Structural behavior of reinforced concrete segments of tunnel linings strengthened by a steel-concrete composite

Structural behavior of reinforced concrete segments of tunnel linings strengthened by a steel-concrete composite

Composites Part B 178 (2019) 107444 Contents lists available at ScienceDirect Composites Part B journal homepage: www.elsevier.com/locate/composites...

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Composites Part B 178 (2019) 107444

Contents lists available at ScienceDirect

Composites Part B journal homepage: www.elsevier.com/locate/composites

Structural behavior of reinforced concrete segments of tunnel linings strengthened by a steel-concrete composite Jiao-Long Zhang a , Xian Liu a ,∗, Tian-Yu Ren a , Yong Yuan a , Herbert A. Mang a,b a b

Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Siping Road 1239, 200092 Shanghai, China Institute for Mechanics of Materials and Structures, Vienna University of Technology, Karlsplatz 13, 1040 Vienna, Austria

ARTICLE Keywords: Layered structures Mechanical properties Analytical modeling Mechanical testing Tunnel segments

INFO

ABSTRACT Structural defects of existing segmental tunnel linings call for strengthening techniques. This was the motivation for conducting real-scale bearing-capacity tests and computational analysis of strengthened segments by means of a steel-concrete composite (SCC). Two load cases were considered: (i) a sagging moment, resulting in intrados and extrados fibers in tension and compression, respectively, and, conversely, (ii) a hogging moment. The experimental observation has shown that, for both load cases, the strengthened segments failed in a ductile fashion. In case of sagging moments, the strengthening effect of the SCC is characterized by the fact that the yield moment and the post-cracking stiffness of the strengthened segments with a 40-mm-thick SCC are by 393 % and 315 %, respectively, larger than those of the unstrengthened segments. As for hogging moments, the strengthening effect of the SCC is considerable, albeit less significant than that for sagging moments. The derived formulae allow for description of the structural behavior of the strengthened segments and for quantification of the tractions at the interface between the steel shell and the concrete. Two main conclusions are drawn from the computational analyses: (i) Concerning the increase of the strengthening effect, in case of a sagging moment, an increase of the thickness of the steel shell is more efficient than that of the new concrete. The converse situation occurs in case of a hogging moment. (ii) For a sagging moment, the interface is in shear-tension, whereas for a hogging moment it is in shear-compression.

1. Introduction Segmental tunnel linings may be subjected to large deformations. The reinforced concrete (RC) segments may have cracks, and there may be leakage. These defects are consequences of the structural degradation with increasing years of service or of exceptional loads, resulting from constructions in the vicinity of tunnels [1]. They endanger the structural safety of the segmental tunnel linings. This explains the need for research on strengthening of segmental tunnel linings. The purpose of such strengthening is basically to improve the bearing capacity and/or the structural stiffness of the linings. Improvement of their structural stiffness is beneficial to restrain their deformations. This is of great importance for tunnel linings, the convergences of which have grown so large that the traffic running through the tunnel is negatively affected. Notably, an improvement of the structural stiffness can be unfavorable to tunnel linings under squeezing condition [4] or in active faulted zones [5]. As regards improvement of both the bearing capacity and the structural stiffness, several strengthening techniques have been used, see

Fig. 1. Epoxy-bonded steel rings are key elements of the most often applied technique, see e.g. [6,7], in China, [8], in Japan, and [9], in England. A real-scale test of a segmental tunnel ring, strengthened by epoxy-bonded steel rings, was carried out by Liu et al. [2]. It has shown that the epoxy-bonded steel rings resulted in both an increase of the structural stiffness and of the bearing capacity of the RC segmental tunnel rings. However, bonding of the steel segments to the inner surface of the tunnel lining requires a hoisting machine, because of the relatively great self-weight of the steel shells. Moreover, welding of steel segments to their neighboring segments is needed in order to obtain closed steel rings. Therefore, from the viewpoint of construction, this strengthening technique is inefficient. This was the motivation for the development of new strengthening methods. Liu et al. [3,10] proposed the use of epoxy-bonded filament wound profiles (FWP), see Fig. 1(b). Such a profile consists of four rectangular steel tubes. By means of welding, these tubes are integrated to a profile. The integrated profile is then cladded with carbon fiber cloth, resulting in a FWP. Its relatively light weight facilitates handling of

∗ Corresponding author. E-mail addresses: [email protected] (J.-L. Zhang), [email protected] (X. Liu), [email protected] (T.-Y. Ren), [email protected] (Y. Yuan), [email protected] (H.A. Mang).

https://doi.org/10.1016/j.compositesb.2019.107444 Received 18 June 2019; Received in revised form 26 August 2019; Accepted 12 September 2019 Available online 14 September 2019 1359-8368/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Photograph of real-scale tests of a segmental tunnel lining after strengthening by (a) epoxy-bonded steel rings [2] and (b) epoxy-bonded filament wound profiles (FWP) [3]; failure of bond at the interface (c) between the RC ring and the steel ring [2] and (d) between the RC ring and the FWP [3].

bridges [15,16], the aim of the present paper is to develop a method for strengthening of segmental tunnel linings with the help of SCC. The steel-concrete composite shown in Fig. 2 consists of the outermost steel plate, the new concrete filled between the old concrete and the steel plate, the rebars partly embedded in the old concrete, and the headed studs welded to the steel plate. Because the steel plate serves as a permanent formwork, this strengthening method is efficient from the viewpoint of construction. It is also quite efficient in the sense of improving the bearing capacity and the stiffness of RC beams, because of the relatively high yield stress and Young’s modulus of the steel plate [17]. In addition, the strengthened RC beams usually fail in a ductile mode. This mode is preferred in case of failure of engineering structures. These conclusions, drawn from research on straight RC beams strengthened by SCC, hold the promise for application of such a strengthening technique to curved beams, as occur e.g., in circular tunnel linings. Given that the structural behavior of curved beams is different from that of straight beams [18], the extension of the strengthening method for straight beams towards curved tunnel segments is not straightforward. This suggests a scientific investigation of tunnel segments strengthened by this method. The present paper is a pioneering work concerning the application of SCC on strengthening of segmental tunnel linings. Notably, SCC segments have been successfully used in new construction of particular segmental tunnel linings. They refer to deep tunnels subjected to high water pressure and ground pressure from outside [19], to hydraulic tunnels subjected to high water pressure from inside [20], and to tunnels with large rectangular cross-sections [21].

the profile. After installation of the FWP onto the inner surface of the segmental tunnel lining, high-strength mortar is grouted into the cavity of the FWP. This strengthening technique is very efficient from the viewpoint of construction. The failure of the tested segmental tunnels rings, strengthened both by epoxy-bonded steel rings [2] and epoxybonded FWP [3], was brittle. It was characterized by failure of the bond at the interface between the RC ring and the steel ring and the FWP ring, respectively. In addition, Jiang et al. [11] proposed the so-called FRP-PCM method in order to strengthen degraded tunnel linings. FRP is the abbreviation of Fiber-Reinforced Plastic and PCM the one of Polymer Cement Mortar. The FRP-PCM method is based on strengthening tunnel linings by FRP grids, embedded in PCM. The feasibility of the FRP-PCM method of strengthening tunnel linings was assessed by means of bending tests of strengthened straight-beams and numerical investigations of strengthened tunnel linings [11]. A similar research process is ongoing with the strengthening method, making use of textile-reinforced concrete (TRC), see e.g. [12]. The authors have carried out tests of columns, subjected to eccentric compression. This is a typical load case of the cross-section of the tunnel linings. It was shown that TRC resulted in a substantial increase of the bending resistance of the columns. As regards the strengthening effect either on individual tunnel segments or on entire segmental tunnel linings, direct experimental evidence from real-scale tests or engineering practice is still lacking. The search for new methods of strengthening of segmental tunnel linings is still a topic of ongoing research. Inspired by the successful application of steel-concrete composites (SCC) on strengthening of straight RC beams in buildings [13,14] and 2

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Fig. 2. Illustration of a beam strengthened by a steel-concrete composite (a) at its bottom surface, mainly increasing the bending resistance, (b) at its side surface, mainly increasing the shear resistance, and (c) with a U-shaped cross-section, increasing both the bending and the shear resistance [13]. Table 1 Mechanical properties of the used materials.

However, making use of SCC for strengthening of existing segmental tunnel linings is a new trail. The method, verified for segmental tunnel linings, can also be used for strengthening of other curved structures, such as e.g. RC arch bridges and vaults of underground space. In this paper, real-scale tests of tunnel segments are described. The segments were a priori strengthened by SCC. In other words, the strengthened RC segments were unloaded. Both the segments and the SCC carry the external loads. This represents a step towards application of SCC to strengthening of segmental tunnel linings in service. In addition, computational analyses of the strengthened segments were performed in order to gain insight into the structural behavior of these segments.

Material

Strength class

Young’s modulus 𝐸 [GPa]

𝑓𝑐 of concrete or 𝑓𝑦 of steel [MPa]

𝑓𝑡 [MPa]

Old concrete New concrete Steel rebar Steel shell Embedded rebar Headed stud Chemical anchor bolt

C55 C60 HRB400 Q295 HRB400 – –

35 36 200 200 200 200 200

35.5 38.5 400 295 400 320 400

2.74 2.85 – – – – –

2.2. Specimens

The highlights of this paper in the area of strengthening of segmental tunnel linings are explained in the following. One of them is the material used for strengthening, i.e. SCC. It is different from the ones mentioned in the literature, see e.g. [2] for the steel plate, [3] for FWP, [11] for FRP-PCM, and [12] for TRC. Another highlight is the experimental specimens, i.e. real-scale tunnel segments. Their curvature has raised the need for scientific research on tunnel segments strengthened by SCC, although SCC has been widely used for strengthening of straight beams and plates.

The pre-strengthened RC segment is a component of a circular tunnel ring. Its radius, 𝑅, is equal to 2.925 m. Only one half of a segment is shown in Fig. 3, since the tested segments are symmetric with respect to the vertical line at midspan. The span of the pre-strengthened segments, 𝐿, amounts to 2.952 m, the width, 𝐵, to 600 mm, and the thickness, 𝐻, to 350 mm, see Fig. 4. A layer of steel-concrete composite is added to the intrados of the RC segments, resulting in strengthened segments. Two sets of altogether eight specimens were tested, see Table 2 for detailed information. The first test set deals with segments subjected to a sagging moment, resulting in tension of the intrados fibers and compression of the extrados fibers. The second test set deals with segments subjected to a hogging moment, resulting in compression of the intrados fibers and tension of the extrados fibers. The specimen number consists of three characters. The first one indicates the mode of bending to which the segment is subjected. ‘‘S’’ stands for a sagging moment and ‘‘H’’ for a hogging moment. The remaining characters of the specimen number, i.e. 00, 40, 60, and 80, represent the thickness of the SCC layer, measured in mm. The strengthening procedure consists of the following steps:

The paper is organized as follows: The experimental program is described in Section 2. Experimental results are presented and discussed in Section 3. Section 4 is devoted to computational analyses of the strengthened segments. Section 5 contains the conclusions drawn from the present study.

2. Experimental program 2.1. Materials

(1) Holes of around 125 mm deep are drilled into the produced fragments and cleared, see Fig. 5(a). (2) The inner surface is roughened and the produced fragments are cleared, see Fig. 5(b). (3) The curved rebars are embedded in the drilled holes by means of filling the remaining space in the holes by epoxy, see Fig. 5(c) for segments after embedding of the rebars. (4) The headed studs are welded to the steel shell, see Fig. 5(d). (5) The steel shell is placed at the intrados surface by means of chemical anchor bolts, see Fig. 5(e). The holes of the steel shell and deep into the RC segment are used to embed the chemical anchor bolts. (6) The gap between the steel shell and the intrados surface of the RC segments is filled by the new concrete, see Fig. 5(f).

The strength class of the old concrete in reinforced segments is C55. According to the specifications in the Chinese code [22], the uniaxial compressive strength, 𝑓𝑐 , the uniaxial tensile strength, 𝑓𝑡 , and Young’s modulus of concrete, 𝐸𝑐 , are equal to 35.5 MPa, 2.75 Pa, and 35 GPa, respectively. The strength class of the new concrete, used in steelconcrete composites, is C60. The uniaxial compressive strength, the uniaxial tensile strength, and Young’s modulus of the new concrete, are equal to 38.5 MPa, 2.85 MPa, and 36 GPa, respectively. As for the reinforcement and the embedded rebars, Chinese hot-rolled steel rebars, with the specification HRB400 are used. The yield stress and Young’s modulus of the rebars, 𝑓𝑦 and 𝐸𝑠 , amount to 400 MPa and 200 GPa, respectively. The Chinese steel grade of the steel shell is Q295. Table 1 contains the mechanical properties of the used materials. 3

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Fig. 3. Sketch of one half of a strengthened segment. 𝑡 denotes the thickness of the strengthening layer, i.e. of the SCC.

Table 2 Specimen numbers, bending scenarios, and thicknesses of the SCC of the tested segments. No.

Bending scenario

Thickness of the SCC 𝑡 = 𝑡𝑠𝑠 + 𝑡𝑛𝑐 [mm]

Thickness of the steel shell 𝑡𝑠𝑠 [mm]

Thickness of the new concrete 𝑡𝑛𝑐 [mm]

S00 S40 S60 S80

sagging sagging sagging sagging

0 40 60 80

0 8 10 10

0 32 50 70

H00 H40 H60 H80

hogging hogging hogging sagging

0 40 60 80

0 8 10 10

0 32 50 70

moment of the segment, 𝑀, experienced in the region between the two loading points, reads as 1 𝑃𝑙 . (1) 2 2 The tests are carried out in a force-controlled fashion. Experimental measurements include the mid-span deflection and the strains of the concrete, the reinforcement, and of the steel shell, see Fig. 7 for the layout of the monitoring points. As for the segments S40 and H40, the relative displacements between the steel shell and the RC segment in the radial and the circumferential direction are monitored at five points with even spacing. In addition, the strains of the two chemical anchor bolts, denoted as CAB1 and CAB2, in the vicinity of the midspan, are measured during the tests. The deflections and the relative displacements are measured by Linear Variable Differential Transformers (LVDTs), with a precision of 0.1 mm. The strains are measured by means of strain gauges, with a precision of 1 × 10−7 . In addition, cracking of concrete and debonding of the new and the old and of the new concrete and the steel shell are observed during the tests. 𝑀=

Fig. 4. Cross-section of strengthened segment; *the number of the chemical anchor bolts is 2 for specimens S40 and H40 and 3 for the others.

2.3. Test set-up and measurement equipment

3. Experimental results and discussions

Two load cases are considered. They refer to the sagging and the hogging moment. The experiments are carried out in the form of four-point bending tests, see Fig. 6 for the test set-up. The hydraulic jack produces a vertical force 𝑃 , imposed, via a steel beam, onto the segments, resulting in two point loads. Thus, the maximum bending

3.1. Structural tests of the segments subjected to a sagging moment 3.1.1. Failure process The failure process of the unstrengthened segment S00 is one of a typical balanced-reinforced concrete beam. It starts with cracking of the 4

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Fig. 5. Illustration of the procedure of strengthening the RC segments: (a) drilling holes (b) roughening the inner surface, (c) RC segment after embedding of the rebars, (d) welding the headed studs to the steel shell, (e) installing the steel shell, and (f) filling the gap between the steel shell and the RC segment by means of casting the new concrete.

Fig. 6. Set-up of four-point bending tests of the segments subjected to (a) a sagging moment and (b) a hogging moment.

intradros concrete at 𝑀 = 61 kNm, see Fig. 8(a), followed by yielding of the intrados reinforcement at 𝑀 = 97 kNm. This is the case when the measured strain exceeds the elastic limit strain 𝜀𝑦 = 𝑓𝑦 ∕𝐸𝑠 = 2 × 10−3 , see the square in Fig. 9(a). The failure process ends with crushing of the extrados concrete at 𝑀 = 130 kNm, see Fig. 8(b). All of the strengthened segments have virtually undergone the same failure process, but different from that of the unstrengthened segment as described above. Without loss of generality, the detailed failure process of segment S40 is described in the following. Bending-induced cracks of the new concrete occurred at midspan, at 𝑀 = 155 kNm. The cracks propagated in the thickness direction through the old concrete, see Fig. 10(a). Then, the debonding-induced crack propagated along the interface of the new concrete and the steel shell, as was observed in the midspan region, at 𝑀 = 171 kNm, see Fig. 10(b). This was followed by a debonding-induced crack, propagating along the interface between the new and the old concrete in the same region, at 𝑀 = 367 kNm, see Fig. 10(c). Thereafter, the steel shell and the intrados reinforcement began to yield successively, at 𝑀 = 461 kNm and 𝑀 = 478 kNm,

respectively. This was indicated by the monitored strains exceeding the elastic limit strain of the steel shell and the reinforcement, i.e. 1.675 × 10−3 and 2 × 10−3 , see the circle and the square in Fig. 9(b). Finally, the bearing capacity of the segment was reached at 𝑀𝑢 = 495 kNm. This was accompanied by fracturing of the chemical anchor bolts in the midspan region, see Fig. 10(d). Notably, this is still a ductile mode of failure. This is of great importance for the safety of the people using the structure and for the structure itself. This also explains why the proposed method for strengthening of tunnel linings is superior to the existing methods, such as e.g. epoxy-bonded steel rings [2] or FWP [3]. The bending moments, associated with progressive failure of the tested segments, are listed in Table Table 3. 𝑀𝑐𝑟 , 𝑀𝑠𝑛 , 𝑀𝑛𝑜 , 𝑀𝑦𝑠𝑠 , 𝑀𝑦𝑠𝑖 , and 𝑀𝑢 denote the bending moments associated with the onset of bending-induced cracking, debonding of the steel shell and the new concrete, debonding of the new and the old concrete, yielding of the steel shell, yielding of the intrados reinforcement, and with the ultimate-limit bearing capacity of the segments, respectively. 5

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Fig. 7. Layout of the points for monitoring of strains and displacements.

Fig. 8. Observed failure modes of the unstrengthened segment S00 subjected to a sagging moment: (a) onset of cracking of the intrados concrete and (b) compressive crushing of the extrados concrete. Table 3 Bending moments, associated with progressive failure of segments subjected to a sagging moment. No.

𝑀𝑐𝑟 [kNm]

𝑀𝑠𝑛 [kNm]

𝑀𝑛𝑜 [kNm]

𝑀𝑦𝑠𝑠 [kNm]

𝑀𝑦𝑠𝑖 [kNm]

𝑀𝑢 [kNm]

S00 S40 S60 S80

61 155 170 186

– 171 170 186

– 367 462 552

– 461 558 644

97 478 611 686

130 495 667 730

3.1.2. Effect of the thickness of the SCC on the structural behavior of the strengthened segments The structural properties of interest are (i) the bending moment 𝑀𝑐𝑟 , corresponding to the initiation of cracking of concrete at the elastic limit of the segments, (ii) the yield moment 𝑀𝑦𝑠 , herein associated with yielding of the intrados reinforcement, referring to the designed bearing capacity of the segments, (iii) the maximum bending moment 𝑀𝑢 , referring to the ultimate-limit bearing capacity of the segments, (iv) the stiffness up to the elastic limit state, denoted as 𝐾1 , and (v) the stiffness from the onset of cracking up to yielding of the reinforcement, denoted as 𝐾2 . 𝐾1 and 𝐾2 are estimated by two simple formulae, based on a bilinear deflection – bending-moment diagram, assumed to hold for the segments. They read as 𝐾1 =

𝑀𝑐𝑟 , 𝛿𝑐𝑟

𝐾2 =

𝑀𝑦𝑠 − 𝑀𝑐𝑟 𝛿𝑦𝑠 − 𝛿𝑐𝑟

,

(3)

where 𝛿𝑐𝑟 and 𝛿𝑦𝑠 denote the midspan deflections, associated with the onset of bending-induced cracking and with yielding of the reinforcement, respectively. The investigated structural properties of the segments, subjected to a sagging moment, are strongly improved by the SCC, see Fig. 11(a) for the deflection – bending-moment diagrams, obtained from the experiments, and Fig. 11(b) for the relative increase of 𝑀𝑐𝑟 , 𝑀𝑦𝑠 , 𝑀𝑢 , 𝐾1 and 𝐾2 of the strengthened segments, as compared to the unstrengthened segment S00. For instance, the values of 𝑀𝑐𝑟 , 𝑀𝑦𝑠 , 𝑀𝑢 , 𝐾1 and 𝐾2 of S40 are 154 %, 393 %, 281 %, 44 %, and 315 % larger than those of S00. This underlines the usefulness of the developed method for strengthening of the tunnel segments subjected to a sagging moment. The strengthening effect of SCC results from two contributions. On the

(2) 6

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Fig. 9. Experimentally measured strains of the reinforcement for (a) S00, and of the reinforcement, the steel shell, and the concrete for (b) S40, (c) S60, and (d) S80, at midspan of the segments subjected to a sagging moment. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the segment to yielding of the chemical anchor bolts, i.e. from 30 % to 60 % of the bearing capacity, results in a rather small increase of the relative displacements between the steel shell and the RC segment, compare Fig. 13(b) with Fig. 13(a). However, the load increment from yielding of the chemical anchor bolts to the ultimate limit state of the segment, i.e. from 60 % to 100 % of the bearing capacity, results in a significant increase of the relative displacements between the steel shell and the RC segment, compare Fig. 13(c) with Fig. 13(b). Debonding of the steel shell reduces the composite effect of the original RC segment and the SCC. In addition, yielding of the chemical anchor bolts occurs earlier than that of the steel shell, as follows from a comparison of Figs. 12(a) and 12(b). The ultimate bearing capacity of the segments is reached when the chemical anchor bolts are fractured. It is concluded that the bearing capacities of the strengthened segments, subjected to sagging moments, are governed by the mechanical behavior of the chemical anchor bolts.

one hand, strengthening by SCC results in an increase of the height of the cross-section of the segments. On the other hand, the steel shell results in an increase of the area of the effective reinforcement in tension. The increase of the strengthening effect from S60 to S80 is mainly resulting from the first contribution, since the thickness of the steel shell of S80 and S60 is constant. The increase of the post-cracking stiffness 𝐾2 with increasing thickness of SCC is more significant than that of the pre-cracking stiffness 𝐾1 , see the violet and the green bars in Fig. 11(b). This is a consequence of the second contribution of the steel shell. 3.1.3. Composite effect between the original RC segment and the SCC The interface between the steel shell and the new concrete as well as the one between the new and the old concrete are weak zones. This was the motivation for using headed studs, connecting the steel shell and the new concrete, embedded rebars, connecting the new and the old concrete, and chemical anchor bolts, connecting all of these three constituents. The role of these connectors, i.e. the headed studs, the embedded rebars, and the chemical anchor bolts, is to transmit the forces between these constituents. This results in a progressive increase of the strains of the steel shell with increasing external load, see Fig. 12(a). The monitored strains of the chemical anchor bolts indicate that they start yielding at midspan of the segment S40 at a value of the bending moment 𝑀 around 300 kNm, when the measured strain exceeds the elastic limit strain 2 × 10−3 , see Fig. 12(b). Yielding of the chemical anchor bolts results in significant debonding of the steel shell. This is underlined by the fact that the load increment from cracking of

3.2. Structural tests of the segments subjected to a hogging moment 3.2.1. Failure process Initial damage of the unreinforced segment H00 is characterized by cracking of the extradros concrete at 𝑀 = 55 kNm. The extrados reinforcement then yields in tension at 𝑀 = 111 kNm. This occurs when the measured strain exceeds its elastic limit 𝜀𝑦 = 2 × 10−3 , see the black circle in Fig. 14(a). Thereafter, the intrados reinforcement yields in tension at 𝑀 = 131 kNm, This is the case when the measured strain exceeds the elastic limit 𝜀𝑦 = 2 × 10−3 , see the blue square in 7

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Fig. 10. Observed failure modes of the strengthened segment S40 subjected to a sagging moment: (a) bending-induced crack of the concrete, propagating through the new and the old concrete, (b) debonding-induced crack, propagating along the interface of the new concrete and the steel shell, (c) debonding-induced crack, propagating along the interface of the new and the old concrete, and (d) fracture of the chemical anchor bolts.

Fig. 11. Comparison of the strengthened to the unstrengthened segments subjected to a sagging moment: (a) deflection – bending-moment diagrams, obtained from experiments, and (b) relative increase of the cracking moment 𝑀𝑐𝑟 , the yield moment 𝑀𝑦𝑠 , the ultimate moment 𝑀𝑢 , the stiffness up to the elastic limit state, 𝐾1 , and the stiffness from the onset of cracking up to yielding of the reinforcement, 𝐾2 , compared to the unstrengthened segment S00. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14(a). Finally, the bearing capacity of the segment is reached when the intrados concrete fails by crushing at 𝑀 = 156 kNm, see Fig. 15(a) for the failure pattern of H00.

All of the three strengthened segments, subjected to a hogging moment, have undergone virtually the same failure process, characterized by a ductile mode. Such a process consists of cracking of concrete and 8

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Fig. 12. Experimentally measured strains of (a) the steel shell and (b) the chemical anchor bolts of the segment S40; S1, S2, S3, S4, S5, S6, and S7 denote the number of the strain-monitoring points of the steel shell, and CAB1 and CAB2 stand for the strain-monitoring points of the chemical anchor bolts, see Fig. 7. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Experimentally measured relative displacements between the steel shell and the RC of the segment S40, as (a) bending-induced cracks occur, (b) the chemical anchor bolts yield, and (c) the chemical anchor bolts are fractured, associated with 30 %, 60 %, and 100 % of the ultimate bearing capacity, respectively; the circles stand for the positions of the displacement-monitored points, see Fig. 7. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 4 Bending moments, associated with progressive failure of the segments subjected to a hogging moment. No.

𝑀𝑐𝑟 [kNm]

𝑀𝑦𝑠𝑒 [kNm]

𝑀𝑦𝑠𝑖 [kNm]

𝑀𝑢 [kNm]

H00 H40 H60 H80

55 65 80 81

111 148 153 153

131 186 186 206

156 216 234 264

3.2.2. Comparison of the strengthening effect of the SCC on the segments for a hogging moment and a sagging moment The SCC results in an improvement of the structural properties of the segments subjected to a hogging moment, see Fig. 16(a). The strengthening effect is evaluated, analogous to the load scenario in case of a sagging moment, see Section 3.1.2, in the form of the percentage increase of the cracking moment 𝑀𝑐𝑟 , the yield moment 𝑀𝑦𝑠 , herein referring to yielding of the extrados reinforcement, and the pre-cracking and the post-cracking structural stiffness, 𝐾1 and 𝐾2 , respectively, see Fig. 16(b). The percentage increase of 𝑀𝑐𝑟 , 𝑀𝑦𝑠 , 𝑀𝑢 , and 𝐾2 of segments subjected to a sagging moment is an order of magnitude larger than that of segments subjected to a hogging moment, as follows from a comparison of the blue, orange, yellow, and green bars in Fig. 11(b) with those in Fig. 16(b). However, the percentage increase of 𝐾1 for both load cases is approximately the same, as follows from a comparison of the violet bars in Fig. 11(b) with those in Fig. 16(b). The magnitude of the differences is a consequence of the influence of the steel shell on the strengthening effect of the SCC. For a sagging moment, the steel shell is in tension, see Fig. 12(a). In this case, it contributes to an increase (i) of the cross-sectional height of the segments and (ii) of the reinforcement in tension, resulting in an increase of the loadcarrying capacity. The first contribution is independent of the load case. Therefore, the percentage increase of the elastic structural-stiffness 𝐾1 of the segments, strengthened by SCC of the same thickness, is the same for both load cases. The second contribution, however, only

yielding of extrados reinforcement, similar to that of the unstrengthened segment H00, yielding of intrados reinforcement, and fracture of extrados reinforcement, see the corresponding moments in Table 4. Without loss of generality, the detailed failure process of the segment H40 is described in the following. At first, the segment is damaged by cracking of the extradros concrete at 𝑀 = 65 kNm, 15 % larger than the cracking moment of H00. Then, the extrados reinforcement yields in tension at 𝑀 = 148 kNm, 33 % larger than the yield moment of H00. This occurs when the measured strain exceeds its elastic limit 𝜀𝑦 = 2 × 10−3 , see the black circle in Fig. 14(b). Thereafter, the intrados reinforcement yields in tension at 𝑀 = 186 kNm. This occurs when the measured strain exceeds the elastic limit 𝜀𝑦 = 2 × 10−3 , see the blue square in Fig. 14(b). The bearing capacity of the segment is reached when the extrados reinforcement fails by fracture at 𝑀 = 216 kNm, 38 % larger than the bearing capacity of H00, see Fig. 15(b) for the failure pattern of H40. The above mentioned percentages indicate the strengthening effect of SCC on the segments subjected to hogging moments. 9

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Fig. 14. Experimentally measured strains of the reinforcement and the concrete for (a) H00 and of the reinforcement, the steel shell, and the concrete for (b) H40, (c) H60, and (d) H80, at midspan of these segments, subjected to a hogging moment. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Observed failure modes of the segments subjected to a hogging moment: failure pattern (a) of the unstrengthened segment H00 and (b) of the strengthened segment H40.

prevails for a sagging moment, because the steel shell is in compression for a hogging moment, see Fig. 17(a). The strengthening effect of the steel shell in tension is more efficient than that in compression. From the above discussion, it is concluded that the effectiveness of the proposed method for strengthening tunnel segments, subjected to sagging moments, is much greater than that in case of hogging moments.

3.2.3. Comparison of the composite effect between the original RC segment and the SCC for a hogging moment and a sagging moment The discussion of the experimental results of the tests for a sagging moment resulted in insight into the stresses transmitted between the SCC and the RC segment. This is of great importance, because considerable debonding of the steel shell is observed, and the ultimate 10

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Fig. 16. Comparison of strengthened to unstrengthened segments subjected to a hogging moment: (a) deflection–bending moment diagrams, obtained from experiments and (b) relative increase of the cracking moment 𝑀𝑐𝑟 , the yield moment 𝑀𝑦𝑠 , the ultimate moment 𝑀𝑢 , the stiffness up to the elastic limit state, 𝐾1 , and the stiffness from the onset of cracking up to yielding of the reinforcement, 𝐾2 , with respect to the unstrengthened segment H00. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 17. Experimental measurements of (a) the strains of the steel shell of the segment H40 and (b) the relative displacement between the steel shell and the RC of the segment, as its ultimate bearing capacity is reached; the circles mark the points at which the displacements are monitored, see Fig. 7. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

bearing capacity is governed by the failure of the chemical anchor bolts, crossing the SCC and the RC segment. However, the relative displacements between the steel shell and the RC of the segment subjected to a hogging moment are very small all the way up to the ultimate bearing capacity, see Fig. 17(b). This is consistent with the experimental observation that the interface between the steel shell and the new concrete as well as the one between the new and the old concrete remains in perfect bond at its ultimate limit state, see Fig. 15(b). These differences provide the motivation for studying the stresses, transmitted between the SCC and the RC segment for both load scenarios, see Section 4.4.

interface between the steel shells and the reinforced concrete segment are quantified by means of equilibrium conditions. 4.1. Assumptions for the derivation of the aforementioned formulae The flexural capacity of the segments, strengthened by the SCC, is evaluated by means of closed-form formulae. Their derivation is based on the following assumptions: • • • •

Plane sections remain plane during deformation. The bond between the RC segment and the SCC is perfect. The RC segments are strengthened before being loaded. The new and the old concrete are treated as a macroscopically homogeneous material with mechanical properties of C55 concrete. • The material behavior assumed for concrete and steel is illustrated in Fig. 18, see Eqs. (A.1) and (A.2) for their constitutive equations.

4. Computational analyses of the flexural capacity of segments strengthened by the SCC The analyses are based on closed-form formulae, which will be derived in the following. They are validated by means of comparing the computational results with the experimental results. With the help of these formulae, the effect of the individual constituents of the SCC on the structural behavior of the strengthened segments is quantified by means of parametric analysis. The forces transmitted across the

The expressions for the bending moment 𝑀 are derived in a curvature-controlled fashion. The prescribed curvature 𝜅 results in a linear distribution of the normal strains along the height of the crosssection, see Figs. 19 and 20. The normal stresses follow from inserting 11

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In Eq. (4), 𝑦 denotes the coordinate axis along the height of the crosssection. The expressions for 𝑁 are given in Appendix A. They are functions of the height of the compression zone of concrete, denoted as ℎ𝑐 in Figs. 19 and 20. ℎ𝑐 is determined from setting 𝑁 = 0 and solving the resulting relation. Finally, the sought bending moment 𝑀 follows from 𝑀=

∫𝐴

𝜎(𝑦) 𝑦 𝑑𝐴.

(5)

The expressions for 𝑀 are also given in Appendix A. 4.2. Comparison of the computational results with the experimental results Specializing Eqs. (4)–(5) for the physical quantities in Table 1 and for geometrical quantities in Table 2 and Fig. 4 allows for computing the bending moment as a function of the curvature, see the blue graphs in Figs. 21 and 22 for the computationally determined diagrams. The red graphs refer to the experimentally determined curvature – moment diagrams, based on strain measurements. In case of a sagging moment the strains of the steel shell, the intrados reinforcement, the extrados reinforcement, and the extrados concrete are measured during the tests, see Fig. 9(b)–(d). The experimentally determined curvatures are obtained by means of linear regression of the strains at the positions of their monitoring. They represent the slopes of the linear distribution of

Fig. 18. Constitutive relations of (a) concrete and (b) steel; 𝑓𝑐 and 𝑓𝑡 denote the compressive and the tensile strength of concrete, respectively, 𝜀𝑐0 = −0.002 and 𝜀𝑡0 = 𝑓𝑡 ∕𝐸𝑐 stand for the elastic-limit strain of concrete in compression and tension, respectively, 𝐸𝑐 denotes Young’s modulus of concrete, and 𝑓𝑦 and 𝜀𝑦 denote the yield stress and the yield strain of steel, respectively.

the normal strains into the constitutive laws, illustrated in Fig. 18. The axial force 𝑁 follows from 𝑁=

∫𝐴

𝜎(𝑦) 𝑑𝐴.

(4)

Fig. 19. Analysis of the normal stress of a segment with a curvature resulting in a sagging moment: (a) structural dimensions, (b) distributions of strains and stresses before cracking of the segment, (c) after its cracking, but before yielding of the steel shell, (d) after yielding of the steel shell, but before yielding of the intrados reinforcement, (e) after yielding of the intrados reinforcement, but before crushing of concrete, and (f) after crushing of concrete. 12

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Fig. 20. Analysis of the normal stress of a segment with a curvature resulting in a hogging moment: (a) structural dimensions, (b) distributions of strains and stresses before cracking of the segment, (c) after its cracking, but before yielding of the extrados reinforcement, (d) after yielding of the extrados reinforcement, but before yielding of the intrados reinforcement, (e) after yielding of the intrados reinforcement, but before crushing of concrete or yielding of the steel shell in compression.

the strains over the height of the cross-section. Analogous to the situation for a sagging moment, the experimentally determined curvatures of the segments subjected to a hogging moment are obtained on the basis of the measured strains of the steel shell, for strengthened segments, or of the extrados concrete, for unstrengthened segments, of the intrados reinforcement, and of the extrados reinforcement, see Fig. 14. The computational results generally agree well, at least qualitatively, with the experimental results, as follows from a comparison of the blue and the red graphs in Figs. 21 and 22. As regards the loading scenario in case of a sagging moment, the computational model overestimates the moment at the onset of yielding of the steel shell, see the circle markers in Fig. 21, with an error up to 13 %, and the moment associated with yielding of the intradros reinforcement, see the squared markers, with an error up to 16 %. These overestimates are the consequence of the deviation of the actual situation from the assumption of perfect bond between the RC segment and the SCC. In reality, however, debonding between the two was observed in the experiments. Nevertheless, this discrepancy is acceptable for engineering design. The computational model underestimates, by 56 %, the cracking moment, see the triangle markers in Fig. 21. The reason for this difference is that the steel shell, located at the intrados side, does not permit observation of the propagation of cracks from the

intrados surface, where the maximum tensile stresses of the concrete occur. The experimentally determined cracking moment is, in fact, associated with the observation of the propagation of cracks on the side surface, see Fig. 10(a). It should be larger than the nominal cracking moment. However, this moment is less important for a strengthened segment subjected to a sagging moment, because cracks that open on the intrados surface are sealed by the steel shell. For a loading scenario in case of a hogging moment, the experimental results imply that the bond between the RC segment and the SCC was perfect up to the ultimate limit state, which is consistent with the assumption for the developed computational model. Therefore, this model provides reliable predictions of the yield moment, associated with yielding of the extrados reinforcement in tension, see the circle markers in Fig. 22. This is of great importance, because, from the perspective of design, the yield moment refers to the bearing capacity of the segment. The model underestimates the yield moment, associated with yielding of the intrados reinforcement in tension, see the square markers, because the strains of the extrados reinforcement were growing so large when the intrados reinforcement started yielding in tension, see Fig. 14, that the strain-hardening effect of the extrados reinforcement became significant. However, the linear-elastic, ideallyplastic material model for steel does not account for this hardening 13

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Fig. 21. Curvature – moment diagrams based on computational analysis and experimental measurements, in case of sagging moments for (a) S40, (b) S60, and (c) S80. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

effect, resulting in conservative estimates of the yield moment, associated with yielding of the intrados reinforcement in tension. Since bending-induced cracks of segments, subjected to a hogging moment, initially occur at the extrados surface, they can be observed without hindrance by the steel shell at the intrados side. Therefore, the prediction quality of the model for the cracking moments of segments, subjected to a hogging moment, is more accurate than that in case of a sagging moment, as follows from a comparison of the triangle markers in Figs. 22 and 21. It is interesting to note that the model-predicted bending moments decline after cracking and then increase with increasing curvature, while the experimentally determined bending moments are continuously increasing, see the blue and the red graphs in Fig. 22. The reason for this is that the computational formulae were derived in a curvaturecontrolled fashion, whereas the experiments were carried out in a force-controlled fashion. The curvature-controlled bending moments can also continuously increase after cracking, as long as the area of the reinforcement in tension is sufficiently great [23], see e.g. the blue graphs in Fig. 21. Debonding between the steel shell and RC segments, subjected to a sagging moment, and strain hardening of the extrados rebars of the segments, subjected to a hogging moment, may require the development of a more advanced computational model. However, the model presented in Section 4.1 remains to be useful from the viewpoint of a practiceoriented engineering mechanics approach. It allows for prediction of (i) yield moments, associated with tensile yielding of the steel shell and of the intrados reinforcement of the segment, subjected to a sagging

moment, and of (ii) cracking moments and yield moments, associated with tensile yielding of the extrados reinforcement of the segment subjected to a hogging moment. The predicted bending moments are of great relevance for the design of the tunnel segments. 4.3. Parametric analysis with respect to the thickness of the individual parts of the steel-concrete composite Two practical problems are considered. The first one refers to a situation where the thickness of the SCC is limited, because the clear diameter of the cross-section of the segmental tunnel ring is reduced below a tolerable limit. Structural sensitivity analyses are performed with respect to the thickness of the steel shell, while the thickness of the SCC is kept constant. Herein, the thickness of the SCC is kept constant at 60 mm. As the thickness of the steel shell is increased from 8 mm, via 10 mm, to 12 mm, the cracking moment of segments subjected to a sagging moment increases from 66 kNm, via 71 kNm, to 76 kNm, see the triangle markers in Fig. 23(a), and the yield moment increases from 516 kNm, via 625 kNm, to 728 kNm, see the circle markers. In other words, a change of the thickness of the steel shell by ±20 % results in a change of the cracking moment by ±8 % and of the yield moment by ±17 %, in case of sagging moments. However, the influence of the increase of the thickness of the steel shell on the structural behavior of segments subjected to a hogging moment is insignificant, see Fig. 23(c). The second practical problem refers to a situation where the thickness of the steel shell is limited because of the limited handling capacity of the hoisting machine. In this case, structural sensitivity analyses 14

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Fig. 22. Curvature – moment diagrams based on computational analysis and experimental measurements in case of hogging moments for (a) H00, (b) H40, (c) H60, and (d) H80. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

concerning the thickness of the new concrete are performed, while the thickness of the steel shell is kept constant. Herein the thickness of the steel shell is kept constant at 10 mm. A change of the thickness of the new concrete by ±20 % results in a moderate change of the cracking moment and the yield moment of the segments for both load scenarios, see Figs. 23(b) and 23(d). The maximum value of this change is limited to ±3 %. As regards the thickness of the steel shell and of the new concrete, there is a trade-off between the clear diameter of the segmental tunnel ring, the handling capacity of the hoisting machine, and the strengthening effect. As far as the latter is concerned, in case of a sagging moment it is more efficient to increase the thickness of the steel shell than that of the new concrete. In case of a hogging moment, however, increasing the thickness of the new concrete is more efficient than that of the steel shell, albeit the strengthening effect remains quite small.

the thickness of the shell is a very good approximation. The equilibrium conditions, involving the forces in the circumferential and the radial direction, read as ( ) 𝑡 𝑑𝜑 𝑑𝜑 + 𝑇𝜑 𝐵 𝑅𝑠𝑠 + 𝑠𝑠 𝑑𝜑 + (𝜎 + 𝑑𝜎) 𝐵𝑡𝑠𝑠 cos = 0, (6) −𝜎𝐵𝑡𝑠𝑠 cos 2 2 2 ( ) 𝑡 𝑑𝜑 𝑑𝜑 −𝜎𝐵𝑡𝑠𝑠 sin + 𝑇𝑟 𝐵 𝑅𝑠𝑠 + 𝑠𝑠 𝑑𝜑 − (𝜎 + 𝑑𝜎) 𝐵𝑡𝑠𝑠 sin = 0. (7) 2 2 2 𝑡 Since 𝑡𝑠𝑠 ≪ 𝑅𝑠𝑠 , it is admissible to set 𝑅𝑠𝑠 + 𝑠𝑠 ≈ 𝑅𝑠𝑠 . Making use of 2 this approximation in the Eqs. (6) and (7) and dividing the resulting relations by 𝐵, yields 𝑑𝜑 + 𝑇𝜑 𝑅𝑠𝑠 𝑑𝜑 + (𝜎 + 𝑑𝜎) 𝑡𝑠𝑠 cos 2 𝑑𝜑 −𝜎𝑡𝑠𝑠 sin + 𝑇𝑟 𝑅𝑠𝑠 𝑑𝜑 − (𝜎 + 𝑑𝜎) 𝑡𝑠𝑠 sin 2 𝑑𝜑 𝑑𝜑 Considering cos ≈ 1 and sin ≈ 2 2 follow from Eqs. (8) and (9) as −𝜎𝑡𝑠𝑠 cos

4.4. Quantification of the forces transmitted across the interface of the steel shell and the concrete

𝑡𝑠𝑠 𝜎, 𝑅𝑠𝑠 𝑡 𝑑𝜎 𝑇𝜑 = − 𝑠𝑠 . 𝑅𝑠𝑠 𝑑𝜑

The headed studs and the chemical anchor bolts result in a strong interaction of the new concrete and the steel shell. Both normal and shear tractions, 𝑇𝑟 and 𝑇𝜑 , respectively, are transmitted from the new concrete to the steel shell. These tractions are computed by means of equilibrium conditions. To this end, an infinitesimal element of the steel shell, see Fig. 24, is considered. The thickness of the shell, 𝑡𝑠𝑠 , is much smaller than the radius of the curvature of the midsurface of the shell, 𝑅𝑠𝑠 . Thus, a uniform distribution of the normal stresses 𝜎 over

𝑇𝑟 =

𝑑𝜑 = 0, (8) 2 𝑑𝜑 = 0. (9) 2 𝑑𝜑 , the tractions, 𝑇𝑟 and 𝑇𝜑 2 (10) (11)

The radial traction 𝑇𝑟 tends to zero as 𝑅𝑠𝑝 → ∞, see Eq. (10). Hence, 𝑇𝑟 is equal to zero for straight SCC plates and beams. The shear traction 𝑇𝜑 , resulting in shear slip, is generally considered, because of its significant influence on the deformations [24] and on the bearing 15

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Fig. 23. Structural sensitivity analysis of SCC-strengthened segments, subjected to (a)–(b) a sagging moment and (c)–(d) a hogging moment, with respect to (a/c) the thickness of the steel shell, for a total thickness of the SCC kept constant at 60 mm, and to (b/d) the thickness of the new concrete, for a thickness of the steel shell kept constant at 10 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

capacity [25,26] of straight SCC plates and beams. Increasing the curvature of the shell results in an increase of the normal interface traction 𝑇𝑟 . Moreover, Eqs. (11) and (10) imply that the tractions result in a biaxial stress state at the interfaces. The normal traction 𝑇𝑟 has the same sign as the normal stress of the steel shell. Therefore, in case of a sagging moment, the normal traction is tensile. In case of a hogging moment, however, it is compressive. Interfaces are much more vulnerable to coupled shearing-tensile loading than to coupled shearing-compressive loading. That is why debonding occurs for segments subjected to sagging moments whereas the interfaces of segments subjected to hogging moments remain perfectly bonded, as discussed in Section 3.2.3. Eqs. (11) and (10) allow for quantification of the stresses transmitted across the interface between the steel shell and the concrete, provided that the normal stresses of the steel shell are known. The strains of the steel shell of the segments S40 and H40 were measured in the tests, see Figs. 12(a) and 17(a). The stresses of the steel shell follow from inserting the values of the measured strains into the constitutive laws, illustrated in Fig. 18(b). Substituting the obtained stresses into Eq. (10) allows for evaluation of the normal interface traction, 𝑇𝑟 (𝜑𝑗 ), where 𝜑𝑗=1, 2 ,…, 7 denote the angular coordinates of the strain-monitoring cross-sections, 𝑆𝑗=1, 2 ,…, 7 , see Figs. 25(a) and 25(c). As regards the shear traction 𝑇𝜑 , the first derivation in Eq. (11) is approximated by means of a central finite difference. Thus, 𝑡 𝜎(𝜑𝑗+1 ) − 𝜎(𝜑𝑗−1 ) 𝑇𝜑 (𝜑𝑗 ) ≈ − 𝑠𝑠 , 𝑗 = 2, 3 , … , 6 , 𝑅𝑠𝑠 2𝛥𝜑

where 𝛥𝜑 denotes the angles enclosed by neighboring cross-sections at which the strains of the steel shell are measured, see Fig. 7. This delivers the shear tractions at 𝜑𝑗=2, 3 ,…, 6 , see Figs. 25(b) and 25(d). The tensile normal tractions are imposed on the steel shell primarily via the chemical anchor bolts. On the one hand, the tensile normal tractions were computed, see Fig. 25(a). On the other hand, the strains of the chemical anchor bolt in the vicinity of the midspan region were measured in the test of S40, see Fig. 12(b). This provided the motivation for checking whether equilibrium of the tensile normal tractions and the tensile stresses of the chemical anchor bolts exists. To this end, the midspan region, i.e. 𝜑 ∈ [−4◦ , 4◦ ], is considered, see Fig. 26(a). The experimentally determined stress of the chemical anchor bolts in this region follows from averaging the measured strains, illustrated in Fig. 12(b) and inserting the value of the averaged strain into the constitutive laws, illustrated in Fig. 18(b), see the blue graphs in Fig. 26(b). The computationally determined stress of the chemical anchor bolts was derived as follows: It was assumed that the normal traction in the region [𝜑𝑗 , 𝜑𝑗+1 ] is linearly distributed in the circumferential direction. Thus, the stress resultant of the normal traction in the midspan region is obtained as [ ] ] 𝛥𝜑 [ 1 𝛥𝜑 3 3 1 𝑇𝑟 (𝜑3 ) + 𝑇𝑟 (𝜑4 ) 𝐵𝑅𝑠𝑠 + 𝑇𝑟 (𝜑5 ) + 𝑇𝑟 (𝜑4 ) 𝐵𝑅𝑠𝑠 . (13) 𝑇 = 4 4 2 4 4 2 The average stress of the chemical anchor bolts in the midspan region is estimated by subdividing the stress resultant 𝑇 by the total crosssectional area of the chemical anchor bolts in this region, 𝐴𝑐𝑎𝑏 , as

(12) 16

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• In case of a sagging moment, use of the SCC results in a significant increase of both the bearing capacity and the bending stiffness of the segments. As for the investigated segment, using the SCC with a thickness of 40 mm results in an increase of the cracking moment by 154 %, the yield moment by 393 %, and of the postcracking stiffness by 315 %. The strengthening effect increases with increasing thickness of the SCC. • Since the steel shell of strengthened segments, subjected to a hogging moment, is in compression, the strengthening effect is less significant than that in case of a sagging moment, resulting in tension of the steel shell. • The ductile failure and the improvement of both the bearing capacity and the bending stiffness of the strengthened segments underlines the usefulness of the proposed method for strengthening of tunnel segments. Analytical formulae were derived in order to describe the structural behavior of the strengthened segments. The following conclusions are drawn from the computational analyses: • Usefulness of the derived formulae is underlined by the agreement of the numerical and the experimental results. • Concerning the increase of the strengthening effect, in case of a sagging moment, an increase of the thickness of the steel shell is more efficient than one of the new concrete. In case of a hogging moment, however, increasing the thickness of the new concrete appears to be more efficient than that of the steel shell. • It is well known that the failure of the interfaces between straight beams and their externally bonded constituents, e.g. a steel plate [27] and FRP sheet [28], is caused by pure shear. However, this is not the case for curved beams with externally bonded constituents, e.g. in case of strengthened tunnel segments. For a sagging moment, the interface is in shear-tension, whereas for a hogging moment it is in shear-compression. The normal stresses and the shear stresses, transmitted across the interfaces, can be quantified by means of the derived formulae in Section 4.4.

Fig. 24. Tractions and stresses acting on an infinitesimal element of the steel shell.

𝑐𝑜𝑚𝑝 𝜎𝑐𝑎𝑏 =

𝑇 , 𝐴𝑐𝑎𝑏

(14)

see the red graphs in Fig. 26(b). Good agreement between the experimentally and the computationally determined stresses of the chemical anchor bolt up to yielding is obtained, compare the blue and the red graphs in Fig. 26(b). This underlines the usefulness of the derived formulae (11) – (14). They are relevant to the design of the connectors across the interface between the steel shell and the concrete, since they allow for quantification of the forces transmitted across this interface.

The proposed strengthening method is based on the concept that the original RC segments and the steel-concrete composite jointly carry the load, because the segments are strengthened before the external loading is applied. In fact, degraded tunnel linings were typically loaded before they were strengthened. A real-scale test of a segmental tunnel ring, strengthened by a steel-concrete composite, considering the loading history of the RC segmental tunnel ring, will be the topic of follow-up research work [29].

5. Conclusions In the present paper a novel method for strengthening segmental tunnel linings was proposed. It makes use of a steel-concrete composite. A series of real-scale tests of strengthened tunnel segments were carried out. From the obtained experimental data and the related discussions, the following conclusions are drawn:

Acknowledgments Financial support by the National Natural Science Foundation of China (Grant No. 51578409) is gratefully acknowledged. The authors are also indebted to Mr. Zijie Jiang for helpful discussions.

• The loading has a significant effect on the failure process of the strengthened segments. As for segments subjected to a sagging moment, the damage process started with (i) cracking of concrete, continued by (ii) debonding between the steel shell and the new concrete and followed by (iii) debonding between the new and the old concrete, (iv) yielding of the chemical anchor bolts in tension, (v) of the steel shell in tension, and (vi) of the intrados reinforcement in tension. The damage process ended with failure of the chemical anchor bolts. As regards a hogging moment, the damage process of the strengthened segments started with (i) cracking of concrete, followed by (ii) yielding of the extrados reinforcement in tension and (iii) of the intrados reinforcement in tension. It ended with fracture of the extrados reinforcement. In the second load case, debonding at the two interfaces was not observed. • The failure mode of the strengthened segments for both load cases is ductile. This is a relatively favorable failure mode of engineering structures.

Appendix A. Closed-form expressions of the stress resultants 𝑵 and 𝑴 The constitutive equations of concrete, illustrated in Fig. 18(a), read as ⎧−𝑓𝑐 , ⎪ [ ( )2 ] ⎪ ⎪−𝑓 2 𝜀𝑐 − 𝜀𝑐 , 𝜎𝑐 = ⎨ 𝑐 𝜀𝑐0 𝜀𝑐0 ⎪ ⎪𝐸 𝑐 𝜀 𝑐 , ⎪ ⎩0 ,

if

𝜀𝑐 < 𝜀𝑐0 ,

if

𝜀𝑐0 ≤ 𝜀𝑐 < 0 ,

if

0 ≤ 𝜀𝑐 < 𝜀𝑡0 ,

if

𝜀𝑐 ≥ 𝜀𝑡0 .

(A.1)

The constitutive equations of steel, illustrated in Fig. 18(b), read as ⎧−𝑓 , ⎪ 𝑦 𝜎𝑠 = ⎨𝐸𝑠 𝜀𝑠 , ⎪𝑓𝑦 , ⎩ 17

if 𝜀𝑠 < −𝜀𝑦 , if − 𝜀𝑦 ≤ 𝜀𝑠 < 𝜀𝑦 , if 𝜀𝑠 ≥ 𝜀𝑦 .

(A.2)

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J.-L. Zhang et al.

Fig. 25. Evaluation of the interface tractions acting on the steel shell of the strengthened segments (a)–(b) for S40 and (c)–(d) for H40: (a/c) normal traction 𝑇𝑟 and (b/d) shear traction 𝑇𝜑 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 26. (a) Sketch of the midspan region of the steel shell and (b) comparison of the stress of the chemical anchor bolt in this region, estimated from the experimentally measured strains of the bolts, see Fig. 12(b), and from the computed normal tractions, see Fig. 25(a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

loading is subdivided into five stages. The mechanical basis of this

A.1. Sagging moment

subdivision is the distribution of the normal stresses over the cross-

Inspired by the findings from the experimental investigation of the flexural behavior of the strengthened segments, the process of their

section, see Fig. 19. The generalized expressions of 𝑁 and 𝑀 read 18

Composites Part B 178 (2019) 107444

J.-L. Zhang et al. Table A.1 Numerical values of 𝛼1, 2, …, 9 , for a sagging moment.

Table A.2 Numerical values of 𝛽1, 2, …, 8 , for a hogging moment.

Loading stage

Stress state

𝛼1

𝛼2

𝛼3

𝛼4

𝛼5

𝛼6

𝛼7

𝛼8

𝛼9

Loading stage

Stress state

𝛽1

𝛽2

𝛽3

𝛽4

𝛽5

𝛽6

𝛽7

𝛽8

I II III IV V

see see see see see

1 1 1 1 1

1 1 1 1 1

1 1 1 0 0

1 0 0 0 0

1 1 0 0 0

0 1 1 1 1

0 0 1 1 1

0 0 0 1 1

0 0 0 0 1

I II III IV

see see see see

1 1 0 0

1 1 1 0

1 1 1 1

1 1 1 1

1 0 0 0

0 1 1 1

0 0 1 1

0 0 0 1

Fig. Fig. Fig. Fig. Fig.

19(b) 19(c) 19(d) 19(e) 19(f)

[

Fig. Fig. Fig. Fig.

20(b) 20(c) 20(d) 20(e)

)2 ] ( 1 𝐵𝐸𝑐 𝜅 −𝐻 − 𝑡𝑛𝑐 + ℎ𝑐 + 𝛽6 3 [ )] ( + 𝛽7 𝑓𝑦𝑠𝑏 𝐴𝑠𝑒 −𝐻 − 𝑡𝑛𝑐 + ℎ𝑐 + 𝑎𝑠𝑒 [ ( )] + 𝛽8 𝑓𝑦𝑠𝑏 𝐴𝑠𝑖 ℎ𝑐 − 𝑡𝑛𝑐 − 𝑎𝑠𝑖 , + 𝛽5

as

)] [ ( [ 𝑓𝑐 𝜅 2 ℎ3𝑐 𝑓𝑐 𝜅ℎ2𝑐 ( )] 𝑁 = 𝛼1 −𝐸𝑠 𝐴𝑠𝑒 𝜅 ℎ𝑐 − 𝑎𝑠𝑒 + 𝛼2 −𝐵 + 𝜀𝑐0 3𝜀2𝑐0 [ [ ( )] ( )2 ] 1 + 𝛼3 𝐸𝑠 𝐴𝑠𝑖 𝜅 𝐻 − ℎ𝑐 − 𝑎𝑠𝑖 + 𝛼4 𝐵𝐸𝑐 𝜅 −𝐻 + ℎ𝑐 + 𝑡𝑛𝑐 2 [ ( ) ] 1 + 𝛼5 𝐸𝑠 𝜅 𝐻 − ℎ𝑐 + 𝑡𝑛𝑐 + 𝑡𝑠𝑠 𝐴𝑠𝑠 2 ) ( 2 ) ( 1 𝐵𝑓𝑡 + 𝛼7 𝑓𝑦𝑠𝑠 𝐴𝑠𝑠 + 𝛼6 2 𝜅𝐸𝑐 [ ( ) ( 𝜀 )] + 𝛼8 𝑓𝑦𝑠𝑏 𝐴𝑠𝑖 + 𝛼9 −𝐵𝑓𝑐 ℎ𝑐 + 𝑐0 (A.3) 𝜅 and [ ( )] [ 𝑓𝑐 𝜅 2 ℎ4𝑐 2𝑓𝑐 𝜅ℎ3𝑐 ( )2 ] 𝑀 = 𝛼1 −𝐸𝑠 𝐴𝑠𝑒 𝜅 ℎ𝑐 − 𝑎𝑠𝑒 + 𝛼2 −𝐵 − 3𝜀𝑐0 4𝜀2𝑐0 [ [ ( )2 ] ( )3 ] 1 + 𝛼3 −𝐸𝑠 𝐴𝑠𝑖 𝜅 𝐻 − ℎ𝑐 − 𝑎𝑠𝑖 + 𝛼4 𝐵𝐸𝑐 𝜅 −𝐻 + ℎ𝑐 + 𝑡𝑛𝑐 3 ( ) [ ] 3 ( )2 1 1 𝐵𝑓𝑡 + 𝛼5 −𝐸𝑠 𝜅 𝐻 − ℎ𝑐 + 𝑡𝑛𝑐 + 𝑡𝑠𝑠 𝐴𝑠𝑠 + 𝛼6 − 2 3 𝜅 2 𝐸𝑐2 [ ( )] 1 + 𝛼7 −𝑓𝑦𝑠𝑠 𝐴𝑠𝑠 𝐻 − ℎ𝑐 + 𝑡𝑛𝑐 + 𝑡𝑠𝑠 2 [ ] ( ) + 𝛼8 −𝑓𝑦𝑠𝑏 𝐴𝑠𝑖 𝐻 − ℎ𝑐 − 𝑎𝑠𝑖 ( )] [ ( 𝜀 𝜀 ) ℎ𝑐 − 𝑐0 , (A.4) + 𝛼9 −𝐵𝑓𝑐 ℎ𝑐 + 𝑐0 𝜅 2 2𝜅

( −

𝐵𝑓𝑡3

)

3𝐸𝑐2 𝜅 2

(A.6)

where 𝛽1, 2, …, 8 are coefficients related to the stress states of segments subjected to hogging moments, see Table A.2 and Fig. 20.

Appendix B. List of abbreviations and symbols

CAB FRP FWF HRB LVDT PCM RC SCC TRC 𝑎𝑠𝑒 𝑎𝑠𝑖

where 𝛼1, 2, …, 9 are coefficients related to the stress states of segments subjected to sagging moments, see Table A.1 and Fig. 19. In Eqs. (A.3) and (A.4), 𝐴𝑠𝑒 , 𝐴𝑠𝑖 , and 𝐴𝑠𝑠 stand for the area of the extrados reinforcement, the intrados reinforcement, and of the steel shell, respectively, 𝑎𝑠𝑒 and 𝑎𝑠𝑖 represent the thickness of the concrete cover of the extrados and the intrados reinforcement, respectively, and 𝑓𝑦𝑠𝑠 and 𝑓𝑦𝑠𝑏 denote the yield stress of the steel shell and the steel bars, respectively.

𝐴𝑐𝑎𝑏 𝐴𝑠𝑒 𝐴𝑠𝑖 𝐴𝑠𝑠 𝛼1, 2, …, 9

A.2. Hogging moment

𝐵 𝛽1, 2, …, 8

By analogy to the situation for a sagging moment, the process of loading of strengthened segments subjected to a hogging moment is subdivided into four stages, see Fig. 20 for the strain and the stress distributions at each stage. The generalized expressions of 𝑁 and 𝑀 read as [ [ ( )] ( )] 𝑁 = 𝛽1 𝐸𝑠 𝐴𝑠𝑒 𝜅 𝐻 + 𝑡𝑛𝑐 − ℎ𝑐 − 𝑎𝑠𝑒 + 𝛽2 −𝐸𝑠 𝐴𝑠𝑖 𝜅 ℎ𝑐 − 𝑡𝑛𝑐 − 𝑎𝑠𝑖 [ ( )] [ ( ) ] 𝑓𝑐 𝜅 2 ℎ3𝑐 𝑓𝑐 𝜅ℎ2𝑐 𝑡𝑠𝑠 + 𝛽3 −𝐸𝑠 𝜅 ℎ𝑐 + 𝐴𝑠𝑠 + 𝛽4 𝐵 + 2 𝜀𝑐0 3𝜀2𝑐0 ( ) 2 [ 𝐵𝑓𝑡 ( ) ( )2 ] 1 + 𝛽5 𝐵𝐸𝑐 𝜅 𝐻 + 𝑡𝑛𝑐 − ℎ𝑐 + 𝛽6 + 𝛽7 𝑓𝑦𝑠𝑏 𝐴𝑠𝑒 2 2𝐸𝑐 𝜅 ( ) + 𝛽8 𝑓𝑦𝑠𝑏 𝐴𝑠𝑖 (A.5)

𝛿𝑐𝑟 𝛿𝑦𝑠 𝛥𝜑 𝐞𝑟 𝐞𝜑 𝜀𝑐 𝜀𝑐0 𝜀𝑠 𝜀𝑡0 𝜀𝑦 𝐸 𝐸𝑐 𝐸𝑠 𝑓𝑐 𝑓𝑡

and

[ ( )2 ] 𝑀 = 𝛽1 −𝐸𝑠 𝐴𝑠𝑒 𝜅 𝐻 + 𝑡𝑛𝑐 − ℎ𝑐 − 𝑎𝑠𝑒 [ ( )2 ] + 𝛽2 −𝐸𝑠 𝐴𝑠𝑖 𝜅 ℎ𝑐 − 𝑡𝑛𝑐 − 𝑎𝑠𝑖 [ [ ( )] ( )2 ] 𝑓𝑐 𝜅 2 ℎ4𝑐 2𝑓𝑐 𝜅ℎ3𝑐 𝑡 + 𝛽3 −𝐸𝑠 𝐴𝑠𝑠 𝜅 ℎ𝑐 + 𝑠𝑠 + 𝛽4 𝐵 + 2 3𝜀𝑐0 4𝜀2 𝑐0

19

Chemical Anchor Bolt Fiber-Reinforced Plastic Filament Wound Profiles Hot-Rolled Steel Linear Variable Differential Transformer Polymer Cement Mortar Reinforced Concrete Steel-Concrete Composite Textile-Reinforced Concrete thickness of the concrete cover of the extrados reinforcement of RC segments thickness of the concrete cover of the intrados reinforcement of RC segments sum of the cross-sectional areas of the chemical anchor bolts in the midspan region cross-sectional area of the extrados reinforcement cross-sectional area of the intrados reinforcement cross-sectional area of the steel shell coefficients related to stress states of segments subjected to sagging moments width of the cross-section coefficients related to stress states of segments subjected to hogging moments midspan deflection corresponding to the crack moment midspan deflection corresponding to the yield moment angle enclosed by neighboring cross-sections at which the strains of the steel shell are measured base vector in the radial direction base vector in the circumferential direction concrete strain elastic limit strain of concrete in compression steel strain elastic limit strain of concrete in tension elastic limit strain of steel Young’s modulus Young’s modulus of concrete Young’s modulus of steel uniaxial compressive strength of concrete uniaxial tensile strength of concrete

Composites Part B 178 (2019) 107444

J.-L. Zhang et al.

𝑓𝑦 𝑓𝑦𝑠𝑏 𝑓𝑦𝑠𝑠 ℎ𝑐 𝐻 𝜅 𝐾1 𝐾2 𝑙1 𝑙2 𝐿 𝑀 𝑀𝑗=1, …, 5 𝑀𝑐𝑟 𝑀𝑠𝑛 𝑀𝑛𝑜 𝑀𝑦𝑠 𝑀𝑦𝑠𝑒 𝑀𝑦𝑠𝑖 𝑀𝑦𝑠𝑠 𝑀𝑢 𝑁 𝑁𝑗=1, …, 5 𝑃 𝜑 𝜑𝑗=1, …, 7 𝑅 𝑅𝑠𝑠 𝜎𝑐 𝑐𝑜𝑚𝑝 𝜎𝑐𝑎𝑏 𝜎𝑠 𝜎𝑠𝑒 𝜎𝑠𝑖 𝜎𝑠𝑠 S1, ⋯, 7 𝑡 𝑡𝑛𝑐 𝑡𝑠𝑠 𝑇 𝑇𝑟 𝑇𝜑

yield stress of steel yield stress of steel rebars yield stress of steel shell height of concrete in compression height of the cross-section of RC segments curvature of segments pre-cracking stiffness of segments post-cracking stiffness of segments horizontal distance between the two loading beams horizontal distance, measured from one of the loading beams to its neighboring support span of circular segments bending moment bending moment corresponding to the 𝑗 th stress stage crack moment bending moment, associated with debonding between the steel shell and the new concrete bending moment, associated with debonding between the new and the old concrete yield moment yield moment, associated with tensile yielding of the extrados reinforcement yield moment, associated with tensile yielding of the intrados reinforcement yield moment, associated with tensile yielding of the steel shell ultimate-limit bending moment axial force axial force corresponding to the 𝑗 th stress stage jack force circumferential coordinate, measured from the crown of the segments circumferential position of the 𝑗 th monitoring point for the strain of the steel shell radius of curvature of the midsurface of the circular RC segments radius of curvature of the midsurface of the steel shell concrete stress computationally determined stress of the chemical anchor bolt steel stress stress of extrados reinforcement stress of intrados reinforcement stress of steel shell numbers of points for monitoring of the strains of the steel shell thickness of the strengthening layer, i.e. of the SCC thickness of the new concrete thickness of the steel shell stress resultant of the normal tractions normal traction shear traction

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