Section Modulus calculation and optimization

Section Modulus caculations and Optimization for Hood design

Aim: To determine the section modulus for the given hood design.

Section modulus: Section modulus is defined as the ratio of moment of inertia os a section about its centroidal axis and to the distance of extreme layer from neutral axis. It is also refered as polar modulus or the torsional section modulus, other geometric properties used in design include area for teension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Equation for section modulus of common shape are given as:

S= I/Y

Where,

S= Section modulus

I=Moment of inertia

Y= Distance from neutral axis to extreme layer

Mainly section modulus are divide into two types,

1.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 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 :-

2.Elastic section modulus

The elastic section modulus is defined as S = I / y, where I is the Second moment 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.

Significance of section modulus

1. Section modulus is the important factor for design of beam and flexural member.

2. Higher the valve of section modulus, higher will be the resistance of member to bending.

3. It is required to calculate stresses in beam.

4. It is used to calculate strength of the steel structure.

5. More the section modulus , it can be withstand more load and it is also considered to be more tougher.

Steps for section modulus for hood

Firstly the intersection curve was taken and than section inertia analysis was done:

Case: 1

As i designed previously hood design we got section modulus and distance has 16.662mm

Section modulus :

MOI(Max) = 9.064931099e+07(mm^4)

MOI (Min) = 2.881834965e+05(mm^4)

Y= 875.4807/2 = 437.74035mm

As we known that,

S=I/Y

S = 2.881834968e+05/437.74035

= 658.343mm^3

Case 2

Here i increased the depth of hood design for up to 5.55mm so i got section modulus as,

Section modulus:

MOI(Max) = 9.095097979e+07 (mm^4)

MOI(Min) = 2.893408944e+05 (mm^4)

S=I/Y

S= 2.893408944e+05/437.74035

= 660.987mm^3

Conclusion:

In the first case section modulus has 658.343mm^3 has a distance 16.6627mm, as in second case section modulus has 660.987mm^3 for distance about 22.0102mm.

So has we increased the depth of emboss we get more Moment of inertia and also increased the section modulus, therefore if we increase the section modulus we get more strength of the hood and can resist the better than previous design.