an analytical solution for axisymmetric tunnel problems in elasto plastic media

7/26/2019 An Analytical Solution for Axisymmetric Tunnel Problems in Elasto Plastic Media

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS.

VOL.

8.

325-342

(1984)

A N ANALYTICAL SOLUTION FOR

AXISYMMETRIC TUNNEL PROBLEMS

IN ELASTO-VISCOPLASTIC MEDIA

P.

FRITZ

Swiss Federal Institute of Technology, Zurich, Switzerland

SUMMARY

An analytical solution is presented

for

the time-dependent stresses and displacements in plane strain

around a circular hole when it

is

loaded by an axisymmetric internal and far-field pressure. The material

is assumed to be elasto-viscoplastic with dilatant plastic deformations according to a non-associated flow

rule. Strain softening is considered by a modified St. Venant slider which is characterized by Mohr-

Coulomb yield conditions

for

both the peak and the residual strengths.

INTRODUCTION

Time-dependent deformations of the rock mass around a tunnel are brought about by the

excavation progress and by time-dependent mechanical properties

of

the rock. In this article,

only the influence

of

time-dependent material properties is considered, i.e. the problem will

be treated as two dimensional. A simple model for the rock, which takes into account time

dependency, could be a visco-elastic model. However, such a model should be applied to

competent rocks only

in

which no serious problems arise (Salamonz5).When serious problems

arise, the behaviour

of

the rock should be approximated by a non-linear model. This is done

in the present work by means of an elasto-viscoplastic model. Since for deeply located tunnels

the initial state of stress may be

in

a first approximation, idealized as hydrostatic and due to

the assumed isotropic medium, the problem is treated as axisymmetric in geometry, load and

material, thus enabling an analytical solution. In tunnelling practice, parametric studies often

have to

be

carried out. Therefore, it is especially advantageous to have such an analytical

solution.

Proposals for the analytical treatment of the axisymmetric plane strain problem of a circular

hole in an infinite medium as well as the associated problem

of

a thick-walled cylinder subjected

to

both axisymmetric internal and external pressures have appeared

in

the literature over many

decades. Early investigations concentrated on the calculation of the state of stress without

consideration of defomational and time effects. Thus St. Venantz4 had already obtained a

solution for the

state

of

stress

in

a

fully plastic, thick-walled cylinder. Terzaghi2' described,

qualitatively, in a very apt manner how, near the excavation for a tunnel, stresses exceed the

rock strength, causing the surrounding rock, which is still in an el'astic state, to carry a part of

the load. However, Hartmann was the first to succeed in calculating the stresses in an

elasto-plastic material assuming a cohesionless soil with friction (see Nadai ). W e ~ t e r ga a r d ~~

determined the state

of

stress with consideration of cohesion. The significanceof these solutions,

0363-9061/84/040325-18$01.80

@ 1984 by John Wiley Sons, Ltd.

Received

11

August

1982

Revised 28 February

1983


2/18

326

P.

FRITZ

in relation to their useful application to tunnelling problems, has been discussed by Fenner

and by Kastner. Proposed models to extend these time-independent solutions,

so

as to be

able to consider the

deformational state,

may be found in the work of Nadai who assumed

the so-called plastic zone to be fully incompressible. With the assumption

of

purely deviatoric

plastic strains, Hill

et aLI3

solved the problem by also considering elastic strains in the plastic

zone using the yield conditions of Tresca and von Mises. Also assuming deviatoric plastic

deformations but applying the yield condition of Mohr-Coulomb, Deist discussed stability

criteria for a gradual onset of strain softening after reaching failure, which continues until the

strength is reduced to zero. Hendron and Aiyer extended this solution in two ways: first,

plastic volumetric strains were considered to be governed by the normality rule and, secondly,

a sudden loss of strength was assumed on reaching the yield stress. The strength drops to a

residual value, depending upon the hydrostatic stress component, by introducing a residual

cohesion

cR.

Egger4 developed an analytical solution for similar rock conditions by considering

progressive strain softening. Both UlgudurZ8 and PanetZ3 obtained closed-form analytical

solutions for this problem, whereby the first author assumed complete strain-softening

analogous

to

Deist and the second author neglected elastic strains in the plastic zone.

SALUSTOWCZ.1958 MADEJSKI. 196

WIERZBICKI.1963 M M W

+ DIMOV. E)66 NONAKA.1978

LOONEN.

1962

PARASCWEWOV. SALAMON.1974 * m

1964

Figure

1 .

Material laws represented by mechanical mode ls used by variou s authors

to

obtain analytical so lutions

Analytical solutions for the treatment of openings in media with

time-dependent properties

are comparatively few. SalustowiczZ6 alculated the state of stress in a rock mass for the von

Mises yield condition, without consideration

of

the deformations. The time-dependent material

model used is shown symbolically in Figure

1

with a combination of Hookean (spring),

Newtonian (dashpot) and St. Venant (slider) elements. Madejski solved the problem of a

thick-walled sphere with a similar material model (Figure l ) ,also by assuming von Mises yield

condition (deviatoric plastic deformations only). LoonenI6 applied the Mohr-Coulomb condi-

tion by assuming the plastic zone to be fully incompressible. Further material models used by

different authors to obtain analytical solutions are presented in Figure

1.

However, all these

solutions have in common that plastic f l ow is assumed to be at constant volume, mostly with

use of the von Mises yield condition. In this work, elasto-viscoplastic material behaviour based

on the model of Madejski (Figure

1)

is investigated. Dilatancy effects, as well as a sudden loss

of strength to a residual value, will be considered. For this purpose, two yield criteria-for

peak and residual strength-of the Mohr-Coulomb type are introduced together with a

non-associated flow rule. The load history is assumed

to

comprise an initial stress

pa

(com-

pression positive) before the hole is excavated. The stresses acting normal to the plane are

a%.Owing to the excavation, secondary stresses and the corresponding radial deformations

(positive in the direction of the opening) are developed.


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AXISYMMETRICAL TUNNEL PROBLEMS 327

ELASTO-PLASTIC MATERIAL

The solution for a time-independent elasto-plastic material behaviour (Figure

2)

will serve as

a boundary condition at time t -

00

for the time-dependent approach. The derivations for this

case follow basically the work of Florence and Schwer.6 As an extension to that paper, a

modified definition

of

the St. Venant element, including residual strength and a non-associated

flow

rule, is used. If a circular tunnel is excavated in an elasto-plastic rock which is subjected

at infinity to an axisymmetric pressure pa in the vicinity of the tunnel, plastic deformations

will develop. The corresponding area, the so-called plastic zone, extends from the boundary

of the hole of radius

Ri

o a radius pm. Outside the plastic zone, i.e. for r > pao,the material

remains elastic. Let the principal stresses in the plane of the tunnel be u: in the radial direction

and a in the tangential direction. Then, equilibrium requires that at an arbitrary distance r

from the tunnel axis

Figure

2.

Elasto-plastic material model

For small

deformations, the kinematic relations, which relate the strains E with the radial

displacements urn are

d(Au")

A&,=-

dr

A U r n

A&,

=

'A'

signifies the change of these values due t o excavation. In the elastic zone, i.e. for

t pm

the generalized Hooke's law is valid. Denoting the modulus

of

elasticity by E and Poisson's

ratio by

v,

it provides for plane strain conditions

l + v

E

&:=- [(

1

-

) A u ? - ~ A u p ]

l + u

E

E;=-[(~-V)AUF- vAu:]

A&:=O

where

(3)


4/18

328 P.

FRITZ

Substitution of the strains

A :

=

A s ,

and

A s :

=

A s t

from equation (2) in Hooke's law (3) and

elimination of

A u m

from the equilibrium condition

1)

leads, together with the boundary

conditions

r = p a o :

u : = p p

r+w:

u:

=

pa

and equations

(4), to

u: =u*

The behaviour in the plastic zone

is

governed mainly by the properties

of

the plastic St. Venant

element in Figure 2. This so-called modified St. Venant element (Fritz*) starts to deform when

its stress reaches, for the first time, the peak strength, defined by a yield condition

Fp.

The

initiated deformational process is then characterized by the residual strength, described by a

second condition F R . The modified St. Venant element can be interpreted as a mass on a

plane, whose sliding friction is smaller than the static one. In the present case, the conditions

for F P and

F R

are formulated analogous to the theory of Mohr-Coulomb.

As

long as the

axial stress lies between the principal stresses in the plane perpendicular to the tunnel axis, i.e.

u, =

const

N~RMACSTRESS

Figure

3.

Stress-strain relation and yield conditions of the modified St. Venant element

this theory relates the principal stresses

u:

and

up

to the strength parameters (cohesion c and

angle

of

friction

4).

Denoting these parameters according to Figure

3

by

c p , d P

and

c R , R

for the peak and residual values, the yield criteria are


5/18

MISYMMETRICAL TUNNEL PROBLEMS

and

with

F~ =

u7-

mRu?-

of:

=0 ( for r


6/18

330

The plastic strain increments

P.

FRITZ

are directed normal to the surface FG. Herein, A is a scalar factor of proportionality. The

incremental denotation can be interpreted as differentiation with respect to the radius of the

plastic zone (Hill et Except for

4G

0 or

m G=

1,respectively, formulation (14) includes

plastic volumetric strains.

Addition of e< and e? from (14), integration with respect to

pm

and taking into account

the boundary condition

pa= r: A s f ) = A ~ ~ ~ , = ~ ~ ~ - h s ~ ( ~ = ~ ~ )

A &? = A E F ( ~ = ~ ~ ) - A E F ( ~ = ~ ; )

A&

=0,

where

As:

and

As:

are given by (3) ,

( 5 )

and ( lo), leads to

As =

0

For plane strain conditions, the deformation of the elasto-plastic model

of

Figure

2

is given

by summing up the deformations of the individual elements

Aer= As:+AsF

A&,= As:+Aef

(16)

As,=AsC,=Ae =O

As before, the elastic components As:,,, of the strains due to excavation are determined by

Hooke's law, which leads, for plane (plastic) strain conditions, to

a:= a%+ ( A o ~ + A a F )

(17)

Owing to the special formulation of the flow rule

(14)

or the potential surface FG (13),

a

is also independent of the deformations.

At this stage, the assumption (6) could be discussed after which

a:

should be intermediate

principal stress, thus excluding stress states at edges of the yield surface in principal stress

space. According to (17),

Au:

is directly proportional to Poisson's ratio v. Considering the

great uncertainties in the determination of

v,

even in the laboratory, from the practical point

of view such a discussion (cf. Florence and Schwer6) is not relevant and is therefore omitted.

In the following, the radial displacements

Aua

shall be derived. Elimination of Aum in the

kinematic relations (2) leads to the compatibility condition

dAe,

dr

r-=Asr-As,


7/18

AXISYMMEXRICAL TUNNEL PROBLEMS 331

Substitution of A , and A E ~ccording to (16), A E : and A s : from Hooke's law (3) , A E ~rom

(15)

and the stresses

u:

and

up

according to

(10)

eads to the differential equation

with the solution

The constant c1 is obtained from the boundary condition

r=po0: A E ~ = A : ( , - ~ ~ , - A : ( , _ ~ ~ ,

and substitution

of

the elastic strains from

(3)

and the unknown stresses

from

equation (5)

and (10).The displacements are derived from the kinematic relation (2) for A E , and expression

(16) for the deformations. Substitution of the elastic part A E ~rom Hooke's law (3) and

introduction of the stresses according

to (10)

eads to

A u m = - r'+

k ,

(

JmR-'

+

k2(

F)

k 3 } (for r~ p,)

where

k l =

[

1

-

) 1+mRmG- ' ] ( * + p p )

mR +m m

k3=pa

- p p -

,

-

2

The special case, that in the plastic zone only elastic and no plastic volumetric strains occur,

is given by

(14)

with

as (18a) with

dc

=o+ mG= 1

ELASTO-VISCOPLASTICMATERIAL

For an elasto-viscoplastic medium (Figure 4), an analytical solution is presented which partly

involves a numerical evaluation (equation

(31)).

Such a material exhibits, at the instant of an


8/18

332

P.

FRITZ

V

Figure

4.

Elasto-viscoplastic material model

arbitrary loading, purely elastic behaviour. Therefore at time t =

0

the problem is described

by equation 9, rovided that

pW

s replaced by the excavation radius R i ,and p,, by the internal

pressure

pi .

This leads to

Cr ;

=

With time, the dashpot relaxes in zones in which the yield condition of the modified St. Venant

element is fulfilled. Excluding a change in the statical system (e.g. the introduction of a lining

for t >0), the final state at t - 00 corresponds to the findingsof the foregoing section. It remains

to determine the time-dependent development. This is governed essentially by the actual stress

in the Newton dashpot, i.e. by the plastic strain velocities. To make an analytical treatment

possible, the plastic strain velocities are defined analogous to Hooke's law

(3)

(Madej~ki'~).

With the designation uWor the stresses at time t

--* 00

the plastic strain velocities are (Fritz7)

l + v

7 )

=

-

( 1

-

p ) (a, u?) p (0 -a; )]

& O

Herein

7 )

is the viscosity

of

the dashpot, u? and

u r

respectively are the radial and tangential

stresses according to (10) and vp is a measure for the plastic volume dilatancy. To fulfil the

normality rule,

up

either should equal

1 / 2

or should depend on time and location

r.

Because

in the first case only deviatoric plastic deformations are accounted for, it does not represent

the general case. However, the assumption of vp =

v p ( t , r )

would hinder an analytical treatment.

Therefore, v,, is assumed to be constant. Hereby the normality rule is no longer fulfilled exactly.

The error involved with this approach will be discussed when deriving the absolute value of

vp

(cf. qua tio n (32)) .

The differentiation in (21) is executed with respect to time. To simplify the equations the

abbreviations

W

cr=

u r

u,

-

0

=

u1

- T I


9/18

AXISYMMETRICAL TUNNEL PROBLEMS 333

are introduced. The total strain velocities are obtained by summation

of

the elastic parts

according to Hooke s law

(3)

and the viscoplastic parts according to the flow rule

(21)

to

l+u) d

1+

E

dt

77

1+u) d l v

E

dt

77

,

=

(1

- ) 6 ,

-

A

(1 -

up)@,

- P6J

E ;

=

l

-

u p t -

u6,]+--9[ 1-

UP)@' - p6, ]

(23)

Elimination

of

the displacement Au in the kinematic relation

(2)

and differentiation with

respect to time leads to

d E ;

.

.

r - = & , - E ,

dr

Because equilibrium condition 1) holds both at time

t - 00

and at time

t ,

it follows that

d6,

dr

- =

6'-6,

(24)

Substitution

of

the strain velocities in the compatibility condition

(24)

by expressions

(23)

and

elimination

of

6'on the right-hand side by means

of

equilibrium condition

(25)

leads to the

differential equation

= O

with

(1

-

u ~ ) E

a =

1

-

2)77

Integration with respect to r yields

d

6,

+

6') a(6, 6')

f3

I

=

0

dt

and with respect to

t

6 6'=f4( ) e- '

+

5 (

The expressionsf3

t ) , f4(

) and

f5 t

are functions of r and t , respectively yet to be determined.

Substitution of

6'

from equilibrium condition

(25),

integration with respect to r and back-

transformation by means of

(22)

provides the general solution

where

f6(t),

f 7 ( t )

and

f 8 ( r )

are functions of

r

and

t

yet to

be

determined. By means

of

the

radial and tangential stresses cr: and u: at time t

= O

from (20) the boundary conditions

t = O : ur=cr :

r = R i :

u r = p i


10/18

334

P.

FRITZ

can be formulated. Taking into account the equilibrium condition

(25),

the stresses result to

It should be observed that these expressions are valid within the initial (at time t = O+) plastic

zone only, i.e. for

r < po

(cf. equation

(29)) .

By adding the elastic and plastic strain velocities

in

(23)

and with the boundary condition

(26)

it was implicitly assumed that the rock at radius

r

is always in the plastic zone.

The displacements for r s po are obtained by integration of the strain velocities E ; in (23)

considering the abbreviations

(22)

and the kinematic relation

(2)

for AE, as

l +

+ V

Au

=-

r [ (

1

-

)a , -

~ a , ] + ~ r

( l -

u ) ( a ,

-

a?)

-

(a,-

a?)]

t

+f9(r)

E

7 )

The function f9(r) is determined from the boundary condition

t = 0:

Au

=Aue

whereby the displacements

Au

at time t = 0 are defined in 20). ubstitution of the stresses

according to

(27)

leads to

l + u

E

-

{[(

1

-

)a:-

va:]

e-a'+[( 1

-

) a T -

ua:]( 1- -a )}r

l + u 1

- - '

+--9[(1-

V,)

a:-a:) - p a :

-

a33

y

7 )

( l + v ) ( 1 - 2 u )

Par

E

(for R is

r

s o )

The radius po of the plastic zone at time t = O+ is obtained by substitution of

a:

and

a:

n the

yield condition (7) according to the flow rule (20) as

As a further assumption, the condition is introduced that the total stresses at r 5 o may fulfil

the yield condition, but that they may never exceed it. This hypothesis influences the time-

dependent development of the stresses and strains, but not the final state at

t

+

00

When the

plastic zone at time t extends to the radius p,, the state in the range p o d r < p I is therefore

described analogous to the elasto-plastic material. Hence the stresses are defined analogous

to (10)

and

(17) ,

the displacements to

(18)

and the radius p , to

(12) ,

when

Ri

is replaced by


11/18

AXISYMMETRICAL TUNNEL PROBLEMS

335

PO of

(29)

and

Pi

is replaced by the stress

a?

at r = p o according to

(27) :

a,

=a%+ ( A a , + A a l )

h = - r

E

" {k,

(

m R - l +

k 2 ( 7 ) m G + 1 +

k3}

The radial stress

a?

at

r

= po

is defined using

(27) .

The state of stress and deformation for

r >

pl follows as for the time-independent case from

( 5 )

whereby

pm

must be replaced by

pr.

The function

f 6 ( t )

in

(27)

and

(28)

is determined by compatibility considerations for the

displacements at r = p o . At this point, the displacements of (28) are equated to those of

(30),

and after substitution of a? by expression (27) one obtains a differential equation for

f 6 ( t )

with the functions

(1 +

Y ) (

1- )

k4

E

4 =


12/18

336

and the boundary conditions

P. FRITZ

f = O :

f6=

f6df=O

Because of its transcende ntal character, equation (31) for f6(f) has to be solved numerically,

e.g. by a sim ple comp uter prog ram. For th e spec ial case of deviatoric plastic strains only

( d G

=

0

in (13), i.e.

up

= 1/2), a dimensionless representation can be found, which, within practical

accuracy requirements, is independent of the material properties E , .rl and the initial stress pa.

In the uppe r part of Figure

5

the related function f6 is shown , an d in the lower pa rt its integral,

both versus a related time f *

=

( E / v ) f . or an arbitrary time and a desired ratio p i / p a , 6 and

f6

can be read and, by means of equations

(27)

nd (30), the stresses and displacements can

be calculated. Figure 5 applies for the indicated values of cohesion and angle of friction.

However, the influence

of

the peak value of the cohesion

c p

on

f6

is relatively small. Thus,

for pi

=

0, the function f6 is indepen dent of

cp ;

for p i / p a

=

5 per cent, a reduction

o t

the cohesion

from th e value indicated in Figure 5 to

c p=

0 causes

f6

at time

f * =

4000 to increase by about

20 per cent. H owever, the influence of th e angle of friction is greater. Figure

5

applies for

4'=

4R

20 and Figure 6 for

4'

=

4 R

=

30 .

It should be noticed that assumption (21) for the plastic strain velocities implies volumetric

strains depending o n up' Although up is properly variable with time and rad ius, it was assumed

to be constant.

To

minimize the deviation from norm ality rule, vp is determined in such a way

that the displacements at the excavation boundary coincide for t + 00 with the time-independent

Figure 5 . Representationof the functions

f

and If from equation (31) for

t$G

=0, i.e.deviatoric plastic deforma tions;

4' =

4'

=

20 ; p

= 1

N/mm2,

c R

=

0; Y =0.3


13/18

AXISYMMETRICAL TUNNEL

PROBLEMS

337

16

- - DETAIL

0.2

I . . .

-

t

2

i o t i d '

9

0.1

Figure

6.

Representation

of

the functionsf and

sf

from equation (31) for

d G

=O, i.e.deviatoric plastic deformations;

4'

=

4'

=

30"; c p=

1

N/mm',

c R

0 ; P

=

0.3

results from

(18).

Herewith the displacements correspond to the ones calculated with the

normality rule outside the initial radius p o

of

the plastic zone at any time and at the excavation

boundary

R j

for

t + m .

For

R i < r < p o

they differ compared with the exact solution. An

illustrative example

to

this at different times t may be found in Fritz.' At

t

+m the difference

follows directly from

(18) .

Usually it amounts at most to a few per cent. Therefore the influence

of a variable up= up t ,

r )

is neglected. The value

of

up may hence be found by equating the

time-dependent displacements at time t

+

according to

(28)

to the time-independent ones

of

(18) .

Substitution

of

the integral

l F f 6 f )

dt from

(31)

and consideration

of

fa(t+m)

= O

and

poo

according to

(12)

leads to


14/18

338

P. FRITZ

For the special case of deviatoric plastic deformations only ( & G =

0

or mG

= 1,

respectively),

vp follows from

(32)

or directly from (21):

vp

=

1/2 (for &G = o ) .

For this case the normality rule is fulfilled all over the domain at any time.

ILLUSTRATIVE EXAMPLE

To demonstrate the applicability of the derived solutions an example from engineering

practice is discussed. For a deeply located tunnel the radial displacements have been measured

as a function of time before the placing of the lining. It was assumed that the statical and the

material model could be idealized according to Figure

7.

The elastic parameters of the Hooke

element were chosen to agree with the results of laboratory tests.

H

Statistical model Materia l model

Overburden H = 6 0 0 m

Modulus

of elasticity E

=

2,000N/mm2

Spec. weight

y

=

28 kN/m3

Poisson's ratio

v = 0 . 3

Vert. pressure

u,

=

yH

Cohesion c;

CR

Horiz. pressure a, = a

Angle

of

friction

4 . 4 R

Intern. pressure p , = 0 . 6 N/mm2

Angle of

flow

rule 4G

Radius

R , = 6 m

Viscosity

7

Figure 7. Model of a deeply located tunnel in Austria

From the measured final deformations at the excavation boundary it was possible to back-

calculate pairs

of

values of cohesion and friction angles from equation

(18).

Different flow

rules

(13)

and different magnitudes of residual strengths lead

to

distinct peak strength param-

eters (Figure

8).

Which

flow

rule fits best to an actual problem cannot be determined from

the deformations at the excavation boundary alone. It would also be necessary, for example,

to have a knowledge

of

the deformations within the rock mass.

It remains to determine the viscosity coefficient

7

of the Newtonian element. It is chosen

to approximate the measured displacements as close as possible. In Figure

9,

the resulting

curve is shown for certain strength parameters.

Of particular interest also is the stress in a lining erected before a stable state of equilibrium

is reached. Depending on the depth of the lining and its time

of

installation, different stresses

are attained at time t - 00 (cf. Figure

10

for the case of an elastic lining and a rock with equal

peak and residual strengths). According to this diagram, the stresses are most unfavourable


15/18

AXISYMMETRICAL TUNNEL PROBLEMS

339

ANGLE OF FRICTION (pp = IN

Figure

8.

Back-calculation of the strength parameters for different rameters

of

the

Row

rule

Np, p :

residual

strength peak strength,

NR, R: P-

Qp,

c R =

O

for a lining with small thickness which is erected immediately at the face. The later a lining is

installed the smaller are its stresses. However, in tunnelling practice, the time span between

excavation and lining is often limited by the amount of admissible deformations or by the need

to finish the tunnel within certain time limits.

In the literature the argument can

be

found (e.g. Pacher ), that if too much time is allowed

before the installation

of

a lining, its stresses can increase or, in other words, that an optimum

time of installation exists, for which the smallest stresses develop. Without discussing the actual

field conditions, only the mathematical aspects shall

be

treated here. According to Figure

10

it is obvious that, for the material model of the rock 'Hooke-(St. Venant/Newton)' with ideal

plastic behaviour (i.e. equal peak and residual strength), the stresses in the lining decrease

with increasing time of installation. Disregarding the phenomenon of loosening pressure, an

optimum time

of

installation as defined above does not exist for this particular model. Therefore

Egger4 discussed a material with strain softening characteristics in the post-failure region.

. .

q=4200 DAVS Nlmm2

P

a

160 260

3bo

460,

TIME H DAYS

Figure 9. Time depe ndent displacements at the excavation boundary


16/18

340

P. FRITZ

A

MATERIAL MODEL

OF ROCK:

N

LINING: E

=

2OOOO N l m z

V - 0.3

TIME

OF

~STALLATION

IN

DAYS

Figure

10.

Stresses in the lining dependin gon ts time

of

installation (support pressure by anchors pp, deviatoric plastic

deform ations), esidual strength= peak strength:

c p

= c R =0 7

N/mm2;

I '= 4 R= 20 ;9 =

6,000

days N/mm2,

---

residual strength < peak strength:

p

=0.5 N/mm2;

c R

= O ; &'= I R =

30 ;

7 =

5,000

days N/mm2

Taking as limiting case of such a behaviour an instantaneous drop down to the residual strength,

it can be seen from equation (18) that for this case also a stable equilibrium is reached (cf.

Figure

10

example residual strength


17/18

AXISYMMETRICAL

TUNNEL

PROBLEMS 341

seems always to be justified. Disregarding big deformations, stability will therefore, theoreti-

cally, always be guaranteed. It seems that for any material model consisting of an arbitrary

combination of Hooke, Newton and modified St. Venant elements, no optimum time of

installation ca n b e found.

RE FE RE N CE S

1. F. H. Deist, A nonlinear continuum approach to t he problem of fracture zones an d rockbursts, J. South

African

Insr. M ia Metallurgy,

502-522,

May

(1965).

2.

I.

Dimov, Eine Methode zur Bestimmung der nicht-elastisch-viskosen Bereiche um isolierte horizontale

Gru ben bau e und der Druck auf den Ausbau, Bericht

6.

Lindertreffen IBG, Abh. Deutsch. Akad. Wissen-

schaften, Berlin, s. 93-100 (1965).

3. 1. Dimov and R. Paraschkewov, Die Entstehung und Entwicklung von plastischen Bereichen um Grubenbau

und die V eranderung des Druck es auf de n Ausbau als rheologisch-plastische Aufgabe. Bergakadem ie, He ft 8,

s. 485-487, August (1964).

4. P. Egger, Einfluss des Post-Failure Verhaltens von Fels auf den Tunnelausbau, Verijffentlichung des Inst. fur

Boden- und Felsmech. Univ. Fridericiana, Karlsruhe, Hef t 57 (1973).

5.

R. Fenner, Untersuchungen

zur

Erkenntnis des Gebirgsdrucks, Diss.

TU

Breslau

(1938).

6.

A. L. Florence and L. E. Schwer, Axisymmetric compression of a Mohr-Coulomb medium around a circular

7. P. Fritz, Numerische Erfassung rheologischer Probleme in der Felsmechanik, Mitteilung 47 des Inst. Strassen-

8.

P. Fritz , Modelling rheological beh aviour

of

rock, 4rh

Inr. Conf. Num.

Merh. Geomech., Edmonton, June

(1982).

9. P. Fritz, Numerical solution of theological p roble ms in rock,

Froc.

Znr.

Symp. Num ericul Mod els in Geomechanics,

Zurich, September (1982).

10. W. Hartmann, Ueber die Integration de r Differential-gleichungendes ebenen Gleichgewichtszustandes fur den

allgemein-plastischen Korper,

UnueroflentlichteHundschrift,

Gijttingen (cf. Nadai, 1928) (1925).

11. H. Hencky, Ueber einige statisch bestimmte Falle des Gleichgewichts in plastischen Korpern,

2. ngew. Math

und Mech,

Band 3 , Heft 4, s. 241-251 (1923).

12. A. J . H end ron and A. K. Aiyer, Stresses and strains around a cylindrical tunnel in an elasto-plastic material with

dilatency, Tech Rep.

No. 10,

Omaha District, Corps

of

Engrs., Contract DACA

45-69-C-0100 (1972).

13.

R. Hill, E. H. Lee and S. J. Tupper, The theory of combined plastic and elastic deformation with particular

reference t o a thick tu be und er internal pressure,

Proc. Roy.

Soc.

London, Series A, Mark

and

Phys. Sci,

191,

No. A 1026, November (1947).

hole,

I n t .

j .

numer.

anal.

methods geomech, 2.

367-379 (1978).

Eisenbahn-Felsbau, Fed. Inst. Technology,

s. 59, (1981).

14. R. Hill, m e Mathematical Theory of Plusriciry, Oxford Univ. Press, Lond on, 1950.

15. H. Kastner. Ueber de n echten G ebirgsdruck beim Bau tiefliegender Tunnel,

Oesrerreich. Bauzeirschrift,

Heft 10,

s. 157-159 und Heft 11, s. 179-183 (1949).

16. H. E. Loonen , Theoretische Berechnung de r um einen zylindrischen H ohlraum in einem

visko-elastisch-pastischen

Medium auftretenden Spannungen und Verschiebungen, Central Profstation

von

de Staatsmijnen in Limburg,

Hoesbroek-Treebeek (1962).

17. J. Madejski, Theory

of

non-stationary plasticity explained on the example of thick-walled spherical reservoir

loaded with internal pressure,

Archiwum Mechaniki

Srosowanej,

5/6. 12, 775-787 (1960).

18.

A. N adai, Das Gleichgewicht ockerer Massen,

Mech. der elasr.

Korper.

Kup.

6. V, pringer Verlag, Berlin (1928).

19.

A. Nadai, Partial yielding in a thick-walled tube, in Plasticity, McGraw-Hill, New

York,

th impression,

1931,

20.

T.Non aka, An elasto-visco-plastic analysis for spherically a nd cylindrically sym metric problems, Ingenieur-Archiv,

21.

T.

Nonaka,

A

time-independent analysis for th e final state

of

an elasto-visco-plastic mediu m with inte rnal cavities,

Inr. J.

Solids Strucrures,

17, 961-967 (1981).

22. F.

Pacher, Deformationsmessungen m Versuchsstollen als Mittel zur E rforschung des Gebirgsverhaltens und zur

Bemessung d es Ausbaues,

Felsmechanik

und

Zngenieur geol.,

Suppl. 1, s. 149-161 (1964).

23. M. Panet, Analyse d e la stabilit6 dun tunnel creus6 dans un massif rocheux e n tenant com pte du compo rtement

aprks la rupture, Rock Mech,

8,

209-223 (1976).

24. M. d e Saint-V enant, Sur Iintensitk des forces ca pa bl e de deform er, avec continuit6 des blocs ductiles, cylindriques,

pleins ou tvidbs, et pl ac b da ns diverses circonstances, Comptes

Rendus des SPunces de IAcud imie de s Sciences,

74,

1009-1015 (1872).

25. M. D. G. Salamon, Rock mechanics of underground excavations,

Proc.

Third

Congr. I n r

Sac.

Rock Mech.,

1,

part B, 994-1000 (1974).

26. A. Salustowicz, Neue Auschauungen uber den Spannungs- und Formanderungszustand im Gebirge in der

Nachbarschaft bergmannischer Hohlraume. Inr.

Gebirgdruckragung, Leipzig,

s. 5-1 1 (1958).

pp.

196-200.

47, 27-33 (1978).


18/18

342

P.

FRITZ

27.

K.

von Terzaqhi, Die Erddruckerscheinungen in ortlich beanspruchten Schuttungen und die Entstehung von

Trgkorpern, Wochenschrifr

fur

den iirfnrL Baudiensr, 25. Jahrgang, Heft 18, s. 206210 , Wien (1919).

28. S.Ulgudur, A non-linear continuum theory of fracture for rocks,

Thesis,

Univ. Newcastle upon Tyne, April, 1973.

29. H.

M.

Westergaard, Plastic state of stress around a deep well, J. Boston Soc.

Civ. Engrs.,

27, 1-5 (1940).

30. T. Wierzbicki,

A

thick-walled elasto-visco-plastic spherical container under stress and displacement boundary

value condition, Arch iwum Mechaniki Srosowaney,2/15, 297-307 (1963).

an analytical solution for axisymmetric tunnel problems in elasto plastic media

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