Support Reactions | Method of Sections

**Method of Joints:** Method of joints is
usually used when we are interested in finding the internal force
in every single member of the truss. As its name implies, it relies
on the application of equilibrium condition at each joint to solve
for the member forces.

To start the analysis, it is best to draw a detailed free-body diagram, or an exploded view,of the truss as shown below. Since the members at each joint are held together by a single frictionless pin, then the exploded view should also include the free-body diagrams of these pins. Each member force is represented by a unique name in blue color. Notice that the external forces are shown acting on the joints and not the members. Also to simplify the analysis, all member forces in the preliminary free-body diagram are shown to be in tension. If the solution produces a negative member force, it indicates that its correct direction is opposite to that assumed.

With all reactions calculated previously, we will consider one joint at a time, and solve for the unknown member forces at that joint. The main factor to remember is the so-called feasibility condition,that is the joint being analyzed has to have at least one known force and at most two unknown forces acting on it. Otherwise, there will be more unknowns than the number of equilibrium equations available to solve for them.

In this example, the analysis can begin from either joint 1 or 4 as both joints meet the feasibility condition. The free-body diagram of joint 1 is shown below

The forces acting on a truss joint form a coplanar, concurrent force system. Therefore, only the equilibrium of forces in the horizontal and vertical directions can be used to solve for the unknowns. For joint 1 these equations are used to solve for F and F that are the axial forces in members 1 and 2.

ΣF_{y} = 0 |
=> | R_{1-y} + [
(y_{6} - y_{1}) / l ] F = 0 |

where l is the length of member 1, and the term inside brackets is the direction cosine associated with F.

ΣF_{x} = 0 |
=> | R_{1-x} +
F = 0 |

With the unknown forces at the starting joint found, the focus shifts to another joint that meets the feasibility condition. In this example, that would be joint 2. We did not pick joint 6 as it violates the feasibility condition with more than two unknown forces. This process is repeated for as many joints as necessary until all member forces are determined.

Once all the member forces are found, then the directions of internal forces determined to be negative can be corrected in the exploded diagram of the truss. Showing member forces in correct directions helps to identify which members are in tension and which are in compression. This difference is crucial as members in compression should be designed mainly against static instability known as buckling, and those in tension should be designed based on the allowable tensile strength of the material.