Tarun Kant

Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory

Analytical formulations and solutions to the natural frequency analysis of simply supported composite and sandwich plates hitherto not reported in the literature based on a higher-order refined theory developed by the first author and already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate - thus modelling the warping of transverse cross-sections more accurately and eliminating the need for shear correction coefficients. In addition, few higher-order theories and the first-order theory developed by other investigators and already available in the literature are also considered for the evaluation. The equations of motion are obtained using Hamiltons principle. Solutions are obtained in closed form using Naviers technique and by solving the eigenvalue equation. The comparison of the present results with the available elasticity solutions and the results computed independently using the first-order and the other higher-order theories available in the literature shows that this refined theory predicts the fundamental and higher frequencies more accurately than all other theories considered in this paper. After establishing the accuracy of present results for composite plates, new results for sandwich laminates using all the theories considered in this paper are also presented which may serve as a benchmark for future investigations.

Estimation of transverse/interlaminar stresses in laminated composites - a selective review and survey of current developments

A review is made on the different methods used for the estimation of transverse/interlaminar stresses in laminated composite plates and shells. Both analytical and numerical methods are considered. In numerical methods while the emphasis is given on finite element methods, other methods like the finite difference method is also briefly discussed. Aspects considered are: effects of variation in geometric and material parameters, transverse shear and normal deformation, interface stress continuity and the interfacial bonding on the accuracy of prediction of transverse/interlaminar stresses. Finally some general conclusions are presented along with future directions of research on the analysis of multilayered composite plates and shells for free-edge effects.

Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory

Abstract Analytical formulations and solutions to the static analysis of simply supported composite and sandwich plates hitherto not reported in the literature based on a higher order refined theory developed by the first author and already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate – thus modelling the warping of transverse cross-sections more accurately and eliminating the need for shear correction coefficients. In addition, a few higher order theories and the first order theory developed by other investigators and already available in the literature are also considered for the evaluation. The equations of equilibrium are obtained using principle of minimum potential energy (PMPE). Solutions are obtained in closed form using Naviers technique by solving the boundary value problem. The comparison of the present results with the available elasticity solutions and the results computed independently using the first order and the other higher order theories available in the literature shows that this refined theory predicts the transverse displacement and the stresses more accurately than all other theories considered in this paper. After establishing the accuracy of present results for composite and sandwich plates, new results for the stretching–bending coupling behaviour of antisymmetric sandwich laminates using all the theories considered in this paper are presented which will serve as a benchmark for future investigations.

Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations

A simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant variation of transverse displacement through the plate thickness. Further, the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not imposed. The validity of the present development(s) is established through, numerical evaluations for deflections/stresses/stress-resultants and their comparisons with the available three-dimensional analyses/closed-form/other finite element solutions. Comparison of results from thin plate. Mindlin and present analyses with the exact three-dimensional analyses yields some important conclusions regarding the effects of the assumptions made in the CPT and Mindlin type theories. The comparative study further establishes the necessity of a higher-order shear deformable theory incorporating warping of the cross-section particularly for sandwich plates.

Numerical analysis of thick plates

Bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory. The theory is based on a higher order displacement model and the three-dimensional Hookes laws for plate material, giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness. It is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency, economy, reliability and accuracy in such applications

A critical review and some results of recently developed refined theories of fiber-reinforced laminated composites and sandwiches

A critical review of literature pertinent to the subject matter of this paper was carried out under the following two broad headings: free vibration and transient dynamics. Each of these groups describes the various theoretical developments in fiber reinforced laminated composite and sandwich plates. The theoretical developments are further classified according to the refinement/accuracy of the theories developed, such as the classical theory, the first-order shear deformation theory, and the three-dimensional elasticity/higher-order shear deformation theories. The present literature review is limited to linear free vibration and transient dynamic analyses, and geometric nonlinear transient response of multilayer sandwich/fiber-reinforced composite plates. A comparative study of recently developed refined theories in conjunction with the C° isoparametric finite element formulation has been made and the conclusions were drawn based on the literature review and the refined theories results. In order to compare the present results with the available results and to provide an easy means for future comparisons by other investigators, the numerical results are presented in tabular form.

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

Abstract 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.

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.

Flexural analysis of laminated composites using refined higher-order C ° plate bending elements

A finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented here. The present higher-order theory incorporates linear variation of transverse normal strains and parabolic variation of transverse shear strains through the plate thickness, and as a result it does not require shear correction coefficients. A nine-noded Lagrangian parabolic isoparametric plate bending element is described. The applications of the element to bending of laminated plates with various loading, boundary conditions, and lamination types are discussed. The numerical evaluations also include the convergence study of the element used. The present solutions for deflections and stresses are compared with those obtained using the three-dimensional elasticity theory, closed-form solutions with another high-order shear deformation theory, and the Mindlins theory. In addition, numerical results for a number of new problems, not available in the literature, are presented for future reference.

Closed-form thermo-mechanical solutions of higher-order theories of cross-ply laminated shallow shells

Closed-form formulations of 2D higher-order shear deformation theories for the thermo-mechanical analysis of simply supported doubly curved cross-ply laminated shells are presented. Formulation includes the Sanders theory for doubly curved shells. Two of the higher-order shear deformation theories account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of the transverse shear deformation. In these developments a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The temperature variation considered in the formulation is uniform or sinusoidal over the surface and linearly varying through the thickness. Numerical results are presented for thermal and mechanical load cases in laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plate or shells are compared with the available 3D elasticity solutions for mechanical loading and it is believed that solutions for thermal loading will serve as bench mark in future.