1. 10.7 Moments of Inertia for an Area about Inclined Axes
In structural and mechanical design,
necessary to calculate the moments and
product of inertia Iu, Iv and Iuv for an area
with respect to a set of inclined u and v
axes when the values
of θ, Ix, Iy and Ixy are known
Use transformation
equations which relate
the x, y and u, v coordinates

2. 10.7 Moments of Inertia for an Area about Inclined Axes
For moments and product of inertia of dA about the u and v axes,

3. 10.7 Moments of Inertia for an Area about Inclined Axes
Integrating,
Simplifying using trigonometric identities,

4. 10.7 Moments of Inertia for an Area about Inclined Axes
Polar moment of inertia about the z axis passing through point O is independent of the u and v axes

5. 10.7 Moments of Inertia for an Area about Inclined Axes
Principal Moments of Inertia
Iu, Iv and Iuv depend on the angle of inclination θ of the u, v axes
To determine the orientation of these axes about which the moments of inertia for the area Iu and Iv are maximum and minimum
This particular set of axes is called the principal axes of the area and the corresponding moments of inertia with respect to these axes are called the principal moments of inertia

6. 10.7 Moments of Inertia for an Area about Inclined Axes
Principal Moments of Inertia
There is a set of principle axes for every chosen origin O
For the structural and mechanical design of a member, the origin O is generally located at the cross-sectional area’s centroid
The angle θ = θp defines the orientation of the principal axes for the area. Found by differentiating with respect to θ and setting the result to zero

7. 10.7 Moments of Inertia for an Area about Inclined Axes
Principal Moments of Inertia
Therefore
Equation has 2 roots, θp1 and
θp2 which are 90° apart and
so specify the inclination of
the principal axes

8. 10.7 Moments of Inertia for an Area about Inclined Axes
Principal Moments of Inertia

9. 10.7 Moments of Inertia for an Area about Inclined Axes
Principal Moments of Inertia
Depending on the sign chosen, this result gives the maximum or minimum moment of inertia for the area
It can be shown that Iuv = 0, that is, the product of inertia with respect to the principal axes is zero
Any symmetric axis represent a principal axis of inertia for the area

10. 10.7 Moments of Inertia for an Area about Inclined Axes
Example 10.9
Determine the principal moments of inertia
for the beam’s
cross-sectional area with
respect to an axis passing
through the centroid.

11. 10.7 Moments of Inertia for an Area about Inclined Axes
View Free Body Diagram
Solution
Moment and product of inertia of the cross-sectional area with respect to the x, y axes have been computed in the previous examples
Using the angles of inclination of principal axes u and v
Thus,

12. 10.7 Moments of Inertia for an Area about Inclined Axes
Solution
For principal of inertia with respect to the u and v axes or

13. 10.7 Moments of Inertia for an Area about Inclined Axes
Solution
Maximum moment of inertia occurs with respect to the selected u axis since by inspection, most of the cross-sectional area is farthest away from this axis
Maximum moment of inertia occurs at the u axis since it is located within ±45° of the y axis, which has the largest value of I

14. 10.8 Mohr’s Circle for Moments of Inertia
It can be found that
In a given problem, Iu and Iv are variables and Ix, Iy and Ixy are known constants
When this equation is plotted on a set of axes that represent the respective moment of inertia and the product of inertia, the resulting graph represents a circle

15. 10.8 Mohr’s Circle for Moments of Inertia
The circle constructed is known as a Mohr’s circle with radius
and center at (a, 0) where

16. 10.8 Mohr’s Circle for Moments of Inertia
Procedure for Analysis
Determine Ix, Iy and Ixy
Establish the x, y axes for the area, with the origin located at point P of interest and determine Ix, Iy and Ixy
Construct the Circle
Construct a rectangular coordinate system such that the abscissa represents the moment of inertia I and the ordinate represent the product of inertia Ixy

17. 10.8 Mohr’s Circle for Moments of Inertia
Procedure for Analysis
Construct the Circle
Determine center of the circle O, which is located at a distance (Ix + Iy)/2 from the origin, and plot the reference point a having coordinates (Ix, Ixy)
By definition, Ix is always positive, whereas Ixy will either be positive or negative
Connect the reference point A with the center of the circle and determine distance OA (radius of the circle) by trigonometry
Draw the circle

18. 10.8 Mohr’s Circle for Moments of Inertia
Procedure for Analysis
Principal of Moments of Inertia
Points where the circle intersects the abscissa give the values of the principle moments of inertia Imin and Imax
Product of inertia will be zero at these points
Principle Axes
To find direction of major principal axis, determine by trigonometry, angle 2θp1, measured from the radius OA to the positive I axis

19. 10.8 Mohr’s Circle for Moments of Inertia
Procedure for Analysis
Principle Axes
This angle represent twice the angle from the x axis to the area in question to the axis of maximum moment of inertia Imax
Both the angle on the circle, 2θp1, and the angle to the axis on the area, θp1must be measured in the same sense
The axis for the minimum moment of inertia Imin is perpendicular to the axis for Imax

20. 10.8 Mohr’s Circle for Moments of Inertia
Example 10.10
Using Mohr’s circle, determine the principle
moments of the beam’s cross-sectional area
with respect to an axis
passing through the
centroid.

21. 10.8 Mohr’s Circle for Moments of Inertia
View Free Body Diagram
Solution
Determine Ix, Iy and Ixy
Moments of inertia and the product of inertia have been determined in previous examples
Construct the Circle
Center of circle, O, lies from the origin, at a distance

22. 10.8 Mohr’s Circle for Moments of Inertia
Solution
With reference point A (2.90, -3.00) connected to point O, radius OA is determined using Pythagorean theorem
Principal Moments of Inertia
Circle intersects I axis at
points (7.54, 0) and
(0.960, 0)

23. 10.8 Mohr’s Circle for Moments of Inertia
Solution
Principal Axes
Angle 2θp1 is determined from
the circle by measuring CCW
from OA to the direction of the
positive I axis

24. 10.8 Mohr’s Circle for Moments of Inertia
Solution
The principal axis for Imax = 7.54(109) mm4
is therefore orientated at an angle θp1 = 57.1°,
measured CCW from the positive x axis
to the positive u axis
v axis is perpendicular
to this axis

25. 10.9 Mass Moment of Inertia
Mass moment of inertia of a body is the property that measures the resistance of the body to angular acceleration
Mass moment of inertia is defined as
the integral of the second moment
about an axis of all the elements of
mass dm which compose the body
Example
Consider rigid body

26. 10.9 Mass Moment of Inertia
For body’s moment of inertia about the z axis,
Here, the moment arm r is the perpendicular distance from the axis to the arbitrary element dm
Since the formulation involves r, the value of I is unique for each axis z about which it is computed
The axis that is generally chosen for analysis, passes through the body’s mass center G

27. 10.9 Mass Moment of Inertia
Moment of inertia computed about this axis will be defined as IG
Mass moment of inertia is always positive
If the body consists of material having a variable density ρ = ρ(x, y, z), the element mass dm of the body may be expressed as dm = ρ dV
Using volume element for integration,

28. 10.9 Mass Moment of Inertia In the special case of ρ being a constant,
When element volume chosen for integration has differential sizes in all 3 directions, dV = dx dy dz
Moment of inertia of the body determined by triple integration
Simplify the process to single integration
by choosing an element volume with
a differential size or thickness in 1
direction such as shell or disk elements

29. 10.9 Mass Moment of Inertia Procedure for Analysis
Consider only symmetric bodies having surfaces which are generated by revolving a curve about an axis
Shell Element
For a shell element having height z, radius y and thickness dy, volume dV = (2πy)(z)dy

30. 10.9 Mass Moment of Inertia Procedure for Analysis Shell Element
Use this element to determine the moment of inertia Iz of the body about the z axis since the entire element, due to its thinness, lies at the same perpendicular distance r = y from the z axis
Disk Element
For disk element having radius y, thickness dz, volume dV = (πy2) dz

31. 10.9 Mass Moment of Inertia Procedure for Analysis Disk Element
Element is finite in the radial direction and consequently, its parts do not lie at the same radial distance r from the z axis
To perform integration using this element, determine the moment of inertia of the element about the z axis and then integrate this result

32. 10.9 Mass Moment of Inertia Example 10.11
Determine the mass moment of inertia of the
cylinder about the z axis. The density of the
material is constant.

33. 10.9 Mass Moment of Inertia Solution Shell Element
For volume of the element,
For mass,
Since the entire element lies at the
same distance r from the z axis, for
the moment of inertia of the element,

34. 10.9 Mass Moment of Inertia Solution
Integrating over entire region of the cylinder,
For the mass of the cylinder
So that

35. 10.9 Mass Moment of Inertia Example 10.12
A solid is formed by revolving the shaded area
about the y axis. If the density of the material is
5 Mg/m3, determine the mass moment of inertia
about the y axis.

36. 10.9 Mass Moment of Inertia Solution Disk Element
Element intersects the curve at the arbitrary point (x, y) and has a mass
dm = ρ dV = ρ (πx2)dy
Although all portions of the
element are not located at the
same distance from the y axis,
it is still possible to determine
the moment of inertia dIy about
the y axis

37. 10.9 Mass Moment of Inertia Solution
In the previous example, it is shown that the moment of inertia for a cylinder is
I = ½ mR2
Since the height of the cylinder is not involved, apply the about equation for a disk
For moment of inertia for the entire solid,

38. 10.9 Mass Moment of Inertia Parallel Axis Theorem
If the moment of inertia of the body about an axis passing through the body’s mass center is known, the moment of inertia about any other parallel axis may be determined by using parallel axis theorem
Considering the body where the z’ axis passes through the mass center G, whereas the corresponding parallel z axis lie at a constant distance d away

39. 10.9 Mass Moment of Inertia Parallel Axis Theorem
Selecting the differential mass element dm, which is located at point (x’, y’) and using Pythagorean theorem,
r 2 = (d + x’)2 + y’2
For moment of inertia of body about the z axis,
First integral represent IG

40. 10.9 Mass Moment of Inertia Parallel Axis Theorem
Second integral = 0 since the z’ axis passes through the body’s center of mass
Third integral represents the total mass m of the body
For moment of inertia about the z axis,
I = IG + md2

41. 10.9 Mass Moment of Inertia Radius of Gyration
For moment of inertia expressed using k, radius of gyration,
Note the similarity between the definition of k in this formulae and r in the equation dI = r2 dm which defines the moment of inertia of an elemental mass dm of the body about an axis

42. 10.9 Mass Moment of Inertia Composite Bodies
If a body is constructed from a number of simple shapes such as disks, spheres, and rods, the moment of inertia of the body about any axis z can be determined by adding algebraically the moments of inertia of all the composite shapes computed about the z axis
Parallel axis theorem is needed if the center of mass of each composite part does not lie on the z axis

43. 10.9 Mass Moment of Inertia Example 10.13
If the plate has a density of 8000kg/m3 and a
thickness of 10mm, determine its mass moment of
inertia about an axis perpendicular to the page and
passing through point O.

44. 10.9 Mass Moment of Inertia Solution
The plate consists of 2 composite parts, the 250mm radius disk minus the 125mm radius disk
Moment of inertia about O is determined by computing the moment of inertia of each of these parts about O and then algebraically adding the results

45. 10.9 Mass Moment of Inertia Solution Disk
For moment of inertia of a disk about an axis perpendicular to the plane of the disk,
Mass center of the disk is located 0.25m from point O

46. 10.9 Mass Moment of Inertia Solution Hole
For moment of inertia of plate about point O,

47. 10.9 Mass Moment of Inertia Example 10.14 The pendulum consists of two
thin robs each having a mass of
100kg. Determine the
pendulum’s mass moment of
inertia about an axis passing
through (a) the pin at point O,
and (b) the mass center G of the
pendulum.

48. 10.9 Mass Moment of Inertia Solution Part (a)
For moment of inertia of rod OA about an axis perpendicular to the page and passing through the end point O of the rob,
Hence,
Using parallel axis theorem,

49. 10.9 Mass Moment of Inertia Solution For rod BC,
For moment of inertia of pendulum about O,

50. 10.9 Mass Moment of Inertia Solution Part (b)
Mass center G will be located relative to pin at O
For mass center,
Mass of inertia IG may be computed in the same manner as IO, which requires successive applications of the parallel axis theorem in order to transfer the moments of inertias of rod OA and BC to G

51. 10.9 Mass Moment of Inertia Solution
Apply the parallel axis theorem for IO,

52. Chapter Summary Area Moment of Inertia
Represent second moment of area about an axis
Frequently used in equations related to strength and stability of structural members or mechanical elements
If the area shape is irregular, a differential element must be selected and integration over the entire area must be performed
Tabular values of the moment of inertia of common shapes about their centroidal axis are available

53. Chapter Summary Area Moment of Inertia
To determine moment of inertia of these shapes about some other axis, parallel axis theorem must be used
If an area is a composite of these shapes, its moment of inertia = sum of the moments of inertia of each of its parts
Product of Inertia
Determine location of an axis about which the moment of inertia for the area is a maximum or minimum

54. Chapter Summary Product of Inertia
If the product of inertia for an area is known about its x’, y’ axes, then its value can be determined about any x, y axes using the parallel axis theorem for product of inertia
Principal Moments of Inertia
Provided moments of inertia are known, formulas or Mohr’s circle can be used to determine the maximum or minimum or principal moments of inertia for the area, as well as orientation of the principal axes of inertia

55. Chapter Summary Mass Moments of Inertia
Measures resistance to change in its rotation
Second moment of the mass elements of the body about an axis
For bodies having axial symmetry, determine using wither disk or shell elements
Mass moment of inertia of a composite body is determined using tabular values of its composite shapes along with the parallel axis theorem

56. Chapter Review

57. Chapter Review

58. Chapter Review

59. Chapter Review