Section Modulus calculation and optimization

AIM:

To calculate the section modulus of a self designed hood and analyzing its strength and also by optimizing the design to identify the bending strength of the hood. Higher the section modulus of as structure, the higher it becomes ressistive towards bending.

SECTION MODULUS:

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. 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.

S = I/y

Where,

S = Section Modulus

I = Moment of Inertia

y = distance between the neutral axis and the extreme end of the object.

Elastic section Modulus:

The elastic section modulus is defined as S = I / y, where I is the second moment of area 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 Modulus:

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an aceptable limit. Design 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'.

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 :-

Shape factor for a rectangular section is 1.5.

FIRST SECTION MODULUS:

Moment of Inertia (MAX): 8.31649x10^5xmm^4

Moment of Inertia (MIN):5.483120x10^3xmm^4

Y = 484.6971/2 = 242.34855mm

S = I/Y = 5.483120x10^3 / 242.34855 = 22.625 mm^3

HOOD DESIGN OPTIMIZATION:

By increasing the depth of the inner panel by 0.5mm

Moment of Inertia (MAX): 8.291809x10^5xmm^4

Moment of Inertia (MIN): 5.669382x10^3xmm^4

Y = 484.6971/2 = 242.34855mm

S = I/Y = 5.669382x10^3 / 242.34855 = 23.3935 mm^3

CONCLUSION:

From the above images the difference in moment of inertia and sectional modulus and can be identified and it is achieved by changing the inner panel depth.