Numerical solution of the two-dimensional unsteady-state problem of the motion of a shell under the action of the products of an axial detonation

A numerical solution is obtained to the problem of the motion of an incompressible cylindrical shell with a charge of explosive, with excitation of the detonation simultaneously along the whole axis of the charge. The strength of the shell is not taken into consideration. A three-term equation of state is adopted for the products of the detonation. In [1] a numerical solution is obtained to the problem of the one-dimensional motion of a shell with the axial detonation of a charge of explosive.     

The distributions of the interlaminar stress in a laminated shell with free end boundary

This paper studies the free edge effect yielded by interlaminar stress in a laminated cylindrical shell made up of fiber reinforced layer [0°], [90°] and the isotropic material layer under axisymmetric thermal load or radial pressure. Both ends of the shell are in free boundary condition. The exact solution of the problem can be obtained by using the three-dimensional theory of elasticity. For illustration, the numerical laminar stresses in a double-layer laminated shell under thermal load or radial pressure are calculated.     

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.     

Hurling of shells by hollow charges

Results of the numerical solution of the problem of one-dimensional hurling of shells by hollow explosive charges are elucidated. The results of the numerical solution are compared with asymptotic formulas. Numerous domestic and foreign papers have been devoted to the question of hurling shells by explosive charges. A numerical solution of the problem of convergence of a ring to the center under the effect of detonation products is presented in [1–3]. The problem of hurling a shell by a hollow explosive charge with an internal lining is considered in [4]; the solution of the problem of hurling a shell by a hollow explosive charge without the cavity lining is presented in [5] on the basis of the energy-balance equations; however, the complete picture of the processes occurring in the detonation products is not considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 161–166, May–June, 1976.     

ON THE GENERAL SOLUTION OF CYLINDRICAL SHELL EQUATIONS

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Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

Constructive symmetry in the stability problem for a cylindrical shell reinforced by stiffening rings under hydrostatic pressure

We use the solution of the problem under study which was earlier obtained by the author [1—3] and which is based on a decomposition of the reinforced shell into separate constructive elements (the ribs and the shell itself) and then on deriving the equilibrium equations and consistency conditions for their strains. Under the assumption that the shell precritical state is momentless, this is a sufficiently exact solution. Its drawback is the significant complexity of the computational algorithm. In the present paper, we show that the laboriousness of the algorithm can be reduced dramatically if the shell under study has elements of design symmetry (identical ribs, their uniform spacing, or both). In addition, we present dependencies determining the stiffening rib rigidity needed to ensure that the shell remains stable for a given critical value of the external hydrostatic pressure.     

Stability of nonlinear local axisymmetrical strains of shells of revolution

A brief review of the results of investigation of the stability of the axisymmetrical strains of elastic shells of revolution is contained in [1, 2]. In [3] the problem was formulated and solved for a round shell, uniformly loaded along its hinged edge by a radial compressive force. Below, this problem is formulated for an arbitrary shell of revolution with a uniformly compressed hinged edge. Results of its solution are given for conical and spherical shells.     

The fundamental solution in the theory of shallow shells

A series representation for the fundamental solution of the shallow shell equations is obtained by means of a plane-wave decomposition of the Dirac δ-function. From this solution we can produce the singular solutions which correspond to concentrated forces, couples and thermal hot spots applied to a shallow shell with an arbitrary quadratic middle surface. The solutions converge for the entire range of the Gaussian curvature. Numerical results are presented for the case of a concentrated normal force acting on infinite shells having positive, zero or negative Gaussian curvature.     

Ultimate deformation behavior of a laminated spherical shell

Summary A solution method is developed for a laminated spherical shell, subject to uniform normal tractions or constant radial displacements on the boundary. Displacements and stresses in the shell can be obtained from the solution formulas, which are as simple as those for homogeneous spherical shells. These formulas are accurate when the thicknesses of the constituent spherical layers are small in comparison with the radii of the shell, and become exact in case these thicknesses tend to zero under a fixed overall shell thickness. The solution is obtained mainly by exactly treating the product of an infinitely large number of matrices which link the displacements and stresses in different constituent spherical layers. A continuous analysis and a number of numerical results are provided to validate this development. Received 4 December 1996; accepted for publication 12 September 1997     

TO SOLVE THE SHALLOW SHELL EQUATIONS WITH THE HELP OF THE PLATE’S EQUATION SOLUTION(LOWERING THE ORDER OF PARTIAL DIFFERENTIAL EQUATION FROM 8 TO 2)

If a plate solution is known which has the same boundary and loading conditions with a shallow shell, the solution of that shell can be reduced to a non-homogeneous Helmholtz’s equation in complex domain. Two examples are given to illustrate our method.     

Jet flow around an elastic shell

In [1] an investigation was made of jet flow around an elastic plate. Below, in an exact nonlinear statement, a study is made of the problem of jet flow around an elastic cylindrical shell, fastened at one end and having the second end free. With certain limitations on the form of the shell, the single-valued solvability of the problem is demonstrated, and a method for its solution is proposed. Some results of calculations are given. A statement and a solution of the inverse problem of static hydroelasticity are also given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 24–32, March–April, 1977.     

Heat conduction between confocal elliptical surfaces using the theory of a Cosserat shell

This paper is concerned with steady-state heat conduction in rigid shell-like interphase regions. By analogy this work may provide insight into related problems of electric, dielectric and magnetic behavior. Although the field equations for three-dimensional linear Fourier heat condition are rather simple, the solution of problems in shell regions is significantly complicated when the shell has a general geometry and variable thickness. Here, the problem of heat conduction between confocal elliptical surfaces is solved within the context of the theory of a Cosserat shell. This problem is of particular interest because the Cosserat solution can be compared with an exact solution and the influences of variable shell thickness and strong variations of the temperature field through the shell’s thickness can be explored independently. The results show that the Cosserat approach is reasonably accurate even for moderately thick shells, moderate ellipticity, and moderately strong variation of the temperature through the shell’s thickness.     

轴对称层合圆柱壳混合状态方程的弱形式及其边值问题

从Ｈｅｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，在柱坐标系中，导出圆柱壳轴对称问题的弱形式混合状态方程和边界条件，联用状态空间法给出强厚度叠层柱壳的解析解，此法使得求解该类问题的形式得以扩大和统一。     

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

A general approach, based on shearable shell theory, to predict the influence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The effects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid finite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear stiffness matrices are derived by exact analytical integration. In part two, the modal coefficients are obtained, using Green's exact strain-displacement relations, for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.     

The general solution for the stress problem of circular cylindrical shells with an arbitrary cutout

I.IntroductionIncircularcylindricaIsheIls.likeaircraftfuselages,deep-diving\-ehicles.pressurevesselsandoiIlines,openingsareinvariabl}'necessaryforavarietyoffunctionalrequirements.Stressconcentrationsonthecontourofcircularornon-circu1arcutoutsreducebearing…     

Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.     

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.