LIU Wencheng, and WANG Shuqing,2),*

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 Axial and hoop stiffness can describe the elastic responses of reinforced thermoplastic pipes (RTPs) subjected to axisymmetric loads, such as tension, compression, pressure, and crushing loads. However, an accurate analytical prediction cannot be provided because of the anisotropy of RTP laminates. In the present study, a stiffness surface method, in which the analytical expressions of the axial and hoop stiffness are derived as two concise formulas, is proposed. The axial stiffness formula is obtained by solving the equilibrium equations of RTPs under a uniaxial stress state based on the homogenization assumption, whereas the hoop stiffness formula is derived from the combination of the elastic stability theory, the classical lamination theory, and NASA SP-8007 formula. To verify the proposed method, three types of RTPs are modeled to conduct the quasi-static analyses of the tension and crushing cases. The consistency between numerical and analytical results verifies the effectiveness of the proposed method on the prediction of the axial and hoop stiffness of RTPs, which also proves the existence of stiffness surfaces. As the axial stiffness is proportional to the radii, the axial stiffness surface consists of a series of straight lines, which can be used to predict both thin-walled and thick-walled RTPs.Meanwhile, the hoop stiffness is more applicable for thin-walled RTPs because the proposed method ignores the proportional relationship between the homogenized hoop elastic moduli and the reciprocal radii in helical structures.

Key words reinforced thermoplastic pipes (RTPs); helical fibers; axial stiffness; hoop stiffness

1 Introduction

In recent years, reinforced thermoplastic pipes (RTPs)have attracted the attention of researchers in the fields of ocean engineering and oil exploration because of their outstanding performance, including pressure and corrosion resistance, low installation and maintenance costs, high stiffness-weight ratio, and thermal insulation (Xiaet al.,2002; Natsukiet al., 2003; Gemiet al., 2009; Sahinet al.,2009; Baiet al., 2014; Tohet al., 2018). As shown in Fig.1,RTPs generally consist of coating, fiber-reinforced laminates, and liner (Baiet al., 2015). Among these layers, the liner and coating, which are usually made of high-density polyethylene (HDPE), are used to protect the reinforced laminates from corrosion by internal and external fluids,whereas the composite laminates made of HDPE and spiral carbon fibers (or glass fibers) are the main layers that bear various loads (Yuet al., 2015). For a well-made RTP,all layers are bonded together so that friction and relative slip do not occur between adjacent layers.

As oil exploration moves toward deep-water areas, analytical research on the elastic properties of RTPs has become more significant to ensure the safety of offshore platforms, even though RTPs have been successfully used in some shallow-water areas of the Middle East and Southeast Asia (Dalmolenet al., 2009). A large number of scholars have conducted far-reaching research on theories about RTPs. Lekhnitskii (1981) investigated the stress distribution of a homogeneous anisotropic cylinder by analyzing the boundary conditions and elastic equilibrium. Based on Lekhnitskii’s work (1981), Xiaet al.(2001) proposed an analytical solution for the stress distribution of RTPs under internal pressure and determined that the stresses and deformations of laminated pipes depend strongly on the stacking sequence. Xiaet al.(2002) also investigated the pure bending case by applying the classical laminated plate theory and proposed an exact solution for the stress distribution in the radial, hoop, and axial directions. Sunet al.(2014a, b) proposed the homogenization method, which could produce nine homogenized elastic constants in the cylindrical coordinate system. This method provided a fast and reliable way to access RTPs with different stacking sequences and radii. Jones (1971, 1999) determined the critical elastic buckling pressure of composite pipes by solving Donnell-type equations (Donnell, 1933), which is more accurate for thin-walled pipes.

Fig.1 Schematic diagram of a reinforced thermoplastic pipe.

As for numerical and experimental studies, Yuet al.(2015) investigated the effect of material nonlinearity on the flexural behavior of RTPs by analyzing cylindrical shells in Abaqus/Standard and determined that the spoolability of RTPs can be improved by using a certain twoangle-ply system. Tarak?io?luet al.(2005) investigated the fatigue failure behavior of glass/epoxy ±55° filament wound pipes under internal pressure by applying experimental methods. During the experimental process, the failure modes, including whitening, leakage, and final failure conditions, were observed and analyzed. Gemiet al.(2009)used a similar method and discussed the progressive failure process of ±75° filament wound pipes under pure internal pressure. They conducted a series of experiments and investigated the failure process of glass/epoxy ±55°filament wound pipes under impact loading (Gemiet al.,2017, 2018b) and axial compression (Gemiet al., 2018a).Moreover, Gemiet al.(2020) conducted far-reaching research on the experimental methods for determining the mechanical properties of pipes, which include hardness,ring tensile, and burst tests. For the crushing case, ?zbeket al.(2020) conducted a radial compression test on glass/carbon intraply hybrid filament wound composite pipes to investigate their energy absorption capability. They also discussed the failure modes and fracture mechanisms of crushed samples and verified the influence of hybridization on crashworthiness parameters through load-displacement response. Eggerset al.(2019) conducted the same test on composite pipes and evaluated the influence of the winding angle, stacking sequence, and diameter-thickness ratios on the mechanical response. During the test, they observed that delamination is the dominant failure mode for the crushing case. Liu and Wang (2019a) simulated the crushing case of RTPs and obtained relatively accurate homogenized hoop elastic moduli by employing the‘infinite’ boundary conditions. Liu and Wang (2019b) also conducted an eigenvalue analysis to simulate the elastic buckling of RTPs and discussed the hoop stress distribution on the cross section when it turns oval-shaped. Geuchy Ahmad and Hoa (2016) conducted pure bending tests on thick-walled RTPs instead of three-point and four-point tests to avoid causing high local deformation and observed that the bending stiffness was consistent with the theoretical results. Zhang and Hoa (2012) proposed a limit-based approach to analyze the stress distribution of cylindrically orthotropic composite cylinders under pure bending. The consistency between the results and those calculated by NASTRAN illustrated the effectiveness of the limit-based approach. Baiet al.(2019) used 3D Hashin criteria to predict the progressive failure damage of composite pressure vessels in numerical simulations, in which the stiffness and strength parameters were obtained through a series of fundamental mechanical tests. The numerical predictions were consistent with the experimental results.

In the present study, to determine the axial and hoop stiffness of RTPs, an analytical method, which derives two readable formulas and describes the relationship between the elastic properties of RTPs and the material constants clearly, is proposed. The axial stiffness formula is in the frame of the homogenization theory, and the hoop stiffness formula combines the classical elastic stability theory and the NASA SP-8007 formula. The tension and crushing cases are simulated by three types of RTPs to verify this method. Compared with the numerical method,the proposed method can produce accurate results. Both methods can predict the stiffness surfaces for each type of RTP, and the features of these surfaces are investigated.

2 Novel Method Based on the Homogenization of RTPs

As shown in Fig.2, RTPs are a mixture of matrix and fibers and can be considered the assembly of multiple concentric cylindrical laminates. The geometric parameters shown in Fig.2,i.e., inner radiusa, outer radiusb, and lengthl, are used to describe the geometric features of RTPs. Moreover,φidenotes the winding angle of fibers in theith layer. For the homogenous liner and coating, the winding angles are seen as 0° in the analytical model because isotropic materials can be considered to have numerous symmetric planes through a point (Daniel and Ishai,2006). For high-quality RTPs, all layers are bonded together without delamination and voids. This section aims to predict the axial and hoop stiffness theoretically based on the homogenization of RTPs.

Fig.2 Geometric description of a RTP.

2.1 Homogenized Axial Elastic Moduli

whereФ(i)is the coefficient matrix (described in detail in Appendix A), which is used to build the kinematic relationships between original RTPs and homogenized pipes.Meanwhile, the strains of theith layer can be calculated as follows:

When homogenized pipes are under the uniaxial stress state, as shown in Fig.3, the homogenized stresses, includingσr,σθ,τθz,τrz, andτrθ, can be considered to be 0.Moreover, the homogenized axial stress can be calculated as follows:

By substituting Eq. (3) into Eq. (1), the stress state of theith layer can be obtained as follows:

wherevzθandEzare the homogenized elastic constants,which are unknowns and can be calculated using the following equilibrium equations of RTPs under the uniaxial stress state (as shown in Fig.3):

In Eq. (7), the first term shows that the total axial force equals the sum of axial tension loaded on each layer,whereas the second term shows that the total force in the hoop direction equals 0. By substituting Eq. (4) into Eq.(7),Ezandvzθare obtained and take the following form:

Finally, by substituting Eq. (8) into Eq. (4), the stresses of each layer can be expressed as follows:

Fig.3 Schematic diagram of a RTP subjected to axial tension.

2.2 Homogenized Hoop Elastic Moduli

The homogenized hoop elastic moduli are also difficult to determine by theoretical derivation. Liu and Wang (2019a)conducted far-reaching research on this subject and proposed an elastic-stability-based method. This method combines the classical elastic stability theory of cylinders (Timoshenko and Gere, 1961), NASA SP-8007 (1968) formula, classical lamination theory, and Donnell-type stability differential equations (Donnell, 1933). This method assumes that the critical pressure of homogenized pipes equals the critical pressure calculated using the classical lamination theory and Donnell-type stability differential equations when buckling collapse occurs. According to the elastic stability theory (Timoshenko and Gere, 1961;Tanget al., 2016), the critical pressure of homogenized pipes can be calculated as follows:

wheretis the thickness of RTPs andRis the mean radius.

Meanwhile, for composite pipes, the critical pressure calculated using the NASA SP-8007 (1968) formula can be expressed as follows:

By combining Eqs. (10) and (11) and considering a modification factor of 0.75 (NASA SP-8007, 1968), the homogenized hoop elastic moduli can be derived as follows (Liu and Wang, 2019a):wheretis the thickness of RTPs. In the classical lamination theory,A22,B22, andD22are the extensional stiffness component, the extensional-bending coupling stiffness component, and the bending stiffness component, respectively(Madenci and ?zütok, 2020). Thus, Eq. (12) can take the extensional response, the extensional-bending coupling response, and the bending response of fiber-reinforced laminates into consideration.

In Eq. (12),A22,B22, andD22take the following form(Madenci and ?zütok, 2017, 2020; Liu and Wang, 2019a):

whereziandzi-1are defined in the basic laminate geometry shown in Fig.4 (Xu, 2013; Liu and Wang, 2019a) andS22is an element of the transformed stiffness matrix that takes the following form (Jones, 1999):

wheres11,s12,s22, ands66are elements of the stiffness matrix (defined in Appendix A) that can be expressed as follows (Jones, 1999; Vinson and Sierakowski, 2004; ?zütok and Madenci, 2017):

where the subscripts of the engineering constants denote the principal material directions in the material coordinate system, as shown in Fig.5.

Eqs. (12) and (15) illustrate that the homogenized hoop elastic moduli are only relevant to the engineering constantsE1,E2,v12, andG12, asv21is dependent onE1,E2,andv12. Moreover, these equations indicate that the homogenized hoop elastic moduli are dependent on the total thickness of pipes, the stacking sequence, and the thickness of each lamina.

Fig.4 Geometric feature of an n-layered laminate.

Fig.5 Principal material directions in the material coordinate system.

2.3 Axial and Hoop Stiffness of RTPs

According to the open literature (Féret and Bournazel,1987; Liu, 2011; Renet al., 2013), given the uniaxial stress state shown in Fig.3, the axial stiffness of RTPs can be defined as follows:

whereKzis the axial stiffness,Fis the axial tension acting on both ends of RTPs, andAis the total cross-sectional area. As shown in Fig.2,lis the length of a RTP, and Δlis the longitudinal elongation.

The hoop stiffness of RTPs can be defined by the crushing case, as shown in Fig.6. When the plane strain condition is ignored, the radial displacement of the top pointw0can be derived according to the classical elastic stability theory, as follows (Timoshenko and Gere, 1961; Liu and Wang, 2019b):

From Eq. (17), given thatA22,B22, andD22are only related to the stacking information, the hoop stiffness is proportional to the third power of 1/R.

The axial stiffnessKzand the hoop stiffnessKθdescribe the elastic responses of RTPs under tension, compression,pressure, and crushing loads. As expressed in Eqs. (16) and(18), the linkages between the cross-sectional mechanical properties (i.e.,KzandKθ) and the material parameters of laminates are established. Notably, the unit of the axial stiffnessKzisN, whereas the unit of the hoop stiffnessKθisPabased on the definition previously presented.

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Fig.6 Schematic diagram of a ring under crushing loads.

Fig.7 Flowchart of the stiffness surface method.

The proposed method is compiled in MATLAB code,and the implementation process of this method is summarized in the flowchart shown in Fig.7. First, the geometric parameters are inputted in the principal material coordinate system to establish the compliance transformation matrix and stiffness transformation matrix. Then, the material parameters are inputted to establish the compliance and stiffness matrices in the principal material coordinate system.Subsequently, in the global coordinate system, the compliance and stiffness matrices are calculated. The axial stiffnessKzcan be obtained by calculating(i)sD, and the hoop stiffnessKθcan be obtained by calculating the reduced stiffness matrix. Subsequently, the winding angles and inner radius are updated to prepare for the next calculation. Finally, the analytical results are used to plot the stiffness surfaces and evaluate the mechanical properties of RTPs until it has a sufficient number of points.

3 Numerical Simulations

3.1 Finite Element Models

In the present study, the commercial software Abaqus(Dassault Systèmes, 2010) was used to conduct the numerical analysis. As shown in Table 1, three sets of finite element models including Type A, Type B, and Type C,were created with different cross-sectional parameters. At the same time, the materials were the same for all models,as listed in Table 2, in which HDPE was selected as the material for the liner and coating, and T300/934 (Sunet al.,2014a) was selected as the composite ply. For fiber- reinforced layers, the winding angles of fibers in adjacent layers were the same but in opposite directions, which means that the fibers in one layer were in the clockwise direction and those in the other layer were in the counterclockwise direction.

For composite laminates, the element type SC8R (eightnode continuum shell elements with reduced integration)was used to simulate composite laminates layer by layer as it is more realistic than the shell element type S4R and more efficient than the solid element type C3D8R (eightnode linear brick elements with reduced integration) (Baiet al., 2016). The liner and coating were simulated with C3D8R because this element type exhibits a good performance in solving contact problems. To predict the axial and hoop stiffness of RTPs, the axial tension and crushing cases were simulated, respectively.

Table 1 Geometric parameters of the finite element models

Table 2 Material parameters of the finite element models

3.2 Simulation of the Axial Tension Case

To minimize the boundary effect, the finite element models used in the axial tension case were 1000 mm long in the longitudinal direction. As shown in Fig.8, two reference points, RP1 and RP2, were created at the centers of both ends and kinematically connected with all nodes on the corresponding cross section by setting the coupling constraint in Abaqus. RP1 was fixed, whereas RP2 could only translate along the longitudinal direction and was loaded with tension. Abaqus/Explicit quasi-static analyses (Ren,2015) were conducted using the finite element models to obtain the axial stiffness. Notably, the ratio between the kinetic energy and the internal energy should be controlled at less than 5% throughout the entire calculation process,which means that the inertial effect can be ignored, and the accuracy of the results can be guaranteed (Ren, 2015).

3.3 Simulation of the Crushing Case

The crushing case was simulated in the present study to predict the hoop stiffness of RTPs (Liu and Wang, 2019a,b), which has been verified by crushing experiments on the pressure armor layer of unbonded flexible risers (Ren,2015). Meanwhile, the crushing test is recommended as one of the five basic tests for pipes by ASTM Specification D3517-04 (2004).

To improve the computational efficiency and save time,the length of RTP models used in the crushing case was set as 150 mm, and the ‘infinite’ boundary conditions were used to eliminate the boundary effect by applying kinematic coupling on the corresponding nodes of two cross sections (Liu and Wang, 2019b). As shown in Fig.9, uniform pressure was applied to the top plate, which could only translate up and down. Meanwhile, the bottom plate was fixed to produce the reaction force. Then, according to Eq. (18), the hoop stiffness could be obtained using the radial displacementw0and the crushing loadP, which is equal to the area of the top plate multiplied by the uniform pressure.

Fig.8 Finite element models.

Fig.9 Schematic diagram of the crushing case.

4 Results and Discussion

4.1 Comparison of the Axial Stiffness Calculated Using Different Methods

Table 3 Comparison between 12-in. Type-A RTPs with different winding angles

The differences between the numerical approach and the proposed method are all less than 6% and even less than 1%, which indicates that the proposed method can predict the axial stiffness of RTPs accurately. As the proposed method is based on formulas and does not require building specific model for each case, it is time-saving. Less than 1 s is needed by the proposed method to predict the axial stiffness of a RTP, whereas many hours are needed by the numerical method for modeling and computation.

To verify the effectiveness of the analytical model for thick-walled RTPs, three types of RTP finite element models with 3-in. diameters and 55°/-55° winding angles were created. The results are listed in Table 4. Notably, for thickwalled pipes, the results obtained using the analytical method are consistent with those obtained using the numerical method.

In Figs.10, 11, and 12, the surfaces are obtained by the proposed method, and the points are predicted by the numerical method. The consistency between numerical and theoretical results shows that a curved axial stiffness surface exists for RTPs. Notably, the axial stiffness decreases with the increase in the winding angles and the elastic moduli in 1direction are greater, which contributes significantly to the axial homogenized moduli when the winding angles are close to 0°.

Fig.10 Axial stiffness surface of Type-A RTPs.

Fig.11 Axial stiffness surface of Type-B RTPs.

Fig.12 Axial stiffness surface of Type-C RTPs.

Table 4 Comparison of the axial stiffness of thick-walled RTPs

Table 5 Change trends of the material parameters for Type-A RTPs

4.2 Features of the Axial Stiffness Surface

For each type of RTPs with the same winding angle,the axial stiffness is proportional to the radii, as shown in Figs.10, 11, and 12. Taking Type-A RTPs as an example,the points on the axial stiffness surface are projected on theY-Zplane, as shown in Fig.13. For RTPs with different winding angles, the results obtained using the proposed method are consistent with those obtained using the numerical method.

Fig.13 The straight lines in the axial stiffness surface of Type-A RTPs.

wherehiis the thickness of theith layer andHis the total thickness of RTPs. From Eq. (19), the radii have few influences on the determination of the area ratioAi; thus,Airemains constant basically. Because of this, the homogenized axial elastic moduli are equal for any type of RTPs with the same winding angle, even if their radii are different.

Second, the axial stiffness can be expressed as follows:

AsHis relatively small in comparison with the inner radiusa,H2can be ignored. According to Eq. (20), the axial stiffness is proportional to the radii for any type of RTPs with the same winding angle, which indicates that the axial stiffness surface consists of a large number of straight lines.

4.3 Hoop Stiffness Surface of RTPs

Similarly, the aforementioned three types of RTPs were modeled with different radii and winding angles of fibers to analyze the effects of thickness-radius ratios and winding angles on the hoop stiffness of RTPs. The hoop stiffness of 10-in. Type-B RTPs with different winding angles of fibers are listed in Table 6. As shown in Table 6, the results obtained by the proposed method are consistent with numerical results, and the differences are close to 10% for most RTPs.

Table 6 Comparison between 10-in. Type-B RTPs with different winding angles

The hoop stiffness of Type-A, Type-B, and Type-C RTPs with different winding angles and radii is shown in Figs.14, 15, and 16, respectively. With the winding angles plotted in theX-axis, the thickness-radius ratio plotted in theY-axis, and the hoop stiffness plotted in theZ-axis, a curved hoop stiffness surface can be observed to exist for each type. The hoop stiffness reaches the peak as the thickness-radius ratio reaches the maximum and the winding angles are close to 90° because the elastic moduli in 1 direction contribute the most to the homogenized hoop elastic moduli when the winding angles are close to 90°. Meanwhile, as expressed in Eq. (18), the hoop stiffness is proportional to the third power of 1/R.

4.4 Effect of Thickness-Radius Ratios on the Hoop Stiffness Surface

As shown in Figs.14, 15, and 16, the differences between numerical and analytical results increase with the increase in the thickness-radius ratios. The main reason is that the homogenized hoop elastic moduli calculated by Eq. (12) are equal for the same type of RTPs with different thickness-radius ratios, as Eq. (12) is only related to the material parameters and thickness of plies. The homogenized hoop elastic moduli have a proportional relationship with the reciprocal of the radius because of the homogenization process (Liu and Wang, 2019a). According to the numerical simulations and experiments in the open literature (Liu and Wang, 2019a; Renet al., 2019), a proportional relationship exists between the homogenized hoop elastic moduli and the reciprocal radii in helical structures, which can be expressed as follows (Liu and Wang,2019a):

Fig.15 Hoop stiffness surface of Type-B RTPs.

Fig.16 Hoop stiffness surface of Type-C RTPs.

whereRis the mean radius. As shown in Fig.17,Kcis cross-sectional bending stiffness,Iis the inertia moment per unit length of the cross section,krelies on the winding angles and inertia moment in the longitudinal section and is a constant, andnis the number of wires or fibers in the layer (S?vik and Ye, 2016).

The proposed method could not consider the proportional relationship. As a result, the hoop stiffness calculated using the proposed method has a linear relationship with the cube of the thickness-radius ratio (t/R)3, whereas the numerical results are proportional to the fourth power of the thickness-radius ratio (t/R)4. It leads to greater differences when the thickness-radius ratio increases and reveals the complexity of the analytical prediction of the hoop stiffness of RTPs.

Fig.17 Description of Keq and Kc.

5 Conclusions

This work focuses on the prediction of the axial and hoop stiffness of RTPs. A new method is proposed, in which the axial stiffness formula is based on the homogenization assumption, and the hoop stiffness formula combines the elastic stability theory and the NASA SP-8007 formula. Two concise formulas are obtained to describe the elastic properties of RTPs, which could improve the computational efficiency and save time in comparison with the numerical method. The tension case of RTPs was simulated by conducting quasi-static analyses to determine the axial stiffness, whereas the crushing case was employed to calculate the hoop stiffness. Compared with the numerical method, the proposed method exhibits a good performance in predicting the axial and hoop stiffness, as the differences of Eq. (16) are less than 6% and the differences of Eq. (18) are close to 10%.

The results predicted by the numerical method and the proposed method are reliable for RTPs with different winding angles because of their consistency, which verifies that axial and hoop stiffness surfaces exist for each type of RTPs. For the RTPs used in the present study, the axial stiffness decreases dramatically when the winding angles range from 0° to 50° and subsequently remain relatively steady, whereas the hoop stiffness increases smoothly with the increase of winding angles. This study also shows that the axial stiffness is proportional to the radii for each type of RTPs with the same winding angle, and the hoop stiffness is proportional to the fourth power of the thicknessradii ratio. It means that the axial stiffness surface consists of a series of straight lines in theZ-Yplane, and the hoop axial stiffness surface consists of a series of biquadratic curves in theZ-Yplane.

The advantages of the proposed method are quite clear:1) It can provide accurate and reliable analytical predictions of the axial and hoop stiffness of RTPs, regardless of different winding angles and thickness-radius ratios. 2)It is time-saving and derives two readable formulas. 3)The stiffness surfaces can be used to analyze and evaluate the cross-sectional mechanical properties of RTPs subjected to tension, compression, pressure, and crushing loads.

Acknowledgements

This work is supported by the National Science Fund for Distinguished Young Scholars, China (No. 51625902),the Offshore Flexible Pipe Project from the Ministry of Industry and Information Technology, China, and the Taishan Scholars Program of Shandong Province, China (No.TS201511016).

Appendix A

The relationship between strain and stress in the principal material coordinate system, as shown in Fig.4, can be expressed as follows (Daniel and Ishai, 2006; Shenet al.,2013; ?zütok and Madenci, 2017; Madenciet al., 2020b):

where the elements of the compliance matrix can be expressed as follows (Daniel and Ishai, 2006):

The stiffness matrixCis the inverse of the compliance matrix and can be expressed as follows:

wheresis the stiffness matrix.

A transformation matrix is used to determine the strainstress relationship in the cylindrical coordinate system, as shown in Fig.2, and it takes the following form (Zhanget al., 2014; Madenciet al., 2020a):

Therefore, the strain-stress relationship in the cylindrical coordinate system can be expressed as the transformed compliance matrix (?zütok and Madenci, 2013):

According to Sunet al.(2014a), in Section 2.1, the variables in the coefficient matrixФ(i)can be calculated as follows: