2.1.6. Prestress Loss Caused by Shrinkage and Creep of Concrete, σ*l*<sup>5</sup>

The shrinkage and creep of the original concrete pipe were involved in the calculation of the prestressed stress of the prestressing wires. The shrinkage and creep of the reinforced pipe have been basically completed before the reinforcement, and, as such, no further repeated calculations were performed for the prestress loss caused by shrinkage and creep of concrete, i.e., σ*l*<sup>5</sup> = 0.

## 2.1.7. Prestress Loss Due to Long-Term Relaxation of the Strand, σ*l*<sup>6</sup>

The deformation of the strand will change with time, and the stress will decrease accordingly when the strand is subjected to a constant external force, which is the prestress loss due to the long-term

relaxation of the strand. The greater the tensile force of the steel strand, the more obvious the stress relaxation effect is. The relaxation generally occurred earlier in the process, without considering the quality of the strands. This effect can be basically completed after one year, and then gradually calmed. The relaxation loss, σ*l*6, is related to the relaxation coefficient, *k*, and can calculated according to the following formula:

$$
\sigma\_{l6} = k \sigma\_{st} \tag{21}
$$

where *k* is the relaxation coefficient and is related to the quality of the steel. For the cold-drawn thick steel bar, *k* is taken as 0.05 for one-time tensioning and 0.035 for ultra-tensioning. As for the steel wires and steel strands, *k* is considered to be 0.07 for one-time tensioning and 0.045 for ultra-tensioning [17]. For low-relaxation steel wires, the value of *k* can be taken to be 0.002 when no data are available, which we have learned from our experience.

The total prestress loss of the prestressed wires can calculate by Equation (22):

$$
\sigma\_{st,l} = \sum\_{1}^{6} \sigma\_{l} \tag{22}
$$

#### *2.2. Calculation of Area of Prestressed Steel Strands*

According to the study by Zarghamee M. [22–25], the cracking of PCCPs under combined loads mainly occurs at (1) the bottom of the inner surface of the concrete core, (2) the top of the inner surface of the concrete core, and (3) the spring-line of the outer surface of the concrete core. Therefore, these three sections are defined as dangerous sections. The area of prestressed steel strands can be determined under the assumption of a complete loss of prestress of prestressing wires.

#### 2.2.1. Stress of PCCPs under Combined Loads

The combined loads acting on the pipe include the vertical earth pressure at the top of the pipe, *Fsv*,*k*, the lateral earth pressure, *Fep*,*k*, the ground pile load, the weight of the pipe, *G*1*k*, the weight of fluid in the pipe, *Gwk*, and the variable load.

The values of *Fsv*,*<sup>k</sup>* and *Fep*,*<sup>k</sup>* are calculated according to Marston's theory [17] and Rankine's earth pressure theory [26], respectively. The variable load can be regarded as the ground stacking load, and its standard value is defined as *qmk* = 10 kN/m.

The weight of the pipe can be written as per Equation (23):

$$G\_{1k} = \pi r\_{\mathbb{G}} (D\_i + h\_c) h\_c \tag{23}$$

where *Di* is the inner diameter of the pipe (m), *hc* is the thickness of concrete core (m), and *rG* is the gravity density of the pipe (kN/m3).

The weight of fluid in the pipe, *Gwk*, can be calculated by Equation (24):

$$G\_{wk} = \frac{r\_W \pi D\_i^2}{4} \tag{24}$$

where *rw* is the gravity density of the fluid in the pipe (kN/m3).

2.2.2. Calculation of the Area of Prestressed Strands, Considering the Concrete Core Compression of Ultimate Limit States

According to Chinese specifications [27,28], the design requirements for the calculation of ultimate limit states under the external soil load, weight of pipe, weight of fluid, and other variable loads are detailed. The design value of the maximum bending moment of the pipe at the spring-line, *Ml max*, can be calculated using Equation (25). The value of *Ml max* is negative, indicating that the outer surface of the concrete core is subjected to tension. The absolute value can be taken when the formula is substituted for reinforcement. The design value of the maximum axial tension of the pipe at the spring-line, *N<sup>l</sup>* , is written as per Equation (26):

$$M\_{\rm max}^{l} = \gamma\_0 r \left[ k\_{\rm vm} \left( \gamma\_{\rm G3} F\_{\rm sv,k} + \psi\_c \gamma\_{\rm Q2} q\_{\rm vk} D\_1 \right) + k\_{\rm hm} \gamma\_{\rm G4} F\_{\rm cp,k} D\_1 + k\_{\rm um} \gamma\_{\rm G2} G\_{\rm uk} + k\_{\rm gm} \gamma\_{\rm G1} G\_{1k} \right] \tag{25}$$

$$N^l = \gamma\_0 \left[ \psi\_c \gamma\_{Q1} P\_d r \times 10^{-3} - 0.5 \left( F\_{sv,k} + \psi\_c q\_{vk} D\_1 \right) \right] \tag{26}$$

where γ<sup>0</sup> is the factor of importance. It varies with the structure and the layout of the pipes. The value of γ<sup>0</sup> is generally 1.1. For side-by-side pipelines, the γ<sup>0</sup> value should be taken as 1.0, in particular. Moreover, the value of γ<sup>0</sup> should also be taken as 1.0 for pipes with storage facilities or those which are used for drainage. *r* is the calculated radius of the pipe (*m*), and *kvm*, *khm*, *kwm*, *kgm* represent the bending moment coefficient of the bending moment at the spring-line of the pipe under the vertical earth pressure, lateral earth pressure, the weight of fluid inside the pipe, and the weight of the pipe, respectively. These factors were determined according to Appendix E [27]. The *kgm* of the arc-shaped soil bedding can be adopted according to the data of the bedding angle of 20◦. γ*Gi* and γ*Qj* are the partial coefficients under the permanent load i and the variable load j. ψ<sup>c</sup> is the combination coefficient of the variable loads and usually takes the value of 0.9. *D*<sup>1</sup> is the outer diameter of the pipe (m). *Pd* is the designed water pressure (N/mm2).

The area of the prestressed strands of ultimate limit states should be calculated by Equation (27):

$$A\_{st} \ge \frac{\lambda\_y}{f\_{pyk}} \left( N^l + \frac{M\_{\text{max}}^l}{d\_0} - A\_{sc} f\_{yy}^{\prime} \right) \tag{27}$$

where λ*<sup>y</sup>* is the comprehensive adjustment factor of the PCCP, *fpyk* is the design strength of prestressed strands (N/mm2), *d*<sup>0</sup> is the distance from the prestressed strand to the center of gravity of the pipe(m), *Asc* is the area of the cylinder per unit length (m2/m), and *f*- *yy* is the design strength of the cylinder (N/mm2).

2.2.3. Checking Calculation of Prestressed Strands Considering the Concrete Core Compression of Serviceability Limit States

The maximum bending moment of the pipe at the top or the bottom, *Mpms*, is calculated as per Equation (28). The value of *Mpms* is negative, indicating that the outer surface of the concrete core is subjected to tension. The absolute value is taken when substituting the following equations. The axial tension of the pipe wall, *Nps*, is written as per Equation (29):

$$M\_{\rm prms} = \gamma\_0 \left[ k\_{\rm vm} \left( F\_{\rm sv,k} + \psi\_c q\_{\rm vk} D\_1 \right) + k\_{\rm lm} F\_{\rm cp,k} D\_1 + k\_{\rm um} G\_{\rm uk} + k\_{\rm gm} G\_{\rm 1k} \right] \tag{28}$$

$$N\_{\rm ps} = \psi\_{\rm c} P\_{\rm d} r \times 10^{-3} \tag{29}$$

where *kvm*, *khm*, *kwm*, and *kgm* represent the bending moment coefficient of the bending moment at the top or the bottom of the pipe under the vertical earth pressure, lateral earth pressure, the weight of fluid inside the pipe, and the weight of the pipe, respectively. These factors can be determined according to Appendix E [27]. The *kgm* of the arc-shaped soil bedding can adopted, according to the data of the bedding angle of 20◦.

The maximum tensile stress at the edge of the pipe at the bottom, σ*ss*, is calculated as per Equation (30).

$$
\sigma\_{\rm ss} = \frac{N\_{\rm ps}}{A\_{\rm II}} + \frac{\mathcal{M}\_{\rm pms}}{\omega\_{\rm c} \mathcal{W}\_{\rm p}} \tag{30}
$$

where *An* is the conversion area of the pipe section (including the cylinder, steel strands, and the mortar coating) (m2/m). ω*<sup>c</sup>* is the conversion coefficient of the elastic resistance moment of the tensioned edge of the pip -wall. *Wp* is the momentary elastic resistance of the unconverted tension edge of the rectangular section of the pipe wall (m2/m).

The effective prestress of the prestressed steel strands after the prestress loss, σ- st, can be written as σ- *st* = σ*st* − σ*st*,*l*.

Therefore, the area of prestressed strands of serviceability limit states should be calculated by Equation (31):

$$A\_{st} \ge \left(\sigma\_{ss} - K\gamma f\_{ty}\right)\frac{A\_{ll}}{\sigma\_{st}'}\tag{31}$$

where *K* is the influence coefficient of concrete in the tension area, γ is the plastic influence coefficient of concrete in the tension area, and *fty* is the standard value of concrete tensile strength.

The area of prestressed strands needs to simultaneously meet the requirements outlined in Equations (27) and (31).

2.2.4. Checking Calculation of the Mortar Coating under Serviceability Limit States

Checking the calculation of mortar at the spring-line of the pipe should be carried out under serviceability limit states.

The maximum bending moment of the pipe at the spring-line, *Ml pms*, can be calculated by Equation (32). The value of *Ml pms*, is negative, indicating that the mortar coating is subjected to tension. The absolute value is taken when the following equations are substituted. The axial tension of the pipe at the spring-line, *N<sup>l</sup> ps*, can be written as per Equation (33):

$$M\_{\rm pms}^{l} = r \left[ k\_{\rm vm} (F\_{\rm sv,k} + \psi\_{\rm c} q\_{\rm vk} D\_1) + k\_{\rm hm} F\_{\rm cp,k} D\_1 + k\_{\rm vm} G\_{\rm wk} + k\_{\rm gm} G\_{1k} \right] \tag{32}$$

$$N\_{ps}^{l} = \psi\_{c} P\_{d} r \times 10^{3} - 0.5 \left( F\_{sv,k} + \psi\_{c} q\_{rk} D\_{1} \right) \tag{33}$$

The maximum tensile stress at the edge of the pipe at the spring-line, σ*<sup>l</sup> ss*, can be calculated as per Equation (34):

$$
\sigma\_{ss}^{l} = \frac{N\_{ps}^{l}}{A\_n} + \frac{M\_{pws}^{l}}{\omega\_m \mathcal{W}\_p} \tag{34}
$$

The maximum tensile stress at the edge of the mortar coating at the spring-line, σ*<sup>l</sup> ss*, should be less than its tensile strength (Equation (35)) under serviceability limit states. If not, Sections 2.2.2 and 2.2.3 should be repeated.

$$
\sigma\_{ss}^{l} \le \alpha\_m \varepsilon\_{mt} E\_m \tag{35}
$$

where α*<sup>m</sup>* is the design parameter of the mortar coating strain, which is equal to 5. ε*mt* is the strain of mortar coating when the strength reaches the tensile strength, and can be given as <sup>ε</sup>*mt* <sup>=</sup> *fmt*,*<sup>k</sup> Em* <sup>≥</sup> 0.52√*fmc*,*<sup>k</sup> Em* .

#### 2.2.5. Checking Calculation of Mortar Coating under Quasi-Permanent Limit States

Checking the calculation of mortar at the spring-line of the pipe should be carried out under quasi-permanent limit states.

The maximum bending moment of the pipe at the spring-line, *M<sup>l</sup> pml*, can be calculated as per Equation (36). The value of *M<sup>l</sup> pml* is negative, indicating that the mortar coating is subjected to tension. The absolute value is taken when substituting the following equations. The axial tension of the pipe at the spring-line, *N<sup>l</sup> ps*, is written as per Equation (37):

$$M\_{pml}^{l} = r \left[ k\_{\rm vm} (F\_{\rm sv,k} + \psi\_{\rm qv} q\_{\rm vk} D\_1) + k\_{\rm lm} F\_{\rm cp,k} D\_1 + k\_{\rm vm} G\_{\rm wk} + k\_{\rm gv} G\_{1k} \right] \tag{36}$$

$$N\_{pl}^{l} = \psi\_{q\overline{w}} P\_d r \times 10^3 - 0.5 \left( F\_{sv\lambda} + \psi\_{q\overline{v}} q\_{vk} D\_1 \right) \tag{37}$$

where ψ*qv*, ψ*qw* is the quasi-permanent coefficient of vertical pressure generated by ground vehicle loads and the internal water pressure, respectively.

The maximum tensile stress at the edge of the pipe at the spring-line, σ*<sup>l</sup> ls*, is calculated as per Equation (38):

$$
\sigma\_{ls}^{l} = \frac{\mathcal{N}\_{pl}^{l}}{A\_{\rm n}} + \frac{\mathcal{M}\_{pml}^{l}}{\omega\_{m}\mathcal{W}\_{p}}.\tag{38}
$$

The maximum tensile stress at the edge of the mortar coating at the spring-line, σ*<sup>l</sup> ss*, should be less than its tensile strength (Equation (39)) under quasi-permanent limit states. If not, we return to Equations (2) and (3).

$$
\sigma\_{ls}^{l} \le \alpha\_m' \varepsilon\_{mt} E\_m \tag{39}
$$

where α- *<sup>m</sup>* is the design parameter of strain for mortar coating and is equal to 4.

Above all, the area of prestressed strands per unit length, *Ast*, should be determined.

The prestressed steel strands are spirally wound at equal intervals. Thus, the center spacing of steel strands can be calculated by Equation (40):

$$d\_{\rm sf} = A \times \frac{1000}{A\_{\rm st}} \tag{40}$$

where *A* is the nominal section area, without polyethylene, of the adopted steel strand.

#### **3. Applications**

In order to verify the feasibility of the deduction, an example calculation of the theory and a prototype test were carried out on the same PCCP with broken wires. The specimen was an embedded prestressed concrete cylinder pipe (ECP) and the calculation process used is illustrated in Section 3.2. The center spacing of steel strands, calculated through the deduction, was then applied to the same pipe in a prototype test (Section 3.3).

### *3.1. Parameters of the Design and Materials*

The theory and prototype tests were carried out on the same pipe. The geometric parameters of the adopted pipe are given as Table 1. Key parameters of the materials, involving the concrete, mortar and cylinder, are shown in Table 2.


**Table 1.** Geometric parameters of the embedded concrete pipe (ECP).


**Table 2.** Key parameters of the materials.

As for the parameters of load, the internal working pressure used was *Pw* = 0.6 N/m2. The internal transient pressure was Δ*Hr* = max(0.4*Pw*, 276 kPa) = 0.276 N/mm2. The internal design pressure was *Pd* = *Pw* + <sup>Δ</sup>*Hr* = 0.876 MPa <sup>≈</sup> 0.9 N/mm2. The thickness of soil above the top of the pipe was *H* = 3 m. The bedding angle was 90◦. The type of installation was trench-type with a positive projecting embankment. The standard value of the ground stacking load was *qmk* = 10kN/m2.

The parameters of environment are shown as follows: The average relative humidity of the storage environment was 70% RH, the time in outdoor storage was *t*<sup>1</sup> = 270 d. Burial time after outdoor storage was *t*<sup>2</sup> = 1080 d.

Key parameters of the adopted strand are given in Table 3.

**Table 3.** Key parameters of the adopted strand.


#### *3.2. Example Calculation*

### 3.2.1. Calculation of Prestress Loss, σ*st*,*<sup>l</sup>*

For the utilized split wedge without jacking force, Δ*lre* = 6 mm (measured in the prototype test [29]). Given μ = 0.1, the calculated radius of the strand wrapped outside the pipe, *rst*, and the retraction length, *lre*, can be known as follows:

$$r\_{st} = \frac{D\_i}{2} + h\_c + \frac{d\_{st}}{2} = 1.1726 \text{ m},\\l\_{rc} = \sqrt{\frac{\Delta l\_{rc} E\_{st} r\_{st}}{\mu \sigma\_{st}}} = 3.4217 \text{ m}.$$

The corresponding angle of the retraction length, θ*re*, is π, which is consistent with the value of θ, indicating that the assumption is reasonable (Equation (10)).

The prestress loss caused by friction resistance, anchorage deformation, elastic compression of concrete during batch tensioning, crack reduction and closure, shrinkage and creep of concrete, and long-term relaxation of the strand is given in Table 4.


**Table 4.** The calculation results of the prestress loss.

### 3.2.2. Stress of PCCP under Combined Loads

The stress of the adopted PCCP under combined loads, involving the vertical earth pressure at the top of the pipe, the lateral earth pressure, the variable load, weight of the pipe, and weight of water in the pipe, is presented in Table 5.


**Table 5.** The calculation results of the stress.

3.2.3. Calculation of Area of Prestressed Strands, Considering the Concrete Core Compression of Ultimate Limit States

Assuming that the area of prestressed strands is *Ast* = 2223 mm2/m, the calculation process of the area of prestressed strands, considering the concrete core compression of ultimate limit states, can be depicted in Table 6.

The value of *Ml max* is negative, indicating that the outer surface of the concrete core is subjected to tension. The absolute value is taken when substituting the formula for reinforcement.

Therefore, *Ast* <sup>≥</sup> <sup>λ</sup>*<sup>y</sup> fpyk <sup>N</sup><sup>l</sup>* <sup>+</sup> *Ml max <sup>d</sup>*<sup>0</sup> − *Asc f*- *yy* = 1069.413 m2/m




3.2.4. Checking Calculation of Prestressed Strands Considering the Concrete Core Compression of Serviceability Limit States

The conversion coefficient of the elastic resistance moment of the tensioned edge of the pipe wall can be obtained by interpolation, where ω*<sup>c</sup>* = 1.017 and ω*<sup>m</sup>* = 0.9932. The checking calculation process of the prestressed strands, considering the concrete core compression of serviceability limit states, is depicted in Table 7.


**Table 7.** The calculation results of the stress.

Therefore, the area of prestressed strands should meet the requirement of *Ast* ≥ <sup>σ</sup>*ss* <sup>−</sup> *<sup>K</sup>*<sup>γ</sup> *fty An* σ- *st* = 2222.300 mm2/m.

Above all, the area of prestressed strands is *Ast* = 2223 mm2/m.

#### 3.2.5. Checking Calculation of Mortar Coating under Serviceability Limit States

The maximum bending moment of the pipe at the spring-line was *Ml pms* = –24.897 kN·m/m. The value of *M<sup>l</sup> pms*, is negative, indicating that the mortar coating was subjected to tension. The absolute value was taken when substituting the following equations. The axial tension of the pipe at the spring-line was *N<sup>l</sup> ps* = 769.388 kN/m The maximum tensile stress at the edge of the pipe at the spring-line was σ*<sup>l</sup> ss* <sup>=</sup> 9.44 N/mm2. The strain of mortar coating was <sup>ε</sup>*mt* <sup>=</sup> *fmt*,*<sup>k</sup> Em* ≥ 0.52√*fmc*,*<sup>k</sup> Em* = 0.0001444. The design parameter of strain for the mortar coating was α*<sup>m</sup>* = 5. Thus, α*m*ε*mtEm* = 17.44 N/mm<sup>2</sup> > σ*l ss* = 9.44 N/mm2 , indicating that the area of prestressed strands is able to meet the tensile requirement of the mortar coating under the serviceability limit states.

#### 3.2.6. Checking Calculation of Mortar Coating under Quasi-Permanent Limit States

ψ*qv* and ψ*qw* are the quasi-permanent coefficient of vertical pressure generated by the ground vehicle loads and the internal water pressure, respectively. Here, ψ*qv* = 0.5 and ψ*qw* = 0.72. The maximum bending moment of the pipe at the spring-line was *Ml pml* = <sup>−</sup>23.422 kN·m/m. The value of *Ml pml* is negative, indicating that the mortar coating was subjected to tension. The absolute value was taken when substituting the following equations. The axial tension of the pipe at the spring-line was *N<sup>l</sup> pl* = 602.911 kN/m. The maximum tensile stress at the edge of the pipe at the spring-line was σ*l ls* <sup>=</sup> 8.21 N/mm2. The design parameter of strain for the mortar coating was <sup>α</sup>- *<sup>m</sup>* = 4. Therefore, α- *<sup>m</sup>*ε*mtEm* = 13.95 N/mm<sup>2</sup> > (σ*<sup>l</sup> ss* = 8.21 N/mm2), indicating that the area of prestressed strands is able to meet the tensile requirement of the mortar coating under quasi-permanent limit states.

Above all, this is reasonable of the calculation result of the area of prestressed strands, which is *Ast* = 2223 mm2/m. The center spacing of steel strands was *lst* = *<sup>A</sup>* <sup>×</sup> <sup>1000</sup> *Ast* = 62.99 mm.

#### *3.3. A Prototype Test*

A prototype test of ECP reinforced by steel strands with the fixed spacing calculated in Section 3.2 was performed in an assembled apparatus (Figure 7). The apparatus was mainly constituted by two ECPs, whose internal diameters were 2000 mm [29]. The adopted pipes were exactly the same as those given in Section 3.1. The entire test process involved five load stages, namely, (1) increasing the internal water pressure to the working pressure (0–0.6 MPa), (2) cutting the prestressing wires manually until the cracks propagated in the concrete core (0.6 MPa), (3) decreasing the internal water pressure to the artesian pressure (0.6–0.2 MPa), (4) performing the tensioning operation after wrapping the strands externally around the pipe (0.2 MPa), and (5) increasing the internal water pressure to the original level (0.2–0.6 MPa). In most of the actual pipe failures modes, most pipes failed at 4 or 8 o- clock, not at the invert, crown, or spring-lines [29]. The position of 8 o- clock was chosen in this test for convenience (Figure 8).

Post-tensioning was designed with the theory conducted in Section 3.2, indicating that the target tensile strength was equal to 1171.8 MPa and the center spacing of steel strands was taken as 62 mm. To prevent a prestress loss due to the retraction of clips and the stress relaxation of strands, excessive stretching is essential here. The tensioning process is divided into six stages, which were 20%, 25%, 50%, 75%, 100%, and 115%. Tensioning was performed simultaneously from both sides and in a symmetrical manner along the pipeline axis.

The statuses of each component of the pipe and the steel strands were measured by resistance strain gauges along the axial direction at inverted (360◦), crown (180◦), and spring-line (90◦, 270◦) orientations (Figure 8). Figure 9 exhibits the hoop strains in the concrete core before and after the reinforcement under the working pressure (0.6 MPa). The strains in the concrete core all showed a drastic drop after the process of tensioning. Moreover, the maximum width of the cracks in the outer concrete core at spring-line reduced from 2.2 mm to 0.1 mm after strengthening, as observed through field observation.

**Figure 7.** The spot photo of the test apparatus.

**Figure 8.** Layout of measuring points of (**a**) the pipe and (**b**) the steel strands.

**Figure 9.** Comparison of strains in core concrete before and after the reinforcement: (**a**) 2.5 m inner concrete core at 0◦; (**b**) 2.5 m inner concrete core at 90◦; (**c**) 2.5 m inner concrete core at 180◦; (**d**)3m inner concrete core at 0◦; (**e**) 3 m inner concrete core at 90◦; (**f**) 3 m inner concrete core at 180◦; (**g**)3m inner concrete core at 270◦; (**h**) 2.5 m outer concrete core at 0◦; (**i**) 2.5 m outer concrete core at 90◦; (**j**) 3 m outer concrete core at 90◦; (**k**) 3 m outer concrete core at 180◦; (**l**) 4 m outer concrete core at 90◦; (**m**) 4 m outer concrete core at 180◦.

The strengthened pipe was capable of sustaining the working pressure and the water tightness property was in a good state. The strains of the steel strands were all below the tensile strain level. The reinforcement of the PCCP with external prestressed steel strands was able to meet the strengthen requirement of the test. The rationality of the derivations in this paper were verified by the effective reinforcement effect with external prestressed steel strands.
