In order to facilitate this implementation, the expressions from Chapter 4 are put into a form that is capable of generating dimensionless stresses and displacements.. Using this form, w
Trang 1Section A.2 Evaluation of the Coefficients in the Potentials 85
(due to the fact that Cauchy’s Integral Theorem, which guarantees path
inde-pendence in a simply-connected region, is not necessarily satisfied for contours
around the hole) It follows that
L
∗
where C c is some complex constant (the same reasoning applies here as was
used to show that the value B cin (A.13) is constant)
Proceeding as before (see the reasoning used to determine equation (A.17)),
the integral of ∗
(z)can be written as
z
z0
∗
(z) dz= C c
2πilog(z − z c )+∗∗ϕ (z), (A.20) where∗∗
ϕ (z)is a single-valued analytic function Substituting (A.20) in (A.18)
and expanding gives
ϕ(z) = A c z log(z − z c ) − A c z c log(z − z c ) − A c (z − z c )
+ C c
2πilog(z − z c )+∗∗ϕ (z) + C. (A.21)
Combining logarithmic terms and incorporating all single-valued analytic terms
in a new analytic function,ϕ(z)∗
, results in
ϕ(z) = A c z log(z − z c ) + γ c log(z − z c )+ϕ(z),∗ (A.22)
where the constant γ c and the new single-valued analytic function ∗
ϕ(z)have been introduced for convenience
The multi-valued nature of the potential ψ(z) can be determined by noting
that (z) = ψ(z)is single-valued (this is because the stresses on the left-hand
side of (2.5) and the function ϕ(z)on the right-hand side of the same equation
are all single valued) With the same reasoning used to determine (A.17) and
(A.20), it can be shown that the integral of (z) can be written as
ψ (z)=
z
z0
(z) dz = γ clog(z − z c )+ψ (z).∗ (A.23)
where γ
cis a complex constant and ∗
ψ (z)is a single-valued analytic function
All that remains for a full determination of the multi-valued nature of the
poten-tials in a region with a hole is to determine the unknown coefficients in (A.22)
and (A.23) This is accomplished as follows
In order to ensure that the displacements associated with the multi-valued
potentials given by (A.22) and (A.23) are single-valued, equation (2.1) must not
Trang 286 Multi-Valued Complex Potentials Appendix A
become multi-valued anywhere in the region Substituting (A.22) and (A.23)
in (2.1) gives
2µ(u + iv) = κA c z log(z − z c ) + γ c log(z − z c )+ϕ(z)∗
− z
A c log(z − z c )+ A c z
z − z c + γ c
z − z c +ϕ∗(z)
− γ
c log(z − z c )−ψ (z).∗
(A.24)
Separating the multi-valued parts from the single-valued parts results in
2µ(u + iv) = (κA c z + κγ c ) log(z − z c )
− (zA c + γ
c ) log(z − z c ) + f (z), (A.25)
where
f (z) = κ ϕ(z)∗ − A c zz
z − z c − zγ c
z − z c − z ϕ∗(z)−ψ (z),∗ (A.26)
which is single-valued Denoting by r zc the distance from z to z c, denoting by
ϑ zc the argument of z − z c(see Figure A.3), and splitting the logarithms into their real and imaginary parts yields
2µ(u + iv) = (κzA c + κγ c )( ln r zc + iϑ zc )
− (zA c + γ
c )( ln r zc − iϑ zc ) + f (z). (A.27)
After collecting terms, we have
2µ(u + iv) = (κzA c + κγ c − zA c − γ
c ) ln r zc + i(κzA c + κγ c + zA c + γ
c )ϑ zc + f (z). (A.28) The function ϑ zc in this expression increases continuously along each circuit
around the point z c The other functions in (A.28) are single-valued It follows
that the coefficient of ϑ zcmust be set to zero in order to ensure single-valued displacements:
A c (κ + 1)z + κγ c + γ
As a result
and
κγ c + γ
The coefficients γ c and γ
ccan be related to the resultant force on the hole through use of (2.6) We take an integration path consisting of a complete circuit
around L hin a clockwise direction (see Figure A.3) Note that this integration path is chosen such that the region is to the left, as specified for (2.6) Denoting
by[ ]L the increase undergone by the expression inside the brackets along the
Trang 3Section A.2 Evaluation of the Coefficients in the Potentials 87
..
..
.. . . .
.
x
y
ϑ zc
r zc
z c
L h
Figure A.3: Integration path
integration path, integrating along L hin (2.6) and substituting equations (A.22)
and (A.23) gives
γ c log(z − z c )+ϕ(z)∗ + zγ c
z − z c + z ϕ∗(z) + γ
c log(z − z c )+ψ (z)∗
L h
= i
L h
(t x + it y ) ds = i( F h x+ iF h y ), (A.32)
whereF h x + i F h y is the resultant force on the hole, and where we have utilized
(A.30) Using the same notation as above, the logarithms can be split into their
real and imaginary parts, resulting in
i(γ c − γ
c )ϑ zc + (γ c + γ
c ) ln r zc + g(z)
L h
= i( F h x+ iF h y ), (A.33) where
g(z)= zγ c
z − z c +ϕ(z)∗ + z ϕ∗(z)+ψ (z).∗ (A.34)
We note that ϑ zcdecreases by an amount 2π upon making a complete circuit
around L h , and that ln r zc and g(z) each return to their original values upon
making a complete circuit around L h The result is that
(γ c − γ
c )( −2π) = F h x+ iF h y (A.35)
Solving for γ c and γ
cusing (A.31) and (A.35), applying (A.30), and substituting the results in (A.22) and (A.23) finally yields
ϕ(z)= −
h
F x+ iF h y
2π(1 + κ) log(z − z c )+ϕ(z),∗ (A.36)
Trang 488 Multi-Valued Complex Potentials Appendix A
and
ψ (z)=κ(
h
F x− iF h y )
2π(1 + κ) log(z − z c )+ψ (z).∗ (A.37)
The potentials ϕ(z) and ψ(z) for a finite plane with multiple holes can be
obtained by a superimposition of the potentials for a single hole (given by (A.36) and (A.36)) This is due to the fact that the coefficients of the logarithms (or more precisely, the integrals (A.13), (A.19), and (A.23) and the resultant force, given by the integral (A.32)) are not effected by the presence of other holes It
follows that the potentials for a finite plane with m holes can be written as
ϕ(z)= −
m
k=1
k
F x+ iF k y
2π(1 + κ)log(z − z k )+ϕ∗n (z), (A.38)
ψ (z)=
m
k=1
κ(
k
F x− iF k y )
2π(1 + κ) log(z − z k )+ψ∗n (z), (A.39)
where the values z kdenote points within their respective holes The functions
∗
ϕ (z)and ∗
ψ (z)denote new arbitrary single-valued analytic functions
Trang 5Appendix B
IMPLEMENTATION IN A COMPUTER PROGRAM
The focus of this appendix is the implementation of the results from Chap-ter 4 in a compuChap-ter program In order to facilitate this implementation, the expressions from Chapter 4 are put into a form that is capable of generating dimensionless stresses and displacements Using this form, we take the deriva-tives of the potentials and use them to write out the equations for the stresses and displacements explicitly
In order to generate dimensionless expressions for the stresses and the
displace-ments, we divide the complex potentials by a constant P that has dimensions
of force per unit length This will generate dimensionless complex potentials
on the basis of (2.1) – (2.5) The constant P will be defined according to the
parameters appearing in the solution of particular problems
We start this process for the results derived in Chapter 4 by dividing the
values of the coefficients in the Laurent expansions (4.27) and (4.28) by P We
make the following definitions:
p k= a k
P , q k = b k
P , r k = c k
P , s k= d k
where a k and b k are the coefficients in the Laurent expansion (4.27), given by
(4.49), (4.50), and (4.55) The coefficients c k and d k in the Laurent expan-sion (4.28) are given by (4.35) – (4.37) Dividing the Laurent expanexpan-sions (4.27)
and (4.28) by P results in
∗
ϕ0(ζ ) = p0+
∞
k=1
p k ζ k+
∞
k=1
∗
ψ0(ζ ) = r0+∞
k=1
r k ζ k+∞
k=1
89
Trang 690 Implementation in a Computer Program Appendix B
where
∗
ϕ0(ζ )=ϕ m0(ζ )
∗
ψ0(ζ )=ψ m0(ζ )
Dividing (4.35) – (4.37) by P gives
r0= B0◦
P − p0−1
2p1−1
r k= B◦−k
P − q k−1
2(k + 1)p k+1+1
2(k − 1)p k−1, k > 0, (B.6)
s k= B◦k
P − p k+1
2(k − 1)q k−1−1
2(k + 1)q k+1, k > 0. (B.7)
On the basis of (4.42) – (4.47) and (4.51), the only parameters in (4.49), (4.50), and (4.55) that are not dimensionless areβ k i13andβ k i13 These parameters have the dimensions of the Fourier coefficientsB◦kandA◦k Dividing equations (4.49)
and (4.50) through by P results in
p k =β k011p0+β k012p0+
k−1
i=0
k
β i13
P k = 1, 2, 3, , (B.8)
q k =β k021p0+β k022p0+
k−1
i=0
k
β i23
P k = 1, 2, 3, (B.9)
The expression for a0, equation (4.55), cannot be implemented in a computer
program unless the limit k → ∞ is replaced by an expression like k = k I, where
k I is some large integer, say 10,000 Dividing (4.55) by P and replacing the
symbol∞ with k Iresults in
p0=
k I
β012
kI−1
i=0
k I
β i13
P −k β I021
kI−1
i=0
k I
β i23 P
k I
β021
k I
β011−k β I022
k I
β012
We now have complete and dimensionless expressions forϕ∗
0(ζ )and ∗
ψ0(ζ )
which can be programmed into a computer
Finally, we convert the complete potentials given by (4.56) and (4.57) to their dimensionless forms This gives
∗
ϕ(ζ )= −κ(
h∗
F x+ iF h∗y )
2π(1 + κ) log(−
i2a
1− ζ )−
h∗
F x+ iF h∗y
2π(1 + κ)log(−
i2aζ
1− ζ ) + p0+∞
k=1
p k ζ k+∞
k=1
q k ζ −k ,
(B.11)
Trang 7Section B.2 Dimensionless Coordinates 91
and
∗
ψ (ζ )=
h∗
F x− iF h∗y
2π(1 + κ) log(−
i2a
1− ζ )+
κ(
h∗
F x− iF h∗y )
2π(1 + κ) log(−
i2aζ
1− ζ ) + r0+∞
k=1
r k ζ k+∞
k=1
s k ζ −k ,
(B.12)
where
h∗
F x+ iF h∗y=
h
F x+ iF h y
∗
ϕ(ζ )= ϕ m (ζ )
∗
ψ (ζ )= ψ m (ζ )
P . (B.13)
The coordinates in the z-plane can be written in dimensionless form by dividing
(4.7) by h, the depth of the tunnel This results in
∗
z=ω(ζ )∗ = −ia∗1+ ζ
where∗
z = z/h and ω(ζ )∗ are the dimensionless coordinates and dimensionless
mapping function, respectively, and where we have defined the parameter
∗
a= 1− α2
This parameter is simply the dimensionless form of (4.8) The dimensionless
coordinates ∗
z can be obtained by using (B.14) instead of (4.7) to map the
coordinates back from the ζ -plane.
It is most convenient to calculate (2.1) and (2.4) – (2.5) in the ζ -plane and then
to relate the resulting values to the dimensionless coordinates∗
zthrough use
of (B.14) In order to perform these calculations we will need expressions for
∗
ϕ(z), ϕ∗(z), andψ∗(z) in terms of ζ The derivative of ϕ(z) with respect to z
in terms of ζ was derived in (4.14) Similar results can be obtained for ∗
ϕ(z) and ∗
ψ(z) They are
∗
ϕ(z)= 1
h
∗
W1(ζ ) ϕ∗(ζ ), ψ∗(z)= 1
h
∗
W1(ζ )∗
ψ(ζ ), (B.16) where we have defined the (dimensionless) function
∗
W1(ζ )= h
ω(ζ ) = i(1− ζ)2
2∗
a
Trang 892 Implementation in a Computer Program Appendix B
The chain rule can be used twice to determine∗
ϕ(z) in terms of ζ :
∗
ϕ(ζ )= d
dζ
∗
ϕ(z)ω(ζ ) = d
dz
∗
ϕ(z)· dz
dζ ·ω(ζ )+ϕ∗(z)ω(ζ ). (B.18)
It follows that
∗
ϕ(z)= 1
h2[W∗1(ζ )]2 ∗ϕ(ζ )− 1
h2
∗
W3(ζ ) ∗
ϕ(ζ ), (B.19)
where we have defined the (dimensionless) function
∗
W3(ζ ) = h2 ω(ζ )
[ω(ζ )]3 = −(1− ζ)3
2∗
a2
The derivatives of (B.11) and (B.12) with respect to ζ are:
∗
ϕ(ζ )= −κ(
h∗
F x+ iF h∗y )
2π(1 + κ) ·
1
1− ζ −
h∗
F x+ iF h∗y
2π(1 + κ)·
1
ζ (1− ζ)
+ ∞
k=1
kp k ζ k−1−∞
k=1
∗
ψ(ζ )=
h∗
F x− iF h∗y
2π(1 + κ)·
1
1− ζ +
κ(
h∗
F x− iF h∗y )
2π(1 + κ) ·
1
ζ (1− ζ )
+
∞
k=1
kr k ζ k−1−
∞
k=1
and
∗
ϕ(ζ )= −κ(
h∗
F x+ iF h∗y )
2π(1 + κ) ·
1
(1− ζ)2+
h∗
F x+ iF h∗y
2π(1 + κ)·
1− 2ζ [ζ(1 − ζ)]2 + ∞
k=1
k(k − 1)p k ζ k−2+∞
k=1
k(k + 1)q k ζ −k−2 . (B.23)
Expressions for the displacements in terms of ζ can be obtained by dividing (2.1) by P and substituting (4.7), (B.16), (B.19), and (B.21) – (B.23) in the
resulting expression This gives
2µ
P (u + iv) = κ ϕ(ζ )∗ −ω(ζ )∗ ∗
W1(ζ )∗
ϕ(ζ )−ψ (ζ ).∗ (B.24) Expressions for the stresses can be obtained by multiplying (2.4) and (2.5) by
h/Pand substituting the same expressions in the result This yields
h
P (σ xx + σ yy )= 2
∗
W1(ζ ) ϕ∗(ζ )+W∗1(ζ ) ϕ∗(ζ )
Trang 9Section B.4 Comparison of Numerical and Analytical Solutions 93
and
h
P (σ yy − σ xx + 2iσ xy )
= 2
∗
ω(ζ )[W∗1(ζ )]2 ∗ϕ(ζ )−ω(ζ )∗ ∗
W3(ζ ) ϕ∗(ζ )+W∗1(ζ )∗
ψ(ζ )
. (B.26)
The dimensionless stresses and displacements have been fully determined in
terms of the mapping parameter ζ The dimensionless coordinates x/ h and y/ h
corresponding to the values calculated using (B.24) - (B.26) can be obtained by
mapping ζ to∗
zwith (B.14)
A computer program implementing the equations in the preceding sections is
approximately 40% slower than code which determines p0, p k and q kby solving
the system of equations (4.38) - (4.39) numerically and examining the behavior
of p k for large values of k and different values of p0, as performed by
Ver-ruijt [42] It should be noted that part of this difference is due to the fact that
in Verruijt’s implementation it is assumed that all the coefficients are purely
imaginary (so that all calculations involve only real variables) whereas the
cal-culations in this implementation are fully complex (making it possible to include
a non-vertical resultant force acting on the tunnel) It may be noted that both
the numerical and the analytical computation of the coefficients of the
Lau-rent series result in extremely small errors in the normalized displacements and
stresses Even so, there are some small differences between the two techniques,
as can be seen in Table B.1, and as will be discussed below
Table B.1: Comparison of solutions for the ground loss problem with ν = 0.3 and with
no resultant forces acting on the boundaries The values have been normalized through
use of the parameter P = 2µu g , where u gis the maximum convergence of the tunnel
Solution r/ h Num Terms Max Error in u/u g Max σsurface·h/(2µu g )
Trang 1094 Implementation in a Computer Program Appendix B
Based on the ground loss problem with no resultant forces acting on the boundaries, the fully analytical solution is approximately one order of
mag-nitude more accurate for common tunnel depths (r/ h < 0.5) and about one order of magnitude less accurate for extremely shallow tunnels (r/ h > 0.8) The surface stresses (σ yy and σ xy) in the new solution are on the average of
one order of magnitude more accurate for all values of r/ h As noted earlier,
however, both solutions are very accurate and the differences discussed here are negligible in comparison to the small errors in the boundary conditions
Trang 11Appendix C
THE OVALIZATION BOUNDARY CONDITION
It is the purpose of this appendix to derive the transformed Fourier expansion
of the boundary conditions for the ovalization of a circular tunnel used in this
thesis The transformation is made from the physical z-plane containing the tunnel to the conformally mapped annulus in the ζ -plane (see Figure C.1) using
the mapping function (4.7) presented in Chapter 4 The results are written in the
form of the given displacement expansion G(ασ )introduced in (4.31) This solution was first published in [35]
The displacements for the ovalization of a tunnel in the z-plane are given in terms of the polar coordinates r and θ (see the left side of Figure C.1) by
where u r and u θ are the radial and tangential displacements along the tunnel
boundary, and where u ois the maximum displacement of the tunnel boundary
.
..
.
.
.....
.
. . .
. . . x y r h θ L h .. ...
.....
.
.
.
.
.
.
ξ
η
σ
1
α ϑ
Figure C.1: Ovalization of a tunnel in the z-plane; conformal mapping the ζ -plane.
95