中文摘要：

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。

英文摘要：

It is very complicated to give the expressions for deflection and internal forces on ladder square foundation plates with four free boundaries because of difficulties in satifying four free boundaries, displacement continuity and smoothness at ladder transitions. Until now, the expressions of deflection and internal forces on ladder square foundation plates under vertical loads are not available. Herein, by dividing the load with bending stiffness, the concept of load stiffness ratio is put forward, and the problems of continuity and smoothness at ladder transitions are cleverly solved. By the Fourier expansions of load stiffness ratio and plate deflection, the Fourier coefficients of plate deflection of satisfying the plate differential equation and four free boundaries are determined. The expressions for deflection and internal forces on ladder square foundation plates are derived. The expressions are verified by FEM. The proposed concept of load stiffness ratio may provide a new method for solving the deflection and internal forces on ladder square foundation plates.