Solving a truss means finding the internal force in every member and determining whether each one is in tension or compression. The process relies on static equilibrium: the idea that a structure at rest has zero net force and zero net rotation. For a two-dimensional truss, that gives you three equilibrium equations for the whole structure and two equations at every joint, which is enough to solve for all unknowns in a statically determinate truss.
There are two main methods, the method of joints and the method of sections, and most problems use one or both. Before either method works, though, you need to handle some preliminary steps.
Assumptions Behind Truss Analysis
Every textbook truss problem rests on a set of simplifying assumptions that make the math tractable. Members are straight and connected at their ends by frictionless pins, meaning the joints transmit forces but not moments. Because of this, every member experiences only axial force along its length, either pure tension or pure compression. External loads are applied only at the joints, never along a member’s span. And the deformations under load are small enough that the geometry of the truss doesn’t change in any meaningful way.
These assumptions let you treat each joint as a point where several straight forces meet, which is exactly the setup needed for simple force-balance equations.
Check Static Determinacy First
Before you start solving, verify that the truss is solvable with equilibrium alone. Count the number of members (m), the number of joints (j), and the number of external reaction forces (r). For a statically determinate truss:
m + r = 2j
Each joint gives you two equilibrium equations (sum of forces in x = 0, sum of forces in y = 0), so the total number of equations is 2j. If m + r equals 2j, you have exactly as many unknowns as equations. If m + r is less than 2j, the truss is unstable. If it’s greater, the truss is statically indeterminate and requires more advanced techniques beyond basic statics.
For example, a truss with 5 joints, 7 members, and 3 reaction forces gives 7 + 3 = 10, and 2 × 5 = 10. That checks out as statically determinate.
Step 1: Find the Support Reactions
Treat the entire truss as a single rigid body. Draw a free-body diagram showing the external shape of the truss, all applied loads, and the reaction forces at the supports. A pin support provides two reaction components (horizontal and vertical), and a roller provides one (perpendicular to the rolling surface), which typically gives you three unknown reactions for a 2D truss.
Start by summing moments about the support that has the most unknowns. This eliminates those unknowns from the equation and lets you solve for a reaction at the other support directly. Then write the sum of forces in the x-direction to find the horizontal reaction, and the sum of forces in the y-direction to find the remaining vertical reaction. At the end of this step, every external force acting on the truss should be known. Double-check your work by confirming that all three equilibrium equations are satisfied simultaneously.
Step 2: Spot Zero-Force Members
Before diving into joint-by-joint calculations, scan the truss for zero-force members. These carry no load under the current loading condition, so you can set their forces to zero immediately and simplify the rest of the analysis.
- Two-member unloaded joint: If only two members meet at a joint with no external load, and the two members are not collinear (not along the same line), both members are zero-force members.
- Three-member unloaded joint: If three members meet at an unloaded joint and two of them are collinear, the third member is a zero-force member. The two collinear members carry equal and opposite forces.
- Two-member loaded joint: If a joint connects exactly two members and has an external load, and one member is aligned with the direction of that load, the other member is a zero-force member.
Identifying these early reduces the number of unknowns you need to solve for and often makes subsequent joints solvable that otherwise would have too many unknowns.
Method of Joints
The method of joints is usually the fastest way to find the force in every member of a truss. You isolate each joint as a free body and apply two equilibrium equations: sum of forces in x equals zero, and sum of forces in y equals zero. Because each equation set has only two unknowns at most (if you pick your starting joint wisely), you can work through the truss joint by joint.
Choosing a Starting Joint
Pick a joint where you already know the external reactions and only two member forces are unknown. This is almost always one of the support joints. Solve for the two unknowns using the two force-balance equations, then move to an adjacent joint that now has at most two unknowns thanks to the forces you just found.
Setting Up Each Joint
Draw a free-body diagram of the joint. Include any external loads or reaction forces acting there. For each connected member, draw a force along the line of that member. A useful convention is to assume every unknown member force is tensile, meaning the arrow points away from the joint. After solving, a positive result confirms tension and a negative result means the member is actually in compression.
Resolve each force into x and y components using the geometry of the truss (rise, run, and member length or the angle). Write the two equations, solve, and record the results with their sign. Then move to the next joint.
Working Through the Truss
Continue joint by joint, always choosing a joint with no more than two unknowns. In a well-ordered truss, each solved joint reveals enough information to make the next joint solvable. By the time you reach the last joint, all member forces are determined. Use that final joint as a check: its equilibrium equations should be satisfied automatically. If they aren’t, there’s an arithmetic error somewhere upstream.
Method of Sections
When you only need the force in one or two specific members, cutting through the entire truss joint by joint is inefficient. The method of sections lets you jump straight to the members you care about.
Making the Cut
Imagine slicing the truss into two pieces with a straight cut that passes through the members you want to solve for. The cut should go through members, not through joints, and should cross no more than three members in a 2D problem. That limit matters because you have three equilibrium equations (sum of forces in x, sum of forces in y, sum of moments) for the section, so three unknowns is the maximum you can handle.
Drawing the Section Free-Body Diagram
Choose whichever half of the cut truss has fewer external forces acting on it. Draw it as a free body, showing all external loads and reactions on that half, plus the internal forces in the cut members. These internal forces act along the lines of their respective members. Again, assume tension (arrows pointing away from the cut face) for a consistent sign convention.
Solving With Moment Equations
The real power of the method of sections comes from choosing your moment point strategically. If two of the three cut members intersect at a point, summing moments about that point eliminates both of them and gives you a single equation with one unknown: the force in the third member. Repeat with different moment points or use force-balance equations to find the remaining unknowns.
This approach can give you the force in a member deep inside the truss in just a few equations, without solving any other joints first.
Interpreting Your Results
Once you have numerical answers, the sign tells you the type of force. If you assumed all members were in tension at the start, every positive value is a tensile force (the member is being pulled apart) and every negative value is a compressive force (the member is being squeezed). When sketching results on the truss, tension is typically shown with arrows pulling away from the joints at each end of the member, and compression with arrows pushing toward the joints.
This distinction matters in real design because tension and compression members fail differently. A tension member needs enough cross-sectional area to avoid being pulled apart, while a compression member can buckle, meaning it bows sideways and collapses at a load well below what would crush the material. Top chords of a truss under downward loading are almost always in compression, which is why their stability against buckling is a primary design concern.
Choosing the Right Method
If a problem asks for the force in every member, start with zero-force members, then use the method of joints. Work systematically from a support joint across the truss. If a problem asks for the force in just one or two specific members, the method of sections is faster. In many real problems you’ll combine both: use the method of joints to solve a few joints near a support, then switch to a section cut to reach a member in the middle of the truss without grinding through every intermediate joint.
Regardless of which method you use, the sequence is always the same: check determinacy, find support reactions, identify zero-force members, then apply equilibrium equations at joints or across sections until every unknown is resolved.