wave propagation in the soil: theoretical background and ... · mulated in multilayered media, as...
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Wave propagation in the soil:theoretical background and application to traffic induced vibrations
Geert DegrandeK.U.Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Heverlee, Belgium
This paper reviews the direct stiffness method for the calculation of harmonic and transient wave propagationin horizontally layered media. Developed in the early eighties by Kausel and Roesset, this method has becomea standard tool for the computation of the Green’s functions in layered media. As it is based on a superpositionof plane harmonic waves in the frequency-wavenumber domain, it is very useful as a didactical tool for elasticwave propagation. The Green’s functions are subsequently used in a boundary element formulation to computethe dynamic impedance of the soil in a subdomain formulation for dynamic soil-structure interaction. This isillustrated for the case of road traffic induced vibrations, where the invariance of the problem domain in thelongitudinal direction is exploited by working in the wavenumber domain. A numerical example demonstratesthe influence of the soil stratification on free field vibrations during the passage of a two-axle truck on a trafficplateau.
1 INTRODUCTION
Wave propagation in the soil plays a crucial role inproblems of rail or road traffic induced vibrations inbuildings as it couples the source (the road or thetrack) and the receiver (the building). Numerical mod-els aiming to predict traffic induced vibration shouldtherefore account for dynamic soil-structure interac-tion at the source and the receiver and incorporate amodel that accurately describes wave propagation inthe soil.
Volume gridding methods as the finite elementmethod, the finite difference method and the spectralelement method allow to model arbitrary layering. Asthe soil is semi-infinite, however, absorbing bound-ary conditions are needed that account for Sommer-feld’s radiation conditions. Local absorbing boundaryconditions preserve the banded structure of the sys-tem matrices, but cause spurious wave reflections atthe boundaries. Therefore, large mesh extensions areneeded at low frequencies. Furthermore, the elementlength should be small enough with respect to thesmallest wavelength propagating through the model,imposing small elements at high frequencies. Bothconditions result in models that may become pro-hibitively large for 3D analyses. Consistent boundaryconditions or the combination with a boundary ele-ment formulation do not produce spurious reflectionsbut couple all degrees of freedom along the boundary,
affecting the banded nature of the system matrices.In this paper, we will briefly review the direct stiff-
ness formulation proposed by Kausel and Roesset(1981) for the calculation of wave propagation in hor-izontally layered elastic media. This method is basedon the decoupling of P-, SV- and SH-waves and thesuperposition of plane harmonic waves in the fre-quency wavenumber domain, which makes it very at-tractive from a didactical point of view.
The method can be applied to various problems for-mulated in multilayered media, as the computation ofamplification ratios, free surface waves and the har-monic and transient response under external loads.An example is the computation of the Green’s func-tions of a layered halfspace, needed in a boundary ele-ment formulation. As these Green’s functions implic-itly account for the traction free condition at the freesurface, as well as displacement continuity and stressequilibrium at layer interfaces, the boundary elementdiscretization can be efficiently restricted to the in-terfaces between the soil and the structure(s). The dy-namic soil-structure interaction problem can thereforebe formulated on interfaces between subdomains, asis the case in the subdomain formulation proposed byAubry and Clouteau (1992).
A formulation in the frequency-wavenumber do-main offers interesting perspectives for the compu-tation of transfer functions in soil-structure interac-
1
tion problems that are invariant (roads) or periodic(tracks, tunnels) in the longitudinal direction. Appli-cation of the dynamic reciprocity theorem to the caseof moving loads allows for an efficient calculation ofthe free field traffic induced vibrations. This will be il-lustrated in the present paper with a numerical modelfor road traffic induced vibrations (Lombaert et al.2000), that has been extensively validated by meansof in situ vibration measurements (Lombaert and De-grande 2001).
2 GOVERNING EQUATIONS2.1 Equilibrium equationsIn the Cartesian frame of reference, the displacementcomponents of the elastic medium are denoted asus
i(x; t). The components of the small strain tensor �s
ij
are computed with the following strain-displacementrelations:
�s
ij=
1
2
�us
i;j+ u
s
j;i
�(1)
The equilibrium equations are:
�ij;j + �sbi = �
s�usi
(2)
where a dot on a variable denotes differentiation withrespect to time. In this equation, �ij denotes the stresstensor, �s the density of the elastic medium and �sbithe body force. For an isotropic linear elastic material,the constitutive equations become:
�ij = �s�s
kkÆij + 2�s�s
ij(3)
with �s and �s the Lame coefficients.These equations are completed by initial condi-
tions, Dirichlet and Neumann boundary conditions, aswell as Sommerfeld’s radiation conditions at infinity.
The Navier equations result from the eliminationof the stress and strain tensors from the equilibriumequations (2) by means of the constitutive equations(3) and the strain-displacement relations (1):
(�s + �s)us
j;ji+ �
sus
i;jj+ �
sbi = 0 (4)
In the following, the influence of body forces will bedisregarded.
2.2 Green’s functionsIn the particular case where a concentrated impul-sive load �bj(x; t) = Æ(x� �)Æ(t)Æij is applied at thesource point � in a direction ei, the solution usj(x; t)of the elastodynamic problem at a receiver x in a di-rection ej is referred to as the fundamental singularsolution or the Green’s function of the medium (Apseland Luco 1983; Luco and Apsel 1983) and denoted asthe second order tensor uG
ij(�;x; t).
On a plane with unit outward normal vector n,Cauchy’s stress principle is applied to calculate the
tractions tsj(x; t) from the second order stress tensor�kj according to �kj(x; t)nk. The Green’s tractions arethen denoted as the second order tensor tG
ij(�;x; t).
It will be outlined later how the Green’s functionscan be computed with a direct stiffness formulation.
2.3 The dynamic reciprocity theoremThe dynamic Betti-Rayleigh reciprocity theoremspecifies a relationship between a pair of elastody-namic states, represented by body forces �b1(x; t) and�b2(x; t), boundary tractions t1(x; t) and t2(x; t) anddisplacements u1(x; t) and u2(x; t). If the displace-ments u and velocities _u vanish in both states fort tending to �1, the dynamic reciprocity theoremreads as follows:Z
S
Zt
�1
t1j(x; t� �)u2j(x; �)d� dS
+ZV
Zt
�1
�b1j(x; t� �)u2j(x; �)d� dV
=ZS
Zt
�1
t2j(x; t� �)u1j(x; �)d� dS
+ZV
Zt
�1
�b2j(x; t� �)u1j(x; �)d� dV (5)
In the frequency domain, this theorem becomes:ZS
t1j(x; !)u2j(x; !)dS
+ZV
�b1j(x; !)u2j(x; !)dV
=ZS
t2j(x; !)u1j(x; !)dS
+ZV
�b2j(x; !)u1j(x; !)dV (6)
Applying the dynamic reciprocity theorem to twoelastodynamic states, where the first is the unknownstate characterized by displacements usj(x; t), trac-tions tsj(x; t) and body forces �bsj(x,t), while the sec-ond is characterized by the fundamental Green’s solu-tions uG
ij(�;x; t) and tG
ij(�;x; t), the representation the-
orem of elastodynamics is obtained. In the frequencydomain, this theorem is written as follows:
usi(�; !) =ZS
tsj(x; !)uGij(�;x; !)dS
�
ZS
tG
ij(�;x; !)usj(x; !)dS
+ZV
�bsj(x; !)uGij(�;x; !)dV (7)
This theorem forms the basis for boundary integralformulations.
2.4 Decomposition of the displacement vectorsApplication of the Helmholtz decomposition us =grad�s + rot s with a scalar function �s and a vec-tor function s, for which div s = 0, to the Navierequations (4) results in the following set of uncoupledpartial differential equations:
(�s + 2�s)r2�s = �s ��s (8)
�sr
2 s = �
s � s (9)
with r2 the 3D Laplace operator. The longitudinalor P-wave is described by the scalar potential �s andpropagates with the velocity Cp =
q(�s + 2�s)=�s. It
decouples from the shear or S-wave, that is describedby the vector potential s and propagates with theshear wave velocity Cs =
q�s=�s.
The contribution of the S-wave to the displacementvector can be decomposed into a component parallel(SH-wave) and normal (SV-wave) to a bounding sur-face. As we are interested in the response of a hori-zontally layered halfspace where the vertical z-axis isperpendicular to the layer interfaces, it is convenientto choose the bounding surface parallel with the hori-zontal (x; y)-plane.
For 2D wave propagation in the (x; z)-plane, thedependence on the y-coordinate can be ignored andthe following decomposition holds:
us = grad�s + rot(ey s) + rot(ez�s) (10)
The following set of hyperbolic PDE is obtained:
(�s + 2�s)r2�s = �s ��s (11)
�sr
2 s = �
s � s (12)
�sr
2�s = �
s ��s (13)
with r2 the 2D Laplace operator. The in-plane mo-tions us
xand us
zdecouple from the out-of-plane mo-
tion usy. The former are described in terms of scalar
wave potentials �s (P-wave) and s (SV-wave). Theout-of-plane motion is described in terms of the scalarwave potential �s (SH-wave).
3D wave propagation in multi-layered media canbe described as the superposition of the solution ofthe in-plane P-SV problem and the out-of-plane SHproblem. In the following, we will restrict to a briefdescription of the P-SV problem and the SH problem.
3 DISPERSION RELATIONSThe in-plane propagation of P-waves is governed bythe hyperbolic PDE (11). The time t is transformedto the circular frequency ! by means of a forwardFourier transformation and followed by a Fouriertransformation of the horizontal coordinate x to thehorizontal wavenumber kx, as it is assumed that thegeometry is invariant in the horizontal direction:
(�s + 2�s)
"�k
2
x+d2 ~�s
dz2
#+ !
2�s ~�s = 0 (14)
The function �s(x; z; t) is decomposed into asuperposition of plane harmonic wave potentials~�s(kx; z; !) in the frequency-wavenumber domain.The solution of equation (14) is equal to:
~�s(kx; z; !) = Is
Pe�ikzpz +R
s
Pe+ikzpz (15)
where the potentials Is
Pand R
s
Prefer to incident
(propagating in the positive z-direction) and reflected(propagating in the negative z-direction) waves. kzp isthe vertical component of the wave propagation vec-tor kp = fkx; kzpg
T of the P-waves and follows fromthe following dispersion relation:
k2
x+ k
2
zp= k
2
p(16)
with kp = !=Cp the magnitude of the wave propa-gation vector kp. Hysteretic material damping in thesolid skeleton is introduced according to the corre-spondence principle, using a complex Lame coef-ficient (�s + 2�s)(1 + 2i�s
p), with �
s
pthe material
damping ratio. This results in a complex P-wave ve-locity Cp and wavenumber kp. Equation (16) enablesto calculate the complex vertical wave number kzpfor each horizontal wavenumber kx and circular fre-quency !. kzp equals (k2
p� k
2x)0:5 if kx � kp, which
corresponds to propagating waves in the z-direction.If kx > kp, kzp equals �i(k2
x� k
2p)0:5 and represents
inhomogeneous waves with exponential decrease orincrease in amplitude with z.
Analogous derivations hold for SV- and SH-waves,which are also decomposed into a superposition ofplane harmonic waves:
~ s(kx; z; !) = Is
SVe�ikzsz +R
s
SVe+ikzsz (17)
~�s(kx; z; !) = Is
SHe�ikzsz +R
s
SHe+ikzsz (18)
where kzs is the vertical component of the wave prop-agation vector ks = fkx; kzsg
T of the S-waves and fol-lows from the following dispersion relation:
k2
x+ k
2
zs= k
2
s(19)
ks = !=Cs is the magnitude of the wave propaga-tion vector ks. Hysteretic material damping in the
solid skeleton is introduced according to the corre-spondence principle, using a complex Lame coeffi-cient �s(1 + 2i�s
s), with �s
sthe material damping ra-
tio. This results in a complex S-wave velocity Cs andwavenumber ks. The potentials Is
SV, Is
SH, Rs
SVand
Rs
SHrefer to incident and reflected waves, respec-
tively.Similar results are obtained for P-SV wave prop-
agation in the axisymmetric case, where the Fouriertransformation of the time t to the circular frequency! is now followed by a Hankel transformation of theradial coordinate r to the radial wavenumber kr. Thelatter is related to the horizontal wavenumbers kx andky as k2
r= k
2x+ k
2y.
4 THE DIRECT STIFFNESS FORMULATIONThomson (1950) and Haskell (1953) have formulatedtransfer matrices in the frequency-wavenumber do-main for the study of wave propagation in layeredmedia. The transfer matrices follow from the exactsolution of the wave equations and express the rela-tionship between the state vectors at both interfacesof a single layer of the medium.
Kausel and Roesset (1981) have presented a formu-lation with stiffness matrices in the transformed do-main as an alternative for the Haskell-Thomson trans-fer matrix approach. Compared to the use of trans-fer matrices, the stiffness matrices have the advantagethat they are symmetric and allow for an efficient nu-merical implementation.
4.1 Dry halfspace elementThe dry halfspace element models the propagation ofwaves in a semi-infinite halfspace. The amplitudes ofthe plane harmonic waves should be non-increasingfunctions with the distance travelled. Therefore, onlythe outgoing wave amplitudes ~aI = fIs
SV; I
s
PgT , prop-
agating in the positive z-direction, are accounted for.
Figure 1: Dry halfspace element (P-SV).
In the 2D P-SV case, the element displacements~ue = f~usx; i~uszg
T and the element tractions ~Te =f�~�zx;�i~�zzg
T at the surface of the layer are writtenin terms of the outgoing wave amplitudes ~aI . Elimi-nation of these amplitudes leads to the derivation of a
complex symmetrical element stiffness matrix ~Ke thatrelates the tractions ~Te to the displacements ~ue:
~Ke~ue = ~Te (20)
Kausel and Roesset (1981) have derived analyticalexpressions for the elements of the 2 by 2 com-plex symmetrical element stiffness matrix ~Ke, whichare functions of the material properties of the elasticmedium, the circular frequency ! and the wavenum-ber kx. Limiting expressions for the case of zero fre-quency (quasi-static solution) and zero wavenumber(1D case) have also been derived. These expressionswill not be repeated herein.
Identical expressions are obtained in the axisym-metric P-SV case if the imaginary units are omitted inthe derivation of the element vectors.
In the 2D SH-case, the element displacement vec-tor reduces to ~ue = f~usyg
T and the element tractionvector is equal to ~Te = f�~�zyg
T , so that the elementstiffness matrix ~Ke reduces to a 1 by 1 complex sym-metrical matrix.
4.2 Dry layer element
Figure 2: Dry layer element (P-SV).
When the in-plane motion in a 2D layer element isconsidered, the element displacements ~ue = f~usx(z =0); i~usz(z = 0); ~usx(z = L); i~usz(z = L)gT and theelement tractions ~Te = f�~�zx(z = 0);�i~�zz(z =0); ~�zx(z = L); i~�zz(z = L)gT at both interfaces of thelayer are written in terms of both the outgoing waveamplitudes ~aI = fI
s
SV; I
s
PgT and the incoming wave
amplitudes ~aR = fRs
SV;R
s
PgT . Elimination of these
unknown wave amplitudes leads to the derivation of acomplex symmetrical element stiffness matrix ~Ke thatrelates the tractions ~Te to the displacements ~ue:
~Ke~ue = ~Te (21)
Analytical expressions for the elements of the 4 by 4complex symmetrical element stiffness matrix ~Ke are
presented by Kausel and Roesset (1981) and will notbe repeated herein.
In the 2D SH-case, the element displacement vec-tor reduces to ~ue = f~usy(z = 0); ~usy(z = L)gT and theelement traction vector is equal to ~Te = f�~�zy(z =
0); ~�zy(z = L)gT . The element stiffness matrix ~Ke re-duces to a 2 by 2 complex symmetrical matrix (Kauseland Roesset 1981).
Similar developments are possible for a poroelasticlayer and halfspace element, where the solid skeletonis saturated with a pore fluid (Rajapakse and Senjun-tichai 1995; Degrande et al. 1998).
4.3 Assembly of equationsThe propagation of waves in a horizontally layeredhalfspace can be modelled with N � 1 layer elementson top of a halfspace element, with N interfaces. Foreach interface i between two layers i � 1 and i, thestress equilibrium and the continuity of displacementsare expressed for each frequency ! and wavenumberkx, resulting in the following system of equations:
~KS~uS = ~TS (22)
where the complex symmetrical and banded stiffnessmatrix ~KS is assembled analogously as in a finite el-ement formulation. Due to the introduction of hys-teretic material damping, the coefficient matrix ~KS isnot singular in the real wavenumber domain (Apseland Luco 1983; Luco and Apsel 1983).
Since the frequency and wavenumber dependenteigenvectors are used as shape functions for the for-mulation of the element stiffness matrices ~Ke, themass distribution is treated exactly without the needof subdividing a layer into smaller elements. Wavepropagation within an element is treated exactly andelements can extend from one interface to another, re-sulting in a system of equations (22) of limited dimen-sion.
Alternatively, the exact solution in the vertical di-rection can be replaced by polynomial shape func-tions, resulting in a thin layer formulation (Kauseland Roesset 1977; Waas 1972). The layer thicknessshould be small with respect to the minimum wave-length propagating through the model. The thin layermethod has originally been formulated for a layeredstratum on a rigid bedrock. Wave propagation in asemi-infinite layered halfspace can be treated usinga hybrid formulation, where a thin layer formulationfor the layers is combined with the aforementionedhalfspace element.
The direct stiffness formulation is suitable for thesolution of a wide variety of problems, such as the siteamplification of incident plane harmonic waves, thedispersion and attenuation of surface waves and theresponse to external loads. Within the frame of this
review, the last two problems are discussed in moredetail.
4.4 Free surface wavesThe free surface waves in a layered halfspace are thenatural modes of vibration and equal to the displace-ments ~uS when the external load vector ~TS in equa-tion (22) equals zero. Non-trivial solutions for ~uS canbe obtained if the coefficient matrix ~KS is singular orif the determinant of ~KS is equal to zero:
det ~KS = 0 (23)
Figure 3: Surface waves.
This equation corresponds to a transcendentaleigenvalue problem in terms of the real frequency !and the complex horizontal wavenumber kx, whichenables the determination of the free surface wavesand their attenuation. This eigenvalue problem hasan infinite number of solutions and must be solvedby search techniques (Draelants 1994). For the caseof a homogeneous halfspace and in the absence ofmaterial damping, equation (23) reduces to the clas-sical cubic equation that was first formulated byRayleigh (1887), revealing a single non-dispersivesurface wave with phase velocity CR.
The solution of the eigenvalue problem (23) is alsoan indispensable step in the spectral analysis of sur-face waves (SASW) method to determine the theoret-ical dispersion curve of a multilayered site.
In the case of a layered soil supported by a rigidstratum, the thin layer method offers an advantage, asthe eigenvalue problem for the natural surface wavemodes is algebraic instead of transcendental.
4.5 Forced vibration problemsA second application involves the calculation of theresponse of a layered halfspace to an external tran-sient loading. At each interface between two layers,the displacements or tractions can be prescribed.
In the in-plane case, the traction is represented by afunction T (x; t) = S(x)F (t), where S(x) and F (t)denote the spatial and temporal distribution of theload, respectively. The right hand side of equation
(22) is the product of the forward Fourier transformsof the functions F (t) and S(x). A special case offorced vibrations is the computation of the Green’sfunctions of a layered halfspace where a Dirac load inspace and time is applied.
Figure 4: Forced vibration problem.
The solution of the system of equations (22) yieldsthe transformed displacements ~usi(kx; z; !) at eachinterface. The response in the space-frequency do-main is obtained by an inverse Fourier transformationfrom the wavenumber to the spatial domain of the fol-lowing general form:
usi(x; z;!)
=1
2�
Z1
�1
~usi(kx; z; !)eikxxdkx (24)
The behaviour of the integrand in this equationpresents some difficulties:
1. The function ~usi(kx; z; !) follows from the solu-tion of the system of equations (22) for differentvalues of ! and kx, which may become expen-sive due to presence of transcendental functions,eventhough the relative increase in computationtime due to an increase in the number of layers isnot particularly large (Xu and Mal 1987).
2. This function also exhibits dense oscillations forparticular values of kx, which are due to the ex-ponential terms in the wave solutions. The pres-ence of very low speed layers is likely to causerapid oscillations in some parts of the integrand,which are irregular as their exact location and na-ture cannot be predicted (Xu and Mal 1987).
3. In the absence of material dissipation, the func-tion ~usi(kx; z; !) becomes zero for certain val-ues of kx, which correspond to the surface wavepoles of the integrand. The introduction of com-plex Lame coefficients shifts the poles from thereal kx-axis into the complex plane and enablesthe use of numerical integration techniques.
4. The kernel functions in the integral transformsintroduce rapid oscillations for large values ofthe horizontal source-receiver distance.
In view of these remarks, an efficient quadraturescheme is needed for the evaluation of the inversewavenumber integrals. According to Xu and Mal(1985), an adaptive algorithm with self-adjusting in-terval �kx, concentrating abscissas around regions ofsharp variations in ~usi(kx; z; !) and taking full advan-tage of previously computed values of the integrand,is useful for the accurate evaluation of the wavenum-ber integral with a minimum number of function eval-uations. The fourth problem is solved using a gener-alized Filon method, where only the kernel-free part~usi(kx; z; !) of the integrand is interpolated.
Another difficulty is that, at low frequencies andwhen the source and the receiver are located at thesame depth, a high upper limit kmax
xof the integration
interval is needed for an accurate evaluation.The inverse transformation from the frequency to
the time is performed by an inverse FFT algorithm.
5 ROAD TRAFFIC INDUCED VIBRATIONSIn the following, a numerical model for the pre-diction of road traffic induced vibrations (Lombaertet al. 2000) is briefly reviewed. Dynamic axle loadsare computed using simple 2D vehicle models. Themodel demonstrates how the Green’s functions of thesoil are used to compute the road’s impedance witha boundary element formulation, as well as to cal-culate the transfer functions between a fixed sourcepoint on the road and a receiver in the free field. It isalso shown how the dynamic reciprocity theorem isused to compute the free field response due to a vehi-cle moving on the road.
5.1 The dynamic axle loadsThe dynamic axle loads are calculated with a 2D ve-hicle model, which is composed of discrete masses,springs and dampers. The vehicle body and the wheelaxles are assumed to be rigid inertial elements, whilethe primary suspension system and the tyres are rep-resented by a spring-dashpot system. The distributionof n axle loads can be written as the summation of theproduct of Dirac functions that determine the positionof the force and a time-dependent function gk(t):
F (x; y; z; t)
=nX
k=1
Æ(x� xS)Æ(y� yk � vt)Æ(z)gk(t) (25)
yk is the initial position of the k-th axle that moveswith the vehicle speed v along the y-axis. As the in-fluence of the road displacements on the dynamic axleloads can be neglected, the frequency content gk(!) of
a single axle load is calculated from the contributionof n vehicle axles and the road surface profile:
gk(!) =nXl=1
hfkul(!)ul
w=r(!) (26)
The FRF hfkul(!) represents the frequency content ofthe axle load at axle k, when a unit impulse excitationis applied to axle l (Hunt 1991). ul
w=r(!) represents
the frequency content of the road unevenness appliedat axle l and is calculated from the wavenumber do-main representation ~uw=r(ky) of the longitudinal roadprofile uw=r(y):
ul
w=r(!) =
1
v~uw=r(�
!
v) exp (i!
yl
v) (27)
For increasing vehicle speed, the quasi-static value ofthe road profile experienced by the vehicle axles de-creases, while the frequency content shifts to higherfrequencies. Equations (26) and (27) are used to rep-resent the contribution of all axles to a single axle loadby a single FRF hfku(!):
gk(!) =1
v~uw=r(�
!
v)
nXl=1
hfkul(!) exp (i!yl
v)
=1
v~uw=r(�
!
v)hfku(!) (28)
5.2 The road-soil transfer functionThe road-soil transfer function hzi(x; y; z; t) repre-sents the fundamental solution at a point of the roador the soil for the displacement in the direction ei dueto a vertical impulse load on the road. Its calculationrequires the solution of two subprobems. First, a dy-namic substructure method is used to calculate thetractions at the road-soil interface (Aubry et al. 1994).Next, the displacements at an arbitrary location arecalculated from these soil tractions.
The road is assumed to be invariant with respectto the longitudinal direction y and to have a rigidcross section, supported by the soil along the inter-face �rs. The first assumption allows to perform aFourier transformation from the longitudinal coordi-nate y to the horizontal wavenumber ky, resultingin an efficient solution procedure in the frequency-wavenumber domain. The second assumption allowsto write the vertical road displacements ~urz(x; ky; !)as a function of the vertical translation ~ucz(ky; !) ofthe cross section’s centre of gravity and the rotation~�cy(ky; !) about this centre:
~urz(x; ky; !) = �r(x) ~�(ky; !) (29)
The displacement modes of the rigid cross section arecollected in a vector �
r(x) = f1; xgT , while the vec-
tor ~�(ky; !) collects the displacement ~ucz(ky; !) andthe rotation ~�cy(ky; !).
The foregoing kinematical assumptions result inthe following equilibrium equations for the road,which govern the longitudinal bending and torsionaldeformations, respectively:�
~Kr � !2 ~Mr +
Z�rs
�r~tsz(~�s)dx
�~� = ~fÆ
c(30)
with ~Kr = diagfEIxk4y;GCk2yg the stiffness matrix
and ~Mr = diagf�A;�Ipg the mass matrix of the road.A is the road’s cross section, Ix the moment of inertiawith respect to x, C the torsional moment of inertiaand Ip the polar moment of inertia; E is the Young’smodulus, G the shear modulus and � the density ofthe road. The road impedance ~Kr � !
2Mr becomessingular for free bending waves with phase veloc-ity Cb =
4
qEIx!
2=�A and free torsional waves with
phase velocity Ct =qGC=�Ip.
The integral on the left hand side of equation (30)denotes the soil impedance ~Ks, and follows from theequilibrium at the road-soil interface �rs. ~tsz(~�s
) isthe frequency-wavenumber domain representation oftsz(�s), the vertical component of the soil tractionsts = �sn on a boundary with a unit outward normal ndue to the displacement mode �s.
A boundary element method is used to calculate thesoil tractions ~tsz(~�s
) at the road-soil interface in thesoil (Aubry et al. 1994; Lombaert et al. 2000). Theboundary element formulation is based on the for-mulation of the boundary integral equations in thefrequency-wavenumber domain, using the Green’sfunctions of a horizontally layered soil. The Green’sfunctions in the frequency-wavenumber domain arecalculated as the inverse transformation of the Green’sfunctions in the frequency-radial wavenumber do-main.
The vector ~fÆc
in equation (30) is the force vectorrelated to the Dirac load applied in a point (xS ;0;0)at time t = 0.
The solution of the system of equations (30) givesthe complex modal coordinates ~�. The soil tractions~tsz(~us) at the road-soil interface are calculated fromthese modal coordinates as ~tsz(~�s) ~�.
The dynamic reciprocity theorem is used to com-pute the road-soil transfer function ~hzi(�1; ky; �3; !)from the soil tractions at the interface. In the load caseconsidered, only the vertical tractions ~tsz(x; ky; z =0; !) have a non-zero resultant. When the loaded areais small compared to the wavelength in the soil, it canbe assumed that the horizontal tractions have a smallinfluence on the free field displacements, so that:
~hzi(�x; ky; �z; !) =Z�rs
~uGzi(�x� x; ky; �z; !)
~tsz(x; ky; z = 0; !)dx (31)
and only the Green’s function ~uGzi(�x; ky; �z; !), repre-
senting the fundamental solution for the displacementin the direction ei due to a vertical impulse load, isneeded.
6 RESPONSE TO MOVING LOADSIn the case where the load moves with a constantspeed v, the dynamic reciprocity theorem allows toderive a relation between the unknown response andthe fundamental response to the fixed impulsive load.
The moving load is represented by a body force�bj(x; t) = Æ(x� xSk(t))gk(t)Æzj in the vertical direc-tion ez, where xSk(t) = fxS; yk+ vt; zSgT denotes thetime-dependent position and gk(t) the time history ofthe moving load.
For a problem geometry that is invariant in thedirection ey of the moving load, the following ex-pression can be derived for the unknown responseusi(�x; �y; �z; t):
usi(�x; �y; �z; t) =Z
t
�1
gk(�)
hzi(�x; �y � yk � v�; �z; t� �)d� (32)
where it is assumed that the transfer functionhzi(�x; �y; �z; t) is the response to a vertical pulse atthe point fxS;0; zSgT of the road surface. The coor-dinates in the argument of the transfer function corre-spond to a receiver that moves in the opposite direc-tion of the source. As the transfer function can be ef-ficiently computed in the frequency-wavenumber do-main according to equation (31), it is advantageousto transform equation (32) also to the frequency-wavenumber domain. The following expression is ob-tained:
~usi(�x; ky; �z; !) = gk(!� kyv)
~hzi(�x; ky; �z; !) exp(+ikyyk) (33)
The response is computed as the product of the trans-fer function and the frequency content of the source,provided that the latter is shifted by kyv. For a limit-ing small velocity v, this shift tends to zero and thesolution for the case of a load at a fixed position isrecovered.
The representation usi(�x; �y; �z; !) of the responsein the frequency domain is obtained as the inversewavenumber domain transform:
usi(�x; �y; �z; !) =1
2�
Z+1
�1
gk(!� kyv)
~hzi(�x; ky; �z; !) exp [�iky(�y � yk]dky (34)
A change of variables according to ky = (! � ~!)=vmoves the frequency shift from the frequency content
of the moving load to the wavenumber content of thetransfer function:
usi(�x; �y; �z; !) =1
2�v
Z+1
�1
gk(~!)
~hzi(�x;!� ~!
v; �z; !)
exp��i
�!� ~!
v
�(�y � yk)
�d~! (35)
The frequency content gk(~!) of the load and the dis-placement usi(�x; �y; �z; !) are coupled through thewavenumber at which the transfer function is evalu-ated. For a limiting large velocity v, the wavenumberky = (!� ~!)=v tends to zero and the solution for the2D case of a line load along the path of the movingsource is obtained.
The solution in the time domain is finally obtainedas the inverse inverse Fourier transformation of thecircular frequency ! to the time t.
7 EXAMPLEIn the following example, the influence of the soilstratification is demonstrated for the free field vibra-tions generated by the passage of a two-axle truck ona traffic plateau (Lombaert et al. 2001). Three casesare considered for the soil stratification:
Case (a): a homogeneous halfspace with �s =
1800kg/m3, Cs = 150m/s, Cp = 300m/s and�s
s= �
s
p= 0:025. The ratio s of the wave veloci-
ties Cs and Cp is equal toq(1� 2�s)=(2� 2�s)
and only depends on the Poisson’s ratio �s; s isequal to 0.5 if �s = 1=3.
Case (b): a layer built in at its base with the mate-rial properties of the halfspace in case (a) and athickness d = 5m.
Case (c): a layer with the same material properties asthe halfspace in case (a) and a thickness d= 5m,supported by a halfspace with �s = 1800kg/m3,Cs = 300m/s and Cp = 600m/s.
First, wave propagation in the soil will be describedby the axisymmetric Green’s functions. Second, itis shown how the dynamic axle loads are calculatedfrom the longitudinal road profile and the vehicletransfer functions. Next, the solution of the dynamicinteraction problem is discussed. Finally, the free fieldvelocities are calculated and the influence of the soilstratification is demonstrated.
7.1 The axisymmetric Green’s functions of the soil
In order to illustrate the dispersion of the body andsurface waves for the three cases considered, the ax-isymmetric Green’s function is briefly discussed.
Figure 5a shows the logarithm of the modulus ofthe axisymmetric Green’s function ~uGaxi
zz(kr; z = 0; !)
at the surface z = 0 of the homogeneous halfspace asa function of the circular frequency ! and the dimen-sionless radial wavenumber kr = krCs=!. The func-tion has been multiplied by the circular frequency ! toelucidate its behaviour at high frequencies. The prop-agation and decay of body waves can be derived fromthe dispersion relations. All waves propagate in theradial direction for all values of kr. Although figure5a only shows results on the surface of the halfspaceand does not allow to observe wave propagation inthe z-direction, the dispersion relations allow to drawsome general conclusions on wave propagation in thez-direction. The dimensionless vertical wavenumberkzp for the P-waves is equal to kzp = (s2 � k
2
r)0:5 if
kr < s and kzp = �i(k2
r� s
2)0:5 if kr � s. The di-mensionless vertical wavenumber kzs for the S-wavesis equal to kzs = (1 � k
2
r)0:5 if kr < 1 and kzs =
�i(k2
r� 1)0:5 if kr � 1. If 0 � kr � s, both P- and
S-waves propagate in the vertical direction. The P-waves become inhomogeneous in the z-direction ifkr � s, while the S-waves become inhomogeneous inthe z-direction if kr � 1. Both events appear as dips inthe surface plot in figure 5a corresponding to valuesof kr that are equal to s and 1. The peak at kR = 1:073corresponds to the pole of the Rayleigh wave. In thecase of a homogeneous halfspace, none of the bodywaves and the Rayleigh wave are dispersive and allphase velocities are frequency independent.
Figure 5b shows the axisymmetric Green’s function~uGaxizz
(kr; z = 0; !) for the case of a layer built in atits base. Wave propagation takes place at frequencieshigher than the natural frequencies of the layer builtin at its base. The peak at 15 Hz corresponds to thefirst vertical eigenfrequency Cp=4d of the layer builtin at its base. The second vertical eigenfrequency isequal to 3Cp=4d = 45 Hz. At limiting high frequen-cies, the wavelength in the soil is small compared tothe thickness of the layer and the response convergesto the response of a halfspace with the same materialproperties as the layer.
Figure 5c shows the axisymmetric Green’s function~uGaxizz
(kr; z = 0; !) for the case of a layer on top of ahalfspace. At low frequencies, the wavelength in thesoil is large with respect to the thickness of the layerand the response corresponds to the response of a ho-mogeneous halfspace with the same material proper-ties as the underlying halfspace. At high frequencies,the wavelength is small compared to the thickness ofthe top layer and the wave propagation in the soil
is determined by the characteristics of the latter. Theprevious examples illustrate that wave propagation ina layered halfspace is dispersive as the wave velocitiesdepend on the frequency.
a. Homogeneous halfspace.
b. Layer built in at its base.
c. Layer on a halfspace.Figure 5: Logarithm of the modulus of the product ofthe axisymmetric Green’s function ~uGaxi
zz(kr; z = 0; !)
and ! as a function of ! and kr for the 3 cases con-sidered.
7.2 The dynamic axle loadsFigure 6a shows the profile uw=r(y) of the trafficplateau with a height H = 0:12m, a length L = 10mand sinusoidal slopes with a length l = 1:20m. Fig-ure 6b shows the representation ~uw=r(ky) of the roadprofile in the wavenumber domain. The separationbetween the lobes in the wavenumber domain is in-versely proportional to the mean length L + l of theplateau.
Figure 7a shows the FRF hf1u for the rear axle load
−10 −5 0 5 100
0.05
0.1
0.15
0.2
y [m]
Roa
d un
even
ness
[m]
a. uw=r(y).
0 5 10 15 200
0.5
1
1.5
Wavenumber [rad/m]
Roa
d un
even
ness
[m2 /r
ad]
b. ~uw=r(ky).Figure 6: The longitudinal road profile of a trafficplateau with sinusoidal slopes (a) as a function of thecoordinate y and (b) in the wavenumber domain.
of a 2D 4DOF vehicle model of a two-axle VolvoFE7 truck. The FRF are dominated by the pitch andbounce modes (1.6 Hz and 1.9 Hz) and the axle hopmodes (9.1 Hz at the front axle and 9.5 Hz at the rearaxle) of the vehicle. The FRF are used to calculate thedynamic axle loads during the passage of the two-axletruck on the traffic plateau at a speed v = 14m/s.
0 10 20 30 40 500
1
2
3
4
5
6
7x 10
6
Frequency [Hz]
FR
F [N
/m]
a. hf1u(!).
0 10 20 30 40 500
1
2
3
4
5
6x 10
4
Frequency [Hz]
Rea
r ax
le lo
ad [N
/Hz]
b. g1(!).
Figure 7: (a) The vehicle FRF hf1u for the rear axleload and (b) the frequency content g1(!) of the rearaxle load.
Figure 7b shows the frequency content of the rearaxle load. From equation (27), it follows that the ve-hicle speed couples the frequency content ul
w=r(!) to
the wavenumber domain representation ~uw=r(ky) ofthe road profile. The separation between the smalllobes in the frequency domain is therefore propor-tional to the vehicle speed. The spectrum of the axleloads is dominated by the pitch and bounce modes.
7.3 The roadThe case is considered where the load is applied at thecenter of the road (xS = 0). As the bending and thetorsional modes are uncoupled, the modal coordinate~�cy of the torsional modes is zero and the discussionis restricted to the bending modes.
The road has a width 2B = 3m; it is composedof a bituminous top layer, a granular subbase and afoundation. This three-layer system can be replacedby a single equivalent bituminous layer with a thick-ness h = 0:14m and a density � = 5910kg/m3 withthe same bending stiffness EI and weight �A as thethree-layer system.
7.4 The soil’s impedance
The soil’s impedance ~Ks is calculated from the soiltractions for the deformation modes of the cou-pled road-soil system. A boundary element method,based on the Green’s functions in the frequency-wavenumber domain, is used for the calculation ofthe soil tractions. The road-soil interface is discretizedinto 32 boundary elements of equal length le = 0.094m. The element size is small compared to the wave-length at the maximum frequency f = 50Hz, but isrequired for an accurate representation of the peaksof the soil tractions at the edges x = �B of the road.
a. Homogeneous halfspace.
b. Layer built in at its base.
c. Layer on a halfspace.Figure 8: Real (left hand side) and imaginary part(right hand side) of the bending term ~Ks(1;1) of thesoil’s impedance as a function of ! and ky for the 3cases considered.
Figure 8a shows the real and the imaginary part ofthe bending term ~Ks(1;1) of the soil’s impedance asa function of the circular frequency ! and the dimen-sionless wavenumber ky for the case of the homoge-neous halfspace. The imaginary part of the impedancerepresents the energy dissipation due to radiation andmaterial damping in the soil and increases for increas-ing frequencies.
The impedance is calculated in the frequency-wavenumber domain after an inverse Fourier transfor-mation of the horizontal wavenumber kx to the hori-
zontal coordinate x perpendicular to the road. In theexpression of the axisymmetric Green’s functions, theradial wavenumber is therefore written as a functionof the horizontal wavenumbers:
k2
r= k
2
x+ k
2
y(36)
with kx = kr cos � and ky = kr sin � where � is theangle of the (plane) wave with the x-axis. The bodywave dispersion relations now become:
k2
x+ k
2
y+ k
2
zp= s
2 (37)
k2
x+ k
2
y+ k
2
zs= 1 (38)
P- and S-waves become inhomogeneous in the verti-cal z-direction if kr � s and kr � 1, respectively.
Considering the dispersion relation for P-wavespropagating in the horizontal (x; y)-plane (kzp=0), kxtends to zero when ky tends to s, corresponding to aP-wave propagating in the y-direction. The real partof the impedance decreases. Considering the disper-sion relation for S-waves propagating in the horizon-tal (x; y)-plane (kzs=0), kx tends to zero when ky
tends to 1, corresponding to a S-wave propagating inthe y-direction.
Whereas in figure 5a the Rayleigh wave only ap-pears at a single wavenumber kr = kR, it now influ-ences the impedance at each horizontal wavenumberky � kR. Although not immediately observable onfigure 8a, these Rayleigh waves propagate in the hori-zontal (x; y)-plane in the direction � = arcsin(ky=kR)with respect to the x-axis. For ky tending to kR, cor-responding to a Rayleigh wave propagating in the y-direction, the real part of the impedance reaches a lo-cal minimum. For larger values of ky, the real partincreases and the imaginary part decreases to a rela-tively low value, determined by material damping.
Figure 8b shows the real and the imaginary partof the soil’s impedance for the layer built in at itsbase. At ky = 0, the real part of the impedance be-comes zero at the natural frequencies of a 2D rigidbeam on top of the layer built in at its base. At fre-quencies lower than the first vertical eigenfrequencyof the layer built in at its base, the imaginary partis small. The same holds for wavenumbers largerthan the Rayleigh wavenumber of a homogeneoushalfspace with the characteristics of the layer. Un-der these conditions, the damping term is mainly de-termined by material damping. For those frequenciesand wavenumbers that correspond to a surface wavepropagating in the y-direction, both the real and theimaginary part exhibit a local minimum.
Figure 8c shows the real and the imaginary partof the soil’s impedance for the case of a layer on ahalfspace. At frequencies lower than the first verticaleigenfrequency of a layer built in at its base, the soil’s
impedance is influenced by the presence of the under-lying halfspace and differs strongly from the case ofthe layer built in at its base. At higher frequencies,the influence of the top layer increases and the soil’simpedance tends to the same value in all cases.
7.5 The road-soil interaction problemThe sum of the road and the soil’s impedance rep-resents the total impedance of the coupled road-soilsystem. The real part is determined by both the roadand the soil’s impedance, while the imaginary part isonly determined by the soil’s impedance, as energy isonly dissipated in the soil.
Since the bending and torsional modes are un-coupled, the modal coordinate ~ucz(ky; !) of thebending modes is equal to the inverse of the totalimpedance for the bending modes. The soil tractions~tsz(x; ky; z = 0; !) at the interface �rs are calculatedas the product of ~ucz(ky; !) and ~tsz(~�s
(1)).
0 10 20 30 40 500
0.5
1
1.5
Frequency [Hz]
k y [ −
]
a. Homogeneous halfspace.
0 10 20 30 40 500
0.5
1
1.5
Frequency [Hz]
k y [ −
]
b. Layer built in at its base.
0 10 20 30 40 500
0.5
1
1.5
Frequency [Hz]
k y [ −
]
c. Layer on a halfspace.Figure 9: Logarithm of the modulus of the product ofthe modal coordinate ~ucz(ky; !) and ! as a functionof ! and ky for the 3 cases considered.
Figure 9a shows, for the case of a homogeneous
halfspace, a contour plot of the logarithm of the mod-ulus of the modal coordinate ~ucz(ky; !), multipliedby the circular frequency !, as a function of ! andky. Large values of the modal coordinate appear atwavenumbers close to the Rayleigh wavenumber kR.These maxima are obtained when the impedance ofthe total system is nearly singular and correspond to awave of the coupled road-soil system, that propagatesat a velocity Crs close to CR.
Figure 9a illustrates that, at a limiting low fre-quency, the wave velocity Crs is equal to CR. Thewavelength in the soil is large and the influence ofthe road becomes negligible. At!rs =C
2R
q12�=E=h,
the velocity Cb of the free bending waves equals CR
and the wave of the coupled system propagates atthe Rayleigh wave velocity. At frequencies lower than!rs, the wave velocity of the coupled road-soil systemis lower thanCR. In this frequency range, the dampingis determined by material damping in the soil. At fre-quencies higher than !rs, the wave velocity is largerthan CR due to the influence of the bending stiffnessof the road; the damping is determined by both mate-rial and radiation damping in the soil.
Figure 9b shows the results for the case of a layerbuilt in at its base. At ky = 0, ~ucz(ky; !) exhibits max-ima at the resonance frequencies for the 2D case of arigid beam on top of the layer built in at its base. Dueto the influence of the road inertia, these resonancefrequencies are slightly lower than the vertical eigen-frequencies of the layer built in at its base. The firstwave of the coupled system emanates at the first ver-tical eigenfrequency of the rigid beam on top of thelayer. At higher frequencies, the wave velocityCrs de-creases due to the influence of the bending wave term.Less dominant maxima correspond to higher road-soilmodes, with a wave velocity close to the values for thesurface wave modes in the soil.
Figure 9c shows the results for the case of a layeron top of a halfspace. At a fixed frequency ! belowthe first vertical eigenfrequency of the rigid beam ontop of the layer, ~ucz(ky; !) reaches a maximum valueat a wavenumber close to the Rayleigh wavenumberof a halfspace with the characteristics of the underly-ing halfspace. In this frequency range, the first wavemode of the coupled system is mainly determinedby the characteristics of the underlying halfspace. Athigher frequencies, an increase of the wave velocitydue to the influence of the top layer is followed by adecrease due to the bending stiffness of the road.
The transfer functions ~hzz(x; ky; z; !) between theroad and the soil are calculated from the soil tractionsat the interface and the Green’s functions of the soilfor receivers located at the surface (z = 0) from x =8m to x = 64m with an increment of 8 m.
Figure 10a shows, for the case of a homoge-neous halfspace, the modulus of the transfer func-
a. Homogeneous halfspace.
b. Layer built in at its base.
c. Layer on a halfspace.Figure 10: Modulus of the product of the transferfunction ~hzz(x = 8; ky; z = 0; !) and ! as a functionof ! and ky for the 3 cases considered.
tion ~hzz(x = 8; ky; z = 0; !), multiplied by the cir-cular frequency !, as a function of ! and ky. At afixed frequency !, the modulus of the transfer func-tion ~hzz(x = 8; ky; z = 0; !) exhibits a maximum atky = kR. For wavenumbers ky � kR, the body andthe surface waves are inhomogeneous and the modu-lus becomes very small. At a fixed wavenumber ky,the modulus decreases for increasing frequencies !,due to material damping in the soil.
Figure 10b shows the results for the case of a layerbuilt in at its base. The modulus of the transfer func-tion exhibits a cut-off frequency at the first verti-cal eigenfrequency of the rigid beam on top of thelayer. At a fixed frequency ! higher than this eigen-frequency, the modulus attains a maximum value at
the Rayleigh wavenumber corresponding to a halfs-pace with the characteristics of the layer. The maxi-mum value decreases for increasing frequencies.
Figure 10c shows the results for the case of a layeron top of a halfspace. Although the transfer functionexhibits no cut-off frequency, it can be noticed thatthe modulus is small at frequencies below the firsteigenfrequency of the beam on top of the layer. Forfrequencies tending to this frequency, the influence ofthe weak top layer increases and, consequently, themodulus increases as well. At higher frequencies, themodulus decreases due to material damping in thesoil. For limiting high frequencies, the modal coordi-nate and the transfer function tend to the same valuein the three cases.
7.6 The free field vibrations
0 20 40 60 80−1
−0.5
0
0.5
1
1.5
2
Distance [m]
Tim
e [s
]
a. Homogeneous halfspace.
0 20 40 60 80−1
−0.5
0
0.5
1
1.5
2
Distance [m]
Tim
e [s
]
b. Layer built in at its base.
0 20 40 60 80−1
−0.5
0
0.5
1
1.5
2
Distance [m]
Tim
e [s
]
c. Layer on a halfspace.Figure 11: Time history of the free field vertical ve-locity as a function of the distance to the road for the3 cases considered.
Figure 11a shows the time history of the free fieldvertical velocity as a function of the distance to thecenter of the road for the case of the homogeneoushalfspace. It can be observed that wave propagation
in the soil delays and attenuates the signals for in-creasing distance to the source. The arrival time in-creases linearly for increasing distances to the road.Figure 11b shows, on the same scale, the results forthe case of the layer built in at its base. The ampli-tude of the ground vibrations is smaller, while theresonance of the beam on top of the layer dominatesthe time history. Figure 11c shows the results for thecase of a layer on a halfspace. The dispersive natureof the wave propagation in the layered halfspace isnoticed when the arrival time is compared for the re-ceivers at different distances from the source. At smalldistances, the arrival times differ more than at largerdistances. This is due to the refracted compressionalwave.
Figure 12a shows the time history and the fre-quency content of the vertical free field velocity at adistance of 8 m to the center of the road for the caseof the homogeneous halfspace. The frequency contentof the ground vibrations is mainly situated below 20Hz and is dominated by the pitch and bounce modes(at approximately 2 Hz) and the axle hop modes (be-tween 9 Hz and 10 Hz) of the vehicle.
−1 0 1 2 3−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Time [s]
Vel
ocity
[m/s
]
8 m
0 10 20 30 40 500
0.5
1
1.5
2x 10
−4
Frequency [Hz]
Vel
ocity
[m/s
/Hz]
8 m
a. Homogeneous halfspace.
−1 0 1 2 3−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Time [s]
Vel
ocity
[m/s
]
8 m
0 10 20 30 40 500
0.5
1
1.5
2x 10
−4
Frequency [Hz]
Vel
ocity
[m/s
/Hz]
8 m
b. Layer built in at its base.
−1 0 1 2 3−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Time [s]
Vel
ocity
[m/s
]
8 m
0 10 20 30 40 500
0.5
1
1.5
2x 10
−4
Frequency [Hz]
Vel
ocity
[m/s
/Hz]
8 m
c. Layer on a halfspace.Figure 12: Time history (left hand side) and frequencycontent (right hand side) for points at a distance of 8m to the center of the road for the 3 cases considered.
Figure 12b shows the vertical free field velocity forthe case of the layer built in at its base. Comparedto the case of the homogeneous halfspace, the peakparticle velocity is much smaller. A cut-off frequency
appears at the first vertical eigenfrequency of the cou-pled system and dominates the frequency content. Thesoil exhibits the characteristics of a high-pass filterand the influence of the eigenfrequencies of the ve-hicle below 10 Hz is no longer observable.
Figure 12c shows the vertical free field velocityfor the case of the layer on a halfspace. At low fre-quencies, the wavelength is large and the velocity islower than in case (a). At frequencies higher than thefirst vertical eigenfrequency of the layer built in at itsbase, the frequency content of the free field velocityin cases (a) and (c) is very similar.
8 CONCLUSIONThe direct stiffness method has been reviewed as atool to compute harmonic and transient wave prop-agation in horizontally layered media. It allows forthe calculation of the Green’s functions, neededin a boundary element formulation to compute theimpedance of the soil. This has been illustrated forthe case of road traffic induced vibrations due to thepassage of a vehicle on an uneven road. The numer-ical results confirm that the dominant frequencies oftraffic induced vibrations are determined by the vehi-cle characteristics, the unevenness profile and, impor-tantly, the soil stratification, as has been confirmed byin situ experiments (Lombaert and Degrande 2001).
Present developments include the dynamic interac-tion of the incident wave field with structures as wellas the development of source models for trains run-ning on surface tracks and in tunnels, but have notbeen treated in the present paper.
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