Thermal buckling and postbuckling of laminated composite beams with temperature-dependent properties

https://doi.org/10.1016/j.ijnonlinmec.2011.11.009Get rights and content

Abstract

The thermal buckling and postbuckling analysis of laminated composite beams with temperature-dependent material properties is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. The differential quadrature method (DQM) as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. A direct iterative method is employed to obtain the critical temperature (bifurcation point) as well as the nonlinear equilibrium path (the postbuckling behavior) of symmetrically laminated beams. The applicability, rapid rate of convergence and high accuracy of the method are established via different examples and by comparing the results with those of existing in literature. Then, the effects of temperature dependence of the material properties, boundary conditions, length-to-thickness ratios, number of layers and ply angle on the thermal buckling and postbuckling characteristic of symmetrically laminated beams are investigated.

Highlights

► Postbuckling of laminated composite beams with temperature-dependent properties is presented. ► The differential quadrature method is employed to discretize the governing equations. ► It is shown that increasing the constraint of beam, the effect of temperature dependence increases. ► Decreasing the length-to-thickness ratio, the effect of temperature dependence increases. ► Property temperature dependency reduces the overall stiffness of beam and cannot be neglected.

Introduction

Laminated composite materials in comparison with conventional materials have superior mechanical and thermal properties. Therefore laminated composites have received wide applications as structural components in civil, mechanical, aerospace and marine engineering such as diaphragms and deck plates in launch vehicles, aerospace vehicles, supersonic/hypersonic flight vehicles, reusable space transportation systems, naval ships and submarine structures. Thermal buckling is one of the primary modes of failure of composite structures when subjected to thermal and severe thermal environments. Thus the thermal buckling and postbuckling behavior of laminated composite structural elements such as beams, plates and shells are the factors governing their design and accurate determinations are of interest to designers [1], [2], [3], [4], [5], [6], [7], [8]. There are some research works on the thermal buckling and postbuckling of laminated beams which are reviewed here.

Abramovich [1] presented analytical solution for the thermal buckling analysis of cross-ply laminated composite beams based on first order shear deformation theory (FSDT). Lee [2] investigated the thermal buckling behavior of laminated composite beams using layerwise beam theory and finite element method (FEM). Lee and Choi [3] investigated the thermal buckling and postbuckling analysis of a laminated composite beam with embedded shape memory alloy (SMA) wires analytically based on the classical beam theory. Khdeir [4] developed exact analytical solutions based on the refined beam theories to obtain the critical buckling temperature of cross-ply laminated composite beams. Aydogdu [5] obtained the critical buckling temperature of cross-ply laminated beams based on a three-degree-of-freedom shear deformable beam theory with different boundary conditions by applying the Ritz method.

Since increasing the environmental temperature causes degradation of the thermoelastic properties, it is necessary to take into account the temperature dependence of these properties for rigorous design [6], [9]. However, to the best of authors' knowledge, the role of material temperature-dependent properties on the thermal buckling and post-buckling behavior of laminated composite beams has not been addressed in literature. More recently, Vaz et al. [9] studied the initial thermo-mechanical post-buckling of isotropic hinged ends beams with temperature-dependent physical properties using the perturbation method and based on the classical beam theory.

The geometrical and physical nonlinearity are two sources that cause the governing equations to become more complex to be solved analytically. Therefore, the approximate methods should be employed to solve this problem. Usually the finite element method [2], [10] or the Ritz methods [5], [11] were employed in previous works. However, in studying the global behaviors of the structural elements such as buckling and free vibration analysis, better convergence behavior together with less computational efforts is observed by the differential quadrature method (DQM) compared with its peer numerical competent techniques such as the finite element method, the finite difference method, the boundary element method and the meshless technique [12], [13], [14], [15], [16]. In addition, since the strong form of the governing equations and the related boundary conditions are discretized in using this method, it is free from the shear locking phenomenon that was occurred in using the FEM for the problems which include the transverse shear deformation. The other advantages and also the disadvantages of the DQ method can be found in the review paper of Bert and Malik [16].

Due to the fact that the critical buckling temperature of thick laminated composite beams is above the melting point of its constituents, the buckling phenomenon can be an important mode of failure for thin-to-moderately thick laminated beams. Hence, the first order shear deformation beam theory seems to be satisfactory for the thermal buckling and post buckling analysis of laminated beams.

Based on the above review, as a first attempt, the thermal buckling and postbuckling behavior of the laminated composite beams with temperature-dependent material properties subjected to different boundary conditions is studied. The differential quadrature method (DQM) is applied to discretize the governing equations and the related boundary conditions, which are based on the FSDT. Then, a direct iterative method is employed to solve the system of nonlinear algebraic equations to obtain the bifurcation point and the thermal postbuckling characteristic of the laminated composite beams. Finally, the effect of temperature dependency of material properties, boundary conditions, number of layers and length-to-thickness ratio on the results are investigated.

Section snippets

Governing equations and solution procedure

Consider a laminated composite beam composed of NL perfectly bounded orthotropic layers of length L and total thickness h (see Fig. 1). It is assumed that the beam is stress free at the constant temperature T0 and the temperature rise is uniform within the beam. As mentioned previously, the buckling phenomenon can be an important mode of failure for thin-to-moderately thick beams and consequently, the plane stress assumption can be used to obtain the axial equilibrium stress component as [17]σxx

Numerical results

In this section, firstly, the formulation and the method of solution are validated by studying its convergence behavior and by comparing the results with those available in the literature. Then, some new results for thermal postbuckling analysis of laminated composite beams with temperature-dependent material properties are presented. In the all solved examples, otherwise specified, the free stress temperature of the beam is assumed to be T0=30 °C.

As a first example, the convergence and accuracy

Conclusion

As a first attempt, the thermal buckling and postbuckling analysis of laminated composite beams with temperature-dependent material properties is investigated. The differential quadrature method as an efficient and accurate numerical method is employed to discretize the nonlinear differential equations and the related boundary conditions. Beams with different boundary conditions subjected to uniform temperature rise are analyzed. A direct iterative method is applied to determine the bifurcation

Cited by (0)

View full text