DISCUSSIONS

The circular tunnel in elastic ground MUIR WOOD, A. M. (1975). Gt%technique 25, No. 1, 115127.

D. J. Curtis, Mott, Hay and Anderson, Croydon Muir Wood presents a simple approach to the problem of proportioning tunnel linings,

which can be useful in many situations. A great amount of mathematical analysis is contained in the Paper, and this contribution questions some of the assumptions and the accuracy of some of the formulae. The extension of the method to include the parameter of time in the context of visco-elastic behaviour of ground and lining is indicated, in a manner which retains the simplicity desired by the Author.

NOTATION

The Author’s notation is followed wherever possible, but some variations and additional symbols are employed. In general, a symbol with subscript c refers to ground behaviour, whilst a symbol without this subscript refers to the lining. All stresses and displacement described occur at the interface between ground and lining, at r=ro.

g H3 N

PW Q Q2

Qlr sn, St TO

gravitational acceleration water head at radius r3 circumferential (hoop) thrust per unit length of lining

pressure on lining due to water Author’s compressibility factor R, analogous to the inverse of Author’s

stiffness factor R,, expressed thus as for a solid lining Qz= Qro2/t2

seepage factor normal and shear stresses time interval between excavation and installation of lining

Ul

uo

ii

Y E

P %, =h

deformation of lining deformation of ground in absence of

a lining deformation of ground inhibited by presence of a lining

rate of creep strain density of water pre-existing vertical and horizontal

stress in undisturbed ground

(%--%=Po) ratio of uniaxial creep strain to immediate (elastic) strain

BENDING IN THE ELLIPTICAL MODE

Using Airy’s stress function, for a circular opening in an infinite ground, deformations at the periphery of the opening can be obtained:

uo = w [(S - 6v)S,,+ (4-6v)S,,] cos 28 . . . . . (Al) 0

uoe = -r’$~+V)[(4-6~)S..+(5-6~)S,1sin 20 . . . . (A2) c

where stresses S,, cos 20 and - S,, sin 28 act normally and tangentially around the opening. When the opening is excavated, the distortional unloading of the ground is

PO T cos 28

PO so that S,, = S,, = T

232

The elastic, or immediate deformation of the ground is

DISCUSSIONS

u o. = ~(3-4v)+cos28 . . . . . . . (A3) C

UOBO = -r”~+“)(3-4~)$sin28 . . . . . . . (A4) C

A thin inextensible lining will deform

Ul = r,4(2S,+S,)cos 28/18EZ . . . . . . - (A5) uel = -ro4(2sn+ S,> sin 28/18EZ . . . . . . . (A6)

producing forces in the lining

Z&$ (S,+2S,) c0s 28 . . . . . . . . (A7)

M = $(2Sn+S,)c0s 28 . . . . . . . . * WV

Making the Author’s assumption of no shear between lining and ground, (i.e. S,=O) we proceed as follows.

(a) The ground will deform radially under the action of % cos 28 according to eqn (A3).

(b) The lining will deform radially under a load S, cos 28, from eqn (A5)

ul = gz2S., cos 28.

(c) The ground will be restrained from deforming radially under the action of a load S,,, cos 28 = -S,, cos 28 from eqn (Al)

no = <~C(l+u)(5-6~)S. cos 28.

(d) Since u. +a0 = ul, we find

s = 3(3 - 4v)P,,/2 n 5_6v+4e2 .

where

From eqns (A7) and (A8)

. . . . . . . 6412)

...... (A9)

...... (A101

...... W 1)

The foregoing analysis may be compared with the Author’s.

(a) Eqn (12) may be directly deduced from section (a) above. (b) Eqns (16) and (A12) are not identical because the Author has made two implicit

assumptions.

Muir Wood does not define the cause of the change in deformational loading PO, but the relationship stated after eqn (14) is equivalent to assuming, as we have, that PO is the stress which would act in the ground at the periphery if the tunnel had not been excavated. In such

DISCUSSIONS 233

Fig. 1

an unperforated mass of ground the stress would be the same no matter at what radius the load P, were applied. However, the further away the load is applied, the greater the deforma- tion of the periphery of a hole in the ground. The bending moment in the lining is, therefore, a function of the location of the cause of the distortional stress changes around the tunnel as well as of their magnitude.

The stress regime in the vicinity of a deep tunnel, before excavation, is usually considered to be uniform, i.e. uV and uh are each constant at all points. At any point on the periphery of the tunnel, the normal and shear stresses are obtained by transformation of axes. The Author assumes that if only normal pressures can be transmitted across the interface between lining and ground, the shear component of PO will not affect the load on the lining. Since the shear component makes a contribution to the radial deformation of the ground, uoC (eqn (A3)) this assumption is incorrect.

The values of bending moment produced by the two equations are compared in Fig. 1.

EFFECT OF SHEAR FORCES BETWEEN GROUND AND LINING

If we assume full shear interaction between lining and ground, and repeat the procedure given in an earlier section of this contribution, but equate both radial and tangential deforma- tions, we find

s = (1- QzPoP

n l+Q,(g 1

. . . . . . . . . . (Al3)

s = (1+2QzPoP

t l+Q,(g 1

. . . . . . . . . . (A14)

N msx _ -+Poro

2 . . . . . (Al%

M max =+/{l+Q2[~]} . . . . . (A 16)

234 DISCUSSIONS

Author’s equation (31: ,

Fig. 2 O.11

It may be noted that Qz usually possesses a numerical value well in excess of unity, and then S, is a negative quantity. Provided that the uniform component of the total load on the lining, which will be a positive number, exceeds S, there is no danger of locally separating the

lining from the ground. BPR Since S, =ji/2, the Author’s equation z = 2 I+R

is invalid. He has assumed that the normal and shearing components of P,, are shared between ground and lining in identical proportions, which by virtue of eqns (A13) and (A14) is not the case.

If the shear stress in the interface is limited to, say

S, = T . . . . . . . . . . (A17)

then

s, = 3(3 - 4~) $‘- {2Qz + (4 - 6v))T

4Qz+5-6~ . . . . . . 64w

Equation (24) should be derived as follows. If we write

Eg = y [(l -“)Os--va,] c

and

u l&u, l O = -+--

r r80

we eventually obtain for direct comparison with eqn (24).

uoo = Cl+4 -- EC [

F~~)$+(L+)T~~] . . . . . (A19)

The Author’s criterion (a) is expressed by eqn (27), in which the mean pressure jj should surely be replaced by the mean normal pressure Ap from his eqn (37).

Criterion (b) is expressed by eqns (A14) and (A17), i.e.

Since

l!=S 2 n . . . . . . . . . . (A21)

DISCUSSIONS

we find from eqns (A13) and (A20)

235

* . . . . . . .

<+fi when Q.$+l . . . . . . . (A23)

Figure 2 shows the difference between eqns (A23) and (31). The general expression for h is obtained from eqn (Al) as

X = 3S,,E,/r,(l +v)[(5 - 6v)&,+ (4- 6v)S,,] . . . . (A24)

Since there is coupling between radial and tangential deformations, this should be reflected by using eqns (Al) and (A2) to derive coupled ground reaction coefficients.

EFFECT OF THICKNESS OF THE LINING

In all the expressions in the first two sections of this contribution, the Author’s distinction between r. and the centroidal radius qr, has been ignored. Comparisons between analogous formulae derived for a continuous thick ring suggest that for bending moments M, Mccp,~2r02, but in the expression for Q2, r. should be retained. For thrusts N, NccPOr,,, but in the expres- sions for Qz or R,, vro should be used for ro.

GROUNDWATER

It is difficult to see why the integration constant A in the Author’s solution should be zero. If the ground is infinite in extent the flow is also zero and therefore B must be zero. We can proceed along lines similar to those of the Author, and indeed extend the method to account explicitly for the effect of the lining. If it is assumed that although the ground is infinite in extent, it is freely permeable beyond radius r3 from the axis of the tunnel where r3 > r. and that the water head at r3 is Ha, the pressure on the lining due to seepage forces through the ground is given by

1 . . . . . . (A25)

where

Q=&&-$yR, -t

k Qk = E Wdro> rl’o

ro-ri The pressure on the lining due to direct water loads is

p N pgH,v/(l + Q><l + Qg) . . . . . . . (A26)

(If the lining is impermeable p = 0 and (p = pgH,/(l + Q)). The hoop thrust in the lining is

N = (p+pW)ro . . . . . . . . (A27)

and the flow into the tunnel

(A281

Over a wide range of ground and lining properties it will be found that the total effect of water loads upon the lining, (eqn (A27)) will be very close to the full hydrostatic head. The steady state simple seepage condition, is, however, a conservative design case, because it

236 DISCUSSIONS

ignores the historical variation in ground properties from the time of excavation of the tunnel to final equilibrium.

The seepage model referred to earlier is not very satisfactory as it will overestimate the water inflow compared with a more realistic model. The stress calculations are reasonably accept- able when compared with finite element analyses.

EXTENSION OF THE METHOD

We can now continue the Author’s search for a simple initial approach to tunnel design, and indicate how easy it is to employ a visco-elastic model. First, the ground movement which is delayed because it is supported by the face at the time the lining is installed is examined. The interaction formulae derived above apply directly and it is merely necessary to modify the modulus values, so that

E’ = E/(1++) . . . . . . . . (A29)

where 4 = E,/E, (Fig. 3)

The ground movement which has occurred before the lining is installed is now considered. (The derivation is given here for distortional loads, but is similar for the uniform load situation).

If the creep strain of a uniaxially stressed column is expressed as

where F is a function of time T, which has a value of unity when T= 0 and zero when T = co, at time To when the lining is installed the ground movements are given by replacing EC in eqns (A3) and (A4) by EC/{1 ++,[l - F(TJ]}.

If no lining had been installed, the movements for infinite time would have been calculated from the same equations, writing for the modulus EC’= EC/(1 +q&).

The load due to creep gradually builds up on the lining. If the final values are S, 20 and S,cos 28 the deformations of the lining are given by eqns (A5) and (A6), substituting E’= E/(1 + 4) if the lining is also a visco-elastic material.

The deformations of the ground are restrained by the lining, and eqns (Al) and (A2) are used for stresses - S, and - St, and substituting EC for the modulus.

Equating deformations and solving for the two unknowns S, and St we find that the resulting expressions are those resulting from the purely elastic analysis, e.g. eqns (A15) and (A16), but with modified modulus values as above and multiplied by F(T&,/( 1 + CO).

We can substitute into eqns (A7) and (A8) to find the forces in the lining. For example, eqn (A16) becomes in the visco-elastic case

M max

where

EC 1 l++ ro3 Qz =BiTviq-121 0

(A30)

Both F (To) and &/(l -+I&) cannot exceed unity. In the simple Kelvin visco-elastic model F(T,J = e - YTo where y is a ground property defining the rate of creep. The parameter 4 defines the amount of creep.

DISCUSSIONS 231

Fig. 3. Kelvin visco-elastic cell

These parameters are all that are required in addition to the linear elastic ground (or lining) properties. In principle they can be obtained from geotechnical studies, just as are the elastic properties.

CONCLUSIONS The simplicity of the visco-elastic relationships provides a means of examining numerically

the sensitivity of the load to be carried by the lining. In tunnelling terms we can assess the circumstances in which it will be profitable to delay the installation of a lining, as well as the implications of the amount of creep expected.

As the Author has emphasized in his Paper, however, the simplicity which is the attraction and usefulness of the analytical relationships presented must not be allowed to distract attention from the difficulty of modelling the behaviour of real ground.

Author’s reply to Curtis

The difference between my eqn (16) and his eqn (A12) stems entirely from my simplification in ignoring initial shear stresses between ground and lining. It is of interest to note (Curtis, Fig. 1) that my solution lies intermediately between the two cases he considers for zero shear and for fully developed shear respectively between ground and lining, the validity of the latter case requiring confirmation in respect of criteria (a) and (b) (p. 119 of the Paper) using Curtis’ eqn (A22) for the latter.

Consideration of seepage forces within a finite cylinder provides a practical first approxi- mation for assessment of inflow and seepage forces, although it is often more valid to consider the familiar form of potential flow towards the tunnel, considered as a sink, from an imaginary source, with the upper surface of the aquifer in which the tunnel is situated constituting a plane of symmetry.

For the extension of the elastic to the visco-elastic case, while the substitution

E’ = E/(1 ++[l --F(T)]}

provides a simple first parametric test, there is only a direct application of this approach where the time effects are long-term in relation to the period during which the tunnel advances through the influence zone. Otherwise, the time base for F(T) cannot be measured from a unique instant and Curtis’ 57, at the time of placing the lining will relate to a lower fraction of total creep strain as the distance from the tunnel increases. Even for an assumed Kelvin model, a valid solution will require a numerical integration in recognition that creep will occur as the stress pattern is varying. Thus the problems of three-dimensional time dependence unfortu- nately persist. However, Curtis’ simplification will generally lead to a conservative result.