Analytical solution of the boundary value problem for Euler-Bernoullie beam using Green functions

Abstract

Analysis of the dynamic behavior of beams is a fundamental task in the construction, transport and mechanical engineering industries, as it plays a key role in ensuring the safety, reliability and durability of engineering structures. The paper investigates the dynamic response of a simply supported Euler-Bernoulli beam, which is simultaneously subjected to a concentrated force and a moment load. Unlike most previous studies, where these factors were considered separately, their combined effect is taken into account here. To solve the problem, the Green's function method was used, which provides an analytical solution in closed form. This approach has a number of significant advantages: it allows you to avoid additional determination of eigenvalues and eigenfunctions, which are required in the series decomposition method, and is also a universal tool for analyzing various types of loads. Since the Green's function describes the response of a beam to a localized disturbance, any external load, represented as an integral of Dirac delta functions, can be taken into account through an integral convolution transformation. In particular, the concentrated force is modeled by the Dirac delta function, and the action of the concentrated moment is described by its first derivative. As a result, an analytical solution of the Euler-Bernoulli boundary value problem in a closed form is obtained, which allows not only to determine the response of the beam, but also to study the behavior of the beam under different load parameters. The constructed graphs illustrate the influence of different values of loads on the shape and nature of the structure. The obtained analytical solution can serve as a reference model for testing numerical methods, be used in structural optimization problems, and also become a basis for further research of more complex systems with different fastening schemes and combined loads.