**5. To propose a new estimation method by considering the factor and using mathematical approach**

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

**Figure 10** shows the bending moment *M* and the axial stress σ<sup>x</sup>

before the maximum moment was reached and sets the value σ<sup>x</sup>

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

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,

1 until the moment becomes the maximum moment. However, the axial compression stress

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,

at point Q at the quarter-web width increases until the moment becomes the maximum

at point P in the middle of the compression flange increases up to the yielding stress

gular tube with *t* = 1.2 mm, *a* = 50 mm, *b* = 150 mm (σbuc-a = 2.8σ<sup>s</sup>

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

stress σ<sup>x</sup>

as denoted in (a).

**Figure 9.** Moment *M* and axial stress σ<sup>x</sup>

380 Numerical Simulations in Engineering and Science

**Figure 10.** Moment *M* and axial stress σ<sup>x</sup>

respectively, as denoted in (a).

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

σs

σx

on cross-section for a rectan-

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

). As shown in **Figure 10(a)**, the

equal approximately to

/σs

on cross-section for a rectangular tube with *t* = 0.5 mm, *a* = 20 mm, *b* = 100 mm are

on cross-section for a rectangular tube with *t* = 1.2 mm, *a* = 50 mm, *b* = 150 mm

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

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 compressive stress σ<sup>x</sup> along the width direction is characterized by two effective widths, *b*e1 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:

$$\begin{cases} \begin{aligned} \bullet \\ b\_{\epsilon1} &= \frac{b\_{\epsilon}}{3 - \psi} \end{aligned} \end{cases} \begin{aligned} \text{with} \\ \begin{aligned} \begin{aligned} b\_{\epsilon1} &= \frac{b\_{\epsilon}}{3 - \psi} \\ b\_{\epsilon2} &= \begin{cases} b\_{\epsilon}/2 & \text{when } \psi \le -0.236 \\ b\_{\epsilon} - b\_{\epsilon1} & \text{when } \psi > -0.236 \end{cases} \end{aligned} \end{cases} \tag{10}$$

In addition, *b*e1 + *b*e2 shall not exceed the compression portion of the web. Here, ψ is ratio of *<sup>f</sup>* 1 ∗ and*f* 2 ∗ . *f* 1 ∗ and*f* 2 ∗ are web stresses shown in **Figure 11(b)**.

$$
\psi = \frac{f\_1^\*}{f\_1^\*} \tag{11}
$$

**Figure 11.** Plate subjected to compression and bending: (a) analyzed model and (b) axial compressive stress σ<sup>x</sup> distribution on E–E cross-section in (a).

λ is defined by

 *<sup>λ</sup>* <sup>=</sup> <sup>√</sup> \_\_\_\_\_ *<sup>σ</sup>*\_\_\_\_*<sup>s</sup> σbuc*−*<sup>b</sup>* (12) of a buckling plate is reduced greatly by imperfections when the buckling stress is close to the yield stress [8]. Therefore, Eq. (16) is desirable for the present model because the influence of imperfections is not taken into consideration here. Moreover, in order to consider continuity of the load capability of a web with λ = 1, for which elastic buckling does not occur because

Eq. (10) is applied to the webs investigated in **Figures 7(b)** and **9(b)**to determine the corresponding effective width; the stress distributions on the web based on the obtained effective width using Eq. (10) are shown in **Figures 12(a)** and **13(a)**. In **Figure 12(a)**, the stress distribution obtained using Eq. (10) is qualitatively corresponding with the redistribution of the compression stress after buckling obtained from the FEM numerical simulation. However, in **Figure 13(a)**, even though there is a fall of the compression stress in the compression portion of the web after buckling as shown by the FEM simulation, the stress distribution obtained from Eq. (10) looks like a straight line because the effective widths *b*e1 and *b*e2 determined by

In fact, when the effective width is determined using Eq. (10), there are many instances in which Eq. (18) is satisfied. **Figure 14** shows various possible values of buckling stress of web, for which Eq. (18) is satisfied, for various assumed stress ratios ψ by solid line, as evaluated in Eq. (10) with ρ defined in Eq. (16). In **Figure 14**, the dashed line shows the corresponding result if *ρ* was calculated using Eq. (17); it is also seen from the dashed line that even if Eq. (17) is used for *ρ* the instances in which Eq. (18) is satisfied still exist. For these instances, the redistribution of the compression stress after buckling cannot be expressed by the effective width obtained from Eq. (10); this means that there is a possibility of giving a too large load

**Figure 12.** Stress distribution of web when the ultimate load is reached for the tube used in **Figure 7(b)**: (a) comparison

<sup>1</sup> <sup>−</sup> *<sup>ψ</sup>* (18)

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226 383

, we apply Eq. (16) to the present study.

Eq. (10) satisfy the following equation:

with Eq. (10) and (b) comparison with Eq. (19).

*be*<sup>1</sup> <sup>+</sup> *be*<sup>2</sup> <sup>=</sup> \_\_\_\_ *<sup>b</sup>*

which means *b*e1 + *b*e2 is equal to the compression portion of the web.

σbuc-b = σ<sup>s</sup>

The elastic buckling stress of web σbuc-b is calculated as follows:

$$
\sigma\_{\text{buc}\to b} = \frac{k\_b \pi^2 E}{12(1 - \nu^2)} \left(\frac{t}{b}\right)^2 \tag{13}
$$

where the buckling coefficient *k*<sup>b</sup> is given by

$$k\_{\flat} = 4 + 2\left(1 - \psi\right)^{3} + 2(1 - \psi)\tag{14}$$

*b*e is given by

*be* = *b* (15)

ρ is called the reduction factor and is given by

$$
\rho = \frac{1}{\lambda} \tag{16}
$$

which is proposed by von Karman et al. [6]. The following formula for ρ:

$$
\rho = \frac{1}{\lambda} \left( 1 - \frac{0.22}{\lambda} \right) \tag{17}
$$

is also proposed by Winter [7] and is well used for design specifications. The reason of Eq. (15) modified to Eq. (17) in actual design is mainly due to the fact that the maximum load capacity of a buckling plate is reduced greatly by imperfections when the buckling stress is close to the yield stress [8]. Therefore, Eq. (16) is desirable for the present model because the influence of imperfections is not taken into consideration here. Moreover, in order to consider continuity of the load capability of a web with λ = 1, for which elastic buckling does not occur because σbuc-b = σ<sup>s</sup> , we apply Eq. (16) to the present study.

Eq. (10) is applied to the webs investigated in **Figures 7(b)** and **9(b)**to determine the corresponding effective width; the stress distributions on the web based on the obtained effective width using Eq. (10) are shown in **Figures 12(a)** and **13(a)**. In **Figure 12(a)**, the stress distribution obtained using Eq. (10) is qualitatively corresponding with the redistribution of the compression stress after buckling obtained from the FEM numerical simulation. However, in **Figure 13(a)**, even though there is a fall of the compression stress in the compression portion of the web after buckling as shown by the FEM simulation, the stress distribution obtained from Eq. (10) looks like a straight line because the effective widths *b*e1 and *b*e2 determined by Eq. (10) satisfy the following equation:

$$b\_{e1} + b\_{e2} = \frac{b}{1 - \psi} \tag{18}$$

which means *b*e1 + *b*e2 is equal to the compression portion of the web.

λ is defined by

*b*e

is given by

*<sup>λ</sup>* <sup>=</sup> <sup>√</sup>

*<sup>σ</sup>buc*−*<sup>b</sup>* <sup>=</sup> *kb <sup>π</sup>*<sup>2</sup> *<sup>E</sup>* \_\_\_\_\_\_\_

ρ is called the reduction factor and is given by

*ρ* = \_\_1

*ρ* = \_\_1

where the buckling coefficient *k*<sup>b</sup>

distribution on E–E cross-section in (a).

382 Numerical Simulations in Engineering and Science

The elastic buckling stress of web σbuc-b is calculated as follows:

is given by

which is proposed by von Karman et al. [6]. The following formula for ρ:

\_\_\_\_\_ *<sup>σ</sup>*\_\_\_\_*<sup>s</sup> σbuc*−*<sup>b</sup>*

**Figure 11.** Plate subjected to compression and bending: (a) analyzed model and (b) axial compressive stress σ<sup>x</sup>

12(1 <sup>−</sup> *<sup>ν</sup>* 2) (

*kb* = 4 + 2 (1 − *ψ*)3 + 2(1 − *ψ*) (14)

*be* = *b* (15)

*<sup>λ</sup>*(<sup>1</sup> <sup>−</sup> \_\_\_\_ 0.22

is also proposed by Winter [7] and is well used for design specifications. The reason of Eq. (15) modified to Eq. (17) in actual design is mainly due to the fact that the maximum load capacity

\_\_*t b*) 2

*<sup>λ</sup>* (16)

*<sup>λ</sup>* ) (17)

(12)

(13)

In fact, when the effective width is determined using Eq. (10), there are many instances in which Eq. (18) is satisfied. **Figure 14** shows various possible values of buckling stress of web, for which Eq. (18) is satisfied, for various assumed stress ratios ψ by solid line, as evaluated in Eq. (10) with ρ defined in Eq. (16). In **Figure 14**, the dashed line shows the corresponding result if *ρ* was calculated using Eq. (17); it is also seen from the dashed line that even if Eq. (17) is used for *ρ* the instances in which Eq. (18) is satisfied still exist. For these instances, the redistribution of the compression stress after buckling cannot be expressed by the effective width obtained from Eq. (10); this means that there is a possibility of giving a too large load

**Figure 12.** Stress distribution of web when the ultimate load is reached for the tube used in **Figure 7(b)**: (a) comparison with Eq. (10) and (b) comparison with Eq. (19).

capability of web from Eq. (10). Therefore, here as a comparison, we also use another solution given by Rusch and Lindner [9] which is given for the same plate shown in **Figure 11(a)** but with one of the two longitudinal edges BC being free. Although the free boundary condition at the longitudinal edge BC is different from the actual situation of web constituting the tube, the effect is assumed to be small because BC is under tension stress.

In Ref. [9], the effective widths *b*e1 and *b*e2 are given by

*be*<sup>1</sup> = *be* − *be*<sup>2</sup>

Here, λ and *ρ* are calculated by Eqs. (12) and (16), respectively, the buckling stress σbuc-b is

*kb* = 1.7 − 5*ψ* + 17.1 *ψ*<sup>2</sup> (21)

**Figures 12(b)** and **13(b)** show the comparisons of stress distributions on the web obtained from FEM and Eq. (19) for the tubes used in **Figures 7(b)** and **9(b)**, from which it is seen that the redistribution of stress after web buckling can be approximately expressed using Eq. (19). Comparing (a) and (b) in **Figure 12**, it is seen that for the stress distribution on the web in the tube used in **Figure 7(b)**, Eq. (19) is inferior in accuracy to Eq. (10). However, as shown in **Figure 13**, which shows the stress distributions on the web for the tube used in **Figure 9(b)**, although the fall of the compression stress in the compression portion of the web after buckling is not expressed by the solution obtained from Eq. (10), it is expressed by the solution from Eq. (19). In fact, it is seen from Eq. (20) that for the stress distribution on the web as obtained from Eq. (18), the length of *b*e1 + *b*e2 is always smaller than the compression portion of the web.

For tubes with large aspect ratio of web to flange, as an effect of web slenderness on the tube collapse, we considered the possible buckling of web and thus investigated the existence of Cases 4 and 5, as shown above. Hereafter, we consider the other effect of web slenderness on

As shown in **Figure 6**, for tubes with *b*/*a* = 3, there is a large discrepancy between Kecman's prediction and the FEM simulation. When the tubes are very thin (e.g., when *t*/*a* < 0.02 for *b*/*a* = 3) it is thought that the error generating is brought about because the web buckling was not taken into consideration in the Kecman's method. However, for the relatively thick tubes, the cause which produces the error is clearly different because buckling does not occur in such tubes. For example, for the tube with *b*/*a* = 3 and *t*/*a* = 0.024 shown in **Figure 10(b)**, even though the buckling stress of flange σbuc-a calculated by Eq. (2) is σbuc-a/σ<sup>s</sup> = 2.8 > 2, the maximum moment *M*max as evaluated by FEM numerical simulation is *M*max /*Mpl* <sup>≅</sup> 0.9, which is not in agreement with Eq. (6) for the case of σbuc-a/σ<sup>s</sup> = 2 in the Kecman's method. Here, buckling does not occur in the web either because σbuc-b/σ<sup>s</sup> = 1.4 > 1. Also, it is seen from **Figure 10(b)** that the stress distribution on the cross-section is different from that shown in **Figure 2(c)** for Case 3 corresponding to the cross-sectional fully plastic yielding. This fact means that the condition for reaching the cross-sectional fully plastic yielding is also related to the web slenderness.

(19)

385

<sup>1</sup> <sup>−</sup> *<sup>ψ</sup>* (20)

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

*b* \_\_\_*e*2 *<sup>b</sup>* <sup>=</sup> \_\_\_\_\_ 0.226 *λ*2

\_\_*e <sup>b</sup>* <sup>=</sup> *<sup>ρ</sup>* \_\_\_\_

determined as follows:

**5.2. Effect of the web slenderness on the cross-sectional fully plastic yielding**

{

*<sup>b</sup>*

the cross-sectional fully plastic yielding.

determined by Eq. (13) with *k*<sup>b</sup>

where

**Figure 13.** Stress distribution of web when the ultimate load is reached for the tube used in **Figure 9(b)**: (a) comparison with Eq. (10) and (b) comparison with Eq. (19).

**Figure 14.** Various possible buckling stress σbuc-b and stress ratio ψ with Eq. (18) satisfied.

In Ref. [9], the effective widths *b*e1 and *b*e2 are given by

$$\begin{cases} b\_{\epsilon1} = b\_{\epsilon} - b\_{\epsilon2} \\ \frac{b\_{\epsilon2}}{b} = \frac{0.226}{\lambda^2} \end{cases} \tag{19}$$

where

capability of web from Eq. (10). Therefore, here as a comparison, we also use another solution given by Rusch and Lindner [9] which is given for the same plate shown in **Figure 11(a)** but with one of the two longitudinal edges BC being free. Although the free boundary condition at the longitudinal edge BC is different from the actual situation of web constituting the tube,

**Figure 13.** Stress distribution of web when the ultimate load is reached for the tube used in **Figure 9(b)**: (a) comparison

the effect is assumed to be small because BC is under tension stress.

with Eq. (10) and (b) comparison with Eq. (19).

384 Numerical Simulations in Engineering and Science

**Figure 14.** Various possible buckling stress σbuc-b and stress ratio ψ with Eq. (18) satisfied.

$$\frac{b\_{\epsilon}}{b} = \frac{\rho}{1 - \psi} \tag{20}$$

Here, λ and *ρ* are calculated by Eqs. (12) and (16), respectively, the buckling stress σbuc-b is determined by Eq. (13) with *k*<sup>b</sup> determined as follows:

$$k\_b = 1.7 - 5\psi + 17.1\,\psi^2\tag{21}$$

**Figures 12(b)** and **13(b)** show the comparisons of stress distributions on the web obtained from FEM and Eq. (19) for the tubes used in **Figures 7(b)** and **9(b)**, from which it is seen that the redistribution of stress after web buckling can be approximately expressed using Eq. (19). Comparing (a) and (b) in **Figure 12**, it is seen that for the stress distribution on the web in the tube used in **Figure 7(b)**, Eq. (19) is inferior in accuracy to Eq. (10). However, as shown in **Figure 13**, which shows the stress distributions on the web for the tube used in **Figure 9(b)**, although the fall of the compression stress in the compression portion of the web after buckling is not expressed by the solution obtained from Eq. (10), it is expressed by the solution from Eq. (19). In fact, it is seen from Eq. (20) that for the stress distribution on the web as obtained from Eq. (18), the length of *b*e1 + *b*e2 is always smaller than the compression portion of the web.

### **5.2. Effect of the web slenderness on the cross-sectional fully plastic yielding**

For tubes with large aspect ratio of web to flange, as an effect of web slenderness on the tube collapse, we considered the possible buckling of web and thus investigated the existence of Cases 4 and 5, as shown above. Hereafter, we consider the other effect of web slenderness on the cross-sectional fully plastic yielding.

As shown in **Figure 6**, for tubes with *b*/*a* = 3, there is a large discrepancy between Kecman's prediction and the FEM simulation. When the tubes are very thin (e.g., when *t*/*a* < 0.02 for *b*/*a* = 3) it is thought that the error generating is brought about because the web buckling was not taken into consideration in the Kecman's method. However, for the relatively thick tubes, the cause which produces the error is clearly different because buckling does not occur in such tubes. For example, for the tube with *b*/*a* = 3 and *t*/*a* = 0.024 shown in **Figure 10(b)**, even though the buckling stress of flange σbuc-a calculated by Eq. (2) is σbuc-a/σ<sup>s</sup> = 2.8 > 2, the maximum moment *M*max as evaluated by FEM numerical simulation is *M*max /*Mpl* <sup>≅</sup> 0.9, which is not in agreement with Eq. (6) for the case of σbuc-a/σ<sup>s</sup> = 2 in the Kecman's method. Here, buckling does not occur in the web either because σbuc-b/σ<sup>s</sup> = 1.4 > 1. Also, it is seen from **Figure 10(b)** that the stress distribution on the cross-section is different from that shown in **Figure 2(c)** for Case 3 corresponding to the cross-sectional fully plastic yielding. This fact means that the condition for reaching the cross-sectional fully plastic yielding is also related to the web slenderness.

In order to consider the effect of the web slenderness on the tube collapse, the condition of σbuc-a > 2σ<sup>s</sup> for Case 3 or for *M*max = *M*pl in the Kecman's method is replaced in the present study by the following condition:

$$\begin{cases} \sigma\_{bu \to u} \ge \mathcal{2}\sigma\_s\\ \sigma\_{bu \to b} \ge \mathcal{2}\sigma\_s \end{cases} \tag{22}$$

*<sup>b</sup>*

one from both values of *bs*

satisfied: σbuc-a = 1.23σ<sup>s</sup>

Case 2, the value of *bs*

ea and *t*

and is given by

*t*

*t*

*t*

*t* ≥ √

*ψ* = −

maximum moment for Cases 4 and 5 is calculated as follows:

*<sup>σ</sup><sup>s</sup> <sup>t</sup>* <sup>=</sup> *<sup>ψ</sup>*[*b*<sup>2</sup> <sup>+</sup> *<sup>d</sup>*<sup>1</sup>

<sup>2</sup> − *d*<sup>2</sup> 2 ] +

*M*max

and is calculated using Eq. (29) if

stress σ<sup>s</sup>

Using *t*

ψ to be

for Case 4:

\_\_\_\_\_

\_\_*s <sup>b</sup>* <sup>=</sup> ⎧ ⎪ ⎨ ⎪ ⎩

and σbuc-b = 1.39σ<sup>s</sup>

is calculated using Eq. (27) if

eb, the condition of Eq. (22) can be rewritten as.

\_\_ 2 *t*

**5.3. Estimation of collapse load for thin-walled rectangular tubes under bending**

<sup>1</sup> <sup>−</sup> <sup>√</sup>

\_\_\_\_\_\_\_\_ 2 − ( \_\_*t t eb*) 2

*eb* = *b* √

1 (*for t* ≥ √

where *t*eb is the web thickness for which the elastic buckling stress σbuc-b is equal to the yielding

Furthermore, we assume that this technique can also be used to evaluate the maximum moment in the case when Eq. (25) is satisfied. That is, *M*max is determined from the smaller

understood from **Figure 16(b)** shown later, in which for the tube with *t*/*a* = 0.016, Eq. (25) is

*ea* > *t*

*ea* < *t*

*ea* and*t* ≥ √

**Figure 15** shows a flow chart of a new method proposed in the present study for predicting the maximum moment of tubes under pure bending. This method includes both the possible buckling at web and the effect of web slenderness on the cross-sectional fully plastic yielding. In the flow chart, σbuc-b,1 and σbuc-b,2 are the buckling stress of web assuming the stress ratio

> *<sup>b</sup>* <sup>−</sup> *<sup>y</sup>* \_\_\_\_1 *y*1

and ψ = −1, respectively. Moreover, it is notable that in calculating the maximum bending moment for Cases 4 and 5 the stress ratio ψ is also unknown, which shall be determined from the conditions of pure bending through trial and error. Using the determined value of ψ, the

= − *ae* <sup>+</sup> *<sup>b</sup>* \_\_\_\_

2(1 <sup>−</sup> *<sup>ψ</sup>*) \_\_\_\_\_\_

<sup>3</sup>*<sup>b</sup>* [*b*<sup>3</sup> <sup>+</sup> *<sup>d</sup>*<sup>1</sup>

<sup>3</sup> − *d*<sup>2</sup> 3

\_\_ 2 *t*

\_\_\_\_\_\_\_ 12(1 − *ν* 2) \_\_\_\_\_\_\_ *ka <sup>π</sup>*<sup>2</sup> √

(*for t*

*eb* < *t* < √

\_\_ 2 *t eb*)

given in Eq. (27) and in Eq. (29). Validity of this assumption can be

. Therefore, when using Eq. (26) to determine *M*max for

*eb* (31)

*eb* (32)

*eb* (33)

*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* (34)

] + *ae b* (35)

\_\_ *σ* \_\_*<sup>s</sup>*

\_\_ 2 *t eb*) 

*<sup>E</sup>* (30)

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

(29)

387

Here, σbuc-b is determined assumed ψ = −1. When Eq. (22) is not satisfied, that is, when

Here,  $\sigma\_{\text{buv}\bullet}$  is aeermineda assumed  $\psi = -1$ . winen Eq. (22) is \*\*n\*\*sätsneed, \*\*nar is, wenn\*\*

$$\begin{cases} \sigma\_s < \sigma\_{\text{buv}\bullet} < 2\,\sigma\_s\\ \sigma\_{\text{buv}\bullet} \ge 2\,\sigma\_s \end{cases} \tag{23}$$

or

$$\begin{cases} \sigma\_{hw \to u} \ge 2\,\sigma\_s\\ \sigma\_s \le \sigma\_{hw \to b} < 2\,\sigma\_s \end{cases} \tag{24}$$

or

or

$$\begin{cases}
\sigma\_s < \sigma\_{h\omega \to l} < 2\,\sigma\_s \\
\sigma\_s < \sigma\_{h\omega \to l} < 2\,\sigma\_s
\end{cases}
\tag{25}$$

the stress on cross-section is expressed by Case 2 shown in **Figure 2(b)**. This fact can be confirmed from **Figure 10(b)** for which Eq. (24) is satisfied.

It is seen from the cross-sectional stress distribution shown in **Figure 10(b)** that the maximum moment in this case is dependent on the plastic yielding region in the web. Denoting the length of this plastic yielding region by *bs* (see **Figure 2(b)**), the maximum moment can be evaluated through the value of *bs* as follows:

for Case 2,

$$M\_{\text{max}} = \sigma\_s t \left[ \frac{1}{6} (2 \, b^2 + 2 \, b b\_s - b\_s^2) + ab \right] \tag{26}$$

Substituting Eqs. (8), (9), and (26) into Eq. (5), *bs* is obtained as

$$\text{Substituting Eqs. (8), (9), and (26) into Eq. (5), } b\_s \text{ is obtained as}$$

$$\frac{b\_s}{b} = \begin{cases} 1 - \sqrt{2 - \left(\frac{t}{t\_\alpha}\right)^2} & \text{(for } t\_\alpha < t < \sqrt{2} \, t\_\alpha\text{)}\\ 1 & \text{(for } t \ge \sqrt{2} \, t\_\alpha\text{)} \end{cases} \tag{27}$$

where *t*ea is the flange thickness for which the elastic buckling stress σbuc-a obtained from Eq. (2) is equal to the yielding stress σ<sup>s</sup> and is given by

$$t\_{ea} = a \sqrt{\frac{12\{1 - \nu^2\}}{k\_s \pi^2}} \sqrt{\frac{\sigma\_s}{E}} \tag{28}$$

Eq. (26) means that the *bs* is determined by the flange slenderness only when Eq. (23) is satisfied. Therefore, when Eq. (24) is satisfied, we also suppose that the *bs* can be determined by the web slenderness only as follows:

$$\begin{aligned} \frac{b\_s}{b} &= \begin{cases} 1 - \sqrt{2 - \left(\frac{t}{t\_\alpha}\right)^2} & \text{(for } t\_\alpha < t < \sqrt{2} \, t\_\alpha\text{)}\\ 1 & \text{(for } t \ge \sqrt{2} \, t\_\alpha\text{)} \end{cases} \end{aligned} \tag{29}$$

where *t*eb is the web thickness for which the elastic buckling stress σbuc-b is equal to the yielding stress σ<sup>s</sup> and is given by

$$t\_{ab} = b \sqrt{\frac{12(1 - \nu^2)}{k\_\circ \pi^2}} \sqrt{\frac{\sigma\_\circ}{E}} \tag{30}$$

Furthermore, we assume that this technique can also be used to evaluate the maximum moment in the case when Eq. (25) is satisfied. That is, *M*max is determined from the smaller one from both values of *bs* given in Eq. (27) and in Eq. (29). Validity of this assumption can be understood from **Figure 16(b)** shown later, in which for the tube with *t*/*a* = 0.016, Eq. (25) is satisfied: σbuc-a = 1.23σ<sup>s</sup> and σbuc-b = 1.39σ<sup>s</sup> . Therefore, when using Eq. (26) to determine *M*max for Case 2, the value of *bs* is calculated using Eq. (27) if

$$t\_{\alpha} \ge t\_{\phi} \tag{31}$$

and is calculated using Eq. (29) if

In order to consider the effect of the web slenderness on the tube collapse, the condition of

*σbuc*−*<sup>a</sup>* ≥ 2 *σ<sup>s</sup> σbuc*−*<sup>b</sup>* ≥ 2 *σ<sup>s</sup>*

*σ<sup>s</sup>* < *σbuc*−*<sup>a</sup>* < 2 *σ<sup>s</sup> <sup>σ</sup>buc*−*<sup>b</sup>* <sup>≥</sup> <sup>2</sup> *<sup>σ</sup><sup>s</sup>*

*σ<sup>s</sup>* < *σbuc*−*<sup>a</sup>* < 2 *σ<sup>s</sup> <sup>σ</sup><sup>s</sup>* <sup>&</sup>lt; *<sup>σ</sup>buc*−*<sup>b</sup>* <sup>&</sup>lt; <sup>2</sup> *<sup>σ</sup><sup>s</sup>*

the stress on cross-section is expressed by Case 2 shown in **Figure 2(b)**. This fact can be con-

It is seen from the cross-sectional stress distribution shown in **Figure 10(b)** that the maximum moment in this case is dependent on the plastic yielding region in the web. Denoting the

<sup>6</sup>(2 *b*<sup>2</sup> + 2 *bbs* − *bs*

(*for t*

2

*ea* < *t* < √

\_\_ 2 *t ea*)

\_\_ *σ* \_\_*<sup>s</sup>*

is determined by the flange slenderness only when Eq. (23) is satis-

\_\_ 2 *t ea*) 

is obtained as

as follows:

\_\_1

\_\_\_\_\_\_\_\_ 2 − ( \_\_*t t ea*) 2

and is given by

*ea* = *a* √

1 (*for t* ≥ √

where *t*ea is the flange thickness for which the elastic buckling stress σbuc-a obtained from Eq. (2)

\_\_\_\_\_\_\_ 12(1 − *ν* 2) \_\_\_\_\_\_\_ *ka <sup>π</sup>*<sup>2</sup> √

(see **Figure 2(b)**), the maximum moment can be

) + *ab*] (26)

*<sup>E</sup>* (28)

can be determined by

*σbuc*−*<sup>a</sup>* ≥ 2 *σ<sup>s</sup> σ<sup>s</sup>* < *σbuc*−*<sup>b</sup>* < 2 *σ<sup>s</sup>*

Here, σbuc-b is determined assumed ψ = −1. When Eq. (22) is not satisfied, that is, when

for Case 3 or for *M*max = *M*pl in the Kecman's method is replaced in the present study

(22)

(23)

(24)

(25)

(27)

σbuc-a > 2σ<sup>s</sup>

or

or

for Case 2,

by the following condition:

386 Numerical Simulations in Engineering and Science

{

{

{

{

length of this plastic yielding region by *bs*

*M*max = *σ<sup>s</sup> t*[

Substituting Eqs. (8), (9), and (26) into Eq. (5), *bs*

\_\_*s <sup>b</sup>* <sup>=</sup> ⎧ ⎪ ⎨ ⎪ ⎩

<sup>1</sup> <sup>−</sup> <sup>√</sup>

fied. Therefore, when Eq. (24) is satisfied, we also suppose that the *bs*

evaluated through the value of *bs*

*<sup>b</sup>*

Eq. (26) means that the *bs*

is equal to the yielding stress σ<sup>s</sup>

*t*

the web slenderness only as follows:

firmed from **Figure 10(b)** for which Eq. (24) is satisfied.

$$t\_{\alpha} \le t\_{\phi} \tag{32}$$

Using *t* ea and *t* eb, the condition of Eq. (22) can be rewritten as.

$$t \ge \sqrt{2} \, t\_a \, \text{and} \, t \ge \sqrt{2} \, t\_{ab} \tag{33}$$

### **5.3. Estimation of collapse load for thin-walled rectangular tubes under bending**

**Figure 15** shows a flow chart of a new method proposed in the present study for predicting the maximum moment of tubes under pure bending. This method includes both the possible buckling at web and the effect of web slenderness on the cross-sectional fully plastic yielding. In the flow chart, σbuc-b,1 and σbuc-b,2 are the buckling stress of web assuming the stress ratio ψ to be

$$
\psi = -\frac{b - y\_i}{y\_1} = -\frac{a\_\circ + b}{a + b} \tag{34}
$$

and ψ = −1, respectively. Moreover, it is notable that in calculating the maximum bending moment for Cases 4 and 5 the stress ratio ψ is also unknown, which shall be determined from the conditions of pure bending through trial and error. Using the determined value of ψ, the maximum moment for Cases 4 and 5 is calculated as follows:

for Case 4:

$$\frac{M\_{\text{max}}}{\sigma\_\ast t} = \psi \left[ b^2 + d\_1^2 - d\_2^2 \right] + \frac{2(1-\psi)}{3b} \left[ b^3 + d\_1^3 - d\_2^3 \right] + a\_\epsilon b \tag{35}$$

**Figure 15.** Flow chart of a new method proposed in the present study for predicting the maximum moment of tubes under pure bending.

for Case 5:

$$\frac{M\_{\text{max}}}{\sigma\_{\ast}^{\prime}t} = \psi \left[ b^2 + d\_1^2 - d\_2^2 \right] + \frac{2(1-\psi)}{3b} \left[ b^3 + d\_1^3 - d\_2^3 \right] + ab \tag{36}$$

In Eqs. (35) and (36).

$$d\_1 = b\_{c2} + \frac{\psi}{\psi - 1} b\_\prime \, d\_2 = \, b - b\_{c1} \tag{37}$$

the yielding stress σ<sup>s</sup>

ment with the FEM results for all of Cases 4 and 5.

, the method 1 gives a too large prediction as compared with the FEM

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226 389

results, reflecting the fact that *b*e1 + *b*e2 given by Eq. (10) may be equal to the compression portion of the web as shown in **Figure 14**. However, for small *t*/*a*, when σbuc-b is very much less than the yielding stress, method 1 is more accurate compared with method 2. Combining the advantages of these two methods, it is seen from **Figure 16(a)**, **(b)**, and **(c)** that the smaller one from both solutions obtained from method 1 and obtained from method 2 is in good agree-

**Figure 16.** Prediction of the maximum bending moment *M*max for rectangular tubes: (a) *a*/*b* = 1; (b) *a*/*b* = 2; and (c) *a*/*b* = 3.
