**Theoretical Study and Application of the Reinforcement of Prestressed Concrete Cylinder Pipes with External Prestressed Steel Strands**

**Lijun Zhao 1,2, Tiesheng Dou 1,2,\*, Bingqing Cheng 1,2, Shifa Xia 1,2, Jinxin Yang 3, Qi Zhang 3, Meng Li 1,2 and Xiulin Li <sup>1</sup>**


Received: 6 November 2019; Accepted: 13 December 2019; Published: 16 December 2019

**Abstract:** Prestressed concrete cylinder pipes (PCCPs) can suffer from prestress loss caused by wire-breakage, leading to a reduction in load-carrying capacity or a rupture accident. Reinforcement of PCCPs with external prestressed steel strands is an effective way to enhance a deteriorating pipe's ability to withstand the design load. One of the principal advantages of this reinforcement is that there is no need to drain the pipeline. A theoretical derivation is performed, and this tentative design method could be used to determine the area of prestressed steel strands and the corresponding center spacing in terms of prestress loss. The prestress losses of strands are refined and the normal stress between the strands and the pipe wall are assumed to be distributed as a trigonometric function instead of uniformly. This derivation configures the prestress of steel strands to meet the requirements of ultimate limit states, serviceability limit states, and quasi-permanent limit states, considering the tensile strength of the concrete core and the mortar coating, respectively. This theory was applied to the reinforcement design of a PCCP with broken wires (with a diameter of 2000 mm), and a prototype test is carried out to verify the effect of the reinforcement. The load-carrying capacity of the deteriorating PCCPs after reinforcement reached that of the original design level. The research presented in this paper could provide technical recommendations for the application of the reinforcement of PCCPs with external prestressed steel strands.

**Keywords:** prestressed concrete cylinder pipe; external prestressed steel strands; theoretical study; wire-breakage

### **1. Introduction**

A prestressed concrete cylinder pipe (PCCP) contains four components, namely, (1) a concrete core, (2) a steel cylinder lined with concrete (LCP) or encased in concrete (ECP), (3) high strength prestressing wires to withstand the internal high water pressure and external load, and (4) a mortar coating to protect the wires and cylinder against corrosion. The promise of the PCCP lies in its high bearing capacity, strong permeability resistance, and cost-effectiveness. Efficiencies in construction and reductions in fabrication costs have led to the extensive use of PCCPs in the USA, Canada, and China, and have also led to the pivotal development of this pipe. However, these pipes may suffer from prestress loss caused by wire-breakage. Wire-breakage or rupturing can result in significant losses to society, making the reinforcement of deteriorating pipes essential.

Reinforcement with external prestressed steel strands is regarded as an efficient way of strengthening bridges and beams that are deteriorating due to increased overloading and progressive structural aging [1–5]. Miyamoto A. [6,7] demonstrated the feasibility of applying this prestressing technique to the strengthening of existing steel bridges. Chen S. [8] proposed a finite element model to investigate the inelastic buckling of continuous composite beams that were prestressed with external tendons. Lou T. [9] also concluded that external prestressing significantly improved the short-term behavior of a composite beam. Tan K. H. [2], Aparicio A. C. [10], Park S. [11], and others have presented a series of prototype tests regarding externally prestressed concrete beams and have verified that external tendons can be used to effectively influence beam behavior.

The reinforcement of a PCCP with external prestressed steel strands involves repairing critical pipes with additional external post-tensioning to increase the longevity of problematic PCCP pipelines. The strands are wrapped outside the pipe with a fixed spacing between each strand, according to the service water pressure [12] (Figure 1). A well-known large-scale application of external prestressed strands is in the Great Man-Made River pipelines in Libya [12]. Most of the pipes in this project have an internal diameter of 4.20 m. Authorities have determined that repair of the critical pipes should proceed, with additional external post-tensioning in areas where pipes had burst. The reinforcement of the external prestressed strands on PCCPs has proven to be effective here. This approach is advantageous due to its ability for construction to proceed with no need to drain the pipeline. However, few theoretical studies have been carried out regarding the prestress losses and the mechanism applying external prestressed strands to strengthen PCCPs.

**Figure 1.** Structural drawing of a prestressed concrete cylinder pipe (PCCP) strengthened with prestressed steel strands.

This study introduces a theoretical derivation and investigates the prestress loss of steel strands applied to PCCPs. The normal stress between the strands and the pipe wall is assumed to be distributed as a trigonometric function, instead of uniformly, to estimate prestress losses. The area of the steel strands is determined to meet the requirements of ultimate limit states, serviceability limit states, and quasi-permanent limit states, considering the tensile strength of the concrete core and the mortar coating, respectively. An example calculation of this theory and a prototype test is calculated on the same PCCP to verify the feasibility of this theory. The load response of the pipe before and after the reinforcement process is analyzed.

#### **2. Theoretical Derivations**

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

The prestress loss persisted during and after the tensioning operation, and can be divided into two categories, namely, instantaneous loss and long-term loss [13–15]. Instantaneous loss, i.e., short-term loss during the tensioning operation, described the prestress losses caused by friction resistance between the surface of the pipe wall and the steel strands, the anchor deformation, the concrete elastic compression induced by stepwise tensioning operation, and cracks closures. Long-term prestress losses included prestress losses [13,16], while taking into account the materials aging, including the effects of shrinkage and the creep losses of concrete, and the long-term relaxation losses of prestressed steel strands. Types of prestress loss of steel strands applied to PCCPs are illustrated in Figure 2. Since the reinforcement of PCCPs with external prestressed steel strands is a post-tensioning method, the impact of temperature can be removed from consideration when considering the reinforcement of PCCPs with external prestressed steel strands.

**Figure 2.** Types of prestress loss of steel strands applied to PCCPs.

2.1.1. Calculation of Retraction Length, *lre*, and Its Corresponding Retraction Angle, θ*re*

As far as we know, the stress distribution along the strand is nonlinear. The anchor influenced the prestressed steel strand within a certain length range due to the static friction caused by the retraction of the strand. This length is called the retraction length, *lre*. Strands within the retraction length showed a displacement opposite to the tension direction, which decreases the prestress. The movement trend is demarcated at point *C*, and the stress is redistributed from *ACB* to *A*-*CB* (Figure 3).

**Figure 3.** Distribution of strand stress caused by retraction.

The circumferential micro-segment of the prestressed steel strand is regarded as the research object, where the corresponding angle is *d*θ (Figure 4). Assuming that the normal stress of the steel strand in the micro-segment is evenly distributed, a differential equation can be established according to the static equilibrium conditions:

$$T \cdot \sin\left(\frac{d\theta}{2}\right) + (T + dT) \cdot \sin\left(\frac{d\theta}{2}\right) - dP = 0\tag{1}$$

where *T* and *P* stand for the tension force and the normal pressure of the strand, respectively.

**Figure 4.** Stress of a micro-segment of a strand.

Higher variables were omitted, taking *<sup>d</sup>*<sup>θ</sup> <sup>2</sup> <sup>=</sup> *sin d*θ 2 . Equation (1) can be simplified to *Td*θ = *dP*. The equation describing the momentary balance for rotation around the center of curvature, *O*, can be written as follows:

$$r\_{st} \cdot \mu dP + r\_{st} dT = 0 \tag{2}$$

where *T* = *T*<sup>0</sup> when θ = 0, thus, *T* = *T*0*e*−μθ. *rst*, where *rst* is the calculated radius of the strand wrapped outside the pipe (m) and μ is the friction coefficient between the prestressed strands and the outer surface of the deteriorating pipe. The influencing factors of μ mainly include the type of steel, the type of lubricating grease, the materials wrapped outside, and the quality control of the construction. Here, μ ranges from 0.08 to 0.12, with a mean value of 0.1.

The stress for an arbitrary cross section is calculated as per Equation (3):

$$
\sigma = \sigma\_{st} \mathfrak{e}^{-\mu 0} \tag{3}
$$

where <sup>σ</sup>*st* is the tension stress of prestressed steel strands (N/mm2). <sup>σ</sup>*st* = *fst*,*t*·α, in which <sup>α</sup> is the control coefficient for the tension of the steel strands (N/mm2). Normally, this value ranges between 0 and 0.75 [17–19]. *fst*,*<sup>t</sup>* is the nominal tensile strength of the prestressed strand (N/mm2).

The stress at the end section of the retraction length can be written as follows:

$$
\sigma\_{n\ell} = \sigma\_{st} e^{-\mu \partial\_{n\ell}}.\tag{4}
$$

The length reduction of the strand caused by the anchor deformation and the clip retraction, Δ*lre*, can be expressed by Equation (5):

$$
\Delta l\_{rc} = \int\_0^{0\_{rt}} \frac{\sigma\_{l2} r\_{st}}{E\_{st}} d\theta = \frac{2r\_{st}\sigma\_{sl} \left| 1 - e^{-\mu 0\_{rt}} (1 + \mu \theta\_{rc}) \right|}{\mu E\_{st}} \tag{5}
$$

where *Est* is the elastic modulus of the adopted steel strand (N/mm2) and *e*−μθ*re* is expanded into a power series according to the Taylor formula. Only the first three terms of the formula have sufficient precision, since μθ*re* is adequately small, which is given by Equation (6):

$$
\epsilon^{-\mu\theta\_{\rm re}} = 1 - \mu\theta\_{\rm re} + \frac{\left(\mu\theta\_{\rm re}\right)^2}{2}.\tag{6}
$$

Equation (7) can be derived by incorporating Equation (6) into Equation (5) and omitting the high micro (μθ*re*) 3 :

$$
\Delta l\_{\rm tr} = \frac{\mu r\_{\rm st} \sigma\_{\rm st} \theta\_{\rm tr}^{-2}}{E\_{\rm st}} \tag{7}
$$

The correspondence between the retraction length, *lre*, and the retraction angle, θ*re*, is represented as follows:

$$l\_{rc} = r\_{st} \theta\_{rc}.\tag{8}$$

Therefore, the retraction length, *lre*, and its corresponding angle, θ*re*, can be given by Equations (8) and (9).

$$l\_n = \sqrt{\frac{\Delta l\_{rt} E\_{st} r\_{st}}{\mu \sigma\_{st}}} \tag{9}$$

The various types of anchorage used with steel strands were classified as plug and cone, straight sleeve, contoured sleeve, metal overlay, and split wedge anchorages. The value of Δ*lre* varies with the type of anchor.

### 2.1.2. Prestress Loss Caused by Friction Resistance, σ*l*<sup>1</sup>

The prestress loss caused by the friction resistance, σ*l*1, can be calculated based on the consideration of two parts, namely, the bending loss and the deviation loss. The radial pressing force, σ*r*, is produced between the strand and the pipe wall by prestressed strands, thereby resulting in extrusion friction. The bending loss accounted for a large proportion of the total friction loss.

Based on the assumption of a rigid body, we hypothesized that the pressure between the strand and the pipe wall would be uniformly distributed [20], and that elastic deformation would occur when the two elastic bodies were pressed into contact with each other. The stress between the contact surfaces is ellipsoidal, and its value can be related to the radius of curvature and the elastic modulus of the contact object. It is not accurate enough to consider the contact stress as uniformly distributed under normal contact pressure due to the large tensile force of the prestressed steel strands.

The scope of the bending loss can be related to the retraction length, *lre*. We can assume that the normal stress between the strands and the pipe wall would be distributed as a trigonometric function [21], as illustrated in Figure 5 and Equation (10).

$$p\_{(a)} = p\_0 \cos^2 \left(\frac{\pi}{\theta} a\right) \tag{10}$$

$$\text{where } \cos^2\left(\frac{\pi}{\theta}a\right) = \begin{cases} & 0 \ a = \frac{\theta}{2} \\ & 1 \ a = 0 \\ & 0 \ a = -\frac{\theta}{2} \end{cases}$$

**Figure 5.** Distribution of the normal stress of the pipe wall excluding the friction.

A balance of forces in the z-direction can be established by Equation (11). From Equation (11), we derived Equation (12). Therefore, the normal stress can be calculated using Equation (13):

$$T \cdot \sin\left(\frac{\theta}{2}\right) + (T + dT) \cdot \sin\left(\frac{\theta}{2}\right) + 2 \int\_0^{\frac{\theta}{2}} p\_{(a)} \cdot \cos \alpha dl = 0\tag{11}$$

$$p\_0 = \frac{\sin\frac{\phi}{2}}{\int\_0^{\frac{\phi}{2}} \cos^2\left(\frac{\pi}{\theta}\alpha\right) \cos\alpha d\alpha} \times \frac{T}{R} \tag{12}$$

where <sup>θ</sup> 2 <sup>0</sup> cos<sup>2</sup> π θ α cos α*d*α = cos<sup>2</sup> π θ α sin α θ 2 <sup>0</sup> <sup>+</sup> <sup>π</sup> θ θ 2 <sup>0</sup> sin <sup>α</sup> sin 2π <sup>θ</sup> α *<sup>d</sup>*<sup>α</sup> <sup>=</sup> <sup>2</sup>( <sup>π</sup> θ ) 2 sin <sup>θ</sup> 2 ( <sup>2</sup><sup>π</sup> θ ) 2 −1 .

$$p\_{(a)} = \left(2 - \frac{\Theta^2}{2\pi^2}\right) \cdot \cos^2\left(\frac{\pi}{\Theta}a\right) \cdot \frac{\sigma\_0}{R} \tag{13}$$

The prestress loss related to the bending loss, *F*, during the tensioning operation is depicted in Equation (14).

$$F = 2\int\_0^{\frac{\theta}{2}} \mu p\_{(\theta)} \cdot dl = \mu \theta \sigma\_0 \left(1 - \frac{\theta^2}{4\pi^2}\right) \tag{14}$$

The deviation loss stems from errors in pipe positioning and installation, which causes friction between the force rib and the pipe material, thereby forming contact friction. The deviation loss occupies a small proportion of the total friction loss. The correction coefficient, *c*1, is involved here, and the deviation loss is not separately calculated in this paper. As a result, the total prestress loss caused by the friction resistance can be calculated, as displayed in Equation (15).

$$
\sigma\_{l1} = \mathfrak{c}\_1 \mathbf{F} \tag{15}
$$

where *c*<sup>1</sup> is the correction coefficient, accounting for the bending loss and the deviation loss, and is usually in the range of 1 to 1.3 [19].
