Research on mechanical performance of longitudinal joints in segmental tunnel linings strengthened by fiber-reinforced plastic grid with polymer−cement−mortar method

Abstract

In this study, we propose the use of a fiber-reinforced plastic grid with polymer−cement−mortar (FRP Grid-PCM) to reinforce segment joints in tunnel shield linings. These joints play a crucial role in determining bearing capacity but are vulnerable to deterioration during operation. To investigate how to enhance the flexural performance of longitudinal shield lining joints, we built eccentric short column specimens by bolting two half-corbel columns together and tested them in the laboratory. The test program comprised two control specimens and three strengthened specimens with FRP grid applied on one side, away from the axial load. The tests varied two main parameters: loading eccentricity and the number of FRP grid layers. We conducted a detailed analysis of the failure process, bearing capacity, and bending stiffness of longitudinal joints under different conditions. Furthermore, we developed an analytical model to predict the flexural bearing capacity of longitudinal joints upgraded with the FRP Grid-PCM method and validated it through experimental results. The research demonstrates that the FRP grid effectively reduces joint opening and rotation angles while enhancing the bearing capacity of the short column, particularly with concurrent increases in loading eccentricity and the number of FRP grid layers. Overall, our findings offer a novel alternative for improving the flexural performance of longitudinal joints in shield tunnels.

Graphical abstract

Keywords

longitudinal joints / flexural performance / eccentric short column / fiber-reinforced grid / experiment / theoretical model

1 Introduction

As of August 31, 2023, China has constructed and opened 297 urban rail transit lines in 54 cities, covering a total of 9771 km. Between 50% and 70% of these lines are shield tunnels [1]. This evidence illustrates that shield tunnels have become an integral part of urban infrastructure within the country [2]. Shield lining structures can experience severe complications and even safety incidents due to initial construction flaws and external environmental disturbances during operation [3,4]. These issues gravely impede the normal functioning of rail transit systems and pose substantial risks to passengers’ lives and property [5]. Shield linings, structures assembled by connected bolts, contain numerous circumferential and longitudinal joints. Various problems, such as concrete cracks and water leakage, occur predominantly at these longitudinal joints [6,7]. Consequently, the longitudinal joints represent the most critical and vulnerable section of shield tunnel lining structures [8–10] and exert a controlling influence on the overall performance of the lining structure [11–13]. This situation has led to an urgent need for strengthening or retrofitting existing shield tunnels, with particular emphasis on reinforcing the longitudinal joints.

In recent years, substantial research has been conducted both domestically and internationally on the strengthening and retrofitting of shield tunnels. Chang et al. [14] examined the deformation and dislocation of the Panchiao Line of the Taipei Mass Rapid Transit, detailing how steel inner lining was utilized to enhance the structure. Huang et al. [15] presented a field case where unexpected dumped soils disrupted the segmental lining’s deformational responses. They comprehensively described the situation and proposed treatment measures, including unloading, plugging, and bonding carbon fiber-reinforced plastic and steel plates to the inner surface. Liu et al. [16] published an article that thoroughly discussed the bearing performance of shield lining strengthened by steel plates on the inner surface through prototype testing. Additionally, they proposed a corresponding simplified nonlinear numerical method for analyzing such strengthened structures [17]. Liu et al. [18] also introduced a refined three-dimensional solid finite element model to explore the mechanical behavior, deformation characteristics, and failure mode of tunnel segments after reinforcement with steel plates.

In research led by Liu and Zhang [19], numerical simulation techniques were utilized to analyze the mechanism by which bonding Aramid Fiber Reinforced Polymer (AFRP) controls the lateral deformation of shield tunnels. The primary parameters investigated were the AFRP bonding duration and the number of layers. Liu et al. [20,21] provided a comprehensive description of a shield tunnel’s structural responses caused by unexpected leakage. They proposed the use of steel plates bonded across circumferential joints to regulate the deformation of shield tunnels. In another study by Liu et al. [22], the full-scale test method was applied to examine the ultimate bearing capacity of longitudinal joints strengthened by epoxy-bonded FRP and filament wound profile plates, respectively. Wu et al. [23] conducted a full-scale experiment on longitudinal joints reinforced with concrete-filled steel tubes, demonstrating that this approach could substantially enhance the mechanical behavior of the joints when subjected to sagging or hogging moments.

Traditional methods for strengthening or retrofitting lining structures often involve the external bonding of steel plates or FRP [24–26]. However, these methods have limitations. Steel plates can be bulky, heavy, challenging to fix, susceptible to corrosion and fatigue, and intrusive to the structural member dimensions [27]. External FRP strengthening techniques also have drawbacks, primarily related to the epoxy resins used to bond or impregnate the fibers. Such issues include unsuitability for high-temperature or moisture-prone applications, low compatibility with substrate concrete, and ineffectiveness in enhancing the tunnel structure’s overall stiffness [28]. These challenges have prompted a search for alternative solutions.

The fiber-reinforced plastic grid with polymer-cement-mortar (FRP Grid-PCM) method is an innovative approach to upgrading and enhancing reinforced concrete (RC) structures using an FRP grid with a polymer cement mortar (PCM). It exhibits commendable performance [29] and can effectively overcome the drawbacks of traditional reinforcement techniques.

In conclusion, there is a limited body of research focusing on enhancing the bending resistance of longitudinal tunnel joints, especially studies concerning strengthening through the FRP Grid-PCM method. In this study, eccentric short-column specimens were built by bolting two half-corbel columns together to simulate the mechanical behavior of longitudinal joints. The flexural behavior of these joints strengthened by the FRP Grid-PCM method was examined experimentally, considering various combinations of loading eccentricity and FRP grid layers. Additionally, an analytical model was developed and validated to predict the flexural bearing capacity of the longitudinal joints upgraded using the FRP Grid-PCM method. The research results enhance our understanding of longitudinal joints strengthened using the FRP Grid-PCM method and provide practical insights for optimizing such applications in engineering, ensuring safe and cost-effective construction practices. The development of an analytical model further aids engineers in predicting the flexural bearing capacity of these strengthened joints, streamlining the design process and potentially saving time and costs in practical projects.

2 Experimental program

2.1 Specimens design

While full-scale experimentation remains the optimal method for analyzing the behavior of longitudinal joints in segmental tunnel lining strengthened by the FRP Grid-PCM method [30], executing such tests can be prohibitively expensive and challenging. As a result, many researchers have turned to simplified models to investigate the relevant mechanical properties of tunnel linings. A prevalent approach has been to model the segmental tunnel lining as an eccentric short column [31–34]. For instance, in a study on the mechanical properties of longitudinal joints, Japanese scholars Murakami and Koizumi [35] used an eccentric short column formed by bolting together two half-corbel columns. By altering the axial load’s eccentricity, they could induce either bending failure at the joint or crushing of the concrete, enabling an examination of the mechanical performance of the longitudinal joints. Therefore, this paper employs a similar model, using an eccentric short column formed by bolting two half-corbel columns, to study the mechanical performance of longitudinal joints in segmental tunnel linings strengthened by the FRP Grid-PCM method (Fig.1).

For this study, a 1/3 scale test was conducted, with a typical tunnel segment structure in Shanghai, China, serving as the prototype project. The eccentric short column was constructed using C55-grade concrete, and its section size was 400 mm × 120 mm, with an overall height of 800 mm. The specimen featured symmetric reinforcement, with the longitudinal reinforcement comprising four HRB400 steel bars 8 mm in diameter. The end of the eccentric short column was designed in the shape of a corbel to accommodate the application of an eccentric load.

The stirrups utilized in the structure were made from HPB300-grade steel bars with a diameter of 6 mm. They were strategically densified at the corbel to prevent local pressure damage at the end of the eccentric short column. A detailed representation of the specimen’s reinforcement is shown in Fig.2. The eccentric short column was fastened using two M10 straight bolts of grade 8.8, with the joint positioned at the half-column height (400 mm). The PCM was applied to encase the FRP grid, acting as a bonding and protective material to form an external overlay on the inner surface. The primary constituents of the PCM included Portland cement, fly ash, silica sand, a water-reducing admixture, polyvinyl alcohol fiber, and other additives. The mechanical parameters of the PCM, as provided by the manufacturer, are listed in Tab.1.

The FRP grids were composed of carbon yarns with a thickness of 0.3 mm. The additional geometric and mechanical properties of the CFRP grid, as supplied by the manufacturer, are detailed in Tab.2. The research parameters for this study were the loading eccentricity and the number of FRP grid layers, leading to the design of five distinct specimens. The specific research parameters for each specimen are shown in Tab.3.

2.2 Specimens preparation

The fabrication process for the test piece was conducted in two distinct stages. The first stage involved the fabrication of the half-corbel short column, while the second stage focused on the assembly of the corbel short column and the creation of the FRP Grid-PCM. In the production of the half-corbel short column, three primary steps were followed. These included the construction of the formwork, the binding of the reinforcement, and the pouring of the concrete. Two Polyvinyl chloride (PVC) pipes 10 mm in diameter were embedded at the designated position in advance to accommodate bolt holes in the design and were removed after the concrete reached its initial setting. The half-corbel short columns were cured under standard laboratory conditions for a period of 28 d.

The columns were then assembled into eccentric short columns. The steps in the FRP Grid-PCM application were as follows. 1) The pre-reinforced surface of the specimen was ground to remove the cover cement layer, exposing the aggregate, and the surface was then cleaned (Fig.3(a)). 2) The two half-corbel short columns were aligned along the bolt holes and connected using bolts to form a corbel short column (Fig.3(b)). 3) The bottom mortar layer was troweled onto the pre-reinforced surface of the assembled specimen (Fig.3(c)). 4) The FRP grid layer was applied and pressed slightly into the mortar (Fig.3(d)). 5) The top mortar layer was applied to completely cover the FRP grid layer (Fig.3(e)–3(f)). 6) The specimens were cured under ambient laboratory conditions for an additional 28 d.

2.3 Measuring instruments and loading

The experiment was conducted using a 50-t servo testing machine. Initially, the specimen was vertically positioned on the testing machine with knife-edge supports placed at both ends, and an eccentric load was applied via a vertical jack. A schematic representation of the loading procedure is shown in Fig.4. The process began with a preload phase to compress the specimen, followed by the placement of measuring instruments. Subsequently, the formal loading phase commenced. The loading was administered in a displacement monotonic mode until the specimen exhibited failure, with a loading rate of 0.01 mm/s (controlling the accuracy to less than 0.5%).

Five linear variable differential transformers were spaced evenly along the tensile side of the specimen, at the middle of the column (joint) and at distances of 170 and 340 mm from the middle, to measure the deflection of the short column on the tensile side. Two draw-wire displacement sensors were placed on the left and right sides of the joint on the tensile side of the specimen to measure the joint opening. Four strain gauges were installed on the spans of the tensile and compressive sides of the joints to measure the longitudinal strains at these locations. Ten strain gauges were distributed at equal intervals along the height of the specimen on both sides of the joint to study the strain distribution along the height of the section. The concrete strain gauge was 100 mm long, and the strain gauges for the bolts and steel bars were 5 mm long. These were designated 5 through 14. Four additional strain gauges, designated 15 through 18, were positioned at the centers of the bolts to measure bolt strain, marked as 15 through 18. A schematic illustration of the measuring instrument arrangement is shown in Fig.5.

3 Experimental results

3.1 Failure process

3.1.1 Unstrengthened specimens

The failure modes of the unreinforced specimens S-120-0 and S-140-0 are depicted in Fig.6 and Fig.7, respectively. Both modes appear fundamentally similar. At the initial stage of loading, the joint, due to the bolts’ preload, was subjected to full-section compression. As the load increased, the joint gradually assumed a ‘Ʌ’ shape, with one side closing and the other opening. Concurrently, the joint section began to form a concrete compression zone. Upon reaching loads of 64.25 and 46.84 kN, respectively, transverse cracks emerged in the concrete near the upper hand holes of specimens S-120-0 and S-140-0, as illustrated in Fig.6(a) and 7(a). As the loading continued, these cracks quickly extended to the compression side, and the joints widened, as seen in Fig.6(b) and 7(b). Ultimately, when the loads reached 95.7 and 75.48 kN, respectively, the upper part of the concrete at the compression side joint of specimen S-120-0 collapsed (Fig.6(c)), and the concrete at the compression side joint of specimen S-140-0 was crushed (Fig.7(c)). The testing procedure was halted once the concrete at the compression side joint of the specimen was found to be crushed.

3.1.2 Strengthened specimens

When the specimen was reinforced with a layer of FRP grid, the joint opening increased gradually as the load was incrementally applied. Transverse cracks appeared on the surface of the FRP grid layer at the lower part of specimen S-120-1 and the upper part of specimen S-140-1 when the loads reached 37 and 28 kN, respectively. As the load continued to increase, the number of transverse cracks on the surface of the FRP grid layer also grew, with the crack width and extension depth rising slowly. Notably, the transverse cracks for specimen S-120-1 occurred in the lower bolt-hole area (Fig.8(a)), whereas those in specimen S-140-1 occurred in the upper bolt-hole area (Fig.9(a)). At loads of 65.9 and 52.1 kN, the rate of joint opening accelerated notably, and debonding began at the interface between the FRP grid layer and the concrete, with the debonded area increasing as the load increased. In specimen S-120-1, the lower part primarily debonded (Fig.8(b)), while in specimen S-140-1, the debonding mainly occurred upward (Fig.9(b)). When the specimens reached peak loads of 111.05 and 92.85 kN, the concrete at the lower part of the compression side joint of S-120-1 was crushed (Fig.8(c)), and the concrete at the compression side joint of S-140-1 was similarly crushed (Fig.9(c)).

In the case of specimens strengthened by two layers of FRP grid, such as S-120-2, no obvious cracks were visible on the surface of the FRP grid layer during the initial stages of loading. However, as the load was progressively increased, cracks began to form on the surface of the FRP grid layer, becoming more distinct. These minor cracks were evenly spread on both sides of the joint, and one crack on the upper side of the joint extended laterally across the specimen (Fig.10(a)). When the load reached 91.29 kN, localized interfacial debonding occurred in the FRP grid layer near the joint. As the loading continued to increase, the interface between the FRP grid layer and the concrete exhibited debonding both above and below the joint, with the downward debonding being more pronounced. At the point when the load reached 132.8 kN, the concrete on the compression side of the specimen was crushed, resulting in a rapid increase in the downward debonding speed of the FRP grid layer until it was entirely debonded (Fig.10(c)). Compared to specimen S-120-1, the joint opening rate of specimen S-120-2 was significantly lower.

Fig.11 illustrates the final failure state of each specimen. Under equivalent load conditions, the joint opening of specimens strengthened by an FRP grid layer was noticeably diminished. In comparison to the unstrengthened specimens, the strengthened specimens that reached the ultimate load exhibited substantial increases in the number of cracks on the surface of the FRP grid layer. The majority of these were minor cracks evenly distributed across the surface. As the number of FRP grid layers increased, the number of cracks gradually increased, and the crack spacing and surface crack width decreased. Notably, specimens strengthened in this manner exhibited a tendency toward interfacial debonding failure at the joints, and this type of failure became more pronounced with an increased number of FRP grid layers. The load on each specimen at each loading stage is shown in Tab.4.

3.2 Strain

3.2.1 Bolt

Fig.12 shows the eccentric load–strain curve for the specimens’ bolts, demonstrating that the bolts were in tension throughout the entire loading process. The bolt strain in specimens with an eccentricity of 140 mm was significantly larger than that in specimens with an eccentricity of 120 mm. Upon reaching the ultimate load, the bolts in the unstrengthened specimens yielded in tension, whereas no yielding was observed in the bolts of the strengthened specimens. Under comparable load conditions, the bolt strain in the strengthened specimens was noticeably less than in the unstrengthened ones. Additionally, the greater the loading eccentricity and the higher the number of FRP grid layers, the more the bolt strain was mitigated. This pattern suggests that the FRP grid is effective in distributing the tensile stress of the bolts, with the proportion of load shared by the FRP grid increasing with greater eccentricity.

3.2.2 Eccentric load

Fig.13 illustrates the variation in concrete strain at various positions along the eccentrically loaded short column for various eccentricities. As the joint gradually opens, the strain in the concrete at the joint increases. The concrete strain on the tensile side rises sharply, while the strain on the compressive side increases slowly. As the load continues to increase, when the concrete on the tensile side reaches a certain strain, it will not increase further. This is because the opening of the joint causes the strain gauge on the tensile side to rupture and fail. Under the same load conditions, as the eccentricity distance increases, the ultimate strain of the concrete decreases accordingly.

Fig.14 shows the strain curves of the compressive-side concrete and the tensile-side FRP grid-reinforced layer for specimens S-120-1 and S-140-1 as they vary with load. In general, the strains in the compressive-side concrete and the tensile-side FRP grid-reinforced layer both continue to increase. The compressive-side strain increases slowly, while the tensile-side FRP grid-reinforced layer strain increases more rapidly. Under the same load conditions with different eccentricities for reinforcement, the strains on the tensile side of the FRP grid-reinforced layer tend to be consistent, while the compressive-side strain increases with greater eccentricity.

As Fig.15 shows, under the same load conditions, as the amount of grid reinforcement increases, both the compressive- and tensile-side strains of the specimens increase, particularly on the tensile side of the FRP grid-reinforced specimens, where the effect is more pronounced.

3.3 Deflection

Fig.16 illustrates the eccentric load–lateral deflection curve for the specimen. In comparison to the unstrengthened specimen, the strengthened specimens exhibited a marked improvement in ultimate bearing capacity, coupled with a notable reduction in lateral deflection under equivalent load conditions. Specifically, the ultimate bearing capacities of specimens S-120-1 and S-120-2 increased by 16.03% and 38.7%, respectively, relative to specimen S-120-0, and the ultimate bearing capacity of specimen S-140-1 increased by 23.01% relative to specimen S-140-0. These results indicate that the application of an FRP grid can effectively enhance the ultimate bearing capacity of eccentric short columns. The enhancement ratio of ultimate bearing capacity grows with the increase in the number of FRP grid layers and the greater loading eccentricity.

3.4 Joint opening

Fig.17 presents the eccentric load-joint opening curves of the specimen. Under identical load conditions, the joint opening of the specimen with an eccentricity of 140 mm was appreciably higher than that of the specimen with an eccentricity of 120 mm. Relative to the unstrengthened specimens, the joint opening of the three strengthened specimens was efficiently controlled, with the effectiveness of this control increasing with greater loading eccentricity and more FRP grid layers.

3.5 Joint rotation angle

The joint rotation angle \theta is closely related to the bending stiffness of the longitudinal joint K, according to the following relationship [36]:

(1)K=\frac{M}{\theta }=\frac{Mh}{V},

where M is the bending moment of the longitudinal joint of the specimen, V is the difference between the amount of opening on the tensile side of the specimen joint and the amount of closing on the compression side, and h is the thickness of the longitudinal joint.

Fig.18 illustrates the bending moment-joint rotation angle curve for the specimen. In contrast to the unstrengthened specimen, the slope of the joint rotation angle curve of the strengthened specimen increased, resulting in a significantly smaller joint rotation angle under equivalent bending moments. This indicates a substantial improvement in the bending stiffness of the longitudinal joint when strengthened by an FRP grid. The changing trends of the bending moment-joint rotation angle curves for specimens S-120-1 and S-140-1 were largely analogous. In scenarios of small bending moments, the longitudinal joint rotation angle increased gradually, and the bending stiffness of the joint was substantial. Due to the full section compression of the longitudinal joint, with the concrete in the elastic compression stage exhibiting high bending stiffness, the joint opened slowly. However, when the bending moment escalated to a specific value, the joint rotation angle augmented markedly, resulting in a significant decrease in joint bending stiffness. This trend suggests that the concrete on the compression side of the joint had begun to enter the plastic stage, as shown in Fig.18(b).

Throughout the entire loading process, the bending moment and rotation angle of the longitudinal joint underwent continuous changes in response to the load. By analyzing the alterations in the slope of the bending moment-joint rotation angle curve across all specimens—reflecting changes in joint bending stiffness—the failure process of the joint within the unstrengthened specimen was categorized into three distinct stages.

Stage I: This initial phase extended from the onset of full-section compression of the joint until the concrete yielded on the compression side. During this stage, the tensile action of the bolts, combined with the compressive effect of the joint’s concrete, maintained a minimal joint opening rate. Consequently, the bending stiffness of the joint diminished gradually, with the stiffness at its maximum when the full section was under compression.

Stage II: Upon yielding of the concrete on the compression side of the joint, along with tensile yielding of the bolts (resulting in a gradual decline in tension capacity), the control effect exerted by the bolts and the compressed concrete on the joint opening progressively weakened. The ongoing reduction in the height of the compressed concrete further exacerbated this trend. As a result, the bending stiffness of the longitudinal joint underwent further reduction.

Stage III: In this final stage, the bolts experienced plastic deformation under tension, while the concrete on the compression side of the joint reached its ultimate compressive strain. Upon the crushing of the concrete, the joint lost its ability to sustain pressure. The bending stiffness of the joint consequently plummeted to its lowest point, as illustrated in Fig.18(a).

These stages elucidate the complex interplay of mechanical forces at work within the joint, highlighting the critical role of various structural elements in governing the joint’s behavior under varying load conditions.

The failure process of the joints in specimens strengthened by an FRP grid exhibits differences compared to those in unstrengthened specimens. This process can be delineated into three distinct stages.

Stage I: During this phase, the joint experiences compression from the initial full section until the joint begins to open. Thanks to the synergistic effect of the FRP grid, bolts, and concrete on the compression side of the joint, the joint opens at a slow pace, with a correspondingly gradual decrease in bending stiffness. At the point where the full section of the joint is under compression, the bending stiffness of the longitudinal joint is at its maximum.

Stage II: This stage is characterized by a slow joint opening rate. The concrete on the compression side reaches its yield point, resulting in a further reduction in the bending stiffness of the strengthened specimen. However, as the bending moment increases, the tensile effect of the FRP grid also grows, causing the bending stiffness of the strengthened specimen to decrease more slowly than that of the unstrengthened specimen.

Stage III: After the concrete on the compression side of the joint attains its ultimate compressive strain, the FRP grid and bolts no longer form an effective resisting moment. The joint rotation angle then increases swiftly, with a sharp escalation in deformation. Upon structural collapse, the bending stiffness of the joint descends to its lowest point, as shown in Fig.18(b).

4 Theoretical analysis

This section presents analytical models for the load-bearing performance of unstrengthened and strengthened eccentric short-column joints based on the experimental results. A material constitutive model and basic assumptions of the analytical models are first presented. Theoretical models for unreinforced and reinforced joints are then presented. Finally, the theoretical model results are verified by comparison with the experimental results.

4.1 Material constitutive model

In the context of the present study, a bifold line elastic–plastic constitutive model has been adopted for the concrete, derived from the simplified Saenz formula [37]. The stress–strain curve for C55 concrete is presented in Fig.19, where {\mathcal{E}}_{\mathrm{b}} is the yield strain, {\mathcal{E}}_{\mathrm{c},\mathrm{u}} is the ultimate strain, and the elastic modulus E_1 of the OB section of C55 concrete is 35.5 GPa. The Poisson’s ratio v is 0.17, hardening modulus E_{\mathrm{r}} for the BC section is 592 MPa, yield strength {\sigma }_{\mathrm{b}} is 25.3 MPa, and ultimate strength {\sigma }_{\mathrm{c},\mathrm{u}} corresponds to an ultimate concrete compressive strain {\mathcal{E}}_{\mathrm{c},\mathrm{u}} of 0.002.

Fig.20 shows an ideal elastic–plastic constitutive model applied to 8.8-grade bolts. In this model [30], the elastic modulus E_{\mathrm{b}} is 200 GPa, the tensile strength {\sigma }_{\mathrm{u}} is 800 MPa, and the yield ratio is 0.8. The behavior of the FRP grids, composed of carbon yarns with an elastic modulus E_{\mathrm{F}\mathrm{R}\mathrm{P}} of 240 GPa and a tensile strength of 2300 MPa, follows a linear elastic constitutive model.

4.2 Basic assumptions

1) During the stress application, the joint remains planar. It is categorized into two distinct zones: a compression zone and a tension zone. These zones intersect at point O [38,39], as shown in Fig.21.

2) The deformation experienced at the joint of the short column is primarily due to the tensile deformation of the bolt and the compressive deformation of the concrete.

3) The strengthening layer exhibits strong adhesion to the concrete surface.

4.3 Theoretical model

Fig.22 is a schematic illustration of the theoretical model for the joint in an eccentrically loaded short column. In this figure, N represents the eccentric load exerted on the joint, h is the thickness of the joint, a_{\mathrm{s}\mathrm{j}} is the distance from the bolt to the farthest edge of the joint under load, h_0 is the distance from the bolt to the edge of the compressed concrete, A_{\mathrm{b}} is the cross-sectional area of a single bolt, and l_{\mathrm{b}} is the effective length of the bolt, which is considered to be a fixed value.

Based on the test results, the ultimate bearing capacity of the eccentric short column is determined by the crushing of the concrete at the joint’s paraxial load side edge. Therefore, it is assumed in this study that the joint attains its maximum load capacity when the concrete at the paraxial side edge reaches the ultimate compressive strain.

4.3.1 Unstrengthened joint model

The eccentric short-column loading test results indicate that the failure process for an unstrengthened short-column joint consists of three distinct stages, described as follows. Stage I: The joint remains unopened, and the entire section of the joint is compressed. A force analysis of the joint is shown in Fig.23(a). In this figure, {\sigma }_{\mathrm{c}1} and {\sigma }_{\mathrm{c}2} are the concrete compressive stresses at the farthest and nearest load edges of the joint, respectively; {\epsilon }_{\mathrm{c}1} and {\epsilon }_{\mathrm{c}2} are the concrete compressive strains at these respective edges; {\epsilon }_{\mathrm{s}\mathrm{j}} is the tensile strain of the bolt; T_{\mathrm{b}} is the tensile force on the bolt; N_{\mathrm{c}} is the resultant force on the compressed concrete; and y is the distance from the point where the concrete is under compression to the edge of the joint on the paraxial load side. At this juncture, the height of the concrete compression zone x_{\mathrm{c}} equals the thickness of the joint h. As the joint remains unopened, with the joint rotation angle at \theta =0, the bending stiffness of the joint is considered equivalent to that of the eccentric short column. Stage II: The joint has opened, the bolts are experiencing elastic tension, and the concrete at the paraxial load side edge has yielded. The force analysis for this stage is presented in Fig.23(b). In this figure, {\sigma }_{\mathrm{c}} is the ultimate compressive stress of the concrete at the paraxial load side edge of the joint. At this point, y={\displaystyle \frac{x_{\mathrm{c}}}{3}}. Stage III: The joint has opened, the bolts have yielded, and the concrete in the compression zone of the joint has reached the ultimate compressive strain, that is, {\epsilon }_{\mathrm{c}}={\epsilon }_{\mathrm{c},\mathrm{u}}. The force analysis for this stage is illustrated in Fig.23(c). In this figure, x_1 is the height of the elastic compression zone of the concrete, x_2 is the height of the compression zone after the concrete has reached the yield stress, {\sigma }_{\mathrm{b}} is the yield stress of the concrete, and {\epsilon }_{\mathrm{c}} is the compressive strain of the concrete at the paraxial load side edge of the joint.

4.3.2 Fiber-reinforced plastic grid strengthened joint model

In the case of the short columns strengthened with an FRP grid, the strengthening process involves applying the FRP Grid-PCM to the paraxial load side edge of the unreinforced short column. The effectiveness of the strengthening largely relies on the proportion of tensile force absorbed by the FRP grid. Building on an understanding of the behavior of an unstrengthened short column and based on the test results, the failure process for a short column joint enhanced with an FRP grid can similarly be divided into three stages.

Stage I: The joint remains unopened, with the entire section of the joint under compression. The FRP grid has not yet begun to exert its bearing capacity, and the joint rotation angle \theta =0. The bending stiffness of the joint is considered equivalent to that of the eccentric short column. During this stage, the stress state of the joint is identical to that of the unstrengthened joint. The force analysis for this stage is presented in Fig.24(a).

Stage II: The joint has opened, and the concrete at the paraxial load side edge has yielded, with the bolts experiencing elastic tension. The force analysis is shown in Fig.24(b). According to the basic equations of equilibrium for the axial load and the bending moment:

(2)\{\begin{array}{l}N+T_{\mathrm{F}\mathrm{R}\mathrm{P}}+n{T}_{\mathrm{b}}={\displaystyle \frac{1}{2}}{\sigma }_{\mathrm{c}}{x}_{\mathrm{c}}b,\\ N\left(e+h_0-{\displaystyle \frac{h}{2}}\right)=T_{\mathrm{F}\mathrm{R}\mathrm{P}}{a}_{\mathrm{s}\mathrm{j}}+{\displaystyle \frac{1}{2}}{x}_{\mathrm{c}}{\sigma }_{\mathrm{c}}b\left(h_0-{\displaystyle \frac{x_c}{3}}\right).\end{array}

According to the assumption of a plane section:

(3)\frac{{\epsilon }_{\mathrm{s}\mathrm{j}}}{{\epsilon }_{\mathrm{c}}}=\frac{x_{\mathrm{c}}-h_0}{x_{\mathrm{c}}},

(4)\frac{{\epsilon }_{\mathrm{F}\mathrm{R}\mathrm{P}}}{{\epsilon }_{\mathrm{c}}}=\frac{x_{\mathrm{c}}-h}{x_{\mathrm{c}}},

(5)k_1{\epsilon }_{\mathrm{s}\mathrm{j}}{l}_{\mathrm{b}}=T_{\mathrm{b}},

(6)T_{\mathrm{F}\mathrm{R}\mathrm{P}}={\sigma }_{\mathrm{F}\mathrm{R}\mathrm{P}}{A}_{\mathrm{F}\mathrm{R}\mathrm{P}},

where b is the width of the joint, A_{\mathrm{F}\mathrm{R}\mathrm{P}} is the cross-sectional area of the FRP grid, {\epsilon }_{\mathrm{F}\mathrm{R}\mathrm{P}} is the tensile strain of the FRP grid, k_1 is the tensile stiffness of the bolt, k_1={\displaystyle \frac{E_{\mathrm{b}}{A}_{\mathrm{b}}}{l_{\mathrm{b}}}}, and E_{\mathrm{b}} is the elastic modulus of the bolt.

The bending stiffness of the joint at this stage can be deduced by combining Eqs. (1)–(6).

Stage III: The joint has opened, and the paraxial load side edge compression of concrete is crushed, that is, {\epsilon }_{\mathrm{c}}={\epsilon }_{\mathrm{c},\mathrm{u}}. The bolts remain in elastic tension. The force analysis is shown in Fig.24(c). According to the basic equations of equilibrium for an axial load and bending moment:

(7)\{\begin{array}{l}N+T_{\mathrm{F}\mathrm{R}\mathrm{P}}+n{T}_{\mathrm{b}}={\displaystyle \frac{1}{2}}{\sigma }_{\mathrm{b}}{x}_{1}b+{\sigma }_{\mathrm{b}}{x}_{2}b+{\displaystyle \frac{1}{2}}\left({\sigma }_{\mathrm{c}}-{\sigma }_{\mathrm{b}}\right)x_{2}b,\\ N\left(e+h_0-{\displaystyle \frac{h}{2}}\right)={\displaystyle \frac{1}{2}}{x}_1{\sigma }_{\mathrm{b}}b\left(h_0-x_{\mathrm{c}}+{\displaystyle \frac{2x_1}{3}}\right)+{\sigma }_{\mathrm{b}}{x}_{2}b\left(h_0-{\displaystyle \frac{x_2}{2}}\right)+{\displaystyle \frac{1}{2}}\left({\sigma }_{\mathrm{c}}-{\sigma }_{\mathrm{b}}\right)b{x}_2\left(h_0-{\displaystyle \frac{x_2}{3}}\right)+T_{\mathrm{F}\mathrm{R}\mathrm{P}}{a}_{\mathrm{s}\mathrm{j}}.\end{array}

According to the assumption of a plane section:

(8){\sigma }_{\mathrm{c}}={\sigma }_{\mathrm{b}}+E_{\mathrm{r}}\left({\epsilon }_{\mathrm{c}}-{\epsilon }_{\mathrm{b}}\right),

(9)\frac{{\epsilon }_{\mathrm{F}\mathrm{R}\mathrm{P}}}{{\epsilon }_{\mathrm{c}}}=\frac{h-x_{\mathrm{c}}}{x_{\mathrm{c}}},

(10)\frac{{\epsilon }_{\mathrm{s}\mathrm{j}}}{{\epsilon }_{\mathrm{c}}}=\frac{h_0-x_{\mathrm{c}}}{x_{\mathrm{c}}}.

The bending stiffness of the joint at this stage can be derived by combining Eqs. (1), (5)–(10).

4.4 Theoretical results

In an effort to validate the feasibility of the theoretical model describing the bearing performance of the eccentric short-column joint, a comparison was made between the theoretical calculations and experimental data pertaining to the ultimate bearing capacity and the joint’s bending stiffness. The findings are delineated in Tab.5 and Tab.6. Tab.5 reveals that across all specimens, the relative error between the theoretical and experimental values of the bearing capacity ranges from 5% to 9%. The current study attributes this relative error to several factors.

1) The bolts were manually preloaded during testing, leading to disparities in the specimens’ bolt preload. Furthermore, the theoretical model does not account for the effect of bolt preload.

2) Localized failure inside the specimens occurred during the loading process, resulting from the interaction between the bolt and the joint’s bolt hole.

3) While the PCM was cast to encapsulate the FRP grid, this bonding material does not guarantee a consistent bond between the FRP grid and the concrete surface of the specimen.

Consequently, the experimental value for the joint’s ultimate bearing capacity proved smaller than the theoretical value. Tab.6 illustrates that, during the three stages of the joint’s failure process, the relative error between theoretical and experimental values for the joint’s bending stiffness does not exceed 10%. Moreover, the relative error in the third stage is noticeably higher than in the first two stages.

In conclusion, the theoretical outcomes for the bearing capacity and bending stiffness of the specimen are fundamentally in alignment with the test results, within an acceptable range of relative error. This coherence underscores the potential applicability and relevance of the theoretical model and bending stiffness improvement theory presented in this study.

5 Conclusions

The innovative FRP Grid-PCM method was used in this study to strengthen longitudinal joints and enhance their mechanical performance. Eccentric short-column specimens were built with and without FRP Grid-PCM reinforcement and tested according to an experiment designed to study the effects of FRP-Grid PCM reinforcement on the joint failure process, ultimate bearing capacity, bending stiffness of the joint, and influence of bolt strain. The experimental results were used to verify a theoretical model for the improvement in joint bending stiffness achieved using FRP Grid-PCM reinforcement. The following conclusions were reached.

1) The failure process of short-column joints strengthened with FRP grids can be divided into three stages: initial full compression of the joint, yielding of the concrete at the paraxial load side edge, and crushing.

2) An FRP grid layer significantly enhances the bearing capacity of an eccentric short column. The ultimate bearing capacity improvement increases with the number of FRP grid layers and the loading eccentricity.

3) A short-column joint strengthened with an FRP grid exhibits improved bending stiffness compared to an unstrengthened column joint. This bending stiffness is at its maximum when the joint is fully compressed and gradually diminishes as the paraxial load side edge approaches the ultimate compressive stress.

4) An FRP grid layer alters the stress distribution in a short column, leading to a notable increase in surface cracks, which are primarily small and evenly distributed. As the number of FRP grid layers increases, the number of cracks increases, and the crack spacing and surface crack width decrease. The use of an FRP grid layer on the tension side effectively manages tensile stress and restricts the increase in the joint opening and rotation angle.

5) A theoretical model for eccentric short column joints reinforced with an FRP grid was established and found to yield good agreement with the experimental results, confirming its validity. This study contributes a novel method for predicting the effects of joints strengthened using the FRP Grid-PCM method that could significantly enhance the flexural performance of longitudinal joints in shield tunnels.

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