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.

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 modulus of common shapes are given below.There are two types of section modulus , the elastic section modulus and the plastic section modulus .

S = I / Y

Here ,

S = Section modulus

| = Moment of Inertia ( Unit mm4 )

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 Layaman'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 .

Elastic section modulus :

For general design , the elastic section modulus is used applying up tp the yield point for most metals and other common materials .

The elastic section modulus is defined as S = 1 / Y , Where I is the second moment of area ( 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 ist the distance form 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 = Sx σy , where σy ist the yield strength of the material .

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 :

Zp = Acyc + ATYT

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 = Fy * 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 :

k = Z/S

shape factor for a rectangular section is 1.5 .

So here we are Calculating Elastic section Modulus . 

We can have the value of Y from the above picture , ie ;

Y = 242.028mm

# DESIGN 1 : Section Modulus of Initial Hood Design

From the above analysis , following minimum and maximum moment of inertia are obtained

Imax = 4766699.998 mm4

Imin = 15559.9696 mm4

y = 440 mm

Smax Imax / y = 4766699.998 / 440 = 10833.40 mm³

Smin = Imin / y = 15559.9696 / 440 = 35.363 mm³

#DESIGN 2 : Section Modulus of Optimized Hood design

What changes I made and Why ?

Section Modulus can also be represented as

S =  miAi² / y where A is the area of the section and M is the mass of the body

From the above equation , it can be observed that the section modulus of the body is directly proportional to the sectional area of the body .

Here the section area is increased by reducing the embossing depth and offsetting the outer panel by 3 mm . This increases the value of section modulus .

From the above analysis , following minimum and maximum moment of inertia are obtained

Imax = 475621.8426 mm4

Imin = 15856.98093 mm4

y = 440 mm

Smax = Imax / y = 475621.8426 / 440 = 1080.95 mm²

Smin Imin / y = 15856.98093 / 440 = 36.038 mm³ .

# CONCLUSION :

• By comparing the values we can say that section modulus of the optimized hood ( #Design 2 ) is increased by 1.87 % .
• MOI increases with the increase of section area .
• MOI increases with the increase of section length .
• If we increase the gap between the inner and outer panel of hood eventually section length will increase thus MOI will increase .