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7330 15 FOUNDATION VIBRATIONS GEORGE GAZETAS, Ph.D., P.E. Professor of Soil Mechanics National Technical University Athens, Greece and State University of New York Buffalo 15.1 INTRODUCTION mass mo rotating with an eccentricity ro at the operational circular frequency = 2nf, where f = frequency in cycles When subjected to dynamic loads, foundations oscillate in a per second (Hz). The forces and moments acting on the way that depends on the nature and deformability of the soil-foundation interface and transmitted into the ground supporting ground, the geometry and inertia of the foundation are of the form cos wt or, using complex notation, and superstructure, and the nature of the dynamic excitation. 2 that is, they vary harmonically with time. Such an excitation may be in the form of support motion due Waves are emitted from the interface and propagate in all to waves arriving through the ground during an earthquake, directions within the deposit. In the presence of the free ground an adjacent explosion, or the passage of a train; or it may result surface and of soil layers with differing stiffnesses these waves from the dynamic forces imposed directly or indirectly on the undergo numerous reflections and refractions, as well as foundation from operating machines, ocean waves, and vehicles transformations into surface waves. Much of the energy imparted moving on the top of the structure. onto the foundation is diffused by such outward- and downward- Since the very important subject of foundation response spreading waves, while a small portion is dissipated by inelastic during earthquake shaking is treated in the next chapter, action in the soil. attention herein will be focused on determining the vibratory As a result, the soil-foundation interface, and with it the response of foundations to applied loads such as those produced foundation block, undergoes harmonic oscillations of the form by a machine. A key step in such response analyses (and + or, using complex notation, exp + hence the main thrust of this chapter) is to estimate the with frequency-dependent amplitude and phase lag, = dynamic "spring" and "dashpot" coefficients of flexibly-supported and Q = The basic goal of the geotechnical design is to foundations. To this end, an engineering procedure is developed, limit the amplitudes of all possible modes of oscillation to based on simple algebraic formulae and dimensionless charts, small enough levels that will neither endanger the satisfactory for surface and shallow foundations, embedded foundations, operation of the machine nor disturb the people working and piles. Note that, in addition to being directly applicable to in the immediate vicinity. Charts like the one depicted in machine-loaded foundations, much of the information presented Figure 15.1b (based on information from Richart, 1975) may could also be used in assessing the dynamic soil-foundation- guide the selection of an appropriate upper limit for a structure interaction during seismic (or any other ground) satisfactory foundation performance. shaking. Of course, in such cases the loading arises from inertial Notice that these limiting displacement amplitudes are (D'Alembert) forces developing in the oscillating typically of the order of a hundredth of a centimeter-compared This chapter also presents information on the pertinent to the several centimeters that is the usual restriction for dynamic soil parameters, and outlines current methods of foundation settlement under static load. A direct consequence measuring them in the laboratory and in the field. Some useful is that soil deformations would in the majority of cases results and concepts from dynamics and wave propagation by quasielastic, involving negligible nonlinearities and no theory are also presented and elucidated when the need arises, permanent deformations. Among the possible exceptions are a throughout the chapter; they provide background information laterally oscillating piled foundation working at low frequencies, and help in developing a better understanding of the methods which may induce strains of the order of 0.02 percent in soft presented. The chapter concludes with a number of illustrative clayey layers; and a rocking shallow foundation may induce realistic examples. large strains directly under its edges. Thus, analyses to predict vibration amplitudes assume linear viscoelastic soil behavior, with hysteretic soil damping to model energy losses at these 15.2 MACHINE FOUNDATION VIBRATIONS: STATEMENT OF THE PROBLEM It has become traditional in dynamics to introduce complex-number A sketch of a typical rigid block foundation carrying rotatory notation, which significantly simplifies the computations. The under- standing, of course, is that at the end the absolute value (amplitude) machinery and supported on a layered soil profile is shown in and phase angle can be recovered from a complex response + iu2; Figure 15.1. The dynamic loading arises from an unbalanced the former being equal to and the latter to tan 553554 Foundation Engineering Handbook FO sin wt morow2 = wt mo w ro Sensitive facility Fig. The machine foundation problem. 1 (a) Estimate magnitude and characteristics of the dynamic loads. The most common types of machines include: Rotating machinery, which produces sinusoidaly varying forces as already explained (examples: turbines, compressors, pumps, fans) Reciprocating machinery, which generates biharmonic loads 0.1 of the form F = + a exp (2iwt)], where a is a geometric constant (examples: steam engines, internal- combustion engines, piston-type compressors and pumps) Impact producing machines, involving intermittent impulsive loading with a nearly triangular variation of applied force versus time (examples: forging hammers, stamping machines, 0.01 presses) Machines with simultaneous impulsive and rotatory forces, in which the former are due to the main function of the machine (hammering) while the latter generate parasitically from unbalanced wear of the hammers (solid-waste shredders, car-shredders, rotatory rock crushers, all kinds of hammer- 0.001 mills) This crucial task will not be further addressed herein, since it has been treated in detail in the first edition of the Foundation Engineering Handbook (Richart, 1975). Additional information may be found in Barkan (1962), Richart et al. (1970), Arya et al. (1979), Major (1980), and Prakash and Puri (1988). 1 10 100 (b) Establish the soil profile and determine the appropriate Frequency, Hz shear modulus and damping, G and B, for each soil layer. In addition to standard geotechnical soil investigation techniques, Fig. 15.1 (b) Typical performance requirements for machine special dynamic procedures are used today to assess these soil foundations. parameters in the field and the laboratory. Section 15.4 presents up-to-date information on this subject. (c) Guided by experience, select the type and trial dimensions small strain amplitudes. The low-strain value of the shear on the foundation, and in cooperation with the client establish modulus (denoted by Go or in the literature) is the key performance criteria such as those of Figure 15.1b. soil parameter that must be assessed for each layer. The design of a machine foundation is a trial-and-error (d) Estimate the dynamic response of this trial foundation, process involving the following main steps (engineering tasks). subjected to the load of step (a) and supported by the soilFoundation Vibrations 555 deposit established in step (b). This key step of the design Dilatational waves, denoted as P waves, propagate with a process usually starts with simplifying and idealizing soil velocity Vp related to the constrained modulus Mc: profile and foundation geometry, and involves selecting the most suitable method of dynamic soil-foundation interaction analysis. To this end, several formulations and computer programs have been developed in recent years. Moreover, for Vp = (15.2) the two key parameters, the dynamic stiffness and damping, numerous solutions have been published in the form of For an elastic material, depends on the shear modulus G parametric dimensionless graphs, applicable to a variety of and the Poisson's ratio of the soil so that idealized situations. The main contribution of this chapter is to present in a concise and comprehensive way a complete set of ready-to-use results for the stiffness and damping ("spring" (15.3) and "dashpot") of foundations on and in several characteristic soil profiles. The relationship Vp/Vs versus from Equation (15.3) is plotted (e) Check whether the estimated response amplitude of in Figure 15.2. step (d) at the particular operation frequency conforms with Therefore, Vs and or G and Mc, or G and are the the performance criteria established in step (c). Repeat steps (c), equivalent pairs of soil parameters relevant to wave propagation (d), and (e) until a (theoretically) satisfactory design is phenomena. Note that waves other than S and P also arise in established. At this stage, two additional checks may be the ground under an oscillating foundation, most notably necessary: First, to ensure that the motions transmitted to Rayleigh and Love waves. All these other waves, however, also nearby structures and underground facilities are within safe relate to G and as they are the outcome of combinations levels for their uninterrupted functioning-a task usually ("interferences") of S and P waves. accomplished with the help of semiempirical energy-attenuation For the small strains (less than about 0.005 percent) usually relationships, and guided by experience. Second, if the subsoil induced in the soil by a properly designed machine foundation, contains soft clays and/or loose sands, to investigate the shear deformations are the result of particle distortion rather potential for accumulation of large permanent deformations- than sliding and rolling between particles. Such deformation is an unlikely event, requiring shear strain amplitudes well in almost linearly elastic: the hysteresis loops that do develop upon excess of 0.01 percent. unloading and reloading are very, very narrow. The actual The design process frequently stops here. However, in case behavior can be simulated quite accurately as that of a linear of important projects one or two additional postconstruction hysteretic solid described through the "tangent-at-the-origin" shear modulus Go and a damping ratio steps are necessary: In fact, the approximation as a linear hysteretic solid (f) Monitor the actual motion of the completed foundation is also employed to describe dynamic behavior at large and compare with the theoretical predictions of step (d). strains. However, as illustrated in Figure 15.3, the appropriate The necessity of this task arises from the several simplifying ("equivalent linear") modulus G is the secant modulus, that is, assumptions that are unavoidably introduced with even the the slope of the line connecting the origin with the tip of the most sophisticated analyses. Furthermore, experience, and hysteresis loop. G is smaller than Go (hence the familiar notation confidence in the advantages of advanced methods of analysis of the latter as Gmax). At the same time, the area of the hysteresis can only be gained through such comparisons of theoretical loop has expanded owing to increased dissipation of energy predictions with reality. Reference is made to Richart et al. resulting from sliding at particle contacts. The equivalent linear (1970), Gazetas and Selig (1985), and Hall (1985) for information hysteretic damping ratio is larger than on instrumentation and field measurements related to machine foundations and to man-induced vibrations. (g) Finally, if the actual performance of the constructed 5 foundation does not meet the aforesaid design criteria (step remedial measures must be devised. These may be, repair of the worn-out parts to minimize unbalanced masses; change of the mass of the foundation or the location of the machinery; 4 4 stiffening of the subsoil through, for example, grouting; increasing the soil-foundation contact surface; construction of piles through the existing foundation mat; and on. Steps (d), (e), and (f) must be repeated until a satisfactory design is finally 3 3 achieved. V This chapter addresses in detail tasks (b) and (d). Vs 2 2 15.3 SOIL MODULI AND DAMPING-FIELD AND V2 LABORATORY TESTING PROCEDURES 1 1 A vibrating foundation emits shear and dilatational waves into the supporting ground. The former, denoted as S waves, propagate with a velocity that is controlled by the shearing 0 stiffness G and the mass density p of the soil: 0.1 0.2 0.4 0.5 = (15.1) Fig. 15.2 Comparison of the actual (Vp and Vs) and "apparent" (VLa) wave velocities used in foundation vibration analyses.556 Foundation Engineering Handbook Go G 1 Go=Gmax 1 Monotonic loading curve Yc Y Yc Yc 10 6 10-5 10- 10-3 2 10 6 10-5 10-4 101 3 10 2 Mexico City Clay G 0.5 0.5 curve for Gmax GRAVELLY soils CLAYS SANDS 0.001 1 0.001 0.01 0.1 1 20 20 B, percent 10 10 SANDS AND CLAYS GRAVELS Ip=100 Mexico City Clay 0.001 0.01 0.1 1 0.001 0.01 0.1 1 percent percent Fig. 15.3 The nonlinear-hysteretic cyclic stress-strain behavior of soils is conveniently represented in terms of modulus decreasing and damping ratio increasing with shear strain amplitude. Apparently, the bigger the cyclic shear strain, the smaller 15.3.1 Shear Modulus Gmax and S-Wave Velocity the "equivalent" modulus G and the larger the "equivalent" damping B. Plots of modulus ratio G/Gmax max and damping ratio as functions of cyclic strain have become the traditional Factors affecting Gmax and From the foregoing way of depicting cyclic stress-strain behavior, following the discussion it is clear that the low-strain shear modulus, pioneering work by Seed and Idriss (1970). Figure 15.3 or the corresponding S-wave velocity = is summarizes published data for clays, sands, and gravels, the single most important soil parameter influencing the encompassing some recently published information. response of machine foundations. Laboratory and field testsFoundation Vibrations 557 have revealed a number of factors on which and Suggested Values of for Equations 15.7 (Seed depend. The following discussion summarizes the most significant and Idriss) findings of these tests. for stress units of (1) The two most important parameters influencing max of all types of soils (granular and cohesive) are the mean confining Soil Type kPa psf effective stress and the void ratio e. From the published Loose sand 8 35 results it appears that Gmax is proportional to where typically Dense sand 12 50 Very dense sand 16 65 n 0.3 to 0.6 for granular and n = 0.5 to 0.9 for silty and clayey Very dense sand and gravel 25 to 40 100 to 150 soils. Experimental tests with large cubic samples of dry sand at the University of Texas (Knox et al., 1982) have revealed (a) that (and Gmax) depend only on the stresses and in the directions of wave propagation and particle motion, 0.5 respectively; they are independent of the stress in the out-of-plane direction. (2) The static-stress prehistory, expressed for instance 0.3 through the overconsolidation ratio, OCR, influences mainly the modulus Gmax of clays. The granular material changes in OCR are adequately accounted by the present void ratio. On the other hand, cyclic prestraining, that is, application of 0,1 moderately large shear strains for a large number of cycles, tends to increase the modulus of granular soils beyond what 0 is anticipated with the increased void ratio. With cohesive soils 0 50 100 the effect of prestraining is not clear. (3) For cohesive soils, geologic age seems to be of great PLASTICITY INDEX importance, as it perhaps controls the creation of "bonds" (b) between the clay platelets or clay clusters. In the laboratory, attempts to simulate the natural process of aging are being Fig. 15.4 Suggested values for the coefficients and u in made by increasing the duration of the initial confining state Equations 15.4 and 15.7. of stress to several days, before applying the cyclic loading. Increases in of the order of 100 percent have often Note, however, that in many actual situations S waves will been reported. Aging may also be important for fine-grained propagate in all directions away from the foundation, and it cohesionless soils that are partly saturated. will not be readily evident which are the directions a and b. (4) For partially saturated (S, 10 to 50 percent) fine Hence it may be as advantageous to use Equation 15.4. granular soils (silty sands) capillary stresses may increase For granular soils Seed and Idriss (1970) developed the by 50 to 100 per cent over the value of Gmax measured in the simpler expression laboratory on completely dry or on fully saturated samples. (15.7) (5) For all soils, cohesionless and cohesive, the frequency, in which the dimensional empirical coefficient is a or the rate of loading, has no practical effect on (at function of the (relative) density of the material (dimension: least within the range of parameters applicable to machine square root of stress) given in Figure 15.4, for both SI and foundations). This means that soil is basically not a viscous, English units. but rather a hysteretic, material. For saturated clays, Gmax relates to undrained shear strength Su: Empirical correlations for Several expressions relating Gmax to other soil parameters have been devised on the basis 2500 (15.8) of laboratory test results. For granular and cohesive soils Hardin (1978) proposed that (The geotechnical engineer should not be surprised at such high values. The value G 100S, reported in soil mechanics (15.4) literature refers to near-failure conditions, that is, at strains in excess of 1 percent.) Use of empirical expressions such as those of Equations 15.4 where Pa = the atmospheric pressure in the same units as to 15.6 may be recommended in practice in several cases: and and u is a function of the plasticity index plotted (1) in feasibility studies and preliminary design calculations, in Figure 15.4. before any direct measurements have been performed in the On the other hand, the aforementioned experimental work field or laboratory; (2) for final design calculations in small at the University of Texas 1982) has concluded that projects, where the cost of proper testing for cannot be justified-unless parameter studies reveal a high sensitivity (15.5) of the response to the "exact" value of modulus; (3) to should be used in place of in Equation 15.4. and are provide an order-of-magnitude check against the experimentally the effective stresses in the directions of wave propagation determined values. and particle motion, respectively). Alternatively, the following Another empirical correlation of interest is between and expression can be used for clean sands: the Standard Penetration Test (SPT) resistance N (blows/ft). Using mostly Japanese data, Seed et al. (1986) have proposed that (15.6) (15.9a)558 Foundation Engineering Handbook or This expression, however, is rather unreliable: small errors in the values of or will lead to substantial errors in (15.9b) On the other hand, shows little sensitivity to soil type, in which the corrected resistance is given by confining pressure, and void ratio, but depends very much on the degree of saturation and the drainage conditions. (15.10) Consequently, it is not very difficult to make a reasonably good prediction of saturation and drainage conditions are known. where = vertical effective overburden stress, and ER = ratio As an example, the following values are given as a guide in of the energy actually transmitted to the rod of the SPT, divided selecting in practical cases. by the theoretical free-fall energy. Several other empirical Saturated clays and sands, beneath the water table correlations between and N values have also been Nearly saturated clays, above the water table v=0.40 proposed in the literature. One that has been frequently quoted Wet silty sands (S, = 50 to 90 percent) in the literature has been proposed by Ohsaki and Iwasaki Nearly dry sands, stiff clays, and rocks (1973): Once has been estimated, Equation 15.3 is used to (kPa) (15.11a) determine unless of course = 0.50 that, as previously (ksf) explained, Equation 15.3 is meaningless. An interesting con- (15.11b) clusion drawn from studies of foundation vibrations is that the However, the reliability of such relations is very low, and they influence of is not of great significance in most cases; an should only be used, if necessary, for crude preliminary estimates exception is vertical and rocking oscillations in soils with of soil stiffness. approaching 0.50. Hence, small errors in assessing the value of would likely be of no practical consequence. 15.3.2 Constrained Modulus and P-Wave Velocity 15.3.4 Damping Ratio Whereas shear (S) waves can propagate only through the mineral skeleton of a soil (fluids offer no shear resistance), The low-strain value of material damping, Bo, depends only dilatational (P) waves can propagate through both the mineral marginally on such variables as the confining stress and the skeleton and the pore water. Since water is far less compressible void ratio. For most soils it ranges between 2 and 6 percent. than any soil skeleton, P-waves in fully saturated soils are Since oscillating foundations generate "radiation" damping essentially transmitted solely through the water phase with a that may be substantially higher than Bo, the precise value velocity that is of the order of, or somewhat larger than, of the latter is usually rather insignificant. (Exceptions are m/sec (or 4900 ft/sec)- velocity of sound waves rotational oscillations at low frequencies, and translational in water. oscillations on a shallow soil stratum, again at low frequencies.) On the other hand, the presence of even small amounts of air in the pores might dramatically increase the compressibility of the water-air phase; only the soil skeleton would then resist 15.3.5 Measurement of Low-Strain Moduli the induced dilatation: would be essentially the same as the P-wave velocity of a dry, but otherwise identical, soil sample For satisfactory design of a machine four dation the geotechnical For a clean sand, Figure 15.5a portrays the sensitivity of engineer must: to variations in the degree of saturation S, As long as saturation Establish the soil profile, including layering and depth to remains below about 99 percent is nearly independent of S,, bedrock, physical characterization and classification of each being a measure of the incompressibility of the soil skeleton. layer, elevation of water table and groundwater conditions, (The small decline from the dry velocity dry at large values and extent of lateral homogeneity of S, is the consequence of increasing mass density, rather than Determine with in-situ or laboratory tests the low-strain value of decreasing constrained modulus in Equation 15.2.) As S, of shear modulus and select proper values for Poisson's approaches 100 percent, jumps to a very high value, ratio and damping ratio that is controlled by the pressure wave velocity in water, For practical purposes, the velocity is independent of the Standard subsurface exploration techniques and field and type of soil, is similar for clays and sands, and shows only a laboratory testing required for static design may provide a slight dependence on and e, as visualized in Figure 15.4b. complete answer to the first of the foregoing tasks. But, with Hence, measuring the P-wave velocity of saturated soils is of few exceptions, estimation of soil parameters for dynamic little if any value in assessing the actual soil stiffness. analyses is presently being done increasingly frequently with The foregoing experimental findings can be qualitatively the help of special "dynamic" procedures in the field and explained with elastic theory. Saturated soil is a practically the laboratory. Only a summary of the best techniques for incompressible material with Poisson's ratio approaching determining is offered herein. More detailed information 0.50. Equation 15.3 would then predict that is far greater may be found in Richart (1975), Woods (1978, 1985), Stokoe than Vs, and, in the limit, = 0.5 and = 00 regardless of (1980), and Drnevich (1985). Note: most dynamic tests provide Vs-that is, regardless of soil stiffness. an indirect evaluation of G max through measurements of the S-wave velocity (15.13) 15.3.3 Poisson's Ratio in which p is the known total mass density of the soil. For soils that are not close to saturation, can be obtained In-situ testing procedures have some distinct advantages from Equation 15.3 once and have been measured: over laboratory techniques. Sample disturbance, for example, may be more deleterious for determining low-strain soil stiffness (15.12) (which reflects the exact particle arrangement- than behavior at large strains and failure (after a rearrangement ofFoundation Vibrations 559 1500 P : 1200 1800 V : P 900 1200 e = 0,42 , = 200 600 600 e = 0.60 = 100 300 0 0 50 100 99.4 99.6 99.8 100 DEGREE OF SATURATION ST : % (a) 3000 saturated 1000 Vp : m/s 300 100 30 20 50 100 200 500 1000 200 : kpa (b) Fig. 15.5 Dependence of P-wave velocity on void ratio, confining effective stress, and degree of saturation (references given in the text). particles has occurred - of the initial "fabric"). 15.3.6 Field Procedures Moreover, simulating in the laboratory the effects of stress prehistory, aging, and capillary stresses is not a routine task. Dynamic in-situ tests induce strains smaller than and In fact, with granular soils even reproducing the in-situ void thereby measure V. max and The list of in-situ testing ratio and geostatic stresses (which control according to procedures includes the following. Equation 15.3), may prove a rather difficult task. With coarse sand and gravel, things get even more complicated. 1. The Crosshole Seismic Survey (or simply crosshole method) As a result, in-situ measured moduli are almost invariably This is probably the best geotechnical method for determining found to exceed those measured in the laboratory-sometimes the variation with depth of in-situ low-strain S-wave velocity, by more than 100 percent. However, when the effects of all Illustrated by a sketch in Figure 15.6, the crosshole the important factors (Figure 15.4) are properly reproduced, method is based on a very simple concept: it generates S waves laboratory test results can closely match the field test data. in a borehole and measures their arrival times at the same Moreover, laboratory tests are valuable for studying the effect elevation in neighboring boreholes. The wave velocity is of various variables on for determining the damping ratio, computed from the travel times and the spacing between the and for obtaining and G at moderate and large strains. boreholes. For the success, however, of a crosshole test there560 Foundation Engineering Handbook (a) plan Receivers 2 4m (13ft) (13 ft) Impact Transducer Transducer wedged in place Assumed Wave Path P and S Waves during SPT 2 1 P S 1 Typical Receiver Records 2 P S Fig. 15.6 Sketches of (a) the crosshole, (b) the downhole, and (c) the seismic cone penetration tests (references in the text).Foundation Vibrations 561 (b) Lateral Impact Path Transducer (c) hammer V5 ( m/s ) (bar) 0 50 100 150 200 250 0 100 S Waves 5 10 Seismic CPT Crosshole562 Foundation Engineering Handbook are several requirements. (a) There should be at least two and and = and the value from Equation 15.13 would preferably three boreholes, which are spaced about 3 to 5 m correspond to a depth of about of the wavelength (the "center" (10 to 15 ft) apart, the verticality of which is instrumentally of the R-wave displacement profile). By progressively decreasing secured. (b) The source must be rich in shear wave generation the frequency f of vibrations, the wavelength would increase and poor in P-wave generation, that detection of S-wave and the R-wave would affect soil at greater depths, having arrivals is unambiguous (torsional sources are the best in this different properties. Equation 15.14 would at every frequency sense, but the SPT offers a good inexpensive solution). (c) The give a different value of CR. From these values the velocity receivers (geophones) must have a proper frequency response profile is constructed as and should be oriented in the direction of the particle motion. Moreover, they must be in "perfect" contact with the (at depth = (15.15) surrounding soil, either directly (in case of stiff cohesive soils) As an example, Figure 15.7b (adapted from Gazetas, 1982) plots or through properly grouted casing (in case of granular and the theoretical variation of R-wave velocity versus frequency soft cohesive soils). Coupling between geophone transducer and for a deposit consisting of an inhomogeneous layer over vertical wall should be accomplished with use of specially bedrock. For the layer = + where H designed packers. (d) The triggering and recording systems must is its thickness. The bedrock velocity Vrock is 8 times We be accurate. Evidently, "crosshole" would not classify among denote by fs the fundamental frequency of the stratum in shear; the most economic in-situ tests, but it is one of the most reliable. fs Notice that at frequencies fr exceeding 15fs See Woods (1978), Hoar and Stokoe (1978), and Woods and the R-wave velocity approaches while at f less than 0.5fs, Stokoe (1985) for more details. CR approaches (Plots like that of Figure 15.7b are called "dispersion" relations.) 2. The Seismic Downhole Survey (or simply the downhole Clearly, this method cannot even in theory produce the method) This is the economic alternative to crosshole testing. accurate and detailed (layer-by-layer) information of the three It is explained with the help of Figure 15.6. It needs only one borehole methods. However, it can provide: (a) the near-surface borehole inside which the receiver(s) is (are) placed at various wave velocity which controls the radiation damping of depths while the source is at the surface, 2 to 5 m (6 to 15 ft) high-frequency machine foundations, as well as the response in away. Travel times of body waves (S or P) between surface and rocking and torsion at all frequencies; (b) an average (over the are recorded, and then travel-time versus depth plots horizontal and vertical direction) wave velocity of a stratum are constructed from which or of all the layers can over bedrock, covering a large area; and (c) with high-power be determined. An effective and economic S-wave source equipment operating at low frequencies, the velocity of deeper consists of a steel-jacketed rigid beam weighted down the strata that could not be reached inexpensively with a borehole. ground and struck horizontally with a sledge hammer. However, if the source is placed too close to the borehole, parasitic waves 5. The Spectral Analysis of Surface Waves This recent develop- are created and S-wave arrivals cannot be easily identified; if ment is a very promising evolution of the foregoing steady-state it is placed too far from the source, the direct wave path may vibration method (Nazarian and Stokoe, 1983). Its goal is to not be a straight line. These problems are largely avoided with determine the detailed profile, as with "crosshole", but crosshole testing. working entirely from the surface. A vertical impact at the surface generates transient Rayleigh (R) waves, which are 3. The Seismic Cone Penetration Test (or simply the seismic recorded by vibration transducers located a known distance cone) This recent development (Robertson et al., 1985) is apart. If the subsoil were very deep and homogeneous (half- sketched in Figure 15.6. It combines the downhole method with space) the signals of the two transducers would have the cone penetration testing. To this end, a small rugged velocity same shape. However, in nonhomogeneous or layered deposits seismometer is incorporated inside the electronic penetrometer the various frequency components generated by the impact and downhole measurements of seismic S-wave velocity are propagate at different speeds (recall Figure 15.7), thereby performed during brief pauses in cone penetration testing. In arriving at different relative times at the two locations; hence, addition to its speed, a significant advantage of the seismic cone the two signals have different shapes. Through a fast Fourier is that with a single sounding test one obtains information for transform spectral analysis of the two signals the "dispersion" the stratigraphy of the site, the low-strain moduli of the various relation (that is, the variation of with frequency) is computed layers, as well as the (static) strength-related parameters for the particular site. The thicknesses and S-wave velocities of (point bearing stress) and fs (sleeve frictional resistance). each and every layer are then back-calculated by use of an Comparisons with the "crosshole" are very encouraging, as analytical "inversion" procedure. The results of the method seen in Figure 15.6. A limitation of the method is that it may seem to be in excellent agreement with crosshole measurements. not be appropriate for some types of soils (such as those Several other field tests are available to the profession but containing coarse gravel). are not discussed herein for various reasons. They include the seismic refraction survey, which is good mainly for preliminary 4. The Steady-State Vibration of the Free Surface This method, surveys covering large areas, and for determining the P-wave requiring no boreholes, is based on the fact that a circular velocities of near surface soft layers and the depth to rock footing vertically oscillating with frequency f generates along (Richart et al., 1970); the resonant footing method, in which the the free surface primarily Kayleigh (R) waves. Their wavelength resonant frequency of a concrete block placed on the surface is the distance between any two successive crests (or troughs) is determined and utilized in conjunction with homogeneous of the vibrating surface (Figure 15.7a), and their velocity CR is halfspace theories to back-calculate the (average) soil modulus calculated from (Moore, 1985); and the standard penetration test, which may CR (15.14) provide indirect crude estimates of moduli (e.g. Eq. 15.9). Measurement of is made by moving a seismic geophone away from the vibrator and locating points that are moving in 15.3.7 Laboratory Procedures phase. If the subsoil were very deep and homogeneous its S-wave velocity would have been unique and roughly equal to Low-strain values of moduli and wave velocities can be obtained 1.06 times CR. With real-life inhomogeneous deposits = with the following laboratory tests.Foundation Vibrations 563 value from Equation Qo sin 2nft out of the wavelength ( profile). By progressive] the wavelength AR soil at greater 15.14 would at ever From these values 3 AR I T AR = = (adapted from 0 R-wave velocity versu an inhomogeneous + velocity rock is 8 time frequency of the stratu (a) Z t at frequencies fr exc 8 es while ke that of Figure 15.71 R-wave not even in theory p y-layer) information ( can provide: (a) the n 6 ntrols the radiation ( 100 m ations, as well as the S (b) an averag on) wave velocity of : area; and (c) with 8 the velocity CR 4 1 inexpensively with Vs Waves This recei on of the foregoing d Stokoe, 1983). Its file, as with "cross ace. A vertical imp 2 yleigh (R) waves, ers located a deep and homogene transducers would nogeneous or layere nts generated by recall Figure 0 5 10 f Hz S at the two locatio apes. Through a (b) two signals the "d.7 (a) A harmonically-oscillating footing generates Rayleigh (R) waves propagating along the surface of a soil deposit, and with frequency) is shg" to a depth of about one wavelength. (b) The R-wave velocity in a nonhomogeneous two-layer stratum decreases with frequency and S-wave of the decreasing wavelength. by The results of th crosshole Resonant Column Test This truly dynamic test is in the axial mode. H = the height of the sample, and E = the ailable to the profetedly the best widely available today for determining Young's modulus of the soil, E = 2(1 + v) G. Equations 15.16 is reasons. They inin the laboratory. It uses solid or hollow cylindrical and 15.17 provide G and E, respectively. Material damping good mainly for prs and subjects them to torsional or axial steady-state ratio can also be estimated either from the free-vibration for determining thnic excitation with the help of an electromagnetic device logarithmic decrement or from the half-power bandwidth of ers and the depthetch in Figure 15.8). The frequency of the input vibration the steady-state response curve. Figure 15.8b plots in dimension- ooting method, in changed until the fundamental resonant condition is less form the (theoretically determined) response curve. The lock placed on thined. The resonant frequency is a function of soil stiffness, distribution of shear (or normal) strains along the sample during with home geometry, and boundary conditions of the apparatus resonance follows a sinusoidal law: the (average) soilyed. For the case of fixed base and free top sketched in enetration test, 15.8a the frequency at first resonance is either (15.18) moduli (e.g. Eq. 1 4H G/p (15.16) To achieve the development of an almost uniform distribution of strains in the sample, Drnevich (1977) adds a mass at the torsional mode, or top as shown in Figure 15.8a. Such a uniformity is highly desirable when at strains exceeding 10- is needed. The velocities can be (15.17) hollow cylinder is also a necessity in such a case, since the 4H distribution of shear strains across the thickness of a solid564 Foundation Engineering Handbook H Z 2 H - at 1st resonance at 1st resonance (a) 100 10 DA 1 0.1 1 3 5 f / (b) Fig. 15.8 Resonant column test. Distribution of rotation amplitude along sample length in two variants of the test, and dynamic amplification of the top motion versus imposed frequency. (Based on Woods, 1978; Drnevich, 1985.) cylindrical sample in torsion is nonuniform, varying from 0 at and are used to determine stress-strain the center to a maximum at the periphery. For more details, hysteresis loops (from which "effective" moduli and damping see Woods (1978) and Drnevich (1985). ratios are deduced, and degradation characteristics are studied). However, in recent years special cyclic triaxial apparatuses have 2. The Ultrasonic Pulse Test Piezoelectric crystals at one end been designed capable of determining moduli at 5 X 10-6 of the soil sample generate dilatational or shear waves, and at (Ladd and Dutko, 1985). the other end record their arrival. From the travel time and the known sample thickness, the appropriate velocity, or is calculated. The identification and recognition of the exact 15.4 HARMONIC VIBRATION OF BLOCK wave arrival, requiring the use of an oscilloscope, is by no means a routine operation. The results of this method are in FOUNDATIONS DEFINITION AND USE OF good accord with resonant column data. An advantage of the IMPEDANCES (DYNAMIC "SPRINGS" AND method is that it can use the same sample to determine both "DASHPOTS") and (and hence Poisson's ratio, or the condition of saturation). Moreover, it can be performed on very soft clays Frequently, machine foundations are constructed as rigid while still retained in the Shelby minimizing reinforced-concrete blocks, whose response to dynamic loads disturbance. arises solely from the deformation of the supporting ground. Like any rigid body, such foundations possess six degrees of 3. Cyclic Load-Deformation Tests In their standard form, freedom, three translational and three rotational: (dynamic) they are appropriate only for medium and large strains displacements along the axes x, y, and z, and (dynamic) rotationFoundation Vibrations 565 y Combining Equations 15.19 and 15.20 leads to = 0 (15.21) Fz from which it is evident that the key to solving the problem is Mz the determination of the impedance that is, of the dynamic force-over-displacement ratio according to Equation 15.20. Note also that, as it is well known from structural dynamics, the steady-state solution to Equation 15.21 for a harmonic excitation = cos wt is also harmonic with the same frequency Theoretical and experimental results reveal that, in Equation 15.20, a harmonic action P, applied on to the ground and the resulting harmonic displacement have the same frequency but are out of phase. That is, if (15.22) then can be expressed in the following two equivalent ways: (15.23a) (wt + a) (15.23b) Fig. 15.9 Rigid foundation block with its six degrees of freedom. around the same axes (Figure 15.9). In this section a general method is presented for computing each of these six dynamic displacements and rotations due to steady-state harmonic excitation (forces and moments). The choice of harmonic oscillations is made not only because many machines usually block mass produce unbalanced forces that indeed vary harmonically with time (rototary or reciprocating engines), but also because nonharmonic forces (such as those produced by punch presses and forging hammers) can be decomposed into a (large) number of sinusoids through Fourier analysis. 15.4.1 Vertical Oscillation Let us explain the method with the help of the easy-to-visualize case of vertical vibrations. Figure 15.10 portrays a rigid foundation block of total mass m, assumed to have a vertical axis of symmetry Z passing through the centroid of the soil-foundation contact surface. The foundation is underlain Fz(t) by a deposit consisting of horizontal linearly deforming soil Applied Force layers. Subjected to a vertical harmonic force along the Z axis, this foundation will experience only a vertical harmonic displacement The question is to determine given To this end, we consider separately the motion of each Inertia Force m "body": the foundation block and the supporting ground (Figure 15.10). The two free-body diagrams are sketched in the figure and include the inertial (D'Alembert) forces. The foundation "actions" on the soil generate equal and opposite Resultant of "reactions", distributed in some unknown way across the Soil Reactions interface and having an unknown resultant Furthermore, since in reality the two bodies remain always in contact, their displacements are identical and equal to the rigid body Pz(t) Resultant of displacement Thus, the dynamic equilibrium of the block Foundation Actions takes the form (15.19) and that of the linearly deforming multilayered ground can be "summarized" as (15.20) in which K is called the dynamic vertical "impedance", determined for this particular system with one of the methods Fig. 15.10 Analysis of the dynamic equilibrium of a vertically described in the sequel. vibrating foundation block.566 Foundation Engineering Handbook where the amplitude Uz and phase angle Q are related to the Equation 15.33 is the equation of motion of a simple inphase, U1, and the 90°-out-of-phase, U2, components according oscillator with mass m, spring "constant" K2, and dashpot to "constant" C2-justifying our previous interpretation. The quotation marks around the word constant are placed delib- (15.24a) erately: in fact, K2 and C2 are not constant but vary with the frequency of oscillation. Nonetheless, Equation 15.33 suggests (15.24b) for the vertical mode of oscillation an analogy between the actual foundation-soil system and the system depicted in We can rewrite the foregoing expressions in an equivalent but Figure 15.11 and consisting of the same foundation but far more elegant way using complex number notation: supported on a "spring" and "dashpot" with characteristic moduli equal to K2 and C2, respectively. (15.25) Once these moduli have been established for a particular (15.26) excitation frequency, is obtained from Equation 15.34: where now and are complex quantities: (15.35a) (15.27) (15.28) and thereby the amplitude of oscillation that is of interest is Equations 15.25 to 15.28 are equivalent to Equations 15.22 to simply 15.24, with the following relations being valid for the amplitudes: (15.35b) (15.29) (15.30) Conclusion: The soil reaction against a vertically oscillating while the two phase angles, a and Q, are properly "hidden" in foundation is fully described with the complex frequency- the complex forms. dependent dynamic vertical impedance or, equivalently, In addition to elegance, it is computational ease that the frequency-dependent "spring" (stiffness) and "dashpot" motivates the adoption of complex notation, as will become (damping) coefficients, (w) and Once these parameters apparent later on. have been obtained for the particular frequency (or frequencies) With P, and being out of phase (Eqs. 15.22 to 15.23) or, of interest, solving the equation of motion yields the desired alternatively, with and being complex numbers (Eqs. 15.25 amplitude of the harmonic vertical displacement. to 15.28), the dynamic vertical "impedance" (force-displacement ratio) becomes 15.4.2 Generalization to All Modes of Oscillation number (15.31) The definition of dynamic impedance given in Equation 15.31 for vertical excitation-response is also applicable to each of the other five modes of vibration. Thus, we define as lateral which may be put in the form: (swaying) impedance K the ratio of the horizontal harmonic force, (t), imposed in the short direction at the base of the (15.32) foundation over the resulting harmonic displacement, in the same direction: in which both K2 and C2 are functions of the frequency They can be interpreted as follows. The real component, K2, termed (15.36) "dynamic stiffness', reflects the stiffness and inertia of the supporting soil; its dependence on frequency is attributed solely to the influence that frequency exerts on inertia, since soil Similarly, properties are to a good approximation frequency-independent Kx = the longitudinal (swaying) impedance (force-displace- The imaginary component, wC2, is the product of (circular) ment ratio), for horizontal motion in the long direction frequency times the "dashpot coefficient" C2, which reflects the Krx = the rocking impedance (moment-rotation ratio), for two types of damping-radiation and material damping- rotational motion about the long axis of the foundation generated in the system, the former due to energy carried by basemat waves spreading away from the foundation, and the latter due Kry = the rocking impedance (moment-rotation ratio), to energy dissipated in the soil due to hysteretic action. for rotational motion about the short axis of the Equation 15.32 is a (theoretical and experimental) fact for foundation basemat all foundation-soil systems. However, the interpretation of K = the torsional impedance (moment-rotation ratio), for and C as dynamic stiffness and dashpot coefficients must be rotational oscillation about the vertical axis justified. This is easy if we substitute Equation 15.32 into Equation 15.21. We are looking for the harmonic response Moreover, in embedded foundations and piles, horizontal forces to the harmonic excitation F2 exp (iwt). Straight- along principal axes induce rotational in addition to translational forward operations lead to oscillations; hence, two more "cross-coupling" horizontal- rocking impedance exist: and K They are usually = (15.33) neglegibly small in shallow foundations, but their effects may become appreciable for greater depths of embedment, owing and to to the moments about the base axes produced by horizontal soil reactions against the sidewalls. In piles the "cross-coupling" - = (15.34) impedances are as important as the "direct" impedances.Foundation Vibrations 567 Rigid and massless 0 0 (a) Rigid and massless = Rigid base (b) Fig. 15.11 (a) A foundation-structure system and the associated rigid and massless foundation. (b) Physical interpretation of the dynamic stiffness and dashpot coefficients for a vertically vibrating footing. Note that throughout this chapter (as in most of the subjected to harmonic external forces and to harmonic base literature) all impedances refer to axes passing through the motion. foundation basemat-soi interface. The eight impedances turn out to be complex numbers and functions of frequency that can be written in the form of 15.4.3 Coupled Swaying-Rocking Oscillation Equation 15.32. Thus, in general, for each mode = + iwC(w) Figure 15.12 portrays a typical rigid block foundation: it has (15.37) equal depth of embedment along all the sides and possesses and the analogy suggested in Figure 15.11 extends to all modes. two orthogonal vertical planes of symmetry, and yz, the Once, for a particular excitation frequency w, the eight intersection of which defines the vertical axis of symmetry, z. dynamic impedances (or the eight pairs of dynamic stiffness or The foundation plan also has two axes of symmetry, x and y. "spring" and "dashpot" coefficients) have been determined by For such a foundation the vertical and torsional modes of following the procedures to be presented in this chapter, by oscillation along and around the Z axis can be treated separately, recourse to the published literature, or by using available as was previously illustrated for vertical oscillation. In other numerical formulations and computer codes, the steady-state words, each of these two modes is uncoupled from all the others. response of a rigid foundation block to arbitrary harmonic On the other hand, swaying oscillation in the y direction external forces can be computed analytically by application of cannot be realized without simultaneous rocking oscillation Newton's laws. Also analytically, one can derive the steady-state about x. This coupling of these two modes is a consequence of response of a flexible structure possessing natural modes and the inertia of the block and the fact that its center of gravity568 Foundation Engineering Handbook For a harmonic excitation: (15.40) (15.41) CG in which the amplitudes and Mx may be either constant, or (more typically) proportional to the square of the operational frequency w = 2nf. and Mx result from the operation of the Zc machine. y The steady-state harmonic response can be written in the form: 0x (15.42) (15.43) in which and are complex frequency-dependent displace- ment and rotation amplitudes at the center of gravity. Note that Equations 15.40 to 15.43 do not by any means imply that the two components of motion and the two components of excitation are all in phase. Instead, the true phase angles are "hidden" in the complex form of each displacement component. Using similar arguments with regard to the soil reactions, one may, without loss of generality, set (15.44) (15.45) The complex amplitudes and Tx are related to the complex y CG displacement and rotation amplitudes through the correspond- ing dynamic impedances. Recalling that the latter are referred to the center of the foundation base, rather than the block center of gravity, one can immediately write (15.46) (15.47) Substituting Equations 15.40 to 15.47 into the governing Equations 15.37 to 15.38 leads to a system of two (coupled) algebraic equations with two unknowns and The solution Fig. 15.12 Coupled swaying-rocking oscillations: definition of is obtained using Kramer's rule: displacement variables. Top: section. Bottom: plan. is above the center of pressure of the soil reactions. Thus, if the (15.48) block is initially being displaced only horizontally, an inertial (15.49) force arises at the center of gravity and produces a net moment at the foundation base-hence rocking is born. Similarly coupled are swaying in the x direction and rocking around y. in which the following substitutions have been made To study the coupled swaying-rocking oscillations of the (15.50a) block in the zy plane, we call S, and the horizontal displacement at the foundation center of gravity and the (15.50b) angle of rotation of the rigid block, respectively. Referring to Figure 15.12 and calling and Mx(t the excitation force and moment at the block center of gravity, one can write the and translational force and rotational moment dynamic equilibrium N = (15.51) as follows: (15.38) Notice that, for a particular frequency w, determination of the motions from Equations 15.48 to 15.51 is a straightforward Tx(t) - = (15.39) operation once the dynamic impedances (or the corresponding where "spring" and "dashpot" coefficients, and Cij) are known. Of course, the computations are somewhat tedious if performed mass by hand, since complex numbers are involved; but with even = mass moment of inertia about a principal small microcomputers the calculations can be done routinely, horizontal axis, parallel to x and passing at minimal cost. through the block center of gravity Therefore, it is proposed that this procedure, in connection P, and Tx = net horizontal force and rocking moment with an appropriate evaluation of impedances at the frequency reactions, acting from the soil against the (or frequencies) of interest, should be used in analysis of machine foundation during swaying and rocking, and foundations vibrating in swaying-rocking. referring to the centroid of the foundation Vibrations in the vertical and torsional mode (each of which basemat is practically uncoupled from all the other modes in the usualFoundation Vibrations 569 case of nearly symmetric foundations), can be respectively Moreover, even when the appropriate sophisticated code is analysed with Equation 15.35 and its torsional counterpart: available, the effort involved in getting one or two sets of usable results may be such that no time/budget is left for the necessary M2 (15.52) parametric studies. Such studies are of course critical for exploring various design options and for evaluating the effects in which = the dynamic "spring" coefficient for torsion, of uncertainties in poorly known parameters (e.g., soil properties, or quality of soil-foundation contact). = the "dashpot" coefficient for torsion, = the moment of An alternative engineering approach has been the develop- inertia of the whole foundation (including the machine) about ment of easy-to-use closed-form expressions and graphs, based the vertical Z axis, and exp (iwt) = the harmonic external on the results of rigorous and approximate formulations. This moment around is the approach taken in this chapter. 15.5 COMPUTING DYNAMIC IMPEDANCES: 15.5.1 Presentation of Tables and Graphs TABLES AND CHARTS FOR DYNAMIC "SPRINGS" AND "DASHPOTS" Six large tables (15.1 through 15.6) present comprehensive and easy-to-use information for dynamic "spring" and "dashpot" Several alternative computational procedures and computer coefficients. The information is in the form of simple algebraic codes are in principle available to the engineer wishing to obtain formulas and dimensionless graphs pertaining to all possible dynamic impedance functions ("springs" and "dashpots") for (translational and rotational) modes of oscillation and covering each specific machine-foundation problem. The choice among a wide range of idealized soil profiles and foundation geometries. these methods depends to a large extent on the required The engineer should be able, by using the tables, to approximate accuracy, which in turn is primarily dictated by the size and with sufficient accuracy the actual problem in many cases. importance of the particular project. Furthermore, the method Figure 15.13 sketches the soil-foundation systems covered to be selected must reflect the key characteristics of the in each table. Specifically: foundation and the supporting soil. Specifically, one may broadly classify soil-foundation systems according to the 1. Table 15.1 and the accompanying set of graphs refer to following material and geometric characteristics: foundations of any solid shape resting on the surface of a homogeneous halfspace. The shape of the foundation (circular, strip, rectangular, 2. Table 15.2 and the related graphs are for foundations with arbitrary) any solid basemat shape partially or fully embedded in a The type of soil profile (deep uniform deposit, deep multi- homogeneous halfspace. layered deposit, shallow stratum on rock) 3. Table 15.3 refers mainly to circular and strip foundations The amount of embedment (surface foundation, embedded on the surface of a homogeneous soil stratum underlain foundation, piled foundation) by bedrock (some results are also given for rectangular Broadly speaking, the various computational methods can foundations). 4. Table refers to circular and strip foundations partially be grouped into four categories, each with its own merits and limitations: or fully embedded in a homogeneous stratum underlain by bedrock. Analytical and semi-analytical methods that can handle 5. Table 15.5 pertains to square and strip foundations on the multi-layered soil deposits and rectangular surface foundations, surface of some inhomogeneous profiles, in which the but cannot treat embedment (e.g., Luco, 1976; Gazetas and modulus increases smoothly with depth according to Roesset, 1976, 1979). (15.53) Dynamic finite-element methods that can treat surface, embedded, and piled foundations on or in layered soil profiles. 6. Table 15.6 is mainly for laterally oscillating single floating Most of these methods are limited to axisymmetric (circular) piles in two inhomogeneous and a homogeneous stratum or or plane-strain (strip) situations, that is, they cannot study halfspace; some information is also given for vertical rectangles and arbitrary shapes; and usually they require the oscillations, and for pile-soil-pile dynamic interaction presence of a rigid bottom boundary (bedrock) at relatively factors. shallow depths (Waas, 1972; Kausel, 1974; Lysmer et al., 1975). Simplicity without any serious compromise in accuracy has Combined analytical-numerical methods that try to take been the prime goal when developing these tables. It is believed advantage of the capabilities of analytical and numerical that, in general, the errors that may result from their use will be well within an acceptable 15 percent. (Use of the approaches. Included in this category are recently developed boundary element methods (Kausel, 1981; Lysmer et al., approximation symbol, however, implies a slightly inferior 1981; Tassoulas, 1981). accuracy.) Approximate techniques that simplify the physics of the The formulas and graphs given in Tables 15.1 and 15.2 for problem and can provide engineering solutions to some very arbitrarily shaped surface and embedded foundations on or in complicated situations (e.g., separation between foundation a homogeneous halfspace have been compiled from some recent sidewalls and backfill) that cannot be treated rigorously publications by the author and his coworkers (Dobry and Gazetas, 1985; Gazetas et al., 1985, 1987; Fotopoulou et al., (Beredugo and Novak, 1972; Meek and Veletsos, 1973; Novak et al., 1978; Nogami, 1979; Gazetas and Dobry, 1984; 1989). They are based on (a) some simple physical models calibrated with results of rigorous boundary-element formulations Wolf, 1985, 1988; Gazetas and Tassoulas, 1987). and (b) data from the literature (most notably from the work Application of most of the rigorous methods and solutions of Lysmer, Veletsos, Luco, and Roesset and Kausel-see to a specific engineering problem usually involves using a references). On the other hand, Tables 15.3 and 15.4 pertaining specialized computer code, which may or may not be available. to surface and embedded foundation on a homogeneous stratum Developing "tailor-made" codes is a very impracticable under- over bedrock are based on results by Kausel (1974), Johnson taking, in view of the mathematical complexity of the problem. et al. (1975), Gazetas and Roesset (1976, 1979), Elsabee and570 Foundation Engineering Handbook 2B Good contact between basemat soil and vertical sidewall base area area Ab 2L Ab 2L y y Plan no contact between Circumscribed soil and rectangle L>B Section aa 2 1 massless rigid foundation D total sidewall soil contact area Aw Homogeneous Halfspace G V Homogeneous Halfspace 3 4 Homogeneous Stratum H H G Stratum rigid formation rigid formation 5 6 Go Go Go E5 Es Inhomogeneous Inhomogeneous or Homogeneous G(z) Deposits Z S Z Fig. 15.13 The six soil-foundation systems studied in this chapter. Numbers 1 through 6 refer to the corresponding tables 15.1 to 15.6 and the associated graphs.Foundation Vibrations 571 Morray (1977), Jakub and Roesset (1977), Kausel and Ushijima 15.5.2 Use of Tables and Graphs; Illustrative (1979), and Chow (1987). The sources of Table 15.5 for surface examples foundation on a number of inhomogeneous deposits include Hadjan and Luco (1977), Gazetas (1983), Booker et al. (1985), 1. Surface Foundation on Halfspace Wong and Luco (1985), Werkle and Waas (1986), and Novak (1987). Finally, the information of Table 15.6 on single piles is For an arbitrarily shaped foundation mat, the engineer must based on Roesset (1980a, b), Sanchez-Salinero (1982), Velez first determine a circumscribed restangle 2B by 2L (L> B) et al. (1983), and Gazetas (1984). The pile-group interaction using common sense as explained in Figures 15.13 and 15.14. factors are from Dobry and Gazetas (1988) and Gazetas and Then, to compute the impedances in the six modes of vibration, Makris (1990), calibrated with results from Nogami (1979), from Table 15.1, all he needs is the values of: Kaynia and Kausel (1982), and Davies et al. (1985). The reader will certainly find useful detailed information in Ibx, Iby and = area, area moments of inertia about these original sources. However, Tables 15.1 to 15.6 and the x, y, and polar moment of inertia about z, of the actual accompanying graphs are sufficient for complete dynamic soil-foundation contact surface. If loss of contact under analyses. part of the foundation (e.g., along the edges of a rocking Ab = 57,6 (614 I bx = 82 m 4 (30) = 188.5 (49') 1.2 (4') 2B=4.3 Plan 1.9 (6') (14') X 1.2 (4') (22.16') 2.7 (9') y FIRST NUMERICAL EXAMPLE 0 G = 192 MPa G = 2,000 tsf Elevation p = 2 p = - 0.33 0,33 Z 8 0,05 0,05 SECOND NUMERICAL EXAMPLE T 6m 4m 4m Elevation soil parameters as above Fig. 15.14 Geometry and material parameters of the two illustrative numerical examples. (Note also that Vs 310 m/s, VLa 500 m/s; 0.26.)572 TABLE 15.1 DYNAMIC STIFFNESSES AND DASHPOT COEFFICIENTS FOR ARBITRARILY SHAPED FOUNDATIONS ON THE SURFACE OF A HOMOGENEOUS HALFSPACE. Dynamic Stiffness K Static Stiffness K Dynamic Stiffness Radiation Dashpot General Shape Coefficient k Coefficient Vibration (foundation-soil contact surface is of area Ab Square shape; C Mode and has a circumscribed rectangle 2L by 2B; > B)* L=B (General Shapes) Vertical, Z 4.54GB C2 = is plotted in Graph a is plotted in Graph C 9GB Horizontal, y (in the lateral is plotted in Graph b direction) is plotted in Graph d Horizontal, X 1 (in the longitudinal direction) Rocking, rx 0.20ao = (around longitudinal with axis) area moment of inertia of the foundation-soil is plotted in Graphs e and f contact surface around the x(y) axis 0.15 Rocking, ry 0.45: (around lateral axis) is plotted in Graph Torsional = = = , with polar moment of the soil-foundation contact surface - 8 (strip footing) the theoretical values of K2 and - the values computed from the two given formulas correspond to a footing with L/B 20.Foundation Vibrations 573 15 GRAPHS ACCOMPANYING TABLE 15.1 e 1 05 2L 0.40 L/B 1 2 4 6 C VERTICAL 1.2 0 a 2 1.1 VERTICAL ROCKING (rx) 2 0 6 10 L/B 0 2 1 09 0 0.5 2 ao k, 4 f 05 2 1 1 10 Fine soft 0 saturated soils ry 0.5 VERTICAL -0.5 0 2 0 0 2 d ROCKING (ry) 0 b 0 1 2 2L 3 g k y 1 2 10 1 L/B 6 6 2 1 Q5 1 SWAYING (y) SWAYING (y) TORSION 0 0 2 2 0 2 wB wB foundation) is likely, the engineer may use his judgment to the values combine hysteretic and radiation damping). discount the contribution of this part. To incorporate such damping, simply add to the foregoing B and L = semi-width and semi-length of the circumscribed C value the corresponding material dashpot coefficient rectangle. G and or and VLa = shear modulus, Poisson's ratio, shear-wave velocity, and "Lysmer's analog" wave velocity. Total C = radiation C + (15.56) The last is the apparent propagation velocity of compression- extension waves under a foundation and is related to according to Numerical Example A numerical example illustrates the use of Table 15.1 and the attached graphs in computing the dynamic (15.54) stiffnesses ("springs") and damping coefficients ("dashpots"), for four modes of vibration. A sketch of the foundation and w = 2nf = the circular frequency (in radians/second) of the lists of all pertinent geometric, material, and load parameters applied force (e.g., frequency of operation of the machine). are included in Figure 15.14. The computations follow in SI units, but the results are also given in English units (in This table as well as all the other tables gives: parentheses). The excitation frequency is f = 30 Hz. The dynamic stiffnesses ("springs"), K = K(w), as a product of the static stiffness, K, times the dynamic stiffness coefficient VERTICAL MODE k = k(w): Static stiffness: (15.55) The radiation damping ("dashpot") coefficients C = C(w). 7.45 [0.73 + These coefficients do not include the soil hysteretic damping, (the only exception is with Table 15.6 (for piles), where 5500 kN/m (3.8 108576 Foundation Engineering Handbook (a) (b) 2L 2L y y 2B 2B 3 3 2B 2 B 2B 2 3 B 2 2 3 plans L 2B L/B=1 L/B 2 My D X 2B elevation Boundary Element V = 0.40 D = B Expression of Table II 6 Cry 4 P LIB=1 2 0 0 1 2 Fig. 15.16 Comparison of the results derived using the simple expressions given in this chapter with the results of rigorous (boundary- element) formulations for two T-shaped foundations (L/B = 1 and 2) embedded in a homogeneous halfspace.Foundation Vibrations 577 He thus decides that the effective height of sidewall-soil contant "Dashpot": is d = 4 m. The computations that follow make use of the results of the previous example for and = VERTICAL MODE + 2.0 34.4 + 2.0 X 500 119.2 = + 2.1 + 11.9 Aw = 2 (14.9 + 4.3) 4.0 = 153.6 D/B = 6/2.15 = 2.79 = 153.6/57.6 = 2.67 Total 0.05 Static and dynamic stiffness ("spring"): ROCKING MODE rx (AROUND THE LONGITUDINAL AXIS) = 5.5 106 kN/m The dynamic stiffness coefficient is obtained by linear inter- = 1.86 polation between the "fully-embedded" value, B/L = 2.15/7.45 0.29 and the value for the foundation placed in an open trench, Static and dynamic stiffness ("spring"): without sidewalls, 1.26 X 1.86 1.24 Thus, 2.7 5.89 107 kN m and = = 9 0.83 = 7.4 106 kN/m Krx,emb 107 0.74 m "Dashpot": "Dashpot": Cz,emb = + 2.0 310 X 153.6 Total = x 0.05 = = 1.0 LATERAL HORIZONTAL MODE (y) = 4.1 + 2.0 500 635.7 1.0 + 2.0 310 h=4m = = 1.86 (236.5 + 235.3) X 1.0 Aw/L2 153.6/7.452 2.77 Static and dynamic stiffness ("spring"): 96.8 X kN.s.m Ky,emb = + + 0.52(1.86 X Total 188.5 0.05 X 1.25 X 2.0 12.0 106 kN/m The dynamic stiffness coefficient is obtained with the help of the graphs that accompany Table 15.2. The "fully-embedded" value is TORSIONAL MODE (AROUND z, PASSING THROUGH BASE ky,d=6m 0.3 CENTROID) obtained by interpolating between the L/B = 2 and L/B = 6 "Spring": plots, for D/B 2.79. Then for the partial embedment: k(2/3) k(1) m = 0.4 =TABLE 15.2 DYNAMIC STIFFNESSES AND DASHPOT COEFFICIENTS FOR ARBITRARILY SHAPED PARTIALLY OR FULLY EMBEDDED IN A HOMOGENEOUS HALFSPACE. Dynamic Stiffness Static Stiffness Kemb For foundations with arbitrarily- Radiation Dashpot Coefficient shaped basemat Ab of circumscribed Cemb(w) rectangle 2L by 2B; total sidewall- Dynamic Stiffness Vibration soil contact area Aw (or constant Coefficient Rectangular Foundation Mode sidewall-soil contact height d) General Foundation Shape 2L by 2B by d Vertical, Z Fully embedded: with and according to Table 15.1 In a trench: obtained from Table 15.1 Aw = actual sidewall-soil contact area; for constant effective contact height d along the Partially embedded: perimeter estimate by interpolating between Aw = (d) (Perimeter), = the two Fully Fully embedded, Horizontal, y or X and can be estimated in Cy,emb = = terms of and d/B for each Aws = = total effective value of ao from the graphs x sidewall area accompanying this table according to Table 15.1 shearing the obtained from Table 15.1 soil Kx.emb similarly computed from Kx.sur = total effective sidewall area compressing the soil 9, angle of inclination of surface Awi from loading direction Cy.sur obtained from Table 15.1 similarly computed from Cx.surExpressions valid for any basemat shape but constant effective contact height d along the perimeter Rocking, (around the longitudinal axis) + with as in the preceding column and according to Table 15.1 Rocking, ry (around the lateral axis) = total moment of inertia about their base axes parallel to of all sidewall surfaces effectively compressing the soil = distance of surface Awcai from the axis Jws = polar moment of inertia about their base axes parallel to of all sidewall surfaces effectively shearing the soil is similarly evaluated from with y replacing and, in the equation for L replacing B Coupling term Swaying-rocking as in the previous column Swaying-rocking Torsional = + with as in the preceding column = total moment of inertia of all and according to Table 15.1 sidewall surfaces effectively compressing the soil about the projection of the Z axis onto their plane Azi = distance of surface Awi from the Z axis 3.4 is the apparent propagation velocity of compression-extension waves. 579580 Foundation Engineering Handbook 2 GRAPHS ACCOMPANYING L/B 1 TABLE 15.2 ky D / B 0 (2L) 0 0 2 2 2 D k, y (surface) kx 2 1 k(d/D) 2 2 3 3 1 0 0 0 2 2 2 2 0 0 D / B 0 1 2 ky y kx 0 ao 1 0 0 0 2 0 2 ao 2 1 k. y 0 L/B = 8 1 0 2 aoFoundation Vibrations 581 "Dashpot": 3. In view of the complexity of the problem of arbitrarily shaped partially embedded foundations, the formulas and graphs of Table 15.2 provide a very simple and complete solution, while allowing the engineer to use his experience and judgment. To give an idea as to how well the formulas of Table 15.2 may compare with rigorous theoretical solutions, Figure 15.16 refers to two foundations having T-shaped basemats and subjected to harmonic rocking oscillations. The circumscribed rectangles have L/B = 1 and 2 and each 4.0 + 4.3 foundation is embedded at depth D = B. The rigorous results are from a dynamic boundary-element solution and are 1.9 4.0 2.7 4.0 plotted as data points. The developed expressions for Crx, given in Table 15.2, yield for each foundation the correspond- X 1.2 X 4.0 ing continuous lines. The agreement is indeed excellent and 451 + 786 + 318 + 19 + 284 indicative of the capabilities of the simple methods utilized in this chapter. Also encouraging are comparisons of the presented solution with small-scale experimental measure- ments. 3. The Presence of Bedrock at Shallow Depth 500 2315 1.17 310 Natural soil deposits are frequently underlain by very stiff material or even bedrock at a shallow depth, rather than 27.1 extending to practically infinite depth as the homogeneous halfspace implies. The proximity of such a stiff formation to 46 kN.s.m the oscillating surface modifies the static stiffnesses, K, the dynamic stiffness coefficients, k(w), and the dashpot coefficients C(w). Specifically, with reference to Table 15.3 and its graphs Total 0.05 we see the following. (1) The static stiffnesses in all modes increase with the relative depth to bedrock H/B. This is evident from the formulas of It is apparent that embedment has produced very substantial Table 15.3, which reduce to the corresponding halfspace stiff- increases for all "springs" (except the horizontal) and all nesses when H/R approaches infinity. "dashpots." We summarize these effects in terms of the dynamic Particularly sensitive to variations in the depth to rock are stiffness ratio and the equivalent damping ratios the vertical effect being far more pronounced (from the previous example) and (from the value of Cemb with strip footings (factor 3.5 versus 1.3). Horizontal stiffnesses computed herein): are also appreciably affected by H/R (factors of 2 for strip and 0.5 for circle), while the rotational stiffnesses (rocking and torsion) are the least affected. In fact, for H/R > 1.5 the response Mode to torsional loads is essentially independent of the layer Semb thickness. Vertical 1.45 38% 80% An indication of the causes of this different behavior Lateral 0.83 23% 100% (between circular and strip footings and, in any footing, between Rocking 5.90 5% 42% the different types of loading) can be obtained by comparing Torsion 4.25 15% 59% the depths of the "zone of influence" (also called the "pressure bulb") in each case. Circular and square foundations on a homogeneous halfspace induce vertical normal stresses O2 along Several conclusions of practical significance emerge from the centerline that become practically negligible at depths Table 15.2 and the illustrative example. exceeding = 5R; with strip foundations practically vanishes 1. Increasing the embedment (in size and quality) may be a only below 15B. The "depths of influence", Zh, for the very effective way to reduce to acceptable levels the anticipated horizontal shear stresses due to lateral loading are of the amplitudes of vibration, especially if these amplitudes arise order of 2R and 6B for circle and strip, respectively. On the due to rocking or torsion. Such an improvement would other hand, for all foundation shapes (strip, rectangle, circle), be effected mainly by the increase in radiation damping moment loading is "felt" down to a depth, of about 2B or produced by waves emanating from the vertical sidewalls. 2R. For torsion, finally, 0.75R or 0.75B. 2. To rely on such a beneficial effect, however, the engineer Apparently, when a rigid formation "cuts" through the must ensure that the quality of sidewall-soil contact is "pressure bulb" of a particular loading mode, it eliminates the indeed high. In reality, unless special construction procedures corresponding deformations and thereby increases the stiffness. are followed, separation ("gapping") and slippage are likely to occur near the ground surface where the initial confining (2) The variation of the dynamic stiffness coefficient with pressures are small. Such effects may jeopardize the increase frequency reveals an equally strong dependence on H/B. On a in damping and must be taken into account in the analysis. stratum, k(w) is not a smooth function, as with a halfspace, To this end, the areas and moments of inertia of the sidewalls but exhibits undulations (peaks and valleys) associated with on which damping and stiffness depend should be given the natural frequencies (in shear and compression) of the suitably reduced values, rather than their nominal ones. stratum. In other words, the observed fluctuations are theTABLE 15.3 DYNAMIC STIFFNESSES AND DASHPOT COEFFICIENTS FOR SURFACE FOUNDATIONS ON HOMOGENEOUS STRATUM OVER BEDROCK (sources are listed in the text). Homogeneous Stratum H formation Circular Foundation of Radius Foundation Shape B=R Rectangular Foundation 2B by 2L (L > B) Strip Foundation 2L 00 B Vertical, Z Lateral, y Static stiffnesses, K Lateral, X Rocking, rx Rocking, ry Torsional, t Vertical, Z kz = is plotted in Graph III-2 for rectangles and strip Dynamic is obtained from Graph III-1 stiffness coefficients, Horizontal, y or X k(w) is obtained from Graph III-1 obtained from Graph III-3 Rocking, rx or ry Torsional Vertical, Z C2 0 at frequencies f regardless of foundation shape Radiation > dashpot coefficients, At intermediate frequencies: interpolate linearly. C(w) Lateral, y or X [ at At intermediate frequencies: interpolate linearly. fs = Similarly for Cx Rocking, or ry ~ 0 at at Similarly for Torsional, t . Not available.GRAPHS ACCOMPANYING TABLE 15.3 RECTANGLE H/B v=0.30 - STRIP 1 III-1 CIRCLE v=0.30 v=0,30 2 1 4 1 8 0 4 H/B 3 v=0.30 H/B=4 2 H/R=2 1 0 2 0 L/B=1 2 2 0 0 1 8 H/B = 2 v=0.30 H/R=2 1 1 k, 2 0 2 H/B=2 Vs 0 1 2 1 0 Vs a. Vs7 Engineering Handbook ce phenomena: waves emanating from the of Table 15.2 (embedded foundation in halfspace) and of 1 reflect at the soil-bedrock interface and Table 15.3 (surface foundation on a stratum). No further source at the surface. As a result, the explanation seems necessary. on motion may significantly increase at natural frequencies of the deposit. Thus, stiffness (being the inverse of displacement) exhibits 5. Effect of G Increasing with Depth which are very steep when the hysteretic damping in the soil is small (in fact, in certain cases, k would be exactly Often, the assumption of homogeneous layer or halfspace may zero if the soil were ideally elastic). not be realistic, as the soil stiffness usually increases with depth, For the "shearing" modes of vibration (swaying and torsion) even in uniform soils. The prime cause is the increase of the the natural fundamental frequency of the stratum, which confining pressure with depth, and the ensuing increase of controls the behavior of k(w), is (for example, according to Equation 15.4). The effects of such an inhomogeneity could be assessed with the help of Table 15.5, which provides information for foundation on a number of (15.60) inhomogeneous deep deposits, including (but not limited to): A deposit with (low-strain) shear modulus while for the "compressing" modes (vertical, rocking) the corresponding frequency is (15.62) (15.61) which is representative of many cohesionless soil deposits, in which the shear modulus is proportional to the square root (3) The variation of the dashpot coefficients with frequency of the confining pressure. Go = the modulus at the ground reveals a twofold effect of the presence of a rigid base at relatively surface, that is, at Z and should not be confused shallow depth. First, the C(w) also exhibit undulations (crests with the low-strain modulus In this chapter we and troughs) due to wave reflections at the rigid boundary. essentially always deal with very small strains and hence all These fluctuations are more pronounced with strip than with G appearing in Tables 15.1 to 15.5 are low-strain circular foundations, but are not as significant as for the values. corresponding stiffnesses k(w). Second, and far more important A deposit with from a practical viewpoint, is that at low frequencies, below the first resonant ("cut-off") frequency of each mode of (15.63) vibration, radiation damping is zero or negligible for all shapes of footings and all modes of vibration. This is due to the fact that no surface waves can exist in a soil stratum over bedrock which is representative of deposits of saturated normally and at such low frequencies; and, since the bedrock also prevents slightly overconsolidated clays. waves from propagating downward, the overall radiation of A deposit with wave energy from the footing is negligible or nonexistent. Such an elimination of radiation damping may have severe consequences for heavy foundations oscillating vertically or (15.64) horizontally, which would have enjoyed substantial amounts of damping in a very deep deposit (halfspace)-recall illustrative examples for Tables 15.1 and 15.2. On the other hand, since which can simulate deposits with a relatively faster increase the low-frequency values of C in rocking and torsion are small of G at large depths, a is a parameter determined by fitting even in a halfspace, operation below the cut-off frequencies may the experimental test results. not be affected appreciably by the presence of bedrock. Note that at operating frequencies f beyond fs or for as The following trends are worthy of note in Table 15.5 and appropriate for each mode, the "stratum" damping C(H/B) the accompanying graphs. fluctuates about the "halfspace" damping The "amplitude" of such fluctuations tends to decrease with 1. For the static stiffnesses, K, use of a single "effective" increasing H/B; moreover, if some wave energy penetrates into modulus, G = Geff, in the formulas for a homogeneous bedrock (as does happen in real life thanks to some weathering halfspace (Table 15.1) would, at best, provide the correct of the upper mass of rock) the fluctuations tend to wither stiffness in only one particular mode. This is because the away-hence the recommendation of Table 15.3. "pressure bulb" of each mode reaches a different depth and is thus affected by different values of modulus. As one might expect, vertical loading (especially on a strip) penetrates the deepest-the "effective depth" Zeff is of the order of one to 4. Foundation Embedded in Stratum two times B, the foundation semiwidth or radius. Moment loading, on the other hand, is the least affected by inhomogeneity, with "effective depths" merely ranging from As can be seen from Table 15.4, embedding a foundation in a to B. shallow stratum, rather than a halfspace, has one additional 2. In strongly inhomogeneous soils, the dynamic stiffness effect over those addressed in Table 15.2: the static stiffnesses coefficients, k, plotted as functions of are always tend to increase thanks to the decrease in the depth of the smaller than those of a homogeneous halfspace with deforming zone underneath the foundation. = The differences, however, are quite small and could It is evident that the results summarized in Table 15.4 can be neglected in practical applications, as suggested in follow from a proper combination of the pertinent results Table 15.5.TABLE 15.4 DYNAMIC STIFFNESSES AND DASHPOT COEFFICIENTS FOR FOUNDATIONS EMBEDDED IN HOMOGENEOUS STRATUM OVER T. d H Stratum rigid formation Foundation Shape Circular Foundation of Radius R Strip Foundation Vertical + Static Horizontal, y or stiffnesses, K Rocking, rx or ry Coupled swaying-rocking - Torsional Dynamic The relationships between Kemb and follow approximately the same pattern as those between embedded and stiffness k(w) surface foundation on a homogeneous halfspace. Therefore, use the results of Table 15.2 as a first approximation. coefficients, Radiation Cemb exceeds Csur by an amount that depends on the geometry of the sidewall-soil contact surface and is practically dashpot independent of the presence or absence of a rigid base at shallow depths. Therefore, use the results of Table 15.2, coefficients, but with Csur corresponding to the layered profile and thus obtained according to Table 15.3 (approximate guideline). . Sources are listed in the text. are the stiffnesses for the corresponding surface foundations, and can be obtained from Table 15.3.TABLE 15.5 DYNAMIC STIFFNESSES AND DASHPOT COEFFICIENTS FOR SURFACE FOUNDATIONS ON "DEEP" INHOMOGENEOUS DEPOSITS. : Examples of G(z) Go G IN z IN Static Stiffnesses K "Equivalent" Depths Vibration Mode Square Square Strip Vertical, Z 1 2 Horizontal, xory 1/2 Rocking, 1/3 Torsional, t The above expressions are only crude approximations based on limited information. They should be used judiciously. Also, one could utilize static computer codes and procedures. The strip stiffnesses are per unit length.Dynamic Stiffness Coefficients k (w) All Modes k decreases only slightly faster with ao in a strongly inhomogeneous than in a homogeneous soil deposit. For practical applications one can make the approximation k(a) with little error for realistically inhomogeneous deposits (i.e., if a is not too large) Radiation Damping Coefficients C (w) General Foundation Base Shape (area inertias Jb) General Expression High-frequency Asymptotic Expression Vertical, Z Horizontal, y Horizontal, Rocking, rx Rocking, ry Torsional, t All the dimensionless coefficients are invariably smaller than the corresponding coefficients for a homogeneous halfspace of shear modulus Go that were given in Table 15.1. Graph V plots vs L/B and ao = for certain combinations of the inhomogeneity parameters a and n.588 Foundation Engineering Handbook GRAPHS ACCOMPANYING TABLE 15.5 G(z) / Go 3 y 0 0 .2 B G(Z) 1 2 Go 2 4 Z B 3 6 2 4 V-1 CIRCLE V-2 STRIP 1 (homogeneous) 2 0.5 0 1 2 1 0 0.5 2 0 2 1 c; 2/3 1 1 0 2 0 1 1 0.025 0.5 0 0 2 1 2 WB WB V50Foundation Vibrations 589 3. The strongest influence of modulus increasing with depth is For three characteristic soil profiles, Table 15.6 presents on radiation damping, especially in the translational modes. simple algebraic expressions for estimating of a circular solid At low frequencies damping remains invariably lower than in pile with diameter d and Young's modulus For each profile, a homogeneous halfspace of modulus Go. (Of course, if the the only soil parameter that affects is the reference Young's homogeneous halfspace had a modulus G = for which modulus, at a depth Z = d. the static stiffnesses coincide, the differences would be For the three lateral impedances ("springs" and "dashpots"), even greater!) At higher frequencies the discrepancies and defined in Figure 15.17, Table 15.6 between damping on the two media (inhomogeneous and presents easy-to-use formulas, which, however, are valid only homogeneous with G = tend to become vanishingly for piles with length small. The cause and consequences of such a behavior are explained below. (15.66) Such piles are described as "flexible" piles in the literature. But A medium with continuously varying wave velocity can note that a good majority of real-life piles, even some with large reflect propagating waves. In this case, even total reflection of diameters, would fall into this category. Among the exceptions the downward-propagating waves is possible owing to the are short piers and caissons. increase of soil velocity with depth. A discontinuity in velocity From a theoretical point of view, most of the formulas in (e.g., bedrock) is not necessary for such a reflection, since the Table 15.6 are reasonably accurate, as they are basically curve rays in inhomogeneous media are not straight lines but curves fits to rigorous numerical results. The real difficulty, however, (e.g., circular arcs for G(z) as in Equation 15.63). Hence, wave is to select the proper profile and modulus for the supporting energy cannot be radiated away as effectively as in homogeneous soil. Even with a uniform top layer, the secant soil modulus media-small radiation damping. In fact, as the degree of will change with the magnitude of induced strains, which inhomogeneity increases, radiation damping decreases, in spite decreases with depth. Other nonlinear phenomena, such as of the greater soil stiffness. development of a gap between pile and soil near the ground On the other hand, at "very high" frequencies the wave- surface, further complicate the problem. One solution might be lengths of the emitted waves are "very small" and the source to conduct a suitable full-scale or small-scale lateral pile load "sees" the transmitting medium as a homogeneous halfspace test in the field. By "suitable" we mean one that produces having modulus and velocities Vso (in the shearing modes) deformations comparable with those anticipated in the final and (in the compressing modes). It appears that small" design. Reference is made to Blaney and O'Neil (1986), and to may not be small in absolute terms, merely comparable with Novak (1985), who have described such field tests, and to the dimensions of the source (the foundation). Consequently, Richart and Chon (1977), who conducted a somewhat similar "very high" frequencies are always within the range of laboratory-scale test, among several others. frequencies of practical interest (ao590 TABLE 15.6 DYNAMIC STIFFNESSES AND DAMPING COEFFICIENTS FOR FLEXIBLE PILES Linear Increase of Soil Modulus with Depth* Parabolic Increase of Soil Modulus with Depth* Es Constant Soil Modulus at All Depths Es Es=constant H length" Natural shear frequency of fs deposit fs = fs where the S-wave velocity at depth Z = H (bottom of stratum) where VSH the S-wave velocity at depth Z = H (bottom of stratum) Static lateral (swaying) stiffness = Lateral (swaying) stiffness KHH 1 coefficient KHH 1 KHH 1 Lateral (swaying) coefficient: DHH + for DHH + CHH = 2KHHDHH/W for { Static rocking stiffness KMM = Rocking stiffness coefficient KMM 1 KMM 1 Rocking dashpot coefficient: KMM 1 { f>ts { for { for Static swaying-rocking cross- KHM = = stiffness Swaying-rocking cross- KHM = KMH? 1 stiffness coefficient KHM = 1 KHM = KMH 1 Swaying-rocking dashpot coefficient: { DHM + for CHM = 2KHMDHM/W { forThe axial stiffness of a pile depends not only on its relative compressibility but also on the slenderness ratio L/d and the tip support conditions (end-bearing versus floating). 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