Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory

Natural frequencies and buckling stresses of cross-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the lowest natural frequency and buckling stress are examined in detail. Numerical results are compared with those of the published existing theories and FEM solutions. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. It is noticed that the present global higher-order approximate theories can predict the natural frequencies, buckling stresses and stresses of thick multilayered composite laminates as accurately as three-dimensional solutions.     

Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory

A two-dimensional global higher-order deformation theory is presented for thermal buckling of cross-ply laminated composite and sandwich plates. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through the principle of virtual work. Several sets of truncated Mth-order approximate theories are applied to solve the eigenvalue problems of a simply supported multilayered plate. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. Numerical results are compared with those of the published three-dimensional layerwise theory in which both in-plane and normal displacements are assumed to be C⁰ continuous in the continuity conditions at the interface between layers. Effects of the difference of displacement continuity conditions between the three-dimensional layerwise theory and the global higher-order theory are clarified in thermal buckling problems of multilayered composite plates.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

An improved in-plane displacement model for the stability analysis of laminated composites with general lamination configurations

This paper describes how the derivatives of lateral displacement are eliminated in the general displacement field of the global–local higher-order theory for stability analysis of laminated composite and sandwich plates with general lamination configurations. In contrast to previous models, the present model is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. Based on known traction forces on the surface boundaries, the derivatives of transverse displacement have been taken out from the general displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To verify the proposed theory, the classical quadratic six-node C⁰ triangular element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Numerical results show that following the proposed theory simple C⁰ finite elements could accurately predict the critical loads of sandwich plate with soft core, which has long been a difficult case for the other global higher-order theories.     

Buckling Analysis of Angle-ply Composite and Sandwich Plates by Combination of Geometric Stiffness Matrix

Buckling response of angle-ply laminated composite and sandwich plates are analyzed using the global-local higher order theory with combination of geometric stiffness matrix in this paper. This global-local theory completely fulfills the free surface conditions and the displacement and stress continuity conditions at interfaces. Moreover, the number of unknowns in this theory is independent of the number of layers in the laminate. Based on this global-local theory, a three-noded triangular element satisfying C¹ continuity conditions has also been proposed. The bending part of this element is constructed from the concept of DKT element. In order to improve the accuracy of the analysis, a method of modified geometric stiffness matrix has been introduced. Numerical results show that the present theory not only computes accurately the buckling response of general laminated composite plates but also predicts the critical buckling loads of soft-core sandwiches. However, the global higher-order theories as well as first order theories might encounter some difficulties and overestimate the critical buckling loads for soft-core sandwich plates.     

An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams

Several displacement-based theories are assessed by analyzing the free vibration and the buckling behaviors of laminated beams with arbitrary layouts as well as soft-core sandwich beams. The equations governing the dynamic response of laminated structures are derived by using Hamilton’s principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier’s technique and solving the eigenvalue equations, analytical solutions based on the global–local higher-order theory used in this paper are first presented in present study. At the same time, the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse shear stress on the global response of both laminated beams and soft-core sandwiches has been also studied. Numerical results show that by increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies and the critical loads of laminated beams with arbitrary layouts and soft-core sandwich beams whereas these global higher-order theories are still less accurate compared to the global–local higher-order theory and the zig-zag theory used herein.     

A C-type higher-order theory for bending analysis of laminated composite and sandwich plates

In this paper, a C⁰-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C¹ interpolation functions. To assess the developed theory, the C¹-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.     

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.     

Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds: Beams,plates and solids

This paper presents an efficient method for predicting the free and transient vibrations of multilayered composite structures with parallelepiped shapes including beams, plates and solids. The exact three-dimensional elasticity theory combined with a multilevel partitioning hierarchy, viz., multilayered parallelepiped, individual layer and layer segment, is employed in the analysis. The continuity constraints on common interfaces of adjacent layer segments are imposed by a modified variational principle, and the displacement components of each layer segment are assumed in the form of orthogonal polynomials and/or trigonometric functions. Numerical studies are given for free vibrations of composite laminated and sandwich beams, plates, and solids. Some in-plane shear vibration modes missed in previous elasticity solutions for multilayered plates are examined. The natural frequencies derived from Reddy’s high-order shear deformation theory and layerwise theory for soft-core sandwich plates show significant deviation from elasticity solutions. Transient displacement and stress responses for angle-ply laminated and sandwich plates under four types of impulsive loads (including rectangular, triangular, half-sine and exponential pulses) are obtained by the Newmark integration procedure. The present solutions may serve as benchmark data for assessing the accuracy of advanced structural theories and new developments in numerical methods.     

A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis

This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C¹ continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories.     

An accurate higher-order theory and C finite element for free vibration analysis of laminated composite and sandwich plates

At present, it is difficult to accurately predict natural frequencies of sandwich plates with soft core by using the C⁰ plate bending elements. Thus, the C¹ plate bending elements have to be employed to predict accurately dynamic response of such structures. This paper proposes an accurate higher-order C⁰ theory which is very different from other published higher-order theory satisfying the interlaminar stress continuity, as the first derivative of transverse displacement has been taken out from the in-plane displacement fields of the present theory. Therefore, the C⁰ interpolation functions is only required during its finite element implementation. Based on the Hamilton’s principle and Navier’s technique, analytical solutions to the natural frequency analysis of simply-supported laminated plates have been presented. To further extend the ranges of application of the proposed theory, an eight-node C⁰ continuous isoparametric element is used to model the proposed theory. Numerical results show the present C⁰ finite element can accurately predict the natural frequencies of sandwich plate with soft core, whereas other global higher-order theories are unsuitable for free vibration analysis of such soft-core structures.     

Bending of sandwich plates with anti-symmetric angle-ply face sheets – Analytical evaluation of higher order refined computational models

The aim of the present study is to assess the accuracy of the few computational models based on various shear deformation theories in predicting the bending behaviour of sandwich plates with anti-symmetric angle-ply face sheets under static loading. Five two-dimensional models available in the literature are used for the present evaluation. The performance of the various models is evaluated on a simply supported laminated plate under sinusoidal loading. The equations of equilibrium are derived using the principle of minimum potential energy (PMPE). Analytical solution method using double Fourier series approach is used in conjunction with the admissible boundary conditions. The accuracy of each model is established by comparing the results of composite plates with the exact solutions already available in the literature. After establishing the correctness of the theoretical formulations and the solution method, benchmark results for transverse displacement, in-plane stresses, moment and shear stress resultants are presented for the multilayer sandwich plates.     

Post-buckling analysis of imperfect laminated plates under combined axial and thermal loads

A post-buckling analysis of composite laminated plates subjected to combined axial compression and uniform temperature loading is presented. The two cases of thermal post-buckling of initially compressed plates and of compression post-buckling of initially heated plates are considered. The initial geometrical imperfection of the plates is taken into account. The analysis uses a perturbation technique to determine buckling loads and post-buckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. Typical results are presented in dimensionless graphical form.     

A new local high-order laminate theory

A new local high-order deformable theory of laminated composite/sandwich plates is presented here. The displacement fields of each discrete layer were assumed in the present theory to be of a high-order polynomial series through layer-thickness. The displacement and traction continuity conditions at the interface between layers and the traction conditions at the outer surfaces were imposed as the constraint conditions, and introduced into the potential energy functional by the Lagrange multiplier method. The equations of motion and admissible boundary conditions were given on the basis of the present theory by using the generalized variational principle. Pagano's 3-D elasticity solutions of generally rectangular laminated composite/sandwich plates, fully simply supported, subjected to transverse sinusoidal loading were used for assessment of the present theory and other theories discussed in previous literature. The present theory was found to agree very closely with 3-D elasticity solutions.     

Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

Natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations and rotatory inertia. The modulus of elasticity of the plates is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded (FG) plates is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Critical buckling stresses of FG plates subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG plates without in-plane stresses is presented. The distributions of modal displacements and modal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2-D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG plates.     

A new shear and normal deformation theory for isotropic,transversely isotropic,laminated composite and sandwich plates

In the present study, a sinusoidal shear and normal deformation theory taking into account effects of transverse shear as well as transverse normal is used to develop the analytical solution for the bidirectional bending analysis of isotropic, transversely isotropic, laminated composite and sandwich rectangular plates. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and traction free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. The displacement field uses sinusoidal function in terms of thickness coordinate to include the effect of transverse shear and the cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal. The kinematics of the present theory is much richer than those of the other higher order shear deformation theories, because if the trigonometric term is expanded in power series, the kinematics of higher order theories are implicitly taken into account to good deal of extent. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. The Navier solution for simply supported laminated composite plates has been developed. Results obtained for displacements and stresses of simply supported rectangular plates are compared with those of other refined theories and exact elasticity solution wherever applicable.     

Analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sandwich plates

Analytical formulations and solutions to the static analysis of simply supported anti-symmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher-order refined theory already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations, which account for the effect of transverse shear deformation and a non-linear variation of in-plane displacements with respect to the thickness coordinate. The transverse displacement is assumed to be constant throughout the thickness. The equations of equilibrium are obtained using principle of minimum potential energy. Solutions are obtained in closed form using Navier's technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already reported in the literature. After establishing the accuracy of the solutions, numerical results with real properties are presented for the multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

Analysis of laminated composite plates using a higher-order shear deformation theory

A higher-order shear deformation theory is used to analyse laminated anisotropic composite plates for deflections, stresses, natural frequencies and buckling loads. The theory accounts for parabolic distribution of the transverse shear stresses, and requires no shear correction coefficients. A displacement finite element model of the theory is developed, and applications of the element to bending, Vibration and stability of laminated plates are discussed. The present solutions are compared with those obtained using the classical plate theory and the three-dimensional elasticity theory.     

Effects of higher-order deformations on in-plane vibration and stability of thick circular rings

Summary The effects of higher-order deformations on natural frequencies and buckling stresses of a thick circular ring with rectangular cross-sections subjected to circumferential tensile and/or compressive stresses are studied. Based on the power series expansion of displacement components, a set of fundamental dynamic equations of a one-dimensional higher-order ring theory is derived through Hamilton's sprinciple. Several sets of truncated approximate theories which can take into account the complete effects of higher-order deformations such as shear deformation and depth change and rotary inertia are applied to solve the eigenvalue problems of a thick circular ring. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress for the flexural and extensional displacement modes of thick rings are examined in detail. It seems that the present approximate theories can predict benchmark data of the natural frequency and buckling stress of thick rings more accurately compared to other existing theories.     

具有局部分层的对称铺设层合板在非保守力作用下的屈曲分析

利用弹性非保守系统自激振动的拟固有频率变分原理,推导出复合材料矩形板受非保守随从力作用的变分方程,进而导出复合材料层合板含分层损伤的有限元基本方程及求解临界荷载力和固有频率的特征方程。用载荷增量法计算了不同边长比的复合材料矩形板在面内受随从力作用且含有分层的临界载荷,分析了不同角铺设方向对临界载荷的影响。分析表明,分层对临界载荷有一定影响,复合材料层合板的角铺设方向对临界载荷有较大影响。