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.
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.