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.