Abstract
The modulus of subgrade reaction is a key parameter used in tunnel design. However, the existing analytical relations for determining the modulus mainly focus on deep tunnels in the homogeneous ground. Therefore, the complex variable method was used to develop analytical solutions for calculating the modulus of subgrade reaction for deep circular tunnels close to bedrock and shallow circular tunnels under uniform and oval deformation patterns. Subsequently, the analytical solutions were validated by comparing them with numerical solutions. At last, parametric studies were conducted to analyze the impacts of the bedrock and ground surface on the modulus of subgrade reaction. The results show that the modulus increases substantially near the bedrock and decreases slightly away from the bedrock, while the ground surface has the opposite effect. In addition, the effects of the bedrock and the ground surface on the radial modulus are more significant than that on the tangential modulus. Furthermore, the results confirm the effect of the tunnel deformation pattern on the modulus of subgrade reaction. The analytical solutions may be used for obtaining the approximate magnitude of the modulus of subgrade reaction in the structure design of shallow tunnels and deep tunnels close to bedrock.
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Avci B, Gurbuz A (2018) Modulus of subgrade reaction that varies with magnitude of displacement of cohesionless soil. Arabian Journal of Geosciences 11:351, DOI: https://doi.org/10.1007/s12517-018-3713-1
Basudhar PK, Yadav SK, Basudhar A (2018) Treatise on winkler modulus of subgrade reaction and its estimation for improved soil-structure interaction analysis. Geotechnical and Geological Engineering 36:3091–3109, DOI: https://doi.org/10.1007/s10706-018-0523-x
Bouzid DA, Bhattacharya S, Dash SR (2013) Winkler Springs (p-y curves) for pile design from stress-strain of soils: FE assessment of scaling coefficients using the Mobilized Strength Design concept. Geomechanics and Engineering 5(5):379–399, DOI: https://doi.org/10.12989/gae.2013.5.5.379
Chen JL, Feng YQ, Shu WY (2016) An improved solution for beam on elastic foundation using quintic displacement functions. KSCE Journal of Civil Engineering 20(2):792–802, DOI: https://doi.org/10.1007/s12205-015-0424-y
Dincer I (2011) Models to predict the deformation modulus and the coefficient of subgrade reaction for earth filling structures. Advances in Engineering Software 42(4):160–171, DOI: https://doi.org/10.1016/j.advengsoft.2011.02.001
Dutta SC, Roy R (2002) A critical review on idealization and modeling for interaction among soil-foundation-structure system. Computers and Structures 80:1579–1594, DOI: https://doi.org/10.1016/S0045-7949(02)00115-3
Elachachi SM, Breysse D, Houy L (2004) Longitudinal variability of soils and structural response of sewer networks. Computers and Geotechnics 31(8):625–641, DOI: https://doi.org/10.1016/j.compgeo.2004.10.003
Ghiasi V, Ghiasi S, Prasad A (2012) Evaluation of tunnels under squeezing rock condition. Journal of Engineering, Design and Technology 10(2):168–179, DOI: https://doi.org/10.1108/17260531211241167
Ghiasi V, Koushki M (2020) Numerical and artificial neural network analyses of ground surface settlement of tunnel in saturated soil. SN Applied Sciences 2:939, DOI: https://doi.org/10.1007/s42452-020-2742-z
Ghiasi V, Mozafari V (2018) Seismic response of buried pipes to microtunnelling method under earthquake loads. Soil Dynamics and Earthquake Engineering 113:193–201, DOI: https://doi.org/10.1016/j.soildyn.2018.05.020
González C, Sagaseta C (2001) Patterns of soil deformations around tunnels. Application to the extension of Madrid Metro. Computers and Geotechnics 28:445–468, DOI: https://doi.org/10.1016/S0266-352X(01)00007-6
Guha I, Randolph MF, White DJ (2016) Evaluation of elastic stiffness parameters for pipeline-soil interaction. Journal of Geotechnical and Geoenvironmental Engineering 142(6):04016009, DOI: https://doi.org/10.1061/(ASCE)GT.1943-5606.0001466
Kog YC, Kho C, Loh KK (2013) Tunnel design and modulus of subgrade reaction. Journal of Performance of Constructed Facilities 29(2): 04014065, DOI: https://doi.org/10.1061/(ASCE)CF.1943-5509.0000537
Jamil I, Ahmad I (2019) Bending moments in raft of a piled raft system using Winkler analysis. Geomechanics and Engineering 18(1):41–48, DOI: https://doi.org/10.12989/gae.2019.18.1.041
Luscher U (1966) Buckling of soil-surrounded tubes. Journal of the Soil Mechanics and Foundations Division ASCE 92(6):211–228, DOI: https://doi.org/10.1061/JSFEAQ.0000920
Matsubara K, Hoshiya M (2000) Soil spring constants of buried pipelines for seismic design. Journal of Engineering Mechanics, ASCE 126(1):76–83, DOI: https://doi.org/10.1061/(asce)0733-9399(2000)126:1(76)
Okeagu B, Abdel-Sayed G (1984) Coefficients of soil reaction for buried flexible conduits. International Journal of Geotechnical Engineering 110:908–922, DOI: https://doi.org/10.1061/(asce)0733-9410(1984)110:7(908)
Prendergast LJ, Gavin K (2016) A comparison of initial stiffness formulations for small-strain soil-pile dynamic Winkler modelling. Soil Dynamics and Earthquake Engineering 81:27–41, DOI: https://doi.org/10.1016/j.soildyn.2015.11.006
Terzaghi K (1955) Evaluation of coefficients of subgrade reaction. Geotechnique 5(4):297–326, DOI: https://doi.org/10.1680/geot.1955.5.4.297
Timoshenko SP, Goodier JN (1951) Theory of elasticity, 2nd edn. McGraw-Hill, New York
Verruijt A (1997) A complex variable solution for a deforming circular tunnel in an elastic half-plane. International Journal for Numerical and Analytical Methods in Geomechanics 21(2):77–89, DOI: https://doi.org/10.1002/(SICI)1096-9853(199702)21:2<77::AID-NAG857>3.0.CO;2-M
Woo KS, Lee DW, Yang SH, Ahn JS (2018) Static behavior of a laterally loaded guardrail post in sloping ground by LS-DYNA. Geomechanics and Engineering 15(5):1101–1111, DOI: https://doi.org/10.12989/gae.2018.15.5.1101
Wood M (1975) The circular tunnel in elastic ground. Geotechnique 25(1):115–127, DOI: https://doi.org/10.1680/geot.1975.25.1.115
Yu Y, Shang YD, Sun HY (2014) A theoretical method to predict crack initiation in stabilizing piles. KSCE Journal of Civil Engineering 18(5):1332–1341, DOI: https://doi.org/10.1007/s12205-014-0063-8
Zhang D, Huang H, Phoon KK, Hu Q (2014) A modified solution of radial subgrade modulus for a circular tunnel in elastic ground. Soils and Foundations 54(2):225–232, DOI: https://doi.org/10.1016/j.sandf.2014.02.012
Zhang Y, Wang X, Wang X, Mang HA (2022) Virtual displacement based discontinuity layout optimization. International Journal for Numerical Methods in Engineering 123(22):5682–5694, DOI: https://doi.org/10.1002/nme.7084
Zhang Y, Zhuang X (2018) Cracking elements: A self-propagating strong discontinuity embedded approach for quasi-brittle fracture. Finite Elements in Analysis and Design 144:84–100, DOI: https://doi.org/10.1016/j.finel.2017.10.007
Zhang Y, Zhuang X (2019) Cracking elements method for dynamic brittle fracture. Theoretical and Applied Fracture Mechanics 102:1–9, DOI: https://doi.org/10.1016/j.tafmec.2018.09.015
Zhang Y, Zhuang X, Lackner R (2018) Stability analysis of shotcrete supported crown of NATM tunnels with discontinuity layout optimization. International Journal for Numerical and Analytical Methods in Geomechanics 42(11):1199–1216, DOI: https://doi.org/10.1002/nag.2775
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This work was supported by the National Natural Science Foundation of China (Grant numbers 52079134, 51991393).
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Zhao, W., Xie, P., Chen, W. et al. Modulus of Subgrade Reaction for Circular Tunnels considering Bedrock and Ground Surface. KSCE J Civ Eng 27, 2593–2602 (2023). https://doi.org/10.1007/s12205-023-2029-1
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DOI: https://doi.org/10.1007/s12205-023-2029-1