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四边自由阶梯方形基础板挠度和内力场的求解极为复杂,它不仅涉及基础板四自由边界条件的难愈满足,更涉及阶梯过渡处位移连续性和光滑性的难愈处理。到目前为止,人们还未给出四边自由阶梯方形基础板在中心垂直荷载作用下挠度和内力场的表达式。本文将载荷与弯曲刚度相除,提出了载刚比的概念,巧妙地解决了基础板阶梯过渡处连续性和光滑性难愈处理的问题,然后将基础板的载刚比展开成双重余弦傅里叶级数,将挠度函数展开成带补充项的双重余弦傅里叶级数,在满足板控制方程和四自由边界的条件下,确定双重余弦傅里叶级数中的待定系数,获得阶梯方形基础板挠度和内力场的级数表达式,所得表达式得到有限元数值仿真结果的验证。文中提出的载刚比概念将为非等厚度基础板挠度和内力场的解析求解提供新思路。
The solution of the deflection and internal force fields of the free-standing square footings on all four sides is extremely complicated. It involves not only the difficult to satisfy the four free boundary conditions of the foundation slab, but also the difficult to cure the displacement continuity and smoothness at the step transition. Up to now, the expressions of the deflection and internal force fields have not been given under the vertical load at the center of the free-standing square base plate. In this paper, the load and bending stiffness are divided, and the concept of load-to-carry ratio is put forward. The problem of the continuity and smoothness of the step transition of the base plate is skilfully solved. Then the bearing ratio of base plate is expanded into double cosine The Fourier series is expanded to a double cosine Fourier series with supplementary terms. Under the condition that the plate control equation and the four free boundaries are satisfied, the undetermined coefficients in the double-cosine Fourier series are determined to obtain a ladder The expressions of the deflection of the base plate and the series of internal force field are obtained. The obtained expressions are verified by the numerical simulation results of the finite element method. The concept of load-to-carry ratio proposed in this paper will provide a new idea for the analytical solution of the deflection and internal force field of non-equal thickness base plate.