1. ERT 146 ENGINEERING MECHANICS
MOMENTS OF INERTIA
MRS SITI KAMARIAH BINTI MD SA’AT
SCHOOL OF BIOPROCESS ENGINEERING
UNIVERSITI MALAYSIA PERLIS

2. Learning Outcomes: At the end of this topics, student should able to:
Develop method for determining the moment of inertia for an area
Determine the mass moment of inertia.

3. Moment of Inertia Or called second moment of area, I
Measures the efficiency of that shape its resistance to bending
Moment of inertia about the x-x axis and y-y axis.

4. Moment of Inertia of an Area
Unit :
m4, mm4 or cm4

5. Moment of Inertia for common solid shapesx b h y

6. Moment of Inertia of common shapesv b h h x x u u n n v b
Rectangle at one edge
Iuu= bh3/3
Ivv= hb3/3
Triangle
Ixx= bh3/36
Inn= hb3/6

7. Moment of Inertia of common shapesy B y h x H x x x b y y
Ixx = (BH3-bh3)/12
Iyy = (HB3-hb3)/12
Ixx= Iyy = πd4/64

8. Parallel-Axis Theorem for an Area
Used to find the moment of inertia of an area about centroidal axis.

9. Example 10-1: Calculate the moment of inertia at z-z axis.
b=150mm;h=100mm; d=50mm d z x

10. Example 10-2: Calculate the moment of inertia about x-x axis x x
400 mm
24 mm
12 mm
d= 212 mm
200 mm

11. Solution: Ixx of web = (12 x 4003)/12= 64 x 106 mm4
Ixx of flange = (200 x 243)/12= 0.23 x 106 mm4
Ixx from principle axes xx = 0.23 x106 + Ad2
Ad2 = 200 x 24 x 2122 = x 106 mm4
Ixx from x-x axis = 216 x 106 mm4
Total Ixx = ( x 216) x106 =496 x 106 mm4

12. Radius of Gyration of an Area
Unit of length
Used in design of columns in structure.

13. Moment of Inertia for Composite Areas
Many cross-sectional areas consist of a series of connected simpler shapes, such as rectangles, triangles, and semicircles.
In order to properly determine the moment of inertia of such an area about a specified axis, it is first necessary to divide the area into its composite parts and indicate the perpendicular distance from the axis to the parallel centroidal axis for each part.
Use the moment of inertia of an area or parallel axis theorem.

14. Example 10-3 (Example 10.5):

15. Solution Subdivide the cross-section into three-part A,B,D
Determine moment of inertia of each part, for rectangular, I=bh3/12.
Use the parallel axis theorem formula for each part.
Summation for entire cross-section.

16. Moment of Inertia for Composite Areas
Try
Fundamental problems
Problems: till

17. Product of Inertia for an Area
To use this method, first determine the product of inertia for the area as well as its moments of inertia for given x, y axes.

18. Product of Inertia for an Area
Units: m4, mm4.
Product of inertia may either +ve, -ve or zero depending on the location and orientation of the coordinate axes.
If the axis symmetry for an area, product of inertia will be zero.

19. Parallel-Axis Theorem
Passing through the centroid of the area.

20. Example 10-3
Determine the product of inertia about the x and y centroid

21. Subdivide the cross-section into three-part A,B,D
Determine product moment of inertia

22. Solution
Total up the product moment of inertia

23. Product of Inertia for an Area
Try
Example 10.7
Problems: 10-71, ,10-82

24. Mass Moment of Inertia
A measure of the body’s resistance to angular acceleration.
Used in dynamics part, to study rotational motion.
Mass moment of inertia of the body:
Where r= perpendicular distance from the axis to the arbitrary element dm.

25. The axis that is generally chosen for analysis, passes through the body’s mass center G
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,
When ρ being a constant,

26. Procedure for Analysis
Shell Element
For a shell element having height z, radius y and thickness dy, volume dV = (2πy)(z)dy
Disk Element
For disk element having radius y, thickness dz, volume dV = (πy2) dz

27. Parallel-Axis Theorem
For moment of inertia about the z axis,
I = IG + md2

28. Radius of Gyration
For moment of inertia expressed using k, radius of gyration,

29. Example 10-4

31. Exercise:
Try:
Problems: ,96,97,99,100

32. Quiz
The definition of the Moment of Inertia for an area involves an integral of the …
SI units for the Moment of Inertia for an area.
The parallel-axis theorem for an area is applied to ….
The formula definition of the mass moment of inertia about an axis is …

33. Calculate the moment of inertia of the rectangle about the x-axis
2cm
3cm x

34. Thank Q
GOOD LUCK !!!