Finite element modelling of load shed and non-linear buckling solutions of confined steel tunnel...
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Finite element modelling of load shed Finite element modelling of load shed and non-linear buckling solutions of and non-linear buckling solutions of
confined steel tunnel linersconfined steel tunnel liners
10th Australia New Zealand Conference on 10th Australia New Zealand Conference on Geomechanics, Geomechanics,
Brisbane Australia, October, 2007Brisbane Australia, October, 2007
Doug Jenkins - Interactive Design ServicesAnmol Bedi – Mott MacDonald
Introduction Introduction
Port Hedland Under Harbour TunnelLined with 250 m thick gasketed precast
concrete segments – now corrodingProposal to reline with steel backgrouted
linerGeotechnical and structural finite element
analysesComparison with analytical solution
TopicsTopicsThe proposed remedial workConfined liner bucklingJacobsen Closed Form Buckling SolutionLinear buckling FEAApplication to the project
– Current stress state in tunnel liner– Future Installation of Steel Liner– Geotechnical FEA results
Conclusions
Port Hedland Under Harbour Port Hedland Under Harbour TunnelTunnel
Material PropertiesMaterial Properties
Material (kN/m3) E (MPa) Cohesion (kPa) Fill 18 25 30 25 0.45 Marine Mud 18 5 30 25 0.45 Red Beds 20 23 55 32 0.3 Upper Conglomerate 22 1000 250 36 0.3 Sandstone 22 100 130 34 0.25
Lower Conglomerate 22 1000 55 32 0.3
Table 1: Material Properties
Closed Form SolutionsClosed Form SolutionsUnrestrained solution similar to Euler
column bucklingRigid confinement restrains initial
bucklingGap 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 BucklingSingle Lobe Buckling
Comparison of buckling theoriesComparison of buckling theoriesBerti (1998) compared theories by Amstutz
and JacobsenAmstutz approach was simpler, but
assumed constants may be unconservativeAlso found that rotary symmetric equations
are unconservative compared with JacobsenComputerised analysis allows the more
conservative Jacobsen method more general use
Jacobsen EquationsJacobsen Equations
sinsin
tansinsin.4
sin
sin.
2sin
sin1 2
m
rt
m
rt
rtm
y
E
Rp
E
Rp
RE
3
2
23
sinsin
149.12
m
rt
E
Rp
Jacobsen EquationsJacobsen EquationsJacobsens Analysis for Calculating Single Lobe Buckling of Circular Steel Tunnel Liner
Thickness t 25 mmRadius r 2062.5 mm
Poisson's Ratio 0.25
Gap 0 mmGap/radius k 0Yield stress y 250 MPa
Young's Mod E 200000 MPaR/t Rrt 82.5
Em 213333.3333 Mpa
Estimated a 0.8
Value ErrorRrt 82.50 0.0000
Solve
Jacobsen EquationsJacobsen Equations
Value Errory 250.0 0.0000
Value ErrorRrt 82.50 0.0000
Jacobsen Solution Unrestrained Buckling Solutiona 0.6755 Radiansb 0.6636 Radiansp 1.4960 MPa Pcr 89.0 kPa
3/3 REIPcr
Parametric StudyParametric Study
Run No Variable Pressure Pipe Deformation,mm 1-3 Pipe deform. Uniform 0, 10, 20 4-6 Pipe deform. Hydro. 0, 10, 20
Run No Variable Pressure Gap Contact Friction
Contact Stiffness
Rock E Surcharge Pressure
mm Factor MN/m GPa Ratio 7-10 Pipe/restraint gap Uniform 0, 1, 2, 5 0.5 10 11-14 Pipe/restraint gap Hydro. 0, 1, 2, 5 0.5 10 15-17 Contact friction Hydro. 2 0.7, 0.5, 0.3 10 18-20 Contact stiffness Hydro. 2 0.5 1, 5, 100 21-25 Rock stiffness Hydro. 2 0.5 100 10,1,0.25,0.1,0.05 26-29 Surcharge press. Hydro. 2 0.5 100 1 0, 0.3, 0.6, 1.2
Unrestrained Buckling ModelUnrestrained Buckling Model
Unrestrained BucklingUnrestrained Buckling
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100
Pressure, kPa
Deflectio
n, m
m
Undeformed 10 mm deformation 20 mm deformation
Undeformed 10 mm deformation 20 mm deformation
Uniform Pressure
Hydrostatic Pressure
Unrestrained BucklingUnrestrained Buckling
Unrestrained BucklingUnrestrained Buckling
FE Model for Restrained BucklingFE Model for Restrained Buckling
FE Model DetailFE Model Detail
FE Model DetailFE Model Detail
Restrained Buckling - deflectionRestrained Buckling - deflection
0
10
20
30
40
50
60
0 200 400 600 800 1000 1200 1400
Pressure, kPa
De
fle
cti
on
, m
m
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
Restrained Buckling - deflectionRestrained Buckling - deflection
Restrained Buckling - gapRestrained Buckling - gap
0
200
400
600
800
1000
1200
1400
1600
0 1 2 3 4 5
Gap, mm
Cri
tical P
ressu
re, k
Pa
FEA-Uniform FEA-Hydrostatic Jacobsen
Effect of contact friction and restraint Effect of contact friction and restraint stiffnessstiffness
500
600
700
800
900
1000
1100
1200
1300
0 1 2 3 4 5
Run No
Cri
tic
al P
res
su
re, k
Pa
Friction Contact Stiffness Rock/Soil Stiffness
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/mRock/Soil stiffness: 1=10, 2=1.0, 3=0.25, 4=0.1, 5 =0.05 GPa
Effect of surcharge pressureEffect of surcharge pressure
-300.0
-200.0
-100.0
0.0
100.0
200.0
300.0
0 500 1000 1500 2000 2500
Pressure, kPa
Str
es
s,
MP
a
0 kPa Top face 30 kPa 60 kPa 120 kPa
0 kPa bottom 30 kPa 60 kPa 120 kPa
Geotechnical Analysis – Current Stress Geotechnical Analysis – Current Stress StateState
Geotechnical Analysis – Elastic Modulus v Geotechnical Analysis – Elastic Modulus v Bending MomentBending Moment
Elastic Modulus v Bending Moment
0
0.5
1
1.5
2
2.5
3
0% 20% 40% 60% 80% 100% 120%
% of in-situ Elastic Modulus
Ben
din
g M
omen
t (kN
m)
K0 = 0.3
ko=3
Geotechnical Analysis –Bending Moment Geotechnical Analysis –Bending Moment transfer to Steel Linertransfer to Steel Liner
Geotechnical Analysis – Axial Load Geotechnical Analysis – Axial Load Distribution in SteelDistribution in Steel Axial Force (Ko=0.3)
450
500
550
600
650
700
0 2 4 6 8 10 12 14
Distance Around Liner [m]
Axi
al F
orc
e [k
N]
Summary – Parametric StudySummary – 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 AnalysisSummary – 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.
ConclusionsConclusions 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.