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

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

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

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

5.4 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”

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

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

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.

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

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

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

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

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

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

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

5.5 Equilibrium in Three Dimensions (FBD)

5.5 Equilibrium in Three Dimensions (FBD)

5.5 Equilibrium in Three Dimensions (FBD)

5.5 Equilibrium in Three Dimensions (FBD)

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

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

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.

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

5.5 Equilibrium in Three Dimensions (FBD) Solution

5.5 Equilibrium in Three Dimensions (FBD) Solution

5.5 Equilibrium in Three Dimensions (FBD) Solution