Mathematical Problems in Engineering

Volume 2015, Article ID 520491, 7 pages

http://dx.doi.org/10.1155/2015/520491

## Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

^{1}Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan^{2}DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

Received 8 May 2015; Revised 20 August 2015; Accepted 23 August 2015

Academic Editor: Bin Jiang

Copyright © 2015 R. Naz and F. M. Mahomed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density , and the applied load denoted by , a function of transverse displacement . The complete Lie group classification is obtained for different forms of the variable lineal mass density and applied load . The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of . For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when is constant with variable applied load . For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

#### 1. Introduction

Daniel Bernoulli and Leonard Euler developed the theory of the Euler-Bernoulli beam problem. Let be the transverse displacement at time and position from one end of the beam taken as the origin, the flexural rigidity, and the lineal mass. The transverse motion of an unloaded thin beam is represented by the following fourth-order partial differential equation (PDE):

Euler-Bernoulli beam equation (1) has been frequently studied in the literature. Gottlieb [1] studied the isospectral properties of this equation and its nonhomogeneous variants with and . Soh [2] considered the equivalence problem for an Euler-Bernoulli beam utilizing the Lie symmetry approach. Later on Morozov and Soh [3] attempted the problem with the aid of Cartan’s equivalence method. Recently, Ndogmo [4] obtained the complete equivalence transformations of the Euler-Bernoulli equation which were initially considered in the work [3] in terms of some undetermined set of functions. Özkaya and Pakdemirli [5], using the symmetry method, investigated the transverse vibrations of a beam moving with time-dependent axial velocity and obtained approximate solutions for an exponentially decaying and harmonically varying problem.

Now let be the elastic modulus, let be the area of inertia, let be the mass per unit length, let be the transverse displacement at time and position , and let be the applied load. The transverse motion of a loaded thin elastic beam is governed by the following dynamic beam fourth-order PDE [6]:where the applied load is a function of . Bokhari et al. [7] studied the following dynamic Euler-beam equation from the symmetry viewpoint with , , as constants and dependent on :A complete group classification was obtained for (3). The symmetry reductions were derived to reduce the fourth-order PDE to fourth-order ordinary differential equations (ODEs). For the power-law load function, compatible initial-boundary value problems corresponding to clamped end and free end beams were formulated and the reduced fourth-order ODEs were determined. The static beam problem was discussed by Bokhari et al. [8].

The dynamic fourth-order Euler-Bernoulli PDE having a constant elastic modulus and area moment of inertia, a variable lineal mass density , and the applied load denoted by , a function of transverse displacement , is given by

In this paper we study dynamic Euler-Bernoulli beam equation (4) from the symmetry point of view.

We give a complete classification of the Lie symmetries for dynamic Euler-Bernoulli beam equation (4). The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type applied loads for different forms of (see Table 1). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed by Bokhari et al. [7] when is a constant with variable applied load . We derive the group invariant solutions for the general power-law and exponential cases. The fourth-order PDE reduces to a fourth-order ODE with the help of similarity transformations. For the power-law applied load case compatible initial-boundary value problems for the clamped and free end beam cases are formulated. We deduce the corresponding fourth-order ODE with appropriate initial and boundary conditions. We show that the solution fails to satisfy the initial or boundary conditions for the exponential and logarithmic cases.