The results shown in Figures 12(a)-(b) and 13(a)-(b) concern the bifurcation behaviour of isotropic elastic and elastic-plastic hat-section beams (uniform major axis bending) with end sections that are locally and globally pinned and may warp freely (Gonsalves and Camotim, 2005). While Fig­ure 12(a) shows the 9 most relevant deformation modes (out of 15 – the cross-section discretisation involved the following intermediate nodes: 3 in each web, 1 in the flange mid-point and 1 in each stiffener free end), Figure 12(b) depicts three GBT-based buckling curves, corresponding to elastic and elastic-plastic (Z2-flow and /2-deformation) beams and providing the variation of the critical moment Mcr with the normalised length L/h (logarithmic scale). In addition, the white dots in Figure 12(b) stand for FEM-based elastic critical moments obtained using the code Adina (Bathe,

2003) .[4] As for Figures 13(a)-(b), they make it possible to compare the Adina (perspective) and GBT-based (in-span cross-section) elastic critical buckling mode shapes of the beams indicated in Figure 12(b): lengths L/h = 5 (six-wave local-plate buckling modes 7 + 9) and L/h = 10 (single­wave distortional-flexural-torsional buckling modes 3 + 4 + 6). Although an in-depth discussion of the results shown in these two figures is beyond the scope of this paper, the following general comments and remarks are appropriate:

(i) The linear elastic Mcr values yielded by the GBT-based analyses practically coincide with the ones obtained using Adina. Moreover, one notices that the critical buckling mode nature varies with L/h as follows: (i) local-plate buckling (modes 7 + 9) for short beams (L/h < 8.8),

(ii) distortional-flexural-torsional buckling (modes 3 + 4 + 6) for intermediate beams (8.8 <


©- Distortional-Flexural-Torsional Mode


Figure 13. ADINA and GBT-based elastic critical buckling mode shapes – beams with (a) L/h = 5 and (b) L/h = 10.

L/h < 20) and (iii) “classical” lateral-torsional buckling (modes 3 + 4) for the longer beams (L/h > 20). Finally, note that the accuracy of the Adina results deteriorate for L/h < 4 (local-plate buckling), due to the occurrence of stress concentrations in the shell element model (in the vicinity of the end sections) – obviously, they lower the Mcr values.

(ii) There is also a virtual coincidence between the elastic critical buckling mode shapes yielded by GBT and Adina obviously, the former concern the most deformed beam cross-sections.

(iii) The elastic-plastic GBT-based results confirm the well-known fact that deformation theory leads to lower critical buckling loads than flow theory. This is particularly true for local-plate buckling, which involves almost exclusively transverse plate bending – recall that the transverse plate bending stiffness 13E22/12 decreases more rapidly for deformation theory (Gonsalves and Camotim, 2004b). The differences are much less relevant for distortional-flexural-torsional buckling and vanish for lateral-torsional buckling (the beams buckle elastically). Due to higher local-plate buckling stresses, the length interval related to critical distortional-flexural-torsional buckling is larger for flow theory than for deformation theory.

Figure 14. Variation of, for (a) 10 < L < 1000 cm and (b) 40 < L < 200 cm, and (c) modal participation diagram of the beam fundamental vibration mode with L and M/Mcr.