THE DIFFERENTIAL EQUATION: Laterally Loaded Pile.

Compatibility
As stated earlier, the problem of the laterally loaded pile is similar to the beam-on-elastic foundation problem. The interaction between the soil and the pile or the beam must be treated quantitatively in the problem solution. The two conditions that must be satisfied for a rational analysis of the problem are,

1. Each element of the structure must be in equilibrium and
2. Compatibility must be maintained between the superstructure, the foundation and the supporting soil.

If the assumption is made that the structure can be maintained by selecting appropriate boundary conditions at the top of the pile, the remaining problem is to obtain a solution that insures equilibrium and compatibility of each element of the pile, taking into account the soil response along the pile. Such a solution can be made by solving the differential equation that describes the pile behavior.

The Differential Equation of the Elastic Curve
The standard differential equations for slope, moment, shear and soil reaction for a beam on an elastic foundation are equally applicable to laterally loaded piles.

The deflection of a point on the elastic curve of a pile is given by y. The x-axis is along the pile axis and deflection is measured normal to the pile-axis.

The relationships between y, slope, moment, shear and soil reaction at any point on the deflected pile may be written as follows.

deflection of the pile = y

where El is the flexural rigidity of the pile material.

The soil reaction p at any point at a distance x along the axis of the pile may be expressed as

where y is the deflection at point x, and Es is the soil modulus. Eqs (16.4) and (16.5) when combined
gives

which is called the differential equation for the elastic curve with zero axial load.

The key to the solution of laterally loaded pile problems lies in the determination of the value of the modulus of subgrade reaction (soil modulus) with respect to depth along the pile.

Fig. 16.2(a) shows a vertical pile subjected to a lateral load at ground level. The deflected position of the pile and the corresponding soil reaction curve are also shown in the same figure. The soil modulus Es at any point x below the surface along the pile as per Eq. (16.5) is

Figure 16.2 The concept of (p-y) curves: (a) a laterally loaded pile, (b) characteristic
shape of a p-y curve, and (c) the form of variation of Es with depth

As the load Pt at the top of the pile increases the deflection y and the corresponding soil reaction p increase. A relationship between p and y at any depth x may be established as shown in Fig. 16.2(b). It can be seen that the curve is strongly non-linear, changing from an initial tangent modulus Esi to an ultimate resistance pμ. ES is not a constant and changes with deflection.

There are many factors that influence the value of Es such as the pile width d, the flexural stiffness El, the magnitude of loading Pt and the soil properties.

The variation of Es with depth for any particular load level may be expressed as

in which nh is termed the coefficient of soil modulus variation. The value of the power n depends upon the type of soil and the batter of the pile. Typical curves for the form of variation of Es with depth for values of n equal to 1/2, 1, and 2 are given 16.2(c). The most useful form of variation of Es is the linear relationship expressed as

which is normally used by investigators for vertical piles.

Table 16.1 Typical values of n, for cohesive soils (Taken from Poulos and Davis, 1980)

Table 16.1 gives some typical values for cohesive soils for nh and Fig. 16.3 gives the relationship between nh and the relative density of sand (Reese, 1975).

Figure 16.3 Variation of nh with relative density (Reese, 1975)