考虑横向剪切变形的扁球壳在集中力作用下的分析

采用适于夹层壳的直线假设扁壳理论，应用三角级数法，导出了扁球壳齐次方程的解析解。进而分析了在顶点作用法向集中力和在偏心集中力作用下的解。计算了在偏心集中力作用下带孔球壳的位移和应力，并将结果与经典理论的结果进行了比较分析，结果表明，在集中力作用处和孔边处两种理论结果明显不同。     

Elasto-plastic behaviour of double-curved shells under a concentrated load

The paper presents the application of the so-called geometrical elements method to the solution of the elasto-plastic behaviour of spherical shells subjected to an axisymmetrical concentrated load. The approach is based on the observation that during large deformations, the shell structure deforms in a nearly isometrical manner. The shell is sub-divided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. Equilibrium of the structure is defined by the stationariness of the total potential energy. The total energy is compared with Pogorelov's result for the same strain energy. The solution obtained defines the large deformation behaviour and motion of the plastic zones on the surface of the shell.A simplified solution for similar problems of the shells with double positive Gaussian curvature is also presented.     

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.     

Analytical solution of cracked shell resting on elastic foundation

The elastostatic problem for cracked shallow spherical shell resting on linear elastic foundation is considered. The problem is formulated for a homogeneous isotropic material within the confines of a linearized shallow shell theory. By making use of integral transforms and asymptotic analysis, the problem is reduced to the solution of a pair of singular integral equations. The stress distribution obtained, around the crack tip, is similar to that of the elasticity solutions. The numerical results obtained agree well with those of previous work, where the elastic supports were neglected. The influences of the shell curvature and the modulus of subgrade reaction on the stress intensity factor are given.     

用Green函数求解扁球壳受环状线载和线偶作用的问题

本文从复变量形式的扁壳基本方程出发,通过建立复Green函数导出了在环状线载和线偶作用下扁球壳的位移和内力分布,通过积分可以求得轴对称的表面受变化分布载荷情况的解答,本文方法还可求得圆饭、圆柱壳等问题的解答,而且适用于各种轴对称边界条件。     

Analysis of shallow spherical shell with circular base under eccentrically applied concentrated loads

In this paper,problems of a shallow sphericalshell with circular base under eccentrically appliedconcentrated loads are discussed.The solutions forsix cases of eccentrically applied concentrated loadsare given,namely:(1)Normal concentrated load,(2)Meridional tangential concentrated load,(3)Circumferential tangential concentrated load;(4)Concentrated moment in the tangential plane,(5)Concentrated moment in the meridional normalplane,(6)Concentrated moment in the circumferentialnormal plane.From the solutions of concentrated loads,thesolutions of distributed line loads in the form ofcosnθalong the circle are obtained.     

Dynamic stability of nearly cylindrical orthotropic shells of revolution subjected to meridional forces

We study the natural vibrations and the dynamic stability of nearly cylindrical orthotropic shells of revolution subjected to meridional forces uniformly distributed over the shell ends. We consider shells of medium length for which the shape of the midsurface generatrix is described by a parabolic function. Using the theory of shallow shells, we obtain the resolving equation for the vibrations of the corresponding prestressed shell. In the isotropic case, this equation differs from the well-known equation [1] by an additional term, which can be of the same order as the other terms taken into account. We consider shells of both positive and negative Gaussian curvature. We assumed that the shell ends are freely supported. The formulas and universal curves describing the dependence of the minimum frequency, the wave generation shape, and the dynamic instability domain boundaries on the orthotropy parameters, the preliminary stress, the Gaussian curvature, and the amplitude of the shell deviation from the cylinder are given in dimensionless form. We find that in the case of prestresses the orthotropy parameters and the shell deviation from the cylinder (of the order of thickness) can significantly change the least frequencies, the wave generation shape, and the dynamic instability domain boundaries of the corresponding prestressed orthotropic cylindrical shell.In this case, we note that for convex shells under preliminary compression the influence of the elastic parameter in the axial direction is stronger than the influence of the elastic parameter in the circular direction, while the situation is opposite in the case of concave shells. In the case of preliminary extension, the leading role of any orthotropy parameter can vary depending on the value of the preliminary stress and the Gaussian curvature.     

The Thermoelastic State of Orthotropic Shells Heated by Concentrated Heat Sources

A problem on thin-walled orthotropic shells of arbitrary Gaussian curvature acted upon by concentrated heat sources is solved by means of the two-dimensional Fourier transformation. The temperature is assumed to distribute linearly throughout the shell thickness. Convective heat exchange with the environment under Newton's law is taken into account. Calculated results are presented. The influence of the curvature on the thermoelastic state of various orthotropic shells is studied     

On problems of optimal design of shallow shell with double curvature on elastic foundation

The present paper discusses a method of optimal design of the shallow shell with double curvature on the elastic foundation Substantially we take the initial flexural function as the control function or design variable which will be found and the potential energy of the external loads as the criterion of quality of the optimal design of the shallow shell with double curvature, therefore the functional of the potential energy will be aim function. The optimal conditions and the isoperimetric conditions belong to the constrained conditions. thus we obtain the necessary conditions of the optimal design for the given problems, at the same time the conjugate function is introduced, then the problems are reduced to the solutions of two boundary value problems for the differential equation of conjugate function and the initial flexual function.     

Deformation of a shallow shell of revolution into a circular plate

If the curvature of a shell changes under the action of an external force, then the shell can enter a strain state in which it acquires the shape of a plate. In the framework of physical and geometrical linearity, we suggest a solution of the axisymmetric problem about the stress-strain state of a shallow shell of revolution transformed into a circular plate.     

Thermoconductivity and thermoelasticity of thin isotropic shells heated by a moving heat pulse

The paper presents a solution to the problem of thermal conduction and thermoelasticity for a thin shallow spherical shell heated by a concentrated or local impulsive heat source moving over the shell surface. It is assumed that temperature is linearly distributed across the shell thickness and that the shell, on its sides, exchanges heat with the environment in accordance with Newton’s law of cooling. The Fourier and Laplace transforms are used to find an analytic solution. The dependence of the temperature field and stress/strain components on the type of heating and the form of heat source is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 85–92, November 2006.     

Effect of transverse shear and material orthotropy in a cracked spherical cap

The elastostatic problem for a relatively thin-walled spherical cap containing a through crack is considered. The problem is formulated for a specially orthotropic material within the confines of a linearized, shallow shell theory. The theory used is equivalent to Reissner's theory of flat plates and hence permits the consideration of all five physical conditions on the shell boundaries separately. The solution of the problem is reduced to that of a pair of singular integral equations and the asymptotic stress state around the crack tips is investigated. The numerical solution of the problem is given for an isotropic shell and for two specially orthotropic shells. The results indicate that the material orthotropy as well as the shell curvature and thickness may have a considerable effect on the stress intensity factors at the crack tips.     

ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

ASYMPTOTIC SOLUTIONS OF THE NONLINEAR STABILITY OF A TRUNCATED SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

In this paper,the nonlinear stability problem of a clamped truncated shallow sphericalshell with a nondeformable rigid body at the center under a concentrated load is studied bymeans of the singular perturbation method.When the geometrical parameter k is large,theuniformly valid asymptotic solutions are obtained.     

Determining the elastic equilibrium of a cylindrical shell by Vekua’s theory based on the classical elasticity theory and the theory of binary mixtures

Using the approach based on separation of variables, an analytic solution of the class of boundary value problems of the shallow cylindrical shell theory is constructed by Vekua’s method. The cylindrical shell is assumed to be rectangular in the plan. Conditions of a free support or sliding fixation are given on the sides of the rectangle; the load on the shell being arbitrary. The solution of boundary value problems is constructed using both a classical elastic medium and the theory of binary mixtures. Analysis of the constructed solutions is carried out.     

Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

A mixed variational principle is derived by Hamilton’s method from the principle of minimum potential energy for thin anisotropic shells of revolution and is then used to derive a normal system of equations with complex coefficients. Discrete orthogonalization is used to solve this homogeneous system and the nonlinear system of equations that describes the precritical state of shells. A shell generated by revolving a circular arc around the axis parallel to its chord is analyzed for stability. The solution is compared with the approximate solution obtained assuming that the precritical state is membrane. It is established that the approximate problem formulation gives incorrect results for shells of negative Gaussian curvature     

CALCULATION OF THE FUNDAMENTAL SOLUTION FOR THE THEORY OF SHALLOW SHELLS CONSIDERING SHEAR DEFORMATION

In this paper, some formulas are derived for the numerical computation of the fundamental solution obtained in ref [1] and relevant computer methods are also discussed in detail. As an application of the fundamental solution, problems of a concentrated normal force acting on infinite shallow shells having positive, zero and negative Gaussian curvatures are calculated according to the numerical methods given in the paper.     

The general solution of spherically isotropic magnetoelectroelastic media and its applications

A general solution of the three-dimensional equilibrium problem of spherically isotropic magnetoelectroelastic media is presented. Base on the obtained general solution, exact and compact form solutions are obtained for (1) a spherically isotropic magnetoelectroelastic cone subjected to concentrated force, concentrated couple, a point charge and a point electric current at its apex; (2) a spherically isotropic magnetoelectroelastic space with a concentrated force at the origin; (3) a spherical shell under spherically symmetric deformation; and (4) stress concentration around a spherical cavity subjected to remote uniform tensile force, electric charge and electric current.     

Fracture mechanics analysis of curved stiffened panels using BEM

A dual boundary element method is developed for a analysis of reinforced cracked shallow shells. Boundary integral equations are derived from the Betti’s reciprocal theorem for a cracked shallow shell with transverse frames and longitudinal stiffeners. The effect of frames and stiffeners are treated as a distribution of line body forces. The radial basis function is used to transform domain integrals to boundary integrals. Stress intensity factors are evaluated from crack opening displacements. The effect of curvature on the stress intensity factors is illustrated by numerical examples. Three examples are presented to demonstrate the accuracy of this method compared with solutions obtained using the finite element method.     

On one iteration method for solving problems on the stress-strain state of thin shells

Decomposition methods are used in solving problems on the stress-starin state of thin shells. The problem is reduced to the iterative solution of a system of equations that have a simpler structure and permit constructing a computing parallel-integration algorithm. The stress-strain problem is solved by an example of a shallow spherical shell. The effect of the curvature of the shell on the number of iterations necessary for reaching the accuracy prescribed is studied. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 98–103, June, 2000.