. Institute of Geotechnical and Underground Engineering, Beijing University of Technology, Beijing 100124, China . Department of Civil Engineering, School of Civil Engineering, Tsinghua University, Beijing 100062, China
Using the complex variable method, an elastic analytical solution of the ground displacement caused by a shallow circular tunneling is derived. Non-symmetric deformation relative to the horizontal center line of the tunnel cross-section is used as a boundary condition. A comparison between the proposed analytical method and the Finite Element Method is carried out to validate the rationality of the obtained analytical solution. Two parameters in the Peck formula, namely the maximum settlement of the ground surface center and the width coefficient of settlement curve, are fitted and determined. We propose a modified Peck formula by considering three input parameters, namely the tunnel depth, tunnel radius, and the tunnel gap. The influence of these three parameters on the modified Peck formula is analyzed. The applicability of the modified Peck formula is further investigated by reference to six engineering projects. The ground surface displacement obtained by the explicit Peck formula is in good agreement with the field data, and the maximum error is only 1.3 cm. The proposed formula can quickly and reasonably predict the ground surface settlement caused by tunnelling.
Just Accepted Date: 16 August 2024Online First Date: 29 October 2024Issue Date: 28 November 2024
Cite this article:
Xiuli DU,Qingtao LIN,Dechun LU, et al. Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel[J]. Front. Struct. Civ. Eng.,
2024, 18(11): 1637-1648.
Fig.1 Nonuniform deformation pattern of a tunnel cross-section.
Fig.2 Analytical mechanical model of a shallow tunnel.
Fig.3 Comparisons between FEM and proposed analytical method: (a) established finite element model; (b) comparison results.
Fig.4 Influence of on the vertical and horizontal displacement of the ground surface: (a) vertical displacement curves; (b) horizontal displacement curves.
Fig.5 Influence of on the ground vertical settlement in different depths.
Fig.6 Ground surface settlement curves for different relative depths (r0/h): (a) 1/4; (b) 1/5.
Fig.7 Relationship between r0/h and smax for different values of .
Fig.8 Comparisons between the predicted values and analytical results.
Fig.9 Influence of on the width coefficient of the settlement curve i.
r0 (m)
ki
R2
2.0
0.797
0.999
2.5
0.792
0.998
3.0
0.786
0.997
3.5
0.779
0.995
4.0
0.773
0.994
4.5
0.769
0.992
5.0
0.761
0.989
5.5
0.752
0.987
Tab.1 Values of ki and R2 for different values of r0
Fig.10 Influence of r0 and h on the width coefficient of the settlement curve i.
Fig.11 Curves of the ground surface settlement resulting from Eq. (25) and analytical solution.
Fig.12 Influence of , r0, and h on the curve of the ground surface settlement: (a) influence of ; (b) influence of tunnel radius; (c) influence of tunnel depth.
Tunnel name
Ground condition
Depth (h/m)
Radius (r0/m)
Deformation parameter (/m)
Ref.
Bangkok tunnel
soft clay; stiff clay; fine sand
18.5
1.33
0.0926
[32]
Docklands light railway lewisham extension (MS-5)
made ground; terrace gravel; Woolwich and Reading bedsground; thanet sand
13.8
2.925
0.00754
[33]
Frankfurt subway tunnel
sand and gravel natural; stiff overconsolidated clay marl
14
3.35
0.0319
[33]
Green park tunnel
sand and gravel; stiff fissured clay
29.4
2.07
0.0389
[32]
Heathrow express trial tunnel
fill ground; terrace gravel; stiff London clay
19
4.25
0.0663
[32]
Thunder bay tunnel
silty sand; soft-to-firm clay; firm-to-stiff clay
10.7
1.235
0.187
[32]
Tab.2 Tunnel geometry parameters and deformation parameters for the case studies
Fig.13 Displacement comparisons between the field data and the proposed method: (a) Bangkok Tunnel; (b) Docklands Light Railway Lewisham Extension (MS-5); (c) Frankfurt Subway Tunnel; (d) Green Park Tunnel; (e) Heathrow Express Trial Tunnel; (f) Thunder Bay Tunnel.
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