design criteria for plain concrete lining in water and power tunnels

RESEARCH

Design Criteria for Plain Concrete Lining in Water and Power Tunnels

Bhawani Singh, G. C. Nayak, Ram Kumar and Gopal Chandra

Abstract--Plain concrete lining is used in the construction of many power tunnels in rock masses having good and ]air qualities under internal pressure (p). This type o/concrete is susceptible to cracking and jointing as a result o/the construction process. The average crack opening (2u) is estimated to be approximately equal to (1 + v) t(ft + p)/E, where E and u are the modulus o[ deformation and Poisson' s ratio o/rock mass, respectively; and t and ft are, respectively, the thickness and ultimate tensile strength o/the concrete lining. In order to ensure that the crack will close over time, this opening (2u) should not be allowed to exceed a permissible limit. Case histories are included to obtain the permissible limits [or crack opening and spacing.

R6sum6--Un revOtement de simple bbton est utilisb dans la construction de beaucoup de tunnels sous pression dans des masses roc heuses ayant de bonnes qualitks lorsque soumis it une pression (p ). Ce genre de bkton est susceptible au craquelage et au [issurage lors de la construction. L ' ouverture moyen d' une ]issure (2u ) est estimke ~tre plus ou moins kgal gz (1 + v) t (ft + p)/E, ou E et u sont r espectivement le module de d'klasticitb et le coe]]icient de Poisson de la roche, alors que t et ft sont respectivement l'kpaisseur et la rksistance en tension du revOtement de bbton. A[in de s" assurer que la ]issure se ]ermera fi long terme, l' ouverture ne dolt pas exckder une certaine valeur limite. Des expbriences passkes sont incluses pour obtenir les valeurs limites d'ouverture et d" espacement des [issures.

Introduction

A Plain cement concrete l in ing for a water power tunnel is likely to crack in a number of places (see

Fig. 1). This paper presents an analysis of a concrete l in ing segment formed by cracks. Construction joints are likely to open up under internal water pressure. Because six construction joints generally are provided, the actual behavior of the l in ing is likely to resemble that of a segmented l in ing having six segments, rather than that of a monoli thic concrete l ining, as has been convention- ally assumed (Jaeger 1972).

Reinforcement hampers the installa- tion of a good, dense cement lining. Furthermore, no reinforcement should be provided in the l in ing of tunnels except at inlet and outlet ends, in distressed areas and in the plug area. Good, compact concrete capable of withstanding high velocities and abra- sion is desirable.

The disadvantage of cracked segmented concrete l in ing is that the flow of water may dislodge the segments from the tunnel l ining. In 1979, a failure of this type occurred in Costa Rica, when a 16-ft-(4.9-m-) diameter pla in concrete l in ing failed while undergoing testing under a water head of 85 psi (0.58 Mpa) (Engineer ing N e w Record 1979). A 2624-ft- (800-m-) long fissure, up to 16in. (0.41m) wide in places, opened up at the top of the l ining, which had been designed for 114 psi (0.78 MPa) water pressure. Less than a

Present address: Dr Bhawani Singh, Professor of Civil Engineering, University of Roorkee, Roorkee 247667, India.

week after the tunnel tests (performed at 85psi [0.58MPa] only) had been completed for commissioning of the plant, dirt, leaves and debris began flowing out of the tunnel. The tunnel was closed immediately and an inspec- tion revealed the crack referred to above.

This example points up the need for caution in using an unreinforced thin l ining, as well as the need to improve our understanding of the behavior of a cracked plain concrete tunnel l in ing (Kumar et al. 1978a).

Finite Element Analysis of Segmented or Cracked Lining

Because the cracks will be distributed symmetrically in the concrete l ining, only half of the segment between two consecutive cracks need be analysed. Figure 2 shows the boundary conditions. Using the above symmetry, the stress distribution in the remaining half of the segment can be obtained.

The following assumptions are made:

(1) There is no relative movement between the rock and the tunnel l ining. This assumption is justified because the mobilized coefficient of friction is lower than the permissible limit.

(2) Because the tunnel l in ing is very long, a plane-strain condit ion is assumed.

(3) The pressure inside the tunnel l in ing is assumed to be uniform. (However, this assumption is not strictly correct, because there is some leakage of water through the cracks.)

(4) The l in ing thickness is one- twelfth of the tunnel diameter.

(5) The cracks are assumed to be radial and to extend through the l ining.

(6) The rock surrounding the tunnel l in ing and the tunnel l in ing material are homogeneous, elastic and isotropic.

(7) There is no gap between the l in ing and the rock mass.

NOTATIONS

a internal radius of lining; .4 a constant; D internal diameter of lining (2a); E modulus of deformation of rock

mass (determined from the loading cycle of uniaxiai jacking tests);

Ec elastic modulus of plain concrete; ]t ultimate tensile strength of con-

crete; F shear force in the segment; KN normal stiffness of rock mass; K T tangential/shear stiffness of rock

mass; M bending moment in the segment; N number of segments or cracks in

the lining due to internal water pressure;

p internal water pressure in the lining;

t thickness of the concrete lining; T hoop force in the segment; u tangential displacement of the

segment; w radial deflection of the segment; Wc radial deflection of segment at

crack end (p/KN); Z section modulus of segment (t~/6); 0 angle between centre line of

segment and radius vector under consideration;

v Poisson's ratio of saturated rock mass;

Vc Poisson's ratio of concrete; qs angle subtended by the segment at

its centre.

Tunnelling and Underground Space Technology, Vol. 3, No. 2, pp. 201-208, 1988 . 0886-7798/88 $3.190 + .00 Printed in Great Britain. Pergamon Press plc 2 0 ]

ROCK HASS

ICRETE LINING

;TRUCTIO N JOINTS, RIB SUPPORT

CONCRETE

. . . . j~.-:.::.::;. .... _'. : :.: ." - : . : ..,.,2..~.1 - - : . . ,

;KS IN THE LINING

Figure 1. Development o[ cracks and opening oJ construction joints in a tunnel lining subjected to internal pressure.

Finite Element Discretization Six patterns of crack distribution in

the tunnel l in ing have been assumed and analysed. The actual crack pattern satisfies the following criteria:

(1) No transmission of stresses in the l in ing across the crack.

(2) An intact rock mass capable of

carrying the shear transfer between the l in ing and the rock mass.

The angle between the two successive cracks for the six cases analysed is assumed to be 5, 10, 15, 20, 30 and 45 degrees, corresponding to cases involving 72, 36, 24, 18, 12 and 8 cracked

segments, respectively. All o1 these cases have been analysed using isoparametri{ eight-noded elements. The ratio of the elastic modulus of the saturated rock mass to that of the concrete is taken to be equal to 0.2, 0.4 and 0.6.

The finite element discretization of the tunnel is shown in Fig. 2. The total number of elements is 20; the total number of nodes, 85. The maximum aspect ratio of elements is 10 when tile angle is 45 degrees.

Other details of finite element formulation and programming are available in the text by Zienkiewicz (1967).

The hoop stresses for the cracked l in ing are obtained by the finite element method for different crack patterns. The variation of stresses at the mid-portion of the cracked l in ing and the rock mass is plotted in Fig. 3. The variation in shear stresses at face B-B' (see Fig. 2) in the rock mass is illustrated in Fig. 4.

Table 1 gives the percentage reduction in maximum hoop stress, with respect to the hoop stress in a solid reinforced lining, as calculated from the theory presented by Jaeger (1972); and the maximum crack opening and maximum coefficient of friction mobilized along the l in ing for E/E~ equal to 0.2, 0.4 and 0.6 and for 12, 18 and 72 segments of lining.

Discussion of Results T e n s i l e s tresses i n the c o n c r e t e l i n i n g

a n d r o c k m a s s . At the mid-portion of the l in ing segment, the tensile hoop stresses

v , y v', y'

u',x'

~--UaX

o / =-lIl.i.,l [ " .

t20 II~I~L20 t Io L ! "w ~ - - I -m

i CONC. LImNG ol2 i ~ i _ .

Figure 2. Finite-element mesh.

,o

ROCK 2-41110

, ,o $ , , , __J

202 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY Volume 3, Number 2, 1988

~> 2 .0 [ / / / ~ [ l E t = 0 . =

= o / / , , , ° ' 1 / Z'V/ Io/,= 0/6 el , ' ~ v I / / / 2 ~ / I

[ -"°ITff '" , - $ . 0 k .

Figure 3. Hoop stresses along the thickness.

12

"=r I , , , , . o . , to,F

. 0 . , r

- 0 " 4 ~

Figure 4. Variation of shear stresses along the outer face B-B'.

in a concrete tunnel l in ing increase a long its thickness. These stresses suddenly drop to approx, one-fifth of the peak values as they enter the rock mass, due to the sudden change in the material properties.

Based on Fig. 2, the fol lowing points may be noted:

(1) As the number of cracks increases, the location of tensile hoop stresses shifts towards the rock mass.

(2) The max imum tensile stress in the rock mass a long the center line of a segment does not occur at the interface, but instead shifts to a posi t ion at a distance 0.5 times the l in ing thickness, i.e. D/24. This phenomenon is caused by the shear stress at the interface.

(3) As the number of cracks increases,

the hoop stresses increasingly deviate from a l inear distribution.

(4) The percentage reduction in hoop stresses increases as the E/Ec ratio increases (see Table 1).

Opening of cracks. Table 1 shows that the crack opening decreases with increasing E/Ec and is inversely propor t ional to the number of cracks.

Shear stress. The shear stress distribu- t ion a long the rock mass circumference B-B' (see Fig. 2) is shown in Fig. 4. The rate of increase in the shear stresses on this face is larger for a smaller number of segments. The variat ion becomes linear for 36 segments, and stresses increase from 0 to a ma x imum value of

Table 1. Results of finite-element analysis.

St. Number of E/Ec % Reduction in No. segments max. hoop stress

1 12 0.2 35 0.4 50 0.6 57

2 18 0.2 50 0.4 57 0.6 69

3 72 0.6 100

Crack opening t outer length p of segment

approx. 1.0 p at the crack level. Thus, as the number of segments increases, the l ikel ihood of s l iding between the l in ing and the rock decreases. Table 1 shows that the shear stress (mobilized coefficient of friction) decreases with increasing E/Eo

The coefficient of friction mobilized is generally equal to 1.0 + .20 (Fig. 4). Experience shows that this appears to be a safe l imit for the concrete-rock interface, since the rock surface is highly irregular. Thus, no slip is likely to occur between the l in ing and the rock mass; and all the cracks are likely to be dis- persed uniformly along the periphery.

Proposed Simple Method of Analysis

An approximate and simple method of analysis is first proposed for a pre l iminary study of a cracked or segmented lining. For detailed study, a rigorous analysis is provided in the Appendix.

The fol lowing assumptions apply: (1) Because the l in ing thickness is

small, i.e. less than one-third the length of the segment, the segment may be assumed to behave approximate ly as a thin shell. The l in ing is circular and of uniform thickness.

(2) The l in ing is subjected to internal water pressure, which also acts a long the surface of the crack.

(3) The rock mass is replaced by normal and tangent ia l /shear springs of stiffnesses KN and Kr.

(4) For a homogeneous, isotropic, elastic rock mass without radial fractures, the normal stiffness of the rock mass is given by

E KN - (1)

(1 +V) a

If radial fractures develop as a result of tangential tensile stresses in the rock mass, the normal stiffness will decrease drastically. Blasting may cause damage to the rock mass to some radial distance from the wall, thereby further decreasing KN.

(5) The modulus of deformation of the rock mass is small compared to that of the l ining.

(6) No sl iding occurs between the rock mass and the lining.

(7) Because the in-situ tangential stress a round the opening is greater than the tangential tensile stress developed in the rock mass, there is no possibi l i ty of radial cracking of the rock mass.

37 x 10 -s 0.96 21 x 10 -s 0.75 Theory 15 x 10 -s 0.66 The radial deflection of the l in ing is 38 x 10 -s 0.92 given by 22 x 10 -s 0.73 16 x 10 -s 0.62 P 15 x 10 -s 0.33 w = K---N- (2)

Volume 3, Number 2, 1988 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY 203

The crack opening of all the cracks will be equal to 2~-w; therefore, the crack opening of each crack will be wO. In other words, the tangential displacement of each crack is

wO PO u - - ( 3 )

2 2KN

The maximum shear stress will occur at the crack and will be equal to

pqJ KT { = KT U = X (4)

2 KN

The maximum hoop tension in the l in ing occurs at the center line of the segment and is approximately equal to the resultant shear stresses at the base of the lining. Thus,

l aqt T = -~ K r u --if- - p t

( 5 )

apO z K r - x - - - p t .

8 KN

The l in ing segment will be subjected to a bending moment that is also at its max imum at the center line of the segment (see Fig. 5). The bending moment is given by

1 aq, t - - X - -

m = 2 K r u 2 2

apt qJ2 KT - - - - X - -

16 K x

0

(6)

L

KNW

Figure 5. Forces on segment o[ tunne l lining.

It is clear from Equations (5) and (6) that the higher the shear stiffness of the rock mass, the more probable will be the development of tensile stresses in the lining, since shear resistance prevents the l in ing from relaxing completely. The ratio KT/KN will depend inversely on the ratio of areas of the loading surfaces. Thus, Kr/K,v will be propor- tional to t/q,.

Because the contact shear stress is not uniformly distributed, the coefficient of proportionality is difficult to assess. However, it can be assessed by comparing the results of finite element analysis (Kumar et al. 1978b) with proposed approximate solutions. Figure 4 suggests that the ratio of maximum shear stress to radial pressure is equal to 1.0 ± .20 and is practically independent of the moduli i of deformation of the concrete and the rock and the number of segments. This assumption is valid for rocks of poor quality (E/Ec <-- 0.20) and a small number of cracks (N < 36). It is generally in such situations that accurate analysis is required. Equation (4) then leads to the following correlation of KT/KN for an isotropic elastic rock mass:

K r 2

KN ql (7)

Thus, the magnitudes of hoop tension and moment at the center line may be determined approximately by the following relationship:

apqJ T = 4 - pt (8)

aptO M = (9)

8

The maximum tensile stress will be at the outer periphery of the l in ing and is given by

ap qt apt 0 6 oo = 4t + 8 x t z - p

(10) apqJ

- p .

t

An attempt has been made to derive an exact solution in the Appendix. For small values of q* and thin l ining, the solution converges to the above equations.

A rock mass is seldom isotropic; rather, it is usually highly anisotropic (Singh 1973), with a shear modulus of the order of one-tenth of its deformation modulus. The presence of joints in the rock mass will drastically reduce the shear stiffness K r of the rock mass. Moreover, the interface sliding that is likely to occur will further decrease the value of Kr. Thus, a realistic value for K T / K N may be as low as one-third of q*;

this would substantially ~educe lhe hoop stresses in the lining.

Some people have expressed concern that cracks in plain concrete l in ing may tend to concentrate in one location. In fact, such an occurrence seems unlikely so long as there is a good-quality bond between the l in ing and rock mass. Such a bond can be ensured by good contact grouting.

Recommendations for Designing a Plain Concrete Lining

Figure 1 shows a crack pattern in a plain concrete lining. The actual number of cracks and the width of the crack opening may be smaller than predicted due to percolation of water inside the rock mass through cracks. The presence of steel sets to support the rock mass may further reduce the number of cracks.

Some important guidelines for the design of plain concrete l in ing for power tunnels and water tunnels are outlined below:

(1) The number of cracks may be obtained from Equation (10), assuming that the crack occurs wherever the tangential stress exceeds the tensile strength. Thus,

2 t rap Jt = N t - P (11)

2 ~rap or N - (12)

t(/t+P)

where/t is equal to the average ultimate tensile strength of the concrete l in ing in bending.

Table 2 compares the results of Equation (12) and a finite-element analysis considering the tangential stress to be equal to the ultimate tensile strength. The agreement is encouraging for the large number of cracks that are likely to actually develop in the lining.

The number of cracks should be limited so that the length of the segment is more than approximately three times the thickness of the l ining, and so that the segment is not easily eroded by the flow of water.

According to Equation (12), the spacing of cracks (S) at the outer periphery will be

S = 2 rr a / N : ([t+P)t/P

> 3 t or 1 .75m. ( 1 3 )

(2) The opening of each crack should be less than the permissible limit (3 mm). In other words, the average crack opening (2u) obtained from Equat ion (3) also should be checked as follows:


Table 2. Comparison of results o] simple analysis and ]inite-element analysis.

E Maximum Ec Tangential

stress (oO/p)

0.2 2.06 2.0 1.49 1.14 0.69

Number of cracks Actual* Predicted by

Equation (12) t

8 14 12 15 18 18 24 20 36 26

Crack opening (2u/t) F.E.M. Predicted by

Equation (14) t

180x10 -s 92 x 10 -s 135x10 -5 92 x 10 -5

91x10-ss 75 x 10 -s 68xl O- 64 x 10 -s 44x10 -5 51 x 10 -5

* That is, number of segments assumed in the FE analysis. t Considering ultimate tensile strength equal to maximum tangential stress.

2 U =

2rrp 27rp t ( / t+p) t(]t+p) m X - - -

NKN KN 2rr ap aKN

t(lt+p) (l+v) a - X

a E

(l +u) t(h+p) or, 2u - < 3 m m . (14)

E

Equat ion (14) is quite reasonable when compared with the results of finite-element analysis on the assumption that the max imum tensile stress (in the tangential direction) is equal to the ultimate tensile strength. The equation is particularly reasonable under conditions involving poor rock and a large number of cracks (see Table 2).

There appears to be a need for a correction factor in Equations (12) and (14) to account for anisotropy of the rock mass, possible slip between the l in ing and the rock mass, percolation of water inside the rock mass and uneven distribution of cracks.

(3) For non-uniform l in ing thickness, t may be taken as the average l in ing thickness. It is preferable that the crack pattern be predicted by finite-element analysis.

(4) The l in ing thickness should be adequate to withstand the net water pressure from outside the l ining, which may develop when the internal water pressure is reduced.

(5) A good bond between the concrete l ining, blocking concrete and rock mass should be ensured by contact grouting under adequate grout pressure.

(6) The internal water pressure should not be more than the allowable bearing pressure of the rock mass and not more than the overburden pressure.

(7) There appears to be no need for bolt ing the l in ing to the rock mass to prevent dislodging of cracked segments as a result of hydraulic forces, provided that the crack opening is smaller than the permissible l imit and there are not too many cracks.

Plain concrete l in ing should be

avoided in rock masses of very poor quality and in areas o[ thick shear zones, even if the rock cover is adequate. Because a pla in l in ing may not be able to withstand the centrifugal forces resulting from flowing water, a plain concrete l in ing should not be used where the tunnel a l ignment is curved.

Reinforcement of the concrete l in ing near the entry and exit points, as well as in areas characterized by poor rock conditions and curved alignment, is recommended as a general practice.

According to Equation (12), the cracks will be spaced too closely together if the internal water pressure is too high. In the case of a 585-mm-thick, M25 plain concrete l ining, the maximum water pressure should not exceed 12.5 kg/cm 2 (1.25 MPa).

(8) Where there is doubt about the actual conditions in the tunnel, it is advisable to conduct pressure tunnel tests in which the increase in tunnel diameter is measured and the actual number of cracks is observed.

(9) It is recommended that water pressure be applied to the tunnel l in ing slowly, not abruptly, in order to minimize the damage to the l ining. The l in ing should be inspected after the tunnel is emptied.

Case Histor ies

The Kotmali power tunnel in Sri Lanka cracked when it was subjected to internal water pressure of 5.7 kg/cm z (0.57 MPa). Because the water pressure had developed very rapidly, the water had no opportuni ty to seep behind the

Table 3. Details of tunnels for various projects in India.

SI. Pr~ect No.

1 Ram Ganga River Project (U.P.)

2 Maneri Bhali Hydel Scheme Stage I (U.P.)

3 Yamuna Hydro-electric scheme Stage II (U.P.)

4 Maneri Bhali Hydel Scheme Stage II (U.P.)

5 Tehri Dam Project (UP) Diversion Tunnel

6 Tehri Dam Project (UP) Head Race Tunnel

Kopli Hydel Project Assam in jointed granite and gneiss

Shape

Circular

Circular but excavation of horseshoe shape)

Circular

Horse Shoe

Horse Shoe

Circular

Circular

Diameter in meters

9.0

4.75

7.0

6.0

11.0

8.0

4.5

Lining thickness at crown

(mm)

750

300-500

300-600

300-500

375-900

600

150-200

Pressure (kg/cm 2)

4.5

1.84-6.22

4.4-6.2

1.5-3.5

4-6.0

2-12.0

16

Ed modulus of (rock)

deformation (kg/cm 2)

8500-35,000

75,000

Crack opening

(mm)

3.1-0.8

0.1-0.25

Crack No. of spacing cracks

(mm)

6.6t

(14.6-5)t

5000-70,000 2.1-0.3 (6.7-5)t

30,0OO-100,000 3.2-0.17 (17.7-8.1)t

50(X)-30,000 2.6-1.1 (7.3-5.2)t

8000-70,000 2.4-0.4 3.1 t

5700-15,000 1.3-0.70 2.6t

9

2-6

7-9

3-5

10-14

17

11


tunnel l in ing and counterbalance the internal water pressure.

The diameter of the l in ing was 3.0 m and approx. 30 cm thick. There was no hoop reinforcement. The tunnel opening was supported by steel ribs that were cast-in-place in the plain concrete l ining. One side of the tunnel was limestone with solution cavities; the other side of the cross-section was breccia.

No plate load tests were conducted. The concrete l in ing was installed with six construction joints. After the tunnel was emptied, a crack was observed in the crown and in both side walls; and the construction joints in the bottom corner opened up. The crack openings varied from 1 to l0 mm; the total sum of the c r a c k openings was approx. 12 ram.

Equat ion 12 predicted 7.5 cracks (a = 150, p = 5.7kg/cm 2, [t = 16kg/cm2). Nine cracks and construction joints were observed. This is an encouraging agreement with the prediction.

Table 3 summarizes case histories of various tunnels in India where plain concrete l in ing has been used (M25 concrete was used in all cases). It may be noted that the calculated value of c r a c k opening using Equation (14) is generally less than 3 mm for safe tunnels. The calculated spacing of cracks is more than three times the thickness of the lining, and 1.75 m for safe tunnels.

It should be mentioned that the 30- cm-thick reinforced concrete l in ing at the Kopli Hydel project in India failed at a water pressure of 6.5kg/cm ~ (0.65 MPa) where the rock cover was less than 31 m. This failure also damaged the plain concrete l in ing which was constructed in a region of higher rock cover. The proposed analysis shows that even plain concrete l in ing may fail at design water pressure of 16kg/cm 2 ( 1.6 MPa), since the spacing of the c r a c k is predicted to be 40 cm only. Under the above conditions, the small l in ing segments are likely to be washed away with the water flow.

Conc lus ions

Based on the above research, the following conclusions are offered with respect to the design criteria for plain concrete l in ing of water and power tunnels:

(1) Although cracking in the plain concrete l in ing may reduce hoop tension drastically, significant bending moments will be generated within the segment due to development of shear stresses along the interface.

(2) Simple expressions for c r a c k opening, hoop tension and bending moment are considered reasonably accurate for the purpose of l in ing design.

(3) The crack spacing (S) at the outer periphery of the l in ing can be determined as follows:

S : (l, + P) t / p .

The spacing (S) should be greater than three times the l in ing thickness, and 1.75 m.

(4) The average crack opening (2u) in a plain concrete l ining is

(l+u) t(h+p) 2u -

E

2u should be less than 3 mm. []

Refe rences

Cracked water tunnel trims energy supply. 1979. Engineering News Record 202 (22):17.

Jaeger, C. 1972. Rock Mechanics in Engineering. Cambridge, England: Cam- bridge University Press.

Kumar, R., Nayak, G. C. and Singh, B. 1978a. Feasibility of plain concrete tunnel lining. In Proc. Syrup. on Economic and Civil Engineering Aspects o[ Hydroelectric Schemes, v- 12-16. Roorkee, India: Univer- sity of Roorkee.

Kumar, R. Nayak, G. C. and Singh, B. 1978b. Stress distribution in cracked concrete tunnel lining. In Proc. Con. on Geo- technical Engineering, Vol. 1, pp. 300- 303. New Delhi, India: Indian Geo- technica l Society.

Singh, B. 1973. Continuum characterization of jointed rock masses, Part 2. Int. ]. Rock Mechanics Mining Sci. 2:337-349.

Zienkiewicz, O. C. 1967. Finite Element Method in Engineering Science. London: McGraw-Hill Publications.

A p p e n d i x . R i g o r o u s a n a l y s i s o f l i n i n g s e g m e n t

The assumptions inherent in a rigorous analysis of l in ing segments are the same as those that apply in the simple analysis presented in the main body of the paper. However, the following analysis takes into account elastic deformations of the segment.

The solutions derived from this analysis are applicable to rocks characterized by a high modulus of deformation, as well as to l inings using longer segments. Only in extreme cases involving no cracking or only one or two cracks is the proposed solution likely to be in error. In practice, the number of construction joints is much greater. It is for such practical cases that this general solution is proposed.

For simplification purposes, the solution has been derived for the plane stress condition. Because the plane- strain condition will prevail in actual cases, it is necessary to substitute modified values of E~ and vc. That is, in all expressions, Ec and Uc should be

Ec Vc replaced by and

(1 - ~c) (1 - Vc) ' respectively. However, when the effect of Ec andvc is of secondary importance compared to the effects of the rock mass, these modifications may be ignored.

Theory

Figure 6 shows the forces acting on an element of the l in ing segment. These forces are as follows:

Hoop tension (T). Bending moment (M). Shear force (F). Pressure from the inside (p) Normal reaction from the rock (KNw) Shearing resistance from the rock (Kru).

The l in ing will have a tendency to flatten outwards due to a reduction in hoop tension.

Moment Equilibrium

Considering the moment of all forces acting on the element about the center of the lining,

F a dO = KT u(a+t)dO~2 + dM

(A1) dJ~ t

or - - ~ = - KTU(a+t) "~ + F a .

Equilibrium in Radial Direction

Considering the equil ibr ium of components of forces in the radial direction,

dO P a dO = KNW(a+t)dO + 2 T - ~ - dF

(A2) dF

or P a = KNw(a+t) + T - d--O

Equilibrium in Tangential Direction

Considering the equil ibrium of forces along the tangential direction,

dO FdO d T + Kru(a+t)dO + (F+dF)--~-__ - - ~ = 0

(AS)

dT dFd0 or ~ = - KTU(a+t) ; neglecting 2

Compatibility Condition

Considering the compatibility of strains at the interface of the rock and the l ining,

T ucp M 1 - - + + - - - -

tEc E~ Z E~

(a+t+w)dO -(u+du) + u - (a+t)dO =

or T -

(a+t)dO (A4)

Ect du Mt (a+t) (w - - ~ - vc pt - --~ .


0

/ /I / / /

/ I I / / / / / /

/ / /

/ / i / p /

/

t . L

H +dH . p

T

K2W ¢ ¢ 7 KT u

Figure 6. Forces act ing on an e lement o[ tunne l l ining.

Differentiating Equation (A4) with dw Kr respect to 0, the following is obtained: or d--0 = K--'~ u . (A8)

d T Ect dw d2u] t d M d-0 = (a+t) ( ~ -~ - " -Z-~'" (AS)

Comparing Equations (A3) and (AS),

Ect ( d w d2u t d M (-;~) , ~ - dO~ ) : - I~ (a+t) u + ~ d--g"

dM Substituting the value of

Equation (A1) in Equation (A6),

E j ( dw d2u

('a+t) d O - dO 2 )

(A6)

from

t t : - K (a+t)u + Z ( -K -~u(a+t)+F a)

(a7) t ~

= - K u(a+t) (l+ ~ ) (approx.).

Combining Equations (A2) and (A3),

d dw dT d2F d'--O (pa) = 0 = KN (a+t) - ~ + d"O- dO "--7

dw d2F = KN(a+t) - ~ - KTU(a+t); neglecting dO ~

dM Eliminating ~ from Equation (A7)

with the help of Equation (A8),

E~t rZ¢~ d~u 1 (a-~) I. ~ u- dO ~ j

o r

t 2 = - Kr(a+t) (i + ~-~) u

d 2 u d O s - B2u, where, O 2

(A9)

t2 Kr(a+t) 2 +Kr = (1 + - ~-~ ) E j KN"

Boundary Conditions

(1) u=0, a t 0 = 0

(2) T = - p t , at 0 = 0 /2

(3) M = 0 , a t 0 = 0 / 2 .

Solution of the differential equation gives

u = A e ~ - A e -he : A [e ° ° - e ~n°]

(al0) = 2A sinh BO.

From Equation (A4), hoop tension at the 0 .

crack end, i.e. at 0 = ~ ~s

Ect du T = (a+t----~ [w~ - ~ ] - ,,~ pt

Ect o r - pt : ~ [wc - 2 #.,1 c o s h - - ]

SOA =

- vc pt (A11)

wc + (1 - vc) p (a+t) / Ec

2 13 cosh (B0/2)

Radial Deflection of the Lining

Equation (A2) gives the following expression for the radial deflection of the lining:

w = pa / KMa+t ) - T .

0 At the crack end, i.e. at 0 = - 2 '

T = - p t (A12)

so wc = pa / KN(a+t) + p t ~-- p / KN.

Crack Opening

The maximum crack opening will take place at the rock face; the minimum opening, at the lining face. 0.

The average crack opening at 0 = -~ is:

2u = (we + (l - vc) p (a+t) / Ec) tanh O'_._0 2

~- Wc 0 (For small value of 0) .

Thus, it is seen that for all practical purposes, the assumed crack opening in Equation (2) of the simple analysis is valid.

Bending Moment

Equation (A1) gives the value of the moment, provided the shear force F is neglible, as:

d M t -dO = - K r (a+t) ~ ,4 (e ns + e -he)

t so M = - Kr(a+t) ~ A (e t~ + e-n°)~[3 + C

=0, a t 0 = 0 /2

t t~0 so c = Kr(a+t) ~3 A cosh -~-

The maximum bending moment can be expressed as follows:

, 7 M = K r ( a + t ) ~ A (cosh ~ - 1). (AI3)

For small values of 0, (cosh ~-~- 1) O~ 02/8


so M ~ KT(a+t)t Bz 02 B 8

w~+ (1 - Vc) p (a+t)/Ec ×

of hoop tension as follows at 0 : 0:

E,,t M t T - [ w o - 2 A / 3 ] - v c p t - - -

(a+t) Z

2B (1+ B 2 02/8) E~t [30

0 .2 = (a+t) cosh(/3 qJ/2) [Wo cosh -~- ~-- Kr(a+t) t 16

(1-vc) p (a+t) M t

x p / K N + (1 - Uc) p (a+t)/Ec - Wc Ec ] - u~pt- --Z

(1+ ~2 ~.2/8 )

E, tWo fl 2 qj2 (1- vc) Ec ~ p KT(a+t) t 16KN (A14) (a+t) 8 (a+t)

The above expression agrees with t p(a+t) M t - v c p t - Equation (A5) of the simple analysis E< Z given in the main body of this paper.

When 0 is small,

Hoop Tension M t Ectw dJ 2 T = - p t +

Equation (A4) gives a maximum value - - -Z- (a+t) 8

KT KT(a+t) 2 (I +t ~) [ K x - - + E,t 2Z I

t 0'2p P.'¢t -- - p t - z KT(a+t)t 16KN + (a+t------~

d/~ K T p p K T 02 t ~ - - - - + (a+t) (1+ 8 K2;v KN -8- ~ )

~02 Ectp KT 0 '~ = - p t + KT(a+t) 8 + (a+t) K2N 8

(A15) 0 2 p K r (a+t) - pt , if t ~ (a+t) :K--7 Y

The above expression tallies with Equation (4) of the simple analysis given in the main body of the paper.

It may again be noted that the shear stiffness of rock mass K r may be assumed on the basis of Equation (6) given in the simple method.


design criteria for plain concrete lining in water and power tunnels

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