Transcript Training

Finite element modelling of load shed
and non-linear buckling solutions of
confined steel tunnel liners
10th Australia New Zealand Conference on
Geomechanics,
Brisbane Australia, October, 2007
Doug Jenkins - Interactive Design Services
Anmol Bedi – Mott MacDonald
Introduction
Port Hedland Under Harbour Tunnel
 Lined with 250 m thick gasketed precast
concrete segments – now corroding
 Proposal to reline with steel backgrouted
liner
 Geotechnical and structural finite element
analyses
 Comparison with analytical solution

Topics
The proposed remedial work
 Confined liner buckling
 Jacobsen Closed Form Buckling Solution
 Linear buckling FEA
 Application to the project

– Current stress state in tunnel liner
– Future Installation of Steel Liner
– Geotechnical FEA results

Conclusions
Port Hedland Under Harbour
Tunnel
Material Properties
Material
 (kN/m3) E (MPa) Cohesion (kPa)
Fill
18
25
30
Marine Mud
18
5
30
Red Beds
20
23
55
Upper Conglomerate
22
1000
250
Sandstone
22
100
130

25
25
32
36
34
Lower Conglomerate
32 0.3
22
1000
Table 1: Material Properties
55

0.45
0.45
0.3
0.3
0.25
Closed Form Solutions
Unrestrained solution similar to Euler
column buckling
 Rigid confinement restrains initial
buckling
 Gap between pipe and surrounding
material allows single or multi lobe
buckling to occur
 Buckling frequently forms a single lobe
parallel to the tunnel

Single Lobe Buckling
Comparison of buckling theories
Berti (1998) compared theories by
Amstutz and Jacobsen
 Amstutz approach was simpler, but
assumed constants may be unconservative
 Also found that rotary symmetric
equations are unconservative compared
with Jacobsen
 Computerised analysis allows the more
conservative Jacobsen method more
general use

Jacobsen Equations
 sin  
1
 y  sin    p.Rrt sin    4 p.Rrt2  sin  sin   tan    



 
Em
2Rrt
Em sin  sin  
 Em sin   


12 p.Rrt3  9 2

  2  1
Em
 4

 sin   




sin



3
Jacobsen Equations
Jacobsen Equations
y
Value
250.0
Error
0.0000
Rrt
Value
82.50
Error
0.0000
Jacobsen Solution
a
0.6755 Radians
b
0.6636 Radians
p
1.4960 MPa
Unrestrained Buckling Solution
Pcr
89.0 kPa
Parametric Study
Run No
1-3
4-6
Run No
7-10
11-14
15-17
18-20
21-25
26-29
Variable
Pipe deform.
Pipe deform.
Variable
Pipe/restraint gap
Pipe/restraint gap
Contact friction
Contact stiffness
Rock stiffness
Surcharge press.
Pressure
Uniform
Hydro.
Pressure
Uniform
Hydro.
Hydro.
Hydro.
Hydro.
Hydro.
Pipe Deformation,mm
0, 10, 20
0, 10, 20
Gap
Contact
Friction
mm
Factor
0, 1, 2, 5
0.5
0, 1, 2, 5
0.5
0.7, 0.5, 0.3
2
2
0.5
2
0.5
2
0.5
Contact
Stiffness
MN/m
10
10
10
1, 5, 100
100
100
Rock E
GPa
Surcharge
Pressure
Ratio
10,1,0.25,0.1,0.05
1
0, 0.3, 0.6, 1.2
Unrestrained Buckling Model
Unrestrained Buckling
400
350
Uniform Pressure
Deflection, mm
300
250
200
Hydrostatic
Pressure
150
100
50
0
0
20
40
60
80
Pressure, kPa
Undeformed
10 mm deformation
20 mm deformation
Undeformed
10 mm deformation
20 mm deformation
100
Unrestrained Buckling
Unrestrained Buckling
FE Model for Restrained Buckling
FE Model Detail
FE Model Detail
Restrained Buckling - deflection
60
Deflection, mm
50
40
30
20
10
0
0
200
400
600
800
1000
1200
Pressure, kPa
0mm Gap, Uniform
1 mm Gap
2 mm Gap
5 mm Gap
0 mm Gap, Hydrostatic
1 mm Gap
2 mm Gap
5 mm Gap
1400
Restrained Buckling - deflection
Restrained Buckling - gap
1600
Critical Pressure, kPa
1400
1200
1000
800
600
400
200
0
0
1
2
3
4
Gap, mm
FEA-Uniform
FEA-Hydrostatic
Jacobsen
5
Effect of contact friction and restraint
stiffness
1300
Critical Pressure, kPa
1200
1100
1000
900
Friction Values: 1=0.7; 2=0.5; 3=0.3
Contact stiffness: 1=100, 2=10.0, 3=5.0, 4=1.0 MN/m
Rock/Soil stiffness: 1=10, 2=1.0, 3=0.25, 4=0.1, 5 =0.05 GPa
800
700
600
500
0
1
2
3
4
Run No
Friction
Contact Stiffness
Rock/Soil Stiffness
5
Effect of surcharge pressure
300.0
200.0
Stress, MPa
100.0
0.0
-100.0
-200.0
-300.0
0
500
1000
1500
2000
2500
Pressure, kPa
0 kPa Top face
30 kPa
60 kPa
120 kPa
0 kPa bottom
30 kPa
60 kPa
120 kPa
Geotechnical Analysis – Current Stress
State
Geotechnical Analysis – Elastic Modulus v
Bending Moment
Bending Moment (kNm)
Elastic Modulus v Bending Moment
3
2.5
2
K0 = 0.3
1.5
ko=3
1
0.5
0
0%
20%
40%
60%
80%
% of in-situ Elastic Modulus
100%
120%
Geotechnical Analysis –Bending Moment
transfer to Steel Liner
Geotechnical Analysis – Axial Load
Distribution in Steel
Axial Force (Ko=0.3)
700
Axial Force [kN]
650
600
550
500
450
0
2
4
6
8
10
Distance Around Liner [m]
12
14
Summary – Parametric Study






FE buckling analysis results in good agreement with
analytical predictions under uniform load for both
unrestrained and restrained conditions.
Under hydrostatic loads the unrestrained critical pressure
was greatly reduced, but there was very little change for
the restrained case.
FE results in good agreement with Jacobsen for gaps up
to 20 mm.
Varying restraint stiffness had a significant effect, with
reduced restraint stiffness reducing the critical pressure.
A vertical surcharge pressure greatly increased the
critical pressure, with the pipe failing in compression,
rather than bending.
Variation of the pipe/rock interface friction had little
effect.
Summary – Geotechnical Analysis



The coefficient of in-situ stress (K0) and the soil or rock
elastic modulus both had an effect on the axial load in the
steel liner.
Since plasticity had developed around the segmental liner
further deterioration of the concrete segments resulted in
only small further strains in the ground.
The arching action of the ground and the small increase
in strain resulted in increased axial load in the concrete
segments and steel liner, but negligible bending moment
transferred to the steel liner.
Conclusions

For the case studied in this paper the Jacobsen
theory was found to be suitable for the design of
the steel liner since:
– It gave a good estimate of the critical pressure under
hydrostatic loading
– Deterioration of the concrete liner was found not to
increase the bending moments in the steel liner
significantly


In situations with different constraint stiffness or
loading conditions the Jacobsen results could be
either conservative or un-conservative.
Further investigation of the critical pressure by
means of a finite element analysis is therefore
justified when the assumptions of the Jacobsen
theory are not valid.