simplified methods in soil dynamics

Simplified methods in Soil Dynamics

Ricardo Dobry n

Rensselaer Polytechnic Institute, Troy, NY, USA

a r t i c l e i n f o

Article history:Received 10 June 2013Accepted 10 February 2014

Keywords:Soil DynamicsSimplified methodsDynamic-soil–structure-interactionRadiation dampingWave propagation

a b s t r a c t

After 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 ofpowerful 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. Simplified Methodsallow the engineer to conduct calculations by hand or with a minimum computational effort, includingparametric variations. In the process, the engineer has the possibility to develop a feel for the physicalmeaning and relative importance of the various factors, with more personal control of calculations anddecisions including use of engineering judgment as needed. A list of simplified procedures proposed bythe author is provided, covering systems that range from the free field and earth dams to shallow anddeep 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 veryhelpful to engineers, including Dynamics and Wave Propagation concepts. Some of this understanding isdeveloped in the paper, with focus on shallow machine foundations and other dynamic soil–structureinteraction applications.

The Lecture concentrates on shallow machine foundations on a half-space subjected to dynamic loadsin any of the six degrees of freedom of the foundation, and the Simplified Methods that have beenproposed over the years to characterize the corresponding equivalent soil springs and dashpots. Thisincludes both frequency-dependent and frequency-independent springs and dashpots. It started with thecircular surface foundation which was studied over much of the 20th Century, until the breakthroughs byLysmer and others in 1966–1971, and continued with the cases of surface and embedded foundationsof 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 embeddedfoundations of arbitrary shape is discussed in some detail, including the contribution of the author in theearly part of the process.

& 2014 Published by Elsevier Ltd.

1. 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 withDr. Nabor Carrillo and his many accomplishments. It is also anhonor to be associated with the people who have been CarrilloLecturers over the years and who have made such giganticcontributions to the geotechnical field. Let me add that I amespecially proud to follow two Carrillo Lecturers who were also myprofessors and who had an extraordinary influence over my career.One of them is Tamez [68], who directed my Master Thesis onSand Liquefaction During Earthquakes at the UNAM in México Citymany years ago, and who inspired me to specialize in SoilDynamics and Earthquake Engineering. The other is Whitman

[71], who unfortunately died this year, and who directed myDoctoral Thesis at MIT, also on Soil Dynamics. I would not be herewithout them, both of them were great teachers and mentors tome, 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 backin México, where I have so many friends and colleagues. Oneof them is Prof. Eulalio Juárez Badillo, who together withProf. Alfonso Rico taught me so well the ABC of soil mechanicsduring my graduate studies at the División de Posgrado of UNAM.

The theme of my presentation today is the Simplified Methodsin Soil Dynamics. This immediately poses two questions: What isSoil Dynamics, and what kind of Simplified Methods are wetalking about?

In his Fifteenth Carrillo Lecture, Whitman [71] defined pro-blems in Soil Dynamics as those in which the inertia force ofthe soil plays a significant role. I would add to this a few othercharacteristics common to most Soil Dynamics problems: (i) theloads tend to act much faster than in typical soil mechanics

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problems; (ii) the loads change direction periodically because theyare associated with vibrations, and therefore produce cyclic ratherthan 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 muchsmaller than those we are familiar with in regular soils testing, like0.1%, 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 SoilDynamics. It includes the problems of machine foundations, earth-quake engineering, pile driving, techniques used to compact sands inthe field, problems of ocean wave loading of offshore structures, etc.

Let me say a couple of things, first about earthquakes and thenabout machine foundations, so as to give a better idea of some ofthe complexities of analyzing Soil Dynamics systems and the needfor Simplified Solutions. Fig. 1 shows the amplification of theearthquake waves by the soft clay in Mexico City in the 1985earthquake, which caused a lot of damage to buildings and killedthousands of people, and which has been studied in detail by anumber of Mexican engineers.

The curves in the figure are acceleration response spectra, andthey measure the maximum lateral force experienced by a buildingthat behaves elastically during the earthquake in number of accel-erations of gravity, or g's, versus the period of the building in seconds.In 1985 essentially all collapsed buildings and fatalities were on soiland not on rock. This happened because the earthquake inertia forceson these assumed elastic buildings due to the shaking, were much

greater on soil than on rock, as much as ten times higher, as can seenin the figure by how much bigger is the recorded accelerationspectrum on soil at the building of the Secretaría de Comunicacionesy Transportes (SCT), compared with the same recorded spectrum onrock at the University (UNAM) [11,63,66].

The way we analyze the earthquake amplification by the soil ina situation like this, is by feeding into a computer program themotions on the rock, together with a dynamic profile of the soilwhich must include for each layer properties like the density ofthe soil, the shear wave velocity Vs, and the internal damping.Then the computer program will calculate the motion on top ofthe soil. This computer program is relatively complex, becomingeven more so if you include 2D and 3D effects due to the presenceof hills nearby, or the effect of inclined or irregular soil layers.

The shear wave velocity of the Mexico City clay is quite low, ofthe order of 70 or 80 m/s, and this low shear wave velocity playeda significant role in the large site amplification during the 1985earthquake. Shear wave velocity is by far the most important soilproperty needed for these earthquake calculations. The shearwave velocities for most soils in the world range from about 60to 800 m/s; a factor of about fifteen. It turns out that to know withsome precision the value of this parameter for your particularproblem is also key to the analysis of most Soil Dynamicsproblems, not only earthquake soil amplification. In fact, shearwave velocity is clearly the single most important soil parameterin the whole of Soil Dynamics, as important as soil shear strengthis for slope stability calculations.

Fig. 2 illustrates another important category of Soil Dynamicsproblems: machine foundations, where a structure on a shallowor deep foundation is excited by dynamic loads above ground,typically due to unbalanced inertia forces caused by operation ofindustrial machinery. The loads can be complicated, ranging fromsinusoidal forces having one amplitude, direction and frequency,to very irregular loads and moments, and combinations of vertical,horizontal, rocking and torsional vibrations. Other parameters thatadd complication to the solution include the type, geometry, mass,degree of embedment, and flexibility of the foundation; andthe soil layering and soil properties of each layer including mostprominently the shear wave velocity.

This machine foundation problem is mathematically verysimilar to other problems that involve dynamic soil–structureinteraction. For example, the dynamic forces and moments acting

Table 1Applications of Soil Dynamics [71].

Applications/Aplicaciones� Machine foundations/Cimentaciones

de Maquinaria� Earthquakes/Temblores� Pile driving/Hincado de pilotes� Dynamic compaction/Compactación

dinamica� Vibratory compaction/Compactación

por vibración� Offshore structures/Estructuras fuera

de costa

� Traffic vibrations/Vibracionesdebidas al tránsito

� Weapons effects/Efecto deproyectiles

� Exploration/Exploración� Blasting/Explosiones� Missile penetration/Penetración de

misiles� Equipment isolation/Aislamiento de

equipos

Fig. 1. Earthquake amplification on the Mexico City soft clay in 1985 [63,66,11].

R. Dobry / Soil Dynamics and Earthquake Engineering 61-62 (2014) 246–268 247

on the pile group in Fig. 2f could originate from ocean wavespushing periodically against the side of an offshore oil platform.These dynamic forces and moments may also arise from theinertia 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 mathe-matical similarity, we often use the solutions developed formachine foundations, to analyze also the dynamic soil–structureinteraction during earthquakes. To a large extent, the differencesbetween 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 ofcyclic strains induced in the soil.

2. The need for simplified methods

Let me address the issue of the Simplified Procedures. For thepurpose of this presentation, I will define a Simplified Procedureas a method that: (i) is derived totally or partially from basictheory; and (ii) can be used to analyze a geotechnical systemeither with a calculator or with minimum computational effort, ofthe type than can be programmed in a spreadsheet.

We constantly use Simplified Methods in Geotechnical Engi-neering for the analysis and design of static loads. Fig. 3 showsthree of them, all very familiar to geotechnical engineers. In fact,I obtained the information for this figure from two standardfoundation engineering textbooks.

Let us take a look at these three methods. The ultimatebearing capacity equation in Fig. 3a is based on an approximateTheory of Plasticity solution developed by Prandtl and Reissner[53,57], that Terzaghi [69] and Meyerhof [49] simplified further,producing the equation at the bottom. The material parametersrequired 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 penetrationtests. The Schmertmann and Hartman [65] method in Fig. 3b,which is used to compute foundation settlement in sand,depends on a triangular stress distribution with depth that is asimplification of the theoretical profile of stress with depthobtained from the Theory of Elasticity Boussinesq solution. Inthis settlement calculation the key material parameter is themodulus, Ez, of each sand layer. And, finally, the popularordinary method of slices with an assumed circular failuresurface, proposed by Fellenius [24], sketched in Fig. 3c, justuses basic equations of static equilibrium together with somesimplifying assumptions, allowing definition of the Factor ofSafety of the slope when the shear strength of the soil variesalong the failure surface.

Therefore, the three methods start from some basic and verygeneral theory, and they add simplifications and assumptionsalong the way until they arrive to a simple mathematical modelthat still contains the main parameters of interest and is broadenough to accommodate the values of these parameters for manypossible systems. Furthermore, the application of any of theseSimplified Methods requires material parameters like c, φ or Ezthat are either measured in the lab or field, or are correlatedempirically to field tests like the CPT or the SPT.

These Simplified Methods have two main characteristics, whichare common to static and dynamic loads: (i) they start with basictheory and they simplify that theory while keeping the relevantfactors; and (ii) they still cover a broad range of possible condi-tions, allowing the engineer to bring into the analysis his/her ownloads, foundation or soil geometries, soil profiles and soil proper-ties. These simplified methods have a number of uses, including:

� They allow the engineer to conduct calculations, either byhand or using a minimum computational effort (hand calcu-lator, spreadsheet).

Fig. 2. Machine foundation vibrations and dynamic soil–structure interaction.

R. Dobry / Soil Dynamics and Earthquake Engineering 61-62 (2014) 246–268248

� They allow the engineer to develop a feel for the physicalmeaning 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 (“realitychecks”). This is a very important function of the simplifiedmethods, as already noted by Ing. Enrique Santoyo in his 20thCarrillo Lecture [64].

It is interesting that until about 30 years ago or so, that beforethe age of powerful accessible computers, there was no need tojustify or defend these simplified methods, as generally there wasnothing else engineers could use. But with the advent of compu-ters, things have changed, and in principle the engineer cananalyze very complicated systems and loadings without the needto simplify the theory. As a result, some people are tempted to goonly that route with the exclusion of more traditional simplifiedmethods, which as noted by Santoyo [64] is not a good idea at all.

Table 2 lists a number of Simplified Solutions and associatedpublications, proposed with the participation of the author overthe years, for a variety of Soil Dynamics systems ranging from thefree field and earth dams to shallow and deep foundations, and forexcitations covering mainly seismic and machine vibrations.

3. The machine foundation problem

The rest of this Lecture describes the development of simplifiedprocedures for shallow machine foundations that took place overmost of the 20th Century. While I played a role on this in the 1980sthrough my collaboration at that time with Prof. George Gazetas, anumber of the key breakthroughs had already taken place by then,through the work of such excellent researchers as Reissner, Reissnerand Sagoci, Arnold et al., Bycroft, Barkan, Lysmer and Richart, Hall,Whitman and Richart, Elorduy et al., Gladwell, Richart et al., Luco andWestmann, Veletsos and Wei, Kausel and Roesset, Johnson et al.,Wong and Luco, Gazetas and Roesset, Dominguez and Roesset, andRoesset [4,5,8,22,23,30,31,37,38,41,42,46,47,55,56,59,60,70,72–74].

Let me repeat again that, although we call it for simplicity themachine foundation problem, we are really solving here all kinds ofsoil–structure interaction problems where the loads may be causednot only by machines but also by earthquakes or ocean waves.

3.1. Vertical vibration of rigid mass

Fig. 4 depicts the original machine foundation problem, whichlooks deceptively simple. Fig. 4a shows the system. It is a perfectlyrigid cylindrical mass M of radius R, located on the surface of anelastic half-space representing the soil, which is the same elastichalf-space we use in static Soil Mechanics to calculate theBoussinesq [7] solution for the stresses under a foundation, or inthe Newmark [51] charts to calculate foundation settlement. Asusual, we need only two elastic parameters to characterize thisisotropic homogeneous material, which we select as being theshear modulus, G, and Poisson's Ratio, μ. In addition, because ofthe inertia forces associated with the dynamic loading, we alsoneed the mass density, ρ, which in practical terms is usually thetotal unit weight of the dry or saturated soil divided by theacceleration of gravity. In the simplest case of vertical vibrationshown in Fig. 4b, the applied vertical load at the top of the massvaries sinusoidally with time, with amplitude Pm and frequency f,say in cycles per second. The question to be solved is to calculatethe vertical displacement of the foundation, w, for given P, f andthe rest of the parameters of the problem.1 This problem, that atfirst sight looks so simple, attracted the attention of top analyticalresearchers during a period spanning 30 years, and was comple-tely solved only in the 1960s when computers became available[47]. The reason why the problem is so difficult to solve analyti-cally, is that it involves a mixed boundary dynamic condition, withthe displacement of the ground surface being constant over the

Fig. 3. Examples of Simplified Methods in Soil Mechanics (modified after Liu and Evett and Das [45,9]).

1 It can be shown that the time history of w is also sinusoidal of the samefrequency of the loading, w¼wm sin (2πft�α), so the problem is reduced to thedetermination of the amplitude, wm, and phase angle, α, of the displacementresponse.


area of the foundation, while the vertical normal stress outside thearea of the foundation is constant and equal to zero.

What would a Simplified Solution look like? As indicated inFig. 5, we may simplistically try to replace 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 the properties of the half-space, G, ρ, μ, of theradius of the foundation, R, and of the frequency of the loading, f.With luck, perhaps we will conclude that k is not very sensitive tothe frequency f. This would be ideal as we want to be able to usethe solution also for loadings which are not sinusoidal. In this casewe would have a Simplified Solution characterized by an equiva-lent 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 massconnected to a spring of Fig. 5. This is a well known system inDynamics called the undamped simple oscillator, or undampedsingle degree-of-freedom system, that for the applied sinusoidalload has the solution for the displacement, w¼(Pm/k) [sin(2πft�α)]/[1�(f/fn)2]2, which becomes infinite when the loadingfrequency, f, becomes equal to the natural frequency of theoscillator, f¼ fn¼(1/2π)(k/M)1/2. On the other hand, all indicationswere that there is no value of f for which the displacement w ofthe foundation in Figs. 4 and 5 become very large, let aloneinfinite. Therefore, a spring is not enough, and some element hasto be added to the equivalent simplified system of Fig. 5 that notonly 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 totake care of the necessary energy dissipation, 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 aforce that is proportional to the displacement, w, of the mass,while the dashpot c generates a force that is proportional to thevelocity of the mass, dw/dt. As a result, the equation of motion ofthe system that allows solving the problem once the values of kand c have been determined is

Md2wdt2

þcdwdt

þkw¼ P ¼ Pm sin ð2πf tÞ ð1Þ

This is, in fact, the correct form of the exact solution for theoriginal problem of a mass on a foundation lying on a half-spaceshown in Fig. 4, and what is left is to determine how these springand dashpot depend on the parameters of the problem. Of course,that is the difficult part that took 30 years to solve.

Table 2Simplified Solutions proposed for various Soil Dynamics problems with participa-tion of the author.

Author(s) Year Problem addressedby Simplified Solution

Roesset et al. [62] 1973 Estimated modal damping of structurewith consideration of dynamic soil–structure interaction

Dobry et al. [20] 1976 Fundamental period of soil profile onrigid rockDobry and Gazetas [14] 1985

Dobry and O’Rourke [17] 1983 Bending moment in pile due to seismickinematic effect

Dobry et al. [19] 1984 Estimation of seismic shear strains inearth dam for evaluation of liquefactionand flow failure

Gazetas and Dobry [29] 1984 Equivalent horizontal spring and dashpotat the top of a pileDobry and Gazetas [14] 1985

Dobry and Gazetas [14] 1985 Springs and dashpots for surfacefoundations of arbitrary shapeDobry and Gazetas [15] 1986

Dobry et al. [18] 1986

Gazetas et al. [35] 1985a Vertical spring and dashpot for embeddedfoundations of arbitrary shapeGazetas et al. [36] 1985b

Dobry and Gazetas [14] 1985

Dobry and Gazetas [16] 1988 Equivalent springs and dashpots offloating pile groups

Dobry [11] 1991a Use of Roesset and Whitman [61]theoretical solution for steady-stateamplification, to provide estimate of peakof Ratio of Response Spectra for soildeposit on flexible rock

Dobry [12] 1991bDobry [13] 1995

Dobry et al. [21] 1995 Decrease with distance to river or lake, oflateral spreading of ground due to sandliquefaction in an earthquake

Fig. 4. Machine foundation problem.


As mentioned before, a number of efforts were made todevelop this equivalent spring and dashpot, forgetting about themass for the time being, and replacing the contact area betweenmass and soil by a massless rigid circular plate welded to thesurface of the half-space (Fig. 7). Finally, in 1966, in his Doctoralthesis at the University of Michigan under the direction of Prof.Richart, Lysmer found the exact solution to the problem with thehelp of this powerful new tool called computers [47]. Then theyproceeded to find that a frequency-independent Simplified Solu-tion was possible, because in this particular case neither the springnor the dashpot were very sensitive to changes in the frequency f.Fig. 7 shows Lysmer's proposed approximate expressions for theequivalent vertical spring, kv, and dashpot, cv. The two expressionsare a marvel of simplicity.

The value of the spring is kvE4GR/(1�μ), which is the same asthe static vertical stiffness for a rigid circular foundation obtainedby integrating the static Boussinesq solution for the half-space.

That is, the selected kv corresponds to f¼0. The expression for thedashpot is even more interesting. It can be expressed either interms of the shear modulus, G, or alternatively in terms of theshear wave velocity of the soil, Vs. That is, cvE[3.4/(1�μ)] (Gρ)1/2

R2¼[3.4/(1�μ)] (ρVs) R2, taking advantage of the fact that Vs and Gare related through the basic elasticity equation:

V s ¼Gp

� �1=2

ð2Þ

Fig. 8 includes the comparison presented by Lysmer and Richartfor the dynamic response curves for the cylindrical mass on the half-space of Fig. 4. The solid line is the exact solution and the dashedline is the Simplified Solution calculated with the frequency-independent spring and dashpot of Fig. 7.

The graph of Fig. 8 plots the normalized amplitude of the massdisplacement, wm, versus the normalized frequency of the loading.

Fig. 5. First too simplistic attempt of a Simplified Solution.

Fig. 6. Second more realistic attempt of a Simplified Solution.


The curves have the typical shape of response of a damped singleoscillator, showing that the system has quite a bit of damping; thisis reflected in the fact that the peaks of the curves are all belowthree. But the most important conclusion from our viewpoint isthat the Simplified Method predicts very well the exact response,so it can be used by engineers with confidence as a basic tool forthese kinds of calculations. And in fact, this Simplified Solutionand corresponding expressions of vertical spring and dashpot for acircular surface foundation, are listed today as standard equationsin a number of textbooks and foundation manuals.

Table 3 summarizes the history of the development of thesolution. Lamb [43] had solved the problem of the concentratedvertical dynamic load at the surface of an elastic half-space, whichis the dynamic counterpart of the Boussinesq [7] solution fora concentrated static load. In the 1930s, Reissner [55] integratedLamb's solution over a circular area assuming a constant pressuredistribution, that is he provided a solution for a perfectly flexiblefoundation rather than a rigid foundation. After various effortscontaining assumptions and approximations by several authors inthe 1950s and early 1960s; finally Lysmer and Richart [47] solvedthe problem numerically using a computer and provided thebeautiful Simplified Solution of Fig. 7, where the half-space belowthe foundation is replaced by a frequency-independent spring

and a frequency-independent dashpot. Table 3 also lists two 1967papers by Richart and Whitman, where they validated the Sim-plified Solution with field tests and developed a design procedure,making the new solution available to the engineering community.

3.2. Horizontal vibration

After Lysmer and Richart solved for the vertical loading bycombining theory with computer calculations, the rest of thesolutions came fast within the next few years for other dynamicexcitations acting on the same surface circular foundation. Fig. 9shows the case of horizontal loading, where again it was possibleto obtain frequency-independent expressions for the horizontalspring and dashpot.

Fig. 7. Frequency-independent Simplified Solution for vertical loading, also labeled “Lysmer's Analog” [47].

Fig. 8. Vertical dynamic response of mass on a half-space: comparison betweenLysmer's Analog and exact solution [47].

Table 3Theoretical and Simplified Solutions to machine foundation problem.

Author(s) Year Contribution

Lamb [43] 1904 Solution for concentrated vertical forceon surface of half-space (DynamicBoussinesq Problem).

Reissner [55] 1936 Solution for flexible circular foundationassuming uniform load.

Quinlan [54] 1953 Approximate solution for rigid circularfoundation assuming static pressuredistribution.

Sung [67] 1953 Solutions for various assumed pressuredistributions.

Bycroft [8] 1956 Simplified Solution by averagingdisplacements over foundation area.

Hsieh [40] 1962 Introduced idea of frequency-dependent equivalent spring anddashpot.

Lysmer andRichart [47]

1966 Obtained exact frequency-dependentspring and dashpot for rigid circularfoundation using computer. Proposedapproximate frequency-independentspring and dashpot as SimplifiedSolution for engineers (Lysmer'sAnalog).

Richart andWhitman [58]

1967 Validated Lysmer's Analog with fieldfooting vibration tests.

Whitman andRichart [72]

1967 Design procedure based onLysmer's Analog.


3.3. Simplified systems for design and equivalent circle

In their 1967 paper, Whitman and Richart summarized all theseSimplified Solutions for surface or very shallow circular foundations,and gave recommendations on how to use them in actual engineer-ing projects. These recommendations included how to produce thenecessary values of soil shear modulus and Poisson's Ratio needed tocalculate the spring (stiffness) and radiation dashpot for vertical,horizontal, rocking and torsional excitations (Fig. 10).

Finally, they also provided recommendations on how to combinethese radiation dashpots with the internal damping associated withthe energy dissipated by the cyclic loading within the soil itself, mostlyin friction. Table 4 lists the expressions for the four static stiffnesses,recommended by Whitman and Richart [72] as the frequency-independent spring constants for the respective Simplified Solutions.2

They also suggested that foundations which do not have a circularshape, like square, rectangular, etc., should be first transformed into anequivalent circle before using those Simplified Solutions.

These Simplified Solutions for the circular surface foundationwere an important breakthrough, and their use for all kindsof foundation shapes through the equivalent circle method hasserved the profession well. However, they still left open the issueof what to do when the foundation is embedded rather than beingat the surface or very close to it, and also how good is theequivalent circle approximation, say, for a very long rectangle ora similar elongated foundation shape.

Today's mathematical and computational techniques are muchmore powerful than those available in the 1960s, and a number ofthese cases have been solved by a combination of analytical andnumerical methods in the last 30–40 years, with many articles, tablesand charts published in research journals and books. Furthermore,powerful dynamic finite elements computer programs than can solveyour specific problem for any shape and any embedment, as well asfor arbitrary soil layering, are now commercially available, and they areroutinely used in very important or critical structures such as nuclearpower plants or large bridges. But in most projects, SimplifiedSolutions continue to be used. Even in projects like a large bridge, afinite element programmay be used to analyze the abutments and the

foundations of the piers, with the simplified methods utilized toanalyze the foundations of the approaches to the bridge.

Therefore, there was clearly a need to extend these SimplifiedSolutions produced by Lysmer, Richart and Whitman, to bothembedded foundations and to noncircular shapes. I will beaddressing these other cases later in this Lecture, but it turns outthat before we can do that, it is necessary to clarify first thephysical origin of the equivalent viscous dashpots shown in Figs. 7and 9 for the vertical and horizontal vibrations. So, let me focusnow on these viscous dashpots.

3.4. Viscous dashpots and radiation damping

The problem can be posed as follows (Fig. 11): the soil isrepresented by a purely elastic material filling the half-space, whichdoes not have any internal damping and therefore has no wayto dissipate energy in the material itself. If the foundationhad been on top of a closed elastic system with rigid boundariessurrounding the soil, the displacement of the foundation would havebeen infinite when vibrating at the natural frequency of the system.But because the system is open instead of closed, energy escapes inthe form of waves propagating in the soil, with this energy nevercoming back, and this is why the displacement of the foundation isnever infinite. This form of elastic energy dissipation in the form ofwaves traveling away from the foundation is called Radiation (orGeometric) Damping, and it is the physical origin of the vertical andhorizontal viscous dashpots I mentioned before, which are justapproximate mathematical representations of the phenomenon.

Which types of waves are these, and what helpful informationcan we obtain from wave propagation theory? Let us take a look.

It is useful to start with the case of horizontal vibrations, which issimpler. Fig. 12 presents again the Simplified Solution for the surfacecircular foundation of Fig. 9. The same equations for kh and ch arerepeated at the bottom of Fig. 12. The viscous dashpot of expression,ch¼[4.64/(2�μ)]ρVsR2, represents the radiation energy carried awayfrom the foundation by the waves propagating in the soil.

Fig. 12 also calculates this expression for two values of Poisson'sRatio, 0.33 and 0.50, which approximately cover the range of interestof this parameter for soils. The dashpot becomes, respectively,2.78ρVsR2 and 3.09ρVsR2. These two expressions are numerically verysimilar, indicating that the dashpot is not very sensitive to the exactvalue of Poisson's Ratio of the soil, Furthermore, and this is veryimportant from a theoretical viewpoint, the numerical coefficients in

Fig. 9. Frequency-independent Simplified Solution for horizontal loading [38,46,70].

2 The equation in Table 4 and Fig. 9 for kho was obtained a few years later byLuco and Westmann and Veletsos and Wei [46,70], and is slightly different from theapproximate expression proposed by Hall [38] and used by Whitman and Richart intheir 1967 paper.


the two expressions are within 10% of the value of π¼3.14. So, whatthe Simplified Solution fitted to the original exact solution is tellingus is that the horizontal viscous dashpot is approximately theproduct of ρVs (which depends only on the properties of the soil),times the area of the circle, πR2 (which depends only on thegeometry of the contact area between soil and foundation). That is,chEρVsA¼(ρVs)(πR2). This is very interesting and has significanttheoretical as well as practical implications.

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

Fig. 13 assumes that we have placed the same massless rigidcircular plate of Fig. 12, but now on the surface of an infinitely long

Fig. 10. Equivalent simplified systems for design based on equivalent circular foundation [72].

Table 4Static stiffnesses of rigid circular foundation on thesurface of an elastic half-space.

Loading Static stiffness

Verticalkv0 ¼

4GR1�μ

Horizontalkh0 ¼

8GR2�μ

Rockingkr0 ¼

4GR3

3ð1�μÞTorsional

kt0 ¼16GR3

3


elastic solid tube of radius R, with the tube in Fig. 13 having the sameproperties of the half-space of Fig. 12. Fig. 13 is an example of one-dimensional elastic wave propagation, in which the horizontalvibration of the plate generates a pure shear wave that propagatesvertically down with a wave speed Vs, while inducing horizontaldisplacements along the tube. It turns out that it is possible toreplace mathematically the tube under the plate by an equivalenthorizontal dashpot, ch¼ρVsA¼(ρVs)(πR2). The equivalent horizontalspring, kh¼0 in this case. It is important to note that this equivalentdashpot, ch¼ρVsA¼(ρVs) (πR2), is not an approximation but is anexact mathematical analog to the infinite tube in every respect. Theproduct ρ Vs is so important in wave propagation and Soil Dynamicsthat it has been given a special name: it is called the Shear Impedanceof the material. This Impedance, ρ Vs, completely controls the relationbetween load and displacement at the interface between themassless rigid plate and the elastic material below for 1D wavepropagation in Fig. 13. The expression, ch¼ρ Vs A, is not restrictedto the case when the load Q in Fig. 13 is sinusoidal, but it is valid forany time history of Q¼Q(t). Also, the expression is still rigorouslyvalid for noncircular shapes of the rigid massless plate andFig. 11. Radiation of energy by waves propagating from foundation [44].

Fig. 12. Horizontal radiation dashpot for two Poisson's ratios.

Fig. 13. Perfect viscous dashpot analog for 1D shear wave propagation.


associated cross-section of the elastic tube, including square andrectangular, with the expression for the dashpot being alwaysρ Vs A, where A is the actual area of the square, rectangle, orother shape.

If the massless plate located on top of the elastic tube wereexcited vertically instead of horizontally, as done in Fig. 14,a compression–extension wave (similar to a sound wave) willpropagate down the tube. This 1D wave propagation model inFig. 14 is relevant to the original problem of vertical excitation ofthe foundation on a half-space of Figs. 4–7.

In Fig. 14, when the massless plate vibrates vertically, as thecompression–extension waves propagate down, the material inthe tube alternately compresses and extends in the verticaldirection, generating vertical displacements along the tube. Thiscompression–extension wave in the tube of Fig. 14 will propagateat a speed greater than the shear wave velocity, with this speed, V,controlled either by the constrained modulus, D, V¼VD¼(D/ρ)1/2,or by the Young's Modulus, E, V¼VL¼(E/ρ)1/2. The actual wavespeed, V, will be either of these two values (or a value in between),and it will depend on how freely can the rod expand or contractlaterally. In one extreme case, if the tube is completely surroundedby a rigid wall and cannot strain laterally at all (similar to thesituation in a soil consolidometer test), the wave velocity will behigh, VD¼(D/ρ)1/2. On the other hand, if the tube is completely freeto expand or contract laterally (similar to a triaxial or unconfinedcompression test), the wave velocity will be lower, VL¼(E/ρ)1/2.

Table 5 lists normalized values of VL and VD for two values ofPoisson's Ratio. But whatever the value of this wave speed for thecompression–extension waves, the whole infinite rod can alwaysbe replaced analytically by a vertical dashpot equal to the Impe-dance 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.

What does this all mean for the original problem of the circularrigid plate on the surface of the half-space? As sketched in Fig. 15,the actual foundation problem is typically a 3D situation, and infirst approximation the waves under the plate do not travelvertically but go out in many directions controlled by the max-imum angle θ shown in the figure. This general picture is true forboth vertical and horizontal excitations. The problem for thevertical vibrations sketched in Fig. 15a is further complicated bythe fact that compression–extension waves predominate only veryclose to the vibrating plate, with other waves including shearwaves appearing at longer distances from it. But from the view-point of this discussion, the rather simplified sketch of Fig. 15a willsuffice, as the dynamic vertical load–displacement relation for theplate depends on the speed of this compression–extension excita-tion 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 can agree that the horizontalvibrations of the plate are mainly shearing the interface with thesoil, so that assuming that the waves sent down into the soil are

Fig. 14. Perfect viscous dashpot analog for 1D compression–extension wave propagation.

Table 5Compression–extension wave velocities relevant to vertical vibration of plate on a half-space.

Poisson' ratio Lysmer's Analogwave velocityVLa¼{[3.4/π(1�μ)]}Vs

Dilatational wave velocity(no lateral straining,VD¼(D/ρ)1/2)

Rod wave velocity(free to strain laterally,VL¼(E/ρ)1/2)

μ VLa/Vs VD/Vs VL/Vs

0.33 1.62 2.0 1.630.50 2.16 1 1.73

Notes: D¼Constrained modulus¼2G(1�μ)/(1�2μ). E¼Young's modulus¼2G(1þμ). G¼Shear modulus. Vs¼Shear wave velocity¼(G/ρ)1/2.


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 shearwaves were going down vertically as 1D waves, the equivalenthorizontal dashpot would be exactly ch¼ρVsA. As we saw beforewhen discussing Fig. 12, the correct solution in this case, while notexactly ch¼ρVsA, is numerically close to it, within 10%, and it is alsoapproximately independent of frequency. This suggests that for thiscase of horizontal vibration of a circular plate in Fig. 15b, it wouldappear as if the angle θ of the waves is actually not far from zero, andthat the problem is surprisingly close to being one-dimensional. Howcan this be? Because of a phenomenon known as destructive waveinterference, the waves going out at angles greater than zero tend tocancel each other, leaving only shear waves that travel down more orless vertically in this particular case.

This is a very important conclusion for the extension of theSimplified Solutions to foundation shapes that are not circular,because if that conclusion was true for any foundation shape, wecould say that the equivalent horizontal dashpot could always becalculated using this expression ρ Vs A, where the area A is just theactual total area of contact between the foundation and the soil.It turns out that things are not so simple, but still, this gives usa starting point for the development of Simplified Solutions fornoncircular shapes.

Let us now turn our attention back to the vertical vibration of acircular foundation in Fig. 15a. We apply the same logic, except thatin this case the compression–extension wave velocity to put in theequation ρ V A is not obvious, because as we saw in Fig. 14, thecorresponding compression–extension waves may travel relativelyslow or faster depending on the lateral straining of the tube.We solved this by inventing a new wave velocity, that we label VLa,defined by the expression, VLa¼3.4Vs/(π(1�μ)). The symbol VLastands for “Lysmer's Analog wave velocity”, and it is simply thevalue of the velocity V¼VLa that, when inserted it in the expressionρVA, gives the correct cv defined by Lysmer in his Simplified Solutionto the original problem, cv¼[3.4/(1�μ)]ρVsR2 (see Fig. 7). If weconsider the range of possible wave speeds for compression–exten-sion waves in soils, the value of this new wave velocity VLa isrelatively low, and close to the value of wave velocity controlled bythe 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 expandlaterally, so the situation in the soil immediately below the founda-tion is closer to a triaxial than to a consolidometer test.

I just spent some time going over the details of these deriva-tions in Fig. 15a and b. But the effort is worth it, because the twoexpressions 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 SimplifiedSolutions, not only for foundations of noncircular shapes, butalso for embedded foundations. In the early 1980s, Prof. GeorgeGazetas and I explored this approach in some detail, and theapproach now has become part of the accepted State of Practicefor 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 thesame case of the cylindrical rigid foundation excited vertically, butnow embedded in the half-space (Fig. 16). We assume that the baseradiates energy in the form of compression–extension waves travel-ing with the Lysmer's Analog wave velocity, so the radiation dashpotassociated with the base is ρVLaAb, where Ab is the area of the base, inthis case πR2. In short, we assume that this dashpot associated withthe base is identical to the one found by Lysmer when the foundationwas at the ground surface and not embedded.

We also assume that the perimeter of the cylinder whenvibrating vertically is sending shear waves into the soil whichpropagate horizontally with the wave speed Vs. This is what ourintuition tells us and it seems reasonable. This gives us a seconddashpot associated with this radiation of energy at the contactbetween soil and foundation sidewall, which is shown here, ofvalue ρVsAw, where Aw is the total area of contact of the sidewall.And because the two dashpots are in parallel, we can just add upthe two values to get the total dashpot for the whole embeddedfoundation. This is done in Fig. 16, providing a simple estimatedexpression for the equivalent vertical radiation dashpot of theembedded foundation cvEρVLaAbþρVsAw.

There is a need to be careful here, as we have jumped a lotahead of a more rigorous analysis, and have made a number ofassumptions based only on our intuition. It turns out that this verysimple expression works well for this case, as I will show you in aminute. But it does not work so well in other cases, and one shouldalways check these simplified models against more rigoroussolutions before applying them with confidence.

On the other hand, once it is shown that a Simplified Solutionlike this works, it provides a tremendous amount of insight toresearchers and engineers. Let me give you two conclusions out ofthis expression in Fig. 16, so you can appreciate better what I amsaying. The first conclusion has to do with the contribution of theembedment 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 itselftends to be much greater than the area of the base. For example, if

Fig. 15. Waves and radiation damping in vertical and horizontal vibrations.


the depth of embedment D¼R, which is not a large embedment,the wall area will be twice that of the base, so already the wall iscontributing roughly as much as the base, and for greater embed-ment it contributes significantly more to the total dashpot thanthe base. Therefore, we should expect that embedded foundationswill have a lot of damping, having less dynamic response at thecritical frequencies than surface foundations, which is good. Butthe counterpart to this, and this is my second conclusion, is thatbefore taking advantage of this beneficial effect of the embedment,you better make sure that you have a good contact between thesidewall and the soil. If you do not, you may have a foundationwith much less damping that you thought you had, and that maybe dangerous. As embedded foundations are often constructedfirst in a trench leaving a gap at the sides, and the gap is filled laterwith a sandy fill that is sometimes difficult to compact because ofthe lack of space, there is always the possibility that the contact isnot so good. So, the engineer may want to do a parametric studyassuming that the second term of the expression in Fig. 16 doesnot exist, it exists, or it is only partially efficient, by multiplyingthis second term of the expression by a factor between 0 and 1.And this is the great advantage of such a Simplified Solution; itallows the engineer to use his/her judgment in the analysis and tokeep control of the situation instead of relying completely on acomplicated computer program he may not understand or control.

Fig. 17 shows a comparison for the dashpot of a circularembedded foundation, between the predictions of this SimplifiedSolution and rigorous dynamic finite element calculations. Thecurves in the figure, corresponding to the Simplified Solution, showa slight effect of frequency because the exact dashpot was used forthe contribution of the base, instead of the simplified frequency-independent dashpot of Fig. 16. The comparisons in Fig. 17 corre-spond to various degrees of embedment up to an embedment equalto the diameter of the foundation. The agreement between Simpli-fied and rigorous solutions is excellent, and confirms that a sig-nificant embedment with good contact between sidewalls andsurrounding soil may provide a total radiation dashpot that is severaltimes the value of the dashpot of the surface foundation.

3.6. Vertical static spring for embedded foundation

Fig. 18 includes the corresponding expression for the staticvertical spring, kv0, of the same circular embedded foundationaddressed in Figs. 16 and 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 surfacefoundation without embedment discussed before. The secondfactor, (1þ0.1D/R), is the “trench coefficient”, which is a smallcorrection, and corresponds to placing the foundation at thebottom of the trench of depth D, but without any contact betweenthe sidewall and the soil. And finally, the third factor, [1þ0.19(Aw/Ab)2/3], which provides a much bigger correction, is associatedwith the actual contact area between the foundation wall and thesoil, Aw. Again, if the engineer has doubts about the quality of thiscontact, he/she should conduct a parametric study which includesreducing this third factor to a value closer to, or equal than one.

3.7. Embedment and dynamic response

It is important to see how this very significant influence of theembedment on the radiation damping translates into a muchreduced dynamic response when subjected to dynamic loads. Bothanalysis and experiments have consistently verified the importanceof the effect. This is illustrated by Fig. 19, that shows the results ofexperiments using three small-scale models conducted by [52],where he excited vertically and horizontally a foundation embeddedin a partially saturated dense loess loam. The effect of embedmentis similar for vertical and horizontal excitations. Let us focus on thevertical excitation, shown in the upper plot of Fig. 19. The firstexperiment for the fully embedded foundation with good contactwith the surrounding soil, labeled “A,” produced a response curve ofdisplacement versus frequency which is very flat, with low values ofthe dynamic displacement. The curve indicates a highly dampedsystem with a very stiff equivalent spring. In the second experiment,labeled “B,” the sidewall contact was weakened by placing arelatively well compacted sandy fill between the foundation andthe surrounding soil; now the response curve is a bit higher and hasa small peak at a frequency of about 3000 revolutions/min. And thenin the third experiment, labeled “C,” the model foundation was justplaced at the bottom of the trench without any contact with the soilat the sides. Clearly in this third experiment there is much lessdamping in the system, with a very high peak and large displace-ments at a frequency of about 2000 revolutions/min. The figureprovides a dramatic illustration on how different the dynamicresponse of a foundation can be without the benefit provided bythe contact with the surrounding soil.

Fig. 16. Simplified vertical radiation dashpot for embedded circular foundation (modified after [35]).


4. Equivalent Springs and Dashpots for noncircular Shapes

I want to use the rest of this Lecture discussing the develop-ment of Simplified Methods for noncircular foundation shapesthat took place in the 1980s, both for surface and embeddedfoundations. Table 6 lists the publications reporting the mainresults as well as the two summaries by [27,28]. Originally it grew

out of a collaboration at the beginning of the 1980s between Prof.George Gazetas and myself, built on some of the ideas I discussedbefore, especially for the radiation damping generated by the baseof the foundation and the foundation sidewalls. While I played arole at the beginning of the process and I am a co-author in thefirst three papers listed in Table 6, Prof. Gazetas was the drivingforce of the whole project, and pursued it systematically througha series of studies and publications over a number of years, untilhe was able to put it all together in two publications listed at thebottom of Table 6. Gazetas [27] is a chapter in a FoundationEngineering Handbook, and Gazetas [28] is an article in the Journalof Geotechnical Engineering of the American Society of CivilEngineers. In these two publications, he provides charts, formulasand numerical examples, ready to use by practicing engineers

Table 6 gives an idea of how magnificent was this project byProf. Gazetas. For each of the six degrees of freedom, vertical,torsional, horizontal in the two directions, and rocking in the twodirections, he compared possible Simplified Solutions for springsand dashpots with rigorous computer results, modifying theSimplified Solutions as needed to fit the rigorous results, andarrived to recommendations that engineers could use. He alsoprovided convincing experimental validation for his calculations.The work was further complicated by the strong coupling betweenhorizontal and rocking in embedded foundations, that he alsoaddresses in his 1990 and 1991 publications. You can appreciatethe complexity of the work involved in getting the correctSimplified Solutions for horizontal, rocking and torsional springsand dashpots of embedded foundations, by noticing that heneeded a total of six papers to present all necessary results.

My main purpose today is to provide you with an introductionto the basic approach used by Prof. Gazetas, as an introduction to

Fig. 17. Vertical radiation dashpot of embedded circular foundation: comparisonbetween simplified (curves) and dynamic finite element results (data points)[10,35].

Fig. 18. Vertical static spring of embedded circular foundation including trench andsoil–wall contact factors [35].

Fig. 19. Experimental verification of soil–wall contact effect on dynamic verticaland horizontal responses of embedded circular foundation [52].


his 1990 and 1991 publications, which may be useful if you need touse them in one of your projects.

4.1. General problem formulation

The general formulation for both surface and embeddedfoundations of arbitrary shape is presented in Fig. 20. The basehas an arbitrary shape and an area, Ab, which is embedded atdepth D, but with the possibility of the actual depth of contact ofthe foundation wall with the soil being smaller, doD, and with theactual total contact area between the sides of the foundation andthe soil being Aw. An important tool of these procedures is arectangle 2L�2B that circumscribes the actual base area. Thisrectangle defines the degree of elongation of the actual area by itsaspect ratio, L/B. For both square and circular foundations theaspect ratio is 1.0.

This surface or embedded foundation is located in a half-spacewhich has the same properties already discussed (G, μ and ρ), towhich it is added now the internal damping ratio of the soil,labeled β. It is not necessary to worry about β through most of thederivations, with all calculated dashpots being radiation dashpots.Later in this Lecture I will provide the general expression usedto increase the values of these radiation dashpots in order toincorporate the effect of β.

The solutions presented by Gazetas [28] generate springs anddashpots for six degrees of freedom, all shown in Fig. 20: verticalloading; horizontal loading in the short direction, that is alongthe y-axis; horizontal parallel to the long direction x; a rockingmoment in the short direction, that is around the x-axis; rockingaround the y-axis; and finally, a torsional moment around the z-axis.It is useful to illustrate some of the complexities that must beconsidered in the formulation of these Simplified Solutions, by

Table 6Simplified Solutions for surface and embedded foundations of arbitrary shape (Gazetas and co-workers [1–3,15,18,25,27,28,32,33,34,39]).

Authors Year Contribution Stiffness Damping Experiments

Dobry and Gazetas [15] 1986 Surface foundations, all six DOF's X XDobry et al. [18] 1986 Experimental verification for circular,

square and rectangular shapes, all six DOF'sX

Gazetas et al. [35] 1985a Embedded foundations, vertical X XGazetas and Tassoulas [32] 1987a Embedded foundations, horizontal XGazetas and Tassoulas [33] 1987b XHatzikonstantinou, Tassoulas, Gazetas,Kotsanopoulos and Fotopoulou [39]

1989 Embedded foundations, rocking X

Fotopoulou, Kotsanopoulos,Gazetas and Tassoulas [25]

1989X

Ahmad and Gazetas [1] 1991 Embedded foundations, torsional X XAhmad and Gazetas [2] 1992a XAhmad and Gazetas [3] 1992b XGazetas [27] 1990 Surface and embedded foundations,

all six DOF's; formulas, charts & numerical examplesX X

Gazetas [28] 1991 X XGazetas and Stokoe [34] 1991 Experimental verification for circular,

square and rectangular shapes, vertical, horizontal, and rockingX

Fig. 20. General formulation for surface and embedded foundations of arbitrary shape [28].


considering the calculation of the horizontal radiation dashpot alongthe long axis x for the embedded foundation of Fig. 20. Thefoundation is moving back and forth horizontally in the x-direction,and the question is: what is happening in terms of the wavesgenerated by the different contact areas? The base is clearly shearingthe soil, so you would expect its contribution to the total radiationdashpot to be proportional to the area Ab and to the shear wavevelocity of the soil Vs. The situationwith the foundationwalls is morecomplicated. As the foundation moves back and forth in the longdirection, the two walls parallel to x in the figure, are also shearingthe soil so their contribution should be proportional to Vs. But thetwo walls perpendicular to x, are pushing back and forth against thesoil behind them, so you would expect that their contribution shouldnot be proportional to Vs but to the other wave velocity we definedbefore for compression–extension waves, the Lysmer's analog wavevelocity that we labeled VLa. And there are also other walls in Fig. 20which are neither parallel nor perpendicular to x, which furthercomplicate the situation.

Fig. 21 presents a more detailed formulation for only the base ofthe surface or embedded foundation, for arbitrary shapes includingcircular, square, rectangular or in fact any shape. The graph showsagain the three axes, x, y and z, all passing by the centroid O of the areaof 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

andMt. The actual area of the foundation is called A in Fig. 21, while itis labeled Ab in Fig. 20 and other plots. There are other parametersassociated with the area A which are also important for the calcula-tions. They are (see Fig. 21): the area moment of inertia around the xaxis, Iax; the same area moment of inertia around the y axis, Iay; andthe polar area moment of inertia around the z axis, J¼ Iaxþ Iay. Thesethree area parameters are the same studied in school in the Strengthof Materials course when looking at sections of beams and columnssubjected to bending or torsion. It turns out that these three areamoments of inertia are needed in the Simplified Method whencomputing the springs and dashpots for rocking and torsional vibra-tions. 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 surfacefoundation of arbitrary shape, reproduced from Dobry and Gazetas

[15]. The charts for the vertical spring in Figs. 22 and 23 arerepresentative of other similar charts and formulas associatedwith equivalent springs for horizontal, rocking and torsionalvibrations presented by [15,28]. The chart in Fig. 22 allowscalculating a dimensionless parameter Sz0, which is used toestimate the static vertical spring, kz0, through the expression,kz0¼Sz0 (2LG)/(1�μ). The parameter along the abscissas is A/4L2,which for the special case of a rectangular shape is equal to thereciprocal of the aspect ratio of the rectangle, (L/B)�1¼B/L. That is,it is A/4L2¼1.0 for a square shape, A/4L2¼0.33 for a rectangle ofaspect ratio 3, etc. Why did we select this strangely looking newparameter instead of simply using B/L? Because it turns out thatthe square and circular shapes, while they have the same aspectratio of 1.0, have different values of Sz0, as shown in Fig. 22 by thecorresponding data points, so we had to invent a new parameterfor the plot to work. The data points in Fig. 22 correspond torigorous elasticity solutions, obtained either mathematically ornumerically, that we retrieved from the literature. This includesthe rigorous solution for the circle previously discussed in thisLecture, the solution for the square, and solutions for a number ofincreasingly elongated rectangles, ranging between L/B¼2 andL/B¼20. The figure also includes a number of data points forelliptical, triangular and other shapes, and finally we just fitted theFig. 21. Main parameters for surface foundation of arbitrary shape [15].

Fig. 22. Vertical static spring of surface foundation of arbitrary shape [15].

Fig. 23. Vertical dynamic spring of surface foundation of arbitrary shape [15].


equation for Sz0 shown on the figure which is the one we proposedfor use in the Simplified Solution for vertical loading.

Fig. 23 presents the variation of the vertical spring kv¼kz withfrequency for a Poisson's Ratio of 1/3. From now on, I will beshowing some plots where springs and dashpots vary withfrequency, as this is an unavoidable part of some of the SimplifiedProcedures. We were lucky before, that for the circular surfacefoundation the vertical and horizontal springs and dashpots do notchange much with frequency; this allowed Lysmer and otherauthors to propose approximate springs and dashpots which areindependent of frequency (Figs. 7 and 9). Fig. 23 confirms this lackof sensitivity of the spring constant to changes in frequency forareas which are not elongated. That is, for squares, circles andshort rectangles of L/B¼1 and 2, the curve in the graph is ratherflat. This is still true for rectangles of L/B¼4, but it is not true at allfor very long rectangles of aspect ratios of 6 or greater, whichincludes the very important case of strip footings, for which thedynamic stiffness increases very fast at low frequencies and thendecreases. So in a case like that, the Simplified Solution mustconsider the effect of the frequency of the applied loading.

4.3. Vertical and horizontal radiation dashpots

Let me switch now my attention to the radiation dashpots offoundations of arbitrary shape. For the time being, we are stilltalking only about surface foundations, and Fig. 24 is essentiallythe same sketch of Fig. 15, of what happens to a surface foundationsubjected to vertical and horizontal vibrations. When discussingFig. 15 before for the case of a circular foundation, it was concludedthat the vertical dashpot was given approximately by the expres-sion, cvEρVLaA, that is the Impedance times the area, where theimpedance ρVLa was controlled by the velocity VLa, the Lysmer'sAnalog 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 Impedancetimes the area of the foundation, but now with the impedancecontrolled by the shear wave velocity of the soil, Vs. Theseexpressions for cv and ch independent of frequency, worked wellfor the circular shape, for which the value of the dashpot is aboutthe same at low and high frequencies.

Unfortunately this is not true anymore for elongated shapeslike long rectangles, where the value of the dashpot changes

dramatically with frequency. But what we did find, first followingour intuition and then through a rigorous demonstration by [26],is that these expressions for the vertical and horizontal dashpotsreproduced at the bottom of Fig. 24 are always true for anyfoundation shape at high frequencies. That is, as the frequencyincreases, the angle θ of the waves in Fig. 24 becomes zero, thewaves propagate vertically down as one-dimensional waves, andthese two expressions become exact whatever the shape of thefoundation. In other words, as f-1:

cv ¼ cz-ρVLaA ð3Þ

ch ¼ cx ¼ cy-ρV sA ð4ÞThis useful behavior of the waves generated by the foundation

happens because of strong destructive wave interference at thehigh frequencies, which cancels all waves traveling at angles, θ40.It turns out that a similar phenomenon is well known in acousticsand is used in the analysis and design of speakers. This is whyin rock concerts, the low frequency sounds are radiated out byspeakers that cover a wide range of directions, while for the highfrequencies, directional speakers are needed that radiate energyonly in one direction more or less as a 1D beam of sound [48,50].

The finding summarized by Eqs. (3) and (4), represented animportant breakthrough in our development of Simplified Solu-tions for two reasons. The first is that it tells us that for vertical andhorizontal dashpots of surface foundations, we should be normal-izing the actual dashpot obtained from rigorous solutions, dividingit by either ρVLaA or by ρVsA, with the expectation that this ratiowill become 1.0 at high frequencies irrespective of the shape of thefoundation. And the second reason is that additional simpletheoretical derivations tell us that for the rocking vibrations,the same thing should be happening at high frequencies of surfacefoundations as for vertical vibrations, because during rockingvibrations the foundations is also pushing and pulling verticallythe soil below, with the controlling wave velocity still being VLa,but replacing the area A in the expression, by the area moment ofinertia of the foundation around the corresponding axis (Iax or Iay).That is, for any arbitrary foundation shape, at high frequencies, therocking dashpots, crxEρVLaIax and cryEρVLaIay. The same is true fortorsional vibrations, which similarly to the horizontal loading alsoshears the soil below, where the expression at high frequencies forct should still be controlled by the shear wave velocity Vs, but withthe area in the expression replaced by the polar moment of inertia

Fig. 24. Vertical and horizontal radiation dashpots of surface foundations of arbitrary shape: the high-frequency asymptotes [15].


of that area, J. In summary, Eqs. (3) and (4) for the three trans-lational degrees of freedom are supplemented by Eqs. (5) to (7) forthe three rotational degrees of freedom, indicating that, asf-1:

crx-ρVLa Iax ð5Þ

cry-ρVLa Iay ð6Þ

ct-ρV s J ¼ ρV s ðIaxþ IayÞ ð7ÞThese ideas were confirmed by Dobry and Gazetas [15] for the

radiation dashpots, first for vertical and horizontal and then forrocking and torsional. Fig. 25 illustrates the results for thehorizontal dashpot in the short direction, versus frequency ofloading. Following the conclusion summarized in Eq. (4), thevariable along the ordinate axis in Fig. 25 is the dashpot cy dividedby ρVsA. The figure confirms the previous conclusion that for non-elongated shapes (circles, squares and rectangles of L/B¼2), theratio cy/(ρVsA) is about constant and close to one at all frequenciesconsidered. On the other hand, for long rectangles and stripfootings, cy is much greater than ρVsA at low frequencies, withthe ratio between the two converging to one at high frequencies,exactly as predicted. So this is the key plot for the radiationdashpot for horizontal loading in the short direction. The situationfor horizontal loading in the long direction of the foundation issimilar, and the plot for the vertical dashpot looks just like Fig. 25,except that VLa is used instead of Vs on the ordinate axis.

4.4. Torsional and rocking radiation dashpots

Fig. 26 includes the corresponding plot for torsional vibrations,where the torsional dashpot ct has been normalized to the productρVsJ, as suggested by Eq. (7). Here you see a different phenomenon,which is typical of the rotational vibrations of surface foundationsincluding not only torsional but also rocking oscillations. Thisphenomenon is that the equivalent radiation dashpots for theserotational vibrations invariably go to zero as the frequencydecreases and goes to zero, because of destructive wave inter-ference which does not allow any energy to leave the neighbor-hood of the foundation when the frequency approaches zero. Thisis true for circular shapes, as shown by the corresponding curve

for L/B¼1, and it is also true for long rectangles. In all cases, thetorsional or rocking radiation dashpot is not constant but increasesrapidly with frequency at the beginning, and then it stabilizesat the theoretical value at high frequencies, with this theoreticalhigh-frequency value for the torsional case of Fig. 26 being theproduct ρVsJ. A very similar pattern to that of Fig. 26, is exhibitedby the rocking dashpots, which also go to zero at low frequenciesand converge at high frequencies to the product ρVLaIax or ρVLaIay.This variation with frequency of rocking and torsional radiationsdashpots certainly complicates the formulation of the SimplifiedMethods, but unfortunately this complication is unavoidable.

4.5. Simplified systems including embedment

Let us move on to embedded foundations of arbitrary shape.Fig. 27 shows a sketch of the different effects contributing to thestatic horizontal stiffness of an embedded foundation: (i) thestiffness of the base of the foundation shearing the soil, which inthe first approximation is equal to the stiffness of the correspond-ing surface foundation; (ii) the trench effect, that is the increase instiffness due to the foundation being placed at the bottom of thetrench instead of at the surface of the soil; and very importantly(iii) the contribution of the contacts between the embeddedfoundation walls and the surrounding soil. These are the samethree factors described before in this Lecture, when discussing thevertical stiffness of an embedded circular foundation.

Fig. 28 includes the expression for the horizontal static springin the short direction, ky0, developed by Gazetas and Tassoulas [32]on the basis of rigorous calculations for several shapes and degreesof embedment. The expression assumes that the surface staticstiffness, ky0,sur, has already been calculated, with the expressiongiving the factors greater than 1.0 that reflect the trench andsidewall effects. Please notice that the solution allows for thepossibility of the foundation walls not being in contact withthe soil near the top of the excavation, and it certainly allowsthe engineer to reduce the contribution of the sidewall contact ifhe/she does not trust the overall quality of the contact betweenthe wall and the soil.

And finally, Fig. 29 presents the basic sketch used by Gazetasand Tassoulas [33] to study the different contributions to the totalhorizontal radiation dashpot of an embedded foundation, of the

Fig. 25. Normalized horizontal radiation dashpot versus frequency of surfacefoundation of arbitrary shape [15].

Fig. 26. Normalized torsional radiation dashpot versus frequency of surfacefoundation of arbitrary shape [15].


various contact areas and types of waves. The area of the basealways generates shear waves. For horizontal vibrations along theshort direction as shown in the figure, the two walls perpendicularto the short direction push and pull against the soil generatingcompression–extension waves, so the contribution of that wall isproportional to the actual area of contact of that wall times ρ VLa.On the other hand, the two walls parallel to the short direction areshearing the soil, so their contribution should be controlled by the

shear wave velocity Vs. Walls in the figure which are neitherparallel nor perpendicular to the direction of motion, generateboth shear (Vs) and compression–extension waves (VLa), as shownon the figure. The situation would seem to be too complicated for aSimplified Method. But [33], after integrating all these contribu-tions, concluded that from the viewpoint of the horizontal dashpotit was only necessary to consider the four walls of the circum-scribed rectangle rather than the walls of the actual foundation,which is much simpler.

Fig. 30 illustrates what I mean. It presents a numerical exampletaken from the summary paper by Gazetas [28], where hecalculates all six sets of springs and dashpots for this embeddedfoundation, which has a slightly irregular shape, and where thewall reaches an embedment depth of 6 m but has no contact withthe soil in the top 2 m. When it comes to computing the totalhorizontal dashpot of this embedded foundation in the shortdirection y, the procedure ignores the actual foundation wallsand replace them by the four walls of the circumscribed rectangleof sides 2L�2B¼16�5 m2. That is, the total area of contact withthe soil of the two long walls of total length 4L¼32 m, is assumedto generate compression–extension waves over the height ofcontact of 4 m, with this contribution controlled by VLa and bythe total area of contact 32�4¼128 m2; while the total area ofcontact of the two short walls of total length 4B¼10 m, is assumedto generate shear waves, with this contribution controlled by Vs

and by the total area of contact 10�4¼40 m2. Then the threecontributions of base area and shear waves, area associated with4L and compression–extension waves, and area associated with 4Band shear waves, are just added up to obtain the total radiationdashpot.

Fig. 31 includes a partial view of the summary table forembedded foundations in Gazetas [28] that provides clear instruc-tions on how to compute different things. The last column ofFig. 31 includes the rules just described, on how to generate the

Fig. 27. Horizontal stiffness of embedded foundation of arbitrary shape: basic sketch [32].

Fig. 28. Horizontal stiffness of embedded foundation of arbitrary shape: equationand correlation for the sidewall contact factor [32].


horizontal dashpots cy and cx for an embedded foundation. In eachcase you have three term. For example, the expression for the totalcy¼cy, emb includes: (i) a first term labeled Cy, which is thecontribution of the base, and is calculated in another tableessentially as the area of the base times ρ Vs, with a slight influenceof frequency; (ii) a second term, 4ρVsBd, which is the contributionof the contact area associated with the two sides of the circum-scribed rectangle that are shearing the soil; and finally, (iii) a third

term, 4ρVLaLd, which is the contribution of the contact areaassociated with the other two sides of the rectangle, which arepushing back and forth against the soil. For the dashpot in theother direction, cx, the two walls that were shearing before arenow pushing and Vs is replaced by VLa, etc. Fig. 31 also includes theexpression for the vertical radiation dashpot of the sameembedded foundation. The situation for vertical is much simpler.The expression is cz,emb¼CzþρVsAw; all sidewalls are shearing the

Fig. 29. Horizontal radiation dashpot of embedded foundation of arbitrary shape: basic sketch [33].

Fig. 30. Embedded foundation having an arbitrary shape and partial embedment: numerical example [28].


soil, and the total sidewall area Aw is the actual area of contactaround the foundation, with the circumscribed rectangle playingno role in this calculation.

5. Effect of internal soil damping

To complete the picture, it is important to add to the viscousdashpots calculated with these Simplified Methods, the contribution

due to the energy dissipated internally in the soil, mainly due tofriction, which is typically characterized by a soil damping ratio, β(Fig. 32). The value of β depends on several factors, like the level ofcyclic shear strain induced in the soil by the dynamic loading, thetype of soil, and the Plasticity Index if the soil is a clay. Depending onthe 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 (20–30%). Fortunately from the viewpointof the Simplified Methods covered in this Lecture, once the radiationdashpot, cradiation, has been calculated at a certain frequency f, the

Fig. 31. Partial view of summary table for embedded foundations of arbitrary shape [28].

Fig. 32. Contribution of soil internal damping, β, to total dashpot [28].


total dashpot, ctotal, including the effect of β can be obtained using

ctotal � cradiationþkπf

� �β ð8Þ

where k is the corresponding stiffness calculated at the samefrequency for the elastic half-space. This simple expression for β isobtained from the Correspondence Principle of the Theory ofViscoelasticity [6], and it is valid for any of the six degrees of freedom(vertical, horizontal, rocking and torsional), as well as for surface andembedded foundations of any shape.

6. Final comments

Simplified Methods will continue to play an important role inSoil Dynamics as they do in the rest of Geotechnical Engineering.While powerful computer simulations can produce more exactand detailed information, Simplified Methods are irreplaceableas a basis for codes and regulations, and as a tool to verify thecomputer results (“reality checks”). Furthermore, they are justbetter suited to many applications, where

� they allow the engineer to conduct calculations by hand orwith a minimum computational effort, including parametricvariations; and

� in the process, the engineer has the possibility to develop a feelfor the physical meaning and relative importance of the variousfactors, with more personal control of calculations and deci-sions including use of engineering judgment as needed.

It is useful for the engineer applying any of these SimplifiedMethods, to have at least a basic understanding of the theory andsimplifications behind the procedure, which in Soil Dynamicsincludes some Dynamics and Wave Propagation concepts. ThisCarrillo Lecture was aimed at providing some of this understand-ing, with focus on shallow machine foundations and otherdynamic soil–structure interaction applications.

Many researchers have proposed useful Simplified Methodsin Soil Dynamics, some of which are referenced in the paper.Methods suggested by the author are listed in Table 2, and theycover systems ranging from the free field and earth dams toshallow and deep foundations, subjected to excitations thatinclude both seismic shaking and machine vibrations.

The main focus of this Carrillo Lecture was on shallow machinefoundations on a half-space subjected to dynamic loads in any ofthe six degrees of freedom of the foundation, and the SimplifiedMethods that have been proposed over the years to characterizethe corresponding equivalent soil springs and dashpots. Thisincluded both frequency-dependent and frequency-independentsprings and dashpots. It started with the circular surface founda-tion which was studied over much of the 20th Century, until thebreakthroughs by Lysmer and others in 1966–1971, and continuedwith the cases of surface and embedded foundations of arbitraryshape that culminated in the two summary publications byGazetas in 1990 and 1991. These solutions for machine founda-tions are also useful for analysis of dynamic soil–structure inter-action during earthquakes.

The development of these simplified equivalent springs anddashpots for both surface and embedded foundations of arbitraryshape was discussed in some detail, including the contribution ofthe author in the early part of the process. This discussion servesto introduce some of the basic dynamic theoretical conceptsbehind the methods, and hopefully also as an introduction to theiruse in actual engineering projects.

Acknowledgments

I am most grateful to George Gazetas for our many exhilaratingdiscussions in the early 1980s, about basic concepts of dynamicsand wave propagation that could be used to develop SimplifiedProcedures for shallow and pile foundations subjected to dynamicloads. I am also grateful to several colleagues with whom I hadthe pleasure to develop and validate Simplified Methods on thevarious problems listed in Table 2: José M. Roesset, RobertV. Whitman, Issa Oweis, Alfredo Urzua, George Gazetas, MichaelJ. O'Rourke, Ramli Mohamad, Panos Dakoulas, Kenneth. H. StokoeII, John L. Tassoulas, Victor Taboada and Lee Liu. Finally, I amextremely grateful to the professors that taught me Soil Mechanicsand Soil Dynamics and mentored my initial research efforts:Arturo Arias at the U. of Chile, Eulalio Juárez Badillo, Alfonso Ricoand Enrique Tamez at the UNAM, and Robert V. Whitman, JoséM. Roesset and John T. Christian at MIT.

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