Effects of approximations in analyses of beams of open thin-walled cross-section - part I: Flexural-torsional stability

In formulating a 6nite element model for the 7exural–torsional stability and 3-D non-linear analyses ofthin-walled beams, a rotation matrix is usually used to obtain the non-linear strain–displacement relationships. Because ofthe coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simpli6ed small rotation matrix is often used in the formulation of 6nite element models for the 7exural–torsional stability analysis of thin-walled beams ofopen cross-section. However, the approximations in the small rotation matrix may lead to the loss ofsome signi6cant terms in the stability sti ness matrix. Without these terms, a 6nite element line model may predict the incorrect 7exural–torsional buckling load ofa beam. This paper investigates the e ects ofapproximations in the elastic 7exural–torsional stability analysis ofthin-walled beams, while a companion paper investigates the e ects ofapproximations in the 3-D non-linear analysis. It is found that a 6nite element line model based on a small rotation matrix may predict incorrect elastic 7exural–torsional buckling loads ofbeams. To perform a correct 7exural–torsional stability analysis ofthin-walled beams, modi6cation ofthe model is needed, or a 6nite element model based on a second-order rotation matrix can be used.