diogo miguel mendes ferreira torcato extended abstract

Seismic behaviour of shallow tunnels in stratified ground

Diogo Miguel Mendes Ferreira Torcato

Extended Abstract

Supervisor: Professor Doutor Jaime Alberto dos Santos Co-Supervisor: Professor Doutor Rui Pedro Carrilho Gomes

October 2010

Seismic behaviour of shallow tunnels in stratified ground using modal analysis

Master Thesis 2

1. INTRODUCTION

The purpose of this study is to evaluate the behaviour of shallow tunnels under seismic action. This evaluation is

done by modal analysis, in a numerical model of the tunnel, based on the finite element method.

It is known that underground structures are less vulnerable than above ground structures though this cannot be

a reason to justify not considering the seismic action. In fact, recent earthquakes that caused tunnel damages –

for instance the 1995 Kobe earthquake – brought this issue to discussion. Several works study the behaviour of

the tunnel in homogeneous medium, such as Wang (1993), Penzien (1998 & 2000), Hashash (2001 & 2005) and

Gomes (1999), but little is available for stratified medium.

This work is focused on the effect of stratification on tunnel seismic behaviour. Therefore, this study can set new

guidelines for the design phase of an underground structure. In order to validate the numerical model, analytical

solutions for homogeneous medium were used as reference.

The study is divided into 6 chapters, according with the following plan

2. DAMAGES IN TUNNELS DUE TO SEISMIC ACTION

As referred, it is known that tunnels, as well as other underground structures, are less vulnerable to earthquakes

than above-ground structures. In spite of this, according to the evidence of damage or collapse of tunnels, the

effect of an earthquake should not be overlooked. According to Wang (2001), the idea of invulnerability has been

shattered, as during in the Taiwan Chi-chi earthquake (1999), of 49 tunnels studied 57 were damaged.

A milestone in the seismic behaviour of underground structures occurred due to Hyogoken-Nambu earthquake

(Kobe, Japan, 1995), which induced damages to a tunnel and metro stations.

7 - Conclusions and further developments

6 – Results of the model; Study of the influence of the parameters

3 – Definition of the seismic action – The action used in the model is described

1 – Introduction – Considerations, objectives, basis and organization of the study

2 - Damages in tunnels – Review of cases that were damaged by earthquakes

5 – Validation and testing of the numerical model – Calibration of the model to simulate stratified medium

4 – Presentation of analytical solutions – A review of the published results for homogeneous medium


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3. DEFINITION OF SEISMIC ACTION

The definition of the seismic action has high importance as it influences the model response. The designer should

apply the building standard law that is available for the country or zone. Note that seismic codes usually define

the seismic action through elastic response spectra. Normally modal analysis uses spectral accelerations to

compute forces and displacements. The spectrum used is a constant uniform and unitary spectrum, as the goal of

the study was to perform a sensitivity analysis,.

4. BEHAVIOUR OF TUNNELS IN HOMOGENEOUS MEDIUM

The approaches to study the tunnel behaviour in homogeneous medium may have variable complexity. The

simplest way to analyse the problem is by considering only the soil, without tunnel or hole, which contrasts with

the possibility to consider the hole of the tunnel but no lining is taken into account, Wang (1993).

Wang (1993) and Penzien (2000) studies describe more complex and accurate solutions for soil-tunnel

interaction in homogeneous medium. Comparing both approaches it is noticeable a convergence of the results in

what deformations and bending moments is concerned. However, in the axial force, results are divergent as

Penzien approach leads to lower results than Wang’s – despite both considering no-slip between soil and lining.

The comparison done is in accordance with the results published by Hashash (2001 & 2005).

5. NUMERIC MODEL – DEFINITION AND TESTS

DEFINITION

INTRODUCTION The purpose of the thesis is to study the tunnel behaviour in stratified medium. The studies published do not

consider stratified mediums. The construction of the model for stratified medium is based on the homogeneous

medium studies. The studies in the homogeneous field are well documented, coming from the analytical

approaches of Wang (1993), Kramer (1996) and Penzien (1998 & 2000), passing through the comparison of

results by Hashash (2001 & 2005), and ending in a modelisation of the problem by Gomes (1999).

The analysis performed, with the finite element method (FEM) code SAP2000 v11, is based on modal analysis

and response spectrum. The modal analysis studies the behaviour of a structure under a vibration excitation –

the earthquake waves, for instance. Therefore, the vibrating frequencies and mode shapes of the structures are

determined. Regarding the response spectrum, the elastic acceleration response spectrum represents the peak

acceleration of the response of a single-degree-of-freedom oscillator with a given frequency. Therefore, for a

certain frequency of vibration, there is an acceleration to be applied according to the spectrum.

Each structure has a certain number of modes which are more relevant. After the first modes, the importance of

the modes decreases, since the participating mass ratios decreases.


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MODELISATION

MODEL BASE AND CROSS-SECTIONS In order to achieve a feasible model, several studies were conducted. The model is defined in Figure 1 and the

cross-sections studied are presented in Figure 2.

D=10mD=5m

Figure 1 – Model scheme for circular cross-section Figure 2 – Tunnel cross-sections

In the upper part of the scheme there is a dimension called nD, that is the size of the interaction region. Outside

the interaction region, the free-field (FF) region intends to represent the vertical propagation of the SH-waves far

away from the tunnel.

SOIL MODELING

The soil was modelled by rectangular plane elements, admitting plane-strain condition with elastic and linear

behaviour. The model considers the vertical propagation of shear waves in uniform damped soil laying on rigid

rock. This model considers the effect of damping, which takes into account the dissipation of energy in the

medium.

Table 1 – Characteristics of soils used in the model

Characteristics/Soil E=10 MPa E=500 MPa

γ [KN/m3] 20 20

υ 0,3 0,3

G [MPa] 3,85 192,31

To accurately describe a sinusoidal wave, 8 to 10 points are required. As it is only considered the influence of the

1st mode, the model is well calibrated for all the soils in analysis.

Table 2 shows the computations of the size of the elements in the direction of wave propagation for the model

with E=500MPa, that has higher frequency. Therefore, the size of the elements shall be 0,50m

Table 2 – Wavelength for the type of soil – recommended size of elements

Soil WL [m] 1/8 WL [m] 1/10 WL [m]

500MPa 4,39 0,55 0,44


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Afterwards, the size of the model (L) was tested, to determine whether the size of the mesh was enough to

ensure good results. The method used was the comparison of the displacements in free-field with the analytic

solution. The results were quite satisfying, but especially for the case of L=100m (deviation of 0,1%).

INTERACTION SOIL-STRUCTURE As referred, the interaction soil-structure is crucial to ensure the accuracy of the model. The objective of the

interaction region, nD, in Figure 1 can be understood by the analysis of Figure 3 where the deformation profile in

function of height is presented.

To have a good transmission of the deformations in the interaction zone, Gomes (1999) showed that the soil

shouldn’t have any constraints. Therefore, the constraints applied for the model without tunnel (blockage of the

vertical movement and the same horizontal movement for the nodes at equal depth) were removed. The zones

that are far from the tunnel, outside the interaction region, are considered to behave as free-field (FF) with the

above-referred constraints.

The interaction region can control the soil amplitude of movement from the free-field to the zone surrounding

the tunnel, according to the stiffness contrast between the medium and the lining. The size of the interaction

region was studied, the dimension is related to the displacements next to the tunnel, until the border. A length of

12D permits to achieve accurate results.

TESTING

SIMPLE SHEAR

The results, considering the action to be simple shear, are quite similar to the Wang’s and Penzien’s approach for

deformations (ΔDlining/ΔDFree-field) as well as structural forces (except axial force for Penzien).

RESPONSE SPECTRUM The analysis done concerning the behaviour of tunnels is mainly based on the study of structural forces (Figure

4) and deformations (Figure 5).

Due to the reduced importance of the modes that follow the 1st, it was only considered the first mode.

Free-field Free-field Interaction region

Figure 3 - Schematic effect of interaction region – displacements profile


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Figure 4 – Maximum bending moment and axial force of analytical solution vs numerical solution – EC8 Response spectrum, 1 mode considered

A good adjustment of the structural forces in the model is observed for Wang’s solution (both for moments and

axial force). When Penzien is analyzed, an accurate adjustment is observed in the bending moments but when it

comes to the axial force, there is clear divergence. This effect in the axial force was already observed by Hashash

(2005). Wang uses a full-slip approach to calculate the bending moment and a no-slip methodology to determine

the axial force, as also does Penzien. Despite Penzien’s approach for no-slip, the results are not close to Wang’s

nor to the model results, that is why it results in this discrepancy (Figure 4).

Figure 5 - Behaviour of circular cross-section with Poisson’s coefficient – EC8 Response spectrum, 1 mode considered

From this results it can be noticed that the using the modal analysis the trend does not follow Wang’s approach.

This result contrasts with the simple shear case, where the results follow closely Wang’s trend. In fact, for F<1,

the ratio ΔDl/ΔDff is higher than the Wang’s approach, whereas for F>1 the ratio is lower. This behaviour is

understood if the deformed shape is observed in Figure 6, for the cases A, B, C and D.


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Case A F=0,07 10MPa D=5 t=0,50

Case B F=0,60 10MPa D=10 t=0,50

Case C F=3,73 500MPa D=5 t=0,50

Case D F=29,82 500MPa D=10 t=0,50

Figure 6 – Deformed shapes and bigger bending moments zones, for EC8 response spectrum and scale factor of 500.

If Figure 6 is analysed for F=0,07 and F=0,60, both F<1, the results are higher than Wang’s curve and the

deformations are more expressive than the cases of F=3,73 and F=29,82 whereas, for F>1 the deformations are

less expressive and the results are under the Wang’s curve.

Considering the difference between the approaches, simple shear leads to a linear deformation in the free-field

whereas, the modal analysis, considering the 1st mode of vibration, results in a distortion that varies with the

height. Despite the behaviour described, the deformations in the tunnel height are nearly the same. Therefore,

the reason to the difference of behaviour between simple shear and modal analysis approach is the variation of

the deformation along the model height, which has an important role in the tunnel’s deformation.

6. RESULTS FOR STRATIFIED MEDIUM

In this stage of the study the results of the stratified medium model, are presented. Several parameters are

analyzed, in order to show its influence. The results are presented step by step, so that the influence of each

parameter can be pointed out.

The phases of this analysis are the following:

- Deformed shape of the cross-section;

- Comparison with the solution for the homogeneous medium;

- Influence of the interface position and stiffness contrast;

- Influence of the cross-section size;

- Influence of the cross-section stiffness (lining thickness).

The analyses performed took into account a unitary constant response spectrum.

DEFORMED SHAPE OF THE CROSS-SECTION With the purpose of having a physical view of the problem, the deformations in the cross-section are presented.

Although the analysis is divided in the three interfaces studied, the type of deformation is the same for all the

cases in a qualitative view. Figure 7 and Figure 8 present the type of behaviour faced by the tunnel lining under

modal analysis in stratified soil for a certain case, as example.


Master Thesis 8

Figure 7 - Deformation due to modal analysis, case A – D=5m, t=0,50m and bottom interface, scale factor

500

Figure 8 – Relative displacement in free-field, case A

Which is synthesized in Figure 9

Figure 9 – Displacements and deformation scheme

Where dv is the vertical displacement and dh is the horizontal displacement.

COMPARISON WITH THE SOLUTION FOR THE HOMOGENEOUS MEDIUM Considering the deformations of the cross-section, a different behaviour is observed when compared with the

formulation of Wang (simple shear). However, the results of the homogeneous medium of the model (modal

analysis) have the same trend of the results of the stratified medium, but are quantitatively different.

Regarding the structural forces in the tunnel lining, through K1, the results have a similar trend and close values

between the model (for homogeneous and stratified soil) and Wang’s formulation. It shall be referred, as

previously stated, that the formulation does not work well for two situations:

- Stiff lining in relation to the medium (F<<1);

- Rigid medium in relation to the lining (F>>1).

INFLUENCE OF THE INTERFACE POSITION AND STIFFNESS CONTRAST

The influence of the interface position and stiffness contrast induces important results in deformations and

structural forces.

Es=10MPa

Ei=50MPa

dv dh

dv dh dv

dh

dv dh

Tunnel before wave motion

Tunnel after wave motion


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The distortion in the tunnel lining is influenced by the following parameters, presented in Table 3.

Table 3 - Distortion in the tunnel lining - Synthesis of the behaviour and Control factor

Interface position

relative to the tunnel

γlining.Ei Control factor

Es/Ei<1 Es/Ei>1

Bottom Decreases Decreases Upper soil

Middle Decreases Decreases Upper soil

Upper Aprox.Constant Aprox.Constant Lower soil

Regarding the bending moments the results are synthesized in Table 4.

Table 4 – Bending moment - Synthesis of the behaviour and control factor

Interface position


Mmax.Ei Control factor

Es/Ei<1 Es/Ei>1

Bottom Decreases Decreases Upper soil

Middle Decreases Decreases Upper soil

Upper Aprox.Constant Aprox.Constant Lower soil

Finally the axial force is resumed in Table 5.

Table 5 – Axial force - Synthesis of the behaviour and control factor

Interface position


Pmax.Es/Ei Control factor

Es/Ei<1 Es/Ei>1

Bottom Increases Increases Es/Ei

Middle Increases Increases Es/Ei

Upper Increases Increases Es/Ei

These results assume a particular relevancy as the interface influence is deeply studied as well as the contrast of

rigidities

INFLUENCE OF THE CROSS-SECTION SIZE

The two circular cross-sections that were studied, are presented in Table 6.

Table 6 – Synthesis of the structural forces in function of the cross-section size

Interface position relative

to the tunnel

Bending Moment Axial Force

Bottom Circular D=5 Higher Lower

Circular D=10 Lower Higher

Middle Circular D=5 Higher Lower


Upper Circular D=5 Higher Lower



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From the studies performed for all the interfaces, concerning the structural forces, the results state that the

circular cross-section diameter 5m has the bigger bending moments, for the same Es/Ei. When it comes to the

axial force, the circular diameter 10m leads to higher axial forces, when compared with the diameter 5m.

INFLUENCE OF THE CROSS-SECTION STIFFNESS (LINING THICKNESS)

In order to study its influence it is considered the circular cross-section with a diameter of 5m, only varying the

thickness of the lining (t=0,25m and t=0,50m).

For a thicker lining the structural forces increase, when comparing to a lining that is thinner;

Regarding the deformations, the results are higher for bottom and middle interface, when F <1, for the thicker

tunnel whereas, for Fi>1 the deformations are lower for the thicker lining;

Concerning the deformations for the upper interface, the results are almost constant, showing that there is a low

influence of the lining stiffness. These results are synthesized in Table 7.

Table 7- Synthesis of the deformations and structural forces in function of the lining thickness

Interface position relative to the tunnel Deformations Bending Moments

F<1 F>1

Bottom t=0,25m Lower Higher Lower

t=0,50m Higher Lower Higher

Middle t=0,25m Lower Higher Lower

t=0,50m Higher Lower Higher

Upper t=0,25m

Aprox.Equal Lower

t=0,50m Higher

7. CONCLUSIONS AND FURTHER DEVELOPMENTS

CONCLUSIONS The initial stages of this study consisted in understanding the basis of the problem, and summarize the damages

reported in tunnels in earthquakes.

Considering the model, the validation is done by comparison of results with the analytical expressions.

Consequently, at this stage the development of a similar model for this type of analysis can now be done easily.

This point is relevant when considering possible time savings, as it corresponds to a phase of

calibration/validation that was clearly time consuming.

Taking into account the results for homogeneous medium, the following results can be drawn: For F<1, the

deformations of the model for modal analysis are higher than Wang’s approach and simple shear. For F>1 the

deformations of the model for modal analysis are lower than Wang’s and simple shear.


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The model developed for stratified medium permitted the evaluation of the deformations and structural forces

by modal analysis. From those results, the influence of some parameters must be highlighted:

- Interface position and stiffness contrast:

The control factors are, in a general view, the upper soil stiffness for bottom and middle interface,

whereas the lower soil stiffness influences the upper interface model.

- Cross-section size:

Regarding structural forces, the circular D=5m cross-section induces higher bending moments.

However, it is the circular D=10m that has higher axial force, for the interfaces studied.

FURTHER DEVELOPMENTS Despite that the objectives were achieved, I believe there is scope for further developments. Those developments

can be, for instance:

i) The medium;

Regarding the medium, the study of other geometry (e.g. slope at surface) and more advanced

constitutive models.

ii) Cross-section shape;

Taking into consideration the cross-section, further studies can be done with rectangular and other

non-circular cross-sections. Actually, the study of other cross-sections could be interesting as it

could make an approximation to current cross-sections used for tunnels (rectangular, rectangular

with central column, horse-shoe section, among others cross-sections).

iii) Type of action.

Finally, the action can be performed, for example, considering different response spectrums. The

response spectrum may be more adequate to the region where the tunnel is situated and/or

considering the influence of the site effects. Regarding the action, more modes of vibration can be

considered in the analysis.

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