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.
Bending Equation Derivation
The following is the equation for simple bending:
M / I = σ /y = E / R
where
I = Moment of Inertia,
E = Modulus of Elasticity,
σ = Stress at any fibre at a distance of y from the neutral axis,
M = Bending moment
R = Radius of curvature.
we take
M / I = σ /y
M=σ*I/Y
M=σ*S
Take S= I/y
Moment of Inertia
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 is a geometrical property of an area that measures how its points are distributed with regard to an arbitrary axis, providing measures of how efficiently the cross-sectional shape can resist bending caused by loading.
The area moment of inertia is also called the second moment of inertia, second moment of area, or second area moment.
It is denoted by I.
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.
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).
Calculation:
Section Modulus :
vertical intersection
Case 1:
>Here we are considered the innerpannel section of the to analysis the section inertia analysis after doing the we got the desired results shown in the above image
I (MAX) = 8.103114877e+06(mm^4)
I (MIN) = 7.849104577e+03 (mm^4)
As per the section modulus equation,
S = I/Y
I = MOI (min) Value is considered
Y = Total distance / 2
Y=487.4152/2
Y=243.70755
Therefore,
S = 7849.104577 /243.70755
S1 = 32.20706mm^3
Case 2:
>Here we are considered an section of inner pannel at the cureved region and by doing the analysis we got the desired results shwon below
I (MAX) = 1.556584551e+07 (mm^4)
I (MIN) = 124578.097 (mm^4)
As per the section modulus equation,
S = I/Y
I = MOI (min) Value is considered
Y = Total distance / 2,
Y=487.4152/2
Y=243.70755
Therefore,
S = 124578.097/ 243.70755mm
S2=511.17865 mm^3
Results:
>After the calculations of the above 2 cases the section modulud of an S1 and S2 the difference is 478.97159
>The section modulus value of case 2 is higher than the case 1
>As we increase the section area, the moment of inertia & section modulus increase which means strength and load material also increase. Which the moment of inertia directly proportional to the strength of the material.