DESIGN OF RC PRESSURE TUNNELS.pdf

CRACK CONTROL

CONSIDERING ROCK

Rui

Vaz Rodrigues

Abstract

The design of circular cross-section Waterways is frequently controlled by the openings of radial

cracks that form when water pressure is

provide economical reinforcement ratios and intelligent design of such elements, the rock

interaction must be considered along with the non

concrete tensile ring. This is done analytically is this paper and

a design chart useful for fast and practical calculation of the required reinforcement for a given

crack opening, steel stresses, rock and reinforced concrete resul

Keywords: Pressure tunnels and shafts, R

1 Rock – Structure Interaction

The pressure tunnel is considered to be surrounded by one layer of fissured rock and a remaining

part of sound rock (Fig.1), accordingly to (ASCE 1989).

using drill-and-blast procedures.

tunnel radius from the rock edge point (

displacement at the rock edge point (indicated at the radius

Eq. 4. Assuming that Er1 = Er2

field measurements during the drilling of the tunnel) and that

can be expressed as a function of the radial deformation of the concrete tunnel (Eq.5), rel

the applied pressure on the rock, at a radius of

top part of (Fig. 2) for various values of the sound rock elastic modulus and is a characteristic of

the rock mass where the tunnel is inser

(Fig. 2)). The reinforced concrete tunnel carries a tensile force

seen from (Eqs. 1 to 3) that the tunnel has to be designed to carry the remaining part of the

pressure that is not absorbed by the rock mass (

between the radial cracks is neglected, then the behavior of the concrete tunnel in tension is

described by (Eq.6). In this equation

tension, b=1.0 m and h the tunnel thickness. The curves are plotted in the lower part of

Es = 200 GPa, the elastic modulus of reinforcement. The use of this chart type to design pressure

tunnels follows the original idea of (Lauffer and Seeber 1961) for steel liners.

fib Symposium PRAGUE 2011

Session 2B-8: Construction Technology

ONTROLLED DESIGN OF RC PRESSURE TUNNELS

RING ROCK-STRUCTURE INTERACTIO

section Waterways is frequently controlled by the openings of radial

cracks that form when water pressure is installed, caused by the tensile ring forces. In order to

provide economical reinforcement ratios and intelligent design of such elements, the rock

interaction must be considered along with the non-linear behavior of the cracked reinforced

ete tensile ring. This is done analytically is this paper and the results are summarized in

design chart useful for fast and practical calculation of the required reinforcement for a given

crack opening, steel stresses, rock and reinforced concrete resultant pressures.

Pressure tunnels and shafts, Rock-structure interaction, Crack opening.

Structure Interaction

he pressure tunnel is considered to be surrounded by one layer of fissured rock and a remaining

, accordingly to (ASCE 1989). This is the case if the tunnel is excavated

blast procedures. For design purposes the zone of the fissured rock extends out one

tunnel radius from the rock edge point (Rrock) Accordingly to the same source the

displacement at the rock edge point (indicated at the radius Rrock as shown at

r2/4 = Er (ASCE, 1989), (more accurate values can be calculated after

field measurements during the drilling of the tunnel) and that Rrock≈R, as indicated in

can be expressed as a function of the radial deformation of the concrete tunnel (Eq.5), rel

the applied pressure on the rock, at a radius of Rrock as shown in (Fig. 1).This equation is plotted in

for various values of the sound rock elastic modulus and is a characteristic of

the rock mass where the tunnel is inserted (the Poisson ratio of the rock νThe reinforced concrete tunnel carries a tensile force Nring, as shown in

seen from (Eqs. 1 to 3) that the tunnel has to be designed to carry the remaining part of the

pressure that is not absorbed by the rock mass (pring). If the contribution of the concrete in tension


described by (Eq.6). In this equation ρ = As/(b⋅h) where As is the total amount of reinforcement in

the tunnel thickness. The curves are plotted in the lower part of

= 200 GPa, the elastic modulus of reinforcement. The use of this chart type to design pressure

ls follows the original idea of (Lauffer and Seeber 1961) for steel liners.

Symposium PRAGUE 2011

8: Construction Technology

595

SSURE TUNNELS

STRUCTURE INTERACTION

section Waterways is frequently controlled by the openings of radial

installed, caused by the tensile ring forces. In order to

provide economical reinforcement ratios and intelligent design of such elements, the rock-structure

linear behavior of the cracked reinforced

the results are summarized in

design chart useful for fast and practical calculation of the required reinforcement for a given

rack opening.

he pressure tunnel is considered to be surrounded by one layer of fissured rock and a remaining

This is the case if the tunnel is excavated

For design purposes the zone of the fissured rock extends out one

) Accordingly to the same source the radial

as shown at (Fig. 1)) is given by

(ASCE, 1989), (more accurate values can be calculated after

, as indicated in (Fig.1), (Eq.4)

can be expressed as a function of the radial deformation of the concrete tunnel (Eq.5), relating it to

.This equation is plotted in

for various values of the sound rock elastic modulus and is a characteristic of

νr is assumed 0.20 in

, as shown in (Fig 1b). It can be

seen from (Eqs. 1 to 3) that the tunnel has to be designed to carry the remaining part of the internal

). If the contribution of the concrete in tension


is the total amount of reinforcement in

the tunnel thickness. The curves are plotted in the lower part of (Fig. 2) for

= 200 GPa, the elastic modulus of reinforcement. The use of this chart type to design pressure



596

pi

Fissured RockSound Rock

R.C.

LinnerRrock

l Rrock=

Radial cracks

pi

prock

N

N

ring

ring

βd

βd2

βd2

R

ds

a) b)

Rrock

ds = dβ · R

-p ·ds + p ·ds - d ·N = rock i ring

β 0

N = (p - p )·R rocki

N = p ·R ring

Er2Er1v r

h

(1)

(2)

(3)

Fig. 1 Pressure tunnel: a) Applied loads, geometry and zone of fissured rock; b) Equilibrium along the

radial direction

( )12

693.01

r

rockrockr

r

rockrockrock

E

Rp

E

RpR

⋅⋅++

⋅=∆ ν (4)

R

REp

r

rrock

∆⋅

++=

772.21 ν (5)

R

R

R

hEp sring

∆⋅⋅⋅= ρ

(6)

Referring to (Fig. 2), it can be seen that the internal pressure (pi) is to be shared between the rock

and the reinforced concrete liner by imposing equal radial deformations (see curves (a) and (b) of

the design example). The radial deformation can also be calculated by pring+prock = pi, with pring and

prock given by (Eqs 5 and 6). After knowing the radial deformation then the pressures pring and prock

are calculated by the same equations. The contribution of concrete between the cracks for the

stiffness of the tunnel should be considered, because it increases its axial stiffness, causing the

tunnel to carry more load. The equation describing the tensile behavior of the tunnel considering

this contribution is given by (Eq. 7). This corresponds to a translation of (Eq. 6) by εcm shown in

(Fig. 2).

+∆

⋅⋅⋅= cmsringR

R

R

hEp ερ (7)

( )

s

s

s

effpe

effp

effct

t

cmEE

fk

σρα

ρε 4.0

1 ,

,

,

≤

⋅+⋅

=

(8)

+∆

⋅= cmssR

RE εσ (9)

It is adopted for the mean strain of concrete between cracks (Eurocode 2, 2010) in (Eq.8). The

parameters are defined in EC2 (clause 7.3.4), being kt=0.4 (long term loading) or kt=0.6 (short term

loading), ρpeff = (As/2)/(hc,eff⋅b) with hc,eff = min (2.5(h-d); h/2) and αe = Es/ Ec. The total amount of

reinforcement in the section is As as indicated in the top part of (Fig.2).The axial force carried by

reinforcement across the cracks is given by replacing (Eq.7) in (Eq.3), which gives the expression



597

indicated in (Eq.9) for the reinforcement stresses. The calculated value of the steel stress should be

used to verify the inequality in (Eq.8). As identified by (Schleiss 1997), limiting the crack openings

is among the design criteria. In order to estimate the crack opening under the reinforcement

stresses, an indirect control based on the reinforcement spacing is performed. For this purpose the

values defined by (Eurocode 2, 2010), Table 7.3N are plotted in the left lower part of (Fig. 2).

100 200 300 400

100

200

300

σσσσ

s[mm]

[MPa]

wk= 0.2 mm

wk= 0.3 mm

wk= 0.4 mm

p rock

[kN/m ]2

p ring

[kN/m ]2

s

R∆∆∆∆

p i

εεεεcm

hA s 2

A s 2

1.00 m

ρ =A s

b

b =

·h

E s = 200 GPa

rν = 0.2

s

(Design example)(a)ρρρρ hr = 1.144 x 10

-3

(a)

(b)

(Design example)

= 311 MPaσσσσs

(d)

(c)

Fig. 2 Design chart for evaluating rock (prock) and reinforced concrete (pring) pressures, radial

deformation(∆R/R) and stresses (σs) and maximal crack widths (wk) in reinforced concrete linings of

pressure tunnels under internal water pressure (pi)

2 Design Example

Consider the reinforced concrete pressure tunnel with external radius Rrock = 4.75 m, wall thickness

h = 0.50 m and circumferential reinforcement composed of top and bottom layers of ∅20 mm bars

spaced at 200 mm plus ∅16 mm bars spaced at 200 mm (As = 51.52 cm2/m with s=100 mm). The

concrete type is C20/25 and the concrete cover 50 mm. The internal water pressure is pi= 1000

kN/m2. The rock elastic modulus is Er = 2.0 GPa.The solution should verify a maximal crack

opening of wk = 0.3 mm.From (Fig. 2 – see Design example (a)), the curves describing the

behavior of the rock (E=2.0 GPa) and reinforced concrete tunnel without the contribution of



598

concrete between the cracks (ρ = 1.03/100 and ρ⋅h/r = 1.144⋅10-3

with r=4.50 m) are identified. The

total water pressure to carry between the rock and reinforcement is 1000 kN/m2, therefore the radial

deformation of the tunnel is ∆R/R = 1.365‰ and the steel stresses σs= 1.365/1000 ⋅ 200⋅103= 273

MPa (see bottom part of Fig.2). The same result can be numerically obtained by from prock+pring =

pi with prock and pring given by (Eqs. 5 and 6). Resultant rock and reinforced concrete pressures are

prock= 687 kN/m2 and pring = 313 kN/m

2. The contribution of concrete between the cracks by (Eq.8),

assuming long term loading with kt = 0.4 is εcm= 0.28‰ with ρpeff = 28.78/(hc,eff⋅100) = 1.72/100,

with hc,eff= 15 cm (see EC2 clause 7.3.4(2). The load carrying behavior of the tunnel considering

the additional stiffness is represented in (Fig.2), corresponding to a translation of εcm = 0.28‰ of

the original curve to the left as indicated. The total water pressure to carry between the rock and

reinforcement is 1000 kN/m2, therefore the radial deformation of the tunnel is ∆R/R = 1.28‰. The

same result can be obtained from prock+pring = pi with prock and pring given by (Eqs. 5 and 7).

Resultant rock and reinforced concrete pressures are prock= 644 kN/m2 and pring = 356 kN/m

2 (see b)

in (Fig.2). The stress in the reinforcement is given by (Eq. 9), which gives σs= 311MPa. The same

result can be obtained from (Fig. 2), see point c). Since the spacing of the reinforcement is s = 100

mm (see point d)), it can be seen that this solution verifies a maximal crack opening of wk = 0.30

mm. If the reinforcement solution does not verify the required crack opening, the amount of

reinforcement should be increased or spacing reduced under reasonable limits.

3 Conclusions

In this paper, an analysis chart is developed for practical calculation of the reinforcement stresses,

rock and reinforced concrete resultant pressures, and crack openings of circular reinforced concrete

pressure tunnels considering rock-structure interaction. The following aspects should also be

considered in the design: i) The tunnel should be designed to carry external loads (water, rock mass

displacements), without water on the inside. ii) The overall stability of the rock mass and minimum

rock cover should be verified.

The author gratefully acknowledges the support of COBA, Consultores de Engenharia e Ambiente.

References

[1] ASCE, P.:Civil Enginnering Guidelines for Planning and Designing Hidroelectrical

Developments, Volume 2 - Waterways . ASCE, 1989.

[2] LAUFFER, H., SEEBER, G.:Design and Control of Linings of Pressure Tunnels and Shafts,

Based on Measurements of the deformability of the Rock. Question 25, Report 91, Seventh

Congress of Large Dams, Rome Italy, 1961.

[3] SCHLEISS, A.J.:Design of reinforced concrete linings of pressure tunnels and shafts. The

International Journal on Hydropower & Dams, Issue Three, Volume Four, 1997.

[4] EUROCODE 2, Design of Concrete Structures. Part 1-1: General Rules and rules for

buildings. EN 1992-1-1, 2010.

Rui Vaz Rodrigues, PhD., C.Eng. � COBA

Consultores para Obras, Barragens e Planeamento, S.A.

Av. 5 de Outubro, 323

1649-011 Lisboa Portugal

� (+351) 210 125 000

� (+351) 210 125 144

☺ [email protected]

URL www.coba.pt

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DESIGN OF RC PRESSURE TUNNELS.pdf