Third Order GBT Terms And Its Expression

Chiu, Rujiang (Manchester School of Engineering, University of Manchester)



First-order Generalized Beam Theory (GBT) describes the behaviour of prismatic structures by ordinary uncoupled differential equations, using deformation functions for bending, torsion and distortion. In secondorder theory, the differential equations are coupled by the effect of deviating forces. By introduced the virtual works of longitudinal membrane bending moment and membrane shear strain into the GBT system, the complete expansions of the third order GBT equation, were obtained in the form of a series of large discretized iterated functions and converted to sets of tangent stiffness matrices for numerical analyses. By deriving out the membrane stresses as the third order terms ijrkVσ and ijrkVt and integrating advanced numerical techniques to find a complete solution, the third order Generalized Beam Theory provides a rigorous and efficient numerical tool to investigate post-buckling large deflection behaviours in thin-walled structures (Davies and Chiu, 2004; Lecce and Rasmussen, 2005).


Generalized Beam Theory (GBT) unifies the conventional theories for the analysis of prismatic thin-walled structural members with a consistent notation. Besides the four rigid body modes with which the conventional beam theory deals with, namely extension, bending about the two principal axes and torsion, the higher order modes of deformation which involve cross section distortion are included in the theory. It provides elegant and economical solutions to a wide range of complex problems and a natural transition from beam theory to folded plate theory. As a unique advantage, especially for the cold formed cassette analysis, GBT is capable to consider a single buckling mode and the combination of any selected buckling modes as specified by the user. This is not possible with any other method. The development of GBT was pioneered by Schardt (1966, 1982, 1984, 1994) and his colleagues at the University of Darmstadt in Germany, work that has extended over more than 20 years.