The Effect of Tunnel Buoyancy on
Ground Surface Settlement in Elastic
Soil
Assaf Klar
Senior Lecturer, Faculty of Civil Engineering, Technion – Israel Institute of Technology, Haifa, Israel
TECHNICAL NOTE
ABSTRACT
Previous published elastic solutions for ground displacement due to tunneling ground loss ignore the effect of tunnel buoyancy. This note discusses this issue and quantifies the buoyancy contribution to the surface ground settlement in elastic soil. Solutions are given both for homogenous and Gibson’s soils.
KEYWORDS:
Tunneling, Settlement, Elastic solution.INTRODUCTION
The use of ground loss, or volume loss, to predict ground settlement profile is very popular. Instead of considering stress release as basic input, the ground loss is used as a basic parameter. For incompressible soil, the ground loss, which may be defined as the horizontal integral of vertical displacement, is constant at any depth above the tunnel, and is also equal to the apparent material deformation into the tunnel void (Fig 1.)
Figure 1. Tunnel contraction (ground loss)
Sagaseta (1987) utilized the ground loss parameter as a basic input for his closed form approximate solution for incompressible soil, based on the method of sinks and sources. Verruijt and Booker (1996) extended this solution to include also an ovalization of the tunnel. Verruijt (1997) gave a rigorous solution for the same problem, using complex variables. These solutions completely ignore the fact that as the tunnel is being excavated, there is a global vertical equilibrium change. That is, for any horizontal line below the tunnel the following relation is fulfilled in these solutions:
(1)
conditions that may be used in the solution of the tunnel problem, and considered the type used later by Bobet to be of little application to geotechnical engineering.
This note attempts to quantify the buoyancy effect on the shape of the tunnel settlement trough in elastic soil.
SOLUTION
The elastic solutions presented in the paper were derived using the finite difference code FLAC (Itasca, 2005). A uniform mesh constructed of squared elements with a size r0/10 was used. A smooth vertical boundary was placed at a distance of 40r0 away from the tunnel centerline, while a smooth horizontal boundary, representing a rigid rock, was placed at different depths below the tunnel. The tunnel boundary condition was mixed, controlled by both displacement and equilibrium requirements. The tunnel contraction was assumed to be radial, with a prescribed value of u0, while the equilibrium requirement was fulfilled by translating the tunnel in the vertical direction, , to obtain a vertical resistance equal to P. The displacement of the tunnel is the supper position of these two movements as described in the following expression representing the boundary condition at the tunnel:
(2)
where h is the depth of the tunnel, and , are changes in normal and shear stress around the tunnel contour (Fig 1). The above equilibrium condition is equivalent to demanding that the integral in Eq. 1 be equal to P. The ground loss, VL, may be related to u0 as:
(3)
displacement ratio increases and theoretically should approach one (this limit case cannot be numerically simulated, but only be approached, due to infinite small element size). This has little physical meaning, and is simply an outcome of the mixed boundary condition. In reality, excavation near rigid rock will not result in a radial ground loss.
Figure 2. Relative tunnel downward movement
Since the solution is elastic, the contribution of buoyancy may be decoupled from the ground loss effect. Fig. 3 shows the contribution of each of these two components to the surface settlement profile. The complete solution is the superposition of the two:
(4)
Figure 3. Surface settlement in elastic soil
From Eq. 4 and Fig. 3 it is clear that with increase of ground loss the relative effect of the buoyancy decreases. The buoyancy effect also decreases with increase in soil stiffness. Broadly, tunnel construction may result in ground loss in the range of 1-3%, while the “engineering” shear stiffness of clays may typically be in the range 5MPa-20MPa. As a result, the buoyancy can significantly affect the settlement trough. For example, Fig. 4 shows a hypothetical case of a 4m diameter tunnel with cover depth of 14m in a Gibson’s soil with a shear stiffness of 10MPa at tunnel level. Ground loss of 3% is considered. Even though the ground loss is relatively large, the buoyancy effect decreases the peak settlement by about 33%. In a homogenous soil the buoyancy effect would be expected to be even greater.
Figure 4. Example of a deep tunnel in Gibson’s soil
CONCLUSIONS
Elastic solutions for surface settlement due to tunneling have been presented. These involve a combined boundary condition of prescribed ground (volume) loss and buoyancy effect. The buoyancy effect, neglected in previous solutions, may have a substantial influence on the ground settlement, although this decreases with increase in soil stiffness and ground loss. Analysis which ignores the buoyancy effect is conservative in estimating maximum settlement.
It should be noted that the soil is rarely elastic and hence the solution should be considered with some reservation. However, as a result of the mixed boundary condition, the ground loss component is not greatly affected by the constitutive model of the soil (as pointed out by Sagaseta, 1998), and the buoyancy component is mostly due to deformation far from the tunnel where the soil may be regarded to behave approximately elastically. On the other hand, the assumption of radial ground loss may be inaccurate, and may affect the settlement trough more than the buoyancy. To include ovalization, one may add Verruijt and Booker’s (1996) expression to the solution.
REFERENCES
Attwell, P.B., S. Yeates, and A.R. Selby (1986) “Soil Movements Induced by Tunnelling and Their Effects on Pipelines and Strcutres,” London Blackie & Son.
Bobet, A. (2001) “Analytical Solutions for Shallow Tunnels in Saturated Ground,” Journal of Engineering Mechanics, Vol. 127(12) pp. 1258-1266 Itasca (2005) “FLAC – user manual,” Minneapolis.
Mindlin, R.D. (1936) “Forces at a Point in the Interior of a Semi-infinite Soild,” Physics 7, pp. 195-202
Pender, M.J. (1980) “Elastic Solutions for a Deep Circular Tunnel,” Geotechique, Vol. 30, pp. 216-22
Sagaseta, C. (1987) “Analysis of Undrained Soil Deformation due to Ground Loss,” Geotechnique, Vol. 37. pp. 301-320
Sagaseta, C. (1998) “Surface Settlements due to Deformation of a Tunnel in an Elastic Half Plane” Geotechnique, Vol. 48, pp. 709-713
Verruijt, A. (1997) “A Complex Variable Solution for a Deforming Circular Tunnel in an Elastic Half-Plane,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 21, pp. 88-90
Verruijt, A., and J.R. Booker (1996) “Surface Settlements due to Deformation of a Tunnel in an Elastic Half Plane,” Geotechnique, Vol. 46, pp.753-756
