Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.     

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.     

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.     

Effects of Boundary Conditions in Laminated Composite Plates Using Higher Order Shear Deformation Theory

This paper extends the applicability of a modified higher order shear deformation theory to accurately determine the in-plane and transverse shear stress distributions in an orthotropic laminated composite plate subjected to different boundary conditions. A simpler, two-dimensional, shear deformable, plate theory accompanied with an appropriate set of through-thickness variations, is used to accurately predict transverse shear stresses. A finite element code was developed based on a higher order shear deformation theory to study the effects of boundary conditions on the behavior of thin-to-thick anisotropic laminated composite plates. The code was verified against three dimensional elasticity results. The study also compared the stresses and deformation results of higher order theory with those obtained using commercial software such as LUSAS, ANSYS and ALGOR. The commercial software are heavily used by designers to design various components/products made of composites. Various combinations of fixed, clamped and simply supported boundary conditions were used to verify a large class of anticipated applications. Results obtained from software are in good agreement for some cases and significantly differ for others. It was found that LUSAS and ANSYS yield better results for transverse deflection and in-plane stresses. But for transverse shear stresses, it is highly dependent on boundary conditions.     

Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings

The objective of this paper is to present an equivalent single-layer shear deformation theory for evaluation of displacements and stresses of cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load. A trigonometric shear deformation theory is used. The in-plane displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Stresses and displacements for orthotropic, two-layer antisymmetric, and three-layer symmetric square cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load are obtained. Numerical results of the present theory for displacement and thermal stresses are compared with those of classical, first-order and higher-order shear deformation plate theories.     

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. Material properties of functionally graded plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as Pasternak foundation. Equations of motion are derived using Hamilton’s principle. Closed-form solution of rectangular plates is derived, and the obtained results are compared well with three-dimensional elasticity solutions and third-order shear deformation theory solutions. Finally, the influences of power law index, thickness ratio, foundation parameter, and boundary condition on the natural frequency of plates have been investigated.     

A Layerwise Shear Deformation Theory for Two-Layered Cross-Ply Laminated Plates

A layerwise trigonometric shear deformation theory for flexural analysis of two-layered laminated plates, taking into account transverse shear deformation effects, is presented. The present theory has only three variables, that is, two variables less than those in the first-order shear deformation theory. The displacement field uses a sinusoidal function in terms of thickness coordinate to represent the shear deformation. The noteworthy feature of the theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with reasonable accuracy, satisfying the shear stress free surface conditions at the top and bottom surfaces of the plate and continuity conditions at interface between the layers. The transverse shear stresses can also be obtained, with better accuracy, by integrating equilibrium equations. The theory obviates the need for a shear correction factor. The governing equations and boundary conditions are obtained using the principle of virtual work. A two-layered cross-ply laminated plate is considered for the numerical study to demonstrate the efficacy of the theory. The results obtained using the present theory are discussed critically with those of other theories and are found to agree well with the exact elasticity results.     

On the simple and mixed first-order theories for plates resting on elastic foundations

This article investigates the bending response of an orthotropic rectangular plate resting on two-parameter elastic foundations. Analytical solutions for deflection and stresses are developed by means of the simple and mixed first-order shear deformation plate theories. The present mixed plate theory accounts for variable transverse shear stress distributions through the thickness and does not require a shear correction factor. The governing equations that include the interaction between the plate and the foundations are obtained. Numerical results are presented to demonstrate the behavior of the system. The results are compared with those obtained in the literature using three-dimensional elasticity theory or higher-order shear deformation plate theory to check the accuracy of the simple and mixed first-order shear deformation theories.     

Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.     

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.     

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

In this study, a simple C₀ isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C₀ formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.     

A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels

Based on the global-local superposition technique 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.], a global-local higher order laminated shell model is proposed for predicting both displacement and stress distributions through the thickness of laminated shells. This shell model satisfies transverse shear stress continuity conditions at interfaces as well as free surface conditions of transverse shear stresses. The merit of this model is that transverse shear stresses can be accurately predicted directly from constitutive equations without smoothing techniques. Cylindrical bending of laminated and sandwich shell panels is chosen to assess the present model wherein the results from several 2D laminated shell models and three-dimensional elasticity solution are available for comparison. In addition, thermal bending and thermal expansion of laminated cylindrical shell panels are also considered in this paper.     

EFFECT OF MESH DISTORTION ON THE ACCURACY OF TRANSVERSE SHEAR STRESSES AND THEIR SENSITIVITY COEFFICIENTS IN MULTILAYERED COMPOSITES

SUMMARY

A study is made of the effect of mesh distortion on the accuracy of transverse shear stresses and their first-order and second-order sensitivity coefficients in multilayered composite panels subjected to mechanical and thermal loads. The panels are discretized by using a two-field degenerate solid element, with the fundamental unknowns consisting of both displacement and strain components, and the displacement components having a linear variation throughout the thickness of the laminate. A two-step computational procedure is used for evaluating the transverse shear stresses. In the first step, the in-plane stresses in the different layers are calculated at the numerical quadrature points for each element. In the second step, the transverse shear stresses are evaluated by using piecewise integration, in the thickness direction, of the three-dimensional equilibrium equations. The same procedure is used for evaluating the sensitivity coefficients of transverse shear stresses. Numerical results are presented showing no noticeable degradation in the accuracy of the in-plane stresses and their sensitivity coefficients with mesh distortion. However, such degradation is observed for the transverse shear stresses and their sensitivity coefficients. The standard of comparison is taken to be the exact solution of the three-dimensional thermoelasticity equations of the panel.     

Comparison of 2D finite element modeling assumptions with results from 3D analysis for composite skin-stiffener debonding

The influence of two-dimensional finite element modeling assumptions on the debonding prediction for skin-stiffener specimens was investigated. Geometrically nonlinear finite element analyses using two-dimensional plane-stress and plane-strain elements as well as three different generalized plane-strain type approaches were performed. The computed skin and flange strains, transverse tensile stresses and energy release rates were compared to results obtained from three-dimensional simulations. The study showed that for strains and energy release rate computations the generalized plane-strain assumptions yielded results closest to the full three-dimensional analysis. For computed transverse tensile stresses the plane-stress assumption gave the best agreement. Based on this study it is recommended that results from plane-stress and plane-strain models be used as upper and lower bounds. The results from generalized plane-strain models fall between the results obtained from plane-stress and plane-strain models. Two-dimensional models may also be used to qualitatively evaluate the stress distribution in a ply and the variation of energy release rates and mixed mode ratios with delamination length. For more accurate predictions, however, a three-dimensional analysis is required.     

INTERLAMINAR STRESSES IN SHELLS OF REVOLUTION

SUMMARY

A comparative study of the interlaminar stresses in shells of revolution has been made between first order shear deformation theory (FSDT), higher order shear deformation theory with thickness stretch (HSDT7), higher order shear deformation theory with higher order inplane displacement terms (HSDT9) and three-dimensional finite element (3D) models. A semi-analytical approach is used for all the models. Interlaminar stresses are evaluated using equilibrium equations in the cases of FSDT, HSDT7 and HSDT9 models as interlaminar stresses obtained from constitutive equations are not correct. For the 3D model an eight-noded quadratic quadrilateral semi-analytic solid element is used whereas for equivalent single layer (ESL) theories a three-noded isoparametric curved element is used. Crossply parabolic and hyperbolic caps subjected to uniform external pressure and a simply supported cylindrical shell subjected to an internal sinusoidal pressure are considered in the present study.     

A state of stress and displacement of elastic plates using simple and mixed shear deformation theories

This paper presents accurate two-dimensional solutions for bending response of four types of single-layer orthotropic rectangular plates. The plates considered are of the type having two opposite sides simply-supported, and two other sides having combinations of simply-supported, clamped, and free-boundary conditions. Analytical solutions for deflections and stresses of rectangular plates are developed by means of the simple (SFPT) and mixed (MFPT) first-order shear deformable plate theories. The present MFPT not only shows improvement on predicting frequencies, critical buckling loads, deflections and in-plane stresses, but also accounts for variable transverse shear stress distributions through the thickness. This puts into evidence the important role played by MFPT in the modeling of homogeneous plate theories, which in contrast to SFPT does not require the incorporation of a shear-correction factor. For illustrative purposes, sample free vibration, stability, and bending problems for simply supported orthotropic plates are considered and comparisons of the obtained results are made with the exact and higher-order shear deformation theory results given in the literature.     

Further results concerning the refined theory of anisotropic laminated composite plates

A simple refined discrete-layer theory of anisotropic laminated composite plates is substantiated. The theory is based on the assumption of a piecewise linear variation of the in-plane displacement components and of the constancy of the transverse displacement throughout the thickness of the laminate. This plate model incorporates transverse shear deformation, dynamic and thermal effects as well as the geometrical non-linearities and fulfills the continuity conditions for the displacement components and transverse shear stresses at the interfaces between laminae. As it is shown in the paper, the refinement implying the fulfillment of continuity conditions is not accompanied by an increase of the number of independent unknown functions, as implied in the standard first order transverse shear deformation theory. It is also shown that the within the framework of the linearized static counterpart of the theory, several theorems analogous to the ones in the 3-D elasticity theory could be established. These concern the energetic theorems, Betti's reciprocity theorem, the uniqueness theorem for the solutions of boundary-value problems of elastic composite plates, etc. Finally, comparative remarks on the present and standard first order transverse shear deformation theories are made and pertinent conclusions about its usefulness and further developments are outlined.     

Bending of cross-ply laminated composites: An accurate and efficient plate theory based upon models of Lekhnitskii and Ren

Bending laminated composites results in a distinctive zig-zag shaped deformation pattern, accordingly jumping transverse shear strains at layer interfaces, but continuous courses of transverse shear stresses there. An accurate representation of this laminate-specific mechanical behavior in terms of plate theories is challenging, even more if computational efficiency is aimed for. Here, an axiomatic equivalent single layer plate theory for cross-ply laminated composites is presented, which is based on the work of Lekhnitskii and Ren and delivers accurate deformation and stress prognoses at the cost of six solution variables. Fulfilling transverse stress continuity, the infinitesimal equilibrium equations are considered in order to derive an appropriate ansatz for the transverse shear stresses including the influence of all plane stress reduced stiffness components. However, the effect of the normal stress σ_zz is neglected, and deflection w is assumed constant across the plate thickness. The equilibrium equations and corresponding boundary conditions of the plate theory are derived by application of the principle of virtual displacements. Numerical results for symmetrical and non-symmetrical composites as well as for typical sandwich plates obtained by the present theory show good agreement with corresponding exact elasticity solutions given by Pagano, even for thick plates.     

A refined higher-order composite box beam theory

A three-dimensional theory is developed to model composite box beams with arbitrary wall thicknesses. The theory, which is based on a refined displacement field, approximates the three-dimensional elasticity solution so that the beam cross-sectional properties are not reduced to one-dimensional beam parameters. Both in-plane and out-of-plane warping are included automatically in the formulation. The model can accurately capture the transverse shear stresses through the thickness of each wall while satisfying stress-free boundary conditions on the inner and outer surfaces of the beam. Numerical results are presented for beams with varying wall thicknesses and aspect ratios. The static results are correlated with available experimental data and show excellent agreement. Results presented for thick-walled box beams show the importance of including transverse shear in the formulation and the difficulty of defining a ‘beam’ twist for the entire cross-section.     

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

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells 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 continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.