Section Modulus calculation and optimization

Report On Section Modulus Calculation Of Hood

Aim

To calculate the section modulus of the previously designed hood for analyzing its strength and also optimizing the design to see the difference in the bending strength of the hood. Higher the section modulus of a structure, the more the resistive it becomes to bending.

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. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus.

S=I/Y

Here,

S = Section modulus

I= Moment of Inertia (Unit mm 4)

Y= Distance from Neutral axis to the extreme end (unit mm)

The Unit of a “section modulus” is (mm3)

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. 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.

Section modulus equations^[3]

Cross-sectional shape	Figure	Equation	Comment
Rectangle		S = b h 2 6	Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		S x = B H 2 6 − b h 3 6 H S x = I x y , with y = H 2	NA indicates neutral axis
doubly symmetric I-section (minor axis)		S y = B 2 ( H − h ) 6 + ( B − b ) 3 h 6 B [4]	NA indicates neutral axis
Circle		S = π d 3 32 [3]	Solid arrow represents neutral axis
Circular hollow section		S = π ( r 2 4 − r 1 4 ) 4 r 2 = π ( d 2 4 − d 1 4 ) 32 d 2	Solid arrow represents neutral axis
Rectangular hollow section		S = B H 2 6 − b h 3 6 H	NA indicates neutral axis
Diamond		S = B H 2 24	NA indicates neutral axis
C-channel		S = B H 2 6 − b h 3 6 H	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, and equal compressive and tensile 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:

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		Z P = b h 2 4 [5][6]	A C = A T = b h 2 , y C = y T = h 4
Rectangular hollow section		Z P = b h 2 4 − ( b − 2 t ) ( h 2 − t ) 2	where: b = width, h = height, t = wall thickness
For the two flanges of an I-beam with the web excluded[7]		Z P = b 1 t 1 y 1 + b 2 t 2 y 2	where: b 1 , b 2 =width, t 1 , t 2 =thickness, 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		Z P = b t f ( d − t f ) + 0.25 t w ( d − 2 t f ) 2	[8]
For an I Beam (weak axis)		Z P = ( b 2 t f ) / 2 + 0.25 t w 2 ( d − 2 t f )	d = full height of the I beam
Solid Circle		Z P = d 3 6
Circular hollow section		Z P = 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 :-

k = Z S

Shape factor for a rectangular section is 1.5.

Though generally section modulus is calculated for the extreme tensile or compressive fibres in a bending beam, often compression is the most critical case due to onset of flexural torsional (F/T) buckling. Generally (except for brittle materials like concrete) tensile extreme fibres have a higher allowable stress or capacity than compressive fibres.

In the case of T-sections if there are tensile fibres at the bottom of the T they may still be more critical than the compressive fibres at the top due to a generally much larger distance from the neutral axis so despite having a higher allowable stress the elastic section modulus is also lower. In this case F/T buckling must still be assessed as the beam length and restraints may result in reduced compressive member bending allowable stress or capacity.

There may also be a number of different critical cases that require consideration, such as there being different values for orthogonal and principal axes and in the case of unequal angle sections in the principal axes there is a section modulus for each corner.

For a conservative (safe) design, civil structural engineers are often concerned with the combination of the highest load (tensile or compressive) and lowest elastic section modulus for a given section station along a beam, although if the loading is well understood one can take advantage of different section modulus for tension and compression to get more out of the design. For aeronautical and space applications where designs must be much less conservative for weight saving, structural testing is often required to ensure safety as reliance on structural analysis alone is more difficult (and expensive) to justify.

First Design Study

The Intersection of the design was taken and then section Inertia Analysis was done-

Moment of inertia(max) = 9.131073763×10^7×mm^4

Moment of inertia(min) = 2.759706637×10^5×mm^4

Y=301.5/2=150.75 mm

S=I/Y

S=2.759706637×10^5/150.75= 1,830.6511 mm^3

Hood Design Optimization

By increasing the depth of the inner panel by 0.5 mm

Here,

Moment of inertia(max) = 9.13972810×10^7×mm^4

Moment of inertia(min) = 2.758916323×10^5×mm^4

Y=301.5/2=150.75 mm

S=I/Y

S=2.758916323×10^5/150.75 mm=1,830.1269 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(depth).