A 3D sliced-soil–beam model for settlement prediction of tunnelling using the pipe roofing method in soft ground

Abstract

The pipe roofing method is widely used in tunnel construction because it can realize a flexible section shape and a large section area of the tunnel, especially under good ground conditions. However, the pipe roofing method has rarely been applied in soft ground, where the prediction and control of the ground settlement play important roles. This study proposes a sliced-soil–beam (SSB) model to predict the settlement of ground due to tunnelling using the pipe roofing method in soft ground. The model comprises a sliced-soil module based on the virtual work principle and a beam module based on structural mechanics. As part of this work, the Peck formula was modified for a square-section tunnel and adopted to construct a deformation mechanism of soft ground. The pipe roofing system was simplified to a three-dimensional Winkler beam to consider the interaction between the soil and pipe roofing. The model was verified in a case study conducted in Shanghai, China, in which it provided the efficient and accurate prediction of settlement. Finally, the parameters affecting the ground settlement were analyzed. It was clarified that the stiffness of the excavated soil and the steel support are the key factors in reducing ground settlement.

Graphical abstract

Keywords

pipe roofing method / soft ground / numerical simulation / settlement prediction / simplified calculation / parametric analysis

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Notation

A	constant in displacement equations
[A_{\mathrm{v}}]	general stiffness matrix of the beam
B	constant in displacement equations
c	effective cohesion
c_{\mathrm{u}}	undrained shear strength of the soft ground
C	thickness of the overlying soil layer
d_{\mathrm{t}}	diameter of pipe i
d_1	outer diameter of a pipe
d_2	inner diameter of a pipe
D	diameter of the circular-section tunnel
D_{\mathrm{t}}	diameter of the equivalent tunnel
E	pipe elastic modulus
E_{\mathrm{s}1-2}	soil compression modulus
E_{50}^{\text{ref}}	reference secant shear modulus
E_{\text{oed}}^{\mathrm{r}\mathrm{e}\mathrm{f}}	reference oedometer modulus
E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}	reference unloading–reloading modulus
G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}	reference shear modulus at very low strains
H	height of the square-section tunnel
i_{\mathit{\text{z}}}	settlement trough width
k_{\mathrm{s}}	foundation reaction coefficient
k_{\mathrm{s}}^1	latticed improvement stiffness
k_{\mathrm{s}}^2	layered improvement stiffness
k_{\mathrm{t}}	inner steel support stiffness
[k]	foundation stiffness matrix that is a combination of k_{\mathrm{t}} and k_{\mathrm{s}}
K_0	coefficient of earth pressure at rest
k_{\mathrm{t}}	parameter in the Peck formula
L	width of the square-section tunnel
m	power for the stress-level dependency of stiffness
p^{\mathrm{r}\mathrm{e}\mathrm{f}}	reference pressure
P_{\mathrm{p}\mathrm{s}}	equivalent supporting pressure from the pipe roofing
P_{\mathrm{s}\mathrm{p}}	earth pressure on the pipe roofing
\left\{q_i\right\}	force acting on node i
R_{\mathrm{f}}	failure ratio
v_{\mathrm{m}}	maximum ground settlement
V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}	soil volume loss
V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}	tunnel section shrinkage
V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}	trial tunnel section shrinkage
\left\{w_i\right\}	displacement of each beam
\mathit{\text{z}}	depth below the ground face
Z_0	central depth of the tunnel
Z_{\mathrm{m}}	maximum depth of the mechanism
\alpha	parameter controlling the shape of the mechanism
\beta	power exponent of the stress–strain power curve
{\phi }^{\prime }	effective internal friction angle
{\gamma }_{0.7}	shear strain corresponding to 0.7G0ref
{\gamma }_{\mathrm{s}}	shear strain
{\gamma }_{\mathrm{s},\mathrm{f}}^{}	shear strain at maximum shear strength
v	Poisson’s ratio of a pipe
v_{\mathrm{u}\mathrm{r}}	Poisson’s ratio of unloading and reloading
{\nu }_{\mathrm{s}}	Poisson’s ratio of the soil
\rho	unit weight of the soil
{\rho }_{\mathrm{s}}	density of a pipe
\tau	shear strength
\psi	dilation angle

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52178342), the Tianjin Natural Science Foundation (No. 21JCZDJC00590), and the Shanghai Excellent Academic/Technical Leader Program (No. 20XD1432500).

Data availability statements

The data sets generated during and/or analyzed in the current study are available from the corresponding author on reasonable request.

Conflict of Interest

The authors declare that they have no conflict of interest.

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