**4. To understand a factor of the discrepancy by using the results of numerical simulations**

We investigate three tubes to which the Kecman's method is not applicable.

**Figure 7** shows the bending moment *M* and the axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 0.4 mm, *a* = 50 mm, *b* = 100 mm (σbuc-a = 0.30σ<sup>s</sup> ). As shown in **Figure 7(a)**, the maximum moment is less than the value obtained from Eq. (4) for Case 1. The maximum value of σ<sup>x</sup> at point P is in good agreement with the elastic buckling stress σbuc-a given by Eq. (2), and the maximum value occurs before the maximum moment. Meanwhile, the axial compression stress σ<sup>x</sup> at point Q at the quarter-web width decreases also before the moment becomes the maximum moment. Moreover, as shown in **Figure 7(b)**, the axial stress in the compression flange is concentrated at the edges when the maximum moment occurs, and the axial stress on the compression web does not change linearly. This suggests that compression buckling also arises at the web. Therefore, the axial stresses on the web at the maximum moment are less than that obtained by the Kecman's method, as indicated by the arrows in **Figure 7(b).**

The above investigation reveals that, in cases when *b*/*a* are large and σbuc-a < σ<sup>s</sup> , the collapse is not only due to buckling at the compression flange but also due to buckling at the compression web. Therefore, the maximum moment cannot be predicted by the Kecman's method. Based on **Figure 7(b)**, the cross-sectional stress distribution under the maximum moment corresponding to this collapse mode can be schematically represented by **Figure 8(a)** and called Case 4 in this chapter.

**Figure 9** shows the bending moment *M* and the axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 0.5 mm, *a* = 20 mm, and *b* = 100 mm (σbuc-a = 2.83σ<sup>s</sup> ). As shown in **Figure 9(a)**, the maximum moment is less than that obtained from Eq. (6) for Case 3. The axial compression stress σ<sup>x</sup> at point P in the middle of the compression flange increases until the moment becomes the maximum moment, and the value σ<sup>x</sup> /σs becomes approximately 1. However, the axial compression stress σ<sup>x</sup> at point Q at the quarter-web width decreases before the moment becomes the maximum moment. Also, as shown in **Figure 9(b)**, the axial stress distribution in the compression flange is almost constant, and the absolute value of σ<sup>x</sup> /σs is approximately 1 when the maximum moment occurs. However, as compared with Case 3 shown in **Figure 2(c)**, it is found that although the buckling stress of the flange σbuc-a obtained from Eq. (2) is higher than twice the yielding stress, σbuc-a = 2.83σ<sup>s</sup> > 2σ<sup>s</sup> , a plastic yielding region is not found in the web. Moreover, the axial stress in the web does not change linearly and decreases greatly in the compression portion of the web. This suggests that compression buckling arises at the web. Therefore, the axial stress distribution at the maximum moment differs greatly from that obtained by the Kecman's method, as indicated by the arrows in **Figure 9(b)**, because in the Kecman's method, the buckling of web is not taken into account.

The above investigation reveals that, in such tubes with large aspect ratio *b*/*a*, even though

**Figure 8.** Schematic representation of axial stress distribution with considering the buckling at web when the maximum

.

and (b) case 5: σbuc-a > σ<sup>s</sup>

stress distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.007 m−1 and *θ*/*L* = 0.016 m−1, respectively,

pression web in a state of plastic yielding at the compression flange. Therefore, the maximum moment cannot be predicted by the Kecman's method. Based on **Figure 9(b)**, the cross-sectional stress distribution under the maximum moment corresponding to this collapse mode can be schematically represented by **Figure 8(b)** and called Case 5 in this

, collapse is not due to plastic yielding at the flange, but rather buckling at the com-

on cross-section for a rectangular tube with *t* = 0.4 mm, *a* = 50 mm, *b* = 100 mm are

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

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226 379

σbuc-a > σ<sup>s</sup>

**Figure 7.** Moment *M* and axial stress σ<sup>x</sup>

moment occurs: (a) case 4: σbuc-a < σ<sup>s</sup>

as denoted in (a).

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

chapter.

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.

the maximum moment is less than the value obtained from Eq. (4) for Case 1. The maximum

Eq. (2), and the maximum value occurs before the maximum moment. Meanwhile, the axial

becomes the maximum moment. Moreover, as shown in **Figure 7(b)**, the axial stress in the compression flange is concentrated at the edges when the maximum moment occurs, and the axial stress on the compression web does not change linearly. This suggests that compression buckling also arises at the web. Therefore, the axial stresses on the web at the maximum moment are less than that obtained by the Kecman's method, as indicated by

not only due to buckling at the compression flange but also due to buckling at the compression web. Therefore, the maximum moment cannot be predicted by the Kecman's method. Based on **Figure 7(b)**, the cross-sectional stress distribution under the maximum moment corresponding to this collapse mode can be schematically represented by **Figure 8(a)** and called

the maximum moment is less than that obtained from Eq. (6) for Case 3. The axial compres-

becomes the maximum moment. Also, as shown in **Figure 9(b)**, the axial stress distribution in

when the maximum moment occurs. However, as compared with Case 3 shown in **Figure 2(c)**, it is found that although the buckling stress of the flange σbuc-a obtained from Eq. (2) is higher

web. Moreover, the axial stress in the web does not change linearly and decreases greatly in the compression portion of the web. This suggests that compression buckling arises at the web. Therefore, the axial stress distribution at the maximum moment differs greatly from that obtained by the Kecman's method, as indicated by the arrows in **Figure 9(b)**, because in the

at point P in the middle of the compression flange increases until the moment

at point Q at the quarter-web width decreases before the moment

/σs

at point P is in good agreement with the elastic buckling stress σbuc-a given by

at point Q at the quarter-web width decreases also before the moment

on cross-section for a rectan-

). As shown in **Figure 7(a)**,

, the collapse is

on cross-section for a rectangu-

becomes approximately 1. However, the

/σs

, a plastic yielding region is not found in the

). As shown in **Figure 9(a)**,

is approximately 1

**4. To understand a factor of the discrepancy by using the results of** 

We investigate three tubes to which the Kecman's method is not applicable.

The above investigation reveals that, in cases when *b*/*a* are large and σbuc-a < σ<sup>s</sup>

**Figure 9** shows the bending moment *M* and the axial stress σ<sup>x</sup>

becomes the maximum moment, and the value σ<sup>x</sup>

than twice the yielding stress, σbuc-a = 2.83σ<sup>s</sup> > 2σ<sup>s</sup>

lar tube with *t* = 0.5 mm, *a* = 20 mm, and *b* = 100 mm (σbuc-a = 2.83σ<sup>s</sup>

the compression flange is almost constant, and the absolute value of σ<sup>x</sup>

Kecman's method, the buckling of web is not taken into account.

**Figure 7** shows the bending moment *M* and the axial stress σ<sup>x</sup>

gular tube with *t* = 0.4 mm, *a* = 50 mm, *b* = 100 mm (σbuc-a = 0.30σ<sup>s</sup>

**numerical simulations**

378 Numerical Simulations in Engineering and Science

value of σ<sup>x</sup>

compression stress σ<sup>x</sup>

the arrows in **Figure 7(b).**

Case 4 in this chapter.

axial compression stress σ<sup>x</sup>

sion stress σ<sup>x</sup>

**Figure 7.** Moment *M* and axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 0.4 mm, *a* = 50 mm, *b* = 100 mm are subjected to pure bending: (a) changes in moment *M* and axial stress σ<sup>x</sup> at points P and Q on cross-section and (b) axial stress distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.007 m−1 and *θ*/*L* = 0.016 m−1, respectively, as denoted in (a).

**Figure 8.** Schematic representation of axial stress distribution with considering the buckling at web when the maximum moment occurs: (a) case 4: σbuc-a < σ<sup>s</sup> and (b) case 5: σbuc-a > σ<sup>s</sup> .

The above investigation reveals that, in such tubes with large aspect ratio *b*/*a*, even though σbuc-a > σ<sup>s</sup> , collapse is not due to plastic yielding at the flange, but rather buckling at the compression web in a state of plastic yielding at the compression flange. Therefore, the maximum moment cannot be predicted by the Kecman's method. Based on **Figure 9(b)**, the cross-sectional stress distribution under the maximum moment corresponding to this collapse mode can be schematically represented by **Figure 8(b)** and called Case 5 in this chapter.

moment. Also, it is seen from **Figure 10(b)** that the axial stress distribution in the compression

mum moment occurs. Moreover, it is also found from a comparison with **Figure 9(b)** that in the web, no buckling occurs, but plastic yielding regions can be observed. However, the plastic yielding is not generated to the entire web, although the buckling stress of the flange

the stress distribution is different from the cross-sectional fully plastic yielding, as indicated by the arrows in the figure. This suggests that even if a compression buckling does not arise at the web, the web slenderness also affects the cross-sectional fully plastic yielding of the tube under bending. That is, the conditions of generating the cross-sectional fully plastic yielding are dependent not only on the flange slenderness but also on the web slenderness. In the Kecman's method, the conditions for the cross-sectional fully plastic yielding are determined by only the

The above investigation reveals that in such tubes with large aspect ratio *b*/*a*, even though

ing to this collapse mode may differ from that of the cross-sectional fully plastic yielding. Therefore, the maximum moment for such tubes cannot be predicted by the Kecman's method.

Bending stress occurs in the web of tube. The problem of web buckling is expressed in **Figure 11**. In **Figure 11(a)**, plate ABCD is defined by the width *b* and thickness *t*. As a boundary condition, displacement in the out-of-plane direction (displacement in the *z* direction) is fixed at both longitudinal edges (BC and DA). The bending and compression are applied through displacement control. For the ultimate loading after buckling, the distribution of

and *b*e2, as shown in **Figure 11(b)**. In the figure, compressive stress is denoted by a positive value. Many studies have been reported on the ultimate loading of a plate after buckling under bending and compression. For example, the effective widths *b*e1 and *b*e2 for a plate under stress gradient shown in **Figure 11** are given in AS/NZS 4600 standard [4] and NAS [5] as follows:

In addition, *b*e1 + *b*e2 shall not exceed the compression portion of the web. Here, ψ is ratio of *<sup>f</sup>*

2 ∗ \_\_ *f* 1

<sup>∗</sup> (11)

along the width direction is characterized by two effective widths, *b*e1

/2 *when <sup>ψ</sup>* <sup>≤</sup> <sup>−</sup>0.236 *be* <sup>−</sup> *be*<sup>1</sup> *when <sup>ψ</sup>* <sup>&</sup>gt; <sup>−</sup>0.236

**5. To propose a new estimation method by considering the factor** 

, the cross-sectional stress distribution under the maximum moment correspond-

σbuc-a obtained from Eq. (2) is higher than twice the yielding stress, σbuc-a = 2.8σ<sup>s</sup> > 2σ<sup>s</sup>

/σs

is approximately 1 when the maxi-

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226

. Thus,

381

(10)

1 ∗

flange is almost constant, and the absolute value of σ<sup>x</sup>

ratio of σbuc-a to σ<sup>s</sup>

compressive stress σ<sup>x</sup>

and*f* 2 ∗ . *f* 1 ∗ and*f* 2 ∗

σbuc-a > 2σ<sup>s</sup>

.

**and using mathematical approach**

**5.1. Effect of the web slenderness on the buckling at web**

⎧ ⎪ ⎨ ⎪ ⎩

*be*<sup>1</sup> <sup>=</sup> *<sup>b</sup>* \_\_\_\_*<sup>e</sup>* 3 − *ψ*

*be*

*be*<sup>2</sup> <sup>=</sup> {

are web stresses shown in **Figure 11(b)**.

*<sup>ψ</sup>* <sup>=</sup> *<sup>f</sup>*

**Figure 9.** Moment *M* and axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 0.5 mm, *a* = 20 mm, *b* = 100 mm are subjected to pure bending: (a) changes in moment *M* and axial stress σ<sup>x</sup> at points P and Q on cross-section and (b) axial stress distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.036 m−1 and *θ*/*L* = 0.048 m−1, respectively, as denoted in (a).

**Figure 10** shows the bending moment *M* and the axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 1.2 mm, *a* = 50 mm, *b* = 150 mm (σbuc-a = 2.8σ<sup>s</sup> ). As shown in **Figure 10(a)**, the maximum moment is less than that obtained from Eq. (6) for Case 3. The axial compression stress σ<sup>x</sup> at point P in the middle of the compression flange increases up to the yielding stress σs before the maximum moment was reached and sets the value σ<sup>x</sup> /σs equal approximately to 1 until the moment becomes the maximum moment. However, the axial compression stress σx at point Q at the quarter-web width increases until the moment becomes the maximum

**Figure 10.** Moment *M* and axial stress σ<sup>x</sup> on cross-section for a rectangular tube with *t* = 1.2 mm, *a* = 50 mm, *b* = 150 mm are subjected to pure bending: (a) changes in moment *M* and axial stress σ<sup>x</sup> at points P and Q on cross-section and (b) axial stress distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.015 m−1 and *θ*/*L* = 0.026 m−1, respectively, as denoted in (a).

moment. Also, it is seen from **Figure 10(b)** that the axial stress distribution in the compression flange is almost constant, and the absolute value of σ<sup>x</sup> /σs is approximately 1 when the maximum moment occurs. Moreover, it is also found from a comparison with **Figure 9(b)** that in the web, no buckling occurs, but plastic yielding regions can be observed. However, the plastic yielding is not generated to the entire web, although the buckling stress of the flange σbuc-a obtained from Eq. (2) is higher than twice the yielding stress, σbuc-a = 2.8σ<sup>s</sup> > 2σ<sup>s</sup> . Thus, the stress distribution is different from the cross-sectional fully plastic yielding, as indicated by the arrows in the figure. This suggests that even if a compression buckling does not arise at the web, the web slenderness also affects the cross-sectional fully plastic yielding of the tube under bending. That is, the conditions of generating the cross-sectional fully plastic yielding are dependent not only on the flange slenderness but also on the web slenderness. In the Kecman's method, the conditions for the cross-sectional fully plastic yielding are determined by only the ratio of σbuc-a to σ<sup>s</sup> .

The above investigation reveals that in such tubes with large aspect ratio *b*/*a*, even though σbuc-a > 2σ<sup>s</sup> , the cross-sectional stress distribution under the maximum moment corresponding to this collapse mode may differ from that of the cross-sectional fully plastic yielding. Therefore, the maximum moment for such tubes cannot be predicted by the Kecman's method.
