What happens when we don’t have enough equations to solve for all unknown reaction supports? Consider the following bar with 2 fixed ends:

We have 2 unknowns (R_{A} and R_{C}) but only one equation (ΣF_{x} = 0). How do we solve this then? Well we can actually get the 2nd equation by considering the deflection at the ends of the bar. The forces of different magnitudes cause varying displacements along the bar, but we know that the deflections at both ends are zero, since they are fixed. Therefore:

What happens when we don’t have enough equations to solve for all unknown reaction supports? Consider the following bar with 2 fixed ends:

We have 2 unknowns (R_{A} and R_{C}) but only one equation (ΣF_{x} = 0). How do we solve this then? Well we can actually get the 2nd equation by considering the deflection at the ends of the bar. The forces of different magnitudes cause varying displacements along the bar, but we know that the deflections at both ends are zero, since they are fixed. Therefore:

Using the elastic deformation formula, we can then express the displacements in terms of the forces:

P_{CB} and P_{AB} can be expressed as the reaction forces, and this gives us our 2nd equation to solve for the 2 unknown reactions. The displacement equation is called the *compatibility equation*.

Let’s look at an example now.

Using the elastic deformation formula, we can then express the displacements in terms of the forces:

P_{CB} and P_{AB} can be expressed as the reaction forces, and this gives us our 2nd equation to solve for the 2 unknown reactions. The displacement equation is called the *compatibility equation*.

Let’s look at an example now.