# Buckling Analysis of Piles in Multi-Layered Soils
## 2 Methodology
### Formulation of the Problem  
Consider a circular pile with a constant cross-section A, moment of inertia I of the cross-section, length L, and elastic modulus E. Modelling the soil as a Winkler's medium, the subgrade reaction K\({}_{\text{h}}\) can be assumed as either constant or linearly variable, depending on the soil type. The same coefficients of the subgrade reaction as those used in laterally loaded piles are considered [22; 23]. A list of coefficients of subgrade reaction for different soils is provided by [2; 24; 25]. For a pile with a constant cross-section and, therefore, constant diameter d, considering an abscissa x with origin at the pile top, the soil reaction is  
\[\text{K(x)}=\text{K}_{\text{h}}(\text{x}).\text{d} \tag{1}\]  
The pile is axially loaded, hinged at its ends, and embedded in a Winkler's soil with varying rigidity K = K(x), (Figure 2). Function K(x) is assumed to be a step function. Therefore, depending on step height and spacing, K(x) can vary almost linearly with K\({}_{\text{h}}\)(x) (depending on soil type) or be constant. Through using the energetic criterium, the potential energy U\({}_{1}\) in the pile added to the potential energy in the Winkler's soil U\({}_{2}\) is equal to the decrease of potential energy U\({}_{3}\) of the external force P applied at the pile top, that is:  
\[\text{U}_{3}=\text{U}_{1}+\text{U}_{2} \tag{2}\]  
Say y is the second-order displacement of the pile subject to the axial force P. In the following, the order of the derivatives of y are indicated through a Roman numeral superscript. The potential energy of the pile and that in the [PERSON]'s soil are, respectively, expressed as:  
\[\mathrm{U}_{1}=\int_{0}^{\mathrm{L}}\frac{\mathrm{EI}}{2}\mathrm{y}^{2}\mathrm{d}x \tag{3}\]  
\[\mathrm{U}_{2}=\frac{1}{2}\int_{0}^{\mathrm{L}}\mathrm{K(x)}\ \mathrm{y}^{2}\ \mathrm{d}x \tag{4}\]  
where \(\mathrm{E}\) and \(\mathrm{I}\) are the elastic modulus and moment of inertia of the pile cross-section, respectively, and \(\mathrm{y}\) is the second-order displacement. Figure 1: Flowchart of the proposed method for buckling analysis of piles in multi-layered soils. The variation of the potential energy of the force P is:  
\[\mathrm{U_{3}}=\frac{\mathrm{P}}{2}\int_{0}^{\mathrm{L}}\mathrm{y^{2}}\mathrm{y^ {2}}\;\mathrm{dx} \tag{5}\]  
By substituting Equations (3)-(5) into Equation (2), we obtain:  
\[\mathrm{P=}\frac{\int_{0}^{\mathrm{L}}\frac{\mathrm{EI}}{2}\mathrm{y^{2}} \mathrm{u^{2}}\;\mathrm{dx}+\frac{1}{2}\int_{0}^{\mathrm{L}}\mathrm{K(x)}\; \mathrm{y^{2}}\;\mathrm{dx}}{\frac{1}{2}\int_{0}^{\mathrm{L}}\mathrm{y^{2}} \mathrm{dx}} \tag{6}\]  
where K(x) is the soil horizontal stiffness along the pile. Therefore, the buckling load P is obtained from Equation (6) by finding the function \(\mathrm{y=y(x)}\) for which the functional P of Equation (6) is a minimum, that is \(\mathrm{P_{cr}=min(P)}\). For this aim, as y and K are variables dependent on the pile's depth, the Fourier series of y and K are developed, considering y and K as odd functions with period 2L in the interval (\(-\)L, L), thus obtaining:  
\[\mathrm{y(x)}=\sum_{k=1}^{m}\mathrm{a_{k}}\sin\left(\frac{\pi k}{L}\mathrm{x}\right) \tag{7}\]  
\[\mathrm{K(x)}=\sum_{i=1}^{n}\mathrm{b_{i}}\sin\left(\frac{\pi i}{L}\mathrm{x}\right) \tag{8}\]  
Coefficients \(\mathrm{a_{k}}\), k = 1,2,... m, are unknown and will be obtained through solving the variational problem of calculating the buckling load. Coefficients \(\mathrm{b_{i}}\), i = 1, 2,..., n, in Equation (8) depend on function K(x) distribution along with the pile. A multi-layered soil with s layers, each one with constant rigidity, is herein considered.