Theory of Thin-Walled Elastic Beams with Finite Displacements

A general, nonlinear theory of thin-walled elastic beams of open cross-section is presented, account being taken of finite displacements and warping rigidity. The beam is regarded as a thin shell, and appropriate geometrical constraints are introduced which constitute a generalization of those employed in Vlasov's linear theory. Using Kolter's general equations of compatibility for shells, expressions for the strains and the strain energy of the thin-walled beam are derived. The rotations of the beam are described by means of finite rotation vectors which are closely associated with W.R. Hamilton's quaternions. A new variational principle is formulated which may be used as a basis of numerical methods for the solution of beam problems. Some applications of the theory are also discussed.