Experimental study on the mechanical performance of unilaterally cracked tunnel lining strengthened with fiber reinforced plastics

Abstract

Adhering fiber reinforced plastics (FRP) is a typical method for reinforcing cracked tunnel linings. Due to the influence of the asymmetric effect, the mechanical response of FRP-strengthened lining with unsymmetric cracks is not known. Investigating the mechanical performance and strengthening mechanisms of FRP-strengthened linings with unilateral cracks is critical in guiding the reinforcing strategy. In this study, a series of model tests were conducted, two unilaterally cracked linings strengthened with FRP for different ranges (unilateral and full-span strengthening), and one intact lining was tested. The deformation field and damage behavior during the test process were monitored by digital image correlation analysis and acoustic emission technology. Results show that the unilaterally strengthened lining exhibits 30% higher bearing capacity and energy-dissipation capability than the full-span strengthened lining. During the damage process, a comparable percentage (50%) of tensile and shear cracks developed in all the lining specimens. However, compared to the unilateral strengthening, the extra FRP strengthening resulted in a higher severity of cracking. In addition, the strengthening mechanism of FRP-strengthened lining was analyzed based on the section-moment and section-stiffness of the linings. The full-span strengthening caused a greater stress concentration near the pre-crack, resulting in greater section moments and stiffness decay rate, contributing to structural failure. For tunnel linings with unilateral cracking, extra FRP strengthening may compromise reinforcement efficiency.

Graphical abstract

Keywords

model test / tunnel lining / unsymmetrical crack / FRP strengthening / mechanical performance / strengthening mechanism

1 Introduction

Due to stress concentration or structural aging, unsymmetrical cracks often occur in curved concrete structures, such as spandrel cracking in tunnel linings [1]. Due to the unsymmetric effect [2], the unilateral cracks greatly harm the load-bearing capacity and stability of tunnel linings. Consequently, it is imperative to employ effective reinforcement techniques to restore or enhance the mechanical performance of cracked linings. As a pasting reinforcement material, fiber reinforced plastics (FRP) has the advantages of high tensile strength, light weight, corrosion resistance, etc., and is scientifically suitable for different curved surfaces [3]. This technology effectively limits crack propagation and prevents spalling, making it a widely adopted method for reinforcing cracked linings.

Through theoretical analysis and numerical simulation, extensive research has been conducted on the mechanical behavior of FRP-strengthened tunnel linings. Numerous theoretical studies based on analytical methods [4,5] have primarily focused on the stress state at the FRP-strengthened concrete interface. Numerical simulations, represented by the finite element method [6–8], have revealed the mechanical performance and debonding evolution process of FRP-strengthened structures. Lee and Shin [9] carried out numerical simulations to verify the feasibility of FRP for the reinforcement of tunnel lining structures. Li et al. [10] performed numerical simulations to investigate the load-bearing compatibility between CFRP and subway tunnel linings. Additionally, Qin et al. [11] simulated a FRP-strengthened water transfer tunnel and found that increasing the elastic modulus and thickness of carbon CFRP could improve the stress state of the lining.

The advancement of loading setup and monitoring technologies has significantly facilitated the development of model tests on FRP-strengthened tunnel linings [12]. Li et al. [13] conducted model tests on the damaged shield segment strengthened with CFRP and obtained the failure modes of the bonding surface created by different bonding techniques. Sahranavard and Aghanoori [14] monitored the CFRP-strengthened lining of Haji-Abad tunnel in Iran and confirmed that the strengthened structure meets the load-bearing requirements under the designed load. Zhang et al. [15] conducted model tests to investigate the strengthening performance of concrete segments strengthened with basalt fiber reinforced plastics (BFRP) grids, and found that the thickness of BFRP positively influences the stiffness of the strengthened structure. Liu et al. [16] monitored the stress and deformation patterns of FRP-strengthened linings in model tests, revealing that FRP enhances mechanical performance by increasing structural stiffness and suppressing crack propagation. In addition, the effects of factors such as impact loading [17] and biased pressure [18] on FRP-strengthened linings were also considered.

Although the effectiveness of FRP strengthening has been proven, and the relationship between the physical parameters of FRP (e.g., elastic modulus, thickness, width, etc.) and reinforcement performance (e.g., load-bearing capacity, deformability, etc.) has been widely discussed [19–21], asymmetric effects caused by asymmetric cracks have rarely been considered. In the unsymmetric effect, that is, taking a central line of the tunnel contour as the boundary, the influence of the crack on the inner force of lining in its locating side is greater than that of the other side [2]. Few studies have focused on asymmetric cracked lining strengthened with FRP, especially on their mechanical properties and strengthening mechanisms. Although the ACI 440 code [22] and Eurocode 2 [23] give guidance on FRP selection, load capacity calculation, and section stress distribution model for FRP-strengthened concrete structures, there is still a gap in the reinforcement strategy for unsymmetrical cracking.

This study investigates the mechanical properties of unilaterally cracked linings strengthened with FRP. Tests were conducted on three types of partial-scale models: asymmetrically cracked linings reinforced with different FRP coverage areas (specifically, unilateral and full-span area of the lining bottom surface) and one intact lining. Acoustic emission (AE) technology and digital image correlation (DIC) analysis [24,25] were employed to monitor internal damage and external deformation of the specimens. Subsequently, the load-displacement response, deformation characteristics, and cracking pattern based on AE parameters were analyzed. Furthermore, the FRP strengthening mechanism was investigated by analyzing the bending moment and stiffness evolution of the crack-section.

2 Test designs

2.1 Experimental device

A hydraulic loading device was developed to apply loads to the local model of the tunnel [26]. As shown in Fig. 1, the test setup is positioned vertically and comprises three main components: the reaction frame, the loading unit, and the control system. The reaction frame provides the necessary counterforce for loading. The three loading points are arranged at 30° intervals on the reaction frame structure, with a maximum loading capacity of 1000 kN per unit. These three loading units can be loaded synchronously or independently. In this study, only one intermediate loading unit was enabled to provide vertical loading. The control system is used to control the loading process and record the load and displacement of the cylinder, and provides two control modes: load control and displacement control. In this study, the displacement control mode was used, with a loading range of 0 to 300 mm and a displacement rate range of 0.5 to 10.0 mm/min.

2.2 Experimental cases

Three local-scale lining models were designed for loading tests, with a scale ratio of 1/4, as shown in Fig. 2. A too large scale ratio introduces challenges in specimen sheetation, whereas a too small scale ratio leads to substantial discrepancies in mechanical response characteristics between the model specimen and prototype [26]. The 1/4 scale ratio has been widely adopted in lining model tests due to its capability to effectively replicate the load-transfer mechanisms and failure characteristics in real tunnels [27,28]. A typical double-lane highway tunnel in China was selected as the prototype for the physical model. The lining of the prototype tunnel was cast using C35 concrete, with HRB335 steel bars employed as reinforcement. Considering that the tunnel vault is prone to cracking under tensile stress, the vault section was selected as the prototype for the physical model. The thickness was 0.5 m, the span width was 11.8 m, and the height was 9.6 m. Cracks were usually located near the 30°location of lining spandrels in real tunnels [29,30]. Thus, the pre-cracks were set on the right spandrel with an angle of 30° to simulate the unsymmetrical cracks. Further, the cracked lining L₁ was strengthened with FRP unilaterally on the cracked side (unilaterally strengthened lining), and the cracked lining L₂ was strengthened on the full bottom surface (full-span strengthened lining). In addition, an intact lining L₀ was set as the control case.

2.3 Similar designs and materials

Similarity theory for model tests is a framework that elucidates the similar relationships between a structural model and its prototype. It guides the design of the model and the conversion of mechanical parameters through the design of similarity ratios for key physical parameters [31,32]. The key physical parameters include geometry size L, stress σ, elastic modulus E, displacement δ, strain ε, Poisson ratio μ, force F, gravity γ, and bending moment M, etc. To ensure that the specimen size is compatible with the test setup, the geometric dimension similarity ratio (C_L) is set as 1/4. To reproduce the stress state of the prototype scale as accurately as possible [27], the gravity similarity ratio (C_γ) is set as 1, meaning that the prototype material was used to prepare the specimens. The similitude law and dimensional analysis method were employed to determine the similarity ratios for other key physical parameters, as detailed in Table 1.

Based on the similarity designs, the experimental materials are as follows.

1) Concrete and steel rebar

The specimens were constructed using the same C35 concrete and HRB335 steel reinforcement as those employed in the prototype tunnel. With reference to the Chinese standard (GB/T 50081-2002) [32], the concrete mix ratio was determined as cement: coarse aggregate: fine aggregate:water = 1:1.6:2.7:0.42. Continuously graded crushed stone with a particle size range of 10–25 mm was used as coarse aggregate, and well-graded fine sand was adopted as fine aggregate. Portland cement P.O 42.5R was utilized. Consistent with the prototype tunnel, a reinforcement ratio of 0.6% was applied, as shown in Fig. 3.

2) Adhesive and FRP sheet

A UV epoxy resin is selected as the adhesive, which has the characteristics of fast curing and high strength [33,34]. The selected FRP material is EM-300 glass fiber sheet [19], a typical reinforcement material for concrete crack repair. Considering the need for rapid construction to minimise disruption to highway traffic during reinforcement of cracked tunnel lining, the glass fiber with lightweight and high-strength properties was chosen. The EM-300 glass fiber sheet demonstrates rapid UV-curable bonding with adhesives due to its excellent light transmittance (curing time < 10 min), which has been applied in reinforcing cracked concrete pipelines and tunnel linings [33–35]. The physical parameters of epoxy resin and FRP sheet are shown in Table 2.

2.4 Specimen preparation

An assembled steel mold was specifically designed for casting the lining, as depicted in Fig. 4. Prior to casting, the rebar mesh was arranged and tied in accordance with the design specifications and subsequently placed within the steel mold. The linings were cast in a single batch and compacted using a hand-held mechanical vibrator to ensure uniformity. After a two-day curing period within the mold, the specimens were demolded and then subjected to an additional 28 d curing process under ambient laboratory conditions. To simulate the unsymmetrical crack, the cured specimens were pre-cracked on the right side with a range of 30°. In the field of tunnel lining reinforcement, non-penetrating cracks are often strengthened by attaching FRP sheets. To characterize typical non-penetrating cracks in tunnel linings, the width of the pre-crack was set to 3 mm, with a depth of 20 mm.

For the reinforcement of the specimens, an FRP composite layer was applied to the inner surface of each specimen following a detailed procedure. Initially, to ensure optimal bonding between the inner lining surface and the FRP layer, the inner surface was chiselled, thoroughly cleaned with water, and allowed to dry under ambient conditions. Following these preparations, the cracked lining specimens were strengthened by bonding and pasting FRP sheets. This process began with the application of a UV resin adhesive layer (200 g/m²) to the chiselled surface. Next, adhere the EM-300 FRP sheet onto the adhesive resin and squeeze to eliminate the interlayer air bubbles. Finally, the strengthened specimens were subjected to rapid curing treatment with a 265 nm wavelength ultraviolet light source. To ensure the curing efficiency, the irradiation time of the ultraviolet light source is 15 min [33].

2.5 Measuring instruments and loading

The model tests were conducted by applying a vertical load at the top of the specimen until the loss of bearing capacity was observed, as illustrated in Fig. 5. To achieve quasi-static loading, a displacement-controlled mode was adopted with a loading rate set at 2 mm/min [26]. The load and displacement were acquired by sensors attached to the top of the loading cylinder. The deflection were monitored using three displacement transducers positioned beneath the bottom of the lining specimens.

The DIC technique determines the surface displacement of a specimen by tracking the positional changes of speckle points on the object’s surface before and after deformation. This method relies on comparing and calculating digital images of the specimen surface before and after deformation [36]. In this study, a high-definition digital camera was employed to track and record the cracking process of the lining specimen. To ensure sufficient contrast, white primer along with random white speckles was sprayed onto the measurement area of the beam, as shown in Fig. 5. Before the test, the camera angle was adjusted to ensure the lens remained perpendicular to the speckled surface, and additional lighting was supplemented as necessary. After loading, the image data were processed using the open-source software Ncorr (based on the MATLAB platform) to calculate the displacement and strain fields.

Additionally, cracking events in concrete materials are typically accompanied by the release of elastic waves [37], enabling the detection of potential damage through AE techniques. The AE technology is widely used for monitoring damage in concrete structures. To ensure the efficiency of sound wave transmission, two RS-15A AE sensors were mounted on the monitoring surface using a coupling agent [38]. A detection threshold of 45 dB was set to eliminate noise interference. Referring to the research of Wang et al. [26], the resonant frequency and resonant threshold were set at 20–400 kHz and 40 dB.

3 Test results

3.1 Load−displacement response

Figure 6 illustrates the load-displacement curves for the linings under a loading rate of 2 mm/min. As depicted, the entire loading process can be divided by points C (first-cracking point), P (peak load point), and F (failure point). The load-displacement responses of the intact lining (L₀) and the strengthened linings (L₁, L₂) exhibit distinct behaviors. For the strengthened linings (L₁, L₂), the load increased linearly with the advancement of distance (δ) until the peak load (P_peak) was reached. However, a sharp decline in load capacity followed immediately after peak load, reducing it nearly to zero. This phenomenon has been attributed to the sudden propagation of cracks during the post-peak stage, as reported in previous studies [24]. In contrast, the load on the intact lining (L₀) increased more gradually before reaching P_peak, and the subsequent decrease in load was more gradual, occurring in a stepwise manner rather than abruptly. Notably, the load-displacement response of the intact lining (L₀) did not display the brittle failure characteristics observed in the strengthened linings. While the load−displacement curves of the strengthened linings (L₁, L₂) appear similar, the unilaterally strengthened lining (L₂) shows a significantly higher load-bearing capacity than the full-span strengthened lining (L₁).

To further discuss the mechanical performance of linings in detail, key parameters such as the first-cracking load (P_cr), peak load (P_peak), toughness, and stiffness were. Toughness was quantified by calculating the area under the P−δ curve before structural failure, reflecting the structure’s energy-dissipation capacity. Stiffness was assessed by determining the slope of the linear segment of the load-displacement curve closest to a straight line. These parameters are comprehensively listed in Table 3.

As shown in Table 3, the first-cracking load of the intact lining (L₀), unilaterally strengthened lining (L₁) and full-span strengthened lining (L₂) were 82.4, 89.5, and 100.0 kN, respectively. Although the P_cr values for the strengthened linings (L₁, L₂) were all lower than those of the intact lining (L₀), the P_cr values for the unilaterally strengthened lining (L₁) were slightly higher than those for the full-span strengthened lining (L₂). The result indicated that the unilaterally strengthened lining expressed better anti-cracking ability. The load-bearing capacity of the unilaterally strengthened lining (L₁) was 30.2% higher than that of the full-span strengthened lining (L₂), which was close to that of the intact lining (L₀). Those indicated that, for unsymmetrically cracked lining, unilateral strengthening could enhance structural load-bearing capacity more significantly than full-span strengthening.

The toughness of the unilaterally strengthened lining (L₁) was found to be 30.6% higher than that of the full-span strengthened lining (L₂), indicating a superior energy-dissipation capacity. When the displacement δ reached about 12 mm, the unilaterally strengthened lining (L₁) failed due to inferior energy dissipation capacity. The observed behavior can be attributed to the stiffness gradient induced by unilateral strengthening, which enabled the unreinforced side of the lining (with lower stiffness) to preferentially dissipate energy by microcrack propagation. In contrast, the full-span strengthened lining (L₂) exhibited concentrated energy release at the top of the pre-crack, consequently inducing worse cracking. Additionally, the unilaterally strengthened lining (L₁) exhibited a stiffness that was 14.7% greater than that of the full-span strengthened lining (L₂). Overall, for spandrel-cracked linings strengthened with FRP, unilateral strengthening demonstrated superior performance in terms of load-bearing capacity, toughness, and stiffness compared to bilateral reinforcement.

3.2 Failure modes and crack patterns

Figure 7 shows the observed failure modes and crack patterns of the linings. The intact lining (L₀) exhibited circular cracking at the base and concrete crushing at the top of the arch. In contrast, the strengthened linings (L₁, L₂) displayed gradual crack development, culminating in extensive shear damage accompanied by FRP debonding. Additionally, the crack patterns, including their distribution and progression, varied significantly. The intact lining (L₀) primarily exhibited diagonal cracks at mid-span and circular cracks at the base, with cracks gradually developing throughout the loading process. For the strengthened linings (L₁, L₂), regardless of the FRP strengthening range, the dominant crack pattern consisted of diagonal shear cracks concentrated above the pre-cracks. These new cracks emerged rapidly from the pre-cracks after loading to P_peak. Although the crack patterns in the strengthened linings (L₁, L₂) were similar, the unilaterally strengthened lining (L₁) showed a more severe degree of cracking, characterized by a greater number of cracks and a larger cracked area.

3.3 Deformation characteristics

The surface strain of the lining measured by DIC technology is shown in Fig. 8, which includes the horizontal strain field (ε_xx) and the vertical strain field (ε_yy) [39,40]. As shown in Fig. 8(a), at the first cracking point (C) and peak load point (P), the maximum ε_xx for the intact lining specimen (L₀) was symmetrically distributed across the surface. In contrast, for the strengthened linings (L₁, L₂), a single maximum ε_xx strain was observed, concentrated at the bottom. However, the locations of the maximum ε_xx differed between the strengthened specimens. In the full-span strengthened lining (L₂), the maximum ε_xx was distributed at mid-span, whereas in the unilaterally strengthened lining (L₁), it was located near the spandrel crack. This indicates that the high-strain zone in the full-span strengthened lining (L₂) tends to occur near the pre-crack before reaching the peak load, potentially making the structure more susceptible to failure. Upon reaching the failure point (F), the intact lining specimen (L₀) exhibited a high-strain zone at mid-span, while the high-strain zones in the strengthened lining (L₁,L₂) were concentrated at the tops of the unilateral pre-cracks. These findings suggest that vertical damage in the strengthened linings (L₁, L₂) is primarily concentrated above the unilateral pre-cracks, regardless of the FRP strengthening range.

Figure 8(b) presents the strain field distribution of ε_yy. The unilaterally strengthened lining (L₁) has no localized high-strain areas prior to the failure point (F), and the strain distribution is uniform across the lining body, indicating that the lining developed mainly scattered microcracks. This demonstrates that unilateral strengthening effectively utilizes the energy dissipation capacity of unreinforced concrete regions, promoting more homogeneous strain distribution and enhanced load-bearing performance. In contrast, for the full-span strengthened lining (L₂), a localized zone of high strain was consistently present at the top of the pre-crack from the point of first cracking (C) and widened significantly at the point of damage (F). This suggests that the full-span strengthening induced a severe stress concentration phenomenon from the very beginning due to the fact that the stiffness at the cracking location was much less than that of the other regions. The energy was concentrated and released at the precast cracks, leading to more severe cracking. To quantify the severity of shear damage, the ratio of the high-strain zone to the observed zone, denoted as Ψ was calculated using a pixel-point method [41]. The results indicated that the severity of shear damage in the full-span strengthened lining (L₂) was 1.63 times greater than that in the unilaterally strengthened lining (L₁).

3.4 Crack pattern and process based on acoustic emission events

3.4.1 Basic parameter analysis

The AE technique can detect two types of signals: 1) AE parameters, including AE counts, rise time, energy, amplitude, frequency, and frequency content; and 2) AE waveforms [42]. Notably, essential parameters such as AE counts serve as indicators of damage severity and timing, allowing for precise assessment of when damage occurs.

The relationship curves of the AE ringing count, cumulative ringing counts and times for each lining are shown in Fig. 9. During the initial stage (O–C), only a few AE signals were detected, indicating the formation of microcracks within the lining. The first visible crack emerged when the AE counts began to densely cluster and reached higher values. In the second stage (C–P), the AE signals fluctuated within a certain range, reflecting the ongoing development of cracks. After reaching the peak load point, the load in the strengthened lining (L₁, L₂) dropped abruptly within a short time, accompanied by a high number of AE counts, signifying severe cracking behavior. In contrast, the intact lining (L₀) displayed more stable AE count fluctuations, indicative of a progressive failure process.

3.4.2 Acoustic emission rise amplitude-average frequency analysis

To further characterize the crack types of the specimen, two parameters, AF and RA, were analyzed in accordance with the recommendations of JCMS III Guide B5706 (2003). These parameters, RA and AF, are extensively used in monitoring the failure processes of rock and concrete materials [43]. RA is defined as the ratio of the rise time to the maximum amplitude of an AE event, while AF represents the ratio of the ringing counts to the duration time.

(1)\begin{array}{rl}& \mathrm{R}\mathrm{A}=\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}/\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e},\\ & \mathrm{A}\mathrm{F}=\mathrm{A}\mathrm{E}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}/\mathrm{A}\mathrm{E}\mathrm{d}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.\end{array}

The RA and AF parameters are commonly employed to distinguish between shear and tensile cracks in materials. Typically, a high AF/RA ratio is indicative of shear cracks, and a low AF/RA ratio suggests tensile cracks. The AF_max/RA_max ratio has been proposed as a criterion for determining crack types. This analytical method has been widely used in concrete model tests and structural health monitoring [44].

The statistical distribution of AF−RA values for each lining is presented in Fig. 10. Generally, the AF−RA values exhibited a similar distribution across all linings, with tensile cracks being slightly more prevalent than shear cracks, accounting for approximately 55.4% to 57.9% of the total. In addition, the AF−RA values are close in density on both sides of the demarcation line, indicating that the lining was subjected to bending pressure and sprouted a comparable number of tensile and shear cracks. Since the crack type is mainly affected by the material and load type of the specimen, FRP strengthening has no influence on the crack type.

3.4.3 The b-value and acoustic emission energy

The AE b value can reflect the cracking process of concrete and which is one of the important precursors of structural fracture. The b-value and energy curves for each specimen are shown in Fig. 11. The variation in the b-value can be characterized by phases of increase, decrease, or fluctuation.

The magnitude distribution of AE events in brittle or quasi-brittle materials, such as rock and concrete, typically adheres to the modified Gutenberg–Richter law [43], as expressed in Eq. (2).

(2)\log N=a-bM,

where M is A_dB/20, A_dB is the peak amplitude of AE events (dB), N is the number of AE events within the magnitude range M + dM, a is an empirical constant, and b is the b-value of the AE. The b-value reflects the proportion of low-amplitude events relative to high-amplitude events.

(3)b=\frac{{\displaystyle \sum _{i=1}^m{M}_i{\displaystyle \sum _{i=1}^m\lg N_i-m{\displaystyle \sum _{i=1}^m{M}_i\lg N_i}}}}{m{\displaystyle \sum _{i=1}^m{M}_i^2-{\left({\displaystyle \sum _{i=1}^m{M}_i}\right)}^2}}.

As the b-value rises, a higher b-value indicates a larger proportion of low-amplitude events in the AE data. During periods of b-value fluctuation, minimal variation suggests stability in the ratio of large to small AE events. The overall trend in b-value changes was consistent across all linings throughout the loading process. In the initial stage (O–C), few AE events were observed within the specimens. As the load increased, the b-value gradually rose, indicating an increase in micro-scale fractures. Near the first cracking point (C), the b-value sharply decreased, reflecting the penetration of microcracks into visible cracks. Following this point, the b-value fluctuated within a certain range, signifying the continuous occurrence of micro-cracks within the specimens. The overall distribution of AE energy was discreet, with similar patterns observed in both the intact lining (L₀) and the strengthened linings (L₁, L₂). Notably, the full-span strengthened lining (L₂) exhibited higher AE energy values, likely due to the extensive cracking that occurred after the peak load point (P).

4 Strengthening mechanism

The bending moment and stiffness influence the structural deformation characteristics, leading to variations in reinforcement performance. Given that the damage zone of the linings is concentrated near the pre-crack, a simplified theoretical analysis based on the bending moment and stiffness at the crack-section is conducted to investigate the FRP strengthening mechanism.

The simplified force diagram of the reinforced lining structure is shown in Fig. 12. The load and boundary conditions are set the same as those in the model test. The simplified stress distribution of the lining cross-section is shown in Fig. 13, with a linear distribution assumed for the tensile stress [45,46]. For the compressive zone, a parabolic stress distribution model recommended by ACI 400 was adopted, which can more accurately reflect the plastic deformation characteristics of concrete under compression [22,47].

(4){\sigma }_{\mathrm{c}}(y)=f_{\mathrm{c}}^{\prime }\left[2\frac{{\epsilon }_{\mathrm{c}}(y)}{{\epsilon }_0}-{\left(\frac{{\epsilon }_{\mathrm{c}}(y)}{{\epsilon }_0}\right)}^2\right],

where σ_t(y) and ε_t(y) represent the compressive stress and strain at position y of the lining section, respectively, and ε₀ denotes the elastic limit strain under compression. For C30 concrete, ε₀ is taken as 0.002 [48]. Considering the effects of crack developing and FRP strengthening, the section moment calculation formula for the vault section (y-section) is derived as follows.

(5)\begin{array}{rl}{M}_0=& {\int }_0^{x_{\mathrm{c}}}{\sigma }_{\mathrm{c}}\left(y\right)ty\mathrm{d}y+{\int }_0^{h-x_{\mathrm{c}}}\frac{u_1}{h-x_c}{\epsilon }_{\mathrm{t}}{E}_{\mathrm{c}}ty\mathrm{d}y\\ & +2{\epsilon }_{\mathrm{s}}{E}_{\mathrm{s}}{A}_{\mathrm{s}}\left(h_0-x_{\mathrm{c}}\right)+u_2{\epsilon }_{\mathrm{F}}{E}_{\mathrm{F}}{A}_{\mathrm{F}}\left(h-x_{\mathrm{c}}\right),\end{array}

where t and h are the width and height of the cross-section, respectively, x_c is the distance from the neutral axis to the top edge of the lining specimen, E_c, E_s, and E_F are the elastic modulus of concrete, rebar and FRP, respectively, ε_s represents the strain at the position of the tensile rebar. A_s and A_F are the cross-sectional areas of steel rebars and FRP sheet respectively. ε_t, ε_F are the strains of the concrete and FRP material at the bottom edge of the lining. u₁ is the cracking factor, and u₂ is the reinforcement factor. The cracking factor u₁ is directly related to the cracking process of concrete structure: before the load reaches the first-cracking load (uncracked), the contribution of tensile stress borne by the concrete must be taken into account in the calculation of the bending moment (u₁ = 1). When the cracks develop upwards, the tensile force of the concrete below the neutral axis is ignored (u₁ = 0). Referring to the established study [19], the expression of u₁ is:

(6)u_1=\{\begin{array}{ll}1,& P\le P_{\mathrm{c}\mathrm{r}},\\ 0,& P>P_{\mathrm{c}\mathrm{r}}.\end{array}

u₂ is the reinforcement factor, in the area strengthened by FRP, u₂ = 1, and in the unreinforced area, u₂ = 0. Also, taking into account the effect of cracking and FRP strengthening on the lining’s force behavior, the neutral axis position x_c is corrected with reference to Eurocode 2 [23]:

(7)x_{\mathrm{c}}=\{\begin{array}{ll}h/2,& P⩽P_{\mathrm{c}\mathrm{r}},\\ {\displaystyle \frac{h_0{\epsilon }_{\mathrm{c}}}{\left({\epsilon }_{\mathrm{c}}+{\epsilon }_{\mathrm{s}}\right)}},& P>P_{\mathrm{c}\mathrm{r}}.\end{array}

Further, according to the established studies [48,49], the moment M_j, shear force Q_j of any cross-section of the lining structure under pressure at the vault can be derived based on the vault cross-section (y-section).

(8)\begin{array}{c}{M}_{\mathrm{j}}=M_0+y_j{N}_0+x_j{Q}_0,\\ Q_{\mathrm{j}}=Q_0\sin {\phi }_j+N_0\cos {\phi }_j,\end{array}

where N₀ and Q₀ denote the axial and shear forces in the y-section, respectively.

(9)\begin{array}{l}{N}_0=F/2\cos {\beta }_0,\\ Q_0=F/2,\end{array}

where F is the implemented load and β₀ is the angle of the lining end foot section. Further, the ratio of section moment to curvature reflects in real time the section stiffness at that moment, and this parameter is widely used to analyze the damage process of concrete lining [50]:

(10)K_j=M_j/{\rho }_j,

where K_j is the section stiffness and ρ_j is the curvature. According to Zhao et al. [19], K_j could be calculated as follows.

(11){\rho }_j=1/R=\frac{2\sin \left(\arctan {\displaystyle \frac{2f}{L_0}}\right)}{L_0},

where L₀ is the distance between the two ends of the lining and f is the deflection of the lining vault. Calculations were performed by inputting the above theoretical equations based on the force, displacement and strain data obtained from the monitoring system. To ensure the efficiency and accuracy of the calculation, the time interval is 1 min (2 mm).

The crack-section bending moments of linings are shown in Fig. 14. For all linings, the crack-section bending moments increase gradually with loading displacement (δ). The growth trend is not a standard linear pattern due to micro-cracks affecting moment redistribution. When loaded to δ ≈ 6 mm, the crack-section moment of the full-span strengthened lining (L₂) begins to be greater than that of the unilaterally strengthened lining (L₁). The excessive bending moment at the crack-section leads to the rapid development of pre-crack, and the cloud map based on DIC analysis (Fig. 8(b)) precisely confirms this conclusion. When loaded to δ ≈ 12 mm, the full-span strengthened lining (L₂) failed due to reaching its ultimate bending moment capacity, followed by an abrupt drop in sectional bending moment. Due to the slower growth ratio of the crack-section moment, the unilaterally strengthened lining (L₁) exhibits better deformability.

The variation of crack-section stiffness with displacement is plotted in Fig. 15. The crack-section stiffness of all the linings degrades gradually with the loading process. The crack-section stiffness of both strengthened linings (L₁ and L₂) was less than that of the intact lining (L₀), which was caused by the pre-cracks. It can be seen that when cracking occurs in the lining, neither unilateral strengthening nor full-span strengthening can fully restore the section stiffness. Although the crack-section stiffness of the strengthened linings (L₁ and L₂) has similar trends, they still show different characteristics. At the beginning of loading (δ = 2 mm), the crack-section stiffness of the full-span strengthened lining (L₂) is larger than that of the unilaterally strengthened lining (L₂), which shows that the larger reinforcement area leads to larger initial section stiffness. However, the decay rate of stiffness for the full-span strengthened lining (L₂) was significantly greater than that of the unilaterally strengthened lining (L₁). When loaded to δ ≥ 10 mm, the crack-section stiffness of the full-span strengthened range lining (L₂) begins to be less than that of the unilaterally strengthened lining (L₁). Due to the excessive rate of stiffness decay, the full-span strengthened range lining (L₂) fails earlier because it cannot withstand the section moment. Although the full-span strengthening resulted in greater initial stiffness, the full-span strengthening was more likely to cause stress concentrations on the pre-cracks than the unilateral strengthening, resulting in greater bending moments and greater decay rates of section stiffness. For tunnel linings with unilateral cracks, it should be noted that excessive FRP strengthening range may adversely affect the reinforcement efficiency. For unilaterally cracked tunnel linings, it is recommended to monitor and analyze the significantly affected areas of the cracks and then apply local reinforcement.

5 Conclusions

This experimental study aims to evaluate the mechanical performance of unsymmetrically cracked linings applied unilateral and full-span FRP strengthening. The failure modes and cracking evolution were analyzed, and the mechanical performance and strengthening mechanism were also discussed. The main conclusions are as follows.

1) For unsymmetrically cracked linings strengthened with FRP, the unilaterally strengthened lining exhibited a 30.2% higher load-bearing capacity. Although the strengthened linings exhibited similar brittle failure characteristics, the unilaterally strengthened lining demonstrated superior performance in terms of toughness. The full-span strengthened lining demonstrated inferior energy dissipation performance due to the over-concentration of energy release at the top of the pre-crack.

2) The intact lining developed arcuate cracks that propagated along the bottom steel rebar. The FRP-strengthened linings exhibited similar cracking patterns, with concentrated cracking zones developing at the top of the pre-cracks. However, the full-span strengthened lining demonstrated approximately 60% greater cracked area compared to the unilaterally strengthened lining. From the AE monitoring data, it can be seen that there is a comparable number of tensile and shear cracks in all the linings.

3) The full-span strengthened lining distributed higher bending moments in the crack-section than the unilaterally strengthened lining, resulting in a smaller ultimate load-bearing capacity. Although full-span strengthening results in greater initial sectional stiffness, stress concentrations lead to excessive stiffness decay rates, resulting in accelerated lining failure. For unilaterally cracked linings, an extra range of reinforcement is detrimental to reinforcement efficiency.

Extensive research on FRP strengthening for unilaterally cracked linings, considering complex loads, humidity and service time, still needs to be carried out based on extensive model specimens.

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