A THEORETICAL SOLUTION OF CYLINDRICAL SHELLS FOR AXISYMMETRIC PLAIN STRAIN ELASTODYNAMIC PROBLEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅａｎａｌｙｓｉｓｆｏｒｄｙｎａｍｉｃｐｒｏｂｌｅｍｏｆｅｌａｓｔｉｃｂｏｄｉｅｓｈａｓｂｅｅｎｓｔｕｄｉｅｄｆｏｒｍａｎｙｙｅａｒｓ.Ｂａｓｅｄｏｎｔｈｅｍｏｍｅｎｔｌｅｓｓｔｈｉｎｓｈｅｌｌｔｈｅｏｒｙ ,ＨｕｔｈａｎｄＣｏｌｅ[1]ｓｔｕｄｉｅｄｔｈｅｓｔｒｅｓｓｗａｖｅｉｎａｓｐｈｅｒｉｃａｌｓｈｅｌｌｐｒｏｄｕｃｅｄｂｙｐｒｅｓｓｕｒｅｌｏａｄｓｉｎ 1 955.Ｍｃｉｖｏｒ[2 ]ｄｉｓｃｕｓｓｅｄｔｈｅｆｌｅｘｕｒａｌｓｔｒｅｓｓａｎｄｍｅｍｂｒａｎｅｓｔｒｅｓｓｉｎａ…     

A theoretical solution of cylindrically isotropic cylindrical tube for axisymmetric plane strain dynamic thermoelastic problem

A dynamic thermoelastic solution of a cylindrically isotropic cylindrical tube or solid cylinder with axisymmetric plane deformations is developed. Since there exist thermal boundary conditions and tractions on the two surfaces of a cylindrical tube, the problem under consideration is with inhomogeneous boundary conditions. Therefore we introduce a special function to transform the inhomogeneous boundary conditions to homogeneous ones for an unknown function. Then by using the method of separation of variables, the unknown function can be expressed as the multiplication series of Bessel functions and unknown time functions. Thirdly, by virtue of the orthogonal properties of Bessel functions, the equations about these unknown time functions are derived and the solutions are obtained. Finally, the displacement is obtained by adding the two parts mentioned above. By means of the present method, integral transform can be avoided. It is suitable for arbitrary thermal loads and mechanical loads. Numerical results are also presented for thermal shocked, cylindrically isotropic cylindrical tube and solid cylinder. Project supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016).     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

On the theory of loaded general cylindrical Cosserat elastic shells

We study the equilibrium of cylindrical Cosserat elastic shells under the action of body loads and tractions and couples distributed along its edges. The shells have arbitrary open or closed cross-sections and are made from an isotropic and homogeneous material. On the end edges, the appropriate resultant forces and resultant moments are prescribed. We consider the problem of Almansi for cylindrical Cosserat shells and obtain a solution expressed in the form of the displacement field.     

Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

基于混合变量条形传递函数方法的圆柱壳屈曲分析

提出了一种分析交各向异性圆柱壳和阶梯圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Fluegge薄壳理论，通过定义广义位移变量和对应的广义力变量，建立了圆柱壳混合变量能量泛函；然后通过引入条形单元，定义混合状态变量和采用传递函数方法对超级壳单元求解，得到具有多种边界条件圆柱壳屈曲问题的半解析解；最后通过位移连续和力平衡条件，可以得到阶梯圆柱壳屈曲问题的解。理论解推导过程表明此方法在引入边界条件和进行阶梯圆柱壳求解时非常方便。算例分析的结果验证了本方法的正确性。     

Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder

A theoretical method for analyzing the axisymmetric plane strain elastodynamic problem of a non-homogeneous orthotropic hollow cylinder is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions and the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to homogeneous ones. By virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally obtained, which can be degenerated in a rather straightforward way into the solution for a homogeneous orthotropic hollow cylinder and isotropic solid cylinder as well as that for a non-homogeneous isotropic hollow cylinder. Using the present method, integral transform can be avoided and it can be used for hollow cylinders with arbitrary thickness and subjected to arbitrary dynamic loads. Numerical results are presented for a non-homogeneous orthotropic hollow cylinder subjected to dynamic internal pressure. The project supported by the National Natural Science Foundation of China (10172075 and 10002016)     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Stability of Orthotropic Corrugated Cylindrical Shells

A technique for stability analysis of cylindrical shells with a corrugated midsurface is proposed. The wave crests are directed along the generatrix. The relations of shell theory include terms of higher order of smallness than those in the Mushtari–Donnell–Vlasov theory. The problem is solved using a variational equation. The Lamé parameter and curvature radius are variable and approximated by a discrete Fourier transform. The critical load and buckling mode are determined in solving an infinite system of equations for the coefficients of expansion of the resolving functions into trigonometric series. The solution accuracy increases owing to the presence of an aggregate of independent subsystems. Singularities in the buckling modes of corrugated shells corresponding to the minimum critical loads are determined. The basic, practically important conclusion is that both isotropic and orthotropic shells with sinusoidal corrugation are efficient only when their length, which depends on the waveformation parameters and the geometric and mechanical characteristics, is small     

Dynamical Behavior of Viscoelastic Cylindrical Shells Under Axial Pressures

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｒｙｏｆｖｉｓｃｏｅｌａｓｔｉｃｉｔｙａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｅｎｂｅｃｏｍｉｎｇａｖｅｒｙａｔｔｒａｃｔｉｖｅｒｅｓｅａｒｃｈｆｉｅｌｄｓｉｎｃｅｖｉｓｃｏｅｌａｓｔｉｃｓｔｒｕｃｔｕｒｅｓｗｅｒｅｗｉｄｅｌｙａｐｐｌｉｅｄｉｎｔｏｖａｒｉｏｕｓｅｎｇｉｎｅｅｒｉｎｇ .Ｂｕｔｔｏｔｈｅａｕｔｈｏｒｓ’ｋｎｏｗｌｅｄｇｅ,ｔｈｅｒｅａｒｅｒａｒｅｆｏｒｐａｐｅｒｓｃｏｎｃｅｒｎｉｎｇｗｉｔｈｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｏｆｖｉｓｃｏｅ…     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

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

ＥＸＡＣＴＳＯＬＵＴＩＯＮＯＦＴＨＥＴＨＩＣＫＬＡＭＩＮＡＴＥＤＯＰＥＮＣＹＬＩＮＤＲＩＣＡＬＳＨＥＬＬＳＷＩＴＨＦＯＵＲＣＬＡＭＰＥＤＥＤＧＥＳＦａｎＪｉａ－ｒａｎｇ（范家让）（ＨｅｆｅｉＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ）ＨｅｆｅｉＤｉ...     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

On the nonlinear stability of symmetrically laminated cylindrically orthotropic shallow spherical shells

In this paper,the large deflection theory of symmetrically laminated cylindricallyorthotropic shallow spherical shells is established.Based on this theory,applying themodified iteration method,the analytic solution for critical buckling loads of the shells withrigidly clamped edges under actions of uniform pressure has been obtained.     

Stability of corrugated composite noncircular cylindrical shells under external pressure

Cylindrical shells consisting of cylindrical panels of smaller radius and subjected to uniform external pressure are analyzed for stability. The geometrical parameters of the shells are approximated by Fourier series on a discrete set of points. The Timoshenko theory of shells is used. The solution is represented in the form of trigonometric series. It is shown that short-and medium-length shells with cylindrical panels are advantageous over circular shells. By selecting appropriate parameters of the panels, keeping the mass of the shell constant, it is possible to achieve a significant gain in critical loads. The shells under consideration are less effective than isotropic shells. Shells with sinusoidal corrugation under external pressure are of no practical interest __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 91–102, December 2007.     

Analytical solution for axisymmetric problem of thick laminated closed cantilever shells in hamilton system

By giving up any assumptions about displacement models and stress distribution, the mixed state Hamilton equation for the axisymmetric problem of the thick laminated closed cantilever cylindrical shells is established. An identical analytical solution is obtained for the thin, moderately thick and thick laminated closed cantilever cylindrical shells. All equations of elasticity can be satisfied, and all elastic constants can be taken into account. This work is supported by the National Natural Science Foundation of China.     

圆柱壳在两种非轴对称同步移动载荷作用下的动力响应

本文根据工程实例计算的需要,研究了有限长弹性圆柱薄壳在两种非轴对称同步移动载荷作用下的动力响应问题。两种非轴对称同步移动载荷作用是指非轴对称移动的集中载荷,以及同步移动且作用范围随移动位置增加的均布载荷的共同作用。建立了在上述两种不同类型载荷作用下的具有对称形式的动力学微分方程组;分别采用Dirac函数与Heaviside函数表示移动的均布载荷与集中载荷,设定位移函数的基础上,应用Galerkin法及Laplace变换,求得了圆柱薄壳应力与位移动态响应的解析解;通过具体算例,将所得到解析解的计算结果与ANSYS数值解进行了对比分析,验证了解析解的可靠性。     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

Circumferentially traveling radial loads on rings and cylinders

The problem of circumferentially traveling radial loads on rings and infinitely long cylindrical shells will be considered in this paper. Since strong transitional excitations are considered, the response involves moderately large rotations. Hence the governing shell equations are non-linear. To facilitate their solution, a modified version of the Linstedt-Poincaré perturbation procedure is employed. Based on this solution, several numerical results are presented. In addition to considering the effects of displacement induced non-linearity, special emphasis is given to the response behavior in load speed zones which mark transitions from sub- to supercritical waveforms.     

Exact solution for laminated continuous open cylindrical shells

I.Introducti0nNowadays,thecurrenttheoriesofplatesandshe1ls,suchasthetheoriesofReissner's,KirchhoffLove'sandAmbartsumyan'setc,areestablishedons0mehypotheses.Forexample-assumethatthemechanicalquantitiesarethepolynomialsofacertaincoordinatevariable.Itisshown…