Computational method for analytical solution with finite elements (CMAS-FE): Deriving approximate analytical solution for an isotropic homogeneous elastic medium with linear finite element method

• A novel methodology to obtain an approximate analytical solution for an isotropic homoge neous elastic medium is proposed. • A specific symbolic global stiffness matrix in terms of Young’s modulus and Poisson’s ratio is established. • Approximate analytical solutions for beam bending and stress intensity factor are present. This study presents a novel methodology to obtain an approximate analytical solution for an isotropic homogeneous elastic medium with displacement and traction boundary conditions. The solution is derived through solving a specific numerical problem under the scope of the linear finite element method (LFEM), so the method is termed computational method for analytical solutions with finite elements (CMAS-FE). The primary objective of the CMAS-FE is to construct analytical expressions for displacements and reaction forces at nodes, as well as for strains and stresses at elemental quadrature points, all of which are formulated as infinite series solutions of various orders of Poisson’s ratios. Like the conventional LFEM, the CMAS-FE forms global sparse linear equations, but the Young’s modulus and Poisson’s ratio remain variables (or symbols). By employing a direct inverse method to solve these symbolic linear systems, an analytical expression of the displacement field can be constructed. The CMAS-FE is validated via patch and bending tests, which demonstrate convergence with mesh and term refinement. Furthermore, the CMAS-FE is applied to obtain the bending stiffness of a beam structure and to estimate an approximate stress intensity factor for a straight crack within a square-shaped plate.