Section Modulus calculation and optimization

Objective :

To calculate the section modulus of designed hood for analyzing its strength and also optimizing the design of hood to see the difference in the bending strength of the hood.

Concept :

Section modulus :

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and secound moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. 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. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

Mathematical represntation of 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 (mm3)

Elastic section modulus :

The elastic section modulus is defined as S = I / y, where I is the secound momment of area 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 (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yeild strenght of the material.

Elastic Section modulus equations

Cross-sectional shape	Figure	Equation
Rectangle
doubly symmetric I-section (major axis)		, with
doubly symmetric I-section (minor axis)
Circle
Circular hollow section
Rectangular hollow section
Diamond
C-channel

Plastic section modulus :

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior 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 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, and equal compressive and tensile 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

Application of section modulus on hood design :

Hood design's section modulus:

Moment of inertia(max) =8.801979837e+05(mm^4) = 8.801979837×10^5(mm^4)

Moment of inertia(min) =8.163086180e+03(mm^4) = 8.163086180×10^3×(mm^4)

Y=2.163059601e+03(mm) =2.163059601x10^3 = 2163.059601/2=1081.5298005 mm

S=I/Y

S=8.163086180×10^3/1081.5298005= 7.547 mm^3

Optimization of Hood design's section modulus :

By increasing the depth of the inner panel by 0.5 mm.

Moment of inertia(max) =8.806023507e+05(mm^4) =8.806023507×10^5(mm^4)

Moment of inertia(min) =8.213657496e+03(mm^4) = 8.213657496×10^3×(mm^4)

Y=2.163059601e+03(mm) =2.162882132x10^3 = 2162.882132/2=1081.441066mm

S=I/Y

S=8.213657496×10^3/1081.441066= 7.654 mm^3

Results :

So , here we can see that by making same slight changes in the design the moment of inertia and section modulus also get some diffrence in value .