Design Calculation of Section Modulus of Car Hood/Bonnet

 

MODULUS OF ELASTICITY: The ratio of stress to strain or the stiffness of the material of a structural member (resistance to deformation).

Essentially, the modulus of elasticity is a more general term regarding the overall stiffness of a member.

E= Modulus of Elasticity. If E increases, then the stiffness increases.

MOMENT OF INERTIA: There are two types of moment of inertia; mass moment of inertia and area moment of inertia.

The moment of inertia of a Mass (I)

Defined as the sum of the products of the mass (m) of each particle of the body the square of its perpendicular distance (r) from the axis is mathematically represented as

I = mr²

The mass distribution of a body of rotating particles from the axis of rotation is represented by the moment of inertia. The value of the moment of inertia is independent of the forces involved and depends only on the body geometry and position from the axis of rotation. The mass moment of inertia for rotation is analogous to mass in linear movement. So, the mass moment of inertia for rotation is treated the same way as the mass in linear motion with features like

 

The angular momentum of a body is given by I.ω. Newton’s Second Law of Motion, when applied to rotating bodies, states that the torque is directly proportional to the rate of change of angular momentum.

When a body with the mass moment of inertia (I) is rotated about any given axis, with an angular velocity ω, then it possesses some kinetic energy of rotation given by = 1/2 Iω2.

 

Moment of Inertia of an Area (I)

It represents the distribution of points in a cross-sectional area with respect to an axis. It is also known as the second moment of area. For an elemental area dA in XY plane, the area moment of inertia is mathematically defined as Ix and Iy

In beam theory, the formula of the moment of inertia is very important. Depending on the cross-section of the object the equation of moment of inertia varies. Note that, the moment of inertia is always positive.

The unit of mass moment of inertia in the SI unit system is kg.m2 and in the FPS unit system is lbf·ft·s2

The unit of an area moment of inertia in SI unit is m4 and in FPS unit system is inches4.

Polar Moment of Inertia

The polar moment of inertia is defined with respect to an axis perpendicular to the area considered. It provides a beam’s ability to resist torsion or twisting. The polar moment of inertia (J) of a circular area is given by J=πd^4/32.

Applications of Moment of Inertia

1.       Mass moment of inertia provides a measure of an object’s resistance to change in the rotation direction.

2.       Area moment of inertia is the property of a geometrical shape that helps in the calculation of stresses, bending, and deflection in beams.

3.       A polar moment of inertia is required in the calculation of shear stresses subject to twisting or torque.

4.       The moment of inertia “I” is a very important term in the calculation of Critical load in Euler’s buckling equation. The Critical Axial load, Pcr is given as Pcr=π^2EI/L^2.

5.       A moment of inertia is required to calculate the Section Modulus of any cross-section which is further required for calculating the bending stress of a beam. Bending stresses are inversely proportional to the Moment of Inertia. The larger the moment of inertia, the greater the moment of resistance against bending.

The Section modulus is usually used in the calculation of stresses in Cross-sections, however, the Moment of inertia is often used instead, probably because designers are more familiar with the Moment of inertia formulations.

SECTION MODULUS: The section modulus is a cross-sectional geometric property of structural elements such as beams, columns, slabs, etc. and it is used to calculate stresses in Cross-sections. In general, it can be said that the greater the dimensions of a cross-section under a given load, the greater the Section modulus and the smaller the bending stress.

S = I / y

Where

I = area moment of inertia, and

y = Distance from the neutral axis to any perpendicular Edge.

Neutral Axis:-

The neutral axis for the beam subjected to bending is a line passing through the cross-section at which the fibers of the beam do not experience any longitudinal stress (compressive or tensile).

Let us calculate the Section Modulus of the Car Hood Designed in NX-Cad.

According to the Section Inertia Analysis of Hood, we have arrived at the following values in 3 different size cross-sections.

The Distance of Neutral axis from one end of the fiber = 490/2=245mm

Case 1:-

Centre of Gravity (C.G) = (0.001236780, 745.432693317, -931.373702297) mm.

Moment of Inertia (MOI) Max =4.915844711*e^6 mm^4

                                                  Min = 1.650744173^e^4 mm^4

Section Modulus = I/y= 16507.44173/245 = 67.37731318 mm^3

Case 2:-

Centre of Gravity (C.G) = (-0.004962977, 748.308500648, -930.871310186) mm.

Moment of Inertia (MOI) Max =5.100180436*e^6 mm^4

                                                  Min = 2.058035014^e^4 mm^4

Section Modulus = I/y= 20580.35014/245 = 84.00142914 mm^3

Case 3:-

Centre of Gravity (C.G) = (-0.010860745, 753.949024466, -926.885933164) mm.

Moment of Inertia (MOI) Max =5.383081735 *e^6 mm^4

                                                  Min = 3.297183411^e^4 mm^4

Section Modulus = I/y= 32971.83411/245 = 134.5789147 mm^3

Section Modulus Analysis
Case No	MOI (I)	Distance (y)	S=I/y
1	16507.44173	245	67.37731318
2	20580.35014	245	84.00142914
3	32971.83411	245	134.5789147

 

Thus we can conclude that with an increase in the overall cross-section of the object the Area Moment of inertia increases which are directly proportional to the section modulus.