All expansion joints are difficult. I have been in this business well over 20 years, worked for a manufacturer (metal/fabric), head of design, software developer, international consultant, training... and there is always new items to learn. Today designed 5 units with drawings for a big tender.

Effective diameter definition as used in specifications is given in EJMA, ASME VIII Div 1 App 26 and EN14917. You can find it there.

Expansion joint design and their use in stress analysis is not for those who think that the result is accurate to 8 decimals. For metal expansion joints spring rates given have easily tolerance of +-30%. What spring rate to use for fabric type is estimation/thump suck. Spring rate in this case has so minor impact that it doesn't matter. Fabric must not be stretched/loaded by pipe movement - maybe a bit. If too much it will tear.

I recently visited a plant where they had expansion joint related problems. One was fabric expansion joint. It keeps failing. It was not difficult to find the reasons. Too tight when cold. Even small duct movement would tear the belt. Bigger problem was that during the shutdown water gets to the insulation material of the expansion joint. On start up they let over 400 C air into the duct. Water changes to steam and local pressure increase destroys the belt.

EJMA has extensive explanations and sample calculations how to calculate anchor loads. Basically the load is a sum of the spring load and pressure thrust. Forces are vectors so directions need to be considered. For anchor design spring loads are small compared to pressure thrust.

There is one special type metal expansion joint that works slightly different. This is thick wall expansion joints, see ASME VIII Div 1 App 5. Those joints are very stiff so in low pressure cases spring force can be meaningful. But those thick elements compress/extend minimal so although spring rate is high the force can still be small. These thick units are used in heat exchangers. There is old manual calculation method for which I have software. Same is included also into PV Elite. However recommended method is given in latest TEMA. It is based on FEA.