GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅲ)-CALCULATION FOR C-SHAPED BELLOWS

ＩｎｔｒｏｄｕｃｔｉｏｎＣ＿ｓｈａｐｅｄｂｅｌｌｏｗｓｏｒｒｅｆｅｒｒｅｄｔｏａｓｓｅｍｉ＿ｃｉｒｃｌｅｂｅｌｌｏｗｓａｒｅｗｉｔｈｃｏｍｐｌｅｘｃｏｎｆｉｇｕｒａｔｉｏｎ(Ｆｉｇ .1) .Ａｔｔｈｅｃｏｎｃａｖｅａｎｄｃｏｎｖｅｘｊｏｉｎｔｓｏｆｔｈｅｓｈｅｌｌ,ｆｏｒｅｘａｍｐｌｅａｔｐｏｉｎｔＣ ,ｔｈｅｌａｔｉｔｕｄｅａｎｇｌｅψｉｎｃｒｅａｓｅｓｂｙ 180°ａｂｒｕｐｔｌｙ .Ｓｏｉｔｉｓｔｈｅｋｅｙｔｏｃａｌｃｕｌａｔｉｏｎｆｏｒｂｅｌｌｏｗｓｔｈａｔｍａｋｉｎｇｕｐｓｏｌｕｔｉｏｎｓｅｘａｃｔ…     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅳ)-CALCULATION FOR U-SHAPED BELLOWS

This is one of the applications of Part (I), in which the angular stiffness, and the corresponding stress distributions of U-shaped bellows were discussed. The bellows was divided into protruding sections, concave sections and ring plates for the calculation that the general solution (I) with its reduced form to ring plates were used respectively, but the continuity of the surface stresses and the meridian rotations at each joint of the sections were entirely satisfied. The resent results were compared with those of the slender ring shell solution proposed earlier by the authors, the standards of the Expansion Joint Manufacturers Association (EJMA), the experiment and the finite element method. It is shown that the governing equation and the general solution (I) are very effective. Contributed by HUANG Qian Biography: ZHU Wei-ping (1962-)     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅳ)-CALCULATION FOR U-SHAPED BELLOWS

ＩｎｔｒｏｄｕｃｔｉｏｎＵ＿ｓｈａｐｅｄｂｅｌｌｏｗｓａｒｅｕｓｕａｌｌｙｓｅｅｎｉｎｔｈｅｐｉｐｅｌｉｎｅｓｙｓｔｅｍ (Ｆｉｇ .1) ,ｗｈｉｃｈａｒｅｕｓｅｄｔｏｃｏｍｐｅｎｓａｔｅｆｏｒｔｈｅａｘｉａｌｍｏｖｅｍｅｎｔ,ｔｈｅｒｏｔａｔｉｏｎａｎｄｔｈｅｄｅｆｌｅｃｔｉｏｎｏｆｔｈｅｐｉｐｅｌｉｎｅｓｄｕｅｔｏｔｈｅｔｈｅｒｍａｌａｃｔｉｏｎ ,ｔｈｅｂａｓｅｕｎｅｑｕａｌｓｅｔｔｌｅｍｅｎｔｓａｎｄｔｈｅｉｎｓｔａｌｌａｔｉｏｎｅｒｒｏｒｓ.Ｔｈｅａｘｉｓｙｍｍｅｔｒｉｃａｌｄｅｆｏｒｍａｔｉｏ…     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅱ)- CALCULATION FOR OMEGA-SHAPED BELLOWS

ＩｎｔｒｏｄｕｃｔｉｏｎＢｅｌｌｏｗｓ[1,2 ]ｉｓａｓｈｅｌｌｏｆｒｅｖｏｌｕｔｉｏｎｗｉｔｈｗｒｉｎｋｌｅｄｍｅｒｉｄｉａｎ ,ｕｓｕａｌｌｙｎａｍｅｄａｆｔｅｒｉｔｓｍｅｒｉｄｉｏｎａｌｓｈａｐｅｉｎａｐｅｒｉｏｄｏｒｉｔｓｐｒｏｆｉｌｅｏｆａｃｏｎｖｏｌｕｔｉｏｎ ,ｓｕｃｈａｓΩ＿ｓｈａｐｅｄｂｅｌｌｏｗｓ,Ｃ＿ｓｈａｐｅｄｂｅｌｌｏｗｓ,Ｕ＿ｓｈａｐｅｄｂｅｌｌｏｗｓａｎｄｓｏｏｎ .Ｔｈｅｙａｒｅｗｉｄｅｌｙｕｓｅｄｉｎｍａｎｙｆｉｅｌｄｓ,ｆｏｒｅｘａｍｐｌｅ ,ｔｈｅｅｌａｓｔｉｃｓｅｎｓｅ…     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅰ)-GOVERNING EQUATION AND GENERAL SOLUTION

The overall bending of circular ring shells subjected to bending moments and lateral forces is discussed. The derivation of the equations was based upon the theory of flexible shells generalized by E.L. Axelrad and the assumption of the moderately slender ratio less than 1/3 (i.e., ratio between curvature radius of the meridian and distance from the meridional curvature center to the axis of revolution). The present general solution is an analytical one convergent in the whole domain of the shell and with the necessary integral constants for the boundary value problems. It can be used to calculate the stresses and displacements of the related bellows. The whole work is arranged into four parts: (Ⅰ) Governing equation and general solution; (Ⅱ) Calculation for Omega-shaped bellows; (Ⅲ) Calculation for C-shaped bellows; (Ⅳ) Calculation for U-shaped bellows. This paper is the first part.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅰ)-GOVERNING EQUATION AND GENERAL SOLUTION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｈｅｏｒｙｏｆｆｌｅｘｉｂｌｅｓｈｅｌｌｓ,ｅｓｔａｂｌｉｓｈｅｄｂｙＥ .Ｌ .Ａｘｅｌｒａｄ[1,2 ]( 1 976,1 987) ,ｉｓａｓｉｍｐｌｉｆｉｅｄｔｈｅｏｒｙｆｏｒｔｈｉｎｓｈｅｌｌｓ,ｉｎｗｈｉｃｈｓｈｅｌｌｓｏｆｒｅｖｏｌｕｔｉｏｎａｒｅｆｏｒｍｕｌａｔｅｄｉｎｍｅｒｉｄｉｏｎａｌｄｉｒｅｃｔｉｏｎｂｙｂｅｎｄｉｎｇｔｈｅｏｒｙｏｆｔｈｅｓｈｅｌｌｓａｎｄｂｙｔｈｅｍｅｍｂｒａｎｅｔｈｅｏｒｙｉｎｔｈｅｃｉｒｃｕｍｆｅｒｅｎｔｉａｌｄｉｒｅｃｔｉｏｎ .ＴｈｅＡｘｅｌｒ…     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅱ)- CALCULATION FOR OMEGA-SHAPED BELLOWS

is one of the applications of (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of Omega-shaped bellows were calculated, and the present results were compared with those of the other theories and experiments. It is shown that the non-homogeneous solution of (Ⅰ) can solve the pure bending problem of the bellows by itself, and be more effective than by the theory of slender ring shells; but if a lateral slide of the bellows support exists the non-homogeneous solution will no longer entirely satisfy the boundary conditions of the problem, in this case the homogeneous solution of (Ⅰ) should be included, that is to say, the full solution of (Ⅰ) can meet all the requirements.     

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, 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 load, and boundary conditions, the general bending problem of the conical shell on the elas     

SOME APPLICATIONS OF THE SINGULAR PERTURBATION METHOD TO THE BENDING PROBLEMS OF THIN PLATES AND SHELLS

This paper presents a review which tackles some nonlinear bending problems of plates and shells in a unified way by means of the technique of undetermined small parameters.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

In order to analyze bellows effectively and practically, the finite-element-displacement-perturbation method (FEDPM) is proposed for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes. The formulations are mainly based upon the idea of perturba-tion that the nodal displacement vector and the nodal force vector of each finite element are expanded by taking root-mean-square value of circumferential strains of the shells as a perturbation parameter. The load steps and the iteration times are not as arbitrary and unpredictable as in usual nonlinear analysis. Instead, there are certain relations between the load steps and the displacement increments, and no need of iteration for each load step. Besides, in the formulations, the shell is idealized into a series of conical frusta for the convenience of practice, Sander’s nonlinear geometric equations of moderate small rotation are used, and the shell made of more than one material ply is also considered.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

ＩｎｔｒｏｄｕｃｔｉｏｎＢｅｌｌｏｗｓ,ａｓｈｅｌｌｏｆｗｒｉｎｋｌｅｄｍｅｒｉｄｉａｎｏｆｒｅｖｏｌｕｔｉｏｎ ,ｈａｓｂｅｉｎｇｉｎｃｒｅａｓｉｎｇｌｙｕｓｅｄｉｎｍｏｄｅｒｎｅｑｕｉｐｍｅｎｔａｓａｎｅｌａｓｔｉｃ＿ｓｅｎｓｉｔｉｖｅｅｌｅｍｅｎｔａｎｄａｆｌｅｘｉｂｌｅｊｏｉｎｔ.Ｔｈｅｆｏｒｍｅｒｉｓｕｓｕａｌｌｙｓｕｂｊｅｃｔｅｄｔｏｌａｒｇｅｄｉｓｐｌａｃｅｍｅｎｔｓａｔｉｔｓｅｎｄｓ,ａｎｄｔｈｅｌａｔｔｅｒｍａｙｐａｒｔｌｙｅｎｔｅｒｉｎｔｏｐｌａｓｔｉｃｓｔａｔｅ .Ｈｅｎｃｅ ,ｂ…     

EQUATION OF AXISYMMETRICAL RING SHELLS WITH VARIABLE WALL THICKNESS IN COMPLEX QUANTITY AND ITS GENERAL SOLUTION

The purpose of this paper is to derive the equation of axisymmetrical ring shells withvariable wall thickness in complex quantity and to give its general solution.     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

THE GENERAL SOLUTION FOR DYNAMIC RESPONSE OF NONHOMOGENEOUS BEAM WITH VARIABLECROSSSECTION

ＴＨＥＧＥＮＥＲＡＬＳＯＬＵＴＩＯＮＦＯＲＤＹＮＡＭＩＣＲＥＳＰＯＮＳＥＯＦＮＯＮＨＯＭＯＧＥＮＥＯＵＳＢＥＡＭＷＩＴＨＶＡＲＩＡＢＬＥＣＲＯＳＳＳＥＣＴＩＯＮＪｉＺｈｅｎ－ｙｉ（纪振义）（ＡｎｈｕｉＡｒｃｈｉｔｅｃｔｕｒａｌＩｎｄｕｓｔｒｙＣｏｌｌ...     

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.     

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１］，ｔｈｅｓｅｒｉｅｓｓｏｌｕｔｉｏｎｓｏｆｔｈｅｃｉｒｃｕｌａｒｐｌａｔｅｗｉｔｈｈｉｎｇｅｄｅｄｇｅａｎｄｗｉｔｈｔｈｅｓｉｍｐｌｙｓｕｐｐｏｒｔｅｄｅｄｇｅｈａｓｙｅｔｂｅｎｃｏｎｓｔｒｕｃｔｅｄ，ｗｈｉｃｈａ...     

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.     

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION ANDTHE ASYMPTOTICS OF SOLUTIONS (Ⅰ)

I.IntroductionThestudyorbendingproblemsforcircularthinplatewithlargedeflectionhasalonghistoryandalotofresearchhasbeendoned~41.Especially,aftertheappearanceoftheChienWeizangmethodthatusestheratioofthecenterdeflectiontothethicknessofP-l'Llteasperturbationparameter,theapplicationofperturbationmethodstothestudyofthebendingproblemsofplatesandshellsbecomesmoreandmorepopular,owingtothet'actthatonlybyusingregularperturbationmethodonecanfinditssufficientlycorrectsolution.OneweaknessinChienWeizangmet…     

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION ANDTHE ASYMPTOTICS OF SOLUTIONS (Ⅰ)

In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. The series solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.     

UNSYMMETRICAL BENDING PROBLEMS FOR THE ANNULAR AND CIRCULAR THIN PLATES UNDER VARIOUS SUPPORTING CONDITIONS(Ⅰ)

In this paper we consider the nonlinear and unsymmetricalbendings of annular and circular thin plates under varioussupports.The uniformly valid asymptotic solutions havebeen derived by means of the perturbation method presentedin[7].