Section Modulus calculation and optimization

AIM

To find and improve the Sectional Modulus of the designed Hood

THEORY

SECTION MODULUS

Section modulus 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. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below.

There are two types of section moduli:

1. Elastic section modulus
2. Plastic section modulus

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,

I = the second moment of area (or area moment of inertia, not to be confused with moment of inertia)

y = the distance from the neutral axis to any given fibre

It is often reported using

y = c,

c = the distance from the neutral axis to the most extreme fibre

It is also often used to determine the yield moment (My), such that

My = S × σy,

Σy = the yield strength of the material

Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		With,	NA indicates neutral axis
doubly symmetric I-section (minor axis)			NA indicates neutral axis
Circle			Solid arrow represents neutral axis
Circular hollow section			Solid arrow represents neutral axis
Rectangular hollow section			NA indicates neutral axis
Diamond			NA indicates neutral axis
C-channel			NA indicates neutral axis

Plastic Section Modulus 

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behaviour 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:

Z_P = A_Cy_C + A_Ty_T

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:

Shape factor for a rectangular section is 1.5.

Description	Figure	Equation	Comment
Rectangular section			,
Rectangular hollow section			where: b=width, h=height, t=wall thickness
For the two flanges of an I-beam with the web excluded			Where, b1, b2 =width, t1, t2, =thickness, y1, y2 =distances from the neutral axis to the centroids of the flanges respectively
For an I Beam including the web
For an I Beam (weak axis)			d = Full height of the I Beam
Solid Circle
Circular hollow section

SIGNIFICANCE OF SECTION MODULUS

1. It is of prime importance when designing beams
2. Direct measure of strength of a beam, as higher the section modulus higher is the resistance to bending

CALCULATION OF SECTION MODULUS

We consider the Minimum moment of Inertial (S_min) for comparision purpose as it gives the Threshold value (i.e. breaking point) wherein the Hood or a another part would start crushing/bending or deforming under load and is the minimum moment of inertia that a section can withstand.

1. Initial Design of Hood

I_max = 5074315.13 mm⁴

I_min = 15986.29 mm⁴

y = 440.8mm

S_max = I_max / y = 5074315.13 / 440.8 = 11511.60 mm³

S_min = I_min / y = 15986.29 / 440.8 = 36.27 mm³

2. Optimized Hood design

Section modulus is defined by

S=I/y, where I is the second moment of inertia and y is the distance from the neutral axis to the most extreme fibre.

The second moment of inertia is directly related to the area of material in the cross-section and the displacement of that area from the centroid.

Thus, Section Modulus is directly proportional to the sectional area of the body and increasing the cross-sectional area increases the section modulus thus increasing the resistance of the flexible part to load and bending.

1. Increase of Outer Panel thickness from 0.75mm to 1mm

I_max = 5090761.07 mm⁴

I_min = 16025.03 mm⁴

y = 440.8mm

S_max = I_max / y = 5090761.07 / 440.8 = 11548.91 mm³

S_min = I_min / y = 16025.03 / 440.8 = 36.35 mm³

2. Along with the increase in the Outer Panel thickness, decreasing emboss distance by 1mm, from 12.5mm to 11.5mm and offsetting the outer panel by 2mm, from 0mm to 2mm

I_max = 5089748.79 mm⁴

I_min = 16617.29 mm⁴

y = 440.8mm

S_max = I_max / y = 5089748.79 / 440.8 = 11546.61 mm³

S_min = I_min / y = 16617.29 / 440.8 = 37.69 mm³

CONCLUSION

• The S_min was increased by 3.9% from 36.27 mm³ to 37.69 mm³ by design optimization of the Hood
• It is also calculated and found that Sectional Modulus increases with increase in cross-sectional area
• Also, the Sectional Modulus of the Hood can be increased by increasing the distance between the outer and inner panel