A general laminated plate theory accounting for continuity of displacements and transverse shear stresses at material interfaces

A general two-dimensional theory, suitable for the static and/or the dynamic analysis of transverse shear deformable laminated plates, is presented. This displacement-based theory is capable of satisfying continuity of both displacements and transverse shear stresses at the plate material interfaces. The derivation of its governing differential equations is based on the application of Hamilton's principle in conjunction with the method of Lagrange multipliers. Moreover, this new theory is capable of accounting for unlimited multiple choices of continuous displacement distributions, through the plate thickness, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. With such a double-infinite freedom, it is concluded that for the analysis of the particular laminated plate considered one may start with the solution of the governing equations of the 5-degrees-of-freedom theory derived for relatively simple choices of through-the-thickness displacement distributions. Then, either increasing the number of the degrees of freedom or reforming, suitably, the aforementioned displacement distributions, one may iteratively improve the efficiency of the theory until a sufficient degree of accuracy is achieved for the results obtained.     

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.     

Analytical solution for bending of cross-ply laminated plates under thermo-mechanical loading

The static thermo-elastic response of symmetric and anti-symmetric cross-ply laminated plates has been investigated by the use of a unified shear deformation plate theory. The present plate theory enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. The validity of the present theory is demonstrated by comparison with solutions available in the literature. A wide variety of results is presented for the static response of simply supported rectangular plates under non-uniform sinusoidal mechanical and/or thermal loadings. The influence of material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the thermally induced response is studied.     

Higher order zig-zag plate theory under thermo-electric-mechanical loads combined

A higher order zig-zag plate theory is developed to refine the predictions of the mechanical, thermal, and electric behaviors partially coupled. The in-plane displacement fields are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses. Artificial shear correction factors are not needed in the present formulation. Thus the proposed theory has only seven primary unknowns and they do not depend upon the number of layers. Through the numerical examples of partially coupled analysis, the accuracy and efficiency of the present theory are demonstrated. The present theory is suitable in the predictions of deformation and stresses of thick smart composite plate under mechanical, thermal, and electric loads combined.     

A cubic triangular finite element for flat plates with shear

This paper presents the development of a straightforward displacement type triangular finite element for bending of a flat plate with the inclusion of transverse (or lateral) shear effects. The element has twenty two degrees of freedom consisting of ten for the lateral displacement of the midplane and six for rotations of the normal to the undeformed midplane of the plate. The latter are taken as independent of the slopes of the deformed midplane in order to include deformation due to transverse shear. The element is fully conforming and may be orthotropic. At interelement boundaries, the element matches adjacent elements both with respect to lateral displacement of the midplane and the rotations of the normal. The result is an efficient ‘linear moment’ triangular element but with transverse shear deformation included. Numerical computations for a number of examples are presented. The results show the element to be more flexible than most other finite element models and agree closely with those from a numerical solution of the three dimensional elasticity equations. The results also converge to those from thin plate theory when the thickness to length ratio becomes small or when the transverse shear moduli are artificially increased.     

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.     

Finite element analysis of laminated composite plates using zeroth-order shear deformation theory

In this paper a generalized finite element model is developed for static and dynamic analyses of laminated composite plates using zeroth-order shear deformation theory (ZSDT). The theory ensures the parabolic distribution of transverse shear stresses across the plate thickness. A four-noded plate element is considered in this model and the generalized nodal variables are expressed using Lagrangian linear interpolation functions and Hermitian cubic interpolation functions. The solutions of the finite element model have been compared with the existing solutions for symmetric and antisymmetric laminated composite plates. The comparison confirms that the ZSDT can be efficiently used for finite element analysis of both thin and thick plates with high accuracy.     

A higher-order finite element for analysis of composite laminated structures

A new six-node higher-order triangular composite layered shell finite element with six degrees of freedom at each node is presented. With respect to the inplane variables, the in-plane and the out-of-plane displacement fields of the element are quadratic and cubic respectively. By using Utku's method (AFFDL-TR-71-160, Air Force Third Conf., Wright Paterson, Ohio, 1971), the transverse shear strain energy is computed directly from the displacement field rather than from the stress couple field. Some typical bending problems for composite laminated beams and plates with different stack sequences are analyzed. Excellent agreements are obtained when compared to the exact solutions, the first order shear deformation theory (FSDT), the higher order shear deformation theory (HSDT) and some other existing finite element models. ‘Shear locking’ is avoided when the plate is thin.     

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.     

Error estimates for a sixth-order theory of plate bending

A refined linear theory for the bending of anisotropic, homogeneous plates which takes account of transverse shear deformation and transverse normal stress is rigorously validated by imbedding it in the linear theory of elasticity. Three-dimensional displacement and stress fields are constructed from the two-dimensional plate theory and shown by the hypersphere theorem to approximate exact elasticity solutions with a relative mean square error proportional to the plate thickness cubed. This improves previous estimates for sixth-order theories involving error bounds proportional to the square of thickness.     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates

In this article, a new five-variable refined plate theory for the free vibration analysis of functionally graded sandwich plates is developed. By dividing the transverse displacement into bending, shear, and thickness stretching parts, the number of unknowns and governing equations of the present theory is reduced, and hence, makes it simple to use. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or more in the case of other shear and normal deformation theories. The theory accounts for hyperbolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using a shear correction factor. Two common types of functionally graded material (FGM) sandwich plates, namely, the sandwich with FGM facesheet and homogeneous core and the sandwich with homogeneous facesheet and FGM core, are considered. The equations of motion are obtained using Hamilton's principle. Numerical results of the present theory are compared with three-dimensional elasticity solutions and other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and efficient in predicting the free-vibration response of functionally graded sandwich 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.     

A third-order triangular multilayered plate finite element with continuous interlaminar stresses

Based on a refined third-order shear deformation plate theory recently proposed by the author, a three-node, fully conforming, multilayered anisotropic plate element of arbitrary triangular shape is developed in this paper. The element incorporates 10 nodal d.o.f., namely the two in-plane displacements, the two shear rotations, the transverse displacement and its first and second derivatives, thus giving a total of 30 d.o.f. The formulation includes extension, bending and transverse shear deformation states; moreover, it fulfils a priori the geometric and stress continuity conditions at the interfaces, and it requires only five generalized displacements to describe the kinematics of the plate deformation. The formulated plate element is assessed for its performance comparing its predictions either with exact solutions from the plate model or with other approximate two-dimensional solutions.     

An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates

A Reissner mixed variational theorem (RMVT)-based third-order shear deformation theory (TSDT) is developed for the static analysis of simply-supported, multilayered functionally graded material (FGM) plates under mechanical loads. The material properties of the FGM layers are assumed to obey either the exponent-law distributions through the thickness coordinate or the power-law distributions of the volume fractions of the constituents. In this theory, Reddy’s third-order displacement model and the layerwise parabolic function distributions of transverse shear stresses are assumed in the kinematic and kinetic fields, respectively, a priori, where the effect of transverse normal stress is regarded as minor and thus ignored. The continuity conditions of both transverse shear stresses and elastic displacements at the interfaces between adjacent layers are then exactly satisfied in this RMVT-based TSDT. On the basis of RMVT, a set of Euler–Lagrange equations associated with the possible boundary conditions is derived. In conjunction with the method of variable separation and Fourier series expansion, this theory is successfully applied to the static analysis of simply-supported, multilayered FGM plates under mechanical loads. A parametric study of the effects of the material-property gradient index and the span-thickness ratio on the displacement and stress components induced in the plates is undertaken.     

An improved zig-zag model for the bending of laminated composite shells

A simple layerwise higher-order zig-zag model is proposed for the bending of laminated composite shells. The model provides a cubic variation of both the in-plane displacements and the transverse shear stresses within each layer. As the displacement model satisfies the zero transverse shear stress conditions at the free surfaces, there is no need for the use of shear correction factors. By imposing the continuity of the in-plane displacements and the transverse shear stresses at the interfaces, the number of variables is shown to be the same as that given by the first-order shear deformation shell theory, irrespective of the number of layers considered. For the sake of consistency, all terms of the order of the thickness coordinate-to-radius ratio have been retained in the derivation of the governing equations. Numerical results for the cylindrical bending of thick, symmetric homogeneous orthotropic and three-layer laminated shells under sinusoidal loading show that the maximum transverse deflections and in-plane stresses are in good agreement with available exact elasticity solutions for radius-to-thickness ratios greater than or equal to four.     

A New Locking Free Higher Order Finite Element Formulation for Composite Beams.

A refined 2-node, 7 DOF/node beam element formulation is presented in this paper. This formulation is based on higher order shear deformation theory with lateral contraction for axial-flexural-shear coupled deformation in asymmetrically stacked laminated composite beams. In addition to axial, transverse and rotational degrees of freedom, the formulation also incorporates the lateral contraction and its higher order counterparts as degrees of freedom. The element shape functions are derived by solving the static part of the governing equations. The element considers general ply stacking and the numerical results shows that the element exhibits super convergent property. The efficiency of the element in capturing both the static and dynamic inter-laminar stresses is demonstrated. The accuracy of the element to capture free vibration and wave propogation responses with small problem sizes is also demonstrated.     

Hygrothermal analysis of antisymmetric cross-ply laminates using a refined plate theory

The effect of hygrothermal conditions on the antisymmetric cross-ply laminates has been investigated using a unified shear deformation plate theory. The present plate theory enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions do not involve the undesirable implications of the transverse shear correction factors. The differential equations of laminated plates whose deformations are governed by either the shear deformation theories or the classical one are derived. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. A wide variety of results is presented for the static response of simply supported rectangular plates under non-uniform sinusoidal hygrothermal/thermal loadings. The influence of material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the hygrothermally induced response is studied.     

An efficient FE model based on combined theory for the analysis of soft core sandwich plate

An efficient C⁰ continuous finite element (FE) model is developed based on combined theory (refine higher order shear deformation theory (RHSDT) and least square error (LSE) method) for the static analysis of soft core sandwich plate. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with a different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements. In order to calculate the accurate through thickness transverse stresses variation, LSE method has been used at the post processing stage. The proposed combine model (RHSDT and LSE) is implemented to analyze the laminated composites and sandwich plates. Many new results are also presented which should be useful for future research.