Section Modulus calculation and optimization

what is section modulus:

Section modulus is the property of the cross-section that is used while the design of beams.

The section modulus is classified into two types:-

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 moment of inertia) and y is the distance from the neutral axis to any given fibre. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often used to determine the yield moment (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yield strength of the material.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle		{\displaystyle S={\cfrac {bh^{2}}{6}}}	Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		{\displaystyle Sx={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}} {\displaystyle Sx={\tfrac {Ix}{y}}}, with {\displaystyle y={\cfrac {H}{2}}}	NA indicates neutral axis
doubly symmetric I-section (minor axis)		{\displaystyle Sy={\cfrac {B^{2}(H-h)}{6}}+{\cfrac {(B-b)^{3}h}{6B}}}[4]	NA indicates neutral axis
Circle		{\displaystyle S={\cfrac {\pi d^{3}}{32}}}[3]	Solid arrow represents neutral axis
Circular hollow section		{\displaystyle S={\cfrac {\pi \left(r_{2}^{4}-r_{1}^{4}\right)}{4r_{2}}}={\cfrac {\pi (d_{2}^{4}-d_{1}^{4})}{32d_{2}}}}	Solid arrow represents neutral axis
Rectangular hollow section		{\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}	NA indicates neutral axis
Diamond		{\displaystyle S={\cfrac {BH^{2}}{24}}}	NA indicates neutral axis
C-channel		{\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}	NA indicates neutral axis

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:

{\displaystyle Z_{P}=A_{C}y_{C}+A_{T}y_{T}}

The Plastic Section Modulus is not the 'First moment of area'. Both relate to the calculation of the centroid, but Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section, while the First moment of area is calculated based on either side of the "considering point" of the cross section and it is different along the cross section and depends on the point of consideration.

Description	Figure	Equation	Comment
Rectangular section		{\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}}[5][6]	{\displaystyle A_{C}=A_{T}={\cfrac {bh}{2}}} , {\displaystyle y_{C}=y_{T}={\cfrac {h}{4}}}
Rectangular hollow section		{\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}-(b-2t)\left({\cfrac {h}{2}}-t\right)^{2}}	where: b = width, h = height, t = wall thickness
For the two flanges of an I-beam with the web excluded[7]		{\displaystyle Z_{P}=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\,}	where: {\displaystyle b_{1},b_{2}} =width, {\displaystyle t_{1},t_{2}}=thickness, {\displaystyle y_{1},y_{2}} are the distances from the neutral axis to the centroids of the flanges respectively.
For an I Beam including the web		{\displaystyle Z_{P}=bt_{f}(d-t_{f})+0.25t_{w}(d-2t_{f})^{2}}	[8]
For an I Beam (weak axis)		{\displaystyle Z_{P}=(b^{2}t_{f})/2+0.25t_{w}^{2}(d-2t_{f})}	d = full height of the I beam
Solid Circle		{\displaystyle Z_{P}={\cfrac {d^{3}}{6}}}
Circular hollow section		{\displaystyle Z_{P}={\cfrac {d_{2}^{3}-d_{1}^{3}}{6}}}

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

SECTION MODULUS OF THE HOOD

Objective: To find the section modulus of the hood and optimisation to improve the overall section modulus.

Formula of Section Modulus: It may be defined as the ration of total moment resisted by the section to the stress in the extreme fibre which is equal to yield stress.

There fore

S = I/Y, Where

S = Section modulus (mm^3)

I = Moment of Inertia of the section about the bending axis. (mm^4)

Y = Distance between the neutral axis and extreme end of the object (mm)

Area moment of Inertia:

The area moment of inertia of a beams cross sectional area measures the beam ability to resist bending. The larger the moment of inertia less will be the bend.

Where,

M = Bending moment

I = Area moment of Inertia

E = Youngs modulus

R = Radius of gyration

Neutral axis: When beam is subjected to pure bending the top most fibres of the beam are subjected to maximum compression, while the bottom most fibres of the beams are subjected to maximum tension.

Material Data: Stainless steel AISI 302 – cold rolled

Where yield strength of stainless steel is 520MPa = 520 N/mm^2

Calculation:

CASE 1: Before Optimization

The Hood which was already created is imported and the section inertia analysis is done.

We have to make sure that the curve obtained is a single/closed curve.

Fig 1.1

Fig 1.2

MOI (max) = 8.928754301e+07 (mm^4)

MOI (min) = 3.196609661e+05 (mm^4)

As per the section modulus equation,

S = I/Y

I = MOI (min) value is considered

Y = Total distance /2, Total distance =875.5680mm

Y = 875.5680 / 2

= 437.784 mm

Therefore, S = 319660.966 mm^4/437.784 mm

S = 730.179 mm^3

CASE 2: After optimization

The hood inner panel’s curve is offset by 0.5mm outwards and section inertia analysis is done same as case 1 for a closed curve.

Fig 1.3

Fig 1.4

MOI (max) = 8.910657812e^07 mm^4

MOI (min) = 3.264954768e^05 mm^4

As per section modulus equation,

S = I / Y

I = MOI (min) value is considered

Y = Total distance /2, Total distance =875.5680mm

Y = 875.5680 / 2

= 437.784 mm

Therefore, S = 326495.4768 mm^4 / 437.784 mm

S = 745.792 mm^3

Conclusion:

Hence when comparing case 1 & case 2,

1. The section modulus value of case 2 is higher than the case 1

2. As we increase the section area, the moment of inertia & section modulus increases which means strength and load material also increase.