Abstract
A novel technique, the method of projection, is applied to the plane strain problems of determining the tractions, and stress intensity factors, at the fixed end of a cantilever beam under tension, bending or flexure at infinity. The method represents a useful alternative to the integral equation method of Erdogan, Gupta and Cook, and possesses certain advantages; in particular it is much easier to extend the present method to the more difficult dynamics case. An unusual feature of the method is that the required tractions are expanded as a series whose terms have the natural role of displacements rather than stresses.
Similar content being viewed by others
References
Adams G. G. and Bogy D. B., The plane solution for bending of joined dissimilar elastic semi-infinite strips. Int. J. Solids and Structures 12 (1976) 239–249.
Benthem J. P., A Laplace transform method for the solution of semi-infinite and finite strip problems in stress analysis. Quart. J. Mech. Appl. Math. 16 (1963) 413–429.
Bogy D. B., Solution of the plane end problem for a semi-infinite strip. Z.A.M.P. 26 (1975) 749–769.
Bryant R. H., An examination of the edge effect in a cantilever beam. J. Engng. Math. 7 (1973) 351–360.
Davis P. and Rabinowitz P., Methods for numerical integration (2nd ed.). Academic Press, New York, 1975.
Erdogan, F., Gupta, G. D. and Cook, T. S., The numerical solutions of singular integral equations. In: Methods of analysis and solutions to crack problems, (G. C. Sih, ed.), Noordhoff, 1972.
Forsythe, G. and Moler, C. B., Computer solution of linear algebraic systems. Prentice Hall, 1967.
Gregory R. D., Green's functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge. J. of Elasticity 9 (1979) 283–309.
Gregory R. D., The traction boundary value problem for the elastostatic semi-infinite strip; existence of solution, and completeness of the Papkovich-Fadle eigenfunctions. J. of Elasticity 10 (1980) 295–327.
Gregory, R. D. and Wan, F. Y. M., Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory. To be published. Accepted by J. of Elasticity.
Gupta G. D., An integral equation approach to the semi-infinite strip problem. J. Appl. Mech. 40 (1973) 948–954.
NAG Library Manual, Mark 8, Numerical Algorithms Group Ltd., Oxford.
Spence D. A., Mixed boundary value problems for the elastic strip: the eigenfunction expansion. Technical Summary Report 1863, Mathematics Research Center, Madison, 1978.
Swan G. W., The semi-infinite cylinder with prescribed end displacements. SIAM J. Appl. Math. 16 (1968) 860–881.
Timoshenko, S. and Goodier, J. N., Theory of elasticity. McGraw-Hill, 1951.
Torvik P. J. The elastic strip with prescribed end displacements. J. Appl. Mech. 38 (1971) 929–936.
Williams M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech. 19 (1952) 526–528.
Stern M. and Soni M. L., On the computation of stress intensities at fixed-free corners. Int. J. Solids and Structures 12 (1976) 331–337.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gregory, R.D., Gladwell, I. The cantilever beam under tension, bending or flexure at infinity. J Elasticity 12, 317–343 (1982). https://doi.org/10.1007/BF00042208
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00042208