Section Modulus calculation and optimization
Section Modulus:
Section Modulus is a geometric property used to calculate the bending stresses in a structural member.
Mathematically,
It is defined as the ratio of the moment of inertia of a section about its centroidal axis and to the distance of the extreme layer from the neutral axis.
It is denoted by Z.
Z = I/ymax
Mathematically By bending stress equation,
M/I = σmax/ymax
M = σmax × (I/ymax)
M = σmax × Z
Where,
I = Moment of Inertia(m⁴)
y = distance from NA to extreme layer(m)
The unit of section modulus is m³.
Types of Section Modulus
There are two types of section modulus:
- Elastic Section Modulus
- Plastic Section Modulus
Elastic Section Modulus
Elastic section modulus is calculated when we are to use the material behavior till the yield point. For general design purposes, we use elastic section modulus.Generally, elastic section modulus is denoted by Zₑ.
So,
Zₑ = I/ymax
Where,
I = Moment of inertia
ymax = Distance from NA to extreme layer
Plastic Section Modulus
The plastic section modulus is used for materials where the elastic yield is acceptable and the plastic behavior is considered to be an acceptable range.
It depends on the location of the plastic neutral axis. The PNA is the axis that bisects the cross-section such that the compressive force from the region in compression is equal to the tension force from the region under tension. So, for sections with a constant yield stress, the area above and below the PNA will be equal, but for mixed sections, this is not necessarily the case.
Mathematically,The plastic section modulus is the sum of the areas of cross-sections on each side of the plastic neutral axis that may or may not be equal and multiplied by the distance from the local centroid of the two regions to the plastic neutral axis.
Zₚ = (Ac × yc) + (Aₜ + yₜ)
Where,
Ac = Area of cross-section under compression.
Aₜ = Area of cross-section under tension.
yc = Distance from the plastic neutral axis to the center of gravity of the areas under compression.
yₜ = Distance from the plastic neutral axis to the center of gravity of areas under tension.
Section Modulus Calculation:
Section 1:Along YZ axis
Area Moment of Inertia MOImin=5.59389936e+03 mm^4
Y=478mm/2=239(Distance from NA axis.Hence,it is divided into halve)
S=MOImin/Y
S=5.59389936e+03/239=23.405436 mm^3
Smax=23.405436mm^3
Section 2:Along YZ axis
By modifying the second emboss,another section is formed.
MOI min=5.480281202e+03
Y=478mm/2=239
S=5.480281202e+03/239=22.9300468
S=22.9300468mm^3.
Strength is reduced while modifying the second emboss by splitting it.
Section 3:Along YZ axis
in section 3,second emboss applied on inner hood is reduced.
MOImin=6.211112989e+03mm^4
Y=478mm/2=239
S=6.211112989e+03/239=25.9879204
S=25.9879204mm^3
Section 4:Along XY axis
MOI min=1.455714115e+04
Y=878mm/2=439
S=1.455714115e+04/439=3.315977
S=3.315977mm3
Conclusion:
Among the all sections,Reduced section of the second emboss gives more strength.Section modulus of the section 3 is higher than other two.From results,We can conclude that area of the section increases section modulus which results in increasing strength.Section modulus can also be increased by reducing distance from the neutral axis fiber.Comparing all these 4 sections,Section along XY axis has more cross sectional but also distance from Netural axis is high.Strength will be less comparatively sectiong along YZ axis.From the analysis,Processes like emboss should be modified along YZ axis of the hood that give more strength to hood.