Equilibrium; Free Body Diagrams; Equations of Equilibrium; Equilibrium of Coplanar Concurrent and Parallel Force

Systems

Lessandro Estelito O. GARCIANO

Associate Professor

Department of Civil Engineering

De La Salle University

Definition and Meaning of Equilibrium

o Equilibrium is the term used to designate the condition where the resultant of a force system is zero.

o The physical meaning of equilibrium, as applied to a body, is that the body is either at rest or is moving along a straight line path with constant velocity.

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Definition and Meaning of Equilibrium

o Both these statements are implied in Newtons first law of motion; namely, a particle acted upon by a balanced force system has no acceleration.

o Problems in mechanics always involve the interaction of bodies upon one another.

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Free-Body Diagrams

o Successful solution of these problems generally requires that the bodies be isolated from one another so that the forces involved may be analyzed and unknown forces determined.

o An isolated view of a body which shows only the external forces exerted on the body is called a free-body diagram (or FBD).

o The external forces are caused either by direct bodily contact or by gravitational or magnetic attraction.

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Steps involved in drawing a FBD

1. Draw a diagram of the body completely isolated from all bodies. The free body may consist of an entire assembled structure or any combination or part of it.

2. Represent the action of each body or support that has been removed by a force (or its components) as given in the previous table

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Steps involved in drawing a FBD

3. Label each force by its magnitude if known, or by a symbol is unknown. If the sense of an unknown force along its line of action should be incorrectly assumed, the solution will give a negative sign, but its magnitude will nevertheless be correct.

4. The FBD should be legibly and neatly drawn, and of sufficient size that all pertinent data can be clearly depicted.

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Sample Problem

In each of the following examples, the body to be isolated is shown in the left-hand diagram, and an incomplete FBD of the isolated body is shown on the right.

Add whatever forces are necessary in each case to form a complete FBD.

The weights of the bodies are negligible unless otherwise indicated.

Dimensions and numerical values are omitted for simplicity.

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Sample Problem

In each of the following examples, the body to be isolated is shown in the left-hand diagram, and either a wrong or an incomplete free-body diagram (FBD) is shown on the right.

Make whatever changes or additions are necessary in each case to form a correct and complete FBD.

The weights of the bodies are negligible unless otherwise indicated.

Dimensions and numerical values are omitted for simplicity.

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Sample Problem

Draw a complete and correct free-body diagram of each of the bodies designated in the statements.

The weights of the bodies are significant only if the mass is stated.

All forces, known and unknown, should be labeled. (Note: The sense of some reaction components cannot always be determined without numerical calculation.)

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Equations of Equilibrium

o Equilibrium is defined as the condition when the resultant of a force system is equal to zero.

o The FBD should be legibly and neatly drawn, and of sufficient size that all pertinent data can be clearly depicted.

o Recall that the most general force system may be reduced to a concurrent force system at any arbitrary point plus a couple which equals the moment sum of the original system about that point.

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Equations of Equilibrium

o When the force-couple system is equal to zero, the basic vector equations of equilibrium therefore are

and

which are equivalent to the following six scalar equations:

and

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Sample Problem

The forces on the gusset plate of a joint in a bridge truss act as shown. Determine the values of P and F to maintain equilibrium of the joint. Ans: F = 800 lb : P = 3342 lb

3000 lb

P

4000 lb

F

45o 15o

60o

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Sample Problem

Two beams supported as shown and separated by a roller at C, jointly carry the indicated loads. On CD, the load varies uniformly from an intensity of 100 lb/ft to an intensity of 200 lb/ft, while AB supports a concentrated load of 500 lb. Determine the values

of the reactions at A, B and D. Ans: AV = 600 lb, AH = 0 lb, BV = 700 lb, DV = 1000 lb, DH = 0 lb

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D

A

100 lb/ft

8

200 lb/ft

B

4 4 4

500 lb

C AH

AV DV

DH

Check Your Understanding

Two cylinders, A of weight 400 lb. and B of weight 200 lb., rest on smooth inclines. They are connected by a bar of negligible weight hinged to each cylinder at its geometric center by smooth pins. Find the force P acting as shown that will hold the system in the given position. P = 107.7 lb

A

B

60o 45o

60o

15o

P

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