Trends in Project Management

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Chapter 3Rigid Bodies Equivalent Systems of
ForcesPart -2
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Couple
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Moment of A Couple

• Moment of the force F,

- F
MO rA x F
F

• Moment of the force -F,

MO rB x (-F)
O

• Moment of the couple,

MO rA x F rB x (-F)
MO (rA - rB) x F
MO r x F
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Moment of A Couple
Where d is the perpendicular distance between the
lines of action of F and -F
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Problem 3.69 on page 116

• A piece of plywood in which several holes are
  being drilled has been secured to a workbench by
  means of two nails. Knowing that the drill
  applies a 12 Nm couple of moment on the plywood,
  determine the magnitude of resulting forces
  applied to the nails if they are located to
• At A and B
• At A and C

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Problem 3.69 on page 116

• Nails are at A and B
• 12 Nm (0.45m) F
• ---gt F 26.67 N

F26.67 N
-F 26.67 N
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Equivalent Couples

• the two couples lie in parallel planes, and

• the two couples have the same sense or the
  tendency to cause rotation in the same direction.

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Equivalent Couples
M
20 N
20 N
6 m
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Non- Equivalent Couples
M
1 m
120 N
20 N
120 N
M
20 N
6 m
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Addition of Couples
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Couples Can Be Represented by Vectors

• A couple can be represented by a vector with
  magnitude and direction equal to the moment of
  the couple.

• Couple vectors obey the law of addition of
  vectors.

• Couple vectors are free vectors, i.e., the point
  of application is not significant.

• Couple vectors may be resolved into component
  vectors.

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Sample Problem 3.6 on page 113
Determine the components of the single couple
equivalent to the couples shown.
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Sample Problem 3.6 on page 113
SOLUTION
compute the sum of the moments of the four forces
about D.
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Sample Problem 3.6 on page 113

• ALTERNATIVE SOLUTION
• Attach equal and opposite 20 lb forces in the x
  direction at A, thereby producing 3 couples for
  which the moment components are easily computed.

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Sample Problem 3.6 on page 113

• The three couples may be represented by three
  couple vectors,

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

• Force vector F can not be simply moved to O
  without modifying its action 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.

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Resolution of a Force Into a Force at O and a
Couple
F
F
MO
r
A
A
O
O
Any force F acting at a point A of a rigid body
can be replaced by a force-couple system at an
arbitrary point O. It consists of the force F
applied at O and a couple of moment MO equal to
the moment about point O of the force F in its
original position. The force vector F and the
couple vector MO are always perpendicular to each
other.
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Sample Problem 3.7 on page 114
Replace the couple and force shown by an
equivalent single force applied to the lever
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Sample Problem 3.7 on page 114
First calculate the moment created by the couple
Secondly resolve the given force into a force at
O and a couple
We move the force F to O and add a couple of
moment about O
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Sample Problem 3.7 on page 114
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Problem 3.80 on page 118

• A 135 N vertical force P is applied at A to the
  bracket shown. The bracket is held by two screws
  at B and C.
• Replace P with an equivalent force-couple system
  at B
• Find the two horizontal forces at B and C that
  are equivalent to the couple of moment obtained
  in part a

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Problem 3.79 on page 118
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Problem 3.79 on page 118
Find the two horizontal forces at B and C that
are equivalent to the couple of moment obtained
in part a
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Reduction of a System of Forces to One Force and
One Couple
Any system of forces can be reduced to a
force-couple system at a given point O.
First, each of the forces of the system is
replaced by an equivalent force-couple system at
O. Then all of the forces are added to obtain
a resultant force R, and all of couples are
added to obtain a resultant couple vector MO.
In general, the resultant force R and the couple
vector MO will not be perpendicular to each
other.
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Reduction of a System of Forces to One Force and
One Couple
First, each of the forces of the system is
replaced by an equivalent force-couple system at
O. Then all of the forces are added to obtain
a resultant force R, and all of couples are
added to obtain a resultant couple vector MO.
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Sample Problem 3.8 on page

• SOLUTION
• Compute the resultant force for the forces shown
  and the resultant couple for the moments of the
  forces about A.

1. Find an equivalent force-couple system at B based
   on the force-couple system at A.

1. Determine the point of application for the
   resultant force such that its moment about A is
   equal to the resultant couple at A.

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Sample Problem 3.8 on page
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Sample Problem 3.8 on page