Fig 1.
Elastic foundation models: (a) Winkler model; (b) Pasternak model.
Fig 2.
Flowchart of the superposition method with locally removed springs.
Fig 3.
Element force analysis.
Fig 4.
q(δ) acting on the Pasternak elastic foundation beams.
Fig 5.
Calculation model: (a)Zoning of the elastic foundation beam model; (b) Segmentation of the existing tunnel.
Fig 6.
Calculation model for vertical displacement of segment BC.
Fig 7.
Calculation process for vertical displacement of segment CD under additional load q(x): (a)Semi-infinite elastic foundation beam model; (b)Extension beam model; (c)Recut at C-section; (d)Application of reaction forces.
Fig 8.
Solution process of the proposed theoretical model.
Fig 9.
Device for underpinning loads.
Fig 10.
Calculation model under the active underpinning load.
Fig 11.
Calculation model under symmetrical underpinning loads.
Table 1.
Settings of main parameters.
Fig 12.
Distribution of vertical displacement and bending moment obtained from different methods: (a) Vertical displacement; (b) Bending moment.
Fig 13.
Distribution of vertical displacement and bending moment under underpinning loads: (a) Vertical displacement; (b) Bending moment.
Fig 14.
Schematic layout of the Metro Line 10 undercrossing the Metro Line 1.
Fig 15.
Comparison of calculated and measured results.
Table 2.
Settings of parameters in the theoretical model.
Fig 16.
3D FEM numerical model: (a) Geometry and mesh; (b) Underpinning loads.
Fig 17.
Comparison of calculated and simulated results.
Fig 18.
Vertical tunnel displacement for various underpinning loads: (a) Vertical displacements; (b) Peak displacements.
Fig 19.
Vertical tunnel displacement for various bending stiffnesses: (a) Vertical displacements; (b) Peak displacements.
Fig 20.
Vertical displacements for various values of K0: (a) Vertical displacements; (b) Peak displacements.
Fig 21.
The distribution of vertical displacements in the tunnel for various excavation zones: (a) Vertical displacement; (b) Peak displacements.