MR17

1

STRUCTURAL AND GEOTECHNICAL ENGINEERING DEPARTMENT

ROCK MECHANICS 2ROCK MECHANICS 2

Giovanni Barla

Politecnico di Torino

(a) Typical analyses in ground engineering drained conditions undrained conditions(b) Boundary Conditions(c) Simulation of excavation and construction stages (d) Stability and ground movements in near surface tunnelling: fundamental concepts

LECTURE 17 - OUTLINETunnelling in Urban EnvironmentApplications of Numerical Methods

FEM ANALYSIS: Classes of problemsDrained and Undrained Behaviour


In this course the ground constitutive behaviour was expressed inthe following forms:

1) []=[Ct] []

2) []=[Cs] []

3) []=[Cep] []

where, in general, the finite elementsolutions have been found by usingrelationships between increments oftotal stress and strain, because theequilibrium equations are expressedin terms of total stress. Also, for CILEmaterials we have: []=[C] []

where: []T = [x, y, z, xy, xz, yz]

[]T = [x, y, z, xy, xz, yz ]

are increments of total stress and strain, with the [C] matrixcontaining the elastic constants

The FEM formulation presented so far can therefore be used toanalyse the following two classes of problems: Fully drained problemsin which there is no change in pore fluid pressure (pf=0). This impliesthat changes in effective and total stress are the same, i.e. [] = [], and that the [C] matrix contains the effective constitutive behaviour. For example, for isotropic linear elastic behaviour [C] will be based on drainedYoungs modulus E and drained Poissons ratio

Fully undrained problemswhere the [Ct], [Cs] and [Cep] matrix are expressed in terms of total stress parameters. For isotropic linear elastic behaviour [C] is based on an undrained Youngs modulus Eu and an undrained Poissons ratio u. In such a case, if the ground is saturated there would be no volume change. For an isotropic elasticsoil this would be modelled by, ideally, setting the undrained Poissons ratio u equal to 0.5. However, as can be seen by inspection of the [C] matrix, thisresults in severe numerical problems as all terms of the [C] matrix becomeinfinite. To avoid such indeterminate behaviour it is usual to set the undrained Poissons ratio to be less than 0.5, but greater than 0.49.



2

BOUNDARY STRESSESBOUNDARY STRESSES

4

1 2

3

Example of stress boundaryconditions

Stress boundary conditions haveto be converted into equivalentnodal forces:

Se [H]eT [fs]edS

In many computer codes thecalculation of the equivalent nodal forces is performed automaticallyfor generally distributed boundary stresses and for arbitrarily shapedboundaries

POINT LOADS AND SIMPLE STRESS DISTRIBUTIONSPOINT LOADS AND SIMPLE STRESS DISTRIBUTIONS

P=0 =20

L

1/30 L 2/30 L

= 0 = 0

= 20 = 0

= 0

= 20

L L L LL

1/20L 1/20L 1/30L 2/30L 1/60L 1/60L2/30L 1/30L2/30L0

Point loading axesi

xpyp

yG

xGGlobal axes

Equivalent nodal forces for element side with 2 nodes and 3 nodes

SIMULATION OF INITIAL CONDITIONSSIMULATION OF INITIAL CONDITIONS

Gravity loading : there are two possible ways to introduce the gravitationalstresses in a model. In most FEM codes the calculation of nodal forcescorresponding to body forces is performed automatically. In particular, this isthe case for the gravitational stresses:

Perform a stress analysis where all the elements in the mesh are subjected togravity loading, with appropriate boundary conditions being introduced as shown in the exampe below:

0

xzxz

v=0

u=0

z=z, x=k0 z

Set the initial stresses in the model as computed directly versus depth z:

i.e. z=z, x=k0 z, xz= 0if the ground surface is horizontal

x

z

Note: the color intensity in the above figureshows the increase in vertical stress z versusdepth z

SIMULATION OF EXCAVATIONSIMULATION OF EXCAVATION

A

B

T

-T

Simulation of excavation

1. Set the initial state of stress and specifythe elements to be excavated in the shaded portion A

2. Determine the equivalent nodal forcesT to be applied to the excavation boundaryto simulate removal of the elements (note: no displacements or changes in stress occur ifmaterial is removed, but replaced by tractions T which are equal to the internalstresses in the soil mass that act on theexcavated surface before A is removed)

Zone to beexcavated

3. Apply the traction -T at the excavationboundary, when material is removed by

deactivating the elements in zone A

3

i

Excavation surface

[R0]i = 1m [R0]jj=1

m I = node on the excavation surfacej = elements around node im= number of elements around node i

[R0]j =Ve [B]Te [C]e [B]e [u]e dV

SIMULATION OF EXCAVATIONSIMULATION OF EXCAVATION SIMULATION OF EXCAVATION - EXAMPLESIMULATION OF EXCAVATION - EXAMPLE

Nodal Forces applied at tunnel boundary

AB

A

B

Radial displacement

Gra

dual

dest

ress

ing

at t

he t

unne

l bou

ndar

y

The simulation of the excavation is performedin multiple stages

hh

Nodal ForcesNodal Forces

SIMULATION OF CONSTRUCTIONSIMULATION OF CONSTRUCTION

Many geotechnical problems involve the placing of new material, such asembankment construction and back filling or tunnel lining installation, etc. Simulation of such activities in a finite element analysis is possible. However the code must be able to accomodate a number of provisions, such as activation or deactivation of elements depending on thesimulation sequence to be simulated. Let us see this with an exampledealing with embankment construction

Layer 3Layer 2Layer 1

A B

1. All elements in the embankment are deactivatedup to the stage when the embankment is to beconstructed

2. In increment 1, layer 1 is to be constructed and therefore, at the beginning of the increment, all the elements in this layer are reactivated (i.e. added tothe active mesh) and assigned a constitutive modelappropriate to the material behaviour during placing.Self weight forces are then assumed for theseelements and the equivalent nodal forces calculatedand added to the incremental right hand side vector. The global stiffness matrix and other boundaryconditions are assemled and a solution found.

h

h

hhh h

h

SIMULATION OF CONSTRUCTIONSIMULATION OF CONSTRUCTION

Layer 3Layer 2Layer 1

A B

h

h

hhh h

h

3. The incremental displacements calculated for the nodes connected to the constructed elements, but not connected to those elements forming the original ground (i.e. all active nodes above line AB) are zeroed. A new constitutive model appropriate to thebehaviour of the fill once placed is then assigned tothe elements just constructed. Any material state parameters are then calculated and any stress adjustments made .

4. The procedure for construction of layers 2 and 3 follows similar steps. The final result is obtained by accumulating the results for each increment of the analysis. Clearly the results of the analysis willdepend on the number and therefore thickness of construction layers.

Procedure for deactivating the elements

The deactivated elements (GHOST ELEMENTS) in the active mesh are given a very low stiffness. These elements do contribute to the elementequations and all degrees of freedom remain active. Their effect on the solution depends on the stiffnessthat they assume. Most software that use thisapproach automatically set low stiffness values forthe ghost elements, or encourage the user to set low values. However care must be taken that the resulting Poissons ratio does not approach 0.5.

4

TUNNELS IN URBAN ENVIRONMENT

Tunnels in urban environment are near surface tunnels and in most casesthey are constructed in soft ground. Given that construction techniques(by open face tunnelling and closed face tunnelling - TBM) have alreadybeen discussed, the following topics will be addressed in the following: Stability Ground Movements

(a) Tunnel heading in soft ground (b) Two dimensional idealization

C

D

P

LINING

D

C

t t

s s

UNDRAINED STABILITYUNDRAINED STABILITY, In low permeability clayey soils undrainedstability is of more importance during tunnel excavation, but in the caseof a standstill drained conditions could become more relevant.For undrained stability conditions, the stability ratio N can be defined as:

762

++

=u

ts

c

DCN

(Broms and Bennemark,1967; Peck,1969)


where: = unit weight of soilz = depth to the tunnel axis (C+D/2)s = surface surcharge pressure (if any)t = tunnel support pressure (if any)cu = undrained shear strength at tunnel axis level

762

++

=u

ts

c

DCN

BasedBased onon thethe stabilitystability ratioratio NN, Cesarin e Mair (1981) geve a set of curvesto compute the displacement at the surface, crown and faceas follows:

(s- t)/cu

In the diagram the followingdisplacements are given:

(s), surface displacement(c ) crown displacement(c ) face displacement

for different N values , given by (s- t)/cu


Stability Ratio

DRAINED STABLITY, DRAINED STABLITY, for for c=cohesion and =friction angle, thestability at the face in terms of t can be evaluated as follows:

Instability Conditions at the face in a cohesiveground (Leca and Dormieux,1990)

( )'cot'2

'cot' +++= cQDQc sst

Derived on the basis of the three-dimensional conditions shown in the figure below, where instabilty phenomena occur ahead of the face and Q e Qs can be computed vs

Q e Qs parameters for evaluating the stabilityconditions vs depth H, for different values of (Leca e Dormieux,1990)


5

Onset of Plastic Zone around a tunnel in urban environment for different Ko valuesand as stress relief takes place around it (the ground is ideally plastic and c=0 and is 35 (Wong and Kaiser,1991)


Mode 1(localised)

Mode 2 (all around)

Mode 3

Mode 1-1 Mode 1-2

Mode 1-1.1 Mode 1-1.2

Mode 2-1 Mode 2-2

Onset of

Plastic Zone

Plastic Zone

at the sidewalls

Plastic Zone

at the crown

STRE

SS R

ELIE

F

Loading Conditions represnted by height Ht,for different RMR values and for increasing overburden C (D=10m, K0=1)

TUNNELS IN URBAN ENVIRONMENTin a Weak Rock Mass (RMR gives the rock mass conditions)

Loading Conditions represented by height Ht,for different RMR values and for increasing overburden C (D=10m, K0=0.5)

Settlement above advancing tunnel heading

Gaussian curve used to describe the transverse settlement trough

z

y

The development of the surface settlement trough above and ahead of the advancing heading has been studied by many authors based on observation and measurements in situ. The TRANSVERSE TRANSVERSE SETTLEMENTSETTLEMENT trough immediately following tunnel construction is well described by a Gaussiandistribution curve given by (Schnidt,1969; Peck,1969):

= 2

2

max 2exp)(

iysys

s(y) = settlementsmax = maximum settlement on the tunnel centre-line (y=0)i = horizontal distance from the tunnel centre-line to the

point of inflexion of the setllement through

TUNNELS IN URBAN ENVIRONMENTGROUND MOVEMENTS

point of inflexion

3D visualisation of settlement above tunnel heading(Attewell et al.,1986; Mair et al.,1993)


6

K ground conditions

0.4 - 0.5 clays0.6 - 0.7 soft clays (cu= 0-20 kPa)0.2 - 0.3 sands and gravels

i = K z0i = K z0

where K is a trough width parameter and z0 is the depth of tunnel (is broadlyindependent of tunnel construction method and of tunnel diameter, exceptfor very shallow tunnels where the cover to diameter ratio is less than 1). Kis as follows for different ground conditions:

PARAMETER i: PARAMETER i: The following relationship is proposed to evaluate thei parameter:


xOriginal ground level

smzx0.5 smzx

Longitudinal settlement profile(cumulative probability form) without face support

Tunnel face

Advancing tunnel

=

2221

00max kz

Xxerfkz

XxerfsS sf

The settlementsettlement alongalong thethe tunneltunnelaxis axis (LONGITUDINAL (LONGITUDINAL SETTLEMENT) SETTLEMENT) can be written as follows:

( ) ( ) dtezerf z t = 02

2

where:

Error function


4

2DVV ls

=

VOLUME OF THE SURFACE SETTLEMENT TROUGH VOLUME OF THE SURFACE SETTLEMENT TROUGH VVss: : The volumeof surface settlement trough (per metre length of tunnel) Vs can be calculatedby integrating the equation for s(y) to give:

Vs= 2 i smax 2.5 i smaxVs= 2 i smax 2.5 i smax

The volume loss (some time referred as ground loss) is the amount of groundlost in the region close to the tunnel, primarily due to one or more of the following components: deformation of the ground towards the face resulting from stress relief; passage of the shield (overcutting edge); tail void (gap between the tailskinof the shield and the lining); deflection of the lining, consolidation. It can be expressed as follows:

The following values are generally assumed for Vl based on experience (Mair,1997): stiff clays (no stabilising pressure at the face): 1-2% stiff clays (with stabilising pressure at the face): 1-1.5% sand (EPB o SS): 0.5-2.0%


Typical Roof Instability for a near surface tunnel Maximum shear strain curves above a tunnelduring excavation (Hansmire e Cording,1985)

NOTE: NOTE: The above considerations and results for surface settlementsdevelopment above a tunnel hold true for all cases where no instabilityphenomena occur as depicted in the figures below


MR17

Documents

Transcript of MR17