Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory

This paper introduces a generalized 5 degrees of freedom (DOF) higher-order shear deformation theory (HSDT) to study the bending and free vibration of plates and shells, which may be used to create other HSDTs. It also introduces a new HSDT for shells that is more accurate than many available HSDTs despite having the same 5DOF, and which is also able to reproduce the well-known Soldatos’ HSDT as special case. The governing equations and boundary conditions of the generalized formulation are derived by employing the principle of virtual work. These equations are solved via Navier-type closed-form solutions. Static and dynamic results are presented for plates and cylindrical and spherical shells with simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. Results from the new and well-known HSDTs introduced and reproduced based on the present generalized 5DOF HSDT are compared with the exact three-dimensional elasticity solution. The present new HSDT for plates and shells is found to be more accurate than the well-known HSDTs developed by other authors, for analyzing the static and free vibration of isotropic and multilayered composite plates and shells.     

Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory

A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.     

Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory

A two-dimensional global higher-order deformation theory is presented for thermal buckling of cross-ply laminated composite and sandwich plates. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through the principle of virtual work. Several sets of truncated Mth-order approximate theories are applied to solve the eigenvalue problems of a simply supported multilayered plate. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. Numerical results are compared with those of the published three-dimensional layerwise theory in which both in-plane and normal displacements are assumed to be C⁰ continuous in the continuity conditions at the interface between layers. Effects of the difference of displacement continuity conditions between the three-dimensional layerwise theory and the global higher-order theory are clarified in thermal buckling problems of multilayered composite plates.     

Post-buckling analysis of composite plates containing embedded delaminations with arbitrary shape by using higher order shear deformation theory

The compressive post-buckling behavior of composite laminates containing embedded delamination with arbitrary shape is investigated analytically. For modeling the embedded delamination, the laminate is divided into three smaller regions. The higher order shear deformation theory is implemented and the formulation is based on the Rayleigh-Ritz approximation technique by the application of the simple/complete polynomial series for each region. The nonlinear equilibrium equations, which are achieved through the application of the principle of Minimum Potential Energy, are solved by employing the Newton-Raphson iterative procedure. Some interesting results are obtained and compared with those achieved by the finite element method of analysis using ANSYS commercial software. A good agreement is seen to exist between the results. This is while for a given level of accuracy in the results, ANSYS requires a markedly larger number of degrees of freedom compared to that needed by the developed method. Moreover, a considerable reduction in the load carrying capacity of laminate is noticed due to the presence of delamination.     

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.     

Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.     

General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels

ABSTRACT

The present article illustrates a general formulation for a higher-order layer-wise theory related to the analysis of the free vibrations of thick doubly-curved laminated composite shells and panels. The theoretical framework relates to the dynamic analysis of shell structures by using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order layer-wise formulation is to propose a kinematic assumption, with an arbitrary number of degrees of freedom. The main aim of this work is to determine the explicit fundamental operators that can be used for the layer-wise (LW) approach. These fundamental operators are obtained for the first time by the author and are related to motion equations of doubly-curved shells described in an orthogonal curvilinear co-ordinate system. The free vibration shell and panel problems are computationally solved using the generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) techniques. The numerical results are compared with recent papers in the literature and commercial finite element codes.     

Flexural analysis of laminated plates based on trigonometric layerwise shear deformation theory

ABSTRACT

A trigonometric layerwise shear deformation theory is developed for the flexural analysis of laminated plates. The present theory achieves in-plane displacement continuity, transverse shear stress continuity, and traction-free boundary condition. Hence, botheration of shear correction coefficient is neglected. The governing differential equation and boundary conditions are obtained from the principle of virtual work. Although the present analytical method is bounded to a corner supported boundary condition, it neglects the numerical and computational error. Like first-order shear deformation theory, the present theory possesses five numbers of unknowns. Several numerical predictions are carried out and results are compared with those of other existing numerical approaches.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.     

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.     

Analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sandwich plates

Analytical formulations and solutions to the static analysis of simply supported anti-symmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher-order refined theory already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations, which account for the effect of transverse shear deformation and a non-linear variation of in-plane displacements with respect to the thickness coordinate. The transverse displacement is assumed to be constant throughout the thickness. The equations of equilibrium are obtained using principle of minimum potential energy. Solutions are obtained in closed form using Navier's technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already reported in the literature. After establishing the accuracy of the solutions, numerical results with real properties are presented for the multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

Low velocity impact analysis of composite sandwich shells using higher-order shear deformation theories

In the present investigation, higher-order and conventional first-order shear deformation theories are used to study the impact response of composite sandwich shells. The formulation is based on Donnell’s shallow shell theory. Nine-noded Lagrangian elements are used for the finite element formulation. A modified Hertzian contact law is used to calculate the contact force. The results obtained from the present investigation are found to compare well with those existing in the open literature. The numerical results are presented to study the changes in the impact response due to the increase of core depth from zero to some specified value and the changes in core stiffness for a particular core depth.     

Dynamic stability of laminated FGM plates based on higher-order shear deformation theory

This paper conducts a dynamic stability analysis of symmetrically laminated FGM rectangular plates with general out-of-plane supporting conditions, subjected to a uniaxial periodic in-plane load and undergoing uniform temperature change. Theoretical formulations are based on Reddys third-order shear deformation plate theory, and account for the temperature dependence of material properties. A semi-analytical Galerkin-differential quadrature approach is employed to convert the governing equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotins method. Free vibration and bifurcation buckling are also discussed as subset problems. Numerical results are presented in both dimensionless tabular and graphical forms for laminated plates with FGM layers made of silicon nitride and stainless steel. The influences of various parameters such as material composition, layer thickness ratio, temperature change, static load level, boundary constraints on the dynamic stability, buckling and vibration frequencies are examined in detail through parametric studies.This work was fully supported by grants from the Australian Research Council (A00104534) and from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1024/01 E). The authors are grateful for this financial support.     

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 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.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter

A study of static deformations and free vibration of shear flexible isotropic and laminated composite plates is presented. A layerwise theory for laminated or sandwich plates is used. The analysis is based on a new numerical scheme, where collocation by radial basis functions is viewed as a pseudospectral method to produce highly accurate results. A cross-validation technique is used to optimize the shape parameter for the basis functions. Numerical results for symmetric laminated composite and sandwich plates are presented and discussed.     

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.     

A new higher order shear deformation theory for sandwich and composite laminated plates

A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.     

A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates

In the present paper, a n-order model for functionally graded and composite sandwich plate is developed. This model uses the n-order polynomial term to represent the displacement field. Zero transverse shear stress boundary conditions at the top and bottom of the plate are satisfied. The third-order theory of Reddy [8] can be considered as a special case of present n-order theory. Natural frequencies of the functionally graded and composite plates with various side-to-thickness ratios, material properties are computed by present n-order theory with different n values. The results are compared with available published results which demonstrate the accuracy and efficiency of present n-order theory.