topographic amplification of earthquakes in puerto rico
TRANSCRIPT
i
Topographic Amplification of Earthquakes in Puerto Rico and its Effects in Residential Construction
University of Puerto Rico at Mayagüez
Department of Civil Engineering, P.O. Box 9041, Mayagüez PR 00681 Tel (787) 265-3815, Fax (787) 833-8260
Email: [email protected] , [email protected]
Final Technical Report
VOLUME I
Numerical Study of the Amplification of the Seismic Ground Acceleration Due to Local Topography
FEMA-1247-DR-PR HMGP PR-0060B
Submitted to:
Lic. Melba Acosta
Governor’s Authorized Representative Commonwealth of Puerto Rico
Hazard Mitigation Office, P.O. BOX 9023434 San Juan, Puerto Rico 00902-3434
Mr. José A. Bravo
Disaster Recovery Manager Federal Emergency Management Agency
P.O. Box 70105 San Juan, Puerto Rico 00936-8105
By
Luis E. Suárez, Principal Investigator
Ricardo R. López, Co-Principal Investigator María Elena Arroyo Caraballo, Graduate Student
Drianfel Vásquez Torres, Graduate Student
Submitted: June 30,2003
i
Topographic Amplification of Earthquakes in Puerto Rico and its Effects in Residential Construction
EXECUTIVE SUMMARY
The objective of this project was to study the amplification of the earthquake
waves caused by topography, and to evaluate what effects should be expected on construction located in areas prone to suffer this phenomenon.
The research was divided in to two parts. The results presented in Volume I are concerned with the amplification of the seismic waves. Volume II deals with the effects on the structures, in particular residential constructions. It was found that most reinforced concrete houses built on slender columns are vulnerable to an earthquake amplified by the topography. A rehabilitation technique based on the addition of reinforced concrete walls is proposed in the recommendations in Volume II.
The research was carried out from November 2000 to May 2003. This investigation
was performed by:
Luis E. Suárez, Principal Investigator Ricardo R. López, Co-Principal Investigator Drianfel Vázquez Torres, Graduate Student María Elena Arroyo Caraballo, Graduate Student
The two volumes include the following information:
I. Volume I: Numerical Study of The Amplification of The Seismic Ground Acceleration Due to Local Topography. This investigation presents a study of the effects of local topography on the ground acceleration produced by earthquakes. The graduate student Maria Elena Arroyo Caraballo developed a Master of Science in Civil engineering thesis based on the subject of the first phase of the project.
II. Volume II: Seismic Behavior and Retrofitting of Hillside and Hilly Terrain
R/C Houses Raised on Gravity Columns. This investigation presents a study, by means of numerical simulation, of the seismic behavior of typical residences located at hills or hillsides and raised on gravity columns. The study takes into account the topographic amplification of the ground motion due to the location of the residences. The attention is focused on the seismic evaluation of the residences with typical geometric parameters obtained from a field survey carried out across Puerto Rico. Non-linear static pushovers and non-linear dynamic transient analyses are performed for the seismic vulnerability evaluation. The results of the analyses are used to select a seismic rehabilitation technique. As part of this investigation, the graduate student Drianfel Vázquez Torres submitted a dissertation in partial fulfillment of the requirements for the degree of Ph. D. in Civil engineering.
i
Abstract
This investigation presents a numerical study of the effects of local topography on the
ground acceleration induced by earthquakes. The attention is focused on the
amplification of the peak acceleration at the surface of the topographic irregularities.
Two-dimensional embankments and hills are studied. Four soil profiles defined in the
UBC-97 are used to define the material properties. The study is performed with the finite
element program QUAD4M. The seismic input is the acceleration time history of two
historic earthquakes with different characteristics. The nonlinear behavior of the soil is
taken into account with the Equivalent Linear Method. Two formulas that provide
amplification factors as a function of the geometry of the escarpment/hill at any point
along their surfaces are derived. The amplification factors found range from 1.00 to 2.35.
The case of an actual group of hills in Puerto Rico is examined, using real parameters and
a site-specific artificial earthquake.
ii
Compendio
En esta investigación se presenta un estudio numérico abarcador sobre los efectos
de la topografía en la aceleración del suelo causada por terremotos. La investigación se
concentra en estimar la amplificación de la aceleración pico en la superficie de
irregularidades topográficas. En particular, se estudiaron taludes y montañas, los cuales
se simularon mediante modelos de elementos finitos en dos dimensiones. Para definir las
propiedades de los materiales, se utilizaron cuatro perfiles de suelo definidos en el código
UBC-97. Para el análisis se utilizó el programa de elementos finitos QUAD4M. Los
datos de aceleración que se utilizaron fueron los historiales en el tiempo de dos
terremotos pasados. Se tuvo en cuenta el comportamiento no lineal del suelo mediante el
Método Lineal Equivalente. Como parte de la investigación se derivaron dos fórmulas
que proveen los factores de amplificación como función de la geometría de los taludes o
montañas en cualquier punto de su superficie. Se encontró que estos factores de
amplificación fluctúan entre 1.00 y 2.35. También se estudió un caso real de un grupo de
montañas en Puerto Rico, para el cual se utilizaron propiedades de los suelos específicos
del área, la geometría real de una sección transversal, y un terremoto artificial
especialmente generado para esta zona.
iii
Contents
List of Figures vi
List of Tables ix
I Introduction 1
1.1 Problem description . . . . . . 1
1.2 Previous works . . . . . . . 2
1.3 Scope of the thesis . . . . . . 7
1.4 General organization of the thesis . . . . 8
II Description of the General Methodology 11
2.1 Introduction . . . . . . . 11
2.2 Finite element concepts . . . . . 11
2.3 Description of the computer program QUAD4M . . 13
2.4 The pre and post processor Q4MESH . . . 17
2.5 The Equivalent Linear Model . . . . . 17
2.6 Finite element models . . . . . . 23
2.7 Seismic excitation . . . . . . 27
2.8 Boundary conditions . . . . . . 29
2.9 Guidelines for soil type categorization . . . 33
III Amplification of Seismic Motion Due to Escarpments 38
3.1 Introduction . . . . . . . 38
3.2 Slope stability analysis . . . . . . 48
3.3 General input data . . . . . . 43
iv
3.4 Selection of total height . . . . . 44
3.5 Escarpment amplification results . . . . 46
3.6 A general equation for the amplification factor . . 54
3.7 Nonlinear behavior of soils . . . . . 60
3.8 Summary . . . . . . . . 63
IV Amplification of Seismic Motion Due to Hills 65
4.1 Introduction . . . . . . . 65
4.2 General considerations . . . . . . 66
4.3 Ridge amplification results . . . . . 69
4.4 A general equation for the amplification factors . . 75
4.5 Frequency analysis . . . . . . 78
4.6 Summary . . . . . . . . 81
V A Case Study in Puerto Rico 82
5.1 Introduction . . . . . . . 82
5.2 Geographic conditions of Puerto Rico . . . 83
5.3 Site location . . . . . . . 85
5.4 Study of soils at the site . . . . . 88
5.5 Seismic excitation . . . . . . 91
5.6 The finite element model . . . . . 94
5.7 Results of the numerical simulation . . . . 95
5.8 Comparison with the proposed simplified methodology . 100
5.9 Summary and final comments. . . . . 100
v
VI Conclusions and Recommendations 102
6.1 Summary and conclusions . . . . . 102
6.2 Suggestions for further studies . . . . 105
References 108
vi
List of Figures
2.1 Linear and nonlinear shear-strain relationship . . . 18
2.2 Secant shear modulus . . . . . . 19
2.3 Shear modulus (a) and Damping ratio (b) curves . . 21
2.4 Stress Strain curve . . . . . . . 23
2.5 Dimensions of the escarpment model . . . . 24
2.6 Example of escarpment finite element mesh . . . 25
2.7 Dimensions of the ridge model . . . . . 26
2.8 Example of a finite element mesh of a ridge . . . 27
2.9 Acceleration time histories and Fourier Spectra for (a) El Centro
and (b) San Salvador earthquakes . . . . 29
2.10 Comparison of results using fixed and transmitting boundaries and extending the
mesh for the escarpment subjected to the El Centro
earthquake . . . . . . . . 31
2.11 Comparison of results (a) using transmitting boundaries (b) without
Transmitting boundaries at the sides and (c) extending the mesh,
for the ridge subjected to the San Salvador earthquake . 32
3.1 An example of an XSTABL result . . . . . 41
3.2 Escarpment heights . . . . . . . 45
3.3 Surface nodal points . . . . . . 47
3.4 Peak acceleration for a 15° slope escarpment . . . 48
vii
3.5 Amplification of the peak acceleration in a soil deposit without
irregularities . . . . . . . . 49
3.6 Comparison of the peak accelerations for a 30° escarpment
caused by both earthquakes . . . . . 50
3.7 Comparison of the peak accelerations for a 40° escarpment
caused by both earthquakes . . . . . 51
3.8 Peak acceleration for a 50° slope escarpment . . . 52
3.9 Peak acceleration for a 65° escarpment . . . . 53
3.10 Identification of parameters . . . . . 56
3.11 Amplification factor as a function of the escarpment’s height ratio
for a 40° slope . . . . . . . 57
3.12 The original line and the proposed equation for the 40° case . 60
3.13 Stress-Strain curve for a typical finite element . . . 62
4.1 Parameters for the equation of the parabola . . . 67
4.2 Nodes at the surface of the model of the hill . . . 70
4.3 Peak ground acceleration for a hill with n = 38 ft . . 70
4.4 Peak ground acceleration for a hill with n = 75 ft . . 71
4.5 Peak acceleration at the top of a soil deposit without irregularities 72
4.6 Peak ground acceleration for a hill with n = 115 ft . . 73
4.7 Peak ground acceleration for a hill with n = 225 ft . . 74
4.8 Parameters identification . . . . . . 75
viii
4.9 An example of a cubic trendline for n/m = 0.095 . . . 77
4.10 The cubic trendline and the general equation for n/m = 0.095 . 78
4.11 Comparison of ridge amplification subjected to El Centro
earthquake . . . . . . . . 79
5.1 Topographic view of Puerto Rico . . . . . 84
5.2 Residences located on the hill . . . . . 85
5.3 Map of the municipalities of Puerto Rico showing the location
of Guánica . . . . . . . . 85
5.4 View of the hill selected for study . . . . . 86
5.5 Topographic map of the case studied . . . . 87
5.6 View from road PR 116 of the Caño Hill . . . . 88
5.7 Soil map of the Valle de Lajas Area . . . . 90
5.8 Seismic activity in America and Caribbean . . . 98
5.9 Acceleration time history (a) and Fourier Spectrum (b) of the
artificial earthquake . . . . . . . 93-94
5.10 Finite element mesh for the Caño Hill . . . . 95
5.11 Coordinates for the Caño Hill mesh . . . . 96
5.12 Acceleration results for Caño Hill . . . . . 97
5.13 Average acceleration at different elevation . . . 98
5.14 Stress distribution . . . . . . . 99
5.15 Acceleration distribution . . . . . . 99
ix
List of Tables
2.1 Important Characteristics of the Earthquakes Selected . . 28
2.2 Soil Profile Types . . . . . . . 36
3.1 Total Height According to Material . . . . 46
3.2 Amplification Factors for an Escarpment:
El Centro earthquake (0.1g) . . . . . 54
3.3 Amplification Factors for an Escarpment:
San Salvador earthquake (0.1g) . . . . . 54
3.4 Amplification Factors for Different Escarpment’s Heights:
El Centro earthquake (0.1g) . . . . . 56
3.5 Most Critical Amplification Factors Considering Both Earthquakes 64
4.1 Maximum Amplification Factors for the Hill . . . 74
4.2 Amplification Factors for Hills of Different Heights . . 76
4.3 Most Severe Seismic Motions for the Analysis . . . 80
4.4 Summary of Maximum Amplification Factors for Different Hills 89
5.1 Materials Properties for the Model of the Caño Hill . . 93
5.2 Characteristics of an Artificial Earthquake for Puerto Rico . 98
5.3 Amplification and Deamplification Factors . . .
1
Introduction
1.1 Problem description
The study and analysis of damage during past earthquakes has shown that the
surface topography surrounding the site of the structures can considerably amplify the
ground motions. Although it is not currently considered, local site effects due to
topographic irregularities should play an important role in earthquake-resistant design.
There was evidence of this phenomenon in the 1976 Friuli and 1980 Irpino earthquakes
in Italy, and in the Chile earthquake of 1985.
Although this phenomenon has been known for several years and despite of its
importance for sites with pronounced surface irregularities, this effect is not considered in
the seismic codes. In particular, it has not been included in the US seismic codes and
thus in the design codes adopted for Puerto Rico. Probably one of the reasons why the
codes such as the Uniform Building Code disregard this phenomenon, is that in the US
mainland, with a few exceptions, the regions with conditions prone to topographic
amplifications are scarcely populated. Nevertheless, this phenomenon could be very
important in seismic prone zones that have this type of surface irregularities such as
Puerto Rico. The geography of the Island, along with the social and economic conditions
that affect the population distribution, makes many regions prone to topographic
amplification. The problem is aggravated by the many residential structures located on
hills and slopes that are constructed with weak first stories consisting of slender columns.
2
In addition, the amplification of the seismic motion can have potentially serious
consequences in terrains that are sensitive to landslides.
The topographic conditions can influence all of the important characteristics of a
strong ground motion, such as amplitude, frequency and duration. A few studies
concluded that, in addition to the magnitude amplification, the irregular topography could
cause a large increase in the duration of the motion. Although it has not received much
attention, this effect is worth of further studies. It is well known that motions of longer
duration increase the likelihood of resonance with the structures built on these regions. It
was also observed that the amplitude and phase of the ground motion vary rapidly along
mountain slopes, giving rise to differential motions. For those residential structures
supported by columns on steep slopes, the differential motion has potentially serious
consequences.
The main objective of the research described in this thesis is to develop a simple
methodology to take into account the amplification of the seismic waves that arrive to
hills or escarpments. In addition, the effect of the local topography of a region in Puerto
Rico on the potential earthquake motions will be used as a cased study.
1.2 Previous works
During the last decades several investigators have evaluated the effects of the
surface topography in the seismic response of the soil. The phenomenon has been
studied using different methodologies, analytical and numerical, as well as from
theoretical and experimental points of view. According to Geli et al (1988), it has been
3
reported that when destructive earthquakes are felt on hilly areas, those buildings at the
top of massive crests suffer more extensive damage than those located at the base.
Recent examples of this situation can be found in the 1976 Friuli and 1980 Irpino
earthquakes in Italy, and in the Chile earthquake of 1985.
Geli and his colleagues presented a brief review of experimental and theoretical
results on the effect of surface topography on seismic motions. They pointed out that the
two sets of results are only consistent on a qualitative basis. They speculate that the
differences are because the theoretical models are not sufficiently sophisticated. They
computed the response of a ridge with a smooth shape due to incoming SH waves, which
produce only out-of-plane displacements. They considered the presence of periodic
neighboring ridges and concluded that they may be responsible for the larger crest/base
amplifications observed. They also included subsurface layers in the model and
mentioned that it is difficult to separate their effects from those solely due to geometry.
They concluded that their results underestimate the actual amplification probably because
the more important SV and P waves were not considered.
Bouchon (1973) presented a study in which the topography was assumed to be
one-dimensional, and the displacements due to plane seismic waves incident on this
topography and coming from any direction were computed. For the analysis Bouchon
used a method developed by Aki and Larner (Bouchon, 1973) for the general case of
scattering of body waves in a layered medium having an irregular interface. A very
idealized seismogram was used to represent the earthquake. Bouchon concluded that the
effect of topography on surface motion appears to be very important when the
4
wavelength of the seismic wave is of the order of dimension of the anomaly, and it can
locally be responsible for both strong amplification and attenuation. The amplification is
very likely to occur at the top of the ridge whereas attenuation is probable at the bottom
of a depression. Bouchon was the first researcher to analyze the effects of incident in-
plane waves on ridges and valleys of an elastic homogeneous halfspace.
Castellani, Peano and Sardella (1982) proposed to use the finite element method
to study the effect of topography on the seismic ground motions because it provides more
realistic solutions and it can consider several factors that are not possible with analytical
techniques. For example, using finite elements one can achieve a detailed description of
the geometry of the topography and the subsurface layering. Moreover, it is possible to
include in the analysis the dissipation characteristics of the soils and their nonlinear
behavior when they are subjected to strong excitations. Castellani and his co-authors
explained how the finite element codes for plane strain can also be used for soil dynamic
problems dealing with the propagation of SH waves.
The effects of a ridge-like surface irregularity on the seismic response were
investigated by Athanasopoulos and Zervas (1993) using a 2-D finite element model.
The study considered the vertical propagation of SV waves due to four recorded
earthquakes. They found that the greatest values of amplification were obtained when the
base length of the ridge is two times the incident seismic wavelength for gentle slopes.
For steep slopes the worst case occurs when the two quantities are equal. The seismic
wavelength is defined as the product of the shear wave velocity vs of the soil on the ridge
and the dominant period Tpeak of the seismic waves. The value of Tpeak is defined as the
5
period with the highest ordinate in the response spectrum. The amplification factors
obtained by the authors range from 1 to 3.
As was mentioned previously, Italy is a country that is prone to suffer topographic
site effects. Sano and Pugliese (1999) reported that after the Ms 5.9 earthquake of
September, 26 1997 that hit the Umbria-Marche (a region in central Italy), the Italian
government decided that the amplification due to local effects had to be taken into
account in the post earthquake repair and reconstruction. The damaged area was located
on the Apennine Mountains, and thus the local soil amplification due to topographic
effects was important. Sano and Pugliese used a two dimensional code, based on the
indirect boundary element method, to investigate the phenomenon and also the effects of
geometric parameter changes. The numerical results show that the motion on a hill can
be highly variable from point to point and in a very short distance, depending on the
shape of the hill. Moreover, it was found that a small variation of the geometry, such as
changing the slope or the horizontal dimension of a hill, only affects the response at high
frequency and the space variability of the motion.
One of the leading authorities in topographic amplification is F. Sánchez-Sesma.
He contributed with a number of works to the study of the phenomenon (Sánchez-Sesma
1990, 1997, Sánchez-Sesma and Campillo 1993). Particularly useful is a chapter that he
prepared for a handbook of earthquake engineering (Sánchez-Sesma 1997). There the
author reviews the effects of local topography and local geology on strong seismic
motions. Sánchez-Sesma divided the various analytical methods proposed to predict site
effects in simplified configurations in three groups. They are: (1) the one-dimensional
6
propagation in a layered structure, (2) the two dimensional scalar wave propagation using
cylindrical eigenfunctions, and (3) some 2-D exact scalar and vector solutions based on
ray theory. Also the same author classified the numerical techniques developed for the
same purpose into two groups: domain and boundary methods. The finite element
method and the finite difference technique are examples of domain methods. These are
the most widely used methods by the geotechnical engineering community. An example
of the boundary methods is the boundary element method. This last technique is until
now mostly used by researchers. Nevertheless, the boundary methods have gained
increasing popularity. Other methods are the asymptotic techniques and the hybrid
methods that combine different techniques or the adaptation of approaches originally
devised to study other problems. To be able to solve the complex equations of motion,
the analytical methods are forced to use a simple geometry and to assume that the soil is
an elastic, homogeneous, isotropic and linear medium. As was mentioned before, due to
these assumptions the results obtained do not always agree with those measured at the
sites. However, these methods have the advantage that in some cases they can yield
closed-form solutions to the problem, which permit estimation of the effect on the
response of varying the different parameters. In other cases the analytical techniques
require the use of computers to evaluate the solution and the main benefit of using them
is lost. In addition, the analytical methods are very elaborate and require a mathematical
and geophysical background to understand them. Consequently, they are not useful for
the purpose of this thesis and they will not be further discussed here.
7
1.3 Scope of the investigation
The goal of the investigation is to assess the effects of topographic irregularities
on the earthquake-induced acceleration at the soil’s free surface such that this
phenomenon can be incorporated in seismic design codes. Two types of topographic
irregularities will be considered: an escarpment or slope and a hill or ridge. The idea is to
account for the effects of surface topography by means of a set of amplification factors.
These factors, when applied to the peak acceleration at the free field without irregularities
should give a conservative but reasonably accurate estimate of the peak acceleration
expected on the surface of the escarpments and hills. To achieve this goal, a parametric
study based on a numerical simulation of the phenomenon will be undertaken. The
numerical simulations will be carried out by means of finite element analyses of models
that include the topographic feature and the soil deposit underneath. Closed-form
equations defined only in terms of the parameters that describe the geometry of the
irregularity will be proposed to determine the amplification factors. In addition, the
amplification factors will be provided in tables. Since it is expected that the results will
be incorporated into guidelines or codes for the design of structures on zones with
topographic irregularities, the values of the amplification factors must cover the most
critical cases. This requires carrying out an extensive parametric study in which all the
major parameters affecting the phenomenon must be varied.
8
1.4 General organization of the investigation
Chapter I contains a general introduction to the investigation. The motivation and
problem description are briefly discussed. The chapter continues with a review of the
most relevant previous works on the subject of topographic amplification of earthquake
motions. The scope and organization of the investigation are also included in the first
chapter.
Chapter II contains a general description of the methodology used later in the
succeeding chapters to carry out the seismic analysis of the soil profiles below
topographic irregularities. There is a brief description of the computer program
QUAD4M for finite element (FE) analysis, which is the main tool used in this research.
There is also a discussion of the pre - and post - processor program Q4MESH, which is
used in conjunction with the previous program to generate the FE mesh and to visualize
the results. The procedure used by the program QUAD4M to account for the behavior of
soils with limited nonlinearities is discussed. The seismic excitations used as the input
for the analysis of the FE models are presented. The finite element model used to
describe the geometry of a soil deposit below a topographic irregularity on top is
introduced. The implications of the use of special boundaries in the FE program to
account for the unlimited extension of the soil in the horizontal directions are examined.
Finally, the chapter concludes with a description of the soil classification adopted for the
research.
In Chapter III begins the actual study of the amplification of the seismic motion
due to the presence of an irregular soil profile. Here the case of an escarpment or slope is
9
considered. The chapter starts with a description of a slope stability analysis carried out
using the program XSTABL. The objective of this analysis was to determine the highest
values of the angles of the slopes that can be naturally maintained with the soils
considered. In the next section, a selection of the total depth of the soil profile that
maximizes the amplification is presented. The core of the chapter that follows this
section, are the results of the amplification study for escarpments with increasing slope
angles. The results are presented in the form of graphs depicting the peak absolute
acceleration at the surface of the slope and at the horizontal level of the ground surface.
These results are next used to derive a general formula to calculate the amplification
factor as a function of the angle of the slope at any elevation. The mathematical
procedure followed to formulate the closed-form expression for the amplification factors
is explained. The chapter winds up with a discussion of the nonlinear behavior of the
analyzed soils when they are subjected to the earthquake ground motions selected for the
study.
Chapter IV is devoted to the study of the amplification of the earthquake shaking
on the surface of hills. The standard shape of the hill used in the FE models is described.
The main content of the chapter is the presentation of the results of the parametric study
done to quantify the amplification of the peak acceleration on the slopes of the hills. A
sample of graphs showing the peak acceleration in the nodes of the FE model at the top
surface is presented and discussed. Using the previous results, a general equation that
allows one to predict the amplification factors at a given height of a hill and as a function
of its aspect ratio is derived. The chapter ends with a discussion of some unexpected
10
results and a possible explanation based on a frequency analysis of the soil system and
the earthquake signals.
Chapter V contains a real case study of topographic amplification in Puerto Rico.
The site of a residential community located on the slopes of a group of hills was selected.
The community, known as Caño, is located in the municipality of Guánica in the south of
Puerto Rico. The elevation of the site was obtained from topographic maps. Using this
information, a two dimensional profile was prepared and modeled with plane strain finite
elements. The mesh was analyzed with the program QUAD4M. The properties of the
soil were defined by using soil maps, and by means of laboratory analyses of samples
taken at the site. The seismic input was obtained from a previous study carried out at the
Department of Civil and Surveying Engineering of the University of Puerto Rico at
Mayagüez. A synthetic earthquake, compatible with a design spectrum developed for the
city of Ponce, was applied at the location of the bedrock. A discussion of the findings
from the study and their implications are provided. The results obtained by applying to
this case the methodology developed in the previous chapter are presented along with a
comparison with the output of the specific numerical simulation.
The investigation ends up with the conclusions and recommendations presented in
Chapter VI. A summary of the main findings and achievements are discussed. A list of
areas and specific topics where it is deemed that more work would be beneficial is
provided.
11
CHAPTER II
Description of the General Methodology
2.1 Introduction
As it was discussed in the previous chapter, the topographic amplification of earthquake
waves has been studied for several years. The differences among these numerous studies
were the methodologies used. This thesis makes use of extensive numerical simulations
by means of a computer program based on the finite element method. The program
performs a finite element analysis of plane soil structures subjected to a horizontal
earthquake excitation at the base. The Equivalent Linear Method is used to
approximately take into account the nonlinear behavior of the soils subjected to strong
seismic motions. To create realistic analytical models it was necessary to simulate, as
accurate as possible, a soil deposit shaken by an earthquake. This includes using the
proper boundary conditions and soil profile, and the earthquake time history applied to
the base of the model. This chapter contains a discussion of all the concepts used in the
proposed methodology.
2.2 Finite element concepts
The finite element method (FEM) is a numerical procedure for analyzing structures and
continuous media. Usually the problem addressed is too complicated to be solved
satisfactorily by classical analytical methods. The problem may concern stress analysis,
heat conduction, or any of several other areas. The finite element procedure produces a
12
set of simultaneous algebraic equations in static cases or coupled ordinary differential
equations in dynamic problems, which are generated and solved on a digital computer.
The FEM models a structure as an assemblage of small parts or elements. Each element
is of simple geometry and therefore is much easier to analyze than the actual structure.
Elements are called “finite” to distinguish them from differential elements used in
analytical methods. The connection or dots between elements are called “nodes”. In
solid mechanics problems, the algebraic or differential equations that describe the finite
element model are solved to determine the displacements of the nodes representing
specific points of the structural system.
The power of the FEM resides principally in its versatility. The method can be
applied to various physical problems. The body analyzed can have arbitrary shape, loads,
and support conditions. The mesh can mix elements of different types, shapes, and
physical properties. User prepared input data controls the selection of problem type,
geometry, boundary conditions, element selection, and so on.
The FEM also has some disadvantages. A specific numerical result is found for
each specific problem: a finite element analysis provides no closed-form solution that
permits an analytical study of the effects of changing various parameters. A computer, a
reliable program, and intelligent use are essential to
obtain meaningful results. A general-purpose FE program has extensive documentation
with which one must be familiar before attempting to use it. Experience and good
engineering judgement are also needed in order to define a good model. Many input data
13
are required and voluminous output must be sorted out and understood (Cook et al,
1989).
2.3 Description of the computer program QUAD4M
The finite element method has shown to be a powerful tool for the solution of
various problems in continuum mechanics. Although its use is not widespread in their
communities, the FEM is a very useful technique for the geotechnical engineers.
However, it has been applied extensively for the evaluation of the seismic response of a
variety of soil deposits and earth structures (Idriss et al. 1973). The displacement-based
formulation of the FEM is typically used for geotechnical applications and the results are
presented in the form of displacements, stresses and strains at the nodal points. To select
a suitable program, the user must consider the implementation of the constitutive model
and the availability of different types of finite elements such as triangular or quadrilateral.
A number of programs to solve geotechnical problems using time domain solutions as
well as frequency domain solutions have been written in the past 30 years.
In 1973 researchers at the Department of Civil Engineering of University of
California at Berkeley developed the program QUAD4 for the seismic response of soil
structures (Idriss et al. 1973). This program is essentially a two dimensional variable
damping finite element procedure. The analytical procedure implemented in the program
permits the use of both strain-dependent modulus and damping ratios for each element in
the finite element representation of a deposit. This was an improved tool to perform a
response analysis of soil deposits having considerable geometric variations and having
14
greatly varying material characteristics. In addition, the formulation allows for the
incorporation of nonlinear stress-strain relationships through the use of equivalent linear
and strain dependent material properties. The equations of motion are solved by a direct
step-by-step numerical method. The output from the program includes the maximum
values of the horizontal and vertical accelerations along with their time of occurrence,
and a time history for the shear strain in each element. In 1994 researchers from the
University of California at Davis modified the original program QUAD4 to include a
compliant base. This version is known as QUAD4M (Hudson et al. 1994).
Likewise the original code, QUAD4M is a dynamic, time domain, equivalent linear two-
dimensional computer program. The implementation of a transmitting base, an improved
time stepping algorithm, seismic coefficient calculations, and a restart capability are the
improvements undertaken in the new program.
The evaluation of the seismic response is carried out by solving in the time domain a
system of equations represented in matrix form as:
[ ] [ ] [ ]{ } { } g
.....uRuKuCuM =+
⎭⎬⎫
⎩⎨⎧+
⎭⎬⎫
⎩⎨⎧
(2.1)
where dots represent differentiation with respect to time and:
{ } vector nt displaceme relative u
u direction one in on accelerati outcropg
..
[ ] matrix mass M
15
[ ] matrix dampingC
[ ] matrix stiffnessK
[ ] R rvector load }]{Μ[− =
{ } tscoefficien influence withvector r
The time stepping method previously used in QUAD4 (the Wilson-θ method) was
changed in QUAD4M. To obtain the displacement, velocity and acceleration at each
time step the program now uses the Trapezoidal rule.
It was mentioned that the new program QUAD4M has absorbing boundaries
implemented to allow for the reduction in the size of the FE mesh. The concept of
absorbing or viscous boundaries for FE models of infinite or semi-infinite domain was
first introduced in 1969 by Lysmer and Kuhlemeyer (Hudson et al. 1994). They
suggested the use of dampers or dashpots with proper constants to absorb the P, S, or
Rayleigh waves impinging on the borders of the model. The viscous dampers are applied
in two orthogonal directions at each of the nodes that make up the base as well as in those
nodes at the sides of the semi-infinite model. To implement these dampers in the
computer code, the damping coefficients of the dashpots are added to the appropriate
diagonal terms of the total damping matrix.
It was mentioned that QUAD4M could also compute the seismic coefficient. This
coefficient is defined as the ratio of the force induced by the earthquake in the block of
16
the mesh with respect to the weight of that block. This parameter is computed for each
time step.
To model the damping in soils, QUAD4M uses the Rayleigh damping
assumption. However, opposed to most well known FE programs, the Rayleigh damping
matrix is defined for each finite element. For each element q the damping matrix is
formulated as follows:
[ ] [ ] [ ]qqqqq KMC β+α= (2.2)
The damping matrix of the full model is constructed by assembling the element
matrices in the usual way. The coefficients α and β are selected by specifying damping
ratios at two frequencies. One frequency is chosen as the fundamental frequency ω1 of
the model. The second frequency is established as a multiplier of the fundamental
frequency. The value ω1 of the system is internally calculated using the formula obtained
from a continuous and homogeneous soil deposit that can only withstand shear
deformation. The output file displays the two frequencies at which the damping ratios
were set.
Additional modifications made to the original QUAD4 program were aimed at
making the new program conform to a structured FORTRAN language and implementing
data structures to describe the elements and nodes (Hudson et al. 1994).
17
2.4 The pre and post processor Q4MESH
The US Army Engineer Waterways Experiment Station (now the Engineering
Research and Development Center) under the sponsorship of some organizations,
developed a pre- and post processor called WINMESH for use with the finite element
program STUBBS. WINMESH is a finite element pre-processor, a post-processor, and a
menu-driven input file builder created as a productivity tool to be used with the
geotechnical finite element program STUBBS (Peters and Kala 1999).
A new pre and post processor, Q4MESH was generated by the Engineering
Research and Development Center. This work was carried out as part of the investigation
reported in this thesis. It is a combination of the WINMESH and QUAD4M programs
into a new one, which has more capabilities and is more user-friendly. The pre processor
has the capability to graphically build a mesh file, assign material properties and
boundary conditions which can be pasted into a text file to be used as input file for
QUAD4M. The post processor can display plots of stresses or strains on the mesh, and
acceleration time histories. The pre and post processor was based on the WINMESH
program. The Q4MESH program is CAD-like and permits the user to select elements
and nodes using the “mouse”. The technique used is called “rubberbanding”. Initially,
only the menu referred to, as FILE is active. The brief description of the input of
Q4MESH is just to provide an overview, because the program is more powerful.
2.5 The Equivalent Linear Model
18
When a strong earthquake affects a soil deposit, it can produce strains in the
material of such magnitude that the soil behaves in a nonlinear fashion. Therefore, to
perform a site response analysis for this case, one must know the stress-strain relationship
of a soil subjected to a shear state. For the case of small strains such as those induced by
a weak to moderate earthquake, the soil follows the Hooke’s law for shear, i.e.:
γ=τ oG (2.3)
For more severe cases of seismic excitation, the Hooke’s law must be replace by a
nonlinear constitutive equation with a form
( )γ=τ f (2.4)
such as the one illustrated in Figure 2.1
Figure 2.1 Linear and nonlinear shear-strain relationship
In 1970, H. B. Seed and I. M. Idriss developed a procedure to consider
approximately the nonlinear behavior of soils in dynamic problems that became known as
the Equivalent Linear Method. The main idea of this method is to obtain similar results
to those obtained from a truly nonlinear analysis by means of a series of linear analyses
γ
τ
τ=f(γ)
τ≈Goγo
19
utilizing ”effective” values of the shear modulus and damping. The effective shear
modulus or secant shear modulus is approximated as (Kramer 1996),
c
csecG
γτ
= (2.5)
where:
amplitude strain shear cγ
amplitude stress shearcτ
Thus, as shown in Figure 2.2, Gsec describes the general inclination of the hysteresis loop.
The effective damping ratio ξeff is the area Ahyst inside the hysteresis loop corresponding
to γc, divided by the elastic energy corresponding to the same deformation level, as
illustrated in Figure 2.2, i.e.:
2
csec
hyst2
csec
hysteff
G2
A
2G
A41
γπ=
γπ=ξ (2.6)
Figure 2.2 Secant shear modulus
20
Of course, the values of γc are not known in advance, since they are calculated from the
solution of the equations of motion. Therefore, it is necessary to implement an iterative
process to calculate the effective shear modulus and effective damping ratio. The process
starts by estimating initial values for the shear modulus and damping ratio. The response
of an equivalent linear system defined using these values can be calculated in the time
domain with modal analysis or numerical integration, or in the frequency domain using
the discrete Fourier transform. Using the assumed initial values of the modulus and
damping, the maximum shear strains for each element in the model are calculated. With
these new values, the previously assumed values of Geff and ξeff are corroborated. The
procedure requires knowing how Geff and ξeff vary with the shear strain. If one adopts a
specific nonlinear constitutive equation, say for instance the hyperbolic model, then it is
possible to obtain analytical expressions relating the effective parameters to γ.
Alternatively, one can provide tables that relate Geff and ξeff to γ at certain values. This is
the approach adopted in QUAD4M. The program uses curves proposed by Seed and
Idriss of Geff and ξeff for different type of soils shown in Figure 2.3.
21
(a)
(b)
Figure 2.3 Shear modulus (a) and Damping ratio (b) curves
Entering these curves with the shear strains in the x-axis, the equivalent Geff and ξeff can
be verified and updated, if needed. In the latter case, these new values are used to
generate new stiffness and damping matrices. The response is again obtained with a
linear analysis, and the shear strains are used to find Geff and ξeff. The process finishes
0100020003000400050006000700080009000
10000
0.0001 0.001 0.01 0.1 1
Shear Strain
Shea
r Mod
ulus
(ksf
)
Clay material Gravel material
0
5
10
15
20
25
30
0.0001 0.001 0.01 0.1 1
Shear Strain
Dam
ping
Rat
io (%
)
Clay material Gravel material
22
when the differences between the most recent values and those from the previous cycle
are acceptable. The most recently updated values of the shear strain, stress or
displacement obtained at the last iterative step are considered to represent a good
approximation of the actual nonlinear solution.
The Equivalent Linear Method was compared on several occasions with real
nonlinear analyses and it was found that the results predicted by the former are
satisfactory provided there is convergence. The Equivalent Linear method, however, is
not exempt of problems. For example, one of the drawbacks associated with the method
is that there is no proof of convergence. Moreover, some difficulties were reported when
it was applied to problems involving deconvolution of the seismic waves, i.e. finding the
signal at the bottom of the deposit when it is known at the surface.
Nevertheless, and despite these problems, the Equivalent Linear method is widely
used in geotechnical earthquake engineering. It is important to bear in mind that a truly
nonlinear analysis requires much more time from the analyst and the computer, and it
also requires knowing more parameters for each soil layer to define the constitutive
model. The diversity of soil properties and the uncertainty in some of them contribute to
increase the popularity of the Equivalent Linear method. Another reason that plays an
important role in the acceptance of the method is its widespread implementation in
computer programs for geotechnical earthquake engineering. For example, in addition to
the computer program used in this research, the popular program SHAKE that evaluates
the seismic response of one-dimensional soil deposits employs the method. Figure 2.4
23
presents an example of the shape of the stress-strain relationship of a soil used in the
QUAD4M program.
Figure 2.4 Stress Strain curve
2.6 Finite element models
Two types of surface topography were considered in this research to quantify the
amplification in the ground acceleration due to the seismic waves. The soil deposit was
first modeled with an escarpment and later with a ridge or hill. To study the escarpment
it was necessary to establish somehow a reduced number of dimensions for the model
among the infinite possible configurations. The mesh of the FE model was created using
the interface program Q4MESH and the analysis was carried out using the finite element
program QUAD4M. Different types of escarpments were analyzed. The depth H of the
block of soil under the escarpment was selected as two times the height of the
escarpment. The total height h+H depends on the type of soil and is selected so that the
amplification is maximized, as explained in Chapter III, Section 3.4. The lateral
0
20
40
60
80
100
120
0 0.005 0.01 0.015 0.02 0.025 0.03
Shear Strain, γ
She
ar S
tress
(ksf
)
24
extension of the finite element mesh far beyond the toe and top of the slope was taken
into account by using viscous boundaries. The horizontal distances to the boundaries a
were selected as two times the horizontal distance of the ridge L, as illustrated in Figure
2.5.
Figure 2.5 Dimensions of the escarpment model
To obtain amplification factors for the many configurations that can be found in real
situations, it was necessary to examine different cases. The angle of the slope was
chosen as the variable parameter that distinguish the different models. Five element
meshes were used to model different inclination angles of the escarpment. This
inclination angle was selected by a slope analysis to avoid static stability failure and
using a correlation between the critical height and the material properties. The angles
considered were 15, 30, 40, 50 and 65 degrees. Four types of soil profiles were also
considered, although some materials do not resist angles of slopes greater than the
friction angle of the soil. These types of soils and the details of the slope stability
α
h
H=2h
a =2L L a =2L
25
analysis are presented in the next chapter. Figure 2.6 presents an example of a typical
escarpment finite element model used in this research.
Figure 2.6 Example of escarpment finite element mesh
For each case the output of the program QUAD4M included peak nodal
accelerations and peak element stresses. As mentioned previously, to carry out the
dynamic analysis QUAD4M performs a number of iterations that is fixed by the user in
the input data. To guarantee that the process converges to a reasonable extent, in all the
cases studied the maximum number of iterations was set equal to ten. At this number of
iterations, the final differences in percent in the shear modulus and damping ratio at two
consecutive steps were approximately zero. A higher number of iterations was not
considered necessary nor justified, especially taken into account that hundreds of
computer runs would be carried out throughout the study.
The other cases analyzed were the topographic irregularities with the shape of
ridges. Evidently, in this case also one can have an infinite number of shapes to describe
the ridge’s geometry. Many of the ridges previously studied in the literature had the
26
shape of a wedge. This was done mostly to facilitate the analysis, but we did not
consider this as a realistic shape. Therefore, it was decided to use a “smooth” shape to
describe the ridge. Since a parabola has the advantage that by varying the three
coefficients in its equations its shape can be easily manipulated, it was chosen as the
geometry of the ridge. The ridges were defined by two parameters: its base length m, and
the total height of the ridge, n. Viscous boundaries are used to take into account the
theoretically infinite lateral extension of the finite element mesh along the left and right
sides. The lateral extension of the FE mesh beyond the ridge’s toe is defined by a and it
was set equal to m/2, as illustrated in Figure 2.7. The depth of the soil deposit under the
hill is H and it was taken equal to a/2.
Figure 2.7 Dimensions of the ridge model
The parameter n was varied during the numerical simulation study and four
models were analyzed. As it was done in the case of the escarpment, a slope stability
analysis was carried out to evaluate the critical height. The stability analysis is discussed
in Chapter IV. The value of m was kept constant and by varying n different aspect ratios
of the ridge was examined. The values of n used for the study were 38, 75, 115 and 225
ft. There is no particular reason for selecting those values. The first value of n used in
2m
ma = m/2 a = m/2
n
H = a/2
27
the preliminary studies was 75 ft and the other values were chosen a posteriori to cover a
reasonable range. The value of m was fixed at 400 ft and thus the aspect ratios of the hill
are approximately equal to 0.1, 0.2, 0.3 and 0.6. The English or fps system of units was
used throughout the analysis. Figure 2.8 illustrates an example of a typical finite element
mesh of a ridge used in the study. Finally, all the finite element models of the ridge and
deposit were analyzed using ten iterations in the Equivalent Linear method to obtain
results with a reasonably accuracy.
Figure 2.8 Example of a finite element mesh of a ridge
2.7 Seismic excitation
The seismic excitation, defined as an acceleration time history, was applied to the
base of the model, that is, to the bedrock. The recorded horizontal accelerogram of two
historical earthquakes, namely the El Centro and San Salvador earthquakes, were used as
input motion for all the finite element models. The El Centro earthquake occurred on
May 18, 1940 and the San Salvador was more recent: it occurred on October 10, 1986
28
causing approximately 1000 deaths. Table 2.1 shows the most important characteristics
of both seismic events (Strong Motion Data Center 1999).
Table 2.1 Important Characteristics of the Earthquakes Selected
Earthquake Magnitude
(Richter Scale)
Peak
Displacement
(cm)
Peak Velocity
(cm/s)
Peak
Acceleration
(cm/s2)
El Centro 7.1 10.87 33.45 341.70
San Salvador 5.5 11.90 80.00 -680.80
The original peak ground acceleration of both seismic records was scaled to 0.1g. This
was done so that the soils would behave linearly or with limited nonlinear excursions.
For a given seismic input, the nonlinear behavior of the soils tends to reduce the absolute
accelerations at the free surface, compared to a soil with a linear response. Therefore,
since we are interested in determining maximum amplification factors and not the actual
accelerations due to the earthquakes selected, it was considered prudent to limit the level
of the seismic excitation.
This hypothesis was proved by running trial cases. The two earthquake records
were selected in order to use seismic input time histories with different characteristics.
Figure 2.9 displays the acceleration time histories and the Fourier spectrum for the two
earthquakes used in the study. The El Centro accelerogram is typical of a broad band
process while the San Salvador represents a narrow band process.
29
(a)
(b)
Figure 2.9 Acceleration time histories and Fourier Spectra for (a) El Centro and (b) San Salvador earthquakes
2.8 Boundary conditions
To increase the computational efficiency, it is desirable to minimize the number
of elements in a finite element model. For many soil response analysis and soil structure
interaction problems, rigid or near rigid boundaries (such as those representing the
0 5 10 15 20 25 30 35 40 45 50
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time [sec]
Acc
eler
atio
n [%
g]
0 10 20 30 40 50 60 700
200
400
600
800
1000
1200
1400
1600
Frequency [rad/sec]
Am
plitu
de
0 1 2 3 4 5 6 7 8 9-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time [sec]
Acc
eler
atio
n [%
g]
0 10 20 30 40 50 60 700
100
200
300
400
500
600
700
800
Frequency [rad/sec]
Am
plitu
de
0 1 2 3 4 5 6 7 8 9-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time [sec]
Acc
eler
atio
n [%
g]
0 10 20 30 40 50 60 700
100
200
300
400
500
600
700
800
Frequency [rad/sec]
Am
plitu
de
30
bedrock) are located at considerable distances, particularly in the horizontal direction,
from the region of interest (Kramer 1996). By doing this, the errors introduced by the
spurious reflections of the seismic waves at the rigid boundaries are minimized.
However, the price to pay is an increase in the number of elements, degrees of freedom
and execution time of the program.
The new version of QUAD4M has the option of using a so-called compliant base in the
soil mass. This is a better way of dealing with infinite field conditions, because it
permits minimizing the number of elements that represent the underlying half space.
When a transmitting base is used, the input motion is a function of the materials
properties of the half space below the mesh and the properties and geometry of the mesh.
In all cases analyzed the soil deposit is assumed to be underlain by a half space having a
shear wave velocity of 3000 ft/sec, a compression wave velocity of 7000 ft/sec and a unit
weight of 135 pcf. To assess whether the dimensions of the meshes to be used later are
acceptable, different boundary conditions were first considered. This was done for both
topographic features, the escarpment and the hill. The analysis consisted in comparing
the responses obtained with an original FE mesh, the same mesh but with transmitting
boundaries, and with another extended mesh without the special boundaries. All the
other properties were kept equal. Figure 2.10 shows the peak accelerations along the free
surface for the case of an escarpment using as input the El Centro earthquake. Note that
in the central region of the plot, i.e. where the escarpment is located, the peak
accelerations were practically the same for the three meshes. This demonstrates that the
31
transmitting boundaries are capable of absorbing the energy of the seismic waves that
arrive at the sides.
Figure 2.10 Comparison of results using fixed and transmitting boundaries and extending the mesh for the escarpment subjected to the El Centro earthquake
The analysis was repeated with a ridge. In this case the accelerogram used was
that of the San Salvador earthquake. From the time history results produced by the
program, the peak accelerations at the free surface were retrieved and are plotted in
Figure 2.11 as a function of the horizontal position. As it is illustrated in Figure 2.11,
after the mesh reaches a sufficient length, extending further the mesh does not change the
peak acceleration in the region of the hill. Therefore, for all the future analysis the FE
mesh to be used will be the original (i.e. that with the shorter length) mesh with
transmitting boundaries.
0
0.05
0.1
0.15
0.2
0 100 200 300 400 500 600 700
Horizontal Distance (ft)
Peak
Acc
eler
atio
n (g
)
w ith transmitting boundaries on the sides
w ithout transmitting boundaries on the sides
respective nodes on the long mesh
32
(a) (b)
(c)
Figure 2.11 Comparison of results (a) using transmitting boundaries (b) without transmitting boundaries at the sides and (c) extending the mesh, for the ridge subjected to
the San Salvador earthquake
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
mpl
ifica
tion
(g)
n=75, total length=800
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800 900
Horizonta l Distance (ft)
Pea
k Am
plifi
catio
n (g
)
n=75 , total length=800
0
0.05
0.1
0.15
0.2
0.25
0 500 1000 1500 2000 2500
Horizontal Distance (ft)
Pea
k A
mpl
ifica
tion
(g)
n=75, total length=2400
33
2.9 Guidelines for soil type categorization
Some researchers pointed out that it is very difficult, if not impossible, to separate
the amplification due to the topographic irregularity from that due to the local geological
conditions. In other words, the effects of the type of soil and the discontinuity in the
geometry are interwoven and they must be considered simultaneously. Therefore, to
carry out the research presented in this thesis, it was necessary to select the types of soil
that will be assumed for the escarpments and hills as well as for the soil deposit
underneath them. Depending on the specific purpose, there are a number of soil
classifications available. Since the objective of the present study is to develop a simple
methodology such that the effect of the local topography can be incorporated into seismic
codes, it was decided to use the classification presented in these documents. Puerto Rico
recently adopted the 1997 edition of the Uniform Building Code (UBC-97) as its official
design and construction code (ICBO 1997). Therefore, it was decided to use the soil
profile types listed in this code to categorize the material of the deposits, slopes and
ridges.
The soil profile types in the UBC-97 are defined in terms of three alternative
parameters. The main one is the average shear wave velocity, that is the average value of
the velocity of propagation of S waves along the different layers of a soil deposit. The
shear wave velocity is characteristic of a given material and, for a given soil, it is a
function of its shear modulus G and mass density ρ.
According to the UBC-97, the average shear wave velocity sV−
is determined as
34
∑
∑
=
=−
=n
1i si
i
n
1ii
s
vd
dv (2.7)
where:
deposit the in layers different of numbern
(m) feet in layer of thickness di i
(m/sec)ft/sec in layer ofvelocity waveshear vsi i
i
isi
GV ρ=
Another alternative parameter used to classify the soils in the UBC-97 is the
average standard penetration resistance−
N. This parameter gives information concerning
the degree of compactness or stiffness of the soil in situ. It is defined as
∑
∑
=
=−
=n
1i i
i
n
1ii
N
d
dN (2.8)
∑=
−=
n
1i i
i
sCH
N
N dd
(2.9)
where:
(mm) feet in layer of thickness di i
35
m) (30.48 feet 100 top the in layers soil sscohesionle of thickness total the ds
standards recognized nationally approved withaccordance
in layer soil of resistance n penetratio standard the Ni i
The term NCH is the standard penetration resistance for cohesionless soil layers.
Cohesionless soils are granular materials such as sand or gravel. In these types of soils,
the resistance depends on the effective overburden pressure. In homogeneous conditions,
it means that the resistance augments as the depth increases. The behavior of this type of
soil requires special attention, and for this reason it is important to evaluate if there exist
layers of cohesionless materials in the soil deposit.
The third and last property used by the UBC-97 to categorize the type of soil is
the average undrained shear strength. This is an important parameter related to porewater
pressures. It is especially important in soft clays and silts under static conditions and in
loose sands under dynamic loading. According to the UBC-97, this parameter is
calculated using the following formula,
∑=
−
=n
1i ui
i
cu
Sd
dS (2.10)
where:
m) (30.48 feet 100 top the in layers soil cohesive of ds) - (100 thickness total the dc
36
(250kPa) psf 5,000 exceed to not standards, recognized nationally approved withaccordance in strength shear undrained the Sui
The formulas given before as well as the range of values provided in Table 2.2 (taken
from the UBC-97) are used to determine the type of soil profile.
Table 2.2 Soil Profile Types (International Conference of Buildings Officials 1997)
1 Soil Profile Type SE also includes any soil profile with more than 10 feet of soft clay defined as a soil with a PI>20, wmc�40% and Su<500 psf .
The material properties required to define the soil in the FE program are the unit
weight, the Poisson ratio and the shear modulus. For a given soil type, it is assumed that
these three parameters have the same values for all cases. As it was pointed out, the
analytical models were evaluated using the soil types in the UBC-97, and the average
shear wave velocity was used to classify them. The soil profile types SA and SF were not
considered in this study. The soil SA is classified as a hard rock (VS > 5000 ft/s) and it is
estimated that this soil is not common in Puerto Rico since it is characteristic of regions
in the eastern US.
The soil classified as SF was not used in the research because it requires a site-
specific evaluation and thus the general results and formulas obtained in this study would
Soil Profile Soil Profile Name/ Shear Wave Velocity, Standard Penetration Test, N [or NCH Undrained Shear Strength,Type Generic Description Vs feet/sec (m/s) for cohesionless soil layers] (blows/foot) Su psf (kPa)
SA Hard Rock >5,000 (1,500) _ _SB Rock 2,500 to 5,000 (760 to 1,500)SC Very Dense Soil and Soft Rock 1,200 to 2,500 (360 to 760) >50 >2,000 (100)SD Stiff Soil Profile 600 to 1,200 (180 to 360) 15 to 50 1,000 to 2,000 (50 to 100)SE
1 Soft Soil Profile <600 (180) <15 ,1,000 (50)SF
Average Soil Properties For Top 100 Feet (30 480 mm) of Soil Profile
Soil Requiring Site-specific Evaluation
37
not be applicable anyway. The values of the shear wave velocity for the different soil
profiles used throughout the present study are the following:
SB: 3750 ft/s
SC: 1850 ft/s
SD: 900 ft/s
SE: 575 ft/s
38
CHAPTER III
Amplification of Seismic Motion Due to Escarpments
3.1 Introduction
This chapter presents a description of the finite element analyses of escarpments
subjected to acceleration time histories at the bedrock. The results obtained from the
studies of many different escarpment configurations of different heights and made up of
several soil types are presented. Using these results, amplification factors that relate the
peak ground acceleration to the peak acceleration at the escarpment’s free surface are
derived. Although the goal of the present study is to examine the effect of surface
irregular topography in the seismic motion, it is important to consider that when soil
deposits are subjected to cyclic loads, they can present some type of failure. The
combined effect of seismic loads and the changes in shear strength will result in an
overall decrease in the stability of slopes. Therefore, before undertaking the seismic
amplification of the escarpments, a slope stability analysis is carried out.
3.2 Slope stability analysis
Approximately 40 percent of the US population are exposed to effects of
landslides. Landslides often are triggered by natural events such as floods, earthquakes
and volcanic eruptions. The slope failures are usually due either to a sudden or gradual
loss of strength by the soil or to a change in geometric conditions. The main items
required to evaluate the stability of a slope are: (1) the shear strength of the soils, (2) the
39
slope geometry, (3) the pore pressures or seepage forces and, (4) the loading and
environmental conditions (Abramson et al 1996).
The first task undertaken before performing the analyses presented in this chapter
was to establish what inclination angles could be realistically used for further time history
studies. A computer program was used for this purpose: the program XSTABL
(Interactive Software Designs 1995). This program consists of two interactive, but
separate portions: (1) the data preparation interfaces and, (2) the slope stability analysis
programs. The program performs a two-dimensional limit equilibrium analysis to
compute the factor of safety for a layered slope.
For the analysis of the stability of a slope it is important to know the geometry
and the subsoil conditions. The result of the analysis, i.e., the factor of safety, is a vital
parameter in the design of slopes. The lower the quality of the site investigation, the
higher the desired factor of safety should be, particularly if the designer has only limited
experience with the materials in question.
A number of slopes were selected to verify their stability. The angles of the
slopes varied from 15 degrees to 75 degrees, in increments of ten. The program
XSTABL permits to construct the slope profile using coordinates, to establish the soil
parameters, to define the water table condition, to select the analysis that the user prefers,
and to apply loads. In all the cases the program performs a pseudo-static analysis to
simulate the effects of an earthquake. An average horizontal seismic coefficient of 0.15
was entered in the data table. This is a typical value for the seismic coefficient, the Corps
of Engineers use in the practice and also the same value generated by Seed (1979). The
40
method of analysis used was the Simplified Bishop Method. This method was used to
identify the critical surface with the lowest factor of safety. This method satisfies vertical
force equilibrium for each slice and overall moment equilibrium about the center of a
circular trial surface. To simulate more realistic conditions, a phreatic surface was
included in the models analyzed. In the program the free groundwater level defines the
phreatic surface, or the phreatic line in two dimensions. Proceeding in this way, the value
of the factor of safety calculated will be conservative.
As mentioned previously, if the designer does not know the characteristics of the
site, for example the soil properties, it is very difficult to determine a factor of safety and
the critical surface. Many cases were studied by changing soil parameters such as the
cohesion and friction angle. Cohesionless and cohesive soils were analyzed using a unit
average weight of 125 pcf. When site investigations are carried out, they usually show
that to find a unique type of material in a region is nearly impossible. A typical cohesive
soil (clay) was studied and the results show that the critical surface is located outside the
slope. In this case, when the critical surfaces are not generated in the face of the slope, it
is due to the component of cohesion in the soil particles. The majority of the cases
examined presented a failure surface that went from the toe of the slope to the top. The
ground water table lies at the toe of the slope and is illustrated in Figure 3.1 as w1. This
behavior is the opposite of the one found for cohesive materials. It means that the
inclination angles are unstable using these kind of material properties.
41
The profile, the free groundwater level, the lower limit and the most critical
surfaces are illustrated in Figure 3.1. This figure shows an example of many cases
considered.
Figure 3.1 An example of an XSTABL result
The output of the program provides ten critical surfaces, the coordinates of the
center of the circle that produces the critical surface, and the factor of safety for this
method. After many cases were considered and the results analyzed, it was decided that
the slopes from 15 to 65 degrees would be studied with the QUAD4M program. The
slope of 75° was eliminated due to the small value of the factor of safety obtained. For
this study, the values of the factor of safety that were considered acceptable were those
greater than 1.1. It is important to have in mind that for seismic design the values should
be higher. However, since the analysis carried out did not consider a mixture of different
materials, values greater than 1.1 were regarded as acceptable.
It is important to mention that not all the amplification factors for slopes are
presented, because other soil parameters are also considered in the stability analysis. For
example, the materials have different properties as cohesion and angle of internal friction.
For slope stability analysis, it is very important to know the maximum slope that the soil
resists. This is measured by the angle of repose, which is the angle between the
42
horizontal and the maximum slope that a soil can assume through natural processes. For
dry granular soils, the effect of the height of slope is negligible; for cohesive soils, the
effect of height of slope is so important that the angle of repose is meaningless.
Therefore, the angle of repose is a critical parameter for non-cohesive materials. Those
angles can reach up to 40° depending on the loose or dense soil condition. The angle of
repose depends on:
1. The size and particle shape - large, angular particles have steeper angle of repose
2. Sorting - well-sorted materials have a higher angle of repose
3. Composition of particles - stronger particles will have a steeper angle of repose
Cohesive materials like clays, silts and rocks are able to maintain very high slope angles
(up to 90° as a cliff).
Other factor that is used to evaluate the stability of slopes is the height. A slope
underlain by clean dry sand is stable regardless of its height, provided that the angle
between the slope and the horizontal is equal to or smaller than the angle of internal
friction. A cohesive material can stand a vertical slope at least for a short time, provided
the height of the slope is less than HC, which is defined as
γ
= uC
S4H (3.1)
where:
strength shear undrainedSu
weightunit soil γ
43
If the height of a slope is greater than HC, the slope is not stable unless the slope
angles β are less than 90°. The greater the height of the slope, the smaller must be the
angle β. If the height is very large compared with HC, the slope will fail unless the slope
angle is equal to or less than the internal friction angle [Terzaghi et al,1996].
3.3 General input data
As discussed previously, the data for the soils used in this study are based on the
UBC 1997 code. The UBC 1997 divides the soil profiles in six categories, depending on
the values of the shear wave velocity, the standard penetration resistance, or the
undrained shear strength. From the six categories listed in the UBC 1997, only four were
selected to carry out the study, namely the SB, SC, SD and SE soil types.
To prepare the input data, specific properties of the material are required. One of
them is the soil unit weight: all calculations were conducted using a value equal to 125
pcf. Another data required is the shear wave velocity of the soil. The values used were
the average values of the limits prescribed in the UBC 1997 for each soil type. Using
these values, the program Q4MESH calculates internally the shear modulus Gmax
according to the following equation,
g
VG2
maxγ
= (3.2)
where:
velocity waveshearV
soil the of weightunit γ
44
gravity to due on accelerati g
Because of the method of analysis used by the computer program, a shear
modulus for the first iteration is also needed. This value is calculated as 80% of the value
Gmax defined by equation 3.2. The other material properties needed are the Poisson’s
ratio for the stress-deformation relationship and the damping ratio. Both parameters have
the same values in all the cases, and are equal to 0.35 and 0.05, respectively.
At last, the curves describing the variation with the shear strain of the shear
modulus and damping ratio must be selected. The curves for different materials from
coarse to fine soil including rock were used. The criterion for electing the curves was the
shear wave velocity used in the UBC 1997 code to describe the different soil profiles.
The input data is completed with the acceleration time history due to the
earthquake. The horizontal motion used for all the escarpment analyses was the El
Centro earthquake and the San Salvador earthquake modified through a scaling factor to
produce a maximum acceleration of 0.1g.
3.4 Selection of total height
The main objective of this research is to consider the effects of the topographic
irregularities on the surface accelerations by means of amplification factors. Therefore,
the most critical cases should be taken into account. These most severe cases are defined
as the configurations that yield the highest value of the amplification factors. To
determine these cases, a study of various soil deposit-escarpment systems with varying
height was performed. For a given type of soil material, a slope at an angle α = 15°
45
subjected to the El Centro seismic motion was selected and evaluated. The study was
repeated using each of the four soil profiles. For each soil type, soil deposits with
different heights H' = h+H as illustrated in Figure 3.2 were evaluated. The soil type in
the deposit of height H and in the escarpment of height h was the same.
Figure 3.2 Escarpment heights
To complete the definition of the geometry of the model we need to know the
relation between h and H. As explained in Chapter II, a ratio h/H = ½ was used. The
amplification factors were calculated in all the cases, their values were compared and the
most critical cases were selected. Six soil deposits of total height H’ equal to 150, 100,
75, 50, 25 and 10 feet were analyzed. The FE meshes were generated considering these
heights and the fixed slope angle. The values of the amplification factors obtained after
all the cases were considered ranged from 0.8 to 1.5. To select the most critical heights a
total of 24 FE models meshes were evaluated for each soil type analyzed. The results are
summarized in the following table:
H
h
H’
α
46
Table 3.1 Total Height According to Material
Also, the results were compared with the value of HC given by equation 3.1 to
verify the stability of the slope. To evaluate this equation, the value of Su used was the
average of the values established by the UBC 1997 and the unit weight γ was 125 pcf. In
all the cases examined it was checked that the critical height Hc was larger than h.
3.5 Escarpment amplification results
This section illustrates the important influence that the presence of an escarpment
has on the characteristics of the surface seismic motions. The soil system formed by the
escarpment and a uniform horizontal soil deposit, shown in Figure 2.5, was modeled
using the 2-D plane strain finite element model. The sizes of the escarpment and soil
deposit analyzed are defined in Figure 2.5. A typical mesh is shown in Figure 2.6. The
absolute ground acceleration was calculated at different points along the escarpment
surface. The results for five different slope angles, two earthquakes with different
characteristics, and the four classifications of the soil profile are presented in the
following sections. The peak surface accelerations obtained from 40 cases of seismic
response analyses were retrieved, stored and processed in the course of the study. The
amplification factors were calculated for all the cases analyzed. The analyses were done
Material Type Total Height, H' (ft)SB 150SC 100SD 100SE 75
47
considering a single homogeneous material for both the soil deposit and the escarpment.
Since the effect of the material type and the earthquake motions show different patterns.
The behavior displayed in the amplification plots shown in this section is typical of all
the cases examined in the investigation. The plots indicate that the amplification of the
acceleration continuously increases from the base or toe to the top of the escarpment.
Figure 3.3 illustrates the nodal points of the FE model at which the ground acceleration
was evaluated.
Figure 3.3 Surface nodal points
The results of amplification obtained in this part of the study are in agreement
with those of other works dealing with ridge topographic amplification, for example the
ones carried out by Athanasopoulos and Zervas [1993] and by Geli et al. [1988].
Results for a 15 degrees slope
The peak ground acceleration obtained for this escarpment configuration is
presented graphically in Figure 3.4. The curve in Figure 3.4 shows how the acceleration
varies along the free surface due to the topographic irregularity. The
48
results in this figure, obtained for the 15° slope, correspond to the most critical case. It
occurs for the El Centro earthquake and for the SC material. The maximum amplification
occurs at the top of the escarpment and the resulting amplification factor there is 1.48.
Figure 3.4 Peak acceleration for a 15° slope escarpment
To gain insight into the behavior of the escarpment-soil deposit system, a
comparison with a model without the escarpment is presented next. The dimensions of
the depth are the same as the higher level of the model with a 15° slope, i.e. 150 ft. In
this case the soil deposit has a depth of 150 ft and length of 933 ft. The object of the
analysis is to evaluate the amplification at the higher (right) side of the escarpment
model. Figure 3.5 presents the results of the peak ground acceleration when the El
Centro earthquake acts at the bottom of the deposit. As expected, the plot shows that the
acceleration at the surface is constant if no irregularities are present. These results agree
with the accelerations computed at the top of the escarpment in Figure 3.4, which was
approximately 0.135g.
0.08
0.1
0.12
0.14
0.16
0.18
0 100 200 300 400 500 600 700
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
Sc material, A.F. = 1.48
49
Figure 3.5 Amplification of the peak acceleration in a soil deposit without irregularities
Results for a 30 degrees slope
We compare here the peak surface accelerations caused by earthquake motions in
a 30° escarpment. It was observed that the most critical condition for the two
earthquakes was obtained with the SC soil or gravel material. As shown in Figure 3.6, the
San Salvador earthquake produces higher ground acceleration than the El Centro
earthquake. Nevertheless, the amplification factor at the very top of the escarpment for
the San Salvador earthquake was slightly less than the one produced by the El Centro
earthquake, as indicated in Figure 3.6. It is seen there that the average peak crest
acceleration for both earthquakes was about 1.5 times the average base acceleration, i.e.
the acceleration at points on the surface and away from the toe.
0.08
0.1
0.12
0.14
0.16
0.18
0 200 400 600 800 1000
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
Sc material
50
Figure 3.6 Comparison of the peak accelerations for a 30° escarpment caused by both earthquakes
Results for a 40 degrees slope
The peak acceleration along the free surface of the soil deposit and escarpment
was calculated using the two earthquakes previously mentioned. Figure 3.7 shows how
the peak acceleration varies at different points along the slope and at the ground along the
bottom and top levels. The results in the figure were obtained for a soil type SC. Here
again the San Salvador earthquake induces higher surface acceleration than the El Centro
earthquake. This may be due to the fact that the dominant frequency in the spectrum of
the San Salvador earthquake is closer to the fundamental frequency of the soil deposit.
However, the amplification factor for the El Centro earthquake is higher (1.64) than for
the San Salvador (1.35).
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0 50 100 150 200 250 300 350
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
A.F. = 1.51, El Centro Earthquake A.F. = 1.50, San Salvador Earthquake
51
Figure 3.7 Comparison of the peak accelerations for a 40° escarpment caused by both earthquakes
Results for a 50 degrees slope
The results of the most critical case for the 50° slope are shown in Figure 3.8.
This case corresponds to the El Centro earthquake and for a soil classified as SC. As in
the previous cases, there is a continuous increase of the amplification of the horizontal
accelerations as one move from the base to the top of the escarpment. The graph shows
an amplification of the crest motion with respect to the base of approximately 1.67.
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 50 100 150 200 250
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
A.F = 1.64, El Centro Earthquake A.F. = 1.35, San Salvador Earthquake
52
Figure 3.8 Peak acceleration for a 50° slope escarpment
Results for a 65 degrees slope
The results for the 65° escarpment presented here are those due to the San
Salvador earthquake and a SE soil which corresponds to a clay material. The variation of
the peak ground acceleration as a function of the horizontal distance is displayed in
Figure 3.9. It is noted that, in fact, the most critical case for the 65° slope was occurred
for the El Centro earthquake with a SC soil. This type of soil was considered as a sand
and it can never maintain stability at 65°. For this reason, this case was not considered.
The maximum amplification factor for the case in Figure 3.7 is 1.57 at the very top of the
slope.
0.08
0.1
0.12
0.14
0.16
0.18
0 20 40 60 80 100 120 140 160
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
Sc material, A.F.= 1.67
53
Figure 3.9 Peak acceleration for a 65° escarpment
From all the previous plots it can be observed that the minimum ground
acceleration is approximately 0.1g, which is the same value than the peak acceleration of
the earthquake at the base of the soil deposit. This means that the effect of the local
geology in the soil deposit with smaller depth is not significant, i.e. there is almost no
amplification due to the so-called site effects.
Examining the previous results one concludes that the maximum value of the
acceleration at the free surface was nearly 0.22g and it occurred for the 65° slope. The
amplification factors calculated in all the cases are defined as the ratios between the peak
accelerations at the top and at the level of the base of the escarpment but away from its
toe. The results presented also show that the higher amplification factors were obtained
for the El Centro earthquake. Nevertheless, the San Salvador earthquake caused higher
peak accelerations along all the surface nodal points.
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0 10 20 30 40 50 60 70
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
SE material, A.F. = 1.57
54
Finally, it must be pointed out that the plots previously presented are just a few
examples of the numerous cases evaluated. The next two tables summarize the
amplification factors obtained for all the soil types and the different slope angles. Table
3.2 corresponds to the El Centro earthquake and Table 3.3 shows the results for the San
Salvador earthquake.
Table 3.2 Amplification Factors for an Escarpment:
El Centro earthquake (0.1g)
Table 3.3 Amplification Factors for an Escarpment: San Salvador earthquake (0.1g)
3.6 A general equation for the amplification factor
The preceding section described the procedure followed to calculate the peak
surface accelerations and thus the amplification factors. In this section, we will develop a
general formula to calculate the amplification factor based on the procedure and results
θ° SB SC SD SE
15 1.00 1.48 1.33 1.4430 1.01 1.51 1.30 1.3840 1.01 1.64 1.18 1.2450 1.01 1.67 1.18 1.1965 1.04 1.75 1.13 1.19
Soil Classification (UBC 1997)
θ° SB SC SD SE
15 1.11 1.31 1.38 1.4030 1.17 1.50 1.20 1.3040 1.12 1.35 1.17 1.2250 1.12 1.40 1.62 1.5765 1.13 1.25 1.45 1.57
Soil Classification (UBC 1997)
55
obtained in the previous sections. The amplification factors can also be displayed by
means of tables. However, if possible, it is more expeditious to have a closed-form
expression that provides directly the amplification factors. This formula is convenient
because it can be easily incorporated into a computer program or into a seismic code, for
example. The formula must be a function of the height of the escarpment and the angle
of the slope. The equation sought must be able to produce accurate but conservative
results, for escarpments with slopes ranging from 0° to 65°. It is realized that the
escarpment and horizontal soil deposits can be formed by a single soil type or by a
combination of them. For instance, the soil of the deposit and the escarpment can be of
type SB, SC, SD or SE. Or, the deposit can have a soil type SB whereas the escarpment can
be SD, etcetera. In principle, one can develop an equation for the amplification factor for
each of these cases. However, this was deemed impractical. It is recalled that the
objective of this work is to come up with a simple (yet accurate and conservative)
methodology that can be incorporated into seismic codes. Therefore, it was decided to
use an unfavorable combination of soil types that can be encountered in practice with a
reasonable probability. The two soil profiles selected were SE for the escarpment and SC
for the deposit, irregardless of the angle of the slope. Figure 3.10 presents a sketch of the
soil system studied with the parameters used to derive the formula.
56
Figure 3.10 Identification of parameters
The escarpment is made up of a soil with a shear wave velocity VS1 equal to 575
ft/sec and the horizontal deposit is formed by a soil with VS2 equal to 1850 ft/sec for the
different slopes (15°, 30°, 40°, 50° and 65°). The amplification factors were calculated at
different ratios of the total height of the escarpment H.
Table 3.4 Amplification Factors for Different Escarpment’s Heights: El Centro Earthquake (0.1g)
y/H 15° 30° 40° 50° 65°0.00 1.00 1.00 1.00 1.00 1.000.13 1.04 1.08 1.09 1.14 1.150.25 1.21 1.19 1.19 1.26 1.260.38 1.41 1.34 1.31 1.38 1.360.50 1.55 1.47 1.42 1.48 1.470.62 1.65 1.60 1.53 1.57 1.630.75 1.79 1.73 1.62 1.88 1.970.88 2.05 2.14 1.94 2.29 2.421.00 2.24 2.31 2.07 2.52 2.77
Amplification FactorsAngle θ
θ
y
h = 50’VS1
VS2
57
Using the data on each column of Table 3.4, plots of the amplification factor as a
function of y/H were prepared. Figure 3.11 illustrates an example of the graph obtained
for a 40° slope. It can be seen in the figure that the relationship between the
amplification factor and the height’s ratio can be approximated with a straight line. Thus,
an equation of the form a * y/H + b was obtained using the data plotted in the figure.
Figure 3.11 Amplification factor as a function of the escarpment’s height ratio for a 40° slope
The process was repeated for each of the five different slopes. With this
information, a mathematical procedure was implemented to obtain the correct
coefficients of a general equation that is valid for all the slopes considered. The
mathematical procedure consists of setting and solving a system of equations where the
coefficients of the general equation are unknown. It begins by calculating the slope mi of
each of the straight lines similar to that in Figure 3.11 for each slope angle. Next, a set of
linear equations with the following form is established:
0.00
0.50
1.00
1.50
2.00
2.50
0 0.2 0.4 0.6 0.8 1
Fraction of escarpment's height
Am
plifi
catio
n fa
ctor
58
mi = a1θi4+a2θi
3+a3θi2+a4θi+a5 ; i = 1,2,3,4,5 (3.3)
where θI are the five angles of the slope in radians, and ai are the unknown coefficients.
Equation (3.3) can be represented in matrix form as follows,
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
θθθθ
θθθθ
5
1
5
1
52
53
54
5
12
13
14
1
m...m
a...a
1...............1
(3.4)
where the vector with the slopes mi is:
{ }
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=
6814.14652.10568.1304.12540.1
m
Each row corresponds to one of the angles studied. To obtain the coefficients a1, a2, a3,
a4, and a5 we need to solve the system of simultaneous equations. The solution vector
contains the coefficients of the slope in the linear equation sought:
{ }
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−
−
−
=
2664.60778.58
4614.1481228.1536040.54
ma
To evaluate the part corresponding to the intercept of the linear equation the
procedure is repeated, but the vector in the right hand side of equation (3.4) now
corresponds to the intercept of the straight lines for the different degrees. This vector is:
59
{ }
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=
8274.08791.09338.08865.09205.0
b
The solution of the second system of simultaneous equations is
{ }
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−
−=
2300.28909.95509.247033.24
6277.8
ba
Using these results one can write a general equation of the form,
)b()m(.F.A +α=
where .F.A is the amplification factor and α is the fraction of escarpment’s height =
y/Ho
The final result is the following equation that yields the amplification factor for any slope
from 0° to 65°:
( ) ( )θθ +α= BA.F.A (3.5)
where:
2664.60778.584614.1481228.1536040.54A 234 −θ+θ−θ+θ−=θ
2300.28909.95509.247033.246277.8B 234 +θ−θ+θ−θ=θ
radians in angle slope θ
For an escarpment with a given slope θ, the amplification factor assumes the following
form:
60
θ = 15° = 0.2618 rad → A.F. = 1.2540α + 0.9205
θ = 30° = 0.5236 rad → A.F. = 1.304α + 0.8865
θ = 40° = 0.6981 rad → A.F. = 1.0568α + 0.9338
θ = 50° = 0.8727 rad → A.F. = 1.4652α + 0.8791
θ = 65° = 1.1345 rad → A.F. = 1.6814α + 0.8274
To corroborate the results produced by equation (3.3) with the linear regression
equations for each slope, they are compared in Figure 3.12 for the case of θ = 40°
previously presented. It is observed that both lines coincide almost perfectly for the
whole range of escarpment’s height ratio.
Figure 3.12 The original line and the proposed equation for the 40° case
3.7 Nonlinear behavior of soils
Since its development by Seed and Idriss (1970) the Equivalent Linear Method
was and still is the most popular technique to calculate the nonlinear dynamic response of
0.000
0.500
1.000
1.500
2.000
2.500
0 0.2 0.4 0.6 0.8 1
Fraction of escarpment's height
Am
plifi
catio
n fa
ctor
s
Line Proposed equation
61
soil deposits subjected to bedrock acceleration. The method is valid for soils showing
small or medium non-linearities. It is an iterative procedure without a rigorous
mathematical foundation associated to it (Arroyo 1999). This method is incorporated in
QUAD4M, the computer program used in the research to evaluate the seismic response of
soil deposits.
To gain insight into the nonlinear behavior of the soils considered throughout the
present study, a special case was examined. An escarpment of 15 degrees and SD
material was subjected to the El Centro earthquake. The system was analyzed using
earthquakes with increasing peak ground accelerations (PGA). The peak accelerations
considered are 0.05g, 0.1g, 0.2g and 0.4g. Two individual elements of the FE model
were selected to study in detail their behavior. Here we present the results for the
element located at the middle height of the slope. The output of the program QUAD4M
gives the peak shear stress at each element. For the type of elements used in the program,
the stresses are constant within each element. The shear strain was obtained by dividing
the stress in the element by the shear modulus calculated at the last iteration where the
percent of difference is almost zero. The procedure was repeated for the four PGAs and
the shear stresses obtained are presented in Figure 3.13. Each diamond represents the
result for an earthquake with increasing peak acceleration. The plot of shear stress vs.
shear strain shows the nonlinear behavior of the soil and gives an idea of the nonlinear
constitutive model implemented in the program QUAD4M.
62
Figure 3.13 Stress-Strain curve for a typical finite element
It is recalled that in all the models studied in the thesis, except in the case study of
Chapter V, the peak acceleration of the earthquake applied at the bedrock is 0.1g.
Therefore, according to the graph in Figure 3.13, the soil is behaving only slightly
nonlinearly. As it was explained in a previous chapter, this is the behavior sought,
because as the soil enters more into the nonlinear range, it dissipates more energy and the
amplification is reduced. There is another reason why it is necessary to limit the
nonlinear excursions. It was mentioned in Chapter II (see Figure 2.3) that the program
QUAD4M uses curves of the effective shear modulus and effective damping ratio as a
function of the maximum shear strain γ to implement the Equivalent Linear method.
These curves, however, are defined for a limited range of γ and it was observed that when
very strong earthquakes were applied to the model, the program experienced convergence
problems. It is suspected that these problems are caused because the values of γ exceed
that range of the curves mentioned before.
0
20
40
60
80
100
120
0 0.02 0.04 0.06 0.08 0.1
Shear Strain
She
ar S
tress
(ksf
)
63
3.8 Summary
This chapter presented the results of the seismic response of an escarpment-soil
deposit system by means of a finite element analysis. Many models of the soil system
were considered, to examine the effect of the different parameters on the peak surface
acceleration. In the next chapter a similar study will be presented, but using a hill-soil
deposit system to complete the evaluation of the effects of the surface topography in the
seismic waves.
The data obtained for all the cases indicate that there is an increase in the ground
acceleration along the surface of the slope compared to the original acceleration at the
bedrock. The results indicate that a gravel material, type SC in the UBC 97 classification,
causes greater amplification for all the slopes considered. Another parameter considered
in the study is the earthquake ground motion acting at the rigid base. Two historic
earthquakes with different characteristics were used as input. The results of the
numerical simulations indicate that the amplification ratios between the top and base
were greater for El Centro earthquake than for the San Salvador earthquake. However,
the San Salvador produced higher peak accelerations, in absolute terms.
The highest value of the amplification factor was obtained for a material type SC,
a cohesionless soil, in an escarpment at a 50° angle. Performing a parametric study of
the effects of topographic irregularities, analyzing the results, and obtaining the most
severe cases so that these effects can be considered, as part of the building codes is the
main objective of the research. A brief summary of the most critical cases for both
seismic motions is presented in Table 3.5. The table lists the highest amplification
64
factors for different types of soils and escarpment’s angles. It is noted that for the SC soil
type, the table does not present any value for 65°. The same happens for a SD soil and
angles higher than 50°. As it was explained in the previous chapter, these materials do
not exhibit stability for those slopes.
Table 3.5 Most Critical Amplification Factors Considering Both Earthquakes
θ° SB SC SD SE
15 1.11 1.48 1.38 1.4430 1.17 1.51 1.30 1.3840 1.12 1.64 1.18 1.2450 1.12 1.67 1.5765 1.13 1.57
Soil Classification (UBC 1997)
65
Chapter IV
Amplification of Seismic Motion Due to Hills
4.1 Introduction
The other surface irregularity to be considered in this study is the presence of a
ridge or hill at the top of the soil deposit. As it was done in the case of the escarpment,
the seismic input will be defined by an accelerogram applied at the bedrock. Only one
horizontal component of the earthquake will be considered. The same historic
earthquakes used in the previous chapter, namely the El Centro and San Salvador ground
motions are employed here. Both are scaled so that their PGA is equal to 0.1g. Because,
as it was discussed in Chapter IV, the type of soil profile plays a significant role in the
amplification, here the same four soils from the UBC-97 code are used in the model of
the system formed by the deposit and the hill. Evidently the number of possible
geometric configurations of hills is infinite. Therefore, it is necessary to choose a general
shape that can represent the different hills. The possible configurations include, for
example, half-sines, triangular shapes, trapezoidal shapes, etc. It was decided that the
most appropriate shape would be a smooth curve that at the same time is simple enough.
A parabola satisfies both conditions and thus it was the shape selected to represent the
cross-sections of the hills. Four hills with increasing heights are studied. The width of
the base of the hill is kept constant, and thus the hills differ in their aspect ratio. One of
the goals of this chapter is to derive a general formula that allows one to calculate the
66
maximum amplification factors for hills with different aspect ratios at an arbitrary point
along the hill.
4.2 General considerations
This chapter completes the study of the effect of topographic irregularities on the
acceleration induced by earthquake acting at the bedrock. The cases studied are hills
with different dimensions and the methodology used is the same than the one presented in
Chapter III. First a stability analysis of the models of the hills based on the angle of
repose was performed. The procedure followed to do this type of study was the same as
the one applied in the previous chapter to escarpments. The maximum angle of the
slopes at the bottom of the hills varies according to the type of material. For example, the
natural slopes for cohesive materials can be very high, whereas those of granular
materials principally depend on their angle of internal friction. These features were taken
into account to establish limits to the aspect ratio of the hills, based on the type of soil.
The hills were modeled as parabolas with tangent equal to zero at the top,
maximum height n and width of the base m (see Figure 4.1). To facilitate the
construction of the finite element mesh, a general equation was derived to obtain the
coordinates of the parabola.
The quadratic equation that describes the contour of the hill is
cx*b2x*ay ++= (4.1)
where the three coefficients a, b, and c are:
2mn*4a −
=
67
21
m)x*2m(n*4
b+
=
111 y
mx1
mx*n*4c +⎟
⎠⎞
⎜⎝⎛ +
−=
where (x1, y1) are the coordinates of the left point at the base of the hill (see Figure 4.1).
Figure 4.1 Parameters for the equation of the parabola
Knowing the points (x1, y1), the maximum height (n) and the length of the base
(m), the coordinates of the points in the FE mesh along the parabola were obtained with
equation 4.1 and a spreadsheet. To minimize the number of cases to be studied, the
dimension m was fixed to 400 ft, and the following four values of n were considered: n =
38 ft, n = 75 ft, n = 115 ft, and n = 225 ft. Only two cases for n = 225 ft were analyzed
due to the problem of the stability of the natural slope. The slopes corresponding to the
values of n and m listed before are 21°, 37°, 49° and 66°, respectively. The soil deposit
depth beneath the hill was taken equal to 100 ft and the lateral extension from the hill’s
toes to the borders was equal to 200 ft in all the cases. The properties of the soil used as
input are the following:
n
m
y
x
x1,y1
68
Unit weight γ = 125 pcf
Poisson’s ratio ν=0.35
Damping ratio ξ = 0.05
The selection of the soil material used in the study was based on the UBC-97
Code as discussed in Chapter III. Only four soil categories were used: SB, SC, SD and SE,
and the shear wave velocities employed were the average of the values used in the code
to separate the different types. Using the shear wave velocity, the soil’s unit weight and
equation 3.2, the Q4MESH program calculates internally the shear modulus. The curves
for the variation of the shear modulus and damping ratio as a function of shear strain
were selected by trying to match the average shear wave velocity obtained from the
UBC-97 code with a suitable curve available in the program QUAD4M.
In the same manner as done with the escarpment, two historic earthquakes scaled
to have a maximum acceleration of 0.1g were used. The accelerograms are those of the
El Centro and San Salvador earthquakes. According to the objective of the study, only
the most critical cases were tabulated and are presented here. The finite element models
were always evaluated with both earthquakes to obtain the maximum amplification. The
final results presented here for the models of the hills with n = 38 and 225 ft were
obtained using the El Centro earthquake whereas those for the models with n = 75 and
115 ft were calculated with the San Salvador earthquake.
69
4.3 Ridge amplification results
This section present the results of the different cases and configurations of hills
examined. The effect of the different soil types and the two earthquakes on the
amplification of the acceleration signal at the hill’s surface is studied. The finite element
models analyzed are those illustrated in Figures 2.7 and 2.8 (section 2.6). By varying the
height of the hill four meshes were generated. These models were evaluated using the
finite element program QUAD4M. The results of the seismic analysis are displayed in
graphical form, as a plot of the peak acceleration along the hill and horizontal free
surface. At the end of this section the maximum amplification factors found as the
function of the height of the hill and the type of soil profile are presented.
The results of the study reported here are based on a horizontal soil deposit and
hill made up of only one type of material. Figure 4.2 shows the nodes of the finite
element model at the surface where the acceleration was calculated. In the following
subsections, the presentation of the results is divided according to the height of hill. The
graphs presented next are the most critical cases for each model. Therefore, in some
cases the seismic input is the El Centro earthquake and in other models the results
presented are those for the San Salvador ground motion.
Finally, the results of the study agree quantitatively with those of other studies
previously mentioned. They indicate that the amplification increases from the base to the
top of the hill.
70
Figure 4.2 Nodes at the surface of the model of the hill
Hill with height n = 38 ft
Figure 4.3 presents the peak acceleration obtained for a hill with a total height of
38 ft and aspect ratio n/m = 0.095. These results were obtained considering that the
combined soil system consists of a SE material or clay and using the El Centro
earthquake. The plot shows how the peak ground acceleration changes due to the
presence of the irregularity. The maximum values occur at the top of the hill and near the
boundaries of the model. The maximum amplification factor for this case is 1.53.
Figure 4.3 Peak ground acceleration for a hill with n = 38 ft
0
0.04
0.08
0.12
0.16
0.2
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
El Centro Earthquake, A.F.=1.53
71
Hill with height n = 75 ft In this subsection the height of the hill was doubled with respect to the previous
case. The San Salvador earthquake was selected to generate the numerical results
because it produced higher values of amplification for this hill. This case was evaluated
for the four soil types and it was found that the SE material (or clay) produces higher
accelerations. Therefore, the curve in Figure 4.4 is for this material. The maximum
value of the absolute acceleration occurred at the top of the hill and is approximately
equal to 0.22g. The points at the base of the hill actually show deamplification although
not very significant: the peak acceleration is slightly less than 0.1g. According to the
graph, the maximum amplification ratio at the top with respect to the toe of the hill is
2.34.
Figure 4.4 Peak ground acceleration for a hill with n = 75 ft
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
San Salvador Earthquake, A.F.=2.34
72
To gain further insight into the behavior of the soil system and to verify the small
deamplification observed, a model of the soil deposit without a hill is presented next.
The dimensions of the soil deposit are equivalent to the model with a hill of n = 75 ft.
The horizontal extension of the soil deposit is 800 ft and its depth is 175 ft, i.e. the same
as the total height of the previous model. Figure 4.5 presents the results for a SE material
and the San Salvador earthquake. As it was expected, the plot shows that the peak
acceleration is constant when no irregularities are present. The graph shows that the
value of the deamplification at the top of the soil deposit agrees very well with the results
at the toe of the hill in the previous case (Figure 4.4).
Figure 4.5 Peak acceleration at the top of a soil deposit without irregularities
Hill with height n = 115 ft
This case was also evaluated for the different soils and the two earthquakes.
However, the SE material and the San Salvador earthquake yielded higher accelerations.
0.09
0.1
0.11
0.12
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
San Salvador Earthquake, SE material
73
The results are presented in Figure 4.6 and they show that there is an amplification of the
acceleration along the surface of the hill. The maximum amplification ratio is 1.60.
Figure 4.6 Peak ground acceleration for a hill with n = 115 ft
Hill with height n = 225 ft
This model was only evaluated for two soil profiles: SB and SE. The soils
classified as SC and SD, granular materials, are not able to maintain stability at this slope,
and for this reason they were not considered. The variation of the peak acceleration for a
SB soil and the scaled El Centro earthquake (the most critical one) is presented in Figure
4.7. At the very top of the hill there is an amplification of 2.35 times the acceleration at
the toe of the hill.
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
San Salvador Earthquake, A.F.=1.60
74
Figure 4.7 Peak ground acceleration for a hill with n = 225 ft
General comments
It is important to note that all the plots previously presented are only examples of
many cases evaluated. The next table summarizes the amplification factors obtained
using the four models with different heights of the hill and the four types of materials.
The amplification factors are defined as the ratio between the maximum acceleration at
the surface of the hill with respect to the peak acceleration at the toe of the hill. The
maximum value of the surface acceleration for all the cases examined was nearly 0.22g
and it occurred in the n = 75 ft model. The maximum amplification factor is 2.35 and it
was obtained for the n = 225 ft model.
Table 4.1 Maximum Amplification Factors for the Hill
0
0.04
0.08
0.12
0.16
0.2
0.24
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
El Centro Earthquake, A.F.=2.35
n (ft) n/m SB SC SD SE
38 95/1000 1.01 1.52 1.46 1.5375 188/1000 1.34 1.93 2.16 2.34115 288/1000 1.48 1.47 1.60225 563/1000 2.35 0.90
Soil Classification (UBC - 97)
75
4.4 A general equation for the amplification factors
Table 4.1 can be used to estimate the maximum amplification factors expected on
hills with four different aspect ratios n/m and a homogeneous soil. It is recommended to
interpolate the values of the table for cases not covered there but reasonably close. To
evaluate other models that cannot be associated to the cases presented in the table of the
previous section, in this section we evaluate a special case to generate a general equation.
With this equation the user can obtain the amplification factor for hills with any aspect
ratio within a certain range. The procedure followed to derive the equation is very
similar to the one followed for the case of the escarpment. The resulting equation is a
function of the ratio between the height, and the width of the hill and the location of the
point, defined as a fraction of height of the ridge. The validity of the equation and thus
the amplification factors is limited to values of n/m from 0.095 to 0.56. The equation was
derived for a soil deposit-hill system formed by two different materials for the deposit
and for the hill. Figure 4.8 shows the parameters involved in the case studied.
Figure 4.8 Parameters identification
n
m
y
Vs1
Vs2
76
The shear wave velocities were taken equal to Vs1 = 575 ft/s and Vs2 = 1850 ft/s,
whose values correspond to soil profiles SE and SC according to the UBC-97
classification. Table 4.2 presents the amplification factors calculated using heights with n
equal to 38, 75, 115 and 225 ft at different elevations expressed as percent of the total
height of the hill.
Table 4.2 Amplification Factors for Hills of Different Heights
The procedure to determine the formula sought begins by plotting the
amplification factors as a fraction of the height of the hill for each model defined by its
ratio n/m. This is illustrated in Figure 4.9 for a hill with height/width ratio of 95/1000.
The graph suggests that a cubic equation could be used to represent reasonably well the
general pattern. Therefore, a cubic equation of the form
ii2
i3
i D*C*B*A.F.A +α+α+α=
y 38 75 115 2250n 1.00 1.00 1.00 1.00
0.22n 1.19 1.32 1.00 1.050.38n 1.32 1.93 1.27 1.330.55n 1.49 2.50 1.59 0.880.67n 1.71 2.42 1.61 0.710.77n 1.84 2.42 1.63 0.920.84n 2.03 2.60 1.77 1.090.91n 2.14 2.72 1.89 1.28
n 2.87 2.85 2.08 1.50
Amplification Factor (m = 400 ft)n (ft)
77
was derived for each of the four hills with different ratio n/m.
Figure 4.9 An example of a cubic trendline for n/m = 0.095
A mathematical procedure similar to that discussed in Chapter III was used to
obtain the appropriate coefficients of a general equation, also a cubic polynomial, which
must be able to cover the four cases. Equation 4.2 is the resulting formula that provides
the amplification factor (A.F.) for hills with height-to-width ratios from 95/1000 to
563/1000.
432
23
1 a*a*a*a.F.A +α+α+α= (4.2)
where:
257.15r*012.163r*294.499r*043.423a 231 +−+−=
454.10r*593.82r*400.158r*404.33a 23
2 −+−=
518.4r*964.105r*780.456r*585.516a 233 −+−=
127.1r*283.2r*491.8r*724.8a 23
4 +−+−=
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 0.2 0.4 0.6 0.8 1 1.2
Fraction of height of ridge
Am
plifi
catio
n Fa
ctor
78
:m/nr = ratio between the height and width of the base of the hill
α = fraction of height of hill
Figure 4.10 compares the curve obtained with the proposed equation (4.2) and
with the original best-fit cubic curve passing through the data points. The latter is the
same curve presented in Figure 4.8. Figure 4.10 highlights the accuracy of the proposed
general equation.
Figure 4.10 The cubic trendline and the general equation for n/m = 0.095
4.5 Frequency analysis
According to the previous works discussed on Chapter I and the results obtained
from the escarpment analysis, it was expected that, as the elevation of the irregularity
increases, so do the acceleration. However, this was not necessarily the case in the study
of hills. As an example, the peak surface acceleration for hills with n = 38 ft and n = 75
ft are compared in Figure 4.11, where the difference with the previous patterns are
0.000
0.500
1.000
1.500
2.000
2.500
3.000
0 0.2 0.4 0.6 0.8 1
Fraction of height of escarpment
Am
plifi
catio
n fa
ctor
Cubic Proposed
79
evident. Note that at the top of the hill a greater peak acceleration was obtained for the
case of n = 38 ft, i.e. for the lower hill.
Figure 4.11 Comparison of ridge amplification subjected to El Centro earthquake
The results in Figure 4.11 are for the case where El Centro earthquake was
applied. It is noted that when the San Salvador earthquake is used as input, the results
obtained are those expected. Two types of soils were used in the case of Figure 4.11: a
soil SC for the bottom deposit and a soil SE for the hill. Explaining the reasons for this
behavior requires looking at the amplification problem from another point of view. The
cause may be related to the frequency content of the earthquake compared to the natural
frequencies of the soil deposit. If some of the frequencies of the dominant components of
the seismic motion coincide with a lower frequency of the soil deposit, the seismic waves
will be further amplified.
To check whether the explanation for the results in Figure 4.11 are due to a
resonant-like condition, the case of a hill with height n = 38 ft was evaluated using the El
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 100 200 300 400 500 600 700 800 900
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
n/m=95/1000 n/m=188/1000
80
Centro ground motion and two soil profiles: a material type SC and next a type SE soil.
The natural frequencies of the soil deposit-hill system were calculated using a finite
element program for two-dimensional problems written in FORTRAN (Arroyo 1997).
The reason for switching programs is that the program QUAD4M does not compute the
natural frequencies of the soil system. The results of the frequency analysis were the
following. The soil system with a SC material has a fundamental frequency of 25.27 rad/s
whereas the frequency diminishes to 7.85 rad/s for the SE material. The frequency
spectrum of the El Centro earthquake shows many peaks since it corresponds to a typical
broad band signal. However, the relative importance of the lower frequencies produces
more amplification in the n = 38 ft case.
Because of the interdependence between the fundamental frequency of the soil
deposit-hill system and the frequency content of the earthquake in the amplification
obtained, it was necessary to determine for each case which one is the most severe
earthquake. Only this earthquake was used to calculate the amplification factors for the
different hills. Table 4.3 summarizes the earthquake used to analyze each of the hills
with different heights.
Table 4.3 Most Severe Seismic Motions for the Analysis
Height of hill n height/width ratio Earthquake(ft) n/m38 95/1000 El Centro75 188/1000 San Salvador115 288/1000 San Salvador225 563/1000 El Centro
81
4.6 Summary
This chapter presented a dynamic finite element analysis of different models of
soil deposit-hill systems to calculate the peak acceleration at the surface nodes. The
pattern of results observed indicate that there is amplification along the hill in all the
cases. All the cases except the one for a hill of height n = 38 ft shows a deamplification of
the absolute acceleration at the toe of the hill. Table 4.4 shows a summary of the highest
amplification factors for the different hill geometries regardless of the type of soil
material. To obtain these values, all the material types were considered, with the
exception of the SC and SD soils in some cases. As it was cited before, for n = 225 ft
these granular materials do not exhibit stability for hills with aspect ratio of 563/1000 or
higher. For an unfavorable combination of soil types, a general formula that yields the
amplification factor at different elevations along the hill was derived. The formula is
rigorously valid for hills with aspect ratio n/m from 95/1000 to 563/1000. Hills with
lower height-to-width ratios show negligible amplifications. Hills with values of n/m
higher than 0.563 are not likely to be the site of residences.
Table 4.4 Summary of Maximum Amplification Factors for Different Hills
n (ft) n/m Amplification Factors38 95/1000 1.5375 188/1000 2.34115 288/1000 1.60225 563/1000 2.35
82
Chapter V
A Case Study in Puerto Rico
5.1 Introduction
The effects of surface topography on the seismic ground response have been
subjected to numerous studies during the last two decades. The present study and others
mentioned in Chapter I verified via numerical simulation the amplification in the ground
accelerations due to surface irregularities. This phenomenon has been observed in the
field several times, especially in countries such as Italy or Greece, in which there are
many seismic regions with an irregular topography. Because of its unique geography,
also in Puerto Rico have many regions prone to experience topographic seismic
amplification.
The objective of this research is to perform a parametric study to account for the
amplification of seismic waves in hills and escarpments. The final goal is to come up
with a simple methodology to account for this effect in the seismic codes. Chapter III
and Chapter IV examined the effects of two irregularities with idealized geometries on
the amplification for two earthquakes with distinct characteristics. In this chapter a case
study of potential topographic amplification in Puerto Rico will be described and
analyzed. This real case differs from the ideal models used in the previous two chapters.
The geometry of the irregularity is rather complex and so is the soil profile at the site
under study. To define them, it was necessary to use topographic maps as well as site
inspections to take soil samples for laboratory testing. Moreover, an artificial earthquake
83
especially generated for the south of Puerto Rico was used as seismic input, instead of the
historical earthquakes used previously.
5.2 Geographic conditions of Puerto Rico
Puerto Rico is part of the Greater Antilles Island chain in the Caribbean Sea. It
lies approximately 100 miles south of the Puerto Rico Trench, a depression of the ocean
floor that reaches depths of 30,249 feet, the deepest depth known in the Atlantic Ocean.
The Island has an approximate rectangular shape that extends about 110 miles from east
to west and 40 miles from north to south. The island has a high population density that
reached 3.8 million inhabitants in the year 2000. Puerto Rico is a mountainous island
with central highland areas that rise to a maximum altitude of about 4,400 feet above sea
level. The Cordillera Central, the Sierra de Luquillo, and the Sierra de Cayey are
generally oriented east-west and dominate the mountainous southern two-thirds of the
island, as illustrated in Figure 5.1. An area of gently dipping limestone that has been
deeply dissected by dissolution forms a wide band of karst topography along most of the
north. Flat-lying coastal plains and alluvial valleys compose a discontinuous belt around
much of the periphery of the island. The coastal plain is especially prominent along part
of the south coast where adjacent streams to form a broad, continuous plain (USGS 1997)
deposited coalescing fan deltas.
84
Figure 5.1 Topographic view of Puerto Rico (Rivera 1995)
Clearly, the geography of Puerto Rico, along with the social and economic
conditions that affect the population distribution, makes a study of the topographic
amplification worthwhile in many regions of the Island. For example, Figure 5.2 shows a
typical view of residential structures located on hills and slopes where topographic site
effects can exacerbate the consequences of a strong earthquake. The picture was taken in
Yauco near the city of Ponce at the south of the island.
85
Figure 5.2 Residences located on the hill
5.3 Site location
A review of the topographic maps of Puerto Rico, indicates that there are many
regions that may be prone to seismic amplification due to topographic irregularities. An
area near the town of Guánica, in the south of the Island was chosen for study. The town
of Guánica is indicated within an ellipse in Figure 5.3. Near this town there are
numerous hills surrounded by the flat land of the Valle de Lajas.
Figure 5.3 Map of the municipalities of Puerto Rico showing the location of Guánica
86
This location was selected to be studied because there are many residences
established on the hill slopes. However, it was later called to our attention that it is
suspected that the zone contains irregular soil deposits. Although it was assumed in the
previous chapters that the soil layers are horizontal, the numerical model used in the
study can handle irregular deposits. The problem is that usually the profiles of these
deposits are unknown. Even if borings are made to obtain the soil properties and depth of
the layers, this information is not enough to determine the geometry of the irregular
layers, unless a large number of borings are done. Figure 5.4 presents a view of one of
these communities. The hill is crowned by a tank of water at its top.
Figure 5.4 View of the hill selected for study
The community in the picture is known as Caño and is located at the intersection
of roads PR-2 and PR-116. Figure 5.5 shows a map of this zone and the section A-A’ or
front view used to prepared the finite element model. This section cuts the hill across the
highest point. The number (1) in the map marks approximately the point of view for the
picture in Figure 5.6.
Water tank
88
Figure 5.6 View from road PR 116 of Caño Hill
5.4 Study of soils at the site
After selecting the area of study, an intensive search was conducted to identify the
soil profile for the Caño hill. Opposed to other models previously studied, in which a
homogeneous soil profile was used, this case required use of a stratified profile. To
evaluate the site conditions a visual inspection was conducted and simple laboratory tests
were prepared. In addition, the soils were identified with the help of the maps of the
National Soil Survey. During the field visit a number of samples of soils at different
elevations of the hill were collected. Those samples presented different textures and
colors. The soil samples were tested in the geotechnical laboratory of the Civil
Engineering and Surveying Department of UPR-M. The tests of Liquid Limit, Plastic
Limit, moisture content were carried out and the unit weight was obtained. Once these
characteristics were available, it was possible to identify the soils according to the UBC
89
97 classification and the proper materials curves available in the Q4MESH computer
program were selected. To verify these properties, the soils in the region of Caño were
also evaluated according to a soil map prepared by the National Soil Survey (USDA
1963). Figure 5.7 shows a soil map of the Valle de Lajas Area with the soils identified
with letters and numbers. The first capital letter is the initial of the soil name. The
second capital letter shows the slope and the final number (2) indicates that the soil is
eroded.
The soil profile of the Caño Hill consists of five types of materials. These types
vary from clay to gravel. In turn, the clay types vary in unit weight and in their plasticity
indices. The hill and the surrounding valley consist mostly of clay, but the top of the hill
is composed of rock fragments, limestone and other dense materials. Table 5.1
summarized the properties of the materials analyzed: unit weight γ, shear wave velocity
VS in ft/s, Poisson’s ratio ν, and damping ratio ξ.
Table 5.1 Materials Properties for the Model of the Caño Hill
Material Identification γ (pcf) VS (ft/s) ν ξDense soil and soft rock 125 1850 0.35 0.05
Clay 90 900 0.35 0.05Clay 100 900 0.35 0.05
High plasticity Clay 90 575 0.35 0.05Dense soil and soft rock 100 1850 0.35 0.05
91
5.5 Seismic excitation
It is known that the crust of the Earth consists of approximately twelve tectonics
plates. The relative movements of the borders of those plates produce
different kinds of earthquakes. The seismic risk of Puerto Rico is due to the fact that it is
located between the plates of the Caribbean and North America, as illustrated in Figure
5.8.
Figure 5.8 Seismic activity in America and Caribbean (Yeats et al 1997)
92
Hence, Puerto Rico is located in an active seismic zone that extends from
America Central to Venezuela and is classified by the UBC 97 as seismic zone 3. The
history of Puerto Rico has witnessed several strong earthquakes that occasioned serious
damages. Those earthquakes occurred in times where the majority of building
constructions was in timber and the population was scarce compared to present levels.
The damages that a strong seismic event can cause nowadays are much greater than in the
past, due to the population increase and the many constructions vulnerable to ground
motions.
Puerto Rico suffered four strong earthquakes since the colonization by the
Spaniards. The most recent one occurred on October 11, 1918, and its epicenter was
located at the northwest of the Island. The magnitude of that earthquake is estimated to
have been approximately 7.5 in the Richter scale. The damages were restricted
principally to the west region due to the proximity to the epicenter. The consequences of
the event were 116 deaths and millions of dollars of losses. Many residential
constructions, public buildings, bridges and other edification suffered significant damage.
Other strong earthquakes affected the Island in 1867, 1787 and 1670, but the data about
them is scant.
The study presented in Chapter III and IV used two historical earthquakes as
seismic input, namely the El Centro and San Salvador. In this part of the research we
will evaluate a case study of topographic amplification in Puerto Rico. Therefore, we
decided to use as input applied at the rigid base rock an artificial accelerogram
compatible with a new design spectrum recommended for Puerto Rico in a recent study
93
(Irizarry 1999). The study, conducted at the Civil Engineering and Surveying
Department of UPR-M, defined ground design spectra for four cities in the Island and a
sample of artificial accelerograms compatible with the spectra. The seismic excitation
used as input motion for the finite element model was scaled to 0.3g. Table 5.2 shows
the main characteristics of the original accelerogram from the study mentioned. Also,
Figure 5.9 illustrates the acceleration time history and Fourier spectrum for the
accelerogram.
Table 5.2 Characteristics of an Artificial Earthquake for Puerto Rico
Duration
(secs)
Peak
Displacement
(cm)
Peak Velocity
(cm/s)
Peak
Acceleration
(cm/s2)
10.02 11.78 55.25 -451.26
(a)
0 1 2 3 4 5 6 7 8 9 10
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Time [sec]
Acc
eler
atio
n [%
g]
94
(b)
Figure 5.9 Acceleration time history (a) and Fourier Spectrum (b) of the artificial earthquake
5.6 The finite element model
As mentioned previously, this chapter presents the topographic amplification
study of a real case, i.e. a hill located in Guánica, P.R. The hill and the underlying soil
deposit were discretized with finite elements using the QUAD4M program. Unlike the
configurations analyzed in previous chapters where the hills were described by perfect
curves, in this study a topographic map was used to evaluate the real natural elevations
from the sea level. Figure 5.6 (section 5.2.1) presents a view of the hill to be studied
from road PR 116. The map in Figure 5.5 shows the section studied and the location
from where the photo in Figure 5.6 was taken.
Figure 5.10 displays the 2-D finite element mesh generated for the Caño Hill.
The model has 539 elements, 584 nodes and a total of 1168 degrees of freedom. The
0 10 20 30 40 50 60 700
500
1000
1500
2000
2500
3000
3500
4000
Frequency [rad/sec]
Am
plitu
de
95
figure shows the different elevations taken from the map and the valley surrounding the
hill. The maximum elevation was 344 ft with respect to the level of the ground surface
(see Figure 5.11). The extension of the plains at both sides was 2952 ft. The figure also
shows the boundary conditions used at the bottom and sides of the model. The bedrock is
considered to begin at the location of the fixed nodes at the bottom.
Figure 5.10 Finite element mesh for the Caño Hill
5.7 Results of the numerical simulation
This section presents the results of the numerical simulation of the topographic
amplification problem. It must be pointed out that this case differs from the theoretical
cases studied in Chapter IV. In the previous chapter, the materials at the site were
assumed to be homogeneous and the ridge was considered to be isolated. In this case,
however, the soil profile is quite different from a simple homogeneous halfspace and
different kinds of materials are used in the FE model. Also, the Caño Hill cross section
has three different heaps and thus it is not an isolated hill as in the cases studied in
Chapter IV. Figure 5.11 displays the coordinates for the mesh generated. The horizontal
coordinates in this figure will help in the interpretation of the results to be presented next.
96
These more general types of topographic feature present additional scattering and
diffraction of seismic waves and there could be amplification or deamplification of the
seismic motion depending on the point considered. This can be clearly seen from the
graph of the peak accelerations presented in Figure 5.12. The plot shows the peak
acceleration at all the surface nodes of the model. The results in Figure 5.12 are peculiar
in the sense that the maximum amplification not necessarily occurs at the top of the hills.
Actually the highest amplification occurs at the valley on the right side of the mesh, and
at an elevation of 40% of the maximum height of the hill. There are many points or
nodes that present a deamplification, in particular those at the tops of the hills where the
material type is gravelly with limestone fragments. The materials where amplification
occurs are either cohesive materials or different types of clays.
Figure 5.11 Coordinates for the Caño Hill mesh
97
Figure 5.12 Acceleration results for Caño Hill
It is known that the seismic waves are amplified on sites where the soil deposits
are soft and with significant thickness. Those areas generally include alluvial valleys and
zones of drained ponds and lakes. During an earthquake, those sites shake with more
intensity and for more time. It is important to mention that Caño Hill is located beside a
river. Indeed, the valley is part of a river basin.
To generate and tabulate the amplification factors for this local topography, it was
necessary to calculate average values of the peak acceleration at all the points at a given
elevation. The reason for presenting the results in this form is because the mesh is not
symmetrical and the surface acceleration obtained is not equal for all the points at a same
elevation. Figure 5.13 presents the variation of the average peak acceleration with
respect to the fraction of the hill’s height. It is recalled that the seismic motion applied at
the bedrock has a PGA = 0.3g. Finally, Table 5.3 illustrates the amplification or
deamplification factors depending on the elevation considered.
0
0.1
0.2
0.3
0.4
0.5
0.6
-4000 -2000 0 2000 4000 6000 8000 10000 12000
Horizontal Distance (ft)
Pea
k A
ccel
erat
ion
(%)
98
Figure 5.13 Average acceleration at different elevation
Table 5.3 Amplification and Deamplification Factors
The next two figures show the distribution of stresses and accelerations in the full
mesh predicted by QUAD4M and plotted using the program Q4MESH. Figure 5.14
illustrates the shear stress distribution where the maximum value is represented by a red
color with a value of 4753 psf and the minimum values are represented by a blue color
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
Average Acceleration (g)
Frac
tion
of H
eigh
t
Fraction of Amplification Deamplification Height of Hill Factor Factor
0.05h 1.000.14h 1.150.29h 1.460.43h 0.910.55h 0.630.62h 0.490.71h 0.450.76h 0.420.81h 0.490.9h 0.350.95h 0.40
h 0.42
99
with a value of 151.4 psf. The figure shows that the maximum stresses occur at the base
of the model.
Figure 5.14 Stress distribution
Figure 5.15 illustrates the distribution of peak accelerations over the model. The
range of values goes from 0.07 to 0.48 g. Here again, the red color and the blue color
represent the maximum and minimum values, respectively. The minimum values are at
the top of the hill where the material type is cohesionless with fragment of rocks. On the
contrary, the maximum accelerations are found at the toe of the hill, where the majority
of the residential structures are located.
Figure 5.15 Acceleration distribution
100
5.8 Comparison with the proposed simplified methodology
It is interesting to compare the results of the Caño hills with those from the
general amplification formula and the data in the table generated in Chapter IV for ideal
hill cases. To use the table or formula, we need to know the ratio between the height of
the hill and the length of its base. For the Caño hills a ratio of 0.095 was used to
calculate the amplification factors. According to Table 5.3, the amplification factor at
14% of the maximum height is 1.15, whereas the equivalent result with the proposed
equation at the same elevation is 1.17. At 29% of the height, the real case (i.e., the study
done in this chapter) shows an amplification of 1.46, while the proposed equation
predicts an amplification factor of 1.28. If we calculate an amplification factor as the
average value of the amplification factors for the four materials and for n/m = 0.095, we
obtain 1.38. It is important to have in mind that the proposed general equation was
derived using only two types of soil materials and the amplification factors listed in Table
4.1 of Chapter IV were generated for soil deposits and hills of uniform materials.
Therefore, the results obtained with the methodology of Chapter V are acceptable
considering the differences in the material profiles and the complicated geometry of the
real case studied.
5.9 Summary and final comments
In this chapter a real case of topographic amplification was described and studied
in detail. The hill or group of hills analyzed is located in Guánica, Puerto Rico. The
conditions of the surface topography are very particular. Firstly, an extended valley or
101
plain surrounds the hill, beside the toe of the hill there exists a river and the soil material
of the zone is not homogeneous. The soils consist of clays with different plasticity
indexes, and the top of the hill consists of a cohesionless material with rock fragments.
Because of these characteristics, the results do not follow the same trend than in the
theoretical cases studied in the previous chapter. In particular, the types of materials play
an important role in this kind of analysis. Nevertheless, the numerical simulation done
with the finite element program showed that there is significant amplification.
The amplification factors predicted with the formula and table presented in
Chapter IV were compared with those of the real case of the Caño hills. The accuracy of
the results obtained with the proposed simplified methodology is acceptable, given the
vast differences in geometry and material composition between the idealized cases used
to derive the formula and table and the actual site studied.
In closing, it is important to underscore that in the case of the Caño community
the highest amplification occurred at almost 0.3 of the maximum height of the hill. This
indicates that there is a clear and present danger in the event of a severe earthquake: the
majority of the residential structures are located at about this elevation. To compound the
problem, many of the houses have slender columns with the rooms located in the second
floor. In fact, they are cases of cantilevered-column building systems, structural
configurations that are particularly vulnerable to seismic motions. The study of the
seismic response these residential structures will be the object of a future investigation.
102
Chapter VI
Conclusions and Recommendations
6.1 Summary and conclusions
This thesis presented a numerical study of the phenomenon known as
“topographic amplification” of the seismic waves. The seismic response of interest was
the absolute acceleration at the free surface of the topographic irregularities. Two
irregularities were studied: an escarpment or embankment and a hill or ridge, both
considered as 2-dimensional, i.e. their profile is the same in any cross-section parallel to
the plane of analysis. Two acceleration time histories of past earthquakes with different
characteristics, the 1940 El Centro and the 1986 San Salvador earthquakes, were used as
excitation. Both accelerogram were scaled so that their peak ground acceleration is 0.1g.
The finite element method was used for modeling the soil system and calculating the
response of the topographic irregularities when they were subjected to a seismic motion
at the underlying bedrock. The program QUAD4M was used along with the interface
program Q4MESH to generate the required finite element meshes. The QUAD4M
program considers in an approximate way the nonlinear behavior of the soil deposits
when they are subjected to severe earthquakes. This is done by means of the Equivalent
Linear Method.
The results obtained for the cases of escarpments indicate that there is an increase
in the ground acceleration as one moves up along the surface of the slope compared to the
original acceleration at the bedrock. In the escarpment analyses, the material type
103
defined as SC in the UBC-97 classification causes the largest amplification. Also, the
studies indicate that the amplification ratios between the top and base were greater for El
Centro earthquake than for the San Salvador earthquake. However, the San Salvador
produced higher peak accelerations in absolute terms.
The models of the hills showed greater amplification factors than the
escarpment’s models. The maximum acceleration for a hill was found when a model
with height n=75 ft was used. However, the maximum amplification factor was obtained
in the hill with height n=225 ft. In the case of the hills a phenomenon not observed in the
escarpment occurred. All the hill cases but one exhibited a deamplification at the toe of
the hill.
The maximum amplification factors were presented in tables and they range from
1.00 to 2.35. Not all the soils in the UBC-97 classification were always used because in
some cases the materials did not exhibit stability for the more pronounced slopes.
In theory, the topographic amplification is strictly caused by the discontinuity in
the geometry of the soil deposit. The diffraction of the seismic waves as well as the
multiple reflections within the boundaries of the irregularity are usually thought to be
responsible for the amplification. Therefore, in principle, the type of material is not
relevant for this problem. However, one of the first peculiarities of the phenomenon that
was observed during the course of the study was that the type of soil material does play
an important role in the occurrence of amplification. Therefore, it was decided that each
model of the soil deposit-hill and soil deposit-escarpment system needed to be analyzed
using different types of soils. The classification provided in the UBC-97 code, adopted in
104
Puerto Rico, was used to select the soil types. There are six soil types in the UBC-97.
Out of these six types, the soils referred to as SA and SF were not used. The type SA is a
very hard rock typical of some regions in the Eastern US and thus it was considered that
it would not represent the situation in Puerto Rico. The soil profile SF requires, according
to the code, site-specific geotechnical studies and thus the general character of the study
carried out here would no be useful for this kind of material. Therefore, only the soils
classified as SB, SC, SD and SE were considered. In some models it was assumed that the
same material formed the escarpment and hill as well as the soil deposit underneath. In
other models, a combination of two different soils that was shown to be unfavorable, i.e.
that leads to higher amplification, was used.
The amplification studies presented in Chapters IV and V are based on highly
idealized geometries of the topographic irregularities. To observe the situation in an
actual case, a site at the south of Puerto Rico in the municipality of Guánica was selected
for a detailed analysis. Using the best data available from maps and site visits, a finite
element model of the cross-section of a group of hills was created. In order to further
enhance the real character of the analysis, a synthetic earthquake specifically developed
for the south of Puerto Rico in a previous study was used as seismic input. The case was
complicated not only because of its geometry but also due to the fact that the soil profiles
at the site were not similar to those studied before. For instance, the valley at the bottom
of one of the hills is near a river basin. The results were interesting because the
maximum amplification did not happen at the top of the hill, but at about 0.3 of the
maximum height. This can pose a problem to the many residential structures built at this
105
elevation. Although the geometry and materials of this real case are quite different from
those considered in the previous chapters, when the tables and general formula developed
therein are applied, they produce reasonable results.
6.2 Suggestions for further studies
Although this work tried to be as comprehensive as possible, there are a number
of topics that were not addressed because of time limitations. It is believed that these
topics are worth, at least, a closer look. They are listed below in no specific order.
1. The same methodology used in this thesis can be applied to study hills with
shapes different than the parabolic case considered here. In addition, the 3-
dimensional effect that may occur in canyons is worth studying.
2. If one can find seismic records recorded at the field, both away from the
topographic irregularity and also on its surface and on the hill or escarpment
itself, they can be very useful to validate the numerical simulation study. In
this case, it is necessary to know at least in an approximate way the properties
of the soil profiles where the accelerogram were collected.
3. Although in this study a case of topographic amplification using real field data
and geometry was used, it is interesting to perform more of these analyses for
other sites in Puerto Rico.
4. The study undertaken here focused on one of the horizontal components of the
earthquake. This was done because, in general, they are the strongest
106
components and those that affect the most the buildings and other structures.
However, it is recommended to extend the analysis so that the vertical
components of the ground motions are also included.
5. The curves in the program QUAD4M that describe the variation of the shear
modulus G and the damping ratio ξ as a function of the shear strain are
limited to a few options. These curves are required by the Equivalent Linear
Method to take into account the nonlinear behavior of the soil under a strong
ground motion. The user is forced to adopt one of the curves that better
matches the material for the case under study. This is not a problem if one can
choose among a variety of curves, but as mentioned before, the options are
limited. If it is possible to incorporate more options to the library of curves of
the program, it will enhance the applicability of the computer program.
6. Two features of the topographic amplification phenomenon were not studied
in this thesis: the possible increase in the duration of the accelerogram and the
occurrence of differential motions along different points on the surface of the
irregularities. These effects should be examined because, even though the
most threatening result of the terrain irregularities is the amplification of the
ground acceleration, they may be also be detrimental for certain structures.
7. The San Salvador and El Centro earthquakes used in the parametric study
were selected because they represent two seismic motions with opposite
characteristics. However, if one wants to increase the confidence in the
108
References
1. Abramson, L.W., Lee, T.S., Sharma, S., and Boyce G.M. 1996. Slope Stability and Stabilization Methods. John Wiley & Sons, New York.
2. Arroyo, J. R. 1997. “UPRSOIL, a computer program”. Civil Engineering
Department, University of Puerto Rico, Mayagüez, P.R. 3. Arroyo, J. R. 1999. Nonlinear Seismic Response of Stratified Soil Deposits Using
Higher Order Frequency Response Functions in the Frequency Domain. Ph.D. Thesis. Civil Engineering Department, University of Puerto Rico, Mayagüez, P.R.
4. Athanasopoulos, G. A. and Zervas C. S. 1993. Effects of Ridge-Like Surface
Topography on Seismic Site Response. Soil Dynamics and Earthquake Engineering VI, Computational Mechanics Publications, Boston, Massachussets. 3-18.
5. Bouchon, M. 1973. Effect of Topographic on Surface Motions. Bulletin of the
Seismological Society of America. 63(3): 615-632. 6. Castellani, A., Peano A. and Sardella, L. 1982. Seismic Response of Topographic
Irregularities. Third International Earthquake Microzonation Conference Proceedings, Seattle, Washington, 2:533-540.
7. Cook, R. D., Malkus, D.S. and Plesha M. E. 1989. Concepts and Applications of
Finite Element Analysis. John Wiley & Sons, New York. 8. Geli, L., Bard, P. and Jullien, B. 1988. The Effect of Topographic on Earthquake
Ground Motion: A Review and New Results. Bulletin of the Seismological Society of America. 78(1):42-63.
9. Hudson, M., Idriss, I. M. and Beikae, M. 1994. “User's Manual for QUAD4M”,
Center for Geotechnical Modeling, University of California, Davis, California.
10. Idriss, I. M., Lysmer, J., Hwang, R. and Seed, H.B. 1973. “User's Manual for QUAD4”, College of Engineering, University of California, Berkeley, California.
11. Interactive Software Designs, Inc. 1995. Reference Manual, XSTABL-An Integrated
Slope Stability Analysis Program for Personal Computers, Version 5, Moscow, Idaho. 12. International Conference of Buildings Officials. 1997. Uniform Building Code 1997.
2, Chapter 16, Division V,Section 1636.2.6. Whittier, California.
109
13. Irizarry, J. 1999. Design Earthquake and Design Spectra for Puerto Rico’s Main
Cities based on Worldwide Strong Motion Records. MSCE Thesis, Civil Engineering Department, University of Puerto Rico, Mayagüez, P.R.
14. Kramer, S. L. 1996. Geothecnical Earthquake Engineering. Prentice-Hall, New
Jersey.
15. Peters, J. and Kala, R. 1999. “Users Guide to WinMesh”, United States Army Engineer Waterways Experiment Station, Vicksburg, Mississippi.
16. Rivera, M. 1995. Welcome to Puerto Rico. http://www.welcome.topuertorico.org,
updated February 2001. 17. Sanchez-Sesma, F. J. 1990. Elementary Solutions for Response of a Wedge-Shaped
Medium to Incident SH and SV Waves. Bulletin of the Seismological Society of America. 80(3):737-742.
18. Sanchez-Sesma, F. J. 1997. Strong Ground Motion and Site Effects. Computer
Analysis and Design of Earthquake Resistant Structures A Handbook. Computational Mechanics Publications, Southampton, UK, 3:201-239.
19. Sanchez-Sesma, F. J. and Campillo, M. 1993. Topographic Effects for Incident P,
SV and Rayleigh Waves. Tectonophysics, 218:113-125. 20. Sano, T. and Pugliese, A. 1999. Parametric Study on Topographic Effects in
Seismic Soil Amplification. Second International Symposium on Earthquake Resistant Engineering Structures. WIT Press, Ashurst, Southampton, UK. pp. 321-330.
21. Strong Motion Data Center, Department of Conservation, Mines & Geology. http://
www.consrv.ca.gov/dmg/csmip/index.htm, 1999. 22. Terzaghi, K., Peck, R. B., and Mesri, G. 1996. Soil Mechanics in Enginerring
Practice. John Wiley & Sons, New York. 23. U.S. Department of Agriculture, Soil Conservation Service. 1963. Soil map of Lajas
Valley Area, Puerto Rico. 24. U.S. Geological Survey. 1997. Ground water atlas of the United States, Segment 13.
Hydrologic Investigations Atlas 730-N, Reston, Virginia, p. N23. 25. Yeats, R. S., Sieh, K., and Allen, C. R. 1997. The Geology of Earthquakes. Oxford
University Press, New York.