1. CE 3305 Engineering FLUID MECHANICS
Lecture 4: Buoyancy and Stability

2. Outline
Forces on curved surfaces
Buoyancy
Stability

3. Forces on Submerged Surfaces
Recall from last time:
Magnitude:
Line of Action:

4. EXAMPLE
Consider the following problem:

5. EXAMPLE
State problem, include sketch

6. EXAMPLE
List Known Properties:

7. EXAMPLE
List Unknown Properties:

8. EXAMPLE
Identify Relevant Governing Principles and Equations:

9. EXAMPLE
Solve for the Unknown(s)

10. EXAMPLE
Solve for the Unknown(s)

11. EXAMPLE Validate/Discuss Results
Problem asks for force/length; then asks for the actual forces on the ties.
Use static force balance, and divide resultant by spacing to get force/unit width.
Use static moment balance to find required forces at top and bottom – this information used to specify bolt strength

12. FORCES ON SUBMERGED OBJECTS
Recall from last time that the force and line of action are determined by
(1) integrating the pressure distribution across the area, and by
(2) integrating the moment distribution about the free surface.
Integral methods will always work, but there is an easier way for reasonably regular geometries.

13. Alternate method
A practical alternate is based on displacement volumes to find components of the resultant force

14. Alternate method
A practical alternate is based on displacement volumes to find components of the resultant force

15. Alternate method
A practical alternate is based on displacement volumes to find components of the resultant force

16. Alternate method
A practical alternate is based on displacement volumes to find components of the resultant force

17. Alternate method
A practical alternate is based on displacement volumes to find components of the resultant force

18. EXAMPLE
Find resultant force on curved surface shown where width is 1 meter into the diagram

19. Example
State problem; include sketch

20. example
List known values

21. example
List unknowns

22. example
Identify Relevant Governing Principles and Equations:

23. Example
Solve for the Unknown(s)

24. Example
Solve for the Unknown(s)

25. Example
Solve for the Unknown(s)

26. Example
Validate/Discuss

27. buoyancy
Objects submerged in a liquid have an upward force applied by the liquid – called buoyant force.
Imagine the block shown,
1 cubic meter block, 1 meter deep in a liquid

28. buoyancy Analyze the forces on the block
1 cubic meter block, 1 meter deep in a liquid
Pressure forces (top, sides, and bottom)
Weight of the block itself

29. buoyancy Analyze the forces on the block
1 cubic meter block, 1 meter deep in a liquid
Pressure forces (top, sides, and bottom)
Weight of the block itself

30. buoyancy
Learn that for any object, the buoyant force will equal the weight of the liquid displaced!
Handy because we only need to understand volumes to compute buoyant force.

31. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

32. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

33. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

34. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

35. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

36. buoyancy
A body floats if the buoyant force equals the weight of the object.
Consider the block shown

37. buoyancy The depth d is called the draft of the vessel.
It (d) is important in shallow water operation of vessels and offshore floating platforms.

38. Stability
Determination of whether an object will remained oriented as placed or not is stability analysis
Important for cruise ships, ferries, submarines, and platforms.
Important for ducks; if a duck is unstable and turns upside down, then it will quack up!
The overturning or righting moment depends on the relative position of the weight and buoyant force lines of action

39. Stability analysis
Weight acts through the center of gravity – CG does not change location relative to the object
Buoyant force acts through the centroid of the submerged section.
The forces create a moment couple that establishes stability

40. Stability analysis
Intersection of a bisector running up through the vessel in the desired condition (a) through the CG and a line of action through the buoyant force vector is called the vessel metacenter.
If the metacenter is above the CG the body is stable
The righting moment is

41. Stability analysis
The metacentric height GM is obtained from the moment of inertia of the wedge that lifts and sinks

42. Stability analysis example
Problem Statement

43. Stability analysis example
Sketch

44. Stability analysis example
Known
3m X 4m X 2m
Draft = 1.2 m
Sea water; SG=1.03
Angle = 8 degrees of arc

45. Stability analysis example
Unknown
MC height
Righting moment

46. Stability analysis example
Equations

47. Stability analysis example
Solution

48. Stability analysis example
Discussion
Use tables for finding geometric properties

49. summary Chapters 1-3 Pressure Hydrostatic equilibrium
Hydrostatic equation
Pressure distributions and forces
Concepts of pressure on curves surfaces
Buoyant force and stability

50. Next Time
Quiz #1,Manometry, Euler’s equation