Section Modulus calculation and optimization

Section modulus calculation and optimization

Aim –

Section modulus-

• Section modulus is the geometric property of the given cross-section of any shaft or beam structure. It is the directly measure of the strength of the shaft section modulus of a structure, the more the resistive it become to bending.
• Section module is a geometric property for a given cross-section used in the design of beams or flexural member. 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.
• Y=I/Y
• Where
• Section modulus(mm^4)
• I = moment of inertia of the section about the bending axis
• Y= distance between the neutral axis and extreme end of the object(mm)

*Moment of inertia- the moment of inertia is the capacity of a cross section to resist bending when it is subjected to an external load.

*moment of inertia is directly proportional to the strength of the material, which can be calculated through section modulus.

• M= Bending moment
• I= Area moment of inertia
• E= Youngs modulus
• R= Radius of gyration

• Neutral axis-

The neutral axis for a beam is defined as the line in the cross-section where there is neither longitudinal compression nor tensile stress. It is commonly accepted in the analysis of wood beams that the neutral axis coincides with the centroid of the beam

• Type of section modulus-

• Elastic section module
• Plastic section module

• Elastic section module-

For general design, the elastic section modulus is used, applicable up to the yield point for most metals and other common materials.

- The elastic section modulus is defined as S = I / y,

The elastic section modulus is defined as S = I / y, where I is the the second moment of area (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 fibre.

- It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often used to determine the yield moment (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yield strength of the material.

• Plastic section module –

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, and equal compressive and tensile 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, M_p, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by M_p = F_y ⋅ 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

�=��

Difference between elastic and plastic modulus

Elastic Section Modulus 1-Ze is the Elastic Section Modulus 2-Resisting moment My = fy x Zy 3-The elastic section modulus assumes the section remains elastic. 4-If the failure is considered till the elastic zone then we consider elastic modulus. 5-Elastic modulus is the steel modulus based on the stress strain curve before yielding. 6-Ze is used to determine the maximum bending moment up to the point where the extreme fiber has yielded elastically. All other fibers will be below their elastic yield stress.

Plastic Section Modulus 1-Zp is the Plastic Section Modulus 2-Resisting moment Mp = fy x Zp 3-The plastic section modulus assumes the entire section yields. 4-If the failure is considered till the plastic zone (breaking point) then we consider plastic modulus 5-The plastic modulus is after yielding. 6-Zp is used to determine the maximum bending moment at the point where all fibers have yielded elastically and the entire section is behaving plastically. This is the point when the so-called plastic hinge has developed.

Steps to calculate section modulus: -

• Step1= find location of centroid of given cross section.
• Step 2= find moment of inertia about neutral axis
• Step 3=find elastic Ymax , Ymin from neutral axis.
• Step 4= find elastic section modulus using the formula provided above.

• Here the section modulus of hood design before optimization, the moment of inertia is calculated by the NX software.
• Imax- 34556548*10^5(mm^4)
• Imin- 3.25225253*10^3(mm^4)
• Y= total length / 2

= 486.13/2

=243.065mm

Section modulus for max=8.34556548*10^5/243.065

=3433.4706

section modulus for min=3.25225253*10^3 /243.065

=179.029

Conclusion: - section modulus of hood of the car was calculated and optimized section modulus, with offset the section area 0.5mm and found some better result.