Nondestructive Dynamic Testing of Soils and Pavements

Tamkang Journal of Science and Engineering, vol.1, No. 2, pp. 61-81 (1998) 61

Nondestructive Dynamic Testing of Soils and Pavements Jose M. Roesset

Director, Offshore Technology Research Center Department of Civil Engineering Texas A&M University, U.S.A.

Abstract

Non destructive testing techniques based on the application of dynamic loads on the surface of a soil deposit or a pavement system, and the measurement of the resulting deflection basins or the phase difference between the motions recorded at various receivers have become powerful tools in civil engineering. In many cases, however, the interpretation of the recorded data relies on static analyses, ignoring entirely dynamic needed for a correct dynamic interpretation using wave propagation theory propagation in an elastic half space, a homogeneous layer of finite thickness resting on a rigid base, and a horizontally layered medium in general are presented with special emphasis on the interpretation of the data collected in the Falling Weight Deflectometer ( FWD ) and the Spectral Analysis of Surface Wave ( SASW ) tests.

1 Introduction

The effects of the soil on the characteristics of earthquake motions (soil amplification) and on the seismic response of structures (soil structure interaction) were problems of great interest in the 1960’s and 70’s. Through a considerable amount of research a number of different formulations were developed and proposed to solve these problems with varying degrees of sophistication. Some of these formulations were based on continuous wave propagation theory, while others relied on discrete models using finite differences, finite elements or boundary elements. The former provide rigorous solutions for cases with relatively simple geometry, such as a homogeneous elastic half-space. These solutions are directly applicable to a number of practical cases, can be used as first estimates for preliminary design purposes in other situations, and always serve as benchmarks to evaluate the accuracy of numerical procedures. The latter allow one to consider more general, arbitrary geometries and nonlinear behavior but are subject to discretization errors which must be controlled by appropriately refining the meshes and increasing the number of degrees of freedom. Clearly a combination of both types of approaches is needed to make progress in the research effort and to solve actual problems with a good degree of confidence. The increase in the accuracy of the analytical and computational solutions developed through

this research have led to a point where our ability to solve the mathematical model far exceeds our knowledge of the significant parameters which must be used as input, such as the characteristics and wave content of the expected earthquake motions on one hand and the appropriate soil properties and their variation with the state of stresses on the other. Seismologists are working actively on the reduction of the uncertainties involved in the selection of the design earthquake. Engineers are working on the development of more reliable techniques to determine the soil properties in the field in a fast and economical way. At the same time that these developments were taking place, it also became clear that while we now have the ability to design new buildings to reasonably withstand a credible earthquake, there is a very large inventory of structures that were designed and built before most of the present knowledge and the ensuing code provisions were available. It is thus necessary to assess reliably and economically the actual condition and capacity of these structures as built much as it is necessary, after a strong earthquake, to determine the condition of buildings which are standing but which may have suffered some degree of damage. More importantly, and independently of the earthquake problem, the realization that our civil infrastructure is aging, and in some cases badly deteriorating, has pointed out the need to assess the condition of highway pavements, bridges,

Tamkang Journal of Science and Engineering, vol.1, No. 2 (1998) 62

pipelines, industrial plants, buildings and their foundations, and to do so in an efficient way. A number of different nondestructive testing techniques have been developed over the last years in response to these needs. Their development is often the result of a great deal of intuition and imagination and relies on relatively simple physical principles. When the behavior of the problem is controlled by a small number of variables, the methods work well and the interpretation of the data tends to be straightforward. There are cases, however, where the situation is more complicated and a larger number of variables affect the results. The methods may then provide erroneous, or at least unreliable, predictions unless more accurate analytical models are used in the processing of the data. It is thus necessary to conduct research to define the range of applicability of each method, the conditions under which simple, fast, and economical procedures can be used to backfigure the desired system properties, and the cases where more elaborate solutions are required. The discussion in this paper will concentrate on nondestructive testing techniques based on the propagation of stress waves due to the application of dynamic loads. The loads may be of a transient nature (short duration impulses) or a steady state harmonic excitation. The quantities of interest may be the amplitudes of the displacements, velocities, or accelerations, at various points and their variation with time or frequency (spectral analysis), first times of arrival of the waves at a point, interarrival times between two or more receivers, or the phase differences between the motions recorded at two points as a function of frequency. The models used to simulate these tests, to find the relation between the system properties and the recorded quantities, and to evaluate the inversion procedures, are basically the same ones developed in the 60’s and 70’s for the study of soil amplification and soil structure interaction problems, but the accuracy requirements tend to be much stricter and there is a stronger need for a solid understanding of the basic physical phenomena involved and for the availability of rigorous benchmark solutions. Two specific applications will be considered: the determination of the elastic moduli of pavement systems, and the in situ determination of soil properties and their variation with depth. 1.1 Nondestructive Testing of Pavements

Dynamic nondestructive testing techniques have been extensively used for years to evaluate the structural capacity and integrity of highway

and airfield pavements. These techniques can be grouped into two general categories: 1. deflection basin tests and 2. wave propagation tests. Deflection basin tests are those in which the deflections are recorded along the surface of a pavement subjected to a steady state harmonic load or a transient dynamic impact. Typical of this group are the Dynaflect and Road Rater tests (steady state loads) and the Falling Weight Deflectometer test (impact load). The second group, wave propagation tests, is constituted by the nondestructive techniques originally developed to determine soil properties in situ, which can be equally applied to pavements. Characteristic of these methods is the Spectral Analysis of Surface Waves (SASW) procedure which will be discussed in the next section. Figure 1 shows schematically the typical configuration for any test. The number of receivers and the position and characteristics of

the applied loads will vary depending on the specific method. Among the deflection basin tests, the Falling Weight Deflectometer (FWD) has seen the most widespread use, in large part because of its ability to impose dynamic loads similar to those induced by truck traffic. The FWD (Fig. 2) consists of a drop weight mounted on a vertical shaft and housed in a trailer that can be towed by most conventional vehicles. The drop weight is hydraulically lifted to predetermined heights ranging from 5 to 50 cm (2 to 20 inches). The weight is dropped on a 30 cm (11.8 inches) diameter loading plate resting on a 5.6 mm (0.22 inches) thick rubber pad. The resulting load is a force impulse with a duration of approximately 30 msec and a peak magnitude ranging from about

Jose M. Roesset: Nondestructive Dynamic Testing of Soils and Pavements 63

9,000 to 90,000N (2,000 to 20,000 lbs.) depending on the drop weight, the drop height and the pavement stiffness. The force and deflection records at various points along the surface are measured by a load cell and a set of vertical velocity transducers. These transducers are placed typically under the load and at distances of 30 cm (1 ft). There are often 7 receivers, the last one at 1.8 m (6 ft) from the load. The actual number and position of the receivers depends, however, on the model or brand (manufacturer) of the device, as does the variation of the applied load with time. The diagram shown in Fig. 2 is a typical load history. When trying to conduct dynamic analyses to backfigure the elastic moduli of the pavement layers from measured data, it is necessary to know and use the actual time histories of the applied load and the measured displacements. For parametric studies of a more generic nature, intended to determine the characteristics, potential and limitations of the procedure (as described later in this paper) it is customary to use simplified load histories corresponding to a half sine or a triangular pulse with typical durations of 30 to 32 msec. Although most makes of falling weight deflectometers can record the time histories of the vertical displacements (or actually velocities) at the different stations, at least for some duration, often only the peak values are maintained and used to backfigure the elastic moduli of the layers. The peak values are used to define a deflection basin which is often assumed to be caused by a static load, although these values do not occur in fact at the same time (there is a small time lag between the peaks at the various receivers). These static analyses neglect dynamic (inertia) effects entirely,

which can be important in some cases. Moreover, it is often assumed that the subgrade is an elastic half-space extending to infinity, neglecting the possibility of having much stiffer rock at some finite depth. The implications of these simplifying assumptions will be further discussed and illustrated later, performing more accurate and realistic dynamic analyses to simulate the actual test conditions. 1.2 In Situ Determination of Soil Properties The downhole and crosshole methods have been extensively used to determine soil properties in the field from the times of first arrival or the interarrival times of body waves. While these methods, and particularly the crosshole technique, are considered highly reliable, they are expensive and time consuming, requiring the drilling of boreholes. An efficient and more economical alternative is provided by the analysis of surface waves. In the original form of this method, known as the steady Rayleigh wave technique, a vibrator acting vertically on the surface of the soil produced a harmonic excitation. A vertically oriented sensor was moved away from the source until the recorded motion was in phase with the excitation. The distance between any two of these successive positions, or between the source and the first one, was assumed to correspond to one wavelength of a Rayleigh wave. By repeating the process for different excitation frequencies, a plot of propagation velocity (product of the frequency by the wavelength) versus frequency (or wavelength) could be obtained. Such a plot is known as a dispersion curve. To estimate the soil properties from the experimental dispersion curve it was finally assumed that the measured propagation velocity from one test corresponded to the shear wave velocity of the material located at a depth of half a wavelength. This last step was later modified to account for the relationship between the shear wave velocity and the Rayleigh wave velocity for an elastic half-space and to consider that the velocity corresponded to the properties of the material at a depth of one third of the wavelength. In the new form of the method, known as the Spectral Analysis of Surface Waves (SASW) technique, an impulsive load is applied on the surface of the soil deposit. A variety of sources can be used to generate the impact, from hand held hammers of different sizes (small hammers are sufficient for high frequency excitation) to drop weights (heavier weights for low frequencies). The passage of the wave train generated by the


impact is monitored by two vertical receivers. In the typical arrangement the distance between receivers is equal to the distance from the source to the first receiver (Fig. 3). The electrical signals recorded by the receivers are digitized and transformed to the frequency domain by a dynamic spectral analyzer using a Fast Fourier Transform algorithm. The analyzer also automatically provides the cross spectrum from which the phase difference between the two signals can be obtained as a function of frequency. The interarrival time and the phase velocity can then be easily computed. For a given arrangement of source and receivers, the test thus provides the dispersion curve over a certain range of frequencies. The complete dispersion curve is obtained by repeating the process for different distances.

The determination of the soil properties and their variation with depth from the experimental dispersion curve is based on the concept that for very high frequencies (very short wavelengths) the apparent velocity of propagation (phase velocity) will be the Rayleigh wave velocity of the material very near the surface. As the wavelength increases (the frequency decreasing) the phase velocity will be affected by the properties of the materials over a depth of one wavelength or so. One can thus try to first compute the properties of the top surficial layer to match the velocity obtained for the highest frequency, then proceed down the profile, finding the properties of the underlying layers to match the

data for smaller frequencies. Using this procedure, or the one proposed earlier for the steady state Rayleigh wave technique, one can obtain a first estimate of the soil properties. Starting with this assumed soil profile a direct analysis is performed to theoretically predict the dispersion curve. One can then compare the analytical dispersion curve to the experimental one, modify the soil properties, based on experience or on an automated least squares minimization algorithm with a gradient search, and repeat the process until a satisfactory agreement is reached. There are two main ways in which the direct analyses can be performed: considering only plane generalized Rayleigh waves (a two-dimensional solution) or accounting for all types of waves involved in a full three-dimensional solution modeling the actual experimental setup. The first method provides a simple and expedient basis to understand the results of the tests, while the second simulates the physical process more realistically. In the first method, the dynamic stiffness matrices of each layer, function of the wave number and the frequency, are formed and assembled to obtain the dynamic stiffness matrix of the complete profile. A determinant search technique is then used to compute the values of the wavenumber that make the determinant zero. For a soil profile with properties increasing with depth there will always be at least a real eigenvalue (wavenumber). In most cases, there will be more than one. A question arises then as to which one of these eigenvalues, if any, corresponds to the wave propagation velocity that would be measured in the field. When the soil properties increase smoothly with depth, the first eigenvalue (smallest wavenumber) is the one of interest. When soil properties vary in a more complex way, however, this may not be always the case and one may find that the measured propagation velocities are in better agreement with the phase velocities of the second, third, or fourth eigenvalue. When the modulus of the underlying half-space or layer is smaller than those of the upper layers (typical situation for a pavement profile) there is a maximum frequency above which there are no real wavenumbers making the determinant zero. A proper solution in this case would require the determination of the complex eigenvalues leading also to complex phase velocities. Simpler alternatives often used are to assume that the half-space is made of air (plate theory) or to select real values of the wavenumber that will make 0 the real part of the determinant. Both of them are approximations. The second, more sophisticated procedure, is to solve the actual problem of a


vertical disk load applied on the surface of a layered half-space.

2 Theoretical Formulation

To understand how a pavement system or a soil deposit respond to dynamic loads applied to the surface, it is helpful to review theoretical studies dealing with the dynamic response of uniform and layered systems. 2.1 Dynamic Loads on an Elastic Half Space Lamb (1904) was the first one to study the effect of a pulse on a uniform elastic half-space. Lamb treated four basic problems: surface line and point load sources, and buried line and point load sources. He derived his solution for these problems through Fourier synthesis of the steady-state propagation solution. For the surface source problem, Lamb evaluated the surface displacements (horizontal and vertical), and pointed out that the largest disturbance in the far field is the Rayleigh surface wave. He noted the nondispersive nature of the solution, and for point-load excitation, that it decays as r where r is the distance from the source. Through the years these problems have taken on the name, “Lamb’s problem.” The first closed-form solution for Lamb’s problem in three-dimensional space was provided by Pekeris (1955) for the particular case of a material with Poisson’s ratio of 0.25. A generalization for arbitrary values of Poisson’s ratio is due to Mooney (1974) and can also be found in Erigen and Suhubi (1975); however, the Green’s functions (in the time domain) for this case are available only for a vertical point pulse with a step time-function acting on the free-surface. The correct solution for a harmonic vertical point load of the form Pexp iωt( ) was obtained by Rucker (1982) for a Poisson’s ratio of 0.25. This solution was extended by Foinquinos (1995) to any value of Poisson’s ratio using Mooney’s approach. The solution can then be written as

w r( ) =Pexp iωt( )

Grw r( )

wherer =ωrVs

is a dimensionless frequency, r is

the horizontal distance to the point load, ω the circular frequency of the excitation and Vs the shear wave velocity of the material, equal to the square root of the shear modulus G divided by the mass density. The function w r( ) can be expressed

in integral form (Foinquinos 1995). The integrals are well-behaved and their numerical evaluation presents no numerical difficulties. Figure 4 shows the amplitude of the vertical surface displacements for an elastic half-space subjected to a harmonic point load as a function of r and of a dimensionless distance obtained by dividing the actual distance r by the wavelength λ with

λ = 2πVs ω = Vs f . the displacements plotted are w r( ). For small values of the dimensionless frequency (less than 2) or of the dimensionless

distance (less than 0.4) the plot is a horizontal line. This implies that the displacement is inversely proportional to the distance r (the product of the displacement by the distance is constant). This is referred to as the near field. For larger values of the dimensionless frequency or distance, the displacement increases almost linearly, in logarithmic scale, with a slope of 0.5. This indicates a variation of the displacement inversely proportional to the square root of the distance r (the product of the displacement by r is directly proportional to the square root of r ). It should be noticed, however, that the actual solution oscillates


slightly around the straight line representing the approximate far field or Rayleigh solutions. The results in Fig. 4 correspond to a Poisson’s ratio of 0.25. Fig. 5 shows the corresponding results versus dimensionless frequency for different values of Poisson’s ratio. It can be seen that as the ratio increases so do the oscillations around the far field approximate solution (or the pure Rayleigh wave assumption). The normal assumption of a variation of the displacement amplitude inversely proportional to the square root of the distance then becomes less and less accurate. Fig. 6 shows the variation of the phase

velocities, computed at one point, as a function of the dimensionless distance. For a null Poisson’s ratio, the phase velocity is essentially that of the theoretical Rayleigh wave for distances larger than 2. For a Poisson’s ratio of 0.25 this assumption is still reasonable but less accurate due to some clear oscillations. The amplitude of these oscillations increases significantly with increasing Poisson’s ratio. In this case, the actual phase velocity will depend on the position of the receiver, and can no longer be assumed to be the Rayleigh wave velocity, independent of distance. This point is further illustrated in Fig. 7, where phase velocities obtained from the phase differences at two receivers (placed at various distances to the source) are plotted vs. Poisson’s ratio and compared to the Rayleigh wave velocity. The agreement with this velocity is very good for receiver spacings of 2 and 4 wavelengths from the source. Fig. 8 shows finally the partition of power between body waves ( P and S waves) and surface waves (Rayleigh waves) as a function of Poisson’s ratio. For a null Poisson’s ratio, the Rayleigh wave has nearly 76% of the total power but it has only about 55% for a Poisson’s ratio of 0.5.


Miller and Pursey (1954) considered the

case of a circular disk vibrating harmonically and normally on the free surface of a half-space. They

found explicit expressions for the displacements at points at large distances from the loaded area. These expressions for the horizontal and vertical (u, w) displacements at the surface of the medium due to a unit disk load are of the form:

f v( ) a2

GωCrr

⋅ e− i ωr

Cr , where a is the radius of the

disk load, G is the shear modulus of the medium, ω is the circular frequency of the excitation, Cr is the Rayleigh wave velocity of the medium and r is the distance to the source. The term f v( ) is a function of Poisson’s ratio. For instance, for v equal to 1/3, f v( ) equals −0.182 2 2 + i 2 2( ) for the horizontal displacement and 0.286 2 2 − i 2 2( ) for the vertical displacement. The exact solution at any distance can be expressed as an integral involving a product of Bessel functions. The integrand has a pole corresponding to the Rayleigh wave. The result is given by two branch line integrals which represent the effect of the body waves plus the residue at the Rayleigh pole, which represents the effect of the surface wave (Foinquinos 1995). Figure 9 shows the amplitude of the vertical displacement normalized again by the factor Ga P as a function of a dimensionless distance r a where a is the radius of the circular harmonic load. Results are shown for different values of a dimensionless frequency

a0 = ωa Vs .

It can be seen that the magnitude and shape of the surface displacements are frequency-dependent. For a value of a0 = 3.5 , which is very close to the first Rayleigh wave critical frequency, the displacements essentially vanish at a distance of about 2.5 times the Rayleigh wave wavelength (equal to 1.65a for this frequency). Fig. 10 shows the partition of power for a uniform circular load. It should be noticed that in this case the fraction of power held by the Rayleigh wave is smaller and that it decreases significantly with increasing values of the dimensionless frequency. P waves carry an important fraction of the power for low values of Poisson’s ratio, particularly at high frequencies, whereas S waves become the predominant ones for values of Poisson’s ratio close to 0.5.


2.2 Dynamic Loads on Layered Media

Consider a pavement system or a soil deposit that consists of horizontal layers. The mass densities and elastic moduli change with depth, from layer to layer, but are (assumed to be) constant over each layer. For a pavement, the top layer could represent the pavement surface layer (assuming that it extends to infinity in both horizontal directions), the second layer would be the base, and the remaining layers the sub-base layer and/or the soil subgrade. Determination of the response of this system to dynamic loads applied on the surface (or at any point within the profile) falls mathematically into the area of wave propagation theory. Formulation of these problems starts normally by considering steady-state harmonic forces and displacements at a given frequency. For the case of the Dynaflect this is all that is needed. For an arbitrary transient excitation (case of the FWD or the SASW), the time history of the specified forces must be decomposed into different frequency components using a Fourier series, or more conveniently a Fourier transform. Results are then obtained for each term of the series (each frequency) and combined to obtain the time history of displacements (inverse Fourier transform). For a single layer with uniform properties and a given frequency ω , the stresses and displacements along the top and bottom surfaces can be expanded in a Fourier series in the circumferential direction and a series of Bessel (or modified Bessel) functions in the radial direction. For each term of the series, corresponding to a given wave number, one can determine closed form analytical expressions in the form of a transfer matrix relating amplitudes of stresses and displacements at the bottom surface to the corresponding quantities at the top (or vice-versa). This approach has served as the basis for most studies on wave propagation through layered media in the last 35 years. An alternative is to relate the stresses at both surfaces to the displacements obtaining a dynamic stiffness matrix for the layer (Kausel and Roesset, 1981), which can be used and understood in much the same way as done in structural analysis. For a half-space, the stiffness matrix relates the stresses at the top to the displacements at the same level. For the particular case at hand, with an axisymmetric load, only one term of the Fourier series is needed (the 0 term) and the vector of radial and vertical displacements u, w can be written as

V =uw

⎧ ⎨ ⎩

⎫ ⎬ ⎭

= k0

∞

∫J1 kr( ) 0

0 J0 kr( )⎡

⎣ ⎢ ⎤

⎦ ⎥ Udk


where k is the wavenumber, r is the radial distance to the center of the loaded area and J0 , J1 are the Bessel functions of zero and first order respectively. U is the vector of displacements in the wave number domain. At the surface where the excitation is applied, the load vector can be expressed in the spatial domain as

P =Pr

Pz

⎧ ⎨ ⎩

⎫ ⎬ ⎭

= q01

⎧ ⎨ ⎩

⎫ ⎬ ⎭ 0 ≤ r ≤ a

where q is the amplitude of the load and a is the radius of the loaded area. In the wavenumber domain, the load can be expressed as

P =12π

rJ1 kr( ) 0

0 J0 kr( )⎡

⎣ ⎢ ⎤

⎦ ⎥ 0

2π

∫0

∞

∫ Pdθdr .

Performing the integration, the only nonzero term of the vector P is the second term which is equal to qaJ1 kr( ) k .

The displacements U and forces P in the wavenumber domain are then related by

KU = P where K is the dynamic stiffness matrix of the profile obtained by assembling the stiffness matrices of the layers and the underlying half-space. If u1 and w1 are the first two terms of the vector U , obtained by solving for a vector P with all components 0 and a 1 as the second term (for every value of k ), the surface displacements as a function of the distance r to the center of the loaded area become

u = qa u10

∞

∫ J1 ka( )J1 kr( )dk

w = qa w10

∞

∫ J1 ka( )J1 kr( )dk

The solution of the problem thus requires assembling the dynamic stiffness matrix K of the layered medium, solving the system of equations for many different values of k and evaluating numerically the integrals above. The numerical integration is performed through shifting the poles of the integrand by including a small attenuation in the materials (for materials with damping, all the poles are complex, so that no singularities are encountered along the real axis of integration). However, for a system with sharp variation in

material properties between layers, the integrands may exhibit considerable waviness, making it difficult to evaluate the integrals. The solution of the equations is time-consuming when there is a large number of layers. The procedure is convenient when dealing with a homogeneous half-space or a small number of layers. An alternative can be obtained by expanding the terms of the dynamic stiffness matrix of a layer in terms of k and keeping terms only up to second-degree (the terms of the transfer or stiffness matrices of each layer are transcendental functions). It can be shown that this is equivalent to assuming that the displacements have a linear variation with depth over each layer using standard finite element techniques to derive the layer matrix. The stiffness matrices of each layer, the half-space, and the total profile can then be expressed in the form

K = Ak 2 + Bk + G −ω 2M where the expressions for the matrices A , B , G , and M can be found in Kausel and Roesset (1981). By computing the in-plane modes of propagation as the solution of a quadratic eigenvalue problem and keeping only the modes propagating outwards, Kausel (1981) has shown that the displacements u , w can be expressed as

u = uii =1

2n+ 2

∑ wik

ki k2 − ki2( )

w = wi2

i=1

2n +2

∑ 1k2 − ki

2( )

for a system with n layers, where ui and wi denote the horizontal and vertical displacements at the surface in the i th mode and ki is the eigenvalue or wave number in the i th mode. By substituting these expressions, the integrals can then be evaluated analytically in closed form. This formulation requires a subdivision of the layers (thin layers are needed to accurately reproduce the variation of displacements with depth with a piece-wise linear approximation). It is particularly convenient when dealing with a large number of layers as is the case when it is desired to obtain a detailed variation of soil properties with depth. Furthermore, since the fundamental solutions (or Green’s functions) are known explicitly, one can determine the displacements or strains at many locations without significant additional computational effort. Both formulations (continuous and discrete) had been implemented at the University of Texas at Austin (Roesset and Shao, 1985) (Roesset,


Stokoe and Foinquinos, 1993) to simulate the FWD and SASW tests. Although a large number of sublayers must be used in the discrete formulation in order to obtain satisfactory results, this formulation has been found in general to be much more efficient computationally than the continuous formulation. The results presented in this paper were obtained using the discrete formulation. A case of some interest is that of a homogeneous layer of finite thickness resting on much stiffer, rock-like material, which can be assumed to be infinitely rigid. Fig. 11 shows the transfer functions for the vertical displacements (amplitude of the displacements as a function of frequency) for the case of a half-space and a finite layer over rigid base. The excitation is uniformly distributed over an area of the typical size of the FWD pad. Results are shown both for the displacement at the center of the loaded area and at a 6 ft. distance. It can be seen that for a half-space, the displacement amplitude under the load is nearly independent of frequency over a range of frequencies, then decays monotonically with increasing frequency. At 6 ft. the displacement increases smoothly over a range of frequencies, then decreases monotonically. For the finite layer, the displacement is still essentially constant for small frequencies, with a value smaller than that of the half-space (notice that this is the value of the static displacement, corresponding to a zero frequency). The displacement next shows a pronounced peak at a frequency corresponding to resonance of the layer (the peak becomes relatively more pronounced at increasing distances). Afterwards, the results are similar to those of the half-space but exhibit a number of oscillations. Fig. 12 shows the phase spectrum (variation of the phase angle with frequency) at the same two points and for the same conditions. It can be seen that for a half-space the phase spectrum is a straight line starting at a value of 0.04 because a damping ratio of 2% was assumed for the material. For the finite layer the phase spectrum is initially horizontal (phase angle independent of frequency). It shows then a sudden jump at the resonant frequency of the layer (the same frequency at which the amplitude exhibited a peak). For increasing frequencies it oscillates slightly around the half-space solution. This indicates that below the resonant or threshold frequency of the layer there is no radiation of waves in the horizontal direction and thus no loss of energy beyond the internal losses due to material damping. It has sometimes been stated that the threshold frequency is the natural frequency of the layer under vertical excitation. This frequency would be

fp = Vp 4H

where Vp is the P wave velocity of the material and H is the layer thickness. The value of Vp increases, however, without bound as Poisson’s ratio approaches 0.5. Fig. 13 shows the variation of the threshold frequency f , divided by the shear frequency fs = Vs 4H as a function of Poisson’s ratio. Shown in the figure are also the values of the so-called Lysmer analog fLa which had been proposed as an estimate of the threshold vertical frequency. It can be seen that for values of Poisson’s ratio below 0.3, the value of f is essentially fp . For larger values of Poisson’s ratio the solution is different. It tends to the Lysmer analog frequency for very high values of Poisson’s ratio (above 0.45).

Fig. 14 shows finally the variation with time of the displacements that would be recorded at the two points for the same cases. The excitation is assumed to be a half sine squared pulse. Under the load, the displacement is essentially a similar pulse. It can be seen, however, that the solution for the


finite layer has some oscillations following the pulse. These oscillations become much more clear with increasing distance (the shape of the pulse also changes, showing first the arrival of body waves, then the surface wave).

3 Fwd Testing

Two generalized pavement profiles, a flexible one and a rigid one, were selected to illustrate the dynamic response of pavement systems to application of FWD and SASW. Because variations in total unit weight ( γ ), Poisson’s ratio (v ), and damping ratio ( D ) have minor effects on the dynamic response (within ranges of logical values) as compared with changes in the stiffnesses of the layers, they were taken to be the same for all the layers; that is γ =120 lb/ft3 (18,850 N/m3), v =0.35 and D=0.02. The elastic properties and thicknesses of the layers in both

profiles are given in Fig. 15. Fig. 16 shows the transfer functions of the

vertical displacement at the center of the loaded area (receiver 1) for different thicknesses of the subgrade. As pointed out earlier, these curves represent the amplitude of the harmonic displacements due to a harmonic uniformly distributed circular load as a function of frequency. As the thickness of the subgrade decreases the amplitude of the peak and the frequency of the peak (the threshold frequency) increase. In the low frequency range where the displacement is essentially independent of frequency with its static value, the displacement amplitude decreases somewhat with decreasing layer thickness. In the high frequency range, the results are independent (or nearly independent) of subgrade thickness. Fig. 17 shows the corresponding results for a receiver situated at 6 ft. from the center of the load (the position often of the 7th and last receiver). The same observations can be made, but the effects (relative amplitude of the peak at the threshold frequency, relative reduction in amplitude of the static, or low frequency, displacements with decreasing subgrade thickness) are much more pronounced. Multiplying the complex transfer function (including the amplitude shown in the two figures as well as the phases) by the Fourier transform of the applied load, one would obtain the Fourier transform of the displacement at the selected receiver. The time history of the displacement is then obtained, applying the inverse Fourier transform. Fig. 18 shows the displacement records at all 7 receivers for the case of an infinite subgrade and a 20 ft. thick subgrade resting on rigid rock. It can be seen again that in the first case the motion consists only of one pulse with the same shape as the load. In the second case, on the other hand, the pulse is followed by free vibration oscillations with the natural period of the subgrade. It should be noticed that the free vibrations are


essentially the same for all the receivers, although on a relative basis their importance is larger for the farthest receivers, where the amplitude of the main pulse is smallest. It can also be seen that in both cases, there is a clear phase lag between the arrival of the pulse at the various receivers. This phase lag can be easily detected looking at the peak of the pulse. From the phase lag one can determine the properties of the subgrade. Fig. 19 shows the displacement basins caused by the applied load for the different thicknesses of the subgrade. The bottom figure shows the results from a full dynamic analysis, whereas the top figure illustrates what the results would have been if the load had been applied statically. In the actual dynamic case as illustrated by Figure 18 previously, the peak displacements do not occur at the same time at all receivers. Yet it is common practice in the interpretation of the FWD tests to plot the maximum displacements as if they occurred simultaneously and use only this information. It is also commonly assumed that the displacements are static, using a static analysis program to backfigure the layer properties. Figure 19 shows that the static displacements depend on the thickness of the subgrade whereas the dynamic ones are nearly independent of this thickness for this pavement system. It is also common to assume in the interpretation of the data that the subgrade extends to infinity. This assumption would be reasonable if the inversion was carried out using dynamic analyses but it can lead to serious errors if static analyses are performed. To further illustrate this point, Fig. 20 shows the ratio of the dynamic displacements to the static ones using the same, actual profile for both analyses. It can be seen that in this case the dynamic displacements are larger than the static ones for relatively small thicknesses of the subgrade (depth to bedrock) varying from less than 15 to less that 40 ft. depending on the station (distance of the receiver to the load), whereas they are smaller for larger thicknesses. This implies that performing static analyses for the inversion but using the correct depth to bedrock would predict values of the elastic moduli smaller than the real ones for shallow depths to bedrock and would on the other hand overpredict the stiffnesses for thicker subgrades. Fig. 21 shows the corresponding results if the static analyses are performed assuming that the subgrade extends to infinity, while the dynamic analysis is performed for the actual profile. This indicates that if the assumption of an infinite subgrade is made and static analyses are performed, one will always overestimate the moduli, since the computed static displacements will always be larger than the true dynamic ones.

Figures 22 and 23 show the corresponding results for the rigid pavement. It should be noticed that in this case the deflection basins are almost straight lines and that the effect of the thickness of the subgrade on the static displacements is even larger. For the dynamic displacements, only a subgrade thickness of 10 ft. or less would result in any noticeable difference. It is also interesting to notice that for both pavements the difference between the static displacements at the various receivers for the various subgrade thicknesses is essentially constant. For the rigid pavement the use of static analyses assuming an infinite depth to bedrock for the inversion would result in even more severe overestimations of the elastic moduli. Fig. 24 shows the sensitivity of the (dynamic) displacement basins to the properties of the layers for the flexible pavement. It can be seen that a variation in the modulus of the top layer by ±50% produces only a noticeable difference in the amplitude of the displacements at the center of the loaded area. There is no variation for distances larger than 2 ft. A similar variation in the elastic modulus of the base (2nd layer) will affect the displacements in the first three receivers (under the load) at 1 ft. and at 2 ft. A variation of ±50% in the properties of the subgrade significantly affects, however, the displacements at all receivers. This implies that the FWD test is particularly well-suited to determine the elastic properties of the subgrade (if an appropriate dynamic inversion is performed) but that it is much more difficult to estimate accurately the properties of the base and even worse those of the surface layer.



The determination of the elastic properties (Young’s modulus) of the various layers is normally performed in a iterative way, assuming a set of properties, applying a direct analysis to compute a theoretical displacement basin, comparing it to the experimental data, changing the assumed properties and iterating until a satisfactory match is obtained. As stated earlier, it is common to use a static analysis program to calculate the theoretical basins and to assume that the subgrade extends to infinity. This is an approximation that can lead to serious errors in some cases. Much better estimations could be performed if one considered the complete time histories of the recorded displacements, rather than only the peaks, and if dynamic analyses were conducted in each cycle of iteration. Chang et al (1992), and Seng (1992), suggested a procedure to estimate the depth to bedrock and the elastic modulus of the subgrade in order to obtain first estimates to initiate the iterative procedure. Foinquinos (1995) suggested a pseudo-dynamic inversion procedure more economical than the full dynamic inversion and almost as accurate based on the elimination of the dynamic effects from the measured displacement basin to obtain the static displacements. This requires, however, a set of displacement records with adequate duration (of the order of 180 msec or more), which is more than what is normally recorded and stored. Two of the main assumptions in the dynamic modeling of the pavement system were that the layers extend to infinity in both horizontal directions and that they are linear elastic. An accurate solution requires the consideration of the finite width of the pavement and possibly nonlinear behavior. Kang et al (1990) studied the effect of the finite width on the dynamic deflections of pavements, and concluded that the loading position with respect to the edge of the pavement can influence the amplitude of the deflections and the shape of the deflection basin obtained with the FWD test. They found, however, that for most pavements, the error committed by assuming that the pavement extends to infinity will not be serious if the load is placed more than 2 ft. (0.6m) from the edge of pavements at level sites or 4 ft. (1.2m) from the edge when the pavement is on an embankment or a ramp with concrete retaining walls. Chang et al (1992) studied nonlinear effects in FWD testing using an approximate nonlinear analysis procedure (a linear iterative approach in the frequency domain) and a true nonlinear incremental analysis with a generalized cap model to reproduce the nonlinear material behavior. They showed that nonlinear behavior can be significant

and localized around the loaded area if testing is performed on a flexible pavement with a rather thin surface layer and a soft subgrade. However, they also showed that nonlinear effects can be neglected for small to intermediate loads for many pavement systems and that very little nonlinearity will normally be generated in thick rigid pavements.

4 Sasw Testing

Fig. 25 illustrates the basic concept behind the Spectral Analysis of Surface Waves (SASW) technique. Assuming the propagation of a surface (generalized Rayleigh) wave, a wave with a short wavelength (high frequency) would only penetrate the first layer and would propagate with the Rayleigh wave velocity of this layer. Calculating the velocity of propagation from the phase difference between the motions at two receivers, one could then obtain the Rayleigh wave velocity of the top layer and assuming a value of Poisson’s ratio its shear wave velocity. Assuming also a value of the mass density, one could finally determine the shear modulus or the Young’s modulus of the material. Considering next a smaller frequency and a wave with a longer wavelength its phase velocity would depend on the properties of the first two layers. Once the modulus of the top layer was known, one could then determine the properties of the second layer that would yield the measured phase velocity. One could then proceed taking smaller and smaller frequencies, computing the properties of the different layers in a sequential way. As the previous discussion on the propagation of waves in a layered medium (or even a half-space) due to a dynamic load distributed over a circular area pointed out, the concept of a single surface wave propagating horizontally is only an approximation, and the actual situation is more complicated. Even so, the basic concept of the method is still valid. The main question is what procedure must be used to determine the phase velocities for a given (assumed) profile to compare them with the experimental data. In the field application of the SASW technique, it is common to place two receivers at a distance between themselves equal to the distance between the source and the first receiver (Fig. 26). The source can vary from a small hammer to a large drop weight depending on the application and the desired range of frequencies (and wavelengths) to be generated. For each position of the receivers, several impacts are applied with the source on one side, then on the other. The motions at the two receivers are recorded and processed by a spectral


analyzer that automatically computes the cross-spectrum and the phase difference between receivers as a function of frequency. The results of the various impacts for a given setup are averaged, yielding a variation of phase with frequency as illustrated in Fig. 27, as well as a coherence function. The results are expected to be valid over the range of frequencies where the coherence is close to unity. The phase difference obtained varies from -180 to 180 degrees. It is necessary to unfold these results as illustrated in Fig. 28 (this process may present difficulties in some cases). Tests are then repeated for different receiver spacings. As the spacing between receivers (and source to first receiver) increases, the results will be valid over a range of smaller frequencies (longer periods and longer wavelengths). The previous discussion on wave propagation due to dynamic loads had indicated that best results were obtained when the spacing between receivers was of the order of two wavelengths. It is not possible, however, to keep changing the spacing in the field for each frequency or wavelength as was done in the original Rayleigh wave method (it would become too time -consuming). One must thus define for each setting a range of frequencies or wavelengths over which the predictions are

expected to be reliable. The results from the various tests (for different receiver spacings) are finally combined to produce a curve giving the variation of phase velocity (assumed to be the Rayleigh wave velocity) versus frequency, or versus wavelength, as illustrated in Fig. 29. This is known as a dispersion curve. Because the ranges of the results for different spacings overlap there will be some scatter in the curve. A smooth or average curve is often drawn fitting the experimental data (Fig. 30). To backfigure the soil or pavement properties and their variation with depth, one must start with an assumed profile. This initial profile can sometimes be obtained from relatively simple procedures (assuming for instance that the phase velocity at a given wavelength corresponds to the Rayleigh wave velocity of the material at a depth equal to one third of the wavelength). Direct analyses are then performed to obtain the theoretical dispersion curve corresponding to the assumed profile. The theoretical curve is compared to the experimental one and the properties of the layers are changed, starting an iterative procedure. The iterations are terminated when the agreement between the theoretical and experimental results is deemed to be satisfactory.


various iterations for an actual soil deposit and Fig. 32 shows the final experimental and theoretical dispersion curves and the resulting profile. The soil properties obtained from the SASW test are compared to those obtained from crosshole measurements at the site.

If the soil were a homogeneous half-space, the dispersion curve would be expected to be a horizontal line as illustrated in Fig. 33. It should be remembered, however, that this will only be so

if the receivers are at a sufficient distance from the source. When dealing with a layer of finite thickness resting on a softer half-space (typical situation of a pavement), the dispersion curve would start with the value of the Rayleigh wave velocity of the top, stiffer layer and then decrease monotonically, approaching the value of the Rayleigh wave velocity of the half-space. On the other hand, when dealing with a soft layer over a stiffer half-space (more typical soil profile), the dispersion curve would be initially horizontal then


increase gradually towards a second horizontal plateau.

Fig. 34 shows the theoretical dispersion curves for the flexible pavement that had been studied earlier, combining the results for different receiver spacings. It can be seen that there is scatter in the results as there would be in the field. Fig. 35 shows the corresponding results for

receiver spacings in terms of the wavelengths (notice that the first situation agrees more closely with what would be done in practice). Fig. 36 shows the effect of the thickness of the subgrade on the dispersion curves. It can be seen that the first part of the curve is independent of the subgrade thickness. It provides the Rayleigh wave velocity of the pavement layer very clearly, and the thickness of the layer can also be estimated. The thickness of the subgrade affects the wavelength at which the dispersion curve starts to increase again if there is a much stiffer rock underneath. Fig. 37 shows the sensitivity of the dispersion curves to the material properties of the pavement layer, the base and the subgrade. Variations in the properties of the pavement layer (to which the FWD displacement basins were very insensitive) can be very clearly recognized with the SASW method. The variations in the properties of the subgrade are also clearly identifiable. On the other hand, the variations in the properties of the base may be harder to estimate accurately. In the simplest possible form, the

determination of the material properties (inversion of the data) is carried out assuming for the direct analyses (for each assumed profile) that one has only a single Rayleigh wave propagating


horizontally. To compute the phase velocity of this wave, one must find the eigenvalue of the dynamic stiffness matrix of the soil profile. It is common to assume that the main contributor to the results is the first Rayleigh mode (smallest eigenvalue). This is often referred to as a 2D solution. It works reasonably well when dealing with soil profiles where the properties vary smoothly, increasing steadily with depth. When a stiff layer is underlain by a more flexible one, the situation becomes more complicated and the eigenvalues may no longer be real. It is hard to determine then which is the one corresponding to the first mode. Moreover, the first mode may not be the predominant one. A more laborious but also more accurate technique is to solve the actual 3D problem computing the displacements due to the dynamic loads following the formulation described earlier and simulating the actual test procedure. Figs. 38 and 39 illustrate the solution for two actual soil profiles where the soil properties do not vary smoothly. The top figure in each case shows the initial assumed profile (using the one-third of the wavelength rule) and the final soil profile after iterations using the complete 3D solution. The second figure shows the experimental dispersion curve and the theoretical curves predicted using the approximate 2D (single Rayleigh mode) and the more accurate 3D solutions for the initial profile. It can be seen that the 2D solution seems to provide a better match for this profile. The bottom figure shows the corresponding dispersion curves for the final soil profile. In this case the 3D solution gives a much better fit.

5 Conclusions

Nondestructive dynamic testing techniques such as the Falling Weight Deflectometer or the Spectral Analysis of Surface Waves can provide economical and reliable measures of the elastic moduli of pavement systems or soil deposits. The interpretation of the experimental data to backfigure the layer properties can be done in some simple cases on the basis of approximate formulations. In most cases, however, it is necessary to try to simulate the complete three-dimensional dynamic problem to get accurate predictions. So for instance the interpretation of FWD data using static analyses and ignoring, therefore, inertial (dynamic) effects can lead to an overestimation of the elastic moduli when combined with the assumption of an infinite subgrade; when accounting properly for the depth


to bedrock static analyses may underestimate or overestimate the moduli depending on the thickness of the subgrade. Similarly, the interpretation of SASW data assuming a single Rayleigh wave mode will yield sensible results only when dealing with soil deposits where the stiffness increases smoothly with depth. When there are stiff layers underlain by softer layers, a complete 3D solution is necessary. It should also be noted that each method has advantages and limitations. The FWD can provide fast and efficient estimates of the properties of the subgrade, even with some rather simple procedures, if one is able to record the time histories of the displacements for a sufficient duration. Results are, on the other hand, very insensitive to the properties of the top (pavement) layer, which makes an accurate determination of these properties very difficult. The SASW can provide very simply, on the other hand, accurate values of the modulus of the top layer as well as its thickness, but may have difficulties estimating with the same degree of accuracy the properties of the base in the case of a pavement system or recognizing soil layers which are thin relatively to the depth at which they are encountered. Other methods such as the impulse response (or impact-echo) method used to detect cavities in concrete elements or for integrity testing of piles, can also provide very easily an estimate of either the modulus of the pavement layer (if its thickness is known) or of its thickness (if the modulus is known). It would appear that a combination of various of these techniques could have great advantages and provide a more reliable

and complete picture of the pavement or soil profile.

Bibliography

1. Chang, D.W., Kang, V.Y., Roesset, J.M., and Stokoe, K.H., II, “Effect of Depth to Bedrock on Deflection Basins obtained with Dynaflect and FWD Tests,” Transportation Research Record 1355, pp. 8-16, 1992.

2. Chang, D.W., Roesset, J.M., and Stokoe, K.H., II, “Nonlinear Effects in Falling Weight Deflectometer Tests,” Transportation Research Record 1355, pp. 1-7, 1992.

3. Eringen, A.C., and Suhubi, S., “Elastodynamics,” Vol. 2, Academic Press, New York, NY, 1975.

4. Foinquinos, R., Roesset, J.M., and Stokoe, K.H., II, “Dynamic Interpretation of FWD Deflection Basins,” XII Congreso Mundial de IRF, Madrid, Spain, 1993.

5. Foinquinos, R., “Dynamic Nondestructive Testing of Pavements,” Report GR 95-4, Civil Engineering Department, The University of Texas at Austin, 1995.

6. Kang, V.K., Roesset, J.M., and Stokoe, K.H., II, “Effect of the Finite Width of Pavements on Deflection Basins obtained with Dynaflect and FWD Tests,” Transportation Research Board, pp. 1-26, 1990.

7. Kausel, E., and Roesset, J.M., “Stiffness Matrices for Layered Soils,” Bulletin of the Seismological Society of America, Vol. 71, No. 6, 1981.

8. Kausel, E., “An Explicit Solution for the Green Functions for Dynamic Loads in Layered Media,” Research Report R81-13, Massachusetts Institute of Technology, Cambridge, MA, 1981.

9. Lamb, H., “On the Propagation of Tremors over the Surface of an Elastic Solid,” Phil. Transactions Royal Society of London, Series A, pp. 203, 1904.

10. Miller, G.F., and Pursey, H., “The Field and Radiation Impedance of Mechanical Radiators on the Free Surface of a Semi Infinte Isotropic Solid,” Proceedings, Royal Society of London A223, 1954.

11. Mooney, H.M., “Some Numerical Solutions for Lamb’s Problems,” Bulletin of the Seismological Society of America, Vol. 64, No. 2, 1974.


12. Pekeris, C., “The Seismic Surface Pulse,” Proceedings, National Academy of Science, pp. 41, 1955.

13. Roesset, J.M., and Shao, K., “Dynamic Interpretation of Dynaflect and Falling Weight Deflectometer Tests,” Transportation Research Record 1022, pp. 7-16, 1985.

14. Seng, C.R., “Effect of Depth to Bedrock on the Accuracy of Backcalculated Moduli obtained with Dynaflect and FWD Tests,” M.Sc. Thesis, The University of Texas at Austin, 1992.

Manuscript Received: Oct. 15, 1998

Revision Received: Jan. 1, 1999 and Accepted: Jan. 1, 1999

Documents

Nondestructive Dynamic Testing of Soils and Pavements