矩形桩竖向受荷三维弹性变分解

矩形桩作为一种典型的横截面非圆形桩，通常采用等效圆形桩的近似方法来分析，未能从理论上考虑桩-土相互作用。针对该问题，结合复变函数保角变换技术提出一种适用于竖向受荷矩形桩的三维弹性变分方法。基于最小势能原理和变分法获得桩位移函数和土竖向位移传递函数的控制微分方程，为克服土体位移传递函数求解域边界形状复杂的困难，应用保角变换将桩-土接触面的复杂边界形状转化为简单边界，并获得了桩位移函数的解析解及土体竖向位移函数的半解析解。使用MATLAB软件编制相应的分析程序，并将分析得到的结果与有限元分析及现有解答进行对比。最后，对影响桩沉降的主要因素进行参数分析。研究结果表明：所得解与有限元分析结果吻合得更好，证明了所提分析方法的正确性；摩擦桩量纲一化的桩顶刚度随桩-土模量比的增加逐渐减小，当桩的长细比增加到某数值时，其对桩顶刚度的影响可忽略不计，即摩擦桩存在有效桩长；相对于摩擦桩，当桩-土模量比较大时，其对端承桩的桩顶刚度影响较小，当端承桩长细比足够大时，可转化为摩擦桩；桩顶刚度随土体泊松比先减小后增加，间接表明其不仅受土体抗剪强度的影响，而且受土体压缩性能的影响；在相同混凝土用量下，当桩-土模量比较小时，与等横截面面积圆桩相比，矩形桩的桩顶刚度明显大于圆桩，当桩-土模量比超过某数值时，二者桩顶刚度逐渐趋于一致；当桩-土模量比较小时，等横截面面积矩形桩随横截面长宽比的增加，桩顶刚度增加。

Three-dimensional Elastic Variational Solution for Vertically Loaded Rectangular Piles

A rectangular pile, as a typical noncircular cross-sectional pile, is often analyzed by the approximate method of an equivalent circular pile, and the pile-soil interaction is not considered theoretically. To solve this issue, a 3D elastic variational solution combined with a complex function conformal mapping technique was developed for a vertically loaded rectangular pile. The governing differential equations of the pile and soil were derived based on the principle of minimum potential energy and the variational method. By applying the conformal transformation, the complex boundary shape of the pile-soil interface was transformed into a simple boundary, which made the solution of the soil control equation easier. The solution of the pile-soil system was obtained using the analytical form for the pile and the semi-analysis solution for the soil. A program was developed based on MATLAB software. The results obtained from the analysis were compared with the existing solution and finite-element analysis, proving the correctness of the proposed theoretical method. Finally, parametric studies focusing on the major factors of the effects of pile settlement were conducted. The results show that the normalized stiffness of the floating pile head decreases with an increase in the pile-soil modulus ratio. When the slenderness ratio of the pile increases to a certain value, its influence on the stiffness of the pile head can be ignored-that is, there is an effective length of the floating pile. Contrast with the floating pile, when the pile-soil modulus ratio is larger, it has little effect on the normalized head stiffness of the end-bearing pile, and it can be transformed into a floating pile if the slenderness ratio of the end bearing pile is sufficiently large. The stiffness of pile head decreases first and then increases with the increase in the Poisson ratio of the soil, which indirectly indicates that it is affected by not only the shear strength of the soil but also the compression property. Under the same amount of pile concrete, when the pile-soil modulus ratio is small, the pile normalized stiffness of the rectangular pile is significantly greater than that of the circular pile with an equal cross-sectional area. When the pile-soil modulus ratio exceeds a certain value, the pile top stiffnesses of the rectangular and circular piles gradually tend to be the same. For rectangular piles with equal cross-sectional areas, when the pile-soil modulus ratio is small, the normalized pile head stiffness increases with an increase in the length-width ratio of the cross-section.