Expressions for deflection and internal forces on ladder square foundation plate

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.