Experimental validation of a numerical prediction model for free field traffic induced vibrations by in situ experiments

Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

www.elsevier.com/locate/soildyn

Experimental validation of a numerical prediction model for free ®eld traf®c induced vibrations by in situ experiments G. Lombaert*, G. Degrande K.U. Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium Accepted 25 February 2001

Abstract This paper deals with the validation of a numerical model for traf®c induced vibrations. Road unevenness subjects the vehicle to vertical oscillations that cause dynamic axle loads, which generate waves propagating in the subsoil. A 2D vehicle model is used for the calculation of the axle loads from the longitudinal road pro®le. The free ®eld soil response is calculated with the dynamic Betti±Rayleigh reciprocity theorem, using a transfer function between the road and the receiver that accounts for dynamic road±soil interaction. The validation relies on the measured response of the vehicle's axles and the soil during the passage of a truck on an arti®cial unevenness with speeds varying from 30 to 70 km/h. The agreement between the numerical and the experimental results is good: the in¯uence of the vehicle speed and the distance from the road is well predicted, while the ratio of the predicted and the measured PPV is less than two. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Traf®c induced situations; Experimental validation; Soil structure interaction; Dynamic soil characteristics; Road unevenness

1. Introduction Traf®c induced vibrations are a common source of environmental nuisance as they may cause malfunctioning of sensitive equipment, discomfort to people and damage to buildings. They are mainly due to heavy lorries that pass at relatively high speeds on a road with an uneven surface. A lot of experimental research related to traf®c induced vibrations has been performed at the Transport and Road Research Laboratory (TRRL) in Crowthorne, Berkshire, UK. Several surveys were carried out to identify the causes of traf®c induced vibrations and to investigate the relation between the degree of nuisance experienced by residents and their exposure to traf®c noise and vibrations [3,29]. Furthermore, the correlation between noise exposure measures and vibration annoyance was studied. A parametric study concluded that the dynamic properties of the vehicle suspension system in¯uence the vibration levels, which tend to increase with vehicle speed and the elevation of the road surface irregularity [25]. Heavy goods vehicles and buses were found to produce the most perceptible vibrations. Other surveys have investigated the possible link between vibrations and damage to heritage buildings but could not attribute the observed structural damage to traf®c * Corresponding author. Fax: 132-16-32-19-88. E-mail address: [email protected] (G. Lombaert).

induced vibrations [26,27]. Traf®c induced vibrations nevertheless cause nuisance to people, which justi®es the development of numerical prediction tools to determine whether a road surface defect is likely to cause vibration nuisance [28]. Based on experimental observations, an empirical method has been developed to predict the peak particle velocity (PPV) from the maximum height or depth of a surface defect, the vehicle speed and the soil characteristics [28]. However, this method does not clarify the mechanisms involved and cannot be used for an extrapolation to more complex situations. Theoretical models for road traf®c induced vibrations start from the calculation of the dynamic axle loads, which are determined by the vehicle dynamics, the road unevenness and the road ¯exibility. As the road is much stiffer than the vehicle's suspension or tyres [4,10,19], the calculation of the dynamic axle loads is, in a ®rst approximation, uncoupled from the calculation of the soil response. For linear vehicle models, vehicle frequency response functions (FRF) facilitate the calculation of the axle loads [4±6,12] from the road unevenness. The calculation of the response to moving loads is often based on the dynamic reciprocity theorem [22], where it is assumed that the road is invariant in its longitudinal direction. An extensive survey of calculation methods for solids or structures under moving loads is given by FryÂba [9]. Grundmann et al. [11] have recently applied the dynamic

0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(01)00017-3

486

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

reciprocity theorem to the calculation of free ®eld vibrations due to a time-dependent load moving along the surface of a layered halfspace. Dynamic soil±structure interaction between the road and the soil can be accounted for by means of a substructuring technique [1], where an analytical beam model is used for the road, while the soil is modelled by means of boundary elements [2]. This approach has been used by the authors in a numerical prediction model for road traf®c induced vibrations in the free ®eld [15,16]. The objective of the present paper is the experimental validation of this numerical model. First, the essential elements of the model are brie¯y recapitulated. It is shown how the free ®eld vibrations are calculated from the dynamic axle loads and a transfer function between the road and the soil. Second, the experimental setup is discussed. On the test circuit of DAF in Sint-Oedenrode (The Netherlands), both the response of the vehicle and the soil have been measured during the passage of a truck on an arti®cial unevenness at vehicle speeds varying from 30 to 70 km/h. Finally, the experimental results are compared with the numerical predictions. Special emphasis goes to the in¯uence of the vehicle speed on the PPV and the frequency content of the response. 2. The numerical model In the following, the numerical model for the prediction of road traf®c induced vibrations [16] is brie¯y recapitulated. First, it is shown how the dynamic axle loads are computed using simple 2D vehicle models. Next, the source±receiver transfer functions between the road and the soil are derived. Finally, these ingredients are used in the dynamic reciprocity theorem to compute the free ®eld response due to a vehicle moving on a road. 2.1. The dynamic axle loads The dynamic axle loads are calculated with a 2D vehicle model, which is composed of discrete masses, springs and dampers. The vehicle body and the wheel axles are assumed to be rigid inertial elements, while the suspension system and the tyres are represented by a spring±dashpot system. The distribution of n axle loads can be written as the summation of the product of Dirac functions that determine the position of the force and a time-dependent function gk(t): F…x; y; z; t† ˆ

n X kˆ1

d…x†d…y 2 yk 2 vt†d…z†gk …t†

…1†

yk is the initial position of the k-th axle that moves with the vehicle speed v along the y-axis. As the in¯uence of the road displacements on the dynamic axle loads can be neglected, the frequency content g^k …v† of a single axle load is calculated from the contribution of n vehicle axles and the road surface pro®le: g^k …v† ˆ

n X lˆ1

h^ fk ul …v†u^lw=r …v†

…2†

Fig. 1. The road±soil interaction problem.

The FRF h^ fk ul …v† represents the frequency content of the axle load at axle k, when a unit impulse excitation is applied to axle l [4,12]. u^lw=r …v† represents the frequency content of the road unevenness applied at axle l and is calculated from the wavenumber domain representation u~w=r …ky † of the longitudinal road pro®le uw=r …y†:     1 v yl l exp iv …3† u^ w=r …v† ˆ u~ w=r 2 v v v The range of road unevenness which is important for vehicle dynamics is characterized by wavelengths l y ˆ 2p/ky between 0.5 and 50 m. Eq. (3) shows that, for increasing vehicle speed, the quasi-static value of the road pro®le experienced by the vehicle axles decreases, while the frequency content shifts to higher frequencies. 2.2. The road±soil transfer function The road±soil transfer function hzi(x, y, z, t) represents the fundamental solution at a point (x, y, z) of the road or the soil for the displacement component i due to a vertical impulse load on the road. Its calculation requires the solution of two subproblems. First, a dynamic substructure method is used to calculate the tractions at the road±soil interface [2]. Next, the displacements at an arbitrary location are calculated from these soil tractions. The road is assumed to be invariant with respect to the longitudinal direction y and to have a rigid cross section (Fig. 1). The road is supported by the soil along the interface S rs. The ®rst assumption allows a Fourier transformation from the longitudinal coordinate y to the horizontal wavenumber ky to be performed. This results in an ef®cient solution procedure in the frequency±wavenumber domain. From the second assumption it follows that the vertical road displacements u~ rz …x; ky ; v† can be written as a function of the vertical translation u~ cz …ky ; v† of the cross section's

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

centre of gravity and the rotation b~ cy …ky ; v† about this centre: ~ y ; v† …4† u~ rz …x; ky ; v† ˆ u~cz …ky ; v† 1 xb~ cy …ky ; v† ˆ fr …x†a…k The displacement modes of the rigid cross section are ~ y ; v† collected in a vector fr ˆ {1, x} T, while the vector a…k collects the displacement u~cz …ky ; v† and the rotation b~ cy …ky ; v†. The latter can be interpreted as unknown modal coordinates. In the following, it is understood that a tilde above a variable denotes its representation in the frequency±wavenumber domain so that the arguments ky and v can be omitted. The foregoing kinematic assumptions result in the following equilibrium equations for the road, which govern the longitudinal bending and torsional deformations, respectively: 02 @4

EIx ky4

0

0

GCky2

3

"

5 2 v2

rA 0

#1( ) 8 ~e 9 ( ) < f cz = 0 u~ 1 A cz ˆ 1 e ; : ~ rIp b cy xS m~ cy

…5† In these equations, A is the road's cross section, Ix the moment of inertia with respect to x, C the torsional moment of inertia and Ip the polar moment of inertia; E is the Young's modulus, G the shear modulus and r the density of the road. The vertical force per unit length f~ecz and the torsional moment per unit length m~ ecy are the forces exerted by the soil on the road along the interface S rs, while the second vector in the right hand side of this equation represents the contribution of the Dirac load applied in a point (xS, 0, 0) at time t ˆ 0. The contribution of the soil follows from the equilibrium at the road±soil interface S rs. The equilibrium equations can be written in matrix±vector notation: Z ~ s †dGŠa~ ˆ f~ cd ~r1 ~ r 2 v2 M fr t~sz …f …6† ‰K S rs

Ä r the mass matrix of the road Ä r the stiffness matrix, M with K and fÄcd the force vector related to the Dirac load. The integral on the left hand side of Eq. (6) denotes the soil impedance ~ s † is ~ s and is proportional to the width 2B of the road. t~sz …f K the frequency±wavenumber domain representation of tsz …fs †, the vertical component of the soil tractions ts ˆ s sn on a boundary with a unit outward normal n for the scattered wave ®eld fs. A boundary element method is used to calculate the soil tractions t~sz …f~ s † at the road±soil interface in the soil [2,14,16]. The boundary element formulation is based on the formulation of the boundary integral equations in the frequency±wavenumber domain, using the Green's functions of a horizontally layered soil [7,13,18]. The solution of the system of Eq. (6) gives the complex ~ The soil tractions t~sz …u~ s † at the road± participation factors a. soil interface are calculated from the participation factors as ~ s †a. ~ t~sz …f The reciprocity theorem is used for the calculation of the road±soil transfer function h~zi …j 1 ; ky ; j 3 ; v† from the soil tractions at the interface. In the load case considered, only the vertical tractions t~sz …x; ky ; z ˆ 0; v† have a non-zero resultant.

487

When the loaded area is small compared to the wavelength in the soil, it can be assumed that the horizontal tractions have a small in¯uence on the free ®eld displacements, so that: h~ zi …j 1 ; ky ; j 3 ; v† ˆ

Z

S rs

u~G zi …j 1 2 x; ky ; j 3 ; v†t~sz …x; ky ; z ˆ 0; v†dG

…7† u~ ziG …j 1 ; ky ; j 3 ; v†,

representing and only the Green's function the fundamental solution for the displacement component i due to a vertical impulse load, is needed for the calculation of the transfer function. 2.3. The response due to the moving dynamic axle loads The dynamic Betti±Rayleigh reciprocal theorem is used to calculate the road or soil response to the moving axle loads [22]. The frequency content of the displacement component u^ si …x; y; z; v† is found as the inverse Fourier transform of the representation in the frequency±wavenumber domain [2]: 1 Z1 1 ~ h …x; ky ; z; v† u^si …x; y; z; v† ˆ 2p 2 1 zi …8† n X g^k …v 2 ky v†exp‰2iky …y 2 yk †Šdky kˆ1

A change of variables according to ky ˆ …v 2 v~ †=v moves the frequency shift from the axle load to the transfer function: ! n X 1 Z1 1 ~ v 2 v~ ; z; v g^ k …v~ † hzi x; u^si …x; y; z; v† ˆ v 2pv 2 1 kˆ1 # " ! v 2 v~ …y 2 yk † dv~ exp 2 i v …9† and illustrates that traf®c induced vibrations are caused by dynamic vehicle loads that cause wave propagation in the soil. The solution in the time domain is ®nally found as the inverse Fourier transformation from the circular frequency v to the time t. 3. The experimental con®guration At the test circuit of DAF in Sint-Oedenrode (The Netherlands) both the soil response and the vehicle response have been measured during the passage of a truck on an arti®cial unevenness (Fig. 2) with speeds varying from 30 to 70 km/h [17]. 3.1. The vehicle The vehicle is a two-axle DAF FT85 truck with a trailer. The truck has a wheel base of 3.80 m, a leaf-spring suspension at the front axle and an air suspension at the rear axle. The trailer has three axles. The distance from the truck's

488

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

Fig. 2. The passage of the truck on the arti®cial unevenness.

rear axle to the middle axle of the trailer is 6.46 m; the distance between the trailer axles is 1.40 m. The total mass of the vehicle is 39,490 kg. A 2D vehicle model with 13 degrees of freedom (DOF) [21] that accounts for one longitudinal half of the vehicle is

used for the calculation of the dynamic axle loads. The contact between the tyres and the road is modelled as a point contact. In the vehicle model, the three axles of the trailer are modelled as a single axle. In the subsequent calculations of the free ®eld vibrations, the calculated axle loads will be multiplied by a factor 2, in order to account for the full vehicle and the single axle load of the trailer will be split into three equal axle loads that are placed at the correct axle positions. Fig. 3 shows the modulus of the frequency response functions (FRF) h^fk ul …v† of the vehicle model. The functions h^ fk uk …v† represent the frequency content of the axle load at axle k for an impulse excitation at the same axle. They have a larger amplitude than the FRF h^ fluk …v† for an excitation at a different axle l. Furthermore, it follows from the reciprocity theorem that h^fk ul …v† ˆ h^ fk uk …v†. Two groups of eigenmodes dominate the frequency content: the pitch and bounce modes (between 1 and 3 Hz) and the axle hop modes (between 8 and 12 Hz). For limiting high frequencies, the FRF h^ fk uk …v† tend to kt 1 ivct where kt and ct are the tyre stiffness and damping coef®cient, respectively, while the FRF h^fk ul …v† tend to zero.

Fig. 3. Modulus of the FRF h^ fk ul …v† for the 2D 13 DOF vehicle model of the truck and the trailer.

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

489

Fig. 5. The longitudinal pro®le of the arti®cial unevenness (a) as a function of the coordinate y along the road and (b) in the wavenumber domain.

clearly be in¯uenced by the vehicle speed in the frequency range of the axle hop modes (between 8 and 12 Hz). This conclusion is in agreement with experimental observations [25]. 3.3. The characteristics of the road Fig. 4. The arti®cial unevenness.

3.2. The arti®cial unevenness The arti®cial unevenness is composed of two three-ply boards and has a total height of 54 mm, a width of 1 m and a total length of 1.70 m (Fig. 4). It is installed on a road with a smooth asphalt surface, using one three-ply board for each wheel path. Fig. 5a shows the longitudinal pro®le uw=r …y† of the arti®cial unevenness when the bottom plate of the pro®le is assumed to have sloping ends, resulting in a base length of 1.90 m and a mean length of 1.60 m. Fig. 5b shows the representation u~w=r …ky † of the road pro®le in the wavenumber domain. The separation between the lobes is inversely proportional to the mean length of the pro®le. The latter has been chosen to control the frequency content of the unevenness experienced by the vehicle axles at varying vehicle speeds, as demonstrated in Fig. 6. As expected from Eq. (3), the quasi-static value of the spectrum decreases for increasing vehicle speed, while the frequency content shifts to higher frequencies. As the frequency content of the axle loads is, according to Eq. (2), obtained as the product of u^ lw=r …v† (Fig. 6) and the vehicle FRF h^ fk ul …v† (Fig. 3), it will

The road has a width 2B ˆ 3.50 m and is composed of an asphalt top layer and a granular subbase (Fig. 7). The bending stiffness of both layers is equivalent to the stiffness of a single asphalt layer with a thickness h ˆ 0.15 m [20]. In the following, a Young's modulus E ˆ 10,000 £ 10 6 N/m 2, corresponding to a mean value for a bituminous top layer, is assumed. The Poisson's ratio n ˆ 0.3 and the density r ˆ 2100 kg/m 3. 3.4. The dynamic soil characteristics The soil at the DAF test site primarily consists of ®ne sand, with a weak presence of loam. A spectral analysis of surface waves (SASW) has been performed to determine the layering and the dynamic soil characteristics of the site [8]. A transient excitation is generated by dropping a mass of 110 kg from a height of 0.9 m on a square (0.7 £ 0.7 m) steel foundation with a mass of 600 kg (Fig. 8). A dashpot is used to control the frequency content of the loading and to prevent rebound of the mass. The vertical response is measured at 10 points at the surface at a distance from 2 to 48 m from the source. The inversion procedure minimizes the difference between the experimental and theoretical dispersion curve of the ®rst surface wave, as illustrated in Fig. 9. The SASW

Fig. 6. The unevenness experienced by the vehicle at a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

490

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

Fig. 8. The dropping weight device and the measurement line.

Fig. 7. The asphalt road at the test circuit.

reveals the presence of a weak top layer with thickness d ˆ 0.47 m and a shear wave velocity Cs ˆ 75.0 m/s and a layer (d ˆ 0.64 m, Cs ˆ 186.0 m/s) on top of a halfspace (Cs ˆ 241.0 m/s). In the past, cone penetration tests have been performed at the same site [24] to determine the soil layering and to estimate the dynamic Young's modulus. These tests have also shown the presence of a weak top layer with a thickness varying from 0 to 2 m and a Young's modulus E between 38 £ 10 6 N/m 2 and 152 £ 106 N/m 2, on top of a stiffer layer with a thickness varying from 2 to 5 m and a Young's modulus E between 76 £ 10 6 N/m 2 and 304 £ 10 6 N/m 2. The soil density was found to vary from 1594 to 1710 kg/m 3. As the road is not located at the surface of the weak top layer, the in¯uence of the latter is discarded in the following. A mean value of the soil density r ˆ 1650 kg/m 3 and a Poisson's ratio n ˆ 1/3 are assumed to calculate the dynamic Young's modulus E from the SASW results. The hysteretic material damping ratio b of the soil has a large in¯uence on the response at distances from the source that are large with respect to the dominant wavelength. However, it has not been included in the inversion process for the determination of the soil characteristics from the SASW data. The material damping has been estimated as 0.050 for the ®rst layer and 0.025 for the halfspace. This corresponds to the experimental observation that material damping decreases with depth. Table 1 summarizes the layer thickness d, the Poisson's ratio n , the density r , the Young's modulus E and the hysteretic material damping ratio b in both shear and volumetric deformation for the soil layers, as they will be used in the numerical model for the computation of the Green's function. The assumed soil layering and the Young's moduli correspond well with estimations from cone penetration tests [24].

4. Experimental results The response of the truck's axles and the soil have been measured during the passage of the truck on the arti®cial pro®le at ®ve speeds varying from 30 to 70 km/h, with an increment of 10 km/h. Two passages have been recorded for each vehicle speed. Close inspection of the signals has revealed good reproducibility of the measurements, so that the following discussion can be limited to a single passage for each vehicle speed. 4.1. The response of the truck's axles Accelerations of both the front and rear axle of the truck at the left and right hand sides have been measured during the passage on the arti®cial pro®le. Fig. 10a shows the time history at the left and right hand sides of the rear axle of the truck during a passage at a speed v ˆ 30 km/h. At t ˆ 0, the front axle of the truck is located at the centre of the arti®cial unevenness. The impact during the ascending and the descending of the unevenness can clearly be observed. Similar results are shown in Fig. 10b and c for a vehicle speed v ˆ 50 and 70 km/h, respectively. At these higher

Table 1 Soil pro®le Layer

d (m)

n [±]

r (kg/m 3)

E ( £ 10 6 N/m 2)

b [±]

1 2

0.64 1

1/3 1/3

1650 1650

152 256

0.050 0.025

Fig. 9. The experimental (solid line) and theoretical (dashed line) dispersion curve.

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

491

Fig. 10. Time history of the vertical acceleration at the left (solid line) and right (dash-dotted line) hand side of the truck's rear axle during the passage on the pro®le at a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

vehicle speeds, the impacts during the ascending and descending can no longer be distinguished. As the longitudinal unevenness pro®les at the left and right wheel paths are the same, the accelerations at both sides of the wheel axle are very similar and the in¯uence of vehicle rolling on the dynamic axle loads is expected to be small. 4.2. The response of the soil The measurement line (Fig. 11) is situated perpendicular to the road. For the de®nition of the measurement directions, a right-handed Cartesian frame of reference is chosen, with its origin in the middle of the three-ply boards, at the centre of the asphalt road. The y-axis is parallel to the road, the xaxis is perpendicular to the road and the z-axis is pointing upwards. Vertical accelerations have been measured at nine locations with PCB seismic piezoelectric accelerometers. The ®rst three accelerometers are located at the centres of the roads adjacent to the asphalt road (Fig. 7), at distances of 5.35, 9.05 and 13.15 m from the centre. The accelerometers are mounted on stiff steel plates with three legs. In the free ®eld, the vertical acceleration is measured at 16, 20, 24, 32, 40 and 48 m. The accelerometers are mounted on steel or aluminium stakes with a cruciform cross section to minimize dynamic soil±structure interaction. Additionally, the horizontal accelerations in the x-

direction perpendicular to the road have been measured at four locations at 16, 24, 32 and 40 m from the centre of the asphalt road. These accelerometers are mounted on the same stakes as used for the measurement of the vertical component. At 16 and 40 m, the signals are disturbed at both low and high frequencies, which is probably due to an inadequate ®xation of the stake in the soil. The measurement of the vertical acceleration is not affected, however. As all accelerometers are located on the x-axis, it is assumed that surface waves travelling in the (x, z) plane will dominate the response and that the horizontal acceleration in the y-direction will be small. This is con®rmed by experiments performed by Taniguchi and Sawada [23]. The horizontal acceleration in the y-direction has, therefore, not been measured. A Kemo VBF 35 system is used as a power supply, ampli®er and anti-aliasing ®lter with a low-pass frequency at 125 Hz. The A/D conversion is performed using a 16-bit Daqbook 216 data acquisition system at a sample rate fs ˆ 500 Hz. 4096 data points, corresponding to a period T ˆ 8.192 s, have been recorded for each passage. The resolution in the frequency domain equals Df ˆ 0.122 Hz. Fig. 12a shows the time history of the vertical velocity as a function of the distance from the centre of the road during the passage of a truck with a speed of 30 km/h, as obtained after integration of the measured accelerations. The peak particle velocity (PPV) at the closest receiver equals

Fig. 11. Location of the measurement points and directions.

492

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

Fig. 12. Time history of the vertical velocity at all locations during the passage of the truck on the pro®le at a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h. All results are shown on the same scale. The PPV at the closest receiver equals 0.9, 1.9 and 1.7 mm/s, respectively.

0.9 mm/s. Accounting for the total distance of 11.66 m between the front axle of the truck and the rear axle of the trailer, as well as the length of the pro®le (1.70 m), a transient signal with a duration of approximately 1.60 s is expected. It can be observed that wave propagation in the soil delays and attenuates the time signals for increasing distance to the source. Fig. 12b and c show similar results on the same scale for a vehicle speed of 50 and 70 km/h. The PPV at the closest receiver equals 1.9 and 1.7 mm/s, respectively. Comparison of the results for both vehicle speeds demonstrates that the duration of the transient signal is much shorter for the highest vehicle speed, while the PPV increases, although not monotonously. Fig. 13 shows, for each vehicle speed, the PPV as a function of the distance to the centre of the road and reveals a large difference of the PPV for the vibrations generated by the truck at 30 km/h, compared to the higher vehicle speeds. It can also be observed that the PPV does not decrease monotonously with increasing distance to the source. This is explained by the fact that the ®rst three measurement points are located on the roads adjacent to the asphalt road. Moreover, the layering of the soil is expected to in¯uence the decrease of PPV for increasing distance.

5. Comparison of experimental and numerical results 5.1. The axle response of the truck Figs. 14 and 15 compare the measured and predicted time history and frequency content of the acceleration measured at the left and right hand sides of the rear axle of the truck for vehicle speeds v equal to 30, 50 and 70 km/h. The peak acceleration is overestimated, while the predicted signal is attenuated more rapidly. The frequency content is dominated by the axle hop modes. As expected from Eq. (3), the frequency content of the pro®le experienced by the vehicle axles shifts to higher frequencies for increasing vehicle speed. The width of the lobes in the spectra is inversely proportional to the time duration of the signal applied to the axles and increases for increasing vehicle speed. Besides a slight overestimation of the predicted values, the agreement between the predicted and the measured accelerations is good. The axle loads of the truck have been measured by DAF in an independent laboratory experiment, where the measured axle accelerations were imposed to the truck. Figs. 16 and 17 compare the predicted and measured time history and frequency content of the dynamic axle load at the truck's rear axle for vehicle speeds v equal to 30, 50 and 70 km/h. The predicted peak axle load is larger for the higher vehicle speeds, while the duration of the transient signal is shorter. The spectrum of the dynamic axle loads is dominated by the pitch and bounce and the axle hop modes and shifts to higher frequencies for increasing vehicle speeds. Fig. 17 reveals that dynamic axle loads are overestimated at both low and high frequencies. 5.2. The response of the soil

Fig. 13. Measured PPV as a function of the distance to the centre of the road.

As both vehicle axles are located symmetrically with respect to the axis of the road, the axle loads are applied at the road's centre of gravity at xS ˆ 0. the bending and torsional deformation modes are uncoupled in this case and ~ s is the participation factor b~ cy ˆ 0. The soil impedance K calculated with 40 boundary elements of equal length le ˆ 0.0875 m. The size of the boundary elements is small

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

493

Fig. 14. Time history of the predicted (solid line) acceleration of the rear axle of the truck and the measured acceleration at the left (dash-dotted line) and the right hand side (dotted line) for a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

Fig. 15. Frequency content of the predicted (solid line) acceleration of the rear axle of the truck and the measured acceleration at the left (dash-dotted line) and the right hand side (dotted line) for a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

Fig. 16. Time history of the predicted (solid line) dynamic axle load at the rear axle of the truck and the measured load at the left (dash-dotted line) and the right hand side (dotted line) for a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

Fig. 17. Frequency content of the predicted (solid line) dynamic axle load at the rear axle of the truck and the measured load at the left (dash-dotted line) and the right hand side (dotted line) for a speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

494

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

Fig. 18. Time history of the predicted (solid line) and measured (dash-dotted line) vertical velocity at 16, 24, 32 and 48 m for a vehicle speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

compared to the minimum wavelength at f ˆ 50 Hz, but is required for an accurate representation of the peaks of the soil tractions at the edges of the road at x ˆ ^B. The computed axle load of the trailer is divided by three and located at the correct axle positions for the calculation of the free ®eld vibrations. Fig. 18 shows the measured and predicted time history of the vertical free ®eld vertical velocity at 16, 24, 32 and 48 m from the centre of the road. The duration of the transient time domain signal is inversely proportional to the vehicle speed. The PPV generally increases for increasing vehicle speeds. Wave propagation in the soil delays and attenuates the time signals for increasing distance to the source. The attenuation with distance and the time delay are well predicted, whereas the PPV is overestimated in the calculations. The in¯uence of the vehicle speed is also well predicted. Fig. 19 shows the measured and predicted frequency content of the vertical free ®eld velocities at 16, 24, 32 and 48 m. The spectrum shifts to higher frequencies for increasing vehicle speeds in a similar way as the frequency content of the axle loads (Fig. 17). The dominant frequency does not change signi®cantly and is mainly determined by the axle hop modes of the vehicle (Fig. 3). The agreement between the measured and the predicted signals is good. At low frequencies, the response is slightly underestimated, while the response is overestimated at high frequencies. The latter is partly due to the overestimation of the dynamic axle loads at high frequencies (Fig. 17). The overestimation

is more pronounced at 16 m, however, and gradually decreases for the receivers at larger distances. This is due to an incomplete knowledge of the material damping of the soil, as the latter affects signi®cantly the response at distances from the source that are large with respect to the dominant wavelength. Fig. 20 shows the measured and predicted time history of the horizontal free ®eld vertical velocity (x-direction) at 24 and 32 m from the centre of the road. The measured horizontal accelerations at 16 and 40 m are disturbed at both low and high frequencies and are, therefore, not presented here. The horizontal velocities in the x-direction have the same order of magnitude as the vertical components. This was also observed by Taniguchi and Sawada [23], who concluded that the Rayleigh wave generated by the vertical component of the interaction force dominates the ground vibrations induced by road traf®c. The quality of the prediction of the horizontal component perpendicular to the road and the vertical component is similar. This con®rms the theoretical assumption that only the vertical component of the tractions at the road±soil interface can be taken into account, provided that the wavelength in the soil is large compared to the width of the road. Fig. 21 shows the measured and predicted frequency content of the horizontal free ®eld velocity (x-direction) at 24 and 48 m. The vehicle speed has an in¯uence on the horizontal component analogous to that on the vertical component. The predicted signals underestimate the frequency content of the response at low frequencies,

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

495

Fig. 19. Frequency content of the predicted (solid line) and measured (dash-dotted line) vertical velocity at 16, 24, 32 and 48 m for a vehicle speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

Fig. 20. Time history of the predicted (solid line) and measured (dashdotted line) horizontal velocity (x-direction) at 24 and 32 m for a vehicle speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

Fig. 21. Frequency content of the predicted (solid line) and measured (dashdotted line) horizontal velocity (x-direction) at 24 and 32 m for a vehicle speed (a) v ˆ 30 km/h, (b) v ˆ 50 km/h and (c) v ˆ 70 km/h.

496

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

overall trends are well predicted by the model and the PPV is overestimated by a factor less than two. The results demonstrate that the free ®eld vibration levels depend on the vehicle speed, the shape of the unevenness, the vehicle characteristics and the soil characteristics. The validation shows that the numerical model can be employed to predict the incident wave ®eld during the passage of a vehicle on an uneven road with reasonable accuracy. In a subsequent step, this incident wave ®eld will be used to calculate the response of nearby structures, using a dynamic soil±structure interaction approach. The model will be used to predict vibration levels in newly built situations as well as to estimate the ef®ciency of vibration-isolating measures. Fig. 22. Predicted PPV as a function of the distance to the centre of the road.

Acknowledgements whereas the response is slightly overestimated at high frequencies. Similar comments made for the vertical components are also valid here. Fig. 22 shows, for each vehicle speed, the vertical PPV as a function of the distance to the road. Compared to Fig. 13, the PPV increases progressively with increasing vehicle speed, although the computed PPV at 50, 60 and 70 km/h do not differ much. The predicted PPV at 5, 9 and 13 m are much higher than the measured PPV, as the roads on which these receivers are located are not included in the numerical model. The vibration levels at 16, 20 and 48 m and at vehicle speeds higher than 30 km/h are only slightly overestimated, while the vibration levels at 32 and 40 m are overestimated by a factor of almost two. 6. Conclusions In this paper, a numerical model for the prediction of free ®eld traf®c induced vibrations has been validated. Free ®eld vibrations are calculated in a two-stage approach, where the calculation of the vehicle response is uncoupled from the solution of the road±soil interaction problem. First, the dynamic axle loads are calculated from a simple 2D vehicle model and the longitudinal road pro®le. Second, the road±soil interaction problem is solved and the Betti±Rayleigh reciprocal theorem is used for the calculation of the soil response to the moving dynamic axle loads. At the test circuit of DAF, simultaneous free ®eld and vehicle response measurements have been made during the passage of a truck on an arti®cial unevenness at vehicle speeds varying from 30 to 70 km/h. These measurements allow the prediction accuracy of the vehicle and free ®eld response to be checked. The frequency content of the axle loads is overestimated at high frequencies, which also affects the predicted free ®eld response. However, as the overestimation of the response at higher frequencies gradually decreases for an increasing distance to the source, the soil characteristics also have an important in¯uence. The

The results presented in this paper have been obtained within the frame of the research project MD/01/040 `The study of determining factors for traf®c induced vibrations in buildings'. The support of the Prime Minister's Services of the Belgian Federal Of®ce for Scienti®c, Technical and Cultural Affairs is gratefully acknowledged. The collaboration of the truck manufacturer DAF in both the experiments and the numerical modelling is kindly acknowledged. References [1] Aubry D, Clouteau D. A subdomain approach to dynamic soil±structure interaction. In: Davidovici V, Clough RW, editors. Recent advances in earthquake engineering and structural dynamics. Nantes: Ouest Editions/AFPS, 1992. p. 251±72. [2] Aubry D, Clouteau D, Bonnet G. Modelling of wave propagation due to ®xed or mobile dynamic sources. In: Chouw N, Schmid G, editors. Workshop Wave '94, Wave Propagation and Reduction of Vibrations. RuÈhr University, Bochum, Germany, December 1994, p. 109±21. [3] Baughan CJ, Martin DJ. Vibration nuisance from road traf®c at fourteen residential sites. Laboratory report 1020, Transport and Road Research Laboratory, 1981. [4] Cebon D. Interaction between heavy vehicles and roads. Sp-951. L. Ray Buckendale Lecture, SAE, 1993. [5] Clauwaert C. Twee rekenmethoden voor een eenassig voertuig ter bepaling van de dynamische krachten op een wegpro®el en hun experimentele veri®catie. Technical Report RV24/84, Opzoekingscentrum voor de Wegenbouw, 1984. [6] Courage WMG. Bronmodel wegverkeer. Technical Report 93-CONR0056-04. Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek, September 1993. [7] de Barros FCP, Luco JE. Moving Green's functions for a layered visco-elastic halfspace. Technical report, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA, May 1992. [8] Dewulf W, Lombaert G, Degrande G. Bepaling van de dynamische grondkarakteristieken met behulp van de SASW methode op de DAF proefbaan te Sint-Oedenrode. Internal report BWM-2000-02. Department of Civil Engineering, Katholieke Universiteit Leuven, February 2000. DWTC Programma Duurzame Mobiliteit, Project MD/01/040. [9] FryÂba L. Vibration of solids and structures under moving loads. 3rd ed. London: Thomas Telford, 1999. [10] Gillespie TD, Karamihas SM, Sayers MW, Nasim MA, Hansen W, Ehsan N, Cebon D. Effects of heavy-vehicle characteristics on

G. Lombaert, G. Degrande / Soil Dynamics and Earthquake Engineering 21 (2001) 485±497

[11] [12] [13] [14]

[15]

[16] [17]

[18] [19]

pavement response and performance. Technical Report 353, NCHRP, Transportation Research Board, Washington, DC, 1993. Grundmann H, Lieb M, Trommer E. The response of a layered halfspace to traf®c loads moving along its surface. Archive of Applied Mechanics 1999;69:55±67. Hunt HEM. Modelling of road vehicles for calculation of traf®cinduced ground vibrations. Journal of Sound and Vibration 1991;144(1):41±51. Kausel E, RoeÈsset JM. Stiffness matrices for layered soils. Bulletin of the Seismological Society of America 1981;71(6):1743±61. Lombaert G, Degrande G. Study of determining factors for traf®c induced vibrations in buildings. Second biannual report BWM1999-04, Department of Civil Engineering, Katholieke Universiteit Leuven, July 1999. DWTC Research Programme Sustainable Mobility, Research Project MD/01/040. Lombaert G, Degrande G, Clouteau D. Deterministic and stochastic modelling of free ®eld traf®c induced vibrations. In: Pereira P, Miranda V, editors. International Symposium on the Environmental Impact of Road Pavement Unevenness, Portugal, March 1999, p. 163±76. Lombaert G, Degrande G, Clouteau D. Numerical modelling of free ®eld traf®c induced vibrations. Soil Dynamics and Earthquake Eng 2000;19(7):473±88. Lombaert G, Teughels A, Degrande G. Trillingsmetingen in het vrije veld op de Daf proefbaan te Sint-Oedenrode. Internal report BWM2000-01, Department of Civil Engineering, Katholieke Universiteit Leuven, January 2000. DWTC Programma Duurzame Mobiliteit, Project MD/01/040. Luco JE, Apsel RJ. On the Green's functions for a layered half-space. Part I. Bulletin of the Seismological Society of America 1983;4:909±29. Mamlouk MS. General outlook of pavement and vehicle dynamics.

[20]

[21] [22] [23] [24]

[25] [26] [27] [28] [29]

497

Journal of Transportation Engineering, Proceedings of the ASCE 1997;123(6):515±7. N.N. Dimensionering van de verhardingsconstructie van de parallelweg op de proefbaan te Sint-Oedenrode voor Van Doorne's Bedrijfswagenfabriek DAF. Rapport w/79/04/18/0021/OKWZ, Wegmeetdienst Regionale Laboratoria, 1979. N.N. Vehicle model for a DAF FT95 truck and trailer. Private communication, DAF, 1999. Payton RG. An application of the dynamic Betti±Rayleigh reciprocal theorem to moving point loads in elastic media. Quarterly of Applied Mathematics 1964;21(4):299±313. Taniguchi E, Sawada K. Attenuation with distance of traf®c-induced vibrations. Soils and Foundations 1979;19(2):15±28. Van Bree MHP. Het bepalen van verdichtingsgraden door middel van C.M.C.-metingen op de proefbaan DAF voor Van Doorne's Automobielfabrieken. Analyse Z7906060244, Zuidelijk Wegenbouw Laboratorium, 1979. Watts GR. Traf®c-induced ground-borne vibrations in dwellings. Research report 102, Transport and Road Research Laboratory, 1987. Watts GR. Case studies of the effects of traf®c-induced vibrations on heritage buildings. Research report 156, Transport and Road Research Laboratory, 1988. Watts GR. The effects of traf®c-induced vibrations on heritage buildings, further case studies. Research report 207, Transport and Road Research Laboratory, 1989. Watts GR. Traf®c induced vibrations in buildings. Research report 246, Transport and Road Research Laboratory, 1990. Watts GR. Vibration nuisance from road traf®c, results of a 50 site survey. Laboratory report 1119, Transport and Road Research Laboratory, 1984.