PY Curves for Laterally Loaded Piles
PY curves are the numerical models used to simulate the response of soil resistance (p, soil resistance per unit length of the pile) to the pile deflection (y) for the piles under lateral loading. With this approach, the soils are conveniently represented by a series of nonlinear springs varying with the depth and soil type in the analysis for laterally loaded piles.
This concept was first developed in the 1940’s and 1950’s when energy companies built offshore structures that had to sustain heavy horizontal loads from waves. An exact publication date of the model is not available since the py curves are still modified and improved today. The earliest recommendations on py curves could be traced back to the 1950s by the works of Skempton and Terzaghi (Ruigrok 2010).
Ideally, py curves should be generated from fullscale lateral load tests on instrumented test piles. In the absence of experimentally derived py curves, it is possible to use empirical py formulations that have been proposed in the literature for different types of soils.
From the 1970s, Further modification and improvement have been given to the py method. Instead of giving inputs for the nonlinear spring constant (i.e., the values of the spring constant as a function of pile deflection), py curves are given as inputs to the analysis in the py method. Different py curves have been developed over the years for different soil types, which give the magnitude of soil pressure as a function of the pile deflection.
In the analysis, the pile is divided into small segments, and for each segment, a py curve is given as input. Depending on the magnitude of the deflection of a pile segment, the correct soil resistance is calculated from the py curve iteratively. With the development of the finite element method, analysis using beam finite elements have been widely adopted in many calculations involving the subgradereaction approach or the py method. Today, the py method is the most widely used method for calculating the response of laterally loaded piles.
PY Curve Models in PileLAT and PileGroup Programs
Cohesive Soils
There are five different PY curves available for modeling cohesive soils:

API Soft Clay. It is recommended in API RP2A 21st Edition (2000). The ultimate resistance of soft clay is determined in the same way as Matlock (1970). The only difference is that the piecewise curves are used for both static and cyclic loading conditions.

Soft Clay (Matlock). This is based on the method established by Matlock (1970) for both static and cyclic loading conditions.

Stiff Clay without Water (Reese). This py curve model for still clay without water is based on the works by Welch and Reese (1972).

Stiff Clay without Water with Initial Modulus. This py curve model is based on the works of Welch and Reese (1972) except for the initial slope follows the recommendations on the model of still clay with water by Reese et al. (1975).

Stiff Clay with Water (Reese). This py curve model is based on the works by Reese et al. (1975) where the lateral ultimate resistance is determined separately for the location near the ground surface and also at deep depth.
Cohesionless Soils
There are three different PY curves available for modeling cohesionless soils:

API Sand. The py curve model follows the recommendations in API RP2A 21st Edition (2000).

Reese Sand. The py curve model is based on the works of Reese et al. (1974) for both static and cyclic loading conditions.

Liquefied Sand. The py curve model is based on the works of Rollins et al. (2005).
Rock
There are four different PY curves available for modeling rock:

Weak Rock  Reese (1997). The py curve model is based on the method established by Reese (1997).

Strong Rock  Turner (2006). The py curve for strong rock is based on the method proposed by Turner (2006).

Massive Rock  Liang et al. (2009). The py curve for massive rock is based on the method proposed by Liang et al. (2009).

Calcareous Rock  Fragio et al. (1985). The py curve for calcareous rocks is based on the method proposed by Fragio et al. (1985).
General Geotechnical Materials
There are two options available for modeling the general geotechnical materials:

ElasticPlastic Model. The soil subgrade modulus is determined by Vesic's method and the ultimate lateral resistance is determined by API clay model for cohesive soils, Reese sand model for granular soils and Weak rock  Reese method for rocks.

Elastic Model. The elastic subgrade modulus is adopted in the method.
Some py curve examples are shown in the figures below.