A Simplified Limit-State Design and Verification for Prestressed Concrete Cylinder Pipes under Internal Water Pressure

Abstract

Ignoring the effect of a concrete core on bearing performance, the current design of prestressed concrete cylinder pipes (PCCPs) under internal water pressure only focuses on the fracture of prestressed steel wire, while the complexity of the AWWA C304 design method leads to a strong dependence on software that cannot be sufficiently mastered by the designers. In view of these issues, a simplified limit-state design process was induced to eliminate a large number of iterative operations and was verified by a three-dimensional finite element model (FEM) with a prototype test of PCCPs under internal water pressure. Meanwhile, the bearing performance of PCCPs was investigated using the parametric simulations of the FEM. The results showed that the cross-sectional area of the prestressed steel wire is higher by about 10% than that designed using the AWWA C304 method. The FEM provides a complete evolution process of the mechanical response of the structural constituents and simulates the strain mutation phenomenon of the prototype test well. The internal water pressure of the PCCPs designed using the simplified limit-state design process has enough safety to reach 4.7 times the working pressure at serviceability and 5.5 times the pressure at the ultimate limit state. A burst in the PCCPs took place under an internal pressure greater than 6.75 times the working pressure. The result of the FEM shows that an increase in the tensile strength of the concrete core is of great significance for improving the bearing performance of the PCCPs.

1. Introduction

In long-distance water diversion projects, prestressed concrete cylinder pipes (PCCPs) are mostly used as the main hydraulic structure. PCCPs are mainly composed of a concrete core, thin-steel cylinder, prestressed steel wire, protective mortar, and an external anti-corrosion coating [1,2]. The bearing performance of PCCPs mainly depends on the prestress of steel wire that is wound to the outside of the concrete core, while the seepage of water in the PCCP pipeline is avoided with the thin-steel cylinder [3,4]. At an early stage, PCCPs were designed using the allowable stress method combined with the empirical formula, which mainly limits the pressure in the pipe to control the tensile stress of the concrete core within its ultimate tensile strength [5]. However, this leads to an excessive use of materials based on the analytical results of hundreds of sets of pipeline test data [6]. Therefore, Zarghamee and Fok [7] proposed a new idea for PCCP design using a multi-layer ring model analysis. Subsequently, the most advanced limit-state design theory was applied to PCCPs. Based on the combination of factored and non-factored design load and internal pressure, the American Water Works Association code C304 (AWWA C304) provides a complete and technically correct design method [8,9]. The problem of the excessive use of materials can be solved by introducing three limit states of the PCCPs at serviceability, elasticity, and strength [10]. The drawback is that without relying on proprietary design software UDP, the AWWA C304 is difficult to understand and follow by the designers [11].

With regard to PCCP engineering accidents due to the pipe bursting [12,13], prototype tests of PCCPs with a diameter of 1.8 to 4.0 m under internal water pressure were performed to identify the failure mode of the broken wire [14], the deformation of the protective mortar [15], the cracking pressure of the concrete core [16], and the deformation of the steel cylinder [17,18]. The point sensors used for most of the tests could not continuously monitor the deformation of the pipeline in time and space [19]; therefore, a distributed optical fiber sensor was applied to measure and visualize the strain and load response of the PCCPs [20]. In view of the fact that the integrity of the prototype test results may not fully meet the expected requirements due to limitations with restricted test equipment and other objective factors, the finite-element model (FEM) was applied to analyze the bearing performance of the PCCPs. The mechanical response of the PCCPs with broken wires under internal water pressure can be analyzed by establishing a three-dimensional FEM [13,14,21]. The length of the broken wires can be determined using a theoretical method that considers the bonding quality of the mortar coatings [22,23]. The effect of the broken steel wire ratio on the bearing capacity of the PCCPs under internal water pressure can estimate the structural integrity of the PCCPs and thereby supports the recommendation for the operation and maintenance of the pipelines [24,25]. Meanwhile, considering the overestimation of damage caused by broken wires due to removing all the broken wire, and the underestimation of damage with simplified contact interactions, a new broken wire FEM was proposed [26,27].

The current research using the FEM for PCCPs mainly focuses on the broken prestressed steel wire. Environmental corrosion or hydrogen embrittlement is the main reason for prestressed steel wire damage [13,28]. However, both damage models of the prestressed steel wire have the characteristic of time accumulation with the premise of ensuring the qualification of production materials. In actual PCCP production, the steel wire put into production is the final product that can be directly used, and its production quality can be guaranteed by the steel wire manufacturer. In contrast, the pouring quality of the concrete core is susceptibly influence by the raw material, pouring environment, production management and worker’s technical level as well as the maintenance status of the molds and the pouring equipment [29]. The existing prototype tests point to the fact that the concrete core or the protective mortar rather than the steel wire is always firstly damaged [15,16,17,18]. This means that the concrete core or the protective mortar, rather than the prestressed steel wire, is more likely to be a weak link for the PCCPs. Under the internal water pressure, the concrete core undergoes a transition from compression to tension [16]. Once a full sectional tension appears on the concrete core, the protective mortar is put under tension. Therefore, the stress states of the concrete core and the mortar are effectively influenced by that of the PCCP. Unfortunately, a lack of research has been performed on this aspect to provide a deeper understanding.

In summary, the current design of PCCPs under internal water pressure only focuses on the damage to the prestressed steel wire, while the effect of the concrete core on the bearing performance is ignored and therefore does not receive enough attention. Moreover, the complexity of the AWWA C304 design method leads to a strong dependence on software that cannot be well mastered by the designers. In view of these issues, a simplified limit-state design process was induced to eliminate a large number of iterative operations [30]. The design results are compared with the AWWA C304 design method and verified using a three-dimensional FEM with a prototype test of the PCCP under internal water pressure. Meanwhile, the effects of the concrete core and the mortar are investigated using a parametric study. This can help designers correctly understand the performance of the PCCPs under internal water pressure when also considering the effects of the concrete core and the mortar.

2. Design Process

2.1. Design Method

A simplified limit-state design method was adopted by the code of China code CECS 140 [30]. The design process is shown in Figure 1. The differences between CECS 140 and AWWA C304 in the design process mainly focus on the load combination, the limit state, and the cracking control of the concrete core and mortar.

2.1.1. Load and Pressure Combination

The effects of pipe weight G_1k, water weight G_wk, and earth load F_sk are considered by the two design methods, as shown in

Table 1. Load and pressure combination for AWWA C304.

Combinations	Fsk	G1k	Gwk	q	Pw	Pt	Pft (1)
	Working Load and Pressure Combinations
W1	1.0	1.0	1.0	/ (2)	1.0	/	/
W2	1.0	1.0	1.0	/	/	/	/
FW1	1.0	1.0	1.0	/	/	/	/
	Working Plus Transient Load and Pressure Combinations
WT1	1.0	1.0	1.0	/	1.0	1.0	/
WT2	1.0	1.0	1.0	1.0	1.0	/	/
WT3	1.0	1.0	1.0	1.0	/	/	/
FWT1	1.1	1.1	1.1	/	1.1	1.1	/
FWT2	1.1	1.1	1.1	1.1	1.1	/	/
FWT3	1.3	1.3	1.3	/	1.3	1.3	/
FWT4	1.3	1.3	1.3	1.3	1.3	/	/
FWT5	1.6	1.6	1.6	2.0	/	/	/
FWT6	/	/	/	/	1.6	2.0	/
	Field-Test Condition
FT1	1.1	1.1	1.1	/	/	/	1.1
FT2	1.21	1.21	1.21	/	/	/	1.21

Note: (1) P_ft is the field test pressure; (2) “/” means that the load or pressure is not considered in this combination.

[1] and

Table 2. Load and pressure combination for CECS 140.

Combinations	Calculation Content	G1k	Gwk	Fsk	Fep	σpe (1)	Fwd (2)	q or Ws	Pgw (3)
I	Anti-floating stability	1.0	/ (4)	1.0	/	/	/	/	1.0
II	Thrust resistance stability	1.0	1.0	1.0	1.0	/	1.0	/	1.0
III	Pipe barrel strength	1.2	1.27	1.27	1.0	/	1.4	1.4	/
IV	Standard combination of controlled cracking	1.0	1.0	1.0	1.0	1.0	1.0	1.0	/
V	Quasi permanent combination of controlled cracking	1.0	1.0	1.0	1.0	/	1.0	1.0	/

Note: (1) σ_pe is the prestress of steel wire after deducting the prestress loss; (2) F_wd is the design pressure; (3) P_gw is the groundwater buoyancy. (4) “/” means that the item is not considered in this condition.

[30]. Both the vehicle live load q and the stacking load W_s are considered by AWWA C304, while the maximum value of q and W_s is considered by CECS 140. In terms of the internal water pressure, AWWA C304 separates it into a working pressure P_w and a water hammer pressure P_t, while CECS 140 specifies a design pressure F_wd. Compared to the five load and pressure combinations specified in CECS 140, twelve load and pressure combinations at the working condition and two combinations at the field-test condition are specified in AWWA C304. This leads to a complexity in the design process of the PCCPs.

2.1.2. Limit States and Control Criteria

The limit states and control criteria of both methods are shown in

Table 3. Limit states and control criteria for CECS 140 and AWWA C304.

Limit State	Control Materials and Location	Purpose	Limit Criteria
			CECS 140	AWWA C304
Serviceability	Concrete core at crown/invert	Microcracking control	/ (2)	εci ≤ 1.5εce (W1)
		Visible crack control	εci ≤ (1.75~3) εce (IV)	εci ≤ 11εce (WT2, WT2, FT1)
	Concrete core at the spring line	Microcracking control	/	εci ≤ 1.5εce (W1)
		Visible crack control	εci ≤ (1.75~3) εce (IV)	εci ≤ 11εce (WT2, WT2, FT1)
		Control of compression level	/	fci ≤ 0.55f’c (W2) fci ≤ 0.65f’c (WT3)
	Protective mortar at the spring line	Microcracking control	/	εmo ≤ 6.4εme (W1)
		Visible crack control	εmo ≤ 4εme (V) εmo ≤ 5εme (IV)	εmo ≤ 8εme (WT2, WT2, FT1)
Elastic Limit	Steel cylinder	Avoid yielding	/	−σfr + nσfc + Δfy ≤ fyy (WT2, WT2, FT1) −σfr + nσfc + Δfy ≤ 0 (WT3)
	Steel wire at the spring line	Avoid exceeding limit stress	/	−σfs + nσfc + Δfs ≤ σcon (FWT1, FWT2, FT2)
	Concrete core at the spring line	Control of compression level	/	fci ≤ 0.75f’c (FWT1, FWT2, FT2)
Ultimate Limit	Wire and cylinder at the spring line	Control wire and cylinder from design strength	As ≥ λy/fs(N (1) + M (1)max − Ayfy) (III)	/
	Full pipe circumference	Prevent pipe floatation	(G1k + Fsv,k)/Ffw,k ≥ Kf (I)	/
		Control thrust force	Fk/Fwp,k ≥ Ks (II)	/
Strength Limit	Steel wire at the spring line	Control wire from yielding	/	−σfs + nσfc + Δfs ≤ fsy (FWT3, FWT4)
	Concrete core at the spring line	Prevent crushing	/	M ≤ Mult (FWT5)

Note: (1) the force and moment after deducting the prestress loss of the prestressed steel wire. (2) “/” means that the item is not considered in this condition.

[1,30]. AWWA C304 includes three limit states at the serviceability, the elasticity and the strength, which considers the nonlinear mechanical response of structural materials. The serviceability limit state avoids the cracking of concrete and mortar. The elastic limit state controls the steel cylinder from yielding and the prestressed steel wire stress from exceeding the limit stress. The strength limit state controls the prestressed steel wire from yielding and the concrete from local crushing. CECS 140 specifies two limit states at the serviceability and the ultimate. The PCCPs are considered as an elastic system, ignoring the redistribution of internal forces caused by plastic deformation [30,31]. The serviceability limit state avoids the cracks that appeared on the concrete core at the crown/invert and the protective mortar at the spring line. The ultimate limit state controls the constituents from exceeding their strengths.

2.1.3. Cracking Control of Concrete and Mortar

AWWA C304 describes the mechanical response of the concrete core at four stages: (1) prestressing; (2) elastic; (3) transition; and (4) strain softening, as shown in Figure 2a. No crack of concrete appears in stages 1 to 3, and the microcrack will grow in stage 4. Points A, B, and C separately indicate the concrete in elastic limit, the initial microcracks, and the critical transferring from microcrack to visible crack, respectively. The widths of the microcrack and the visible crack are 0.025 mm and 0.05 mm, respectively [32]. The occurrence of visible cracks indicates that the PCCP reaches the serviceability limit state. For the protective mortar, stage 1 is not considered, while stages 2 to 4 are similar to those of concrete, as shown in Figure 3a. The control strain to visible crack of the concrete and the mortar is 11 ε_ce and 8 ε_me, respectively.

Compared to AWWA C304, CECS 140 neglects stage 3 to describe the mechanical response of the concrete core and the mortar, as shown in Figure 2b and Figure 3b. The control strain to visible crack of the concrete and the mortar is only 3 ε_ce and 4 ε_me, respectively.

2.2. Design Process

2.2.1. Earth Load

The vertical earth load is calculated as following according to the specifications in CECS 140 and AWWA C304 [30,33] and marked as F_sk,CECS and F_sk,AWWA, respectively.

$$F_{\mathrm{sk},\mathrm{CECS}}=1.4{\gamma }_{\mathrm{s}}HD$$

(1)

$$F_{\mathrm{sk},\mathrm{AWWA}}=\left\{\begin{array}{c}\frac{e^{\pm 2K\mu \left(\frac{H}{B_{\mathrm{c}}}\right)}-1}{\pm 2K\mu }{\gamma }_{\mathrm{s}}{D}^2(H\le H_{\mathrm{e}})\\ \left(\frac{e^{\pm 2K\mu \left(\frac{H_{\mathrm{e}}}{B_{\mathrm{c}}}\right)}-1}{\pm 2K\mu }+\left[\left(\frac{H}{B_{\mathrm{c}}}\right)-\left(\frac{H_{\mathrm{e}}}{B_{\mathrm{c}}}\right)\right]e^{\pm 2K\mu \left(\frac{H_{\mathrm{e}}}{B_{\mathrm{c}}}\right)}\right){\gamma }_{\mathrm{s}}{D}^2(H>H_{\mathrm{e}})\end{array}\right.$$

(2)

2.2.2. Maximum and Minimum Wire Area

The minimum center distance d_s,min shall not be less than twice the diameter (d) of the steel wire, and the maximum center distance d_s,max shall not be greater than 38 mm. The minimum cross-sectional area A_s,min and maximum cross-sectional area A_s,max are calculated as follows:

$$A_{\mathrm{s},\min }=\frac{250\pi d^{2}n}{d_{\mathrm{s},\max }}$$

(3)

$$A_{\mathrm{s},\max }=\frac{250\pi d^{2}n}{d_{\mathrm{s},\min }}$$

(4)

2.2.3. Prestress

In this study, the prestress loss is attributed to the elastic compression of the concrete core (σ₁), the shrinkage and creep of the concrete core (σ₂), and the relaxation of prestressed steel wire (σ₃) [1,30]. Figure 4 shows the theoretical trend of the prestress change with the storage time. The time t₀ to t₁ is the storage time of the PCCP. When the prestressed steel wire is wound around the concrete core, the elastic compression of concrete produces the radial deformation of the concrete core, resulting in the prestress loss σ₁. The radial deformation distributes in law of larger at the middle and smaller at both ends along the axial of the PCCP [34], as shown in Figure 5. With the prestress losses σ₂ and σ₃, the effective prestress of the steel wire decreases continuously.

For CECS 140, the σ₁ and σ₃ are calculated using Formulas (5) and (6), and the effective prestress σ_pe is calculated using Formula (7):

$${\sigma }_1=0.08{\sigma }_{\mathrm{con}}{\varphi }_{\mathrm{t}}\varphi$$

(5)

$${\sigma }_3=0.5n_{\mathrm{s}}\rho {\sigma }_{\mathrm{con}}$$

(6)

$${\sigma }_{\mathrm{pe}}={\sigma }_{\mathrm{con}}-{\sigma }_1-{\sigma }_2-{\sigma }_3$$

(7)

The σ₂ is determined by a ratio of the normal precompression stress σ_p on the concrete core to the standard value f′_cu of the cube compressive strength of concrete. To ensure the convergence of concrete shrinkage and creep, σ_p ≤ 0.5 f′_cu [30]. For a single-layer winding of steel wire, σ₂ is obtained from

Table 4. Prestress loss caused by concrete shrinkage and creep.

σp/f′cu	Stress Level
	0.1	0.2	0.3	0.4	0.5
σ3 (MPa)	20	30	40	50	60

[30].

For AWWA C304, the prestress losses are calculated based on the previous research [35,36] and the shrinkage and creep model of concrete in the code ACI 209.2R [37]. The effective prestress σ_pe is calculated using Formula (8):

$${\sigma }_{\mathrm{pe}}={\sigma }_{\mathrm{con}}-{\sigma }_1-{\sigma }_2-{\sigma }_3={\sigma }_{\mathrm{con}}-n_{\mathrm{i}}{\sigma }_{\mathrm{ic}}-R{\sigma }_{\mathrm{con}}-\frac{A_{\mathrm{c}}\left({\sigma }_{\mathrm{ic}}{\varphi }_{\mathrm{c}}{n}_{\mathrm{r}}+E_{\mathrm{s}}{s}_{\mathrm{c}}\right)-R{A}_{\mathrm{s}}{\sigma }_{\mathrm{con}}{n}_{\mathrm{r}}\left(1+{\varphi }_{\mathrm{c}}\right)}{A_{\mathrm{c}}+\left(n_{\mathrm{r}}{A}_{\mathrm{s}}+{n^{\prime }}_{\mathrm{r}}{A}_{\mathrm{y}}\right)\left(1+{\varphi }_{\mathrm{c}}\right)}$$

(8)

The initial prestress σ_ic of the concrete core is calculated using Formula (9):

$${\sigma }_{\mathrm{ic}}=\frac{A_{\mathrm{s}}{\sigma }_{\mathrm{con}}}{A_{\mathrm{c}}+n_{\mathrm{i}}{A}_{\mathrm{s}}+{n^{\prime }}_{\mathrm{i}}{A}_{\mathrm{y}}}$$

(9)

Therefore, the final prestress σ_fc of the concrete core is separately calculated using Formulas (10) and (11) for CECS 140 and AWWA C304.

$${\sigma }_{\mathrm{fc},\mathrm{CECS}}=\frac{A_{\mathrm{s}}{\sigma }_{\mathrm{pe}}}{A_{\mathrm{c}}+n_{\mathrm{y}}{A}_{\mathrm{y}}+n_{\mathrm{s}}{A}_{\mathrm{s}}}$$

(10)

$${\sigma }_{\mathrm{fc},\mathrm{AWWA}}=\frac{{\sigma }_{\mathrm{ic}}\left(A_{\mathrm{c}}+n_{\mathrm{r}}{A}_{\mathrm{s}}+{n^{\prime }}_{\mathrm{r}}{A}_{\mathrm{y}}\right)-\left(A_{\mathrm{s}}{E}_{\mathrm{s}}+A_{\mathrm{y}}{E}_{\mathrm{y}}\right)s_{\mathrm{c}}-A_{\mathrm{s}}R{A}_{\mathrm{s}}{\sigma }_{\mathrm{con}}}{A_{\mathrm{c}}+\left(n_{\mathrm{r}}{A}_{\mathrm{s}}+{n^{\prime }}_{\mathrm{r}}{A}_{\mathrm{y}}\right)\left(1+{\varphi }_{\mathrm{c}}\right)}$$

(11)

2.2.4. Decompression and Burst Pressures

Under the prestress of steel wire, the concrete core has a certain precompression strain [16]. The precompression strain will decrease and turn into a tension with the increase in internal water pressure. This induces a decompression point which represents the disappearing of the precompression stain in the concrete core, corresponding to the decompression pressure P₀, as shown in Figure 6. The strain will suddenly increase when the concrete core reaches the elastic limit, which means the cracking of the concrete core.

The decompression pressure is, respectively, calculated using Formulas (12) and (13) for the CECS 140 and the AWWA C304, and it is marked as P_0,CECS and P_0,AWWA:

$$P_{0,\mathrm{CECS}}=\frac{A_{\mathrm{s}}{\sigma }_{\mathrm{pe}}}{1000D_{\mathrm{y}}}$$

(12)

$$P_{0,\mathrm{AWWA}}=\frac{{\sigma }_{\mathrm{fc}}\left(A_{\mathrm{c}}+n_{\mathrm{r}}{A}_{\mathrm{s}}+{n^{\prime }}_{\mathrm{r}}{A}_{\mathrm{y}}\right)}{1000D_{\mathrm{y}}}$$

(13)

The burst pressure P_b is calculated using Formula (14):

$$P_{\mathrm{b}}=\frac{2\left(A_{\mathrm{y}}{f}_{\mathrm{yu}}+A_{\mathrm{s}}{f}_{\mathrm{su}}\right)}{D_{\mathrm{y}}}$$

(14)

2.2.5. Moment and Thrust

Based on the rectangular distribution theory, CECS 140 divides the soil load distribution into a vertical earth pressure F_sk and a horizontal earth pressure F_ep, as shown in Figure 7a. The PCCPs tend to horizontal deformation under the F_sk, while F_ep can inhibit this trend. However, AWWA C304 distributes the earth pressure into an Olander’s bulb form, as shown in Figure 7b.

Table 5. Calculation of axial force and moment for the two design methods.

Calculation Content	CECS 140	AWWA C304
Moment at invert/crown (M1)	r0[kvm(Fsk + ψcqD) + khmFepD + kwmGwk + kgmG1k]	R[Cm1e(Fsk + q) + Cm1pG1k + Cm1fGwk]
Moment at spring line (M2)	γ0r0[kvm(γG3Fsk + γQ2ψcqD) + khmγG3FepD + kwmγG2Gwk + kgmγG1G1k]	R[Cm2e(Fsk + q) + Cm2pG1k + Cm2fGwk]
Thrust at invert/crown (N1)	ψc Fwd,k r0 × 10−3	0.5DyP − [Cn1e(Fsk + q) + Cn1pG1k + Cn1fGwk]
Thrust at spring line (N2)	γ0[ψc γQ1Fwd,k r0 × 10−3 − 0.5(Fsk + ψcqD)]	0.5DyP − [Cn2e(Fsk + q) + Cn2eG1k + Cn2eGwk]
Moment redistribution at spring line (M2r)	Not considered	M1 + M2 − M1cap

lists the formulae for calculating the moment and axial force of the PCCPs [1,30]. The moment redistribution is considered by AWWA C304. In both design methods, the assumed cross-sectional area of prestressed steel wire can be adopted only when all working combinations meet the control criteria of the corresponding limit state.

2.3. Design Parameters

Table 6. Geometry and design parameters for PCCP.

Inner Diameter (mm)	External Diameter of Steel Cylinder (mm)	Working Pressure (MPa)	Design Pressure (MPa)	Underground Burial Depth (m)	Thickness (mm)			Wire Diameter (mm)	Initial Winding Stress (MPa)
					Concrete Core	Protective Mortar	Steel Cylinder
3200	3343	0.4	0.6	5	245	25	1.5	7	1099

shows the geometry and design parameters of the PCCP, which is installed in a positive embankment with the arc soil foundation. The central angle of the pipe foundation is 120°. The following standard values were used: the unit weight of the backfill soil γ_s = 20 kN/m³, the pipe G_1k = 75.64 kN/m, and the fluid G_wk = 80.42 kN/m. The additional load W_s = 10 kN/m², and the vehicle live load q = 5.2 kN/m². According to the codes [38,39], the design parameters of materials are presented in

Table 7. Material properties for design.

Material	Standard Compressive Strength (MPa)	Modulus of Elasticity (MPa)	Standard Tensile Strength (MPa)	Design Tensile Strength (MPa)	Design Yield Strength (MPa)	Ultimate Tensile Strength (MPa)
Concrete	55	35,500	2.74	/	/	/
Mortar	45	24,165	3.49	/	/	/
Cylinder	/	206,000	/	215	235	370
Steel wire	/	205,000	/	1110	1177.5	1570

Note: “/” means that the item is not considered in this condition.

. The groundwater and the anti-floating stability are not required. The total outdoor laying time of the PCCPs is 270 d, and the time from the burial to the water supply is 90 d.

2.4. Design Results

The design results of the PCCP are shown in

Table 8. Design results for PCCP.

Parameters		CECS 140	AWWA C304	CECS 140/AWWA C304
External load (kN/m)		525.6 (Fsk)	488.7	Not
		32.92 (Fep)
Prestressing loss (MPa)	σ1	30.44	67.97	0.4
	σ2	35.20	65.23	0.5
	σ3	87.92	88.24	1.0
Effective prestress σpe (MPa)		945.4	877.7	1.1
Initial prestress in concrete core σic (MPa)		9.72	8.47	1.1
Final prestress in concrete core σfc (MPa)		8.36	6.71	1.2
Decompression pressure P0 (MPa)		1.33	1.08	1.2
Burst pressure Pb (MPa)		2.54	2.25	1.1
Cross-sectional area of wire As (mm2/m)		2350	2102	1.1
Wire spacing ds (mm)		16.4	18.3	0.9
Control limit state		Serviceability	Serviceability	Not
Control working condition		IV	W1	Not
Control location		Invert/crown	Invert/crown	Not
Control criterion		εci ≤ 3εce (Visible crack)	εci ≤ 1.5εce (Micro crack)	Not

. The prestress losses σ₁ and σ₂ of AWWA C304 were 2.3 times and 1.9 times that of CECS 140, respectively. The difference of σ₁ is caused by the higher tensile strength and lower elastic modulus of concrete adopted by AWWA C304. The difference of σ₂ is attributed to the adopted shrinkage and creep models of concrete. The prestress loss σ₃ is very close for the two methods. CECS 140 leads to an effective prestress of 7.7% higher than AWWA C304. This leads to the CECS 140 producing a concrete core with 1.1 times the initial prestress and 1.2 times the final prestress compared to AWWA C304, respectively.

Based on the design results, the cross-sectional area of prestressed steel wire designed by CECS 140 is about 10% higher than that by AWWA C304. This shows that CECS 140 can ensure the safety of the PCCP while reducing the calculation complexity and the dependence on software. Moreover, the cost of the prestressed steel wire was about 18% and 16% of the total manufacture cost of the PCCP designed using CECS 140 and AWWA C304, respectively. Therefore, the PCCP designed using the simplified method will increase cost less than 2% of the total manufacture cost because of the increase of about 10% prestressed steel wire under the working condition in this study.

3. Full-Scale Test and FE Model

3.1. Full-Scale Test

The test PCCP was designed by CECS 140 and was fabricated in a precast plant; for details, see the published paper [16]. The ages of the concrete core and protective mortar were 43 days and 38 days when the experimental study was carried out, respectively. Three group cubic specimens with dimensions of 150 mm were made during the concrete core pouring, which were respectively used to measure the concrete strength f_12h at demolding, the strength f_2d at the beginning of winding prestressed steel wire, and the strength f_28d at a standard curing age of 28 days. After the completion of the internal water pressure test, the PCCP was sampled with cylinder specimens of Φ100 mm × 100 mm to measure the actual strength (f_43d) [40]. The compressive strength results are shown in Figure 8.

The sample of steel cylinder and prestressed steel wire was tested by using the standard methods [41,42,43]. The yield and ultimate tensile strength of the steel cylinder were 300 MPa and 470 MPa. The ultimate tensile strength of prestressed steel wire was 1620 MPa. Because the test values of the f_2d and the ultimate tensile strength of prestressed steel wire are higher than the design value shown in Table 7, some related parameters are corrected based on the formulas (7), (9), (10), (12), and (14); the results are shown in

Table 9. Results after revised.

Parameter	σpe (MPa)	σic (MPa)	σfc (MPa)	P0 (kN)	Pb (kN)
Revised value	951.9	9.76	8.45	1.34	2.67

.

The positions of the strain monitoring device and experiment process are shown in Figure 9. For more details, refer to the published paper [16].

3.2. Finite Element Model (FEM)

The eight-node hexahedral-reduced integral linear solid element (C3D8R) was used to simulate the concrete core and protective mortar. A linear truss element (T3D2) and four-node shell element (S4R) were used to describe the prestressed steel wire and steel cylinder, respectively. The mesh results for all parts are shown in Figure 10. The model has a total of 27,840 solid elements, 24,240 truss elements, and 4800 shell elements.

3.2.1. Material Parameters

Generally, the probability of concrete strength no less than the standard design strength f_cu,k is considered in the design, as shown in Figure 11.

The f_cu,m and f_cu,max are the average and the upper limit of concrete strength. The δ is the standard deviation of concrete strength. Based on the standard normal distribution, the functional relationship P(t) between P and probability t can be expressed as shown below:

$$P\left(t\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_t^{+\infty }{e}^{-\frac{t^2}{2}}}dt$$

(15)

$$t=\frac{f_{\mathrm{cu},\mathrm{m}}-f_{\mathrm{cu},\mathrm{k}}}{\delta f_{\mathrm{cu},\mathrm{m}}}$$

(16)

Therefore, the f_cu,k, f_cu,m, and f_cu,max were adopted in FEM to consider the dispersion of concrete strength. Based on the test results, the average strength can be converted into a standard value and maximum value according to the Formulas (17) and (18) [44]. The calculation results of mortar and concrete strength used in FEM are shown in

Table 10. Strength calculation results of concrete and mortar (unit: MPa).

Materials	Parameters	Cubic Compressive Strength	Axial Compressive Strength	Axial Tensile Strength	Elastic Modulus
Concrete	Maximum	72.2	46.8	3.80	37,305
	Average	61.1	39.6	3.22	36,128
	Standard	50.0	32.4	2.64	34,554
Mortar	Average	56.1	36.9	3.13	25,817
	Standard	45.0	29.6	2.51	24,165

. Due to the particularity of the roll-casting process, the upper limit of strength is not considered for the protective mortar.

$$f_{\mathrm{cu},\mathrm{k}}=f_{\mathrm{cu},\mathrm{m}}\left(1-1.645\delta \right)$$

(17)

$$f_{\mathrm{cu},\max }=f_{\mathrm{cu},\mathrm{m}}\left(1+1.645\delta \right)$$

(18)

Based on the test results, the yield strength of steel cylinder and the ultimate tensile strength of prestressed steel wire are 300 MPa and 1620 MPa, respectively. The yield strength of prestressed steel wire is taken as 1200 MPa, which is 75% of the ultimate tensile strength according to China code GB/T 5223 [43]. According to the codes [38,39,41,43], the elastic modulus of the steel cylinder and prestressed steel wire are 206 GPa and 195 GPa, respectively. Moreover, Poisson′s ratio of concrete and mortar is set at 0.2, and that of steel materials is set at 0.3.

3.2.2. Stress–Strain Relationships

The nonlinear stress–strain curve of concrete under compression is expressed by Formula (19) [38]. A bilinear constitutive relationship is used for the tensile stress–strain of concrete. These relationships at different strengths of concrete are shown in Figure 12. A double linear model with the control cracking strain of 4ε_me is adopted for the tensile stress–strain relationship of the mortar.

$${\sigma }_{\mathrm{m}}=\left(1-d_{\mathrm{c}}\right)E_{\mathrm{c}}{\epsilon }_{\mathrm{m}}$$

(19)

The stress–strain relationship of the Abaqus CDP model is a function related to the scalar damage degradation variables d′_t and d′_c, as shown in Formulas (20) and (21) [45].

$${\sigma }_{\mathrm{t}}=\left(1-d_{\mathrm{t}}^{\prime }\right)E_0\left({\epsilon }_{\mathrm{t}}-{\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pl}}\right)$$

(20)

$${\sigma }_{\mathrm{c}}=\left(1-d_{\mathrm{c}}^{\prime }\right)E_0\left({\epsilon }_{\mathrm{c}}-{\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}}\right)$$

(21)

The calculated nominal stress and strain need to be converted into the real stress and strain during the calculation. Assuming that the elastic deformation is incompressible, the expression of true stress can be obtained from the constant volume, as shown in Formulas (22) and (23) [46]. Then, d′_t and d′_c are calculated according to Formulas (24) and (25), respectively. Furthermore, the damage parameters of concrete in the CDP model can be obtained, as shown in Figure 13.

$${\epsilon }_{\mathrm{true}}=\ln \left(1+\epsilon \right)$$

(22)

$${\sigma }_{\mathrm{true}}=\sigma \left(1+\epsilon \right)$$

(23)

$${d^{\prime }}_{\mathrm{t}}=\frac{1-\left({\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)}{\left({\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)+0.5\left({\epsilon }_{\mathrm{true}}-{\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)}$$

(24)

$${d^{\prime }}_{\mathrm{c}}=\frac{1-\left({\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)}{\left({\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)+0.3\left({\epsilon }_{\mathrm{true}}-{\sigma }_{\mathrm{true}}/E_{\mathrm{c}}\right)}$$

(25)

The plastic flow of the CDP model is assumed to be a non-correlated flow. The yield function is shown in Formula (26) [47]:

$$F=\frac{1}{1-\alpha }\left(\overline{q}-3\alpha \overline{p}+\left(\frac{{\overline{\sigma }}_{\mathrm{c}}\left({\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}}\right)}{{\overline{\sigma }}_{\mathrm{t}}\left({\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pl}}\right)}\left(1-\alpha \right)-\left(1+\alpha \right)\right)〈{\widehat{\overline{\sigma }}}_{\max }〉-\frac{3\left(1-K_{\mathrm{c}}\right)}{2K_{\mathrm{c}}-1}〈-{\widehat{\overline{\sigma }}}_{\max }〉\right)-{\overline{\sigma }}_{\mathrm{c}}\left({\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}}\right)=0$$

(26)

In this study, α = 0.12, and K_c = 0.667.

An ideal elastic–plastic model of the stress–strain relationship is selected for the steel cylinder, and a two-stage linear constitutive model is adopted for the prestressed steel wire, as shown in Figure 14.

3.2.3. Boundary Condition and Material Interaction

The axial and circumferential displacement constraints were applied to the top and bottom of the PCCP, while radial displacement was allowed. In this study, the “embedded” command was used to define the interaction between the steel cylinder to concrete and the prestressed steel wire to concrete, which assumed that the steel cylinder and prestressed steel wire are perfectly in contact with the concrete core. The “Tie” command was used to define the interaction of mortar to concrete, which assumed that no delamination exists between mortar and concrete.

3.2.4. Load Application and Working Conditions

The analysis steps of FEM are shown in

Table 11. Analysis step settings.

Analysis Step	Step 1	Step 2	Step 3	Step 4	…	Step 29
Load or pressure application	Gravity	Dead mortar and apply prestress	Activate mortar and add pressure to 0.1 MPa	Add pressure to 0.2 MPa	…	Add pressure to 2.7 MPa

. The final pressure is set to the burst pressure (2.7 MPa). Six groups of FEM models are established, in which the CAV + MSV group uses the same design parameters as the actual PCCP for the control, as shown in

Table 12. Combination of working conditions.

Combinations	Mortar: Average Value	Mortar: Standard Value
Concrete: max. value	CMV + MAV	CMV + MSV
Concrete: average value	CAV + MAV	CAV + MSV (Control)
Concrete: standard value	CSV + MAV	CSV + MSV

.

3.2.5. Prestress Exerting

The equivalent temperature reduction method is used to simulate the prestress of prestressed steel wire [27]. The expression of the relationship between temperature and stress is shown below:

$$\sigma =\Delta t{E}_{\mathrm{s}}{\alpha }_{\mathrm{t}}$$

(27)

In this study, α_t = 1 × 10⁻⁵/°C.

The prestress simulation results of the concrete core and the steel wire under CAV + MSV working conditions are shown in Figure 15. The design and simulation results of all working conditions are listed in

Table 13. Comparison of prestress results.

Combinations	Values of Concrete Core (MPa)			Values of Prestressed Wire (MPa)
	Designed	Simulated Max.	Simulated Min.	Designed	Simulated Max.	Simulated Min.
CMV + MAV	8.48	9.18	8.29	952.7	952.9	952.5
CMV + MSV		9.18	8.29		952.9	952.5
CAV + MAV	8.45	9.16	8.27	951.9	951.7	951.3
CAV + MSV		9.16	8.27		951.7	951.3
CSV + MAV	8.41	9.13	8.24	950.7	950.0	949.6
CSV + MSV		9.13	8.24		950.0	949.6

. This shows that the simulation results are in good agreement with the design results.

3.3. Validation of FEM with Test Results

Figure 16 shows the FEM simulation results compared with the test results, in which the test 1 m, test 2.5 m, and test 4 m indicate the distance of the monitoring point away from the spigot ring of the PCCP. The linear characteristics present to the strains of concrete, mortar, and prestressed steel wire, when the internal water pressure is lower than 1.8 MP. When the internal water pressure exceeds 1.8 MPa, the concrete strain suddenly increases. The FEM simulates the characteristic of strain mutation well with a higher accuracy in the linear stage of the materials while achieving the expected aim of simulating the strain abrupt phenomenon.

4. Discussion of Analytical Results

4.1. Effect of the Strength Variations of Concrete and Mortar

Figure 17 shows the mechanical response of the concrete core under internal water pressure influenced by the strength variations of concrete and mortar. The red horizontal line represents the strain corresponding to the theoretical decompression pressure. With the axial compressive strength of concrete increased from 32.4 to 46.8 MPa, the decompression pressure increases by 13.7~14.3% and 13.6~14.4%, respectively, for the inner and outer concrete cores. However, the decompression pressure of the concrete core only increases by a maximum of 1.6% with the increasing compressive strength of the mortar. This indicates that the compressive strength of concrete has a dominant effect.

The cracking pressure of the concrete core influenced by the strength variations of concrete and mortar is shown in Figure 18. The red horizontal line expresses the cracking strain. The cracking pressure of the concrete core increases from 1.90 to 2.08 MPa with the concrete strength. This is attributed to the increase in the tensile strength of concrete, which is very significant to the safe operation of the pipeline.

The effect of the strength variations of concrete on the mechanical response of steel cylinder and prestressed steel wire is shown in Figure 19. With the tensile strength of concrete increased from 2.64 to 3.80 MPa, the internal water pressure at the yield of the steel cylinder increases from 2.35 to 2.43 MPa, and that at the yield of the prestressed steel wire increases from 2.30 to 2.38 MPa. Moreover, the tensile stress of the prestressed steel wire in the CSV and CAV series fully reaches the ultimate strength under the internal water pressure of 2.7 MPa, while that in the CMV series only reaches the ultimate strength at the top and bottom of the PCCP. This indicates that the increase in concrete tensile strength can reduce the stress level of prestressed steel wire under working conditions.

In summary, the decompression pressure of the concrete core can increase by increasing the compressive strength of concrete. The cracking of the concrete core and the yielding of the steel cylinder can be delayed, while the stress level of the prestressed steel wire can be reduced by improving the tensile strength of concrete. Concrete rather than mortar has priority in the measures to improve the structural performance of the PCCP. However, the protection to prestressed steel wires can be ensured by improving the crack resistance and impermeability of the protective mortar.

4.2. Cracking Priority of the Concrete Core

The strain simulation results of the concrete core at the mutation point are shown in Figure 20. The strain of the inner concrete core was higher than that of the outer concrete core, which is consistent with the monitoring results reported [18,20]. Furthermore, the comparison of tensile stress changes of the inner and outer concrete cores at the mutation point are shown in Figure 21. For the series of CSV, CAV, and CMV, the falling of the tensile stress curve of the inner concrete core is prior to that of the outer concrete core. This is consistent to the previous test that a longitudinal crack was found on the inner concrete core before the strain of the outer concrete core suddenly changed [16]. Therefore, the strain mutation of the outer concrete core may be attributed to the cracking of the inner concrete core. Additionally, the crack location is affected by the concrete pouring process and the circumferential crack location on the inner core.

4.3. Mechanical Response of Steel Cylinder and Prestressed Steel Wire after Concrete Cracking

The residual stress of the concrete core is not normally considered after cracking. In this study, the mechanical response of the steel cylinder and prestressed steel wire after concrete cracking is simulated using FEM, as shown in Figure 22. With the increase in internal water pressure, the steel cylinder first reaches the design strength, and it subsequently reaches the yield strength after the yield of prestressed steel wire. However, before the fracture of prestressed steel wires, the yielded prestressed steel wire still provides a “hoop effect” on the steel cylinder that gradually decreases with the increase in internal water pressure. This indicates that an excessive free deformation cannot take place in the steel cylinder before the fracture of the prestressed steel wire. Therefore, the PCCP may burst at any time after the prestressed steel wire breaks since the cracked concrete is not able to restrain the free deformation of the steel cylinder.

4.4. Bearing Capacity of the PCCP

The designed and simulation values of the decompression pressure P₀ as well as the cracking pressure of the concrete or mortar of the CSV, CAV, and CMV models are listed in

Table 14. Comparison of the values obtained from theoretical design and numerical simulation.

Parameter	CSV + MSV		CAV + MSV (Control)			CMV + MSV
	Design	Simulation	Design	Test	Simulation	Design	Simulation
P0 (Inner core)	1.34	1.29	1.34	/	1.38	1.34	1.47
P0 (Outer core)	1.34	1.46	1.34	1.40	1.56	1.34	1.67
Pt (Mortar cracking)	1.74	1.89	1.85	/	1.98	1.95	2.09
Pt (Concrete cracking)	1.74	1.88	1.85	1.90	1.98	1.95	2.08

Note: “/” means that the item is not considered in this condition.

. The designed cracking pressure is calculated as follows [2].

$$P_{\mathrm{t}}=\frac{A_{\mathrm{s}}{\sigma }_{\mathrm{pe}}+1.06f_{\mathrm{tk}}{A}_{\mathrm{n}}}{1000r_0}$$

(28)

It can be seen from Table 14 that the same design value of P₀ for a PCCP model does not reflect the influence of the inner or outer concrete core, while the same design value of P_t does not differentiate the influence of the cracking of concrete or mortar. The simulation result shows that P₀ is corresponds more to the decompression of the outer concrete core, while P_t is not influenced by the cracking of the concrete or mortar. P_t of the concrete core increases by 10.6% with the increase in concrete tensile strength. For the CAV model, the values of P₀ and P_t are 11% and 4.2% higher than the test values. This is attributed to the use of a homogeneous material model in the FEM without congenital defects. The P_t value of the CSV group is closer to the test. The CAV and CMV models may overestimate the bearing capacity of the PCCP under internal water pressure. Therefore, the lower limit strength of concrete should be adopted for the numerical simulation using the FEM.

The tensile stress of the PCCP is mainly caused by the steel cylinder and the prestressed steel wire after cracking of the concrete core. The PCCP will access the ultimate limit state when the tensile stresses of the steel cylinder and the prestressed steel wire reach the design strength. The simulation results of the CSV group show that the PCCP reaches the ultimate bearing capacity under an internal water pressure of 2.2 MPa, and it bursts under an internal water pressure over 2.7 MPa.

Based on the FEM simulation results, the cracking pressure is 4.7 times the working pressure for the PCCP designed using CECS 140. The concrete core undergoes irreversible damage to access the serviceability limit state. When the internal water pressure is 5.5 times the working pressure, the PCCP reaches the ultimate limit state. Finally, the PCCP will be damaged under an internal water pressure that is more than 6.75 times the working pressure. In summary, the PCCP designed using the simplified limit-state design process is sufficient in safety for actual operation.

5. Conclusions

(1) Using the design conditions in this study, the simplified limit-state design process gives about a 10% higher cross-sectional area of the prestressed steel wire than using AWWA C304. This ensures the operation safety of PCCPs using a simplified calculation with no dependence on specialist design software.

(2) The finite element model not only accurately simulates the bearing behavior of PCCPs in the linear phase of materials but also achieves the expected aim regarding the abrupt point. The parametric simulation results show that the decompression pressure of the concrete core increases with the compressive strength of concrete, while the bearing resistance to internal water pressure increases with the tensile strength of concrete.

(3) The PCCP designed using the simplified limit-state design process can be respectively subject to 4.7 and 5.5 times the working pressure at the serviceability and the ultimate limit state. The PCCP will burst under an internal water pressure greater than 6.75 times the working pressure.

(4) The simulation of a finite element model makes up for the deficiency of the prototype test of the PCCP, which exhibits a complete mechanical response of the structural constituents of PCCPs under internal water pressure. This assists with comprehensively understanding the bearing performance of PCCPs.

(5) Because only case studies are carried out in this study, further verification should be conducted to ensure the rationality of the simplified limit-state design process. Moreover, improving the crack resistance and impermeability of protective mortar and increasing the tensile strength of concrete need to achieve enough attention in the design process.