Two-force and three-force members
Some members are loaded at only two or three points. Recognising them gives extra information for free, often turning a hard problem into an easy one.
Two-force members
A two-force member has forces applied at only two points and no couples. For equilibrium, the two forces must be equal, opposite, and collinear, acting along the line joining the two points. Links, struts, hydraulic cylinders pinned at both ends, and truss members are two-force members.
Why the direction is known
If the two forces were not along the line joining their points, they would form a couple and the member would rotate. So the direction is fixed; only the magnitude, and whether it is tension or compression, remains unknown. This cuts two unknowns at each end down to one for the whole member.
Curved two-force members
A two-force member need not be straight. A curved link pinned at both ends still carries its force along the straight line between the pins, not along its own curve, which is why curved members can bend under load.
Three-force members
A member loaded at exactly three points with no couples is a three-force member. For equilibrium, the three forces must either be concurrent, passing through a single point, or all parallel. Otherwise, one force would have an unbalanced moment about the intersection of the other two.
Using concurrency
If two of the three forces' lines of action are known, their intersection fixes the direction of the third. A ladder resting on a smooth wall and a rough floor is a three-force member: its weight and the wall's horizontal reaction meet at a point, so the floor's reaction must also pass through that point.
Graphical solution
For a three-force member, the three forces form a closed triangle when drawn head to tail, since they sum to zero. Engineers historically solved such problems with scale drawings and protractors, and the triangle still gives a quick check today.
Identifying member types
Check each member: if it has pins at only two points and no loads between them, it is a two-force member, so draw its forces along the line between the pins. This step is crucial in frames and machines, where many members are two-force links.
Worked example
A sign of weight 400 N hangs from the end of a horizontal 2 m beam, pinned to a wall at A and held by a strut BC pinned at both ends, reaching the beam at its midpoint at 45 degrees. The strut is a two-force member, so its force acts along BC. Moments about A: F sin 45 x 1 = 400 x 2, so F is about 1,131 N of compression in the strut.
Key takeaways
A two-force member carries equal, opposite forces along the line joining its two load points, in tension or compression, even if the member is curved. A three-force member's three forces must be concurrent or parallel, so two known lines fix the third's direction, and the forces form a closed triangle. Spotting two-force links reduces unknowns dramatically, which is essential for trusses, frames, and machines.
Flashcard
Loaded at two points only
Check your understanding
The forces on a two-force member act: