Post-Buckling

The results presented here concern the distortional post-buckling behaviour of two identical FRP laminated plate lipped channel columns (i) having locally/globally pinned and free-to-warp end sections, (ii) with the material properties and cross-section dimensions displayed in Figure 11(b) (identical to the ones considered in the vibration analyses presented in the previous subsection),

(iii) containing critical-mode initial geometrical imperfections with amplitudes v0 = ±0.15 ■ t (t is the wall thickness and v0 is the outward/inward motion of the flange-lip corners at mid-span),

(iv) with length L = 40 cm and (v) discretised into (vi) 6 natural and 17 intermediate (5 in the web and flanges and 1 per lip) nodes and (v2) 8 finite elements (Silvestre and Camotim, 2004e). It is worth noting that the adopted column geometry ensures that (i) bifurcation occurs in a single-wave distortional mode and that (ii) local-plate/distortional mode interaction effects are not relevant – the ratio between the minimum local-plate and distortional buckling loads is 1.6 (Silvestre and Camotim, 2004c).

Figure 16(a) shows the post-buckling equilibrium paths a/acr. D vs. v/t (acr. D = 38.6 MPa is the column critical buckling load and v is the additional outward/inward flange-lip motion) of the two columns, henceforth termed “outward” and “inward”. Also included are the results yielded by the shell FEA performed in Abaqus (HKS, 2002). Figure 16(b), on the other hand, displays diagrams that provide the contributions of the various GBT deformations modes (depicted in Figures 2, 9 and 10) to various column deformed configurations located along their equilibrium paths. Finally, Figure 16(c) provides the post-buckling evolution of the mid-span normal stress distribution along

the mid-lines of the outward and inward column outer layers (longitudinally aligned fibres).6 After

a close observation of the above figures, one is led to the following conclusions:

(i) First of all, one instantly recognises the important role played by the initial imperfection “sign”. Indeed, the inward column post-buckling stiffness and strength are larger than their outward column counterparts by a non-negligible amount. Like in the case of the isotropic members (Silvestre and Camotim, 2003), this distortional post-buckling asymmetry stems mostly from the different contributions of the shear modes 15 + 19 + 23 to the outward and inward column deformed configurations.

(ii) Then, attention should be drawn to the virtual coincidence between the GBT-based equilibrium paths and the post-buckling results yielded by Abaqus. The fact that the GBT analyses never involved more than 450 degrees of freedom provides a clear assessment of the high compu­tational efficiency of this approach. The corresponding (and similarly accurate) FEM results required the discretisation of the thin-walled columns by means of very refined shell element meshes.

(iii) In the initial pre-buckling stages, the normal stress distribution is uniform, since mode 1 dom­inates. As post-buckling progresses and regardless of the v0 sign (outward or inward motions), the normal stress distribution becomes non-linear in the web and flanges, mostly due to the shear modes 15 + 19 + 23.

(iv) In the outward column post-buckling stages, the contributions of modes 5 and 15 account for the fact that the compressive stresses (iv1) increase near the web-flange node and (iv2) decrease in the vicinity of the flange-lip corner. Moreover, for a/acrD > 1.0, tensile stresses start to develop around the flange-lip node and rather high compressive stresses appear close to the lip free ends. Conversely, the inward column exhibits a compressive stress increase near the flange-lip corner (all the flanges are under compression) and high tensile stresses develop in

Due to symmetry, only one half of the normal stress distributions are represented in Figure 16(c).

the neighbourhood of the lip free ends – however, these tensile stresses are lower than the com­pressive ones appearing in the outward column. This last fact is due to the relevant participation of mode 19, reinforcing mode 5 in the outward column and opposing it in the inward one.