Chapter 7 Moment of Inertia

Learning Objectives Introduction Moment of Inertia of a Plane Area Units of Moment of Inertia Methods for Moment of Inertia Moment of Inertia by Routh’s Rule Moment of Inertia by Integration Moment of Inertia of a Rectangular Section Moment of Inertia of a Hollow Rectangular Section Theorem of Perpendicular Axis Moment of Inertia of a Circular Section Moment of Inertia of a Hollow Circular Section Theorem of Parallel Axis Moment of Inertia of a Triangular Section Moment of Inertia of a Semicircular Section Moment of Inertia of a Composite Section Moment of Inertia of a Built-up Section

Introduction We have already discussed in Art. 3.2 that the moment of a force (P) about a point, is the product of the force and perpendicular distance (x) between the point and the line of action of the force (i.e. P.x). This moment is also called first moment of force. If this moment is again multiplied by the perpendicular distance (x) between the point and the line of action of the force i.e. P.x (x) = Px2, then this quantity is called moment of the moment of a force or second moment of force or moment of inertia (briefly written as M.I.). Sometimes, instead of force, area or mass of a figure or body is taken into consideration. Then the second moment is known as second moment of area or second moment of mass. But all such second moments are broadly termed as moment of inertia. In this chapter, we shall discuss the moment of inertia of plane areas only.

Moment of Inertia of a Plane Area Consider a plane area, whose moment of inertia is required to be found out. Split up the whole area into a number of small elements. Let a1, a2, a3, ... = Areas of small elements, and r1, r2, r3, ... = Corresponding distances of the elements from the line about which the moment of inertia is required to be found out. Now the moment of inertia of the area, I = a1 r12 + a2 r22 + a3 r32 + ... = ∑a r2 Units of Moment of Inertia As a matter of fact the units of moment of inertia of a plane area depend upon the units of the area and the length. e.g., If area is in m2 and the length is also in m, the moment of inertia is expressed in m4.   If area in mm2 and the length is also in mm, then moment of inertia is expressed in mm4.

Moment of Inertia By Routh’s Rule The Routh’s Rule states, if a body is symmetrical about three mutually perpendicular axes*, then the moment of inertia, about any one axis passing through its centre of gravity is given by: I = A (or M ) X S ... (For a Square or Rectangular Lamina) 3 I = A (or M ) X S ... (For a Circular or Elliptical Lamina) 4 I = A (or M ) X S ... (For a Spherical Body) 5 where A = Area of the plane area M = Mass of the body, and S = Sum of the squares of the two semi-axis, other than the axis, about which the moment of inertia is required to be found out.

Moment of Inertia By Integration The moment of inertia of an area may also be found out by the method of integration as discussed below: Consider a plane figure, whose moment of inertia is required to be found out about X-X axis and Y-Y axis as shown in Fig 7.1. Let us divide the whole area into a no. of strips. Consider one of these strips. Let dA = Area of the strip x = Distance of the centre of gravity of the strip on X-X axis and y = Distance of the centre of gravity of the strip on Y-Y axis. We know that the moment of inertia of the strip about Y-Y axis = dA . x2 Now the moment of inertia of the whole area may be found out by integrating above equation. i.e.,  IYY = ∑ dA . x2 Similarly IXX = ∑ dA . y2  In the following pages, we shall discuss the applications of this method for finding out the moment of inertia of various cross-sections. Fig. 7.1. Moment of inertia by integration.

Moment of Inertia of a Rectangular Section Consider a rectangular section ABCD as shown in Fig. 7.2 whose moment of inertia is required to be found out. Let b = Width of the section and d = Depth of the section. Now consider a strip PQ of thickness dy parallel to X-X axis and at a distance y from it as shown in the figure .: Area of the strip = b.dy We know that moment of inertia of the strip about X-X axis, = Area × y2 = (b. dy) y2 = b. y2. dy Now *moment of inertia of the whole section may be found out by integrating the above equation for the whole length of the lamina i.e. from Fig. 7.2. Rectangular section.

Example Find the moment of inertia of a hollow rectangular section about its centre of gravity if the external dimensions are breadth 60 mm, depth 80 mm and internal dimensions are breadth 30 mm and depth 40 mm respectively. Solution Given: External breadth (b) = 60 mm; External depth (d) = 80 mm ; Internal breadth (b1) = 30 mm and internal depth (d1) = 40 mm.  We know that moment of inertia of hollow rectangular section about an axis passing through its centre of gravity and parallel to X-X axis,

Theorem of Perpendicular Axis It states, If IXX and IYY be the moments of inertia of a plane section about two perpendicular axis meeting at O, the moment of inertia IZZ about the axis Z-Z, perpendicular to the plane and passing through the intersection of X-X and Y-Y is given by: IZZ = IXX + IYY Proof Consider a small lamina (P) of area da having co-ordinates as x and y along OX and OY two mutually perpendicular axes on a plane section as shown in Fig. 7.4.  Now consider a plane OZ perpendicular to OX and OY. Let (r) be the distance of the lamina (P) from Z-Z axis such that OP = r. From the geometry of the figure, we find that r2 = x2 + y2 We know that the moment of inertia of the lamina P about X-X axis, IXX = da. Y2 ...[ I = Area × (Distance)2] Similarly, IYY= da. X2 and IZZ = da. r2 = da (x2 + y2) ...( r2 = x2 + y2) = da. x2 + da. y2 = IYY + IXX Fig. 7.4. Theorem of perpendicular axis

Moment of Inertia of a Circular Section Consider a circle ABCD of radius (r) with centre O and X-X' and Y-Y' be two axes of reference through O as shown in Fig. 7.5. Now consider an elementary ring of radius x and thickness dx. Therefore area of the ring, da = 2 ԉx. dx and moment of inertia of ring, about X-X axis or Y-Y axis = Area × (Distance)2 = 2 ԉ x. dx × x2 = 2 ԉ x3. dx Now moment of inertia of the whole section, about the central axis, can be found out by integrating the above equation for the whole radius of the circle i.e., from 0 to r. Fig. 7.5. Circular section

Example Find the moment of inertia of a circular section of 50 mm diameter about an axis passing through its centre. Solution Given: Diameter (d) = 50 mm  We know that moment of inertia of the circular section about an axis passing through its centre,

Moment of Inertia of a Hollow Circular Section Consider a hollow circular section as shown in Fig.7.6, whose moment of inertia is required to be found out. Let D = Diameter of the main circle, and d = Diameter of the cut out circle. We know that the moment of inertia of the main circle about X-X axis Fig. 7.6. Hollow circular section

Example A hollow circular section has an external diameter of 80 mm and internal diameter of 60 mm. Find its moment of inertia about the horizontal axis passing through its centre. Solution Given : External diameter (D) = 80 mm and internal diameter (d) = 60 mm. We know that moment of inertia of the hollow circular section about the horizontal axis passing through its centre,

Theorem of Parallel Axis It states, If the moment of inertia of a plane area about an axis through its centre of gravity is denoted by IG, then moment of inertia of the area about any other axis AB, parallel to the first, and at a distance h from the centre of gravity is given by: I AB = IG + ah2 where IAB = Moment of inertia of the area about an axis AB, lG = Moment of Inertia of the area about its centre of gravity a = Area of the section, and h = Distance between centre of gravity of the section and axis AB. Proof Consider a strip of a circle, whose moment of inertia is required to be found out about a line AB as shown in Fig. 7.7. Let da = Area of the strip y = Distance of the strip from the centre of gravity the section and h = Distance between centre of gravity of the section and the axis AB.

We know that moment of inertia of the whole section about an axis passing through the centre of gravity of the section = da. Y2 and moment of inertia of the whole section about an axis passing through its centre of gravity,  IG = å da. Y2 Moment of inertia of the section about the axis AB, It may be noted that å h2 . da = a h2 and å y2 . da = IG [as per equation (i) above] and å da.y is the algebraic sum of moments of all the areas, about an axis through centre of gravity of the section and is equal to a. y , where y is the distance between the section and the axis passing through the centre of gravity, which obviously is zero.

Moment of Inertia of a Triangular Section Consider a triangular section ABC whose moment of inertia is required to be found out. Let b = Base of the triangular section and h = Height of the triangular section. Now consider a small strip PQ of thickness dx at a distance of x from the vertex A as shown in Fig. 7.8. From the geometry of the figure, we find that the two triangles APQ and ABC are similar. Therefore, Fig. 7.8. Triangular section

Example A hollow triangular section shown in Fig. 7.9 is symmetrical about its vertical axis. Find the moment of inertia of the section about the base BC. Solution Given : Base width of main triangle (B) (b) = 120 mm; Height of main triangle (H) = 100 mm and = 180 mm; Base width of cut out triangle height of cut out triangle (h) = 60 mm. We know that moment of inertia of the triangular, section about the base BC, Fig. 7.9.

Moment of Inertia of a Semicircular Section Consider a semicircular section ABC whose moment of inertia is required to be found out as shown in Fig. 7.10. Let r = Radius of the semicircle. We know that moment of inertia of the semicircular section about the base AC is equal to half the moment of inertia of the circular section about AC. Therefore moment of inertia of the semicircular section ABC about the base AC, Fig. 7.10. Semicircular section ABC.

Example A hollow semicircular section has its outer and inner diameter of 200 mm and 120 mm respectively as shown in Fig. 7.11. What is its moment of inertia about the base AB ? Fig. 7.11.

Moment of Inertia of a Composite Sections Solutions Given: Outer diameter (D) = 200 mm or Outer Radius (R) = 100 mm and inner diameter (d) = 120 mm or inner radius (r) = 60 mm. We know that moment of inertia of the hollow semicircular section about the base AB, IAB = 0.393 (R4 – r4) = 0.393 [(100)4 – (60)4] = 34.21 × 106 mm4 Moment of Inertia of a Composite Sections The moment of inertia of a composite section may be found out by the following steps : First of all, split up the given section into plane areas (i.e., rectangular, triangular, circular etc., and find the centre of gravity of the section). Find the moments of inertia of these areas about their respective centres of gravity.  

Now transfer these moment of inertia about the required axis (AB) by the Theorem of Parallel Axis, i.e., IAB = IG + ah2 where IG = Moment of inertia of a section about its centre of gravity and parallel to the axis. a = Area of the section, h = Distance between the required axis and centre of gravity of the section. The moments of inertia of the given section may now be obtained by the algebraic sum of the moment of inertia about the required axis. Example A rectangular hole is made in a triangular section as shown in Fig. 7.19. Determine the moment of inertia of the section about X-X axis passing through its centre of gravity and the base BC.

Solution As the section is symmetrical about Y-Y axis, therefore centre of gravity of the section will lie on this axis. Let y be the distance between the centre of gravity of the section and the base BC.

Moment of Inertia of a Built –Up Section A built-up section consists of a number of sections such as rectangular sections, channel sections, I-sections etc., A built-up section is generally made by symmetrically placing and then fixing these section by welding or riveting. It will be interesting to know that a built-up section behaves as one unit. The moment of inertia of such a section is found out by the following steps: Find out the moment of inertia of the various sections about their respective centres of gravity as usual. Now transfer these moments of inertia about the required axis (say X-X axis or Y-Y axis) by the Theorem of Parallel Axis. Example A built up section is made by needing too stable and two channel sections as shown in Fig. 7.22.

Determine moment of inertia of a built up section about X-X axis passing through centre of gravity of the section. Solution As the section is symmetrical about X-X axis and Y-Y axis therefore centre of gravity of the section will coincide with the geometrical centre of section.   We know that the moment of inertia of one top or bottom plate about an axis through its centre os gravity and parallel to X-X axis,

Now moment of inertia of the whole built-up section about an axis through its centre of gravity and parallel to X-X axis,  IXX = (6.5 × 106) + (3.0 × 106) + (0.6 × 106) = 10.1 × 106 mm4 Ans.