Solving ordinary thin plate bending problem in engineering, only a few analytical solutions with simple boundary shapes have been proposed. When using numerical methods (e.g. the variational method) to solve the problem, the trial functions can be found only it exhibits a simple boundary shape. The R-functions can be applied to solve the problem with complex boundary shapes. In the paper, the R-function theory is combined with the variational method to study the thin plate bending problem with the complex boundary shape. The paper employs the R-function theory to express the complex area as the implicit function, so it is easily to build the trial function of the complex shape thin plate, which satisfies with the complex boundary conditions. The variational principle and the R-function theory are introduced, and the variational equation of thin plate bending problem is derived. The feasibility and correctness of this method are verified by five numerical examples of rectangular, I-shaped, T-shaped, U-shaped, and L-shaped thin plates, and the results of this method are compared with that of other literatures and ANSYS finite element method (FEM). The results of the method show a good agreement with the calculation results of literatures and FEM.

解决工程中普通薄板弯曲问题，仅提出了少数具有简单边界形状的解析解。当使用数值方法（例如变分法）解决问题时，试函数只能找到它表现出简单的边界形状。R 函数可用于解决具有复杂边界形状的问题。本文将R函数理论与变分方法相结合，研究了具有复杂边界形状的薄板弯曲问题。本文采用R函数理论将复杂面积表示为隐函数，因此容易建立满足复杂边界条件的复杂形状薄板的试函数。介绍了变分原理和R函数理论，并推导出薄板弯曲问题的变分方程。通过矩形、I形、T形、U形、L形薄板5个数值算例验证了该方法的可行性和正确性，并将该方法的结果与其他文献和ANSYS的结果进行了比较有限元法 (FEM)。该方法的结果与文献和有限元法的计算结果具有很好的一致性。