Bending,buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory

The bending, buckling and free vibration problems of non-homogeneous composite laminated cylindrical shells are considered. Hamilton–Reissner's mixed variational principle is used to deduce a consistent first-order theory of composite laminated cylindrical shells with non-homogeneous elastic properties. The governing equations with their required boundary conditions are derived without introducing any shear correction factors. Numerical results for the transverse deflections, stresses, natural frequencies and critical buckling loads are presented to show the advantages of this theory. The influences of the non-homogeneity and thickness ratio on the shell structural response are investigated. The study concludes that the inclusion of the non-homogeneity effect is required, even if it is weak, for predicting the actual structural response of the shells.     

A higher-order shear deformation theory of laminated elastic shells

A higher-order shear deformation theory of elastic shells is developed for shells laminated of orthotropic layers. The theory is a modification of the Sanders' theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The Navier-type exact solutions for bending and natural vibration are presented for cylindrical and spherical shells under simply supported boundary conditions.     

On the transient response of cross-ply laminated circular cylindrical shells

Transient response of simply-supported circular cylindrical shells is investigated using a higher-order shear deformation theory (HSDT). The theory is a modification of the Sanders' shell theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the bounding surfaces of the shell. The results obtained using the classical shell theory (CST) and the first-order shear deformation theory (FSDT) are compared with those obtained using the higher-order theory. The state-space approach is used to develop the analytical solutions to the equations of motion of the three theories.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. 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 deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

An improved zig-zag model for the bending of laminated composite shells

A simple layerwise higher-order zig-zag model is proposed for the bending of laminated composite shells. The model provides a cubic variation of both the in-plane displacements and the transverse shear stresses within each layer. As the displacement model satisfies the zero transverse shear stress conditions at the free surfaces, there is no need for the use of shear correction factors. By imposing the continuity of the in-plane displacements and the transverse shear stresses at the interfaces, the number of variables is shown to be the same as that given by the first-order shear deformation shell theory, irrespective of the number of layers considered. For the sake of consistency, all terms of the order of the thickness coordinate-to-radius ratio have been retained in the derivation of the governing equations. Numerical results for the cylindrical bending of thick, symmetric homogeneous orthotropic and three-layer laminated shells under sinusoidal loading show that the maximum transverse deflections and in-plane stresses are in good agreement with available exact elasticity solutions for radius-to-thickness ratios greater than or equal to four.     

Admissible boundary conditions and solutions to internally pressurized thin arbitrarily laminated cylindrical shell boundary-value problems

Admissible boundary conditions are derived for an arbitrarily laminated internally pressurized cylindrical shell of finite length, under the framework of Donnell’s, Love–Timoshenko’s and Sanders’ kinematic relations, and the CLT (based on Love’s first approximation theory). Closed-form solutions for the same cylindrical shell are presented for Love–Timoshenko’s theory, with two sets of asymmetrically placed prescribed boundary conditions. As the first example, internally pressurized thin hybrid general (asymmetric) four-layer cylindrical shells with RS2-C4 boundary conditions, made of glass and carbon fiber reinforced composite layers, are numerically investigated. In the second example, the numerical results for two-layer asymmetrically laminated cylindrical shells, with RS2-SS1 boundary conditions, are compared with those, computed using triangular finite elements based on the layer-wise constant shear-angle theory (LCST), in order to evaluate the limit of applicability of the CLT.     

Buckling of moderately thick laminated cylindrical shells: a review

The present paper is a review article on the problem of buckling of moderately thick, laminated, composite shells subjected to destabilizing loads. The loads consist of uniform axial compression, uniform lateral pressure and torsion applied individually or in combination. In all the works reported in the literature, the analysis is based on higher-order shear deformation (HOSD) shell theory and/or first-order shear deformation (FOSD) shell theory with or without a shear correction factor. Results obtained by these two shell theories and by employing classical thin shell theory are compared to determine the range of applicability of each in predicting critical conditions. The effect of stacking sequence, radius-to-thickness ratio and length-to-radius ratio is assessed. Typical numerical results are presented in tabular form. Moreover, some limited results, which are based on limit point analysis are also presented (imperfection sensitivity studies).     

Asymptotic differential quadrature solutions for the free vibration of laminated conical shells

 Three-dimensional (3-D) elasticity solutions for the free vibration analysis of laminated circular conical shells are presented by means of an asymptotic approach. The formulation begins with the 3-D equations of motion in circular conical coordinates. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple time scales is used to eliminate the secular terms and make the asymptotic expansion feasible. The method of differential quadrature (DQ) is adopted for solving the problems of various orders. The present asymptotic formulation is applicable to the analysis of laminated cylindrical shells by vanishing the semivertex angle (α). The natural frequencies, modal stresses of cross-ply cylindrical and conical shells with simply supported – simply supported (S-S) boundary conditions are studied to demonstrate the performance of the present asymptotic theory. It is shown that the asymptotic DQ solutions of the present study converge rapidly. The present convergent results are in good agreement with the accurate solutions obtained from the approximate 2-D shell theories in the cases of thin shells. Furthermore, these present results may serve as the benchmark solutions for assessment of various 2-D shell theories in the cases of moderatively thick shells. Received 11 August 1999     

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.     

Dynamic response of cross-ply laminated circular cylindrical shells with various boundary conditions

Summary Closed-form solutions of the dynamic response of cross-ply laminated circular cylindrical shells are developed for arbitrary boundary conditions and under arbitrary loadings. The equations of motion of the classical, first-order and third-order theories are converted into a single-order system of equations by using state variables. To solve for the dynamic response, the biorthogonality conditions of principle modes of the original and adjoint eigenfunctions are used to decouple the state space equation. The study reveals that the disagreement between shear deformation theories in much less than the disagreement between them and the classical theory.     

Nonlinear vibrations of angle-ply laminated circular cylindrical shells: Skewed modes

Skewed modes in geometrically nonlinear forced vibrations of angle-ply laminated circular cylindrical shells are investigated in the present study by using the Amabili–Reddy higher-order shear deformation theory. An harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retains dissipation. Numerical results are obtained by using the pseudo-arclength continuation method and bifurcation analysis.     

A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories

The main objective of this paper is a theoretical unification of most of the variationally consistent classical and shear deformable cylindrical shell theories available in the literature. This is achieved by introducing into the shell displacement approximation certain general functions of the transverse coordinate which account for the incorporation of the transverse shear deformation effects. Avoiding having to provide a single choice of the forms of these ‘shear deformation shape functions’ befor or during the variational formulation of the general theory, the present formulation leaves open possibilities for a multiple, a-posteriori specification of particular shear deformable shell theories. As a result, the classical Donnell-, Love- and Sanders-type shell theories as well as their well known uniform and parabolic shear deformable analogues are obtained as particular cases. Moreover, a generalized ‘zig-zag’ displacement model is presented which gives further multiple freedom in achieving continuous distributions of interlaminar stresses through the thickness of an unsymmetric cross-ply laminated cylindrical shell.     

A solution of laminated cylindrical shells using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piece-wise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, the displacement functions remain to be unconstrained. This theory is very useful for the finite element analysis because it requests only C⁰ continuity for the assumed displacement fields. By comparing with three-dimensional elasticity theory for laminated orthotropic cylindrical shell, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for the thin and moderately thick shells.     

Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings

The objective of this paper is to present an equivalent single-layer shear deformation theory for evaluation of displacements and stresses of cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load. A trigonometric shear deformation theory is used. The in-plane displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Stresses and displacements for orthotropic, two-layer antisymmetric, and three-layer symmetric square cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load are obtained. Numerical results of the present theory for displacement and thermal stresses are compared with those of classical, first-order and higher-order shear deformation plate theories.     

Exact analysis for static response of cross ply laminated smart shells

Models and analytical solutions are formulated and developed for the static behavior of cross ply smart laminated shells with extension piezoelectric laminae. The models are based on a rigorous first order shell theory. The state space approach is used to find exact solutions for the static response of cross ply spherical, cylindrical and doubly curved shells with various boundary conditions. The smart shells possess two parallel edges simply supported and the remaining ones having any possible combination of boundary conditions: free, clamped or simply supported. Deflections of cross ply laminated shells incorporating piezoelectric layers are determined. Numerical results of six layer laminates are generated to investigate the shell static behavior. The exact solutions for deflections can be used as benchmarks for approximate solutions such as Rayleigh–Ritz and finite element methods.     

层壳考虑横向剪切效应的自由振动分析

本文采用一个分层的剪切变形理论分析层壳的自由振动。假定层壳各层横向剪切应变彼此线性相关,从而未知函数个数与一阶剪切变形理论相同,但控制微分方程的阶数为十二阶,且不含剪切修正因子。文中计算了一个短圆柱壳与两种扁壳的自由振动频率,数值结果与经典层合理论、一阶剪切变形理论及其他剪切变形理论的计算结果进行了比较。      

Computational approach of piezoelectric shells by the GDQ method

A piezoelectric laminated cylindrical shell with shear rotations effect under the electromechanical loads and four sides simply supported boundary condition was studied by using the two-dimensional generalized differential quadrature (GDQ) computational method. The typical hybrid composite shells with 3-layered cross-ply [90°/0°/90°] graphite–epoxy laminate and bounded PVDF layers are considered under the sinusoidal pressure loads and electric potentials on the shell. The governing partial differential equation with first-order shear deformation theory in terms of mid-surface displacements and shear rotations can be expressed in series equations by the GDQ formulation. Thus we obtain the GDQ numerical solutions of non-dimensional displacement and stresses at center position of laminated piezoelectric shells. Displacement is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential.     

Exact analysis for static response of cross ply laminated smart shells

Models and analytical solutions are formulated and developed for the static behavior of cross ply smart laminated shells with extension piezoelectric laminae. The models are based on a rigorous first order shell theory. The state space approach is used to find exact solutions for the static response of cross ply spherical, cylindrical and doubly curved shells with various boundary conditions. The smart shells possess two parallel edges simply supported and the remaining ones having any possible combination of boundary conditions: free, clamped or simply supported. Deflections of cross ply laminated shells incorporating piezoelectric layers are determined. Numerical results of six layer laminates are generated to investigate the shell static behavior. The exact solutions for deflections can be used as benchmarks for approximate solutions such as Rayleigh–Ritz and finite element methods.