Section Modulus calculation and optimization

Section Modulus is a geometrical property which is used to calculate the bending stresses in a structural member or we can say it is the direct measure of the strength of a structural member. Bending a structural member which has a greater section modulus compared to another will be harder and stronger.

Mathematically,

It is defined as the ratio of the moment of inertia of a section about its central axis to the distance of the extreme end from the neutral axis. And is denoted by ‘S’. Also, the Section modulus is directly proportional to the moment of inertia of a structural member.

where_, I = moment of inertia (m⁴)

ymax = distance from the neutral axis to the extreme end (m)

There are two types of section modulus they are,

1. Elastic section modulus
2. Plastic section modulus

Elastic section modulus is used till the yield point of the metal behaviour. Generally, for design purposes, we use elastic section of modulus. And for different cross-section moment of inertia will be different and thus differs in section modulus.

The plastic section of modulus is used until elastic yield is acceptable and plastic behaviour is considered acceptable.

In our case plastic behaviour is not acceptable. Hence, for design purposes, we use an elastic section of modulus.

Case Study

Case-1- Cross-section of hood design

Length= 457.4152 mm

y= 457.4152/2

y= 228.7076 mm

Max MOI= 1.565762 e+07 (mm⁴)

Min MOI= 1.308955 e+05 (mm⁴)

S_max = I / Y

= 1.565762 e+07 / 228.7076

S_max= 68461.30 mm³

S_min = I / Y

= 1.308955 e+05 / 228.7076

S_min= 572.32 mm³

Case-2- By increasing the cross section by 2mm,

Length= 457.4152 mm

y= 457.4152/2

y= 228.7076 mm

Max MOI= 1.564907 e+07 (mm⁴)

Min MOI= 1.485469 e+05 (mm⁴)

S_max = I / Y

= 1.564907 e+07 / 228.7076

S_max= 68423.91 mm³

S_min = I / Y

= 1.485469 e+05 / 228.7076

S_min= 649.50 mm³

Conclusion

Thus, comparing cases 1 and 2 we can clearly see by increasing the cross-section by 2mm there is an increase in section modulus S_min. Hence, the Section modulus is directly proportional to the moment of inertia of a structural member.

*Note-Images attached for reference.