6.4: Method of Joints

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Key Questions

• What are the important components to include on a free-body diagram of a joint in a truss?
• How are the solutions found at one joint used to create an accurate free-body diagram of another joint?
• How do we ensure that tension or compression in a member is properly represented?

The method of joints is a process used to solve for the unknown forces acting on members of a truss. The method centers on the joints or connection points between the members, and it is most useful when you need to solve for all the unknown forces in a truss structure.

The joints are treated of particles subjected to force by the connected members and any applied loads. As the joints are in equilibrium and the forces are concurrent, \(\Sigma \vec{F} = 0\) can be applied, but the \(\Sigma M=0\) equation provides no information.

For planar trusses, each joint yields two scalar equations, \(\Sigma F_x=0\) and \(\Sigma F_y=0\text{,}\) and so two unknowns can be found. Therefore, a joint can be solved when there are one or two unknowns forces and at least one known force acting on it.

Forces are transferred from joint to joint by the connecting members, so when unknown forces on a joint are found, the corresponding forces on adjacent joints are also found.

Procedure

The procedure is straightforward application of rigid body and particle equilibrium

1. Determine if the structure is a truss and if it is determinate. See Subsection 6.3.2
2. Identify and remove all zero-force members. This is not required, but will eliminate unnecessary computations. See Subsection 6.3.4.
3. Determine if you need to find the external reactions. If you can identify a solvable joint immediately, then you do not need to find the external reactions.

   A solvable joint includes one or more known forces and no more than two unknown forces. If there are no joints that satisfy this condition then you will need to find the external reactions before proceeding, using a free-body diagram of the entire truss.

4. Identify a solvable joint and solve it using the methods of Chapter 3. When drawing free-body diagrams of joints you should
   • Represent the joint as a dot.
   • Draw all known forces in their known directions with arrowheads indicating their sense. Known forces are the given loads, and forces determined from previously solved joints.
   • Assume the sense of unknown forces. A common practice is to assume that all unknown forces are in tension, i.e. pulling away from the free-body diagram of the pin, and label them based on the member they represent .

   Finally, write out and solve the force equilibrium equations for the joint. If you assumed that all forces were tensile earlier, negative answers indicate compression.

5. Once the unknown forces acting on a joint are determined, carry these values to the adjacent joints and repeat step four until all the joints have been solved. Take care when transferring forces to adjoining joints to maintain their sense — either tension or compression.
6. If you solved for the reactions in step two, you will have more equations available than unknown forces when you reach the last joint. The extra equations can be used to check your work.

Rather than solving the joints sequentially, you could write out the equations for all the joints first and solve them simultaneously using a matrix solution, but only if you have a computer available as large matrices are not typically solvable with a calculator.

The interactive below shows a triangular truss, loaded at the top and supported by a pin a \(A\) and a roller at \(B\text{.}\) You can see how the reactions and internal forces adjust as you vary the load at \(C\text{.}\) You can solve it by starting at joint \(C\) and solving for \(BC\) and \(CD\text{,}\) then moving to joint \(B\) and solving for \(AB\) Joint \(A\) can be used to check your work.

Figure 6.4.1. Internal and external forces of a simple truss.