On the application of oblique co-ordinates to problems of plane elasticity and swept-back wing structures

Abstract

Summary.--The object of this report is two-fold. On the mathematical side it seeks to illustrate the use of oblique co-ordinates in applications to Elasticity and Structure Theory. On the practical side it seeks to provide methods by which designers can solve problems of stress distribution and deflection for the case of swept-back wing structures, whose ribs lie parallel to the direction of flight. The report is divided into three parts. In Part I the mathematical basis is developed. Formulae are derived which express the fundamental concepts and relations of Geometry, Kinematics, Statics and Plane Elasticity in terms of vector components in oblique co-ordinates. In Part II, the results obtained in Part I are applied to a uniform, symmetrical, rectangular section, swept-back box. A complete theory of stress distribution and deflections is obtained for the case of loading by \\'normal\\' forces and couples (forces whose directions and couples whose planes are normal to the plane of sweep-back) applied to the ends of the box. Some consideration is also given to problems of constraint against warping. In Part III the main results of Part II are generalised to cover the case of a more representative wing structure. This represents an extension of the usual Engineer\\'s Theory of Bending and Torsion to cover the case of swept-back wings with ribs parallel to the flight direction. Practical procedures based upon this extension are laid down for stress distribution and deflection calculations. These will have the same validity for swept-back wings, as the usual design approximations have for the unswept case. An appendix reproduces tables and graphs of certain functions useful in the application of the theory, from a paper by S. R. Lewis.