Xiao et al 2024a - Revision history

Rimni at 13:17, 9 April 2024

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Rimni at 12:23, 9 April 2024

2024-04-09T12:23:49Z

Rimni at 12:07, 9 April 2024

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Rimni at 11:48, 9 April 2024

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Rimni at 11:47, 9 April 2024

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Rimni at 11:35, 9 April 2024

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Rimni at 11:32, 9 April 2024

2024-04-09T11:32:02Z

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 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] The Institution of Civil Engineers (Japan). Standard specification for tunnels (shield) and explanations, 2001.
+[1]  Standard specification for tunnels (shield) and explanations. The Institution of Civil Engineers, Japan, 2001.
 [2] Zhu H.H., Yang L.D., Chen Q.J., et al. Two force design models for the lining system of tubular joints in shield tunnels.Proceedings of the Fifth National Academic Conference on Structural Engineering. Tsinghua Publishing Press, 400-404, 1996.

@@ Line 536: / Line 536: @@
 ==References==
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] The Institution of Civil Engineers [Japan].Standard specification for tunnels (shield) and explanations,2001.
+[1] The Institution of Civil Engineers (Japan). Standard specification for tunnels (shield) and explanations, 2001.
-[2] Zhu H.H., Yang L.D., Chen Q.J.,et al. Two force design models for the lining system of tubular joints in shield tunnels.Proceedings of the Fifth National Academic Conference on Structural Engineering.Tsinghua Publishing Press,400-404,1996.
+[2] Zhu H.H., Yang L.D., Chen Q.J., et al. Two force design models for the lining system of tubular joints in shield tunnels.Proceedings of the Fifth National Academic Conference on Structural Engineering. Tsinghua Publishing Press, 400-404, 1996.
-[3] Zhu H.H, Cui M.Y., Yang J.S. Design model for shield lining segments and distribution of load.Chinese Journal of Geotechnical Engineering,22(2): 190-194,2000.
+[3] Zhu H.H, Cui M.Y., Yang J.S. Design model for shield lining segments and distribution of load. Chinese Journal of Geotechnical Engineering, 22(2):190-194, 2000.
-[4] Peng Y.C., Ding W.Q, Zhu H.H., Zhao W., Jin Y.L. Shell-joint model for lining structures of shield-driven tunnels.Chinese Journal of Geotechnical Engineering,35(10): 1823-1829,2013.
+[4] Peng Y.C., Ding W.Q, Zhu H.H., Zhao W., Jin Y.L. Shell-joint model for lining structures of shield-driven tunnels. Chinese Journal of Geotechnical Engineering, 35(10):1823-1829, 2013.
-[5] Xie J.C., Wang J.C, Huang W.M. Nonlinear structural analysis on cracking behavior of shield tunnel segment under surface loading.Journal of Railway Science and Engineering,18(01):162-171,2021.
+[5] Xie J.C., Wang J.C, Huang W.M. Nonlinear structural analysis on cracking behavior of shield tunnel segment under surface loading. Journal of Railway Science and Engineering, 18(01):162-171, 2021.
-[6] Yang Y.B., Xie X.Y. Breaking mechanism of segmented lining in shield tunnel based on fracture mechanics.Chinese Journal of Rock Mechanics and Engineering,34(10):2114-2124,2015.
+[6] Yang Y.B., Xie X.Y. Breaking mechanism of segmented lining in shield tunnel based on fracture mechanics. Chinese Journal of Rock Mechanics and Engineering, 34(10):2114-2124, 2015.
-[7]  Liu H.Q., Liu H.B. Numerical investigation on the mechanical behaviour of shield tunnel segment and their longitudinal joint.Chinese Journal of Underground Space Engineering,15(06):1800-1810+1873,2019.
+[7]  Liu H.Q., Liu H.B. Numerical investigation on the mechanical behaviour of shield tunnel segment and their longitudinal joint. Chinese Journal of Underground Space Engineering, 15(06):1800-1810+1873, 2019.
-[8] Zhang L., Feng K., He C.,et al. Three-dimensional refined numerical simulation of segmental joint of shield tunnel.Tunnel Construction,40(8):1169,2020.
+[8] Zhang L., Feng K., He C., et al. Three-dimensional refined numerical simulation of segmental joint of shield tunnel. Tunnel Construction, 40(8):1169, 2020.
-[9] Ge S.P., Xie D.W., Ding W.Q., Ouyang W.b. Simplified numerical simulation method for segment joints of shield tunnels.Chinese Journal of Geotechnical Engineering,35(09):1600-1605,2013.
+[9] Ge S.P., Xie D.W., Ding W.Q., Ouyang W.b. Simplified numerical simulation method for segment joints of shield tunnels. Chinese Journal of Geotechnical Engineering, 35(09):1600-1605, 2013.
-[10] Science and Technology Committee of Shanghai Municipal Construction Commission.Metro Line 1 Project.Shanghai Science and Technology Press,1998.
+[10] Science and Technology Committee of Shanghai Municipal Construction Commission. Metro Line 1 Project. Shanghai Science and Technology Press, 1998.
-[11] Zhou H.Y. Theoretical study and test on the mechanical behavior of lining segment in shield tunnel.Dalian University of Technology,2011.
+[11] Zhou H.Y. Theoretical study and test on the mechanical behavior of lining segment in shield tunnel. Dalian University of Technology, 2011.
-[12] Liu X., Zhang H.l., Lu L., Wang X.Z. Experimental study on loade bearing capacity of shield tunnel structure under overload condition.Underground Engineering and Tunnels,(04):10-15+59,2013. (in Chinese)
+[12] Liu X., Zhang H.l., Lu L., Wang X.Z. Experimental study on loade bearing capacity of shield tunnel structure under overload condition (in Chinese). Underground Engineering and Tunnels, (04):10-15+59, 2013.
-[13] Bi X.l., Liu X., Wang X.Z., Lu L., Yang Z.H. Experimental investigation on the ultimate bearing capacity of continuous-jointed segmental tunnel linings.China Civil Engineering Journal.47(10):117-127,2014.
+[13] Bi X.l., Liu X., Wang X.Z., Lu L., Yang Z.H. Experimental investigation on the ultimate bearing capacity of continuous-jointed segmental tunnel linings. China Civil Engineering Journal, 47(10):117-127, 2014.
-[14] Lu L., Sun Y.F,Liu X., Wang X.Z., Wang W.P. Full-ring Experimental Study on the Ultimate Bearing Capacity of the Lining Structure of the Metro Shield Tunnel.Structural Engineers.28(06):134-139+180,2012.
+[14] Lu L., Sun Y.F, Liu X., Wang X.Z., Wang W.P. Full-ring experimental study on the ultimate bearing capacity of the lining structure of the metro shield tunnel. Structural Engineers, 28(06):134-139+180, 2012.
-[15] Liu X., Dong Z.B., Bai Y., et al. Investigation of the structural effect induced by stagger joints in segmental tunnel linings: First results from full-scale ring tests. Tunnelling and Underground Space Technology. 66 (1), 1-18, 2017.
+[15] Liu X., Dong Z.B., Bai Y., et al. Investigation of the structural effect induced by stagger joints in segmental tunnel linings: First results from full-scale ring tests. Tunnelling and Underground Space Technology, 66(1):1-18, 2017.
-[16] LI Q.T. Numerical modeling analysis of structural defects of shield tunnel in soft soil area.Railway Standard Design.65(01):95-103,2021.
+[16] Li Q.T. Numerical modeling analysis of structural defects of shield tunnel in soft soil area. Railway Standard Design, 65(01):95-103, 2021.
-[17] Huang H.W., Xu L., Yan J.L., Yu Z.K. Study on transverse effective rigidity ratio of shield tunnels.Chinese Journal of Geotechnical Engineering.(01):11-18,2006.
+[17] Huang H.W., Xu L., Yan J.L., Yu Z.K. Study on transverse effective rigidity ratio of shield tunnels. Chinese Journal of Geotechnical Engineering, (01):11-18, 2006.
-[18] Wang T., Mo H.h., Chen J.S, Yang C.S. The impact of staggered joint and soil stiffness on transverse effective rigidity ratio of shield tunnel.Railway Standard Design.59(09):136-140,2015.
+[18] Wang T., Mo H.h., Chen J.S, Yang C.S. The impact of staggered joint and soil stiffness on transverse effective rigidity ratio of shield tunnel. Railway Standard Design, 59(09):136-140, 2015.
-[19] Yang X., Luo Y.P, Zhang Z.B., Wang Z.Y. A model-test-based study of the anti-bending effectiveness of the longitudinal and transverse rigidity of shield-drilled tunnels.Traffic Engineering and Technology for National Defence.18(04):39-42,2020.
+[19] Yang X., Luo Y.P, Zhang Z.B., Wang Z.Y. A model-test-based study of the anti-bending effectiveness of the longitudinal and transverse rigidity of shield-drilled tunnels. Traffic Engineering and Technology for National Defence, 18(04):39-42, 2020.
-[20] Chen H.C. Study on transverse mechanical behavior of shield tunnel based on model tests and its application in surface surcharge .South China University of Technology,2020.
+[20] Chen H.C. Study on transverse mechanical behavior of shield tunnel based on model tests and its application in surface surcharge, South China University of Technology, 2020.
-[21] Li X.J, Huang B.Q., Yang Z.H., Li X.X. Lateral equivalent stiffness of large diameter shield tunnel lining structure at various buried depths.Journal of tongji University(Natural Science).43(08):1159-1166,2015.
+[21] Li X.J, Huang B.Q., Yang Z.H., Li X.X. Lateral equivalent stiffness of large diameter shield tunnel lining structure at various buried depths. Journal of Tongji University (Natural Science), 43(08):1159-1166, 2015.
-[22] Feng K., He C., Xia S.L. Prototype tests on effective bending rigidity ratios of segmental lining structure for shield tunnel with large cross-section.Chinese Journal of Geotechnical Engineering.33(11):1750-1758,2011.
+[22] Feng K., He C., Xia S.L. Prototype tests on effective bending rigidity ratios of segmental lining structure for shield tunnel with large cross-section. Chinese Journal of Geotechnical Engineering, 33(11):1750-1758, 2011.

@@ Line 293: / Line 293: @@
 |}
-where, <math display="inline">{C}_{1}</math>, <math display="inline">{C}_{2}</math>, and <math display="inline">{C}_{3}\,</math> are constants; <math display="inline">{E}_{0}</math> is the elastic modulus of the joint; <math display="inline">\, \eta</math> is  the transverse stiffness efficiency. The constant values <math display="inline">C_1 </math>, <math display="inline">C_2 </math>, <math display="inline">C_3 </math> in Eq.(2) are given in [[#tab-1|Table 1]].
+where, <math display="inline">{C}_{1}</math>, <math display="inline">{C}_{2}</math>, and <math display="inline">{C}_{3}\,</math> are constants; <math display="inline">{E}_{0}</math> is the elastic modulus of the joint; <math display="inline">\, \eta</math> is  the transverse stiffness efficiency. The constant values <math display="inline">C_1 </math>, <math display="inline">C_2 </math>, and <math display="inline">C_3 </math> in Eq.(2) are given in [[#tab-1|Table 1]].
 <div id='img-13'></div>

@@ Line 281: / Line 281: @@
 ==4. Empirical formula of the elastic modulus for weakening joint==
-In the simulation of single-ring and three-rings lining, the elastic modulus of the joint is 50MPa. Although the bearing capacity, deformation of joint are basically consistent with the experiment results, there  are still some differences. And the stiffness of the tunnel lining is related to the type of assemblage too, we try to find an empirical formula of the elastic modulus for the weakening joint no matter how the lining is assembled. Considering the transverse stiffness efficiency reflects the influence of joint stiffness on the lining ring stiffness, the formula of elastic modulus to the transverse stiffness efficiency is found, which can provide a good reference on the determination of elastic modulus for the joint weakening simulation. Taking the three-rings lining as the research object, the relationship between the transverse stiffness efficiency and the elastic modulus of the joint under the four different assemblages which are straight joint, stagger joint-22.5°, stagger joint-45° and stagger joint-180° is analyzed, as shown in [[#img-13|Figure 13]]. It can be seen that the elastic modulus varies nonlinearly with the transverse stiffness efficiency. The curve is fitted by exponential function, that is :
+In the simulation of single-ring and three-rings lining, the elastic modulus of the joint is 50 MPa. Although the bearing capacity, deformation of joint are basically consistent with the experiment results, there  are still some differences. And the stiffness of the tunnel lining is related to the type of assemblage too, we try to find an empirical formula of the elastic modulus for the weakening joint no matter how the lining is assembled. Considering the transverse stiffness efficiency reflects the influence of joint stiffness on the lining ring stiffness, the formula of elastic modulus to the transverse stiffness efficiency is found, which can provide a good reference on the determination of elastic modulus for the joint weakening simulation. Taking the three-rings lining as the research object, the relationship between the transverse stiffness efficiency and the elastic modulus of the joint under the four different assemblages which are straight joint, stagger joint-22.5°, stagger joint-45° and stagger joint-180° is analyzed, as shown in [[#img-13|Figure 13]]. It can be seen that the elastic modulus varies nonlinearly with the transverse stiffness efficiency. The curve is fitted by exponential function, that is :
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 361: / Line 361: @@
-In the above three-rings lining simulation, the elastic modulus of the joint is 50MPa. According to the deformation in the experiment and the simulated homogeneous ring, the transverse stiffness efficiency is about 0.43. The elastic modulus of the joint should be 23 MPa calculated from Eq.(2). A new model with 23 MPa weakening joint is built again for the same three-rings lining. The vertical convergence-load curve is shown in [[#img-14|Figure 14]]. It can be seen that the model built according to the value of <math display="inline">E_0 </math> calculated by Eq.(2) is more consistent  with the experiment results than the former model in the elastic period and the elastic-plastic stage after the lining joint fails. In the failure stage of the joint, there is a bigger difference between the simulation and the experiment. The reason is that the transverse stiffness efficiency depends on the deformation value in the elastic period. In general, when the elastic modulus of the joint in three-ring lining joint is adjusted to 23 MPa, the simulation results are closer to the test results than the former elastic modulus 50 MPa, which means that the above formula is good for the determination of elastic modulus of the joint.
+In the above three-rings lining simulation, the elastic modulus of the joint is 50 MPa. According to the deformation in the experiment and the simulated homogeneous ring, the transverse stiffness efficiency is about 0.43. The elastic modulus of the joint should be 23 MPa calculated from Eq.(2). A new model with 23 MPa weakening joint is built again for the same three-rings lining. The vertical convergence-load curve is shown in [[#img-14|Figure 14]]. It can be seen that the model built according to the value of <math display="inline">E_0 </math> calculated by Eq.(2) is more consistent  with the experiment results than the former model in the elastic period and the elastic-plastic stage after the lining joint fails. In the failure stage of the joint, there is a bigger difference between the simulation and the experiment. The reason is that the transverse stiffness efficiency depends on the deformation value in the elastic period. In general, when the elastic modulus of the joint in three-ring lining joint is adjusted to 23 MPa, the simulation results are closer to the test results than the former elastic modulus 50 MPa, which means that the above formula is good for the determination of elastic modulus of the joint.
 <div id='img-14'></div>

@@ Line 82: / Line 82: @@
 ===2.3 result analysis===
-The variation of joint deformation with axial force under positive and negative bending moments are shown in [[#img-3|Figure 3]], in which there are joint opening curve and joint compression curve. It can be seen that the tendency of joint deformation in opening and in compression coincides with the experiment results.  The change of the elastic modulus of the joint under positive bending moment has much more influence on the curve of joint deformation than under negative bending moment, which means that the value of elastic modulus affects the behavior of joint heavily when the joint is in the positive bending moment. From the [[#img-3|Figure 3]], it is known that the difference between the numerical analysis results and the experiment results is the smallest when the elastic modulus of the joint is 50M Pa.
+The variation of joint deformation with axial force under positive and negative bending moments are shown in [[#img-3|Figure 3]], in which there are joint opening curve and joint compression curve. It can be seen that the tendency of joint deformation in opening and in compression coincides with the experiment results.  The change of the elastic modulus of the joint under positive bending moment has much more influence on the curve of joint deformation than under negative bending moment, which means that the value of elastic modulus affects the behavior of joint heavily when the joint is in the positive bending moment. From the [[#img-3|Figure 3]], it is known that the difference between the numerical analysis results and the experiment results is the smallest when the elastic modulus of the joint is 50 MPa.
 <div id='img-3'></div>

← Older revision		Revision as of 11:47, 9 April 2024
Line 50:		Line 50:


−	The axial force <math display="inline">N_1</math> is equal to and opposite to the horizontal force <math display="inline"> N </math>, which is 1500 kN, joint bending moment <math display="inline">M</math> is <math display="inline">\pm 225</math> kN·m (eccentricity e is <math display="inline">\pm 0.15</math> m). The relationship between vertical load <math display="inline">P_0</math> and horizontal axial force <math display="inline"> N </math> is:	+	The axial force <math display="inline">N_1</math> is equal to and opposite to the horizontal force <math display="inline"> N </math>, which is 1500 kN, joint bending moment <math display="inline">M</math> is <math display="inline">\pm 225</math> kN·m (eccentricity <math display="inline"> e </math> is <math display="inline">\pm 0.15</math> m). The relationship between vertical load <math display="inline">P_0</math> and horizontal axial force <math display="inline"> N </math> is:

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 148: / Line 148: @@
-It can be known that the performance of the joint in the simulation is most similar to the experiment result when the elastic modulus of the joint is 50MPa from the above analysis. The elastic modulus of the joint in the below single-ring and multi-ring lining will be taken as 50 MPa too.
+It can be known that the performance of the joint in the simulation is most similar to the experiment result when the elastic modulus of the joint is 50 MPa from the above analysis. The elastic modulus of the joint in the below single-ring and multi-ring lining will be taken as 50 MPa too.
 ==3. Verification of the applicability of the elastic modulus <math display="inline">E=50</math> MPa==
@@ Line 361: / Line 361: @@
-In the above three-rings lining simulation, the elastic modulus of the joint is 50MPa. According to the deformation in the experiment and the simulated homogeneous ring, the transverse stiffness efficiency is about 0.43. The elastic modulus of the joint should be 23MPa calculated from Eq.(2). A new model with 23 MPa weakening joint is built again for the same three-rings lining. The vertical convergence-load curve is shown in [[#img-14|Figure 14]]. It can be seen that the model built according to the value of <math display="inline">E_0 </math> calculated by Eq.(2) is more consistent  with the experiment results than the former model in the elastic period and the elastic-plastic stage after the lining joint fails. In the failure stage of the joint, there is a bigger difference between the simulation and the experiment. The reason is that the transverse stiffness efficiency depends on the deformation value in the elastic period. In general, when the elastic modulus of the joint in three-ring lining joint is adjusted to 23 MPa, the simulation results are closer to the test results than the former elastic modulus 50 MPa, which means that the above formula is good for the determination of elastic modulus of the joint.
+In the above three-rings lining simulation, the elastic modulus of the joint is 50MPa. According to the deformation in the experiment and the simulated homogeneous ring, the transverse stiffness efficiency is about 0.43. The elastic modulus of the joint should be 23 MPa calculated from Eq.(2). A new model with 23 MPa weakening joint is built again for the same three-rings lining. The vertical convergence-load curve is shown in [[#img-14|Figure 14]]. It can be seen that the model built according to the value of <math display="inline">E_0 </math> calculated by Eq.(2) is more consistent  with the experiment results than the former model in the elastic period and the elastic-plastic stage after the lining joint fails. In the failure stage of the joint, there is a bigger difference between the simulation and the experiment. The reason is that the transverse stiffness efficiency depends on the deformation value in the elastic period. In general, when the elastic modulus of the joint in three-ring lining joint is adjusted to 23 MPa, the simulation results are closer to the test results than the former elastic modulus 50 MPa, which means that the above formula is good for the determination of elastic modulus of the joint.
 <div id='img-14'></div>

@@ Line 519: / Line 519: @@
 ==5. Conclusions==
-Taking full-scale two segments including longitudinal joint of the experiment as the research object, the local weakening method is used to simulate the uneven stiffness caused by the connection between segments. By comparing the simulation results with the test results, the appropriate elastic modulus E<sub>0</sub> of the joint which is lower than the segment is selected, that is the weakening on the stiffness of the joint. Then, the full-scale single-ring lining structure and a three-rings lining structure of experiments are analyzed, in which the  local weakening method  is used to simulate the joint between segments and rings, and the numerical simulation results are compared with the test to verify the applicability of the method in the simulation of whole ring structure. Finally, the empirical formula for the determination of elastic modulus E<sub>0</sub> of the weakening joints in lining ring under different assemblage for different tunnel linings are given. The following conclusions are drawn:
+Taking full-scale two segments including longitudinal joint of the experiment as the research object, the local weakening method is used to simulate the uneven stiffness caused by the connection between segments. By comparing the simulation results with the test results, the appropriate elastic modulus <math> E_0 </math> of the joint which is lower than the segment is selected, that is the weakening on the stiffness of the joint. Then, the full-scale single-ring lining structure and a three-rings lining structure of experiments are analyzed, in which the  local weakening method  is used to simulate the joint between segments and rings, and the numerical simulation results are compared with the test to verify the applicability of the method in the simulation of whole ring structure. Finally, the empirical formula for the determination of elastic modulus <math> E_0 </math> of the weakening joints in lining ring under different assemblage for different tunnel linings are given. The following conclusions are drawn:
-(1) The varying of elastic modulus E<sub>0</sub> of the joint affects the deformation of the joint under positive bending moment, but has little influence on the deformation of the connection under negative bending moment. When the elastic modulus<math> E_0 </math>  is 50 MPa, the simulation results agree with the experiment results well.
+(1) The varying of elastic modulus <math> E_0 </math> of the joint affects the deformation of the joint under positive bending moment, but has little influence on the deformation of the connection under negative bending moment. When the elastic modulus<math> E_0 </math>  is 50 MPa, the simulation results agree with the experiment results well.
 (2) In the analysis of the full-scale sing-ring lining and three-rings lining using the local weakening method, it is found that the weakening method has high effectiveness for the simulation of the joints in Shanghai typical tunnel lining. But the elastic modulus <math> E_0 </math>  isn't unchanged 50 MPa. The elastic modulus <math> E_0 </math>  is affected by the assemblages of the tunnel lining.

@@ Line 517: / Line 517: @@
 |}
-==5. conclusion==
+==5. Conclusions==
-<span id='_Hlk127791228'></span>Taking full-scale two segments including longitudinal joint of the experiment as the research object, the local weakening method is used to simulate the uneven stiffness caused by the connection between segments. By comparing the simulation results with the test results, the appropriate elastic modulus E<sub>0</sub> of the joint which is lower than the segment is selected, that is the weakening on the stiffness of the joint. Then, the full-scale single-ring lining structure and a three-rings lining structure of experiments are analyzed, in which the  local weakening method  is used to simulate the joint between segments and rings, and the numerical simulation results are compared with the test to verify the applicability of the method in the simulation of whole ring structure. Finally, the empirical formula for the determination of elastic modulus E<sub>0</sub> of the weakening joints in lining ring under different assemblage for different tunnel linings are given. The following conclusions are drawn:
+Taking full-scale two segments including longitudinal joint of the experiment as the research object, the local weakening method is used to simulate the uneven stiffness caused by the connection between segments. By comparing the simulation results with the test results, the appropriate elastic modulus E<sub>0</sub> of the joint which is lower than the segment is selected, that is the weakening on the stiffness of the joint. Then, the full-scale single-ring lining structure and a three-rings lining structure of experiments are analyzed, in which the  local weakening method  is used to simulate the joint between segments and rings, and the numerical simulation results are compared with the test to verify the applicability of the method in the simulation of whole ring structure. Finally, the empirical formula for the determination of elastic modulus E<sub>0</sub> of the weakening joints in lining ring under different assemblage for different tunnel linings are given. The following conclusions are drawn:
-(1) The varying of elastic modulus E<sub>0</sub> of the joint affects the deformation of the joint under positive bending moment, but has little influence on the deformation of the connection under negative bending moment. When the elastic modulus E<sub>0</sub> is 50MPa, the simulation results agree with the experiment results well.
+(1) The varying of elastic modulus E<sub>0</sub> of the joint affects the deformation of the joint under positive bending moment, but has little influence on the deformation of the connection under negative bending moment. When the elastic modulus<math> E_0 </math>  is 50 MPa, the simulation results agree with the experiment results well.
-(2) In the analysis of the full-scale sing-ring lining and three-rings lining using the local weakening method, it is found that the weakening method has high effectiveness for the simulation of the joints in Shanghai typical tunnel lining. But the elastic modulus E<sub>0</sub> isn't unchanged 50MPa. The elastic modulus E<sub>0</sub> is affected by the assemblages of the tunnel lining.
+(2) In the analysis of the full-scale sing-ring lining and three-rings lining using the local weakening method, it is found that the weakening method has high effectiveness for the simulation of the joints in Shanghai typical tunnel lining. But the elastic modulus <math> E_0 </math>  isn't unchanged 50 MPa. The elastic modulus <math> E_0 </math>  is affected by the assemblages of the tunnel lining.
-(3) According to the relationship  between the transverse stiffness efficiency and the elastic modulus E<sub>0</sub>, the empirical formula of the elastic modulus E<sub>0</sub> is given, which can provide a reference for the determination of the elastic modulus of the weakening joint not only in the general tunnel lining (outer diameter is about 6m) but in the large-diameter (outer diameter is 10m) and super-large-diameter (outer diameter is 15m) tunnel lining.
+(3) According to the relationship  between the transverse stiffness efficiency and the elastic modulus <math> E_0 </math>, the empirical formula of the elastic modulus <math> E_0 </math> is given, which can provide a reference for the determination of the elastic modulus of the weakening joint not only in the general tunnel lining (outer diameter is about 6 m) but in the large-diameter (outer diameter is 10 m) and super-large-diameter (outer diameter is 15 m) tunnel lining.
 ==Acknowledgement==

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 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-! rowspan='3' style="vertical-align: top;" |Tunnel type !!  rowspan='3' style="vertical-align: top;" |Assembling mode !!   colspan='9' | Transverse stiffness efficiency <math display="inline">\, \eta</math>
+! rowspan='3' style="vertical-align: top;text-align:left;" |Tunnel type !!  rowspan='3' style="vertical-align: top;" |Assembling mode !!   colspan='9' | Transverse stiffness efficiency <math display="inline">\, \eta</math>
 |-style="text-align:center"
 !! colspan='3'  |0.2-0.5 !!   colspan='3'  |0.5-0.8 !! colspan='3' |0.8-0.9