The p-y method, in full view

The governing equation

A laterally loaded pile behaves like a beam-column on a bed of nonlinear springs. The equilibrium of a slice of the pile gives a fourth-order differential equation:

EI·y⁗ + Pₓ·y″ + Eₚᵧ·y − W = 0

where

• y is lateral deflection and EI the pile's flexural rigidity (which may vary with depth);
• Pₓ is the axial load, contributing a second-order (P-Δ) term;
• Eₚᵧ = p / y is the secant soil modulus from the nonlinear p-y curve; and
• W is any distributed lateral load.

How it's solved

The equation is discretized by central finite differences into a pentadiagonal system. Because the soil modulus Eₚᵧ depends on the very deflection we're solving for, the solution is obtained by Picard iteration: guess a deflection field, read the secant modulus off each p-y curve, solve the linear system, and repeat (with under-relaxation) until the deflection field stops changing. From the converged deflection, slope, bending moment (M = EI·y″), shear and soil reaction follow directly.

A built-in equilibrium check integrates the soil reaction and compares it to the applied load. For a converged solution the residual is essentially zero — and PileCalc shows it to you, so you can see the answer is balanced.