§ 5.3 Comparison to Yu’s Solution The results of the calculation have also been compared to the solution byYu [45] for a rigid, buoyant tunnel in an infinite plane.. Figure 5.3: Comparis
Trang 1Section 5.3 Comparison to Yu’s Solution 37
-5
-4
-3
-2
-1
0
-5 -4 -3 -2 -1 0
x/ h
- y/ h
(the horizontal constraint is imposed along the vertical axis of symmetry) For
now, it is assumed that the stresses along the surface more than five times the
tunnel depth away from the origin will be small enough to result in a higher
so-called small strain stiffness of the ground (see, for example, [1]) which would
severely limit the deformations beyond these points It should be emphasized
that the stresses are not affected by the choice of rigid body motion [23].
The computer program which has generated Figure 5.1 and Figure 5.2 can
also be used to calculate the maximum errors in the normalized stresses along
the surface, which are on the order of 1 × 10 −12 The maximum errors in the
normalized displacements along the tunnel are on the order of 1 × 10 −7 These
values correspond to the problem shown in Figure 5.1 and Figure 5.2 In
addi-tion, a numerical integration of the tractions along the tunnel can be performed
by the program This integration has been performed after the superposition of
the gravitational stresses (5.7) – (5.9) This results in a buoyant force (in terms
of γ h 2 ) after deformation which is on the order of 1 × 10 −17 , indicating that
the complete solution is indeed in equilibrium, within computational accuracy.
§ 5.3 Comparison to Yu’s Solution
The results of the calculation have also been compared to the solution byYu [45]
for a rigid, buoyant tunnel in an infinite plane Comparisons of the isotropic
stress 1
2 (σ xx + σ yy ) and deviatoric stress
1
4 (σ yy − σ xx ) 2 + σ 2
xy are shown in Fig-ure 5.3 and FigFig-ure 5.4, respectively It is clear from these plots that the stresses
in the ground and around the tunnel are heavily influenced by the presence of
a stress-free surface The stress free surface is included in the exact solution
presented in this chapter, but is absent from Yu’s solution It is also apparent
Trang 2-3 -2 -1 0 1 2 3
-3
-2
-1
0
-3 -2 -1
0
x/ h
- y/ h
.
. . .
Figure 5.3: Comparison between the exact solution (left-hand side) and Yu’s solution (right-hand side) for the incremental isotropic stresses in the case that ν = 0.5 and r/ h = 0.25 The contours are given in increments of 0.0125γ h. -3 -2 -1 0 1 2 3 -3 -2 -1 0 -3 -2 -1 0 x/ h - y/ h
.
.
.
.
.
.
Figure 5.4: Comparison between the exact solution (left-hand side) and Yu’s solution
that both the isotropic and deviatoric stresses are very small at distances along the surface larger than three times the tunnel depth This is in line with the
choice for the fixed point in the solution at x/ h = ±5.
It is interesting to note the magnitude of the deviatoric stresses on the sides
of the tunnel Plastic effects are likely to occur in these areas since Mohr’s Circle will be large there and will be located near the origin (as can be seen by noting that the zero stress lines intersect the tunnel near its axis Figure 5.3) It
Trang 3Section 5.3 Comparison to Yu’s Solution 39
should be noted, however, that Mohr’s circle will be shifted by addition of the
initial stresses.
The normalized displacements of the two solutions are shown in Figure 5.5.
Again, the influence of the surface is apparent in the exact solution As can be
expected, the buoyancy force causes more movements along the surface in this
case This is due to the relative reduction of stiffness above the tunnel resulting
from the absence of material above the surface.
-3
-2
-1
0
-3 -2 -1
0
x/ h
- y/ h
.
Figure 5.5: Comparison between the exact solution (left-hand side) and Yu’s solution
γ h 2
The stresses on the tunnels in the two solutions should converge for very
deep tunnels This is indeed the case, but the difference does not vanish until
r/ h < 0.01 Apparently, the effect of the surface is quite pronounced in these
solutions The differences between the solutions are shown for two different
relative tunnel depths in Figure 5.6, from which it can be seen that the two
solutions indeed converge for deeper tunnels, but not very fast, as for r/ h = 0.1
the difference is still some 10%.
In Figure 5.6 positive (tensile) stresses are drawn outside the circle and
negative (compressive) stresses are drawn inside the circle (this convention will
be adopted for all plots of the stresses on the tunnel).
It appears that the radial stress σ rr has tensile values above the tunnel for
shallow tunnels in the exact solution Some insight in this phenomenon can
be gained by observing that there is a tensile zone of stress near the surface
in the isotropic stresses generated by the exact solution depicted in Figure 5.3.
This tensile zone moves closer to the tunnel for shallower tunnels and comes
in contact with the tunnel for r/ h values around 0.5 for ν = 0.5 and around
0.4 for ν = 0 This effect is not present in Yu’s solution, where the stresses are
Trang 4
. . . .
. σrr ν = 0.5 r/ h = 0.8
. . .
σrr ν = 0.5 r/ h = 0.1 Figure 5.6: Two comparisons between exact solutions (left-hand sides) and Yu’s solu-tions (right-hand sides) for a shallow tunnel and a deep tunnel unaffected by the radius to depth ratio of the tunnel and where all the stresses above the tunnel are compressive § 5.4 Stresses on and around the Tunnel The radial and shear stresses σ rr and σ rt acting on the tunnel are shown in Figure 5.7 and Figure 5.8 (for the exact solution) It appears that the value of Poisson’s ratio ν has only a moderate effect on the stresses on the tunnel, especially for deeper tunnels In fact, for very deep tunnels the differences in the radial and shear stresses for different values of ν vanish entirely For shallow tunnels, smaller values of ν generate larger values for the stresses acting on the tunnel This is most apparent for the radial stresses shown in the upper left-hand corner of Figure 5.7.
. .
. . .
.
.
. . . .
.
.
σrr ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.7
. . . .
.
σrr
ν = 0.5
ν = 0.25
ν = 0
r/ h = 0.1
Figure 5.7: The radial stresses acting on the tunnel as a function of ν.
Trang 5Section 5.5 Displacements along the Surface 41
.
. .
.
.
.
.
.
. . .
.. .
.. .
. . .
.
. . . . . . . . .
.. .
.. .
.. .
. .
.
.
σrt ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.7
. .
.. .
. . . . . . . . . . . .
. . ...
.. .
. .
.. .
. . . . . . . . .
.
. . ...
.. .
. .
σrt ν = 0.5 .ν = 0.25 ν = 0 r/ h = 0.1 Figure 5.8: The shear stresses acting on the tunnel as a function of ν The hoop stress σ t t around the tunnel shows more sensitivity to Poisson’s ratio ν, as is shown in Figure 5.9 It is interesting to note that for ν = 0 the hoop stress vanishes completely for all ratios r/ h of the tunnel radius to depth This may be due to the fact that the boundary conditions for a completely rigid tunnel do not directly affect the hoop stress, as would be the case if the tunnel deformed In the case of a rigid tunnel it appears that the hoop stress is only indirectly related to the boundary conditions via Poisson’s ratio It is also
. . .
.
. .
σt t ν = 0.5 ν = 0.25 ν = 0 r/ h = 0.7
.
.
. .
.
. ...
. . .
σt t
ν = 0.5
.ν = 0.25
ν = 0
r/ h = 0.1
Figure 5.9: The hoop stress around the tunnel as a function of ν.
apparent from Figure 5.9 that the value of Poisson’s ratio plays a much larger
role in the magnitude of the hoop stress than the relative tunnel depth.
§ 5.5 Displacements along the Surface
The amplified vertical displacements v along the surface are shown in
Fig-ure 5.10 and FigFig-ure 5.11 (for the exact solution) The amount of heave appears
to depend strongly on the value of Poisson’s ratio: the amount of heave for
ν = 0 is roughly 50% more than for ν = 0.5 It would appear that this is due to
Trang 6an effective reduction in stiffness of the ground above the tunnel for low values
of Poisson’s ratio, as can also be observed in the reduction of the hoop stress
for low values of ν in Figure 5.9.
0 1 2 3 4
0 1 2 3 4
4.3µ
γ h 2 v
x/ h
.
. . .
. .
.
.
.
.
.
.
ν = 0 .ν = 0.25 ν = 0.5 Figure 5.10: Surface heave as a function of ν for r/ h = 0.7. -5 -4 -3 -2 -1 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 4 140µ γ h 2 v x/ h
.
. . .
. . .
.
.
.
.
ν = 0
ν = 0.25
ν = 0.5
The change in the shape of the displacement curves along the surface is not strongly affected by Poisson’s ratio Changes in Poisson’s ratio tend only to amplify the displacements The shape of the displacement curve is, however, somewhat affected by the relative depth of the tunnel, as can be seen by compar-ing Figure 5.10 and Figure 5.11 The heave appears to be of a more localized nature (relative to the depth of the tunnel) for deeper tunnels, and has much smaller values.
The maximum heave v max as a function of the relative tunnel depth is given
in Figure 5.12 for two different choices of the point of vertical constraint (which fixes the rigid body motion) in the solution.
It is apparent from these plots that the amount of heave is strongly dependant
on the choice of the point at which the vertical displacements are assumed to vanish For the two choices in Figure 5.12 the difference is approximately
a factor of two for tunnels with r/ h < 0.5 Based on the moderate extent
Trang 7Section 5.5 Displacements along the Surface 43
r/ h
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
2µ
γ h 2 v max
.
.
.
. .
ν = 0
ν = 0.25
ν = 0.5
r/ h
0.00
0.25
0.50
0.75
1.00
2µ
γ h 2 v max
.
.
.
.
. .
ν = 0
ν = 0.25
ν = 0.5
Figure 5.12: Maximum heave as a function of r/ h and ν In the upper graph the point of
Trang 8of the incremental isotropic and deviatoric stresses shown in Figure 5.3 and
Figure 5.4, the choice of a point of vertical constraint at x/ h = 2 may well be
more realistic than the choice at x/ h = 5, as the stresses have largely dissipated
at twice the tunnel depth The extent of influence of the isotropic and deviatoric stresses is, however, somewhat dependant on the relative depth of the tunnel.
In addition, the choice for the point of vertical constraint may be dependant on other modes of deformation and other parameters as well (see the discussion
in Chapter 8) It is clear, however, that moving the point of vertical constraint closer to the tunnel will result in smaller amounts of heave In addition, it will also affect the relationships between the amounts of heave for different values
of ν, especially for lower values of r/ h, as is evident from the graph on the
bottom of Figure 5.12.
It is also noted from Figure 5.12 that there is a nonlinear relationship between
the normalized maximum heave and r/ h, and that the maximum amount of
heave is clearly dependant on the value of Poisson’s ratio.
§ 5.6 Comparison to a Finite Element Calculation
The solution presented in this chapter has been compared to a Finite Element Method (FEM) calculation, using the program Plaxis, version 8.0 [3] This comparison is not intended as a validation of the solution or as a validation of the program Plaxis, but is merely an illustration of the differences that may be encountered A linearly elastic material model has been chosen for the ground in the FEM calculation in order to afford for a more direct comparison to the exact solution The elements chosen are 15-noded triangular elements The tunnel is a stiff, but weightless, solid linearly elastic cylinder, also modeled with 15-noded triangular elements For the particular details of the FEM implementation in the program Plaxis, see [3].
The buoyant, rigid tunnel problem was calculated in Plaxis by computing the initial stresses with the same elastic material inside the tunnel as outside the tunnel The staged construction module was then used to replace the material inside the tunnel with a very stiff, but weightless, solid circular cylinder The values of the parameters used are given in Table 5.1.
Table 5.1: Parameters used in the FEM calculation for a rigid, buoyant tunnel The
abbreviations grnd and tunl refer to parameters for the ground and tunnel, respectively.
Trang 9Section 5.6 Comparison to a Finite Element Calculation 45
In the Table 5.1, the values of x max and y max define the range of the FEM mesh
in the horizontal and vertical directions, respectively Note that the relative
mesh size is larger than commonly used in geotechnical calculations This has
been done in order to minimize the effect of the boundaries on the stresses and
displacements near the tunnel.
The results of the calculations for the displacements are given in Figure 5.13,
and the results for the stresses are given in Figure 5.14.
-5
-4
-3
-2
-1
0
1 2 3 4 5 -5 -4 -3 -2 -1 0 x/ h - y/ h
Figure 5.13: Comparison between the exact solution (left-hand side) and a Finite
Ele-ment Calculation (right-hand side) for the actual displaceEle-ments, magnified by a factor
of 5 The parameters used in the solutions are given in Table 5.1.
Along the right and left boundaries the FEM mesh is constrained horizontally
but is free to move vertically The mesh is completely fixed along its lower
boundary It can be expected that the best overall match of the displacements
will be achieved by choosing a point of vertical constraint in the exact solution
which corresponds to the center of the fixed boundary in the FEM mesh In the
exact solution shown in Figure 5.13 a rigid body motion was added such that
the vertical displacements are zero at the point (x/ h = −2.5, y/h = −5.0).
The effects of the horizontal and vertical constraints in the FEM solution can
be clearly seen be comparing the displacements of the two solutions near the
boundaries However, aside from these discrepancies, the solutions appear to
match well.
It should be emphasized that depth of the fixed mesh at the bottom of the
FEM solution has a similar effect on the vertical displacements, or heave, as the
choice of the point of vertical constraint in the exact solution This must be taken
into consideration when constructing FEM meshes for excavation problems, and
meshes will usually be less deep than the one used here Several methods can
be considered to achieve realistic results, such as using constitutive models for
Trang 10-5 -4 -3 -2 -1
-5
-4
-3
-2
-1
0
. . . . . .
. .
.
.
.
.
1 2 3 4 5 -5 -4 -3 -2 -1 0 x/ h - y/ h
.
Figure 5.14: Comparison between the exact solution (left-hand side) and a Finite
contours intersecting the right and left edges have values of zero The parameters used
in the solutions are given in Table 5.1.
ground which allow a stiffer E modulus to be used for unloading (for example,
the hardening soil model in Plaxis, see [3]) It is also possible to add layers of increasing stiffness in the lower parts of the mesh or to use different material models for different parts of the underground (see, for example, [32]).
Contours of the horizontal shear stress σ xy are shown in Figure 5.14 From these contours it is clear that the solutions match well for points close to the tunnel, but progressively diverge for points further away from the tunnel This again seems to be an effect of the restrained boundaries of the FEM mesh.
§ 5.7 Chapter Summary
In this chapter an analytical solution for a rigid buoyant tunnel in a linearly elastic half-plane has been obtained and validated In the solution the tunnel lining is considered to have negligible weight, and the buoyancy force is set equal to the weight of the excavated material When the solution is compared
to a previously published solution for a very deep tunnel, it appears that the stress-free surface has a large effect on the displacements and stresses around the tunnel, even for relatively deep tunnels The solution exhibits singular displacements at infinity, a property which is an artifact of the buoyancy force acting on the hole This property of the solution means that the displacements are dependant on an arbitrary rigid body motion, which is set by choosing a point
in the plane at which the vertical displacements are constrained The choice
of this point has a large effect on the amount of heave along the surface An