What if the calculated expansion stress range is too high? How can we reduce it?
Consider the initial example, with the pipe fully anchored at its ends. What would happen in real life? The restraints would probably bend some under the enormous load, allowing some piping expansion, which would then reduce the internal load (since expansion loads are self limiting). What happens if the bending of a support — i.e., its flexibility — is explicitly considered in the analysis? (Normally a pipe stress program by default considers a restraint to be "infinitely rigid". For example, CAESAR It's default restraint stiffness is in the range of 1E12 pounds per inch.)
If the restraint actually has a lateral stiffness of 10,000 pounds per inch (instead of 1E12), the thermal growth is partially absorbed by the pipe and partially absorbed by the restraint:
This significantly reduces the stress range (from the previous value of 17,700 psi) — not through any actual modification, but simply through a refinement of the model. From this we can gain two insights:
It is sometimes a good idea to provide actual restraint (and nozzle) stiffnesses in the model — the closer to reality the model becomes, the more accurate are the results. Refinement of the model may save the cost of modifying piping systems which initially appear to be over stressed.
If a system really is over stressed, a potential fix may be the introduction of flexibility at the restraints, either by removing restraint or by providing less than infinitely-rigid restraints (or gaps).
Restraint stiffnesses may be calculated through any means and then entered by hand, or simulated in the piping model through the use of structural or piping elements. Vessel nozzle stiffnesses may be calculated manually using Welding Research Council Bulletin 297 or some equivalent. Modeling of restraints using CAESAR II's structural modeler and use of WRC Bulletin 297 are discussed in Modeling And Analysis Of The Piping System
Note that it is best not to selectively enter flexibilities for some restraints and not for others. This will result in the inaccurate distribution of loads, resulting in non-conservative results.