Analysis of shear-deformable composite beams in postbuckling

The nonlinear response of composite beams modeled according to higher-order shear deformation theories in postbuckling is investigated. The beam ends are restrained from axial movement, and as a result the contribution of the midplane stretching is considered. The equations of motion and the boundary conditions are derived using Hamilton’s principle. The shear deformation effect on the critical buckling load and static postbuckling response is introduced using classical, first-order, and higher-order shear deformation theories. This paper presents an exact solution for the static postbuckling response of a symmetrically laminated simply supported shear-deformable composite beam. The shear effect is shown to have a significant contribution to both the buckling and postbuckling behaviors. Results of this analysis show that classical and first-order theories underestimate the amplitude of buckling while all higher-order theories, considered in this study, yield very close results for the static postbuckling response.     

Analytical solutions of thermal buckling and postbuckling of symmetric laminated composite beams with various boundary conditions

Based on the first-order shear deformation beam theory, considering geometric nonlinearity, the governing equations for symmetric laminated composite beams subjected to uniform temperature rise are derived by using Hamilton’s principle, and then three solving methods are presented to deal with it. By introducing an auxiliary function, which is shown in method one, the governing equations are reduced to be a single fourth-order integral-differential equation, and the exact solutions for the thermal buckling and postbuckling of symmetric laminated composite beams with combination of in-plane immovable simply supported and clamped boundary conditions are presented for the first time. On the basis of the results given in the method one, the explicit solutions for the thermal buckling and postbuckling of the beams are presented by giving accurate displacement functions (method two) and Ritz method (method three), respectively. Then, the effects of the transverse shear effects and boundary conditions on the thermal buckling and postbuckling of the beams are qualitatively discussed. What is more, a preliminary discussion on the probability and difference of extending the giving methods to the higher-order shear deformation beam theory with various boundary conditions is conducted. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and efficiency of the present analysis and numerical results. And then the symmetric cross-ply laminated composite beam (0/90/0) is taken as an example to numerically evaluate the effects of the length-to-thickness ratio, beam theories, and boundary conditions on the thermal buckling and postbuckling of symmetric laminated composite beams. Some meaningful conclusions have been drawn.     

Transverse Shear and Normal Deformation Theory for Bending Analysis of Laminated and Sandwich Elastic Beams

The influence of transverse normal strain on bending analysis of cross-ply laminated and sandwich beams is presented. A higher-order shear deformation beam theory is developed. Euler-Bernoulli classical, Timoshenko first-order and simple higher-order theories have been also used in the analysis. The governing equations for a beam composed of orthotropic layers and subjected to any given mechanical load distribution are derived. Making use of Navier-like approach, exact solutions are obtained for cross-ply laminated and sandwich beams subjected to arbitrary loadings. Numerical results for beams with the simply-supported boundary conditions are presented. The effects due to transverse normal strain, transverse shear deformation and number of layers on the static response of the beams are investigated.     

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.     

A review of meshless methods for laminated and functionally graded plates and shells

This review focuses mainly on the developments of element-free or meshless methods and their applications in the analysis of composite structures. This review is organized as follows: a brief introduction to shear deformation plate and shell theories for composite structures, covering the first-order and higher-order theories, is given in Section 2. A review of meshless methods is provided in Section 3, with main emphasis on the element-free Galerkin method and reproducing kernel particle method. The applications of meshless methods in the analysis of composite structures are discussed in Section 4, including static and dynamic analysis, free vibration, buckling, and non-linear analysis. Finally, the problems and difficulties in meshless methods and possible future research directions are addressed in Section 5.     

Buckling of moderately thick laminated cylindrical shells: a review

The present paper is a review article on the problem of buckling of moderately thick, laminated, composite shells subjected to destabilizing loads. The loads consist of uniform axial compression, uniform lateral pressure and torsion applied individually or in combination. In all the works reported in the literature, the analysis is based on higher-order shear deformation (HOSD) shell theory and/or first-order shear deformation (FOSD) shell theory with or without a shear correction factor. Results obtained by these two shell theories and by employing classical thin shell theory are compared to determine the range of applicability of each in predicting critical conditions. The effect of stacking sequence, radius-to-thickness ratio and length-to-radius ratio is assessed. Typical numerical results are presented in tabular form. Moreover, some limited results, which are based on limit point analysis are also presented (imperfection sensitivity studies).     

Analysis of symm≐tric cross-ply laminated elastic plates using a higher-order theory: Part I—Stress and displacement

A simple theory for bending of composite anisotropic plates that are laminated symmetrically about their mid-plane is presented. This theory incorporates transverse shear deformation and transverse normal stress as well as the higher-order effects and fulfills the static conditions on the external boundary planes. Further on, by using Lévy-type solutions considered in conjunction with the state space concept, the state of stress and displacement of rectangular plates for a variety of edge conditions is determined and the results are compared to their first-order shear deformation and classical counterparts, obtained by using the same state-space technique.     

Thermomechanical postbuckling analysis of symmetric and antisymmetric composite plates with imperfections

In the present paper the postbuckling response of symmetrically and antisymmetrically laminated composite plates subjected to a combination of uniform temperature distribution through the thickness and in-plane compressive edge loading is presented. The structural model is based on a higher-order shear deformation theory incorporating von Karman nonlinear strain displacement relations. Adopting the Galerkin procedure, the governing nonlinear partial differential equations are converted into a set of nonlinear algebraic equations, which are solved using the Newton–Raphson iterative procedure. The critical buckling temperature is obtained from the solution of the linear eigenvalue problem. Postbuckled equilibrium paths are obtained for symmetrically laminated cross-ply, quasi-isotropic and antisymmetric angle-ply composite plates with and without initial geometric imperfections. Modal participation of each mode in the postbuckling deflection is reported using a multi-term Galerkin procedure.     

On the transient response of cross-ply laminated circular cylindrical shells

Transient response of simply-supported circular cylindrical shells is investigated using a higher-order shear deformation theory (HSDT). The theory is a modification of the Sanders' shell theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the bounding surfaces of the shell. The results obtained using the classical shell theory (CST) and the first-order shear deformation theory (FSDT) are compared with those obtained using the higher-order theory. The state-space approach is used to develop the analytical solutions to the equations of motion of the three theories.     

Effect of cross-sectional warping of anisotropic sandwich laminates due to dynamic loads using a refined theory and C° finite elements

An attempt has been made to study the effect of cross-sectional warping in the symmetrically laminated anisotropic composite sandwich plates for transient loads. A higher-order shear deformation theory (HOST) is used in conjunction with the simple displacement based C° finite element method (FEM). As is well-known, the classical first-order theories hitherto considered were inadequate to describe the propagation of waves in the highly orthotropic sandwich laminates. The present theory, which is more accurate than the Reissner-Mindlin theory, is applied herein, for the evaluation of plate response to different types of dynamic loads. An explicit central difference scheme is employed for the integration of dynamic equations of equilibrium with a diagonalized mass matrix obtained by a special procedure applicable to quadrilateral isoparametric elements. The numerical results of the present investigation have been compared with the first-order shear deformation theory (FOST) and the differences between HOST and FOST are examined. The results presented here should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.     

Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates

The classical and shear deformation beam and plate theories are reformulated using the nonlocal differential constitutive relations of Eringen and the von Kármán nonlinear strains. The equations of equilibrium of the nonlocal beam theories are derived, and virtual work statements in terms of the generalized displacements are presented for use with the finite element model development. The governing equilibrium equations of the classical and first-order shear deformation theories of plates with the von Kármán nonlinearity are also formulated. The theoretical development presented herein should serve to obtain the finite element results and determine the effect of the geometric nonlinearity and nonlocal constitutive relations on bending response.     

Postbuckling Analysis of Laminated Composite Plates Using a Higher-Order Zig-Zag Theory

This study examines the effects of incorporating zig-zag kinematics in the postbuckling analysis of laminated composite plates. A higher-order zig-zag plate element for nonlinear analysis was developed based on works of Averill and Yip. Their zig-zag element is especially suitable for a nonlinear structural laminate analysis due to its high accuracy and a low, constant number of degrees of freedom regardless of the number of layers. The article examines global postbuckling response as well as local displacement and stress fields of various laminated plates. The results derived from higher-order zig-zag theory are compared with predictions of first-order shear deformation theory (FSDT). Significant differences between these two theories are obtained for laminated plates with drastically different transverse stiffness properties with length-to-thickness aspect ratios L / t = 30 and 50. FSDT leads to good predictions of global and local behavior only for L / t = 50 and 100 with a typical layup in which the adjacent plies do not have very different transverse stiffness properties. Results presented in this article indicate that the zig-zag theory is required to predict accurately stresses and in-plane displacements through the thickness in moderately thick plates in the postbuckled state.     

复合材料双曲率壳屈曲和后屈曲的非线性有限元研究

基于 ABAQUS软件分析平台 , 采用非线性有限元法研究了横向载荷作用下复合材料双曲率壳的屈曲和后屈曲行为。通过在有限元模型中引入 Tsai2Wu失效准则 , 预测了复合材料双曲率壳的初始失效及渐进破坏过程 , 数值结果和试验数据吻合较好 , 表明了该模型的合理有效性 , 并详细讨论了各种参数对屈曲和后屈曲行为的影响。经分析复合材料双曲率加筋壳在均布压载和剪力联合作用下的屈曲和后屈曲行为 , 得到了屈曲载荷的拟合曲线 , 研究表明顺剪力的存在有利于提高屈曲载荷。     

剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。该方程与相应的边界条件构成了微分特征值问题。直接求解该问题,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams

An efficient zigzag one-dimensional (1D) theory of laminated beams is assessed by comparison of analytical solutions of simply supported beam with exact 2D elasticity solutions for static patch load, natural frequencies, harmonic transverse load with sinusoidal longitudinal variation and buckling under axial load. These results for a highly inhomogeneous test beam, symmetric composite beam, anti-symmetric composite beam and sandwich beam are also compared with first order (FSDT) and third order (TOT) shear deformation theories. The effect of the laminate lay-up and the thickness-to-span ratio on the accuracy is assessed. The zigzag theory yields much more accurate results than the TOT and FSDT.     

On the analysis of the buckling behaviour of laminated composite plates and shells

In general, plates and shells out of composite laminates are more susceptible to transverse shear than those out of homogeneous isotropic material. By means of different shear flexible theories the effect on buckling and postbuckling behaviour is studied. It turns out that there is a rather limited range of plates where the transverse shear is of considerable influence. Results obtained with Mindlin- or Reissner-type theories prove almost as adequate as those obtained with Reddy's theory. For the buckling analysis of shells the Kirchhoff–Love theory is precise enough. Snap-through buckling and imperfection sensitivity appear in composite shells, too, especially if they are optimized with respect to high bifurcation buckling loads.     

A critical review and some results of recently developed refined theories of fiber-reinforced laminated composites and sandwiches

A critical review of literature pertinent to the subject matter of this paper was carried out under the following two broad headings: free vibration and transient dynamics. Each of these groups describes the various theoretical developments in fiber reinforced laminated composite and sandwich plates. The theoretical developments are further classified according to the refinement/accuracy of the theories developed, such as the classical theory, the first-order shear deformation theory, and the three-dimensional elasticity/higher-order shear deformation theories. The present literature review is limited to linear free vibration and transient dynamic analyses, and geometric nonlinear transient response of multilayer sandwich/fiber-reinforced composite plates. A comparative study of recently developed refined theories in conjunction with the C° isoparametric finite element formulation has been made and the conclusions were drawn based on the literature review and the refined theories results. In order to compare the present results with the available results and to provide an easy means for future comparisons by other investigators, the numerical results are presented in tabular form.     

DYNAMIC INELASTIC RESPONSE AND BUCKLING OF METAL MATRIX COMPOSITE INFINITELY WIDE PLATES DUE TO THERMAL SHOCKS

SUMMARY

A micromechanical approach is combined with a structural analysis in order to investigate the thermally induced response and dynamic thermal buckling of infinitely wide plates composed of elastic-viscoplastic matrix reinforced by elastic fibres. The micromechanical analysis relies on the thermoelastic and inelastic properties of the constituents of the composite, and provides the instantaneous effective relation for the metal matrix composite at any point of the structure. The structural analysis employs the classical and higher-order plate theories in conjunction with a spatial finite difference and temporal Runge-Kutta integrations to generate the dynamic response of the plate. Results are given that illustrate the effects of the metallic matrix inelasticity, shear deformation, fibre orientation and initial geometrical imperfection on the dynamic response of the metal matrix composite plate subjected to various types of thermal shocks.     

Postbuckling and free vibrations of composite beams

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.