International Journal of Fatigue 137 (2020) 105670
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International Journal of Fatigue
journal homepage: www.elsevier.com/locate/ijfatigue
Enhanced crack sizing and life estimation for welded tubular joints under
low cycle actions
Liuyang Feng, Xudong Qian
T
⁎
Department of Civil and Environmental Engineering, Centre for Offshore Research and Engineering, National University of Singapore, Singapore 117576, Singapore
A R T I C LE I N FO
A B S T R A C T
Keywords:
Low-cycle fatigue
Tubular joints
Fatigue crack propagation
Local energy indicator
Continuum damage mechanics
This study examines the low-cycle fatigue of circular hollow section (CHS) X-joints made of S355 steels, through
a combined experimental and numerical investigation, assisted by non-destructive crack sizing methods, namely
the Alternating Current Potential Drop and Ultrasonic Phased Array. The comparison of the two methods reveals
the discrepancy in the crack sizing, and illustrates the fatigue crack initiation and propagation in CHS joints. This
study extends the Improved Modified Neuber’s Rule to estimate the local energy indicator for CHS X joints. The
numerical analysis implements a continuum damage mechanics method, which estimates fatigue lives of test
specimens with reasonable accuracy.
1. Introduction
As an important structural component, welded tubular joints in the
offshore structures, bridges and large span structures often experience
cyclic actions caused by the current, wind and traffic loads [1–3]. The
cyclic actions with fairly large magnitudes usually initiate low-cycle
fatigue cracks at the geometrically discontinuous locations of the tubular joints [4,5], for example, near the weld toes. These fatigue cracks,
with relatively short fatigue lives, impose potential threats to the safety
of structures, which can cause unstable failure in the entire structure.
Therefore, accurate fatigue life assessments of the welded tubular joints
based on reliable crack size measurement are critical and essential for
the integrity and life cycle analysis of engineering structures. Previous
research works focus primarily on the experimental investigation and
numerical analysis of the high-cycle fatigue behavior of welded tubular
joints [6–12]. Recently, some research works [13,14] have explored the
extreme low-cycle fatigue of tubular connections. However, limited
research efforts [15,16] on the low-cycle fatigue of welded tubular
joints reflect a need to enhance the current understanding and to improve the assessment procedures.
The local material near the weld toe experiences noticeable plastic
deformations under low-cycle fatigue loads, which lead to different
crack initiation and propagation in the tubular joints in contrast to
those under high-cycle actions. Experimental investigations provide
direct evidence to understand the fatigue crack initiation and propagation, and therefore, represent the necessary benchmarks for improved
approaches and their numerical implementations. Non-Destructive
⁎
Techniques (NDTs) have found wide applications in the health monitoring of engineering structures, and are convenient tools to quantify
the fatigue crack initiation and propagation in welded tubular joints.
The common NDTs include Alternating Current Potential Drop (ACPD)
[17–20], Acoustic Emission (AE) [21,22], X-ray based approaches
[23,24], Ultrasonic Phased Array (UPA) [25–27], Eddy Current
[28,29], etc. Among these different NDTs, ACPD and UPA are relatively
easy to deploy in measuring fatigue cracks in welded tubular joints with
sufficient accuracy. Relying on the strong skin effect of steel, ACPD
measures the crack depth through the difference in the current potentials between the cross-crack probes and the reference intact probes
[19]. In UPA, the high-frequency ultrasonic energy produced by the
electric device penetrates into the material through a couplant. Upon
intersecting a flaw or crack, the ultrasonic wave gets reflected or
transmitted to the receiving electric transducer. Based on the ultrasonic
wave traveling time and intensity of the received energy, the ultrasonic
phased array determines the location, size, and orientation of the reflector, which can be a crack, a flaw or a defect. Based on these two
NDTs (ACPD and UPA), this study characterizes the low-cycle fatigue
crack initiation and propagation in the welded tubular joints.
The low-cycle fatigue assessment for tubular joints often employs a
quick S-N type approach or a detailed numerical analysis to describe the
fatigue damage evolution under the cyclic actions. Among the different
indicators driving the low-cycle fatigue failure [19,30–33], the local
energy indicator emerges as an accurate parameter to quantify the fatigue driving force. Feng and Qian [19] have developed an Improved
Modified Neuber’s rule (IMNR) to derive the local energy indicator at
Corresponding author.
E-mail address: qianxudong@nus.edu.sg (X. Qian).
https://doi.org/10.1016/j.ijfatigue.2020.105670
Received 13 March 2020; Received in revised form 17 April 2020; Accepted 23 April 2020
Available online 25 April 2020
0142-1123/ © 2020 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Nomenclature
Achord
C
D
E
F
Ibrace
KN
KNe
L
Lc
N
N0
Nf
Ni
P
Q∞
Sc
Sr
Vc
Vr
Wc
a
b
c
ci
d0
d1
g
l0
l1
t0
t1
Δw
Δw0
α
α′
γ
Δεeq
Δεij
Δεije
Δεnom
Δσij
Δσije
Δσnom
Δσpressure
Δσeq
εa
ε′f
ε pl
ρ
σ
σ′
σa
σ ′f
σy
σyo
δ
cross-section area of chord
kinematic hardening parameter
scalar damage variable
Young’s modulus
Elastic-plastic notch reduction factor in the IMNR
moment of inertia of the brace member
stress and strain concentration factor
elastic stress and strain concentration factor
characteristic length in the finite element model
damage related parameter
fatigue cycles
cycles to initiate the damage
fatigue life
fatigue initiation life
external load
limit surface of the isotropic hardening
distance between the probes crossing the crack
distance between the reference probes
voltage between the probes crossing the crack
voltage between the reference probes
critical value of the energy at the yielding point
crack length
fatigue strength exponent
fatigue ductility exponent
damage related parameters (i = 1, 2, 3, 4 )
diameter of the chord member
diameter of the brace member
isotropic hardening parameter
length of the chord
length of each brace
thickness of the chord
thickness of the brace
accumulated hysteresis energy per cycle
accumulated hysteresis energy per cycle in the first cycle
backstress tensor
deviatoric backstress tensor
kinematic hardening parameter
the equivalent strain at the weld toe
strain range tensor
elastic strain range tensor
nominal strain range
stress range tensor
elastic stress range tensor
nominal stress range
pressure on the chord by MTS
equivalent stress range at the weld toe
strain amplitude
fatigue ductility coefficient
effective accumulative plastic strain
position angle along the brace-to-chord intersection
stress tensor
deviatoric stress tensor
stress amplitude
fatigue strength coefficient
yield strength
yield strength at zero plastic strain
displacement
P
Chord
(a)
t0
l1
Grinding side
l0
d1
75 mm
t1
Brace
d0
Welded plate
Grinding side
(c)
(b)
XJ6
XJ1
Fig. 1. (a) Geometric configuration of the CHS X-joints; and the experimental set-up for (b) XJ1; and (c) XJ6.
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International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
shown in Fig. 1b and c, in a load-controlled scheme. This generates an
in-plane bending action on the joint. The loading scheme of the CHS Xjoints includes: (1) preload the joint specimen for one to two cycles; (2)
apply a constant-amplitude, low-cycle fatigue load with a load ratio
around 0.1; (3) stop fatigue test when the maximum crack depth
reaches about 70% of the chord wall thickness (based on the continuous
ACPD measurement in the test), and re-apply the constant-amplitude,
low-cycle fatigue load with the same load ratio around 0.1 after rotating the X-joint by 180° in the plane of the joint; (4) stop the second
fatigue test when the maximum crack depth reaches 70% of the chord
wall thickness and push the joint to failure. The subsequent text denotes
the second fatigue test in step 3 as the test after the flip-over. The above
loading procedure allows generation of two fatigue life data points
through one large-scale specimen, and facilitates the subsequent investigation on the residual capacity of the cracked joint.
The fatigue test prior to the flip-over introduces plastic deformations under compressive stresses near the top crown position, which
experiences tensile cyclic loadings after the flip-over. The two consecutive fatigue loadings on the same specimens allow the examination
of the loading history effect on the low-cycle fatigue of CHS X-joints.
The frequency of the load-controlled test remains fixed at 0.5 Hz. To
ensure that the crack initiates from one side of the brace-to-chord intersection, the other side of the brace-to-chord intersection undergoes a
grinding treatment along the weld toe to reduce the stress concentration, as shown in Fig. 1a. A pre-test numerical analysis indicates that
the maximum load in the cyclic test leads to severe plastic deformations
near the weld toe, which anticipates a low-cycle fatigue failure. A series
of strain gauges attached to the weld toe monitors the strain values
during the tests. The dynamic oscilloscope DL850 records the experimental data at a sampling rate of 20 data points per second.
The experimental scheme adopts two non-destructive techniques
including the ACPD and UPA to measure the fatigue-induced cracks
during the test. Fig. 2a and b show the ACPD instrument and ultrasonic
phased array instrument. The ACPD records the crack depth at every
eight seconds in eight locations along the weld line. The ultrasonic
phased array instrument captures the crack size manually every 500 to
1500 cycles. Fig. 2c shows the arrangement of the ACPD probes and
ultrasonic probes in the tension zone. Fig. 2d illustrates the angular
position, ρ, along the brace-to-chord intersection, measured from the
side view. After the final fracture failure, this study reproduces the
failure surface of the X-joints using the silicone-replica, which allows
direct measurements of the final crack size.
To calibrate the base material properties, this experimental study
includes fourteen standard smooth specimens fabricated from the chord
member at 400 mm away from the brace-to-chord intersection, with a
13 mm × 20 mm cross-section, as shown in Fig. 3a. Two of these specimens undergo monotonic tension tests and the other twelve experience fully reversed strain-controlled low-cycle fatigue tests. The strain
amplitude of the low-cycle fatigue tests covers the strain amplitude of
0.3%, 0.5%, 0.75% and 1.0%, with three duplicates for each strain
amplitude. The experimental procedure utilizes the 1000 kN MTS
servo-hydraulic machine and follows the ASTM E606 [37] for the
strain-controlled tests, as shown in Fig. 3b. The calibrated extensometer
of 50 mm gauge length, fastened on the standard specimens, helps to
monitor the total strain amplitude under the strain-controlled low-cycle
fatigue tests, as presented in Fig. 3c. The monotonic tensile test has the
same experimental set-up with that of low-cycle fatigue tests but is
the weld toe of welded plate joints through a simplified approach.
However, the applicability of the IMNR for tubular joints requires further verifications. The costly and time-consuming low-cycle fatigue
experimental tests motivate researchers [6–12] to develop a high fidelity model, which is capable of describing the damage growth in the
material under cyclic, plastic deformations. The elastic-plastic fracture
mechanics approach relies on a series of built-in cracks, in the highly
detailed finite element models [34], which are time-consuming and
intractable in an engineering office. Continuum damage mechanics
defines the failure when a damage parameter reaches a critical value,
and thus estimates the crack growth automatically [35]. Previous studies [19,33] have developed a convenient and efficient method to calibrate the damage-related parameters for standard coupon specimens
and welded plate joints. However, the verification of the proposed
damage-based approach in large-scale structural components, such as
welded tubular joints, requires further experimental supports.
This study couples the NDT crack sizing with the low-cycle fatigue
tests on large-scale welded circular hollow section (CHS) joints. Based
on the improved crack sizing by applying a previously proposed algorithm to the UPA S-scan image, the experimental study examines the
initiation and propagation of fatigue cracks generated in CHS X-joints.
To facilitate fatigue assessments, this study develops a simplified approach to estimate the local stress and strain at the weld toe of CHS
joints under low-cycle fatigue actions. In addition, this study implements a continuum damage mechanics approach using material parameters calibrated from the coupon specimens to describe the fatigue
damage evolution and to estimate the total fatigue life for CHS X-joints.
The structure of this paper is organized as follows. Section 2 introduces the experimental program including the low-cycle fatigue tests
of CHS X-joints and the standard coupon specimens. Section 3 summarizes the experimental results including the load-displacement relationship and the variation in the global stiffness under cyclic loads, as
well as the fatigue crack measured by the ACPD and UPA for the CHS Xjoints. Based on the cyclic elastic-plastic material properties calibrated
from the standard coupon specimens, this study compares the stress,
strain and local energy indicator around the weld toe in Section 4.
Section 5 summarizes the fatigue life assessment of CHS X-joints by the
experimental S-N approach and the numerical simulation based on the
continuum damage mechanics. Finally, Section 6 concludes the main
findings of this paper.
2. Experimental details of the CHS tubular joints
The experimental program focuses on determining the fatigue life
and fatigue crack propagation during the low-cycle fatigue tests of CHS
X-joints under in-plane bending loads. Fig. 1a shows the geometric
configurations of the CHS X-joints. Two braces welded to the main
chord are pin supported at the two ends. Table 1 lists two different sets
of dimensions for the X joints, with five specimens in total. The weld
profile along the brace-to-chord intersection follows the complete penetration welds (CJP) in AWS [36], with a post-welding ultrasonic
examination. The CHS X-joint utilizes S355 steels for both the chord
and brace members. The brace members are simply supported at 75 mm
away from the two ends with a plate welded inside the brace tube, as
shown in Fig. 1a.
The CHS X-joints experience a cyclic axial load at the top end of the
chord member imposed by the 500 kN MTS servo-hydraulic machine, as
Table 1
Dimensions for the two sets of CHS X-joints.
t1
t0
Specimen
d 0 (mm)
t0 (mm)
d1(mm)
t1(mm)
l 0 (mm)
l1(mm)
τ=
XJ1/XJ2/XJ3
XJ5/XJ6
273
324
10
16
219
219
8
12.5
1920
1920
1075
1075
0.8
0.78
3
β=
0.80
0.68
d1
d0
γ=
d0
2t 0
13.65
10.13
α=
2l0
d0
14.07
11.85
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 2. (a) ACPD instrument; (b) GE Phasor XSTM UPA instrument; (c) the arrangement of the ACPD probes and UPA probes; (d) the position angle measured in the
side view.
Fig. 3. (a) Dimensions of the coupon specimens; and the experimental set-up for (b) strain-controlled low-cycle fatigue tests; and (c) monotonic tension tests.
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International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Table 2
Experimental loading scheme and total fatigue life for CHS X-joints.
Preload (kN)
XJ1
XJ2
XJ3
XJ5
XJ6
LCF load (kN)
Min
Max
Min
Max
0
0
0
0
0
105
105
105
146
180
10.46
10.46
8.56
10
1.5
104.6
104.6
85.6
160
190
LCF cycles
Flip over
Preload (kN)
22000
17000
40486
25512
15675
LCF load (kN)
LCF cycles
Min
Max
Min
Max
0
0
105
105
10.4
10.4
104.6
104.6
7251
4000
0
0
180
200
2.5
4.5
175
190
9757
7389
-
overall displacement shows a slight increase compared to the beginning
of the test. The overall stiffness of the joint shown in Fig. 5c first decreases in the first few cycles due to the initial stabilization in the
testing frames. Beyond the first few cycles, the overall stiffness remains
almost constant, implying that the fatigue crack sizes are sufficiently
small without causing adverse effects on the joint stiffness. As the crack
size increases substantially towards the final stage of the fatigue test,
the joint stiffness experiences a noticeable decrease.
To monitor the fatigue crack initiation and propagation under lowcycle fatigue loadings, this study deploys two non-destructive techniques (ACPD and UPA) to measure the evolution of cracks in the X-joints.
The ACPD is an electromagnetic crack measurement technique. The
fatigue crack initiation and propagation affect the path of the alternating current due to the skin effect in the steel. The crack depth has a
quantitative relationship with the voltages between the reference points
and the cross-crack points,
under displacement-control with a stable deformation rate of 0.2 mm/
min.
3. Experimental results of CHS X-joints
Table 2 summarizes the loading details and the fatigue lives of five
CHS X-joints before and after the flip-over. The maximum load for XJ1
and XJ2 is 105 kN. The specimen XJ3 has a maximum load of 85.6 kN.
For XJ5 and XJ6, the maximum loads are 160 kN and 190 kN before the
flip-over, respectively. After the flip-over, the maximum loads for XJ5
and XJ6 become 175kN and 190 kN. Fig. 4a and b present the final
failure mode of XJ1 and XJ6 joints. Fig. 4c shows the crack-front profile
after the final fracture test along the position angle ρ. This study employs the silicone-replica (the Provil Novo putty soft fast set) to reveal
the final crack surface, as shown in Fig. 4d. The replica of the crack
profile indicates clearly the crack profile at the end of the low-cycle
fatigue test.
Fig. 5a presents the load-displacement curve at different fatigue
cycles of XJ6 before and after the flip-over, which exhibits a predominantly linear relationship. The X-joint specimen experiences apparent ratcheting under the load-control cyclic actions, caused by the
accumulative plastic strain near the weld toe and the crack initiation
and propagation. Fig. 5b shows the variation of the displacement range
of XJ6 before and after the flip-over. At the end of the fatigue test, the
a=
Sr ⎛ Vc
S
− c⎞
2 ⎝ Vr
Sr ⎠
⎜
⎟
(1)
where Vc and Sc represent the voltage and distance between the probes
crossing the crack; Vr and Sr denote the voltage and distance between
the reference probes at the intact areas, as shown in Fig. 6a. This study
positions the ACPD probes in the tension sides (both the grinding and
non-grinding side) of the X-joints.
Fig. 4. Final failure of: (a) XJ1; and (b) XJ6; (c) failure surface near the weld toe; (d) silicone-replica of the final failure surface.
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International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 5. (a) Load-displacement relationship near the beginning and end of fatigue tests; (b) variation in the displacement range with increasing cycles before and after
flip-over; (c) variation in the joint stiffness with increasing cycles; for XJ6.
some distance away from the surface crack at the weld toe. An important UPA parameter is the number of legs. The number of legs indicates the number of reflections of the ultrasonic wave in the transmission phase. The angle of the focused ultrasonic wave varies from 20°
to 88° in this study. The digital gain is an important parameter in the
initiation set-up. A too large or a too small gain cannot differentiate the
flaws or intact surface from the crack. After a few rounds of tuning, this
The ultrasonic instrument deployed in the study is the GE Phasor
XSTM ultrasonic phased array (UPA). The velocity of the shear ultrasonic wave equals 2337 m/s in mild steel. The ultrasonic probe, numbered 115-100-004, has 32 electric elements, 10.0 mm elevation,
0.5 mm pitch and 5.0 MHz frequency. The angle of the wedge is 36°.
The probe emits and receives ultrasonic wave signals. Since the crack
initiates at the surface, the probe of the ultrasonic instrument locates at
Fig. 6. (a) ACPD measurement scheme; (b) typical UPA S-scan image; (c) graphical illustration of A-scan; and (d) graphical illustration of S-scan.
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International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
propagation before flip-over. The cracks in the grinding areas reach
around 3 mm while the deepest crack in the non-grinding areas propagates to 11 mm. In contrast, the crack first initiates from the grinding
sides and propagates faster than that at the non-grinding sides after flipover. This phenomenon also exists in other CHS X-joints in this study.
ACPD requires a normalization procedure before detecting the crack,
and tends to neglect the existing small cracks in the structural components.
Fig. 8a and b compares the crack depth evolution at two deepest
points in XJ6 measured by UPA and ACPD before and after flip-over,
respectively. The UPA_COR represents the crack size measured by the
UPA based on the improved algorithm in [38]. Before flip-over, when
the crack size is small, the value from the ultrasonic measurement is
larger than that from the ACPD. When a crack grows deeper, the
agreement in the measured crack size by ACPD and UPA improves. The
ACPD utilizes the strong ‘skin effect’ in the steel material to depict the
crack depth. In the early stage of the fatigue test, the crack profile along
the weld line is not continuous and consists of multiple small cracks,
which subsequently coalesce into a final large crack. The alternating
current in ACPD thus flows along the shortest path in the material
bypassing the cracks, and may not detect the crack initiation accurately.
The UPA, on the other hand, detects all cracks covered in the scan area,
and is more accurate in detecting small cracks. With increasing crack
sizes, the accuracy of the ACPD is enhanced and comparable with the
UPA. After flip-over in Fig. 8b, the UPA detects the existing cracks while
the ACPD detection assumes that there are no existing cracks. Fig. 8c
and d compares the final crack profile in XJ6 before and after the flipover. After the flip-over, the final failure surface is also measured using
the reproduced fracture surface from silicone-replica. The difference
between the ACPD and ultrasonic measurement in the final failure
surface is small, with UPA being more accurate than ACPD, as shown by
the comparison with the measurement from the replica in Fig. 8d.
Table 3 illustrates the fatigue initiation cycles detected by ACPD, UPA
and visual inspection. The UPA tends to detect the crack initiation
study adopts a gain value of 16.4 dB for the UPA measurement.
Fig. 6b presents a typical original ultrasonic S-scan image. The
distance between the head of the ultrasonic probe and the weld toe is
around 30 mm. Part A in the image illustrates the A-scan information
along an incident angle. Part B combines all A-scan results under different incident angles in one sector shape image. The color of each pixel
in the ultrasonic images indicates the echo intensity. The flaws and
cracks usually show stronger echo signals. If the strength of the echo
exceeds the threshold gate (30% in this study), the corresponding pixel
represents a crack or defect. The ultrasonic probe in the A-scan emits
and receives the ultrasonic wave at one fixed location (Fig. 6c). The Sscan generates an ultrasonic phased-array image by emitting a series of
focused ultrasonic beams with different incident angles. The scanned
area ahead of the ultrasonic probe envelops a sector shape in two dimensions, as shown in Fig. 6d. Two limitations exist in the ultrasonic
phased array measurement: (1) the receiving transducer at one fixed
location does not always capture the reflected ultrasonic wave by the
crack due to the limited coverage of the probe; (2) the number of receiving legs can be different from the number of emitting legs. This
leads to some errors in the default calculations of the crack size and
orientation in the UPA instrument, which assumes equal numbers of the
emitting and receiving legs. To alleviate these two limitations, this
study adopts the measurement scheme and algorithm proposed in a
previous study [38], by placing the UPA probe at three locations
(10 mm, 20 mm and 30 mm) away from the weld toe and reformulating
the crack size using the actual numbers of legs determined from image
processing.
Fig. 7a and b show a typical evolution of the crack depths along the
brace-to-chord intersection measured by the ACPD probes for XJ6 before flip-over at different loading cycles for the non-grinding side and
grinding side. Fig. 7c and d present the crack depths of XJ6 after flipover at different loading cycles for the non-grinding and grinding side.
From the ACPD results, the grinding procedure alleviates the initial
defects at the weld toe and thus delays the crack initiation and
Fig. 7. Variation in the fatigue crack depth along the position angle for XJ6 measured by ACPD for (a) non-grinding side before flip-over; (b) grinding side before flipover; (c) non-grinding side after flip-over; and (d) grinding side after flip-over.
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International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 8. Evolution of the crack depth at the deepest crack-front positions in XJ6: (a) the non-grinding side; and (b) the grinding side; and the final crack depth along
the position angle: (c) before flip-over and (d) after flip-over.
where σ and σ ′ denote the stress tensor and deviatoric stress tensor; α
and α ′ represent the backstress tensor and the deviatoric backstress
tensor, while σy denotes the current size of the yield surface.
With increasing cumulative plastic strain ε pl , the size of the yield
surface evolves, which represents the isotropic hardening property. A
typical isotropic hardening relationship follows,
Table 3
Fatigue initiation life cycles by ACPD, UPA and visual inspection for CHS Xjoints.
XJ1
XJ2
XJ3
XJ5
XJ6
Visual
inspection
ACPD
UPA
17,000
14,000
19,000
8000
3000
15,000
14,000
21,000
7310
3200
12,000
7000
19,000
6000
2000
Flip-over
Visual
inspection
ACPD
UPA
100
0
–
700
0
1000
200
–
900
500
0
0
–
0
0
σy = σyo + Q∞ (1 − exp(−gε pl ))
(3)
where σyo denotes the yield stress at the zero plastic strain, g represents
the evolution rate of the size of yield surface with changing cumulative
plastic strains and Q∞ refers to the limit value of the yield surface at a
large cumulative plastic strain.
The nonlinear kinematic hardening of the model defines the center
of the yield surface, relying on the backstress tensor. The following
equations outline the relationship between the change in the backstress
tensor and the incremental plastic strain,
earliest and the ACPD shows a delay in detecting the crack initiation.
4. Numerical simulation
4.1. Elastic-plastic material properties
2
Cdεijpl − γαdε pl
3
(4a)
1 2
2
dε pl = ⎛ dεijpl: dεijpl ⎞
⎝3
⎠
(4b)
dα =
Fig. 9a presents the two monotonic tensile true stress-strain curves
of S355 steel measured from two coupon specimens. The cyclic elasticplastic material parameters derive from the standard low-cycle fatigue
coupon tests. This study considers the constitutive material model originally proposed by Armstrong and Frederick [39] and subsequently
modified by Chaboche [40,41]. Following the definition in ABAQUS
[42], the kinematic and isotropic hardening relationship follows,
f (σ − α ) − σy = 0
(2a)
3
f (σ − α ) = ⎛ (σ ′ − α ′): (σ ′ − α ′) ⎞
⎠
⎝2
(2b)
σ′ = σ −
1
Tr (σ ) I
3
(2c)
α′ = α −
1
Tr (α ) I
3
(2d)
where γ denotes the rate at which the kinematic hardening modulus
decreases with increasing plastic deformations and C represents the
initial kinematic hardening modulus.
Fig. 9b presents a typical stress amplitude variation with increasing
cycles of the fatigue coupon specimen. The stress amplitude approaches
quickly to a stable value within a few cycles. This implies that the
isotropic hardening of S355 remains insignificant under the given strain
amplitude in the low-cycle fatigue test, as also reflected by the yield
plateau in Fig. 9b. The calibration of the kinematic strain hardening
thus excludes the isotropic hardening in the numerical model. Based on
the experimental stress-strain relationship of the coupon specimens
under the strain amplitude of 0.75% at 0.5Nf cycle, this study follows
the procedure in [33] to calibrate the kinematic hardening parameters
8
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 9. (a) Monotonic true stress-strain curve; (b) variation of the stress amplitude with increasing cycles; (c) comparison of the numerical and experimental stressstrain curves; (d) ε-N curve for coupon specimens.
weld leg length (wl ) at the crown and the saddle equals 1.5t1 and 1.75t1,
respectively. At the intermediate points between the crown and the
saddle, the weld leg length follows the size from linear interpolation.
The parameter (wh) indicates the location of the brace weld toe relative
to the weld root and equals the brace wall thickness (t1) . This study
adopts an automatic mesh generation script in Patran [55]. Fig. 11
presents a quarter symmetric finite element model, utilizing 20-node
solid elements with reduced integration (C3D20R in ABAQUS [42]).
The mesh size near the weld toe is highly refined for an accurate estimation of the local stress and strain values. The mesh size increases
gradually for regions away from the weld toe with a total 15,552 elements and 70,890 nodes in the finite element model.
Figs. 12 and 13 describe the range of local equivalent stress,
equivalent strain and energy indicator at the weld toe for the two sets of
tubular X joints. The range of local equivalent stress and strain follows
Table 4
Kinematic hardening properties for S355 steel.
εa
E (MPa)
C1 (MPa)
γ1
α1 (MPa)
0.75%
18,000
24,684
221.17
−109.56
using one kinematic hardening law. Table 4 lists the calibrated kinematic hardening parameters. Fig. 9c illustrates a satisfying agreement
between the experimental and numerical stress-strain variations in the
first few cycles with the strain amplitude of 0.75%.
The low-cycle fatigue tests of the standard coupon specimens provide data required in an S-N curve. Based on Basquin-Coffin-Manson
relationships [43–45], the fatigue indicator accounting for the strain
amplitude describes the total fatigue life in the following form,
εa =
σ′ f
E
(2Nf )b + ε′f (2Nf )c
(Δσxx − Δσyy )2 + (Δσxx − Δσzz )2 + (Δσyy − Δσxx )2
(5)
2
2
2
)
+ 6(Δσxy
+ Δσxz
+ Δσyz
where Nf represents the fatigue life, σ ′f andε′f denote the fatigue strength
and ductility coefficient, b and c define the fatigue strength and ductility exponent. Fig. 9d presents the S-N curve data, the fitting curve and
the coefficient of determination. Table 5 lists the corresponding fitting
coefficients in Eq. (5).
Δσeq =
4.2. Stress and strain concentration
For both two sets of X-joints, the stress concentration factor decreases with increasing external loads. For a small traction applied on
the chord end, e.g., 1 MPa, the material deforms elastically. At large
remote tractions, the plastic deformation accumulates near the weld toe
and the difference between the equivalent stress range diminishes due
Δεeq =
Based on the calibrated monotonic and cyclic material properties,
this study conducted a series of elastic-plastic numerical simulations to
reveal the behavior of the CHS X-joints in the experiment. The mesh
refinement in this study follows the suggestion in [10]. The local weld
notch geometry entails a rounded radius of 1 mm, recommended and
adopted by the International Institute of Welding [46] and other research studies [19,47–53]. The weld size follows the AWS complete
joint penetration (CJP) welding details [54]. The local dihedral angle
(ψ) describes the angle formed by tangents to the brace and the chord
outer surface perpendicular to the weld line, as shown in Fig. 10. The
(6a)
2
2
3
2
2
2
2
2
2
3(Δγxy
) + 6(Δεxy
)
+ Δγxz
+ Δγyz
+ Δεxz
+ Δε yz
(6b)
4
Table 5
Parameters of Basquin-Coffin-Manson equation for S355 steel in Eq. (5).
9
E (MPa)
σ ′f (MPa)
b
ε′f
c
209,000
1354
−0.0906
0.3159
−0.65
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 10. (a) Geometric details of the brace-to-chord intersection; (b) typical CJP welds.
and strains under the remote nominal stress range Δσnom and the
nominal strain range Δεnom . For linear-elastic materials, the elastic
concentration factor becomes,
KNe =
Δσije Δεije
(8)
Δσnom Δεnom
where Δσije and Δεije refer to the range of local stresses and strains with a
linear-elastic material, respectively.
The nominal stress and strain range derive from the applied load,
(9)
Δεnom =
Δσnom
E
(10)
KN = KNe for Δσnom Δεnom ⩽
to the material strain hardening. This implies that the stress range may
not be an effective fatigue driving force indicator to differentiate the
low-cycle fatigue life. In contrast, the equivalent strain range and local
energy indicator (Δσij Δεij ) show apparently noticeable differences under
both the small and large loadings. For both two sets of X-joints, all three
indicators drop rapidly when the location angle ρ exceeds 60°. The
variation of the local energy indicator is more significant in comparison
with the equivalent stress and strain range along the weld line, especially for XJ5 and XJ6. For both sets of X-joints, the maximum local
energy indicator occurs at ρ = 45o , which corresponds to the crack initiation locations observed in the experiment. The experimental procedure monitors the crack initiation location through three different
methods: (1) the physical visualization; (2) the ACPD approach; and (3)
the Ultrasonic phased approach. Based on the crack size measurement
at every 500 to 1500 cycles, the crack usually initiates at 10° and 45°
according to the physical visualization and UPA. The crack size at the
initiation has a small depth of 0.5 mm.
This study adopts the local energy indicator to assess the fatigue life.
In this study, the concentration factor of the local energy indicator
follows,
Wc
KNe
(11a)
1
Δσnom Δεnom ⎞ ⎞ ⎫ e
⎧
KN = 1 − (1 − F ) ⎜⎛1 − exp ⎜⎛ e −
·KN for Δσnom
⎟⎟
⎨
K
Wc
N
⎝
⎠⎠⎬
⎝
⎩
⎭
W
Δεnom ⩾ ec
KN
(11b)
where F defines the ratio of the converging value of KN over the elastic
energy concentration factor KNe . The value of Wc in Eq. (11b) represents
the critical energy at the yielding of the material (Wc = 0.89σy2 E ) under
the nominal stress ratio of 0.1. The above equations estimate directly
the concentration factor KN at any nominal load. Determination of F
requires a large plastic deformation in the weld toe, which ensures that
the energy-based concentration factor locates at the third phase in
Fig. 14a. (the F ·KNe dominated phase). In this study, the F value corresponds to a large nominal stress of 20 MPa applied on the CHS Xjoint, which makes the value of Δσnom Δεnom Wc large enough to obtain a
convergence value of F .
Table 6 illustrates the KNe and F for the specimens XJ1-XJ3 and XJ5XJ6. Relying on Eq. (11), Fig. 14b compares the local energy indicator
of the X-joints by the IMNR and numerical results. A good agreement
between the analytical values and finite element values confirms the
applicability of the IMNR in the low-cycle fatigue of CHS X-joints.
Δσij Δεij
Δσnom Δεnom
0.25Δσpressure·Achord ·d1
ΔMy
=
Ibrace
Ibrace
where the σpressure represents the pressure applied on the chord end by
the MTS hydraulic machine; Achord denotes the area of the chord crosssection and Ibrace indicates the moment of inertia for the brace.
Feng and Qian [19] have described the evolution of the energy
concentration factor KN by an improved modified Neuber’s rule
(IMNR), which consists of three different phases (KNe dominated zone,
transition phase and the F ·KNe dominated zone). Fig. 14a shows the
curve of the IMNR, which follows,
Fig. 11. A one-quarter finite element model of CHS X-joints.
KN =
Δσnom =
(7)
where Δσij and Δεij denote the range of the local elastic-plastic stresses
10
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 12. Variations in: (a) equivalent stress range; (b) equivalent strain range; and (c) local energy range along the weld line for XJ1-XJ3.
5. Fatigue life assessment
cycles when the crack depth reaches 70% of the chord thickness. The
black curve represents the S-N curve based on the nominal stress range
recommended by the AWS DT [54], which is adopted for comparison
purpose as the global response of the X-joint exhibits a predominantly
linear behavior. The fatigue life after the flip-over decreases substantially in comparison with that before the flip-over, since the fatigue
5.1. S-N curve
Fig. 15a presents the fatigue life of the five CHS X-joints based on
the nominal stress range. All the fatigue life is quantified by the fatigue
Fig. 13. Variations in: (a) equivalent stress range; (b) equivalent strain range; and (c) local energy range along the weld line for XJ5 and XJ6.
11
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 14. (a) Illustration of IMNR; and (b) comparison of the local energy estimated by IMNR and by FE.
Table 6
IMNR Parameters in Eq. (11) for the two sets of X-joints.
Wc
0.4005
Table 7
Four damage-related material parameters in Eq. (14) for S355 steel.
XJ1 – XJ 3
XJ5 – XJ6
K Ne
F
K Ne
F
32.16
0.68
34.52
0.7
c1
c2
Lc
c4
14364.15
−1.58
1240.6745
0.542
fatigue initiation life over the total fatigue life for all test data before
the flip-over. The horizontal axis represents the local energy indicator
values. As shown in Fig. 15c, the fatigue initiation life decreases with
increasing local energy indicator. The fatigue initiation consumes less
than 60% of the total fatigue life, which is consistent with the conclusions from [19] for the low-cycle fatigue of cruciform welded plate
joints.
crack occurs in both the compression and tension sides of the joint after
the flip-over. In addition, the plastically deformed material near the
weld toe in compressively loaded zones before the flip-over accumulates a tensile residual stress field [7]. After the flip-over, the compressive zone experiences tension, and the presence of a tensile residual
stress leads to the early fatigue damage. All the fatigue data locate
above the normal stress S-N curve in AWS [54]. Fig. 15b plots the fatigue data based on the local energy indicator Δσij Δεij .
Fig. 15c presents the fatigue initiation life and the percentage of the
5.2. Numerical estimation based on CDM
Darveaux [56] and Lau et al. [57] have described the fatigue life
Fig. 15. Comparison of the S-N type assessment based on: (a) nominal stress range; and (b) local energy indicator; and (c) fatigue initiation life and its percentage in
total fatigue life X-joint specimens.
12
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
Fig. 16. (a)(b) Final failure estimated by CDM analyses; (c) the test fatigue life versus the CDM prediction; (d) comparison of the crack depth evolution with the nondestructive measurements.
N0 = c1 Δw0 c2
(12a)
dD
c Δw c4
= 3
dN
L
(12b)
(13)
Δw = Δw0
using the damage indicator relating to the hysteresis energy under
cyclic loadings, which is implemented in ABAQUS [42] for the lowcycle fatigue analysis. This offers an approach to compute the total
fatigue life of the CHS X-joints from the finite element analysis which
relies on the continuum damage mechanics theory. ABAQUS [42]
adopts the direct cyclic analysis to alleviate the computational cost of
performing a cycle by cycle analysis, which constructs a displacement
function using the Fourier series during the load cycle. By solving for
corrections to the displacement Fourier coefficients, the updated displacement solution is imported into the next iteration to obtain the
displacements at each instance until the convergence is achieved.
The initiation of fatigue crack depends on the initial hysteresis energy per cycle. After the damage occurs, the accumulated plastic hysteresis energy per cycle defines the rate of the damage evolution per
cycle [42]. The expressions of the damage initiation life and damage
evolution rate follow,
Following the calibration procedure recommended by Feng and
Qian [19,33], the final fatigue life of the standard coupon specimens
derives from,
Nf = c1 Δw0 c2 +
Lc
L
,
= Lc
Δw0 c4 c3
(14)
Table 7 lists the four damage parameters in Eq. (14) based on the
standard coupon specimens. This study implements the damage mechanics model using these calibrated parameters to estimate the fatigue
life of CHS X-joints. The characteristic length L in the thickness direction near the welds of the X-joint model in Fig. 11 equals 4.51 mm for
XJ1-XJ3 and 4.44 mm for XJ5-XJ6. Fig. 16a shows a typical fatigue
failure mode. The crack grows in the thickness direction in the tensions
side of the chord. The compression side of the chord experiences severe
plastic deformations, as shown in Fig. 16b.
Fig. 16c compares the fatigue life measured in the tests and those
estimated from the continuum damage mechanics simulation. The total
fatigue life of CHS X-joints before flip-over demonstrates close agreement with the numerical prediction. The initial condition for the X-joint
after flip-over, including the residual stress and the presence of cracks
in the compression side, causes large discrepancies in the predicted and
measured fatigue lives. Fig. 16d compares the evolution of the crack
depth at the deepest crack front by the experimental measurement
(UPA and ACPD), and the numerical estimation for XJ6 before flip-over.
The numerically estimated crack propagation agrees reasonably well
with the UPA measurement when the crack size remains small, but
increases beyond the experimental measurement at a large crack size.
where N0 denotes the number of cycles to initiate the damage, Δw0
represents the cyclic hysteresis energy in the first cycle, D measures the
damage in the element, which quantifies the degree of stiffness loss,
ci (i = 1, 2, 3, 4) are the four material damage constants, defining the
initiation and evolution rate of damage, L is the characteristic length
associated with the distance between adjacent integration points in the
finite element model, Δw denotes the hysteresis energy in the subsequent cycles. For the coupon specimen shown in Fig. 9b, the stress
amplitude remains approximately constant during the entire fatigue
test with a constant strain amplitude. Therefore,
13
International Journal of Fatigue 137 (2020) 105670
L. Feng and X. Qian
6. Summary and conclusions
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This study investigates the low-cycle fatigue behavior of CHS Xjoints under the brace in-plane bending through experimental tests and
numerical simulation. The nondestructive techniques (ACPD and UPA)
help to measure the fatigue crack initiation and propagation, and allow
a comparison on their accuracy. The material testing on a series of
standard coupon specimens under low-cycle actions enables calibration
of the isotropic and kinematic hardening material properties and the
damage-related parameters. This paper presents two different approaches in estimating the fatigue life, the rapid S-N type estimation
using the energy indicator based on the IMNR and the continuum damage mechanics model using the parameters calibrated from coupon
specimens. The above results support the following conclusions.
(1) Before the flip-over, the grinding treatment delays the fatigue crack
initiation, and therefore, prolongs the total fatigue life. After the
flip-over, the fatigue initiates almost immediately due to the tensile
residual stress induced by the large plastic deformation accumulated before the flip-over and the material damage caused by severe
plastic deformations under cyclic compressive actions. The X-joints
after flip-over demonstrates accelerated fatigue crack propagation.
This demonstrates the adverse effects on the fatigue life caused by
prior plastic deformations generated by cyclic compressive actions.
Such loading history effect in the low-cycle fatigue behavior for
welded CHS joints requires further investigation.
(2) ACPD and UPA show differences in the measured fatigue crack
initiation and crack size. The difference is significant when the
crack size is small and gradually decreases with the increasing fatigue crack size. At small crack sizes, UPA demonstrates enhanced
accuracy as ACPD entails a strong skin effect and tends to bypass
the small cracks.
(3) Relying on the previously proposed IMNR, this study calculates the
KNe and F to estimate the local energy indicator analytically, which
agrees closely with the numerically computed local energy for CHS
X-joints. This approach allows a rapid estimation of the fatigue life,
once sufficient fatigue test data become available to develop an
energy-based S-N curve.
(4) Based on the damage parameters calibrated from coupon specimens, the calculated fatigue life agrees reasonably well with the
experimentally measured fatigue life for CHS X-joints before flipover. Without considering the prior loading effect, the CDM based
numerical approach overestimate the fatigue life after the flip-over.
The crack depth evolution estimated from the CDM simulation
presents a reasonable agreement with the UPA measurement when
the crack size is small.
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.
Acknowledgment
The authors would like to acknowledge the financial contribution
provided by the Singapore Maritime Institute (SMI) (SMI-2015-OF-06),
as well as the supports provided by American Bureau of Shipping
(Singapore), Eddyfi Europe SAS, and Ashtead Technology Pte Ltd. (UK).
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