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Section 6.3 Method of Joints

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 useful if you need to solve for all the unknown forces in a truss structure.

The process used in the method of joints is outlined below:

  1. First, determine if the structure is a truss and if it is determinate. See Subsection 6.2.2

  2. Identify and eliminate all zero-force members. This is not required, but can minimize your computations. See Subsection 6.2.4 above.

  3. Next, determine if you need to solve for the reactions first using the full section free-body diagram, or if you can solve for them as part of the Method of Joints. Joints are treated as a series of particles. As all forces on joints are concurrent, these forces can be solved with equilibrium force equations, but moment equations do not add any information. Thus each 2D joint yields two equations (\(\Sigma F_x=0\) and \(\Sigma F_y=0\)). To be able to solve for the unknowns at a joint in a step-by-step fashion, you’ll need ≤ 2 unknowns at each joint and ≥ 1 known force at each joint. If there are no joints that satisfy this condition and you would like to solve for the unknown forces in a step-by-step manner, then you will need to solve for the truss reactions first before starting the Method of Joints.

    Note that you could write out all the equations for each free-body diagram and solve them all simultaneously with a linear algebra matrix solution, but only if you have a computer available as large matrices are not typically solvable with a calculator.

  4. Next, represent the forces at each joint with a free-body diagram. It is simplest to:

    • Draw one joint free-body diagram at a time
    • Always draw known forces in their known direction and value (whether external, reaction, or interaction),
    • Draw unknown forces in assumed directions and uniquely label them. A common practice for trusses is to assume that all unknown forces are in tension (or pulling away from the free-body diagram of the pin) and label them based on the member they represent (like\(F_{AB}\)).

  5. Finally, write out and solve the force equilibrium equations for each of the joint free-body diagrams. If you assumed that all forces were tensile earlier, remember that negative answers indicate compressive forces in the members.

Figure 6.3.1. Interactive showing how the method of joints creates free-body diagrams of each joint for you to solve for the internal tension and compression forces. In this interactive forces are labeled \(\vec{F}\text{,}\) internal tension forces \(\vec{T}\text{,}\) and internal compression forces \(\vec{C}\text{.}\) To meet the ≤ 2 unknowns at each joint and ≥ 1 known force at each joint criteria, you could either solve for the reactions at \(B\) and \(D\) first or go straight into the method of joints, starting at \(C\text{,}\) then moving to \(D\text{,}\) and finally to \(B\) (where you could check your work).