Thermal Postbuckling of Laminated Composite Plates with Temperature Dependent Properties

The paper deals with the theoretical investigation of the postbuckling of laminated composite rectangular plates subjected to uniform in-plane temperature. An analytical method based on Chebyshev polynomial is employed. The formulation is based on Reissner–Mindlin plate theory and von Kármán nonlinear kinematics. The resulting nonlinear coupled differential equations are linearized using quadratic extrapolation technique. Double Chebyshev finite series is used to discretize the differential equations. An incremental iterative approach is employed for the solution. The effects of temperature dependent mechanical and thermal properties on the limiting/critical temperature and the postbuckling response are studied. The numerical results for different boundary conditions and lamination schemes are presented. Analysis results indicate that temperature dependent properties reduce the critical/limiting temperature and postbuckling strength.     

Thermomechanical Postbuckling of Cross-Ply Laminated Rectangular Plates

Postbuckling analysis is presented for shear deformable cross-ply laminated composite rectangular plates subjected to the combination of in-plane edge compressive mechanical loading and thermal loads due to a linearly varying temperature across the thickness. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. The analysis uses a quadratic extrapolation technique for linearization and Chebyshev polynomials for spatial discretization. An incremental iterative approach is employed to estimate the critical load. The boundary conditions consisting of clamped, simply supported, free edge, and their combinations are considered. The effects of the thinness ratio, aspect ratio, lamination scheme, the number of layers, and the modulus ratio on the critical load/limit load and postbuckling behavior are studied.     

Analytical and Numerical Models of Postbuckling of Orthotropic Symmetric Plates

Two analytical perturbation methods which give approximate solutions of postbuckling behavior of orthotropic simply supported plates are considered in this work: the method of Chandra and Raju and the method of Shen and Zhang. The reproduction of the algebraic developments of these methods by the Mathematica symbolic manipulator program has revealed that there are errors in the formulas included in the original paper by Chandra and Raju. After a revision and correction of these errors, the analytical results of both methods for a set of 23 orthotropic plates are compared, an excellent agreement being found for a wide range of values of geometrical and mechanical parameters in which many actual plates lie. A numerical simulation performed on a reduced sample of six plates using finite-element code ABAQUS has validated analytical results. The present work is intended as a first step in the investigation of the possibility of using reliable analytical formulas in the design of composite plates.     

Thermal Postbuckling Behavior of Composite Sandwich Plates

By considering the total transverse displacement of a sandwich plate as the sum of the displacement due to bending of the plate and that due to shear deformation of the core, a 72 degrees of freedom high precision high order triangular-plate element is developed for the thermal postbuckling analysis of rectangular composite sandwich plates. Due to an uneven thermal expansion coefficient in the two local material directions, the buckling mode of the plate can be changed from one mode to another as the fiber orientation or aspect ratio of the plate is varied. By examining the local minimum of total potential energy of each mode, a clear picture of buckle pattern change is presented. Numerical results show that for a sandwich plate with cross-ply laminated faces, buckle pattern change may occur when the plate has a long narrow shape. However, for sandwich plates with angle-ply laminated faces, the buckling mode is dependent on the fiber orientation and aspect ratio of the plate. The effect of temperature gradient on the postbuckling behavior of the sandwich plate is limited except for angle-ply laminated sandwich plates with fiber angle greater than 70° or less than 20°.     

Buckling of Laminated Composite Rectangular Plates

The present study estimates the critical/buckling loads of laminated composite rectangular plates under in-plane uniaxial and biaxial loadings. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. Chebyshev series is used for spatial discretisation and quadratic extrapolation is used for linearization. An incremental iterative approach is used for estimation of the critical load. Different combinations of simply supported, clamped and free boundary conditions are considered. The effects of plate aspect ratio, lamination scheme, number of layers and material properties on the critical loads are studied.     

Bounds on Flexural Properties and Buckling Response for Symmetrically Laminated Composite Plates

Nondimensional parameters and equations governing the buckling behavior of rectangular symmetrically laminated plates are presented that can be used to represent the buckling resistance, for plates made of all known structural materials, in a very general, insightful, and encompassing manner. In addition, these parameters can be used to assess the degree of plate orthotropy, to assess the importance of anisotropy that couples bending and twisting deformations, and to characterize quasi-isotropic laminates quantitatively. Bounds for these nondimensional parameters are also presented that are based on thermodynamics and practical laminate construction considerations. These bounds provide insight into potential gains in buckling resistance through laminate tailoring and composite-material development. As an illustration of this point, upper bounds on the buckling resistance of long rectangular orthotropic plates with simply supported or clamped edges and subjected to uniform axial compression, uniform shear, or pure in-plane bending loads are presented. The results indicate that the maximum gain in buckling resistance for tailored orthotropic laminates, with respect to the corresponding isotropic plate, is in the range of 26–36% for plates with simply supported edges, irrespective of the loading conditions. For the plates with clamped edges, the corresponding gains in buckling resistance are in the range of 9–12% for plates subjected to compression or pure in-plane bending loads and potentially up to 30% for plates subjected to shear loads.     

Postbuckling Response Simulations of Laminated Anisotropic Panels

Numerical simulations are used in conjunction with experiments to study the buckling and postbuckling responses and failure initiation of flat, unstiffened composite panels. The numerical simulations are conducted using two‐dimensional shear‐flexible finite elements. The effect of the laminate stacking sequence on the buckling and postbuckling responses is studied. Correlation between numerical and experimental results is good through buckling, but the numerical models overestimate the postbuckling stiffness of the panels when nominal values of the material properties are used. To explain the discrepancies in the postbuckling stiffnesses, analytic sensitivity derivatives are calculated and used to study the sensitivity of the buckling and postbuckling responses to variations in different material and lamination parameters. Experimental results indicate that failure occurs along a nodal line. Numerical results show that the location of failure initiation corresponds to that of the maximum transverse shear‐strain energy density in the panel, which occurs at the edge of the panel at a nodal line. However, the transverse shear deformation has a negligible effect on the global response characteristics of the panel.     

Postbuckling Analysis of Plates Resting on a Tensionless Elastic Foundation

The buckling and large deflection postbuckling behavior of plates laterally constrained by a tensionless foundation and subjected to in-plane compressive forces are investigated. A nonlinear finite-element formulation based on Marguerre’s nonlinear shallow shell theory, modified by Mindlin’s hypothesis, is employed to model the plate response. To overcome difficulties in solving the plate–foundation equilibrium equations together with the inequality constraints due to the unilateral contact condition, two different approaches are used: (1) the unilateral constraint is accounted for indirectly by a bilinear constitutive law and (2) the problem is formulated as a mathematical programming problem with inequality constraints from which a linear complementarity problem is derived and solved by the Lemke algorithm. To obtain the nonlinear equilibrium paths, the Newton–Raphson algorithm is used together with path-following strategies. Plate–foundation interaction leads to interesting deformation sequences, characterized by the variation of the contact and noncontact zones along the postbuckling path, leading sometimes to sudden changes in the deformation pattern. The results have a remarkable dependence on the plate aspect ratio, foundation stiffness, and buckling shape. The effects of geometric imperfections on the nonlinear response of the plate are also investigated. From these results, a number of insightful conclusions regarding the behavior of such plate–foundation systems are drawn.     

Analysis of Hybrid Laminated Composite Plates

Hybrid laminated composite plates are analyzed using a nine‐noded isoparametric plate finite element based on Mindlin's theory. The shear flexibility is included in the finite element modeling. Shear flexibility is of importance, especially when different materials are used in the laminate design. Hybrid laminates consisting of graphite∕epoxy and kevlar∕epoxy plies are considered for illustration. The study indicates that hybrid laminates provide stiffnesses that are intermediate to the values obtained for single‐material laminates. The minimum deflection is achieved at different fiber orientation for thick plates compared to thin plates. The deflection behavior of hybrid laminates seems to be less affected by outer‐ply stiffness in the case of thick plates. Thick plates show less variation in the first natural frequency with fiber orientation but hybridization changes the natural frequency considerably. The first natural frequency of the hybrid laminate can be made higher than the stiffer single‐material laminate.     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

Free-Vibration Analysis and Material Constants Identification of Laminated Composite Sandwich Plates

Free vibration of symmetrically laminated composite sandwich plates with elastic edge restraints is studied via the Rayleigh–Ritz approach. The proposed Rayleigh–Ritz method is constructed on the basis of the layer-wise linear displacement theory. The accuracy of the method in predicting natural frequencies of composite sandwich plates with different boundary conditions is verified by the results reported in the literature or the experimental data obtained in this study. The proposed method is then applied to the material constant identification of free composite sandwich plates using the first six theoretical natural frequencies of the plates. In the identification process, trial material constants are used in the present method to predict the theoretical natural frequencies, a frequency discrepancy function is established to measure the sum of the squared differences between the experimental and theoretical natural frequencies, and a stochastic global minimization algorithm is used to search for the best estimates of the material constants by making the frequency discrepancy function a global minimum. Applications of the material constant identification technique are demonstrated by means of several examples.     

Novel Formulation to Study Thermal Postbuckling of Circular Plates with Edges Elastically Restrained against Rotation

A novel formulation is used to study the thermal postbuckling behavior of circular plates, with the edges supported to not have lateral deflection and elastically restrained against rotation. The elastic restraint is mathematically represented by an elastic rotational spring. The circular plate is subjected to a uniform edge compressive radial load, developed because of a uniform temperature rise. The formulation is on the basis of on the radial tensile load developed in the plate because of the large deflections of the plate with edges immovable in the plane normal to the edge and the linear buckling load corresponding to the uniform edge radial compressive load. The developed radial tensile load is obtained by using Berger’s approximation. The numerical results obtained from the present investigation in terms of the ratios of the postbuckling to the buckling loads for several rotational spring stiffness values compare well with those obtained by using the versatile finite-element analysis.     

Higher-Order Finite Strip Method for Postbuckling Analysis of Imperfect Composite Plates

Postbuckling analysis is essential to predict the capacity of composite plates carrying considerable additional load before the ultimate load is reached, and manufacturing-induced geometric imperfections often reduce the load-carrying capacity of composite structures. A higher-order finite strip method based on the higher-order shear deformation plate theory is developed for postbuckling analysis of laminated composite plates with initial geometric imperfection subjected to progressive end shortening. The arbitrary nature of initial geometric imperfection induced during manufacturing is accounted for in the analysis. Nonlinear equilibrium equations are solved by a Newton-Raphson procedure. Examples of postbuckling analyses of unsymmetric cross-ply, angle-ply, and arbitrary laminates are presented, and the accuracy and performance of the method are examined. The numerical higher-order finite strip method presented can be used as an accurate and efficient tool for postbuckling analysis of imperfect composite plates.     

Postbuckling of Pressure-Loaded Functionally Graded Cylindrical Panels in Thermal Environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to lateral pressure in thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher-order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. The two straight edges of the panel are assumed to be simply supported and two curved edges are either simply supported or clamped. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflection in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of simply supported, pressure-loaded, perfect and imperfect, functionally graded cylindrical panels with two constituent materials under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections, are studied.     

Thermal Buckling Analysis of SMA Fiber-Reinforced Composite Plates Using Layerwise Model

Thermal buckling analysis of laminated smart composite plates subjected to uniform temperature distribution has been presented. Shape memory alloy (SMA) fibers whose material properties depend on temperature have been used as a smart material. A three-dimensional layerwise plate model has been employed in developing the system equations using variational approach. Finite-element method has been adopted for discretization of the laminate. Lagrangian interpolation functions have been used to approximate the displacement components along the thickness as well as in the in-plane direction. The actual variation of prebuckling stresses has been accounted for in the derivation of the geometric stiffness matrix of the laminates. An incremental load technique has been used in the analysis to take into account the nonlinearity in the material properties of the SMA arising due to their temperature dependence. The effects of thickness ratio, orthotropic ratio, fiber orientation, aspect ratio, stacking sequence and various boundary conditions on the critical buckling temperature have been examined in details. The results have been validated with those available in the literature.     

Postbuckling of Shear Deformable Laminated Cylindrical Shells

A postbuckling analysis is presented for a shear deformable laminated cylindrical shell of finite length subjected to compressive axial loads. The governing equations are based on Reddy’s higher-order shear deformation shell theory with a von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, cross-ply laminated cylindrical shells. The effects of transverse shear deformation, shell geometric parameters, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Natural Vibrations of Laminated and Sandwich Plates

An exact analytical solution based on the propagator matrix method and a semianalytical solution based on a higher-order mixed approach (displacement and stress interpolation) have been presented in this paper to evaluate the natural frequencies as well as the stress and displacement mode shapes of simply supported, cross-ply laminated and sandwich plates. Continuity of the transverse stresses and displacements has been maintained at the laminae interfaces. Results have been presented for orthotropic plates, symmetric as well as nonsymmetric cross-ply composite and sandwich laminates. Results from the propagator matrix agree well with the published results for frequencies as well as displacement and stress mode shapes. Furthermore, the frequencies and displacement and stress eigenvectors obtained from the proposed layerwise mixed method are in excellent agreement with those obtained by three-dimensional elasticity theory. Results obtained from the present equivalent single layer theory are in good agreement with those obtained from the displacement based higher order methods. The high accuracy of the present methods is further confirmed by comparing the response of a sandwich plate with significantly different layer properties for which the conventional displacement based formulations yield inaccurate solutions.     

Thermomechanical Postbuckling Analysis of Cross-Ply Laminated Cylindrical Shell Panels

The postbuckling analysis of symmetric and antisymmetric cross-ply laminated cylindrical shell panels subjected to thermomechanical loading is examined in this paper. The formulation is based on an extension of Reissner’s shallow shell simplifications and accounts for parabolic distribution of transverse shear strains. Adopting a multiterm Galerkin’s method, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations. The nonlinear equilibrium paths through limit points are traced using the Newton–Raphson method in conjunction with Riks approach. Numerical results are presented for symmetric [?start0/90/0end?] and antisymmetric [?start0/90end?] cross-ply laminated cylindrical shell panels, that illustrate the influence of mechanical edge loads, lateral distributed load, initial imperfection, and temperature field on the limit loads and snap-through behavior.     

Buckling of Simply Supported Rectangular Reissner–Mindlin Plates Subjected to Linearly Varying In-Plane Loading

This paper is concerned with the buckling analysis of simply supported rectangular Reissner–Mindlin plates subjected to linearly varying edge loads. An analytical solution is developed and the effect of load intensity variation on the critical load is investigated. The solution is verified with the commercial computer code ANSYS. It is observed that the results of the present solution are in excellent agreement with those of ANSYS. The inaccuracy of the currently cited data in some design handbooks is highlighted.     

Electroelastic Response of a Laminated Composite Plate with Piezoelectric Sensors and Actuators

The paper deals with the fully coupled response characteristics of a multilayered composite plate with piezoelectric layers. The response quantities of the plate are coupled by the mechanical field and the electric field. Based on the three-dimensional linear piezoelectricity and the first-order shear deformation theory, the fundamental unknowns, such as the displacements and the electric potential, are assumed to be expandable through the plate thickness coordinate. The governing equations of motion of the plate are presented in terms of the unknown displacement and electrical potential coefficients. When the boundary conditions and electromechanical inputs are specified, the double Fourier series is used to obtain the response of the simply supported multilayered plates. Numerical results for the static and dynamic response of the laminated composite plates with different lamination schemes and having a PIC-151 piezoelectric material layer are obtained. The effects of the plate thinness ratio, plate aspect ratio, lamination scheme, fiber orientations, and piezoelectric coupling on the static and dynamic response are presented.