Section Modulus calculation and optimization

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. Higher the section modulus of a structure, the more the resistive it becomes to bending.

INTRODUCTION

• 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 moment of inertia and polar moment of inertia for stiffness.

Defination of Section Modulus

• Ratio of total moment resisted by the section to the stress in the extreme fibre.
• It gives us an idea about how much of load the given structure can take.
• If the section modulus increases the moment of resistance increases and hence its load bearing capacity.

Formula 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)

Moment OF Intertia

The second moment of area is also known as the moment of inertia of a shape. The second moment of area is a measure of the 'efficiency' of a cross-sectional shape to resist bending caused by loading.

Symbol is I. Units are mm⁴

Both beams have the same area, but the beam is much stronger on its edge than its side. This is explained by the second moment of area (I).

Second Moment of Area of a cross-section is found by taking each mm² and multiplying by the square of the distance from an axis.
Then add them all up.

Second Moment of Area (Definition)
I =  (A * d²)    
Units are mm⁴

So the best way to get a high Second Moment of Area is to get as much area as possible the longest distance from the centre axis (Called the centroidal axis or neutral plane).

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 amd improves the strength.

Other Types Section Modulus

Elastic section modulus:-

The elastic section modulus is defined as the ratio of the second moment of area (or moment of inertia) and the distance from the neutral axis to any given (or extreme) fiber. It can also be defined in terms of the first moment of area.

S=I/Y

Significance

It is also often used to determine the extreme fiber elastic stresses as well as the yield moment (My">My) of elastic sections such that My=S×σy">My=S×σy, where σy">σy is the yield strength of the material. It is a commonly used property in the design of flexural members of steel, post-tensioned concrete, and timber sections.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		,	NA indicates neutral axis
doubly symmetric I-section (minor axis)			NA indicates neutral axis
Circle			Solid arrow represents neutral axis
Circular hollow section			Solid arrow represents 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 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 can also be called the 'First moment of area'

Description	Figure	Equation	Comment
Rectangular section			,
Rectangular hollow section			where: b=width, h=height, t=wall thickness
For the two flanges of an I-beam with the web excluded			where:  are the distances from the neutral axis to the centroids of the flanges respectively.
For an I Beam including the web
For an I Beam (weak axis)			d = Full height of the I Beam
Solid Circle
Circular hollow section

The plastic section modulus is used to calculate the plastic moment, M_p, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by M_p=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 :-

HOOD SECTION MODULUS CALCULATION

Step-1

• First intersection taken from existing Hood design.
• Then to caluculate the moment of inertia we have create surface from the intersectian in CATIA . 
• After that calculate using Measure inertia for to get maximum and minmum moment of inertia for the Hood design.

Moment of inertia(Max)=7.343*10^5mm^4

Moment of inertia(Min)=1.622*10^8mm^4

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

y=480.989/2=240.4945mm

S=I/y

S=1.622*10^8/240.4945=674,443.69mm^3

Step-2

Hood Design Optimization:-

By increasing the depth of the inner panel by 0.5 mm

Moment of inertia(Max)=7.718*10^5mm^4

Moment of inertia(Min)=1.666*10^8mm^4

y=480.989/2=240.4945

S=I/y

S=1.666*10^8/240.4945=692,739.33mm^3

CONCLUSION

So, here the difference can be seen in the moment of inertia and the change in the section modulus can be seen by the change in the design(depth).