Comparisons of buckling-to-weight behavior of cylindrical shells with composite and metallic stiffeners

The present research deals with the study of buckling pressure-to-weight ratio of cylindrical shells with laminated ring stiffeners. The research introduces a strain energy formulation for laminated rings, in which each ring stiffener may be treated individually and the geometrical dimensions and material properties of the stiffeners may be different from one another. In a parametric study, the impacts of various geometrical parameters of rings on the buckling pressure-to-weight ratio of shells for composite and metallic stiffeners are compared. The results show that the buckling-to-weight behavior of the shells with Steel stiffeners generally differs from the shells with stiffeners made of Aluminum, E-glass/epoxy, Graphite/epoxy, or Kevlar49/epoxy.     

Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid

This paper presents the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. The originality of this work lies (i) in the development of a variational formulation for the fully coupled fluid/piezoelectric structure system, and (ii) in the finite element implementation of an inexpensive and accurate axisymmetric adaptive laminated conical shell element. Various modal results are presented in order to validate and illustrate the efficiency of the proposed fluid–structure finite element formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Vibration of composite-material cylindrical shells with ring and/or stringer stiffeners

Free vibration is analyzed for thin-walled circular cylindrical shells constructed of composite materials and provided with ring and/or stringer stiffeners. Numerical results are presented for graphite-epoxy shells: unstiffened and stiffened with rings only, stringers only, and both rings and stringers.     

Axisymmetric vibration analysis of laminated hollow cylinders with ring stiffeners

This paper presents an axisymmetric vibration analysis of laminated hollow cylinders composed of monoclinic layers and stiffened by ring stiffeners. A successive approximation approach, which is based on a transfer matrix and then an equivalent stiffness matrix formulation, is used to enable three-dimensional solutions to be found. It is assumed that the ring stiffeners are attached to the lateral surfaces of the cylinder and only provide elastic supports in radial direction. These constraints are imposed by using Lagrange multipliers to couple the responses of a number of vibration modes of corresponding cylinders without stiffeners. Using this method the natural frequencies of a stiffened cylinder are found to be the eigenvalues of a constraint matrix and the predictions can be arbitrarily close to the exact three-dimensional solutions.     

Three-dimensional piezoelasticity solution for dynamics of cross-ply cylindrical shells integrated with piezoelectric fiber reinforced composite actuators and sensors

A benchmark three-dimensional (3D) exact piezoelasticity solution is presented for free vibration and steady state forced response of simply supported smart cross-ply circular cylindrical shells of revolutions and panels integrated with surface-bonded or embedded monolithic piezoelectric or piezoelectric fiber reinforced composite (PFRC) layers. The effective properties of PFRC laminas for the 3D case are obtained based on a fully coupled iso-field model. The governing partial differential equations are reduced to ordinary differential equations in the thickness coordinate by expanding all entities for each layer in double Fourier series in span coordinates, which identically satisfy the boundary conditions at the simply-supported ends. These equations with variable coefficients are solved using the modified Frobenius method, wherein the solution is constructed as a product of an exponential function and a power series. The unknown constants of the general solution are finally obtained by employing the transfer matrix method across the layers. Results for natural frequencies and the forced response are presented for single layer piezoelectric and multilayered hybrid composite and sandwich shells of revolution and shell panels integrated with monolithic piezoelectric and PFRC actuator/sensor layers. The present benchmark solution would help assess 2D shell theories for dynamic response of hybrid cylindrical shells.     

压电复合材料三维壳体简化数值建模研究

为有效分析三维压电复合材料壳体结构非线性、 单向耦合压电弹性问题, 基于变分渐近方法(VAM)建立了壳体结构在机械和电场作用下的简化模型。推导了基于旋转张量分解概念的压电复合材料三维壳体能量表达式; 利用变分渐近法将三维壳体严格拆分为二维壳体线性分析和沿法线方向的一维非线性分析; 进行了降维后近似能量推导及Reissner-Mindlin形式转换; 提供了三维场重构关系以得到沿厚度方向的准确应力分布。通过对由4层压电复合材料构成的壳体柱形弯曲算例分析表明: 基于该理论和重构过程开发的变分渐近程序VAPAS重构生成的三维应力场精确性较一阶剪切变形理论和古典层合理论更好, 与三维有限元精确解相吻合, 表明该压电复合材料壳体模型的有效性。      

Buckling of thin-walled composite structures with intermediate stiffeners

The interactive buckling of prismatic, thin-walled composite columns with open sections, reinforced with intermediate stiffeners and with edge reinforcements, has been considered. The columns are assumed to be simply supported. The nonlinear problem has been solved with the Koiter’s asymptotic theory within the first order approximation. The asymptotic theory of the first order nonlinear approximation allows for simultaneous evaluation of the effect of imperfections and interactions of various modes of buckling on the behaviour of thin-walled structures. This evaluation can be only the lower bound estimation of the load carrying capacity. Detailed calculations have been made for several cases of columns.     

Nonlinear dynamic stability of laminated composite shells integrated with piezoelectric layers in thermal environment

This paper reports the nonlinear dynamic stability characteristics of laminated composite cylindrical (CYL) and spherical (SPH) shells integrated with piezoelectric layers using the finite element method. The shells are subjected to a thermal environment in addition to the in-plane periodic load and the electric load. The theoretical formulation considers Sanders?? approximation for doubly curved shells, and von Kármán type nonlinear strains are incorporated into the first-order shear deformation theory (FSDT). The formulation includes the effects of transverse shear, in-plane and rotatory inertia. The in-plane periodic load is taken as the parametric excitation in the governing equation. The nonlinear matrix amplitude equation is obtained by employing Galerkin??s method. The correctness of the formulation is established by comparing the authors?? results with those available in the published literature. Detailed parametric studies are carried out to investigate the effects of different parameters on the dynamic stability characteristics of laminated composite shells.     

Axisymmetric nonlinear analysis of shallow spherical shells by a combined boundary element and finite difference method

This paper is concerned with the development of the mixed boundary element method and finite element method for the analysis of spherical annular shells under axisymmetric loads. The boundary element techniques are used to solve the equilibrium equation of shells and the central difference operator is adopted to deal with the compatibility equations. Iterative techniques are used throughout the analysis procedure. A number of numerical examples are given in the paper to illustrate the validity of the present approach.     

压电复合材料层合板的混沌动力学研究

压电复合材料具有独特的机电耦合性、比强度高、比刚度大和抗疲劳性能好等优点,越来越广泛地被应用于航空航天等工程领域中。根据Reddy的三阶剪切变形层合板理论,研究了受面内横向外激励力、面内纵向参数激励和面外参数激励以及压电参数激励联合作用下四边简支矩形复合材料层合板在1:2:4内共振情况下的六维平均方程。考虑平均方程存在一对双零特征值和两对纯虚特征值的情况,利用规范形理论进行化简,得到与原方程拓扑等价的最简规范形,然后在此基础上利用能量相位法从理论上分析得到系统会产生同宿的多脉冲跳跃现象;同时,基于平均方程,通过数值仿真,发现系统会产生混沌运动,具有跳跃现象。     

Cylindrical and spherical shells with cracks

The symmetric problem for the cylindrical and spherical shells containing a meridional crack is considered. The problem is solved for a uniform membrane load and a uniform bending moment applied to the surface of the crack. The extensional and bending components of the stress intensity factor ratio are obtained as functions of shell parameter and are tabulated. The results are also plotted in order to compare them with the existing asymptotic solutions.

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Zusammenfassung Das symmetrische Problem für zylindrische and sphärische Schalen, die einen meridionalen Riß haben, wurde behandeft. Das Problem wurde für eine einheitliche Membranlast and ein einheitliches Biegungsmoment auf der Oberfläche des Risses angewandt, gelöst.Die ausdehnenden und biegenden Komponente des Druckstärkefaktorenradius wurden als Funktionen des Schalenparameters erhalten and tabuliert. Die Ergebnisse wurden auch aufgezeichnet, um sie mit den bestehenden asymmetrischen Lösungen zu vergleichen.

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Résumé On considère le problème symétrique d'enveloppes cylindriques et sphériques comportant une fissure qui se développe suivant un méridien. La solution est apportée dans le cas d'une contrainte de membrane uniforme et d'un moment de flexion uniformément appliqué aux lèvres de la fissure.Ces composantes de traction et de flexion qui agissent sur le facteur d'intensité des contraintes sont determinées en fonction des paramètres de l'enveloppe, et présentées sous forme de tableaux. Les résultats ont 6t6 également portés en diagrammes, de manière à les comparer aux solutions asymptotiques existantes.

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This work was supported by the National Aeronautics and Space Administration under the Grant NGR 39-007-011.     

Analysis of active damping in composite laminate cylindrical shells of revolution with skewed PVDF sensors/actuators

Active damping in a FRP composite cylindrical shell with collocated piezoelectric sensors/actuators is studied. The electrode on the sensors/actuators are spatially shaped to reduce spillover between circumferential modes. A three noded, isoparametric, semianalytical finite element is developed and used to model the cylindrical shell. The element is based on a mixed piezoelectric shell theory which makes a single layer assumption for the displacements and a layerwise assumption for the electric potential. The effects of location of patch of collocated piezoelectric sensors/actuators, percentage length of the shell covered with these patches, fiber angle of the laminae in the composite laminate, stacking sequence of laminae in a laminate and skew angle of the sensor/actuator piezoelectric material, on the system damping for various modes is studied.     

An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.     

Suppression of nonlinear composite panel flutter with active/passive hybrid piezoelectric networks using finite element method

For the suppression of nonlinear panel flutter, a new optimal active/passive hybrid control design with piezoceramic actuators is proposed using finite element methods. This approach has the advantages of both active (high performance, feedback action) and passive (stable, low power requirement) systems. Piezoceramic actuators are connected in series with an external voltage source and a passive resonant shunt circuit which consists of an inductor and resistor. The shunt circuit should be tuned correctly to suppress the flutter effectively with less control effort as compared to purely active control. To obtain the best effectiveness, active control gains are simultaneously optimized together with the value of the resistor and inductor through a sequential quadratic programming method. The governing equations of the electromechanically coupled composite panel flutter are derived through an extended Hamilton’s principle, and a finite element discretization is carried out. The adopted aerodynamic theory is based on the quasi-steady first-order piston theory, and the von Kármán nonlinear strain–displacement relation is used. Nonlinear modal equations are obtained through a modal reduction technique. Optimal control design is based on linear modal equations of motion, and numerical simulations are based on nonlinear-coupled modal equations. Using the Newmark integration method, suppression results of a hybrid control and a purely active control are presented in the time domain.     

A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 John Wiley & Sons, Ltd.     

一种基于Layerwise理论的压电复合材料层合圆柱壳机电耦合有限单元

基于Reddy的Layerwise理论,对含压电铺层的复合材料层合壳的静力响应特性进行了理论研究。基于Layerwise理论,推导了含压电层的复合材料层合壳的应变分量与电场强度表达式。利用Hamilton原理和变分法,推导了压电智能层合壳的欧拉-拉格朗日方程,并采用有限元解法,建立了相应的有限元控制方程及其机电耦合刚度矩阵。通过算例结果与文献中的精确解和试验值进行了对比,表明相较于传统的经典层合板壳理论,本文理论方法的有效性和优势性;并分析了径厚比等参量对两端简支压电智能层合壳静力响应值的影响规律。      

Non-linear coupled multi-field mechanics and finite element for active multi-stable thermal piezoelectric shells

In this paper, a coupled multi-field mechanics framework is presented for analyzing the non-linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations in thermal environments. The mechanics incorporate coupling between mechanical, electric and thermal fields and encompass geometric non-linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and are combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. A finite element methodology and an eight-node coupled non-linear shell element are developed. The discrete coupled non-linear equations of motion are linearized and solved, using an extended cylindrical arc-length method together with a Newton–Raphson technique, to enable robust numerical predictions of non-linear active shells transitioning between multiple stable equilibrium paths. Validation and evaluation cases on laminated cylindrical strips and cylindrical panels demonstrate the accuracy of the method and its robust capability to predict non-linear response under thermal and piezoelectric actuator loads. Moreover, the results illustrate the capability of the method to model piezoelectric shells undergoing large shape changes by actively jumping between stable equilibrium states and quantify the strong relationship between shell curvature, applied electric potential, applied temperature differential and induced shape change. Copyright © 2008 John Wiley & Sons, Ltd.     

An investigation of the buckling behavior of composite elliptical cylindrical shells with piezoelectric layers under axial compression

The paper is focused on the elastic buckling behavior of piezocomposite elliptical cylindrical shell finite element formulation. The formulation is based on the shear deformation theory, and the serendipity quadrilateral eight-node element is used to study the elastic behavior of elliptical cylindrical shells. The strain-displacement relations are accurately accounted for in the formulation. The contributions of work done by the applied load are also incorporated. A constant gain displacement control algorithm coupling the direct and inverse piezoelectric effect is applied to provide active control of composite non-circular shells in a self-monitoring and self-controlling system. The governing equations obtained using the principle of minimum potential energy are solved through an eigenvalue approach. The influences of elliptical cross-sectional parameter and displacement feedback gain (G _d ) values on the critical buckling loads of elliptical cylindrical shells are examined.     

Prediction of energy absorption in composite stiffeners

The energy absorption behavior of composite stiffeners subjected to axial compression has been investigated. Flat plate, angle, and channel specimens were fabricated of T650-35/F584 graphite/epoxy plain-weave fabric and were crush tested under axial compression. A nonlinear finite element approach was used to model the sustained crushing of the flat plate specimens, and a progressive failure model was implemented as part of the finite element analysis to enable investigation of the fundamental mechanisms involved in the crushing behavior. The progressive failure model was based on linear elastic fracture mechanics for prediction of crack growth and a set of failure criteria for predicting fiber/matrix failures that occurred as a result of large deformations. Friction between the specimen and the crushing surface was included in the model. A semi-empirical analysis methodology was developed for prediction of the energy absorption capability of composite stiffeners based on crush tests of flat plate specimens and an understanding of the fundamentals of the energy absorption process.