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Burst Strength of RTP Pipeline
27 Burst Strength of RTP Pipeline Chapter Outline 1. Introduction 611 2. Experimental Analysis
612
Material Properties 612 Burst Tests 613
3. Analytical Analysis
614
Introduction 614 Coordinate Systems 615
4. Finite Element Analysis 616 5. Results and Comparison 617
1. Introduction Due to its high cost effectiveness, excellent corrosion resistance, and ease of installation, reinforced thermoplastic pipe (RTP) is now increasingly being used for onshore and offshore applications. Normally, RTP consists of one polyethylene liner, two layers of reinforced tape overwrapping the liner, and one outer polyethylene coating, as shown in Figure 27.1. The inner liner pipe and outer coating pipe are made of high density polyethylene (HDPE). The mechanical response of filament-wound structures under internal pressure has been studied by many researchers. Xia et al. [1] performed a stress analysis of multilayered filament-wound composite pipes under an internal pressure based on a 3D anisotropy elasticity theory. Kruijer et al. [2] developed a multilayer “generalized plane strain” model based on a plane strain characterization for RTP under hydrostatic pressure. Kobayashi et al. [3] proposes an elastic-plastic analysis model on the filament-wound carbon-fiber reinforced composite pipes by applying partially plastic thick-walled cylinder theory. Zheng et al. [4] presents an analytical procedure to predict the short-term burst pressure of PSP (plastic pipes reinforced by cross helicalwound steel wires) based on 3D anisotropic elasticity and the maximum stress failure criterion. An elastic solution procedure based on Lekhnitskii’s theory was developed by Onder et al. [5] to predict the burst failure pressure of the pressure vessels using the Tsai-Wu failure criterion, the maximum strain, and stress theories. A material degradation model based on the ply-discounting approach was implemented in a UMAT subroutine and applied to analyze the progression failure of a two-dimensional plate by Knight [6]. A 3D parametric finite of the cylindrical part Subsea Pipeline Design, Analysis, and Installation. http://dx.doi.org/10.1016/B978-0-12-386888-6.00027-4 Copyright Ó 2014 Elsevier Inc. All rights reserved.
612
Qiang Bai and Yong Bai
Figure 27.1 Cross section of RTP. (For color version of this figure, the reader is referred to the online version of this book.)
of a composite vessel was established by Xu et al. [7] to explore the failure evolution behavior and the failure pressure of composite vessels. Different failure criteria, such as the maximum stress by Hoffman, Tsai-Hill, and Tsai-Wu, are integrated in the main program. A comprehensive review on the recent developments of the damage and failure evolution using finite element analysis of composite laminates was reported by Liu et al.[8]. In this chapter, RTP is considered a thick cylinder, and the stress distribution is characterized as a generalized plane strain. Since the thicknesses of two reinforced tapes are relatively thin compared to that of the pipe, an assumption of uniform stresses through the thickness of the two layers is made to simplify the analysis. The material of the two reinforced tapes is considered as transverse isotropic. It is assumed that the strains in the reinforcing layers are equal to the strains in the isotropic material. The fiber failure and the matrix failure, the two failure modes used as the failure criterion, are used to determine the failure pressure of RTP. A 3D finite element RTP model is established to evaluate the relationship between the mechanical properties and the final failure pressure. To predict the damage evolution and the failure strength of RTP, the behavior of the reinforcing tape is simulated using a model proposed by Linde et al. [9], which is implemented in a user subroutine (UMAT) of the ABAQUS-standard nonlinear finite element analysis tool. The failure pressure calculated from finite element method and theoretical method are compared with the experimental burst pressure of RTP.
2. Experimental Analysis Material Properties In this analysis, two kinds of HDPEs are used for the liner, coating, and matrix. The reinforced tapes are manufactured by embedding the aramid strands in the HDPE. The HDPEs are modeled as linear elastic materials. Table 27.1 lists the secant
Burst Strength of RTP Pipeline
613
Table 27.1 Mechanical Properties of PE
Material
PE100 (k [ 1)
PE100 (k [ 4)
PE100 (k [ 2, 3)
Secant modulus, Ek (MPa) Poisson’s ratio, mk
350(E1) 0.4(m1)
390(E4) 0.4 (m4)
460 (Em) 0.4 (mm)
modulus and Poisson’s ratio used for modeling HDPEs. The secant modulus is determined based on the loading speed, temperature, and maximum strain during the burst test. Poisson’s ratio is assumed to be 0.4 according to Kruijer et al. [10]. The reinforced tapes were modeled to be linear-elastic transverse isotropic. The five independent constants (EL, ET, GLT, GTT, mLT, where E denotes Young’s modulus, G denotes the shear modulus, m denotes Poisson’s ratio, subscript L denotes the longitudina, fiber, direction, and subscript T denotes the two transverse directions orthogonal to the longitudinal direction) are listed in Table 27.2. The damage initiation properties of the reinforced tapes are shown in Table 27.3.
Burst Tests Burst tests are carried out according to ASTM D 1599-99 “Standard Test Method for Resistance to Short-Time Hydraulic Pressure of Plastic Pipe, Tubing, and Fittings” [11]. As per procedure A, a short-time loading process was applied, such that the time to failure for all the specimens was between 60s and 70s. The RTP samples are 760 mm long and terminated by steel swage fittings. The RTPs are free to deform during the tests. In addition, the pressure applied to the internal pressure of RTP is increased uniformly and continuously until the specimen fails. The experimental temperature was controlled at 23w24 C. Table 27.4 summarizes the measured burst pressures of RTP pipes. The average burst pressure of 35.3 MPa is obtained for the RTP samples. It was noted that failure of each tested specimen occurred at the RTP close to the end fittings (Figure 27.2). From Table 27.2 Transverse Isotropic Elastic Properties of Reinforced Tapes EL (MPa)
ET (MPa)
GLT (MPa)
GTT (MPa)
mLT
mTT
20,390
170
160
60
0.38
0.4
Table 27.3 Damage Initiation Properties of Reinforced Tapes sLf ;t (MPa)
sTf ;t (MPa)
770
6.0
614
Qiang Bai and Yong Bai
Table 27.4 Measured Burst Pressure of RTP Samples
Specimen no. 1 2 3 4 5 Average S-SD
Time to Failure (min)
Burst Pressure (MPa)
31 30 28 31 27
37.0 37.2 32.2 33.5 36.9 35.3 1.98
Figure 27.2 Burst samples after testing. (For color version of this figure, the reader is referred to the online version of this book.)
the burst tests, it was observed that the failure of fiber strands of the inner reinforced layer (layer 2) occurs first, which causes catastrophic damage to the entire pipe. The orientation of the crack is almost parallel to the outer reinforced layer (layer 3).
3. Analytical Analysis Introduction In the analytical analysis, RTP was considered as a thick cylinder. The deformation in the axial direction was assumed to be uniform. The stresses and strains (except for εzz
Burst Strength of RTP Pipeline
615
and szz) of the two isotropic layers are not uniform, but a function of the radius of the layer. The analysis is carried out using the following procedure: l
l
l
First, the strain and stress functions of the two isotropic layers are deduced using elastic mechanics theory. Second, the strain and stress functions can be obtained through the strain continuum condition. Finally, the unknown constants in the strain equations are determined by interface conditions and equilibrium equations.
Coordinate Systems A cylindrical coordinate system, as shown in Figure 27.3, is used for the analytical analysis. The coordinate axis r, q, and z denote the radial, circumferential, and axial directions of RTP pipe, respectively. The local material coordinate system of the reinforced tape layers is designated as (L, T, r), where L is the wound direction, T is the direction perpendicular to the aramid wire in plane, and r is the normal direction, same as in the cylindrical coordinate system. The term a is the wound angle of reinforced layer, the angle between L direction and z direction. The mathematical solutions are based on the model developed by Zheng et al. [4] for predicting the short-term burst pressure of PSP, by applying the 3D anisotropic elasticity and the maximum stress failure criterion, to calculate the short-term burst pressure of RTP pipe, except for the maximum strain failure criterion. Provided that the interfaces between the fiber yarn and PE are perfectly bonded, the strain of the aramid wire and PE in the aramid-wound direction can be considered to be equal. Because Young’s modulus of the aramid fiber is far greater than that of PE, the stresses in the aramid fiber are much greater than those in the PE. When the z r,r
θ
z
T
r
θ
α
L
σ zz
σ rr σθθ
(L,T,r) : Local material coordinate system (z,θ ,r) : Cylindrical coordinate system
Figure 27.3 Coordinate systems. (For color version of this figure, the reader is referred to the online version of this book.)
616
Qiang Bai and Yong Bai
RTP is subjected to internal pressure, aramid wires first reach their strength limits and break, resulting in the RTP losing the reinforcement of the fibers yarn and bursting in the short term. The detailed mathematical equations for the strains of each layers are found in the Bai et al. paper [12].
4. Finite Element Analysis A finite element model is also developed to simulate the mechanical properties of RTP under internal pressure. The damage of reinforced tapes is simulated using a model proposed by Linde et al. (2004) [11], which is implemented in the ABAQUS subroutine UMAT. Failure initiation and damage progression law are two major parts of the UMAT model. The failure initiation criterion is the same as failure criterion used in the theoretical analysis. In the following section, the damage progression law is described. When a fiber failure initiates, the fiber damage variable, df, evolves according to the following equation: df ¼ 1
f ;t f ;t f ;t ε11 c e½C11 ε11 ðff ε11 ÞL =Gf ; ff
[27.1]
where Lc is the characteristic length associated with the location. The evolution law of the matrix damage variable, dm, is expressed as dm ¼ 1
f ;t f ;t f ;t ε22 c e½C22 ε22 ðfm ε22 ÞL =Gm fm
[27.2]
The reinforced layers are assumed to be transversely isotropic. When the damage is progressive, the effective elasticity matrix is reduced by the two variables df and dm, as shown in Table 27.5. To improve the convergence speed, a technique based on the viscous regularization is used to regularize the damage variables. The regularized damage variables are stored as solution-dependent variables, SDV3 and SDV4, respectively.
Table 27.5 Degradation Factors Components of the Elasticity Matrix
Degradation Factors
C11, C22, C12, C33,
1 – df 1 – dm (1 – df)(1 – dm) 1
C13, and C23, and C21, and C55, and
C31 C32 C44 C66
Burst Strength of RTP Pipeline
617
Figure 27.4 Fiber damage evolution of layer 2 (SDV3 contour plot). (For color version of this figure, the reader is referred to the online version of this book.)
A 760 mm long finite element model was built in a cylindrical coordinate system using the ABAQUS/standard nonlinear finite element analysis software. The eightnode linear brick, reduced integration element C3D8R was adopted to mesh the RTP pipe. The HDPE was modeled as an isotropic elastic material. The failure of HDPE is not considered in the analysis because the reinforced layers are damaged first. The mesh of the model includes 10,000 liner elements, 20,000 composite elements, and 10,000 coating elements. Kinematic coupling was used to constrain all six degrees of freedom of each ends at a reference point. For a perfect FEA model without defect, the failures normally initiate near the boundary of the model, and many failure modes initialize at the same time. It is difficult to determine which mode causes the failure. To study the failure mode, an initial defect was added to the failure initiation area of the perfect FEA model before the analysis. For the FEA model with a defect, initial failure and failure progression of the finite element model occurred at the area around the defect. The fiber damage of the reinforce tape layer 2 in the longitudinal direction happened first, then a matrix damage of the reinforce tape layer 3 in the transverse direction happened. Compared with the experimental results, it can be concluded that the failure mode of RTP can be predicted from the fiber damage evolution, as shown in Figure 27.4, and the matrix damage evolution, as shown in Figure 27.5. The damage angle of the fibere and the matrix is close to 45 , which is similar to the failure angles of the tested specimens, as shown in Figure 27.2.
5. Results and Comparison The mechanical behavior of RTP before the initiation of failure is evaluated by analytical analysis and finite element analysis. The predicted tangential strains, axial strains, and torsion angles are presented in Figure 27.6. The calculated values from the analytical and finite element analysis agree with each other when pressure is lower
618
Qiang Bai and Yong Bai
Figure 27.5 Matrix damage evolution of layer 3 (SDV4 contour plot). (For color version of this figure, the reader is referred to the online version of this book.)
Figure 27.6 Comparison of analytical analysis with FE analysis for RTP under internal pressure. (For color version of this figure, the reader is referred to the online version of this book.)
Table 27.6 Burst Pressure Obtained from Different Methods
Methods
Analytical Analysis
Finite Element Analysis
Burst Test
Results (MPa)
44.7
41.8
35.4
Burst Strength of RTP Pipeline
619
than 10 MPa. The deviations increase when the internal pressure increases, which may be due to the different boundary conditions and the definition of initial defect used in the finite element model, also they may be due to the assumptions used in the analytical analysis. Although the inner and outer reinforced layers generate a positive and a negative torsion moments, respectively, they are not balanced with each other, as indicated by the existence of torsion angle. Table 27.6 shows a comparison of the burst pressures from analytical analysis, FE analysis, and experimental data. Burst pressure predicted from analytical analysis is 26% higher than that from the experimental data. However, the burst pressure from FE analysis is much closer to the experimental results, only 18% larger than the experimental results. The deviation between the analytical analysis and the test data may be caused by the boundary condition used in the analytical analysis, which is not consistent with the experimental condition. In the FE method analysis, six degrees of freedom at the both model ends are constrained, which is more similar to the actual experimental condition. The analytical analysis shows that the fiber of layer 2 fails first. The same behavior is observed from the both FE method and experimental data. FE analysis can also predict the progression of crack after the failure initiation. The orientation of the crack predicted from FE method is close to the experimental results, indicating that the FEA model is capable of predicting the post failure behavior.
References [1] Xia M, Takayanagi H, Kemmochi K. Analysis of multi-layered filament-wound composite pipes under internal pressure. Composite Structures 2001;53:483–91. [2] Kruijer MP, Warnet LL, Akkerman R. Modelling of the viscoelastic behavior of steel reinforced thermoplastic pipes. Composites, Part A 2006;37:356–67. [3] Kobayashi S, Imai T, Wakayama S. Burst strength evaluation of the FW-CFRP hybrid composite pipes considering plastic deformation of the liner. Composites, Part A 2007;38:1344–53. [4] Zheng JY, Li X, Xu P. Analysis on the short-term mechanical properties of plastic pipe reinforced by cross helically wound steel wires. J Pressure Vessel Technology 2009;131. [5] Onder A, Sayman O, Dogan T, Tarakcioglu N. Burst failure load of composite pressure vessels. Composite Structures 2009;89:159–66. [6] Knight NF. User-defined material model for progressive failure analysis. NASA/CR2006–214526, Virginia; 2006. [7] Xu P, Zheng JY, Liu PF. Finite element analysis of burst pressure of composite hydrogen storage vessels. Materials Design 2009;30:2295–301. [8] Liu PF, Zheng JY. Recent Developments on damage modeling and finite element analysis for composite laminates: A review. Materials Design 2010;31:L3825–34. [9] Linde P, Pleitner J, Boer H, Carmone C. Modelling and simulation of fibre metal laminates. ABAQUS Users’ Conference 2004:421–39. [10] Kruijer MP, Warnet LI, Akkerman R. Analysis of the mechanical properties of a reinforced thermoplastic pipe (RTP). Composites, Part A 2005;36:291–300.
620
Qiang Bai and Yong Bai
[11] ASTM D 1599-99. Standard test method for resistance to short-time hydraulic pressure of plastic pipe, tubing, and fittings. Philadelphia: American Society for Testing Materials; 2003. [12] Bai Y, Xu F, Cheg P, Badaruddin MF, Ashri M. Burst capacity of reinforced thermoplastic pipe under internal pressure. OMAE 2011-49325. Rotterdam, The Netherlands: OMAE; 2011.
612
Material Properties 612 Burst Tests 613
3. Analytical Analysis
614
Introduction 614 Coordinate Systems 615
4. Finite Element Analysis 616 5. Results and Comparison 617
1. Introduction Due to its high cost effectiveness, excellent corrosion resistance, and ease of installation, reinforced thermoplastic pipe (RTP) is now increasingly being used for onshore and offshore applications. Normally, RTP consists of one polyethylene liner, two layers of reinforced tape overwrapping the liner, and one outer polyethylene coating, as shown in Figure 27.1. The inner liner pipe and outer coating pipe are made of high density polyethylene (HDPE). The mechanical response of filament-wound structures under internal pressure has been studied by many researchers. Xia et al. [1] performed a stress analysis of multilayered filament-wound composite pipes under an internal pressure based on a 3D anisotropy elasticity theory. Kruijer et al. [2] developed a multilayer “generalized plane strain” model based on a plane strain characterization for RTP under hydrostatic pressure. Kobayashi et al. [3] proposes an elastic-plastic analysis model on the filament-wound carbon-fiber reinforced composite pipes by applying partially plastic thick-walled cylinder theory. Zheng et al. [4] presents an analytical procedure to predict the short-term burst pressure of PSP (plastic pipes reinforced by cross helicalwound steel wires) based on 3D anisotropic elasticity and the maximum stress failure criterion. An elastic solution procedure based on Lekhnitskii’s theory was developed by Onder et al. [5] to predict the burst failure pressure of the pressure vessels using the Tsai-Wu failure criterion, the maximum strain, and stress theories. A material degradation model based on the ply-discounting approach was implemented in a UMAT subroutine and applied to analyze the progression failure of a two-dimensional plate by Knight [6]. A 3D parametric finite of the cylindrical part Subsea Pipeline Design, Analysis, and Installation. http://dx.doi.org/10.1016/B978-0-12-386888-6.00027-4 Copyright Ó 2014 Elsevier Inc. All rights reserved.
612
Qiang Bai and Yong Bai
Figure 27.1 Cross section of RTP. (For color version of this figure, the reader is referred to the online version of this book.)
of a composite vessel was established by Xu et al. [7] to explore the failure evolution behavior and the failure pressure of composite vessels. Different failure criteria, such as the maximum stress by Hoffman, Tsai-Hill, and Tsai-Wu, are integrated in the main program. A comprehensive review on the recent developments of the damage and failure evolution using finite element analysis of composite laminates was reported by Liu et al.[8]. In this chapter, RTP is considered a thick cylinder, and the stress distribution is characterized as a generalized plane strain. Since the thicknesses of two reinforced tapes are relatively thin compared to that of the pipe, an assumption of uniform stresses through the thickness of the two layers is made to simplify the analysis. The material of the two reinforced tapes is considered as transverse isotropic. It is assumed that the strains in the reinforcing layers are equal to the strains in the isotropic material. The fiber failure and the matrix failure, the two failure modes used as the failure criterion, are used to determine the failure pressure of RTP. A 3D finite element RTP model is established to evaluate the relationship between the mechanical properties and the final failure pressure. To predict the damage evolution and the failure strength of RTP, the behavior of the reinforcing tape is simulated using a model proposed by Linde et al. [9], which is implemented in a user subroutine (UMAT) of the ABAQUS-standard nonlinear finite element analysis tool. The failure pressure calculated from finite element method and theoretical method are compared with the experimental burst pressure of RTP.
2. Experimental Analysis Material Properties In this analysis, two kinds of HDPEs are used for the liner, coating, and matrix. The reinforced tapes are manufactured by embedding the aramid strands in the HDPE. The HDPEs are modeled as linear elastic materials. Table 27.1 lists the secant
Burst Strength of RTP Pipeline
613
Table 27.1 Mechanical Properties of PE
Material
PE100 (k [ 1)
PE100 (k [ 4)
PE100 (k [ 2, 3)
Secant modulus, Ek (MPa) Poisson’s ratio, mk
350(E1) 0.4(m1)
390(E4) 0.4 (m4)
460 (Em) 0.4 (mm)
modulus and Poisson’s ratio used for modeling HDPEs. The secant modulus is determined based on the loading speed, temperature, and maximum strain during the burst test. Poisson’s ratio is assumed to be 0.4 according to Kruijer et al. [10]. The reinforced tapes were modeled to be linear-elastic transverse isotropic. The five independent constants (EL, ET, GLT, GTT, mLT, where E denotes Young’s modulus, G denotes the shear modulus, m denotes Poisson’s ratio, subscript L denotes the longitudina, fiber, direction, and subscript T denotes the two transverse directions orthogonal to the longitudinal direction) are listed in Table 27.2. The damage initiation properties of the reinforced tapes are shown in Table 27.3.
Burst Tests Burst tests are carried out according to ASTM D 1599-99 “Standard Test Method for Resistance to Short-Time Hydraulic Pressure of Plastic Pipe, Tubing, and Fittings” [11]. As per procedure A, a short-time loading process was applied, such that the time to failure for all the specimens was between 60s and 70s. The RTP samples are 760 mm long and terminated by steel swage fittings. The RTPs are free to deform during the tests. In addition, the pressure applied to the internal pressure of RTP is increased uniformly and continuously until the specimen fails. The experimental temperature was controlled at 23w24 C. Table 27.4 summarizes the measured burst pressures of RTP pipes. The average burst pressure of 35.3 MPa is obtained for the RTP samples. It was noted that failure of each tested specimen occurred at the RTP close to the end fittings (Figure 27.2). From Table 27.2 Transverse Isotropic Elastic Properties of Reinforced Tapes EL (MPa)
ET (MPa)
GLT (MPa)
GTT (MPa)
mLT
mTT
20,390
170
160
60
0.38
0.4
Table 27.3 Damage Initiation Properties of Reinforced Tapes sLf ;t (MPa)
sTf ;t (MPa)
770
6.0
614
Qiang Bai and Yong Bai
Table 27.4 Measured Burst Pressure of RTP Samples
Specimen no. 1 2 3 4 5 Average S-SD
Time to Failure (min)
Burst Pressure (MPa)
31 30 28 31 27
37.0 37.2 32.2 33.5 36.9 35.3 1.98
Figure 27.2 Burst samples after testing. (For color version of this figure, the reader is referred to the online version of this book.)
the burst tests, it was observed that the failure of fiber strands of the inner reinforced layer (layer 2) occurs first, which causes catastrophic damage to the entire pipe. The orientation of the crack is almost parallel to the outer reinforced layer (layer 3).
3. Analytical Analysis Introduction In the analytical analysis, RTP was considered as a thick cylinder. The deformation in the axial direction was assumed to be uniform. The stresses and strains (except for εzz
Burst Strength of RTP Pipeline
615
and szz) of the two isotropic layers are not uniform, but a function of the radius of the layer. The analysis is carried out using the following procedure: l
l
l
First, the strain and stress functions of the two isotropic layers are deduced using elastic mechanics theory. Second, the strain and stress functions can be obtained through the strain continuum condition. Finally, the unknown constants in the strain equations are determined by interface conditions and equilibrium equations.
Coordinate Systems A cylindrical coordinate system, as shown in Figure 27.3, is used for the analytical analysis. The coordinate axis r, q, and z denote the radial, circumferential, and axial directions of RTP pipe, respectively. The local material coordinate system of the reinforced tape layers is designated as (L, T, r), where L is the wound direction, T is the direction perpendicular to the aramid wire in plane, and r is the normal direction, same as in the cylindrical coordinate system. The term a is the wound angle of reinforced layer, the angle between L direction and z direction. The mathematical solutions are based on the model developed by Zheng et al. [4] for predicting the short-term burst pressure of PSP, by applying the 3D anisotropic elasticity and the maximum stress failure criterion, to calculate the short-term burst pressure of RTP pipe, except for the maximum strain failure criterion. Provided that the interfaces between the fiber yarn and PE are perfectly bonded, the strain of the aramid wire and PE in the aramid-wound direction can be considered to be equal. Because Young’s modulus of the aramid fiber is far greater than that of PE, the stresses in the aramid fiber are much greater than those in the PE. When the z r,r
θ
z
T
r
θ
α
L
σ zz
σ rr σθθ
(L,T,r) : Local material coordinate system (z,θ ,r) : Cylindrical coordinate system
Figure 27.3 Coordinate systems. (For color version of this figure, the reader is referred to the online version of this book.)
616
Qiang Bai and Yong Bai
RTP is subjected to internal pressure, aramid wires first reach their strength limits and break, resulting in the RTP losing the reinforcement of the fibers yarn and bursting in the short term. The detailed mathematical equations for the strains of each layers are found in the Bai et al. paper [12].
4. Finite Element Analysis A finite element model is also developed to simulate the mechanical properties of RTP under internal pressure. The damage of reinforced tapes is simulated using a model proposed by Linde et al. (2004) [11], which is implemented in the ABAQUS subroutine UMAT. Failure initiation and damage progression law are two major parts of the UMAT model. The failure initiation criterion is the same as failure criterion used in the theoretical analysis. In the following section, the damage progression law is described. When a fiber failure initiates, the fiber damage variable, df, evolves according to the following equation: df ¼ 1
f ;t f ;t f ;t ε11 c e½C11 ε11 ðff ε11 ÞL =Gf ; ff
[27.1]
where Lc is the characteristic length associated with the location. The evolution law of the matrix damage variable, dm, is expressed as dm ¼ 1
f ;t f ;t f ;t ε22 c e½C22 ε22 ðfm ε22 ÞL =Gm fm
[27.2]
The reinforced layers are assumed to be transversely isotropic. When the damage is progressive, the effective elasticity matrix is reduced by the two variables df and dm, as shown in Table 27.5. To improve the convergence speed, a technique based on the viscous regularization is used to regularize the damage variables. The regularized damage variables are stored as solution-dependent variables, SDV3 and SDV4, respectively.
Table 27.5 Degradation Factors Components of the Elasticity Matrix
Degradation Factors
C11, C22, C12, C33,
1 – df 1 – dm (1 – df)(1 – dm) 1
C13, and C23, and C21, and C55, and
C31 C32 C44 C66
Burst Strength of RTP Pipeline
617
Figure 27.4 Fiber damage evolution of layer 2 (SDV3 contour plot). (For color version of this figure, the reader is referred to the online version of this book.)
A 760 mm long finite element model was built in a cylindrical coordinate system using the ABAQUS/standard nonlinear finite element analysis software. The eightnode linear brick, reduced integration element C3D8R was adopted to mesh the RTP pipe. The HDPE was modeled as an isotropic elastic material. The failure of HDPE is not considered in the analysis because the reinforced layers are damaged first. The mesh of the model includes 10,000 liner elements, 20,000 composite elements, and 10,000 coating elements. Kinematic coupling was used to constrain all six degrees of freedom of each ends at a reference point. For a perfect FEA model without defect, the failures normally initiate near the boundary of the model, and many failure modes initialize at the same time. It is difficult to determine which mode causes the failure. To study the failure mode, an initial defect was added to the failure initiation area of the perfect FEA model before the analysis. For the FEA model with a defect, initial failure and failure progression of the finite element model occurred at the area around the defect. The fiber damage of the reinforce tape layer 2 in the longitudinal direction happened first, then a matrix damage of the reinforce tape layer 3 in the transverse direction happened. Compared with the experimental results, it can be concluded that the failure mode of RTP can be predicted from the fiber damage evolution, as shown in Figure 27.4, and the matrix damage evolution, as shown in Figure 27.5. The damage angle of the fibere and the matrix is close to 45 , which is similar to the failure angles of the tested specimens, as shown in Figure 27.2.
5. Results and Comparison The mechanical behavior of RTP before the initiation of failure is evaluated by analytical analysis and finite element analysis. The predicted tangential strains, axial strains, and torsion angles are presented in Figure 27.6. The calculated values from the analytical and finite element analysis agree with each other when pressure is lower
618
Qiang Bai and Yong Bai
Figure 27.5 Matrix damage evolution of layer 3 (SDV4 contour plot). (For color version of this figure, the reader is referred to the online version of this book.)
Figure 27.6 Comparison of analytical analysis with FE analysis for RTP under internal pressure. (For color version of this figure, the reader is referred to the online version of this book.)
Table 27.6 Burst Pressure Obtained from Different Methods
Methods
Analytical Analysis
Finite Element Analysis
Burst Test
Results (MPa)
44.7
41.8
35.4
Burst Strength of RTP Pipeline
619
than 10 MPa. The deviations increase when the internal pressure increases, which may be due to the different boundary conditions and the definition of initial defect used in the finite element model, also they may be due to the assumptions used in the analytical analysis. Although the inner and outer reinforced layers generate a positive and a negative torsion moments, respectively, they are not balanced with each other, as indicated by the existence of torsion angle. Table 27.6 shows a comparison of the burst pressures from analytical analysis, FE analysis, and experimental data. Burst pressure predicted from analytical analysis is 26% higher than that from the experimental data. However, the burst pressure from FE analysis is much closer to the experimental results, only 18% larger than the experimental results. The deviation between the analytical analysis and the test data may be caused by the boundary condition used in the analytical analysis, which is not consistent with the experimental condition. In the FE method analysis, six degrees of freedom at the both model ends are constrained, which is more similar to the actual experimental condition. The analytical analysis shows that the fiber of layer 2 fails first. The same behavior is observed from the both FE method and experimental data. FE analysis can also predict the progression of crack after the failure initiation. The orientation of the crack predicted from FE method is close to the experimental results, indicating that the FEA model is capable of predicting the post failure behavior.
References [1] Xia M, Takayanagi H, Kemmochi K. Analysis of multi-layered filament-wound composite pipes under internal pressure. Composite Structures 2001;53:483–91. [2] Kruijer MP, Warnet LL, Akkerman R. Modelling of the viscoelastic behavior of steel reinforced thermoplastic pipes. Composites, Part A 2006;37:356–67. [3] Kobayashi S, Imai T, Wakayama S. Burst strength evaluation of the FW-CFRP hybrid composite pipes considering plastic deformation of the liner. Composites, Part A 2007;38:1344–53. [4] Zheng JY, Li X, Xu P. Analysis on the short-term mechanical properties of plastic pipe reinforced by cross helically wound steel wires. J Pressure Vessel Technology 2009;131. [5] Onder A, Sayman O, Dogan T, Tarakcioglu N. Burst failure load of composite pressure vessels. Composite Structures 2009;89:159–66. [6] Knight NF. User-defined material model for progressive failure analysis. NASA/CR2006–214526, Virginia; 2006. [7] Xu P, Zheng JY, Liu PF. Finite element analysis of burst pressure of composite hydrogen storage vessels. Materials Design 2009;30:2295–301. [8] Liu PF, Zheng JY. Recent Developments on damage modeling and finite element analysis for composite laminates: A review. Materials Design 2010;31:L3825–34. [9] Linde P, Pleitner J, Boer H, Carmone C. Modelling and simulation of fibre metal laminates. ABAQUS Users’ Conference 2004:421–39. [10] Kruijer MP, Warnet LI, Akkerman R. Analysis of the mechanical properties of a reinforced thermoplastic pipe (RTP). Composites, Part A 2005;36:291–300.
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[11] ASTM D 1599-99. Standard test method for resistance to short-time hydraulic pressure of plastic pipe, tubing, and fittings. Philadelphia: American Society for Testing Materials; 2003. [12] Bai Y, Xu F, Cheg P, Badaruddin MF, Ashri M. Burst capacity of reinforced thermoplastic pipe under internal pressure. OMAE 2011-49325. Rotterdam, The Netherlands: OMAE; 2011.