Tunnels – MSc - SZIE - Mott MacDonald

Tamás MEGYERI

Load cases

Worst credible case ?

…. overall factor of safety of 1.0

EC2-EC3-EC7-EC8

Underground openings are just partially covered by the above

Geotechnical category: III

Factors of safety

Loads

Temporary Permanent

1.2 1.4

gf

Moderately conservative case

Problem statement

Structural models

Opening set

Material behaviour

• Elasticity (linear)

• Non-linearity

• Plasticity

of stress strain behaviour

s

e

Time dependent behaviour

Creep

Shrinkage

Ageing

s

t

e

Simple models for complex problems

Construction sequence

Discuss spatial layout

Design tools

• Empirical methods

• Analytical methods

• Numerical methods

Empirical methods

Simple

Quick

Cheap

Unknown factor of safety

Limits of applicability ?

Typical in mining practice

+ -

Analytical methods

Simple

Quick

Cheap

Detailed results

Simplifications - geometry,

behaviour, time effects

+ -

Numerical methods

Powerful because

you can model

more features

explicitly

Complex - to do & to

interpret

Time-consuming

Expensive

+ -

Even the most complex model is still

only an approximation of reality

Sources of error in design

Aspect of model Example

Geometry 2 D analyses

Construction method “Wished in place analyses”

Constitutive modelling & parameter selection

Using a linear elastic model for shotcrete lining

Theoretical basis Modelling discontinuous ground as a continuum

Interpretation Application of factors of safety

Human error Errors in input data

Error reduction

• Checking - by yourself (inputs & outputs); by

another person; by another calculation

method

• Calibration

• Sensitivity studies

• Engineering judgement

• Avoid: trial and error approaches!!!

• Simple methods!!! (single brick, simple beam…etc)

“sanity check”

Empirical methods

• Q-system

• Q-TBM

• RMR - rock mass rating

Mainly used for rock tunnelling

Design support charts

Q

Analytical methods

• Lining loads - “closed form solutions”, bedded beam

models

• Stability - of face or blocks

Simple equations

Numerical methods

• Finite element

• Finite difference

• Discrete element

• Boundary element

Hybrid methods

Continua

= soil or massive rock

Discontinua

= fractured rock

2D vs 3D

• Axisymmetric

• Plane strain

• Plane stress

• Full 3D

There are several means of approximating

the 3D stress redistribution in 2D analyses

Components

• Discretization

• Boundary conditions

• Material behaviour

• Groundwater

• Construction

sequence

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x x xx x xx x x x x x

Modelling strategy

• Plan carefully

• Start simple & develop

• Calibrate / validate

• Sensitivity study

• Perform a “sanity check”

Physical methods

• Trial tunnels

• Large-scale laboratory tests

• Centrifuge tests

Observational method ?

Concluding remarks

• Stress / strength .

• Groundwater .

• Joints

• Time-dependence

• Rock burst;

squeezing

• Inflow; stability;

swelling

• Block stability

• Creep

Tunnelling Course 2007

Mechanisms of behaviourRock

A new state of equilibrium

du

Stress relief at faceCross-section

new equilibrium

pi pi

…. a gradual process

tunnel pi

du

Deformations occur over

a finite time period

pi

ground

reaction

curve

Timing & stiffness of support

pi

du or tdeformation

or time

support

pre

ssure

p/u = E stiffness

support curve

F = m a zero ?

Vertical displacement

Horiz. displacement

0

60

mm

0

-20

20

Failure criterion in 3D stress space

-s1

-s2 -s3 back

back

Design Considerations

Complex soil-structure interaction

Complex geometry

Mode of behaviour of ground

Material behaviour (of ground & support)

Time effects (of ground & support)

Other design criteria - e.g. for durability

Design tools

• Empirical methods

• Analytical methods

• Numerical methods

Bedded beam models

• Advantages

– Simple & quick

– Freedom in application of loads

• Limitations

– Determining spring stiffness

– Superseded by 2D numerical models

Numerical methods

• Finite element

• Finite difference

• Discrete element

• Boundary element

Hybrid methods

Continua

= soil or massive rock

Discontinua

= fractured rock

Discontinua

• Block stability : Phases2 …. with DIPS

• Rock mass : Discrete Element Method

• Faults : Boundary Element Method

Continua

• Finite element method

• Finite difference method

Why FLAC?

Good for nonlinearity & ground modelling

Good for large displacements / instability

FISH language

Experience

FDM - Theory

Előnyök/hátrányokexplicit / implicit

Explicit, time marching Implicit, static

1. Non linearity can be handled directly. (sigma-

epszilon available at each step)

Iteration required.

2. Computational time: N3/2 N2 or even N3

3. Not sensitive for geological instabilities. (‚any

number”)

Problematic to model.

4. Matrix is only partially stored during solving.

Computaionally less expensive.

Large RAM or SWAP file requirement.

5. Large starin / failure / sliding modes can be

computed effectively

Hints needed.

Continua – an expamle – uniaxially loaded

beam

FLACnon-linear strain softening soil model for OC LC

2.0

1.6

1.2

0.8

0.4

0.00.001 0.01 0.1 1

Test

Jardine

εA [%]

Eu/Cu Jardine et al. 1986

Eu es

Cu C= A + B cos ( (lg ( ) )

g

Anizotropy

Izotróp paramétek

Eu ev

Cu T= R + S cos ( (lg ( ) )

μ

FLACMeshHexa vs Tetra – mixed discretization

Modelling strategy

• Plan carefully

• Start simple & develop

• Calibrate / validate

• Sensitivity study

• Perform a “sanity check”

What are you looking for ?

• Far field movements ?

– Small strain behaviour of ground ?

– Anisotropy ?

• Near field & tunnel lining ?

– Construction methodology

– Plasticity in the ground

2D vs 3D

• Axisymmetric

• Plane strain

• Plane stress

• Full 3D

There are several means of approximating

the 3D stress redistribution in 2D analyses

Correction factors for 3D

• Stress reduction method - b or

• Stiffnes reduction method

• Gap parameter (for TBM’s)

• Hypothetical Modulus of Elasticity (for SCL tunnels)

Components

• Discretization

• Boundary conditions

• Material behaviour

• Groundwater

• Construction

sequence

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x x xx x xx x x x x x

Mesh

X = 3 * Zo

Y = 12 * R

Z = 3 * Zo

R = radius

of tunnel

FLAC (Version 3.40)

LEGEND

13-Dec-99 11:12

step 53777

Cons. Time 2.4706E+03

-9.000E+00 <x< 9.000E+00

1.110E+02 <y< 1.290E+02

Grid plot

0 5E 0

Displacement vectors

Max Vector = 1.619E-03

0 5E -3

1.120

1.140

1.160

1.180

1.200

1.220

1.240

1.260

1.280

(*10^2)

-8.000 -6.000 -4.000 -2.000 0.000 2.000 4.000 6.000 8.000

JOB TITLE : Mission Valley East LRT-SDSU Tunnel-Opt 2 new layout-non linear soil

Itasca Consulting Group, Inc.

Minneapolis, Minnesota USA

FLAC (Version 3.40)

LEGEND

13-Dec-99 11:12

step 53777

Cons. Time 2.4706E+03

-9.000E+00 <x< 9.000E+00

1.110E+02 <y< 1.290E+02

Y-displacement contours

-1.00E-03

-5.00E-04

0.00E+00

5.00E-04

1.00E-03

1.50E-03

Contour interval= 5.00E-04

Moment on

Structure Max. Value

# 1 (Beam ) 2.302E+01

# 2 (Beam ) -2.302E+01

1.120

1.140

1.160

1.180

1.200

1.220

1.240

1.260

1.280

(*10^2)

-8.000 -6.000 -4.000 -2.000 0.000 2.000 4.000 6.000 8.000

JOB TITLE : Mission Valley East LRT-SDSU Tunnel-Opt 2 new layout-non linear soil

Itasca Consulting Group, Inc.

Minneapolis, Minnesota USA

Stress distribution in the lining

1.0

0.0

Detailed Design for ILW 10mDia chamber and headwall /

dewatering facility in crystalline rock formation

ILW - evaluation of plate jacking test (evaluation of

load vs displacement measurements)

0

200

400

600

800

1000

1200

0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0

Elmozdulás (mm)

Teh

er

(t)

4m

2m

1m

0.5m

0m

ILW - DEM vs Continuum model

Primary lining forces - discrete element vs. continuum models:

Conceptual Design: Engineering barrier for ILW - Clay

gauge / crushed rock composite in clay filled thrust fault

in crystalline rock formation

3DEC capabilities

DEM

Hydroproject - Kemano

UDEC – Metro station in HK

104

106

108

110

112

114

116

-8 -6 -4 -2 0 2 4 6 8

Ele

vatio

n (m

ATD

)

X coordinate (m)

Element numbering (-) - Stage: 16 (28 MPa strength)

Special aspects / tailor made solutions

Concrete grade

28.01 MPa Fibres ? No FALSE C28.01/35.0125 D 0.5 S 0

500 MPa [Yes / No] after BS EN 14487-1

205 GPa

Design Flexural Tensile Strength (fctd, fl) 0.00

0.00 kh = 0.53

0.000 fR.k1 = 1.26 MPa

0.000 fR.k4 = 0.98 MPa

Effective Tensile Strain Limit if Unreinforced -0.50% Material factors

434.78 SFRC 1.5

1000 mm Conc 1.5

500 mm Steel 1.15

Diameter Spacing Cover to bar

0 150 120 mm (closest to the surface)

0 150 120 mm

380 mm

380 mm

0 mm2

0 mm2

8 150 50 mm

0 150 50 mm

54 mm

50 mm

335 mm2

0 mm2

500 mm

Actual 0.00%

Minimum for Reinforced 0.14%

Depth (h)

Tension Bar 1

Factored Yield Stress

Reinforcement

Width of section (b)

RILEM ValuesTensile s1

MPa

Minimum residual s3

Maximum residual s2

Steel Elastic Modulus (Es)

Steel (fy)

Section Data

Concrete Grade (fck)

Tension Bar 2

d-1

d-2

d'-1

Ast1

Ast2

Compression Bar 1

Compression Bar 2

N.B. Cover-to-bar is to the tension or

compression bar itself, not just the specified

minimum cover to rebar generally.

d'-2

Asc1

Asc2

Section Effective depth d

Tension Steel RatiosSection classed as unreinforced, capacity

calculated to Section 12.6

-1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

0 50 100 150 200 250 300 350 400

Axia

l F

orc

e (kN

/m)

Bending Moment (kNm/m)

Moment Interaction Diagram / hoop direction / FF - at 672 hours age (28.01 MPa strength) RTypD

Capacity curve

Prim_16_RTypD

104

106

108

110

112

114

116

-8 -6 -4 -2 0 2 4 6 8

Ele

vati

on

(mA

TD)

X coordinate (m)

Bending moment capacity (%) - Stage: 16 (28 MPa strength)

-37 299

104

106

108

110

112

114

116

-8 -6 -4 -2 0 2 4 6 8

Ele

vati

on

(m

AT

D)

X coordinate (m)

Axial force distribution (kN/m) - Stage: 16 (28 MPa strength)

157 1466

Rigorous checking procedure

Thanks for listening! Wish you successful exams!!

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