变地基模量上圆板弯曲问题的研究

本对中心受集中力P作用的自由放置在弹性地基上的圆板，在考虑其地基系数为抛物线变化的情况下，利用Ritz法获得了该问题的弯曲解答。在特殊情况下得到地基系数为常数的结果。     

Higher order bending and membrane responses of thin linearly elastic plates

The limit behaviors of three-dimensional displacements in thin linearly elastic plates, as the half-thickness ε tends to zero, is now known for various lateral boundary conditions (see [1], [5]). In the generic case one obtains that the leading term of the asymptotic series u⁰ + ge¹+ε²u² +… of the scaled displacement is a Kirchhoff-Love field. In this Note we investigate the case where this leading term vanishes, giving the structure of the first non-vanishing term u^k and an error estimate for its deviation from the scaled solution u(ε) multiplied by ε^−k. There are essentially only three new cases (uncoupling in membrane and bending). Finally, in these situations a boundary layer term of the same order as the actual leading term appears in a generic way.     

Axisymmetric vibrations of circular plates of linearly varying thickness

Summary Free axisymmetric vibrations of circular plates of linearly varying thickness have been studied on the basis of classical theory of plates. The differential equation governing the motion of such plates have been solved by Frobenius method. The transverse displacement has been expressed as an infinite power series in the radial coordinate. Frequency parameters have been computed for the first three modes of vibration for clamped as well as simply supported circular plates corresponding to different values of taper constant.

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Zusammenfassung Auf Grund der klassischen Scheibentheorie werden die freien axialsymmetrischen Schwingungen der Kreisscheibe mit linear veränderlicher Dicke studiert. Die Differentialgleichung der Bewegung wird mit Hilfe der Methode von Frobenius gelöst. Die transversale Verschiebung wird als eine Reihe der radialen Koordinaten ausgedrückt. Die Schwingungsparameter werden für die ersten drei Eigenschwingungen sowohl für die eingespannte als auch für die aufgelegte Kreisscheibe für verschiedene Werte des Anzugs berechnet.

     

Cylindrical bending of an elastic rectangular sandwich plate on a deformable foundation

Cylindrical bending of an elastic rectangular sandwich plate having a rigid filler and resting on an elastic foundation is considered. To describe the kinematics of the plate, asymmetric across its thickness, the hypotheses of broken normal are assumed. The reaction of the foundation is described by the Winkler model. A system of equilibrium equations is derived, and its exact solution is obtained in terms of displacements. A numerical analysis of the solution is presented.     

On the expansion of a deformed cylindrical elastic membrane

In the present note we consider the effect of radially expandingone end of a cylindrical elastic membrane that is also subjectedto longitudinal extension coupled with a twist. In doing thiswe start with the results of Tait, Steigmann and Zhong who studiedthe extension and twist of a cylinder with fixed end radii.We use tension field theory to provide a model of the membranein wrinkled regions giving rise to explicit analytic solutionswithin these regions. Elsewhere we rely on a numerical schemeto provide graphical results which are displayed for variousquantities of interest.     

Dispersion of normal waves in a linearly orthotropic circular cylinder with a clamped lateral surface

We propose a method of obtaining the dispersion equation for normal waves in an orthotropic cylinder from the boundary conditions on the rigidly clamped boundary using a system of exponential particular solutions of the three-dimensional equations of its stationary wave motions. We compute the real and imaginary branches of the dispersion spectrum for a waveguide made of monocrystalline strontium sulfate. On figure. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 96–99.     

State of stress around a circular opening in orthotropic spherical shells with linearly varying thickness

The state of stress of a thin spherical shell of orthotropic material of linearly varying thickness weakened by a circular opening at the pole is investigated. A solution in Bessel functions is constructed for the differential equations describing the state of stress of the shell. Graphs are presented for the dependence of the stress concentration coefficients at the edge of the opening and the uniformity of the state of stress on the coefficient characterizing the variation of the thickness. It is shown that by a suitable choice of this coefficient it is possible to obtain a shell in which the stresses at the edge of the opening and on the equator are equal.     

Exact solution for Transient bending of a circular plate integrated with piezoelectric layers

This paper presents the exact, explicit solution for the transient motion of a circular plate surface bonded by two piezoelectric layers, based on Kirchhoff plate model. The distribution of eclectic potential along the thickness direction is simulated by a quadratic function so that the Maxwell static electricity equation is satisfied. The piezoelectric layers are electrically grounded over the edge and electrodes at the two surfaces of the piezoelectric layers are shortly connected. The differential equations of motion are solved for simply supported and clamped boundary conditions. The solutions are expressed by elementary Bessel functions and obtained via exact inverse Laplace transform.     

Elastoplastic equilibrium of an isotropic strip with a circular hole under pure bending

The range of variation of external forces over which the plastic region in a bent strip develops from initiation to complete envelopment of a circular aperture is determined. A perturbation method is used to construct a solution for these interactions. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhnika, No. 30, pp. 105–111, 1999.     

Wrinkle-free solutions of circular membrane problems

Axisymmetric deformations of a plane circular membrane subjected to an axial surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering the small-finite-deflection theory of Föppl-Hencky as well as a simplified version of Reissner's nonlinear theory of thin shells of revolution, the determination of the principal stresses in the membrane is shown to reduce to the solution of a nonlinear second-order ODE. In the Föppl case, the basic equation becomes singular when the membrane edge is free of traction. Nevertheless, existence and uniqueness of nonnegative solutions for zero edge traction is proved in both Föppl and Reissner models. Thereby, a limit curve for the radial displacement at the boundary is induced in the Reissner case, which subdivides the actual parameter range into complementary domains of existence and nonexistence of tensile solutions. The limit curve is studied, analytically and numerically. Finally, a maximum principle is established in order to determine the more restricted subdomain of those parameters which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.     

Fundamental frequency of a circular membrane with a strip of small length

A circular membrane with an arbitrarily placed internal strip of small length is concerned in this article. A two-term asymptotic expansion for the fundamental frequency of the membrane, as the length of the strip approaching to zero, is specified. Comparing it with the one [8] derived for the membrane with an internal circular core, it is found that the position of the internal constraint has more effect than the shape of the internal constraint on the fundamental frequency. The asymptotic approximation is also compared with the numerical data computed by the dual boundary element method [2] for a circular membrane of radius 1 with a radially placed internal strip of length 2c. These two sets of data are in good agreement. The relative error is less than 3 % as c is less than or equal to 0.1, for all positions of the strip. Moreover, the relative error is less than 1 % as c is less than or equal to 0.01.     

Fundamental frequency of a circular membrane with a strip of small length

A circular membrane with an arbitrarily placed internal strip of small length is concerned in this article. A two-term asymptotic expansion for the fundamental frequency of the membrane, as the length of the strip approaching to zero, is specified. Comparing it with the one [8] derived for the membrane with an internal circular core, it is found that the position of the internal constraint has more effect than the shape of the internal constraint on the fundamental frequency. The asymptotic approximation is also compared with the numerical data computed by the dual boundary element method [2] for a circular membrane of radius 1 with a radially placed internal strip of length 2c. These two sets of data are in good agreement. The relative error is less than 3 % as c is less than or equal to 0.1, for all positions of the strip. Moreover, the relative error is less than 1 % as c is less than or equal to 0.01.     

Solution of the bending problem of a circular plate with a free edge by using paired equations

Used in the investigation of the bending of a plate with a free edge is a method including the solution of non-standard paired equations that are trigonometric series, which results in a quasi-completely-regular system of linear algebraic equations. The nature of the plate state of stress and strain is investigated. One of the methods reducing to the solution of paired summation equations is elucidated in /1/ in application to the problem under consideration. A combination of support and clamping is examined in the majority of papers while the case of the free edge is inadequately studied.     

Interaction between two neighboring circular holes in a prestretched simply supported orthotropic strip under bending

In this work, the influence of initial stretching of a simply supported plate-strip containing two circular holes on the stress concentration around the holes caused by bending of the strip is examined using the finite-element method. The mathematical formulation of the corresponding boundary-value problem is presented within the frame work of the three-dimensional linearized theory of elasticity (TDLTE) under a plane strain state. The material of the plate-strip is linearly elastic, homogeneous, and orthotropic. The numerical results obtained in investigating the influence of the initial stretching and the location of holes on the stress concentration are presented. In particular, it is established that the initial stretching significantly decreases the stress concentration at some characteristic points on the contour of the holes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 827–838, November–December, 2008.     

Hoop stress calculation for a nearly circular hole in a finitely deformed plate under plane strain and uniaxial tension

This work applies the theory of small deformations superposed on an exact finite solution (see Green/Zerna [2], Graham [7]) to discuss the plane strain deformation of a plate of incompressible Mooney-Rivlin material containing an almost circular hole and which is expanded by a radial pressure at this hole and uniaxial tension at infinity.As a specific example, we consider the theory as it applies to an elliptic hole of small eccentricity. The resulting differential equations are solved using Colsys [12, 13], a recent numerical technique for handling boundary value problems for families of ordinary differential equations, and graphs of hoop stress are drawn against finite extension for various values of eccentricity and uniaxial tension.

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Zusammenfassung Die vorliegende Arbeit wendet die Theorie der kleinen, einer exakten Lösung überlagerten, Deformationen an. um die ebene Verzerrung einer Platte aus inkompressiblem Mooney-Rivlin-Material zu diskutieren, die ein fast rundes Loch enthält und beansprucht wird durch Radialdruck an diesem Loch und einachsige Spannung im Unendlichen.Als spezielles Beispiel betrachten wir die Anwendung der Theorie auf ein elliptisches Loch mit kleiner Exzentrizität. Die resultierenden Differentialgleichungen werden durch Colsys [12, 13] gelöst. Dies ist eine neue numerische Methode, um Randwertprobleme für Familien von gewöhnlichen Differentialgleichungen zu lösen. Es werden Graphen der Tangentialspannung in Funktion der endlichen Dehnungen für verschiedene Werte von Exzentrizität und einachsiger Spannung aufgezeichnet.

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Visited Simon Fraser University during the academic year 1982/83.