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5.1 · Recognize two-force and three-force members

Learn to recognize two-force and three-force members through clear examples and targeted practice.

University of Alberta ENGG 130: Engineering Mechanics: Statics

Frames and Machines

Classifying an isolated member by its external forces and their lines of action

A member is a body or component that we isolate to study the forces acting on it. To recognize whether it is a two-force or three-force member, count the external forces on the isolated member—not the forces elsewhere in the structure. Include support reactions and applied forces in that count. A pin may exert horizontal and vertical components, but those components combine into one resultant force at the pin. The useful questions are how many external forces act on the member and how their lines of action are arranged. The recognition rules follow from planar equilibrium: the net force and net moment must both be zero.

What you will learn

Identify whether an isolated member has a two-force or three-force pattern.
Explain why two forces on a member in equilibrium must be equal, opposite, and collinear.
Recognize the line-of-action conditions for a three-force member in equilibrium.
Use a free-body diagram and force and moment checks to confirm a classification.

1. Isolate the member before counting forces

Choose one member as the system and imagine it separated from everything around it. Replace each connection to another body with the force or forces that the connection exerts on the isolated member. Include all applied loads. A force is counted as one external resultant, even when it is resolved into two components for calculations.

A free-body diagram (FBD) shows the isolated member and the external forces acting on it. It helps prevent missed forces and makes each force’s line of action clear. A force’s line of action is the straight line through its point of application in the force’s direction.

For this lesson, treat the member as a rigid body in planar static equilibrium. A two-force member has exactly two external forces and no other external load or couple. A three-force member has exactly three external forces and no other external load or couple. If there are additional forces or an applied couple, do not apply these classifications as stated.

Isolate one member and count its external resultants.
A pin’s force components together represent one force at the pin.
Check for every applied load and any applied couple before classifying.

2. The two-force-member pattern

Suppose the only external forces on a member are at its ends. Force equilibrium requires their vector sum to be zero, so they must have equal magnitudes and opposite directions. But that is not enough: if their lines of action were different, they would create a net turning effect. Moment equilibrium therefore requires the forces to be collinear, meaning they lie on the same straight line.

Thus, a two-force member in equilibrium has exactly two external forces that are equal, opposite, and collinear. A pin-connected link with no other load is a common example. The forces may pull away from the member or push toward it; the pattern identifies the force directions, but does not by itself determine which case applies.

Use right and upward as the positive coordinate directions, and counterclockwise moments as positive. In planar statics, check the horizontal and vertical force sums and the moment sum. When the two forces are collinear, their moments about a point on their common line are zero.

\sum F_x=0,\quad \sum F_y=0,\quad \sum M=0

Exactly two external forces, with no other load or couple.
The forces must be equal, opposite, and collinear.
Collinearity follows from moment equilibrium as well as force equilibrium.

3. The three-force-member pattern

A three-force member has exactly three external forces and no external couple. If the three forces are not all parallel, their lines of action must meet at a common point for the member to be in equilibrium. To understand why, take the intersection of two force lines as the moment centre. Those two forces have zero moment about that point. For total moment to be zero, the third force must also have zero moment there, so its line of action must pass through the same point.

There is another possible arrangement: all three forces may be parallel. For that case, use the force and moment equations directly rather than looking for a finite common intersection. If two force lines are parallel and the third is not, the three lines cannot be concurrent, so that arrangement cannot satisfy equilibrium for a three-force member.

Concurrency is a necessary line-of-action condition for the nonparallel case, not proof of equilibrium by itself. The force magnitudes and directions must also give zero net force. A force line can pass through the common point and still have the wrong magnitude or sense.

\sum F_x=0,\quad \sum F_y=0,\quad \sum M_O=0

Exactly three external forces, with no external couple.
If the force lines are not all parallel, they must be concurrent in equilibrium.
If all three forces are parallel, check force and moment balance directly.

4. A reliable recognition routine

First, name the isolated member and draw its FBD. Mark each force, its point of application, and its direction, including unknown reactions. Count forces as resultants, not as vector components. If there are exactly two, test whether they are equal, opposite, and collinear. If there are exactly three, check whether the lines are concurrent or all parallel, then use equilibrium to confirm the arrangement.

Choose a point for the moment check that makes the calculation simple. A force whose line of action passes through that point has zero moment about it. Otherwise, the moment magnitude is the force magnitude multiplied by the perpendicular distance to the line of action. Use a consistent sign for clockwise and counterclockwise moments, and keep force units such as newtons separate from moment units such as newton-metres.

If a calculated force is negative, it acts opposite to the direction initially assumed. The force-counting rule is unchanged. Recognition is the starting point; the FBD and equilibrium checks confirm whether the stated force pattern can actually be in equilibrium.

M=Fd_\perp

Draw, count, classify, and then verify with equilibrium.
A force through the moment centre has zero moment about that point.
Force balance and moment balance are separate checks.

Worked example

1. A pin-connected link with only end forces

A horizontal link

AB

is pinned at each end and has no other applied load. The isolated link has a 240 N force to the right at

A

and a 240 N force to the left at

B

. Classify it and verify equilibrium.

Isolate and count
The system is the link alone. It has two external forces, one at each pin, and no other load or couple. It is a candidate two-force member.
Check force balance
Take right and upward as positive. The horizontal forces cancel, and there are no vertical force components.
$\sum F_x=240-240=0,\quad \sum F_y=0$
Check moment balance
The forces are collinear along $AB$ . Their perpendicular distances from a point on that line are zero, so their moments about such a point are zero.
$\sum M_A=0\,\mathrm{N\cdot m}$
Classify
The two forces are equal, opposite, and collinear, so the link meets the two-force-member conditions.

Answer: The link is a two-force member. Its two forces are equal, opposite, and collinear.

Check: The horizontal and vertical force sums are zero, and the moment about A is zero. Force is measured in N and moment in N·m.

Worked example

2. Three concurrent forces that do not balance

A member carries three forces whose lines of action meet at point

O

. The first force is 6 kN to the right, the second is 8 kN upward, and the third is 10 kN directed down and left at an angle of

216.87^\circ

from the positive horizontal axis. Determine whether the forces are in equilibrium and classify the member.

Identify the force pattern
The isolated member has exactly three external forces and no external couple. All three are applied at $O$ , so each has zero moment about $O$ .
$\sum M_O=0\,\mathrm{kN\cdot m}$
Resolve the third force
The down-left direction follows the component proportions of a 6-8-10 right triangle. Its horizontal component is 8 kN left and its vertical component is 6 kN down.
$\mathbf{F}_3=(-8\mathbf{i}-6\mathbf{j})\,\mathrm{kN}$
Check force balance
The horizontal and vertical components do not cancel. Concurrency makes the moment sum zero, but it cannot correct a nonzero resultant force.
$\sum F_x=6-8=-2\,\mathrm{kN},\quad \sum F_y=8-6=2\,\mathrm{kN}$
Classify and conclude
The member has a three-force pattern by force count, but the stated forces do not produce equilibrium. Concurrency is necessary for this nonparallel arrangement, not sufficient.

Answer: The member has a three-force pattern, but the specified forces are not in equilibrium.

Check: The resultant force is

(-2\mathbf{i}+2\mathbf{j})\,\mathrm{kN}

, so force balance fails. All three moments about O are zero.

Worked example

3. Three parallel forces on a member

A horizontal member has three vertical forces: 4 kN upward at its left end, 4 kN upward at its right end, and 8 kN downward at its midpoint. The end-to-end length is 2 m. Determine whether it is in equilibrium and identify its force pattern.

Count and choose signs
The isolated member has three external forces, all vertical, and no couple. Take upward forces and counterclockwise moments as positive. This is the parallel-force case.
Check force balance
The two upward forces total 8 kN, balancing the 8 kN downward force. There are no horizontal forces.
$\sum F_y=4-8+4=0\,\mathrm{kN},\quad \sum F_x=0$
Check moment balance
Take moments about the left end. The left force has zero moment. The midpoint force creates a clockwise moment, while the right-end force creates a counterclockwise moment.
$\sum M_A=-(8\,\mathrm{kN})(1\,\mathrm{m})+(4\,\mathrm{kN})(2\,\mathrm{m})=0\,\mathrm{kN\cdot m}$
Classify
Both force and moment balance hold. The member is a three-force member with parallel forces; their lines do not meet at one finite common point.

Answer: The member is in equilibrium and is a three-force member with all three forces parallel.

Check: The resultant force is zero, and the moments about the left end sum to zero in kN·m.

Common mistakes and how to avoid them

Counting the two components of a pin force as two separate forces.

Correction: Treat the components together as one resultant force acting at the pin when counting forces.

Calling any member with two forces a two-force member, even if their lines of action are not collinear.

Correction: A two-force member in equilibrium requires equal, opposite, collinear forces. Check moment balance as well as force balance.

Assuming that concurrent forces must be in equilibrium.

Correction: Concurrency makes the moment about the common point zero, but the vector sum of the forces must also be zero.

Applying the concurrency rule to three parallel forces.

Correction: For parallel forces, check force balance and moment balance directly.

Lesson summary

Isolate the member, draw its FBD, and count external forces as resultants.
A two-force member has exactly two external forces that are equal, opposite, and collinear.
For a three-force member in equilibrium, nonparallel force lines must be concurrent; alternatively, all three forces may be parallel.
A recognizable force pattern alone does not prove equilibrium. Verify both force and moment balance.

Check your understanding

Question 1

An isolated link has only two pin forces and no other load. Which condition must hold for equilibrium?

The forces are equal, opposite, and collinear.
The forces have equal magnitude but point in the same direction.
The forces are perpendicular.
The forces act at the same point but may have any magnitudes.

Show answer and explanation

The forces are equal, opposite, and collinear.

Force balance requires equal and opposite forces, and moment balance requires their lines of action to coincide.

Question 2

A member has exactly three nonparallel external forces and no applied couple. What line-of-action pattern is required for equilibrium?

The lines of action must meet at one common point.
The lines of action must all be perpendicular to the member.
Exactly two lines must be parallel.
The forces must have equal magnitudes.

Show answer and explanation

The lines of action must meet at one common point.

For three nonparallel forces, moment equilibrium requires their lines of action to be concurrent.

Question 3

Three vertical forces on a member sum to zero, but their moments about the left end do not. Is the member in equilibrium?

Yes, because the resultant force is zero.
Yes, because the forces are parallel.
No, because moment equilibrium also has to hold.
No, because parallel forces cannot act on one member.

Show answer and explanation

No, because moment equilibrium also has to hold.

A member in planar equilibrium must have zero net force and zero net moment.

Key terms

Member: A body or component isolated for analysis.
External force: A force exerted on the isolated member by another body or by an applied load.
Line of action: The straight line through a force’s point of application in its direction.
Collinear: Located on the same straight line.
Concurrent forces: Forces whose lines of action pass through one common point.
Free-body diagram: A drawing of an isolated body showing the external forces acting on it.

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