Truss analysis is the process of finding the internal forces in the straight members of a pin-connected structure. This cheat sheet covers the method of joints and the method of sections, two standard tools used in introductory engineering statics. Students need these methods to decide whether truss members are in tension or compression and to check if a structure can safely carry loads.
The sheet is designed as a quick reference for solving homework, lab, and design problems.
Key Facts
- A truss is commonly modeled as pin-connected members with loads applied only at the joints, so each member is a two-force member.
- For the whole truss in static equilibrium, use sum of Fx = 0, sum of Fy = 0, and sum of M = 0 to find support reactions.
- In the method of joints, isolate one joint at a time and apply sum of Fx = 0 and sum of Fy = 0.
- Assume unknown member forces are in tension by drawing them pulling away from the joint, and a negative answer means the member is in compression.
- In the method of sections, cut through no more than three unknown members when possible and apply sum of Fx = 0, sum of Fy = 0, and sum of M = 0 to one side of the cut.
- Choose a moment center where two unknown cut-member forces intersect so their moments are zero and the remaining unknown can be solved directly.
- A zero-force member occurs at an unloaded joint with two non-collinear members, or at an unloaded joint with three members where two are collinear and the third is zero.
- For a planar truss, a common determinacy check is m + r = 2j, where m is members, r is reaction components, and j is joints.
Vocabulary
- Truss
- A structure made of straight members connected at joints and designed to carry loads mainly through axial forces.
- Method of Joints
- A truss analysis method that solves forces by applying equilibrium equations to individual joints.
- Method of Sections
- A truss analysis method that cuts through selected members and applies equilibrium equations to one part of the truss.
- Two-Force Member
- A member with forces acting only at its two ends, so the forces are equal, opposite, and along the member axis.
- Tension
- An axial force that pulls a member outward and tends to stretch it.
- Compression
- An axial force that pushes inward on a member and tends to shorten it.
Common Mistakes to Avoid
- Forgetting to solve support reactions first is wrong because joint and section equations need the external reaction forces for equilibrium.
- Treating member forces as vertical or horizontal when the member is diagonal is wrong because a member force acts along the member and must be resolved into x and y components.
- Using too many unknowns at one joint is wrong because a 2D joint only provides sum of Fx = 0 and sum of Fy = 0, so it can solve at most two unknowns.
- Cutting through more than three unknown members in the method of sections is usually wrong because one free-body diagram gives only three independent equilibrium equations.
- Changing the sign convention halfway through a problem is wrong because negative values only have meaning if the assumed tension or compression direction stays consistent.
Practice Questions
- 1 A simply supported truss has a pin at A, a roller at B, span AB = 6 m, and a 12 kN downward load at the midpoint. Find the vertical reactions at A and B.
- 2 At a joint, two unknown member forces act along a horizontal member and a 3-4-5 diagonal member. If the diagonal force is 10 kN in tension, what are its horizontal and vertical components?
- 3 A section cut passes through members DE, DF, and EF. If the lines of action of DE and DF meet at joint D, which equilibrium equation is best for solving force EF directly?
- 4 Why is it important that truss loads are applied at joints when using the two-force member assumption?
Understanding Truss Analysis Method of Joints & Sections
The ideal truss model works because it simplifies a complicated frame into separate force paths. A real bridge or roof truss has gusset plates, bolts, welds, member weight, and joints with some stiffness. In a basic statics problem, these details are replaced by perfect pins and straight members.
This means a member can only pull or push along its own length. It does not resist bending in the model. The geometry matters greatly.
A triangular arrangement stays rigid because changing its shape requires a member to change length. A four-sided shape can distort unless it has a diagonal brace. This is why triangles appear so often in cranes, towers, bicycle frames, and roof supports.
Before finding any member force, draw a careful free body diagram of the entire structure. Include every external load, the support reactions, and the important distances. A roller support usually supplies one reaction in its allowed direction.
A pin support supplies horizontal and vertical reactions. Moment equations are especially useful because a force passing through the chosen point creates no turning effect about that point. Check units throughout the work.
If loads are in kilonewtons and lengths are in metres, moments are in kilonewton metres. A wrong support reaction will spread errors through every later calculation, so it is worth checking that all horizontal forces, vertical forces, and turning effects balance.
When working joint by joint, choose a joint with no more than two unknown member forces after known loads and reactions are included. Resolve each sloping member force into horizontal and vertical parts. Use the angle shown in the diagram, not an angle guessed from the page.
A common mistake is mixing up sine and cosine. The component next to the angle uses cosine, while the component opposite the angle uses sine. Keep one sign convention for the whole problem.
For example, take rightward and upward forces as positive. The first result often provides the known force needed at a neighboring joint. Move through the truss in an orderly path rather than jumping between joints.
A section is most efficient when the problem asks for only a few members far from the supports. Imagine cutting the selected members, then draw one side as a separate free body diagram. Each cut member must be shown with its force acting along the original member direction.
Choosing the smaller side usually reduces the number of loads to include. The best moment center is often the point where two cut-member lines meet. Their effects disappear from the moment equation even if that intersection lies outside the physical truss.
After finding one force, use horizontal or vertical force balance for the others. This approach reflects a useful engineering idea. Good equation choices reduce work before any arithmetic begins.
Zero-force members deserve attention because they reveal structure rather than merely saving calculation time. They may be included to stabilize the truss during construction, support loads applied in a different case, reduce vibration, or provide redundancy if another member is damaged. Their force is zero only for the particular loading and ideal conditions being analyzed.
A joint that is unloaded in one diagram may carry a load in another. Finally, do not treat the determinacy count as proof that a truss is safe or stable.
It only gives a first check on whether statics equations may be enough. Member buckling, material strength, joint design, and deflection require further analysis in real structures.