Section Modulus calculation and optimization

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 second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question.

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

ELASTIC SECTION MODULUS :-

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, 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.

Cross-sectional shape	Figure	Equation	Comment
Rectangle		�=�ℎ26	Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		��=��26−�ℎ36� ��=���, with �=�2	NA indicates neutral axis
doubly symmetric I-section (minor axis)		��=�2(�−ℎ)6+(�−�)3ℎ6�[4]	NA indicates neutral axis
Circle		�=��332[3]	Solid arrow represents neutral axis
Circular hollow section		�=�(�24−�14)4�2=�(�24−�14)32�2	Solid arrow represents neutral axis
Rectangular hollow section		�=��26−�ℎ36�	NA indicates neutral axis
Diamond		�=��224	NA indicates neutral axis
C-channel		�=��26−�ℎ36�	NA indicates neutral axis

PLASTIC SECTION MODULUS OF INERTIA :-

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:

��=����+����

First Design:-

The Intersection of the design was taken and then section Inertia Analysis was done-

1. Maximum Moment Of Inertia - 9.222X10^7 mm^4
2. Minimum Moment Of Inertia - 2.9187X10^05 mm^4

SECTION MODULUS (S):- I/y

I - Moment of inertia 

y- Distance from Neutral axis to the extreme end

Y=875.33/2=437.675

SECTION MODULUS (S)=I/Y=2.9187X10^05 mm^4/437.675=666.75