Stresses around cavities caused by in situ or residual stresses in the rock have already been dealt with (sections 5.4 and 10.2). The stresses to be ana- lysed in this chapter are those induced in the rock masses by the hydrostatic (or hydrodynamic) pressure p of the water or fluid (gas) filling the tunnel gallery or cavity.
Analysis of these stresses usually starts with the theory of stresses in thick pipes, which is then extended to the case of the unlined tunnel and finally to the case of the lined tunnel (Jaeger, 1933,1949ft). The theory of stress-strain measurements in circular tunnels under a radial load (Seeber, 1961) applied along the whole circumference of a tunnel (Austrian technique on rock testing) can easily be related to the equations obtained for the pressurized tunnel. Similarly, these theories can be used to establish safe overburden conditions above a tunnel under a pressure p (Jaeger, 19616). Finally, the basic equations to be developed in this chapter are used in the theory of water hammer waves which develop in pressure tunnels and shafts.
10.3.1 Theory of thick elastic pipes
The internal and external radius of the pipe is respectively b and c (fig. 10.4).
Fig. 10.4 Deformation of a thick pipe.
R is the radius of a small element Rd<f>dR inside the pipe, with b < R < c.
When the pipe is filled with a liquid or gas under pressure, the radius R increases by u and the circumference increases from 2TTR to 2n(R + u). The specific length increase in a circumferential direction is
_ 2TT(R + u) — ITTR __ u
d
In a radial direction R increases by u and becomes R + u; similarly, dR increases by dw and becomes dR + du = dR[l + (du/dR)]. The specific radial increase in length is
_ dR + du - dR _ du
r~ dR ~ dR*
On the other hand 8t and 8r depend on at and ar as follows (m = 1/v):
. du I
8> = dR = E or
mE / u dw\
H ) (1)
£" being the modulus of elasticity. It can be shown that u is a solution of the equation:
R' + R % 0 ( 3>
the integral of which is
u = BR + ~ (4)
A rapid checking yields:
du n C d2u 2C
B d ( 5 )
which, when introduced in equation (3) give 0 = 0. As u is known the stresses at and aT are now:
mE
mE mEC nf C
G — 2 A = B A (7)
1 m-\ ^(m+l)R2 R2 W
The constants B and C depend on the boundary condition for R = b and A thick pipe is loaded by internal static pressue
Pi and external pressure pe (fig. 10.5).
For R = b:
For i? = c:
Fig. 10.5 Thick pressure pipe.
232 Underground excavations This yields:
ằ, =
c2-b2 Therefore, fori* = b:
1 ib2 (Pe-Pd 2 = b2 + c2
2 c2 b2 Pi c2 b2
^ = 2c2pe
1 c2 -b2 c2 -b2 Pi c2 - b2 c2 - b2 For R = c:
Pe^-pfi2 (pe - Pi) u2 a'= c*-b* --^Z-tfb =?*
_PeC2-Pib2 (Pe-Pi)h2_ n 2b2 C2 + b2
10.3.2 Case of a pressure tunnel in sound rock (fig. 10.6) The boundary conditions on a wall of a tunnel in sound rock are:
f o r R = b, Pi— P-
Fig. 10.6 Pressure tunnel in sound rock.
The other boundary conditions are:
for R = oo, o> = pe = 0.
This second condition yields for R = oo:
or = B' = 0, B = 0 and at = 0 and for any value 0 < R < oo:
mEC
For R = b:
mEC , pb\m + 1) a' = -J^W=p and c=~—
+ At any point inside the rock:
mEC
The stresses ar and at decrease rapidly inside the rock. At a distance R = 2b, they are only 25 % of what they are on the tunnel wall.
10.3.3 Concrete-lined pressure tunnel
(1) Unfissured sound rock (fig. 10.7). The internal radius of the tunnel lining is b, its external radius is c. If p is the hydrostatic pressure inside the
Fig. 10.7 Concrete-lined pressure tunnel in sound rock.
tunnel a certain load, pc = Xp, is transmitted from the concrete lining to the rock. Furthermore, it is assumed that there is no gap between concrete and rock. For c < R < oo, the stress ar is given by:
m2Hi2 o2
the subscript '2' for m2, E2, B2, C2 referring to rock. The condition ar = 0 for R = oo yields 2?2 = 0 and or = — at for the whole mass of rock.
For R = c on the rock side;
and
m2E2C2
ar = — — = pc = Ap, m2 + 1 c2
pcc2(m2 + 1 )
C2 = — ' , with B2 = 0.
E2m2
234 Underground excavations
The radial displacement u for R = c in the rock is URmC = B2K +
In the concrete when b < R < c, we introduce subscript T to represent the concrete.
For R = b:
For R = c:
and
Equating the elastic displacements for R = c:
ôằ-c = B1c + — = BC C2c + — c c
mi - 1 (fe3 - Ac2) , mx + 1 cb2 gt _m2+l c
m.E, c* -b* CP + ~^ET ^ T ^ 1 " W ~ —^-J
Out of this equation we get:
2b2
p m2 + 1 (m, - l)c2 + (wit + \)b2 m2E2 ++
The stresses in the concrete lining are:
f o r * = 6:
_ c2 + b2 - 2Xc2 for R*= c:
arc = Ap,
2fc2 _ ^(C2 + £2)
(2) The concrete lining is fissured. If the concrete lining the tunnel walls were uniformly fissured in a radial direction, a pressure:
Po = -p,b
would be directly transmitted to the rock and the stresses on the rock surface would be
b
(With water penetrating the radial cracks the pressure transmitted could be as high as pc =/?.)
(3) The fissures penetrate in the rock to depth d (fig. 10.8). Along the rock surface the radial pressure is
Pc = (b/c)p and at = 0; arc = pc = (6/c)/?.
limit of radiai fissures
Fig. 10.8 Lined tunnel in radially fissured rock (R < d).
At any depth R < d inside the fissured rock mass at = 0; o> = (b/R)p.
At the limit of the sound rock the pressure ispd = (bjd)p. Inside sound rock (d<R< oo):
bd^_bd ar- at- p -R 2- R2p.
10.3.4 Steel-lined pressure tunnels and shafts
(1) In sound rock (fig. 10.9). The hydrostatic pressure inside the tunnel is p.
A pressure pb < p is transmitted from the steel shell to the concrete and a pressure pc < pb from the concrete to the rock.
The elastic deformation ub of the steel lining is
236 Underground excavations
Fig. 10.9 Steel-lined tunnel.
where E = modulus of elasticity of the steel plate, e = thickness of the steel plates and pb = Xxp. The deformation ub of the steel shell must be equal to the elastic deformation of the internal face of the concrete lining and the deformation of the external face of the concrete must equal the yielding of the rock surface.
Detailed calculations (Jaeger, 1933) show that:
* - 7
b2/Ee
+ (mx + 1X1 - and
A* = ~ = -
2 Pb (m2 + \)jm2E2 + [{m1 — l)c2
(2) Rock fissured radially (fig. 10.8). The deformation ub of the steel plate is now:
ub = — = p(\ — A3) — with pb = A3/>
and of the concrete lining and rock {d = length of radial rock fissures):
(Pb ~t~ Pd) (jb — d) d{rn2 + 1)
ub = ~ J— ^ ^ •" P*' Equating the two values of ub yields:
- [id2 - b2)l2dEx] + [(ma + l)6/m2£2] 10.3.5 Boundary conditions
Implicitly the theory developed in 10.3.3 and 10.3.4 assumes perfect adher- ence of concrete to rock, or steel lining to concrete. Equating rock radial
deformations to concrete radial deformations is then possible. In most practical cases, rock has already deformed elastically or plastically before the concrete support is able to bear some radial load. Shrinkage of the con- crete has also to be considered. Some designers cause special boundary conditions to occur at the contact of concrete and rock when grouting the contact area, with the purpose of creating a radial pressure on the concrete lining to balance stresses due to inside hydraulic pressure in the gallery.
Lombardi (1974) claims that the correct assessment of boundary conditions is more important than sophisticated calculating methods using the finite element method techniques. His approach will be dealt with in section 10.10.
10.3.6 Steel liner buckling
Natural cleft water pressure or pressure from water infiltrating from the upper reaches of a shaft (surge tank area) can cause heavy outside pressure on the steel liner, pushing it inwards. When the pressure shaft is full of pressurized water it can be assumed that the inside water pressure balances the outside cleft water pressure. When the steel-lined shaft is emptied, inward buckling of the liner may occur. Similar conditions may occur during grouting of the rock behind the lining.
These problems have been considered by Jaeger (1955a, and discussion of the paper). Amstutz (1950, 1953) has dealt extensively with the theory of the stability of steel liner against buckling.