The analytical predictions on displacement and stress around shallow tunnels subjected to surcharge loadings

Wang, H.N.; Chen, X.P.; Jiang, Mingjing; Song, Fei; Wu, Lei

doi:10.1016/j.tust.2017.09.015

Cited by 72 publications

(22 citation statements)

References 31 publications

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“…In Step 2 of the i ‐th iteration (Figure D), the redundant tractions, whose expressions are as follows:

\begin{matrix} centerfalse \\ Horizontal \\ component \end{matrix} : \begin{matrix} centertrue \\ x_{1} o_{1} y_{1} : T_{s - x_{1}}^{((), i)} (x_{1}) = τ_{x_{1} y_{1}}^{i, 1} (x_{1} + hi) \\ x_{2} o_{2} y_{2} : T_{s - x_{2}}^{((), i)} (x_{2}) = T_{s - x_{1}}^{((), i)} (x_{1}) \end{matrix}; \begin{matrix} centertrue \\ Vertical \\ component \end{matrix} : \begin{matrix} centertrue \\ x_{1} o_{1} y_{1} : T_{s - y_{1}}^{((), i)} (x_{1}) = σ_{y_{1}}^{i, 1} (x_{1} + hi) \\ x_{2} o_{2} y_{2} : T_{s - y_{2}}^{((), i)} (x_{2}) = T_{s - y_{1}}^{((), i)} (x_{1}) \end{matrix} .

are exerted in reverse on the surface of the semi‐infinite ground without any holes. The additional stresses and displacements in this step can be addressed by Flamant's solution and its modification with respect to the variables x 2 and y 2 as follows:

\begin{matrix} lefttrue \\ Normal stresses \\ in the x_{2} and y_{2} directions : σ_{x_{2}}^{i, 2} = \frac{2}{π} \int_{} \end{matrix}

…”

Section: Analytical Solutions Of the Ground Displacement And Stress Fmentioning

confidence: 99%

Analytical study of ground responses induced by the excavation of quasirectangular tunnels at shallow depths

Zeng

Wang

Jiang

2019

Num Anal Meth Geomechanics

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Summary The construction of quasirectangular tunnels at shallow depths is becoming increasingly common in urban areas to efficiently utilize underground space and reduce the need for backfilling. To clarify the mechanical mechanism of the stresses and displacements around the tunnels, this study proposes analytical solutions that precisely account for quasirectangular tunnel shapes, the ground surface, the tunnel depth, and the ground's elastic/viscoelastic properties. The Schwarz alternating method combined with complex variable theory is employed to derive the elastic solution, and convergent and highly accurate solutions are obtained by superposing the solutions in the alternating iterations. Based on the solution and the extended corresponding principle for the viscoelastic problem, the time‐dependent analytical solutions for the displacement are obtained for the ground assuming any viscoelastic model. The analytical solutions agree well with the finite element method (FEM) numerical results for models that are completely consistent, and qualitatively agree with field data. Furthermore, based on the stress solution combined with the Mohr‐Coulomb failure criterion, the predicted initial plastic zone and propagation directions around the tunnels are qualitatively consistent with those determined by the limit analysis. A parametric study is performed to investigate the influences of the rectangular/quasirectangular tunnel shape, burial depth, and supporting pressure on the ground stresses and displacements.

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“…In Step 2 of the i ‐th iteration (Figure D), the redundant tractions, whose expressions are as follows:

\begin{matrix} centerfalse \\ Horizontal \\ component \end{matrix} : \begin{matrix} centertrue \\ x_{1} o_{1} y_{1} : T_{s - x_{1}}^{((), i)} (x_{1}) = τ_{x_{1} y_{1}}^{i, 1} (x_{1} + hi) \\ x_{2} o_{2} y_{2} : T_{s - x_{2}}^{((), i)} (x_{2}) = T_{s - x_{1}}^{((), i)} (x_{1}) \end{matrix}; \begin{matrix} centertrue \\ Vertical \\ component \end{matrix} : \begin{matrix} centertrue \\ x_{1} o_{1} y_{1} : T_{s - y_{1}}^{((), i)} (x_{1}) = σ_{y_{1}}^{i, 1} (x_{1} + hi) \\ x_{2} o_{2} y_{2} : T_{s - y_{2}}^{((), i)} (x_{2}) = T_{s - y_{1}}^{((), i)} (x_{1}) \end{matrix} .

\begin{matrix} lefttrue \\ Normal stresses \\ in the x_{2} and y_{2} directions : σ_{x_{2}}^{i, 2} = \frac{2}{π} \int_{} \end{matrix}

…”

Section: Analytical Solutions Of the Ground Displacement And Stress Fmentioning

confidence: 99%

Analytical study of ground responses induced by the excavation of quasirectangular tunnels at shallow depths

Zeng

Wang

Jiang

2019

Num Anal Meth Geomechanics

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“…However, the slight discrepancies shown in Figures 6 to 10 also appear in other papers. 11,25,28 The reasons are considered to be the fact that "the numerical model is subjected to a finite geometric size, while the analytical solution is derived in an infinite half plane." 25…”

Section: Analytical Solution Verificationmentioning

confidence: 99%

“…The analytical solutions can provide the displacement field of the whole ground through strict mathematical derivation, with respect to all of the concerned parameters. 11 There are four methods to obtain the elastic analytical solution: Airy stress function solution [12][13][14] ; the bipolar coordinate method [15][16][17] ; virtual image technique method [18][19][20][21] ; and complex variable solution. [22][23][24][25][26][27][28] The complex variable method can determine displacements and stresses simultaneously throughout the entire half-plane by an infinite power series in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

Displacement analytical prediction of shallow tunnel based on unified displacement function under slope boundary

Kong

et al. 2018

Num Anal Meth Geomechanics

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Summary For slope condition of ground surface, the asymmetrical deformation about the vertical center line and the horizontal center line of the tunnel cross section can be formed. A unified displacement function expressed by the Fourier series is presented to express the asymmetrical deformation of the tunnel cross section. Five basic deformation modes corresponding to the expansion order 2 are a complete deformation mode to reflect deformation behaviors of the tunnel cross section under slope boundary. Such this complete displacement mode is implemented into the complex variable solution for analytically predicting tunneling‐induced ground deformation under slope boundary. All of these analytical solutions are verified by good agreements of the comparison between the analytical solutions and finite element method results. A parameter study is carried out to investigate the influence of deformation modes of the tunnel cross section, geometrical conditions of the tunnel and the slope angle, and “Buoyancy effect” on the displacement field. Finally, the proposed method is consistent with measured data of the Hejie tunnel in China qualitatively. The presented solution can provide a simplified indication for evaluating the ground deformation under slope condition of ground surface.

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“…Most viscoelastic work focuses on deep tunnels so far (Sulem et al, 1987;Dai et al, 2004;Nomikos et al, 2011;Fahimifar et al, 2010;Wang et al, 2017) and the viscoelastic analytical solutions of shallow tunnels have been rarely covered. Wang et al, (2018) obtained analytical solution of elastic and viscoelastic when the ground surface of shallow tunnel and the tunnel cross-section subject to the stress loads. However, a viscoelastic analytical solution of a shallow tunnel is rarely considered when the tunnel cross-section boundary is the displacement.…”

Section: Introductionmentioning

confidence: 99%

“…al., 2006;Wang et al, 2018). elastic analytical solution of displacement can be obtained in section 3.1.…”

mentioning

confidence: 99%

Viscoelastic analytical solution for shallow tunnel considering time-dependent displacement boundary

Shen

Kong

et al. 2020

JGS Special Publication

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A time-dependent unified displacement function is proposed to capture main deformation responses of the tunnel cross-section. When the expansion order n equals 2, this complete displacement mode is used as displacement boundary condition (DBC) of the tunnel cross-section. Complex variable method is adopted in order to obtain an elastic analytical solution. Considering the time effect of ground mechanical parameters and tunnel cross-section deformation, a viscoelastic analytical solution of ground displacement is obtained. By parametric analyses, time effects of settlement trough curve, ground displacement and the settlement at the center point of ground surface are analyzed.

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The analytical predictions on displacement and stress around shallow tunnels subjected to surcharge loadings

Cited by 72 publications

References 31 publications

Analytical study of ground responses induced by the excavation of quasirectangular tunnels at shallow depths

Analytical study of ground responses induced by the excavation of quasirectangular tunnels at shallow depths

Displacement analytical prediction of shallow tunnel based on unified displacement function under slope boundary

Viscoelastic analytical solution for shallow tunnel considering time-dependent displacement boundary

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