Responses of in-service shield tunnel to overcrossing tunnelling in soft ground

Abstract

In dense urban areas, a new shield tunnel frequently crosses over in-service shield tunnels due to limited underground space. Construction of a new tunnel leads to relief of ground stress and soil displacement, which will inevitably result in a series of adverse impacts on the existing shield tunnels, such as tunnel heave, the dislocation between segmental rings, distortion of tunnel track. In this paper, a simplified analytical method was proposed to predict the responses of the in-service shield tunnel associated with overcrossing tunnelling in soft ground. The existing shield tunnel is treated as a Timoshenko beam. The tunnel-ground interaction is considered using the Pasternak foundation. The two-stage analysis method is used to divide the problem into two connected steps. First, overcrossing tunnelling-caused unloading loads imposing on the top surface of the existing tunnel are computed based on the Mindlin’s solution. Second, the tunnel deformation owing to the unloading loads is solved numerically by means of the finite difference method. The applicability of the proposed method is validated by two case histories in literature. The tunnel heaves predicted by the proposed method are in good agreement with the field measurements. According to the parametric analyses, it is found that when a new tunnel crosses over existing shield tunnel obliquely or parallelly, the induced tunnel heave is greater than that induced by perpendicularly crossing tunnelling. At given clearance between the new tunnel and the underlying existing shield tunnel, large-diameter tunnel excavation above causes greater tunnel heave and dislocation between adjacent rings than small-diameter tunnel. Improve the ground elastic modulus will effectively reduce tunnel heave when the ground elastic modulus is relatively low. To increase the equivalent shear stiffness will remarkably reduce dislocation between adjacent rings, however, its effects on reducing the tunnel heave is negligible.

Appendices

Appendix A. Matrices and vectors in Eq. (13)

$$[K_{1} ]{ = }\frac{1}{{l^{4} }}\left[ {\begin{array}{*{20}c} {A_{1} } & {A_{2} } & 2 & {} & {} & {} & {} & {} & {} \\ {A_{3} } & 5 & { - 4} & 1 & {} & {} & {} & {} & {} \\ 1 & { - 4} & 6 & { - 4} & 1 & {} & {} & {} & {} \\ {} & 1 & { - 4} & 6 & { - 4} & 1 & {} & {} & {} \\ {} & {} & \ddots & \ddots & \ddots & \ddots & \ddots & {} & {} \\ {} & {} & {} & 1 & { - 4} & 6 & { - 4} & 1 & {} \\ {} & {} & {} & {} & 1 & { - 4} & 6 & { - 4} & 1 \\ {} & {} & {} & {} & {} & 1 & { - 4} & 5 & {A_{3} } \\ {} & {} & {} & {} & {} & {} & 2 & {A_{2} } & {A_{1} } \\ \end{array} } \right]_{(n + 1) \times (n + 1)}$$

(23)

$$[K_{2} ] = \frac{{\Phi_{{{\text{eq}}}} G_{{\text{c}}} D + kD\left( {EI} \right)_{{{\text{eq}}}} }}{{\left( {EI} \right)_{{{\text{eq}}}} \left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} B \\ 1 \\ {} \\ \end{array} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ { - 2} \\ 1 \\ \end{array} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ 1 \\ { - 2} \\ \end{array} } \\ \ddots \\ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ {} \\ 1 \\ \end{array} } \\ \ddots \\ {\begin{array}{*{20}c} 1 \\ {} \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ \ddots \\ {\begin{array}{*{20}c} { - 2} \\ 1 \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ {} \\ {\begin{array}{*{20}c} 1 \\ { - 2} \\ {} \\ \end{array} } \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \\ {} \\ {\begin{array}{*{20}c} {} \\ 1 \\ B \\ \end{array} } \\ \end{array} } \right]_{(n + 1) \times (n + 1)}$$

(24)

$$[K_{3} ] = \frac{{kD\Phi_{{{\text{eq}}}} }}{{\left( {EI} \right)_{{{\text{eq}}}} \left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)}}\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ {} \\ \end{array} } & {\begin{array}{*{20}c} {} \\ 1 \\ \end{array} } & {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } \\ {\begin{array}{*{20}c} {} & {} \\ \end{array} } \\ \end{array} } \\ {} & {} & \ddots & {\begin{array}{*{20}c} {} & {} \\ \end{array} } \\ {} & {} & {} & {\begin{array}{*{20}c} 1 & {} \\ \end{array} } \\ {} & {} & {} & {\begin{array}{*{20}c} {} & 1 \\ \end{array} } \\ \end{array} } \right]_{(n + 1) \times (n + 1)}$$

(25)

$$\{ Q_{1} \} = \frac{{D\Phi_{{{\text{eq}}}} }}{{\left( {EI} \right)_{{{\text{eq}}}} \left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)}}\left\{ {q_{0} ,q_{1} , \ldots ,q_{i} , \ldots ,q_{n - 1} ,q_{n} } \right\}^{{\text{T}}}$$

(26)

$$\{ Q_{2} \} = \frac{D}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}\left[ {\begin{array}{*{20}c} {q_{1} - 2q_{0} + q_{ - 1} } \\ {q_{2} - 2q_{1} + q_{0} } \\ \vdots \\ {q_{n} - 2q_{n - 1} + q_{n - 2} } \\ {q_{n + 1} - 2q_{n} + q_{n - 1} } \\ \end{array} } \right]_{(n + 1) \times 1}$$

(27)

$$\{ Q_{3} \} = \left\{ {C_{1} ,C_{2} ,0,0, \ldots ,0,0,C_{3} ,C_{4} } \right\}^{{\text{T}}}$$

(28)

where $A_{1} = \frac{{k^{2} D_{{}}^{2} l^{4} }}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)^{2} }} + 2$; $A_{2} = - \frac{{2kDl^{2} }}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)}} - 4$; $A_{3} = \frac{{kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}} - 2$; $B = \frac{{kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}}$; $C_{1} = \frac{{2Dq_{0} }}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }} + \frac{D}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}\left( {q_{1} - q_{ - 1} } \right)$; $C_{4} = \frac{{2Dq_{n} }}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }} + \frac{D}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}\left( {q_{n + 1} - q_{n - 1} } \right)$; $C_{2} = - \frac{D}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}q_{0}$; $C_{3} = - \frac{D}{{\left( {\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D} \right)l^{2} }}q_{n}$.

Appendix B. Expressions of matrices and vectors of Eq. (16)

$$[W_{1} ] = - \frac{{(EI)_{{{\text{eq}}}} (\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D)}}{{2\Phi_{{{\text{eq}}}} l^{3} }}\left[ {\begin{array}{*{20}c} {E_{1} } & {E_{2} } & {} & {} & {} & {} & {} & {} & {} \\ {E_{3} } & 1 & { - 2} & 1 & {} & {} & {} & {} & {} \\ { - 1} & 2 & { - 2} & 1 & {} & {} & {} & {} & {} \\ {} & \ddots & \ddots & \ddots & \ddots & {} & {} & {} & {} \\ {} & {} & { - 1} & 2 & { - 2} & 1 & {} & {} & {} \\ {} & {} & {} & \ddots & \ddots & \ddots & \ddots & {} & {} \\ {} & {} & {} & {} & { - 1} & 2 & { - 2} & 1 & {} \\ {} & {} & {} & {} & {} & { - 1} & 2 & { - 1} & { - E_{3} } \\ {} & {} & {} & {} & {} & {} & {} & { - E_{2} } & { - E_{1} } \\ \end{array} } \right]_{(n + 1) \times (n + 1)} ;$$

(29)

$$[W_{2} ] = \frac{{(EI)_{{{\text{eq}}}} kD}}{{2\Phi_{{{\text{eq}}}} l}}\left[ {\begin{array}{*{20}c} {G_{1} } & 2 & {} & {} & {} & {} & {} & {} & {} \\ { - 1} & 0 & 1 & {} & {} & {} & {} & {} & {} \\ {} & { - 1} & 0 & 1 & {} & {} & {} & {} & {} \\ {} & {} & {} & \ddots & \ddots & {} & {} & {} & {} \\ {} & {} & {} & { - 1} & 0 & 1 & {} & {} & {} \\ {} & {} & {} & {} & {} & \ddots & \ddots & {} & {} \\ {} & {} & {} & {} & {} & { - 1} & 0 & 1 & {} \\ {} & {} & {} & {} & {} & {} & { - 1} & 0 & 1 \\ {} & {} & {} & {} & {} & {} & {} & { - 2} & { - G_{1} } \\ \end{array} } \right]_{(n + 1) \times (n + 1)} ;$$

(30)

$$\left[ {Q_{4} } \right] = - \frac{{(EI)_{{{\text{eq}}}} D}}{{2\Phi_{{{\text{eq}}}} l}}\left\{ {\begin{array}{*{20}c} {q_{1} - q_{ - 1} } \\ {q_{2} - q_{0} } \\ \vdots \\ {q_{i + 1} - q_{i - 1} } \\ \vdots \\ {q_{n} - q_{n - 2} } \\ {q_{n + 1} - q_{n - 1} } \\ \end{array} } \right\}_{1 \times (n + 1)} ;$$

(31)

$$[Q_{5} ] = \frac{{(EI)_{{{\text{eq}}}} }}{{\Phi_{{{\text{eq}}}} }}\left\{ {\begin{array}{*{20}c} {F_{1} } \\ {F_{2} } \\ \vdots \\ {F^{\prime}_{2} } \\ {F^{\prime}_{1} } \\ \end{array} } \right\};$$

(32)

where $E_{1} = \frac{{ - 2kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}} - \frac{{k^{2} D^{2} l^{4} }}{{(\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D)^{2} }}$; $E_{2} = \frac{{2kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}}$; $E_{3} = - \frac{{kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}}$; $F_{1} = - \frac{{D(q_{1} - q_{ - 1} )}}{2l}$; $F_{1}^{\prime } = \frac{{D(q_{n - 1} - q_{n + 1} )}}{2l}$; $F_{2} = \frac{D}{2l}q_{0}$; $F^{\prime}_{2} = - \frac{D}{2l}q_{n}$; $G_{1} = - 2 - \frac{{kDl^{2} }}{{\Phi_{{{\text{eq}}}} + G_{{\text{c}}} D}}$.