an experimental study to predict ground response curve for tunnels by singh

1

1. INTRODUCTION Closed form solutions are available for predicting the closure of a circular tunnel in an isotropic elastic rock mass [1]. It is, however, a challenging task to assess the rock mass modulus, which is an important input parameter in the analysis. Geological formations at tunnel sites contain joints, and the deformability properties of jointed rocks (or rock masses) are greatly influenced by the characteristics of the joints. Kulatilake et al. [2] discussed various approaches to assess the modulus of these rock masses. It was emphasised that joint geometry plays an important role in defining this parameter. A concept was presented [2] which enables the reduction in the modulus of a rock mass to be defined in terms of the geometrical characteristics of the joints. The concept involves a weakness coefficient, referred to as Fracture Tensor, which is defined in terms of these characteristics. A similar approach was presented by Ramamurthy et al. [3-7] who defined a Joint Factor coefficient.

These two studies considered jointed rocks (or rock masses) to be in unconfined and confined states. Negligible studies have been carried out to consider the applicability of these approaches when used to predict ground response curves for tunnels. The current paper highlights an experimental study, in which experimental ground response curves are obtained from laboratory data, enabling the applicability of the Fracture Tensor and Joint Factor concepts to be considered. Only non-squeezing elastic ground conditions are considered. Squeezing ground conditions, involving large plastic deformations, will be discussed in a subsequent publication.

2. WEAKNESS COEFFICIENTS 2.1. Fracture Tensor Kulatilake et al. [2, 8] defined the Fracture Tensor weakness coefficient for fully persistent smooth joints as:

AN EXPERIMENTAL STUDY TO PREDICT GROUND RESPONSE CURVE FOR TUNNELS

Singh, Mahendra Department of Civil Engineering, IIT Roorkee- 247667, INDIA Choudhari, J. AF-Colenco Ltd., Noida 20130, UP, INDIA

Samadhiya, N.K. Department of Civil Engineering, IIT Roorkee- 247667, INDIA

ABSTRACT: The closure analysis of a tunnel is carried out with the help of ground response curves. A correct understanding of rock mass deformability properties is essential to carry out this analysis. The deformability properties of rock masses are greatly influenced by the characteristics of joints present in the rock mass. It is, therefore, necessary to incorporate the joint characteristics in tunnel-support interaction analysis. Attempts have been made in the past to define rock mass modulus in terms of intact rock modulus and a weakness coefficient. Examples of these approaches are Fracture Tensor and Joint Factor concepts. The rock mass modulus is then determined on the basis of data obtained from laboratory tests on intact rock specimens and from joint mapping in the field. There has been little effort in the laboratory or field, to validate the applicability of these studies to predict the ground response curves for tunnels. The present paper highlights an experimental study in which the ground response curves for jointed rock masses are obtained in the laboratory and the validity of the weakness coefficients concept for predicting the ground response curves is studied. Based on the outcome of the study, correlations have been proposed to obtain rock mass modulus to be used in tunnel - support interaction analysis.

Keywords: Tunnels, Ground response curve, Rock mass, Modulus

Proceedings of the International Conference on Rock Joints and Jointed Rock Masses, Tucson, Arizona, USA, January 7-8, 2009

2

( )=

=N

1mm

22xx sinrF (1)

where is the joint density (average number of joints per unit area); r is the joint size; and is the angle between the joint plane and the horizontal axis (assuming the major principal stress to be vertical). Kulatilake et al. [2] suggested the following relationship between the rock mass modulus Em in any direction and the fracture tensor component in that direction as:

1F726.0EE 613.0

k

im

+= (2)

where Ei is the intact rock modulus; Em is the rock mass modulus; and Fk is the Fracture tensor component in the appropriate direction. Equation 2 may be used to estimate the anisotropic rock mass modulus in any direction. 2.2. Joint Factor Ramamurthy et al. [3-7] suggested that the most important characteristics of joints which affect the rock mass modulus are frequency, orientation and shear strength. The Joint Factor concept, which considers these parameters, was derived from the results obtained from uniaxial compression tests strength tests on jointed cylindrical specimens of artificial and natural rocks. The orientation of the joints, joint frequency and surface roughness were varied during the tests. An empirical parameter called the joint inclination parameter, n (Table 1), was defined to represent the effect of the orientation of the joints on the response of the jointed rocks to loading. The Joint Factor was defined as:

rn

JJ nf = (3)

where Jn is the number of joints per metre length in the direction of the major principal stress; n is the joint inclination parameter (Table 1) and r is the joint strength parameter. Direct shear tests between the mating joint surfaces were recommended to compute the latter value defined as: r = tan j (4) where j is the joint friction angle at low normal stress levels.

Based on extensive laboratory test results, Ramamurthy [4] defined the rock mass modulus in terms of the intact rock modulus as:

)J0115.0exp(EE fim = (5) where Ei is the intact rock modulus and Em is the rock mass modulus.

Table 1: Inclination parameter, n [4] Orientation of joint o

Inclination parameter n

Orientation of joint o

Inclination parameter n

0 10 20 30 40

0.810 0.460 0.105 0.046 0.071

50 60 70 80 90

0.306 0.465 0.634 0.814 1.000

: Angle between joint plane and loading direction

Singh [6, 7] conducted uniaxial compressive strength tests on 76 large specimens of a jointed block mass and varied the orientation, interlocking level and spacing of joints within the test specimens to achieve various modes of failure. Four modes of failure were observed to be dominant i.e. (i) splitting of intact material, (ii) shearing of intact material, (iii) rotation of elemental blocks and (iv) sliding along the critical joints. It was shown that the deformation modulus of the rock mass also depends on the failure mode in addition to the frequency, orientation and joint shear strength. The following expressions were suggested for rock mass modulus: Em = Ei exp (-0.020 Jf), for splitting/shearing (6) Em = Ei exp (-0.035 Jf), for sliding (7) Em = Ei exp (-0.040 Jf), for rotation (8) The approaches discussed above for estimating the rock mass modulus were developed from studies conducted on rocks under uniaxial loading conditions. However, the rock mass around the periphery of an underground opening is partly constrained and hence does not have same degree of freedom to dilate. The mobilized value of rock mass modulus may, therefore, differ from that derived from uniaxial tests. Direct applicability of these approaches is therefore doubtful, and there is a need to validate their applicability through physical model tests. An experimental study was, therefore, planned and executed in which the closure of an opening in a simulated jointed rock mass was monitored to predict the ground response curves.


3

3. EXPERIMENTAL PROGRAMME The experimental study involved the use of elemental blocks to simulate the specimens of jointed rock mass. The rock mass specimens contained two sets of joints and a D-shaped opening supported by rigid supports. The rock mass specimens were loaded in two directions thereby causing incremental closure as the supports deformed. Variations in the support pressure due to closure of the opening were monitored. The experimental programme involved the following steps:

(i) Design and fabrication of a large sized reaction frame capable of applying stresses in two directions.

(ii) Selection and characterization of a suitable model material to represent a low strength rock (Table 2).

(iii) Preparation of rock mass specimens: The rock mass specimens were formed from elemental blocks. The size of the specimens was 750 x 750 x 150 mm, and that of the elemental blocks was 25 x 25 x 75mm. Approximately1800 to 2000 elemental blocks were required to form one specimen of rock mass (Figure 1).

(iv) Formation of a D-shaped tunnel: Several trials were undertaken to simulate a tunnel being excavated in the field. This procedure was, however, unsuccessful as it was difficult to control and measure the closure of the drilled opening. Finally, it was decided to leave a cavity (including wall and roof supports) in the mass while assembling the elemental blocks. The tunnel was 150 mm wide x 215 mm high. The ratio of opening size to the rock mass specimen size was 1:5. The rigid supports contained load cells to measure support pressure. A nut-bolt arrangement allowed the rigid supports to deform.

(v) Application of stresses and development of experimental response curves as explained in subsequent sections.

(vi) The tests were conducted for different combinations of joint orientations and stress ratios (Table 3). A total of 10 specimens were tested.

(vii) 3.1. Application of Stresses The stresses were generated by applying loads simultaneously in both directions using hydraulic

jacks. The desired horizontal and vertical stresses (ph and pv) were reached in five to six load increments. At each increment, the horizontal and vertical deformations of the specimen were monitored using dial gauges. The loads were maintained at constant level until the dial gauge readings stabilised. Generally it took approximately one hour to achieve stabilisation. These readings represented the deformations of the specimen applicable to the applied stresses. The supports were kept stationary while the stresses were applied and the support pressures (internal pressures) were monitored.

Test Specimen

Reaction frame

VD1VD2

HD1

HD2

HydraulicJack

Loading platen

Load

ing

plat

enVLc

HLc

WS

RS

Reaction frame

Reac

tion

fra

me

Reac

tion

fram

e

Spac

er

VD1 - vertical dialgauge 1 VD2 - vertical dialgauge 2HD1 - horizontal dialgauge 1 HD2 - horizontal dialgauge 2

RS - roof of opening WS - wall of opening

VLc - Vertical Load Cell HLc. - Horizontal Load Cell

openingmodel

HydraulicJack

Figure1. Rock mass specimen with tunnel model

Table 2: Engineering and physical properties of the model material

Sr. No. Property Value

1 Dry density, (kN/m3) 18.5 2 Porosity (%) 36.2 3 Specific gravity 2.5 4 UCS, ci (MPa) 7.0 5 Brazilian strength, ti (MPa) 1.3 6 Intact rock cohesion, ci (MPa) 2.0 7 Intact rock friction angle, i 33.0 8 Tangent modulus, Ei (MPa) 2200.0 9 Friction angle along the joints, j 39.0 10 Deere and Miller classification [9] EM


4

Table 3: Joint configurations and in-situ stress conditions

Sr. n

o.

Join

t o

rien

tatio

ns

Ho

rizo

nta

l in-

situ

stre

ss, p h

(MPa

) V

ertic

al in

-

situ

stre

ss, p v

(MPa

) St

ress

ra

tio

p h/p

v

1 00/90 1.72 1.72 1.00 2 00/90 2.13 1.60 1.33 3 00/90 2.13 1.07 2.00 4 00/90 1.54 0.21 7.33 5 45/45 2.22 2.22 1.00 6 45/45 2.13 1.60 1.33 7 45/45 1.78 0.89 2.00 8 30/60 2.13 2.13 1.00 9 30/60 2.13 1.60 1.33 10 30/60 2.13 1.07 2.00

The subsequent increments of stress were then applied and the corresponding deformations recorded. These steps were followed until the maximum values of the stresses, ph and pv had been achieved. The strains in the specimen were then computed. Typical stress-strain plots from test-1 (00/90; SR-1.0) are presented in Figure 2. These plots were used to obtain the rock mass modulus applicable to the loading condition. Plane stress conditions were assumed to be applicable. Deformations under such conditions may be obtained by modifying the modulus and Poissons ratio values.

0

0.4

0.8

1.2

1.6

2

0.00 0.20 0.40 0.60 0.80Strain, %

Stre

ss, M

Pa

HorizontalVertical

Figure 2. In-situ stress-strain plots for test-1 (00/90; SR-1.0)

3.2. Experimental Ground Response Curves The final stresses (ph and pv) were maintained for 24 hours, and the internal support pressure values (pih and piv) were monitored during this period. The latter values stabilised within 24 hours. These

values were assumed to represent the initial internal pressure values corresponding to zero closure of the tunnel. Small incremental closures were then achieved in both the horizontal and vertical directions by displacing the supports using the nut and bolt arrangement. The internal support pressures instantly fell as a result of these closures, however, they increased with time and then stabilised within approximately 24 hours. The resulting support pressures were lower than the initial values as the supports had had displaced. The support pressures were recorded as were the respective closures in the wall and roof of the tunnel. Subsequent increments of closures were then allowed in both directions simultaneously, and the stabilised internal support pressures were again recorded after 24 hours. These steps were repeated until the support pressures had reduced to zero i.e. equivalent to the condition of an unsupported opening. Generally 15 to 20 days were required to complete one test. The results of these tests were summarised in the form of ground response curves. A typical set of curves from test -1 (00/90; SR-1) is presented in Figure 3.

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Closure, %

Pres

sure

, M

Pa

WallRoof

Figure 3. Ground response curves for test-1 (00/90; SR-1.0)

4. THEORETICAL GROUND RESPONSE CURVES 4.1. Closed Form Solutions Closed form solutions to estimate closure of circular openings subjected to far field stresses and internal pressure in elastic ground conditions are available in Bray [1] and Singh and Goel [10]. These


5

solutions were used to predict theoretical ground response curves for the model tested in this study. Assume a circular underground opening is subjected to far field stresses, ph and pv, and uniform internal pressures, pih and piv in horizontal and vertical directions respectively. Also assume that the rock mass behaves elastically and the principles of superposition of stresses and strains are valid. The resulting circumferential stress at the wall ( = 0) and the roof ( = 90) may be obtained as:

hvvivihw pp3p4p +pi

= (9)

vhhihivr pp3p4p +pi

= (10)

where w is the circumferential stress at the wall;

r is the circumferential stress at the roof; h is half of the angle subtended at the centre of the opening by the arc of the circle at which the internal horizontal pressure pih is acting and v is half of the angle subtended at the centre by the arc of the circle at which the internal vertical pressure piv is acting. The resulting radial deformations at the wall or roof of the opening may be estimated by summing the deformations due to far field stresses and, horizontal and vertical internal pressures. The radial deformation due to the far field stresses may be obtained as:

[ ][ ]++

++=

2cos)1(4)2cos1)(1(E2

ap

2cos)1(4)2cos1)(1(E2

apu

2

m

v

2

m

hr

(11) where ur is the radial deformation at the periphery of the opening; is the direction in which radial deformation is required (0 for wall and 90 for roof); a is the radius of opening; is Poissons ratio of the rock mass, and Em is the rock mass modulus. Radial deformations due to the horizontal and vertical internal pressures may be obtained from the solution presented by Bray [1]. If a distributed load (internal pressure) acts in a horizontal direction at the periphery of the opening, and the loaded arc subtends an angle 2h at the centre, the radial closure of the opening at the wall and the roof, under plane strain condition, may be defined as:

( )

+

+

pi= h

h

hh

2

m

ihwall 2

cos1cos1

sin1E

ap2u (12)

( ) hm

ihroof sin21E

)1(apu

+= (13)

where uwall and uroof are the radial closure of the wall and the roof respectively. The radial closures of the roof and the wall due to the internal vertical pressure can be obtained by modifying Equations (12) and (13). The total radial closure of the wall or roof may be obtained by algebraically summing the individual closures due to the far field stresses and the internal horizontal and vertical pressures respectively. It should be noted that the parameters Em and should be applicable to plane strain or plane stress condition whichever the case may be. 4.2. Pressure Dependency of Rock Mass Modulus In the closed form solutions discussed above, the rock mass modulus, Em corresponds to uniaxial loading condition. There is experimental evidence [11] that the modulus is pressure dependent. Duncan and Chang [12] showed that the variation of modulus with confining pressure takes the following form:

=

a

3ap p

kpE (14)

where Ep is the pressure dependent modulus; k is the modulus number; is a modulus exponent; 3 is the minor principal stress and pa is atmospheric pressure (0.105 MPa). An empirical expression is suggested in the current study to account for the pressure dependency of the elastic modulus. It assumes that the modulus depends on the internal pressure, pi in the respective direction and is expressed as:

m

a

aip Ep

ppE

+= (15)

where Ep is the pressure dependent modulus in the respective direction; pi is the internal pressure in the respective direction; Em is the rock mass modulus under uniaxial compression (i.e. at confining pressure equal to the atmospheric pressure); pa is 0.105 MPa and is an empirical modulus exponent to be determined from back analysis.


6

4.3. Pressure Dependency of Rock Mass Strength The strength of a confined rock mass at the applicable internal pressure is required when generating ground response curves. A parabolic strength criterion as proposed by Singh et al. [13, 14] was assumed for this purpose. The strength criterion is expressed as: 1 = A (3)2 + (12Aci)3 + cj; 3 ci (16) where 3 and 1 are the minor and major principal stresses at failure; ci and cj are UCS values of intact and jointed rock respectively; and A is an empirical parameter expressed as: A = -1.23 (ci)-0.77 (17) The UCS of jointed rock, cj was obtained from the correlation suggested by Singh and Rao [15] as:

63.0

i

mcicj E

E

= (18)

where Em is the rock mass modulus under uniaxial loading condition. The in-situ stress vs. strain plots (Figure 2) were used to obtain the value of Em for the current study. Equation 16 reduces to the form [16]:

2331 275.085.4545.1 += (19)

where 3 and 1 are expressed in MPa. 4.4. Application of Closed Form Solutions to the

Present Study An equivalent circular opening was determined to apply the closed form solutions to the tunnel models being considered in this study. The size of the equivalent opening was obtained by equating its surface area with the surface area of the model tunnel. Plane stress conditions were assumed. The rock mass modulus, Em was obtained using the Fracture tensor and Joint Factor concepts. The pressure dependent modulus was obtained using Equation 15. The value was used for Em when computing the radial closure. The following procedure was then applied to predict the ground response curves: Assume the ground response curve is required

for the walls of the equivalent circular opening. The closure of the walls was determined for the various values of internal pressures observed during the experimental ground response curve.

Consider the initial values of horizontal and vertical internal pressures in the equivalent circular opening.

Compute circumferential stress at the wall. Use a weakness coefficient (Fracture Tensor or

Joint Factor) to obtain the rock mass modulus, Em under uniaxial loading conditions. Adjust the parameters of the weakness coefficient so that the resulting value for Em is nearly equal to the experimental Em.

Assign an initial value for the modulus exponent, and determine pressure dependent modulus accordingly using Equation 15.

Compute the confined strength of the rock mass in the wall by setting 3 equal to the internal horizontal pressure, pih. The strength should be more than the circumferential stress for application of the present elastic analysis.

Compute the radial closure due to the far field stresses by setting equal to zero in Equation 11. Compute the radial closures of the wall due to the horizontal and vertical internal pressures. The total closure is taken as the algebraic sum of the three components. During laboratory study, the closure corresponding to the initial internal pressure was considered to be zero. The total closure obtained above, provides the correction to be applied in each calculated closure value in subsequent steps.

Consider the next pair of internal pressures and calculate the total radial closure resulting from the three components. Apply the correction as mentioned in the previous step to get the corrected closure.

The procedure is continued for subsequent pairs of internal pressure values until the computed strength of the rock mass at the periphery is exceeded by the circumferential stress. This stage represents the failure of the rock mass and development of a plastic zone. The analysis is now complete.

Several trial values of modulus exponent, for computing the pressure dependent modulus, were used. The resulting ground response curves were compared with the experimentally obtained curves to ascertain the most appropriate value of the exponent.

This procedure was used to predict the ground response curves applicable to the wall and the roof of the tunnel for all the tests conducted for hydro-static stress condition. The experimental and the


7

predicated ground response curves for Test-1, Test-5 and Test-8 respectively, are shown in Figures 4 through 9. It can be seen that the curves can be predicted reasonably by using the weakness coefficients. Based on the results of the present study, the following expressions are suggested for rock mass modulus, Em under uniaxial loading conditions: (i) Fracture Tensor concept:

1F726.0EE 71.0

k

im

+= (20)

(ii) Joint Factor concept: )J009.0exp(EE fim = (21)

0

0.5

1

1.5

2

0.0 0.1 0.2 0.3Wall closure, %

Inte

rna

l pre

ssu

re, M

Pa

PredictedExperimental

Figure 4. Observed and predicted closure of walls for Test-1.

0

0.5

1

1.5

2

0.0 0.1 0.2 0.3Roof closure, %

Inte

rnal

pre

ssur

e, M

Pa PredictedExperimental

Figure 5. Observed and predicted closure of roof for Test -1.

0

0.5

1

1.5

2

0.0 0.1 0.2 0.3 0.4

Wall closure, %

Inte

rnal

pr

essu

re, M



It is suggested that the pressure dependent modulus, Ep be obtained from the following expression:

m

625.0

a

aip Ep

ppE

+= (22)

Equations 20 and 21 are similar to those suggested by Kulatilate et al. [2] and Ramamurthy [4] respectively. It is therefore inferred that the Fracture Tensor and Joint Factor concepts may be applicable for predicting the ground response curves for hydrostatic far field stresses.

0

0.5

1

1.5

2

0.0 0.1 0.2 0.3Roof closure, %

Inte

rnal

pr

essu

re, M



0

0.5

1

1.5

2

0.0 0.1 0.2 0.3 0.4Wall closure, %

Inte

rnal

pr

essu

re, M



0

0.5

1

1.5

2

2.5

0.0 0.1 0.2 0.3 0.4Roof closure, %

Inte

rnal

pr

essu

re, M




8

5. CONCLUSIONS Deformational behaviour of jointed rocks and rock masses is primarily governed by the characteristics of the joints. The present study presents a method for estimating the closure of a tunnel as a function of the joint characteristics in terms of Fracture Tensor and Joint Factor concepts. Elastic non-squeezing ground conditions and hydrostatic stress field are considered. An experimental programme was carried out in which rigid tunnel supports were allowed to displace and the closure of simulated tunnel was monitored. Response curves were obtained from the tests and compared with theoretically based response curves obtained using closed form solutions. It is recommended that pressure dependent modulus value be used if Fracture Tensor and Joint Factor concepts are applied. By doing so, the joint characteristics can be considered when estimating the rock mass modulus. It is concluded that the ground response curves can be estimated using these concepts. Modified correlations have also been suggested for deriving rock mass modulus values. ACKNOWLEDGEMENTS The authors are thankful to Dr. Bhawani Singh, Professor (Retd.), and Dr. M. N. Viladkar, Professor, IIT Roorkee, Roorkee for their guidance and technical inputs during the experimental and theoretical investigations of this research.

REFERENCES 1. Bray J. W. (1987) Some applications of elastic theory, In

Analytical and Computational Methods in Engineering Rock Mechanics, ed. E. T. Brown, 32-94.

2. Kulatilake P.H.S.W., Wang S. and Stephansson O. (1993) Effect of finite size joints on the deformability of jointed rock in three dimensions. Int. J. Rock Mech. Min. Sci., 30:5: 479501.

3. Arora V.K. (1987) Strength and Deformational Behaviour of Jointed Rocks, Ph.D. Thesis, IIT Delhi, India.

4. Ramamurthy T. (1993) Strength and modulus response of anisotropic rocks, In Comprehensive Rock Engg., ed. J.A. Hudson, 313-329.

5. Ramamurthy T. and Arora V.K. (1994) Strength prediction for jointed rocks in confined and unconfined states, Int. J. Rock Mech. Min. Sci., 13:1:9-22.

6. Singh M. (1997) Engineering Behaviour of Jointed Model Materials, Ph.D. Thesis, IIT, New Delhi, India.

7. Singh M., Rao K.S. and Ramamurthy T. (2002) Strength and Deformational Behaviour of Jointed Rock Mass, Rock Mech. & Rock Engg, 35:1:45-64.

8. Kulatilake P.H.S.W., Liang J. and Gao H. (2001) Experimental and numerical simulation of jointed rock block strength under uniaxial loading. Journal of Engg. Mech.., 127:12:1240-1247.

9. Deere D.U. and Miller R.P. (1966) Engineering classification and index Properties for intact rock, Technical Report No. AFNL-TR-65-116. Air Force Weapons Laboratory, New Mexico.

10. Singh B. and Goel R.K. (2006) Tunnelling in Weak Rocks, Elsevier Geo-Engineering book series, Vol. 5, Pub. Elsevier, U.K.

11. Kulhawy F. H. (1975) Stress deformation properties of rock and rock discontinuities, Engineering Geology, 9:4:327-350.

12. Duncan J. M. and Chang C. Y. (1970) Nonlinear analysis of stress and strain in soils, Journal of the Soil Mechanics and Foundation Division, ASCE, 96:SM5:1629-1653.

13. Singh M. and Singh B. (2004) Critical state concept and a strength Criterion for rocks, Proceedings 3rd Asian Rock Mechanics Symposium: Contribution of Rock Mechanics to the New Century, eds. Ohinishi and Aoki, 877-880, Kyoto, Japan.

14. Singh M. and Rao K.S. (2005) Bearing capacity of shallow foundations in anisotropic non Hoek-Brown rock masses, ASCE Journal of Geotechnical and Geo-environmental Engineering, 131:8:1014-1023.

15. Singh M. and Rao K.S. (2005) Empirical methods to estimate the strength of jointed rock masses, Engineering Geology, 77:127-137.

16. Choudhari J. (2007) Closure of Underground Openings in Jointed Rocks, Ph.D. Thesis, IIT Roorkee, India.


an experimental study to predict ground response curve for tunnels by singh

Documents