Section Modulus calculation and optimization
Section modulus
Section Modulus is a geometric property used to calculate the bending stresses in a structural member.
Mathematically is defined as the ratio of the moment of inertia of a section about its centroidal axis and to the distance of the extreme layer from the neutral axis.
Generally, section modulus is denoted by S
So,
S= I/y
Where,
I = Moment of Inertia(m⁴)
y = distance between the neutral axis and the extreme end of the object (mm)
The unit of section modulus is m³.
Types of section modulus
there are two types of section modulus:
- Elastic Section Modulus
- Plastic Section Modulus
Elastic Section Modulus
Elastic section modulus is calculated when we are to use the material behavior till the yield point.
For general design purposes, we use elastic section modulus.
Generally, elastic section modulus is denoted by Sₑ.
So,
Sₑ = I/ymax
Where,
I = Moment of inertia
ymax = distance between the neutral axis and the extreme end of the object
Plastic section modulus
The plastic section modulus is used for materials where the elastic yield is acceptable
and the plastic behavior is considered to be an acceptable range.
It depends on the location of the plastic neutral axis.
Sₚ = (Ac × yc) + (Aₜ + yₜ)
Where,
Ac = Area of cross-section under compression.
Aₜ = Area of cross-section under tension.
yc = Distance from the plastic neutral axis to the center of gravity of the areas under compression.
yₜ = Distance from the plastic neutral axis to the center of gravity of areas under tension.
Area moment of inertia
Area Moment of Inertia or Moment of Inertia for an Area - also known as Second Moment of Area - I, is a property of shape that is used to predict deflection, bending and stresses in beam.
The Area Moment of Ineria for a rectangular section can be calculated as
Ix = b h3 / 12
Iy = b3 h / 12
where
b = width
h = height
Calculating the section modulus of hood (without changing)
The hood is taken for the section modulus calculation without changing anything.
S= I/y
Area moment of inertia (I) = 4.788764743×10^6 mm^4
Distance between the neutral axis and the extreme end of the object (Y) = 440 mm
S= 4.788764743×10^6/440
Section modulus (S) = 1.08835562×10^4mm^3
Calculating the section modulus of hood (After applying changes )
The innner hood section is offsetted 0.4mm downwards and additional curves are applied in the projected curve for calculating the section modulus. This additional changes is used to increase the area moment of inertia.
S= I/y
Area moment of inertia (I) = 4.812541280×10^6 mm^4
Distance between the neutral axis and the extreme end of the object (Y) = 440 mm
S= 4.812541280×10^6/440
Section modulus (S) = 1.09375938×10^4mm^3
| 1.Section modulus of hood (without changing) | 2.Section modulus of hood (After changes) |
| The hood is taken for the section modulus calculation without changing anything. |
The innner hood section is offsetted 0.4mm downwards and additional curves are applied in the projected curve for calculating the section modulus. This additional changes is used to increase the area moment of inertia.
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Area moment of inertia (I) = 4.788764743×10^6 mm^4
Distance between the neutral axis and the extreme end of the object (Y) = 440 mm
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Area moment of inertia (I) = 4.812541280×10^6 mm^4
Distance between the neutral axis and the extreme end of the object (Y) = 440 mm
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S= 4.788764743×10^6/440
Section modulus (S) = 1.08835562×10^4mm^3
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S= 4.812541280×10^6/440
Section modulus (S) = 1.09375938×10^4mm^3
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Conclusion
The section modulus is used to calculate the bending stresses in the structural member. Here I calculated the section modulus for my previous project hood design.
With this I can understand the section modulus is increased when the area moment of inertia is increased.The section modulus is directly proposional to the area moment of inertia. I can undestand the difference between the two section of modulus shown above the two section is same but the small difference in the second section is offset of 0.4mm and additional area is included. That area difference leads to increase the section of modulus in second section.