[1]

“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.

[2]

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 cross-

section.

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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 non-

relative 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)

[4]

(

(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 decreases

Figure 3-2 Relation between F-K1

0

0,5

1

1,5

2

2,5

0,1 10 1000 100000

K1

F

v=0,3 v=0,1 v=0,5

[5]

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 (

(a) (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

length of 3 diameters are a good approach.

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

0

5

10

15

20

25

30

0 10

Am

plif

icat

ion

fac

tor

Frequency (Hz)

|H| - Valor teorico

Freq - SAP

0,00

0,20

0,40

0,60

0,80

1,00

1,20

-2,00 -1,00 0,00 1,00 2,00

He

igh

t (m

)

Displacement (m)

Modo 1

Modo 2

Modo 3

0,0065

0,0066

0,0067

0,0068

0,0069

0 50 100 150

Dis

pla

cem

en

ts (

m)

Distance to the center(m)

U1 - 2D

U1 - 3D

U1 - 4D

U1 - 5D

[7]

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.

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”.

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.

113

113,5

114

114,5

115

0 2 4

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Vibration modes

M 1º Modo

M 1º+2ºModo

M 1º+2º+3ºModo

Mtotal

-40

-20

0

20

40

-15 -10 -5 0 5 10 15

Force (KN)

Displacement (m) Takeda Kinematic

[8]

(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

moment the distortion degree was considered between 10-5

and 10-2

.

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.

Figure 5-1Bending moment for the two critical points due to seismic action

-40

-30

-20

-10

0

10

20

30

40

-0,15 -0,1 -0,05 0 0,05

Force (KN)

Displacement (m)

0,1m 0,2m 0,4m 1m

0

5

10

15

20

25

30

35

-2 -1 0

Force (KN)

Displacement (m) 0,1m 0,2m 0,4m 1m

-100

-50

0

50

100

0 10 20 30 40

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time (s) Link 3

-200

-100

0

100

200

0 20 40

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time(s) Link 9

[9]

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

(a) (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

Figure 5-5 Bending moments; Left – Middle points for dynamic impulse; Right – External points for seismic action

-200

0

200

0 10 20 30

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time (s)

Mced=71,8KN.m

-20

0

20

40

60

80

0 10 20 30

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time (s)

Mced

-100

-50

0

50

100

150

200

0 10 20 30

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time(s)

Mced=71,8KN.m

-200

-100

0

100

200

300

0 10 20 30

Be

nd

ing

mo

me

nt

(KN

.m/m

)

Time (s)

Mced t=0,25

[10]

6. Conclusions and Future Perspectives

6.1 Conclusions

According to the results above, we can deduce that for the stiffness variation of the cross-

section 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.