Statics · Module 6 of 11 · Intermediate
Structural Analysis
Trusses, frames, and machines are solved by exposing the forces inside members.
Readiness check
From Module 5. Tick only what you can do closed-notes.
- Find support reactions of a simply supported structure quickly and correctly.
- Explain what a two-force member is and why its force direction is known.
- Apply ΣFx = 0 and ΣFy = 0 at a single point (particle equilibrium).
- Take a moment about a chosen point to eliminate unknowns.
- Work with the slope of a line (rise over run) to get force direction cosines.
The core idea
Every truss member is a two-force member: pure tension or pure compression.
Joints: ΣFx = 0, ΣFy = 0Sections: add ΣM = 0Method of joints: walk the truss one pin at a time, at most two unknowns per joint. Method of sections: cut through three or fewer members and use a moment equation to reach a deep member directly.
The method
Reactions first: treat the whole truss as one rigid body.
Scan for zero-force members before any algebra.
Isolate a joint with at most two unknowns; assume tension.
Walk joint to joint, or cut a section for a deep member.
Worked example: method of joints on a simple truss
A triangular truss: joint A(0, 0) on a pin, B(4 m, 0) on a roller, apex C(2 m, 2 m). A 12 kN load acts straight down at C. Find the force in members AB and AC.
- ProblemFind the forces in members AB and AC of the truss in Figure 1.
- Given / findGeometry above, 12 kN at C. Find FAB and FAC, tension or compression.
- AssumptionsFrictionless pins, weightless members, load at the joint only: the ideal truss.
- ModelReactions first: by symmetry Ay = By = 6 kN, Ax = 0. Then isolate joint A (Figure 2). Member AC runs at 45° (2 m up over 2 m across); assume both member forces in tension.
- EquationsJoint A, ΣFy = 0: 6 + FAC sin 45° = 0 Joint A, ΣFx = 0: FAB + FAC cos 45° = 0
- SolveFAC = −6/0.7071 = −8.49 kN, so 8.49 kN compression. Then FAB = −(−8.49)(0.7071) = +6 kN tension.
- CheckSigns make physical sense: the sloped top member is squeezed by the load, and the bottom chord is stretched holding the supports together. Joint B mirrors A by symmetry.
- ConclusionAC must be checked against buckling (compression member); AB against yielding (tension member). Same force scale, completely different failure modes. That is why the sign matters.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Treating a frame member as a two-force member | Force directions assumed along the member when they are not | "Is this member loaded at exactly two points and nowhere else?" | If a member carries a load between its ends or at three points, draw its full FBD with both components at each pin. |
| Abandoning the tension-positive convention mid-solution | Random sign flips; compression reported as tension | "Did I draw every unknown pulling away from the joint?" | Always assume tension. A negative result means compression. Never redraw mid-problem. |
| Missing zero-force members | Extra algebra; sometimes wrong propagated values | "Two collinear members plus one off-axis member, no load. What is the off-axis force?" | Learn the two zero-force rules and scan the truss before solving anything. |
| Cutting four members with one section | Four unknowns, three equations: dead end | "How many members does my cut cross?" | A section may cross at most three members with unknown forces. Re-route the cut. |
Practice ladder
At an unloaded joint, two members are collinear and a third meets them at an angle. What is the force in the third member, and why?
Show answer
Zero. Equilibrium perpendicular to the collinear pair has only one force component, the angled member's, so it must vanish. The collinear members then carry equal forces.
For the worked-example truss, find the force in member BC and confirm joint C balances.
Show answer
By symmetry FBC = FAC = 8.49 kN compression. Joint C check: the two compressions push up and inward on C; vertical components 2 × 8.49 sin 45° = 12 kN balance the load.
A flat Pratt-style truss panel: you need the force in one bottom-chord member near midspan, far from the supports. Which method do you choose, where do you cut, and which equation do you write first?
Show answer
Method of sections. Cut vertically through three or fewer members including the target chord; take moments about the joint where the other two cut members intersect. The target force is then the only unknown in one equation.
Find a real truss (roof, pedestrian bridge, crane boom, transmission tower). Photograph it, identify the triangulation pattern, mark which members you expect in tension and compression under gravity load, then verify your intuition with a quick joints analysis of a 5-member simplification.
What good work looks like
Annotated photo, simplified line model, predicted and computed senses for at least 4 members, and a note on which compression member looks most buckling-critical (longest and most slender).
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Design and analyze a small truss bridge (spaghetti, popsicle sticks, or CAD): pick a span and load, compute all member forces by joints, identify the critical compression member, and document the predicted failure load. Build and test it if materials allow, then compare prediction to reality.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What two idealizations define a simple truss?
Members joined by frictionless pins, and loads applied only at the joints (members weightless or weight split to joints).
2. Why is every ideal truss member in pure tension or compression?
It is a two-force member: forces at only two points must be equal, opposite, and along the line joining them.
3. State the two zero-force member rules.
(1) Two non-collinear members at an unloaded joint: both are zero. (2) Two collinear members plus one angled member at an unloaded joint: the angled one is zero.
4. When does the method of sections beat the method of joints?
When you need one specific member deep inside the truss: a single cut and a moment equation reach it without solving every joint.
5. How does frame and machine analysis differ from truss analysis?
Frame members are multi-force members: dismember the structure, draw a full FBD of each part, and use Newton's third law at every connection.
Textbook mapping
| Item | Mapping |
|---|---|
| Main textbook | R.C. Hibbeler, Engineering Mechanics: Statics, Chapter 6, Structural Analysis |
| Core sections | 6.1 Simple Trusses · 6.2 The Method of Joints · 6.3 Zero-Force Members · 6.4 The Method of Sections · 6.6 Frames and Machines |
| Recommended problems | Fundamental Problems F6-1 onward (partial solutions in the back). Do joints problems until you stop redrawing arrows; then sections; then two frame problems minimum. |
| Skip on first pass | 6.5 Space Trusses (3D); return after the 3D equilibrium sections of Chapter 5. |
| Read next | Chapter 7, sections 7.1 to 7.2 before opening Module 7. |