J. Eberhardsteiner, H.A. Mang, P. Torzicky:
"Hybrid BE-FE Stress Analysis of the Excavation of a Tunnel Bifurcation on the Basis of a Substructuring Technique";
in: "Advances in Boundary Element Techniques", J.H. Kane, G. Maier, N. Tosaka, S.N. Atluri (Hrg.); Springer, 1993, S. 105 - 128.
A realistic mechanical simulation of the driving of a tunnel under complex geological conditions and an economic design of non-regular cross-sections of tunnels, as occuring at a station or at tunnel bifurcations, require the use of three-dimensional (3D) numerical analysis procedures . One way to analyze such complicated geotechnical problems is to use the conventional finite element method (FEM). However, in order to approximate the boundary conditions at infinity adequately, because of the large spatial dimensions of the analysis models for solving geotechnical problems a realistic FE model requires a relatively large numer of three-dimensional finite elements, resulting in very large requirements of computing time. In order to reduce these requirements, for three-dimensional elasto-plastic stress analysis of the driving of a tunnel by means of the New Austrian Tunnelling Method, consisting of a sequence of excavation steps and of securing these partial excavations by a shotcrete shell, the coupling of boundary element (BE) and Fe discretizations is suggested. The use of such a hybrid analysis technique enables exploiting the complementary advantages of the BEM and the FEM. Hence, the "near-field", including the interior of the tunnel, the shotcrete shell and its outer vicinity, where stress concentrations and plastic deformations may occur, is discretized by the FEM. The boundary of the elastic "far-field" and the coupling surface of the BE subregion are discretized by the BEM. The coupling of the two discretizations is expected to provide a more efficient solution technique for stress analysis of the driving of tunnels.
The hybrid BE-FE discretization technique used int his paper was developed by Li, Han, Mang and Torzicky . The basic idea of coupling BE and FE discretization of solids goes back to Zienkiewicz, Kelly and Bettess . In the field of stress analysis of tunnels by means of a coupled BE-FE approach, work by Beer and Swoboda  as well as by Dallmann  should be mentioned. Chen, Hofstetter, Li, Mang and Torzicky  as well as Eberhardsteiner, Mang and Torzicky  extended the coupling method, proposed in , considering the inhomogeneity and the transverse isotropy of the rock mass into which the tunnel is driven.
In the followingm the theoretical fundamentals of the developed hybrid BE-FE method will be presented with an emphasis on the special situation resulting from the excavation, that is, from changes of the original domain. Thereafter, a detailed comment on the mechanical inconsistency of symmetrizing the unsymmetric coupling laod stiffness matrix resulting form hybrid BE-FE discretizations will be made. This topic is important insofar as little experience about the influence of such a symmetrization on the solution of "nonacademic" problems is available. After presentation of the theoretical fundamentals, a detailed description of a substructuring technique on the basis of a multistep condensation of nodal degrees of freedom is given. The condensation strategy is used for the numerical computations. The numerical investigation contains characteristic results from a 3D elasto-plastic hybrid BE-FE stress analysis of the driving of a tunnel bifurcation in the area of a station of Vienna's underground line U3. Moreover, results from comparative numerical studies are presented to quantify the influence of three parameters on the deformation or on the stress state of the investigated tunnel. These parameters are the location of the coupling interface, the development of the stiffness of the shotcrete used for the shotcrete shell, and the symmetrization of the coupling stiffness matrix.
 Duddeck, H.: Application of numerical analyses for tunnelling. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 15, 223-239, 1991.
 Li, H.-B.; Han, G.-M.; Mang, H.A.; Torzicky, P.: A new method for the coupling of finite-element and boundary-element discretized subdomains of elastic bodies. Computer Methods in Applied Mechanics and Engineering, Vol. 54, 161-185, 1986.
 Zienkiewicz, O.C.; Kelly, D.M.; Bettess, P: The coupling of the finite element method and boundary solution procedures. International Journal for Numerical Methods in Engineering, Vol. 11, 355-376, 1977.
 Beer, G.; Swoboda, G.: Application of advanced boundary element and coupled methods in geomechanics. Proceedings of the IUTAM Symposium on Advanced Boundary Element Methods, San Antonio, USA, 1987 (Ed.: T.A. Cruse), 19-23, Springer, Berlin, 1988.
 Dallmann, R.: Kopplung von Rand- und Finite-Elemente-Methode beim Tunnelvortrieb im Voll- und Halbraum. Dissertation, Institut für Statik, Report No. 88-56, Technical University of Braunschweig, West Germany, 1989.
 Chen, Z.-S.; Hofstetter, G.; Li, Z.-K.; Mang, H.A.; Torzicky, P: Coupling of FE- and BE-discretizations for 3D-stress analysis of tunnels in layered anisotropic rock. Proceedings of the IUTAM/IACM Symposium on Discretization Methods in Structural Mechanics, Vienna, Austria, 1989 (Eds.: G. Kuhn, H.A. Mang), 427-436, Springer, Berlin, 1990.
 Eberhardsteiner, J.; Mang, H.A.; Torzicky, P.: Three-dimensional elasto-plastic hybrid BE-FE stress analysis of the driving of tunnels. Proceedings of teh 5th International Conference on Computational Engineering with Boundary Elements, Newark, Delaware, USA, 1990 (Eds.: A.H.D. Cheng, C.A. Brabbia, S. Grilli), Vol. 2, 139-159, Computational Mechanics Publication, 1990.