Section Modulus calculation and optimization

INTRODUCTION:-

WHAT IS 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 the design include an area for tension and shear, the radius of gyration for compression, and the moment of inertia, and polar moment of inertia for stiffness.

Types of section modulus:

Elastic section modulus(S).
Plastic section modulus(Z).

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 a 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 (My) such that My = S × σy, where σy is the yield strength of the material.

Cross-sectional shape	Figure	Equation
Rectangle
doubly symmetric I-section (major axis)		,
doubly symmetric I-section (minor axis)
Circle
Circular hollow section
Rectangular hollow section
Diamond
C-channel

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'

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			where: =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 the elastic limit of the material. This could be shown mathematically with the formula:-

The shape factor for a rectangular section is 1.5.

Procedure:

1. Create Intersection:

Create an intersection for the outer and inner panel by using the Intersection tool. Where the first element is the element you want to cut and the second element is the cutting element or plane.

2. The boundary for Cross-sectional area:

In order to calculate the Moment of Inertia, we have to create a 2D cross-section plane, for creating. select outermost intersection line of the outer panel and innermost intersection line of the inner panel. Fill in gaps by using Spline and make sure that there is no gap or discontinuity. Join all the lines and create a surface by using Fill. shown in fig.

3. Calculating the section modulus: To calculate section modulus by S = I/y. where I = moment of inertia, y = distance from the centroid to top.

We need I = moment of inertia, y = distance from the centroid to top

Moment of Inertia: by using measure inertia tools we can find the Moment of Inertia of any shape of 2D geometry.

From the fig., we have Min and Max of I which is M1 and M2.

M1 = 6.109 X 10^7 mm^4

M2 = 1.321X 10^8 mm^4

y = distance between the neutral axis and the extreme end of the object = 243.78mm

So

S1 = I/y

S1= 6.109X10^7 /243.78

S1= 2.505x10^5 mm

S2= 1.321X10^8 /243.78

S2= 5.418x10^5mm

Similarly by changing the section area.

offset inner panel side curve by 0.5mm outside and create a close loop and 2d surface.

rom the fig. we have Min and Max of I which is M1 and M2.

M1 = 6.799 X 10^7 mm^4

M2 = 1.406X 10^8 mm^4

y = distance between the neutral axis and the extreme end of the object = 243.78mm

So

S1 = I/y

S1= 6.799X10^7mm4 / 243.78

S1= 2.788x10^5 mm

S2= 1.406X10^8 /243.78

S2= 5.767X 10^5mm

CONCLUSION:

Thus the section modulus of the Front Hood of a car was calculated and compared with the optimized hood and found the better result. Hence I'm concluding that when the sectional area of the hood increases the stiffness and resistance of the hood will also be increased.