Trusses: the method of joints

Engineering · First-year engineering · about 12 min

A truss is a frame made of straight members joined at their ends, like a bridge or roof frame. Each member is only pulled (tension) or pushed (compression) along its length. The method of joints finds the force in every member by treating each joint as a point in equilibrium.

The key idea

At every joint, the member forces and any loads balance in both directions:

\sum F_x = 0 \qquad \sum F_y = 0

Two equations per joint means you can solve a joint with at most two unknown member forces. Start at one of those, then move from joint to joint.

Try it yourself

Change the load and geometry and see which members go into tension or compression.

Tension and compression

Assume every unknown member force is tension, pulling away from the joint. Solve the equations:

a positive answer means the member really is in tension;
a negative answer means it's in compression, pushing on the joint.

Assuming tension everywhere means you never have to guess directions.

Two horizontal straight members, one above the other, each with a round pin at both ends. Top member: a dark red arrow at each end pointing away from the member, outwards. Bottom member: a dark red arrow at each end pointing into the member, inwards. — A member in tension is pulled and a member in compression is pushed.

The method, step by step

Find the support reactions for the whole truss first (moments and equilibrium, as for a beam).
Pick a joint with no more than two unknown members, usually a support.
Resolve each member force into $x$ and $y$ parts using the member's geometry: a member rising $3$ m over $2$ m has length $\sqrt{13}$ m, so its force has components $\tfrac{2}{\sqrt{13}}F$ and $\tfrac{3}{\sqrt{13}}F$ .
Solve $\sum F_y = 0$ and $\sum F_x = 0$ .
Carry the known forces to the next joint and repeat.

Zero-force members

At a joint where only two non-parallel members meet and there's no load, both members carry zero force. At a joint where three members meet, two of them in a straight line, and there's no load, the third member carries zero force. Spotting these first saves a lot of work.

Worked example: a simple triangular truss

A triangular truss has a pin at $A(0, 0)$ , a roller at $B(4, 0)$ and its apex at $C(2, 3)$ , in metres. A $10$ kN load acts downwards at $C$ . Find the forces in members $AC$ and $AB$ .

A triangular truss with joints A (0, 0), B (4, 0) and C (2, 3) in metres. Exactly three members: AB, AC and BC. A on a pin support, B on a roller support. A downward force arrow above C. Dimension of 4 m under the base, and 3 m on a vertical line on the far right showing the height. — The truss in this example.

Show the solution

Reactions. The load is central, so $A_y = B_y = 5$ kN (and $A_x = 0$ ).

Joint $A$ . Member $AC$ rises $3$ m over $2$ m, so its length is $\sqrt{2^2 + 3^2} = \sqrt{13}$ m. Assume both members in tension.

\sum F_y: \; 5 + F_{AC}\frac{3}{\sqrt{13}} = 0 \;\Rightarrow\; F_{AC} = -\frac{5\sqrt{13}}{3} = -6.01 \text{ kN}

\sum F_x: \; F_{AB} + F_{AC}\frac{2}{\sqrt{13}} = 0 \;\Rightarrow\; F_{AB} = \frac{10}{3} = 3.33 \text{ kN}

Answer: $AC$ carries $6.01$ kN in compression (negative), and $AB$ carries $3.33$ kN in tension. By symmetry, $BC$ is also $6.01$ kN in compression.

Common mistakes

Using the member length instead of its components

Resolve each force with the ratios $\frac{\Delta x}{L}$ and $\frac{\Delta y}{L}$ from the member's own geometry.

Starting at a joint with three unknowns

Two equations can only solve two unknowns. Start at a support or any joint with two unknown members.

Reading a negative answer as a mistake

A negative force just means compression when you assumed tension. Keep the sign when you carry it to the next joint.

Check your understanding

1. You assume a member is in tension and get a negative force. What does that mean?

2. How many unknown member forces can you solve at a single joint?

3. A member rises $4$ m over a horizontal run of $3$ m and carries $10$ kN. What is the vertical component of its force?

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