A unified quasi-3D HSDT for the bending analysis of laminated beams

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Abstract

This paper presents a static analysis of laminated cross-ply beams by using an unavailable unified higher order shear deformation theory (HSDT) for composite beams which includes the thickness stretching effect. The generalized governing equations are derived by employing the principle of virtual work. Navier-type closed-form solution is obtained for several beams subjected to several kinds of loads (sinusoidal, uniform, linear and point load). Infinite HSDTs for beams can be further developed just by modifying the shear strain shape functions. Several hybrid type shear strain shape functions were introduced to evaluate the generality of the proposed theory. The stretching effects for the accurate prediction of the transverse stresses are analyzed. Numerical results of some HSDTs for beams are compared with the elasticity solutions and other HSDTs. Convergence studies were carried out to determine the necessary half-wave number for the different models.

Introduction

Laminated composite beams have representative transverse shear deformation and this affects the transverse displacement significantly. The elementary theory for beams or Euler–Bernoulli theory (EBT) is inadequate to analyze this type of beam because it assumes that planes initially normal to the midplane remain plane and normal after deformation, neglecting transverse shear deformations. Nowadays, higher order shear deformation theories (HSDTs) are used to account for this effect.

An improvement over the Euler–Bernoulli theory is the first order shear deformation theory (FSDT) or commonly known as Timoshenko beam theory. However, this theory is limited in use because it assumes a constant transverse shear deformation through the thickness of the beam. Therefore it requires a shear correction factor to correct the strain energy of deformation. To overcome the limitations of Euler–Bernoulli theory and FSDT, there has been a development of HSDTs for laminated beams including effects of transverse shear deformation. Khdeir and Reddy [1] presented the analysis for the bending of cross-ply laminated beams using the classical beam theory, FSDT and two HSDTs. The HSDTs presented by Khdeir and Reddy have a simple formulation and are reasonably close to the exact elasticity solution presented by Pagano [2], and thus are widely used for comparison purposes. However, these theories lack stretching effect, that is, the variation of the transverse displacement across the thickness is neglected and transverse stresses are inaccurate. For this reason, other HSDTs that include stretching effect have been developed to improve the accuracy of the results.

The available HSDTs consider a higher variation of displacements as a function of the thickness, which can be linear, parabolic, trigonometric, or any other arbitrary function. Some HSDTs applied to thick beams are presented in Refs. [3], [4], [5], [6]. HSDTs developed to model composite beams are common, and many models have been developed to consider the stress continuity in laminate composites. The laminated beam can be modeled with increasing accuracy and complexity using an equivalent single layer, zig–zag or layer-wise theory, and some HSDTs developed for this purpose solved by analytical methods are given in Refs. [7], [8], [9], [10], [11], [12]. Finite element solutions using a variety of HSDTs are presented in Refs. [13], [14], [15], [16], [17], [18].

A unified formulation for the development of HSDTs known as Carrera's Unified Formulation (CUF) has been presented by Carrera [19]. This formulation has been applied to consider thermal loads, analyze piezoelectric structures and general multifield problems as given in Refs. [20], [21], [22]. Different classes of 1D models using a variety of functions such as trigonometric and exponential have been proposed and applied for static and dynamic analysis, as presented in Refs. [23], [24], [25], [26], [27], [28], [29].

Beam theories have been used for special applications and to analyze engineering structures. Lanc et al. [30] presented a finite element model for the buckling analysis of laminated beams. Turkalij et al. [31] presented a beam formulation for the analysis of composite frames accounting for the semi-rigid connection behavior. Pagani et al. [32] modeled aerospace structures using beam assumptions in the framework of CUF. Senjanović et al. [33] modeled thin-walled girders using a beam finite element for the vibration analysis of ships.

In this paper, a unified HSDT theory including stretching effect for the bending analysis of laminated beams is presented. Four shear strain shape functions which were arbitrarily selected were evaluated. The beam governing equations are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for laminated beams subjected to transverse load for simply supported boundary conditions. The results are compared with 3D elasticity solution and with Reddy's HSDT for beams. It is important to remark that Reddy's HSDT lacks stretching effect, and thus transverse displacements and transverse stresses obtained by the present theory are included to show the variation of displacement across the thickness. Overall, infinite HSDTs can be derived by the present unified formulation and further optimization studies can be done to improve the accuracy of the present theory. The present unified HSDT can be used to model single and sandwich laminated beams by accurately predicting transverse displacements across the thickness, which is of special importance when transverse stresses are considered in the analysis.

Section snippets

Beam under consideration

An cross-ply laminated beam of length L, width b and a total thickness h is considered in the present analysis. The beam occupies the following region:0xL;b/2yb/2;h/2zh/2 The displacements are assumed to be small, and the body forces are neglected. The beam is subjected to lateral load only, and two dimensional constitutive laws are used (x and z).

Theoretical displacement field

The general displacement field of the HSDT considered is given as:u(x,z)=u0(x)+zα(x)+f(z)ϕ(x)w(x,z)=w0(x)+g(z)θ(x) where u and w are the

Solution procedure

For simply-supported boundary conditions, the Navier solution is assumed to be:u0(x)=nUncos(αx)w0(x)=nWnsin(αx)θ(x)=nΘnsin(αx)ϕ(x)=nΦncos(αx) where α=nπ/L. Substituting Eq. (18) and the appropriate expression for the load as given in Eqs. (13)–(17) in the governing equations, Eqs. (8) to (11), yields the four algebraic equations from which the unknown coefficients of the Fourier expansions are obtained. The material properties are taken to be the same as the properties used by

Shear strain shape functions

The results are obtained for cross-ply laminated beam choosing suitable shear strain shape functions f(z) and g(z). The results from the elementary theory of beams (EBT), Timoshenko beam theory (FSDT), and the third order theory developed by Reddy (HSDT) were reproduced for comparison purposes. When it is available, the exact solution (indicated as elasticity) given by Pagano [2] is also provided. The displacements and stresses obtained are presented in the following non-dimensional form:u¯=E2bu

Conclusions

An unavailable and unified HSDT with 4-unknowns with thickness stretching effect is presented in this paper to analyze laminated beams. The governing equations are derived by employing the principle of virtual work. A Navier-type closed-form solution is obtained for cross-ply laminated beams subjected to loads for simply supported boundary conditions. A convergence study is done to compare the convergence of the models evaluated. The important conclusions that emerge from this paper can be

Conflict of interest statement

There are no conflicts of interest with any person or entity associated with the research reported in this paper.

References (36)

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