A p-y curve describes how much lateral resistance the soil offers at one depth as the pile pushes into it. The p is soil reaction per unit length of pile (kN/m) and the y is the lateral deflection of the pile at that depth (m). The curve is nonlinear: resistance builds quickly at small deflection, then flattens as the soil approaches its ultimate capacity, and in some soils it falls away again once the ground has been remoulded or scoured. Every soil layer has its own curve, and the curve at one depth differs from the curve at another in the same layer because both the confining stress and the failure mechanism change with depth.

Why model soil as springs rather than as a continuum?

A full continuum model (finite elements in three dimensions) is more faithful but needs constitutive parameters most projects never measure, takes far longer to build and solve, and is very sensitive to choices a designer cannot verify. The p-y method replaces the continuum with independent nonlinear springs whose behaviour was calibrated directly against full-scale pile load tests. It gives up the shear transfer between soil layers and any true three-dimensional effect, and in exchange it is fast, repeatable and anchored to measured pile behaviour rather than to assumed soil parameters. That trade is why it became standard practice.

Where do the p-y curves come from?

From instrumented full-scale lateral load tests. Matlock (1970) derived the soft clay curves from tests on 324 mm pipe piles at Lake Austin and Sabine River. Reese, Cox and Koop derived the sand curves from a 610 mm pile at Mustang Island (1974) and the stiff clay curves from tests at Manor, Texas (1975). In each case the researchers measured bending moment along the pile, differentiated it twice to recover the soil reaction, integrated it twice to recover the deflection, and fitted a curve to the resulting p and y pairs. That derivation history is also the method's main limitation: the curves carry the diameter, soil and installation method of the piles they came from.

What is the difference between a short pile and a long pile?

A short pile rotates essentially as a rigid body: the whole pile moves, the tip kicks back the other way, and capacity is governed by the soil failing along the full length. A long pile bends: only the upper few diameters deflect appreciably, the lower part stays effectively fixed, and adding more length changes nothing for lateral load. The transition depends on the ratio of pile stiffness to soil stiffness rather than on length alone. You can see which case you have from the deflection diagram - if the tip is still moving, the pile is behaving as a short pile.

Why does the ultimate resistance change with depth?

Because the failure mechanism changes. Near the surface the soil fails by pushing a wedge upward and outward toward the free ground surface, and the resistance available depends on how much overburden is holding that wedge down - so it grows with depth. Deep enough, the wedge cannot reach the surface and the soil instead flows horizontally around the pile, a mechanism whose resistance no longer depends on depth. Every p-y model takes the lesser of the two, which is why the ultimate resistance rises with depth and then plateaus. The depth where they cross is the transition depth.

What does epsilon-50 mean and why does clay need it?

Epsilon-50 is the strain at which a clay sample reaches half its maximum stress in a laboratory compression test. It sets how quickly the p-y curve mobilises: the reference deflection is y50 = 2.5 times epsilon-50 times the pile diameter, and the curve reaches half its ultimate resistance there. A soft, high-plasticity clay strains a long way before mobilising resistance and has a large epsilon-50; a hard clay mobilises quickly and has a small one. Two clays with the same undrained shear strength but different epsilon-50 give visibly different pile deflections, which is why the parameter matters as much as the strength.

How is the governing equation actually solved?

The pile is a beam column on an elastic foundation, so the governing equation is EI times the fourth derivative of deflection, plus the axial thrust times the second derivative, minus the soil reaction, equals zero. Because the soil reaction depends on the deflection it is solving for, the equation is nonlinear and cannot be solved in closed form. The pile is divided into a few hundred stations, the derivatives are replaced by finite differences, and the system is solved repeatedly: each pass evaluates the soil springs at the current deflection, solves the resulting linear system, and feeds the new deflection back in until it stops changing.

Why use the secant soil stiffness rather than the tangent?

Two of the published curves have a falling branch - stiff clay with free water, and sand past its plateau - where the tangent stiffness is negative. Feeding a negative stiffness into the system destroys the property that makes it stable to solve, and the iteration diverges. The secant stiffness, meaning the soil reaction divided by the deflection, is never negative no matter what the curve is doing. Using it converges more slowly but it converges reliably even past the peak of the curve, which is exactly where a heavily loaded pile is operating.

What are p-multipliers and when do they matter?

A pile in a closely spaced group attracts less soil resistance than an isolated one, because neighbouring piles are leaning on the same soil - a shadowing effect that is strongest for a pile directly behind another in the loading direction. The usual treatment is to scale the whole p-y curve by a p-multiplier less than one, chosen from the spacing and the position within the group. Multipliers around 0.4 to 0.8 are common at three-diameter spacing. This calculator analyses a single isolated pile, so group effects must be applied separately.

Tiếng Việt

Lateral Pile Analysis and the p-y Method - Theory

Q: Why does cyclic loading reduce the resistance?

Repeated loading remoulds clay and breaks down the structure that gave it strength, and where free water is present it lets water circulate in the gap that opens behind the pile, scouring the soil away. Both effects mean the same deflection mobilises less resistance after many cycles than on the first push. Matlock and Reese measured the difference and published separate cyclic curves for it. The degradation is worst near the surface, where the gap can form and water can reach, and disappears at depth where the soil flows around the pile instead.

The theory behind this calculator: why a laterally loaded pile is modelled as a beam on nonlinear soil springs, how each published p-y curve is constructed from the soil strength, how the governing beam-column equation is solved by finite differences, and how to judge whether the answer is trustworthy.

🪨Open the interactive lateral pile analysis calculator→

Run this method in the free calculator and export the result to Excel (.xlsx) or PDF.

Why a laterally loaded pile is a hard problem

A pile carrying vertical load is a comparatively simple thing: the load goes down the shaft and into the ground, and the pile barely moves. Push the same pile sideways and the problem changes character completely. The pile bends, the soil beside it deforms, and the resistance the soil offers depends on how far the pile has already moved into it. The soil is stiff at first and softens as it approaches failure, so the stiffness of the support changes continuously as the load is applied. There is no linear solution to fall back on, and the answer depends on the load level.

Worse, the resistance is not the same at every depth. Near the ground surface the soil can escape upward, so it fails at a low load. Deep down it is confined on all sides and must flow horizontally around the pile, which takes far more force. The same soil, described by the same strength, offers very different resistance at 1 m and at 10 m.

The p-y method handles both difficulties by replacing the soil with a set of independent nonlinear springs, one at each depth, each following a curve calibrated against full-scale pile load tests. It is the basis of standard practice for laterally loaded piles and drilled shafts, and it is what this calculator implements.

The model: a beam on nonlinear springs

The pile is treated as a beam column. The soil is discarded as a continuum and replaced by springs attached along the shaft. Each spring is nonlinear and independent of its neighbours: it knows only the deflection of the pile at its own depth, not what the soil above or below is doing. The relationship between the reaction it develops and that deflection is the p-y curve.

Both symbols are per unit length of pile. $p$ is the soil reaction in force per metre of pile (kN/m), and $y$ is the lateral deflection at that depth (m). The word "curve" matters - the relationship is never a straight line except in the elastic-subgrade idealisation.

The governing equation

Equilibrium of a beam column on an elastic foundation gives a fourth-order differential equation. With depth $z$ measured downward from the pile head:

EI\dfrac{d^{4y}}{dz^{4}} + P\dfrac{d^{2y}}{dz^{2}} - p(y,z) = 0

where:

EI

bending stiffness of the pile section (kN·m²)

y

lateral deflection at depth z (m)

P

axial thrust, which contributes a second-order (P-delta) moment as the pile bends (kN)

p(y,z)

soil reaction per unit length, the p-y curve at that depth (kN/m)

The term that makes this hard is the last one. Because $p$ depends on $y$ - the very quantity being solved for - the equation is nonlinear and has no closed-form solution. Only the special case of a constant spring stiffness, the elastic subgrade model, can be solved analytically; that solution is useful mainly as a way to verify a numerical implementation, which is exactly how this engine was checked.

How a p-y curve is built

Every published p-y model is assembled from two ingredients: an ultimate resistance that caps the curve, and a backbone shape that says how quickly that resistance is mobilised.

Ultimate resistance and the two failure mechanisms

Near the surface, soil in front of the pile fails by lifting a wedge upward and outward toward the free ground surface. The resistance depends on how much overburden holds that wedge down, so it grows with depth. Below a certain depth the wedge can no longer reach the surface and the soil instead flows horizontally around the shaft, a mechanism whose resistance does not increase with depth. The governing value is the lesser of the two. For Matlock's soft clay:

p_u = \min\left[\left(3 + \dfrac{\gamma' z}{c_u} + \dfrac{Jz}{D}\right)c_u D,\; 9\,c_u D\right]

where:

p_u

ultimate soil resistance per unit length (kN/m)

c_u

undrained shear strength of the clay (kPa)

\gamma'

effective unit weight of the soil (kN/m³)

D

pile diameter or width (m)

J

empirical factor, 0.5 for soft clay and 0.25 for stiff clay

z

depth below ground level (m)

The first expression is the wedge; the second, $9c_{uD}$ , is the flow-around value. Setting them equal gives the transition depth at which the mechanism changes:

z_r = \dfrac{6D}{\dfrac{\gamma' D}{c_u} + J}

For a 0.6 m pile in a soft clay of 40 kPa this lands at roughly 3 to 4 m - a few diameters, which is also where the maximum bending moment usually occurs. That is not a coincidence: the moment peaks where the soil has finished reversing the shear, and the soil can only do that once it is strong enough to push back hard.

The backbone shape

Ultimate resistance alone says nothing about stiffness. The backbone supplies that, and for clay it is anchored to a reference deflection derived from a laboratory strain:

y_{50} = 2.5\,\varepsilon_{50}\,D

where:

y_{50}

deflection at which half the ultimate resistance is mobilised (m)

\varepsilon_{50}

strain at 50% of the maximum stress in a laboratory compression test

D

pile diameter (m)

Matlock's soft clay curve is then a cubic parabola which reaches the ultimate resistance at eight times that deflection:

\dfrac{p}{p_u} = 0.5\left(\dfrac{y}{y_{50}}\right)^{1/3} \quad (p \le p_u)

Reese and Welch's stiff clay curve has the same ultimate resistance but a fourth-root backbone, which is stiffer at small deflection and takes sixteen times $y_{50}$ to reach $p_u$ . That single difference in exponent is why two clays of equal strength can give visibly different pile deflections, and why $\varepsilon_{50}$ deserves as much attention as the strength itself.

Sand: a different construction

Sand has no undrained strength to anchor to, so Reese, Cox and Koop built their curve from the friction angle and the effective overburden instead. It has four segments: an initial straight line of slope $k\,z$ , a fitted parabola, a straight segment, and a plateau at the ultimate value. The ordinates of the characteristic points come from empirical coefficients $A$ and $B$ read against the normalised depth $z/D$ .

The API RP 2A formulation simplifies all of that into a single smooth expression, which is why offshore practice generally prefers it - there are no digitised chart values to get wrong, and no falling branch to destabilise a solver:

p = A\,p_u \tanh\!\left(\dfrac{k\,z\,y}{A\,p_u}\right)

where:

A

factor for static or cyclic loading, 3 - 0.8z/D but not less than 0.9

k

initial modulus of subgrade reaction, from the friction angle (kN/m³)

p_u

lesser of the shallow wedge and deep flow-around resistance (kN/m)

Note that $E_s = k\,z$ in sand: the initial stiffness grows linearly with depth rather than being a property of the material alone. This is the single biggest conceptual difference between the sand and clay models.

The soil models and where each applies

Model	Source	Use it when
Soft clay	Matlock (1970), OTC 1204	Clay below roughly 50 kPa, on or offshore. The most widely used curve in the whole method.
Stiff clay with free water	Reese, Cox & Koop (1975), OTC 2312	Stiff clay below the water table where water can circulate in the gap behind the pile. Includes a falling branch for scour.
Stiff clay without free water	Reese & Welch (1975)	Stiff clay above the water table, or where no free water can reach the pile. Stiffer response, no scour.
Sand	Reese, Cox & Koop (1974), OTC 2080	Any sand onshore. Four-segment curve from the published A and B charts.
Sand (API)	API RP 2A-WSD	Offshore work, or where a code requires it. Smooth, robust, generally the more conservative.
Elastic subgrade	Classical (Hetenyi 1946)	First estimates, sensitivity studies, and verifying an implementation against a closed-form answer.

A single analysis can mix models freely - a profile of sand over soft clay over stiff clay uses three different formulations at once, each applying over its own layer.

Cyclic loading

Repeated loading degrades soil. In clay the repeated straining remoulds the material and destroys the structure that gave it strength. Where free water is present, a gap opens behind the pile on each cycle and water circulating in it scours soil away. The result is that the same deflection mobilises less resistance after many cycles than on the first push.

Matlock and Reese measured this and published separate cyclic curves. In soft clay the cyclic curve follows the static one up to about 0.72 of the ultimate resistance and then degrades, with the reduction largest near the surface and vanishing below the transition depth - which makes physical sense, because that is exactly where the gap can form and the water can reach.

The practical consequence is a useful sanity check: a cyclic analysis must predict more deflection and usually more moment than the equivalent static one. If it does not, something in the model is wrong.

Solving the equation

With nonlinear springs there is no closed-form answer, so the equation is solved numerically. The pile is divided into a few hundred stations at uniform spacing, the derivatives are replaced by central finite differences, and the boundary conditions at the head and tip replace the equations at the outermost nodes.

Finite differences rather than finite elements is a deliberate choice here: it is the formulation the original p-y literature and the reference programs use, so results are directly comparable, and it produces the soil reaction at each node rather than as an element average.

The solution is then iterative. Each pass evaluates every spring at the current deflection to get a stiffness, assembles and solves the resulting linear system, and feeds the new deflection back in. The cycle repeats until the deflection stops changing to within a tolerance.

Why the secant stiffness, not the tangent

The obvious choice is the tangent to the p-y curve, which converges quadratically. But two of the published curves - stiff clay with free water, and sand past its plateau - have a falling branch where the tangent is negative. A negative spring stiffness destroys the diagonal dominance of the system matrix and the iteration diverges precisely where a heavily loaded pile is operating.

E_{s,\text{secant}} = \dfrac{p(y)}{y} \;\ge\; 0 \quad \text{always}

The secant stiffness is never negative regardless of the curve shape, so the solve stays stable well past the peak. It converges linearly rather than quadratically, which is a fair price for robustness, and it is why the solver also applies under-relaxation: blending each new deflection with the previous one damps the oscillation that a softening curve would otherwise produce.

Boundary conditions at the pile head

Two conditions are needed at the head, and the choice materially changes the answer.

Condition	Models	Effect
Shear + moment	A free head - a single pile with a pinned connection	Largest deflection, moment peaks below ground
Shear + slope	A fixed head - a pile in a rigid cap (slope = 0)	Much smaller deflection, moment peaks at the head
Shear + rotational stiffness	An elastic connection, the realistic middle ground	Between the two extremes
Displacement + moment	A prescribed movement, e.g. back-analysing a load test	Solves for the load that caused it
Displacement + slope	A fully prescribed head	Used for imposed-movement studies

Real pile caps are neither perfectly pinned nor perfectly rigid. When the distinction matters, run both extremes: they bracket the true behaviour, and the difference in deflection is often a factor of two or three.

Piles that stand above the ground

Jetty piles, bridge piers, pile bents and sign posts have a free length between the load and the ground. That length has no soil beside it, so it contributes no resistance at all - it acts as a cantilever and adds a lever arm to everything below. The governing equation is unchanged; the stations above ground simply have no spring.

The effect is strongly asymmetric. The head deflection grows very quickly with free length, because the cantilever term dominates, while the groundline deflection grows much more slowly, because only the extra moment reaches the soil. This is why published load tests quote the groundline deflection: it is the part that reflects the soil-structure interaction rather than the geometry of the test rig.

Getting the soil parameters

A p-y model needs undrained shear strength or friction angle, effective unit weight, the strain factor $\varepsilon_{50}$ and the modulus of subgrade reaction $k$ . Very few projects measure all of them, so most are derived from whatever the site investigation produced.

From	Gives	Typical correlation
SPT blow count	Friction angle, undrained strength	Peck, Hanson & Thornburn (sand); Stroud / Terzaghi & Peck (clay), after energy and overburden correction
CPT cone resistance	Friction angle, undrained strength, unit weight	Kulhawy & Mayne (sand); cu = (qt - sigma_v)/Nkt (clay); Robertson & Cabal (unit weight)
Laboratory testing	Strengths directly	None needed - only the curve-shape parameters are still derived
Soil description	Everything, as typical ranges	Handbook values - the lowest-confidence route, for preliminary work only

These correlations carry real scatter. Stroud's $c_u/N$ ratio alone varies by roughly a factor of two with plasticity, and the cone factor $N_{kt}$ ranges from about 10 to 20. That is why the calculator shows every derived value with the correlation that produced it, and lets you override any of them: a number a reviewer cannot trace is a number they cannot check.

Broms' method - the ultimate-capacity cross-check

Everything above answers one question: how far does the pile move under the load you apply? It does not answer the other one - what load would make the pile fail at all. Those are different questions, and a p-y analysis does not directly give the second. Broms' method does, and this calculator offers it as a second analysis method for exactly that reason.

Broms (1964a, 1964b) replaces the whole nonlinear problem with a rigid-plastic idealisation: the soil is everywhere at its ultimate resistance, and the pile is rigid until it forms a plastic hinge. That throws away the deflection information the p-y method exists to produce, and in exchange it gives a closed-form collapse load from hand-checkable statics.

The assumed soil resistance

In clay, Broms takes zero resistance over the top 1.5 diameters and full flow-around resistance below it:

p_u = 0 \quad (z < 1.5D), \qquad p_u = 9\,c_u D \quad (z > 1.5D)

The zero-resistance zone is the signature of the method. A gap opens behind the pile near the surface and the remoulded clay heaves rather than resisting, so Broms discards it entirely. This is the sharpest difference from p-y, where resistance grows continuously from the ground surface, and it is deliberately conservative.

In sand, the resistance grows linearly with depth:

p_u(z) = 3\,K_p\,\gamma' z\,D, \qquad K_p = \tan^{2}\!\left(45^\circ + \dfrac{\varphi'}{2}\right)

The factor of three on the plane Rankine value accounts for the three-dimensional wedge that forms in front of a discrete pile rather than a continuous wall.

Two mechanisms, and the lesser one governs

A laterally loaded pile can fail in two quite different ways, and which one happens depends on whether the pile or the soil gives way first.

Short pile. The pile is strong relative to the ground, so it stays essentially straight and rotates rigidly about a point at depth. The soil fails over the full length. Capacity follows from statics alone, and it is set by the ground: more embedment buys capacity, a stronger section does not.
Long pile. The pile cannot rotate rigidly because a plastic hinge forms first at the point of maximum moment. Capacity is then set by the section: a stronger pile buys capacity, extra length does not.

Broms computes both and takes the lesser - whichever failure happens first is the one that happens. The classification is therefore an output, not something you have to decide in advance. For a free-headed long pile the hinge condition is

M_y = H_u\,(e + z_m)

where:

M_y

plastic moment of the pile section (kN·m)

H_u

ultimate lateral load at the pile head (kN)

e

load eccentricity above ground - the free-standing length (m)

z_m

depth below ground to the maximum moment (m)

A pile with a fixed head (a rigid cap) forms two hinges, one at the head and one in the span, so the right-hand side carries $2M_y$ and the capacity is substantially higher. The calculator picks the fixed-head form automatically from the head condition you set.

A worked comparison

The calculator's default profile is a 0.6 m steel pipe pile, 20 m long, in ground that averages to a clay with $c_u = 63$ kPa. The section has $M_y = 1938$ kN·m. The resistance below the 0.9 m gap is $9 \times 63 \times 0.6 = 340$ kN/m. Broms then gives:

Mechanism	Ultimate load Hu	What it means
Short pile (soil fails)	2522 kN	The ground could carry this much
Long pile (hinge forms)	882 kN	The section yields first - this governs
Governing capacity	882 kN	Hinge at 3.49 m below ground

The pile is comfortably in the long-pile regime, so its capacity is a property of the steel section rather than of the clay. Driving it deeper would not help; a thicker wall would.

Reading the two methods together

The two answers are not competing estimates of the same quantity, and comparing them naively is the easiest mistake to make here. The p-y diagrams are drawn at the load you applied; the Broms diagrams are drawn at the collapse load. On the default profile that is 150 kN against 882 kN, so the Broms moments look about an order of magnitude larger - which is arithmetic, not disagreement.

The meaningful comparison is at the sameload. Re-running the p-y analysis at 882 kN gives a peak moment of about 2640 kN·m against Broms' 2494 - a ratio of 0.95, which is close agreement between two methods that share no equations. A large divergence is worth investigating: the usual cause is a strongly layered profile, since Broms is single-layer by construction and the calculator has to average the strata into one equivalent layer before it can run.

Two cautions on using the number. Broms himself recommended a factor of safety of 2 to 3 on the computed capacity, reflecting the scatter in the underlying idealisation. And for most real structures the pile is governed by deflection long before it approaches failure - a serviceability limit of 10 or 25 mm at the head is typically reached at a small fraction of Hu. Broms tells you how much margin exists against collapse; it does not tell you the pile is acceptable.

Limitations worth knowing

The springs are independent. No shear is transferred between soil layers, so a thin weak layer between two strong ones is modelled more crudely than it behaves.
The curves carry their calibration. They came from driven piles roughly 300 to 1200 mm in diameter. Applying them to a 6 m monopile is extrapolation, and the offshore wind industry developed separate formulations for exactly that reason.
Group effects are excluded. A pile in a close group attracts less resistance than an isolated one; that is handled with p-multipliers applied to the whole curve, not by the single-pile analysis.
Installation matters and is not modelled. A driven pile densifies the sand around it; a bored pile may loosen it. The published curves largely reflect driven piles.
Some things are outside the method. Liquefaction, lateral spreading, downdrag, sloping ground and soil movement imposed from outside all need separate treatment.

Against published field tests, a p-y analysis generally lands within about a factor of 1.5 on deflection. For a soil calculation that is good, but it is an estimate rather than a precise answer - and the sensible response is to check how much your conclusion would change if the soil were half as stiff.

References

Matlock, H. (1970). "Correlations for Design of Laterally Loaded Piles in Soft Clay." Offshore Technology Conference, OTC 1204.
Reese, L.C., Cox, W.R. & Koop, F.D. (1974). "Analysis of Laterally Loaded Piles in Sand." Offshore Technology Conference, OTC 2080.
Reese, L.C., Cox, W.R. & Koop, F.D. (1975). "Field Testing and Analysis of Laterally Loaded Piles in Stiff Clay." Offshore Technology Conference, OTC 2312.
Reese, L.C. & Welch, R.C. (1975). "Lateral Loading of Deep Foundations in Stiff Clay." ASCE Journal of the Geotechnical Engineering Division, 101(GT7).
Broms, B.B. (1964a). "Lateral Resistance of Piles in Cohesive Soils." ASCE Journal of the Soil Mechanics and Foundations Division, 90(SM2), 27-63.
Broms, B.B. (1964b). "Lateral Resistance of Piles in Cohesionless Soils." ASCE Journal of the Soil Mechanics and Foundations Division, 90(SM3), 123-156.
Reese, L.C. & Van Impe, W.F. (2001). Single Piles and Pile Groups Under Lateral Loading. Balkema.
American Petroleum Institute (2014). API RP 2A-WSD, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms. 22nd edition.
Hetenyi, M. (1946). Beams on Elastic Foundation. University of Michigan Press.
Skempton, A.W. (1986). "Standard Penetration Test Procedures and the Effects in Sands of Overburden Pressure, Relative Density, Particle Size, Ageing and Overconsolidation." Geotechnique, 36(3), 425-447.
Stroud, M.A. (1974). "The Standard Penetration Test in Insensitive Clays and Soft Rocks." Proceedings of the European Symposium on Penetration Testing, Stockholm.

Frequently asked questions

Ready to analyse a pile? Build a layered soil profile from SPT, CPT or laboratory data and get deflection, bending moment, shear and soil reaction against depth.

🪨Open the interactive lateral pile analysis calculator→

Rate this

No ratings yet

Loading…