“Irregularities” influence on underground structures
seismic behaviour
Catarina Francisca Noronha Bragança Vaz
Department of Civil Engineering and Architecture, IST, Technical University of Lisbon
1. Introduction
Underground structures have an important role in modern cities. In general, this type of
structure is modelled as symmetric and continuous structures.
This work is about “irregularities”, in the cross-section, and their influence on the structures’
global behaviour, under seismic action using Finite Elements method.
2. Underground structures’ damages
According to Owen and Scholl (1981), when an earthquake occurs, the tunnel behaviour is
conditioned by the soil around. His deformation can be three types: axial, longitudinal curvature and
cross-section distortion, which one is caused by seismic waves that propagate in the perpendicular
direction (St Jonh and Zahrah, 1987) (Figure 2-1 and Figure 2-2).
(a)
(b)
Figure 2-1 Longitudinal tunnel deformation: (a) Axial; (b) longitudinal curvature (Owen & Scholl, 1981)
(a)
(b)
Figure 2-2 Cross-section deformation: (a) ovaling; (b) racking (Wang, 1991).
After, knowing the possible deformation cases, it is time to understand the soil’s rupture
mechanisms. This can happen in different ways. According to Owen and Scholl (1981) four mechanisms
were identified.

Fault movements – Happen when the structure intercepts an active fault. The damages are,
normally, near to the fault zone and it depends on his displacement.

Soil collapse – Is related to the soil liquefaction, ground displacements and rocks. Most of the
damages occur on the tunnel’s portal and on shallow structures.
[1]

Vibrating movement – Here and according to st. Jonh and Zahrah (1987), the soil will lose
stiffness when submitted to vibrating movements. This fact will transfer additional loads to the
tunnel causing damages, which are diversified.

Different stiffness – Includes the transition tunnel-station or the difference of soil geological
characteristics. These cause different behaviours along the structure.
About seismic vulnerability, two major studied can be highlighted: Dowding and Rozen, 1978;
and Sharma and Judd, 1990.
While Dowding and Rozen related the damages to the Peak Ground Acceleration (PGA) and
Peak Ground Velocity (PGV), Sharma and Judd linked the tunnels’ damages to: the depth of the
structure, the soil proprieties, PGA, earthquake magnitude and the epicentre distance. Off all the
parameters studied by Sharma and Judd, the earthquake magnitude, the epicentre distance and the
depth of the structure are the ones that affect more the vulnerability.
2.1 Examples of “irregularities”
Some of the “irregularities” that can be found in underground structures as well as ways to
mitigate them, are:

High soil’s stiffness contrast – Since soil is heterogeneous it can be found two material with high
stiffness contrast. This will obligate the structure to adjust to the different deformations. To
improve the structure behaviour it is used isolation layer that can be from rubber to steel. This
layer will become the tunnel more flexible. It is also possible to improve the geological
proprieties of the soil through jet-grouting.

Active fault intersection – Again, this situation triggers differential displacements that the
structures have to accommodate. So, to avoid the structure collapse, it should be done a study
to turn the tunnel more flexible. It is also possible to use the isolation layer, above referred.

Cross-section geometry changes – Usually, this happen when there are a connection between
tunnel and station or the underground structure and the building. This sudden change of the
cross-section will generate some critical points that should be study more carefully, when
submitted to earthquake loading.

Water surge and soil liquefaction – The liquefaction, is a type of soil rupture and it causes the
soil to behave like liquid. So it is possible to observe the floating of the tunnel. One way to
prevent this type of structure collapse is by using cut-off walls (Schmidt e Hashash, 1999),
which will decrease the water pressure.

Tunnel Portal – This is caused by the falling of rocks or landslide, which will block the entrance
of the tunnel

Difference stiffness in the top of the tunnel – One of common “irregularities” is the thickness
variation on the top of the tunnel due to construction. This creates a weak point on crosssection.
[2]

Segmental tunnels – This consists on precast segments that are putting together on the
construction. The weak point is the link between the segments.
3. Seismic Response
The underground structure behaviour, which includes interaction soil-structure, depends on
the behaviour of the surrounding soil, so it is important to identify the soil’s geological composition.
There are many ways to approach this problem, either by dimensionality (1D, 2D, 3D) or by considering
linear/non-linear analysis.
For geotechnical engineering, knowing the behaviour of soil and his response, when submitted
to seismic action, is one of the most intriguing problems.
3.1 Linear elastic analysis
Here, it is exposed the linear elastic analysis. This is associated to one-dimensional approach,
viscoelastic behaviour of the soil, all layers are horizontal and homogeny and the response is caused by
vertical shear wave’s propagation, like Figure 3-1 show (Kramer, 1996).
Figure 3-1 Uniform, damped soil on rigid rock
Since the dynamic soil’s proprieties remain constant, when is done a linear analysis, it can be
used transfer function for uniform damped soil. These functions can relate the movement of two points,
in the soil. The transfer function is written like equation 3.1
|
3
|
|
⁄
|
(3.1)
The transfer function’s peaks are the natural frequencies of each vibration mode and the
maximum is the fundamental frequency (n=1).
(
(
)
(3.2)
3.2 Soil-structure interaction approach
Soil-structure interaction is about shear stress, it can be two types: No-slip – There aren’t nonrelative displacements between soil and structure; or Full slip – There aren’t any transfer of shear stress
between soil and structure. The real situation is between both of them.
Two adimensional parameters relate the stiffness of the tunnel with the stiffness of the soil:
Flexibility and Compressibility ratio, which were suggested by Hoeg (1968) e Peck (1972)
[3]
(
(3.3)
(
(
)
(3.4)
Where Em, Soil’s Young modulus; El, Tunnel’s Young modulus; νl, Tunnel’s Poisson ratio; F,
Flexibility ratio; C, Compressibility ratio , t, thickness of the lining and R, tunnel radius.
According to Wang’s study the axial force and bending moment are:

Full-slip:
(
(3.5)
(
(3.6)
(
(3.7)

No-slip:
(
(3.8)
[
]
[
]
(
)
(
(3.9)
For Wang, the full-slip case only should be considered for very soft soil or high intensity seismic
action. However, for current situations it can happen partial or no-slip. So it should be taken the worst
situation.
Figure 3-2 relates the
coefficient with Flexibility ratio, F for a full-slip situation and different
values of Poisson coefficient, ν .It is revealed that, for high flexibility structures the
2,5
2
K1
1,5
1
0,5
0
0,1
10
v=0,3
F
v=0,1
1000
100000
v=0,5
Figure 3-2 Relation between F-K1
[4]
decreases
4. Simuation and Calibration of the numerical model
4.1 Modulation
The model has 250m of width and is divided in two zones: Free-Field zone and interaction zone.
In the Free-Field (FF) zone, the behaviour is like a soil column, while in the interaction zone the soil
behaviour is influence by the structure. The height is 40 meters. It was used rectangular and triangular
plane finite elements with plane-strain deformation and linear elastic behaviour.
The mesh has 20600 nodes and 90 beam elements, which were used to represent the 30 cm
thickness, tunnel lining. The radius of the tunnel is 5 meters.
The boundary conditions were: free movements of the nodes in the interaction area; in the FF
zone, vertical displacements were restricted and to represent the rigid rock is was used fixed supported
restraints. It was also used a Diaphragm constraints to uniform the FF displacements on both sides.
Figure 4-1 Longitudinal mesh.
Figure 4-2 Detail of the mesh
Another important aspect, in the mesh, is the dimension of the elements in wave propagation’s
direction (vertical). This dimension is related to the lower wavelength. Knowing that, the vertical
dimension of the element is 0,5m for the range frequencies:
[
]
.
4.2 Validation
To make sure that the results given by the model are consisted with reality. It is necessary to
validate. Therefore, we start with one-dimensional problem – Soil column.
4.2.1
One dimensional model
This model has the same boundary condition as well as the type of the elements (plain-strain),
mentioned above. However, it’s only about Free-Field behaviour (Figure 4-3)
Figure 4-3 Soil column
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After this, it was imposed an acceleration on the bottom of the model. Applying the Fourier
transfer function to the absolute displacements, it is possible to discover analytically the range of
frequencies. The peaks of the function are the same frequencies that are calculated by FE program.
Besides this, the normalized displacements should also be analysed (
1,20
|H| - Valor teorico
Freq - SAP
25
1,00
Height (m)
Amplification factor
30
20
15
10
0,80
Modo 1
Modo 2
Modo 3
0,60
0,40
0,20
5
0
0
-2,00
10
Frequency (Hz)
(a)
0,00
-1,00 0,00
1,00
Displacement (m)
2,00
(b)
Figure 4-4 (a)Fourier Transform function; (b) normalized displacement
4.2.2
Bi-dimensional model
The first of all, it was done a bi-dimensional model of, just, Free-Field, which implies that the
soil behaves like the soil column. After this, it was drawn the interaction area, where the length must be
studied because of the soil-structure interaction. So this length should well represent this interaction.
To do the analysis it was used the displacements of the first vibration mode and a very soft soil
(Es=5KPa) so it will be highlighted the interaction (Figure 4-5). Looking at Figure 4-5 we can conclude that
Displacements (m)
length of 3 diameters are a good approach.
0,0069
U1 - 2D
0,0068
U1 - 3D
0,0067
U1 - 4D
0,0066
U1 - 5D
0,0065
0
50
100
Distance to the center(m)
150
Figure 4-5 Interaction soil-structure area length
With this we have the final bi-dimensional model represented in Figure 4-6
Figure 4-6 Bi-dimensional model
Now, we have all the conditions to analyse the interaction soil-structure results with Wang
approach. In fact, while the normal stress has minor error, for solution of the bending moment there is a
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little difference, which is caused by the differences of the boundary conditions Wang’s approach, where
the structure is in pure shear.
Finally, to understand the influence of the dynamic mode it was studied the bending moment in
the tunnel, for each one. For that, it was applied a response spectrum load that will activate just the
modes in study. The result is present in Figure 4-7 where we can concluede that the first mode is the
most important one.
Bending moment
(KN.m/m)
115
M 1º Modo
114,5
M 1º+2º
Modo
M 1º+2º+3º
Modo
Mtotal
114
113,5
113
0
2
4
Vibration modes
Figure 4-7 Bending moments- dynamic modes relation
4.3 NLlinks
The link element will be used to simulate “irregularities”, so it is necessary to understand their
behaviour. The study was done using a rigid cantilevered beam. Two types of analysis were done: the
type of non-linear model (Kinematic or Takeda) and the length of the link.
In fact, the theories for the non-linear behaviour due to cyclical load have the same course in
the first cycle and are symmetrical. After the first one, the major difference is geometrical. While
Kinematic theory is more rectangular the Takeda is more stretched. This difference occurs because of
the path for loading and unloading, and so the stiffness loss (Figure 4-8). Since Takeda model is used for
concrete material behaviour, it was adopted this theory for simulate “irregularities”.
40
Force (KN)
20
0
-15
-10
-5
0
5
10
15
-20
-40
Displacement (m)
Takeda
Kinematic
Figure 4-8 Takeda vs kinematic theory
The length of the link affects the bending moments in the link and is more important for cyclic
load then monotonic loads. Nevertheless, the global structure bending moment isn’t changed. So it is a
local influence.
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35
30
25
20
15
10
5
0
40
Force
30
(KN)
20
10
0
-0,15
-0,1
-0,05 -10 0
0,05
-20
-30
-40
Displacement (m)
0,1m
0,2m
0,4m
1m
-2
Force
(KN)
-1
0
Displacement (m)
0,2m
0,4m
0,1m
1m
(a)
(b)
Figure 4-9 Link length influence
5. Results analysis
It was applied the link element with non-linear behaviour to the structure, in order to assume
the “irregularities” behaviour. The “irregularities” studied were the segmental tunnel and the stiffness
variation of the tunnel lining.
It was applied three types of loads, one static, one high dynamic impulse and one seismic action
consistent with regulatory action. The parameters studies were diameter variation and yield bending
-5
-2
moment the distortion degree was considered between 10 and 10 .
5.1 Segmental tunnel
The segmental tunnel is a tunnel constructed by precast concrete segments. The joining
between those segments is done in the construction work and is a fragile point. This connection is the
aim of the study. It was used Janssen (1983) theory to simulate their behaviour.
For the static loads it was shown that the variation of the diameter and the bending moment
has similar course, in fact non-linear behaviour is achieved for some links. Now, for the dynamic impulse
all the links reached non-linear behaviour. However, the same points that first started the non-linear
behaviour are the same that in the static load have non-linear behaviour. The same happens to seismic
action.
200
Bending moment
(KN.m/m)
Bending moment
(KN.m/m)
100
50
0
-50
100
0
-100
-100
-200
0
10
Link 3
20 Time
30(s)
40
0
Link 9
20
Figure 5-1Bending moment for the two critical points due to seismic action
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40
Time(s)
5.2 Stiffness variation
Here it was done two different modelling. The first just consider this variation in one link, the
other has a soft thickness decreasing by using more links.
(a)
(b)
Figure 5-2 The two models for stiffness variation (a) one link; (b) more links
For the first one, just one link, we could observe that only for the high dynamic impulse the
non-linear behaviour is achieved, which means that the yield bending moment is reached, like
80
Mced=71,8KN.m
Bending moment
(KN.m/m)
Bending moment
(KN.m/m)
200
0
-200
60
Mced
40
20
0
-20
0
10
20
30
Time (s)
0
10
20
(a)
30
Time (s)
(b)
Figure 5-3 bending moments; (a) Dynamic impulse, (b) Seismic action
Finally, for the other model, it was seen that only the external points (link 1 and 7) were more
critical and the middle ones didn’t reach non-linear behaviour (link 3, 4 and 5). However, the non-linear
behaviour only appeared for the dynamic impulse, which means that for static and seismic action we
just have linear behaviour.
Figure 5-4 Detail: stiffness variation and the location of the points
300
Mced=71,8KN.m
150
Bending moment
(KN.m/m)
Bending moment
(KN.m/m)
200
100
50
0
200
Mced t=0,25
100
0
-100
-50
-200
-100
0
10
0
20 Time(s) 30
10
20
Time (s)
30
Figure 5-5 Bending moments; Left – Middle points for dynamic impulse; Right – External points for seismic
action
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6. Conclusions and Future Perspectives
6.1 Conclusions
According to the results above, we can deduce that for the stiffness variation of the crosssection there isn’t non-linear behaviour. Which means that the global behaviour of the structure isn’t
affected and the elastic linear approach generally used give us a good result. Another way to confirm
this structure’s response is by looking at the tunnel deformation. In fact the tunnel, for this “irregularity”
has a deformed shape near the elastic one.
Still, for the segmental tunnels there are some points that should have our attention, during the
design. We were able to identify two links (the one’s that make 30⁰ with the horizontal and the vertical
direction) that achieve the non-linear behaviour for distortion around 1%.
6.2 Future Perspectives
It is recalled that this sensitive analysis was done using finite elements, which means that it
could be complemented by experimental tests. Besides, these results are workable for very soft and
uniform soil and a tunnel with circular cross-section, so an analysis considering stratified, better soil and
a different geometry for the tunnel cross-section, could be done.
Finally, there were only two “irregularities” studied, thus there are others “irregularities” that
can be analysed, as well as others finite elements programs for using and at the same time simulate
soil’s non-linear behaviour.
7. Bibliography

Dowding, C. H., & Rozen, A. (1978). Damage to rock tunnels for earthquake shaking. Journal of
the Geotechnical, 175-191.

Gomes, R. C. (Novembro de 1999). Dissertação para obtenção do grau de Mestre.
Comportamento de Estruturas subterrâneas submetidas à acção sísmica. Lisboa: Instituto Superior
Técnico.

Judd, S. S. (13 de June de 1990). Underground opening damage from eatrhquake. Engineering
Geology, 30, 263-276.

Kramer, S. L. (1996). Geotechnical Earthquake Engineering. USA: Prentice-Hall, Inc.

Luttikholt, A. (2007). Ultimate Limit State Analysis of a Segmented Tunnel Lining. Tese de
Mestrado, University of Delft, Delft.

Owen, G. N., & Scholl, R. E. (1981). Earthquake engineeringof large underground structures.
Prepared for Federal Highway Administration.

St Jonh, C. M., & Zahrah, T. F. (1987). Aseismic design of underground structures. Tunnelling
and underground space technology, 165-197.

Wang, J. N. (1993). Seismic Design of Tunnels. Parsons Brinckerhoff Mobograph 7.
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