Section Modulus calculation and optimization

Brief explanation about Section Modulus:

Section Modulus: Geometric property

It is defined as the ratio moment of inertia of an section about its centroidal axis and to the distance of extreme layer from neutral axis.

It is give idea about how much load take by structure.

Section modulus increase, Momentum of resistance increases and load capacity.

Important of section modulus:

• Safety
• Economically

Section modulus is also expressed as the ratio of bending moment (M) to the bending stress(fmax) of a given object within the elastic limit.

2.Section modulus (Z) of Hollow Rectangular cross section of outer and inner breadth B,b and depth D,d as shown in figure.

3.Section modulus (Z) of a solid circular cross section of diameter d

4.Section modulus (Z) of a Hollow circular cross section of outer diameter D and internal diameter d

Mathematically:

Z = I/y Where,

Z = Section Modulus

I = Moment of Inertia

y = distance between the neutral axis and the extreme end of the object

Example Q) A simply supported beam of 8mm effectively carry uniform distribution load 2knm, permissible bending stress 160 mpa, then suitable size.

• a) Solid circle area of cross section 9500 Mm² & section modulus 1306603 Mm³
• b) Solid rectangle area of cross section 6000 Mm² & section modulus 10000 Mm³
• c) T area of cross section 3900 Mm ² & section modulus 105780 Mm³
• d) I area of cross section 2900 Mm² & section modulus 129100 Mm³

1.Calculation ratio of maximum bending moment.

M=WL/8

Where L=lenght of beam 8mm

W=Load 2knm

M=2x8²/8

M=16kn

2.Calculation of section modulus

Z=M/Fmax

Fmax=Permissible bending stress 160 mpa

Z=16x10⁶/160

Z=10⁵MM²

Where Z value indicates safety value (Decrease in Z value less safety & Increase in Z value more safety)

From: From above option except (b) all other option passes safety criteria.

3.Decide shape based on cross section decrease in area value for economically and same section should pass above or equal to Z value for safety requirement.

Result: I area of cross section 2900 Mm² & section modulus 129100 Mm³

CASE NO 1: Use the Section from your Hood design and calculate the section modulus

Y=877.2/2= 438.6 mm

Moment of inertia(max) = 4.83524×10^5×mm^4

Moment of inertia(min) = 1.52899×10^3×mm^4

Z=I/Y

Z=1.52899×10^3/438.6= 3.48606mm^3

CASE NO 2: After optimization by increasing distance between inner panel & outer panel.

Moment of inertia(max) = 5.86788×10^5×mm^4

Moment of inertia(min) = 1.82400×10^3×mm^4

Z=I/Y

Z=1.82400×10^3/438.6= 4.15868mm^3

CONCLUSION:-

It is observed that increasing the gap between the inner and outer panels has resulted to improve the section modulus and strength of the material is directly proportional to the moment of inertia.

( Momentum of inertia increases then strength of hood panel increases, Area of cross increases then moment of inertia & section modulus also increases)

Note: Strength of the material also depends upon the composition of material and thickness of an object.