The cross-section of a thin-walled member is classified according to its mid-line, which may be
(i) open or closed and (ii) branched or unbranched (see Figure 4). As mentioned earlier, the conventional GBT is valid only for folded-plate members, i. e., members with open unbranched crosssections. Thus, in order to extend its application to all other cross-section shapes, one must modify
Figure 5. (a) Determination of the additional deformation modes accounting for shear deformation and (b) the 10 most relevant deformation modes of a narrow rectangular hollow cross-section.
the cross-section analysis procedure – since the member analysis remains unaltered, both system (1) and its boundary conditions retain their forms. This task has now been completed for any conceivable cross-section shape and comprised the following three stages:
(i) The first extension concerned (unbranched) single-cell closed cross-sections and involves the inclusion of four additional deformation modes, which (ii) are related to the shear deformation of the cross-section mid-line and (i2) stem from the imposition of unit transverse displacements in each wall, while preventing all the other ones, as shown in Figure 5(a) (Gonsalves and Camotim, 2004a). Moreover, one must incorporate the term fs GtVi Vjds in the analysis, to account for the virtual work associated with the shear strains. The 10 most relevant deformation modes of a narrow rectangular hollow section are displayed in Figure 5(b) and one notices that, besides the global (1-4) and local-plate (6-10) modes, similar to the ones yielded by the conventional GBT, a novel distortion (not distortional) mode 5 appears – together with mode 4, it models the cross-section shear deformation (Gonsalves and Camotim, 2004b).
(ii) Next, a methodology that can handle arbitrarily branched open cross-sections was developed, thus overcoming difficulties related to (ii1) the proper selection of the elementary warping and flexural functions and (ii2) the solution of the statically indeterminate folded-plate problem (Dinis et al., 2006). One must view the cross-section as a combination of an unbranched subsection and an ordered sequence of branches, which leads to the straightforward identification of the dependent natural nodes, i. e., the natural nodes where the warping displacements cannot be imposed (they must be calculated) – the number of such nodes is equal to Y, (mwi — 2), with the summation extending to all branching nodes and mwi being the number of walls emerging from branching node i. These concepts are illustrated in Figure 6, where (ii1) the cross-section depicted in Figure 6(a) has six dependent nodes (see Figure 6(e)) and (ii2) two possible combinations of an unbranched sub-section and the corresponding branch sequence are displayed in Figures 6(b)-(d) (note that in both cases one must go up to second-order branches).
(iii) Finally, Gonqalves et al. (2006) have just developed the “definitive” formulation, in the sense that it is applicable to fully arbitrary cross-sections, namely those combining closed cells with open branches. A brief description of the steps and procedures involved in this formulation is presented next and illustrated through the cross-section depicted in Figure 7(a): an I-shaped section with closed cells separating the web from the unequal flanges – it has 13 walls, 2 closed cells and 12 natural nodes (6 branching ones):
Figure 6. Illustrative branched section (a) geometry and two possible (b) unbranched sub-sections, (c) first-order branches, (d) second-order branches and (e) independent/dependent natural nodes.
Figure 7. (a) Cross-section geometry and (b) dependent and independent natural nodes.
(111.1) Choice of the dependent natural nodes – Figures 7(b1)-(b2) show two possible choices for the 6 dependent and 6 independent natural nodes to be considered in the analysis.
(111.2) Determination of the “warping initial shape functions”, by imposing elementary functions at each independent natural node and assuming that Vlassov’s hypothesis holds in all walls,
i. e., following the methodology developed by Dinis et al. (2006).
(111.3) Determination of the “local-plate initial shape functions”, by imposing elementary flexural functions at each intermediate node – 13 intermediate nodes were included in the illustrative example (mid-points of each internal wall and free ends of the external ones).
(111.4) Identification of the conventional deformation modes, yielded by the simultaneous diagon- alisation of the stiffness matrices [Cik] and [Bik]. In the case of the illustrative example, one identifies 19 deformation modes, 11 of which are shown in Figure 8: 5 warping (2-6) and 6 local-plate (13-18) – note that modes 2-4 are “rigid-body” ones and mode 1 (axial extension) has been omitted.
(111.5) Sequential imposition of unit membrane shear strains in each wall belonging to a closed cell, keeping all remaining walls free of those strains, and determination of the corresponding “initial shear shape functions”. In the example, 6 unit shear strains are imposed (3 per closed cell), by combining u and v displacements – enforcing these unit strains may be a quite cumbersome task and some guidelines on how to carry it out can be found in the work by Gonqalves et al. (2006).
Figure 8. In-plane shapes of the 17 most relevant warping (2-6), shear (7-12) and local-plate (13-18) modes.
(iii.6) Identification of the shear deformation modes, again by simultaneously diagonalising matrices [Cik] and [Bit] – in this case, one obtains the 6 shear deformation modes (7-12) shown in Figure 8.