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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 77
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 17
Dynamics of a Tunnel: Coupling of Finite Element (FEM) and Integral Transform Techniques (ITM) H. Grundmann and K. Müller
Department of Engineering Mechanics, Technische Universität München, Munich, Germany Full Bibliographic Reference for this paper
, "Dynamics of a Tunnel: Coupling of Finite Element (FEM) and Integral Transform Techniques (ITM)", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 17, 2003. doi:10.4203/ccp.77.17
Keywords: halfspace dynamics, tunnel dynamics, integral transform technique.
Summary
In the dynamic calculation of tunnels subjected to time depending moving loads
one has to consider the tunnel structure itself with its finite dimensions
in the lateral and its infinite extension in the longitudinal direction. For
the soil, a system which extends to infinity in both horizontal directions and
in the positive z-direction, an excavation for the tunnel has to be taken into
account. This situation requires a description which allows to represent the
"local" effects as well as the effects of the infinity of the tunnel in
the longitudinal and of the soil additionally in the lateral and vertical
direction. Correspondingly it is advantageous to apply a FEM/BEM or a FEM/ITM
coupling [1,2,3]. In the paper the tunnel and a
portion of the
surrounding soil shall be modelized by Finite Elements in a Fourier
transformed wavenumber/frequency domain (regarding the longitudinal direction
and the time) in order to take account of the respective infinity. The soil as
a whole will be described in a domain transformed additionally in regard of
the second horizontal direction, the y-direction. (The chosen approach leads
to a favourable description particularly in view of the possibility of a
coupling to moving vehicles.)
Only a very small FEM mesh (in the Fourier transformed domain) has to be considered, if it is coupled at its outer boundary to a stiffness matrix in the same transformed domain representing the infinite halfspace exterior to the excavation for the FEM mesh.
The stiffness matrix is derived by the aid of ITM starting with the halfspace
without any excavation: A sufficiently large set of properly selected shape
functions for dynamic loadings is applied along a preselected internal closed
surface surrounding the tunnel. This surface is narrower than the surface of
the later FEM excavation. The dynamic reaction of the full halfspace (without
excavation) subjected to these fictitious loadings which are
After the stiffness matrix is applied at the boundary, the interaction problem
can be solved by a 2D FEM calculation in
the transformed ( The application shall be illustrated for a simple situation to show the possibilities and the efficiency of the dynamic FEM/ITM coupling. References
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