Section Modulus calculation and optimization

Section Modulus Calculation and Optimization

Section Modulus:

Section Modulus is generally calculated for the extreme tensile or compressive fibers in a bending body, compression is often the most critical case due to onset of flexural torsional (F/T) buckling. Generally (except for brittle materials like concrete) tensile extreme fibers have a higher allowable stress or capacity than compressive fibers.

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 in Car Hood Design:

1. Section Modulus will give idea of volume of body hence helps in optimum use of raw material or sheet.
2. 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.{Can be done by calculating elastic yield moment for beam:M_y = S ⋅ σ_y, where σ_y is the yield strength of the material and M_y is yield moment}.
3. 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 is used to calculate the plastic moment, M_p, or full capacity of a cross-section.

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)

Calculation of Section Modulus of designed car hood:

Trim the body to take view of cross section of Car Hood

Cross-section of Car Hood

Section report using command Section Inertia Analysis of NX CAD

Calculation of Section Modulus

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

so,

Imax = 8.0881*10^5 [mm^4]

Imin = 4.0322*10^3 [mm^4]

y = 234.0016 mm

Zmax = Imax/y

Zmax = 3456.429 [mm^3]

Zmin = Imin/y

Zmin = 17.2315 [mm^3]

Optimization of Design of Hood:

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

For that purpose, we will offset the outer panel of our hood by 2mm to increase the sectional area of our hood and thereby increase in section modulus.

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.

Sectional Report for Optimized Car Hood Design

Calculation of Section Modulus for Optimized Design

Imax = 8.8142*10^5 [mm^4]

Imin = 5.6566*10^3 [mm^4]

y = 236.0016 mm

Zmax = Imax/y

Zmax = [mm^3]

Zmin = Imin/y

Zmin = [mm^3]

Conclusion from optimization of Car Hood

Hence the section modulus of our hood has been increased as compared with the previous value which is 3456.429 [mm^3] and the new value is 3734.8052[mm^3].

Percentage increase in section Moudulus= 8.05%.

1. 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.
2. 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.
3. Importance of section modulus when it comes to the selection of right material or right design which will have more strength.
4. Factors affecting Section modulus and how to enhance the strength of a material by using section modulus as a check parameter.