Resources for Structural Engineers and Engineering Students

## What is compatibility?

Compatibility, like equilibrium, is one of the primary tools that we can use to solve structural analysis problems. As discussed in the previous section, if there are only three unknown forces in a 2D structure or system, then we can typically solve for those three unknowns using the equilibrium expressions in Equation \eqref{eq:Equil2D}.

$$\label{eq:Equil2D}\tag{1} \boxed{ \sum_{i=1}^{n}{F_{xi}} = 0; \sum_{i=1}^{p}{F_{yi}} = 0; \sum_{i=1}^{q}{M_{zi}} = 0 }$$

If there are more unknowns, then we may have to rely additionally on compatibility to provide extra information about the system. The extra information provided by compatibility may or may not be enough to fully solve for all of the forces in the structure.

Compatibility is related to the shape of a structure. This includes deformations, location of reaction points and the way that a structure is allowed to bend and deform. Depending on the type of structure, there are some things that we know about its compatibility. For example, we may know that a structure may not move at certain locations in certain directions (called reactions) or we may know that a beam or column is continuous, and so its slope cannot change abruptly. Alternatively, if we were to apply a specific amount of displacement to a structure at a certain location (as opposed to applying a force), we would be affecting the compatibility at that location.

The concept of compatibility will become vitally important in the second half of this book. For now it will be enough to know that it relates to the shape.