Statics

The Method of Sections: Cut Through and Solve One Member

Cut a section to expose bars, one side as a rigid-body FBD, the ΣM moment-center trick, member-force solve, sections vs joints

To find one member deep inside a truss, you don't have to crawl joint by joint - cut an imaginary line straight through it and balance the whole piece. One section, often a single moment equation, one answer.

Slide the cut across the truss. A good section slices right through the member you want, crossing at most three bars with unknown forces.

Keep either side as one rigid body. The cut bars become external forces on it - at most three unknowns against the three equilibrium equations.

Here is the trick. Take moments about the point where two of the cut bars cross - they drop out, and the third member falls straight out of ΣM=0.

Drag the load. With the moment center chosen, the bottom-chord force solves in a single equation and scales right with P.

Compare the two tools. Sections leap straight to a chosen member in one equation; the method of joints must walk bar by bar to reach it.

In PracticeThe method of sections cuts an imaginary line through a truss, crossing at most three unknown bars, and treats one side as a rigid body under ΣFx=0, ΣFy=0, ΣM=0. Taking moments about the point where two cut bars intersect isolates the third member in a single equation - the section's signature move. Reach for sections when you need only a few specific members deep in the truss, and for the method of joints when you need them all. Next, the same free-body cut extends to frames and machines, where members carry more than axial force.

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