Nonlinear bending analysis of arbitrarily shaped plates using discrete Ritz method and virtual springs

Discrete Ritz method (DRM) is combined with virtual spring technique, first-order shear deformation theory (FSDT), and Newton–Raphson method, for the first time, to analyze the nonlinear bending problem of arbitrarily shaped plates under different geometric boundary conditions. DRM constructs a rectangular domain enclosing the geometric domain of the plate, and then, by using Gauss points in the rectangular domain for discretization associate with variable stiffness properties, the plate geometry is numerically simulated by cutouts within the rectangle. The global displacement field of the plate is approximated by Legendre polynomials based first-order shear deformation theory. The geometric nonlinearity is considered in terms of the von Kármán nonlinear theory. Virtual spring technique is used to simulate spring stiffness coefficient of distinct boundary conditions on complex geometric domain and embedded into the global stiffness matrix. DRM is combined with Newton-Raphson method to solve the nonlinear bending equations of plate with complex geometries. Numerical examples and comparisons with the results in the literature and FEM demonstrate that DRM based on constraint springs can be used to analyze the geometric nonlinearity of arbitrarily shaped plates under different boundary conditions with good feasibility and accuracy.