INTRODUCTION
As important structural members offshore, marine risers that connect the surface
floating facilities and the subsea wells are generally made of heavy duty girth
welded tubulars. These girth welds are often the key parts and the limiting
factor in marine risers design. As fracture critical components, marine risers
require high performance reliability due to the unacceptable consequences caused
by any potential failure. For this reason, fatigue test on marine risers is
important to examine their performance.
Over the past years, several fatigue test methods have been used to evaluate
the fatigue life of marine risers in different ways. Axial tension fatigue test
can directly find out the stress according to the sections and axial load of
specimen to validate the performance of material, but cannot work for the full
scale specimen (McMaster et al., 2007). The process
of the fourpoint bending test is simple, but lasts a long testing time with
low efficiency (Van Wittenberghe et al., 2011).
The Resonant Translational Bending (RTB) technique was proven as a quick and
efficient method of fatigue performance evaluation for girth welds and mechanical
joints in tubing pipe, casing pipe and drilling pipe (Kerkhof
et al., 1990; Varma et al., 1997)
and is going to be a trend to test marine risers for economical and efficiency
reasons (Buitrago et al., 2003). But the previous
works still have left unresolved issues: (1) These works mainly focused on the
fatigue test results analysis (Hasegawa et al.,
2002; Varma et al., 2002). The principle
and process of RTB test was barely mentioned; (2) A finite element model was
generally used to predict and guide the test (Bertini et
al., 2008), did not study the relationship between the relative parameters
in different test conditions, such as the effect of end mass on the natural
frequencies of testing specimen.
In this study, a fatigue test was performed on a marine riser with flange joint.
The fatigue test system which produces a resonant translational bending consisting
of a test setup part and a data measurement part was described. A mathematical
model was built to control and obtain the required natural frequencies and the
response of testing specimen to guide towards the test execution. Furthermore,
this article has a greater promotional value for design of fatigue test system
and test of other marine risers.
FATIGUE TEST SYSTEM DESIGN
The fatigue test system was based on the principle of resonant translational
bending.

Fig. 1: 
Layout of fatigue test setup of marine riser 

Fig. 2: 
Components of test setup part 
In order to produce enough bending load to meet the test condition, a testing
specimen of marine riser consists of two long pipe bodies, one midjoint (in
this study is the flange joint) and welds. During test, the specimen was applied
an excitation with a frequency close to its natural frequency, which causes
the specimen come into resonance. As shown in Fig. 1, the
fatigue test system consists of a setup part and a data measurement part.
Test setup: An overview of the test setup is shown in Fig.
2. The specimen is supported by two supports. The drive housing is clamped
to one end of the specimen, in which a rotating eccentric mass connected to
the variable speed electromotor driven via a cardan shaft. The electromotor
is used to generate the frequency of excitation. It drives the eccentric mass
to load the specimen near its first natural frequency by adjusting its RPM.
Eccentricity of the eccentric mass plays an important role to create and control
excitation force. The dead weight housing is clamped to the other end of the
specimen to balance the assembly. Both the housings can be changed to be applicable
for the marine risers with different diameters and wall thicknesses. As the
eccentric mass is replaceable, some eccentric masses with different weight and
eccentricity were prefabricated to meet a wide range of test requirement.
Data measurement: The data measurement system consists of strain gages,
Industrial Personal Computer (IPC), Uninterrupted Power Supply (UPS), data acquisition
software, frequency inverter, weight sensor, pressure switch, as shown in Fig.
1. Axial strain gages on the outer surface of the pipe body are used to
monitor the bending strains during test. Both bending strains and number of
cycles are monitored and recorded by the data acquisition software which could
obtain 16 groups data at the same time. Each minute the data acquisition software
records the average maximum and minimum strains for each strain gage along with
the number of cycles. The weight sensor is placed below the flanges to stop
the test if a bolt fails and drops out of the flange. When the internal pressure
drops due to a throughwall crack, the pressure switch installed on the dead
weight housing will stop the test. Frequency inverter is used to provide power
to electromotor and control its output frequencies to meet test condition. As
long last the test, even one day, a UPS is used to provide emergency power to
ensure continuity of the test.
Specimen for modeling and test: In order to explain the principle and
process of test conveniently, a specimen of marine riser with flange joint will
be modeled and tested. As Fig. 3 shows, the total length of
the specimen is about 8229.6 mm. Its Outer Diameter (OD) and Internal Diameter
(ID) are 533.4 and 501.6 mm, respectively. Its weight is 2590 kg. The specimen
contains a made up flange joint in center, two pipetopipe welds (W1, W4) and
two flangetoriser pipe welds (W2, W3). The purpose of the test is to evaluate
the fatigue performance of welds undergo the same load cycle during one excitation
cycle. Because the weld and wall mismatch will cause local distorted strains,
the gages should be placed far enough from the welds to obtain pipe body stresses.
Four sections (S1 to S4) are located on either side of the welds. Sixteen strain
gages are installed on circumferential locations 0°, 90°, 180° and
270° around the pipe body at each section. The test will be performed at
room temperature within about 30 Hz at the specified stress range resulting
in 1.72.5 million cycles a day.
MATHEMATICAL MODEL
Fundamental equations: In RTB test, the specimen is applied an excitation
by a typical electromotor with about 20 to 40 Hz range of frequency. Because
of its long size, the natural frequency (generally is above 40 Hz) of the specimen
is difficult to get within the frequency range of electromotor. The natural
frequency mainly depends on bending stiffness and mass. Therefore, the natural
frequencies of specimen can be lowered by attaching proper masses at both ends
of the pipe body. In this work, the drive housing and dead weight housing were
used as end mass.
The specimen was assumed to be a model of free massive beam (with total mass
m_{s} uniformly distributed on the length L, two masses m_{f}
at both ends of the riser pipe, l_{f}, the distance from the centers
(x direction) of end masses to the riser pipe ends) to calculate its natural
frequencies and natural modes, as shown in Fig. 4.
The dynamic behavior of the specimen can be expressed using a partial differential
equation:
where, u(x,t) is the lateral displacement of the specimen, E is Young’s
modulus, I is area moment of inertia and m_{s}/L is the mass per meter
of the specimen.

Fig. 3: 
Location of strain gages and welds 

Fig. 4: 
Dynamic model of a free massive beam 
By solving this partial differential equation, the general solution of Eq.
1 can be written as Eq. 2:
where, X(x) can be given by Eq. 3 with arbitrary constants
A_{1}, A_{2}, A_{3}, A_{4} and λ which
is written as Eq. 4:
In order to obtain the arbitrary constants, boundary conditions at of the specimen
ends are substituted into the partial differential equation, as shown from Eq.
57:
where, J_{f} is the mass moment of inertia of end mass, F_{e}(t)
is an excitation force caused by spinning the eccentric mass me in the drive
housing. It will induce rotating bending with high amplification near the resonance
and can be written as Eq. 6, where r_{e} is the eccentricity
of rotating mass and ω_{e} is the angular excitation frequency.

Fig. 5: 
The first three nature modes of the free massive beam 
In the boundary conditions, EI∂^{2}u/∂x^{2}, m_{f}l_{f}∂^{2}u/∂t^{2}
and (J_{f}+m_{f}l_{f})∂^{3}u/∂x∂t^{2}
are the bending moments of the specimen, linear acceleration motion of m_{f}
and angular acceleration motion of m_{f }respectively, EI∂^{3}u/∂x^{3},
m_{f}∂^{2}u/∂x^{2}, m_{f}l_{f}∂^{3}u/∂x∂t^{2}
are the corresponding shear forces.
Control of natural frequency: When F_{e}(t) = 0, the angular
natural frequencies ω_{n} and natural modes are found. In Fig.
5, the first three nature frequencies and corresponding nature modes of
the specimen are calculated by Matlab^{TM} ver. 7.01, where f = ω/2π,
E = 201Gpa, m_{s} = 2590 kg, m_{f} =210 kg, l_{f} =
60.25 mm.
When F_{e}(t) = 0 and m_{f} = 0, the first natural frequencies
and natural modes of the specimen without end mess can also be found to analyze
the effect of end mess. As shown in Fig. 6, adding end mass
is an effective way to affect the first natural model of the specimen. With
210 kg end mess, the maximum lateral displacement of the specimen increased
from 0.72 to 0.85 mm and the first natural frequency of the specimen is controlled
from 40.52 Hz down to 29.75 Hz, which meets the test condition of about 30 Hz.
This indicates that the attached end masses can improve the excitation efficiency
and the reduction of the natural frequencies of the specimen produced by the
end masses is particularly remarkable.
As the specimen is assumed to a free massive beam, no constraint condition
is applied. The specimen will be simply supported by two supports.

Fig. 6: 
Effect of end mess on the first natural mode of the specimen 
In Fig. 6, it is found that even the weight of end mass is
increased from 0 kg up to 263 kg continuously, the location of the two null
displacement points will never change. Therefore, the supports will be located
at the two points where lateral displacement is null.
Control of natural frequency: When the angular excitation frequency
ω_{e} (close to ω_{n}), m_{e} and r_{e}
are chosen, lateral displacement of the specimen can be calculated by Eq.
(8):
where, ω_{e} is the rotational speed of eccentric masses generated
by electromotor and could be expressed by excitation frequency f_{e}.
The bending moment M(x) and the resulting stress σ_{B}(x) at the
outer surface of the specimen can be written by Eq. 9 and
10:
In the mathematical model, no damping effect is considered. Based on this assumption,
the excitation frequency should be equal to the natural frequency of the specimen.
However, when the excitation frequency moves up on the resonance status, the
damping effect on the test results becomes more significant and results in a
strongly amplified in practice. Therefore, the excitation frequency should be
chosen within an overlap area of subresonant area of the specimen and working
frequency range of the electromotor. In Fig. 7, the maximum
theoretical bending stress of the specimen (at the midjoint) is calculated
by Eq. 10 and is plotted versus the first excitation frequency.

Fig. 7: 
Excitation frequency range induced by resonance 
For the specimen with 210 kg end mass, the excitation frequency range is chosen
between 20 and 30 Hz. As the natural frequency of specimen is 29.75 Hz, the
excitation frequency should be closed to it. If the stress amplitude requires
a value of 120 MPa, the excitation frequency is chosen with about 27.5 Hz.
TEST RESULTS AND ANALYSIS
Verification of mechanical model: In Fig. 8, the lateral
displacement and bending moment of the excitated specimen by electromotor are
predicted by Eq. 8 and 9, respectively.
With m_{e} = 35.8 kg, r_{e} = 33 mm, the lateral displacement
has an amplitude of 0.38 mm and the bending moment reaches a value of 436 kNm
at the flange joint. The location x = 1830 mm and x = 6360 mm are the point
where the specimen should be supported in the test system.
In the RTB test, a 28.2 Hz excitation frequency that was about 0.95 times the
natural frequency of the specimen generated by electromotor. Figure
9 shows the theoretical and test results of bending stress at the OD of
specimen, on which the stress distribution along the specimen was calculated
by Eq. 10. Test results are the mean value at section S1,
S2, S3 and S4 (with 4 strain gages at one section). The bending stress developed
in the specimen are close to zero at the ends and reaches about 136.5 MPa at
the flange joint. The maximum theoretical stress of welds is 124.1 MPa at W3.
Among the four sections, the maximum error is +4.93% at S4. From S1 to S4, the
relative error increases continuously, as shown in Table 1.
The reason could be that (1) The specimen is simply supported by two couples
of rubber wheels (Fig. 2). The friction at the supporting
wheels will affect the excitation from electromotor.

Fig. 8: 
Displacement and bending moment with excitation 

Fig. 9: 
The theoretical and test bending stress at the OD of specimen 
When the excitation frequency moves up to the natural frequency of the specimen,
this influence will be strongly amplified and (2) The excitation is transmitted
from S1 to S4, which accumulates the relative error of the four sections. The
theoretical results are accurate and agree very well with the test results.
It can be concluded that theoretical results at the four sections were verified
through strain gauges measurements during tests and the mathematical model could
satisfactorily calculate the behaviors of the specimen applied an excitation.
In Fig. 10, the stress amplitude of the sixteen strain gauge
locations over time is shown. The vibration direction of strain gauge location
is in zdirection on 0° and 180°
and is in ydirection on 90° and
270°. It can be seen that the
ystress and zstress have the almost equal amplitude at the same section, but
that the ystress has a phase delay of 90°.
This means that the standing wave of excitation transmitted in a clockwise circular
direction.
Fatigue curve for marine risers: The fatigue design is based on the
use of SN curves, which are obtained from fatigue tests.
Table 1: 
Comparison between theoretical and test results 

*Relative error = (Theoretical resultstest results)/test
resultsx100% 
Fatigue curves specifically for marine risers have not yet been adopted. However,
fatigue curves published by BSI (1993) and DET NORSKE
VERITAS (DNV, 1984) for offshore structures have been
used to design and assess the fatigue performance of marine risers. Single sided
girth welds such as those used for marine riser fabrication are generally classified
as F2 for the purpose of fatigue analysis. A survey of a wide range of test
results on single sided girth welds (Dickerson, 1997)
shows that in almost all cases the performance of the welds exceeds that of
an E fatigue detail classification (BSI, 1993; DNV,
1984). This gives a fatigue resistance some 2 to 3 times that of an F2 curve.
The weld test specimens considered are representative of welds used for pipeline
construction. Riser welds are expected to be as good if not of better quality
than those of the weld test specimens reviewed. Based on this review, it is
reasonable that as an Eclass fatigue detail is used for the fatigue life prediction
of marine riser coupling welds and an F2class fatigue detail is used for the
fatigue test.
Test results: The specimen was tested twice. The W3 weld cracked after
316,306 cycles. The crack ground out with the length of about 52 mm until it
was no longer visible, as shown in Fig. 11. After patching
the crack by repairing welding, the specimen was loaded back into the test system
and cycled at the original stress range until it cracked at the repair weld
and leaked after a total of 442,637 cycles (including the initial 316,306 cycles).
In Fig. 12, the test results are presented and superimposed
on the design fatigue curves of E and F2 in BS7608 of BSI. The fatigue results
of the cracked W3 weld are plotted in the SN curve.

Fig. 10: 
Stress amplitude of strain gauge locations 

Fig. 11: 
Crack on W3 weld of the specimen 

Fig. 12: 
Marine riser fatigue test results 
The pipetoflange weld W3 did not pass the E design curve but passed the target
life of F2 design curve, indicating that the cracked weld is of generally acceptable
quality and meets the design requirements, but its fatigue life could not satisfy
the test requirements.
CONCLUSION
A RTB test system consists of the setup part and data measurement part was
established. A mathematical model was built to guide test. End mass is an important
effect factor to control natural frequencies of the specimen to meet the test
condition. Based on this test system, the fatigue performance of a specimen
with flange joint was evaluated. Both theoretical and test results could obtain
the bending stress distribution characteristics on the outer surface of the
specimen with a good agreement. The specimen cracked at pipetoflange weld
W3 with about 124.1 Mpa stress. This study is also bringing promotional values
for other marine risers with different tubulars and connections.