3.64
3 Couple and equivalent systems of forces
Couple, scalar
Let’s consider two equal, opposite, non-collinear forces. Their
sum is equal to zero, hence they have no tendency to produce
linear motion. However, their moment is not equal to zero.
Thus, they will have tendency to rotate the body.
Definition: Couple is a moment produced by two equal,
opposite, non-collinear forces.
Note: the line of action of the two
forces is parallel.
̅ - without a subscript
Notation: 𝑴
Units: 𝑵�∙𝒎� or 𝒍�𝒃�∙𝒇�𝒕�.
Computation: Scalar (mostly 2D)
Moment about 𝑶:�𝑀𝑜 = 𝐹(𝑎 + 𝑑) − 𝐹(𝑎) = 𝐹𝑑 = 𝑀
Note: Important there is no reference to 𝑶�!
�
(
) − 𝐹(𝑎′ ) = 𝐹𝑑 = 𝑀𝑜
Moment about 𝑶′: 𝑀𝑜′ = 𝐹 𝑎′ + 𝑑
= 𝑀
Note: as can be seen from the equivalence of the couples with
regards to different points of application, couple is a free vector
with no importance of point of application
Sign: usual convention for moments (counter clockwise is
positive).
3.65
Couple, Vector
Computation:
Vector Algebra (2D or 3D)
Moment about 𝑶:
̅̅̅̅
̅ + ̅𝒓̅̅𝑩̅ × (−𝑭
̅ ) = (̅̅̅
̅ = 𝒓̅ × 𝑭
̅
𝑴𝒐 = ̅̅̅
𝒓𝑨 × 𝑭
𝒓𝑨 − ̅𝒓̅̅𝑩̅) × 𝑭
̅,
Direction: perpendicular to 𝒓̅ and 𝑭
Sense: Right hand rule or screwdriver rule.
Magnitude:
̅ | = 𝒓 ∙ 𝑭 ∙ 𝒔𝒊𝒏(𝜽) = 𝒅 ∙ 𝑭
|̅̅̅̅
𝑴𝒐 | = |𝒓̅ × 𝑭
Note: 𝒓̅�is a free (no reference to O!) vector between two
arbitrary, conveniently selected points on two lines of action.
Some properties of a couple
1. Couples can be added using vector algebra, like moments
2. For the same 𝑭� and 𝒅�, force direction doesn’t matter:
3. The above is true for couples contained in the same or in
parallel planes. As long as the moment of a couple is the
same, the couples are equivalent.
3.66
4. Couple moment (rotational tendency) can be increased by
increasing force magnitude or arm d:
 Addition of couples: distributive (similar to moments). Can
be shown using Varignon’s theorem.
Couple, Computation in Problems
2D
- scalar (𝒅�∙𝑭�)
- components (2 couples formed by 𝑭𝒙 and 𝑭𝒚 )
̅)
- vector (𝒓̅ × 𝑭
3D
̅)
- almost exclusively vector (𝒓̅ × 𝑭
Couple, Example 1
3.67
Example 2
Example 3
3.68
Example 4
3.69
Force-Couple System, Single Force
A single force applied to a rigid body creates a tendency for a
linear motion and a rotational motion, with respect to an
arbitrary point B.
Let’s shift its line of action so it passes through the point B. The
tendency for creation of linear motion will not change.
However, we will need a couple to maintain tendency for a
rotational motion.
This can be done by adding two opposite and collinear forces
̅ and passing through B:
with the line of action parallel to 𝑭
Note: All 3 systems are equivalent – they lead to the same
result.
The combination of a force and a couple is called a force-couple
system.
3.70
Force-Couple System, 3D
Similarly, force-couple can be computed for 3D body:
̅ applied at 𝑨� is equivalent to the force
where original force 𝑭
and a couple at point 𝑶�.
̅ = 𝒓̅ × 𝑭
̅ is a free vector (not attached to 𝑷�) and 𝒓̅ is a
Here 𝑴
̅ , i.e.
position vector from 𝑶� to any point on a line of action of 𝑭
not restricted to ̅̅̅̅̅.
𝒓𝑶𝑨
Applications of Force-Couple Systems
Force to force-couple equivalency is very important in statics,
since it allows us to shift the line of action of a force and :
1) To analyze the effect of action of a force with respect to an
arbitrary (critical) point of interest within a structure
2) To (sometimes) simplify calculations
3) To compute resultants of systems of forces at particular
points
3.71
System of Forces-Couples
A system of forces-couples is equivalent to a force-couple
̅ �acting
system at an arbitrary point 𝑶�, where resultant force 𝑹
at 𝑶� is:
̅ = ∑𝑭
̅
𝑹
̅ is the sum of all couples plus the sum
and a resultant couple 𝑴
of moments of the original forces about 𝑶�:
̅̅̅̅̅̅
̅ + ∑ ̅̅̅̅̅
̅ + ∑ 𝒓̅ × 𝑭
̅
𝑴𝑹𝑶 = ∑ 𝑴
𝑴𝑶 = ∑ 𝑴
Where the first additive in the last sum is a sum of all couples
and the second one is a sum of all moments about 𝑶�.
Force-Couple Systems, Example 1
3.72
Example 2
Example 3
3.73
Example 4
3.74
Elimination of a Couple
The process used to shift the line of action of a force can be
reversed: line of action can be shifted, so that a force-couple
system is reduced to just a force.
𝑴
̅ �at the new location 𝑨�
The distance = 𝑭 . so that the force�𝑭
̅ about the old location 𝑩�.
should create the moment 𝑴
Since any system of forces and couples in 2D can be reduced to
a force and a couple resultants, by shifting the line of action of
the resultant force one can eliminate the resultant couple.
2D Resultants
Direction and Magnitude of a resultant force can be found by
the vectorial addition of forces (tail-to-head). If force was a free
vector, that’d be all.
However, force is a sliding vector, hence the line of action also
matters.
Direction, Magnitude, and Line of Action can be found by one
the two basic methods.
3.75
 Shifting the Line of Action of a Force
2D Resultants, Example 1
3.76
Example 2
 Sequential Two-Force Addition
To add two forces apply Principle of Transmissibility (i) and
Parallelogram Law of Vector Addition (ii).
Notes:
Sequence of forces doesn’t
matter.
Seldom used for more than 3
forces.
3.77
 Two-Force Addition via Force-Couple equivalency
Use of force-couple equivalency with respect to an arbitrary
reference point.
1) Shift lines of action to 𝑶� and introduce
couples for equivalency.
̅ = ∑𝑭
̅ and
2) Calculate resultant force 𝑹
couple
𝑴 = ∑𝑴 = ∑𝑭 ∙ 𝒅
3) Replace resultant force-couple system by
a single force (eliminate couple by
̅ ).
shifting line of action of 𝑹
̅ ≠ 𝟎. When 𝑹
̅ = 𝟎 you can’t
Note: Can do that as long as 𝑹
eliminate couple.
Similar operation can be performed in 3D. However, only
̅ perpendicular to 𝑹
̅ �can be eliminated by
component of 𝑴
̅ . Remaining system of parallel 𝑹
̅ �and�𝑴
̅ �is called
shifting 𝑹
“wrench” (see page 130 in the book). System of parallel 3D
forces can be, however, be reduced to a single resultant force,
̅.
𝑹
3.78
Force Addition, Example 1
Example 2
3.79
Example 3
Example 4
3.80