MOMENT OF A COUPLE
In

Class activities
:
•
Check Homework
•
Reading Quiz
•
Applications
•
Moment of a Couple
•
Concept Quiz
•
Group Problem Solving
•
Attention Quiz
Today’s Objectives
:
Students will be able to
a) define a couple, and,
b) determine the moment of a couple.
READING QUIZ
1. In statics, a couple is defined as __________ separated by a
perpendicular distance.
A) two forces in the same direction
B) two forces of equal magnitude
C) two forces of equal magnitude acting in the same direction
D) two forces of equal magnitude acting in opposite directions
2. The moment of a couple is called a _________ vector.
A) Free
B) Spin
C) Romantic D) Sliding
APPLICATIONS
A torque or moment of 12 N
∙ m is required to rotate the wheel.
Why does one of the two grips of the wheel above require less
force to rotate the wheel?
APPLICATIONS
(continued)
Would older vehicles without power steering have
larger or smaller steering wheels?
When you grip a vehicle’s steering wheel with both
hands, a couple moment is applied to the wheel.
MOMENT OF A COUPLE
The moment of a couple is defined as
M
O
= F
d (using a scalar analysis) or as
M
O
=
r
F
(using a vector analysis).
Here
r
is any position vector from the line of action of
–
F
to the line of action of
F
.
A couple is defined as two
parallel forces with the same
magnitude but opposite in
direction separated by a
perpendicular distance d
.
MOMENT OF A COUPLE
(continued)
Moments due to couples can be added
together using the same rules as adding
any vectors.
The net external effect of a couple is that
the net force equals zero and the
magnitude of the net moment equals F *d.
Since the moment of a couple depends
only on the distance between the
forces, the moment of a couple is a
free vector
. It can be moved anywhere
on the body and have the same
external effect on the body.
EXAMPLE

SCALAR APPROACH
1) Resolve the forces in x and y directions so they can
be treated as couples.
2) Add the two couples to find the resultant couple.
Given
: Two couples act on the
beam with the geometry
shown.
Find:
The resultant couple
Plan:
No! Only the 240 lb components create a couple. Why?
EXAMPLE

SCALAR APPROACH
The x and y components of the
upper

left 300 lb force are:
(4/5)(300 lb) = 240 lb vertically up
(3/5)(300 lb) = 180 lb to the left
Do both of these components form
couples with their matching
components of the other 300 force?
The net moment is equal to:
+
M =
–
(240 lb)(2 ft)
–
(150 lb)(cos 30
º)(2 ft)
=
–
480
–
259.8 =

739.8 ft∙lb
EXAMPLE

SCALAR APPROACH
Now resolve the lower 150 lb force:
(150 lb) (sin 30
°
)
, acting up
(150 lb) (cos 30
°
), acting to the left
Do both of these components create
a couple with components of the
other 150 lb force?
EXAMPLE
–
VECTOR APPROACH
1) Use
M = r
F
to find the couple moment.
2) Set
r
=
r
AB
and
F
= {35
k
} N .
3) Calculate the cross product to find
M
.
Given
:
A 35 N force couple
acting on the rod.
Find
: The couple moment
acting on the rod in
Cartesian vector notation.
Plan:
= {(
–
8.75
–
0)
i
–
(0
–
0)
j
–
(0
–
0
)
k
} N
∙
m
= {
–
8.75
i
+
0
j
+
0
k
}
N
∙
m
M
=
r
AB
F
=
N
∙
m
i
j
k
0
–
0.25 0.1443
0 0 35
EXAMPLE
–
VECTOR APPROACH
r
AB
= { 0
i
–
(0.25)
j
+ (0.25 tan 30
°
)
k
} m
r
AB
= {
–
0.25
j
+ 0.1443
k
} m
F
= {0
i
+
0
j
+
35
k
} N
CONCEPT QUIZ
2.
If three couples act on a body, the overall result is that
A) The net force is not equal to 0.
B) The net force and net moment are equal to 0.
C) The net moment equals 0 but the net force is not
necessarily equal to 0.
D) The net force equals 0 but the net moment is not
necessarily equal to 0 .
1.
F
1
and
F
2
form a couple. The moment
of the couple is given by ____ .
A)
r
1
F
1
B)
r
2
F
1
C)
F
2
†
r
1
D)
r
2
†
F
2
F1
r1
F2
r2
GROUP PROBLEM SOLVING
–
SCALAR APPROACH
1) Use definition of a scalar couple to find P
and F.
2) Determine the net moment (couple).
3) Equate the net moment to zero to find d.
Given
: Two couples act on the
beam. The resultant
couple is zero.
Find
: The magnitudes of the
forces P and F and the
distance d.
PLAN:
GROUP PROBLEM SOLVING
–
SCALAR APPROACH
Now solve this equation for d.
d = (0.6
) N
∙m
/ (4) N =
0.15 m
It was given that the net moment equals zero. So
+
M = (2)(0.3)
–
(4
)(d) = 0
From the definition of a
couple:
P = 2 kN
F = 4 kN
Determine the net moment
+
M = (2)(0.3)
–
(4
)(d)
GROUP PROBLEM SOLVING
–
VECTOR APPROACH
1) Use
M = r
F
to find the couple moment.
2) Set
r
=
r
AB
and
F
= {15
k
} N .
3) Calculate the cross product to find
M
.
Given
:
F
= {15
k
} N and
–
F
= {
–
15
k
} N
Find
: The couple moment
acting on the pipe
assembly using
Cartesian vector
notation.
Plan:
= {( 7.5
–
0 )
i
–
(1.5
–
0)
j
+
k
(0) } N
∙ m
= {
7.5
i
–
1.5
j
} N ∙ m
GROUP PROBLEM SOLVING
–
VECTOR APPROACH
r
AB
= { (0.3
–
0.2 )
i
+ (0.8
–
0.3)
j
+ (0
–
0)
k
} m
= { 0.1
i
+
0.5
j
} m
F
= {15
k
} N
i
=
M
=
r
AB
F
N
∙
m
j
k
0.1 0.5 0
0 0 15
ATTENTION QUIZ
1. A
couple
is applied to the beam as shown. Its moment equals
_____ N
∙m.
A) 50
B) 60
C) 80
D) 100
3
4
5
1m
2m
50 N
2. You can determine the couple
moment as
M
=
r
F
If
F
= {

20
k
} lb, then
r
is
A)
r
BC
B)
r
AB
C)
r
CB
D)
r
BA
SIMPLIFICATION OF FORCE AND COUPLE SYSTEMS
& FURTHER SIMPLIFICATION OF A
FORCE AND COUPLE SYSTEM
Today’s Objectives
:
Students will be able to:
a)
Determine the effect of moving a
force.
b) Find an equivalent force

couple
system for a system of forces and
couples.
In

Class Activities
:
•
Check Homework
•
Reading Quiz
•
Applications
•
Equivalent Systems
•
System Reduction
•
Concept Quiz
•
Group Problem
Solving
•
Attention Quiz
READING QUIZ
1. A
general system
of 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
APPLICATIONS
What are the resultant effects on the person’s hand
when the force is applied in these four different ways?
Why is understanding these difference important when
designing various load

bearing structures?
APPLICATIONS
(continued)
Several forces and a couple moment
are 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 one
couple moment at point O with the
same external effect? How will
you do that?
 
??
SIMPLIFICATION OF FORCE AND COUPLE SYSTEM
(Section 4.7)
When a number of forces and couple
moments 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 systems
since they
have the same
external
effect on the
body.
MOVING A FORCE ON ITS LINE OF ACTION
Moving a force from A to B, when both points are on the
vector’s 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).
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 be
applied at any point on the body.
B
SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM
When several forces and couple moments
act on a body, you can move each force
and its associated couple moment to a
common point O.
Now you can add all the forces and
couple moments together and find one
resultant force

couple moment pair.
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)
W
R
= W
1
+ W
2
(M
R
)
o
= W
1
d
1
+ W
2
d
2
FURTHER SIMPLIFICATION OF A FORCE AND COUPLE
SYSTEM
(Section 4.8)
=
=
In three special cases,
concurrent, coplanar, and parallel
systems
of forces, the system can always be reduced to a single force.
If
F
R
and
M
RO
are perpendicular to each other, then the system
can be further reduced to a single force,
F
R
, by simply moving
F
R
from O to P.
EXAMPLE #1
1) Sum all the x and y components of the forces to find F
RA
.
2) Find and sum all the moments resulting from moving each
force component to A.
3) Shift F
RA
to a distance d such that d = M
RA
/F
Ry
Given:
A 2

D force system
with geometry as shown.
Find:
The equivalent resultant
force and couple
moment acting at A and
then the equivalent
single force location
measured from A.
Plan:
EXAMPLE #1
(continued)
+
F
Rx
= 150 (3/5) + 50
–
100 (4/5)
= 60 lb
+
F
Ry
= 150 (4/5) + 100 (3/5)
= 180 lb
+ M
RA
=
100 (4/5) 1
–
100 (3/5) 6
–
150(4/5) 3 =
–
640 lb∙ft
F
R
= ( 60
2
+ 180
2
)
1/2
=
190 lb
= tan

1
( 180/60) =
71.6
°
The equivalent single force F
R
can be located at a distance d
measured from A.
d = M
RA
/F
Ry
= 640 / 180 =
3.56 ft
.
F
R
EXAMPLE #2
1) Find
F
RO
=
F
i
= F
Rzo
k
2) Find
M
RO
=
(
r
i
F
i
) = M
RxO
i
+ M
RyO
j
3) The location of the single equivalent resultant force is given
as x =
–
M
RyO
/F
RzO
and y = M
RxO
/F
RzO
Given:
The slab is subjected to
three parallel forces.
Find:
The equivalent resultant
force and couple
moment at the origin O.
Also find the location
(x, y) of the single
equivalent resultant
force.
Plan
:
EXAMPLE #2
(continued)
F
RO
= {100
k
–
500
k
–
400
k
} =
–
800
k
N
M
RO
= (3
i
)
(100
k
) + (4
i
+ 4
j
)
(

500
k
)
+
(4
j
)
(

400
k
)
= {
–
300
j
+ 2000
j
–
2000
i
–
1600
i
}
= {
–
3600
i
+
1700
j
}N
∙m
The location of the single equivalent resultant force is given as,
x =
–
M
Ryo
/ F
Rzo
= (
–
1700) / (
–
800) =
2.13 m
y = M
Rxo
/ F
Rzo
= (
–
3600) / (
–
800) =
4.5 m
CONCEPT QUIZ
2. Consider
two couples
acting 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
•
•
•
•
GROUP PROBLEM SOLVING
1) Sum all the x and y components of the two forces to find F
RA
.
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 M
RA
.
Given:
A 2

D force and couple
system as shown.
Find:
The equivalent resultant
force and couple
moment acting at A.
Plan:
GROUP PROBLEM SOLVING
(continued)
+
F
x
= (5/13) 26
–
30 sin 30
°
=
–
5 kN
+
F
y
=
–
(12/13) 26
–
30 cos 30
°
=
–
49.98 kN
+ M
RA
= {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.
F
RA
= ( 5
2
+ 49.98
2
)
1/2
=
50.2 kN
and
= tan

1
(49.98/5)
=
84.3
°
Summing the force components:
GROUP PROBLEM SOLVING
(continued)
Given
: Forces
F
1
and
F
2
are
applied to the pipe.
Find
: An equivalent resultant
force and couple moment
at point O.
Plan
:
a) Find
F
RO
=
F
i
=
F
1
+
F
2
b) Find
M
RO
=
M
C
+
(
r
i
F
i
)
where,
M
C
are any free couple moments in CVN (
none
in this example).
r
i
are the position vectors from the point O to any point on the line
of action of
F
i
.
GROUP PROBLEM SOLVING
(continued)
F
1
= {
–
20
i
–
10
j
+ 25
k
} lb
F
2
= {
–
10
i
+ 25
j
+ 20
k
} lb
F
RO
= {
–
30
i
+ 15
j
+ 45
k
} lb
r
1
= {1.5
i
+ 2
j
} ft
r
2
= {1.5
i
+ 4
j
+ 2
k
} ft
Then,
M
RO
=
(
r
i
F
i
) =
r
1
†
F
1
+
r
2
F
2
= {(50
i
–
37
.
5
j
+ 25
k
) + (30
i
–
50
j
+ 77.5
k )
} lb
∙ft
= {
80
i
–
87
.
5
j
+
102.5
k
}
lb
∙ft
M
RO
= {
} lb
∙ft
i j k
1.5 2 0

20

10 25
i j k
1.5 4 2

10 25 20
+
ATTENTION QUIZ
1. For this force system, the equivalent system at P is
___________ .
A) F
RP
= 40 lb (along +x

dir.) and M
RP
= +60 ft
∙
lb
B) F
RP
= 0 lb and M
RP
= +30 ft
∙
lb
C) F
RP
= 30 lb (along +y

dir.) and M
RP
=

30 ft
∙
lb
D) F
RP
= 40 lb (along +x

dir.) and M
RP
= +30 ft
∙
lb
P
1'
1'
30 lb
40 lb
30 lb
•
x
y
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.
REDUCTION OF A SIMPLE DISTRIBUTED LOADING
In

Class Activities
:
•
Check Homework
•
Reading Quiz
•
Applications
•
Equivalent Force
•
Concept Quiz
•
Group Problem Solving
•
Attention Quiz
=
Today’s Objectives
:
Students will be able to determine an
equivalent force for a distributed load.
READING QUIZ
2. The line of action of the distributed load’s equivalent force
passes through the ______ of the distributed load.
A) Centroid
B) Mid

point
C)
Left edge
D) Right edge
1. The resultant force (F
R
) due to a
distributed load is equivalent to
the _____ under the distributed
loading curve, w = w(x).
A) Centroid B) Arc length
C) Area
D) Volume
x
w
F
R
Distributed load curve
y
APPLICATIONS
To analyze the load’s effect on the steel beams, it is often
helpful to reduce this distributed load to a single force.
How would you do this?
There is a bundle (called a bunk) of 2” x 4” boards
stored on a storage rack. This lumber places a
distributed load (due to the weight of the wood) on
the beams holding the bunk.
APPLICATIONS
(continued)
The uniform wind pressure
is acting on a triangular
sign (shown in light
brown).
To be able to design the joint
between the sign and the sign
post, we need to determine a
single equivalent resultant force
and its location.
DISTRIBUTED LOADING
In such cases, w is a function of x
and has
units of force per length
.
In many situations, a surface area
of a body is subjected to a
distributed load. Such forces are
caused by winds, fluids, or the
weight of items on the body’s
surface.
We will analyze the most common
case of a distributed pressure
loading. This is a uniform load
along one axis of a flat rectangular
body.
MAGNITUDE OF RESULTANT FORCE
The net force
on the beam is given by
+
F
R
=
L
dF =
L
w(x) dx = A
Here
A is the area under the loading
curve w(x)
.
Consider an element of length dx.
The force magnitude dF acting on it is
given as
dF = w(x) dx
LOCATION OF THE RESULTANT FORCE
The total moment about point O is
given as
+ M
RO
=
L
x dF =
L
x w(x) dx
Assuming that F
R
acts at , it will produce
the moment about point O as
+ M
RO
= ( ) (F
R
) =
L
w(x) dx
x
x
x
The force dF will produce a moment of
(x)(dF) about point O.
LOCATION OF THE RESULTANT FORCE
(continued)
Comparing the last two equations,
we get
You will learn more detail later, but
F
R
acts through a point “C,” which is
called the geometric center or
centroid of the area under the loading
curve w(x).
EXAMPLES
Until you learn more about centroids, we will consider only
rectangular and triangular loading diagrams whose centroids are
well defined and shown on the inside back cover of your textbook.
Look at the inside back cover of your textbook. You should find
the rectangle and triangle cases. Finding the area of a rectangle and
its centroid is easy!
Note that triangle presents a bit of a challenge but still is pretty
straightforward.
EXAMPLES
The rectangular load: F
R
= 400
10 = 4,000 lb and = 5 ft.
x
The triangular loading:
F
R
= (0.5) (600) (6) = 1,800 N and = 6
–
(1/3) 6 = 4 m.
Please note
that the centroid in a right triangle is at a distance
one third the width of the triangle as
measured from its base
.
x
Now lets complete the calculations to find the concentrated loads
(which is a common name for the resultant of the distributed load).
CONCEPT QUIZ
2. If F
1
= 1 N, x
1
= 1 m, F
2
= 2 N
and x
2
= 2 m, what is the location
of F
R
, i.e., the distance x.
A) 1 m B) 1.33 m C) 1.5 m
D) 1.67 m E) 2 m
F
R
x
F
2
F
1
x
1
x
2
1. What is the location of F
R
, i.e., the
distance d?
A) 2 m
B) 3 m
C) 4 m
D) 5 m
E) 6 m
F
R
B
A
d
B
A
3 m
3 m
GROUP PROBLEM SOLVING
1)
The distributed loading can be divided into three parts. (one
rectangular loading and two triangular loadings).
2) Find F
R
and its location for each of these three distributed loads.
3) Determine the overall F
R
of the three point loadings and its
location.
Given
:
The loading on the
beam as shown.
Find
:
The equivalent force
and its
location
from point A.
Plan:
GROUP PROBLEM SOLVING
(continued)
For the left triangular loading of
height 8 kN/m and width 3 m,
F
R1
= (0.5) 8 kN/m
3 m = 12 kN
x
1
= (2/3)(3m) = 2 m from A
For the top right triangular loading of height 4 kN/m and width 3 m,
F
R2
= (0.5) (4 kN/m) (3 m) = 6 kN
and its line of action is at =
(1/3)(3m) + 3 =
4 m from A
x
2
For the rectangular loading of height 4 kN/m and width 3 m,
F
R3
= (4 kN/m) (3 m) = 12 kN
and its line of action is at =
(1/2)(3m) + 3 =
4.5 m from A
x
3
GROUP PROBLEM SOLVING
(continued)
For the combined loading of the three forces, add them.
F
R
= 12 kN + 6 kN + 12 kN =
30 kN
+ M
RA
= (2) (12) + 4 (6) + (4.5) 12 =
102 kN
• m
Now, (F
R
x) has to equal M
RA
= 102 kN
• m
So solve for x to find the equivalent force’s location.
Hence, x = (102
kN
• m) / (30 kN) =
3.4 m from A
.
ATTENTION QUIZ
1. F
R
= ____________
A) 12 N
B) 100 N
C) 600 N
D) 1200 N
2. x = __________.
A) 3 m
B) 4 m
C) 6 m
D)
8 m
F
R
100 N/m
12 m
x
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