**3. To point out a case in which the estimation method is not applicable by using the results of numerical simulations**

both edges of the flange due to a corner constraint at the edges. Just after buckling, the stress increment at both edges is greater than the stress decrement in the middle of the compression flange, and thus the total force on the compression side increases and the moment increasing continuously. It is also noted that the stress on the web changes almost linearly; this suggests that buckling does not occur at the web. Therefore, the axial stress distribution when the maximum moment occurs is in good agreement with that obtained by the Kecman's method using the effective width of the compression flange, as shown by the solid line in the figure.

to buckling at the compression flange, and the maximum moment can be predicted by the

moment is in good agreement with the value obtained from Eq. (5) for Case 2. The maximum

Moreover, as shown in **Figure 5(b)**, the absolute value of the axial stress at phase (2), for which the maximum moment occurs, is greater than the value at phase (1) for all cross-sectional

exist plastic yielding regions in the webs. In **Figure 5(b)**, Kecman's stress distribution when the maximum moment occurs is obtained by linear interpolation of two theoretical stress distributions corresponding to *Mel* and *Mpl*, respectively, and is shown by a solid line. It is seen from **Figure 5(b)** that the axial stress distribution when the maximum moment occurs obtained from numerical simulation is in good agreement with Kecman's stress distribution.

distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.025 m−1 and *θ*/*L* = 0.065 m−1, respectively, as

, collapse is due

on cross-section for a square

at point P becomes 1.

, and there also

). As shown in **Figure 5(a)**, the maximum

/σs

on cross-section for a square tube with *t* = 0.9 mm, *a* = *b* = 50 mm are subjected

at points P and Q on cross-section and (b) axial stress

The above investigation confirms that for such tubes with b/a = 1 and σbuc-a < σ<sup>s</sup>

at point P and Q occurs in the maximum moment and σ<sup>x</sup>

positions. At phase (2), the stress at the flanges is equal to the yield stress σ<sup>s</sup>

**Figure 5** shows the bending moment *M* and the axial stress σ<sup>x</sup>

tube with *t* = 0.9 mm, *a* = *b* = 50 mm (σbuc-a = 1.52σ<sup>s</sup> > σ<sup>s</sup>

Kecman's method for Case 1.

376 Numerical Simulations in Engineering and Science

**Figure 5.** Moment *M* and axial stress σ<sup>x</sup>

denoted in (a).

to pure bending: (a) changes in moment *M* and axial stress σ<sup>x</sup>

value of σ<sup>x</sup>

In order to investigate the accuracy of the Kecman's method for predicting the maximum moment *Mmax* of tubes under bending, **Figure 6** shows the maximum bending moment of FEM numerical simulations for tubes with aspect ratios *b*/*a* = 1, 2, and 3. Eqs. (4), (5) and (6) are also shown as a comparison. As shown in the figure, the prediction of the Kecman's method is well in agreement with the results of FEM numerical simulations when the relative thickness *t*/*a* is not very small and the aspect ratio of web to flange *b*/*a* is not large, for example, when the tube relative thickness is about *t*/*a* > 0.008 for *b*/*a* = 1 and is about *t*/*a* > 0.016 for *b*/*a* = 2. However, for large aspect ratios, there is a large discrepancy between the values of maximum moment obtained from the Kecman's method and the FEM numerical results. This means that tubes with cross-section of a large aspect to which the Kecman's method does not apply are found

**Figure 6.** Comparison of the Kecman's method and the FEM numerical results.

to exist. Therefore, it is necessary to investigate the bending collapse mechanism of rectangular tubes in order to give an effective method for predicting the maximum moment of tubes.
