support pressure estimation for based on a parametric ... · support pressure estimation for...

The Journal of The South African Institute of Mining and Metallurgy VOLUME 106 REFEREED PAPER FEBRUARY 2006

Introduction

Underground structures are subject to aprimary state of stress with no deformationbefore any excavation is carried out. Thisnatural stress state is disturbed by anexcavation and so deformation of the rockstarts to occur ahead of the opening face. Withthe advance of the opening, deformation of therock around the opening gradually increaseswith decreasing face restraint behind the face.Support pressure provided by the face allowsenough time for installing the support. Inaddition to face restraint, deformation of theopening depends mainly on the properties of

rock mass, stress state, dimensions and shapeof the opening. All these factors could beincorporated into a simple curve showing therelation between deformation of the openingand support pressure for a particular problem.This curve is commonly known as a groundresponse curve and could be used forpreliminary estimation of necessary supportpressure to stabilize an opening at anacceptable deformation level.

Researchers to calculating the groundresponse curve have presented a wide range ofsolutions. A detailed review of these solutionsproposed in the literature can be found inBrown et al. (1983) and Alonso et al. (2003)references. Solutions are generally based upona two-dimensional plane strain model with acircular shape opening excavated inhomogenous, isotropic and elasto-plasticmaterial subject to a hydrostatic stress field.Differences mainly arising in these solutionsare how they treat elasto-plastic materialbehaviour in the post-failure region and theyielding behaviour, which is generallyassumed according to Mohr-Coulomb failurecriterion (Brady and Brown, 1993 ; Hoek etal., 1995) or Hoek-Brown failure criterion(Hoek and Brown, 1980; Brown et al., 1983;Wang, 1996; Carranza-Torres and Fairhurst,1999). After yielding, the behaviour of rockmaterial is assumed to be perfectly plastic(Hoek et al., 1995; Carranza-Torres andFairhurst, 1999 ), perfectly brittle (Hoek andBrown, 1980; Brown et al., 1983; Brady andBrown, 1993; Wang, 1996) or strain softening(Brown et al., 1983; Duncan Fama, 1993;Brady and Brown, 1993; Alonso et al., 2003).Compared to other work, Hoek (2000) andHoek and Marinos (2000) proposed moresimple ground response curve estimationmodels for circular shape openings under a

Support pressure estimation forcircular and non-circular openingsbased on a parametric numericalmodelling studyby H. Yavuz*

Synopsis

Closed form solutions assume opening shape to be circular tosimplify the calculation of the ground response curve for the designof underground openings. However, in mining and civil engineeringworks, non-circular openings are commonly excavated. Therefore,there is a need to estimate the ground response curve for suchopenings. In this respect, a simple model for circular, arch andrectangular shape underground openings subject to a hydrostaticstress field is the aim of this study. A comprehensive numericalmodelling, using the finite difference method, based on numericalmodelling code FLAC version 4.0, was carried out. In addition toshape of an opening, the effect of rock mass strength, opening size,in situ stress on strain (tunnel deformation to tunnel radius) of theopening in response to applied support pressures was investigatedparametrically by developing models for each specific condition. Thestrain results for each excavation shape obtained from this largenumber of models against the normalized strength of rock mass andinternal support pressure with in situ stress magnitude wereevaluated by multiple regression analysis. As a result, a model thatwas well correlated to data was developed for an overall estimate ofthe ground response curve depending on relevant effectingparameters. The regression coefficients of the model were given forestimating both roof and sidewall response curve of each excavationshape. The model was verified by statistical tests and by ananalytical solution. Diagrams were also established to highlight theinfluence of relevant parameters and to estimate the supportpressure for a stable opening staying within the acceptable limits ofstrain. Strains occurring in a rectangular opening are found to belarger than those in the arch and circular openings for the otherfixed parameters. In the mean time, larger strains occur around anarch shape opening than a circular opening.

* Department of Mining Engineering, SuleymanDemiral University, Isparta, Turkey.

© The South African Institute of Mining andMetallurgy, 2006. SA ISSN 0038–223X/3.00 +0.00. Paper received Jan. 2005; revised paperreceived Jun. 2005.

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Support pressure estimation for circular and non-circular openings

uniform stress field. These models are statistically derivedfrom the results of parametric studies. Hoek (2000)investigated the effect of parameters on the strain state of acircular opening by means of a closed form analyticalsolution, while Hoek and Marinos (2000) investigated it bymeans of a finite element model. Asef et al. (2000) modelleda circular tunnel, 10 m in diameter, subject to hydrostaticstress conditions using the finite difference code ‘FLAC’ andproposed a ground response curve prediction chart for RMRvalues ranging from 20 to 80 for a specific problem.

Opening cross-sections are commonly non-circular inpractice, for example, mining galleries in Turkey, which aremostly either arch or rectangular in shape with differentcross-sectional areas depending on their purpose (Biron andArioglu, 1983). As highlighted before, closed form analyticalsolutions, including the numerical modelling studies carriedout for determining the ground response curve, assume theopening shape to be circular.

To estimate the ground response curve for a non-circularopening from the present closed form solutions, Curran et al.(2003) propose a method of predicting support pressure byapproximating the non-circular cross-section with anequivalent circular tunnel of the same cross-sectional area.Although this method could be used for an overallpreliminary prediction of support measure, its application toopenings deviating greatly from a circular shape could resultin an erroneous prediction (Curran et al., 2003).

The aim of this study is to develop a simple groundresponse curve estimation model for circular, horseshoe andrectangular shape openings under a hydrostatic stress field.For this purpose, a large number of numerical modelsinvestigating the effect of relevant parameters on the strainstate of the opening were carried out. Strain data obtainedfrom these parametric models were then evaluated for eachopening shape and two different rock mass classes by meansof multiple regression analysis.

Method of investigation

The effect of the opening shape, together with the otherparameters influencing the deformation response of theopening, was investigated through a parametric numericalmodelling program. The reason behind the selection ofnumerical modelling was its advantages over empiricaldesign methods or closed form solutions due to its capacity toexplicitly simulate any aspect of the problem (Coulthard,1999). All the computations were carried out by the finitedifference based two dimensional continuum modellingprogram FLAC version 4.0 (Fast Lagrangian Analysis ofContinua) (ITASCA, 2000).

This code is based on an explicit solution technique, inwhich the evolution of a system is computed by means of atime-stepping numerical integration of Newton's equations ofmotion for grid points within the model (ITASCA, 2000). Akey advantage of this approach is that the underlyingequations for the mechanics are not based on an initialassumption that the system is in equilibrium, as in implicitmethods (ITASCA, 2000). The numerical solution is thusable to follow the development of a failure and even theultimate collapse in a system. In cases of quasi-static loading,damping is added to allow the numerical solution to comesmoothly to equilibrium or to a steady state of failure(ITASCA, 2000; Coulthard, 1999).

Models were used to perform a parametric study,providing insight into possible range of strain responses ofan opening, given the likely ranges for the variousparameters. The tunnel strain data produced from the modelswere then evaluated by multiple regression analysis toestablish a ground response curve estimation model forcircular and non-circular openings surrounded by a weakquality rock mass.

Modelling parameters

In the determination of ground response curve for thepreliminary design of underground openings based on rock-support interaction analysis, models should take geometryand dimensions of the opening, initial stress field andproperties of the rock mass into account.

Geometry and dimensions of modelled openings

Three different opening shapes, commonly practised inmining and civil engineering underground constructions,were modelled. These are circular, horseshoe and rectangularin shape. The diameter of the circular openings modelledranged from 2 to 10 metres in 2-m intervals. In studying thedimensions of the horseshoe and rectangular shapeopenings, commonly practised dimensions of mining galleriesserving the extraction of ore reserves in Turkey wereconsidered. The idealized excavation geometry anddimensions of horseshoe shaped openings for the models,coded according to their cross-sections as B-5, B-8, B-10, B-14 and B-18 (Biron and Arioglu, 1983), are given in Figure 1, while the rectangular shape opening modelled hascommonly practised dimensions, 4.8 m in width and 2.4 m inheight.

Initial stress field

Two main in situ stress parameters, which are vertical stressand stress ratio (horizontal stress / vertical stress), should beconsidered in parametric modelling. In the current work, thenumber of stress parameters considered assumes excavatinga tunnel in a hydrostatic stress field. The far-field stressmagnitude applied to the boundary of each model was in therange 5 to 30 MPa in 5 MPa intervals. These are equivalentto depths ranging from 185 m to 1110 m for an average unitweight 0.027 MN/m3 of the rock mass overlying theexcavation. For shallower depths with weak rock massconditions, rock support interaction analysis may not be thecorrect method due to the possible extent of the failure zoneup to the surface. For these cases, rock load calculation couldgive better results.

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130 FEBRUARY 2006 VOLUME 106 REFEREED PAPER The Journal of The South African Institute of Mining and Metallurgy

Figure 1—Dimensions of horseshoe cross-section openings modelled

Rock properties

In analysing the tunnel deformation under differentconditions, estimating the properties of rock mass is veryimportant. For systematic data production, the most widelyaccepted system proposed by Hoek and Brown was used(Hoek and Brown, 1997). The general form of the failurecriterion for the determination of rock mass properties isgiven by the equation:

[1]

Where σ1 and σ3 are maximum and minimum principlestresses, σc is the uniaxial compressive strength of intactrock, mb is the rock mass material constant determined frommaterial constant mi for intact rock and Geological StrengthIndex (GSI), s and a are constants which depend upon thecharacteristics of the rock mass.

The deformation behaviour of weak rock massessurrounding the opening are of interest here due to thestability problems encountered mostly in excavationssurrounded by these type of rock masses. Lower boundquality ratings for poor and fair quality rock masses with GSIvalues of 23 and 44 (Hoek and Brown, 1988), respectively,were adopted in the calculation of rock mass parameters fromthe Hoek-Brown failure criterion. The rock mass was treatedas a homogenous, isotropic, elastic-perfectly plastic materialyielding according to the Mohr-Coulomb failure criterion withno volume change. This assumption could be made forrelatively weak rock masses (Hoek and Brown, 1997) due toits relative simplicity and the rather small number of easilyinterpretable parameters appearing in it (Anagnostou andKovari, 1993).

In the determination of rock mass properties, the GSI,uniaxial compressive strength of intact rock (σc) and materialconstant (mi) of intact rock are essential parameters to bedetermined. According to intact rock classification, theuniaxial compressive strength of intact rocks stays within therange of 25 to 5 MPa for weak rock and 50 to 25 MPa formedium strong rocks (Hoek and Brown, 1997). The materialconstant mi changes from 7 to 25 depending on the type andcharacteristics of intact rock (Hoek and Brown, 1988). Theseparameters were incorporated into the generation of rockmass data in a way that it was possible to evaluate the effectof rock mass strength from worst to best condition on thedeformation of the tunnel (see Table I). The calculation ofequivalent Mohr-Coulomb strength parameters from theHoek-Brown failure criterion was based on the procedure

suggested by Hoek and Brown (1997) and the cohesion (c)and internal friction angle (φ) values found in this way aregiven in Table I. A Poisson ratio of 0.25 was consideredrepresentative for most rocks, while the deformation modulusof rock mass (Em) was calculated from the following equation(Hoek and Brown, 1997):

[2]

The global strength of rock mass (σm) for each data set,determined by applying the following well-known transfor-mation yielded a data range for the modelling of 9.6 MPa to0.5 MPa (Table I).

[3]

Model specification

A half section of openings was modelled due to symmetry. Itwas assumed in models that rock properties did not changein longitudinal direction and a two-dimensional plane straincondition with no strain occurring perpendicular to the modelplane was applicable. This assumption is adequate forground response curve determination in this study (Hoekand Marinos, 2000; Hoek, 2001; Asef, 2000). However, 3-dimensional modelling is necessary to determine the grounddeformations that occur before the support installation inorder to establish the reaction curve of the support in rock-support interaction analysis (Carranza-Torres and Fairhurst,1999; Curran et al., 2003). Boundary conditions andgenerated mesh were the same for all opening shapes and areshown in Figure 2 for a horseshoe shape opening. Thenumber of zones was 10 032 in all models. The aspect ratioof the zones was kept close to unity, with increasing sizeaway from the boundary of the opening. The initial stressfield was assumed to be hydrostatic and uniform. At the far-field boundaries, the applied stresses are equal to the initialstress (po) and kept constant during calculations for eachmodel. A symmetrical boundary was fixed so as not todeform the width of openings in the radial half. Excavation ofthe tunnel was simulated through a gradual unloading of thetunnel boundary from its initial stress value to null in orderto generate ground response curve for the specifiedconditions. The support pressure (pi) was assumed to beuniform along the boundary of the opening and so uniformlydistributed radial support pressure along the openingboundary was applied.

σφφm

c=

−21

cossin

Emc GSI= −( )σ

10010 10/40

σ σ σσσ1 3= + +

c b

3

c

a

m s

Support pressure estimation for circular and non-circular openingsTransaction

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131The Journal of The South African Institute of Mining and Metallurgy VOLUME 106 REFEREED PAPER FEBRUARY 2006 ▲

Table I

Numerical modelling input data for rock mass properties

Rock mass class GSI σc (MPa) mi c (MPa) φ (Degrees) Em (MPa) σm (MPa)

50 25 2.35 38 5000 9.644 17 1.88 35 4696 7.2

Fair quality 44 38 15 1.59 33 4364 5.932 10 1.21 30 4000 4.225 7 0.87 27 3540 2.825 25 0.73 31 1057 2.621 17 0.53 29 969 1.8

Poor quality 23 18 15 0.43 27 897 1.414 10 0.29 24 791 0.910 7 0.18 21 668 0.5

Model results

For six different far-field stress parameters and ten differentmaterial property parameters (see Table I), 300 models forcircular openings with diameters which are 2, 4, 6, 8 and 10m, 300 models for horseshoe shape openings withdimensions shown in Figure 1, and 60 models for arectangular opening were arranged. In all these models,initial support pressure applied to the boundary of theopening was equal to the far-field stress magnitude and wasthen gradually reduced, in steps of 0.2 MPa, to zero. In eachsupport pressure step, the final deformation satisfying theequilibrium condition of the model was recorded automat-ically by implementing this procedure to FLAC through anad-in programming language FISH. This procedure allowedthe generation of a ground response curve for each modelcase that differed from other cases by one parameter.

The extent of the failure zone and displacements alongthe boundary of the circular opening were uniform anddisplacements were radial to the opening wall. However, thenon-circular opening cross-section caused a non-uniformextent of the failure zone and distribution of displacements.This can be seen from Figures 3a and b showing the directionand magnitude of the displacement vectors around anunsupported horseshoe and rectangular opening excavated inrock mass having 5.9 MPa compressive strength (see Table I)subject to 20 MPa field stress, respectively. In all model casesof rectangular and horseshoe shape openings, the extent ofthe yielding zone and the deformations on the roof is largerthan those on the sidewall with the decrease in strength ofrock mass. Generation of a single ground response curve waspossible for circular opening, whereas determination of afamily of curves was required for the other opening shapesbecause the variation in deformation magnitude occurredalong the opening boundary at various support pressures.

Given the shape of an opening and the quality of thesurrounding rock mass, the deformation obtained from modelopenings was investigated according to the followingparameters,

[4]

where u is the deformation of the opening, σm is thecompressive strength of the rock mass, po is the in situ stressmagnitude, pi is internal support pressure and r is the radius

of the opening. Analysis of output deformation data showedthat the deformation of the opening increased linearly withthe opening size when the other parameters were taken asequal. So the strain of an opening (u/r) could be determinedby applying the following transformations to the independentparameters (Hoek, 2000 and Hoek, 2001)

[5]

Where σm/po is the competency factor of the rock mass,which is also commonly used as a squeezing indicator forrock mass (Barla, 2001), and this input data varied from0.52 to 0.03 for poor quality rock and from 1 to 0.1 for fairquality rock mass. pi/po is the ratio of support pressure to insitu stress and was gradually reduced in all models from 1 forinitial case to 0 for unsupported case with no face restraint.u/r is the resulting strain of the opening depending on theσm/po and pi/po ratios and varied in the results of thenumerical models from zero for pi/po =1 to 1 for unsupportedopening excavated in very weak rock mass. In the calculationof strain (u/r) from the deformation data of numericalmodelling for horseshoe and rectangular openings, anapproximation of the non-circular cross-section with anequivalent circular opening of the same cross-sectional areawas made. The radius of the equivalent circular opening (r)was calculated from the relationship, r = √A/π, where A is thearea of horseshoe or rectangular opening.

By taking the natural logarithm of both the u/r and σm/podata, a smooth data distribution was observed despite thewide range of parameters evaluated in models, as shown inFigures 4 and 5 for all shape openings surrounded by eitherfair or poor quality rock masses, respectively.

Regression analysis of strain data

By regression analysis of data shown in Figures 4 and 5 witha large number of equations, the following exponentialfunction model with the Taylor series polynomial exponentwas found to be the best one describing the strain changedepending on competency factor and the ratio of supportpressure to in situ stress. The equation representing themodel can be written in the following form:

[6]

where β0, β1, β2, β3, β4, β5, β6, β7, β8, and β9 are regressioncoefficients. This equation could be converted into linearform with the following transformations in order to estimatethe regression coefficients from the least square method(Snedecor, 1989):

[7]

where Y = ln (u/r), X1 = ln (σm/po), X2 = pi/po, X3 =(ln(σm/po))2, X4 = (pi/po)2, X5 = ln(σm/po)(pi/po), X6 =(ln(σm/po))3, X7 = (pi/po)3, X8 = ln(σm/po)(pi/po)2, X9 =(ln(σm/po))2(pi/po).

For the 9 variables in n data points, the model equationfor the ith observation could be written as follows:

Y X X X X

X X X X Xo 2

6

= + + + + +

+ + + +

β β β β β

β β β β β1 1 2 3 3 4 4

5 5 6 7 7 8 8 9 9

u

re

om

o

i

o

m

o

i

o5

m

o

i

o

m

o

p

p

p p

p

p p

p

p

p

p

=

+

+

+

+

+

+

+

β βσ

β βσ

β βσ

βσ

β

1 2 3

2

4

2

6

3

7

ln ln ln

ln ii

o8

m

o

i

o

m

o

i

op p

p

p p

p

p

+

+

3 2

9

2

βσ

βσ

ln ln

u

rf

p,

p

pm

o

i

o

=

σ

u f , p , p ,rm o i= ( )σ


▲


Figure 2—Details and conditions for models



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Figure 3—Examples of deformations occurring around non-circular openings (a) horseshoe shape (b) rectangular shape

Figure 4—Data distribution and regression curve for strain response of openings excavated in fair quality rock mass (a) on the roof and sidewall of circularshape (b) on the roof of horseshoe shape (c) on the sidewall of horseshoe shape (d) on the roof of rectangular shape

(a)

(b)

(c)

(d)

0.00 0.500 1.000 1.500 2.000 2.500 3.00

LEGEND

step 80000-4.881E-01 <x< 3.430E+002.809E+01 <y< 3.201E+01

Boundary plot

0 1E 0Displacement vectorsMax Vector= 4.281E-02

0 1E-1

Horseshoe shape opening (B-10), σm = 5.9 MPa, po = 20 MPa

FLAC (Version 4.00) 3.175

3.125

3.075

3.025

2.975

2.925

2.875

2.825

0.00 0.500 1.000 1.500 2.000 2.500 3.00

LEGEND

step 80000-2.516E-01 <x< 3.047E+002.839E+01 <y< 3.169E+01Boundary plot

0 1E 0Displacement vectorsMax Vector= 5.415E-02

0 1E-1

Rectangular shape opening (width = 4.8 mm height = 2.4 m), σm = 5.9 MPa, po = 20 MPa

FLAC (Version 4.00) 3.150

3.100

3.050

3.000

2.950

2.900

2.850

OPENING SHAPE: CIRCULARSTRAIN SITE: ROOF AND SIDEWALL

OPENING SHAPE: HORSESHOESTRAIN SITE: ROOF

OPENING SHAPE: HORSESHOESTRAIN SITE: SIDEWALL

OPENING SHAPE: RECTANGULARSTRAIN SITE:ROOF

(a)

(b)


▲


(e)

(e)

Figure 4 (continued)—Data distribution and regression curve for strain response of openings excavated in fair quality rock mass (e) on the sidewall ofrectangular shape

Figure 5. Data distribution and regression curve for strain response of openings excavated in poor quality rock mass (a) on the roof and sidewall ofcircular shape (b) on the roof of horseshoe shape (c) on the sidewall of horseshoe shape (d) on the roof of rectangular shape (e) on the sidewall ofrectangular shape

(a)

(b)

(c)

(d)

OPENING SHAPE: RECTANGULARSTRAIN SITE:SIDEWALL

OPENING SHAPE: CIRCULARSTRAIN SITE:ROOF AND SIDEWALL

OPENING SHAPE: RECTANGULARSTRAIN SITE:ROOF

OPENING SHAPE: RECTANGULARSTRAIN SITE:SIDEWALL

OPENING SHAPE: HORSESHOESTRAIN SITE:ROOF

OPENING SHAPE: HORSESHOESTRAIN SITE:SIDEWALL

[8]

There are n such equations, one for each data point. Themethod of least squares estimates the coefficients, whichminimize the sum of squared deviations between the fittedand actual strain data, from the following equation(Snedecor, 1989):

[9]

where Y is a 9x1 matrix and X is an n x 10 matrix for thisproblem. That is,

[10]

The best estimate of a coefficient is the value minimizingthe residual sum of squares in the regression model to bringthe curve close to the data points. From the scattered dataillustrated in Figures 4 and 5, regression coefficientsproviding a best fit surface curve to data—as shown in thesame figures for circular, horseshoe and rectangular

openings , quality of rock mass and site of the deformation(roof or sidewall)—are given in Table II.

Statistical testing for the model validity

Equation [6 ]was found to be the best model with the loweststandard error of estimate (S) and higher coefficient ofmultiple determination (R2) among the large number ofequations fitted to the data. S, which is an important measurefor showing how close the actual data points fall to thepredicted values on the regression curve, and multiple R2,which measures the proportion of variation in the data pointsexplained by regression model, are given in Table IIseparately for each opening shape, rock mass quality anddeformation site around the opening. The S values incomparison to the mean of the predicted values of the strainsfor each case are less than 10% of the mean. The R2 of themodel is higher than 0.959 for describing the strain data.These measures show that the model is useful to describe thestrain of openings against the σm/po and pi/po withregression coefficients given in Table II for different shape ofopening, rock mass quality and deformation site.

F statistics was performed for testing the globalusefulness of the model. This test is widely used inregression and analysis of variance (Snedecor, 1989). Thenull hypothesis for this test is Ho: β1 = β2 = ..... = β9 = 0against the alternative hypothesis H1: at least one of β1, β2,....., β9 not equal to zero. The calculated F ratios of the modelfor different parameters (opening shape, rock mass quality,etc.) are given in Table II. For a significance level of 1%, thetabulated F ratio with the numerator degrees of freedom of 9and with the denominator degrees of freedom, larger than350 (e.g. rectangular shape opening) is 2.41. In comparisonto the tabulated F ratio, computed F ratios of the model in allcases are quite large enough to reject the null hypothesis (see

Y

Y

Y

Y

X

X X X X X X

X X Xn n n n

=

=

1 11

21

1

and

1 1 . . . . . . . . . . . . . . . . . . . .1

2

12 19

22 29

2 9

.

.

.

. .. .

.

.

.

.. .

β

β

β

0

1

1

9

.

.

.

= ( )−X X X YT T

Y X X X X

X X X X Xi o i 2 i i i

i 6 i i i i

= + + + + +

+ + + +

β β β β β

β β β β β1 1 2 3 3 4 4

5 5 6 7 7 8 8 9 9



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Table II

Results from regression analyses of data


Table II). From this it can be concluded that model is validand significant to calculate the strain of an opening againstthe σm/po and pi/po.

Hypothesis testing on the significance of individualcoefficients was performed by t-test. The null hypothesis forthis test is Ho: βi = 0 for each coefficient against thealternative hypothesis Ho: βi ≠ 0. If one of the parameters isnot important in the model then the b value of that parameteris found to be zero from this test. The t-ratio for eachestimated value of βi found by dividing the βi to its standarddeviation (Sβi) are given in Table II. The tabulated t-ratio for1% significance level was found to be 2.326 from the tableand was same for all cases. As could be seen from Table II,the absolute values of computed t-ratios are significantlylarger than the tabulated ratio, suggesting that all the coeffi-cients in the model are not equal to zero. It can be concludedfrom this that the parameters incorporated into the model areall relevant and significant for the calculation of openingstrain and the model is valid.

Comparison of the model with an analytical solution

The validity of the current model described by Equation [6]was checked with an analytical solution suggested by Hoeket al. (1995). Comparison was made just for circular openingdue to there being no available solution for non-circularopenings to calculate the ground response curve. In thederivation of the equations predicting the ground responsecurve, this solution assumed similar conditions to this work,e.g. a circular tunnel surrounded by weak rock mass underhydrostatic stress, elastic-perfectly plastic material with novolume change and Mohr-Coulomb yielding behaviour.Comparison was made for two different material propertiesand stress conditions. Firstly, a circular tunnel of 9 m indiameter surrounded by a fair quality rock mass with 5.9MPa compressive strength subjected to a 28 MPa stress field,and secondly, a circular tunnel of 2 m in diametersurrounded by a poor quality rock mass with 1.8 MPacompressive strength subjected to a 15 MPa stress field, wereconsidered. Results both from this solution and model areillustrated in Figure 6, showing that the support pressure-deformation relation suggested by the model is in closeagreement with the analytical solution.

Effect of parameters on the deformation response ofthe opening

Model results show that the lower rock mass strength, thehigher the in situ stress, the larger the opening diameter, andthe lower the support pressure conditions, then the greaterdeformations around the opening. On the other hand, theopening shape analysed and its effect contained in the modelwith coefficients had a significant effect on the deformationlevel. Figure 7 illustrates this effect with the plotted groundresponse curves generated for the roof of circular, horseshoeand rectangular openings excavated in fair or poor qualityrock mass. As expected, a rectangular opening causes theoccurrence of more deformation than a horseshoe shapeopening relative to a circular shape opening. However, itseffect is not more significant than the quality of rock massand σm/po ratio, as can be seen from Figure 7.

Support pressure estimationOne useful way of predicting the necessary support pressurewithin an acceptable level of tunnel strain and for seeing the

effect of different parameters, is to construct dimensionlessplots. These dimensionless plots were established from themodel and are illustrated in Figures 8 and 9 for the roof of anopening excavated in fair or poor quality rock massconditions, respectively. For a given shape of an opening,rock mass quality, and rock mass strength to in situ stressratio, the support pressure to in situ stress ratio required tomaintain per cent strain can be estimated from thesediagrams. The support pressure can then be readilycalculated from the known in situ stress. Type and amount ofsupport needed for the stability of opening can be estimatedfrom the work of Hoek (1998) using the diameter of acircular opening or equivalent diameter of horseshoe andrectangular openings in conjunction with the supportpressure readily available from these diagrams.

It is important to point out that allowing the tunnel toundergo deformations extensively to reach the minimumlevels of support pressures is not possible due to the gravityeffect, leading to unstable failure propagation on the roof.Therefore, there should be an allowable limit of strain beforesetting up the support. Field observations and measurementsperformed by Sakurai (1983) and followed by Chern et al.(1998) show that strain of the opening in excess of 1% isassociated with the onset of tunnel instability and withdifficulties in providing adequate support. Hoek (2001)reports that a 1% limit of strain is only an indication of

▲


Figure 6—Comparison between model and an analytical solution

Figure 7—Effect of opening shape, rock mass quality and pi/po ratio onthe strain response of the opening

analytical solution (Hoek et al., 1995)model

Fair rock mass, σm = 5.9 MPa, r = 4.5 m, po = 28 MPa

Poor rock mass, σm = 1.8 MPa, r = 1 m, po = 15 MPa

0 20 40 60 80 100 120

Radial deformation (u), mm

Sup

port

pres

sure

(pi),

MP

a

Opening excavated in fair quality rock mass, σm / po = 0.2Opening excavated in poor quality rock mass, σm / po = 0.2

C : Circular shape openingH : Horseshoe shape openingR : Rectangular shape opening

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 1 2 3 4 5 6 7 8 9 10 11

Strain (%) = (u/r)*100

p i/ p

o

30

25

20

15

10

5

0



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increasing difficulty to install the support and it should notbe assumed that sufficient support should be installed tolimit the opening strain to 1%. Based on measurementscarried out in some tunnels, Hoek (2001) suggests that it isdesirable to allow the tunnel to undergo strains of as muchas 5%, in some cases, before activating the support. Thesetunnels were successfully completed without having stabilityproblems, but construction problems increased withincreasing strain levels.

A similar observation is published by Singh et al. (1997)for tunnels in India. They define the critical strain, beyondwhich the necessary support pressure increases sharply, to be

5–6% due to unstable failure propagation initiated at thisvalue, and the rock suddenly loses its cohesion. Thiscorresponds to a jump of the ground response curve from therock with the original cohesion to zero cohesion. Thesevalues yield first estimations of the optimal value of thetunnel convergence and can only be used only for apreliminary design purpose.

For an unsupported opening subject to hydrostatic stress,deformations increase very significantly when the ratio ofrock mass strength to in situ stress level is less than 0.2 forcircular shape, 0.25 for horseshoe shape and 0.3 forrectangular shape openings, respectively. According to theabove field experiences, preliminary supporting of theopening should be done as close to the face as possible, aswithin 1% of tunnel strain, further deformation of theopening up to 5–6% strain level may be allowed under thecontrol of support to reduce support pressure if the final sizeof the opening is not a major issue. Very weak rock masssurrounding the opening may be improved by grout injection,placement of grouted pipe forepoles, or reinforcement withfibreglass dowels if very high pressure is necessary for astable condition on the roof of the tunnel near to the face andthe final size of the opening is important.

Conclusions

A parametric numerical modelling study was carried out toestablish a ground response curve prediction model forcircular, horseshoe and rectangular shape openings.Parameters for the purpose of the developed models areshape and dimensions of opening, rock mass quality andstrength, in situ stress magnitude for hydrostatic stresscondition, and the roof and sidewall deformation response tothe opening with respect to support pressure. Supportpressure applied on a model opening boundary was uniformand reduced from in situ stress magnitude to zero in steps ofequilibrium condition reached in the system. In spite of thewide range of parameters incorporated into the models,opening strain data (u/r) depended on dimensionlessparameters (σm/po and pi/po) exhibiting a smooth data distri-bution for each opening shape evaluated separately for eachrock mass quality. For horseshoe and rectangular shapes,calculation of strain was done by the radius of the openingequivalent in area to the circular section. As a result ofregression analysis and the statistical testing process, anexponential function model with the Taylor series polynomialexponent was found to be the best one for calculating theground response curve. Regression coefficients of this modelfor each opening shape, rock mass quality and site of thedeformation are given in Table II. Compared to an analyticalsolution, the ground response curve calculated from themodel for a specific example proved that the model is valid.With the same conditions and an equal area, a rectangularsection causes more deformation of the opening wall than ahorseshoe section, whereas a horseshoe section causes moredeformation than a circular section. On the other hand, thelower the rock mass strength, the higher the in situ stress,the larger the opening diameter, and the lower the supportpressure conditions, then the greater deformations aroundthe opening. Diagrams were also created to highlight theimportance of each parameter and to quickly estimate thenecessary support pressure for the tunnel roof within theacceptable limits of tunnel strain, which are up to 1% before

(a)

0.1 0.2 0.4 0.6 0.8 1σm / po

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Figure 8—Support pressure estimation diagrams for the roof ofopenings excavated in fair quality rock mass (a) circular shape (b)horseshoe shape (c) rectangular shape

(c)0.1 0.2 0.4 0.6 0.8 1

σm / po

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support installation and up to 5–6% after installation of thesupport. In using these diagrams, previous fieldmeasurements should not be forgotten in that the openingroof requires a sharp increase in support pressure after 5–6%strain due to excessive material loosening and continuingincrease in the upward extent of failure zone. For the worstground conditions, ground improvement may be necessary toreduce the pressure imposed on supports and to provide asafe and stable tunnel condition.

Acknowledgement

The author would like to express his sincere gratitude to TheScientific and Technical Research Council (TUBITAK) of

Turkey for providing financial support for this researchstudy.

References

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Figure 9—Support pressure estimation diagrams for the roof ofopenings excavated in poor quality rock mass a) circular shape b)horseshoe shape c) rectangular shape

0.04 0.06 0.08 0.1 0.2 0.4σm / po

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support pressure estimation for based on a parametric ... · support pressure estimation for...

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