ROCK TUNNELS | SQUEEZING GROUND


2.3 Model of Advancing Tunnel Heading The model simulates the transient processes during ongoing mechanised excavation and lining installation, as well as during a TBM standstill. It presupposes negligible TBM weight, and thus uniform tunnel support and overcut, as well as backfilling around the lining. The construction process for the model is simulated


as continuous with an average advance rate v, as shown to be a sufficiently accurate simplification (Leone et al. 2023). The step-by-step method is adopted with steps of length s = 1m; (Figure 1b), equal to lining is installed immediately behind the shield. The tunnel face is considered unsupported, which


Above, figure 2: Evolution of radial displacement normalised by the tunnel radius over time in the plane strain problem (η = 1000 MPa d, k = 10–9 m/s)


two-phase porous medium, according to Terzaghi’s


principle of effective stresses. Seepage flow is modelled on Darcy’s law. The solid grains and pore water are assumed incompressible in relation to the rock.


Below, figure 3:


Average rock pressure developing on the shield (normalised by the in situ stress) as a function of the normalised advance rate v* (p = 4 MPa)


2.2 Model of Cross Section Far Behind the Advancing Face Far behind the face, plane strain conditions can be assumed. The plane strain model (Figure 1a) simulates the ground response to excavation via an instantaneous unloading of the tunnel boundary. The tunnel boundary subsequently remains unsupported and evolution of its radial displacement is monitored. Due to rotational symmetry, the problem can be


analysed as 1D - a single strip, discretised with a finite element (FE) mesh of four-noded, linear, quadrilateral, axisymmetric elements.


is reasonable considering that open shield TBMs are employed in most practical cases of mechanised tunnelling through squeezing rocks. The shield of length L is modelled with non-linear radial springs, which consider no loading (zero stiffness) for convergences below the radial overcut ΔR, and a linear elastic stiffness Ks


for the portion of convergences that exceeds that


angular gap ΔR. The lining is modelled with elastic radial springs of , assuming direct contact with the ground


stiffness Kl


immediately upon installation due to backfilling. The ground unloading behind the shield tail and its reloading over the lining are considered via distinct installation points. The computational domain was a structured FE mesh


of 11,532 four-noded, linear, quadrilateral, axisymmetric element, their sizes increasing in the radial direction, and along the tunnel axis remaining constant, equal to step length - which is sufficiently small to ensure enhanced prediction accuracy. An excavation length of 10R0


(60


excavation steps) was simulated. With the model enabling a determination of the


longitudinal profile of rock pressure, the average value was evaluated over the shield length, the simulated excavation length being long enough for the average value to become constant.


3 Time-Dependent Contraction of a Tunnel Cross Section Far Behind the Face The radial displacement of the unsupported tunnel boundary depends, in general, on all independent problem parameters, i.e., in-situ stress, tunnel radius, material constants, time, and, in the case of creep, viscosity. In the case of consolidation, the permeability, in-situ


pore pressure, unit weight of the pore water, and the size (far-field radius) of the seepage flow domain are considered in the numerical model. Figure 2 shows the normalised radial displacement


as a function of the normalised time for creep (black line) and consolidation (dashed and solid red lines, for two values of the in-situ pore pressure, 1MPa and 4MPa, respectively). Very low values of normalised time correspond to


rapid excavation, where instantaneous ground response can be considered; very high values correspond to


12 | December 2025


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