Truss: method of joints: practice problems with step-by-step solutions

Also called: method of sections · truss analysis

Free Truss: method of joints practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: symmetric, asymmetric, vertical + horizontal load.

Free practice problemEasy
ABCDEFGH15 kN25 kN4 × 3 m3 m

Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.

B: 15 kN ↓D: 25 kN ↓
Next problem →
Practise Truss: method of joints for freePractise Truss: method of joints now5 free problems every day. No card needed.

What you practise

  1. Easysymmetric
  2. Mediumasymmetric
  3. Hardvertical + horizontal load

How to solve Truss: method of joints problems

  1. Find the support reactions for the whole truss.
  2. Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
  3. Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
  4. Assume tension; a negative answer means compression.

Common mistakes

  • Starting at a joint with three unknowns.
  • Forgetting the angle components of inclined members.
  • Swapping tension and compression.

Key formulas

Method of joints
ΣFx = 0, ΣFy = 0 at each joint
Method of sections
ΣM = 0 about a cut point
Member force
positive = tension, negative = compression

Learn Truss: method of joints from scratch

Before this: Beam support reactions

1What a truss is

A truss is a frame of straight bars joined at their ends to form triangles. You see them in roofs, cranes, pylons and many bridges.

ACBPAy​Cy​

The bars are called members and the connection points are joints. In the simple model, every joint is a pin and loads act only at the joints.

Then each member is pulled or pushed only along its own length. It carries a single axial force:

  • tension: the member is pulled and gets longer. We count this as positive.
  • compression: the member is pushed and gets shorter. We count this as negative.

Triangles are used because a triangle cannot change shape without changing the length of a side. That makes a truss stiff while using little material.

Self-check A member force comes out as −12 kN. What does that mean?
  1. The load points upwards
  2. The member carries no force
  3. The member is in compression, 12 kN
  4. The member is in tension, 12 kN

Answer: The member is in compression, 12 kN. With tension counted positive, a negative force is compression: the member is pushed and shortens.

2Forces at a joint

Each joint is in equilibrium on its own. So at every joint two equations hold:

ΣFx = 0 ΣFy = 0

Two equations means you can find at most two unknown member forces at one joint. Always start at a joint where only two are unknown.

Draw each unknown member force as tension: an arrow pointing away from the joint, along the member. If the answer is negative, the member is in compression.

FAB​FAC​Ay​αA

An inclined member force has two components. If the member makes an angle α with the horizontal:

Fx = F · cos α Fy = F · sin α

You rarely need the angle itself. From the member’s horizontal run a, height h and length ℓ = √(a² + h²): cos α = a/ℓ and sin α = h/ℓ.

Self-check A member runs 4 m horizontally and 3 m vertically, so it is 5 m long. It carries 10 kN. What is the vertical component?
  1. 10 kN
  2. 6 kN
  3. 7.5 kN
  4. 8 kN

Answer: 6 kN. sin α = 3/5 = 0.6, so Fy = 10·0.6 = 6 kN. 8 kN is the horizontal component (cos α = 0.8).

3The method step by step

  1. Find the support reactions from the whole truss: ΣMA = 0 gives Cy, then ΣFy = 0 gives Ay (and ΣFx = 0 gives Ax if there is a horizontal load).
  2. Pick a joint with at most two unknown member forces, usually a support joint.
  3. Draw the joint with all unknown member forces as tension (away from the joint).
  4. Write ΣFy = 0. If only one member has a vertical component, it gives that force directly.
  5. Write ΣFx = 0 for the second force.
  6. Move to the next joint, using the forces you already know. A negative result means compression.

The method of sections is an alternative: cut through the members you need and take moments about a point where the other cut members meet.

4A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

The truss spans 8 m and is 3 m high, with joint B above midspan. A vertical load of 30 kN acts at B. Find the forces in members AB and AC.

1

Look at the truss

ACB30 kN8 m3 m4 m

A is a pin and C is a roller. Each inclined member has a horizontal run of 4 m and a height of 3 m, so its length is √(4² + 3²) = 5 m.

2

Support reactions

The truss and the load are symmetric, so each support carries half the load:

Ay = Cy = 30 / 2 = 15 kN
3

Joint A: vertical balance

FAB​FAC​15 kNαA

sin α = 3/5 = 0.6 and cos α = 4/5 = 0.8. Only AB has a vertical component at A.

15 + FAB · 0.6 = 0 ⇒ FAB = −25 kN

Negative, so AB is in compression.

4

Joint A: horizontal balance

FAC + FAB · 0.8 = 0 ⇒ FAC = −(−25) · 0.8 = 20 kN

Positive, so AC is in tension. It ties the feet of the truss together.

5

Check the answer

By symmetry FBC = −25 kN as well. At joint B the two members push up with 2 · 25 · 0.6 = 30 kN, which balances the 30 kN load.

Answer

FAB = −25 kN (compression) and FAC = 20 kN (tension).

5Common mistakes

  • Starting at a joint with three unknown forces. Two equations cannot solve three unknowns.
  • Swapping sin and cos. Check which side of the triangle is vertical.
  • Drawing some unknowns as compression and others as tension, and then misreading the signs. Draw all of them as tension.
  • Forgetting the support reaction at the joint, or giving it the wrong direction.
  • Forgetting the horizontal load in the moment equation for the reactions.
Self-check At roller C the reaction is Cy = 9 kN upwards. Member BC is the only member with a vertical component there, with sin γ = 0.6. What is FBC?
  1. +15 kN (tension)
  2. −11.25 kN (compression)
  3. −15 kN (compression)
  4. −5.4 kN (compression)

Answer: −15 kN (compression). ΣFy = 0: 9 + FBC·0.6 = 0, so FBC = −15 kN. +15 kN loses the sign; −5.4 multiplies by sin γ instead of dividing; −11.25 uses cos γ = 0.8.

6How exam questions look

  • Easy: symmetric. B is above midspan and the load is vertical, as in the example. The reactions are half the load each.
  • Medium: asymmetric. B is not above midspan, so AB and BC have different angles. Find Cy from moments about A first.
  • Hard: a vertical and a horizontal load at B. The pin at A gets a horizontal reaction too. Joint C is still the easiest start, because the roller gives only Cy.

You are asked for one member force: AB, BC or AC. Give tension as positive and compression as negative.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
ABBCACACB45 kN6 m3 m2 m

Use the method of joints to find the force FAB in member AB. Give tension as positive and compression as negative.

span = 6 mheight = 2 mB is 3 m from Avertical load = 45 kN

Find FAB (kN)

Show hints
  1. Find the support reactions first from the whole truss.
  2. At joint A, only member AB has a vertical component. ΣFy = 0 gives FAB directly.
  3. Ay = 22.5 kN.
Show the answer and the step-by-step solution

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB45 kNAy​Cy​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·6 = 45·3 → Cy = 22.5 kN
ΣFy = 0: Ay = 45 − 22.5 = 22.5 kN

Joint A

Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.

sin α = 2/3.606 = 0.5547
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −22.5/0.5547 = −40.56 kN

Answer

FAB = −40.56 kN (compression)
Example 2Medium
ABBCACACB45 kN6 m1 m2 m

Use the method of joints to find the force FBC in member BC. Give tension as positive and compression as negative.

span = 6 mheight = 2 mB is 1 m from Avertical load = 45 kN

Find FBC (kN)

Show hints
  1. Find the support reactions first from the whole truss.
  2. Go to joint C. Only BC has a vertical component there, so start with ΣFy = 0.
  3. Cy = 7.5 kN.
Show the answer and the step-by-step solution

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB45 kNAy​Cy​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·6 = 45·1 → Cy = 7.5 kN
ΣFy = 0: Ay = 45 − 7.5 = 37.5 kN

Joint C

The roller gives only a vertical reaction, so joint C has just two unknown member forces. Assume both in tension.

sin γ = 2/5.385 = 0.3714, cos γ = 5/5.385 = 0.9285
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −7.5/0.3714 = −20.19 kN

Answer

FBC = −20.19 kN (compression)
Example 3Hard
ABBCACACB60 kN30 kN5 m1 m3 m

Use the method of joints to find the force FBC in member BC. Give tension as positive and compression as negative.

span = 5 mheight = 3 mB is 1 m from Avertical load = 60 kNhorizontal load = 30 kN ←

Find FBC (kN)

Show hints
  1. Find the support reactions first from the whole truss.
  2. Go to joint C. Only BC has a vertical component there, so start with ΣFy = 0.
  3. Cy = −6 kN.
Show the answer and the step-by-step solution

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB60 kN30 kNAy​Cy​Ax​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·5 = 60·1 − 30·3 → Cy = −6 kN
ΣFy = 0: Ay = 60 − −6 = 66 kN
ΣFx = 0: Ax = 30 kN

Joint C

The roller gives only a vertical reaction, so joint C has just two unknown member forces. Assume both in tension.

sin γ = 3/5 = 0.6, cos γ = 4/5 = 0.8
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −−6/0.6 = 10 kN

Answer

FBC = 10 kN (tension)

Questions and answers

What is Truss: method of joints?

A truss is a frame of straight bars joined at their ends to form triangles. You see them in roofs, cranes, pylons and many bridges.

The bars are called members and the connection points are joints. In the simple model, every joint is a pin and loads act only at the joints.

How do I solve Truss: method of joints problems step by step?
  1. Find the support reactions for the whole truss.
  2. Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
  3. Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
  4. Assume tension; a negative answer means compression.
What are common mistakes in Truss: method of joints?
  • Starting at a joint with three unknowns.
  • Forgetting the angle components of inclined members.
  • Swapping tension and compression.
What do the Truss: method of joints exercises cover?
  • Easy: symmetric
  • Medium: asymmetric
  • Hard: vertical + horizontal load

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Learn first

Beam support reactions

Continue with

Frames & machines

Calculators

Truss calculator (method of joints)

Courses with this topic

StaticsStructural AnalysisFE Civil exam practice

More topics in Civil

Shear force & bending momentBeam deflectionCentroid & second moment of areaColumn bucklingEffective stress in soilConsolidation settlementHydrostatic forceOpen-channel flow (Manning)

Printable worksheet

Truss: method of joints worksheet with answers
Practise Truss: method of joints for freePractise Truss: method of joints now5 free problems every day. No card needed.