LIU Wencheng, WANG Shuqing, , BU Jiarun, and DING Xindong

1)College of Engineering, Ocean University of China, Qingdao 266100, China

2)Shandong Provincial Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao 266100, China

Abstract The accurate prediction of bending stiffness is important to analyze the buckling and vibration behavior of reinforced thermoplastic pipes (RTPs)in practical ocean engineering. In this study, a theoretical method in which the constitutive relationships between orthotropic and isotropic materials are unified under the global cylindrical coordinate system is proposed to predict the bending stiffness of RTPs. Then, the homogenization assumption is used to replace the multilayered cross-sections of RTPs with homogenized ones. Different from present studies, the pure bending case of homogenized RTPs is analyzed, considering homogenized RTPs as hollow cylindrical beams instead of using the stress functions proposed by Lekhnitskii. Therefore, the bending stiffness of RTPs can be determined by solving the homogenized axial elastic moduli and moment of inertia of cross sections. Compared with the existing theoretical method, the homogenization method is more practical, universal, and computationally stable. Meanwhile, the pure bending case of RTPs was simulated to verify the homogenization method via conducting ABAQUS Explicit quasi-static analyses. Compared with the numerical and existing theoretical methods, the homogenization method more accurately predicts the bending stiffness and stress field. The stress field of RTPs and the effect of winding angles are also discussed.

Key words reinforced thermoplastic pipes; bending stiffness; pure bending case; homogenization assumption; stress analysis

1 Introduction

Reinforced thermoplastic pipes (RTPs)have attracted the attention of many researchers in ocean engineering because of their excellent properties, including corrosion and pressure resistance, thermal insulation, and high stiffnessto-weight ratio (Dalmolenet al., 2009; Martinset al., 2013;Baiet al., 2014; Tohet al., 2018; Louet al., 2020; Liuet al., 2021; Wanget al., 2021). As shown in Fig.1, RTPs are composed of a liner, fiber-reinforced composite laminates, and coating. The liner and coating made of isotropic materials are used to protect composite laminates from corrosion by transported fluids and saline water in the marine environment, whereas the composite laminates are the main layers that bear various loads (Yuet al., 2015a; Wanget al., 2020).

To understand the mechanical response of RTPs subjected to bending loads, many researchers have conducted far-reaching research into theories of hollow composite cylinders in the past decades. Lekhnitskii (1981)proposed the theory of elasticity of an anisotropic body, in which he used differential equations to determine the stresses of single-layered composite cylinders under the pure bending case. Jolicoeur and Cardou (1994)extended Lekhnitskii’s work to a more general situation of multilayered composite cylinders, in which the theoretical expression of bending stiffness was derived based on the pure bending case. The theoretical expression of bending stiffness considered the continuity condition at the interface of each layer. However, for laminates with 0? or 90? winding angles, the theoretical expression of bending stiffness resulted in numerical instability during matrix calculation because unknowns were more than equations (Zhang and Hoa,2012). To overcome this problem, Zhang and Hoa (2012)established a limit-based approach, in which the Taylor series expansion is used to find and replace equations.However, this method was only designed for composite cylinders with cylindrical orthotropy [90?/0?], and its effectiveness on RTPs with different winding angles and materials was not proven. Meanwhile, Sunet al.(2014b)constructed the homogenization theoretical model of RTPs subjected to bending loads based on the frameworks of Lekhnitskii (1981)and Jolicoeur and Cardou (1994). Similarly, for composite grating panels, Rafieeet al.(2021)and Yazdanparast and Rafiee (2020)developed a homogenization method for replacing the lattice structure with an equivalent orthotropic homogeneous plate. The reliability of this method was provenviabending tests. Menshykova and Guz (2014)analyzed the stress distribution of pipes with a thin homogeneous inner layer and obtained the theoretical expression of bending stiffness, which was based on the stress function proposed by Natskuiet al.(2003). Presently, most theoretical expressions of bending stiffness and stress distribution take a complex form and are derived from the differential stress functions. However, these theoretical expressions would lead to numerical instability during matrix calculation when RTPs have different winding angles and materials. Therefore, these theoretical expressions are not easy to use.

Fig.1 Reinforced thermoplastic pipe.

For experimental research, Zhang (2018)conducted fourpoint bending tests on pipes to predict the maximum residual cross-sectional deformation. However, four- and threepoint bending tests would lead to high local deformation because highly concentrated transverse forces were used to introduce bending moments (Saga, 2007; Derisi, 2008),thereby influencing the final result. To address this problem, Shadmehri (2012)and Geuchy Ahmad and Hoa (2016)used the pure bending test rig to smoothly introduce bending moments. This large machine was particularly designed for pure bending tests. For numerical simulation, Yuet al.(2015a, 2015b)created a finite element model with a free end and fixed end using ABAQUS. The rotation angle was applied to the free end of RTPs to introduce bending moments and predict the structural response. Wanget al.(2018)applied the same simulation method and used the UMAT subroutine to investigate the effect of plasticity. Arjomandi and Taheri (2012)conducted linear eigenvalue analyses of pipes to predict the buckling response of pipes and used nonlinear analyses to calculate the post-buckling response.In the aforementioned open literatures, compared with theoretical and experimental methods, numerical simulations could provide reliable predictions for pipes subjected to bending loads. Meanwhile, because of their ease of use and high maturity, numerical simulations are widely used to solve this problem.

Furthermore, many excellent studies in the open literature focus on other cross-sectional properties aside from bending stiffness. The novel methods and ideas in these studies can also provide a basis for analyzing bending stiffness. Rafiee and Habibagahi (2018a, 2018b)assessed the stiffness of glass fiber-reinforced plastic (GFRP)pipes subjected to compressive transverse loading using finite element modeling, which considered the large deformation and inelastic behavior of materials. The transverse compressive loading tests were also conducted. The numerical and experimental studies agreed well with each other. Malekiet al.(2019)investigated the effect of delamination on the stiffness reduction of composite pipes using the cohesive zone method and determined that the low angles of the delamination center lead to stiffness reduction. Rafiee and Ghorbanhosseini (2020, 2021)analyzed the long-term properties of GFRP pipes under compressive transverse loading. Their study was conducted on the macro and micro levels, which includes four stages,namely, modeling stress analysis at the macro level, longterm creep analysis in pure resin, stress analysis at the micro level to extend the creep behavior of resin to the lamina, and updating the constitutive relationship between each layer.

In this study, to obtain a readable theoretical expression of bending stiffness, a theoretical method, in which the homogenization assumption is used to replace complex cross sections with homogenized ones, is proposed. Different from studies, the bending stiffness is determined by analyzing the pure bending case of homogenized RTPs based on the Mechanics of Materials framework instead of using the stress functions proposed by Lekhnitskii (1981).Meanwhile, the stress distribution of each layer is obtained based on the connection between RTPs and homogenized pipes. Compared with the existing theoretical method established by Jolicoeur and Cardou (1994), the proposed theoretical expression of bending stiffness takes a more concise form and does not lead to numerical instability even when winding angles include 0? and 90?. To verify the accuracy of the homogenization method, the pure bending case of RTPs is simulated by conducting ABAQUS Explicit quasi-static analyses, in which symmetrical boundary conditions are applied to the middle cross sections.Compared with the existing theoretical method and the numerical method, the homogenization method exhibits universality, computational stability, and good performance in predicting bending stiffness and stress distributions. The stress distribution of each layer and the effect of winding angles are also discussed.

2 Novel Method Based on Homogenization

RTPs consist of orthotropic laminates with three directions. Normally, adjacent layers of RTPs are bonded without friction and voids. In the proposed model, the following geometric parameters, as shown in Fig.2, are used to describe the cross-sectional feature of theith layer: the internal radius,ri; the winding angle of fibers,φi; the total number of layers,n; the internal radius of RTPs,a; and the external radius,b(Liuet al., 2021).

Fig.2 Cross-sectional geometry of RTPs.

2.1 Constitutive Relationship Between Orthotropic Composites

According to the generalized Hooke’s law, for orthotropic materials, the strain-stress relationship can be expressed as a compliance matrix, which has the following form (Daniel and Ishai, 2006):

whereCis the compliance matrix. As shown in Fig.3,subscript 1 denotes the fibers’ direction and subscripts 2 and 3 denote the transverse directions in the principal material coordinate system. The elements of the compliance matrixCtake the following forms:

in whichE1,E2,E3,v21,v31,v23,G23,G13, andG12are the nine engineering constants of orthotropic materials.

As different laminates may have different winding angles, the strain-stress relationship should be transformed into the global cylindrical coordinate system, as follows(Zhanget al., 2014):

whereS(i)is the compliance matrix of theith layer in the global cylindrical coordinate system. The relationship betweenS(i)andCcan be expressed as follows (Zhanget al.,2014):

In Eq. (4), the transformation matrixT(i)can be expressed as follows (Zhanget al., 2014; Liu and Wang, 2021a):

Fig.3 Principal material coordinate system.

whereφiis the winding angles of theith layer.

Because isotropic materials have numerous planes of material symmetry through a point, the strain-stress relationship between isotropic materials could also be expressed in the form of the aforementioned matrix (Daniel and Ishai, 2006; Liu and Wang, 2021b). The winding angles of isotropic materials are assumed to be 0 when using the transformation matrix. Therefore, the liner and coating are considered in the analytical method.

2.2 Homogenization Assumption

The homogenization assumption proposed by Sunet al.(2014a, 2014b)provides a fast and efficient approach to evaluate RTPs using nine homogenized engineering constants. The homogenization assumption states that composite pipes with complex cross sections can be replaced with homogenized ones. In this assumption, all stresses and strains are assumed to be continuous. Based on the strain compatibility, the stresses of theith layer can be expressed as follows (Sunet al., 2014a):

According to Sunet al.(2014a), the variables in the coefficient matrix can be calculated using the following formula:

whereD(i)andSs(i)are related to the transformed compliance matrix of theith layer, which can be expressed as follows:

In Eq. (8),Eθ,Ez,vθr,vzr,vzθ,andGθzare homogenized engineering constants of RTPs.

Based on this assumption, RTPs with complex cross sections can be replaced with homogenized sections. Meanwhile, based on the aforementioned connection between RTPs and homogenized pipes, the stresses of each layer can be predicted once the stress distributions of homogenized pipes are obtained.

2.3 Solution of Bending Stiffness

Based on the works of Lekhnitskii (1981)and Jolicoeur and Cardou (1994), Sunet al.(2014b)combined the aforementioned homogenization assumption and stress function to analyze the pure bending case of composite pipes.The obtained formulas were determined to be the same as Eqs. (43.16)to (43.18)proposed by Lekhnitskii (1981). In this study, the homogenization assumption is combined with the Mechanics of Materials framework (Liu, 2011)instead of the differential stress functions.

In Section 2.2, the connection between original RTP models and homogenized pipes is established. Therefore,this section mainly focuses on analyzing the homogenized pipes subjected to pure bending considering homogenized pipes as beams with annular cross-sections. The homogenized pipes are analyzed based on the Mechanics of Materials framework (Liu, 2011).

When homogenized pipes are subjected to pure bending loads, as shown in Fig.4, according to the Mechanics of Materials framework (Liu, 2011), the uniaxial stress assumption is established. The uniaxial stress assumption states that the local elements of homogenized pipes subjected to pure bending loads are only compressed and stretched along the longitudinal direction. Meanwhile, no other normal and shear stresses are detected. Therefore, the homogenized axial stressσzcan be expressed as follows:

in whichEzis the homogenized axial elastic modulus andεzis the homogenized axial strain.

Meanwhile, according to the plane section assumption,the homogenized axial strainεzcan be expressed as follows:

whereρis the curvature radius and 1/ρis the curvature.As shown in Fig.4,ydenotes the distance to the longitudinal axis. Notably,yis positive on the stretched side and negative on the compressed side. Combining Eqs. (9)and(10), the homogenized axial stressσztakes the following form:

As shown in Fig.5, the homogenized axial stress is also positive on the stretched side and negative on the compressed side.

Fig.4 Schematic diagram of the homogenized pipe under the pure bending case.

Fig.5 Stress distribution on the cross-section of homogenized pipes.

Moreover, the curvature of the homogenized pipes can be expressed as follows (Liu, 2011):

whereMis the bending moment,Izis the moment of inertia of the cross section, andEzIzis the bending stiffness.For a round cross-section,Iztakes the following form:

According to Eq. (11), the axial stressσzof homogenized pipes is proportional to the ordinatey. Combining Eqs. (11)and (12)and transformingyinto the cylindrical coordinate system, the homogenized axial stress can be expressed as follows:

In Eq. (14),r(i)denotes the middle radius of theith layer.When homogenized pipes are subjected to pure bending loads, all homogenized stresses, except for the homogenized axial stressσz, are equal to 0. Therefore, according to Eqs. (6 – 8), for RTPs, the stresses of theith layer can be expressed as follows:

whereEzandvzθare derived from the axial tension case proposed by Sunet al.(2014a)and are expressed as follows:

In Eq. (16), as shown in Fig.2,Aiis the area ratio of theith layer andtiis the thickness ratio. According to Eq.(15), the hoop stress, axial stress, andθ-zshear stress of each layer along the hoop direction take a sine curve form.As the middle radius of theith layerr(i)is different for each lamina, the stress values of reinforced layers are not equal, which indicates that the stress gap could be considered in the homogenization method. Meanwhile, according to Eqs. (12)and (13), the bending stiffness derived from the homogenization theoretical model can be expressed as follows:

2.4 Flowchart of the Homogenization Method

As shown in Fig.6, this study focuses on providing a theoretical solution for the bending stiffness and stress field of RTPs subjected to pure bending loads. Because of material anisotropy, the stress field cannot be easily predicted. Therefore, the homogenized pipes are analyzed using the homogenization method to obtain the final results. The homogenization method was compiled as MATLAB code in this study. First, according to Sunet al.(2014a), the homogenized axial elastic modulus can be obtained based on the equivalence between RTPs and homogenized pipes. The details of the equivalence between RTPs and homogenized pipes are presented in the study of Sunet al.(2014a). Then, as expressed in Eq. (7), the coefficient matrixФ(i)can be calculated using the compliance matrix and homogenized axial elastic modulus.Further, the homogenized pipes are analyzed based on the Mechanics of Materials framework (Liu, 2011). According to the uniaxial stress and plane section assumptions,the homogenized stresses and expression of bending stiffness can be obtained. According to Eq. (6), the real stresses of each lamina can be obtained using the coefficient matrix and homogenized stresses.

Fig.6 Flowchart of the homogenization method.

3 The Existing Theoretical Method

Lekhnitskii (1981)obtained the solution of bending stiffness of single-layered orthotropic cylindrical shell under bending moments using a system of partial differential equations. He used free boundary and load conditions to determine the unknowns in the equations of the stress field.Using the continuity conditions of stress and displacement at the interface of adjacent layers, Jolicoeur and Cardou(1994)extended the solution to a more general case of multilayered hollow orthotropic cylinders and derived the bending stiffness as follows:

whereis the characteristic root of the stress function,is the constant calculated using the material parameters and stress functions,andare the constants determined based on the material properties, andis a parameter derived based on the boundary condition of each layer. For(Jolicoeur and Cardou, 1994):

whereis the ele ment of the reduced compliance matrix of theith layer, which can be expressed as follows(Lekhnitskii, 1981):

For:

The derivation of Eq. (18), which involves the theory of anisotropic elasticity and the solution of Cauchy-Euler differential equations, is complicated and not repeated in this study. The details are presented in the study of Jolicoeur and Cardou (1994). For ann-layered RTP, Eq. (18)has 14 ×nunknowns. In particular, to obtain, a 4n×4nmatrix must be solved. When the winding angles are 0?, this matrix would be prone to singularity, which leads to numerical instability (Zhang and Hoa, 2012). When the winding angles of isotropic materials are considered to be 0?, obtaining the bending stiffness using the method proposed by Jolicoeur and Cardou would be difficult (Zhang and Hoa, 2012). Zhang and Hoa (2012)proposed a limitbased approach, which is creative and productive, to determine. However, this limit-based approach is only designed for a particular case of cylindrical orthotropy[90?/0?]. As RTPs have more different winding angles,uniting the calculations of homogeneous and composite layers into one would be difficult. Therefore, compared with the existing theoretical method, the homogenization method has a more concise form and is more practical, universal, and computationally stable.

4 Numerical Simulations

As the composite laminates are between the liner and coating, their stress condition is difficult to observe using experimental methods. In this research, to verify the proposed model and determine the stress distributions of each layer, the commercial software ABAQUS (Dassault Systèmes, 2010)is used to simulate the pure bending case.

4.1 Finite Element Models

Three finite element models of RTPs, namely, Types A,B, and C, were created to predict the bending stiffness. As shown in Fig.7, all RTP models consisted of a liner, fiber-reinforced composite laminates, and coating. Polyethylene (PE)was used as the material of the liner and coating, whereas 60% glass fiber-reinforced PE tapes were wrapped around the composite laminates. For adjacent composite laminates, the winding angles of fibers were 55?/-55?. For each RTP model, the length and internal radius are 1000 mm and 2 in, respectively. The detailed geometric and material parameters are listed in Tables 1 and 2, respectively. Among them, the material parameters were provided by the RTP manufacturer who conducted axial tension tests (Liuet al., 2021). As shown in Fig.7, in a single lamina, 1 is the fiber orientation, 2 is the in-plane transverse direction, and 3 is perpendicular to the 1 – 2 plane.

Fig.7 Finite element models.

Table 1 Geometric parameters of the finite element models

Table 2 Material parameters of the finite element models

4.2 Simulation of the Pure Bending Case

The application of bending moments to the models using highly concentrated transverse loads in the three- and four-point bending tests would lead to large local deformation and stress concentration (Geuchy Ahmad and Hoa,2016). Therefore, in this study, a different method is used to simulate the pure bending case of RTPs. Two reference points, namely, RP1 and RP2, were created at the center of both cross sections, as shown in Fig.7, and kinematically coupled with all nodes on the corresponding cross section. Bending moments were loaded onto RP1 and RP2 with the same value but in opposite directions. Thus,the loads could be applied to the two end cross sections uniformly.

As shown in Fig.4, two ends of the theoretical model are free. To ensure consistency with the theoretical model,the boundary conditions of the numerical model were the same as that of the theoretical model. Furthermore, to avoid rigid motion and produce uniform cross-sectional deformation, symmetric boundary conditions were applied to the middle cross sections of the RTP models (Renet al.,2019). As shown in Fig.8, all nodes on the vertical axis could only move up and down, whereas all nodes on the horizontal axis could only move right and left (Renet al.,2019). As all layers were assumed to be perfectly bounded,in the numerical model, adjacent layers shared the same nodes on the interface between them (Baiet al., 2014; Tohet al., 2018; Liu and Wang, 2019a). C3D8R (eight-node linear brick elements with reduced integration)was used to determine the element type of isotropic layers. Meanwhile, SC8R (eight-node continuum shell elements with reduced integration)was used to simulate composite laminates. SC8R follows the layer-wise theory, enabling the accurate determination of ply-level stresses (Reddy, 1993).ABAQUS Explicit quasistatic analyses were conducted to obtain the bending stiffness and stress distributions of each layer (Renet al., 2013; Liu and Wang., 2019b). To reduce the inertial effect and maintain the energy ratio (ALLKE/ALLIE)at less than 5% during the analysis, a smooth step was used to control the loading rate.

Fig.8 Symmetric boundary conditions on the middle cross section.

4.3 Numerical Solution of Bending Stiffness

In this section, the bending stiffness of RTPs was calculated numerically based on the pure bending case. To ensure easy understanding of readers, the derivation process is briefly described.

According to Eq. (12), the bending stiffness can be calculated as follows:

In the numerical simulation, when the bending moment is known, the bending stiffness could be obtained once the curvature radiusρis calculated. The numerical model was obtained based on the geometric relationships between deformed and undeformed models, as illustrated in Fig.9. This study aims to provide a theoretical solution of the bending stiffness and stress field of RTPs under pure bending loads, which is limited to the elastic phase. Therefore, the numerical simulation does not involve the nonlinear behavior of RTPs under ultimate bending. The deformation is in the elastic phase and small. To describe the geometric relationships between deformed and undeformed models clearly, the deformation of RTPs in Fig.9 was enlarged.

In Fig.9, as the rotation angle of the end cross sectionθ1equals the bending angleθ2, the curvature radius takes the following form:

wherelis the longitudinal length of the finite element model andθ1can be obtainedviasetting the history output request in ABAQUS.

Therefore, the numerical solution of bending stiffness can be expressed as follows:

Fig.9 Deformation relationship in the bending case.

5 Results and Discussion

5.1 Bending Stiffness Calculated by Different Methods

In this section, the reliability of the numerical method was first verified using a two-layered composite cylinder in the open literature (Zhang and Hoa, 2012). Zhang and Hoa (2012)predicted the bending stiffness of a two-layered composite cylinder using the existing theoretical method. The two-layered composite cylinder is made of graphite/polymer with a 25-mm internal radius, and 7-mm-thick layers. The longitudinal length of the two-layered composite cylinder is 600 mm. The detailed material parameters are presented in Zhang and Hoa (2012).

The bending stiffness obtained using the existing theoretical and numerical methods is shown in Table 3. Notably, the result obtained using the numerical method is close to that obtained using the existing theoretical method, with a difference of 0.3%, which illustrates the effectiveness of the numerical method in calculating bending stiffness. When using the existing theoretical method to calculate the bending stiffness of RTPs, the matrix used to solveis prone to singularity, and the bending stiffness could not be obtained. Therefore, the numerical method is used to verify the homogenization method further in the subsequent paragraph.

The comparison of the bending stiffness of the three types of RTPs mentioned in Section 4.1 is shown in Table 4. Notably, the differences between the homogenization and numerical methods are less than 5% for the three types of RTPs. The aforementioned two-layered composite cylinder is made of graphite/polymer and only has [90?/0?]winding angles. Compared with the two-layered composite cylinder, the three types of RTPs consist of a liner, a coating, and fiber-reinforced composite laminates, which have more complicated material components and fiber orientations. Therefore, the effectiveness and computational stability of the homogenization method in predicting bending stiffness are verified. Meanwhile, compared with the existing theoretical method, the homogenization method has a more readable form and is more practical and applicable for RTPs with complex cross sections.

Table 3 Comparison with the existing theoretical method

5.2 Stress Analysis

When RTPs are under pure bending loads, as expressed in Eq. (15),,, andtake sine curve forms along the hoop direction and,, andof each lamina equal 0. Therefore, in this section,,, andare mainly discussed. As shown in Fig.10, when Type A RTPs are under 300 N m bending moments, the stress paths of each layer are created along the circumference,and the aforementioned stresses are compared.

The stress comparison between the homogenization and numerical methods is shown in Figs.11 – 16. Notably, the results obtained using the homogenization method are consistent with those obtained using the numerical method for the liner, coating, and fiber-reinforced composite laminates.This finding indicates that the homogenization method could predict the stress field of RTPs with high accuracy.As expressed in Eq. (15), the numerical and analytical results both show that the axial, hoop, and shear stresses of each layer all take a sine curve form along the circumference. The maximum stress occurs when the angleθin Fig.10 equals 90? or 270?. Whenθequals 0? and 180?, all stresses reach the lowest point and equal 0.

Fig.10 Stress path.

The axial stress is greater than the hoop and shear stresses for each layer, which indicates that the axial stress is the main stress when RTPs are subjected to pure bending loads.Particularly, the axial stresses of composite laminates vary slightly but are greater than those of the liner and coating.This finding indicates that the composite laminates play the main role in resisting loads when RTPs are under pure bending loads.

Fig.11 Stresses of the liner.

Fig.12 Stresses of the first composite lamina.

Fig.13 Stresses of the second composite lamina.

Fig.14 Stresses of the third composite lamina.

For each layer, the hoop stress is close to 0, which indicates that the oval deformation of the cross-section can be ignored at this time. However, the hoop stresses of homogeneous and composite layers have different directions.On the stretched side, the liner and coating are stretched in the hoop direction, whereas the composite laminates are compressed. On the compressed side, the liner and coating are compressed, whereas the composite laminates are stretched in the hoop direction.

Fig.15 Stresses of the fourth composite lamina.

Fig.16 Stresses of the coating.

The comparison also shows that there is no shear stress in the liner and coating. However, in the composite laminates, the shear stresses are at the same level.

Notably, the homogenization method could consider the stress gap between different reinforced layers even when the stress values shown in Figs.11 – 16 are close. The thickness of a single fiber-reinforced lamina is only 0.25 mm,which is smaller than the middle radius of theith layer.Therefore, according to Eq. (15), the stress values of adjacent layers predicted by the homogenization method would also be close.

5.3 Effects of Winding Angles

According to Eqs. (4), (8), and (17), the winding angles of fibers have a significant effect on the calculation of bending stiffness as they would influence the orientation of material orthotropy in the global cylindrical coordinate system. To investigate the effect of winding angles, the numerical and analytical models of the three types of RTPs are modeled with different winding angles. Because of the limitation of production technology, layers with winding angles less than 25?/-25? cannot be easily applied during filament winding. In this section, considering the technological limitation discussed in the open literature, the winding angles of fibers used for the pipes ranged from 25?/-25? to 90? (Rafiee and Reshadi, 2014; Rafiee and Amini,2015).

The results of Type A RTPs are listed in Table 5. The differences are all less than 5%, which indicates that the homogenization method can be applied to the prediction of the bending stiffness of RTPs with different winding angles. The universality and reliability of the homogenization method are further verified.

The change trend of bending stiffness along different winding angles is shown in Fig.17. Generally, the bending stiffness of RTPs decreases with the increase in the winding angles. For each type of RTP, the geometric parameters, including the internal and external radii are constant;thus, the moment of inertia of the cross sectionIzwould be fixed. For composite materials, the elastic modulus in Direction 1 is greater than that in Directions 2 and 3. As expressed in Eq. (17), the elastic modulus in Direction 1 has a substantial contribution to the homogenized axial elastic modulusEzwhen the winding angles are large.When the winding angles are greater than 60?, the bending stiffness decreases slowly and reaches the lowest level,which indicates that the elastic modulus in Direction 1 has only a slight contribution to the homogenized axial elastic modulus. In other words, fibers play less important roles in resisting the bending loads at this time.

Table 5 Bending stiffness of Type A RTPs with different winding angles

Fig.17 Change trend of bending stiffness along different winding angles.

As the composite laminates are the main load-bearing layers and the liner is the thickest layer of Type-A RTPs,in this section, the stresses of the composite layers and liner are mainly discussed. As discussed in Section 5.2, for each lamina, axial stress is the most important one among the different types of stresses. Therefore, for RTPs with different winding angles, their axial stresses are mainly compared and discussed.

As shown in Figs.18 and 19, the change trend of axial stress along different winding angles is consistent with that of bending stiffness. For composite layers, the greater the winding angles are, the smaller the peak values of axial stress. This finding indicates that composite layers play a less important role in resisting bending loads when the winding angles are large. When the winding angles are greater than 60?, the peak values of axial stress increase slowly and remain constant. This finding indicates that,when the winding angles are greater than 60?, the influence of fibers on resisting bending loads is limited. Fig.19 shows that the axial stress of the liner exhibits the opposite trend. This is mainly because the more bending loads the composite layers resist, the less the homogeneous layers would resist.

Fig.18 Axial stresses of the third composite layer when the RTPs have different winding angles.

Fig.19 Axial stresses of the liner when the RTPs have different winding angles.

6 Conclusions

This work focuses on solving the bending stiffness and predicting the stress field of RTPs under pure bending loads. In this study, a novel homogenization method is proposed to predict the bending stiffness and stress field.The homogenization method can be mainly divided into three steps. First, the constitutive relationships between orthotropic and isotropic materials are unified under the global cylindrical coordinate system. Then, the homogenization assumption and unified constitutive relationship are used to replace the complex cross-sections of RTPs with homogenized ones and establish the connection between original RTPs and homogenized pipes. Finally, different from previous studies, the pure bending case of homogenized RTPs is analyzed based on the Mechanics of Materials framework instead of using the stress functions proposed by Lekhnitskii (1981),i.e., homogenized RTPs are considered hollow cylindrical beams. Therefore,the bending stiffness of RTPs can be determined by solving the homogenized axial elastic moduli and moment of inertia of cross sections. Compared with the existing theoretical method proposed by Jolicoeur and Cardou (1994),the proposed theoretical expression of bending stiffness takes a more readable form and does not lead to numerical instability even when the winding angles include 0?.Therefore, the homogenization method is practical, universal, and computationally stable.

To verify the homogenization method, the pure bending case of RTPs is simulated by conducting ABAQUS Explicit quasi-static analyses, in which symmetrical boundary conditions are applied to the middle cross-sections.Compared with the numerical method, the homogenization method more accurately predicts the bending stiffness and stress distributions. For the three types of RTPs,the differences in bending stiffness are all less than 5%.

The stress analysis shows that the axial stresses of the composite layers are greater than those of the liner and coating. The comparison of hoop stress illustrates that the homogeneous and composite layers on the same side have opposite deformation trends in the hoop direction.

The effect of winding angles on the bending stiffness and stress distributions is also investigated. For RTPs with different winding angles, the homogenization method can provide an accurate prediction of bending stiffness. The comparison of the stress distributions reflects the change of the roles of the liner and composite laminates in resisting bending loads.

Acknowledgements

The authors would like to express their gratitude for the financial support from the National Science Fund for Distinguished Young Scholars, China (No. 51625902), the Taishan Scholars Program of Shandong Province, China (No.TS201511016), the Offshore Flexible Pipe Project from the Ministry of Industry and Information Technology, China,and the National Natural Science Foundation of China (No.51879249).