1 Introduction
Shield tunneling is currently the most commonly employed method for constructing urban rail transit tunnels [
1–
4]. This technique utilizes high-strength bolts to connect precast concrete segments, resulting in quick and convenient construction with minimal external disruption [
5–
8]. However, the presence of longitudinal joints can negatively impact structural integrity, causing significant deformation when subject to external disturbances [
9–
13]. Furthermore, the joint, being the weakest part of the tunnel lining, plays a critical role in both potential failure and the strengthening effectiveness [
14–
16]. That is, strengthening longitudinal joint is the core concern of shield tunnel strengthening [
17,
18].
Ultra-high performance concrete (UHPC), characterized by excellent compressive and tensile strength, as well as exceptional impact and fatigue resistance, has been widely used for strengthening concrete structure [
19–
22]. Experimental studies demonstrate that incorporating UHPC with steel plates substantially enhances the bearing capacity and stiffness of segment linings [
23,
24]. However, a comprehensive investigation into the mechanical response of the joint strengthened by the steel-UHPC composite remains unexplored. The mechanical response of longitudinal joint features multi-stage characteristics, owing to construction measures such as inside and outside filleting concrete [
25,
26]. Previous studies have employed experimental and analytical methods to analyze the multi-stage deformation characteristics of un-strengthened longitudinal joints [
27–
31]. Nonetheless, these analytical methods are based on observed experimental phenomena to divide multiple stages. Less research has been conducted on autonomously judging stages within the theoretical calculation process, enabling the continuous computation of mechanical responses.
The mechanical response of un-strengthened joints differs under sagging and hogging moments, mainly depending on the bolts position and the compressed concrete height [
30,
31]. This disparity will be particularly pronounced in strengthened longitudinal joint, with steel-UHPC composites positioned on the tension and compression sides under sagging and hogging moments, respectively. Steel plates and UHPC undergo tension stress under sagging moments and compression stress under hogging moments, inevitably exacerbating differences in the strengthening effect [
23]. Also, the effects of steel plate thickness and UHPC thickness regarding the strengthening effect remain unclear. Previous research has indicated that axial force has a notable impact on increasing the bearing capacity of longitudinal joints, implying that the strengthening effect differs to some extent under different axial force levels [
27]. Despite several available strengthening methods for shield tunnels and extensive full-scale testing conducted based on these methods [
17,
18,
32,
33], a clear theoretical explanation regarding the strengthening mechanisms is still lacking. Additionally, the differences and similarities in strengthening effect under sagging and hogging moments require further theoretically explained. In short, due to the absence of theoretical model of the strengthened longitudinal joints, current research lacks a comprehensive understanding of strengthening mechanisms and effectiveness.
Addressing the aforementioned issue, a mechanical model for strengthened shield tunnel joint was introduced first and verified by numerical models. The significance of this research lies in the following aspects: 1) continuous load and deformation responses of joints under various external loads were captured using proposed method; 2) strengthening effect was clearly quantified and compared; 3) a parameter analysis of the strengthening parameters and axial force level was conducted; 4) the strengthening mechanics under sagging and hogging moment were investigated and compared.
2 Analytical solution of longitudinal joint
2.1 Strengthened longitudinal joint
Fig.1 shows the longitudinal joint strengthened by a steel UHPC composite. The steel plate is anchored on the interior of the longitudinal joints, with UHPC then poured into the gap between the steel plate and the lining. The detail of strengthened longitudinal joint is shown in Fig.1, which includes waterproof gaskets, outside filleting concrete, core concrete, inside filleting concrete, UHPC layer and steel plate. The joint has a radius of 3.1 m, a center angle of 45° , a thickness of 350 mm, and a width of 1500 mm. The diameter of the bolt is 30 mm. The yield stress of the bolt is 640 MPa. The segment is designed with concrete strength grade C50. The rebars used in the segment consists of HRB400 Chinese hot-rolled steel bars with an expected yield stress of 400 MPa. A steel plate with the specification Q345 is used. The welding studs are made of Q195 with a 13 mm diameter, while the anchor screws are 12.9-grade with a 12 mm diameter.
2.2 Assumption of analytical solution
As shown in Fig.2, un-strengthened and strengthened joints are simplified into single-layer and three-layer models, respectively. Concrete is mainly divided into outside filleting concrete, core concrete and inside filleting concrete. The gasket is neglected in the model in a consideration of negligible influence on the mechanical behavior of the joint [
30,
31]. The analytical solution derivation is predicated on the following fundamental assumptions:
1) the cross-sectional strain remains planar;
2) the curvature of the cross-section increases throughout the deformation process;
3) the connection between the strengthening layer and joint is robust;
4) the constitutive models of materials are simplified as depicted in Fig.3. The specific expressions are provided in Appendix A in Electronic Supplementary Material.
2.3 Derivation of analytical solution
As shown in Fig.4, under the action of bending moments, the longitudinal joint undergoes deformation characterized by a strain distribution that can be characterized by the curvature of the cross-section (κ) and the height of the compressed zone (hc). With a constant axial force, the deformation progresses with the increasing bending moment, indicating a monotonic relationship between the two. This relationship allows the back-calculation of bending moments from the observed continuous deformation response.
In fact, structural mechanical response is governed by four variables: bending moment (M), axial force (N), hc, and κ. The axial force is given constant, and the cross-section curvature is a given increasing quantity. Using axial force equilibrium equation, compressed zone height can be resolved. Based on the plane section strain assumption, the strain distribution across the section can be determined by the curvature and the corresponding compressed zone height. Then, all material strains are monitored and compared with critical strains according to their constitutive relationships. Once the strain of the material exceeds the critical strain, the stress response of the material begins to change. Subsequently, material stress states are adjusted to represent various stress response stages, enabling continuous calculation of mechanical response. Finally, using bending moment equilibrium equations, the bending moment is calculated.
Fig.5 shows the detailed calculation process. First, the physical, geometry properties and initial value of state coefficients of each material are input. The cross-section is assumed to undergo deformation with curvature (κ). The initial value of curvature is 0 m−1. The maximum curvature value is set at 0.1 m−1 with the increment set as the maximum curvature value divided by 5000. The stress states of various materials can be expressed by the known curvature and constitutive equations and the unknown height of the compression zone. Then the height of the compression zone is solved according to the equilibrium of the axial force, expressed by
where y denotes the coordinate axis along the height of the cross-section. Then, the stress of various materials is determined by known curvature and resolved height of the compression zone. By incorporating the stress states of various materials into the moment equilibrium equation, the bending moment exerted on the cross-section can be accurately calculated, as follows from Eq. (2).
Fig.6–Fig.9 show the strain and stress responses of the un-strengthened joint subjected to a sagging moment, un-strengthened joint subjected to a hogging moment, strengthened joint subjected to a sagging moment, and strengthened joint subjected to a hogging moment, respectively. The corresponding structural failure process in Fig.6–Fig.9 is determined according to the strain monitoring and comparison of each material, with details provided in Tables B1–B4 in Electronic Supplementary Material in Appendix B in Electronic Supplementary Material. The corresponding general analytical expressions for the axial force and bending moment of the above-mentioned joints models are provided by Eqs. (C1)–(C8) in Appendix C in Electronic Supplementary Material. Material state coefficient, described in Tables C1–C4 in Electronic Supplementary Material, are introduced to characterize the different stress stages of material. The coefficients take values of 1 and 0, where the former indicates material participation in a certain stress stage, and the latter signifies non-participation. Utilizing the calculated height of the compression zone and the curvature of the cross-section, the strain state of each material is calculated. State coefficients are adjusted according to the comparation between calculated material strains and critical strains in the constitutive relationship. When the curvature is below the maximum, curvature is increased by one increment for next round calculation. Otherwise, the calculation ends.
The compressive deformation of concrete at the outer edge of compression zone, Δ
c, can be related to the concrete strain using the following equation [
27,
29–
31]
where lef is influence depth of compressive strain at outer edge of compression zone. The calculation is as follows
where
H is the height of the segment lining,
η is an empirically determined coefficient between 0.55 and 0.70 [
30,
31]. In this paper, the coefficient takes a value of 0.7. Then the joint rotation is calculated by [
27,
29]
Take un-strengthened longitudinal joint under sagging moment as an example, the specific calculation process is as follows. Fig.6(b) shows distribution of strains and stress before the bolt is in tension, corresponding to material state coefficient of stage I in Table B1 in Electronic Supplementary Material. As curvature increases, once the monitored bolt strain exceeds 0, the distribution of strains and stress in shown in Fig.6(c), with material state coefficients transitioning into stage II-A. Once the monitored concrete compressive strain surpasses hardening strain, the strains and stress distribution remain as shown in Fig.6(c), but the material state coefficients shift to stage II-B. Then, when the monitored bolt strain exceeds yield strain, the distribution of strains and stress in depicted in Fig.6(d), with material state coefficients advancing to stage III. When the monitored Δc is greater than Δc_initial, the strains and stress distribution is shown in Fig.6(c), with material state coefficients moving to stage IV.
3 Verification
3.1 Numerical model
Fig.10 shows the 3D finite element model of strengthened longitudinal joint. The arrangement of concrete, steel rebar, bolts, screws, and steel plates in the model are strictly consistent with Fig.1. The thickness of UHPC layer and steel plate layer are selected as 36 and 4 mm, respectively. The model has a total of 62412 nodes and 48766 elements. Concrete Damaged Plasticity model is applied to simulate the mechanic behavior of C50 concrete and UHPC. As listed in Tab.1, The parameters are determined based on the full-scale experiment of strengthened segment lining [
23]. Mechanical properties of steel rebars, steel plate and bolts are listed in Tab.2.
As shown in Fig.11(a), the boundary condition was set as follows. Reference points were set at the arch foot to constrain the displacement of
Y direction. Moreover, displacement of the longitudinal joint in
Z direction was constraint by roller supports. Reference points were coupled with the end face through “coupling” to apply a horizontal force
N on the center of the joint cross-section. The axial force is set to 750 kN for sagging moments and 1100 kN for hogging moments, respectively [
23,
24]. The calculation cases are listed in Tab.3. Reference points were set on both sides of the span at a distance of 400 mm from the joint, which is coupled with the loading area to apply
Fy. The horizontal axial force
N was applied first, followed by the application of the vertical force
Fy.
Fig.11(b) shows interactions in this model. The interface between the longitudinal joint of the segment lings, the interface between the bolt and the bolt hole, the interface between concrete-UHPC and the interface between the waterproof rubber pad are all set with “Surface to Surface” contact, and the parameters are set as shown in Tab.4. The “cohesive” attribute is used to simulate bonding between concrete-UHPC, and the parameters are shown in Tab.5 [
23,
34,
35]. The interaction between waterproof gasket and sealing groove is simulated by “tie” constraint. The interaction between steel bar and concrete segment and the interaction between anchoring screw and assembly are simulated by embedded constraint. In this model, the welding stud on the steel plate is omitted.
3.2 Comparison between analytical solution and numerical model results
Fig.12 shows the moment−rotation curves for both strengthened and un-strengthened longitudinal joints subjected to sagging and hogging moments. The curves obtained through numerical model and analytical solution exhibit consistency in terms of trend and values of bending moment. For specimen S-J-0-0-750 kN, as shown in Fig.12(a), four stages of structural response are observed. Stage I exhibits nearly zero rotation angle with no forces acting on the bolts. Transitioning to Stage II, bolts gradually begin to experience loading and eventually reach yielding, leading to rotation angle development and a rapid increase in cross-sectional bending moment. Stage III is characterized by bolt yielding in tension, causing a swift rotation angle development. The bearing capacity remains constant, indicating preliminary joint failure. Finally, outside filleting concrete comes into contact, resulting in slowed rotation angle development and a slight increase in bearing capacity.
For specimen S-J-4-4-750 kN, as shown in Fig.12(b), the mechanical response exhibits five stages. Stage I shows a rapid increase in bearing capacity prior to UHPC cracking, with minimal rotation angle development. In Stage II, the rotation angle gradually initiates, accompanied by a substantial increase in cross-sectional bending moment. The steel plate remains below its yield point. In Stage III, as the bolts approach yield in tension, the rotation angle accelerates. After bolt yielding in Stage IV, the cross-sectional bending moment reaches a plateau, and the rotation angle undergoes rapid development. Finally, in Stage V, as the outside filleting concrete makes contact, the rate of rotation angle development begins to decelerate. The bearing capacity of joint exhibits a gradual, albeit slower, increase once again.
For specimen H-J-0-0-1100 kN (Fig.12(c)) and specimen H-J-4-4-1100 kN (Fig.12(d)), the mechanical response can be divided into three stages. Stage I features a high compression zone, where the bolts are not yet loaded. Here, the rotation angle remains undeveloped. In Stage II, as the bolts approach yielding, the cross-sectional bending moment continues to grow, accompanied by an accelerated rotation angle. Moving into Stage III, after the bolts yield, the bearing capacity of the structure reaches a plateau, with no further increase.
It is worth emphasizing that yielding of bolts signifies structural failure, regardless of joint strengthening or not, under sagging or hogging moments. When bolts yield, the bearing capacity either remains unchanged or only slightly increases. Therefore, this paper focuses on comparing the calculation deviations associated with load of bolt yielding, as listed in Tab.6. The deviation range is within plus or minus 10%, meaning that the analytical solution accurately reflects the mechanical response of the joint.
4 Strengthening effect
4.1 Bearing capacity and stiffness
The initial strength indicators, S1, and the final strength indicators, S2, are determined by selecting the bending moments at the onset of bolt tension and at the yield point of the bolts, denoted as Mb and Myb, respectively. The initial flexural stiffness, K1, and the overall flexural stiffness, K2, are defined as the ratio of the bending moment to the corresponding rotation angle at the onset of bolt tension and at the yield point, respectively. They read as
where θb and θyb denotes the rotation angle at the onset of bolt tension and at the yield point of the bolts, respectively. The increase rates are determined by comparing the performance indicators of the strengthened joints to those of the un-strengthened joints. Fig.13 shows the performance indicators increase percentages under sagging and hogging moments. Under sagging moments, the strengthened longitudinal joint exhibits significant enhancements in all mechanical performances. S1 and S2 increased by 5.42 and 4.14 times, respectively, signifying a notable boost in strength. Additionally, K1 and K2 increased by of 2.71 and 2.93 times, respectively, reflecting an improved ability to withstand deformation. These findings clearly show that the strengthening significantly enhances the structural strength, surpassing the improvement on stiffness. For hogging moments, the improvement in structural bearing capacity is moderate, with S1 and S2 increasing by 1.33 and 1.65 times, respectively. However, the ability to resist deformation shows substantial enhancement, as K1 and K2 increase by 5.66 and 4.46 times. This emphasizes the much greater enhancement of structural stiffness. Strengthened longitudinal joints under sagging and hogging moments using steel-UHPC method yields favorable results, albeit with slightly different emphases. The former mainly improves structural strength, and the latter emphasizes enhancing structural stiffness.
4.2 Parameter analysis
In the parameter analysis, the strength grades are consistent with the numerical model: concrete (C50), steel plate (Q345), bolt (yield strength 640 MPa), and UHPC (compressive strength 140 MPa). The analyzed parameters primarily include the thickness of the steel plate, the thickness of the UHPC, and the level of horizontal axial force. The specimens number consist of five components: S and H denote sagging and hogging moments, respectively; J represents longitudinal joints; the third component is the thickness of the strengthening layer in centimeters; the fourth component is the thickness of the steel plate in millimeters; and the final component indicates the level of axial force. The specimen numbering scheme is consistent with the content in Tab.4.
4.2.1 Thickness of steel plate
Fig.14 shows the moment−rotation curves of joints under sagging and hogging moments, strengthened with different steel plate thicknesses. As shown in Fig.14(a), increasing the steel plate thickness, while keeping the strengthened layer thickness constant, significantly enhances the bearing capacity. However, the improvement in bearing capacity under hogging moments is minimal, see Fig.14(b). For the strengthened joints under sagging moments, with a 4 mm steel plate thickness as the basis, each 1 mm increase in steel plate thickness leads to a relative increase of 12.94% to 21.05% in S1, 11.54% to 12.78% in S2, 2.20% to 3.20% in K1, and a decrease of 1.85% to 7.69% in K2, see Fig.15(a). These findings underscore the sensitivity of steel plate thickness in enhancing structural strength, but it necessitates careful consideration of its impact on overall stiffness. Extreme thickness should be avoided. For the strengthened joints under hogging moments, with a 4 mm steel plate thickness as the basis, each 1 mm increase in steel plate thickness results in a relative increase of 0.20% to 1.18% in S1, 0.32% to 0.53% in S2, 1.49% to 9.82% in K1, and 1.27% to 3.43% in K2. This implies that the enhancement in overall stiffness is modest, which prompts caution given the significant cost of steel plates.
4.2.2 Thickness of ultra-high performance concrete
Fig.16 shows the moment−rotation curves of longitudinal joints strengthened with different UHPC thicknesses under sagging and hogging moments. Clearly, increasing the UHPC thickness, with a fixed steel plate thickness, leads to improved bearing capacity for both moment conditions. Fig.17 provides the corresponding performance indicators increase percentage. For sagging moments, with a 4 cm strengthening layer thickness as the basis, each 1 cm increment in UHPC thickness leads to a relative increase of 23.36% to 31.75% in S1, 5.88% to 6.17% in S2, 4.62% to 6.80% in K1, and 2.33% to 3.07% in K2, see Fig.17(a). These findings highlight the positive influence of UHPC thickness, particularly on initial strength improvement. Under hogging moments, with a 4 cm strengthening layer thickness as the basis, each 1 cm increase in UHPC thickness yields a relative increase of 0.66% to 1.60% in S1, 4.85% to 5.92% in S2, 13.92% to 19.79% in K1, and 12.72% to 14.20% in K2, see Fig.17(b). This emphasizes the beneficial impact of augmenting UHPC thickness on overall stiffness enhancement.
4.2.3 Axial force
Fig.18 shows the moment-rotation curves of strengthened joints with varying axial force levels. Clearly, increasing the axial force level improves the bearing capacity of the joints, especially under hogging moments. Fig.19 shows the corresponding performance indicators increase rate. Under sagging moments, with zero axial force as the basis, a 750 kN increase in axial force yields a relative enhancement of 31.09% to 63.91% in S1, 3.31% to 5.30% in S2, and a relative decrease of 7.82% to 23.43% in K1, and 7.76% to 12.32% in K2, see Fig.19(a). Elevating the axial force level significantly bolsters the structural strength, particularly in terms of initial strength. But it exhibits a minor sacrifice in structural stiffness. Furthermore, under hogging moments, with 1100 kN as the basis, a 1100 kN rise in axial force results in a relative improvement of 98.77% to 100.13% in S1, 55.15% to 58.07% in S2, 3.36% to 93.51% in K1, and 36.05% to 53.59% in K2, see Fig.19(b). Increasing the axial force level yields outstanding improvements in stiffness and strength under hogging moments, far surpassing the benefits obtained from increasing the steel plate or UHPC thickness. Indeed, the significant disparity in stiffness values between joints under sagging and hogging moments presents an important consideration. Sacrificing some stiffness in the joints under sagging moments is justifiable when compared to the substantial enhancement in mechanical performance observed under hogging moments.
5 Strengthening mechanism
5.1 Sagging moment
Fig.20 shows the percentage of the moment borne by the various material, under sagging moments, before and after strengthening. Prior to strengthening, the percentage of moment carried by the bolts gradually increases and stabilize around 78.84% as the rotational angle develops. Conversely, the contribution of concrete decreases and stabilizes around 21.16%. After strengthening, the percentage of moment carried by the bolts and UHPC eventually stabilizes around 11.04%, while the contribution of steel plate stabilizes around 57.57%. This substantial contribution from the steel plate is the key factor behind the increased bearing capacity. Additionally, Fig.21 reveals that the steel plate and UHPC handle approximately 60.77% and 12.68% of the tensile axial forces, respectively, effectively sharing the tensile stress of the bolts. Furthermore, as shown in Fig.22, the increased height of the compressed region after strengthening suggests that the concrete mechanical performance can be further utilized. In summary, the strengthening mechanism for joint subjected to a sagging moment involves sufficiently utilizing the tensile performance of the steel plate and UHPC. This also further leverages the mechanical performance of concrete in the compressed region. The simultaneous strengthening of the tensile and compressed regions achieves a more favorable mechanical match.
5.2 Hogging moment
Fig.23 shows the moment distribution percentage under hogging moments before and after strengthening. In the H-J-0-0-1100 kN specimens, the bending moment borne by the bolts gradually increases and stabilizes at approximately 43.86% as the joint angle developed. While the contribution of concrete shows the opposite trend, gradually decreasing and stabilizing at around 56.14%. After strengthening, the moment carried by the bolts gradually increases and eventually stabilizes at approximately 84.83%. Meanwhile, the moment borne by the steel plates and UHPC initially increases, then slowly decreases, finally stabilizes at approximately 11.36% and 3.81%, respectively. This indicates that the improved bearing capacity primarily stems from sufficient utilization of the bolts bending performance. Furthermore, as shown in Fig.24(a), UHPC accounts for approximately 27.52% of the axial tension force, partially sharing the axial tension force of the bolts. As shown in Fig.24(b), the steel plates and UHPC bear all the axial compression forces in the later stage, with the former accounting for about 86.72%. However, due to the further reduction in the height of the compressed zone, as shown in Fig.25, the compressive resistance of the UHPC could not be fully utilized. The strengthening mechanism of the joint subjected to a hogging moment involves further reducing the height of the compressed zone by utilizing the ultra-high compression performance of the steel plates and UHPC. This also fully taps into the bending performance of the bolts, and thereby improvs the bearing capacity to a certain extent. The strengthening effect is weaker under hogging moments than under sagging moments. The main reason is the limited improvement resulting from further utilization of the bolt bending performance, which falls significantly short of fully utilizing the strengthening materials, namely steel plate and UHPC.
6 Conclusions
This study presents a mechanical model for longitudinal joints strengthened by a steel plate-UHPC composite. A novel continuous calculation method was proposed and verified well with numerical result. Strengthening effect and mechanism under both sagging and hogging moment were further discussed. Parameter analysis was conducted to investigate the influences of strengthening layer thickness and axial force on the bearing capacity and stiffness. The main conclusions are as follows.
1) The proposed calculation method precisely captures the mechanical response of the joint before and after strengthening. For joints under sagging moments, the un-strengthened joint response can be divided into four stages, which becomes five stages after strengthening. For joints under hogging moments, both before and after strengthening, structural response consists of three stages. In either case, the yielding of the bolts clearly signifies structural failure.
2) The strengthening effect differs under different types of moments. Under sagging moments, the strengthened joints exhibit substantial improvements in the performance indicators S1, S2, K1, and K2, by 5.42, 4.14, 2.71, and 2.93 times, respectively. Strengthening significantly heightens structural bearing capacity. While, under hogging moments, the performance indicators S1, S2, K1, and K2 improve by 1.33, 1.65, 5.66, and 4.46 times, respectively. It notably bolsters structural stiffness. Overall, the strengthening effect of joints under sagging moments outperforms that under hogging moments.
3) The strengthening effect is influenced differently by the thickness of steel plates and UHPC, as well as the axial force levels. Parameter analysis shows that increasing steel plate thickness benefits bearing capacity under sagging moments, while UHPC thickness optimally improves overall joint stiffness. For hogging moments, increasing axial force level efficiently addresses insufficient bearing capacity improvement. Increasing UHPC thickness and axial force level significantly improve structural stiffness. Optimal strengthening involves avoiding excessive steel plate thickness, appropriately increasing UHPC thickness, and ensuring sufficient axial force level. This provides guidance for further harmonizing the mechanical performance differences of joints under sagging and hogging moments.
4) The strengthening mechanism differs between sagging and hogging moments. The strengthening mechanism under sagging moments is to fully utilize the tensile performance of the strengthening materials and exploits the mechanical properties of the compressed zone concrete. This simultaneous enhancement of tensile and compressive zone results in remarkably strengthening effects. The strengthening mechanism under hogging moments emphasizes further exploring the bending resistance of the bolts, with maximizing the utilization of compressive performance of local UHPC and steel plate.
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