Elsevier

Composite Structures

Volume 152, 15 September 2016, Pages 306-315
Composite Structures

Free vibration and buckling of laminated beams via hybrid Ritz solution for various penalized boundary conditions

https://doi.org/10.1016/j.compstruct.2016.05.037Get rights and content

Abstract

This paper presents an analytical solution for the buckling and free vibration analysis of laminated beams by using a refined and generalized shear deformation theory which includes the thickness expansion. The eigenvalue equation is derived by employing the Rayleigh quotient, and the Ritz method is used to approximate the displacement field. The functions used in the Ritz method are chosen as either a pure polynomial series or a hybrid polynomial-trigonometric series. The hybrid series is used due to the superior convergence and accuracy compared to conventional pure polynomial series for certain boundary conditions. The boundary conditions are taken into account using the penalty method. Convergence of the results is analyzed, and numerical results of the present theory are compared with other theories for validation. Nondimensional natural frequencies and critical buckling loads are obtained for a variety of stacking sequences. The effect of the normal deformation on the fundamental frequencies and critical buckling loads is also analyzed.

Introduction

Composite materials have ample use in aerospace and shipbuilding due to their better properties than other common materials. However, an accurate prediction of the structural capabilities of composite materials is more difficult, due to their anisotropic properties. The elastic behavior and characterization of multidirectional laminates is detailed in Ref. [1]. Beam structural elements made from composite materials are used in many applications due to their superior performance. However, a laminate composite beam has a considerable transverse shear deformation, increasing the difficulty of the analysis since the elementary theory for beams (Euler–Bernoulli) is inadequate.

An improvement over the Euler–Bernoulli theory for beams is obtained by considering the transverse shear deformation. For this purpose, a first order beam theory (FOBT), second order beam theory (SOBT) and a third order beam theory (TOBT) have been developed. The results obtained using these theories are widely used for comparison purposes. Analysis of bending, free vibration and buckling of laminated beams using FOBT, SOBT and TOBT has been developed by Khdeir and Reddy in Refs. [2], [3], [4]. Pagano [5] presented elasticity-based solutions for composite beams in cylindrical bending, and Plankis et al. [6] compared the free vibration frequencies using common beam theories and those obtained using a elasticity-based solution.

Many other higher shear deformation theories (HSDTs) have been proposed and evaluated by the finite element method or using analytical solutions. The static analysis of laminated beams using HSDTs is given in Refs. [7], [8], [9], [10], [11], [12], [13]. In order to consider arbitrary boundary conditions in the free vibration and buckling analysis, the Ritz method is commonly used. The Ritz method is described in Refs. [14], [15], [16]. Free vibration and linear buckling analysis of laminated beams using the Ritz method is given in Refs. [17], [18], [19]. Thermal buckling and postbuckling are primary modes of failure in environments with thermal loads, and their analysis is presented in Refs. [20], [21], [22], [23]. Other methods used besides the Ritz method are the finite element method [24], [25], the dynamic stiffness method [26], [27], [28], the Navier solution [29], meshless approximations [30], and the spectral finite element method [31].

Beams with composite materials with special properties or for specialized use such as piezoelectric or magneto-electro-elastic beams are analyzed in Refs. [32], [33], [34], [35], [36], [37], [38]. Due to their structural importance, free vibration and buckling analysis of thin-walled composite beams have been developed, as presented in Refs. [39], [40], [41], [42], [43], [44]. Curved composite beams are structural members of great importance, and the analysis of this member is given in Refs. [45], [46], [47], [48], [49], [50], [51].

A unified formulation known as Carrera’s Unified Formulation (CUF) has been used to analyze structural elements using theories with arbitrary order. This unified formulation is described in Refs. [52], [53], [54]. Carrera et al. [55] used this formulation to develop beam elements with arbitrary cross-section geometry. The static, free vibration and buckling analysis of laminated beams have been developed in Refs. [56], [57], [58], [59], [60]. The use of dynamic stiffness elements in the framework of CUF for the analysis of laminated beams has been presented by Pagani et. al [61]. Analysis of single and multi-cell laminated box beams has been developed by Carrera et. al [62].

In this paper, a refined and generalized shear deformation theory is used for the free vibration and buckling analysis of laminated beams. The eigenvalue equation is derived by using the Rayleigh quotient and the Ritz method, using different functions chosen according to the boundary conditions in order to improve convergence. To impose the boundary conditions the penalty method is used. The results are compared with solutions from Reddy [3], [4] and Aydogdu [11]. Convergence of results depending on the number of terms in the Ritz series and the stiffness penalty parameter is also analyzed. The nondimensional frequencies and critical buckling loads are obtained for a variety of laminated beams. The effect of the normal deformation is also analyzed.

Section snippets

Beam under consideration

A 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. Two dimensional constitutive laws are used.

Beam displacement field

The general displacement field is given as:u(x,z)=u0(x)+zα(x)+f(z)wbx(x)w(x,z)=wb(x)+ws(x)+g(z)θ(x)where u and w are the displacement components in the X and Z axis respectively, and u0, wb, ws and θ are four

Numerical results and discussion

The shear strain shape functions are chosen as follows:f(z)=4z33h2,g(z)=1-4z2h2

To evaluate numerical results the following material properties are used:E1E2=40,E3E2=1,G12E2=G13E2=0.6G23E2=0.5,v12=v13=v23=0.25

The nondimensional natural frequencies, nondimensional critical buckling loads and nondimensional penalty stiffness parameter are defined as:ω¯=ωL2hρE2,N¯CR=NCRL2bh3E2,p¯S=kiL2E1Iwhere I is the second moment of area of the cross section of the beam.

Conclusions

A refined and generalized shear deformation theory is used in this paper to obtain the analytical solutions of the free vibration and buckling of laminated beams. The eigenvalue equation is derived by using the Rayleigh quotient and the Ritz method, and the boundary conditions are imposed using the penalty method. A hybrid series has been used to approximate the displacement variables, with superior convergence and accuracy than other commonly used functions. The numerical results have been

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