Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
1/8

Organization (S): EDF-R & D/MMC

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
Document: V3.02.105

SSLP105 - Excavation of a circular tunnel in
a linear elastic solid mass

Summary:

This test constitutes an example of implementation of a total methodology for simulation
two-dimensional of the digging and the supporting of a circular gallery in an underground solid mass with
Code_Aster.

To validate the step on the basis of simple analytical solution, one is brought to make assumptions
restrictive on the geometry of the problem, the behavior of materials (elastic linear) and the field of
constraint initial (isotropic). The reference solution is given by the method known as “convergence
containment “, traditional for this type of modeling 2D.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
2/8

1
Problem of reference

1.1 Geometry

It is about a circular tunnel of section, covered by a concrete ring, which one excavates in one
solid mass of ground. The two materials are supposed to be elastic linear.

Z
18,20 m
y
B
0,30 m
1,50 m
X
20 m
With
20 m


1.2
Properties of material

The materials are elastic linear.

1.2.1 Ground

Es = 4 GPa
S = 0,3

1.2.2 Concrete

Eb = 20 GPa
B = 0,2
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
3/8

1.3
Initial conditions, boundary conditions and loadings

The constraints in the solid mass are supposed initially isotropic (xx = yy = zz = 0).
method used to simulate the excavation and the pose of supporting is the method known as
“convergence-containment” presented for example in [bib1] and [bib2].

The guiding principle rests on a reduction in the nodal reactions generated at the edge of the future one
gallery by the initial state of stresses. This operation is indicated by name
“déconfinement”. When déconfinement the value reached which corresponds to the conditions of
building site which one wishes to model, one carries out the simulation of the pose of concrete supporting
at the edge of the gallery.


Solid mass of ground
Solid mass of ground
Excavation of
Pose
Initialization of
gallery
coating concrete
constraints
Calculation of the reactions
nodal
1
2
3
4

The boundary conditions and the loading are summarized in the following table. Phases
correspond to those of the diagram above, the edges are composed with the nodes identified on
the diagram of the paragraph [§ 3.1] and between brackets the name of the groups of mesh or node of
file .comm).

Edges
Phase 1
Phase 2
Phase 3
Phase 4
N0N1 (in
DY = 0
DY = 0
-
-
no_bas1)
N1N2
DY = 0
DY = 0
-
DY = 0
(bas_bet)
N2 N3
DY = 0
DY = 0
DY = 0
DY = 0
(no_bas2)
N3N4
DX = 0
DX = 0
DX = 0
DX = 0
(no_droit)
N4N5
=
MPa
=
MPa
=
MPa
=
MPa
(ma_haut)
yy
5
-
yy
5
-
yy
5
-
yy
5
-
N5N6
DX = 0
DX = 0
DX = 0
DX = 0
(no_left2)
N6N7
DX = 0
DX = 0
-
DX = 0
(no_left_bet)
N7N0 (in
DX = 0
DX = 0
-
-
no_left1)
N6N2 (edge)
-
-
Nodal reactions
-
corresponding to
déconfinement
N7N1 -
-
-
Free
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
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2
Reference solution

2.1
Method of calculation

2.1.1 Behavior of the ground

That is to say the rate of déconfinement, which represents the relative position of the section of tunnel considered
compared to the coal face. In the method “convergence - containment”, one replaces the future
ground excavated by a tensor of the constraint are equivalent, which one cause a drop in the intensity via for
to simulate the digging and the distance of the coal face.


Coal face
Tunnel
= 0
0
= 1
(-).
0
=


The solution of the problem is thus similar to that of the infinitely thick tube charged by a pressure
intern of intensity (1) .0 and by an external pressure of intensity 0 (see [bib3] for the detail of
calculations).

Constraints radial, orthoradiale as well as radial displacement with the wall of the tunnel in
springy medium subjected toa rate of déconfinement are as follows



R
. 2

0
R = 1
.


2

R




R
. 2
= 1+
0


2.


R



R2
0


UR =.
.

R
G
2

E
G is the modulus of rigidity given by the following relation: G =
.
2 (1+)
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
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2.1.2 Behavior of supporting

Supporting will be opposed to the movement convergence natural of the tunnel and thus will apply one
artificial containment with the rock.

Either Ks the stiffness of supporting, it is given by the following relation if it is considered that it
supporting is comparable to a thin tube (B is the Poisson's ratio of the concrete):

= (E
K
B
S

- 2
1 b) E R

K
If K
S
S =
represent the relative rigidity of the concrete compared to the solid mass and
2 G
D
the rate of
déconfinement with the installation of supporting, then radial constraints and orthoradiales thus
that radial displacement in wall are given by [bib1]:


= ks
0
R
(1 - D)

1+ ks


K
=
S
(1+
0
D)

1+ ks

0

1+ D

R =
K
U
S
R


1+ ks
2 G

2.2
Sizes and results of reference

One tests the following sizes on the level of the wall at points A and B of the figure of the 1.1, at the moment
where déconfinement is total:

· radial constraint: yy of A or zz out of B;
· constraint orthoradiale: zz of A or yy out of B;
· radial displacement: uy of A or uz out of B.

2.3
Uncertainties on the solution

None. Exact analytical result.

2.4 References
bibliographical

[1]
The calculation of the tunnels by the method convergence-containment, Mr. Panet, Presses of
the ENPC 1995
[2]
How to simulate the digging of a tunnel with Code_Aster? Principle of the method,
implementation and validation, A. Courtois, R. Saidani, P. Sémété, note EDF HT-25/02/045/A -
2002
[3]
Mechanics of the continuous mediums, volume 2, J. Salençon, ED. Ellipses - 1988
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
6/8

3 Modeling
With

3.1
Characteristics of modeling

Modeling 2D in plane deformations.


y
N4
N5
N6
N7
X
N0
N1
N2
N3


3.2
Characteristics of the grid

A number of nodes:
8477
A number of meshs:
3304 of type QUAD 8

3.3 Functionalities
tested

The objective of this case test is to test a method more than one quite precise functionality of
Code_Aster, also the following table presents the principal commands which structure the file of
commands.

Commands

Comments
CREA_CHAMP
Initialization of the constraints geostatics (here isotropic 5 MPa in
compression)
STAT_NON_LINE Blocage of the nodes of the gallery for calculation of the reactions
nodal to inject to simulate déconfinement
CREA_CHAMP
Recovery of the nodal reactions
STAT_NON_LINE D-injection of the nodal reactions
Intermediate STAT_NON_LINE Calcul to pass from a model without mesh
representing the voussoirs concrete with a model with meshs them
representative (see [bib2])
STAT_NON_LINE progressive Déconfinement of the solid mass

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
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3.4
Sizes tested and results

After the pose of the coating (urgent final), one tests the components and with the nodes N2 and N6
xx
yy
as well as radial displacement in these points (DX for N2, DY for N6).

Reference
Aster Difference
(%)
Node N2



xx
- 1,52821.106 - 1,53154.106 0,218
yy
- 8,47179.106 - 8.52772.106 0,660
DX - 1,6925.10-3 - 1,6684.10-3 - 1,422
N6 node



xx
- 8,47179.106 - 8,41147.106 - 0,712
yy
- 1,52821.106 - 1,52943.106 0,080
DY - 1,6925.10-3 - 1.7184.10-3 1,529

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Code_Aster ®
Version
6.5
Titrate:
SSLP105 - Excavation of a circular tunnel in an elastic solid mass
Date:
09/09/03
Author (S):
A. COURTEOUS Key
:
V3.02.105-A Page:
8/8

4
Summary of the results

The values obtained with Code_Aster are in agreement with the values of the analytical solution of
reference.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-26/03/023/A

Outline document