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
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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………… … ……
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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
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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.
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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.
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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
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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
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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
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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
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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
Figure 3.22. Comparison between EERA and QUAD4M. Acceleration response spectra.
Rigid halfspace. Maximum Credible Earthquake.............................................................54
Figure 3.23. Comparison between EERA and QUAD4M. Acceleration response spectra.
Halfspace with Vs=1500m/s. Chi-Chi Taiwan Earthquake, 1999....................................55
Figure 3.24. Comparison between EERA and QUAD4M. Acceleration response spectra.
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
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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
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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
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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.
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Chapter 1. General information
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
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Chapter 1. General information
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.
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Chapter 1. General information
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.
• Response spectra at several depths.
• 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.
• Response spectra at several depths.
• 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
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Chapter 1. General information
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
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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
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Chapter 2. Method of Analysis
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
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Chapter 2. Method of Analysis
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.
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Chapter 2. Method of Analysis
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.
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Chapter 2. Method of Analysis
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.
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Chapter 2. Method of Analysis
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.
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Chapter 2. Method of Analysis
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
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Chapter 2. Method of Analysis
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.
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Chapter 2. Method of Analysis
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
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Chapter 2. Method of Analysis
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%
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Chapter 2. Method of Analysis
-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
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Chapter 2. Method of Analysis
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
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Chapter 2. Method of Analysis
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
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Chapter 2. Method of Analysis
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
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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
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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
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Chapter 2. Method of Analysis
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
Chapter 2. Method of Analysis
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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.
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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
Chapter 2. Method of Analysis
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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
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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;
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(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
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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.
Chapter 2. Method of Analysis
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• 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
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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 .
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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)
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(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)
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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
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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)
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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%.
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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
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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
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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
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(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
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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
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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
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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.
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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+
Chapter 3. Case study
3 4 D i i f h fi i l d l
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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
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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
Chapter 3. Case study
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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.
Chapter 3. Case study
3 6 Results
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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
Halfspace Vs=2500m/s
Rigid halfspace Vs=25000m/s
Rigid halfspace+fic. elements Vs=1500m/s
Rigid halfspace+fic. elements Vs=2500m/s
Chapter 3. Case study
M i l ti d th
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Maximum acceleration vs. depth
-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 )
Halfspace Vs=1500m/s
Halfspace Vs=2500m/s
Rigid halfspace
Rigid halfspace+fic. elements Vs=1500m/s
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.
Chapter 3. Case study
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Halfspace 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
Rigid halfspace+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
Halfspace Vs=2500m/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
Rigid halfspace+fic. elements Vs=2500m/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
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
Chapter 3. Case study
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Halfspace 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 )
Chichi24m
50m
76m
122m
175m
Rigid halfspace+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 )
Chichi24m
50m
76m
122m
175m
Halfspace Vs=2500m/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
Rigid halfspace+fic. elements Vs=2500m/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
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
Chapter 3. Case study
3.6.3 Comparison between the results of the programs
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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
Maximum acceleration vs. depth
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
Chapter 3. Case study
From Figures 3.19 y 3.20, it can be concluded that the maximum accelerations obtained by
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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.
Chapter 3. Case study
A l ti t Acceleration response spectra
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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
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 24m
EERA 24m
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Chapter 3. Case study
Acceleration response spectra Acceleration response spectra
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54
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Chapter 3. Case study
Acceleration response spectra Acceleration response spectra
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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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Chapter 3. Case study
Acceleration response spectra Acceleration response spectra
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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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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
Acceleration response spectra
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.
Chapter 3. Case study
3.6.4 Transfer functions
In Figure 3 25 and Figure 3 26 the transfer functions corresponding to the different cases are
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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
Rigid halfspace+fic. elements Vs=1500m/s
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
Rigid halfspace+fic. elements Vs=1500m/s
Chapter 3. Case study
Dam crest settlement (MCE)
300
settlement(cm)
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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)
Halfspace Vs=1500m/s
Rigid halfspace
Rigid halfspace+fic. elements Vs=1500m/s
Rigid halfspace+fic. elements Vs=2500m/s
Halfspace Vs=1500m/s+fic. elements Vs=1500m/s
Chapter 3. Case study
Dam crest settlement
settlement(cm)
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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
Chapter 3. Case study
Dam crest settlement20
settlememt(cm)
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0
5
10
15
20
1 2 3 4wedge
Halfspace Vs=1500m/s
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
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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.
Chapter 4. Final remarks and conclusions
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
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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
Chapter 4. Final remarks and conclusions
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
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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
Chapter 4. Final remarks and conclusions
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
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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
Chapter 4. Final remarks and conclusions
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
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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
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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.
7/31/2019 Dissertation2009 Ordonez[1]
http://slidepdf.com/reader/full/dissertation2009-ordonez1 79/80
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