Section Modulus calculation and optimization

SECTION MODULUS

Definition : Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. The elastic modulus is denoted by "S". 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.

Significance : 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.

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

Types : There are two types of section modulus. They are :

• Elastic section modulus
• Plastic section modulus

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

S = section modulus

I = moment of inertia

Y = is the distance from the neutral axis to the most extreme fibre

Plastic section modulus:The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behaviour is assumed to be an acceptable limit. 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.

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.

Relation between Plastic Section Modulus & Elastic Section Modulus :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 elastic limit of material. This could be shown mathematically with the formula:

K = Z/S

SECTION MODULUS CALCULATION

Steps to Create Section for Calculation:

• Create a datum plane along the centre of a curve that divides the hood into two symmetrical halves.
• Take an intersection of both the inner and outer panels with the datum plane.
• Then project the intersection onto a sketch for editing and create a closed cross section.

Distance from Neutral Axis :

Since the distance between the two extremes is 878.3069mm, we have to divide it by 2 to get the distance from the neutral axis.

Distance from Neutral Axis, Y = 878.3069 /2

= 439.15345 mm

CASE 1: ORIGINAL HOOD MODEL

Cross Section :

Section Inertia Analysis Report :

Section Modulus :

S = I / Y

Since Moment of Inertia is the resistance to angular acceleration, the minimum value from the section inertia report has to be considered.

I = 1.306255468 x 10⁴ mm⁴

Y = 439.15345 mm

S = 1.306255468 x 10⁴ / 439.15345

= 29.745 mm³

CASE 2 : MODIFIED HOOD MODEL

The initial hood design was modified to incorporate embossment into the section.

Cross Section :

Section Inertia Analysis Report :

Section Modulus :

S = I / Y

I = 1.406964056 x 10⁴ mm⁴

Y = 439.15345 mm

S = 1.406964056 x 10⁴ / 439.15345

= 32.038 mm³

INFERENCE

So as to improve the Section Modulus, we have to increase the Moment of Inertia or reduce the Distance from Neutral Axis. Since the hood overall dimensions need to be the same, we should increase the Moment of Inertia.

This can be achieved through increasing the cross sectional area which can be done by:

• Adding embossments to the section.
• Increasing the distance between the inner & outer hood panels.