Series solutions for clamped peridynamic beams using fourth-order eigenfunctions

TL;DR

This paper introduces an analytical series solution for nonlocal Euler--Bernoulli beams based on fourth-order eigenfunctions, demonstrating higher efficiency and accuracy compared to Fourier sine series.

Contribution

The paper develops a novel series solution method using fourth-order eigenfunctions for nonlocal beam equations, outperforming traditional Fourier series in convergence and accuracy.

Findings

Series in fourth-order eigenfunctions require ten times fewer terms than Fourier sine series.

Eigenfunction coefficients decay faster, leading to higher accuracy with fewer terms.

The approach effectively solves nonlocal Euler--Bernoulli beam problems with high precision.

Abstract

We propose an analytical approach to solving nonlocal generalizations of the Euler--Bernoulli beam. Specifically, we consider a version of the governing equation recently derived under the theory of peridynamics. We focus on the clamped--clamped case, employing the natural eigenfunctions of the fourth derivative subject to these boundary conditions. Static solutions under different loading conditions are obtained as series in these eigenfunctions. To demonstrate the utility of our proposed approach, we contrast the series solution in terms of fourth-order eigenfunctions to the previously obtained Fourier sine series solution. Our findings reveal that the series in fourth-order eigenfunctions achieve a given error tolerance (with respect to a reference solution) with ten times fewer terms than the sine series. The high level of accuracy of the fourth-order eigenfunction expansion is due to the fact that its expansion coefficients decay rapidly with the number of terms of the series, one order faster than the Fourier series in our examples.