Interlaminar stresses in antisymmetric angle-ply cylindrical shell panels

Layerwise theory of Reddy is utilized for investigating free-edge effects in antisymmetric angle-ply laminated shell panels under uniform axial extension. Following some physical arguments, governing displacement field is divided into local and global parts. The former is discretized through the shell thickness by a zig-zag interpolation function while the latter is calculated by a first-order shear deformation theory. Local equilibrium equations are then solved through a state space approach. Accuracy of the proposed technical solution is subsequently verified by a novel analytical elasticity solution. For this end, the problem is analytically solved for specific boundary conditions along the edges. The numerical results show excellent agreement between two theories for various composite shell panels.     

Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels

A hitherto unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doubly-curved panel of rectangular plan-form, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the HSDT-based laminated shell analysis, with the C4-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of the deflection and moment of a cross-ply spherical panel, and also by comparison with the available FSDT (first-order shear deformation theory) based analytical solution. Additionally, numerical results pertaining to flat symmetric and antisymmetric cross-ply plates with the same boundary conditions have also been reproduced. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to shell geometry (cylindrical and spherical), lamination, lamina material property, and thickness effects, as well as their interactions. Comparison with their SS2 counterparts demonstrates the effect of end clamping on the deflections and moments of thin to thick singly- and doubly-curved cross-ply panels.     

Buckling analysis of cross-ply laminated conical panels using GDQ method

The buckling analysis of cross-ply laminated conical shell panels with simply supported boundary conditions at all edges and subjected to axial compression is studied. The conical shell panel is a very interesting problem as it can be considered as the general case for conical shells when the subtended angle is set to 2π and also cylindrical panels and shells when the semi-vertex angle is equal to zero. Equations were derived using classical shell theory of Donnell type and solved using generalized differential quadrature method. The results are compared and validated with the known results in the literature. The effects of subtended angle, semi-vertex angle, length, thickness and radius of the panel on the buckling load and mode are investigated.     

Thermally induced vibrations of cross-ply laminated shallow shells

Summary An exact analytical solution of the dynamic response of cross-ply laminated shallow shells subject to rapid heating is presented. The classical theory (based on Love-Kirchhoff assumption), involving three coupled partial differential equations, is used. The solution is applicable to shells whose parallel edges are simply supported and the remaining ones are clamped. A generalized modal approach is used to obtain the solution. The equations of motion are converted into a single-order system of equations by using state variables. The biorthogonality conditions of principal modes of the original and adjoint eigenfunctions are used to decouple the state space equation. Histories of deflection of graphite-reinforced aluminum shell panels are presented through numerical examples.     

A three-dimensional asymptotic analysis of anisotropic inhomogeneous and laminated cylindrical shells

Summary An asymptotic for anisotropic inhomogeneous cylindrical shells is formulated on the basis of three-dimensional elasticity withouta priori assumptions. The inhomogeneities are considered varying in the radial direction, thus including the important case of laminated composite shells in which the material properties are piecewisely constant through the thickness. Reformulation and nondimensionalization of the basic three-dimensional elasticity equations reveal that the problem can be treated by means of asymptotic expansion. By using asymptotic expansion and successive integration, we show in a natural and consistent manner that the asymptotic theory gives rise to two-dimensional shell equations, in forms similar to those in the classical laminated shell theory (CST), at each level of approximation. Thus the three-dimensional asymptotic solution for the problem is no more difficult to obtain than that based on CST. The asymptotic solution can be improved by determining the higher-order modifications in a systematic way. Various through-thickness effects can be accounted for hierarchically. The performance of the theory is illustrated by comparing the asymptotic solutions with the available numerical results for a homogeneous and for a multilayered cross-ply cylindrical shell under lateral loads.     

Static and free vibration analysis of sandwich cylindrical shell based on theory of elasticity and using DQM

In this paper, static and free vibration analysis of a sandwich cylindrical shell is performed using theory of elasticity formulation. The core layer is made of functionally graded material with material properties varying along the thickness direction according to a simple power law. For the case of simply supported boundary conditions, equations of motion and equilibrium equations are solved analytically by applying a state-space technique along the radial direction and Fourier series expansion along the axial and circumferential direction. When boundary conditions are not simply supported, a semi-analytically solution is performed by using the differential quadrature method along the axial direction. The present approach is validated by comparing the obtained numerical results with those published in the available literature. Moreover, effects of boundary conditions, graded direction, mid-radius to thickness and length to mid-radius ratios on bending and vibration behavior are considered.     

Flutter of delaminated cross-ply laminated cylindrical shells

In this study, the instability of delaminated cross-ply thin laminated cylindrical shells and panels when subjected to supersonic flow parallel to its length edge is investigated. The delamination is parallel to the shell reference and it extends along the entire length of the cylindrical shell. The Love’s shell theory and Von-Karman–Donnell type of kinematic relations along with first-order potential theory have been employed to construct the aeroelastic equations of motion. The effects of several parameters such as length to radius ratio, delamination position, size and thickness on the critical values are discussed in the details. The results indicate that the presence of delamination reduced the overall stiffness of the structure and thereby decreases the flutter critical boundaries.     

A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels

Based on the global-local superposition technique proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212.], a global-local higher order laminated shell model is proposed for predicting both displacement and stress distributions through the thickness of laminated shells. This shell model satisfies transverse shear stress continuity conditions at interfaces as well as free surface conditions of transverse shear stresses. The merit of this model is that transverse shear stresses can be accurately predicted directly from constitutive equations without smoothing techniques. Cylindrical bending of laminated and sandwich shell panels is chosen to assess the present model wherein the results from several 2D laminated shell models and three-dimensional elasticity solution are available for comparison. In addition, thermal bending and thermal expansion of laminated cylindrical shell panels are also considered in this paper.     

Free vibration response of composite sandwich cylindrical shell with flexible core

This study deals with the free vibration analysis of composite sandwich cylindrical shell with a flexible core using a higher order sandwich panel theory. The formulation uses the classical shell theory for the face sheets and an elasticity theory for the core and includes derivation of the governing equations along with the appropriate boundary conditions. The model consists of a systematic approach for the analysis of sandwich shells with a flexible core, having high-order effects caused by the nonlinearity of the in-plane and the vertical displacements of the core. The behavior is presented in terms of internal resultants and displacements in the faces, peeling and shear stresses in the face–core interface and stress and displacement field in the core. The accuracy of the solution is examined by comparing the results obtained with the analytical and numerical results published in the literatures. The parametric study is also included to investigate the effect of geometrical properties such as radius of curvature, length and sector angle of the shell.     

On the stress analysis of cross-ply laminated plates and shallow shell panels

This paper extends the applicability of a new stress analysis method (Soldatos KP, Watson PA. Acta Mech 1997;123:163–186) towards the accurate prediction of stresses within cross-ply laminated doubly curved shell segments having a rectangular plan-form. The method is based on the successful incorporation of three-dimensional elasticity information for stress distributions into a two-dimensional five-degrees-of-freedom shallow shell theory. This successful matching is achieved by means of a set of two shape functions, which are incorporated within the two-dimensional shell model whereas their form depends on the particular problem considered. In the present case, two different sets of shape functions are developed and tested, one of which is more accurate than the other is, the later being however simpler than the former.     

双层横观各向同性圆柱壳体的弹性理论解

本文是在文献[10]的基础上,进一步分析了双层横观各向同性圆柱壳体在承受轴对称分布载荷时的三维弹性理论解。文中假定壳体的两端为简单支承,研究了壳体的物理参数与几何参数对层间应力分布的影响。本文还给出了双层横观各向同性圆柱壳体在有横向均布压力与轴向等应变(ε₀)拉伸同时作用下的数值计算结果。     

Double orthogonal set of solution functions for cross-ply laminated shear flexible cylindrical/doubly curved panels

An alternate set of solution functions in the form of orthogonal double Fourier series to analyse boundary-value problems of shear flexible general anti-symmetric cross-ply laminated cylinder/doubly curved panels is presented. Two types of boundary conditions are considered, simply supported, and all-edge clamped. The shell deformation characteristics are defined by five highly coupled partial differential equations in second order, according to the shell theory laid by Sanders. The solution functions that define the displacement behaviour of the shell presented in the form of double Fourier series are considered of mixed type, in contrast to the one prescribed earlier by Chaudhuri and Kabir. Efficiencies in the form of convergence and computational time are observed in the present method with improvement in the solutions. The numerical results, thus presented for various parametric effects can be considered as valuable tools to check the numerical solutions obtained from approximate methods.     

Analysis of simply-supported laminated circular cylindrical shell roofs

An elasticity solution has been presented for the analysis of a laminated circular cylindrical shell roof with simply-supported edges, and the displacements and stresses of the solution are expressed in terms of infinite series. A solution according to the classical shell theory (CST) is also developed. Computations are made for various ratios of midsurface radii to thicknesses and the results of the CST have been examined in the light of the elasticity solution results.     

Analysis of interlaminar stresses in general cross-ply laminates with distributed piezoelectric actuators

This paper represents an analytical solution to determine the interlaminar stresses of general cross-ply laminates with piezoelectric layers as actuators under transverse mechanical loads. The three-dimensional constitutive equations of piezoelectricity are considered and the governing equations are derived within the framework of second-order shear-deformation plate theory as a set of partial differential equations. These equations are solved for two kinds of boundary conditions and the unknown assumed functions of displacement field are found. Numerical results show that this approach can generally predict the behavior of interlaminar stresses. Also, they clearly indicate the singular behavior of interlaminar normal and shear stresses in the boundary region near the edges of the laminate.     

粘弹性圆柱壳在轴向恒压下的动力稳定性

基于Timoshenko-Mindlin假设,得到考虑粘弹性的各向同性圆柱壳及纤维增强正交铺设层合圆柱壳在轴向恒压下的动力学方程。文中对两端简支的圆柱壳进行了分析,依Laplace变换,导出动力稳定的特征方程,由Routh-Hurwitz判据建立动力稳定性条件,对两类圆柱壳讨论了横向剪切变形的影响。     

Analysis of laminated composite beams using layerwise displacement theories

Within the displacement field of a layerwise theory, two laminated beam theories for beams with general lamination are developed. In the first theory, an existing layerwise laminated plate theory is adapted to laminated beams. The procedure used in the second theory is simple and straightforward and similar to the one used in the development of plate and shell theories. These theories can also be used in developing simpler theories such as classical, first, and higher-order shear deformation laminated beam theories. Equations of motions are obtained by using Hamilton’s principle. For the assessment of the accuracy of these theories, analytical solutions for static bending and free vibration are developed and compared with those of an existing three-dimensional elasticity solution of cross-ply laminates in cylindrical bending and with the three-dimensional finite element analysis for angle-ply laminates.     

On stress-strain state in thick-walled cylindrical shells bounded by corrugated surfaces

Applying the second variant of the boundary-shape perturbation method and using three-dimensional equations and relations of the elasticity theory including the first three approximations the author has obtained an analytical solution to the problem of elastic equilibrium for noncanonical near-cylindrical shells subjected to fluctuating axial external forces. The influence of individual shapes of noncanonical shells, amplitude and parameter of oscillation of boundary surfaces in the axial or circumferential directions on the value of maximum stresses has been clarified. An almost numerical convergence is demonstrated, based on contribution (in absolute magnitude) of each of the approximations to the sum total. For a shell in the form of bellows, which is under a permanent external pressure, a comparison is made between the hoop stress values as found by the boundary-shape perturbation method and the data obtained by the finite-difference method using equations of the Vekua theory of shells.     

Buckling of functionally graded conical panels under mechanical loads

This paper presents an analytical approach to investigate the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure and the combination of these loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and linear stability equations in terms of displacement components for conical panels are derived by using the classical thin shell theory. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling loads. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of conical panels.     

Levy type Fourier analysis of thick cross-ply doubly curved panels

A hitherto unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doubly curved panel of rectangular plan-form, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the HSDT-based general cross-ply shell analysis, with the SS2-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of a moderately thick cross-ply spherical panel. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, lamina material property, and thickness and curvature effects, as well as their interactions.     

A new theory of laminated plate

For the analysis of cross-ply composite laminated plates an assumption based on the theory of composite beams is presented. Under the assumption the in-plane displacements between layers are continuous and the transverse shear stresses between layers are also continuous. Equilibrium equations and boundary conditions which are similar to those of the classical plate theory are obtained. Exact closed-form solutions are compared with the three-dimensional elasticity solutions. The results of the present theory agree very closely with the exact solutions.