Matrix-Form Green’s Function Method for Unified Static Analysis of Uniform Euler–Bernoulli Beams Under Arbitrary External Loads and Boundary Conditions

This paper provides a novel unified method, termed the unified computational method (UCM), for calculating the static responses of uniform Euler–Bernoulli beams with arbitrary external loads and boundary conditions. The proposed method uses generalized functions to describe arbitrary external loads and bending moments in a unified expression. By treating the end restraints as multi-directional elastic end springs, a general description covering arbitrary boundary conditions is realized, avoiding the need for categorical discussions when confronting various end restraints. Based on Green’s function method, the obtained closed-form solution can be represented in a unified integral expression, with the integrand being the convolution of Green’s function and the loading function. Green’s function can be rewritten in a concise matrix form, where there is a parametric matrix inside that depends only on the boundary conditions. For different idealized end restraints, the corresponding parametric matrices are derived and tabulated for reference. To uniformly compute the parametric matrix for any end restraints, a generalized parametric matrix expressed in terms of dimensionless restraint parameters is proposed, which exhibits better convergence compared to actual end spring stiffnesses. This is because using the dimensionless restraint parameters can reasonably quantify the degree of end restraints. Lastly, with the proposed method, a programming example is implemented to verify the feasibility of the UCM-modified finite element method (FEM) in reducing the number of elements.