Analytical Solution to Orthotropic Sector
Publication: Journal of Engineering Mechanics
Volume 110, Issue 4
Abstract
A solution for the bending of polar orthotropic pie and ring‐sector plates is presented for three cases of boundary conditions along the straight edges. The classical method of separation of variables is employed and, the basic function in the angular direction satisfies the boundary conditions of the radial edges. Satisfaction of the governing differential equation of the plate is achieved by the derivation of appropriate radial functions. The deflection expression is generated. Isotropic and Orthotropic sector plates of various end conditions have been analyzed and compared with other methods wherever comparisons have been possible.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Bassali, W. A., and Gorgui, M. A., “Flexural Problems of Circular Ring Plates and Sectorial Plates I,” Proceedings of the Cambridge Philosophical Society, Vol. 56, Pt. 1, June, 1960, pp. 75–95.
2.
Bassali, W. A., and Gorgui, M. A., “Flexural Problems of Circular Ring Plates and Sectorial Plates II,” Proceedings of the Cambridge Philosophical Society, Vol. 56, Pt. 4, Oct., 1960, pp. 414–424.
3.
Bassali, W. A., and Halim, E., “Bending of Uniformly Loaded Sectorial Plates,” Journal of the Mechanics and Physics of Solids, Vol. 9, No. 4, Nov., 1961, pp. 230–241.
4.
Bassali, W. A., and Halim, E., “The Transverse Flexure of Normally Loaded Thin Sectorial Plates,” Journal of the Mechanics and Physics of Solids, Vol. 11, No. 2, Mar., 1963, pp. 79–96.
5.
Ben‐Amoz, M., “Note on the Deflections and Flexural Vibrations of Clamped Sectorial Plates,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 26, No. 1, Mar., 1959, pp. 136–137.
6.
Benson, P. R., and Hinton, E., “A Thick Finite Strip Solution for Static, Free Vibration and Stability Problems,” International Journal for Numerical Methods in Engineering, Vol. 10, 1976, pp. 665–678.
7.
Bhattacharaya, A. P., “Bending of a Sectorial Plate Having Clamped Straight Edges,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 42, Mar., 1975, pp. 229–230.
8.
Buchacek, K., “Statics of an Orthotropic Plate in the Form of a Sector of an Annulus,” Acta Technica, C.S.A.V., Vol. 18, No. 3, 1973, pp. 242–265.
9.
Carrier, G. F., “The Bending of the Clamped Sectorial Plate,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 66, 1944, pp. 134–139.
10.
Chapkis, R. L., and Williams, M. L., “Stress Singularities for a Sharp‐Notched Polarly Orthotropic Plate,” Proceedings of the Third U.S. National Congress of Applied Mechanics, (held at Brown University, June, 1958), American Society of Mechanical Engineers, New York, 1958, pp. 281–286.
11.
Cheung, M. S., and Chan, M. Y. T., “Static and Dynamic Analysis of Thin and Thick Sectorial Plates by the Finite Strip Method,” Computers and Structures, Vol. 14, No. 1–2, 1981, pp. 79–88.
12.
Cheung, Y. K., “The Analysis of Cylindrical Orthotropic Curved Bridge Decks,” International Association for Bridges and Structural Engineering, Vol. 29, No. 2, 1969, pp. 41–51.
13.
Cheung, Y. K., King, I. P., and Zienkiewicz, O. C., “Slab Bridges with Arbitrary Shape and Support Conditions: A General Method of Analysis Based on Finite Elements,” Proceedings, Institution of Civil Engineers, Vol. 40, May, 1968, pp. 9–36.
14.
Conway, H. D., and Huang, M. K., “The Bending of Uniformly Loaded Sectorial Plates with Clamped Edges,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 19, No. 24, Mar., 1952, pp. 5–8.
15.
Coull, A., and Das, P. C., “Analysis of Curved Bridge Decks,” Proceedings, Institution of Civil Engineers, Vol. 37, May, 1967, pp. 75–85.
16.
Coull, A., and Ergin, A. S., “Analysis of Bridge Slabs Curved in Plan,” Civil Engineering and Public Works Review, Vol. 60, Dec., 1965, pp. 1745–1749.
17.
Deverall, L. I., and Thorne, C. J., “Bending of Thin Ring‐Sector Plates,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 18, No. 4, Dec., 1951, pp. 359–363.
18.
Dey, S. S., and Samuel, A. T., “Static Analysis of Orthotropic Curved Bridge Decks,” Computers and Structures, Vol. 12, No. 2, Aug., 1980, pp. 161–166.
19.
Dey, S. S., “Finite Strip Method of Analysis for Orthotropic Curved Bridge Decks,” Proceedings, Institution of Civil Engineers, Part 2, Vol. 69, June, 1980, pp. 511–519.
20.
Flugge, W., Handbook of Engineering Mechanics, McGraw‐Hill Book Co., New York, N.Y., 1962.
21.
Harik, I. E., “Bending of Transversally Loaded Orthotropic Rectangular and Sector Plates,” thesis submitted to Wayne State University, Detroit, Mich., in 1982, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
22.
Hasse, H. R., “The Bending of a Uniformly Loaded Clamped Plate in the Form of a Circular Sector,” Quarterly Journal of Applied Mathematics, Vol. 3, Pt. 3, 1950, pp. 271–278.
23.
Heins, C. P., and Hails, R. L., “Behavior of Stiffened Curved Plate Model,” Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Vol. 95, No. ST11, Nov., 1969, pp. 2353–2370.
24.
Heins, C. P., Bending and Torsional Design in Structural Members, Lexington Books, Lexington, Mass., 1975.
25.
Horvay, G., and Hanson, K. L., “The Sector Problem,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 24, No. 30, Dec., 1957, pp. 574–581.
26.
Jurney, W. H., “Displacements and Stresses of a Laterally Loaded Semi‐circular Plate with Clamped Edges,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 26, No. 59, June, 1959, pp. 224–226.
27.
Kuhn, R., “Beitrag zur Berechnung der Halbkreisplate,” Ingenieur‐Archiv, Vol. 12, 1941, pp. 307–319.
28.
Kuno, R., “Strength of a Semi‐Circular Plate,” Japanese Society of Mechanical Engineers, Nihon Kikai Gakkaishi Journal, Vol. 37, 1934, pp. 101–103.
29.
Lekhnitskii, S. G., Anisotropic Plates, Gordon and Breach, Science Publishers, Inc., New York, N.Y., 1968.
30.
Marguerre, E., and Woernle, H. T., Elastic Plates, Blaisdell Publishing Co., 1969.
31.
Morley, L. S. D., “Variational Reduction of the Clamped Plate to Two Successive Membrane Problems with an Application to Uniformly Loaded Sectors,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 16, Pt. 4, 1963, pp. 451–471.
32.
Muggenburg, H., “Einflussflachen fur die am bogerformiger Rand Eingespannte und am Geraden Rand Freie Halbkreisplatte,” Ingenieur‐Archiv, Vol. 24, 1956, pp. 308–316.
33.
Muster, D. F., and Sadowsky, M. A., “Bending of a Uniformly Loaded Semicircular Plate Simply Supported Around the Curved Edge and Free Along the Diameter,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 7, No. 7, Sept., 1956, pp. 329–335.
34.
Nadai, A., “Uberdas Ausbulen von Kreisformigen Platten,” Zeitschcrift Vereines Deutscher Ingenieure, Vol. 59, Pt. 1, No. 9, Feb., 1915, pp. 169–174.
35.
Pardoen, G. C., and Marienthal, M. J., “Finite Strip Method in Polar Coordinates,” Journal of the Engineering Mechanics Division, ASCE, Vol. 104, No. EM3, June, 1978, pp. 587–604.
36.
Ramachandra, R. B. S., and Sridhara, J. K., “Bi‐Orthogonality Relation for Clamped Sector Plates,” Journal of Engineering Mathematics, Vol. 4, No. 4, Oct., 1970, pp. 361–367.
37.
Ramachandra, R. B. S., and Kolathaya, V., “Bending of a Uniformly Loaded Clamped Sector Plate,” Applied Science Research, Vol. 26, July, 1972, pp. 381–388.
38.
Rubin, C., “General Solution to Bending of Orthotropic Sectors,” Journal of the Engineering Mechanics Division, American Society of Civil Engineers, Vol. 109, No. 1, Feb., 1983, pp. 168–174.
39.
Rubin, C., “Application of Orthotropic Plate Theory to Windmill Blade Design,” Journal of Mechanical Design, American Society of Mechanical Engineers, Vol. 103, No. 4, Oct., 1981, pp. 892–894.
40.
Rubin, C., and Blech, J. J., “Bending of Cylindrically Anisotropic Sector Plates,” American Institute of Aeronautics and Astronautics Journal, Vol. 13, No. 8, Aug., 1975, pp. 1103–1105.
41.
Rubin, C., “Bending of Ring and Pie‐Shaped Sectors,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 42, June, 1975, pp. 492–494.
42.
Sawko, F., and Merriman, P. A., “Finite Element Analysis of Bridges Curved in Plan,” Developments in Bridge Design and Construction, Edited by K. C. Rockey et al., Crosby Lockwood, London, England, 1971, pp. 334–346.
43.
Sekiya, T., and Saito, A., “Bending of Sectorial Plates under a Concentrated Load,” Proceedings of the Fourth Japanese Congress for Applied Mechanics, 1954, No. 40, pp. 195–198.
44.
Smith, D. M., “Deflection and Stresses in Built‐Up Half Diaphram,” Proceedings of the Fifth Congress for Applied Mechanics, (held at Harvard University and Massachusetts Institute of Technology), Cambridge, Mass., Sept., 1938, pp. 62–65,
American Association of State Highway and Transportation Officials, Washington, D.C., 1973.
45.
Szilard, R., Theory and Analysis of Plates, Prentice‐Hall, Inc., Englewood Cliffs, N.J., 1974.
46.
Timoshenko, S., and Woinowsky‐Krieger, S., Theory of Plates and Shells, 2nd Ed., McGraw‐Hill Book Co., Inc., New York, N.Y., 1959.
47.
Tolotti, C., “Applicazione di un Nuovo Metodo di M. Picone all' Integrazione delle Equazioni dell'Elasticita in un Parallelepipedo Rettangolo,” Accademia d'Italia. Classe di Scienze Fishiche, Matematiche e Naturali. Atti. Rendiconti, (S. 7 of R. Accademia dei Lincei), Vol. 1, 1940, pp. 512–524.
48.
Tolotti, C., “Sul Calcolo delle Piastre Elastiche a Forma di Settore Annulare,” Accademi d'Italia. Classe di Scienze Fishiche, Matematiche e Naturali. Atti. Rendiconti, (S. 7 of R. Accademia dei Lincei), Vol. 2, 1940, pp. 517–525.
49.
Ufliand, Y. S., “The Integral Equation of Bending of Sectorial Plate with Fixed Radial Edges,” Doklady Akademiia Nauk S.S.S.R., Vol. 72, 1950, pp. 251–254.
50.
Ufliand, Y. S., “The Bending of Sectorial Plates with Fixed Edges,” Prikladneia Matematika I Mekhanika, Vol. 15, 1951, pp. 515–518.
51.
Wahl, A. M., “Strength of Semicircular Plates and Rings Under Uniform External Pressure,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 54, No. 28, 1932, pp. 311–320.
52.
Weber, J. W., “Bending of Sectorial Plates with Clamped Edges,” Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, Vol. 99, No. EM2, Proc. Paper 9641, Apr., 1973, pp. 440–444.
53.
Weber, J. W., “Uniformly Loaded Sectorial Plates,” Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Vol. 103, No. ST1, Jan., 1977, pp. 274–278.
54.
Williams, M. L., “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 19, Dec., 1952, pp. 526–528.
55.
Williams, M. L., “Surface Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates under Bending,” Proceedings of the First U.S. National Congress of Applied Mechanics (held at the Illinois Institute of Technology, June, 1951), American Society of Mechanical Engineers, New York, N.Y., 1952, pp. 325–329.
56.
Williams, M. L., Closures to “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 20, Dec., 1953, p. 59.
57.
Williams, M. L., and Owens, R. H., “Stress Singularities in Angular Corners of Plates Having Linear Flexural Rigidities for Various Boundary Conditions,” Proceedings of the Second U.S. National Congress of Applied Mechanics, (held at the University of Michigan, June, 1954), American Society of Mechanical Engineers, New York, N.Y., 1954, pp. 407–411.
58.
Woinowsky‐Krieger, S., “Clamped Semicircular Plate Under Uniform Bending Load,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 22, No. 1, Mar., 1955, p. 129.
59.
Woinowsky‐Krieger, S., “Uber die Verwendung von Bipolarkoordinaten zur Lozung eineger Probleme der Plattenbiegung,” Ingenieur‐Archiv, Vol. 24, 1956, pp. 47–52.
60.
Yonezawa, H., “Moments and Free Vibrations in Curved Girder Bridges,” Journal of the Engineering Mechanics, Proceedings of the American Society of Civil Engineers, Vol. 88, No. EM1, Feb., 1962, pp. 1–21.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 110 • Issue 4 • April 1984
Pages: 554 - 568
Copyright
Copyright © 1984 ASCE.
History
Published online: Apr 1, 1984
Published in print: Apr 1984
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.