Section Modulus calculation and optimization

Aim

To calculate the section modulus of the previously designed hood for analyzing its strength and also optimizing the design to see the difference in the bending strength of the hood. Higher the section modulus of a structure, the more the resistive it becomes to bending.

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members it is also referred to as the Polar Modulus or the Torsional Sectional Modulus. 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 (S) and the plastic section modulus (Z).

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 moment of inertia) and Y is the distance from the neutral axis to any given fiber.

S=I/Y

Here,

S = Section modulus

I= Moment of Inertia

Y= Distance between the neutral axis and extreme end of the object.

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 fiber. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fiber, 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 yields strength of the material.

There are two different side the beam load acting, the difference is shown in the following images.

In this images two different sides of beam load acting , there is a more material from axis to the specific direction to resist the load which makes less bend and improve the strength.

Plastic Section Modulus

The plastic section modulus is used for material where elastic yielding is acceptable and plastic behaviour is assumed to be an acceptable limit, design generally strive to ultimately remain below the plastic limit to avoid permanent deformations ofte 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 thge 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 section with constant yielding stress, the area above and below the PNA will be equal, but for composite section this is not necessary the case.

The plastic section modulus is the sum of the area of the cross section on each side of the PNA multiplied by the distance from the local centroids of the two areas to the PNA Zp = Acyc + ATyT

The plastic section modulus is also called as first moment of inertia.

First Design Study

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

Moment of inertia(max) = 9.123807131×10^7×mm^4

Moment of inertia(min) = 2.463344470×10^5×mm^4

Y=878.0252/2=439.0126mm

S=I/Y

S=2.463344470×10^5/439.0126= 561.11 mm^3

Hood Design Optimization

By increasing the depth of the inner panel by 0.5 mm

Here,

Moment of inertia(max) = 9.10726×10^7×mm^4

Moment of inertia(min) = 2.509628×10^5×mm^4

Y=878.0252/2=439.0126mm

S=I/Y

S=2.509628×10^5/439.0126=571.65 mm^3

Conclusion :

So, here the difference can be seen in the moment of inertia and the change in the section modulus can be seen by the change in the design (depth).