impedance of surface footings on layered ground

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Computers and Structures 86 (2008) 72–87

Impedance of surface footings on layered ground

L. Andersen a,*, J. Clausen b

a Department of Civil Engineering, Aalborg University, Aalborg, Denmarkb Esbjerg Institute of Technology, Aalborg University, Esbjerg, Denmark

Received 26 November 2006; accepted 10 May 2007Available online 27 June 2007

Abstract

Traditionally only the static bearing capacity and stiffness of the ground is considered in the design of wind turbine foundations. How-ever, modern wind turbines are flexible structures with resonance frequencies as low as 0.2 Hz. Unfortunately, environmental loads andthe passage of blades past the tower may lead to excitation with frequencies of the same order of magnitude. Therefore, dynamic soil–structure interaction has to be accounted for in order to get an accurate prediction of the structural response. In this paper the particularproblem of a rigid foundation on a layered subsoil is discussed. Based on the Green’s function for a stratified half-space, the impedanceof a surface footing with arbitrary shape is computed. A wind turbine foundation is analysed in the frequency range 0–3 Hz. Analysesshow that soil stratification may lead to significant changes in the impedance related to both rocking and translation at frequencies closeto the first resonance frequency of an offshore wind turbine.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Domain-transformation method; Boundary elements; Wind turbine; Foundation; Footing; Layered soil; Dynamic stiffness

1. Introduction

Over the last decades wind turbine towers and bladeshave increased significantly in height and length, respec-tively, with only a small increase in weight. Thus, modernwind turbines are flexible structures showing a dynamicresponse—even at low frequencies. The terminology‘‘eigenmodes’’ does not apply, since mechanical energy dis-sipates from the base of the turbine tower into the sur-rounding soil. Further, the rotation of the rotor leads toparametric excitation. However, strong resonance of thestructure is identified at a number of frequencies in therange 0.2–3 Hz. The first circular resonance frequency ofthe wind turbine, x1, and the circular frequency of bladespassing the tower, 3X, are of special interest. Typicallyx1 and 3X are close to each other. Hence, even a slightchange in one of these frequencies may change the dynam-

0045-7949/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2007.05.030

* Corresponding author. Fax: +45 9814 2555.E-mail address: [email protected] (L. Andersen).

ical properties of the system significantly, possibly leadingto large amplitudes of vibration.

The aim of this paper is to investigate to which extentsoil stratification may lead to dynamic soil–structure inter-action which is significantly different from the situation fora homogeneous half-space. The dynamic response of foot-ings has been studied by several researchers. Torsionalvibrations of a rigid circular footing on a homogeneoushalf-space were considered by Luco and Westmann [1]and by Veletsos and Nair [2] who studied the influence ofmaterial damping. Approximate closed-form solutions forthe torsional impedance of circular embedded foundationshave been reported by Novak and Sachs [3] and Aviles andPerez-Rocha [4]. A closed-form solution for the verticalimpedance of a flexible circular foundation on a homoge-neous half-space was presented by Krenk and Schmidt[5], while Yong et al. [6] considered a circular footing ona layered soil. Rocking and horizontal sliding of circularfootings on a homogeneous half-space was considered byVeletsos and Wei [7], and Luco [8] presented a solutionfor a rigid circular footing on the surface of a stratum.

mailto:[email protected]

Fig. 1. Model with three layers overlaying a half-space.

L. Andersen, J. Clausen / Computers and Structures 86 (2008) 72–87 73

More recently, Vostroukhov et al. [9] gave a solution for ahorizontal force applied uniformly over a circular area inthe interior of a layered half-space.

Wong and Luco [10] gave a solution for the impedanceof rigid massless square foundations resting on layered vis-coelastic soil, while Mita and Luco [11] evaluated theimpedance of a square foundation embedded in a homoge-neous half-space by means of a hybrid approach. Later,Vrettos [12] studied the impedance of a rigid rectangularfooting on a half-space with continuously increasing shearstiffness over depth, while Ahmad and Rupani [13]employed a boundary-element (BE) model to the analysisof a square foundation embedded in a layered half-space.Emperador and Domınguez [14] applied the boundary-ele-ment method to the analysis of the dynamic response ofaxisymmetric embedded foundations, and recently Liing-aard et al. [15] analysed a flexible circular suction caissonby means of a coupled boundary-element/finite-elementmodel. Finally, the dynamics performance of piles in lay-ered soil has been investigated by Senm et al. [16], employ-ing the Green’s-function formalism proposed by Kausel[17].

A further summary of the work concerning rocking andsliding of foundations was given by Bu and Lin [18], andboth this and the present review indicate that most solu-tions are limited to circular or rectangular foundations.In particular, so far no analyses have been carried outfor the hexagonal or octagonal footings that are often uti-lised for wind turbines, e.g. at the Nysted Offshore WindFarm at Rødsand in Denmark. Numerical models basedon, for example, the boundary-element method (BEM) orfinite-element method (FEM) may be applied to evaluateof the impedance of footings of an arbitrary geometryand resting on a layered ground. However, such analysesare time consuming and, hence, inadequate for parametricstudies.

As an alternative to the BEM or FEM, a semi-analyticalmodel based on the domain-transformation method andthe Green’s function for a stratified ground [19,20] is pro-posed in the present work which is an updated and revisedversion of the conference paper [21]. The contact stressesbetween the footing and the subsoil are modelled by a num-ber of distributed loads, each applied with radial symmetryaround a point on the soil–foundation interface. Thisallows a fast transformation from wavenumber domainto space domain, because the inverse Fourier transforma-tion only involves numerical evaluation of a line integral.However, the method is applicable to surface footings ofan arbitrary geometry.

The present method is compared with a BEM solution[22] for a homogeneous viscoelastic half-space in order totest the computation speed and accuracy. Whereas theBEM solution based on the full-space Green’s functiontakes considerably longer when a stratified half-space isconsidered, the present method is expected to provide onlya small increase in computation time since it employs theGreen’s function for a layered half-space. This is impor-

tant, both when parameter studies are to be carried out,and when the impedance of a footing has to be includedin the aero-elastic codes utilised in the wind turbine indus-try. Subsequently, the influence of soil stratification is stud-ied for a massless hexagonal footing. The aim of thisanalysis is to investigate whether the impedance may beinfluenced significantly at low frequencies close to the firstresonance frequency of a wind turbine.

2. Response of a layered half-space

In order to allow a semi-analytical solution for the wavepropagation problem, the ground is considered as a linearviscoelastic stratum overlaying a homogeneous half-spaceor bedrock. As illustrated in Fig. 1, the surface and allinterfaces are assumed to be horizontal, and layers arenumbered from the top of the ground, i.e. Layer 1 is thetop layer. Accordingly, the origin of the Cartesian(x1,x2,x3)-frame of reference is located on the surface ofthe ground with the x3-axis pointing downwards.

Let u10i ðx1; x2; tÞ ¼ uiðx1; x2; 0; tÞ, i = 1,2,3, denote the

displacement at the surface of the ground in time domainand in Cartesian space. Likewise, the surface traction isdenoted p10

i ðx1; x2; tÞ ¼ piðx1; x2; 0; tÞ. The double super-script 10 indicates that the quantities belong to the top ofLayer 1, whereas the bottom of Layer 1 is identified bythe superscript 11. Quantities at the top and bottom ofLayer j are given the superscripts j0 and j1, respectively.Continuity at interfaces involves that u11

i ¼ u20i , p11

i ¼ p20i ,

and so on, as illustrated in Fig. 2 for the displacements.Note that the traction is defined as positive in the coordi-nate directions rather than the direction of the outwardnormal.

Further, let gij denote the Green’s function, or funda-mental solution, relating the displacement in direction i

at the observation point (x1,x2,0) and time t to the tractionapplied in direction j at the source point (y1,y2,0) and times. Both points are situated on the surface of the ground.Given that the response is linear, the Green’s function is

Fig. 2. Continuity of displacements at interfaces between adjacent layers.

Fig. 3. Definition of the global and local coordinates for Layer j with thedepth hj. The (x1,x2,x3)-coordinate system has the origin O, whereas thelocal ðx1; x2; x

j3Þ-coordinate system has the origin Oj.

74 L. Andersen, J. Clausen / Computers and Structures 86 (2008) 72–87

invariant to space and time translation. Hence, gij =gij(x1 � y1,x2 � y2, t � s), and the total displacement atthe observation point is found as

u10i ðx1; x2; tÞ ¼

Z t

�1

Z 1

�1

Z 1

�1gijðx1 � y1; x2 � y2; t � sÞ

� p10j ðy1; y2; sÞdy1 dy2 ds; ð1Þ

where summation is applied over repeated indices. The dis-placement at any point on the surface of the ground and atany instance of time may be evaluated by means of Eq. (1)given that gij is available. However, an analytical solutioncannot be established for a layered ground; hence, in prac-tice, the space–time domain solution expressed by Eq. (1) isinadequate.

Assuming a linear response of the stratum, the analysismay be carried out in the frequency domain. The Fouriertransformation with respect to time is defined as

U 10i ðx1; x2;xÞ ¼

Z 1

�1u10

i ðx1; x2; tÞe�ixt dt: ð2Þ

Similar definitions apply to the surface load, i.e. betweenthe quantities p10

i ðx1; x2; tÞ and P 10i ðx1; x2;xÞ, and to the

Green’s function, that is between gij(x1 � y1,x2 � y2, t � s)and Gij(x1 � y1,x2 � y2,x). It then follows that

U 10i ðx1; x2;xÞ ¼

Z 1

�1

Z 1

�1Gijðx1 � y1; x2 � y2;xÞ

� P 10j ðy1; y2;xÞdy1 dy2: ð3Þ

Next, if only horizontal interfaces are present, a transfor-mation can be carried out from Cartesian coordinates intothe horizontal wavenumber domain. The following defini-tion of the double Fourier transformation over spaceapplies,

U 10i ðk1; k2;xÞ ¼

Z 1

�1

Z 1

�1U 10

i ðx1; x2;xÞe�iðk1x1þk2x2Þdx1 dx2:

ð4ÞDefining similar transformations for the surface tractionand the Green’s function, Eq. (3) achieves the form

U 10i ðk1; k2;xÞ ¼ Gijðk1; k2;xÞP 10

j ðk1; k2;xÞ: ð5Þ

This equation has the advantage when compared to theprevious formulation in space and time domain, that noconvolution has to be carried out. Thus, the displacementamplitudes in the wavenumber–frequency domain are re-lated directly to the traction amplitudes for a given set ofthe circular frequency x and the horizontal wavenumbersk1 and k2 via the Green’s function tensor Gijðk1; k2;xÞ.The response in space–time domain is derived from thewavenumber–frequency solution by means of the inverseFourier transformations

U 10i ðx1;x2;xÞ ¼

1

4p2

Z 1

�1

Z 1

�1U 10

i ðk1;k2;xÞeiðk1x1þk2x2Þdk1 dk2;

ð6Þ

u10i ðx1;x2; tÞ ¼

1

2p

Z 1

�1U 10

i ðx1;x2;xÞeixt dx: ð7Þ

The present analysis is carried out in the frequency domain;hence, use will be made of Eq. (6), whereas the inverse Fou-rier transformation provided by Eq. (7) will not be utilised.

2.1. Transfer matrix for a single layer

A stratum with J layers is considered. The layers havethe depths hj, j = 1,2, . . . ,J, and a local ðx1; x2; x

j3Þ-coordi-

nate system is defined such that xj3 2 ½0; hj� (see Fig. 3).

Each layer is assumed to be homogeneous and isotropicwith the mass density qj. Further, the soil is idealised as alinear elastic material, well-knowing that soil exhibits anonlinear response as further discussed below. Hence, inthe space–time domain the wave propagation in Layer j

is governed by the Navier equations:

ðkj þ ljÞ o�j

oxiþ lj o2uj

i

oxkoxk¼ qj o2uj

i

ot2: ð8Þ


Here kj and lj are the Lame constants of the material inLayer j, and Dj ¼ Djðx1; x2; x

j3; tÞ is the dilatation,

Dj ¼ oujk

oxk; j ¼ 1; 2; . . . ; J : ð9Þ

The phase velocities of compression and shear waves, orP- and S-waves, in Layer j are identified as

cP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkj þ 2ljÞ=qj

q; cS ¼

ffiffiffiffiffiffiffiffiffiffiffilj=qj

p; ð10Þ

respectively. Accordingly, when the response varies har-monically with time with the circular frequency x, P- andS-waves in Layer j are associated with the wavenumberskj

P and kjS. These are defined as

fkjPg

2 ¼ x2=fcjPg

2; fkj

Sg2 ¼ x2=fcj

Sg2; ð11Þ

respectively. Introducing the parameters ajP and aj

S as thelarger of the roots to

fajPg

2 ¼ k21 þ k2

2 � fkjPg

2; faj

Sg2 ¼ k2

1 þ k22 � fk

jSg

2 ð12Þ

and applying the Fourier transformations (2) and (4) to theNavier Eqs. (8) for the layer, the governing equations in thewavenumber–frequency domain achieve the form

ðkj þ ljÞik1Dj þ lj d2U j

1

dx23

� fajSg

2Uj1

� �¼ 0; ð13aÞ

ðkj þ ljÞik2Dj þ lj d2U j

2

dx23

� fajSg

2Uj2

� �¼ 0; ð13bÞ

ðkj þ ljÞ dDj

dx3

þ lj d2U j3

dx23

� fajSg

2Uj3

� �¼ 0: ð13cÞ

Here Dj ¼ Djðk1; k2; xj3;xÞ is the amplitude of the dilata-

tion, which is given in terms of the three complex displace-ment amplitudes

U ji ¼ Uj

iðk1; k2; xj3;xÞ; i ¼ 1; 2; 3;

Dj ¼ ik1U j1 þ ik2U j

2 þdUj

3

dx3

: ð14Þ

In the frequency domain, hysteretic material dissipation inthe soil is introduced by application of complex Lameconstants,

ba

Fig. 4. Material behaviour of soil during cyclic loading: (a) backbone curvhysteresis; (c) shear modulus, G, and loss factor, g, as functions of the strain cystrain, c.

kj ¼ mjEjð1þ i signðxÞgjÞð1þ mjÞð1� 2mjÞ ; j ¼ 1; 2; . . . ; J ; ð15aÞ

lj ¼ Ejð1þ i signðxÞgjÞÞ2ð1þ mjÞ ; j ¼ 1; 2; . . . ; J : ð15bÞ

Here Ej and mj are Young’s modulus and Poisson’s ratio,respectively, and gj is the loss factor. Accordingly, thephase velocities and wavenumbers defined by Eqs. (10)and (11), respectively, are complex with an imaginary partrepresenting material dissipation. It is noted that Eq. (15)cannot be interpreted directly in the time domain. Further-more, theoretically the hysteretic material damping modelis invalid, since it provides a non-causal response. How-ever, for all practical purposes useful results are obtainedfor loss factors gj < 0.1, which are typical for soil [23]. Afurther note should be made with regard to the nonlinearbehaviour of soil. Thus, at large strains the soil exhibitsmaterial hysteresis. This has not been considered in thepresent work, but in the linear model hysteresis may be ac-counted for by utilisation of an equivalent secant stiffnessand loss factor based on a so-called backbone curve [24],see Fig. 4.

Eqs. (13) are ordinary differential equations in xj3. The

boundary conditions at the top and the bottom of the jthlayer are:

Uj0i ðk1; k2;xÞ ¼ Uj

iðk1; k2; 0;xÞ; ð16aÞUj1

i ðk1; k2;xÞ ¼ Ujiðk1; k2; h

j;xÞ; ð16bÞ

P j0i ðk1; k2;xÞ ¼ P j

iðk1; k2; 0;xÞ; ð17aÞP j1

i ðk1; k2;xÞ ¼ P jiðk1; k2; h

j;xÞ: ð17bÞ

The quantities defined in Eqs. (16) and (17) may advanta-geously be stored in common vector form as

Sj0 ¼ Uj0

Pj0

" #; Sj1 ¼ Uj1

Pj1

" #; ð18Þ

where Uj0 ¼ Uj0ðk1; k2;xÞ is the column vector with thecomponents U j0

i , i = 1,2,3, etcetera. With this notation,the full solution for displacements, Uj, and tractions, Pj,in the layer may be written as

c

e with typical variation of material stiffness; (b) stress–strain loop withcle width, Dc. The curves are plotted in terms of shear stresses, s, and shear


Sj ¼ Uj

Pj

" #¼ AjEjbj; ð19aÞ

bj ¼ aj1 bj

1 cj1 aj

2 bj2 cj

2

� �T; ð19bÞ

where Ej is a matrix of dimension (6 · 6), which has the

diagonal terms Ej11 ¼ eaj

Pxj

3 , Ej22 ¼ Ej

33 ¼ eajSxj

3 , Ej44 ¼ e�aj

Pxj

3 ,Ej

55 ¼ Ej66 ¼ e�aj

Sxj

3 , and is otherwise empty. Further, Aj isa matrix of dimension (6 · 6), the derivation of whichcan be found, for example, in the work by Sheng et al.[20]. For completeness, the matrix is given in AppendixA. It is noted that the solution for a static force, i.e. forthe frequency x = 0, is fundamentally different from thedynamic solution. The integration constants aj

1 and aj2 rep-

resent P-waves moving up and down in Layer j. Similarly,bj

1 and bj2 represent S-waves polarised in the x1-direction

and moving up and down in the layer, respectively, whilecj

1 and cj2 represent contributions from S-waves polarised

in the x2-direction and travelling up and down in the layer,respectively.

Finally, the displacements and tractions at the bound-aries of Layer j can be found:

Sj0 ¼ Aj0bj; Aj0 ¼ Aj; ð20aÞ

Sj1 ¼ eajP

hjAj1bj; Aj1 ¼ Aj0Dj: ð20bÞ

Dj is the matrix e�ajP

xj3 Ej evaluated at xj

3 ¼ hj, i.e. a (6 · 6)matrix with the diagonal terms Dj

11 ¼ 1, Dj22 ¼ Dj

33 ¼eða

jS�aj

PÞhj

, Dj44 ¼ e�2aj

Phj

, Dj55 ¼ Dj

66 ¼ e�ðajPþaj

SÞhj

, and zerosat all other positions. Eqs. (20a) and (20b) may be com-bined in order to eliminate vector bj. This provides a trans-fer matrix for the layer as proposed by Thomson [25] andHaskell [26],

Sj1 ¼ eajP

hjAj1½Aj0��1

Sj0 ð21Þ

forming a relationship between the displacements and thetractions at the top and the bottom of a single layer.

2.2. Transfer matrix for a stratum

Continuity of the displacements and of the tractions atthe interfaces between adjacent layers involve that Sj0 ¼Sj�1;1, j = 2,3, . . . ,J, i.e. the quantities at the top of Layerj are equal to those at the bottom of Layer j � 1. Proceed-ing in this manner, Eq. (21) for the single layer may berewritten for a system of J layers,

SJ1 ¼ evAJ1½AJ0��1AJ�1;1½AJ�1;0��1 � � �A11½A10��1

S10;

v ¼XJ

j¼1

ajPhj:

ð22Þ

Introducing the transfer matrix T,

T ¼ AJ1½AJ0��1AJ�1;1½AJ�1;0��1 � � �A11½A10��1

; ð23Þ

Eq. (22) may in turn be written as SJ1 ¼ evTS10, or

UJ1

PJ1

" #¼ ev T11 T12

T21 T22

� �U10

P10

" #; v ¼

XJ

j¼1

ajPhj; ð24Þ

where T11, etc. are the 3-by-3 submatrices of T. Thisformulation is due to Thomson [25] and Haskell [26] andestablishes a relationship between the traction and the dis-placements at the free surface of the half-space and theequivalent quantities at the bottom of the stratum.

2.3. Green’s function for a homogeneous or stratifiedviscoelastic half-space

A homogeneous half-space is identified by the super-script J + 1. Assuming that no external forces act in theinterior of the half-space, only waves propagating down-wards can exist. Dividing the matrices Aj and Ej for a layerof finite depth, cf. Eq. (19), into four quadrants, and thecolumn vector bj into two sub-vectors,

Aj ¼ Aj11 Aj

12

Aj21 A

j22

" #; Ej ¼ Ej

11 Ej12

Ej21 E

j22

" #; bj ¼ bj

1

bj2

" #;

ð25Þit is therefore evident that only half of the solution appliesto the half-space, i.e.

SJþ1 ¼ UJþ1

PJþ1

" #¼ AJþ1

12

AJþ122

" #EJþ1

22 bJþ12 ; ð26aÞ

bJþ12 ¼ aJþ1

2 bJþ12 cJþ1

2

Th iT

: ð26bÞ

The terms including the integration constants aJþ11 , bJþ1

1

and cJþ11 are physically invalid as they correspond to waves

coming in from xJþ13 ¼ 1, i.e. from infinite depth.

From Eq. (26), the traction on the interface between thebottommost layer and the half-space may be expressed interms of the corresponding displacements,

UJþ1 ¼ AJþ112 ½A

Jþ122 �

�1PJþ1: ð27Þ

The matrix EJþ122 reduces to the identity matrix of order 3,

since all the exponential terms are equal to 1 for xJþ13 ¼ 0.

Firstly, if no layers are present in the model of the stra-tum, J = 0. Hence, it immediately follows from Eq. (27)that Eq. (5), written in matrix form, becomes

U10 ¼ GhhP10; Ghh ¼ A1012½A

1022��1: ð28Þ

Here Ghh ¼ Ghhðk1; k2;xÞ is the Green’s function for thehomogeneous half-space in the horizontal wavenumber–frequency domain.

Secondly, if J layers overlay the homogeneous half-space, continuity of the displacements, equilibrium of thetraction and application of Eq. (27) provide

UJ1 ¼ UJþ1;0 ¼ AJþ112 ½A

Jþ122 �

�1PJ þ 1;0

¼ AJþ112 ½A

Jþ122 �

�1PJ1: ð29Þ

Insertion of this result into Eq. (24) leads to the followingsystem of equations:

Fig. 5. Definition of the three angles u, h and #.


UJ1

PJ1

" #¼

AJþ112 ½A

Jþ122 �

�1PJ1

PJ1

" #¼ ev

T11 T12

T21 T22

" #U10

P10

" #:

ð30Þ

From the bottommost three rows of the matrix equation,an expression for PJ1 is obtained, which may be insertedinto the first three equations. This leads to the solution

U10 ¼ GlhP10; ð31Þ

where Glh ¼ Glhðk1; k2;xÞ is the Green’s function matrixfor the layered half-space,

Glh¼ AJþ112 ½A

Jþ122 �

�1T21�T11

�1

� T12�AJþ112 ½A

Jþ122 �

�1T22

:

ð32Þ

It is noted that the exponential function ev vanishes in thisrepresentation, which is a great advantage from a compu-tational point of view [20]. In the following, no distinctionis made between Ghh and Glh, and the common denotationG is applied.

2.4. Evaluation of the response in cylindrical coordinates

The transformation from the horizontal wavenumberdomain into the Cartesian space involves a double inverseFourier transformation. Numerical treatment of Eq. (6) bymeans of discrete (fast) inverse Fourier transformation is atime-consuming operation. However, a great reduction incomputation time is achieved by an evaluation in polarcoordinates as explained in the following.

Firstly, as pointed out by Sheng et al. [20], the evalua-tion of Aj, and therefore also G, is particularly simple alongthe line defined by k1 = 0. For any other combination ofthe wavenumbers, the Green’s function matrix can befound as

Gðk1; k2;xÞ ¼ RðuÞbGða;xÞ½RðuÞ�T: ð33Þ

Here bGða;xÞ ¼ Gð0; a;xÞ, and R(u) is the transformationmatrix defined by

k1

k2

x3

26643775 ¼ RðuÞ

c

a

x3

26643775; RðuÞ ¼

sin u cos u 0

� cos u sin u 0

0 0 1

26643775:ð34Þ

This corresponds to a rotation of the (k1,k2,x3)-basis bythe angle u � p/2 around the x3-axis as illustrated inFig. 5. It follows from Eq. (34) that Rij(u) = Rji(p � u),that is [R(u)]T = R(p � u).

Since the propagation of horizontally polarised S-waves(i.e. SH-waves) in the stratum is decoupled from the prop-agation of P-waves and vertically polarised S-waves (i.e.

SV-waves), bGða;xÞ simplifies to the form [19]

bGða;xÞ ¼bG11 0 0

0 bG22bG23

0 bG32bG33

26643775: ð35Þ

Due to reciprocity, the off-diagonal terms of the matrixbGða;xÞ are antisymmetric, i.e. bG32 ¼ �bG23 [27].

Similarly to the transformation of the horizontal wave-numbers from (k1,k2) into (c,a) provided by Eq. (34), theCartesian coordinate system is rotated around the x3-axisaccording to the transformation (see Fig. 5)

x1

x2

x3

264375 ¼ RðhÞ

q

r

x3

264375; RðhÞ ¼

sin h cos h 0

� cos h sin h 0

0 0 1

264375:ð36Þ

The displacements and tractions along the r-axis aredenoted bUðr; x3;xÞ ¼ Uðq; r; x3;xÞ and bPðr; x3;xÞ ¼Pðq; r; x3;xÞ, respectively. With q = 0, the following rela-tionship applies:

x1 ¼ r cos h; x2 ¼ r sin h; ð37aÞ

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22

q; tan h ¼ x2

x1

: ð37bÞ

Accordingly, the complex amplitudes of displacement andtraction follow by the transformation

Uðx1; x2; x3;xÞ ¼ RðhÞbUðr; x3;xÞ; ð38aÞPðx1; x2; x3;xÞ ¼ RðhÞbPðr; x3;xÞ: ð38bÞ

For the purpose of double inverse Fourier transformationin polar coordinates, the coordinate transformations (34)and (36) are conveniently combined by introducing theangle

# ¼ p=2þ u� h ð39Þ

defining the rotation of the wavenumbers (c,a) relatively tothe spatial coordinates (q, r). The resulting transformationis illustrated in Fig. 5. Evidently R(u) = R(h)R(#), andthe wavenumbers (k1,k2) in the original Cartesian frameof reference may be obtained from the rotated wavenum-bers (c,a) by either of the transformations


k1

k2

x3

264375 ¼ RðuÞ

c

a

x3

264375 ¼ RðhÞRð#Þ

c

a

x3

264375; ð40aÞ

Rð#Þ ¼sin# cos# 0

� cos# sin# 0

0 0 1

264375: ð40bÞ

This identity is easily proved by a combination of Eqs. (34),(36) and (39).

Secondly, by application of the coordinate transforma-tion (36) in Eq. (6), the response at the surface of the stra-tum may be evaluated by a double inverse Fouriertransform in polar coordinates, here given in matrix form

bU10 ¼ 1

4p2

Z 1

0

Z 2p

0

Rð#ÞbG½Rð#Þ�T � bP10eiar sin# d#ada;

ð41Þ

where ar sin# = k1x1 + k2x2 is identified as the scalar prod-uct of the two-dimensional vectors with lengths a and r,respectively, and p/2 � # is the plane angle between thesevectors as given by Eq. (39). In accordance with Eq. (38),the load amplitudes given in terms of x3 and the horizontalwavenumbers (kq,kr) are found from the correspondingload amplitudes in (k1,k2,x3)-space by means of the trans-

formation bPðkq; kr; x3;xÞ ¼ ½RðhÞ�TPðk1; k2; x3;xÞ. Further-more, transformation of the displacement amplitudes from(q, r,x3)-coordinates into (x1,x2,x3)-coordinates providesthe double inverse Fourier transformation

U10¼ 1

4p2RðhÞ

Z 1

0

Z 2p

0

Rð#ÞbG½Rð#Þ�T½RðhÞ�T�P10eiar sin#d#ada:

ð42ÞIn the general case P10 depends on both the angle # and thewavenumber a. However, if the complex amplitudes of theload are independent of #, i.e. if the load is applied withrotational symmetry around the x3-axis, the vector P10

may be taken outside the integral over # in Eq. (42), i.e.

U10 ¼ 1

2pRðhÞ

Z 1

0

eG½RðhÞ�TP10ada; ð43aÞ

eG ¼ 1

2p

Z 2p

0

Rð#ÞbG½Rð#Þ�Teiar sin # d#: ð43bÞ

Generally, Eq. (43a) has to be evaluated numerically. How-ever, the components of eG defined by Eq. (43b) are identi-fied as integral representations of Bessel functions. Thus,

1

2p

Z 2p

0

eiar sin # d# ¼ J 0ðarÞ; ð44aÞ

1

2p

Z 2p

0

sin2 #eiar sin # d# ¼ J 0ðarÞ � 1

arJ 1ðarÞ; ð44bÞ

1

2p

Z 2p

0

sin#eiar sin # d# ¼ iJ 1ðarÞ; ð44cÞ

1

2p

Z 2p

0

cos2 #eiar sin# d# ¼ 1

arJ 1ðarÞ: ð44dÞ

Here, Jn(ar) are the Bessel functions of the first kind andorder n. These may be computed in an efficient mannerby their series expansions [28]. Note that the remaining ker-nels of the integrals in Eq. (43b) are odd functions of # onthe interval # 2 [�p;p] and therefore vanish.

Application of the Bessel functions in accordance withEq. (44) and further taking into account that the Green’sfunction tensor is skew symmetric, see Eq. (35), the compo-nents, eGij ¼ eGijða; r;xÞ, of the integral in Eq. (43b) become

eG11 ¼ J 0ðarÞ � 1

arJ 1ðarÞ

� �bG11 þ1

arJ 1ðarÞbG22; ð45aÞ

eG22 ¼1

arJ 1ðarÞbG11 þ J 0ðarÞ � 1

arJ 1ðarÞ

� �bG22; ð45bÞ

eG12 ¼ eG13 ¼ eG21 ¼ eG31 ¼ 0; ð45cÞeG23 ¼ �eG32 ¼ iJ 1ðarÞbG23; eG33 ¼ J 0ðarÞbG33: ð45dÞ

This result was obtained by Aursch [19] for a concentratedforce. However, the derivation above shows that the formu-lation is also valid for a distributed load with radial symme-try. Further, it is noted that Aursch [19] employed acoordinate transformation leading to non-standard Besselfunctions, whereas the present formulation is based on theBessel functions Jn(ar). As ar! 0, the terms eG11ða; r;xÞand eG22ða; r;xÞ approach the limit

limar!0

eG11ða; r;xÞ ¼ limar!0

eG22ða; r;xÞ ¼bG11 þ bG22

2: ð46Þ

Eqs. (43) only involves numerical integration in one dimen-sion. Hence, it provides an efficient evaluation of thecomplex amplitudes of the surface displacements, U10 =U10(x1,x2,x). The drawback is that only axisymmetric loaddistribution are allowed. However, any other distributionof the contact stresses at the soil–foundation interfacemay be approximated by a linear combination of suchloads as further discussed in the next section.

3. Impedance of rigid foundations

A rigid footing has three translational and three rota-tional degrees of freedom as shown in Fig. 6. In the fre-quency domain, these are related to the correspondingforces and moments via the impedance matrix C(x),

CðxÞZðxÞ ¼ FðxÞ; ð47aÞZðxÞ ¼ V 1 V 2 V 3 H1 H2 H3½ �T; ð47bÞFðxÞ ¼ Q1 Q2 Q3 M1 M2 M3½ �T: ð47cÞ

In the most general case, the impedance matrix C(x) is full,i.e. all the rigid-body motions of the footing are interre-lated. However, in the present case the footing rests onthe surface of a horizontally layered stratum. Further,assuming that the stress resultants act at the centre of thesoil–foundation interface, the torsional and vertical dis-placements are completely decoupled from the remaining

a

b

Fig. 6. Degrees of freedom for a rigid surface footing: (a) displacementsand rotations and (b) forces and moments.

a

c

b

d

Fig. 7. Definition of axes for different geometries of a footing: (a) circular,(b) square, (c) hexagonal and (d) octagonal footing. The horizontal planeis considered, and all the footings have the same characteristic length, R0.


degrees of freedom. Thus, the impedance matrix simplifiesto

CðxÞ ¼

C11 C12 0 C14 C15 0

C12 C22 0 C24 C25 0

0 0 C33 0 0 0

C14 C24 0 C44 C45 0

C15 C55 0 C45 C55 0

0 0 0 0 0 C66

266666666664

377777777775: ð48Þ

A further simplification of C(x) is obtained if the momentof inertia around a given horizontal axis is invariant to arotation of the footing around the z-axis. This is the casefor the gravitation foundations that are typically utilisedfor wind turbines, i.e. circular, square, hexagonal andoctagonal footings. With reference to Fig. 7, the momentsof inertia are Ix1

¼ Ix2¼ In ¼ I f, where f is an arbitrary

horizontal axis. As a result of this, C11 = C22, C44 = C55

and C15 = �C24, and the coupling between sliding in thex1-direction and rocking in the x2-direction (and vice versa)vanishes, i.e.

CðxÞ ¼

C11 0 0 0 �C24 0

0 C22 0 C24 0 0

0 0 C33 0 0 0

0 C24 0 C44 0 0

�C24 0 0 0 C55 0

0 0 0 0 0 C66

266666666664

377777777775: ð49Þ

In order to compute the non-zero components of theimpedance matrix C(x), the distribution of the contactstresses at the interface between the footing and the grounddue to given rigid-body displacements has to be deter-

mined. However, Eq. (42) provides the displacement fieldfor a known stress distribution. Generally this implies thatthe problem takes the form of an integral equation. For theparticular case of a circular footing on a homogeneoushalf-space, Krenk and Schmidt [5] derived a closed-formsolution for the vertical impedance, whereas both the trans-lation and rocking impedances of a circular footing on alayered half-space were derived by Luco [8]. In both anal-yses, relaxed contact was assumed at the soil–foundationinterface, i.e. the interface is assumed frictionless in thecomputation of the vertical and rocking impedances, andno normal stresses occur at the interface when a horizontalforce is applied. By contrast, welded contact is assumed inthe present analysis.

More recently, Yong et al. [6] proposed that the totalcontact stress be decomposed into a number of simple dis-tributions obtained by a Fourier series with respect to theazimuthal angle and a polynomial in the radial direction,e.g.

P 10r ðr; #;xÞ ¼

XM

m¼1

XN

n¼1

amnrn cosðm#Þ ð50Þ

for the component in the r-direction and a symmetric con-tact stress distribution. Similar expressions were given forthe components in the q- (or #-) and x3-direction and forthe antisymmetric case. The response to each of the contactstress distributions can be computed, and the coefficientsamn are determined so that the prescribed rigid-body dis-placements are obtained.

However, for arbitrary shapes of the footing it may bedifficult to follow this idea, i.e. an orthogonal set ofglobal symmetric and antisymmetric load distributionscannot be established in the general case. Hence, in this


study a different approach is taken which has the follow-ing steps:

(1) The displacement corresponding to each rigid-bodymode is prescribed at N points distributed uniformlyat the interface between the footing and the ground.

(2) The Green’s function matrix is evaluated in the wave-number domain along the a-axis and Eq. (43b) isevaluated by application of Eq. (45).

(3) The wavenumber spectrum for a simple distributedload with unit magnitude and rotational symmetryaround a point on the ground surface is computed.

(4) The response at point n to a load centred at point m iscalculated for all combinations of n,m = 1,2, . . . ,N.This provides a flexibility matrix for the footing.

(5) The unknown magnitudes of the loads appliedaround each of the points are computed. Integrationover the contact area provides the impedance.

In particular, if the surface traction vector in the wave-number–frequency domain takes the form P 10

i ðk1; k2;xÞ ¼Dðk1; k2ÞeP iðxÞ, i = 1,2,3, where Dðk1; k2Þ is a stress distri-

bution with unit magnitude and eP iðxÞ is an amplitude,Eq. (43a) may be computed as

U10 ¼ RðhÞbG½RðhÞ�TeP; ð51aÞ

bG ¼ bGðr;xÞ ¼ 1

2p

Z 1

0

eG bDada: ð51bÞ

Here it is noted that bD ¼ bDðaÞ ¼ Dð0; aÞ, since an axisym-metric distribution is assumed. The choice of contact stressdistribution and various discretization aspects are dis-cussed below. Alternatively, a boundary-element modelbased on the Green’s function for the layered half-spacemay be employed. However, this involves some additionalwork since the Green’s function for traction has to beevaluated.

3.1. Simple distributed loads with rotational symmetry

In Cartesian coordinates, a concentrated point force act-ing in direction i with magnitude eP i on the surface of thehalf-space may be expressed as

P 10i ðx1; x2;xÞ ¼ eP idðx1Þdðx2Þ: ð52Þ

Double Fourier transformation with respect to the hori-zontal coordinates provides the load spectrum in wave-number domain P 10

i ðk1; k2;xÞ ¼ eP i. The fact that there isno decay of P 10

i in the high wavenumber regime is disad-vantageous from a computational point of view. In orderto evaluate Eq. (43a) accurately, numerical integrationmust be carried out for a great wavenumber range. Thisproblem can be circumvented by application of a distrib-uted load. Here two different load distributions are sug-gested, which are both axisymmetric implying that P10

may be taken outside the integral over # in Eq. (42) as dis-cussed previously.

Firstly, consider a surface load applied uniformly indirection i over a circular area centred at the origin andwith radius r0. In cylindrical coordinates the amplitudefunction reads [20]

bP 10i ðr;xÞ ¼

eP i=ðpr20Þ for r 6 r0;

0; else:

(ð53Þ

Double Fourier transformation with respect to the polarcoordinates (r,h) yields

bP 10i ða;xÞ ¼

eP i

pr20

Z r0

0

Z 2p

0

e�iar sin# d#r dr )

bP 10i ða;xÞ ¼

2eP i

ar0

J 1ðar0Þ:ð54Þ

Here a is the radial wavenumber, and # = p/2 + u � h isthe angle between the wavenumber and the radius vectorsin polar coordinates (a,u) and (r,h), respectively. As dis-cussed above, ar sin# corresponds to the scalar product be-tween the two vectors with the length a and r, respectively.

Note that bP 10i ða;xÞ ! eP i for ar0! 0.

Secondly, a ‘‘bell-shaped’’ surface load is considered,provided by means of a double Gaussian distribution withmagnitude eP i in direction i,

bP 10i ðr;xÞ ¼

eP i

4pr21

e��

r2r1

�2

: ð55Þ

Here r1 is the standard deviation. Double Fourier transfor-mation of bP 10

i ðr;xÞ with respect to the polar coordinates(r,h) yields the load in wavenumber domain,

bP 10i ða;xÞ ¼

eP i

4pr21

Z 1

0

Z 2p

0

e��

r2r1

�2

e�iar sin# d#r dr )

bP 10i ða;xÞ ¼ eP ie

�a2r21 :

ð56ÞFor small values of r0 or r1, the distributed loads defined byEqs. (54) and (56) approach the delta spike, i.e. the repre-sentation of a concentrated point force. However, for finitevalues of r0 or r1, the load spectra decay rapidly with a. The‘‘bell-shaped’’ load has the advantage that the wavenumberspectrum is a monotonic decreasing function of a, which isnot the case for the spectrum provided by Eq. (54) due tothe wavy nature of the Bessel function. This results in aload that is particularly adequate for numerical evaluationof the inverse Fourier transformation (43a).

3.2. Discretization considerations

In order to achieve an accurate and efficient computa-tion of the impedance matrix for a footing with the presentmethod, a number of issues need consideration:

(1) Eq. (51) has to be evaluated numerically. This

requires a computation of bGða;xÞ for a number ofdiscrete wavenumbers. All peaks in the wavenumber

Table 1Material properties for homogeneous half-space

Material no. G (MPa) m q (kg/m3) g

Material 1 50 0.250 2000 0.05Material 2 50 0.400 2000 0.05Material 3 50 0.495 2000 0.05

b

a

Fig. 8. Models used for computation of the impedance: (a) Presentmethod with ‘‘bell-shaped’’ loads applied at 37 points under the footingand (b) boundary-element model with quadratic interpolation. The shadedelements comprise the footing.


spectrum must be represented well, demanding a finediscretization in the low wavenumber range—in par-ticular for a half-space with little material damping.

(2) No significant contributions may exist from the prod-

ucts bDðaÞbGijða;xÞ, i, j = 1,2,3, for wavenumbersbeyond the truncation point in the numerical evalua-tion of the integral in Eq. (51).

(3) Enough points should be employed at the soil–struc-ture interface in order to provide a good approxima-tion of the contact stress distribution.

Concerning item 1 it is of paramount importance todetermine the wavenumber below which the wavenumberspectrum may have narrow-banded peaks. Here use canbe made of the fact that the longest wave present in ahomogeneous half-space is the Rayleigh wave. An approx-imate upper limit for the Rayleigh wavenumber is providedby the inequality aR = x/cR < 1.2x/cS for m 2 [0;0.5]. For astratum with J layers overlaying a homogeneous half-space, the idea is now to determine the quantity

a1 ¼ 2x=min c1S; c

2S; . . . ; cJþ1

S

�; ð57Þ

where index J + 1 refers to the underlying homogeneoushalf-space. In a stratum, waves with wavenumbers higherthan a1 are generally subject to strong material dissipationsince they arise from P- or S-waves being reflected multipletimes at the interfaces between layers. Only if the loss fac-tor is gj = 0 for all layers, undamped Love waves may exist;but this situation is not likely to appear in real soils.

Concerning item 2 it has been found by numerical exper-iments that the integral of Eq. (51) may be truncatedbeyond the wavenumber a2 determined as

a2 ¼ max 5a1; 20a0f g; a0 ¼ 2p=R0: ð58Þ

Here R0 is a characteristic length of the foundation, e.g. theradius of a circular footing. For strata with gj > 0.01 for alllayers it has been found that accurate results are typicallyobtained by Simpson integration with 2000 points in thewavenumber range a 2 [0;a1] and 500 points in the rangea 2 [a1;a2]. The numerical evaluation of the integral inthe range a 2 [a1;a2] is particularly efficient for the ‘‘bell-shaped’’ load distribution discussed in Section 3.1, sincebDðaÞbGijða;xÞ, i, j = 1,2,3, are all monotone functions be-yond a1 in this case as demonstrated in the example below.

Finally, concerning item 3, the number of points perS-wavelength should be at least 4–5. Five points are thusrequired in the radial direction of a circular footing forx 6 6cS/R0. The displacement amplitude U10 must be eval-uated for all combinations of receiver and source points. Inthe present analyses, bGðr;xÞ is computed at 250 points onthe r-axis from r = 0 to r = rmax, where rmax is the maxi-mum distance between two points in the discretization ofthe soil–foundation interface. Subsequently U10 is foundby Eq. (51) using linear interpolation of bGðr;xÞ. This pro-vides a fast solution of satisfactory accuracy.

3.3. A circular footing on a homogeneous half-space

The impedance of a circular footing with radius R0 =5 m is computed. The footing rests on a homogeneous lin-ear viscoelastic half-space consisting of one of the materialslisted in Table 1. Here G is the shear modulus, m is the Pois-son ratio, q is the mass density, and g is the loss factor.Note that G is a real number in contrast to l = (1 + ig)Gwhich defines the complex shear stiffness, cf. Eq. (15).The three materials may resemble very soft rock, drainedsandy soil and undrained soil, respectively.

The circular frequency of the excitation is normalisedwith respect to the quantity x0 = cS/R0, where cS is the shearwave velocity in the half-space computed according to Eq.(10). A model with N = 91 uniformly distributed pointsunder the footing has been employed in the present method.The maximum distance between two such points is rmax =2R0 and at least five points are present per S-wavelengthat circular frequencies below x = 6x0. The contact stressis discretized into ‘‘bell-shaped’’ loads applied at these N

points with r1 ¼ R0=ffiffiffiffiffiffiffi4Np

, cf. Eq. (55). Alternatively, for aload applied uniformly over a circular area, i.e. accordingto Eq. (53), the natural choice is r0 ¼ R0=

ffiffiffiffiNp

, implying thatthe areas over which the N loads act add up to the total areaof the footing. In any case, for point m the contribution indirection i is eP m

i Dðx1 � xm1 ; x2 � xm

2 Þ. The principle has beenillustrated in Fig. 8a for a vertical ‘‘bell-shaped’’ load, how-ever for a model with only 37 points.


The results of the present method are compared withthose of a three-dimensional boundary-element (BE) modelemploying the full-space Green’s function [22,29] andquadrilateral elements with biquadratic interpolation ofthe displacement and traction fields. Use has been madeof the fact that the contact stress distribution due to arigid-body motion of the circular footing is either symmet-ric or antisymmetric. Thus, only half the footing is includedin the model, see Fig. 8b.

Fig. 9 shows the wavenumber spectra for the ‘‘bell-shaped’’ load and the components of the Green’s functionmatrix for Material 1 in terms of the normalised wavenum-ber a0 = 2p/R0 at the circular frequency x = 6x0. Similarresults are obtained at other frequencies and for the other

materials listed in Table 1. The components of bGða;xÞhave all been normalised with respect to the quantity

G0 ¼ bG33ð0;xÞ.Pronounced peaks are present in the wavenumber spectra

for Green’s function components in Fig. 9 at the P-wave-number and, in particular, at the Rayleigh wavenumber.However, as indicated in the previous subsection, both theload and the Green’s function components are monotonefunctions for wavenumbers beyond the Rayleigh wavenum-ber. Hence, only a few points are necessary in order to

Fig. 9. Wavenumber spectra for the ‘‘bell-shaped’’ load and the compo-nents of the Green’s function matrix for Material 1 (see Table 1) and forx = 6x0. The vertical lines indicate a/a0, where a0 = 2p/R0.

obtain an accurate discrete inverse Fourier transformationof the wavenumber range beyond a1.

Next, the impedance matrix C(x), cf. Eq. (48), is com-puted for a number of frequencies in the range x0 2 [0;6].All components are normalised with respect to the staticvertical stiffness C0 = C33(0), and the results are plottedin Figs. 10–12 for the materials listed in Table 1. It is notedthat the impedances for rocking and torsion, i.e. C44 andC66, have been divided by R0 in order to get results ofthe same order of magnitude for all components of C(x).Further, since the circular footing is doubly symmetric,C11 = C22 and C44 = C55.

From Figs. 10–12 it is concluded that the presentmethod provides results of great accuracy for the compo-nents C22 and C33, i.e. horizontal and vertical translation.On the other hand there are some discrepancies in the rock-ing and torsion impedances predicted with the two meth-ods. Whereas the same phase angle is achieved with thetwo methods, the amplitude predicted by the presentmethod is approximately 10% smaller than the amplitudeobtained with the BE model. However, it is worthwhileto mention that the present method generally provides stiff-nesses that are too small, whereas the opposite is the casefor the BE solution. A convergence study indicates that a

Fig. 10. Impedance of a circular footing on a homogeneous half-space ofMaterial 1 (see Table 1): present method (—) and boundary-elementsolution (––).

Fig. 11. Impedance of a circular footing on a homogeneous half-space ofMaterial 2 (see Table 1): present method (— 0) and boundary-elementsolution (––).

Fig. 12. Impedance of a circular footing on a homogeneous half-space ofMaterial 3 (see Table 1): present method (—) and boundary-elementsolution (––).


better agreement between the present method and the BEmodel is achieved when more degrees of freedom are usedin both models.

In addition to this, the computation time for 121 fre-quencies is 52 s with the present method on a 1.6 GHz P4laptop computer, whereas the BE solution took about100 s per frequency, or roughly three hours for 121 fre-quencies. For a layered half-space the difference becomesmore pronounced. Andersen and Jones [30] concluded thatthe BE computation mesh must be truncated farther awayfrom the load than is necessary in the case of the homoge-neous half-space. Further, compared to the model of ahalf-space, a better discretization is needed due to the exis-tence of waves propagating in the layers. Thus, in compar-ison to the model illustrated in Fig. 8b, a BE model withmore than twice as many degrees of freedom has to beapplied for the analysis when a half-space overlaid by a sin-gle layer. As the system matrices in the BE formulation arefully populated [22] the solution time scales with the cubeof the number of degrees of freedom using Choleskydecomposition. Hence, the BE analysis for a single layerover a half-space may take as long as a week for 121 fre-quencies. On the other hand, with the present model the

computation time only increases from 52 s to 60 s, i.e. byabout 15%.

4. A wind turbine foundation on a layered ground

In this section, the impedance of a wind turbine founda-tion is studied for frequencies in the range 0–3 Hz. This fre-quency range is relevant for the first vibration modes ofboth the tower and the rotor blades, which are typicallyclose to 0.25 Hz and 1.0 Hz, respectively, for a 3 GW windturbine. The foundation is modelled as a regular hexagonalfooting with the characteristic length R0 = 10 m resting ona subsoil with a single layer overlying a homogeneous half-space. This corresponds approximately to the situation atthe Nysted Offshore Wind Farm at Rødsand in Denmark.The discretization of the soil–foundation interface is illus-trated in Fig. 13. As discussed above, the components ofthe impedance matrix for the hexagonal footing is invariantto a rotation of the footing around the x3-axis. The toplayer is assumed to consist of soft undrained soil, and theunderlying half-space consists of a stiffer material. Thematerial properties of the top layer and the half-space arelisted in Table 2. These properties may be representative

Fig. 13. Discretization of the hexagonal footing.

Table 2Material properties for layered half-space

Layer no. h (m) l (MPa) m q (kg/m3) g

Layer 1 20–80 5 0.495 2000 0.05Half-space 1 500 0.250 2500 0.01

Fig. 14. Impedance of a hexagonal footing on a subsoil with soft sandysoil overlaying a stiffer homogeneous half-space: h1 = 20 m.


of, for example, a sandy or clayey deposit overlying lime-stone or bedrock.

The results of the analyses are shown in Figs. 14–17, andseveral interesting observations can be made. The mainconclusions are:

(1) A comparison of Figs. 10–12 with Fig. 17 shows thatthe impedance of the footing on the 80 m deep layerover the half-space resembles that of a footing on ahomogeneous half-space, in particular at the higherfrequencies. This was to be expected, since the toplayer in this situation is deep compared to the radiusof the footing. Thus, the normalised frequencyx/x0 = 3 corresponds to an S-wavelength less than1/5 of the layer depth.

(2) As an exception to the general observation that adeep layer resembles a half-space, the horizontalimpedance C22 does not increase monotonously athigh frequencies. Local tips and dips continue tobe present in both the normalised amplitude andthe phase angle—even in the case of the 80 m deeplayer, cf. Fig. 17. These tips and dips correspond tonegative and positive interference, respectively,between the outgoing SH-waves generated at the





footing and the incoming SH-waves reflected fromthe interface between the layer and the underlyinghalf-space.

(3) For a homogeneous half-space with a Poisson ratiolower than approximately 0.4, the phase angles of allimpedance components are monotone increasingfunctions of the frequency with the limit p/2 as thefrequency goes to infinity, cf. Fig. 10. For higherPoisson ratios, the phase angles associated with ver-tical motion and rocking increase to a global maxi-mum and then decrease to the asymptotic value p/2,cf. Figs. 11 and 12. The maximum value of thephase shift approaches p for Poisson ratios near1/2, since in this case the half-space becomes infi-nitely stiff with regard to the propagation ofP-waves. On the other hand, for the layered half-space the phase angle varies significantly with thefrequency. As the depth of the top layer is increased,so is the number of tips and dips in the phase anglesof the impedance components within the consideredfrequency range.

(4) With increasing layer depth, the static stiffness relatedto rocking and torsion increases relatively more thanthose related to vertical or horizontal translation.

(5) The magnitude of the translation impedances of a foot-ing on a homogeneous half-space increases monoto-nously with increasing frequency. Figs. 14–17 showthat this is not the case for a footing on a layeredhalf-space. In particular it is noted that the 60 and80 m deep layers give rise to a decrease in the imped-ance of about 30% at 0.20–0.30 Hz, i.e. close to the firstresonance frequency of a modern wind turbine.

(6) In the case of a homogeneous half-space, there is aslight decrease of the rotational impedances with fre-quency for low frequencies, cf. Figs. 10–12. Thisdecrease is more pronounced when the half-space isoverlaid by a layer with h1 = 20 or 40 m, in particularwith respect to the rocking impedance. For h1 = 20 mthe decrease is most significant. However, for h1 =40 m the dip is located at a lower frequency, whichmay be critical for the response of a wind turbine.

Based on these findings it is concluded that stratificationof the subsoil may lead to changes of both the translationand rotation impedances of a wind turbine foundation inthe frequency range that is critical to the structural response.In this regard, it should be noted that not only a reduction ofthe dynamic stiffness compared to the stiffness provided by a


simple model (e.g. assuming a homogeneous half-space)may be critical. Any change of the impedance of the footingcauses a shift in the resonance frequencies of the superstruc-ture, and as mentioned in the introduction this may lead tocoalescence with the excitation frequencies that was notexpected with a simple model of the soil–foundation system.Further, it may be argued that a stiffer footing issues anincrease in the loads on the structure. Eventually, this mayresult in premature fatigue failure of a wind turbine.

5. Conclusions

A method has been presented which allows a fast evalu-ation of the impedance of a footing with arbitrary shapeand resting on a layered viscoelastic half-space. TheGreen’s function for the displacements induced by a dis-tributed load applied at the surface of the half-space isfound in the wavenumber–frequency domain. Adopting asimple load distribution with rotational symmetry arounda source point, the transformation to space domain onlyinvolves numerical integration in one dimension, i.e. alonga line. The response at a point on the surface of the strat-ified half-space to a load applied at any other point and inany given direction is found by a simple coordinatetransformation.

Firstly, the method has been tested against a boundary-element solution for a rigid circular footing on a viscoelastichomogeneous half-space. Generally there is a good agree-ment between the results obtained with the two methods.However, computation time is much smaller with the pres-ent method than with the boundary-element method—evenfor the homogeneous half-space. Thus, computation timefor a single frequency of excitation is more than 200 timesfaster than a similar boundary-element solution in the caseof a homogeneous half-space. This is particularly usefulwhen parametric studies are to be carried out. For a layeredhalf-space, the difference in computation time becomes evenmore pronounced. Fortunately, the implementation of thepresent method in a computer code is relatively simple com-pared with the implementation of a standard boundary-ele-ment or finite-element scheme [22,31] that can be used tosimulate wave propagation in stratified soil.

Secondly, the influence of soil stratification on theimpedance of a rigid hexagonal footing for a wind turbinehas been studied with the present method. The resultsclearly demonstrate that soil layering has to be taken intoconsideration when the dynamic response of the structureis to be determined. Thus, even in the low-frequency range0–3 Hz, changes in the impedance of more than 50% rela-tively to the static stiffness can be expected for subsoils withrealistic properties. For low frequencies around 0.25 Hz,corresponding to the first resonance frequency of a3 MW wind turbine, changes in the impedance are strongerfor deep soft layers overlying a stiff half-space. However, athigher frequencies a shallow top layer may be critical.

In any case, if a simple ground model based on a homo-geneous half-space is employed, the fatigue lifespan may be

incorrectly predicted. Thus, implementation of the layered-ground model into the aero-elastic codes utilised in thewind-turbine industry may be crucial. This may be achievedby coupling of the turbine model with a lumped-parametermodel [32] fitted from the frequency-domain solution pre-sented in this paper.

Appendix A. Fundamental matrix for an elastic layer

The fundamental matrix for a single layer, i.e.

Aj(k1,k2,x), may advantageously be computed in horizon-tal wavenumber domain along the line defined by k1 =c = 0, k2 = a, x 5 0. In particular, at the top of Layer j,the matrix is evaluated as bAj0 ¼ Aj0ð0; a;xÞ. Accordingto Sheng et al. [20],

bAj0 ¼

0 1 0 0 1 0bAj021 0 1 bAj0

21 0 1bAj031 0 bAj0

33 �bAj031 0 �bAj0

33

0 bAj042 0 0 �bAj0

42 0bAj051 0 bAj0

53 �bAj051 0 �bAj0

53bAj061 0 bAj0

63bAj0

61 0 bAj063

2666666666664

3777777777775; ðA:1Þ

wherebAj021 ¼ �ia=fkj

Pg2; bAj0

31 ¼ �ajP=fk

jPg

2;bAj0

33 ¼ �ia=ajS;

bAj042 ¼ aj

Slj;bAj0

51 ¼ �2iljajPa=fk

jPg

2; bAj0

53 ¼ ljða2=ajS þ aj

SÞ;bAj061 ¼ �lj fkj

Sg2 þ 2faj

Sg2

=fkj

Pg2; bAj0

63 ¼ �2ilja:

At the bottom of Layer j, the matrix is evaluated asbAj1 ¼ bAj0Dj; ðA:2Þwhere Dj is a diagonal matrix with the terms Dj

11 ¼ 1,

Dj22 ¼ Dj

33 ¼ eðajS�aj

PÞhj; Dj

44 ¼ e�2ajP

hj;

Dj55 ¼ Dj

66 ¼ e�ðajPþaj

SÞhj:

Due to the simple structure of the matrices bAj0 and bAj1,matrix inversion can be carried out analytically.

In accordance with the derivations in Section 2.3, bAj0

and bAj1 are utilised to ‘‘assemble’’ the Green’s functionmatrix bGða;xÞ ¼ Gð0; a;xÞ. Subsequently, as described inSection 2.4, the Green’s function matrix for any other com-bination of the horizontal wavenumbers than k1 = c = 0and k2 = a may be found by a simple coordinate transfor-mation. Finally, it is noted that a different solution appliesin the static case, i.e. when x = 0. However, a good approx-imation is achieved with Eq. (A.1) for a low frequency,x � 0.

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