Section Modulus calculation and optimization

                                                    SECTION MODULUS CALCULATION AND OPTIMIZATION

 

AIM:

To calculate the section modulus of the previously designed hood for analysing 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.

OBJECTIVE:

To 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

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)

 

 

 

 

 

• Changing the position can show the difference in bending which is because of the centroid distance from extreme fibre end.
• In Layman's terms, in this position there is more material from axis to the specific direction, to resist the load which makes less bend and improves the strength.

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.

 

Section modulus equations:
Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents neutral axis
doubly symmetric I-section (major axis)			NA indicates neutral axis
doubly symmetric I-section (minor axis)			NA indicates neutral axis
Circle			Solid arrow represents neutral axis
Circular hollow section			Solid arrow represents neutral axis
Rectangular hollow section			NA indicates neutral axis
Diamond			NA indicates neutral axis
C-channel			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:

                         

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:

                                        

shape factor for a rectangular section is 1.5.

So here we are Calculating Elastic section Modulus

 

SECTION MODULUS CALCULATION:

Moment of inertia (MAX) =5.1191 mm^4

Moment of inertia (MIN) =1.7667 mm^4

                                             Y=440mm

                                     

   Section Modulus            S= (1.7667*10^3)/440

                                              S=4.015mm^3

 

OPTIMISED HOOD DESIGN:

The depth of the hood inner panel is increased with 0.5 mm in order to optimise the design for section modulus.

 

Moment Of Inertia (MAX) =9.247mm^4

Moment Of Inertia (MIN) =2.774mm^4

Distance of neutral axis to the extreme end, Y= 880/2

                                                                Y=440mm

 

Section modulus,       S=2.774*10^3/440

                                 S =6.3045 mm^3

  

OBSERVATION:

• Section of modulus for the hood design without optimisation is                        S=4.0154 mm^3
• Section of modulus for the hood design after optimisation (increased depth) is, S =6.3045 mm^3
• It is observed that after increasing the depth of the inner panel of the hood the moment of inertia value is increased which results in the increase in the section modulus
• From this observation it is found that with the increase in cross sectional area the section modulus value can be increased

CONCLUSION:

 The strength of the material is directly proportional to the moment of inertia, hence increase in sectional area there is have increase in overall mass of an object ,hence have increase in strength.