Look up at a railway bridge, a stadium roof, or the lattice arm of a tower crane and you are looking at a truss — a frame of straight members pinned together at their ends. It looks complicated, a web of triangles carrying load along paths that are not obvious. Yet the engineer who designed it sized every single member with nothing more than the equations of static equilibrium and a clear head.

The trick is to stop looking at the whole structure at once. This article explains the method of joints, the workhorse technique of truss analysis, works through a loaded joint by hand, and points out the assumptions and slips that catch people out.

Why this calculation matters

A truss earns its keep by being efficient. Triangulated members carry load mostly as pure tension or compression — straight along their length — rather than as bending. Axial load uses material far better than bending does, which is why trusses span long distances with remarkably little weight. But that efficiency only holds if every member is sized for the force it actually carries, and those forces are rarely what intuition suggests.

Get the force in a member wrong and the consequences split two ways. Underestimate a tension member and it may yield or tear. Underestimate a compression member and it may buckle, often long before the material yields. You cannot size any member until you know its force and its sign — tension or compression. Truss analysis is the step that produces both, and it is the foundation for every check that follows.

The core method

A truss is built so that each member is a two-force member: loads are applied only at the joints, the joints behave as frictionless pins, and each member therefore carries a single force directed along its own axis. That idealization is what makes a truss tractable.

The method of joints applies static equilibrium to one joint at a time. Each pin joint in a plane truss is a particle in equilibrium, so the forces meeting there must balance in two directions:

sum of F_x = 0
sum of F_y = 0

Two equations per joint. That means you can solve a joint as long as it has no more than two unknown member forces. The strategy is simple: start at a joint where you already know enough — often a support or a loaded joint with only two members — solve its two unknowns, then move to an adjacent joint where those now-known forces leave only two new unknowns. Work across the truss joint by joint until every member force is found.

A useful sign convention: assume every unknown member force is tension, pulling away from the joint. If the equilibrium equations return a positive number, the member really is in tension. A negative number means the member is in compression, pushing on the joint instead. Letting the algebra tell you the sign is far safer than guessing it up front.

For the geometry, a member inclined at an angle theta contributes a horizontal component F times cos(theta) and a vertical component F times sin(theta). Resolving each member force into those components and summing is the whole of the method.

A worked example

Take a single joint inside a loaded truss. A vertical downward load of 10 kN acts at the joint, which is held by two members: one horizontal member and one diagonal member inclined at 45 degrees. We want the force in each.

Step 1 — vertical equilibrium. Only the diagonal member has a vertical component, and it must balance the 10 kN downward load. With the diagonal inclined at 45 degrees:

F_diag * sin(45 deg) = 10 kN
F_diag = 10 / 0.7071 = 14.14 kN

The diagonal must push up on the joint to support the load, so it is in compression. A compression member pushes on the joints at both its ends.

Step 2 — horizontal equilibrium. The diagonal also has a horizontal component, and the only other member that can balance it is the horizontal one:

F_horiz = F_diag * cos(45 deg) = 14.14 * 0.7071 = 10.0 kN

The horizontal member pulls on the joint, so it is in tension.

So at this joint the diagonal carries 14.14 kN in compression and the horizontal member carries 10.0 kN in tension. Notice the structure of the work: each joint gives exactly two equilibrium equations, and from them come exactly two member forces. Repeat the process joint by joint and the entire truss is solved — no bending, no deflection, just equilibrium applied patiently.

Common mistakes

Starting at the wrong joint. The method only works at a joint with two or fewer unknown member forces. Begin at a joint with three unknowns and you have two equations and three unknowns — unsolvable. Find a support or a simple loaded joint first, then chain inward.

Dropping the sign. A negative result is not an error; it is information. It means you guessed tension and the member is actually in compression. Carry the sign through and report tension or compression explicitly for every member.

Confusing the angle. The components are F times cos(theta) and F times sin(theta), but which is which depends on how theta is measured. Always sketch the member, mark the angle from a clear reference, and confirm that the vertical component uses the sine of the angle to the horizontal.

Forgetting that loads must act at joints. The method of joints assumes loads and reactions are applied only at the pins. A load hanging from the middle of a member breaks the two-force-member assumption — that member now carries bending, and a different analysis is needed.

Skipping the determinacy check. Before solving, confirm the truss is statically determinate. If members equal twice the number of joints minus the support reactions, equilibrium alone is enough; otherwise you need compatibility as well.

Try the interactive NovaSolver calculator

Working a small truss by hand builds the right intuition, but for anything larger it pays to let a solver carry the bookkeeping. The 2D Truss Analyzer on NovaSolver uses the direct stiffness method to solve planar trusses and color-codes every member by its axial force — tension in red, compression in blue. You can start from preset geometries such as Pratt, Warren, and K-trusses, set the modulus and cross-section area, edit the nodal loads, and read back the maximum tension, maximum compression, maximum displacement, and the node and member counts. It is a fast way to see how load redistributes when you change the layout.

Related calculators

• Statics truss — focus on the equilibrium side, working member forces the way the method of joints does by hand.
• Bridge truss — apply the method to span-type structures and see how load travels to the supports.
• FEM truss — go deeper into the stiffness-matrix approach behind larger, computer-solved frames.

The full set lives in the structural tools hub.

Closing note

Truss analysis looks formidable until you accept its central idea: do not solve the structure, solve one joint. Two equilibrium equations per joint, two member forces at a time, chained across the frame — that is the whole method. Keep your sign convention honest so tension and compression sort themselves out, check determinacy before you start, and respect the two-force-member assumption. Master one joint and the whole bridge becomes a sequence of small, solvable problems.