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A three-dimensional explicit finite element analysis of rolled joint process by varying the roller path
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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A three-dimensional explicit finite element analysis of rolled joint process by varying the roller path
T
Ritu J. Singha,∗, K.N. Jonnalagaddab, P.J. Guruprasadb, H.S. Kushwahaa a b
Atomic Energy Regulatory Board, Mumbai, 400094, India Indian Institute of Technology, Bombay, Mumbai, 400076, India
ARTICLE INFO
ABSTRACT
Keywords: Rolled joint Residual stresses Contact pressure Pressure tube Strain path
In pressurized heavy water reactors, pressure tube to end fitting rolled joint is crucial for safe operation. The joint requires sufficient contact pressure for leak tightness and lower undesirable residual stress in the transition region of the tube. The objective of this work is to assess various paths of roller for roll expansion of tube such that the residual tensile stress is reduced while maintaining the required leak tightness. Consequently, three dimensional, explicit finite element simulations, of the roll expansion process are carried out. Several roller paths are considered in the simulation to vary the strain path of deformation. Results showed that with increase in number of rotations of the roller, the residual stresses decreased. In addition, the contact pressure is found to be within acceptable range for leak tightness. But, beyond a certain number of rotations, the stresses are found to saturate, hence putting an upper bound to their number.
1. Introduction
1.1. Roller expansion process
In the nuclear industry, rolled joints have been used to join two dissimilar metals, which cannot be welded due to the formation of brittle intermetallic compounds such as shown in Fig. 1a where Zr 2.5% Nb pressure tubes is joined to stainless steel (SS 403) end fittings. The end fitting has three circumferential grooves which are machined on the inner surface of the end fitting [1]. During the roll expansion, the rollers rotate and radially displace in a spiral path (Fig. 1b), deforming the pressure tube. The localized expansion of the tube is continued until the desired wall thickness reduction is achieved. At the same time, the pressure tube also extrudes into the grooves on the end fitting. At the conclusion of the expansion process, the pressure tube and end-fitting spring back (elastic recovery) such that a residual contact pressure is developed between the two components. The expanded tube can be divided into three zones. These zones are uniformly expanded zone (also known as rolled region), transition zone, and unexpanded zone (also known as unrolled region) as shown in Fig. 1c. During the process, in addition to the development of compressive residual stress field in the expanded zone, the tensile residual stress field also gets developed within the tube. This tensile residual stress field within the tube is highest in the transition region [2,3]. Therefore, the transition region may be susceptible to delayed hydride cracking.
In mechanical roll expansion, a rolling tool is used which plastically deform the pressure tube. This deformation of the pressure tube forms an interference fit between pressure tube and end fitting. Typically, five equal, circumferentially spaced rollers, are expanded radially by a tapered mandrel and at the same time, these rollers also move in a planetary motion inside the tube. The radial feed of the rollers is a function of the taper angle of the mandrel, which moves axially along the cylindrical axis. The forward and reverse movement of the mandrel leads to loading and unloading of the tube. The speed of rotation of the mandrel controls the rotary speed of rollers. Therefore, the speed and radial feed of the rollers control the strain path taken by each point in the pressure tube. These strain paths in turn influence the residual stress development. In the past, a number of researchers [4–9] have analysed tube–tubesheet roller expansion by reducing it to an axisymmetric quasi-static problem. In 1993, Updike et al. [4] developed a mathematical model of tube to tube sheet joint assuming axisymmetric deformation to evaluate residual stress in the transition zone of the tube. Williams et al. [7] used numerical studies for prediction of residual stress field in mechanically expanded steam generator tube plugs. Initially, they used an axisymmetric 2D idealisation, and later they carried out 3D simulation owing to absence of loading symmetry [10]. The 3D simulation employed a
Corresponding author. E-mail addresses: [email protected] (R.J. Singh), [email protected] (K.N. Jonnalagadda), [email protected] (P.J. Guruprasad), [email protected] (H.S. Kushwaha). ∗
https://doi.org/10.1016/j.ijpvp.2019.103975 Received 15 May 2019; Received in revised form 6 August 2019; Accepted 28 August 2019 Available online 03 September 2019 0308-0161/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 1. a Schematic diagram of pressure tube-end fitting rolled joint b Representative sketch of spiral path of the each roller c Section of tube showing zones of the expanded tube.
scheme of applying displacements on various nodes to simulate the expansion and unloading. Barring a few publications, most of the existing work on mechanical roll expansion has been simulated with the underlying assumption of axisymmetry. Moreover, some researchers [11–13] have focussed on the residual stress in the rolled region and hence analysed roll expansion with 2D plane stress/strain assumption. They evaluated stress field in the rolled region but not in the transition region. Aboodi et al. [11] simulated mechanical rolling process using 2D plane stress model in order to overcome the lack of loading symmetry in a mechanical rolling process. Further Merah et al. [3] performed 3-D finite element (FE) simulations of heat exchanger tube roll expansion to investigate the influence of large initial clearance and strain hardening of tube material. In this simulation, the rollers were modelled as rigid lines and they were subjected to outward radial displacement and rotation to simulate the expansion kinematics. It was assumed that each roller deforms a group of nodes. A sinusoidal function is used for describing the radial movement of each node. Julien et al. [14] simulated the roll expansion process in steam generators to evaluate the residual stresses. Work by Metzger et al. [15] gave a description of evolution of stresses with time using 3D simulation of a steam generator rolled joint. Manu et al. [16] performed numerical simulations of roll expansion of calandria tube joint for design optimisation where roller motion is implemented by defining radial and tangential velocity of rollers. Banwatt et al. [17] also analysed calandria tube to tube-sheet rolled joint to describe qualification of fabrication of joint using a predictive model and limited experiments. They also simulated roll expansion by radial displacement of rollers along with imposed rotation.
simulation of rolled joint process is effective in predicting the residual stress development in pressure tube. It offers the opportunity to assess the effect of various parameters on rolled joints. It needs to be emphasized that numerical modelling of the rolled joint is a challenging problem due to three dimensional nature of the problem. It involves large strain plastic deformation and loading-unloading cycle along with moving contact load. Limited work is available so far in open literature on 3D simulation of rolled joint between pressure tube to end fitting. The main motivation of the current work is to perform 3D simulations of the process of forming the roll joint between pressure tube and end fitting, by varying the roller path, in order to achieve lower undesirable residual stress. This work focuses on residual stress in the axial and hoop directions in the expanded and transition region of the tube. 2. Numerical simulation of the roller expansion process 2.1. Material model Pressure tube is manufactured from Zr 2.5% Nb alloy, which exhibits orthotropic material behavior [18]. In the current analysis the material is assumed to be homogeneous and isotropic. Tension-compression asymmetry in the flow behavior is ignored in the analysis. Considering that the process is quasi-static in nature, the strain rate effect of the Zr 2.5%Nb pressure tube is not accounted in the analysis. To calibrate power law strain hardening scalar constitutive equation, room temperature uniaxial tensile stress strain curve in longitudinal direction for Zr 2.5%Nb obtained from an experiment is employed. The curve is fitted for two different plastic strain ranges as given below,
1.2. Motivation for the current work Since residual stress is one of the most important characteristic in a rolled joints. Experimental test methods have been used to measure the residual stresses. However, these stress measurement methods inherently have some uncertainty and limited accessibility. Numerical
= 1257.8
p
0.1252,0.7%
= 1015.2
p
0.0648,
p
p
2.9 %
2.9%
True stress-plastic strain plot is linearly interpolated between 0 and 0.7% plastic strains. The initial yield stress considered in the analysis of 2
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rollers move outwards in the spiral path, and the unloading phase, when the rollers are radially retracted. Rollers move in spiral paths about the center of the rotation i.e. for each incremental rotation of the roller, the radial displacement increases linearly. The rotational speed of the rollers is fixed at 600 rpm. Maximum radial displacement at the end of the loading phase is fixed at 0.5 mm. The x and y displacement of the roller is described at the reference node of the roller. The rollers are restrained from moving axially. The rotation of the roller about its own axis is given at each reference node based on the kinematics of the roller motion and assuming no slip condition at the line of contact between roller and pressure tube. To simulate different roller paths, the maximum radial displacement is achieved for varying number of rotations. This corresponds to 0.5 mm radial displacement applied in 3, 6, 10, 20 and 40 rotations of the roller. This is to simulate the usage of mandrels with different slopes for applying, the desired radial displacement per rotation. The time taken, for applying the desired radial displacement of 0.5 mm, varies from 0.3 s to 4 s, depending on the number of rotations. Retraction of the roller is simulated by gradually reducing the radial displacement from the maximum value such that the roller and the pressure tube are separated. Furthermore, to damp out dynamic effects quickly after unloading, and to reach quasi static equilibrium in minimal number of increments, viscous pressure loading is defined. It is commonly used to damp out kinetic energy associated with structural motion. The viscous pressure load is defined as
Fig. 2. True stress -true plastic strain for pressure tube and end fitting material.
Zr 2.5%Nb is 580 MPa. The end fitting is manufactured from AISI 403 martensitic steel. The tensile properties evaluated by Dubey et al. [19] are used in the analysis. The stress-strain data is extrapolated linearly up to the strains encountered in the analysis, with initial yield stress of 680 MPa. The fitted true stress vs. plastic strain curves for pressure tube and end fitting material is shown in Fig. 2. The end fitting material is modelled with linear isotropic hardening behavior.
p=
c v v. n
where, c v is the viscous coefficient, v is the velocity of the point on the surface and n is the unit outward normal to the element at the same point. The viscous pressure coefficient is set at 2% of the ρCd, where ρ is the density of the material and Cd is the dilatational wave speed of material. Contact is defined between the rollers, pressure tube surfaces and end fitting inner surface using a general contact algorithm in ABAQUS/ Explicit [20], which uses penalty method to enforce constraints. The finite sliding formulation of this algorithm allows for arbitrary separation, sliding, and rotation of surfaces in contact. The spring stiffness that relates the contact force to the penetration distance is chosen such that its effect on stable time increment is minimal. Default contact stiffness is used with the end fitting as the master surface. The isotropic coulomb friction model is used to define the frictional behavior. Based on previous work on similar material [16], the coefficient of friction used between pressure tube and end fitting is taken as 0.2, and pressure tube and roller is taken as 0.1. Separate studies were done to understand the effect of coefficient of friction.
2.2. Finite element idealisation In the current work, the inner diameter and wall thickness of the tube are 83 mm and 3.3 mm, respectively. The end fitting inner diameter is such that there is zero clearance between the end fitting and pressure tube outer surface. The end fitting thickness considered in the analysis is 26.5 mm. The pressure tube and end fitting are discretized using 8-noded iso-parametric solid element with reduced integration and combined hourglass control. The rollers are modelled using analytical rigid surfaces. The profile of the roller (Fig. 1a) is rotated about an axis to form the three dimensional surface. The rigid surface is associated with rigid body reference node, whose motion governs the motion of the roller. Pressure tube is discretized with element dimension of 0.664 mm in thickness direction (i.e. 5 elements in the thickness direction), 0.648 mm in circumferential direction and 0.5 mm to 1.005 in the axial direction. Finer mesh is used in the transition region. End fitting is discretized at inner surface with element dimension of 1 mm in thickness direction, 0.707 mm in circumferential direction and varied from 0.5 mm to 1 mm in the axial direction. Mesh transition was used in end fitting to have finer mesh at inner surface and coarser mesh at outer surface. The model consisted of 302325 elements in pressure tube and 473200 elements in end fitting.
2.4. Numerical method In the current work, the explicit dynamic analysis is chosen for solving the problem. Explicit dynamic method is based on central difference scheme. This method is efficient in solving quasi static problems like deep drawing, plate rolling which involves large deformation and complex contact conditions [21]. Both the stages (the expansion stage as well as spring back) are simulated using dynamic explicit procedure in ABAQUS. As the natural timescale of the event is not practical to simulate, the loading rate (i.e., rotations per minute (RPM) of the roller) is increased such that the dynamic effects are negligible in the analysis. The dynamic effects can be neglected only if the kinetic energy of the deformable bodies is limited to 5–10% of its internal energy. In the current work, the kinetic energy achieved in all the simulations is well below 1% of internal energy and hence, quasi-static solution can be assumed. During the analysis, different roller speeds are assumed to fix the most suitable speed of the rollers. A range of 600–3000 rpm was analysed and it was observed that largely the solution was independent of RPM of the roller. However, under certain roller speeds, there were
2.3. Boundary conditions and loading The end fitting face ‘a’ in Fig. 3 is considered at a distance of 20 mm beyond pressure tube edge ‘b’. The end fitting extended portion provides the space for free extrusion of pressure tube edges. The end fitting face ‘a’ is fixed in all degree of freedom as shown in Fig. 3. There are no external constraints defined on the pressure tube. The loading of the pressure tube from inside is simulated by the prescribed outward radial displacement of the rollers. The complete loading cycle is divided into two phases, i.e., loading phase, when the 3
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 3. Boundary conditions used in the analysis.
peaks in the kinetic energy indicating resonance. The results at 600 rpm did not result in any resonance peaks of kinetic energy. And, therefore, the roller speed of 600 rpm is selected for the current study.
Another important parameter in the simulation is the effect of coefficient of friction (COF). The effect of COF is studied by considering the COF as 0.2 and 0.5 between the pressure tube and the end fitting. The frictional force acts against the relative movement between surfaces of pressure tube and end fitting. As the COF increase, a higher frictional force acts against the relative movement of pressure tube. This is also evident from the axial extrusion of PT. As the COF increases, the axial extrusion of pressure tube decreases. It is observed that higher frictional force leads to higher radial strain and hoop strain in the tube. And due to the increase in strains, the residual compressive axial and hoop stress increases. So evidently, it is observed that residual stresses are dependent upon COF between pressure tube and end-fitting. These stresses increase when COF increases. It is important to note that, end fitting remains elastic during the process for COF 0.2. But it showed some plastic yielding at its inner surface as the COF is increased to 0.5. Also it is observed that contact pressure increased marginally with COF (2.3% increase in the contact pressure). The observations further discussed in the current work concerning the effect of number of rotations consider COF as 0.2 and are deliberated in the following sections.
3. Results and discussion During the simulations, the increase in the number of rotations of the roller leads to change in the amount of displacement applied at the inner surface of pressure tube per unit time. Larger number of rotation means lower radial displacement per unit time. The effect of increasing the number of rotations for a given maximum displacement is investigated. The output quantities (displacement and stresses) are expected to vary in all the directions spatially, i.e., axial, circumferential and radial, with respect to time. And that's why, observations are extracted and reported at selected locations in the assembly. Primarily, these locations are inner surface of pressure tube (PTID), outer surface of pressure tube (PTOD) and inner surface of end fitting (EFID). These outputs are extracted at two most important regions, one at location ‘a’, within the rolled region, and at location ‘b’, in the transition region (see Fig. 4). The circumferential variation of the parameters is carefully recorded by selection of nodes on a circumferential path at location ‘a’ or ‘b’, by varying angle θ from 0° to 360° (see Fig. 4). To compare the effect of number of rotations, output variables are averaged around the circumference. The output variables are extracted at two most important stages, one at the end of loading (i.e., when the rollers are at maximum radial displacement) and second, when the rollers are fully extracted. At the stage when rollers are fully extracted, the output variables are termed as residual quantities.
3.1. Residual displacements Out of all the observations, residual displacements are directly measurable affecting the process. The radial displacements of nodes (at PTID and EFID) w.r.t time, are shown in Fig. 5a. For better observation, these nodes are selected at zero degree at location ‘a’ (Fig. 4). The loading phase (when rollers apply the displacements) is simulated as the percentage of time from 0 to 100%. The unloading phase (when the
Fig. 4. Nomenclature of rolled joint. 4
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 5. a Radial displacement with time b Average residual radial displacement for different rotations.
rollers are retracted) is simulated during the time beyond 100%. It is important to note that although, the final radial expansion of PTID at the end of loading, is same for two number of rotations (3 number of rotations and 10 numbers of rotations) but the path to reach the final displacement is different. And this difference is captured very clearly through the plot (Fig. 5a). It is observed in the plot that the displacement value are the same at the end of loading for different rotations (3 and 10) but they are different after unloading. The final residual displacements are different for two number of rotations. It is also clearly observed from the plot that the final residual displacement of inner surface of pressure tube is lower for lower rotations and higher for higher rotations. At the same time, the final displacement of inner surface of end fitting is higher for lower rotations and lower for higher rotations. To take it forward and strengthen the outcome, residual displacement of all nodes on the circumferential path are collected and shown in Fig. 5b. The residual radial displacement of PTID increases as the number of rotations increases. This can also be represented as spring back (see Fig. 5a). Moreover, this also can be observed from the plot in Fig. 5b that the residual radial displacement of the PTOD and EFID decrease as the number of rotations increases.
various locations is also important for the analysis. Fig. 6a shows the von Mises stress which is computed along the circumference of PTID for 3 and 10 rotations at the end of loading phase (when the maximum displacement is applied). Also some sharp peaks are observed in the plot. These peaks are due to presence of rollers at the instant of time. The plot shows that the average stress, developed at the PTID between any two roller positions for 3 and 10 rotations, is 750 MPa and 422 MPa, respectively. It is important to note that von Mises stress is lower for higher number of rotations. The residual von Mises stress after unloading, is shown in Fig. 6b. It is seen that higher number of rotations also results in lower residual stress. This observation can be explained by the nature of loading applied by the rollers. Each roller applies a line load such that the stress under the roller is highest and it decays in both direction similar to hertz contact stress in contacting cylinders. The magnitude of the stress depends on the applied displacement. As the number of rotations increases, the displacement for each rotation reduces thereby reducing the load on the pressure tube. Therefore, the incompatibility created with the elastic end fitting also gets reduced and this results in lower residual stresses. Fig. 7a shows the residual hoop stress developed at PTID along the circumference for 3 and 10 rotations. The residual stress for 3 rotations varied between −850 MPa and −630 MPa with an average value equal of −733 MPa. For 10 rotations, the stresses varied from −545 MPa to −415 MPa with an average of −492 MPa. Further, the range of stress
3.2. Residual stresses in rolled region Along with the, displacements, the observation of the stresses at
Fig. 6. a von Mises Stress across the circumference at PTID at end of loading phase b after retraction of rollers. 5
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 7. a Residual hoop stress across circumference b Average residual hoop stress for different rotations c Average contact pressure for different rotations d Average residual axial stress for different rotations.
variation reduced from 220 MPa for 3 rotations to 130 MPa for 10 rotations. These simulations indicate that as the number of rotations increases, the residual compressive hoop stress reduces and moreover it becomes more uniform. Fig. 7b brings out the circumferentially averaged residual hoop stress developed at PTID, PTOD, and EFID, for different number of rotations. It can be observed that as the number of rotations increases the residual compressive hoop stress reduces from −730 MPa to −360 MPa at PTID and from −750 MPa to −650 MPa at PTOD. The variation of the stress in terms of standard deviation is indicated by bars. The plot shows that as the number of rotations increases, the stress becomes more uniform. It is important to note that the residual stress at the inner surface end fitting changes from a compressive in nature to tensile in nature. Also, it can be seen that the hoop stress saturates as the number of rotation increases. So, for a given thickness of the sheet, after a certain number of fixed increment rotations, it is expected that hoop stress saturates and varies smoothly around the circumference. This phenomenon is beneficial, given the axisymmetric loading during normal operation of the reactor. The residual contact pressure, developed between the pressure tube and end fitting, is also an important parameter. This is directly related to the leak tightness and the pull-out strength of the joint. Fig. 7c shows the effect of number of rotations on the circumferentially averaged contact pressure. The contact pressure reduces with the number of rotations and also shows a saturation effect. The
reduction of the contact pressure with rotation can be explained by compatibility of strain at the interface. The displacement load, applied by the rollers, depends on number of rotations. This implies that for given Δt, the load applied for 3 rotation is higher compared to 20 rotations per unit Δt. This leads to development of higher strains in 3 rotation case compared to 20 rotations case. The total hoop strain at end fitting inner surface is higher for 3 rotations. After unloading, the end fitting (being elastic) springs back and tries to recover the original diameter. However, due to condition of strain compatibility with the plastically deformed pressure tube, contact pressure is developed. This pressure is higher if higher strains are developed as in case of 3 rotations. As the rotations increase, the contact pressure will reduce due to lower strains developed at interface. Fig. 7d shows variation of averaged residual axial stress at PTID, PTOD and EFID with number of rotations. The residual compressive axial stress at PTID and EFID reduces from −200 MPa and −110 MPa to −40 MPa and −50 MPa respectively. The stress value also shows a saturating effect with increase in rotations. The average axial stress at PTOD is tensile in nature and increases from 170 MPa to 260 MPa with increase in rotation. However, the axial stress also indicates a saturating effect with number of rotations. The standard deviation of stress, denoted using the bars shows that the stress becomes more uniform as the number of rotation increases. And the highest stress at large number of rotations is within the variation found after 3 rotations.
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International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 8. a Residual Axial Stress along circumference for different rotations b Average and maximum residual axial stress for different rotations.
3.3. Residual stresses in transition region
The maximum hoop stress changes from 80 MPa for 3 rotations to 38 MPa for 40 rotations at PTID. Similarly the maximum stress changes from 82 MPa for 3 rotations to 65 MPa for 40 rotations at PTOD. The average hoop stress along with standard deviation is also shown in the plots. The residual hoop stress at PTID and PTOD reduce with number of rotations. It gives an important insight that as a lower value of the tensile residual stress is desirable, increase in number of rotations is beneficial to reduce the tensile residual hoop stress. Therefore, it can be inferred that number of rotations to achieve the final displacement, significantly affects the stress field in the pressure tube. The effect can be summarised in terms of impact on contact pressure which is correlated to the leak tightness and tensile residual stresses in the transition region. The contact pressure reduces by 35% (Fig. 7c). This will affect the leak tightness. However in pressure tube to end fitting joint 3 circumferential grooves are present. The extrusion of pressure tube material in the grooves during rolling substantially improves the leak tightness and pull-out strength. Therefore reduction in contact pressure is compensated by the presence of the grooves. Further, the uniform tensile residual stresses in the PTOD of the roller region add to the pull-out strength. The tensile residual stress in the transition region especially the hoop stress reduces by 55% and the
Fig. 8a shows the residual axial stress along the circumference in the transition region of PTID for different number of rotations. The maximum axial stress of 125 MPa is obtained in case of 3 rotations and the minimum axial stress of −30 MPa is obtained for 40 rotations. Further, it varies from 125 MPa to −50 MPa along the circumference for 3 rotations. However, the stress varies from −50 MPa to −83 MPa along the circumference for 40 rotations. It indicates that when the number of rotations increases, the maximum and average stress reduces and becomes more uniform along the circumference. Fig. 8b shows average (along circumference) and maximum value of the axial stress plotted with number of rotations. The variation of stress in terms of standard deviation is also shown as bars in the plot. The plot shows that the standard deviation reduces with the increase in rotations and this makes the stress becomes more uniform circumferentially. In fact, the residual axial stress in the transition region becomes compressive in nature. This is very favourable as susceptibility of PTID for failure will reduce due to this compressive stress. Fig. 9 shows the maximum and circumferentially averaged value of residual hoop stress in the transition region in both PTID and PTOD.
Fig. 9. a Average and maximum residual hoop stress in transition region for different no.number of rotations at PTID b at PTOD. 7
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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measurements, contact pressure and hoop stress in the rolled region is considered for comparison. It should be highlighted that deformed shape of the pressure tube and end fitting is well captured in the simulation. The deformed profile of the pressure tube (Fig. 10a) shows that the extrusion at inner surface of the pressure tube is higher compared to extrusion at its outer surface. This observation is also reflected in the simulation results (Fig. 10b). The total extrusion of pressure tube spool piece is observed to be ~8–10 mm in experiments whereas the simulation results for 20 rotations gives ~10.7 mm total extrusion. The average contact pressure measured in the rolled region between the pressure tube spool and end fitting spool is estimated to be ~30–40 MPa whereas the numerical results gives average contact pressure of 50 MPa. The measured hoop stress in the rolled region has been observed to be ~ -400 to −500 MPa. The calculated hoop stress in the rolled region at 40 mm from the edge is −500 MPa. The difference in simulation and experimental results can be attributed to simplified material model considering isotropic material properties. The absence of grooves, in simulation, results in higher axial expansion compared to experiments. This can be attributed to the restrain in the axial expansion due to presence of grooves. However, the deformed shape of the tube is very well captured in the simulation including the taper at the free edge of the pressure tube and length of transition region. 4. Conclusion A 3D explicit finite element analysis of pressure tube –end fitting rolled joint process is carried out. Different number of roller rotations are used to achieve the maximum radial expansion of the pressure tube. The effect of various roller paths or number of rotations used for roll expansion is investigated. The following observations can be made from the numerical analysis:
Fig. 10. Comparison of a experimental b simulation deformed profile of PT post rolling (all dimension in mm).
axial stress at PTID becomes compressive, which makes the stress field very favourable from the point of delayed hydride cracking susceptibility. The improvement in the stress field achieved by increasing the rotations show a saturating behavior, i.e., increasing number of rotations beyond 40 or even 20 rotations may not be incrementally very beneficial.
• Lower number of rotations, i.e., larger displacement per rotation • •
3.4. Comparison with experimentally observed data
•
Rolled joints are routinely tested at Bhabha Atomic Research Institute, Mumbai. The rolled joints for testing are formed by rolling pressure tube spool pieces of ~120 mm length and martensitic steel sleeve having three circumferential grooves. These rolled joints are subjected to stress measurement using sleeve removal, slit cutting and hole drilling method and dimensional measurement of the deformed pressure tube. The residual stress measurements and dimensional measurements are used to compare the numerical results of the rolled joint simulations. It is important to highlight the variability associated with the test program. The rolled joint formed in workshop has variability in input conditions, like the clearance between the pressure tube and end fitting, yield strength of the components, positioning of the rollers etc. Further, uncertainty also exists in stress measurement technique. It is important to note that simulations considered for comparison are modelled with zero clearance between the components and material model considers isotropic yield and isotropic hardening behavior instead of the anisotropic behavior of the pressure tube. The grooves present on the end fitting inner surface are not assumed to be present in the simulation. In view of the above discussion, it should be clarified that the results can only be compared in an average sense. The simulation results for 20 rotation case is compared with rolled joint measurements. This case is decided because the actual rolling operation takes around 22 rotations. The maximum applied radial displacement of the simulation is fixed so as to match the thickness reduction achieved in the actual rolled joint. The dimensional
• •
leads to larger yielding per rotation and hence higher residual stresses. With increase in the number of rotations, the compressive stress in the rolled region reduces. The variation of the residual stress components along the circumference reduces and becomes uniform. The residual hoop and axial stress in the transition region, maximum as well as average value shows a decreasing trend as the number of rotations increases Similar to the rolled region, increase in the number of rotations make the residual stress in transition region more uniform, i.e., amplitude of residual stresses across the circumference reduces. The contact pressure developed between the pressure tube and the end fitting reduces as the number of rotation increases. However it is compensated by the presence of three grooves in the actual joint. The coefficient of friction between the end fitting and pressure tube affects the stresses developed in the pressure tube and the endfitting.
Declarations of interest None. Acknowledgements The author wish to acknowledge support from Atomic Energy Regulatory Board, Mumbai for carrying out the work, National Metallurgical laboratory, Jamshedpur for providing uniaxial stress strain data, Shri S. K. Sinha, Head, Reactor Engineering Division, BARC for providing all the support towards understanding the rolled joint formation process and discussions related to comparison with experimental findings. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 8
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[1] K. Madhusoodanan, R.K. Sinha, Experimental development of rolled joints for Zr2.5 wt% Nb pressure tubes for use in Indian PHWRs, 12th International Conference on Structural Mechanics in Reactor Technology, 1993, p. F07. [2] L. Cizelj, On the Estimation of the Steam Generator Maintenance Efficiency by the Means of Probabilistic Fracture Mechanics, Dissertation University of Ljubljana, Slovenia, 1993. [3] N. Merah, A. Al-Aboodi, A.N. Shuaib, Y. Al-Nassar, 3-D finite element analysis of roller-expanded heat exchanger tubes in over-enlarged tubesheet holes, Appl. Petrochem. Res. 1 (2012) 45–52. [4] D.P. Updike, A. Kalnins, S.M. Caldwell, Residual stresses in transition zones of heat exchanger tubes, ASME J. Press Vessel Technol. 114 (2) (1992) 149–156. [5] D.K. Williams, Comparison of residual stresses in the mechanical roll expansion of HX tubes into TEMA grooves, ASME J. Press Vessel Technol. 129 (2) (2007) 234–241. [6] N. Merah, A. Al-Zayer, A. Shuaib, A. Arif, Finite element evaluation of clearance effect on tube-to-tubesheet joint strength, Int. J. Press. Vessel. Pip. 80 (12) (2003) 879–885. [7] D.K. Williams, Prediction of residual stress field in mechanically expanded 0.750" diameter steam generator tube plugs: Part 1-2-D solution, Proc. ASME Pressure Vessel Piping Conf. 327 (1996) 173–180. [8] A. Al-Aboodi, N. Merah, N.Y. Shuaib, Y. Al-Nassar, Effects of friction on roller expanded tube–tubesheet joint strength, Int. J. Material Form. 3 (4) (2010) 253–257. [9] A.S. Al-Aboodi, N. Merah, A.R.N. Shuaib, Y. Al-Nassar, FE evaluation of effect of groove geometry on roller and hydraulically expanded joint strength, Int. J. Precis. Technol. l (2) (2009) 201–207. [10] D.K. Williams, Prediction of residual stress field in mechanically expanded 0. 750 double prime diameter steam generator tube plugs. Part 2: 3-D solution, Am. Soc. Mech. Eng. Press. Vessel. Pip. Div. PVP 354 (1997) 17–28. [11] Abdulaziz Alaboodi, Kinematic simulation of three rollers in circular motion using
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Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
A three-dimensional explicit finite element analysis of rolled joint process by varying the roller path
T
Ritu J. Singha,∗, K.N. Jonnalagaddab, P.J. Guruprasadb, H.S. Kushwahaa a b
Atomic Energy Regulatory Board, Mumbai, 400094, India Indian Institute of Technology, Bombay, Mumbai, 400076, India
ARTICLE INFO
ABSTRACT
Keywords: Rolled joint Residual stresses Contact pressure Pressure tube Strain path
In pressurized heavy water reactors, pressure tube to end fitting rolled joint is crucial for safe operation. The joint requires sufficient contact pressure for leak tightness and lower undesirable residual stress in the transition region of the tube. The objective of this work is to assess various paths of roller for roll expansion of tube such that the residual tensile stress is reduced while maintaining the required leak tightness. Consequently, three dimensional, explicit finite element simulations, of the roll expansion process are carried out. Several roller paths are considered in the simulation to vary the strain path of deformation. Results showed that with increase in number of rotations of the roller, the residual stresses decreased. In addition, the contact pressure is found to be within acceptable range for leak tightness. But, beyond a certain number of rotations, the stresses are found to saturate, hence putting an upper bound to their number.
1. Introduction
1.1. Roller expansion process
In the nuclear industry, rolled joints have been used to join two dissimilar metals, which cannot be welded due to the formation of brittle intermetallic compounds such as shown in Fig. 1a where Zr 2.5% Nb pressure tubes is joined to stainless steel (SS 403) end fittings. The end fitting has three circumferential grooves which are machined on the inner surface of the end fitting [1]. During the roll expansion, the rollers rotate and radially displace in a spiral path (Fig. 1b), deforming the pressure tube. The localized expansion of the tube is continued until the desired wall thickness reduction is achieved. At the same time, the pressure tube also extrudes into the grooves on the end fitting. At the conclusion of the expansion process, the pressure tube and end-fitting spring back (elastic recovery) such that a residual contact pressure is developed between the two components. The expanded tube can be divided into three zones. These zones are uniformly expanded zone (also known as rolled region), transition zone, and unexpanded zone (also known as unrolled region) as shown in Fig. 1c. During the process, in addition to the development of compressive residual stress field in the expanded zone, the tensile residual stress field also gets developed within the tube. This tensile residual stress field within the tube is highest in the transition region [2,3]. Therefore, the transition region may be susceptible to delayed hydride cracking.
In mechanical roll expansion, a rolling tool is used which plastically deform the pressure tube. This deformation of the pressure tube forms an interference fit between pressure tube and end fitting. Typically, five equal, circumferentially spaced rollers, are expanded radially by a tapered mandrel and at the same time, these rollers also move in a planetary motion inside the tube. The radial feed of the rollers is a function of the taper angle of the mandrel, which moves axially along the cylindrical axis. The forward and reverse movement of the mandrel leads to loading and unloading of the tube. The speed of rotation of the mandrel controls the rotary speed of rollers. Therefore, the speed and radial feed of the rollers control the strain path taken by each point in the pressure tube. These strain paths in turn influence the residual stress development. In the past, a number of researchers [4–9] have analysed tube–tubesheet roller expansion by reducing it to an axisymmetric quasi-static problem. In 1993, Updike et al. [4] developed a mathematical model of tube to tube sheet joint assuming axisymmetric deformation to evaluate residual stress in the transition zone of the tube. Williams et al. [7] used numerical studies for prediction of residual stress field in mechanically expanded steam generator tube plugs. Initially, they used an axisymmetric 2D idealisation, and later they carried out 3D simulation owing to absence of loading symmetry [10]. The 3D simulation employed a
Corresponding author. E-mail addresses: [email protected] (R.J. Singh), [email protected] (K.N. Jonnalagadda), [email protected] (P.J. Guruprasad), [email protected] (H.S. Kushwaha). ∗
https://doi.org/10.1016/j.ijpvp.2019.103975 Received 15 May 2019; Received in revised form 6 August 2019; Accepted 28 August 2019 Available online 03 September 2019 0308-0161/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Pressure Vessels and Piping 177 (2019) 103975
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Fig. 1. a Schematic diagram of pressure tube-end fitting rolled joint b Representative sketch of spiral path of the each roller c Section of tube showing zones of the expanded tube.
scheme of applying displacements on various nodes to simulate the expansion and unloading. Barring a few publications, most of the existing work on mechanical roll expansion has been simulated with the underlying assumption of axisymmetry. Moreover, some researchers [11–13] have focussed on the residual stress in the rolled region and hence analysed roll expansion with 2D plane stress/strain assumption. They evaluated stress field in the rolled region but not in the transition region. Aboodi et al. [11] simulated mechanical rolling process using 2D plane stress model in order to overcome the lack of loading symmetry in a mechanical rolling process. Further Merah et al. [3] performed 3-D finite element (FE) simulations of heat exchanger tube roll expansion to investigate the influence of large initial clearance and strain hardening of tube material. In this simulation, the rollers were modelled as rigid lines and they were subjected to outward radial displacement and rotation to simulate the expansion kinematics. It was assumed that each roller deforms a group of nodes. A sinusoidal function is used for describing the radial movement of each node. Julien et al. [14] simulated the roll expansion process in steam generators to evaluate the residual stresses. Work by Metzger et al. [15] gave a description of evolution of stresses with time using 3D simulation of a steam generator rolled joint. Manu et al. [16] performed numerical simulations of roll expansion of calandria tube joint for design optimisation where roller motion is implemented by defining radial and tangential velocity of rollers. Banwatt et al. [17] also analysed calandria tube to tube-sheet rolled joint to describe qualification of fabrication of joint using a predictive model and limited experiments. They also simulated roll expansion by radial displacement of rollers along with imposed rotation.
simulation of rolled joint process is effective in predicting the residual stress development in pressure tube. It offers the opportunity to assess the effect of various parameters on rolled joints. It needs to be emphasized that numerical modelling of the rolled joint is a challenging problem due to three dimensional nature of the problem. It involves large strain plastic deformation and loading-unloading cycle along with moving contact load. Limited work is available so far in open literature on 3D simulation of rolled joint between pressure tube to end fitting. The main motivation of the current work is to perform 3D simulations of the process of forming the roll joint between pressure tube and end fitting, by varying the roller path, in order to achieve lower undesirable residual stress. This work focuses on residual stress in the axial and hoop directions in the expanded and transition region of the tube. 2. Numerical simulation of the roller expansion process 2.1. Material model Pressure tube is manufactured from Zr 2.5% Nb alloy, which exhibits orthotropic material behavior [18]. In the current analysis the material is assumed to be homogeneous and isotropic. Tension-compression asymmetry in the flow behavior is ignored in the analysis. Considering that the process is quasi-static in nature, the strain rate effect of the Zr 2.5%Nb pressure tube is not accounted in the analysis. To calibrate power law strain hardening scalar constitutive equation, room temperature uniaxial tensile stress strain curve in longitudinal direction for Zr 2.5%Nb obtained from an experiment is employed. The curve is fitted for two different plastic strain ranges as given below,
1.2. Motivation for the current work Since residual stress is one of the most important characteristic in a rolled joints. Experimental test methods have been used to measure the residual stresses. However, these stress measurement methods inherently have some uncertainty and limited accessibility. Numerical
= 1257.8
p
0.1252,0.7%
= 1015.2
p
0.0648,
p
p
2.9 %
2.9%
True stress-plastic strain plot is linearly interpolated between 0 and 0.7% plastic strains. The initial yield stress considered in the analysis of 2
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rollers move outwards in the spiral path, and the unloading phase, when the rollers are radially retracted. Rollers move in spiral paths about the center of the rotation i.e. for each incremental rotation of the roller, the radial displacement increases linearly. The rotational speed of the rollers is fixed at 600 rpm. Maximum radial displacement at the end of the loading phase is fixed at 0.5 mm. The x and y displacement of the roller is described at the reference node of the roller. The rollers are restrained from moving axially. The rotation of the roller about its own axis is given at each reference node based on the kinematics of the roller motion and assuming no slip condition at the line of contact between roller and pressure tube. To simulate different roller paths, the maximum radial displacement is achieved for varying number of rotations. This corresponds to 0.5 mm radial displacement applied in 3, 6, 10, 20 and 40 rotations of the roller. This is to simulate the usage of mandrels with different slopes for applying, the desired radial displacement per rotation. The time taken, for applying the desired radial displacement of 0.5 mm, varies from 0.3 s to 4 s, depending on the number of rotations. Retraction of the roller is simulated by gradually reducing the radial displacement from the maximum value such that the roller and the pressure tube are separated. Furthermore, to damp out dynamic effects quickly after unloading, and to reach quasi static equilibrium in minimal number of increments, viscous pressure loading is defined. It is commonly used to damp out kinetic energy associated with structural motion. The viscous pressure load is defined as
Fig. 2. True stress -true plastic strain for pressure tube and end fitting material.
Zr 2.5%Nb is 580 MPa. The end fitting is manufactured from AISI 403 martensitic steel. The tensile properties evaluated by Dubey et al. [19] are used in the analysis. The stress-strain data is extrapolated linearly up to the strains encountered in the analysis, with initial yield stress of 680 MPa. The fitted true stress vs. plastic strain curves for pressure tube and end fitting material is shown in Fig. 2. The end fitting material is modelled with linear isotropic hardening behavior.
p=
c v v. n
where, c v is the viscous coefficient, v is the velocity of the point on the surface and n is the unit outward normal to the element at the same point. The viscous pressure coefficient is set at 2% of the ρCd, where ρ is the density of the material and Cd is the dilatational wave speed of material. Contact is defined between the rollers, pressure tube surfaces and end fitting inner surface using a general contact algorithm in ABAQUS/ Explicit [20], which uses penalty method to enforce constraints. The finite sliding formulation of this algorithm allows for arbitrary separation, sliding, and rotation of surfaces in contact. The spring stiffness that relates the contact force to the penetration distance is chosen such that its effect on stable time increment is minimal. Default contact stiffness is used with the end fitting as the master surface. The isotropic coulomb friction model is used to define the frictional behavior. Based on previous work on similar material [16], the coefficient of friction used between pressure tube and end fitting is taken as 0.2, and pressure tube and roller is taken as 0.1. Separate studies were done to understand the effect of coefficient of friction.
2.2. Finite element idealisation In the current work, the inner diameter and wall thickness of the tube are 83 mm and 3.3 mm, respectively. The end fitting inner diameter is such that there is zero clearance between the end fitting and pressure tube outer surface. The end fitting thickness considered in the analysis is 26.5 mm. The pressure tube and end fitting are discretized using 8-noded iso-parametric solid element with reduced integration and combined hourglass control. The rollers are modelled using analytical rigid surfaces. The profile of the roller (Fig. 1a) is rotated about an axis to form the three dimensional surface. The rigid surface is associated with rigid body reference node, whose motion governs the motion of the roller. Pressure tube is discretized with element dimension of 0.664 mm in thickness direction (i.e. 5 elements in the thickness direction), 0.648 mm in circumferential direction and 0.5 mm to 1.005 in the axial direction. Finer mesh is used in the transition region. End fitting is discretized at inner surface with element dimension of 1 mm in thickness direction, 0.707 mm in circumferential direction and varied from 0.5 mm to 1 mm in the axial direction. Mesh transition was used in end fitting to have finer mesh at inner surface and coarser mesh at outer surface. The model consisted of 302325 elements in pressure tube and 473200 elements in end fitting.
2.4. Numerical method In the current work, the explicit dynamic analysis is chosen for solving the problem. Explicit dynamic method is based on central difference scheme. This method is efficient in solving quasi static problems like deep drawing, plate rolling which involves large deformation and complex contact conditions [21]. Both the stages (the expansion stage as well as spring back) are simulated using dynamic explicit procedure in ABAQUS. As the natural timescale of the event is not practical to simulate, the loading rate (i.e., rotations per minute (RPM) of the roller) is increased such that the dynamic effects are negligible in the analysis. The dynamic effects can be neglected only if the kinetic energy of the deformable bodies is limited to 5–10% of its internal energy. In the current work, the kinetic energy achieved in all the simulations is well below 1% of internal energy and hence, quasi-static solution can be assumed. During the analysis, different roller speeds are assumed to fix the most suitable speed of the rollers. A range of 600–3000 rpm was analysed and it was observed that largely the solution was independent of RPM of the roller. However, under certain roller speeds, there were
2.3. Boundary conditions and loading The end fitting face ‘a’ in Fig. 3 is considered at a distance of 20 mm beyond pressure tube edge ‘b’. The end fitting extended portion provides the space for free extrusion of pressure tube edges. The end fitting face ‘a’ is fixed in all degree of freedom as shown in Fig. 3. There are no external constraints defined on the pressure tube. The loading of the pressure tube from inside is simulated by the prescribed outward radial displacement of the rollers. The complete loading cycle is divided into two phases, i.e., loading phase, when the 3
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Fig. 3. Boundary conditions used in the analysis.
peaks in the kinetic energy indicating resonance. The results at 600 rpm did not result in any resonance peaks of kinetic energy. And, therefore, the roller speed of 600 rpm is selected for the current study.
Another important parameter in the simulation is the effect of coefficient of friction (COF). The effect of COF is studied by considering the COF as 0.2 and 0.5 between the pressure tube and the end fitting. The frictional force acts against the relative movement between surfaces of pressure tube and end fitting. As the COF increase, a higher frictional force acts against the relative movement of pressure tube. This is also evident from the axial extrusion of PT. As the COF increases, the axial extrusion of pressure tube decreases. It is observed that higher frictional force leads to higher radial strain and hoop strain in the tube. And due to the increase in strains, the residual compressive axial and hoop stress increases. So evidently, it is observed that residual stresses are dependent upon COF between pressure tube and end-fitting. These stresses increase when COF increases. It is important to note that, end fitting remains elastic during the process for COF 0.2. But it showed some plastic yielding at its inner surface as the COF is increased to 0.5. Also it is observed that contact pressure increased marginally with COF (2.3% increase in the contact pressure). The observations further discussed in the current work concerning the effect of number of rotations consider COF as 0.2 and are deliberated in the following sections.
3. Results and discussion During the simulations, the increase in the number of rotations of the roller leads to change in the amount of displacement applied at the inner surface of pressure tube per unit time. Larger number of rotation means lower radial displacement per unit time. The effect of increasing the number of rotations for a given maximum displacement is investigated. The output quantities (displacement and stresses) are expected to vary in all the directions spatially, i.e., axial, circumferential and radial, with respect to time. And that's why, observations are extracted and reported at selected locations in the assembly. Primarily, these locations are inner surface of pressure tube (PTID), outer surface of pressure tube (PTOD) and inner surface of end fitting (EFID). These outputs are extracted at two most important regions, one at location ‘a’, within the rolled region, and at location ‘b’, in the transition region (see Fig. 4). The circumferential variation of the parameters is carefully recorded by selection of nodes on a circumferential path at location ‘a’ or ‘b’, by varying angle θ from 0° to 360° (see Fig. 4). To compare the effect of number of rotations, output variables are averaged around the circumference. The output variables are extracted at two most important stages, one at the end of loading (i.e., when the rollers are at maximum radial displacement) and second, when the rollers are fully extracted. At the stage when rollers are fully extracted, the output variables are termed as residual quantities.
3.1. Residual displacements Out of all the observations, residual displacements are directly measurable affecting the process. The radial displacements of nodes (at PTID and EFID) w.r.t time, are shown in Fig. 5a. For better observation, these nodes are selected at zero degree at location ‘a’ (Fig. 4). The loading phase (when rollers apply the displacements) is simulated as the percentage of time from 0 to 100%. The unloading phase (when the
Fig. 4. Nomenclature of rolled joint. 4
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Fig. 5. a Radial displacement with time b Average residual radial displacement for different rotations.
rollers are retracted) is simulated during the time beyond 100%. It is important to note that although, the final radial expansion of PTID at the end of loading, is same for two number of rotations (3 number of rotations and 10 numbers of rotations) but the path to reach the final displacement is different. And this difference is captured very clearly through the plot (Fig. 5a). It is observed in the plot that the displacement value are the same at the end of loading for different rotations (3 and 10) but they are different after unloading. The final residual displacements are different for two number of rotations. It is also clearly observed from the plot that the final residual displacement of inner surface of pressure tube is lower for lower rotations and higher for higher rotations. At the same time, the final displacement of inner surface of end fitting is higher for lower rotations and lower for higher rotations. To take it forward and strengthen the outcome, residual displacement of all nodes on the circumferential path are collected and shown in Fig. 5b. The residual radial displacement of PTID increases as the number of rotations increases. This can also be represented as spring back (see Fig. 5a). Moreover, this also can be observed from the plot in Fig. 5b that the residual radial displacement of the PTOD and EFID decrease as the number of rotations increases.
various locations is also important for the analysis. Fig. 6a shows the von Mises stress which is computed along the circumference of PTID for 3 and 10 rotations at the end of loading phase (when the maximum displacement is applied). Also some sharp peaks are observed in the plot. These peaks are due to presence of rollers at the instant of time. The plot shows that the average stress, developed at the PTID between any two roller positions for 3 and 10 rotations, is 750 MPa and 422 MPa, respectively. It is important to note that von Mises stress is lower for higher number of rotations. The residual von Mises stress after unloading, is shown in Fig. 6b. It is seen that higher number of rotations also results in lower residual stress. This observation can be explained by the nature of loading applied by the rollers. Each roller applies a line load such that the stress under the roller is highest and it decays in both direction similar to hertz contact stress in contacting cylinders. The magnitude of the stress depends on the applied displacement. As the number of rotations increases, the displacement for each rotation reduces thereby reducing the load on the pressure tube. Therefore, the incompatibility created with the elastic end fitting also gets reduced and this results in lower residual stresses. Fig. 7a shows the residual hoop stress developed at PTID along the circumference for 3 and 10 rotations. The residual stress for 3 rotations varied between −850 MPa and −630 MPa with an average value equal of −733 MPa. For 10 rotations, the stresses varied from −545 MPa to −415 MPa with an average of −492 MPa. Further, the range of stress
3.2. Residual stresses in rolled region Along with the, displacements, the observation of the stresses at
Fig. 6. a von Mises Stress across the circumference at PTID at end of loading phase b after retraction of rollers. 5
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Fig. 7. a Residual hoop stress across circumference b Average residual hoop stress for different rotations c Average contact pressure for different rotations d Average residual axial stress for different rotations.
variation reduced from 220 MPa for 3 rotations to 130 MPa for 10 rotations. These simulations indicate that as the number of rotations increases, the residual compressive hoop stress reduces and moreover it becomes more uniform. Fig. 7b brings out the circumferentially averaged residual hoop stress developed at PTID, PTOD, and EFID, for different number of rotations. It can be observed that as the number of rotations increases the residual compressive hoop stress reduces from −730 MPa to −360 MPa at PTID and from −750 MPa to −650 MPa at PTOD. The variation of the stress in terms of standard deviation is indicated by bars. The plot shows that as the number of rotations increases, the stress becomes more uniform. It is important to note that the residual stress at the inner surface end fitting changes from a compressive in nature to tensile in nature. Also, it can be seen that the hoop stress saturates as the number of rotation increases. So, for a given thickness of the sheet, after a certain number of fixed increment rotations, it is expected that hoop stress saturates and varies smoothly around the circumference. This phenomenon is beneficial, given the axisymmetric loading during normal operation of the reactor. The residual contact pressure, developed between the pressure tube and end fitting, is also an important parameter. This is directly related to the leak tightness and the pull-out strength of the joint. Fig. 7c shows the effect of number of rotations on the circumferentially averaged contact pressure. The contact pressure reduces with the number of rotations and also shows a saturation effect. The
reduction of the contact pressure with rotation can be explained by compatibility of strain at the interface. The displacement load, applied by the rollers, depends on number of rotations. This implies that for given Δt, the load applied for 3 rotation is higher compared to 20 rotations per unit Δt. This leads to development of higher strains in 3 rotation case compared to 20 rotations case. The total hoop strain at end fitting inner surface is higher for 3 rotations. After unloading, the end fitting (being elastic) springs back and tries to recover the original diameter. However, due to condition of strain compatibility with the plastically deformed pressure tube, contact pressure is developed. This pressure is higher if higher strains are developed as in case of 3 rotations. As the rotations increase, the contact pressure will reduce due to lower strains developed at interface. Fig. 7d shows variation of averaged residual axial stress at PTID, PTOD and EFID with number of rotations. The residual compressive axial stress at PTID and EFID reduces from −200 MPa and −110 MPa to −40 MPa and −50 MPa respectively. The stress value also shows a saturating effect with increase in rotations. The average axial stress at PTOD is tensile in nature and increases from 170 MPa to 260 MPa with increase in rotation. However, the axial stress also indicates a saturating effect with number of rotations. The standard deviation of stress, denoted using the bars shows that the stress becomes more uniform as the number of rotation increases. And the highest stress at large number of rotations is within the variation found after 3 rotations.
6
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Fig. 8. a Residual Axial Stress along circumference for different rotations b Average and maximum residual axial stress for different rotations.
3.3. Residual stresses in transition region
The maximum hoop stress changes from 80 MPa for 3 rotations to 38 MPa for 40 rotations at PTID. Similarly the maximum stress changes from 82 MPa for 3 rotations to 65 MPa for 40 rotations at PTOD. The average hoop stress along with standard deviation is also shown in the plots. The residual hoop stress at PTID and PTOD reduce with number of rotations. It gives an important insight that as a lower value of the tensile residual stress is desirable, increase in number of rotations is beneficial to reduce the tensile residual hoop stress. Therefore, it can be inferred that number of rotations to achieve the final displacement, significantly affects the stress field in the pressure tube. The effect can be summarised in terms of impact on contact pressure which is correlated to the leak tightness and tensile residual stresses in the transition region. The contact pressure reduces by 35% (Fig. 7c). This will affect the leak tightness. However in pressure tube to end fitting joint 3 circumferential grooves are present. The extrusion of pressure tube material in the grooves during rolling substantially improves the leak tightness and pull-out strength. Therefore reduction in contact pressure is compensated by the presence of the grooves. Further, the uniform tensile residual stresses in the PTOD of the roller region add to the pull-out strength. The tensile residual stress in the transition region especially the hoop stress reduces by 55% and the
Fig. 8a shows the residual axial stress along the circumference in the transition region of PTID for different number of rotations. The maximum axial stress of 125 MPa is obtained in case of 3 rotations and the minimum axial stress of −30 MPa is obtained for 40 rotations. Further, it varies from 125 MPa to −50 MPa along the circumference for 3 rotations. However, the stress varies from −50 MPa to −83 MPa along the circumference for 40 rotations. It indicates that when the number of rotations increases, the maximum and average stress reduces and becomes more uniform along the circumference. Fig. 8b shows average (along circumference) and maximum value of the axial stress plotted with number of rotations. The variation of stress in terms of standard deviation is also shown as bars in the plot. The plot shows that the standard deviation reduces with the increase in rotations and this makes the stress becomes more uniform circumferentially. In fact, the residual axial stress in the transition region becomes compressive in nature. This is very favourable as susceptibility of PTID for failure will reduce due to this compressive stress. Fig. 9 shows the maximum and circumferentially averaged value of residual hoop stress in the transition region in both PTID and PTOD.
Fig. 9. a Average and maximum residual hoop stress in transition region for different no.number of rotations at PTID b at PTOD. 7
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measurements, contact pressure and hoop stress in the rolled region is considered for comparison. It should be highlighted that deformed shape of the pressure tube and end fitting is well captured in the simulation. The deformed profile of the pressure tube (Fig. 10a) shows that the extrusion at inner surface of the pressure tube is higher compared to extrusion at its outer surface. This observation is also reflected in the simulation results (Fig. 10b). The total extrusion of pressure tube spool piece is observed to be ~8–10 mm in experiments whereas the simulation results for 20 rotations gives ~10.7 mm total extrusion. The average contact pressure measured in the rolled region between the pressure tube spool and end fitting spool is estimated to be ~30–40 MPa whereas the numerical results gives average contact pressure of 50 MPa. The measured hoop stress in the rolled region has been observed to be ~ -400 to −500 MPa. The calculated hoop stress in the rolled region at 40 mm from the edge is −500 MPa. The difference in simulation and experimental results can be attributed to simplified material model considering isotropic material properties. The absence of grooves, in simulation, results in higher axial expansion compared to experiments. This can be attributed to the restrain in the axial expansion due to presence of grooves. However, the deformed shape of the tube is very well captured in the simulation including the taper at the free edge of the pressure tube and length of transition region. 4. Conclusion A 3D explicit finite element analysis of pressure tube –end fitting rolled joint process is carried out. Different number of roller rotations are used to achieve the maximum radial expansion of the pressure tube. The effect of various roller paths or number of rotations used for roll expansion is investigated. The following observations can be made from the numerical analysis:
Fig. 10. Comparison of a experimental b simulation deformed profile of PT post rolling (all dimension in mm).
axial stress at PTID becomes compressive, which makes the stress field very favourable from the point of delayed hydride cracking susceptibility. The improvement in the stress field achieved by increasing the rotations show a saturating behavior, i.e., increasing number of rotations beyond 40 or even 20 rotations may not be incrementally very beneficial.
• Lower number of rotations, i.e., larger displacement per rotation • •
3.4. Comparison with experimentally observed data
•
Rolled joints are routinely tested at Bhabha Atomic Research Institute, Mumbai. The rolled joints for testing are formed by rolling pressure tube spool pieces of ~120 mm length and martensitic steel sleeve having three circumferential grooves. These rolled joints are subjected to stress measurement using sleeve removal, slit cutting and hole drilling method and dimensional measurement of the deformed pressure tube. The residual stress measurements and dimensional measurements are used to compare the numerical results of the rolled joint simulations. It is important to highlight the variability associated with the test program. The rolled joint formed in workshop has variability in input conditions, like the clearance between the pressure tube and end fitting, yield strength of the components, positioning of the rollers etc. Further, uncertainty also exists in stress measurement technique. It is important to note that simulations considered for comparison are modelled with zero clearance between the components and material model considers isotropic yield and isotropic hardening behavior instead of the anisotropic behavior of the pressure tube. The grooves present on the end fitting inner surface are not assumed to be present in the simulation. In view of the above discussion, it should be clarified that the results can only be compared in an average sense. The simulation results for 20 rotation case is compared with rolled joint measurements. This case is decided because the actual rolling operation takes around 22 rotations. The maximum applied radial displacement of the simulation is fixed so as to match the thickness reduction achieved in the actual rolled joint. The dimensional
• •
leads to larger yielding per rotation and hence higher residual stresses. With increase in the number of rotations, the compressive stress in the rolled region reduces. The variation of the residual stress components along the circumference reduces and becomes uniform. The residual hoop and axial stress in the transition region, maximum as well as average value shows a decreasing trend as the number of rotations increases Similar to the rolled region, increase in the number of rotations make the residual stress in transition region more uniform, i.e., amplitude of residual stresses across the circumference reduces. The contact pressure developed between the pressure tube and the end fitting reduces as the number of rotation increases. However it is compensated by the presence of three grooves in the actual joint. The coefficient of friction between the end fitting and pressure tube affects the stresses developed in the pressure tube and the endfitting.
Declarations of interest None. Acknowledgements The author wish to acknowledge support from Atomic Energy Regulatory Board, Mumbai for carrying out the work, National Metallurgical laboratory, Jamshedpur for providing uniaxial stress strain data, Shri S. K. Sinha, Head, Reactor Engineering Division, BARC for providing all the support towards understanding the rolled joint formation process and discussions related to comparison with experimental findings. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 8
International Journal of Pressure Vessels and Piping 177 (2019) 103975
R.J. Singh, et al.
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