## ELASTIC SETTLEMENT OF PILES

The elastic settlement of a single pile depends on the relative stiffness of the pile and the soil, the lengthto- diameter ratio of the pile, the relative stiffness of the soil at the base and of the soil over the pile length, and the distribution of elastic modulus of the soil along the pile length. Laboratory and fi eld soil test results rarely duplicate the installation effects, so you need to be cautious in using soil stiffness from these tests. The relative stiffness of the pile to the soil is Kps 5 Ep/Eso, where Ep is the elastic modulus of an equivalent solid cross section of the pile and Eso is the elastic modulus of the soil. The elastic modulus of the soil at the base or tip of the pile will be denoted by Esb. Usually, the secant elastic modulus is used in design practice. For short-term loading in fi ne-grained soils, (Eso)u is used, where the subscript u denotes undrained condition. Various analyses have been proposed to calculate the settlement of single piles and pile groups. Poulos (1989) provided an excellent discussion on the various numerical procedures to calculate settlement of piles. The settlement consists of three components—skin friction, end bearing, and elastic shortening. Skin friction tends to deform the soil near the shaft, as illustrated in Figure 13.5a. The deformation

mode near the shaft is analogous to simple shear strain, and the shear strain, gzx, is

where G is the shear modulus, t is shear stress, and fs is the skin frictional stress. The shear strains can be integrated over the pile length to give the elastic settlement (res) resulting from skin friction; that is,

where (z) means that the parameter in front of it varies with depth. To solve Equation , we need to know how G and t vary with depth, but this we do not know. Therefore, we have to speculate on their variations and then solve Equation . A further complication arises in that the boundary conditions for a pile problem are complex. Thus, we have to solve Equation using numerical procedures. For example, we can assume that

where F(z, i) is a stress function and t(i) is the shear stress at an ordinate i. We can use fi nite element or boundary element to solve Equations .Stress functions for a point load within a half-space were developed by Mindlin (1936). For a homogeneous soil (Eso is constant with depth), a solution of Equation (13.51) using Equation for the elastic settlement of a single pile is

where I is an infl uence factor and Qaf is the design load transferred as skin friction. An approximate equation for I is

Soft soils tend to have elastic moduli that vary linearly with depth; that is,

where m is the rate of increase of Eso with depth (units: kPa/m). For soft soils, the elastic settlement is

Poulos (1989) developed another solution for the elastic settlement. He showed that res for a fl oating pile can be determined from

where Ir is an infl uence factor that depends on the L/D ratio and Kps, as shown in Figure. The soil mass under a pile is subjected to compression from end bearing. We can use elastic analyses to determine the elastic compression under the pile. For friction piles, the settlement due to end bearing is small in comparison to skin friction and is often neglected. The elastic shortening of the pile (rp) is found for column theory, which for a soil embedded in a homogeneous soil is