Challenge

Section Modulus calculation and optimization

BBharathbhushan A24 Jun 2023

Report On Section Modulus Calculation and optimization Of Hood

OBJECTIVE:

The aim of the project is to find the section modulus using the formula S=I/Y of the designed hood and optimize it to get an improved section modulus by making necessary changes 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 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 (Unit mm 4)

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

The Unit of a “section modulus” is (mm3)

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.

SECTION MODULUS

Section Modulus is a geometric property used to calculate the bending stresses in a structural member.

Mathematically,

It is defined as the ratio of the moment of inertia of a section about its centroidal axis and to the distance of the extreme layer from the neutral axis.

It is denoted by Z.

Z = I/ymax

Mathematically By bending stress equation,

M/I = σmax/ymax

M = σmax × (I/ymax)

M = σmax × Z

Where,

I = Moment of Inertia(m⁴)

y = distance from NA to extreme layer(m)

The unit of section modulus is m³.

TYPES OF SECTION MODULUS

There are two types of section modulus:

Elastic Section Modulus
Plastic Section Modulus Elastic Section Modulus

Elastic section modulus is calculated when we are to use the material behavior till the yield point.

For general design purposes, we use elastic section modulus.

Generally, elastic section modulus is denoted by Zₑ.

So,

Zₑ = I/ymax

Where,

I = Moment of inertia

ymax = Distance from NA to extreme layer

Elastic Section Modulus of different sections

There are following different elastic section modulus:

RECTANGULAR SECTION

As we know the moment of inertia of the rectangular section,

I = bd³/12

And the distance from the neutral axis,

ymax = d/2

So,

Zₑ = I/ymax

Zₑ = (bd³/12)/(d/2)

After calculating them,

Section modulus of rectangular section,

Zₑ = bd²/6

HOLLOW RECTANGULAR SECTION

As we know the moment of inertia of the hollow rectangular section,

I = (BD³/12) - (bd³/12)

ymax = D/2

After putting these values,

Zₑ = I/ymax

(BD³/12) - (bd³/12)/(D/2)

Zₑ = (BD³ - bd³)/6D

SOLID CIRCULAR

For solid Circular moment of inertia,

I = (π/64) × d⁴

ymax = d/2

Then,

Zₑ = {(π/64) × d⁴}/(d/2)

After calculating then find values of section modulus of solid circular

Zₑ = (π/32) × d³

HOLLOW CIRCULAR

The moment of inertia of hollow Circular,

I = (π/64) × (D⁴ - d⁴)

And,

ymax = D/2

After putting these values,

Zₑ = (π/64) × (D⁴ - d⁴)/(D/2)

Zₑ = π/32 × {(D⁴ - d⁴)/D}

SYMMETRIC I AND C SECTION

Moment of inertia for the I & C section,

I = (BD³/12) - {(B - b)d³}/12

ymax = D/2

So,

Zₑ = [(BD³/12) - {(B - b)d³}/12]/(D/2)

Zₑ = {BD³ - (B - b)d³}/6D

DIAMOND

Elastic section modulus for diamond is,

Zₑ = bd²/24

PLASTIC SECTION MODULUS

The plastic section modulus is used for materials where the elastic yield is acceptable and the plastic behavior is considered to be an acceptable range.

It depends on the location of the plastic neutral axis. The PNA is the axis that bisects the cross-section such that the compressive force from the region in compression is equal to the tension force from the region under tension. So, for sections with a constant yield stress, the area above and below the PNA will be equal, but for mixed sections, this is not necessarily the case.

Mathematically,

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.

Zₚ = (Ac × yc) + (Aₜ + yₜ)

Where,

Ac = Area of cross-section under compression.

Aₜ = Area of cross-section under tension.

yc = Distance from the plastic neutral axis to the center of gravity of the areas under compression.

yₜ = Distance from the plastic neutral axis to the center of gravity of areas under tension.

Since on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ

And,

yc = yₜ

PLASTIC SECTION MODULUS OF DIFFERENT SECTIONS

There are following section modulus for different section:

RECTANGULAR SECTION

As we know,

on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ

And,

yc = yₜ

Ac = Aₜ = bd/2

yc = yₜ = d/4

So,

Zₚ = (Ac × yc) + (Aₜ + yₜ)

Zₚ = (bd/2 × d/4) + (bd/2 × d/4))

After calculating

Zₚ = bd²/4

Similarly, we can calculate the section modulus of the rectangular hollow section,

Zₚ = (bd²/4) - (b- 2t) × {(b/2) - t}²

CIRCULAR SECTION

As we know, on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ = πd²/8

yc = yₜ = 2d/3π

Zₚ = (πd²/8 × 2d/3π) + (πd²/8 × 2d/3π)

After calculating,

Zₚ = d³/6

Similarly, we can calculate the section modulus of the circular hollow section,

Circular hollow section,

Zₚ = (D³ - d³)/6

Hood assembly - Sectioin Modulus:

Here below image is my hood assembly.

Here first i would check section modulus from the front side impact strength check, it does mean load wil be apply from front side so i have made datum plane accourdingly,

To create cross section of the hood assembly, I did use curve on datum then made profile help of tool curve intersection.

Then I have made sketch with help of projection, and studio spline for gap filling by poles.

Then with help of section inertia analysis tool via found the below value of my hood.

First Design Study

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

Moment of inertia(max) = 8.987229981e⁺⁷×mm^4

Moment of inertia(min) = 2.665423367e⁺⁵×mm^4

NOTE: we always consider minimum Moment of inertia(MOI) always considering factor of safety.

Y=876.3740/2=438.187 mm

Z1=I/Y

Z1=2.665423367e⁺⁵/438.187 = 395.58/438.187 = 0.9027mm³

Hood Design Optimization

By increasing the depth of the inner panel by 0.5 mm

Moment of inertia(max) = 8.969271065e⁺⁷×mm^4

Moment of inertia(min) = 2.707515053e⁺⁵×mm^4

NOTE: we always consider minimum Moment of inertia(MOI) always, considering factor of safety.

Y=876.3740/2=438.187 mm

Z2=I/Y

Z2=2.707515053e⁺⁵/438.187 = 401.830/438.187 = 0.917mm³

Z2=0.917mm³ > Z1=0.9027mm³

So we can improve our hood via strengthen our hood inner panel by increaseing the deep of the emboss deep / Hood inner panel deep.

Here I have checked the section modulus for load applying the front side of the vehicle in this same way we can check the section modulus to applying th load from side of the vehicle.

Current hood:

Moment of inertia(min) = 2.154e-06m4

NOTE: we always consider minimum Moment of inertia(MOI) always, considering factor of safety in CATIA M1 is minimum moment of inertia.

Y=477.513mm/2=238.7565 mm=0.2387565 m

Z1=I/Y

Z1=2.154e-06/0.2387565 = 0.005339/0.2387565 = 0.02236m³

Now we going to improve the moment of inertia as much as better and then will check again to find out differences how our hood improved.

New improved hood:

Moment of inertia(min) =2.23714e-06 m4

Y=477.513mm/2=238.7565 mm=0.2387565 m

Z2=I/Y

Z2=2.23714e-06/0.2387565 = 0.005545/0.2387565 = 0.023224m³

TREE STRUCTURE:

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).

Section Modulus calculation and optimization

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Ship work like thisin your own course.

Talk to a career counsellor

Ship work like this
in your own course.