Section Modulus calculation and optimization

Aim :

To Use the Section from Hood design and calculate the Section Modulus using the formula

ObJectives :

Come up with a new section that has an improved Section Modulus compared to the previous one and mention what changes you made which resulted in the increased strength.

Procedure :

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 second moment of area and polar second moment of area 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.

S = I/y

Where,

S = Section Modulus

I = Moment of Inertia

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

Lets Discuss In Detail,

Section Modulus

Section Modulus is a geometric property used to calculate the bending stresses in a structural member.

Mathematically,

It is defined as the ratio of the moment of inertia of a section about its centroidal axis and to the distance of the extreme layer from the neutral axis.

It is denoted by Z.

Z = I/ymax

Mathematically By bending stress equation,

M/I = σmax/ymax

M = σmax × (I/ymax)

M = σmax × Z

Where,

I = Moment of Inertia(m⁴)

y = distance from NA to extreme layer(m)

The unit of section modulus is m³.

Types of Section Modulus

There are two types of section modulus:

• Elastic Section Modulus
• Plastic Section Modulus

Elastic Section Modulus

Elastic section modulus is calculated when we are to use the material behavior till the yield point.

For general design purposes, we use elastic section modulus.

Generally, elastic section modulus is denoted by Zₑ.

So,

Zₑ = I/ymax

Where,

I = Moment of inertia

ymax = Distance from NA to extreme layer

Elastic Section Modulus of Different Section

There are following different elastic section modulus:

Rectangular Section

As we know the moment of inertia of the rectangular section,

I = bd³/12

And the distance from the neutral axis,

ymax = d/2

So,

Zₑ = I/ymax

Zₑ = (bd³/12)/(d/2)

After calculating them,

Section modulus of rectangular section,

Zₑ = bd²/6

Hollow Rectangular Section

As we know the moment of inertia of the hollow rectangular section,

I = (BD³/12) - (bd³/12)

ymax = D/2

After putting these values,

Zₑ = I/ymax

(BD³/12) - (bd³/12)/(D/2)

Zₑ = (BD³ - bd³)/6D

Solid Circular

For solid Circular moment of inertia,

I = (π/64) × d⁴

ymax = d/2

Then,

Zₑ = {(π/64) × d⁴}/(d/2)

After calculating then find values of section modulus of solid circular

Zₑ = (π/32) × d³

Hollow Circular

The moment of inertia of hollow Circular,

I = (π/64) × (D⁴ - d⁴)

And,

ymax = D/2

After putting these values,

Zₑ = (π/64) × (D⁴ - d⁴)/(D/2)

Zₑ = π/32 × {(D⁴ - d⁴)/D}

Symmetric I & C Section

Moment of inertia for the I & C section,

I = (BD³/12) - {(B - b)d³}/12

ymax = D/2

So,

Zₑ = [(BD³/12) - {(B - b)d³}/12]/(D/2)

Zₑ = {BD³ - (B - b)d³}/6D

Diamond

Elastic section modulus for diamond is,

Zₑ = bd²/24

Plastic Section Modulus

The plastic section modulus is used for materials where the elastic yield is acceptable and the plastic behavior is considered to be an acceptable range.

It depends on the location of the plastic neutral axis. The PNA is the axis that bisects the cross-section such that the compressive force from the region in compression is equal to the tension force from the region under tension. So, for sections with a constant yield stress, the area above and below the PNA will be equal, but for mixed sections, this is not necessarily the case.

Mathematically,

The plastic section modulus is the sum of the areas of cross-sections on each side of the plastic neutral axis that may or may not be equal and multiplied by the distance from the local centroid of the two regions to the plastic neutral axis.

Zₚ = (Ac × yc) + (Aₜ + yₜ)

Where,

Ac = Area of cross-section under compression.

Aₜ = Area of cross-section under tension.

yc = Distance from the plastic neutral axis to the center of gravity of the areas under compression.

yₜ = Distance from the plastic neutral axis to the center of gravity of areas under tension.

Since on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ

And,

yc = yₜ

Plastic Section Modulus of Different Section

There are following section modulus for different section:

Rectangular Section

As we know,

on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ

And,

yc = yₜ

Ac = Aₜ = bd/2

yc = yₜ = d/4

So,

Zₚ = (Ac × yc) + (Aₜ + yₜ)

Zₚ = (bd/2 × d/4) + (bd/2 × d/4))

After calculating

Zₚ = bd²/4

Similarly, we can calculate the section modulus of the rectangular hollow section,

Zₚ = (bd²/4) - (b- 2t) × {(b/2) - t}²

Circular Section

As we know, on the neutral axis, the area under compression is equal to the area under tension.

So,

Ac = Aₜ = πd²/8

yc = yₜ = 2d/3π

Zₚ = (πd²/8 × 2d/3π) + (πd²/8 × 2d/3π)

After calculating,

Zₚ = d³/6

Similarly, we can calculate the section modulus of the circular hollow section,

Circular hollow section,

Zₚ = (D³ - d³)/6

Moment of inertia :

It is also known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.

Based On Profiles we have 'n' number of formulaes but in case of our hood design we can get through the Nx Cad Software.

However Remember that

The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section.The moment of inertia I is also defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, that is

Lets Make The Section Modulus On Hood :

The Intersection of the design was taken From Hood and then section Inertia Analysis As Shown in Figure Given Below.

Finding y :

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

Moment of inertia(max) = 4.8592*10^6 mm^4

Moment of inertia(min) = 1.3656×10^4 mm^4

Y=874.446/2=437.223 mm

S=I/Y

S=1.3656×10^4/437.223= 31.22 mm^3

Hood Design Optimization :

Here We can Also Change the Emboss Or To Incrase the Emboss Height We Can Get Some better results lets see...