A. R. C. Markl investigated the phenomenon of fatigue failure of piping during the 1940's and 1950's, and published his results in papers such as "Piping Flexibility Analysis", published in 1955. He tested a number of configurations (straight pipe, and various fittings, such as pipe elbow, miter bend, unreinforced fabricated tee, welding tee, etc.) by using cyclic displacements to apply alternating bending stresses. Plotting the cycles to failure for each applied displacement, he found that the results of his experiments followed the form of fatigue curves.
If an initially applied displacement load causes the pipe to yield, it results in plastic deformation, producing a pre-stress in the system, which must be overcome by subsequent stress applications, resulting in lower absolute stresses during later load cycles. Because of the system "relaxation", the initial values of the thermal stress are allowed to exceed the material yield stress, with the aim being that the system "self-spring" during the first few cycles and then settle into purely elastic cycling. This "self-springing^' is also called Elastic Shakedown. As shown in Figure 1-20, the maximum stress range may be set to 2S (Yield) (or more accurately, the sum of the hot and the cold yield stresses) in order to ensure eventual elastic cycling.
Based upon this consideration, the initial limitation for expansion stress design was set to the sum of the hot and the cold yield stresses — the maximum stress range which ensured that the piping system eventually cycled fully within the elastic stress range. Incorporating a factor of safety, this resulted in the following criterion:
Se ≤ F (Syc + Syh)
Se = expansion stress range, psi
F = factor of safety, dimensionless
Syc = material yield stress at cold (installed) temperature, psi
Syh = material yield stress at hot (operating) temperature, psi
Introduction to Pipe Stress Analysis