Cones to model foundation vibrations: incompressible soil and axi-symmetric embedment of arbitrary shape

RESEARCH REPORT C:1761

STRUCTURAL MECHANICS GROUP

SCHOOL OF CIVIL & RESOURCE ENGINEERING

THE UNIVERSITY OF WESTERN AUSTRALIA

CONES TO MODEL FOUNDATION VIBRATIONS:

INCOMPRESSIBLE SOIL AND AXI-SYMMETRIC EMBEDMENT

OF ARBITRARY SHAPE

JOHN P. WOLF1 and ANDREW J. DEEKS2

ABSTRACT

The recently streamlined strength-of-materials approach using cones to calculate vibrations of

foundations embedded in layered half-spaces and full-spaces is applied to incompressible and nearly-

incompressible soil and to axi-symmetric embedments of arbitrary shape. For incompressible soil the

axial-wave velocity in the cones is limited to twice the shear-wave velocity and a trapped mass for the

vertical motion and a trapped mass moment of inertia for the rocking motion moving as a rigid body

with the under-most disk of an embedded foundation are introduced. In the case of a fully embedded

foundation, a mass and a mass moment of inertia are also assigned to the upper-most disk. For an axi-

symmetric embedment of arbitrary shape, the disks have varying radii. No modifications to the

formulation are, however, required. For these two extensions the strength-of-materials approach using

cones leads to the same sufficient engineering accuracy as is achieved in other more conventional

cases. This is demonstrated in a vast study. Thus the same other advantages also apply: physical

insight with conceptual clarity, simplicity and sufficient generality.

Keywords: cone model, dynamic stiffness, embedded foundation, embedment of arbitrary shape,

foundation vibration, incompressible, layered half-space.

1Civil Engineering, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne, Switzerland.

2School of Civil & Resource Engineering, The University of Western Australia, Crawley, Western

Australia 6009, email: [email protected], phone: +61 8 9380 3093, fax: +61 8 9380 1018.

(Corresponding author.)

mailto:[email protected]

Research report C:1761 Department of Civil and Resource Engineering, UWA

CONES TO MODEL FOUNDATION VIBRATIONS:

INCOMPRESSIBLE SOIL AND AXI-SYMMETRIC EMBEDMENT OF

ARBITRARY SHAPE

JOHN P. WOLF and ANDREW J. DEEKS

1. INTRODUCTION

To analyse the vibrations of a foundation on the surface of or embedded in a layered half-space or full-

space, an approach using conical bars and beams, called cones, exists. The complicated exact

formulation of three-dimensional elastodynamics is replaced by the simple one-dimensional description

of the theory of strength of materials, postulating the deformation behaviour (“plane sections remain

plane”).

The half-space with linear elastic behaviour and hysteretic material damping (Fig. 1) can consist of any

number of horizontal layers of depth dj, shear modulus Gj, Poisson’s ratio νj, mass density ρj and

material damping ratio ζj either overlying a half-space or fixed at its base. A cylindrical foundation of

embedment depth e and radius r0 is shown in Fig. 1, with the wall and base of the embedded foundation

assumed to be rigid.

In the frequency domain (for harmonic excitation of frequency ω), the dynamic behaviour of the

foundation is described by the dynamic-stiffness coefficients S(ω) relating, for each degree of freedom,

the force amplitude P(ω) to the displacement amplitude u(ω) through the equation

P(ω) = S(ω) u(ω) ................................................................................................................................ (1)

(For seismic waves, in addition the so-called effective foundation input motion is required to allow

formulation of the equations of motion, which is based on the same concept as the dynamic-stiffness

coefficients. This aspect is not addressed in this paper.)

Only approximations of the one-dimensional strength-of-materials approach based on wave propagation

in cones apply. No other assumptions are made. For each degree of freedom only one type of body

wave exists: for the horizontal and torsional motions S-waves propagating with the shear-wave velocity;

and for the vertical and rocking motions P-waves propagating, in principle, with the dilatational-wave

velocity. The corresponding displacements can be formulated directly in closed form as a function of

the depth of the site, without any spatial Fourier transformation into the wave number domain.

The use of cone models does indeed lead to some loss of precision compared to applying rigorous

methods based on three-dimensional elastodynamics. However, this is more than compensated by the

many advantages. The strength-of-materials approach using cones leads to physical insight with

conceptual clarity, and is simple to use and solve, as the mechanical formulation is simplified, provides

sufficient generality (layered site, embedment, all frequencies) and acceptable engineering accuracy.

The accuracy of any analysis is limited anyway, because of the many uncertainties, some of which can

never be eliminated (for instance the definition of the dynamic loads and the values of the dynamic soil

Research report C:1761 School of Civil & Resource Engineering, UWA

properties in the analysis of a machine foundation). Parametric studies are thus essential. The analysis

with cones fits the size and economics of practical engineering projects. Thus, cone models should be

applied for foundation vibration and dynamic soil-structure-interaction analysis in a design office

whenever possible.

A just published text book exists [1], where the one-dimensional strength-of-materials approach using

cones is developed from scratch and applied to practical foundation vibration problems using either the

provided executable program or the corresponding MATLAB routines. Recent key papers include Refs

[2] and [3], where wave reflection and refraction, each occurring in their own cones, at a material

discontinuity corresponding to a (single) layer interface is addressed and Ref. [4] where an embedded

foundation is modelled with a stack of disks. For a complete description of cone models up to 1994

including other simple physical representations such as spring-dashpot-mass models, the book [5] can

be consulted.

Only very recently, from 2001 onwards, the physically appealing concept of one-dimensional wave

propagation in cone segments with reflections and refractions occurring at material interfaces as

mentioned above is generalised to a multiple-layered half-space without introducing additional

assumptions. It is possible to track the reflection and refraction of each incident wave sequentially and

determine the resulting wave pattern up to a certain stage by superposition. This important concept is

derived in Ref. [6] where the accuracy is also evaluated in a parametric study, and is shown to be more

than sufficient. An efficient computer implementation with recursive evaluation of the reflected and

refracted waves is discussed in Ref. [7]. Summarising, the reader interested in becoming familiar with

today’s state of the art of the analysis of foundation vibrations using cones can concentrate on Ref. [1].

This paper has two goals. The first is to address a procedure to analyse incompressible and nearly-

incompressible soil for vertical and rocking motions limiting the (dilatational) wave velocity and

introducing a trapped mass. This concept, developed in Ref. [8] for a surface foundation on a

homogeneous half-space, is applied to foundations embedded in a layered half-space. The accuracy is

comparable to that of the compressible layered half-space. The second is to address axi-symmetric

foundations of arbitrary embedded shape with a vertical axis, either partially or fully embedded (Fig. 2).

Unlike a cylindrical foundation, the wall does not have to be vertical. Again, performance is similar to

that of a cylindrical foundation. The main thrust of the paper thus consists of demonstrating that the

strength-of-materials approach using cones leads to results of sufficient accuracy in an extended range

of practical situations. In addition, certain features (simplicity with physical insight) are demonstrated

in an elementary way, which should motivate the reader to study the procedure in depth.

Section 2 presents a summary of the computational scheme. Section 3 addresses the modifications

necessary to consider the vertical and rocking motions of foundations embedded in a layered half-space

of incompressible and nearly-incompressible soil. Section 4 discusses the caution which must be

exercised when analysing foundations of arbitrary embedded shape. Section 5 evaluates the accuracy of

the extensions discussed in Sections 3 and 4, demonstrating clearly that cone models can also be applied

to foundations embedded in a layered half-space of incompressible and compressible soil and to axi-

symmetric embedments of arbitrary shape. Section 6 contains concluding remarks.


2. SUMMARY OF THE COMPUTATIONAL SCHEME

It is appropriate to summarise the computational scheme to calculate the dynamic stiffness of a

cylindrical foundation embedded in a layered half-space using cones. (This will not be sufficient for the

reader who has not studied systematically in depth this one-dimensional strength-of-materials approach,

which can easily be done by consulting Ref. [1].)

For illustration, the vertical dynamic stiffness of a cylindrical foundation embedded in a layered half-

space is addressed (Fig. 3). The left-hand side of Fig. 3a shows a cylindrical soil region with radius r0

extending to a depth e into the layered half-space. This soil region, which will later be excavated, is

viewed as a sandwich consisting of rigid disks separated by soil layers (slices). The disks will coincide

with the interfaces of the half-space. In addition, further disks are selected to adequately represent the

dynamic behaviour. At least 10 nodes (disks) per wave length are recommended. On the disks on the

surface of and embedded in the layered half-space, unknown vertical forces Pi are applied, which can be

regarded as redundants. The primary system, which is indeterminate, addresses the vertical

displacements uj of the disks on the surface of and embedded in the layered half-space without

excavation, that is, the so-called free field. As in the force method of structural analysis, the

relationship between the displacements of the disks and the forces acting on the disks is established.

This is based on wave propagation in cone segments considering layering. To determine the

displacements caused by the force Pi applied on a disk of radius r0 shown in Fig. 3b, each disk

embedded in the full-space is modelled by two truncated semi-infinite cones (called a double cone) with

an opening angle determined by the aspect ratio z0/r0. For the vertical degree of freedom and ν ≤ 1/3

(compressible case), the wave velocity is cp, the dilatational-wave velocity. Two possibilities exist to

calculate z0/r0: either the static stiffness of two disks on half-spaces stuck together, as presented in Fig.

3c, or the static stiffness of a disk embedded in a full-space, as shown in Fig. 3d, is set equal to the static

stiffness of the double cone, yielding the z0/r0 specified in Table 1. The shear wave velocity is denoted

as cs = G/ρ. It turns out that z0/r0 is a function of ν only. (The case of 1/3 < ν ≤ 1/2 and the line

labelled ‘Trapped Mass’ will be discussed in Section 3.) The two semi-infinite cones in Fig. 3b, each

resisting half the force Pi/2, model the outward wave propagation occurring from the disk. The

sectional property of these initial cones increases in the direction of wave propagation. The left-hand

side of Fig. 3e presents the lower radiating cone. When the outward propagating wave in the initial

cone, denoted as the incident wave, encounters a material discontinuity at a layer interface, a reflected

wave and a refracted wave, each propagating in its own cone, are created, as shown on the right-hand

side of Fig. 3e. Enforcement of compatibility of displacement and equilibrium at the fictitious disk

located at the interface permits the reflected and refracted waves to be expressed as a function of the

incident wave. (Free surfaces and fixed boundaries are treated as special cases. Analytical expressions

describing the wave propagation along a cone segment (Fig. 3e) and the wave mechanisms at the

interface are summarised in the Appendix.) The reflected and refracted waves generated at an interface


will also encounter material discontinuities as incident waves at a later stage, yielding additional

reflections and refractions. By tracking the reflections and refractions sequentially, the superimposed

wave pattern can be established for a layered site up to a certain stage. The resulting wave pattern

shown in Fig. 3f permits the displacement at disk j to be calculated, yielding the dynamic flexibility

matrix. Inversion of this displacement − force relationship yields the dynamic stiffness for the disks on

the surface of and embedded in the layers site, expressing the forces (redundants) as a function of the

displacements. Referring to the left-hand side of Fig. 3a, the disks and the soil trapped between them

are constrained to execute rigid-body motion, as the walls of the foundation are rigid. For the vertical

degree of freedom this is enforced by equating the vertical displacements of all disks. This also results

in the force acting on the rigid foundation being equal to the sum of all the vertical forces acting on the

disks. The trapped soil is analytically “excavated” by subtracting the mass of the soil times acceleration

of the rigid interior cylinder of soil from the force, as indicated in the centre part of Fig. 3a. This leads

to the dynamic stiffness of the rigid foundation embedded in a layered half-space illustrated on the

right-hand side of Fig. 3a.

In a nutshell, the computational procedure (applicable also for the other degrees of freedom) can be

characterised as follows. In this one-dimensional strength-of-materials approach, waves of one type for

each degree of freedom propagate in cone segments outwards with reflections and refractions occurring

at layer interfaces. The sectional property of the cone segments increases in the direction of wave

propagation, downwards as well as upwards. The restrictions of tapered bar and beam theory only

apply. No other assumptions are enforced. For a surface foundation the computational procedure is

analytical. For an embedded foundation that part of the half-space which will be excavated is

discretised with a stack of disks, and thus a numerical approximation is required.

3. INCOMPRESSIBLE AND NEARLY-INCOMPRESSIBLE LAYERED HALF-SPACE

For the horizontal and torsional degrees of freedom, S-waves are present. As the corresponding shear-

wave velocity cs remains finite for all Poisson’s ratios ν, the construction of the cone model presented in

Table 1 applies for all ν.

For the vertical and rocking degrees of freedom, axial distortions occur in the cone models. For low

values of ν, P-waves propagating with the dilatational-wave velocity cp are present. For ν approaching

1/2, cp = cs 2(1-ν)/(1-2ν) tends to infinity, and a cone model cannot be constructed in the usual way.

Special features are necessary for this case.

Thus, for the vertical and rocking degrees of freedom, the procedure depends on ν. In the range 0 ≤ ν ≤

1/3 (compressible case), the wave velocity c is equal to cp and z0/r0 follows from Table 1. In the range

1/3 < ν < 1/2, called the nearly-incompressible case, and for ν = 1/2 (the incompressible case), two

features are enforced: first, the axial-wave velocity c is limited to 2 cs; and second, a trapped mass for

the vertical degree of freedom and a trapped mass moment of inertia for the rocking degree of freedom

are introduced. This is illustrated in Fig. 4 for various cases.


For a disk on the surface of a half-space (Fig. 4a) a trapped mass of soil beneath the disk is introduced,

which moves as a rigid body in phase with the disk. For the vertical degree of freedom the trapped

mass is

∆M = 2.4

ν −

13 ρ A0 r0 ................................................................................................................. (2)

with A0 = π r02, and for the rocking degree of freedom the trapped mass moment of inertia is formulated

as

∆Mϑ = 1.2

ν −

13 ρ I0 r0 .................................................................................................................. (3)

with I0 = π r04/4. According to these formulae, the inclusion of trapped mass begins at ν = 1/3 and

increases linearly with ν. Although these quantities are determined by simple curve fitting, they have a

physical justification. (In engineering practise it has been observed that blunt-ended piles may be

driven just as easily as piles with pointed tips. The reason is that the pile creates its own pointed tip in

incompressible soil.) Fig. 4a illustrates the trapped mass ∆M for the vertical degree of freedom, but also

applies with the trapped mass moment of inertia ∆Mϑ. The procedure to construct the cone model is

summarised in Table 1.

For a disk embedded in a full-space modelled as a double cone (Fig. 4b), each of the two truncated

semi-infinite cones yields a trapped mass. Twice the values applicable for a disk on a half-space thus

apply for the disk embedded in a full-space.

For a cylindrical foundation embedded in a half-space (Fig. 4c), the trapped mass need only be included

for the cone of soil under the base. As sketched in the computational procedure of Section 2, the

intermediate disks of the stack are separated by thin slices of soil. In the course of the analysis,

however, the mass of the slices is subtracted out (excavated) thus resulting in only the trapped mass on

one side of the under-most disk forming the base of the embedded foundation.

For a cylindrical foundation embedded in a full-space (Fig. 4d), besides the trapped mass term for the

under-most disk representing the cone of soil under the base, that of the upper-most disk modelling the

cone of soil above the fully-embedded foundation is included. Again, the mass of the excavation does

not appear. (Of course, as the walls and base of the foundation are rigid, the two contributions of the

trapped mass are added.)

For a layered half-space, the same concept applies. For instance, for the fully-embedded foundation of

Fig. 4d, the contribution to the trapped mass is calculated with the mass density of the soil adjacent to

the disk.

In a soil-structure-interaction analysis, it is convenient to assign the trapped mass and trapped mass

moment of inertia to the base-mat of the structure. They are thus processed as part of the structure.

4. AXI-SYMMETRIC FOUNDATIONS OF ARBITRARY EMBEDMENT SHAPE

When modelling cylindrical foundations, all the disks in the stack located in the region of the soil which

will later be excavated will have the same radius. However, the strength-of-materials approach using


cones is sufficiently general to allow variation of the radii of the disks, permitting axi-symmetric

foundations (with a vertical axis) of arbitrary embedded shape to be analysed, as shown in Fig. 2.

Some caution must be exercised when generating a cone model for a foundation of varying radius. It is

implicit in the strength-of-materials approach that the radii of the disks modelling the foundation are

always less than the radius of the cone segment in which an incident wave propagates. When all disks

have the same radius, this is guaranteed (Fig. 5a). However, if two adjacent disks are significantly

different in radius but close together in the vertical direction, this assumption may be violated. This is

the case when a foundation in the shape of a spherical cap (Fig. 5b) is modelled.

To avoid generating a model which significantly violates this strength-of-materials assumption, certain

modifications are introduced. In the zone where the radius of the spherical cap measured horizontally

from the vertical axis of symmetry is small, the (curved) cap is replaced by a (plane) disk of substantial

radius. The vertical distance ∆e between two neighbouring disks follows as usual from the requirement

that, for instance, 10 nodes (that is disks) per wave length must be present, resulting in

∆e ≤ π5

cω ........................................................................................................................................... (4)

with the wave velocity c, in general, equal to cs. The radii of the disks above are selected somewhat

larger than would follow from the geometry. The reader is referred to the examples addressing hemi-

ellipsoid and sphere foundations in Sections 5.5 and 5.6 where engineering accuracy is still achieved,

although modifications to the geometry of the foundation are introduced.

5. EVALUATION OF ACCURACY

To gain confidence in applying cone models for incompressible soil and for axi-symmetric embedded

foundations of arbitrary shape, a systematic evaluation of the accuracy is performed. Besides

addressing the basic cases of a disk embedded in a homogeneous full-space, cylindrical foundations

embedded in a homogeneous half-space and in a layered half-space, both incompressible, are examined

for the vertical and rocking degrees of freedom. As examples of axi-symmetric embedments of

arbitrary shape, a hemi-ellipsoid embedded in a homogeneous half-space and a sphere embedded in a

homogeneous full-space are discussed. The effect of the two calibration methods for the cone aspect

ratio is also considered. Analytical solutions are used for comparison whenever available, together with

some rigorous numerical results based on a very fine discretisation.

The dynamic-stiffness coefficient S(ω) defined in Eq. 1 is addressed. Introducing the dimensionless

frequency

a0 = ω r0

cs ............................................................................................................................................... (5)

with r0 representing a characteristic length of the foundation, for instance the radius, and cs the shear-

wave velocity of some layer, the dynamic-stiffness coefficient is decomposed as

S(a0) = K [k(a0) + i a0 c(a0)] ............................................................................................................... (6)


with a suitably chosen static-stiffness coefficient K. k(a0) represents the dimensionless spring

coefficient and c(a0) the dimensionless damping coefficient. In general, the comparison is performed

for k(a0) and c(a0). Note that while the real part of S(a0) is represented by k(a0), the imaginary part is

obtained by multiplying c(a0) by a0. In the higher-frequency range (a0 >> 1), c(a0) is more important

than k(a0). When only one quantity is to be addressed when checking the accuracy, the magnitude

|S(a0)| = K k2(a0) + [a0 c(a0)]2 ......................................................................................................... (7)

is chosen, representing the largest value of the force P(a0) for a unit u(a0).

5.1. Disk embedded in homogeneous full-space

The basic case which is applied at the start of the computational procedure (Fig. 3b) consists of a disk

embedded in a homogenous full-space. To establish the opening angle determined by z0/r0 either the

half-space calibration (Fig. 3c) or the full-space calibration (Fig. 3d) can be applied. The equations are

specified in Table 1. For the incompressible case (ν = 0.5) both calibration methods yield the same

z0/r0.

The agreement of the double cone model and the effect of the calibration method are addressed. For the

vertical degree of freedom the dynamic-stiffness coefficient is specified in Eq. A.2, multiplying the

right-hand side by 2 to represent the double cone. For the vertical (Fig. 6a) and rocking degrees of

freedom (Fig. 6b), the magnitude of the dynamic-stiffness coefficient (Eq. 7), non-dimensionalised by

the exact (analytical) static-stiffness coefficient of the disk embedded in the full-space is determined as

a function of a0. This leads to results of the cone model with full-space calibration that are exact for the

static case a0 = 0. For the vertical degree of freedom Poisson’s ratio is selected as 1/3, permitting direct

comparison with the exact results of Ref. [9]. For the two calibrations the dynamic stiffnesses are

plotted as continuous lines in Fig. 7, while the ‘exact’ results are indicated as discrete points. In the

case of the vertical degree of freedom use of the full-space calibration technique does not yield any

significant improvement (Fig. 7a), with the exception of the very small frequency range.

For the rocking degree of freedom Poisson’s ratio is selected as 0.3 and 0.5, allowing comparison with

Ref. [10]. The results are plotted in Figs 7b and 7c. In this case, when ν = 0.3, use of the full-space

calibration technique significantly improves the agreement between the computed dynamic stiffness and

the exact values (Fig. 7b). When ν = 0.5, both calibration methods yield the same aspect ratio, and the

results are in close agreement with the exact solution (Fig. 7c).

5.2. Disk embedded in homogeneous half-space

The calculation of the dynamic-stiffness coefficient of a disk of radius r0 embedded with depth e in a

homogeneous half-space also serves to illustrate the computational procedure (Section 2). The vertical

degree of freedom is addressed first with embedment ratios (e/r0) of 2 and 5 and Poisson’s ratios (ν) of

0.25 and 0.5. The embedded disk is modelled with a double cone (Fig. 3b). The relationship between

the initial disk displacement with amplitude u−(ω) and the force with amplitude P(ω) is (Eqs A.1, A.2

and 2)


u−(ω) = 21

ρ c2 πr02

z0 − ω2 ∆M + i ω ρ c πr0

2

P(ω) .................................................................................. (8)

where the term with ∆M is present only for the incompressible case ν = 0.5 (Fig. 4b). The first term in

the denominator equals the static-stiffness coefficient, which for a half-space calibration (Fig. 3c) is

equal to the exact static-stiffness coefficient of a disk on the surface of a half-space 4 ρ cs2 r0/(1-ν).

The wave propagating downwards in the initial cone below the disk (with apex denoted 1 in Fig. 8) does

not lead to any reflections, and need not be processed any further. The wave propagating upwards in

the initial cone (Fig. 3 and Fig. 22a) above the disk (with apex denoted as 2) impinges on the free

surface as an incident wave with the amplitude (Eq. A.4 replacing d by e)

f(ω) = z0

z0 + e e−i

ωc e

u(ω) ................................................................................................................... (9)

The reflected wave from this surface (Eq. A.5) is

g(ω) = −α(ω) f(ω) = z0

z0 + e e−i

ωc e

u(ω) ....................................................................................... (10)

since −α(ω) = +1 for a free boundary (Eq. A.7 with c′ = 0). This reflected wave propagates in the new

initial cone with apex denoted as 3 in Fig. 8. When it reaches the depth of the disk its amplitude has

decreased to (analogous to Eq. A.3 considering the geometry of the cone)

f(ω) = z0 + e

z0 + 2e e−i

ωc e

z0

z0 + e e−i

ωc e

u(ω) = z0

z0 + 2e e−i

ωc 2e

u(ω) ........................................... (11)

Since the half-space is homogeneous, this wave continues to propagate towards infinity in the truncated

semi-infinite cone, and no further reflections occur. The total displacement at the disk is obtained by

superposing this wave (Eq. 11) and the incident generating wave (Eq. 8), yielding the amplitude

u(ω) =

1 + z0

z0 + 2e e−i

ωc 2e

u(ω) ................................................................................................ (12)

This corresponds to the description of the downwards and upwards wave propagation (Fig. 3f). Eq. 12,

after substituting Eq. 8 defines the dynamic-flexibility coefficient. Inversion of this relationship leads

to the force − displacement relationship

P(ω) = S(ω) u(ω) .............................................................................................................................. (13)

with the dynamic-stiffness coefficient S(ω) of the disk embedded in the half-space

S(ω) = 2 1

1 + z0

z0 + 2e e−i

ωc 2e

ρ c2 πr0

2

z0 − ω2∆M + i ω ρ c πr0

2 ................................................... (14)

The dynamic-stiffness coefficient (Eq. 6) is normalised with respect to the static-stiffness coefficient of

the same disk on the surface of the half-space K = 4 ρ cs2 r0/(1-ν) for both embedment ratios. The


spring coefficient k(a0) and damping coefficient c(a0) are plotted together with the exact results of Ref.

[11] in Fig. 9. Impressive agreement is evident.

The torsional degree of freedom leading to twisting of the embedded disk is also investigated. The

dynamic-stiffness coefficients with embedment ratios (e/r0) of 1 and 2 are plotted in Fig. 10, normalised

with respect to the exact static-stiffness coefficient of the same disk on the surface. Exact results from

Ref. [12] are plotted as discrete points for comparison.

5.3. Cylinder embedded in incompressible homogeneous half-space

Turning to the modelling of incompressible soil (Section 3), the dynamic-stiffness coefficients of a rigid

cylindrical foundation of radius r0 with an embedment ratio e/r0 of 1 (Fig. 11) in an incompressible

homogeneous half-space (Poisson’s ratio ν = 1/2) are calculated for vertical and rocking motion. A

hysteretic damping ratio ζ of 5% is present. The results are determined for dimensionless frequencies

a0 from 0 to 4.

The maximum vertical distance ∆e follows from Eq. 4, which can be expressed as a function of a0 (Eq.

5) as

∆e ≤ π5

r0

a0 ......................................................................................................................................... (15)

The maximum distance ∆e follows with a0 = 4 as (π/20)r0. ∆e = 0.125r0 is selected. This leads to 8

slices yielding 9 disks in the domain of the soil which will later be excavated (Fig. 3a). For the under-

most disk (Fig. 4c), a trapped mass ∆M (Eq. 2) or trapped mass moment of inertia ∆Mϑ (Eq. 3) is

included.

The dynamic-stiffness coefficients are decomposed using the static-stiffness coefficients computed by

the cone model. The dynamic-stiffness coefficients using cones for the vertical and rocking motions are

plotted in Fig. 12a and Fig. 12b. Also plotted in these figures are rigorous solutions presented in Ref.

[13] using a very fine mesh for the thin-layer method, which are denoted as exact and plotted as discrete

points. Note that k(a0) exhibits a downward tendency for increasing a0, caused by the trapped mass

leading to a contribution of −ω2 ∆M and −ω2 ∆Mϑ to the dynamic-stiffness coefficients. The solutions

of the cone models agree impressively with the exact results, especially for the damping coefficient at

higher dimensionless frequencies.

5.4. Cylinder embedded in incompressible layered half-space

The accuracy for a cylindrical foundation embedded in an incompressible multi-layered half-space for

the vertical and rocking degrees of freedom is evaluated. The rather extreme site (Fig. 13) consists of

three layers with the shear modulus decreasing with depth, which could correspond to a typical landfill

situation. A fixed base exists, leading to total reflection of all waves. Waves will still propagate

towards infinity, but in the horizontal direction only. (This is a stringent test, as this wave propagation

towards infinity does not occur for low frequencies for vanishing material damping, yielding a


vanishing damping coefficient in this range.) The dynamic-stiffness coefficients are calculated up to a0

= 6 (= ωr0/cs1, with the shear velocity of the first layer cs1).

The number of disks in the stack in that part of the half-space which will be excavated is determined

first. Applying Eq. 4 with a0 = 6, ∆e = (π/30)r0, yielding 10 slices in the first layer. The shear-wave

velocity in the second layer is lower, yielding ∆e = (π/(30 2))r0 and 4 slices in the second layer

(rounded up), which results in 10+4 = 14 slices and 15 disks in total.

The accuracy of the computed dynamic-stiffness coefficients is determined by comparison with results

obtained from Ref. [13]. These results were calculated rigorously using the thin-layer method, and will

be denoted as exact. The decomposition of Eq. 6 is applied, with the exact static-stiffness coefficient K

of a disk of radius r0 on the surface of a homogeneous half-space with the material properties of the first

layer determined for the corresponding degree of freedom. Fig. 14a plots the dynamic-stiffness

coefficient for vertical motion. The agreement is satisfactory, and, in the case of the magnitude (Eq. 7),

engineering accuracy is obtained over most of the frequency range. Fig. 14b plots the dynamic-stiffness

coefficient for rocking motion. Although some discrepancy is apparent for the spring coefficient,

particularly at high dimensionless frequencies, over the entire frequency range the magnitude of the

dynamic-stiffness coefficient retains engineering accuracy.

5.5. Hemi-ellipsoid embedded in homogeneous half-space

The modelling of axi-symmetric foundations of arbitrary embedded shape with a stack of disks of

variable radius (Section 4) is addressed next. The dynamic-stiffness coefficients for a range of hemi-

ellipsoids embedded with depth e in a homogeneous half-space, as illustrated in Fig. 15, are calculated

for torsional motion. Corresponding exact solutions are available in Ref. [14].

The radius of the hemi-ellipsoid (measured horizontally from the vertical axis of symmetry) varies

continuously from the initial radius at the surface r0 to zero at the depth of embedment e, which will be

termed the toe. Near the toe the radius changes sharply with depth. To avoid generating a model that

significantly violates the strength-of-materials assumption discussed in Section 4, the following

procedure is used. The minimum number of slices required in the vertical direction is determined in the

usual way (Eq. 4). ∆e = 0.1r0 is selected. The radius of the disk on each interface is determined by

vertical projection of the intersection of the hemi-ellipsoid with the interface immediately above the

current interface. This is illustrated on the left-hand side of Fig. 16 for a hemi-sphere. The resulting

model boundary, shown on the right-hand side of Fig. 16 for a hemi-sphere, is slightly different from

the original foundation near the toe. However, there is little variation at the top, and adequate results

can be obtained.

The dynamic-stiffness coefficients for the hemi-ellipsoids illustrated in Fig. 15 are plotted in Fig. 17,

along with the exact solution of Ref. [14]. The decomposition of Eq. 6 is used, with the static-stiffness

coefficient K taken as that of a hemi-sphere embedded in a half-space in torsional motion, K = 4π G r03.

Over the entire range of e/r0 ratios calculated, the accuracy of the spring coefficient k(a0) is quite

remarkable. The damping coefficient c(a0) is reasonably accurate, although showing a consistent error


of between +15% and +20%. However, engineering accuracy is maintained across the frequency range,

and the ratio of the damping coefficients between different hemi-ellipsoids in the sequence is predicted

very well.

The overestimation of the damping coefficient is directly linked to the larger surface of the boundary of

the model than that of the hemi-ellipsoid. In the high-frequency limit c(a0) will be proportional to the

surface area. Using 10 layers the model surface area for a hemi-sphere is 10% larger than 2πr02.

5.6. Sphere embedded in homogeneous full-space

To further evaluate the accuracy of the method using cones for a foundation of varying radius embedded

in a homogeneous full-space, the case of a sphere is considered. Due to the point symmetry of the

problem, only two types of motion exist, torsional and rectilinear (Fig. 18). Exact solutions are

available for each type of motion (Refs [15, 16]). The material properties of the full-space are selected

as the shear modulus G, Poisson’s ratio ν = 0.25 and mass density ρ, with the radius of the sphere

denoted as r0.

The modelling procedure for the sphere is essentially identical to that described for hemi-ellipsoids in

Section 5.5. A slice thickness of ∆e = 0.1r0 is used, and the disk radii are calculated as illustrated in

Fig. 18, resulting in a slightly different geometry from that of a true sphere.

Fig. 19a plots the dynamic-stiffness coefficient calculated using the half-space calibration method (Fig.

3c, Table 1) for rectilinear motion. The decomposition of Eq. 6 is used with the exact static-stiffness

coefficient for a sphere in a full-space in rectilinear motion, K = 24π G r0 (1−ν)/(5−6ν). Both the spring

and damping coefficients are about 20% too small, causing the magnitude (Eq. 7) to deviate by a similar

amount. Although within engineering accuracy, these results are a little disappointing.

However, when the full-space calibration method is used (Fig. 3d, Table 1), the dynamic-stiffness

coefficient plotted in Fig. 19b results. In this case the performance is entirely satisfactory, illustrating

the importance of the alternative calibration method in fully-embedded problems.

For torsional motion both calibration methods yield identical models, and the dynamic-stiffness

coefficient plotted in Fig. 19c is obtained. In the decomposition the exact static-stiffness coefficient for

a sphere embedded in a full-space in torsional motion, K = 8π G r03, is used. Excellent agreement with

the exact solution is achieved, although the calculated damping is slightly too large. The same

explanation as discussed in Section 5.5 applies.

In the model shown in Fig. 18, the axes of the cones coincide with the vectors of the two motions

(vertical). This leads for the rectilinear motion to axial deformations in the cones (dilatational-wave

velocity cp) and for the torsional motion to shear deformations in the cones (shear-wave velocity cs).

Since shear deformations are also present in the exact solution for the latter motion, this approach is

appropriate. However, for the former motion, the axes of the cones can also be chosen at a right angle

to the vector describing the motion (Fig. 20). This leads for the rectilinear motion to shear deformations

in the cones (shear-wave velocity cs). The dynamic-stiffness coefficients for this model are plotted for

the two calibrations in Fig. 21. Although still within engineering accuracy, agreement is not quite as


close as is achieved with the cone axes aligned with the direction of motion. Nevertheless, the adequate

performance of the two alternative cone models further increases confidence in the approach.

6. CONCLUDING REMARKS

1. The stream-lined one-dimensional strength-of-materials approach using cones to analyse

foundation vibrations is applied to a) foundations embedded in incompressible and nearly-

incompressible homogeneous and layered half-spaces, and b) axi-symmetric foundations of

arbitrary embedded shape.

2. For an embedded disk, a trapped mass representing the soil material above and below is added to

the double cone model for vertical motion, and, analogously, a trapped mass moment of inertia is

added for rocking motion. These terms increase linearly from zero at a Poisson’s ratio of 1/3, to

maximums at a Poisson’s ratio of 1/2 (incompressible), as indicated in Table 1. In addition, the

wave velocity in the cones is limited to twice the shear-wave velocity for this range of Poisson’s

ratio and these motions. No additional degree of freedom is introduced, as the trapped mass and

trapped mass moment of inertia move as a rigid body with the disk. For an embedded foundation

only the under-most disk will be effected, representing the soil material below this disk. For a

fully-embedded foundation, only the trapped mass and trapped mass moment of inertia for the

upper-most and under-most disks are included, modelling the soil above the upper-most disk and

that below the under-most disk.

3. For an axi-symmetric foundation of arbitrary embedded shape, the disks in the stack will have

varying radii. No modifications in the formulation are necessary as this case is fully covered in the

strength-of-materials approach. However, some caution should be exercised to generate results of

acceptable accuracy. It is desirable that the radii of the disks modelling the foundation are less than

the radius of the cone (segment) in which an incident wave propagates. To avoid generating a

model which significantly violates this requirement, certain variations of the geometry of the

foundation can be introduced if necessary.

4. The two applications to foundations in incompressible soil and to axi-symmetric foundations of

arbitrary embedded shape lead to the same engineering accuracy (deviation ±20% from the exact

result based on three-dimensional elastodynamics) as in other cases. Thus the same other

advantages also apply to these cases: physical insight with conceptual clarity, simplicity and

sufficient generality.

Acknowledgement

The authors are indebted to Professor John L. Tassoulas of The University of Texas at Austin, who

calculated on our request the results based on the thin-layer method used for comparison. Without this

support a systematic evaluation of the accuracy for foundations in incompressible soil would not have

been possible.


References

1. Wolf JP, Deeks AJ. Foundation Vibration Analysis: A Strength-of-Materials Approach. Elsevier: Oxford, 2004.

2. Meek JW, Wolf JP. Insights on cutoff frequency for foundation on soil layer. Earthquake Engineering and Structural Dynamics 1991; 20:651-665, also in Proceedings of the 9th European Conference on Earthquake Engineering, EAEE, Moscow 1990; vol. 4-A, 34-43.

3. Wolf JP, Meek JW. Cone models for a soil layer on a flexible rock half-space. Earthquake Engineering and Structural Dynamics 1993; 22:185-193.

4. Meek JW, Wolf JP. Cone models for an embedded foundation. Journal of the Geotechnical Engineering Division, ASCE 1994; 120(1):60-80.

5. Wolf JP. Foundation Vibration Analysis Using Simple Physical Models. Prentice-Hall: Englewood Cliffs, NJ, 1994.

6. Wolf JP, Preisig M. Dynamic stiffness of foundation embedded in layered half-space based on wave propagation in cones. Earthquake Engineering and Structural Dynamics 2003; 32:1075-1098.

7. Deeks AJ, Wolf JP. Recursive procedure for Greens function of disk embedded in layered half-space using strength-of-materials approach. Submitted to Earthquake Engineering and Structural Dynamics for review and possible publication. Also available as UWA Civil Engineering Research Report C:1760.

8. Meek JW, Wolf JP. Cone models for nearly incompressible soil. Earthquake Engineering and Structural Dynamics 1993; 22:649-663.

9. Selvadurai APS. The dynamic response of a rigid circular foundation embedded in an isotropic medium of infinite extent. International Symposium on Soils under Cyclic and Transient Loading, Swansea 1980; 597-608.

10. Selvadurai APS. Rotary oscillations of a rigid disc inclusion embedded in an isotropic elastic infinite space. International Journal of Solids and Structures 1981; 17:493-498.

11. Pak RYS, Gobert AT. Forced vertical vibration of rigid discs with arbitrary embedment. Journal of Engineering Mechanics, ASCE 1991; 117:2527-2548.

12. Pak RYS, Abedzadeh F. Forced torsional oscillation from the interior of a half-space. Journal of Sound and Vibration 1993; 160(3):401-415.

13. Tassoulas JL. Personal communication 2002. 14. Aspel RJ, Luco JE. Torsional response of rigid embedded foundation. Journal of the Engineering

Mechanics Division, ASCE 1976; 102(6):957-970. 15. Chadwick P, Trowbridge EA. Oscillations of a rigid sphere embedded in an infinite elastic solid I.

Torsional oscillations. Proceedings of the Cambridge Philosophical Society 1967; 63:1189-1205. 16. Chadwick P, Trowbridge EA. Oscillations of a rigid sphere embedded in an infinite elastic solid II.

Rectilinear oscillations. Proceedings of the Cambridge Philosophical Society 1967; 63:1207-1227.


APPENDIX – WAVE PROPAGATION AND REFLECTION/REFRACTION

The concept of wave propagation in a layered half-space is based on two building blocks: outward wave

propagation along the axis of a cone (segment) and wave reflection and refraction at a material

discontinuity corresponding to an interface between two layers (Fig. 3e). It is appropriate to list the

analytical expressions of this one-dimensional strength-of-materials approach for the translational cone,

addressing the vertical degree of freedom (Fig. 22). For a detailed derivation also addressing the

rotational cone, the reader is referred to Ref. [1].

Outward wave propagation

A disk of radius r0 with a (vertical) force of amplitude P(ω) applied, embedded in a half-space with

Poisson’s ratio ν, wave velocity c (for ν<1/3, c=cp) and mass density ρ is modelled as a double cone

(Fig. 3b). The index i is omitted for conciseness. The amplitude of the disk’s displacement u(ω)

follows from the force − displacement relationship (analogous to Eq. 1)

u(ω) = 1

S(ω) P(ω)

2 ......................................................................................................................... (A.1)

with the dynamic-stiffness coefficient of the single semi-infinite cone with apex 1 (Fig. 22a)

S(ω) = ρ c2 πr0

2

z0 + i ω ρ c πr0

2 ..................................................................................................... (A.2)

The apex height z0 (Table 1) follows with the half-space calibration (Fig. 3c) from equating the

(vertical) static-stiffness coefficients of the semi-infinite cone ρ c2 πr02/z0 and of the disk on a

homogeneous half-space 4 ρ cs2 r0/(1-ν) (exact solution of three-dimensional elastostatics).

The displacement amplitude u(z, ω) of the outward propagating wave (downwards in the positive z-

direction, Fig. 22a) is

u(z, ω) = z0

z0 + z e− i

ωc z

u(ω) ........................................................................................................... (A.3)

Note that u(z, ω) diminishes with the distance of the apex from the disk z0+z present in the denominator.

Wave reflection and refraction

At a distance d a material discontinuity occurs with the properties of the lower layer denoted as ν′, c′, ρ′

(Fig. 22a). The amplitude of the incident wave f(ω) follows from Eq. A.3 as

f(ω) = u(d, ω) = z0

z0 + d e− i

ωc d

u(ω) ............................................................................................... (A.4)

The interface with a discontinuity in material properties represents the source of a disturbance. It can be

modelled as a fictitious disk of radius r which is not loaded. Enforcing compatibility of the

displacements and equilibrium of the disk requires that an additional reflected wave with amplitude

g(ω) propagating upwards in the first material, and an additional refracted wave with amplitude h(ω)

propagating downwards in the second material are generated. (The arrows shown in Fig. 22a indicate


the direction of wave propagation.) Both waves propagate away from the source of disturbance in their

own radiating cones with apexes 3 and 2 as new incident waves (Figs 22c and 22b). z0′/r follows from

Table 1 using ν′ and c′. The area of these cones increases in the direction of wave propagation. This

yields

g(ω) = −α(ω) f(ω) ........................................................................................................................... (A.5)

h(ω) = (1 − α(ω)) f(ω) .................................................................................................................... (A.6)

with the reflection coefficient

−α(ω) =

ρ c2 π r2

z0 + d

1 + i

ω(z0 + d)c −

ρ' c'2 π r2

z0'

1 + i

ωz0'c'

ρ c2 π r2

z0 + d

1 + i

ω(z0 + d)c +

ρ' c'2 π r2

z0'

1 + i

ωz0'c'

....................................................... (A.7)

The first and second terms in the denominator of the expression on the right-hand side represent the

dynamic-stiffness coefficients (Eq. A.2) of the cones with the reflected and refracted waves at the

interface with radius r. The reflection coefficient is thus equal to the ratio of the difference and the sum

of these values.


Motion Horizontal Vertical Rocking Torsional

Aspect Ratio z0

r0

(half-space calibration)

π8 (2−ν) π

4 (1−ν)

c

cs

2

9π32 (1−ν)

c

cs

2

9π32

Aspect Ratio z0

r0

(full-space calibration)

π32

7-8ν1−ν π

16 3-4ν1−ν

c

cs

2

9π128

3-4ν1−ν

c

cs

2

9π32

Poisson’s Ratio ν all ν ν ≤

13

13 < ν ≤

12 ν ≤

13

13 < ν ≤

12 all ν

Wave Velocity c cs cp 2cs cp 2cs cs

Trapped Mass ∆M ∆Mϑ

0 0 2.4

ν−

13 ρA0r0 0 1.2

ν−

13 ρI0r0 0

Table 1 Cone models for disk on surface of homogeneous half-space.


G1 ν1 ρ1 ζ1

Gj νj ρj ζj

Gn νn ρn ζn

r0

d1

d2

dj

e

1

2

3

j

j+1

n

P(ω) u(ω)

Fig. 1 Cylindrical foundation embedded in layered soil half-space with degrees of freedom.


1

n

a)

b)

Fig. 2 a) Foundation with axi-symmetric embedment of arbitrary shape in soil layers fixed at base. b)

Fully-embedded foundation in layered soil half-space.


ujPj

u0

u1P1

uiPi

u0

u0P0

me

r0a)

unPn

r0

z0

r0

z0z0

b)

z0

Pi/2

r0

z0

Pi/2z0

c) d)

Pi

f)

ij

Pi /2

REFLECTED WAVE

REFRACTED WAVE

INCIDENT WAVE MATERIALINTERFACE

e) 1

2

3

1

i

j


Fig. 3 Concept of strength-of-materials approach using cones for vertical degree of freedom. a) Stack of disks with redundants embedded in layered

half-space, enforcement of rigid-body displacement and excavation of trapped mass between disks. b) Initial double cone with outward wave

propagation modelling disk embedded in full-space. c) Calibration of opening angle of cone using static-stiffness coefficient of disk on half-space.

d) Calibration of opening angle of cone using static-stiffness coefficient of disk embedded in full-space. e) Incident wave resulting in reflected and

refracted waves at material discontinuity at interface propagating in their own cones. f) Downwards and upwards wave propagation for embedded

disk in layered half-space.


∆M

c = 2cs

r0

z0

z0

b)

2∆M

c = 2cs

c = 2cs

r0

z0

a)

∆M

c = 2cs

r0

c)

e

c = 2cs

d)c = 2cs

r0

e

∆M

∆M

Fig. 4 Trapped mass for vertical degree of freedom, nearly-incompressible and incompressible soil. a) Disk on

surface of half-space. b) Disk embedded in full-space. c) Cylindrical foundation embedded in half-space. d)

Cylindrical foundation embedded in full-space.


a) b)

Fig. 5 Cones of initial wave propagating from under-most disk. a) Disks of constant radius contained by upper

cone. b) Disks of varying radius extending outside upper cone.

a)

z0

u(ω)

r0

z0

P(ω)

b)

z0

r0

z0

ϑ(ω)

M(ω)

∆Mϑ

Fig. 6 Disk embedded in homogeneous full-space modelled as double cone. a) Vertical. b) Rocking (trapped

mass moment of inertia for ν = 0.5).


Fig. 7 Dynamic-stiffness magnitude of disk embedded in homogeneous full-space. a) Vertical, ν = 1/3. b)

Rocking, ν = 0.3. c) Rocking, ν = 0.5.

P

z0+e

z0

z0

e r0

u

1

2

3

Fig. 8 Wave propagation from disk embedded in homogeneous half-space.


Fig. 9 Dynamic-stiffness coefficient of disk embedded in homogeneous half-space in vertical motion.

a) Embedment ratio = 2. b) Embedment ratio = 5.


Fig. 10 Dynamic-stiffness coefficient of disk embedded in homogeneous half-space in torsional motion.

O

r0

e

Fig. 11 Cylindrical foundation embedded in homogeneous half-space.


a)

b)

Fig. 12 Dynamic-stiffness coefficient of cylindrical foundation embedded in incompressible homogeneous half-

space. a) Vertical. b) Rocking.


r0

G0 ν = 0.5ρ0 ζ = 5%

0.5G0 ν = 0.5

0.2G0 ν = 0.5 0.89ρ0 ζ = 5%

ρ0 ζ = 5%

r0

0.5 r0

r0

e = 1.25r0

Fig. 13 Cylindrical foundation embedded in layered half-space consisting of three layers fixed at base.

Fig. 14 Dynamic-stiffness coefficients of cylinder embedded in layered half-space consisting of three

incompressible layers fixed at base. a) Vertical. b) Rocking.


r0

e/r0 = 0.2

e/r0 = 0.6

e/r0 = 1.0

e/r0 = 1.5

e/r0 = 2.0

e/r0 = 2.5

e/r0 = 0.0

Fig. 15 Hemi-ellipsoid foundations embedded in homogeneous half-space.

r0

HEMISPHEREBOUNDARY

MODELBOUNDARY

Fig. 16 Modelling of embedded hemisphere foundation with 11 disks (10 slices).


Fig. 17 Dynamic-stiffness coefficients of hemi-ellipsoid foundations embedded in homogeneous half-space for

various radii ratios. a) Spring coefficient. b) Damping coefficient.


r0

SPHEREBOUNDARY

MODELBOUNDARY

Fig. 18 Modelling of sphere foundation embedded in homogeneous full-space.


Fig. 19 Dynamic-stiffness coefficients of sphere foundation embedded in full-space.

a) Rectilinear motion with half-space calibration. b) Rectilinear motion with full-space calibration. c) Torsional

motion.


Fig. 20 Alternative modelling of sphere foundation embedded in homogeneous full-space with cones at right

angles to direction of motion.

Fig. 21 Dynamic-stiffness coefficients of sphere foundation embedded in full-space for rectilinear motion

calculated with cones at right angles to motion. a) Half-space calibration. b) Full-space calibration.


zu0 r0ν c ρ

ν’ c’ ρ’

r

z0

z0+d

z0’

1

2

3

d

u(d) = f g

h

u(z)

u0 = h

2

u0 = g

3

a) b) c)

uh(z)

ug(z)

Fig. 22 Outward wave propagation and wave mechanism at material discontinuity interface. a) Incident wave

impinging on interface generating reflected and refracted waves. b) Refracted wave as new incident wave. c)

Reflected wave as new incident wave.


Cones to model foundation vibrations: incompressible soil and axi-symmetric embedment of arbitrary shape

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