le from the supplier Instron 2015 (reference manual). The setup is shown in Fig. 10.
The compression tests were force controlled. The tests had a power increase of 500 N/s up to the set maximum limit, which depended on the specimen geometry, and the critical buckling force calculated from the finite element model (Abaqus).
In order to identify the critical force for sideways movement, the experimental data need to be analyzed. The sideways movement is initiated by exceeding a critical force. In order to identify the onset of buckling, change in strain to force was analyzed by use of the following equation related to flexibility. Flexibility is derivative of displacement with respect to force, and this equation is correlated to flexibility by a constant factor since displacement is replaced by strain.
The number of datapoints for each specimen is typically above 4500, so the change from point to point is small. Therefore, the derivative gives a good indicator for changes both the force and the strain. The absolute value gives a more readable figure. A typical example of the critical force is given in Fig. 11. The geometry of this specimen can be seen in Table 2, with a smaller diameter of 7 mm and a radius of curvature of 56 mm.
The recorded data are somewhat noisy and no data filters are used. Here, the derivative shows the first clear peak which is interpreted as the critical force for onset of sideways movement. A closer look at the recorded data shows that the maximum peak of the derivative is caused by a change in the measured longitudinal strain not followed by a corresponding change in force. This explains how the sideways movement appears. The derivative of the strain to force curve in Fig. 11 shows a clear peak at 22600 N. The corresponding force-displacement curve in Fig. 12 does not show any clear sign of onset of buckling.
Results and discussion
The model showed a reasonable correlation with the experimental results, as shown in Table 3. The deviation between the model and the experiments is maximum 11% between average of experiments and model. The model shows a significant better correlation for the larger diameter specimen. One can see from Table 3 that a larger diameter (d) gives a higher measured buckling force as expected. It is also clear that the smaller ratio R/d between the specimen diameter (d) and the radius of curvature (R) gives a higher buckling force.
A large diameter gives a smaller deviation between the experiment and the model. The model allows the whole bottom grip of the specimen to move freely in the X-Z plane. This is a simplification that does not include the stiffness with respect to sideways movement of the lower elongation rod of the test machine as can be seen from the photo in Fig. 4. This simplification is expected to give lower buckling force values in the model than in the experiments. In general, it is observed that the calculated values from the model are smaller than the measured values from the experiments.
This investigation was adopting a 3D model perfectly straight with the resultant of the force applied coinciding with the model’s axis. The onset of buckling is normally caused by a perturbation for a perfectly straight model. In the current investigation, the perturbation is expected to be caused by a numerical effect in the calculations. Other factors influencing the onset of buckling could be tilting of the specimen or geometric imperfections in the shape of the specimen. By adapting a 0.1 and a 1° tilting of the specimen to the compression axis, the used model gave a lower buckling force than was found in the experiments. The determination of the degree of tilting will therefore cause a substantial uncertainty in the model. The same goes for the determination of the value of machine stiffness for predicting the onset of sideways movement of the specimen. Both these factors will influence the predicted buckling force.
The observed phenomenon named buckling in this work is principally different from a classic buckling. In those cases, the onset of sideways movement will 