Section Modulus calculation and optimization

SECTION MODULUS

Definition of 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. If a section modulus increases the moment of resistance increases hence it impacts load bearing capacity.

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 S.

S = I/y

S=Section Modulus

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 behaviour till the yield point. For general design purposes, we use elastic section modulus.

Generally, elastic section modulus is denoted by Sₑ. So,

S = I/ymax

Were,

I = Moment of inertia

ymax = Distance from NA to extreme layer

Elastic Section Modulus of Different Section

There are following different elastic section modulus:

1. Rectangular Section

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

I = bd³/12

And the distance from the neutral axis. So,

ymax = d/2

Sₑ = I/ymax

Sₑ = (bd³/12)/(d/2)

After calculating them, Section modulus of rectangular section,

Sₑ = bd²/6

2. 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,

Sₑ = I/ymax

(BD³/12) - (bd³/12)/(D/2)

Sₑ = (BD³ - bd³)/6D

3. Solid Circular

For solid Circular moment of inertia,

I = (π/64) × d⁴

ymax = d/2

Then,

Sₑ = {(π/64) × d⁴}/(d/2)

After calculating then find values of section modulus of solid circular

Sₑ = (π/32) × d³

4. Hollow Circular

The moment of inertia of hollow Circular,

I = (π/64) × (D⁴ - d⁴)

ymax = D/2

After putting these values,

Sₑ = (π/64) × (D⁴ - d⁴)/(D/2)

Sₑ = π/32 × {(D⁴ - d⁴)/D}

6. Symmetric I & C Section

Moment of inertia for the I & C section. So,

I = (BD³/12) - {(B - b)d³}/12

ymax = D/2

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

Sₑ = {BD³ - (B - b)d³}/6D

7. Diamond

Elastic section modulus for diamond is,

Sₑ = bd²/24

2. Plastic Section Modulus

The plastic section modulus is used for materials where the elastic yield is acceptable and the plastic behaviour 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.

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.

Sₚ = (Ac × yc) + (Aₜ + yₜ)

Were,

Ac = Area of cross-section under compression.

Aₜ = Area of cross-section under tension.

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

yₜ = Distance from the plastic neutral axis to the centre 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

yc = yₜ

2. Plastic Section Modulus of Different Section

There are following section modulus for different section:

1. Rectangular Section

As we know,

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

Ac = Aₜ

yc = yₜ

Ac = Aₜ = bd/2

yc = yₜ = d/4

So, Sₚ = (Ac × yc) + (Aₜ + yₜ)

Sₚ = (bd/2 × d/4) + (bd/2 × d/4))

After calculating

Sₚ = bd²/4

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

Sₚ = (bd²/4) - (b- 2t) × {(b/2) - t}²

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

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

After calculating,

Sₚ = d³/6

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

Sₚ = (D³ - d³)/6

Procedure to Calculate Section Modulus for a Hood:

Step 1:

First, we need a closed 2D sketch of a section from the hood design. The best place is at the intersection of two half of the inner and outer panel assembly.

Using Intersection Curve tool, we can the get the outline of the intersecting Inner and Outer panel.

Step 2:

Create a sectional plane and Project the 2 individual curves from Inner and Outer plane. Join the curve to form single curve and join the edges,

Type 1:

Formula to calculate section modulus,

S = I/y

I = Minimum Moment of Inertia = 6.111046126e+03(mm^4)

Y= distance from Neutral axis: 487.411/2 = 243.82mm

S=6.111046126e+03(mm^4) / 243.82mm

So, Moment of Inertia of First type of section is

S=25.063760mm^3

Hood Design Optimization:

By increasing the depth of the inner panel by 0.5 mm

S = I/y

I = Minimum Moment of Inertia = 8.130141765e+03(mm^4)

Y= distance from Neutral axis: 487.411/2 = 243.82mm

S=8.130141765e+03(mm^4) / 243.82mm

So, Moment of Inertia of Modified section is

S=33.34mm^3

Conclusion

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