Trusses: method of sections

By the end of this topic you will be able to

Choose a section that cuts no more than three members with unknown forces.
Apply moment and force equilibrium to the free body on one side of the section.
Find the force in a single chosen member without solving the whole truss.

Why cut a truss?

The method of joints finds every member force, which is wasteful when you only need one or two, perhaps in the middle of a long truss. The method of sections (also called Ritter's method) goes straight to the members you want.

Imagine passing a section through the truss, cutting the members you want. Each cut member is replaced by the internal force it was carrying, and the part of the truss on one side must be in equilibrium on its own. A rigid body in a plane gives three equations, so the section should cut no more than three members whose forces are unknown.

Method

Find the support reactions for the whole truss.
Choose a section that cuts the member(s) you need and at most two others.
Draw one side of the truss (whichever has fewer loads), with the reactions, loads, and the cut member forces. Assume the cut members are in tension.
Choose equations that find a single unknown at a time: - Take moments about the point where two cut members intersect. Those two members have no moment about that point, so one unknown remains. - Use $\sum F_y = 0$ (or $\sum F_x = 0$ ) for a member that is not horizontal (or vertical).
A negative result means compression.

Interactive modelChoose a panel to cut. The cut members' forces keep the left part in equilibrium.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Worked example

In the Warren truss (8 m span, 1.8 m deep, 20 kN at joint C), find the force in the top chord member GH and the bottom chord member BC.

Reactions. $R_A = R_E = 10$ kN.

Section through GH, GC and BC. Cut the truss just left of joint C so that the members cut are the top chord, a diagonal and the bottom chord. Take moments about C (where the diagonal and the bottom chord meet). Only GH has a moment about C, with a lever arm of 1.8 m, and so does $R_A$ with a lever arm of 4 m:

\sum M_C = 0:\quad 10\times4 + F_{GH}\times1.8 = 0 \;\Rightarrow\; F_{GH} = -22.2 \text{ kN (compression)}

Member BC. Take moments about G (3, 1.8) where GH and the diagonal meet: $10\times3 - F_{BC}\times1.8 = 0$ , so $F_{BC}$ is 16.7 kN. As the model confirms, BC is in tension.

Use the model's section selector to see the same forces and the equilibrium check.

Practice problems

Why must the section cut at most three members?

Show answer

The cut part gives only three independent equations of equilibrium, so at most three unknown forces can be found.
In the model (Warren truss, load 20 kN at C), choose the panel between joints B and C. Which members are cut, and what is the equilibrium check reading?

Show answer

The bottom chord BC, the top chord FG and the diagonal BG. The check ( $\sum F_y$ on the left part) reads about 0.
When is the method of joints better than the method of sections?

Show answer

When you need the force in every member, or members near a support where joints have only two unknowns.
A truss has a 12 kN load at the apex and the section cuts a member inclined at 60°. If the vertical component of that member must resist 6 kN, find the member force.

Show answer

$F\sin60^\circ = 6$ , so $F = 6.93$ kN.

Key formulae

Moment equilibrium about a point on the section: $\sum M_{\text{point}} = 0$
Vertical force equilibrium of the cut part: $\sum F_y = 0$
Number of unknowns on the cut: $\text{at most } 3$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (45 min). 10 min: recap joints on the Warren truss and why it is slow for a mid-span member; 10 min: the idea of a cut and choosing the moment point; 15 min: worked example on the board; 10 min: model with different panels.

Activity. Students choose a panel and calculate the chord forces by hand, then check with the model.

Discussion. How do the chord forces compare with the bending moment diagram of the equivalent beam? (Chord force ≈ M/h.)

Assessment idea. A Pratt truss with two loads; ask for three named members using sections only.