Section Modulus calculation and optimization
SECTION MODULUS
Section modulus is termed as the ratio of second moment of area and distance from N.A (Neutral axis) to the extreme fiber. 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:
Here, I is “moment of inertia” and y is distance from “neutral axis” to top or bottom of fiber.
There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.
1. Elastic Section Modulus
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.
2. Plastic Section Modulus
he 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. This could be shown mathematically with the formula :-
- Section modulus is the important factor for design of beam and flexural member
- Higher the value of section modulus, higher will be the resistance of member to bending
- It is required to calculate stresses in beams.
- It is used to calculate strength of the steel structure
- More the section modulus, it can withstand more load and it is also considered to be more tougher.
SECTION MODUTUS CALCULATIONS AND COMPARISON
1. HOOD ORIGINAL DESIGN:
From the above image, the following data can be noted:
I(max) = 557749.28mm^4
I(min) = 164950.22 mm^4
y = 862/2 mm = 431 mm
Therefore the maximum and minimum section modulus is calculated below:
S(max) = 557749.28/431 = 1294.08mm^3
S(min) = 164950.22/431 = 382.71 mm^3
2. Hood changed design
What change made to the original design?
Moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section .
So from the above statement the MOI will increase if the mass or the length of the body is increased.
In order to increase the strength of the hood the minimum section modulus should be high, as moment of inertia is directly proportional to section modulus then the valus of MOI should increase, to increase the MOI the cross sectional area should be increased.
The minimum moment of inertia indicates the minimum load the hood can withstandor the start of the breaking point of the hood.
From the above observation, to increase the cross section of the hood, the hood inner panel will be offseted in the downwards direction by a value of 0.5 mm.
By performing the above operation we can note the moment of inertia value again.
By performing the inertial calculation again for the hood we can observe that the minimum moment of inertia value increases.
CALCULATIONS
MOI(max) = 550124.91 mm^4
MOI(min) = 166538.20 mm^4
y = 862/2 = 431 mm
Section Modulus calculations
S (max) = MOI(max)/y =550124.91 mm^4 /431 = 1276.39 mm^3
S (min) = MOI(min)/y = 166538.20 mm^4/431 = 386.39 mm^3
| HOOD DESIGNS | MAXIMUM SECTION MODULUS | MINIMUM SECTION MODULUS | |
| ORIGINAL DESIGN | 1294.08 mm^3 | 382.71 mm^3 | |
| CHANGED DESIGN | 1276.39 mm^3 | 386.39 mm^3 |
SECTION MODULUS
Section modulus is termed as the ratio of second moment of area and distance from N.A (Neutral axis) to the extreme fiber. 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:
Here, I is “moment of inertia” and y is distance from “neutral axis” to top or bottom of fiber.
There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.
1. Elastic Section Modulus
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.
2. Plastic Section Modulus
he 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. This could be shown mathematically with the formula :-
- Section modulus is the important factor for design of beam and flexural member
- Higher the value of section modulus, higher will be the resistance of member to bending
- It is required to calculate stresses in beams.
- It is used to calculate strength of the steel structure
- More the section modulus, it can withstand more load and it is also considered to be more tougher.
SECTION MODUTUS CALCULATIONS AND COMPARISON
1. HOOD ORIGINAL DESIGN:
From the above image, the following data can be noted:
I(max) = 557749.28mm^4
I(min) = 164950.22 mm^4
y = 862/2 mm = 431 mm
Therefore the maximum and minimum section modulus is calculated below:
S(max) = 557749.28/431 = 1294.08mm^3
S(min) = 164950.22/431 = 382.71 mm^3
2. Hood changed design
What change made to the original design?
Moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section .
So from the above statement the MOI will increase if the mass or the length of the body is increased.
In order to increase the strength of the hood the minimum section modulus should be high, as moment of inertia is directly proportional to section modulus then the valus of MOI should increase, to increase the MOI the cross sectional area should be increased.
The minimum moment of inertia indicates the minimum load the hood can withstandor the start of the breaking point of the hood.
From the above observation, to increase the cross section of the hood, the hood inner panel will be offseted in the downwards direction by a value of 0.5 mm.
By performing the above operation we can note the moment of inertia value again.
By performing the inertial calculation again for the hood we can observe that the minimum moment of inertia value increases.
CALCULATIONS
MOI(max) = 550124.91 mm^4
MOI(min) = 166538.20 mm^4
y = 862/2 = 431 mm
Section Modulus calculations
S (max) = MOI(max)/y =550124.91 mm^4 /431 = 1276.39 mm^3
S (min) = MOI(min)/y = 166538.20 mm^4/431 = 386.39 mm^3
| HOOD DESIGNS | MAXIMUM SECTION MODULUS | MINIMUM SECTION MODULUS | |
| ORIGINAL DESIGN | 1294.08 mm^3 | 382.71 mm^3 | |
| CHANGED DESIGN | 1276.39 mm^3 | 386.39 mm^3 |
CONCLUSION:
By comparing the values of the minimum moment of inertia for the original design and the changed design we can observe that the Moment of inertia increases if the cross sectional area of the hood increases.
The changed hood will be have higher strength so it will be able to withstand more load.