Nonlinear analysis of doubly curved symmetrically laminated shallow shells with rectangular planform

Summary A multi-mode solution to the dynamic Marguerre-type nonlinear equations is presented for the nonlinear free vibration of doubly curved, symmetrically laminated, imperfect shallow shells of rectangular planform on a Winkler-Pasternak elastic foundation. The shell edges are assumed to be transversely supported and the variation of rotational stiffness is identical along opposite edges. Generalized double Fourier series with time-dependent coefficients and the method of harmonic balance are used in the solution. The boundary condition for the varying rotational stiffness is fulfilled by replacement of bending moments along the four edges by an equivalent lateral pressure. Based on a single-mode approximation numerical results for the amplitude-frequency response of doubly curved isotropic, orthotropic, cross-ply and angle-ply shallow shells with square planform are presented for various boundary conditions, material properties, curvature ratios, initial imperfections, edge tensions, and moduli of the elastic foundation. Graphical results for postbuckling behavior of an imperfect angle-ply cylindrical panel are also presented as a special case.

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Nichtlineares Verhalten zweifach gekrümmter, symmetrisch geschichteter dünner Schalen mit rechteckigem Grundriß

Übersicht Vorgestellt wird eine Mehrfachwellenform-Lösung der dynamischen Gleichungen vom Marguerre-Typ für die nichtlinearen Eigenschwingungen zweifach gekrümmter dünner Schalen aus symmetrischen Schichtungen, die geometrisch imperfekt sind und über einen rechteckigen Grundriß auf einer elastischen Winkler-Pasternak-Bettung lagern. Angenommen wird eine gelenkige Lagerung der Plattenränder mit einer veränderlichen Drehbehinderung, die auf gegenüberliegenden Rändern gleich ist. Zur Lösung werden verallgemeinerte zweifache Fourier-Reihen mit zeitabhängigen Koeffizienten und die Methode der harmonischen Balance benutzt. Die Randbedingung für die veränderliche Drehbehinderung wird dadurch erfüllt, daß die Randmomente durch einen äquivalenten Vertikaldruck ersetzt werden. Mit der Näherung nur einer Wellenform werden numerische Ergebnisse für die Beziehung zwischen Amplitude und Frequenz angegeben; variiert werden dabei die Randbedingungen, die anfänglichen Imperf ektionen, die Krümmungsverhältnisse, die Bettungszahlen und die Materialkonstanten, wobei die Platte entweder isotrop oder bei recht- und schiefwinkliger Schichtung orthotrop ist. Als Sonderfall wird auch das Nachbeulverhalten einer imperfekten, schiefwinklig geschichteten Zylinderschale graphisch dargestellt.

     

Asymptotic analysis of laminated plates and shallow shells

It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

Nonlinear dynamic buckling of symmetrically laminated cylindrically orthotropic shallow spherical shells

Summary In this paper, a model of cusped catastrophe at nonlinear dynamic buckling of a symmetrically laminated cylindrically orthotropic shallow spherical shell is presented. The shell is subjected to an axisymmetrical load. Effects of transverse shear are taken into account. Effects of the shear modulus, geometry and parameters of the material on the nonlinear dynamic buckling are discussed. Received 2 April 1997; accepted for publication 27 November 1997     

Application of the method of split rigidities to anisotropic laminated shallow shells

In this paper, according to the method stated by Hu Haichang in [3], on the basis of [1], the method of split rigidities is generalized for the purpose of solving the problems of lateral deflection, stability and lateral vibration for anisotropic laminated shallow shells, and a simple and practical approximate method is obtained, in which the errors and computing work are comparatively small.     

Exact solution of the thick laminated open cylindrical shells with four clamped edges

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Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

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On the vibration of laminated nonhomogeneous orthotropic shells

In this paper, an analytical procedure is given to study the free vibration of the laminated homogeneous and non-homogeneous orthotropic conical shells with freely supported edges. The basic relations, the modified Donnell type motion and compatibility equations have been derived for laminated orthotropic truncated conical shells with variable Young’s moduli and densities in the thickness direction of the layers. By applying the Galerkin method, to the basic equations, the expressions for the dimensionless frequency parameter of the laminated homogeneous and non-homogeneous orthotropic truncated conical shells are obtained. The appropriate formulas for the single-layer and laminated complete conical and cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the non-homogeneity, the number and ordering of layers and the variations of the conical shell characteristics on the dimensionless frequency parameter are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Nonlinear vibration of symmetrically laminated composite skew plates by finite element method

Here, the large amplitude free flexural vibration behavior of symmetrically laminated composite skew plates is investigated using the finite element method. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Kármán's assumptions is introduced. The nonlinear matrix amplitude equation obtained by employing Galerkin's method is solved by direct iteration technique. Time history for the nonlinear free vibration of composite skew plate is also obtained using Newmark's time integration technique to examine the accuracy of matrix amplitude equation. The variation of nonlinear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, fiber orientation and boundary condition.     

Nonlinear free vibration analysis of piezoelastic laminated plates with interface damage

This paper presents a nonlinear model for piezoelastic laminated plates with damage effect of the intra-layers and inter-laminar interfaces. Discontinuity of displacement and electric potential on the interfaces are depicted by three shape functions. By using the Hamilton variation principle, the three-dimensional nonlinear dynamic equations of piezoelastic laminated plates with damage effect are derived. Then, by using the Galerkin method, a mathematical solution is presented. In the numerical studies, effects of various factors on the natural frequencies and nonlinear amplitude-frequency response of the simply-supported peizoelastic laminated plates with interfacial imperfections are discussed. These factors include different damage models, thickness of the piezoelectric layer, side-to-thickness ratio, and length-to-width ratio.     

Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates

A meshless method based on thin plate spline radial basis functions and higher-order shear deformation theory are presented to analyze the free vibration of clamped laminated composite plates. The singularity of thin plate spline radial basis functions is eliminated by adding infinitesimal to the zero distance. Convergence characteristics of the present thin plate spline radial basis functions for the vibration analysis of the clamped laminated plates are investigated. The frequencies computed by the present method agree well with the available published results.     

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric (FGP) layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton’s principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different parameters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.     

R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells

Nonlinear free vibration of functionally graded shallow shells with complex planform is investigated using the R-functions method and variational Ritz method. The proposed method is developed in the framework of the first-order shear deformation shallow shell theory. Effect of transverse shear strains and rotary inertia is taken into account. The properties of functionally graded materials are assumed to be varying continuously through the thickness according to a power law distribution. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. Admissible functions are constructed by the R-functions theory. To implement the proposed approach, the corresponding software has been developed. Comprehensive numerical results for three types of shallow shells with positive, zero and negative curvature with complex planform are presented in tabular and graphical forms. The convergence of the natural frequencies with increasing number of admissible functions has been checked out. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural and nonlinear frequencies is brought out. For simply supported rectangular FG shallow shells, the results obtained are compared with those available in the literature. Comparison demonstrates a good accuracy of the approach proposed.     

An analysis of shallow spheroidal shells by a semi-inverse contour method

This paper presents a linear analysis of a shallow prolate spheroidal shell with a planar elliptical boundary. The shell is subjected to a uniform load, q, and clamped along the boundary. The theory used in this paper is characterized by the well known Mushtari-Donnell-Vlasov equations which consist of a compatibility equation and an equilibrium equation where the normal displacement, w, and a stress function, φ, are the dependent variables.The method employed for the solution of this problem is developed in three major stages. The first stage involves the determination of w, under the assumption that the contours of w be ellipses concentric to the boundary. The second stage is devoted to the determination of a stress function φ, which, together with w, satisfies the MDV compatibility equation exactly. The third stage of the development is concerned with the computation of a loading, q^*, which, together with w and φ, satisfies the equilibrium equation exactly and which is nearly equal to the desired uniform loading q.     

The finite deflection equations of anisotropic laminated shallow shells

In this paper, basing on ref. [1] we improved and extended that which is concerned with a view of investigating the finite deflection equations of anisotropic laminated shallow shells subjected to static loads, dynamic loads and thermal loads. We have considered the most general bending-stretching couplings and the shear deformations in the thickness direction, and derived the equilibrium equations, boundary conditions and initial conditions. The differential equations expressed in terms of generalized displacements u₀, ₀ and are obtained. From them, we could solve the problems of stress analysis, deformation, stability and vibration. For some commonly encountered cases, we derived the simplified equations and methods.     

Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method

This paper focuses on the free vibration analysis of thick, rotating laminated composite conical shells with different boundary conditions based on the three-dimensional theory, using the layerwise differential quadrature method (LW-DQM). The equations of motion are derived applying the Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying the DQM in the meridional direction. This study demonstrates the applicability, accuracy, stability and the fast rate of convergence of the present method, for free vibration analyses of rotating thick laminated conical shells. The presented results are compared with those of other shell theories obtained using conventional methods and a special case where the angle of the conical shell approaches zero, that is, a cylindrical shell and excellent agreements are achieved.     

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitude-frequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.