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4.6 Moment due to Force
Couples
A Couple is defined as two Forces having the
same magnitude, parallel lines of action, and
opposite sense separated by a perpendicular
distance.
In this situation, the sum of the forces in each
direction is zero, so a couple does not affect
the sum of forces equations A force couple will however tend to rotate the
body it is acting on
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Moment Due to a Force Couple
By multiplying the magnitude of one Force by the
distance between the Forces in the Couple, the
moment due to the couple can be calculated.
M = Fdc The couple will create a moment around an axis
perpendicular to the plane that the couple falls in.
Pay attention to the sense of the Moment (Right
Hand Rule)
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Moment of a Couple
Two couples will have equal moments if
2211
dFdF
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.
Do
example
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Why do we use Force Couples?
The reason we use Force Couples to analyze
Moments is that the location of the axis the
Moment is calculated about does not matter
The Moment of a Couple is constant over the
entire body it is acting on
Equivalent couplestwo couples that
produce the same magnitude and direction(example).
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Couples are Free Vectors
The point of action of a Couple does notmatter
The plane that the Couple is acting in does
not matter All that matters is the orientation of the plane
the Couple is acting in
Therefore, a Force Couple is said to be aFree Vector and can be applied at any pointon the body it is acting
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Example: Force Couple
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4.7 Simplification of a Force and
Couple System (Resolution of
Vectors)
The Moment due to the Force Couple is
normally placed at the Cartesian Coordinate
Origin and resolved into its x, y, and z
components (Mx, My, and Mz).
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Vector Addition of Couples
By applying Varignons Theorem to the Forces in the
Couple, it can be proven that couples can be added
and resolved as Vectors.
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Force Couple System
Two opposing force can be added to a rigid bodywithout affecting the equilibrium of it.
If there is a force acting at a distance from an axis,
two forces of equal magnitude and oppositedirection can be added at the axis with out affectingthe equilibrium of the rigid body.
The original force and its opposing force at the axismake a couple that equates to a moment on therigid body.
The other force at the axis results in the same forceacting on the body
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Force Couple Systems
As a result of this it can be stated that any
force (F) acting on a rigid body may be
moved to any given point on the rigid body as
long as a moment equal to moment of (F)
about the axis is added to the rigid body.
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Resolution of a System of
Forces in 3D
Each Force can be Resolved into a Force
and Moment at the point of interest using the
method just discussed.
The Resultant Force can then be found by
Vector Addition.
The Resultant Moment must also be found
using Vector Addition.
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SIMPLIFICATION OF FORCE AND COUPLE SYSTEMS
Objectives:
:
a) Determine the effect of moving a
force.
b) Find an equivalent force-couplesystem for a system of forces and
couples.
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READING QUIZ
1. A general systemof forces and couple moments acting on a
rigid body can be reduced to a ___ .
A) single force
B) single moment
C) single force and two moments
D) single force and a single moment
2. The original force and couple system and an equivalent
force-couple system have the same _____ effect on a body.
A) internal B) external
C) internal and external D) microscopic
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APPLICATIONS
What are the resultant effects on the persons handwhen the force is applied in these four different ways?
Why is understanding these difference important when
designing various load-bearing structures?
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APPLICATIONS (continued)
Several forces and a couple momentare acting on this vertical section of
an I-beam.
For the process of designing the I-
beam, it would be very helpful if
you could replace the various forces
and moment just one force and onecouple moment at point O with the
same external effect? How will
you do that?
| | ??
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SIMPLIFICATION OF FORCE AND COUPLE SYSTEM
(Section 4.7)
When a number of forces and couplemoments are acting on a body, it is
easier to understand their overall effect
on the body if they are combined into a
single force and couple moment having
the same external effect.
The two force and couple systems are
called equivalent systemssince they
have the same externaleffect on thebody.
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MOVING A FORCE ON ITS LINE OF ACTION
Moving a force from A to B, when both points are on the
vectors line of action, does not change the external effect.
Hence, a force vector is called a sliding vector. (But the
internal effect of the force on the body does depend on where
the force is applied).
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MOVING A FORCE OFF OF ITS LINE OF ACTION
When a force is moved, but not along its line of action, there is
a change in its external effect!Essentially, moving a force from point A to B (as shown above)
requires creating an additional couple moment. So moving a
force means you have to add a new couple.
Since this new couple moment is a free vector, it can beapplied at any point on the body.
B
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SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM
When several forces and couple moments
act on a body, you can move each forceand its associated couple moment to a
common point O.
Now you can add all the forces and
couple moments together and find oneresultant force-couple moment pair.
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If the force system lies in the x-y plane (a 2-D case), then the
reduced equivalent system can be obtained using the following
three scalar equations.
SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM
(continued)
WR = W1 + W2(MR)o= W1 d1+ W2 d2
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EXAMPLE #1
1) Sum all the x and y components of the forces to find FRA.
2) Find and sum all the moments resulting from moving each
force component to A.
3) Shift FRAto a distance d such that d = MRA/FRy
Given: A 2-D force system
with geometry as shown.
Find: The equivalent resultant
force and couple
moment acting at A and
then the equivalentsingle force location
measured from A.
Plan:
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EXAMPLE #1
(continued)
+ FRx
= 150 (3/5) + 50100 (4/5)
= 60 lb
+ FRy = 150 (4/5) + 100 (3/5)
= 180 lb
+ MRA = 100 (4/5) 1100 (3/5) 6
150(4/5) 3 =640 lbft
FR= ( 602+ 1802)1/2 = 190 lb
= tan-1( 180/60) = 71.6
The equivalent single force FRcan be located at a distance d
measured from A.
d = MRA/FRy = 640 / 180 = 3.56 ft.
FR
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CONCEPT QUIZ
2. Consider two couplesacting on a body. The simplest possible
equivalent system at any arbitrary point on the body will have
A) One force and one couple moment.
B) One force.
C) One couple moment.
D) Two couple moments.
1. The forces on the pole can be reduced to
a single force and a single moment at
point ____ .
A) P B) Q C) R
D) S E) Any of these points.
R
Z
S
Q
P
X
Y
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GROUP PROBLEM SOLVING
1) Sum all the x and y components of the two forces to find FRA.
2) Find and sum all the moments resulting from moving each
force to A and add them to the 45 kN m free moment to find
the resultant MRA.
Given:A 2-D force and couple
system as shown.
Find: The equivalent resultant
force and couple
moment acting at A.
Plan:
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GROUP PROBLEM SOLVING (continued)
+ Fx = (5/13) 2630 sin 30
=5 kN
+ Fy =(12/13) 2630 cos 30
=49.98 kN
+ MRA = {30 sin 30(0.3m)30 cos 30(2m)(5/13) 26 (0.3m)
(12/13) 26 (6m)45 } = 239 kN m
Now find the magnitude and direction of the resultant.
FRA= ( 52+ 49.98 2)1/2= 50.2 kN and = tan-1(49.98/5)
= 84.3
Summing the force components:
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ATTENTION QUIZ
1. For this force system, the equivalent system at P is
___________ .
A) FRP= 40 lb (along +x-dir.) and MRP= +60 ft lb
B) FRP= 0 lb and MRP= +30 ft lb
C) FRP= 30 lb (along +y-dir.) and MRP= -30 ft lb
D) FRP= 40 lb (along +x-dir.) and MRP= +30 ft lb
P
1' 1'
30 lb40 lb
30 lb
x
y
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ATTENTION QUIZ
2. Consider three couples acting on a body. Equivalent
systems will be _______ at different points on the body.
A) Different when located
B) The same even when located
C) Zero when located
D) None of the above.
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Examples: Simplification of a Force and
Couple System:
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