APPROXIMATE SOLUTION FOR BENDING OF RECTANGULAR PLATES Kantorovich-Galerkin’s Method

This paper derives the cubic spline beam function from the generalized beam differential equation and obtains the solution of the discontinuous polynomial under concentrated loads, concentrated moment and uniform distributed by using delta function. By means of Kantorovich method of the partial differential equation of elastic plates which is transformed by the generalized function (δ function and σ function), whether concentrated load, concentrated moment, uniform distributed load or small-square load can be shown as the discontinuous polynomial deformed curve in the x-direction and the y-direction. We change the partial differential equation into the ordinary equation by using Kantorovich method and then obtain a good approximate solution by using Glerkin’s method. In this paper there ’are more calculation examples involving elastic plates with various boundary-conditions, various loads and various section plates, and the classical differential problems such as cantilever plates are shown.     

Free rigid/plastic plane bending of a slender beam

Summary Large rigid/plastic plane bending of a slender structural element as e.g., a beam or a sheet metal strip is examined, in view of its applications to metal forming. Loads, i.e. forces and moments, are admitted only at the ends of the element; the loads or the corresponding displacements and rotations may be prescribed. Using the normal force, the transversal force and the bending moment as generalized stresses acting at the cross sections the differential equations of the problem are set up for an arbitrary, strain-hardening and/or rate-sensitive material. As an example a homogeneous, ideally plastic beam is considered to be plastified by means of the bending moment only. It is shown that it can be brought into any shape provided the end conditions are adequately controlled. Circular bending as a special case becomes possible in two different ways i.e., by pure bending (without end forces) or by localized bending (generated by a moving yield hinge).

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Ebenes, freies biegen von schlanken, starrplastischen trägern

Übersicht Es wird die große, ebene, starrplastische Biegung schlanker Träger, also beispielsweise von Balken oder Blechen, im Hinblick auf Anwendungen in der Umformtechnik von Metallen untersucht. Lediglich an den Enden der Träger greifen Lasten (Kräfte, Momente) an; diese Lasten oder die entsprechenden Verschiebungen und Neigungen kann man vorgeben. Mit der Normalkraft, der Querkraft und dem Biegemoment als generalisierte Spannungen in den Querschnitten werden die Differentialgleichungen des Problems für ein beliebig verfestigendes und/oder geschwindigkeitsabhängiges Material formuliert und speziell auf einen homogenen, idealplastischen Träger angewendet, dessen Plastifizierung nur vom Biegemoment abhängt. Es wird gezeigt, daß man ihn in jede beliebige Gestalt bringen kann, vorausgesetzt, die Bedingungen an den Enden werden angemessen gesteuert. Eine kreisförmige Biegung erreicht man zum Beispiel auf zwei verschiedenen Wegen: Durch reine Biegung (ohne Kräfte an den Enden) oder durch lokalisierte Biegung infolge eines wandernden Fließgelenkes).

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Herrn Prof. Dr. Dr. h.c. H.-P. Stüwe, Montan-Universität Leoben, zum 60. Geburtstag am 14. Sept. 1990 gewidmet.     

Kantorovich solution for the problem of bending of a ladder plate

Based on the Kantorovich approximation solution for a rectangular plate in bending, this paper deals with the solutions for the ladder plate with various boundary conditions. The deflection of the plate is expressed in a first-order displacement function w(x,y)=u(x,y)v(y) where the u(x,y) in x direction is the generalized beam function. By making use of the principle of least potential energy, the variable coefficients differential equations for v(y) may be established. By solving is, these differential eugations and making use of the boundary conditions, the accurate solutions of v(y) in y direction may be obtained. Then the displacement function w(x,y) is the solution for the problem of the bending of the ladder plate with a better degree of approximation.     

各向异性矩形薄板弯曲问题的一般解

给出了各向异性矩形薄板弯曲问题微分方程的一般解。可以求解任意载荷作用下各种边界的弯曲问题。以四边固支的正方形板为例进行了数值计算。     

An analytic solution to asymmetrical bending problem of diaphragm coupling

Because rigidity of either hub or rim of diaphragm coupling is much greater than that of the disk, and asymmetrical bending is under the condition of high speed revolution, an assumption is made that each circle in the middle plane before deforma- tion keeps its radius unchanged after deformation, but the plane on which the circle lies has a varying deflecting angle. Based on this assumption, and according to the principle of energy variation, the corresponding Euler＇s equation can be obtained, which has the primary integral. By neglecting some subsidiary factors, an analytic solution is obtained. Applying these formulas to a hyperbolic model of diaphragm, the results show that the octahedral shear stress varies less along either radial or thickness direction, but fluctuates greatly and periodically along circumferential direction. Thus asymmetrical bending significantly affects the material＇s fatigue.     

Approximate solution of the three-dimensional inverse problem for nonlifting bodies with optimum cavitation characteristics

A formulation and method of solving the three-dimensional inverse problem of ideal fluid mechanics are proposed and examples of their application to the design of bodies of revolution with optimum cavitation characteristics for given overall dimensions are presented.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 92–100, May–June, 1994.     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Numerical solution of the curved rigid line problem in an infinite plate

Summary This paper deals with the problem of evaluating the stress singularity coefficients at the tips of a rigid line in an infinite plate. If the traction difference between the upper and lower borders of the rigid line is taken as an unknown function and the displacement of the curved rigid line as the right-hand term of the integral equation, then a new integral equation for the curved rigid line problem is obtained which is presented in this paper. The newly obtained integral equation has a logarithmic singular kernel. To solve the integral equation an interpolation equation for the traction difference functions (the undetermined functions in the integral equation) is proposed. A numerical examination is carried out to demonstrate the efficiency of the proposed technique. Also two numerical examples are given in this paper.

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Numerische lösung des problems einer gekrümmten linie als starrer einschlu\ in einer unendlichen platte

Übersicht Vorgeführt wird die Bestimmung von Spannungssingularitätsfaktoren an den Enden einer starren Kurve in einer unendlichen Platte. Wenn die Spannungsdifferenz an beiden Seiten der starren Linie als unbekannte Funktion und die Verschiebung dieser Linie als rechte Seite der Integralgleichung gewählt werden, erhält man eine neue Integralgleichung, die hier vorgestellt wird. Diese enthält einen logarithmisch singulären Kern. Um sie zu lösen, wird eine Interpolationsfunktion für die unbekannten Funktionen der Spannungsdifferenzen vorgeschlagen und ein numerischer Test zur Demonstration der Leistungsfähigkeit der Methode durchgeführt. Zwei numerische Beispiele werden vorgestellt.

     

Falkner-Skan方程的近似解析解

研究了粘性流体绕流楔型物体的Falkner-Skan边界层方程求解问题.利用Adomian拆分方法,通过引入Crocco变量变换将无穷区间的边界值问题转为初值问题并利用Padé逼近技巧确定初值,给出了一种有效的解析分解方法.进一步,本文设计了一种数值解法,将本文得到的近似解析解及数值结果与早期研究者Hartree等人的结果进行了比较,证明了本文提出的解法的有效性和可靠性.     

Twin-characteristic-parameter solution of axisymmetric dynamic plastic buckling for cylindrical shells under axial compression waves

In the present investigation on the dynamic plastic buckling of cylindrical shells under axial compression waves, the critical axial stress and the exponential parameter of inertia terms in stability equations are treated as a couple of characteristic parameters. The criterion of transformation and conservation of energy in the process of buckling initiation is used to derive the supplementary restraint equation of buckling deformation at the fronts of axial elastic and plastic compression waves. The supplementary restraint equation, stability equations, boundary conditions and continuity conditions constitute the necessary and sufficient conditions of determining the two characteristic parameters. Two characteristic equations are derived for the two characteristic parameters. The critical axial stress or the critical buckling time, the exponential parameter of inertia terms and the initial modes of buckling deformation are calculated quantitatively from the solution of the characteristic equations.     

Approximate solution of the problem of the bottom zone in a rarefied gas

The problem of the steady-plane monatomic rarefied gas flow around a semiinfinite bar is considered (the plane stationary case of the problem about the bottom zone). The problem is solved numerically at the level of the Krook relaxation model [1, 2]. A depenence of the gas density, velocity, and temperature in the whole flow domain on the space coordinates is obtained for supersonic and subsonic gas streams flowing around a body by using calculations on an M-20 electronic calculator.Khar'kov. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 139–143, January–February, 1972.     

Approximate solution to nonself-similar problem of motion of a piston after an impact

An approximate solution is obtained to the problem of the motion of a piston after an impact and under the influence of gas pressure under the assumption that the parameter = u_o/a _o, where u_o is the initial velocity of the piston anda _o is the velocity of sound in the gas at rest, is small. Functions that determine the law of motion of the piston and the shock wave, and also the gas flow in the disturbed region are found explicitly to terms of order ³ Translated from Izvestiya Akadeinii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1982.     

Dynamic bending of polygonal plastic slabs

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 149–157, July–August, 1988.     

Numerical solution of the dynamic problem of elasticity for bodies with concentrated masses

On the basis of the virtual-displacement and the DAlembert principles, a new initial-boundary-value problem is formulated for the system of dynamic equilibrium equations of two bodies contacting with friction over a constant area. Their masses are assumed to be distributed with certain densities over the contact surfaces. An equivalent generalized problem is formulated. Classes of discontinuous functions of the finite-element method are used to construct computational schemes of high order of accuracy.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 65–75, December 2004.     

On the solution of the dynamic Eshelby problem for inclusions of various shapes

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions.