# Buckling Analysis of Piles in Multi-Layered Soils
## 3 Discussion and Results
### Applications and Validations
Figure 6: Examples of second-order displaced shapes and buckling load of piles in multi-layered soil conditions with: (**a**) two not-embedded segments, i.e., stilt houses; (**b**) bottom of the pile is not embedded in hard rock; (**c**,**d**) same configuration of soil layers with only an inversion of the sedimentation order of the two soft soils; (**e**) presence of clay and peat layers surrounding the pile without crossing any caves; (**f**) soft clay with no cavities and peat layers. The buckling load of the pile in Figure 6a is mainly affected by the not-embedded part of the pile between the foundation and the first soil layer. The pile also deviates from the straight configuration when the lower cave is passed through, but this deviation is so slight that it can be identified only by calculations, while it is not distinguishable in the plot. Figure 6b shows that when a pile, instead of being embedded in the limestone rock, is only embedded in the soft clay layer at the cave base, its bottom end is free to rotate, thus causing the buckling load to decrease dramatically and become much less than Npl. Figure 6c,d show that if a pile passes through a limestone cave with two layers of soft soil with low but different stiffnesses (peat and very soft clay) sedimented over the bottom of the cave, buckling load is highly affected by the inversion of the sedimentation order of the two soft soils. Figure 6e,f show that a micropile may buckle in soft soil layers with c\({}_{\text{u}}\) even higher than 15 kPa, which is more than the limit proposed by some design codes for checking pile buckling [11; 12]. Even if there are no cavities or peaty soils where buckling can more easily occur, the low stiffness of these types of soils can make a pile prone to buckle. In fact, Figure 6f shows that buckling can also occur in soft soils without peat, such as very soft clays and silty sands. Buckling load results increased with the absence of peat, but were still lower than Npl. The buckling behaviour of one more pile was analysed to validate the proposed method through comparison with the experimental results obtained by testing a real micropile in situ [9]. The micropile characteristics are presented in Table 2[27], and the soil layers specifications are described in Figure 7. The pile load-bearing capacity is Npl = A\({}_{\text{s}}\)-f\({}_{\text{y}}\) = 2750 kN. The flexural stiffness of the grout was neglected, as in the previous buckling analyses. Loads were applied in successive steps of AF = 325 kN over a period of about five hours. The experimental result shows that the micropile experienced buckling between the fifth and sixth load steps, which is between 1625 to 1923 kN. In addition, the authors implemented in ABAQUS a finite element model (FEM) for which buckling occurred with P\({}_{\text{cr}}\) = 2000 kN, which is decreased to 1740 kN accounting for the eventual effect of imperfections. Through the variational approach proposed in this paper, P\({}_{\text{cr}}\) = 1923 kN was obtained. Moreover, the same pile (details in Table 2) was modelled in SAP2000 [28] to confirm the approach's accuracy. The buckling factor was obtained from the modal analysis, using horizontal springs on the pile representing Winkler's medium. For the sake of precision, spring spacing was 25 cm, and spring stiffness was selected according to the coefficient of the subgrade reaction in this interval. The output of the analysis showed a buckling factor equal to 1895 kN with the identical deformed shape. It must be noticed that while the variational method was a continuum-based analytical method, the FEA was performed on a discrete-spring model. Even if the experimental evaluation of the buckling load cannot be precise, the results obtained through the proposed method were shown to fit very well with those obtained from finite element analyses and quite well with the experimental results.