Section Modulus calculation and optimization

Section Modulus.

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

It 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.

There are two types of section moduli, the elastic section modulus and the plastic 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 moment of inertia) and y is the distance from the neutral axis to any given fibre.

It is also often used to determine the yield moment (M_y) such that M_y = S × σ_y, where σ_y is the yield strength of the material which is maximum allowable load in a mechanical component.

The Plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an acceptable limit.

Formula for calculating Section Modulus:

S or Z = I/y

S or Z - Section modulus of material

I- Moment of inertia

y- Length of extreme fiber from the neutral axis

For symmetrical sections the value of Z is the same above or below the centroid.

For asymmetrical sections, two values are found: Z max and Z min.

Significance of Section Modulus:

• The section modulus of the cross-sectional shape is of significant importance in designing beams. It is a direct measure of the strength of the beam. A beam that has a larger section modulus than another will be stronger and capable of supporting greater loads.
• In case there are two beams having same shape and probably made up of same material then how we can decide which to beam should be used. Here we can evaluate them by calculating the section modulus of both as the section modulus will provide us the strength of each beam. The beam having more section modulus will carry more load in bending i.e provides more resistance to bending.

As, we know that

`

where:

σ_max is the maximum stress at the farthest surface from the neutral axis (it can be top or bottom)

M is the bending moment along the length of the beam where the stress is calculated

if the maximum bending stress is required then M is the maximum bending moment acting on the beam

I_x is the moment of inertia about x (horizontal) centroidal axis

c is the maximum distance from the centroidal axis to the extreme fiber (again, this can be to the top or bottom of the shape)

Z_x is called section modulus and is a term that combines the moment of inertia and the distance to the extreme fiber (Z_x = I_x / c).

#Section 1: Section modulus of initial Hood.

As hood shape is asymmetrical there will be two values of section modulus,

Imax =5.65061 X 10^6 mm^4

Imin= 2.001388 X 10^4 mm^4

y= 465.3885 mm

Zmax= Imax/y

= 5.65061 X 10^6/465.3885

= 12141.70526 mm^3

Zmin= Imin/y

=2.001388 X 10^4/465.3885

= 43.004672 mm^3

This is the result of section modulus our initial hood design.

#Section 2: Section modulus of modified Hood.

To improve this value of section modulus we need to optimize our present hood design.

The reason behind providing offset to the outer panel

As we know that section modulus

S= Σmi*Ai^2/y

where 'A' is the area

'M' is the mass of the body

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

That's why we can increase the section modulus by increasing the sectional area by offsetting the outer panel and reducing the emboss depth.

As the section area will increase Moment of inertia will also increase thereby Section Modulus of our hood will increase.

as we can see the offset has increased in this section.

Here in the above sectional report we can see that section area is increased as compared to previous section area in old design of our hood.

Also the Moment of inertia has been significantly changed as it is function of section area.

Now we will calculate the section modulus

Imax= 6.06017 X 10^6 mm^4

Imin= 2.5307 X 10^4 mm^4

y= 465.3885 mm

Zmax= Imax/y

= 6.06017 X 10^6/465.3885

= 13021.7442 mm^3

Zmin= Imin/y

=2.5307 X 10^4/465.3885

= 54.3782 mm^3

Hence the section modulus of our hood has been increased as compared with the previous value which is 12141.70526 mm^3 and the new value is 13021.7442 mm^3.

CONCLUSION:

• Hence, from the above study, it is clear that optimization helps to improve the results in a design as we have increased the Section Modulus of our hood.
• Gain crucial insight about Section Modulus its relationship with bending stress, Moment of inertia and got an idea of how the section modulus changes with respect to change in section area of a body and its effect on the value of section modulus.
• Importance of section modulus when it comes to the selection of right material or right design which will have more strength.
• Factors affecting Section modulus and how to enhance the strength of a material by using section modulus as a check parameter.