (9 for a smooth pile, 12 fully rough), where α is the pile–soil adhesion factor. The wedge-to-flow-around transition completes at roughly 14.5–17 diameters — far deeper than Matlock's few-diameter transition — which is why ISO diverges from API/Matlock most for large-diameter piles. With the gap open (a crack can form behind the pile) the overburden term σ′ᵥ/su is added; with the gap closed the wedge resistance doubles, both capped at the deep value.
Two modeling details differ from Matlock. The strength su is entered on a Direct Simple Shear (DSS) basis — the ISO-preferred test — so convert triaxial/UU values before entry. And there is no ε₅₀: the backbone is a normalized p/pu vs. y/b curve, fully mobilized at y/b = 0.25.
Stiff clay without free water, using k
The classic Welch & Reese (1972) stiff-clay curve, p = 0.5·pu·(y/y₅₀)^0.25, has a mathematical quirk: its slope is infinite at zero deflection, so the predicted response at small loads is unrealistically stiff. The modified model (Reese & Van Impe, 2011 — LPILE's “stiff clay without free water using k”) caps the initial branch with a subgrade-reaction line:
At working loads the k-line governs near the origin and the Welch–Reese backbone takes over as deflection grows, so ultimate resistance is unchanged — only the small-load stiffness is made finite and realistic. That matters most for serviceability checks and for the head-stiffness springs, which are secants near the origin. In PileCalc, entering a nonzero modified-model k on a stiff-clay-above-water layer switches it to this form; leave it 0 for the classic curve.
User-defined p-y curves
When you have better information than any published criterion — p-y curves back-calculated from a site-specific lateral load test, curves supplied by a geotechnical report, or a material none of the built-in models covers — enter them directly. A user-defined layer holds one or more depth-tagged tables of (y, p) points; the solver interpolates linearly within each table and between tables in depth, so supply at least a curve at the top and bottom of the layer (a single curve applies uniformly). Points must be nondecreasing in y; resistance is held constant beyond the last point.
Soil parameters
Below is every parameter the soil models use, with what it is and why it changes the result. Each layer also carries its top and bottom depth, which must tile the profile without gaps.
The unit weight used to build the effective vertical stress profile that drives p-y resistance with depth.
Why it matters. Higher effective stress means stronger soil and a stiffer response. Use total unit weight above the water table and buoyant (effective) unit weight below it. Typical total ~18–20 kN/m³; buoyant ~8–10 kN/m³.
The undrained shear strength of the clay — the cohesion that resists pile movement.
Why it matters. It scales the ultimate soil resistance pu directly: stronger clay carries more lateral load before the soil yields. Guide values — soft 12–25, medium 25–50, stiff 50–100 kPa.
Source: Matlock (1970)
The strain at one-half the maximum stress difference in an undrained triaxial test.
Why it matters. It sets y₅₀ = 2.5·ε₅₀·b, the deflection that mobilizes half of pu — so it controls how soft or stiff the clay p-y curve is near the surface. Soft clay ≈ 0.02, medium ≈ 0.01, stiff ≈ 0.005–0.007.
Source: Matlock (1970)
An empirical factor in the depth term of the soft-clay ultimate resistance.
Why it matters. It sets how quickly pu grows with depth in the shallow wedge zone. 0.5 is standard; 0.25 suits stiffer, more brittle clays.
Source: Matlock (1970)
The initial modulus of subgrade reaction — the starting slope of the p-y curve, increasing with depth.
Why it matters. Governs near-surface stiffness for sand and below-water stiff clay. This is one of the few inherently unit-dependent inputs — match it to your unit system (see Units). Loose sand ~5,400, medium ~16,300, dense ~34,000 kN/m³; lower below water.
Source: Reese, Cox & Koop (1974)
The drained angle of internal friction of the sand.
Why it matters. Controls the passive-wedge size and the ultimate resistance pu — a few degrees materially change capacity and deflection. Loose 28–30, medium 32–36, dense 38–42°.
Source: API / O'Neill & Murchison (1983)
The uniaxial (unconfined) compressive strength of the intact rock.
Why it matters. Sets the ultimate side resistance of the weak-rock p-y curve. Weak rock is roughly 0.5–5 MPa.
Source: Reese (1997)
The initial modulus of the rock mass.
Why it matters. Sets the initial stiffness of the weak-rock p-y curve before yielding. Often on the order of 100–500× qu for weak rock.
Source: Reese (1997)
The percentage of an intact rock core recovered in pieces ≥ 100 mm — a fracturing index.
Why it matters. Reduces the rock-mass strength via aᵣ = 1 − ⅔(RQD/100); lower RQD means more fractured, weaker rock. Weak / fractured rock is often 25–75%.
Source: Reese & Nyman (1978)
A constant relating the rock-mass modulus to the deflection at which resistance mobilizes.
Why it matters. Larger k_rm softens the rock p-y curve. Reese suggests a narrow band, 0.0005 to 0.00005; use the lower end for stiffer rock.
Source: Reese (1997)
For the elastic (Winkler) model, the linear spring modulus p = E_py·y, specified at the top and bottom of the layer.
Why it matters. Defines a linear, deflection-independent spring. Set the top value to 0 and the bottom to n_h · depth for a modulus that increases linearly with depth. Back-calculate it from a known stiffness.
Source: Terzaghi subgrade reaction
Linear in-layer variation
Real deposits rarely change strength in steps — a clay strengthens with depth, a sand densifies. Each layer's advanced parameters let the key properties vary linearly from the top of the layer to its bottom: enter a bottom-of-layer value for cohesion (cBottom), subgrade modulus (kBottom), friction angle (phiBottom), or ε₅₀ (e50Bottom), and the solver interpolates the property at each node's depth when building its p-y curve. This matches how LPILE and RSPile take a layer's top/bottom strengths, and it avoids slicing a thick layer into artificial sub-layers just to capture a gradient. Leave a bottom value at 0 (blank) for a constant layer.
Head boundary conditions
The pile head needs two prescribed quantities. Which two you set is the boundary condition, and it has a first-order effect on the answer: a free head deflects most, while fixing rotation cuts deflection and shifts the peak moment up to the head.
The horizontal load applied at the pile head — the primary demand.
Why it matters. Deflection and bending moment grow with it, nonlinearly, because the soil softens as it loads. Present in every head condition.
The bending moment applied at the pile head.
Why it matters. An eccentric load or a partially fixed cap applies head moment, increasing near-surface bending and deflection. Use 0 for a free, concentrically loaded head.
The prescribed rotation dy/dx of the pile head.
Why it matters. Set 0 to model a head fully fixed against rotation — for example, embedded in a rigid cap — which markedly reduces deflection.
The rotational restraint provided by a partially fixed connection.
Why it matters. Bridges the free and fixed extremes — a stiffer connection rotates less and sheds more moment into the pile. Large → fixed, 0 → free.
A prescribed lateral displacement of the pile head (paired with a head moment).
Why it matters. Use when the head movement is known — a serviceability target, say — and you want the force and moment that develop to produce it.
Loading & options
Whether the p-y curves are built for static (monotonic) or cyclic (repeated) loading.
Why it matters. Cyclic loading degrades soil resistance — especially in clay — giving larger deflections and moments for the same load. Use static for one-off loads; cyclic for wind, wave, or traffic.
How layered soil is handled. Direct builds each layer's curve at its true depth (original COM624P). Georgiadis shifts lower layers to an equivalent depth that accounts for the resistance of the soil above.
Why it matters. Georgiadis is what LPILE and RSPile use for layered profiles and generally improves agreement; choose it for multi-layer soils. Direct reproduces the original COM624P behavior.
Source: Georgiadis (1983)
The number of finite-difference segments the pile is divided into.
Why it matters. More increments improve accuracy but cost a little compute. About 100 is plenty for typical piles; 100–200 is a good range.
Axial load carried down the pile (compression positive), included as a second-order term.
Why it matters. Axial load acting through the lateral deflection adds P-Δ moment, which increases deflection — important for slender piles with significant axial load. Set 0 to ignore it.
A factor that scales the soil resistance p of every curve, leaving the deflection axis unchanged — the standard way to model group shadowing, where the failure wedge of one pile overlaps the soil that its neighbors rely on.
Why it matters. A trailing-row pile in a closely spaced group mobilizes far less resistance than a single pile at the same deflection. From the full-scale group tests of Brown, Morrison & Reese (1988), typical row values at a 3-diameter spacing are ≈ 0.8 for the lead row, ≈ 0.4–0.5 for middle rows, and ≈ 0.3 for trailing rows; use 1.0 for a single pile. Enter one scalar for the whole pile, or switch on the depth-varying profile to define fₘ as a piecewise-linear function of depth — group interaction is strongest where the wedges overlap near the surface, so a profile like 0.75 at grade tapering to 0.3 at depth captures reductions a single constant misses.
Source: Brown, Morrison & Reese (1988)
Tip boundary conditions
By default the pile tip is free — zero moment and zero shear at the toe, the standard COM624P condition. Three inputs change that. A prescribed tip shear and tip moment replace the zero-force boundary conditions directly — use them when something below the model applies a known force to the toe. More useful in practice is the tip-shear curve: a table of mobilized base shear versus tip movement that acts as a restoring spring at the toe, solved iteratively alongside the p-y curves.
Why it matters. On a long pile the tip barely moves and none of this matters. On a short or intermediate pile the tip translates, and an enlarged or belled base then develops lateral shear against the soil below — a real resistance the standard model ignores. Feeding in a base shear–displacement curve (from a base load test, or an estimate of base friction under the vertical load) measurably reduces the computed head deflection for short piles. Leave it off for slender piles, where it has no effect.
Load sweep (pushover)
Because the soil springs soften as they load, a single solution tells you the response at one load level only. The load sweep option additionally solves the pile at a series of proportionally scaled head loads — from a fraction of the target up to the full max shear / max moment (which default to the head condition's own loads) — and returns the whole pushover curve in one run. Set the number of steps (2–50; about 10 is typical) and read the results on the Sweep tab: head deflection vs. shear and maximum moment vs. shear. The curvature of those plots is the nonlinearity itself — a flattening deflection curve means the soil is approaching full mobilization, and points that fail to converge mark the practical capacity of the system. Each sweep point is an independent p-y solution, so a 20-step sweep costs about 20 single runs of compute but only one analysis.
Cracked section (nonlinear EI)
A constant EI is only right while the section stays elastic. With the cracked section option you describe the structural section itself — solid circular, pipe, or rectangular, with its steel modulus E, yield stress Fy, and optional hardening ratio — and the solver replaces the pile EI node-by-node with the stiffness read off the section's fiber-integrated moment–curvature (M–φ) relation, the same approach LPILE uses for nonlinear EI. Where the computed moment exceeds the yield moment My, the stiffness drops and deflections grow beyond what a constant-EI run predicts. The results report the section's elastic EI, My, and plastic moment Mp, and flag the run when the maximum moment reaches My — a sign the structural section, not the soil, is becoming the limit.
Round and rectangular reinforced-concrete sections are supported alongside the steel shapes: round sections use circular cages, while rectangular sections use straight-sided perimeter reinforcement with explicit face counts and clear cover. Optional permanent casing and steel core apply to round sections, and prestress is supported for both reinforcement layouts — see moment–curvature for the section inputs. You can also supply your own M–φ table directly instead of a section description.
Plastic-moment capacity
A constant or even nonlinear EI still lets the computed moment grow without bound. Specifying a plastic-moment capacity Mcap makes the pile elastic–plastic: wherever the moment demand reaches Mcap, a plastic hinge forms — the moment is capped there and the pile rotates instead of resisting more. This is the mechanism behind pushover analysis of a pile bent: past hinge formation, added head load produces rapidly growing deflection rather than moment, and the response flattens toward a mechanism.
Why it matters. Without a cap, a run can report a maximum moment the section could never carry, silently overstating the system's stiffness at that load. With Mcap set — typically the section's plastic moment, on the order of Fy·Z for steel — the results show where hinges form and what deflection the mechanism implies. In the UI Mcap is a single scalar (leave 0 for an elastic pile); through the API you can supply a capacity-vs-depth table instead, for sectioned or tapered piles whose capacity varies along the shaft.
LRFD load combinations
Load and Resistance Factor Design doesn't change the pile analysis — it changes how loads are entered and how the answer is checked. With the LRFD panel enabled you enter unfactored head loads by type — dead, live, wind, earthquake, snow, and so on, each with its shear, moment, and axial components. The tool then runs the ACI 318 strength combinations (1.4D; 1.2D + 1.6L + 0.5Lr; 1.2D + 1.6Lr + 1.0L; 1.2D + 1.0W + 1.0L + 0.5Lr; 1.2D + 1.0E + 1.0L + 0.2S), each as an independent p-y solution at the factored loads — correctly capturing the nonlinearity, since factoring the load does not factor the response.
Each combination is checked against the factored capacities: |Mmax| ≤ φ_b·Mn for flexure and, when a shear capacity is given, |Vmax| ≤ φ_v·Vn. The nominal moment Mn defaults to the plastic moment of the structural section when one is supplied; the shear capacity Vn is always a user input (LPILE likewise computes no standardized Vn), and the shear check is simply skipped when it's omitted. The resistance factors default to φ_b = 0.9 (tension-controlled flexure) and φ_v = 0.75 per ACI 318. The LRFD results tab lists every combination with its factored loads, demand/capacity ratios, and pass/fail — the governing case is the one closest to (or past) unity.
Reading the results
The tool reports four summary quantities and four response profiles.
Summary quantities
- Head deflection — the lateral movement at the top of the pile, compared against the structure's allowable displacement. Usually the governing serviceability check.
- Max moment and its depth — drives the structural (reinforcement / section) design of the pile.
- Max shear — checked against the section's shear capacity.
- Equilibrium residual — applied load minus integrated soil reaction; ≈ 0 for a good solution, large if the model failed to balance.
Response profiles
Deflection, bending moment, shear, and soil reaction are plotted against depth, drawn with depth increasing downward. These shapes are what engineers reason about: where the pile crosses zero deflection, where the moment peaks, how deep the soil reaction develops. The Data tab gives the node-by-node table for export.
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