Section Modulus calculation and optimization
Aim:
To find the section modulus of the hood and optimization to improve the overall section modulus.
Objective:
Use the section from your Hood design and calculate the section modulus using the formula S= I/Y.
Section Modulus:
The section modulus (S) is geometry property of the cross section used for designing beams and flexural members. It does not represent anything physically.
Formula of section Modulus: It may be defined as the ratio of total moment resisted by the section to the stress in the extreme fibre which is equal to yield stress.
Therefore
S=I/Y, Where
S= Section modulus (mm^3)
I = Moment of inertia of the section to the stress in the extreme end of the object (mm^4)
Y= Distance between the neutral axis and extreme end of the object (mm).
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation. Or in more simple terms, it can be described as a quantity that decides the amount of torque needed for a specific angular acceleration in a rotational axis. Moment of Inertia is also known as the angular mass or rotational inertia. The SI unit of moment of inertia is kg m2.
Moment of inertia is usually specified with respect to a chosen axis of rotation. It mainly depends on the distribution of mass around an axis of rotation. MOI varies depending on the axis that is chosen.
In General form,
Moment of Inertia is expressed as I = m × r2
where,
m = Sum of the product of the mass.
r = Distance from the axis of the rotation.
Area Moment of Inertia:
The area moment of inertia of a beam cross section area measures the beam ability to resist bending. The larger the moment of inertia less will be the bend.
Where,
M = Bending Moment
I = Area moment of Inertia
E = Young modulus
R = Radius of gyration
Neutral Axis: When beam is subjected to pure bending the top most fibres of the beam are subjected to maximum compression, while the bottom most fibre of the beams are subjected to maximum tension.
- This picture contains the following scene, Diagram of a beam on its narrow edge supported at both ends. A man is walking on a straight beam. Diagram of a beam resting on its flat side supported at both ends. A man is walking on the straight beam and its bending in the middle.
Calculation:
Section Modulus :
Case 1:
I (MAX) = 9.010852204 x 10^07(mm^4)
I (MIN) = 2.976273428 x 10^05 (mm^4)
As per the section modulus equation,
S = I/Y
S = Section Modulus
I = MOI (min) Value is considered
Y = Total distance / 2
Total distance or Distance between the neutral axis and the extreme end of the object = 876.4582 mm
Y= 876.4582 / 2
Y= 438.2291 mm
Therefore,
S = 29762.73428 mm^4 / 438.2291 mm
S1 = 66.9159 mm^3
'S1' value for case 1 is 66.9159 mm^3.
Case 2:
I (MAX) = 8.994325259 x 10^07(mm^4)
I (MIN) = 3.017350898 x 10^05 (mm^4)
As per the section modulus equation,
S = I/Y
S = Section Modulus
I = MOI (min) Value is considered
Y = Total distance / 2
Total distance or Distance between the neutral axis and the extreme end of the object = 876.4582 mm
Y= 876.4582 / 2
Y= 438.2291 mm
Therefore,
S = 30173.50898 mm^4 / 438.2291 mm
S2 = 68.8532 mm^3
'S2' value for case 1 is 68.8532 mm^3.
Results:
- After the calculations of the above 2 cases the section modulud of an S1 and S2 the difference is 1.9373 mm^3.
- The section modulus value of case 2 is higher than the case 1.
Conclusion:
Strength of material is directly proportional to the moment of interia, thus increasing the sectional area increase the increases strength of component.