The case of a complex underground tunnel system, in urban environment, is analyzed next. The goal of the simulation is to identify surface settlements induced by a multi-stage excavation/construction sequence, associated with the enlargement and modification of an existing tunnel system. The difficulties to overcome are: a complex 3D geometry, accounting properly for a complex initial state, and modeling of an arch-pipes-parasol system, among others. A wide range of options available in the finite element software Z_SOIL 3D are illustrated by the example.
The analyzed tunnel system, embedded at about 10m underground, consists initially of two existing circular
tunnels with a diameter of 4m each. A relatively
large building is already constructed on the surface above the considered
system and therefore a careful estimation of surface settlements is
necessary. The numerical simulation of the proposed excavation/construction
works is rigorously modeled with a 3D finite element model according
to the assumed time schedule and applied technology, which will be shortly
described below.
The final geometry of the tunnel system is shown
in Fig. 25. The old tunnel in both, left and right, branches is enlarged
in a zone located between the sections A-A and B-B and the cross-section
geometry is designed as a set of circular segments with variable radii
and centers; the most interesting part, is
the tunnel junction
Fig.25. New tunnelingsystem with junctions
The excavation/construction
scheme, the same for both left (LB) and right (RB) tunnel branches, consists
of several steps, illustrated in the accompanying figures. The procedure
starts with the installation of a special arch support system (ARCH),
which is installed above the planned elevation of the new enlarged tunnel
and consists of
Fig. 26. Enlargement and ARCH installation
Fig. 27. Excavation of pilot gallery
In further steps another excavation procedure is applied due to significantly larger tunnel dimensions. First the existing tunnel is fully filled with concrete and then a temporary pilot gallery, protected first by an ARCH system, is excavated and then strenghtened by a concrete shell. The procedure is such that first ARCH pipes are driven, then excavation takes place and a new pilot gallery lining, inside the excavated part, is installed. These steps are shown in Figure 27.
After stabilization of the deformation, the tunnel will be successively enlarged up to its design dimensions, first by an excavation of the crown and then, when the crown excavation front is at least 12m away from the front of the pilot gallery, by an excavation of the invert fill (see Fig. 28).
The
crown is excavated in 6 steps and the upper part of the enlarged tunnel
is successively protected with a
Fig.28. Excavation of the invert fill
In subsequent excavation steps the new bifurcating tunnel is excavated and, simultaneously, the concrete lining is installed; in the end, the remaining part of the existing tunnel, previously filled with concrete, is reopened, reaching the stage shown in Fig. 25. The same excavation/construction procedure is applied to both branches of the tunnel system.
This complex problem shows a large variety of the problems encountered by the analyst during generation of the model. The key issue is a 3D finite element mesh, which should fit assumed geometry up to required level of accuracy and which has to take into account the presence of the excavation/construction fronts of assumed geometry. In order to increase computational efficiency, low order 3D elements are applied and mostly, if only possible, eight-node bricks. For these elements the effect of volumetric locking can easily be overcome if BBAR or Enhanced Assumed Strain or Stabilized approaches are applied; all are available within the Z_SOIL 3D environment. This point is important as standard low order bricks would exhibit substantial locking effects, which would lead to an overestimation of the limit loads, the safety factors and an underestimation of the settlements.
The next aspect is the proper modeling of the tunnel lining behavior. In the considered case we assume that the lining behavior is represented by MITC Q4 shell elements and the stress-strain relation, for the further purpose of dimensioning, is assumed to be linearly elastic. The last point is questionable as the excavation/construction time schedule indicates that an effect of concrete aging should be included. This is possible within Z_SOIL by the application of the so-called “aging concrete” material model, which reproduces stiffness evolution in time and creeping/relaxation phenomena as well. If needed, it is also possible to apply explicitly an elastic modulus varying in time, but such an approach may result in stress-resultant oscillations in time.
In this example we do not include an effect of soil-structure interaction between tunnel and neighboring soil although contact can be easily incorporated into the model by the application of a Coulomb friction law and an augmented Lagrangian approach, both options being available in Z_Soil.
In most complex numerical calculations it is not possible to include all effects, like soil nonlinearity, contact, consolidation, creep, seepage etc., from the beginning, as analysis of results can be very difficult and the analyst cannot easily identify which effect is dominant. Usually we start with an elastic calculation and then, step-by-step, we add additional phenomena and significant constitutive nonlinearties. Such a hierarchical analysis serves as a base for proper understanding of the analyzed problem and may detect potential user mistakes made in data preparation.
Proper representation of the excavation/construction process, corresponding to the assumed time schedule, plays a significant role as far as result accuracy is concerned. For this reason, in Z_SOIL, with every finite element, we can associate a so called existence function attribute which indicates at which time the element becomes active and when it is deleted. This option is of primary importance in tunneling software and therefore, in Z_SOIL 3D mesh generators, we have a wide collection of methods which simplify this process by application of Import/Export to/from Excel files; in addition, robust object selection tools are available.
In the case considered, an arch pipe system is modelled with the aid of 3D Timoshenko beam elements, which are embedded within solid elements by means of so called kinematic constraints. It is obvious that we cannot analyse all the local interaction effects between soil and steel pipes as this happens on a smaller scale as compared with dimensions of the structure, but we can roughly represent the stiffness of this support system assuming that pipes continuously deform with the neighboring soil. In such a case, it is possible, in the Z_SOIL environment, to impose a continuity condition by imposing appropriate constraints via a penalty formulation.
The tunnel system is embedded in an over-consolidated clay layer with an in situ Ko coefficient equal to K0=1. As we do not know the initial state after construction of the existing two tunnels we have to compute it. Within Z_SOIL.PC there exists a special, incremental, predictor-corrector procedure to evaluate the initial state. The assumption is that, for actual external forces applied earlier to the system and gravity, we determine initial stresses corresponding to an undeformed state by incremental superposition of loads and corresponding initial stresses, completed by K0 confinement effects.
The over-consolidated clay layer is represented by an elasto-plastic Mohr-Coulomb model which reproduces only a few major features of soil behavior. The assumed values of material properties are as folows: E=80000 kPa, n=0.3, c=30 kPa, f =200, y=00. A more comprehensive material model, like Cap or Duncan-Chang, also available in Z_SOIL PC, could be used here too but would require a definitely larger set of material properties.
In this example we assume that concrete behaves as a linearly elastic material with a constant Young modulus. As we have already mentioned this assumption is questionable and further analyses with an “aging concrete” material model should be carried out. The analysis with this more sophisticated model is, however, more costly as each time step, generated by an excavation/construction event, must be split at least in 4 to 5 steps in order to achieve a certain level of accuracy in the stress integration process.
The whole computation was carried out in about
70 excavation/construction steps. The final settlements of the surface
due to tunneling are shown in Fig.29. We can see that for the given material
properties the maximum vertical displacement is about
Fig.29. Settlements at final stage of tunneling
The example described here shows that the analysis of complex tunneling problems in complex urban areas can be successfully carried out using modern numerical software designed for PC platforms.