Lecture 11- 1page

7/27/2019 Lecture 11- 1page

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Vector Mechanics for Engineers: StaticsE i gh t h

E d i t i on

3 - 1

Resolution of a Force Into a Force at O and a Couple

• Force vector F can not be simply moved to O without modifying itsaction on the body.

• Attaching equal and opposite force vectors at O produces no net

effect on the body.

• The three forces may be replaced by an equivalent force vector and

couple vector, i.e, a force-couple system.





E d i t i on

3 - 2

Resolution of a Force Into a Force at O and a Couple

• Moving F from A to a different point O’ requires the

addition of a different couple vector M O’

F r M Or

r

r

×′='

• The moments of F about O and O’ are related,

( )F s M

F sF r F sr F r M

O

Or

r

r

r

r

r

r

r

rr

r

r

r

×+=

×+×=×+=×= ''

• Moving the force-couple system from O to O’ requires the

addition of the moment of the force at O about O’.





E d i t i on

3 - 3

Sample Problem 3.6

Determine the components of the

single couple equivalent to the

couples shown.

SOLUTION:

• Attach equal and opposite 20 lb forces in

the + x direction at A, thereby producing 3

couples for which the moment componentsare easily computed.

• Alternatively, compute the sum of the

moments of the four forces about an

arbitrary single point. The point D is a

good choice as only two of the forces will

produce non-zero moment contributions..





E d i t i on

3 - 4

Sample Problem 3.6

• Attach equal and opposite 20 lb forces in

the + x direction at A

• The three couples may be represented bythree couple vectors,

( )( )

( )( )

( )( ) in.lb180in.9lb20

in.lb240in.12lb20

in.lb540in.18lb30

⋅+=+=

⋅+=+=

⋅−=−=

z

y

x

M

M

M

( ) ( )

( )k

ji M r

rrr

in.lb180

in.lb240in.lb540

⋅+

⋅+⋅−=





E d i t i on

3 - 5

Sample Problem 3.6

• Alternatively, compute the sum of the

moments of the four forces about D.

• Only the forces at C and E contribute tothe moment about D.

( ) ( )

( ) ( )[ ] ( )ik j

k j M M Dr

rr

rrrr

lb20in.12in.9

lb30in.18

−×−+

−×==

( ) ( )

( )k

ji M r

rrr

in.lb180

in.lb240in.lb540

⋅+

⋅+⋅−=

Lecture 11- 1page

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