CAPÍTULO II. MATRIZ AUMENTADA DE HACHTEL
2.3. Implementación de las técnicas en MATLAB
Several 2D DEM analyses were performed to identify the potential failure mechanisms occurring in 690
the excavation of a circular unsupported tunnel in a rock mass intersected by three independent sets 691
of joints. The results of our simulations show that i) failure does not always occur along the steepest 692
joint set, which is rather counterintuitive; ii) nor does failure occur along just any particular two joint 693
sets; iii) it is possible for all the joint sets to participate in the failure mechanism; iv) the failure 694
mechanisms are numerous. Therefore, it emerges that the prediction of the exact failure location is 695
unlikely to be successful in practice since the failure location is highly sensitive to the joint patterns 696
and there could be more than one failure mechanism taking place at the same time. So a qualitative 697
description of the failure patterns using specific descriptors is unattainable. Hence, the study 698
underpins the conventional geotechnical practice of supporting a large circumference of the tunnel 699
opening for moderately to heavily jointed rock masses. 700
It has been shown that DEM analyses can be employed in the design phase of tunnel 701
supports in order to determine the main parameters of supports consisting of rock bolts, one lining 702
or a combination of them. A comprehensive parametric analysis was run investigating the effect of 703
bolt bonded length, bolt spacing, bolt length, bolt pretension, bolt stiffness, lining thickness on 704
tunnel convergence for different in-‐situ stresses. The use of numerical methods for the design of the 705
tunnel support represents a significant step forward from traditional empirical or analytical methods 706
which use rock mass classification systems (Barton et al., 1974; Bieniawski, 1983) to characterise the 707
mechanical properties of the rock which is treated as a continuum and therefore cannot adequately 708
capture the behaviour of the rock mass especially in cases where the geology deviates significantly 709
from the dataset of site investigations employed to establish the ratings of the classification systems. 710
The results of the DEM simulations were compared with elastic solutions. To the authors’ 711
knowledge, this is the first verification of DEM calculations for supported openings against available 712
elastic solutions. For the purpose of comparison with elastic solutions, the equivalent shear 713
modulus of the jointed rock mass and rock bolt elastic modulus had to be determined. Estimates for 714
the shear modulus were obtained using standard solutions to make a consistent comparison. No 715
attempt was made to modify the assigned input to improve the comparison. The comparison was 716
encouraging when sufficient support was provided. Although the models employed here are simple, 717
this is believed to be an important milestone, because it suggests that future work using more 718
complex models is feasible. 719
Finally, we proposed the use of an interaction diagram to evaluate the relative effectiveness 720
of rock bolts and lining thickness in the design of the tunnel support. The engineer can use this 721
interaction diagram to make the optimal design decision given a target maximum allowable 722
convergence. The choice among the possible combinations of input values giving rise to the same 723
amount of convergence, i.e. the amount of rock bolt versus lining thickness, will be done by the 724
engineer accounting for the costs required for rock bolt and lining installation. Also the diagram can 725
help the engineer to gauge the effectiveness of introducing an “increment” of either rock bolt 726
support or lining support on an existing support. So this framework sets out opportunities for 727
investigating important problems in rock tunnelling, in which the composite support system (rock 728
bolts and lining) is considered to be one of the governing variables. 729
The concepts and approach proposed in this paper have only been illustrated in 2-‐D plane-‐ 730
strain and for the case of a wished in place support. Therefore, the analyses presented are 731
consistent with the practice of most design standards (Federal Highway Association 2009; British 732
Tunnelling Society 2003), in which well-‐known solutions proposed by Muir Wood (1975) and Einstein 733
& Schwartz (1979) are still widely used to-‐date in the preliminary design stage. Therefore, a direct 734
comparison with these analytical solutions can be readily made by practitioners. Like the closed-‐ 735
form solutions of (Muir Wood, 1975; Einstein & Schwartz, 1979), the analyses presented in this 736
article are plane-‐strain, with 3-‐D effects being neglected. This implies for instance the neglect of the 737
progressive loss of strength of the rock mass as the excavation proceeds, and rock movements due 738
to the 3D shape of the blocks. However, the approach and concepts discussed in this article could 739
be developed further employing, for the final design stage, 3-‐D DEM models able to capture the 740
influence of different excavation stages and rock joint orientations relative to the tunnel axis 741
realistically. 742
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