Design of reinforced concrete Iinings of pressure tunnels and shafts rrof. Dr. A.J. Schleiss Laboratory of Hydraulic Constructions Civil Engineering Department Swiss Federal Institute of Thchnology Lausanne, Switzerland
Reprinted from THE INTERNATIONAL JOURNAL ON
HYDKOPOWEK
&DAMS lssue Three, Volume Four, 1997
Design of reinforced concrete Iinings of pressure tunnels and shafts A.J. Schleiss, Laboratory of Hydraulic Constructions, EPFL Switzerland An interactive computational method for reinforced concrete linings of pressure tunnels and shafts is presented, which considers the effect of the seepage forces in the lining and rock as well as the deformation-dependent permeability of the lining. The history of the development of cracks and its influence on the stress distribution in the hoop reinforcement is also taken into account. Design criteria for reinforced, concrete lined pressure tunnels and shafts are discussed.
he development of cracks in the concrete lining of a pressure tunnel cannot be prevented by rcinforcing the concrete. The main purpose of the reinforcement is to increase the number of cracks and to limit their width. If, compared with the rock, the permeability of the concrete lining is reduced in rhis way. water losses from the tunnel will also be diminishcd. In any case, reinforcement prevcnts unconrrolled cracking of the lining and local high water losses, which can cause the washing out of joint fil!íngs and instability of slopes abo ve thc tunncl. In practicc. reinforced concrete linings are often designed oo thc assumption that water pressure acts only on the inncr surface of the concrete linlng [Kastner, 1962 1]. This corresponds to the assumption that, from the static point of view, thc lining is tight. In fact this is not the case, because, under interna! pressure, seepage flow into thc rock will develop through the cracks in the concrete lining. Depending on the head loss through the cracks, a certain portion of the interna! water pressure is al so present on the outsíde of the concrete lining. The seepage flow Joads both the liníng and rock by scepage tOrces which must not be ncglected when calculating deformations and stresses in the reinforcement and the rock. Furthermore, mechanical-hy(iraulic interaction will occur because the width of the cracks in the lining and. therefore, its permeability are changed by the deformation of the rock [Schleiss. 19R82 J. Moreovcr, the crack pattem in the lining and, correspondingly, the distribution ofrhe stresses in the reinforcement are modified when the internal pressure is increased [Schleiss. 19903 ].
T
Fir;. J. Cradt:d, ránforccd concrete !ininr;.
Pressure el seep;ge í\r-w
l l
R-~
Rnck E,,' ,,k 1
Ralirel rei~forcement lcnmtud;nal re:níorcBri~nt
In this paper, a computation method for reinforced, concrete-lined pressure tunnels is presented which takes into account the permeability of the lining and rock, and the effects of mechanical-hydraulic interaction, as well as the history of cracking of the lining during increasing of thc internal pressure.
1. Computation model and assumptions In view of the static and hydraulic behaviour, thc folJowíng three zones have to be considered separately (Fig. l ): crackcd concrete lining; • the rock mass affected by seepage tlow; and, • the rock mass not affected by scepagc t1ow. Both the defonnation and the permeability of the cracked concrete líning are intlucnced by the reinforcement. Besidcs radial symmetrical behaviour of the above zones, the computation method is based on the following assumptions: • Deformability and permeability of the rock mass are homogeneous and isotropic. Only elastic behaviour of tlle rock is considered under interna! water pressure. • Seepage takes place only through the cracks; the permeability of the concrete between the cracks is neglected. • Regarding the load-sharing between the reinforced concrete lining and the roe k mass, the reinforcement is smeared to an cquivalent ·rhickncss of a steel lincr. However, when thc distribution of cracks in the Jining ís determined. the srress pauem in thc reintOrcement between two cracks is assumed to be parabolic [Bírkenmaier, 1983~ j. The stress level in the steeJ bars between two cracks, that is, in the uncracked section, is int1ucnced by the history of cracking (Fig. 2). • FuJI loading eífect (11 = 100 per cent) of the secpage pressure in the cracked concrete lining and in the rock mass is assumed [Schleiss, 1986 5·6 ]. To determine the load takcn by each of the three zones, the mcchanical stresses and the water prcssures at thcir boundarics havc to be known. The unknown water pressures are derived from continuity conditions. The boundary stresses can be determined by applying compatibility condítions.
2. Initial cracking of tbe Iining The lining wíll crack as sotm as the tangential stresses in thc concrete excced íts tensílc strength. Considcring the scepage forces, the maximum tangential stresses in Hydropower & Dams
lssue Three. 1997
crete is small, the assumptions R = 2 r(i in the case of rather pcrvious rock {k,- ;-::: l 00 kJ and R = 1O ra in the case of tight rock (k,- ~ k-) give sufficiently correct results. The boundary pressure between concrete and rock is given by Schleiss [19865 ]:
Spacing nf cracks d Number uf cracks n
cy _......-Stresses in steai /a,
r _P.,) .¡-2(2. -v)!((r./r,)' -ll+T¡'.
¡(P.
0
¡
]
1 Reinfurcement bar
L+(l-2v)!(1-é/r,,)
. [-3(E,(l+v,)/E,(1+v,))p, p,(r;,)~ í . '
3 :2(1-v)!((~,/r,t-I)+
l
Spacing of cracks 1 12 d Number of cracks 2n
JI J
1.
+ E,.(l +v,)!E,.(l +v,)+ l- 2 v, J ... (3)
The condition for the formation of the initial cracks is: ... (4)
e \
e
1
\,-~(~
QJJ
Jnserting p,, = p" · g · b ( groundwater present) or p, according to Eq.(2) (no groundwater tablc present) and pr{r..) according to Eq.(3) in Eq.(1), the critica! interna] pressure p;,r at which initial cracks occur in the lining can be calculated using Eq.(4). In the case of a tunnel within the groundwater, Eq.(4) gives the effectivc interna! pressurc exceeding the external groundwatcr pressure. Thus, the cracking pressure is PrcT + Pw· g·b.
Spacing uf cracks 1 14 d Number of cracks 4n
\,\
\
3. Head loss of seepage flow across the cracked lining and seepage losses
'
~
Fig. 2. Development ofcracks wul disrribution ofstresses in the steel bars (slurwn schematicalf.v).
the uncracked concrete lining due to interna! water pressure are given by Schleis;-ll986 5 j: a,,,,~
(P,, -Pi) (2 -V, ) 3(1-v,.) .
3.1 Seepage losses through cracked concrete lining
.,~.l+(r
Jr;At-1
First, the water pressure acting on thc outer side of the concrete lining, that is, thc head loss of the secpage flow through the cracks, has to be determined. For reasons of continuity, the losses through the cracked concrete lining and into the rock mass must be the same.
1-(r,!r;.)
Assuming laminar, parallcl flow in the cracks and knowing the width of the cracks, the water Josses through the cracked concrete lining can be calculated using the following equation:
+ 2Pr{r:.)
1-(¡¡/rj
... (1) ... (5)
lf the tunncl or the shaft is situated within the groundwatcr tablc then, as a reasonable approximation. the acting water prcssure p" on the outside of the uncrackcd !íning is equal to groundwatcr pressurc p,. g. h. If the tunnel or the shaft is abo ve the the groundwater table, the acting water pressure on the outside of the lining as a result of the seepagc can be derived from: ~--..
12)
Since the inHuence of p,, on the stresses in the conHydropower & Dams
lssue Three, 1997
The water losses through the rock mass for the various cases considercd (Fig. 3) are given by the following equations: • For a tunnel within the groundwater rabie [Rat. 1973': Schleiss. 1985 8 ]:
(p.Jp,
r
qoo
P,_
l+(k,. ln{¡;,!r,))!(k, ln(R/1;,)) ..
3.2 Seepage losses through the rock mass
In
Lb/r;
g-
h) 21t k, .~
(1+yl ; 2 /b')]
.. (6)
• For a tunnel abovc groundwater leve! [Bouvard, 1975''j:
4.1 Deformation of the reinforcement The radial deformation of the reinforcement can be calculated from its strain as follows:
////
u,,(rJ:;:;:E1 ·r, ::::m·€, 2 -r,
:;::;:;f11·0', 2 ·r1 1E,
... ( 1O)
1 /
\
where the tensile force in the cracked section is Z = cr_, 2 • A_,. The associated steel stress is:
\ . / 1 '-.,
"'-
\
'
l.
:
./
/
•
~-Y~-:.~,:
)' /
.¡.
1
'" ( ll)
"
With a rcdw..:tion factor m, it is considered that the
\
\ 3 \ q 1~ ·In 4 ') 2rr ·k, 1t k, ¡;,
Fig. 3. Fiow parte m
ofsecpage out of/lm1/el or siwfl into rock: ieft. tunnd within gnmndwater tab/e: and rigl!i, tmme! above gmundwutr!r
strain E, and the steel stress as in the reinforcement are not constant, but have a parabolíc distribution and are dependent on the history of cracking (Fig. 2). The factor m should be selected according to the sequencc of formation of cracks:
'- r
p,, ·g
''' (7)
• lst series of cracks: m = l/3 (average stecl stress
cr, ~ cr,, + l/3(cr,,- cr")) • For a vertical shaft within the groundwater table [Schleiss, !985
8
2nd series of cracks: m = 2/3 3rd series of cracks: m = 5/6
}:
rabie.
nth series of cracks: m = l ' '(8)
lf no groundwater table is prcsent around thc shaft, then b:::: O has to be used in Eq.(8). For a cracked concrete lining, the reach of the radial-symctrical seepage tlow can be assumed as follows: R = lO r, in the case of rathcr pervious rock (k, ;::::IOO k) and R = lOO ro in the case of tight rock (kr::;; kc).
3.3 Acting water pressure at the outer side of the concrete lining The water pressure on the outside of the concrete lining can be derived frorn the continuity condition, that is, Eq.(5) equal to Eq.(6), (7) or (8).
Considering the water pressure in the cracks, thc radial stress in the cracked, pervious concrete lining at the position of the reinforcemcnt is [Schlciss, I986 5 J:
(J (
r
r,
)~ r,(p,-pJ(l-(·/·.l') 2(.-r - ) ~~ l,, a
r,
'" (!2)
4.2 Deformation of the cracked lining The total compression of the cracked concrete lining betwccn the inner smtace and the reinforccment is given by thc sum of the following two values [Schleiss, 19865 ]:
3.4 Water losses Knowing the water pressure on the outside of the liníng, the losses per unit !ength of the tunncl or shaft can
wherc:
be determincd from Eqs.(6) or 0) or (8), depending on
thc case considered.
[r.:- r,1 -2r,2 ·In (1;,/r,)j
4. Load carried by the reinforcement
'" (14)
Tite loading on thc reinforcement can be obtained from a compatibility condition. To detemüne the load raken by the reinforcement, it is regarded statically as a steel lincr with equivalent thickness. This corresponds to the assumption that. like a steel liner, the reinforcement exerts a uniform pressure on the concrete rschleiss, 19861 This unif(mn pressure, p,-{r), can be derived from the following compalibility condition:
u,(r,) u (r,)+u,(1;,)
'" (9)
The sum of the radial deformation of the cracked concrete lining and of the rock mass has to be identical to the radi~J deformatíon of the reinforcement. These radial defonnations are derived below. For the case of no surroundíng groundwatcr, the dcpth of the groundwatcr table h is assumed to be zero.
Assuming linear distribution of the water pressure in the cracks (laminar ilow), the water pressure at the location of the reínforccment is cqual to:
P. '" (15)
4.3 Deformation of tbe rock The radial deformation of the roe k zone influenced by scepage is given by the theory of pervious, thickwalled cylindcrs [Schlcis~, 1986'-,(l
u,{c)~(p,-h p,
R)C,-p,(R)
e, -(p,(R) a,(¡;,)) e, Hydropower & Dams
'" (16)
lssue Three, 1997
where:
ri e~ ,;,(I+"J_¡ ·
2E,(I-v,)
!'.
,
l
2v,+(RirJ_+ (RII;J-I !.' ' \
1
+( I- 2v ) ! l+ .
I-v
.
\j'!
'
In (RII;,))
"
. ( I7)
4.4 Pressure between reinforcement and concrete lnserting Eqs.(!O). (!3) and (24) into the compatibiiity condition as given by Eq.(9), the pressurc transmitted by the reinforcemcnt lO the concrete can be obtained: ... (25) p, (r,) ~ D 1 /D 2 where: D,
e _,;,(I+v,.)(I-2v,.) ,E
. (!8)
"e
~m-a,(r,){r, 2 1E, ·A,)-u""
-(p,
-b
p,. g)[e, -e,(e, +C,)j
-I12(p,- p,)(I + r,ll;,) [e,-
,,'(I +V,) (I- 2v, + (Rll;, )')
e3 ~ ---"'e--ce-~--_!_
D2 ~m· r,2 !(E, AJ + [(!- v;) IE,jr, ·
E,(R'-r;)
. ( I9)
The external radius of the rock zone affectcd by seepage is assumed to be the shortcst, vertical reach of the seepagc tlow above the tunnei[Schieiss, I9865·6 ]: • Tunnel within goundwater table: R = b • Tunncl above groundwater level: R : : : a 8 (In 2)/rr where OB :::::: qlk-. In the case of a vertical shaft. the reach of the seepage flow can be assumed to be as given in scction 32. Besides the water pressure (pa) outsíde the lining, the mechanical boundary pressures at the inner and outer surface of the rock zonc influenccd by the seepage, a,(r.,) andp,(R), have also to be considered in Eq.(l6). The following radial stress is transmitted by the cracked concrete lining to the rock [Schleiss. !9865 ]:
cr,(¡;,) p,(¡J = ~ l/2 (p,- p,,) (I + ljl,,) + +p,(r,) rA,
. (20)
The boundary pressure pr(R) between the rock zone which is influenced by secpage and that which is not, is obtained from another compatibility condition;
where:
C.
~.
'
. I
'¡
(r 1R)' + ( R' -
2(I-v,)L"
e, (e,+ c,)j
¡¡:) (1- v,)
2R 2 In (RII;,)
l
j
. (22)
In rJr, + (r,lé,) [e,- e, (e,+ e1 )j
5. Width of cracks in the concrete lining 5.1 General Without knowing the width of the cracks in the concrete lining, the head loss of the secpage flow through the lining (that is, p,,) cannot be calculatcd with the formulae given in section 3. The question is how the width and the spacíng of cracks are influcnced by the reinforcement. Severa! attcmpts to salve this very compiex problem have been based on experiments with reinforced concrete beams and the empiricallaw of bonding between concrete and sted bars. Esscntially, the average spacing of the cracks is a function of stresses in the reinforcement in cracked conditions, the concrete strength, zone of int1uence of the reinforcement the thickness and spacing of steel bars, the concrete cover and thc bond between the concrete and reinforcement bars .
5.2 Determination of width and spacing of the cracks Fig. 4 shows a reinforced concrete Jining which ts crackcd. According to the calculation model of Birkenrnaier [1983 4), the width and spacing of the cracks are given as a function of the tensile stresscs in the reinforccment and concrete and of the concretereinforcement bond stress. With increasing distance from thc crack. the stresscs in the reinforcement are decreased by rhe bond stress betwcen the reinforccment and the concrete (see Fig. 4). The reduction of the steel stresses is given by the following equilibrium condítion: cr,:::::: a,::::::: as1 + t (dls)
.. (26)
.(23) Takíng into account Eqs.(20) and (21), the radial deformation of the rock on the outside of the lining_ according Eq.( 16) is:
u,(,;,)=(p,-h p, g)[c, -e,(e, +e}]+ +
l/2 (p, - P,) (I + é 1é,) [e, -e, (e, +e)]+
1- p
The maximum sted stress (a_~ 1 ) between any two cracks of the first series is withín the rangc:
O<
a_,<~,· E, lE.
.. (27)
Assurning a linear (triangular) Jistríbution of the steel-concrete bond, the distribution of rhe stresses in the sted bar between two cracks will be parabolic fBirkenrnaier, l98J1 j. Thus, the width of thc cmck is:
(r,) (rJé,) [e,- e, (C.+ c,)j ... (24)
Hydropower & Dams
lssue Three, 1997
.. (28)
6. Calculation procedure With the theoretical bases given in sections 3, 4 and 5, water lo:;ses, stresses in the reinforcement and the width of crack.s can be computed for a given percentage of reinforcement, but not directly because of the mechanícal-hvdraulk interaction. Furthermore, the cakulation model has to be modified (by changing factor m in Eqs.( lO) and (2.5)). when a further series of cracks (2nd crack; 3rd crack: and so on) ís fonned. Therefore, after determining the critica} internal pressure with Eqs. ( 1) to (4). the following step-by-stcp calculation procedure is proposed: (A) Assume water pressure on rhe outside of thc conCJ'ete liníng to be somewhat below internal pressure (p,, < p,). (B) Calculate the pressure transmitted from the reínforcement to the concrete lining by using Eq.(25) and taking into account Eqs.( 12), ( 14), ( 17), íl 8), (19). (22) and (23).
Bwd stresses steel-cancrete
Fig. 4. {orccmcnt ú111Í crmaete,
concrete 5tt>~:'l-concrete
stresses in reinhond stress.
It has bcen sllown by experimcnts that the ~tcel-con crete bond stress increases llnearly with the compressi ve strength of the concrete [Martín and Noakowski, 1981 11 ]. The empirical relatíonship for steel bars with a normal surfucc profile is: .. (29)
when (2a) ís exprcssed in millimetres. Beginníng with the critica! interna! pressure (see section 2) and using Eqs.(26) to (29), the spacing and width of the f:irst serie~ of cracks can be computed by tria! and error. Thcn the interna! water prcssure has to be increased un ti! the second series of cracks is forming. This ls the case as soon as the strcsses in the conI.Tetc between two cracks excccd the tensile strength of thc conLTete (~ 2 :::::: 1 to 2 N/mm 2): < ~'
... (30)
After the t!rst series of cracks has formed. the spac~ ing of the crack~ is known (see Fig. 2) and the width of the cracks can be determined directly from Eq.(2B). However. it has ro be noted that, for every ncw series of cracks, the steel stress Jistríbution as well as the spacing of cra<:ks has to be modified (by changing factor m and ha! vi ng the spacíng), whereas the relatíonship betwcen cr,. anda,~- can be assumed as follows: 2nd series of cracks: cr,: = 112 cr<~ 3nd series í)f cracks: G-. 1 :::::: 314 cr,, nth <,crics of cracb;:
(C) Determine the stresses in the reinforcement from Eq.(ll) and considering Eqs.( 12) and (25 ). (D) Compute width and spacing ofcracks in the lining for the first series of cracks from Eqs.(26) to (:29), by tria! and error - for the second series of cracks and the following series directly from Eq. (28). (E) CalcuJate the water pressure (pa) on the outside of the lining using Eqs.(5) and (6) or(7) or (8). (F) With the above water pressure Pa repeat the calculaüon steps B to E until the water pressure fh on
the outside of the lining remains constant. (G) Increase internal water pressure and control the
stresses in the concrete lining in the section between two cracks with Eq.(30). As soon as the tensile strength of the concrete is exceeded, the next series of cracks will be formed. Repeat the calculation steps (A) to (F) for the second series of cracks (that is, the following series), whereas the spacing of thc crw:.:ks is half of thc spacing of the first series of cracks (that is, the preceding series) and the factor m has to be increased according to 4. l. (H) Repeat the calculatíon procedure until the actual intemal water pressure is reachcd.
7. Effect of the reinforcement on the distribution of the cracks In the foiJowing, the ini1uence of the reinforcement on the history of development of cracks, their width and the steel stresscs shall be discussed in an example. The pressurc tunnel considered is si!uatcd above thc groundwater table and characterized by the followíng parameters: r, = 1.8 m; r, : : : l. 9 m; r" = 2.1 m: E, = 20 GPa,: E, = 4 GPa; E. = 200 GPa, v, = v. = 0.2: k.= Ht' mis: k. = l(p mis (uncracked): ~' = 1 N/mm': ~,. = 30 N/mm'. In Fig. 5. the effcct of the spacing of stcel bars for a certain percentage of reinf(xccment on the width of thc cracks and rhcír development with increasíng interna! water pressurc is shown. Since the number of cracks is increased by reducing thc spacing of the steel bars, the width of thc cracks is reduccd accordingly. Thus, the spacing of the bars should be made as small as is practically possible. On thc contrary. the development of cracks is intluenced only toan insignificant degrce by the spacing of the bars. The second series of cracks form~ with thc ~ame internal pressure ( 11 bars) for all bar spacings ccmsidcred. The 3rd series of Hydropower & Dams
lssue Three, 1997
0.35
-~
--o-s=26 mm, a;~ 34 cm (n :.241 --o- s~20 mm, d~ ~ 20 cm {n "'29) 16cm\n~32) ---o- s~18 mm,
0.301
---e- s ~ 20 mm, d~ ~ 20 cm {n ~ 29) - - - s~26 mm, ds ~20 cm (n~25)
1
'
0.30
¡
0-25
1 1
'
'
0.25
0.20 "
1 1
e
0.20 1
É
~ w
1
'
~
OJ5
1
1
1
0.10
w
0.10 ---·
1
'
¡
1
~
~
0.05
1
30
10
Interna! water pressure (bar)
40
50 1
_j
Fig. 5. Widrh of cracks as ajimaion of interna! pressure for d!fferenr diameters and spacings ofsteel hars at the same percentage ojreinforcement.
cracks develops in the case of smaller spacing of the bars, somewhat earlicr (42 bars instead of 44 bars). The crack history is mainly influenced by the amount of reinforcement as can be seen from Fig. 6. If the percentage of reinforcement is increased by keep~ ing the bar spacing unchanged, the 2nd and 3rd series of cracks form at lower interna! pressure. Then, as a resulr of their larger number, the widths of the cracks are smaller in the case of a higher amount of reinforcement (see Fig. 6 at interna! pressure 30 bar). Howcver. for the sarne series of cracks. the cracks are even wider for a higher percentage of reinforcement (see Fig. 6 at interna! pressure 20 bar). The reason is that a stronger reinforced concrete lining is stiffer
8. Design criteria for the reinforcement The design of the reinforcement, that is, the spacing and the diameter of the steeJ bars. is governed by the following criteria: (1) Limit stresses in the reinforcement; and. (JI) Lirnit width of cracks in thc líning; and/or, (III) Limit water Josses from the pressure tunncl. In rnost cases. the stresses in the steel hner have to be lower than the permissiblc values (for example 240 N/mm: for Stcd 56) hecause critcria II or III is governinE. Jf th~ stability of the tunnel or shaft is endangered by the washing out of joint fillings as a result of hígh con~ centrated water Josses, the width of the cracks has ro be limited fSchleiss, 19B7 10]. Such erosion should not Hydropower & Dams
~
1
0.05
o o
0.15
~
1
~
I
lssue Three, 1997
1
o o
1
lnternal water pressure {bar)
Fig. ó. Width qf'cracks as afunction ofinterntlf pressurefor different percentagcs of" reinforcement at the same .ljHJcinx of steel bars. 400
350
300
- - - s~26 mm, dt ~20 cm in~25) --o- s~26 mm. d, -34 cm {n~24) -:>-- s-20 mm, ds ~20 cm ln~29) - - - & - s~ 18 mm, ds~ 16 cm ln~32)
)_. ·-·-i
1
1 =
i
052 ¡¡er een!)
1
250
1
200
j 1 1
~
"
J
1
150 1
1
50
1
'
1 'L_
o o
1
i
100
1
T
~¡-
" Ni
10
1
20
50
1
Interna! water ¡¡ressure lbar) 1
Fig 7. Stresses in the steel bars as afunction of"interna! presl"ltn.' for differenr diametcrs ami spacings of m:cl bars.
occur provided the width of the crack>; is less than 0.3 mm. In spite of this crack width limitation of 0.3 mm, if thc tunncl is above thc groundwater tabie and ín the case of high roe k mass penneability (kr > w-n m/s) water Josses rnay ~tiii be too high. Besides thc purcly cconomíc aspect. the Jeakage out of the tunncl or shaft, that is. thc saturated rock zone. should not extend to thc natural ground surface. Seepage Hows reaching poorly drainíng surface deposits may induce landslides [Schleiss, 1987"']. The extension of the saturatcd rock zone aí-> a rcsult of seepage is a function of thc permeability of the cracked concrete lining and can therefore be iní1u-
cnccd by the desígn of the reinforcement. Nevcrtheless. increasing overburden is normally more successfuL The three criteria mcmioned above refer to thc mechanical and hydraulic bchaviour of the hning. In addition, the overall stability ofthe tunneL namely. the bearing capacity of the roe k mass. also has to be controlled. especially in the case of low rock cover [Schleiss, l987 10 j.
9. Conclusions Pressure nmncls \vith reinforced concrete linings are pervious under rhe effect of interna! pressure. Thus, seepage tlow into the rock mass occurs through the cracks in the concrete lining. The lining and rock mass are loaded by these seepage forces. which are a function of the interna! pressure and the permeabilíty of the líning and the roe k mass. Because the width of the cracks and therefore the permeability of the lining are influenced by the deformatíon of the rock mass. mechanical-hydraulic interactions exist. The effect of these intcractions and the previously mentioned seepage forces, as well as the hístory of cracking, ha ve to be considered in the computations to allow for an analysis of the behaviour of reinforced concrere lined pressure tunnels which is as realistic as possible. This al so guaramees a more economical design of the reinforccment. Thc crack widths in a concrete lining are influenced much more by a reduction ín the diarneter. and above all, by the spacing of the steel bars, than by increasing the percentage of reinforcement. Heavy reinforcement consisting of thick steel bars at wide spacing normally results in a very unfavourable crack pattem and, consequently, in a larger width of cracks in the concrete líning. Thus. the pcrcentage of reínforcemcnt and the :.pacing of the steel bars should be made as s~all as possible from thc static and practica} points of VlCW.
The governing design criteria for reinforced, concrete lined pressure tunnels are: límitíng the width of crach. <;tresses in the reinforcement and water Josses. Besides thesc criteria, consideration of thc ovcrall stability of the pressure tunneL that is. the bearing capacity of the rock mass, is also essential. O
References !. Kastncr, H., "Statik des Tunne!- und Stollenb
9. Bouvard, M .. ·'Le'> fuite<, des galerie~ en charge en terrain sec. Róle du rcví:!ement. des injection. du terrain". La f!ouilfe B!anche. No.4; 1975. 1O. Schlciss, A•. "Dcsign criteria for pcrvious and unlined pressure tunne!s". ProccedinRS. lnternational Conferences on Hydropower, Oslo. ~orway. Vol. 2: 1987. 11. Martin, H.; Noakowski, P.. "Verbundverhalten von BetonsU:ih!en. Untersuchung auf Grundlage von Ausziehversuchen··. Deutscher Aussdmss für Stahlbeton, No. 319; 1981.
= chm·acteristic value of scepage pattcm IBouvard, 1975''1 1 = area of radial reinforcement per unit length of tunne! '¡ (2a) =average crack width ) b = depth of the tunnd be!ow groundwater leve! 1 d =average spacmg: between cracks d, = spadng between the stee! bars E = modulus of e!astidty , = subscript for rock 1 = ~ubscript for concrete 1 = subscript for ~teel ¡; = acceleration due to gravity 1 k, = permeability of unnacked concrete 1 k, = permeability of rock mass ih
A,
1,
·, ,,' .·.
= munber of cracks (= 2nr,/d) ::::interna! water pressure =water pressure on th~:: outer ~ide of the concrete !ining p, = water pressure at the position of the reinforcement p,(r_) = boundary pressure between stecl and concrete p/r.J= boundary pressure between concrete and rock Prlr")= houndary pressure between concrete and rock (ín the case of uncracked !ining) 1 p,(R) = boundary pressure between saturated and unsaturated rock zones . = interna! r;¡dius of lining 1 =externa! radíus of !ining i = radiu'l of reinforcement =externa! mdius of the rock zone atfected by the seepage = diameter of steeJ bar q =water !osses per unit !ength of tunne! u,(r,) =radial deformatíon of reinforcemcnt 1 ¡ u,(r,) =radial dcfonnation of concrete lining u,(rcJ= radial dcformation of rock , u"'' =total compression of the crackcd concrete lining , = tensile force in the reinforcernent , B.. = compressive strength of concrete 1 P = tensile strength of concrete '.: 11, ""loading cffect of seepage prcssure [assumed as 100 per cent 1 1 11 = pen:cntage of reinforcement (= n:(s/2)~/(r, - r )Id) = Poisson ratio (subscripts as for E) ' v v,_ = kinematic viscüsity of water 1, p" = density of water (J, "" tensile stress in !he concrete betwecn two cracks a,.(r) =radial stress in the cracked concrete !ining 1 a.¡ = ~ted stress in the uncracked concrete section , a,: e:::: stee! strcs:-. in the cracked concrete section 1 G,-,- = maximum tang~·ntia! stresses in the uncruckcd J concrete linin<' hond stre~s be'tv,-een steel bar and concrete
1, 1
1
~
IR
. l
L
Prot'. Anton .J. Schleiss gruaduated in Civil Engincering frorn the Swi~s Federal lmtitute of Tcchno!ogy (ETH) in Zurich. Swit;;-crland. in l97R. After joining the Laboratory of Hydraulic,~. Hydrology and Glacio!ogy at ETH as a research as~ociate and senior a~>sistant, he obtained a Do<.:torate of Technical Sciences on the subjcct of pressure tunnel dcsign in 1986. After that he worked for 1! vcars for Electrowatt Engincering Ltd and was invo!veJ in. the design of many hydropowcr projecrs around the world as an expert on hydnmlics and underground waterways, Un!il recently he wa¡, Head of !he Hydraulíc Structurcs Se..::tion in the Hydropowcr Department ;Jt Electrow
86, !9K6. 6. SchJeiss. A .. "Design of pavitm~ prcssure tunn.:b" Húrcr Po'Ni!r & !Jam Consirunion. M ay 19B6.
Rat, M .. ''Ecoulem<:nt d répartition de~ pres~ion\ interstitic!!es autour de<, wnnds .. Bu!!. Liaison Lahormoire des Ponts u Chaussées, NoVccmber am1 Deccmber 1973. X Schleiss, A.. "Bcmes~ung von J)ruchtollen. Te !JI: Litcr~uur. Fclshydraulik imbcsondcre Sickerqrimwng
Notation
1
!
Ecok Po!ytcchnique fédérale de Lausannc. DGC-LCH. , CH- lO J 5 Lm~;mn.c. Switzerland.
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L Hydropower & Dams
/ssue Three, 1997
