Section Modulus calculation and optimization

SECTION MODULAS CALCULATION AND OPTIMIZTION

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.

It includes the idea that most of the work in bending is being done by the extreme fibres of the beam, i.e. the top and bottom fibers of the section. The distance of the fibers from top to bottom is therefore built into the calculation.

According to bending equation-

Where,

Z = Section modulus of the section

M= Moment of resistance

σ = Bending stress

y = the distance to the extreme fiber from the centroid

It is known that the stress in a fibre is proportional to its distance from the neutral axis. If y max is the distance of the extreme fiber from neutral axis then

Section modulas for reactangular section:

Explanation:

The modulus of section may be defined as the ratio of moment of inertia to the distance to the extreme fiber. It is denoted by Z

For rectangular section,

                                                        I = BD³/12 &

                                                        y = D/2.

                                                        Z_XX = BD²/6

Hence, if the maximum stress offered by the section is known we can easily compute the moment of resistance that can be offered by the section. hence for a beam of given material the greatest moment of resistance the beam section can offer is given by

                                                        M = f (safe).Z

Where, f(safe) = maximum bending stress which occurs at the point most distant from the neutral axis.

It may be defined as the ratio of total moment resisted by the section to the stress in the extreme fibre which is equal to yield stress.

Section modulus of a beam having circular cross-section:

I_XX = Area moment of inertia of the circular cross-section about the XX axis

I_YY = Area moment of inertia of the circular cross-section about the YY axis

                                                    I_XX = I_YY = ПD⁴/64

y = distance of the outermost layer of the section from its neutral axis or centroidal axis

                                                    y = D/2

Section modulus of the circular cross-section about XX axis and YY axis could be secured as mentioned here

                                                   Z_XX = Z_YY = ПD³/32

 Inelastic bending, the stress distribution in the beam is as follows:

How is section modulus used?

The section modulus of the cross-sectional shape is of significant importance in designing beams. It is a direct measure of the strength of the beam. A beam that has a larger section modulus than another will be stronger and capable of supporting greater loads.

Section modulus calculations of hood:

From The above design section, minimum moment of inertia (I) = 1.455207× 10⁴  mm⁴

And the   Y = 879.6/2

                 = 439.8 mm

For section modulus,  S   =I/Y

                                     =1.455207× 10⁴/439.8

                                     = 33.08 mm³

In order to increase resistant to bending of Hood, we have to improve the section modulus of hood. From the above equation it can be observe that the section modulus of body is directly proportional to the area moment of inertia.                           

So for improving section modules we have to increase the moment of inertia of the section. 

Changes that I have made in my hood model are:

• Increased the offset distance of outer panel by 0.5 mm.
• Increased the offset distance of hem flange by 0.5 mm.

So the new hood section shown in below:

 From the above design section,

 Minimum moment of inertia (I) = 1.482687× 10⁴ mm⁴

                                           Y  = 439.8 mm

For section modulus,

                   S= I/Y 

                     =1.482687× 10⁴ /439.8

                     =33.71 mm³

From the above calculation it is observed that we improve the section modulus value by 0.63mm³.

So that we made improvement in bending resistant of hood.