**2. To understand the validity of existing estimation methods by using the results of numerical simulations**

### **2.1. Numerical simulation method**

**Figure 1(a)** shows the simulated rectangular tubes, in which one end of the rectangular tube was fixed to a rigid wall, and pure bending was applied from the other end by modeling a lid rotating about the *z* axis under rotary control *θ*. Bending moment *M* can be derived from the rigid wall as reaction moment. **Figure 1(b)** shows a deformation shape and bending angle *θ*. Until buckling occurs, axial strain *ε*<sup>x</sup> can be defined by

$$
\varepsilon\_x = \frac{\theta}{L} y \tag{1}
$$

**2.2. Kecman's method for predicting the maximum bending moment of** 

on the value of buckling stress σbuc-a of the compression flange

three cases are distinguished, as shown in **Figure 2**. In Eq. (2), *k*<sup>a</sup>

*<sup>σ</sup>buc*−*<sup>a</sup>* <sup>=</sup> *ka <sup>π</sup>*<sup>2</sup> *<sup>E</sup>* \_\_\_\_\_\_\_

shape; and (c) axial strain distribution on cross-section at *θ*/*L* = 0.01 m−1.

*ka* <sup>=</sup> 5.23 <sup>+</sup> 0.16 \_\_*<sup>a</sup>*

Kecman focused on slenderness corresponding to buckling stress of the compression flange and proposed a formula to predict the collapse load or the maximum moment *M*max. Depending

**Figure 1.** Numerical simulation model: (a) rectangular tube to which a pure bending moment is applied; (b) deformed

12(1 <sup>−</sup> *<sup>ν</sup>* 2) (

\_\_*t a*) 2

(2)

is the buckling coefficient,

*<sup>b</sup>* (3)

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226 373

**rectangular tubes**

which Kecman assumed to be

**Figure 1(c)** shows the axial strain distribution *ε*<sup>x</sup> on cross-section for a square tube with *t* = 0.4 mm, *a* = *b* = 50 mm at *θ*/*L* = 0.01 m−1. As shown in **Figure 1(c)**, the axial strain distribution *ε*x of FEM is in good agreement with the value obtained from Eq. (1). The effects of various geometric parameters were investigated under bending collapse. These parameters were tube thickness *t*, width of the flange *a*, and width of the web *b*. In order to prevent torsional behavior, a rigid lid was adopted as suggested by Guarracino [3]. In particular, the lid thickness *t f* was set to five times of *t*. In the simulation, a homogeneous and isotropic elastic perfectly plastic material was employed for the tube material. As a yield condition, von Mises yield conditions were adopted. In this chapter, the material mechanical properties are set as follows. Young's modulus *E* is set as 72.4 GPa, the yield stress σ<sup>s</sup> is set as 72.4 MPa, and Poisson's ratio ν is set as 0.3.

In this chapter, in order to formulate the geometric nonlinear behavior and solve the nonlinear equation, the updated Lagrange method, algorithm based on the Newton–Raphson method, and return-mapping method were used. The rectangular tubes were meshed using four-node quadrilateral thickness shell elements (Element type 75) with five integration points across the thickness. A convergence test on element size was conducted, and the adopted divide method was that the wall width divided into at least 20 sublengths, and the wall length divided as the elements become almost square.

In order to neglect the influence of the boundary conditions, the ratio of the length and width *L/a*, *L/b* was set to *L/a* > 6, *L/b* > 6. It means that the length of tubes was assumed to be large enough.

• To propose a new estimation method by considering the factor and using mathematical

• To understand the validity of the new estimation method by comparing with the results of

We selected "Collapse load for thin-walled rectangular tubes under bending" as the subject of these topics. The research content of this chapter is based on our recent paper [2], and this

**Figure 1(a)** shows the simulated rectangular tubes, in which one end of the rectangular tube was fixed to a rigid wall, and pure bending was applied from the other end by modeling a lid rotating about the *z* axis under rotary control *θ*. Bending moment *M* can be derived from the rigid wall as reaction moment. **Figure 1(b)** shows a deformation shape and bending angle *θ*.

can be defined by

*θ*

*t* = 0.4 mm, *a* = *b* = 50 mm at *θ*/*L* = 0.01 m−1. As shown in **Figure 1(c)**, the axial strain distribution

to five times of *t*. In the simulation, a homogeneous and isotropic elastic perfectly plastic material was employed for the tube material. As a yield condition, von Mises yield conditions were adopted. In this chapter, the material mechanical properties are set as follows. Young's modu-

In this chapter, in order to formulate the geometric nonlinear behavior and solve the nonlinear equation, the updated Lagrange method, algorithm based on the Newton–Raphson method, and return-mapping method were used. The rectangular tubes were meshed using four-node quadrilateral thickness shell elements (Element type 75) with five integration points across the thickness. A convergence test on element size was conducted, and the adopted divide method was that the wall width divided into at least 20 sublengths, and the wall length divided as the

In order to neglect the influence of the boundary conditions, the ratio of the length and width *L/a*, *L/b* was set to *L/a* > 6, *L/b* > 6. It means that the length of tubes was assumed to be large enough.

a rigid lid was adopted as suggested by Guarracino [3]. In particular, the lid thickness *t*

 of FEM is in good agreement with the value obtained from Eq. (1). The effects of various geometric parameters were investigated under bending collapse. These parameters were tube thickness *t*, width of the flange *a*, and width of the web *b*. In order to prevent torsional behavior,

*<sup>L</sup> y* (1)

is set as 72.4 MPa, and Poisson's ratio ν is set as 0.3.

on cross-section for a square tube with

*f* was set

**2. To understand the validity of existing estimation methods by** 

chapter shows the results of numerical simulations in detail.

**using the results of numerical simulations**

approach.

numerical simulations.

372 Numerical Simulations in Engineering and Science

**2.1. Numerical simulation method**

Until buckling occurs, axial strain *ε*<sup>x</sup>

lus *E* is set as 72.4 GPa, the yield stress σ<sup>s</sup>

elements become almost square.

*ε*x

*ε<sup>x</sup>* = \_\_

**Figure 1(c)** shows the axial strain distribution *ε*<sup>x</sup>

**Figure 1.** Numerical simulation model: (a) rectangular tube to which a pure bending moment is applied; (b) deformed shape; and (c) axial strain distribution on cross-section at *θ*/*L* = 0.01 m−1.

### **2.2. Kecman's method for predicting the maximum bending moment of rectangular tubes**

Kecman focused on slenderness corresponding to buckling stress of the compression flange and proposed a formula to predict the collapse load or the maximum moment *M*max. Depending on the value of buckling stress σbuc-a of the compression flange

$$
\sigma\_{bu\to u} = \frac{k\_s \pi^2 E}{12(1 - \nu^2)} \left(\frac{t}{a}\right)^2 \tag{2}
$$

three cases are distinguished, as shown in **Figure 2**. In Eq. (2), *k*<sup>a</sup> is the buckling coefficient, which Kecman assumed to be

$$k\_a = 5.23 \div 0.16 \frac{a}{b} \tag{3}$$

**Figure 2.** Schematic representation of axial stress distribution is used in the Kecman's method: (a) case 1: σbuc-a < σ<sup>s</sup> ; (b) case 2: σ<sup>s</sup> < σbuc-a < 2σ<sup>s</sup> ; and (c) case 3: σbuc-a > 2σ<sup>s</sup> .

The maximum moment *M*max for the rectangular tube is given by

116: назишиш полиени  $M\_{\text{max}}$  ио ше теслациан чисе з фучен ю 
$$M\_{\text{max}} = \sigma\_s tb^2 \frac{2a + b + a\left(3\frac{a}{b} + 2\right)}{3(a+b)}\tag{4}$$

For Case 1

$$M\_{\text{max}} = M\_{cl} + \left(M\_{pl} - M\_{cl}\right) \frac{\sigma\_{hc^{cu}} - \sigma\_{s}}{\sigma\_{s}} \tag{5}$$

For Case 2, and

$$M\_{\text{max}} = M\_{pl} \tag{6}$$

σbuc-a given by Eq. (2) are also shown as a comparison. As shown in **Figure 4(a)**, the maximum

**Figure 3.** Flow chart of the Kecman's method for predicting the maximum moment of tubes under pure bending.

web width keeps increasing after buckling occurs at point P in the middle of the compression

as shown in **Figure 4(b)**, although the axial compression stress in the middle of the compression flange decreases due to buckling at the flange, the axial compression stress increases at

distribution on cross-section at phases (1) and (2) corresponding to *θ*/*L* = 0.012 m−1 and *θ*/*L* = 0.038 m−1, respectively, as

at point P is in good

at point Q at the quarter-

Collapse Load for Thin-Walled Rectangular Tubes http://dx.doi.org/10.5772/intechopen.71226 375

at point Q occurs in the maximum moment. Moreover,

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

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

moment of FEM is in agreement with Eq. (4). The maximum value of σ<sup>x</sup>

agreement with Eq. (2). In addition, the axial compression stress σ<sup>x</sup>

flange, and the maximum value of σ<sup>x</sup>

**Figure 4.** 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>

For Case 3. In the above equations

$$a\_{\epsilon} = a\left(0.7\frac{\sigma\_{hwu}}{\sigma\_{s}} + 0.3\right) \tag{7}$$

and *M*el and *M*pl are the maximum elastic moment

$$M\_d = \sigma\_s tb \left( a + \frac{b}{3} \right) \tag{8}$$

and the cross-sectional fully plastic bending moment

$$M\_{pl} = \sigma\_s tb \left( a + \frac{b}{2} \right) \tag{9}$$

respectively.

**Figure 3** shows a flow chart of the Kecman's method for predicting the maximum moment of tubes under pure bending.

### **2.3. The applicability of the Kecman's method for square tubes**

**Figure 4** shows that the bending moment *M* and the axial stress σ<sup>x</sup> on cross-section for a square tube with *t* = 0.4 mm, *a* = *b* = 50 mm are subjected to pure bending (σbuc-a = 0.31σ<sup>s</sup> < σ<sup>s</sup> ). In order to better understand the bending collapse, Eq. (4) for Case 1 and the elastic buckling stress

**Figure 3.** Flow chart of the Kecman's method for predicting the maximum moment of tubes under pure bending.

The maximum moment *M*max for the rectangular tube is given by

.

<sup>2</sup>*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* <sup>+</sup> *ae*(<sup>3</sup> \_\_*<sup>a</sup>*

**Figure 2.** Schematic representation of axial stress distribution is used in the Kecman's method: (a) case 1: σbuc-a < σ<sup>s</sup>

*M*max = *Mpl* (6)

\_\_\_\_ *buc*−*<sup>a</sup> σs*

**Figure 3** shows a flow chart of the Kecman's method for predicting the maximum moment of

to better understand the bending collapse, Eq. (4) for Case 1 and the elastic buckling stress

tube with *t* = 0.4 mm, *a* = *b* = 50 mm are subjected to pure bending (σbuc-a = 0.31σ<sup>s</sup> < σ<sup>s</sup>

*<sup>b</sup>* <sup>+</sup> <sup>2</sup>) \_\_\_\_\_\_\_\_\_\_\_\_

> *<sup>σ</sup>buc*−*<sup>a</sup>* <sup>−</sup> *<sup>σ</sup>* \_\_\_\_\_\_\_*<sup>s</sup> σs*

3(*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>*) (4)

+ 0.3) (7)

<sup>3</sup>) (8)

<sup>2</sup>) (9)

on cross-section for a square

). In order

(5)

; (b)

*M*max = *σ<sup>s</sup> tb*<sup>2</sup>

374 Numerical Simulations in Engineering and Science

For Case 3. In the above equations

*M*max = *Mel* + (*Mpl* − *Mel*)

; and (c) case 3: σbuc-a > 2σ<sup>s</sup>

*ae* <sup>=</sup> *<sup>a</sup>*(0.7 *<sup>σ</sup>*

and *M*el and *M*pl are the maximum elastic moment

*Mel* <sup>=</sup> *<sup>σ</sup><sup>s</sup> tb*(*<sup>a</sup>* <sup>+</sup> \_\_*<sup>b</sup>*

and the cross-sectional fully plastic bending moment

*Mpl* <sup>=</sup> *<sup>σ</sup><sup>s</sup> tb*(*<sup>a</sup>* <sup>+</sup> \_\_*<sup>b</sup>*

**2.3. The applicability of the Kecman's method for square tubes**

**Figure 4** shows that the bending moment *M* and the axial stress σ<sup>x</sup>

For Case 1

For Case 2, and

case 2: σ<sup>s</sup> < σbuc-a < 2σ<sup>s</sup>

respectively.

tubes under pure bending.

σbuc-a given by Eq. (2) are also shown as a comparison. As shown in **Figure 4(a)**, the maximum moment of FEM is in agreement with Eq. (4). The maximum value of σ<sup>x</sup> at point P is in good agreement with Eq. (2). In addition, the axial compression stress σ<sup>x</sup> at point Q at the quarterweb width keeps increasing after buckling occurs at point P in the middle of the compression flange, and the maximum value of σ<sup>x</sup> at point Q occurs in the maximum moment. Moreover, as shown in **Figure 4(b)**, although the axial compression stress in the middle of the compression flange decreases due to buckling at the flange, the axial compression stress increases at

**Figure 4.** Moment *M* and axial stress σ<sup>x</sup> on cross-section for a square tube with *t* = 0.4 mm, *a* = *b* = 50 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* = 0.012 m−1 and *θ*/*L* = 0.038 m−1, respectively, as denoted in (a).

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.

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

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

the maximum moment can be evaluated by Eq. (5) for Case 2.

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

is not due to buckling at the compression flange, but rather plastic yielding at the flange, and

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

, the collapse

377

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

The above investigation confirms that for such tubes with b/a = 1 and σbuc-a < σ<sup>s</sup> , collapse is due to buckling at the compression flange, and the maximum moment can be predicted by the Kecman's method for Case 1.

**Figure 5** shows the bending moment *M* and the axial stress σ<sup>x</sup> on cross-section for a square tube with *t* = 0.9 mm, *a* = *b* = 50 mm (σbuc-a = 1.52σ<sup>s</sup> > σ<sup>s</sup> ). As shown in **Figure 5(a)**, the maximum moment is in good agreement with the value obtained from Eq. (5) for Case 2. The maximum value of σ<sup>x</sup> at point P and Q occurs in the maximum moment and σ<sup>x</sup> /σs at point P becomes 1. 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 positions. At phase (2), the stress at the flanges is equal to the yield stress σ<sup>s</sup> , and there also 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.

**Figure 5.** Moment *M* and axial stress σ<sup>x</sup> on cross-section for a square tube with *t* = 0.9 mm, *a* = *b* = 50 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* = 0.025 m−1 and *θ*/*L* = 0.065 m−1, respectively, as denoted in (a).

The above investigation confirms that for such tubes with b/a = 1 and σbuc-a > σ<sup>s</sup> , the collapse is not due to buckling at the compression flange, but rather plastic yielding at the flange, and the maximum moment can be evaluated by Eq. (5) for Case 2.
