A Unified Solution to Linear Beam Bending Problems

Abstract

Determining the static deflection of slender members subjected to transverse loading is a classical problem in engineering mechanics. In undergraduate engineering courses, the treatment of the topic is commonly restricted to the linear Euler-Bernoulli beam model covered by almost all textbooks on solid mechanics or strength of materials. Students are typically requested to determine solutions to various support and loading conditions by applying either the direct integration method or the method of superposition. Even though this process of repeated application undoubtedly trains procedural skills, it appears to be a rather cumbersome way towards conceptual understanding. The contribution at hand proposes a unified solution based on Taylor series expansion in combination with Macaulay's method of handling discontinuities, which is shown to be easily applicable to both statically determined and statically over-determined beam problems. Mechanics lecturer involved in undergraduate engineering education may regard the presented solution framework as an effective way to provide a profound insight into linear beam theory.