'Rept of Test on Seismic Vibration Testing of One 14' Dresser … · )Test Report No. No. of Pages...

) Test Report No. No. of Pages

Bepept of Test o'n

SEISMIC VIBRATION TESTING OF ONE14" DRESSER STYLE 38 COUPLINGFOR DRESSER MANUFACTURING DIV.OF DRESSER INDUSTRIES, INC.UNDER PURCHASE ORDER NO. 37443

I KNVlRONMENTAL

ACTQMTESTlNQ

CO RPORATl ON

November 15, 1978

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Preca"ed~

Checked ApproveaM.L.To1f

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lt 'll'i t' il i "g f 014" Style 38 Coupling.

Dresser Mfg. Div. Dresser Industries, Inc.Bradford, PA. 16701

14" OD Style 38 CouplingDresser Orawing No. 66668

~ ~

Dresser Purchase Order No. 37443

One {1)

Uncl assified

November 3, 1978

P.ltcDermott

Returned to Dresser.

Refer to the results section herein.

14333Page

Rev 1

ACTQN

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1 . 0 TEST REQUIREMENTS

It was required to perform a seismic vibration test on theDresser Mfg. 14" Style 38 coupling in the manner specified inDresser Purchase Order No. 37443. The coupling was to be vibratedwith 225 psi water pressure within the pipe section 1" DA

9 10.0 Hz for 30 seconds. The coupling was to be examined forleaks before, during and after the seismic simulation.

2.0 TEST PROCEDURES

The test item was setup in the test fixture shown .in Dressere

Drawing 66388 Revision A. The test item assembly was welded tothe floor, in the reinforced floor area of the vibration testfacility. The AETC 35 KIP actuator was attached to a bulkhead inline with the simulated pipe run. Dresser personnel present atthe time of the test torqued the coupling bol ts to the recom-mended torque after lubricating the gaskets.

The pipe section was filled with water and pressurized to atleast 225 psig. A checkout run was performed 9 1.0 Hz 1" DA.

During this run, welds cracked on the piston backstop. The

cracks were rewelded and an 8" I-Beam was welded between thebackstop and the test fixture as a stiffener. Another 1.0 Hz 1" DA

checkout run was performed and everything checked out. A testrun was performed (Test 1) 9 10 Hz 1" DA for 30 seconds. Three(3) runs, 5 sec. each, were performed 9 10 Hz 1" DA with no

pressure for photographic purposes. The coupling was replacedwith another one and the cycling test was repeated (Test 2)910 Hz 1" DA for 30 seconds pressurized.

The output of the displacement transducer was displayed on an

Report No.Rev 1

ACTUSPage 2

oscillographic recorder. The oscillographic recordings made

at the time of test are included with this test report.

3.0 TEST RESULTS

There was no evidence of damage or deterioration to the couplinggaskets as a result of the test as specified in para. 2.0 above

with the exception of slight chaffing of the gaskets due torelative motion between the aaskets and the piping. The pipingsection was pressurized to 230 psia during Test 1 and to 240 psigduring the Test 2. Internal pressure decayed to 195 psig afterTest 1 and to 190 psig after Test Z. The internal pressure decay

may be accounted for by leakage through the regulator. Therewas no visible evidence of water leakage from the couplina duringor after the test specified in para.. 2.0.

14333R»N

TESTINGACTDN

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Advantages of Dresser Couplings

DRESSER SEISMICALLYQUALIFIED COUPLINGS provide a lower cost fiexible piping systemcompared to rigid welded systems. On site welding, heat treating, radiograph, or liquid penetranttests are avoided.

DRESSER COUPLINGS can accommodate up to 6 of deflection and up to 3/8" of axial movement.

DRESSER COUPLINGS are available in sizes up to 405" in diameter. Couplings can be provided forup to 2,000 psi line pressure, and temperature to 1500 F.

DRESSER COUPLINGS can be coated to provide corrosion/abrasion protection.

DRESSER COUPLINGS have a load force of 1,000 pounds per inch of pipe diameter. This is onlya fraction of the load force applied to pipe anchors on welded pipe systems.

DRESSER COUPLINGS are less expensive than welding. Shlled labor is not required for installation.Pipe alignment is not as critical and the installation (pipe joining time) is considerably reduced.

DRESSER COUPLINGS have been seismically tested to 5.1g acceleration for 30 seconds;

DRESSER COUPLINGS provide pipe dampening to reduce resonance coming from pumps.

DRESSER COUPLINGS,when used in conjunction with valves and pumps, facilitate easy installationalignment and, thereafter, provide quick access to the inside of pumps and valves.

DRESSER COUPLINGS are a fast and economical uniform method of joining most all types of pipe.Pipe materials include steel, cast iron, asbestos, cement, plastics, and various metal alloys andcompositions..

DRESSER COUPLINGS are available for transitions from one pipe material to another. Alsoreductions with transitions, as flange adapters and special configurations on request. Both insulatingand non-insulating couplings can be provided in various lengths, materials, and thickness of middlerings. The Dresser 440 joint harness can be used for pull-out protection. Standard gasket materialsare Buna S, Buna N, EPT, EPDM, Butyl, Viton, and asbestos.

DRESSER COUPLINGS have demonstrated 100 years of reliability on countless piping installationsthroughout the world.

ASME SECTION IIICOUPLINGS

SCOPE

This speciQcation outlines the material performance and quality assurance requirementsfor a mechanical type steel coupling for joining underground steel piping used onnuclear service water piping. The coupling shall be of the wedg~keted Qared sleeve

type with diametexs to Qt the pipe. Each coupling shaH consist of one steel middle ring,two steel followers, two wedge-shaped zubbexwompounded gaskets, and steel bolts.

. A. Middle Ring, Followers, Bolts, and Nuts

The middle zing, followers, bolts, and nuts of the coupling shall be produced ofmaterials manufactured in accordance with ASME Section IIIand NCA 3800.

B. Gaskets

Gaskets shall be of a zubbezwompounded matexial that willnot deteriorate from ageor exposure to water under normal use conditions and which have been proven in "

'eneraluse in couplings for at least 10 years. The compound shall meet the follow-ing minimum physical zecoxixements to provide long-time sealing pexfoxmance.

1. Physical Requirements

C lore ~ ~ ~ ~ s ~ ~ ~ ~ ~ o ~ ~ ~ ~ ~ ~ e o ~ o o BlackColoDurometer Hardness.......75+ 5 Shore A*Compression Set....4% Max. - 30 Minutes

3% Max. - 3 Houzs

*Compression set to be detexmined upon a ~/i" diameter x 'A" thick disc cutfrom blocks subjected to 600 PSI pressure for 48 hours at zoom temperature.AH other test procedures shall be as speciQed in ASTM D395 Method A usingthe external loading device.

IILPERFORMANCE

A. The mechanical coupling shall be capable of absorbing movements normallyencountered under buxied piping conditions. The coupling shall be capable ofabsorbing axial piping movements, deQection movements, and vibrations noxmaHyencountered in buried piping under seismic conditions. The recommended aHow-.ances shall be determined by the coupling manufacturer.

B. The design of the method of providing a seal on the mecluuncal coupHng shall becapable of passing the foHowing tests that simulate extreme seismic conditions thatoccur in buried piping.

0

1. Assemble the coupling on a steel test pipe in accordance with the manufacturer'spublished instaHation instructions. HydrostaticaHy pressurize the test specimento a magnum of 1'A times the system's design pzesstzze.

Move the pipe axially from the neutral position a minimum of 'A" inward travelback to the neutral position and 'A" in.the outward position and back to aozmalposition at 10Hz per second for a minimum of 30 seconds. Observe sealing per-formance during the complete test cycle. The structural poztions of the couplingshaH not be affected by the above test and no leakage shaH be observed before,duziag, or after the test is performed.

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IV. QUALMYASSURANCE

'The procedure for quality.assurance shaH meet the coupHng manufacturer's qualitycontrol manual as amended for nuclear ap plicatioas surveyed.and approved by anappHcable certificate holder or the requirements of ASMF. Sections, NCA 3800.

V. REPORTS AND'TESTING

'The vendor shall provide the consulting engineer with asimuiated,seismic test'eportfrom an independent laboratozy. The test report shall show that a representativecoupling has been tested under pressure to a minimum of 5 g's acceleration for 30sec onds or as speuQed hezem. --"-' --'- — - - - " 4

'V'he

vendor shaH provide the consultiag engineer with a usezs list demonstrating aot lessthaa two previous installations for service intended for ASM'ection 1II, Class 1, 2, & 3fabrication. Letters of previous audit shall also be requized.

The vendor shall submit one copy of his quality control program for review prior toshipment. MillRun test certificatioas shall also be required; Prior to shipment, eachcoupling shall be hydzostaticaQy tested to 1'h times the working presazze.

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DRESSER COUPLING USERS LIST

Cleveland Electric IlluminatingPerry Nuclear Units 1 & 2

South Carolina Electric & GasVirgilSummer Nuclear Station

Government of YugoslaviaKrysko Nuclear Station

Gilbert/Commonwealth Consulting EngineersReading, Pennsylvania.

Pullman Kellogg Nuclear Power PipingWilliamsport, Pennsylvania

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DeLaval Turbine and Compressor DivisionNuclear Power Plant ProductsOakland, California

Westinghouse PPSDPittsburgh, Pennsylvania

TUBCO Nuclear Power PipingBrooklyn, Ne~York

AttachmentPG&E Letter DCL-97-191

ATTACHIVIENT3-1

Letter from Robert Pyke, Consulting Engineer, to PG&E, dated May 28, 1996,

Subject: Dynamic Soil Properties for Analysis ofASW Piping System Bypass,Diablo Canyon Nuclear Power Plant

AttachmentPG8cE Letter DCL-97-xm

ATTACHIVlENT3-1

Letter from Robert Pyke, Consulting Engineer, to PG&E, dated May 28, 1996,Subject: Dynamic Soil Properties for Analysis ofASW Piping System Bypass,

Diablo Canyon Nuclear Power Plant

Robert Pyke, Consulting Engineer230266

May 28, 1996

Eric M. S. FujisakiPacific Gas Ec Electric Company245 Market Street, Room 823B,P.O. Box 770000,San Francisco CA 94177

Dynamic SoB Properties forAnalysis of ASW Piping System BypassDiablo Canyon Nuclear Power Plant

Dear Eric,

In association with= W.- Andrew Herlache of Harding'awson Associates (HLA), San

Francisco, I have reviewed previous HLA reports related to this subject, attended the meeting heldat your offices on December 18, 1995, and reviewed the additional site investigation and laboratorytesting work that has been completed by HLA.

HLA's recent investigation indicates that the backfill adjacent to and above the circul:itingwater intake conduits, in which the ASW bypass willbe installed, consists mainly of the followingthree material types: (1) a mixture of varying quantities of gravel, sand and clay that was obtainedpredominantly from on-site sources; (2) poorly-graded gravel with clay and sand that appears to beimported material; and (3) poorly-graded sand that appears to be imported bedding material. Theclays are relatively low in plasticity, with measured values of the plasticity index (Pl) in the orderof 20.

These findings confirm my preliminary recommendations made at the December 18 meeting:

(1) That the assumption made in the 1978 HLA analyses that the low strain shear moduluswas strongly a function of confining pressure and decreased to zero at zero confining pressure is

questionable, and that the shear wave velocities measured at that time, which show values decreasingbelow 800 feet per second at shallower depths, are also questionable for the material typesencountered in the recent HLA investigation. Based on the recent field observations andmeasurements in similar fillmaterials at other sites, I believe that it would be more appropriate toassume a constant shear wave velocity in the order of 800 feet per second in the current analyses.In the absence of new site specific measurements, I would suggest that the uncertainty in the lowstrain modulus values might be addressed by checking the effect ofvarying the assumed shear wavevelocity profiles with values ranging from a low of 650 feet per second to a high of 1200 feet persecond.

1076 Carol Lane, Suite 136, Lafayette, CA 94549Telephone 510/283-6765 FAX 510/283-7614

RECEIVEO

MAY 30 j996CLAIM

SUF%0$ tT SERVICCSIIMS

~30266

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(2) That, while it was consistent with practice at that time, the shear modulus reduction curveused in the 1978 HLA analyses was too "soft" for a fillcontaining clays of low plasticity. On thebasis of extensive work that is reported in detail in the Electric Power Research Institute report"Guidelines for Determining Design Basis Ground Motions" (EPRI TR-102293,. Project 3302,November 1993), I recommend that shear modulus reduction and damping curves corresponding toa reference strain of 0.1 percent, as shown on the attached figures, be used in the current analyses.As shown on the attached figures, the curves corresponding to a reference strain of 0.1 percent aregenerally similar to the upper bound curve for modulus reduction and the lower bound curve fordamping of sands from the standard work by Seed and Idriss. In recent years this combination ofthe Seed and Idriss curves has been widely accepted as a better estimate of the modulus reduction and

damping curves for low plasticity natural soils than the original Seed and Idriss curves for clayswhich were given more weight by HLA in their previous studies. While I believe that it is unlikelythat a modulus reduction curve any "softer" than the cilrve for'a reference strain'of 0:1 percent wouldbe applicable in this case, it is possible that slightly stiffer curves may be applicable in the moreclayey material and I therefore suggest that the current analysis might be checked for their sensitivityto a shift in the stiffer direction by using the modulus reduction and damping curves for a referencestrain of 0.2 percent.

Sincerely,

I.Robert Pyke Ph.D, E.

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THDCP - Version 3.03Copyright, 1992, Robert Pyke / TAGACompiled by rmp on 09/12/92 using theMicrosoft FORTRAN Optimizing Compiler v.5 ~ 1

Reference strain = 0.1%

Gmav = 1.000000E+07Taumax = 10000.000000Gmax/Taumax = 999.999900Reference strain = 1.000000E-03

FOR 5 CYCIES AT 1.00 HERTZ

YcjIC SHEAR STRAINPERCENT

.0001

.0002

.0003

.0005

.0010

.0020

.0030

.0050

.0100

.0200

.0300

.0500

.1000

.2000

.3000

.5000

.70001.0000

G/Gmax

1.00001.00001.00001.0000

.9965

.9840

.9697

.9409

.8751

.7723

.6962

.5892

.4380

.2989

.2300

.1592-1223.0911

DAMPING RATIOPERCENT

.950

.965

.9841.0221.1231.3221.5171.8852.7064.0495.1566.954

10.07113.75716 '6819.00520.87822.750

eference strain = 0 ~ 2+0

Gmax = 1.000000E+07Taumax = 20000.000000Gmax/Taumax = 500.000000Reference strain = 2.000000E-03

FOR 5 CYCLES AT 1.00 HERTZ

CYCLIC SHEAR STRAINPERCENT

F 0001.0002.0003.0005.0010.0020.0030.0050.0100.0200.0300.0500.aooo.2000.3000.5000.7000

a.oooo

G/Gmax

1.00001.00001.00001.00001.0000

.9965

.9906

.9768

.9407

.8750

.8194

.7315

.5891

.4379

.3538

.2596

.2066

.1591

DAMPING RATIOPERCENT

.939

.944

.951~ 969

1 ~ 017,1.1171.217a.4151.8802.7003.4104.6196.948

10.06512.16915.01516,95219.000

AttachmentPGAE Letter DCL-97-191

ATTACHMENT5A-1

Gazetas, Gary, "Analysis ofMachine Foundation Vibrations: State of the Art,"published in Soil D namics and Earth uake En ineerin, Vol. 2,No. 1,

January 1983, pp 2 to 42; and associated 1984 errata sheet

Ol nam) CS

a uaen Ineefln

VOLUME 2, NUMBER 1,JANUARY 1983, ISSN 0261 7277

(

The Journal aims to encourage and enhance the role of mechanics andother disciplines as they relate to earthquake engineering by providingopportunities for the publication of the work of applied mathematicians,engineers and other applied scientists involved in solving problemsclosely related to the field of earthquake and geotechnical engineering.Emphasis is placed on new concepts and techniques, but case historieswill also be published if they enhance the presentation and understand.ing of new technical conce pts.

Fields covered:~ Seismology and geology relevant to earthquake problems~ Elastodynamics: wave propagation and scattering soil and rock

dynamics~ Oynamic constitutive behaviour of materials~ Mathematical methods:.system, methodology and identification in ~

soil dynamics~ Practical methods relevant to earthquake phenomena~ Probabilistic methods in

SeismologyGeotechnical earthquake engineeringRisk analysisEarthquake engineering reliabilityInteraction problems: Soil structure interaction

F luidwond interactionInstrumentation and experimental methodsInelastic and nonlinear problemsFinite element analysis in dynamics and elastodynamicsSeismic design criteriaEarthquake case historiesTsunamis

Soil Dynamics and Earrhqueke Engineering is published quarterly(January, April,July, October) byC.M.L. PublicationsAs hurst LodgeAshurstSoutharnoton SO4 2AATel: (042129) 3223

Annual Subscription: UK 552.00, elsewhere E58.00 ($ 125)Orders to C.M.L. Publications at the above address

Springer.Ver(ag New York Inc. )s the distributor for Soil Dynamics andEarthquake Engineering in North America (including Canada andMexico). Subscriptions are entered with prepayment only and shouldbe sent to Springer-Verlag New York Inc., 44 Hertz Way, Secaucus,NJ 07094, USA. Tel: (201) 348<033. Annual subscription rate $ 131.00including postage and handling.

All rights reserved

No part of this publication may be reproduced, stored in a retrievalsystem or be transmitted, in any form by any means electronic,mechanical, photocopying, recording or otherwise without the writtenpermission of the Publisher.

Permission to photocopy for internal or personal use of specificclients is granted by CML Publications for libaries and otherregistered users registered with the Copyright Clearance Centre(CCC), provided that the stated fee per copy is paid directly to theCCC, 21 Congress Street, Salem, MA01970. Special requests shouldbe addressed to CML Publications, Ashurst Lodge, Ashurst, South-ampton SO4 2AA, England. Tel: (042129) 3223.

O C.M.L. Publications 1983Typset by Unicus Graphics Ltd, Horsham. Printed by Hobbs thePrinters of Southarnpton, England

Contents

Analysis of machine foundations: state of the artGeorge Gazetas

Vibration of hammer foundationsM. Novak and L. El Hifnawy

Foundations for auto shreddersF. E. Richart, Jr and R. D. Woods

New books

Calendar

UNIVERSITY OF CALIFORN!.-.Earthquake Engineering Research Cep

JUL 8 1983

LIBRARY

VifARMNGTHIS MAT:RIALMAYB

PROTECTEO BY COPYR! -HT LA'vV

(Tit)e 17 U.B. Code)

II54

60

60

~"

Analysis of machine foundation vibrations: state ofthe art

GEORGE GAZETAS

Rensselaer Polytechnic Insritute, Troy, New York, USA

The paper reviews the state-of-the.art of analysing the dynamic response of foundations subjectedto machine-type loadings. Following a brief outline of the historical developments in the field, theconcepts associated with the definition, physical interpretation and use of the dynamic impedancefunctions of foundations are elucidated and the available analytical/numerical methods for theirevaluation are discussed. Groups of crucial dimensionless problem parameters related to the soilprofile and the foundation geometry are identified and their effects on the response are studied.Results are presented in the form of simple formulae and dimensionless graphs for.both the staticand dynamic parts of impedances, pertaining to surface and embedded foundations having circular,strip, rectangular or arbitrary plan shape and supported by three types of idealized soil profiles: thehalfspace, the stratum-over-bedrock and the layer-over-halfspace. Consideration is given to the effectsof inhomogeneity, anisotropy and non.linearity of soil. The various results are synthesized in a case

study referring to the response of two rigid massive foundations, and practical recommendations are

made on how to inexpensively predict the response of foundations supported by actual soil deposits.

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INTRODUCTION

The basic goal in the design of a machine foundation is to' " ' limit its motion to amplitudes which'will neither endanger

the satisfactory operation of the machine nor will theydisturb the people working in the immediate vicinity. Thus,a key ingredient to a successful machine foundation design

is the careful engineering analysis of the foundation responseto the dynamic loads from the anticipated operation of themachine. Furthermore, when excessive motions of an

existing foundation obstruct the operation of the sup-

ported machinery, analysis is necessary in order to under-stand the causes of the problem and hence to guideappropriate remedial action.

'Iiie theory of analysing the forced vibrations of shallowand deep foundations has advanced remarkably in the last15 years and has currently reached a mature state ofdevelopment. A number of formulations and computerprograms have been developed to determine in a rationalway the dynamic response in each specific case. Numerousstudies have been published exploring the nature of associ ~

ated phenomena and shedding light on the role of several

key parameters influencing the response. Solutions are also

presently available in the form of dimensionless graphs andsimple mathematical expressions from which one canreadily estimate the response of"surface, embedded and pilefoundations of various shapes and rigidities, supported bydeep or shallow layered soil deposits. Clearly, the currentstate. of.the.art of analysing machine foundation vibrationshas progressed substantially beyond the state of the art ofthe late 1960s which had been reviewed by Whitman andRichart in 1967'nd by McNeil in 1969.a

In addition to the selection and application of analysisprocedures to predict the response, the design of a machinefoundation involves (I) the establishment of performancecriteria, (2) the determination of dynamic loads, and (3)

'Presented at the International Conference on Soil Dynasnics andEarthquake Engineering, held at the University of Southampton,Enghnd, 13-1 5 July 1982.

the establishment of the soil profile and evaluation ofcritical soil properties. Great progress has also been made incurrent years in developing in situ and laboratory testingprocedures to obtain representative values of dynamic soilparameters; a comprehensive review of the available experi-mental methods has been presented by Woods,s whileOzaydin et al.,'oods and Richarte have summarizedthe present knowledge on the factors influencing thedynamic soil parameters. These developments in determin-ing material properties complement the advances inanalysing foundation vibrations, and provide considerablejustification for the use of sophisticated numerical formula.tions in the design of machine foundations.

On the other hand, little if any progress has been made

in reliably estimating dynamic machine loads and improving(through calibration with field data) the available perform.ance criteria. The state-of-the-art in these two areas has

remained essentially unchanged during the last decade;reference is made to hfcNeil and Richart, Woods and HaQc

for comprehensive reviews of these subjects.An additional and often overlooked step in machine

.foundation design is the post. construction observation ofthe foundation performance and its comparison with the

predicted foundation behavior. Such comparisons are

needed to calibrate new analysis procedures —an essential

task in view of the simplifying assumptions on which even

sophisticated formulations are based.In the final analysis, confidence in the advantages pro-

vided by the use of advanced methods of analysis can onlybe gained if these are shown to have the capability to pre.dict the field performance of actual machine foundauons.Unfortunately, only a limited number of case histories has

so far been published evaluating state-of-the art methods ofanalysis through detailed field observations.

The objective of this paper is to review the present state-of.the-art of determining the dynamic response of founda-tions subjected to machine. type loadings. The outline ofthe paper follows the chronology of historical develop-

ments: from the dynamics of circular footings resting on

the surface of an elastic halfspace to the behavior of cir-

2 Soil Dynamics and Earthquake Engineering, 1983, Vol.. No. 1

0261-7277/83/010002-41 $ 2.00C 1983 CML Publications

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l1lluljdlk VJ llWQI~ jljQ j+NI4luiaVa~ ~ ve» v ~ woaolv V] 4 ~ sw «r ~ ~ V V4

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cular and non-circular foundations embedded in a layeredsoil deposit and, finally, to the response of piles. Particularemphasis is accorded to the effects of dimensionless groupsof geometric and material parameters on the dynamicstiffness functions and on the response of massive founda-tions. Normalized graphs and simple formulas are presentedfor a variety of idealized soil profiles and foundation geo-metries. The use of such data to estimate to translationaland rotational motions of actual foundations in practiceis clearly demonstrated and the various results are syn-thesized by means of a case study. Practical recommenda-tions are then made on how to approximately obtaindynamic stiffness and damping coefficients for actualfoundations, accounting only for the most crucial para-meters of the problem.

Since the limiting motion for satisfactory performanceof a machine foundation usually involves displacementamplitudes of a few thousandths or even ten.thousandths ofan inch, soB deformations are quasi-elastic, involving negli-gible nonlinearity and no permanent deformations. Thus,most of the solutions reported herein assume linear iso-tropic viscoelastic soil behavior, with a hysteretic soildamping to model energy losses at those small strainamplitudes. However, some consideration is also given tothe effects of soil nonlinearity on the vibration of stripfootings under. strong horizontal and rocking excitation.Moreover, the importance of soil anisotropy and soilinhomogeneity are also considered.

OLDER METHODS OF ANALYSIS

In the past, machine foundations were frequently designedby rules. of-thumb without any analysis of the expectedvibration amplitudes. For instance, one such design rulecalled for a massive concrete foundation of a total weightequal to at least three to five times the weight of the sup-ported machine(s). Although such a proposition may atfirst glance seem logical. it is in fact an obsolete one sinceit ignores the effect on the motion of all the other variablesof the problem (e.g. type of excitation, nature of support-ing soil, and so on). For one thing, increasing the mass ofa foundation decreases the resonant frequency of thesystem and, perhaps more importantly, reduces its effectivedamping.'bviously, this is not what those applying therule had in mind.

Following the pioneering experimental studies carriedout by the German Degebo in the early 1930s, a number ofempirical analysis procedures were developed and usedextensively at least until the 1950s. These methods focusedon determining only the 'natural frequency'f a founda-tion. To this end, the concepts of 'in-phase mass'nd'reduced natural frequency'ere developed. The formerassumes that a certain mass of soB immediately below thefooting moves as a rigid body, in.phase with the foundation.The latter postulates that the 'natural frequency's solelya function of the contact area, the soil bearing pressure andthe type ofsoil.

Physical reality contradicts the concept of an 'in.phasemass'. No soil mass moves as a rigid body with the founda-tion. Instead, shear and dBational waves emanate from thefooting.soB interface into the soil, causing oscillatingdeformations at the surface and carrying away some of theinput energy. The factors that have an influence on thesephenomena cannot be possibly accommodated throughsuch an artiTicial concept. Indeed, the early attempts toobtain specific values of the 'in-phase mass'ere frustrated

by the sensitivity of this 'mass'o the foundation weight,mode of vibration, type of exciting force, contact area, andnature of the underlying soil. Apparently, there is absolutelyno value in this concept and its use in practice may verywell mislead the designer.

Tschebotarioff's 'reduced natural frequency'ethod,based on the results of a few case histories, went a stepbeyond the original 'in.phase mass'ethods.s The 'reducednatural frequency'as defined as the 'natural

frequency'ultiplied

by the square-root of the average vertical contactpressure and was given graphically as a function of the typeof soil and of the contact area. Although this method wasnot without merit, it was often interpreted to mean that'the single most important factor in machine-foundationdesign was the soil bearing pressure'.~ Thus, in more thanone occasion, the design was based on soil bearing capacityvalues taken from local building codes!

In addition to the aforementioned drawbacks, these oldrules were only concerned with the resonant frequency,providing no information about vibration amplitudes thatare primarily needed for design purposes. As a consequence,such rules are now obsolete and will not be furtheraddressed in this paper. Reference is made to Richart eral.7for more details on the subject.

Dynamic lankier modelThis model was introduced as an extension of the well

known 'Winkler'r 'elastic subgrade reaction'ypothesis,which is still rather succ™essfully employed ln some

static'oil-foundationinteraction problems.v In order to simulatethe stiffness characteristics of the actual system, the modelreplaces the supporting soil by a bed of independent elasticsprings resting on a rigid base. Plate bearing tests, con-ducted in the field, form the basis for evaluating the springconstants (often called 'coefficients of subgrade reaction').On the basis of field measurements in the USSR,

Barkan'as

presented tables and empirical formulae with which onecan readBy estimate design values of the coefficient forseveral types of soil, for each possible mode of vibration(translational or rotational). He has also shown that, in eachcase, the dynamic coefficient is approximately equal to theratio of applied pressure increment to the resulting displace.ment during static repeated loading tests. In these testsstatic loads 'similar'o the combined dead and Bve load ofthe actual foundation are first imposed, foUowed byrepeated slow loading, at frequencies of the order of 0.001

cps, i.e. much slower than those expected in reality.It is evident that this model can at least give some

reasonable information on the low-frequency (near. static)response of a foundation. But since no radiation damping isincluded, the amplitude of motion at frequencies nearresonance cannot be realistically estimated. It has been

argued that by neglecting damping one obtains conservativeestimates of the response and very good estimates ofnatural frequencies. In fact, this is the procedure currentlyincorporated into the 1970 'Indian Standard Code of Prac-

tice for Design of Machine Foundations'." There is littlemerit in this argument, however. For instance, the highdamping values present in the translational modes of vibra-

tion (of the order of 50io of critical) do affect the'resonant'requencies, in addition to drastically reducingamplitudes. Moreover, avoiding 'resonance'by a safetyfactor of 2) in such cases is an unfortunate design recom-

mendation which may lead to an overly conservative solu-

tion. In other cases, especially when the rotational modes

are of main concern, an unsafe design is quite possible since

ciwn Ils nnsiii c nitif s'nrtltnlint'n p'»cr6vunvitecr 10c'0 i'ni ~ .xn

lj

the actual foundation stiffness at high frequencies may veryweU be appreciably smaller than the static stiffness used inthe analysis (see, for example, Fig. 5).

An improved version of the dynamic Winkler model(caUed 'Winkler-Voigt'odel) places a set of independentviscous dampers in parallel with the independent elasticsprings to provide the 'dynamic subgrade reaction'. Accord-ing to Barken and llyichev,'a this model forms the basis ofthe 1971 USSR machine-foundation code. Again, however,the model itself provides no information on its spring anddashpot coefficients. These are instead backfigured fromdynamic plate. load tests conducted in the field. Both theobserved amplitude and frequency at resonance are utilizedto backfigure the two coefficients. Analyzing the results ofnumerous field tests, Barkan and his co-workers found adiscrepancy between the spring constants backfigured fromresonance plate tests and from static repeated loading tests(described previously). They, thus, resorted to the 'in-phasesoil mass'oncept to essentially match the model constantsobtained from the two types of tests. This added soil masswas found to depend on the size and embedment of thefoundauon and on the nature and properties of the soildeposit, for a given mode of vibration.

It therefore appears that the 'Winkler-Voigt'odel isa purely empirical one, requiring field static and dynamicplate-load tests for each particular situation. Such testsare not only very expensive and difficult to successfullyconduct, but, moreover, they yield results which cannot be

...readUy interpreted and. extrapolated. to prototype.condi-.tions. IfI may slightly rephrase Gibson s

'The model conspicuously lacks what aU modelsshould possess —predictive

power.'he

only possible explanation for the present.day use ofdynamic Winkler models in machine-foundation analysis isthe accumulation in some countries of a wealth of pertinentfield data. Such data, often available in the form of

tables,'an

be directly utilized in practice, thus avoiding theburden of performing plate. load tests. Again, one should bevery careful in picking up values for the coefficients frompublished field data. For it is practicaUy impossible toensure a similarity in aU the crucial physical and geometricresponse parameters of the new prototype and of the oldmodel foundation schemes.

FUNDAMENTALSOF CURRENT METHODS OFVIBRATIONANALYSIS

Hisrorical perspecriveModern methods of analysis of foundation osciUations

attempt to rationally account for the dynamic interactionbetween the foundation and the supporting soil deposit.Cornerstone of the developed methods is the theory ofwave propagation in an elasuc or viscoelastic solid.(con-tinuum). This theory has seen a remarkable growth since1904, when Lamb published his study on the vibration ofan elastic semi infinite soUd (half-space) caused by a

concentrated load ('dynamic Boussinesq'roblem). Numer-ous applications, primarily in the fields of seismology andapplied mechanics, have given a great impetus in the

~ ~

~

development of the 'elastodynamic'heory. Reissner in1936" attempted what is considered to be the first engin-

'eering application; his publication on the response of a

verticaUy loaded cyUndrical disk on an elastic halfspacemarked the beginning of modern soil dynamics. The solu.tion was only an approximate one since a uniform distri~

bution of contact stresses was assumed for mathematicalsimplification. Nonetheless, Reissner's theory offered a

major contribution by revealing the existence of radiationdamping —a phenomenon previously unsuspected buttoday clearly understood. Every time a foundation moves

against the soil, stress waves originate at the contact surfaceand propagate outward in the form of body and surfacewaves. These waves carry away some of the energy trans-mitted by the foundation on to the soil, a phenomenonreminiscent of the absorption of energy by a viscous

damper (hence the name).For many massive foundations the assumption of a

uniform contact stress distribution is an unrealistic one, forit yields a non.uniform pattern of displacements at the soil-footing interface. To closer approximate the rigid bodymotion of such foundations, a number of authors in themiddle 1950s assumed contact stress distributions whichproduce uniform or linear displacements at the interface,under statically applied force or moment loadings, respec-

tively. Thus, Sung'nd Quinian'resented results forvertically oscillating circular and rectangular foundationswhile Arnold er al.'nd Bycroft'tudied both horizontaland moment loading of a circular foundation. These solu-

tions are only approximate: in reality the pressure distribu-tions required to maintain uniform or linear displacementsare not constant but vary with the frequency ofvibration.

'The first 'rigorous'olutions appeared about ten yearslater when the vibrating soil-foundation system was

...analysed.as a. mixed. boundary:.value problem, with pre-scribed patterns of displacements under the rigid footingand vanishing stresses over the remaining portion of thesurface. Introducing some simplifying assumptions regard-

ing the secondary contact stresses ('elaxed'oundary),Awojobi er al.'tudied aU possible modes of osciUation ofrigid circular and strip footings on a halfspace, by recourseto integral transform techniques. On the other hand,Lysmer"- obtained a solution for the vertical axisymmetricvibration by discretizing the contact surface into concentricrings of uniform but frequency-dependent vertical stresses

consistent with the boundary conditions. A conceptuallysimilar approach was followed by Elorduy er al."-'or ver-

ticaUy loaded rectangular foundations.Perhaps equally important with the aforementioned

theoretical developments of this period was the discovery

by Hsieh- and by Lysmer that the dynamic behavio of a

verticaUy loaded massive foundation can be represented bya single-degree. of-freedom 'mass-spring-dashpot'scillatorwith frequency-dependent stiffness and damping coeffi-cients. Lysmer'0 went a step farther by suggesting the use

of the following frequency. independent coefficients toapproximate the response in the low and medium frequencyrange:

4GR 3.4RaE~ = —;C„= —~GpI-v I-v

in which: E„=spring constant (stiffness), C, =dashpotconstant (damping), R = radius of the circular rigid loadingarea, G and v = shear modulus and Poisson's ratio of the

homogeneous halfspace (soil), and p = mass density of soU.

Note that the expression for E„ in equation (I) is identicalwith the expression for the static stiffness of a verucaUyloaded rigid circular disk on a halfspace.

The success of Lysmer's approximation (often called'Lysmer's Analog') in reproducing with very good accuracythe actual response of the system had a profound effect on

the further development and engineering applications of the

4 Soil Dvnamics and Earrhauake Zngineering, 1983, Vol. 2, No. 1

~ '

.

(

s

'halfspace'heories. Richart and Whitman~ extendedLysmer's Analog by demonstrating that all modes ofvibration can be studied by means of lumped-parametermass. spring-dashpot systems having properly selectedfrequency-independent parameters. The axisymmetric (ver-tical and torsional) oscQlations of a cylindrical foundationcan be represented by a I-degreecf-freedom (1-dof)system described by:

mx+ Cx+Kx=P(r) (2)

in which x, x and x= the displacement, velocity and

acceleration, respectively, of the vertically oscillating mass;

P(r) = the external dynamic force arising from the opera-tion of the machine(s). The lumped parameters are theequivalent mass, m, the effective damping, C, and theeffective stiffness K. (For torsional oscillations m should bereplaced by Iz, the effective mass polar moment of inertiaand x should be interpreted as the angle of rotation aroundthe vertical axis of symmetry.) On the other hand, the twoantisymmetric modes of oscillation (horizontal translationand rocking) o'f a cylindrical foundation are coupled andcan be represented by a 2-dof system characterized by theeffective mass and mass moment of inertia, the twoeffective values of damping (for swaying and rocking), andthe two effective values of the stiffness (for swaying androcking).

Different values of the inertia, stiffness and dampingparameters are needed for each one of these four modes ofexcitation."Whitman and Richart~ suggested the choice of-stiffnesses appropriate for low frequencies, and of averagedamping values over the range of frequencies at whichresonance usually occurs. In order to obtain a good aaree-

ment between the resonant frequencies of the lumped-parameter and the actual system, they recommended thata fictitious mass (or mass moment of inertia) be added tothe actual foundation mass (or mass moment of inertia).The need for such a recommendation stemmed not fromthe existence of any identifliable soil mass moving in.phasewith the foundation, but rather from the fact that inreality the stiffnesses decrease with increasing frequency(see Figs. 5 and 7), instead of remaining constant and equalto the stare stiffnesses, as the model assumes. In otherwords, instead of decreasing K, the lumped-parameter ~

model increases m to keep the resonant frequency, co„.unchanged. Recall that ter is proportional to the square.root oi(K/m).

Whitman and Richart s and later Richart, Woods andHall7 and Whitman'4 presented expressions for these para-meters for all four vibration modes. Table I displays theseexpressions, which have enjoyed a significant popularityover the last decade.

Priniarily because of its simplicity, the lumped-para-meter approximation had a great impact on the applicationof the 'half-space'heory. It demonstrated that this rationaltheory can be cast into a tractable, simple engineering form,which can be used by the profession with hardly anygreater difficultythan the older empirical procedures.

Motivated to a large extent by the need to understandthe phenomena associated with seismic soil-suucture inter-action, the analysis of the dynamic response of foundationshas been a subject of considerable interest throughout the1970s. A significant amount of related research has led tothe development of new formulations and computerprograms, while numerous publications have studied theimportance of critical foundation, soil and loading para-meters and have presented graphs, tables and simple ex-

Table l. Equivalent lumped parameters for analyziz ofcircularfoundations on elastic hal

fspaee'ode

Vertical Horizontal Rocking lozsion

Stiffness:4GR1-v

SGR 8GR'6GR'-

v 3(1- v) 3

Mass ratio m:m(1- v) m(2- v) 31„(l-v) I4pR'pR'pRs pRs

0.425 0.29 0.15 0.50Damping zatio: mus mus (1+ m) m"'+ 2m

Fictitious addedmass:

0.27m 0.095m 0.24lz 0.24Im m m m

Iz, Ia = mass moments of inezth around a horizontal, vertical axis,respectively; damping ratio = C/C«where C«= 2(Km)'" orC«2(KI)u'or translational or rotational modes of vibration,withI= I„orIz for rocking or torsion, respectively.

pressions, suitable for direct use in practical applications.It is worth mentioning some of the most important contri-butions to the current state of the art.

Newly developed (mid-1960s) mathematical techniquesto solve mixed boundary-value elastodynamic problemswere utilized by Luco er al." and Karasudhi et al.~ toobtain 'exact'umerical solutions for aG modes of-vibration—-of strip footings on a halfspace, and by Luco er al.'andVeletsos er al.'"zv to extend the available halfspace solu.tions for circular foundations to the high frequency ranae

and, also, to a viscoelastic material with linear hystereticdamping. The development of dynamic finite-elementformulations with energy absorbing ('viscous'nd 'consist-ent') lateral boundaries prompted the study of the responseof surface and embedded foundations supported by a

layered soil stratum.' Only plane-strain and axisym-metric geometries could be handled with these finiteelement formulations, however, and the presence at a

relatively shallow depth of a non.compliant rock-likematerial underlying the stratum was an unavoidable require-ment regardless of whether such rock did actually exist.

On the other hand, Lucoss and Gazetas~ presentedanalytical solutions for circular, strip and rectangularfoundations on the surface of a layered halfspace or a

layered stratum (i.e. with or without a rigid rock as thelast layer, respectively). Utilizing these formulations theyoffered results which bridged the gap between the twopreviously studied extreme proflles-the halfspace and thestratum. on-rigid-base. At about the same titne,

Novak'btained

approximate analytical solutions for circularfoundations embedded in a halfspace, by deriving closed-

form expressions for the dynamic stiffness and dampingcoefficients along the vertical sides of the foundation.Later, on this method was easily adapted to study the

dynamic response of piles.4'~'n

more recent years research efforts have been pri~

marily directed to determining solutions: (a) for rigidfoundations of rectangular and arbitrary shapes: 'b) forfoundations of finite flexural rigidity "(c} for founda-tions on inhomogeneous and on anisotropic soils 'nd(d) for foundations on nonlinear (Ramberg-Osgood)soils. Furthermore, a very substantial amount of research

work has been devoted to the dynamic behavior of single

(floating and end-bearing) piles embedded in homogeneous,

Soil Dvnamics and Earthttuake Engineering, 1989, Vol. ". h'o 1

1"

Aruuysts oJ nracrurre Joundanon vibranonst state oJ'the art: (; f'a=eras

inhomogeneous or layered soil deposits, and the firstattempts have already been made to obtain solutions fordynamic loaded pile groups. For comprehensive lists ofrelated references, see Dobry et al.,'7 Kagawa er al.,~ andNovak.s9

tImpedance and compliance fimctionst definition andphysical interpretation

An important step in current methods of dynamicanalysis of rigid massive machine foundations is the deter-mination (using analytical or numerical methods) of thedynamic impedance functions, E(u),'f an

'associated'igid

but massless foundation, as a function of the excita-tion frequency, ~. As shown in Fig. I the

'associated'oundation.

soil system is identical (in both material prop-erties and geometry) with the actual system, except thatthe foundation mass is taken equal to zero. It will beexplained in the following section how, once the harmonicresponse of such a massless foundation has been deter-mined, the steady-state response of the massive foundation,or of any structure supported on it, may be evaluatedusing standard procedures. In addition, the transientresponse to non-harmonic machine forces can also beevaluated by recourse to Fourier analysis and synthesistechniques.

For each particular harmonic excitation with frequencyos, the dynamic impedance is defined as the ratio betweenthe steady state force (or moment) and,the resulting.dis-placement (or rotation) at the base of the massless founda-tion. For example, the vertical impedance of a foundationwhose plan has a center of symmetry is defined by:»

l

R„(t)Ko=-v(t)

(3)

in which R,(t) = R„exp(i~t) is the harmonic vertical forceapplied at the base of the disk, and v(t) =

v exp (i~t) is theuniform harmonic settlement of the soil foundation inter-face. It is evident that R, is the total soil reaction againstthe foundation; it is made up of the normal stresses againstthe basemat plus, in case of embedded foundations, the

~ shear stresses along the vertical side walls, as illustrated inFig. I.

Similarly one may define the torsional impedance, K,.from the torsional moment and rotation: the horizontalimpedances, Er„ from the horizontal forces and displace-ments along the principal axes of the base; and the rockingimpedances, E„ from the moments and rotations aroundthe same horizontal principal axes. However, since hori-zontal forces along the principal axes produce rotations inaddition to horizontal displacements, cross. horizontal-rotational impedances K,h may also be defined; they are

r5514, 0555155$5»trr55510n

usually negligibly small in case of surface and very shallowfoundations, but their effect may become appreciable forgreater depths of embedment.

Referring to equation (3), it is interesting to note thatdynamic force and displacement are generally out of phase.In fact, any dynamic displacement can be resolved into twocomponents: one in phase and one 90'ut of phase withthe imposed harmonic load. It is convenient then to intro-duce complex notation to represent forces and displace-ments. As a consequence, impedances may also be writtenin the form: »

(4)Ea(u) = ESSt(w) + iESSz(~)

a=v,h,r,hr,t; i=~iThe real and imaginary components are both functions

of the vibrational frequency ca. The real component reflectsthe stiffness and inertia of the supporting soil; its depen-dence on frequency is attributed solely to the influencewhich frequency has on inertia, since soil properties areessentially frequency independent. The imaginary com-ponent reflects the radiation and material damping of thesystem. The former, being the result of energy dissipationby waves propagating away from the foundation, is fre-quency dependent; the latter, arising chiefly from thehysteretic cyclic behavior of soiil, is practically frequency

massless foundation. soil system has been drawn by Roesset.Assuming a harmonic excitation P(t) =Poexp(icut), thesteady-state response x(r) = xo exp (lent) of the I-dofoscil-lator may be obtained by substitution into equation (2);

P(t)(K—mw') + iCu =— (5)

x(r)Contrasting equations (5) and (3) prompts the definition

of a dynamic impedance function for the I-dof mass-spring. dashpot system:

K = (K—mma) + i Cor (6)

and, by comparison with equation (4):

K, =K-muaE~ = C4)

In other words, the dynamic impedance of our familiarI-dof oscillator is indeed a complex number with a fre.quency dependent real part representing the stiffness andinertia characteristics of the system, and a frequencydependent imaginary part expressing the energy loss in thesystem. Therefore, it is quite natural to express the dynamicimpedance of soil.footing systems in a complex form, as

done in equation (4).Having, thus, established the analogy between I-dofand

massless footing.soil systems, let equation (6) for theI-dofbe rewritten as:

independent.A very instructive analogy between the dynamic response

of a simple -I-dof oscillator-and-of a three4mensional- ».-

~ ~

~ \ ~~ ~ \' ~ ) ~

~ '

~ ~ ~ 1 ~ \ % ~ ~ \ ~

\ 0 L ~ ~

~ ~~, ~~ '

~ ~ i ~

~ I ~

55 5 ~r '

' ~ r~

~ ~ ~

E=E I-—. —:12P— (9a)

~ ~ ~ ~ r ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~

~~ ~ ~ ~~ ~ ~ y ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ \ ~ ~ ~ t

Figure 1. Machine foundanon and the associated rigid'assless foundanon

'old letters are used in the text for impedanccs, compiianccs andsome suffncss and damping coefficients (equation (17)); in thefigures, caUigraphic characters are used for these quantities.

or

E =E {k+iucs)in which the critical viscous damping ratio is:

C Cf1=-Ccr -"Elean

(9b)

(10)

b

18"n

Cs to support the mass of the simple oscillator. This damperis described through a hysteretic damping ratio, $ . Duringeach cycle of motion it dissipates an amount of energyproportional to the maximum strain energy, W, of thesystem:

the natural frequency ~a„= (K/m)"',k = (I- ta /can) andc,= C/K. Equation (9b) implies that the dynamic imped-ance of a I-dof simple oscillator may be expressed as aproduct of the spring constant K, which happens to be thestatic stiffness of the system, times a complex numberk+itac„which encompasses the dynamic characteristicsof the system (inertia and viscous damping) and is here-after called 'dynamic part'f the impedance. At zerofrequency the dynamic part becomes a real number, equalto I, and the impedance coincides with the static stiffnessK of the simple system. k and c, are named respectivelystiffness and damping coefficients and their variation withfrequency for the I-dof's is plotted in Fig. 2. Notice thatk decreases as a second degree parabola with increasingta, whereas c, remains constant.

It should not surprise the reader that the actual varia-tion with ta of the stiffness and damping coefficients, k„and c». of a vertically vibrating circular disk on an elastichalfspace is indeed very similar to the variation of thek and c, of the I-dof system! (To see this similarity justcompare Fig. 2 to Fig. 5(a).) However, in general, k and c,of a foundation-soil system may vary in a rather compli-cated manner with ~, depending primarily on the modeof vibration, the geometry, rigidity and embedment of thefoundation, and, finally. the profile and properties of thesupporting soil deposit. Figures 5, 8, 9, 10 and 20 may bepreviewed to confirm this statement. Nonetheless, in aUcases, the dynamic impedance functions can be expressedas products of a static and a dynamic part, as describedby equation {9b). Alternatively, a dimensionless frequencyfactor is often introduced:

u8ac=-Vs

in which: B= a critical foundation dimension like, e.g.,the radius of a circular foundation or half the width of astrip or a rectangular foundation; and Vs = a characteristicshear wave velocity of the soil. Combining equations (9b)and (11) allows the impedance to be case in the form:

K = K(k+Iallc) (12)

with

VsC=C (13)

Since both ao and c are dimensionless quantities, equation(12) is strongly preferred to equation (9b) in presenting theresults of dynamic analyses.

Let it now be assumed that a 'hysteretic damper'sadded in.paraUel with the spring and the 'viscous

damper'/Mq

I LS 1 I LS 1

Figure Z Dynamic sriffness and damping coefjVcienrs ofa 1-dofsimple oscillaror

AWE = 4ngW (14)

in which W =(~~) Kxo~. On the other hand, during a cycleof motion the viscous damper has consumed an amountof energy equal to:

BWv = nCtaxo

= 4nP —W (15)tan

so that the total dissipated energy, d W= BWi, +5W„, as

a function of W is:

—=4n P—+$ (16)

This expression suggests that the simple addition rule,

f+Pu/upped

may be used to obtain the 'effective'ampingratio of a system possessing both viscous, P, and hysteretic,P, damping. A vibrating foundation-onsoil is one suchsystem, with its radiation damping being of a viscous naturewhile the material damping is of the hysteretic type.

The presence of material damping in the soil affects boththe stiffness and damping coefficients, k and c. In an

"attempt to isolatr the-effects of hysteretic material. damping, an alternative expression to equation (12) is oftenused for the dynamic impedance:

K = K(k+iaec) (I+2i)) (17)

RecaUing the so.caUed 'correspondence principle,~'nemay anticipate that the new coefficients, k and c, areindependent of material damping. If this were true, itwould then be sufficient to obtain solutions for a purelyelastic soil and then extrapolate the results to soils withany hysteretic damping ratio by multiplying the undampedimpedances by I+2ig. Indeed, for very deep soil depositswhich can be modeled as a halfspace the above

'principle's

reasonably accurate and has been repeatedly utilized toobtain solutions for damped soUs."v ~1's However, in thecase of a shallow stratum on rigid rock both k and c arefairly sensitive to the assumed material darnpina ratio (seeFig. 9, for example); this discredits to a large extent the'correspondence principle', as Kausel" had first noticed.

None the less, it is convenient to express the impedancefunctions in the form of equation (17), and this practice isfrequently followed in the sequeL Alternatively, however,equation (12) is also used in some cases.

Dvnamic compliance funcrionsAlso given the names dynamic 'displacement'unctions

and dynamic 'flexibiUty'unctions, they are essentially theratios between dynamic displacements (or rotations) andthe dynamic reactive forces {or moments) at the base of a

foundation. They were first introduced by Reissner."FoUowing the previous discussion, it is convenient toexpress each compiiance using complex notation:

F =F i(u)+ iF z(ui) (18)

a = v,h,r,hr,rThe real and imaginary parts represent the disphcementcomponents which are in.phase and 90'nut.of.phase'with

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, iV'o. I 7

the reactive force, respectively, and they both are functionsof frequency, as discussed in detail previously. For a

foundation which in plan has a center of symmetry, thevertical and torsional compliances are simply the inverseof the vertical and torsional impedances:

1

Fa= —', b=v,r (19a)Kg,

However, due to the coupling between rocking andswaying motions, the corresponding compliances shouldbe obtained by inverting the matrix of impedances:

[lr ler]

The following alternative form to equation (18) is alsofrequently used in presenting compliance functions:

1

F. =—(f.t(~)+ if.z(c )j (2o)K~

where K, is the corresponding static stiffness.

Computational procedures fordeterminingimpedance juncrions

Several alternative computational procedures are pre-sently avaHable to obtain dynamic impedance functions foreach speciTic machine foundation problem. The choiceamong these methods depends to a large extent on therequired accuracy, which in turn is primarily. dictated by.the size and importance of the particular project. Further-more, the method to be selected must reflect the keycharacteristics of the foundation and the supporting soil.Specifically, one may broadly classify soil foundationsystems according to the following material and geometrycharacteristics:

1. The shape of the foundation (circular, strip, rect-angular, arbitrary).

2. The type of soH profie (deep uniform deposit, deeplayered deposit, shaHow layered stratum on rock).

3. The amount of embedment (surface foundation,embedded foundation, deep foundation).

4. The flexural rigidity of the foundation (rigid founda-tion, flexible foundation).

Two computationaHy different approaches have beenfoHowed over the years to obtain the dynamic impedancesof foundations wiih various characteristics: a

'continuum'pproach,

which led to the development of analytical andsemi. analytical formulations, and a 'discrete'pproach,which resulted in the development of finite-difference and,primarily, finiteclement models. In the past (mid-1970s),considerable controversy was held about the relativemerits and deficiencies of each approach and some extremeand unjustified positions were advocated. Today, it is quiteclear that both procedures, if correctly understood andimplemented, are very useful tools in analysing the behaviorof dynamicaHy loaded foundations. Moreover, they yieldvery similar results if they are appropriately used to solvethe same problem. Hadjian er al.~ and Jakub er a1.4s havepresented excellent discussions and comparative studies onthis subject. The foHowing paragraphs intend to ratherbriefly introduce the most important analytical, semi-analytical and numerical procedures which are currently

. available to the machine-foundation analyst. The list is byno means exhaustive, and the emphasis is on discussing thestrong and weak points of each method.

'Connnuum'ethods. Starting point of aH the devel.

oped formulations is the analytical solution of the pertinentwave equations governing the imposed deformations in each

uniform soil layer or halfspace. However, the boundaryconditions at the soH-footing interface are handled differ-ently by the various methods. In that respect, one may verybroadly classify the available continuum formulations intoanalytical and semi-analytical solutions.

The known analytical solutions simplify the mechanicalbehavior of the soil-footing contact surface by assuming a

'relaxed'oundary. That is, no frictional shear tractions candevelop during vertical and rocking vibrations, while forhorizontal vibrations the normal tractions at the interfaceare assumed to be zero. This assumption has been necessaryto avoid the more complex mixed boundary conditionsresulting from the consideration either of a perfect attach.ment between foundation and soil ('rough'oundation) orof a contact obeying Coulomb's friction law (an even morerealistic idealization).

By recourse to integral transform techniques (involv-ing Hankel or Fourier transforms for axisymmetric orplane. strain geometries, respectively) the relaxed boundaryconditions yield sets of dual integral equations for eachmode of vibration. Each set is then reduced to a Fredholmintegral equation which is finally solved numericaHy.

Such analytical solutions have so far been published forsurface circular and strip foundations of infinite flexularrigiditysupported by an elastic or viscoelastic halfspacep 'v

for. circular foundations on a layered elastic or viscoelastic---'".soil deposit " 4 for circular foundations of finite flexuralrigidity supported on a halfspace v for circular foundationson a cross anisotropic halfspace; 'nd even for verticallyloaded rigid rectangular foundations on a halfspace."

The semi-analytical type solutions are based on thedetermination of the displacements at any point within thefooting-soil interface, caused by a unit normal or sheartime-harmonic force applied at another point of the sameinterface. Then, by properly discretizing the contact sur.face, the matrix of dynamic influence or Green's functionsis assembled and the problem is solved after imposing therigid-body motion boundary conditions. Several differenttechniques (in essence different integration procedures)have been formulated to carry out these steps of theanalysis. For example, Elorduy et al." and Whittakerer al.~ utilized Lamb's solution for a point loaded half-space; Luco er al.'7 obtained pairs of Cauchy type integralequations which they numerically solved after reducing iocoupled Fredholm equations; Gazetas ~ and Gazeias er af.utilized a fast Fourier transform algorithm; Wong andWong er al.~ used the solution for a uniformly loadedrectangle; and so on.

For the purpose of this discussion, one may list as a

semi-analytical solution the formulation of Dominguez andRoesset,4'ho applied the socaHed 'boundary integralequation'r, more simply, 'boundary element'ethodto obtain dynamic impedance functions of rectangularfoundations at the surface of, or embedded in a halfspace.To this end, they utiTized the closed-form solution to the'dynamic Keivin'roblem of a concentrated load in aninfinite medium,6~ and discretized either only the contactsurface, in the case of surface footings with

'relaxed'oundaries,or both the contact and the surrounding soilsurfaces, in the cases of embedded footings and of surfacefootings 'adhesively'ttached to the soi1

So far rigorous semi-analytical solutions have been pub-lished for rigid strip foundations on the surface of a layered

8 Soil Dynamics and Earrhquake Engineering, 1983, Vof. 2, Po. 1

A

+c

1

(i

'

halfspace or stratum. on rock;s" s"'v ~ for rigid rectangularfoundations on a halfspace; '~ ~ ~ ~~ " a 'or rect-

angular foundations of finite flexural rigidity " for rigidrectangular foundations embedded in a halfsgace;4'nd,finally, for rigid foundations ofarbitrary shape.

Note that approximate semi-analyrical procedures have

already been developed to obtain the impedances of cylin-drical embedded foundations and circular piles."~' '~These procedures assume that only horizontally propa-gating waves generate at the vertical foundation soil inter-face, and they neglect the coupling between forces anddisplacements at various points. Instead, they only computethe displacement at the point of application of the load.Thus, in effect, the soil is modeled as a Winkler medium,the spring and dashpot characteristics of which are esti-mated from realistic, albeit simplified, wave propagationanalyses.

Finally, several similar approximate analytical formula-tions have been developed, again for deeply embeddedcylindrical foundations and end-bearing piles in soilstrata.~ 's These procedures attempt to analytically solvethe governing wave equations for the stratum, by neglectingthe secondary component of displacement (i.e. the verticalcomponent for lateral vibrations or the radial one forvertical vibrations). The boundary conditions at the soil-

pile interface are analytically enforced by expanding thecontact pressure distribution to an infinite series in terms ofthe natural modes of vibration of the soil layer.

'Discrete'models. Dynamic finite difference and finiteelement models have been developed'for problems ofcomplicated geometry which are not easily amenable toanalysis with continuum type, analytical or semi. analyticalformulations. Today, finite difference formulations suchas those proposed by Ang et al.,~ Agabein et al.,~ Krizeket al.,s'nd Tseng er al., 'ind very Iinle ifany applicationin solving foundation vibration problems, and, therefore,will not be further addressed in this paper. On the otherhand, several finite element formulations and computerprograms are presently widely available and frequentlyused in analysing foundation oscillations.

The use of finite elements in dynamic foundation prob-lems is different from other applications of finite elementsin statics and dynamics in that soil strata of infinite extentin the horizontal and even in the vertical direction must be

represented by a model of a finite size. Such a finite modelcreates a fictitious 'box'ffect, trapping the energy of thesystem and distorting its dynamic characteristics. To avoidthis problem, wave absorbing lateral boundaries are intro-duced to account for the radiation of energy into the outerregion not included in the model. Two main types of suchboundaries are available. The approximate 'viscous'oun-dary proposed by Lysmer et al ss and extended by Valliappaner al.~ must be placed at some distance from the founda-tion. The alternative 'consistent'oundary developed byWaassi and extended by Kausel" is very effective in accur-

ately reproducing the physical behavior of the system, andit also results in considerable economy by being placeddirectly at the edge of the foundation. This

'consistent'oundary

provides a dynamic stiffness matrix for themedium surrounding the plane or cylindrical vertical cavitywhich is assumed to occupy the central region under thestrip or circular foundation. This matrix correspondsexactly to the boundary stiffness matrix that would beobtained from a continuum type formulation.

Unfortunately, 'consistent'oundaries have been devel-

oped only for plane. strain and axisymmetric (cylindrical)geometries. No such boundary is available for truly three.

dimensional (3D) geometries, in cartesian coordinates.

Thus, to solve 3D problems a finite<lement model must

resort to 'viscous'r elementary boundaries placed far

away from the loaded area In this way the fictitiouslyreflected waves are dissipated through hysteresis and fric-

tion (material damping) in the soil before they return to

the foundation region. However, the cost of such analyses

is prohibitive and truly 3D solutions are very rarely used

in practice. An attempt has been made to modify a 2D

computer program by adding viscous dashpots to the

lateral faces of its plane-strain elements, in order to simu.

late the radiation damping of 3D situations. 'otwith.standing the popularity enjoyed by this pseudo3D model,its only difference from the 2D model is that it introduces

an artificial increase in damping, which cannot possiblyreproduce all aspects of the true 3D behavior. In fact, insome cases the actual 3D radiation damping in rocking is

overestimated rather than underestimated by a 2D model;~thus by adding viscous dashpots the situation may worsen

instead of improving.~ es

Consequently, today, two types of finite<lement models

are practically available: plane-strain 2D models appropriatefor strip footings or elongated rectangular structures;and 3D axisymmetric.geometry models appropriate forcylindrical foundations and nearly square structures.

't

is noted that embedded foundations and layered soil

strata can be routinely handled with all the finite.element

formulations. On the other hand, the presence of a fixedbottom boundary is required by most of the available

"codes. This is hardly a drawback'if a stiff,'ock-like stratum--:---does exist at a relatively shallow depth. Otherwise, when

the supporting soil deposit is very deep, the cost of a

realistic finite.element analysis may become substantiaLConclusion. With the available analytical, semi-analytical

and finite<lement computer programs the foundation vibra.

tion analyst may obtain solutions for foundations of various

shapes, surface or embedded, supported by deep or shallowsoil deposits. In selecting the most appropriate code foreach specific situation, attention should first focus on the

depth of embedment and the nature of the underlying soil.

When dealing with very shallow footings on deep deposits

which can be weil reproduced by a small number of layers

with different properties, continuum type analytical orsemi. analytical formulations are clearly more advantageous;

the choice of the most appropriate among them will be

mainly dictated by the shape of the footing (strip, circular,rectangular, arbitrary) and the desired degree of accuracy.

On the other hand, for embedded foundations in a shallow

stratum or whenever a large number of layers with sharplydifferent properties exists below the footing, finite elementmodels are particularly appropriate.

Furthermore, attention should be accorded to the opera-

tional frequencies of the machine and the inertia character-istics of the foundation. At very high frequencies of vibra-

tion, f, discrete models may become very costly; because,

in order to transmit high frequencies, a large number ofsufficiently small. sized elements must be used. For instance,

it is usually recommended that the maximum dimension ofan element should not exceed 'A/8, where X = V/f is the

wavelength in a particular soil layer having shear wave

velocity V. Therefore, with high frequencies, analyticalmodels may become advantageous. Notice, though, that the

computer costs of semi. analytical formulations may also be

adversely affected by a large increase in the operationalfrequency, since they, too, discretize the contact area or

the whole uppermost surface.Regarding the inertia characteristics of the foundation,

Soil Dynamics and Earthquake Engineering, 1989, Vol. 2, JVo. 1 9

the author and Roessets9 have demonstrated that for heavyfoundations (i.e. with high mass ratios) smail errors inmodeling the different soil layers are unimportant and onecan safely base the design on available halfspace solutionsor on the results of analytical type computer programs.On the other hand, relatively light foundations are quitesensitive to the existence of competent rock at a shallowdepth and of different soil layers beneath the footing, thusrequiring a good soil exploration followed by finite~lementanalyses. These conclusions are further illustrated andgeneralized in a later secuon of this paper.

ln addition to the existing computer programs numeroussolutions have been published in the literature in the formof dimensionless graphs, tables and simple formulae forimpedance and compliance functions of foundations withseveral different geometries, depths of embedment andstiffness characteristics, supported by various idealized soilprofiles (halfspace, stratum, etc.). These solutions can givevery satisfactory results in many practical cases and areespecially valuable in conducting preliminary analyses andparameter sensitivity studies. One of the goals of this state-of-the-art paper is to present and discuss the most signi-ficant of these available solutions. Before doing this,however, it is expedient to illustrate how the impedancefunctions may be utilized to obtain the dynamic responseof rigid massive foundations.

CROSS-SECT IOH

PLAN

)CGII

I lI

I~

I

h

r4l1.

I. 7I

Use ofimpedance functions: response ofmassive machinefoundarions

The first step in analysing the response of a massivemachine foundation is to evaluate the pertinent dynamicimpedances at the anticipated frequency, or range of fre.

~~

quencies, of the machine. This is done either by utilizingexisting discrete or continuum type formulations, or byresortina to published solutions available in the soil dyn-amics literature. The use of dynamic impedance to obtainthe response is illustrated herein.

Figure 3 portrays a massive, rigid foundation having equaldepth of embedment along all the sides and possessing twoorthogonal vertical planes of symmetry, the intersection ofwhich defines a vertical axis of symmetry. The foundationplan, having two axes of symmetry, may be of any axi-symmetric or orthogonal shape, including the infinitelylong strip (2D geometry). For such foundations, verticaland torsional oscillations are uncoupled, while horizontalforces and moments along and around the principal axesproduce displacements and rotations only along and aroundthe same axes. Thus, vrlth the notation of Fig. 3, the equa-tions of motion in vertical translation v(t), torsional rota-tion 8(r), and coupled horizontal translation h(t) androcking r(t), all referred to the center of gravity of themachine. foundation system, are respectively:

fr)+R,(t) = Q,(t) (21)

I, 8(t)+T,(r) =M,(t) (22)

m h(r)+Rt,(r) = Q>(r) (23)

Io„.r(t)+T,(t)-R>(t) z =M,(t) (24)

in which: m = total foundation mass; lo„= mass momentof inertia about a principal horizontal axis passing throughthe center of gravity; I = mass moment of inertia aroundthe vertical axis of symmetry; Rv, T,, Rt, and T„= vertical,torsional, horizontal and rocking reactions of the soil actingat the center of the foundation base (remember Fig. lb);Q„M„Q» and M,= vertical, torsional, horizontal and

Figure 3. Definirion ofdeformation variables

P, = arctan (v>/vi) (31)

*~+ ~ ~ ~

rocking exciting forces and moments, acting at the centerof gravity and resulting from the operation of the machine.

As already mentioned, only the steady. state response

due to a harmonic excitation is of interest here. Not onlybecause most machines usually produce unbalanced forceswhich indeed vary harmonically with time (rotary or recip-rocating engines), but also because non-harmonic forces

(such as those, for example produced by punch presses and

forging hammers) can be decomposed into a large numberof sinusoids through Fourier analysis. Therefore, the excita.tions may be written as:

Qa Qa exp [i(~t+4a)) a = v, h (25)

Ma Ma exp ti(ut+4a)l a ze r ( 6)

in which the amplitudes Q, and Iii, are either constants or(more frequently) proportional to the square of the opera.tional frequency u = 2",.f; p, are the phase angles of thefour excitations, v, h, r and z.

With the excitation forces described by equations (25)-(26), the steady-state motions may be cast in the form:

v(t) =v exp(i~at); v= v,+iv.. (27)

8(t) =8 exp(itat); 8 = 8,+i8> (28)

h(t) =h exp(i~at); h =h,+ih> (29)

r(r) = r.exp(itat); r =r,+ir, (30)

in which: v, 8, h and r are complex, frequency-dependentdisplacement and rotation amplitudes at the center ofgravity. Note that equations (27+30) do nor by anymeans imply that the four components of motion are ail

in phase, nor that the phase. angles between the corre-

sponding excitations and motions are equal to p, (equations(25)-(30)). instead, the true phase angles yi, are 'hidden'n

the complex form of each displacement component. Forinstance, the vertical motion will exhibit:

10 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2. lVo. 1

in which vi and v: are the real and imaginary parts of v

(equation (27)), while its amplitude is of a magnitude:

lv) —(v" q va) u2 (32)

Also, since Q, and iV, in equations (25)-(26) are real quan.

tities, the phase lags between excitations and motions willbe simply equal to 4,-p,.

Using similar arguments with regard to the soil reactions,

one may, without loss of generality, set:

R,=R,.exp(lent) a=v,h (33)

T, = T, exp(iut) a = zp r (34)

whereby the complex amplitudes R, and T, are related tothe complex displacement and rotation amplitudes throughthe corresponding dynamic impedances E„a = v, h, r, hr,t (see equations (3)-(4)). Recalling that the latter are

referred to the center of the foundation base, one can

promptly write:

~ ~H:',v ~

~ ~~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~

bedrock

halfspace~ ~ ~ ~

H~ ~

(b)

~'

~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~

Rv =Eo'v (35)

Tc =Er'e (36)

Rh = Kri ~ (h -zcr)+Khr'r (37)

Tr = Kr'r+Ku.(h -zcr) (38)

Substituting equations (25)-(30) and (33)-(38) into thegoverning equations of motion (21)-(24) and solving theresulting system of four algebraic equations yields thefollowing complex-valued displacement and rotationamplitudes at the center of gravity':

Q„exp (id.„)

E,(ca) -musN .exp(inc)8— (40)Er(~)-I ~'

= (K,'.Qr, exp(i') -Kr',r hf,exp(ip,)).N (41)

r = (Kri Mrexp(iver)-Kr'„Qh exp(iver,)) A'42)in which the following substitutions have been performed:

Kr", = Kh(~) -mu"- (43)

Ehr Khr(~) Eh(~) sc (44)

Kr = Er(m) —loxw"+Kg(w) -c -Khr(~) "c (45)

and, finally,N= (KriKr E ~) i (46)

Notice that, for a particular frequency ~, determination ofthe motions from equations (39)-(42) is a straightforwardoperation once the dynamic impedances are known. Ofcourse, the computations are somewhat tedious if per-formed by hand, since complex numbers are involved; buteven with small microcomputers the calculations can be

done routinely, at a minimal cost.Therefore, the author proposes that this procedure

(equations (39)-(42), in connection with an appropriateevaluation of impedances at the frequency(ies) of interest,should be used in machine foundation analysis in place ofthe currently popuhr 'equivalent lumped frequency-inde ndent- aiameter' roach.P PP

'1 PRESENTATION OF RESULTS FOR SURFACE ANDEMBEDDED FOUNDATIONS

The subsequent four sections of the paper present a com-prehensive compilation of characteristic numerical results

D 2L

2B 2B

for the dynamic impedances (or compliances) of massless

foundations, pertaining to all possible (translational and

rotational) modes ofvibration. These results can be directlyused in equations (40)-(43) to make satisfactory and inex-

pensive predictions of the dynamic behavior of machinefoundations in many practical cases, without the need toresort to costly computer programs for evaluating theimpedances; this should be of especially great value inpreliminary design calculations.

A second, equally important objective of the presenta.tion is to assess the significance of various phenomena and

to illustrate the role of key dimensionless geometric and

mategal parameters on the response. It is thus hoped thatthe reader can gain a valuable insight into the mechanics

of foundation vibrations.Results are presented for three categories of idealized

soil profiles (Fig. 4): the halfspace, the uniform stratumon rigid base and the hyer on top of a halfspace. These

models represent a wide spectrum of actually encounteredsoil profiles and are simple enough for their geometry to be

described in terms of a single quantity, namely, the thick-ness H of the uppermost layer. (For the halfspace H~~.)For most problems considered, the following groups ofdimensionless parameters which appreciably influence the

dynamic impedances have been identified:

(a) the ratio HJB of the top layer thickness, H, over a

critical foundation-plan dimension, B; the lattermay be interpreted as the radius, R, of a circularfoundation or half the width of a rectangular or a

strip foundation(b) the embedment ratio D/B, where D is the depth

from the surface to the horizontal soil.footing inter-face

Figuie 4. (aj 7Tie'three soil profiles studied; (b J definition=-ofgeometric parameters

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, iso. I I I

4V

(c) the shape of the foundation plan: circular, strip,rectangular, circular ring; in the last two cases theplan geometry may be defined in terms of thelength-towidth or 'aspect'atio, L/B,or the internal-to-external radii rati, RI/R, respectively

(d) the frequency factor ao = os'/Vs, where Vs is acharacteristic shear wave velocity of the soII deposit

(e) the ratio G,/G, of the shear moduli correspondingto the upper soil layer and the underlying halfspace,respectively; this ratio may attain values rangingfrom 0, in case of a uniform stratum on rigid base,to I, in case of a uniform halfspace

(f) the Poisson's ratio(s) v of the soil layer(s)(g) the hysteretic critical damping ratio(s) $ of the soil

layer(s)(h) the factors n and m which express the 'degree'f

anisotropy and the 'rate'f inhomogeneity, respec-tively; n = EH/Ey, where Eff and Err are the hori-zontal and vertical Young's moduli of a cross-anisotropic soil; while m, for a certain type ofinhomoOgeneit, describes the change of shearmodulus from the surface to a depth equal to B

(i) the relative fiexural rigidity factor RF=(ZgE,)(I -vf') (r/B)'here E~, vf and f are, respectively,the Young's modulus, Poisson's ratio and thicknessof the foundation raft; RF ranges from ~, for aperfectly rigid foundation, to 0, for an ideallyflexible mat.

torsional response is totally independent of v at all fre-quencies. It thus appears that the importance of Poisson'sratio increases when the relative contribution of generateddilational (P) waves increases. Indeed, in the vertical and

rocking modes P waves are significant; in the horizontalmode P waves are of secondary importance; and in thetorsional mode only SH waves are generated and P waves

play no role in the response.2. The coefficients kI„cp, and ce are essentially inde-

pendent of frequency and can be considered constantwithout any appreciable error. On the other hand, ke, lc„cr and cr exhibit a strong sensitivity to variations in thefrequency parameter, while kr shows an intermediatebehavior. Of particular interest is the rapid decrease of thevertical and rocking stiffness coefficients ke and ic, withincreasing ao, for values of Poisson's ratio close to 0.5»

(typical for saturated clays). In fact, ko and k„become

RIGID SURFACE FOUNDATIONS ON HOMOGENEOUSHALFSPACE

Rigid circular foundationWhen dealing mth a deep and relatively uniform soil

deposit, it makes engineering sense to model it as a homo-geneous halfspace. This idealization, primarily because ofits simplicity, has been mdely employed to determinestresses and deformations in soils, and its use in soildynamics has led to results in qualitative agreement withobservations. From a practical point of view, perhaps thegreatest value of the model has been in explaining impor-tant features associated v:ith foundation vibrations.

The dynamic impedance functions for a rigid cir-cular foundation on the surface of a homogeneous half-space have been tabulated by Veletsos er al.as and Lucoet al.;ar'aLe'ig. 5 presents their results in the form ofequation (li), ~ith zero hysteretic damping ratio. (Obvi-ously. in this case. I' k and c = c.) The values of k and ccorresponding to non.zero values of internal damping are,for all practical purposes, very similar to those plotted inFig. 5, in accord with the

correspondenceprinciple. Refer-

ence is made to Veletsos er al:, Lucoe and Lysmer fora more detailed discussion on this subject. Notice that onlythe diagonal elements of the impedance matrix are shownin the figure, as the cross swaying-rocking impedance isessentially zero.

It is evident from Fig. 5 that the normalized impedancesK,/GR and KI,/GR, where a refers to thc translationalmodes ti and h and b to the rotational modes r and r,depend only on the Poisson's ratio rp of the halfspace andthe frequency factor ao. The following trends are worthy

~~

~~

of note in Fig. 5.I. The vertical and rocking stiffness, K, and dynamic

stiffness coefficients, k, are the most sensitive to variationsin Poisson's ratio. On the other hand, the horizontal imped.ance function has a small dependence on v, while the

't is noted that although for saturated soft clays under static un.drained loading one should use v=0.50, udrh dynamic loadingi = 0.50 leads io infinite dihiailonal wave velocity, which is noiobserved In the hboratory; instead the Blot-Ishihara theory forporoelastlc media yields a maximum value of v sllghdy less than0.50.

negative for values of ao greater than 2.5 and 5, respec-tively. Some years ago it appeared that use of 'addedmasses'ould adequately account for the decrease withao of the stiffness coefficients, in the range of low fre-quencies. Such 'masses'ould in effect produce dynamicstiffness coefficients of the form k-m~a-a reasonableapproximation indeed for low frequencies, which formedthe basis of the 'lumped. parameter'odel, described ina preceding section of the paper. Unfortunately, as isevident from Fig. 5, this approximation may lead to sub.stantial errors for larger frequencies. Moreover, the conceptof 'added mass'as all too often been confused with the" " 'physically incorrect notion ofan 'in-phase s'oil mass', whichat much earlier times had found considerable use in thedesign practice.

3. While the damping coefficients of the translationalmodes, c, and ch, attain large and nearly constant valuesthroughout the frequency range 0< ao 4 8, the coefficientsc„and c, of the two rotational modes are very sensitive tovariations in frequency in the low range of uo, tending tozero as ao approaches zero. At larger frequencies (ao greaterthan about 3) cr and c, are essentially frequency-indepen.dent, but their values both equal to about 0.30, aresignificantly smaller than the corresponding values ofc,~0.95 and cp,~0.60. These differences imply that a

smaller radiation of wave energy takes place during rockingand torsional than during venical and horizontal oscilla-tions. It seems that the dynamic stress and strain fieldsinduced in the soil by the two types of rotational loadingsare of limited extent, with the generated waves decayingvery rapidly away from the loading area due to 'construc-tive interference'. These phenomena will become moreevident in connection with the behavior of footings onlayered or inhomogeneous soil deposits.

In any case, the practical implication of the existence ofonly a small amount of radiation damping in the rockingand torsional modes ofoscillation is that a realistic estimateof the response may be obtained by incorporating theeffects of material (hysteretic) damping in the soil. On thecontrary, material damping is insignificant for horizontaland, especially, vertical oscillations and, with little loss inaccuracy, it may be neglected in the presence of the muchhigher radiation damping.

12 Soil Dynamics and Earrhquake Engineering, 1983, Vol. 2, Plo. 1

1/2

———1/3 ~~ ~~ e

as

K =~4~v 1-V

0 0

~~ ~ ~

~ ~ ~ ~ ~

osh K =—SGR

h 2-V'W ~ OW ~ ~ ~4A 4 ~ W

0

1

K SGRr 3(1-V )

~—. ~'e„~ ~

~~'

k, os

K 16GRt 3

08 0

Bp

Figure 5. Impedance funcrions of n'gid circular foorings on homogeneous halfspace

4Gp

Soil Dynamics and Earrhquake Engineering, 1983, Vol. 2. iVo. I 13

"v

I

Clearly, soil deposits having a constant G and extendingto practically infinite depths, as the homogeneous halfspacemodel assumes, do not abound in nature. In addition,circular foundations are rather rarely constructed. Nonethe-less, the results of Fig. 5 for a circular foundation on a half-space are of great value in understanding the phenomenaassociated with foundation vibrations. From a practicalpoint of view, however, the shape and trends of theseimpedance functions are more important than their exactvalues.

Rigid strip foundationWhen dealing with long and narrow foundations, the

length of which is larger than their width by a factor of 5or greater, it is a common practice to idealize their shape asan infinitely long strip. If, moreover, the dynamic loading

is reasonably uniform along the longitudinal direction,plane-strain conditions prevail throughout and 2D analysesare sufficient to obtain the response.

Figure 6 displays the dynamic impedance of a rigid stripfoundation on the surface of a homogeneous halfspace.

. These results were obtained by the semi-analytical pro-cedure of Gazetas~ and Gazetas and Roessetse and are inagreement with the results of Karasudhi er al.~~ It is notedthat in this case the impedance functions are presented inthe form described by equation (4), and not in one of themost usual forms of equations (12) or (I7). The necessityfor this change stemmed from the fact that the static ver-tical and horizontal stiffnesses of a'n infinite strip on a

halfspace are zero, in agreement with the classical theory ofelasticity. This is at variance with the behavior of circularfoundations, whose (nonzero) static stiffnesses can be

V

1/2—~——1/3 4=o

i

0 J. l ~J-

CVI

0

0.5 0.5Bp

Figure 6. Impedance funcrions of rigid strip foonngs on homogeneous halfspace

1

Gp1.5

I 4 Soil Dvnamics and Earthquake Engineering, 19B3. Vol. 2. 1Vo. I

I

found from the expressions included in Fig. 5. The infinitedisplacement of a strip loaded halfspace arise from the largedepths of the corresponding 'zones of influence'. In otherwords, the static stresses induced by the strip surface loadsdecay slowly with depth and, thus, cause appreciablestraining of even remote soil elements; accumulation ofthese strains yields infinite displacements.

On the other hand, the stress and strain fields inducedby moment loading are confined to the near surface soilonly,'hereby producing small surface displacements andnon-zero static stiffnesses. For a rigid strip foundation, anexpression for the static rocking stiffness is included inFig. 6.

A few other trends are worthy of note in Fig. 6. First,one should notice that there are only three possible modesof vibration of a strip (vertical, horizontal and rocking) as

compared to the four modes of a circular footing. Appar-ently, torsional osciHations involve out-of-plane motionsand hence are impossible with strip footings.

In general, the dependence of the dynamic impedanceson the Poisson's ratio of soil is very similar for strip andcircular foundations. Thus, the discussion of the precedingsection on the sensitivity of circular impedance functionsto v, is also applicable to the present case.

Regarding the variation of impedances with frequency,on the other hand, there are some differences betweencircular and strip footings, although clearly the generaltrends are similar. Thus, in the very low frequency range,the real parts K,i and Kqt of the two translational modesincrease with"increasing'nd they attain peak values of-ao ranging from about 0.25 to about 1.0, depending pri-marily on the Poisson's ratio and the type of oscillation.This impHes that 'constructive interference'f various Pand S waves originating at the soil. foundation interfacereduces the depth of the 'zone of influence', this resultsinto finite displacements and non-zero dynamic stiffnesses.

Beyond their peak values, K i and Khi behave much liketheir circular coumerparts. Notice, however, that atPoisson's ratios close to 0.50 the vertical strip stiffnessbecomes negaiive at ao values greater than 1.3, as comparedwith the corresponding value of 2.5 which was observedfor circular footings in Fig. 5.

The imaginary parts Kvz and KI,z of the vertical andhorizontal modes increase almost linearly with ao, thusindicating qualitatively similar. radiation damping character-istics of strip and circular foundations. (Notice that thedamping coefficients c in the latter case are proportionalto the slopes of the imaginary component of impedance-versumo curves; hence a constant c implies a linearlyvarying Kz.)

FinaHy, the rocking stiffness and damping terms ofbothstrip and circular foundations exhibit essentially identicaltrends. Evidently, rocking induced static or dynamicstresses influence only the near. surface soil under bothplane. strain and axisymmetric loading conditions.

Rigid re'mngular foundarionResults are now available for the complete dynamic

impedance matrix of rigid rectangular foundations withvarying aspect ratios L/B, over the low and medium fre-quency range." For the vertical, horizontal and rockingmodes, in particular, results are available even for moder-ately high values ofuo.~'~'s

Again, in presenting the variation with frequency andaspect ratio of impedances it is convenient to express themin the form of equation (17), with ao = ~B/V, where 2B

is the width of the smallest side of the foundation. Resultsfor the static stiffnesses are presented first.

It has been known for some time that the static stiffnessof a typical rectangular foundation can be approximatedwith reasonable accuracy by the corresponding stiffness of'equivalent'ircular foundations. For the translationalmodes in the three principal directions (x, y and z) theradius Ro of the 'equivalent'ircular foundation is obtainedby equating the areas of the contact surfaces; hence:

(47)

For the rotational modes around the three principal axes,the 'equivalent'ircular foundations have the same areamoments of inertia around x, y and z, respectively, withthose of the actual foundation. Thus, the equivalent radiiare:

Ro„= (16L Bs/3ir)u

for rocking around the x.axis;

Roy = (16B Ls/3ii)u

for rocking around they-axis; and

1 6B L (B +L )""

Ro =6n

(48)

(49)

(50)

for torsion around the z-axis.The,results of recent parametric studies have confirmed

the similar static'behavior "of rect'angul'ar "and" equivalent'ircularfoundations. Table 2 is a synthesis of the results of

several such investigations. It presents theoretically'exact'ormulae

for aH the translational and rotauonal static stiff-nesses of rigid rectangular foundations having a wide rangeof aspect ratios. These formulae are cast in the form:

K = Ko(Ro).J(L/b) (51)

in which: K= the actual static stiffness; Ko(Ro) = the corre.sponding stiffness of the equivalent circular foundation,obtained from Fig. 5; Ro = the radius of the

'equivalent'ircle;

and J(L/B) = a 'correction'actor, function of theaspect ratio, L/B. IfJ(L/B) were equal to I for aH aspectratios, the static equivalence between the two types offootings would have been perfect. Conversely, the larger thedifference is between J(L/B) and 1, the less accurate itwould be the approximate a rectangular with a circularfooting.

It may first be noted that only smaH discrepancies existin the values of the 'correction'unctions computed fromthe results of several authors. These discrepancies are dueto either the assumed soil.footing interface behavior('smooth'ersus 'adhesive'ontact), or the employeddifferent numerical solution schemes. In practice, however,in view of the small magnitude of these differences, onemay safely use for J(L/B) the average of the values pre-sented in Table 2, for each particular aspect ratio.

The following conclusions are evident from this Table.1. Even for aspect ratios, 1./B, as high as 8, the 'equi.

valent'ircular foundations yield stiffnesses which are

within 3(Fza of the corresponding stiffness of the actualrectangular foundation. This is by no means a large error,in view, for example, of the uncertainty in estimating thesoil modulus in practice.

2. For aspect ratios, L/B, less than 4 the'equivalent'tiffnesses

are in very good agreement with the actual

Soil Dynamics and Earrhquake Engineering, 1989, Vol. 2, Po. 1 15

rp

i

i

Table 2 Sraric sriffnesscs for rectangular rigid foundarion

l. Ver:ical stiffness

K- RK Kv=—JU(L/B)4GR,1-v

'Correction'actor J„

Gorbunov-L Posadov BarkanB (1961) (1962)

Dominguez c ral. (1978)Savidls(1977) 'Adhesive'Smooth'

1.023 0.953 0.944 1.052 1.0812 1.025 0.975 0.973 1.063 1.1304 1.108 1.077 1.072 1.107 1.1966 1.197 1.1528 1.266 1.196 1.200

10 1.313 1.25020 1372

2. Horizontal stiffnesses

K„=—'„(L/B) K =—'(L/B)

8GR, 8GR,2 v y 2 v y

'Correction'actor J» 'Correction'actor Jy

Dominguez DommguezL Barkan er al. Barkan er al.B (1962) (1978) (1962) (1978)

1

468

10

0.993 1.0350.983 1.0441;000

'1.085'.055

1.1321.191

0.993 1.0351.008 1.105, ...„

1.221

3. Rocking stiffnesses

Kr»-—'" J~(L/B) Kr =~ J .(L/B)8GR o» 8GRo'v

» 3(l —v) 3(l-v)

'Coerce:ion'actor Jr» 'Correction'actor J>.

Gorbunov- Gorbunov-L Posadov Dominguez Posadov DominguezB er cl. (1961) er al. (1978) er al. (1961) er cl. (1978)

1

4

810

0.9911.0341.04881.1781.281

0.9651.0391.117

0.991 0.9651.035 1.0311.P72 1 14P1 'l161.319

4. Torsional stiffness

Kr ~Kt = GRor's(L/B)z 3

'Correction'actor JrLB Dominguez er al. (1978) Roesset et al. (1977)

0.9501.0001.0161.166

1.0332

ones. Typically, the error is within IFic and, hence, it isinsignificant for aH practical purposes.

3. The greatest differences are observed between actualand 'equivalent'tiffnesses for torsion (K,) and for hori-

zontal displacement in the y direction (Ky). For L/B = 4,the error in K, is about 17'nd in Ky about 2%c. It isworthy of note that whereas for a circular foundationK»o Kyo SGRo/(2 —v), where Ro is given by equation(47), a rectangular foundation with the larger side 2Lnormal to they-axis (Fig. 4) is characterized by:

/LK ~K +'GB~--I» 2 (52)

for typical values of Poisson's ratio.

Varr'aria with ao. Figure 7 portrays the dependence ofthe dynamic stiffness and damping coefftcients, k and c, onthe frequency factor ao and the aspect ratio L/B. Theseresults were obtained with the Boundary Element Methodby Dominguez and Roesset,42 for a single value of Poisson's

ratio, v=~2. Only the coefficients of the six diagonalcomponents of the impedance matrix are shown, theycorrespond to the translational modes of vibration (x,yand g) along each of the three principal axes, and to therotational modes (r„, ry and r,) around each of the same

three principal axes. The two cross-swaying-rocking (coup-ling) impedances, corresponding to the xry and yr„modes,are negligibly smaH for surface foundations, and are thusomitted from this presentation. Also shown in Fig. 7 as

circles are the predictions of the 'equivalent'ircularfoundations, computed from Fig. 5 in conjunction withequations (47)-(50). One may notice the following trendsi'n ig.

1. The terms k„and c„of the impedance against motionnormal to the smaller side 2B are insensitive to variations inao. Moreover, k„ is essentiaHy independent of the aspect

ratio, L/B, while c„ increases almost in proportion to thesquare. root of L/B. RecaH that c„must be multiplied byao= os/V, to obtain the imaginary component of thedynamic part of the impedance (equations (12) or (17)),in which 2B is the width of the smallest side of the footing.On the other hand, the frequency f'actor aoo of the 'equi-valent'ooting equals oaRo/Vs, with:

u2

Ro= —B- (47a)~sr B

i.e. aoo is proportional to the square-root of L/B. Hence,

plotted in Fig. 7, both stiffness and damping coefttcientsof the 'equivalent'ooting are in excellent agreement withthe corresponding coefficients of the actual rectangularfooting, for aH aspect ratios studied (L/B = I —4}, at least

in the frequency range, 0<ao 41.5.2. The variation of the vertical stiffness and damping

coefficients, k„and c„, has a similar shape with the varia.

tion of k„and c„. In this case, however, the two coeffi~

cients are more sensitive to variations in ao and L/B andthe damping term c„ is always larger than c„. Moreover, theagreement between actual and 'equivalent'oefficients is

reasonably good, for aH practical purposes.3. The coefficients ky and cy, for a motion parallel to

the smaHer side 2B, show a greater sensitivity to both ao

and L/B. Furthermore, the discrepancies between 'equi-valent'nd actual values for these coefficients are appreci ~

able, increasing with the aspect ratio. In fact, footingswith a large L/B ratio (e.g. h4) tend to behave more likestrip rather than circular footings, as a comparison between

Figs. 5, 6 and 7 indicates.4. The stiffness coefficient kr» for rocking around the

longest axis, x, exhibits no sensitivity to the aspect ratio,

16 Soil D> namics and Eanhquake Engineering, 1989, Vol. 2, /Vo. /

LJB; moreover, its variation as a function of ao is nearlyidentical with the variation of the corresponding stiffnesscoefficient of both the 'equivalent'ircular footing and a

strip footing with the same width B (Fig. 6). The dampingcoefficient c,„attains negligible values in the low frequencyrange and increases approximately in proportion to the

fourth-root of LIB at high frequencies. Recalling that thefrequency factor of the 'equivalent'ircular footing is

proportional to:

(4Sa)

1 ~————-20

e ~ ~ 3 Q~ ~ ~ 0 4 6

82

000

0Q

0

0

k, .5

0* «« .'«««e ~r«««

i

Cx

1

Iall L/B ratios

0

0 .Q.

0

Cy8

5'

0

all L/B racios

.5

0 .5 1 0 .5Gp 80

Figure 7. Dynamic coefficients ofrigid rectangular foorings on homogeneous halfspace 7 (circles obtained by this aurhorfor equivalent 'circular foo ring)

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, Eo. 1 17

Figurc 7 - continued

0 0

k, .5

8~I~I~ ~

~~'

I/r'

0 0' g y Ql ~ Q~Qy '4f 'f 0 A '~Wl%H~R NA &A

i .5

.5ap

.5ap

whereas the term c,„ is multiplied simply b> ao = ~B/V, inequations (1") or (17), one can directly unveil the veryclose proximity between the actual and 'equivalent'amp-ing coefficients.

5. The stiffness coefficients k> and k, for rockingaround the shortest axis and torsion, respectively, show asomewhat similar dependence on L/B and exhibit somefluctuations with ao as L/B increases. The two coefficientsare predicted only with small accuracy by the

'equivalent'ircular

footings. On the other hand, the two dampingcoefficients c> and c, grow rapidly with both frequencyand aspect ratio. In this regard, it is interesting to noticethat, for instance, the frequency factor for the rz mode isproportional to:

(49a)

which reveals a much stronger increase of c> with L/B, ascompared with the corresponding increase of c,„(a powerof 4~for c> versus, for c,„). Again, the values of the twocoefficients may be reasonably well predicted by the'equivalent'ircular foundation.

In conclusion, ~t'th the help of the formulae of Table 2and the graphs of Fig. 7, the dynamic behavio of'ect-angular foundations with essentially any aspect ratio can beobtained. Furthermore, the 'equivalent'ircular footingsdescribed through equations (47)-(50), yield reasonablygood estimates of the response for values of L/B less thanabout 4 and frequency factors at least up to 1.5. For largervalues of L/B, the static stiffnesses of Table 2 can beutilized in conjunction with the dynamic coefficient ofan equal. width strip foundation (Fig. 6). More parametricstudies are, however, necessary to obtain results in the highfrequency range (1.5 < ao 4 8).

RIGID SURFACE FOUNDATIONS ON AHOMOGENEOUS SOIL STRATUM

Natural soil deposits very rarely have uniform propertieswithin large depths from the loaded surface. More typicalis the presence of a stiffer material or even bedrock at a

relatively shallow depth. The response of a foundation ona soil stratum underlain by such a stiffer medium can besubstantially different from the response of an identical

18 Soil D~ namics and Earrhouake Ene'ineerine. 1983. Vol. 2. Po. 1

(V

QE

foundation resting on a uniform halfspace. It is, thus,imperative to study the dynamics of massless foundationson such soil deposits. Specifically two types of idealized

soil profiles are considered in this section:

(a) a homogeneous soil stratum over a rigid base, and

(b) a homogeneous soil stratum over a homogeneoushalfspace.

Results for non-homogeneous soil strata, with modulicontinuously increasing or decreasing with depth, will be

presented in a subsequent section of the paper.In addition to the four dimensionless parameters which

control the behavior of rigid footings on a halfspace,namely, ao, v, $ and L/B, the ratio H/B (or H/R) is ofcrucial importance in the response of footings on a homo-geneous stratum. Its effect is, thus, studied throughout thissection. Furthermore, the moduli ratio Gt/Gz is of interestwhenever the soil stratum is underlain by a non.rigid base

(halfsp ace).

Circular foundarion on snarum over a rigid base

Results for the dynamic impedance functions of a rigidcircular disk at the surface of a stratum-on. rigid-base are

presented in Table 3 and in Figs. 8 and 9. SpeciTically,Table 3 offers simple and quite accurate formulae for the

determination of the static stiffnesses; Fig. 8 studies theeffect of the H/B ratio on the dynamic stiffness and damp-

ing coefficients,,k, and. c, for„,a.single value„of,.hystereticdamping ratio, $ =0.05; and Fig. 9 shows the sensitivityof k and c to variations in $ , for a single value of the

ratio, H/B = 2. These results have been derived byKausel and Kausel er al. ~ and have been discussed byRoesset.c" e'everal significant conclusions may be drawnfrom this data.

Static stiffitesses. It is evident from the formulae ofTable 3 that the existence of rigid bedrock at a relativelyshaUow depth may drastically increase the static stiffnessesof a rigid surface foundation. The four expressions reduceto the corresponding halfspace stiffnesses when HIR tendsto infinity,but their values increase with decreasing H/R.

Vertical stiffnesses are particularly sensitive to variationsin the depth to bedrock (notice the 1.28 factor). Hori~

zontal stiffnesses are also appreciably affected by H/R(factor of 0.5) while the rotational stiffnesses (rockingand torsion) are the least affected. In fact, for H/R)1.5the response to torsional loads is practically independentof the layer thickness.

An indication of the causes of this different behavior ofa circular footing to the four different types of loading can

be obtained by observing the depths of the 'zone of influ-ence'known as 'pressure bulb'ver since Terzaghi) in each

case. Thus, from Gerrand and Harrison,~ in a homogeneous

halfspace, the vertical normal stress, tr„along the centerline

of a vertically loaded rigid circular disk becomes less than

10% of the average, applied pressure at depths greater than

z,a 4R; the horizontal shear stress, r„, becomes less than

IÃ/o of the average applied shear traction at depth greater

than ghee 2R. From Gazetasp the horizontal shear stresses

r,8 and rP due to linearly distributed torsional surface

stresses become less than 10%%ua of the maximum appliedshear traction at z)zt ~ 0.75R. Finally, moment loadingwith a linear distribution of normal tractions varying from0 to p yields zr~ 1.25R, below which a, is less than

0.10p.

Vanarion with ao, H/R and P. The variation of the

dynamic stiffness and damping coefficients with frequencyreveals an equally strong dependence on H/B. On a stratum,both k and c are not as smooth functions as on a halfspace,but exhibit undulations (peaks and valleys) associated withthe natural frequencies (in shear and dilation) of the soillayer. In other words, the observed fluctuations are the

outcome of resonance phenomena: waves emanating fromthe oscillating foundation reflect at the soil.bedrock inter-face and return back to their source at the surface. As a

- result, the-amplitude&-foundation-*motion may signi--: -"ficantl increase at specific frequencies of vibration, which,as shown subsequently, are close to the natural frequenciesof the deposit. Thus, the stiffness coefficients exhibitvalleys which are very steep when the hysteretic damping inthe soil is small (in fact, in certain cases, k would be exactlyzero ifthe soil were ideally elastic); on the other hand, withlarge amounts of hysteretic damping ($ = 0.10-0.20) thevalleys become less pronounced (Fig. 9). They also become

less pronounced as the relative thickness of the layer, H/R,increases (Fig. 8).

Another important phenomenon is revealed through thevariation with ao of the damping coefficients. At low fte-quencies, below the first resonant frequency, radiationdamping is zero. This is due to the fact that no surfacewaves can be physically created in a soil stratum at such

frequencies and, since the bedrock prevents waves frompropagating downward, geometrical spreading of wave

energy is negligible. The small values of the damping in thisrange (Fig. 9) just reflect the energy loss through hystereticdamping; for a purely elastic soil c would be zero

Table 3. Static stiffnc'stet ofrigidcircular foundation on a stratutn-over rigid-base

Type of loading Static stiffness Range of validityt Soil profile

Vertical: Kv =—1+ 1.28— HIR >2

Horizontal:

Rocking:

Kh =—1+—„ HIR > 1

4>HIR >1

H. G,v

Torsion: Kt = —GR'6t HIR > 1.25

'dapted from Kausel" and Kausel ct al.'4

t For HIR < 2 or 1 these expressions would still provide reasonable estimates of the actual static stiffnesses

Snil Dynamics and EartItttuaI;e Engineering. 1983, Vol -. P'o I

I

1'l

Analysis ofmachine foundation vibrarionst state of the arts G. Gazetas

oooooooooo CO

34

)H/R

k„.5

ooo ooo ~ doing~ ooo oo

j

0

n

oo+V<< i ooooooo

Ch

eooooo~rP I X /

=0 0

h,.5 Og~ oooo 0 0 op ~ +oooo

0

'

.3 0> ++ ~

.5

0

Bp apFigure 8. Dynamic coefficients ofrigid circular footing on srratumwver-bedrock; effecr ofH/R ratio (v = 1/3,

0.0a) s3, sg w

The phenomena described in the two preceding para-graphs are observed to a larger or lesser degree in all fourmodes of vibration. However, there exist marked differ-ences among the dynamic coefficients of vertical, swaying,rocking and torsional osciUations. Specifically:

1. For rocking and torsion, k and c are relatively smoothfunctions ofao, rapidly approaching the corresponding half-

space curves as the layer thickness increases beyond 3R.Thus, H/R exerts only a small influence on the variation ofthese two coefficients. On the other hand, for vertical andhorizontal translation, k and c display some very pro-nounced fluctuations with ae. Both the location and theshape of the resonant valleys are quite sensitive to varia-tions in H/R, and only for H/R values larger than 8 do

5)020

i, 5

0

{ M h ~ ~A~& ~'l~l IW% A~~. C W ~

k„.5

0

k,~ ~ ~ ~ c

.3

.5

0 0

Bp pFigure 9. Dynamic coefficients ofrigidcircular footing on strarumwver-bedrock; effecr of $ (v = 1/3, H/R = 2) '

k(a<) and c(ao) approach the corresponding halfspacecurves, if $ = 0.05. These results are consistent with theconclusions derived previously regarding the depth of the'pressure bulb'r 'influence zone'f a statically loadedfoundation. Under dynamic loads, 'constructive interfer-ence'f downward propagating waves leads to a shallowdynamic 'pressure bulb'in both rocking and torsion.

2. The resonant frequencies of horizontal (swaying)oscillations are in remarkable agreement with the natural

frequencies of the stratum. As an example, the funda-

mental frequency of the stratum in vertical shear waves,

f,, equals V,/4H and, thus:tr R

(53)~Hwhich is equal to m/4, for HJR = 2. As seen in Fig. 9,this value of ae essentially coincides with the firs resonantfrequency in swaying. It is not difficult to explain how the

Soil Dynamics and Eanhquake Engineering, 1983, Vol. 2, Eo. 1 21

simple one-dimensional wave propagation theory can sosuccessfully predict the first resonant frequency of a three.dimensional problem: at values of ao below resonanceessentially only shear waves exisr in rhe srrarum, propa-gating verucaUy between foundation and bedrock. There-fore, when this first resonance occurs we have a none-dimensional 'standing'ave and, in addition, little damp-ing and thus high response. Of course, as it may be inferredfrom Figs. 8-9, the situation becomes a little more involvedat higher resonant frequencies. Thus, the second 'reson-ance'ccurs at about the fundamental natural frequencyof the stratum in dilational waves, and the third 'resonance't

about the second natural frequency in shear waves. Inboth cases, however, some non-vertical waves also partici.pate in the motion, as evidenced by the existence of non-zero radiation damping. Due to multiple wave reflections,P, S and Rayleigh waves are also generated and, hence, theonedimensional theory predicts with smaller accuracy thepertinent swaying resonant frequencies of the soil-founda-tion system.

On the other hand, vertical and rocking foundationoscBlationsinduce mabrly P bur also S waves in the srrarum.The relative importance of each type of wave depends tosome extent on the Poisson's ratio of the soil. Recall thatthe ratio between the two wave velocities and between thecorresponding natural frequencies of the stratum is given by:

V f )(] v) u2

n = 1, 2, 3,... (54)V, f, „ I-2v

t.

which, for v= ~3 yields a ratio of 2. Figures 8-9 clearlyshow that the first resonant frequencies for both verticaland rocking osciUations are reasonably close to the funda.mental frequency of the stratum in vertical P-waves

(aop t =r/ forHIR = 2). Higher resonances, however, canhardly be predicted by the simple one-dimensional wavepropagation theory since, apparently, they involve a mix-ture ofP-, S- and Rayleigh (R) waves.

Referring to Fig. 9, it is observed that k and c are quitesensitive to variations in material damping, especially atfrequencies near resonance. This is contrary to the so called'correspondence principle'hich assumes that the imped.ances derived for an undamped but otherwise identicalmedium by a simple multiplication with the factor1+21). Remember, however, that this 'principle'orksreasonably well for a homogeneous halfspace.

The effect of Poisson's ratio is not studied in detailherein and reference is made to Kausel er al.a9 for arigorous assessment of its importance in swaying androcking. Note, nonetheless, that the variation of the dyn-amic coefficients with frequency may be sensitive to

this parameter, because of its influence on Vp and fp„as previously explained (equation (54)). Thus, vertical androcking coefficients are highly sensitive to v, especiallywith shaUow layers; but swaying and torsional coefficientsare practically independent ofv.

Strip foundation on slrarum over a rigid base

Table 4 and Figs. 10 and 11 present the results forvertical, horizontal and rocking osciUations of a massless

rigid strip footing which rests on the surface of a homo-geneous soil layer overlying bedrock. These results wereobtained with the formulation of Gazetas and Roesset'"

'nd

are h exceUent agreement with the results of Chang-Liang.a7 Additional numerical studies can be found inJakub er al.~'6s and Gazetas.~

Sraric behavior. Simple expressions ofsufficient accuracyfor practical purposes have been derived for the three staticstiffnesses and these are listed in Table 4. Evidently, thepresence of (infinitely rigid) bedrock at shallow relativedepths has a dramatic effect on the static behavior of stripfoundations. Vertical and horizontal stiffnesses, being nolonger zero as in the case of a halfspace, are strongly in.creasing functions of B/H. Rocking stiffness also increaseswith BIH. Two noteworthy conclusions may be drawn bycontrasting the expressions of Table 4 to those of Table 3:

1. The effect of Poisson's ratio on the static stiffnessesis the same for both strip,and„circular rigid foundations..The effect is greatest for vertical and rocking loading[factor (1 —v)] and smallest for horizontal loading [factor(2 —v)].

2. Layer depth is substantiaUy more important for stripthan for circular foundations, especially with the two trans-lational modes (factors of 3.5 and 2 in the vertical androcking expressions for a strip, as compared with 1.28 and0.5 in the corresponding expressions for a circle). This is a

natural consequence of the much deeper 'pressure bulb'na cominuum subjected to plane-suain rather than axi-symmetric surface loading, as it has already been iUustratedin preceding sections.

3. Vertical stiffness is far more sensitive to variations inBIH (factor of 3.5) than horizontal and rocking stiffnessesare (factors of 2 and 0.20, respectively). The explanationlies again in the much greater 'depth of influence'f thevertical loads. On the other hand, moment loading inducesstresses which decay very rapidly with depth; because onany horizontal plane, small normal stresses at largedistances from the centerline contribute much to equi-librating the applied moment. Thus, rocking stiffnessesextubit about the same smaU sensitivity to layer depth for

Table 4. Srarie sriffrresses ofrigid srrip founderion on a errarum~ver-rigid. base

Type of loadingStatic stiffness

(pcr unit length) Range of validity~ Soil profile

Vcrticah Ku = 1+35 1 (HIB (10

Horizontal:

Rocking:

2.1GI B 4Kh =—'(1 + 2 -)2-.l H J

1 (HIB (8

1 (HIB (3

Outside this range the proposed expressions would still provide reasonable estimates of thc actual static stiffncsscs

22 Soil Dynamics and Earrhquake Engineering, 1983, Vol. 2, ¹. J

nn

also obvious.Qne first notices in

nf'edrock both the

respectively).

~ ~

ERRATA

Anal'ts of 'Machine Foundation Vibrations: State of the Art. Soil DInamics andYo. l by George Gazetas.

Figs. 10-11 that due to the presencein-phase (real) and the 90'-out-of-

~ant (r n|TI

Earthquake Engineering, i9S4 Vol

ln Fig. lo. page 23. there is a mistake in the ordinates where the shear modulus G has been inadvertedly used In place of'I'oung's modulus E l =3G). The correct Figure is shown below:

3

0

I Ii I

IXt

E.,

IN-PHASE COHPOXEXTS

I II

"I ...I

l

—90 -OUT-OF-PHASE COHPOXEXTS

I I

N~SAWVt~ WW ~ WbA W, %J' l~~NA44 Vl

2

IIIII

'

g ~

I 'I.

I

IiIII

I

I

I-i' ~

cv QoLQ

~ /

8 p ao

Fig. IO. Compliance fiuIctions of rigid strip fooring on stratum-orer-bedrock: equi'ecr of HIB ratio (v=0.49. ~ =00-)

Also. in Table S. page 32. two brackets have been erroneously omitted in the expression for the vertical static sti<nesPThe correct expression is:

—l —: i.2S — I —— i+ O.S5-0.28-

a-

~ ~ g V) ~ ~ h ~ ~ ~ Oh gh ~ & ~

both strip and circular footings (factors of 1/5 and 1/6,respectively).

Dynamic behavior. Figures 10 and 11 portray the varia-tion with frequency of the dimensionless compliance func-tions GF~ where a = t/ or h, and GBsF,. Specifically, Fig.10 intends to show the effect ofH/B, and Fig. ll the effectof v. The results of Fig. 10 were obtained for v = 0.49 and

P = 0.05, with four different values of H/B, i.e. 1,3, 8 and~; the hst value corresponds to the homogeneous halfspaceand is included for a comparison, Figure 11 shows theeffect of v on vertical and rocking compliances only, for a

layer with H/B=2 and a homogeneous halfspace; theeffect of v on swaying, being of secondary importance, isnot studied herein.

The same general trends observed in the dynamicbehavior of circular foundations can now be seen in the

response of strip footings, although some differences are

also obvious.One first notices in Figs. 10-11 that due to the presence

of bedrock both the in.phase (real) and the 90'-out-of-phase (imaginary) components of displacement (com-pliance) are not smooth and monotonically decreasing

functions of frequency, as on a halfspace. Instead, theyexhibit peaks and valleys at frequencies related to thenatural frequencies of the stratum. Note that, in general,the peaks of a compliance function correspond to valleysin the impedance function.

The major differences between strip and circular founda-tions stem from the much greater sensitivity of the verticaland swaying oscillation of a strip to variations in H/B.Even for H/B=g, relatively high amplitude peaks are

observed in the two compliance functions of the strip,for the case $ = 0.05; their difference from the halfspace

H/Bco

31

TN-PHASE COMPONENTS

II"I...1

1

90 -OUT-OF-PHASE COMPONENTS

0

r,/h/

II/III

I

l'II

0

ap ap

Figure 10. Compliance funcrions ofrI'gr'd srrip foonng on srrarumwver.bedrock; effecr ofH/B rario (v = 0.49, $ = 0.05)

Soil Dvnamics and Earrha//ake E/r/i/Ieerins. 1983. Vol. 2. Nn. 1 23

compliances is substantial. On the other hand, rockingvibrations of a strip exhibit very similar trends with rockingof a circular plate; beyond H/R = 3 the presence of bed-rock is hardly noticeable.

In the case of vertical loading, the resonant peaks are notas sharp as those of the horizontal displacements. In fact,on very shallow deposits (H/B = I) only a single flat reson-ance takes place, which is characteristic of a highly dampedsystem. A possible explanation of such a behavior has beensuggested by Gazetas and Roesset': at frequencies belowthe first resonance some 'leakage'f energy occurs in theform of laterally propagating P-, S- and R-waves. Evidencein favor in this explanation comes from the fact that thefirst resonant frequency, ae„ lies in between the funda-mental natural frequencies of the stratum in vertical S-waves, ae„„and in vertical P-waves, a>,. For example,Fig. 11 shows that, for H/B=2 and v=0.40,a<,~1.30compared to aug f 0.785 and aep i 1.90. Recall thatfor the circular foundation a<, was much closer to aep

No extensive numerical results for rigid rectangularfoundations supported by a soil stratum have been foundin the literature.

Foundarion on srrarum over a halfspaceThe homogeneous halfspace and the stratum. over-rigid-

base are two idealizations of extreme soil profiles. A moregeneral soil model, the stratum-over-halfspace, is studied inthis subsection. Besides the H/R or H/B ratio, the moduliratio G,/G> is needed to describe such a soil model WhenGi/Gi tends to 0, the stratum. on-rigid base is recovered;when it becomes equal to I, the model reduces to a homo-geneous halfspace. Thus, the results presented in thissection help in bridging the gap between 'halfspace'nd'stratum'olutions to which we have restricted our atten-tion unt0 now (Figs. 5-11).

Numerical solutions for a uniform layer over a halfspacehave been published by Hadjian and Lucos'ho studiedthe dynamic of circular foundations, and by Gazetas andRoesset'" who studied the response of strip footings.

Based on the results provided by Hadjian and Luco,s7the author has derived simple but reasonably accurateformulae for the static stiffnesses of a rigid circular disk, interms of H/R and G,/G~. Table 5 displays these formulae,which are valid for the usual case in which G, CG>, i.e.a halfspace stiffer than the layer. At the lower limit,G,/Ga~0, these expressions reduce to those of Table 3for a layer-on-rigid-base; at the upper limit, G,/Ga =I,the halfspace expressions of Fig. 5 are recovered. At inter.mediate values, as the rigidity of the supporting halfspacedecreases, the static stiffnesses of the foundation decrease,apparently due to increasing magnitude of strains in thehalfspace. The results are intuitively obvious and need nofurther explanation.

For circular footings, no results are presented here ondynamic stiffness and damping coefficients, but referenceis made to the original publication by Hadjian and Luco.s7

The variation of the dynamic compliances of a stripfooting with ap and G,/G> is portrayed in Fig. 12 for alayer with H=2B, v=0.40 and )=0.05. Shallower aswell as deeper layers have been examined by Gazetas andRoesset.'"'~

An inspection of Fig. 12 indicates that the effects oflayering increase with increasing contrast between Gi and

..G~;. these„effects are. extreme,fot:.alayer. on. rigid. bedrock(G,/G, =0) and, naturally, disappear in the case of ahomogeneous halfspace (G,/Ga = I). There are two maineffects of increasing the softness of the halfspace. First,even for small positive values of Gi/G., i.e. as long as wedo not deal with an infinitely rigid bedrock, the statictranslational displacement tends to infinity, although at a

.4G ~vi

.4

0

8 G B,3-~1 .8

4GB Q

8 p ap

Figure 11. Compliancefuncrions ofrigid strip fooring on srrarumwver-bedrock: effecr ofv (H/B = 2, $ = 0.05)

24 Soil Dynamics and Earrhquake Engineering, 1983, Vol. 2, Po. I

e

Table S. Sranc sriffncsscs ojcircular foundations on asrrarumwvcr.halfspacc'ype

of loading

Vertical:

Horizontal:

Ro*ing:

Static stiffness'

1+ 1.28—4GQ H

1-v, R G~1+1.28 —~

'1R8GR 2H2-v, 1R G,1+--

-'HG,

1R8G,R'H3(1-vs) 1 R G,I

6HG,

Range of validity

H1<- <5R

H1<-<4R

H0.75 < —<2

R

Profile

R

G, ~

G,

0< —<1G,

G,

'erived by the author on the basis of results provided by Hadjian and Luco"

much slower rate compared to the halfspace displacements.Thus, in the very low frequency range the in-phase (real)components of the displacement's (compliances) are largerthan in the case of rigid bedrock.

On the other hand, at any specific frequency, the radia.tion damping of the system increases due to partial trans-mission of body-waves in the halfspace and the existence

of suiface waves ai all frequencies. Consequently, theresulting variation of displacements with ao is smootherthan in the rigid rock case.

The effects of decreasing stiffness and increasing radia-

tion damping are of major importance at frequencies equalto 'or lower than the first resonant frequencies of thesystem. With G</Gz ascending from 0 (rigid bedrock)towards 1 (homogeneous halfspace), the aforementionedresonant peaks become shorter and flatter and the corre.sponding resonant frequencies shift to lower values.

Higher resonant peaks also decrease substantially and

may in some cases be completely suppressed. An example:the third resonant peak in swaying (which, we recall, occursat the second natural frequency of the stratum in S-waves)disappears as soon as G>/Gz exceeds 0.10.

Finally. it is hardly surprising that the vertical dynamiccompliances are most sensitive to variations in G>/Gz,while rocking compliances are least sensitive. The conceptof a 'dynamic pressure bulb'roves again very convenientin explaining these differences. The depth of the

'bulb'ttains

relatively large values in case of vertical vibrations,somewhat smaller values for. swaying and very small valuesfor rocking.

SOME RESULTS FOR RIGID SURFACEFOUNDATIONS OF 'ARBITRARY'HAPE

Only a few numerical results are available for foundationshaving 'arbitrary'eometries, i.e. plan shapes other thanstrip, circular or rectangular. One reason for the lack ofinterest is that foundations of such 'arbitrary'hape are notconstructed very frequently. Moreover, substantial corn.putational effort must be expended to obtain dynamicsolutions for such foundation geometries. The followingpresentation is divided into two parts: one dealing withvertically loaded footings of various 'solid'hapes and onewith the complete response of annular footings.

Verrt'cally loaded foundarions ofvarious 'solid'shapes

Analytical expressions for the static stiffnesses of rigidfoundations supported to an elastic halfspace and havingseveral different shapes (but without internal holes) can be

derived from the results of Borodachev~ (see also Selva-

duraiv). It is convenient to cast these expressions into ourfamiliar.form:

4GRoKv = 'Jv (55)I-v

in which: Ro= ~A/m is the radius of the 'equivalent'ir-cular foundation, A being the area of the soil-footingcontact surface; J„ is a shape-depended correction factor,numerical values of which have been tabulated in Table 6

for numerous plan shapes.Table 6 in conjunction with Table 2 (part 1) can be used

for determining the vertical static stiffnesses of a variety offoundations with very good accuracy. Moreover, the follow-ing trends are worthy of note:

1. The circular disk yields the smallest stiffness of allfootings with a given contact area.

2. Of all rigid footings with an n.sided polygon shaped

phn of a given area, the reguhr n-sided polygon yields thesmallest stiffness.

3. 'Ihe correction factor depends primarily on the'aspect'atio of the foundation, being surprisingly insensi.tive to the details of each particular shape. By

'aspect'atio

we somewhat loosely mean the ratio between largestand smallest critical foundauon dimensions. Thus, forexample, a rhombus, a rectangle and an ellipse having thesame aspect ratio, equal to 4, yield very similar correctionfactors ofabout 1.12.

In conclusion it seems that, by means of equation (55)and Tables 2 and 6, very good estimates can be routinelymade of the vertical static stiffnesses of arbitrary-shapedrigid foundations on homogeneous halfspace.

No information is available regarding the variation withfrequency of the dynamic stiffness coefficient k„. However,

inspection of Figs. 5 and 7 reveals that the'equivalent'ircular

footing can successfully predict the actual k„ofrectangular footings with aspect ratios up to 4, at least inthe low and medium frequency range (ao 4 1.5). Hence,

and in view of the observed insensitivity of the static stiff-

,C'nn fivnnmirs meed Earihauake Eneineerine, l983, Vol.. h'o. l 25

w vg I isaiah'gvunuallvn vlarallolts: state oj tne art: G. Gazetas

.4

GiK

G) /G20

0.0 6---—-- 0 25~ ~ r ~ ~ ~ ~ o ]+0

4~ ~

I% ~~

.8 .8

,4

.82

GiB g„i.8

G B2+

4

0 2Bp Bp

Figure 12. Effecr of G,/Gs ratio on compliance functions of rigid strip foorings on soil layerwver-halfspac'e (H/R =2,v =0.40, $ =0.0'

ness to the details of the foundation shape, it is proposedthat the variation of k„with ae for an 'arbitrary'-shapedfoundation be estimated from Fig. 5 using the

'equivalent'adius,

Re = ~A(n.On the other hand, the damping coefficient c„ is practic-

aUy independent of frequency, as it is evident from Figs. 5and 7. For an arbitrary-shaped foundation, moreover,Dobry et al." have recently derived expressions for the(radiation) damping coefficients in vertical and swayingvibrations, based on simple but realistic physical approxi-mauons. For the vertical damping coefficient of a surfacefoundation their expression reduces to:

O.g5cv = (56)

Jv

in which J„= the shape correction factor to be read fromTable 6 or Table 2. Consequently, the vertical dynamicimpedance of an arbitrary-shaped rigid foundation on a

homogeneous halfspace can be directly and reliably esti-mated using the provided information.

For the other translational and rotational modes ofvibration of arbitrary. shaped rigid foundations, much less

information is presently available. The'equivalent-circle'pproximation

appears to be a simple and reasonable choice.

Rigid annular foundation on soi! srratum

It appears that the conclusions of the preceding sub-

section cannot be extended to foundations containinginternal holes, like annular and crossed. beam foundations.

CitdcRcydar hexagonSemicircleEquihtctal triangleTriangle with angles, 45', 45',

90'rianglewith angles, 30', 60',90'llipsewith a/b = 2,

Ellipse with c/b = 3Ellipse with a/b = 4El!ipse with c/b = 6Rhombus with an angle of

60'hombuswith an angle of45'hombuswith an angle of30'ectanglewith L/B =2

Rectangle with L/B = 4Rectangle with L/B = 8

1.001.011.051.071.101.121.031.071.131.211.071.141.271.031.131.23

Table d. Values of shape depended concction factor for verticalstatic stiffn

esses'hape

of foundation phn

sensitive not only to Ri/R but to H/R as well. An example:increasing Ri from 0 to 0.95R reduces K„ to 70% of itsoriginal value for H/R =2; for a halfspace the corre-

sponding value is 77%. But, again, for values ofRi/R up to0.5, K, remains practically equal to its original value,

4GR(1+1.28R/H)/(I -v).Figure 14 depicts the variation with ao of the dynamic

stiffness and damping coefficients, k and c. Four values ofRi/R are considered, 0, 0.5, 0.8 and 0.90, with the firstvalue corresponding to a solid circular foundation. It is

clear that: (I) there is little change in k and c with Ri/R;(2) the effect ofRi/R is largest for vertical vibrations; and

(3) the differences in the four sets of curves occur in the

high frequency range (ao 0 1. 5).

'ased on Botodachcv'"~

t a, b are the mslot, minor axes of the clUpsc

For example, the vertical static stiffness of such founda-tions does not increase in proportion to the square-root ofthe contact area, /I, as equation (55) implies. In otherwords, . e -equivalent-circle'pproximation is no longervalid

O

V

C

tcX

HIR =m

3

5 .5

Resu, 'ie static displacements of a rigid circular i

! ring on ." have been Published: by Egorov and Figure 13. Sraric sriffnesses of a rigidannularfoundarion96

".-'. —.*Dhawan":—"—'l loading by=Dhawan for moment~~ IR - -- --; = pcs;s7—jTci = infernal raa'tus)

loading; and ~ awan~ for torsional loading. Wong andLuco studied the dynamic vertical response of a rigidsquare foundation with a square internal hole. Recently,Tassoulas~ presented a comprehensive parametric investi-gation of the dynamic behavior of rigid circular-ringfoundations on a homogeneous stratum-over-rigid base.

All modes of vibration were considered and the effect ofthe dimensionless parameters Ri/R, H/R and ao = os/Vswas grapiucaliy illustrated. The following discussion is

2

tt„1

C„

s. tI:1,'III

based primarily on the results of Tassoulas,~ although someresults from Dhawan,~7 ~ are also included for comparison.

Figure 13 plots the variation of all static stiffnesses of a

circular ring versus Ri/R, where Ri is the internal radius. Asexpected, all stiffnesses invariably decrease as the size ofthe hole increases, while the radius R remains constant. Inthe limit, when Ri becomes equal to R, the sufinessesvanish (concentrated ring load). However, the sensitivity ofstiffnesses to increases in the Ri/R ratio is surprisinglysmall. Particularly insensitive are the rocking and torsionalstiffnesses. For values of Ri/R up to 0.50, they are prac-tically equal to the corresponding stiffnesses of the circularfoundation with radius R; for Rt/R = 0.95, K, and K, are

respectively equal to 86% and 83%, of the circular stiff-nesses in torsion and rocking (while the contact area has

been reduced to only 10% of the original circle). Theexplanation is rather obvious: the large shear or normalstresses which develop near the outside edge of the footing,i.e. at large distances from the center, contribute substan.tially to equilibrating the applied torsion or rockingmoments. In other words, the central foundation 'core's

'underutilized'nd, hence, its 'removal's of little conse-

quence. Notice also that the variation of K, and K, withRi/R is independent of H/R —a result consistent with theshallow 'pressure bulb'f moment loading discussed inpreceding sections (e.g. Table 3).

The horizontal stiffness is only slightly more sensitive

to Ri/R. In contrast, the vertical stiffness is relatively

kh1$

,4

../> i;!Ib

I

II

: I

,'..r'Cr

k,.75

.52 4 0 2 4

Figure 14. Dynamic coefficienrs ofa rigiannular founda-non~

Soil Dynamics and Eanhquake Engineering, 1989, Vol. 2, Po. 1 27

C

F'

THE INFLUENCE OF INHOMOGENEITY,ANISOTROPYAND NONLINEARITYOF SOIL

The results presented so far have been based on the simpli-fying assumption that the soil can be modeled as a homo-geneous, isotropic and linearly visco.elastic stratum orhalfspace. However, real soil strata frequently increase inrigidity with depth as a reflection of the increase in over-burden pressure, while in some other cases weatheredcrusts, in which rigidity decreases with depth, overlaydeposits of softer clay. Furthermore, laboratory tests showthat soils deform differently in the vertical and horizontaldirections —a manifestation of anisotropic fabric acquiredduring natural formation and subsequent loading. Finally,when subjected to large enough stresses, soils respond asnonlinear and inelastic materials.

This section of the paper presents characteristic resultsand important conclusions from a number of recent studiesaimed at assessing the influence of soil inhomogeneity,anisotropy and nonlinearity on the behavior of dynamicaiiyloaded surface foundations.

Effect ofsoil inhomogeneityExisting dynamic finite-element formulations can easily,

albeit approximately, simulate a continuous variation of soilproperties, by dividing the deposit into a number ofhomo-geneous layers of increasing or decreasing stiffness. Yet,such formulations have not been adequately exploited to

- parametrically study the'dynami'c behavior of foundations.Thus, most of the available solutions have been derivedusing analytical and semi-analytical methods.

Numerous studies have been published for the verticalstatic problem. Prominent among them is the work ofGibson and his co-workers '~' who studied theresponse to arbitrary surface loads of a halfspace or stratumwhose moduli increase linearly with depth, i.e. in the formG=Go+m(/R). where Go and m are the moduli at the

halfspace, regardless of foundation geometry. The surfacesettlement, on the other hand, being quite sensitive to theassumed soil profile, becomes directly proportional to theapplied normal pressure when Go = 0, independent of thesize and shape of the loaded area and of the thickness, H,of the soQ layer on a rigid but frictionless (smooth) base.Thus, such a soil behaves like a Winkier medium rather thana homogeneous halfspace, its spring constant being simplyequal to 2in/R. Expressions for the vertical static stiffnessesof surface foundations of several shapes supported by sucha soil deposit (frequently referred to as 'Gibson soil') areshown in Table 7.

'Hiis behavior remains only qualitatively true whendrained soil behavior is taking place (i.e. y(0.50). Thus,with increasing degree of inhomogeneity (e.g. increasingm) normal and shear stresses affect the soil at greatervertical and lesser horizontal distances, in agreement withintuition that expects stiffer material to attract largerstresses. On the other hand, surface displacements, beingmoderately sensitive to v, do tend to become propor-tional to the applied local pressures as m increases. It is,thus, generally concluded that an inhomogeneous depositleads to more uniform distribution of stresses under rigidfoundations than the simple elastic theory (homogeneoushalfspace) predicts.

This general behavior of vertically loaded surface founda-tions on an inhomogeneous soil deposit has been recentlyshown to be applicable to torsionally loaded circular—footings.vs'he

static and dynamic vertical, horizontal and rockingbehavior of a rigid strip foundation supported by a half-space or a stratum whose wave velocities increase linearlywith depth, has been studied by the author.~ Some resultsof that study are presented here for a halfspace consistingof soil with a constant mass density, a constant Poisson'sratio, v = 0.25, and a constant hysteretic damping, $ = 0.05,and an S-wave velocity varying with depth according to:

surface and at a one. radius (or one-semimdth) depth. Thesestudies revealed that for an incompressible medium, i.e.with Poisson's ratio of 0.50, the stress distribution is hardlyinfluenced by the degree of inhomogeneity; in the parti-

Ve=Vp I+X— (57)

cular case of zero surface modulus (Go = 0) this distribu-tion is identical with the distribution in a homogeneous

in which: Vo=surface velocity; 2B= foundation width;and X = the dimensionless rate of inhomogeneity.

Table 7. Sian'c zriffnesses ofrigid foundarions on /nhomogeneous and cross~nisorropicsoils'ype

of loading

Vertical, on foundation ofany shape

Verncai, on rigid strip

Horizontal, on rigid strip

Static stiffness

m2 —A

BA = contact area

A = contact area

45Zy

1 ~ 3g [4 nl (i/6)(H/B)(

HsBJ

5B

5 4.10-n(H/B)4'4

Range of validity

Undrained loading conditions

H1 c- c4

B

03 cn c28

H1 c-c6

B

05 c n c 2.5

Soil profile

Cross-anisotropic'Gibson'alfspace

obeying equation (60),with a modulus GyH = m (=/B)

General cross~tropic'Gibson'alfspace

(i.e. noi obeyingequation (60)) with a modulusGyH = m(./B)

Shallow cross-anisotropicundrained layer, soil propertiesare uniform throughout thelayer and they satisfyequation (60)

H

'ased on results by Gibson" and Gazetas"

28 Soil Dynamics and Earrhauake Engineerine. 1 983. Vnl. 2. iVn. l

L

0

Figure 15 portrays the dependence of X of the normal-ized vertical, horizontal and rocking stiffnesses. As onemight expect, the vertical stiffness exhibits the largestsensitivity to X and the rocking stiffness the smallest-another manifestation of the difference in the 'pressurebulbs'f the three types of loading.

The effect of soil inhomogeneity on the three dynamiccompliance functions is shown in Fig. 16. Two values of theparameter X are considered: 0 and 1.5. The former valuecorresponds to a homogeneous halfspace, the wave velocityof which, V,ff was selected to be the same with the wavevelocity of the inhomogeneous halfspace at a depth equalto the foundation halfwidth, B; i.e.:

V,tr = Vo(I+X) (58)

The choice of such a homogeneous halfspace for thecomparison has been motivated by the frequent use inpractice of solutions developed for homogeneous soils,with an effecnve modulus equal to the actual modulusat a depth equal to B or R, to approximate the actualresponse.

It is evident from the comparison of Fig. 16 that, in thelow frequency range examined, th» inhomogeneousmedium yields vertical and horizontal displacements (bothin-phase and 90'-out-of-phase components) which are,indeed, of about the same average level with those of the'equivalent'omogeneous halfspace. However, the rockingmotions on .the inhomogeneous deposit are seriouslyunderpredicted by the chosen homogeneous halfspace

4 M

v IIV

3f

«IICl

G < q'IVI ~P)

LS LS I I I

Figure 15. Static stiffnesses of a rigid sm'p foundanon onan inhomogeneous halfspace (v = 0.25 jss

model; to yield comparable rotation levels the two media'usthave the same moduli at a depth of about B/2, or

somewhat less.Furthermore, a substantial difference between the

'X =1.5'nd 'X = 0'ompliance functions may be noted.Namely, the former are not smoothly varying functions ofao, as are the latter, but exhibit peaks and valleys which are

apparently the result of resonance phenomena. In the verylow frequency range the imaginary components of the'X = 1.5'ompliances attain quite small values, increasingalmost linearly with ao.

These phenomena are reminiscent of the dynamicbehavior of foundations supported by a stratum-over.a-rigid-base (Figs. 10-11). In this case, total reflection of thedownward propagating waves is possible due to the increas-ing soil velocity with depth. A discontinuity in velocity isnot necessary for such a reflection, since the wave rays ininhomogeneous media with linear velocity profile are notstraight lines but circular arcs. As a result, however, theresonant peaks on inhomogeneous soils are very flat andthe radiation damping is never zero. In contrast, thepresence of stiff rock. like material at some depth beneaththe surface leads to very sharp and pronounced displace-ment peaks, occurring at well separated frequencies (see

Fig. 8-11).

Deposits with a weathered crust. The dynamics of a rigidstrip foundation on an idealized soil deposit consisting ofa homogeneous stratum or halfspace overlain by a topstiffer layer m which the shear modulus decreases as a

" '

seconddegree parabola (Fig. 17) has been recently studiedby the author." Also recently, Rowe and Booker,' pre.sented comprehensive parametric results pertaining tovertical static uniform loading, both plane-strain and axi ~

symmetric, on several realistic inhomogeneous deposits,including a homogeneous layer with a weathered crust.

Figure 17'5 illustrates the effect of the reduced crustthickness D«/B on the three normalized dynamic imped.ance functions of a rigid strip. The soil profiles are charac-terized by a shear modulus ratio, G«/G, equal to 4, andrealistic values of the Poisson's ratios, v«and v, equal to0."5 and 0.45, respectively. Note that the ratios G«/B and

.2Ica

0

,4

<> ~ »4'I CO~viis1$

0

55 2

9» 0'v 09 fllASI OOOO,II%15

I~I 15

paL5

D«/B may be considered as indexes of the degree anddepth ofweathering.

It is evident that the presence of the crust has a pro.nounced effect on aU impedances. Especially sensitive tochanges in D«/B are the horizontal impedances, whereasthe vertical and rocking ones are somewhat less affected.Variations in the assumed moduli ratio (not depicited inFig. 17) have been shown to have a similar effect.

Furthermore, the weathering effects exhibit a strongdependence on frequency. For example, at low frequencyfactors vertical impedances are relatively indiffetent tovariations (within realistic limits) in either stiffness or depthof the crust. This is understandable in view of the fact thatvertical surface strip loading affects the soil at great depths,of the order of 8B, as discussed previously; thus, a stiffcrust with D«CB can only be of secondary importance.

Figure 16.tion on an

-.2

2 .4 5 0 2 .4 .6a a

Compliance funcnons ofa rigid strip founda-inhomogeneous halfspace (v = 0.25, g = 0.05)"

This picture, however, changes at higher frequency'factors,i.e. lower wavelength. thickness ratios, as may be seen inFig. 17. Greater participation of surface (Rayleigh) wavesin the motion and stronger reflection of the body waves

emanating from the foundation by the soft layer interface,may be part of the explanation.

It may also be noticed that rocking impedances showabout the same sensitivity to weathering throughout the

Soil Dynamics and Eanhquake Engineering, 1983, VoL 2, Pfo. 1 29

4

~ ~ ~ ~ ~ ve ovrw vassals vg ae ~ o vl ~ ~ v, vQ

>-2 0

-40

)~ 80

cc rcI

Dcr/81

ILS

II2

Dc(81

as

G

vccthcccd'S vcc

64

2

0

OQ /81

Its

02

< 20

D /81

osOl

oocchcii» coocolidcccdclogI r

tation makes the clay a cross.anisotropic material with a

vertical axis of symmetry. Similarly, fabric anisotropy insands arises from the influence of gravity forces and particleshape on the deposition process, while in rocks the aniso.

tropy may result from the anisotropy of forming mineralsand micro. or macro-fabric features.

While an isotropic elastic material is characterized byonly two independent elastic constants (e.g. shear modulusand Poisson's ratio), five parameters are needed to describethe stress-strain relationships of an elastic cross-anisotropicmaterial: a Young's modulus Ev in the vertical direction; aYoung's modulus EH in the horizontal direction (EH =nEv); a Poisson's ratio vvH for the effect of vertical onhorizontal strain; a Poisson's ratio vHH for the effect ofhorizontal on complementary horizontal strain; and a shearmodulus GvH = GHv for distortion in any vertical plane,i.e. any plane paraHe] to the vertical axis of material sym-metry. Note that isotropic materials are just a particularclass (subset) of cross.anisotropic materials characterizedby rr = 1 (i.e. EH = Ev =-E, vvH = >HH =- v and GvH = G =E/2(1 + v)).

The condition of incompressibility, appropriate for un-drained loading conditions, requires that:

nvvH = 0.50, vHH = 1— (59)

4

2I 0

-8

Dcr/ 8I

24

16

8

D $ 8

1

02

frequency range examined, and that, in general, the imagin-ary parts of all three impedance functions exhibit only asmall dependence on either Ddr/B or G«/G.

Reference is made to the original publication by theauthor for a more complete parametric assessment of thedynamic effects of 'weathering'n strip foundations.The author sees a definite need to extend these studies todynamically loaded circular foundations.

Effect af'so i7 anisa tropyNumerous experimental studies have shown that most

natural soils and rocks possess anisotropic deformationalcharacteristics." ' 'Dd This anisotropy stems from thefact that soil fabric is intimately related to the mechanicalprocesses occumng during formation, which involves aniso-tropic stress systems. Thus, for example, natural claydeposits formed by sedimentation and, subsequently, one-dimensional consolidation over long periods of time acquirea fabric that is characterized by particles or particle clustersoriented in a horizontal arrangement. This preferred orien-

I 2 3 0 I 2 3

ao aa

Figure 1 7. Impedance funcnons of a rigid strip faunda-n'on on deep soil deposit with a weathered crust (G«/G = 4,Vdr 0.25, v = 0.45, $ = 0.05)

and, thus, reduces the number of independent materialconstants to three. Moreover, utilizing the results of sqxeralexperimental investigations, the author has recentiy shownsd

that, in many clays, the shear modulus GvH is closelyrelated to the other four material constants. Under un-drained conditions, for example, with a reasonable accuracy:

EvGvH =— (60)4-n

Thus, the number of independent material constantsreduces to two, under undrained conditions, and to four,under drained conditions.

Results for statica0y loaded rigid foundations on crass-anisotropic soils have been presented by Gerrard andHarrison, " 'ibson'nd Gazetas.sc Solutions fordynamicaliy loaded foundations on cross.anisotropic homo-geneous soil deposits whose elastic constants satisfy equa-tion (60) (or its 'drained'ounterpart, not given here) havebeen presented by Kirkner'i for circular foundations onhalfspace and by the authors ~ '09 for strip foundationson homogeneous stratum or on halfspace. Table 7 andFig. 1 8 offer some characteristic results from the men.tioned publications.

Speciflically, Table 7 displays simple but fairly accurateformulae for the vertical static stiffness of arbitrary. shapefoundations on a cross-anisotropic, incompressible andinhomogeneous halfspace ('ibson'oil), and for thevertical and horizontal static stiffnesses on a homogeneousand incompressible cross.anisotropic shallow soil stratum-on-rigid base. Notice that on an anisotropic 'Gibson'alf-space abeying equation (60), the degree of anisotropy has

no influence on the vertical stiffness. In all other cases,

however, the stiffnesses increase substantially withn = EH/Ev. In fact, for n «4 all stiffnesses tend to infinity,since the strain energy of such a material is zero for allpossible applied stress systems.'

Regarding the sensitivity of the dynamic response tovariations in the degree of anisotropy, n, under undrainedconditions, the main conclusions of the aforementionedstudies are summarized as follows.

QA .Unit Acinn~i~c ~n8 r-nrthnunke Fneineerine 1 Orr ~ Vnr

kaI

Ip

N

N I

I>.AA5s eovocons se CUT o/ INA5s C~(NS

0,

I g

// 1

//\ r

E„

E„

f/31

nI ~

3

-2

/'III iI r

1 I

I 1

/ ~I

0 0

2 0 1 2

8 I~E„le

Figure 18..„Effect. ofsoil anisorropy on compliance func;tions of a rigid stri foundation on srratum-over-bedrock(undrained conditions, H/B =3, $ = 0.05)~

0 I

Effect ofso/7 nonlinearityIn current soil-structure interaction practice the non-

linear plastic soil behavior is usually approximated througha series of iterative linear analyses, using soil properties(moduli and damping ratios) that are consistent with thelevel of shearing strains resulting from the previous anal-ysis.~'" These analyses may uulize a wealth of availableexperimental soil data relating the decrease in (secant)

For the two compressional modes of vibration, i.e.vertical and rocking, the influence of anisotropy is appreci-able but seems to decrease as the thickness of the stratum-on. bedrock decreases, with the shear modulus Gs 0 remain-ing constant. The effect of anisotropy on the shearingmode of vibration, i.e. swaying, is similar with the effectof anisotropy on

Gita'nd

independent of the layer thick-ness; in other words, two soils with identical Gt H and Hbut with different Young's moduli and n, will yield verysimilar undrained dynamic displacements.

Figure 18 portrays the dependence on n of the threecompliance functions of frequency, for a layer with H= 3Band constant Ev. It is concluded that, in the low andmedium frequency ranee, by increasing n the dynamicdisplacements decrease and the resonant frequencies shiftto the right, roughly in proportion to (4-n) "'. Obviously,the corresponding decrease in the static stiffness may beheld responsible for this effect. Atcertain higher frequencies,however, rocking and vertical displacements increase,instead of decreasing, with n. Nonetheless, the practicalsignificanc of such a reversal will probably be small, inview of the small displacement amplitudes at such fre-quencies.

In conclusion, anisotropy exerts its main effect throughthe static stiffnesses of the soil-foundation system.

shear modulus and the increase in (effective) damping ratiowith increasing amplitude of shear strain.

Soil nonlinearities are not usually of a significant magni-tude in machine foundation problems, for the reasonsmentioned in the introduction. (In contrast, the responseof soil-foundation systems to strong earthquakes is verysensitive to deviations from linear-elastic soil behavior.)Nonetheless, even with small amplitudes of vibration,it is almost certain that some soil elements will undergoplastic deformations. For instance, under the edges ofrocking shallow foundations, large concentration of stressesand low confining pressures will invariably lead to yieldingofsoil.»

An interesting parametric investigation of the effects ofsoil nonlinearities on the dynamic impedance functions of arigid strip foundation has been conducted by Jakub andRoesset.~ In their studies the soil was modeled as a homo-geneous or inhomogeneous stratum-over-rigid-base withreduced thicknesses H/B = 1, 2 and 4. A Ramberg-Osgoodmodel was used to simulate the nonlinear constitutiverelation ofsoil and iterative linear analyses were performed.One of the two parameters of the Ramberg-Osgood model,r, was kept constant equal to 2, while the second one, e,was varied so as to cover a wide range of typical soQ stress-strain relations. For such a model the variation of secantmodulus and effective damping ratio with stress amplitudeis given by:

G 1

Go I+<(</Gory)and

2 G(62)

3/t Go Gory

in which: Go = the initial shear modulus for low levels ofstrain; yv= a characteristic shear strain, typically rangingfrom 0.00017o to 0.0IFo, and r = the amplitude of theinduced shear stress.

From these studies Jakub and Roesset~' concludedthat a reasonable approximation to the swaying and rockingimpedances of a rigid strip may be obtained from the avail.able linear viscoelastic solutions (e.g. Table 4 and Figs.10-11), provided that 'effective'alues of G and $ areestimated from equations (61)-(62) with:

i rc (63)

where so is the statically induced shear stress at a depthequal to 0.50B, immediately below the foundation edge.

While more studies would be necessary to improve thereliabQity of this simple rule, its use in machine foundationanalyses can be safely recommended, in view of the smalllocal nonlinearities that usually develop.

RIGID EMBEDDED FOUNDATIONS

The response of embedded foundations to static anddynamic loads has received considerable attention. As a

result, several finite<lement as well as approximatecontinuum-type formulations have been developed, whileparametric studies have explored the relative significanceof the depth and 'type'f embedment. Reference is madeto the work of Lysmer et al.,~ Novak etal.,"'eredugo

Evidence of such yielding has been presented by Richari e/al.'Figs.10 and 26), while recent experimental work at the University

of Michigan revealed a similar phenomenon under torsionally excitedfooungs '"

Soil Dynamics and Eanhquake Engineering, 1983, Vol. 2, Pfo. 1 31

et al.,~ Waas et al., 'ausel et al.,a9 Chang-Liang, John-son et al.,'" Luco,~ Dominguez et al.," Harada et al.~and Tassoulas,~ among several others.

Results have been presented for circular, strip and rect-angular foundations and a variety of idealized soil profiles,including tive halfspace, stratum-over-bedrock and stratum-over-halfspace. In each case, the new key dimensionlessproblem parameter, in addition to thc parameters control-ling the response of surface foundations, is the relativeembedment, DjB or D/R. Moreover, the assumed interfacebehavior at the contact between vertical sidewaHs andbackfill is of crucial importance. Most of the aforementionedstudies assume that waHs and soQ remain in full contactduring vibrations, as if they were welded at their interface.ln reality, however, no tensile stresses can be sustainedbetween the two media, while the magnitude of developingshear tractions cannot violate Coulomb's friction law.Hence, separation and sliding are likely to occur betweensidewaHs and backfill, depending primarily on the modeof vibration and the nature and method of placement ofthe soil. Field evidence, documented by Stokoe andRichart," seems to indicate that separation and sliding aremore likely with clayey than with sandy soils, in accordwith intuition. Furthermore, it is expected that separationwill be more significant with the two antisymmetric modesof vibration (swaying and rocking}, whereas sliding will beof greater importance in the two symmetric modes (verticaland torsional}. Ideally 'welded'oundations are studiedtrst., a w s~ ~ -»»a 'w tanya»

f'll/elded'ylindrical foundations in a homogeneous stratumThe results to be presented are based on the work of

Rauscl" and are strictly applicable to foundations havinginfinitely rigid sidewaHs and mat, which are aH in perfectcontact with the soil. Moreover, the backfill must be ofvery eood quality and have the same properties with thesoil beneath the mat. These are rather extreme conditionsand, thus, yield an upper bound of the possible effect ofembedment.

Table 8 dispiays five simple and sufficiently accurateformulae for the static stiffnesses of cylindrica founda-tions, perfectly embedded in a homogeneous soH layeroverlying bedrock. It is evident that embedment increasesthe values of the static stiffnesses substantially. The increasein D/R is especially beneficial to the two rotational modes,rocking and torsion; the two translational modes, verticaland horizontal, are considerably less affected (factors of

I+1.85(1-v)(D/R)co ~ 0.85

1+ sz (D/R)

The increase of the two damping coefficients mth D/Ris reflected in the much larger coefficients they are multi~

(65)

1/2 and 2/3 for vertical and horizontal loading, ascomparedto 2 and 2.67 for rocking and torsion).

In contrast, the effect of D/H is more visible in thevertical and horizontal modes, appreciably less importantin rocking, and negligible in torsion; this is consistent withthe expected depths of the corresponding 'pressure bulbs',discussed in the preceding sections.

Note that with embedded foundations the crosseouplingstiffness, JCb„, can no longer be neglected, being approxi-mately equal to OAEhD.

The effect of embedment on the frequency variation ofthe dynamic stiffness and damping coefficients is demon-strated in Fig. 19. We notice that k is not very sensitive toD/R. In fact, Hsabee etal.'" recommended that the actualfrequency variation of k of an embedded foundation beapproximated by the variation of the corresponding surfacefoundation. This seems to be very reasonable for aH vibra-tion modes at low frequencies. For rocking and torsion, inparticular, the approximation will for aH practical purposesbe good throughout the frequency range examined; in otherwords, the beneficial effect of increasing D/R on the staticrotational stiffnesses is preserved even at higher values ofao, at least for not very large D/R ratios. However, beyondthe first resonant frequency, vertical and swaying vibrationsexhibit undulations in k which cannot be well reproducedwith the results of surface foundations.

AH damping coefficients increase substantially withincreasing embedment, although below the first resonance,aor-,they-remain smaHr It-has been recommended~ "4 that~-for ao)ao„c be taken equal to a constant value, corre-sponding to the average value of c of a foundation em.bedded in a halfspace. To estimate this latter value of c,use may be made of the simple expressions derived byDobry et al.es on the basis of simple but realistic physicalapproximations. For the two translational modes, the fre-quency-independent damping coefflcients for cylindricalfoundations embedded in a halfspace are approximated by:

n(2 —v) 1+1.3(D/R) I 1+ (3.6/tt(I -v))jch (64)

8 1+ 2a (D/R)

Table 8. Statfc stif/nesscs ojrfgidembedded cyffndrfcaf joundatfons 'welded'into a homogeneous soil stratumwvcr.bedrock

Type of loading

Vertical

Horizontal

Rocking

Coupled horizontal-rocking

Torsion

Static stiffness

4GRi Rl I I DV D DIH—(1+1.28 —) (1+- -P 1+ 0.8S -0.28——

1-v( H)( 2RA R 1-DjH/

1+- — 1+ 2- 1+0.7—

0 40K/,D

—GR'[1+ 2.67 -JRJ

ProSe

4I

D'

I

' // ///// /ll///Range of va1idity:

D—<2R

DH—c 08'

From Elsabcc et al "'nd Kausel ct al."t For foundation with deeper embedment the formulae undctptcdict the 'actual'ncrease in the stiffncsscs

32 Soil Dvnamics and Earthauake Fneiner rior lQRP Vnl 7 Mn l

0

0

.3

.5

0 0Bp Bp

Figure 19. Effect of embedment on dynamic coefficients of a rigid cylindrical foundation on stratum<ver-bedrock(H/R = 3, v = 1/3, P

= 0.05) s'~ ~

plied with in the numerator than in the denominator; e.g.for v = 0.40, ci, is proportional to (1+3.8D/R)/(1+0.67D/R) and cv is proportional to (I +1.1D/R)/(1+ 0.5D/R).Expressions similar to those of equations (64)-(65) havenot been developed for rocking and torsion.

It is finally noted that, with very good accuracy, onemay set for the cross-coupling impedance:

ctr =0 (66)

Imperfeet contact berween sidewall and backfillTwo recent studies have addressed the question of the

dynamic response of embedded foundations, the sidewallsof which are not perfectly bonded to the backfill.~'"In both studies, the nonlinear contact phenomena associ-ated with separation and sliding are modeled in an approxi-mate way. Thus, Tassoulas assumes that no contact existsbetween sidewall and backfill near the ground surface but

Soil Dvnamics and Earthquake Engineering, 1983. Vol. 2, Po. I 33

P

that a perfect contact is effective over a height equal to dabove the basement. By allowing d to vary between 0 andD ail cases between the extremes of 'no-contact'nd'welded~ontact'ould be studied. On the other hand,Novak et al. considers the sidewalls to be in contact notwith the undisturbed soil but with a cylindrical zoneconsisting ofsofter material. By allowing the shear modulusof this zone to take values between the shear modulus ofthe backfill and zero, various qualities of contact could beconsidered. Note that a similar parametric study for static-ally loaded foundations, the sidewalls of which are sur-rounded by a soft cylindrical zone, has been presented byJohnson et al." Only results from Tassoulas~ are shownherein.

The sensitivity of the static stiffnesses to variation in thecontact-height over embedment ratio, d/D, is graphicallydisplayed in Fig. 20. The effect is essentia!ly independentof 8/R and D/R; hence only one curve is plotted for eachmode. Consistent with the observations made in theprevious subsection, the effects of d/D are very significantfor rochng and torsional loading, substantial for horizontalloading and secondary for vertical loading. For instance,the 'welded~ontact'tiffnesses (d/D =1) are 2.74, 2.33,1.60 and 1.30 times larger than the 'n~ontact'tiffnesses-(d/D= 0) for rocking, torsional, horizontal and verticalloading, respectively.

Figure 21 portrays the effect of d/D on the variation ofk and c versus a<. The stiffness coefficients are only slightlyaffected by d/D at low,frequencies;,at,higherZrequencies„however, the sharpness of the resonant valleys decreases as

d/D increases. On the other hand, the damping coefficientsshow a substantial decline as the 'welded. contact'eight, d,between sidewalls and backfill, decreases. Exception: c„which is less affected by d/D as well as by D/R (see equa-tion (65) and Fig. 19). Notice also that the influence ofd/D on c, depends strongly on the particular frequencyof oscillation.

Embedded strip foundanonsDynamic compliance functions of rigid strip foundations

embedded in a homogeneous soil stratum overlying bedrockhave been obtained by Chang-Liang.a'erfect contact isassumed between the two sidewalls and the backfill, andthe results are cast in the form of equation (20) (i.e. dyn-amic compliances normalized with the static stiffnesses).

Jakub and Roesset,~'y utilizing the results of an

extensive parametric study, developed simple expressionsfor the static horizontal and rocking stiffnesses, which are

displayed in Table 9. It is evident that the influence ofembedment is much smaller for strip than it is for circularfoundations. In fact, the two coefiicients multiplying D/Bin Table 9 (1/3 and 1) are exactly one-half of those multi-plying D/H in Table 8 (2/3 and 2, respectively). Intuitively,these results appear to be very reasonable since a stripfoundation has sidewalls along two sides only. Thus, perunit length, the ratio of sidewall area to basement area isequal to 2D/2B = D/B. Whereas, for a circular foundationthe ratio of the two areas is 2nRD/nRa= 2(D/R)! Thisseems to imply that the influence of D/R or D/B is pro-portional to the sidewall-over-basemat area ratio.

The two normalized compliance functions, ft,i+ift,aand f,i+f~, show practical!y no sensitivity to the D/Bratio and hence are not reproduced herein. Reference ismade to the original publications7 for more detailedinformation.

Rectangular foundarions embedded in halfspace

Dominguez and Roesset'7 developed a boundary elementformulation on the basis of which they derived uniqueresults for embedded rectangular foundations perfectlybonded into a homogeneous halfspace. Figure 22 presentsa few of their results for a foundation with an aspect ratioL/B=2 and three embedment ratios, D/B=O, 2/3 and4/3™Gnly.the stiffness and'damping coefficients"are plotfedin Fig. 22 versus ao.

Results for the static stiffnesses are not shown here. Itappears, however, that the sensitivity ofmost stiffnesses onD/B is not as strong as in the case of circular foundations„but is quite stronger than that of a strip footing. Note thatthe sidewall.basemat area ratio in this case becomes equal to4(B+L)D/(2B.2L)=1.5(D/B), which is in between the1 and 2 times the embedment ratio of the previous twocases!

The dependence on D/B of the k and c versus ao curves,shown in Fig. 2, reveals the following trends.

1. In the frequency range examined the sensitivity ofthe'stiffness coefficients to iarge variations in D/B is quitesmall. For all modes, the decline of k with ao at low fre.quencies becomes sharper as the level of embedinentincreases.

2ohC

// 10// X 4/// gN $'6/' / 'AK

0 0.5

d/oFigure 20, Static stiffnesses ofcylindrical foundations with different d/D ratios (H/R = 3, D/R = 1, v = 1/3/~

34 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

Analysis ofmachinefoundarion vibrarionst state of the art: G. Gazetas

/ Ml

0

IIII

II

//I

h

l/ 1

I/I

/I

II/

dlo

0 ------ 05

0,5

III

I // 4

J~~y ~

C„

0

04

I

//'--r-~ /\ ///

0

ap GpFigure 21. Dependence ofdynamic coefjicients of cylindrical foundations on height ofsidewall-backfill contact (H/R = 3.D/R =1, v =1/3, ) =0.05)~~

"„II'

«4

pa

~ g llJ I C JIJNI lA44ltln b IululjVuh aluie VJ Illa unt u. ua eras

Table 9. Srarle srijjnesses ojrigidembedded srrip joundarions 'vvelded'into a homogeneous srrarumwver bedrock

Type of loading Static stiffness Profile

Horizontal

Rod:ing

'rom Jakub and Roesset"'"

1+-- 1+- I+-- IHis~2D/E c 2/3

2. All the damping coefficients increase substantiallywith increasing D/B. The effect is particularly important forthe rotational modes. Indeed, for rocking and torsion cdoes not tend to zero in the low frequency range when thefoundation is embedded. The practical significance of thisphenomenon is obvious, especially in cases involving smallamounts ofhysteretic damping in the soil.

SYNTHESIS: COMPARATIVE STUDY ANDPRACTICAL RECOMMENDATIONS

(67)

(68)

(69)

and the solution can be derived from equations (39), (41)and (42) by substituting: Q, =modoeo p 90 Qh =modooa gh = O,M, = o Qh and P, = 0.

Four different sets (1, 2, 3 and 4) of dynamic imped-ance functions, E expressed in the form of equation (17),are considered. Set 1 corresponds to a surface foundationon a halfspace (Fig. 5). Set 2 corresponds to a 'surfacefoundation on a stratum-over. bedrock with H/R = 2 (Fig.8). Sets 3 and 4 correspond to a foundation embeddedin a stratum with H/R = 3 and D/R = I; 'welded'idewall-

The previous sections have studied the effects of crucialproblem parameters, related to the soil proflle and thefoundation geometry, on the dynamic response of masslessrigid foundation plates. It is interesting, however, to alsoinvestigate the influence of these parameters on theresponse of a massive foundation, and thus deveIop"a betterperspective of the role of some of these parameters. Wenote that, in such a study, equations (39)-(46) can bedirecfly utilized to obtain amplitudes of steady. statemotion, once the dynamic impedance functions have beenevaluated.

The goal of the comparative study described here is toinvestigate the sensitivity of the response of massive founda-tions to the exact variation with frequency of the dynamicstiffness and damping coefficients, k and c. To this end,two different foundations, both circular in plan, are con-sidered. Foundation A is a relatively heavy one, having aradius R =2m, a mass m =40pR'nd a central massmoment of inertia lo„=m(0.75R)a. Foundation B is arelatively light one, having R = I m, m = 5 pR and Io„=mR'. The center of gravity of the machine. foundationsystem is located in both cases at a distance e= 1.10Rabove the base. Both foundations support a machine withan unbalanced mass mo rotating with an eccentricity do atfrequencies eo; the center of rotation is located at a distancezo =R above the center of gravity of the system, in eachcase. Thus, the excitation forces, refer ed to the center ofgravity, are:

Q, = modoca'xp [i(tot+90')]Qh=modoeo exp(ieot)

Mr= Qh zo

backfill contact is assumed for set 3, no contact for set 4(Fig. 21). Material (hysteretic) damping is invariably takenequal to 0.05.

In our desire to isolate the effects of the dynamic partsof the impedance, k+iaoc, from the effects of the staticstiffnesses, E, the latter are assumed to be the same in allfour sets. Thus, the four cases differ only in the corre-sponding k and c values. In reality, of course, the staticstiffnesses of each set differ substantially from the corre-sponding stiffnesses of the other sets. For instance, thehorizontal stiffnesses corresponding to sets 1, 2, 3 and 4,are in the ratio of 1:1.25:2.76:1.725, respectively. Theappreciable influence of these static stiffnesses on thefoundation response is well known, however, and requiresno further demonstration. After all, the profession candetermine static dis~lacements with sufficient confidence„.and the numerous closed-form expressions offered in thispaper make. very simple the task of reliably estimating thestatic stiffnesses of essentially arbitrary foundations on/ina variety of soil profiles.

The question then which we try to answer in this sectionis the following: After having properly determined the staticstiffnesses of a foundation, how important is it to alsoaccurately determine the dynamic stiffness and dampingcoefficients at the frequency range of interest?

Figure 22 compares the four response spectra of founda-tions A and B, corresponding to the aforementioned casesI, 2, 3 and 4. Plotted in this figure is the variation with aoof the normalized amplitude of the horizontal displacement,lu,l, experienced by the highest point of each foundation,at a distance z,=1.2R above the center of gravity. Thefollowing trends are worthy of note in Fig. 22.

1. For frequency factors ao) I, no differences existbetween the four response curves, of either the heavy or thelight foundation. In fact, the four displacement curvesattain a nearly constant value which is apparently con-trolled by the static stiffnesses of each foundation. (Re-member that in our study these stiffnesses do not changefrom case to case.) Such a behavior is consistent with thehigh. frequency response of a 1-dof oscillator under a

rotating. mass-type excitation.7 The implication is clear:at relatively high frequenc'y factors, the motion of a rigidmassive foundation is controlled by its static stiffnesses andit is not influenced by the exact variation of k and c withao', therefore, one can safely use for k and.c the valuesobtained for surface foundations on halfspace, regardlessof the actual soil profile and depth of embedment!

2. In the low frequency range ao(1, the responsecurves depend on the assumed dynamic coefficients as wellas the inertia characteristics of the foundation.

The 'heavy'oundation experiences two resonant peaks.The first occurs at a frequency ao ~ 0.15 regardless of theexact values of k and c. The only difference from case tocase is in the maximum displacement amplitude, which is

'K

~ ~ ~ ~ '~

~, ~ ~

~ ~ ~ ~'

. ~

~ ~

~ ~

~ ~ I ~ ~ ~ ~

~ ~

~ ~ ~ . ~ ~ ~

~'

I ~ ~

~ I ~ I ~

~ ~ ~ ~

~ ~ . ~ ~ ~'

~ ~ ~ I ~

~ ~ ~ . ~

~, ~ ~ ~ ~

~ ~ ~ ~ ~'

~ ~

~ ~'

~ ~

~ . ~ ~ . ~

~ ~ ~ . ~ ~ ~, ~

~ ~ '~

~ ~

t ' '

~ ~

~ ~ I I ~ ~

l

uy»~<ui ~uliiless ano Qaillplllg coelllclenis, K aflQ c; c canbe assumed to be equal to:

0 forf+f,c=<halfzpace fOrf~fl (70)

SOME OTHER TOPICS

The dynamic behavior of pile foundations, the effects of a

finite flexular mat rigidity, and the dynamic interactionbetween adjacent foundations, are three topics that havereceived considerable attention in recent years. However,present knowledge and understanding of the phenomenarelated to these problems is more limited than for (single)rigid shallow foundations. Research is currently underwayin several institutions, aimed at filling the existing gaps ofknowledge in these three areas. This section is restrictedto a brief general discussion of these topics and a listing ofpertinent references for a more detailed study.

Dynamic impedances ofpilesResults have been presented by numerous authors for

end.bearing and floating single piles subjected to vertical,horizontal, rocking and torsional loading. One may broadlyclassify the developed formulations within three categories:(a) dynamic Wink!el-foundation type formulations, whichneglect the coupling between forces and displacements atvarious points along the pile soil interface.ai-ai.s7,ila,lie

(b) analytical continuum-type formulations, which neglectthe secondary components of deformation and enforce theboundary conditions at the soil.pile interface by expand-

C. ~ Ci'l NC~ I)t la

1~

C.CS

I ~

zt

sou aoevlvsI

z ~ z~la

h

I U I u'o ao

Figure 24. Lateral dynamic coefficients of single pile inan inhomogeneous stratum '""

where fl = VJ4H is the first resonant frequency of the soil-foundation system for each particular mode; k can beapproximated with the values obtained for surface founda-tions on a stratum-on. rigid-base; if, however, such solutionsare not avaBable, use can be made of the halfspace values ofk provided that the latter are approximately corrected atand near the fundamental natural frequencies of the actualstratum, using as a guide the results of Fig. 8.

The above conclusions and recommendations are strictlyapplicable to rigid massive foundations carrying rotating-mass-type machines. For constant-force-type excitationsthe recommendations are still reasonably accurate. Frame-foundations, however, may be quite sensitive to the exactvariation of k and c at frequencies around the fundamentalfrequency of the superstructure.

mg tne contact pressure distribution to an inimiie seriesin terms of the natural modes of vibration of the soBlayer~ '" and (c) finite-element formulations."

Figure 24 presents a typical variation of the horizontalimpedance E> versus ao, for an end-bearing pile withlengthier-diameter ratio, H/D, equal to I5. The soil-stratum consists of material with moduli increasing linearlywith depth and a constant Poisson's ratio of0.40, which istypical for normally consolidated clays. The pBe is ofcircular cross-section and has a Young's modulus E =

P8000E„where E, is the soil modulus at a depth z =H/4.This figure has been adapted from a recent study by Velezet al.,"a who utBized the finite<lement formulation ofBlaney et al." The dynamic impedance is expressed in theform:

<a =<a(kt +»Pt) (7I)where Et, = static horizontal stiffness, kt, = dynamic stiff-ness coefficient, and Pi, = equivalent critical damping ratio.

It is evident from this figure, that the general character-istics of the pBe behavior are similar to those of a shallowfoundation on a soB stratum. The first resonance occursalmost precisely at the fundamental frequency of the in-homogeneous stratum in vertical shear waves, and noradiation damping occurs below this frequency. At higherfrequencies, kh attains an essentiaBy constant value; thesecond resonance is barely noticeable, and hence of minorimportance, despite the relatively small amount of assumedhysteretic damping (0.05).

Reference is made to the aforementioned publicationsfor-detailed studies of the-influence of the main problemparameters on the response of single piles.

In the last few years, interest has switched to the dyn.amies of groups of pBes, a substantiaBy more complexproblem than that of a single pile. The first results, basedon a rigorous formulation,'ss indicate that the dynamicstiffness and damping coefficients of a large group ofclosely-spaced pBes may be drastically different from thecoefficients obtained by a simple superposition of theresults for a single pBe. More extensive parametric studiesare, however, needed before definitive conclusions can bedrawn and before simple formulae and dimensionless graphsofdirect applicability are developed for practical use.

Effects offinitefoundation rigr'dityThe in.plane (membrane) rigidity of mat foundations is

practically infinitely large, when compared to the deform-ability of soils; hence, for horizontal and torsional loadingmost foundations clearly qualify as 'rigid', and the resultsof the preceding sections of this paper are thus pertinent.However, in many practical situarions, the foundationresponse to vertical and rocking loading cannot be properlypredicted without accounting for the finite out-of-plane(fiexural) rigidity of the mat.

A few studies have appeared lately on the dynamicbehavior of flexible circular and rectangular plates restingon a homogeneous halfspace.'~ 'he additional dimension.less parameter which in this case controls the foundationresponse is the relative flexural rigidity factor RF= (Ey/E,).(I—vf) (t/8), where E~, v~ and t are, respectively, theYoung's modulus, Poisson's ratio and thickness of thefoundation raft. In addition, moreover, the exact distri-bution of the applied loading influences appreciably thebehavior, especiaUy ofvery flexible foundations.

The results of the aforementioned studies indicate a

reduction in the vertical and rocking damping coefficients

38 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

~ s

C

4

4'4

"~ 4e

(are ol rfie alv v va~ela>~fl"If'0IIvJ lflacfufleioufluauvflI'IvraIEINII:0

C~ and C, as the relative rigidity of the plate decreases.On the other hand, for small RF values, the stiffnesscoefficients kv and k„do not exhibit as steep a decay withau as the one observed with rigid foundations (Fig. 5, forv 0 I/3). Although additional parameter studies are neededto draw definitive conclusions, the author believes that themain influence of a decreasing RF on the response of amachine foundation materializes through the correspondingdecrease of the static stiffnesses; in other words, the effectof the changes in k and c can be neglected, at least forreaHstic values of the RF factor. The results of the com-parative study offered in the preceding section clearlysupport such a recommendation.

An idea of how sensitive the static stiffnesses, Eo andK„are to changes in the relative rigidity factor, RF, can beobtained from the results of Table 10. The vertical staticstiffness of a circular mat supported by a homogeneoushalfspace and loaded by either a uniformly or a paraboHc-aHy distributed load, are expressed in the form of equation(55). The 'correction'actor J„, which accounts for the matflexural rigidity, is given as a function ofRF. The J, versusRF relationship was computed on the basis of some recentresults by the author I~ the average of the center and edgesettlements were used in deriving the stiffness of theflexible mat.

Dynamic interacnon between adjacent foundanons

increases. But even for distances as small as SB (or SR) thepresence of the second mass will in most cases be ofsecondary importance, in view of the many other uncer-tainties of the problem. It is noted, however, that thenatural frequencies of the soH-foundation system may alsochange due to the interaction.

2. The motions induced in the 'passive'oundation arelarger than the motion changes due to interaction effects onthe 'active'oundation. This is a quite logical result sincewaves emanating from the 'active'oundation excite the'passive'oundation before they are 'reflected'ack to the'active'ne. Typically, one may expect the motions in thesecond foundation to be about 20% of those experiencedby the excited mass, for distances of the order of SR andhysteretic damping ratio in the soil of 5%. However, forstrip foundations (plane-strain problem) on deep soildeposits, the above value may increase to about 50%.

To protect sensitive structures from the vibrationsinduced by a nearby machine foundation, 'active'nd/or'passive'solation measures may frequently be necessary.Results of experimental and theoretical investigations onthe effectiveness of several isolation schemes have beenpublished by Barkan,'ichart et al. and

Haupt.'ONCLUSION

The stateef-the-art of analysing the. forced osciHations ofshaUow and deep foundations has advanced remarkably inthe last 15 years and has reached a mature stage of develop-ment. Several formulations and computer programs have

The vibration of a machine foundation may sometimesappreciably affect a nearby structure; conv'ersely,' the"presence of such a structure may influence the response ofthe machine foundation itself. This 'coupling'n the motionof two adjacent structures through the soil is referred to as

'structure-soil-structure'nteraction, and was first studiedanalyticaUy by Warburton er al.,'n connection withcylindrical rigid foundations on a halfspace. More recently,comprehensive studies have been presented by Chang-Liangs'ho considered two rigid strip foundations on astratum. over-bedrock and by Roesset et al." for two rigidrectangular massive foundations or two structures idealizedas simple I-dof systems and also resting on the surface of a

homogeneous stratum. The following conclusions may bedrawn from the results of these studies.

I. The presence of a nearby ('passive') mass has a rathersmall overall influence on the motion of the foundationcarrying the machine ('active'). Perhaps the most importanteffect from a design point of view is the appearance ofrocking motions, even under vertical excitation; this isapparently the result of waves that are reflected by the'passive'oundation. These effects increase when the massesof the two foundations increase, when the distance betweenthem decreases, and when the thickness of the soil stratum

been developed to determine in a rational way the responseof foundations having various shapes and supported on/inany kind of soil deposit. Numerous studies have beenpublished exploring the nature of associated phenomenaand shedding light on the role of the key parametersinfluencing the response. This paper has reviewed thesedevelopments and presented results in the form of simpleformulae and dimensionless graphs for the dynamic imped-ance functions of circular, strip, rectangular and arbitrary-plan-shape foundations. The various results have beensynthesized in a case study referring to two massivefoundations, and practical recommendations have beenmade on how to inexpensively predict the response offoundations in pracuce.

This progress in developing new methods of analysisfor machine foundations has been paralleled by an equallyimpressive progress in our understanding of the dynamicbehavior of soHs and the development of excellent irr situand laboratory procedures to obtain representative valuesof dynamic soil parameters.

The author believes that, at present, there is a great needto calibrate our analytical procedures by means of actualcase histories. Systematic post. construction observations ofactual foundation performances are the key to this soimportant task. After aH, confidence in advanced methodsof analysis can only be g'ained if these are proved capableofpredicting the field behavior of actual foundations.

Analytical work is also needed to improve the presentknowledge and understanding of, among other topics, thedynamic behavior ofgroups ofpiles, including the influenceof the pileeap; the response of flexible mats founded on asoil stratum; the dynamic characteristics of foundationsconsisting of multiple isolated footings; and the effects ofthe non-uniform initial distribution of static stresses in thesoil, arising from the weight of the structure.

Table 10. Static verticul stiffness of Jlexible rircular mat onhalfspucet

Ju(RF)General

ex pressioiiUniform

loadParabolic

load

0.014GR 0.1

KII= —JQ(RF) I10100

0 ~ 800 ~ 860 '60 '8F 00

0.950 '80 '9F 00F 00

~irv nnrl Fnrrhnunl a hnoinoorinn 1OQV t~nl 0 Vn t 'iO

Finally. Table 10. page 39, should be replaced with the correct on

I g

rh

4g

10

II12

13

14

15

16

17

18

19

20

21

23

24

Whitman, L V. and Richart, F. E Design procedures fordynamically loaded foundations, J. Soil Mech. Fdn. Engrg.,AS', 1967, 93, SM6, 169McNeB, R. L Machine Foundations: The State-oi'-the-Art.Proc, Soi7 DyrL Spec, Sess. 7th ICSMFE, pp. 67-1 QQ, 1969Woods, L D. Measurement of Dynamic SoB Properties. Proc,Eanhq. Engrg. Soi7 Dyn., ASCE, Pasadena, 1978, I, 91Ozaydin, K., Richart, F. E, Dobry, R., Ishihara, K. andMarcuson, W. F. 111. Dynamic Properties and Behavior of

. SoBs. Proc. 7th 4'CEE, State-of-the-Art Volume, istanbul,1980Woods, R. D. Parameters affecting elastic properties, Dyn.Merh. Soil Rock Mech., 1977, I, 37Richaxt F. E Field and'Iaboiaibzy m'easuiements of dy'namicsoil properties, Dyn. hferh. Soil Rock Mech., 1977, 1, 3Richart, F. E., Woods, R. D. and Hall, J. R. Vibrations ofSoils and foundations, Prentice-HaB, 1970Tschebotarioff, G. P. and Ward, E. R. The Response ofMachine Foundations and the Soil Coefficiems WhichAffect It,Doc. 2nd ICSMFE, 1948, I, 309Selvadurai, A. P. S. h7azric Analysis of Sor7.FoundarionJnreracrion, Elsevier ScientiTic Publishing Co., 1979Bazkan, D. D. Dynamics ofBases and Foundations, McGraw-Hill (tzanslated3, 1962Prakash, S. Soil Dynamics, McGzaw-HBI. 1981, pp. 361-7Barkan, D. D. and llyichev, V. A. Dynamics oi'ases andFoundations, Proc. 9th ICShfFE, Tokyo, 1977, 2. 630Gibson, R. E. The analytical method in soil mechanics, Geo-rechnique, 1974, 24, No. 2, 115Reissner, E. Siationare, axialsymmetxische, durch cineschut.telnde Masse erregte Schwingungen eines homogenenelastischen Halbraumes. Ing. Arch., 1936, 7, 381Sung, T. Y. Vibration in Semi-infinite Solids Due to PeriodicSurface Loading. Sc.D. rhesis. Hazvaxd University, 1953Quinlan, P. M. The Elastic Theory of SoB Dynamics. Symp.on Dyn. Test. ofSoils, 1953, ASTM STP No. 156, 1953, 3-34Arnold, R. N. Bycroft, G. N. and Warburton, G. B. Forcedvibrations of a body on an infinite elastic soBd, J. Appi.Nc'ch. ASME, 1955, 22, 391Bycroft, G. N. Forced vibration of a rigid circular plate on asemi-infinite elastic space and an elastic stratum, Phil. Bans.Royal Soc. Lend., 1956, A248, 327Awojobi, A. D. and Grootenhuis, P. Vibration of RigidBodies on Elastic Media, Proc. Royal Soc. Lorrd., 1965,A287, 27Lysmer, J. Vertical Motions of Rigid Footings, Ph.D. rhesis,University ofMichigan, Ann Arbor, 1965Eiozduy, J., Nieto, J. A. and Szekely, E.M. Dynamic Responseof Bases of Arbitrary Shape Subjected to Periodical VerticaLoading, Roc. Inr. Symp. tr~ave Prop. 4 Dyn. Rop. EanhMar., University of New Mexico, Albuquerque, 1967, 105-21.Hsieh, T. K. Foundation Vibrations, Proc. Insr. Civil Eiigrs.,1962, 22, 211Richaxt, F. E. and Whitman, R. V. Comparison of footingvibration tests with theory, J. Soil Mech. Fdn. Engrg. Div,1967Whitman, R. V. SoB.platform Interaction. Proc. Conj onBehav. ofOffshore Srrucr., NGI, Oslo, 1976, I 817

k

ACKNOWLEDGEMENTS

I would like to acknowledge financial support by thc USNational Science Foundation (Grant CEE82-00955) andby the Renssclacr Polytechnic Institute (BUILDprogram).I am also pleased to express my gratitude to my mentorJose M. Rocsset, who encouraged my interest in the subjectof dynamic soil.foundation interaction. During the courseof this work I had fruitful discussions with Ricardo Dobrywho, also, read a draft of the paper and offered valuablecomments. Professor Frank E. Richart, Jr., reviewed thepaper and offered many useful suggestions. Finally, mythanks are extended to the Organizing Committee of theInternational Conference on Soil Dynamics and EarthquakeEngineering for the invitation to prepare and present thisstate. of-the-art paper.

25 Luco, J. E. and Westmann, R. A. Dynamic response of azigid footing bonded to an elastic halfspace, J. AppL Mech.ASME, 1968, 35E, 697

26 Karasudhi, P., Keer, L. M. and Le», S. L Vibratoxy motionof a body on an elastic half'lane, J. AppL Mech. ASME,1968, 35E, 697

27 Luco, J. E and Westmann, R. A. Dynamic response ofcircular foothigs, J. Engng. Nech. Div., ASCE, 1971, 97iEM5, 1381

28 Veletsos, A. S. and Wei, Y. T. Lateral and rocking vibrationsof footings, J. Sor7 Mech. Found. Div., ASCZ, 1971, 97,SM9, 1227

29 Veletsos, A. S. and Verbic, B. Vibration of viscoelasticfoundations, Inr. J. Eanhq. Engrg. Srrucr. Dyn., 1973,2,87

30 Kuhlemeyer, R. Vertical Viiirations of Foothigs Embeddedin Layered Media, Ph.D. thesis, University of California,Berkeley, 1969

31 Waas, G. Analysis Method for Footing Vibrations ThroughLayered Media, Ph.D. rhesix, University of California,Berkeley, 1972

32 Chang-Liang, V. Dynamic Response, of Structure in LayeredSoBs, Ph.D. rhesfx, MIT,1974

33 Kausd, E. Forced Vibrations of CircuLtr Foundations onLayered Media, Research Rap. R74-11, MIT, 1974

34 Lysmer, J„Udaka, T, Seed, H. B. and Hwang, R. LUSH-A Computer Pxogram for Complex Response Analysis ofSoB-Structure Systems, Report No. EERC 74-4, UniversityofCalifornia, Berkeley, 1974

35 Luco, J. E. Impedance functions for a rigid foundation on a

layered medium, NucL Err grg. Des., 1974, 31, 204 ~

36 Gazetas, G. Dynamic Stiffness Functions of Strip and Rect-angular Footings on Layered Soil, Shf. 77rexix, MIT, 1975

37 Hadjian, A. H. and Luco, J. E. On the Importance of Layer-ing on Impedance Functions, Proc. 6th IvCEE, Neu Delhi,

.191738 Gazctas, G. and Roesset, J. M. Forced vibrations of strip

footings on layered soils, Neth. Srrucr. AnaL, ASCE, 1976,I e 115

39 Gazetas, G. and Roesset, J. M. Vertical vibration of rnachinefoundations, J. Georech. Engng. Div„ASCE, 1979, 105,GT12, 1435

40 Novak, M. Vibrations of Embedded Footings and Structures,ASCE JVarL Srrucr. Engrg. Neer., 1973, Reprint 2Q29

41 Novak, M. Dynamic stiffness and damping of piles, CanadGeorech. J., 1974, 11, 574

42 Novak, M., Nogami, T. and Aboul.Ella, F. Dynamic SoBReactions for Plane Strain Case, Res. Rep. BLtr'TI-?7,University of Western Ontario, 1977

43 Novak, M. and Aboul-EBa. F. Impedance functions for pilesin layered media, J. Engrg. hfech. Div. ASCE, 1978

44 Wong, H. L. and Luco, J. E. Dynamic response of rigidfoundations of arbitrary shape, Eanhq. Engrg. Srrucr. Dyn.,1976, 4, 579

45 Awojobi, A. O. and Tabiowo, P. H. Vertical vibzation oi'rigidbodies with rectangular bases on elastic media, Eanh. Zngrg.Srrucr. Dyn., 1976, 4, 439

46 Kitamura, Y. and Sakurai, S. Dynamicstiffnessforrectanguiarrigid foundations on a semi-infinite elasuc medium, Ini. J.AnaL Plum. Meth. Geomech., 1979

47 Dominguez, J. and Roesset, J ~ M. Dynamic Stiffness ofRectangtdar Foundations, Research Rcport R?8.20, MIT,1978

48 Savidis, S. A. Analytical methods for the computation ofwavefields, DyrL Mesh. Soil Rock Mech., 1977, I, 225

49 Lin, Y. J. Dynamic response of circular plates on visco-eiastlc halfspace, J. AppL Mech., ASME, 1978, 45E, 379

50 Whittaker, W. L. and Christiano, P. Dynamic response ofplate on elastic halfspace, J. Zngrg. Mech. Div., ASCE, 1982,108, EM1, 133

51 IgucM, M. and Luco, J. E. Dynamic response of flexiblerectanguLir foundations on an elastic halfspace, Eanhq.Errgrg. Srrucr. Dyn., 1981, 9, 239

52 Awojobi, A. O. Viliration of a rigid circular foundation onGibson soil, Georechnique, 1972, 22, No. 2, 333

53 Gazetas, G. Static and dynamic displacements of foundationson heterogeneous multilayered soils, Georechnique, 1980,30, No. 2, 159

54 Gazetas, G. Strip I'oundations on cross-anisotzopic soil layersubjected to static and dynamic loading, Georechnique,198la, 31, No. 2, 161

40 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, PJo. l

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107

108

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110

boundary for anisotzopic material, IVum. Mech. Geomech.,ASCZ, 1975, 2, 1013I.ysmer, J., Udaka, T., Tsai, C.F. and Seed, H. B. FLUSH-A Computer Program for Approximate 3.D Analysis of Soil-Structure Interaction Problems, Reporr IVo. EERC 75.30,University ofCalifornia, Berkeley, 1975Luco, J. E. and Hadjian, A. H. Two-dimensional approxima-tions to the three-dimensional soil-structure interactionproblem, IIucLEngrg. Des., 1974, 31, 195Chang-Liang, V. Dynamic Response of Structure in LayeredSoBs. Ph.D. rhesfx, MIT, 1974Tassoufas, J. L. Elements for the Numerical Analysis of WaveMotion in Layered Media, Research Rap. R81-2, MIT, 1981Kausel, E. and Roesset, J. M. Dynamic stiffness of circularfoundations, J. Engrg. Mech. Div.. ASCE, 1975, 101, EM12,771Kausel, E. and Ushijima, R. Vertical and Torsional StiffnessofCylindrical Footings, Research Rap. R76.6, MIT, 1979Roesset, J. M. The use of simple models in soB-structureinteraction, Clv. Engrg. chic iVucL Power, ASCE, 1980b, II,10(3(1-25Gerrard, C. M. and Harriso, W. J. Circular Loads Applied zoa Cross-Anisozropic Halfspace. Paper 8. CommonwealthScientific and Industrial Research Organization: Div. ApplMech., Australia, 1970aGazetas, G. Torsional dispiacements and stresses in non-homogeneous soQ, Georechnique, 198le, 31, No. 4, 487Borodachev, N. M. Determination of the settlement on rigidplates, Soll Mich. Fdn. Engrg. (USSR/, 1964, I, 210Dobry, R. and Gazetas, G. Stiffness and Damping ofArbitrary-shaped Embedded Foundations, Research Rep. CE42-04,RPI, 1982Egorov, K. E. Calculation of Bed for Foundation udth RingFooting, Proc. 6th ICSMFE, 1965, 2, 41Dhawan, G. K. A transversely-isotropic halfspace indentedby.a flat,annular rigid stamp, Acra Mechanica„1979,.31291Dhawan, G. K. An asymmetric mixed boundary value prob-lem of a transversely-isotropichalfspace subjected to momentby an annular rigid punch, Acra hiechanica, 19&la, 38, 257Dhawan, G. K. A mixed boundary value problem of a trans-versely-isotropic halfspace under torsion by a flat annularrigid stamp, Acra Nechanlca, 19&lb,41, 289Brow+, P. T. and Gibson, R. E. Surface settlement of a deepelastic stratum whose modulus increases lineaxly with depth,Can. Geozech. J., )972, 9,467Gibson, R. E. and Kalsi, G. S. The surface settlement of a

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NOTATION

The following symbols are frequently used in the paper.

Related to geometryB = half-width of a strip footing or the shortest

half-width of a rectangular footingD = depth of embedmentd = height ofperfect sidewall.backfill contact above

the foundation base

HL

RRo

c

thickness of soil stratumone-half of the longest side of a rectangularfoundationradius of a cylindrical foundationradius of 'equivalent'ircular foundation(equations (47)-(50))distance of center of gravity of a machine-foundation system above the base.

Related to maten'al propertiesG = shear modulus of soilm = increase of shear modulus from the surface to

a depth equal to R or B (applicable to inhomo.geneous soil deposits)

n = EH/Ey, where E~ and Etr are the horizontaland vertical Young's moduli of a cross.aniso-tropic soil

v = Poisson's ratio of soQ= hysteretic critical damping ratio ofsoil

Subscriptsv ~ = vertical (also designated by z)h = horizontal (also x,y)r = rocking (also r„, ry)r = torsion (also r,)hr = coupled horizontal-rocking (also xry,yr„)

Related to foundation impedancesE = static stiffness referred to the base of the

foundation (Fig. I)E = dynamic impedance function of frequency; it

may be expressed in one of the followingalternative forms:

= Et(co) + iE2(co),.= A'(k+iaoc).(l.+.2ig),,= K(k+iaoc)

Calligraphic characters are used on the figures in place ofthe boldK,k and c.k and k = (dynamic) stiffness coefficients, functions of toc and c = (dynamic) damping coefficients, functions of ca

ao = taB/V~ or caR/V, (dimensionless frequencyfactor)

F = dynamic compliance function of ca; it may be ~

expressed in one of the foQowing alternativeforms:

= Ft(co)+iFz(ca)I

= —Vt(ca)+ ifz(~)l

,!

42 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2. No. 1

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