Theoretical solutions for NATM excavation in soft rock with non-

hydrostatic in-situ stresses

Nagasaki University

Z. Guan

Y. Jiang

Y.Tanabasi1. Philosophy and construction process2. Key problem: convergence released before and after supporting installation

1. Vertical in-situ stress Pv and horizontal in-situ stress Ph are apparently different from each other in most occasions

1. Constitutive law: strain-softening model2. Three zones: elastic zone, strain-softening zone and plastic-flow zone

1. Introducing some assumption2. Relatively simple without numerical method involved and useful for primary design

Background--NATM

Philosophy of NATM Construction process Key problem in the

design of supporting

Philosophy of the research

Analytical model for cross section

Take face effect (longitudinal effect) into account

Figure1 Schematic representation of NATM Bac

k

Analytical model for cross section

Plane strain problem Strain-softening

deformation characteristic Non-hydrostatic in-situ

stresses

Figure2 Plane strain analytical model for cross section

p'3ε

ε 1p'

ε 3p

p1ε

f

h

+

3ε

ε 1e1ε f

1ε

E'E

*

c

= eε 1fε 1

pε 1eε 1

3=constant

1ε

1- 3

1

3 3

1

c

13 dhd 333111 ; dd ee

Constitutive law for soft rock

Figure3 Typical stress and strain curves under triaxial tests

311'

1 )( pe

c KE

eccE1

*'

)1(

Relationship between 1 and

3Mohr-Coulomb Criterion

Plastic Poisson Ratio h

Relationship between 1 and 3

Back

p'3ε

ε 1p'

ε 3p

p1ε

f

h

+

3ε

ε 1e1ε f

1ε

E'E

*

c

= eε 1fε 1

pε 1eε 1

3=constant

1ε

1- 3

1

3 3

1

c

3*

1 Pc K 333111 ; dd ff

13 dfd

Constitutive law for soft rock

Figure3 Typical stress and strain curves under triaxial tests

Relationship between 1 and

3Mohr-Coulomb Criterion

Plastic Poisson Ratio

Relationship between 1 and 3

Back

Angle-wise approximation assumption

20hv PP

P

20

hv PPS

1

2cos42)( 00

p

ci K

SPP

So that the stress state at the inner boundary could verify Mohr-coulomb criterion exactly.

rpct K

Figure4 Classical problem in elasticity

loadings Vertical far field stress Pv horizontal far field stress Ph Inner pressure Pi() varying with

azimuth

Angle-wise approximation

assumption approximate its solution in elastic zone to the classical one mentioned above

The essence of this assumption is to neglect shear deformation in rock mass

1

2cos42 00

p

cer K

SP

)2cos)14((1

00

SPE

er

er

)2cos)43((1

00

SPE

er

et

eet

e Ru

Figure5 Approximation for an infinitesimal azimuth

At elastic boundary (r=Re)

Analytical solutions in strain-softening zone

Equilibrium equation

0

rdr

d trr

Geometry equation

r

u

dr

dutr

et

er h

r

uh

dr

du

Displacement governing equation

Stress governing equation

r

E

rr

K

dr

d ettssc

rpr )(1 '

et

er

h

eet

erss r

Rh

h

ru

1

)1(

et

er

h

eet

errss r

Rhh

h

1

)1(

1

et

er

h

eet

ertss r

Rh

h

1

)1(

1

)1(

0

)1(

00

1

pk

e

h

e

pp

crss r

RC

r

R

Kh

Z

K

Z

rsspettssctss KE )('

h

EZ

er

et

1

)('0

pp

cer Kh

Z

K

ZC

000 1

Analytical solutions in plastic-flow zone

Displacement governing equation

Stress governing equation

ft

fr h

r

uf

dr

du

rr

K

dr

d cr

pr*1

ft

fr

f

fft

frpf r

Rf

f

ru

1

)1(

)1(

**pK

ffrp

frccrpf r

RK

rpfpctpf K *

u, and in all three zones could be expressed as the functions of radius r, with two parameters Re and Rf unknown

Equilibrium equation

0

rdr

d trr

Geometry equation

r

u

dr

dutr

Determination of Re and Rf

Continuum condition of tangent stress t at Rf boundary

ff RrtRrt ||

Continuum condition of radial stress r at tunnel wall boundary

22

22

)()21(

)(

)1(caac

caa

ac

cc

tRR

tRR

R

EK

Kc is Radial stiffness of lining

ra is the interaction force between rock mass and lining

ua is the tunnel wall convergence

acar uK

Set up an analytical solutions for cross section model

u, and in all three zones are totally determined

Equivalent series stiffness hypothesis

Before supporting The face carry the loading

partly Pre-released displacement

occurs

After supporting and face advancing away

The supporting together with rock mass carry the full load

Displacement release goes on, until to the ultimate convergence

Kc

Lining stiffness in reality

ciniequ KKK

111

Back-analyze Kini

initial stiffness due to face

Pre-released displacement

Ultimate convergence

Kequ uaEquivalent series stiffness

Forward-analyze

Equivalent series stiffness

Figure6 Physical significance of Kequ

Summary of theoretical solutions

Introduce angle-wise approximation assumption to simplify non-hydrostatic in-situ stresses

Introduce equivalent series stiffness hypothesis to take pre-released displacement into account

For every infinitesimal azimuth , search for proper Re and Rf that verify all the boundary and continuum conditions

To determine all the state variables (u, and ) in three zones, especially ultimate convergence (ua)

Basic case: Pv=2.5, Ph=1.5

Both of two zones connected

Solution implementation

Parameters employed in the basic case

Calculation results

E (Mpa) h f () c (Mpa) c* (Mpa)

2000 0.3 1.33 1.88 0.41 25 1 0.65

Pv (Mpa) Ph (Mpa) Ra (m) Ec (Mpa) c tc (m)  

2.5 1.5 5 20000 0.25 0.1 0.3  

Pv=2.75, Ph=1.25

Only s-s zones connected

Pv=3.0, Ph=1.0

Both of two zones separated

Case studies

The object Reveal the influence of different parameters on the

supporting effect in NATM Provide primary design and suggestion for NATM

The evaluation indices Re (the range of strain-softening zone), ua (the ultimate

convergence of tunnel wall) and Eng (energy stored in equivalent lining)

Dimensionless indices, Re/Re0, ua/ua0 and Eng/Eng0 are employed in case studies to standardize and highlight the variation of them

2

2

1aequuKEng

Influence of rock mass properties

c and c* influence both Re and ua greatly

c and c* determine the energy storage capability of rock mass

E influences ua drastically, whereas takes little effect on Re

E only change the energy storage proportion between elastic zone and lining

Influence of supporting properties

In theory Kc play identical role to

Kini

In practice Kini vary hundred times according

to It is difficult to control

Suggestion Pay more attention to and Kini It is better that make Kini equal

to Kc

Conclusions

Establish a set of solutions and implementation for NATM excavation in soft rock with non-hydrostatic in-situ stresses

After case studies, it is clarified that these solutions could predict the state of NATM excavation well, and useful for primary design of supporting