Questions: Two-Force and Three-Force Members

Question 1

A slender link is pinned at both ends A and B with no loads applied between the pins. What do you immediately know about the force in this link?

• A. The force magnitude equals the weight of the link
• B. The force direction is horizontal, since pins can only exert horizontal reactions
• C. The force is directed along the line AB — the direction is determined by geometry alone
• D. Nothing — you must apply ΣF = 0 and ΣM = 0 to determine both direction and magnitude

Answer: (C)

A link pinned at both ends with no intermediate loads is a two-force member. By the two-force member theorem, the forces at A and B must be equal, opposite, and collinear — directed along the line connecting A and B. This direction is determined purely from geometry before any equations are written. This recognition reduces the unknown from two force components to one scalar magnitude, which is why identifying two-force members first is critical in truss and frame analysis.

Question 2

In a three-force member, two of the three lines of action are known. How do you find the direction of the third force?

• A. The third force must be perpendicular to the resultant of the other two forces
• B. The third force must be parallel to the resultant of the first two forces
• C. Extend the lines of action of the two known forces until they meet; the third force must pass through that intersection
• D. The third force direction is indeterminate until the magnitudes are known

Answer: (C)

For a three-force member in equilibrium, all three forces must be concurrent (or all parallel). The method: extend the lines of action of the two known forces until they intersect. For the net moment about that intersection to be zero, the third force must also pass through it — otherwise it alone creates an unbalanced moment. Since the third force also passes through its own application point on the body, both points are known and the direction is fully determined geometrically, without solving any simultaneous equations.

Question 3

In a two-force member, knowing only the locations of the two force application points is sufficient to determine the direction of the forces.

• T. True
• F. False

Answer: True

This is the central insight of the two-force member theorem. The forces must be collinear along the line connecting the two application points — the only direction satisfying both translational equilibrium (equal and opposite) and rotational equilibrium (zero net moment about every point). The direction is completely determined by geometry. The magnitude remains unknown until additional equilibrium equations are applied to a larger system, but the direction is geometrically certain.

Question 4

Two forces acting on a body can hold it in static equilibrium even if they are not collinear, provided they are equal in magnitude and opposite in direction.

• T. True
• F. False

Answer: False

Equal and opposite forces that are not collinear form a couple — a pure moment with no net force but a nonzero rotational effect. The body satisfies ΣF = 0 but not ΣM = 0, so it is not in static equilibrium. For complete equilibrium, the forces must be equal in magnitude, opposite in direction, AND collinear (acting along the same line of action). The collinearity requirement is the condition most easily overlooked — it is what eliminates the net moment.

Question 5

Explain why the forces in a two-force member must be collinear, not just equal and opposite.

Model answer: Equal and opposite forces in the same line satisfy both ΣF = 0 and ΣM = 0. If the forces were equal and opposite but offset (not collinear), they would cancel as a net force but produce a couple — a pair of non-collinear parallel forces with a nonzero net moment. In a two-force member, nothing else exists to balance that couple: there are forces at only two points and no other loads. The body would therefore rotate, violating rotational equilibrium. Collinearity is the additional constraint, beyond equal-and-opposite, that eliminates the net moment.

This is why the two-force member theorem is more powerful than it initially appears: it does not just say the forces are equal and opposite (which follows from ΣF = 0 alone) — it specifies their direction from geometry. This reduces the unknowns from four (two components each) to one (a single magnitude along the known axis).