Section Modulus calculation and optimization
SECTION MODULUS
Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members, it is also referred to as the Polar Modulus or the Torsional Sectional Modulus. 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.
It is termed as the ratio of second moment of inertia and distance from N.A (Neutral axis) to the extreme fibre. Also it is the measure of strength of given member. The stress in the outermost section of beam is computed with the help of section modulus. It is indicated by S.
Section modulus is indicated as follows:
`S = I/y`
Here, I is “moment of inertia” and y is distance from “neutral axis” to top or bottom of fibre.
Unit of a “section modulus” is (mm3).
Section modulus depends only on the cross section shape of the beam. Cross section shapes like rectangular, square, circular, I section and T, composite section etc.
For symmetric sections, the value of “section modulus” is for both above and below of centroid of the fibre.
An asymmetric section has two values for top and bottom of the fibre due to different distance from centroid to the top and bottom of the material, the maximum “section modulus” is S m ax and the minimum section modulus is S min.
Importance:
• It is significant for beam and flexural member design.
• More the section modulus, more the member is resistant to bending of the beam.
• It is easier to calculate stresses in beam and an exact measure of strength of steel.
• if two beams are made of same material and comparing the “section modulus” of two beams, the beam with bigger section modulus will be tougher and more capable to withstand larger loads.
There are two types of section moduli, the elastic section modulus and the 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, 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 (My) such that My = S × σy, where σy is the yield strength of the material.
1)The elastic section modulus assumes the section remains elastic.
2)If the failure is considered till the elastic
zone then we consider elastic modulus.
3)Elastic modulus is the steel modulus based on the stress strain curve before yielding.
4)Ze is used to determine the maximum bending moment up to the point where the extreme fiber has yielded elastically. All other fibers will be below their elastic yield stress.
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:
the Plastic Section Modulus can also be called the 'First moment of area'
The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, Fy, by Mp=Fy*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
1)The plastic section modulus assumes the entire section yields.
2)If the failure is considered till the plastic zone
(breaking point) then we consider plastic modulus
3)The plastic modulus is after yielding.
4)Zp is used to determine the maximum bending moment at the point where all fibers have yielded elastically and the entire section is behaving plastically. This is the point when the so-called plastic hinge has developed.
OBJECTIVE
Use the Section from your Hood design and calculate the Section Modulus using the formula
S = I/y
Where,
S = Section Modulus
I = Moment of Inertia
y = distance between the neutral axis and the extreme end of the object
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.
DESIGN STUDY
For calculating the section modulus
S = I/y
we have to calculate Moment of Inertia and distance between the neutral axis and the extreme end of the object .for calculating the moment of inertia the outer most skin of the hood inner panel and outer panel is taken(ignoring the thickness).
the intersection of the design is taken for calculation
for calculating the section inertia analysis requires a closed model. the gap between the inner and outer panel is closed using line
for calculating the distance between the neutral axis and the extreme end of the object .the diatance between two extreme end is calculated
`y =483.3090/2 =241.6545mm`
calculating moment of inertia using section inertia analysis in NX
Here the moment of inertia min is taken
I =5.270710846e+03 `mm^4`
y=241.6545mm
section modulus `S = I/y=(5.270710846e+03)/241.6545` = 21.8109361`mm^3`
OPTIMISATION
the section modulus depends upon the moment of inertia.for improving the section modulus
For improving the moment of inertia the depth between the inner panel varied
1) the depth of inner paner is increased by .5mm by ofsetting the curve and the gap is joined
the depth of inner panel is increased by .5mm and moment of inertia is calculated using section inertia analysis
Here the moment of inertia min is taken
I =5.475325802e+03 `mm^4`
y=241.6545mm
section modulus `S = I/y=(5.475325802e+03)/241.6545` = 22.6576612`mm^3`
2)the depth of inner panel is increased by .8mm and moment of inertia is calculated using section inertia analysis
Here the moment of inertia min is taken
I =5.600821885e+03 `mm^4`
y=241.6545mm
section modulus `S = I/y=(5.600821885e+03)/241.6545` = 23.1769816mm^3
CONCLUSION
| study | optimisation | moment of inertia | distance y | section modulus |
| no-1 | 5.270710846e+03 | 241.6545 | 21.8109361 | |
| no-2 | depth increase by .5mm | 5.475325802e+03 | 241.6545 | 22.6576612 |
| no-3 | depth increase by .8mm | 5.600821885e+03 | 241.6545 | 23.1769816 |
from S=I/y the section modulus increase with increase of moment of inertia
From the study by increasing the depth of inner panel the moment of inertia increases, thereby increasing the section modulus and increase in strength