Calculating and Optimization of Section Modulus of Hood Design.

Aim: 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. The higher the section modulus of a structure, the more resistive it becomes to bending.

Objective: Use the Section from your Hood design and calculate the section modulus. Come up with a new section that has an improved Section Modulus compared to the previous one and mention what changes you made which resulted in the increased strength.

Theory:

• 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 the design include an area for tension and shear, the radius of gyration for compression, and the 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.

S = I/y

Here,

S = Section Modulus

I = Moment of Inertia (mm^4)

y = distance between the neutral axis and the end of the object (mm)

The Unit of Section Modulus is (mm^3)

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

In Layman's terms, in this position there is more material from the 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 the 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 x σy, where ay is the yield strength of the material.

Cross-Sectional Shape	Figure	Equation	Comment
Rectangle			The solid arrow represents the neutral axis.
Doubly symmetric I- section (Major Axis)			NA Indicates neutral axis.
Doubly symmetric I- section (Minor Axis)			NA Indicates neutral axis.
Circle			The solid arrow represents the neutral axis.
Circular hollow section			The solid arrow represents the neutral axis.
Rectangular hollow section			NA Indicates neutral axis.
Diamond			NA Indicates neutral axis.
C-channel			NA Indicates neutral axis.

• 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 anent 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 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, Fy, by Mp=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 the elastic limit of the material. This could be shown mathematically with the formula:

The shape factor for a rectangular section is 1.5.

So here we are Calculating Elastic Section Modulus.

Procedure:

Total Length Span of hood section= 875.8367mm

Y = 875.836/2 = 437.918 mm

• Initial Section Inertia Analysis:

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

Maximum Moment of Inertia, M1 = 4.772390958x10^6 x mm^4

Maximum Moment of Inertia, M2 = 1.336536353x10^4 x mm^4

We are going to take the Minimum MOI.

Y = 875.836/2 = 437.918 mm

We know that Section Modulus, S = I/Y

Initial S (min) = M2/Y = (1.336536353x10^4)/(437.918)

S = 30.52 mm^3

• Hood Design Optimization:

In order to optimize the Hood Section, I offset the whole section which we created below by 0.5mm and then calculated the inertia for the offset section.

Here,

Maximum Moment of Inertia, M1' = 4.731453339x10^6 x mm^4

Maximum Moment of Inertia, M2' = 1.396016984x10^4 x mm^4

We are going to take the Minimum MOI.

Y = 875.836/2 = 437.918 mm

We know that Section Modulus, S = I/Y

Optimized S (min) = M2'/Y = (1.396016984x10^4)/(437.918)

S = 31.818 mm^3

• Comparing both values:

Initial Section Modulus	Optimized Section Modulus
S = 30.52 mm^3	S = 31.818 mm^3

Conclusion: Hence by offsetting the section modulus by 0.5mm below, I have obtained an improved value of section modulus, thus strengthening the hood design.