The literature review shows that accurate closed form solutions may not
be found to analyse the ratchetting behavior of the pressurized pipes
under cyclic bending loading which can be caused by seismic loads. However,
approximate solutions have been developed by Mahbadi and Eslami (2006),
Abdel-Karim (2005), Chen et al. (2005), Johansson et al.,
(2005), Chen and Jiao (2004), Bari and Hassan (2000, 2001, 2002), Abdel-Karim
and Ohno (2000), Chaboche (1991, 1994) and Beaney (1990, 1991) which can
be used to calculated the induced incremental plastic strains caused by
ratchetting. Experimental works to study the ratchetting of the straight
pipes have also been carried out by the EPRI (Ranganath et al.,
1989; English, 1988).
The kinematic hardening theory of plasticity based on the Armstrong-Frederick
model is used by Mahbadi and Eslami (2006) to evaluate the cyclic loading
behavior of thick cylindrical vessels. The results found from their numerical
analysis shows that the when the stress range is more than twice the yield
stress, kinematic hardening theory with the Armstrong-Frederick model
excluding creep, predicts ratchetting for load controlled cyclic loading
while shakedown is predicted for deformation controlled cyclic loading.
Kinematic hardening theory with the Prager model predict shakedown for
load and deformation controlled cyclic loading of thick vessels (Eslami
and Mahbadi, 2001).
The work reported here is based on a series of tests conducted using
specimens, having 8≤Dm/t≤28. There is notable dearth
of information available which seeks to compare experimental data such
as that reported in references (Moreton et al., 1994, 1996, 1998a,
b), with finite element computations. This is surprising since it is well
known that analytical solutions (such as those presented in references
(Beaney, 1990) differ with experimental data by several orders of magnitude.
It is of paramount importance to establish reliable theoretical methods
for predicting ratchetting rates and the use of Finite Element (FE) codes
would seem to be a logical way forward. Therefore, in this study a finite
element analysis with the nonlinear isotropic/kinematic (combined) hardening
model is used to evaluate ratchetting behavior of plain stainless steel
pressurized cylinders subjected to dynamic bending moment.
MATERIALS AND METHODS
In this study, a finite element code, ABAQUS, was used to study the ratchetting
of plain stainless steel pressurized pipes subjected to cyclic bending
Firstly, a series of tests has been undertaken subjecting pressurized
pipe specimens to rising amplitude dynamic (5 Hz, the resonant frequency)
bending moments.Secondly, by conducting a series of finite element runs
based on nonlinear isotropic/kinematic hardening model using the ABAQUS,
the experimental tests have been modeled and ratchetting data obtained.
The two sets of results are compared with each other and with the analytical
models of references (Beaney, 1990).
The isotropic and kinematic hardening models are used to simulate the
inelastic behavior of materials that are subjected to cyclic loading.
The use of plasticity material models with isotropic type hardening is
generally not recommended since they will continue to harden during cyclic
loading. The isotropic hardening model always predicts shakedown behavior,
if creep is not considered. Kinematic hardening plasticity models are
proposed to model the inelastic behavior of materials that are subjected
to repeated loading. For example, The Armstrong-Frederick kinematic hardening
model is suggested for the nonlinear strain hardening materials. The results
of these models are discussed for structures under various types of cyclic
loads in references (Mahbadi and Eslami, 2006; Eslami and Mahbadi, 2001;
A kinematic hardening model or a (combined) nonlinear isotropic/kinematic
hardening model may be used to simulate the behavior of materials that
are subjected to cyclic loading. The evolution law in these models consists
of a kinematic hardening component which describes the translation of
the yield surface in stress space. An isotropic component which describes
the change of the elastic range is added for the nonlinear isotropic/kinematic
Isotropic hardening model: The isotropic hardening model which
describes the change of the elastic range is discussed here. Isotropic
hardening means that the yield surface changes size uniformly in all directions
such that the yield stress increases in all stress directions as plastic
According to the isotropic hardening model, the yield function is generally
expressed as (Lubliner, 1990):
where, ξi is the internal variable tensor. If we assume
that the material constant are independent of temperature, Eq.
1 may be rewritten in the form:
where, k is a non-negative constant being function of the internal variable
tensor ξi. The mathematical meaning of Eq.
2 is that the center of yield surface in the stress space will not
change as the result of loading or unloading, but its distance from the
coordinate origin change as the internal variable ξi change.
The flow rule associated with the yield function (2) has the general
form (Mendelson, 1960)
where, dλ is a non-negative constant and function of the internal
variable ξi. The constant dλ is defined in terms
of the effective stress and the increment of effective plastic strain.
For Von-Mises yield criterion Eq. 3 is (Mendelson,
where, Sij is the deviator stress tensor, dεP
is the increment of effective plastic strain and σe is
the effective stress defined as (Mendelson, 1960):
Kinematic hardening model: The classical linear kinematic hardening
rule and different nonlinear kinematic hardening models are available
for the plastic analysis of structures. The nonlinear kinematic hardening
model was first proposed by Armstrong-Frederick (1966). Nonlinearities
are given as a recall term in the Prager rule. So that the transformation
of yield surface in the stress space is different during loading and unloading.
This is done by assuming different hardening modulus in loading and unloading
The yield function for rate independent plasticity is expressed as:
f (σij, T, ξk)
= 0i,j = 1,2,3k = 1,2,3...,m
with the evolution equations of
T, ξk) = 0i,j = 1,2,3k = 1,2,3
where, ξk is the internal variable tensor. Assuming that
the material constants are independent of temperature, Eq.
8 for the Von-Mises yield criterion may be rewritten in the form (Mahbadi
and Eslami, 2006)
where σij and αij are the stress and
back stress tensors and sij and aij are the stress
and back stress deviatoric tensors in the stress space. If the plastic
and the back stress tensor αij are assumed as
the internal variables, the evolution equations are:
(I) Flow rule:
(II) Armstrong-Frederick kinematic hardening model:
where, C and γ are two material constants in the Armstrong-Frederick
kinematic hardening model and they will be found from the uniaxial strain
controlled stable hysteresis curve. Assuming that during the loading dεP
is positive, since dax = dσx, Eq.
12 for the uniaxial load changes to the following form:
Solution of differential Eq. 13 for αx
where, c0 is the constant of integration. The trace
of αx begins from -C/γ at .
Which corresponds to the starting point of the strain controlled stable
hysteresis loop (Armstrong and Frederick, 1966). Using the condition,
the constant c0 is found and Eq. 14 for
the positive (tensile
plastic strain increment) may be rewritten as:
If during loading the value of
is negative (compressive plastic strain increment), then Eq.
13 and 15 changes to:
However, this model provides an anisotropy effect in tension/compression
curve due to the nonlinearity of the trace of αx.
Nonlinear isotropic/kinematic (combined) hardening model: In the
kinematic hardening models the center of the yield surface moves in stress
space due to the kinematic hardening component. In addition, when the
nonlinear isotropic/kinematic hardening model is used, the yield surface
range may expand due to the isotropic component. These features allow
modeling of inelastic deformation in metals that are subjected to cycles
of load or temperature, resulting in significant inelastic deformation
and, possibly, low-cycle fatigue failure.
The evolution law of this model consists of two components: a nonlinear
kinematic hardening component, which describes the translation of the
yield surface in stress space through the backstress α and an isotropic
hardening component, which describes the change of the equivalent stress
defining the size of the yield surface, as σ0 a function
of plastic deformation.
The kinematic hardening component is defined to be an additive combination
of a purely kinematic term (linear Ziegler hardening law) and a relaxation
term (the recall term), which introduces the nonlinearity. When temperature
and field variable dependencies are omitted, the hardening law is:
where, C and γ are material parameters that must be calibrated from
cyclic test data. C is the initial kinematic hardening modulus and γ
determines the rate at which the kinematic hardening modulus decreases
with increasing plastic deformation. The kinematic hardening law can be
separated into a deviatoric part and a hydrostatic part; only the deviatoric
part has an effect on the material behavior. When C and γ are zero,
the model reduces to an isotropic hardening model. When γ is zero,
the linear Ziegler hardening law is recovered.
The isotropic hardening behavior of the model defines the evolution of
the yield surface size, σ0 as a function of the equivalent
plastic strain .
This evolution can be introduced by specifying σ0 as a
by using the simple exponential law
where, σ|0 is the yield stress at zero plastic strain
and Q∞ and b are material parameters. Q∞
is the maximum change in the size of the yield surface and b defines the
rate at which the size of the yield surface changes as plastic straining
develops. When the equivalent stress defining the size of the yield surface
remains constant σ0 = σ|0, the model reduces
to a nonlinear kinematic hardening model.
REVIEW OF BEANEY METHOD
Beaney undertook experimental work on both plain pipes and a range of
pipework components. Only his work on plain pipes will be discussed here.
His test pipes (austenitic and ferritic) having diameters between 25.4
and 89 mm were shaken at each end by hydraulic actuators at a fundamental
natural frequency of about 5 Hz. Most of the pipes were pressurized to
their nominal design pressure of about 2/3 of the yield. The pipes were
strain gauged and data acquisition and processing system provided. Beaney
(1991) found that the most notable observation from his tests was that
the level of resonant vibration was self-limiting. As the input amplitude
increased, yielding started to occur in the pipe wall and this yielding
absorbed energy and hence limited the bending moment that the pipe could
sustain. It was noted that unpressurized pipework did not fail even when
excessively high input accelerations were applied at the supports. The
predicted level of input accelerations during an earthquake is typically
about 0.25 g and accelerations as high as 5 g were used in these tests.
The pressurized pipes exhibited hoop ratchetting both in the plain pipe
and at areas of local stress concentration. In some tests failure occurred
in as few as 40 cycles when the hoop strains which developed exhausted
the ductility of the material (Fig. 1).
In 1986 Beaney attempted to quantify the ratchet strain by modifying
the Edmunds and Beer relationship by using the more accurate Mises yield
criterion but retaining the assumption of elastic/perfectly plastic material
||Von mises yield surface
Edmunds and Beer gave the ratchet strain in the hoop direction as:
where, σθ is the hoop stress (due to pressure), εφ
is the applied dynamic strain amplitude, σy is the yield
stress and E is the Young`s modulus.
Clearly one problem in using this relationship is in estimating the applied
dynamic strain amplitude. Beaney (1990) suggested that this equation could
be written as:
where, σφ is the axial stress amplitude. For,
Eq. 21 reduces to:
Beaney commented that it was important to use the dynamic yield stress.
However, for strain-hardening materials he found the static 0.2% proof
stress a good substitute. For ferritic steels a good estimate of the dynamic
yield stress could be made by noting that as soon as yielding occurs so
does ratchetting. By observing the axial strain amplitude at the onset
of ratchetting the maximum elastic stress σ could be calculated and
thus the yield stress σy calculated, for various pipe
pressures, using the expression he derived earlier (Moreton et al.,
1994). That is:
where, σθ is the hoop stress (due to pressure),
σy is the dynamic yield stress and σ is the maximum
elastic axial stress.
REVIEW OF EXPERIMENTAL ARRANGEMENT
The experimental arrangements used for testing plain cylinders and other
pressurized piping components have been detailed previously (Yahiaoui
et al., 1992). It is sufficient to give a brief outline of the
Cylindrical specimens were machined from stainless steel bar stock to
the form illustrated in Fig. 2. In order to minimize
any residual stresses, these specimens were machined 1.5 mm oversize on
all dimensions. These blanks were stress relived at 650°C (1 h + furnace
cool). The bores of all specimens were reamed to 30 mm diameter and the
outside surface profiled using a CNC (computer numeric controlled) lathe
while holding the specimen on a mandrel.
||Specimen 8≤Dm/t≤28 ratios
||The test specimen-All dimensions in mm
Six holes were provided in each
end flange using the powered axial tooling of the CNC lathe. A total of
six specimens were machined in this way and the dimension A was varied
to provide the Dm/t ratios detailed in Table
Strain gauges were bonded to the top and bottom surfaces using M-Bond
AE 10 curing for 4 h at 30°C and 2 h at 100°C. Two-element, 90
rosettes were used to provide strain measurement in the hoop and axial
directions. The gauge type selected was EA-06-125TM-120 from Micro measurements.
Tensile test specimens were taken axially from the bar stock. These were
subjected to the same oversize machining, heat treatment and final machining
stages as the cylindrical specimens. Tensile test showed that the linear
part of the curve extended up to about 180 MPa and then strain-hardened
significantly up to 76% strain with an ultimate stress of 565 MPa. Using
the ASME III, Boiler and Pressure Vessel Code (Section III, Subsection
NB), the allowable design stress intensity Sm was determined
A typical stress-strain curve is included in Fig. 3.
It should be noted that all values of stress given above and in Fig.
3 are engineering stress. The rig used to provide simulated seismic
bending is illustrated in Fig. 4. This is a 250 kN servohydraulic
testing machine fitted with a fatigue module. The test specimen was attached
to extension limbs (via the flanged connections) and mounted in roller
bearing supports outboard of the flanged connections.
||Material properties obtained by tensile test
||Stress-strain curve for the stainless steel used to
manufacture the tubular specimens
||The seismic test rig
Tuning weights were
added to the ends of the extension limbs which were supported on constant
force springs to eliminate any gravity stresses. Excitation of the test
machine crosshead thus caused large-amplitude vibration of the test pipework.
Frequency sweeps at elastic amplitudes allowed the natural frequency to
be established and to confirm the amplitude of vibration to be the same
on each side of the rig. The bending moment experienced by the test specimen
was extrapolated from moment measurements made in the elastic region of
the extension limbs.
The design pressure for each cylinder was calculated using the ASMI
E II code which gives:
with Sm = 161 MPa (Table 2) and y = 0.4
All specimens were tested using a rising amplitude technique; i.e., having
mounted the specimen in the test rig, tuned the natural frequency and
applied the test pressure, where small input amplitude was applied and
maintained for about 20 sec. During this time a high-speed data logger
was used to record the input displacement, all strain gauge readings and
the output acceleration provided by an accelerometer positioned on one
of the tuning weights. Having completed such a test, the amplitude of
vibration was increased and the process repeated. At high input amplitudes
the duration of the test was reduced because of the limited stored hydraulic
capacity of the testing machine.
FINITE ELEMENT ARRANGEMENT
For all specimens the nonlinear finite element code, ABAQUS, was used
to study ratchetting of straight pressurized pipe subjected to simulated
seismic bending moments.
The cylindrical specimen model under pressure and cyclic bending moment
is shown in Fig. 5. The simulation assembly was a 2.30
m long pipework modeled by 23 elements. The central test section was 3
elements long and the lateral extension limbs 10 elements long. Each element
used 18 integration points around the pipe and four Fourier (or ovalization)
modes. In the radial direction, 7 and 9 integration points through the
thickness were used for the thin- and thick- walled models, respectively.
The latter numbers of integration points were decided after a series of
solution convergence runs. In the analysis, the load reactions were simulated
by applying boundary conditions at nodes 11 and 14 of the model. The latter
nodes are the ends of the three elements making up the central test section.
The displacements in all three directions and twisting about the pipe
axis were prevented at these nodes. The additional boundary condition
along the pipe axis was to simulate the closed end axial force reaction
due to the internal pressure.
The most accurate element in the ABAQUS code for this type of structural
system considering beam elements, pipe elements and elbow elements is
the elbow element. Four types of elbow elements are available in the ABAQUS
library, of which the two-noded element ELBOW 31B was found to give the
best results. Although these elbow elements appear like beam elements,
they are actually elements where shell theory is used to model the behavior.
Element type ELBOW 31B is cheaper (in computational time) than the standard
ELBOW 31 and ELBOW 32 elements.
||Cylindrical specimen model under pressure and cyclic
It uses a simplified formulation where
only ovalization is considered. Both warping and axial gradients of the
ovalization are neglected.
The loading was applied in two stages. First the internal pressure, set
at the design value of the pipe, was applied and held constant for the
remainder of the analysis. Next, the dynamic load to induce the cyclic
bending was applied at the end nodes of the simulation model. It was specified
as a sinusoidal force with a circular frequency as obtained from the simulation
test. Because of the dynamic nature of the analysis which induces different
inertia loads due to the distributed weight of the lateral extension limbs
as the vertical displacement frequency and amplitude were increased, the
amplitude of the excitation had to be carefully adjusted until an equivalent
moment equal to the value obtained during testing was achieved.
The results gained experimentally and from FE analyses using nonlinear
isotropic/kinematic (combined) hardening model (with C = 1.907 GPa and
γ = 5.78) are detailed below. Also, included is the result obtained
using the Beaney (1990) analysis.
EXPERIMENTAL AND THEORETICAL RESULTS
Detailed results will be presented for two of the specimens tested (SS2
and SS6) and summary results will be given for all tests conducted. It
is perhaps useful to present, firstly, the bending moment response obtained
experimentally (Fig. 6a) and by the FE analysis (Fig.
6b). These clearly show that a reasonably stable bending moment response
has been achieved for the duration of the test- in this case up to 10
Strain gauges and a high-speed data capture system were used to record
the experimental hoop and axial strains developed on the top and bottom
surfaces of the seismic specimens. A FORTRAN routine was written to reconstruct
the form of the strain signal and this is presented for the top surface
hoop strain gauge of specimen SS2 in Fig. 7a.
||(a) Experimental dynamic bending moment responses and
(b) FE analysis dynamic bending moment responses
(a) Experimental hoop strain data for the top surface
of specimen SS2 at a dynamic bending moment of 582.45 Nm (M/M0.2)
= 1.15 and (b) Hoop strain data using FE analysis for the true stress-strain
curve and combined hardening for the top surface of specimen SS2 at
a dynamic bending moment of 582.45 Nm (M/M0.2) = 1.15
obtained axial strain was found to be constant throughout the test and
thus no data for these gauges has been included. The hoop strain data
extracted from the FE analysis for the top surface of the cylinder are
presented in Fig. 7b.
All of the experimental ratchetting results for the tests and from FE
analysis on specimens SS1 to SS6 are plotted in Fig. 8a
and b, respectively. Here, the strain for each cycle
has been calculated as the average over the period of the test and plotted
against M/M0.2. Figure 8a and b
show the data recorded for the bottom surface.
The response of the specimens during these tests is illustrated in Fig.
9. The dynamic bending moment experienced by the specimen has been
plotted against the input displacement for each of the specimens SS1 to
Although there is some evidence in this plot that the dynamic bending
moments do approach a self-limiting value, this is much less distinct
than has been seen in previous works (Moreton et al., 1996; Beaney,
A typical set of results for specimen SS6 is shown in Fig
10, which includes results from the FE analysis using with the combined
hardening model. Here, the ratchet strain per cycle averaged over the
first 20 sec of excitation has been plotted against increasing M/M0.2
ratios for the experimentally obtained data, the finite element data and
the Beaney (1990) analytical solution (M0.2 is moment based
on proof stress σ0.2 = 242 MPa). For both experimental
data and the finite element results, the averages of the top and bottom
surface ratchet strains are shown. The same information obtained for specimen
SS2 is illustrated in Fig. 11.
Figure 12 illustrates the nature of this failure for
typical specimen of mild steel which appears to be a combination of hoop
ratcheting (leading to the gross local swelling of the cylinder) and fatigue
causing the hoop crack. Figure 13 shows the nature
of this failure for typical stainless steel specimen. The ductility of
the stainless steel used is very much greater than mild steel. Fig.
12 and 13 shown that the stainless steel specimens
do not have any greater fatigue life than the mild steel specimens. Swelling
in stainless steel is less than the mild steel.
It is evident from Fig. 7a, b, 10
and 11 that the hoop strain ratcheting rates predicted
by the Beaney analysis and the FE analysis are very much greater than
that found experimentally.
(a) Experimental ratchetting data recorded on the bottom
surface of specimens SS1 to SS6 and (b) Ratchetting data using FE
analysis on the bottom surface of specimens SS1 to SS6
||Dynamic bending moment against the input displacement
for each of the specimens SS1 to SS6
||Experimental, FE (Combined hardening) and Beaney ratchet
strains for specimen SS6
||Experimental, FE (combined hardening) and Beaney ratchet
strains for specimen SS2
||Swelling in mild steel specimen
In Table 3, the ratchet strains
found experimentally over a 20 sec test period and by FE analysis, for
the same period, for specimen SS6 are summarized. The strains recorded
on the top and bottom surfaces were found to be significantly different.
||Experimental, analysis and FE ratchetting data for specimen
Columns 4, 5 are the average of the top and bottom surface strains, Data obtained
for P = Pd
= 11.21 MPa, M0.2
was based on
proof stress σ0.2
= 242 MPa Figures in parentheses
in column five indicate the duration of the test in seconds at a
testing frequency of 5.76 Hz
||Experimental, analysis and FE ratchetting data for specimen
Columns 4, 5 are the average of the top and bottom surface strains,
Data obtained for P = Pd = 26.10 MPa (P/P0.2 = 0.66), M0.2 was based on proof stress σ0.2 = 242 MPa, Figs. in parentheses in column five indicate the duration
of the test in seconds at a testing frequency of 7.26 Hz
||Swelling in stainless steel specime
In Table 3, the average of these two surface strains
is presented. Table 4 gives the equivalent information
for the specimen SS2.
RESULTS AND DISCUSSION
The experimental work reported here provides reliable data which can
be used to judge the value of the available analytical solution (Beaney,
1990) and FE analysis using the ABAQUS package. However, it should be
noted that the experimental work used a rising amplitude technique which
may effectively reduce the ratchet strain at any particular dynamic bending
moment. It is possible that those tests conducted at low amplitude will
harden the material sufficiently to reduce the ratchet strains observed
at higher amplitudes. It is not possible to quantify the possible magnitude
of this effect. This possible effect would not have influenced the dynamic
bending moment at which ratchetting was first observed. Typical data obtained
experimentally and from FE model for specimens SS1 to SS6 on the bottom
surface are shown in Fig. 8a and b.
Also Fig. 8a and b indicate that
some ratchetting was observed at ratios of M/M0.2 between 0.8
and 1.0 for the thinnest specimen SS6. Complete sets of data for specimens
SS6 and SS2 are plotted in Fig. 10 and 11,
The important conclusion of this paper is to show the properties of nonlinear
isotropic/kinematic hardening model to predict the cyclic loading behavior
of the structures. Both experimental results and the FE analysis agree
that ratcheting is influenced by the material stress-strain curve and
load history. The rate of ratchetting depends significantly on the magnitude
of the internal pressure and tangent modulus of the bilinear material.
The results show that initial the rate of ratchetting is large and then
it decreases with the increasing of cycles. The FE model predicts the
hoop strain ratchetting rate to be greater than that found experimentally.
The ratchetting rate predicted by the Beaney equation for all specimens
overestimates the experimental rate. The results also show that the FE
method and analytical solutions give over estimated values comparing with
the experimental data.
Appreciation is expressed to the technical staff of the Applied Mechanics
Division of the Department of Mechanical Engineering at the University
of Tabriz (Iran) and University of Liverpool (UK) for their assistance
with the work.
||Cylinder mean diameter
||Cylinder outside diameter
||Dynamic bending moment
||Allowable design stress intensity
||Thickness correction factor = 0.4
||Ratchet strain in the hoop direction