Vibration of multilayered beams using sinus finite elements with transverse normal stress

Abstract

A family of sinus models is presented for the analysis of laminated beams in the framework of free vibration. A three-noded finite element is developed with a sinus distribution with layer refinement. The transverse shear strain is obtained by using a cosine function avoiding the use of shear correction factors. This kinematic accounts for the interlaminar continuity conditions on the interfaces between the layers, and the boundary conditions on the upper and lower surfaces of the beam. A conforming FE approach is carried out using Lagrange and Hermite interpolations. It is important to notice that the number of unknowns is independent of the number of layers. Vibration mechanical tests for thin and thick laminated and sandwich beams are presented in order to evaluate the capability of these new finite elements to give accurate results with respect to elasticity or finite element reference solutions. Both convergence velocity and accuracy are discussed and these new finite elements yield very accurate results at a low computational cost for various boundary conditions. In particular, the two models including the transverse normal effect have the capability to take into account the thickness mode shape.

Introduction

Considering the increasing applications of composite and sandwich structures in the industrial field due to their high specific strength and stiffness, it is important to develop advanced and accurate models to design. In this framework, accurate knowledge of deflection and stresses is required to take into account effects of the transverse shear deformation due to the low ratio of transverse shear modulus to axial modulus, or failure due to delamination… In fact, they can play an important role on the behaviour of structures in services, which leads to evaluate precisely their influence on local stress fields in each layer or on natural frequencies.

The aim of this paper is to construct and evaluate a family of finite elements to analyze laminated beams in elasticity in relation to small displacements, so as to obtain the accurate predictions of frequencies in free vibration case. In particular, we put the emphasis on the influence of transverse normal effects, especially for very thick beams.

According to published research, various theories in vibrational mechanics for composite or sandwich structures (beams and plates for the present scope) have been developed. The following classification is associated with the dependency on the number of degrees of freedom (dofs) with respect to the number of layers:

• The Equivalent Single Layer approach (ESL): the number of unknowns is independent of the number of layers, but the transverse shear and normal stresses continuity on the interfaces between layers are often violated. The first work for one-layer isotropic plates was proposed in [1]. Then, we can distinguish the classical laminate theory [2] (it is based on the Euler–Bernoulli hypothesis and leads to inaccurate results for composites and moderately thick beams, because both transverse shear and normal strains are neglected), the first order shear deformation theory [3], [4], [5], [6], [7], [8], [9], and higher order theories [10]. In this later, hyperbolic [11], exponential [11], [12], parabolic [13], cubic [14], [15], [16], [13], [17], polynomial functions [18], [19], [20] are used in the expansion of the in-plane displacement component. Some of these theories also include the transverse normal effect with non-constant polynomial expressions of the out-of-plane displacement [21], [19], [22], [17]. Most of these approaches are based on a displacement formulation; nevertheless, mixed formulations are also carried out. In this framework, finite element [14], [21], [10], [16], [13], analytical [11], [15], [17], [12], or state-space [5], [23] solution are carried out.

• The Layerwise approach (LW): The number of dofs depends on the number of layers. This theory aims at overcoming the restriction of the ESL concerning the discontinuity of out-of-plane stresses on the interface layers. This approach was introduced in [24], [25], and also used in [26], [27], [28]. In recent contributions, various orders of expansion for the in-plane displacement are chosen: trigonometric [29], linear [30], quadratic [31], cubic [32], [33]. Some of them take into account the transverse normal effect [32] with a mixed approach. Special studies dedicated to the sandwich beam can be cited [34], [35]. The transverse displacement is constant in faces and quadratic in core. See also [36].

In this framework, refined models have been developed in order to improve the accuracy of ESL models avoiding the additional computational cost of LW approach. Based on physical considerations and after some algebraic transformations, the number of unknowns becomes independent of the number of layers. [27] has extended the work of [37] for symmetric laminated composites with arbitrary orientation and a quadratic variation of the transverse stresses in each layer. A family of models, denoted zig-zag models, was first employed in [38], then in [39], [31], [40]. More recently, it was also modified and improved by some authors [41], [42], [43], [44], [45], [46] with different order of kinematics assumptions, taking into account the transverse normal strain.

This above literature deals with only some aspects of the broad research activity about models for layered structures and corresponding finite element formulations. An extensive assessment of different approaches has been made in [47], [48], [49], [50], [51]. About the particular point of the evaluation of transverse normal stresses, see [52], [53]. A survey of developments in the vibration analysis of laminated composite beams is compiled in [54], see also [55].

In this work, a family of finite elements for rectangular laminated beam analysis is built, in order to have a low cost tool, efficient and simple to use. In fact, our approach is associated with the ESL theory. These elements are totally free of shear locking and are based on a refined shear deformation theory [56] avoiding the use of shear correction factors for laminates. Our elements are based on the sinus model [57] and the important feature is the capability of the model to include the transverse normal effect. So, the transverse displacement is written under a second order expansion which avoids the Poisson locking mechanism (see [58]). For the in-plane displacement, the double superposition hypothesis from [59] is used: three local functions are added to the sinus model. Finally, this process yields to only six or seven independent generalized displacements. It should be noted that all interface and boundary conditions are exactly satisfied for displacements and transverse shear stress. Therefore, this approach takes into account physical meaning.

As far as the interpolation of these finite elements is concerned, our elements are ${\text{C}}^0$-continuous except for the transverse displacement associated with bending which is ${\text{C}}^1$.

In this article, the mechanical formulation for the different models is described. For each of these approaches, the associated finite element is given. They are illustrated by numerical tests which have been performed upon various laminated and sandwich beams. A parametric study is given to show the effects of different parameters such as length to thickness ratio and number of degrees of freedom. The accuracy of computations are also evaluated by comparisons with an exact 2D elasticity solution, two-dimensional computations using commercial finite element software, three other sinus models [60] and also results available in literature. Different boundary conditions are also evaluated. Through these examples, the influence of transverse normal effect is highlighted.