Analysis of Plane Trusses
Definition
A plane truss is a two-dimensional structure composed of straight members joined at their ends by pin joints, with loads and reactions acting only at the joints, such that each member carries only axial force, either tension or compression.
Analysis of plane trusses is the process of determining the internal axial forces in the truss members and the support reactions using the equations of static equilibrium.
Main Content
1. Basic Assumptions and Idealization of Plane Trusses
Members are straight and slender
- Each member is assumed to be long compared to its cross-sectional dimensions, so bending effects are neglected and only axial force is considered.
Joints are ideal pin connections
- The connections are treated as frictionless pins, allowing rotation without transfer of moments. This means members do not resist bending at the joints.
Loads act only at joints
- External loads and support reactions are assumed to be applied only at the joints, ensuring each member behaves as a two-force member.
Self-weight is neglected or lumped at joints
- In basic analysis, the weight of members is ignored or converted into equivalent joint loads for simplicity.
Members are in either tension or compression
- Since bending and shear are neglected, each member develops only an axial force along its length.
Why these assumptions matter
- They make the structure statically solvable by using equilibrium equations, which is the foundation of plane truss analysis.
2. Conditions for Stability and Determinacy
Stable structure requirement
- A truss must be sufficiently constrained so that it does not move as a rigid body under loading. If it is under-supported or improperly shaped, it becomes unstable.
Perfect truss condition
- For a plane truss, the relation is used to identify a perfect truss, where is the number of members and is the number of joints.
Statically determinate truss
- If the number of unknown member forces and reactions can be found entirely from equilibrium equations, the truss is statically determinate.
Statically indeterminate truss
- If there are more unknowns than equilibrium equations, additional compatibility relations are needed, making the truss indeterminate.
Importance in design
- Before analysis, the engineer checks whether the truss is stable and determinable, because an unstable or indeterminate truss requires different methods.
Example of determinacy
- A simple triangular truss is perfect and stable because it satisfies the member-joint relation and can be solved using equilibrium alone.
3. Methods of Analysis of Plane Trusses
Method of Joints
- Each joint is isolated as a free body, and the equilibrium equations and are applied to determine member forces one joint at a time.
Method of Sections
- An imaginary cut is made through the truss to expose selected member forces, and equilibrium equations are applied to one part of the truss to directly find forces in specific members.
Zero-force members
- Some members may carry no load under a given loading condition. These can often be identified quickly using joint geometry and equilibrium rules, simplifying the analysis.
Sign convention
- A member force assumed to pull away from the joint is taken as tension; if the calculated result is negative, the member is actually in compression.
Choice of method
- The method of joints is best for finding all member forces systematically, while the method of sections is efficient when only a few specific member forces are required.
Example
- In a roof truss, if the force in a central diagonal is needed, the method of sections can determine it faster than analyzing every joint.
Working / Process
1. Determine support reactions
- Draw the whole truss as a free-body diagram.
- Replace supports by reaction components.
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Apply global equilibrium equations:
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Solve for the reactions at the supports before moving to individual members.
2. Analyze joints or sections
- Use the method of joints by selecting a joint with not more than two unknown member forces.
- Or use the method of sections by cutting through members whose forces are required.
- Apply equilibrium equations to the selected joint or truss portion.
- Continue systematically until all required member forces are found.
3. Interpret member forces and verify results
- Identify whether each member is in tension or compression based on the force direction.
- Check for zero-force members and confirm equilibrium at each joint.
- Ensure the results are physically reasonable and consistent with truss geometry and loading.
Advantages / Applications
Efficient load transfer
- Plane trusses carry loads effectively through axial forces, reducing bending moments and material stress.
Lightweight and economical
- Because members mainly resist tension or compression, trusses use less material than solid beams for long spans.
Wide structural use
- They are commonly used in bridges, roof frames, towers, transmission line supports, cranes, and industrial sheds.
Easy analysis and design
- Under ideal assumptions, trusses can be analyzed using basic equilibrium principles, making them suitable for engineering calculations.
Long-span capability
- Trusses can span large distances with minimal deflection when properly designed.
Modular construction
- Truss members can be prefabricated and assembled quickly at the site.
Example application
- Railway bridges often use trusses because they can support heavy moving loads while remaining structurally efficient.
Summary
- Plane trusses are two-dimensional frameworks designed to carry loads mainly through axial forces in members.
- Their analysis depends on equilibrium, stability, and idealized pin-joint behavior.
- The method of joints and method of sections are the main tools used to find member forces.
- A plane truss is efficient, economical, and widely used in real engineering structures.