CONE MODELS FOR HOMOGENEOUS SOIL. I

By Jethro W. Meek,1 Associate Member, ASCE, and John P. Wolf,2 Member ASCE

ABSTRACT: For dynamic excitation, it is convenient to idealize homogeneous soil under a base mat by a semi-infinite truncated cone. It is easy to analyze the cone model for vertical and horizontal translation, as well as for rocking and torsional rotation. The accuracy by comparison to rigorous half-space solutions is quite adequate for practical applications. Time-domain computational methods for trans-lational and rotational motions are described in both the stiffness and flexibility formulations and elucidated by examples. The infinite cone is dynamically equiv­alent to a discrete element representation of the soil, consisting of an intercon­nection of a small number of masses, springs, and dashpots. As an alternative to the physical-component model, the response may be determined directly by simple recursive numerical procedures. The recursive methods are exact and particularly well suited for hand calculations of short-duration excitations.

INTRODUCTION

During the past 20 years, sophisticated analytical and numerical tech­niques, such as the finite-element and the boundary-element methods, have been developed to solve foundation-dynamics problems (Veletsos and Wei 1971). Characteristic of all these rigorous procedures is the idealization of the soil as an elastic half-space, with or without layers, excavations, or other such inhomogeneous features.

As an alternative to the rigorous approach, the soil may be modeled approximately in simple cases as a semi-infinite truncated cone. The cone model for translational response was introduced nearly half a century ago (Ehlers 1942), the cone model for rotational considerably later (Meek and Veletsos 1974; Veletsos and Nair 1974). By comparison to rigorous solu­tions, the cone models originally appeared to be such an oversimplification of reality that they were used primarily to obtain qualitative insights. For example, the surprising fact that the cones are dynamically equivalent to an interconnection of a small number of masses, springs, and dashpots with frequency-independent coefficients encouraged a number of researchers to match discrete element representation of exact solutions in the frequency domain by curve fitting (Veletsos and Verbic 1973; Wolf and Somaini 1986; de Barros and Luco 1990). Proceeding in another direction, Gazetas (1984 b), and Gazetas and Dobry (1984) employed wedges and cones to elucidate the important phenomenon of radiation damping in two and three dimen­sions.

In the meantime, a large compendium of exact solutions has accumulated, and many qualitative aspects of the response have been clarified. Thus the "primitive" cone models would seem to have become superfluous.

'Engr., Fr. Hoist Constr. Co. , Ellerholzweg 14, D-2102 Hamburg 93, Germany. 2Engr., Dept. of Civ. Engrg., Inst, of Hydr. and Energy, Swiss Federal Inst, of

Tech., CH-1015 Lausanne, Switzerland. Note. Discussion open until September 1, 1992. Separate discussions should be

submitted 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 November 28, 1990. This paper is part of the Journal of Geotechnical Engineering, Vol. 118, No. 5, May, 1992. ©ASCE, ISSN 0733-9410/92/0005-0667/$1.00 + $.15 per page. Paper No. 956.

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However, besides being theoretically complicated and expensive to cal­culate, the exact solutions have an additional noteworthy disadvantage: at some stage of the analysis they almost invariably require Fourier transfor­mation into the abstract frequency domain. This precludes general accept­ance of such exact solutions by practicing engineers, who prefer to imple­ment dynamic analyses exclusively in the familiar time domain of everyday experience. In addition, time-domain analysis makes it possible to consider nonlinear behavior of the structure.

Cone models are potentially attractive because the calculations may be performed purely in the time domain. They are also based on sound physical approximations and do not violate the principles of wave propagation. Ap­propriate methods are described in this paper, taking advantage of recently developed recursive procedures for numerical convolution [Wolf (1988) sec­tions 6.8-6.9 and Meek (1990)].

The use of cone models in engineering practice presupposes, of course, that their quantitative accuracy may be verified. The necessary confirmation is provided by comparison with exact results. The accuracy is quite adequate without resorting to any sort of modification, such as curve fitting.

The goal of this paper is to enable the practitioner to evaluate the dynamic response to slab footings (base mats) on the surface of, or barely embedded in, unlayered soil. The extension to layered soil is treated in the companion paper (Meek and Wolf 1992). The cone models are regarded as an appro­priate means to this end. The body of the text is application-oriented; for the theoretically inclined reader, the mathematical solutions of cone models are summarized in appendices.

For excitations of short duration, all necessary computations may be performed with a hand calculator. For longer excitations, such as earth­quakes, any general-purpose computer program for structural dynamics may be employed. Since general-purpose programs do not include semi-infinite cones in their libraries of standard elements, the cones must be replaced by dynamically equivalent discrete-parameter systems consisting of a small number of springs, dashpots, and (sometimes) masses with frequency-inde­pendent coefficients. For the more complicated rotational cone, the appro­priate discrete parameter system is not uniquely defined; and the advantages and disadvantages of various alternative representations are discussed in detail.

CONE MODELS

To a first approximation, unlayered soil may be idealized as a linearly elastic, semi-infinite medium with mass density p. The soil is regarded to possess no material damping. The stress-strain relationship is specified by two independent constants. In elementary soil mechanics, it is traditional to choose as these elastic properties the constrained modulus of compression M and Poisson's ratio v. In soil dynamics it is more customary to specify the shear modulus G and Poisson's ratio v. When working with cone models, it is convenient to select yet another pair of fundamental properties, the propagation velocities cs of shear waves and c of dilatational waves. The various elastic properties may be computed from one another using the relationships

G = pel (la)

M = pel (lb)

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fey-v = — ^ — ( l c ) 2 f e ) -

In the substructure approach to soil-structure interaction, the structure and the soil are first viewed separately, then coupled. The structural portion, which includes the mass of the base mat, may be modeled by an intercon­nection of masses, dashpots, and (possibly nonlinear) springs—or, equiv-alently, by finite elements. The formulation of the structural model is well understood and need not be treated herein. The following section addresses the model of the soil, which begins at the bottom surface of the base mat and extends downwards. Conceptually, it is convenient to regard the soil-structure interface as a massless rigid footing that imposes the displacement of the underside of the base mat. The area of the fictitious massless footing is A0; its area moment of inertia about the axis of rotation is I0 (for torsion I0 = polar moment of inertia).

The soil is idealized to be a semi-infinite elastic cone with apex height z0, as shown in Fig. 1. Depending on the nature of the deformation, it is necessary to distinguish between the translational cone for vertical and horizontal motion [Fig. 1(a)] and the rotational cone for rocking and torsion [Fig. 1(b)]. As indicated in Table 1, the appropriate wave-propagation ve­locities are c = cs for the horizontal and torsional cones, which deform in shear, and c = cp for the vertical and rocking cones, which deform axially.

FIG. 1. Cone Models: (a) Translational Cone; and (b) Rotational Cone

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Fig. 1 includes typical free-body diagrams for incremental slices of the translational cone (vertical motion) and the rotational cone (rocking mo­tion). The equations of dynamic equilibrium are seen to be

— = PAU (2fl) OZ

for the translational cone and

£ - " * <=») for the rotational cone. The relationships A = A0(z/z0)

2; I = I0(z/z0)4; N

= pc2Adu/dz; M = pc2Id%lBz are substituted into (2a) and (2b), yielding the governing partial differential equations of motion

d2U 2 du U , . .

^2 + -zTz = e (3fl)

for the translational cone, and

B2% 4 3-ft % —- + = — (3b) dz2 z dz c2 y ' for the rotational cone. In all these formulas, z = depth below the apex; u = translational displacement; and •& = angle of rotation. The same quan­tities with subscript 0 refer to the respective values at the surface. An additional surface parameter is the equivalent radius r0 of a circular disk with the same area as the footing (for translation) or the same moment of inertia (for rotation); for convenience, formulas for r0 are presented in Ta­ble 1.

Although the preceding discussion has dealt with the translational cone in vertical motion and the rotational cone in rocking, it should be apparent that the same expressions would be derived for the translational cone in horizontal motion and the rotational cone in torsion, provided that the wave velocity c is taken to be the shear-wave velocity cs, and the d'Alembert inertial forces shown in Fig. 1 are considered to act horizontally instead of vertically.

SOLUTIONS OF GOVERNING EQUATIONS: STIFFNESS FORMULATION

The key to the solution of the governing equations [(3a) and (3b)] is to exploit their similarity to the familiar one-dimensional wave equation

^L) = 0 . . . .(4)

To convert the equations for the cone to this simple form, appropriate changes of variables are introduced that eliminate the d/dz-terms in (3). The mathematics is summarized in Appendix I for the more complicated rota­tional cone.

For practical applications, it is not necessary to compute explicitly the displacement field propagating into the unbounded soil. One may view the soil as a "black box" and restrict attention to the force-displacement rela­tionships of the massless footing. It is convenient to consider first the stiffness formulation (force as a function of displacement) of the translational cone

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TA

BL

E

1.

Pro

pe

rtie

s of

Con

e M

odel

s

Con

e ty

pe

(1)

tran

slat

iona

l

K

=

PC

UQ

/ZO

;

C =

pc

A0

rota

tion

al

K,

=

3pc2/„

/z0;

C

8 =

pc

/ 0;

h ~

P2o4

Mot

ion

(2)

vert

ical

hori

zont

al

rock

ing

tors

ion

c (3)

CP

c s

CP

c s

(4)

VIo

Tn

VA

OT

H

V4

/ 0/T

T

VI^

A

Gen

eral

for

mul

a (5

)

TT/8

(c p

/c,)

V[(

c p/c

s)2

" 1]

TT/1

6 [3

(^/^

)2

- 2]

/ l(

c p/c

s)2

- 1]

9W

64

(cpl

c syi

[(c f

lcsf

-1

]

9w/3

2

Com

pres

sibl

e So

il

c plc

s =

V

2 (6

)

1.57

1

0.78

5

1.76

7

0.88

4

v =

0

(7)

TT/2

ir/4

9ir/

16

9ir/

32

AS

PE

CT

RA

TIO

z„

lr„

c plc

s =

2.

0 (8

)

2.09

4

0.65

4

2.35

6

0.88

4

v =

1/

3 0)

2i

r/3

5TT

/24

3ir/

4

9TT

/32

Cp/

C,

=

2.26

4 (1

0)

2.01

2

—

2.26

4

—

Inco

mpr

essi

ble

Soil

v =

1/

2 (1

1)

512/

81-ir

—

64/9

-IT

—

(12)

—

0.58

9

—

0.88

4

v =

1/

2 (1

3)

—

3ir/

16

—

9TT

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P0 = KuQ + Cu0 (5)

in which K = pc2A0/z0; and C = pcA0. From the simple form of (5), it follows that K and C may be interpreted as an ordinary spring (the static stiffness) and an ordinary dashpot. This discrete model for the translational cone, shown in Fig. 2 was derived nearly half a century ago (Ehlers 1942). It is quite remarkable that from outside it is impossible to ascertain whether or not the black-box model of the soil contains an unbounded cone with an infinite number of internal degrees of freedom or just a spring and a dashpot with no internal degrees of freedom. To take translational soil-structure interaction into account, it is only necessary to attach to the underside of the structural model the spring K and the dashpot C. Then, the complete system may be analyzed directly with a general-purpose computer program. For earthquake excitation, the ground motion is applied not at the soil-structure interface, but rather at the far end of K and C. (This base node does not represent deep rock at infinity. The motion of the base node is properly interpreted as the surface motion of the massless footing in the absence of the structure).

It seems reasonable to expect that the moment-rotational relationship of the rotational cone should involve a rotational spring K% as the static stiff­ness, augmented by a rotational dashpot C0. This is indeed the case; how­ever, the results are somewhat more complicated. From Appendix I

M0 = K^0 + C A - *i * C«*o . , . , ( 6 )

M0 M0

in which K$ = 3pc2I0/z0; and Cd = pcl0. The first two terms in (6) account for that portion of the moment due to the present values of rotation $0 and angular velocity -&0. Wolf (1988) denotes this instantaneous contribution to the response as the singular part M0. The last term in (6), denoted by a star and preceded by a minus sign, is a convolution (Duhamel's integral)

-h, * C A = -_[ ' h,(t - T ) C ^ 0 ( T ) d-r (la)

involving an impulse-response function

hi(t) = - e~c"za for t > 0 {lb)

hx(t) = 0 for t < 0 (7c)

The convolution depends on all previous values of angular velocity h0 and

Po.Ui

TrrmrmrmrmT

FIG. 2. Discrete-Element Model for Translation

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may be regarded as the system's memory of the past. Wolf (1988) names this lingering portion of the response the regular part Mj.

It is computationally inefficient to evaluate the convolution -hx -k CQ$Q from its definition, {la). The calculations may be expediated by re­cursive procedures, discussed later. Alternatively, one may employ discrete physical models that incorporate the convolution implicitly; two such models are shown in Figs. 3(a) and 3(b). In Fig. 3(a), besides the spring K$, an additional rotatory mass moment of inertia 7a = pz0I0 is introduced, together with an associated internal degree of freedom •&! located, so to say, within the black box and attached across the dashpot Ca to the footing. The equiv­alence of the mathematical model to (6) is demonstrated straightforwardly (Meek and Veletsos 1974; Wolf 1988).

The mechanical model captures the primary energy-dissipation effect in unbounded soil, radiation damping. The discrete elements may be easily modified to incorporate the additional influence of material damping [Wolf (1988) page 41-48], not considered herein.

The discrete model in Fig. 3(a), first proposed by Meek and Veletsos (1974) with the mass-damper interconnection as a sort of monkey tail, is not easy to comprehend physically; intuitively one might fear that the mass would eventually fall off. This conceptual disadvantage is avoided by using, instead of the monkey tail, the spring-damper combination in Fig. 3(b), recently suggested by Wolf (1988). Again, an associated internal degree of freedom is introduced. The appropriate values of the discrete elements are indicated in the figure.

Unfortunately, there also is a conceptual difficulty associated with the model in Fig. 3(b): negative springs and dashpots do not exist in reality. It is, of course, no problem to handle them mathematically. In practical ap­plications, either of the discrete mechanisms shown in Fig. 3(a) or 3(b) may be attached to the underside of the structural system as an exact equivalent of the semi-infinite rotational cone. Then, the complete system may be analyzed directly with a general-purpose computer program.

The expressions for the dashpots, C = pcA0 and Cd = pdQ, are inde­pendent of the apex height of the cones, z0. The same relationships may be derived simply by assuming that every surface in contact with the soil pos­sesses an inherent amount of radiation damping equal to pc per unit area. This very important elastodynamical property has been ascertained quite generally (Gazetas 1984b) and appears to reflect an underlying law for high-frequency excitation characterized by small alternating displacements with high velocity.

In contrast to the dashpots, the expressions for the springs, K = pc2A0/ z0 and K$ — 3 pc2I0/z0 are dependent on the cone's apex height z0. The

Mo, do ^ - x ° ' d o

(a) (b)

FIG. 3. Discrete-Element Models for Rotation: (a) Monkey-Tail Model; (b) Spring-Damper Model

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specification of z0 according to the formulas presented in Table 1 makes the various static stiffnesses of the cone equal to those of the elastic half-space. It is interesting to observe that the vertical and rocking cones, in which dilational waves predominate, are slender (z0/r0 > 1), whereas the horizontal and torsional cones, in which shear waves predominate, are squatty (z0/r0 < 1).

Matching the static stiffness of the cone to that of the elastic half-space results in a doubly asymptotic approximation, correct both for zero fre­quency (the static case) and the high-frequency limit dominated by the radiation dashpots C and Ca. This explains the accuracy of cone models, even though they transmit axially propagating body waves only. In reality, nonrepresentable Rayleigh surface waves, which propagate perpendicular to the cones' axis, transmit the major portion of energy away from a vibrating footing. The convolution implicit in the rotational cone [term -h1*Cf)-b0 in (6)] is only important in the intermediate frequency range within these asymptotic states.

It is worth mentioning that the rocking cone introduced in this paper [Fig. 1(b)] deforms in compression-extension and is not identical to the original rocking cone (Meek and Veletsos 1974), which deforms in shear. The new compression-extension cone is more accurate for values of cjcs £ 2 (Pois-son's ratio v ^ 1/3). The original shear cone does, however, yield the proper solution for the perfectly incompressible case. In the limit v = 1/2, the radiation dashpot of the rocking shear cone becomes

C* = !) Pc»r° (8a)

This expression may be set equal to that of the rocking compressional cone

Q = pcp/0 = - pcpr4

0 (86)

and solved for an upper limit of the velocity ratio

c 64 ^ < ^ = 2.264 (8c)

When working with the compressional cone, the dilatational velocity cp should never be assumed to be greater than 2.264 cs; otherwise, the radiation damping will be overestimated. This conclusion, derived for the compres­sional cone in rocking, is also valid for the translational cone in vertical motion and has been accounted for in Table 1. The values of z0/r0 in column 10 of the table are chosen to match the static stiffness of the vertical and rocking cones to the rigorous half-space solutions for v = 1/2.

For compressional cones in vertical and rocking motion Gazetas and Dobry (1984) advocate the use of the so-called Lysmer apparent velocity cLa instead of cp

3.4 c, TT(1 - v)

(9)

In the incompressible case (v = 1/2), cLa = 2.165 cs, in good agreement with the upper limit 2.264 cs proposed previously. For compressible soil, however, the use of cp instead of cLa cannot be recommended. Rigorous half-space solutions for the vertical and rocking cases (v ^ 1/3) show that

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the radiation damping per unit area always converges in the high-frequency limit to pcp, not pcLa.

SOLUTIONS OF GOVERNING EQUATIONS: FLEXIBILITY FORMULATION

In previous equations [(5) and (6), stiffness formulations], the displace­ments and rotations are regarded as the input, the corresponding forces and moments as the output. In a flexibility formulation, the roles of input and output are reversed. Known forces and moments serve as the excitation; unknown displacements and rotations are the desired response. Flexibility formulations are particularly well suited for transient machine-vibration problems; these are amenable to simple hand calculation.

The fundamental solution required for a flexibility formulation is the response (displacement or rotation) due to a Dirac-delta impulse (normal­ized force PJK or normalized moment MJK^}. It is customary to denote the Dirac-impulse response by the symbol h(t). For the flexibility formu­lation of the translational cone, the impulse response turns out to be the same simple exponential function

hAt) = - e-c"z« for t > 0 (10a)

h^t) = 0 for t < 0 (106)

previously encountered [(7b and c)] in the stiffness formulation of the ro­tational cone. This fortuitous result is a mathematical coincidence. For the flexibility formulation of the rotational cone, the impulse response is a more complex damped sinusoid

c I V3 ct V3 cA hJt) = - e-i-5«/zo 3 cos — V3 sin —- - for t > 0 . . . (10c)

z0 \ 2 z0 2 z0J h2{t) = 0 for t < 0 (10d)

The impulse responses hx and h2 may be derived by solving the equations of motion [(3)] in the time domain. It is, however, much easier to obtain the results by using Fourier or Laplace transform techniques in order to invert the stiffness relationship (5) and (6). Although space limitations pro­hibit inclusion of the derivations, the correctness of the impulse-response functions given by (10) is verified in Fig. 4, which compares them with rigorous results (Veletsos and Verbic 1974) for the disk on the elastic half-space with v = 1/3. The initial peak values at time t = 0 are exact because the cones enable an error-free representation of the radiation dashpots C and CV

Formally, the flexibility formulation expresses the resulting displacement and rotation as convolutions (Duhamel's integrals) of the force and moment with the respective impulse responses h1 and h2, i.e.,

Ko(0 = *1 * J = jo W - T) ^ ^ ( l l f l )

for the translational cone, and

v o = k2 • = r h2(t ~ T) a, (lw) A f l Jo A e

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TIME PARAMETER Cst/ro 0 1 2

TIME PARAMETER cp t / r „ FIG. 4. Impulse-Response Functions (Poisson's Ratio v = 1/3): (a) Horizontal Translation; (to) Vertical Translation; and (c) Rocking

for the rotational cone. The response is purely of the lingering (regular) type, without an instantaneous (singular) part.

RECURSIVE EVALUATION OF CONVOLUTION

As noted previously in conjunction with (7), it is computationally inef­ficient to perform convolution directly by numerical quadrature of Duha-mel's integral. A more elegant approach is to use recursion formulas, a technique developed in digital signal processing, but just as applicable to civil engineering dynamics. The recursion method may be described in gen­eral terms by considering an excitation x(t) with equally spaced samples x„ = x(nAt) and a response y(t) with equally spaced samples y„ = y(n&t), related by the convolution y = h it x. Whereas the Duhamel integral extends over the entire time history from 0 to t, the recursive evaluation of the cone response involves only a few discrete samples of the immediate past

y„ = a i ^ - i + a2yn-2 + b0x„ + blx„.1 + b2xn. (12)

The computational savings are obviously enormous for long excitations such as earthquakes. For free motion (x„ = 0) the recursion constants ar and a2 capture the lingering memory of the cone. They are given by

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fll = e-^t/zo (13fl)

a2 = 0 (13b)

for convolutions with hx and

V3cAf fll = 2e-» 5cA"20cos — — (13c)

2 z0

a2 = _e-3cA,/zo (1 3 d)

for convolutions with h2. The constants b0, bx, and fo2, which account for the excitation, may be evaluated in the straightforward manner described in Appendix II.

The recursive algorithm is self-starting from quiescent initial conditions at time t = 0. The quiet past implies that in the evaluation of (12) for n = 1, the samples y0, _y_1; x0, and x_ t are all zeros, so that yx = bgx^ If the ongoing excitation sequence xu x2, x3, . . . happens to be TV-point periodic, the first N response samples yu y2, . . . , y„ will not yet have attained the periodic steady state. Convergence from the initial start-up phase to steady-state behavior occurs quite rapidly, however, because of the large damping inherent in the cone models. After several ,/V-length periods of excitation have been processed, the y„-values will indeed repeat N-point periodically.

It should be emphasized that the recursion relationship (12) is not an approximate numerical integration formula. The results are exact, subject only to the assumption that the excitation function x(t) is a polygon formed by joining the samples x„ by straight lines. The recursive algorithm is un­conditionally stable for all values of the time increment At; its applicability is, by no means, restricted to the cones treated herein. Any linear system that may be idealized exactly or approximately by a finite number of springs, dashpots, and masses is amenable to recursive analysis. For discrete models more complicated than those shown in Fig. 3, additional constants a„ and b„, n > 2, are required. They may be computed by general procedures described elsewhere (Wolf and Motosaka 1989; Meek 1990).

EXAMPLE

The various computational procedures for cone models, in particular the recursive techniques, may be demonstrated by calculating the vertical and rocking response to a rounded triangular pulse of prescribed displacement or rotation

2-TTt

1 - cos —

"o = "cw 2 ~' Q<t<To (14«)

2ttt 1 - cos —

#o = flew — 2 ~ 0<t<TQ (Ub)

with a corresponding velocity history

/ IT \ . 2-nt ,. . >. "o = "(w I Y J sin — (14c)

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\ = *«w (£) sin jr (1*0

These excitation quantities are plotted in Figs. 5(a) and 5(b). The choice of the time increment At = TQ/8 allows the velocity to be adequately rep­resented by a polygon joining the sample points (implicit assumption for the recursive evaluation of convolutions). Numerical values of u0 (or •&0) and u0At (or $0At) are listed in columns 3 and 4 of Table 2. To simplify the captions in the table, the common factors M0max and ^omax are omitted, or if the reader prefers, set equal to 1 m and 1 rad, respectively.

Suppose that the duration of the pulse, T0, expressed in terms of r0 and c, happens to be 4r0/c. Accordingly, the time increment At is 0.5 r0lc, and the associated dimensionless parameter cAt/z0, which specifies the recursion constants, is equal to 0.5/(Z(Ao)- From Table 1, the appropriate aspect ratios zQ/r0 are 2.094 for vertical motion and 2.356 for rocking, leading to cAt/zQ = 0.2387 and 0.2122, respectively.

0 0.5 1 1.5 2 TIME PARAMETER NTo

FIG. 5. Example: (a) Displacement Pulse; (b) Associated Velocity History; (c) Vertical Force; and (d) Rocking Moment

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6Z9

C7\CyiAOJfOMO^OOO-vJO\UiAWN)MO

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to i^ to to w to -O. NO --J -O NO -O. --J N) -O. vj M -J -O. -J -J -O. -J O

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ct 3

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In the stiffness formulation for vertical motion, the force is determined from (5), rearranged as

po,Ua + -sfu (15) K

Since K = pc2A0/z0, and C = pcA0, the denominator of the second term is identical to the aforementioned parameter cAt/z0 = 0.2387, and, thus,

^ = u0 + 4.1888 H0Af (16) K as listed in column 5 of Table 2 and plotted in Fig. 5(c).

In the flexibility formulation for vertical motion, the force samples PJK are regarded to be the input. Ideally, the convolution u0 = hl ~k P0/K should retrieve the original time history of displacement. To effect this convolution, the appropriate recursion constants are evaluated as described in Appendix II: ax = 0.787626; b0 = 0.110408; b, = 0.101966. Using these, column 6 of Table 2 is calculated via (12). A typical value is computed as follows: for n = 4

w0 = 0.787626-0.8228 + 0.110408-1.0000

+ 0.101966-2.0167 = 0.9641 (17)

The results in column 6 are in good agreement with the original values in column 3. The small amount of error (3.5%) is due to the fact that the recursive algorithm implicitly assumes the samples PJK to be joined to a polygon instead of interpolated by a smooth curve.

In the stiffness formulation for rocking, the singular part of the moment is determined by (6), rearranged as

^ - o ^ T 4 * - («>

* ®" Since K^ = 3 pc2/0/z0; and Ca = pcl0, the denominator of the second term in now equal to three times the parameter cAt/z0 = 3-0.2122 = 0.6366, and, thus

M" —- = $0 + 1.5708 ft0At (19)

as listed in column 7 of Table 2. Next, the regular part of the moment is addressed. Just as for the singular

part, the expression for MrJK^ also includes the factor 1.5708

Mr

-rr = K * (-1.5708 •froA/) (20)

The appropriate recursion constants from Appendix II are ax = 0.808798; b0= -1.5708-0.098980= - 0.155478; bx = -1.5708-0.092222= -0.144862. Note that the constant factor —1.5708 is multiplied a priori with the bs.

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This enables column 4 of Table 2 to be used directly as input to the recursive algorithm.

The sum of columns 7 (singular part) and 8 (regular part) of Table 2 yields the total moment MJK% listed in column 9 and shown as a solid line in Fig. 5(d). It may be observed that the contribution of the regular part (dotted curve and the entire lingering tail for t > T0) is small, but not negligible.

To conclude the example, the moment samples M0IK% are regarded as input for the flexibility formulation. The convolution %Q = h2 -k M0/K$ should retrieve the original time history of rotation. The reader is encour­aged to consult Appendix II, evaluate the recursion constants ax = 1.430258; a2 = -0.529078; b0 = 0.276710; bx = 0.016322; b2 = -0.194212, and recompute the numerical values in column 10 of Table 2. The error between columns 3 and 10 is practically negligible (<2%) because the solid curve in Fig. 5(d) is very well approximated by the dashed polygon.

CONCLUSIONS

For all components of motion, a mat footing (base mat) on unlayered soil may be modeled by semi-infinite truncated cone. The accuracy by com­parison to rigorous solutions is quite adequate, even though the cones cannot represent surface waves, which, in reality, transmit the major portion of energy away from a vibrating foundation.

The cone models are dynamically equivalent to simple discrete systems consisting of a small number of springs, dashpots, and (sometimes) masses with frequency-independent coefficients. If the soil part of a soil-structure interaction problem is idealized by these discrete elements, the coupled system can be analyzed with a general-purpose computer program.

Alternatively, and particularly for hand calculations, the cones may be analyzed by simple numerical procedures based on recursion relationships.

The various computational techniques, which work exclusively in the time domain, are illustrated by an example that incorporates both stiffness and flexibility formulations.

The mathematical solutions for cone models are summarized in appen­dices to the main text. Again, all derivations are performed in the time domain.

APPENDIX I. DERIVATION OF FORCE-DISPLACEMENT RELATIONSHIP FOR ROTATIONAL CONE MODEL

The governing equation of the rotational cone

d2$ 4<H> -& dz2 Z dz 2+ZTZ-~i = 0 (21)

must first be modified by introducing a dimensionless potential function ip such that

* = £°*E (22) z dz K '

Substitution of (22) into (21) results in

dz3 z dz2 z2 dz c2 dz l '

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Eq. (23) may be rewritten as

d_

dz K>™ (zip) A2

dz 2 (Z (P) -(zip)

A solution of (24) is apparent by inspection

d2(zy) (zip)

dz2 = const -z

(24)

(25)

The left-hand side in brackets is recognized as the wave equation in zip. It turns out to be convenient to specify the constant on the right-hand side to be 3®/zo- Then the solution of this modified wave equation takes the form

z<P = -Zo\f(a) ~ F] 0 z 3

2zl <$>z (26)

with additional constants of integration 3> and F. The minus sign preceding the term in brackets is chosen for convenience; f(a) is any arbitrary function of the argument a = t - (z - z0)/c for outward-propagating waves. The correctness of the solution is verified by insertion of (26) into (25). Note that F is nothing more than a constant shift and may be incorporated into f(a). With this simplification, the potential function becomes

. ,*>•!£ + $

The angle of rotation •& is retrieved via (22)

^=(f)3/(«)+f (|°)2/'(a) + 0 ....

(27)

(28)

Without loss of generality, the constant shift 0 may be incorporated into -5 (i.e., set equal to zero). The expression for the derivative is computed via the chain rule

-dft

dz 2 z0

Z0\ ,., \ . ~> I Zt

z! m + z f(a) 2 o / -° ) /"(«)

At the surface (z = z0) (28) and (29) simplify to

o = / + - " /

(29)

(30)

for the rotation, and

-dfto = 2 / f zof dz z0 V c J + / + ? / - f (31)

for its derivative, whereby, the latter expression may be written in terms of its predecessor as

-3tf0

dz z0 c f c

(32)

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To eliminate / , (30) is differentiated once with respect to time

*•->+?£ <33)

The general solution of this first-order equation in. / may be found in any text on ordinary differential equations; it is the convolution

/ = hy • tf0 = Jo K(t - T)d0(T) or (34)

in which h^t) = {clz0)e-c,lz0. Insertion of (34) into (32) yields

-afto 3 •&„ hx * ftp . . = — Un + (35)

dz z0 c c v '

Finally M0 = -pc2/03-&0/3z; i.e.

M0 = ^ 2 tf0 + pc/0 \ ~ fti • pc/0 ^o ^ 1 _ S ^ Sr> (36)

which is the desired force-displacement relationship, (6). In certain instances, in particular for the layered systems considered in

the companion paper (Meek and Wolf 1992), it is necessary to know how the angle of rotation decreases with depth. Substitution of (30) (with argu­ment a instead of t and / ' instead of / ) into (28) (with 0 = 0) yields

%{z,t) = ( | ° ) *0(fl) + Z0 \ (Zo

/ («) (37) z,

in which, by analogy to (34)

f=h1*\ (38) At the surface (z = z0), the contribution of the second term vanishes and -&(0, t) = $0(t). In the underground region, however, part of the rotation is due to the convolution / ; and the amplitude decays in inverse proportion to a combination of the square and the cube of depth.

APPENDIX II. DERIVATION OF RECURSIVE CONSTANTS

The response to a Dirac impulse of excitation is denoted as h(t). The second integral of the Dirac impulse is the ramp function.

x{t) = t for t a 0 (39a)

x{t) = 0 for t < 0 {2,9b)

Consequently, the second integral of h(t), denoted as s(t), is the response to the ramp. In particular

Si(0 = t - - (1 - e~d/z°) (40a)

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Sl(t) = t - ^ -° e"1-**'" sin ^ - (406) 3 c Z Zg

for * > 0; and .^(f) = .?2(f) = 0 for f < 0. The reader may verify that the second derivatives of sx and s2 are indeed identical to ft, [(10a)] and h2 [(10c)].

Samples of the ramp excitation are JC_X = x0 = 0; ^ = Af; *2 = 2At; and #3 = 3Af. Corresponding samples of the ramp response, computed from (40a) or (40b), are y ^ = y0 = Q;yx = s(Af);y2 = s(2Af);andy3 = s(3Af).

Next postulate the existence of a recursion formula

yn = fll^-l + a2yn-2 + V » + M/.-1 + &2*«-2 (41)

and evaluate it for n = 1

*(A0 = b0M (42a)

for n = 2

s(2Af) = a^(A0 + 2b0Af + btAf (42b)

for « = 3

s(3Af) = ax5(2A0 + a2s(At) + 3b0At + 2bxAf + b2At (42c)

These three equations may be solved in turn for b0, b l5 and b2

b0 = ^ (43a)

= <2At) - (2 + flt)a(AQ ( 4 3 f c )

fe = ^(3AQ - (2 + gl)j(2AQ + (1 + 2at - a2)s(M) ^

The recursion constants a2 and b2 are both zero for convolutions involving the simpler impulse response h1.

APPENDIX III. REFERENCES

de Barros, F. C. P., and Luco, J. E. (1990). "Discrete models for vertical vibrations of surface and embedded foundations." Earthquake Engrg. Struct. Dyn., 19(2), 289-303.

Ehlers, G. (1942). "The effect of soil flexibility on vibrating systems" (in German). Beton und Eisen, Berlin, Germany, 41(21/22), 197-203.

Gazetas, G. (1984a). "Rocking of strips and circular footings." Proc. Int. Symp. Dynamic Soil-Structure Interaction, D. E. Beskos et al., eds., A. A. Balkema, Rotterdam, the Netherlands, 3-11.

Gazetas, G. (1984b). "Simple physical methods for foundation impedances." Dy­namic behaviour of foundations and buried structures (Developments in soil me­chanics and foundation engineering, Vol. 3, P. K. Banerjee and R. Butterfield, eds., Elsevier Applied Science, London, England, 45-93.

Gazetas, G., and Dobry, R. (1984). "Simple radiation damping model for piles and footings."/. Engrg. Mech. Div., ASCE, 110(6), 937-956.

Meek, J. W. (1990). "Recursive analysis of dynamical phenomena in civil engi­neering" (in German). Bautechnik, Berlin, Germany, 67(6), 205-210.

Meek, J. W., and Veletsos, A. S. (1974). "Simple models for foundations in lateral

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and rocking motion.'' Proc. 5th World Conf., on Earthquake Engrg., IAEE, Rome, Italy, 2, 2610-2613.

Meek, J. W., and Wolf, J. P. (1992). "Cone models for soil layer on rigid rock." J. Geotech. Engrg., ASCE, 118(5), 686-703.

Veletsos, A. S., and Nair, V. D. (1974). "Torsional vibration of viscoelastic foun­dations." J. Soil Mech. Found. Div., ASCE, 100(3), 225-245.

Veletsos, A. S., and Verbic, B. (1973). "Vibrations of viscoelastic foundations." Earthquake Engrg. Struct. Dyn., 2(1), 87-102.

Veletsos, A. S., and Verbic, B. (1974). "Basic response functions for elastic foun­dations." /. Engrg. Mech. Div., ASCE, 100(2), 189-202.

Veletsos, A. S., and Wei, Y. T. (1971). "Lateral and rocking vibrations of footings." J. Soil. Mech. Found. Div., ASCE, 97(9), 1227-1248.

Wolf, J. P. (1988). Soil-structure-interaction analysis in time domain. Prentice-Hall, Englewood Cliffs, N.J.

Wolf, J. P., and Motosaka, M. (1989). "Recursive evaluation of interaction forces of unbounded soil in the time domain from dynamic-stiffness coefficients in the frequency domain." Earthquake Engrg. Struct. Dyn., 18(3), 365-376.

Wolf, J. P., and Somaini, D. R. (1986). "Approximate dynamic model of embedded foundation in time domain." Earthquake Engrg. Struct. Dyn., 14(5), 683-703.

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