Distortional Modes of Thin-Walled Beams

The classic thin-walled beam theory for open and closed cross-sections can be generalized by including distortional displacement modes. The introduction of additional displacement modes leads to coupled differential equations, which seems to have prohibited the use of exact shape functions in the modelling of coupled torsion and distortion.

However, if the distortional displacement modes are chosen as those which decouple the differential equations as in non proportionally damped modal dynamic analysis then it may be possible to use exact shape functions and perform analysis on a reduced problem. In the recently developed generalized beam theory (GBT) the natural distortional displacement modes are determined on the basis of a quadratic eigenvalue problem.

However, as in linear modal dynamic analysis of proportionally damped structures this problem has been solved approximately using linear eigenvalue analysis of modified sub problems. This seems to have worked well for open cross sections but not for closed. It will be shown that it is possible to solve the distortional quadratic eigenvalue problem and find the natural distortional displacement modes using a method equivalent to that used for non proportionally damped (linear) dynamic modal analysis.

The beam displacement field is separated into a sum of products of the cross-section displacement modes and their axial variation. This displacement field will then be constrained to follow the shear assumptions made in Vlassov beam theory, which condenses the problem considerably and reduces the number of possible eigen modes.