3-D shell analysis of structure in portal section of mountain tunnel under seismic SH wave action
Introduction
Tunnel portals are likely to be damaged by earthquake because they often locate in rather weak ground where earthquake motion is amplified and the ground deforms to a large extent. The damage to tunnel portal sections includes the cracks in portal part structure and lining (Genis, 2010). There are numerous reports on the damage to tunnel portals caused by earthquakes. Chi-Chi earthquake (Wang et al., 2002, Uenishi et al., 1999), Düzce earthquake (1999) and Wenchuan earthquake (Chen et al., 2012a, Chen et al., 2012b, Li, 2012) cause extensive failures in portal section of mountain tunnel, reviving the interest in the associated analysis and design methods.
Three methods are mainly used for the seismic study of underground structures: model tests, theoretical analysis and numerical analysis. By limitation of monitoring method, sometimes data are not enough in model test. Further, it takes much cost. Numerical analysis is aimed at case-specific analyses and complex problem. Nevertheless, theoretical analysis can enrich and replenish effectively to model test, which is essential in exploring physical nature of the problem, and also can be used for checking the accuracy of numerical methods.
The seismic study of underground structure in theoretical analysis way has increased considerably in the past three decades. The analytical solutions are divided into fluctuation method based on wave equation and interaction method based on motion equation. Free field deformation approach is a simper type of fluctuation method, which is widely used in current design guidelines, and can provide a first-order estimate of the anticipated deformation of the structure (Hashash et al., 2001). The method is based on the theory of wave propagation in homogeneous, isotropic, elastic media, which ignore the interaction between the structure and soil (Newmark, 1968).
In order to provide design seismic strains and stresses of underground structure due to body and surface waves through the previous method, Newmark (1968) and Kuesel (1969) calculated ground strains due to a wave propagating at a given angle of incidence in a medium. St John and Zahrah (1987) used Newmark’s approach to develop solutions for these strain components due to P-waves, S-waves and R-waves. Furthermore, axial, hoop and shear strains are presented through 3-D shell theory, regarding long cylindrical underground structures subjected to seismic shear (S) and Rayleigh wave excitation (Kouretzis et al., 2006, Kouretzis et al., 2011). Some other analytical solutions simulate soil-structure interaction effects, by employing the beam-on-elastic foundation approach or modeling the underground structure as a cylindrical shell embedded in an elastic half-space, and account for slippage at the soil-structure interface (Wong et al., 1986, Luco and de Barros, 1994).
The presented researches mainly calculate structure strains in half-space and infinite space. To the authors’ knowledge, no studies have discussed on seismic response of structures embedded in inclined plane based on free field deformation approach.
However, dynamic response of lining at the tunnel portal section is mainly affected by slope. Different angle of slope will cause different extent, types and directions of reflected wave. The waves are superimposed in the vicinity of slope forming a complex wave field (Shi et al., 2008). Mechanics characteristics of lining under complex wave field caused by slope reflection are not well studied. Single free face slope is built to model entrance section of a mountain tunnel. The theory of elastic wave propagation is used to deduce the solution of free field ground motion on the axis of tunnel under incident SH wave, while ignoring the existence of tunnel structure. Then the 3-D elastic shell theory is employed in order to provide strains by the ground motion. Due to the relative complexity of the mathematics, the resulting relations are verified against results from shaking table test.
Section snippets
Basic assumptions
The derivation is based on following assumptions:
- (1)
Ignoring the impact of presence of tunnel on dynamic response of ground.
- (2)
Ignoring interaction between tunnel and rock.
- (3)
Simplifying tunnel portal section as single free face slope, while the grading angle of slope is less than 45°.
- (4)
In many cases, severely-weathered, broken and loose rock mass are distributed in portal section of tunnel. This kind of rocks would be classified into class V according to the BQ system in China Code for Design of Road
Parametric analyses
The analytical relation for tunnel portal section derived above is validated by comparison with shaking table tests. So firstly, the strains of tunnel portal section through above theoretical solutions under the propagation of SH wave from bedrock are derived.
The displacement induced by an vertically propagating SH wave from bedrock can be expressed as f(t) = A sin (2πft), the peak particle displacement of the seismic motion is 0.2 m, with a propagation velocity of C = 200 m/s, a period of T = 1 s.
The
Model test overview
The test aimed at the dynamic response of lining at tunnel portal section, so we can validate the law of dynamic strain response of lining by above parametric analyses.
Axial strain, hoop strain of lining, and relative displacement of lining along axial direction are measured in the test. The position of strain gauge along lining cross section is illustrated in Fig. 8. Relative displacement sensors arranged outside model box are used to measure the displacement of lining relative to model
Conclusions
Mountain tunnel in portal section is simplified to single free surface slope, and the theory of elastic wave propagation is used to derive the solution of free field ground motion on the axis of tunnel under incoming SH wave. The 3-D elastic shell theory is employed in order to provide strains of structure. The resulting analytical solutions are verified against the results of shaking table test. Meanwhile, dynamic mechanics characteristics of lining under vertically propagating SH-Waves is
Acknowledgements
The authors gratefully acknowledge the support of National Natural Science Foundation of China (Grant Nos. 41202221, 51038009, 41272337), and Opening fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Grant No. SKLGP2012K002), Beijing Natural Science Foundation (Grant No. KZ200910005009).
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