Design of reinforced concrete linings of pressure tunnels and shafts

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


2/8


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:


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


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)


6/8

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

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Documents

Design of reinforced concrete linings of pressure tunnels and shafts