VOLUME 19 NUMBER 2 (July to December 2026)

Solving beam deflection and rotation using higher-order moments of force

Traditional methods for determining beam deflection, such as the double-integration and moment-area method, involve multiple steps including indefinite integration, diagram construction, and evaluation of constants. This paper introduces a novel analytical framework based on higher-order moments of force to streamline the analysis of beam rotation and deflection. In some fields, higher-order moments, such as area moment of inertia and mass moment of inertia, are well-established, yet no formal definition currently exists in structural engineering for higher-order moments of force, particularly in the context of elastic beam analysis. This study defined and applied second- and third-order moments of force to derive closed-form expressions for slope and deflection without requiring auxiliary diagrams or segment-by-segment integration. The method was grounded in Euler–Bernoulli beam theory and was applicable to a wide range of loading types, including concentrated loads, distributed loads, and concentrated moments. Though the proposed higher-order moments lacked physical interpretation, their mathematical validity was demonstrated through rigorous derivation and consistency with classical solutions. This work offered a generalizable, efficient alternative to traditional methods and represented an initial effort to formalize the use of higher-order moments of force in structural engineering analysis.