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 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.

Though generally section modulus is calculated for the extreme tensile or compressive fibres in a bending beam, often compression is the most critical case due to onset of flexural torsional (F/T) buckling. Generally (except for brittle materials like concrete) tensile extreme fibres have a higher allowable stress or capacity than compressive fibres.

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:

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.

AIM:

To Come up with a new section that has an improved Section Modulus compared to the previous one and mention what changes have made to increase the strength.

Use the section from the Hood design and calculate the section modulus

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

CASE STUDY

• Case 1

In this case we are using the section from our normal hood design

Fig 1.1

Fig 1.2

Moment of inertia (max) =4.876847053+06mm^4

Momemt of inertia (min) = 1.674830518e+04mm^4

Total distance of the Hood =881.8182mm

Distance between the neutral axis to the extreme end of the object Y =D/2

= 881.8182/2

Y = 440.9091 mm^3

S =I(min)/Y

S = 16,748.30518/440.9091

S = 37.9858mm^3

• Case 2

In order to increase the resistance to bending of HOOD, We have to make an improvement in section modulus of HOOD

So I have made some changes in my HOOD design and its section is given as follows,

I have increased the offset of inner panel to 1mm and included an extra emboss

Fig 2.1

Fig 2.2

Moment of inertia (max) = 4.939016256e+06mm^4

Moment of inertia (min) = 1.747054272e+04mm^4

Distance between the neutral axis to the extreme end of the object Y = 440.9091 mm^3

S =I(min)/Y

S =17,470.54272 /440.9091

S = 39.6239 mm^3

From the above case study I have made a improvement in section modulus of about 1.6381mm^3 which is a great variation and will definitely improve the strength of my Hood

So the variation which i gave the Hood had Yeilded me a positive result