Asymptotic solutions for shells with general boundary curves

Author:
T. R. Logan

Journal:
Quart. Appl. Math. **31** (1973), 93-114

DOI:
https://doi.org/10.1090/qam/99709

MathSciNet review:
QAM99709

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: The influence of arbitrary edge loads on the stresses and deformations of thin, elastic shells with general boundaries is studied by means of asymptotic expansions of a general tensor equation. Expansions are made in terms of an exponential or an Airy function and a series in powers of a small-thickness parameter. Most of the steps in the procedure are effected by using the dyadic form of the tensors. Solutions are obtained that are valid in the large, with no restrictions on the loading or on the boundary geometry.

**[1]**C. R. Steele,*A geometric optics solution for the thin shell equation*, Int. J. Eng. Sci.**9**, 681-704 (1971) MR**0302022****[2]**T. R. Logan,*Asymptotic solutions for shell with general boundary curves*. Ph.D. Thesis, Stanford University, July 6, 1970**[3]**A. L. Goldenveizer,*Theory of elastic thin shells*, London, Pergamon Press, 1961 MR**0135763****[4]**A. van der Neut,*Bending at the oblique end section of cylindrical shells*, in*Proceedings of the Symposium on the Theory of Thin Elastic Shells*, W. T. Koiter, ed., Amsterdam, North Holland, 1960, pp. 247-269. MR**0120882****[5]**F. Kitching and M. P. Bond,*Flexibility and stress factors for mitred bends under in-plane bending*, Int. Journal Mech. Sci.**12**, 267-285 (1970)**[6]**D. E. Johnson,*Stresses in a spherical shell with a nonradial nozzle*, J. Appl. Mech. 299-307 (1967)**[7]**A. K. Naghdi and A. C. Eringen,*Stress distribution in a circular cylindrical shell with a circular cutout*, Ingenieur-Archiv.**34**, 161-172 (1965)**[8]**G. N. Savin and A. N. Guz,*Stress around curvilinear holes in shells*(original in Russian, available in English as National Aeronautical Lab., Bangalore, Report TT-2, Sept. 1967 (NASA No. N68-23539))**[9]**A. I. Lure,*Concentration of stress in the vicinity of an aperture in the surface of a circular cylinder*(original in Russian, available in English as New York Univ. Report IMM-NYU 280, Nov., 1960 (AD250308))**[10]**Peter Van Dyke,*Stresses about a circular hole in a cylindrical shell*, A.I.A.A. Journal**3**, 1733-1742 (1965)**[11]**B. M. N. Rao, and T. Ariman,*On the stresses around an elliptic hole in a cylindrical shell*, Report UND-70-5, Defense Documentation Center; presented at the Sixth U. S. National Congress of Applied Mechanics, Harvard University. June, 1970**[12]**P. M. Naghdi,*Foundations of elastic shell theory*, in*Progress in Solid Mechanics*4, North Holland, 1963, pp. 1-90 MR**0163488****[13]**P. M. Naghdi,*On the differential equations of the linear theory of elastic shells*, in*Proceedings of the XI International Congress of Applied Mechanics*, Munich, 1964, pp. 262-269**[14]**V. V. Novozhilov,*Thin shell theory*, The Netherlands: Noordhoff Ltd., 1964 MR**0208886****[15]**J. L. Sanders,*An improved first approximation theory for thin shells*, NASA Report 24, 1959**[16]**W. T. Koiter,*A consistent first approximation in the general theory of thin elastic shells*, in*Proceedings of the Symposium on the Theory of Thin Elastic Shells*, W. T. Koiter, ed., Amsterdam, North Holland, 1960 MR**0142241****[17]**Robert M. Lewis and Joseph B. Keller,*Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell's equation*. New York Univ. Report No. EM-194, 1964**[18]**C. R. Steele,*An asymptotic fundamental solution of the reduced wave equation on a surface*, Q. App. Math. 509-524 (1972) MR**0408397****[19]**Robert M. Lewis et al.,*Uniform asymptotic theory of creeping waves*, Comm. Pure Appl. Math.,**20**, 295-328 (1967) MR**0213101****[20]**C. R. Steele,*A systematic analysis for shells of revolution with nonsymmetric loads, Proceedings, 4th U. S. Nat. Congress of App. Mech.*, pp. 783-792**[21]**George B. Airy,*On the intensity of light in the neighborhood of a caustic*, Trans. Cambridge Philos. Soc.**VI**, part III, 379-402 and plate 7 (1838)**[22]**Donald Ludwig,*Uniform asymptotic expansions at a caustic*, Comm. Pure Appl. Math.**19**, 215-250 (1968) MR**0196254****[23]**Joseph B. Keller,*A geometrical theory of diffraction*, in*Calculus of variations and its application*, New York, McGraw-Hill, 1958, pp. 27-52 MR**0094120**

Additional Information

DOI:
https://doi.org/10.1090/qam/99709

Article copyright:
© Copyright 1973
American Mathematical Society