# Buckling Analysis of Piles in Multi-Layered Soils
## 1 Introduction  
The design problem of pile buckling is not usually faced in civil engineering practice because its occurrence is infrequent. However, this problem is becoming more important with the increasing use of micropiles, especially for retrofit interventions, and indeed elastic buckling of micropiles surrounded by soft soils is possible and is therefore worth investigating. In the literature, different approaches have been adopted for studying pile buckling. Following [PERSON] and [PERSON] [1] study on buckling of piles in an elastic continuum and supported at the bottom by a rigid base, some other authors studied rod buckling through modelling the soil as an elastic continuum. [PERSON] and [PERSON] [2] demonstrated that the soil models using [PERSON]'s medium or elastic continuum show identical behaviour. [PERSON] and [PERSON] [3] proposed a variational approach to study the elastic buckling of a bar on an elastic foundation, hinged at its ends and axially loaded on top. [PERSON]'s study's conclusion is similar to the Engesser formula for calculating the buckling load of an elastically embedded beam. [PERSON] et al. [4] developed a pile buckling model assuming a general power distribution of the soil's horizontal subgrade reaction to represent various soil conditions. The minimum potential energy method was adopted to develop the model using the Rayleigh-Ritz method to select a suitable deflection function. The effects of nine different boundaryconditions on the buckling capacity and the equivalent buckling length as well as the effects of the distribution of the horizontal subgrade reaction were examined; the effect of the subgrade reaction was considered as fully embedded or partially embedded, so could only be used to evaluate two layers of soil [4]. [PERSON] [5] compared pile buckling loads obtained from a semiempirical relationship developed by [PERSON] [6] with the allowable loads permitted by building codes and design guidelines [7]. A new formula (P\({}_{\text{cr}}\) = \(\beta\) (c\({}_{\text{u}}\)EI)\({}^{0.5}\), where c\({}_{\text{u}}\) is undrained shear strength of the soil, EI is flexural rigidity, and \(\beta\) is a factor for regulation) derived from Engesser was derived to calculate the critical global buckling load. While the formula factor \(\beta\) was improved to calculate the pile imperfection, the method was for homogeneous soils [5]. Using the linear small-angle bending theory, [PERSON] [8] provided a theoretical relationship between critical buckling load ratio and lateral deflection of a micropile subjected to vertical axial load and embedded in a weak homogeneous soil. Although the study covered different boundary conditions, it merely studied fully embedded piles. [PERSON] et al. [9] extended buckling analysis of micropiles in homogeneous soils to micropiles driven into different soil layers. They based their assumptions on experimental buckling tests carried out on 4 m long micropiles embedded in soft clay. The research studied three types of pile-buckling calculations: 1. Code design calculations [10; 11], 2. Experimental tests, and 3. FEM calculations using ABAQUS software. Despite the fact that the study considered multi-layered soil, the calculations were done for each layer separately, and the softest soil layer was the only candidate for the buckling position. [PERSON] et al. [12] highlighted how buckling failure can occur at higher undrained shear strengths in slender piles (i.e., micropiles) despite the codes' assumptions. In this case, the eccentricity of the pile was included in the momentum equilibrium of the forces under vertical load. Thus, this method considered pile imperfection in the derivations. Nevertheless, this method is only valid for single-layered soil. Piles instability in liquefied soils is due to the absence of lateral support [13]. Buckling failure of piles, considering vertical and seismic loads, require a more critical regard of the undefined effects of forces [14].