# Buckling Analysis of Piles in Multi-Layered Soils
## 2 Methodology
### Conditions for Minimum P  
Equation (22) synthetically shows that, for a given geometry and various mechanical properties of the pile driven into the multi-layered soil, the load P can vary through varying the coefficients \(\mathrm{a}_{1}\), \(\mathrm{a}_{2}\),..., \(\mathrm{a}_{k}\),..., \(\mathrm{a}_{m}\) that define the Fourier series of the deformed shape of the pile. To find the buckling load P\({}_{cr}\) and the values of the coefficients \(\mathrm{a}_{k}\) defining the related pile deformed shape, the Rayleigh-Ritz method is used. It is a so-called direct variational method that allows for finding of a solution for the minimum of the functional P[y(x), y\({}^{\prime\prime}\)(x), y\({}^{\prime}\)(x), x] defined in Equation (6) and re-arranged in Equation (22) after developing through a Fourier series the functions y(x) and K(x). Since for Equation (7), the function y(x) is a linear combination of m sinusoidal functions through the m unknown constants \(\mathrm{a}_{k}\), k = 1,..., m, then, according to the Rayleigh-Ritz method, the functional P[y(x), y\({}^{\prime\prime}\) (x), y\({}^{\prime}\)(x), x] can be considered as a function P = P(\(\mathrm{a}_{1}\), \(\mathrm{a}_{2}\),..., \(\mathrm{a}_{k}\),..., \(\mathrm{a}_{m}\)) of m unknowns \(\mathrm{a}_{k}\). The stationary condition for this function is then imposed through the stationary conditions for P = P(\(\mathrm{a}_{1}\), \(\mathrm{a}_{2}\),..., \(\mathrm{a}_{k}\),..., \(\mathrm{a}_{m}\)), that is:  
\[\frac{\partial\mathrm{P}}{\partial\mathrm{a}_{k}}=0,\;\mathrm{k}=1,\;2,\; \ldots,\;\mathrm{m} \tag{23}\]  
thus allowing us to find the m coefficients \(\mathrm{a}_{k}\) and therefore the related buckling deformation shape y(x). Of course, the higher the number of equations and therefore of coefficients \(\mathrm{a}_{k}\) used to describe the function y = y(x), the better is the accuracy of the determination of y(x) and therefore of the buckling load P\({}_{cr}\). Since in Equation (6) P is a ratio between the numerator \(\mathrm{N}\left(\int_{0}^{\mathrm{L}}\;\frac{\mathrm{EI}}{2}y^{\prime\prime 2 }\;\mathrm{dx}+\frac{1}{2}\int_{0}^{\mathrm{L}}\mathrm{K}(\mathrm{x})\;y^{2} \;\mathrm{dx}\right)\) and the denominator D (\(\frac{1}{2}\int_{0}^{\mathrm{L}}\mathrm{y}^{\prime 2}\;\mathrm{dx}\)), the stationary conditions can be rewritten as:  
\[\frac{\partial\mathrm{N}}{\partial\mathrm{a}_{k}}-\mathrm{P}_{\mathrm{cr}} \frac{\partial\mathrm{D}}{\partial\mathrm{a}_{k}}=0 \tag{24}\]  
After deriving Equation (24), we can obtain:  
\[\mathrm{a}_{1}\frac{\mathrm{L}^{2}}{2\pi^{3}\mathrm{k}^{2}}\left(\sum_{i} \mathrm{b}_{i}\phi_{i1\mathrm{k}}\right)+\ldots+\mathrm{a}_{k}\left[\frac{ \mathrm{L}^{2}}{2\pi^{3}\mathrm{k}^{2}}\left(\sum_{i}\mathrm{b}_{i}\phi_{i \mathrm{kk}}\right)+\frac{\mathrm{EI}\pi^{2}\mathrm{k}^{2}}{\mathrm{L}^{2}}- \mathrm{P}_{\mathrm{cr}}\right]+\ldots+\mathrm{a}_{m}\frac{\mathrm{L}^{2}}{2 \pi^{3}\mathrm{k}^{2}}\left(\sum_{i}\mathrm{b}_{i}\phi_{i\mathrm{mk}}\right)=0 \tag{25}\]  
By varying k, this is an equation system that defines an eigenvalue-eigenvector problem. It can be, in fact, noted that Equation (25) defines a characteristic system of equations, that is:  
\[\mathrm{A}-\mathrm{P}_{\mathrm{cr}}\;\mathrm{I}=0 \tag{26}\]  
where I is the unit matrix, and the square matrix \(\mathrm{A}\) is the sum of a matrix \(\mathrm{A}_{1}\) is:  
\[\mathrm{A}_{1}=\left[\begin{array}{cccc}\frac{\mathrm{L}^{2}}{2\pi^{3} \mathrm{I}^{2}}\sum_{i=1}^{n}(\mathrm{b}_{i}\phi_{i11})&\cdots&\frac{\mathrm{L}^ {2}}{2\pi^{3}\mathrm{I}^{2}}\sum_{i=1}^{n}(\mathrm{b}_{i}\phi_{i1\mathrm{k}}) &\cdots&\frac{\mathrm{L}^{2}}{2\pi^  
and of a matrix A\({}_{2}\) with all terms zero except the m terms on the diagonal is:  
\[\mathbf{A_{2}}=\left[ \begin{array}{ccccc}\frac{\mathrm{EI}}{\mathrm{L}^{2}}\; \pi^{2}1^{2}&\cdots&0&\cdots&0\\ \vdots&&\vdots&&\vdots\\ 0&\cdots&\frac{\mathrm{EI}}{\mathrm{L}^{2}}\;\pi^{2}\mathrm{k}^{2}&\cdots&0\\ \vdots&&\vdots&&\vdots\\ 0&\cdots&0&\cdots&\frac{\mathrm{EI}}{\mathrm{L}^{2}}\;\pi^{2}\mathrm{m}^{2} \end{array}