Finite element analysis of laminated composite plates using a higher-order displacement model

A C° continuous displacement finite element formulation of a higher-order theory for flexure of thick arbitrary laminated composite plates under transverse loads is presented. The displacement model accounts for non-linear and constant variation of in-plane and transverse displacement model eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with nine degrees-of-freedom per node. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form, the theory of elasticity and the finite element solutions with another higher-order displacement model by the same authors. A computer program has been developed which incorporates the realistic prediction of interlaminar stresses from equilibrium equations.     

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.     

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

Abstract

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C⁰-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.     

Postbuckling analysis of stiffened laminated composite panels,using a higher-order shear deformation theory

The investigation aims at: (i) constructing a modified higher-order shear deformation theory in which Kirchhoff's hypotheses are relaxed, to allow for shear deformations; (ii) validating the present 5-parameter-smeared-laminate theory by comparing the results with exact solutions; and (iii) applying the theory to a specific problem of the postbuckling behavior of a flat stiffened fiber-reinforced laminated composite plate under compression.The first part of this paper is devoted mainly to the derivation of the pertinent displacement field which obviates the need for shear correction factors. The present displacement field compares satisfactorily with the exact solutions for three layered cross-ply laminates. The distinctive feature of the present smeared laminate theory is that the through-the-thickness transverse shear stresses are calculated directly from the constitutive equations without involving any integration of the equilibrium equations.The second part of this paper demonstrates the applicability of the present modified higher-order shear deformation theory to the post-buckling analysis of stiffened laminated panels under compression. to accomplish this, the finite strip method is employed. A C ²-continuity requirement in the displacement field necessitates a modification of the conventional finite strip element technique by introducing higher-order polynomials in the direction normal to that of the stiffener axes. The finite strip formulation is validated by comparing the numerical solutions for buckling problems of the stiffened panels with some typical experimental results.     

An evaluation of the edge solution for a higher-order laminated plate theory

An analytical study was conducted to determine the edge effects upon a higher-order laminated plate theory due to Touratier (Int. J. Engng Sci., 29 (1991) 911-16). The problem considered is a sandwich plate simply supported along two opposite edges and clamped along its two others. The plate is loaded in statics by a pressure upon its top surface and the load is ‘sinusoidal-uniform’.

Results are presented for the transverse displacement and the stresses, using the theories of Kirchhoff-Love, Touratier and three-dimensional finite element computations. The results indicate clearly the importance of including transverse shear effects. Analytical calculations from the higher-order plate theory are in good agreement with three-dimensional finite element computations. Near to the clamped edges of the sandwich plate, a particular effect is shown upon the in-plane stresses, i.e. the Kirchhoff-Love theory.     

Finite element analysis of stiffened laminated plates under transverse loading

A finite element model is developed to study the behavior of stiffened laminated plates under transverse loadings. Transverse shear flexibility is incorporated in both beam and plate displacement fields. A laminated plate element with 45 degrees of freedom is used in conjunction with a laminated beam element having 12 degrees of freedom for the bending analysis of eccentrically-stiffened laminated plates. The validity of the formulation is demonstrated by comparing with the available solutions in the literature. The numerical results are presented for eccentrically-stiffened layered plates having various boundary conditions and with stiffeners varying in number.     

An improved isoparametric quadratic element based on refined zigzag theory to compute interlaminar stresses of multilayered anisotropic plates

An improved eight-noded isoparametric quadratic plate bending element based on refined higher-order zigzag theory (RHZT) has been developed in the present study to determine the interlaminar stresses of multilayered composite laminates. The C⁰ continuous element has been formulated by considering warping function in the displacement field based on the RHZT. Shear locking phenomenon is avoided by considering substitute shear strain field. The continuity of transverse shear stresses cannot be ensured by the proposed zigzag formulation directly, and hence, the continuity conditions of transverse shear stresses have been established by using the three-dimensional (3D) stress equilibrium equations in the present study. The transverse shear stresses are computed in a simplified manner using the differential equations of stress equilibrium. A finite element code is developed by using MATLAB software package. The performance of the present finite element model is validated by comparing the results with 3D elasticity solutions. The superiority of the proposed element in view of computational efficiency, simplicity, and accuracy has been examined by comparing the present solutions with those available in published literature using other elements.     

A higher-order polynomial shear deformation theory for geometrically nonlinear free vibration response of laminated composite plate

In this paper, a nonlinear analysis for large amplitude free vibration of laminated composite plates is developed using higher-order shear deformation theory. The effect of all higher-order terms arising from nonlinear strain-displacement relations are included in the formulation and present plate theory exhibits traction-free surface of the laminated plate in von-Karman sense. A finite element procedure considering a C° continuous isoparametric nine-node rectangular element is implemented for nonlinear model. The accuracy of the theory is validated with some available theory for different aspect ratio, modular ratio, number of layers, ply orientations, etc. through some numerical examples.     

A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates

A C⁰-type global-local higher order theory including interlaminar stress continuity is proposed for the cross-ply laminated composite and sandwich plates in this paper, which is able to a priori satisfy the continuity conditions of transverse shear stresses at interfaces. Moreover, total number of unknowns involved in the model is independent of number of layers. Compared to other higher-order theories satisfying the continuity conditions of transverse shear stresses at interfaces, merit of the proposed model is that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To verify the present model, a C⁰ three-node triangular element is used for bending analysis of laminated composite and sandwich plates. It ought to be shown that all variables involved in present model are discretized by only using linear interpolation functions within an element. Numerical results show that the C⁰ plate element based on the present theory may accurately calculate transverse shear stresses without any postprocessing, and the present results agree well with those obtained from the C¹-type higher order theory. Compared with the C¹ plate bending element, the present finite element is simple, convenient to use and accurate enough.     

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.     

An adhesively laminated plate element for PZT smart plates

An adhesively laminated element taking into consideration peel stress is developed for a piezoelectric smart plate. In this novel finite element analysis formulation, a four node piezoelectric element is firstly derived, and an adhesive element of finite thickness with both shear and peel stiffness is sandwiched between two collocated four node plate elements to form an adhesively laminated element for a piezoelectric smart plate. In this framework of finite element analysis, because the displacement filed in this adhesively laminated element is continuous and a plate element is derived based on the Reissner–Mindlin plate theory, and thus it can be accurately applied to a thin or moderately thick host plate with bonded or debonded piezoelectric actuators and sensors. The formulation is performed for an isotropic host plate and a fiber reinforced laminate plate. Numerical results are presented to compare with those of the exact solutions for smart beams, and validate with the experimental results of the isotropic and composite host plates available in the literature. Using the present finite element analysis formulation, energy transfer stresses in the adhesive and equivalent forces induced in the host plate are investigated. The present formulation is demonstrated to allow debondings of piezoelectric patches and the debonding detection.The authors are grateful to the support of the Australian Research Council via a Discovery Projects grant (grant No: DP0346419).     

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.     

A partial hybrid stress model for the refined C1 higher‐order plate theory

Abstract

The partial hybrid stress model is applied to the refined C ¹ higher‐order plate theory in this paper. The displacement model is adopted in the flexural part and the hybrid stress model in the transverse shear part. The plate concept is introduced and the governing equations of plate are derived variationally from the modified Hellinger‐Reissner principle. This new plate element is demonstrated to be more accurate than displacement formulation in the analysis of orthotropic thick laminated plates. Moreover, the through thickness distribution of transverse shear stress is precisely predicted.     

Analysis of laminated composite beams using layerwise displacement theories

Within the displacement field of a layerwise theory, two laminated beam theories for beams with general lamination are developed. In the first theory, an existing layerwise laminated plate theory is adapted to laminated beams. The procedure used in the second theory is simple and straightforward and similar to the one used in the development of plate and shell theories. These theories can also be used in developing simpler theories such as classical, first, and higher-order shear deformation laminated beam theories. Equations of motions are obtained by using Hamilton’s principle. For the assessment of the accuracy of these theories, analytical solutions for static bending and free vibration are developed and compared with those of an existing three-dimensional elasticity solution of cross-ply laminates in cylindrical bending and with the three-dimensional finite element analysis for angle-ply laminates.     

A new shear flexible cubic spline plate element for vibration analysis

Here, a new cubic B‐spline plate element is developed using field consistency principle, for vibration analysis. The formulation includes anisotropy, transverse shear deformation, in‐plane and rotary inertia effects. The element is based on a laminated refined plate theory, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the plates. The lack of consistency in the shear strain field interpolations in its constrained physical limits produces poor convergence and results in unacceptable solutions due to locking phenomenon. Hence, numerical experimentation for the evaluation of natural frequencies of plates is carried out to check this deficiency with a series of assumed shear strain functions, redistributed in a field consistent manner. Copyright © 2002 John Wiley & Sons, Ltd.     

Buckling of Laminated Composite Plates by a New Element Based on Higher Order Shear Deformation Theory

The simple higher-order shear deformation theory proposed by Reddy has been successfully implemented in a triangular element recently developed by the authors. In this paper the element is applied to buckling of composite plates to study its performance. In this plate theory the transverse shear stress has parabolic through thickness variation and it is zero at top and bottom surfaces of the plate. Moreover, it does not introduce any additional unknown in the formulation. Thus, the plate theory is quite simple and elegant but it cannot be implemented in most of the elements, as the plate theory demands C 1 continuity of transverse displacement along the element edges. This has inspired the authors to develop this new element, which has shown an excellent performance in static analysis of composite plates. To demonstrate the performance of the element in the problem of buckling, examples of isotropic and composite plates under different situations are solved. The results are compared with the analytical solutions and other published results, which show the precision and range of applicability of the proposed element in the present problem.     

A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates

A new 4-node quadrilateral finite element is developed for the analysis of laminated composite plates containing distributed piezoelectric layers (surface bonded or embedded). The mechanical part of the element formulation is based on the first-order shear deformation theory. The formulation is established by generalizing that of the high performance Mindlin plate element ARS-Q12, which was derived based on the DKQ element formulation and Timoshenko’s beam theory. The layerwise linear theory is applied to deal with electric potential. Therefore, the number of electrical DOF is a variable depending on the number of plate sub-layers. Thus, there is no need to make any special assumptions with regards to the through-thickness variation of the electric potential, which is the true situation. Furthermore, a new “partial hybrid”-enhanced procedure is presented to improve the stresses solutions, especially for the calculation of transverse shear stresses. The proposed element, denoted as CTMQE, is free of shear locking and it exhibits excellent capability in the analysis of thin to moderately thick piezoelectric laminated composite plates.     

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.     

Partial hybrid stress element for the analysis of thick laminated composite plates

A new element—a partial hybrid stress element—is proposed in this paper for the analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger–Reissner principle through dividing six stress components into a flexural part (σ_x, σ_y, σ_xy, σ_z) and a transverse shear part (τ_xy, τ_yz). The element stiffness matrix can be formulated by assuming a stress field only for transverse shear stresses, while all the others are obtained from an assumed displacement field. Consequently, this new element combines the benefits of the conventional displacement method and the hybrid stress method. A twenty-node hexahedron element is employed in each layer for the displacement field. For the assumed transverse shear stress field, only the traction-free boundary conditions and interface traction continuity are satisfied. The equilibrium equation is enforced by the variational principle. Hence, the complicated work of searching an equilibrating stress field for all the six stress components in the hybrid stress method can be avoided. Furthermore, the interlaminar traction discontinuity, especially transverse shear, encountered by the conventional displacement method and higher-order plate element for laminated plate analysis can also be overcome. Examples are illustrated to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element.     

Stability Analysis of Laminated Soft Core Sandwich Plates Using Higher Order Zig-Zag Plate Theory

C₀ finite element model based on higher order zig-zag plate theory is used to study the stability analysis of laminated sandwich plates. The in-plane displacement field is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and constant in the faces. The conditions regarding transverse shear stress at layer interfaces and top and bottom are satisfied. Numerical examples covering different features of laminated sandwich plates are presented to illustrate the accuracy of the model.