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
klar@technion.ac.il
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 construction of tunnels may cause damage to existing structure such as buildings
and buried pipelines. To evaluate the risk one may conduct a finite element analysis
which considers both the existing structures and the tunneling process. In this case,
the ground movements are implicitly involved in the solution. On the other hand, one
may use the green field ground settlement profile as an input for a direct soil structure
interaction on the exiting structures (e.g. Attewell et al., 1986; Klar et al., 2005). In
this case, the ground settlement profile is a basic input parameter to the analysis,
which needs to be defined explicitly for the solution.
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)
where v is the change of vertical stress. In reality, as the tunnel is being excavated,
a weight of r02 (where  is the unit weight of the soil, and r0 is the radius of the
tunnel) is removed from the continuum. Hence, the integral in Eq. 1 should evaluate
to -r02. In other words, the given solutions disregard the buoyancy effect of the
tunnel. Note that in incompressible media, the buoyancy effect does not induce
ground changes. Bobet (2001) gave a solution of the elastic problem, accurately
considering stress equilibrium. However, the boundary conditions are such that
distant boundaries are loaded with in-situ stresses while the center of the tunnel
remains in place. While the stress distribution derived from Bobet’s solution is
accurate, the displacement is not, since the problem does not correspond to stress
release from a prestressed medium. Pender (1980) discussed the different boundary
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)
It is known, from Verruijt’s (1997) half space solution, that when P is equal to zero
(case of no buoyancy), the tunnel moves downwards. Fig. 2 shows a comparison
between Verruijt’s solution for tunnel translation to the present numerical results for
this condition of P=0. The uc value plotted in the figure is not exactly equal to
,
but is the difference between vertical displacement at the tunnel and at the surface at
the far end boundary. If the vertical boundary could have been positioned at an
infinite distance from the tunnel then
. Also given in the figure is the
averaged vertical displacement along the tunnel circumference as obtained from
Sagaseta’s solution. The solutions of Verruijt and Sagaseta are for an incompressible
elastic half space (i.e. Poisson’s ratio =0.5) while the numerical results are for a
nearly incompressible soil =0.49) on a rigid layer. It can be seen that as the depth of
the rigid layer, ZB, increases, the solution approaches that of Verruijt. Note that as the
base of the tunnel approaches the rigid rock (i.e.
) the vertical
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)
where 1 is the value taken from the dashed lines representing the contribution of the
ground loss, and 2 taken from the continuous lines repressing the buoyancy effect.
Fig. 3 shows solutions for homogenous and Gibson’s soils (i.e. linear increase of
stiffness with depth). In the case of Gibson’s soil the shear stiffness G, used in Eq. 4,
is the value at tunnel depth h. As can be seen, for the homogenous elastic soil, the
buoyancy effect on the settlement trough increases with increase of rigid rock depth.
This is a direct outcome of the elastic solution of plane strain problems in
homogenous soil, where for a half space (i.e. rigid rock at infinite depth) loading
results in an infinite displacement (e.g. Mindlin’s (1936) solution). This does not
occur in Gibson’s soil. It is interesting to note that except for the case of ZB/r0=10, the
contribution of ground loss to the settlement trough is generally very similar in the
homogenous and Gibson’s soils, both of them similar to Sagaseta’s (1987) solution.
This supports the remark made by Sagaseta (1998) that his solution of radial ground
loss is valid for any incompressible soil, irrespective of the constitutive model or
homogeneity.
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 5MPa20MPa. 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.
Klar, A., T.E.B. Vorster, K. Soga, and R.J. Mair (2005) “Soil-Pipe
Interaction due to Tunnelling: Comparison Between Winkler and Elastic
Continuum Solutions,” Geotechnique Vol. 55, pp 461-466
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
©
2006
ejge