Eccentric Circle Research Articles

Transverse bending of a thin circular plate loaded normally over an eccentric circle

ABSTRACTComplex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the loadover an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .For a clamped boundary the solution is obtained in finite terms.

Bending of an elastically restrained circular plate under a linearly varying load over an eccentric circle

ABSTRACTThe complex potentials and deflexion at any point of a thin circular plate with a normal linearly varying load over an eccentric circle are determined under a general boundary condition including the usual clamped and hinged boundaries.

Bending of a circular plate with an eccentric circular patch symmetrically loaded with respect to its centre

ABSTRACTThe complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.

The Eccentric Circle of Boscovich

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show latcr on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

The Eccentric Circle of Boscovich

Consideration of the special case in which the centre of the eccentric circle lies on the straight line whose points of intersection with the conic are required.In what follows it will be convenient to abandon the system of lettering adopted by Boscovich which we have hitherto followed, and as far as possible to use that common to modern text books.

Scientific Serials

The Mathematical Gazette, No. 1, April 1893, 8 pp. (London: Macmillan.)—It is now maiter of ancient history that a correspondence in the columns of NATURE resulted in the formation, in January 1871, of the Association for the Improvement of Geometrical Teaching. The original objects of the association were threefold: to collect and distribute information as to the prevailing methods of instruction in geometry practised in this and other countries, to use its influence to induce examining bodies to frame their questions in geometry without reference to any particular text-book, and to stamp with its approval some text-book already submitted, or to bring out a new one under its own auspices. Ten years later, viz. in January 1881, the association widened its basis, though after some discussion it retained its name. The objects it had in view were now sought to be carried out by the reading of papers and raising discussions, and by the appointment of committees to report on existing defects in the usual methods, order, range, &c, in teaching special subjects—all branches of elementary mathematics and mathematical physics being included in the widened basis. Now that the association has passed its majority, it is thought that it owes its continued existence to a “widespread desire on the part of teachers of mathematics to become acquainted with the methods of other teachers.” The editor of the Gazette, Mr. E. M. Langley, to whose long-continued and enthusiastic advocacy of its aims the Association for the Improvement of Geometrical Teaching owes so much as late secretary, hopes, through the agency of its columns, to extract from experienced teachers MSS. which have long been lurking in desk or pigeonhole for want of a suitable organ for making them known. This new venture, which has been started in consequence of a resolution passed at the association's annual meeting in January last, is proposed to be “a terminal journal for students and teachers.” The editor has to feel his way: words of encouragement have come from the far East and West, as well as from many teachers in this country. The number before us opens with a short paper, by the editor, on the eccentric circle of Boscovich. We borrow from Dr. C. Taylor's classical book on Conies the following verdict on Bosovich's work:—It is “a clear and compact treatise, which for simplicity, depth, and suggestiveness will not readily be surpassed.” Dr. j. S. Mackay abstracts the first book of Gino Loria's treatise on “the exact sciences in ancient Greece,” viz. that on the Greek geometers before Euclid. In addition to the works cited by Dr. Mackay, we may call attention to three notes on the history of mathematics by the Danish mathematician, H. G. Zeuthen (which have recently been published in the Bulletin de l'Acadénie Royale des Sciences de Danemark (1893). Prof. A. Lodge gives some useful approximations and reductions. Then follow some elegant solutions of examination questions, and a select number of questions for solution. A commendation of the new French journal, l'Intermédiare des Mathematiciens closes this No. 1. The size of the page, the clear type, and the excellent paper, should secure for the Gazette far more than a mere succés d'estime. The figures are lithographed on a separate sheet. We note one little slip—Adam's property for Adams' (p. S.)

The Eccentric Circle of Boscovich

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.