Yogesh M. Desai

Analytical solutions for vibrations of laminated and sandwich plates using mixed theory

Abstract A semi-analytical method has been presented in this paper to evaluate the natural frequencies as well as displacement and stress eigenvectors for simply supported, cross-ply laminated and sandwich plates by using higher order mixed theory. Models based on equivalent single layer as well as layerwise (LW) theories have been formulated. By assuming a non-linear variation of axial displacements through the plate thickness, the warping of the transverse cross-section has been considered. Hamilton’s principle has been employed to derive the equilibrium equations. The proposed LW model fulfills a priori the continuity of displacements as well as the transverse and the normal stress components at each interface between two adjacent layers. Results obtained by present higher order mixed theory have been found in good agreement with those obtained by three-dimensional elasticity solutions. After establishing the accuracy of present results for orthotropic plates, new results for thin and thick sandwich plates have been presented which can serve as benchmark solutions for future investigations.

Free vibrations of laminated beams using mixed theory

An analytical method for evaluating the natural frequencies of laminated composite and sandwich beams has been developed in this paper using higher-order mixed theory. The stiffness and mass matrices for the laminate are computed by evaluating the integrands explicitly. Each lamina of the laminate is assumed to be orthotropic and in a two-dimensional state of plane stress. Hamiltons principle has been employed to derive the equilibrium equations. Numerical experiments on laminated composite and sandwich beams of thin and thick sections are performed. The results obtained are compared with those of the first-order shear deformation theory and are validated by comparing them with the published results of analytical methods incorporating displacement-based higher order theories.

Stability of sandwich plates by mixed, higher-order analytical formulation

Abstract A unified mixed, higher-order analytical formulation has been presented in this paper to predict general buckling as well as wrinkling of a general multi-layer, multi-core sandwich plate having any arbitrary sequence of stiff layers and cores. Assumptions of thin stiff layers and anti-plane core, which are usually made in the analysis of sandwiches, have been eliminated in the present formulation. Displacements as well as the transverse stress continuities have been enforced in the formulation by incorporating them as the degrees-of-freedom. The modal transverse stresses have been obtained directly as eigen vectors and thus their separate calculations have been advantageously avoided. Two sets of mixed models have been proposed on the basis of individual layer as well as equivalent single layer (ESL) theories by selectively incorporating non-linear components of Green’s strain tensor. Solutions from the models have been shown to be in excellent agreement with the available three-dimensional elasticity solutions as well as with the available experimental results. It has been demonstrated that the ESL theories cannot accurately evaluate the overall buckling as well as the wrinkling loads of sandwiches. Limitations of the typical simplifying assumptions have also been highlighted.

Dynamic analysis of laminated composite plates using a layer-wise mixed finite element model

Abstract This paper deals with an accurate, three-dimensional, higher order, mixed finite element (FE) modeling for the free vibration analysis of multi-layered thick composite plates. An 18-node, three-dimensional mixed FE model has been developed by using Hamilton’s energy principle. Continuity of the transverse stress and the displacement fields has been enforced through the thickness of laminated composite plate. Further, in addition to the displacement components, the transverse stress components ( σ z , τ xz , and τ yz , where z is the thickness direction) have been invoked as the nodal degrees-of-freedom by applying elastic equations between stress and displacement fields. Thus, the present FE model has a novel feature of maintaining the fundamental elastic relationship throughout an elastic continuum. Natural frequencies of laminated composite plates obtained through the present formulation have been shown to be in good agreement with the available elasticity and closed form solutions. Stress mode shapes have been plotted for the fundamental natural vibration of plates with various lamination schemes. Strain variations through the thickness of laminated plates have also been presented graphically to show the magnitude of discontinuity in the variation of transverse strains at the layer interfaces.

AN EFFICIENT SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES SUBJECTED TO THERMAL LOAD

A simple, semi-analytical model with mixed (stresses and displacements) fundamental variables starting from the exact three dimensional (3D) governing partial differential equations (PDEs) of laminated composite and sandwich plates for thermo-mechanical stress analysis has been presented in this paper. The plate is assumed simply supported on all four edges. Two different temperature variations through the thickness of plates are considered for numerical investigation. The accuracy and the effectiveness of the proposed model are assessed by comparing numerical results from the present investigation with the available elasticity solutions. Some new results for sandwich laminates are also presented for future reference.

On Accurate Stress Analysis of Composite and Sandwich Narrow Beams

A simple, semi-analytical methodology defining a two-point boundary value problem (BVP) governed by a set of linear first-order ordinary differential equations (ODEs) with mixed primary variables whose number equals the order of partial differential equations (PDEs) for narrow layered beams is proposed in this paper. These primary quantities consist of displacements ( u , w ) and the corresponding transverse stresses ( τ xz , σ z ). Continuity of transverse stresses and displacements through the thickness of a laminated beam is enforced naturally in the formulation. The accuracy and the effectiveness of the proposed methodology is addressed by comparing numerical results from the present formulation with available closed-form analytical and finite element (FE) solutions.

Comparisons of displacement-based theories for waves and vibrations in laminated and sandwich composite plates

A semi-analytical method incorporating various displacement-based formulations has been developed to investigate propagation of time harmonic waves and vibrations in fiber reinforced polymer composite laminated and sandwich plates. Various displacement-based models starting from the first order shear deformation theory to the fourth order theory have been developed using combinations of linear, quadratic, cubic and quartic variation of axial and transverse displacements through the thickness of a lamina or a mathematical sub-layer. These displacement-based formulations have been validated by comparing their results with the analytical solutions available in the literature. Results of all the displacement models have been compared with those obtained by displacement model using quartic variation of in-plane and transverse displacements for vibration problem. Higher order displacement-based theory using cubic variation of in-plane and transverse displacements through the thickness of sub-layer has been found to yield converging results for wave propagation in laminated composite plates as well as for vibration problems. All the investigations performed indicate the importance of higher order theories for analysis of wave propagation and vibrations in composite laminated and sandwich plates.

First Ply Failure of Laminated Composite Plates - A Mixed Finite Element Approach

First-ply failure (FPF) analyses of composite laminated plates have been presented in this paper. A 3-D layer-wise mixed finite element model developed by the authors (Ramtekkar, G.S., Desai, Y.M. and Shah, A.H. (2002). Mixed Finite Element Model for Thick Composite Laminated Plates, Mechanics of Advanced Materials and Structures, 9(2): 133-156.) has been employed for the computation of components of stress and strain. FPF has been predicted by using the maximum stress, the maximum strain, Tsai-Hill, Tsai-Wu and Hoffman failure theories. It has been observed that the 3-D failure theories have yielded approximately the same failure load as their 2-D counterparts because the failure of a lamina (ply) under flexure is primarily associated with the in-plane stress state. It has been demonstrated that the present methodology predicts the FPF load for cross-ply laminated plates very well. Results have been presented for cross-ply-angle-ply, symmetric-anti-symmetric and thin-thick plates to show important observations.

Effects of Partial Edge Loading and Fibre Configuration on Vibration and Buckling Characteristics of Stiffened Composite Plates

In this work, the influence of uniaxial and biaxial partial edge loads on buckling and vibration characteristics of stiffened laminated plates is examined by using finite element method. As the initial pre-buckling stress distributions within an element are highly non-uniform in nature for a given loading and edge conditions, the critical loads are evaluated by dynamic approach. Towards this, a nine-node heterosis plate element and a compatible three-node beam element are developed by employing the effect of shear deformation for both the plate and the stiffeners respectively. In the structural modeling, the plate and the stiffener elements are treated separately, and then the displacement compatibility is maintained between them by using a transformation matrix. Effect of different parameters such as loaded edge width, position of loads, boundary conditions, ply-orientations and stiffener factors are considered in this study. Buckling results show that the uniaxial loaded stiffened plate with around (+30o/-30o)2 layup can withstand higher load irrespective of boundary conditions and loading patterns, whereas the maximum load resisting layup for the bi-axially loaded stiffened plate is purely dependent on edge conditions and loading patterns.

Stress Analysis of Laminated Composite and Sandwich Beams using a Novel Shear and Normal Deformation Theory

A novel Normal and Shear Deformation Theory (NSDT) for analysis of laminated composite and sandwich beams, taking into account shear deformation as well as normal deformation, is developed. The paper investigates flexural behaviors of thick laminated and sandwich beams under plane stress conditions using NSDT. A generalized displacement-based refined formulation is elucidated with inclusion of various warping functions in terms of thickness coordinates to represent shear and normal deformation effects. These effects become pronounced in thick laminated beams and particularly in sandwich beams with transversely flexible core. Present formulation satisfies the shear stress free surface conditions at the top and bottom surfaces of the beam with realistic through-the-thickness variation of transverse shear stresses. The results obtained are compared with higher order theories available in literature. It is observed that NSDT predicts displacement and stresses accurately compared to other higher order theories.