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Section 6.2 Interactions between members

When analyzing structures we are dealing with multi-body systems, and need to recall Newton’s 3rd Law, “For every action, there is an equal and opposite reaction.”

This law applies to multi-body systems wherever one body connects to another. At any interaction point, forces are transferred from one body to the interacting body as equal and opposite action-reaction pairs. These forces cancel out and are invisible when the structure is intact. Only when we cut through a member or joint in the isolation step of creating a free-body diagram, do we expose the interaction forces. When drawing free-body diagrams of the components of structures, it is critically important to represent these action-reaction pairs consistently. You may assume either direction for one, but the other must be equal and opposite.

For example, look at the members and joints in the truss below. Diagram (a) shows the truss members held together by pins at \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\) The forces holding the parts together cancel and are not shown. In the ‘exploded’ view (b), the parts have been separated and the action-reaction pairs are exposed. Member \(AB\) is in tension, and the forces acting on it, also called \(AB\text{,}\) oppose each other and tend to stretch the member. These stretching forces are accompanied by equal and opposite forces, also called \(AB\) acting on pins \(A\) and \(B\text{.}\) Tensile forces \(BC\) and compressive forces \(CA\) behave similarly.

(a) Whole Truss
(b) Exploded
Figure 6.2.1. External load and global reactions in red. Internal action-reaction pairs in blue.

When a multipart structure is in equilibrium, each part of the structure is also in equilibrium. For example in the truss below, each member of the truss, each joint, and each portion of the truss is also in equilibrium. This continues all the way down to the atoms of the structure. This universal equilibrium across spatial scales is one of the governing principles which allows us to break multi-body systems into smaller solvable parts.

(a) Complete Truss
(b) Member \(AD\)
(c) Truss joints
(d) Section
Figure 6.2.3. Possible free-body diagrams

You will see in this chapter that we have the freedom to isolate free-body diagrams at any scale to expose our target unknowns.

Load Paths.

Load paths can help you think about structural systems. Load paths show how applied forces like the floor load in the image below pass through the interconnected members of the structure until they end up at the fixed support reactions. All structural systems, whether non-moving frames or moving machines have some sort of load path. When analyzing all structures, you computationally move from known values through the interconnected bodies of the system, following the load path, solving for unknowns as you go.

Figure 6.2.4. Load paths