Section Modulus calculation and optimization

#Question :

Use the Section from your Hood design and calculate the Section Modulus using the formula S = I/y ?

Where,

S = Section Modulus

I = Moment of Inertia

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

#Answer :

Theory

Section Modulus :

What is Section Modulus ?

• Ratio of total moment resisted by the section to the stress in the extreame fibre.
• It gives us an idea about how much of load the given structure can take.
• If the section modulus increases the moment of resistance increases and hence its load bearing capacity.

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. Any relationship between these properties is highly dependent on the shape in question. 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. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

It 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,

1. Elastic section modulus
2. Plastic section modulus

1. 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.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		,	NA indicates neutral axis
doubly symmetric I-section (minor axis)			NA indicates neutral axis
Circle			Solid arrow represents neutral axis
Circular hollow section			Solid arrow represents neutral axis
Rectangular hollow section			NA indicates neutral axis
Diamond			NA indicates neutral axis
C-channel			NA indicates neutral axis

2. 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 is not the 'First moment of area'. Both relate to the calculation of the centroid, but Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section, while the First moment of area is calculated based on either side of the "considering point" of the cross section and it is different along the cross section and depends on the point of consideration.

Description	Figure	Equation	Comment
Rectangular section			,
Rectangular hollow section			where: b = width, h = height, t = wall thickness
For the two flanges of an I-beam with the web excluded			=width, =thickness, are the distances from the neutral axis to the centroids of the flanges respectively.
For an I Beam including the web
For an I Beam (weak axis)			d = full height of the I beam
Solid Circle
Circular hollow section

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 :-

Procedure :

• Open the new model and paste the hood inner pannel and outer pannel into it and name as section modulus.
• Then go to the view option and click on edit section

• Then save the copy of section curve, as shown in below image

• After getting this section we have to obtain sigle curve which is closed, for that click on Extract Geometry and select the upper curce of outer panel and lower curve of inner pannel.

• Open the skech and project this curve and joint this upper and lower curve by the line and obtain the closed curve

• Click on the Section Inertia Analysis for finding the moment of Inertia.

Section 1 :

Section 2 :

Calculations :