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.     

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.     

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.     

Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells

The improved third order zigzag theory and its smeared counterpart (without the zigzag effect), recently developed by the authors for static analysis of piezoelectric laminated cylindrical shells, are extended to dynamics. The piezoelectric layers are considered as radially polarized to make use of the extension actuation mechanism that is best suited for effective actuation and sensing. The zigzag theory accounts for the layerwise variation of inplane displacements and includes the transverse normal extensibility under electric field, and also satisfies the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces of the shell. Yet, the number of primary displacement variables is only five, same as its smeared counterpart. The two theories are critically assessed for their accuracy by direct comparison with the three dimensional piezoelasticity solutions for free and forced vibration response of simply supported smart angle-ply infinite-length and cross-ply finite-length shells, with a variety of heterogeneous composite and sandwich laminates. It is shown that the zigzag theory, in spite of being computationally efficient, is very accurate even for shells with highly inhomogeneous laminates. In contrast, the smeared third order theory is grossly inadequate for smart shells made of inhomogeneous composite and sandwich substrates.     

复合材料层板层间缺陷分析——剪切滑移

根据三维弹性平衡方程和层间剪切滑移条件,导出了一个复合材料层板层间剪切滑移模型。本文模型具有一般形式的二维板壳理论的位移场及其平衡方程,但因引入了能反映层板界面粘贴情况及板面条件的剪切变形函数,模型因而简单又精确。层板的弯曲问题和屈曲问题被考虑,层间弱粘贴的影响被讨论。数值结果与精确解比较,表明了本文模型的高精度。      

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.     

Exact Solutions for the Analysis of Piezoelectric Fiber Reinforced Composites as Distributed Actuators for Smart Composite Plates

Performance of a layer of piezoelectric fiber reinforced composite (PFRC) material as the distributed actuator for smart composite plates has been investigated in this paper. The investigation is performed by finding the exact solutions for static analysis of simply supported symmetric and anti-symmetric cross-ply laminated plates integrated with a layer of PFRC material. The results suggest the potential use of PFRC materials for the distributed actuators of smart structures with both thick and thin substrate composite plates.     

Exact Solutions for the Analysis of Piezoelectric Fiber Reinforced Composites as distributed Actuators for Smart Composite Plates

Performance of a layer of piezoelectric fiber reinforced composite (PFRC) material as the distributed actuator for smart composite plates has been investigated in this paper. The investigation is performed by finding the exact solutions for static analysis of simply supported symmetric and anti-symmetric cross-ply laminated plates integrated with a layer of PFRC material. The results suggest the potential use of PFRC materials for the distributed actuators of smart structures with both thick and thin substrate composite plates.     

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     

Static,free vibration and thermal analysis of composite plates and shells using a flat triangular shell element

Finite element static, free vibration and thermal analysis of thin laminated plates and shells using a three noded triangular flat shell element is presented. The flat shell element is a combination of the Discrete Kirchhoff Theory (DKT) plate bending element and a membrane element derived from the Linear Strain Triangular (LST) element with a total of 18 degrees of freedom (3 translations and 3 rotations per node). Explicit formulations are used for the membrane, bending and membrane-bending coupling stiffness matrices and the thermal load vector. Due to a strong analogy between the induced strain caused by the thermal field and the strain induced in a structure due to an electric field the present formulation is readily applicable for the analysis of structures excited by surface bonded or embedded piezoelectric actuators. The results are presented for (i) static analysis of (a) simply supported square plates under doubly sinusoidal load and uniformly distributed load (b) simply supported spherical shells under a uniformly distributed load, (ii) free vibration analysis of (a) square cantilever plates, (b) skew cantilever plates and (c) simply supported spherical shells; (iii) Thermal deformation analysis of (a) simply supported square plates, (b) simply supported-clamped square plate and (c) simply supported spherical shells. A numerical example is also presented demonstrating the application of the present formulation to analyse a symmetrically laminated graphite/epoxy laminate excited by a layer of piezoelectric polyvinylidene flouride (PVDF). The results presented are in good agreement with those available in the literature.The work was partly sponsored by a grant (DAAHO4-95-1-0175) from the army research office with Dr. Gary Anderson as the grant monitor.     

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.     

Active control of functionally graded laminated cylindrical shells

An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO₃ and PZT-5A) and various loading forms on the active vibration control are described in the numerical results.     

METHOD OF INITIAL FUNCTIONS FOR THE ANALYSIS OF LAMINATED CIRCULAR CYLINDRICAL SHELLS UNDER AXISYMMETRIC LOADING

The method of initial functions has been used for the static analysis of an infinite and simply supported, orthotropic, and laminated, circular cylindrical shell of revolution subjected to axisymmettic load. In this method the three-dimensional state equations for an individual ply of a laminated shell are established without making any a priori assumptions regarding the distribution of stresses and displacements across the thickness of the shell By using the continuity conditions of displacements and stresses on each interface between adjacent layers, the state equation far the laminate is obtained. Using the Cayley-Hamilton theorem, the transfer matrix that maps the initial state vector into the field is evaluated explicitly, leading to an exact solution of the problem (MIF—exact). Alternatively, depending on the number of terms retained in the series expansion of the transfer matrix, different-order theories of MIF are derived. The results of different-order MIF theories, classical theories, and shear deformation shell theories are compared with the results of MIF—exact to assess their accuracy and limitations.     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

Mesh‐free models for static analysis of smart laminated composite beams

This paper is concerned with the development of mesh‐free models for the static analysis of smart laminated composite beams. The overall smart composite beam is composed of a laminated substrate composite beam and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the distributed actuator layer of the smart beam. A layer‐wise displacement theory and an equivalent single‐layer theory have been used to derive the models. Several cross‐ply substrate beams are considered for presenting the numerical results. The responses of the smart composite beams computed by the present new mesh‐free model based on the layer‐wise displacement theory excellently match with those obtained by the exact solutions. The mesh‐free model based on the equivalent single‐layer theory cannot accurately compute the responses due to transverse actuation by the piezoelectric actuator. The models derived here suggest that the mesh‐free method can be efficiently used for the numerical analysis of smart structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Three dimensional analysis of piezoelectric circular cylindrical shell of finite length

Summary An exact elasticity solution for a simply supported piezoelectric circular cylindrical shell of finite length is presented. Different boundary conditions are considered, which correspond to the respective situations when it is acted as sensor or actuator. The stresses, displacements and electric potential distributions are shown. Our results show that the distributions depend strongly on the boundary conditions and can not be justified by purely elastic structures. The presented results can be used not only to assess various approximate theories but also to enhance the understanding of the response behavior of piezoelectric structures.     

Influences of elastic foundations and boundary conditions on the buckling of laminated shell structures subjected to combined loads

This article presents to study the stability of laminated orthotropic cylindrical and truncated conical shells resting on elastic foundations and subjected to combined loads with the clamped and simply supported boundary conditions. Here, axial tensile loads separately applied to the small and large bases of a laminated truncated conical shell, respectively. The basic relations, the modified Donnell type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells on the Pasternak type elastic foundation. Applying Galerkin method, the critical combined loads of laminated orthotropic conical shells on the Pasternak type elastic foundation with different boundary conditions are obtained. The appropriate formulas for single-layer and laminated cylindrical shells on the Pasternak type elastic foundation made of orthotropic and isotropic materials are found as special cases. Finally, influences of the boundary conditions, the elastic foundation, the number and ordering of the layers and variations of the shell characteristics on the critical combined loads are investigated. The results are compared with their counterparts in the literature.     

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.     

Asymmetric postbuckling of symmetrically laminated cross ply,short cylindrical panels under compression

Buckling and initial postbuckling behavior of symmetrically laminated, thin cross ply cylindrical panels under axial compression are investigated. The panels are simply supported at all four edges. Closed form solutions are obtained for the buckling loads. The initial asymmetric postbuckling behavior is demonstrated by computing the postbuckling coefficients within the context of Koiter's theory of elastic stability. Parameter studies involving the flatness parameter, the length-to-width ratio, number of layers and Young's moduli ratio are presented for typical cross ply cylindrical panels likely to be encountered in practice.     

The behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces in cylindrical bending

The behavior of a simply supported angle-ply laminated cylindrical shell in cylindrical bending with viscoelastic interfaces is studied. The Kelvin–Voigt model is adopted to represent the character of interfaces. State-space formulations are developed based on the exact elasticity equations, and in particular, a variable substitution technique is used to derive the state equations with constant coefficients. Since the behavior of this structure under static loading is time-dependent, the power series expansion technique is used to approximate the variations of physical variables with time. The response for a laminated shell with viscous interfaces is also investigated as a particular case. Results show that the displacements as well as the maximum stresses in the panel increase rapidly with time. Thus, the imperfect bonding properties should be considered carefully in the design of laminated structures.