Investigation of the Load-Sharing Theory of the RC Pipes Rehabilitated with Slip Liners

Slip-lining is a preferred rehabilitation approach in the departments of transportation in China. Although the method is the most common rehabilitation technique, few research studies have been conducted on the mechanical behavior of a rehabilitated reinforced concrete pipe (RCP). A series of experiments were conducted on RCPs rehabilitated with a corrugated steel pipe (CSP), a steel pipe, a high-density polyethylene (HDPE) pipe, and a shape steel bracket. -e RCP rehabilitated with the CSP showed an increase in both the load-carrying capacity (3.46 times greater than the RCP) and the stiffness (5.35 times greater than the RCP). -e RCP rehabilitated with the steel pipe, HDPE pipe, and steel bracket exhibited an increase in the load-carrying capacity (1.23, 1.50, and 1.31 times greater than the RCP, respectively), and the stiffness of these three pipes was not markedly changed. -e sliplined pipe acts as a “pipe within a pipe” system. A “load-sharing” theory was proposed in this study and provides estimates of the load-carrying capacity of the slip-lined pipes.


Introduction
Slip-lining, which involves inserting a liner into an existing culvert and grouting the space between them [1], is a preferred technique by the engineers [2], not only does it reduce reconstruction costs but also will not cause traffic disruption. A variety of pipe materials can be used as the liners such as high-density polyethylene (HDPE) pipes and corrugated steel pipes (CSPs) [3,4].
is rehabilitation technique is commonly used in China, but few researches have been performed when an reinforced concrete (RC) culvert is used. Zhao and Daigle [5] used a cast-iron pipe rehabilitated with a liner to conduct a two-point loading experiment and found that the existing pipe, grout, and liner acted independently; an approach was proposed to estimate the service life of a sliplined pipe. McAlpine [6] used a rehabilitated concrete sewer to investigate a slip-lined pipe and found that a composite model could be used to estimate the effect of the enhancement. Other researches are devoted to explore the mechanical properties of the rehabilitated pipes. SnapTite [7] considered that the compressive strength of the grout was not important, but Smith et al. [8] considered that a higher compressive strength of the grout resulted in a higher load-carrying capacity of the rehabilitated pipe. Garcia and Moore [9] found that the steel culverts rehabilitated with spray-on liners responded like semirigid structures after rehabilitation. Moore and García [10] found that full interaction and partial interaction both occurred between the corrugated steel pipelines and spray on cementitious. Simpson et al. [11] found that the stiffness of the corrugated steel culverts rehabilitated with grouted HDPE pipe was larger than the pre-rehabilitated pipe and the negative arching of the soil had increased. Simpson et al. [12] found that the existing pipe carried most of the load, the grout and RC pipe (RCP) were bonded, and the ultimate load-carrying capacity of the pipes depended on the bearing capacity of the unpaved ground surface. Tetreault et al. [13] concluded that the level of corrosion had no impact on the structural behavior and that paving the invert improved the structural performance. However, there are no guidelines for designing a slip-lined RC culvert and the load-sharing theory of these culverts, so there is an urgent need to investigate the performance and load-sharing theory of the rehabilitated RC culvert.
In the current research, an RCP was used as a culvert and four liners were used to investigate the performance of the culvert before and after rehabilitation. A series of experiments was conducted on RCPs rehabilitated with a grouted steel pipe, CSP, HDPE, and shape steel bracket liners. e objectives of this research were to determine (1) the loadcarrying capacity of the five specimens (including an RCP, an RCP rehabilitated with grouted steel pipe, an RCP rehabilitated with grouted CSP, an RCP rehabilitated with grouted HDPE pipe, and an RCP rehabilitated with shape steel bracket, respectively), (2) the load-deformation curves of five specimens, (3) failure characters and the cracks distribution of the specimens, and (4) the load-sharing theory of the slip-lined pipes.

Laboratory Tests
Five specimens were used in this experiment to investigate the load-sharing mechanism of the slip-lined RCPs, including (1) an RCP, (2) an RCP rehabilitated with a CSP (RGC hereafter), (3) an RCP rehabilitated with a steel pipe (RGS hereafter), (4) an RCP rehabilitated with a HDPE pipe (RGH hereafter), and (5) an RCP rehabilitated with a shape steel bracket (RGB hereafter). e RCPs had an internal diameter of 1200 mm and a wall thickness of 120 mm. Double-layer cold-stretched steel bars (φ 6 , HRB400) were arranged in the RCPs at a spacing of 50 mm. e strength grade of the RCPs is C60 with a compressive strength of 59.73 ± 2.77 MPa and an elastic modulus of 36 GPa, the strength grade of the steel bars is HRB400 Cold-stretched steel bar with a tensile strength of 575 ± 9 MPa and an elastic modulus of 210 GPa. e pitch of the CSP was 200 mm with a depth of 50 mm with an intact wall thickness of 3 mm. e steel pipe had an internal diameter of 1100 mm and a wall thickness of 10 mm. e specification of the shape steel bracket is ∟70 × 5 with a leg length of 70 mm and thickness of 5 mm. e designations of the steel pipe, CSP, and shape steel bracket are Q235, which has a minimum yield strength of 235 MPa and a minimum tensile strength of 370 MPa and an elastic modulus of 210 GPa. e HDPE pipe (DN/ID 1000, SN 8) had an internal diameter of 1000 mm, a ring stiffness of 8.2 kN/m 2 , a tensile strength of 16 MPa, and a modulus of 800 MPa. All pipes were 1000 mm long. e spacing of the shape steel bracket is 200 mm, and short shape steel was used to connect the shape steel bracket longitudinally.
Two types of grout were used, including (1) C40 concrete for RGC and RGS and (2) high-performance grouting material for the RGH. e C40 concrete has a compressive strength of 49.93 ± 4 MPa and an elastic modulus of 32.5 GPa, and the high-performance grouting material has a compressive strength of 87.7 ± 3 MPa and an elastic modulus of 38 GPa. No grout was used for the RGB, and the shape steel bracket is directly contacted with the RCP. e grout rings of RGC has a minimum thickness of 50 mm (from the crest of the CSP to the inner of the RCP), that of RGS had a thickness of 50 mm, and that of the RGH had a thickness of 20 mm. e RGB has no grout.
A two-point loading experiment was used in this study, and the load was applied to the specimens using a 2500 kN hydraulic actuator, which was attached to a reaction frame over the pipes. A distributing girder and two base plates were used to ensure that the concentrated load could not cause a deterioration of the specimens or a stress concentration. e specimens were loaded to the ultimate state with a loading rate of 15 kN/min, and the loading was paused at various stages to observe the experimental phenomena. e schematic of loading frame is shown in Figure 1, and the specimens are shown in Figure 2.
Four string potentiometers with an accuracy of 0.1 mm were used to measure the vertical and horizontal diameter changes. Two string potentiometers were installed inside the rehabilitated pipe at the crown and invert, and the other two were installed outside of the rehabilitated pipe at the springlines. Because the RCP, grout, and liners are in close contact with each other at the crown and invert, the diameter changes of the three pipe materials should be equal; in addition, the base plates were installed outside the pipe, making it impossible to install the string potentiometers outside of the pipe. erefore, the string potentiometers were installed inside the rehabilitated pipe. However, if the RCP, grout, and liners were to separate from each other at the springlines, the diameter change of the RCP would be larger than that of the other components; considering the most unfavorable situation; the string potentiometers were, therefore, installed outside of the rehabilitated pipe to monitor the diameter change of the RCPs. Figure 3 shows the results of the applied loads versus the diameter changes for the unrehabilitated pipe (RCP) and the rehabilitated pipe (RGC, RGS, RGH, and RGB). e vertical and horizontal diameter changes for each pipe are of similar magnitude but have opposite directions. It can be seen from Figure     . e sti ness was also di erent for these specimens as shown in Figure 3.

Loads versus Diameter Changes.
e RCP had an initial sti ness of 109.3 kN/mm and a secant sti ness of 7.9 kN/mm at the ultimate state; the RGC had a much higher initial sti ness than the RCP (217.6 kN/mm versus 109.3 kN/mm) and a much higher secant sti ness (42.3 kN/mm versus7.9 kN/ mm); e RGS had an initial sti ness of 167 kN/mm and a secant sti ness of 9.73 kN/mm; the RGH had an initial sti ness of 129 kN/mm and a secant sti ness of 8.16 kN/mm; e RGB had an initial sti ness of 48 kN/mm and a secant sti ness of 8.12 kN/mm. From the load-carrying capacity, diameter change, and sti ness enhancement in the twopoint loading tests, it can be inferred that the CSP improved the load-carrying capacity of the RCP because of a signi cant increase in its sti ness and capacity but reduced the ductility of the rehabilitated pipe (reduction in the ultimate diameter change); the steel pipe, HDPE pipe, and shape steel bracket improved the load-carrying capacity of the RCP slightly, but the sti ness of the RCP has not been changed much by these liners.

Cracks Distribution.
e crack distribution at ultimate state is shown in Figure 4. e crack distribution of the RCP re ects the ductility of the specimens, i.e., if the cracks are distributed over a wide range with equal spacing and the maximum width of the crack is small, it indicates that the specimens have good ductility. e cracks of RCP distributed over a wide range and the maximum width is 5 mm; the cracks of RGC distributed over a smaller range than that of RCP and the maximum width is 18 mm, and the steel bar is broken; the cracks of RGS distributed a small range and the maximum width is 17 mm, the steel bar is broken; the cracks of RGH distributed over a wide range and the maximum width is 8 mm; the cracks of RGB distributed over a smaller range than that of RCP and the maximum width is 3 mm. e distribution of the cracks illustrated that the RCP, RGH, and RGB have better ductility than that of RGC and RGS. e cracks at the ends of the specimens can be used to illustrate the combination of RCP, grout, and liners (Figure 5). Circumferential cracks appeared on the interface between grout and RCP, it showed that there is a circumferential slip between them, and they were not completely bonded together (Figures 5(b)-5(d)). e RGB, on the other hand, only contains RCP and liner, no bonding existed between them at all, slip occurred on the contact surface, and the shape steel bracket was buckling (Figures 5(e) and 5(f )). is phenomenon is particularly clear in liners and grouts, liners, and grouts separated with each other especially at the springlines. Figure 5 shows that RCP, grout, and liners are likely to act independently, rather than a composite system.

Load-Sharing Mechanism
If RCP, grout, and liner act as a pipe within a pipe system, the vertical de ections of RCP, grout, and liner should be equal. De ection of the pipe subjected to the concentrated load F is shown in Figure 6. e subscripts 1, 2, and 3 represent the liner, grout, and RCP, respectively, and the vertical de ections of each pipe under two-point loading are expressed as [14] Δ i F i r 3 i ((π/4) −(2/π)) where Δ i is the vertical decrease in the diameter of each pipe, F i is the concentrated load shared by each pipe, r i is the mean radius of each pipe, E i is the modulus of elasticity, and I i is the moment of inertia of the wall's cross section per unit length of the pipe. erefore, the vertical de ections are equal (Δ 1 Δ 2 Δ 3 Δ) and the following equation is obtained: (2) e concentrated load shared by each pipe F i should satisfy where F is the concentrated load carried by the slip-lined pipe, F 1 is the concentrated load carried by the liner, F 2 is the concentrated load carried by the grout, and F 3 is the concentrated load carried by the RCP.
If a sti ness factor φ, as EI/r 3 , is de ned, equation (2) can be changed into the following equation:

Advances in Civil Engineering
where φ 1 is the stiffness factor of the liner, φ 2 is the stiffness factor of the grout, and φ 3 is the stiffness factor of the RCP. Equation (4) indicates that the concentrated load F is shared among the pipe materials and that the distribution of the load depends on the stiffness factor φ. It should be noted that concrete has different short-term and long-term stiffness and the long-term stiffness decreases over time [5], and the term E 3 I 3 is not applicable. In assessing the load sharing and stresses, the value of the long-term stiffness taken at the expected design life should be used for the expected loads and the value of the short-term stiffness should be used for loading conditions of a transient nature. e stiffness of a concrete component can be calculated using the Code for Design of Concrete Structures [15]. For this experiment, a short-term stiffness was used to calculate the load-carrying capacity and the expression is as follows: where B s is the short-term stiffness of the RCP, E s is the elastic modulus of the steel bar, A s is the area of the tensile steel bar, h 0 is the effective thickness of the RC pipe, ψ is the nonuniform coefficient of the strains, α E is the ratio of the elastic modulus of the steel bar to the elastic modulus of the

Advances in Civil Engineering
concrete, ρ is the reinforcement percentage of the tensile steel bar, c f ′ is the ratio of ange area to web area, should be ZERO for a rectangular section, f tk is the characteristic value of the concrete tensile strength, ρ te is the e ective reinforcement percentage, and σ sq is the tensile stress of the steel bar at the crack section. At this point, φ 3 is changed into the following expression: If equations (7) and (4) were substituted into (3), the load-carrying capacity of the slip-lined pipe can be calculated by the following formula: where F 3 is the load carried by the RCP, E 1 I 1 , and E 2 I 2 are the pipe sti ness of the liner and grout, respectively, and r 1 , r 2 , and r 3 are the mean radii of the liner, grout, and RCP, respectively. When the load-carrying capacity of the RCP is known, equation (8) can be used to calculate the rupture load of the rehabilitated pipe. e external load (concentrated load F in this experiment) was borne by the RCP directly; once the RCP was damaged after rehabilitation, the load shared by the grout and liner suddenly increased, resulting in damage. As a result, the load-carrying capacity of the slip-lined pipe indirectly depended on the RCP. erefore, the load-carrying capacity of the sliplined pipe should be calculated based on the RCP (equation (8)). e load-carrying capacity of the RCP (280 kN) was already obtained by this two-point loading experiment, taken as the load-carrying capacity of the pre-rehabilitated pipe (F 3 ); B s could be calculated by equation (5), where E s should be 2.1 × 10 5 MPa, A s should be 650.9 mm 2 , h 0 should be 110 mm, ψ should be 0.707, α E should be 5.83, ρ should be 0.0054, f tk should be 2.85 MPa, ρ te should be 0.011, and σ sq should be 575 MPa; the mean radii of the RCP (r 3 ) should be 660 mm, B s should be 1376053836.208 MPa mm 3 , and φ 3 should be 4.786 MPa. e moment of inertia of the grout could be calculated with the help of drawing software, such as AUTOCAD, since their cross sections are irregular. e sti ness factors φ 1 of the liners can usually be querying in the parameters supplied by the manufacturer. Sti ness factors are summarized in Table 1. e calculation results are shown in Table 2. e percent error for RGC and RGS (8% and 13.4%) is considerably higher than that for RGH and RGB (0.6% and 0.5%), and this may due to the thickness of the slip-lined pipe wall and the buckling of the steel pipe. e RGC has a thickest wall than the other pipes, this will cause the load to di use in the pipe wall tremendously, so that the range of the loads act on the liners will be larger than that act on the RCP, and this indirectly improves the load-carrying capacity of the liners. Since the theoretical analysis did not consider the di usion of the loads and assumed the ranges of the loads act on the RCP, grout, and liner are same, this will make the calculated value is lower than the tested value. e RGS, on the other hand, takes the steel pipe as the liner; the steel pipe is easy to buckle under pressure, and this will make the actual loadcarrying capacity lower than that when it yields, while the theoretical analysis assumed the steel pipe could yield which make the calculated value higher than the tested value. Overall, it can be seen from Table 2, the maximum di erence between the theoretical and experimental results is less than 13.4%, the minimum di erence between them is 0.5%; the calculated results are so close to the experimental results.
is proves that the above theory is very reasonable.
In the theoretical derivation, the short-term sti ness is used to consider the nonlinear e ects of the RCP, and elastic sti ness is used for the grout and liners. e liners, on the other hand, have formed plastic hinge at the crown, invert, and springlines, when the slip-lined pipes reached the ultimate state. Once the plastic hinge formed, the liners should be regarded as destructive [16]; the maximum sti ness that the liners can provide is the elastic sti ness, which should be used in load-sharing calculation. Foamed cement banking, cement mortar, and ne aggregate concrete are always used as grout; once these materials cracked, they cannot carry any loads, but only when they work alone. In a slip-lined system, the radial pressures applied by the RCP and liners will constrain the radial deformation of the grout. Moreover, friction exists on the contact surface of RCP, grout, and liners, though it will not prevent slipping, it will reduce the tensile stress of grout. e friction is always opposite to the tensile stresses caused by the bending moments, and this is similar to an imaginary force that resists the tensile stress of grout at the crack, which allows the grout to act like an elastic body and to exert an imaginary pull. If the elasticity hypothesis is wrong, the calculated results in Table 2 cannot be as good as the experimental results, which indirectly shows that the elasticity hypothesis is correct.
e results also indicate that the plasticity approach and composite behavior method used by Smith et al. [8] are likely not appropriate for rehabilitating RCPs given the behavior demonstrated by these specimens.

Conclusions
is paper presented a load-sharing theory to estimate the load-carrying capacity of the slip-lined pipes. e CSP, steel pipe, HDPE pipe, and shape steel bracket were used as liners to rehabilitated RCPs. Including an RCP, there are 5 pipes were tested in two-point loading experiments. e current  6 Advances in Civil Engineering investigation was undertaken to examine the effect of different liners on the performance and load-carrying capacity of rehabilitated RCPs. e following key conclusions can be drawn from this work: (1) e liners increased both the load-carrying capacity and the stiffness of the RCP. Among them, the CSP increased the load-carrying capacity of the RCP most (3.46 times greater than RCP), followed by HDPE pipe, shape steel bracket, and steel pipe (1.51, 1.31, and 1.23 times greater than RCP, respectively). Although the CSP increased the load-carrying capacity of the RCP, it reduced the ductility of the RCP, whereas HDPE pipe, shape steel bracket, and steel pipe did not. (2) Cracks distributions of the specimens imply that the RCP, grout, and liners act independently, and a loadsharing theory was proposed based on compatibility of vertical deformation. e load-sharing capacity of the RCP, grout, and liners depends on the stiffness factors (φ) of the pipes. When the liners were used to rehabilitate the RCPs, the liners should not be treated as new pipes for the structural analysis, and the loadcarrying capacity of the RCP and grout should also be considered. Once the RCP is damaged after rehabilitation, the loads distributed by the grout and liner will suddenly increase so that they cannot withstand a large load and damage will occur. (3) A "load-sharing" theory was proposed in this study, and equation (8) provides estimates of the load-carrying capacity of the slip-lined pipes (the maximum difference between the theoretical and experimental results is less than 13.4%, and the minimum difference between them is 0.5%). Since different liners were used to compare the calculated results and experimental results, this theory is highly reliable.
While the experiments were undertaken under twopoint loading, the RCPs were buried in the soil in practical engineering. e surrounding soil would influence the vehicle loads distribution and make an interaction between the soil and RCP, and the stress state of the RCP will be very different from these experiments. erefore, the surrounding soil should be considered in the future research.

Abbreviations
RCP: Reinforced concrete pipe RGC: Reinforced concrete pipe rehabilitated with a corrugated steel pipe RGS: Reinforced concrete pipe rehabilitated with a steel pipe RGH: Reinforced concrete pipe rehabilitated with a highdensity polyethylene pipe RGB: Reinforced concrete pipe rehabilitated with a shape steel bracket Δ i : Vertical decrease in the diameter of each pipe Δ 1 : Vertical decrease in the diameter of the liner Δ 2 : Vertical decrease in the diameter of the grout Δ 3 : Vertical decrease in the diameter of the RCP F i : Concentrated load shared by each pipe F 1 : Concentrated load shared by the liner F 2 : Concentrated load shared by the grout F 3 : Concentrated load shared by the RCP r i : Mean radius of each pipe r 1 : Mean radii of the liner r 2 : Mean radii of the grout r 3 : Mean radii of the RCP E i : Modulus of elasticity of each pipe E 1 : Modulus of elasticity of the liner E 2 : Modulus of elasticity of the grout E 3 : Modulus of elasticity of the RCP I i : Moment of inertia of the wall's cross section per unit length of the pipe I 1 : Moment of inertia of the liner I 2 : Moment of inertia of the grout I 3 : Moment of inertia of the RCP F: Concentrated load carried by the slip-lined pipe φ: Stiffness factor, as EI/r 3 φ 1 : Stiffness factor of the liner, as E 1 I 1 /r 3 1 φ 2 : Stiffness factor of the grout, as E 2 I 2 /r 3 2 φ 3 : Stiffness factor of the RCP, as B s /r 3 3 B s : Short-term stiffness of the reinforced concrete, N·mm 2 E s : Elastic modulus of the steel bar, MPa A s : Area of the tensile steel bar, mm 2 h 0 : Effective thickness of the RCP, mm ψ: Nonuniform coefficient of the strains  Advances in Civil Engineering 7 α E : Ratio of the elastic modulus of the steel bar to the elastic modulus of the concrete ρ: Reinforcement percentage of the tensile steel bar f tk : Characteristic value of the concrete tensile strength, MPa ρ te : Effective reinforcement percentage σ sq : Tensile stress of the steel bar at the crack section, Mpa ρ 2 : Density of the grout, kg/m 3 f 2 : Compressive strength of the grout, MPa F c : Calculated load-carrying capacity of the slip-lined pipe F t : Tested load-carrying capacity of the pipe 1: Liner 2: Grout 3: RCP.
Data Availability e (experimental results) data used to support the findings of this study are included in the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.