The airy stress function in curvilinear coordinates with application to the uniform flexure of a naturally curved spiral beam
✍ Scribed by Karl S. Pister
- Publisher
- Elsevier Science
- Year
- 1954
- Tongue
- English
- Weight
- 442 KB
- Volume
- 257
- Category
- Article
- ISSN
- 0016-0032
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✦ Synopsis
In problems of plane elasticity, in the absence of body forces, the stresses are derivable from a scalar function known as the Airy stress function. By expressing this relation as a tensor equation, the use of the Airy function is generalized for any plane, orthogonal coordinate system. A general expression for the compatibility equation in terms of the stress function is given. It is found that considerable simplification results if isometric coordinates are used. The geometry of the isometric, curvillnear coordinate system is determined by a conformal mapping of the rectangular, Cartesian coordinate plane upon the curvilinear coordinate plane. For a given coordinate system the components of the metric tensor are expressible in terms of the mapping function.
As an application of the preceding theory the uniform flexure of a beam whose edges are bounded by logarithmic spirals is discussed. It is found that all spiral beams may be specified by two dimensionless shape parameters. Stress distributions in a typical beam are exhibited. As limiting cases of the solution for a spiral beam, the uniform flexure of a wedge acted upon by a moment at the vertex and the uniform flexure of a sector of a circular ring are obtained.
PART I. THE AIRY STRESS FUNCTION IN CURVILINEAR COORDINATES
* This paper is a condensation of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Theoretical and Applied Mechanics in the Graduate College of the University of Illinois.