This is the paper we have written with my dear wife Pinar Akdogan Demir, based on our experience from Istanbul tunnels. I was already considering uploading the paper here, but when I saw that my paper was not inside the proceedings USB due to a mistake (which also was almost preventing me from presenting - because they simply forgot us), it was a must. So, here it is. As always, get in touch if you have any comments.
Due to comprehensive soil-structure interaction during and after tunnelling, estimation of structural forces on the linings depends on many factors which can be categorized into ground properties, structural properties, and loading properties. Estimation of ground properties has been discussed in the literature in detail and proper modelling of tunnels or any other structure that interacts with ground requires in-depth knowledge of ground properties. In geotechnical engineering, structural properties are thought to be rather well known, however, due to the strictly time-dependent excavation and loading process, even structural components cannot be represented by simpler constitutive or structural models.(Neuner, Cordes, et al., 2017; Neuner, Gamnitzer, et al., 2017; Schädlich et al., 2014; Schädlich & Schweiger, 2014) Static loadings on the tunnels are also discussed in detail and it is either represented by stage-by-stage modelling using finite element method (FEM) or finite difference methods (FDM) or empirical formulations such as Prodotyakonov, Terzaghi, and others to be used on beam-on-foundation solutions. (Celada & Bieniawski, 2019; Széchy, 1967; Terzaghi, 1943) Seismic forces, on the other hand, may impose the greatest load on the tunnels based on the seismicity of the project location. (Kontoe et al., 2008; Roy & Sarkar, 2017; Z. Z. Wang & Zhang, 2013; Zhang et al., 2018) Behaviour of underground tunnels differs significantly from above-ground structures due to complex interaction with the ground around it. (Hashash et al., 2001; Tsinidis et al., 2020)
Although theoretical studies present a great deal of material to estimate seismic forces on the tunnels, there are still important points to be considered in the daily design. In this paper, practical seismic analyses of tunnels will be discussed along with recommendations for practice.
There are several up-to-date methods to calculate the structural forces on tunnel lining which are summarized in the table below.
Full dynamic analysis is, currently, the most advanced method that is available to both practitioners and academics. In this method, earthquake excitation is applied from the bottom boundary of the model and this boundary usually extended up to bedrock to reduce the steps to move the earthquake from bedrock to an upper level using an additional step of site response analysis. Since dynamic analyses are more sensitive to model conditions, the effects of boundary distance and mesh size are significant. (Fabozzi, 2017; Tsinidis, 2015) It is also advisable to use advanced constitutive models such as Hardening Soil Small Strain model which takes the stiffness-strain degradation relationship and damping into account. Analytical methods are very easy to use and proven to provide a reasonable estimation of earthquake induced forces on the lining. (Hashash et al., 2001; J. N. Wang, 1993) However, analytical methods require crude simplifications such as circular geometry and monolithic lining. To account for joints of segmental lining of TBM tunnels, following simplified approach can be used. (Wood, 1975)
$$ I_{eqv}=I_{joint}+I_{seg} (4/n)^2 $$
Ieqv is the equivalent moment of inertia of the tunnel ring composed of joints and segments, Iseg is the moment of inertia of the segments (full section), Ijoint is the moment of inertia of the joints and n is the number of joints. Joint thickness is the clear concrete thickness after gasket. Compared to full dynamic analyses and analytical methods, pseudo-static methods are in between of these methods in terms of both advantages and disadvantages. Pseudo static methods can be divided into both deformation-based methods and force-based methods. Pseudo-static force-based methods such as prescribed acceleration can be a fast approach to seismic problems. However, since ovalling is the main reason of the seismic loading of underground structures, uniform acceleration profile may result in incorrect loads. Therefore, it is advised to check the response of model without tunnels to ensure the strain profile is correct.
Pseudo static methods can be used with reasonable accuracy to estimate seismic tunnel lining forces. The main idea behind the pseudo-static deformation method is imposing ovalling deformation to whole model without any prior assumption about racking coefficient.
Concept of pseudo-static deformation method is developed by Newmark (1968) and simplified derivation of strain due to waves is described below.
In this equation, PGV is the peak ground velocity (m/s) and Vs is the shear wave velocity (m/s). Different notations (Vs for PGV and Cs for Vs) are available in the literature. Since PGV depends on the depth and shear wave velocity depends on the level of strain, effective parameters can be used for better notation:
