Let's start with something simpler - say a fish scale. You catch a fish and put it on the scale and it reads some number N. At that moment the spring is balanced with the weight of the fish. If you pull down on the tail of the fish (or push up), the reading on the scale will no longer be N. The spring is balanced by the weight of the fish plus or minus your change in load.

In a piping system the objective is for the spring to carry the weight load of the system (or that portion of the system associated with the hanger). But what is the weight load considering that at temperature the status of non-linear restraints can change from the installed state? This boundary condition change causes a weight load redistribution in the system and (typically) a change in the load at the hanger locations.

Therefore the load at the hanger locations, determined in the "restrained weight" load case is assumed to be the hot load, so that the spring is balanced in the hot position of the piping system. (In CAESAR II, if during the "free thermal" hanger load cases a non-linear restraint changes status the program changes that restraint and re-performs the restrained weight load case. This because the objective is to balance the spring with the load seen in the hot (operating) load case.) As soon as something else happens to the system, a snow or ice load for example, the spring is no longer balanced with the weight load of the piping (just like pulling on the fish tail).