Why is the reduction applied to the p-y curve rather than to the load?

Because it is the soil that is weakened, not the pile. Scaling the resistance curve by a p-multiplier means the trailing pile still reaches the same deflection under less load, which is what the field tests show. Reducing an applied load instead would leave the stiffness wrong and give the wrong deflection.

Why does a rigid cap make the piles carry different loads?

The cap forces every pile head to the same displacement. Since the trailing piles are supported by weaker soil, they develop less force at that shared displacement - so load redistributes toward the leading row. If the piles could move independently they would each carry the same share; it is the compatibility imposed by the cap that makes the distribution uneven.

How does a pile group resist an overturning moment?

Mainly by axial push and pull, not by bending. The cap rotates slightly, driving the front row down and lifting the back row, and that force couple acts over the plan dimension of the group - metres. A pile bending against its own diameter has a lever arm of a fraction of a metre, so its contribution is much smaller. This is why the axial stiffness of the piles matters as much as their bending stiffness.

What does group efficiency mean, and why is it not a fixed number?

Efficiency compares the stiffness of the group with the same number of isolated piles. It is not constant because the p-y curves are nonlinear: at small load the piles sit in the near-linear part of the curve where initial stiffness dominates and efficiency approaches one, while at working load the multipliers bite and it falls sharply. Quoting a single efficiency figure without stating the load level is close to meaningless.

Why can a pile end up in tension when the whole group is pushed down?

The vertical load is shared roughly equally, but the overturning moment adds compression at the front and tension at the back. If the moment couple exceeds the share of vertical load a back pile carries, the net force is uplift. That is a genuinely different design case: it needs shaft resistance in tension and a connection able to transfer it into the cap.

Tiếng Việt

Pile Group Analysis - Theory

Q: What is the shadowing effect?

When a group is pushed sideways, the leading pile displaces the soil in front of it. The pile behind then meets ground that has already been disturbed and partly unloaded, so it mobilises less resistance for the same movement. The effect compounds down the group, which is why the third row is weaker than the second. It is a soil effect, not a structural one - the piles themselves are identical.

The theory behind this calculator: why a pile in a group attracts less soil resistance than an isolated one, how the AASHTO p-multiplier table quantifies that, how a rigid cap ties every pile into one rigid-body motion, and why most of an overturning moment is carried by axial force rather than bending.

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

Why a group is not nine single piles

Analyse one pile, multiply by nine, and you will get the wrong answer twice over. Two things change when piles are built close together and tied by a cap, and they pull in opposite directions.

The first is the soil. When the group is pushed sideways, the leading pile displaces the ground in front of it. The pile behind then meets soil that has already been disturbed and partly unloaded, so it mobilises less resistance for the same movement. The effect compounds down the group. This is the shadowing effect, and it makes the group softer than the sum of its parts.

The second is the cap. A rigid cap forces every pile head to move together, which stops the piles from finding their own equilibrium independently and redistributes load toward whichever piles are stiffest. It also gives the group a way to resist moment that a single pile does not have.

The p-multiplier

The standard treatment of shadowing is disarmingly simple: take the whole p-y curve for an isolated pile and scale it by a factor less than one.

p_{group}(y) = P_m \cdot p_{single}(y)

where:

p_{single}(y)

the published p-y curve for an isolated pile (kN/m)

P_m

the p-multiplier for the pile’s row (-)

y

lateral deflection at that depth (m)

Note what is scaled and what is not. The resistance is reduced at every deflection, so the trailing pile is both weaker and softer. The deflection axis is untouched, so the shape of the curve, and the deflection at which the soil reaches its limit, are unchanged. Scaling the applied load instead would give the right force and the wrong stiffness.

The AASHTO table

This calculator uses AASHTO LRFD Table 10.7.2.4-1, which tabulates the multiplier by row position and by centre-to-centre spacing in pile diameters:

Row	3D spacing	5D spacing
Row 1 (leading)	0.80	1.00
Row 2	0.40	0.85
Row 3 and beyond	0.30	0.70

Three properties of that table matter in use. It is discrete - two spacings only, so intermediate spacings are interpolated and nothing is extrapolated outside 3D to 5D. It is row-based, so every pile in a row shares one multiplier regardless of its position across the group. And beyond five diameters the piles are treated as isolated, which is where the table stops rather than a claim that interaction vanishes exactly there.

Rows depend on the load direction

Row 1 is the row the load pushes toward. A group that is symmetric in plan is therefore not symmetric in its multipliers: reverse the load and the leading row becomes the trailing one. This calculator derives rows from the plan layout and the load direction rather than asking you to number them, so the numbering cannot drift out of step with the geometry.

The rigid cap

A pile cap is stiff compared with the piles beneath it, so it is idealised as infinitely rigid: the whole group moves as one body with three degrees of freedom in two dimensions.

u_i = u, \qquad w_i = w + \theta\,x_i

where:

u

cap translation along the load direction (m)

w

cap settlement, positive downward (m)

\theta

cap rotation, positive when the leading row goes down (rad)

x_i

in-line coordinate of pile i from the group centroid (m)

The second equation is the one that matters. Rotation moves each pile head vertically in proportion to its distance from the centroid, so the front of the group is driven down while the back is pulled up. That is how the group resists moment.

Equilibrium

The three unknowns are found by requiring that what the piles carry equals what was applied:

\sum V_i = V, \qquad \sum N_i = N, \qquad \sum \left(M_i + N_i x_i\right) = M

Each pile's lateral response is nonlinear, because the p-y springs are, so this is solved by Newton iteration rather than in closed form. The axial response is treated as linear, which is a deliberate simplification discussed below.

Where the moment actually goes

It is tempting to picture a cap moment being carried by the piles bending. It is not. Compare the two lever arms: the axial couple works over the plan dimension of the group, which is metres, while a pile bending against its own section has a lever of a fraction of a metre. The axial path dominates by an order of magnitude.

On the calculator's default 3x3 group under 2,400 kN·m of overturning, the leading row takes 2,114 kN of compression while the trailing row goes into 114 kN of tension. Remove the axial stiffness and the same moment rotates the cap more than twenty times as far.

Two consequences follow. First, the axial stiffness of a pile matters as much as its bending stiffness in a group, which is not true of a single pile. Second, a back-row pile can finish in tension even though the group as a whole is pushed down - and a pile in tension is a different design case, needing shaft resistance and a connection able to transfer it into the cap.

Axial stiffness

The axial head stiffness used here is an elastic column in series with the soil supporting it:

\dfrac{1}{k_{ax}} = \dfrac{1}{A E / L} + \dfrac{1}{Q_{ult} / (0.01 D)}

The first term is exact - it is just the pile shortening. The second is the uncertain half, based on the practical rule that shaft friction is fully mobilised at roughly one percent of the pile diameter. Because they act in series, the combined stiffness is always softer than either alone: a soft soil cannot stiffen an elastic column.

Real axial stiffness varies by a factor of several with installation method, soil type and time. Treat the estimate as a starting point and a pile load test as the reliable source - the calculator states its basis and lets you override the value directly.

Group efficiency depends on the load

Group efficiency compares the stiffness of the group with the same number of isolated piles. It is often quoted as a single figure, and that is misleading, because the p-y curves are nonlinear.

At small load the piles sit in the near-linear part of the curve, initial stiffness dominates, and the multipliers barely register - on the default profile efficiency is about 0.95 at 1.5 mm of cap deflection. At working load the springs mobilise and the reduction bites: the same group is about 0.33 at 23 mm. An efficiency figure quoted without a load level does not mean much.

What this method does not do

No axial capacity check. The analysis distributes load; whether a pile can carry the axial force it ends up with is a separate calculation.
No settlement, no block failure. Group settlement and the block failure mode of a tightly spaced group are different problems entirely.
The axial response is linear. Fine for a working-load distribution, wrong near failure.
The cap is rigid and its strength is not checked. Punching shear and cap reinforcement are a separate design.
The multipliers carry real scatter. Measured values from full-scale group tests spread widely around the AASHTO table, which is a design simplification of a body of test data rather than a measurement.

None of this makes the answer useless - load distribution is exactly what a group analysis is for, and it is the input the subsequent capacity checks need. It does mean the output is a distribution, not a verdict.

References

AASHTO (2020). LRFD Bridge Design Specifications, Table 10.7.2.4-1 - P-Multipliers for Pile Groups. 9th edition, American Association of State Highway and Transportation Officials.
Reese, L.C. & Van Impe, W.F. (2001). Single Piles and Pile Groups Under Lateral Loading. Balkema.
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.
American Petroleum Institute (2014). API RP 2A-WSD, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms. 22nd edition.
Randolph, M.F. & Wroth, C.P. (1978). "Analysis of Deformation of Vertically Loaded Piles." ASCE Journal of the Geotechnical Engineering Division, 104(GT12), 1465-1488.

Frequently asked questions

Ready to analyse a group? Lay out the piles, apply the cap loads and get the shear, axial force and bending moment carried by every row.

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