The method of joints
The method of joints treats each joint as a particle in equilibrium. Working joint by joint, every member force in a determinate truss can be found.
The idea
Each joint of a truss is a point where member forces and external loads meet, so it can be treated as a particle: the forces must satisfy sum Fx = 0 and sum Fy = 0. With two equations per joint, choose joints with no more than two unknown member forces.
Getting started
First find the support reactions using the whole truss as a rigid body. Then begin at a joint, often a support, with only two unknown members. Assume every unknown member is in tension, pulling away from the joint; a negative answer then means compression.
A triangular truss
A truss has supports at A(0, 0) and C(4, 0) m and an apex B at (2, 2) m, with 10 kN downward at B. By symmetry each support carries 5 kN. At joint A, member AB runs at 45 degrees: sum Fy gives FAB sin 45 + 5 = 0, so FAB is about -7.07 kN, meaning 7.07 kN compression. Sum Fx gives FAB cos 45 + FAC = 0, so FAC = 5 kN tension.
Reading the results
In that truss, the sloping members push outward and down on the supports, and the bottom chord AC acts like a tie, holding the supports together in tension. This is how a simple pitched roof works: the rafters are in compression and the ceiling joist in tension.
Zero-force members
Some members carry no force under a particular load. At an unloaded joint with only two non-collinear members, both are zero-force members. At an unloaded joint with three members, two of them collinear, the third is a zero-force member. Spotting these first saves work.
Why zero-force members exist
Zero-force members are not useless: they brace compression members against buckling, keep the shape stable, and carry load when the loading changes, such as when a vehicle moves across a bridge or wind blows from another direction.
Checking the work
The last joint analysed provides spare equations, because the reactions were already found from the whole truss. If those extra equations are satisfied, the earlier answers are very likely correct. Tabulating member forces with T or C labels keeps results organised.
Worked example
In the triangular truss above, joint B carries the 10 kN load and members BA and BC. With both sloping members at 45 degrees and each in 7.07 kN compression, their vertical components push up on B with 7.07 x 0.707 x 2, about 10 kN, balancing the load. The horizontal components, 5 kN each, cancel. Joint B checks out.
Key takeaways
The method of joints applies particle equilibrium, sum Fx = 0 and sum Fy = 0, at each joint. Find support reactions first, then start at a joint with at most two unknowns, assuming tension so negative answers mean compression. In a simple roof truss, rafters are in compression and the tie in tension. Identify zero-force members at unloaded joints with two non-collinear members, or three members with two collinear. Use the final joint's equations as a check.
Flashcard
Particle equilibrium at each joint
Check your understanding
At an unloaded joint with exactly two non-collinear members, the member forces are: