Section Modulus calculation and optimization
The section modulus is a geometric property of a cross-sectional shape of a structural member, typically used in mechanical and civil engineering. It represents the ability of a member to resist bending or flexure, and is defined as the ratio of the moment of inertia of the cross-sectional area about the neutral axis to the distance from the neutral axis to the most distant point in the section.
The section modulus is often denoted by the symbol "S" and is calculated using the formula:
S = I / y
where I is the moment of inertia of the cross-sectional area about the neutral axis, and y is the distance from the neutral axis to the most distant point in the section. The unit of section modulus is typically expressed in cubic millimeters (mm^3) or cubic inches (in^3) in the US customary system.
The section modulus is an important parameter in the design and analysis of structural members, as it provides a measure of the resistance of a member to bending and helps to ensure that the member can support the applied loads without failure.
S=I/Y
Here,
S = Section modulus
I= Moment of Inertia (Unit mm 4)
Y= Distance from Neutral axis to the extreme end (unit mm)
The Unit of a “section modulus” is (mm3)
LOAD CHANGE POSITION :
Changing the position can show the difference in bending which is because of the centroid distance from extreme fiber end.
In Layman's terms, in this position there is more material from axis to the specific direction, to resist the load which makes less bend amd improves the strength.
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 (My) such that My = S × σy, where σy 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:
the Plastic Section Modulus can also be called the 'First moment of area'
The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, Fy, by Mp=Fy*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 :-
DESIGN STUDY 1
The Intersection of the design was taken and then section Inertia Analysis was done.
CALCULATON
Moment of inertia (max) = 4.87174*10^6*mm^4
Moment of inertia (min) = 1.54807*10^4*mm^4
Y =876.633/2
= 438.317
S = I/Y
S = 1.54807*10^4/438.317
S=35.318 mm^3
DESIGN STUDY 2
The Intersection of the design was taken and then section Inertia Analysis was done with off-setting 0.5mm surface.
CALCULATON
Moment of inertia (max) = 4.86390*10^6*mm^4
Moment of inertia (min) = 1.57116*10^4*mm^4
Y =876.633/2
= 438.317
S = I/Y
S = 1.57116*10^4/438.317
S=35.867 mm^3