Line element-less method (LEM) for bending analysis of beams

Thelineelement-less method (LEM) is presented as an analytical framework for the elastic analysis of beams under arbitrary loading and boundary conditions. Building on previous developments originally conceived for torsion of beams and plate deflection, this study reformulates and specializes the LEM for one dimensional structural members, highlighting its capability to exactly reproduce the classical Euler–Bernoulli beam solution within a purely line integral framework. In the proposed formulation, the deflection field is expressed as the sum of a homogeneous solution, composed of polynomial functions, and a particular solution that depends on the applied load. The unknown coefficients are determined by minimizing squared boundary functions under equivalence constraints, ensuring full compatibility with the imposed boundary conditions. This approach provides closed-form solutions for a broad class of beam configurations, as well as systems composed of multiple substructures. Several benchmark examples are presented to validate the method and to demonstrate its accuracy and flexibility. The LEM yields results that coincide with classical analytical solutions while maintaining a compact and computationally efficient formulation. The method thus emerges as a robust and versatile analytical tool for beam mechanics, with potential extensions to composite, layered, and multi-body structural systems.