Keywords: finite element modelling, modified Riks algorithm, cold-formed steel, Z-purlin, imperfections, failure modes.

Roofs of industrial buildings often consist of thin corrugated metal sheeting supported by thin-walled cold-formed steel purlins. These slender purlins have point or single symmetrical cross-section and because the line of action of the external load does not pass through the shear centre, the purlins are subjected to unsymmetric bending and torsion. However, the purlin is connected to the sheeting, by means of self-tapping screws, which are applied from top of the sheeting through the upper flange of the purlin. Because of this connection, the upper flange is restrained from moving laterally. The sheeting also provides a rotational restraint at the upper flange. The ultimate load that a purlin can withstand is a function of numerous geometric parameters and material properties of the purlin-sheeting system. The influence of these parameters on the ultimate load can be investigated with a finite element model of the purlin.

Since the research is to obtain the maximum load and post-buckling behaviour of a purlin, the modified Riks algorithm is used. This method is implemented in the finite element package ABAQUS and is able to take into account a decrease in load after having reached a limit or a bifurcation point.

As described in literature [1,2,3,4,5] and as follows from the research, these thin-walled members are very sensitive to imperfections. Because imperfections are present in a real purlin anyhow and lead to a lower ultimate load than the one for a perfect purlin, imperfections are to be introduced in the model.

Finding the maximum load of the structure discussed is not so straightforward with the Riks method because convergence problems occur in the neighbourhood of the maximum load. Experience shows that convergence is achieved much easier when initial geometrical imperfections are implemented in the model. Different types of imperfections were investigated: local buckling in the web and compressed flange (obtained from an eigenvalue extraction) and global shape imperfections (corresponding with eigenmodes and self-defined ones based on EN10162 [6]). These shape imperfections are applied with different amplitudes and their influence on the ultimate load is presented. The research shows that not every type of imperfection has the same effect on both the computational effort and the ultimate load. Local imperfections gave in the studied cases the best results: they always led to a stable analysis and a decreased ultimate load, which is not always the case for global imperfections. However, different tolerances exist for global and local imperfections. When taking into account the different maximum amplitudes obtained from EN10162, the local imperfections still lead to the lowest ultimate loads.

The conclusions of this contribution are that implementing initial geometric imperfections in the finite element model for the purlin can reduce the encountered numerical problems when trying to find the ultimate load for the purlin. It has been shown that imperfections based on eigenmodes of the purlin corresponding with buckles to the web provide the best solutions. Out of these, the imperfections corresponding with the lowest eigenvalue lead to the lowest ultimate load in all the studies. This is a conservative approach.That global imperfections do not lead to a stable Riks analysis, can be explained by the fact that the general instability is induced by a local instability in the web.

Global imperfections, whether they are derived from geometric tolerances or from general instability eigenmodes, cannot always force the modified Riks method to find the right equilibrium path. Besides, those imperfections can even result in a non-conservative approach since a larger ultimate load, compared to the one of the perfect purlin, is sometimes found.

Finally it is concluded that eigenvalue analysis cannot be used to predict the ultimate load of these structures: non-linear effects seem to have an important influence on both failure load and failure mode.

1

A. Garstecki, W. Kakol, K. Rzeszut, "Classification of local-sectional geometric imperfections of steel thin-walled cold-formed sigma members", Foundations of civil and environmental engineering, 1, 87-96, 2002.

2

E. S. Bernard, R. Coleman, R. Q. Bridge, "Measurement and assessment of geometric imperfections in thin-walled panels", Thin-Walled Structures, 33, 103-126, 1999. doi:10.1016/S0263-8231(98)00043-3

3

B. W. Schafer, T. Peköz, "Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses", Journal of Constructional Steel Research, 47, 193-210, 1998. doi:10.1016/S0143-974X(98)00007-8

4

D. Dubina, V. Ungureanu, "Effect of imperfections on numerical simulation of instability behaviour of cold-formed steel members", Thin-Walled Structures, 40, 239-262, 2002. doi:10.1016/S0263-8231(01)00046-5

5

B. K. J. Hiriyur, B. W. Schafer, "Yield-line analysis of cold-formed steel members", International Journal of Steel Structures, 5, 43-54, 2005.

6

CEN, "EN 10162 Cold rolled steel sections - Technical delivery conditions - Dimensional and cross-sectional tolerances", April, 2003.

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