Section Modulus calculation and optimization

SECTION MODULUS CALCULATION AND OPTIMIZATION

OBJECTIVE:-

Use the Section from Hood design and calculate the Section Modulus using the formula

S = I/y

Optimization with a new section that has an improved Section Modulus compared to the previous one and mention what changes made which resulted in the increased strength.

SECTION MODULUS:-

The section modulus is geometric property of the cross section used for designing beams and flexural members. It does not represent anything physically. 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, radious 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 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.

Formula of Section Modulus:-

It may be defined as the ration of total moment resisted by the section to the stress in the extreme fibre which is equal to yield stress.

S = I/y

Where,

S = Section Modulus (Unit mm^3)

I = Moment of Inertia (unit mm)

y = distance between the neutral axis and the extreme end of the object (unit mm^4)

Area moment of Inertia:

The area moment of inertia of a beams cross sectional area measures the beam ability to resist bending. The larger the moment of inertia less will be the bend.

Where,

M = Bending moment

I = Area moment of Inertia

E = Youngs modulus

R = Radius of gyration

Neutral axis:

The neutral axis is an axis in the cross section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid. All fibers on one side of the neutral axis are in a state of tension, while those on the opposite side are in compression.

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 inertia (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 (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yield strength of the material.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents natural axis
doubly symmetric I-section (major axis)		,	NA indicates natural axis
doubly symmetric I-section (minor axis)			NA indicates natural axis
Circle			Solid arrow represents natural axis
Circular hollow section			Solid arrow represents natural axis
Rectangular hollow section			NA indicates natural axis
Diamond			NA indicates natural axis
C-channel			NA indicates natural axis

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 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 is not the 'First moment of area'. Both relate to the calculation of the centroid, but Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section, while the First moment of area is calculated based on either side of the "considering point" of the cross section and it is different along the cross section and depends on the point of consideration.

CASE 1 STUDY:-

The Intersection of the design was taken and then section Inertia Analysis was done-

Moment of inertia(max) = 4.877407904×10^6×mm^4

Moment of inertia(min) = 1.371260121×10^4×mm^4

Y = 485.8270/2= 242.9135 mm

SO

S=I/Y

S= 1.371260121×10^4/242.9135

S= 56.4505 mm^3

CASE 2 STUDY DESIGN OPTIMIZATION:-

By increasing the depth of the inner panel by 0.5 mm

Moment of inertia(max) = 4.86835587×10^6×mm^4

Moment of inertia(min) = 1.386925919×10^4×mm^4

Y = 485.8270/2= 242.9135 mm

SO

S=I/Y

S= 1.386925919×10^4/242.9135

S= 57.0954 mm^3

Conclusion:-

When comparing case 1 & case 2,

1. The section modulus value of case 2 is higher than the case 1
2. As we increase the section area, the moment of inertia & section modulus increases which means strength and load material also increase.