Method of Joints and Method of Sections
Definition
The Method of Joints is a truss analysis technique in which each joint is treated as a particle in equilibrium, and the forces in the connected members are determined using the equations of equilibrium:
The Method of Sections is a truss analysis technique in which the truss is imagined to be cut through selected members, and one part of the truss is analyzed using equilibrium equations to find the internal forces in the cut members.
These methods are applicable mainly to plane trusses that are statically determinate, meaning all unknown forces can be found using equilibrium alone.
Main Content
1. Method of Joints
- In this method, the truss is analyzed joint by joint. Each joint is isolated as a free-body diagram, and all forces meeting at that joint are considered. Since a joint in a plane truss is a concurrent-force system, only two equilibrium equations are required:
- The method is usually started at a joint where only two unknown member forces are present, because with two equations, only two unknowns can be solved at a time. After finding those forces, the solution moves to adjacent joints systematically.
- This method is ideal for finding forces in all members of a truss, but it can be time-consuming for large structures. The nature of each member force is identified as:
- Tension if the member pulls away from the joint
- Compression if the member pushes toward the joint
For example, in a simple triangular truss with a load at the top joint, the support reactions are first found using whole-truss equilibrium. Then one joint is analyzed to find member forces, and the results are carried from joint to joint.
2. Method of Sections
- In this method, the truss is cut through the members whose internal forces are required. The cut should pass through no more than three unknown members in a plane truss, because the body after cutting is treated as a rigid body and only three independent equilibrium equations are available:
- The portion of the truss on one side of the cut is then isolated and analyzed. By taking moments about a point where two cut-member forces intersect, one unknown can be eliminated, making the calculation easier.
- This method is especially useful when the force in only one or two members is needed, because it avoids analyzing every joint in the truss. It is a quick and powerful method for large trusses.
- Like the method of joints, the results indicate whether a member is in tension or compression depending on the assumed direction of the cut-member force.
For example, if the force in one diagonal member of a bridge truss is needed, a section can be drawn through that diagonal and two other members. Equilibrium of one side of the cut can then directly give the internal force in the diagonal member.
3. Comparison, Assumptions, and Sign Convention
- Both methods are based on important assumptions of truss theory:
- Members are straight and connected by frictionless pin joints
- Loads act only at joints
- Members are two-force members, carrying only axial force
- The truss is stable and statically determinate
- A correct sign convention is essential. Usually, forces are assumed to be in tension initially, pulling away from the joint or cut section. If the calculated value is positive, the assumption is correct; if negative, the actual force is opposite and therefore compression.
- The key difference is efficiency:
- Method of joints is better for finding all member forces
- Method of sections is better for finding selected member forces quickly
- In practical engineering, both methods are often used together. Support reactions are found first from whole-truss equilibrium, then the required member forces are obtained using the method that is most convenient.
A common mistake is applying the method to trusses that are not statically determinate or not planar, where additional analysis methods are required.
Working / Process
1. Determine support reactions
- Begin by treating the entire truss as a single free body.
- Apply global equilibrium equations:
- Find all external reactions at supports before analyzing internal member forces.
2. Choose the appropriate method and isolate the structure
- For the method of joints, select a joint with at most two unknown member forces and draw its free-body diagram.
- For the method of sections, draw a cut through the truss that crosses no more than three unknown members, then isolate one side of the cut.
- Show all external loads, reactions, and member forces acting on the isolated part.
3. Apply equilibrium equations and solve
- For the method of joints, apply and at the chosen joint.
- For the method of sections, apply , , and especially to solve for the cut member forces.
- Continue joint-by-joint if using joints, or calculate only the required members if using sections.
- Interpret the final values:
- Positive result with assumed tension means the member is in tension
- Negative result means the member is in compression
Advantages / Applications
- The method of joints is highly useful for determining the force in every member of a truss, making it suitable for complete structural analysis in classroom problems and design verification.
- The method of sections is faster when only a few specific member forces are required, which saves time in large trusses such as bridge trusses, roof trusses, and crane frameworks.
- Both methods help engineers ensure safe and economical design by identifying members under high tension or compression, allowing proper sizing of structural elements.
Summary
- The method of joints analyzes a truss one joint at a time using force equilibrium.
- The method of sections analyzes a cut portion of the truss using force and moment equilibrium.
- Both methods are essential tools for finding internal member forces in statically determinate trusses.