2.3 · Analyze couples and equivalent couple moments

University of Alberta ENGG 130: Engineering Mechanics: Statics

Force-System Resultants

ENGG 130 Engineering Mechanics: Statics — Study topic 2.3

What you will learn

• Recognize a couple as two equal, opposite, parallel forces acting at different locations.
• Calculate a couple moment using a perpendicular distance or a planar vector moment.
• Determine the direction and units of a couple moment using a consistent sign convention.
• Replace one or more couples with an equivalent couple moment.

1. Recognize a couple

• The two forces of a couple have zero vector sum.
• A couple has a nonzero moment when its lines of action are separated.
• Use a signed moment to record whether the couple turns clockwise or counterclockwise.

2. Calculate the moment and its direction

• For components, calculate $M_z=r_xF_y-r_yF_x$.
• A moment has units of force times distance, such as N·m.
• Check the rotation direction rather than relying only on a memorized sign.

3. Combine couples and test equivalence

• Add signed moments to find the equivalent couple.
• For couples alone, the resultant force is zero.
• For equilibrium, check both $\sum F_x=0$, $\sum F_y=0$, and $\sum M=0$.

Worked example

Vertical force pair on a bar

1. Identify the system
   Take the bar and the two applied forces as the planar system. The forces are equal, opposite, and parallel, so they form a couple.
2. Choose the sign
   The left upward force and right downward force both tend to turn the bar clockwise. With counterclockwise positive, the couple moment is negative.
3. Calculate the couple moment
   The separation is perpendicular to the vertical force lines, so it is the required couple arm. Multiply by the force magnitude and attach the clockwise sign.
   $$M=-Fd=-(45\,\mathrm{N})(0.32\,\mathrm{m})=-14.4\,\mathrm{N\cdot m}$$
4. Verify the force balance
   The vertical forces cancel, and neither force has a horizontal component. The nonzero moment is the couple being sought, not an equilibrium condition for the bar by itself.
   $$\sum F_x=0,\qquad \sum F_y=45-45=0$$

Worked example

Oblique forces with horizontal separation

1. Set axes and position vector
   Use positive $x$ to the right, positive $y$ upward, and counterclockwise moment as positive. Point from the left force to the right force, so the separation vector is horizontal.
   $$\mathbf{r}=(0.50\,\mathrm{m},0)$$
2. Resolve the right force
   Its vertical component is the part that creates moment about the left application point. The horizontal component is parallel to the separation and contributes no moment.
   $$F_x=100\cos 30^\circ,\qquad F_y=100\sin 30^\circ=50\,\mathrm{N}$$
3. Find the couple moment
   Use the signed planar moment expression. The result is positive, so the equivalent couple is counterclockwise.
   $$M_z=r_xF_y-r_yF_x=(0.50)(50)=25\,\mathrm{N\cdot m}$$
4. Verify the force balance
   The left force is the exact negative of the right force, including both components. Therefore their net force is zero, while their separated lines of action leave the calculated moment.
   $$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M=25\,\mathrm{N\cdot m}$$

Worked example

Replace several couples with one

1. Choose signed moments
   Take counterclockwise as positive. The first and third couples are positive; the clockwise couple is negative.
   $$M_1=18\,\mathrm{N\cdot m},\qquad M_2=-7\,\mathrm{N\cdot m},\qquad M_3=4\,\mathrm{N\cdot m}$$
2. Add the moments
   The algebraic sum gives one couple with the same turning effect as all three applied couples together.
   $$M_{\mathrm{eq}}=18-7+4=15\,\mathrm{N\cdot m}$$
3. Check equilibrium
   Each couple has zero resultant force, so the total force is zero. The remaining moment is nonzero, so these couples alone do not satisfy rotational equilibrium.
   $$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M=15\,\mathrm{N\cdot m}$$

Common mistakes and how to avoid them

Lesson summary

• A couple is made of equal, opposite, parallel forces on separated lines of action.
• Its magnitude is force times perpendicular separation; its sign records rotation direction.
• For planar components, use $M_z=r_xF_y-r_yF_x$.
• Equivalent couples have the same signed moment, and multiple couples combine by signed addition.
• For equilibrium, both the resultant force and resultant moment must be zero.

Check your understanding

Key terms

Couple

A pair of equal, opposite, parallel forces whose separated lines of action create a moment.

Couple moment

The turning effect of a couple, equal in magnitude to force times perpendicular separation.

Equivalent couple

A single couple with the same signed moment as a collection of couples.

Perpendicular distance

The shortest distance between the parallel lines of action of a couple's forces.

Continue through ENGG 130

View the complete ENGG 130 University of Alberta ENGG 130: Engineering Mechanics: Statics curriculum and lessons

• 2.1 · Calculate the moment of a force about a point
• 2.2 · Use the cross product for force moments
• 2.4 · Reduce a planar force system to a force and couple
• 2.5 · Replace a simple distributed load with an equivalent resultant
• 1.1 · Use mechanics models, units, significant figures, and assumptions
• 1.2 · Resolve planar forces into Cartesian components

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