Shock and Vibration

Volume 2019, Article ID 2724768, 15 pages

https://doi.org/10.1155/2019/2724768

## Free Vibrations of an Elastically Restrained Euler Beam Resting on a Movable Winkler Foundation

^{1}Research Center for Wind Engineering, Southwest Jiaotong University, Chengdu, China^{2}College of Civil Engineering, Shanghai Normal University, Shanghai, China

Correspondence should be addressed to Qiang Zhou; moc.liamg@3201nalim

Received 12 March 2019; Accepted 28 April 2019; Published 13 June 2019

Academic Editor: Lorenzo Dozio

Copyright © 2019 Qiang Zhou and Tong Wang. 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

The traditional theory of beam on elastic foundation implies a hypothesis that the elastic foundation is static with respect to the inertia reference frame, so it may not be applicable when the foundation is movable. A general model is presented for the free vibration of a Euler beam supported on a movable Winkler foundation and with ends elastically restrained by two vertical and two rotational springs. Frequency equations and corresponding mode shapes are analytically derived and numerically solved to study the effects of the movable Winkler foundation as well as elastic restraints on beam’s natural characteristics. Results indicate that if one of the beam ends is not vertically fixed, the effect of the foundation’s movability cannot be neglected and is mainly on the first two modes. As the foundation stiffness increases, the first wave number, sometimes together with the second one, firstly decreases to zero at the critical foundation stiffness and then increases after this point.

#### 1. Introduction

In most previous studies on transverse vibrations of beams, end supports have usually been simplified as rigid restraints such as hinged, fixed, free, or sliding, neglecting the elasticity of end supports. This assumption is certainly convenient for establishing and analyzing the beam theory, while may bring errors or mistakes in some cases. As is known to all, all structural materials possess elasticity to some extent. So, only when the stiffness of end supports is much larger than that of the beam can the assumption of rigid restraints be reasonable. However, in practice, this prerequisite usually cannot be satisfied very well, for example, a submarine pipeline laid on a soft seabed [1] and a girder bridge supported on highly flexible piers [2], for which the elasticity of end supports has to be considered. Till now, a considerable amount of research has been performed on vibration analysis of beams with elastic restraints (e.g., [3–10]), and the influence of stiffness of flexible restraints on beam vibrations has been found significant.

Structural elements that can be represented as a beam supported on an elastic foundation have wide applications in numerous aspects of engineering. Weaver et al. [11] proposed a vibration model for a rigidly restrained beam resting on elastic foundations and derived its general solution. This together with other works sparked off a surge of research in this area that continues in various guises to this day (e.g., [12–18]). And, more recently, the flexibility of end supports of beams on elastic foundation has aroused universal concern of researchers [19–22] and was also found non-negligible in engineering practice. However, the elastic foundation was usually considered to be fixed with the inertia reference frame in all aforementioned studies of rigidly or elastically restrained beams supported on elastic foundations. To the best of the authors’ knowledge, there is hardly any work done for elastically restrained beams resting on movable elastic foundations, at which some structural elements in the field of engineering can be represented.

We consider in this paper a Euler beam, for simplicity, supported on a movable massless Winkler foundation and with ends elastically restrained by two vertical and two rotational springs, which is one of the most common structures in building and bridge engineering. In Section 2, a general vibration model is developed for the beam. Then, the frequency equations and corresponding mode shapes of the model are analytically derived in Section 3 and can degenerate into many other classical ones. Section 4 presents some numerical examples to study the effects of the movable Winkler foundation as well as elastic restraints on natural characteristics. Finally, some conclusions are summarized in Section 5.

#### 2. Formulation of the Vibration Model

Let us consider a uniform straight Euler beam supported on a movable elastic foundation as shown in Figure 1, where *L*, *A*, *I*, *ρ*, and *E* are the span, cross-sectional area, cross-sectional moment of inertia, beam density, and Young’s modulus of elasticity, respectively. Ends of the Euler beam are elastically restrained by two vertical and two rotational springs with *k*_{1} and *k*_{2} being the left- and right-end transverse (or vertical) restraint stiffness, respectively, and *k*_{3} and *k*_{4} the left- and right-end bending restraint stiffness, respectively. The movable Winkler foundation is represented as a rigid massless beam together with distributed springs with a stiffness of *k* and is plotted above the beam for a clear visibility. As shown in Figure 1, both ends of the rigid beam are hinged to the ones of the Euler beam. So, the Winkler foundation will move vertically and/or rotationally together with the ends of the Euler beam, while has constraint on neither the Euler beam nor the elastic supports. The coordinate system is also shown in Figure 1, where the *x* axis is defined as the centroidal axis of the Euler beam at its original or static position, and the left end of the beam is located at *x* = 0. *y*(*x*, *t*) denotes the absolute transverse displacement of the beam relative to the *x* axis.