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Strain ratcheting of steel tubulars with a rectangular defect under axial cycling: A numerical modeling
Journal of Constructional Steel Research 67 (2011) 1872–1883
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Journal of Constructional Steel Research
Strain ratcheting of steel tubulars with a rectangular defect under axial cycling: A numerical modeling M. Zeinoddini, M. Peykanu ⁎ Department of Civil and Environmental Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 26 February 2011 Accepted 31 May 2011 Available online 2 July 2011 Keywords: Ratcheting Cyclic loading Steel pipe Wrinkling Tubular Rectangular defect
a b s t r a c t Cyclic axial loads in tubular steel sections might lead to local buckling, wrinkling and accumulation of plastic strains in the tube. For example, this can be caused by repetitive start-up/shutdown and temperature changes in an offshore pipeline which generates cycles of axial compression/relaxing in the line. During their life time steel tubes may also experience material loss due to corrosion or wall thinning. The current paper reports the result of a numerical modeling of ratcheting behavior of steel tubes with a rectangular defect under cyclic axial loadings. The tubes have been initially subjected to monotonic axial compression beyond initiation of small amplitude wrinkles and subsequently to persistent axial cyclic loads. A nonlinear isotropic/ kinematic (combined) hardening model has been adopted for the material, which its parameters have been obtained from cyclic tests conducted on small coupon specimens. The results of the numerical simulation have been compared with experimental data. In general, a reasonable agreement has been noticed between the experimental and the numerical results for the ratcheting behavior of the tubes. It is shown that surface imperfections have a very pronounced effect on the ratcheting response of the defected tubes, as compared to the monotonic responses. The model has also been used to study effects of some key factors such as the initial strain level, the stress amplitude, the mean stress, the loading regime, wall thinning and the material hardening properties on the ratcheting response and on the progressive plastic buckling of steel tubes with a rectangular defect. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Steel tubulars axially compressed into the plastic range experience small amplitude wrinkles. If the tubular subsequently becomes subject to persistent cyclic loading, it may face ratcheting or accumulation of compressive strain which eventually results in the member collapse. For example, this can be caused by repetition of start-up/shutdown and temperature changes in an offshore pipeline. Pipelines laid down on the seabed are essentially axially restrained. Because of the restraints a temperature change, caused by the passage of hot hydrocarbons coupled with high internal pressure, can plastically deform the pipe/tube. In some cases the compression is high enough to initiate axial wrinkling. Imperfections due to small misalignments at girth welds, heat-affected regions around the welds, hard spots at connections with other equipment etc., can all enhance the onset of wrinkling. During a lifetime of say 20–30 years, pipelines experience many start-up and shutdown cycles (of the order of hundreds). A question arises as to whether wrinkles formed as a result of such stress rises can grow (ratchet) and if so what are the consequences [1].
⁎ Corresponding author. Tel.: + 98 9126135985. E-mail addresses: [email protected] (M. Zeinoddini), [email protected] (M. Peykanu). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.05.010
On the other hand, as pipeline ages, it may suffer from internal and external corrosions and defects, which may lead to loss of integrity and finally collapse of the pipeline. The aforementioned problems of plastic buckling of circular tubes and effect of corrosion on the strength of steel tubulars, have already been extensively studied in the literature. The problem of inelastic bending and collapse of tubes in the presence of internal pressure was experimentally and analytically investigated by Limam et al. [2]. Plastic buckling of circular tubes under persistent axial cyclic loading, that leads to structural instability and collapse of pipelines, was studied by Jiao and Kyriakides [1]. Chang and Pan [3] reported the results from an experimental investigation on the degradation and buckling of circular tubes subjected to cyclic bending. Paquette and Kyriakides [4] experimentally and analytically studied the plastic buckling and collapse of long steel cylinders under combined internal pressure and axial compression. Bardi and Kyriakides [5,6] presented the results of an experimental study on stainless steel specimens with diameter-to-thickness ratios of 23 to 52. The evolution of wrinkles during the testing was monitored using a special surface-scanning device. Miyazaki et al. [7] examined carbon steel pipes (grade 100A) with local wall thinning under cyclic pure bending loads to evaluate their low cycle fatigue strength. The strength of corroded pipelines has also received considerable previous attentions in the literature. The criteria adopted by the ASME B31G Code [8], are still widely employed for evaluating the residual
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strength of corroded pipelines. More recently, DNV-RP-F101 [9] provided recommended practices for assessing the strength of corroded pipelines under combined internal pressure and longitudinal compressive stress. Based on large and small scale experimental test results and numerical simulations, some semi-empirical formulae were proposed. These take into account single, interacting and complex shaped defects. The axial loads and internal/external pressures considered in these codes are of a monotonic nature and cyclic load effects have not been taken into consideration. Netto [10,11] studied the effect of narrow and long corrosion defects on the collapse pressure of offshore pipelines. Results from small-scale experiments and nonlinear finite element analyses were reported in these studies. Netto et al. [12] Sakakibara et al. [13] studied the effect of internal corrosion or erosion defects on the collapse of pipelines under external pressure. Xue [14] presented a non-linear finite-element analysis for the steady-state buckle propagation phenomenon in subsea corroded pipelines subjected to external hydrostatic pressures. Xue and Fatt [15] tried to derive a closed form analytical solutions for this problem. The subject of ratcheting, wrinkling and collapse of corroded or defected steel tubes under cyclic axial loadings, however, has not received due attention previously. The present paper deals with the ratcheting of corroded pipelines subjected to axial cyclic loadings. A numerical approach has been chosen for this study. The corrosion shape in the steel tube is simplified as a rectangular defect on the tube wall. It reports some results from the finite elements analysis and experimental studied on the subject. 2. Methods In the current study, an advanced finite element program [16] has been used to simulate the ratcheting response of carbon steel tubes.
Fig. 2. Geometry of the specimens tested (in millimeters).
The numerical model has been implemented in order to reproduce a series of laboratory tests conducted on small-scale tubes. These tests were carried out by the authors on intact and defected tubes, in which wrinkling and ratcheting behavior under axial monotonic and cyclic loads were studied. A nonlinear isotropic/kinematic hardening model has been employed to represent the cyclic behavior of the material. The verified model has then been used for a parametric study on ratcheting behavior of the defected tubes under cyclic axial loading. Details of the experimental program and its results will be reported in a separate paper and here a summary is presented. 3. Experimental program The cylindrical specimens tested were geometrically very similar to those used in the cyclic tests carried out by Limam et al. [2], Jiao and Kyriakides [1] and monotonic tests conducted by Paquette and Kyriakides [4]. These researchers, however, did not include corrosion or geometrical defects in their experiments. The specimens were machined (see Fig. 1) from two different carbon steel seamless tube stocks (59.4 mm outside diameter and 5 mm wall thickness), the dimensions of which are given in Fig. 2. The overall specimen length was 280 mm with the following variations in thickness:
Fig. 1. Machined specimen.
• Two thicker parts at far ends of the specimen which were left at the as-received diameter (52 mm long). • The test section in the middle of the specimen (76 mm long and machined down to 2 mm wall thickness). • Two linear tapers which connect the test section to the thicker end segments (50 mm long).
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Material type
Yield stress (MPa)
Ultimate stress (MPa)
Strain at failure
1 2
Carbon steel X42/B Carbon steel X56
297 400
450 550
3.8% 3.2%
Fig. 3. Illustration of irregular and the equivalent rectangular defects (DNV-RP-F101, [9]).
The tapers were long enough to minimize the thickness discontinuity effects on the axial stresses. This choice of specimen geometry implies that the onset and growth of wrinkling would approach that expected in a long uniform tube. The physical shape caused by the metal loss in a corroded tube is irregular in depth and in surface. For engineering purposes such as the evaluation of the residual strength of a corroded pipeline, it is very common to represent these irregular defects with an equivalent rectangular shape (Fig. 3). An equivalent defect is expected to provide a conservative estimation for the residual strength of a corroded pipe line, as the defect size is larger than that from irregular or parabolic shaped defects. Defect depths greater than 85% of the original wall thickness (i.e. remaining ligament is less than 15% of the original wall thickness) are not usually considered in the residual strength evaluation (DNV-RP-F101 [9]). Corrosion, erosion or wear grooves wider than 60° are not also common in offshore pipelines (Sakakibara et al. [13]). In the experiments reported in this paper the corrosion defects were simplified by external machine grooves which were constant in depth, uniform in axial direction and circumferentially between 45 and 60° wide. It is acknowledged that an equivalent rectangular defect brings sharp geometric discontinuities in the tube body, causing high stress concentration around the corners of the defect. Impact of the stress concentrations on the overall static behavior of the tube might be assumed negligible. These, however, will almost certainly have a remarkable effect on the fatigue strength of the defected tube as stress
tends to severely intensify near the edges of the defected area. The shape, width and depth of the equivalent corrosion pits will affect the fatigue strength. Rectangular shaped defects were introduced to the specimens using a CNC machine. The main dimensions of the defects considered in the experiments are: 76 mm long, 1 mm deep and 45 to 60° wide (see Fig. 4). The specimens were first axially pre-compressed and then subjected to cyclic axial loads. Specimens made from two different steel materials were considered in this experimental study. Steel material properties were similar to that used in previous studies [17,18]. Table 1 gives a summary of the material properties and the tests considered in the experimental program are listed in Table 2. 4. Material cyclic properties Ratcheting is defined as the accumulation of plastic strain during cyclic loading in the presence of a non-zero mean stress. Many efforts have been made to determine the cyclic characteristics of materials in uni-axial and multi-axial loadings (Pajand and Sinaie [19]). Rateindependent plasticity is characterized by the irreversible straining that occurs in a material once a certain level of stress is reached. The plastic strains are assumed to develop instantaneously, that is, independent of time. Five different models (rules) are commonly used for the description of the cyclic behavior of metallic material (Lee. et al. [20]) and are as follows: Table 2 Summary of the tests considered in the experimental program. Specimen Material
Specimen condition
Load condition
IM1(I)
steel
Intact
Monotonic –
–
steel
Intact
Monotonic –
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
–
steel
With defect With defect With defect With defect With defect With defect With defect With defect
Monotonic 1
60
Cyclic
1
60
Cyclic
1
60
Cyclic
1
60
Monotonic 1
60
Monotonic 1
45
Cyclic
1
60
Cyclic
1
60
IM2(П) IC1(I) IC2(I) IC3(I) IC4(I) IC5(П) CM1(I) CC1(I) CC2(I) CC3(I) CM2(П)
Section A-A
Side view
CM3(П) CC4(П)
Fig. 4. A side view and cross section of a specimen with an artificial equivalent rectangular defect.
CC5(П)
Carbon X42/B Carbon X56 Carbon X42/B Carbon X42/B Carbon X42/B Carbon X42/B Carbon X56 Carbon X42/B Carbon X42/B Carbon X42/B Carbon X42/B Carbon X56 Carbon X56 Carbon X56 Carbon X56
steel steel steel steel steel steel steel
Defect Defect width depth (mm) (degrees)
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Fig. 5. Four small bars subjected to different symmetric strain cycles.
• • • • •
Bilinear Isotropic Hardening rule (BISO) Bilinear Kinematic Hardening rule (BKIN) Non-linear Isotropic Hardening rule (NLISO) Non-linear Kinematic Hardening rule (Chaboche) Combination of NLISO and Chaboche rules
The isotropic/kinematic hardening rules are also used to simulate the ratcheting behavior of metallic components, when they become subject to cyclic loadings. The evolution law in the above models consists of a kinematic hardening component which describes the translation of the yield surface in the stress space. An isotropic component, which describes the change of the elastic range, is added for the nonlinear isotropic/kinematic hardening model. For example, the Armstrong–Frederick [21] kinematic hardening model is suggested for the nonlinear strain hardening materials. Many constitutive models have then been presented based on the Armstrong–Frederick nonlinear kinematic hardening rule to simulate the uni-axial and multi-axial ratcheting of materials characterized by cyclic hardening or cyclic stable behaviors (Zakavi et al. [22]). Choosing an appropriate material hardening model will be indispensable to proper numerical simulation of the cyclic loading on metallic component. To obtain correct kinematic/isotropic hardening parameters, it is recommended that the hardening model to be
Fig. 7. A typical finite element mesh.
calibrated against experimental data in strains close to the strain ranges and loading history expected to occur in the actual application (Zakavi et al. [22]). In the current study, the necessary stress–strain data has been collected from tests on small bars (see Fig. 5), each subjected to several stabilized stress cycles. Different symmetric strain cycles have been considered for each testing bar. The calibration procedure consisted of five bar tests, one of which was subjected to monotonic tension until necking and others (four specimens) were
500 450 250
Axial Stress σ (MPa)
400
(Δσ/2)-k
200 150 100
Experimental data Trendline
50
350 300 250 200 150
Experiment
100
Numerical Model
50 0
0
0.02
0.04
0.06
0.08
ΔεP/2
0.1
0.12
0.14
Fig. 6. Evaluating the material hardening parameters by regressing Eq. (2) to results from fully reversed symmetric strain-controlled uni-axial loading tests on small coupon specimens.
0
0
0.02
0.04
0.06
0.08
Fig. 8. Experimental and numerical axial stress–strain curves for an intact tube (specimen IM1(I) in Table 2) under axial monotonic loading.
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Fig. 9. Deformed shapes of an intact tube subjected to monotonic axial load; right: from experiment (specimen IM1(I) in Table 2), left: from the numerical simulation.
put under symmetric strain-controlled cycles with different strain levels. The strain levels were chosen according to practical data which the main model (see Section 5) would experience. During these calibration tests, the stress state essentially remained uni-axial. The material parameters for the isotropic hardening exponential law k, Q and b can be derived from monotonic standard tensile test data, where k is the initial size of the yield surface and Q and b are two material coefficients. The material parameters C and γ for the kinematic hardening model have to be determined by conducting separate tests. Three different approaches are usually used for providing experimental data for evaluating these two parameters: • Half-cycle test data, • Single stabilized cycle data. • Test data obtained from several stabilized cycles.
Fig. 11. Deformed shapes of a tube with a rectangular defect subjected to monotonic axial load; right: from experiment (specimen CM1(I) in Table 2), left: from the numerical simulation.
lent plastic strain has been obtained. This equals the summation of the absolute value of the change in longitudinal plastic strains: εp = ∑j Δε p ðiÞj = ∑ jΔε i −Δσ exp Ej
where εp is the plastic strain, εi total strain, σexp the measured stress and E is the elastic modulus. The equivalent back stress, X, is equal to one-half of the difference in the yield stress between the end of the tensile loading and the first yield of the subsequent compressive loading. The resulting data pairs (X, εp), are plotted in Fig. 6. The kinematic hardening parameters, C and γ, could now be estimated by correlating Eq. (2) to the coupon test data in Fig. 6. X=ν
In the current study the latter approach has been used, for which stress–strain data have been obtained from several stabilized cycles on specimens subjected to symmetric strain cycles. From these fully reversed symmetric strain-controlled uni-axial loading, the equiva-
ð1Þ
i
c + γ
h i c X0 −ν exp −νγ εp −εp0 γ
ð2Þ
where ν = ±1 accounts for the direction of flow, and εp0 and X0 are the initial values. The material parameters for the kinematic hardening model have so been estimated as C = 2420 MPa and γ = 10.
450 450
400
400
Axial Stress σ (MPa)
Axial Stress σ (MPa)
350 300 250 200 150
Experiment
100
Numerical Model
50 0
350 300 250 200 150
Experiment
100
Numerical Model
50 0
0.02
0.04
0.06
0.08
Fig. 10. Experimental and numerical (FE) axial stress–strain curve for a defected tube (specimen IM1(I) in Table 2) under axial monotonic loading.
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fig. 12. Experimental and numerical axial stress–strain curves for an intact specimen under axial cyclic loading (model no. 3 in Table 3).
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Table 3 Ratcheting strain data for intact tubes under cyclic loading. Model no.
1 2 3
Total number of load cycles Simulation
Experiment
150 100 7
150 100 4
Initial axial strain
Ratcheting strain rate (strain/cycle) Simulation
Experiment
2% 0. 3% 4%
0.0034% 0.004% 0.1%
0.004% 0.004% 0.32%
As it was mentioned, results from monotonic standard tensile tests were used to derive data for the equivalent stress versus equivalent plastic strains up to the point of necking. An empirical equation has been regressed to these data to define the equivalent stress at arbitrarily high equivalent strains. Using the experimental data corresponding to one-half of the necking strain, the monotonic hardening curve (Eq. (3)) has been regressed to the test data using a nonlinear least-squares regression: εp n σ = σy 1 + m
ð3Þ
In this equation, σ is the equivalent stress, σy is the initial uni-axial yield stress, and m and n are material constants. By fitting Eqs. (2) and (3) to the experimental data, σ and X at any equivalent plastic strain are estimated. The isotropic component of the hardening, σ 0, have then been defined as a function of equivalent plastic strain by: 0
σ ð εpÞ = σ ðεpÞ−X ð εpÞ
ð4Þ
The isotropic material parameters, Q and b, can be determined by regressing Eq. (4) to the results obtained from Eq. (5), using leastsquares nonlinear regression. h i σ0 = k + Q 1−exp −bεp
ð5Þ
The results have yielded Q = 95 MPa and b = 14.8 5. Numerical model A commercially available non-linear finite element program ABAQUS (SIMULIA, [16]) has been used to simulate the experimental tests, described in Section 3, and to study the ratcheting behavior of intact and defected steel tubulars under cyclic loadings. Eightnodded, solid elements type C3D8R, hourglass control and reduced integration points have been used to model the specimen body. In order to reproduce the specimen boundary conditions in the experiments, two lateral translational degrees of freedom at both ends of the numerical models, have been constrained. The longitudinal degree of freedom at the bottom end has been constrained while it remained free at the top end. A fine element mesh has been considered along the middle part of the model (Fig. 7). The element spacing has been selected based on the Jiao and Kyriakides [1] experimental results. They carried out
1 2
Total number of load cycles Simulation
Experiment
7 200
4 200
wrinkling and ratcheting tests on intact steel tubes. Dimensions for the specimen in their study were very close to those from the experimental and numerical modelings reported in this paper. They reported around seven wrinkles being formed along the middle part of their specimens. In other words, the wave length for the wrinkles was around 10 mm. So in the current study, 100 elements were considered along the middle part of the model to insure that wrinkles can be properly replicated (see Fig. 7). The axial loading has been introduced to the model in two stages. At first, the model has been axially compressed, in a displacement control step, to attain an axial strain (in its middle part), comparable to the initial non-linear strain measured during the experiments. The second loading stage has been a load control step in which a cyclic axial load, with a constant load range and a fixed mean value has been used. Mechanical properties such as, yield stress, modulus of elasticity, kinematic hardening parameters C, γ and isotropic hardening parameters k, b, Q corresponded to those obtained from tests, as described in Section 4. 6. Simulation of the experiments 6.1. Monotonic loading tests Fig. 8 compares the numerical and experimental axial stress–strain results obtained for an intact tube (specimen IM1(I) in Table 2) subjected to axial monotonic loading. Fig. 10 gives similar results for a tube (specimen CM1(I) in Table 2) with a rectangular defect. The ordinates in Figs. 8 and 10 present the overall strains. This is the ratio of the overall axial shortening in the specimen to its full length. The abscissa in the figures represents the mean axial stress in the specimen (model). This is the ratio of the axial load to the cross
Table 5 The axial cyclic load amplitude effects: the loading condition; the mean plastic ratcheting strain rate and the number of load cycles to failure.
Table 4 Ratcheting strain data for defected tubes under cyclic loading. Model no.
Fig. 13. Deformed shapes of a defected specimen subjected to cyclic axial load at an axial strain of 5.2% from the experiment (right) and the numerical simulation (left).
Initial axial strain
Ratcheting strain rate (strain/cycle) Simulation
Experiment
2% 0.3%
0.15% 0.02%
0.22% 0.025%
Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
1 2 3 4
2% 2% 2% 2%
175 175 125 125
157 175 225 204
0.001% 0.009% 0.02% 0.0024%
3500 400 180 1270
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0.04
Axial Ratcheting Strain
0.03 0.03 0.02 0.02 0.01
Model No.2(in Table 5) Model No.1(in Table 5)
0.01 0.00
0
10
20
30
Number of Cycles N Fig. 14. Effects of the axial cyclic load amplitude on the plastic ratcheting rates: numerical results for model nos. 1 and 2 (in Table 5).
section area at mid length of the specimen. Figures show a reasonable agreement between the experimental and the numerical stress–strain results. Deformed shapes of an intact tube (specimen IM1(I) in Table 2), from the experiment and the numerical approaches, are given in Fig. 9. The numerical prediction corresponds to at an axial strain of 5.6% and the experimental photo is taken at an axial strain of 6.4%. As it can be seen virtually axisymmetric wrinkles appeared in the middle (main) part of the specimen. At the beginning they had small and constant amplitudes in radial direction. Further increase in the axial load caused the wrinkle amplitude to grow. Radial amplification of the wrinkles was leading to degradation of the axial rigidity. Fig. 11 presents the deformed shapes of a defected tube (specimen CM1(I) in Table 2). The numerical prediction corresponds to an axial strain of 5.3% and the experimental photo is taken at an axial strain of 6.6%. With a defected model, the wrinkles have been non-axisymmetric. They observed to first initiate from the defected region and then gradually expand to the entire circumference. Fig. 11 shows that wrinkles developed in the experimental and numerical models are relatively similar in shape, location and wavelength. 6.2. Cyclic loading tests Simulating the ratcheting behavior of the specimens used in the experimental study (Section 3) under cyclic loadings, required assigning proper isotropic/kinematic hardening characteristics to
0.04
Axial Ratcheting Strain
0.04
Zone II
Zone
Fig. 16. A tertiary ratcheting behaviour reported for experiments on chromium ferrite steel from Kang et al. [24].
the numerical model. For this, the isotropic/kinematic parameters for the steel material have been obtained from coupon tests specially carried out for this numerical simulation (as described in Section 4). Loading regimes for the numerical model, such as the initial axial non-linear strain, the stress cycle amplitude and the mean stress have been kept identical to that used in the experiment. The numerical predictions and experimental results for an intact specimen (IC1(I) in Table 2) subjected to axial cyclic loading are shown in Fig. 12. The figure presents the overall axial strain (overall shortening divided by the full tube length) against mean axial stress (axial load divided by the net cross sectional area at mid-height of the specimen) in the models. With this case, the mean stress and the stress amplitude were 235 MPa and 165 MPa, respectively. The cyclic load started at an initial axial strain of 4%. Numerical and experimental results, for intact tubes under different cyclic loadings, are summarized in Table 3. Table 4 gives similar results for defected tubes. The numerical models were first subjected to a monotonic displacement control loading to attain an initial plastic axial strain equal to that applied in the experiments. They were then subjected to cyclic loading until the tube collapse. The tables report the initial strain values and the total number of load cycles. Mean ratchet strain rates given in these tables are the ratio between the total ratchet strain to the total number of load cycles. Post failure deformed shapes of a defected specimen (model no. 1 in Table 4), after being subjected to axial cyclic loads, are depicted in Fig. 13.
0.03 Zone III
7. Parametric studies
0.03
The numerical model, as reported in the previous section, provided acceptable simulations on the ratcheting response of steel tubes tested under monotonic and cyclic axial loads. The model was subsequently used for a parametric study on the behavior of the intact and defected tubes. The geometrical and material properties for parametric study remained similar to those reported in Sections 3 and 6. Effects of the following parameters have been examined:
0.02 Zone II
Zone
0.02
Model No.3 (in Table 5) Model No.4 (in Table 5)
0.01 0.01 0.00
0
10
20
30
40
50
60
70
80
Number of Cycles N Fig. 15. Effects of the axial cyclic load amplitude on the plastic ratcheting rates: numerical results for model nos. 3 and 4 (in Table 5).
• • • • •
Cyclic stress amplitude Cyclic load regime Cyclic mean stress Hardening defect Defect effects
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Table 6 Cyclic load regime effects: the loading condition; the mean plastic ratcheting strain rate and the number of load cycles to failure.
Loading regime
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
Compression-compression Compression-zero Compression-tension
2.8% 2.8% 2.8%
258 175 125
92 175 225
0.004% 0.009% 0.02%
895 400 15
7.1. The stress amplitude effects
0.032
Intact models have been considered for this parametric study. They have all been subjected to monotonic axial compression to attain an initial axial strain of ε ≈ 2%. The subsequent cyclic axial load has been oscillating around a constant mean value. The load amplitude remained constant for each model and varied for different models. The initial non-linear axial strain, the mean axial stress and the axial stress amplitude for the four models examined are summarized in Table 5. The mean plastic ratcheting strain rate and the total number of load cycles before overall buckling are also reported. Table 5 shows that the stress amplitude (σa) had a major effect on the mean plastic ratcheting strain rate and the number of cycles to failure. Empirical findings from other studies corroborate these results. Jiao and Kyriakides [1] reported results from some ratcheting tests carried out under axial cycling loads. The intact steel tubular specimens considered in their studies all had an average compressive initial strain of 2 to 3% but the cycle amplitudes were different. They reported that cyclic amplitude had played a significant role on the ratcheting strain rate. Jiao and Kyriakides also showed that larger initial wrinkles tend to increase the rate of ratcheting and decrease the number of cycles before collapse. Fig. 14 depicts the axial plastic ratcheting strains against the load cycle number for model nos. 1 and 2 (see Table 5). From this figure, a tertiary ratcheting behavior can be recognized in which the response may be subdivided in three distinct zones. In the early stages of loading, the cyclic strain follows a non-linear decelerating path (see zone I in Fig. 14) and then a stabilized response, with a nearly constant ratcheting strain rate (zone II in Fig. 14). The cyclic strain then greatly accelerates and follows an exponential path (zone III in Fig. 14). These three zones are also marked on Fig. 15, which gives the numerical results for model nos. 3 and 4 (see Table 5). This type of tertiary ratcheting behavior has also been reported by Kang [23] who examined the ratcheting response for two different steel materials, by Jiao and Kyriakides[1]. For example, Fig. 16 shows a similar behavior in experiments made on chromium ferrite steel carried out by Kang et al. [24].
Compression-zero Compression-Tension Compression-Compression
Axial Ratcheting Strain
0.031 0.031 0.030 0.030 0.029 0.029 0.028 0.028
0
20
40
60
80
100
Number of Stress Cycles N Fig. 17. Effects of the cyclic load regime on the plastic ratcheting rates: numerical results for compression–zero, compression–tension and compression–compression load regimes (see Table 6).
with all models also remained constant but the mean value and the amplitude of cyclic load varied. The loading data for these models are given in Table 6. Fig. 17 depicts the axial ratcheting strains against the load cycle number. As it can be noticed, the load regime has had a substantial effect on the ratcheting strain response. These effects are reflected in the initial transient zone (zone I), the stabilized ratcheting or the linear trajectory (zone II) and the onset of the exponential trajectory (zone III). The mean plastic ratcheting strain rate and the final number of load cycles to the overall buckling in the models are summarized in Table 6. It can be seen that the change in load regime from compression–tension to compression-zero and then to compression–compression conditions has caused a significant decrease in the mean plastic ratcheting strain rate.
7.2. The cyclic load regime effects In this part of the parametric study, the effect of the cyclic load regime on the ratcheting behavior of intact tubes has been studied. The regimes considered for the load cycling are i) compression– compression, ii) compression-zero, and iii) compression–tension. Models examined in these parametric studies had a constant initial plastic compression strain of 2.8%. The maximum compressive load
7.3. The mean stress effects For a parametric study on the axial mean stress effects on the ratcheting behavior of the intact tubes, three numerical models have
Table 7 The axial cyclic mean stress effects: the loading condition; the number of load cycles applied to each model; and mean plastic ratcheting strain per cycle. Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
1 2 3
2% 1.7% 1.5%
125 120 114
225 225 225
0.02% 0.009% 0.001%
180 400 3600
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0.0450
500
Axial Stress (MPa)
400 350 300 250 200 Experiment
150
Simulation: with a Nonlinear Kinematic/Isotropic Hardening Rule
100
Simulation: with a Nonlinear Kinematic Hardening Rule
0
0.01
0.02
0.03
0.04
0.05
0.06
0.0400
0.0350
0.0300 Simulation: with a Nonlinear Kinematic/Isotropic Hardening Rule
0.0250
50 0
Axial Ratchteing Strain
450
0.07
Axial Strain
Simulation: with a Nonlinear Kinematic Hardening Rule
0.0200 0
20
40
60
80
100
Number of Stress Cycles N Fig. 18. Experimental and numerical axial stress–strain curves for an intact specimen under axial monotonic loading.
been considered. The amplitude for the cyclic load remained constant but the mean stress varied. A compression–compression loading regime was considered. Table 7 gives a summary of the results. It can be seen that the mean plastic ratcheting strain rate has considerably increased by the mean stress increase. An experimental study was carried out by Kang et al. [24] on the strain cyclic characteristics and ratcheting of 316L stainless steel intact tubes subjected to uni-axial and multi-axial cyclic loadings. The strain ratcheting responses, reportedly, showed substantial dependency on the values of mean stress, stress amplitude and their histories.
7.4. Hardening rule To evaluate the sensitivity of the numerical models to the material hardening rule, two different numerical models have been considered. One model incorporates a non-linear isotropic/kinematic hardening rule and the second employs a non-linear kinematic hardening rule. The material hardening parameters were obtained from small coupon tests described in Section 4. The tubes in both models have been assumed to be intact. They have been studied under incremental monotonic as well as cyclic axial loadings Fig. 18 shows the stress–strain results from numerical simulation of tubes subjected to monotonic axial loading, with the corresponding experimental data. As can be seen, the numerical model which employed a non-linear isotropic/kinematic hardening rule has more accurately reproduced the experimental stress–strain results, as compared to the model which utilized a non-linear kinematic hardening rule.
Fig. 19. Sensitivity of the ratcheting simulations to the material hardening rule.
Fig. 19 shows the results from simulating an intact steel tube subjected to asymmetrical axial cyclic stressing within plastic deformation regimes. It depicts the axial ratcheting strains against the cycling numbers, from numerical models which incorporated different material hardening rules. The mean stress σm and the stress amplitude σa have been 100 and 300 MPa, respectively. The tube has had an initial axial strain of 2% and been examined under a compression–tension load regime. Fig. 19 indicates that the results have been very sensitive to the hardening rules. The models have predicted substantially different ratcheting responses. The mean axial ratcheting strain rate is around 0.0028%/cycle for the model which incorporated a non-linear isotropic/kinematic hardening. It is around 0.005%/cycle for the model which employed a non-linear kinematic hardening. A parametric study has also been performed to evaluate the impacts from different material hardening parameters on the ratcheting response. In total, seven models of intact tubes have been considered. Model no. 1 incorporates a non-linear isotropic/kinematic hardening rule utilizing those material parameters obtained from cyclic tests on small coupon specimens (see Section 4). With model nos. 2 to 7, the material parameters C and γ for the kinematic hardening as well as Q and b for the isotropic hardening have been varied around 10% with respect to those in model no. 1. Table 8 gives a summary of the results. The table also presents the total number of load cycles to the overall buckling for each model. Table 8 shows that with an increase of around 10% in C value, the number of cycles to failure has changed from 10 (model no. 1) to 40 (model no. 3). With a reduction of around 10% in γ value, the number of cycles to failure has changed from 10 (model no. 1) to 19 (model no. 2). So, the ratcheting response appears to have been very sensitive to
Table 8 Impacts from different material hardening parameters on the ratcheting response of an intact tube. Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
C (MPa)
γ
Q (MPa)
b
Number of cycles to failure
1 (Main model) 2 3 4 5 6 7
2%
100
300
2442 2442 2662 2442 2442 2442 2442
10 9 10 10 10 10 10
95 95 95 85 104 95 95
14.8 14.8 14.8 14.8 14.8 13.32 16.28
10 19 40 9 14 9 13
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Table 9 The ratcheting response in the intact and defected tubes. Loading regime
Axial Ratchteing Strain
Compression-compression Compression-zero
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Mean ratcheting rate (strain/cycle) Intact model
Corroded model (in deffected zones)
Corroded model (in perfect zones)
2% 2%
125 175
225 175
0.004% 0.0014%
0.11% 0.022%
0.02% 0.009%
0.06
7.5. Defect effects
0.05
In this section, effects of a geometrical defect in the tube on the ratcheting response have been numerically investigated. For this, two identical intact and defected steel tubular specimens were considered. The rectangular defect on the tube wall was 1 mm deep, 60° wide and 76 mm long. The initial inelastic compressive axial strain, the mean axial stress and the axial stress amplitude remained the same in both the intact and the defected models. It should be mentioned that for this, the average axial stresses at mid-height of both tubes were kept similar. It means that the mean axial load and the load amplitude in the defected specimen were comparatively lower than those in the intact model. The decrease was proportional to the area loss in the cross section of the defected tube at its mid-height. Both a compression-zero and a compression–tension load regime were considered. A summary of the numerical results are given in Table 9. Fig. 20 presents a typical ratcheting response from the intact and defected tubes. The results in this figure correspond to a compression–tension load regime. As it can be seen, two curves are presented for the defected tube. One curve reports the local axial strain in the damaged part while, the second curve gives the local axial strain in the perfect part of the tube. The difference between the ratcheting responses of the defected tube in its damaged and perfect parts can be more clearly noticed in Fig. 21. The figure reports the stress–strain curves in the damaged and perfect parts of a defected tube under a compression-zero load regime. The strains/stresses in Fig. 21 are local values in the mid-height of a tube, recorded at the center axis of the damaged and perfect parts. It can be noticed that, while cyclic stresses in the perfect part of the tube are moved to compression–compression, the damaged part was locally experiencing compression–tension stresses. The stress range in the damaged part was around 30% higher than that in the perfect part of the
0.04 0.03 0.02
Ratchet Strain in Damaged Part of a Defected Tube Ratchet Strain in Perfect Part of a Defected Tube
0.01
Ratchet Strain in an Intact Tube
0.00
0
10
20
Number of Stress Cycles N Fig. 20. Effects of a rectangular defect on the plastic ratcheting strains: numerical results for the intact and defected tubes under a cyclic compression–tension load regime.
the kinematic hardening parameters. This implies that for proper numerical simulation of the strain ratcheting response of tubular models, choosing accurate values for the material kinematic hardening parameters is essential. On the other hand the results show that the ratcheting response has been less sensitive to the changes in isotropic hardening parameters. With a variation of around ±10% in the Q and b values, the number of cycles to failure has changed from 10 (model no. 1) to a maximum of 14 (model no. 5) and a minimum of 9 (model nos. 4 and 6).
0.050 300
0.045
Axial Ratchteing Strain
Axial Stress (MPa)
250 200 150
Local Stress-Strain in the Damage Part
100
Local Stress-Strain in the Perfect Part
50
0.040 0.035 0.030 0.025 0.020
Ratchet Strain in Damaged Part of a Defected Tube
0.015
Ratchet Strain in Perfect Part of a Defected Tube
0.010
Ratchet Strain in an Intact Tube
0.005 0 0 -50
0.01
0.02
0.03
0.04
0.05
Axial Strain
Fig. 21. Numerical results for the local axial stress–strain behaviour in the intact and defected tubes under a cyclic compression–zero load regimes.
0.000
0
10
20
30
Number of Stress Cycles N Fig. 22. Effects of a rectangular defect on the plastic ratcheting strains: numerical results for the intact and defected tubes under a cyclic compression–zero load regimes.
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tube. This can be one reason for the high ratcheting strain rates obtained in the damaged part of the tubes (Table 9). Fig. 20 and Table 9 show that the ratcheting strain rates in the defected tube have significantly increased, in comparison to that in the intact tube. Fig. 20 demonstrates that the ratcheting strains and strain rates in the damaged parts of the defected tube have been noticeably magnified as compared to the corresponding values in the perfect parts of this tube. Unlike intact tubes, the wrinkling in the defected tubes has been noticed to be non-axisymmetric. It first was shaped in the damaged part of the tube. This localized deformation then circumferentially moved towards the perfect areas and gradually extended to the entire perimeter. As the load cycling number increased, a second wrinkle was formed once again in the damaged part, in another location along the tube. It then extended circumferentially towards the perfect areas to encircle the tube. This type of non-axisymmetric wrinkling has also had some effects on the tube ratcheting response. Fig. 20 shows that when the ratcheting strains in the damaged part reach the onset of the exponential trajectory (zone III), the ratcheting strains in the perfect part still remain in the linear trajectory (zone II). Fig. 21 demonstrates that the plastic strains in the damaged parts have been higher and ratchet faster than their corresponding values in the perfect part of the tube. Therefore, the possibility of the initiation and development of wrinkling in the damaged area is higher than in the perfect area. Fig. 22 shows the ratcheting response of a defected tube under an axial cyclic loading with a compression–zero load regimes. Results for a corresponding intact tube are also given in the figure. Fig. 22 also confirms the above mentioned interpretations about the wrinkling and ratcheting behaviors of the defected steel tubes. As it was already mentioned, no attention was previously paid to the axial ratcheting of corroded steel tubes under cyclic loading. For that reason it was not possible to compare the results from the current study to other published literature. 8. Conclusions Strain ratcheting behavior of the intact and defected steel tubes has been investigated. A finite element numerical modeling approach has been considered. It takes into account the combined nonlinear isotropic/kinematic hardening characteristics of the material. Stress– strain data for the material hardening rules have been obtained from several symmetric stabilized stress cycle tests on small coupon specimens. The numerical model has been implemented in order to reproduce a series of laboratory tests already carried out by the authors on small scale intact and defected steel tubes. The experimental and numerical results showed a reasonable degree of agreement. The numerical model has then been used to investigate the effects of the mean stress, stress amplitude, loading regime, hardening parameters and geometrical defects on the ratcheting response of steel tubes. It has been noticed that: a) The ratcheting strain rate was governed by (i) the initial nonlinear strain in the tube, (ii) the stress amplitude and (iii) the mean stress, respectively. This was in agreement with the empirical findings from previous researchers who addressed the axial ratcheting behavior of intact steel tubes. b) A non-linear isotropic/kinematic hardening rule provided more accurate ratcheting predictions, as compared to those from a kinematic hardening rule. c) The strains in the defected tubes were ratcheting at significantly higher rates, in comparison to those in the intact tubes. They more rapidly turned exponential. d) In defected tubes the local wrinkling first initiated from the damaged part. This local buckling then gradually proceeded to the entire circumference. The ratcheting strains in the defected area were very
rapidly turning exponential, while the ratcheting strains in the perfect zone still remained in the linear trajectory. e) In defected tubes the strain ratcheting behavior in the damaged part was distinctively different from the ratcheting behavior in the perfect zones. The ratcheting strains and strain rates in the damaged part of the tube were considerably higher than the corresponding values in the perfect zones. The main contributions of the current study can be summarized as: 1. It addressed the axial ratcheting response of steel pipes having geometrical defects, a subject which has not received due attention previously. 2. It employed a numerical model for simulating the experimental testing. The model was well able to reproduce the experimental results with a reasonable accuracy. 3. It examined in details the effects from the material hardening rules on the simulation results. It demonstrated that selecting an appropriate and empirically founded hardening rule is a key prerequisite for accurate prediction of the ratcheting response of steel tubes. 4. It showed that surface corrosion imperfections had a very pronounced effect on the ratcheting response of the defected tubes, as compared to their monotonic response. 5. The wrinkles in the defected tubes were non-axisymmetric and initiated from the damaged part of the tube. 6. In a corroded tube the local ratcheting response in the damaged and perfect parts of the tube were vividly different. References [1] Jiao R, Kyriakides S. Ratcheting, wrinkling and collapse of tubes under axial cycling. Int J Solids Struct 2009;46:2856–70. [2] Limam A, Lee LH, Corona E, Kyriakides S. Inelastic wrinkling and collapse of tubes under combined bending and internal pressure. Int J Mech Sci 2010;52:637–47. [3] Chang KH, Pan WF. Buckling life estimation of circular tubes under cyclic bending. Int J Solids Struct 2008;46:254–70. [4] Paquette JA, Kyriakides S. Plastic buckling of tubes under axial compression and internal pressure. Int J Mech Sci 2006;48:855–67. [5] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression — part I: experiments. Int J Mech Sci 2006;48:830–41. [6] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression — part II: analysis. Experiments. Int J Mech Sci 2006;48(842):854. [7] Miyazaki K, Nebu A, Ishiwata M, Hasegawa K. Fracture strength and behavior of carbon steel pipes with local wall thinning subjected to cyclic bending load. J Nucl Eng Des 2001;214:127–36. [8] ASME B31G. Manual for determining the remaining strength of corroded pipelines. A supplement to ANSI/ASME B31 code for pressure piping; 1991. [9] DNV Offshore Standard. DNV-RP-F101, “corroded pipeline”, Det Norske Veritas; 2009. [10] Netto TA. A simple procedure for the prediction of the collapse pressure of pipelines with narrow and long corrosion defects — correlation with new experimental data. J Appl Ocean Res 2010;32:132–4. [11] Netto TA. On the effect of narrow and long corrosion defects on the collapse pressure of pipelines. J Appl Ocean Res 2009;31:75–81. [12] Netto TA, Ferraz US, Botto A. On the effect of corrosion defects on the collapse pressure of pipelines. Int J Solids Struct 2007;44:7597–614. [13] Sakakibara N, Corona E, Kyriakides S. Collapse of partially corroded or worn pipe under external pressure. Int J Mech Sci 2008;50:1586–97. [14] Xue J. A non-linear finite-element analysis of buckle propagation in subsea corroded pipelines. J Finite Elem Anal Des 2006;42:1211–9. [15] Xue J, Hoo Fat M. Symmetric and anti-symmetric buckle propagation modes in subsea corroded pipelines. J Mar Struct 2005;18:43–61. [16] SIMULIA. ABAQUS analysis and theory manuals. SIMULIA, the Dassault Systèmes, Realistic Simulation, Providence, RI, USA; 2007. [17] Zeinoddini M, Parke GAR. Dynamic shakedown and degradation of elastic reactions in laterally impacted steel tubes. Int J Damage Mech 2011;20:400–22. [18] Zeinoddini M, Parke GAR, Harding JE. Interface forces in laterally impacted steel tubes. Int J Exp Mech 2008;48:265–80. [19] Pajand M, Sinaie S. On the calibration of the Chaboche hardening model and a modified hardening rule for uni-axial ratcheting prediction. Int J Mech Sci 2009;46:3009–17. [20] Lee JH, Kim SK, Lee JB, Yang YS, Yoo MJ. A numerical simulation model of cyclic hardening behaviour of AC4C-T6 for LNG cargo pump using finite element analysis. J Loss Prev Process Ind 2007;22:889–96. [21] Armstrong PJ, Frederick CO. A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N 731, Central Electricity Generating
M. Zeinoddini, M. Peykanu / Journal of Constructional Steel Research 67 (2011) 1872–1883 Board. The report is reproduced as a paper: 2007. Mater High Temperatures 1966;24(1):1–26. [22] Zakavi SJ, Zehsaz M, Eslami MR. The ratcheting behavior of pressurized plain pipework subjected to cyclic bending moment with the combined hardening model. J Nucl Eng Des 2010;240:726–37.
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[23] Kang G. Ratcheting: recent progresses in phenomenon observation, constitutive modeling and application. Int J Fatigue 2007;30:1448–72. [24] Kang G, Kan Q, Zhang J. Experimental study on the uni-axial cyclic deformation of 25CDV4.11 steel. J Masret Sci Technol 2004;21.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Strain ratcheting of steel tubulars with a rectangular defect under axial cycling: A numerical modeling M. Zeinoddini, M. Peykanu ⁎ Department of Civil and Environmental Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 26 February 2011 Accepted 31 May 2011 Available online 2 July 2011 Keywords: Ratcheting Cyclic loading Steel pipe Wrinkling Tubular Rectangular defect
a b s t r a c t Cyclic axial loads in tubular steel sections might lead to local buckling, wrinkling and accumulation of plastic strains in the tube. For example, this can be caused by repetitive start-up/shutdown and temperature changes in an offshore pipeline which generates cycles of axial compression/relaxing in the line. During their life time steel tubes may also experience material loss due to corrosion or wall thinning. The current paper reports the result of a numerical modeling of ratcheting behavior of steel tubes with a rectangular defect under cyclic axial loadings. The tubes have been initially subjected to monotonic axial compression beyond initiation of small amplitude wrinkles and subsequently to persistent axial cyclic loads. A nonlinear isotropic/ kinematic (combined) hardening model has been adopted for the material, which its parameters have been obtained from cyclic tests conducted on small coupon specimens. The results of the numerical simulation have been compared with experimental data. In general, a reasonable agreement has been noticed between the experimental and the numerical results for the ratcheting behavior of the tubes. It is shown that surface imperfections have a very pronounced effect on the ratcheting response of the defected tubes, as compared to the monotonic responses. The model has also been used to study effects of some key factors such as the initial strain level, the stress amplitude, the mean stress, the loading regime, wall thinning and the material hardening properties on the ratcheting response and on the progressive plastic buckling of steel tubes with a rectangular defect. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Steel tubulars axially compressed into the plastic range experience small amplitude wrinkles. If the tubular subsequently becomes subject to persistent cyclic loading, it may face ratcheting or accumulation of compressive strain which eventually results in the member collapse. For example, this can be caused by repetition of start-up/shutdown and temperature changes in an offshore pipeline. Pipelines laid down on the seabed are essentially axially restrained. Because of the restraints a temperature change, caused by the passage of hot hydrocarbons coupled with high internal pressure, can plastically deform the pipe/tube. In some cases the compression is high enough to initiate axial wrinkling. Imperfections due to small misalignments at girth welds, heat-affected regions around the welds, hard spots at connections with other equipment etc., can all enhance the onset of wrinkling. During a lifetime of say 20–30 years, pipelines experience many start-up and shutdown cycles (of the order of hundreds). A question arises as to whether wrinkles formed as a result of such stress rises can grow (ratchet) and if so what are the consequences [1].
⁎ Corresponding author. Tel.: + 98 9126135985. E-mail addresses: [email protected] (M. Zeinoddini), [email protected] (M. Peykanu). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.05.010
On the other hand, as pipeline ages, it may suffer from internal and external corrosions and defects, which may lead to loss of integrity and finally collapse of the pipeline. The aforementioned problems of plastic buckling of circular tubes and effect of corrosion on the strength of steel tubulars, have already been extensively studied in the literature. The problem of inelastic bending and collapse of tubes in the presence of internal pressure was experimentally and analytically investigated by Limam et al. [2]. Plastic buckling of circular tubes under persistent axial cyclic loading, that leads to structural instability and collapse of pipelines, was studied by Jiao and Kyriakides [1]. Chang and Pan [3] reported the results from an experimental investigation on the degradation and buckling of circular tubes subjected to cyclic bending. Paquette and Kyriakides [4] experimentally and analytically studied the plastic buckling and collapse of long steel cylinders under combined internal pressure and axial compression. Bardi and Kyriakides [5,6] presented the results of an experimental study on stainless steel specimens with diameter-to-thickness ratios of 23 to 52. The evolution of wrinkles during the testing was monitored using a special surface-scanning device. Miyazaki et al. [7] examined carbon steel pipes (grade 100A) with local wall thinning under cyclic pure bending loads to evaluate their low cycle fatigue strength. The strength of corroded pipelines has also received considerable previous attentions in the literature. The criteria adopted by the ASME B31G Code [8], are still widely employed for evaluating the residual
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strength of corroded pipelines. More recently, DNV-RP-F101 [9] provided recommended practices for assessing the strength of corroded pipelines under combined internal pressure and longitudinal compressive stress. Based on large and small scale experimental test results and numerical simulations, some semi-empirical formulae were proposed. These take into account single, interacting and complex shaped defects. The axial loads and internal/external pressures considered in these codes are of a monotonic nature and cyclic load effects have not been taken into consideration. Netto [10,11] studied the effect of narrow and long corrosion defects on the collapse pressure of offshore pipelines. Results from small-scale experiments and nonlinear finite element analyses were reported in these studies. Netto et al. [12] Sakakibara et al. [13] studied the effect of internal corrosion or erosion defects on the collapse of pipelines under external pressure. Xue [14] presented a non-linear finite-element analysis for the steady-state buckle propagation phenomenon in subsea corroded pipelines subjected to external hydrostatic pressures. Xue and Fatt [15] tried to derive a closed form analytical solutions for this problem. The subject of ratcheting, wrinkling and collapse of corroded or defected steel tubes under cyclic axial loadings, however, has not received due attention previously. The present paper deals with the ratcheting of corroded pipelines subjected to axial cyclic loadings. A numerical approach has been chosen for this study. The corrosion shape in the steel tube is simplified as a rectangular defect on the tube wall. It reports some results from the finite elements analysis and experimental studied on the subject. 2. Methods In the current study, an advanced finite element program [16] has been used to simulate the ratcheting response of carbon steel tubes.
Fig. 2. Geometry of the specimens tested (in millimeters).
The numerical model has been implemented in order to reproduce a series of laboratory tests conducted on small-scale tubes. These tests were carried out by the authors on intact and defected tubes, in which wrinkling and ratcheting behavior under axial monotonic and cyclic loads were studied. A nonlinear isotropic/kinematic hardening model has been employed to represent the cyclic behavior of the material. The verified model has then been used for a parametric study on ratcheting behavior of the defected tubes under cyclic axial loading. Details of the experimental program and its results will be reported in a separate paper and here a summary is presented. 3. Experimental program The cylindrical specimens tested were geometrically very similar to those used in the cyclic tests carried out by Limam et al. [2], Jiao and Kyriakides [1] and monotonic tests conducted by Paquette and Kyriakides [4]. These researchers, however, did not include corrosion or geometrical defects in their experiments. The specimens were machined (see Fig. 1) from two different carbon steel seamless tube stocks (59.4 mm outside diameter and 5 mm wall thickness), the dimensions of which are given in Fig. 2. The overall specimen length was 280 mm with the following variations in thickness:
Fig. 1. Machined specimen.
• Two thicker parts at far ends of the specimen which were left at the as-received diameter (52 mm long). • The test section in the middle of the specimen (76 mm long and machined down to 2 mm wall thickness). • Two linear tapers which connect the test section to the thicker end segments (50 mm long).
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Material type
Yield stress (MPa)
Ultimate stress (MPa)
Strain at failure
1 2
Carbon steel X42/B Carbon steel X56
297 400
450 550
3.8% 3.2%
Fig. 3. Illustration of irregular and the equivalent rectangular defects (DNV-RP-F101, [9]).
The tapers were long enough to minimize the thickness discontinuity effects on the axial stresses. This choice of specimen geometry implies that the onset and growth of wrinkling would approach that expected in a long uniform tube. The physical shape caused by the metal loss in a corroded tube is irregular in depth and in surface. For engineering purposes such as the evaluation of the residual strength of a corroded pipeline, it is very common to represent these irregular defects with an equivalent rectangular shape (Fig. 3). An equivalent defect is expected to provide a conservative estimation for the residual strength of a corroded pipe line, as the defect size is larger than that from irregular or parabolic shaped defects. Defect depths greater than 85% of the original wall thickness (i.e. remaining ligament is less than 15% of the original wall thickness) are not usually considered in the residual strength evaluation (DNV-RP-F101 [9]). Corrosion, erosion or wear grooves wider than 60° are not also common in offshore pipelines (Sakakibara et al. [13]). In the experiments reported in this paper the corrosion defects were simplified by external machine grooves which were constant in depth, uniform in axial direction and circumferentially between 45 and 60° wide. It is acknowledged that an equivalent rectangular defect brings sharp geometric discontinuities in the tube body, causing high stress concentration around the corners of the defect. Impact of the stress concentrations on the overall static behavior of the tube might be assumed negligible. These, however, will almost certainly have a remarkable effect on the fatigue strength of the defected tube as stress
tends to severely intensify near the edges of the defected area. The shape, width and depth of the equivalent corrosion pits will affect the fatigue strength. Rectangular shaped defects were introduced to the specimens using a CNC machine. The main dimensions of the defects considered in the experiments are: 76 mm long, 1 mm deep and 45 to 60° wide (see Fig. 4). The specimens were first axially pre-compressed and then subjected to cyclic axial loads. Specimens made from two different steel materials were considered in this experimental study. Steel material properties were similar to that used in previous studies [17,18]. Table 1 gives a summary of the material properties and the tests considered in the experimental program are listed in Table 2. 4. Material cyclic properties Ratcheting is defined as the accumulation of plastic strain during cyclic loading in the presence of a non-zero mean stress. Many efforts have been made to determine the cyclic characteristics of materials in uni-axial and multi-axial loadings (Pajand and Sinaie [19]). Rateindependent plasticity is characterized by the irreversible straining that occurs in a material once a certain level of stress is reached. The plastic strains are assumed to develop instantaneously, that is, independent of time. Five different models (rules) are commonly used for the description of the cyclic behavior of metallic material (Lee. et al. [20]) and are as follows: Table 2 Summary of the tests considered in the experimental program. Specimen Material
Specimen condition
Load condition
IM1(I)
steel
Intact
Monotonic –
–
steel
Intact
Monotonic –
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
–
steel
Intact
Cyclic
–
–
steel
With defect With defect With defect With defect With defect With defect With defect With defect
Monotonic 1
60
Cyclic
1
60
Cyclic
1
60
Cyclic
1
60
Monotonic 1
60
Monotonic 1
45
Cyclic
1
60
Cyclic
1
60
IM2(П) IC1(I) IC2(I) IC3(I) IC4(I) IC5(П) CM1(I) CC1(I) CC2(I) CC3(I) CM2(П)
Section A-A
Side view
CM3(П) CC4(П)
Fig. 4. A side view and cross section of a specimen with an artificial equivalent rectangular defect.
CC5(П)
Carbon X42/B Carbon X56 Carbon X42/B Carbon X42/B Carbon X42/B Carbon X42/B Carbon X56 Carbon X42/B Carbon X42/B Carbon X42/B Carbon X42/B Carbon X56 Carbon X56 Carbon X56 Carbon X56
steel steel steel steel steel steel steel
Defect Defect width depth (mm) (degrees)
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Fig. 5. Four small bars subjected to different symmetric strain cycles.
• • • • •
Bilinear Isotropic Hardening rule (BISO) Bilinear Kinematic Hardening rule (BKIN) Non-linear Isotropic Hardening rule (NLISO) Non-linear Kinematic Hardening rule (Chaboche) Combination of NLISO and Chaboche rules
The isotropic/kinematic hardening rules are also used to simulate the ratcheting behavior of metallic components, when they become subject to cyclic loadings. The evolution law in the above models consists of a kinematic hardening component which describes the translation of the yield surface in the stress space. An isotropic component, which describes the change of the elastic range, is added for the nonlinear isotropic/kinematic hardening model. For example, the Armstrong–Frederick [21] kinematic hardening model is suggested for the nonlinear strain hardening materials. Many constitutive models have then been presented based on the Armstrong–Frederick nonlinear kinematic hardening rule to simulate the uni-axial and multi-axial ratcheting of materials characterized by cyclic hardening or cyclic stable behaviors (Zakavi et al. [22]). Choosing an appropriate material hardening model will be indispensable to proper numerical simulation of the cyclic loading on metallic component. To obtain correct kinematic/isotropic hardening parameters, it is recommended that the hardening model to be
Fig. 7. A typical finite element mesh.
calibrated against experimental data in strains close to the strain ranges and loading history expected to occur in the actual application (Zakavi et al. [22]). In the current study, the necessary stress–strain data has been collected from tests on small bars (see Fig. 5), each subjected to several stabilized stress cycles. Different symmetric strain cycles have been considered for each testing bar. The calibration procedure consisted of five bar tests, one of which was subjected to monotonic tension until necking and others (four specimens) were
500 450 250
Axial Stress σ (MPa)
400
(Δσ/2)-k
200 150 100
Experimental data Trendline
50
350 300 250 200 150
Experiment
100
Numerical Model
50 0
0
0.02
0.04
0.06
0.08
ΔεP/2
0.1
0.12
0.14
Fig. 6. Evaluating the material hardening parameters by regressing Eq. (2) to results from fully reversed symmetric strain-controlled uni-axial loading tests on small coupon specimens.
0
0
0.02
0.04
0.06
0.08
Fig. 8. Experimental and numerical axial stress–strain curves for an intact tube (specimen IM1(I) in Table 2) under axial monotonic loading.
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Fig. 9. Deformed shapes of an intact tube subjected to monotonic axial load; right: from experiment (specimen IM1(I) in Table 2), left: from the numerical simulation.
put under symmetric strain-controlled cycles with different strain levels. The strain levels were chosen according to practical data which the main model (see Section 5) would experience. During these calibration tests, the stress state essentially remained uni-axial. The material parameters for the isotropic hardening exponential law k, Q and b can be derived from monotonic standard tensile test data, where k is the initial size of the yield surface and Q and b are two material coefficients. The material parameters C and γ for the kinematic hardening model have to be determined by conducting separate tests. Three different approaches are usually used for providing experimental data for evaluating these two parameters: • Half-cycle test data, • Single stabilized cycle data. • Test data obtained from several stabilized cycles.
Fig. 11. Deformed shapes of a tube with a rectangular defect subjected to monotonic axial load; right: from experiment (specimen CM1(I) in Table 2), left: from the numerical simulation.
lent plastic strain has been obtained. This equals the summation of the absolute value of the change in longitudinal plastic strains: εp = ∑j Δε p ðiÞj = ∑ jΔε i −Δσ exp Ej
where εp is the plastic strain, εi total strain, σexp the measured stress and E is the elastic modulus. The equivalent back stress, X, is equal to one-half of the difference in the yield stress between the end of the tensile loading and the first yield of the subsequent compressive loading. The resulting data pairs (X, εp), are plotted in Fig. 6. The kinematic hardening parameters, C and γ, could now be estimated by correlating Eq. (2) to the coupon test data in Fig. 6. X=ν
In the current study the latter approach has been used, for which stress–strain data have been obtained from several stabilized cycles on specimens subjected to symmetric strain cycles. From these fully reversed symmetric strain-controlled uni-axial loading, the equiva-
ð1Þ
i
c + γ
h i c X0 −ν exp −νγ εp −εp0 γ
ð2Þ
where ν = ±1 accounts for the direction of flow, and εp0 and X0 are the initial values. The material parameters for the kinematic hardening model have so been estimated as C = 2420 MPa and γ = 10.
450 450
400
400
Axial Stress σ (MPa)
Axial Stress σ (MPa)
350 300 250 200 150
Experiment
100
Numerical Model
50 0
350 300 250 200 150
Experiment
100
Numerical Model
50 0
0.02
0.04
0.06
0.08
Fig. 10. Experimental and numerical (FE) axial stress–strain curve for a defected tube (specimen IM1(I) in Table 2) under axial monotonic loading.
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fig. 12. Experimental and numerical axial stress–strain curves for an intact specimen under axial cyclic loading (model no. 3 in Table 3).
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Table 3 Ratcheting strain data for intact tubes under cyclic loading. Model no.
1 2 3
Total number of load cycles Simulation
Experiment
150 100 7
150 100 4
Initial axial strain
Ratcheting strain rate (strain/cycle) Simulation
Experiment
2% 0. 3% 4%
0.0034% 0.004% 0.1%
0.004% 0.004% 0.32%
As it was mentioned, results from monotonic standard tensile tests were used to derive data for the equivalent stress versus equivalent plastic strains up to the point of necking. An empirical equation has been regressed to these data to define the equivalent stress at arbitrarily high equivalent strains. Using the experimental data corresponding to one-half of the necking strain, the monotonic hardening curve (Eq. (3)) has been regressed to the test data using a nonlinear least-squares regression: εp n σ = σy 1 + m
ð3Þ
In this equation, σ is the equivalent stress, σy is the initial uni-axial yield stress, and m and n are material constants. By fitting Eqs. (2) and (3) to the experimental data, σ and X at any equivalent plastic strain are estimated. The isotropic component of the hardening, σ 0, have then been defined as a function of equivalent plastic strain by: 0
σ ð εpÞ = σ ðεpÞ−X ð εpÞ
ð4Þ
The isotropic material parameters, Q and b, can be determined by regressing Eq. (4) to the results obtained from Eq. (5), using leastsquares nonlinear regression. h i σ0 = k + Q 1−exp −bεp
ð5Þ
The results have yielded Q = 95 MPa and b = 14.8 5. Numerical model A commercially available non-linear finite element program ABAQUS (SIMULIA, [16]) has been used to simulate the experimental tests, described in Section 3, and to study the ratcheting behavior of intact and defected steel tubulars under cyclic loadings. Eightnodded, solid elements type C3D8R, hourglass control and reduced integration points have been used to model the specimen body. In order to reproduce the specimen boundary conditions in the experiments, two lateral translational degrees of freedom at both ends of the numerical models, have been constrained. The longitudinal degree of freedom at the bottom end has been constrained while it remained free at the top end. A fine element mesh has been considered along the middle part of the model (Fig. 7). The element spacing has been selected based on the Jiao and Kyriakides [1] experimental results. They carried out
1 2
Total number of load cycles Simulation
Experiment
7 200
4 200
wrinkling and ratcheting tests on intact steel tubes. Dimensions for the specimen in their study were very close to those from the experimental and numerical modelings reported in this paper. They reported around seven wrinkles being formed along the middle part of their specimens. In other words, the wave length for the wrinkles was around 10 mm. So in the current study, 100 elements were considered along the middle part of the model to insure that wrinkles can be properly replicated (see Fig. 7). The axial loading has been introduced to the model in two stages. At first, the model has been axially compressed, in a displacement control step, to attain an axial strain (in its middle part), comparable to the initial non-linear strain measured during the experiments. The second loading stage has been a load control step in which a cyclic axial load, with a constant load range and a fixed mean value has been used. Mechanical properties such as, yield stress, modulus of elasticity, kinematic hardening parameters C, γ and isotropic hardening parameters k, b, Q corresponded to those obtained from tests, as described in Section 4. 6. Simulation of the experiments 6.1. Monotonic loading tests Fig. 8 compares the numerical and experimental axial stress–strain results obtained for an intact tube (specimen IM1(I) in Table 2) subjected to axial monotonic loading. Fig. 10 gives similar results for a tube (specimen CM1(I) in Table 2) with a rectangular defect. The ordinates in Figs. 8 and 10 present the overall strains. This is the ratio of the overall axial shortening in the specimen to its full length. The abscissa in the figures represents the mean axial stress in the specimen (model). This is the ratio of the axial load to the cross
Table 5 The axial cyclic load amplitude effects: the loading condition; the mean plastic ratcheting strain rate and the number of load cycles to failure.
Table 4 Ratcheting strain data for defected tubes under cyclic loading. Model no.
Fig. 13. Deformed shapes of a defected specimen subjected to cyclic axial load at an axial strain of 5.2% from the experiment (right) and the numerical simulation (left).
Initial axial strain
Ratcheting strain rate (strain/cycle) Simulation
Experiment
2% 0.3%
0.15% 0.02%
0.22% 0.025%
Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
1 2 3 4
2% 2% 2% 2%
175 175 125 125
157 175 225 204
0.001% 0.009% 0.02% 0.0024%
3500 400 180 1270
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0.04
Axial Ratcheting Strain
0.03 0.03 0.02 0.02 0.01
Model No.2(in Table 5) Model No.1(in Table 5)
0.01 0.00
0
10
20
30
Number of Cycles N Fig. 14. Effects of the axial cyclic load amplitude on the plastic ratcheting rates: numerical results for model nos. 1 and 2 (in Table 5).
section area at mid length of the specimen. Figures show a reasonable agreement between the experimental and the numerical stress–strain results. Deformed shapes of an intact tube (specimen IM1(I) in Table 2), from the experiment and the numerical approaches, are given in Fig. 9. The numerical prediction corresponds to at an axial strain of 5.6% and the experimental photo is taken at an axial strain of 6.4%. As it can be seen virtually axisymmetric wrinkles appeared in the middle (main) part of the specimen. At the beginning they had small and constant amplitudes in radial direction. Further increase in the axial load caused the wrinkle amplitude to grow. Radial amplification of the wrinkles was leading to degradation of the axial rigidity. Fig. 11 presents the deformed shapes of a defected tube (specimen CM1(I) in Table 2). The numerical prediction corresponds to an axial strain of 5.3% and the experimental photo is taken at an axial strain of 6.6%. With a defected model, the wrinkles have been non-axisymmetric. They observed to first initiate from the defected region and then gradually expand to the entire circumference. Fig. 11 shows that wrinkles developed in the experimental and numerical models are relatively similar in shape, location and wavelength. 6.2. Cyclic loading tests Simulating the ratcheting behavior of the specimens used in the experimental study (Section 3) under cyclic loadings, required assigning proper isotropic/kinematic hardening characteristics to
0.04
Axial Ratcheting Strain
0.04
Zone II
Zone
Fig. 16. A tertiary ratcheting behaviour reported for experiments on chromium ferrite steel from Kang et al. [24].
the numerical model. For this, the isotropic/kinematic parameters for the steel material have been obtained from coupon tests specially carried out for this numerical simulation (as described in Section 4). Loading regimes for the numerical model, such as the initial axial non-linear strain, the stress cycle amplitude and the mean stress have been kept identical to that used in the experiment. The numerical predictions and experimental results for an intact specimen (IC1(I) in Table 2) subjected to axial cyclic loading are shown in Fig. 12. The figure presents the overall axial strain (overall shortening divided by the full tube length) against mean axial stress (axial load divided by the net cross sectional area at mid-height of the specimen) in the models. With this case, the mean stress and the stress amplitude were 235 MPa and 165 MPa, respectively. The cyclic load started at an initial axial strain of 4%. Numerical and experimental results, for intact tubes under different cyclic loadings, are summarized in Table 3. Table 4 gives similar results for defected tubes. The numerical models were first subjected to a monotonic displacement control loading to attain an initial plastic axial strain equal to that applied in the experiments. They were then subjected to cyclic loading until the tube collapse. The tables report the initial strain values and the total number of load cycles. Mean ratchet strain rates given in these tables are the ratio between the total ratchet strain to the total number of load cycles. Post failure deformed shapes of a defected specimen (model no. 1 in Table 4), after being subjected to axial cyclic loads, are depicted in Fig. 13.
0.03 Zone III
7. Parametric studies
0.03
The numerical model, as reported in the previous section, provided acceptable simulations on the ratcheting response of steel tubes tested under monotonic and cyclic axial loads. The model was subsequently used for a parametric study on the behavior of the intact and defected tubes. The geometrical and material properties for parametric study remained similar to those reported in Sections 3 and 6. Effects of the following parameters have been examined:
0.02 Zone II
Zone
0.02
Model No.3 (in Table 5) Model No.4 (in Table 5)
0.01 0.01 0.00
0
10
20
30
40
50
60
70
80
Number of Cycles N Fig. 15. Effects of the axial cyclic load amplitude on the plastic ratcheting rates: numerical results for model nos. 3 and 4 (in Table 5).
• • • • •
Cyclic stress amplitude Cyclic load regime Cyclic mean stress Hardening defect Defect effects
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Table 6 Cyclic load regime effects: the loading condition; the mean plastic ratcheting strain rate and the number of load cycles to failure.
Loading regime
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
Compression-compression Compression-zero Compression-tension
2.8% 2.8% 2.8%
258 175 125
92 175 225
0.004% 0.009% 0.02%
895 400 15
7.1. The stress amplitude effects
0.032
Intact models have been considered for this parametric study. They have all been subjected to monotonic axial compression to attain an initial axial strain of ε ≈ 2%. The subsequent cyclic axial load has been oscillating around a constant mean value. The load amplitude remained constant for each model and varied for different models. The initial non-linear axial strain, the mean axial stress and the axial stress amplitude for the four models examined are summarized in Table 5. The mean plastic ratcheting strain rate and the total number of load cycles before overall buckling are also reported. Table 5 shows that the stress amplitude (σa) had a major effect on the mean plastic ratcheting strain rate and the number of cycles to failure. Empirical findings from other studies corroborate these results. Jiao and Kyriakides [1] reported results from some ratcheting tests carried out under axial cycling loads. The intact steel tubular specimens considered in their studies all had an average compressive initial strain of 2 to 3% but the cycle amplitudes were different. They reported that cyclic amplitude had played a significant role on the ratcheting strain rate. Jiao and Kyriakides also showed that larger initial wrinkles tend to increase the rate of ratcheting and decrease the number of cycles before collapse. Fig. 14 depicts the axial plastic ratcheting strains against the load cycle number for model nos. 1 and 2 (see Table 5). From this figure, a tertiary ratcheting behavior can be recognized in which the response may be subdivided in three distinct zones. In the early stages of loading, the cyclic strain follows a non-linear decelerating path (see zone I in Fig. 14) and then a stabilized response, with a nearly constant ratcheting strain rate (zone II in Fig. 14). The cyclic strain then greatly accelerates and follows an exponential path (zone III in Fig. 14). These three zones are also marked on Fig. 15, which gives the numerical results for model nos. 3 and 4 (see Table 5). This type of tertiary ratcheting behavior has also been reported by Kang [23] who examined the ratcheting response for two different steel materials, by Jiao and Kyriakides[1]. For example, Fig. 16 shows a similar behavior in experiments made on chromium ferrite steel carried out by Kang et al. [24].
Compression-zero Compression-Tension Compression-Compression
Axial Ratcheting Strain
0.031 0.031 0.030 0.030 0.029 0.029 0.028 0.028
0
20
40
60
80
100
Number of Stress Cycles N Fig. 17. Effects of the cyclic load regime on the plastic ratcheting rates: numerical results for compression–zero, compression–tension and compression–compression load regimes (see Table 6).
with all models also remained constant but the mean value and the amplitude of cyclic load varied. The loading data for these models are given in Table 6. Fig. 17 depicts the axial ratcheting strains against the load cycle number. As it can be noticed, the load regime has had a substantial effect on the ratcheting strain response. These effects are reflected in the initial transient zone (zone I), the stabilized ratcheting or the linear trajectory (zone II) and the onset of the exponential trajectory (zone III). The mean plastic ratcheting strain rate and the final number of load cycles to the overall buckling in the models are summarized in Table 6. It can be seen that the change in load regime from compression–tension to compression-zero and then to compression–compression conditions has caused a significant decrease in the mean plastic ratcheting strain rate.
7.2. The cyclic load regime effects In this part of the parametric study, the effect of the cyclic load regime on the ratcheting behavior of intact tubes has been studied. The regimes considered for the load cycling are i) compression– compression, ii) compression-zero, and iii) compression–tension. Models examined in these parametric studies had a constant initial plastic compression strain of 2.8%. The maximum compressive load
7.3. The mean stress effects For a parametric study on the axial mean stress effects on the ratcheting behavior of the intact tubes, three numerical models have
Table 7 The axial cyclic mean stress effects: the loading condition; the number of load cycles applied to each model; and mean plastic ratcheting strain per cycle. Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Ratcheting rate (strain/cycle)
Number of cycles to failure
1 2 3
2% 1.7% 1.5%
125 120 114
225 225 225
0.02% 0.009% 0.001%
180 400 3600
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0.0450
500
Axial Stress (MPa)
400 350 300 250 200 Experiment
150
Simulation: with a Nonlinear Kinematic/Isotropic Hardening Rule
100
Simulation: with a Nonlinear Kinematic Hardening Rule
0
0.01
0.02
0.03
0.04
0.05
0.06
0.0400
0.0350
0.0300 Simulation: with a Nonlinear Kinematic/Isotropic Hardening Rule
0.0250
50 0
Axial Ratchteing Strain
450
0.07
Axial Strain
Simulation: with a Nonlinear Kinematic Hardening Rule
0.0200 0
20
40
60
80
100
Number of Stress Cycles N Fig. 18. Experimental and numerical axial stress–strain curves for an intact specimen under axial monotonic loading.
been considered. The amplitude for the cyclic load remained constant but the mean stress varied. A compression–compression loading regime was considered. Table 7 gives a summary of the results. It can be seen that the mean plastic ratcheting strain rate has considerably increased by the mean stress increase. An experimental study was carried out by Kang et al. [24] on the strain cyclic characteristics and ratcheting of 316L stainless steel intact tubes subjected to uni-axial and multi-axial cyclic loadings. The strain ratcheting responses, reportedly, showed substantial dependency on the values of mean stress, stress amplitude and their histories.
7.4. Hardening rule To evaluate the sensitivity of the numerical models to the material hardening rule, two different numerical models have been considered. One model incorporates a non-linear isotropic/kinematic hardening rule and the second employs a non-linear kinematic hardening rule. The material hardening parameters were obtained from small coupon tests described in Section 4. The tubes in both models have been assumed to be intact. They have been studied under incremental monotonic as well as cyclic axial loadings Fig. 18 shows the stress–strain results from numerical simulation of tubes subjected to monotonic axial loading, with the corresponding experimental data. As can be seen, the numerical model which employed a non-linear isotropic/kinematic hardening rule has more accurately reproduced the experimental stress–strain results, as compared to the model which utilized a non-linear kinematic hardening rule.
Fig. 19. Sensitivity of the ratcheting simulations to the material hardening rule.
Fig. 19 shows the results from simulating an intact steel tube subjected to asymmetrical axial cyclic stressing within plastic deformation regimes. It depicts the axial ratcheting strains against the cycling numbers, from numerical models which incorporated different material hardening rules. The mean stress σm and the stress amplitude σa have been 100 and 300 MPa, respectively. The tube has had an initial axial strain of 2% and been examined under a compression–tension load regime. Fig. 19 indicates that the results have been very sensitive to the hardening rules. The models have predicted substantially different ratcheting responses. The mean axial ratcheting strain rate is around 0.0028%/cycle for the model which incorporated a non-linear isotropic/kinematic hardening. It is around 0.005%/cycle for the model which employed a non-linear kinematic hardening. A parametric study has also been performed to evaluate the impacts from different material hardening parameters on the ratcheting response. In total, seven models of intact tubes have been considered. Model no. 1 incorporates a non-linear isotropic/kinematic hardening rule utilizing those material parameters obtained from cyclic tests on small coupon specimens (see Section 4). With model nos. 2 to 7, the material parameters C and γ for the kinematic hardening as well as Q and b for the isotropic hardening have been varied around 10% with respect to those in model no. 1. Table 8 gives a summary of the results. The table also presents the total number of load cycles to the overall buckling for each model. Table 8 shows that with an increase of around 10% in C value, the number of cycles to failure has changed from 10 (model no. 1) to 40 (model no. 3). With a reduction of around 10% in γ value, the number of cycles to failure has changed from 10 (model no. 1) to 19 (model no. 2). So, the ratcheting response appears to have been very sensitive to
Table 8 Impacts from different material hardening parameters on the ratcheting response of an intact tube. Model no.
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
C (MPa)
γ
Q (MPa)
b
Number of cycles to failure
1 (Main model) 2 3 4 5 6 7
2%
100
300
2442 2442 2662 2442 2442 2442 2442
10 9 10 10 10 10 10
95 95 95 85 104 95 95
14.8 14.8 14.8 14.8 14.8 13.32 16.28
10 19 40 9 14 9 13
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Table 9 The ratcheting response in the intact and defected tubes. Loading regime
Axial Ratchteing Strain
Compression-compression Compression-zero
Initial axial strain
Mean stress σm (MPa)
Stress amplitude σa (MPa)
Mean ratcheting rate (strain/cycle) Intact model
Corroded model (in deffected zones)
Corroded model (in perfect zones)
2% 2%
125 175
225 175
0.004% 0.0014%
0.11% 0.022%
0.02% 0.009%
0.06
7.5. Defect effects
0.05
In this section, effects of a geometrical defect in the tube on the ratcheting response have been numerically investigated. For this, two identical intact and defected steel tubular specimens were considered. The rectangular defect on the tube wall was 1 mm deep, 60° wide and 76 mm long. The initial inelastic compressive axial strain, the mean axial stress and the axial stress amplitude remained the same in both the intact and the defected models. It should be mentioned that for this, the average axial stresses at mid-height of both tubes were kept similar. It means that the mean axial load and the load amplitude in the defected specimen were comparatively lower than those in the intact model. The decrease was proportional to the area loss in the cross section of the defected tube at its mid-height. Both a compression-zero and a compression–tension load regime were considered. A summary of the numerical results are given in Table 9. Fig. 20 presents a typical ratcheting response from the intact and defected tubes. The results in this figure correspond to a compression–tension load regime. As it can be seen, two curves are presented for the defected tube. One curve reports the local axial strain in the damaged part while, the second curve gives the local axial strain in the perfect part of the tube. The difference between the ratcheting responses of the defected tube in its damaged and perfect parts can be more clearly noticed in Fig. 21. The figure reports the stress–strain curves in the damaged and perfect parts of a defected tube under a compression-zero load regime. The strains/stresses in Fig. 21 are local values in the mid-height of a tube, recorded at the center axis of the damaged and perfect parts. It can be noticed that, while cyclic stresses in the perfect part of the tube are moved to compression–compression, the damaged part was locally experiencing compression–tension stresses. The stress range in the damaged part was around 30% higher than that in the perfect part of the
0.04 0.03 0.02
Ratchet Strain in Damaged Part of a Defected Tube Ratchet Strain in Perfect Part of a Defected Tube
0.01
Ratchet Strain in an Intact Tube
0.00
0
10
20
Number of Stress Cycles N Fig. 20. Effects of a rectangular defect on the plastic ratcheting strains: numerical results for the intact and defected tubes under a cyclic compression–tension load regime.
the kinematic hardening parameters. This implies that for proper numerical simulation of the strain ratcheting response of tubular models, choosing accurate values for the material kinematic hardening parameters is essential. On the other hand the results show that the ratcheting response has been less sensitive to the changes in isotropic hardening parameters. With a variation of around ±10% in the Q and b values, the number of cycles to failure has changed from 10 (model no. 1) to a maximum of 14 (model no. 5) and a minimum of 9 (model nos. 4 and 6).
0.050 300
0.045
Axial Ratchteing Strain
Axial Stress (MPa)
250 200 150
Local Stress-Strain in the Damage Part
100
Local Stress-Strain in the Perfect Part
50
0.040 0.035 0.030 0.025 0.020
Ratchet Strain in Damaged Part of a Defected Tube
0.015
Ratchet Strain in Perfect Part of a Defected Tube
0.010
Ratchet Strain in an Intact Tube
0.005 0 0 -50
0.01
0.02
0.03
0.04
0.05
Axial Strain
Fig. 21. Numerical results for the local axial stress–strain behaviour in the intact and defected tubes under a cyclic compression–zero load regimes.
0.000
0
10
20
30
Number of Stress Cycles N Fig. 22. Effects of a rectangular defect on the plastic ratcheting strains: numerical results for the intact and defected tubes under a cyclic compression–zero load regimes.
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tube. This can be one reason for the high ratcheting strain rates obtained in the damaged part of the tubes (Table 9). Fig. 20 and Table 9 show that the ratcheting strain rates in the defected tube have significantly increased, in comparison to that in the intact tube. Fig. 20 demonstrates that the ratcheting strains and strain rates in the damaged parts of the defected tube have been noticeably magnified as compared to the corresponding values in the perfect parts of this tube. Unlike intact tubes, the wrinkling in the defected tubes has been noticed to be non-axisymmetric. It first was shaped in the damaged part of the tube. This localized deformation then circumferentially moved towards the perfect areas and gradually extended to the entire perimeter. As the load cycling number increased, a second wrinkle was formed once again in the damaged part, in another location along the tube. It then extended circumferentially towards the perfect areas to encircle the tube. This type of non-axisymmetric wrinkling has also had some effects on the tube ratcheting response. Fig. 20 shows that when the ratcheting strains in the damaged part reach the onset of the exponential trajectory (zone III), the ratcheting strains in the perfect part still remain in the linear trajectory (zone II). Fig. 21 demonstrates that the plastic strains in the damaged parts have been higher and ratchet faster than their corresponding values in the perfect part of the tube. Therefore, the possibility of the initiation and development of wrinkling in the damaged area is higher than in the perfect area. Fig. 22 shows the ratcheting response of a defected tube under an axial cyclic loading with a compression–zero load regimes. Results for a corresponding intact tube are also given in the figure. Fig. 22 also confirms the above mentioned interpretations about the wrinkling and ratcheting behaviors of the defected steel tubes. As it was already mentioned, no attention was previously paid to the axial ratcheting of corroded steel tubes under cyclic loading. For that reason it was not possible to compare the results from the current study to other published literature. 8. Conclusions Strain ratcheting behavior of the intact and defected steel tubes has been investigated. A finite element numerical modeling approach has been considered. It takes into account the combined nonlinear isotropic/kinematic hardening characteristics of the material. Stress– strain data for the material hardening rules have been obtained from several symmetric stabilized stress cycle tests on small coupon specimens. The numerical model has been implemented in order to reproduce a series of laboratory tests already carried out by the authors on small scale intact and defected steel tubes. The experimental and numerical results showed a reasonable degree of agreement. The numerical model has then been used to investigate the effects of the mean stress, stress amplitude, loading regime, hardening parameters and geometrical defects on the ratcheting response of steel tubes. It has been noticed that: a) The ratcheting strain rate was governed by (i) the initial nonlinear strain in the tube, (ii) the stress amplitude and (iii) the mean stress, respectively. This was in agreement with the empirical findings from previous researchers who addressed the axial ratcheting behavior of intact steel tubes. b) A non-linear isotropic/kinematic hardening rule provided more accurate ratcheting predictions, as compared to those from a kinematic hardening rule. c) The strains in the defected tubes were ratcheting at significantly higher rates, in comparison to those in the intact tubes. They more rapidly turned exponential. d) In defected tubes the local wrinkling first initiated from the damaged part. This local buckling then gradually proceeded to the entire circumference. The ratcheting strains in the defected area were very
rapidly turning exponential, while the ratcheting strains in the perfect zone still remained in the linear trajectory. e) In defected tubes the strain ratcheting behavior in the damaged part was distinctively different from the ratcheting behavior in the perfect zones. The ratcheting strains and strain rates in the damaged part of the tube were considerably higher than the corresponding values in the perfect zones. The main contributions of the current study can be summarized as: 1. It addressed the axial ratcheting response of steel pipes having geometrical defects, a subject which has not received due attention previously. 2. It employed a numerical model for simulating the experimental testing. The model was well able to reproduce the experimental results with a reasonable accuracy. 3. It examined in details the effects from the material hardening rules on the simulation results. It demonstrated that selecting an appropriate and empirically founded hardening rule is a key prerequisite for accurate prediction of the ratcheting response of steel tubes. 4. It showed that surface corrosion imperfections had a very pronounced effect on the ratcheting response of the defected tubes, as compared to their monotonic response. 5. The wrinkles in the defected tubes were non-axisymmetric and initiated from the damaged part of the tube. 6. In a corroded tube the local ratcheting response in the damaged and perfect parts of the tube were vividly different. References [1] Jiao R, Kyriakides S. Ratcheting, wrinkling and collapse of tubes under axial cycling. Int J Solids Struct 2009;46:2856–70. [2] Limam A, Lee LH, Corona E, Kyriakides S. Inelastic wrinkling and collapse of tubes under combined bending and internal pressure. Int J Mech Sci 2010;52:637–47. [3] Chang KH, Pan WF. Buckling life estimation of circular tubes under cyclic bending. Int J Solids Struct 2008;46:254–70. [4] Paquette JA, Kyriakides S. Plastic buckling of tubes under axial compression and internal pressure. Int J Mech Sci 2006;48:855–67. [5] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression — part I: experiments. Int J Mech Sci 2006;48:830–41. [6] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression — part II: analysis. Experiments. Int J Mech Sci 2006;48(842):854. [7] Miyazaki K, Nebu A, Ishiwata M, Hasegawa K. Fracture strength and behavior of carbon steel pipes with local wall thinning subjected to cyclic bending load. J Nucl Eng Des 2001;214:127–36. [8] ASME B31G. Manual for determining the remaining strength of corroded pipelines. A supplement to ANSI/ASME B31 code for pressure piping; 1991. [9] DNV Offshore Standard. DNV-RP-F101, “corroded pipeline”, Det Norske Veritas; 2009. [10] Netto TA. A simple procedure for the prediction of the collapse pressure of pipelines with narrow and long corrosion defects — correlation with new experimental data. J Appl Ocean Res 2010;32:132–4. [11] Netto TA. On the effect of narrow and long corrosion defects on the collapse pressure of pipelines. J Appl Ocean Res 2009;31:75–81. [12] Netto TA, Ferraz US, Botto A. On the effect of corrosion defects on the collapse pressure of pipelines. Int J Solids Struct 2007;44:7597–614. [13] Sakakibara N, Corona E, Kyriakides S. Collapse of partially corroded or worn pipe under external pressure. Int J Mech Sci 2008;50:1586–97. [14] Xue J. A non-linear finite-element analysis of buckle propagation in subsea corroded pipelines. J Finite Elem Anal Des 2006;42:1211–9. [15] Xue J, Hoo Fat M. Symmetric and anti-symmetric buckle propagation modes in subsea corroded pipelines. J Mar Struct 2005;18:43–61. [16] SIMULIA. ABAQUS analysis and theory manuals. SIMULIA, the Dassault Systèmes, Realistic Simulation, Providence, RI, USA; 2007. [17] Zeinoddini M, Parke GAR. Dynamic shakedown and degradation of elastic reactions in laterally impacted steel tubes. Int J Damage Mech 2011;20:400–22. [18] Zeinoddini M, Parke GAR, Harding JE. Interface forces in laterally impacted steel tubes. Int J Exp Mech 2008;48:265–80. [19] Pajand M, Sinaie S. On the calibration of the Chaboche hardening model and a modified hardening rule for uni-axial ratcheting prediction. Int J Mech Sci 2009;46:3009–17. [20] Lee JH, Kim SK, Lee JB, Yang YS, Yoo MJ. A numerical simulation model of cyclic hardening behaviour of AC4C-T6 for LNG cargo pump using finite element analysis. J Loss Prev Process Ind 2007;22:889–96. [21] Armstrong PJ, Frederick CO. A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N 731, Central Electricity Generating
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