International Journal of Fatigue 139 (2020) 105732 Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue A simpliﬁed structural strain method for low-cycle fatigue evaluation of girth-welded pipe components T ⁎ Xianjun Peia, Pingsha Donga, , Myung Hyun Kimb a b Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor 48109, USA Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, South Korea A R T I C LE I N FO A B S T R A C T Keywords: Structural stress Structural strain Low-cycle fatigue Girth welds Fatigue testing Markl’s pseudo-elastic stress method has been the basis for low cycle fatigue (LCF) evaluation of girth-welded piping connections for decades, which underpins several international codes and standards. Here, we reexamine the applicability of Markl’s method in correlating some recent LCF fatigue data on girth-welded pipes. To mitigate its deﬁciencies, a simpliﬁed structural strain method is proposed for achieving an eﬀective correlation of the same test data. The proposed method can be conveniently used for LCF evaluation of girth-welded pipe components either as an experimental measurement technique or a post-processing procedure of elastic-plastic ﬁnite element results. 1. Introduction Piping components are extensively used in power generation and petrochemical plants. A piping system can contain a large number of circumferential welded (or girth-welded) connections which must be evaluated at the design stage for assessing their fatigue capacity under cyclic start up and shut down, and potential seismic loading conditions [1]. Fatigue failures attributed to relatively low stress amplitudes with little macro-scale plastic deformation are typically classiﬁed as highcycle fatigue (HCF). Low cycle fatigue (LCF), however, is often associated with local plastic deformation that cannot be ignored. These situations include process equipment and piping systems subjected to some unexpected start up and shut down conditions or seismic events. Girth welds widely used in tubular structures in civil, oﬀshore, and wind-energy harvesting installations must also be evaluated for demonstrating their low cycle fatigue capacity under rare severe weather and seismic loading conditions [2]. Generally speaking, LCF might stem from the events that cause signiﬁcant plastic deformation including but not limited to the common operational risks such as misuse, sudden changes of temperature or pressure, etc. It is well established that fatigue behaviors of welded joints in HCF regime is governed by the stress concentration at welds, which can be treated either empirically through weld classiﬁcation approach [3], or more quantitatively, through hot-spot extrapolation approach [4,5], eﬀective notch stress/critical distance method [6–10] and traction structural stress based master S-N curve approach [11–13]. In the latter, ⁎ a large amount of well-known test data including pressure vessel and welded piping components had been considered in supporting its development, but mostly focusing in HCF regime. This is mostly due to the lack of an eﬀective method at that time for treating the plastic deformation eﬀects associated with LCF loading events. Instead, a provisional LCF treatment procedure was introduced in the interim by adopting the notch stress based Neuber’s rule [14] in the context of the structural stress [15], which has been shown to provide fatigue life estimates often with excessive conservatism in LCF regime. Some recent eﬀorts [16–18] have showed that the traction structural stress method can be conveniently reframed as a structural strain method for incorporating plastic deformation eﬀects present in LCF regime. In a subtle way, the structural strain method can be related to the classical low cycle fatigue method developed by Markl [19] for girth-welded piping components. Markl [19] has been credited for the development of a pseudoelastic stress method, dating back to 1950 s. Markl and his colleagues [19–22] demonstrated that a ﬁctitious elastic bending stress could be used to correlate a large amount of fatigue data of full-scale girthwelded pipe components. The ﬁctitious stress, also referred to as pseudo-elastic stress, can be calculated using a pseudo-elastic load obtained by extrapolating the linear part of a measured load–displacement curve at a speciﬁed cyclic displacement amplitude. This method has been adopted by ASME Codes [1] since 1960s and has continued to serve as the basis for assessing welded pipe components to this date. However, some recent experimental investigations by following the Corresponding author at: 2600 Draper Drive, Ann Arbor, MI 48109, USA. E-mail addresses: xpei@umich.edu (X. Pei), dongp@umich.edu (P. Dong), kimm@pusan.ac.kr (M.H. Kim). https://doi.org/10.1016/j.ijfatigue.2020.105732 Received 6 April 2020; Received in revised form 21 May 2020; Accepted 22 May 2020 Available online 27 May 2020 0142-1123/ © 2020 Elsevier Ltd. All rights reserved. International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Nomenclature LCF HCF FE PVRC PRG Pa δa Pnl Z L l t R ΔSs Δσs Δσm Δσb r m Ri Ro ε εe εp mRO α σ0 σprop Low cycle fatigue High cycle fatigue Finite element Pressure Vessel Research Council Paulin Research Group Pseudo elastic load amplitude (see Fig. 1c) Applied displacement amplitude Applied load amplitude (see Fig. 1c) Section modulus of the pipe cross section Geometry dimension deﬁned in Fig. 1a Geometry dimension deﬁned in Fig. 1b Thickness of pipe wall Fatigue test load ratio Equivalent structural stress range Traction structural stress range Membrane traction structural stress range Bending traction structural stress range Bending ratio Master S-N curve parameter,m = 3.6 Inner radius of the pipe Outer radius of the pipe Total strain Elastic strain Plastic strain Material Ramberg Osgood parameter Material Ramberg Osgood parameter rprop σY E δY εs k b l (y ) c F M ε¯p I γ εs ' ε3 Ro t ε' s, i ε' s, o rg εo θa εEP εE Material Ramberg Osgood reference stress Material proportional limit, used in Ramberg Osgood equation σprop/ σ0 , used in Ramberg Osgood equation Material yield strength Material Young’s Modulus Characteristic defection at which the pipe outer ﬁber stress reaches to material yield strengthσY Structural strain Slope to determine structural strain Intercept to determine structural strain Chord length of pipe cross-section, as deﬁned in Fig. 8. Elastic core size Internal resultant force Internal resultant moment Equivalent plastic strain Moment of inertia of pipe Plastic multiplier Structural strain calculated by extrapolation Strain at 3 Ro t Intrados structural strain calculated by extrapolation Extrados structural strain calculated by extrapolation Global bending ratio with respect to pipe section Strain at outer ﬁber Applied remote angular rotation Outer ﬁber strain calculated by elastic–plastic model Outer ﬁber strain calculated by elastic model same Markl’s method [23–25] have showed that a signiﬁcant discrepancy exists in correlating the fatigue data obtained from low cycle four-point bending and cantilever bending tests. In all these more recent tests, Markl’s method tends to underestimate the pseudo-elastic stresses at fatigue failure locations, potentially leading to unsafe fatigue design, particularly under cantilever bending conditions [26]. In this paper, we start with a brief discussion in Section 2 on Markl’s pseudo-elastic stress method in the context of its applications in analyzing two recent sets of LCF data collected from girth-welded pipe components under four-point bending and cantilever bending conditions. Elastic-plastic ﬁnite element (FE) analysis results are presented to elucidate the diﬀerences between Markl’s assumptions and elastic–plastic deformation behaviors in the pipe specimens. Once the regime of applicability of Markl’s method is identiﬁed, a simpliﬁed structural strain method is introduced in Section 3, which is shown applicable both for postprocessing FE resultsand experimental measurements. The simpliﬁed structural strain method in its pseudo-elastic structural stress form is consistent with the basis of Markl’s method, but possessing the advantage of being more broadly applicable for dealing with piping components subjected to complex loading conditions. 2. Markl’s pseudo-elastic stress method 2.1. Pseudo-elastic stress deﬁnition When performing displacement-controlled LCF fatigue tests on girth welded pipe components, Markl’s pseudo-elastic stress method can be simply described as follows under either four-point bending (Fig. 1a) or cantilever bending (Fig. 1b) condition: (a) Establish a stable load–displacement (P − δ ) curve covering displacement amplitudes of interest, e.g., δa as shown in Fig. 1c (b) Perform fatigue test under constant amplitude δa until a throughwall crack is detected, typically through the detection of pipe through-wall leakage with pre-ﬁlled water Fig. 1. Illustration of Markl’s pseudo-elastic stress method for analyzing low cycle fatigue test data of girth-welded pipes: (a) four-point bending, (b) cantilever bending, and (c) determination of pseudo-elastic load by extending the linear part of measured load–displacement curve. 2 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. testing conditions according to Markl’s procedure (see Fig. 1) were used with displacement-based loading ratio of R = −1. The corresponding applied force was also recorded for each test. Further details can be found in [23]. (c) Present test data in a S-N plot in log–log scale in terms of: P ×l σa = a Z (1) in which σa represents a pseudo-elastic outer ﬁber stress corresponding to an extrapolated pseudo-elastic load Pa from the linear part of P − δ curve (Fig. 1c) and Z is the section modulus of the pipe cross section. 2.2.3. Data correlation with ASME’s master S-N curve To examine if the two sets of fatigue test data are consistent with the ASME’s master S-N curve scatter band [12,27], the pseudo-elastic load Pa corresponding to displacement amplitude δa for each test can be directly used to compute the pseudo-elastic structural stress at weld toe and compared with the master S-N curve scatter band in terms of the equivalent structural stress range versus cycles to failure, as shown in Fig. 4. In Fig. 4, the equivalent structural stress range ΔSs is deﬁned as. 2.2. Test data correlation For understanding the applicability of or limitations in Markl’s method described above, here we consider two sets of well-known fatigue tests within ASME Code community on girth-welded pipes carried out by Scavuzzo et al. [25] under the auspice of Pressure Vessel Research Council (PVRC) and by Hinnant [23] at Paulin Research Group (PRG). The former tests were conducted under four-point bending while the latter were under cantilever bending conditions. ΔSs = Δσs t ∗ (2 − m)/(2m) I (r )1/ m t ∗ = t / tref r= 2.2.1. Four-point bending fatigue tests The pipes were made of carbon steel of Type A53F [25], with pipe inner and outer radii of 19.05 mm and 24.13 mm, respectively. All the pipes were welded using a direct current gas metal arc welding with 70–90 amps and 14–18 V, with argon gas shield. The four-point bending test ﬁxture used is illustrated in Fig. 2, with which Instron Material Testing System (Model 1321) was used for conducting the cyclic displacement-controlled fatigue testing. Four yokes were used to provide four simply-supported conditions of the pipe specimens. The two center yokes were ﬁxed to a strong beam at the top, which in turn, was connected to the load cell of the Instron actuator. With respect to the deﬁnitions used in Fig. 1, it can be seen that l = 381 mm and L = 1219.2 mm . A fully reversed sinusoidal cyclic displacement at a speciﬁed δa in amplitude was applied with a displacement-based loading ratio of R = −1. The corresponding peak load Pnl (see Fig. 1c) was also recorded through Instron’s load cell during each test. The fatigue lives in tests were deﬁned as the development of through-wall cracking as indicated by detection of leakage of the preﬁlled water (with negligible pressure) inside the pipe test specimens. All fatigue lives were less than 1000 cycles, suggesting that these fatigue tests were indeed well within LCF regime, as intended. Further information about the testing procedures and specimen details can be found [25]. |σ b | |σm| + |σ b | I (r )1/ m = 0.0011r 6 + 0.0767r 5 − 0.0988r 4 + 0.0946r 3 + 0.0221r 2 + 0.014r + 1.2223 (2) Here, Δσs = Δσm + Δσb stands for the structural stress range, based on a statically equivalent membrane (Δσm) and bending (Δσb ) decomposition along the hypothetical crack plane at weld toe and calculated corresponding to the pseudo-elastic load Pa in Fig. 1c. In addition, m = 3.6, derived through a two stage fatigue crack growth model [28], and tref is a reference thickness, setting as 1mm in [12,27]. The through wall thickness bending ratio r given in Eq. (2) can be more conveniently expressed for pipe components by assuming that beam theory applies, i.e., r= 1⎛ R 1 − i⎞ 2⎝ Ro ⎠ ⎜ ⎟ (3) in which Ri and Ro are inner and outer radius of the pipe, respectively. The eﬀectiveness of ΔSs in Eq. (2) has been demonstrated by its ability in collapsing about 1000 full-scale and large-scale fatigue test data into a narrow band [27,29]. Most recent investigations [18,30] (see “recent data” in Fig. 4) have further conﬁrmed the applicability of the master SN curve approach in correlating test data spanning both LCF and HCF regimes. As can be seen, the four-point bending test data from Scavuzzo et al. [25] (labeled as “PVRC”) are situated near the lower bound deﬁned by the “mean -2σ ” where σ is the standard deviation of the master S-N curve adopted by ASME Div 2 [12,27]. While the cantilever bending data from Hinnant [23] (labeled as “PRG”) are approximately aligned with the “mean − 3σ ” line of the master S-N curve scatter band. The discrepancies seen in Fig. 4 demonstrate the needs for a more quantitative assessment of the application limit of Markl’s pseudo-elastic stress method, which seems to provide un-conservative fatigue life 2.2.2. Cantilever bending fatigue tests A series of Cantilever bending fatigue tests were carried out by Hinnant and Paulin at PRG [23] on girth-welded ASME SCH 40 pipe made from S/A-106 Grade B steel, which have the inner and outer radii of 51 mm and 57.15 mm, respectively. The girth butt welds were completed using the gas metal arc welding (GMAW), ﬂux-cored arc welding (FCAW) and shielded metal arc welding (SMAW) process [23]. To ensure that fatigue failures in these tests consistently occur at the target test weld location, a strong tapped pipe ﬁtting was used at the ﬁxed end, as shown in Fig. 3. Again, displacement-controlled fatigue Fig. 2. Fixture setup for four-point bending fatigue tests carried out by Scavuzzo et al. funded by PVRC [25]. 3 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Fig. 3. Experimental setup for performing cantilever bending fatigue tests at PRG [23]. applied strains and Young’s modulus wherever they exceed the yield point, they are obviously ﬁctitious”. This statement implies two basic assumptions inherent in Markl’s pseudo-elastic stress approach, i.e., (i) the stress calculated from Eq. through the extrapolation procedure was assumed to be equal to E × εo in which εo is the outer ﬁber strain of the pipe at a fatigue prone location of interest; (ii) the pseudo-elastic stress calculated by the product of the outer ﬁber strainεo and Young’s modulus, i.e., E × εo , aka pseudo-elastic stress, can be used as a fatigue parameter for correlating the LCF test data for welded pipe components. To quantitatively evaluate the above assumptions, here we further examine the four-point bending and cantilever bending conditions used in [25] and [23], respectively, by performing detailed elastic–plastic ﬁnite element analyses to relate pipe outer ﬁber strain to applied displacement amplitude corresponding to the fatigue test conditions discussed in Section 2.2. The cyclic stress–strain curves for the carbon steel pipes tested by Scavuzzo et al. [25] and Hinnant et al. [23] are shown in Fig. 5, respectively. These cyclic stress–strain curves were obtained iteratively based on the test data given in [23,25] and modeled by a modiﬁed Ramberg-Osgood equation (see Eq.(4)) based on arecent publications by the same authors [17]. The corresponding Ramberg Osgood parameters are given in Table 1. Fig. 4. Master S-N (ΔSs − N ) representation of cantilever bending (PRG) and four-point bending (PVRC) fatigue test data and comparison with ASME’s scatter band. (The dispersion and standard derivation values shown are with respect to the 4-point bending and cantilever bending pipe fatigue data). estimates in LCF regime. If such a limit can be understood, an alternative technique can be developed for supporting LCF fatigue evaluation needs for pressure vessels and piping components. εe = ⎧ ε= ⎨ εe + εp = ⎩ σ E σ + α E0 ⎡ ⎣ σ E (σ ⩽ σprop) σ mRO σ0 ( ) − r0mRO⎤ (σ > σprop) ⎦ (4) 2.3. Elastic-plastic pipe behaviors 2.3.1. Four-point bending Two types of FE analysis are considered here for investigating the diﬀerence in linear elastic and nonlinear elastic–plastic deformation behaviors under exactly the same displacement-controlled four-point Markl [19] stated that “for loads causing plastic ﬂow, the load P is taken from a straight-line extrapolation of the elastic portion of the curve. In eﬀect, this deﬁnes the computed stress strictly as products of the Fig. 5. Stress–strain curves for pipe components tested in PRG and PVRC: (a) cantilever bending tests [23] (b) four-point bending tests [25]. 4 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Table 1 Material properties used in fatigue test of pipe structures. Materials Test group σprop (MPa) r0 α σ0 (MPa) mRO Carbon Steel S/A106 (Grade B) Carbon Steel A53 type F PRG cantilever bending PVRC four-point bending 310 190 0.86 0.68 0.542 23.45 360 280 55 7 increases, e.g., from δa = 1.57δY to δa = 2.30δY . By comparing the results shown in Fig. 7b with those in Fig. 6b, the degree of under-estimation in the outer ﬁber strain at the weld location (x = 0) becomes much more signiﬁcant in cantilever bending than in four-point bending. The results seem to explain why the cantilever bending fatigue test results are situated further below the four-point tending tests in Fig. 4. bending conditions used by Scavuzzo et al. [25]. The elastic FE analysis was performed under linear elastic conditions with pipe steel Young’s modulus and Poisson ratio while the elastic–plastic analysis was carried out using the true stress–strain curve given in Fig. 5b corresponding to carbon steel A53F used in [25]. The results are shown in Fig. 6. In Fig. 6a, the abscissa represents the normalized distance x by the well-recognized pipe characteristic length parameter Ro t . The ordinate represents the pipe deﬂection normalized by a characteristic defection δY at which the pipe outer ﬁber stress reaches to material yield strength σY . For four-point bending case, δY is given by: δY = σY l (3L − 4l) 6ERo 3. A simpliﬁed structural strain method 3.1. Structural strain deﬁnition (5) Consistent with Markl’s pseudo-elastic stress deﬁnition E × εo discussed in Section 2.3, a generalized structural strain deﬁnition was ﬁrst introduced by Dong et al. [31] and Pei et al. [18], for low cycle fatigue evaluation of welded plate components as a plastic deformation correction procedure built upon the mesh-insensitive traction structural stress method [11]. Pei and Dong [17] recently developed a numerical procedure for computing the structural strain across pipe section under more general strain-hardening conditions, e.g., by incorporating a in which l and L are deﬁned in Fig. 1a and E is material Young’s modulus and Ro is the outer radius of the pipe. For the four-point bending tests considered in Fig. 6, δY = 5.47 mm . As such, δa/ δY provides an indication of the extent of displacement-based loading beyond linear elastic limit. In Fig. 6b, the ordinate represents a normalized pipe outer ﬁber strain by material yield strain (εY = σY / E ), indicating the extent of strain development beyond yielding. Two displacement amplitudes at δa = 6.5δY and δa = 11.6δY (corresponding to R = −1) are considered here to bound the test conditions used in [25]. Although displacement distributions are in a reasonable agreement (Fig. 6a) between the linear elastic and elastic–plastic analysis results, the strain (outer ﬁber) distributions are signiﬁcant different (Fig. 6b). Of a particular interest is the constant strain regions shown in Fig. 6b, within which linear elastic analysis clearly underestimates the strain level developed under elastic–plastic deformation conditions. Such an under-estimation becomes more signiﬁcant as the applied displacement amplitude increases from 6.5δY to 11.6δY . The results seem to provide an explanation why this set of test results is situated at the lower bound of the master S-N curve in Fig. 4 in which linear elastic analysis with the pseudo-elastic load Pa (as described in Fig. 1c) was used to compute the pseudo-elastic structural stress range given in Eq. (2) under the applied displacement conditions. 2.3.2. Cantilever bending Both linear elastic and nonlinear elastic–plastic FE analysis results are compared in Fig. 7 under the cantilever bending fatigue test conditions, as described in [23]. The elastic–plastic analysis incorporated the true stress–strain curve given in Fig. 5a at two applied displacement amplitude of δa = 1.57δY and δa = 2.30δY , which bound the test conditions in [23]. Here, δY has the same meaning as that in Eq. , but taking an expression for cantilever bending condition, as: δY = σY l 2 3ERo (6) in which l is deﬁned in Fig. 1b. Note that δY = 8.27 mm for the pipe components tested by Hinnant et al. [23] at PRG. Similar to the results shown in Fig. 6, the pipe deﬂection curves shown in Fig. 7a based on linear elastic analysis are not signiﬁcantly diﬀerent from those based on nonlinear elastic–plastic analysis, especially when the applied displacement is relatively small, e.g., δa = 1.57δY . However, the strain distributions in Fig. 7b become signiﬁcantly diﬀerent between linear elastic and nonlinear elastic–plastic analysis results, particularly when the girth weld position (x = 0) is approached. As a result, elastic analysis tends to signiﬁcantly underestimate the pipe outer ﬁber strain, more so as the applied displacement Fig. 6. Comparison between linear elastic and nonlinear elastic–plastic analysis results under four-point bending conditions at two applied displacement amplitudes: (a) pipe deﬂection curves; (b) pipe outer ﬁber strain distributions. 5 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. (1) Perform linear elastic analysis (e.g. linear elastic FE analysis) under speciﬁed remote load Pnl (for the present cases, beam bending theory can be directly used). It is worth emphasizing that in the structural strain analysis, we use the actual load Pnl rather than the extrapolated load Pa as in Markl’s method. (2) Extract nodal loads and nodal moments at a location of interest, e.g., at weld toe of a girth welds from all the elements situated on one side of the weld (for the present cases, pipe section moments can be directly obtained by beam bending theory) (3) Calculate traction structural stresses in terms of σm and σb from line force and moments converted from nodal forces and nodal moments (for the present cases, pipe bending stress σb can be directly calculated according to beam bending theory and σm = 0 ) (4) Express the pipe cross section strain in terms of a linear expression (see also Fig. 8b), which is decomposed into elastic and plastic parts: εs = εe + εp = ky + b (7) Here εs , εe , and εp represent the structural strain, elastic strain, and plastic strain, respectively. (5) Impose equilibrium conditions in terms of σm and σb , σ = Eεe = E (εs − εp) = E (ky + b − εp) (8) R ∫−Roo σ (x , y ) l (y ) dy = F = σm π (Ro2 − Ri2 ) R ∫−Roo σ (x , y ) l (y ) ydy = M = σb I Ro = σb π · Ro 4 (Ro2 − Ri2) (9) (6) Calculate the plastic strain εp using return-mapping algorithm given in [17,32,33], by imposing yield criterion: 1/ m Eε¯p f (σ , ε¯p) = |σ| − σ0 ⎛ + r m⎞ ⩽0 ⎝ ασ0 ⎠ ⎜ ⎟ (10) and following plasticity ﬂow rule: Fig. 7. Comparison between linear elastic and nonlinear elastic–plastic analysis results under cantilever bending conditions at two applied displacement amplitudes: (a) pipe deﬂection curves; (b) pipe outer ﬁber strain distributions. dεp = dγ × sign (σ ) dε¯p = dγ modiﬁed Ramberg-Osgood strain hardening model. Fig. 8 schematically demonstrates the structural strain deﬁnition for pipe components. If the stress in a pipe section caused by remote loading in either displacementor load-controlled conditions exceeds material yield limit (Fig. 8b), the resulting stress distribution across a pipe section becomes nonlinear. However, according to the classical beam theory and Navier's hypothesis, a pipe cross-section plane should still remain as a plane even under elastic–plastic deformation conditions. This linear strain distribution through a pipe cross-section is referred as the structural strain with its outer ﬁber value deﬁned as εo , having a rather similar meaning to that used by Markl [19]. Two major diﬀerences are worth noting between the structural strain method and Markl’s εo usage: (1) the structural strain method also takes into account of the eﬀects of strain gradient k in terms of bending ratio given in Eq. (3) in addition to its peak value or outer ﬁber value ε0 ; (2) the structural strain method can be used for fatigue evaluation of plate structures as well, in which it is deﬁned as linearly distributed strain across a plate thickness at a location of interest [17,18]. (11) in which dγ is the plastic multiplier. (7) Solve (k , b) iteratively such that both equilibrium condition Eq. (9) and material yield criterion Eq. (10) are satisﬁed simultaneously [17]. By looping through Steps (5) to (7), the parameters k , b in Eq. (7) can be solved in an iterative manner. Then, the structural strain corresponding to elastically calculated σm and σb is given by Eq.(7). Note that the solution procedure described here does not take account of the biaxial stress state eﬀects on yielding, e.g., those caused by internal pressurization of a pipe. For a more general structure strain solution procedure, one can refer to Ref. [17] for more details. Once the structural strain given in Eq. (7) is obtained, the corresponding pseudo-elastic structural stress can be directly calculated using σs = E × εs and the bending ratio described in Eq. (3) can be expressed, as: σs = E (kRo + b) kRo 2(kRo + b) (1 − ) Ri Ro 3.2. Direct computation method r= Consider the displacement controlled LCF fatigue procedure illustrated in Fig. 1c. The actual cyclic peak load value Pnl which is typically available through load cell measurement during fatigue tests can be directly used in performing linear elastic analysis. The direct computation procedure for the structural strain deﬁned in Fig. 8b (further details can be found in [17] and will be not be given here in detail, due to space limitation) can be outlined as follows: Note the similarity between σs and E × εo used by Markl (see Section 2.3). The results corresponding to the fatigue tests [23,25] analyzed here are summarized in Table 2. The master S-N curve based representation of the fatigue data using the structural strain based pseudo-elastic structural stresses is given in Fig. 9 which now shows a signiﬁcant improvement in the data correlation over Markl’s pseudoelastic stress method. 6 (12) International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Fig. 8. Structural strain deﬁnition for welded pipe component: (a) Pipe under both remote tension and bending; (b) longitudinal pipe cross-section section and stress/ structural strain distributions; (c) transverse pipe cross-section. 3.3.1. Basis and assumptions Consider the nonlinear elastic–plastic analysis results shown in Fig. 7b corresponding cantilever bending conditions. More detailed outer ﬁber strain distributions corresponding to a number of diﬀerent applied displacement amplitudes within the range used in the PRG test conditions are replotted in Fig. 10, in terms of the normalized distance from the girth weld location (x = 0 ). The distance from girth weld position (x = 0 ) is normalized by Ro t . Fig. 10a shows that plastic deformation eﬀects to a large extent are conﬁned with a region of about 10 Ro t from the weld location. Upon further examination, Fig. 10b shows that there exists a regime of approximately linear strain distribution between about 3 Ro t and 5 Ro t within which a linear extrapolation to weld toe can be used for determining its structural strain. 3.3. A linear extrapolation method The direct structural strain computation method described in the previous section typically involves an iterative solution procedure when dealing with nonlinear hardening material behaviors as given in Fig. 5. For applications in pipe components, the structural strain method can be signiﬁcantly simpliﬁed thanks to their simple geometry and loading conditions. The resulting method can be used either as a structural strain measurement technique when performing fatigue testing or as a strain post-processing technique when nonlinear elastic–plastic FE analysis results are available. The method is described below. Table 2 LCF pipe fatigue data analyzed in this study. Test Life Ro (mm) t (mm) δa (mm) Pa (kN) Pnl (kN) Markl's Extrapolation (MPa) E × εs (MPa) Cantilever bending (PRG) 242 240 700 544 608 3088 249 304 615 836 461 786 386 202 708 791 57.15 6.17 24.13 5.08 24.77 25.15 18.16 18.54 18.67 11.56 43.18 38.1 30.48 58.42 26.67 33.02 27.94 35.56 63.5 48.26 51.44 52.23 37.72 38.51 38.77 24.00 23.64 20.86 16.69 31.98 14.60 18.08 15.30 19.47 34.76 26.42 27.93 27.95 27.27 27.36 27.38 22.62 10.62 10.51 10.30 10.89 10.17 10.38 10.21 10.44 10.96 10.72 2032.20 2063.47 1490.28 1521.55 1531.97 948.36 2875.78 2537.45 2029.96 1776.22 2199.13 1860.80 2368.29 3214.11 1691.64 1522.47 3571.48 3570.38 2385.47 2470.78 2470.78 945.18 3807.79 3337.12 2614.01 2227.22 2864.21 2355.34 3104.50 4293.98 2103.96 1859.14 4-point bending (PVRC) 7 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. strain state at the weld location of interest. The latter situation can be readily justiﬁed for dealing with complex pipe conﬁgurations (e.g., pipe bend, branch connections, etc.) and loading conditions. 3.3.2. Procedure With the above observations and considerations, the structural strain at weld toe can then be determined through a simple linear extrapolation scheme, between 3 Ro t and 5 Ro t for simplicity (or using any two positions that are situated within a linear strain distribution regime), as: 5 ε3 2 εs′ = Ro t − 3 ε5 2 Ro t (13) Here ε3 Ro t and ε5 Ro t represent the outer ﬁber strains at 3 Ro t and 5 Ro t , respectively. By observations from Fig. 6b corresponding to the four point bending conditions, the above equation works equally well without exception, since the strain is actually constant between the two loaded positions. Under bending-dominated loading conditions, i.e., F ≈ 0 in Fig. 8, the through pipe wall thickness (t ) bending ratio r can be calculated directly by using Eq. (3). Both the four-point bending and cantilever bending conditions investigated in this paper are in this category. Otherwise, Eq. (13) needs to be applied with respect to both pipe intrados and extrados at a pipe section of interest. Then, the through pipe wall thickness cross-section bending ratio can be expressed, in terms of through-pipe section (2Ro) strain gradient rg , as: Fig. 9. Correlation of PRG and PVRC LCF fatigue test data with ASME master SN curve scatter band for comparing Markl’s and the structural strain based methods (The dispersion and standard derivation values shown are with respect to the 4-point bending and cantilever bending pipe fatigue data). r= rg 2 rg = (1 − ) Ri Ro ε′s, o − ε′s, i ε′s, o + ε′s, i (14) where εs′, o and εs′, i represents structural strains measured at pipe extrados and intrados at a pipe cross section of interest though the linear extrapolation technique described in Eq. (13). It should be pointed out that although Eq. (13) shares a great deal of similarity to the hot spot stress extrapolation methods widely used for tubular joints [34–37], it diﬀers in that the extrapolation positions (i.e., 3 Ro t and 5 Ro t ) are developed based on the existence of the linear strain distribution regime under elastic–plastic deformation conditions. It should be noted in the context of pipe mechanics that a distance of about 3 Ro t from the girth weld essentially fully contains local plastic deformation eﬀects [38,39] which should be avoided when using Eq. (13) by deﬁnition. This also implies that for shorter pipe components, e.g., a pipe length less than 5 Ro t measured from a weld location of interest, the simpliﬁed structural strain method by means of Eq. (13) is no longer applicable. Then, the direct ﬁnite element computation method described in Section 3.2 may become the only option. 3.3.3. Data correlation With the linearly extrapolated structural strains at girth weld toe according to Eq. (13), the same four-point bending and cantilever bending test data are plotted in Fig. 11 after converting the structural strains to the pseudo-elastic structural stresses through Δσs = E × Δεs' for plugging into Eq. (2). The results are rather similar to those calculated using the direct computation method described in Section 3.2. The main advantage of the method described in Eq. (13) is that it can be readily implemented as an experimental measurement technique by positioning two strain gauges at the ﬁxed distances from a weld toe. If the linear regime is uncertain for some complex pipe conﬁgurations, such as at a pipe bend or near a branch connection, a multi-gauge stripe with, say, 6 or 12 strain gauges pre-installed in series, can be used for establishing a strain distribution from which a linear regime can be determined. Fig. 10. Normalized pipe outer strain distributions – cantilever bending (weld location: x = 0 ): (a) over entire pipe length; (b) a local view. Within a region of x < 3 Ro t , both FE computation results and experimental strain measurements are expected become increasingly nonlinear and uncertain as the weld location is approached, while beyond 5 Ro t , strain values can become increasingly less relevant to the 8 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. 4.2. Applicability of Markl’s method Under four-point bending conditions as shown in Fig. 6b, although there exists a region of constant strain within ± 20 Rt corresponding to the two load application positions (see Fig. 2), localized plastic strain developments at the two loading locations introduce non-proportional relationship between the strain at the weld position and the applied displacement δa . Such a relationship can be more clearly illustrated in Fig. 13 by plotting a strain ratio εEP / εE corresponding elastic–plastic (εEP ) and pure elastic (εE ) analyses as a function of δa/ δY in Fig. 13a and of relative elastic core size c / Ro in Fig. 13b. The deﬁnition of elastic core size 2c is given in Fig. 8b, which measures the extent of plastic deformation into a pipe cross section from its outer surface. As a point of reference, the analysis results corresponding to elastic-perfectly-plastic material behavior (i.e., no strain hardening) are also included in Fig. 13. The curve labeled as “Ramberg-Osgood” corresponds to the elastic–plastic analysis results incorporating the actual strain hardening as shown by the stress–strain curve given in Fig. 5b. It is worth noting that with the signiﬁcant nonlinear relationship between εEP / εE and δa/ δY shown in Fig. 13a, the linear elastic analysis consistent with Markl’s method can still provide an estimation of εEP within an error of about 15%. (Note that the 15% deviation from the elastic–plastic solution is chosen here only for demonstration purpose.) This is the case even under the extreme condition corresponding to the elastic–plastic analysis results under elastic-perfectly-plastic conditions (i.e., no strain hardening), as long as the applied displacement δa is within a certain limit. The corresponding upper limit in terms of applied displacement is δa/ δY ≤ 2 (see Fig. 13a) and the lower limit in terms of relative elastic core size is c / Ro ≥ 0.5 (see Fig. 13b). The latter indicates the plastic deformation depth from pipe outer surface can be as much as 50% of its outer radius Ro . Under cantilever bending conditions (see Fig. 7b), the nonlinear relationship at weld location ((x = 0 ) between the strains obtained from linear elastic and elastic–plastic analyses is more signiﬁcant than those seen for the cases under four-point bending conditions shown in Fig. 6b. A more detailed illustration of such behaviors can be seen in Fig. 14. To maintain a 15% underestimation in strain when using elastic analysis method (or Markl’s method), a much smaller applied displacement must be imposed here, e.g., δa/ δY ≤ 1.3 (see Fig. 14a) with the corresponding relative elastic core size about c / Ro ≥ 0.8 (see Fig. 14b). Fig. 11. Correlation of PRG and PVRC LCF fatigue test data with ASME master S-N curve scatter band based on linearly extrapolated structural strain (see E × εs' ) (The dispersion and standard derivation is calculated only based on the 4-point bending and cantilever bending pipe fatigue data). 4. Discussions 4.1. Basis of Markl’s method In light of the elastic–plastic analysis results presented in Section 2.3, Markl’s pseudo-elastic load extrapolation approach as described in Fig. 1c seems to underestimate the strain development (therefore the resulting pseudo-elastic stress) in elastic–plastic deformation regime to a varying degree from four-point bending to cantilever bending test conditions (see Figs. 6b and 7b). Upon a further examination of these cases, it can be inferred that if a pipe specimen is subjected to a constant strain over the pipe length, e.g., under either remote tensile displacement δa (Fig. 12a) or a remote angular rotation θa (Fig. 12b), Markl’s method becomes exact, as long as there exists no plastic deformation instability involved. This is because, as illustrated in Fig. 12, there exists a clearly deﬁned one to one proportional relationship between the strain developed in the pipe and the applied displacement-controlled loading, either in terms of δa or θa . These relationships simply become, εo = δa L 4.3. Simpliﬁed structural strain method (15) To mitigate the above limitations regarding Markl’s method as described in Fig. 1c or elastic FE analysis by prescribing applied displacement of δa or Pa , the simpliﬁed structural strain method described for a pipe under applied remote tension of δa , as illustrated in Fig. 12a, and εo = Ro θa L (16) for a pipe under applied remote angular rotation θa , as illustrated in Fig. 12b, respectively. As indicated in Eqs. (15) and (16), since there are no pipe material properties involved in relating the strain at a location of interest to the applied displacement or angular rotation, any computed strain for such displacement-controlled loading conditions is independent of any material mechanical properties considered, e.g., by assuming either linear elastic or nonlinear elastic–plastic material constitutive behaviors. Then, the corresponding pseudo-elastic stress becomes simply E × εo , as postulated by Markl (see Section 2.3). In actual Markl type experiments, as illustrated in Fig. 1, only elastic properties, E and ν , contribute to the slope of the linear part of P − δ curve. The linearly extrapolated Pa at δa , as illustrated in Fig. 1c, gives the same value as E × εo through Eq. (1). The nonlinear material response corresponding to δa in Fig. 12a or θa in Fig. 12b contributes to the reaction load at the prescribed displacement location, i.e., Pnl in Fig. 1c, which is not used in Markl’s method. Fig. 12. Illustration of constant strain distributions over a pipe specimen length under displacement-controlled loading conditions: (a) remote tensile displacement loading of δa ; (b) remote angular rotation loading of.θa 9 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Fig. 13. Comparison of strain ratio εEP / εE using linear elastic and elastic–plastic material models under four-point bending conditions: (a) as a function of applied relative displacement δa / δY ; (b) as a function of elastic core size.c / R o documented discrepancies in fatigue data correlation between fourpoint bending and cantilever bending conditions. By deﬁnition, the former test conditions produce only a ﬁnite region of constant strain while the latter generate linearly varying strain over the entire pipe length. (2) An axially varying strain/stress distribution over the pipe length shall lead to plastic deformation localization at high stress location subjected to LCF loading. As the extent of the nonlinear relationship between applied displacement and strain increases, an increased deviation from Markl’s original assumption will occur as a result. This further explains why fatigue test data obtained from fourbending conditions tend to show relatively more consistency than those obtained under cantilever bending conditions (3) A simpliﬁed structural strain method is proposed in this paper and proven eﬀective, not only for achieving a good data correlation among the two diﬀerent test conditions, but also with the ASME master S-N curve scatter band. This method can be used for extracting pipe ﬁber strains at a weld location of interest either as an experimental measurement technique or a post-processing method when elastic–plastic FE results are available. in Section 3.3 can be used both in the context of FE analysis or experimental strain measurements through the linear extrapolation method given in Eq. (13). To relate the structural strain either through FE analysis or experimental measurements to the master S-N curve or its scatter band, Eq. (14) can be directly used. As such, the limitations discussed above can be removed, as demonstrated in Fig. 13 for the deformation regime as high as about δa/ δY = 12 and relative elastic core size as small as c / Ro = 0.2 under four point bending conditions, while δa/ δY = 3 and c / Ro = 0.1 under cantilever bending conditions (Fig. 14). In most of LCF fatigue applications such as those stipulated for pressure vessels and piping components in ASME Div 2 [12], a relative elastic core size is typically restricted to c / Ro = 0.5 for preventing plastic collapse under general loading conditions. 5. Conclusion The classical low cycle fatigue evaluation procedure based on Markl’s pseudo-elastic stress method has been critically assessed in detail within the context of two popular fatigue test methods, i.e., fourpoint bending and cantilever bending for evaluation of girth-welded piping components under displacement-controlled conditions. Detailed ﬁnite element analyses were performed for elucidating the plastic deformation conditions under which Markl’s method works reasonably well and beyond which a simpliﬁed structural strain method can be eﬀectively used. Speciﬁcally, the following ﬁndings are worth noting: Declaration of Competing Interest The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. (1) Based on Markl’s deﬁnition, the pseudo-elastic stress is calculated based on the linear extrapolated pseudo-elastic load from the linear part of experimentally measured load–displacement curve. This pseudo-elastic stress works perfectly, as intended, only for pipe components that exhibit constant strain distributions over the entire pipe length. This is the key reason why there exist the well- Acknowledgement The authors gratefully acknowledge the support of this work in part the National Research Foundation of Korea (NRF) Gant funded by the Korea government (MEST) through GCRC-SOP at University of Fig. 14. Comparison of strain ratio εEP / εE using linear elastic and elastic–plastic material models under cantilever bending conditions: (a) as a function of applied relative displacement δa / δY ; (b) as a function of elastic core size.c / R 10 International Journal of Fatigue 139 (2020) 105732 X. Pei, et al. Michigan under Project 2-1: Reliability and Strength Assessment of Core Parts and Material System. 1952;74:287–303. [20] Markl A. Fatigue tests of welding elbows and comparable double-mitre bends. Trans. ASME 1947;69:869–79. [21] Markl A, George H. Fatigue tests on ﬂanged assemblies. Trans. ASME 1950;72:77–87. [22] Markl A. Piping-ﬂexibility analysis. Trans. ASME 1955;77:12743. [23] Hinnant C, Paulin T. Experimental Evaluation of the Markl Fatigue Methods and ASME Piping Stress Intensiﬁcation Factors. ASME 2008 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers; 2008. p. 97–113. [24] Hinnant C, Paulin T, Becht C, Becht IV C, Lock WS. Experimental Evaluation of the Markl Fatigue Methods and ASME Piping Stress Intensiﬁcation Factors: Part II. Pressure Vessels and Piping Conference. American Society of Mechanical Engineers; 2014. pp. V003T003A025. [25] Scavuzzo R, Srivatsan T, Lam P. Report 1: Fatigue of Butt-Welded Pipe. WRC Bull 1998:1–56. [26] Wang W, Pei XJ. An Analytical Structural Strain Method for Steel Umbilical in Low Cycle Fatigue. J Oﬀshore Mech Arct 2019;141. [27] Dong P, Hong J, Osage D, Dewees D, Prager M. The Master SN Curve Method an Implementation for Fatigue Evaluation of Welded Components in the ASME B&PV Code, Section VIII, Division 2 and API 579–1/ASME FFS-1. Welding Research Council Bulletin; 2010. [28] Dong P, Hong JK, Cao Z. Stresses and stress intensities at notches: 'anomalous crack growth' revisited. Int J Fatigue 2003;25:811–25. [29] Dong P, Hong JX, De Jesus AMP. Analysis of recent fatigue data using the structural stress procedure in ASME Div 2 Rewrite. Pres Ves P 2005:253–61. [30] Pei X, Dong P, Song S, Osage D. A Comprehensive Structural Strain Method Incorporating Strain-Hardening Eﬀects: From LCF to Ratcheting Evaluations. ASME 2018 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers; 2018. pp. V01AT01A004-V001AT001A004. [31] Dong P, Pei X, Xing S, Kim MH. A structural strain method for low-cycle fatigue evaluation of welded components. Int J Pres Ves Pip 2014;119:39–51. [32] Simo JC, Hughes TJ. Computational inelasticity. Springer Science & Business Media; 2006. [33] Dunne F, Petrinic N. Introduction to computational plasticity. Oxford University Press on Demand; 2005. [34] Shao Y-B, Du Z-F, Lie S-T. Prediction of hot spot stress distribution for tubular Kjoints under basic loadings. J Constr Steel Res 2009;65:2011–26. [35] Karamanos SA, Romeijn A, Wardenier J. SCF equations in multi-planar welded tubular DT-joints including bending eﬀects. Mar Struct 2002;15:157–73. [36] Chatziioannou K, Karamanos SA, Huang Y. Ultra low-cycle fatigue performance of S420 and S700 steel welded tubular X-joints. Int J Fatigue 2019;129:105221. [37] Varelis GE, Papatheocharis T, Karamanos SA, Perdikaris PC. Structural behavior and design of high-strength steel welded tubular connections under extreme loading. Mar Struct 2020;71:102701. [38] Song Shaopin, et al. A full-ﬁeld residual stress estimation scheme for ﬁtness-forservice assessment of pipe girth welds: Part II–A shell theory based implementation. Int J Press Vessels Piping 2015;128:8–17. https://doi.org/10.1016/j.ijpvp.2015.01. 005. [39] Dong Pingsha, et al. An IIW residual stress proﬁle estimation scheme for girth welds in pressure vessel and piping components. Weld World 2016;60:283–98. https:// doi.org/10.1007/s40194-015-0286-4. References [1] Becht C. Process piping: the complete guide to ASME B31. 3. ASME Press; 2004. [2] Zhang Y, Karr DG. Determining ice pressure distribution on a stiﬀened panel using orthotropic plate inverse theory. J Struct Eng 2017;143:04017003. [3] Standard B. Guide to fatigue design and assessment of steel products, BS 7608: 2014. London, UK: The British Standards Institution; 2014. [4] Niemi E, Fricke W, Maddox SJ. Structural hot-spot stress approach to fatigue analysis of welded components. IIW doc 2018;13. 1819–1800. [5] Niemi E, Fricke W, Maddox SJ. Fatigue analysis of welded components: Designer’s guide to the structural hot-spot stress approach. Woodhead Publishing; 2006. [6] Karakaş Ö. Application of neuber’s eﬀective stress method for the evaluation of the fatigue behaviour of magnesium welds. Int J Fatigue 2017;101:115–26. [7] Karakas Ö. Consideration of mean-stress eﬀects on fatigue life of welded magnesium joints by the application of the Smith–Watson–Topper and reference radius concepts. Int J Fatigue 2013;49:1–17. [8] Karakas O, Zhang G, Sonsino CM. Critical distance approach for the fatigue strength assessment of magnesium welded joints in contrast to Neuber's eﬀective stress method. Int J Fatigue 2018;112:21–35. [9] Karakaş Ö, Tüzün N. State of the art review of the application of strain energy density in design against fatigue of welded joints. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 2019;25:462–7. [10] Karakas Ö, Morgenstern C, Sonsino C. Fatigue design of welded joints from the wrought magnesium alloy AZ31 by the local stress concept with the ﬁctitious notch radii of rf= 1.0 and 0.05 mm. Int J Fatigue 2008;30:2210–9. [11] Dong P. A structural stress deﬁnition and numerical implementation for fatigue analysis of welded joints. Int J Fatigue 2001;23:865–76. [12] Boiler A, Committee PV. ASME Boiler and Pressure Vessel Code. Section 3, Rules for Construction of Nuclear Power Plant Components. American Society of Mechanical Engineers. [13] Wang P, Pei X, Dong P, Song S. Traction structural stress analysis of fatigue behaviors of rib-to-deck joints in orthotropic bridge deck. Int J Fatigue 2019;125:11–22. [14] Karakas O, Tuzun N. Evaluation of fatigue behaviour of magnesium welded joints using energy methods according to neuber’s method. J. Achiev. Mater. Manuf. Eng 2015;73:100–5. [15] P. Dong, Z. Cao, J. Hong, Low-cycle fatigue evaluation using the weld master SN curve, in: ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference, American Society of Mechanical Engineers, 2006, pp. 237-246. [16] Dong PS, Pei XJ, Xing SZ. A Structural Strain Method for Fatigue Evaluation of Welded Components. 33rd International Conference on Ocean, Oﬀshore and Arctic Engineering, 2014 2014;vol. 5. [17] Pei XJ, Dong PS. An analytically formulated structural strain method for fatigue evaluation of welded components incorporating nonlinear hardening eﬀects. Fatigue Fract Eng M 2019;42:239–55. [18] Pei X, Dong P, Xing S. A structural strain parameter for a uniﬁed treatment of fatigue behaviors of welded components. Int J Fatigue 2019. [19] Markl A. Fatigue tests of piping components. Transactions of the ASME 11