A derivation of the beam equation

TL;DR

This paper provides a clear derivation of the Euler-Bernoulli beam equation starting from energy principles, bridging discrete and continuum descriptions, and includes numerical solutions for various loads.

Contribution

It offers a simplified, step-by-step derivation of the beam equation suitable for students, connecting energy methods with discrete and continuum models.

Findings

Successful numerical solutions for different load cases

Clear explanation bridging particle, discrete, and continuum models

Enhanced understanding of the derivation process for engineering students

Abstract

The Euler-Bernoulli equation describing the deflection of a beam is a vital tool in structural and mechanical engineering. However, its derivation usually entails a number of intermediate steps that may confuse engineering or science students at the beginnig of their undergraduate studies. We explain how this equation may be deduced, beginning with an approximate expression for the energy, from which the forces and finally the equation itself may be obtained. The description is begun at the level of small "particles", and the continuum level is taken later on. However, when a computational solution is sought, the description turns back to the discrete level again. We first consider the easier case of a string under tension, and then focus on the beam. Numerical solutions for several loads are obtained.