1b conf. carrillo by dobry- mass

8/12/2019 1b Conf. Carrillo by Dobry- MASS

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Simplified methods in Soil Dynamics1

Dr. Ricardo Dobry XXI Nabor Carrillo Lecture

SOCIEDAD MEXICANA DE INGENIERA GEOTCNICA A.C.

MTODOS SIMPLIFICADOS EN LA DINMICA DE SUELOS

RICARDO DOBRYProfesor Institucional, Instituto Politcnico de Rensselaer,

Troy, Nueva York, E.U.A.

RESUMENDespus de una breve descripcin de las caractersticas principales que definen la Dinmica deSuelos y sus aplicaciones a la ingeniera, se discute la importancia de los Mtodos Simplificados.An con la actual disponibilidad de poderosas simulaciones por computadora, los MtodosSimplificados continuarn desempeando un papel importante en la Dinmica de Suelos ascomo lo han hecho en el resto de la Ingeniera Geotcnica. Los Mtodos Simplificados permitenal ingeniero realizar clculos manuales o con un mnimo de apoyo computacional, facilitando aslos estudios paramtricos. En el proceso, el ingeniero tiene la posibilidad de desarrollar unasensibilidad sobre el significado fsico y la importancia relativa de los factores involucrados,adquiriendo un mayor control sobre los clculos y las decisiones, incluyendo el uso de su criteriocomo ingeniero basado en su experiencia personal. Se presenta una lista de procedimientossimplificados desarrollados por el autor, considerando sistemas que varan desde el campo libre ylas presas de tierra, hasta cimentaciones superficiales y profundas sujetas tanto a excitacionescausadas por movimientos ssmicos, como por vibraciones de maquinaria. Un conocimientobsico de la teora fundamental y de las idealizaciones detrs de los procedimientos simplificados

pueden ser muy tiles a los ingenieros, incluyendo especialmente conceptos de la Dinmica y dela Propagacin de Ondas. Parte de este conocimiento se presenta en el documento, con aplicacina las cimentaciones superficiales para maquinaria y a la interaccin dinmica suelo-estructura.

La Conferencia se centra en las cimentaciones superficiales para maquinaria sobre unsemiespacio sujetas a cargas dinmicas en cualquiera de los seis grados de libertad, y losMtodos Simplificados que se han desarrollado a travs del tiempo para caracterizar a loscorrespondientes resortes y amortiguadores equivalentes del suelo. Esto incluye el caso generalen que estos resortes y amortiguadores dependen de la frecuencia de excitacin, as como lasimplificacin lograda en algunos casos en que estos resortes y amortiguadores se consideranindependientes de la frecuencia. Inicia con el caso de la cimentacin superficial circular que fue

estudiada durante gran parte del siglo XX, hasta las contribuciones seminales desarrolladas porLysmer y otros autores entre 1966 y 1971, y continua con los casos de cimentacionessuperficiales y enterradas con geometra de base arbitraria, concluyendo con las dospublicaciones sintetizadas escritas por Gazetas en 1990 y 1991. El desarrollo de estos mtodossimplificados con resortes y amortiguadores equivalentes para cimentaciones superficiales yenterradas de forma arbitraria se presenta en detalle, incluyendo las contribuciones del autor alinicio de este proceso.


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2XXVI RNMSeIG, Cancn, Quintana Roo, 14-16 noviembre 2012


SIMPLIFIED METHODS IN SOIL DYNAMICS

RICARDO DOBRYInstitute Professor, Rensselaer Polytechnic Institute,

Troy, New York, U.S.A.

ABSTRACTAfter a brief description of the main characteristics that define Soil Dynamics and its engineeringapplications, 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 animportant role in Soil Dynamics as they do in the rest of Geotechnical Engineering. SimplifiedMethods allow the engineer to conduct calculations by hand or with a minimum computationaleffort, including parametric variations. In the process, the engineer has the possibility to developa feel for the physical meaning and relative importance of the various factors, with more personalcontrol of calculations and decisions including use of engineering judgment as needed. A list ofsimplified procedures proposed by the author is provided, covering systems that range from thefree field and earth dams to shallow and deep foundations, subjected to excitations that includeboth seismic shaking and machine vibrations. Basic understanding of the basic theory andsimplifications behind the simplified procedure can be very helpful to engineers, includingDynamics and Wave Propagation concepts. Some of this understanding is developed in the paper,

with focus on shallow machine foundations and other dynamic soil-structure interactionapplications.

The Lecture concentrates on shallow machine foundations on a half-space subjected to dynamicloads in any of the six degrees of freedom of the foundation, and the Simplified Methods thathave been proposed over the years to characterize the corresponding equivalent soil springs anddashpots. This includes both frequency-dependent and frequency-independent springs anddashpots. It started with the circular surface foundation which was studied over much of the 20 thCentury, until the breakthroughs by Lysmer and others in 1966-1971, and continued with thecases of surface and embedded foundations of arbitrary shape that culminated in the twosummary 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 isdiscussed in some detail, including the contribution of the author in the early part of the process.


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1. INTRODUCTION

It is a great honor for me to be asked to present the Twenty-first Nabor Carrillo Lecture, and tobe associated this way with Dr. Nabor Carrillo and his many accomplishments. It is also anhonor 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 especiallyproud to follow two Carrillo Lecturers who were also my professors and who had anextraordinary influence over my career. One of them is Prof. Enrique Tamez (1992), whodirected my Master Thesis on Sand Liquefaction During Earthquakes at the UNAM in MxicoCity many years ago, and who inspired me to specialize in Soil Dynamics and EarthquakeEngineering. The other is Prof. Robert V. Whitman (2000), who unfortunately died this year, andwho directed my Doctoral Thesis at MIT, also on Soil Dynamics. I wouldnt be here without

them, both of them were great teachers and mentors to me, and this is a good opportunity to sayThanks to both of them.

Finally, let me say that it is just a pleasure to be once again back in Mxico, where I have somany friends and colleagues. One of them is Prof. Eulalio Jurez Badillo, who together with

Prof. Alfonso Rico taught me so well the ABC of soil mechanics during my graduate studies atthe Divisin de Posgrado of UNAM.

The theme of my presentation today is the Simplified Methods in Soil Dynamics. Thisimmediately poses two questions: What is Soil Dynamics, and what kind of Simplified Methodsare we talking about?

In his Fifteenth Carrillo Lecture, Prof. Whitman (2000) defined problems in soil dynamics asthose in which the inertia force of the soil plays a significant role. I would add to this a few othercharacteristics common to most soil dynamics problems: (i) the loads tend to act much faster thanin typical soil mechanics problems; (ii) the loads change direction periodically because they are

associated with vibrations, and therefore produce cyclic rather than monotonic stresses andstrains in the soil; and finally (iii) many of the problems that worry us most in soil dynamics, areassociated with shear strains in the soil which are much smaller than those we are familiar with inregular 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 mostimportant practical applications of Soil Dynamics. It includes the problems of machinefoundations, 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 asto 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 thesoft clay in Mexico City in the 1985 earthquake, which caused a lot of damage to buildings andkilled thousands of people, and which has been studied in detail by a number of Mexicanengineers.

The curves in the figure are acceleration response spectra, and they measure the maximum lateralforce experienced by a building that behaves elastically during the earthquake in number ofaccelerations of gravity, or gs, versus the period of the building in seconds. In 1985 essentiallyall 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 muchgreater on soil than on rock, as much as ten times higher, as can seen in the figure by how muchbigger is the recorded acceleration spectrum on soil at the building of the Secretara deComunicaciones y Transportes (SCT), compared with the same recorded spectrum on rock at theUniversity (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 (Romoand Seed, 1987; Seed, 1987; Dobry, 1991a).


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The way we analyze the earthquake amplification by the soil in a situation like this, is by feedinginto a computer program the motions on the rock, together with a dynamic profile of the soilwhich must include for each layer properties like the density of the soil, the shear wave velocityVs, and the internal damping. Then the computer program will calculate the motion on top of thesoil. This computer program is relatively complex, becoming even more so if you include 2D and3D 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, andthis low shear wave velocity played a significant role in the large site amplification during the1985 earthquake. Shear wave velocity is by far the most important soil property needed for theseearthquake calculations. The shear wave velocities for most soils in the world range from about60 to 800 m/s; a factor of about fifteen. It turns out that to know with some precision the value ofthis parameter for your particular problem is also key to the analysis of most Soil Dynamicsproblems, not only earthquake soil amplification. In fact, shear wave velocity is clearly the singlemost important soil parameter in the whole of Soil Dynamics, as important as soil shear strengthis 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 loadscan be complicated, ranging from sinusoidal forces having one amplitude, direction andfrequency, to very irregular loads and moments, and combinations of vertical, horizontal, rockingand 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 andsoil 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 involvedynamic soil-structure interaction. For example, the dynamic forces and moments acting on thepile group in Fig. 2f could originate from ocean waves pushing periodically against the side of anoffshore oil platform. These dynamic forces and moments may also arise from the inertia forcesdeveloped 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 mathematicalsimilarity, we often use the solutions developed for machine foundations, to analyze also thedynamic soil-structure interaction during earthquakes. To a large extent, the differences betweenthe 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 ratherin 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 basictheory; and (ii) can be used to analyze a geotechnical system either with a calculator or withminimum computational effort, of the type than can be programmed in a spreadsheet.

We constantly use Simplified Methods in Geotechnical Engineering for the analysis and designof static loads. Figure 3 shows three of them, all very familiar to geotechnical engineers. In fact, Iobtained the information for this figure from two standard foundation engineering textbooks.

Figure 3. Examples of Simplified Methods in Soil Mechanics (modified after Liuand Evett, 1998; Das, 1999).


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Let us take a look at these three methods. The ultimate bearing capacity equation in Fig. 3a isbased 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. Thematerial parameters required are the soil cohesion and friction angle, which are obtained fromlaboratory tests, or, in the case of sands, the friction angle may be estimated from fieldpenetration tests. The Schmertmann and Hartman (1978) method in Fig. 3b, which is used tocompute foundation settlement in sand, depends on a triangular stress distribution with depth thatis a simplification of the theoretical profile of stress with depth obtained from the Theory ofElasticity Boussinesq solution. In this settlement calculation the key material parameter is themodulus, Ez, of each sand layer. And, finally, the popular ordinary method of slices with anassumed circular failure surface, proposed by Fellenius (1936), sketched in Fig. 3c, just usesbasic equations of static equilibrium together with some simplifying assumptions, allowingdefinition of the Factor of Safety of the slope when the shear strength of the soil varies along thefailure surface.

Therefore, the three methods start from some basic and very general theory, and they addsimplifications and assumptions along the way until they arrive to a simple mathematical modelthat still contains the main parameters of interest and is broad enough to accommodate the valuesof these parameters for many possible systems. Furthermore, the application of any of theseSimplified Methods requires material parameters like c, or Ezthat are either measured in thelab 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 anddynamic loads: (i) they start with basic theory and they simplify that theory while keeping therelevant factors; and (ii) they still cover a broad range of possible conditions, allowing theengineer to bring into the analysis his/her own loads, foundation or soil geometries, soil profilesand soil properties. These simplified methods have a number of uses, including:

They allow the engineer to conduct calculations, either by hand or using a minimumcomputational effort (hand calculator, spreadsheet).

They allow the engineer to develop a feel for the physical meaning and relativeimportance 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). Thisis a very important function of the simplifiedmethods, 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 therewas 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 theneed to simplify the theory. As a result, some people are tempted to go only that route with theexclusion of more traditional simplified methods, which as noted by Santoyo (2010) is not a goodidea at all.

Table 2 lists a number of simplified solutions and associated publications, proposed with theparticipation of the author over the years, for a variety of soil dynamics systems ranging from the


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3. THE MACHINE FOUNDATION PROBLEM

The rest of this Lecture describes the development of simplified procedures for shallow machinefoundations that took place over most of the 20thCentury. While I played a role on this in the1980s 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 asReissner (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), Gazetasand Roesset (1976, 1979), Dominguez and Roesset (1978), and Roesset (1980). Let me repeatagain that, although we call it for simplicity the machine foundation problem, we are reallysolving here all kinds of soil-structure interaction problems where the loads may be caused notonly 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 usein static Soil Mechanics to calculate the Boussinesq (1885) solution for the stresses under afoundation, or in the Newmark (1942) charts to calculate foundation settlement. As usual, weneed only two elastic parameters to characterize this isotropic homogeneous material, which weselect as being the shear modulus, G, and the Poissons Ratio, . In addition, because of theinertia forces associated with the dynamic loading, we also need the mass density, , which inpractical terms is usually the total unit weight of the dry or saturated soil divided by theacceleration 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 andfrequency f, say in cycles per second. The question to be solved is to calculate the verticaldisplacement of the foundation, w, for given P, f and the rest of the parameters of the problem 1.This problem, that at first sight looks so simple, attracted the attention of top analyticalresearchers during a period spanning 30 years, and was completely solved only in the 1960s

when computers became available (Lysmer and Richart, 1966). The reason why the problem is sodifficult to solve analytically, is that it involves a mixed boundary dynamic condition, with thedisplacement of the ground surface being constant over the area of the foundation, while thevertical 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 toreplace the whole elastic half-space by an equivalent elastic spring, k, selecting the value of k sothat it gives us the right w for a given P. In principle, this value of k will be a function of theproperties of the half-space, G, , , of the radius of the foundation, R, and of the frequency ofthe 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 equivalentvertical spring, k, which is frequency-independent.

The problem with this is that with such a frequency-independent spring, what we have is thesystem of a mass connected to a spring of Fig. 5. This is a well known system in Dynamics calledthe undamped simple oscillator, or undamped single degree-of-freedom system, that for theapplied sinusoidal load has the solution for the displacement, w = (Pm/k) [sin(2ft )]/[1 (f/fn)

2]2, which becomes infinite when the loading frequency, f, becomes equal to the naturalfrequency of the oscillator, f = fn= (1/2)(k/M)

1/2. On the other hand, all indications were thatthere is no value of f for which the displacement w of the foundation in Figs. 4-5 becomes verylarge, 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 alsodissipates energy, hence avoiding infinite values for w.

The researchers added a linear viscous dashpot to the system to take care of the necessary energydissipation, as shown in Fig. 6, which transforms the equivalent system into a damped simpleoscillator.

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 thedashpot c generates a force that is proportional to the velocity of the mass, dw/dt. As a result, theequation of motion of the system that allows solving the problem once the values of k and c havebeen determined is:

(1)

This is, in fact, the correct form of the exact solution for the original problem of a mass on afoundation lying on a half-space shown in Fig. 4, and what is left is to determine how these

1It can be shown that the time history of w is also sinusoidal of the same frequency of the loading, w = wmsin

(2ft-), so the problem is reduced to the determination of the amplitude, wm, and phase angle, , of the

displacement response.


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spring and dashpot depend on the parameters of the problem. Of course, that is the difficult partthat 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 anddashpot, forgetting about the mass for the time being, and replacing the contact area betweenmass 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 frequency-independent Simplified Solution was possible, because in this particular case neither the springnor the dashpot were very sensitive to changes in the frequency f. Figure 7 shows Lysmersproposed approximate expressions for the equivalent vertical spring, kv, and dashpot, cv. The twoexpressions are a marvel of simplicity.

Figure 7. Frequency-independent Simplified Solution for vertical loading, alsolabeled Lysmers Analog (Lysmer and Richart,1966).

The value of the spring is kv 4GR/(1-), which is the same as the static vertical stiffness for arigid 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 moreinteresting. It can be expressed either in terms of the shear modulus, G, or alternatively in termsof the shear wave velocity of the soil, V s. That is, cv [3.4/(1-)] (G)

1/2R2 = [3.4/(1-)] (Vs)R2, taking advantage of the fact that Vsand G are related through the basic elasticity equation:

()

(2)

Figure 8 includes the comparison presented by Lysmer and Richart for the dynamic responsecurves for the cylindrical mass on the half-space of Fig. 4. The solid line is the exact solution andthe dashed line is the simplified solution calculated with the frequency-independent spring anddashpot of Fig. 7.


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Figure 8. Vertical dynamic response of mass on a half-space: comparisonbetween Lysmers 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 dampedsingle oscillator, showing that the system has quite a bit of damping; this is reflected in the factthat the peaks of the curves are all below three. But the most important conclusion from ourviewpoint is that the Simplified Method predicts very well the exact response, so it can be usedby engineers with confidence as a basic tool for these kinds of calculations. And in fact, thissimplified solution and corresponding expressions of vertical spring and dashpot for a circularsurface foundation, are listed today as standard equations in a number of textbooks andfoundation manuals.

Table 3 summarizes the history of the development of the solution. Lamb (1904) had solved theproblem 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 the1930s, Reissner (1936) integrated Lambs solution over a circular area assuming a constantpressure distribution, that is he provided a solution for a perfectly flexible foundation rather thana rigid foundation. After various efforts containing assumptions and approximations by severalauthors in the 1950s and early 1960s; finally Lysmer and Richart (1966) solved the problemnumerically using a computer and provided the beautiful Simplified Solution of Fig. 7, where thehalf-space below the foundation is replaced by a frequency-independent spring and a frequency-independent 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 thenew 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 ofhalf space (Dynamic Boussinesq Problem).

Reissner 1936 Solution for flexible circular foundation assuminguniform load.

Quinlan 1953 Approximate solution for rigid circular foundationassuming static pressure distribution.

Sung 1953 Solutions for various assumed pressure distributions.

Bycroft 1956 Simplified solution by averaging displacements overfoundation area.

Hsieh 1962 Introduced idea of frequency-dependent equivalentspring and dashpot.

Lysmer and Richart 1966 Obtained exact frequency-dependent spring anddashpot for rigid circular foundation using computer.

Proposed approximate frequency-independent springand dashpot as Simplified Solution for engineers(Lysmers Analog).

Richart and Whitman

Whitman and Richart

1967

1967

Validated Lysmers Analog with field footing vibration

tests.

Design procedure based on Lysmers Analog.

3.2 Horizontal vibration

After Lysmer and Richart solved for the vertical loading by combining theory with computercalculations, the rest of the solutions came fast within the next few years for other dynamicexcitations acting on the same surface circular foundation. Figure 9 shows the case of horizontalloading, where again it was possible to obtain frequency-independent expressions for thehorizontal 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 surfaceor very shallow circular foundations, and gave recommendations on how to use them in actualengineering projects. These recommendations included how to produce the necessary values of

soil shear modulus and Poissons Ratio needed to calculate the spring (stiffness) and radiationdashpot for vertical, horizontal, rocking and torsional excitations (Fig. 10).

Finally, they also provided recommendations on how to combine these radiation dashpots withthe internal damping associated with the energy dissipated by the cyclic loading within the soilitself, mostly in friction. Table 4 lists the expressions for the four static stiffnesses, recommendedby Whitman and Richart (1967) as the frequency-independent spring constants for the respectiveSimplified 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 usingthose 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 servedthe profession well. However, they still left open the issue of what to do when the foundation isembedded rather than being at the surface or very close to it, and also how good is the equivalentcircle 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 circularfoundation (Whitman and Richart, 1967).


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Table 4. Static stiffnesses of rigid circular foundation on the surface of an elastichalf-space.

Todays mathematical and computational techniques are much more powerful than thoseavailable in the 1960s, and a number of these cases have been solved by a combination ofanalytical and numerical methods in the last 30-40 years, with many articles, tables and chartspublished in research journals and books. Furthermore, powerful dynamic finite elementscomputer programs than can solve your specific problem for any shape and any embedment, aswell as for arbitrary soil layering, are now commercially available, and they are routinely used invery important or critical structures such as nuclear power plants or large bridges. But in mostprojects, simplified solutions continue to be used. Even in projects like a large bridge, a finiteelement program may be used to analyze the abutments and the foundations of the piers, with thesimplified 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 beaddressing these other cases later in this Lecture, but it turns out that before we can do that, it isnecessary to clarify first the physical origin of the equivalent viscous dashpots shown in Figs. 7and 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 elasticmaterial filling the half-space, which does not have any internal damping and therefore has noway 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 havebeen infinite when vibrating at the natural frequency of the system. But because the system isopen instead of closed, energy escapes in the form of waves propagating in the soil, with thisenergy 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 foundationis called Radiation (or Geometric) Damping, and it is the physical origin of the vertical andhorizontal viscous dashpots I mentioned before, which are just approximate mathematicalrepresentations of the phenomenon.


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Figure 11. Radiation of energy by waves propagating from foundation (Lambeand Whitman, 1969).

Which types of waves are these, and what helpful information can we obtain from wavepropagation theory? Let us take a look.

It is useful to start with the case of horizontal vibrations, which is simpler. Figure 12 presentsagain the Simplified Solution for the surface circular foundation of Fig. 9. The same equationsfor kh and ch are repeated at the bottom of Fig. 12. The viscous dashpot of expression, c h =[4.64/(2-)] Vs R

2, represents the radiation energy carried away from the foundation by thewaves propagating in the soil.

Figure 12 also calculates this expression for two values of the Poissons Ratio, 0.33 and 0.50,which approximately cover the range of interest of this parameter for soils. The dashpot becomes,respectively, 2.78 VsR

2and 3.09 VsR2. These two expressions are numerically very similar,

indicating that the dashpot is not very sensitive to the exact value of the Poissons Ratio of thesoil, Furthermore, and this is very important from a theoretical viewpoint, the numericalcoefficients in the two expressions are within 10% of the value of = 3.14. So, what thesimplified solution fitted to the original exact solution is telling us is that the horizontal viscous

dashpot is approximately the product of Vs(which depends only on the properties of the soil),times the area of the circle, R2 (which depends only on the geometry of the contact areabetween soil and foundation). That is, ch VsA = ( Vs) (R

2). This is very interesting and hassignificant theoretical as well as practical implications.

It is useful at this point to look at some basic results of wave propagation theory relevant to theoriginal system of a plate on a half-space of Fig. 12. This is done with the help of Fig. 13.


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Figure 12. Horizontal radiation dashpot for two Poissons 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 nowon the surface of an infinitely long elastic solid tube of radius R, with the tube in Fig. 13 havingthe same properties of the half-space of Fig. 12. Figure 13 is an example of one-dimensionalelastic wave propagation, in which the horizontal vibration of the plate generates a pure shearwave that propagates vertically down with a wave speed Vs, while inducing horizontaldisplacements 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= VsA = (Vs) (R2). The equivalent

horizontal spring, kh= 0 in this case. It is important to note that this equivalent dashpot, c h= VsA = (Vs ) (R

2), is not an approximation but is an exact mathematical analog to the infinitetube in every respect. The product Vsis so important in wave propagation and Soil Dynamicsthat it has been given a special name: it is called the Shear Impedance of the material. This

Impedance, Vs, completely controls the relation between load and displacement at the interfacebetween the massless rigid plate and the elastic material below for 1D wave propagation in Fig.13. The expression, ch = Vs A, is not restricted to the case when the load Q in Fig. 13 issinusoidal, but it is valid for any time history of Q = Q(t). Also, the expression is still rigorouslyvalid for noncircular shapes of the rigid massless plate and associated cross-section of the elastictube, including square and rectangular, with the expression for the dashpot being always VsA,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 ofhorizontally, as done in Fig. 14, a compression-extension wave (similar to a sound wave) willpropagate 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 wavepropagation.

In Fig. 14, when the massless plate vibrates vertically, as the compression-extension wavespropagate down, the material in the tube alternately compresses and extends in the verticaldirection, generating vertical displacements along the tube. This compression-extension wave inthe 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/)

1/2 , or by the YoungsModulus, E, V = VL= (E/)

1/2. The actual wave speed, V, will be either of these two values (or a


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value in between), and it will depend on how freely can the rod expand or contract laterally. Inone extreme case, if the tube is completely surrounded by a rigid wall and cannot strain laterallyat all (similar to the situation in a soil consolidometer test), the wave velocity will be high, V D=(D/)1/2. On the other hand, if the tube is completely free to expand or contract laterally (similarto a triaxial or unconfined compression test), the wave velocity will be lower, V L= (E/)

1/2. Table5 lists normalized values of VLand VDfor two values of the Poissons Ratio. But whatever thevalue of this wave speed for the compression-extension waves, the whole infinite rod can alwaysbe replaced analytically by a vertical dashpot equal to the Impedance of the material, V, timesthe 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 ofplate on a half-space.

What does this all mean for the original problem of the circular rigid plate on the surface of thehalf-space? As sketched in Fig. 15, the actual foundation problem is typically a 3D situation, andin first approximation the waves under the plate do not travel vertically but go out in manydirections controlled by the maximum angle shown in the figure. This general picture is true forboth 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 veryclose to the vibrating plate, with other waves including shear waves appearing at longer distancesfrom it. But from the viewpoint of this discussion, the rather simplified sketch of Fig. 15a willsuffice, as the dynamic vertical load-displacement relation for the plate depends on the speed ofthis compression-extension excitation generated in the soil very close below the plate, rather thanon 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 canagree that the horizontal vibrations of the plate are mainly shearing the interface with the soil, sothat assuming that the waves sent down into the soil are mainly shear waves makes intuitivesense. We know from theory (Fig. 13) that if the angle was zero in Fig. 15, that is if all theshear waves were going down vertically as 1D waves, the equivalent horizontal dashpot would beexactly ch= VsA. As we saw before when discussing Fig. 12, the correct solution in this case,while not exactly ch= VsA, is numerically close to it, within 10%, and it is also approximatelyindependent of frequency. This suggests that for this case of horizontal vibration of a circularplate in Fig. 15b, it would appear as if the angle 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 aphenomenon known as destructive wave interference, the waves going out at angles greater thanzero tend to cancel each other, leaving only shear waves that travel down more or less verticallyin 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 foundationshapes that are not circular, because if that conclusion was true for any foundation shape, wecould say that the equivalent horizontal dashpot could always be calculated using this expressionVsA, where the area A is just the actual total area of contact between the foundation and thesoil. It turns out that things are not so simple, but still, this gives us a starting point for thedevelopment of simplified solutions for noncircular shapes.

Let us now turn our attention back to the verticalvibration of a circular foundation in Fig. 15a.We apply the same logic, except that in this case the compression-extension wave velocity to putin the equation V A is not obvious, because as we saw in Fig. 14, the correspondingcompression-extension waves may travel relatively slow or faster depending on the lateralstraining of the tube. We solved this by inventing a new wave velocity, that we label VLa, definedby the expression, VLa = 3.4 Vs /((1-). The symbol VLa stands for Lysmers Analog wavevelocity, and it is simply the value of the velocity V = VLa that, when inserted it in theexpression V A, gives the correct cv defined by Lysmer in his Simplified Solution to theoriginal problem, cv= [3.4/(1-)] VsR

2 (see Fig. 7). If we consider the range of possible wavespeeds for compression-extension waves in soils, the value of this new wave velocity VLa isrelatively low, and close to the value of wave velocity controlled by the Youngs Modu lus of thematerial (see Table 5). This makes sense, because it would intuitively seem that the soil under thefoundation, when compressed vertically as in Fig. 15a, is relatively free to expand laterally, so the


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situation in the soil immediately below the foundation is closer to a triaxial than to aconsolidometer test.

I just spent some time going over the details of these derivations in Figs. 15a and 15b. But theeffort is worth it, because the two expressions for cvand chas functions of VLaand Vsin 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 1980s, Prof.George Gazetas and I explored this approach in some detail, and the approach now has becomepart of the accepted State of Practice for the approximate calculation of equivalent foundationdashpots.

3.5 Vertical radiation dashpot for embedded foundation

An obvious first application of this simplified concepts is to the same case of the cylindrical rigidfoundation excited vertically, but now embedded in the half-space (Fig. 16). We assume that thebase radiates energy in the form of compression-extension waves traveling with the LysmersAnalog wave velocity, so the radiation dashpot associated with the base is VLaAb, where Abis

the area of the base, in this case R2. In short, we assume that this dashpot associated with thebase is identical to the one found by Lysmer when the foundation was at the ground surface andnot embedded.

Figure 16. Simplified vertical radiation dashpot for embedded circularfoundation (modified after Gazetas et al., 1985a).

We also assume that the perimeter of the cylinder when vibrating vertically is sending shearwaves into the soil which propagate horizontally with the wave speed V s. This is what ourintuition tells us and it seems reasonable. This gives us a second dashpot associated with thisradiation of energy at the contact between soil and foundation sidewall, which is shown here, of


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value VsAw, where Awis the total area of contact of the sidewall. And because the two dashpotsare in parallel, we can just add up the two values to get the total dashpot for the whole embeddedfoundation. This is done in Fig. 16, providing a simple estimated expression for the equivalentvertical radiation dashpot of the embedded foundation cv VLaAb + VsAw.

There is a need to be careful here, as we have jumped a lot ahead of a more rigorous analysis, andhave made a number of assumptions based only on our intuition. It turns out that this very simpleexpression works well for this case, as I will show you in a minute. But it does not work so wellin other cases, and one should always check these simplified models against more rigoroussolutions before applying them with confidence.

On the other hand, once it is shown that a Simplified Solution like this works, it provides atremendous amount of insight to researchers and engineers. Let me give you two conclusions outof this expression in Fig. 16, so you can appreciate better what I am saying. The first conclusionhas to do with the contribution of the embedment to the total vertical dashpot. While the value ofVLaacting at the base is typically 50% to 100% greater than the V sacting on the sidewalls (see

Table 5), the area of the sidewall itself tends to be much greater than the area of the base. Forexample, if the depth of embedment D = R, which is not a large embedment, the wall area will betwice that of the base, so already the wall is contributing roughly as much as the base, and forgreater 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 dynamicresponse at the critical frequencies than surface foundations, which is good. But the counterpartto this, and this is mysecond conclusion, is that before taking advantage of this beneficial effectof the embedment, you better make sure that you have a good contact between the sidewall andthe soil. If you do not, you may have a foundation with much less damping that you thought youhad, and that may be dangerous. As embedded foundations are often constructed first in a trenchleaving 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 sogood. So, the engineer may want to do a parametric study assuming that the second term of theexpression in Fig. 16 does not exist, it exists, or it is only partially efficient, by multiplying thissecond term of the expression by a factor between 0 and 1. And this is the great advantage ofsuch a Simplified Solution; it allows the engineer to use his/her judgment in the analysis and tokeep control of the situation instead of relying completely on a complicated computer program hemay not understand or control.

Figure 17 shows a comparison for the dashpot of a circular embedded foundation, between thepredictions of this Simplified Solution and rigorous dynamic finite element calculations. Thecurves in the figure, corresponding to the Simplified Solution, show a slight effect of frequencybecause 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 variousdegrees of embedment up to an embedment equal to the diameter of the foundation. Theagreement between Simplified and rigorous solutions is excellent, and confirms that a significantembedment with good contact between sidewalls and surrounding soil may provide a totalradiation dashpot that is several times the value of the dashpot of the surface foundation.


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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 samecircular embedded foundation addressed in Figs. 16-17. As indicated in Fig. 18, the value of kv0is obtained by the multiplication of three factors. The first factor, 4GR/(1-), is just theexpression for the stiffness of the surface foundation without embedment discussed before.Thesecond factor, (1+0.1D/R), is the trench coefficient, which is a small correction, andcorresponds to placing the foundation at the bottom of the trench of depth D, but without anycontact between the sidewall and the soil. And finally, the third factor, [1+0.19(Aw/Ab)

2/3], whichprovides a much bigger correction, is associated with the actual contact area between thefoundation wall and the soil, Aw. Again, if the engineer has doubts about the quality of thiscontact, he/she should conduct a parametric study which includes reducing this third factor to avalue closer to, or equal than one.


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Figure 18. Vertical static spring of embedded circular foundation includingtrench 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 radiationdamping 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 isillustrated by Fig. 19, that shows the results of experiments using three small-scale modelsconducted by Novak (1970), where he excited vertically and horizontally a foundation embeddedin a partially saturated dense loess loam. The effect of embedment is similar for vertical andhorizontal 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 surroundingsoil, 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 avery stiff equivalent spring. In the second experiment, labeled B, the sidewall contact wasweakened by placing a relatively well compacted sandy fill between the foundation and thesurrounding soil; now the response curve is a bit higher and has a small peak at a frequency ofabout 3000 revolutions/minute. And then in the third experiment, labeled C, the modelfoundation was just placed at the bottom of the trench without any contact with the soil at thesides. Clearly in this third experiment there is much less damping in the system, with a very highpeak and large displacements at a frequency of about 2000 revolutions/minute. The figureprovides a dramatic illustration on how different the dynamic response of a foundation can bewithout the benefit provided by the contact with the surrounding soil.


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Figure 19. Experimental verification of soil-wall contact effect on dynamicvertical and horizontal responses of embedded circular foundation (Novak, 1970).

4. EQUIVALENT SPRINGS AND DASHPOTS FOR NONCIRCULARSHAPES

I want to use the rest of this Lecture discussing the development of Simplified Methods fornoncircular foundation shapes that took place in the 1980s, both for surface and embeddedfoundations. Table 6 lists the publications reporting the main results as well as the two summariesby Gazetas (1990, 1991). Originally it grew out of a collaboration at the beginning of the 1980sbetween 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 foundationsidewalls. While I played a role at the beginning of the process and I am a co-author in the firstthree papers listed in Table 6, Prof. Gazetas was the driving force of the whole project, andpursued 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 theJournal of Geotechnical Engineering of the American Society of Civil Engineers. In these twopublications, he provides charts, formulas and numerical examples, ready to use by practicingengineers


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Table 6 gives an idea of how magnificent was this project by Prof. Gazetas. For each of the sixdegrees of freedom, vertical, torsional, horizontal in the two directions, and rocking in the twodirections, he compared possible simplified solutions for springs and dashpots with rigorouscomputer results, modifying the simplified solutions as needed to fit the rigorous results, andarrived to recommendations that engineers could use. He also provided convincing experimental

validation for his calculations. The work was further complicated by the strong coupling betweenhorizontal and rocking in embedded foundations, that he also addresses in his 1990 and 1991publications. You can appreciate the complexity of the work involved in getting the correctsimplified solutions for horizontal, rocking and torsional springs and dashpots of embeddedfoundations, 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 needto use them in one of your projects.

Table 6. Simplified Solutions for surface and embedded foundations of arbitraryshape (Gazetas and co-workers, 1985-91).


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4.1 General problem formulation

The general formulation for both surface and embedded foundations of arbitrary shape ispresented in Fig. 20. The base has an arbitrary shape and an area, Ab, which is embedded at depthD, but with the possibility of the actual depth of contact of the foundation wall with the soil beingsmaller, 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 theactual base area. This rectangle defines the degree of elongation of the actual area by its aspectratio, L/B. For both square and circular foundations the aspect ratio is 1.0.

Figure 20. General formulation for surface and embedded foundations ofarbitrary shape (Gazetas, 1991).

This surface or embedded foundation is located in a half-space which has the same propertiesalready discussed (G, and ), to which it is added now the internal damping ratio of the soil,labeled . It is not necessary to worry about through most of the derivations, with all calculateddashpots being radiation dashpots. Later in this Lecture I will provide the general expression usedto increase the values of these radiation dashpots in order to incorporate the effect of .

The solutions presented by Gazetas (1991) generate springs and dashpots for six degrees offreedom, all shown in Fig. 20: vertical loading; horizontal loading in both the short direction, thatis along the y-axis; horizontal parallel to the long direction x; a rocking moment in the shortdirection, that is around the x-axis; rocking around the y-axis; and finally, a torsional momentaround the z-axis. It is useful to illustrate some of the complexities that must be considered in theformulation of these Simplified Solutions, by considering the calculation of the horizontalradiation dashpot along the long axis x for the embedded foundation of Fig. 20. The foundation ismoving 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 areaAband to the shear wave velocity of the soil Vs. The situation with the foundation walls is morecomplicated. As the foundation moves back and forth in the long direction, the two walls parallelto x in the figure, are also shearing the soil so their contribution should be proportional to V s. But

the two walls perpendicular to x, are pushing back and forth against the soil behind them, so youwould expect that their contribution should not be proportional to Vs but to the other wavevelocity we defined before for compression-extension waves, the Lysmers analog wave velocitythat we labeled VLa. And there are also other walls in Fig. 20 which are neither parallel norperpendicular to x, which further complicate the situation.

Figure 21 presents a more detailed formulation for only the base of the surface or embeddedfoundation, for arbitrary shapes including circular, square, rectangular or in fact any shape. Thegraph 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, Hyand V parallel to the three axes, as well as the three moments aroundeach of the three axes, Mx, Myand Mt. The actual area of the foundation is called A in Fig. 21,

while it is labeled Abin Fig. 20 and other plots. There are other parameters associated with thearea A which are also important for the calculations. They are (see Fig. 21): the area moment ofinertia around the x axis, Iax; the same area moment of inertia around the y axis, Iay; and the polararea moment of inertia around the z axis, J = Iax+ Iay. These three area parameters are the samestudied in school in the Strength of Materials course when looking at sections of beams andcolumns subjected to bending or torsion. It turns out that these three area moments of inertia areneeded in the Simplified Method when computing the springs and dashpots for rocking andtorsional vibrations. And finally, the length and width of the circumscribed rectangle, L, B, andthe 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 arerepresentative of other similar charts and formulas associated with equivalent springs forhorizontal, 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 toestimate the static vertical spring, kz0, through the expression, kz0 = Sz0 (2LG)/(1-). Theparameter along the abscissas is A/4L2, which for the special case of a rectangular shape is equalto the reciprocal of the aspect ratio of the rectangle, (L/B)-1= B/L. That is, is A/4L2= 1.0 for asquare shape, A/4L2= 0.33 for a rectangle of aspect ratio 3, etc. Why did we select this strangelylooking 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 Sz0, asshown in Fig. 22 by the corresponding data points, so we had to invent a new parameter for theplot to work. The data points in Fig. 22 correspond to rigorous elasticity solutions, obtained eithermathematically or numerically, that we retrieved from the literature. This includes the rigoroussolution for the circle previously discussed in this Lecture, the solution for the square, andsolutions for a number of increasingly elongated rectangles, ranging between L/B = 2 and L/B =20. The figure also includesa number of data points for elliptical, triangular and other shapes,and finally we just fitted the equation for Sz0shown on the figure which is the one we proposedfor use in the Simplified Solution for vertical loading.


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Figure 21. Main parameters for surface foundation of arbitrary shape (Dobryand Gazetas, 1986).

Figure 22. Vertical static spring of surface foundation of arbitrary shape (Dobryand Gazetas, 1986).


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Figure 23 presents the variation of the vertical spring kv= kzwith frequency for a Poissons Ratioof 1/3. From now on, I will be showing some plots where springs and dashpots vary withfrequency, as this is an unavoidable part of some of the Simplified Procedures. We were luckybefore, that for the circular surface foundation the vertical and horizontal springs and dashpots donot 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 thislack of sensitivity of the spring constant to changes in frequency for areas which are notelongated. That is, for squares, circles and short rectangles of L/B = 1 and 2, the curve in thegraph is rather flat. This is still true for rectangles of L/B = 4, but it is not true at all for very longrectangles of aspect ratios of 6 or greater, which includes the very important case of stripfootings, for which the dynamic stiffness increases very fast at low frequencies and thendecreases. So in a case like that, the Simplified Solution must consider the effect of the frequencyof the applied loading.

Figure 23. Vertical dynamic spring of surface foundation of arbitrary shape(Dobry and Gazetas, 1986).


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4.3 Vertical and horizontal radiation dashpots

Let me switch now my attention to the radiation dashpots of foundations of arbitrary shape. Forthe time being, we are still talking only about surface foundations, and Fig. 24 is essentially thesame 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 wasconcluded that the vertical dashpot was given approximately by the expression, cv VLaA, thatis the Impedance times the area, where the impedance VLawas controlled by the velocity VLa,the Lysmers Analog wave velocity, reflecting the speed of the compression-extension wavestraveling down vertically below the foundation. And the horizontal dashpot, ch, was also given bythe Impedance times the area of the foundation, but now with the impedance controlled by theshear 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 lowand high frequencies.

Figure 24. Vertical and horizontal radiation dashpots of surface foundations ofarbitrary shape: the high-frequency asymptotes (Dobry and Gazetas,1986).

Unfortunately this is not true anymore for elongated shapes like long rectangles, where the valueof the dashpot changes dramatically with frequency. But what we did find, first following ourintuition and then through a rigorous demonstration by Gazetas (1987), is that these expressionsfor 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 of thewaves 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, asf :

(3) (4)

This useful behavior of the waves generated by the foundation happens because of strongdestructive wave interference at the high frequencies, which cancels all waves traveling at angles, > 0. It turns out that a similar phenomenon is well known in acoustics and is used in theanalysis and design of speakers. This is why in rock concerts, the low frequency sounds areradiated 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 1Dbeam of sound (Morse and Ingard, 1968; Massa, 1972).

The finding summarized by Eqs. 3-4, represented an important breakthrough in our developmentof Simplified Solutions for two reasons. The firstis that it tells us that for vertical and horizontaldashpots of surface foundations, we should be normalizing the actual dashpot obtained fromrigorous solutions, dividing it by either VLaA or by VsA, with the expectation that this ratiowill become 1.0 at high frequencies irrespective of the shape of the foundation. And the secondreason is that additional simple theoretical derivations tell us that for the rocking vibrations, thesame thing should be happening at high frequencies of surface foundations as for verticalvibrations, because during rocking vibrations the foundations is also pushing and pullingvertically the soil below, with the controlling wave velocity still being VLa, but replacing the areaA in the expression, by the area moment of inertia of the foundation around the correspondingaxis (Iax or Iay). That is, for any arbitrary foundation shape, at high frequencies, the rocking

dashpots, crx VLa Iax and cry VLa Iay. The same is true for torsional vibrations, whichsimilarly to the horizontal loading also shears the soil below, where the expression at highfrequencies for ctshould still controlled by the shear wave velocity V s, but with the area in theexpression replaced by the polar moment of inertia of that area, J. In summary, Eqs. 3-4 for thethree translational degrees of freedom are supplemented by Eqs. 5-7 for the three rotationaldegrees of freedom, indicating that, as f :

(5) (6)

(7)These ideas were confirmed by Dobry and Gazetas (1986) for the radiation dashpots, first forvertical and horizontal and then for rocking and torsional. Figure 25 illustrates the results for thehorizontal dashpot in the short direction, versus frequency of loading. Following the conclusionsummarized in Eq. 4, the variable along the ordinate axis in Fig. 25 is the dashpot c ydivided by VsA. The figure confirms the previous conclusion that for non-elongated shapes (circles, squaresand rectangles of L/B = 2), the ratio cy /( Vs A) is about constant and close to one at allfrequencies considered. On the other hand, for long rectangles and strip footings, cy is much


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In all cases, the torsional or rocking radiation dashpot is not constant but increases rapidly withfrequency at the beginning, and then it stabilizes at the theoretical value at high frequencies, withthis theoretical high-frequency value for the torsional case of Fig. 26 being the product VsJ. Avery similar pattern to that of Fig. 26, is exhibited by the rocking dashpots, which also go to zeroat low frequencies and converge at high frequencies to the product VLa Iaxor VLa Iay. This

variation with frequency of rocking and torsional radiations dashpots certainly complicates theformulation of the Simplified Methods, but unfortunately this complication is unavoidable.

Figure 26. Normalized torsional radiation dashpot versus frequency of surfacefoundationof 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 thedifferent effects contributing to the static horizontal stiffness of an embedded foundation: (i) thestiffness of the base of the foundation shearing the soil, which in first approximation is equal tothe stiffness of the corresponding surface foundation; (ii) the trench effect, that is the increase instiffness due to the foundation being placed at the bottom of the trench instead of at the surface ofthe soil; and very importantly (iii) the contribution of the contacts between the embedded


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foundation walls and the surrounding soil. These are the same three factors described before inthis Lecture, when discussing the vertical stiffness of an embedded circular foundation.

Figure 27. Horizontal stiffness of embedded foundation of arbitrary shape: basicsketch (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 severalshapes 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 thetrench and sidewall effects. Please notice that the solution allows for the possibility of thefoundation walls not being in contact with the soil near the top of the excavation, and it certainlyallows the engineer to reduce the contribution of the sidewall contact if he/she does not trust theoverall 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 thedifferent contributions to the total horizontal radiation dashpot of an embedded foundation, ofthe 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 wallsperpendicular to the short direction push and pull against the soil generating compression-extension waves, so the contribution of that wall is proportional to the actual area of contact ofthat wall times VLa. On the other hand, the two walls parallel to the short direction are shearingthe soil, so their contribution should be controlled by the shear wave velocity Vs. Walls in thefigure which are neither parallel nor perpendicular to the direction of motion, generate bothshear(Vs) and compression-extension waves (VLa), as shown on the figure. The situation would seemto be too complicated for a Simplified Method. But Gazetas and Tassoulas (1987b), afterintegrating all these contributions, concluded that from the viewpoint of the horizontal dashpot itwas only necessary to consider the four walls of the circumscribed rectangle rather than the wallsof the actual foundation, which is much simpler.


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Figure 29. Horizontal radiation dashpot of embedded foundation of arbitraryshape: 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 embeddedfoundation, which has a slightly irregular shape, and where the wall reaches an embedment depthof 6 m but has no contact with the soil in the top 2 m. When it comes to computing the totalhorizontal dashpot of this embedded foundation in the short direction y, the procedure ignores theactual foundation walls and replace them by the four walls of the circumscribed rectangle of sides2L x 2B = 16 x 5 m. That is, the total area of contact with the soil of the two long walls of totallength 4L = 32 m, is assumed to generate compression-extension waves over the height of contactof 4 m, with this contribution controlled by VLaand by the total area of contact 32 x 4 = 128 m

2;while the total area of contact of the two short walls of total length 4B = 10 m, is assumed togenerate shear waves, with this contribution controlled by Vs and by the total area of contact 10x 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 addedup 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 7includes the rules just described, on how to generate the horizontal dashpots c y and cx for anembedded foundation. In each case you have three term. For example, the expression for the totalcy = cy, emb includes: (i) a first term labeled Cy, which is the contribution of the base, and iscalculated in another table essentially as the area of the base times Vs, with a slight influence offrequency; (ii) a second term, 4VsBd, which is the contribution of the contact area associatedwith the two sides of the circumscribed rectangle that are shearing the soil; and finally, (iii) athird term, 4VLaLd, 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 theother direction, cx, the two walls that were shearing before are now pushing and V sis replaced byVLa, etc. Table 7 also includes the expression for the vertical radiation dashpot of the sameembedded foundation. The situation for vertical is much simpler. The expression is cz,emb= Cz+ Vs Aw; all sidewalls are shearing the soil, and the total sidewall area A w is the actual area ofcontact around the foundation, with the circumscribed rectangle playing no role in thiscalculation.


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Figure 30. Embedded foundation having an arbitrary shape and partialembedment: numerical example (Gazetas, 1991).


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they allow the engineer to conduct calculations by hand or with a minimumcomputational effort, including parametric variations; and

in the process, the engineer has the possibility to develop a feel for the physical meaningand relative importance of the various factors, with more personal control of calculationsand decisions including use of engineering judgment as needed.

Figure 31. Contribution of soil internaldamping, , to total dashpot (Gazetas,1991).


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REFERENCES

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Ahmad, S. and Gazetas, G. (1992a). Torsional Stiffness of Arbitrarily-Shaped Embedded

Foundations,Journal of Geotechnical Engineering, ASCE,118(8), 1168-1185.Ahmad, S. and Gazetas, G. (1992b). Torsional Damping of Arbitrarily-Shaped Embedded

Foundations,Journal of Geotechnical Engineering, ASCE, 118(8), 1186-1199.Arnold, R. N., Bycroft, G. N. and Warburton, G. B. (1955). Forced Vibrations of a Body on an

Infinite Elastic Solid,Journal of Applied Mechanics,77, 391-401.Barkan, D. D. (1962). Dynamics of Bases and Foundations (translated from Russian). McGraw-

Hill Book Co., New York.Bland, D. R. (1960). The Theory of Linear Viscoelasticity,Pergamon Press, New York, NY.Boussinesq, J. (1885). Application des Potentials a LEtude de LEquilibre et du Mouvement

des Solides Elastiques, Gauthier-Villars, Paris, France.Bycroft, G. N. (1956). Forced Vibration of a Rigid Circular Plate on a Semi-Infinite Elastic

Space and an Elastic Stratum, Philosophical Transactions of the Royal Society. London,Series A 248, 327-386.

Das, B. M. (1999). Principles of Foundation Engineering, 4thEdition,PWS Publishing.Day, S. M. (1977). Finite Element Analysis of Seismic Scattering Problems, Ph.D. Thesis.

University of California, San Diego.Dobry, R. (1991a). Soil Properties and Earthquake Response, Invited Paper, Proceedings of X

European Conference of Soil Mechanics and Foundation Engineering, Florence, Italy,May 26-30, 4, 1171-1187.

Dobry, R. (1991b). Soil Properties and Earhquake Ground Response, Invited Paper,Proceedingsof IX Panamerican Conference on Soil Mechanics and Foundation Engineering,4, 1557-1604. Sociedad Chilena de Geotecnia, Santiago, Chile. 1994.

Dobry, R. (1995). Simple Model to Evaluate Maximum Spectral Amplification of Clay Sites,Proceedings of International Symposium on Civil Engineering After 10 Years of the 1985Earhquake in Mexico City, September 18-19, 91-98.

Dobry, R. and Gazetas, G. (1985). Dynamic Stiffness and Damping of Foundations by SimpleMethods, Vibration Problems in Geotechnical Engineering (G. Gazetas and E. T. Selig,eds.), ASCE, New York, NY, 77-107.

Dobry, R. and Gazetas, G. (1986). Dynamic Response of Arbitrarily Shaped Foundations,Journal of Geotechnical Engineering, ASCE,112(2), 109-135.

Dobry, R. and Gazetas, G. (1988). Simple Method for Dynamic Stiffness and Damping ofFloating Pile Groups, Geotechnique,38(4), 557-574.

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Dobry, R., Gazetas, G. and Stokoe, K. H., II. (1986). Dynamic Response of Arbitrarily ShapedFoundations: Experimental Verifications Journal of Geotechnical Engineering, ASCE,112(2), 136-149.

Dobry, R., Mohamad, R., Dakoulas, P. and Gazetas, G. (1984). Liquefaction Evaluation ofEarth Dams A New Approach, Proceedings of 8thWorld Conference on EarthquakeEngineering, San Francisco, CA, 3, 333-340.


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Dobry, R., Oweis, I. and Urzua, A. (1976). Simplified Procedures for Estimating theFundamental Period of a Soil Profile, Bulletin of a Seismological Society of America,66(4), 1293-1321.

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Dominguez, J. and Roesset, J. M. (1978). Dynamic Stiffness of Rectangular Foundations,Research Report R78-20, Department of Civil Engineering, MIT.

Elorduy, J., Nieto, J. A. and Szekely, E. M. (1967). Dynamic Response of Bases of ArbitraryShape Subjected to Periodic Vertical Loading, Proceedings of International Symposiumon Wave Propagation and Dynamic Properties of Earth Materials. Albuquerque, NM,August, 105-121.

Fellenius, W. (1936). Calculation of the Stability of Earth Dams, Trans. 2ndCongress on LargeDams,Washington, 4, 445.

Fotopoulou, M., Kostanopoulos, P., Gazetas, G. and Tassoulas, J. L. (1989). Rocking Dampingof Arbitrarily Shaped Embedded Foundations, Journal of Geotechnical Engineering,ASCE,115(4), 473-490.

Gazetas. G. (1987). Simple Physical Methods for Foundation Impedances,Dynamic Behavior ofFoundations and Buried Structures,Elsevier Applied Science, London, England, 45-93.

Gazetas, G. (1990). Chapter 15: Foundation Vibrations, Foundation Engineering Handbook, 2ndEdition, (Hsai-Yang Fang, ed.), Chapman & Hall Publishing, New York, NY.

Gazetas, G. (1991). Formulas and Charts for Impedances of Surface and Embedded Foundations,Journal of Geotechnical Engineering, ASCE,117(9), 1363-1381.

Gazetas, G. and Dobry, R. (1984). Horizontal Response of Piles in Layered Soils, Journal ofGeotechnical Engineering, ASCE,110(1), 20-40.

Gazetas, G. and Roesset, J. M. (1976). Forced Vibrations of Strip Footings on Layered Soils,Methods of Structural Analysis, ASCE,1, 115-131.

Gazetas, G. and Roesset, J. M. (1979). Vertical Vibrations of Machine Foundations, Journal ofthe Geotechnical Engineering Division, ASCE,105(GT12), 1435-1454.

Gazetas. G. and Tassoulas, J. L. (1987a). Horizontal Stiffness of Arbitrarily Shaped EmbeddedFoundations,Journal of Geotechnical Engineering, ASCE,113(5), 440-457.

Gazetas. G. and Tassoulas, J. L. (1987b). Horizontal Damping of Arbitrarily Shaped EmbeddedFoundations,Journal of Geotechnical Engineering, ASCE,113(5), 458-475.

Gazetas, G. and Stokoe, K. H., II. (1991). Vibration of Embedded Foundations: Theory VersusExperiment,Journal of Geotechnical Engineering, ASCE,117(9), 1382-1401.

Gazetas. G., Dobry, R. and Tassoulas, J. L. (1985a). Vertical Response of Arbitrarily ShapedEmbedded Foundations,Journal of Geotechnical Engineering, ASCE,111(6), 750-771.

Gazetas, G., Tassoulas, J. L., Dobry, R. and ORourke, M. J. (1985b). Elastic Settlement of

Arbitrarily Shaped Foundations Embedded in Half Space, Geotechnique, XXXV(3),

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1b conf. carrillo by dobry- mass

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