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.     

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.     

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.     

Dynamic analysis of clamped laminated curved panels

The free vibration and dynamic response of clamped laminated curved panels subjected to the step, triangular and explosive loadings are investigated. The case of rectangular panels having fixed boundary conditions is also considered. The Rayleigh-Ritz method is employed to obtain the natural frequencies of the clamped laminated curved panels by using a set of simple polynomials as admissible functions. The normal mode superposition method is then used in the analysis of the dynamic response. Numerical results of the symmetric angleply, symmetric and antisymmetric cross-ply laminated curved panels under three kinds of dynamic loadings are presented.     

Effects of geometric imperfections on vibration of compressed shear deformable laminated composite curved panels

Summary The vibrational behavior of geometrically imperfect single and multilayered composite double-curved shallow panels subjected to a system of tangential compressive/tensile edge loads in the pre- and postbuckling ranges is investigated. The effects of transverse shear deformations, lamination, the character of in-plane boundary conditions, and of transverse normal stress are incorporated and their influence is emphasized.Numerical illustrations enabling one to compare the obtained results based on higher order and first order shell theories with their classical counterparts, based on the Love-Kirchhoff model are presented and conclusions related to their range of applicability are outlined.     

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.     

Asymptotic finite strip analysis of doubly curved laminated shells

On the basis of the Hellinger–Reissner (H–R) principle, an asymptotic finite strip method (FSM) for the analysis of doubly curved laminated shells is presented by means of perturbation. In the formulation the displacements and transverse stresses are taken as the functions subject to variation. Imposition of the stationary condition of the H–R functional, the weak formulation associated with the Euler–Lagrange equations of three-dimensional (3D) elasticity is obtained. Upon introducing a set of appropriate dimensionless scaling and bringing the transverse shear deformations to the stage at the leading-order level, the weak formulation is asymptotically expanded as a series of weak-form equations for various orders. An asymptotic FSM according to the present formulation is then developed where the field variables are interpolated as a finite series of products of trigonometric functions and crosswise polynomial functions independently. Through successive integration, the present formulation turns out that three mid-surface displacement degrees-of-freedom (DOF) and two rotation DOF for each node in a strip element are taken as the independent unknowns in the system equations for various orders. The solution procedure for the leading-order level can be repeatedly applied level-by-level in a consistent and hierarchic way. Application of the asymptotic FSM to a benchmark problem is demonstrated.     

A semi-analytical solution for stress analysis of moderately thick laminated cylindrical panels with various boundary conditions

Stress analysis of moderately thick laminated cylindrical panels with different loading and boundary conditions is presented. Boundary conditions include clamped, simply supported and free while uniform and sinusoidal distributed loadings are considered. Assuming effects of shear deformation and initial curvature, governing equations of the problem are derived. The governing partial differential equations (PDEs) in terms of three displacement components, two rotations and ten stress resultants include a system of 15 first order PDEs. Application of the extended Kantorovich method (EKM) to the governing equations yields to a double set of algebric-differential equations in terms of x and θ. The resulted systems are then solved iteratively with very fast convergence. It is demonstrated that the method converges rapidly independent of initial guess functions. Comparisons of the EKM predictions with other analytical and FEM analyses are in close agreement. More results for panels with particular boundary conditions are presented for future studies.     

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.     

Geometrically nonlinear analysis for some axially compressed laminated cylindrical panels

Mainly, this paper deals with cylindrical laminated panels, axially compressed under the constraint of four simply-supported edges.By imposing uniform increasing end-shortening to the two opposite circular edges, characteristic equilibrium paths have been obtained in correspondence to numerical evaluations with finite element models subject to geometric non linearity. A peculiar link among different branches of the total path, each of which is connected to a proper deformed configuration, is pointed out in the axial load/end-shortening space.Convergence problems and mesh sensibility are also examined.     

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。     

Analytical solution for bending of cross-ply laminated plates under thermo-mechanical loading

The static thermo-elastic response of symmetric and anti-symmetric cross-ply laminated plates has been investigated by the use of a unified shear deformation plate theory. The present plate theory enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. The validity of the present theory is demonstrated by comparison with solutions available in the literature. A wide variety of results is presented for the static response of simply supported rectangular plates under non-uniform sinusoidal mechanical and/or thermal loadings. The influence of material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the thermally induced response is studied.     

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.     

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.     

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.     

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.     

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.     

Postbuckling analysis and imperfection sensitivity of metal matrix laminated cylindrical panels

The postbuckling behaviour of metal matrix composite (MMC) laminated cylindrical panels under quasistatic in-plane loading is investigated. A micro-to-macro analysis is used to obtain the response of the composite structure. The micromechanical analysis allows us to establish the overall instantaneous elastic-viscoplastic behaviour of the MMC composite at each load increment. The macromechanical analysis provides the response of the geometrically imperfect cylindrical panel to the applied external loading.

Results are presented for unidirectional and antisymmetric angle-ply SiC/Ti composite panels, subjected to uniaxial compressive loadings. The effects of the panel curvature, initial imperfections and rate of loading on the postbuckling response are illustrated. Comparisons with the response of the corresponding perfectly elastic panels are shown.     

Global optimization of laminated cylindrical panels based on fundamental natural frequency

This investigation presents an optimization of laminated cylindrical panels based on fundamental natural frequency. Also, trends of change in optimum stacking sequence while the proportions of structures vary, are studied which can be insightful for design purposes. A displacement based finite element model is used, in order to extract fundamental natural frequencies of T300/5208 Carbon/Epoxy cylindrical panels. To obtain optimum designs, the Globalized Bounded Nelder–Mead (GBNM) algorithm is employed. Predictions are compared with the results of Genetic Algorithm (GA) method and show faster and more accurate convergence to the global optimum, while variables are continuous in GBNM and discrete in GA. Moreover, verification of novel convergence criteria to ameliorate local searcher in GBNM is examined.