A new finite element method for analysing symmetrically loaded thin shells of revolution

An essential feature of finite element methods of analysis for symmetrically loaded shells of revolution is the setting up of equations representing the response of a short ‘ring’ element to edge loading. In this paper the axisymmetric behaviour of a short elastic cylindrical shell element under edge loading is described in a new way by means of a matrix which is a combination of stiffness, flexibility and ‘neutral’ sub-matrices. The coefficients of the matrix are derived direct from the equations of the problem, which involves a trivial amount of work in comparison with conventional methods. The corresponding matrix for a short section of an arbitrary shell of revolution is set up with little additional effort, and its use is described for calculation of edge response coefficients for portions of spherical shells. Finally, the method is used to study by iteration the behaviour of a thin spherical shell of viscous material containing a rigid boss which is loaded radially inwards: changes in meridional profile are followed as deformation proceeds. Results are presented for both linear and non-linear viscous material.     

功能梯度材料薄球壳热屈曲问题

研究了由金属和陶瓷组成的功能梯度材料薄球壳热屈曲问题。用张量方法推导得到轴对称球壳稳定性方程。将热本构方程应用到球壳稳定性方程中,得到以位移表示的球壳热屈曲方程组。分别考虑均布外压和温度作用,采用伽辽金法计算分析简支球壳的热屈曲问题,给出薄球壳厚度、物性参数变化、内外表面温差变化引起的临界温度变化趋势和临界压力变化趋势。     

Stress-Strain State of a Corrugated Cylindrical Shell

By using the relations of the Kirchhoff–Love theory of shells and the method of distortion of the shape of the boundary in the process of axisymmetric heating through the thickness of shells according to a linear law, we construct a solution of the problem of statics for a corrugated cylindrical shell in an axially symmetric temperature field under the action of internal pressure.     

On the imperfection sensitivity of complete spherical shells

The imperfection sensitivity of elastic complete spherical shells under external pressure is studied for axisymmetric deformations and qualitatively different types of imperfections by means of a numerical analysis of the Reissner shell equations. It is shown that strong reductions of the critical load are obtained for small deviations of the middle surface of the shell from the perfect spherical configuration whereas imperfections of the shell thickness do not have a substantial influence on the critical load.     

A stochastic approach to the problem of stability of a spherical shell with initial imperfections

A probabilistic method is applied to the problem of stability of a spherical shell with initial imperfections, subjected to uniform external pressure. The distribution law of the normal deflections of the shell is obtained following the Smoluchovsky equation, at a given density of the initial deflections. On this basis the probability for the deflections to be in a given interval is found. A relation for the probability for a jump transition to a new stability form is obtained for a spherical shell with a given probability distribution of the initial imperfections. The relations obtained can be used for solving different kinds of reliability problems, as well as problems for prediction of the shell stability forms together with the probability for their realization.     

Buckling of functionally graded conical panels under mechanical loads

This paper presents an analytical approach to investigate the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure and the combination of these loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and linear stability equations in terms of displacement components for conical panels are derived by using the classical thin shell theory. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling loads. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of conical panels.     

Three-dimensional nonlinear thermo-elastic analysis of functionally graded cylindrical shells with piezoelectric layers by differential quadrature method

Three-dimensional nonlinear thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers under the effect of asymmetric thermo-electro-mechanical loads is carried out. The strain–displacement relations are based on the nonlinear Lagrangian strain–displacement relations; that is, nonlinear terms containing derivatives of the displacement in the radial direction are included. Material properties of the shell are assumed to be graded in the radial direction according to a power law but the Poisson’s ratio is assumed to be constant. Cylindrical shells are assumed to be under the effect of pressure loading in cosine form, ring pressure loads, electric and temperature fields. Numerical results of stress, displacement, electric and thermal fields are obtained by using two versions of the differential quadrature methods, namely polynomial and Fourier quadrature methods. The convergence of the solution is studied, and results of the axisymmetric loadings are verified with reported results for a cylindrical shell with material properties obeying a power law. Effects of the grading index of material properties, the temperature difference, the ratio of the mean radius to the thickness of the shell, boundary conditions, the thickness of piezoelectric layers and electric excitation on stress, displacement, electric and temperature fields are presented.     

Dynamics of a thick-walled dielectric elastomer spherical shell

The nonlinear vibration response of a thick-walled spherical shell subjected to the mechanical pressure and electric field is studied in this paper. When subject to an electric field through the thickness of the spherical shell, the material expands in plane and contracts in thickness. The dielectric elastomer is assumed to be isotropic and neo-Hookean. Based on simple geometrical and spherical capacitor assumptions, we deduce an explicit analytical equation of motion of the dielectric elastomer spherical shell. The dynamic behaviors of the spherical shell under a constant electric loading and periodic electric loading are analyzed. In addition, the critical voltage is calculated in terms of various loading.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

The nonlinear stability of axisymmetric functionally graded material annular spherical shells under thermo-mechanical load

The authors of this article investigate the nonlinear stability of axisymmetric functionally graded annular spherical shells with temperature-dependent material properties subjected to thermo-mechanical loads and resting on elastic foundations. Equilibrium and compatibility equations are derived by using the classical thin shell theory in terms of the shell deflection and the stress function. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain the closed-form of load–deflection paths. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the stability of the shells.     

刚塑性扁球壳受弹体冲击时动力学响应的近似解

研究扁球壳在弹体局部冲击载荷作用下的响应问题.首先,在球壳材料为刚塑性模型的假设下,基于现有球壳理论的研究基础,对球壳的变形能模型进行改进,仅考虑凹陷区域变形能和棱区膜变形能,根据能量守恒原理,得到了球壳在冲击载荷下的理论结果;其次,在棱区的宽度和凹陷区半径之间引入近似关系式l≈0.05a,以此为基础给出了刚塑性球壳在低速弹体冲击下的显式近似理论解;进一步,将近似解与实验的各种情况进行对比,误差均小于10%;最后,采用LS-DYNA 有限元软件进行数值模拟,也取得了与近似解一致的结果.     

Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions

This paper describes a general formulation for free, steady-state and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions. The formulation is derived by means of a modified variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first-order shear deformation shell theory. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. Fourier series and polynomials are applied to expand the displacements and rotations of each shell segment. The versatility of the formulation is demonstrated through the application of the following polynomials: Chebyshev orthogonal polynomials, Legendre orthogonal polynomials, Hermite orthogonal polynomials and power polynomials. Numerical examples are given for the free vibrations of functionally graded cylindrical, conical and spherical shells with different combinations of free, shear-diaphragm, simply-supported, clamped and elastic-supported boundary conditions. Validity and accuracy of the present formulation are confirmed by comparing the present solutions with the existing results and those obtained from finite element analyses. As to the steady-state and transient vibration analyses, functionally graded conical shells subjected to axisymmetric line force and distributed surface pressure are investigated. The effects of the material power-law distribution, boundary condition and duration of blast loading on the transient responses of the conical shells are also examined.     

Free vibration of deep spherical sandwich shells

Summary This paper deals with axisymmetric vibrations of deep sandwich spherical shells. The shell consists of a thick core and two face sheets of the same isotropic material with equal thickness. The appropriate differential equations have been obtained in terms of two new arbitrary functions which replace the meridional displacement components. The solution is in terms of Legendre functions from which numerical results are computed by using an iterative technique. Effects of geometric and elastic properties of the core and the face sheets are presented in the form of graphs.     

薄壁扁球壳在撞击载荷下的动态响应和吸能特性研究

对薄壁扁球壳结构撞击刚性板的动态响应和吸能特性进行了研究。通过不同材料特性和撞击速度下半径R与厚度t比值较大（R／t〉250）的扁球壳与刚性板的撞击试验，分析了此类球壳结构的动态响应和吸能特性。以能量守恒为基础，结合大变形薄壳理论和塑性铰方法，建立了撞击力与变形关系、撞击初速度与最大压缩位移的理论分析模型。通过理论计算和试验结果的对比，验证了理论分析模型的正确性，可以满足工程分析的需要。     

Buckling deformation in axially compressed elastic-plastic cylindrical shells initiated by local axisymmetric imperfections

Summary A solution for growth of a perturbation describing buckling deformation initiated by a local axisymmetric imperfection in an axially compressed elastic-plastic cylindrical shell is obtained in simple closed form for small time. The constitutive relations for the plastic strains are based on simple J ₂ flow theory. A comparison is made with bifurcation analysis, and localization of the buckling deformation is examined.     

土中浅埋层合扁球壳的非线性动力响应

本文建立了考虑横向剪切影响、纤维增强对称正交铺设层合扁球壳的非线性动力方程。应用一维波阻法分析了土与结构的相互作用,对冲击波作用下土中浅埋周边弹性支承层合扁球壳的非线性动力响应问题进行了探讨。数值计算中考虑了各种材料参数及横向剪切变形对动力响应的影响。      

Nonlinear response of a shear deformable S-FGM shallow spherical shell with ceramic-metal-ceramic layers resting on an elastic foundation in a thermal environment

This article presents an analytical approach to investigate the nonlinear stability of thick, functionally graded material (FGM) shallow spherical shells resting on elastic foundations, subjected to uniform external pressure and exposed to thermal environments. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents. Using the first-order shear deformation theory and the Galerkin method, the effects of materials, geometry, elastic foundation parameters, and temperature on the nonlinear response of the thick S-FGM shells are analyzed and discussed in detail.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.     

Thermoelastic solution for static deformations of functionally graded cylindrical shell bonded to thin piezoelectric layers

We study infinitesimal axisymmetric deformations of a functionally graded shell with piezoelectric layers perfectly bonded to its inner and outer surfaces, and the hybrid structure subjected to thermo-electro-mechanical loads. The material properties of the shell are assumed to be graded in the radial direction according to a power law but Poisson’s ratio is assumed to be constant. For simply-supported and grounded edges kept at a constant temperature, the problem is analyzed analytically by assuming a Navier type solution for the governing equations and the state space method for solving the resulting ordinary differential equations. Numerical results are given to illuminate influences of the mechanical and the electrical boundary conditions, the exponent of the power law variation, and the radius to thickness ratio.