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.     

A Set of Strain-Displacement Relations in Nonlinear Theories of Rods and Shells

The direct approach of Reissner and of Simmonds and Danielson is used to derive polynomial approximations to the strain-displacement relations in nonlinear theories of rods and shells via the representation of the finite rotational displacement matrix in terms of the exponential of an antisymmetric matrix.     

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.     

Forced Radial Motions of Nonlinearly Viscoelastic Shells

This paper contains an extensive global treatment of radial motions of compressible nonlinearly viscoelastic cylindrical and spherical shells under time-dependent pressures. It furnishes a variety of conditions on a general class of material properties and on the pressure terms ensuring that there are solutions existing for all times, there are unbounded globally defined solutions, there are solutions that blow up in finite time, and there are solutions having the same period as that of the pressure terms. The shells are described by a geometrically exact 2-dimensional theory in which the shells suffer thickness strains as well as the standard stretching of their base surfaces. Consequently their motions are governed by fourth-order systems of semilinear ordinary differential equations. This work shows that there are major qualitative differences between the nonlinear dynamical behaviors of cylindrical and spherical shells.      

Nonlinear shell-type to beam-type fea simplifications for composite frp poles

This paper presents a semi-analytical finite element analysis of pole-type structures with circular hollow cross-section. Based on the principle of stationary potential energy and Novozhilov’s derivation of nonlinear strains, the formulations for the geometric nonlinear analysis of general shells are derived. The nonlinear shell-type analysis is then manipulated and simplified gradually into a beam-type analysis with special emphasis given on the relationships of shell-type to beam-type and nonlinear to linear analyses. Based on the theory of general shells and the finite element method, the approach presented herein is employed to analyze the ovalization of the cross-section, large displacements, the P-Δ effect as well as the overall buckling of pole-type structures. Illustrative examples are presented to demonstrate the applicability and the efficiency of the present technique to the large deformation of fiber-reinforced polymer composite poles accompanied with comparisons employing commercial finite element codes.     

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     

基于一种简化Green应变的薄壳大挠度方程

将正交曲线坐标下的格林应变张量引入到薄壳大变形分析,通过建立恰当的基本假设,直接导出了用格林应变张量表示的壳体变形几何方程,将该方程代入到本构方程,由能量原理得到了小应变非线性变形平衡方程、内力方程和边界条件,在此基础上提出了大应变变形的简化分析方法．文中导得的方程涵盖薄板壳大、小变形的全部方程,推导过程简捷、系统,所得结果规则、清晰,与此前有关分析方法的结果完全吻合．     

双曲率壳的力学研究进展

双曲率壳是飞机、汽车以及船舶上常见的薄壁结构，其中性面可看作是一条动曲线沿着另一条曲线扫掠所形成的曲面．双曲率壳的非线性理论不断更新推动着双曲率壳力学行为的研究．随着工程实际应用的不断改进，如功能梯度材料(FGM)，加筋壳，弹性地基模型等的引入，双曲率壳在强度、变形和稳定性等方面的研究得到了进一步促进．本文首先回顾了双曲率壳结构非线性力学基本理论发展过程，主要阐述了经典的二维板壳理论，如Donnell 薄壳理论，一阶剪切变形壳理论，高阶剪切变形壳理论，和三维板壳理论的理论体系及基本公式，并对几种理论之间的联系和应用进行了总结和讨论，简述了近几十年来国内外学者在双曲率壳非线性弯曲、稳定性和振动等方面的最新研究成果，最后对双曲率壳体研究目前的局限性和未来的研究方向进行了探讨．     

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the perturbation method of multiple scales. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

Elastoplastic stress analysis for cylindrical shell with flat strip imperfection

Based on unequidistant B-spline function, generalized spline subdomain displacement mode of rotational shell is obtained by taking double-direction interpolation of spline. The elastoplastic constitutive equation of shells is established by using the endochronic theory.According to the initial deflection theory of shells, the elastoplastic stress analysis of cylindrical shells with flat strip geometrical imperfection is studied. Numerical results show that the geometrical imperfection has a great effect on the stress distribution of shells.     

Postbuckling analysis and imperfection sensitivity of general shells by the finite element method

Buckling and imperfection sensitivity are the primary considerations in analysis and design of thin shell structures. The objective here is to develop accurate and efficient capabilities to predict the postbuckling behavior of shells, including imperfection sensitivity. The approach used is based on the Lyapunov–Schmidt–Koiter (LSK) decomposition and asymptotic expansion in conjunction with the finite element method. This LSK formulation for shells is derived and implemented in a finite element code. The method is applied to cylindrical and spherical shells. Cases of linear and nonlinear prebuckling behavior, coincident as well as non-coincident buckling modes, and modal interactions are studied. The results from the asymptotic analysis are compared to exact solutions obtained by numerically tracking the bifurcated equilibrium branches. The accuracy of the LSK asymptotic technique, its range of validity, and its limitations are illustrated.     

Axisymmetric shells and plates on tensionless elastic foundations

This paper first describes a finite element method for the large deflection analysis of axisymmetric shells and plates on a nonlinear tensionless elastic foundation. Through the use of discrete data points, any form of nonlinear elastic foundation behaviour can be easily modelled. The analysis is then validated by comparison with existing results for circular plates and beams as the only existing results for shells on tensionless foundations are found to be in error. Following this verification, the analysis is applied to investigate the behaviour of shallow spherical shells subject to a central concentrated load on tensionless linear elastic foundations. A number of insightful conclusions regarding the behaviour of such structure-foundation systems are drawn. The numerical results for shells are believed to be the first correct results, which may be useful in benchmarking results from other sources in the future.     

A perturbation method for nonlinear vibrations of imperfect structures: Application to cylindrical shell vibrations

A perturbation method is used to analyse the nonlinear vibration behaviour of imperfect general structures under static preloading. The method is based on a perturbation expansion for both the frequency parameter and the dependent variables. The effects on the linearized and nonlinear vibrations caused by geometric imperfections, a static fundamental state, and a nontrivial static state are included in the perturbation procedure.The theory is applied in the nonlinear vibration analysis of anisotropic cylindrical shells. In the analysis the specified boundary conditions at the shell edges can be satisfied accurately. The characteristics of the analysis capability are shown through examples of the vibration behaviour of specific shells. Results for single mode and coupled mode nonlinear vibrations of shells are presented. Parametric studies have been performed for a composite shell.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature.     

NONLINEAR DYNAMICAL BIFURCATION AND CHAOTIC MOTION OF SHALLOW CONICAL LATTICE SHELL

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.     

碳纳米管增强复合材料圆锥薄壳非线性自由振动

考虑碳纳米管复合材料作为功能梯度材料的不均匀性,基于连续介质理论以及哈密尔顿变分原理,建立了功能梯度碳纳米管增强复合材料开口圆锥薄壳结构的非线性运动偏微分控制方程,然后利用Galerkin法,将非线性偏微分控制方程转化为常微分控制方程,进而采用谐波平衡法求解了开口圆锥壳的非线性自由振动问题,并探讨了圆锥薄壳几何参数、碳纳米管参数对结构非线性自由振动的影响．数值研究表明结构的无量纲非线性自由振动频率与线性自由振动频率的比值随圆锥薄壳长厚比的增大而变小、并随圆锥角的增大而变大．     

复合材料层合板壳非线性力学的研究进展

复合材料层合板壳是由多种组分材料组合而成.与单一材料的板壳结构相比,它无明确的材料主方向,各层间材料间断和不连续,具有明显的几何非线性和材料非线性等新的特点.其失效模式也远比单一材料的情况复杂,具有如基体开裂、脱胶、分层、分层裂纹偏转、多分层以及分层传播等多种模式.各国学者基于不同的考虑,提出了多种方法研究复合材料层合板壳的失效.首先,在简要介绍了层合板壳线性力学基本理论的基础上,重点回顾了层合板壳结构非线性力学几种基本理论发展的过程,主要阐述了经典大挠度非线性理论、一阶剪切变形理论、高阶剪切变形理论、锯齿理论、广义分层理论的理论体系及基本公式,并对几种理论之间的联系和差异进行了总结;其次,介绍了当前层合结构非线性领域的研究进展,综述了典型复合材料板壳结构的失效机理及优化设计、复合材料板壳结构在复杂环境下的破坏机理、复合材料板壳结构的物理非线性、含脱层纤维增强复合材料板壳结构的破坏机理等各研究热点的最新研究成果;最后,对该领域未来的研究方向进行了展望.