# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
[PERSON] [23] conducted an in-depth study of the ultimate bearing capacity of foundation piles under inclined load based on the principle of joint action of pile and soil and elasticity theory, using numerical methods and experimental means. By experiments and numerical simulations, some researchers have investigated the flexural behavior of the pile under combined axial and lateral loads. [PERSON] and [PERSON] [24] studied pile behavior by in-situ tests, without considering the influence of the pile length above ground. [PERSON] et al. [25] deduced the elastoplastic solutions for partially embedded piles, in which the coefficient of subgrade reaction was supposed to be a constant. By assuming that the coefficient of subgrade reaction increases linearly with depth, [PERSON] and [PERSON] [26] obtained the variational solutions for the transversely isotropic pile. The subgrade reaction method is widely used because of its simplicity and reasonable accuracy. However, the coefficient of subgrade reaction is not a fundamental soil property but a model parameter,and the soil resistance is modeled with independent springs without accounting for the continuity of soil [27]. [PERSON] and [PERSON] [28] put forward the elasticity approach for single piles under lateral loads, in which the soil was assumed to be an ideal, homogeneous, isotropic, semi-infinite elastic material, and the differential solutions were obtained. The research proposed a simplified analysis method for power series calculation based on the m method. However, this method has a large number of calculation parameters and its formulae are lengthy and complex, limiting its application to some extent. Due to the complexity of pile-soil interaction and the nonlinearity of pile perimeter resistance and displacement, single-parameter methods such as the m or c method often struggle to produce results that accurately reflect both pile displacement and internal force calculations. As a result, [PERSON] [29] proposed a comprehensive stiffness principle and two-parameter method for calculating thrust piles, which established a power series solution. However, this solution does not account for the longitudinal and transverse bending that occurs in bridge piles or high bearing piles under horizontal and vertical (or inclined) loads, known as the gravity second-order effect. Obtaining an analytical solution for power series that accounts for longitudinal and transverse bending is challenging. In summary, there are relatively few research results available on elastic pile foundations in viscoelastic media. In other area outside of soil mechanics, it is imperative to consider the post-buckling behavior when the supporting spring system exhibits nonlinearity or softening effects. In such scenarios, subcritical and unstable post-buckling behavior with localized failure modes may occur, rendering the failure mode highly sensitive to imperfections. Consequently, linear buckling analysis may provide limited guidance in predicting the ultimate load. An example of this phenomenon is the buckling of railways subjected to thermal strain [30; 31; 32]. And in other area outside of civil engineering, the viscoelastic-elastic complex system is still widely received by researchers [33; 34; 35; 36; 37]. In this paper, the comprehensive stiffness principle and two-parameter method are extended to the calculation of longitudinally and transversely bent piles, analogous to longitudinally and transversely bent beams in mechanics of materials [38; 39]. The deflection differential equation of a composite loaded pile is established using this method and the corresponding finite difference equation is derived. The finite difference iterative format for pile deformation and internal force is then obtained based on the pile top load, pile end restraint conditions, and the continuity of deformation and internal force at the interface between the free and embedded solid sections of the pile.