1. Moment of Inertia, CE 3003 notes in English, Mechanics of Materials notes in English

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1. Moment of Inertia

1.1 Moment of Inertia (M.I.)

1.1.1 Definition

The Moment of Inertia (M.I.) of a body is a property that quantifies the rotational inertia or resistance to angular acceleration when a torque is applied. It depends on the shape of the object and the distribution of mass around an axis of rotation.

Mathematically, the Moment of Inertia is given by:

$$I = \int r^2 dm$$

where:

• $I$ is the moment of inertia,
• $r$ is the perpendicular distance from the axis of rotation to an element of mass,
• $dm$ is the differential mass element.

In simple terms, the moment of inertia tells us how hard it is to rotate an object around a given axis.

1.1.2 M.I. of Plane Lamina

For a plane lamina (a flat, thin plate), the Moment of Inertia is the sum (or integral) of the moments of inertia of each small mass element of the lamina. The formula depends on the shape and axis of rotation.

1.1.3 Radius of Gyration

The radius of gyration (denoted as $k$) is the distance from the axis of rotation to a point where the entire mass of the body can be assumed to be concentrated without changing the Moment of Inertia. It is given by:

$$k = \sqrt{\frac{I}{m}}$$

where:

• $I$ is the moment of inertia,
• $m$ is the total mass of the body.

1.1.4 Section Modulus

The section modulus (Z) is a measure of the strength of a cross-section, especially in bending. It is defined as:

$$Z = \frac{I}{y_{\text{max}}}$$

where:

• $I$ is the moment of inertia of the section about the neutral axis,
• $y_{\text{max}}$ is the distance from the neutral axis to the furthest point of the section.

1.1.5 Parallel and Perpendicular Axes Theorems (without derivations)

Parallel Axis Theorem

This theorem states that the moment of inertia about any axis parallel to an axis through the centroid of the object is the sum of the moment of inertia about the centroidal axis and the mass of the object times the square of the distance between the axes:

$$I = I_{\text{centroid}} + md^2$$

where:

• $I$ is the moment of inertia about the new axis,
• $I_{\text{centroid}}$ is the moment of inertia about the centroidal axis,
• $m$ is the mass of the object,
• $d$ is the distance between the centroidal axis and the new axis.

Perpendicular Axis Theorem

For a flat, planar object, the perpendicular axis theorem states that the sum of the moments of inertia about two perpendicular axes in the plane of the object is equal to the moment of inertia about an axis perpendicular to the plane:

$$I_z = I_x + I_y$$

where:

• $I_z$ is the moment of inertia about the perpendicular axis,
• $I_x$ and $I_y$ are the moments of inertia about two perpendicular axes in the plane.

1.1.6 M.I. of Common Sections (with derivations)

Below are the moments of inertia for common shapes, derived with respect to their centroidal axes:

Rectangle (about centroidal axis)

$$I = \frac{1}{12} b h^3$$

where:

• $b$ is the base,
• $h$ is the height.

Square (about centroidal axis)

$$I = \frac{1}{12} a^4$$

where $a$ is the side of the square.

Circle (about centroidal axis)

$$I = \frac{\pi r^4}{4}$$

where $r$ is the radius.

Semi-circle (about centroidal axis)

$$I = \frac{\pi r^4}{8}$$

where $r$ is the radius.

Quarter-circle (about centroidal axis)

$$I = \frac{\pi r^4}{16}$$

where $r$ is the radius.

Triangle (about centroidal axis)

$$I = \frac{1}{36} b h^3$$

where $b$ is the base, and $h$ is the height.

1.2 Moment of Inertia of Various Sections

1.2.1 Symmetrical I-section

The moment of inertia for a symmetrical I-section can be calculated using the formula:

$$I = I_{\text{flange}} + I_{\text{web}}$$

For the individual flange and web, you can use standard moment of inertia formulas.

1.2.2 Channel Section

For a channel section, the M.I. can be computed as the sum of the individual sections of the channel. Use the Parallel Axis Theorem for calculating M.I. about the centroid.

1.2.3 T-section

For a T-section, the moment of inertia is found by summing the moments of inertia of the flange and the web, using the Parallel Axis Theorem.

1.2.4 Angle Section

The moment of inertia for an angle section can be derived by treating the angle as a combination of two perpendicular legs, using the Parallel Axis Theorem.

1.2.5 Hollow Sections

The moment of inertia for hollow sections is computed similarly to solid sections but accounting for the difference between outer and inner radii (or dimensions).

1.2.6 Built-up Sections about Centroidal Axes and Other Reference Axes

For built-up sections, the M.I. is calculated by breaking the section into simpler shapes and then using the Parallel Axis Theorem to combine their contributions.

1.3 Polar Moment of Inertia of Solid Circular Sections

The polar moment of inertia ($J$) is the moment of inertia about an axis perpendicular to the plane of the object. For a solid circular section, it is given by:

$$J = \frac{\pi r^4}{2}$$

where:

• $r$ is the radius of the circle.

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Diagrams (Conceptual Examples):

Moment of Inertia of a Rectangle about Centroidal Axis:

  ________
 |        |  h
 |        |
 |        |
 |________|
      b

For a rectangle with base $b$ and height $h$, the moment of inertia about its centroidal axis is $I = \frac{1}{12} b h^3$.

Moment of Inertia of a Circle:

   ________
  /        \
 /          \
 \          /
  \________/
       r

For a circle with radius $r$, the moment of inertia about its centroidal axis is $I = \frac{\pi r^4}{4}$.