8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
1/8
esign
o
reinforced
concrete Iinings
o
pressure tunnels
and shafts
rrof
Dr
A J Schleiss
Laboratory of Hydraulic onstructions
Civil Engineering
Department
Swiss Federal Institute
of
Thchnology
Lausanne Switzerland
Reprinted from
THE INTERNATIONAL JOURNAL ON
HY KOPOWEK
DAMS
lssue Three Volume Four 1997
8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
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8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
3/8
Spacing
nf
cracks d
Number
uf
cracks n
cy
Spacing of cracks
1
12
d
Number
of cracks
2n
\ 1
\ , - ~ ~
Q
Spacing uf
cracks
114 d
Number of
cracks
4n
'
\ \
1
0
e
_...... Stresses in steai
/a
Reinfurcement
bar
e
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:
P,,
-Pi)
(
2
-V )
3(1-v,.) .
. , ~ . l r < r , J
+ li'r,(l+v,,) (r;,(2-v.))]+
Jr;At-1 1-(r, r;.)
+ 2Pr{r:.)
1- /rj
... (1)
l f
the tunncl
or
the shaft is situated within the
groundwatcr tablc then, as a reasonable approxima
tion. the acting wate r prcssure
p
on the outside
of
the
uncrackcd ning
is
equal
to
groundwatcr pressurc
p . g h.
If the tunnel or the shaft is abo ve the the groundwa
ter table, the acting water pressure on the outside of
the lining as a result of the seepagc can be derived
from:
~ - - . . P,_
l+(k,. ln{;, r,)) (k,
ln(R/1;,))
.. 12)
Since the inHuence of p,, on the stresses in the con-
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
[1986
5
]
r _
.-2 2.
-v) ((r./r,)'
-ll+T .
P.
P.,)
] L+(l-2v) (1-/r,,)
J
. [-3(E,(l+v,)/E,(1+v,))p, J
p , ( r ; , ) ~ . ' 1
: 2 ( 1 - v ) ( ( ~ , / r , t - I ) + .
l
E,.(l
+v,) E,.(l
+v,)+ l - 2v, J
... (3)
The condition for the formation of the initial cracks
is:
... (4)
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
gb.
3. Head loss of seepage flow
across the cracked lining and
seepage losses
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 rea
sons of continuity, the losses through the cracked con
crete lining and into the rock mass must be the same.
3.1 Seepage losses through cracked
concrete lining
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:
... (5)
3.2 Seepage losses through the rock mass
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, h) 21t k,
qoo
r
In
Lb r;
(1+yl ;
2
/b )]
. .
(6)
For a tunnel abovc groundwater leve [Bouvard, 1975''j:
8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
4/8
1
/
\ \ . / 1
-., \ l /
'-
'
: /
~ - Y ~ - : . ~ , :
/
"
..
\
1 \
Fig.
3.
Fiow parte m
ofsecpage out of/lm-
1/el
or siwfl into rock:
ieft.
tunnd within
gnmndwater tab/e:
and
rigl i, tmme
above gmundwutr r
rabie.
3 \
q
' -
r 1
In
p,, g
4 )
2rr
k,
1t
k,
;,
(7)
For
a vertical shaft within the groundwater table
[Schleiss,
985
8
}:
' (8)
lf no groundwater table is prcsent around thc shaft,
then
b::::
Ohas to be used in Eq.(8).
For
a cracked con
crete 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 lin
ing can be derived frorn the continuity condition, that
is, Eq.(5) equal to Eq.(6),
(7) or
(8).
3.4 Water losses
Knowing the water pressure on the outside of the lin
ng,
the losses per unit ength
of
the tunncl
or
shaft can
be
determincd from Eqs.(6)
or 0 or
(8),
depending on
thc case considered.
4.
Load
carried by the
reinforcement
Tite
loading
on
thc reinforcement can be obtained from
a compatibility condition.
To
detemne 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 identi
cal
to
the r a d i ~ J deformaton
of
the reinforcement.
These radial defonna tions are de rived below. For the
case of
no surroundng groundwatcr, the dcpth
of
the
groundwatcr table h is assumed to be zero.
4.1 Deformation of the reinforcement
The radial deformation
of
the reinforcement can be
calculated
from its
strain
as
follows:
u,,(rJ:;:;:E
1
r, ::::m,
2
-r,
:;::;:;f110 ,
2
r
1
1E,
...
( 1O)
where
the tensile
force in the cracked section
is
Z
=
cr_
2
A_ .
The associated steel stress is:
'" ( l l)
With a
rcdw..:tion
factor
m, it is
considered that the
strain
E,
and the steel stress
as
in the reinforcement are
not constant, but have a parabolc distribution and are
dependent
on
the history
of
cracking (Fig.
2).
The fac
tor
m
should
be
selected according to the sequencc
of
formation
of
cracks:
lst series
of
cracks:
m = l/3
(average stecl stress
cr,
cr,,
+ l/3(cr,,- cr"))
2nd series
of
cracks:
m = 2/3
3rd series
of
cracks:
m
=
5/6
nth series
of
cracks:
m
= l
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,(p,-pJ(l-(/.l )
r r
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,
1986
5
]:
wherc:
[r.:- r
1
-2r,
2
In 1;,/r,)j
' (14)
Assuming linear distribution
of
the water pressure in
the
cracks (laminar ilow), the water pressure at the
location
of
the renforccment 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, thick
walled cylindcrs
[ S c h l c i s ~ 1986'-,(l
u , { c ) ~ p , - h p, R)C,-p, R)
e,
- p, R) a, ;,)) e,
'
16)
Hydropower Dams lssue Three, 1997
8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
5/8
where:
ri 2v,+(RirJ_+
l
e ~ ,;,(I+"J_
(RII;J-I
1
2E,(I-v,) . . .' I-v \j
+(
I- 2v
) l+ '
, ' \
In (RII;,))
" . ( I7)
e
_,;,(I+v,.)(I-2v,.)
E
e
. ( 8)
,,'(I
+V,)
( - 2v, + (Rll;, )')
e - - - " ' e - - c e - ~ - - _ _
3
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, I986
5
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 seep
age flow can be assumed
to be
as given in scction
32.
Besides the water pressure (pa) outsde 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. 986
5
]:
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'
-
:) (1-
v,)
l
' 2(I-v,)L"
2R
2
In (RII;,)
j
.
(22)
.(23)
Takng 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,)
(
+
1 ,) [e,
-e,
(e, +e)]+
1-
p r,)
rJ,) [e,- e,
(C.+
c,)j
... 24)
Hydropower Dams lssue Three, 1997
4.4 Pressure between reinforcement and
concrete
lnserting Eqs.( O). ( 3) and (24) into the compatibiii
ty
condition as given by Eq.(9), the pressurc transmit
ted by the reinforcemcnt
lO
the concrete can
be
obtained:
p,
r,) D
1
/D
2
... (25)
where:
D, ~ m - a , r , ) { r , 1 E ,
A,)-u
- p,
b p,.
g)[e,
-e, (e, +C,)j
- I12 p -
p,)(I + r,ll;,)
[e,-
e, (e,+
c,)j
D
2
~ r
2
(E, AJ
+
[( -
v;)
IE,jr,
In rJr, +(r,l,)
[e,-
e, (e,+ e
1
)j
5. Width
of cracks in the
concrete lining
5.1 General
Without knowing the width
of
the cracks in the con
crete 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 spacng 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 concrete
reinforcement 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 condtion:
cr,:::::: a,::::::: as1 + t dls) .. (26)
The maximum
sted
stress between any two
cracks of
the first series
is
withn the rangc:
O< a _ < ~ E, lE. .. 27)
Assurning a linear (triangular) Jistrbution of the
steel-concrete bond, the distribution
of
rhe stresses
in
the
sted
bar between two cracks will
be
parabolic
fBirkenrnaier, l98J
1
j.
Thus, the width
of
thc cmck is:
.. 28)
8/10/2019 Design of reinforced concrete linings of pressure tunnels and shafts
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Bwd
stresses
steel-cancrete
Fig. 4. concrete stresses
in
rein
{orccmcnt 111 crmaete 5 t t > ~ : l - c o n c r e t e hond stress.
It has bcen sllown by experimcnts that the ~ t c e l - c o n
crete bond stress increases llnearly with the compres
si ve strength of the concrete [Martn and Noakowski,
1981 ].
The empirical relatonship for steel bars with
a normal surfucc profile
is:
. . 29)
when (2a) s exprcssed in millimetres.
Beginnng with the critica interna pressure (see sec
tion
2)
and using Eqs.(26) to
29),
the spacing and
width of the f:irst s e r i e ~ of cracks can be computed by
tria and error. Thcn the interna water prcssure has to
be increased
un
i the second series of cracks is form
ing. This ls the case as soon as the strcsses in the con
I.Tetc
between two cracks excccd the tensile strength
of thc conLTete 1 to 2 N/mm
2
: