How are the section moduli about the y- and z-axis calculated in RSECTION 1, SHAPE‑THIN, and SHAPE‑MASSIVE?

Q: How are the section moduli about the y- and z-axis calculated in RSECTION 1, SHAPE&#8209;THIN, and SHAPE&#8209;MASSIVE?

SHAPE-THIN/SHAPE-MASSIVE In SHAPE‑THIN and SHAPE‑MASSIVE, the section moduli are calculated according to the following formulas:With the section moduli calculated in this way, it is not possible to directly calculate the maximum and minimum normal stress from the bending moment My or Mz for asymmetric cross-sections. RSECTION In RSECTION, the section moduli are determined in such a way that they enable the direct calculation of extreme normal stresses from the bending moments My/Mz. The section modulus about the y-axis at a point i on the outer contour of the cross-section is determined as follows:The section modulus about the z-axis at point i on the outer contour of the cross-section is determined as follows:The minimum section modulus about the y- or z-axis is the maximum negative section modulus of the i-points. The maximum section modulus about the y- or z-axis is the smallest positive section modulus of the i-points.The governing section moduli about the y- or z-axis are calculated as follows: Example Image 01 shows a rectangular cross-section inclined by 10° with the dimensions of w/h = 50/10 mm.In SHAPE‑THIN or SHAPE‑MASSIVE, the following section moduli Wy,min, Wy,max, Wz,min, and Wz,max are obtained (see Image 02):In RSECTION, the following section moduli Wy,min, Wy,max, Wz,min, and Wz,max are obtained (Image 03):

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Sonja von Bloh, M.Sc.

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Section Moduli About Axes y, z

How are the section moduli about the y- and z-axis calculated in RSECTION 1, SHAPE‑THIN, and SHAPE‑MASSIVE?

Answer:

SHAPE-THIN/SHAPE-MASSIVE

In SHAPE‑THIN and SHAPE‑MASSIVE, the section moduli are calculated according to the following formulas:

W_{y, \min} = \frac{I_{y}}{e_{z, \min}}

I_y	Second moment of area about the y-axis
e_z,min	Maximum edge distance in the negative direction of the centroidal axis z

Minimum Section Modulus About y-Axis

W_{y, \max} = \frac{I_{y}}{e_{z, \max}}

I_y	Second moment of area about the y-axis
e_z,max	Maximum edge distance in the positive direction of the centroidal axis z

Maximum Section Modulus About y-Axis

W_{z, \min} = \frac{I_{z}}{e_{y, \min}}

I_z	Second moment of area about the z-axis
e_y,min	Maximum edge distance in the negative direction of the centroidal axis y

Minimum Section Modulus About z-Axis

W_{z, \max} = \frac{I_{z}}{e_{y, \max}}

I_z	Second moment of area about the z-axis
e_y,max	Maximum edge distance in the positive direction of the centroidal axis y

Maximum Section Modulus About z-Axis

With the section moduli calculated in this way, it is not possible to directly calculate the maximum and minimum normal stress from the bending moment M_y or M_z for asymmetric cross-sections.

RSECTION

In RSECTION, the section moduli are determined in such a way that they enable the direct calculation of extreme normal stresses from the bending moments M_y/M_z.

The section modulus about the y-axis at a point i on the outer contour of the cross-section is determined as follows:

W_{y, i} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{z} \cdot z_{i} - I_{yz} \cdot y_{i}}

I_y	Second moment of area about the y-axis
I_z	Second moment of area about the z-axis
I_yz	Centrifugal moment about the y- and z-axes
y_i	y-coordinate of Point i in the axis system y,z
z_i	z-coordinate of Point i in the axis system y,z

Section Modulus at Point i About y-Axis

The section modulus about the z-axis at point i on the outer contour of the cross-section is determined as follows:

W_{z, i} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{y} \cdot y_{i} - I_{yz} \cdot z_{i}}

I_y	Second moment of area about the y-axis
I_z	Second moment of area about the z-axis
I_yz	Centrifugal moment about the y- and z-axes
y_i	y-coordinate of Point i in the axis system y,z
z_i	z-coordinate of Point i in the axis system y,z

Section Modulus at Point i About z-Axis

The minimum section modulus about the y- or z-axis is the maximum negative section modulus of the i-points. The maximum section modulus about the y- or z-axis is the smallest positive section modulus of the i-points.

The governing section moduli about the y- or z-axis are calculated as follows:

W_{y} = \min (|W_{y, \min}|, |W_{y, \max}|)

W_y,min	Maximum negative section modulus about the y-axis
W_y,max	Minimum positive section modulus about the y-axis

Section Modulus About y-Axis

W_{z} = \min (|W_{z, \min}|, |W_{z, \max}|)

W_z,min	Maximum negative section modulus about the z-axis
W_z,max	Minimum positive section modulus about the z-axis

Section Modulus About z-Axis

Example

Image 01 shows a rectangular cross-section inclined by 10° with the dimensions of w/h = 50/10 mm.

1 Rectangular Section w/h = 50/10 mm Inclined by 10°

In SHAPE‑THIN or SHAPE‑MASSIVE, the following section moduli W_y,min, W_y,max, W_z,min, and W_z,max are obtained (see Image 02):

2 Section Moduli About Axes y, z

W_{y, \min} = \frac{I_{y}}{e_{z, \min}} = \frac{10.12}{- 2.5488} = - 3.97 {cm}^{3}

Minimum Section Modulus About y-Axis

W_{y, \max} = \frac{I_{y}}{e_{z, \max}} = \frac{10.12}{2.5488} = 3.97 {cm}^{3}

Maximum Section Modulus About y-Axis

W_{z, \min} = \frac{I_{z}}{e_{y, \min}} = \frac{0.72}{- 0.9265} = - 0.78 {cm}^{3}

Minimum Section Modulus About z-Axis

W_{z, \max} = \frac{I_{z}}{e_{y, \max}} = \frac{0.72}{0.9265} = 0.78 {cm}^{3}

Maximum Section Modulus About z-Axis

In RSECTION, the following section moduli W_y,min, W_y,max, W_z,min, and W_z,max are obtained (Image 03):

3 Section Moduli About Axes y, z

W_{y, \min} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{z} \cdot z_{6} - I_{yz} \cdot y_{6}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{0.72 \cdot (- 2.549) - (- 1.71) \cdot (- 0.058)} = - 2.25 {cm}^{3}

Minimum Section Modulus About y-Axis (Point 6)

W_{y, \max} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{z} \cdot z_{4} - I_{yz} \cdot y_{4}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{0.72 \cdot 2.549 - (- 1.71) \cdot 0.058} = 2.25 {cm}^{3}

Maximum Section Modulus About y-Axis (Point 4)

W_{z, \min} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{y} \cdot y_{5} - I_{yz} \cdot z_{5}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{10.12 \cdot (- 0.927) - (- 1.71) \cdot 2.375} = - 0.82 {cm}^{3}

Minimum Section Modulus About z-Axis (Point 5)

W_{z, \min} = \frac{I_{y} \cdot I_{z} - {I_{yz}}^{2}}{I_{y} \cdot y_{7} - I_{yz} \cdot z_{7}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{10.12 \cdot 0.927 - (- 1.71) \cdot (- 2.375)} = 0.82 {cm}^{3}

Maximum Section Modulus About z-Axis (Point 7)

Author

Ms. von Bloh provides technical support for our customers and is responsible for the development of the SHAPE‑THIN program as well as steel and aluminum structures.

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