The method of sections
When only a few member forces are needed, the method of sections cuts the truss and uses rigid-body equilibrium on one piece. A single moment equation often gives an answer directly.
Cutting the truss
Imagine slicing the truss through the members of interest, generally no more than three with unknown forces. Either piece is then a rigid body in equilibrium under its external loads, reactions, and the cut member forces, which act along the members.
Three equations, three unknowns
The cut piece gives three equations: sum Fx = 0, sum Fy = 0, and sum M = 0. With at most three unknown member forces, they can all be found without analysing any joints.
Clever moment points
Take moments about the point where two of the cut members' lines of action meet. Those two forces then drop out, leaving one unknown in one equation. The intersection point may lie outside the piece, which is fine.
A bridge example
A truss 12 m long and 3 m high is simply supported and carries 30 kN at the bottom joint at midspan, so each support carries 15 kN. Cutting through the top chord, a diagonal, and the bottom chord just left of midspan, take moments about the top joint above the load: 15 x 6 = F x 3, so the bottom chord carries 30 kN tension. The top chord carries an equal 30 kN of compression.
Chord forces and bending
The truss behaves like a deep beam: the top chord is in compression and the bottom chord in tension, like the top and bottom of a bent beam. Chord forces are largest near midspan, where the bending effect of the loads is greatest; diagonals carry the shear, largest near the supports.
Choosing a method
Use the method of joints to find all member forces, and the method of sections to find a few specific ones, especially in the middle of a long truss. Engineers often combine them, and computer programs now use matrix methods for large trusses.
Real-world consequences
Truss analysis errors can be catastrophic. The 1907 collapse of the Quebec Bridge during construction, which killed 75 workers, was traced to compression members whose loads had been underestimated and which buckled. Careful analysis and checking are part of engineering ethics.
Worked example
In the 12 m truss, deepening it from 3 m to 4 m changes the midspan chord force to 15 x 6 / 4 = 22.5 kN. A deeper truss has smaller chord forces, because the internal couple has a longer lever arm, which is why long-span bridges use deep trusses.
Key takeaways
The method of sections cuts a truss through at most three unknown members and applies three rigid-body equilibrium equations to one piece. Taking moments about the intersection of two cut members isolates the third. A truss acts like a deep beam: top chord in compression, bottom chord in tension, largest near midspan, with diagonals carrying shear. Deeper trusses have smaller chord forces. Use joints for all members and sections for a few. Underestimated compression led to the 1907 Quebec Bridge collapse.
Flashcard
Cut the truss, analyse one piece
Check your understanding
In the method of sections, you usually take moments about: