First-order Transverse Shear Deformation Theory Research Articles

Nonlinear forced vibration analysis of laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects

Assessment of the geometrically nonlinear forced vibration of the doubly-curved laminated composite shallow shell panels enriched by nanocomposites have been carried out in this paper. The proposed shells incorporate some layers composed of fibres and nanocomposite-reinforced resin, resting on the two types Winkler and Pasternak elastic foundations. Since the thin and moderately thick shells are investigated, the first-order transverse shear deformation theory (FSDT) is employed for extending the governing equations. The achieved system of differential equations incorporating the geometrical nonlinear terms is solved by applying the principles of the Galerkin method. This approach yields a closed-form semi-analytical solution for motion differential equation of the shell which is solved by fourth-order Runge-Kutta method. The consequence of the present approach is a strong and robust tool without any time-consuming computational process. The applicability of the proposed method is verified by considering some benchmark problems from the literature and comparing the relevant results with those are achieved in this study. Finally, by solving several examples, the effects of various factors on the forced-vibration of proposed shells are evaluated.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

Free vibration and stability of hybrid nanocomposite-reinforced shallow toroidal shells using an extended closed-form formula based on the Galerkin method

This paper presents an analytical solution for free-vibration and stability analyses of hybrid-composite toroidal shells. The considered hybrid material incorporates the composition of fibers and nanocomposite-reinforced matrix (NCRM) such as carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs). The governing differential equations are extended based on the first-order transverse shear deformation theory (FSDT) and are solved by employing the approximation of Fourier series and Galerkin method. The considered hybrid composite material in the parametric studies are consisted of three layers in which the polymeric matrix are reinforced by CNTs in some layers and reinforced by GPLs in the other layers.

Analysis and Optimization of Composite Kagome Grid Panels Subjected to the Low Velocity Impact

By the present paper, an optimal volume technique of stiffened laminated panel subjected to Low Velocity Impact (LVI) is developed using an analytical model that couples a genetic algorithm optimization technique. Using the first-order transverse-shear deformation theory (FSDT) for anisotropic plates, the stiffness of the Kagome grid stiffeners and the plate are estimated by considering both effects of moments and forces in a basic repetitive cell. Where, the stiffness of the equivalent plate is computed by superposing stiffness the plate and Kagome stiffeners. The contact impactor/panel is idealized by the mass/spring model. The convergence of the impact force and the transverse displacement response at the contact point are plotted. Using the genetic algorithm (GA), the minimization of the panel volume is achieved in two stages. In the first stage, the design variables are the mechanical properties of the equivalent plate and the impactor. But, in the second stage, the stacking sequences were considered as design variables. Finally a parametric study is made and major conclusions are deduced regarding to the panel volume, time and impact velocity on behavior of the stiffened panel. The results show that the increase in stiffeners thickness is traduced by smaller transverse displacement involving a better impact performance, best optimization and panel integrity.

Effects of shear deformation on Mechanical and thermo-mechanical nonlinear stability of FGM shallow spherical shells subjected to uniform external pressure

In the present paper, the asymmetrical nonlinear response of a clamped functionally graded shallow spherical shell is subjected to uniform external pressure. It considers the effects of thermal stresses by both of the theories, Classical Laminate Theory, CLT and First-Order Shear Deformation Theory FSDT. Material properties are graded in the thickness direction according to the power-law distribution in terms of the volume fraction of the constituents. Mechanical and thermo-mechanical properties are assumed to be temperature-independent and linear elastic. All of the governing equations are derived by aid of first-order transverse shear deformation theory considering geometrical nonlinearity. The nonlinear differential equation system is solved employing Galerkin method. Buckling and post-buckling analysis have been done according to one-term deformation mode by the closed form relation of load-deflection that shows the equilibrium path. Parametric studies are conducted to bring out the effects of shear deformation on the equilibrium path in different geometries and boundary conditions. Numerical results are presented in graphical arrangement, showing that the geometrical nonlinear equilibrium paths. The effects of shear deformation on the equilibrium path are considered by comparing the results from FSDT and CLT and was verified by nonlinear finite-element method.

Developing an element free method for higher order shear deformation analysis of plates

A procedure is presented to eliminate transverse shear locking in analysis of laminated composite plates using Element Free Galerkin (EFG) method based on higher-order transverse shear deformation theory (HSDT). In the procedure, derivatives of the transverse displacement are introduced as independent variables. Thus, a formulation requiring C0 continuity shape functions for approximation is proposed for higher-order transverse shear deformation analysis of plates. Shear locking is avoided considering reduced integration for shear stiffness matrix; a method which is implemented in FEM. Moving Least Squares (MLS) method is utilized for shape functions and the penalty method is implemented to impose approximate boundary conditions. The present solutions are verified with other higher-order shear deformable studies. Moreover, a comparison between the present solutions with those obtained by EFG procedure based on First-order transverse Shear Deformation Theory (FSDT) is performed.

Non-linear thermal effects on the bending response of cross-ply laminated plates using refined first-order theory

Based on the representation of the displacement field proper to the first-order transverse shear deformation theory, and using Reissner’s mixed variational principle modified as to include thermoelastic effects, an equivalent single-layer model of composite laminated plates is presented including the geometric non-linearity. In contrast to the usual approach of such models, the present model has the advantage of eliminating the need of the inclusion of a shear correction factor. Moreover, this model predicts continuous stress distributions through the laminate thickness. The equations of motion, the constitutive equations and the required boundary conditions are obtained. Based on the obtained equations, linear and non-linear numerical results on the static state of stresses and displacements of cross-ply symmetric and antisymmetric laminated plates are supplied. To assess the present model, some of the obtained linear results are compared with their counterparts in the literature. The effect of the geometric non-linearity on the thermal response of the laminate is investigated.

Vibration of Laminated Plates Having Elastic Edge Flexibilities

The first known investigation on the problem of free vibration of symmetric cross-ply laminated plates having elastically restrained edges is considered. The Ritz method is used in combination with a variational formulation and a first-order transverse shear deformation theory. The Ritz functions consist of polynomials that include a basic function that imposes the free edge boundary conditions. The elastic edge flexibilities are considered by both the linear elastic rotational and translational supports to act simultaneously. Numerical results are obtained to investigate the effects of elastic restraint stiffness, material properties, and geometric parameters upon the natural frequencies of flexural vibration.

Free vibration of laminated anisotropic shallow shells including transverse shear deformation by the boundary-domain element method

Free vibration of laminated anisotropic shallow shells is considered, based on the first-order transverse shear deformation theory. The effects of both in-plane and rotary inertias on the natural frequencies are included. An integral equation formulation of free vibration of laminated anisotropic shallow shells has been established by the use of the method of weighted residuals and taking the static fundamental solutions as the weighted function. As there are not only boundary integrals but domain integrals in the present formulation, the integral equation presented is reduced to a standard algebraic eigenvalue problem by the use of boundary-domain elements. In the present numerical implementation, constant domain elements are used. Some examples are analysed using boundary-domain elements. The numerical results are compared with the exact solutions and the other solutions available in the literature. The agreement is found to be very good.

A refined two-dimensional theory of anisotropic plates

Based on the representation of the displacement field proper to the first-order transverse shear deformation theory, and using Reissner’s mixed variational principle modified as to include thermoelastic effects, an equivalent single-layer model of composite laminated plates is presented including the geometric non-linearity. In contrast to the usual approach of such models, the present model has the advantage of eliminating the need of the inclusion of a shear correction factor. Moreover, this model predicts continuous stress distributions through the laminate thickness. The equations of motion, the constitutive equations and the required boundary conditions are obtained. Based on the obtained equations, linear and non-linear numerical results on the static state of stresses and displacements of cross-ply symmetric and antisymmetric laminated plates are supplied. To assess the present model, some of the obtained linear results are compared with their counterparts in the literature. The effect of the geometric non-linearity on the thermal response of the laminate is investigated.

Formulation of an elastodynamic theory of laminated shear-deformable flat panels

A simple higher order shear deformation theory (HSDT) of cross-ply laminated flat panels is formulated. The composite panel, built up of an even number of orthotropic layers, exhibits geometric symmetry but material property antisymmetry with respect to its mid-plane. For the sake of generality, the case of different materials for the various symmetric laminae pairs is considered. Within this theory the shortcomings proper to the first order transverse shear deformation theory (FSDT) are eliminated. Furthermore, all the attributes of FSDT related to the number of the involved unknown quantities and to the order of the governing equations are preserved. Several special cases of the developed theory are explicitly presented, while other ones are indicated.

Random response of antisymmetric angle-ply laminated plates

The random response of an antisymmetric angle-ply composite plate subjected to random lateral load on its surface is obtained. The first-order transverse shear deformation theory, involving five coupled partial differential equations, is used. To evaluate the effect of shear deformation, anisotropy and other problem parameters on the response, the numerical results obtained with this theory are compared with the results obtained by the classical plate theory. Significant differences are observed in the two sets of results obtained by the two theories especially for thick plates, thus demonstrating the importance of shear deformations and need for utilization of plate theories which properly account for such deformations.

Response of moderately thick laminated cross-ply composite shells subjected to random excitation

This study deals with the dynamic response of transverse shear deformable laminated shells subjected to random excitation. The analysis encompasses the following problems: (1) the dynamic response of circular cylindrical shells of finite length excited by an axisymmetric uniform ring loading, stationary in time, and (2) the response of spherical and cylindrical panels subjected to stationary random loadings with uniform spatial distribution. The associated equations governing the structural theory of shells are derived upon discarding the classical Love-Kirchhoff (L-K) assumptions. In this sense, the theory is formulated in the framework of the first-order transverse shear deformation theory (FSDT).

Dynamic stability of transversely isotropic viscoelastic plates

This paper deals with a dynamic stability analysis of transversely isotropic viscoelastic plates subjected to in-plane biaxial edge load systems. In deriving the associated governing equations a Boltzmann hereditary constitutive law was used and, in addition, transverse shear deformation, transverse normal stress and rotatory inertia effects were incorporated. The integro-differential equations governing the stability of simply supported plates are solved by using the integral transform technique. The solution concerns the determination of the critical in-plane edge loads associated with the asymptotic instability of plates. While studying this problem the general dynamic stability solutions are compared with those based on the first-order transverse shear deformation theory and with their quasi-static counterparts. Numerical applications are presented and pertinent conclusions are formulated.

Dynamic response of antisymmetric angle-ply laminated plates subjected to arbitrary loading

The transient response of simply supported antisymmetric angle-ply rectangular plates subjected to arbitrary loading is investigated. The state variable technique is used to solve exactly the equations of motion of the first-order transverse shear deformation theory (FSDT) as well as the classical laminate theory (CPT). The solutions of these two theories are considered to bring out the influence of the transverse shear deformation, the degree of anisotropy, and the number of layers.