Section Modulus calculation and optimization

Objective:

To calculate the section modulus of the Hood for analyzing its strength and also optimizing the design to see the change in the bending strength of the hood. The higher the section modulus of a structure, the more resistive it becomes to bending.

Section Modulus:

Section Modulus is the geometric property of the given cross-section of any shaft or beam structure. It is the direct measure of the strength of the shaft section or the beam section. The higher the section modulus of a structure, the more resistive it becomes to bending.

Section modulus is defined as the ratio of polar moment of inertia to the radius of the shaft or the distance from the neutral axis to the outer fibres.

S = I/y

S = Section Modulus

I = Moment of Inertia

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

Changing the position of the body results in a change in resistance to bending

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}}}	NA indicates neutral axis
Circle		{\displaystyle S={\cfrac {\pi d^{3}}{32}}}	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 can also be called the 'First moment of area'

Study of Hood design:

Initial Design

Moment of inertia(max) = 4.8942939×10^6×mm^4
Moment of inertia(min) = 1.690147224×10^3×mm^4
Y=435.7740/2=217.8862 mm
S=I/Y
S= 1.690147224×10^3/217.8862= 7.75 mm^3

Changing the depth of the inner panel :

Moment of inertia(max) = 4.883904069×10^6×mm^4
Moment of inertia(min) = 1.71905905×10^3×mm^4
Y=435.7740/2=217.8862 mm
S=I/Y
S= 1.71905905×10^3/217.8862= 7.8897mm^3

Changing the geometry of the section:

Moment of inertia(max) = 4.915949392×10^6×mm^4
Moment of inertia(min) = 1.680660530×10^3×mm^4
Y=435.7740/2=217.8862 mm
S=I/Y
S= 1.680660530×10^3/217.8862= 7.713 mm^3

Conclusion:

So, here the difference can be seen in the moment of inertia and the change in the section modulus can be seen by the change in the design i.e., change in-depth and section of the panel