atkinson and potts 1977

ATKWSON, J. H. & Porrs, D. M. (1977). Ge’otechnique 27, No. 2, 203-215

Stability of a shallow circular tunnel in cohesionless

J. H. ATKINSON* and D. M. PO’ITSt

soil

This Paper investigates theoretically and experiment- ally the stability of a circular tunnel in a cohesionless soil with support conditions similar to those found during construction. The experimental investigation consisted of small-scale model tests in the laboratory and on the Cambridge University large-diameter centrifuge. All the model tests were carried out in plane strain using Leighton Buzzard sand and tests with and without surcharge loading are reported. The theoretical studies are based on the upper and lower bound theorems and predictions of collapse pressures from these theoretical solutions are shown to bracket closely thevaluesobserved in themodel tests.

Cette etude examine theoriquement et experimentale- ment la stabilite d’un tunnel circulaire dans un sol pulverulent soutenu dans des conditions semblables a celles trouvees pendant la construction. L’etude experimentale consista en essais sur modele a petite 6chelle en laboratoire et sur la centifugeuse a grand diametre de l’Universit6 de Cambridge. Tous les essais sur modele furent mis a execution sous deformation plane en utilisant du sable de Leighton Buzzard, et l’on rend compte des essais avec et sans surcharge additionelle. Les etudes theoriques sont basees sur les theoremes limites suptrieures et inferieures et on montre que des previsions de pressions de rupture a partir de ces solutions sont encadrees Ctroitement par les valeurs observees en essais sur modtles.

INTRODUCTION

As the face of a tunnel is advanced, a means of supporting the ground close to the face may be needed. In competent rock, the tunnel will be stable without support, but in soft ground,

with immediate support provided by such means as a shield, the use of compressed air or clay slurry under pressure may be required; otherwise collapse may occur as a result of gross plastic deformation of the soil, possibly accompanied by flooding. Once a primary lining, usually of concrete or cast-iron segments, is in place, instability is unlikely unless the fabric of the lining deteriorates or the ground in the vicinity of the tunnel is disturbed.

Ground movements during excavation of soft ground tunnels have been discussed recently by Atkinson and Potts (1976). The present Paper is concerned with the of radial pressures to be

supported by compressed air, bentonite slurry, a shield or by other means in order to achieve stability. One solution would be to allow for ground pressures close to the average total stress in the soil before excavation. This would maintain stability and prevent excessive ground movements and flooding, but the use of such large pressures at depth may be expensive and, if air pressure is used as the support, may constitute a health hazard.

The support provided by air or bentonite pressure is equivalent to a uniform normal total stress acting on the exposed periphery and face of the tunnel. The state of stress in the soil around a shield is rather more complicated but the support provided by a smooth circular shield may be approximated to a uniform normal total stress provided bending moments in the

Discussion on this Paper closes 1 September, 1977. For further details see inside back cover. * Department of Civil and Structural Engineering, University College, Cardiff. t Royal Dutch Shell Research Laboratories, Rijswijk, Netherlands.

204 J. H. ATKINSON AND D. M. POTTS

shield are not large. Thus the state of stress in the soil around a newly excavated circular

tunnel supported by air or bentonite pressure or by a smooth shield, neglecting end effects, may be approximated to the state of stress associated with a uniform fluid pressure contained within the heading. The problem is to estimate the magnitude of the pressure required just to cause the tunnel to collapse and an appropriate margin of safety may then be allowed.

The stability of a shallow circular tunnel in a weightless soil to which a uniform surface surcharge pressure was applied was investigated by Atkinson and Cairncross (1973). The present Paper extends this work for cohesionless soils to cover the more important problems of shallow circular tunnels in soils whose self weight forces may themselves cause collapse. The approach to the problem will be through the bound theorems of plasticity theory together with the results of tests on model tunnels in a sand soil.

MODEL TUNNEL TESTS

A number of model test to examine the behaviour of shallow tunnels in sand have been carried out recently at Cambridge University. Details of the apparatuses and of the tests have been given elsewhere (Atkinson et al., 1975; Atkinson et al., 1977; Potts, 1976) and only the essential features will be described here.

It is important to appreciate that these model tests were not intended to reproduce precisely to scale a real tunnel during construction together with all the details of the methods of excavation and support. Instead their purpose was to illustrate the way in which soil around a circular cavity deforms as a fluid pressure within the cavity is reduced, since this approximates to the stress condition within a tunnel during construction. The results of tests of this kind provide a source of experimental data which may be used to examine the accuracy of stability calculations; examination of the mechanisms of deformation observed in such tests may lead to new calculations for estimating the stability and deformations of shallow soft ground tunnels.

In all the model tests the boundary conditions were the same and are illustrated in Fig. 1. The block of soil containing the tunnel was itself contained within an apparatus with rigid boundaries; the front and back faces imposed a condition of plane strain on the soil and the bottom and side faces were sufficiently far removed from the tunnel not to influence the behaviour of the sand around and above the tunnel. The sand surface may be unstressed with

us = 0 but in some tests a uniform surface surcharge us was applied by fluid pressure contained within a flexible rubber membrane. Surface surcharge loadings may occur in

practice where tunnels are driven below structures on flexible foundations or when a tunnel is driven through dense sand overlain by a very soft deposit.

The tunnels were approximately 60 mm in diameter and were constructed completely through the soil sample; consequently the models were strictly examining the plane section of a tunnel removed from its face. The actual diameters of the tunnels varied slightly according to the

methods of construction but each was measured directly. The tunnels were lined with a thin cylindrical rubber membrane of negligible stiffness and strength and the tunnel pressure uT applied a uniform radial total stress to the tunnel wall. The tunnel pressure a, is equivalent to an excess air or bentonite pressure in a tunnel heading during construction or to a circumfer- ential thrust T = uT R per unit length in a smooth and moment-free tunnelling shield or tunnel lining of radius R.

Most of the model tests were conducted on the laboratory floor under normal gravitational accelerations but one series of tests was completed in which the models were accelerated in the Cambridge large-diameter centrifuge. ‘In a centrifuge the model is accelerated and stresses due to body weight forces increase accordingly. If a model is accelerated to ng (where g is the acceleration due to the earth’s gravity) the stresses in the model are equal to those in a

STABILITY OF A SHALLOW CIRCULAR TUNNEL IN COHESIONLESS SOIL 205

Fig. 1. Boundary conditions and dimensions of model tunnel tests

prototype structure n times larger than the model. In effect, so far as stress and strain are concerned, a centrifugal mode1 becomes apparently n times larger when accelerated to ng. The centrifugal mode1 tests were conducted at constant accelerations of 75g; consequently, all other things being equal, the behaviour of the 60 mm diameter model tunnels should be the same as the behaviour of a 4.5 m diameter tunnel in Earth’s gravitational field.

The sand used for these mode1 tests was the fraction of dry Leighton Buzzard sand passing a No. 14 sieve and retained on a No. 25 sieve. This sand has been extensively examined in laboratory tests at Cambridge in the past and its engineering behaviour is well documented.

Since the sand was dry, pore pressures were everywhere atmospheric, so that total and effective stresses were everywhere equal. For all tests, samples were prepared by pouring the dry sand around a pre-formed tunnel; the sand was poured in the direction of the tunnel axis to produce a homogeneous sample which was isotropic in planes normal to the tunnel axis. A specified sand density may be obtained by pouring the sand at a predetermined rate. All the results discussed were taken from mode1 tests in which sand was dense with a voids ratio of approximately 0.52 and all were conducted in a similar fashion.

The tunnels were constructed with the tunnel pressure approximately equal to the vertical stress in the soil at the level of the tunnel axis; thus the initial conditions were

UT = y(C+R)+a,

where C is the cover above the tunnel crown as shown in Fig. 1 and y is the unit weight of the soil. During a particular test the tunnel pressure was reduced in decrements until the tunnel collapsed; this was, in all cases, a sudden and well defined event accompanied by very large soil displacements (Atkinson et al., 1975; Potts, 1976). In most tests lead shot were buried in the sand or markers placed against a transparent front face, and after each decrement of tunnel pressure a radiograph or a photograph was taken of the apparatus. By observing the movements of the lead shot or of the markers on successive radiographs or photographs the complete field of displacement and strain in the soil around the tunnel may be calculated (Potts, 1976). Although soil deformations and strains are not of specific concern here, such observations may be of considerable importance when they suggest mechanisms of collapse or fields of stress necessary for stability calculations.

Three series of model tunnel tests have been completed in which such parameters as the depth of cover C, the initial stress conditions, sand density, stress level and type of loading were varied. These tests are described in detail by Potts (1976). Further tests on model tunnels in


0 0 0

0.5 9

I Q 1.5 CIZR

Fig. 2. Variation of tunnel pressure at collapse with depth of burial for model tunnels with crs =210 kN/m2

IO-

0 0

0 0 I 0 200 400 600

Qs : kN/m2

Fig. 3. Variation of tunnel pressure at collapse with surface pressure for model tunnels with C/2R= 0.5

sand have been carried out in which the tunnels were lined with thin metal tubes and strain gauges used to measure the loads in the tubes (Potts, 1976) but, for the present, only those tests in which tunnels contained thin rubber liners will be considered.

For all the model tests the tunnel pressures at collapse are shown plotted against depth of burial in Figs 2 and 4, and against surcharge pressure ~a in Fig. 3.

THEORETICAL ANALYSIS

The stability of a tunnel may be examined theoretically by making use of the upper and lower bound theorems for perfectly plastic materials. Strictly these. theorems are true only for materials whose flow rule is associated and v=$ where Y is the angle of dilation (Hansen, 1958) and + is the maximum angle of shearing resistance. In general the flow rule for real soils is non-associated and Y < $, and application of the bound theorems to problems in soil mechanics must be treated with some caution. Nevertheless Davis (1968) has shown that an upper bound calculated assuming Y = 4 is also an upper bound for the case where Y < 4. Palmer (1966) has suggested that a lower bound solution based on the Coulomb failure criterion may still be valid even if the flow rule is non-associated. Application of the bound theorems involves, firstly, calculation of an admissible stress field, in which case the external loads cannot cause collapse and, secondly, selection of a mechanism of collapse together with an appropriate work rate calculation, in which case the external loads must cause collapse. Selection of a realistic stress field and a realistic collapse mechanism may often be assisted by examination of soil deformations observed during model tests.

Lower bound solution

If a statically admissible stress field can be found which nowhere violates the failure criterion for the soil, the loading is a lower bound to the true collapse load; it may be noted that such a solution will lead to a safe or high value for the tunnel pressure.


Fig. 4. Variation of tunnel pressure at collapse with depth of burial for model tunnels without surface surcharge

z

0,

I.%* _l

e

(b)

Fig. 5. States of stress in plane elements: (a) close to the tunnel; (b) far from the tunnel

For a cohesionless soil the failure criterion may be written

where

al=/N3 . . . . . . . . . . (1)

l+sin+ C1=l_sin+

and 4 is the maximum angle of shearing resistance. The state of stress in a small plane element in the vicinity of the tunnel is illustrated in Fig.

5(a) and the conditions of equilibrium are

au, GJr - 4 1 aT,e ar+r -rae+yCOSe=O . . . . . . .

~~-!$!-$??-~sin~ = 0 . . . . . . * (3)

where y is the unit of the soil. In general radial and tangential stresses are not principal stresses but, in order to obtain a solution, we assume that T,~ = 0, in which case the equilibrium

conditions become

. . . . . . . . (4)

z--yrsinB=O . . . . . . . . * (5)

and the failure criterion becomes

1 0, = PO, or ue=-a, . . . . . . . . (6)

P

208 J. H. ATKINSON AND D. M. PO-ITS

Examination of soil strains observed during model tests indicates that it is reasonable to assume that, in the vicinity of the tunnel, principal strain rates are radial and tangential and so, ac- cepting coaxiality of the directions of principal stress and principal strain rates, it appears reasonable to assume that principal stresses are also radial and tangential, and hence shear stresses T,~ may be assumed to be zero.

The state of stress in a small plane element far from the tunnel is illustrated in Fig. 5(b) and the conditions of equilibrium are

. . . (7) $+2=-o ......

%+%=y ........

In general vertical and horizontal stresses are not principal stresses, but, in order to obtain a

solution, we assume 7XQ =O, in which case the conditions of equilibrium become

afJx -= 3X

0 . . . . . . . . . .

au z- ,&-y . . . . . . . . . . (10)

and the failure criterion becomes

1 0% = P.? or OX=-a, . . . . . . . . (11)

P

Consider the field of stress illustrated in Fig. 6. Within the region bounded by the circular

arc AFE principal stresses are radial and tangential, and beyond AFE they are vertical and

horizontal. At the point A the failure criterion is just satisfied and the principal stresses

are tangential and radial; hence, if the tunnel pressure uT is reduced, the stresses at A are

(Jr = (Ta = ug

erg = 01 = jL*a,

At the point E the failure criterion is satisfied and the tangential stress may be found by integrating equation (5) around a circular arc just inside AFE; hence at E the stresses are

The failure criterion is satisfied along EC and, substituting for (T@=~L(T~, equation (5) may be

integrated along EC with the limit uI = uT at r = R. The result may be expressed in the form

uB = u1 = pa,+2yrs

2yrs UT = cr’3 = as+-

CL

UT = ~{(~)~-z[~+~)(~-2,+Y] -Y} . . . . (12)

Equation (12) relates the tunnel pressure to the surface pressure, the tunnel geometry and the soil properties.

We have shown so far that when the tunnel pressure is given by equation (12) the states of stress at A and along EC are statically admissible. By integrating equation (5) around cir-

cular trajectories starting from EC and i.ptegrating equation (4) along radii it may be shown


Fig. 6. Possible states of stress in the soil around a shallow tunnel

(Potts, 1976) that the stress field is admissible everywhere within the region of soil contained by the circular arc AFE and that the soil just reaches limiting equilibrium around AFE.

In the region beyond AFE horizontal and vertical stresses are related by

(JZ = Xa, . . . . . . ._. . (13)

and the failure criterion is not violated when

A2lorl\<p . . . . . . . . . . (14) p

From equations (10) and (13) the state of stress in an element at a depth z is

a_. = a,+yz

a, = ; (0s + V)

Consider the change in the state of stress along a radius across the arc AFE at a point such as F in Fig. 6. We imagine that the arc AFE represents a discontinuity in the stress field of finite width containing a large number of fan zones; the arc AFE is not itself a strong discontinuity but the state of stress changes smoothly over a small distance across AFE. The states of stress across such a discontinuity are statically admissible when

S, < e26tane

K‘ . . . . . . . . . . (1%

where S=+(U~ + u3) either side of the fan zones, S, > S_ and 6 is the change in the direction of the major principal stress. Noting that S, 2 S_, we see that equation (15) is satisfied for all 6, including 6 =O, when S, = S_ ; hence the states of stress across the discontinuity are admissible everywhere around AFE when the values of S=$(ul + u3) just inside and just outside the dis-

continuity are equal.


CL?R

Fig. 7. Form of the lower hound solution of equation (16) for tunnels in a weightless soil

At the point F in Fig. 6 [z=r,(l -cos 0)] the values of S just inside and just outside the

discontinuity are given by

si = 3(p+ 1) C us+? (1 -cos 0)

I

s, = 3 1 +i [a,+yrs (l-cos e)] ( 1

and the stress field is admissible when

s, = s,

To continue, it is convenient to examine the two limiting cases when aa=0 and when ~a is large and y may be assumed to be zero.

Firstly, when y=O, S,= S, when X= l/p and, from equation (14), the failure criterion is not

violated. From equation (12) the lower bound or safe tunnel pressure is given by

rs 0 1-p

cry. = as - R

or, since rs = C-k R

. . . . . . . . (16)

This result was obtained1 in slightly different form by Atkinson and Cairncross (1973). How- ever, they did not examine the stress field in the whole region of soil and the present analysis confirms their result as a true lower bound within the limits of the assumptions made here.

The form of equation (16) is shown in Fig. 7 as +/us plotted against depth of burial defined by C/2R, where C is the cover above the tunnel crown. When C= 0 and the crown and surface

coincide uT = us, which must be correct; as the tunnel gets deeper the safe tunnel pressure

reduces rapidly. Secondly, when us = 0, Si = So when

r+l -= P

1,;

1 It should be noted that Atkinsonand Cairncross(l973) definep=(o,-uo,)/(o,+o,)and not p= ol/oa as in this Paper.

STABKITY OF A SHALLOW CIRCULAR TUNNEL IN COHESIONLESS SO11 211

Fig. 8. Form of the lower hound solution from equations (18) and (22) for tunnels without surface surcharge loading

i.e. when X=tL

and, from equation (14) the failure criterion is not violated. From equation (12) the lower bound or safe tunnel pressure is given by

$=&)[($@-z(3-;)-l] . . . . . . . (17)

and, as before rs= C-t R. Finally, we should note that we have not allowed the possibility that the radial stress in the

vicinity of the tunnel may be the major rather than the minor principal stress. The critical location will be at the tunnel crown, and we examine the state of stress in the soil adjacent to the tunnel around BDC in Fig. 6. The tunnel pressure is a principal stress, hence for r= R 7T8 = 0 and, from equation (3),

$--yRsinB=O . . . . . . . . . (18)

At the invert the failure criterion is satisfied, and the tunnel pressure is the minor principal stress; hence the stresses at C are

At the crown the failure criterion is satisfied and the tunnel pressure is the major principal

stress; hence the stresses at B are

(3, = a1 = 0.r.

u8B = u3 = +T/p

Equation (18) may be integrated around BDC giving

uBC-uBB = 2yR . . . . . . . . . (19)

Hence the failure criterion is satisfied simultaneously at the crown and invert when

and a lower bound or safe tunnel pressure is given by

$=(p:_l) . . . . . . . . . (20)


Fig. 9. Collapse mechanism for an upper bound solution

For the case when y = 0, the solution uT > 0 is trivial, but for the case when us = 0 and y > 0 the lower bound or safe tunnel pressure given by equation (20) may exceed that given by equation

(17). The forms of equations (17) and (20) are shown in Fig. 8 as a,/2yR plotted against depth of

burial defined by C/2R; safe tunnel pressures must lie on or above the hatched line. The relationship between the two solutions illustrated in Fig. 8 suggests that stresses close to the tunnel may be critical for deep tunnels.

Upper bound solution

An upper bound to the the true collapse load can be found by selecting any kinematically possible collapse mechanism and performing an appropriate work rate calculation. It may be noted that the solution will lead to an unsafe or low value for the tunnel pressure. The accuracy of an upper bound calculation will depend, in part, on the closeness of the assumed mechanism to the real one, and it is in the selection of a collapse mechanism that the model tests have proved to be of value.

By observing the behaviour of the soil around the tunnels in a series of tests conducted in the laboratory without surcharge loading a collapse mechanism similar to that shown in Fig. 9 suggested itself (Atkinson et al., 1975). The wedge ABC in Fig. 9 moves downwards but dilates at a rate sufficient to prevent separation along AB and BC. To satisfy compatibility requirements the planes AB and BC must make an angle 2~ at B, where v is the angle of dilation.

Noting that the rate of work dissipated in a perfectly plastic frictional material with an associated flow rule v = + is zero, the appropriate work rate calculation involves only the loads due to the tunnel pressure and to the self weight of the soil. The upper bound or unsafe tunnel pressure is then given by

f$=&&-&++;) . . . . . . . provided that C/R 2 1 /sin 4 - 1. This limitation is necessary to ensure that the apex of the sliding wedge at B is below the soil surface. It may be noted that the upper bound, or unsafe, collapse pressure given by equation (21) is independent of the depth of burial and the magni-

tude of any surface pressure.

COMPARISON BETWEEN THE THEORETICAL AND EXPERIMENTAL COLLAPSE PRESSURES

Equations (16), (17), (20) and (21) lead to values of the tunnel pressure at collapse which, in theory, bound the true values and these may be compared with the collapse pressures observed in the model tests and contained in Figs 2-4.


NOW : upper bound ci= 046 kN/ml

0

0 I.0 2.0 3.0 C/2R

(4

IO f

Note : Upper bound UT= 046 kN/ml

OS: kN/m’

6)

Fig. 10. Comparisons between model test results and theoretical predictions for the collapse of shallow circular tunnels for +=SO”: (a) a,=210 kN/m’; (b) C/2R=05; (c) u,=O

The magnitude of the maximum angle of shearing resistance 4 depends on the magnitude of

the state of stress for a particular soil. The sand used in the model tests has been examined in simple shear tests by Stroud (1971), and from his results a value of 4 of approximately 50” is appropriate for the stress levels in the soil around the model tunnels at collapse. This relatively large value for + is a consequence of the very low stress levels in the model tests. Equa- tions (16), (17) (20) and (21) have been evaluated with d= 50” for the various series of mode1 tests and the bounds are shown together with the appropriate experimental results in Fig. 10. For +=50° the value of u,/2yR from equation (20) exceeds that given by equation (17) for all depths of burial and hence, in Fig. 10(c) only the lower bound given by equation 20 is shown.

In Figs 10(a) and (b) the experimental results seem to fall very close to the lower bound solutions. In some instances they lie above, or on the unsafe side of, the theoretical bound, and this may be due to neglect of y in obtaining equation (16) or to an inappropriate value of += 50” for mode1 tests in which the stress levels were raised owing to surface surcharge pressures; if a more conservative value of $= 45” is chosen all the model test results lie between the


upper and lower bound solutions. Nevertheless, the lower bound calculation of equation (16) seems to predict the very small tunnel pressures at collapse with reasonable accuracy.

For the cases where u5=0, the experimental points in Fig. 10(c) obtained from laboratory and centrifuged models fall between the appropriate upper and lower bounds and, moreover, the bounds solutions are relatively close to each other. It can be seen that for 4 = 50” the lower

bound given by equation (17) applies only for tunnels at very small depths of burial and the bound given by equation (20) predominates. This is convenient since equation (20) was obtained without any assumptions about the shear stresses in soil adjacent to the tunnel.

DISCUSSION

Figure 10 clearly indicates the ability of the simple upper and lower bound solutions to predict the stability of the model tunnels with satisfying accuracy.

The work described in this Paper has been limited to plane strain and thus represents the conditions in the plane tunnel section away from the tunnel heading. However, recent model tests conducted by Orme (1975), Argyle (I 976) and Aspden (1976) on headings in sand indicate that collapse occurs in the cylindrical section of the tunnel away from the heading itself. In tests where the cylindrical section was lined collapse could only occur at the tunnel face, in which case the collapse pressure was slightly lower than it was when collapse occurred in the

cylindrical section of unlined headings. In both cases collapse pressures predicted by the bound calculations presented in this Paper were found to be in good agreement with the experimental results. The calculations therefore seem to be applicable to the three-dimensional as well as the plane strain situation.

So far only dry sand has been considered, but the theoretical solutions, which are based on effective stress, may be applied to saturated sands provided that the pore water is stationary and the pore-water pressures around the tunnel are known. The collapse pressure of a tunnel in saturated sand is then simply the sum of the pressures predicted by either the upper or lower bound solutions and the pore-water pressure. The situation will, however, be rather more

complicated if there is a steady state, or transient, seepage. Clearly the values of the angle of shearing resistance + applicable to the sand used in the

present investigations may not be appropriate for granular soils in the field. To obtain satis-

factory estimates of tunnel stability from the analyses given here a realistic value for 4 for the field soil at the stress level in the tunnel at collapse should be found from laboratory tests.

It should be noted that while the Paper is entirely concerned with stress at failure, the theory assumes that the tunnel support can readily adapt itself to the radial deformations of the ground for compliance with the mechanisms of the upper bound solution and the states of stress of the lower bound solution. Such deformations cannot be predicted from the theory, but from the lead shot measurements carried out during the model tests, it was found that radial displacements prior to collapse remained less than 5% of the original tunnel diameter

(Potts, 1976).

ACKNOWLEDGEMENTS

The research described in the Paper was carried out in the Cambridge University Engineer- ing Department and was supported by a Transport and Road Research Laboratory research contract for investigations into the behaviour of tunnels and tunnel linings in soft ground; Dr Potts was supported by an award from the Science Research Council. The Authors are in- debted to Dr E. T. Brown for contributions made to the research, and to Professors A. N. Schofield and J. A. Cheney and to Mr C. Collinson for collaboration in the use of the Cambridge centrifuge.


REFERENCES

Argyle, D. M. (1976). An investigation into the collapse of tunnel headings in dense sand. Engineering Tripos Part II Research Report; Cambridge University.

Aspden, R. (1976). Collapse of unlined tunnels with headings in dense sand. Engineering Tripos Part II Research Report; Cambridge University.

Atkinson, J. H., Brown, E. T. & Potts, D. M. (1975). Collapse of shallow unlined circular tunnels in dense sand. Tunnels & Tunnelling I, 81-87.

Atkinson, J. H. & Cairncross, A. M. (1973). Collapse of a shahow tunnel in a Mohr-Coulomb material. Proc. Symp. Role of plasticity in soil mechanics, Cambridge, 202-206.

Atkinson, J. H. & Potts, D. M. (1976). Subsidence above shallow circular tunnels in soft ground. Internal Report CUED/C-Soils/TR 27, Cambridge University.

_ -

Atkinson. J. H.. Potts. D. M. & Schofield. A. N. (1977). Centrifugal model tests on shallow tunnels in sand. Tunnek & Tunnelling 9, No. 1, 59-64. ’ . ’

Davis, E. H. (1968). Theories of soilplasticity and the failure of soil masses. Soil Mechanics: Selected Topics, (Ed. Lee, I. K.) Butterworth, 341-380.

Hansen, Bent (1958). Line ruptures regarded as narrow rupture zones: basic equations based on kinematic considerations. Proc. Conf. Earth pressure problems, Brussels 1, 39-48.

Orme, G. (1975). Unpublished research report. Cambridge University. Palmer, A. C. (1966). A limit theorem for materials with non-associated flow rules. J. Me’cantque 5, No. 2,

June, 217-222. Potts, D. M. (1976). Behaviour of lined and unlined tunnels in sand. PhD thesis, Cambridge University. Stroud, M. A. (1971). Sand at low stress levels in the SSA. PhD thesis, Cambridge Univeristy.

atkinson and potts 1977

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