Section Modulus calculation and optimization
Section Modulus calculation and optimization on Hood
AIM: To calculate the section modulus of the designed hood and optimize the design of hood until to get a better section modulus.
Section Modulus: Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus.
It is termed as the ratio of the second moment of inertia and distance from Neutral Axis to the extreme fibre. Also, it is the measure of the strength of a given member. The stress in the outermost section of the beam is computed with the help of section modulus. It is indicated by S.
Section Modulus S = I/Y
S = Section Modulus
I = Moment of Inertia
Y = Distance between the neutral axis and the extreme end of the project.
Elastic Section Modulus:
For general design, the elastic section modulus is used, applying up to the yield point for most metals and other common materials.
The elastic section modulus is defined as S = I / y, where I is the second moment of inertia (or area moment of inertia, not to be confused with moment of inertia) and y is the distance from the neutral axis to any given fiber. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fiber, as seen in the table below. It is also often used to determine the yield moment (My) such that My = S ⋅ σy, where σy is the yield strength of the material.
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 elastic limit of material. This could be shown mathematically with the formula.
K = Z/S
Shape factor for a rectangular section is 1.5.
CASE: 1
Calculation of Section Modulus to Initial hood design:
Moment of Inertia (Max) = 1.749366249e+08 [mm^4]
Moment of Inertia (Min) = 5.516342850e+05 [mm^4]
2Y = 877.3895 mm, Y = 877.3895/2 = 438.69475 mm
S = I/Y = 5.516342850e+05/438.69475 = 15.006 mm^3
S = 15.006 mm^3
CASE: 2
Calculation of Modulus by increasing the 1.5mm depth/offsetting to 1.5mm at outward direction to the initial hood design:
Increased the distance between Outer hood panel and Inner hood panel.
Moment of Inertia (Max) = 1.374036916e+08 [mm^4]
Moment of Inertia (Min) = 4.157929022e+05 [mm^4]
2Y = 877.3895 mm, Y = 877.3895/2 = 438.69475 mm
S = I/Y = 4.157929022e +05/438.69475 = 11.313 mm^3
S = 11.313 mm^3
Conclusion: Section Modulus is directly dependent on the area moment of inertia and the distance from the neutral axis. Hence to improve the geometry of the carry more bending load, the section modulus can be improved by increasing the area of Moment of inertia of a given section. Thus, the strength of the Car Hood has been increased by increasing its depth.