Theoretical solutions for NATM excavation in
soft rock with non-hydrostatic in-situ stresses
Nagasaki University
1.
1.
2.
2.
Philosophy
Constitutive
and
law:
construction
strain-softening
process
modelin-situ
Introducing
Vertical
in-situ
some
stress
assumption
Pv and horizontal
Key
Three
problem:
zones:
elastic
convergence
zone,different
strain-softening
released
before
Relatively
stress
Ph are
simple
apparently
without
numerical
from
method
each
and
zone
after
and
supporting
plastic-flow
installation
zone
other in most
involved
and useful
occasions
for primary design
Z. Guan
Y. Jiang
Y.Tanabasi
Background--NATM






Figure1 Schematic representation of 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
Back
Analytical model for cross section
Pv



a
Ph
Tunnel
Opening
Elastic Zone
S-S Zone
P-F Zone
a
Figure2 Plane strain analytical model for cross section
Plane strain problem
Strain-softening deformation
characteristic
Non-hydrostatic in-situ stresses
Constitutive law for soft rock
1
1-3
3
3
e
ε1
+
c
ε1
f
ε1
p
1
E'
E
ε
1
3=constant
h
p
3
ε
c*
e
1
p'
1
ε
ε1
p
1
ε
ε
e
f
ε1= ε1
ε3
p'
3
ε
f
Back
Figure3 Typical stress and strain curves under triaxial tests
Relationship between
1 and 3
Mohr-Coulomb  1   c  E *(1  1 )  K p 3
 c
Criterion
E'  c
e
(  1) 1
'
e
Relationship between 1 and 3
Plastic Poisson
Ratio h
1  1e d1 ;  3   3e d 3
d 3  hd1
Constitutive law for soft rock
1
1-3
3
3
e
ε1
+
c
ε1
f
ε1
p
1
E'
E
ε
1
3=constant
h
p
3
ε
c*
e
1
p
1
ε
ε
e
f
ε1= ε1
p'
1
ε
ε1
ε3
p'
3
ε
f
Back
Figure3 Typical stress and strain curves under triaxial tests
Relationship between
Mohr-Coulomb
Criterion
1 and 3
 1    K P 3
*
c
Relationship between 1 and 3
1  1 f d1 ;  3   3 f d 3
Plastic Poisson
Ratio
d 3   f d1
Angle-wise approximation assumption

loadings

Vertical far field stress Pv
horizontal far field stress Ph

Inner pressure Pi() varying with azimuth 

P0 
Pv  Ph
2
Pi ( ) 
Figure4 Classical problem in elasticity
S0 
Pv  Ph
2
2 P0  4S 0 cos 2   c
Kp 1
So that the stress state at the inner
boundary could verify Mohr-coulomb
criterion  t   c  K p r exactly.
Angle-wise approximation assumption
approximate its solution in elastic zone
to the classical one mentioned above
At elastic boundary (r=Re)
 re 
2P0  4S 0 cos2   c
K p 1
1 
e
( r  P0  S 0 (4  1) cos 2 )
E
1


e
te 
( r  P0  S 0 (3  4 ) cos 2 )
E
re 
u e   t Re
e
Figure5 Approximation for an infinitesimal azimuth
The essence of this assumption
is to neglect shear deformation
in rock mass
Analytical solutions in strain-softening zone


Geometry equation
du
u
r  
t  
dr
r

dr
Displacement
governing equation

1 h

r  e
 Re 
e
e
e
u ss 


h






 
 r
t
r
t 
(1  h) 
 r 


 rss
1 h
1  e
 Re 
e
e
e

 r  h t  h    r   t
(1  h) 
 r 
 tss
1 h

1  e
 Re 
e
e
e



h








 r
t
r
t 
(1  h) 
 r 







r
0
Stress governing
equation
 c E ' ( tss   t )
d r 1  K p

r 

dr
r
r
r
du
u
e
e
 h    r  h t
dr
r

Equilibrium equation
d r  r   t
e
 rss 
 c  Z0
1 K p
Z 0  Re 

 
h Kp  r 
(1 h )
R 
 C0  e 
 r 
(1 k p )
 tss   c  E ' ( tss   t e )  K p rss
e
e
  Z0
Z0
E ' ( t   r )
e
C0   r  c

Z0 
1 h
1 K p h  K p
Analytical solutions in plastic-flow zone


Geometry equation
du
u
r  
t  
dr
r



r
0
Stress governing
equation

d r 1  K p

r  c
dr
r
r
du
u
f
f
 f     r  h t
dr
r
u pf

dr
Displacement
governing equation
1 f
 Rf 
r  f
f
   r f   t f
 r  f t  

(1  f ) 
 r 

Equilibrium equation
d r  r   t





*
 rpf   c   c   r  K p r
*
*
 tpf   c *  K p rpf
f
f

 Rf

 r



(1 K p )
u,  and  in all three zones could be expressed as the functions of
radius r, with two parameters Re and Rf unknown
Determination of Re and Rf

Continuum condition of tangent
stress t at Rf boundary
 t | r  R    t |r  R 
f

f
Continuum condition of radial stress
r at tunnel wall boundary
 r a  Kcua
ra is the interaction force between rock mass and lining
ua is the tunnel wall convergence
Ra  ( Ra  t c ) 2
Kc is Radial stiffness of lining K c 
(1   c ) Ra (1  2 c ) Ra 2  ( Ra  t c ) 2
Ec
2
Set up an analytical solutions for cross section model
u,  and  in all three zones are totally determined
Equivalent series stiffness hypothesis
Before supporting



Back-analyze
Pre-released
displacement
First Stage
After supporting and face advancing away
The face carry the loading
partly
Pre-released displacement
occurs
h
Kini



The supporting together with rock mass
carry the full load
Displacement release goes on, until to the
ultimate convergence
Kc
Kini
Lining stiffness in reality
initial stiffness
due to face
Kc
Second Stage
Equivalent series stiffness
1
1
1


K equ K ini K c
Figure6 Physical significance of Kequ
Kequ
Forward-analyze
Equivalent series stiffness
ua
Ultimate convergence
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)
Solution implementation


Parameters employed in the basic case
E (Mpa)


h
f
w (8)
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)
h
2.5
1.5
5
20000
0.25
0.1
0.3
Calculation results
Basic case: Pv=2.5, Ph=1.5
Pv=2.75, Ph=1.25
Pv=3.0, Ph=1.0
Both of two zones connected
Only s-s zones connected
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)
1
2
Eng  K equ u a
2
Dimensionless indices, Re/Re0, ua/ua0 and Eng/Eng0 are employed in
case studies to standardize and highlight the variation of them
Influence of rock mass properties
2.00
Re/Re0
Ua/Ua0
1.25
Eng/Eng0
1.00
B asic case point
0.75
R e=10.7 ua=16.5
Relative Ra tio
Relative Ra tio
1.50
Re/Re0
1.75
1.50
Ua/Ua0
B asic case point
Eng/Eng0
R e =10.7 ua=16.5
1.25
1.00
0.75
0.50
0.25
0.00
0.50
0.40
0.60
0.80
1.00
1.20 1.40
1.60
1.80
1.0
2.00
c and c* influence both Re and

ua greatly

c and c* determine the energy
storage capability of rock mass
2.5
3.0
3.5 4.0
4.5
5.0 5.5
6.0
E (G pa)
c (M pa)

1.5 2.0

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
1.50
B asic case point
R e =10.7 ua=16.5
1.25
1.00
0.75
0.50
Re/Re0
Ua/Ua0
0.25
Eng/Eng0
Kequ/Kini
Relative Ra tio
Relative Ra tio
1.50
R e =10.7 ua=16.5
1.25
1.00
R e/R e0
0.75
U a/U a0
K equ/K c
E ng/E ng0
0.50
0.25
0.00
0.00
22
43
65
87
109
130 152
174 195
0.1
217
In theory

Kc play identical role to Kini

0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
R elease C oefficient h
Lining S tiffness K c

B asic case point

In practice

Kini vary hundred times according to h

It is difficult to control h
Suggestion

Pay more attention to h 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