5.4 : Alternative Sets of Equilibrium Equations

When analyzing the behavior of structures, engineers often rely on the concept of equilibrium. This refers to the state where all forces and moments acting on a system balance each other, resulting in no net movement or rotation. In many cases, equilibrium can be described by a set of standard equations. However, in some situations, alternative sets of equilibrium equations must be used to describe the system's behavior accurately.

One example of such a situation can be observed in a rectangular plate supported at points A, B, and C and subjected to external forces, as shown in the figure.

In this case, the force and couple moment acting on the plate can be reduced to the equivalent resultant force and resultant couple moment acting at point A. However, the line passing through points A and B is not parallel to the y-axis. This means that standard equilibrium equations will not suffice to describe the plate's behavior. Instead, alternative sets of equilibrium equations must be used.

The first equation states that the resultant force acting on the plate must have no x-component and be parallel to the y-axis. This ensures that there is no horizontal movement of the plate. The second equation states that the resultant moment at point A must be zero. This ensures that there is no rotation of the plate around point A. The third equation states that the moment at point B must be zero. This implies that the y-component of the resultant force on the system is zero, which is necessary for equilibrium. Together, these three equations accurately describe the behavior of the plate and ensure that it remains in a state of equilibrium.

Another set of alternative equilibrium equations that can be used in certain situations states that the sum of the moments at points A, B, and C must be zero.

This set of equations is satisfied when these points are not collinear, meaning they do not all fall on the same line. In cases where the points are collinear, the equations reduce to the standard set of equilibrium equations.