# Improvement and Implementation of Calculation Method for Prestressed Concrete Cylinder Pipes

Authors: Zhu Zhang; Tongchun Li; Hongen Li; Huijun Qi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Prestressed concrete cylinder pipe (PCCP) is a type of composite pipes frequently used in water diversion projects. However, the existing methods for structural analysis of such pipes are complex; thus, very few standards are available for use in practical design. In this paper, the new straightforward technique for analyzing PCCP based on available standards is proposed in order to address the shortcomings of the existing standard and improve the convergence of subsequent calculations. The verification and comparison of case study given in the main standards showed that the proposed method is rational and useful. In addition, an advanced finite element model was utilized in practical design case that considered proper constitutive relationships, contact model, prestress of wires, and practical aspects of manufacturing, installation, and operation. Meantime, the results obtained by nonlinear FEA and the improved method are assessed and compared, which showed that nonlinear FEA based on corresponding intrinsic model and the improved method are recommendable to aid in practical design projects.


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## Body

## 1. Introduction

Prestressed concrete cylinder pipes (PCCPs) are widely used in long distance water transmission projects and urban pipe networks. They are characterized by large diameters, high internal and external pressure resistance, excellent seismic performance, and cost efficiency. They were first manufactured in France in 1939 and then introduced into America in 1942. Nowadays, more than 22,000 miles of PCCPs have been installed in North America [1]. In recent years, they have been widely applied in water diversion projects in China, such as the South-to-North Water Diversion Project and Shanxi Wanjiazhai Yellow River Diversion Project, which employed PCCPs with a diameter of 3 m. PCCP is a composite pipe made of four materials: high-strength concrete, steel for the cylinder, high-strength prestressing wires, and mortar for coating. Each constituent material or part plays an important role in guaranteeing the optimal management and operation of a pipeline. The high-strength concrete is the main material of the pipe and gives the smooth inner surface that is beneficial to the unobstructed flow of water. The steel cylinder is mainly used for seepage control. The helical single or multiple-layer wrapping of prestressing wires produces uniform prestress in the tube core that resists the tension stresses caused by internal water pressure and external loads. The mortar coating is essential to protect wires from corrosion caused by contact with soil or groundwater. There are two main types of PCCPs: the lined cylinder pipes (LCPs) and embedded cylinder pipes (ECPs) [2]. During manufacturing of LCPs, concrete is centrifugally cast inside a cylinder and high-strength prestressing wires are wrapped around the tube core made of a steel cylinder and concrete. In ECPs, concrete is vertically cast around the steel cylinder, and then wires are wrapped around the outer concrete surface, as shown in Figure 1. ECPs are mainly used in water transmission projects where large diameters are required, such as the Great Man-Made River in Libya and China’s water conservancy projects mentioned above; thus researchers try to investigate the structural integrity of ECPs by using the methods of structural mechanics and numerical simulations.Figure 1
ECP components in finite element model.The structural mechanics approach combined with materials mechanics is adopted in standards and specifications [2–6]. The structural mechanics approach is mainly based on calculating external loads, solving for internal forces, and finally obtaining the stress and strain fields. It includes working stress method, limit state method, and multilayered model method [4]. The working stress method adopted by standard AWWA C01-99 [3] is a semiempirical approach to pipe structural design that ensures that the stress generated in the core under various internal pressures and external loads is controlled such that it does not exceed the prestress [5]. This method is essentially based on the tension-free stress criterion of the prestressing theory and large safety factors, which leads to excessive use of steel, large concrete creep, and serious cracking problems in locally restraint zones [7].The limit state method is used in three specifications or standards including AWWA C304-14 [2], Chinese water industry standard Technical Specifications for Presteessed Concrete Cylinder Pipes SL 702–2015 [6], and standard CECS140 : 2002 [5]. Specification AWWA C304-14 adopts the limit state method proposed by Zarghamee et al. [8, 9] based on partial prestress theory and takes deformations as design criteria, while allowing controllable cracks to occur under certain conditions. In standard AWWA M9-95 [10], the external load effects are calculated based on Marston’s theory [11], considering Earth pressure, pipe weight, water pressure, and transient and live load produced by various vehicles. Combining the working pressure, field-test pressure, and transient pressure, it considers fourteen load cases. Afterwards, according to the theory developed by Olander [12] and Paris [13], various load distributions are determined in order to obtain the axial forces and bending moments at critical cross sections, such as the crown, invert, and the springline of the PCCP. Finally, the optimal strain gradient and strain value balanced with the axial force and bending moment can be determined by iterative calculations. Then, the constitutive relationship of the material is introduced to determine the stress-strain distribution in the crown, invert, and springline of the PCCP. Standard SL 702–2015 was compiled from Chinese basic specifications and ANSI/AWWA C304-07, which is the latest edition of AWWA C304-14. Its structural design part is mainly based on American specifications, but the material parameters and basic loads are calculated according to the domestic Chinese standards.A limit state calculation method based on probability theory is adopted in standard CECS140 : 2002. The method is firstly used to calculate the standard values of various loads. Then, it considers five different load cases using these standard load values and eventually the design load values are calculated according to Code for Design of Concrete Structures [14]. Then, the axial forces and bending moments in critical cross sections are obtained and parameters, such as the converted area, converted elastic resistance moment, and converted coefficient at the cross sections, are also calculated. This approach has a disadvantage of regarding all materials as linear elastic. After performing a detailed review of standard C304-14, Roller et al. [15] simplified some of the overly complex provisions incorporated in the document without any significant consequences to the final overall design and provided alternative or simplified provisions for a number of them. The design equations were slightly modified, though the key problem of solution algorithm was not settled.Results produced by the multilayered model proposed by Zarghamee [16] for structural responses due to an internal water pressure and external loads were compared to experimental ones. In this model, a balanced differential control equation for a tube wall section was established based on the theory of thin shells. The pipe wall was divided into several layers according to the type of material. Then, the bending stiffness and axial stiffness were added for each layer considering the corresponding nonlinear constitutive relationships. Finally, the residual internal forces were used as the convergence criterion to solve the problem. This method was used by You [17, 18] to analyze a pipe with broken wires.Numerical simulations using finite element analysis (FEA) are the other method of investigating the mechanical behavior of pipes. 2D and 3D models were developed as discussed in [4, 19, 20]. Numerical simulation methods can solve the problem of obtaining the stress-strain distribution of integral structure that could not be addressed by the structural mechanics methods, but problems still exist in numerical simulation methods, such as incorrect application of prestressing, neglecting stiffness of prestressing wires, incorrect methods of simulating prestress, and failure to consider nonlinear contacts between the pipe and the soil or between two pipes. In short, the existing FEA models are too simplistic and must be improved.In this paper, we give a detailed description of standards used for PCCP design and develop an improved analytical method based on main standards in order to address the shortcomings of the standard computational algorithms. Then, we verify our approach using the case study given in the main standards and apply the improved method in a practical case study. Finally, considering the actual manufacturing, installation, and operation of ECPs, we divide these stages into seven steps in FEA. In the FEA, we also consider nonlinear mechanical behavior of concrete and soil, the interaction between soil and pipe, and the bond-slip in reinforced concrete (RC) and develop a proper prestress simulation method. The results obtained by nonlinear FEA and the improved method are assessed and compared.


## 2. Theory and Methods

### 2.1. Brief Introduction to Standard AWWA C304-14 Solution Method

Calculations show that extreme internal forces in pipes usually occur at the invert, crown, and springline sections. Thus, the three sections are generally analyzed as the stress-strain critical sections. In addition, under combined loads the longitudinal and radial stresses of the tube are far less than the circumferential stress. Therefore, the circumferential stress is the critical stress of the cross section. In calculating the stress and strain in the critical sections, the iterative algorithm from standard AWWA C304-14 is adopted. A strain coefficient,ν2 , at the edge of tensile zone in concrete and a gradient, k , of strain distribution are first assumed based on the assumption of linear distribution of stress and strain in the section. Thus, the stress and strain at each point can be calculated according to the linear geometric relationship. Then, the sectional axial force and bending moment are obtained from the calculated stress values by integration. Finally, iterative calculations are performed until the difference between internal forces of sections obtained by integration of stress and those calculated from external loads is within an acceptable margin of error. The calculation flowchart is shown in Figure 2. It should be noted that the contribution of mortar layer to the sectional internal force is not taken into account in the springline section.Figure 2
Calculation flowchart of standard AWWA C304-14.It can be seen from the calculation principles of the stress and strain of the pipe wall section discussed above that the stress-strain analysis method adopted in standard AWWA C304 is an iterative algorithm based on several theoretical assumptions. The key to apply this method in practical design is to define the iteration range of two parameters,ν2 and k. According to the results of structural analysis, there are tension and compression zones at the same time or full compression in the section. For the former situation, the range of values of k can be determined as 0≤k≤1 , while for the latter situation, k≤0. The range of values of ν2 if the section has tension and compression zones simultaneously can be obtained as follows. According to the assumption of the stress at the edge in the tension zone, the strain at the edge of the tension zone should be greater than 0; i.e., the strain is tensile strain. Further, according to the design criteria, visible cracks cannot appear in the tension zone; i.e., the maximum tensile strain in the tension zone, expressed as ε , is ε≤11εt′, where 11εt′ is the cracking strain and εt′ is the peak strain. Thus, the range of values of ν2 is (−1, 10). In the case of a general section under compression, the range of ν2 is ν2〈0 . The stress-strain relationship for concrete in tension adopted by standard AWWA C304-14 is shown in Figure 3 and the constitutive equations for concrete are as follows:(1)σ=1+ν2ft′ν2≤01−ν2ν0<ν2≤ν0ν2〉ν,where ν is the ratio of ultimate cracking strain to peak strain, which is 11 for concrete and 8 for mortar, respectively, ν2 is the strain coefficient; and ft′ is the peak stress of concrete.Figure 3
Stress-strain relationship for concrete in tension.With the help of modern computing technology, the above standard calculation method is easy to program. But practical case studies showed that programs compiled according to this method did not easily converge and were inefficient. In some cases, the error of iterative solution was very large. For example, the maximum moment balance error in the example given in Chinese standard SL 702–2015 reaches about 9 kN m and thus makes the calculation result inaccurate. In addition, the formula used in the standard is incompatible with other softening curves of concrete. In order to address the shortcomings of the standard, we propose an improved iteration algorithm in the following section.


### 2.2. Improved Method

The nonlinear behavior of materials in PCCPs is considered in standards AWWA C304-14 and SL 702–2015. To solve the nonlinear problem, we propose a new idea of calculating the stress at the three key sections based on the solutions of nonlinear algebraic equation using FEA, standard AWWA C304-14 [2], and the improved equation proposed in [15].There are three methods for solving material nonlinearity problems in FEA: incremental, iterative, and hybrid that combines incremental an iterative approach. The incremental method divides the load into many steps and applies them in succession. During load increment, the stiffness is assumed to be constant but may take different values corresponding to the stress-strain relationship at different load levels. In the iterative method, at each step all loads are applied, but the displacement and strain are gradually modified to satisfy the nonlinear stress-strain relationship. The hybrid method uses both the incremental and the iteration method; thus the load is also divided into increments, but with a less number, and the iterative method is applied to each load increment. The nonlinear stress-strain relationship adopted in standards AWWA C304 and SL 702–2015 is simple and the tangent stiffness can be easily obtained. Therefore, we chose the Newton method for iterative calculations. The detail process of the Newton method is described below.The global equilibrium equations form a system of linear equations as follows:(2)Ψ=Kδδ−P=0,where Ψ is the unbalanced force vector; δ is the nodal displacement vector; Kδ is the stiffness matrix; and P is the nodal loading vector. Equation (2) can be expressed as follows:(3)Ψ=Kδδ−P=Fδ−P=0,where Fδ is the nodal force vector.Supposeδn is the nth approximate solution. In general, we have the following:(4)Ψn=Kδnδn−P=Fδn−P≠0,where Ψn is nth-iteration unbalance force vector; δn is nth-iteration nodal displacement solution vector; and Kδn is nth-iteration stiffness matrix. Equation (4) becomes a linear problem using the Taylor expansion:(5)Ψ=Ψn+Ktnδ−δn=0,where Ktn is nth-iteration tangent stiffness matrix. The (n + 1)th approximate solution can be obtained as follows:(6)δn+1=δn−Ktn−1Ψn=δn−Ktn−1Fδn−P,where δn+1 is (n + 1)th approximate solution vector; and Fδn is the nth-iteration nodal force vector.The flowchart of the improved calculation method proposed in this paper is also shown as Figure2. The difference is that when calculating the internal force of each material, the formula is changed to σ=Eεε, where σ is the material stress and Eε is the elastic modulus that changes with the material strain ε [15]. We divide the pipe cross section into five layers: the outer concrete layer, steel cylinder layer, inner concrete layer, steel wire layer, and mortar layer. The mater layer with thick shell is supposed to be many tiny layers. Similarly, assuming that the entire interface is linearly distributed along the radius direction in the cross section [2] and regarding the strain value at both sides of the tube core as unknown variable, the stress of each layer of material is obtained by the nonlinear constitutive relationship from standard [2] and study [16], and the internal forces are calculated from the stress. If the difference between the internal force and the external load is within the tolerance after several iterations or the maximum number of iterations is reached, the calculations end immediately.The algorithm is as follows:(1)
Set stepk = 1. There are two steps: in the first step, balance the axial force expressed as N0 and N0 resulting from prestress; in the second step, balance the axial force expressed as N1 and bending moment expressed as M1, where N1 and M1 are calculated from the external loads.(2)
Set iterationn = 1. For the first step, k = 1, it is easy to obtain the strain distribution of the whole cross section, because the pipe is under compression at this step. For the second step, k = 2, assume δ1=0, which comprises the strain values at both sides of the tube core. Compute Fδ1 and Kt1−1 with using the nonlinear stress-strain relationships of materials. P is the loading vector made of N1 and M1.(3)
Terminate the iteration process ifFδ1−P is smaller than a given error or the maximum number of iterations is reached. Otherwise, set n = n + 1 and return to equation (6). If the iteration is finished, set k = k + 1.(4)
Obtain the final values of strain and stress of the whole cross section.


## 2.1. Brief Introduction to Standard AWWA C304-14 Solution Method

Calculations show that extreme internal forces in pipes usually occur at the invert, crown, and springline sections. Thus, the three sections are generally analyzed as the stress-strain critical sections. In addition, under combined loads the longitudinal and radial stresses of the tube are far less than the circumferential stress. Therefore, the circumferential stress is the critical stress of the cross section. In calculating the stress and strain in the critical sections, the iterative algorithm from standard AWWA C304-14 is adopted. A strain coefficient,ν2 , at the edge of tensile zone in concrete and a gradient, k , of strain distribution are first assumed based on the assumption of linear distribution of stress and strain in the section. Thus, the stress and strain at each point can be calculated according to the linear geometric relationship. Then, the sectional axial force and bending moment are obtained from the calculated stress values by integration. Finally, iterative calculations are performed until the difference between internal forces of sections obtained by integration of stress and those calculated from external loads is within an acceptable margin of error. The calculation flowchart is shown in Figure 2. It should be noted that the contribution of mortar layer to the sectional internal force is not taken into account in the springline section.Figure 2
Calculation flowchart of standard AWWA C304-14.It can be seen from the calculation principles of the stress and strain of the pipe wall section discussed above that the stress-strain analysis method adopted in standard AWWA C304 is an iterative algorithm based on several theoretical assumptions. The key to apply this method in practical design is to define the iteration range of two parameters,ν2 and k. According to the results of structural analysis, there are tension and compression zones at the same time or full compression in the section. For the former situation, the range of values of k can be determined as 0≤k≤1 , while for the latter situation, k≤0. The range of values of ν2 if the section has tension and compression zones simultaneously can be obtained as follows. According to the assumption of the stress at the edge in the tension zone, the strain at the edge of the tension zone should be greater than 0; i.e., the strain is tensile strain. Further, according to the design criteria, visible cracks cannot appear in the tension zone; i.e., the maximum tensile strain in the tension zone, expressed as ε , is ε≤11εt′, where 11εt′ is the cracking strain and εt′ is the peak strain. Thus, the range of values of ν2 is (−1, 10). In the case of a general section under compression, the range of ν2 is ν2〈0 . The stress-strain relationship for concrete in tension adopted by standard AWWA C304-14 is shown in Figure 3 and the constitutive equations for concrete are as follows:(1)σ=1+ν2ft′ν2≤01−ν2ν0<ν2≤ν0ν2〉ν,where ν is the ratio of ultimate cracking strain to peak strain, which is 11 for concrete and 8 for mortar, respectively, ν2 is the strain coefficient; and ft′ is the peak stress of concrete.Figure 3
Stress-strain relationship for concrete in tension.With the help of modern computing technology, the above standard calculation method is easy to program. But practical case studies showed that programs compiled according to this method did not easily converge and were inefficient. In some cases, the error of iterative solution was very large. For example, the maximum moment balance error in the example given in Chinese standard SL 702–2015 reaches about 9 kN m and thus makes the calculation result inaccurate. In addition, the formula used in the standard is incompatible with other softening curves of concrete. In order to address the shortcomings of the standard, we propose an improved iteration algorithm in the following section.


## 2.2. Improved Method

The nonlinear behavior of materials in PCCPs is considered in standards AWWA C304-14 and SL 702–2015. To solve the nonlinear problem, we propose a new idea of calculating the stress at the three key sections based on the solutions of nonlinear algebraic equation using FEA, standard AWWA C304-14 [2], and the improved equation proposed in [15].There are three methods for solving material nonlinearity problems in FEA: incremental, iterative, and hybrid that combines incremental an iterative approach. The incremental method divides the load into many steps and applies them in succession. During load increment, the stiffness is assumed to be constant but may take different values corresponding to the stress-strain relationship at different load levels. In the iterative method, at each step all loads are applied, but the displacement and strain are gradually modified to satisfy the nonlinear stress-strain relationship. The hybrid method uses both the incremental and the iteration method; thus the load is also divided into increments, but with a less number, and the iterative method is applied to each load increment. The nonlinear stress-strain relationship adopted in standards AWWA C304 and SL 702–2015 is simple and the tangent stiffness can be easily obtained. Therefore, we chose the Newton method for iterative calculations. The detail process of the Newton method is described below.The global equilibrium equations form a system of linear equations as follows:(2)Ψ=Kδδ−P=0,where Ψ is the unbalanced force vector; δ is the nodal displacement vector; Kδ is the stiffness matrix; and P is the nodal loading vector. Equation (2) can be expressed as follows:(3)Ψ=Kδδ−P=Fδ−P=0,where Fδ is the nodal force vector.Supposeδn is the nth approximate solution. In general, we have the following:(4)Ψn=Kδnδn−P=Fδn−P≠0,where Ψn is nth-iteration unbalance force vector; δn is nth-iteration nodal displacement solution vector; and Kδn is nth-iteration stiffness matrix. Equation (4) becomes a linear problem using the Taylor expansion:(5)Ψ=Ψn+Ktnδ−δn=0,where Ktn is nth-iteration tangent stiffness matrix. The (n + 1)th approximate solution can be obtained as follows:(6)δn+1=δn−Ktn−1Ψn=δn−Ktn−1Fδn−P,where δn+1 is (n + 1)th approximate solution vector; and Fδn is the nth-iteration nodal force vector.The flowchart of the improved calculation method proposed in this paper is also shown as Figure2. The difference is that when calculating the internal force of each material, the formula is changed to σ=Eεε, where σ is the material stress and Eε is the elastic modulus that changes with the material strain ε [15]. We divide the pipe cross section into five layers: the outer concrete layer, steel cylinder layer, inner concrete layer, steel wire layer, and mortar layer. The mater layer with thick shell is supposed to be many tiny layers. Similarly, assuming that the entire interface is linearly distributed along the radius direction in the cross section [2] and regarding the strain value at both sides of the tube core as unknown variable, the stress of each layer of material is obtained by the nonlinear constitutive relationship from standard [2] and study [16], and the internal forces are calculated from the stress. If the difference between the internal force and the external load is within the tolerance after several iterations or the maximum number of iterations is reached, the calculations end immediately.The algorithm is as follows:(1)
Set stepk = 1. There are two steps: in the first step, balance the axial force expressed as N0 and N0 resulting from prestress; in the second step, balance the axial force expressed as N1 and bending moment expressed as M1, where N1 and M1 are calculated from the external loads.(2)
Set iterationn = 1. For the first step, k = 1, it is easy to obtain the strain distribution of the whole cross section, because the pipe is under compression at this step. For the second step, k = 2, assume δ1=0, which comprises the strain values at both sides of the tube core. Compute Fδ1 and Kt1−1 with using the nonlinear stress-strain relationships of materials. P is the loading vector made of N1 and M1.(3)
Terminate the iteration process ifFδ1−P is smaller than a given error or the maximum number of iterations is reached. Otherwise, set n = n + 1 and return to equation (6). If the iteration is finished, set k = k + 1.(4)
Obtain the final values of strain and stress of the whole cross section.


## 3. Method Verification

We present the verification results of standard SL 702-2015 in Table1. Because the stress condition at the invert is more dangerous than that at the crown, we present the results at the invert and springline sections. The comparison shows that the results of the two methods are close to each other.Table 1
Comparison of standard and improved method.
Limits and locationLimit criterionApplicable load combinationCalculation valueLimit valueStandard methodImproved methodServiceability at invertInside core tensile strain limit:εci≤1.5εt′W10.0002110.0002060.000211Inner core tensile strain limit:εci≤εk′=11εt′WT10.0005980.0005840.00155WT20.0002140.000208FT10.0006420.000625Inner core-to-cylinder radial tension limit:σr≤0.082MPaFW1−0.114 MPa-0.116 MPa0.082 MPaWT3−0.154 MPa-0.157 MPaServiceability at springlineOuter core tensile strain limit:εco≤1.5εt′W10.0000650.0000690.000211Outer core tensile strain limit:εco≤εk′=11εt′WT10.0001580.0001620.00155WT20.0000660.000071FT10.0001620.000167Outer coating tensile strain limit:εmo≤0.8εkm'=6.4εtm'W10.0005430.0005470.000907Outer coating tensile strain limit:εmo≤εkm'=8εtm'WT10.0006360.0006390.001134WT20.0005440.000548FT10.0006500.000661Inner core compression limit:fci≤0.55fc′W219.814 MPa20.009 MPa26.4 MPaInner core compression limit:fci≤0.65fc′WT319.868 MPa20.063 MPa31.2 MPaElastic limit at invertCylinder stress reaching yield:-fyr+n′fcr+Δfy≤fyyWT1−82.459 MPa MPa MPa−83.92 MPa227.5 MPaWT2−135.35 MPa−135.77 MPaFT1−79.625 MPa−81.358 MPaCylinder stress reaching yield:-fyr+n′fcr+Δfy≤0WT3−180.05 MPa−180.46 MPa0Elastic limit at springlineWire stress limitfsg:-fsr+nfcr+Δfs≤fsgFWT11068.1 MPa1064.8 MPa1177.5 MPaFWT21040.8 MPa1039.1 MPaFT21069.8 MPa1067.3 MPaCore compression limit:fci≤0.75fc′FWT112.674 MPa12.108 MPa36.0 MPaFWT215.206 MPa14.715 MPaFT214.27 MPa13.754 MPaStrength at springlineWire stress limitfsy:-fsr+nfcr+Δfs≤fsyFWT31125.9 MPa1133.7 MPa1334.5 MPaFWT41065.4 MPa1063.6 MPaUltimate limit:M≤MultFWT5145580.4 kN m145560.3 kN m428109.8 kN m


## 4. Method Application and Comparison

We present the verification results obtained by nonlinear FEA and the improved method in a practical design case.


### 4.1. Improved Method

The ECP considered in the study has a diameter of 4 m and working pressure of 0.8 MPa, and the whole wrapping area of wire is 5497 mm2 with two wrapping layers. The thickness of the steel cylinder is 2 mm. The wire diameter is 7 mm. The wire-relaxation factor of the outer layer is 0.127 and that of the inner layer is 0.108, respectively. The shrinkage and creep factors of tube core are 6.6617 × 10−5 and 1.177, respectively. The results calculated by the improved method are shown in Table 2.Table 2
Improved method applied to ECP case with 4 m diameter.
Limit state and locationLimit state criterionApplicable load combinationCalculation valueLimit valueServiceability at invertInside core tensile strain limit:εci≤1.5εt′W10.0001990.000211Inner core tensile strain limit:εci≤εk′=11εt′WT10.0002760.00155WT20.000203FT10.000349Inner core-to-cylinder radial tension limit:σr≤0.082MPaFW1−0.114 MPa0.082 MPaWT3−0.154 MPaServiceability at springlineOuter core tensile strain limit:εco≤1.5εt′W1−0.0000100.000211Outer core tensile strain limit:εco≤εk′=11εt′WT10.000040.00155WT2−0.00001FT10.00006Outer coating tensile strain limit:εmo≤0.8εkm'=6.4εtm'W10.00050.00091Outer coating tensile strain limit:εmo≤εkm'=8εtm'WT10.000560.00113WT20.00050FT10.00058Inner core compression limit:fci≤0.55fc′W219.8 MPa26.4 MPaInner core compression limit:fci≤0.65fc′WT320.1 MPa31.2 MPaElastic limit at invertCylinder stress reaching yield:-fyr+n′fcr+Δfy≤fyyWT1−117.9 MPa227.5 MPaWT2−131.7 MPaFT1−109.5 MPaCylinder stress reaching yield:-fyr+n′fcr+Δfy≤0WT3−158.8 MPa0Elastic limit at springlineWire stress limitfsg:-fsr+nfcr+Δfs≤fsgFWT11033.3 MPa1177.5 MPaFWT21023.8 MPaFT21024.1 MPaCore compression limit:fci≤0.75fc′FWT120.5 MPa36 MPaFWT222.5 MPaFT220.7 MPaStrength at springlineWire stress limitfsy:-fsr+nfcr+Δfs≤fsyFWT31053.3 MPa1334.5 MPaFWT41041.5 MPaUltimate limit:M≤MultFWT5811910.5 kN m1104296.6 kN m


### 4.2. Nonlinear Finite Element Analysis

To evaluate the structural integrity of ECPs, numerical simulations are usually conducted. Thus, we also performed a 3D nonlinear FEA. Similar to the structural mechanics method, the stress-strain states at key cross sections of the pipe under 14 load combinations should be obtained. However, in the 3D nonlinear FEA we considered the installation process and nonlinearity of the materials and contacts. Therefore, the special load combinations were ignored and we selected the load combination WT1 for analysis, which includes internal pressure, transient water hammer pressure, and normal external permanent loads.


#### 4.2.1. FEA Model and Material Constitutive Models

In this model, the height of backfill soil is 10 m and the thickness of tube core is 350 mm. To reduce the boundary effect, we modeled five pipes along the pipeline direction and then chose the middle pipe from the left pipeline pipes for analysis. A total of 123,488 elements were used in the model, including hexahedral elements for concrete, mortar coating, and soil. The prestressing wires were modeled as bar elements and steel cylinder as plate elements. The entire model had a length of 25 m along pipeline direction, a width of 53.2 m, and a height of 29.756 m. For boundary conditions, the top surface of the model was assumed free and the other five sides as fixed in the normal directions. The pipe model is shown in Figure4.Figure 4
Finite element model of ECP.There are four methods for modeling prestress in wires in FEA: the equivalent load method, initial strain method, equivalent cooling method, and wire-wrapping model [21]. In the equivalent load method used in most studies [18, 22–30], it does not consider the stiffness of wires and as a result it is difficult to simulate the stress increment in steel wires under the external loads in pipes with broken wires. In the strain method proposed by Lin [31], applying an initial strain to steel wires was used to model the prestressing force; however, the loss of prestressing force cannot be considered in this method because the loss of prestressing force is represented by stress. In the equivalent cooling method proposed by researchers [32–37], stiffness of wires can be considered. It is actually an equivalent deformation method achieving the required stress by cooling the steel wire based on its temperature characteristics, the physical concept of which is simple and easy to control. However, cooling induces compressive strain on the steel wire, which does not meet the initial tension state of the steel wire. Meantime, if the nonlinearity intrinsic behavior of material and contact are considered while using the cooling method, it is a complex nonlinear coupled problem, which increases the difficulty of solving the finite equations and is not easy to converge to the subsequent solution. The wire-wrapping model proposed by Xiong et al. [38] is very complex and difficult to apply in FEA. To avoid the shortcomings mentioned above, we chose a new prestress simulation method referred to as the initial stress method. The specific idea is to directly apply the wrapping prestress considered as a loss to the Gauss points of wire elements by self-written FEA program called Gehomadrid. This is easy and direct to simulate with good convergence. Most importantly, the loss of prestressing force can easily be considered by this approach.A large number of damage models for concrete may be found in [39, 40]. Wei et al. [41] presented a new four-parameter failure criterion in strain space based on the Hsieh-Ting-Chen four-parameter failure criterion. The new criterion can model the concrete strength precisely not only under uniaxial but also under multiaxial stresses; details can be found in [39, 41]. In this paper, we adopted the new four-parameter failure criterion in strain space combined with the stress-strain curve from [2, 38] as the concrete and mortar damage constitutive model to simulate damage.The nonlinear hyperbolic Duncan-Chang soil model formulas [42] are obtained from conventional triaxial test and include seven parameters. This model is relatively simple and its engineering application is frequent. In the year of 1980, Duncan and Wong et al. replaced Poisson’s ratio with volume compression modulus and presented the improved E-B model; details can be found in [43]. In this study, we adopted the Dun-Chang E-B model for soil.The study by Zhao [39] presented a novel single spring joint element method based on mixed coordinate system and the traditional double spring element method, in which the difficulty of choosing the normal stiffness coefficient was avoided. In this paper, we selected the single spring joint element to model the bond-slip interaction in RC structures. Because of the lack of experimental bond-slip curves between high-strength concrete and plain prestressing wires, we adopted the nonlinear bond-slip constitutive relationship developed by Di [44] for normal concrete and crescent rib wires given as follows:(7)τ=6.59×102s−2.13×104s2+0.22×106s3,where τ is the bond stress (MPa); and s is the slip displacement (mm). Considering that plain wires are used in this study, we assumed that the bond stress was only one-tenth of the value obtained from Di’s formula. The nonlinear stress-strain relationship of wires was taken from standard AWWA C304-14.The common contact interface element used in engineering applications is the Goodman joint element with zero thickness [45]; thus we selected it in this paper. We also adopt the Von Mises yield criterion and an ideal elastoplastic constitutive model for the cylinder steel with complete bond-slip with concrete.


#### 4.2.2. Material Properties

The main material properties adopted in the study are listed in Table3.Table 3
Properties of pipe materials.
MaterialGradeDensityYoung’s modulusPoisson’s ratioCompressive strength/MPaYield strength/MPaTensile strength/MPaCohesion force/NInternal friction angle/°/kg·m−3ConcreteC60232328.6 (GPa)0.16748.0—4.03——Wires—7833191 (GPa)0.29—1334.51570.0——Cylinder—7833207 (GPa)0.29—227.5310.0——Foundation—203980.0 (MPa)0.25———5.0E330Bedding—203940.0 (MPa)0.29———3.0E330Backfill—203950.0 (MPa)0.35———4.0E330The Dun-ChangE-B constitutive model coefficients of backfill soil are listed in Table 4.Table 4
Parameters of Dun-Chang E-B soil constitutive model.
SoilNondimensional parameter (K)Nondimensional index (n)Failure ratio (rf)Experimental parameter (Kb)Experimental parameter (m)Cohesion force (unit: N)Internal friction angle (unit: °)Foundation215.50.920.613151.010.755.0 × 10330Bedding232.60.900.613165.050.753.0 × 10330Backfill198.40.890.613132.270.754.0 × 10330


#### 4.2.3. FEA Results

The FEA considering manufacturing, installation, and operation was divided into seven steps:(1)
Wrapping of inner layer.(2)
Wrapping of outer layer.(3)
Curing the pipe, with long-term creep, shrinkage, and wire-relaxation modeled. The creep, shrinkage, and wire-relaxation were calculated by the method described in [2] and used in the proposed improved method.(4)
Laying bedding soil, installing pipes, and laying bottom backfill soil.(5)
Laying middle backfill soil.(6)
Laying final layer of backfill soil.(7)
Applying working water pressure and water hammer pressure.After wrapping wires and curing the pipe, the range of circumferential stress in concrete was from 10.82 MPa to 12.63 MPa, and the circumferential stress was distributed uniformly, as shown in Figure5.Figure 5
Stress in ECP after wire wrapping and curing.The stresses and strains in the pipe after applying working water pressure and water hammer pressure are shown in Figure6.Figure 6
Stresses and strains in pipe: (a) 1st principal stress, (b) 3rd principal stress, (c) equivalent strain, (d) yield value, (e) wire stress, and (f) cylinder stress.
(a)(b)(c)(d)(e)(f)According to Figure6, the maximum damage value of mortar was 0.4165 at the springline, while the damage of outer concrete only started and its value was 0.17 also at the springline. The 3rd principal stress of inner concrete was 17.3 MPa. The equivalent strain was 0.00035 in mortar and 0.000157 in outer concrete, respectively. The maximum stress of outer layer wires was 1039.7 MPa, while the stress in cylinder at the invert was -138.9 MPa.


#### 4.2.4. Comparison of Two Methods

We present the results of the improved method and FEA in Table5. The results show that the two methods produced close circumferential stresses, 3rd principal stresses in mortar coating and concrete, and stresses in wires and the cylinder. However, large errors exist for mortar coating strains, because the improved method assumed that mortar coating had zero prestress while the FEA showed that it was in fact -3.88 MPa, due to pipe shrinkage and creep. Thus the result yielded by the improved method was larger than that of the FEA. Similarly, the circumferential stress in outer concrete was 10.82 MPa after curing and 12.63 MPa in inner concrete, although the improved method regarded them as equal in the whole cross section of concrete with a value of 12.09 MPa, so the prestress of outer concrete is exaggerated in specifications and standards. Thus, the strain in outer concrete obtained by the improved method was smaller than that of the FEA. Generally, considering the rationality of the finite element model, constitutive models of materials, and contact model, the 3D nonlinear FEA shows great consistency with the improved method in simulating ECPs.Table 5
Main analysis results of pipe.
MethodCircumferential stress in concrete after curing (MPa)Strain in mortar coatingStrain in outer concrete3rd principal stress in inner concrete (MPa)Stress in wires (MPa)Stress in cylinder (MPa)Improved method−12.090.000560.0000419.71024.1−117.9FEA−10.82～−12.630.000350.00015717.31039.7−138.9


## 4.1. Improved Method

The ECP considered in the study has a diameter of 4 m and working pressure of 0.8 MPa, and the whole wrapping area of wire is 5497 mm2 with two wrapping layers. The thickness of the steel cylinder is 2 mm. The wire diameter is 7 mm. The wire-relaxation factor of the outer layer is 0.127 and that of the inner layer is 0.108, respectively. The shrinkage and creep factors of tube core are 6.6617 × 10−5 and 1.177, respectively. The results calculated by the improved method are shown in Table 2.Table 2
Improved method applied to ECP case with 4 m diameter.
Limit state and locationLimit state criterionApplicable load combinationCalculation valueLimit valueServiceability at invertInside core tensile strain limit:εci≤1.5εt′W10.0001990.000211Inner core tensile strain limit:εci≤εk′=11εt′WT10.0002760.00155WT20.000203FT10.000349Inner core-to-cylinder radial tension limit:σr≤0.082MPaFW1−0.114 MPa0.082 MPaWT3−0.154 MPaServiceability at springlineOuter core tensile strain limit:εco≤1.5εt′W1−0.0000100.000211Outer core tensile strain limit:εco≤εk′=11εt′WT10.000040.00155WT2−0.00001FT10.00006Outer coating tensile strain limit:εmo≤0.8εkm'=6.4εtm'W10.00050.00091Outer coating tensile strain limit:εmo≤εkm'=8εtm'WT10.000560.00113WT20.00050FT10.00058Inner core compression limit:fci≤0.55fc′W219.8 MPa26.4 MPaInner core compression limit:fci≤0.65fc′WT320.1 MPa31.2 MPaElastic limit at invertCylinder stress reaching yield:-fyr+n′fcr+Δfy≤fyyWT1−117.9 MPa227.5 MPaWT2−131.7 MPaFT1−109.5 MPaCylinder stress reaching yield:-fyr+n′fcr+Δfy≤0WT3−158.8 MPa0Elastic limit at springlineWire stress limitfsg:-fsr+nfcr+Δfs≤fsgFWT11033.3 MPa1177.5 MPaFWT21023.8 MPaFT21024.1 MPaCore compression limit:fci≤0.75fc′FWT120.5 MPa36 MPaFWT222.5 MPaFT220.7 MPaStrength at springlineWire stress limitfsy:-fsr+nfcr+Δfs≤fsyFWT31053.3 MPa1334.5 MPaFWT41041.5 MPaUltimate limit:M≤MultFWT5811910.5 kN m1104296.6 kN m


## 4.2. Nonlinear Finite Element Analysis

To evaluate the structural integrity of ECPs, numerical simulations are usually conducted. Thus, we also performed a 3D nonlinear FEA. Similar to the structural mechanics method, the stress-strain states at key cross sections of the pipe under 14 load combinations should be obtained. However, in the 3D nonlinear FEA we considered the installation process and nonlinearity of the materials and contacts. Therefore, the special load combinations were ignored and we selected the load combination WT1 for analysis, which includes internal pressure, transient water hammer pressure, and normal external permanent loads.


### 4.2.1. FEA Model and Material Constitutive Models

In this model, the height of backfill soil is 10 m and the thickness of tube core is 350 mm. To reduce the boundary effect, we modeled five pipes along the pipeline direction and then chose the middle pipe from the left pipeline pipes for analysis. A total of 123,488 elements were used in the model, including hexahedral elements for concrete, mortar coating, and soil. The prestressing wires were modeled as bar elements and steel cylinder as plate elements. The entire model had a length of 25 m along pipeline direction, a width of 53.2 m, and a height of 29.756 m. For boundary conditions, the top surface of the model was assumed free and the other five sides as fixed in the normal directions. The pipe model is shown in Figure4.Figure 4
Finite element model of ECP.There are four methods for modeling prestress in wires in FEA: the equivalent load method, initial strain method, equivalent cooling method, and wire-wrapping model [21]. In the equivalent load method used in most studies [18, 22–30], it does not consider the stiffness of wires and as a result it is difficult to simulate the stress increment in steel wires under the external loads in pipes with broken wires. In the strain method proposed by Lin [31], applying an initial strain to steel wires was used to model the prestressing force; however, the loss of prestressing force cannot be considered in this method because the loss of prestressing force is represented by stress. In the equivalent cooling method proposed by researchers [32–37], stiffness of wires can be considered. It is actually an equivalent deformation method achieving the required stress by cooling the steel wire based on its temperature characteristics, the physical concept of which is simple and easy to control. However, cooling induces compressive strain on the steel wire, which does not meet the initial tension state of the steel wire. Meantime, if the nonlinearity intrinsic behavior of material and contact are considered while using the cooling method, it is a complex nonlinear coupled problem, which increases the difficulty of solving the finite equations and is not easy to converge to the subsequent solution. The wire-wrapping model proposed by Xiong et al. [38] is very complex and difficult to apply in FEA. To avoid the shortcomings mentioned above, we chose a new prestress simulation method referred to as the initial stress method. The specific idea is to directly apply the wrapping prestress considered as a loss to the Gauss points of wire elements by self-written FEA program called Gehomadrid. This is easy and direct to simulate with good convergence. Most importantly, the loss of prestressing force can easily be considered by this approach.A large number of damage models for concrete may be found in [39, 40]. Wei et al. [41] presented a new four-parameter failure criterion in strain space based on the Hsieh-Ting-Chen four-parameter failure criterion. The new criterion can model the concrete strength precisely not only under uniaxial but also under multiaxial stresses; details can be found in [39, 41]. In this paper, we adopted the new four-parameter failure criterion in strain space combined with the stress-strain curve from [2, 38] as the concrete and mortar damage constitutive model to simulate damage.The nonlinear hyperbolic Duncan-Chang soil model formulas [42] are obtained from conventional triaxial test and include seven parameters. This model is relatively simple and its engineering application is frequent. In the year of 1980, Duncan and Wong et al. replaced Poisson’s ratio with volume compression modulus and presented the improved E-B model; details can be found in [43]. In this study, we adopted the Dun-Chang E-B model for soil.The study by Zhao [39] presented a novel single spring joint element method based on mixed coordinate system and the traditional double spring element method, in which the difficulty of choosing the normal stiffness coefficient was avoided. In this paper, we selected the single spring joint element to model the bond-slip interaction in RC structures. Because of the lack of experimental bond-slip curves between high-strength concrete and plain prestressing wires, we adopted the nonlinear bond-slip constitutive relationship developed by Di [44] for normal concrete and crescent rib wires given as follows:(7)τ=6.59×102s−2.13×104s2+0.22×106s3,where τ is the bond stress (MPa); and s is the slip displacement (mm). Considering that plain wires are used in this study, we assumed that the bond stress was only one-tenth of the value obtained from Di’s formula. The nonlinear stress-strain relationship of wires was taken from standard AWWA C304-14.The common contact interface element used in engineering applications is the Goodman joint element with zero thickness [45]; thus we selected it in this paper. We also adopt the Von Mises yield criterion and an ideal elastoplastic constitutive model for the cylinder steel with complete bond-slip with concrete.


### 4.2.2. Material Properties

The main material properties adopted in the study are listed in Table3.Table 3
Properties of pipe materials.
MaterialGradeDensityYoung’s modulusPoisson’s ratioCompressive strength/MPaYield strength/MPaTensile strength/MPaCohesion force/NInternal friction angle/°/kg·m−3ConcreteC60232328.6 (GPa)0.16748.0—4.03——Wires—7833191 (GPa)0.29—1334.51570.0——Cylinder—7833207 (GPa)0.29—227.5310.0——Foundation—203980.0 (MPa)0.25———5.0E330Bedding—203940.0 (MPa)0.29———3.0E330Backfill—203950.0 (MPa)0.35———4.0E330The Dun-ChangE-B constitutive model coefficients of backfill soil are listed in Table 4.Table 4
Parameters of Dun-Chang E-B soil constitutive model.
SoilNondimensional parameter (K)Nondimensional index (n)Failure ratio (rf)Experimental parameter (Kb)Experimental parameter (m)Cohesion force (unit: N)Internal friction angle (unit: °)Foundation215.50.920.613151.010.755.0 × 10330Bedding232.60.900.613165.050.753.0 × 10330Backfill198.40.890.613132.270.754.0 × 10330


### 4.2.3. FEA Results

The FEA considering manufacturing, installation, and operation was divided into seven steps:(1)
Wrapping of inner layer.(2)
Wrapping of outer layer.(3)
Curing the pipe, with long-term creep, shrinkage, and wire-relaxation modeled. The creep, shrinkage, and wire-relaxation were calculated by the method described in [2] and used in the proposed improved method.(4)
Laying bedding soil, installing pipes, and laying bottom backfill soil.(5)
Laying middle backfill soil.(6)
Laying final layer of backfill soil.(7)
Applying working water pressure and water hammer pressure.After wrapping wires and curing the pipe, the range of circumferential stress in concrete was from 10.82 MPa to 12.63 MPa, and the circumferential stress was distributed uniformly, as shown in Figure5.Figure 5
Stress in ECP after wire wrapping and curing.The stresses and strains in the pipe after applying working water pressure and water hammer pressure are shown in Figure6.Figure 6
Stresses and strains in pipe: (a) 1st principal stress, (b) 3rd principal stress, (c) equivalent strain, (d) yield value, (e) wire stress, and (f) cylinder stress.
(a)(b)(c)(d)(e)(f)According to Figure6, the maximum damage value of mortar was 0.4165 at the springline, while the damage of outer concrete only started and its value was 0.17 also at the springline. The 3rd principal stress of inner concrete was 17.3 MPa. The equivalent strain was 0.00035 in mortar and 0.000157 in outer concrete, respectively. The maximum stress of outer layer wires was 1039.7 MPa, while the stress in cylinder at the invert was -138.9 MPa.


### 4.2.4. Comparison of Two Methods

We present the results of the improved method and FEA in Table5. The results show that the two methods produced close circumferential stresses, 3rd principal stresses in mortar coating and concrete, and stresses in wires and the cylinder. However, large errors exist for mortar coating strains, because the improved method assumed that mortar coating had zero prestress while the FEA showed that it was in fact -3.88 MPa, due to pipe shrinkage and creep. Thus the result yielded by the improved method was larger than that of the FEA. Similarly, the circumferential stress in outer concrete was 10.82 MPa after curing and 12.63 MPa in inner concrete, although the improved method regarded them as equal in the whole cross section of concrete with a value of 12.09 MPa, so the prestress of outer concrete is exaggerated in specifications and standards. Thus, the strain in outer concrete obtained by the improved method was smaller than that of the FEA. Generally, considering the rationality of the finite element model, constitutive models of materials, and contact model, the 3D nonlinear FEA shows great consistency with the improved method in simulating ECPs.Table 5
Main analysis results of pipe.
MethodCircumferential stress in concrete after curing (MPa)Strain in mortar coatingStrain in outer concrete3rd principal stress in inner concrete (MPa)Stress in wires (MPa)Stress in cylinder (MPa)Improved method−12.090.000560.0000419.71024.1−117.9FEA−10.82～−12.630.000350.00015717.31039.7−138.9


## 4.2.1. FEA Model and Material Constitutive Models

In this model, the height of backfill soil is 10 m and the thickness of tube core is 350 mm. To reduce the boundary effect, we modeled five pipes along the pipeline direction and then chose the middle pipe from the left pipeline pipes for analysis. A total of 123,488 elements were used in the model, including hexahedral elements for concrete, mortar coating, and soil. The prestressing wires were modeled as bar elements and steel cylinder as plate elements. The entire model had a length of 25 m along pipeline direction, a width of 53.2 m, and a height of 29.756 m. For boundary conditions, the top surface of the model was assumed free and the other five sides as fixed in the normal directions. The pipe model is shown in Figure4.Figure 4
Finite element model of ECP.There are four methods for modeling prestress in wires in FEA: the equivalent load method, initial strain method, equivalent cooling method, and wire-wrapping model [21]. In the equivalent load method used in most studies [18, 22–30], it does not consider the stiffness of wires and as a result it is difficult to simulate the stress increment in steel wires under the external loads in pipes with broken wires. In the strain method proposed by Lin [31], applying an initial strain to steel wires was used to model the prestressing force; however, the loss of prestressing force cannot be considered in this method because the loss of prestressing force is represented by stress. In the equivalent cooling method proposed by researchers [32–37], stiffness of wires can be considered. It is actually an equivalent deformation method achieving the required stress by cooling the steel wire based on its temperature characteristics, the physical concept of which is simple and easy to control. However, cooling induces compressive strain on the steel wire, which does not meet the initial tension state of the steel wire. Meantime, if the nonlinearity intrinsic behavior of material and contact are considered while using the cooling method, it is a complex nonlinear coupled problem, which increases the difficulty of solving the finite equations and is not easy to converge to the subsequent solution. The wire-wrapping model proposed by Xiong et al. [38] is very complex and difficult to apply in FEA. To avoid the shortcomings mentioned above, we chose a new prestress simulation method referred to as the initial stress method. The specific idea is to directly apply the wrapping prestress considered as a loss to the Gauss points of wire elements by self-written FEA program called Gehomadrid. This is easy and direct to simulate with good convergence. Most importantly, the loss of prestressing force can easily be considered by this approach.A large number of damage models for concrete may be found in [39, 40]. Wei et al. [41] presented a new four-parameter failure criterion in strain space based on the Hsieh-Ting-Chen four-parameter failure criterion. The new criterion can model the concrete strength precisely not only under uniaxial but also under multiaxial stresses; details can be found in [39, 41]. In this paper, we adopted the new four-parameter failure criterion in strain space combined with the stress-strain curve from [2, 38] as the concrete and mortar damage constitutive model to simulate damage.The nonlinear hyperbolic Duncan-Chang soil model formulas [42] are obtained from conventional triaxial test and include seven parameters. This model is relatively simple and its engineering application is frequent. In the year of 1980, Duncan and Wong et al. replaced Poisson’s ratio with volume compression modulus and presented the improved E-B model; details can be found in [43]. In this study, we adopted the Dun-Chang E-B model for soil.The study by Zhao [39] presented a novel single spring joint element method based on mixed coordinate system and the traditional double spring element method, in which the difficulty of choosing the normal stiffness coefficient was avoided. In this paper, we selected the single spring joint element to model the bond-slip interaction in RC structures. Because of the lack of experimental bond-slip curves between high-strength concrete and plain prestressing wires, we adopted the nonlinear bond-slip constitutive relationship developed by Di [44] for normal concrete and crescent rib wires given as follows:(7)τ=6.59×102s−2.13×104s2+0.22×106s3,where τ is the bond stress (MPa); and s is the slip displacement (mm). Considering that plain wires are used in this study, we assumed that the bond stress was only one-tenth of the value obtained from Di’s formula. The nonlinear stress-strain relationship of wires was taken from standard AWWA C304-14.The common contact interface element used in engineering applications is the Goodman joint element with zero thickness [45]; thus we selected it in this paper. We also adopt the Von Mises yield criterion and an ideal elastoplastic constitutive model for the cylinder steel with complete bond-slip with concrete.


## 4.2.2. Material Properties

The main material properties adopted in the study are listed in Table3.Table 3
Properties of pipe materials.
MaterialGradeDensityYoung’s modulusPoisson’s ratioCompressive strength/MPaYield strength/MPaTensile strength/MPaCohesion force/NInternal friction angle/°/kg·m−3ConcreteC60232328.6 (GPa)0.16748.0—4.03——Wires—7833191 (GPa)0.29—1334.51570.0——Cylinder—7833207 (GPa)0.29—227.5310.0——Foundation—203980.0 (MPa)0.25———5.0E330Bedding—203940.0 (MPa)0.29———3.0E330Backfill—203950.0 (MPa)0.35———4.0E330The Dun-ChangE-B constitutive model coefficients of backfill soil are listed in Table 4.Table 4
Parameters of Dun-Chang E-B soil constitutive model.
SoilNondimensional parameter (K)Nondimensional index (n)Failure ratio (rf)Experimental parameter (Kb)Experimental parameter (m)Cohesion force (unit: N)Internal friction angle (unit: °)Foundation215.50.920.613151.010.755.0 × 10330Bedding232.60.900.613165.050.753.0 × 10330Backfill198.40.890.613132.270.754.0 × 10330


## 4.2.3. FEA Results

The FEA considering manufacturing, installation, and operation was divided into seven steps:(1)
Wrapping of inner layer.(2)
Wrapping of outer layer.(3)
Curing the pipe, with long-term creep, shrinkage, and wire-relaxation modeled. The creep, shrinkage, and wire-relaxation were calculated by the method described in [2] and used in the proposed improved method.(4)
Laying bedding soil, installing pipes, and laying bottom backfill soil.(5)
Laying middle backfill soil.(6)
Laying final layer of backfill soil.(7)
Applying working water pressure and water hammer pressure.After wrapping wires and curing the pipe, the range of circumferential stress in concrete was from 10.82 MPa to 12.63 MPa, and the circumferential stress was distributed uniformly, as shown in Figure5.Figure 5
Stress in ECP after wire wrapping and curing.The stresses and strains in the pipe after applying working water pressure and water hammer pressure are shown in Figure6.Figure 6
Stresses and strains in pipe: (a) 1st principal stress, (b) 3rd principal stress, (c) equivalent strain, (d) yield value, (e) wire stress, and (f) cylinder stress.
(a)(b)(c)(d)(e)(f)According to Figure6, the maximum damage value of mortar was 0.4165 at the springline, while the damage of outer concrete only started and its value was 0.17 also at the springline. The 3rd principal stress of inner concrete was 17.3 MPa. The equivalent strain was 0.00035 in mortar and 0.000157 in outer concrete, respectively. The maximum stress of outer layer wires was 1039.7 MPa, while the stress in cylinder at the invert was -138.9 MPa.


## 4.2.4. Comparison of Two Methods

We present the results of the improved method and FEA in Table5. The results show that the two methods produced close circumferential stresses, 3rd principal stresses in mortar coating and concrete, and stresses in wires and the cylinder. However, large errors exist for mortar coating strains, because the improved method assumed that mortar coating had zero prestress while the FEA showed that it was in fact -3.88 MPa, due to pipe shrinkage and creep. Thus the result yielded by the improved method was larger than that of the FEA. Similarly, the circumferential stress in outer concrete was 10.82 MPa after curing and 12.63 MPa in inner concrete, although the improved method regarded them as equal in the whole cross section of concrete with a value of 12.09 MPa, so the prestress of outer concrete is exaggerated in specifications and standards. Thus, the strain in outer concrete obtained by the improved method was smaller than that of the FEA. Generally, considering the rationality of the finite element model, constitutive models of materials, and contact model, the 3D nonlinear FEA shows great consistency with the improved method in simulating ECPs.Table 5
Main analysis results of pipe.
MethodCircumferential stress in concrete after curing (MPa)Strain in mortar coatingStrain in outer concrete3rd principal stress in inner concrete (MPa)Stress in wires (MPa)Stress in cylinder (MPa)Improved method−12.090.000560.0000419.71024.1−117.9FEA−10.82～−12.630.000350.00015717.31039.7−138.9


## 5. Conclusions

The proposed improved method showed good convergence and efficiency in the case study verification and will be useful for practical design applications to ECPs compared to popular design standards. The results obtained by this method met the specification requirements. Furthermore, a nonlinear FEA was conducted to simulate the mechanical behavior of ECPs considering manufacturing, installation, and operation in seven steps. We also adopted proper elements, constitutive models, and way of simulating prestress in the advanced analysis. The analysis results obtained by this method and FEA showed strong consistency and we also demonstrated that the results of the advanced nonlinear FEA met the specification requirements. Moreover, the FEA showed that the prestress of mortar coating was -3.88 MPa due to shrinkage and creep of ECP. However, this method and specifications regarded it as zero and thus the stress-strain relationship of mortar adopted in specifications and standards proved conservative. Conversely, the prestress of outer concrete is exaggerated in specifications and standards posing risk of inaccurate results when they are applied in practical design. Therefore, when subjected to load combination WT1, the pipe was safe within current design limits. In addition, a nonlinear finite element analysis based on the abovementioned intrinsic structure model is recommended to aid in practical design projects.



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