**6. To understand the validity of the new estimation method by comparing with the results of numerical simulations**

In **Figure 16**, the maximum moment predicted by the present method is compared with that obtained from the FEM numerical simulation with *a*/*b* = 1, 2, and 3 for **Figures 16(a)**, **16(b)**, and **16(c)**, respectively. As a result of the prediction method proposed in this chapter, "method 1" uses Eq. (10) and "method 2" uses Eq. (19) for calculating the effective width.

The case number of the collapse corresponding to each thickness is also shown in the figures. In Case 2, there are two possible subcases: (1) *t*ea > *t* eb as shown in **Figure 16(a)** and **(b)** and (2) *t*ea < *t*eb as shown in **Figure 16(c)**; the maximum moment is determined by Eq. (27) for the former and by Eq. (29) for the latter. As shown in these figures, Eqs. (27) and (29) give good prediction to the corresponding subcase, respectively.

For Cases 4 and 5, although each result obtained from methods 1 and 2 is approximately in agreement with the FEM results of numerical simulations, it is found that there is a gap in the results between methods 1 and 2. When the buckling stress of the web σbuc-b is close to

for Case 5:

under pure bending.

\_\_\_\_\_

388 Numerical Simulations in Engineering and Science

In Eqs. (35) and (36).

*M*max

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

In Case 2, there are two possible subcases: (1) *t*ea > *t*

prediction to the corresponding subcase, respectively.

*<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>

**comparing with the results of numerical simulations**

<sup>2</sup> − *d*<sup>2</sup> 2 ] +

**6. To understand the validity of the new estimation method by** 

uses Eq. (10) and "method 2" uses Eq. (19) for calculating the effective width.

In **Figure 16**, the maximum moment predicted by the present method is compared with that obtained from the FEM numerical simulation with *a*/*b* = 1, 2, and 3 for **Figures 16(a)**, **16(b)**, and **16(c)**, respectively. As a result of the prediction method proposed in this chapter, "method 1"

The case number of the collapse corresponding to each thickness is also shown in the figures.

(2) *t*ea < *t*eb as shown in **Figure 16(c)**; the maximum moment is determined by Eq. (27) for the former and by Eq. (29) for the latter. As shown in these figures, Eqs. (27) and (29) give good

For Cases 4 and 5, although each result obtained from methods 1 and 2 is approximately in agreement with the FEM results of numerical simulations, it is found that there is a gap in the results between methods 1 and 2. When the buckling stress of the web σbuc-b is close to

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

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

<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

*<sup>ψ</sup>* <sup>−</sup> <sup>1</sup> *b*, *d*<sup>2</sup> = *b* − *be*<sup>1</sup> (37)

eb as shown in **Figure 16(a)** and **(b)** and

] + *ab* (36)

**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.

the yielding stress σ<sup>s</sup> , the method 1 gives a too large prediction as compared with the FEM 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 agreement with the FEM results for all of Cases 4 and 5.
