Section Modulus calculation and optimization
Aim: To Calculate the Section Modulus of the Hood section.
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.
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).
Steps to calculate elastic section modulus:-
Step 1:-Find location of centroid of given cross section.
Step 2:-Find moment of inertia about neutral axis
Step 3:-Find Ymax from neutral axis.
Step 4:-Find elastic section modulus using the formula provided above.
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:
The centermost layer of any section of a component is called the Neutral axis.
Neutral Axis doesn't get affected by any kind of load that is perpendicular to it which means there is no stress or strain on the neutral axis.
The range of stress varies with respect to the distance from the neutral axis.
The below figure shows the Neutral axis(dotted line) and in the Load situation the stress on either side of the neutral axis.
SECTION MODULUS:
Section modulus is defined as the ratio of the moment of inertia of a section about the neutral axis to the distance of the outermost layer from the neutral axis. It is denoted by the symbol S. Hence mathematically section modulus is given by,
S=I/Y-max
where
I=Moment of Inertia about the neutral axis and
Y-max = Distance of the outermost layer from the neutral axis.
we know Max Bending Moment(M)/ moment of inertia(I) = Stress(σ)/strain(Y)
The stress σ will be maximum when y is maximum. Hence above equation can be written as
M/ I = (σ-max)/(Y-max)
M= σ max*(I/Y-max)
But
I/(Y-max)=Z
M = σ max * Z
In the above equation, M is the maximum bending moment (or moment of resistance offered by the section). Hence moment of resistance offered by the section is maximum when section modulus Z is maximum. Hence section modulus represents the strength of the section.
Note: Section Modulus varies with respect to the shape of the cross-section.
Factors that influence the section modulus:
Area - Area is directly proportional to section modulus
Thickness of sheet metal- Thickness is also directly proportional to the section modulus
Distance between the outermost layer to the neutral axis called Y. Y is inversely proportional to section modulus.
Cross-sectional depth of the component - It is directly proportional to the sectional modulus.
Section Modulus Analysis using UGNX:
Procedure for an increase in the area:
Case-1: Increasing the area by offsetting the outermost layer.
At Zero Offset:
- Select the Part body on which we have to do analysis and create a cross-section of the part and make sure that the cross-section sketch is a closed profile.
- Perform Section Inertia Analysis over the section that we have created and there we can find the result as shown below
- Create a CG point by using the point command and measure the distance between the point on the outermost layer and the CG point by using the measure command in analysis.
- Here the reason to take Y parallel to the x-axis is if we take the loading condition perpendicular to the x-axis then the Y value also increases as the Area increases but we have to keep one parameter constant for that we took the loading condition parallel to X-axis(this condition can be taken on a side impact).
- The Distance we got here is 879/2=439.5mm
- According to the formula For section modulus
- S = I/Y
- Where,
- S = Section Modulus
- I = Moment of Inertia
- y = distance between the neutral axis and the extreme end of the object
- MOI min/Y = (1.559999889x10^4)/439.5 = 35.4948780205mm^3
- MOI max/Y = (4.818702145x10^6)/439.5 = 10964.0549374 mm^3
- Section Modulus at MOI min = 35.4948780205 mm^3
- Section Modulus at MOI max = 239726.396161 mm^3
- At 0.5 mm Offset:
- MOI min/Y = (1.583204158x10^4)/439.5 = 36.0228477361 mm^3
- MOI max/Y = (4.891673592x10^6)/439.5 = 11130.0878089 mm^3
- Section Modulus at MOI min = 36.0228477361 mm^3
- Section Modulus at MOI max = 11130.0878089 mm^3
- At 1 mm Offset:
- MOI min/Y = (1.602591273x10^4)/439.5 = 36.463965256 mm^3
- MOI max/Y = (4.802549497x10^6)/439.5 = 10927.3026098mm^3
- Section Modulus at MOI min = 36.463965256 mm^3
- Section Modulus at MOI max = 10927.3026098 mm^3
-
S.No Case Y(mm) Section modulus at min MOI (mm^3) 1 At Zero Offset 439.5 35.49487802 2 At 0.5 mm Offset 439.5 36.01019638 3 At 1 mm Offset 439.5 36.46396526
Result: From the above analysis we can say that the section modulus increases with an increase in offset value nothing but an increase in Cross-sectional area.