A piping system, when heating up, normally tries to expand against its restraints, resulting in internal forces, moments and stresses:
The axial force generated in the above configuration can be estimated to be the axial force required to compress the pipe back to its original length after it has been allowed to grow freely. Its free growth is:
The axial force required to compress that growth is:
Considering a rather benign operation — a 12-inch diameter, standard wall pipe (A = 14.58 square inches, E = 29E6 psi) operating at 350°F ( a = 1.88E-3 in/in) — the axial load is calculated as:
P = 14.58 x 29E6 x 1.88E-3 = 800,000 pounds
From the point of view of most piping codes, there is no stress, since no moment is produced in the axial run (although the codes do state that the possibility of buckling must be considered); however, this is not a good design.
An alternate is no restraint at one end, allowing the pipe to grow unimpeded; therefore no load develops. However this is not good design either, since the pipe must normally attach to some relatively fixed piece of equipment, and cannot usually be floating in space.
What is the solution to this problem? It is necessary to have some restraint on the system, but too much may cause excessive forces, moments, and stresses. Looking at the examples above, allowing no movement produces a force of about 800,000 pounds. Allowing 100% of the pipe's desired free movement causes no force. Interpolating, if we could devise a means by which the piping system remained intact, yet allowed 99.8% of the pipe's desired free movement, the developed force would be approximately:
(1.0 - 0.998) x 800,000 = 1600 pounds
This is a much more manageable situation.