1. 5.4 Two- and Three-Force Members
Simplify some equilibrium problems by recognizing embers that are subjected top only 2 or 3 forces
Two-Force Members
When a member is subject to
no couple moments and forces
are applied at only two points
on a member, the member is
called a two-force member

2. 5.4 Two- and Three-Force Members
Two-Force Members
Example
Forces at A and B are summed to obtain their respective resultants FA and FB
These two forces will maintain translational and force equilibrium provided FA is of equal magnitude and opposite direction to FB
Line of action of both forces is known and passes through A and B

3. 5.4 Two- and Three-Force Members
Two-Force Members
Hence, only the force magnitude must be determined or stated
Other examples of the two-force members held in equilibrium are shown in the figures to the right

4. 5.4 Two- and Three-Force Members
If a member is subjected to only three forces, it is necessary that the forces be either concurrent or parallel for the member to be in equilibrium
To show the concurrency
requirement, consider a body
with any two of the three forces
acting on it, to have line of
actions that intersect at point O

5. 5.4 Two- and Three-Force Members
To satisfy moment equilibrium about O, the third force must also pass through O, which then makes the force concurrent
If two of the three forces parallel,
the point of currency O, is
considered at “infinity”
Third force must parallel to
the other two forces to insect
at this “point”

6. 5.4 Two- and Three-Force Members
Bucket link AB on the back hoe is a typical example of a two-force member since it is pin connected at its end provided its weight is neglected, no other force acts on this member
The hydraulic cylinder is pin connected at its ends, being a two-force member

7. 5.4 Two- and Three-Force Members
The boom ABD is subjected to the weight of the suspended motor at D, the forces of the hydraulic cylinder at B, and the force of the pin at A. If the boom’s weight is neglected, it is a three-force member
The dump bed of the truck operates by extending the hydraulic cylinder AB. If the weight of AB is neglected, it is a two-force member since it is pin-connected at its end points

8. 5.4 Two- and Three-Force Members
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.

9. 5.4 Two- and Three-Force Members
View Free Body Diagram
Solution
FBD
Short link BD is a two-force member, so the resultant forces at pins D and B must be equal, opposite and collinear
Magnitude of the force is unknown but line of action known as it passes through B and D
Lever ABC is a three-force member

10. 5.4 Two- and Three-Force Members
Solution
FBD
For moment equilibrium, three non-parallel forces acting on it must be concurrent at O
Force F on the lever at B is equal but opposite to the force F acting at B on the link
Distance CO must be 0.5m since lines of action of F and the 400N force are known

11. 5.4 Two- and Three-Force Members
Solution
Equations of Equilibrium
Solving,

12. 5.5 Equilibrium in Three Dimensions (FBD)
Consider types of reaction that can occur at the supports
Support Reactions
Important to recognize the symbols used to represent each of these supports and to clearly understand how the forces and couple moments are developed by each support
As in 2D, a force that is developed by a support that restricts the translation of the attached member

13. 5.5 Equilibrium in Three Dimensions (FBD)
Support Reactions
A couple moment is developed when rotation of the attached member is prevented
Example
The ball and socket joint prevents any translation of connecting member; therefore, a force must act on the member at the point of connection
This force have unknown magnitude Fx, Fy and Fz
Magnitude of the force is given by

14. 5.5 Equilibrium in Three Dimensions (FBD)
Support Reactions
The force’s orientation is defined by the coordinate angles α, β and γ
Since connecting member is allow to rotate freely about any axis, no couple moment is resisted by a ball and socket joint
Single bearing supports, single pin and single hinge are shown to support both force and couple moment components

15. 5.5 Equilibrium in Three Dimensions (FBD)
Support Reactions
However, if these supports are used with other bearings, pins or hinges to hold a rigid body in equilibrium and the supports are properly aligned when connected to the body, the force reactions at these supports alone may be adequate for supporting the body
Couple moments become redundant and not shown on the FBD

16. 5.5 Equilibrium in Three Dimensions (FBD)

17. 5.5 Equilibrium in Three Dimensions (FBD)

18. 5.5 Equilibrium in Three Dimensions (FBD)

19. 5.5 Equilibrium in Three Dimensions (FBD)

20. 5.5 Equilibrium in Three Dimensions (FBD)
Ball and socket joint provides a connection for the housing of an earth grader to its frame
Journal bearing supports the end of the shaft

21. 5.5 Equilibrium in Three Dimensions (FBD)
Thrust bearing is used to support the drive shaft on the machine
Pin is used to support the end of the strut used on a tractor

22. 5.5 Equilibrium in Three Dimensions (FBD)
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.

23. 5.5 Equilibrium in Three Dimensions (FBD)
View Free Body Diagram
Solution

24. 5.5 Equilibrium in Three Dimensions (FBD)
Solution

25. 5.5 Equilibrium in Three Dimensions (FBD)
Solution

26. 5.5 Equilibrium in Three Dimensions (FBD)
Solution