## Torsion and flexure of composite sections

1950
Payne, Lawrence
##### Organizational Units
Organizational Unit
Mathematics
##### Abstract

In the solution of the Saint Venant problem for composite sections, the general method of attack is the introduction of the conjugate torsion and flexure functions and the reduction of the problem to a Dirichlet type problem. For the orthotropic portion of the section this first involves a transformation of variables and the definition of function harmonic over the transformed variables;In the flexure problem the concept of the center of elasticity is introduced. This center of elasticity plays the same role in the flexure problem as the geometric centroid for the completely isotropic section. It has been pointed out that loading along a geometric axis of symmetry of the section does not insure the absence of a twisting effect. The Young's moduli in the z direction of the respective portions of the section are the only elastic constants entering into the definition of the center of elasticity. If the portions of the section have the same Young's modulus in the z direction regardless of what the other elastic properties of the isotropic or orthotropic portion may be, the center of elasticity will correspond to the geometric center;Solutions have been obtained for composite sections whose boundaries are concentric circles, similar ellipses, confocal ellipses, eccentric circles and rectangles. The values of the torsional rigidities for a number of composite sections have been compared with those for completely isotropic and completely orthotropic sections possessing the same external boundaries;There are a number of other sections which may be solved by the methods developed here. Also it is a simple matter to formulate the problem corresponding to sections composed of two or more different orthotropic materials. Another common type of anisotropy which one might consider a curvilinear anisotropy;Sections of only two different materials have been considered. The methods developed may easily be extended in case of three or more materials. It is sufficient to require that the displacements and tractions be continuous across any common boundary;The torsion and flexure solutions for composite sections suggest the possibility of treating the bending of plates posed of different types of materials. Analogous problems in electrostatics or hydrodynamics might also be formulated and solved.

##### Keywords
Torsion, Elasticity, Girders