The solution of a rectangular thin plate model of working face is directly affected by its boundary conditions. Its analytical solution can be solved only by a semi-analytical method under the conditions of two opposite edges simply supported without a unified and reasonable theoretical method. Due to the advance of working face

the thin plate bending model can generate three boundaries

i.e. fully clamped

two opposite edges simply supported and fully simply supported. From the governing equations of the thin plate bending problem

the basic mechanical parameters are used as the dual variables to construct the Hamilton system

giving the symplectic solutions rationally using the symplectic geometry method. This approach can deal with the above mentioned different boundary conditions with a unified analytical solution. With the characteristics of symplectic orthogonality for all eigenvectors

the proposed algorithm is proved to be stable and convergent fast. The case study shows that the calculation is correct

which provides a new way for the main roof weighting calculation.