1. Chapter 6: Center of Gravity and Centroid

2. Chapter 6.1 C.G and Center of Mass

3. Chapter 6.1 C.G and Center of Mass..2
The same way you can find the centroid of the line and the volume.

4. Example 6-2 (pg.251, sections 6.1-6.3)
Locate the centroid of the area shown in figure
Solution I
Differential Element. A differential element of thickness dx is shown in the Figure. The element intersects the curve at the arbitrary point (x,y), and so it has a height of y.
Area and Moment Arms. The area of the element is dA=y dx, and its centroid is located at

5. Con’t Example 6-2 (pg. 251, Sections 6.1-6.3)
Integrations. Applying Equations 6-5 and integrating with respect to x yields

6. Con’t Example 6-2 (pg. 251, Sections 6.1-6.3)
Solution II
Differential Element. The differential element of thickness dy is shown in Figure. The element intersects the curve at the arbitrary point (x,y) and so it has a length (1-x).
Area and Moment Arms. The area of the element is dA = (1-x) dy, and its centroid is located at

7. Con’t Example 6-2 (pg. 251, Sections 6.1-6.3)
Integrations. Applying Equations 6-5 and integrating with respect to y, we obtain

8. Chapter 6: In-class exercise

9. Example 6-6 (pg. 261, Sections 6.1-6.3)
Locate the centroid C of the cross-sectional area for the T-beam
Solution I The y axis is placed along the axis of symmetry so that To obtain we will establish the x axis (reference axis) thought the base of the area.l The area is segmented into two rectangles and the centroidal location for each is established.

10. Con’t Example 6-6 (pg. 261, Sections 6.1-6.3)
Solution II
Using the same two segments, the x axis can be located at the top of the area. Here
The negative sign indicates that C is located below the origin, which is to be expected. Also note that from the two answers in in =13.0 in., which is the depth of the beam as expected

11. Con’t Example 6-6 (pg. 261, Sections 6.1-6.3)
Solution III
It is also possible to consider the cross-sectional area to be one large rectangle less two small rectangles. Hence we have

12. Problem 6-30 (pg. 264, Sections 6.1-6.3)
Determine the distance to the centroid of the shaded area.
Solution

13. 6.6 Moments of Inertia For Areas
By definition moments of inertia with respect to any axis (i.e. x and y) are
Polar moment of inertia
Always positive value
Units:

14. Geometric Properties of An Area and Volume Geometric Properties of An Area and Volume (page 786-787)

15. Geometric Properties of An Area and Volume Geometric Properties of An Area and Volume (page 786-787)

16. Con’t 6.6 Moments of Inertia For Areas
Example 6.14
Lets proof for rectangular area
Solution
Part (a)

17. Example 6-15 (pg. 287 Sections 6.8-6.9)
Determine the moment of inertia of the shaded area about the x axis.
Solution I (CASE I)
A differential element of area that is parallel to the x axis is chose for integration.
dA =(100-x) dy.
Limits of integration wrt y, y=0 to y=200 mm

18. 6-7 Parallel-Axis Theorem
If we know Moment of Inertia of a given axis, we can compute M.I about another parallel axis

19. Con’t 6-7 Parallel-Axis Theorem
As a result:

20. Con’t 6-7 Parallel-Axis Theorem

21. Con’t Example 6-15 (pg. 287 Sections 6.8-6.9)
Solution II (CASE 2)
A differential element parallel to the y axis is chosen for integration.
Use the parallel-axis theorem to determine the moment of inertia of the element with respect to this axis.
For a rectangle having a base b and height h, the moment of inertia about its centroidal axis is

22. Con’t Example 6-15 (pg. 287 Sections 6.8-6.9)
Solution II (CASE 2)
For the differential element , b = dx and h = y, and thus
Since the centroid of the element is at from the x axis, the moment of inertia of the element about this axis is

23. Example 6-18

24. Con’t Example 6-18

25. Example 6-17 (pg. 291, Sections 6.8-6.9)
Determine the moment of inertia of the cross-sectional area of the T-beam about the centroidal axis.
In class workout

26. Problem 6-87 (pg. 294, Sections 6.8-6.9)
Determine the moment of inertia of the shaded area with respect to a horizontal axis passing through the centroid of the section
Solution:

27. Problem 6-92 (pg. 294, Sections 6.8-6.9)
Determine the moment of inertia of the beam’s cross-sectional area about the y axis
Solution:

28. Problem 6-8 (pg. 257, Sections 6.1-6.3)
Determine the location of the centroid of the quarter elliptical plate.
Solution

29. Con’t Problem 6-8 (pg. 257, Sections 6.1-6.3)
Con’t Solution

30. Chapter 6: Concludes….