The method of joints is a core statics technique for finding the internal forces in the members of a pin-jointed truss. It matters because bridges, roof frames, cranes, and towers often use triangular truss patterns to carry loads efficiently. By solving one joint at a time, an engineer can tell which members are in tension, which are in compression, and which carry no force.
This helps designers choose safe member sizes and understand how loads travel through a structure.
The method works by treating each joint as a particle in equilibrium, so the sum of forces in the x-direction and y-direction must both be zero. Support reactions are usually found first by applying equilibrium to the entire truss, then joints with no more than two unknown member forces are solved. Member forces are assumed to pull away from the joint, meaning a positive result indicates tension and a negative result indicates compression.
Zero-force member rules can simplify the analysis before calculations begin.
Key Facts
- For a stable planar truss in equilibrium, sum Fx = 0, sum Fy = 0, and sum M = 0 for the whole structure.
- At each pin joint, use sum Fx = 0 and sum Fy = 0 because the joint is treated as a particle.
- Assume unknown member forces are in tension by drawing them pulling away from the joint.
- A positive solved member force means tension, and a negative solved member force means compression.
- A joint with two non-collinear members and no external load or support reaction has both members as zero-force members.
- For a determinate simple planar truss, m + r = 2j, where m is members, r is reaction components, and j is joints.
Vocabulary
- Truss
- A truss is a structure made of straight members connected at joints, usually arranged in triangles to carry loads mainly through axial force.
- Method of joints
- The method of joints is a truss analysis technique that applies force equilibrium to one joint at a time to solve member forces.
- Tension
- Tension is an axial force that pulls a member apart, with the member pulling away from each connected joint.
- Compression
- Compression is an axial force that pushes a member together, with the member pushing into each connected joint.
- Zero-force member
- A zero-force member is a truss member that carries no axial force for a particular loading condition.
Common Mistakes to Avoid
- Solving joints before finding support reactions is wrong because unknown reactions act like external forces on the truss and affect joint equilibrium.
- Mixing up tension and compression signs is wrong because a positive force under the tension assumption means tension, while a negative value means the member is actually in compression.
- Choosing a joint with three or more unknown member forces is inefficient because one joint in a planar truss only provides two independent equilibrium equations.
- Forgetting to resolve angled member forces into components is wrong because sum Fx = 0 and sum Fy = 0 require horizontal and vertical force components, not the original diagonal force alone.
Practice Questions
- 1 A triangular truss has joints A and B at the base and joint C above the midpoint. Supports are a pin at A and a roller at B. A 12 kN downward load is applied at C. If the span AB is 6 m and the truss is symmetric, find the vertical reactions at A and B.
- 2 At a joint, a known 8 kN horizontal force acts to the right, a vertical member force FV is unknown, and a diagonal member force FD acts along a 3-4-5 direction upward-left from the joint. Assuming FD is in tension away from the joint, use sum Fx = 0 and sum Fy = 0 to find FD and FV.
- 3 A joint has three connected members, two of which are collinear, and there is no external load or support reaction at that joint. Explain which member is a zero-force member and why this rule is useful before doing numerical calculations.