Section Modulus calculation and optimization

Section Modulus

Section Modulus is the geometric property of the given cross-section of any shaft or beam structures. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape. It is the direct measure of the strength of the shaft section or the beam section. Higher the section modulus of a structure, the more the resistive it becomes to bending.

Section modulus is defined as the ratio of polar moment of inertia to the radius of the shaft or the distance from the neutral axis to the outer fibres. There are two types of section moduli, the elastic section modulus and the plastic section modulus:

1. Elastic Section Modulus

The elastic section modulus is defined as S = I / y, where I is the second moment of area (or area moment of inertia, not to be confused with moment of inertia) and y is the distance from the neutral axis to any given fibre. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre. It is also often used to determine the yield moment (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yield strength of the material.

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behaviour is assumed to be an acceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparing the plastic capacity against amplified forces or stresses.

2. Plastic Section Modulus

The plastic section modulus depends on the location of the plastic neutral axis (PNA). The PNA is defined as the axis that splits the cross section such that the compression force from the area in compression equals the tension force from the area in tension

The plastic section modulus is the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:

Calculation of Section Modulus of Hood

We are taking the hood which we have created previously, and we will be doing the section modulus of that hood and find the maximum and minimum moment of inertia.

So, to do the calculation for section modulus of the hood we have to take a section from the hood and then make it a closed curve.

The intersection curve obtained for this hood will look like as shown in the figure below:

For section modulus Calculation:

Go to Analysis > Section Inertial Analysis > Select the curve.

After selecting the curve and clicking okay the result will be shown:

The values for Moment of Inertia are as follows:

MOI (Max):8.109543811e+05[mm⁴]

MOI (Min):3.379605510e+03[mm⁴]

According to section modulus formula

S = I / Y

Where,

Y = Total distance / 2

Total distance = 483.3191mm

Therefore, Y = 483.3191/2 = 241.159mm

I = 8.109543811e+05[mm⁴]

Therefore, S = 8.109543811e+05[mm⁴] / 241.159mm

S = 3362.74 mm³

Case – 2

Now we will take the hood which we have created newly, and we will be doing the section modulus of this newly created or optimized hood and find the maximum and minimum moment of inertia.

So, to do the calculation for section modulus of the hood we have to take a section from the hood and then make it a closed curve.

The intersection curve obtained for this hood will look like as shown in the figure below:

For section modulus Calculation:

Go to Analysis > Section Inertial Analysis > Select the curve.

After selecting the curve and clicking okay the result will be shown:

The values for Moment of Inertia are as follows:

MOI (Max): 9.367213983e+05[mm⁴]

MOI (Min): 4.152654846+03[mm⁴]

Section Area: 42.504412444[mm²]

According to section modulus formula

S = I / Y

Where,

Y = Total distance / 2

Total distance = 483.3191mm

Therefore, Y = 483.3191/2 = 241.159mm

I = 9.367213983e+05[mm⁴]

Therefore, S = 9.367213983e+05[mm⁴] / 241.159mm

S = 3884.25mm³

Conclusion:

Therefore, as we increase the section area, the moment of Inertia & section modulus increases which means the strength and load capacity of material also increases. Thus, optimized hood will have more moment of intertia and enhance section modulus.