Soil Subgrade Reaction in Pile Foundations

Introduction

Soil subgrade reaction in pile foundations is mostly based on theory of elasticity. Piles are subjected to vertical and lateral loads as well as moment loads. Therefore, the subgrade reaction incorporates two components, a vertical and a lateral, thus two discrete methods for deriving those components will be analyzed. It should be noted that the analyses presented deal with a single pile and not a group of piles. In the latter case, more complex models are required.

Vertical Subgrade Reaction in Pile Foundations

To derive the vertical subgrade reaction in pile foundations, a method that is based on theory of elasticity developed by Poulos and Davis (1980) will be utilized. The initial purpose of the method was to derive the settlements of a pile foundation; thus, the subgrade reaction is determined using the following formula:

where K_V is the vertical coefficient of subgrade reaction, P the applied vertical load, and y the pile settlement. The equation to derive the subgrade reaction is presented below:

where E stands for the modulus of elasticity of the soil (MPa), D is the diameter of the pile (m), and I is a correction factor.

The calculation of the correction factor is slightly different depending on the type of the pile. Piles are fundamentally divided into two categories:

1. End-bearing piles
2. Friction piles

End-bearing piles distribute the largest portion of the vertical load to the toe of the pile. They operate in the same manner as a pillar of a structure. On the contrary, the bearing capacity of friction piles derives from shear stresses that develop at the sides of the pile.

Usually, end-bearing piles are used to transfer the load to a harder soil layer or rock whereas friction piles are utilized when this is not possible.

In their analysis to derive the subgrade coefficient (Poulos and Davis, 1980), a homogenous soil mass with a constant Poisson ratio ν, and Young’s modulus E_S is assumed; nonetheless, the bearing soil may have different parameters E_b and v_b.

The correction factors for end-bearing and friction piles are calculated using Equations 3 and 4, respectively:

The aforementioned factors are thoroughly assessed in the following sections.

I₀: Settlement-influence factor

The settlement-influence factor depends on the ratio of the tip diameter d_b to the top diameter of the pile d (for 1 < d_b/d < 3; usually d_b/d=1) and the ratio of length of the pile L to its diameter d (L/d).

I₀ can be derived via the diagram of Figure 2.

R_k: Compressibility correction factor

The compressibility correction factor, R_k, depends on the L/d ratio and the pile stiffness factor K which practically measures the relative compressibility between the pile and the substratum:

where E_P is the pile’s modulus of elasticity, and R_A is the ratio of the bottom pile section area A_p to the cross-sectional area of the pile A (R_A=A_P/A) . Usually R_A factor is equal to 1.

An example showing the geometry that impacts the R_A factor is shown in Figure 3. The Compressibility correction factor, R_k, is then derived via the diagram presented in Figure 4.

R_b: Base modulus correction factor (End-bearing piles)

The base modulus correction factor, R_b, is applicable in end-bearing piles. R_b depends on the L/d ratio, the pile stiffness factor K, and the ratio of the bearing stratum modulus of elasticity, E_b, to the soil’s modulus of elasticity, E_S, (E_b/E_S). Its value may be obtained from the corresponding diagrams illustrated in Figure 5.

R_h: Depth correction factor (Friction Piles)

The depth correction factor, R_h, is used in friction piles and also depends on the L/d ratio. An incompressible layer is assumed beneath the examined soil layer (E_s, ν) at a certain depth h. The h/L or L/h ratios are utilized to define the value of R_h (Figure 6).

R_v: Poisson ratio correction factor

Finally, the Poisson ratio correction factor, R_v, takes into account the Poisson ratio of the soil layer ν_s, and the pile stiffness factor K. It is derived using the diagram shown in Figure 7.

Consequently, the vertical subgrade reaction for end-bearing and friction piles can be derived using equations [2], [3] and [4], taking into consideration the aforementioned correction factors as:

References

Poulos H.G., Davis E.H. (1980). Pile foundation analysis and design. The University of Sydney. Rainbow-Bridge Book Co.-554786. pp. 71-90.