College of Engineering | University of Nebraska–Lincoln



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.

Notation: [pic] - without a subscript

Units: ������∙������ or ������������∙������������.

Computation: Scalar (mostly 2D)

Moment about [pic]:[pic]

Note: Important there is no reference to ������!

Moment about [pic]:[pic]

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).

Couple, Vector

Computation:

Vector Algebra (2D or 3D)

Moment about [pic]:

[pic]

Direction: perpendicular to [pic] and [pic],

Sense: Right hand rule or screwdriver rule.

Magnitude:

[pic]

Note: [pic]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 ������ and ������, force direction doesn’t matter:

[pic]

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.

4. Couple moment (rotational tendency) can be increased by increasing force magnitude or arm d:

[pic]

• Addition of couples: distributive (similar to moments). Can be shown using Varignon’s theorem.

Couple, Computation in Problems

2D - scalar (������∙������)

- components (2 couples formed by [pic] and [pic])

- vector ([pic])

3D - almost exclusively vector [pic]

Couple, Example 1

[pic]

Example 2

[pic]

[pic]

Example 3

[pic]

[pic]

Example 4

[pic]

[pic]

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 with the line of action parallel to [pic] and passing through B:

[pic]

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.

Force-Couple System, 3D

Similarly, force-couple can be computed for 3D body:

[pic]

where original force [pic] applied at ������ is equivalent to the force and a couple at point ������.

Here [pic] is a free vector (not attached to ������) and [pic] is a position vector from ������ to any point on a line of action of [pic], i.e. not restricted to [pic].

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

System of Forces-Couples

A system of forces-couples is equivalent to a force-couple system at an arbitrary point ������, where resultant force [pic] acting at ������ is:

[pic]

and a resultant couple [pic] is the sum of all couples plus the sum of moments of the original forces about ������:

[pic]

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 ������.

Force-Couple Systems, Example 1

[pic]

Example 2

[pic][pic]

Example 3

[pic]

[pic]

Example 4

[pic]

[pic]

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.

[pic]

The distance [pic] . so that the force [pic] at the new location ������ should create the moment [pic] about the old location ������.

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.

➢ Shifting the Line of Action of a Force

[pic]

2D Resultants, Example 1

[pic]

[pic]

Example 2

[pic]

[pic]

➢ 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.

➢ 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 ������ and introduce couples for equivalency.

2) Calculate resultant force [pic] and couple

[pic]

3) Replace resultant force-couple system by a single force (eliminate couple by shifting line of action of [pic]).

Note: Can do that as long as [pic]. When [pic] you can’t eliminate couple.

Similar operation can be performed in 3D. However, only component of [pic] perpendicular to [pic] can be eliminated by shifting [pic]. Remaining system of parallel [pic] and [pic] is called “wrench” (see page 130 in the book). System of parallel 3D forces can be, however, be reduced to a single resultant force, [pic].

Force Addition, Example 1

[pic]

Example 2

[pic]

Example 3

[pic]

[pic]

Example 4

[pic]

[pic]

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