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.     

Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads

In this paper, the viscoelastic damped response of cross-ply laminated shallow spherical shells is investigated numerically in a transformed Laplace space. In the proposed approach, the governing differential equations of cross-ply laminated shallow spherical shell are derived using the dynamic version of the principle of virtual displacements. Following this, the Laplace transform is employed in the transient analysis of viscoelastic laminated shell problem. Also, damping can be incorporated with ease in the transformed domain. The transformed time-independent equations in spatial coordinate are solved numerically by Gauss elimination. Numerical inverse transformation of the results into the real domain are operated by the modified Durbin transform method. Verification of the presented method is carried out by comparing the results with those obtained by the Newmark method and ANSYS finite element software. Furthermore, the developed solution approach is applied to problems with several impulsive loads. The novelty of the present study lies in the fact that a combination of the Navier method and Laplace transform is employed in the analysis of cross-ply laminated shallow spherical viscoelastic shells. The numerical sample results have proved that the presented method constitutes a highly accurate and efficient solution, which can be easily applied to the laminated viscoelastic shell problems.     

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.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

粘弹性复合材料层合板壳的动力稳定性分析

分析面内周期激励下粘弹性层合平板以及轴向周期荷载作用下粘弹性层合圆柱壳的动力稳定性。设粘弹性复合材料服从Boltzmann积分型本构关系，其松弛模量由Prony—Dirichlet级数表示，基于薄板与薄壳理论，分别得到对称正交铺设层合板与层合圆柱壳的微分-积分型动力学方程，并应用谐波平衡法直接求解，忽略积分运算所产生的衰减项，导出确定动力不稳定区域边界的特征方程。分析结果表明，主要动力不稳定区域的缩小与材料的粘性参数以及结构横向振动的基频密切相关。     

The non-linear buckling analysis of cross-ply laminated orthotropic truncated conical shells

In this study, the non-linear buckling behavior of cross-ply laminated orthotropic truncated conical shells under axial load has been investigated. The basic relations of the cross-ply laminated orthotropic truncated conical shells are derived using the von Karman–Donnell-type of kinematic non-linearity. Then modified Donnell type non-linear stability and compatibility equations are obtained and are solved. Finally, the influences of the number and ordering of layers and the variations of the conical shell characteristics on the non-linear axial buckling load are investigated. Comparison with available results is satisfactorily good.     

Stress analysis of axisymmetric shear deformable cross-ply laminated circular cylindrical shells

A generalized mixed theory for bending analysis of axisymmetric shear deformable laminated circular cylindrical shells is presented. The classical, first-order and higher-order shell theories have been used in the analysis. The Maupertuis–Lagrange (M–L) mixed variational formula is utilized to formulate the governing equations of circular cylindrical shells laminated by orthotropic layers. Analytical solutions are presented for symmetric and antisymmetric laminated circular cylindrical shells under sinusoidal loads and subjected to arbitrary boundary conditions. Numerical results of the higher-order theory for deflections and stresses of cross-ply laminated circular cylindrical shells are compared with those obtained by means of the classical and first-order shell theories. The effects, due to shear deformation, lamination schemes, loadings ratio, boundary conditions and orthotropy ratio on the deflections and stresses are investigated.     

Boundary value problems of thin, moderately-deep cross-ply laminated cylindrical shells

An unavailable analytical solution to the boundary value problems of thin moderately-deep cross-ply laminated shells of rectangular planform, subjected to transverse loads, is presented. Love-Kirchhoff theory-based Sanders' kinematic relations that represent moderately-deep shell deformation behavior are considered. These kinematic relations yield highly coupled two third-order and one fourth-order partial differential equations with constant coefficients. The equations are solved together with the prescribed geometric and natural boundary conditions by utilizing a double Fourier series approach, an approach that manipulates ordinary discontinuities present in the solution functions and/or their derivatives. The numerical results presented, for SS2-type simply supported boundary conditions for various parametric effects, should serve as base-line solutions for future comparison of such popular approximate numerical techniques as finite element and finite difference.     

复合材料层合扁锥壳的冲击屈曲

研究了计及横向剪切的对称铺设正交异性复合材料层合扁锥壳在三角脉冲载荷作用下的非线性动力屈曲问题；通过在复合材料层合扁锥壳非线性稳定性的基本方程中增加横向转动惯量项并引入三角脉冲冲击力，得到了层合扁锥壳的冲击控制方程，采用Galerkin方法得到以顶点挠度表达的冲击响应方程，并用Runge—Kutta方法进行数值求解，应用B—R准则确定冲击屈曲的临界荷载；讨论了壳体几何参数、物理参数和边界条件对复合材料层合扁锥壳冲击屈曲的影响。     

Vibration analysis of cross-ply laminated truncated conical shells using a spline method

Free vibration of symmetric and antisymmetric cross-ply composite laminated truncated conical shells using the spline function technique is studied. The equilibrium equations for a truncated conical shells are formulated including first-order shear deformation theory. The equations of motion are derived in terms of displacement functions and rotational functions using stress–strain and strain–displacement relationships. The coupled differential equations are solved using Bickley-type splines to obtain the generalized eigenvalue problem by combining suitable boundary conditions. The convergence and comparative results are presented. Both symmetric and anti-symmetric cross-ply shells are considered using various types of material properties. Parametric studies are made to investigate the effect of transverse shear deformation on the frequency parameter with respect to the thickness ratio, length ratio, cone angle, and circumferential mode number using different numbers of layers under various types of boundary conditions.     

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.     

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.     

Thermally induced deformation of laminated composite structures

The implications of somewhat unfamiliar thermo-elastic phenomena in laminated fibre composites are discussed in physical terms. By this approach certain simplified relationships are developed and the attendant physical interpretations are encouraged. Specifically, laminated flat plates exhibiting finite, negative thermal expansion coefficients and hoop-wound composite rings subjected to temperature changes are treated so as to emphasize the importance of the phenomena. It is also shown that the translaminar expansion coefficient is an important parameter in the hoop-wound ring which develops significant circumferential stresses when subjected to uniform changes in temperature.     

正交异性扁薄球壳的非线性轴对称振动

给出了一种研究圆柱正交异性扁薄球壳非线性轴对称自由振动的新的时间模态方法。首先,使用变分法导出了决定振动频率的一对代数-微分方程。然后,利用作者提出的改进的修正迭代法求得了渐进解。这使此种壳体的非线性振动的进展获得扩充研究。     

Dynamic stability analysis of cross-ply laminated cylindrical shells using different thin shell theories

Summary The dynamic stability of thin, laminated cylindrical shells under combined static and periodic axial forces is studied here using three common thin shell theories, namely Donnell's, Love's and Flügge's shell theories. A normal-mode expansion of the equations of motion yields a system of Mathieu-Hill equations the stability of which is examined using Bolotin's method. The present study examines and compares the effects of the use of the various shell theories on the dynamic stability analysis.     

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.     

Asymptotic solutions of laminated composite shallow shells with various boundary conditions

Summary The refined asymptotic theory developed recently is applied to three-dimensional analyses of laminated composite shallow shells with various boundary conditions. Problems of cross-ply doubly curved laminated shells subjected to transverse loads are considered. The edge conditions of the shell are such that one pair of the opposite edges is simply supported and the other may be combinations of free, clamped or simply supported edges. On the basis of the refined asymptotic formulation, a systematic and adaptive method is developed by means of separation of variables and a state-space approach, with which analytical solutions to the system of differential equations in the asymptotic theory are determined for laminated shells with various edge conditions.     

Active damping of geometrically nonlinear vibrations of doubly curved laminated composite shells

This paper deals with the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of doubly curved laminated composite shells. Vertically/obliquely reinforced 1–3 piezoelectric composite (PZC) and active fiber composite (AFC) materials are used as the materials of the constraining layer of theACLD treatment. The Golla–Hughes–McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The first-order shear deformation theory (FSDT) and the Von Kármán type non-linear strain displacement relations are used for analyzing this coupled electro-elastic problem. A three dimensional finite element (FE) model of doubly curved laminated smart composite shells integrated with ACLD patches has been developed to investigate the performance of these patches for controlling the geometrically nonlinear transient vibrations of the shells. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the doubly curved laminated cross-ply and angle-ply shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1–3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1–3 PZC and the AFC constraining layers on the control authority of the ACLD patches have also been investigated.     

Active damping of geometrically nonlinear vibrations of laminated composite shallow shells using vertically/obliquely reinforced 1-3 piezoelectric composites

This paper addresses the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of laminated thin composite cylindrical shallow shells using vertically reinforced 1-3 piezoelectric composite (PZC). The constraining layer of the ACLD treatment is considered to be made of this 1-3 PZC material. The Golla–Hughes–McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The Von Kármán type non-linear strain displacement relations and the first-order shear deformation theory (FSDT) are used for deriving this electromechanical coupled problem. A three dimensional finite element (FE) model of smart composite shallow shells integrated with a patch of such ACLD treatment has been developed to demonstrate the performance of the patch on enhancing the damping characteristics of thin laminated cylindrical shells, in controlling the geometrically nonlinear transient vibrations. The numerical results indicate that the ACLD patch significantly improves the damping characteristics of the shells for suppressing the geometrically nonlinear transient vibrations of the shells. The effect of variation of fiber orientation in the PZC material on the control authority of the ACLD patch has also been investigated.     

The buckling of cross-ply laminated non-homogeneous orthotropic composite conical thin shells under a dynamic external pressure

Summary.  The subject of this investigation is to study the buckling of cross-ply laminated orthotropic truncated circular conical thin shells with variable Young's moduli and densities in the thickness direction, subjected to a uniform external pressure which is a power function of time. After obtaining the dynamic stability and compatibility equations we reduce both of them to a time dependent ordinary differential equation with variable coefficient by using Galerkin's method. The critical dynamic and static loading, the corresponding wave numbers, the dynamic factors, critical time and critical impulse are found analytically by applying the Ritz type variational method. The dynamic behavior of cross-ply laminated truncated conical shells is investigated with: (a) lamina that present variations in the Young's moduli and densities, (b) different numbers and ordering of layers, (c) variable semi-vertex angles, and (d) external pressures which vary with different powers of time. It is concluded that all these factors contribute to appreciable effects on the critical parameters of the problem in question. Received June 13, 2002; revised December 10, 2002 Published online: May 8, 2003 This research project was initiated with the support of DAAD when Dr. Sofiyev was in Germany for a research project. The project was successfully accomplished with generous support of TUBITAK (The Scientific and Technical Research Council of Turkey).