The method of sections is a fast way to find forces in selected members of a truss without solving the entire structure joint by joint. It is especially useful for bridge trusses, roof trusses, and crane booms where only a few internal member forces are needed for design checks. The main idea is to slice through the truss, isolate one side, and treat the cut member forces as unknown external forces on a free-body diagram.
This connects structural geometry directly to the equilibrium equations used throughout engineering mechanics.
A valid section cut usually passes through no more than three unknown members because a planar rigid body has three independent equilibrium equations. After cutting, engineers apply sum of forces in x equals zero, sum of forces in y equals zero, and sum of moments equals zero to the isolated piece. Choosing a moment center where two unknown member lines intersect can eliminate them and solve the third force directly.
A positive result in the assumed direction confirms the assumed tension or compression, while a negative result means the actual force acts opposite the assumption.
Understanding Engineering: Truss Analysis by Method of Sections
A truss analysis begins with a structural model, not the cut itself. Engineers usually assume that members meet at pin joints, loads act only at joints, and each member carries force along its own length. Under these conditions, a member is either being stretched or squeezed.
It does not resist bending in the ideal model. Real bridges have welded connections, bolts, gusset plates, and member weight.
Those details can create bending or secondary forces. The simple truss model remains useful when the structure is designed so that these extra effects are small compared with the main axial forces.
Before selecting a section, find the external reactions if they are needed on the side you plan to keep. Start with the whole truss because internal forces cancel when the complete structure is treated as one body. A pin support can provide a horizontal reaction and a vertical reaction.
A roller support usually provides one reaction normal to its surface. For vertical loading on a level roller, that reaction is vertical. Check the reaction results against the loading.
The upward reactions must balance the total downward load. Their moments must balance too. A wrong reaction will make every force found later unreliable.
The best cut is chosen for convenience, not just because it crosses the target member. Keep the smaller side when possible, since it usually has fewer loads and reactions to include. Draw every force that acts on that isolated part.
This includes applied loads, support reactions, and the forces exposed by the cut. For an unknown cut member, it is common to assume tension first. Draw its arrow pulling away from the cut face along the member.
The final sign tells you whether that assumption worked. When using moments, use the perpendicular distance from the pivot to a force line.
This distance is not always a horizontal or vertical measurement. For a sloping member, careless distance measurements are a common source of errors.
Geometry strongly affects the answer. A shallow diagonal can need a large axial force to provide a modest vertical component. This helps explain why deep trusses are often efficient for long spans.
The top chords of a loaded bridge truss are commonly in compression, while the bottom chords are commonly in tension, though the exact pattern depends on loading and support conditions. Students can see the same ideas in roof frames, bicycle frames, transmission towers, and construction cranes. In each case, triangles help prevent shape change.
When checking work, verify that force directions follow the member axes, units remain consistent, and the isolated section satisfies force balance after the member forces are found. A result with an unexpectedly huge value can be real, but it should prompt a careful check of the diagram, distances, and assumed load paths.
Key Facts
- Planar equilibrium equations: ΣFx = 0, ΣFy = 0, and ΣM = 0.
- A section cut should pass through no more than 3 unknown member forces in a simple planar truss.
- Method of sections is best for finding a few specific member forces quickly.
- Taking moments about the intersection of two unknown member forces eliminates both from the moment equation.
- Tension pulls away from a joint or cut surface, while compression pushes toward it.
- For a simply supported truss with vertical loads only, support reactions often start with ΣMA = 0 and ΣFy = 0.
Vocabulary
- Truss
- A structure made of straight members connected at joints, designed so members mainly carry axial tension or compression.
- Method of Sections
- A truss analysis method that cuts through selected members and applies equilibrium to one isolated part of the truss.
- Free-Body Diagram
- A diagram that shows one isolated body with all external forces, reactions, and unknown cut forces acting on it.
- Axial Force
- A force that acts along the length of a truss member, either pulling it in tension or pushing it in compression.
- Support Reaction
- A force or moment supplied by a support to keep a structure in static equilibrium.
Common Mistakes to Avoid
- Cutting through too many unknown members, which makes the section impossible to solve with only three planar equilibrium equations.
- Forgetting to include support reactions before using the section, which gives incorrect force balance for the isolated truss piece.
- Assuming every unknown member force is vertical or horizontal, which is wrong because each member force must act along the member's actual angle.
- Labeling a negative answer as an error, when it usually means the assumed tension or compression direction is opposite to the real direction.
Practice Questions
- 1 A simply supported truss has a pin at A, a roller at F, a span of 12 m, and a 30 kN downward load at joint C located 4 m from A. Find the vertical reactions at A and F.
- 2 A section cut through members BC, CD, and DE leaves the left side of a truss isolated. If taking moments about point C gives 6 m times F_DE minus 48 kN m equals 0, find F_DE and state whether it is tension if it was assumed pulling away from the cut.
- 3 A section cut passes through three unknown members, and two of their lines of action intersect at joint B. Explain why taking moments about joint B is a useful strategy for solving the third member force.