## Abstract

The effects of manufacturing process on mechanically lined pipe structural performance are investigated. Alternative manufacturing processes are considered, associated with either purely hydraulic or thermo-hydraulic expansion. The problem is solved numerically, accounting for geometric nonlinearities, local buckling phenomena, inelastic material behavior and contact between the two pipes. A three-dimensional model is developed, which simulates the manufacturing process in the first stage of the analysis and, subsequently, proceeds in the bending analysis of the lined pipe. This integrated two-stage approach constitutes the major contribution of the present research. Thermo-hydraulically expanded lined pipes are examined, with special emphasis on the case of partially heated liners, and reverse plastic loading in the liner pipe wall has been detected during depressurization. Furthermore, the numerical results show that the thermo-mechanical process results in higher mechanical bonding between the two pipes compared with the purely mechanical process and that this bonding is significantly influenced by the level of temperature in the liner pipe. It is also concluded that the value of initial gap between the two pipes before fabrication has a rather small effect on the value of liner buckling curvature. Finally, numerical results on imperfection sensitivity are reported for different manufacturing processes, and the beneficial effect of internal pressure on liner bending response is verified.

## 1 Introduction

Hydrocarbon mixtures may contain several corrosive ingredients, such as hydrogen sulfide (H_{2}S) or hydrogen chloride (HCl), carbon dioxide (CO_{2}), and water. In order to ensure the structural integrity of the pipeline against corrosion, bi-metallic pipes are produced, containing a thick-walled low-carbon steel (“outer pipe”) providing strength, and a thin layer (“liner pipe”) from a corrosion-resistant alloy (CRA) material, which is fitted inside the outer pipe, resulting in a cost-effective solution instead of producing pipelines from stainless steel or nickel alloy. Bi-metallic pipes can be fabricated by a thermo-mechanical or purely mechanical manufacturing process [1,2]. The former method results in hoop compression of the liner and “mechanical bonding” of the two pipes, whereas in the latter method, bonding of the lined bi-metallic pipe depends on the amount of expansion of the two pipes. Experimental work, investigating the corrosion resistance and sustainability between metallurgically and mechanically bonded pipes in sour environment, was conducted by Chen and Petersen [3] indicating deterioration of the corrosion performance of the metallurgically bonded pipes due to heat treatment. Additional experimental results are published on the corrosive performance of thermo-hydraulically expanded lined pipes, for different liner pipe materials, in a period of exposure [4].

The manufacturing process of hydraulically expanded lined pipes has also been simulated analytically [5–9]. In these works, the problem was solved as two-dimensional, assuming for simplicity (a) hydraulic expansion of both pipes up to a final elastic deformation in the outer pipe, (b) elastic perfect-plastic liner material, and (c) a relatively small-initial radial gap of both pipes in order to achieve mechanical bonding after the depressurization. These analyses result in analytical expressions of the contact pressure of both pipes after the manufacturing process, with respect to the applied internal pressure. However, in such a manufacturing process, a residual radial gap has been reported by industrial suppliers in the lined pipe after the depressurization, in contrast with previous publications [10,11] referring to mechanical bonding of elastically expanded lined pipes. In addition, the effect of different temperature level of the lined pipe, due to the operational conditions, on contact pressure of lined steel pipes was investigated by Zeng et al. [12]. Furthermore, analytical expressions for the diameter change and the hoop stress in the liner, at the end of each step of the thermo-hydraulic manufacturing process, have been presented by Focke et al. [13] assuming plane strain conditions with bi-linear stress–strain response of the liner pipe and have been compared with numerical results.

In several instances, lined pipes are installed in deep-water using the reeling method, in which the pipeline is subjected to severe bending into the plastic region, resulting in significant stresses and strains, as well as cross-sectional ovalization, which may cause local buckling of the thin-walled liner pipe. Experimental work on the mechanical behavior of lined pipes subjected to “reeling” loading conditions has been conducted by Focke [14] on lined pipes with and without pre-stressing. Additional experiments and numerical results on liner wrinkling of lined pipes under axial compression and four-point bending have been presented in Refs. [15–18], whereas further experimental and numerical results have been reported by Tkaczyk et al. [19] and Toguyeni and Banse [20]. A detailed numerical investigation of liner pipe buckling, identifying two sequential bifurcations and examining the effect of several geometric and material parameters, has been presented by Vasilikis and Karamanos [21,22]. In these publications, the manufacturing process was considered by applying an initial compressive hoop stress on the model to simulate with a simpler manner the residual stresses induced by the manufacturing process and result on the final mechanical bonding. This assumption does takes into account the effects of the manufacturing process on the material properties and the corresponding severe plastic deformation of the liner pipe in an indirect manner. Further results on the mechanical response of lined pipes have been presented in Refs. [23,24]. In these works, the manufacturing process (hydraulic expansion up to a plastic deformation level of the outer pipe) has been simulated using a separate axisymmetric model. Subsequently, the obtained stress/strain state of each pipe is averaged through its thickness and is inserted as initial condition in the bending finite element model.

The present work is aimed at extending the work in Refs. [21–24], through the development of an efficient finite element model that incorporates the simulation of the manufacturing process of mechanically bonded lined pipes and proceeds to the modeling of their bending behavior as the second stage of the analysis. The proposed model accounts for two methodologies of lined pipe fabrication. The first methodology consists of hydraulic expansion of both pipes up to elastic or plastic deformation in the outer pipe, whereas the second methodology involves a thermo-hydraulic process, leading to the so-called “tight-fit pipe” (TFP). Upon completion of the simulation of the fabrication process, the present analysis proceeds in monotonic bending of the lined pipe, using the same finite element model, until structural failure of the liner occurs in the form of wrinkling. More specifically, the present analysis offers an integrated approach that employs a single finite element model, which includes the manufacturing process and the structural bending in subsequent stages. Special emphasis is given on the material model of the liner and the outer pipe. Both materials are described using advanced plasticity models obeying non-linear kinematic hardening, capable of accounting for reverse plastic loading (RPL) effects, and are calibrated with available experimental data. Parametric analyses are also conducted, considering the effect of initial radial gap of both pipes, different heating temperatures during the thermo-hydraulic expansion, geometric imperfections, and the presence of internal pressure during bending. The effect of different temperature levels, accounting for either temperature-dependent or temperature-independent material of the liner pipe, is also investigated during the thermo-hydraulic expansion.

## 2 Lined Pipe Material Properties and Numerical Modeling

The numerical simulations were conducted using general-purpose finite element program abaqus [25]. The analysis accounts for geometric non-linearities of the two pipes, whereas the materials of both pipes are considered through elastic-plastic *J*_{2} (von Mises) plasticity model with non-linear kinematic/isotropic hardening to simulate efficiently the mechanical behavior, taking into account the Bauschinger effect due to reverse loading that may occur during manufacturing or bending.

The model, shown in Figs. 1(a) and 1(b), is capable of simulating the manufacturing processes of the lined pipe, and in subsequent step, the mechanical behavior of the lined pipe subjected to bending. The model is three-dimensional simulating a lined pipe segment of length *L*. The analysis considers the half cross section of the lined pipe using symmetry with respect to the *y*-*z* plane of bending, as shown in Fig. 1. In addition, *x*-*y* plane symmetry is assumed in the *z* = 0 plane allowing only in-plane motion on the corresponding nodes, while in the *z* = *L* plane, a reference node is imposed, in which the applied rotation is coupled with the nodes of the lined pipe cross section, so that these nodes can slide on the rotated plane. The reference node at *z* = *L* is simply supported, so that it is free to move in the *z*-direction and rotate about the *x*-axis.

The thin-walled liner pipe is modeled with four-node reduced-integration shell elements (S4R), while the outer pipe is modeled with 20-node reduced-integration solid elements (C3D20R). The half-circumference of the liner pipe contains one hundred elements, whereas the outer pipe contains 50 elements around the half-circumference and two elements through its wall thickness. Seventy-five and three hundred elements of outer and liner pipe in the axial direction, respectively. A surface-to-surface interaction of the two pipes is also considered, with finite-sliding contact formulation, allowing for arbitrary separation, sliding and rotation between the surfaces, and zero friction, an assumption also used in previous works [21–24].

A lined pipe, typical for offshore pipeline applications is considered, similar to lined pipes investigated in previous publications [14,17,21,23]. It consists of a thick-walled outer pipe, made of X70 steel grade, and a thin layer inner pipe, made of stainless steel 316L. The outside diameter (*D*_{o}) and wall thickness (*t*_{o}) of the outer pipe are equal to 12.75 in. (323.85 mm) and 15.9 mm, whereas the outside diameter (*D*_{l}) and thickness (*t*_{l}) of the liner are 289.25 mm and 2.8 mm, respectively, corresponding to an initial radial gap (*g*_{0}) between the liner and the outer pipe equal to 1.4 mm (50% of *t*_{l}). In the following sections, to examine the effect of initial radial gap, the outer diameter of the liner pipe is re-adjusted accordingly. Introducing the dimensionless length parameter $\chi =(L\u2212z)/Dm,ltl$ (where *D*_{m,l} is the liner pipe mean diameter and *z* ranges from zero to *L*, shown in Fig. 1), the normalized lined pipe length is *χ* = 15. The stress–strain curve of the X70 steel material of the outer pipe is shown in Fig. 2 and is obtained from the test results reported by Herynk et al. [26]. Upon first yielding, the material exhibits negligible hardening up to 4.75%, while the Bauschinger effect occurs during reverse plastic loading. Furthermore, Young’s modulus *E*_{o} is equal to 210 GPa, the Poisson’s ratio *ν* is equal to 0.3, and the yield stress *σ*_{y,o} is 498 MPa. To simulate the material response, a cyclic plasticity model with non-linear kinematic/isotropic hardening is employed, which accounts for both the plastic plateau upon initial yielding, and the Bauschinger effect, using an appropriate definition of the hardening modulus, as described in Refs. [27,28]. The model is implemented in a user-subroutine (UMAT) for abaqus/standard, using an integration methodology proposed in Ref. [29], appropriately enhanced to account for the plastic plateau. The stress–strain curve of the liner material is shown in Fig. 3 and is based on experimental data [30]. The Young’s modulus *E*_{l} is equal to 193 GPa, the Poisson’s ratio *ν* is equal to 0.3, and the yield stress of the liner material *σ*_{y,l} is 260 MPa. To simulate the behavior of this material, a non-linear kinematic/isotropic hardening plasticity model is used, built-in into abaqus/standard. In Fig. 2, softening of the outer pipe material occurs, while in Fig. 3, the liner pipe material hardens per cycle, indicated by the numbers next to the arrows →. Finally, in the majority of numerical analyses, the material properties of both pipes are assumed to be temperature-independent, while the influence of temperature-dependent material of the liner pipe on the bending response is investigated in Sec. 5, as an attempt of refinement of the numerical model.

## 3 Simulation of Manufacturing Process

The present section of the paper investigates contact pressure changes (or the bonding stresses) due to different manufacturing processes and the effect of these processes on the mechanical behavior of the lined pipe under monotonic bending. Two different procedures of manufacturing are examined: (a) hydraulic expansion of both pipes up to elastic or plastic deformation in the outer pipe and (b) complete thermo-mechanical process of TFPs. Different values of initial radial gap (*g*_{0}) between the two pipes are examined. For simulating the manufacturing process only (without analyzing its effect on bending), a shorter version of the model is employed for reducing computational cost. In this model, a small value of *L* is employed, equal to 2% of the liner pipe diameter. The two versions of the model provided identical results for the manufacturing process.

The elastically expanded process consists of inserting the liner into the outer, followed by the application of internal pressure (*P*_{in}) up to 80% (41.1 MPa) of the plastic pressure of the outer pipe (*P*_{y,o} = 2*σ*_{y,o}*t*_{o}/*D*_{m,o}, where *D*_{m,o} is the mean diameter of the outer pipe, *D*_{m,o} = *D*_{o} − *t*_{o}). This manufacturing process will be referred to as “elastically expanded” in the sense that the word “elastically” refers to the outer pipe. As shown in Fig. 4, the liner expands initially elastically and then plastically (⓪ → ①), establishing contact with the outer pipe. The response during manufacturing is axisymmetric, and therefore, no sliding occurs between the two pipes. Subsequently, both pipes expand together (① → ②: the outer expands only linearly), followed by depressurization of both pipes (② → ③). Considering initial radial gap (*g*_{0}) values ranging from 35% to 75% of the liner wall thickness (*t*_{l}), and simulating the elastic hydraulic expansion manufacturing process, the residual radial gap (*g*_{r}) at the end of the process is calculated. It is interesting to notice that the value of *g*_{r} is smaller by one order of magnitude, compared with the value of *g*_{0}, as shown in Fig. 5, verifying the observation of industrial suppliers [30].

A variation of the above manufacturing process, consists of applying internal pressure during pressurization of the lined pipe that exceeds the plastic pressure of the outer pipe, equal to 59.9 MPa (117% of *P*_{y,o}). This manufacturing process will be referred to as “plastically expanded.” As shown in Fig. 6, the liner expands initially elastically and then plastically (⓪ → ①), until the liner comes in contact with the outer pipe. Then, both pipes expand together (① → ②), and at the end of stage 2, the outer pipe is also deformed plastically. This is followed by depressurization of both pipes (② → ③). Due to the larger elastic deformation in the outer pipe, compared with the one in the liner pipe, after depressurization, the two pipes remain in mechanical bonding, and the contact pressure depends on the initial gap. During depressurization, the liner pipe does not exhibit any wrinkles.

Furthermore, the so-called TFP process is considered in the present study, which is a thermo-mechanical process. The thermal expansion coefficients are assumed *α*_{l} = 1.62 × 10^{−5} K^{−1} [31] and *α*_{o} = 1.3 × 10^{−5} K^{−1} [14] for the liner and the outer pipe, respectively. In addition, the temperature through the thickness of both pipes is considered constant, during the thermal expansion. The outer pipe is heated first up to *T*_{o} = 680 K (⓪ → ①), as presented in Fig. 7. On this step, the liner pipe is slightly elongated, due to coupling between the outer and the liner pipe at *z* = *L* through the reference node, resulting in a small diameter reduction. Subsequently, the liner is pressurized internally (① → ②) up to *P*_{in}/*P*_{y,l} = 5.94 (31.4 MPa) (where *P*_{y,l} = 2*σ*_{y,l}*t*_{l}/*D*_{m,l} is the plastic pressure of the liner pipe), comes in contact with the outer pipe (as also presented in Fig. 7), followed by expansion of both pipes, while the temperature of the outer pipe is assumed constant in the current step. In this step, the liner pipe is heated, due to the contact with the outer pipe, up to the same temperature with the outer pipe (*T*_{l} = *T*_{o}, also referred to as the 100% case). The thermal hoop strain increases significantly, tending to increase the diameter of the liner pipe (*ε*_{h,T} = (Δ*R*_{l}/*R*_{l})|_{T}). Nevertheless, due to lateral confinement by the outer pipe, the liner hoop tension gradually decreases and hoop compression develops. This is represented by the sharp drop of the hoop stress of the liner shown in Fig. 7. The analysis also shows that RPL occurs in the liner wall during the pressurization step, due to thermal hoop expansion and the lateral confinement, and it is denoted by the horizontal arrow on the curve before the end of stage ②. After depressurization (③), there is residual hoop compression in the liner pipe, due to confinement by the outer pipe.

An alternative process of a TF pipe is also investigated (as shown in Fig. 8), in which during the pressurization step (① → ②), the liner is partially heated (denoted as TFP PH) up to *T*_{l} = 388 K, which is 57% of the outer pipe’s temperature (the term “partially” implies uniform heating of the liner pipe up to a lower temperature level from the *T*_{o}). In this case, the liner pipe exhibits less thermal hoop expansion, resulting in a smaller drop of the hoop stress, as shown in Fig. 8. RPL occurs during the depressurization step, as denoted with the horizontal arrow before the end of stage ③. At the end of the fabrication process (③), the liner pipe is in higher hoop compression stress, compared with the fully heated process (denoted as TFP FH). The initial temperature and the temperature after the depressurization step is *T*_{i} = 298 K for both pipes, for the thermo-hydraulic expansion cases.

In more detail, the residual compressive hoop stress of the liner pipe after the manufacturing process is presented in Fig. 9, showing that the TFP manufacturing process with (full or partial) heating results in higher liner compression compared with the plastic expansion process. In addition, the fully heated TFP results in lower compression stress than the partially heated TFP. In this case, during the pressurization step, the total hoop strain of the liner pipe is the sum of a mechanical part (*ε*_{h,M}), due to hydraulic expansion, and a thermal part (*ε*_{h,T}), due to the contact with the heated outer pipe (*ε*_{h,Tot} = *ε*_{h,T} + *ε*_{h,M}). The total hoop strain of the liner pipe is governed by the outer pipe due to confinement. In this case, an increase of the thermal hoop strain results in a decrease of the mechanical hoop strain. The above results indicate that the level of mechanical bonding depends on the temperature level of the liner pipe and the mechanical hoop strain at the end of the pressurization step.

An important observation refers to the material model used in the simulation of the manufacturing process. The results show that reverse plastic loading may occur during the manufacturing process, and therefore, consideration of the Bauschinger effect in material models is necessary. This implies that the use of isotropic hardening for modeling the liner pipe material may not be adequate for simulating the manufacturing process. During the manufacturing process, the maximum hoop strain in the liner at stage ② before depressurization is 1.12%, 1.51%, 1.61%, and 1.57% for elastically, plastically, fully and partially heated TF pipes, for an initial radial gap equal to 50% of liner thickness. For a gap equal to 75% of liner thickness, the maximum hoop strain is 1.61%, 1.77%, 2.10%, and 2.05%, respectively. During the pressurization, the liner pipe is subjected to tension in the hoop direction, while after the depressurization results in zero stress and elastic hoop compression for the elastically and plastically expanded lined pipes (as shown in Figs. 4 and 6). In the case of the fully heated TF pipes, during the pressurization step, and after contact between the two pipes is established, the liner pipe is gradually compressed, resulting in RPL, as indicated by the arrow (→) in Fig. 7, whereas after depressurization, the liner is elastically compressed in the hoop direction. In the case of partially heated TF pipes, after depressurization, the liner pipe exhibits reverse compressive plastic loading (RPL) as indicated in Fig. 8.

## 4 Bending Analysis of Lined Pipes

In this section, the mechanical behavior of steel lined pipes under monotonic bending is investigated, using the numerical model shown in Figs. 1(a) and 1(b), considering the effect of different manufacturing processes. The manufacturing process is simulated in the first stage of the analysis, as described in Sec. 3, and it is followed by monotonic bending in the second stage of the analysis. The following results refer to imperfection-free lined pipes, whereas imperfect lined pipes are considered in Sec. 5. The lined pipe normalized length, as already described in Sec. 2, is equal to 15, as shown in Fig. 10. In this figure, the model is mirrored (from *χ* = 0 to *χ* = −15) for visualization purposes. Previous publications [21,23] have demonstrated that, after applying bending, the liner pipe gradually detaches from the outer pipe, followed by the formation of a uniform wrinkling pattern at the compression zone leading to localized buckling with further increase of the curvature. Uniform wrinkling has been identified by Vasilikis and Karamanos [21] as the first bifurcation, followed by a second bifurcation at higher curvatures leading to a main buckle (A), with four adjacent minor buckles (B), as shown in Fig. 10, a result also verified by Yuan and Kyriakides [23,24] and more recently by Gavriilidis and Karamanos [32].

In the following, liner deformation is presented in terms of its detachment, which expresses its relative displacement with respect to the outer pipe. The maximum normalized detachment (Δ) of the liner pipe occurs at *χ* = 0 location at the main buckle (A). It is shown in Fig. 11 normalized by the wall thickness of the liner pipe (*t*_{l}), with respect to normalized curvature (*κ* = *k*/*k*_{o}; *k* = *ϕ*/*L*, where *ϕ* is the rotation applied on the reference node at *z* = *L*; $ko=to/Dm,o2$), for the different manufacturing processes analyzed earlier and for the case of initial radial gap (*g*_{0}) equal to 50% of liner thickness. The TFP with partial heating of the liner results in higher hoop compression of the liner, compared with the fully heated TFP process, also noticed in Sec. 3. This observation may explain the results shown in Fig. 11, where the abrupt detachment of the liner pipe in partially heated TFP occurs at higher curvature. In addition, despite that the plastically expanded lined pipe results in mechanical bonding, the abrupt detachment of the liner from the outer pipe occurs earlier when compared with the elastically expanded lined pipe, in which a residual radial gap (*g*_{r}) is observed. The abrupt detachment of liner pipe shown in Fig. 11 is also associated with the drop of bending moment carried by the liner pipe. The latter is represented by normalized moment *m*_{l}, defined as *m*_{l} = *M*_{l}/*M*_{o}, where $Mo=\sigma y,oDm,o2to$. This behavior is shown in Fig. 12, for each manufacturing process, for initial radial gap equal to 50% liner thickness, indicating the onset of local buckling of the liner pipe.

The “critical” curvature (*κ*_{cr}) is defined as the curvature at which the slope of the detachment-curvature diagram (*d*Δ/*dκ*) reaches its maximum, a definition introduced by Gavriilidis and Karamanos [32]. As shown in Fig. 11, the detachment of the liner pipe increases rapidly after a specific curvature value, leading to local buckling of the liner pipe. The value of “critical” curvature from the above definition corresponds to the abrupt drop of moment of the liner pipe, as shown in Fig. 12. Based on the above failure criterion, Fig. 13 presents the critical curvature of the lined pipe from each manufacturing process and for different values of initial radial gap (*g*_{0}). The results show that the TFP manufacturing process with partial heating of the liner pipe results in higher critical curvature values when compared with all the other manufacturing processes. In addition, the plastically expanded lined pipe exhibits the lowest critical curvature values, due to the excessive plastic deformation of the liner. In more detail, the critical buckling curvature of partially heated TFP is 17%, 28%, and 57% higher than the corresponding value for fully heated TFP, elastically expanded, and plastically expanded lined pipes, respectively, assuming a 50% value of initial radial gap. The results are summarized in Table 1, including the values of normalized detachment (Δ_{cr}), radius of curvature (*ρ*_{cr}), and global bending strain (*ε*_{cr} = *D*_{o}/2*ρ*_{cr}) at buckling, for each fabrication process.

In addition to the critical bending curvature (*κ*_{cr}), the manufacturing process may affect the buckling wavelength (*L*_{hw}). The results herein are reported in normalized form ($lhw=Lhw/Dm,ltl$). For the fully and partially heated TF pipes, the normalized half-wavelength is equal to 1.375, while for elastically expanded and plastically expanded pipes, the corresponding values are somewhat smaller, equal to 1.3 and 1.225, respectively. As the plastification of the liner increases, the normalized half-wavelength is further reduced. Furthermore, considering the TFP manufacturing process, the normalized half-wavelength has a somewhat lower value when compared with the value of 1.425 reported in previous works [21,32]. It should be noted though that in these works, the manufacturing process has been taken into account only indirectly, assuming an initial hoop stress. The normalized half-wavelength of the liner pipe, for each manufacturing process, is presented in Fig. 14.

## 5 Parametric Analysis

### 5.1 Imperfection Sensitivity Analysis.

In the present section, the mechanical behavior of lined pipes in the presence of geometrical initial imperfections of the liner pipe is investigated for elastically expanded, plastically expanded, and fully heated TF pipes. The shape of the initial geometric imperfection of the liner pipe is based on the buckling pattern of the imperfection-free lined pipe of the corresponding manufacturing process, normalized by the liner pipe wall thickness (*t*_{l}). The shape of the imperfection is presented in Fig. 15 has the form of buckling mode of the corresponding “perfect” pipe. Therefore, it is expected that the buckled configuration of each lined pipe (elastically expanded, plastically expanded, and fully heated TF pipes) simulates the worst case scenario for the imperfection sensitivity analysis.

In the case of elastically expanded lined pipes, after depressurization, a residual imperfection with amplitude Δ_{r} is observed, as shown in Fig. 16, in terms of the initial imperfection amplitude, while for the other two cases, the residual imperfection after the manufacturing process is negligible for the entire range of initial imperfection amplitude considered. The effect of the initial geometric imperfection, assuming the maximum value of amplitude ($\Delta 0=10%$), is presented in Figs. 17 and 18, showing its influence on the liner pipe detachment (Δ) from the outer pipe and the liner pipe moment (*m*_{l}). The solid curves correspond to imperfection-free liner, while the dashed curves refer to imperfect liner, respectively. In Fig. 19, the critical curvature values of the different examined types of manufacturing processes for the case of 50% initial radial gap (*g*_{0}) are presented. For the case of the elastically expanded and plastically expanded pipes, a significant reduction of the critical curvature value occurs with increasing initial imperfection Δ_{0}, an observation that is consistent with the results reported in previous publications [21,22]. This reduction is more significant for small values of Δ_{0} and less pronounced for larger values of Δ_{0}. It is interesting to note that a 10% initial imperfection results in a 36% and 26% decrease in critical curvature for the case of the elastically expanded and plastically expanded pipes, respectively. On the other hand, the mechanical behavior of the fully heated TF pipes is less sensitive to initial geometric imperfections, resulting in a 9% decrease for the same imperfection size (10%). The critical curvature (*κ*_{cr}) for different values of initial geometric imperfection amplitude and fabrication cases is summarized in Table 2.

### 5.2 Internally Pressurized Lined Pipes.

The effect of internal pressure (*P*_{in}) on the bending response of mechanically bonded lined pipes has been investigated by Gavriilidis and Karamanos [32], and its beneficial role on the critical curvature has been demonstrated. In the present study, lined pipes fabricated through different manufacturing procedures (namely, plastically expanded, partially heated, and fully heated TF pipes), and initial radial gap equal to 50%, are subjected to pressurized bending. The pressure is applied after the manufacturing process and is held constant during the monotonic bending process. The pressure level is 10% of the liner pipe yield pressure (*P*_{y,l} = 2*σ*_{y,l}*t*_{l}/*D*_{m,l}, where *D*_{m,l} is the mean diameter of the liner pipe). A tensile force (*F*_{p}) is also applied on the reference node in the *z* = *L* plane, equal to the applied internal pressure times the internal cross section of the liner (*F*_{p} = *P*_{in}*π*(*D*_{l} − 2*t*_{l})^{2}/4) in order to simulate the force at the two capped ends due to the internal pressure. This force is referred to as “capped-end force” and remains constant during bending, but follows the orientation of reference node (follower force).

The normalized amplitude of detachment (Δ) of the liner pipe is presented in Fig. 20, showing that in the presence of internal pressure the abrupt detachment of the liner occurs at higher values of curvature for all three fabrication procedures. In Fig. 21, the normalized moment of the liner pipe is presented, showing that the sharp drop of each case occurs at higher curvature values a result compatible with the one in Fig. 20. The critical curvature (*κ*_{cr}) of the plastically expanded, fully and partially heated TF pressurized pipes is increased by 27%, 143%, and 115%, respectively. As already reported by Gavriilidis and Karamanos [32], despite the low level of internal pressure, the critical curvature (*κ*_{cr}) increases significantly. In the case of plastically expanded lined pipes, the increase of critical curvature is considerably less pronounced, when compared with the other two manufacturing cases, due to the excessive plastic deformation of the liner pipe. The results are summarized in Table 3, including the values of normalized detachment, radius of curvature and global bending strain (*ε*_{cr}) at the buckling stage, for each fabrication process.

### 5.3 Effect of Liner Pipe Temperature in Tight-Fit Pipes.

The effect of the temperature level of the liner pipe on the final state of stress of the partially heated TF pipes is also investigated. In the previous results, the partially heated liner pipe is examined only for temperature equal to 57% (*T*_{l} = 388 K) of the fully heated TF pipes. In the present section, other temperatures levels ranging from 57% to 100% of the temperature used in the fully heated TF pipe (where the temperature of the liner pipe is equal to the outer pipe’s *T*_{o}) are investigated. In the following, the temperature will be denoted as a percent, referring to the percent of the liner temperature with respect to the temperature of the outer pipe. In this section, two types of analysis are adopted, considering (a) temperature-independent material properties of the liner pipe (as described in Sec. 2) and (b) temperature-dependent material properties of the liner pipe. The latter case (b) should be regarded as an attempt for refinement of the present model.

Figure 22 presents the normalized hoop compression of the liner pipe of partially heated TF pipes after the manufacturing process, with respect to different temperature levels of the liner on the pressurization step. The results are obtained with the assumption of temperature-independent material properties. For temperatures varying from 57% to 92% of *T*_{o}, the hoop compression of the liner decreases slightly as the temperature increases, while the residual compressive stress is higher than the yield stress (*σ*_{y,l}) of the liner. These cases follow corresponding paths of hoop stress of the liner pipe with respect to diameter change as shown in Fig. 8. For 96% and 100% temperature levels, the hoop compressive stress decreases significantly resulting in stress smaller than the yield stress. As explained previously in Sec. 3, during the pressurization step, the thermal part of the hoop strain of the liner pipe increases significantly while the mechanical part decreases resulting in plastic hoop compression. After the depressurization, the hoop stress of the liner pipe is in the elastic region, as shown in Fig. 7, justifying the sharp drop of hoop compressive stress of the liner pipe in Fig. 22. Figures 23–25 present the detachment and the corresponding moment of the liner pipe, showing the abrupt liner detachment and the corresponding sharp drop of the bending moment of the liner, in higher curvature values as the maximum temperature of the liner decreases. The results are summarized in Table 4, including the values of normalized detachment, radius of curvature, and global bending strain (*ε*_{cr}) at the buckling stage for each fabrication process.

All previous results on TF pipes have been obtained under the assumption that the material properties are not affected by the level of temperature. In the following, the mechanical behavior of the liner pipe is also investigated assuming temperature-dependent Young’s modulus (*E*_{l}), yield stress (*σ*_{y,l}), and thermal expansion coefficient (*α*_{l}) of the liner pipe, in an attempt to refine of the numerical model. The dependency of Young’s modulus, yield stress, and thermal expansion coefficient on the temperature adopted in the present study is the one proposed in Refs. [31,33,34], shown in Table 5. In Fig. 26, the normalized hoop stress of the liner pipe, after the manufacturing process, is presented for partially heated TF pipes, taking into account the temperature-dependent material properties of the liner pipe. The residual hoop compression of the liner pipe, when temperature-dependent liner pipe material properties are considered, is higher compared with the temperature-independent liner material, for temperature levels *T*_{l} varying from 57% to 92% of *T*_{o}, as the Young’s modulus and yield stress of the liner pipe decreases. However, for temperature level *T*_{l} near the 100% level, the residual hoop compression of the liner decreases, resulting in lower compression values compared with the temperature-independent case. For this high-temperature range, the thermal expansion coefficient (*α*_{l}) also increases, resulting in higher thermal hoop strain, and lower mechanical bonding.

Figure 27 shows the critical curvature (*κ*_{cr}) with respect to different temperature levels of the partially heated liner pipe, considering temperature-dependent liner pipe material properties. For temperature levels ranging from 57% to 96% of *T*_{o}, the critical curvature values are higher when compared with the corresponding *κ*_{cr} values obtained assuming temperature-independent liner pipe material properties. For the case of temperature-dependent properties, the values of Young’s modulus and yield stress of the liner are lower when compared with the temperature-independent case, resulting in higher residual hoop compression of the liner pipe, as shown in Fig. 26, and critical curvature values, respectively. For fully heated liner pipe, the critical curvature is lower in the temperature-dependent case. This is attributed to the fact that the residual hoop compression, after the manufacturing process, is 52% lower than the corresponding value obtained by the temperature-independent case.

Finally, the difference in the critical curvature value obtained considering temperature-dependent and temperature-independent properties ranges from 0.27% to 3.08%, leading to the conclusion that the assumption of temperature-independent material of the liner yields very reasonable results and can be used for obtaining reliable estimates of liner buckling curvature.

## 6 Conclusions

In the first part of the paper, purely mechanical and thermo-mechanical processes for lined pipe fabrication are simulated. In the case of elastically expanded lined pipes, and for initial radial gap ranging from 35% to 75% of the liner pipe wall thickness, a residual radial gap is observed at the end of the manufacturing process, resulting in a lined pipe with no mechanical bonding. On the other hand, plastically expanded lined pipes as well as fully and partially heated TFP result in contact between the liner and the outer pipe, associated with hoop compression of the liner pipe. The results indicate that the TFP process induces higher hoop stress than the plastically expanded lined pipe, while the mechanical bonding of partially heated TFP is higher than the one in fully heated TFP. In thermo-hydraulic expanded pipes, reverse plastic loading of the liner material has been detected during the manufacturing process, implying that a cyclic-plasticity model should be used in the finite element model.

In the second part, the bending behavior of lined pipes is investigated, accounting for their manufacturing process. The main purpose of this analysis is the calculation of the critical curvature of the liner. The results show that TF pipes buckle at higher curvature compared with the hydraulically expanded pipes, especially when partial heating of the liner occurs. In the case of plastically expanded lined pipes, the critical curvature value is lower than that of the corresponding elastically expanded lined pipes. With increasing plastic deformation of the liner, the critical curvature and the corresponding half-wavelength of the wrinkled liner decrease, implying that, during manufacturing, the plastic deformation of the liner pipe has a significant effect on its structural performance.

In initially imperfect elastically expanded lined pipes, a residual geometric imperfection is observed after manufacturing, while for plastically expanded and fully heated TF pipes, the residual imperfection is negligible. The critical curvature of hydraulically expanded pipes (both elastic and plastic) is imperfection sensitive especially for small values of imperfection amplitude. The imperfection sensitivity of fully heated TF pipes is less pronounced.

The beneficial effect of internal pressure on bending response has been verified. This benefit has been more significant in fully and partially heated TF pipes, but it is less pronounced in plastically expanded lined pipes, due to excessive plastic deformation of the liner pipe induced by the manufacturing process.

Finally, the influence of different levels of liner temperature on the bending response of partially heated TF pipes is investigated, assuming both temperature-independent and temperature-dependent material properties of the liner pipe. Under both assumptions, the residual compression and the corresponding critical curvature of the liner pipe decrease as the maximum temperature of the liner pipe increases. On the other hand, the difference in critical curvature between the two approaches is rather small, concluding that the assumption of temperature-independent properties of the liner pipe material leads to reasonable results and can be used for analyzing lined pipes.

## Acknowledgment

The present work was supported by a Ph.D scholarship awarded to the first author from the UK Engineering and Physical Sciences Research Council (EPSRC)–DTP. The authors would like also to thank Dr. Daniel Vasilikis, Rigid Pipelines Development Engineer TechnipFMC, Aberdeen, Scotland, UK, for providing the experimental data of the liner pipe, and Mr. Konstantinos Chatziioannou, Ph.D student of the University of Edinburgh, for his assistance in cyclic modeling issues.