Section Modulus calculation and optimization

AIM:To find the section modulus of the hood and optimisation to improve the overall section modulus and explanation about section modulus

SECTION MODULUS:

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

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

Mathematically,

• `Z=I/Y`

Sectional Modulus (Z): It is the ratio of Moment of Inertia (I) of the beam cross-section about the neutral axis to the distance (ymax) of extreme fiber from the neutral axis.

Unit of Sectional Modulus(Z)=mm^3,cm^3 or m^3

TYPES OF SECTION MODULUS:

There are two types of section modulus,

1. Elastic Section Modulus
2. Plastic Section Modulus

• ELASTIC SECTION MODULUS

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

Elastic section modulus is denoted by Zₑ.

Zₑ = I/ymax

Where,

I = Moment of inertia

ymax = Distance from NA to extreme layer

There are following different elastic section modulus,

Rectangular Section

 

          

          

          

          

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

 

There are following section modulus for different section,

 

Rectangular section

           

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

          

FIRST CASE STUDY:

In This,

Moment of inertia(max) = 8.16459×10^5×mm^4

Moment of inertia(min) = 5.24885×10^3×mm^4

Y=486.81/2=243.4 mm

Sectional Modulus (Z)=I/Y=5.24885×10^3/243.4=21.56mm^3

SECOND CASE:

By increasing the depth of the inner panel by 0.5 mm

Moment of inertia(max) = 8.30440×10^5×mm^4

Moment of inertia(min) = 5.453063×10^3×mm^4

Y=486.81/2=243.4 mm

Sectional Modulus (Z)=I/Y=5.453063×10^3/243.4=22.4mm^3

CONCLUSION:

Thus sectional modulus for hood design is calculated and optimized design is also done to increase the section modulus of the hood.