A Seminal Publication...
In the history of the ASME B31 pressure piping codes (and ASME III Div.1), there is a handful of publications that have set the design rules for generations to come. One such publication is A.R.C. Markl’s “Piping-Flexibility Analysis”, Transactions of the ASME, February 1955.
Other publications by Markl and his associates are listed here, but the 1955 publication is where the design rules were fully set and explained:
Markl tested full-size pipe fittings and straight butt welds, under cyclic displacement-controlled bending moments. In each test, the specimen was filled with water but unpressurized (Figures 1 and 2). The cyclic displacement was applied until a crack finally appeared, and cycling continued until the crack leaked, i.e. until the crack had propagated through the wall.
Markl and his colleagues then correlated the applied stress, S, to the number of cycles, N, required to achieve a through-wall fatigue crack.
Figure 1 – Reproduction of Markl’s Test Set-up for the Fatigue Testing of Elbows, In-Plane and Out-of-Plane Compared to Butt-Welded Straight Pipe (Markl, 1951)
Figure 2 – Recent Test Set-Up for Markl-Type Tests of a B16.9 Tee. Ref. Experimental Evaluation of the Markl Fatigue Methods and ASME Piping Stress Intensification Factors Part 2, Proceedings of the ASME 2014 Pressure Vessels & Piping Conference, PVP 2014, July 20-24, 2014, Anaheim, CA, USA, PVP2014-28268, C. Hinnant et. al.
Measuring the distance L from the imposed displacement D to the crack, Markl calculated a nominal stress amplitude Snom.ampl
The moment Mampl is correct for elastic cycling, but becomes a fictitious moment if the cycling is plastic under large displacement, D, as Markl noted in his paper: “Where the stress amplitude applied in the tests exceeded the yield strength in bending, a fictitious moment based on a straight-line extension of the elastic moment-deflection curve was computed to conform with usual calculation practice.” In other words, in the plastic regime, Markl’s fictitious stress is S = E × etrue where E is the modulus of elasticity of the pipe, S is the elastically calculated stress, and etrue is the true total strain.
For pipe fittings (such as elbows, reducers, branches, or tees) the general form for the nominal stress is:
For the ideal case of a cantilever beam:
In his 1955 paper, Markl concludes “While this is not strictly true, test data conform reasonably well to a law expressed by:
where i designates the stress-intensification factor, S the nominal endurance strength (cyclic moment applied at point of failure divided by section modulus of matching pipe, rather than fitting), N the number of stress reversals to failure, and C a materials constant.” Markl should have clarified that the cyclic stress S in equation  is a stress amplitude corresponding to the displacement amplitude (zero-to-+Δ).
Markl reports C = 245,000 for Grade B carbon steel, C = 281,000 for Type 316 stainless steel at room temperature, and C = 183,500 for type 347 stainless steel at 1050oF.
The stress intensification factor i (SIF) is defined in the Markl 1955 paper as follows: “… the stress-intensification factor will be defined here as the ratio of the bending moment producing fatigue failure in a given number of cycles in a straight pipe of nominal dimensions, to that producing failure in the same number of cycles in the part under consideration.”
In the more recent publication “Background of SIFs and Stress Indices for Moment Loadings of Piping Components”, EPRI report 1012078, June 2005, Ev Rodabaugh and Ed Wais describe “The SIFs actually are “fatigue correlation factors” that compare the fatigue life of piping components (tees, branch connections, etc.) to that of girth butt welds in straight pipe subjected to bending moments.”
Markl makes an interesting observation regarding the validity of the iSN-0.2=C for low-cycle loading (such as seismic loads) or high-cycle loading.
“… it appears pertinent to point out that the author has found Equation  [iSN-0.2=C] to be as valid for the determination of stress-intensification factors at 20 as at 2,000,000 cycles.”
In fact, most of Markl’s tests were performed in the range of 1000’s of cycles, not 10’s or 1,000,000’s. More recent tests performed by Chris Hinnant and Tony Paulin extended the range of cycles, and resulted in the following improved fatigue correlation, which is published in “Experimental Evaluation of the Markl Fatigue Methods and ASME Piping Stress Intensification Factors”, Proceedings of the ASME 2014 Pressure Vessels & Piping Conference, PVP 2014, July 20-24, 2014, Anaheim, CA, USA, PVP2014-28268, C. Hinnant, T. Paulin, C. Becht IV, C. Becht V, W.S. Lock, Parts 1 and 2. The Hinnant fatigue correlation is
The differences in slope between Markl’s 1950’s stress-cycle correlation and the more recent Hinnant-Paulin correlation indicates that the Markl curve becomes progressively unconservative in the high cycle regime, perhaps above 2,000,000.
Under low-cycle/high-stress loading, when the elastically calculated stress exceeds 2Sy, ASME VIII Div.2 and ASME III Div.1 NB (Class 1) correct the elastically calculated stress by a plasticity multiplication factor Ke. However, according to Hinnant et. al. 2014, “The Hinnant curve matches the fatigue data for the cantilever in both the elastic region (below 2Sy) and in the reverse plasticity region (where the failure stress >> 2Sy and Kemax=1.8) without Ke adjustment. This shows that the Ke adjustment is implicitly used in the Hinnant curve for carbon steel”. Plasticity effects are inherently considered in the Markl fatigue testing process.
On a final note, keep in mind that under low-cycle/high-stress loading, the component may fail by plastic instability, before developing a fatigue crack.
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George Antaki, Fellow ASME, has over 40 years of experience in nuclear power plants and process facilities, in the areas of design, safety analysis, startup, operation support, inspection, fitness for services and integrity analysis, retrofits and repairs. George has held engineering and management positions at Westinghouse and Washington Group International, where he has performed work at power and process plants, and consulted for the Department of Energy (DOE), the Nuclear Regulatory Commission (NRC) and the Electric Power Research Institute (EPRI).
The Markl "Piping-Flexibility Analysis" paper is the one technical paper regarding piping design that should be read by all piping engineers, because so much of what is said in the paper has found it's way into the form and content of the ASME piping code's design requirements, both B31 and SC III. One additional comment regarding the statement "keep in mind that under low-cycle/high-stress loading, the component may fail by plastic instability, before developing a fatigue crack." The probability of plastic instability increases as the piping D/t increases, particularly above a D/t of 100, for which the ASME piping codes claim the validity of the stress intensification and flexibility factor data has not been demonstrated. On the other hand, plastic instability requires considerable bending deformation which is limited by the redundancy of piping supports which are necessary to limit sustained (B31) or primary (SC III) stresses.
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