Rock Mechanics as a Guide for Efficient Utilbation of Natural Resources, Khair (&.) 0 1989 Batkema, Rotterdam. ISBN 90 6191 871 5

Application of physical and mathematical modelling in underground excavations

Bhdrarama, S.Naguleswary & A.S.Balasubramaniam Asian Institute of Technology, Bangkok, Thailand

ABSTRACT: The scope of this paper is to introduce a mathematical model based on matrix algebra in order to determine similitude quantities, which can be arranged in specific formats to simulate the field conditions and associated behaviour. The formulation of a typical mathematical model applicable to Geomechanics is demonstrated here. The examples provided are intended to facilitate comprehension and application of the proposed model in practice.

1 INTRODUCTION

The methods of investigation of the engineering behaviour of geologic media mainly include numerical techniques, analytical models, field and laboratory tests, physical models and empirical methods. Although many advances have been made in Geomechanics, current problems in the design of underground structures in rock have reflected the need for more appropriate simulation methods beyond the capabilities of current conventional modelling techniques. In this paper, the authors present a method of combining physical simulation techniques with aspects of mathematical modelling. This proposed model can incorporate many variable parameters, hence would provide a fundamental input base for analytical and numerical computations and also for experimental design.

2 DIMENSIONAL ANALYSIS

The original concept of dimensional analysis was proposed by Buckingham (1914) in the early twentieth century. This paper presents an extension of this valuable tool as relevant to Geomechanics utilising matrix algebra. Buckingham's theorem states that if an equation is dimensionally homogeneous, it can be reduced to a relationship among a complete set of dimensionless products. The dimensionally homogeneous equation does not depend on the units of measurement. In most cases the governing equation is not precisely known, but it can be considered as homogeneous, provided that all the pertinent variables are included in the analysis. Therefore, the selection of variables is an important step which can be based upon theory or experiments or both. After selecting the appropriate variables, the dimensional matrix can

be formed. The dimensions of any variable Uj can be written as a function of the basic terms, mass(M) , length(L) , and time(T) as given below:

Hence dimensions of a set of variables Ul,U2, .... Un may bb described by the exponents aij. The first subscript i, indicates the dimension and the second subscript j, indicates the variable. In this way an exponent ai specifies the dimension and the variable associated with it. dimensional matrix can be represented by:

The number of independent dimensionless products which can be formed from the above matrix is equal to the total number of variables (i.e. columns,n) minus the rank of [aijl.

3.1 Material similitude parameters

In the case of modelling stress-strain behaviour, similitude of material properties is of paramount importance. The following matrix analysis (Table 1) demonstrates briefly the method of computing the relevant dimensionless terms to establish stress-strain similarity between two materials.

Table 1 : Dimensional mat,rix for material similitude

Dimension 0 v 'J c at E

Mass (MI 0 0 1 1 1'

Time (TI 0' 0 -2 -2 - 2

Stress and Young's modulus have the same dimension M L - ~ T - ~ and they are shown accordingly in the table. Poisson's ratio and friction angle are dimensionless, so they are represented by zeros. The five parameters (0, v , oc, ot and E ) are defined in the Appendix.

The rank of the matrix shown in Table 1 is unity. Therefore there should be at least four independent dimensionless terms to represent complete similitude. The following independent dimensionless terms can be regarded as appropriate :

I . Poisson'$ ratio: v 2. Ptictian angle : 9

3. Critical strain : tc=a,/E 4. Uniaxial strength ratio: at/oc

Poisson's ratio and friction angle are selected automatically because of their dimensionless form. Critical strain has been considered as an

important parameter in the development and utilisation of the elastic, brittle plastic model for rock (Zndraratna C Kaiser,1988). The application of these material similitude parameters can,be

elucidated by the development of an artificially simulated rock (Gystone) which satisfies the material {imilitude quantities. 'Gystone' is characterized by the properties; 6=32 , v =0.25, tc=0.24%, ot/oc=7.4%, which closely resemble an array of sedimentary rocks. For instanae, Fig.1 illustrates the triaxial behaviour (normalised form of 'Gystone' in comparison with several sedimentary rocks. In such a dimensionless representation, it is evident that the above artificial material resembles the stress-strain behaviour of some prototype rocks, particularly sandstone. The relevance of the uniaxial strength ratio (atlac) has already been

recogaised in stability analysis (bifurcation theory). The theory of bifurcation and the associated failure mechanisms for a given material is illustrated in Fig.2 (Vardoulakis,l984). The vertical axis represents the ratio of tensile strength to compressive strength (n)of an uniaxial strength specimen, and the horizontal axis represents the hardening parameter (N) which is determined from stress-strain relationships. The stability curve separates distinct regions where shear failure predominates in zone B, surface instability predominates in zone S and only surface instability occurs in zone D. Rocks while exhibit a Mohr- Coulomb behaviour generally have a magnitude of (ot/oc) between 0.05 and 0.10, and a hardening parameter close to unity, hence indicating predominant shear failure characteriatics, 'Gystone' (N = 1 ,and rl =0.074 :see Fig.23 falls on zone B, and as expected has revealed predominant shear fracture during failure.

3.2 Modelling of strass-strain bJwiour

The stress-strain alNt kinetic siailjtucie can :$e wined for the guspose of deformation analysis. The essence of thi-8,@onasept can bl?.rbrisfly stated as follows,

a. Stress- strain behaviour: E = .a[$ 1 . , - b. Kinetic behaviour: F = Ms.ac, where Ms = p . ~ 3 .

0 = FIA = (Ms.ac)/A, where A = ~2

Benw o =p.L.g (for ac = g) 5 &

The condition of strain similarity for model and prototype material: Em'Ep. j :,

From (a) U ~ I ~ = uplEp

Hence, P m-gls- Lm P p-gp.Lp - - -

Em EP

(Symbols are listed in Appendix)

The above criterion indicates that the geometric similitude (Lm/L is only a function of the material properties for the same gravity iield (gm = gp)-

Similar concepts have been discussed also by Clark(1981) by a different approach for centrifuge modelling.

3.3 Modelling of failure criterion

A model for post-peak deformation (dilation) of yielded rock is illustrated in Fig.3. It indicates in terms of dimensionless parameters, the occurence of radial(trP) and tangential(ogP1 post peak strains (plastic flow) with respect to d normalised linear failure criterion. The principal stresses in the plastic zone can be given by the dimensionless quantities :

Where m@= (l+sin6)/(l-sin0) and 0 ( S < 1

The parameter S is a measure of the degree of strength loss occuring immediately after the peak strength is reached. Post peak dilation in yielded rock can be modelled by the following flow rule :

The dilation coefficient a represents the ratio of the tangential to radial plastic strains. In order to estabilish similitude of post peak behaviour, S and a must be of the same order for both the prototype and the model (synthetic) material.

4 APPLICATIONS IN UNDERGROUND EXCAVATIONS

4.1 Tunnel stability

Table 2 indicates the dimensional matrix for six independent geotechnical parameters relevant for tunnelling in rock. Fig.4 and the Appendix define these quantities.

Table 2 : Dimensional matrix for tunnel stability

Dimension as ac y D L h

Mass (M) 1 1 1 0 0 0

Time (T) -2 -2 -2 0 0 0

It can be shown that the rank of the above matrix is two. Therefore.

the number of minimum independent (dimensionless) terms required for a comprehensive mpt hematical model is given by;

1. oS/oc -.suppn$ to strength iatio 2. y D/ac - overaq 3. h/D - cover to diameter ratio 4. L/D - unlined length to diameter ratio

The above proposed terms can be combined to give the following equations:

Hence, Y D/ac = (av/ac) .f2 (h/D)

Combining (i) and (ii) :

The factor (av-as)/ac can be regarded as an indicator of tunnel wall stability. These parameters can also be -used in soft ground tunnelling

(undrained) by replacing the uniaxial compressive strength (oC) by the undrained shear strength(cu). The dimensionless numbers for soft ground will then be: os/cu, YD/cU, h/D and L/D.

Some of these parameters have been used by several investigators in the past. Brosas & Ikennermark (1967) have discussed an empirical design factor for soft ground tunnel stability given by Iq-as~,1cu ( 6, which is the same factor determined by the analysis. giaurg ,et,..al (1981) have also used the same dimensionless factor as shovn by Fig.5, where N=(o~-o~)/c~.

4.2 Tunnel convergence

The pertinent variables tabulated in Table 3 are defined in the Appendix.

Table 3 : Dimensional matrix for tunnel convergence - - - -

Dimension 00 os E ua a 0

Mass (MI 1 1 1 0 0 0 0

Time (T) -2 -2 -2 0 0 0 0

Since the rank of the above matrix is two, the minimum nuinber of independent quantities required for modelling is five. The proposed non- dimensional quantities are ua/a, as/uo, as/E, u and 0. In elasto plastic analysis, 0 is an important parameter, whereas, for predominantly elastic behaviour the role of 0 is not considered. The quantities ua/a and as/ao

are prudent in the design of underground excavations. For instance, these parameters have been recognised in the representation of ground reaction curves (Fig.6). The vertical axis represents the ratio of tunnel support pressure to field stress ratio (as/ao). The horizontal axis represents the ratio of tunnel convergence to the radius of the tunnel (ua/a), which shows that the normalised support pressure decreases with increasing normalised tunnel convergence. The fundamental similitude laws for plane strain problems in tunnelling

and underground excavations in heavily fractured rock have been discussed elsewhere (Nagules~ary~l989).

5 CONCLUSIONS

In Mining and Geotechnical Engineering, normalised terms have been widely used mainly in an intutive manner rather than deriving them from a theoretical basis. The authors have demonstrated the applicability of simple mathematical modelling based on matrix algebra to derive dimensionless quantities (normalised factors) which are useful for design purposes. This paper has also presented a mathematical model in conjunction with

physical modelling techniques (e.g. use of synthetic materials) in developing appropriate similitude quantities which are prudent in predicting the behaviour of real rocks.

REFERENCES

Broms,B.B. & Bennermark,H. 1967. Stability of clay at vertical openings. ASCE J. Soil mech. & Foundations division, Vol 93, SM1:71-94

Buckingham,E. 1914. On physically similar systems:illustrations of the use of dimensional equations, Phys.Rev., Ool IV, No.4.

Clark,G.B. 1981. Geotechnical centrifngea for model studies and physical property testing of to& and rock structuwes Qoartely, Vo176 no4, Colarado school of mines.

Kimura,T. & Mair,R.J. 1981. Centrifugal testing of model tunnels in soft clay. Proc. 10th Int. conference on soil mech. and Foundation Eng., Stockholm, vol 1:319-322.

Indraratna,B. & Kaiser,P.K. 1988. Design of grouted rock bolts based on convergence control method. submitted to Int. J. of Rock Mech. and min. sci. & Geomech. Abstr.

Naguleswary,S. 1989. Fundamental criteria for geomechanical simulation of static and dynamic behaviour of geotechnical structures. Forthcoming M.Eng Thesis, Geotechnical Engineering Division, Asian Institute of Technology (AIT) , Bangkok.

Vardoulakis,I. 1984. Rock bursting as a surface instability phenmnon. Int. J. of Rock Mech. and Min. Sci. & Gaoaech. Abstr., Vol 21, No.3:137-144.

y - unit weight p - density s - friction angle

ae - tangential stress a - tunnel radius ac - acceleration

u - poisson's ratio E - Youngs modulus E - strain F - Force o - stress g - gravitational acceleration oc - uniaxial compressive strength L - modelling dimension oo - field stress m - model a, - radial stress Ms - mass as - support pressure p - prototype at - tensile strength ua - tunnel convergence av - vertical stress

0

t

Fig.1 Hoek & Brown Failure Representation

- GI 'Gypstme'

Fig.2 Stability Curve for Failure Mechanisms

(after Vardoulakis, 1984)

9

<

,# \ \ zone D

F. '. zone S \

\ \

'c \

q = q=r

0-5 I < zone B typstone

Row Rule

qP + - 0

1 ~ a s m

Post-puk S m g l h o, - m-U, + 0,

Fig.3 F a i l u r e C r i t e r i o n and Flon Rule Fig. 4 Tunnel S e c t i o n

Fig.5 I n f l u e n c e o f Depth L Unl ined Length o n S t a b i l i t y R a t i o a t F a i l u r e ( a f t e r Kinura et a l , 1981)

Fig.6 Ground R e a c t i o n Curve