Class 4 Sections Computing Section Properties and Stresses Due to the Bimoment

1. Introduction

Elastic Theory

The classical approach to determining section properties is presented including sectorial coordinates using centroidal coordinates y_c and z_c see Figure 1 (a) and (b).

$$I_y=\int z_c^{2}dA={\displaystyle \frac{1}{3}}\sum \left(z_{c,i}^2+z_{c,i}{z}_{c,i-1}+z_{c,i-1}^2\right)A_i$$

(1)

$$I_z=\int y_c^{2}dA={\displaystyle \frac{1}{3}}\sum \left(z_{ci}^2+z_{c,i}{z}_{c,i-1}+z_{c,i-1}^2\right)A_i$$

(2)

$$I_{yz}=\int y_c{z}_{c}dA={\displaystyle \frac{1}{6}}\sum \left(2y_{c,i-1}{z}_{c,i-1}+y_{c,i-1}{z}_{c,i}+y_{c,i}{z}_{c,i-1}+2y_{c,i}{z}_{c,i}\right)A_i$$

(3)

$$I_{y\omega }=\int y_c{\omega }_{c}dA={\displaystyle \frac{1}{6}}\sum \left(2y_{c,i-1}{\omega }_{c,i-1}+y_{c,i-1}{\omega }_{c,i}+y_{c,i}{\omega }_{c,i-1}+2y_{c,i}{\omega }_{c,i}\right)A_i$$

(4)

$$I_{z\omega }=\int z_c{\omega }_{c}dA={\displaystyle \frac{1}{6}}\sum \left(2z_{c,i-1}{\omega }_{c,i-1}+z_{c,i-1}{\omega }_{c,i}+z_{c,i}{\omega }_{c,i-1}+2z_{ci}{\omega }_{c,i}\right)A_i$$

(5)

$${\omega }_{c,i}={\omega }_{c,i-1}+y_{c,i-1}\left(z_{c,i}-z_{c,i-1}\right)-z_{c,i-1}\left(y_{c,i}-y_{c,i-1}\right)$$

(6)

$$={\omega }_{c,i-1}+y_{c,i-1}{z}_{c,i}-y_{c,i}{z}_{c,i-1}\mathrm{where}{\omega }_{c,0}=0$$

(7)

(${\omega }_{c,0}$ can be chosen arbitrarily, typically taken as 0)

$$y_{sc}={\displaystyle \frac{I_z{I}_{z\omega }-I_{yz}{I}_{y\omega }}{I_y{I}_z-I_{yz}^2}}$$

(8)

$$z_{sc}={\displaystyle \frac{I_y{I}_{y\omega }-I_{yz}{I}_{z\omega }}{I_y{I}_z-I_{yz}^2}}$$

(9)

$${\omega }_{s,i}={\omega }_{s,i-1}+\left(y_{c,i-1}-y_{sc}\right)\left(z_{c,i}-z_{c,i-1}\right)-\left(z_{c,i-1}-z_{sc}\right)\left(y_{c,i}-y_{c,i-1}\right)\mathrm{where}{\omega }_{s,0}=0$$

(10)

(${\omega }_{s,0}$ can be chosen arbitrarily, typically taken as 0)

$${\omega }_{n,i}={\omega }_{s,i}-{\omega }_{s,mean}$$

(21)

$${\omega }_{s,mean}={\displaystyle \frac{1}{A}}\int {\omega }_{s}dA={\displaystyle \frac{1}{A}}\sum {\displaystyle \frac{{\omega }_{s,i-1}+{\omega }_{s,i}}{2}}{A}_i$$

(12)

$$I_w=\int {\omega }_n^{2}dA={\displaystyle \frac{1}{3}}\sum \left({\omega }_{n,i}^2+{\omega }_{n,i}{\omega }_{n,i-1}+{\omega }_{n,i-1}^2\right)A_i$$

(13)

$$B=\int \sigma {\omega }_{n}dA$$

(14)

$${\sigma }_w=B{\omega }_n/I_w$$

(15)

$$B_{el}=f_y{I}_w/{\omega }_{n,max}$$

(16)

It is convenient computationally to use any arbitrary starting coordinate system (y, z, ω) and the parallel axis theorem to obtain the centroidal properties. For sectorial coordinates with respect to any pole,

$$\begin{gathered} {\omega }_i={\omega }_{i-1}+y_{i-1}\left(z_i-z_{i-1}\right)-z_{i-1}\left(y_i-y_{i-1}\right) \\ ={\omega }_{i-1}+y_{i-1}{z}_i-y_i{z}_{i-1}\mathrm{where}{\omega }_0=0 \end{gathered}$$

(17)

The normalized sectorial coordinates about the shear center can then be determined using

$${\omega }_{n,i}={\omega }_i-{\omega }_{mean}+z_{sc}{y}_{c,i}-y_{sc}{z}_{c,i}$$

(18)

$${\omega }_{mean}={\displaystyle \frac{1}{A}}\int \omega dA={\displaystyle \frac{1}{A}}\sum {\displaystyle \frac{{\omega }_{i-1}+{\omega }_i}{2}}{A}_i$$

(19)

$$\sigma =\left[{\displaystyle \frac{N}{A}}+{\displaystyle \frac{\left(M_y{I}_z+M_z{I}_{yz}\right)}{\left(I_y{I}_z-Iy{z}^2\right)}}z-{\displaystyle \frac{\left(M_z{I}_y+M_y{I}_{yz}\right)}{\left(I_y{I}_z-Iy{z}^2\right)}}y+{\displaystyle \frac{B}{I_w}}w\right]$$

(20)

Once the stresses are computed the effective width of each is plate is computed, the effective width depend on the Axial force N, bending moments My, Mz and Bimoment B combination according to EN 1993-1-1:2022 [1] and EN 1993-1-5:2006 [2] Lee [3].

2 methods can be used to compute the stresses:

Method 1: Consider each internal force independently and compute its stresses with the effective section.

Method 2: Consider all the internal forces simultaneously and compute the stress distribution to altogether and the effective width related to those stresses.

Now taking into account the effective width the effective section properties are computed starting with:

- Centroid, second moment of area about the y and z axes and product of inertia (I_yeff, I_zeff, I_yzeff))

- Shear center, warping coordinates and warping constant (I_weff).

The warping with respect to any pole can be obtained as usual, the mean value w_mean,eff takes into account that only the effective areas are considered, as well to compute the shear center position of the effective section I_yw,eff and I_zw,eff have to be computed for the effective section, finally the warping constant I_w,eff of the effective section is computed.

- Due to possible shift of the centroid there will be a change in the bending moments (M_yeff,M_zeff).

Following the stresses in the effective section are computed:

$$\sigma =\left[{\displaystyle \frac{N}{A_{eff}}}+{\displaystyle \frac{\left(M_{yeff}{I}_{zeff}+M_{zeff}{I}_{yzeff}\right)}{\left(I_{yeff}{I}_{zeff}-{I_{yzeff}}^2\right)}}{z}_{eff}-{\displaystyle \frac{\left(M_{zeff}{I}_{yeff}+M_{yeff}{I}_{yzeff}\right)}{\left(I_{yeff}{I}_{zeff}-{I_{yzeff}}^2\right)}}{y}_{eff}+{\displaystyle \frac{B}{I_{weff}}}{w}_{eff}\right]$$

(21)

The coordinates y_eff and z_eff are referred to the effective centroid, while the warping w_eff is referred to the new shear center.

2. Numerical Examples

The following software is used by Agüero [4] :

https://labmatlab-was.upv.es/webapps/home/thinwallsectionmaterials_class4.html

In this software 10 fibers are used per element, more accurate in case the number of fibers is increased.

Following examples to compute section properties in class 4 sections comparing the results for the gross and effective section when a Bimoment is applied. The material used is steel S355

2.1. I Shape Section 2 Axes of Symmetry

Section dimensions b=30cm h=30cm t=0.4cm

Section properties and stresses Figure 2 and 3:

Gross section:

Iw=405000cm⁶

Centroid and shear center in the intersection 2 axes of symmetry

Effective section:

Iweff=169282cm⁶

Centroid and shear center (the effective section is point symmetric)

To obtain the effective width [2] is used:

k _σ=0.57 therefore λp=2.1496 , ρ=0.424 and the effective width of the compression is flange=0.424*15=6.36cm

2.2. Monosymmetric Section

Channel section stresses Figure 4:

Section dimensions b=15cm h=30cm t=0.4cm

Section properties:

Gross section:

Iw=88593cm⁶

The origin of the reference system is at the intersection bottom flange and web.

Centroid position in the axis of symmetry: Zg=15 Yg=3.75

and shear center in the axis of symmetry: Zsc=15 Ysc=-5.625

Effective section:

To compute the effective part of the bottom flange. Taking into account that ψ=-0.6 , k_σ=0.72, λp=1.91 , ρ=0.47 , the part of that flange under tension is 5.624cm and the effective part under compression is ρ*bc=4.4cm, the remaining part of the flange is 15-4.4-5.624=4.96cm length that has to be removed.

Iweff=53730cm⁶

Centroid position: Ygeff=3.06 Zgeff=16.12

and shear center: Zsceff=9.63 Ysceff=-7

2.3. Non Symmetric Section

Channel non symmetric section:

Section dimensions top flange width bu=15cm, bottom flange width bl=30cm tf=0.4cn h=30cm tw=0.2cm

Section properties and stresses Figure 5:

Gross section:

Iw=135000cm⁶

The origin of the reference system is at the intersection bottom flange and web.

Centroid position in the axis of symmetry: Zg=11.25 Yg=9.375 and shear center in the axis of symmetry: Zsc=5 Ysc=-7.5

Effective section:

Iweff=88379cm⁶

Centroid position: Zgeff=13.27 Ygeff=6.44 and shear center: Zsceff=-0.637 Ysceff=-7.83

3. Conclusions

The present paper deals with warping torsion, bending and axial forces. Presenting a method to compute the shear center of the effective section, the warping constant and warping coordinates needed to compute the stresses due to the bimoment. The stresses are compared when a Bimoment only is applied for the sake of simplicity but any combination with axial and biaxial bending moment is allowed. This paper fills the gap of the current codes to deal with bimoment in Class 4 sections.