# Buckling Analysis of Piles in Multi-Layered Soils
## 3 Discussion and Results
### Applications and Validations  
In general, elastic buckling may occur in foundation piles with lower flexural rigidity, as in micropiles for foundation underpinning, especially if surrounded by soft soils or crossing caves and voids. Next, some applications are described. First, a comparison of the critical buckling load of micropiles in a single-layered (homogeneous) soil calculated both through the variational approach herein presented and through the Engesser [26] approach was made (Figure 5). The outcomes showed identical results for any soil stiffnesses. However, since this method can also be suitably applied to piles driven in more than one soil layer, next, some examples of pile buckling in multi-layered soils were analysed (Figure 6). In addition, the outcome of the experimental test of a pile driven into multi-layered soil [9] was compared to validate the proposed method. The critical buckling load of six micropiles made of steel hollow bars, length 12 m, embedded in different multi-layered soils were analysed. The geometrical and mechanical characteristics of the pile hollow bars are briefly described in Table 1[27]. The micropile tube is usually surrounded by mortar. However, in this study, the flexural stiffness of the mortar was neglected due to the uncertain contribution in flexural resistance because of cracks forming in the mortar. Figure 6 illustrates the soil stratigraphy and the pile deformed shapes. The load-bearing capacity of the micropiles under consideration would be \(\mathrm{N_{pl}}=\mathrm{A_{s}}\). \(\mathrm{f_{y}}=676\) kN, unless buckling occurs before \(\mathrm{N_{pl}}\) is attained, depending on the mechanical characteristics of the different soil layers and on soil stratigraphy, in particular, if soft layers and caves are present. This is what happens in the following examples. The soil stiffness was calculated with two different formulas based on the soil type. For cohesive soils, \(\mathrm{K_{h}}=60\cdot\mathrm{c_{u}/d}\), where \(\mathrm{K_{h}}\) is the subgrade reaction, and \(\mathrm{c_{u}}\) is the undrained shear strength  
Figure 4: Development of Fourier series with 100 terms of the function \(\mathrm{K}\) (x) of soil with four layers (\(\mathrm{K_{1}}=2\) MPa, \(\mathrm{K_{2}}=1\) MPa, \(\mathrm{K_{3}}=4\) MPa, \(\mathrm{K_{4}}=8\) MPa). Figure 3: An example of convergence to \(\mathrm{P_{cr}}\) of the minimum of the eigenvalues as the number of equations increases in a pile with EI = 100 \(\mathrm{k}\mathrm{N}\mathrm{m}^{2}\). of the soil; for cohesionless soils, \(K_{h}=n_{h}\cdot z/d\), where \(n_{h}\) is the coefficient of the subgrade reaction, \(z\) is the soil depth, and \(d\) is the diameter of the pile. Note that when the stratigraphy includes soft soils and rock layers, high precision in the evaluation of rock rigidity is not required because the buckling load is mainly influenced by the presence of soft layers rather than by the variation of rock rigidity, that in any case is very high compared to the rigidity of soft soil. Moreover, the stiffness of the reinforced concrete foundation at the pile top end is given to be of the same magnitude as that assigned to the rock layers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometry and Material Properties** & **Units** & \\ \hline Length (L) & m & 12 \\ Outer diameter (D) & mm & 49 \\ Inner diameter (d) & mm & 26 \\ Cross-section area (A) & mm\({}^{2}\) & 1352 \\ Modulus of elasticity (E) & GPa & 210 \\ Yield strength (f\({}_{y}\)) & MPa & 500 \\ Cross-section moment of inertia (I) & mm\({}^{4}\) & 250,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical and mechanical characteristics of Titan 52/26 hollow bars for piles. Figure 5: Comparison of critical buckling load calculation of piles (L = 4 m) in single-layered soil and different undrained shear strength: (**a**) Pile I [12] EI = 152.7, (**b**) Pile II [12] EI = 37.3 kN.m\({}^{2}\). Figure 6: _Cont._  
Figure 6: _Cont_.