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Ishik University / Sulaimani
Civil Engineering Department
Chapter -4-
Force System Resultant 1
Ishik
Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
4.3 MOMENT OF A COUPLE
Couple
- two parallel forces.
- same magnitude but opposite direction.
- separated by perpendicular distance d.
Resultant force = 0
Tendency to rotate in specified direction.
Couple moment = sum of moments of both couple forces about any arbitrary
point.
Example;
Position vectors rA and rB are directed from O to A and B, lying on the line of
action of F and –F.
Couple moment about O
M = rA X (-F) + rB X (F)
Couple moment about A
M = r X F
since moment of –F about A = 0
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
2
2
A couple moment is a free vector
- It can act at any point since M depends only on the position vector
r directed between forces and not position vectors rA and rB,
directed from O to the forces.
- Unlike moment of force, it do not require a definite point or axis.
Scalar Formulation Magnitude of couple moment
M = Fd
Direction and sense are determined by
right hand rule.
In all cases, M acts perpendicular to
plane containing the forces.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
3
4.3 MOMENT OF A COUPLE
Vector Formulation
For couple moment,
M = r X F
If moments are taken about point A, moment of –F is zero about this
point.
r is crossed with the force to which it is directed.
Resultant Couple Moment
Couple moments are free vectors and
may be applied to any point P and added
vectorially.
For resultant moment of two couples at
point P,
MR = M1 + M2
For more than 2 moments,
MR = ∑(r X F)
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
4
4.3 MOMENT OF A COUPLE
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5
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Equivalent Couples
• Two couples are equivalent if they produce the same moment.
• Since moment produced by the couple is always perpendicular to the plane
containing the forces, forces of equal couples either lie on the same plane or
plane parallel to one another.
Ishik U
niversity-Su
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i A
ssistant Lectu
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aa Ab
du
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d
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Example 4.6 : Force Couple
4
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niv
ersity-S
ula
imani
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rer - A
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aje
ed
7
Example 4.7 : Force Couple
Ishik
Univ
ersity
-Sula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
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5
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Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
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9
Example 4.8
Determine tile resultant couple moment acting on the beam.
Ans:
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Univ
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-Sula
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10
Determine the resultant couple moment acting on the triangular
plate.
Example 4.9
Ans:
6
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Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
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11
Example 4.10
If F=200 Ib, determine the resultant couple moment. Ish
ik U
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ersity
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Solution;
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Ishik
Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
13
Ishik
Univ
ersity
-Sula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
14
Example 4.11
Determine the required magnitude of force F if the resultant
couple moment on the frame is 200 Ib.ft, clockwise.
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Ishik
Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
15
Solution;
16
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
4.4 Equivalent System
A force has the effect of both translating and rotating a body.
The extent of the effect depends on how and where the force
is applied.
We can simplify a system of forces and moments into a
single resultant and moment acting at a specified point O.
A system of forces and moments is then equivalent to the
single resultant force and moment acting at a specified point
O.
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17
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Point O is on the Line of Action
Consider body subjected to force F applied to point A.
Apply force to point O without altering external effects on
body.
- Apply equal but opposite forces F and –F at O.
4.4 Equivalent System
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Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Point O is on the Line of Action
- Two forces indicated by the slash across them can be
cancelled, leaving force at point O.
- An equivalent system has be maintained between each of
the diagrams, shown by the equal signs.
4.4 Equivalent System
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Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Point O is on the Line of Action
- Force has been simply transmitted along its line of action from point A to point O.
- External effects remain unchanged after force is moved.
- Internal effects depend on location of F.
4.4 Equivalent System
20
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
4.4 Simplification of a Force and Couple System
Simplifying any force and couple system,
FR = ∑F
MR = ∑MC + ∑MO
If the force system lies on the x-y plane and any couple moments are perpendicular to this plane,
FRx = ∑Fx
FRy = ∑Fy
MRo = ∑MC + ∑MO
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21
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
4.4 Simplification of a Force and Couple System
MOVING A FORCE OFF OF ITS LINE OF ACTION
22
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Univ
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ula
imani
Assistan
t Lectu
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aa Abdulm
ajeed
Example 4.12
12
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Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
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Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
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25
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.13
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Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
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27
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
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Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.14
Replace the force system acting on the beam by an equivalent
force and couple moment at point A.
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Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
solution:
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Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.15
Replace the force system acting on the beam by an equivalent
force and couple moment at point A.
16
31
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
solution: