Seismic response of underground structures under undrained loading with excess pore pressures accumulation

https://doi.org/10.1016/j.tust.2019.103255Get rights and content

Abstract

The seismic response of tunnels under undrained loading with excess pore pressures accumulation is poorly understood. Two-dimensional dynamic numerical analyses are conducted to assess the seismic response of deep circular tunnels located far from the seismic source under undrained loading conditions. For comparison purposes, analyses under drained loading are also performed. It is assumed that the liner remains elastic and that plane strain conditions apply. A cyclic nonlinear elastoplastic constitutive model is used to simulate the nonlinear behavior of the ground and excess pore pressures accumulation with cycles of loading.

It is found that the tunnel’s response is determined by the loading of the liner, or by the distortions of the cross section, depending on the flexibility ratio (F). For stiff structures, with F 2, important thrust (axial) forces and bending moments are produced in the liner, with larger magnitudes for the undrained case, while the distortions of the cross section are small. When the tunnel becomes more flexible, i.e., F increases, the loading on the liner decreases, but the distortions of the cross section become important. For very flexible structures, with F 10, performance is largely determined by the distortions of the cross section, while the thrust forces and bending moments are almost negligible. Such distortions are larger for drained loading than for undrained loading early during the earthquake; however, the distortions drastically increase with the increase of excess pore pressures and can be even larger for undrained than for drained loading, late during the earthquake.

Introduction

Underground structures must be able to support static overburden loads as well as to accommodate additional deformations imposed by seismic motions. For most tunnels, perhaps with the exception of submerged tunnels, it seems well established that, due to ground shaking, the larger effect on the tunnel is caused by shear waves traveling perpendicular to the tunnel axis (Bobet, 2003, Wang, 1993, Merritt et al., 1985, Hendron and Fernández, 1983). These shear waves cause distortions of the cross section (ovaling for a circular tunnel, and racking for a rectangular tunnel) that result in thrust (axial) forces and bending moments. Previous work has shown that the most important parameter determining the distortions of a cross section of a tunnel is the relative stiffness between the medium and the liner (expressed by the flexibility ratio, F), and that the depth and shape of the structure have second-order effects (Bobet, 2010, Wang, 1993).

In more recent work, analytical solutions to investigate the longitudinal seismic response of tunnels have been proposed (Yu et al., 2019, Yu et al., 2018); three-dimensional numerical analyses to asses the seismic response of long tunnels (Yu et al., 2013), or the interaction between underground structures, surrounding soil and adjacent buildings, have also been performed (Wang et al., 2017); experimental shaking table tests to investigate the response of long tunnels to ground shaking (Yu et al., 2018, Yuan et al., 2018), or centrifuge tests to evaluate the seismic response of deep tunnels have also been conducted (Lanzano et al., 2012). These researches have been fundamental to gain insight into the load-transfer interaction mechanisms between the ground and the structure during seismic events. However, these studies have in common that a dry/drained condition has been assumed. There is little information regarding the behavior of buried structures placed in nonlinear ground under undrained loading conditions, i.e., when excess pore pressures are generated and accumulate during the earthquake. When the nonlinear behavior and excess pore pressures in the ground are considered, the ground is expected to deform more, and thus differences in the tunnel response are anticipated.

Due to the rate of the loading during an earthquake, excess pore pressures may accumulate in soils. This phenomenon has been widely discussed and verified for fine-grained soils and fine sands (Mortezaie and Vucetic, 2016, Boulanger and Idriss, 2004, Matasović and Vucetic, 1995, Byrne, 1991). Even for sand-gravel composites, Kong et al. (2007) reported excess pore pressures accumulation during the Haicheng (1975) and Tangshan (1976) earthquakes in China, and during the Borah Peak (1983) earthquake in the United States. For soils with larger grain sizes (for purely gravelly soils), numerical investigations have shown that at depths of 10 m or less, excess pore pressures up to 50% or 60% of the initial effective confinement stress (σmo=po) can be reached in loose to medium deposits, with hydraulic conductivities between 0.001 and 0.01 m/s (Pender et al., 2016). Thus, it seems that soils, or even weathered rock, with moderate to large permeabilities and grain sizes, may develop excess pore pressures during earthquakes, particularly in shallow deposits.

The paper provides results of 2D full dynamic numerical analyses conducted to investigate the undrained seismic response of deep circular tunnels located far from the seismic source, subjected to ground shaking. (A tunnel is considered deep when there is not or negligible effect of the stress gradient with depth on the tunnel’s response. Dowding (1985) defined a point “far” from the epicenter, for distances higher than 10 km, where the seismic loading usually has a frequency content between 0.1 and 10 Hz). For comparison purposes, drained analyses are also performed. The commercial package FLAC 7.0 (Itasca, 2011) has been used in all the simulations. For the analyses, the liner is assumed to remain elastic; for the ground, a nonlinear elastoplastic constitutive model with excess pore pressures accumulation is used.

As stated above, the paper describes the tunnel’s response to ground shaking, at which, well designed underground structures should experience small to moderate strains. Thus, the model is intended to provide accurate results for small to moderate strains (i.e., we exclude phenomena such as liquefaction, landslides or other mechanisms that involve large deformations). The results are presented in terms of distortions of the cross section and of thrusts (axial) forces and bending moments of the liner. It must be noted that for near motions, with vibration frequencies larger than 10 Hz (source distances smaller than 10 km), the results discussed in this paper may not be applicable, because some dynamic amplification may occur due to the impinging of the waves on the cavity.

Section snippets

Cyclic nonlinear elastoplastic model

The constitutive model used is based on the work by Jung (2009), who investigated the seismic response of retaining walls and from Khasawneh et al. (2017), who used it to capture the response of integral abutment bridges due to thermal cyclic loading. The model is rate-independent, as it is generally assumed for the seismic response of most geomaterials, and is defined within the small-deformation framework for incremental plasticity theory. The model incorporates the well-known Masing’s rules,

Tunnel deformations

The effect of the flexibility ratio on the distortions of the tunnel is investigated for drained loading and for undrained loading at different levels of excess pore pressures. The definition of flexibility ratio provided by Peck et al. (1972), given in Eq. (17), is used.F=E/1+νm6EsIs/(1-νs2)/R3where, E is the Young’s modulus of the ground, Es is the Young’s modulus of the liner, t is the cross-sectional thickness of the liner, νs is the Poisson’s ratio of the liner, and Is is the moment of

Liner loading

Fig. 4 shows thrusts (axial) forces for the cases discussed above, and shown in Fig. 2, Fig. 3. That is, for normalized excess pore pressures in the free field (Δuff/po) ranging between 0.02 and 0.53. The thrust forces are normalized by the product of the input shear stress and the radius of the tunnel, following the proposal by Einstein and Schwartz (1979). Again, the white and dark markers correspond to drained and undrained loading, respectively. The figure shows that, for flexible tunnels (

Discussion

The presence of the tunnel modifies the soil response during dynamic loading, and thus the demand on the tunnel changes depending on the relative stiffness between the liner and the ground (similar observations were made by e.g., Bobet, 2010, Wang, 1993, for linear-elastic ground), as well as on the type of loading (drained or undrained) and on the magnitude of the excess pore pressures, as already discussed. All this is rooted in the interplay that exists between tunnel and ground, and, more

Summary and conclusions

The paper presents results of full dynamic numerical analyses conducted to investigate the drained and undrained response of deep circular tunnels. The liner is assumed to remain linear-elastic, while the ground is modeled as non-linear, hysteretic, elasto-plastic with pore pressure accumulation with cycles of loading due to the coupling of shear and volumetric strains.

The larger excess pore pressures and stiffness degradation that are produced around flexible tunnels, when excess pore

Acknowledgements

The research work presented in this paper was partially funded by the Colombia-Purdue Institute for Advanced Scientific Research (CPI), Universidad del Valle (Colombia) and Purdue University.

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