Equilibrium of Systems of Bodies. Method of Disassembly

A system of bodies consists of several bodies connected by pins, contacts, members, or other constraints. To determine not only external reactions but also interaction forces between the parts, the system often has to be separated and the equilibrium of each body analyzed individually.

External and internal forces

For the complete system, forces of interaction between its bodies are internal. They occur in action-reaction pairs and cancel when the system is considered as a whole. Reactions from external supports and applied loads are external forces.

Method of disassembly

1. Consider the complete system. When useful, first determine external support reactions from the equilibrium equations of the entire system.
2. Separate the system. Conceptually isolate individual bodies at internal pins, contacts, or other connections.
3. Replace interactions by forces. Show the interaction forces exerted by the other bodies on each isolated body.
4. Maintain action-reaction pairs. Interaction forces between two bodies have equal magnitudes and opposite directions and, for an idealized point contact or pin, act along the same line.
5. Draw separate free-body diagrams. Include all external loads, reactions, and interaction forces for each part.
6. Write equilibrium equations. Use the independent equations appropriate to the planar or spatial model of each body.
7. Solve the combined system. Determine external and internal reactions systematically and check their consistency.

Forces at an internal pin

In a planar model, an ideal internal pin can transmit two force components but no moment. If the pin forces on the first body are denoted by $H_x$ and $H_y$, the second body is subjected to $-H_x$ and $-H_y$. These internal forces should not also be included as external forces on the free-body diagram of the complete system.

Why analyze the complete system first

The equilibrium equations of the entire structure often determine some external reactions without introducing internal forces. The system can then be separated to obtain additional equations for pin and connection reactions. This order usually reduces the number of unknowns in each equation.

Example: two beams connected by an internal pin

Suppose beams $AC$ and $CB$ are connected by an internal pin at $C$ and have external supports at $A$ and $B$. After analyzing the complete system, separate the beams at $C$. The beam $AC$ is subjected to pin-force components $C_x$ and $C_y$, while beam $CB$ is subjected to equal and opposite components $-C_x$ and $-C_y$. Separate equilibrium equations are then written for each beam.

Two-force members

If a straight member is pin-connected only at two points and carries no other loads, it is a two-force member. The forces at its ends must be collinear, equal in magnitude, and opposite in direction. This allows two unknown Cartesian components to be replaced by one unknown axial force.

Checking the solution

After solving, verify that interaction forces on paired free-body diagrams have equal magnitudes and opposite directions. It is also useful to substitute the calculated external reactions back into the equilibrium equations of the complete system.

Common mistakes

• treating internal forces as external when analyzing the complete system;
• drawing the same direction for an internal pin force on both isolated bodies;
• adding a reaction moment at an ideal internal pin;
• failing to use the two-force-member property when applicable;
• writing equations for a separated part without a complete free-body diagram.