This paper presents a matrix formulation and explicit expressions for the first- and second-order stiffness matrix and loading vector of a prismatic pile with generalized rotational and lateral translation end-constraints on homogeneous or non-homogeneous Pasternak soil. The expressions for the load vector account for any external transverse load that fits a quadratic function. The proposed approach can be used to study a variety of soil-structural interaction problems such as a) beam-columns on non-homogeneous soil, b) piles subjected to external lateral earth pressures, c) lateral behavior of piles fully or partially embedded in layered soils, d) buckling analysis of piles fully or partially embedded in layered soils, e) second-order effects in piles, among others. The proposed formulation is easy to incorporate into an existing matrix structural analysis code. Two examples are provided to show the simplicity and practicality of the approach.

中文翻译：

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非均质土上桩元的一阶和二阶刚度矩阵及荷载矢量

本文给出了均质或非均质帕斯捷尔纳克土壤上具有广义旋转和侧向平移端约束的棱柱桩的一阶和二阶刚度矩阵以及荷载矢量的矩阵表示和明确表达式。载荷矢量的表达式考虑了适合二次函数的任何外部横向载荷。所提出的方法可用于研究各种土壤-结构相互作用问题，例如a）非均质土壤上的梁柱，b）承受外部侧向土压力的桩，c）完全或部分埋入其中的桩的侧向性能d）完全或部分嵌入层状土壤中的桩的屈曲分析，e）桩中的二阶效应等。提议的公式很容易合并到现有的矩阵结构分析代码中。提供了两个示例来说明该方法的简单性和实用性。