Section Modulus calculation and optimization

Objective:

To calculate the section modulus of the previously designed hood for analyzing its strength and also optimizing the design to see the difference in the bending strength of the hood. Higher the section modulus of a structure, the more the resistive it becomes to bending.

Section modulus

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members, it is also referred to as the Polar Modulus or the Torsional Sectional Modulus. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus.

S=I/Y

Here,

S = Section modulus

I= Moment of Inertia (Unit mm^4)

Y= Distance from Neutral axis to the extreme end (unit mm)

The Unit of a “section modulus” is (mm^3)

Changing the position can show the difference in bending which is because of the centroid distance from extreme fiber end.

In Layman's terms, in this position there is more material from axis to the specific direction, to resist the load which makes less bend and improves the strength.

Elastic section modulus
For general design, the elastic section modulus is used, applying up to the yield point for most metals and other common materials.

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, as seen in the table below. It is also often used to determine the yield moment (My) such that My = S × σ_y, where σ_y is the yield strength of the material.

Plastic section modulus

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.

The plastic section modulus is the sum of the areas of cross-sections on each side of the plastic neutral axis that may or may not be equal and multiplied by the distance from the local centroid of the two regions to the plastic neutral axis.

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. So, for sections with constant yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.

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:

the Plastic Section Modulus can also be called the 'First moment of area'

The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by Mp=F_y*Z. Plastic section modulus and elastic section modulus are related by a shape factor which can be denoted by 'k', used for an indication of capacity beyond elastic limit of material. This could be shown mathematically with the formula: -

k = Z/S

First case study:

The intersection of datum with hood front was taken consideration as the cross section of the analysis:

The analysis values came for the above

Maximum momentum of Inertia = 8.982759e+7 mm^4

Minimum momentum of Inertia = 2.782997 e+5 mm^4

Elastic section modulus (S) = I/Y

Total length of the cross section is 879.5357mm and the value of Y is 879.5357/2 = 439.76785mm

Elastic section modulus (s) = 2.782997e+5/439.76785 = 632.833 mm^3

Second case study:

The depth of inner panel is increased by 0.5mm and analysis performed for same cross section the results as follows:

Maximum momentum of Inertia = 8.966427 e+7 mm^4

Minimum momentum of Inertia = 2.820247030 e+5 mm^4

Elastic section modulus (S) = I/Y = 2.820247030 e+5 / 439.76785 = 641.30359 mm^3

Conclusion:

From the above both cases we can observe that the increase of the depth of the inner panel cause of increase in the section modulus. We can also increase the section modulus value by adding more embosses to the inner panel.