ELSEVIER GEO-ENGINEERING BOOK SERIES VOLUME 5 Part 9 pdf

Critical state rock mechanics and its applications “All things by immortal power near or far, hiddenly to each other are linked.” Francis Thompson English Victorian Post 29.1 GENERAL Bar

Integrated method of tunnelling 441

The short-term support pressure in roof may be assessed by the following correlation(equation (4.6)) for arch opening given by Goel and Jethwa (1991)

proof =7.5B

0.1H0.5− RMR

20 RMR = 7.5 × 160.1× 1650.5− 73

The ultimate support pressure is read by the chart (Fig 5.2) of Barton et al (1974)

as follows (the dotted line is observed to be more reliable than correlation)

proof = 0.9 × 1 × 1 kg/cm2 or 0.09 MPa

(The rock mass is in non-squeezing ground condition (H < 350 Q1/3) and so f′ = 1.0

The overburden is less than 320 m and so f = 1.0.)

It is proposed to provide the steel fiber reinforced shotcrete (SFRS) [and no rock bolts

for fast rate of tunnelling] The SFRS thickness (tfsc) is given by the following correlation(equation (28.1))

tfsc=0.6Bproof

2qfsc =0.6 × 1600 × 0.09

= 16 cm (near portals)

The tensile strength of SFRS is considerd to be about one-tenth of its UCS and so its

shear strength (qsc) will be about, 2 × 30/10 = 6.0 MPa, approximately 5.5MPa (UTS isgenerally lesser than its flexural strength) Past experience is also the same

The life of SFRS may be taken same as that of concrete in the polluted environmentthat is about 50 years Life may be increased to 60 years by providing extra cover of SFRS

of 5 cm If SFRS is damaged latter, corroded part should be scrapped and new layer ofshotcrete should be sprayed to last for 100 years So recommended thickness of SFRS is

tfsc= 13 cm

= 21 cm (near portals)

The width of pillar is more than the sum of half-widths of adjoining openings in thenon-squeezing grounds The width of pillar is also more than the total height of the larger

of two caverns (18 m), hence proposed separation of 20 m is safe

The following precautions need to be taken:

(i) The loose pieces of rocks should be scrapped thoroughly before shotcreting forbetter bonding between two surfaces

(ii) Unlined drains should be created on both the sides of each tunnel to drain out theground water and then should be covered by RCC slabs for road safety

(iii) The tunnel exits should be decorated by art and arrangement should be madefor a bright lighting to illuminate well the tunnels to generate happy emotionsamong road users.

REFERENCES

Barton, N (2002) Some new Q-value correlations to assist in site characterisation and tunnel design

Int J Rock Mech and Min Sci., 39, 185-216.

Barton, N., Lien, R and Lunde, J (1974) Engineering classification of rock masses for the design

of tunnel support Rock Mechanics, Springer-Verlag, 6, 189-236.

Bischoff, J A., Klein, S J and Lang, T A (1992) Designing reinforced rock Civil Engineering,

Goel, R K and Jethwa, J L (1991) Prediction of support pressure using RMR classification

Proc Indian Getech Conf., Surat, India, 203-205.

IS 15026:2002, Tunnelling Methods in Rock Masses-Guidelines Bureau of Indian Standards,

New Delhi, India, 1-24

Samadhiya, N K (1998) Influence of anisotropy and shear zones on stability of caverns.

PhD thesis, Department of Civil Engineering, IIT Roorkee, India, 334

Singh, Bhawani,Viladkar, M N., Samadhiya N K and Sandeep (1995) A Semi-empirical method

for the design of support systems in underground openings Tunnelling and Underground Space Technology, 10(3) 375-383.

Zhidao, Z (1988) Waterproofing and water drainage in NATM tunnel Symp Tunnels and Water,

Ed: Serrano, Rotterdam, 707-711

Critical state rock mechanics

and its applications

“All things by immortal power near or far, hiddenly to each other are linked.”

Francis Thompson English Victorian Post

29.1 GENERAL

Barton (1976) suggested that the critical state for initially intact rock is defined as thestress condition under which Mohr envelope of peak shear strength reaches a point of zerogradient or a saturation limit Hoek (1983) suggested that the confining pressure mustalways be less than the unconfined compression strength of the material for the behavior

to be considered brittle An approximate value of the critical confining pressure may,therefore, be taken equal to the uniaxial compressive strength of the rock material

Yu et al (2002) have presented a state-of-the-art on strength of rock materials andsuggested a unified theory The idea is that the strength criterion for jointed rock massmust account for the effect of critical state in the actual environmental conditions.The frictional resistance is due to the molecular attraction of the molecules in contactbetween smooth adjoining surfaces It is more where molecules are closer to each other due

to the normal stresses However, the frictional resistance may not exceed the molecularbond strength under very high confining stresses Hence, it is no wonder that there is asaturation or critical limit to the frictional resistance (Prasad, 2003) There should be limit

to everything in the nature

Singh and Singh (2005) have proposed the following simple parabolic strengthcriterion for the unweathered dry isotropic rock materials as shown in Fig 29.1

σ1− σ3= qc+ Aσ3−Aσ

2 3

2qc for 0 < σ3 ≤ qc (29.1)

Tunnelling in Weak Rocks

B Singh and R K Goel

qc = average uniaxial compressive strength of rock material at σ3= 0.

It may be proved easily that deviator strength (differential stress at failure) reaches asaturation limit at σ3= qcthat is,

∂(σ1− σ3)

Unfortunately, this critical state condition is not met by the other criteria of strength

It is heartening to note that this criterion is based on single parameter “A” which makes

a physical sense Sheorey (1997) has compiled the triaxial and polyaxial test data fordifferent rocks which are available from the world literature The regression analysis wasperformed on 132 sets of triaxial test data in the range of 0 ≤ σ3≤ qc

The values of the parameter A, for all the data sets were obtained These values were

used to back-calculate the σ1values for the each set for the given confining pressure Thecomparison of the experimental and the computed values of σ1is presented in Fig 29.2

It is observed that the calculated values of σ1are quite close to the experimental values

An excellent coefficient of correlation, 0.98, is obtained for the best fitting line betweenthe calculated and the experimental values

For comparing the predication of the parabolic criterion with those of the others,Hoek and Brown (1980) criterion was used to calculate the σ1values The coefficient ofcorrelation (0.98) for Hoek–Brown predictions is observed to be slightly lower and poor forweak rocks, when compared with that obtained for the criterion proposed in this chapter

Critical state rock mechanics and its applications 445

10 100 1000 10000 10

100 1000

In addition to the higher value of coefficient of correlation, the real advantage of proposed

criterion, lies in the fact that only one parameter, A is used to predict the confined strength

of the rock and A makes a physical sense.

A rough estimate of the parameter A may be made without conducting even a single triaxial test The variation of the parameter, A, with the uniaxial compressive strength (UCS), qc is presented in Fig 29.4 A definite trend of A with UCS (qc) is indicated by

this figure and the best fitting value of the parameter A may be obtained as given below:

A ∼= 7.94

q0.10 c

Fig 29.3 compares experimental σ1 values with those predicted by using tion (29.3) without using the triaxial data A high coefficient of correlation of 0.93 isobtained Thus, the proposed criterion appears to be more faithful to the test data thanHoek and Brown (1980) criterion This criterion is also better fit for weak rocks as thecritical state is more important for rocks of lower UCS The law of saturation appears to

equa-be the cause of non-linearity of the natural laws

29.2 SUGGESTED MODEL FOR ROCK MASS

The behavior of jointed rock mass may be similar to that of the rock material at cal confining pressure, as joints then cease to dominate the behavior of the rock mass

criti-10 100 1000 10000 10

100 1000

0.0

400

Fig 29.4 Variation of parameter A/2qcwith UCS of the intact rocks (qc)

Critical state rock mechanics and its applications 447

Therefore, one may assume that the deviator strength will reach the critical state at σ3= qc

As such, this critical confining pressure may be independent of the size of specimen.Perhaps the deviator strength may also achieve a critical state when intermediate princi-pal stress σ2∼= qc(equation (8.2)) Thus an approximate simple parabolic and polyaxialcriterion is suggested for the underground openings as follows,

qcmass = uniaxial compressive strength of the rock mass,

γ = unit weight of rock mass in gm/cc or t/m3,

Q = post-construction Barton’s rock mass quality just before

supporting a tunnel,

φp = the peak angle of internal friction of a rock mass and

qc = average uniaxial compressive strength (UCS), upper bound of UCS for

anisotropic rock material of jointed rock masses under actual environment

It may be verified by differentiating equation (29.4) that

parameter A was computed by the least square method as was done for the rock materials.

An approximate correlation between A and qcwas found as given below

The polyaxial tests on cubes of jointed rocks at IIT Delhi suggest that the mode offailure at high σ2is brittle and not ductile as expected This is seen in tunnels in medium

to hard rocks

The angle of internal friction (φp) in equation (29.6) may be chosen from the correlation

of Mehrotra (1993) (cited by Singh & Goel, 2002), according to RMR both for the nearlydry and saturated rock masses (Fig 29.5) It is based on the extensive and carefullyconducted block shear tests at various project sites in the Himalaya It may be seenthat φpis significantly less for the saturated rock mass than that for the nearly dry rock

mass for the same final RMR So the parameter A will be governed by the degree of

Rock Mass Rating (RMR)

Fig 29.5 Relationship between rock mass rating (RMR) and angle of internal friction (φp)(Mehrotra, 1992) [nmc: natural moisture content]

saturation in equation (29.7) Hoek and Brown (1997) have developed a chart betweenfriction angle φpand geological strength index (GSI = RMR − 5 for RMR ≥ 23) for the

various values of rock material parameter mr It is seen that φpincreases significantly

with increasing value of mrfor any GSI In the case of rock mass with clay-coated joints,equation (29.13) may be used to estimate φpapproximately Equation (29.13) takes intoaccount approximately the seepage erosion and piping conditions in the weak rock masses(Barton, 2002) Seepage erosion (flow of soil particles from joints due to the seepage,especially during rainy seasons) rapidly deteriorates the rock mass quality (Q) with time.Seepage may be encountered at great depths even in granite unexpectedly, due to the

presence of a fault “Uncertainty is the law of nature.”

It should be mentioned that Murrell (1963) was the first researcher who predicted thatmajor principal stress (σ1) at failure increases with σ2significantly, but it reduces when

σ2is beyond σ1/2 and σ3 = 0 The three sets of polyaxial test data cited by Yu et al.(2002) shows a negligible or small trend of peaking in σ1 when σ2 ≫ qc The attemptwas made to fit in the proposed polyaxial strength criterion (equation (29.4)) in the abovetest data for rock materials (Dunham dolomite, trachite and coarse gained dense marble).The recent polyaxial test data on tuff (Wang and Kemeny, 1995) was also analyzed Theequation (29.4) was found to be fortunately rather a good fit into all the polyaxial test data

at σ2< qcand σ3< qc

Critical state rock mechanics and its applications 449

Kumar (2002) has collected data of 29 km NJPC tunnel in gneiss in Himalaya as

mentioned in Table 29.1 It may be noted that the rock mass strength (qcmass) is too lessthan the expected tangential stress (σθ) along the tunnel periphery The qcmassfrom linearcriterion is some what less than σθ It is interesting to know that the parabolic polyaxial

criterion predicts the rock mass strength (q′′

cmassin equation (29.8)) in the range of 0.64

to 1.4 σθ generally In one situation, the rock mass was found to be in the critical statelocally It matches with the failure conditions in the tunnel beyond overburden of 1000 mwhere (mild rock burst or) spalling of rock slabs was observed So the proposed simplepolyaxial strength criterion (equation (29.4)) fits in the observations in the tunnels in thecomplex and fragile geological conditions in a better way than other criteria Better fitalso suggests that the peak angle of internal friction of rock mass may be nearly the same

as that for its rock material in the case of unweathered rock mass

The equation (29.4) suggests the following criterion of failure of rock mass aroundtunnels and openings (σ3= 0 on excavated face and σ2= Poalong tunnel axis),

σθ > q′′cmass= qcmass+A · Po

2 o

4 · qc ≤ qcmass+A · qc

where q′′cmassis the biaxial compressive strength of rock mass, corrected for greater depths

29.3 RESIDUAL STRENGTH

Mohr’s theory will be applicable to residual failure as a rock mass would be reduced

to non-dilatant soil like condition Thus, residual strength (σ1− σ3)r of rock mass islikely to be independent of the intermediate principal stress So, the following criterion issuggested

(σ1− σ3)r= qcr+ Ar· σ3−Ar· σ

2 3

No H (m) Tunnelling (MPa) (deg) = 2 sin φp

1−sin φ p 7γQ1/3(MPa) Po= γH qcmass+APo

Critical state rock mechanics and its applications 451

cross-check between the proposed theory and the observed support pressures in the ing ground conditions except in a few cases Thus residual cohesion was back-analyzed

squeez-to be 0.1 MPa approximately and zero where deviasqueez-toric strain exceeded 10 percent ilarly, the residual angle of friction was inferred to be about 10◦less than the peak angle

Sim-of internal friction but more than 14◦

This model explains the likely mode of failure of rock mass in the deep tunnels For

example, severe rock burst condition may be encountered where A or φpis high (where

Jr/Ja > 0.5) It is because the peak deviator strength or differential stress (σ1− σ3) isvery high compared to the residual deviator strength (σ1− σ3)r The locked-up strainenergy may be dissipated in the form of seismic waves On the other hand, the squeezing

condition or plastic failure may develop where A or φpis very low, as there may not beany significant difference in the peak and residual deviatoric strengths

As such, the following criteria for the heavy rock burst in deep tunnels in the hardrocks is suggested

Thus severe rock burst conditions may develop in hard rocks which has entered into the

critical state (Po> qc) and where the overburden (H ) exceeds the limit of equation (29.16).

It is assumed that the ratio of in situ minimum principal stress and overburden pressure (K )

is about 0.4 ± 0.10 at great depths in equation (29.16)

The thermic zones (of high temperatures) are also likely to be encountered at greaterdepths So the tunnel face may have to be air-conditioned like in very deep mines (KolarGold Field, India) The efficiency of the workers is very low and they cannot work formore than a few hours under high temperatures In addition, the thermic zone may also

be in critical state and the tunnelling hazard may be doubly serious So it will be better torealign the tunnel to by-pass the rock mass in at least the critical state or in thermic zone

or both, to be on the safe side and avoid severe tunnelling hazards

In case of rock mass in the critical state, pre-tensioned rock bolts or resin grouted boltsmay be effective, as radial stresses will be released It is also suggested that one may try touse thick SFRS (steel fiber reinforced shotcrete) lining which has good bond with the rockmass and its compressive strength is higher than that of the rock material Resin anchorsmay make rock mass ductile

The understanding of critical state rock mechanics is essential for (i) deep tunnelsfor the underground nuclear waste disposal and (ii) also in the petroleum engineering

Very deep drill-holes may be unstable in the weak rock layers (shale) in the critical

state (H > Hcr)

29.4 EFFECT OF CONFINING PRESSURE ON FRICTION ANGLE

Sometimes a fault or thrust passes through deep tunnels A non-linear analysis is preferred

by the incremental method The Mohr’s envelope based on equation (29.1) gives the slope

of angle of friction (φ) at any initial stress condition The shear strength criterion for fault

in terms of increase in shear strength (∆τ) for expected rise in the effective normal stress(∆σ) is simply as follows:

where ∆u is increase in seepage pressure.

If ∆τ exceeds the incremental shear strength of weaker rock adjoining to a fault underhigh confining stresses, earthquakes may occur in that area

The friction angle (φ) along any point of deep-seated fault may be predicted accuratelyfrom the following empirical equation for tan φo < 2 and any value of UCS > 10 MPa

which is derived from equation (29.1) (Singh et al., 2004)

in the large earth plates and so lesser are the chances of great earthquakes in that area.Infact, highest earthquake of only M7 on Richter’s scale had taken place in the Tibetanplateau Thus there is a balancing mechanism in the nature to avoid too high intensity ofearthquakes in a planet

In weak rocks, the high confining stress may reduce porosity of rock material and so

increase its UCS (qcin equation (29.18)) For all practical purposes, the coefficient of tion (∆τ/∆σ) may be assumed to be negligible beyond a confining stress of UCS at σ3= qc

fric-REFERENCES

Arora, V K (1987) Strength and deformational behaviour of jointed rocks PhD thesis, IIT Delhi,

India

Critical state rock mechanics and its applications 453

Barton, N (1976) Rock mechanics review: The shear strength of rock and rock joints Int J Rock Mech Min Sci & Geomech Abstr., 13, 255-279.

Barton, N (2002) Some new Q-value correlations to assist in-situ characterisation and tunnel

design Int J Rock Mechanics and Mining Sciences, 39(2), 185-216.

Brown, E T (1970) Strength of models of rock with intermittent joints J of Soil Mech & Found Div., Proc ASCE, 96(SM6), 1935-1949.

Brown, E T and Trollope, D H (1970) Strength of a model of jointed rock J of Soil Mech & Found Div., Proc ASCE, 96(SM2), 685-704.

Einstein, H H and Hirschfeld, R C (1973) Model studies on mechanics of jointed rock J of Soil Mech & Found Div Proc ASCE, 90, 229-248.

Hoek, E (1983) Strength of jointed rock masses Geotechnique, 33(3), 187-223.

Hoek, E (1980) An empirical strength criterion and its use in designing slopes and tunnels in

heavily jointed weathered rock Proc 6th Southeast Asian Conf on Soil Engg, 19-23, May,

Ladanyi, B and Archambault, G (1972) Evaluation of shear strength of a jointed rock mass

Proc 24th Int Geological Congress, Montreal, Section 13D, 249-270.

Mehrotra, V K (1992) Estimation of Engineering Parameters of Rock Mass PhD Thesis, IIT

Roorkee, India, 267

Murrell, S A K (1963) A criterion for brittle fracture of rocks and concrete under triaxial stress and

the effect of pore pressure on the criteria Vth Sym on Rock Mech., University of Minnesota,

Ed: C Fairhurst, Oxford, Pergamon, 563-577

Prasad, Rajendra (2003) Personal communication, IIT Roorkee.

Roy, N (1993) Engineering Behaviour of Rock Masses Through Study of Jointed Models.

PhD thesis, IIT Delhi, India

Singh, Bhawani and Goel, R K (2002) Software for Engineering Control of Landslides and Tunnelling Hazards Chapter 6, A A Balkema (Swets & Zetpinger), The Netherlands, 344.

Singh, Mahendra and Singh, Bhawani (2005) A strength criterion based on critical state mechanics

for intact rocks Rock Mechanics and Rock Engineering, 38(3), 243-248.

Shankar, Daya, Kapur, N and Singh, Bhawani (2002) Thrust-wedge mechanics and doeval opment of normal and reverse wedge faults in the Himalayas J Geological Society, London,

devel-159, 273-280

Sheorey, P R (1997) Empirical Rock Failure Criteria A A Balkema, The Netherlands, 176.

Singh, Bhawani, Shankar, D., Singh, Mahendra, Samadhiya, N K and Anbalagan, R N (2004)

Earthquake rick reduction by lakes along active faults 3rd Int Conf on Continental Earthquakes, III ICCE, APEC, Beijing, China, July, 11-14.

Wang, R and Kemeny, J M (1995) A new empirical failure criterion under polyaxial compressive

stresses, Eds: J J K Daemen and A Richard, Schultz, Proc 35th U.S Symposium on Rock Mechanics, A A Balkema, 950.

Yaji, R K (1984) Shear strength and deformation response of jointed rocks PhD thesis,

IIT Delhi, India

Yu, Mao-Hong, Zan, Yzee-Wen, Zhao, Jian and Yoshimine, Mitsutoshi (2002) A unified strength

criterion for rock material Int Journal of Rock Mechanics and Mining Sciences, 39,

975-989

The following assumptions are made in the analysis (Fig AI.1):

(i) The rock mass is homogeneous, isotropic, dry, linearly elastic and infinite medium

(ii) The tunnel is circular in shape having radius a.

(iii) The in situ stress field is homogeneous and non-hydrostatic The vertical stress is P and the horizontal stress is λP The vertical stress P is generally assumed equal to the overburden pressure (γH) The horizontal in situ stress is higher than vertical

stress in most of the cases in civil engineering projects

(iv) The modulus of elasticity (or deformation Ed) is the same in loading and unloadingcondition In other words, there is no stress induced anisotropy

Terzaghi and Richart (1952) have derived the solution with respect to the stressedstate The compressive stresses are positive The radial displacement is positive in thedirection of radius vector The following solution is for plane stress case

Tunnelling in Weak Rocks

B Singh and R K Goel

45° θP

λP

(3λ-1)P

(3-λ)P

1 2

Fig AI.1 Stress concentration around circular tunnel in non-hydrostatic state of in situ stress

σr= P

2

(1 + λ)



1 − a2

r2

+ (1 − λ)



1 + 3a4

r4 −4a

2

r2

cos 2θ



1 + a2

r4

cos 2θ



1 − a2

r2cos 2θ

+ 4(1 − ν2) cos 2θ



1 +a2

It is interesting to note that the stress distribution is independent of modulus of

deformation (Ed), as expected It should be noted that the radial displacement ur is half

of that in opening in stress-free state Sometimes extensometers are installed to monitor

Tunnel mechanics 457

relative displacements u12 between point 1 at r = a and point 2 at r = r2(Fig AI.1)

Equation (AI.5) gives the expression for u12

Let us suppose that the tunnel is internally pressurized by radial stress pi The solutionfor induced additional stresses is simple as follows:

It may be seen that tangential stresses are tensile in nature

Since the principle of superposition is valid in the elastic body, the final solution forinternally pressurized tunnel is the sum of equations (AI.1) and (AI.6) for radial stresses;and equations (AI.2) and (AI.7) for tangential stresses

In the special case of hydrostatic in situ stresses, the final solution is simple as shown

in Fig AI.2 It is fascinating to note that the sum of radial and tangential stresses is equal

to twice the vertical in situ stress everywhere in the rock mass

σr = P



1 −a2

Recently, Carranza and Fairhurst (2000) presented the following empirical equation

for estimation of radial displacement urat a distance x ahead of tunnel face (x ≤ 0) and

behind the face (x > 0) towards portal in the case of circular tunnel within a hydrostatic

in situ stress field

Tunnel mechanics 459

AI.2 PROPOSED ELASTO-PLASTIC THEORY

OF STRESS DISTRIBUTION IN BROKEN ZONE

IN SQUEEZING GROUND

Fig AI.3 shows a concentric circular broken zone or plastic zone within squeezingground The following assumptions are made to get approximate solutions for the stressdistribution

(i) Rock mass is isotropic and homogeneous and dry Tunnel is circular having

Elastic Zone

λP λP

P

Fig AI.3 Stress distribution in squeezing ground

α = 2 sin φr

1 − sin φr

qcr = 2 crcos φr

1 − sin φrtan φp= (Jr/Ja) + 0.1

cr= 0.1 MPa

φr= φp− 10◦≥ 14◦

(iv) Broken zone is circular and concentric with the tunnel, and the gravity is assumed

to act radially for simplifying analysis The tunnel is supported uniformly andthere are no rock bolts

(v) In situ principal stress along tunnel axis is Po The vertical in situ stress is P and horizontal in situ stress is λP.

(vi) Tunnel supports are provided and they exert support pressure pvand phin thevertical and horizontal directions, respectively

(vii) There is no rock burst or brittle failure [(Jr/Ja) < 0.5]

The proposed analysis is more rigorous than those suggested by other researchers tothe best knowledge of the authors It is partially verified by case histories in Himalaya

Similarly, the roof support pressure is obtained from equations (AI.19, AI.24) (r = a):

pv=
(3λ − 1) P − qcmass− Po· A/2

2 + (A/2) + crcot φr

a b

α−1

− 1



(AI.30)

The equations (AI.25b) and (AI.28) are same as derived by Daemen (1975) for λ = 1

except the expression for radial stress pbat the outer boundary of the broken zone It is

easy to derive the complicated expression for pb using Hoek and Brown’s criterion.The same may be substituted in equations (AI.24), (AI.25a) and (AI.29) to get supportpressures It may be noted that γ is negative at the bottom of a tunnel

AI.3 SHORT-TERM SUPPORT PRESSURE ON CLOSELY

SPACED TUNNELS IN SQUEEZING GROUND CONDITION

Sometimes several tunnels are required in a hydelproject to carry a given quantity ofwater because the size of a single tunnel becomes too large to be economical Besides theconsiderations of economy, these tunnels may have to be closely spaced due to constraints

of topography on a river side It is, therefore necessary to analyze stresses on the rockpillar between two adjacent tunnels (or road tunnels) Jethwa (1981) derived the followingclosed form elasto-plastic solution

The following assumptions have been made:

(i) The tunnels are circular in shape and their center to center spacing is 2b (Fig.

AI.4)

(ii) The rock mass around the tunnels has failed due to overstressing and the thickness

of the broken zone of rock mass is equal to half the pillar width (Fig AI.4).The stress distribution has been assumed to be axi-symmetric around each tunnel

(iii) The short-term support pressure on the support is phoin the horizontal direction.(iv) The effect of gravity is neglected

(v) There are no cross openings connecting the tunnels

The relationship between the tangential stress σbro

rbroken zone according to Coulomb’s law is given by,

σbroθ = σbror 1 + sin φr

1 − sin φr + 2 · cr

cos φr

1 − sin φr

(AI.31)

Tunnel mechanics 463

Broken Zone Broken Zone

P Elastic Zone

Pillar

a ba

Fig AI.4 Stresses on the pillar between two adjacent tunnels in squeezing ground condition

The radial stress within the broken zone (Daemen, 1975) is given by equation (AI.23)

The strength (Sp) of the pillar per unit length may be determined by integrating thetangential stress σbro

Equation (AI.42) may be used to determine the short-term horizontal support pressure

(Pho) for the design of supports in the case of two adjacent tunnels using an appropriatefactor of safety (about 3) for the pillar Alternately, the factor of safety may be calculated

if Pho is known Equation (AI.42) has actually been adopted in the design of the main

AI.4 SEISMIC SUPPORT PRESSURES

Shotcrete lining is found to fail due to horizontal seismic support pressure within a clayeyfault zone Equation (AI.28) suggests that the horizontal body force αh· γ may act towardstunnel center and a vertical body force may act vertically downwards during earthquake.Thus the additional seismic support pressures may be of the following order,

phseismic= αh· γ · Mγ· (b − a) (AI.43)

pvseismic= αv· γ · Mγ· (b − a) (AI.44)where

phseismic = horizontal seismic support pressure for squeezing ground,

pvseismic = vertical seismic support pressure for squeezing ground,

αh = coefficient of horizontal peak ground acceleration at level of tunnel during

earthquake and

αv = coefficient of vertical peak ground acceleration at level of tunnel during

earthquake

REFERENCES

Carranza Torres, C and Fairhurst, C (2000) Application of the convergence-confinement method

of design to rock masses that satisfy Hoek-Brown failure criterion Tunnelling and Underground Space Technology, 15, 187-213.

Jaeger, J C and Cook, N.G.W (1969) Fundamentals of Rock Mechanics Methuen and Co Ltd.

Art.5.3, 513

Terzaghi, K and Richart, F E (1952) Stresses in rock around cavities Geotechnique, 3, 57-99 Jethwa, J L (1981) Evaluation of rock pressures in tunnels through squeezing ground in lower Himalayas PhD thesis, Department of Civil Engineering, IIT Roorkee, India, 272 Daemen, J J K (1975) Tunnel Support Loading caused by Rock Failure PhD thesis, University

of Minnesota, Minneapolis, U.S.A