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

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



Two-Force MembersExample 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



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



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



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”



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



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



Example 5.13The 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.



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

View Free Body Diagram



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



Solution Equations of

Equilibrium

Solving,

kNF

kNF

FF

F

NFF

F

A

A

y

A

x

32.1

07.1

045sin3.60sin

;0

040045cos3.60cos

;0

3.604.0

7.0tan 1

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



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 by222zyx FFFF



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



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











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



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



Example 5.14Several 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.



Solution

View Free Body Diagram



Solution



Solution



Solution

Documents

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