Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Biharmonic eigenvalue problem of the semi-infinite strip


Author: Gabriel Horvay
Journal: Quart. Appl. Math. 15 (1957), 65-81
MSC: Primary 73.2X
DOI: https://doi.org/10.1090/qam/85734
MathSciNet review: 85734
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Abstract: A basic problem in the evaluation of residual stresses in simple elastic structures concerns the determination of the stress and deformation state produced by self-equilibrating, but otherwise arbitrary, normal and shear tractions acting on the edge $ x = 0$ of a semi-infinite elastic strip $ \left( {0 \le x \le \infty , - 1 \le y \le 1} \right)$ which is free along the edges $ y = \pm 1$. This strip is known to experience, in accordance with St. Venant's principle, inappreciable stresses at distances $ x\mathop > \limits_\sim 2$ from the loaded edge, in spite of the very large stresses it may experience in the vicinity of the edge. An earlier paper [The end problem of rectangular strips, J. Appl. Mech. (1953)] based on the variational principle, established approximate eigenfunctions (modes of response) and eigenvalues (laws of oscillation and decay) for the various possible self-equilibrating end tractions. In this paper we give a rigorous solution of the end problem. This solution is obtained in two steps. First we solve the two ``mixed'' end problems: the parallel edges $ y = \pm 1$ of the strip are free, and along the vertical edge $ x = 0\left( a \right)$ the shear displacement is given, the normal stress is zero, (b) the normal displacement is given, the shear stress is zero. These two problems are solved by extending the strip to the left, to $ - \infty $, and finding the tractions that must be applied at $ y = \pm 1\left( {x < 0} \right)$ and at $ x = - \infty $, so that one have $ {\sigma _x} = 0,\tau = 0$, respectively, at $ x = 0$, while the edge values of the displacements (more specifically, of $ dv/dy$ and $ u$) are orthogonal polynomials in $ y$ (Horvay-Spiess polynomials and Legendre polynomials, respectively). The corresponding stress functions $ {K_n}\left( {x,y} \right), {J_n}\left( {x,y} \right)$ are found in the form of Fourier integrals plus polynomial terms. For $ x \ge 0$ they may be rewritten as real parts of $ \sum {C_{Knk}}{\Phi _k}, \\ \sum {C_{Jnk}}{\Phi _k}$, where $ {\Phi _k} = z_k^{ - 2}{e^{ - zkx}}\left( {\cos {z_k}y - y\cot {z_k}\sin {z_k}y} \right)$ or $ z{}_k^{ - 2}{e^{ - zkx}}\left( {\sin {z_k}y - y\tan \\ {z_k}\cos {z_k}y} \right)$, and $ \sin 2{z_k} \pm 2{z_k} = 0$. An alternate procedure for determining the coefficients $ {C_{Knk,}}{C_{Jnk}}$, based on a formula of R. C. T. Smith, which bypasses the extension of the strip to $ x = - \infty $, is also furnished. The second phase of the solution of the ``pure'' end problem--along the short edge (a) the shear stress is given, normal stress is zero, (b) the normal stress is given, shear stress is zero--consists in recombining the biharmonic eigenfunctions $ {K_{n}}, {J_n}$ within each class into functions $ {H_n}\left( {x,y} \right), {G_n}\left( {x,y} \right)$, so that the $ x = 0$ values of $ {H_{n,xy}}, {G_{n,yy}}$, constitute two complete orthonormal sets of (transcendental) functions in $ y$ into which the given boundary stresses may be expanded.


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  • [1] G. Horvay and F. N. Spiess, Orthogonal edge polynomials in the solution of boundary value problems, Quart. Appl. Math. 12 (1954), 57–69. MR 0065270, https://doi.org/10.1090/S0033-569X-1954-65270-0
  • [2] G. Horvay and J. S. Born, Some mixed boundary-value problems of the semi-infinite strip, J. Appl. Mech. 24 (1957), 261–268. MR 0087321
  • [3] G. Horvay, The end problem of rectangular strips, J. Appl. Mech. 20 (1953), 87–94. MR 0053735
  • [4] G. Horvay, Thermal stresses in rectangular strips, Proceedings of the Second U. S. National Congress of Applied Mechanics, Ann Arbor, 1954, American Society of Mechanical Engineers, New York, 1955, pp. 313–322. MR 0078160
  • [5] J. Fadle, Die Selbstspannungs-Eigenwertfunktionen der quadratischen Scheibe, Ing.-Arch. 11 (1940), 125–149 (German). MR 0002307
  • [6] P. R. Papkovitch, Über eine Form der Lösung des biharmonischen Problems für das Rechteck, C. R. (Doklady) Acad. Sci. USSR 27, 337 (1940)
  • [7] G. Horvay, Thermal stresses in rectangular strips, Proceedings of the Second U. S. National Congress of Applied Mechanics, Ann Arbor, 1954, American Society of Mechanical Engineers, New York, 1955, pp. 313–322. MR 0078160
  • [8] G. Horvay and J. S. Born, The use of self-equilibrating functions in solution of beam problems, Proc. Second U.S. Natl. Congr. Appl. Mech., ASME, 267, 1954
  • [9] G. Horvay and J. S. Born, Tables of self-equilibrating functions, J. Math. and Phys. 33 (1955), 360–373. MR 0068305, https://doi.org/10.1002/sapm1954331360
  • [10] R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Research. Ser. A. 5 (1952), 227–237. MR 0061543
  • [11] G. Horvay, Discussion of the paper, Approximate stress functions for triangular wedges, by I. K. Silverman, J. Appl. Mech. 22, 600 (1955)
  • [12] G. Horvay, Problems of mechanical analysis in reactor technology, Preprint 355 (by AIChE), Nuclear Eng. and Sci. Congr., Cleveland, 1955
  • [13] G. Horvay, Stress relief obtainable in sectioned heat generating cylinders, Proc. Second Midwestern Conference on Solid Mechanics, in press
  • [14] G. Horvay, Saint Venant's principle: a biharmonic eigenvalue problem, submitted to J. Appl. Mech.

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DOI: https://doi.org/10.1090/qam/85734
Article copyright: © Copyright 1957 American Mathematical Society


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