A Novel Back-Analysis Approach for the External Loads on Shield Tunnel Lining in Service Based on Monitored Deformation

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Introduction
Te intense urbanization demands the large development of underground infrastructure. Te utility tunnels in urban areas, taking metro tunnels as examples, are mainly constructed by shield machines given the soil ground conditions. With the huge construction of metro tunnels, the maintenance of in-service shield tunnels has become a critical concern for the regions with soil ground. Existing and potential defects will reduce the performance of inservice shield tunnels [1,2]. Load condition is critical for evaluating the structural performance of shield tunnels when considering the maintenance scheme of the shield tunnel structure [3,4].
According to previous research on the structural performance of shield tunnels [2,5,6], the load-structure method is widely employed for the evaluation of shield tunnels, where diferent load-distribution patterns such as linear patterns and parabolic patterns are adopted. Furthermore, the stratum-structure method is used to calculate the loads on the lining structure [7,8], in which the interaction between the surrounding ground and structure is taken as the external load on the lining. During the operational phase of shield tunnels, the load on the lining is generally infuenced by multiple factors, including adjacent construction [9,10], surcharge load [11], vehicle vibration [12,13], and long-term water leakage [14]. Tese uncertain factors during service will result in structural deformation in both longitudinal and transverse directions [15,16]. However, the load-structure method and stratum-structure method are limited when considering the variation of the load distribution on the lining structure [17,18]. Hence, the load models for both the load-structure method and stratum-structure method should be updated when evaluating the performance of in-service shield tunnels. While in most cases, it is not feasible to obtain the external loads of shield tunnels by using either pressure gauges or the classic earth pressure formulas [18].
Back analysis is considered an efective approach for determining unknown parameters in geotechnical engineering [19][20][21]. For the external loads on the shield tunnel lining, the back-analysis methods can be classifed into two types, namely strain-based and deformation-based methods. Te strain-based methods are widely used for the estimation of external loads during tunnelling, with promising accuracy [22] while the strain of the lining will be required as the input, which is costly and hard to be collected for shield tunnels in service. Hence, the application of strain-based methods on the in-service shield tunnels is not practical [23,24]. Te deformation-based methods provide feasible perspectives for in-service shield tunnels, given the accessibility of deformation monitoring. Te existing deformation-based methods are mainly proposed and applied for rock tunnels [25], the structures of which are quite diferent from shield tunnels. Te property and in-situ stress of surrounding rock can be estimated by performing back analysis of matching numerical modelling results with the measured tunnel convergence [26,27]. Furthermore, the long-term creep and water efects on rock tunnel lining can be back-analyzed based on deformation [28]. It is noted that most of the existing back-analyzed parameters are generalized, heavily dependent on the selected theoretical models, and afected by limited or incomplete monitoring data [29][30][31]. As for the back analysis of external loads, full-ring load distribution is not taken into consideration in related research currently [18,[32][33][34]. In general, the application of deformation-based methods on shield tunnel loads remains to be solved, also due to the difculty in the interpretation of deformation patterns [35]. Recently, meta-model methods based on numerical simulation have been applied in the back analysis of tunnel engineering [36][37][38], where complex optimization algorithms with expensive computational costs are required. Te load distribution on the shield tunnel lining in service is rarely investigated due to the limitations of the existing methods.
In this work, the back-analysis method for the external load on the lining structure in service is investigated from the perspective of Betti's theorem. Te physical models required in Betti's theorem can be replaced by the corresponding numerical models. Ten, a novel back-analysis approach is proposed to calculate the load on tunnel lining in the operational phase based on Betti's theorem and smallstrain assumption. Tis paper is composed of the following 5 parts: (1) the principle and the relation between Betti's theorem, i.e., Maxwell-Betti reciprocal work theorem, and the proposed back-analysis approach are illustrated; (2) the loads on the shield tunnel lining in service were analyzed by conventional load-structure models and scanned deformation; (3) the back-analysis approach was proposed based on Betti's theorem and the load-structure model; (4) the feasibility of the proposed approach was investigated theoretically; (5) the proposed back-analysis approach was verifed by comparison with the model test results and applied for practical engineering.

Betti's Teorem.
Betti's theorem is one of the laws used for structural analysis [39]. Te deformation is one of the most important parameters for structural analysis. And Betti's theorem is widely used in the deformation calculation of structures in the design phase [40]. Te energy from external forces will be transformed into the strain energy of the elastic body according to the Law of Conservation of Energy. Tis principle can be employed for calculating the stress and strain of the deformable body subjected to external loads. It is usually assumed that other kinds of energy losses (e.g., acoustic energy and thermal energy) are small enough to be ignored during the deformation of the elastic body.
As shown in Figure 1, if an elastic body is subjected to a set of external forces, i.e., the body force (f x , f y , f z ) and surface force (q x , q y , q z ), the strain can be expressed in the Cartesian coordinate system as follows: where δ x , δ y , and δ z are the displacement caused by the external forces along the three Cartesian coordinate orientations, ε x , ε y , and ε z are the three normal strains in the elastic body, and c xy , c yz , and c xz are the three shear strains. For the second set of external forces, as illustrated in Figure 1, the induced strain by these forces can be expressed as follows: where (ε x ′ , ε y ′ , ε z ′ ) and (c xy ′ , c yz ′ , c xz ′ ) are the normal and shear strains induced by the set of external forces (f x ′ , f y ′ , f z ′ ) and (q x ′ , q y ′ , q z ′ ), respectively. Te energy (ε 12 ) from the frst set of forces associated with the deformation caused by the second set of forces can be expressed as follows:

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Likewise, the energy (ε 21 ) from the frst set of forces associated with the deformation caused by the second set of forces can be expressed as follows: According to Betti's theorem, the following work balance equation is obtained: Without considering the body inertia forces, the following expression is obtained: Tis matrix-form equation can be written as follows: It can be seen from equation (7) that either set of external forces can be determined based on another three items in the equation if the three items are provided. Te property of the theorem enables the determination of loads based on the measured deformation [41,42]. Hence, the theorem can be applied in the health monitoring of shield tunnels to solve the challenges induced by the uncertainty of the changing load conditions during service. Te illustrated derivation process is restricted to the background of linear-elastic material while Betti's theorem has been expanded to more generalized backgrounds [43], including elastic-plastic material, which is the main constitutive model applied in tunnel engineering.

Te Load-Deformation Model.
Te shield tunnel lining is positioned underground by shield machines. A ring of tunnel lining is a cylindrical structure assembled by prefabricated segments. One of the most widely adopted load patterns of the tunnel lining during construction is illustrated in Figure 2. Te external loads acting on the tunnel lining are usually assumed to be linearly or quadratically distributed around the shield tunnel lining [34], including the vertical pressure at the tunnel crown, the horizontal ground pressure, and the ground reaction at the tunnel bottom. However, the distribution modes of external load on the lining in service are usually way more complex than the simplifed assumption.
Te laser scanning data of lining rings with typical deformation in a metro line are displayed in Figure 3, denoted as Example I, Example II, and Example III, where the deformation is scaled 10 times for demonstration. Te arbitrary and random patterns of lining deformation can be found. Furthermore, the location distribution of the maximum convergence involving successive 1100 lining rings is provided in Figure 4, where the uncertainty of deformation induced by various load patterns is displayed clearly. It is noted that only the ring-wise deformation is discussed here. Te variation of deformation patterns may be greater if the segment-wise deformation is taken into consideration. Te lining ring deformation is dominated by stochasticity due to the unclear pressure of soil and water. According to the conventional assumption depicted in Figure 2, the deformation patterns should be symmetric and similar to each other while it can be observed that the deformation patterns of the shield tunnel lining rings in service are irregular and diferent from each other. Hence, it is unreasonable if the conventional assumption is used for mechanical analysis of the life-cycle performance. It is found that the lining ring of shield tunnels may be subjected to eccentric or irregular load patterns during service due to various factors [44,45], including water leakage, vehicle vibration, adjacent excavation, and surface loading. It is indicated that the description of the load patterns should cover diferent cases.     Structural Control and Health Monitoring

Te Back-Analysis Model for External Loads.
As plotted in Figure 5(a), the arbitrarily distributed ground-structure interaction on the shield tunnel lining can be simplifed as a series of loads distributed around the tunnel lining (denoted as x i (i � 1, 2, . . . , n)), where x i denotes the external load on the lining. Te corresponding structural deformation is denoted by w i (i � 1, 2, . . . , n). Te set of loads is denoted as Load Case I. And the lining ring is assumed to be subjected to another designed set of loads F i (i � 1, 2, . . . , n), denoted as Load Case II and shown in Figure 5(b), with corresponding structural deformation v i (i � 1, 2, . . . , n). Load Case I is the real load condition on the structures, and Load Case II is an assumed load condition. Te loads in both load cases can be generalized loads, including distributed force, concentrated force, and moment. Te distribution of the loading positions in Load Case II depends on the deformation measuring points. According to Betti's theorem, if the material of the tunnel lining is linear elastic, the following equation between Case I and Case II can be obtained: where v n is the structural deformation along the direction of x n in Load Case II (subjecting to the load set f n ). Likewise, w n is the lining deformation along the direction of f n in Load Case I (subjecting to the load set x n ). As shown in equation (8), the load set x n can be calculated based on the other three items, i.e., f n , v n , and w n . In order to solve the unknown load set x n , a total number of n load cases should be created frst, which constitutes the equation set shown as follows:  Figure 4. Varieties of methods and corresponding instruments have been proposed for deformation measurement in shield tunnels, such as total station, laser convergence gauge, tilt sensor, and optical-fbre sensors [46]. However, the deformation derived by the above methods is incomplete. Tese methods are designed for isolated measuring points and usually rely on instruments installed on the lining while the development of laser scanning technology enables the efcient acquisition of large point cloud data during the life cycle of shield tunnels [16], in which detailed three-dimensional space information is included. Te accuracy of deformation measurement based on laser scanning is proved to archive the submillimetre level with proper data processing [47]. Te fullring deformation can be measured based on the point cloud data by the following steps, including preprocessing the point cloud, extracting the tunnel axis, performing the coordinate transformation, performing noise reduction, and ftting the deformation [48]. Te automatic workfow of the steps has also been developed [49]. It is indicated that the application of laser scanning in deformation measurement has the advantages of comprehensiveness and efciency [16]. Te overall workfow of the back-analysis method is demonstrated in Figure 6. In general, there are two main steps in the back analysis of the external load on the tunnel lining during service. Te frst main step is to build and calculate the numerical model of the prototype shield tunnel lining. Ten, the back analysis can be carried out by the equations derived from equation (9) while it is noted that the existence and uniqueness of the solution should be investigated before further study.

Existence and Uniqueness of the Solution.
Segments of shield tunnels are assembled by steel bolts, which form the joints between segments. Although the mechanical behaviour of joints may be nonlinear and complex, it is indicated that the connecting bolts can be regarded as a linear-elastic material under the small-strain condition [50]. In general, the material of the lining rings can also be assumed to be linear elastic under the small-strain condition [51]. Te proposed back-analysis approach can be used for determining the external load on the shield tunnel lining since the requirements of applying Betti's Teorem are satisfed.
For a nonhomogeneous linear equation Ax � b, the uniqueness of the solution is required, which means that the matrix A will be full rank and can be expressed as follows: Te matrix A is built by n groups of load cases. Tus, the rationality of the load cases constructed in numerical simulations is of great importance. If the load cases are not rational, the matrix might be ill-posed which results in an incorrect solution [52]. Te linear equation sets are solved by the singular value decomposition method using MATLAB in this work.

Physical Model.
Te external load on tunnel lining is hard to monitor during service due to the low durability of pressure sensors. Terefore, a full-scale model test on the lining structure [53] is employed to verify the accuracy and feasibility of the proposed back-analysis method. As shown in Figure 7, a single-layer tunnel lining with a thickness of 0.48 m and a width of 1.50 m is adopted in the model test, which is cast with concrete C55. Te outer diameter of the lining ring is 6.80 m. Segments are connected by bolts of M36 steel, of which the positions are shown in Figure 7. Te segments are fxed to steel support at the end and are subjected to the vertical load P and the horizontal load N.

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Te vertical load was given by vertical hydraulic jacks through a set of stif steel distribution beams. At the end of the tunnel segment specimen, the horizontal load was provided by hydraulic jacks. Te mechanical behaviours of segments under the efect of both positive and negative bending moments were investigated in the model test. Te direction of vertical load P in the case of positive bending moment was downward and was upward otherwise.
Te positions of deformation measuring points are also marked in Figure 7. Te vertical displacement of the segments was measured using two vertical displacement sensors D1 and D2. Te deformation of joints at both the external side and internal side was recorded by a pair of displacement transducers V1 and V2.

Numerical Model.
For the proposed approach, the numerical model is required for building the equations to calculate the actual external load on the shield tunnel lining. Te physical model is adopted as the prototype of the numerical model. Te fnite element software ANSYS10.0 is employed to carry out the 3D mechanical   analysis. Te idea of simulation adopted is based on the strip method. Te parts of the fnite element model for further analysis are shown in Figure 8, corresponding to the model test illustrated in Figure 7. Te joint with a width of 110 mm is modelled by the material with the reduced Young's modulus. Te friction between the surfaces of concrete is not considered in the simulation for lower computational cost. Te shear force on the joint is assumed to be mainly undertaken by the connecting bolts, of which the rationality has been proved [54]. Te connecting bolts were modelled according to the actual length, with both ends fxed. Te element solid65 and element beam188 are used for lining segments and connecting bolts, respectively. Te mechanical properties of diferent materials in the fnite element model are demonstrated in Table 1.
To verify the rationality of the numerical model, the vertical displacement and joint opening of the specimens are compared with the results of the model test, displayed in Figure 9. For the same value of loads in the model test, the greatest value of deformation is selected for comparison. Te test data in the unloading phase are not considered since the purpose of this work is not to investigate elaborated models. It can be found that the simulation results ft the model test well, especially in the phase of small loads. It is indicated that the numerical model is rational to be used for the back analysis of the external loads on the specimens of the fullscale model test.

Back Analysis of the Model Test.
Te external loads on the segment specimens in the model test are denoted as Load Case I, i.e., [X i ] 4×1 (i � 1, 2, 3, 4), where X 1 , X 2 is equal to the vertical load P and X 3 , X 4 is equal to horizontal load N. Te corresponding displacement at the loading position is denoted as [W i ] 4×1 (i � 1, 2, 3, 4), which was calculated geometrically based on the monitored deformation in the model test.
Based on the physical model, the numerical model was created to calculate the deformation under the efect of Load Case II [F ij ] 4×4 (i, j � 1, 2, 3, 4), where F ii denotes the assumed load, and other elements are equal to 0. Te loads are assumed to act at the position of measuring points D1, D2, V1, and V2, including concentrated force and concentrated moment. Te corresponding deformation calculated in the numerical model is denoted as [V ij ] 4×4 (i, j � 1, 2, 3, 4), where V ij is the displacement at the position of the j-th load of Load Case I induced by the i-th load of Load Case II. Ten, the back-analysis equations based on Betti's theorem can be expressed as follows: Tere were 50 sets of data collected in the loading phase of the model test, which were adopted for validation. Te comparison between the back-analysis results and the loads monitored in the model test is presented in Figure 10. In general, the back-analysis results achieved promising accuracy for both vertical and horizontal loads. Te average errors are 76.01 kN for vertical loads and 50.55 kN for horizontal loads. Te errors may become greater with larger loads, which is similar to the errors of the numerical simulation shown in Figure 9. Te back-analysis errors are acceptable given that the numerical simulation was performed based on the assumption of small deformation and elastic materials. It is indicated that the proposed backanalysis approach can be used for calculating the external loads on the lining structure.

Practical Application.
Te numerical simulation models for the practical application were built to carry out the fullring mechanical analysis, displayed in Figure 11. Te concrete segment, steel reinforcement, joints, and connecting bolts are all considered in the simulation model. Te two load cases adopted here are the same as illustrated in Figure 5. Te number of loads in the load cases, which is 6 here, depends on the demands of mechanical analysis. For both Load Case I and Load Case II, the 6 loads are designed at the centre of each segment. Te full-ring load value can be calculated via spline interpolation. With the simulation results, the equation for the back analysis of load on the lining can be derived from equation (9), written as follows: Te point clouds of 3 lining rings, namely Ring I, Ring II, and Ring III, in a shield tunnel project, were selected for the demonstration of the practical application. Te deformation of the selected lining rings is plotted in Figure 12, of which the maximum convergence is around 0.5%, 1.0%, and 1.5% of the 6.2 m outer diameter, respectively. It is noted that the deformation in Figure 12 is scaled 10 times for easier understanding. With the scanned deformation as the input, the loads on the lining can be calculated by equation (9). Te back-analysis results of the loads with spline interpolation Structural Control and Health Monitoring    Structural Control and Health Monitoring processing are plotted in Figure 13. Te irregular patterns of the loads can be found in the results, which are compatible with the scanned deformation.

Discussion
Te proposed method provides a novel way for tunnel lining load back-analysis, without the time-consuming optimization process. Because of the irregularity in the load distribution mode, the more detailed deformation data will bring in more accurate load back-analysis results. Hence, the proposed method is suitable to be combined with laser scanning, which has the strength in acquiring a huge amount of deformation data. Te proposed back-analysis method is validated by the comparison with test results. Te accurate results of back-analysis have shown promising efectiveness and feasibility. Due to the nonaccessibility of in-situ load monitoring data, the proposed method is to be applied in the practical project for further validation. Te corresponding simulation model has been built to analyze the mechanical performance of the full-ring lining. For Structural Control and Health Monitoring demonstrating the practical application of the proposed back-analysis method, the scanned full-ring deformation data are used to calculate the loads on a shield tunnel in service. Te results prove that the proposed method can be used to reveal irregular load patterns, which are closer to reality. Te proposed back-analysis method could be further applied in exploring the behaviour of load evolution for in-service shield tunnels. Te background of this work is in the feld of shield tunnel engineering. Te theory of the proposed method is not in confict with other tunnel structures and conditions. With proper numerical modelling methods, the proposed back-analysis method can be applied to other types of tunnels, including immersed-tube tunnels. Te assumption of earth pressure, however, should be carefully considered when it comes to more complicated ground situations. It is also noted that the infuencing factors of deformation are more than external loads. Te deformation induced by the evolution of the material and structure [55] should be further considered to pursue better accuracy.
Te cost time of the proposed methods can be divided into two parts, i.e., the ofine stage and the online stage. In the ofine stage, plenty of simulation results will be required to establish the matrixes and equations. Hence, the cost time in the ofine stage involves the time of numerical computation, which depends on the demanded accuracy and the levels of the numerical models. In the online stage, the cost time is less, which involves the calculation of equations only.

Conclusions
Based on Betti's theorem, the back-analysis approach is proposed in this work to study the external loads on the lining structure in the operational phase. Te external load on the tunnel structure can be calculated by using the in-situ monitoring data and numerical simulations. For the in-situ monitoring, the deformation of the tunnel should be measured. According to the numerical simulation results, deformations of tunnel structure can be obtained for multiple virtual load cases. Based on the actual deformation, virtual deformation, and virtual load, the actual loads acting on the tunnel lining can be back-analyzed by using the singular value decomposition method performed by MATLAB. Te proposed methods and workfows can be applied with the combination of laser scanning or other deformation measuring methods. Te proposed backanalysis method allows the calculation of external loads directly instead of soil properties. Te back analysis of the external loads on the lining does not involve complex optimization methods, which will minimize the computational cost of back analysis.
Te data of a model test were used to verify the feasibility of the back-analysis approach. Meanwhile, the fnite element model is built according to the physical model of the model test. Te simulation result shows satisfying agreement with the test data. It is noted that since the rationality of the backanalyzed results depends highly on the accuracy of the numerical simulation, the model should be calibrated carefully before the load back analysis. Te practical application with the scanned full-ring deformation is carried out successfully. It is indicated the back-analysis approach can be used for analyzing the loading condition on tunnel lining in operation based on the deformation data measured in practice.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.