Advances in Acoustics and Vibration

Volume 2017, Article ID 4740851, 25 pages

https://doi.org/10.1155/2017/4740851

## A Discrete Model for Nonlinear Vibration of Beams Resting on Various Types of Elastic Foundations

Equipe des Etudes et Recherches en Simulation, Instrumentation et Mesures (ERSIM), Université Mohammed V, Ecole Mohammadia des Ingénieurs, Avenue Ibn Sina, BP 765, Rabat, Morocco

Correspondence should be addressed to A. Khnaijar; moc.liamg@rajianhk

Received 18 October 2016; Accepted 20 December 2016; Published 15 March 2017

Academic Editor: Kim M. Liew

Copyright © 2017 A. Khnaijar and R. Benamar. 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

This paper presents a discrete physical model to approach the problem of nonlinear vibrations of beams resting on elastic foundations. The model consists of a beam made of several small bars, evenly spaced. The bending stiffness is modeled by spiral springs, and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. Finally, the nonlinear effect is presented by the axial forces in the bars, assumed to behave as longitudinal springs, due to the change in their length induced by the Pythagorean Theorem. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass matrix, the stiffness matrix, and the nonlinearity tensor to be made separately. Therefore, once the model is established, various practical applications may be performed without the need of going through all the formulation again. The study of the nonlinear behavior makes the solution of the movement equation rise in complexity. By considering this discrete model and using the linearization method, one can achieve an idealized approach to this nonlinear problem and obtain quite easily approximate solutions.

#### 1. Introduction

The analysis of the vibration of beams resting on elastic foundations wears a practical and theoretical interest in many fields such as civil, mechanical, and transportation engineering. Analytical and numerical methods applied to the modeling of such a problem are extensively addressed in the literature. However, the combination between the emergence of high-performance computers and problems complexity made the discrete methods more appealing.

A discrete method such as the Finite Element Method (FEM) is the first to address the problem numerically [1–3]. The Differential Quadrature Method (DQM) is also employed for the solution to similar engineering problems involving beam vibrations and foundations [4–6]. Therefore, Malekzadeh and Karami [7] gather the advantages of both previous methods (DQM and FEM) to perform the free vibration and buckling analysis of thick beams on two-parameter elastic foundations.

On the other hand, soil behavior wears a great complexity, because of its nonlinear nature. In that order, literature has investigated multiple soil models starting from the idealized elastic theories modeled by Winkler to the more complex viscoelastic theories of Pasternak and Hetinyi. Similarly, Kerr [8] carries a study showing the evolution of soil theories with their extensions.

The aim of this work is the development of a flexible general discrete model for linear and nonlinear vibrations of beams resting on elastic foundations, via the adaptation and the extension of the approach presented in [9]. Accordingly, the process of discretization carried out is intended to allow efficient variation of beam and soil geometrical and mechanical characteristics, in order to make it easy to perform multiple parametric studies.

For nonlinear vibrations, the multimode analysis leads to a coupled nonlinear differential system involving the contribution to various modes. Using the harmonic balance method, Benamar [18] and Benamar et al. [19] reduce the nonlinear free vibration problem to a set of nonlinear algebraic equations. Consequently, the objective here is the investigation into linear and nonlinear vibrations, using the discrete physical model developed for various types of soil stiffness, and beam end conditions.

This study establishes a simplification to the nonlinear problem induced by large deflections. However, the nonlinearity tensor obtained becomes heavy in terms of calculation for high values of the discretization parameter (presenting the number of masses in the physical model in Figure 2). In fact, we addressed this complexity and proposed a simplification of the calculation, leading to an efficient algorithm.

The structure of this paper is as follows: Section 2.1 presents the general formulation, where the continuous and discrete models are detailed, in order to show their theoretical similarities. After the establishment of the discrete model, the following Section 2.2 is concerned with determination of the discrete model parameters and the nondimensional formulation. Finally, Section 2.3 introduces a solution to the multimodal nonlinear problem using a linearization procedure.

The physical discrete model validation is done in Section 3, devoted to details and comments on the results obtained in the cases considered. An examination is made of how far the results deviate from the linear and the nonlinear continuous theories, and a discussion is given about their range of validity under the different assumptions adopted.

The main purpose of this work is achieved in Section 4, in which straight applications of the theory are developed and validated via comparisons with previous references. The first application deals with a partially supported beam on elastic foundations, where the springs presenting the soil reaction are applied to only a limited portion of the beam span. In the second application, the beam examined is resting on a variable elastic foundation, where the distribution of soil spring stiffness is linear or parabolic.

#### 2. General Theory

A nonlinear discrete model of a beam, made of extensional bars, concentrated masses and rotational springs located at the bar ends, is introduced in [9] to approach the continuous theory using different assumptions. In the present work, a beam resting on an elastic soil is undergoing the same discretization process by extending and completing the previous model.

##### 2.1. General Formulation

This section presents the discrete and continuous models for nonlinear vibrations of beams resting on elastic foundations. Both models are based on the Lagrangian principle, leading to a formally identical description. The linear behavior of the mechanical system examined is determined by the classical mass and rigidity matrices and . On the other hand, the nonlinear behavior, presented by nonlinearity tensor due to the axial forces induced by the large vibration amplitudes in the continuous case, is established by the application of the Pythagorean Theorem in the discrete physical model.

###### 2.1.1. Continuous Model

Before introducing the extended formulation of the discrete model, a quick review of the theory behind the continuous model [18] is introduced. Let us consider transverse vibrations of a beam, shown in Figure 1, resting on elastic foundations, having the characteristics (*, **, **, **, **, **,* and ) defined in Nomenclature. The total beam strain energy can be written as the sum of the strain energy due to bending denoted as and the strain energy induced by Winkler springs plus the axial strain energy due to the axial load induced by large deflections. Thus , , , and the kinetic energy are as follows [20]:where is the beam transverse displacement. Using a generalized parameterization and the usual summation convention used in [18], the transverse displacement can be described as [20]