Section Modulus calculation and optimization

Aim: Use the section from your hood design & calculate the section modulus using the formula S=I/Y

Description

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.

There are two types of section modulus, the elastic section modulus and the plastic section modulus.

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 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 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:

Z P = A C y C + A T y T {\displaystyle Z_{P}=A_{C}y_{C}+A_{T}y_{T}}

the Plastic Section Modulus can also be called 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

SECTION MODULUS OF THE HOOD:

Objective: To find the section modulus of the hood and optimisation to improve the overall section modulus.

Formula of Section Modulus: It may be defined as the ration of total moment resisted by the section to the stress in the extreme fibre which is equal to yield stress.

There fore

S = I/Y, Where

S = Section modulus (mm^3)

I = Moment of Inertia of the section about the bending axis. (mm^4)

Y = Distance between the neutral axis and extreme end of the object (mm)

Area moment of Inertia:

The area moment of inertia of a beams cross sectional area measures the beam ability to resist bending. The larger the moment of inertia less will be the bend.

Where,

M = Bending moment

I = Area moment of Inertia

E = Youngs modulus

R = Radius of gyration

TOTAL DISTANCE BETWEEN NEUTRAL AXIS AND EXTREME END POINT

SECTION MODULUS 1:

MOI (max)= 4.861141614e+(mm^4)

MOI (min) = 1.349912027e+05 (mm^4)

As per the section modulus equation,

S = I/Y

I = MOI (min) value is considered

Y = Total distance /2

Total distance = 876.2395/2

= 438.16975

S = I/Y

S = 1.349912027e+05/438.16975

S =524.204

SECTION MODULUS 2 :

we have to do a offset curve of the inner panel with 0.5

MOI (max)= 4.843461226e+(mm^4)

MOI (min) = 1.367561550e+05 (mm^4)

As per the section modulus equation,

S = I/Y

I = MOI (min) value is considered

Y = Total distance /2

Total distance = 876.2395/2

= 438.16975

S = I/Y

S = 1.367561550e+05/438.16975

S =525.204