Imperfections

According to EN-1993-1-5 ANNEX C both geometric and structural imperfections should be included in the FE-model. Imperfections may either be obtained from a refined buckling analysis, or from equivalent values. Any type of imperfection should be taken such that the lowest resistance is obtained. In this section various types of imperfections and methods were evaluated

Geometric imperfection

A geometrical imperfection consists of a combination of an imperfection shape and amplitude. The shape can be equal to one or multiple eigenmodes, or an equivalent shape according to Figure C.1. EN-1993-1-5 ANNEX C as illustrated in Figure 29. Both methods are used to provide node coordinates of the deflected shape, that are subsequently superimposed on the node coordinates of the GMNIA.

(a) (b)

Figure 29- Imperfections for flange(a) and web(b) according to Table C.2. EN-1993-1-5 ANNEX C

To obtain the eigenmodes of the beam a Linear Buckling Analysis(LBA) with a subspace eigensolver was used. Even though this method is widely adopted, a problem with the LBA was encountered for models loaded in M-V interaction. The graph in Figure 30 indicates an increased bending moment capacity for V/Vpl=0,4 to V/Vpl=0.6. Further inspection into these analyses indicated a discrepancy in the shape between the first 14 eigenmodes and the failure mode in the GMNIA, as illustrated in Figure 32 and Figure 31.

Figure 30- M-V interaction graph with imperfections obtained from a LBA λf =10 λw =83 S235 0

0,2 0,4 0,6 0,8 1 1,2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

M/Mpl[-]

V/V_pl[-]

Figure 31- First eigenmode of λf =10 λw =83 S235 V/Vpl=0.4 Figure 32- Failure mode of λf =10 λw =83 S235 V/Vpl=0.4

The imperfection obtained from the LBA was not located at the point of failure in the GMNIA; the mismatch resulted in the increased bending moment in Figure 30.

The cause of the mismatch was found when the displacements over the increments of the GMNIA was considered in Figure 33. When stresses are elastic the displacements correspond to the eigenmode of the LBA, subsequently, when the compression flange starts to yield the flange fails premature to buckling of the web. The assumption the member fails in plasticity or elastic buckling seems to be incomplete; elasto-plastic buckling could occur and seems to be sensitive to the applied imperfections.

Obviously, non-linear material behavior is not incorporated in the LBA, and therefore it might provide imperfections in other sections where failure would occur with non-linear material behavior. To check whether the eigenmode is appropriate as input for the imperfection the eigenvalue should be below the elastic moment resistance, however, if not, the researcher cannot switch between methods in one test population.

Increment 9 Mises:

167 N/mm²

Increment 27 Mises:

240 N/mm² Increment 70 Mises:

242 N/mm²

Figure 33- Evolution of displacements U1 over the increments of the GMNIA

Imperfections should be kept constant to prove causality between slenderness, the influence of shear stress, and moment capacity. This poses another challenge when using eigenmodes as input for the GMNIA analyses. The Abaqus manual recommends applying multiple eigenmodes when the corresponding eigenvalues are close together, however, if the eigenmodes overlap on a single node the imperfections are added up. Therefore, to maintain a constant amplitude the multiple eigenmodes should be selected very carefully. Nonetheless, a constant amplitude for the imperfection for every model is nearly impossible. Even though the LBA is used widely, the difference between linear and non-linear behavior makes it an unsuitable method for many GMNIA analyses.

To obtain the node coordinates of the equivalent shapes a static general step is used. Imposed displacements equal to the amplitude illustrated in Figure 29 were applied to points on the flange and web of the beam.

2 Types of imperfections are applied to the sections in M-V interaction: one shape corresponding to local buckling. one shape corresponding to shear buckling. The stress distribution in a 4-point bending test is identical in both sections, therefore, for any M-V interaction the beam can fail in the lowest failure mode; shear or axial buckling. The amplitude of the imperfection in the outer sections is set to 95% of the middle imperfection, to induce local buckling in the middle and prevent erratic behavior for pure bending.

The buckling shape for shear- and local buckling were recreated from observed failure modes from multiple GMNIA analysis. The double sine is used as buckling shape as it resulted in the lowest bending moment capacity in the GMNIA, as is showed in Table 6.

Table 6- Bending moment capacity for single and double sine imperfection obtained from λf=10 λw=83-S235

Single sine 0.96 M/Mpl

Double sine 0.93 M/MplF

The locations for the points that are displaced were derived from the exact solution for the buckling of members under compression.

Figure 34- Imperfections in Abaqus according to equivalent buckling shapes of Figure 29

The advantage of this method is a constant imperfection for every model in contrary to the LBA, this assures less parameters to consider and a better comparison between the results. Using the equivalent buckling shape, no mismatch was obtained between the applied imperfection and failure mode of the GMNIA. Consequently, no increased bending moment capacity as was illustrated in Figure 30 was recorded. Therefore, the LBA was discarded and the static general step was used to obtain imperfections.

Size of geometrical imperfections

The amplitudes of the imperfections for flange and web are illustrated in Figure 29. The sensitivity of the model to this imperfection was tested by scaling the amplitude. The results are illustrated in Figure 35, and show a linear decline of the model for deviations up to a factor 2. It was expected deviations of the imperfection would not exceed that limit, therefore, the behavior of the model was considered reliable.

Figure 35- Sensitivity of the Abaqus model to imperfections 0,7

0,75 0,8 0,85 0,9 0,95 1

0 0,5 1 1,5 2 2,5

M/Mpl[-]

n · imperfection amplitude [-]

Structural imperfection

The influence of residual stress on the maximum bending capacity was evaluated for shear- and local buckling. The residual stress was applied according to (Technical Committee 8–Structural Stability Technical Working Group 8.2–System Publication No. 33, 1984). The flange and web were partitioned into 32 sections over which the residual stress was applied in gradient, as illustrated in Figure 36.

Figure 36- approximation of residual stress in principle(left) and Abaqus(right)

The results in Figure 37 for a bending- and shear dominated case indicated that the residual stress affects the point of yielding, however, the maximum bending capacity was maintained. The required results focused on the maximum capacity; the residual stress was not incorporated in the model.

Figure 37- Influence of residual stress on the maximum moment capacity of shear- and bending dominated 4-point bending test.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 2 4 6 8 10

M/Mpl[-]

Displacement[mm]

Residual stress included BD Residual stress excluded BD Residual stress included SD Residual stress excluded SD