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&#8209;THIN and SHAPE&#8209;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 allow for the direct calculation of the extreme normal stresses from the bending moments My and Mz. You can determine the section modulus about the y-axis at a point i on the outer contours of the cross-section as follows: The section modulus about the z-axis at point i on the outer contours 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&#8209;THIN or SHAPE&#8209;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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005400

Sonja von Bloh, M.Sc.

2023-07-27

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, m i n} = \frac{I_{y}}{e_{z, m i n}}

m_j	Story masses in kg
λ	Distribution coefficient for each story

Minimum Section Modulus About y-Axis

W_{y, m a x} = \frac{I_{y}}{e_{z, m a x}}

γ_M0	Partial factor for cross-section resistance
γ_M2	Partial factor for cross-section resistance in case of failure due to tension

Maximum Section Modulus About y-Axis

W_{z, m i n} = \frac{I_{z}}{e_{y, m i n}}

γ_M2	Partial factor for cross-section resistance in case of failure due to tension
γ_M2	Partial factor for cross-section resistance in case of failure due to tension

Minimum Section Modulus About z-Axis

W_{z, m a x} = \frac{I_{z}}{e_{y, m a x}}

t	Sheet thickness
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 allow for the direct calculation of the extreme normal stresses from the bending moments M_y and M_z.

You can determine the section modulus about the y-axis at a point i on the outer contours of the cross-section as follows:

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

r	Distance to element dA
T_c	Temperature change
ΔT	Temperature gradient
F_N	Axial force perpendicular to the support
p₀	Air pressure on the fluid surface

Section Modulus at Point i About y-Axis

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

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

ρ	Adjusted reduction factor
N_Ed	design value of acting axial force
l	free length
M₀₁, M₀₂	algebraic values of first order moments at both ends of element
M_0Ed	ultimate moment of first order under combination of design loads (including geometric imperfections)

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} = m i n (|W_{y, m i n}|, |W_{y, m a x}|)

α_m	reduction coefficient relating to number of elements where m is number of vertical elements contributing to total effect
h	height of straight section in bending plane

Section Modulus About y-Axis

W_{z} = m i n (|W_{z, m i n}|, |W_{z, m a x}|)

f_cd	design value of compressive strength of concrete
μ_Ed	bending moment reduced in G₀

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.

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):

Section Moduli About Axes y, z

W_{y, m i n} = \frac{I_{y}}{e_{z, m i n}} = \frac{10.12}{- 2.5488} = - 3.97 c m^{3}

Minimum Section Modulus About y-Axis

W_{y, m a x} = \frac{I_{y}}{e_{z, m a x}} = \frac{10.12}{2.5488} = 3.97 c m^{3}

Maximum Section Modulus About y-Axis

W_{z, m i n} = \frac{I_{z}}{e_{y, m i n}} = \frac{0.72}{- 0.9265} = - 0.78 c m^{3}

Minimum Section Modulus About z-Axis

W_{z, m a x} = \frac{I_{z}}{e_{y, m a x}} = \frac{0.72}{0.9265} = 0.78 c m^{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):

Section Moduli About Axes y, z

W_{y, m i n} = \frac{I_{y} \cdot I_{z} - {I_{y z}}^{2}}{I_{z} \cdot z_{6} - I_{y z} \cdot y_{6}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{0.72 \cdot (- 2.549) - (- 1.71) \cdot (- 0.058)} = - 2.25 c m^{3}

Minimum Section Modulus About y-Axis (Point 6)

W_{y, m a x} = \frac{I_{y} \cdot I_{z} - {I_{y z}}^{2}}{I_{z} \cdot z_{4} - I_{y z} \cdot y_{4}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{0.72 \cdot 2.549 - (- 1.71) \cdot 0.058} = 2.25 c m^{3} ()

Maximum Section Modulus About y-Axis (Point 4)

W_{z, m i n} = \frac{I_{y} \cdot I_{z} - {I_{y z}}^{2}}{I_{y} \cdot y_{5} - I_{y z} \cdot z_{5}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{10.12 \cdot (- 0.927) - (- 1.71) \cdot 2.375} = - 0.82 c m^{3}

Minimum Section Modulus About z-Axis (Point 5)

W_{z, m i n} = \frac{I_{y} \cdot I_{z} - {I_{y z}}^{2}}{I_{y} \cdot y_{7} - I_{y z} \cdot z_{7}} = \frac{10.12 \cdot 0.72 - {(- 1.71)}^{2}}{10.12 \cdot 0.927 - (- 1.71) \cdot (- 2.375)} = 0.82 c m^{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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