A circular shaft intersected by two horizontal arched tunnels is analyzed using a three-dimensional elastic-plastic finite element code. The stress concentration around the shaft-tunnel-tunnel intersection is compared with the two-dimensional plane strain analyses of each tunnel and shaft. A large difference is observed between the two-dimensional and three-dimensional models particularly around the intersection, but diminishes at the mesh boundary. The comparison also indicates that the two-dimensional model overestimates the local factor of safety near the intersection.


On a analyse un puit rond qui s"intersecte par deux demi-circulaires tunnels horizontaux en usant L"element fini elastique-plastique à trois-dimensions. Le contrainte concentre autour de l"intersection du puit-tunnel-tunnel s"est compare avec l"analyse du deformation plan à deux-dimensions de chaque tunnel et puit. Une grande difference est observee entre le modèle 2-D et 3-D paticulièrement autour d"une intersection, mais diminue au filet borne. La comparaison aussi indique que le modèle 2-D surestime le coefficient de securite près de l"intersection.


Eine kreisförmige Schacht mit zwei horizontalen Tunneln werden analisiert, die durch drei Dimensionen als elastische-plastische FE Program eingefuehrt worden. Die Spannungskonzentrationen als Tunnel und Schacht sind rundherum das Schacht-Tunnel-Tunnel Kreuzung die durch zwei Dimensionen analisiert und vergleicht werden. Zwischen Model 2-D und 3-D kannte man besonders einen großen Unterschied an dem gekrenzten Punkt wargeben, aber an der Maschgrenze wird die Spannung kleiner. Diese Vergleichung zwischen 2-D und 3-D wurde die Sicherheit von Model 2-D ueberschaft.


The methods of numerical modelling used in rock mechanics design of underground excavations can be grouped into two-dimensional (2-D) and three-dimensional (3-D) models from the viewpoint of dimension. Although a 2-D idealization such as a plane stress, plane strain or axisymmetric analysis demands less efforts and costs compared to the 3-D analysis, a 3-D model is essential for an accurate and reliable assessment of safety in a number of practical engineering designs, particularly when:

  • the strikes of geologic structures such as joints, faults and bedding planes do not coincide with the long axis of an underground opening:

  • the axes of anisotropy of rock and rock masses do not coincide with the axes of excavations:

  • the long axis of an opening is neither parallel nor perpendicular to the principal directions of in-situ premining stresses:

  • the shape of a mine opening or pillar is near equi-dimensional, such that the end effects are no more negligible, and

  • multiple openings cross each other, such that the interaction between adjacent openings is significant near the intersection.

Comparison study between the 2-D and 3-D models is not new. In the case of mine pillar design, detailed comparisons have been made by doing a 3-D finite element (FE) analysis, a 2-D FE analysis either in plane stress or plane strain, and an augmented 2-D analysis which gives the 3-D effects in an engineering accuracy (Pariseau & Sorensen 1919, Pariseau 1981). In the case of tunnel excavation, Pan and Hudson (1988) have compared the plane strain analysis with the axisymmetric 3-D analysis of a multi-step excavation process.

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