I want to know how can I do an analytical solution for stresses distribution due to tunnel excavation in an expansive soil to compare with a numerical simulation. and which method I can use?
It depends on many parameters such as tunnel cross section, the behaviour of the soil (elastic-elastoplastic), tunnelling method (e.g. TBM, NATM), support load etc. For example for circular tunnel there is Kirsch's solution for elastic soils and you can look at this reference for more detail and solutions (see Poulos G. H. and Davis H. E. (1974). Elastic Solutions for Soil and Rock Mechanics, John Wiley & Sons). A much more advanced (and interesting) method is complex variable analysis (see Jaeger C. J., Cook W. N. G., Zimmerman W. R. (2007). Fundamentals of Rock Mechanics, 4th Revised Edition, Blackwell, Verruijt A. (1996). Complex Variable Solutions of Elastic Tunneling Problems, TU Delft, Netherlands, Strack E. O. (2002). Analytic Solutions of Elastic Tunneling Problems, PhD Thesis, TU Delft, Netherlands). I would advise that in order to validate your solution make a comparison of the analytical one with a numerical one, also compare a shallow tunnel and a deep tunnel because the deeper you do the tunnelling the greater the convergence. Keep in mind also the presence of water is of major significance.
Hello Ioannis Ntaliakouras, thank you very much for your answer and the details. Actually the Complex variable method right now I have read about it and I have gotten some ideas about this method, but I am not sure if I can use it because my soil is an expansive soil, the tunnel is a circular and the tunneling method is TBM, and I still miss some information about this method.
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