Quantitatively analyzing deformation of a tunnel with lining anomalies for structural safety evaluation is essential for sustainable tunnel engineering. However, a complete theory and associated methodology accounted for displacement of a built tunnel is absent. This manuscript introduces a novel characteristic matrix method proposed by the authors which is an integrated procedure that analyzes displacement of rock tunnels in operation. First, establish a numerical model expressing a specific state of the tunnel in interest. Derive the characteristic displacement modes by a unit disturbances and check their independency. Once the characteristic displacement modes are proved to be orthogonal (independent), they are ready for use. Given a tunnel displacement, modal decomposition returns the magnitude of each characteristic displacement modes that together consist the given displacement. By realizing the definite value of horizontal and vertical translation, rotation, uniform compression/expansion, elliptic deformation, triangular deformation…etc. for a tunnel displacement, one can distinguish tunnel displacement and deformation types from their modal composition. A circular tunnel with elastic structure and surrounded by elastic rock mass is chosen to demonstrate the outcome of characteristic modes and how they work. The influences of tunnel shape, elastic modulus ratio between rock mass and tunnel structure, and numbers of monitoring points on a tunnel profile were also discussed. Identifying possible tunnel deformation causes is attainable if further connections between mode combination and force increments were built.


Analytical approaches of tunnel engineering began with pursuing the stress-strain relationships around a hole in an elastic media. Started from close-form solution (Kirsch, 1898), the development of underground excavation support design method has been nearly mature when it came to convergence confinement method (Rabcewicz, 1964). The existing methods were mostly proposed by European, where geological conditions are fairly good, and tunnel damages occur during excavation rather than after completion. Thus, it is generally recognized that tunnels are stable after construction. Understanding of tunnel mechanical behavior reaches only to the end of excavation, no rigorous theories aims to analyze possible deformation and lining anomalies of tunnels in operation.

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