Bending analysis of antisymmetric angle-ply laminated plates including transverse shear effects

In this paper an approximate analysis is presented for the bending of rectangular, simply supported, antisymmetric angle-ply laminated plates subjected to sinusoidal transverse loads. A published refined shear deformation theory is used to obtain a closed-form solution. Numerical results are compared with those given by three-dimensional elasticity solutions, first-order shear deformation theory solutions and classical lamination theory solutions. The present theory gives a better estimate of the stresses and deflections.     

Nonlinear analysis of laminated composite plates and shells including the effects of shear and normal deformation

A nonlinear higher-order theory for laminated composite plates and shells with an arbitrary number and sequence of layers is presented. The theory takes into account both transverse shear and normal deformation and considers the elasto-plastic behaviour of the composite materials. The results presented illustrate first the importance of modelling the nonlinear behaviour of the material especially at high levels of loading, and secondly the importance of modelling both transverse shear and normal compression.     

Influence of transverse shear on stochastic instability of symmetric cross-ply laminated plates

The dynamic instability problem of symmetrically laminated cross-ply plates, compressed by time-dependent stochastic membrane forces, is investigated. The effect of shear deformation is included in the formulation. By using the direct Liapunov method, bounds of the almost sure instability of cross-ply plates are obtained. Furthermore, it is shown how the viscous damping coefficient, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, and cross-ply ratio influence the instability regions. A special attention is given to the proper choice of two shear correction factors of the laminated plate. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ² as well as for the harmonic process with an amplitude A.     

Inelastic bending of beam including transverse shear

An incremental force–displacement relation is derived using the finite element matrix method to study the inelastic bending of beams including transverse shear. Solutions are obtained for the case of a cantilever beam with a point load and a simply supported beam with distributed loading considering and also ignoring the effect of transverse shear. They show good agreement for those cases where solutions are available. Also, the effect of transverse shear is found to be quite significant.     

Nonlinear analysis of concrete shells including effects of normal and transverse shear stresses

The previously developed numerical model of the authors for the analysis of conventional reinforced and prestressed concrete shells under short‐term and long‐term loading was improved by including the effects of transverse shear stresses on the shell failure. The 9‐node degenerated shell element with the layered material model through the thickness of the shell was used. The reinforcement was modelled as a separate layer. To include the effect of transverse shear stresses on the shell failure, the failure criterion for concrete and longitudinal reinforcement was defined by a relation of transverse shear stresses and normal stresses in two mutually perpendicular vertical planes. The total transverse shear bearing capacity of the shell cross‐section is obtained by summing up the concrete and reinforcement contributions. The developed numerical model and appropriate software were verified based on experimental tests.     

On vibration and buckling of symmetric laminated plates according to shear deformation theories

Summary The frequency and buckling equations of rectangular plates with various boundary conditions are developed within the third-order and the first-order shear deformation plate theories. The third-order theories account for a quadratic distribution of the transverse shear strains through the thickness of the plate. In the first part of this paper, Levinson's third-order theory, derived as a special case from Reddy's third-order theory, is used to study a plate laminated of transversely isotropic layers. The relationship between the original form of the governing equations and the interior and the edge-zone equations of the plate is closely examined and the physical insights from the latter equations are established. In the second part of the paper, the first-order shear deformation theory and the third-order theory of Reddy are studied for vibration and buckling.     

Damping optimization of symmetrically laminated plates with transverse shear deformation using lamination parameters

This paper deals with the damping characteristics of symmetrically laminated plates with transverse shear deformation. First, the effect of laminate configuration on the damping characteristics is investigated for cantilevered laminated plates based on the Reissner–Mindlin’s first-order shear deformation theory. To examine the effect of laminate configuration, the concept of specific damping capacity is introduced and the damping characteristics are represented on the lamination parameter plane, where the damped stiffness invariants in transverse shear are newly proposed in this paper. Next, the optimal laminate configurations for the cantilevered laminated plates with maximal damping are determined taking into account the transverse shear effect by using differential evolution in which lamination parameters are used as intermediate design variables. The relation between the laminate configurations and the damping characteristics is discussed based on the concept of lamination parameters.     

Vibrations of axially compressed laminated orthotropic cylindrical shells,including transverse shear deformation

Summary The free vibrations, buckling and the effect of initial prestress upon the frequency spectrum of orthotropic composite cylindrical shells are examined in the context of a theory that includes transverse shear deformation. Results obtained are compared with the predictions of refined Love-type theory and simplified Donnell-type theory that do not consider shear deformation.The calculated examples indicate that transverse shear deformation can be significant not only for short composite shells but even for longer shells possessing low shear moduli.

---

Schwingungen axial gedrückter, laminierter, orthotroper zylindrischer Schalen einschließlich Querschubverformung

Zusammenfassung Es werden die freien Schwingungen, das Beulen und der Einfluß einer Anfangsvorspannung auf das Frequenzspektrum orthotroper, kompositer, zylindrischer Schalen im Rahmen einer Theorie, die die Querschubverformung mitberücksichtigt, untersucht. Die erhaltenen Resultate werden mit den Vorhersagen der verbesserten Loveschen Theorie und der vereinfachten Donnellschen Theorie, welche keine Querschubverformung in Betrach ziehen, verglichen.Die durchgerechneten Beispiele zeigen, daß die Querschubverformung nicht nur für kurze, komposite Schalen sondern sogar für längere Schalen mit niederem Schubmodul Bedeutung gewinnen kann.

---

Nomenclature a radius of reference surface of cylindrical shell - A _i amplitudes of displacements - A _ij elastic area - B _ij elastic statical moment - D _ij elastic moment of inertia - E _ij elastic stiffness modulus - h shell thickness - k ₄₄,k ₅₅ shear correction factors - l length of cylindrical shell - L _ij functional operator - m number of half-waves in axial direction - M ,M ,M axial, circumferential and twisting moments, respectively - n number of circumferential waves - N ⁰ axial compression force - N ,N ,N axial, circumferential and shear forces, respectively - Q ,Q transverse shear forces - R ₀,R ₁,R ₂ inertia terms - t time - T matrix defined in Eq. (25) - u, v, w displacements in axial, circumferential and radial directions, respectively - U _i time-independent displacement or rotation - nondimensional axial wave parameter - , , axial, circumferential and shear strains, respectively - radial coordinate, taken positive inward - polar angle - , , curvature changes - circular frequency - nondimensional axial coordinate - density - , , , , stress components - , rotations of normal to undeformed midsurface With 6 Figures     

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.     

Vibration analysis of symmetric laminated composite plates with attached mass

In this study, vibration of symmetrically laminated composite plates with attached mass is studied. The Ritz method with algebraic polynomial displacement field is used. The plates with at least two adjacent free edges are considered in the formulations. The effect of various parameters (number of layers, ratio of attached mass to the plate mass, position of attached mass, fiber orientation) upon the frequencies and mode shapes is investigated. It is found that attachment of a mass has important effects on the vibration characteristics of composite laminated plates.     

Non-linear bending analysis of laminated sector plates using Generalized Differential Quadrature

Non-linear static analysis of laminated sector plates with any combination of clamped, simply supported and free edges is presented using Generalized Differential Quadrature (GDQ) method. Particular interest of this study is large deformation of asymmetric sector plates with free edges. Based on the first-order shear deformation theory and von Kármán type non-linearity, the governing system of equations include a system of 13 partial differential equations (PDEs) in terms of unknown displacements, forces and moments. Successive application of the GDQ technique to the governing equations resulted in a system of non-linear algebraic equations. The Newton–Raphson iterative scheme is then employed to solve the system of non-linear equations. Illustrative examples are presented to demonstrate accuracy and rapid convergences of the method with small number of grid points. Predictions of the presented method show very good agreement with other numerical studies available in the literature. Further results for asymmetric laminated sector plates with free edges are also presented for future references.     

Dynamic plastic behaviour of annular plates with transverse shear effects

The dynamic response of a rigid-plastic annular plate clamped at the outer edge and free at the inner edge is considered.     

Analysis of the nonlinear dynamic response of viscoelastic symmetric cross-ply laminated plates with transverse matrix crack

Based on the Schapery’s 3-D constitutive relationship and the Von Karman’s plate theory, the constitutive equations and the nonlinear dynamic governing equations for viscoelastic symmetric cross-ply laminated plates with transverse matrix cracks are derived. By using the finite difference method and Newmark-β scheme, the unknown functions are separated and these equations are iterated to seek solutions. Numerical calculating results show that the effect of damage on the nonlinear dynamic responses of structures is significant.     

Buckling of shear deformable laminated composite plates

A shear deformable finite element is developed for the buckling analysis of laminated composite plates. The finite element formulation is based on Mindlin's theory in which shear correction factors are derived from the exact expressions for orthotropic materials. A variety of problems on uniaxial and shear bucklings of laminated composite plates are solved. The effects of material properties, plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle on the buckling loads of symmetrically and antisymmetrically laminated composite plates are investigated. Optimal lamination arrangements of layers for maximizing the buckling loads of the plates are determined.     

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.     

Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading

This paper present the second ordered statistics of first-ply failure response of laminated composite plate with random material properties under random loading. The basic formulation is based on higher order shear deformation plate theory (HSDT) with the geometrically nonlinearity in the von-Karman. The direct iterative based C⁰ nonlinear finite element method combined with mean centered first order perturbation technique developed by the authors are extended and successfully applied nonlinearity for failure problem with a reasonable accuracy to predict the second order statistics (standard deviation) of first-ply failure response using Tsai-Wu and Hoffman failure criterion with macroscopic analysis. Typical numerical results for various combinations of boundary conditions, plate thickness ratios, aspect ratios, laminates scheme and layers, elastic modulus ratios have been presented to illustrate the application of developed procedure. Some new results are presented and examined which clearly demonstrated the importance of the randomness in the system parameters in the failure response of the structures subjected to transverse loadings.     

Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method

Asymmetric free vibrations of annular cross-ply circular plates are studied using spline function approximation. The governing equations are formulated including the effects of shear deformation and rotary inertia. Assumptions are made to study the cross-ply layered plates. A system of coupled differential equations are obtained in terms of displacement functions and rotational functions. These functions are approximated using Bickley- type spline functions of suitable order. Then the system is converted into the eigenvalue problem by applying the point collocation technique and suitable boundary conditions. Parametric studies have been made to investigate the effect of transverse shear deformation and rotary inertia on frequency parameter with respect to the circumferential node number, radii ratio and thickness to radius ratio for both symmetric and anti-symmetric cross-ply plates using various types of material properties.     

Two-dimensional analysis of magnetoelectric effects in multiferroic laminated plates

Two-dimensional equations for coupled extensional, flexural, and thickness-shear motions of laminated plates of piezoelectromagnetic layers are obtained from 3-dimensional equations. The equations are used to analyze magnetoelectric couplings in the extensional deformation of a laminated plate of piezoelectric and piezomagnetic layers. Magnetoelectric effects in 4 specific configurations of laminates are calculated and examined.     

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.