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The Quarterly Journal of Mechanics and Applied Mathematics 1958 11(2):142-158; doi:10.1093/qjmam/11.2.142
© 1958 by Oxford University Press
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SECOND-ORDER EFFECTS IN THE TORSION AND BENDING OF TRANSVERSELY ISOTROPIC INCOMPRESSIBLE ELASTIC BEAMS

W. S. BLACKBURN

( Dept. of Math., University of Durham, King's College Newcastle upon Tyne )

General formulae are obtained for the second-order effects in the deformation of incompressible transversely isotropie clastie bodies.

The problem of the second-order torsion and extension of a homogeneous incompressible cylinder transversely isotropie with respect to its generators is reduced to the solution of the classical torsion and flexure boundary-value problems together with another boundary-value problem involving two complex potential functions. Without solving these a formula is obtained for the fractional elongation of the cylinder. When the cross-section is bounded by a single closed curve the equation satisfied by the potential functions reduces to that obtained by Green and Shield for the case of torsion of an isotropic cylinder, and their general method of solution applies.

The problem of the second-order bending of such a eylinder by terminal couples is also reduced to the solution of a single boundary-value problem for two complex potential functions and the classical boundary-value problem for torsion. A general formula is found for the change of length of the line of centroids and the boundaryvalue problem is solved for the case of right circular cylinders.


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