Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
1
MÉTODOS SIMPLIFICADOS EN LA DINÁMICA DE SUELOS
RICARDO DOBRY
Profesor Institucional, Instituto Politécnico de Rensselaer,
Troy, Nueva York, E.U.A.
RESUMEN
Después de una breve descripción de las características principales que definen la Dinámica de
Suelos y sus aplicaciones a la ingeniería, se discute la importancia de los Métodos Simplificados.
Aún con la actual disponibilidad de poderosas simulaciones por computadora, los Métodos
Simplificados continuarán desempeñando un papel importante en la Dinámica de Suelos así
como lo han hecho en el resto de la Ingeniería Geotécnica. Los Métodos Simplificados permiten
al ingeniero realizar cálculos manuales o con un mínimo de apoyo computacional, facilitando así
los estudios paramétricos. En el proceso, el ingeniero tiene la posibilidad de desarrollar una
sensibilidad sobre el significado físico y la importancia relativa de los factores involucrados,
adquiriendo un mayor control sobre los cálculos y las decisiones, incluyendo el uso de su criterio
como ingeniero basado en su experiencia personal. Se presenta una lista de procedimientos
simplificados desarrollados por el autor, considerando sistemas que varían desde el campo libre y
las presas de tierra, hasta cimentaciones superficiales y profundas sujetas tanto a excitaciones
causadas por movimientos sísmicos, como por vibraciones de maquinaria. Un conocimiento
básico de la teoría fundamental y de las idealizaciones detrás de los procedimientos simplificados
pueden ser muy útiles a los ingenieros, incluyendo especialmente conceptos de la Dinámica y de
la Propagación de Ondas. Parte de este conocimiento se presenta en el documento, con aplicación
a las cimentaciones superficiales para maquinaria y a la interacción dinámica suelo-estructura.
La Conferencia se centra en las cimentaciones superficiales para maquinaria sobre un
semiespacio sujetas a cargas dinámicas en cualquiera de los seis grados de libertad, y los
Métodos Simplificados que se han desarrollado a través del tiempo para caracterizar a los
correspondientes resortes y amortiguadores equivalentes del suelo. Esto incluye el caso general
en que estos resortes y amortiguadores dependen de la frecuencia de excitación, así como la
simplificación lograda en algunos casos en que estos resortes y amortiguadores se consideran
independientes de la frecuencia. Inicia con el caso de la cimentación superficial circular que fue
estudiada durante gran parte del siglo XX, hasta las contribuciones seminales desarrolladas por
Lysmer y otros autores entre 1966 y 1971, y continua con los casos de cimentaciones
superficiales y enterradas con geometría de base arbitraria, concluyendo con las dos
publicaciones sintetizadas escritas por Gazetas en 1990 y 1991. El desarrollo de estos métodos
simplificados con resortes y amortiguadores equivalentes para cimentaciones superficiales y
enterradas de forma arbitraria se presenta en detalle, incluyendo las contribuciones del autor al
inicio de este proceso.
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SIMPLIFIED METHODS IN SOIL DYNAMICS
RICARDO DOBRY
Institute Professor, Rensselaer Polytechnic Institute,
Troy, New York, U.S.A.
ABSTRACT
After a brief description of the main characteristics that define Soil Dynamics and its engineering
applications, the role of Simplified Methods is discussed. Despite the current wide availability of
powerful computer simulations, it is concluded that Simplified Methods will continue to play an
important role in Soil Dynamics as they do in the rest of Geotechnical Engineering. Simplified
Methods allow the engineer to conduct calculations by hand or with a minimum computational
effort, including parametric variations. In the process, the engineer has the possibility to develop
a feel for the physical meaning and relative importance of the various factors, with more personal
control of calculations and decisions including use of engineering judgment as needed. A list of
simplified procedures proposed by the author is provided, covering systems that range from the
free field and earth dams to shallow and deep foundations, subjected to excitations that include
both seismic shaking and machine vibrations. Basic understanding of the basic theory and
simplifications behind the simplified procedure can be very helpful to engineers, including
Dynamics and Wave Propagation concepts. Some of this understanding is developed in the paper,
with focus on shallow machine foundations and other dynamic soil-structure interaction
applications.
The Lecture concentrates on shallow machine foundations on a half-space subjected to dynamic
loads in any of the six degrees of freedom of the foundation, and the Simplified Methods that
have been proposed over the years to characterize the corresponding equivalent soil springs and
dashpots. This includes both frequency-dependent and frequency-independent springs and
dashpots. It started with the circular surface foundation which was studied over much of the 20th
Century, until the breakthroughs by Lysmer and others in 1966-1971, and continued with the
cases of surface and embedded foundations of arbitrary shape that culminated in the two
summary publications by Gazetas in 1990 and 1991. The development of these simplified
equivalent springs and dashpots for both surface and embedded foundations of arbitrary shape is
discussed in some detail, including the contribution of the author in the early part of the process.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
1.
XXI Nabor Carrillo Lecture
3
INTRODUCTION
It is a great honor for me to be asked to present the Twenty-first Nabor Carrillo Lecture, and to
be associated this way with Dr. Nabor Carrillo and his many accomplishments. It is also an
honor to be associated with the people who have been Carrillo Lecturers over the years and who
have made such gigantic contributions to the geotechnical field. Let me add that I am especially
proud to follow two Carrillo Lecturers who were also my professors and who had an
extraordinary influence over my career. One of them is Prof. Enrique Tamez (1992), who
directed my Master Thesis on Sand Liquefaction During Earthquakes at the UNAM in México
City many years ago, and who inspired me to specialize in Soil Dynamics and Earthquake
Engineering. The other is Prof. Robert V. Whitman (2000), who unfortunately died this year, and
who directed my Doctoral Thesis at MIT, also on Soil Dynamics. I wouldn’t be here without
them, both of them were great teachers and mentors to me, and this is a good opportunity to say
Thanks to both of them.
Finally, let me say that it is just a pleasure to be once again back in México, where I have so
many friends and colleagues. One of them is Prof. Eulalio Juárez Badillo, who together with
Prof. Alfonso Rico taught me so well the ABC of soil mechanics during my graduate studies at
the División de Posgrado of UNAM.
The theme of my presentation today is the Simplified Methods in Soil Dynamics. This
immediately poses two questions: What is Soil Dynamics, and what kind of Simplified Methods
are we talking about?
In his Fifteenth Carrillo Lecture, Prof. Whitman (2000) defined problems in soil dynamics as
those in which the inertia force of the soil plays a significant role. I would add to this a few other
characteristics common to most soil dynamics problems: (i) the loads tend to act much faster than
in typical soil mechanics problems; (ii) the loads change direction periodically because they are
associated with vibrations, and therefore produce cyclic rather than monotonic stresses and
strains in the soil; and finally (iii) many of the problems that worry us most in soil dynamics, are
associated with shear strains in the soil which are much smaller than those we are familiar with in
regular soils testing, like 0.1%, or 0.01% or even smaller.
Table 1, reproduced from that same Carrillo Lecture by Whitman, lists some of the most
important practical applications of Soil Dynamics. It includes the problems of machine
foundations, earthquake engineering, pile driving, techniques used to compact sands in the field,
problems of ocean wave loading of offshore structures, etc.
Let me say a couple of things, first about earthquakes and then about machine foundations, so as
to give a better idea of some of the complexities of analyzing Soil Dynamics systems and the
need for simplified solutions. Figure 1 shows the amplification of the earthquake waves by the
soft clay in Mexico City in the 1985 earthquake, which caused a lot of damage to buildings and
killed thousands of people, and which has been studied in detail by a number of Mexican
engineers.
The curves in the figure are acceleration response spectra, and they measure the maximum lateral
force experienced by a building that behaves elastically during the earthquake in number of
accelerations of gravity, or g’s, versus the period of the building in seconds. In 1985 essentially
all collapsed buildings and fatalities were on soil and not on rock. This happened because the
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earthquake inertia forces on these assumed elastic buildings due to the shaking, were much
greater on soil than on rock, as much as ten times higher, as can seen in the figure by how much
bigger is the recorded acceleration spectrum on soil at the building of the Secretaría de
Comunicaciones y Transportes (SCT), compared with the same recorded spectrum on rock at the
University (UNAM) (Seed, 1987; Romo and Seed, 1987; Dobry, 1991a).
Table 1. Applications of Soil Dynamics (Whitman, 2000).
Figure 1. Earthquake amplification on the Mexico City soft clay in 1985 (Romo
and Seed, 1987; Seed, 1987; Dobry, 1991a).
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
5
The way we analyze the earthquake amplification by the soil in a situation like this, is by feeding
into a computer program the motions on the rock, together with a dynamic profile of the soil
which must include for each layer properties like the density of the soil, the shear wave velocity
Vs, and the internal damping. Then the computer program will calculate the motion on top of the
soil. This computer program is relatively complex, becoming even more so if you include 2D and
3D effects due to the presence of hills nearby, or the effect of inclined or irregular soil layers.
The shear wave velocity of the Mexico City clay is quite low, of the order of 70 or 80 m/s, and
this low shear wave velocity played a significant role in the large site amplification during the
1985 earthquake. Shear wave velocity is by far the most important soil property needed for these
earthquake calculations. The shear wave velocities for most soils in the world range from about
60 to 800 m/s; a factor of about fifteen. It turns out that to know with some precision the value of
this parameter for your particular problem is also key to the analysis of most Soil Dynamics
problems, not only earthquake soil amplification. In fact, shear wave velocity is clearly the single
most important soil parameter in the whole of Soil Dynamics, as important as soil shear strength
is for slope stability calculations.
Figure 2 illustrates another important category of soil dynamics problems: machine foundations,
where a structure on a shallow or deep foundation is excited by dynamic loads above ground,
typically due to unbalanced inertia forces caused by operation of industrial machinery. The loads
can be complicated, ranging from sinusoidal forces having one amplitude, direction and
frequency, to very irregular loads and moments, and combinations of vertical, horizontal, rocking
and torsional vibrations. Other parameters that add complication to the solution include the type,
geometry, mass, degree of embedment, and flexibility of the foundation; and the soil layering and
soil properties of each layer including most prominently the shear wave velocity.
Figure 2. Machine foundation vibrations and dynamic soil-structure interaction.
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This machine foundation problem is mathematically very similar to other problems that involve
dynamic soil-structure interaction. For example, the dynamic forces and moments acting on the
pile group in Fig. 2f could originate from ocean waves pushing periodically against the side of an
offshore oil platform. These dynamic forces and moments may also arise from the inertia forces
developed in a building during earthquake shaking, due to the arrival of the seismic waves
traveling in the ground, sketched in Fig. 2g in a very simplified way. Due to this mathematical
similarity, we often use the solutions developed for machine foundations, to analyze also the
dynamic soil-structure interaction during earthquakes. To a large extent, the differences between
the solutions for these different forms of dynamic soil-structure interaction (machine foundations,
ocean wave loading, earthquakes), lie not so much in the physical origin of the loading, but rather
in its duration and frequency as well as in the level of cyclic strains induced in the soil.
2.
THE NEED FOR SIMPLIFIED METHODS
Let me address the issue of the Simplified Procedures. For the purpose of this presentation, I will
define a Simplified Procedure as a method that: (i) is derived totally or partially from basic
theory; and (ii) can be used to analyze a geotechnical system either with a calculator or with
minimum computational effort, of the type than can be programmed in a spreadsheet.
We constantly use Simplified Methods in Geotechnical Engineering for the analysis and design
of static loads. Figure 3 shows three of them, all very familiar to geotechnical engineers. In fact, I
obtained the information for this figure from two standard foundation engineering textbooks.
Figure 3. Examples of Simplified Methods in Soil Mechanics (modified after Liu
and Evett, 1998; Das, 1999).
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
7
Let us take a look at these three methods. The ultimate bearing capacity equation in Fig. 3a is
based on an approximate Theory of Plasticity solution developed by Prandtl (1920) and Reissner
(1924), that Terzaghi (1943) simplified further, producing the equation at the bottom. The
material parameters required are the soil cohesion and friction angle, which are obtained from
laboratory tests, or, in the case of sands, the friction angle ϕ may be estimated from field
penetration tests. The Schmertmann and Hartman (1978) method in Fig. 3b, which is used to
compute foundation settlement in sand, depends on a triangular stress distribution with depth that
is a simplification of the theoretical profile of stress with depth obtained from the Theory of
Elasticity Boussinesq solution. In this settlement calculation the key material parameter is the
modulus, Ez, of each sand layer. And, finally, the popular ordinary method of slices with an
assumed circular failure surface, proposed by Fellenius (1936), sketched in Fig. 3c, just uses
basic equations of static equilibrium together with some simplifying assumptions, allowing
definition of the Factor of Safety of the slope when the shear strength of the soil varies along the
failure surface.
Therefore, the three methods start from some basic and very general theory, and they add
simplifications and assumptions along the way until they arrive to a simple mathematical model
that still contains the main parameters of interest and is broad enough to accommodate the values
of these parameters for many possible systems. Furthermore, the application of any of these
Simplified Methods requires material parameters like c, ϕ or Ez that are either measured in the
lab or field, or are correlated empirically to field tests like the CPT or the SPT.
These Simplified Methods have two main characteristics, which are common to static and
dynamic loads: (i) they start with basic theory and they simplify that theory while keeping the
relevant factors; and (ii) they still cover a broad range of possible conditions, allowing the
engineer to bring into the analysis his/her own loads, foundation or soil geometries, soil profiles
and soil properties. These simplified methods have a number of uses, including:
 They allow the engineer to conduct calculations, either by hand or using a minimum
computational effort (hand calculator, spreadsheet).
 They allow the engineer to develop a feel for the physical meaning and relative
importance of the different factors.
 They often serve as the basis for codes and regulations.
 In this day and age, they also allow the engineer to verify the results of more complicated
computer analyses (“reality checks”). This is a very important function of the simplified
methods, as already noted by Ing. Enrique Santoyo in his 20th Carrillo Lecture (Santoyo,
2010).
It is interesting that until about 30 years ago or so, that it before the age of powerful accessible
computers, there was no need to justify or defend these simplified methods, as generally there
was nothing else engineers could use. But with the advent of computers, things have changed,
and in principle the engineer can analyze very complicated systems and loadings without the
need to simplify the theory. As a result, some people are tempted to go only that route with the
exclusion of more traditional simplified methods, which as noted by Santoyo (2010) is not a good
idea at all.
Table 2 lists a number of simplified solutions and associated publications, proposed with the
participation of the author over the years, for a variety of soil dynamics systems ranging from the
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free field and earth dams to shallow and deep foundations, and for excitations covering mainly
seismic and machine vibrations.
Table 2. Simplified Solutions proposed for various Soil Dynamics problems with
participation of the author.
Author(s)
Year
Problem Addressed by Simplified Solution
Roesset, Whitman and Dobry
1973
Estimated modal damping of structure with
consideration of dynamic soil-structure interaction
Dobry, Oweis and Urzua
1976
Fundamental period of soil profile on rigid rock
Dobry and Gazetas
1985
Dobry and O’Rourke
1983
Bending moment in pile due to seismic kinematic
effect
Dobry, Mohamad, Dakoulas and
Gazetas
1984
Estimation of seismic shear strains in earth dam for
evaluation of liquefaction and flow failure
Gazetas and Dobry
1984
Dobry and Gazetas
1985
Equivalent horizontal spring and dashpot at the top
of a pile
Dobry and Gazetas
1985
Dobry and Gazetas
1986
Dobry, Gazetas and Stokoe
1986
Gazetas, Dobry and Tassoulas
1985a
Gazetas, Tassoulas, Dobry and
O’Rourke
1985b
Springs and dashpots for surface foundations of
arbitrary shape
Vertical spring and dashpot for embedded
foundations of arbitrary shape
1985
Dobry and Gazetas
Dobry and Gazetas
1988
Dobry
1991a
Equivalent springs and dashpots of floating pile
groups
Dobry
Use of Roesset and Whitman (1969) theoretical
solution for steady-state amplification, to provide
1991b estimate of peak of Ratio of Response Spectra for
soil deposit on flexible rock
1995
Dobry, Taboada and Liu
1995
Dobry
Decrease with distance to river or lake, of lateral
spreading of ground due to sand liquefaction in an
earthquake
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
3.
XXI Nabor Carrillo Lecture
9
THE MACHINE FOUNDATION PROBLEM
The rest of this Lecture describes the development of simplified procedures for shallow machine
foundations that took place over most of the 20th Century. While I played a role on this in the
1980’s through my collaboration at that time with Prof. George Gazetas, a number of the key
breakthroughs had already taken place by then, through the work of such excellent researchers as
Reissner (1936), Reissner and Sagoci (1944), Arnold et al. (1955), Bycroft (1956), Barkan
(1962), Lysmer and Richart (1966), Hall (1967), Whitman and Richart (1967), Elorduy et al.
(1967), Gladwell (1968), Richart et al. (1970), Luco and Westmann (1971), Veletsos and Wei
(1971), Kausel and Roesset (1975), Johnson et al. (1975), Wong and Luco (1976, 1978), Gazetas
and Roesset (1976, 1979), Dominguez and Roesset (1978), and Roesset (1980). Let me repeat
again that, although we call it for simplicity the machine foundation problem, we are really
solving here all kinds of soil-structure interaction problems where the loads may be caused not
only by machines but also by earthquakes or ocean waves.
3.1
Vertical Vibration of Rigid Mass
Figure 4 depicts the original machine foundation problem, which looks deceptively simple.
Figure 4a shows the system. It is a perfectly rigid cylindrical mass M of radius R, located on the
Figure 4. Machine foundation problem.
surface of an elastic half-space representing the soil, which is the same elastic half-space we use
in static Soil Mechanics to calculate the Boussinesq (1885) solution for the stresses under a
foundation, or in the Newmark (1942) charts to calculate foundation settlement. As usual, we
need only two elastic parameters to characterize this isotropic homogeneous material, which we
select as being the shear modulus, G, and the Poisson’s Ratio, μ. In addition, because of the
inertia forces associated with the dynamic loading, we also need the mass density, ρ, which in
practical terms is usually the total unit weight of the dry or saturated soil divided by the
acceleration of gravity. In the simplest case of vertical vibration shown in Fig. 4b, the applied
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vertical load at the top of the mass varies sinusoidally with time, with amplitude Pm and
frequency f, say in cycles per second. The question to be solved is to calculate the vertical
displacement of the foundation, w, for given P, f and the rest of the parameters of the problem1.
This problem, that at first sight looks so simple, attracted the attention of top analytical
researchers during a period spanning 30 years, and was completely solved only in the 1960’s
when computers became available (Lysmer and Richart, 1966). The reason why the problem is so
difficult to solve analytically, is that it involves a mixed boundary dynamic condition, with the
displacement of the ground surface being constant over the area of the foundation, while the
vertical normal stress outside the area of the foundation is constant and equal to zero.
What would a simplified solution look like? As indicated in Fig. 5, we may simplistically try to
replace the whole elastic half-space by an equivalent elastic spring, k, selecting the value of k so
that it gives us the right w for a given P. In principle, this value of k will be a function of the
properties of the half-space, G, ρ, μ, of the radius of the foundation, R, and of the frequency of
the loading, f. With luck, perhaps we will conclude that k is not very sensitive to the frequency f.
This would be ideal as we want to be able to use the solution also for loadings which are not
sinusoidal. In this case we would have a Simplified Solution characterized by an equivalent
vertical spring, k, which is frequency-independent.
The problem with this is that with such a frequency-independent spring, what we have is the
system of a mass connected to a spring of Fig. 5. This is a well known system in Dynamics called
the undamped simple oscillator, or undamped single degree-of-freedom system, that for the
applied sinusoidal load has the solution for the displacement, w = (Pm/k) [sin(2πft – α)]/[1 –
(f/fn)2]2, which becomes infinite when the loading frequency, f, becomes equal to the natural
frequency of the oscillator, f = fn = (1/2π)(k/M)1/2. On the other hand, all indications were that
there is no value of f for which the displacement w of the foundation in Figs. 4-5 becomes very
large, let alone infinite. Therefore, a spring is not enough, and some element has to be added to
the equivalent simplified system of Fig. 5 that not only stores energy, as the spring does, but also
dissipates energy, hence avoiding infinite values for w.
The researchers added a linear viscous dashpot to the system to take care of the necessary energy
dissipation, as shown in Fig. 6, which transforms the equivalent system into a damped simple
oscillator.
Due to the dashpot, the displacement w is never infinite, whatever the frequency of the loading.
The spring k generates a force that is proportional to the displacement, w, of the mass, while the
dashpot c generates a force that is proportional to the velocity of the mass, dw/dt. As a result, the
equation of motion of the system that allows solving the problem once the values of k and c have
been determined is:
𝑀
𝑑2 𝑤
𝑑𝑡 2
+𝑐
𝑑𝑤
𝑑𝑡
+ 𝑘𝑤 = 𝑃 = 𝑃𝑚 sin(2𝜋𝑓𝑡)
(1)
This is, in fact, the correct form of the exact solution for the original problem of a mass on a
foundation lying on a half-space shown in Fig. 4, and what is left is to determine how these
1
It can be shown that the time history of w is also sinusoidal of the same frequency of the loading, w = w m sin
(2πft-α), so the problem is reduced to the determination of the amplitude, w m, and phase angle, α, of the
displacement response.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
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spring and dashpot depend on the parameters of the problem. Of course, that is the difficult part
that took 30 years to solve.
Figure 5. First too simplistic attempt of a Simplified Solution.
Figure 6. Second more realistic attempt of a Simplified Solution.
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As mentioned before, a number of efforts were made to develop this equivalent spring and
dashpot, forgetting about the mass for the time being, and replacing the contact area between
mass and soil by a massless rigid circular plate welded to the surface of the half-space (Fig. 7).
Finally, in 1966, in his doctoral thesis at the University of Michigan under the direction of Prof.
Richart, Lysmer found the exact solution to the problem with the help of this powerful new tool
called computers (Lysmer and Richart, 1966). Then they proceeded to find that a frequencyindependent Simplified Solution was possible, because in this particular case neither the spring
nor the dashpot were very sensitive to changes in the frequency f. Figure 7 shows Lysmer’s
proposed approximate expressions for the equivalent vertical spring, kv, and dashpot, cv. The two
expressions are a marvel of simplicity.
Figure 7. Frequency-independent Simplified Solution for vertical loading, also
labeled “Lysmer’s Analog” (Lysmer and Richart,1966).
The value of the spring is kv ≈ 4GR/(1-μ), which is the same as the static vertical stiffness for a
rigid circular foundation obtained by integrating the static Boussinesq solution for the half-space.
That is, the selected kv corresponds to f = 0. The expression for the dashpot is even more
interesting. It can be expressed either in terms of the shear modulus, G, or alternatively in terms
of the shear wave velocity of the soil, Vs. That is, cv ≈ [3.4/(1-μ)] (Gρ)1/2 R2 = [3.4/(1-μ)] (ρVs)
R2, taking advantage of the fact that Vs and G are related through the basic elasticity equation:
𝐺 1/2
𝑝
𝑉𝑠 = ( )
(2)
Figure 8 includes the comparison presented by Lysmer and Richart for the dynamic response
curves for the cylindrical mass on the half-space of Fig. 4. The solid line is the exact solution and
the dashed line is the simplified solution calculated with the frequency-independent spring and
dashpot of Fig. 7.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
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Figure 8.
Vertical dynamic response of mass on a half-space: comparison
between Lysmer’s Analog and exact solution (Lysmer and Richart, 1966).
The graph of Fig. 8 plots the normalized amplitude of the mass displacement, wm, versus the
normalized frequency of the loading. The curves have the typical shape of response of a damped
single oscillator, showing that the system has quite a bit of damping; this is reflected in the fact
that the peaks of the curves are all below three. But the most important conclusion from our
viewpoint is that the Simplified Method predicts very well the exact response, so it can be used
by engineers with confidence as a basic tool for these kinds of calculations. And in fact, this
simplified solution and corresponding expressions of vertical spring and dashpot for a circular
surface foundation, are listed today as standard equations in a number of textbooks and
foundation manuals.
Table 3 summarizes the history of the development of the solution. Lamb (1904) had solved the
problem of the concentrated vertical dynamic load at the surface of an elastic half-space, which is
the dynamic counterpart of the Boussinesq (1885) solution for a concentrated static load. In the
1930’s, Reissner (1936) integrated Lamb’s solution over a circular area assuming a constant
pressure distribution, that is he provided a solution for a perfectly flexible foundation rather than
a rigid foundation. After various efforts containing assumptions and approximations by several
authors in the 1950’s and early 1960’s; finally Lysmer and Richart (1966) solved the problem
numerically using a computer and provided the beautiful Simplified Solution of Fig. 7, where the
half-space below the foundation is replaced by a frequency-independent spring and a frequencyindependent dashpot. Table 3 also lists two 1967 papers by Richart and Whitman, where they
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validated the Simplified Solution with field tests and developed a design procedure, making the
new solution available to the engineering community.
Table 3. Theoretical and Simplified Solutions to machine foundation problem.
Author(s)
Year
Contribution
Lamb
1904
Solution for concentrated vertical force on surface of
half space (Dynamic Boussinesq Problem).
Reissner
1936
Solution for flexible circular foundation assuming
uniform load.
Quinlan
1953
Approximate solution for rigid circular foundation
assuming static pressure distribution.
Sung
1953
Solutions for various assumed pressure distributions.
Bycroft
1956
Simplified solution by averaging displacements over
foundation area.
Hsieh
1962
Introduced idea of frequency-dependent equivalent
spring and dashpot.
Lysmer and Richart
1966
Obtained exact frequency-dependent spring and
dashpot for rigid circular foundation using computer.
Proposed approximate frequency-independent spring
and dashpot as Simplified Solution for engineers
(Lysmer’s Analog).
Richart and Whitman
1967
Whitman and Richart
1967
3.2
Validated Lysmer’s Analog with field footing vibration
tests.
Design procedure based on Lysmer’s Analog.
Horizontal vibration
After Lysmer and Richart solved for the vertical loading by combining theory with computer
calculations, the rest of the solutions came fast within the next few years for other dynamic
excitations acting on the same surface circular foundation. Figure 9 shows the case of horizontal
loading, where again it was possible to obtain frequency-independent expressions for the
horizontal spring and dashpot.
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Figure 9. Frequency-independent Simplified Solution for horizontal loading
(Hall, 1967; Luco and Westmann, 1971; Veletsos and Wei, 1971)).
3.3
Simplified systems for design and equivalent circle
In their 1967 paper, Whitman and Richart summarized all these Simplified Solutions for surface
or very shallow circular foundations, and gave recommendations on how to use them in actual
engineering projects. These recommendations included how to produce the necessary values of
soil shear modulus and Poisson’s Ratio needed to calculate the spring (stiffness) and radiation
dashpot for vertical, horizontal, rocking and torsional excitations (Fig. 10).
Finally, they also provided recommendations on how to combine these radiation dashpots with
the internal damping associated with the energy dissipated by the cyclic loading within the soil
itself, mostly in friction. Table 4 lists the expressions for the four static stiffnesses, recommended
by Whitman and Richart (1967) as the frequency-independent spring constants for the respective
Simplified Solutions2. They also suggested that foundations which do not have a circular shape,
like square, rectangular, etc., should be first transformed into an equivalent circle before using
those simplified solutions.
These simplified solutions for the circular surface foundation were an important breakthrough,
and their use for all kinds of foundation shapes through the equivalent circle method has served
the profession well. However, they still left open the issue of what to do when the foundation is
embedded rather than being at the surface or very close to it, and also how good is the equivalent
circle approximation, say, for a very long rectangle or a similar elongated foundation shape.
2
The equation in Table 4 and Fig. 9 for kho was obtained a few years later by Luco and Westmann (1971) and
Veletsos and Wei (1971), and is slightly different from the approximate expression proposed by Hall (1967) and
used by Whitman and Richart in their 1967 paper.
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Figure 10. Equivalent simplified systems for design based on equivalent circular
foundation (Whitman and Richart, 1967).
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Simplified methods in Soil Dynamics
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Table 4. Static stiffnesses of rigid circular foundation on the surface of an elastic
half-space.
Today’s mathematical and computational techniques are much more powerful than those
available in the 1960’s, and a number of these cases have been solved by a combination of
analytical and numerical methods in the last 30-40 years, with many articles, tables and charts
published in research journals and books. Furthermore, powerful dynamic finite elements
computer programs than can solve your specific problem for any shape and any embedment, as
well as for arbitrary soil layering, are now commercially available, and they are routinely used in
very important or critical structures such as nuclear power plants or large bridges. But in most
projects, simplified solutions continue to be used. Even in projects like a large bridge, a finite
element program may be used to analyze the abutments and the foundations of the piers, with the
simplified methods utilized to analyze the foundations of the approaches to the bridge.
Therefore, there was clearly a need to extend these simplified solutions produced by Lysmer,
Richart and Whitman, to both embedded foundations and to noncircular shapes. I will be
addressing these other cases later in this Lecture, but it turns out that before we can do that, it is
necessary to clarify first the physical origin of the equivalent viscous dashpots shown in Figs. 7
and 9 for the vertical and horizontal vibrations. So, let me focus now on these viscous dashpots.
3.4
Viscous dashpots and radiation damping
The problem can be posed as follows (Fig. 11): The soil is represented by a purely elastic
material filling the half-space, which does not have any internal damping and therefore has no
way to dissipate energy in the material itself. If the foundation had been on top of a closed elastic
system with rigid boundaries surrounding the soil, the displacement of the foundation would have
been infinite when vibrating at the natural frequency of the system. But because the system is
open instead of closed, energy escapes in the form of waves propagating in the soil, with this
energy never coming back, and this is why the displacement of the foundation is never infinite.
This form of elastic energy dissipation in the form of waves traveling away from the foundation
is called Radiation (or Geometric) Damping, and it is the physical origin of the vertical and
horizontal viscous dashpots I mentioned before, which are just approximate mathematical
representations of the phenomenon.
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Figure 11. Radiation of energy by waves propagating from foundation (Lambe
and Whitman, 1969).
Which types of waves are these, and what helpful information can we obtain from wave
propagation theory? Let us take a look.
It is useful to start with the case of horizontal vibrations, which is simpler. Figure 12 presents
again the Simplified Solution for the surface circular foundation of Fig. 9. The same equations
for kh and ch are repeated at the bottom of Fig. 12. The viscous dashpot of expression, ch =
[4.64/(2-μ)] ρVs R2, represents the radiation energy carried away from the foundation by the
waves propagating in the soil.
Figure 12 also calculates this expression for two values of the Poisson’s Ratio, 0.33 and 0.50,
which approximately cover the range of interest of this parameter for soils. The dashpot becomes,
respectively, 2.78 ρVs R2 and 3.09 ρVs R2. These two expressions are numerically very similar,
indicating that the dashpot is not very sensitive to the exact value of the Poisson’s Ratio of the
soil, Furthermore, and this is very important from a theoretical viewpoint, the numerical
coefficients in the two expressions are within 10% of the value of π = 3.14. So, what the
simplified solution fitted to the original exact solution is telling us is that the horizontal viscous
dashpot is approximately the product of ρVs (which depends only on the properties of the soil),
times the area of the circle, π R2 (which depends only on the geometry of the contact area
between soil and foundation). That is, ch ≈ ρ Vs A = (ρ Vs) (πR2). This is very interesting and has
significant theoretical as well as practical implications.
It is useful at this point to look at some basic results of wave propagation theory relevant to the
original system of a plate on a half-space of Fig. 12. This is done with the help of Fig. 13.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
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XXI Nabor Carrillo Lecture
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Figure 12. Horizontal radiation dashpot for two Poisson’s ratios.
Figure 13. Perfect viscous dashpot analog for 1D shear wave propagation.
Figure 13 assumes that we have placed the same massless rigid circular plate of Fig. 12, but now
on the surface of an infinitely long elastic solid tube of radius R, with the tube in Fig. 13 having
the same properties of the half-space of Fig. 12. Figure 13 is an example of one-dimensional
elastic wave propagation, in which the horizontal vibration of the plate generates a pure shear
wave that propagates vertically down with a wave speed Vs, while inducing horizontal
displacements along the tube. It turns out that it is possible to replace mathematically the tube
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under the plate by an equivalent horizontal dashpot, ch = ρ Vs A = (ρ Vs) (π R2). The equivalent
horizontal spring, kh = 0 in this case. It is important to note that this equivalent dashpot, ch = ρ Vs
A = (ρ Vs ) (π R2), is not an approximation but is an exact mathematical analog to the infinite
tube in every respect. The product ρ Vs is so important in wave propagation and Soil Dynamics
that it has been given a special name: it is called the Shear Impedance of the material. This
Impedance, ρ Vs, completely controls the relation between load and displacement at the interface
between the massless rigid plate and the elastic material below for 1D wave propagation in Fig.
13. The expression, ch = ρ Vs A, is not restricted to the case when the load Q in Fig. 13 is
sinusoidal, but it is valid for any time history of Q = Q(t). Also, the expression is still rigorously
valid for noncircular shapes of the rigid massless plate and associated cross-section of the elastic
tube, including square and rectangular, with the expression for the dashpot being always ρ Vs A,
where A is the actual area of the square, rectangle, or other shape.
If the massless plate located on top of the elastic tube were excited vertically instead of
horizontally, as done in Fig. 14, a compression-extension wave (similar to a sound wave) will
propagate down the tube. This 1D wave propagation model in Fig. 14 is relevant to the original
problem of vertical excitation of the foundation on a half-space of Figs. 4-7.
Figure 14. Perfect viscous dashpot analog for 1D compression-extension wave
propagation.
In Fig. 14, when the massless plate vibrates vertically, as the compression-extension waves
propagate down, the material in the tube alternately compresses and extends in the vertical
direction, generating vertical displacements along the tube. This compression-extension wave in
the tube of Fig. 14 will propagate at a speed greater than the shear wave velocity, with this speed,
V, controlled either by the constrained modulus, D, V = VD = (D/ρ)1/2 , or by the Young’s
Modulus, E, V = VL = (E/ρ)1/2. The actual wave speed, V, will be either of these two values (or a
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XXI Nabor Carrillo Lecture
21
value in between), and it will depend on how freely can the rod expand or contract laterally. In
one extreme case, if the tube is completely surrounded by a rigid wall and cannot strain laterally
at all (similar to the situation in a soil consolidometer test), the wave velocity will be high, VD =
(D/ρ)1/2 . On the other hand, if the tube is completely free to expand or contract laterally (similar
to a triaxial or unconfined compression test), the wave velocity will be lower, VL = (E/ρ)1/2. Table
5 lists normalized values of VL and VD for two values of the Poisson’s Ratio. But whatever the
value of this wave speed for the compression-extension waves, the whole infinite rod can always
be replaced analytically by a vertical dashpot equal to the Impedance of the material, ρV, times
the actual area of the plate, A, where V is the actual speed of the wave traveling in the tube.
Table 5. Compression-extension wave velocities relevant to vertical vibration of
plate on a half-space.
What does this all mean for the original problem of the circular rigid plate on the surface of the
half-space? As sketched in Fig. 15, the actual foundation problem is typically a 3D situation, and
in first approximation the waves under the plate do not travel vertically but go out in many
directions controlled by the maximum angle θ shown in the figure. This general picture is true for
both vertical and horizontal excitations. The problem for the vertical vibrations sketched in Fig.
15a is further complicated by the fact that compression-extension waves predominate only very
close to the vibrating plate, with other waves including shear waves appearing at longer distances
from it. But from the viewpoint of this discussion, the rather simplified sketch of Fig. 15a will
suffice, as the dynamic vertical load-displacement relation for the plate depends on the speed of
this compression-extension excitation generated in the soil very close below the plate, rather than
on the more distant waves that develop in the soil in the far field.
Consider first the case of the horizontal excitation in Fig. 15b, which is simpler. By now we can
agree that the horizontal vibrations of the plate are mainly shearing the interface with the soil, so
that assuming that the waves sent down into the soil are mainly shear waves makes intuitive
sense. We know from theory (Fig. 13) that if the angle θ was zero in Fig. 15, that is if all the
shear waves were going down vertically as 1D waves, the equivalent horizontal dashpot would be
exactly ch = ρ Vs A. As we saw before when discussing Fig. 12, the correct solution in this case,
while not exactly ch = ρ Vs A, is numerically close to it, within 10%, and it is also approximately
independent of frequency. This suggests that for this case of horizontal vibration of a circular
plate in Fig. 15b, it would appear as if the angle θ of the waves is actually not far from zero, and
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that the problem is surprisingly close to being one-dimensional. How can this be? Because of a
phenomenon known as destructive wave interference, the waves going out at angles greater than
zero tend to cancel each other, leaving only shear waves that travel down more or less vertically
in this particular case.
Figure 15. Waves and radiation damping in vertical and horizontal vibrations
This is a very important conclusion for the extension of the Simplified Solutions to foundation
shapes that are not circular, because if that conclusion was true for any foundation shape, we
could say that the equivalent horizontal dashpot could always be calculated using this expression
ρ Vs A, where the area A is just the actual total area of contact between the foundation and the
soil. It turns out that things are not so simple, but still, this gives us a starting point for the
development of simplified solutions for noncircular shapes.
Let us now turn our attention back to the vertical vibration of a circular foundation in Fig. 15a.
We apply the same logic, except that in this case the compression-extension wave velocity to put
in the equation ρ V A is not obvious, because as we saw in Fig. 14, the corresponding
compression-extension waves may travel relatively slow or faster depending on the lateral
straining of the tube. We solved this by inventing a new wave velocity, that we label V La, defined
by the expression, VLa = 3.4 Vs /(π(1-μ). The symbol VLa stands for “Lysmer’s Analog wave
velocity,” and it is simply the value of the velocity V = VLa that, when inserted it in the
expression ρ V A, gives the correct cv defined by Lysmer in his Simplified Solution to the
original problem, cv = [3.4/(1-μ)] ρVs R2 (see Fig. 7). If we consider the range of possible wave
speeds for compression-extension waves in soils, the value of this new wave velocity VLa is
relatively low, and close to the value of wave velocity controlled by the Young’s Modulus of the
material (see Table 5). This makes sense, because it would intuitively seem that the soil under the
foundation, when compressed vertically as in Fig. 15a, is relatively free to expand laterally, so the
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XXI Nabor Carrillo Lecture
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situation in the soil immediately below the foundation is closer to a triaxial than to a
consolidometer test.
I just spent some time going over the details of these derivations in Figs. 15a and 15b. But the
effort is worth it, because the two expressions for cv and ch as functions of VLa and Vs in Fig. 15,
are the key to the development of a whole new family of Simplified Solutions, not only for
foundations of noncircular shapes, but also for embedded foundations. In the early 1980’s, Prof.
George Gazetas and I explored this approach in some detail, and the approach now has become
part of the accepted State of Practice for the approximate calculation of equivalent foundation
dashpots.
3.5
Vertical radiation dashpot for embedded foundation
An obvious first application of this simplified concepts is to the same case of the cylindrical rigid
foundation excited vertically, but now embedded in the half-space (Fig. 16). We assume that the
base radiates energy in the form of compression-extension waves traveling with the Lysmer’s
Analog wave velocity, so the radiation dashpot associated with the base is ρVLa Ab, where Ab is
the area of the base, in this case πR2. In short, we assume that this dashpot associated with the
base is identical to the one found by Lysmer when the foundation was at the ground surface and
not embedded.
Figure 16.
Simplified vertical radiation dashpot for embedded circular
foundation (modified after Gazetas et al., 1985a).
We also assume that the perimeter of the cylinder when vibrating vertically is sending shear
waves into the soil which propagate horizontally with the wave speed Vs. This is what our
intuition tells us and it seems reasonable. This gives us a second dashpot associated with this
radiation of energy at the contact between soil and foundation sidewall, which is shown here, of
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value ρVs Aw, where Aw is the total area of contact of the sidewall. And because the two dashpots
are in parallel, we can just add up the two values to get the total dashpot for the whole embedded
foundation. This is done in Fig. 16, providing a simple estimated expression for the equivalent
vertical radiation dashpot of the embedded foundation cv ≈ ρ VLa Ab + ρ Vs Aw.
There is a need to be careful here, as we have jumped a lot ahead of a more rigorous analysis, and
have made a number of assumptions based only on our intuition. It turns out that this very simple
expression works well for this case, as I will show you in a minute. But it does not work so well
in other cases, and one should always check these simplified models against more rigorous
solutions before applying them with confidence.
On the other hand, once it is shown that a Simplified Solution like this works, it provides a
tremendous amount of insight to researchers and engineers. Let me give you two conclusions out
of this expression in Fig. 16, so you can appreciate better what I am saying. The first conclusion
has to do with the contribution of the embedment to the total vertical dashpot. While the value of
VLa acting at the base is typically 50% to 100% greater than the Vs acting on the sidewalls (see
Table 5), the area of the sidewall itself tends to be much greater than the area of the base. For
example, if the depth of embedment D = R, which is not a large embedment, the wall area will be
twice that of the base, so already the wall is contributing roughly as much as the base, and for
greater embedment it contributes significantly more to the total dashpot than the base. Therefore,
we should expect that embedded foundations will have a lot of damping, having less dynamic
response at the critical frequencies than surface foundations, which is good. But the counterpart
to this, and this is my second conclusion, is that before taking advantage of this beneficial effect
of the embedment, you better make sure that you have a good contact between the sidewall and
the soil. If you do not, you may have a foundation with much less damping that you thought you
had, and that may be dangerous. As embedded foundations are often constructed first in a trench
leaving a gap at the sides, and the gap is filled later with a sandy fill that is sometimes difficult to
compact because of the lack of space, there is always the possibility that the contact is not so
good. So, the engineer may want to do a parametric study assuming that the second term of the
expression in Fig. 16 does not exist, it exists, or it is only partially efficient, by multiplying this
second term of the expression by a factor between 0 and 1. And this is the great advantage of
such a Simplified Solution; it allows the engineer to use his/her judgment in the analysis and to
keep control of the situation instead of relying completely on a complicated computer program he
may not understand or control.
Figure 17 shows a comparison for the dashpot of a circular embedded foundation, between the
predictions of this Simplified Solution and rigorous dynamic finite element calculations. The
curves in the figure, corresponding to the Simplified Solution, show a slight effect of frequency
because the exact dashpot was used for the contribution of the base, instead of the simplified
frequency-independent dashpot of Fig. 16. The comparisons in Fig. 17 correspond to various
degrees of embedment up to an embedment equal to the diameter of the foundation. The
agreement between Simplified and rigorous solutions is excellent, and confirms that a significant
embedment with good contact between sidewalls and surrounding soil may provide a total
radiation dashpot that is several times the value of the dashpot of the surface foundation.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
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Figure 17. Vertical radiation dashpot of embedded circular foundation:
comparison between simplified (curves) and dynamic finite element results (data
points) (Gazetas et al., 1985a; Day, 1977).
3.6
Vertical static spring for embedded foundation
Figure 18 includes the corresponding expression for the static vertical spring, kv0, of the same
circular embedded foundation addressed in Figs. 16-17. As indicated in Fig. 18, the value of kv0
is obtained by the multiplication of three factors. The first factor, 4GR/(1-μ), is just the
expression for the stiffness of the surface foundation without embedment discussed before. The
second factor, (1+0.1D/R), is the “trench coefficient,” which is a small correction, and
corresponds to placing the foundation at the bottom of the trench of depth D, but without any
contact between the sidewall and the soil. And finally, the third factor, [1+0.19(Aw/Ab)2/3], which
provides a much bigger correction, is associated with the actual contact area between the
foundation wall and the soil, Aw. Again, if the engineer has doubts about the quality of this
contact, he/she should conduct a parametric study which includes reducing this third factor to a
value closer to, or equal than one.
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Figure 18. Vertical static spring of embedded circular foundation including
trench and soil-wall contact factors (Gazetas et al., 1985a).
3.7
Embedment and dynamic response
It is important to see how this very significant influence of the embedment on the radiation
damping translates into a much reduced dynamic response when subjected to dynamic loads.
Both analysis and experiments have consistently verified the importance of the effect. This is
illustrated by Fig. 19, that shows the results of experiments using three small-scale models
conducted by Novak (1970), where he excited vertically and horizontally a foundation embedded
in a partially saturated dense loess loam. The effect of embedment is similar for vertical and
horizontal excitations. Let us focus on the vertical excitation, shown in the upper plot of Fig. 19.
The first experiment for the fully embedded foundation with good contact with the surrounding
soil, labeled “A,” produced a response curve of displacement versus frequency which is very flat,
with low values of the dynamic displacement. The curve indicates a highly damped system with a
very stiff equivalent spring. In the second experiment, labeled “B,” the sidewall contact was
weakened by placing a relatively well compacted sandy fill between the foundation and the
surrounding soil; now the response curve is a bit higher and has a small peak at a frequency of
about 3000 revolutions/minute. And then in the third experiment, labeled “C,” the model
foundation was just placed at the bottom of the trench without any contact with the soil at the
sides. Clearly in this third experiment there is much less damping in the system, with a very high
peak and large displacements at a frequency of about 2000 revolutions/minute. The figure
provides a dramatic illustration on how different the dynamic response of a foundation can be
without the benefit provided by the contact with the surrounding soil.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
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XXI Nabor Carrillo Lecture
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Figure 19. Experimental verification of soil-wall contact effect on dynamic
vertical and horizontal responses of embedded circular foundation (Novak, 1970).
4.
EQUIVALENT SPRINGS AND DASHPOTS FOR NONCIRCULAR
SHAPES
I want to use the rest of this Lecture discussing the development of Simplified Methods for
noncircular foundation shapes that took place in the 1980’s, both for surface and embedded
foundations. Table 6 lists the publications reporting the main results as well as the two summaries
by Gazetas (1990, 1991). Originally it grew out of a collaboration at the beginning of the 1980’s
between Prof. George Gazetas and myself, built on some of the ideas I discussed before,
especially for the radiation damping generated by the base of the foundation and the foundation
sidewalls. While I played a role at the beginning of the process and I am a co-author in the first
three papers listed in Table 6, Prof. Gazetas was the driving force of the whole project, and
pursued it systematically through a series of studies and publications over a number of years,
until he was able to put it all together in two publications listed at the bottom of Table 6. Gazetas
(1990) is a chapter in a Foundation Engineering Handbook, and Gazetas (1991) is an article in the
Journal of Geotechnical Engineering of the American Society of Civil Engineers. In these two
publications, he provides charts, formulas and numerical examples, ready to use by practicing
engineers
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Table 6 gives an idea of how magnificent was this project by Prof. Gazetas. For each of the six
degrees of freedom, vertical, torsional, horizontal in the two directions, and rocking in the two
directions, he compared possible simplified solutions for springs and dashpots with rigorous
computer results, modifying the simplified solutions as needed to fit the rigorous results, and
arrived to recommendations that engineers could use. He also provided convincing experimental
validation for his calculations. The work was further complicated by the strong coupling between
horizontal and rocking in embedded foundations, that he also addresses in his 1990 and 1991
publications. You can appreciate the complexity of the work involved in getting the correct
simplified solutions for horizontal, rocking and torsional springs and dashpots of embedded
foundations, by noticing that he needed a total of six papers to present all necessary results.
My main purpose today is to provide you with an introduction to the basic approach used by Prof.
Gazetas, as an introduction to his 1990 and 1991 publications, which may be useful if you need
to use them in one of your projects.
Table 6. Simplified Solutions for surface and embedded foundations of arbitrary
shape (Gazetas and co-workers, 1985-91).
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4.1
XXI Nabor Carrillo Lecture
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General problem formulation
The general formulation for both surface and embedded foundations of arbitrary shape is
presented in Fig. 20. The base has an arbitrary shape and an area, Ab, which is embedded at depth
D, but with the possibility of the actual depth of contact of the foundation wall with the soil being
smaller, d < D, and with the actual total contact area between the sides of the foundation and the
soil being Aw. An important tool of these procedures is a rectangle 2L x 2B that circumscribes the
actual base area. This rectangle defines the degree of elongation of the actual area by its aspect
ratio, L/B. For both square and circular foundations the aspect ratio is 1.0.
Figure 20. General formulation for surface and embedded foundations of
arbitrary shape (Gazetas, 1991).
This surface or embedded foundation is located in a half-space which has the same properties
already discussed (G, μ and ρ), to which it is added now the internal damping ratio of the soil,
labeled β. It is not necessary to worry about β through most of the derivations, with all calculated
dashpots being radiation dashpots. Later in this Lecture I will provide the general expression used
to increase the values of these radiation dashpots in order to incorporate the effect of β.
The solutions presented by Gazetas (1991) generate springs and dashpots for six degrees of
freedom, all shown in Fig. 20: vertical loading; horizontal loading in both the short direction, that
is along the y-axis; horizontal parallel to the long direction x; a rocking moment in the short
direction, that is around the x-axis; rocking around the y-axis; and finally, a torsional moment
around the z-axis. It is useful to illustrate some of the complexities that must be considered in the
formulation of these Simplified Solutions, by considering the calculation of the horizontal
radiation dashpot along the long axis x for the embedded foundation of Fig. 20. The foundation is
moving back and forth horizontally in the x-direction, and the question is: What is happening in
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terms of the waves generated by the different contact areas? The base is clearly shearing the soil,
so you would expect its contribution to the total radiation dashpot to be proportional to the area
Ab and to the shear wave velocity of the soil Vs. The situation with the foundation walls is more
complicated. As the foundation moves back and forth in the long direction, the two walls parallel
to x in the figure, are also shearing the soil so their contribution should be proportional to Vs. But
the two walls perpendicular to x, are pushing back and forth against the soil behind them, so you
would expect that their contribution should not be proportional to Vs but to the other wave
velocity we defined before for compression-extension waves, the Lysmer’s analog wave velocity
that we labeled VLa. And there are also other walls in Fig. 20 which are neither parallel nor
perpendicular to x, which further complicate the situation.
Figure 21 presents a more detailed formulation for only the base of the surface or embedded
foundation, for arbitrary shapes including circular, square, rectangular or in fact any shape. The
graph shows again the three axis, x, y and z, all passing by the centroid O of the area of the base,
and the three loads Hx , Hy and V parallel to the three axes, as well as the three moments around
each of the three axes, Mx, My and Mt. The actual area of the foundation is called A in Fig. 21,
while it is labeled Ab in Fig. 20 and other plots. There are other parameters associated with the
area A which are also important for the calculations. They are (see Fig. 21): the area moment of
inertia around the x axis, Iax; the same area moment of inertia around the y axis, Iay; and the polar
area moment of inertia around the z axis, J = Iax + Iay. These three area parameters are the same
studied in school in the Strength of Materials course when looking at sections of beams and
columns subjected to bending or torsion. It turns out that these three area moments of inertia are
needed in the Simplified Method when computing the springs and dashpots for rocking and
torsional vibrations. And finally, the length and width of the circumscribed rectangle, L, B, and
the aspect ratio of the foundation, L/B, are also listed in Fig. 21.
4.2
Vertical spring for surface foundation
The next few figures show some selected results for the surface foundation of arbitrary shape,
reproduced from Dobry and Gazetas (1986). The charts for the vertical spring in Figs. 22-23 are
representative of other similar charts and formulas associated with equivalent springs for
horizontal, rocking and torsional vibrations presented by Dobry and Gazetas (1986) and Gazetas
(1991). The chart in Fig. 22 allows calculating a dimensionless parameter Sz0, which is used to
estimate the static vertical spring, kz0, through the expression, kz0 = Sz0 (2LG)/(1-μ). The
parameter along the abscissas is A/4L2, which for the special case of a rectangular shape is equal
to the reciprocal of the aspect ratio of the rectangle, (L/B)-1 = B/L. That is, is A/4L2 = 1.0 for a
square shape, A/4L2 = 0.33 for a rectangle of aspect ratio 3, etc. Why did we select this strangely
looking new parameter instead of simply using B/L? Because it turns out that the square and
circular shapes, while they have the same aspect ratio of 1.0, have different values of S z0, as
shown in Fig. 22 by the corresponding data points, so we had to invent a new parameter for the
plot to work. The data points in Fig. 22 correspond to rigorous elasticity solutions, obtained either
mathematically or numerically, that we retrieved from the literature. This includes the rigorous
solution for the circle previously discussed in this Lecture, the solution for the square, and
solutions for a number of increasingly elongated rectangles, ranging between L/B = 2 and L/B =
20. The figure also includes a number of data points for elliptical, triangular and other shapes,
and finally we just fitted the equation for Sz0 shown on the figure which is the one we proposed
for use in the Simplified Solution for vertical loading.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
Figure 21.
XXI Nabor Carrillo Lecture
Main parameters for surface foundation of arbitrary shape (Dobry
and Gazetas, 1986).
Figure 22. Vertical static spring of surface foundation of arbitrary shape (Dobry
and Gazetas, 1986).
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XXVI RNMSeIG, Cancún, Quintana Roo, 14-16 noviembre 2012
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Figure 23 presents the variation of the vertical spring kv = kz with frequency for a Poisson’s Ratio
of 1/3. From now on, I will be showing some plots where springs and dashpots vary with
frequency, as this is an unavoidable part of some of the Simplified Procedures. We were lucky
before, that for the circular surface foundation the vertical and horizontal springs and dashpots do
not change much with frequency; this allowed Lysmer and other authors to propose approximate
springs and dashpots which are independent of frequency (Figs. 7 and 9). Figure 23 confirms this
lack of sensitivity of the spring constant to changes in frequency for areas which are not
elongated. That is, for squares, circles and short rectangles of L/B = 1 and 2, the curve in the
graph is rather flat. This is still true for rectangles of L/B = 4, but it is not true at all for very long
rectangles of aspect ratios of 6 or greater, which includes the very important case of strip
footings, for which the dynamic stiffness increases very fast at low frequencies and then
decreases. So in a case like that, the Simplified Solution must consider the effect of the frequency
of the applied loading.
Figure 23. Vertical dynamic spring of surface foundation of arbitrary shape
(Dobry and Gazetas, 1986).
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
4.3
XXI Nabor Carrillo Lecture
33
Vertical and horizontal radiation dashpots
Let me switch now my attention to the radiation dashpots of foundations of arbitrary shape. For
the time being, we are still talking only about surface foundations, and Fig. 24 is essentially the
same sketch of Fig. 15, of what happens to a surface foundation subjected to vertical and
horizontal vibrations. When discussing Fig. 15 before for the case of a circular foundation, it was
concluded that the vertical dashpot was given approximately by the expression, cv ≈ ρ VLa A, that
is the Impedance times the area, where the impedance ρ VLa was controlled by the velocity VLa,
the Lysmer’s Analog wave velocity, reflecting the speed of the compression-extension waves
traveling down vertically below the foundation. And the horizontal dashpot, ch, was also given by
the Impedance times the area of the foundation, but now with the impedance controlled by the
shear wave velocity of the soil, Vs. These expressions for cv and ch independent of frequency,
worked well for the circular shape, for which the value of the dashpot is about the same at low
and high frequencies.
Figure 24. Vertical and horizontal radiation dashpots of surface foundations of
arbitrary shape: the high-frequency asymptotes (Dobry and Gazetas,
1986).
Unfortunately this is not true anymore for elongated shapes like long rectangles, where the value
of the dashpot changes dramatically with frequency. But what we did find, first following our
intuition and then through a rigorous demonstration by Gazetas (1987), is that these expressions
for the vertical and horizontal dashpots reproduced at the bottom of Fig. 24, are always true for
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any foundation shape at high frequencies. That is, as the frequency increases, the angle θ of the
waves in Fig. 24 becomes zero, the waves propagate vertically down as one-dimensional waves,
and these two expressions become exact whatever the shape of the foundation. In other words, as
f → ∞:
𝑐𝑣 = 𝑐𝑧 → 𝜌𝑉𝐿𝑎 𝐴
(3)
𝑐ℎ = 𝑐𝑥 = 𝑐𝑦 → 𝜌𝑉𝑠 𝐴
(4)
This useful behavior of the waves generated by the foundation happens because of strong
destructive wave interference at the high frequencies, which cancels all waves traveling at angles,
θ > 0. It turns out that a similar phenomenon is well known in acoustics and is used in the
analysis and design of speakers. This is why in rock concerts, the low frequency sounds are
radiated out by speakers that cover a wide range of directions, while for the high frequencies,
directional speakers are needed that radiate energy only in one direction more or less as a 1D
beam of sound (Morse and Ingard, 1968; Massa, 1972).
The finding summarized by Eqs. 3-4, represented an important breakthrough in our development
of Simplified Solutions for two reasons. The first is that it tells us that for vertical and horizontal
dashpots of surface foundations, we should be normalizing the actual dashpot obtained from
rigorous solutions, dividing it by either ρ VLa A or by ρ Vs A, with the expectation that this ratio
will become 1.0 at high frequencies irrespective of the shape of the foundation. And the second
reason is that additional simple theoretical derivations tell us that for the rocking vibrations, the
same thing should be happening at high frequencies of surface foundations as for vertical
vibrations, because during rocking vibrations the foundations is also pushing and pulling
vertically the soil below, with the controlling wave velocity still being VLa, but replacing the area
A in the expression, by the area moment of inertia of the foundation around the corresponding
axis (Iax or Iay). That is, for any arbitrary foundation shape, at high frequencies, the rocking
dashpots, crx ≈ ρ VLa Iax and cry ≈ ρ VLa Iay. The same is true for torsional vibrations, which
similarly to the horizontal loading also shears the soil below, where the expression at high
frequencies for ct should still controlled by the shear wave velocity Vs, but with the area in the
expression replaced by the polar moment of inertia of that area, J. In summary, Eqs. 3-4 for the
three translational degrees of freedom are supplemented by Eqs. 5-7 for the three rotational
degrees of freedom, indicating that, as f → ∞:
𝑐𝑟𝑥 → 𝜌𝑉𝐿𝑎 𝐼𝑎𝑥
(5)
𝑐𝑟𝑦 → 𝜌𝑉𝐿𝑎 𝐼𝑎𝑦
(6)
𝑐𝑡 → 𝜌𝑉𝑠 𝐽 = 𝜌𝑉𝑠 (𝐼𝑎𝑥 + 𝐼𝑎𝑦 )
(7)
These ideas were confirmed by Dobry and Gazetas (1986) for the radiation dashpots, first for
vertical and horizontal and then for rocking and torsional. Figure 25 illustrates the results for the
horizontal dashpot in the short direction, versus frequency of loading. Following the conclusion
summarized in Eq. 4, the variable along the ordinate axis in Fig. 25 is the dashpot cy divided by ρ
Vs A. The figure confirms the previous conclusion that for non-elongated shapes (circles, squares
and rectangles of L/B = 2), the ratio cy /(ρ Vs A) is about constant and close to one at all
frequencies considered. On the other hand, for long rectangles and strip footings, cy is much
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
35
greater than ρ Vs A at low frequencies, with the ratio between the two converging to one at high
frequencies, exactly as predicted. So this is the key plot for the radiation dashpot for horizontal
loading in the short direction. The situation for horizontal loading in the long direction of the
foundation is similar, and the plot for the vertical dashpot looks just like Fig. 25, except that VLa
is used instead of Vs on the ordinate axis.
Figure 25. Normalized horizontal radiation dashpot versus frequency of surface
foundation of arbitrary shape (Dobry and Gazetas, 1986).
4.4
Torsional and rocking radiation dashpots
Figure 26 includes the corresponding plot for torsional vibrations, where the torsional dashpot ct
has been normalized to the product ρ Vs J, as suggested by Eq. 7. Here you see a different
phenomenon, which is typical of the rotational vibrations of surface foundations including not
only torsional but also rocking oscillations. This phenomenon is that the equivalent radiation
dashpots for these rotational vibrations invariably go to zero as the frequency decreases and goes
to zero, because of destructive wave interference which does not allow any energy to leave the
neighborhood of the foundation when the frequency approaches zero. This is true for circular
shapes, as shown by the corresponding curve for L/B = 1, and it is also true for long rectangles.
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In all cases, the torsional or rocking radiation dashpot is not constant but increases rapidly with
frequency at the beginning, and then it stabilizes at the theoretical value at high frequencies, with
this theoretical high-frequency value for the torsional case of Fig. 26 being the product ρ Vs J. A
very similar pattern to that of Fig. 26, is exhibited by the rocking dashpots, which also go to zero
at low frequencies and converge at high frequencies to the product ρ VLa Iax or ρ VLa Iay. This
variation with frequency of rocking and torsional radiations dashpots certainly complicates the
formulation of the Simplified Methods, but unfortunately this complication is unavoidable.
Figure 26. Normalized torsional radiation dashpot versus frequency of surface
foundationof arbitrary shape (Dobry and Gazetas, 1986).
4.5
Simplified systems including embedment
Let us move on to embedded foundations of arbitrary shape. Figure 27 shows a sketch of the
different effects contributing to the static horizontal stiffness of an embedded foundation: (i) the
stiffness of the base of the foundation shearing the soil, which in first approximation is equal to
the stiffness of the corresponding surface foundation; (ii) the trench effect, that is the increase in
stiffness due to the foundation being placed at the bottom of the trench instead of at the surface of
the soil; and very importantly (iii) the contribution of the contacts between the embedded
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
37
foundation walls and the surrounding soil. These are the same three factors described before in
this Lecture, when discussing the vertical stiffness of an embedded circular foundation.
Figure 27. Horizontal stiffness of embedded foundation of arbitrary shape: basic
sketch (Gazetas and Tassoulas, 1987a).
Figure 28 includes the expression for the horizontal static spring in the short direction, ky0,
developed by Gazetas and Tassoulas (1987a) on the basis of rigorous calculations for several
shapes and degrees of embedment. The expression assumes that the surface static stiffness, ky0,sur,
has already been calculated, with the expression giving the factors greater than 1.0 that reflect the
trench and sidewall effects. Please notice that the solution allows for the possibility of the
foundation walls not being in contact with the soil near the top of the excavation, and it certainly
allows the engineer to reduce the contribution of the sidewall contact if he/she does not trust the
overall quality of the contact between the wall and the soil.
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Figure 28. Horizontal stiffness of embedded foundation of arbitrary shape:
equation and correlation for the sidewall contact factor (Gazetas and Tassoulas,
1987a).
And finally, Fig. 29 presents the basic sketch used by Gazetas and Tassoulas (1987b) to study the
different contributions to the total horizontal radiation dashpot of an embedded foundation, of
the various contact areas and types of waves. The area of the base always generates shear waves.
For horizontal vibrations along the short direction as shown in the figure, the two walls
perpendicular to the short direction push and pull against the soil generating compressionextension waves, so the contribution of that wall is proportional to the actual area of contact of
that wall times ρ VLa. On the other hand, the two walls parallel to the short direction are shearing
the soil, so their contribution should be controlled by the shear wave velocity Vs. Walls in the
figure which are neither parallel nor perpendicular to the direction of motion, generate both shear
(Vs) and compression-extension waves (VLa), as shown on the figure. The situation would seem
to be too complicated for a Simplified Method. But Gazetas and Tassoulas (1987b), after
integrating all these contributions, concluded that from the viewpoint of the horizontal dashpot it
was only necessary to consider the four walls of the circumscribed rectangle rather than the walls
of the actual foundation, which is much simpler.
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
39
Figure 29. Horizontal radiation dashpot of embedded foundation of arbitrary
shape: basic sketch (Gazetas and Tassoulas, 1987b).
Figure 30 illustrates what I mean. It presents a numerical example taken from the summary paper
by Gazetas (1991), where he calculates all six sets of springs and dashpots for this embedded
foundation, which has a slightly irregular shape, and where the wall reaches an embedment depth
of 6 m but has no contact with the soil in the top 2 m. When it comes to computing the total
horizontal dashpot of this embedded foundation in the short direction y, the procedure ignores the
actual foundation walls and replace them by the four walls of the circumscribed rectangle of sides
2L x 2B = 16 x 5 m. That is, the total area of contact with the soil of the two long walls of total
length 4L = 32 m, is assumed to generate compression-extension waves over the height of contact
of 4 m, with this contribution controlled by VLa and by the total area of contact 32 x 4 = 128 m2;
while the total area of contact of the two short walls of total length 4B = 10 m, is assumed to
generate shear waves, with this contribution controlled by Vs and by the total area of contact 10
x 4 = 40 m2. Then the three contributions of: base area and shear waves, area associated with 4L
and compression-extension waves, and area associated with 4B and shear waves, are just added
up to obtain the total radiation dashpot.
Table 7 includes a partial view of the summary table for embedded foundations in Gazetas (1991)
that provides clear instructions on how to compute different things. The last column of Table 7
includes the rules just described, on how to generate the horizontal dashpots cy and cx for an
embedded foundation. In each case you have three term. For example, the expression for the total
cy = cy, emb includes: (i) a first term labeled Cy, which is the contribution of the base, and is
calculated in another table essentially as the area of the base times ρ Vs, with a slight influence of
frequency; (ii) a second term, 4ρVsBd, which is the contribution of the contact area associated
with the two sides of the circumscribed rectangle that are shearing the soil; and finally, (iii) a
third term, 4ρVLaLd, which is the contribution of the contact area associated with the other two
sides of the rectangle, which are pushing back and forth against the soil. For the dashpot in the
other direction, cx, the two walls that were shearing before are now pushing and Vs is replaced by
VLa, etc. Table 7 also includes the expression for the vertical radiation dashpot of the same
embedded foundation. The situation for vertical is much simpler. The expression is cz,emb = Cz + ρ
Vs Aw; all sidewalls are shearing the soil, and the total sidewall area Aw is the actual area of
contact around the foundation, with the circumscribed rectangle playing no role in this
calculation.
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Figure 30. Embedded foundation having an arbitrary shape and partial
embedment: numerical example (Gazetas, 1991).
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
41
Table 7. Partial view of summary table for embedded foundations of arbitrary
shape (Gazetas, 1991).
5.
EFFECT OF INTERNAL SOIL DAMPING
To complete the picture, it is important to add to the viscous dashpots calculated with these
Simplified Methods, the contribution due to the energy dissipated internally in the soil, mainly
due to friction, which is typically characterized by a soil damping ratio, β (Fig. 31). The value of
β depends on several factors, like the level of cyclic shear strain induced in the soil by the
dynamic loading, the type of soil, and the Plasticity Index if the soil is a clay. Depending on the
circumstances, β can be as low as 0.02 or 0.03 (that is 2 or 3%) and as high as 0.20 or 0.30 (2030%). Fortunately from the viewpoint of the Simplified Methods covered in this Lecture, once
the radiation dashpot, cradiation, has been calculated at a certain frequency f, the total dashpot, ctotal,
including the effect of β can be obtained using Eq. 8:
𝑘
𝑐𝑡𝑜𝑡𝑎𝑙 ≈ 𝑐𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 + (𝜋𝑓) 𝛽
(8)
where k is the corresponding stiffness calculated at the same frequency for the elastic half-space.
This simple expression for β is obtained from the Correspondence Principle of the Theory of
Viscoelasticity (Bland, 1960), and it is valid for any of the six degrees of freedom (vertical,
horizontal, rocking and torsional), as well as for surface and embedded foundations of any shape.
6.
FINAL COMMENTS
Simplified Methods will continue to play an important role in Soil Dynamics as they do in the
rest of Geotechnical Engineering. While powerful computer simulations can produce more exact
and detailed information, Simplified Methods are irreplaceable as a basis for codes and
regulations, and as a tool to verify the computer results (“reality checks”). Furthermore, they are
just better suited to many applications, where:
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XXVI RNMSeIG, Cancún, Quintana Roo, 14-16 noviembre 2012
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

they allow the engineer to conduct calculations by hand or with a minimum
computational effort, including parametric variations; and
in the process, the engineer has the possibility to develop a feel for the physical meaning
and relative importance of the various factors, with more personal control of calculations
and decisions including use of engineering judgment as needed.
Figure 31. Contribution of soil internal damping, β, to total dashpot (Gazetas,
1991).
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
43
It is useful for the engineer applying any of these Simplified Methods, to have at least a basic
understanding of the theory and simplifications behind the procedure, which in Soil Dynamics
includes some Dynamics and Wave Propagation concepts. This Carrillo Lecture was aimed at
providing some of this understanding, with focus on shallow machine foundations and other
dynamic soil-structure interaction applications.
Many researchers have proposed useful Simplified Methods in Soil Dynamics, some of which are
referenced in the paper. Methods suggested by the author are listed in Table 2, and they cover
systems ranging from the free field and earth dams to shallow and deep foundations, subjected to
excitations that include both seismic shaking and machine vibrations.
The main focus of this Carrillo Lecture was on shallow machine foundations on a half-space
subjected to dynamic loads in any of the six degrees of freedom of the foundation, and the
Simplified Methods that have been proposed over the years to characterize the corresponding
equivalent soil springs and dashpots. This included both frequency-dependent and frequencyindependent springs and dashpots. It started with the circular surface foundation which was
studied over much of the 20th Century, until the breakthroughs by Lysmer and others in 19661971, and continued with the cases of surface and embedded foundations of arbitrary shape that
culminated in the two summary publications by Gazetas in 1990 and 1991. These solutions for
machine foundations are also useful for analysis of dynamic soil-structure interaction during
earthquakes.
The development of these simplified equivalent springs and dashpots for both surface and
embedded foundations of arbitrary shape was discussed in some detail, including the contribution
of the author in the early part of the process. This discussion serves to introduce some of the
basic dynamic theoretical concepts behind the methods, and hopefully also as an introduction to
their use in actual engineering projects.
ACKNOWLEDGMENTS
I am most grateful to George Gazetas for our many exhilarating discussions in the early 1980’s,
about basic concepts of dynamics and wave propagation that could be used to develop Simplified
Procedures for shallow and pile foundations subjected to dynamic loads. I am also grateful to
several colleagues with whom I had the pleasure to develop and validate Simplified Methods on
the various problems listed in Table 2: José M. Roesset, Robert V. Whitman, Issa Oweis, Alfredo
Urzua, George Gazetas, Michael J. O’Rourke, Ramli Mohamad, Panos Dakoulas, Kenneth. H.
Stokoe II, John L. Tassoulas, Victor Taboada and Lee Liu. Finally, I am extremely grateful to the
professors that taught me Soil Mechanics and Soil Dynamics and mentored my initial research
efforts: Arturo Arias at the U. of Chile, Eulalio Juárez Badillo, Alfonso Rico and Enrique Tamez
at the UNAM, and Robert V. Whitman, José M. Roesset and John T. Christian at MIT.
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Simplified methods in Soil Dynamics
Dr. Ricardo Dobry
XXI Nabor Carrillo Lecture
45
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