Dissertation2009 Ordonez[1]

7/31/2019 Dissertation2009 Ordonez[1]

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Istituto Universitario

di Studi Superiori

Università degli

Studi di Pavia

EUROPEAN SCHOOL FOR ADVANCED STUDIES INREDUCTION OF SEISMIC RISK

ROSE SCHOOL

INFLUENCE OF THE BOUNDARY CONDITIONS ON THE

SEISMIC RESPONSE PREDICTIONS OF A ROCKFILL DAM

BY FINITE ELEMENT METHOD

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

ENGINEERING SEISMOLOGY

by



The dissertation entitled “Influence of the boundary conditions on the seismic response

predictions of a rockfill dam by finite element method”, by Ivan Ordonez, has been approved

in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering.

Name of Reviewer 1 …… … ………Dr. Carlos A. Prato

Name of Reviewer 2………… … ……



Abstract

ABSTRACT

One of the most important issues in soil dynamics and engineering seismology is the assessment of thedynamic response of the soil subjected to seismic loading. Among others, a seismic response analysis

is performed for assessing the influence of the soils deposits on the motions at the surface. Regarding

this issue, it is well known that the presence of a structure founded on the soil deposit, may

considerably modify the dynamic response of the system, since the motion of the waves excites the

structure, which in turn modifies the input motion due to its movement relative to the soil at the same

time.

This thesis focuses on evaluating the effect of the seismic interaction of the “Los Caracoles” Concrete

Face Rockfill Dam (CFRD), recently built in the high seismicity area of San Juan (Argentina), and the

rock underlying the alluvial stratum of the foundation of the dam. For achieving that, finite element

software is used to represent the dynamic behavior of the dam and the foundation, including

alternatives for the representation of the rock that were not considered explicitly in the design.

The underlying rock, which is assumed to be homogeneous and unbounded, in terms of geometry, is

represented by means of a row of fictitious elements at the rock-alluvial interface with given

mechanical features that allow to represent the influence of the flexibility of the rock on the dynamic

response or the dam. In that way, the input motion at the interface rock-alluvial is modified by the



Abstract

In the same way, the flexibility of the foundation rock and radiation damping in the dynamic response

as obtained with QUAD4M and the LEM has considerable effect in the prediction of maximumaccelerations and permanent displacements at the crest as compared with those given by ADINA (MC)

elastoplastic analysis. The amount of the reduction in the expected acceleration and deformation at the

crest depends on the assumed flexibility of the foundation rock. Compared to the case of rigid

foundation rock, considering flexible foundation rock can lead to a reduction up to 28% in the

acceleration and more than 300% of reduction in the settlement.

Keywords: seismic soil-structure interaction; halfspace; flexibility; maximum acceleration, settlement.



Acknowledgements

ACKNOWLEDGEMENTS

A number of people have helped directly and indirectly in the preparation of this thesis. Particularly I

am most grateful to Dr. Carlos Prato and his family for allowing me to spend time in Argentina along

with them and for his permanent guidance and support during the work.

I am also grateful to my family for their constant concern about my projects and to Paola for her

support and encouragement during my studies.

I am also indebted to Mr. Andres Prato for introducing me to his family, for his continuous support

and for showing me some tricks playing football and other sports.



Index

TABLE OF CONTENTS

Page

ABSTRACT .......................................... ............................................. ............................................. ........i

ACKNOWLEDGEMENTS...................................................................................................................iii

TABLE OF CONTENTS ............................................ .............................................. ............................ iv

LIST OF FIGURES...................................... ............................................... .......................................... vi

LIST OF FIGURES...................................... ............................................... ........................................... x

1. GENERAL INFORMATION............................................................................................................1

1.1 Introduction................................................................................................................................1

1.2 Background information ........................................ .............................................. ......................2

1.3 Methodology ........................................... ............................................... .................................... 3

1.4 Scope and objectives of the thesis..............................................................................................5

1.5 Organization...............................................................................................................................5

2. METHOD OF ANALYSIS ........................................... ............................................... ..................... 6

2.1 Dynamic behavior of soils and rockfill......................................................................................6

2.1.1 Linear equivalent model ..................................... ............................................... ..............6

2.1.2 Cyclic nonlinear models ..................................... ............................................... ..............9

2.2 Mechanical properties of the materials ........................................... ......................................... 10

2.3 State of stress .......................................... ............................................... .................................. 11



Index

2.6.1 One-dimensional model based on the shear beam approach ......................................... 19

2.6.2 Finite element plane models ........................................ ............................................. .....20 2.6.3 Three dimensional models ........................................... ............................................. .....20

2.7 Boundary conditions of the 2D finite element model .......................................... ....................22

2.7.1 Boundary conditions for gravitational forces.................................................................22

2.7.2 Boundary conditions for evaluating seismic response...................................................22

2.8 Settlements at the crest. Newmark Method ........................................... .................................. 28

2.8.1 Wedge method (analytical expression) ........................................ .................................. 28 2.8.2 Newmark method...........................................................................................................32

3. CASE STUDY.................................................................................................................................35

3.1 Location of the Project.............................................................................................................35

3.2 Description of the materials .......................................... ............................................... ............35

3.2.1 Geotechnical model ..................................... .............................................. .................... 38

3.2.2 Geomechanical properties of the materials....................................................................38 3.2.3 Damping curve and modulus reduction curve ........................................... .................... 39

3.3 One-dimensional modeling......................................................................................................40

3.3.1 EERA (Equivalent-linear Earthquake site Response Analyses of layered soil deposits)40

3.4 Description of the finite element model...................................................................................44

3.4.1 DSUN-GID Module (preprocessor)...............................................................................44

3.4.2 QUAD4M ..................................... ............................................... .................................. 44

3.5 Boundary conditions ...................................... ............................................... ........................... 44

3.6 Results......................................................................................................................................47

3.6.1 Maximum acceleration-depth cross sections ............................................. .................... 47

3.6.2 Acceleration response spectra........................................................................................48

3.6.3 Comparison between the results of the programs .............................................. ............51

3.6.4 Transfer functions ........................................ .............................................. .................... 57

3.6.5 Dam crest settlement......................................................................................................57

4. FINAL REMARKS AND CONCLUSIONS...................................................................................61

5. REFERENCES........................................ ............................................... ......................................... 66



Index

LIST OF FIGURES

Page

Figure 1.1. Flowchart displaying the methodology for carrying out the work………………..4

Figure 2.1. Dynamic behavior of soils. Hysteresis loops ..........................................................6

Figure 2.2. Modulus reduction curves for fine-grained soils of different plasticity. (After

Kramer, 1996).....................................................................................................................8

Figure 2.3. Influence of mean effective confinig pressure on modulus reduction curves for a

non plastic soil PI=0. (After Kramer,1996) ........................................................................9

Figure 2.4. Variation of damping ratio of fine-grained soil with cyclic shear strain amplitude

and plasticity index (After Kramer, 1996)........................................................................10

Figure 2.5. Effective stresses at the end of the construction. (After Techint-Panedile, 2005) 11

Figure 2.6. Points selected for checking the confining pressure..............................................12

Figure 2.7. Confining pressure distribution within the model computed by means of Plaxis v

7.2......................................................................................................................................13

Figure 2.8. Propagation of a seismic wave from the epicenter to the site. (After Kramer,1996)

...........................................................................................................................................13

Figure 2.9. Displacements fields for plane P and Suaves propagating in the x-z plane

containing the source and the receiver, where the z-axis is vertical. The P-wave

displacement is along the wave vector k. The S wave can be decomposed into two

polarizations, SV and SH, perpendicular to the wave vector. The SH displacement is



Index

Figure 2.13. Acceleration response sprectrum- Chi-Chi (Taiwan). 1999................................16

Figure 2.14. Mode shapes for (a) first mode and (b) second mode of earth dam response.(Alter Dakoulas and Gazetas) ...........................................................................................18

Figure 2.16. Finite element models for footing on halfspace (AfterLysmer& Kuhlemeyer,

1969) .................................................................................................................................24

Figure 2.17. Dimensions of the structure used for computing the properties of the fictitious

elements ............................................................................................................................25

Figure 2.18. Lumped Representation of Structure Foundation Interaction (After Richard et

al. 1970) ............................................................................................................................26

Figure 2.19. Constants and for rectangular bases. (After Richart, F. E et al. Vibrations of

Soils and Foundations, Prentice-Hall, Inc, 1970) .............................................................26

Figure 2.20. Wedge method, with the forces acting on the wedge (After Day, 2002) ...........28

Figure 2.21. Wedges studied in the downstream slope............................................................29Figure 2.22. Diagram of the studied wedges in the downstream slope ...................................30

Figure 2.23. Analogy between (a) potential landslide and (b) block resting on inclined plane.

After Kramer (1996) .........................................................................................................33

Figure 2.24. Diagram illustrating the Newmark method. a) Acceleration vs. time; b) Velocity

vs. time for the darkened portions of the acceleration pulses; c) the corresponding

downslope displacement versus time in response to the velocity pulses. (After Wilson

and Keefer)........................................................................................................................34

Figure 3.1. Grain size distribution for Material 3B. (After Techint-Panedile, 2005) ..............36

Figure 3.2. Grain size distribution for Material 3L. (After Techint-Panedile, 2005) ..............36

Figure 3.3. Grain size distribution for the alluvial material forming the foundation of the dam.

(After Techint-Panedile , 2005) ........................................................................................37Figure 3.4. Modulus reduction curves. G/Gmáx vs. γ (%)..........................................................39

Figure 3.5. Damping curves. β(%) vs. γ (%).............................................................................40

Figure 3.6. Sections within he body of the dam analyzed using EERA..................................43

Fi 3 7 S l h i h i hi h b d f h d D i id d h b f



Index

Figure 3.12. Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick

layer of fictitious elements with shear wave velocity Vs=1500m/s .................................46Figure 3.13. Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick

layer of fictitious elements with shear wave velocity Vs=2500m/s. ................................46

Figure 3.14. Half-space with shear wave velocity Vs=1500m/s and a 20m-thick layer of

fictitious elements with shear wave velocity Vs=1500m/s...............................................46

Figure 3.15. Maximum acceleration vs. depth. Maximum Credible Earthquake....................47

Figure 3.16. Maximum acceleration vs. depth. Chi-Chi Taiwan earthquake, 1999 ................48

Figure 3.17. Acceleration response spectra for several depths within the body of the dam.

Maximum Credible Earthquake........................................................................................49

Figure 3.18. Acceleration response spectra for several depths within th body of the dam. Chi-

Chi Taiwan earthquake, 1999 ...........................................................................................50

Figure 3.19. Comparison between EERA and QUAD4M. Maximum acceleration vs.depth.MCE........................................................................................................................51

Figure 3.20. Comparison between EERA and QUAD4M. Maximum acceleration vs. depth.

Chichi-Taiwan, 1999.........................................................................................................51

Figure 3.21. Comparison between EERA and QUAD4M. Acceleration response spectra.

Halfspace with Vs=1500m/s. Maximum Credible Earthquake ........................................53


Rigid halfspace. Maximum Credible Earthquake.............................................................54


Halfspace with Vs=1500m/s. Chi-Chi Taiwan Earthquake, 1999....................................55


Rigid halfspace. Chi-Chi Taiwan earthquake,1999. .........................................................56Figure 3.25. Transfer functions (Sa-crest/Sa input). Maximum Credible Earthquake............57

Figure 3.26. Transfer functions (Sa-crest/Sa input). Chi-Chi Taiwán earthquake ..................57

Figure 3.27. Dam crest settlement for wedges at different depths. Comparison between

ADINA and rigid halfspace (QUAD4M) Maximum Credible Earthquake 58



Index

Figure 3.30. Dam crest settlement for wedges at different depths computed using EERA and

the Newmark method. Maximum Credible Earthquake ...................................................59Figure 3.31. Dam crest settlement for wedges at different depths computed using EERA and

the Newmark method. Chichi-Taiwan earthquake ...........................................................60



Index

LIST OF TABLES

Table 2-1. K2 for different soils ................................................................................................8

Table 2-2. Comparison between the values of confining pressure within the model. .............12

Table 2-3. Values of βn for first five modes of vibration o an earth dam ...............................17

Table 2-4. Yield acceleration for the wedges ..........................................................................31

Table 3-1. Geotechnical parameters of the materials...............................................................38

Table 3-2. Geomechanical properties of the materials ............................................................41

Table 3-3. Acceleration at the crest ........................................................................................47



Chapter 1. General information

1. GENERAL INFORMATION

1.1 Introduction

One of the very important issues in soil dynamics and engineering seismology is the

assessment of the dynamic response of the soil subjected to seismic loading. Usually, seismic

response analyses are carried out for predicting movements at the surface of the soil deposit,

for establishing design spectrum, for evaluating induced stresses and deformations with the

purpose of estimating the liquefaction potential, permissible settlements and eventually fordetermining induced forces that may lead to instability in the portion of soil or in existing

structures.

Ideally, a complete seismic response analysis encompasses the following issues:

a. Characterization of the seismic sources that may affect the area of influence. Specifically,

a comprehensive understanding of the types of faults that controls the nature of the

earthquake, the activity and the recurrence laws are sought.

b. Estimation of the wave’s propagation’s mechanism from the source up to the bedrock

underlying the site. For achieving that, attenuation relationships are used.

c. Seismic hazard analyses for evaluating the probability of occurrence of an earthquake

with given some features.

d. Seismic site response analyses for assessing the influence of the soils deposits on the

motions at the surface. .

Regarding the last item, it is well known that the presence of a structure founded on the soil

deposit, may considerably modify the dynamic response of the system, since the motion of the

waves excites the structure, which in turn modifies the input motion due to its movement

relative to the soil at the same time.




This program represents the dynamic behaviour of the dam and the foundation, including

alternatives for the representation of the rock that were not considered explicitly in the design.

The underlying rock, which is assumed to be homogeneous and unbounded, in terms of

geometry, is represented by means of a row of fictitious elements at the rock-alluvial interface

with given mechanical features that allow to represent the influence of the flexibility of the

rock on the dynamic response or the dam. In that way, the input motion at the interface rock-

alluvial is modified by the presence of the dam, through its stiffness and mass.

In the present work the systems are modelled with different values base rock stiffness by

means of fictitious elements to account for the effect of the flexibility of the foundation rock.

In addition a comparison between the results from one-dimension and two-dimension

modelling is performed.

1.2 Background information

Concrete Face Rockfill Dams (CFRD) have been used worldwide in seismically active areas

since the body of the dam, under ordinary conditions, is not saturated and due to that the

liquefaction phenomena is not possible. Furthermore, they have proven to be an advisableoption when sufficient amount of clay material is not available. In addition to the seismic

performance, many times this sort of dams has demonstrated to be the most economical

option.

Seismic response analyses on CFRD dams have received only limited attention (Uddin and

Gazetas, 1995). Various researchers (e.g. Sherard and Cooke, 1987) suggest that CFRD dams

are inherently safe seismically since the rockfill remains dry, making improbable the increase

in the pore pressure and because the water acting on the concrete face slab in the upstreamslope, founded on drains, contributes to the stability.

The document “Aprovechamiento Hidroeléctrico “ Los Caracoles”- Rediseño de la sección de

la presa-Análisis dinámico y post-sismico” (in Spanish), is the main reference regarding the

geometry the materials making up the dam considered in this thesis. It presents the results of

the dynamic and post seismic analyses of the Los Caracoles dam in order to fulfill the design

criteria adopted for the assessment of the safety of the dam.

In that way, two seismic records with peak ground acceleration (PGA)=1,02g, response

spectra and duration, defined by the proprietor of the project, were utilized. For the present

analyses, the dam is represented by means of the modified version (1994) of the finite element

program QUAD4 which allows an evolutive modeling starting from the construction and

filling up of the reservoir to the subsequent application of the seismic loads




structure is quite heavy and rigid, as may be the case of a dam, the input motion and the

overall response of the system is different from the hypothesis of rigid rock.

By reviewing the available literature and background information, it is found that there is no

evidence of considerations of this type in similar projects. For this reason, in this work a

simple procedure for evaluating the soil-structure interaction and its influence on the response

of the system is carried out. According to the available information, the software currently

utilized worldwide does not take into account the soil-structure interaction problem in this

type of structures, and its therefore the main purpose of this thesis to explore the eventual

influence of this effect in the evaluation of seismic response.

1.3 Methodology

In order to accomplish the proposed objective, the activities presented in the flowchart shown

in Figure 1.1 are given. In a general way, initially the problem to be evaluated is defined and

basic information for performing the analyses is gathered. Essential aspects such as geometry,

seismic records and material properties are specified to perform the analysis.

Secondly, working models are set up, defining material types and dynamic properties of thematerials, and input motions at the rock-foundation interface are selected. Furthermore,

properties of the fictitious elements representing the flexibility and radiation damping of the

foundation are defined. These elements are employed for estimating the influence of the

flexibility and damping of the rock on the dynamic response of the dam.

In addition, modeling by means of two-dimension finite element software is performed,

varying the boundary conditions and the input motion. From the analysis, maximum

acceleration-depth cross sections, within the body of the dam are obtained; response spectra atseveral depths, dam crest settlement, and transfer functions are also computed. In addition and

as a complement to the foregoing study, a one dimensional model is carried out by using a

program able of simulating the 1D-propagation of waves.

Once the modeling is completed, a comparative analysis of the results is performed. For

achieving so, these results are compared to the available bibliography and the reference

design document.




4

Definition of theproblem. Settingup the workingmodel andmethodolgy of

analysis.

Setting up of the working models

• Definition of the material types.

• Definition of the dynamic properties of the materials.

• Selection of the seismic records.

• Definition of the properties of the

fictitious elements (springs).

Modeling by Finite Element SoftwareQUAD4M

• Variation of the conditions in the halfspace.

• Variation of the seismic input motion.

• Acceleration time histories at several pointswithin the domain of the model.

• Maximum acceleration vs. depth plots withinthe dam.

• Response spectra at several depths.

• Computation of the settlement of the crest.• Transfer functions corresponding to di fferent

boundary conditions.

Preliminary interpretation of the results• Verification of the validity of the results.

Comparison with availbale references

• Comparison with the reference designdocument.

• Comparison of the results obtained by

using the different programs.

Preparation of the definitive document

END

Conclusive results?NO

YES

Gathering of the information

• Geometry of the dam.

• Seismic records

• Material properties

• Properties of the fictitious

elements (springs). Modeling by one-dimensional propagation of

travelling shear waves

• Variation of the conditions in the halfspace.

• Variation of the seismic input motion.

• Acceleration time histories at several depthswithin the one-dimensional columns.

• Maximum acceleration vs. depth plots withinthe dam.


• Computation of the settlement of the crest.

• Transfer functions corresponding to di fferent

boundary conditions.

Preparation of plots and tables for being

presented

• Maximum acceleration vs. depth plotswithin the dam.


• Computation of the settlement of thecrest.

• Comparison between the results

obtained by using the differentprograms.

• Transfer functions

Start

Figure 1.1. Flowchart displaying the methodology for carrying out the work




1.4 Scope and objectives of the thesis

This work seeks to analyze by means of numerical modeling, the influence of the boundaryconditions on the dynamic response of a concrete face rockfill dam (CFRD). Currently

available software is used for considering, in an approximate way, the effect of the soil-

structure interaction phenomena on the seismic input motion at the rock-foundation interface.

The results obtained herein are compared with those obtained by the designer of the dam, who

employed the ADINA software for modeling the dam. This software uses the Mohr-Coulomb

elastoplastic model for evaluating the seismic behavior of the system.

The main objectives of the thesis are:

• Determine the influence of the boundary conditions on the seismic response in a Concrete

Face Rokfill Dam (CFRD), in terms of response spectra, maximum acceleration within the

body of the dam and settlements at the crest.

• Compare the results of the two dimensional modeling using the equivalent linear model,

with the results obtained by the designer of the project who employed the Mohr-Columb

model.

• Perform a parametric study on the influence of the stiffness of the halfspace on the seismic

response of the system model.

• Carry out a comparison between the results obtained by using the ADINA software which

employs the Mohr Coulomb model and simpler models such as the one dimensional

model (EERA software) and two-dimensional model (QUAD4M).

1.5 Organization

This work has been assembled in the following manner:

Chapter 1 refers to the general information of the work, states the scope and objectives

and the methodology followed.

Chapter 2 reviews a theoretical background of the main characteristics of the dynamic

behavior of soils, the elements that have the most influence on the modeling (mechanicalproperties, confining pressure), seismic records, the available options for analyzing the

problem, the boundary conditions and the methodology for computing the settlements at

the crest.

Chapter 3 displays the properties of the model performed. Geometry of the model,

material properties boundary conditions main features of the software handled and the



Chapter 2. Method of Analysis

2. METHOD OF ANALYSIS

2.1 Dynamic behavior of soils and rockfill

The mechanical behavior of soils can be rather complex under static and seismic conditions.

Geotechnical engineers seek to characterize the most important aspects of cyclic soil behavior

as accurately and simple as possible.

Despite of having limited ability for describing certain aspects, equivalent linear model (Seed,1970) is the simplest and most commonly used approach in current design practice of

embankment dams. Although there are available advanced constitutive models that allow

representing more closely dynamic soil behavior, their complexity and difficulty of

calibration often renders them well beyond practical applications for many common

geotechnical earthquake engineering problems.

2.1.1 Linear equivalent model

Typically a soil subjected to symmetric cyclic loading exhibit a hysteresis loop of the typeshown in Figure 2.1




In general terms, the most important characteristics of the shape of a hysteresis loop are itsinclination and its enclosed area. The inclination of the hysteresis loop depends on the

stiffness of the soil, which can be described at any point during the loading process by the

tangent shear modulus Gtan.. Since Gtan varies throughout a cycle of loading an average value

over the entire loop can be approximated by the secant shear modulus Gsec. (Eq. 2-1):

c

cGγ

τ =sec ( 2-1)

where γ c y τc are the shear stress and shear strain amplitudes, respectively.

The area of the hysteresis loop is related to the area, which is a measure of energy dissipation

and can be described by the camping ratio (Eq. 2-2)

2

sec21

4 c

loop

s

D

G A

W W

γ π π ξ == ( 2-2)

where WD is the dissipated energy, Ws the maximum strain energy, and Aloop the area of the

hysteresis loop. If dealing with viscous damping, the area of the hysteresis loop depends on

the frequency of cyclic loading, while when dealing with linear damping the area is frequency

independent.

Because some of the most commonly used methods of ground response analysis are based onthe use of equivalent linear properties, considerable attention is given to the adequate

characterization of Gsec and ξ for different soils.

It is worthy to highlight that, since it is only an approximation of the non-linear behavior of

the soils, the linear equivalent model cannot be used for predicting permanent deformation or

slides since it assumes that the strain is elastic and returns to zero after cyclic loading, and

therefore it can not represent neither limiting strength nor failure. Nevertheless, the

assumption of linearity, with iterative procedures over the stiffness parameters and damping,allows solving as a first approximation, a wide range of earthquake engineering problems in a

very efficient way and moreover it is supported by laboratory and field observations.

a) Maximum Shear Modulus, Gmax

Si i i h i l i d h i l h 3 10-4

% h d




The use of measured shear wave velocities is generally the most reliable means of evaluating

the in situ value of Gmax . However, when shear wave velocity measurements are not

available, it can be determined using empirical relationships.

In coarse materials the shear modulus can be estimated as follows (Seed and Idriss, 1970):

)(1000 '

2max mK G σ = (2-4)

where K2 is determined from the void ratio or relative density (Table 2-1) and'

mσ is in lb/ft2

b) Modulus reduction, G/GmaxAlter reviewing experimental results from a broad range of materials Dobry and Vucetic

(1987) and Sun et al. (1988) concluded that the shape of the modulus reduction curve is

influenced more by the plasticity index than by the void ratio and presented curves of the type

shown in Figure 2.2. The curve corresponding to a plasticity index PI=0 is commonly used in

sands.

Table 2-1. K2 for different soils

Material K2

Loose sand 35

Dense sand 50

Very dense sand 65

Very dense sands and gravels 100 a 150

Dense rockfill 150 a 200

Modulus reduction behavior is also influenced by effective confining pressure, particularly for

soils of low plasticity (Iwasaki et al., 1978; Kokoshu, 1980), as shown in Figure 2.3.




Figure 2.3. Influence of mean effective confinig pressure on modulus reduction curves for a non plastic

soil PI=0. (After Kramer,1996)

c) Damping ratio

Theoretically, no histeretic dissipation of energy takes place at strains below the linear cyclicthreshold shear strain. Experimental evidence, however, shows that some energy is dissipated

even at very low strain levels; therefore the damping ratio is never zero. Above the threshold

strain, the breadth of the hysteresis loops exhibited by a cyclically loaded soil increases with

increasing strain amplitude, which indicates that the damping ratio increases with increasing

strain amplitude.

In the same way, modulus reduction behavior is influenced by plasticity characteristics

(Kokushu et al., 1982; Dobry and Vucetic, 1987; Sun et al., 1988). Damping ratios of highlyplastic soils are lower than those of low plasticity soils at the same cyclic strain amplitude.

The PI=0 damping curve is usually adopted for coarse-grained soils. (Figure 2.4)

2.1.2 Cyclic nonlinear models

The nonlinear stress-strain behavior of soils can be represented more accurately by cyclic

non-linear models. Such models are able to represent the shear strength of the soil and, with

an appropriate pore pressure generation model, changes in effective stress during undrained

cyclic loading

A variety of cyclic nonlinear models are currently available; all of them are characterized by a

backbone curve and a series of rules that govern unloading-reloading behavior, stiffness

degradation, and other effects.




Figure 2.4. Variation of damping ratio of fine-grained soil with cyclic shear strain amplitude andplasticity index (After Kramer, 1996)

2.2 Mechanical properties of the materials

As mentioned in section 2.1.1, the parameters that define the linear equivalent model, are the

shear modulus and the damping and shear modulus reduction curves.

In this work, the numerical modeling of a CFRD will be performed. For achieving so, it is

necessary to determine the mechanical properties mentioned above.

According to the data employed by the designer, the maximum shear modulus is computed as

indicated in Eq (2-4) using the following values for K2.

K2 = 80 for the foundation

K2 =100 for other materials making up the body of the dam.

For utilizing the previous expression, it is indispensable to determine the confining pressure

within the domain of the model and therefore, it is required to know the unit weight of the

materials. According to the document “Aprovechamiento Hidroeléctrico “Los Caracoles”-

Rediseño de la sección de la presa-Análisis dinámico y post-sismico”(in spanish), in situ

density tests were performed obtaining the results that will be presented later on in Table 3-1.




2.3 State of stress

For performing the modeling of the dam, the designer employed the ADINA (Automatic

Dynamic Incremental Nonlinear Analysis) software which allows evolutive analysis from the

beginning of the construction and filling up of the reservoir until the subsequent application of

earthquake dynamic loads.

In Figure 2.5 an illustration of the effective confining pressure distribution after finishing the

construction is shown. Based on it, the static confining pressure in the materials of the dam

and foundation is determined. From this distribution, the values of shear modulus are

computed as indicated in Eq (2-4)

With the purpose of verifying the values obtained by the designer using the ADINA software,

for three points within the body of the dam, the confining pressure at the end of the

construction were computed by means of the following expression (Eq. 2-5):

( )

3

21'

'ovm

K +=σ

σ ( 2-5)

In Eq (2-5)'

vσ

, is computed by multiplying the unit weight of the material by the depth of the

point and Ko is the at rest coefficient of earth pressure which, for coarse materials, can be

assumed as K0=0,5. In Figure 2.6 the points selected for carrying out the comparison are

displayed.




Figure 2.6. Points selected for checking the confining pressure

In Table 2-2 the results obtained from the comparison are shown and it can be concluded that

the confining pressure obtained by ADINA can be employed for computing the material

properties since the values are within the range of accuracy that allows Figure 2.5.

Table 2-2. Comparison between the values of confining pressure within the model.

Point

Depth from

the crest (m)

Computed

confining

pressure (Kpa)

Visualized

confining

pressure (Kpa)

1 50 760 600

2 121 1863 1500

3 175 2695 2250

On the other hand, a finite element model, using the Plaxis v 7.2 (Finite Element Code for

Soil and Rock Analyses), was developed with the purpose of comparing the shape of the

diagram of confining pressure at the end of the construction. This software utilizes finite

elements for analyzing two-dimensional problems in geotechnical engineering. It allows

incorporating advanced constitutive models for simulating the non linear time dependent and




Effective mean stresses

Extreme effective mean stress -4,84*103

kN/m2

kN/m2

-5000.000

-4750.000

-4500.000

-4250.000

-4000.000

-3750.000

-3500.000

-3250.000

-3000.000

-2750.000

-2500.000

-2250.000

-2000.000

-1750.000

-1500.000

-1250.000

-1000.000

-750.000

-500.000

-250.000

0.000

Figure 2.7. Confining pressure distribution within the model computed by means of Plaxis v 7.2

It can be observed that the results obtained by using Plaxis are similar to those presented by

the designer, thus it can be concluded that the adopted confining pressure distribution is in

agreement for various numerical models.

2.4 Seismic waves and earthquake records employed

When a fault ruptures below the surface of the earth, body waves travel away from the source

in all directions. As they reach boundaries between different geologic materials, they are

reflected and refracted. In Figure 2.8 an illustration of the propagation is shown.

There are four main types of seismic waves. From them, two types, the compressional and

transverse are called body waves because they travel through the interior of the crust in zones

far away from the surface and discontinuities. Compressional waves travel through solids,

liquids or gases. Transverse waves which require shear stiffness for being transmitted only

can propagate through solids. Love waves and Rayleigh waves are two types of waves that are

restricted to a media with shear stiffness in the vicinity of the free surface.




Compressional waves are analogous to sound waves. They are also known as longitudinal

waves. P-wave is the fastest of the body waves, thus it is the first arrival of the seismic action

to the site.

Transverse waves or shear waves are analogous to the light wave o the transverse vibration of

a rope. The motion of an individual particle is perpendicular to the direction of s-wave travel.

The velocity of the s-wave is approximately 60% of the velocity of the p-wave.

As displayed in Figure 2.9, at any station, the vertical component of acceleration is the result

of the propagation of the P-wave, whereas the horizontal motion is due to the s-waves.

Regarding numerical modeling, it is customary to consider that the seismic action can be

described as a SH-wave propagating vertically, and producing the horizontal components in

the bedrock, whilst the vertical motions are associated with the propagation of a P-wave.

Figure 2.9. Displacements fields for plane P and Suaves propagating in the x-z plane containing thesource and the receiver, where the z-axis is vertical. The P-wave displacement is along the wave vector k.

The S wave can be decomposed into two polarizations, SV and SH, perpendicular to the wave vector. The

SH displacement is purely horizontal (in the y direction, out of the page) whereas the SV displacement isin the x-z plane. (After Stein and Wysession, 2003)

With the purpose of comparing the results obtained in this work, with those presented by the

designer of the project, the same seismic loads were utilized for the design. In the next section

the main features of these records are summarized.

2.4.1 Maximum Credible Earthquake

According to the International Commision of Large Dams (ICOLD), the Maximum Credible




project is exhibited and, in Figure 2.11 the acceleration response spectra, for a level of

damping assumed ζ=5% of critical damping.

2.4.2 Additional earthquake record

With the purpose of verifying the dynamic behavior of the dam and checking the conclusions

of the foregoing analysis, an additional earthquake record, based on an actual earthquake

(Chi-Chi, Taiwan Earthquake, Sept 21st, 1999, Station TCU068), scaled for getting the same

absolute maximum value of acceleration as the MCE, was used. In Figure 2.12 and Figure

2.13 the acceleration time history and the acceleration response spectrum for this record are

displayed

-1,5

-1

-0,5

0

0,5

1

1,5

0 5 10 15 20 25

time (s)

a ( g )

Figure 2.10. Acceleration time history -Maximum Credible Earthquake-(MCE)

1 0

1,5

2,0

2,53,0

3,5

4,0

4,5Sa(g)

ζζζζ= 5%




-1,5

-1

-0,5

0

0,5

1

1,5

0 5 10 15 20 25 30 35 40 45 50

time (s)

a ( g )

Figure 2.12. Acceleration time history-Checking Earthquake-Chi-chi (Taiwan), 1999

0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

T(s)

Sa(g)

ζζζζ= 5%

Figure 2.13. Acceleration response sprectrum- Chi-Chi (Taiwan). 1999




2.5.1 Shear beam approach

One of the earliest approaches to the dynamic analysis of two-dimensional geotechnical

systems is the shear beam analysis, applied to earth dams by Mononobe et al (1936). The

shear beam approach is based on the assumption that a dam deforms in simple shear, thereby

producing only horizontal displacements. This approach also assumes that either shear

stresses or shear strains are uniform across horizontal planes. The theory allows the two-

dimensional dam section to be represented as a one-dimensional system.

Gazetas (1982) developed solutions to the shear beam wave equation for the case where the

shear modulus increases as a power function of depth according to (Eq. 2-6)

( ) ( )m

b H zG zG / = ( 2-6 )

Where Gb is the average shear modulus at the base of the dam. For such conditions, the nth

natural frequency (assuming h/H=1) is given by

( )( )mm H

V ns

n−+= 24

8

__

β ω

( 2-7 )

Where Vs is the average shear wave velocity of the soil in the dam and n is the nth root of a

period relation (Dakoulas and Gazetas, 1985) tabulated in Table 2-3 for the first five modes

of vibration

Table 2-3. Values of ββββn for first five modes of vibration o an earth dam

1 2 3 4 5

0 2,404 5,52 8,654 11,792 14,931

1/2 2,903 6,033 9,171 12,31 15,451

4/7 2,999 6,133 9,273 12,413 15,544

2/3 3,142 6,283 9,525 12,566 15,7081 3,382 7,106 10,174 13,324 16,471

n

m

Equation 2-7 produces a fundamental period of (Eq 2-8)

( )( ) sV

H

mmT

1

124

16

β

π

−+= ( 2-8 )

Ch 2 M h d f A l i




k q

k

k

q

x

k qk x J

2

0 2)1(!

)1()(

+∞

=∑

++Γ

−= (2-10)

Where ( )•Γ is the gamma function, which is tabulated.

The first and second mode shape are shown in Figure 2.14 for various values of stiffness

parameter, m,

In the preceding derivation, the soil was assumed to be linear and undamped. Nevertheless,

camping can be easily included by repeating the derivation with the soil characterized by a

complex stiffness.

.(a) (b)

Figure 2.14. Mode shapes for (a) first mode and (b) second mode of earth dam response. (Alter Dakoulas

and Gazetas)

2.6 Numerical models for the dynamic analysis of the dam

Dynamic analysis essentially involves the determination of the deformation behavior of the

dam using the finite element or finite difference method. The results of such analyses aresensitive to the input seismologic parameters and engineering properties. Due to that, a pre-

requisite for using these procedures is a thorough seismotectonic assessment and detailed site

and material characterization.

Seed (1979) and Finn et al. (1986) summarize procedures for dynamic analyses of dams.

Chapter 2 Method of Anal sis




b. To evaluate the dynamic soil behavior from in-situ and cyclic laboratory tests for

determining input soil properties required in the dynamic analyses.

c. For the numerical model developed in Step a, to determine the dynamic response

of the dam and foundation using a set of plausible input bedrock motions. The input

bedrock motions should include appropriate accelerograms representing earthquakes

of magnitude and peak acceleration similar to those of the design earthquakes

recorded in a similar geologic environment.

d. The stress-strain models used in the dynamic analysis should reasonably represent

the following aspects of material behavior:

• Non-linearity

• Stress and strain dependence

• Inherent anisotropy

• Strain rate dependence.

e. To evaluate deformations on the basis of strain potential for the individual

elements, which corresponds to the strain that would be experienced if the element

were not constrained by surrounding soil.f. To calculate total embankment deformation on the basis of gravity loads and

softened material properties to determine whether they are within the acceptable

limits.

2.6.1 One-dimensional model based on the shear beam approach

Prato and Delmastro (1988) developed a procedure that combines the shear wedge approach

and the linear equivalent method, allowing in a simplified manner, modeling of

nonhomogeneous cross section dams.

Despite being in good agreement with 2D or 3D analyses, the authors pose the following

limitations:

It can only be applied to homogeneous sections.

Modal shapes that are higher than the fundamental differs considerably between the

different approaches.

The fundamental period of a homogeneous wedge is To=2,59H/Vs and the stratum is

To=4H/Vs being H=heigth, and Vs shear wave velocity of the material. By using a fictitious

value of V it is possible to approximate the fundamental period of the wedge with the

Chapter 2 Method of Analysis




scaled in a proportion of (4/2,59)2 for approximating the period of the triangular wedge and

the horizontal stratum.

As a result of the comparison, they found a reasonable agreement between the proposed

procedure and QUAD4, whereas predictions with SHAKE significantly underestimate

accelerations in the upper half of the dam.

In the same way, they found that the results are in good agreement in the range of frequencies

from zero to approximately 1,3 times the fundamental frequency.

2.6.2 Finite element plane modelsConsidering the geometry to be modeled, the most advisable approximation would consist of

employing a 3D model for evaluating seismic response. Nevertheless, by carrying out 2D

models that allow considering the boundary conditions and the site topography, a good

approach to the solution of the problem can be achieved. In addition, 3D models are

commercially quite scarce and the geotechnical parameters required are more difficult for

measuring.

With the purpose of comparing the results, in this work the same mesh employed by thedesigner was used. However, the size of the elements was reviewed for avoiding a

length/width ratio larger than 3.0 within the body of the dam and its foundation, and a vertical

dimension of the elements lower than 15% of the seismic wavelength for preventing

dispersion or reflection waves in the continuum. In other words, the following criterion is

fulfilled (Eq 2-11)

λ /10 < s < λ /5 (2.11)

where:

Vs: shear wave velocity of the layer (m/s)

Ts: period of the seismic record (s)

λ: wavelength (m)

s: thickness of the layer

λ = Vs * Ts

The mesh employed is consists of 509 elements and 575 nodes and it will be displayed later

Chapter 2 Method of Analysis




Every aspect of the three-dimensional behavior of the dam (state of stress, stability, motion

and stability) is controlled not only by the absolute values of the dimensions of the structure

and site but also by its relation (site coefficient). It means that, for characterizing the 3D siteconditions it is necessary to indicate the site coefficient and a characteristic dimension such

height or length of the crest.

In earth dams taller than 70-80m with site coefficient Ks≈4, three-dimensional effects on

generating vertical stresses can be neglected and can be estimated from results of two-

dimensional analyses.

Within the dam occur opposite movements of the soil masses toward the central section. Thiseffect is more evident when the slope of the face is 45º and decreases drastically by increasing

the angle over 60º or decreasing the angle below 30º.

The increment in the transverse stresses leads the soil to a state close to uniform compression,

which reduces the deformation within the dam and increases its stability. The increment in the

stability is more significant for a site coefficient no lower than 4.

The effect of the 3D conditions in the body of the dam is rather significant, diverse and occursfor very wide sites. This aspect must me kept in mind when optimal design of earth dams,

based on the analysis of the stress-strain state and its stability, are sought.

Prato and Matheu (1991) developed a procedure for incorporating the effect of the

geometrical configuration of the canyon walls in the seismic response analysis of

embankment dams. The purpose of the work was to expand the procedure developed by Prato

and Delmastro (1988)

The analysis considers only the horizontal components of the seismic record and assumes that

there is a plane of symmetry in the canyon cross section.

The lateral restraint provided by the canyon walls to the wedge contained in the vertical plane

of symmetry can be evaluated assuming that the horizontal strips, in the horizontal plane and

unit height act as a shear beam.

By setting up a dynamic equation of equilibrium for the vertical wedge and transforming itinto the frequency domain, it is obtained an expression that can be evaluated numerically by

using piecewise linear between discrete levels of integration along the height of the dam. For

performing each iteration of the equivalent linear method with this procedure, effective values

of the shear modulus and damping ratio for all layers must be determined. In the first

i i h l b i d i ll h l f h i





Earthquake-induced shear stresses can also be obtained from calculated values of shear strains

and the adjusted secant shear modulus which is computed by multiplying the shear strains at a

particular location within a discretized layer, by the secant shear modulus associated to thatlevel of shear strains.

With the purpose of assessing the applicability of the proposed method, they compared the

results with those achieved by means of a three-dimensional modeling of the Long Valley

dam (Mejia et al. 1982, Lai and Seed, 1985) and records of seismic events occurred in the

dam. The following aspects were concluded.

The pseudoacceleration response spectra at the crest obtained with the proponed method isfound to be in good agreement with the in situ values.

A marked discrepancy between measured and computed values of the maximum

acceleration at the center of the crest is found. The estimated values by the finite element

models and the proposed model are approximately 39% higher than the recorded values.

2.7 Boundary conditions of the 2D finite element model

2.7.1 Boundary conditions for gravitational forces

As mentioned previously, dynamic properties of the materials depend, among others, on the

confinement pressure acting on them. Due to that, it is necessary to determine the stress

distribution within the body of the dam. For achieving that, a finite element model, was

carried out. In the model, the boundary conditions depicted in Figure 2.15 are imposed. At

the base and sides of the model all the displacements are restrained; whereas no other restraint

is imposed for other nodes in the model.The nodes along the base and side

boundary of the mesh are restrained

against movement in all directions

Figure 2.15. Imposed Boundary conditions for evaluating earth pressure distribution undergravitational forces




p y

In order for a two-dimensional finite mesh to represent the response of an infinite field

condition, the artificial reflection of seismic waves from side boundaries, as well as from the

underlying half-space, should be minimized. To accomplish this Lysmer and Kuhlemeyer(1969) introduced a simple procedure using dampers as illustrated in Figure 2.16.

The implementation of these dampers involves adding damping at each of the nodes that

make up the base and sides of the finite model. The base dampers are more essential to

incorporate than the side dampers because the finite element system under consideration will

always be placed over a half-space. The effects of side boundaries can be readily minimized

by increasing the extent of the finite model.

To implement these dampers, the parts of the applicable element matrices have the

transmitting boundary damping term added to the diagonal terms. This produces an unknown

force in the x and y direction proportional to the velocity of the specified nodes. The

coefficients added to the diagonal terms are obtained as

Term for direction perpendicular to the boundary : ρVpL

Term for direction parallel to the boundary: ρVsL

The velocity of the P or S waves, and the density, ρ are used for the material in the half space

below the finite element model. The “tributary width” of the node, L, is the length

corresponding to half of the distance to the next node on both sides.

When a transmitting boundary is used, the input motion is a function of the material

properties of the half-space below the mesh, and the properties and geometry of the mesh.

This is the correct choice for a boundary condition when the input motion represents anoutcrop acceleration, recorded at an outcrop of the half-space material. If an infinitely stiff

( ∞→sV ) rock is specified under the underlying stratum, then the input motion will not be

affected by the mesh above.




Figure 2.16. Finite element models for footing on halfspace (AfterLysmer& Kuhlemeyer, 1969)

a) Side nodes

At the side nodes, the horizontal displacement is allowed while the vertical displacement is

restrained. In this way, a boundary condition equivalent to a roller support is accomplished.

b) Transmitting base

Taking into consideration the geometrical configuration, the model analyzed can be

considered as a footing founded on half-space and, in that way its dynamic behavior can be

studied.

The dynamic behavior of a vertically loaded footing can be evaluated by using a Single

Degree Of Freedom (SDOF) oscillator with frequency-dependent stiffness and damping

coefficients. These coefficients represent the dynamic stiffness that governs the behavior of

the footing. Conceptually, the dynamic stiffness or impedance functions of the soil-foundation

system are defined as the ratio between the exciting force (moment) and the corresponding

displacement (rotation) resulting along the direction of the force for a rigid and massless

footing and harmonically excited.

The mathematical representation of the dynamic stiffness is a frequency-dependent complex

function. The real part represents the stiffness and inertia of the soil, idealized by means of

springs, and the imaginary part represents the damping of the material. For representing the




where2

0

2

1ω

ω −=k and

0

02

ω

ς =c being 0ω , the natural frequency and 0ς the damping of the

oscillator, which represents the percentage of damping with respect to the critical damping. k

and c are known as stiffness and damping coefficients, respectively.

Eq (2-11) implies that the dynamic stiffness K ~

can be expressed as the static stiffness K

multiplied by and a complex dynamic factor )( cik ω + which considers the inertia and

damping characteristics of the system.

With the purpose of estimating the influence of the boundary conditions on the seismic

response of the model, several cases for the transmitting base were analyzed, varying the

stiffness of the half-space and the stiffness of the fictitious layer of elements placed under the

line that defines the contact rock-soil, which allows to consider the stiffness induced by the

presence of the dam. The main features are described below.

A simplified illustration of the geometry is depicted in Figure 2.17 and the procedure

followed takes into account the soil-structure interaction for computing the stiffness in the

direction of the movement.

In Figure 2.18 the expressions that allow calculating the stiffness of a rectangular footing

subjected to different types of loading are shown. In this work, only horizontal displacement

is considered and the equation used is (Eq. 2.13)

( ) BLGk x x β υ += 12 (2.13)

In Eq 2.13, G and ν are the shear modulus and the Poisson´s ratio of the foundation,

respectively. B and L are illustrated in Figure 2.19.

b(m) = 10

L (m) = 634

B(m) = 620

h(m) = 140

W( m) = 30




Figure 2.18. Lumped Representation of Structure Foundation Interaction (After Richard et al. 1970)

Figure 2.19. Constants and for rectangular bases. (After Richart, F. E et al. Vibrations of Soils and

Foundations, Prentice-Hall, Inc, 1970)

Values for the density and shear wave velocity commonly found in rocks were used (Kramer,

1996) The procedure followed is as follows;




(Eq.2.13) ( ) ( ) KN/m8463972471620*624)1(56250002,01212 =+=+= BLGk x x β υ

Defining a stiffness per unit depth (Eq. 2-14)

/m/m13651568KN620

8463972471===

B

K k x

unit x (2.14)

To define an equivalent shear modulus, Geq,, a 20m thick layer of fictitious elements was

assumed for representing the stiffness of the elastic half-space an the following expression

was employed (Eq. 2-15)

KPa L

hk G

layer elements fictitiousunit x

eq 430649634

2013651568=

∗=

∗= (2.15)

It is worth mentioning that the value obtained for the modulus is proportional to the thickness

of the layer of fictitious elements introduced in the model. In such a 2D model it is normal to

consider of unit length.

Several alternatives for the transmitting base were analyzed with the purpose of performing aparametric study on the influence of the characteristics of the foundation and fictitious

elements on the seismic response (accelerations, response spectra and vertical displacements).

These elements are adopted for representing the flexibility and energy radiation through the

foundation rock.

The cases analyzed are described below:

• Very rigid half-space: considering a shear wave velocity extremely high,Vs=25000m/s. This case corresponds to the hypothesis adopted by the designer of the

dam.

• Half-space with shear wave velocity Vs=1500m/s which allows, according to the

features of the program used, incorporating dampers at the base of the model as stated

in section 2.7.2

• Half-space with shear wave velocity Vs=2500m/s. This case is identical to theprevious one but increasing the shear wave velocity of the foundation rock, with the

purpose of estimating its influence on the results.

• Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick layer of

fi i i l i h h l i V 1500 / Wi h hi i i




previous one but increasing the shear wave velocity of the layer of fictitious elements,

with the purpose of estimating its influence on the results.

• Half-space with shear wave velocity Vs=1500m/s and a 20m-thick layer of fictitious

elements with shear wave velocity Vs=1500m/s. With this model it is sought to study

the combined effect of the dampers, incorporated by the program and the fictitious

elements

Despite the fact that the dynamic representation of foundations normally considers dampers

and springs, arranged in a parallel configuration, In this case, only is sought presenting a

general trend of the combined effect of this sort of elements arranged in series configuration.

2.8 Settlements at the crest. Newmark Method

For producing displacement in slope due to seismic loads, it is necessary to reach a threshold

of acceleration which depends on the material properties making up the slope.

In dams, it is essential to determine the maximum vertical displacement (settlement) which

may be expected at the crest under seismic load, with the purpose of providing a sufficient

freeboard in the design able of guarantee the safety in the operation of the reservoir.

2.8.1 Wedge method (analytical expression)

The simplest type of slope stability analysis is the wedge method. Figure 2.20 illustrates the

free-body diagram for the wedge method. In this figure the failure wedge has a planar slip

surface inclined at an angle α to the horizontal. Although the failure wedge passes through the

toe of the slope, the analysis could also be performed for the case of the planar slip surface

intersecting the face of the slope.




• W = weight of failure wedge. Usually a two-dimensional analysis is performed based

on an assumed unit length of slope. Thus the weight of the wedge is calculated as the

total unit weight γ t , times the cross-sectional area of the failure wedge.

• Fh =k hW=horizontal pseudostatic force acting through the centroid of the sliding mass,

in an out-of-slope direction.

• N = normal force acting on the slip surface.

• T = shear force acting along the slip surface. The shear force is also known as the

resisting force because it resists failure of the wedge. Based on the Mohr-Coulombfailure law, the shear force is equal to the following:

´´tan´ φ N LcT += (2.16)

where

For an effective stress analysis:

L = length of the planar slip surface.

c´ y φ´= shear strength parameters in terms on an effective stress analysisN´= effective normal force acting on the slip surface.

The assumption in this slope stability analysis is that there will be movement of the wedge in

a direction that is parallel to the planar slip surface. Thus the factor of safety of the slope can

be derived by summing forces parallel to the slip surface and it is as follows (Eq. 2-17):

α α φ α α

α α φ

cossin´tan)sincos(´

cossin´´tan´

h

h

h F W F W Lc

F W N Lc

forcedriving forcereistingFS

+−+=

++== (2.17)

For the present study this methodology will be employed for evaluating the value of Kh that

causes failure in different wedges of the downstream slope. In this case, the slip surface does

not intersect the toe of the slope but, on the contrary, it reaches the surface at different depths

from the crest as shown in Figure 2.21. In addition, all the wedges cut two types of materials.

Mat 4

Mat 3L




By using the same reasoning described previously, Figure 2.22 displays a wedge made up of

two types of materials

Mat 4

φ1, γ 1

Mat 3Lφ2, γ 2

N1

FR1

W1

N2

FR2

W2

FhTOTAL

α

Figure 2.22. Diagram of the studied wedges in the downstream slope

For estimating the weight of the wedge, two areas delimited by the vertical cut (indicated bythe red and blue colors) are defined and the summation of forces acting on the slip surfaces is

set out. In this way the following expressions are derived for estimating the Factor of Safety

of the wedge. (Eq 2.18 to Eq. 2.20). By assigning a unit value for the Factor of Safety the

horizontal acceleration that produces failure in the wedge can be determined. This

acceleration will be employed in the Newmark analysis that will be explained subsequently.

( ) α CosF SinW W F hTOTALareaareaactiuantes ++= 21 (2.18)

22

211

121 tantan φ α α φ α α

⋅−+

⋅−=+= SinF

W

W CosW SinF

W

W CosW F F F hTOTAL

T

areaareahTOTAL

T

areaarea R Rresist

(2.19)

( ) α α

φ α α φ α α

CosF SinW W

SinF W

W CosW SinF

W

W CosW

FShTOTALareaarea

hTOTAL

T

area

areahTOTAL

T

area

area

++

⋅−+

⋅−

=21

2

2

21

1

1 tantan

(2.20)

In the previous analysis it is assumed that the seismic force is distributed in proportion of the

weight of each area.

In this way, the termsT

area

W

W 1 andT

area

W

W 2 come up, multiplying the term Fh total .




Table 2-4. Yield acceleration for the wedges

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

Wedge y/h material γ γγ γ (KN/m3)A

material(m2)φφφφ (º)

W(KN)

per unit

depth

tan φφφφWarea/

WTFh ind Fh(KN)

Driving

forces

(KN)

Resisting

forces (KN)FS ay(g)

4 19,6 91,4 50 1791 1,19 0,75 937

4 19,6 5,7 50 112 1,19 0,05

3B 23,1 21,6 44 498 0,97 0,21

t ot al 2 40 1

4 19,6 96,7 50 1895 1,19 0,58 921

4 19,6 10,2 50 199 1,19 0,06

3B 23,1 51,4 44 1187 0,97 0,36

t ot al 3 28 1

4 19,6 109,5 50 2146 1,19 0,46 979

4 19,6 13,5 50 264 1,19 0,06

3B 23,1 99,2 44 2292 0,97 0,49

t ot al 4 70 2

4 19,6 159,9 50 3135 1,19 0,25 1´552

4 19,6 109,7 50 2150 1,19 0,17

3B 23,1 305,3 44 7052 0,97 0,57 total 12336

4´556

319

674

1´166

4 0,29

3 0,25

1 0,15

2 0,2

6108

1,0 0,523

0,486

0,456

1256 2034 2032

2146 3775 3772 1,0

1595 2695 2695 1,0

9846 9843 1,0 0,495

Details of each one of the columns in Table 2-4 are given below:

(1)-Wedge: corresponds to the number of the wedge

(2)-y/h: this column provides information about the depth of the wedge from the crest. y is the

depth of the wedge and h is the height of the dam.

(3)-Material: due to the fact that the wedges cut different materials, it is necessary to perform

the analysis by defining vertical slices (Figure 2.22) and thus the weight and forces acting on

each one of the areas, are computed. Therefore, in this column the material that comes up

from the division in vertical slices is indicated.

(4)- γ t(KN/m3): it is the unit weight of the materials making up the wedge

(5)- A material (m2): it is the area of the different materials making up the wedge aftersplitting the vertical slices.

(6)-φ (º): it is the friction angle (Mohr-Coulomb criterion) of the materials making up the

wedge (Eq 2 16)




(10)-Fhind: it is the horizontal force (KN) necessary for the Factor of Safety, shown in column

(3) of the soil mass to be equal to 1. It is obtained by multiplying the ratio computed in

column (9) by the horizontal seismic force (column (11)) necessary for the Factor of Safety of the entire wedge to be equal to 1.

(11)-Fh (KN): it is the horizontal seismic force necessary for the Factor of Safety of the entire

wedge to be equal to 1. It is obtained by iterations on the Eq 2.20, in which the driving forces

and the resisting forces are compared.

(12)-Driving forces (KN): these are the forces acting on the slip plane of the wedge computed

by means of (Eq. 2.18).

(13)-Resisting forces (KN): these are the resisting forces acting on the slip plane of the

wedge. (Eq. 2.19)

(14)-FS: it is the Factor of Safety of the wedge.

(15)-ay(g): it is the yield acceleration necessary for the Factor of Safety (column (14)) of the

entire wedge to be equal to 1.

2.8.2 Newmark method

The pseudostatic method of analysis, like all limit equilibrium methods, provides an index of

stability (the factor of safety) but no information on deformations associated with slope

failure. Since the serviceability of a slope after an earthquake is controlled by deformations,

analyses that allow predicting slope displacements provide a more useful indication of seismic

slope stability. Since earthquake-induced accelerations vary with time, the pseudostatic factor

of safety will vary throughout an earthquake. If the inertial forces acting on a potential mass

become large enough that the total (static and dynamic) driving forces exceed the available

resisting forces, the factor of safety will drop below 1.0.

The purpose of the Newmark (1965) method is to estimate the slope deformation for those

cases where the pseudostatic factor of safety is less than 1.0 (i.e., the failure condition). The

method assumes that the slope will deform only during those portions of the earthquake when

the out-of-slope earthquake forces cause the pseudostatic factor of safety to drop below 1.0.

When this occurs, the slope will no longer be stable, and it will be accelerated downslope. Thelonger that the slope is subjected to a pseudostatic factor of safety below 1.0, the greater the

slope deformation. On the other hand, if the pseusostatic factor of safety drops below 1.0 for a

mere fraction of a second, then the slope deformation will be limited.

The situation is analogous to that of a block resting on an inclined plane (Figure 2 23)




Figure 2.23. Analogy between (a) potential landslide and (b) block resting on inclined plane. After

Kramer (1996)

It is only the out-of-slope accelerations that cause downslope movement, and thus only the

acceleration that plots above the zero line is considered in the analysis. In Figure 2.24a, the

dashed line corresponds to the horizontal yield acceleration, which is designated ay. The

horizontal yield acceleration ay is considered to be the horizontal earthquake acceleration that

results in a pseudostatic factor of safety that is exactly equal to 1.0. According to the method,

it is the darkened portions of the acceleration pulses that will cause lateral movement of the

slope.

Figure 2.24b and presents the corresponding horizontal velocity and slope displacement that

occur in response to the darkened portions of the two acceleration pulses.

The magnitude of the slope displacement depends on the following factors:

• Horizontal yield acceleration, a y: The higher the horizontal yield acceleration, ay, the

more stable the slope is for a given earthquake.

• Peak ground acceleration, amax: The peak ground acceleration, amax, represents the

highest value of the horizontal ground acceleracion. In essence this is the amplitude of

the maximum acceleration pulse. The grater the difference between the peak ground

acceleration amax and ay, the larger the downslope movement.

• Length of time: The longer the earthquake acceleration exceeds the horizontal yield

acceleration ay, the larger the downslope deformation. It can be concluded that thelarger the shaded area in Figure 2.24.a, the greater the downslope movement.

• Number of acceleration pulses: The larger the number of acceleration pulses that

exceed the horizontal yield acceleration, ay, the greater the cumulative downslope




Figure 2.24. Diagram illustrating the Newmark method. a) Acceleration vs. time; b) Velocity vs. time for

the darkened portions of the acceleration pulses; c) the corresponding downslope displacement versustime in response to the velocity pulses. (After Wilson and Keefer)

Chapter 3. Case study



3. CASE STUDY

3.1 Location of the Project

The “Los Caracoles” reservoir is a hydroelectric project located in the San Juan province, at

the border between the Ullum and Zonda deparments, in Argentina.

The “Los Caracoles” dam performs mainly three functions. The first is generating

hydroelectric energy; the second is to store water for irrigated lands and to provide a new

tourist place for the province. It is located on the San Juan River and it has the capacity of generating 125MW accumulating an annual generation of 715GWh.

The dam is founded 53km west of the San Juan capital city. It is able of storing 565 hm3

approximately. It is a huge embankment (10´200.000 m3) made up of compacted gravels,

136m height and the length of the crest is 620m.

3.2 Description of the materials

Based on the reference document “Aprovechamiento Hidroeléctrico “Los Caracoles”-

Criterios de Diseño de las Obras Bajo Acciones Sísmicas” (in Spanish), a description of the

materials making up the dam is given below.

• Material 3B: In Figure 3.1 the grain size distribution of Material 3B is shown. It displays

the upper limit and the lower limit of the grain size distribution for an integral sample of

the quarry and a sample prepared through a 1 ¼” sieve for performing triaxial tests. The

blue curve is considered to be representative of the material. Based on this grain sizedistribution, and according to the Unified Soil Classificaction System (USCS), the

material can be classified as poorly graded gravel, GP, since more than 50% of coarse

fraction is retained on No.4 sieve and there is a percentage of sand larger than 15%.




0

10

20

30

40

50

60

70

80

90

100

0.00.11.010.0100.01000.0Tamaño del grano (mm)

% p a s a e n p e s o

Granulometría

integral del

Yacimiento DNV

Muestra preparada en IMS de

la UNSJ

Muestra preparada

en el IDIEM de U. deChile

Figure 3.1. Grain size distribution for Material 3B. (After Techint-Panedile, 2005)

0

10

20

30

40

50

60

70

80

90

100

0.00.11.010.0100.01000.0 Tamaño del grano (mm)

% p a s a e n

p e s o

Granulometría

integral del

yacimientoMuestra preparada en

el IMS de la UNSJ

Muestra preparada

en el IDIEM U de

Chile

Figure 3.2. Grain size distribution for Material 3L. (After Techint-Panedile, 2005)

• Alluvial Materials making up the foundation: In Figure 3.3 the grain size distribution of

the alluvial materials making up the foundation is shown It displays the upper limit and




0

10

20

30

40

50

60

70

80

90

100

0.00.11.010.0100.01000.0Tamaño del grano (mm)

%

p a s a e n p e s o

Granulometría de lafundación de lapresa

Muestra preparada en elIMS para ensayos

triaxiales

Figure 3.3. Grain size distribution for the alluvial material forming the foundation of the dam. (AfterTechint-Panedile , 2005)

• Drainage: it is made up of the crushed rock from the excavation of the dam, resulting in

angular particles.

• Material 3D: According to the reference document ““Aprovechamiento Hidroeléctrico

“Los Caracoles”- Criterios de Diseño de las Obras Bajo Acciones Sísmicas” (in Spanish),

material 3D is a mixture of material 3C (Rockfill) and material 3L. For performing the

dynamic analysis, shear strength parameters of the Material 3L are considered to be

representative for material 3D.

• Material 2: it provides support to the face slab and consists of a material with a fine

content between 2% and 12% and a plasticity index below 7%. Compared to the size of

the dam, this is a quite narrow zone, therefore is has no significant influence on the overall

stability of the dam.

• Material 4: It is a narrow zone made up of large rock dozed to the downstream face. The

maximum particle size is 0,6m.

The mechanical parameters employed for the materials forming the body of the dam were

obtained from the analysis of the results of the triaxial tests performed of Materials 3B and 3L

in the National University of San Juan. The results are shown in Table 3-1




Table 3-1. Geotechnical parameters of the materials

φφφφ residualUnit weight

Plane strain Minimum

value

(KN/m3)

Foundation 40° 21,9

3B 44° 23,1

3L 43° 22,8

Drainage (3B´and D) 38° 18,6

2 43° 22,8

4 50° 19,63D 43° 22,8

Material

3.2.1 Geotechnical model

For analyzing the seismic response of the dam, it is necessary to develop a geotechnical

model in which all the materials forming the dam are properly characterized assigning

relevant geotechnical properties such as those explained in chapter 2.

The soil stress state is one of the most important factors controlling the dynamic behavior of soils. It has a remarkable influence on the maximum shear wave modulus and the damping

and modulus reduction curves. Taking into account that the dimensions of the dam are large

enough that the confinement pressure can reach values of 2300KPa in some points of the

foundation below the dam axis, it is necessary to split the materials according to the

confinement pressure acting on them. For doing that, the stress distribution illustrated in

Figure 2.5 was used for gathering the materials every 50KPa. It means that a derived

material, for every 50KPa of difference in the confinement pressure, was considered. In this

way, a model made up of 66 materials, in which the shear modulus varies as a function of the

confinement pressure, was developed, but on the other hand, the same damping curve and

modulus reduction curve was assigned for each global material forming the dam. The

geomechanical properties and dynamic curves employed are explained in the next section.

3.2.2 Geomechanical properties of the materials

Table 3-2 illustrates the properties of each one of the materials introduced.

(1)-Material: it is the name given to the material, depending on the confinement pressure. As

an example, Material 3B 750, corresponds to the material 3B subjected to a confinement

pressure of 750KPa

(2) Unit weight (KN/m3): it is the unit weight (KN/m3) of each material




(6)- φ triaxial: it is el value of the residual friction angle (º) for each material, according to the

triaxial tests performed on soils samples of the materials.

(7)- Poisson´s ratio: it is the Poisson ratio of each material, computed by means of the

following expression (Das, 1977):

−

−+=

oo

o

2545

253,01,0 t

φ υ (3.1)

where φt correspond to the column (6).

(8)- Vs (m/s): it is the shear wave velocity of each material computed by means of Eq (2.3)

3.2.3 Damping curve and modulus reduction curve

As mentioned previously, no laboratory tests for determining the damping and shear modulus

reduction curves (e.g cyclic triaxial test, bender element test, resonant column test) were

carried out on the materials making up the body of the dam. Therefore it was necessary to

define them based on the visual characterization and grain size distribution curve and looking

up in available international references.

The damping curve and modulus reduction curve of the materials are depicted in Figure 3.4 y

Figure 3.5.

G/Gmax vs Strain

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,0001 0,001 0,01 0,1 1 10

G / G m a x

Material 3B, 3L, 3D, Foundation

Material 4

Drainage

Material 2




Damping vs Strain

0

5

10

15

20

25

30

0,0001 0,001 0,01 0,1 1 10γ(%)

β ( % )

β ( % )

β ( % )

β ( % )

Material 3B, 3L, 3D, Foundation

Material 4

Drainage

Material 2

Figure 3.5. Damping curves. ββββ(%) vs. γ γγ γ (%)

According to the description provided in the previous section, materials 3B, 3L, 3D and

foundation can be considered as a material made up of sands and gravels and therefore, it is

possible to adopt the curves proposed by Seed et al. (1994

In the same way, since material 4 is made up of a rockfill acting as a protection on

downstream face, the dynamic curves proposed by Nose and Naba (1981) can be employed in

the analysis.

Correspondingly, based on the description of the drainage material, the dynamic curves for

crushed rock, proposed by Kokusho et al (1981) are used.

Finally, and taking into account that material 2 contains a higher fine content than the other

materials, the curves proposed by Rollins (1998) for gravels with fines, are employed.

3.3 One-dimensional modeling

3.3.1 EERA (Equivalent-linear Earthquake site Response Analyses of layered soil

deposits)

The EERA software, developed in the University of Southern Calinfornia, computes the

response in a horizontally layered soil rock system subjected to transient and vertical




Table 3-2. Geomechanical properties of the materials

(1) (2) (3) (4) (5) (6) (7) (8)

MaterialUnit wewight

γ γγ γ t (kN/m3)K2

σσσσ conf

(KN/m2)

Shear modulus

Gmáx (kPa)

φφφφ

triaxial

Poisson's

ratio

µµµµ

Vs (m/s)

Material 3B 50 23,1 100 50 154705 38 0,295 259

Material 3B 100 23,1 100 100 218786 38 0,295 308

Material 3B 150 23,1 100 150 267957 38 0,295 341

Material 3B 300 23,1 100 300 378948 38 0,295 405

Material 3B 450 23,1 100 450 464115 38 0,295 448

Material 3B 600 23,1 100 600 535914 38 0,295 482

Material 3B 750 23,1 100 750 599170 38 0,295 509

Material 3B 900 23,1 100 900 656358 38 0,295 533

Material 3B 1050 23,1 100 1050 708947 38 0,295 554

Material 3B 1200 23,1 100 1200 757897 38 0,295 573

Material 3B 1350 23,1 100 1350 803871 38 0,295 590

Material 3B 1500 23,1 100 1500 847354 38 0,295 606

Material 3L 50 22,8 100 50 154705 40 0,325 260

Material 3L 100 22,8 100 100 218786 40 0,325 310

Material 3L 150 22,8 100 150 267957 40 0,325 343

Material 3L 300 22,8 100 300 378948 40 0,325 408

Material 3L 450 22,8 100 450 464115 40 0,325 451

Material 3L 600 22,8 100 600 535914 40 0,325 485

Material 3L 750 22,8 100 750 599170 40 0,325 513

Material 3L 900 22,8 100 900 656358 40 0,325 537

Material 3L 1050 22,8 100 1050 708947 40 0,325 558

Material 3L 1200 22,8 100 1200 757897 40 0,325 577

Material 3L 1350 22,8 100 1350 803871 40 0,325 594Material 3L 1500 22,8 100 1500 847354 40 0,325 610

Foundation 50 22,0 80 50 123764 39 0,31 237

Foundation 100 22,0 80 100 175029 39 0,31 282

Foundation 150 22,0 80 150 214366 39 0,31 312

Foundation 300 22,0 80 300 303159 39 0,31 371

Foundation 450 22,0 80 450 371292 39 0,31 411

Foundation 600 22,0 80 600 428731 39 0,31 441

Foundation 750 22,0 80 750 479336 39 0,31 467

Foundation 900 22,0 80 900 525086 39 0,31 489

Foundation 1050 22,0 80 1050 567158 39 0,31 508

Foundation 1200 22,0 80 1200 606317 39 0,31 525

Foundation 1350 22,0 80 1350 643097 39 0,31 541

Foundation 1500 22,0 80 1500 677883 39 0,31 555Foundation 1650 22,0 80 1650 710970 39 0,31 568

Foundation 1800 22,0 80 1800 742584 39 0,31 581

Drainage 50 18,6 100 50 154705 37 0,28 288

Drainage 100 18,6 100 100 218786 37 0,28 343

Drainage 150 18,6 100 150 267957 37 0,28 380

Drainage 300 18,6 100 300 378948 37 0,28 451

Drainage 450 18,6 100 450 464115 37 0,28 500

Drainage 600 18,6 100 600 535914 37 0,28 537

Drainage 750 18,6 100 750 599170 37 0,28 568

Drainage 900 18,6 100 900 656358 37 0,28 594

Drainage 1050 18,6 100 1050 708947 37 0,28 617

Drainage 1200 18,6 100 1200 757897 37 0,28 638

Drainage 1350 18,6 100 1350 803871 37 0,28 657Drainage 1500 18,6 100 1500 847354 37 0,28 675

Material 3D 600 22,8 100 600 535914 39 0,31 485

Material 3D 750 22,8 100 750 599170 39 0,31 513

Material 3D 900 22,8 100 900 656358 39 0,31 537

Material 3D 1050 22,8 100 1050 708947 39 0,31 558

Material 3D 1200 22,8 100 1200 757897 39 0,31 577

Material 3D 1350 22,8 100 1350 803871 39 0,31 594

M t i l 2 50 22 8 100 50 154705 45 0 4 260




The following considerations are assumed:

The soil is considered to extend infinitely in the horizontal direction.

Each layer is completely defined by the shear modulus, damping, unit weight and

thickness. All these properties are assumed to be frequency-independent.

The response of the system is caused by vertical traveling shear waves from the

bedrock to the surface.

In this work, 5 sections within the body of the dam were defined. For each one of them, one-

dimensional seismic response analyses were carried out. These sections are shown in Figure

3.6 The results obtained were compared to the modeling performed using QUAD4M for

several heights from the bottom of the foundation material as displayed in Figure 3.7.

In the modeling performed, two approaches were considered:

• Employing the unit weight, shear wave velocity, damping curve and modulus reduction

curve according to sections 3.2.2 and 3.2.3 for the materials forming the soil column.

• Employing the unit weight, damping curve and modulus reduction curve according to

sections 3.2.2 and 3.2.3 but increasing the shear wave velocity of the materials according

to section 2.6.1 with the purpose of considering the difference between the fundamental

period of a triangular wedge and a one dimensional soil column. It means that the shear

wave velocities are increased in a proportion of

= ss V V

59,2

4__

. The factor

“4”corresponds to the analytical solution of the fundamental period of a one dimensional

soil column, whereas the factor “2,59”, corresponds to the analytical solution of the period

of vibration of a triangular wedge.




Figure 3.6. Sections within he body of the dam analyzed using EERA

175m

122m

76m50m

24m

Figure 3.7. Several heights within the body of the dam. Datum is considered at the base of the foundation

material.

For the previous cases, the results (acceleration records) obtained were adjusted based on the

tributary area of each column. The following expression was used for weighting theaccelerograms representatives of each one of the heights analyzed. (Eq 3.2)

Weighted accelerogram = Accelerogram section1

damtheofArea

tionareaTributary 1sec+


3 4 D i i f h fi i l d l



3.4 Description of the finite element model.

3.4.1 DSUN-GID Module (preprocessor)

It consists of an interface of the GID software (UPC-Barcelona Tech) which includes a

preprocessor and a postprocessor of the finite element mesh for utilizing in QUAD4M. The

module was developed by students of the Bachelor of Science in Civil Engineering of the

National University of Colombia (2002)

The DSUN module was used for designing the finite element mesh, employing quadrilateral

elements, taking into account the geometry properties of the different types of materials. In

addition, it allows introducing the material properties and input acceleration records forperforming the modeling in QUAD4M. Subsequently it provides a simple way of represent

graphically the results.

In this work, it is used the same finite element mesh employed by the designer. It is made up

of 509 elements and 575 nodes. (See Figure 3.8).

Figure 3.8. Finite element mesh

3.4.2 QUAD4M

(A Computer Program to Evaluate the Seismic Response of Soil Structures Using Finite

Element Procedures and Incorporating a Compliant Base). It is a program worldwide known

since 1973, developed by the University of California at Davis.

It is used for studying 2D problems in plane-strain state, by representing the continuum using

quadrilateral and triangular finite elements. The linear equivalent method is used for

evaluating the dynamic behavior of the soil. The latest version (1994) includes a transmittingbase so that the half-space beneath a mesh can be modeled and the need to assume a rigid

foundation can be eliminated.

3.5 Boundary conditions




Figure 3.9. Very rigid halfspace. Vs=25000m/s

• Half-space with shear wave velocity Vs=1500m/s which allows, incorporating

dampers at the base of the model as stated in section 2.7.2. Figure 3.10 illustrate the

model.

Figure 3.10. Half-space with shear wave velocity Vs=1500m/s

• Half-space with shear wave velocity Vs=2500m/s. This case is identical to the

previous one but increasing the shear wave velocity of the foundation rock, with the

purpose of estimating its influence on the results. Figure 3.11 displays the model.

Figure 3.11. Half-space with shear wave velocity Vs=2500m/s




Figure 3.12. Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick layer of

fictitious elements with shear wave velocity Vs=1500m/s

• Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick layer of

fictitious elements with shear wave velocity Vs=2500m/s. This case is identical to the

previous one but increasing the shear wave velocity of the layer of fictitious elements,

with the purpose of estimating its influence on the results. See Figure 3.13

Figure 3.13. Very rigid half-space with shear wave velocity Vs=25000m/s and a 20m-thick layer of

fictitious elements with shear wave velocity Vs=2500m/s.

• Half-space with shear wave velocity Vs=1500m/s and a 20m-thick layer of fictitious

elements with shear wave velocity Vs=1500m/s. With this model it is sought to study

the combined effect of the dampers, incorporated by the program and the fictitious

elements. In Figure 3.14 an illustration of the model is shown.


3 6 Results



3.6 Results

3.6.1 Maximum acceleration-depth cross sections

Table 2-1 displays a comparison between ADINA and QUAD4M, in terms of maximum

acceleration at the crest. It is worthy to remark that the ADINA software was used by the

designer of the project.

Due to the features of the models and programs, only the case corresponding to the very rigid

base can be compared

Table 3-3. Acceleration at the crest

Earthquake record ADINA

QUAD4M Rigid

base % difference

MCE 2,39 1,41 41%

Chi-Chi Taiwan 2,36 1,52 36%

Maximum Acceleration at the crest

According to Table 3-3 the variation between the programs is substantial and seems not to

depend on the input seismic motion.

In Figure 3.15 and Figure 3.16 the maximum acceleration-depth cross sections obtained by

modeling with QUAD4M. The different curves correspond to the cases explained before.

Maximum acceleration vs. depth

40

60

80

100

120

140

160

180

D e p t h (

m

)

Halfspace Vs=1500m/s


Rigid halfspace Vs=25000m/s

Rigid halfspace+fic. elements Vs=1500m/s



M i l ti d th




-20

0

20

40

60

80

100

120

140

160

180

0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40 1,60

Amax(g)

D e p t h ( m )



Rigid halfspace


RIgid halfspace+fic. elements Vs=2500m/s

Halfspace Vs=1500m/s+fic. elements Vs=1500m/s

Figure 3.16. Maximum acceleration vs. depth. Chi-Chi Taiwan earthquake, 1999

By looking Figure 3.15 and Figure 3.16 it can be concluded that considering a non rigid

halfspace reduce the maximum acceleration within the body of the dam. In the same way, it isconfirmed that the more rigid the halfspace, the closer the values of maximum acceleration at

the interface soil-rock, to the peak ground acceleration of the input motion, as was mentioned

in section 2.7.2

In addition, it can be seen that the effect of the fictitious elements is more evident at the crest

of the dam, where a reduction of approximately 28% in the maximum acceleration can be

observed, with respect to the rigid halfspace.

On the other hand, it can be observed that, the curves corresponding to the rigid halfspace and

halfspace with Vs=1500m/s and Vs=2500m/s are more scattered in the MCE than in the Chi-

Chi Taiwan earthquake.

In the same way, it is observed that the curves corresponding to the fictitious elements with

different shear wave velocities produce nearly the same maximum acceleration at the crest

3.6.2 Acceleration response spectraIn Figure 3.17 and Figure 3.18 the acceleration response spectra, for several depths within

the body of the dam, obtained by modeling with QUAD4M. The different curves correspond

to the cases explained before.





0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m

Rigid halfspace

0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m

Rigid halfspaceVs=1500m/s+fic. elements Vs=1500m/s

0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

24m

50m

76m

122m

175m

Fi 3 17 A l ti t f l d th ithi th b d f th d M i





0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a

( g )

Chichi

24m

50m

76m

122m

175m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi

24m

50m

76m

122m

175m

Rigid halfspace

0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi

24m

50m

76m

122m

175m

Halfspace Vs=1500m/s + fic. elements Vs=1500m/s

0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi

24m

50m

76m

122m

175m

Fi 3 18 A l ti t f l d th ithi th b d f th d Chi Chi T i


3.6.3 Comparison between the results of the programs



Figures 3.19 y 3.20 display a comparison in terms of maximum acceleration. It is performed

between the results obtained by using the programs QUAD4M and EERA. Due to thelimitations imposed by the EERA program, is not possible to consider the fictitious element

and therefore the comparison was performed for the cases corresponding to the rigid halfspace

and the halfspace with Vs=1500m/s. The comparison was performed in terms of maximum

acceleration-depth cross sections and acceleration response spectra for several depths within

the body of the dam. See Figure 3.21 to Figure 3.24

Maximum acceleration vs depth

0

20

4060

80

100

120

140

160

180

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0

Amax(g)

D e p t h ( m )

QUAD Halfspace Vs=1500m/s

QUAD Rigid halfspace

EERA Halfspace Vs=1500m/s

EERA Rigid Halfspace

Figure 3.19. Comparison between EERA and QUAD4M. Maximum acceleration vs. depth.MCE


20

40

60

80

100

120

140

160

180

D e p

t h ( m ) QUAD Halfspace Vs=1500m/s

QUAD Rigid base

EERA Halfspace Vs=1500m/s

EERA Rigid halfspace


From Figures 3.19 y 3.20, it can be concluded that the maximum accelerations obtained by



modeling with EERA and QUAD4M, are similar, except at the crest, where differences can

reach 28% for the MCE and 35% for the Chi-Chi, Taiwan earthquake. This result is in

agreement with the conclusion obtained by Prato and Delmastro (1988), where it is mentioned

that the one-dimensional approach results in an underestimation of the maximum acceleration

at the crest.

By analyzing the previous figures, it can be said that QUAD4M deamplifies the response for

periods below 1,0 and amplifies for longer periods, with respect to EERA. These results can

be explained by the difference in the damping scheme. In EERA, damping is frequency-

independent whereas in QUAD, there is a range of frequencies between the natural frequency,ω1, and the frequency ω2=n ω1 where the system is under-damped.

n is the closest odd integer greater than ωi/ ω1 (ωi is the predominant frequency of the input

earthquake motion)

This scheme allows the model to respond to the predominant frequencies of the input motion

without experiencing significant over-damping.

Since the earthquake records employed exhibit a significant high frequency content, the value

of n becomes larger and thus, the range of under-damping is wider and for that reason, the

results from QUAD are higher that the results from EERA for mostly of the periods.

Figures 3.21 to 3.24, display a comparison in terms of acceleration response spectra. It is

performed between the results obtained by using the programs QUAD4M and EERA

It is observed that the more the height of a given point within the body of the dam, the more

the dispersion of the acceleration response spectra. Such behavior is more evident for the Chi-

Chi Taiwan earthquake.


A l ti t Acceleration response spectra



53

Acceleration response spectra

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,54,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 0m

EERA 0m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,54,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 24m

EERA 24m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 50m

EERA 50m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 76m

EERA 76m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a

( g )

MCE EQ

Quad 122m

EERA 122m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a

( g )

MCE EQ

Quad 175m

EERA 175m

Figure 3.21. Comparison between EERA and QUAD4M. Acceleration response spectra. Halfspace with Vs=1500m/s. Maximum Credible Earthquake


Acceleration response spectra Acceleration response spectra



54


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,54,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 0m

EERA 0m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 24m

EERA 24m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 50m

EERA 50m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 76m

EERA 76m


0,0

0,5

1,0

1,52,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 122m

EERA 122m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

MCE EQ

Quad 175m

EERA 175m

Figure 3.22. Comparison between EERA and QUAD4M. Acceleration response spectra. Rigid halfspace. Maximum Credible Earthquake





55

p p

0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQQuad 0m

EERA 0m

p p

0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQQuad 24m

EERA 24m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQ

Quad 50m

EERA 50m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQ

Quad 76m

EERA 76m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S

a ( g )

Chichi EQ

Quad 122m

EERA 122m


0,0

0,5

1,0

1,52,0

2,5

3,0

3,5

4,0

4,5

0,0 1,0 2,0 3,0 4,0 5,0T(s)

S

a ( g )

Chichi EQ

Quad 175m

EERA 175m

Figure 3.23. Comparison between EERA and QUAD4M. Acceleration response spectra. Halfspace with Vs=1500m/s. Chi-Chi Taiwan Earthquake, 1999





56

0,0

0,5

1,0

1,5

2,02,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQ

Quad 0m

EERA 0m

p p

0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

T(s)

S a ( g )

Chichi EQ

Quad 24m

EERA 24m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQ

Quad 50m

EERA 50m


0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S a ( g )

Chichi EQ

Quad 76m

EERA 76m


0,0

0,5

1,0

1,52,0

2,5

3,0

3,5

4,0

4,5

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

S

a ( g )

Chichi EQ

Quad 122m

EERA 122m


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

T(s)

S

a ( g )

Chichi EQ

Quad 175m

EERA 175m

Figure 3.24. Comparison between EERA and QUAD4M. Acceleration response spectra. Rigid halfspace. Chi-Chi Taiwan earthquake,1999.


3.6.4 Transfer functions

In Figure 3 25 and Figure 3 26 the transfer functions corresponding to the different cases are



In Figure 3.25 and Figure 3.26 the transfer functions corresponding to the different cases are

shown. Such curves are obtained by dividing spectral accelerations at the crest by spectral

acceleration of the input motion.

It is evident the importance of incorporating the fictitious elements in the models, since

amplifications at the crest of the dam are reduced 3 times for the Chi-Chi Taiwan earthquake.

In both figures, it can be observed clearly two groups of curves that make clear the difference

in the results when the fictitious elements are incorporated in the model.

It can be seen that, the results for the different models are quite similar for periods below 1,5s

On the other hand, for longer periods; the two groups mentioned above can be distinguished.

Transfer functions

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0T(s)

Sa output

Sa input

HalfspaceVs=1500m/s

Halfspace Vs=1500+fic. elements Vs=1500m/s

Rigid halfspace


Figure 3.25. Transfer functions (Sa-crest/Sa input). Maximum Credible Earthquake

Trasnfer functions

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

Sa output

Sa input

Halfspace vs=1500m/s

HalfspaceVs=1500m/s+fic. elements Vs=1500m/s

Rigid halfspace



Dam crest settlement (MCE)

300

settlement(cm)



025

5075

100125150

175200225250275300

1 2 3 4wedge

Rigid halfspace ADINA

Figure 3.27. Dam crest settlement for wedges at different depths. Comparison between ADINA and rigid

halfspace (QUAD4M). Maximum Credible Earthquake

It is observed that the results of the settlements obtained by the designer, using the ADINA

software, are smaller than those obtained by modeling in QUAD4M, contradicting theprevious results regarding the maximum acceleration at the crest. This can be explained by

taking into account that the linear equivalent method, employed by QUAD4M, filters

effectively high frequencies but these frequencies does not contribute considerably to the

settlement, when Newmark method is used.

Figure 3.28 and Figure 3.29 display the settlement of the different wedges analyzed for the

cases explained before.

In the same way, a one-dimensional analysis was performed with the purpose of comparing

the results. See Figure 3.30 and Figure 3.31

Dam crest settlements

125

150

175

200

225

250

275

300settlement(cm)


Rigid halfspace



Halfspace Vs=1500m/s+fic. elements Vs=1500m/s


Dam crest settlement

settlement(cm)



0

25

50

75

100

125

150175

200

225

250

275

300

325

350375

1 2 3 4wedge

settlement(cm)Halfspace Vs=1500m/s

Rigid halfspaceHalfspace Vs=15000m/s+fic. elements Vs=1500m/sRigid halfspace+fic. elements Vs=2500m/sHalfspace Vs=1500m/s+fic. elements Vs=1500m/s

Figure 3.29. Dam crest settlement for wedges at different depths computed using QUAD4M and

Newmarkk method. Chichi-Taiwan earthquake

Dam crest settlement

0

5

10

15

20

1 2 3 4

settlementcm)

Halfspace Vs=1500m/sRigid halfspace


Dam crest settlement20

settlememt(cm)



0

5

10

15

20

1 2 3 4wedge


RIgid halfspace

Figure 3.31. Dam crest settlement for wedges at different depths computed using EERA and the

Newmark method. Chichi-Taiwan earthquake

By analyzing Figure 3.28 and Figure 3.29, it can be observed the important effect of

incorporating the fictitious elements since the settlement can be reduced up to 4 times for the

Maximum Credible Earthquake and 6 times for the Chi-Chi Taiwan earthquake in the critical

wedge which is the wedge with a ratio y/h=0,25. This situation represents an important

reduction in the response and the possibility of saving costs in the construction, since the

freeboard, which is supposed to be at least 3 times the computed maximum settlement, can be

diminished.

In the same way, it can be concluded that the stiffness introduced for the fictitious elements

does not have a significant influence on the settlement at the crest since the values are

practically equal when introducing values of 1500m/s and 2500m/s for the shear wave

velocity of the fictitious layer of element.

On the other hand by observing Figure 3.30 and Figure 3.31 it can be seen that EERA

underestimates the settlement as well, which is in agreement with the underestimation in the

acceleration produced by the program. Nevertheless a significant reduction (up to 15 times forthe MCE) of the settlement can be achieved by considering a non rigid halfspace. However,

the Newmark method is strongly dependent on the yield acceleration and the number of times

the acceleration record crosses this value, theferore underestimating the accelerations in a

percentage close to 28% according to Figure 3 19 and Figure 3 20 can lead to an

Chapter 4. Final remarks and conclusions



4. FINAL REMARKS AND CONCLUSIONS

Final comments:

The linear equivalent model provides a simple and efficient alternative for evaluating

the dynamic response of soils. Nevertheless, it must be taken into account the

important limitations it has, such as the impossibility of modeling the failure in the

materials. This fact may lead to significant errors specially when dealing with strong

motion records with high values of peak ground acceleration, like the records

employed in this work.

One of the most important advantages provided by the linear equivalent model is the

easiness in the characterization of the parameters defining it. Nevertheless,

determining the damping curve based on the hysteretic cycles may be inaccurate when

the material exhibits a dynamic behavior where the hysteretic cycles are not accurately

defined. When dealing with coarse materials, performing dynamic tests (e.g., triaxial

tests, bender element, resonant column, etc) may become cumbersome and costly.

Therefore, it is necessary to select the dynamic curves based on general data and

previous experience

When characterizing dynamic curves of the materials (damping and modules reduction

curves) it is necessary to carry out tests able of covering all the range of shear strain

that may experience the material when is subjected to the earthquake loading. This is

an indispensable condition for assuring the stability in the numerical routine

developed by the finite element software

Both earthquake records exhibit peaks up to 4g, at low periods (below 0,5s). For

longer periods there are not significant peaks but the spectral acceleration is, in

average larger than 1,0g. In general, the spectra do not exhibit accelerations below 0,5

g in the range of interest for this sort of structures.


It is important to take into account the natural period of the input seismic records for

designing the finite element mesh in such a way that the vertical dimension of the



elements is small enough, assuring that the traveling wave excites properly the finite

element.

When incorporating boundary conditions in the models, it is necessary to have a

comprehensive understanding of the movement of the structure to be analyzed. Since

it is sought to study the horizontal motion, for the dynamic analysis, a boundary

condition equivalent to a roller support was incorporated.

In the same way, for studying the static condition, at the base and sides of the model

all the displacements are restrained; whereas no other restraint is imposed for other

nodes in the model.

It is necessary to obtain precise information regarding the properties of the rock

making up the halfspace since the results depend on the value of the shear wave

velocity introduced.

Even though the fictitious elements provide an approach for considering the stiffness

provided by the presence of the dam, the model utilized is not the most accurate

representation of the dynamic behavior of such a problem, since the elements

representing springs are located in series configuration along with the dampers

automatically incorporated by the software. Theoretically, the springs and dampers

should be arranged in a parallel configuration but to accomplish such a configuration a

modification in the code would have been necessary, which is beyond the scope of the

work.

Despite the fact of being simple and easy, the Newmark method has an important

limitation since it considers that the slope deforms only when the acceleration induced

by the earthquake, acting on the wedge, exceeds the yield acceleration. Such a

behavior is valid for slopes that deform as a rigid block. In this case, this is an unlikely

hypothesis since due to the lack of cohesion, the materials tends to deform rather than

doing it as a rigid block.

Due to the fact that the wedges analyzed were made up of two different materials, itwas necessary to derive an expression for computing the yield acceleration.

Nonetheless, by dividing the wedge in vertical stripes, a satisfactory approach is

accomplished since reasonable results were achieved and the equilibrium at the slip

plane is fulfilled


Despite the fact of being a cumbersome procedure, dividing the materials depending

on the confinement pressure is necessary for achieving an adequate model since soil

d i b h i i t l d d t th fi t d th i



dynamic behavior is strongly dependent on the confinement pressure and there is a

wide variation of it within the body of the dam.

Despite the fact of being simple and fast, modeling by means of one-dimensional

sections lead to important discrepancies with respect to the two-dimensional

modeling, especially at depths near the dam crest. This is quite relevant since the

freeboard is computed, among others, based on the maximum settlement which can be

underestimated.

When performing parametric studies, it can be quantified the influence of differentfactors on the modeling. In this case, it was verified that the boundary conditions have

strong influence on the results which can lead to either important savings or serious

miscalculations. Despite the fact of being an approach, the procedure followed

suggests that accelerations and settlement can be considerably reduced due to the

presence of the structure.

The differences, in terms of accelerations, between ADINA and QUAD4M, are

considerable, and do not depend on the seismic input motion, since the percentages are

similar for the records used.

The fact of considering the halfspace with values of shear wave velocities, typical for

rocks, reduces the maximum acceleration within the body of the dam. It can be

verified that the more rigid the halfspace, the closest to the peak ground acceleration

will be the maximum acceleration at the rock-soil interface.

It can be observed that the effect of the fictitious elements on the seismic response

becomes more evident at the dam crest where the reduction can reach 28% with

respect to the rigid halfspace.

The results obtained by modeling with the programs used are similar except at the dam

crest where the difference reaches 35%, which agrees with the results displayed by

Prato and Delmastro (1988) in terms of the underestimation at the dam crest.

By comparing EERA and QUAD4M it can be said that QUAD4M deamplifies the

response for periods below 1,0 and amplifies for longer periods, with respect to

EERA. These results can be explained by the difference in the damping scheme. In

EERA, damping is frequency-independent whereas in QUAD, there is a range of


Incorporating the fictitious elements in the modeling reduce even 3 times (for the Chi-

Chi Taiwan earthquake) the amplification at the dam crest. By reviewing the transfer

functions two groups of curves can be clearly identified distinguishing the presence of



functions, two groups of curves can be clearly identified distinguishing the presence of

the fictitious elements. It means that the cases where the largest amplification isachieved are those where the fictitious elements are not incorporated.

The results of the settlements obtained by the designer using ADINA, are below the

results by modeling with QUAD. This behavior is opposite to the results in terms of

maximum acceleration presented where the values given by QUAD4M where even

41% smaller than those given by ADINA. This can be explained by taking into

account that the linear equivalent method, employed by QUAD4M, cuts effectively

high frequencies but these frequencies does not contribute considerably to the

settlement, when Newmark method is used.

Conclusions

As result of the foregoing study the following conclusions may be drawn in relation to the

proposed objectives set for this thesis

- For the case of rigid foundation:

The seismic analysis of the Los Caracoles CFRD Dam performed here based on the Linear

Equivalent Method (LEM) by means of the QUAD4M Program leads to accelerations up to

41% lower at the crest of the dam as compared with those obtained by elastoplastic analysis

with ADINA and Mohr-Coulomb (MC) constitutive model carried out at design stage. This

conclusion also applies for the 1-D analyses performed in the present study where a reductionup to 35% was obtained with respect to the results from QUAD4M. However, permanent

displacement of the crest obtained with the Nerwmark Method by the LEM and by the MC

elastoplastic analysis are in closer agreement, leading to the conclusion that the lower

accelerations given by the LEM are due to a filtering effect of higher frequencies of response

that are kept in the MC model that do not have appreciable influence in the accumulated

permanent displacement due to the earthquake.

-For the case of flexible foundation rock :

The effect of the flexbility of the foundation rock and radiation damping in the dynamic


concluded that more detailed analysis of the foundation rock characteristics at design stage

may appreciably reduce the amount of crest settlement to be expected as a consequence of a

given major earthquake in a CFRD dam In the same way it can be concluded that further



given major earthquake in a CFRD dam. In the same way, it can be concluded that further

research is needed in developing programs able of analyzing this sort of models in such a waythat the dampers and springs would be arranged in parallel instead of series configuration.

.

References



5. REFERENCES

• American Society of Civil Engineers [1986]. “Seismic Analysis of Safety-Related Nuclear

Structures and Commentary. ASCE Standard 4-86”

• Belyakov [1988]. “Three-dimensional behavior of an earth dam at wide site”. Power

Technology and Engineering. 22 (12), 718-725

• Das B.M [1997] Advanced soil mechanics, Taylor and Francis Inc, USA

• Day, R. W [2002]. Geotechnical Earthquake Engineering Handbook, Mc Graw Hill, USA

• Hudson, Idriss, Beikae [1994]. “QUAD4M: A computer program to evaluate the seismicresponse of soil structures using finite procedures and incorporating a compliant base”,

University of California.

• Instituto de Investigaciones Antisísmicas (IDIA) – Ing. Aldo Bruschi [2003]” Informe,

Amenaza Sísmica en las Ubicaciones en el Río San Juan de las Presas Los Caracoles y

Punta Negra – Actualización de su Valoración”.

• International Committee on Large Dams (ICOLD) [2005] “Concrete Face Rockfill Dams.

Concepts for Design and Construction”.• Kramer, S. L. [1996] Geotechnical Earthquake Engineering, Prentice Hall, USA

• Makdisi, Falz and Seed H. [1977] “Simplified Procedure for Estimating Dam and

Embankment Earthquake Induced Deformations”. ASCE, Journal of Geotechnical

Engineering., Vol 104. No.GT7

• Nose y Naba, [1983] “Curvas Dinámicas para enrocados” Jornadas Geotécnicas-Presas en

Colombia. Bogotá, Colombia

• Palacios C., Vargas C., [2002], “Diseño e Implementación de una Interfase GID para

problemas geotécnicos modelados mediante elementos finitos, Módulos DSUN y

PLANUN”, Tesis pregrado, Facultad de Ingeniería, Universidad Nacional de Colombia.

• Prato C A and Delmastro E [1987] “1-D seismic analysis of embankment dams”

References

• Seed H.. Idriss I. et al. [1994] “Moduli and Damping Factors for Dynamic Analises of

Cohesionless Soils”. Earthquake Engineering Research Center. University of California.

Report No UCB/EERC-84/14. Berkeley.



p y

• Sherard, J. L., and Cooke, J.B [1987]. “Concrete-face rockfill dam: I. Assessment and II.

Design”. Journal of Geotechnical Engineering (113)10 1096-1132

• Techint-Panedile [2005] “Aprovechamiento Hidroeléctrico “Los Caracoles”- Criterios de

Diseño de las Obras Bajo Acciones Sísmicas”. Memoria de cálculo, Gobierno de San

Juan E.P.S.E

• Techint-Panedile [2005] “Aprovechamiento Hidroeléctrico “Los Caracoles”- Rediseño de

la sección de la presa-Análisis dinámico y post-sismico”. Memoria de cálculo, Gobierno

de San Juan E.P.S.E

• Uddin, N., Gazetas, G. [1995], “Dynamic response of concrete-faced rockfill dams to

strong seismic excitation”, Journal of Geotechnical Engineering., Vol 121. No.2.

• United States Army Corps of Engineers (USACE) [1995]. “ Earthquake Design and

Evaluation for Civil Works Projects. ER-1110-2-1806”

MSc Dissertation 2009 Influence Of The Boundary Conditions On The Seismic Response Ivan Ordonez



MSc Dissertation 2009 Influence Of The Boundary Conditions On The Seismic Response Ivan Ordonez

Predictions Of A Rockfill Dam By Finite Element Method

Dissertation2009 Ordonez[1]

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