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 ashallow shell,the solution of that shell can be reduced to a non-homogeneous Helmholtz’sequation in complex domain.Two examples are given to illustrate our method.     

THE SCHRDINGER 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 shellwith constant curvature are classified as the same several solutions of Schr(?)dingerequation,and we show clearly that its form in axisymmetric problem;(B)For example for the small deflection problem,we extract the general solution ofthe vibration problem of thin spherical shell with equal thickness by the force in centralsurface 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 shellare classified as the solutions of AKNS equations and in it the one-dimensional problem isclassified as the solution of simple Schr(?)dinger equation for eigenvalues problem,and wetransform the large deflection of shallow shell from nonlinear problem into soluble linearproblem.     

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.     

FORCED VIBRATIONS OF ELASTIC SHALLOW SHELL DUE TO THE MOVING MASS

This paper discusses the forced vibrations of the elastic shallow shell due to the movingmass by means of the variational method. In the text mentioned above a series of problemssuch as the forced vibrations,resonance conditions and critical speed and so forth.     

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of planeproblems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with fouredges built-in as the basic system and then find displacement expressions of the actualsystem by using the reciprocal theorem between the basic system and actual system withvarious edge conditions.When only displacement edge conditions exist,obtaining displacement expressions bymeans of the method of reciprocal theorem is actual.But in other conditions,when staticforce edge conditions or mixed ones exist,the obtained displacements are admissible.Inorder to find actual displacement,the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple,convenient andgeneral one for the solution of plane problems of elasticity of the rectangular plates withvarious edge conditions.Evidently,it is a new method.     

SOLUTION OF NONLINEAR HEAT CONDUCTION EQUATION AND IMAGE METHOD

In this paper,the author proves that, for a nonlinear heat conduction equation, there is no discontinuous solution. Some methods of solution for a nonlinear conduction equation are depicted. For a plane interface, the reflection and transmission of a heat wave are given by the method of images. The 1st order of approximation of this method is proved. Lastly, the prevention of superheated electrons is laser implosion of deuterium tritium gas sphere with a shell made of high Z material is interpreted.     

AN APPROACH TO DETERMINE THE STABLE INTERVAL OF THE CONSTANT TERM OF A CHARACTERISTIC EQUATION

In this paper,an approach is introduced to de-termine the stable interval of the constant term ofa characteristic equation by using the theory ofextended graphical representation of polynomials.Because the constant a_11 itself is not takeninto account and because this method is to get astable interval of a_n,not merely to make a stabi-lity test for a set of known coefficients,this me-thod of stability criteria has some advantages overthe others.The interval of a_n can be obtainedfrom the calculation of some algebraical expressionswhen n≤10,where n is the degree of the character-istic equation.It is very convenient to calculatefor the cases n=5 and n=6.When n≥11,thisinterval can be found only by the method of numeri-cal solutions.     

UNIFORM CONVERGENCE FOR THE DIFFERENCE SCHEME IN THE CONSERVATION FORM OF ORDINARY DIFFERENTIAL EQUATION WITH A SMALL PARAMETER

In this paper,we consider a singular perturbation boundary problem for a self-adjointordinary differential equaiton.We construct a class of difference schemes with fittedfactors,and give the sufficient conditions under which the solution of difference schemeconverges uniformly to the solution of differential equation.From this we propose severalspecific schemes under weaker conditions,and give much higher order of uniformconvergence.     

APPLICATION OF THE METHOD OF THE RECIPROCAL THEOREM TO FINDING DISPLACEMENT SOLUTIONS OF CUBES

In this paper the method of reciprocal theorem is extended to find solutions of three-Dproblems of elasticity.First we give the basic solution of the cube with six surfaces fixed as the basic systemand then using the reciprocal theorem between the basic system acted on by unitconcentrated loads and the actnal system with Precribed surface displacements,we finddisplacement solution of the actual system.     

A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary valueproblem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. It isproved that the solution is equal to the result of separation of variables. As a result, the non-linearcharacteristic equations resulting from the method of separation of variables are transformed into poly-nomial equations that can provide a foundation for approximate computation and asymptotic analysis.     

Solution of nonlinear heat conduction equation and image method

In this paper, the author proves that, for a nonlinear heat conduction equation, there is no discontinuous solution. Some methods of solution for a nonlinear conduction equation are depicted. For a plane interface, the reflection and transmission of a heat wave are given by the method of images. The 1st order of approximation of this method is proved. Lastly, the prevention of superheated electrons is laser implosion of deuterium tritium gas sphere with a shell made of high Z material is interpreted.     

Weak formulation of mixed state equation and boundary value problem of laminated cylindrical shell

IntroductionBasedonthree-dimensionalelasticitytheory,exactsolutionofhomogeneousisotropic,orthotropic,andlaminatedplatesandshellshasbeenstUdiedll]-[61,respeChvely.Butallofabovepapersadoptedrigorousequilibriumandboundarycondihons,andtheirsolutioncanbereliedonlyonspecialtechnique.Thusthosemethodswouldbedifficulttobepopularized.Ref.[7]clarifiedtheimportanceofdriedstateequationofelasticity,andfirStgaveHndltoncanonicalequationbymodifyingHellinger-Reissuervariationalprinciple.AtthesametimeTangL'I…     

Approximated-asymptotic solution of the complex variable equation of toroidal shells under axial symmetric loads

The series-approach and the asymptotic-approach are usually used to solve the complex variable equation of the toroidal shells under axial symmetric loads. As is known, the convergence of the series-solution is good only for small values of . On the other hand, the convergence of the asymptotic solution is good only for large values of μ. In this paper, based on an earlier work^[1], a new approach which may be called the approximated-asymptotic solution has been developed and it is valid for both small and large values of μ. It is shown that the results of the approximated-asymptotic solution for toroidal shell with μ=0.5 coincide very well with those of the series-solution, while the results of the asymptotic solution for this value of μ are not as good, and the results of the approximated-asymptotic solution for μ=15 agree with those of the asymptotic solution. This work has been carried out under the direction of Professor Zhang Wei.     

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.     

On the general solution of stress equation for the membrane theory of shells of revolution

Summary A new method is presented to solve the equation of Neményi-Truesdell, in the membrane theory of thin shells of revolution. It is an ordinary linear differential equation of the second order, for which resolution methods of Truesdell and Mitrinovitch are known. The proposed method is based on the transformation of the given equation into a linear integral equation of Volterra type. This allows to write the general solution of stress equation only in terms of equation of the meridian generating the shell.

---

Sommario Si presenta un nuovo metodo per risolvere l'equazione di Neményi-Truesdell, nella teoria delle membrane dei gusci sottili di rivoluzione. Si tratta di un'equazione differenziale lineare ordinaria del secondo ordine, della quale sono noti metodi risolutivi dovuti a Truesdell ed a Mitrinovitch. Il metodo proposto è basato sulla trasformazione dell'equazione data in un'equazione integrale lineare di Volterra e consente di esprimere l'integrale generale dell'equazione degli sforzi esclusivamente in termini della equazione del meridiano generatore del guscio.

     

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.     

On the problems of sandwich shells having the form of a surface of revolution and face layers of non-equal thicknesses

Fundamental equations which govern the behavior of an elasticsandwich shell having the form of a surface of revolution andface layers of non-equal thicknesses are derived,with the so-lution of Refs.[3]and[4]as special examples.The problems of the shell under the action of symmetricalloads are reduced to the solution of a displacement-function4～,where4αsatisfies a differential equation of sixth order.     

Solution of the deformation problem for flexible noncircular cylindrical shells subject to bending moments at the edges

An exact analytical solution is found to a nonlinear boundary-value deformation problem for a long noncircular cylindrical shell of variable curvature. The shell is subject to bending moments at the edges. The dependence of the stress-strain state of the shell on the curvature is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 93–100, November 2006.     

Analysis of the surface-probe equation

An integrating surface-probe is a tool for the determination of local fluxes from surface-integrated measurements. For an axisymmetric system, the data analysis involves the numerical solution of a severely ill-conditioned Volterra integral equation of the first kind. This study establishes the existence of a unique solution under certain conditions and considers solution techniques based on product integration and also evaluation of the inversion integral. The methods are tested for convergence against exact input data, and the performance of the best methods against simulated experimental data is studied.     

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…