Section Modulus calculation and optimization

Section Modules Calculation

Aim: - To calculate and compare the section modulus of the hood design.

Introduction

Section modulus is a geometric property for a given cross-section used in the design of beam or flexural member, it I 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.

S = Section Modulus Section modulus can be calculate using a formula

S=I/Y

Where,

I = Moment of Inertia

Y = Distance from the neutral axis to the most extreme end of the object

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.

Section Modules are normally two types

1. Elastic Section Modulus
2. Plastic Section Modulus

1. Elastic 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 often reported using y = c, where c is the distance from the neutral axis to the most extreme 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.

1. 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.

Zp = Acyc + A

Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section

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.

k = Z/S

Case 1: Initial Hood Design

S = I/Y

Imax = 8.2869 x 10^5 mm^4

Imin = 6.1289 x 10^3 mm^4

Y = 479.42 mm

S (max) = 8.2869x10^5/479.42 = 1728.53 mm^3

S (min) = 6.1289x10^3/479.42 = 12.78 mm^3

Case 2: Hood design optimization

By increasing the gap between outer and inner panel gap is 2 mm.

S = I/Y

Imax = 8.6904 x 10^5 mm^4

Imin = 6.9656 x 10^3 mm^4

Y = 479.42 mm

S (max) = 8.6904 x 10^5 /479.42 = 1812.69 mm^3

S (min) = 6.9656 x 10^3/479.42 = 14.52 mm^3

Case 3: Increasing the gap between outer and inner panel gap is 3 mm as per the actual master section.

S = I/Y

Imax = 8.8946 x 10^5 mm^4

Imin = 7.4239 x 10^3

Y = 479.42 mm

S max = 8.8946 x 10^5 /479.42 = 1866.96 mm^3

S min = 7.4239 x 10^3/479.42 = 15.48 mm^3

Conclusion

In the above study can see that moments of inertia are changed then the section modulus is changed. As show that increasing the gap between the inner and outer panel then Moment of inertia and section modulus is increase.