In this paper, conventional GBT designates the formulation intended to perform stability (bifurcation) analyses of thin-walled members with unbranched (folded-plate) thin-walled members and made of linear elastic isotropic materials (e. g., most cold-formed steel profiles) – this designation stems from the fact that both (i) the vast majority of Schardt’s publications and (ii) all of Davies’s work concern members with these characteristics. Moreover, all the recently developed GBT formulations can be viewed, to a smaller or larger extent, as modifications or extensions of this conventional one.
A conventional GBT analysis involves (i) a cross-section analysis, leading to the GBT deformation modes and corresponding modal mechanical properties, and (ii) a member linear stability analysis, to obtain the member bifurcation stress resultants and associated buckling mode shapes (e. g., Davies, 1998; or Schardt, 1994a). In the case of the arbitrary q-walled member shown in Figure 1(a) and for the cross-section discretisation depicted in Figure 1(b) (q + 1 natural and m intermediate nodes), the performance of the cross-section analysis leads to the system of q + m + 1 GBT equilibrium equations (one per deformation mode)
ЕСїкФк, хххх &&їкФк, хх + ЕВікфк + Wj.0Xjik^k ,xx — (1)
where (i) (■),x = d(-)/dx, (ii) фк(х) are modal amplitude functions, (iii) E, G are Young’s and shear moduli, (iv) WJo are uniform (usually single-parameter) pre-buckling stress resultants and
(v) the various matrix/tensor components are related to the cross-section stiffness (Сік, Бік, Вік) and geometric effects (Xjiк). Together with its boundary conditions, system (1) defines a standard eigenvalue problem, the solution of which (i) can be obtained by means of several standard methods (e. g., finite differences, finite elements or Galerkin’s method) and (ii) yields the member bifurcation stress resultants and buckling modes. The latter are combinations of the GBT deformation modes, illustrated in Figure 2 for the case of a lipped channel cross-section (q — 5 and m — 7).