Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions

The concept of variable angle tow (VAT) placement is explored for enhancing the buckling resistance of composite plates subjected to axial compression loading. The problem is relatively difficult to solve because of varying stiffness properties and requires prior prebuckling analysis to determine the non-uniform stress variation followed by the buckling analysis of VAT plates. A stress function formulation for in-plane analysis and displacement formulation for buckling analysis was employed to derive the governing differential equations of VAT plates based on classical laminated plate theory. The Differential Quadrature Method (DQM) is applied to solve the differential equations. The novel aspect of the present work is the use of Airy’s stress function to model the prebuckling analysis of VAT plates which considerably reduces the problem size, computational effort and provides generality to model pure stress and mixed boundary conditions. DQM was applied first to solve the prebuckling problem of VAT plates subjected to cosine distributed compressive load/ uniform end shortening. Then, DQM was applied to solve the buckling problem of rectangular VAT plates subjected to axial compression under different plate boundary conditions. Results were validated with detailed finite element analysis and the relative accuracy and efficiency of the DQM approach is discussed.     

Buckling and Postbuckling Behavior of Functionally Graded Nanotube-Reinforced Composite Plates in Thermal Environments

This paper investigates the buckling and postbuckling of simply supported, nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The nanocomposite plates are assumed to be functionally graded in the thickness direction using single-walled carbon nanotubes (SWCNTs) serving as reinforcements and the plates' effective material properties are estimated through a micromechanical model. The higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity is used to model the composite plates and a two-step perturbation technique is performed to determine the buckling loads and postbuckling equilibrium paths. Numerical results for perfect and imperfect, geometrically mid-plane symmetric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC plate, which is a special case in the present study, are compared with those of the FG-CNTRC plate. The results show that the buckling loads as well as postbuckling strength of the plate can be significantly increased as a result of a functionally graded nanotube reinforcement. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC plates.     

Postbuckling analysis of unilaterally constrained laminated thin plates

Postbuckling responses of laminated thin plates supported by a tensionless foundation and subjected to in-plane compressive edge loads are investigated. The postbuckling analysis presented here is based on the classical laminated plate theory, and includes the plate-foundation interaction, for which the foundation reacts in compression only. The plate is assumed to be simply supported at four edges, while the two longitudinal unloaded edges may be either movable or immovable. The initial geometric imperfection of the plate is taken into account. The analysis uses a two step perturbation technique to determine the postbuckling response of the plate. An iterative scheme is developed to obtain numerical results without any prior assumption for the shape of the contact region. The numerical examples concern the postbuckling behavior of antisymmetric angle-ply and symmetric cross-ply laminated plates with unilateral constraints, from which results for the same unconstrained plates are obtained as comparators. The numerical results show that the unilateral constraint has a significant effect on the postbuckling response of laminated thin plates.     

Thermal postbuckling analysis of laminated composite plates by the finite element method

The thermal postbuckling behavior of composite laminated plates subjected to a nonuniform temperature field is investigated by the finite element method. Based on the principle of minimum potential energy, the nonlinear stiffness matrix and geometry matrix are derived. The assumed displacement state over the middle surface of the plate element is expressed as a product of one-dimensional, first-order Hermitian polynomials. An iterative method is employed to determine the thermal postbuckling load. The results of the computations reveal that the thermal postbuckling behavior of composite laminated plates is influenced by lamination angle, plate aspect ratio, modulus ratio and the number of layers.     

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.     

Buckling and postbuckling of anisotropic laminated cylindrical shells under combined axial compression and torsion

A postbuckling analysis is presented for an anisotropic laminated cylindrical shell of finite length subjected to combined loading of axial compression and torsion. The governing equations are based on classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity and including the extension–twist, extension–flexural and flexural–twist couplings. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, anisotropic laminated cylindrical shells for different values of load-proportional parameters. The results show that the postbuckling characteristics depend significantly upon the load-proportional parameter. The results reveal that in combined loading cases the postbuckling equilibrium path is unstable and the shell structure is imperfection-sensitive.     

Postbuckling Analysis of Shear-Deformable Composite Laminated Plates on Two-Parameter Elastic Foundations

Postbuckling analysis is presented for a simply supported, shear-deformable, composite laminated plate subjected to uniaxial compression and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Two cases of in-plane boundary conditions are considered. The formulations are based on Reddy’s higher-order shear-deformation plate theory, including plate–foundation interaction. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performance of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, the character of the in-plane boundary conditions, plate aspect ratio, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Buckling characteristics of nanocrystalline nano-beams

The buckling and the postbuckling characteristics of nanocrystalline nano-beams with/without surface stress residuals are investigated. A hybrid model is proposed where a non-classical beam model is incorporated with a size-dependent micromechanical model. The micromechanical model has the merit of accounting for the beam material structure effects, i.e. the grain size and the grain boundary effects. To account for the beam size effects, the couple stress theory is implemented where some measures are added to capture the grain rigid rotation effects. The proposed hybrid model is harnessed to derive the governing equations of a nano-beam subjected to an axial compressive load accounting for the mid-plane stretching according to von-Karman kinematics and the surface stress residuals. Analytical solutions for the prebuckling and postbuckling configurations and natural frequencies as functions of the applied compressive axial load are derived. The effects of the beam material structure and the beam size on the beam’s prebuckling characteristics and the postbuckling configurations and natural frequencies are studied. The obtained results reveal that both the size and the material structure of nanobeams have great impacts on their buckling characteristics.     

Thermal Postbuckling Analysis of Imperfect Reissner-Mindlin Plates on Softening Nonlinear Elastic Foundations

A thermal postbuckling analysis is presented for a simply supported, moderately thick rectangular plate subjected to uniform or nonuniform tent-like temperature loading and resting on a softening nonlinear elastic foundation. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reissner-Mindlin plate theory considering the first-order shear-deformation effect, and including plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine the thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, thermal load ratio and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form and exhibit interesting imperfection sensitivity.     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates

Thermal buckling and postbuckling behavior is presented for functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to in-plane temperature variation. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates and uniformly distributed CNTRC plates under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. It is found that in some cases the CNTRC plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

复合材料补片加固含椭圆孔钢板刚度分析

针对复合材料补片加固损伤钢板的受力特点, 建立三维弹性力学模型, 对复合材料补片双面加固轴向受拉平板进行了应力和变形分析。研究了粘贴界面的剪应力分布规律, 钢板和补片内轴向拉力的分布规律, 分析了加固后平板相对加固前的相对刚度。研究了含椭圆孔钢板经复合材料补片加固后相对加固前以及相对完好钢板的刚度。结果表明, 复合材料胶接修补含孔钢板的刚度随着补片厚度的增加而增加, 且增强的趋势逐渐趋于缓和。当补片层数大于5层后, 单纯依靠增加补片的厚度并不能有效提高修复效果。解析法与有限元法的计算结果及试验结果吻合得比较好。      

Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT

The finite element dynamic stability analysis of laminated composite skew structures subjected to in-plane pulsating forces is carried out based on the higher-order shear deformation theory (HSDT). The two boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for square and skew plates with or without central cutout are in good agreement with those reported by other investigators. The new results for laminated skew plate structures containing cutout in this study mainly show the effect of the interactions between the skew angle and other various parameters, for example, cutout size, the fiber angle of layer and thickness-to-length ratio. The effect of the magnitude of the periodic in-plane load on the dynamic instability index is also investigated.     

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.     

Buckling and postbuckling of soft-core sandwich plates with composite facesheets

Finite element analysis is conducted to investigate the effect of coupling between the stiff facesheets and the flexible core on the stability of sandwich plates. The facesheets and the core are treated separately as three-dimensional solids. Initially, linear buckling analysis was carried out to determine the buckling loads for a wide range of the facesheet stiffness and core stiffness. Numerical results indicate that, for sandwich plates with a soft core such as PVC foam, the model that smears the facesheets and core overestimates the initial buckling load, demonstrating the importance of treating the facesheets and the core separately. Subsequently, postbuckling analysis was carried out. Numerical results show that, for sandwich plates with a soft core, the postbuckling behavior is unstable in that static equilibrium cannot be maintained once the buckling initiates. Received 20 August 1999     

On the global large deflection and postbuckling of symmetric angle-ply laminated plates

This paper presents studies of the global nonlinear deflection and postbuckling behavior of symmetric angle-ply laminated plates. A new transformation between on-axis and off-axis modulus is established to make a link between the stiffness of angle-ply plates and the global constants in orthotropic transformed space. All the linear and nonlinear terms in the plate equations are properly nondimensionalized, such that the results in this paper are generic rather than specific. The results indicated that with increasing generalized rigidity ratio D^*, the deflection is reduced and the buckling coefficient is increased, and naturally the postbuckling deflection is decreased. The generalized Poisson's ratio only has a small effect on the problems. The information presented in the text should be helpful to whoever is seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

《开孔对矩形板动力性能影响分析》

以超椭圆方程描述开孔边界,利用ANSYS有限元软件,数值分析开孔的形状、面积（尺寸）、位置及转动对典型矩形板前二阶固有频率的影响。研究结果发现开孔能显著改变矩形板的固有频率,其影响效果受板的边界约束条件和开孔形状设计影响很大。实际工程结构中,开孔形状设计应当引起足够的重视。     

Thermal Postbuckling of Imperfect Shear-Deformable Laminated Plates on Two-Parameter Elastic Foundations

Thermal postbuckling analysis is presented for a simply supported, shear-deformable composite laminated plate subjected to uniform or nonuniform parabolic temperature loading and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Reddy's third-order shear-deformation plate theory with von Karman nonlinearity is used. The governing equations also include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, plate aspect ratio, fiber orientation, thermal load ratio, and initial geometric imperfections, is studied. Typical results are presented in dimensionless graphical form.     

Nonlinear stability analysis of a perforated FGM plate under thermal load

The nonlinear thermal stability analysis of a perforated Ni/Al₂O₃ FGM plate with temperature-dependent thermoelastic material properties is carried out through a finite element approach based on the first-order shear deformation theory and the von-Karman's nonlinear kinematics. The thermoelastic material properties of the FGM plate are varied in the thickness direction, and the Mori-Tanaka homogenization scheme is applied to evaluate these properties. After validating the results of the current formulation, parametric studies are conducted to examine the effects of temperature-dependent material properties, material in-homogeneity, cutout shape and size, edge boundary conditions, aspect and slenderness ratios on the thermal buckling and postbuckling response of perforated FGM plate.     

The dynamic behaviour of postbuckled composite plates under acoustic excitation

The Rayleigh-Ritz method was used to find the postbuckling static displacement pattern of a composite plate (CFRP) under uniaxial in-plane compression of uniform edge-shortening. The resonance frequencies and mode shapes at various postbuckled states are then evaluated by eigenvalue analysis of the dynamical matrix equation consisting of up-dated tangential stiffness matrix at corresponding static configuration. The theoretical results are compared with experimental results obtained in 16-layered CFRP laminate of aspect ratio 1·5. The resonance frequencies and mode shapes obtained are used to interpret the multimodal and nonlinear strain responses to high level of acoustic excitation. The dominance of second-mode contribution and softening-spring behaviour are found in the strain response of postbuckled plates.