This boundary condition, having been derived from the Kirsch solution for a stress-free cavity, corresponds the case in which zero shear stresses, σ rt can be expected along the tunnel p
Trang 1Chapter 7
OVALIZATION OF A CIRCULAR TUNNEL
The application of uneven horizontal and vertical stresses may cause a circular tunnel with a flexible lining (or no lining) to become oval in shape Such uneven stresses are often found in the underground, where the horizontal stresses can often be expressed as a factor multiplied by the vertical stresses (see Chapter 5) The ovalization effect due to these uneven stresses can be derived from the basic Kirsch solution [37] for a stress-free cavity with unequal stresses applied
at infinity, as shown by Pender [25], who deduced the incremental displacements for this case The stresses on tunnel linings due to ovalization had already been discussed by Morgan [21] and Muir Wood [22].
The components of ovalization along the tunnel wall derived from the Kirsch
solution contain a radial component u r = u o cos 2θ and a tangential component
u θ = −u o sin 2θ , where u o is the maximum displacement along the tunnel cav-ity This boundary condition, having been derived from the Kirsch solution for
a stress-free cavity, corresponds the case in which zero shear stresses, σ rt can
be expected along the tunnel periphery This corresponds to the case in which the tunnel lining can be assumed to be frictionless Uriel and Sagaseta [39] extended Sagaseta’s original ground loss solution to include these ovalization components, and obtained surface settlements which are valid for incompress-ible soils.
Verruijt and Booker [40] extended Uriel and Sagaseta’s work by using an elastic model for the underground They obtained expressions for the displace-ments in the entire field for all values of Poisson’s ratio, and they obtained them for both the ground loss problem and for an ovalization of a tunnel For their so-lution Verruijt and Booker focused on the related mode of ovalization in which
zero tangential displacements are assumed along the tunnel periphery, u θ = 0 This corresponds to the case in which the tunnel lining is rough relative to the surrounding ground and little or no sliding occurs between the ground and the lining.
In this chapter, the ovalization boundary condition used by Verruijt and Booker is applied to the solution obtained in Chapter 4 in order to obtain the exact solution for an ovalizing tunnel in an elastic half-plane These results, first
61
Trang 2obtained by Strack and Verruijt [35], will be analyzed and compared to Verruijt and Booker’s [40] solution It should be noted that the ovalization boundary condition for which the shear stresses are assumed to vanish was applied inde-pendently by Pinto [26] to Verruijt’s original [42] solution in order obtain the displacements and stresses for an ovalizing tunnel with a frictionless lining.
§ 7.1 Solution of the Problem
The boundary condition for the ovalization of a tunnel with a rough lining, originally proposed by Verruijt and Booker [40], can be written in terms of the
polar coordinates r and θ (see Figure 7.1) as
where u r and u θ are the radial and tangential displacements along the tunnel
boundary, u o is the maximum displacement of the tunnel boundary, and θ is the
polar angle (shown in Figure 7.1).
.
.
.
.
.
. .
.
.
x y
r
h θ
Figure 7.1: Ovalization of a tunnel.
In order to use the general solution presented in Chapter 4 to find the so-lution for an ovalizing tunnel, the boundary condition along the tunnel given
by (7.1) must be mapped to the ζ -plane and then expanded in a Fourier series This Fourier series must then be used to determine the function G (ασ ) given
in (4.31) As this derivation is somewhat involved and has been previously published (see [35]), the details have been relegated to Appendix C The final result is given by
G (ασ ) = ∞
k =−∞
Trang 3Section 7.2 Validation of the Solution 63
where
A k = iµu o
α k −3 ( 1 − α 2 ) 2 [−3 + (k + 1)(1 − α 2 ) ] if k ≥ 2.
(7.3)
The stresses and strains associated with the ovalization of a circular tunnel can
be determined by using the coefficients given in (7.2) to determine the Laurent
expansions in (4.56) and (4.57).
In order to focus on the incremental stresses and displacements caused by the
ovalization of a tunnel, the buoyancy force in the general solution of Chapter 4
is set to zero, as was done in the previous chapter The result is that the solution
consists entirely of the Laurent expansions of the complex potentials ϕ and
ψ The stresses and displacements due to buoyancy can be added later by
superimposing the solution for the buoyant, rigid tunnel discussed in Chapter 5,
and the gravitational stresses can be superimposed through use of (5.7) – (5.9).
In order to focus on the incremental stresses and displacements caused by
the ovalization of a tunnel, the resultant force on the tunnel due to buoyancy
effects is set to zero for this solution, as was also done in the previous chapter
The result is that the solution consists entirely of the Laurent expansions of the
complex potentials ϕ and ψ.
As is also the case in the ground loss problem (Chapter 6), the displacements
generated tend towards a constant value for large values of z After adding a
rigid body motion to the solution such that the displacements vanish at infinity,
the tunnel moves downward as a whole However, the degree of settlement is
much smaller than for the ground loss problem.
§ 7.2 Validation of the Solution
As is the case for the solutions discussed in previous chapters, the solution
for the ovalization of a circular tunnel has been implemented in a computer
program which can be used to generate plots of the normalized stresses and
displacements The plots can be visually inspected to confirm that the stresses
vanish along the surface (see Figure 7.2) and that the tunnel does indeed ovalize
(see Figure 7.3).
It is noted that in Figure 7.2, as in all plots of stress contours in this thesis, the
solid thin curves denote negative (compressive) stresses, the thick solid curves
denote contours of zero stress, and the finely dotted curves denote positive
(tensile) stresses It is emphasized that in most cases such tensile stresses will
be eliminated through superposition of the stresses due to gravity, (5.7) – (5.9).
The plots in this chapter only show the incremental stresses due ovalization.
The initial stresses should be added to obtain the total stresses.
Trang 4-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
-5 -4 -3 -2 -1 0
x/ h
- y/ h
. . .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
Figure 7.2: The stresses σ yy (left-hand side) and σ xy (right-hand side) for ν = 0 and r/ h = 0.7 The contour interval is 0.4µu o / h -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 x/ h - y/ h
.
.
. .
. . . . . . . . . . . . . . . . .
.
.
Figure 7.3: Displacements, scaled by 0.15 u
o for ν = 0 and r/h = 0.7.
The programmed solution yields maximum errors in the normalized stresses
σ yy h/ 2µu o and σ xy h/ 2µu o along the surface ranging from ±2 × 10 −13 for
shallow tunnels (r/ h = 0.9) to ±3×10 −11 for deep tunnels (r/ h = 0.01) The maximum errors of the components of the normalized displacements u t /u o and
v t /u o along the tunnel boundary range from ±3×10 −7 for very shallow tunnels
to ±5 × 10 −5 for deep tunnels These values are for ν = 0.0 The maximum errors for other values of ν appear to be within one order of magnitude of the
errors given here.
Trang 5Section 7.3 Comparison to Verruijt and Booker’s Solution 65
§ 7.3 Comparison to Verruijt and Booker’s Solution
Verruijt and Booker [40] presented an approximate solution for the ovalization
of a circular tunnel in an elastic half-plane which consists of three parts: the first
is a singular solution corresponding to an ovalization of a tunnel in an infinite
plane, the second is an image of the first (obtained by reflecting the first solution
across the surface of the half-plane), and the third solution is a Boussinesq stress
distribution added to the surface to ensure that normal stresses along the surface
of the half-plane are zero The complete solution is approximate in the sense
that the addition of the third part affects the displacements along the tunnel
boundary.
The influence of the third part of the solution can be clearly seen in Figure 7.4
in which the exact solution is compared to Verruijt and Booker’s solution It
-3
-2
-1
0
-3 -2 -1
0
x/ h
- y/ h
.
.
. .
. .
. .
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
Figure 7.4: Comparison between the exact solution (left-hand side) and Verruijt and
Booker’s solution (right-hand side) for the displacements in the case that ν = 0.5 and
r/ h = 0.7 The displacements have been scaled by 0.15/u o
appears that the surface load added by the Boussinesq solution has the effect of
adding an additional compression the tunnel, as can be expected.
For very deep tunnels the effect of the Boussinesq part of the approximate
solution should become negligible and the complete solution should agree with
the solution presented in this chapter For incompressible ground (ν = 0.5) this
is indeed the case, as can be observed in Figure 7.5, where a deeper tunnel with
an r/ h ratio of equal to 0.3 is shown.
For values of Poisson’s ratio unequal to 0.5, however, the solutions do not
tend toward each other when the tunnel radius to depth ratio becomes smaller.
Upon inspection it appears that the boundary condition in theVerruijt and Booker
solution depends on ν This is evident in Figure 7.6, where the displacements
Trang 6-2 -1 0 1 2
-2
-1
0
-2 -1
0
x/ h
- y/ h
..
.
.
.
.
. . .
.
.
.
Figure 7.5: Comparison between the exact solution (left-hand side) and Verruijt and Booker’s solution (right-hand side) for the displacements in the case that ν = 0.5 and r/ h = 0.3 The displacements have been scaled by 0.06/u o along the surface and the boundary of the tunnel are compared for a Poisson’s ratio of 0 and an r/ h ratio equal to 0.3. It appears that the difference is in the original expression for the singular solution in Verruijt and Booker’s paper [40]: this expression does not meet the boundary conditions unless ν = 0.5 The problem can be remedied by replacing -1 0 1 -1 0 -1 0 x/ h - y/ h
.
Figure 7.6: Comparison between the exact solution (solid lines) and Verruijt and Booker’s solution (finely dotted lines) for the surface settlements and deformation of
the tunnel in the case that ν = 0.0 and r/h = 0.3 The displacements have been scaled
by 0.06/u
Trang 7Section 7.4 Stresses around the Tunnel 67
equation (1) in [40] with
u x = −εR 2
x
r 1 2 + x
r 2 2
+ δR 2
m + 1 + R 2
r 1 2
x 3
r 1 4 −
m − 1 + 3 R 2
r 1 2
xz 2 1
r 1 4
+
m + 1 + R 2
r 2 2
x 3
r 2 4 −
m − 1 + 3 R 2
r 2 2
xz 2 2
r 2 4
(7.4)
and replacing equation (2) in [40] with
u z = −εR 2
z 1
r 1 2 + z 2
r 2 2
+ δR 2
m − 1 + 3 R 2
r 1 2
x 2 z 1
r 1 4 −
m + 1 + R 2
r 1 2
z 3 1
r 1 4
+
m − 1 + 3 R 2
r 2 2
x 2 z 2
r 2 4 −
m + 1 + R 2
r 2 2
z 2 3
r 2 4
.
(7.5)
All of the variables in (7.4) and (7.5) are defined as in [40] Unfortunately (7.4)
and (7.5) are much more complicated than the original expressions They will
also lead to a more involved derivation of the Boussinesq solution which must
be added to (7.4) and (7.5) in order to complete the analysis.
§ 7.4 Stresses around the Tunnel
The isotropic stresses 1
2 (σ xx + σ yy ) and deviatoric stresses
1
4 (σ yy − σ xx ) 2 + σ 2
xy
are shown in Figure 7.7 From the contours in Figure 7.7 it appears that the
stresses associated with the ovalization of the tunnel are of a local nature, and
it seems that most of the stresses have dissipated at a distance of two times the
tunnel depth For deeper tunnels the stresses are of an even more local nature
(which is evident from plots not shown here) It can also be seen from the
deviatoric stresses contoured in Figure 7.7 that any plastic effects that may occur
would be concentrated nearly symmetrically around the tunnel boundary, with
the most plasticity occurring where the lines of zero isotropic stress intersect
the tunnel The plastic zones will, however, be affected by the superposition of
the initial stresses.
Finally, it appears from Figure 7.7 that the deviatoric stresses around the
tunnel are not strongly affected by the presence of the stress-free surface, but
that the isotropic stresses are In fact, the presence of the surface seems to allow
for a development of compressive stresses at the top of the tunnel – the stresses
at the bottom of the symmetrically deforming tunnel are tensile.
Trang 8-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
-5 -4 -3 -2 -1 0
x/ h
- y/ h
.
.
Figure 7.7: The Isotropic stress (left-hand side) and deviatoric stress (right-hand side) for the case that ν = 0 and r/h = 0.7 The contour interval is 0.25µu o / h § 7.5 Stresses on the Tunnel The radial and shear stresses σ rr and σ rt acting on the tunnel lining due to ovalization are shown for different values of Poisson’s ratio and different relative tunnel depths in Figure 7.8 and Figure 7.9 It is apparent from these plots that
.
. .
.
..
.
.. .
.. .
. .. .
. . .
. .
. . .
. . .
.
.. .
..
.. .
σ rr ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.7
.. . .
..
.
. . .
.. .
. .. .
.
. . .
. . .
. . .
. ...
.. .
.. .
. . .
σ rr
ν = 0.5
ν = 0.25
ν = 0
r/ h = 0.1
Figure 7.8: The radial stresses acting on the tunnel as a function of ν.
the radial and shear stresses acting on the tunnel lining are only moderately affected by the relative depth of the tunnel The strongest effects in this regard are a moderate reduction in the radial stress along the top of the tunnel and the relatively larger shear stress along the tunnel boundary in comparison to the case for a deeper tunnel.
Trang 9Section 7.5 Stresses on the Tunnel 69
.
. .
.
.
. .
.
.
.
σ rt ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.7
.
.
.
.
. .
.
σ rt ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.1 Figure 7.9: The shear stresses acting on the tunnel as a function of ν. It is also apparent that the radial and shear stresses are only moderately effected by Poisson’s ratio ν. It should be noted that the lack of a symmetrical stress distribution in the radial stresses on the tunnel lining indicates that pure, symmetrical ovalization of the tunnel is only likely to occur if other effects, such as pressurized grouting along the top of the tunnel, account for a more symmetrical loading of the lining The tangential hoop stress σ t t is shown in Figure 7.10 for different values of Poisson’s ratio and different relative tunnel depths It is apparent from these
.
.
.
. . .
.
.
.
.
σ t t ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.7
. .
. . .
.
.
.
. .
.
.
.
σ t t
ν = 0.5
ν = 0.25
ν = 0
r/ h = 0.1
Figure 7.10: The hoop stress around the tunnel as a function of ν.
plots that the hoop stress around the tunnel is much more dependent on the
relative depth of the tunnel and much more dependant on the value of Poisson’s
ratio than the radial and shear stresses are.
The hoop stress for ν = 0.5 is substantially different for relatively shallow
tunnels than for relatively deep tunnels In the case of the shallow tunnel the
hoop stress contains a compressive zone at the top of the tunnel A related
compressive zone in the isotropic stresses can be seen at the top of the tunnel in
Figure 7.7 In the case of the deep tunnel, the hoop stress vanishes for ν = 0.5.
This interesting effect is in contrast to the effect seen in the hoop stress for the
rigid tunnel and ground loss solutions In those cases, the hoop stress around
the tunnel vanishes for values of ν = 0.
Trang 10§ 7.6 Displacements along the Surface
The scaled settlements, denoted by v/u o , for the ovalization of a tunnel are shown in Figure 7.11 for different values of Poisson’s ratio One of the more
-2 -1 0
-2 -1
0 2
u o
v
x/ h
.
. .
..
ν = 0.5
ν = 0.25
ν = 0
Figure 7.11: Surface settlements as a function of ν for r/ h = 0.7.
prominent features of these curves is that there are not only settlements of the surface, but also a certain amount of heave This is due to the fact that there is no loss of volume along the tunnel boundary due to ovalization For incompressible soils this means that there must also be zero volume loss along the surface For compressible soils the volume change along the surface is variable, but remains near enough to zero that the surface exhibits heave.
It can also be observed from Figure 7.11 that the trough width, defined here as the distance between the points of intersection of the trough with the horizontal axis, is approximately twice the depth of the tunnel This is also the case for
other values of r/ h, as can be seen in Figure 7.12.
-2 -1 0
-2 -1
0 10
u o
v
x/ h
.
.
.
ν = 0.5
ν = 0.25
ν = 0
Figure 7.12: Surface settlements as a function of ν for r/ h = 0.1.
From the curves in Figure 7.11 it appears that for shallow tunnels Poisson’s ratio has only relatively minor effects on the shape of the settlement trough.
The trough seems to get increasingly narrower for smaller values of ν, but the
trough depth appears to be relatively constant This latter effect is not the case for deeper tunnels, as can be seen in both Figure 7.12 and Figure 7.13, where Poisson’s ratio has a much larger effect on the trough depth for deeper tunnels This is especially evident in Figure 7.13, where the scaled maximum settlement,
denoted by vmax /u o , has been drawn versus r/ h for different values of Poisson’s
ratio.