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.
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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].
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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].
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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.
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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
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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.
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