DYNAMIC RESPONSE OF ARBITRARILY

SHAPED FOUNDATIONS

By Ricardo Dobry1 and George Gazetas,2 Members, ASCE

ABSTRACT: The paper presents a method to compute the effective dynamic stiffnesses (K) and dashpots (C) of arbitrarily shaped, rigid surface machine foundations placed on reasonably homogeneous and deep soil deposits. The method is based on a comprehensive compilation of a number of analytical results, augmented by additional numerical studies and interpreted by means of simple physical models. All results are offered in the form of easily under­stood dimensionless graphs and formulas, covering the six modes of vibration and a wide range of frequencies, for both saturated and unsaturated soils. Comparisons are made with the widely used equivalent circle approximation. The proposed method is applicable to a variety of area foundation shapes, rang­ing from circular to strip and including rectangles of any aspect ratio as well as odd shapes differing substantially from rectangle or circle. The results con­firm that both frequency and foundation shape may significantly affect K and C. Insight is gained into the mechanics of radiation damping and a general conclusion is drawn regarding its magnitude at high frequencies for different foundation shapes and vibrational modes. The practical application of the method is illustrated with a specific example, while a companion paper (7) presents supporting experimental evidence from model tests.

INTRODUCTION

The development in the 1950s and 1960s of the lumped parameter models for circular rigid foundations resting on the surface of i homo­geneous elastic half-space, summarized in Refs. 30, 31, and 39, consti­tuted a significant advance in the art of analyzing and designing ma­chine foundations. In these models, once the characteristics of the rigid mass supported by the half-space are known, the actual system is re­placed for each vibration mode by the three parameters of a one-degree-of-freedom system: a mass (or mass moment of inertia), a spring, and a dashpot. The parameters of these 1-DOF systems or "analogs" are as­sumed to be frequency-independent, and thus are functions only of the properties of the half-space, the radius of the equivalent circular foun­dation area, and the total mass or mass moment of inertia of the foun­dation block plus machinery. This makes the analogs simple and attrac­tive for practical use. The use of these analogs for circular rigid foundations has been supported by a number of studies (4,9,20,35-38). Other effects such as foundation embedment and the influence of a rock base unde r the soil have also been investigated (9,19).

Machine foundations are generally not circular in shape; however, the difficulty of finding solutions for other shapes has forced the engineers into Using the circular solutions irrespective of the actual foundation shape. The usual practice for transforming any shape is to take an equivalent

'Prof, of Civ. Engrg., Rensselaer Polytechnic Institute, Troy, NY. 2Assoc. Prof, of Civ. Engrg., Rensselaer Polytechnic Institute, Troy, NY. Note.—Discussion open until July 1, 1986. Separate discussions should be sub­

mitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on June 25, 1984. This paper is part of the Journal of Geotechnical Engineering, Vol. 112, No. 2, February, 1986. ©ASCE, ISSN 0733-9410/86/0002-0109/$01.00. Paper No. 20370.

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circle of the same area (translational modes) or equal moment of inertia (rotational modes) (30,39). However, an inspection of recent analytical and experimental investigations, both of rectangular foundations having aspect ratios up to L/B = 10, and of strip footings (L/B = °°), suggests that this procedure is not necessarily correct, and that the aspect ratio L/B can have an important influence on dynamic stiffness and damping. In fact, an increasing number of solutions have been published for dif­ferent foundation shapes; unfortunately, they have often been reported in a form not easily accessible to practicing engineers. These solutions include the vertical static stiffness for a variety of shapes (3,33); and dy­namic stiffness and damping of square, rectangular (8,32,38) and strip (12,13,15,18,21) footings. In addition, vibration tests on models of cir­cular and rectangular foundations with aspect ratios up to L/B = 6 have been performed by Chae (4) and Erden (9).

In the original report of this investigation (6) and in the rest of this paper, analytical solutions available for dynamic stiffness and damping, and corresponding to solid area shapes ranging from circle or square (L/ B = 1) to strip (L/B = °°), have been collected, studied, and organized into dimensionless charts and equations, ready for use in practical en­gineering applications. Charts and equations are presented for each one of the six vibration modes. The dimensionless parameters have been carefully selected to provide: (1) Simple charts applicable to a wide va­riety of solid area foundation shapes (areas with holes, such as ring foundations, are excluded); (2) a framework for a better understanding of the role of the various factors; and (3) an evaluation of the equivalent circle approximation. A numerical example of the proposed method for determining the dynamic stiffness and damping coefficients of a ma­chine foundation is presented at the end of the paper. The companion paper (7) includes comparisons between analytical predictions using this proposed method, and measurements on model foundations reported in Ref. 9. The method has been extended elsewhere to embedded foun­dations subjected to vertical vibration (16).

BASIC APPROACH

Fig. 1 defines the main parameters. A massless rigid disk of arbitrary shape and contact area A lies on the surface of an elastic homogeneous half-space, and is excited by six dynamic forces and moments of the form V = V0 e

Mt, where a> = circular frequency in rad/s, t = time, and i = V - l . These dynamic forces and moments are V (vertical), Hx and Hy (horizontal), Mx and My (rocking), and Mt (tprsion). For each of these six vibration modes it is possible to define an equivalent effective dy­namic spring, K, and a radiation (wave propagation) dashpot coefficient, C. For example, for the vertical mode, K = K2 and C = Cz, and:

V Kz + mC2 = - • • • . ( ! )

where uz = complex vertical displacement of the actual foundation. Five expressions similar to Eq. 1 define K and C for the other modes. The symbols assigned herein to these pairs of K and C for all modes are listed

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(b)

' Massless M , , - ^ Rigid Disk

H 2L H

I v = v Massless - Rigid Disk

c 2 M | K Z |

/777777777T77777777777777777-

X-

FIG. 1.—Basic Approach and Parameter Definition

in Table 1. K and C are functions of the frequency, co, as well as of the characteristics of both the foundation-soil contact surface area and of the elastic half-space. In addition to the area, A, other parameters of the foundation required are the dimensions of the circumscribed rectangle, 2L and 2B [Fig. 1(a)], and the moments of inertia of the area A around

TABLE 1.—Definition of Basic Symbols and Parameters Required

Vibration mode 0)

Vertical

Horizontal (short direction)

Horizontal (long direction)

Rocking (short direction)

Rocking (long direction)

Torsion

Axis (2)

2

V

X

X

V

z

Exciting force or

moment of frequency

CO

(3)

V

«»

Hx

Mx

My

Mt

K" (4)

Kz

K,

Kx

K„

K-ry

K,

Ch

(5)

cz

c,

cx

Cn

c„,

c,

PARAMETERS REQUIRED

General for all modes

(6)

Frequency of excitation, to

Foundation area: Area, A

L B

Homogeneous half-space

Mass density, p Vs = (G/P)1 '2

Poisson's ratio, v Material damping

ratio, p

Specific of mode

(7)

Area A

Area A

Area A

Moment of inertia lx

Moment of inertia ly

Polar moment of inertia /

Expressions for:

Circle R = B

(8)

A = t rB 2

A = TTB2

A = TTB2

, 1 K4

4

, l n* 1,—fl

/ - j i r B *

Rectangle 2L x 2B

0) A = 4LB

A = 4LB

A = 4LB

4 , 3

4 , l„ — I J " 3

4 , , / = - L B ( L 2 + B2)

3

*K = effective dynamic stiffness (p = 0); K(|3) = K - coCp (Eq. 2). bC = Radiation dashpot (p = 0); C(p) = C + (2K/co) p (Eq. 3).

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the three axes of symmetry, x, y and z, with origin at the centroid (point 0) of the contact area. These parameters, plus the properties of the soil (half-space), as listed in Table 1, were used to construct the dimension-less charts presented throughout this paper for dynamic stiffness K and radiation dashpot C.

All values of K and C included herein assume perfectly elastic soil with zero material damping. However, even at small strain levels soil has a material, hysteretic damping, usually specified through a frequency-in­dependent damping ratio, p. For most soils and for well-designed ma­chine foundations causing little straining of the soil mass, p ranges typ­ically from 0.02 to 0.05 (31). Once K and C have been determined for an elastic medium with p = 0, the effect of material damping is incorpo­rated, approximately, using the correspondence principle of viscoelas-ticity (23). The corrected dynamic stiffness, K(P), and effective damping coefficient, C(p), to use in design are given for all vibration modes by:

K(P) = K - coCp (2)

2K C(p) = C + — p (3)

The shapes the writers considered in developing the charts and equa­tions for K and C (presented in Ref. 6 and summarized here) were mainly: circle, square, rectangles of several aspect ratios, L/B, and strip foun­dation. However, for the vertical static stiffness case, results for other shapes, such as regular hexagon, triangles, ellipses and rhombuses were also gathered. Results from a variety of analytical solutions published as curves or tables of numerical values in fifteen articles and reports (1,3,8,12,13,17,18,20,21,32,33,35,36-38) were considered. Some supple­mentary numerical analyses were also kindly provided by Tassoulas (personal communication) or performed by the writers. The published solutions were obtained by the respective researchers using different as­sumptions (such as "rough" or "relaxed" boundary conditions); using a variety of analytical, semi-analytical and numerical procedures; and using calculations performed with various degrees of precision. As a re­sult, there is some • unavoidable scatter in the numbers obtained from different sources for what may appear to be the "same" situation. This scatter approximates that usually present in experimental results, where different answers are obtained from apparently "identical" tests. There­fore, it was decided to treat these analytical results as one would usually do with experimentally measured values: i.e., to plot them as raw "data points" in dimensionless charts, and to fit curves to these data points. This is done consistently throughout the paper.

STIFFNESSES K (STATIC) and K (DYNAMIC)

Fig. 2 presents dimensionless charts for the effective dynamic stiffness K in all modes of circular and strip foundations, for the dimensionless frequency range, 0 < a0 = <aB/Vs £ 4, and for Poisson's ratios of the soil v = 0.333 and v = 0.50, typical of unsaturated and saturated soil. The information in Fig. 2 was obtained from the literature, and from addi­tional calculations performed by the writers using the computer program

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*,-K

0.5

0

0.5

-1

-1.5

-?

Ki 4GB

\ _ = 1/2

\

\ -c- , / 3

-4 irGB

aw

\

°,f

FIG. 2.—Static (K) and Dynamic (K) Stiffnesses of Circular and Strip Foundations

described in Ref. 10, so as to complete the strip foundation curves in the high frequency range.

Available closed-form solutions for the static stiffnesses of the circle are included in Fig. 2, which also incorporates an expression for the rocking static stiffness (per unit length) of the strip, Krx/2L, published by Muskhelishvili (28):

2 X

Kr •nGB' 1 +

In (3 - 4v) (4)

2L 2(1 - v)

For each one of the cases in which the static stiffness, K ¥= 0, is available, the variation of dynamic stiffness with frequency is presented in Fig. 2 as a curve of k = K/K versus a0. In these cases, the dynamic stiffness, K, is obtained from:

K = K-k. (5)

Eq. 5 is not applicable to the vertical (Kz) and horizontal (Ky) dynamic stiffnesses of the strip, because the static Kz = Ky _= 0 in bothcases. Instead, the dimensionless coefficients Sz = (1 - v)Kz/2LG and Sy = (2

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- v)Ky/2LG are plotted in Fig. 2(b). The behavior of Sz and Sy versus a0 is generally similar: Sz = Sy = 0 at a0 = 0, both Sz and Sy increase rapid­ly with a0 at low frequencies, and they stabilize at high frequencies (ex­cept that for v = 0.5, Sz decreases and even becomes negative for a0 ss 1.4). It is convenient to define pseudostatic values Sz and Sy, repre­senting the low frequency range, a0 = 0.05 to 0.2, to be used later in defining the influence of foundation shape on static and dynamic stiff­ness. The writers arbitrarily selected the values Sz = 0.8 and Sy = 2.24 for this purpose, as shown in Fig. 2(b).

The original report of this research (6) presents the results of a detailed study performed for K and K of circular, square, rectangular, strip and other shapes. That information is summarized in the rest of this section, while Figs. 3 to 8 reproduce some of the original results of the study. (The key for the data points in Figs. 3 to 8 is given in Table 3.)

Vertical Stiffness.—The study of vertical dynamic stiffness Kz is pre­sented in Table 2 and Figs. 3-4. Table 2 presents values of the dimen-sionless static stiffness parameter Sz = [(1 - v)/2LG] Kz, obtained for a wide variety of rigid foundation shapes from several references. The shape is represented by A/4L2 (with A/AL2 = B/L for rectangular shapes). The

TABLE 2.—Static Stiffness Parameter S2 for Different Rigid Foundation Base Shapes

Shape (1)

Square

Circle

Hexagon (regular) Rectangle

Triangle (equilateral) Ellipse Semicircle Ellipse Triangle (45745790°) Rhombus (60°) Rectangle Triangle (30760790°) Rhombus (45°) Ellipse Rhombus (30°) Ellipse Rectangle Rectangle Rectangle Strip

L/B (2)

1

1

1 2

1.15 2 2 3 2 2 4 2.3 2.4 4 3.7 6 8

10 20

CO

A/(4L2) (3)

1

0.785

0.65 0.5

0.44 0.39 0.39 0.26 0.25 0.25 0.25 0.22 0.21 0.20 0.14 0.13 0.125 0.10 0.05 0

1 - v Sz = Kz

2GL (4)

2.28 (2.13-2.44)

2.01

1.84 1.65

(1.55-1.80) 1.59 1.46 1.49 1.24 1.24 1.21 1.27 1.18 1.18 1.13 1.06 0.99 0.98 0.91 0.78 0.80

Ref. for K, (5)

1,8,17,33

4 GB Kz =

1-v 3 3

3 3 3 3 3 3 3 3 3 3 3 3

3,17 1,17 3,34

a

*SZ - 0.80 for the strip foundation is a "pseudo static" value selected for the purposes of this paper. Strictly speaking, S, = 0 for this case [see Fig. 2(b)].

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A/(4L 2>

FIG. 3.—Vertical Static Stiffness Parameter Sz versus Base Shape (the Analytical Data Points for the Different Shapes were Obtained from Table 2). Note that for the Rectangular Shapes, A/(4L2) = B/L

parameter S2 is the same used for the strip foundation in Fig. 2(b), and the pseudostatic S2 = 0.8 has been included in Fig. 3 for A/4L2 = 0. In the approximate equivalent circle method, Kz for any shape is obtained from the circular foundation equation, Kz = 4GR/(1 - v), with the radius R of the equivalent circle being R = VA/TT. Therefore, the equivalent circle approximation, plotted in Fig. 3 as a dashed line, predicts S2 = (4/V^)(A/4L2)1/2. All "data points" from Table 2 (see also Ref. 16 for examples of determination of A/4L2 for non-rectangular shapes) are graphically depicted in Fig. 3, including ranges of S2 for the square and for the rectangle of B/L = A/4J/ = 0.5. The data points plot consistently, thus confirming the validity of using the parameters A/4L2 and Sz. The equivalent circle is very good for A/4L2 a 0.3 to 0.5 (corresponding to L/B S 2 to 3 for rectangles), but the agreement deteriorates as the foun­dation becomes longer. The following equations are proposed for prac­tical use:

A S2 = 0.8 for —r<0.02

4L2

(A\°-75 A S2 = 0.73 +1.54 (^—j for —2 > 0.02

Fig. 4 presents the variation of the dynamic coefficient kz = KJKZ versus a0, for both unsaturated (v « 0.33) and saturated (v = 0.5) soil. The "pseudostatic" expression Kz = 0.80 [2LG/(1 - v)] was used for all strip foundation data points. The figure shows clearly the influence of frequency and foundation shape on kz and Kz. For squares, circles and rectangles of moderate aspect ratio (L/B s 4) on unsaturated soil, Kz in Fig. 4(a) is almost constant or decreases somewhat as frequency in­creases; however, for long foundations Kz can be as much as 30% or 40% larger than Kz. For saturated soil [Fig. 4(b)] the variation of Kz with frequency in long foundations is even more dramatic, with kz first in-

(6)

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VERTICAL

(a| UNSATURATED SOIL ( f = 0.33)

FIG. 4.—Vertical Dynamic Stiffness Kz versus Frequency for Different Base Shapes

creasing and then rapidly decreasing to zero and becoming negative at higher frequencies.

If the equivalent circle approach were consistently used in conjunction with the curves for L/B = 1 in Fig. 4 to obtain kz of a long foundation, the error would be even larger than suggested by the corresponding solid lines in Fig. 4. For example, for v — 0.50, a rectangle of L/B = 6

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4

3

2

1

K=S-^£

,S,»2.24 (or

[ -4=40.16 m /

/ /

(a) Horizontal (y)

/ A\o^e , - S v M . 5 -==; 1

1 A ^ 4 L ' ^ - - 1 (or-==20.16 ^ > ^

/ S» ./r>v 4L-(Equivalent Circle

Approximation)

h — 2 L — ^

2B K"AREA A ^ J

|y " J

0 0.2 0.4 0.6 0.8 A/(4L2 )

1-B/L 0.75- v

FIG. 5.—Horizontal Static Stiffnesses Ky and K, versus Base Shape

and a0 = a)B/Vs = 0.8, fc2 = 0.93 in Fig. 4(b), and the curve for L/B = 1 gives a similar kz = 0.89. In fact, the actual error when using the equiv-alent circle is much larger, because once the equivalent circle radius, R = y/A/'n, is obtained, the original width B is disregarded in subsequent calculations. Therefore, in the previous example, a0 = oiR/Vs = 2.76 ooB/ Vs = 2.2 would be mistakenly used instead of 0.8 to enter the L/B = 1 curve, and kz = 0.25 would be estimated [see also Fig. 2(a)] instead of the actual kz = 0.93. Therefore, in this example, the equivalent circle would underestimate kz and Kz by 70-80%. Fig. 4 includes dashed line curves showing the values of kz which would be predicted for L/B = 6 by the equivalent circle method. The comparison between these lines and the corresponding solid lines for L/B = 6 in the same figure indi­cates that very large errors would be committed, and clearly shows that the equivalent circle procedure must not be used to obtain the vertical dynamic stiffness of long foundations.

Horizontal Stiffnesses.—The horizontal dynamic stiffnesses, Kx and Ky, are presented in Figs. 5-6. Fig. 5(a) presents the dimensionless static stiffness parameter, Sy = [(2 - v)/2LG] Ky, versus foundation shape factor A/4L2. Again, the pseudostatic value Sy = 2.24 from Fig. 2(a) has been included in Fig. 5(a) for the strip foundation, A/4L2 = 0. All data points from square to strip obtained from several sources plot consis­tently in Fig. 5(a). The equivalent circle approximation predicts Sy = (8/VTT)(A/4L2) , and this expression is plotted as a dashed line. As ex­pected, the agreement with the data points is very good for square and short rectangular foundations, with A/4L2 a 0.3 to 0.5 (L/B ss 2 to 3 for rectangles), but deteriorates for longer shapes. The following equations are proposed for practical use:

Sy = 2.24

S„ = 4.5 V4L;

0.38

for -^ r<0 .16 4L2

for —r>0.16 4L2

(7)

The writers also performed a study of Kx based on available results

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HORIZONTAL ly)

FIG. 6.—Horizontal Dynamic Stiffness Ks versus Frequency for Different Base Shapes

for circle, square and rectangular foundations of aspect ratios, L/B, up to 10, and for soil Poisson's Ratios between 0.25 and 0.49. It was found that the difference Ky - Kx increases as both L/B and v increase. The corresponding correlation is plotted in Fig. 5(b); it allows calculating Kx

once Sy and K¥ are obtained from Fig. 5(a) or Eqs. 7. The corresponding expression for Kx is;

y 0.75 - v \ L (8)

Fig. 6 presents the variation with a0 of the ratio ky = Ky/Ky. Again, the pseudostatic Ky = 2.24 [2LG/(2 - v)] from Fig. 2 was used for the strip foundation. This figure shows clearly the influence of frequency and foundation shape on ky, as well as the effect of Poisson's ratio for the case of long foundations. For L/B = 1, Ky = Ky, but for long foun­dations Ky can be as much as 50% higher than Ky . Therefore, the curve for L/B = 1 approximately represents the value of ky that would be pre­dicted by the equivalent circle method for any foundation shape; the error is quite significant for long foundation and high frequencies.

All available analytical evidence suggests that Kx is unaffected by fre­quency and hence Kx = Kx for horizontal dynamic motions in the long direction of the foundation, irrespective of foundation shape (8). There­fore, kx= lis adopted here.

Rocking Stiffnesses.—Ref. 6 presents the study of dynamic rocking stiffnesses, Krx and K^ based on results from a number of sources (8,10,12,13,18,20,32,38) and on Eq. 4 for Krx of the strip foundation. The dimensionless static stiffness parameters for rocking in the short arid long directions of the foundation, S„ and Sn,, are:

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g , 0 . 2 S ^

(9) _ 1 - v w

1 - v

where K„, K„, = static rocking stiffness around axes x, y; and Ix, ly = foundation area moments of inertia around the x, y axes. If the equiv­alent circle approximation were valid, with Rx = (4IX/TT)025 and Ry = (4IV/IT)0'25 being the radii of the equivalent circles, then it would be Srx = 3.2(B/L)015 and S„, = 3.2.

In Ref. 6, Srx and Sry were correlated with the shape factor B/L, instead of A/4L2 utilized previously for the vertical and horizontal modes, for two main reasons: (1) No significant difference was found between the Srx (or S^) values of circular and square foundations, which have the same B/L but different A/4L2; and (2) the rest of the available data points collected in Ref. 6 correspond to rectangles and strip foundation, for which B/L = A/4L2.

The following equations were obtained in Ref. 6 for Srx and are sug­gested for practical applications:

B Srx = 2.54 for - < 0.4

/B\°-25 B * ( 1 0 )

Srx = 3.2 I-J for - > 0 . 4

The data collected in Ref. 6 for the static stiffness coefficient for rock­ing in the long foundation direction, and for L/B s 5, indicated that the equivalent circle approximation represents well the data points, and its use is suggested:

B Sny = 3.2 for - > 0 . 2 (11)

The variations with frequency of the dynamic stiffness ratios, krx = Krx/Krx and fc^ = K^/Kn, are reproduced in Fig. 7. Notice that k„ de­creases moderately with frequency, this decrease being essentially in­dependent of L/B. The dashed line in Fig. 7(a) indicates the error that would be committed if krx were calculated using the equivalent circle method for L/B = 6. Fig. 7(b) indicates some influence of L/B on k^. Note that the curve for 2 < L/B < 5 in Fig. 7(b) is based on values of Poisson's Ratio between 0.25 and 0.33, which are appropriate for un­saturated soils; additional information is needed before this figure can be used with complete confidence for fully saturated soils (v = 0.50).

Torsional Stiffness.—Ref. 6 presents the study of dynamic torsional stiffness, Kt, based on results for square, circle and rectangular shapes up to L/B = 4, obtained from Refs. 8, 20 and 36. The following expres­sion was adopted for the dimensionless static torsional stiffness param­eter: S, = (Kt)/[G(]f-7% where Kt = static torsional stiffness and J = Ix

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( 0 ) ROCKING (AROUND x-AXIS}

uB

FIG. 7.—Rocking Dynamic Stiffnesses Krx and K^ versus Frequency for Different Base Shapes

* - ^ 1 0.5

' " K 7

v5

FIG. 8.—Torsional Dynamic Stiffness K, versus Frequency for Different Base Shapes

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+ Iy is the polar moment of inertia of the foundation area. If the equiv­alent circle approximation were valid for torsion, with R = (2//IT)0-25 being the radius of the circle, it would be S, = 3.8 independently of foundation shape. The writers found that his constant St is appropriate for short foundations, L/B s 2 to 3, but S, increases rapidly for longer foundation shapes (6). The following expression was fitted to the data points:

S, = 3.8+ 10.7 1 - - f o r - S 0 . 2 5 (12) V LJ L

Fig. 8 presents the variation of kf with frequency for L/B = 1 and L/ B = 4; in both cases kt decreases slightly with frequency.

RADIATION DASHPOT COEFFICIENTS C

Basic Considerations: Translational Modes.—It is useful to establish first a general framework for the radiation dashpots C in the transla­tional (horizontal and vertical) modes, to guide the selection of the cor­responding dimensionless parameters. It is well known (i.e., 14, 22, 29) that, for a 1-D or plane body wave propagating with speed V in an in­finitely long elastic medium, a boundary condition which exactly sim­ulates this infinite extent for any wave shape is provided by the viscous dashpot C = pVA, where p = mass density of the medium and A = cross-sectional area normal to the direction of propagation. Therefore, for this 1-D case, C = pVA is a perfect dashpot analog for radiation damping. This suggests that the dimensionless parameters to be selected for C in vertical and horizontal vibrations of the foundation should have the form C/pVA, where C = Cz, Cx or Cy depending on the mode of vibration; A = foundation area; p = mass density of the half-space (soil); and V = appropriate wave velocity in the soil for that vibration mode. As sketched in Fig. 9(a), vertical vibration generates mainly compres­sion-extension waves in the soil near the foundation, while Fig. 9(b) sug­gests that shear waves are mainly induced by the horizontal vibration. Therefore, the values of V to be used must be different for the horizontal and vertical modes.

If the foundation vibration phenomena in the translational modes were purely 1-D, it would be C/pVA = 1 for all foundation shapes and fre­quencies. However, the phenomenon is 2-D or 3-D, as sketched in Fig. 9. For a purely 1-D situation, 0 = 0 and waves would be generated only perpendicular to the foundation area, while in actuality waves usually

a) Vertical Excitation b) Horizontal Excitation

Compression-Extension K Shear Waves t e \ Waves

FIG. 9.—Wave Propagation and Radiation Damping in Translational Modes

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spread out in other directions, and 6 > 0; thus, generally C/pVA 5̂ 1. Note that a dimensionless parameter similar to C/pVA is used in

acoustics for the analysis and design of speakers, modeled as a vibrating piston mounted on an extended rigid surface or baffle. There, it has been found that C/p VA is different from 1 and 9 is large at low excitation frequencies, while C/pVA approaches 1 and the angle 6 tends to 0 at high frequencies. At these high frequencies all points on the surface of the piston behave as independent sources radiating 1-D waves perpen­dicular to the piston. The waves spreading out from these different point sources in nonperpendicular (6 > 0) directions cancel each other due to destructive phase interference (24,27). Therefore, it could be hypothe­sized that the use of dimensionless parameters of the form C/pVA or machine foundations could have the advantage of C/p VA converging to a value of about 1 at high frequencies, irrespective of foundation shape.

Horizontal Dashpots.—The shear wave velocity of the soil, Vs, is se­lected for the two horizontal dashpots Cx and Cy, and the two dimen­sionless parameters, cx = Cx/pVsA and cy = Cy/pV$A are defined. Fig. 10 presents the variation of cy versus a0 = o>B/Vs, for a variety of foun­dation shapes. The key for the data points in Fig. 10 and subsequent damping figures is given in Table 3. For (L/B) = 1 to 2, cy — 1 irre­spective of frequency, but for very long foundations (L/B a 10), cy is significantly larger than 1 at low frequencies. In any case, and as ex­pected, cy = 1 at high frequencies irrespective of foundation shape. Also, the value of cy for a given a„ and L/B is not signficantly influenced by the Poisson's Ratio of the soil.

Available analytical evidence suggests that for horizontal dynamic mo­tions in the long direction of the foundation, cx = CJpVsA is close to 1 irrespective of frequency and foundation shape (8). Therefore, the curve labelled "L/B = 1" in the cy plot of Fig. 10 is suggested for cx for all foundations having L/B < 3; for L/B > 3, cx = 1 is recommended at all frequencies.

Vertical Dashpot.—The selection of the appropriate wave velocity, V,

0 I < 1 1 ! . 1 1 I 1 1 1 1 1 1

0 0.5 1 1.5 „«§

FIG. 10.—Horiiontai Radiation Dashpot Cy versus Frequency for Different Base Shapes

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TABLE 3.—Symbols for Data Points Used in Figs. 4 to 13

Symbol

(1)

•

O

m

m

• •

^X <^

•

A

V

O

Shape

(2)

Circle

Circle

Strip Square

Square

Square Rectangle

Rectangle Rectangle

Rectangle

Strip

Strip

Strip

L/B

(3)

1

1

o:

1

1

1 2 - 4

2 - 3 - 4 4 2 6 6

2 -6 -10

2 10

3 - 4 - 1 0 2-10 2-5

2 - 5 - 1 0 2 - 3 - 5

00

00

CO

Fig.(s) (4)

4 -11 5

6-7-10-13(0) 8-12 4 - 6 7-13(a) 8-12

10-11 10-11 4 - 8

10-11-12-13(») 5

7-13(«) 7-13(fl)

4 - 6 - 7 - 1 0 - 1 1 5-12-13(b)

8 13(a) 5 4 5

7-13 4 - 6 - 1 0 - 1 1

5 7(a) 7(b)

13(«) 13(b)

4-6-7(«)-13(a) 10-11 4 - 6

10-11 7(a)-13(«) 7(«)-13(«)

10-11

V

(5)

0.33-0.50 All

0.33-0.50 All 0.33

0.25-0.33-0.50 All 0.33 0.48 0.33

0.40

0.33 , 0.25 0.33 0.33 0.33 0.33 0.49 1

0.33-0.49 1

0.40 1

0.33 0.25 0.25 0.25 0.25 0.25 0.25

0.33-0.48 ' 0.33

0.33-0.50 0.33

0.25-0.33-0.50 0.25-0.50

0.50

Ref. (6)

37 Eq. in Fig. 2(a)

35 36 38 20 20 38

10,12,13 8

Tassoulas, J. L. (personal communi­cation)

38 32

8 8 8 8

Tassoulas, J. L. (personal communica­tion)

38 32 32 32 32 32 32

10,12,13 10,12,13

18 18 18 21 18

to use in this case, is not obvious. One possibility, which has been sug­gested by other authors in a slightly different context for embedded foundations and piles (2,5), is to use the dilatational wave velocity Vp = ys[2(l - v)/(l - 2v)]1/2. This would be the answer if the foundation area were of infinite extent, covering the whole surface of the half-space (22). However, for realistic foundation sizes and shapes, C2 cannot be pro­portional to Vp, as the latter tends to <» for v -> 0.5, while in all solutions obtained by different authors, Cz remains always finite. In addition, a study by Miller and Pursey (25,26,31) revealed that, for a vertical circular foundation and v = 0.25, 93% of the energy is carried away in the far field by surface (Rayleigh) waves and shear waves; and only 7% is trans­ported by P-waves with velocity Vp. Therefore, for vertical foundation

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vibrations, P-waves and Vv are simply not very relevant. What is needed is the speed V of compression-extension waves propagating close to the foun­dation, which truly reflects the energy dissipation characteristics of the problem determining C2. Compression-extension waves travel in an elastic medium with different speeds depending on the possibility the medium has of developing lateral strains in directions normal to propagation. If lateral strains are zero, then, indeed, V = Vv = VD/p, where D = the constrained modulus of the material. If the soil can freely develop lateral strain in only one direction perpendicular to wave propagation, V = Vc = V2/(l - v) Vs. If lateral strains can be fully developed in all perpen­dicular directions, V = F, = VE/p = V2(l + v) Vs = rod wave velocity. Another wave velocity of interest is "Lysmer's Analog Wave Velocity," VLa = 3.4 VS/[TT(1 - v)]. This expression for Vu, defined by the writers, is based on Lysmer's frequency-independent analog for a circular rigid foundation of radius r0 vibrating vertically on the surface of a half-space (22,31): C2 = 3.4 rlpVs/(l - v). By equating this expression with C2 = pVuA = pVia Tsr2

0, the previous expression for VL„ is obtained.

These three velocities, Vc, V, and Vu , increase moderately with v, and they also have similar values in the range v = 0.25 to 0.50 (6). At v = 0.25, Vu = 1.44V„, Vc = 1.637s and V, = 1.58Vs, and Vp = 1.73VS is also in the same range. However, at v = 0.50, VLa = 2.16VS, Vc = 2VS and Vi = 1.73VS, while Vp = °°. Thus, Lysmer's analytical results and Vu derived from them are related to the actual wave speed and lateral strain conditions for the compression-extension waves escaping imme­diately beneath the foundation area. Therefore, the authors decided to use V^ as the appropriate compression-extension wave velocity under a foundation subjected to vertical vibration.

The corresponding plot of c2 = Cz/pVLllA versus a0 is presented in Fig. 11. As expected from Lysmer's results, c2 ~ 1 for circular and square foundations (L/B = 1); furthermore, c2 is also close to 1 and nearly in­dependent of frequency for aspect ratios as high as L/B = 4. But in keep­ing with what was observed for cy in Fig. 10, for very long foundations

FIG. 11.—Vertical Radiation Dashpot Cz versus Frequency for Different Base Shapes

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£z is significantly higher than 1 at low frequencies; in all cases c2 ap­proaches 1 at high frequencies. The fact that, at high frequencies, cz = 1 (Fig. 11), cy — 1 (Fig. 10), and cx — 1, irrespective of foundation shape and of Poisson's Ratio of the soil, confirms the validity of the theoretical framework previously developed for the translational modes.

Basic Considerations: Rotational Modes.—The results discussed above for horizontal and vertical dashpots at high frequencies immediately suggest a corresponding set of dimensionless parameters for the rota­tional radiation dashpots, C,, Crx and C„,. In these, the torsional dash-pot C, is associated with the horizontal dashpots Cx and Cy, while the rocking dashpots Crx and C„, are associated with the vertical dashpot Cz .

For torsion, we assume that, at high frequencies, any small element of foundation area, dA, is supported by a spring and by a dashpot dC, = pVsdA. This is equivalent to assuming that the half-space can be re­placed by a dynamic Winkler medium having identical distributed dash-pots for horizontal and torsional vibration. This is .not generally true, but it is reasonable at high frequencies (small wavelengths), at which the different points of the foundation do act as independent sources, radiating 1-D waves which propagate perpendicular to the foundation contact area. The corresponding equivalent rotational dashpot for tor­sional excitation can thus be obtained by integration over the whole con­tact area, A, of the moments produced by all these elementary dashpots dCt = pVsdA, around the vertical axis z. From this integration Ct = pVJ is obtained, where / = area moment of inertia of the foundation area around the z axis. Therefore, for torsion, we select the dimensionless parameter ct = Ct/pVs], expecting that c, will approach 1 at high fre­quencies irrespective of foundation shape.

The procedure for the rocking dashpots, Crx and C^ is similar, except that dCrx = dCry = pV^dA, the integration is done around the horizontal axes x and y, and, hence, the area moments of inertia, Ix and Iy, are obtained instead of /. Therefore, for rocking, the dimensionless param­eters crx = Crx/pVialx and c^ = C^/pVialy are introduced. Again, it is expected that both parameters crx and c^ will approach 1 at high fre­quencies, for all foundation shapes.

Torsional Dashpot.—Fig. 12 presents the variation of c, versus a0 = «B/y s . For a0 = 0, ct = 0, due to the complete destructive phase wave interference associated with antisymmetric vibration at wave lengths very long compared with the foundation dimensions. As the frequency in­creases, c( also increases monotonically, and, indeed, it invariably ap­proaches 1 at high frequencies. The values of ct are significantly influ­enced by L/B at low frequencies. For the larger aspect ratio included in the figure (L/B = 4), ct ~ 1 at a0 = 1.5, while for L/B = 1, ct = 1 at a0 =* 5 (see small graph on top of Fig. 12). This is reasonable, because for long foundations the torsional dissipation of energy is controlled by waves originating from the ends of the foundation, located at distances of about L from the z axis. As L/B increases, the value of a0 at which the two ends start behaving as independent wave sources, and at which the high frequency (short wavelength) approximation, ct = 1, is valid, should de­crease. This is exactly what Fig. 12 shows. Furthermore, the same argument can be extended to demonstrate that the larger the value of

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FIG. 12.—Torsional Radiation Dashpot C, versus Frequency for Different Base Shapes

FIG. 13.—Rocking Radiation Dashpots C„ and C„, versus Frequency for Different Base Shapes

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L/B, the closer to a0 = 0 is the frequency at which the small wavelength approximation is valid; therefore, in the limit, for L/B = » (strip foun­dation), c, = 0 at a0 = 0 and c, = 1 for a0 > 0. This jump for L/B = oo has been indicated in Fig. 12, and can be useful in practical applications as an upper bound for the values of ct of very long foundations (L/B > 4).

Rocking Dashpots.—Fig. 13 presents the variation of crx = Crx/pVi„Ix and c-ry = Cry/pVialy, respectively, versus a0. The trends in the plots of Figs. 13(a), 13(b) and 12 are very similar. In Fig. 13, the dimensionless dashpot parameters crx and c^ axe equal to zero for a0 = 0, increase monotonically with a0, and are both = 1 at high frequencies. Both crx and c^ approach 1 more rapidly when L/B is large; this can be explained using the argument discussed before for ct. Furthermore, c^ approaches 1 much faster than crx • Again, this is so because for c^ corresponding to rocking in the long foundation direction, the dissipation of energy is essentially controlled by the two foundation ends, located at distances of about L from the rocking axis, and these ends start acting as inde­pendent 1-D wave sources at relatively small values of a0. On the other hand, the corresponding distance for crx is B instead of L, and a higher value of a0 (a shorter wave length) is required for the ends to start acting independently.

The influence of L/B on the results presented in both plots of Fig. 13 is very significant in the frequency range 0 < a0 < 1.5. However, for rocking in the short direction, c„ becomes essentially independent of L/ B for L/B > 10.

Finally, an argument for c^ similar to that already presented for ct is valid for very long footings. That is, if L/B = oo, cn/ = 0 for a„ = 0, and c^ = 1 for a0 > 0. This jump for L/B = oo has been indicated in Fig. 13(b), and can be useful in applications involving long footings.

ENGINEERING METHOD

Table 4 summarizes the main findings of this paper for the dynamic stiffness K and radiation dashpot C of the six modes of vibration and for arbitrary foundation shapes, as well as for the modified values K((3) and C(0) which incorporate the soil material damping p. In practical ap­plications, Table 4, in conjunction with the corresponding figures as mentioned in the Table, and with the help of Table 1 and Fig. 1, can be used as a self-contained set of recommendations to obtain the values of K,K=K-k,C, X(P) and C(P).

Comparisons with available measurements performed on circular, square and rectangular foundation models having aspect ratios up to L/B = 6 also confirm the validity of the equations and charts presented here for K and C. These comparisons, which show good agreement be­tween experimental and analytical results, are included in the compan­ion paper (7).

There are two ways in which the method presented here can be used. In the simplest case, the operational frequency(ies) of the machine are specified, and all K and C values can be directly obtained for that fre­quency. In this case, only the foundation area parameters listed in Table 1 are required. A second case arises if the resonance characteristics of

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TABLE 4.—Proposed Engineering Method for Dynamic Stiffnesses K and Radi-Space (p = 0)

Vibration mode

(1)

Kt-

s2 =

General shape (2)

2LG

1 - v A

= 0.8 for -(4L2)

= 0.73 + 1.54

A f o r — > 0 . 0 2

4L2

See

V

Ss =

See

Kx-

Kx-

See

Fig. 3 2LG

2 — v

= 2.24 for—-4L2

/ A \°'38

= 4.5 — \4LV

Fig. 5(«) 2LG

= sx 2 - v

<0.02

( A \°'75

ULV

<0.16

A for—->0.16

4L2

0.21LG / B\ y 0 . 7 5 - v \ L/

Fig. 5(b)

DYNAMIC STIFF

Static or Pseudo

Circle (3)

Vertical

Horizontal y (short direction)

Horizontal x (long direction)

Rocking x (short direction)

Rocking y (long direction)

Torsion

' 1 - v ihf

2.54 B S„ = ^ for - < 0.4

S„ = 3 .2 fo r ->0 .4 L

•K„, — S„

S„ = 3.2 ' 1 - v

(*y ) a

K. = S(G(/)°-

S, = 3.8 + 10.7 (1 - -

K, = -4 GB

1 - v

8 GB

2 - v

8 GB

2 - v

K„ = -8GB3

3(1 - v)

K„ — • 8GB3

3(1 v)

K,-16 GB3

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ation Dashpots C for Arbitrarily Shaped Machine Foundations on Elastic Half-

NESSK = K-k

static K

Strip (4)

Kz U.8G

2L 1 - v

Ky 2.24G

1L 2-v

K„

2L

IT GB2

2(1 - v) "In (3 - iv)'

)

—

Dynamic k or K (5)

See Fig. 2 and Fig. 4

See Fig. 2 and Fig. 6

See Fig. 2 and Fig. 7(a)

See Fig. 2 and Fig. 7(b)

See Fig. 2 and Fig. 8

Radiation dashpot C (6)

See Fig. 11. At high frequency:

3.4 2 it > P V ^

TT(1 - v)

See Fig. 10. At high frequency:

Cv = pVsA

L For - < 3: Use C* =

B c^pKA, withe* = cy

L for - = 1 in Fig. 10.

B 6

L For - > 3: Use

B C.t = pV„A

See Fig. 13(a). At high frequency:

3.4 C„ = pVJx

TT(1 - V)

See Fig. 13(b). At high frequency:

3.4

TT(1 - v)

See Fig. 12. At high frequency:

C, = pV,/

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Foundation Area Parameters

L / B ' 3 . 5 B/L= 0.286

A = 6 l 4 f t 2

A/(4L8 ) = 0.256 I x = 9,309 ft* I y = 104,594 f t4

J =113,903 I f Centroidri. ' 22 .16 f t

J VV0e'»' Mx = M0e~'

| i _ i h = »ae"",~\ I

Soil Parameters

G = 4 x | 0 6 l b - f r 2

y =l251b- f t~ 3

f = 125/32.2 = 3.88 l b - s e c 2 - I t " 4

V s =(4X I0 8 / 3 .88 ) " 2 = 1,015 I t -sec"1

v =0.333 0=0.05

FIG. 14.—Illustrative Numerical Example (Calculations in Appendix I)

the system are to be calculated (natural frequencies as well as the dash-pot coefficients at those frequencies for the different modes). In this sec­ond case, in addition to the parameters listed in Table 1, information about the mass and mass moments of inertia of the foundation block plus machinery will also be required. In general, the computation of these natural frequencies will involve iterations and repeated use of the charts of k versus frequency.

ILLUSTRATIVE EXAMPLE

A numerical example illustrates the use of the proposed method for computing stiffness and damping of a specific surface foundation. A sketch of the foundation and lists of all foundation, excitation frequency and soil parameters' used in the example are included in Fig. 14. Appendix II shows the calculations of stiffness and dashpot coefficients, K((3) and C(P) of the foundation at the excitation frequency / = 30 Hz (to = 188.5 rad/sec) for all six modes of vibration. To follow the calculations in Appendix I, there is no need to go back to the text of this paper. The only materials needed for these and similar calculations are: (a) A sketch and the lists of parameters included in Fig. 14; (b) Fig. 1 and Table 1 for the definition of the different K and C, and the general Eqs. 2 and 3 at the bottom of Table 1 to obtain K((3) and C((3) from K and C; (c) Table 4 for the detailed calculations of all K and C; and (d) the figures presented throughout the paper, to be used as directed in Table 4.

CONCLUSIONS

The analytical results presented for the various vibration modes, fre­quencies and foundation shapes, provide a generally consistent picture of both effective dynamic stiffness and damping coefficients of rigid

2B = I4

1 - J I I 4 '

Frequency Porometers

f = 1,800/60 = 30 hz »=(2,rH30)= 188.5 rod-secH

a„=(IB8.5)(7l / l ,OI5 = l.3

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foundations on deep soil deposits. These results can be used directly to compute K(0) and C(P) of arbitrarily shaped foundations in engineering applications, with X((3) and C(P) incorporating the stiffness character­istics and Poisson's Ratio of the deposit, as well as radiation and material damping effects.

The results presented herein indicate that K and C are affected to var­ious degrees by frequency and foundation shape, with the effect being greater for long foundations and saturated soils. For the translational modes, the validity of the assumption of frequency-independent K and C made by the Lysmer and Hall analogs was confirmed for a wide range of frequencies for circular and square foundations (L/B = 1) supported by unsaturated soil. However, the same results strongly suggest that, for circular foundations on saturated soil, as well as for long foundations on any soil, the influence of frequency should not be ignored. Further­more, the use of the equivalent circle approach to compute dynamic stiffness and radiation damping can cause large errors, which tend to increase as L/B and v increase. The error committed when using the equivalent circle to compute the dynamic stiffness of a long foundation is generally larger at high frequencies; for radiation damping the reverse is true, with the error being larger at low frequencies.

The dimensionless plots for the damping coefficients C provide con­siderable insight into the mechanics of the radiation damping phenom­enon. In particular, at high frequencies the values of C for all foundation shapes and vibration modes are frequency-independent. These fre­quency-independent dashpots can be obtained by assuming, for each element of area dA, an independent, one-dimensional dashpot dC = p VdA, and by appropriate integration of those elementary dashpots over the whole foundation area A. The velocity V of the waves escaping the foun­dation-soil interface to be used in this integration is either equal to the shear wave velocity of the soil, Vs, for horizontal and torsional vibration, or to the fictitious "Lysmer's Analog" wave velocity, VUl for vertical and rocking vibration.

ACKNOWLEDGMENTS

Several analytical results were kindly provided to the writers by John L. Tassoulas. The writers are also grateful to Jose M. Roesset, Mishac K. Yegian, Robert V. Whitman, Emmanuel Petrakis and Ramli Moha­mad, who critically reviewed various drafts of the paper and offered valuable comments.

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APPENDIX I.—EXAMPLE CALCULATIONS FOR MACHINE FOUNDATION OF FIG. 14

sz K2

kz

MP) s» Ky

. ^ KM)

sx

Kx

Kx

KM)

sn K„

fc-TX

KrM) Sry Kry

fcnf

Kn,(P)

s, Kt

k, KM)

Stiffnesses

= 0.73 + (1.54)(0.256)ore

= (1.28)(2)(24.5)(4 x 106)/(1 - 0.333) = 0.93 >̂ Kz = (0.93)(3.76 x 108) = 3.50 x 10s - (188.5)(4.32 x 106)(0.05) = (4.5)(0.256f4

= (2.61)(2)(24.5)(4 x 106/(2 - 0.333) = 1.2 >̂ Ky = (1.2)(3.07 x 108) = 3.68 x 108 - (188.5)(2.42 x 106)(0.05)

= 2.61 - - (1 - 0.286)

= (2.34)(2)(24.5)(4 x 106)/(2 - 0.333) = K, = 2.75 x 108 - (188.5)(2.42 x 106)(0.05) = 2.54/(0.286f25

= (3.47)(4 x 106)(9,309)°'75/(1 - 0.333) = 0.73 => K„ = (0.73)(1.97 x 1010) = 1.44 x 1010 - (188.5)(3.75 x 107)(0.05) = 3.2 = (3.2)(4 x 106)(104,594)°-7S/(1 - 0.333) = 0.66 => K,y = (0.66)(11.16 x 10lc) = 7.37 x 1010 - (188.5)(6.22 x 108)(0.05) = 3.8 + (10.7)(1 - 0.286)10

= (4.17)(4 x 106)(113,903)0'75

= 0.86 => K, = (0.86)(10.34 x 1010) = 8.89 x 1010 - (188.5)(40.4 x 107)(0.05)

= 1.28 = 3.76 x 108 lb-ft-1

= 3.50 x 108 lb-fr1

= 3.1 x 108 lb-fr1

= 2.61 = 3.07 x 10a lb-ft-1

= 3.68 x 108 lb-fr1

= 3.4 x 108 lb-fr1

= 2.34

= 2.75 x 108 lb-ft"1

= 2.75 x 108 lb-fr1

= 2.5 x 108 lb-fr1

= 3.47 = 1.97 x 1010 lb-ft = 1.44 x 1010 lb-ft = 1.4 x 1010 lb-ft

= 11.16 x 1010 lb-ft = 7.37 x 1010 lb-ft = 6.8 x 1010 lb-ft = 4.17 = 10.34 x 1010 lb-ft = 8.89 x 1010 lb-ft = 8.5 x 1010 lb-ft

APPENDIX II.—REFERENCES

1. Barkan, D. D. (1962), Dynamics of Bases and Foundations (translated from Rus­sian), McGraw-Hill Book Co., New York.

2. Berger, E., Makin, S. A., and Pyke, R. (1977), "Simplified Method for Eval­uating Soil-Pile Structure Interaction Effects," Proceedings of the 9th Offshore Technology Conference, OTC Paper 2954, Houston, TX, pp. 589-598.

3. Borochadev, N. M. (1964), "Determination of the Settlement on Rigid Plates," Soil Mechanics and Foundation Engineering (USSR), 1, 210.

4. Chae, Y. S. (1969), "Vibrations of Noncircular Foundations," Journal of the Soil Mechanics and Foundations Division, Vol. 95, No. SM6, Nov., pp. 1411-1430.

5. Day, S. M. (1977), "Finite-Element Analysis of Seismic Scattering Problems," Ph.D. Thesis in Earth Sciences, Univ. of California at San Diego.

6. Dobry, R., and Gazetas, G. (1985), "Dynamic Response of Arbitrarily Shaped Foundations," Research Report, Dept. of Civ. Engrg., Rensselaer Polytechnic Institute.

7. Dobry, R., Gazetas, G., and Stokoe, K. H., II (1986), "Dynamic Response of Arbitrarily Shaped Foundations: Experimental Verification," Journal of Geo-technical Engineering, ASCE, 112(2), Feb., 136-154.

8. Dominguez, J., and Roesset, J. M. (1978), "Dynamic Stiffness of Rectangular Foundations," Research Report R78-20, Dept. of Civ. Engrg., M.I.T.

9. Erden, S. M. (1974), "Influence of Shape and Embedment on Dynamic Foun-

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Dashpots

Vu = (3.4)(1,015)/[TT(1 - 0.333)] = 1,647 ft/sec cz = 1.1

C2 = (1.1)(3.88)(1,647)(614) = 4.32 x 106 lb-sec-fr1

C2(p) = 4.32 x 106 + (2)(3.50 x 108)(0.05)/188.5 = 4.5 x 106 lb-sec-ft"1

cy = 1.0 Cy = (1.0)(3.88)(1,015)(614) = 2.42 x 10" lb-sec-fr1

C„(p) = 2.42 x 106 + (2)(3.68 x 108)(0.05)/188.5 = 2.6 x 10" lb-sec-fr1

Cx = (3.88)(1,015)(614) = 2.42 x 106 lb-sec-ft"' C»(P) = 2.42 x 106 + (2)(2.75 x 108)(0.05)/188.5 = 2.6 X 106 lb-sec-ft"1

c„ = 0.63 C„ = (0.63)(3.88)(1,647)(9,309) = 3.75 x 107 lb-sec-ft

C„(P) = 3.75 x 107 + (2)(1.44 x 1010)(0.05)/188.5 = 4.5 x 107 lb-sec-ft

C„, = 0.93 C„ = (0.93)(3.88)(1,647)(104,594) = 62.2 x 107 lb-sec-ft

C,(P) = 62.2 x 107 + (2)(7.37 x 1010)(0.05)/188.5 = 66 x 107 lb-sec-ft

c, = 0.90 C, = (0.90)(3.88)(1,015)(113,903) = 40.4 x 107 lb-sec-ft

C,(P) = 40.4 x 107 + (2)(8.89 x 1010)(0.05)/188.5 = 45 x 107 lb-sec-ft

dation Response," Ph.D. Thesis, Dept. of Civ. Engrg., Univ. of Massachu­setts.

10. Gazetas, G. (1982), "ANILAY—A Computer Program for Dynamic Response of Strip Footings on Layered Cross-Anisotropic Soils," Research Report CE82-05, Dept. of Civ. Engrg.

11. Gazetas, G. (1983), "Analysis of Machine Foundation Vibrations: State of the Art," Journal of Soil Dynamics and Earthquake Engineering, Vol. 2, No. 1, pp. 2-42.

12. Gazetas, G., and Roesset, J. M. (1976), "Forced Vibrations of Strip Footings on Layered Soils," Proceedings of the ASCE Specialty Conference on Methods of Structural Analysis, Madison, WI, pp. 115-131.

13. Gazetas, G., and Roesset, J. M. (1979), "Vertical Vibrations of Machine Foun­dations," Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT12, pp. 1435-1454.

14. Gazetas, G., and Dobry, R. (1984), "Horizontal Response of Piles in Layered Soils," Journal of Geotechnical Engineering, ASCE, Vol. 110, No. 1, Jan., pp. 20-40.

15. Gazetas, G., and Dobry, R. (1984), "Simple Radiation Damping Model for Piles and Footings," Journal of Engineering Mechanics, ASCE, Vol. 110, No. 6, June, pp. 937-956.

16. Gazetas, G., Dobry, R., and Tassoulas, J. L. (1985), "Vertical Response of Arbitrarily Shaped Embedded Foundations," Journal of Geotechnical Engineer­ing, ASCE, Vol. I l l , No. 6, June, pp. 750-771.

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17. Gorbunov-Possadov, M. I., and Serebrajanyi, V. (1961), "Design of Struc­tures Upon Elastic Foundations," Proceedings of the 5th International Conference on Soil Mechanics and Foundation Engineering, held in Paris, France, Vol. I, pp. 643-648.

18. Karasudhi, P., Keer, L. M., and Lee, S. L. (1968), "Vibratory Motion of a Body on an Elastic Half Plane," Journal of Applied Mechanics, ASME, 35E, pp. 697-705.

19. Kausel, E., and Roesset, J. M. (1975), "Dynamic Stiffness of Circular Foun­dations," Journal of the Engineering Mechanics Division, ASCE, Vol. 101, No. EM6, pp. 771-785.

20. Luco, J. E., and Westmann, R. A. (1971), "Dynamic Response of Circular Footings," Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM5, pp. 1381-1395.

21. Luco, J. E., and Westmann, R. A., (1972), "Dynamic Response of a Rigid Footing Bonded to an Elastic Half Space," Journal of Applied Mechanics, ASME, June, pp. 527-534.

22. Lysmer, J., and Richart, F. E., Jr. (1966), "Dynamic Response of Footing to Vertical Loading," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 92, No. SMI, pp. 65-91.

23. Lysmer, J. (1980), "Foundation Vibrations with Soil Damping," Proceedings, 2nd ASCE Conference on Civil Engineering and Nuclear Power, Knoxville, TN, Vol. II, Paper 10-4, pp. 1-18.

24. Massa, F. (1972), "Radiation of Sound," American Institute of Physics Handbook, Third Edition, McGraw-Hill Book Co., New York, pp. 3-139.

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26. Miller, G. F., and Pursey, H. (1955), "On the Partition of Energy Between Elastic Waves in a Semi-Infinite Solid," Proceedings of the Royal Society, London, A, Vol. 233, pp. 55-69.

27. Morse, P. M., and Ingard, K. U. (1968), Theoretical Acoustics, McGraw Hill Book Co., NY.

28. Muskhelishvili, N. I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, Groningen, P. Noordhoff.

29. O'Rourke, M. J., and Dobry, R. (1982), "Spring and Dashpot Coefficients for Machine Foundations on Piles," Foundations for Equipment and Machinery, Publication SP-78, American Concrete Institute, pp. 177-198.

30. Richart, F. E., Jr., and Whitman, R. V. (1967), "Comparison of Footing Vi­bration Tests with Theory," J. Soil Mechanics and Foundations Div., ASCE, Vol. 93, No. SM6, pp. 143-168.

31. Richart, F. E., Jr., Hall, J. R., Jr., and Woods, R. D. (1970), Vibrations of Soils and Foundations, Prentice-Hall International, Inc., NJ.

32. Riicker, W. (1982), "Dynamic Behavior of Rigid Foundations of Arbitrary Shape on a Half Space," Earthquake Engineering and Structural Dynamics, Vol. 10, pp. 675-690.

33. Savidis, S. A. (1977), "Analytical Methods for the Computation of Wave-fields," Dynamic Methods in Soil and Rock Mechanics, 1, 225.

34. Selvadurai, A. P. S. (1979), Elastic Analyses of Soil-Foundation Interaction, El­sevier Scientific Publishing Co.

35. Veletsos, A. J., and Wei, Y. T. (1971), "Lateral and Rocking Vibration of Footings," /. Soil Mechanics and Foundation Div., ASCE, Vol. 97, No. SM9, pp. 1227-1249.

36. Veletsos, A. J., and Nair, V. V. (1974), "Torsional Vibration of Viscoelastic Foundations," f. Geotechnical Engineering Div., ASCE, Vol. 100, No. GT3, pp. 225-246.

37. Veletsos, A. J., and Verbic, B. (1974), "Basic Response Functions for Elastic Foundations," /. Engineering Mechanics Div., ASCE, Vol. 100, No. EM2, pp. 189-201.

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38. Wong, H. L., and Luco, J. E. (1976), "Dynamic Response of Rigid Founda­tions of Arbitrary Shape," Earthquake Engineering and Structural Dynamics, Vol. 4, pp. 579-587.

39. Whitman, R. V., and Richart, F. E., Jr. (1967), "Design Procedures for Dy­namically Loaded Footings," /. Soil Mechanics and Foundations Div., ASCE, Vol. 93, No. SM6, pp. 169-193.

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