The present research work was addressed to the study of the influence of the local buckling phenomena on the behaviour of aluminium members in compression or in bending. So that, starting from the main results on the theories of the plate stability in the elastic, the theory of plastic buckling has been developed according to the J2 deformation theory of plasticity to define the plate differential equation by introducing the variability of Poissonโs ratio as a function of the stress level.
Subsequently, the stub column tests and the three point bending tests, carried out on the same SHS aluminium members, have been presented.
Starting from the theoretical and experimental results, previously described, different methodologies have been proposed to evaluate the ultimate behaviour of aluminium members under uniform compression and the beams under moment gradient.
In particular, the plate differential equation at the onset of buckling has been integrated referring to plates under uniform compression and applied to analyse the interactive buckling occurring in the case of box sections, like SHS and RHS, and of H-shaped sections. The boundary conditions accounting for plate elements interaction have been properly derived and the buckling criterion has been defined by imposing the conditions assuring a non-trivial solution of the relevant equation system, so that a fully theoretical approach has been proposed to evaluate the behaviour of columns under uniform compression.
The non-linear behaviour of aluminium alloy beams subject to non-uniform bending has been investigated by means of a finite element model including both geometrical and mechanical non-linearity and initial geometrical imperfections. In particular, a wide parametric analysis has been carried out by varying the non-dimensional geometrical parameters describing the structural scheme and the accuracy of the finite element model, carried out using ABAQUS software. Moreover, in
Finally, the effective thickness approach, currently adopted in the Eurocode 9 for evaluating the ultimate resistance of the fourth-class sections affected by the local buckling, has been extended accounting for the mechanical non-linearity of aluminium material and the interaction of the plate elements constituting the cross sections. In particular, a simplified procedure under displacement control has been performed to determine the inelastic response of the box-shaped sections and H-shaped sections in compression or in bending.
To evaluate the accuracy of the previous methodologies, the experimental results presented in this work and provided in the technical literature have been compared with those obtained by the theoretical and empirical procedures. The comparison of these methodologies is summarised in Table 7.1 and Table 7.2. In the case of members under compression, both the theoretical approach (DTP) and the effective thickness method (ETM) accurately estimate the maximum compressive load.
Instead, as regard the estimated of the normalised deformation, the ETM method represents a very conservative approach, although simpler to apply than the DTP method.
As for the beams under non-uniform bending, the empirical formulas provide the results closer to the experimental one in comparison to the effective thickness method and their applications are very simple. However, the mathematical formulas can not be applied to evaluate the behaviour of aluminium beams in the elastic region and, generally, they have to be applied according to specific ranges of validity of the non-dimensional parameters, previously mentioned. The prediction of the rotational capacity, corresponding to the maximum bending moment, is more conservative than the values obtained by the three point bending tests. Furthermore, high values of standard deviations ๐ suggest the scattering and thus the rotational predictions are considerate of low reliability in relation to the low number of cases analysed.
For the sake of completeness, the comparison has been carried out also with the values of the resistance obtained by the Eurocode 9. In particular, the maximum compressive strength is computed according to Section 6.2.4 of EN 1999-1-1, as depicted in Eq. (1.1), while the maximum flexural strength is evaluated according to Section 6.2.5 of EN 1999-1-1, as provided in Eqns. (1.2) and (1.3).
The safety coefficient ๐พ is assumed equal to 1.00 for both cases, because the estimation is carried out from the point of view of maximum resistance prediction not from that of the safety control as reported in the design code provisions. The values computed according to Eurocode 9 are, respectively, reported in Annex A and in Annex B. From the comparison of the results, it is evident that the ultimate strength values computed according to the design code approach are more
They have been proposed within the framework of the activities of the project teams of CEN/TC250/SC9 encharged of the revision of Eurocode 9.
Finally, the following future developments can be suggested. The fully theoretical procedure, developed for the box-shaped and H-shaped sections, could easily be extended to the other aluminium cross-section (channel section, angle section stiffened rectangular section, ecc).
In reference to the study of the ultimate behaviour of beams in bending, there is still a substantial gap of knowledge, especially, regarding the H-shaped sections. So that, an extensive experimental campaign could be carried out on the H-shaped beams, by varying the width-to-thickness ratios, the shear length and the aluminium alloy. Moreover, for extending the validation of the proposed mathematical formula, the parametric analysis, developed in Chapter 5, could be performed on the other different aluminium alloys possibly belonging to the 5000 and 7000 series. Another aspect that might be worth investigating is related to evaluate accurately the maximum rotational capacity. In fact, the estimation by means of Eq. (6.26) provided very different values respect to the experimental results and for this reason, they have not been depicted. One reason could be related to the inaccurate assessment of the length of the plastic hinge. So that, a future study could be devoted to improving the evaluation of the plastic length through the experimental tests and finite element analysis.
Table 7.1. Comparison between the deformation theoretical procedure (DTP) and the effective thickness method (ETM).
๐๐๐๐๐๐๐ Design Code
EN 1999-1-1
Deformation ๐๐ก๐๐จ๐ซ๐๐ญ๐ข๐๐๐ฅ ๐๐ซ๐จ๐๐๐๐ฎ๐ซ๐ (๐๐๐)
๐๐๐๐๐๐ญ๐ข๐ฏ๐ ๐๐ก๐ข๐๐ค๐ง๐๐ฌ๐ฌ ๐๐๐ญ๐ก๐จ๐ (๐๐๐)
๐๐๐, ๐๐๐ ๐ต๐.๐ฌ๐ช๐
๐ต๐.๐๐๐
๐ต๐.๐ซ๐ป๐ท ๐ต๐.๐๐๐
๐บ๐.๐ซ๐ป๐ท ๐บ๐.๐๐๐
๐ต๐.๐ฌ๐ป๐ด ๐ต๐.๐๐๐
๐บ๐.๐ฌ๐ป๐ด ๐บ๐.๐๐๐
Mean
[ฮผ] 0.93 1.02 0.99 0.96 0.81
Standard deviation
[ฯ] 0.09 0.09 0.18 0.07 0.22
๐๐๐๐๐๐๐ Design Code
EN 1999-1-1
Deformation ๐๐ก๐๐จ๐ซ๐๐ญ๐ข๐๐๐ฅ ๐๐ซ๐จ๐๐๐๐ฎ๐ซ๐ (๐๐๐)
๐๐๐๐๐๐ญ๐ข๐ฏ๐ ๐๐ก๐ข๐๐ค๐ง๐๐ฌ๐ฌ ๐๐๐ญ๐ก๐จ๐ (๐๐๐)
๐, ๐ ๐ต๐.๐ฌ๐ช๐
๐ต๐.๐๐๐
๐ต๐.๐ซ๐ป๐ท ๐ต๐.๐๐๐
๐บ๐.๐ซ๐ป๐ท ๐บ๐.๐๐๐
๐ต๐.๐ฌ๐ป๐ด ๐ต๐.๐๐๐
๐บ๐.๐ฌ๐ป๐ด ๐บ๐.๐๐๐ Mean
[ฮผ] 0.90 1.01 0.97 0.93 0.97
Standard deviation
[ฯ] 0.14 0.08 0.17 0.11 0.28
Table 7.2. Comparison between the empirical regressions (ER) and the effective thickness method (ETM).
๐๐๐๐๐๐๐ Design Code
EN 1999-1-1 ๐๐ฆ๐ฉ๐ข๐ซ๐ข๐๐๐ฅ
๐๐๐ ๐ซ๐๐ฌ๐ฌ๐ข๐จ๐ง๐ฌ (๐๐)
๐๐๐๐๐๐ญ๐ข๐ฏ๐ ๐๐ก๐ข๐๐ค๐ง๐๐ฌ๐ฌ ๐๐๐ญ๐ก๐จ๐ (๐๐๐)
๐๐๐, ๐๐๐, ๐, ๐ ๐ด๐.๐ฌ๐ช๐ ๐ด๐.๐๐๐
๐ด๐.๐ฌ๐น ๐ด๐.๐๐๐
๐ฝ๐.๐ฌ๐น ๐ฝ๐.๐๐๐
๐ด๐.๐ฌ๐ป๐ด ๐ด๐.๐๐๐
๐ฝ๐.๐ฌ๐ป๐ด ๐ฝ๐.๐๐๐ Mean
[ฮผ] 0.91 1.02 0.92 0.97 0.90
Standard deviation
[ฯ] 0.06 0.14 0.15 0.15 0.12