Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam

Ratchata Theinchai,Weera Yukunthorn,Siriwan Chankan

doi:10.1155/2016/5052194

Ratchata Theinchai, Weera Yukunthorn + Show 1 more

Open Access

https://doi.org/10.1155/2016/5052194

Copy DOI

Abstract

We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.

Highlights

The Euler-Bernoulli beam theory states that the action load produces the bending moment M(x) ∈ C([0, L]) which is proportional to deflection characteristics of the beam
In a case of small deformation, we assume that y󸀠(x) is infinitesimal
Our method proposes to reform problem (3) in a sense of fractional calculus without linearization

Summary

Introduction

The Euler-Bernoulli beam theory states that the action load produces the bending moment M(x) ∈ C([0, L]) which is proportional to deflection characteristics of the beam. The equation of this law can be written as follows: y󸀠󸀠. The deformed length of beam is verified by the integral ∫0l[1 + (y󸀠)2]1/2dx, where l = L − Δ It was shown in [1] that the slope of deflection curve represents the following equation y󸀠(x) = G(x)/[1 − G2(x)]1/2, where G(x) = (∫xl M(s)ds)/(EI) for the known function M. Where M(x) = Px. If the slope is very small, the linear EulerBernoulli beam theory [2] governs the problem d4y EI dx. We use Adomian polynomial to approximate a nonlinear term and derive a semianalytical solution by use of Laplace transform for the initial value problem (6)

Preliminaries

Semianalytical Solution via LADM

Conclusion