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4.7 Simplification of a Force and Couple System

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Engineering Mechanics: Statics in SI Units, 12e

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Chapter Objectives

• Concept of moment of a force in two and three dimensions

• Method for finding the moment of a force about a specified axis.

• Define the moment of a couple.

• Determine the resultants of non-concurrent force systems

• Reduce a simple distributed loading to a resultant force having a specified location

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Chapter Outline

• Moment of a Force – Scalar Formation

• Cross Product

• Moment of Force – Vector Formulation

• Principle of Moments

• Moment of a Force about a Specified Axis

• Moment of a Couple

• Simplification of a Force and Couple System

• Further Simplification of a Force and Couple System

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4.7 Simplification of a Force and Couple System

• An equivalent system is when the external effects are the same as those caused by the original force and couple moment system

• External effects of a system is the translating and rotating motion of the body

• Or refers to the reactive forces at the supports if the body is held fixed

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4.7 Simplification of a Force and Couple System

• Equivalent resultant force acting at point O and a resultant couple moment is expressed as

• If force system lies in the x–y plane and couple moments are

perpendicular to this plane,

M M

M

F F

O O R R

M M

M

F F

F F

O O R

y y R

x x R

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Procedure for Analysis

• Establish the coordinate axes with the origin located at point O and the axes having a selected orientation

• Force Summation

• Moment Summation

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Example 4.16

A structural member is subjected to a couple moment M and forces F1 and F2. Replace this system with an equivalent

resultant force and couple moment acting at its base, point O.

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Solution

Express the forces and couple moments as Cartesian vectors.

m N k j

k j

M

N j j i

i

r N r u

N F

N k F

CB CB CB

. } 300 400

3 { 4 500

500

} 4 . 166 6

. 249 ) {

1 . 0 ( )

15 . 0 (

1 . 0 15

. 300 0

) 300 ( )

300 (

} 800 {

2 2

2 1





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Force Summation.

m N k j

i

k j

i k

X k k

j

F X r F X r M M

M M

N k j

i

j i

k F

F F

F F

B C

O C

Ro R R

. } 300 650

166 {

0 4 . 166 6

. 249

1 1

. 0 15

. 0 )

800 (

) 1 ( ) 300 400

(

} 800 4

. 166 6

. 249 {

4 . 166 6

. 249 800

;

2 1

2 1

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Concurrent Force System

A concurrent force system is where lines of action of all the forces intersect at a common point O

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Coplanar Force System

• Lines of action of all the forces lie in the same plane

• Resultant force of this system also lies in this plane

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Parallel Force System

Consists of forces that are all parallel to the z axis

Resultant force at point O must also be parallel to this axis

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Reduction to a Wrench

• 3-D force and couple moment system have an equivalent resultant force acting at point O

Resultant couple moment not perpendicular to one another

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Example 4.18

The jib crane is subjected to three coplanar forces. Replace this loading by an equivalent resultant force and specify where the resultant’s line of action intersects the column AB and boom BC.

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Force Summation

 

 

 

 

 

 

N kN

kN N

F

F F

kN kN

kN kN

F

F F

Ry

y Ry

Rx

x Rx

60 . 2 60

. 2

6 . 5 0

5 4 . 2

; 25 . 3 25

. 3

75 . 5 1

5 3 . 2

;

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Solution

For magnitude of resultant force,

For direction of resultant force,

16 . 4

) 60 . 2 ( ) 25 . 3 ( )

( )

( 2 2 2 2

Rx Ry

R

kN

F F

F

7

. 38

25 . 3

60 . tan 2

tan 1 1

÷

÷÷



Rx Ry

F

F

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Solution

Moment Summation

 Summation of moments about point A,

m y

m kN

m kN

m kN

m kn

kN y

kN

M MRA A

458 . 0

) 6 . 1 5 ( 50 4

. 2 ) 2 . 2 5 ( 50 3

. 2

) 6 . 0 ( 6 . 0 ) 1 ( 75 . 1

) 0 ( 60 . 2 ) ( 25 . 3

;

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Solution

Moment Summation

 Principle of Transmissibility

m x

m kN

m kN

m kN

m kn

x kN m

kN M MRA A

177 . 2

) 6 . 1 5 ( 50 4

. 2 ) 2 . 2 5 ( 50 3

. 2

) 6 . 0 ( 6 . 0 ) 1 ( 75 . 1

) ( 60 . 2 ) 2 . 2 ( 25 . 3

;

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