# Influence of Pile Foundation Stiffness on Column Design in One-Story Reinforced Concrete Frames
## 3 Method for Calculating Pile Foundation Settlement
The non-linear behavior of the pile was represented by functions describing the change in the value (degradation) of the shear modulus as a function of mobilized soil resistance or strain. In the method's algorithm, the pile described by [PERSON]'s modulus of elasticity is digitized through the use of the adopted load transfer function. For the pile shaft, the  
Figure 4: Settlement patterns of pile pair \(i\) and \(j\). Figure 5: Example of load–settlement curves for piles in a group. characterization of the supports is based on the following solution proposed by [PERSON] and [PERSON] [20]:  
\[s_{s}=\frac{\tau_{0}R_{0}}{G_{s}}ln\left(\frac{R_{\max}}{R_{0}}\right) \tag{8}\]  
where  
\(s_{s}\)--displacement of the nodal point of the pile sidewall;  
\(\tau_{0}\)--tangent stress on the side of the pile;  
\(R_{0}\)--radius of the pile shaft;  
\(R_{\max}\)--range of influence of the pile;  
\(G_{s}\)--shear modulus of the soil in the vicinity of the pile shaft. The radius of influence of the pile \(R_{\max}\) can be calculated according to [PERSON] and [PERSON]'s equation [33], which is as follows:  
\[R_{\max}=2(1-\
u)L_{p}\left(\frac{3}{2}-\frac{z}{L_{p}}\right) \tag{9}\]  
where  
\(L_{p}\)--length of the pile in the subsoil;  
\(z\)--depth below ground level;  
\(\
u\)--Poisson's ratio. The deflection of the support under the pile base (base settlement) \(s_{b}\) under the force \(P_{b}\) based on the theory of elasticity is described by the following equation:  
Figure 2:0  
where  
\(R_{b}\)--pile base radius;  
\(G_{b}\)--shear modulus of the soil under the base of the pile;  
\(\
u_{b}\)--[PERSON]'s ratio;  
\(\mu_{d}\)--impact factor of base depth, \(\mu_{d}\) = 0.5. In the above equations, the shear modulus \(G\) decreases according to the degradation function adopted. Initially, the method described here adopted a modified form of the hyperbolic modulus degradation function \(G\) according to [PERSON] [21] as proposed by [PERSON] [32]:  
Figure 2:1  
where  
\(G_{\max}\)--initial shear modulus;  
\(\tau\)--current, mobilized soil resistance;  
\(\tau_{f}\)--limit soil resistance (at failure);  
\(R_{f}\)--hyperbolic curve constant, \(R_{f}\) = 0.5 \(\div\) 0.9. Subsequent modifications were used, among other functions:  
* according to [PERSON] and [PERSON] [33]: \[G=\left\{ \begin{array}{r@{\quad\quad}l}G_{\max}\quad\mathrm{gdy}\quad&\gamma \leq 10^{-5}\\ -G_{\max}(0.3\cdot\log\gamma+0.5)\quad&\mathrm{gdy}\quad&\gamma\in\left(10^{- 5},10^{-2}\;\right)\\ 0.1\cdot G_{\max}\quad\mathrm{gdy}\quad&\gamma\geq 10^{-5}\end{array} \right.\] (12)* according to [PERSON] and [PERSON] [34]: \[G=G_{\max}\frac{\left(1-\left(\frac{\tau}{t_{f}}\right)^{w}\right)^{2}}{1-(1-w) \left(\frac{\tau}{\tau_{f}}\right)^{w}}\] (13) where \(G_{\max}\)--initial shear modulus; \(w\)--equation parameter. Figure 6 shows the selected functions on a graph for comparison. The application of the numerical procedure allows for the load-settlement curve for a single pile to be determined. In addition, the results make it possible to divide and analyze the total load into the part carried by the sidewall and the base for successive load degrees on the pile head. The computational model of the subsoil is characterized by geotechnical layers, which are described by the initial shear modulus \(G_{\max}\), limit shear resistance at the shaft \(\tau_{f}\) and limit resistance under the pile base \(q_{f}\). These parameters can be determined, for example, by an indirect method based on the results of a CPT probe test [35; 36]. The selection of a suitable degradation model requires extensive analytical studies with the use of geotechnical test results and settlement curves from pile test loads [37]. Test interpretation issues for determining the initial shear modulus \(G_{\max}\) and its degradation are presented in [38]. In the examples presented below, the degradation curves for the shaft Equation (13) and for the base Equation (12) are used.