Method of joints

Find the force in every member of a truss, one joint at a time

Last updated 2026-09-25

In Support reactions you solved a triangle one joint at a time. That's the method of joints, and it works on any truss that's built from triangles. Here it is as a recipe, worked through on a bigger truss.

The recipe

1. Find the support reactions.
2. Pretend every member you don't know yet is in tension, pulling away from the joint.
3. Pick a joint where only two member forces are still unknown.
4. Balance it: ΣFx = 0 and ΣFy = 0. Two equations, two unknowns.
5. Move to the next joint where only two forces are still unknown, and repeat.
6. Read the signs: a positive answer means tension, a negative one means compression.

The example: a Warren truss

Three triangles side by side, a shape called a Warren truss. Every sloped member goes 3 m across and 4 m up, so it's 5 m long, and its force splits the same way: 3/5 sideways and 4/5 up or down (see force components). An 8 kN load hangs from C, in the middle of the bottom.

The units don't change anything. Every step only uses how lengths compare to each other (3 out of 5, 4 out of 5, 6 m out of 12 m), so the same truss measured in feet gives exactly the same answers. The forces simply come out in whatever unit the load is in: make it 8 kips instead of 8 kN, and every answer below is in kips, with the same numbers.

Step 1: Support reactions

Moments about A, then the vertical forces:

ΣM_A = 0: 8 kN × 6 m − R_Ey × 12 m = 0 → R_Ey = 4 kN

ΣFy = 0: R_Ay + 4 kN − 8 kN = 0 → R_Ay = 4 kN

The load is in the middle, so each support takes half. Nothing pushes sideways, so the pin's sideways reaction is 0.

Step 2: Joint A

Two unknowns, F_AB and F_AC, and the 4 kN reaction pushing up. Only AB has an up-down part, so start with ΣFy:

ΣFy = 0: 4 kN + F_AB × 45 = 0 → F_AB = −5 kN

Negative, so AB is in compression: 5 kN. Now sideways:

ΣFx = 0: F_AB × 35 + F_AC = 0 → F_AC = 3 kN

Positive, so AC is in tension: 3 kN.

Step 3: Joint B

AB is known now. It's in compression, so it pushes on B: 4 kN up and 3 kN to the right. Two unknowns are left, F_BC and F_BD. BD is flat, so again start with ΣFy:

ΣFy = 0: 4 kN − F_BC × 45 = 0 → F_BC = 5 kN (tension)

ΣFx = 0: 3 kN + F_BC × 35 + F_BD = 0 → F_BD = −6 kN (compression)

Step 4: Joint C

AC and BC are known and both in tension, so both pull on C: AC 3 kN to the left, BC 3 kN to the left and 4 kN up. The load pulls down 8 kN. Two unknowns are left, F_CD and F_CE:

ΣFy = 0: 4 kN + F_CD × 45 − 8 kN = 0 → F_CD = 5 kN (tension)

ΣFx = 0: −3 kN − 3 kN + F_CD × 35 + F_CE = 0 → F_CE = 3 kN (tension)

Step 5: The rest, and a free check

The truss and its load are the same on both sides, so DE mirrors AB: 5 kN compression. That's every member.

The last joint, E, is a free check. Its forces have to balance without anything new to find:

Joint E: ΣFy = 4 kN − 4 kN = 0 ✓ ΣFx = 3 kN − 3 kN = 0 ✓

If the last joint doesn't balance, there's a mistake somewhere before it.

The top chord is pushed together and the bottom chord is pulled apart, just like a beam bending under a load. The diagonals take turns.

Tips

• Always assume tension. Don't guess which way a force points. The sign of the answer tells you.
• Carry known forces over correctly. A member in compression pushes on the joints at both of its ends. A member in tension pulls on both (see Tension and compression).
• Start with the easy direction. If only one unknown has an up-down part, ΣFy gives it straight away.
• Some members carry nothing. Move the load from C to the top joints (8 kN on B and 8 kN on D), and BC and CD come out 0. They're still needed: without them the truss would fold.

Common mistakes

• Starting at a joint with three unknowns. Two equations can't find three numbers. Solve a neighbouring joint first.
• Drawing a known force the wrong way. At the next joint, a compression member pushes in, it doesn't pull.
• Using the member force instead of its components. A 5 kN diagonal puts 4 kN into ΣFy, not 5 kN.
• Skipping the last check. It costs one line and catches almost every sign mistake.

How Truzme does it

Truzme doesn't go joint by joint. It uses the direct stiffness method: it treats every member as a stiff spring, writes the whole truss as one big table of numbers (a matrix), and solves it all at once. That's a job for a computer, not for paper, and it also handles trusses with more members than balance alone can solve.

For a truss like this one, both ways give exactly the same forces. By hand, the method of joints is the one to use: every step is a force you can draw, and you can see why each number comes out the way it does.

Check it in Truzme

Build this truss in Truzme yourself: five nodes, seven members, a pinned support at A, a roller at E and an 8 kN load at C. The Getting started guide shows how to draw one. Or open it ready-made, and check your numbers against every member force on the canvas: