Three-force member equilibrium – angle between two equal inclined forces A jointed member is balanced at its end by two other members that carry equal forces and are symmetrically inclined. For equilibrium of the three-force member, what must be the angle between the two inclined bars?

Introduction / Context:
Many truss and frame joints are three-force members, meaning only three non-collinear forces act on the pin. Recognizing the geometry of force equilibrium for equal forces is a fast way to size members and check joint stability.

Given Data / Assumptions:

• Three concurrent forces act at a joint.
• Two of the forces are equal in magnitude and act along the two inclined bars.
• The third force acts along the balanced member.
• No external moments; pin (frictionless) joint.

Concept / Approach:
For three forces to be in equilibrium, they must be concurrent and their vector sum must be zero. With two equal forces, the force polygon becomes an equilateral triangle when all three forces are equal in magnitude and separated by 120° angles. The resultant of two equal forces F at an included angle alpha is R = 2 F cos(alpha/2). For zero net force, the third force must be equal and opposite to R. Symmetry and equality lead to alpha = 120°.

Step-by-Step Solution:
Let two equal forces be F and F with included angle alpha.Resultant magnitude: R = 2 F cos(alpha/2).For all three forces to be equal in magnitude (common in symmetric arrangements) and sum to zero, require R = F.Solve: F = 2 F cos(alpha/2) → cos(alpha/2) = 1/2 → alpha/2 = 60° → alpha = 120°.

Verification / Alternative check:
Construct a force triangle with three equal sides; internal angles are 60°, which correspond to 120° separations between the line-of-action directions at the joint.

Why Other Options Are Wrong:
60° gives R = 2F cos 30° = 1.732F ≠ F. 90° gives R = 1.414F. 45° and 3° are inconsistent with equilibrium for equal member forces.

Common Pitfalls:
Forgetting that at a joint the angles between the force directions are 120° when the magnitudes are equal; mixing up internal angles of the force polygon with angular separations at the joint.

Final Answer:
120°