The method of joints for truss analysis is one of the two essential solutions to solve for the internal forces of any truss structure (the other being the method of sections).
Key Idea: Disassemble the Truss
Let's recall that the truss is simply an assemblage of members. The fundamental principle for this method is to break down this structure into its components and analyze the free body diagrams (FBD) for each joint and member. We can then solve for the member force using key analysis concepts.
General Outline
The following is a general outline of how to use the method of joints (for plane structures):
Analyze Joint Connection Forces with Two Unknowns First
After breaking the truss into its components, we first start analyzing the free-body diagram of a joint with a maximum of two unknown forces (for 2D truss) or three unknown forces (for 3D truss). It doesn't matter which joint as long as we follow this requirement.
This number corresponds to the number of equilibrium equations we can use for each joint. The key idea here is to use these equations to solve for the forces acting on the joint. We cannot start this method with a connection with unknowns greater than the possible number of equilibrium equations.
Analyze Adjacent Members and Joints
After analyzing one joint, we can interpret the adjacent free-body diagrams and determine the internal bar force for each member.
Using Newton's Third Law, the force on the joint must exert something similar of equal magnitude and opposite direction on the connecting end of the adjacent member. For the whole member to be in equilibrium, it must have a force equal in magnitude but opposite in direction at the other end of the member. This load is the internal bar force of the truss member.
Part of the analysis is also to know whether the axial force is either tension or compression.
- If the two opposing forces cause the member to shorten, the internal bar force in the member is a compressive axial force.
- If these two end forces cause the member to elongate, then the axial force is tension.
After analyzing the member, we repeat the process of transferring the force from the member to the next adjacent joint and solve again for these unknowns as we did in the first step.
Repeat Analysis
At this point, you may already see a general pattern for the method of joints. You're moving your computed bar forces from one joint to member to the next joint. It is a repetition of steps 1 and 2 until we finally solve all internal bar force members:
Tips
- When we don't consider the weight of the members or external forces applied to it (plus, the members of the truss are all straight lines), we don't usually anymore analyze the free body diagram of the member itself. Most references only examine the joints' FBD and interpret the internal bar force from the joint connection force. As an observation, if the joint connection force is acting away from the joint, the bar force is in tension, while if it is working towards it, it is a compressive force.
- Some joints will require you to simultaneously solve two unknowns which is possible since you have two equilibrium equations for plane trusses. For space trusses, you can simultaneously solve three.
- A joint may have more than two unknowns. In these instances, you can proceed to other joints instead and work your way back toward that joint with more than two unknowns. In this way, you're eliminating the number of unknowns for that specific joint.
- Nearing the end of your analysis, you'll realize that you have solved every internal bar force without analyzing one last joint. It is always a good idea to double-check your computations by examining this joint and ensuring that all components' sum equals zero; otherwise, your solution might need to be corrected.
Summarise Your Results
After analysis, it is always a good idea to summarise your results using a table.
Application
- Let's illustrate how to use the method of joints using this truss example.
Summary
Let's summarize:
The method of joints for truss analysis is one of the two essential solutions to solve for the internal forces of any truss structure (the other being the method of sections).
The fundamental principle for this method is to break down this structure into its components and analyze the free body diagrams (FBD) for each joint and member.
We analyze a joint with two unknowns first, then analyze adjacent components. We continually repeat this pattern until we have thoroughly examined all bar forces.
