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.
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 KV 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:
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 ES is assumed; nonetheless, the bearing soil may have different parameters Eb and vb.
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.
The settlement-influence factor depends on the ratio of the tip diameter db to the top diameter of the pile d (for 1 <
I0 can be derived via the diagram of Figure 2.
The compressibility correction factor, Rk, 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 EP is the pile’s modulus of elasticity, and RA is the ratio of the bottom pile section area Ap to the cross-sectional area of the pile A (RA=AP/A) . Usually RA factor is equal to 1.
An example showing the geometry that impacts the RA factor is shown in Figure 3. The Compressibility correction factor, Rk, is then derived via the diagram presented in Figure 4.
The base modulus correction factor, Rb, is applicable in end-bearing piles. Rb depends on the L/d ratio, the pile stiffness factor K, and the ratio of the bearing stratum modulus of elasticity, Eb, to the soil’s modulus of elasticity, ES, (Eb/ES). Its value may be obtained from the corresponding diagrams illustrated in Figure 5.
The depth correction factor, Rh, is used in friction piles and also depends on the L/d ratio. An incompressible layer is assumed beneath the examined soil layer (Es, ν) at a certain depth h. The h/L or L/h ratios are utilized to define the value of Rh (Figure 6).
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