2. Chapter Objectives Chapter Outline Rigid body diagram FBD and RBD
Equilibrium of RBD
Chapter Outline
Conditions for Rigid-Body Equilibrium
Free-Body Diagrams
Equations of Equilibrium
Two and Three-Force Members
Free Body Diagrams (3D)
Equations of Equilibrium (3D)
Constraints and Statical Determinacy

3. 5.1 Conditions for Rigid-Body Equilibrium
For equilibrium when considering about O
For moment about any point, or in this case, point A
For any point

4. 5.2 Free Body Diagrams Support Reactions
Support reaction is when there is contact and the reaction is coming from the floor or the wall
The moment at the support is due to the wall
Different types of supports will have different types of forces

5. 5.2 Free Body Diagrams
cable
weightless link
roller

6. 5.2 Free Body Diagrams roller or pin in confined smooth slot rocker
smooth connecting
surface
member pin connected
To collar on smooth rod

7. member fixed connected
5.2 Free Body Diagrams
smooth pin or hinge
member fixed connected
To collar on smooth rod
fixed support

8. Supports for 2D system
1D Forces
2D Forces
2D forces + 1D moment

9. 5.2 Free Body Diagrams Internal forces
For a rigid body, there is both external and internal forces
For FBD, internal forces within the FBD boundary is not needed
For outside the boundary FBD, it is external forces

10. 5.2 Free Body Diagrams Weight and Center of Gravity
A rigid body are composed of weight of individual smaller rigid bodies, and the self-weight of the rigid body can be represented by a single weight W
The location of the weight W is at center of gravity

11. 5.2 Free Body Diagrams Procedure for Drawing a FBD 1. Sketch a body
Consider if the body is free from all the connections
2. Draw all forces and moments
Draw all external forces and moments acting on the body
Draw self-weight if considered
3. Indicate the magnitude and the directions of forces and moments
Indicate all the magnitude and directions of forces and moments if known
Split all components in x, y or a convenient reference
Display all known distance

12. Example 5.1
Draw the free-body diagram of the uniform beam. The beam has a mass of 100kg.

13. Solution
FBD

14. 5.3 Equations of Equilibrium
For equilibrium of a rigid body in 2D,
∑Fx and ∑Fy represent sums of x and y components of all the forces
∑MO represents the sum of the couple moments and moments of the force components

15. 5.3 Equations of Equilibrium
Sets of Equilibrium Equations
For coplanar equilibrium problems, ∑Fx = 0; ∑Fy = 0; ∑MO = 0
2 alternative sets of 3 independent equilibrium equations,
∑Fx = 0; ∑MA = 0; ∑MB = 0 (A, B is not perpendicular to x)
∑MA = 0; ∑MB = 0 ; ∑MC = 0 (A, B, C are not on the same line)

16. 5.3 Equations of Equilibrium
Procedure for Analysis
Free-Body Diagram
For unknown forces or moments, display the line of action
Put known distances
Equations of Equilibrium
∑MO = 0 and ∑F = 0
Indicate x, y system that simplified calculations
If the result is negative, the direction of forces or moments in FBD in incorrect, hence negate the direction of the forces

17. Example 5.5
Determine the horizontal and vertical components of reaction for the beam loaded. Neglect the weight of the beam in the calculations. (A is a roller and B is a hinge support.)

18. Solution

19. Solution
Equations of Equilibrium

20. 5.4 Two- and Three-Force Members
Two-Force Members
is parts that contain on forces acting at two points, no moments
The two forces are the same in magnitude but opposite directions and lie in the same line of action
Not O.K.
Not O.K.
O.K.
The two forces are in the same line of action

21. 5.4 Two- and Three-Force Members
Forces acting at 3 points
The forces are either meet at a single point in space or they are parallel

22. Example 5.13
The lever ABC is pin-supported at A and connected to a short link BD. If the weight of the members are negligible, determine the force of the pin on the lever at A.

23. Solution Free Body Diagrams BD is a two-force member
Lever ABC is a three-force member
Equations of Equilibrium
Solving,

24. 5.5 Free-Body Diagrams (3D)
Support Reactions
For 3D problems
Reactions forces are at the supports
Some supports may have moments
The directions of forces are indicated by angles α, β and γ

25. 5.5 Free-Body Diagrams

26. 5.5 Free-Body Diagrams

27. Example 5.14
Several examples of objects along with their associated free-body diagrams are shown. In all cases, the x, y and z axes are established and the unknown reaction components are indicated in the positive sense. The weight of the objects is neglected.

29. 5.6 Equations of Equilibrium (3D)
Equilibrium equations in vector notation
2 equations for rigid body
∑F = ∑MO = 0
In vector Cartesian form
∑F = ∑Fxi + ∑Fyj + ∑Fzk = 0
∑MO = ∑Mxi + ∑Myj + ∑Mzk = 0

30. 5.7 Constraints for a Rigid Body
Redundant Constraints
There are more forces and moments from the supports than equilibrium equations
Statically indeterminate: there are more unknown than equations

31. 5.7 Constraints for a Rigid Body
Improper Constraints
There are not enough constraints, hence the system are not in equilibrium

32. 5.7 Constraints for a Rigid Body
Procedure for Analysis
Free Body Diagram
Sketch all bodies
Draw all forces and moments
Put all unknown forces
Display all the known distances
Equations of Equilibrium
Apply equilibrium equations
Apply appropriate reference system, choose the locations where many forces pass

33. Example 5.15
The homogenous plate has a mass of 100kg and is subjected to a force and couple moment along its edges. If it is supported in the horizontal plane by means of a roller at A, a ball and socket joint at B, and a cord at C, determine the components of reactions at the supports.

34. Solution Free Body Diagrams Five unknown reactions acting on the plate
Each reaction assumed to act in a positive coordinate direction
Equations of Equilibrium

35. Solution
Equations of Equilibrium

36. Solution
Solving,