Equation in complex variable of axisymmetrical deformation problems for a general shell of revolution

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tölke(1938)^[3], R.A.Clark(1950)^[4]and V.V.Novozhilov(1951)^[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)^[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.     

The ring shells under gravitative loads

This paper gives Novozlov’s equation (ref. [1]) in a simple style of ring shells with equal thickness under the actions of gravitative loads. By means of Fourier series, the special solution of the equation is established. Using the results of the homogeneous solution in ref. [2], we find out the general solution of the problem, and derive the expressions of stress and displacement.Two examples are given as the application of the above results.     

PROBLEMS OF U-SHAPED BELLOWS WITH NONLINEAR DEFORM TION OF LARGE AXISYMMETRICAL DEFLECTION(Ⅰ)—COUNTING NONLINEAR DEFORMATIONS OF RING SHELLS AND COMPRESSED ANGLE OF BELLOWS

This paper follows the work of[1,2].There are some progress in dealing withmoderately small rotations(the squares of rotation angles are the order of magnitudeof strains)of middle surface normals of inside and outside ring shells and compressedangle of bellows.Calculation results agree with experiments well.To bellow design,the method given in this paper is of practical value and the discussion of the influenceof compressed angle on characteristic relation is helpful.     

Problems of U-shaped bellows with nonlinear deformation of large axisymmetrical deflection (I)-counting nonlinear deformations of ring shells and compressed angle of bellows

This paper follows the work of [1,2]. There are some progress in dealing with moderately small rotations (the squares of rotation angles are the order of magnitude of strains) of middle surface normals of inside and outside ring shells and compressed angle of bellows. Calculation results agree with experiments well. To bellow design, the method given in this paper is of practical value and the discussion of the influence of compressed angle on characteristic relation is helpful.     

On stability of orthotropic cylindrical shells

In contrast to [1–3], the present paper obtains a system of stability equations and the corresponding resolving equation for orthotropic cylindrical shells of any but very short length in the case where the precritical stress state cannot be treated as the zero-moment state. These equations are a generalization of the results obtained in [4]. On the basis of these equations, one can obtain both the well-known formulas [1–3] and, for medium-length shells, some new expressions of the critical load in longitudinal compression and that under the joint action of torsionalmoments, normal pressure, and longitudinal compression. Some estimates are performed and the determination of the domain of application of some formulas given in [2] and in the present paper is attempted. For an orthotropic shell, a relationship between the elastic parameters and the shear modulus is established for axisymmetric and nonaxisymmetric buckling mode shapes in longitudinal compression.     

Complex equations of flexible circular ring shells overall-bending in a meridian plane and general solution for the slender ring shells

ＩｎｔｒｏｄｕｃｔｉｏｎＥｑｕａｔｉｏｎｓｆｏｒｃｉｒｃｕｌａｒｒｉｎｇｓｈｅｌｌｓａｒｅｄｉｆｆｉｃｕｌｔｔｏｓｏｌｖｅ．Ｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｉｓｐｒｏｂｌｅｍｓｔａｒｔｅｄａｔｔｈｅｂｅｇｉｎｎｉｎｇｏｆｔｈｉｓｃｅｎｔｕｒｙ．Ｉｎｔｈｅｌａｔｅ１９７０ｓａｎｄｅａｒｌｙ１９８０ｓ,Ｗ．Ｚ．Ｃｈｉｅｎ（１９７９,１９８０,１９８１）［１～３］ｒｅｂｕｉｌｔｔｈｅｃｏｍｐｌｅｘｅｑｕａｔｉｏｎｓｏｆａｘｉｓ＿ｓｙｍｍｅｔｒｉｃａｌｌｙｌｏａｄｅｄｒｉｎｇｓｈｅｌｌｓｐｒｅｓｅｎｔｅｄ…     

A rigorous solution for the axisymmetrical buckling of simply supported cylindrical sandwich shells under axial load

In this paper,based on ref[1],the axisymmetrical buckling of simply supportedcylindrical sandwich shells under the action of uniform axial load is solved by a rigorousmethod.The classical theory of shells is used for the two face sheets and the core isconsidered as a three-dimensional elastic body.A series of transcendental equations areobtained,from which the critical loads can be calculated by numerical methods.Numericalexamples are given to compare with the solutions of sandwich shell theories.     

Stressed state of a composite cylindrical shell with an elliptical hole

The results of investigations of the stress-strained state (SSS) of isotropic cylindrical shells with an elliptical hole are represented in monograph [4]. The modified method of expansion in terms of minor parameter [3] is suggested for calculation of orthotropic shells. The method does not consider, however, lateral shear strains introducing a significant contribution in SSS of composite shells. The procedure for solving problems of SSS calculation near curvilinear holes in shells of arbitrary shape with variable geometrical and physical characteristics is suggested in [1] on the basis of variational-difference method (VDM). Here the relations of the linear theory of anisotropic inhomogeneous shells and the hypothesis of a straight line are taken as the initial ones for all the packet of laminated composite shell as a whole. In the present work we present the numerical results obtained according to the procedure given in [1] for an orthotropic cylindrical shell with an elliptical hole loaded by the axial force and complaint for the lateral shear.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 11, pp. 57–62, November, 1993.     

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.     

NONLINEAR THEORY OF MULTILAYER SANDWICH SHELLS AND ITS APPLICATION (Ⅱ)──FUNDAMENTAL EQUATIONS FOR ORTHOTROPIC SHALLOW SHELLS

This paper applied the simplified theory for multilayer sandwich shells undergoing moderate/small rotations in Ref. [1] to shallow shells. The equilibrium equations and boundary conditions of large deflection of orthotropic and the special case, isotropic shells, are presented.     

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 Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

NONLINEAR THEORY OF MULTILAYER SANDWICH SHELLS AND ITSAPPLICATION (Ⅱ)──FUNDAMENTAL EQUATIONS FOR ORTHOTROPIC SHALLOW SHELLS

InPartl,anonlineartheoryformultilayersand`"ichshellsundergoingsmallstrainsandmoderaterotatiol1sisgiven-inwhichtranst'ersesheardeformationisconsidered.Thesimplifiedtheoryfortheshellsundergoingmoderateormoderate/smallrotationsisalsoPresented.Theexpressionsi…     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

Axisymmetry problems of ring shells under arbitrary distributed loads

Extending Novozhilov's transformation, this paper simplifies successfully the axisymmetry problems of ring shells under arbitrary distributed loads, and the equation of the problems in complex form which is similar to Novozhilov's is obtained. The particular solution is obtained. Combining with Chien's general solution of homogeneous equation, this paper gives the general solution for the general symmetry problems of ring shells. Various examples of useful loads and closed ring shells are discussed respectively.     

The method of matrices conjoint multiplication for the problem of circular arc corrugated diaphragm

The circular are corrugated diaphragins are taken in this paper and structures of several sections of the ring shells and a central cireular plale I matrices and link matrices are derived by using Prof. Chuen Hei-zang’s general the ring shell[1] and perturbation theory of the circular thin plates. Throngh the meined of matrices conjoint multiplication, the linear exact solution and nonlinear soluaen are obtained. The resutts agree with that of the experiments presented by W. A. Wildhack..     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

Theory of limit equilibrium of bimetallic shells of revolution and circular plates

Bimetallic shells and plates are widely used in technology (see [1, 2]). An investigation into the flexure and stability of thin shells and various types of loading within the limits of elasticity has been carried out in [3]. An investigation into the load-carrying capacity of cylindrical bimetallic shells made of materials which equally resist tension and compression was carried out in [4]. In many cases the materials of the base and plating layers of bimetallic constructions possess substantially different plastic resistance under tension and compression [5]. The given paper is devoted to the investigation of the load-carrying capacity of bimetallic axisymmetric shells which are made of materials that have different resistances to tension and compression; it is also devoted to the assessment of their economy in comparison with homogeneous shells.     

General solution for u-shaped bellows overall-bending problems

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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.