Section Modulus calculation and optimization
Section Modulus Calculation and Optimization of Hood
Objective:
Calculate the section modulus of the previously designed car hood by taking a cross-section and analyzing its strength. Optimizing the design that would increase/improve the section modulus of the existing car hood.
Introduction:
Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. The section modulus of the cross-sectional shape is of significant importance in designing beams or flexural members. It is a direct measure of the strength and signifies that a larger section modulus than another will be stronger and capable of supporting greater loads. Other geometric properties used in the design include an area for tension and shear, the radius of gyration for compression, and the moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape in question.
There are two types of section moduli, the elastic section modulus, and the plastic section modulus.
1. Elastic Section Modulus
The elastic section modulus is defined as,
It is defined as,
S = I / y,
Where S is the Section modulus (or Elastic Section Modulus), 'I' is the moment of inertia (or the second moment of area), and 'y' is the distance from the neutral axis to any given fiber (distance between the neutral axis and the extreme end of the object).
For symmetrical sections, the value of S is the same above or below the centroid.
For asymmetrical sections, two values are found: S max and S min.
2. Plastic section modulus
The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an acceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparing the plastic capacity against amplified forces or stresses.
The plastic section modulus depends on the location of the plastic neutral axis (PNA). The PNA is defined as the axis that splits the cross-section such that the compression force from the area in compression equals the tension force from the area in tension. So, for sections with constant yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.
The plastic section modulus is the sum of the areas of the cross-section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:
The Plastic Section Modulus is not the 'First moment of area'. Both relate to the calculation of the centroid, but Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross-section, while the First moment of area is calculated based on either side of the "considering point" of the cross-section and it is different along the cross-section and depends on the point of consideration.
The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, Fy, by Mp = Fy ⋅ Z. Plastic section modulus and elastic section modulus are related by a shape factor which can be denoted by k, used for an indication of capacity beyond the elastic limit of the material. This could be shown mathematically with the formula,
The shape factor for a rectangular section is 1.5.
Calculation:
y = 887.9788/2 = 443.9894 mm
Case 1:
MOI (max) = I (max) = 4.984844997e+06 mm^4 = 4.984844997x10^6 mm^4.
MOI (min) = I (min) = 1.443924216e+04 mm^4 = 1.443924216x10^4 mm^4.
We are going to take the MOI (min).
y = 443.9894 mm
S = I/y
= I (min)/ y
= 1.443924216x10^4/443.9894 mm^3
S = 32.5215 mm^3
Case 2:
To optimize and check the value of section modulus, we will offset the inner panel of the hood by 0.5 mm.
MOI (max) = I (max) = 4.975453023e+06 mm^4 = 4.975453023x10^6 mm^4.
MOI (min) = I (min) = 1.462499590e+04 mm^4 = 1.462499590x10^4 mm^4.
We are going to take the MOI (min).
y = 443.9894 mm
S = I/y
= I (min)/ y
= 1.461379430x10^4 /443.9894 mm^3
S = 32.9399 mm^3
Case 3:
To optimize and check the value of section modulus, we will offset the inner panel of the hood by 0.5 mm and the embossed region of the inner panel by 1 mm.
MOI (max) = I (max) = 4.970725392e+06 mm^4 = 4.970725392x10^6 mm^4.
MOI (min) = I (min) = 1.461377385e+04 mm^4 = 1.461377385x10^4 mm^4.
We are going to take the MOI (min).
y = 443.9894 mm
S = I/y
= I (min)/ y
= 1.493889108x10^4 /443.9894 mm^3
S = 32.9146 mm^3
Case 4:
To optimize and check the value of section modulus, we will offset both the inner and outer panels of the hood by 0.5mm.
MOI (max) = I (max) = 5.009449905e+06 mm^4 = 5.009449905x10^6 mm^4.
MOI (min) = I (min) = 1.493889108e+04 mm^4 = 1.493889108x10^4 mm^4.
We are going to take the MOI (min).
y = 443.9894 mm
S = I/y
= I (min)/ y
= 1.493889108x10^4 /443.9894 mm^3
S = 33.6469 mm^3
Results:
Case 1, S = 32.5215 mm^3
Case 2, S = 32.9399 mm^3
Case 3, S = 32.9146 mm^3
Case 4, S = 33.6469 mm^3
We obtained two optimized cases with higher section modulus.
Conclusion:
The section modulus was calculated and optimized. An increase in section modulus was obtained. As section modulus is the direct measure of the strength of the hood(Steel/Aluminium). Bending a Steel/Aluminium section that has a larger section modulus than another will be stronger and harder to bend.