Engineering Framework for the Self-Consistent Analysis of FWD Data (revised manuscript) Thomas M. Westover Graduate Research Assistant Department of Civil Engineering University of Minnesota 500 Pillsbury Drive S.E. Minneapolis, MN, USA 55455 Tel: (612) 626-1538 Fax: (612) 626-7750 Email: [email protected]Bojan B. Guzina Associate Professor Department of Civil Engineering University of Minnesota 500 Pillsbury Drive S.E. Minneapolis, MN, USA 55455 Tel: (612) 626-0789 Fax: (612) 626-7750 Email: [email protected]November 15 th , 2006 Total number of words in the paper (excluding title page) = 4,474 + 12*250 = 7474 Submitted to the Transportation Research Board for publication in Transportation Research Record
Transcript
Engineering Framework for the SelfConsistent Analysis of FWD Data
(revised manuscript)
Thomas M. WestoverGraduate Research Assistant
Department of Civil EngineeringUniversity of Minnesota500 Pillsbury Drive S.E.
Minneapolis, MN, USA 55455 Tel: (612) 6261538Fax: (612) 6267750
Total number of words in the paper (excluding title page) = 4,474 + 12*250 = 7474
Submitted to the Transportation Research Board for publication inTransportation Research Record
Westover, T.M. and Guzina, B.B. 2
ABSTRACT
In nondestructive testing of flexible pavements, the Falling Weight Deflectometer (FWD) test is a common method to evaluate the mechanical characteristics of layered pavement structures. Elastostatic backcalculation remains the norm in the interpretation of FWD data, even though its underlying (static) analysis is inconsistent with the dynamic nature of the FWD test. Due to wave propagation effects, especially in the presence of a stiff layer, the peak pavement deflections induced by the FWD loading can differ significantly from their static counterparts, thus compromising the conventional backcalculation of pavement moduli. In this study a frequencydomain based, preprocessing procedure is developed to extract the static pavement response from the transient FWD records, thus providing a more consistent input for the elastostatic recovery of pavement profiles. To ensure the fidelity of the educed static deflections, the key drawbacks in typical (field) FWD data, namely the baseline offset and the lowfrequency noise pollution, are examined and remedied. For pavement engineering applications, the above preprocessing procedure for extracting the static deflections from dynamic FWD records is implemented in a userfriendly graphical environment GopherCalc. By comparing the methodologies using both syntheticallygenerated (elastodynamic) data and field records, it was found that the proposed procedure has the potential of mitigating the errors associated with the dynamic nature of the FWD test while still retaining the computationallyeffective elastostatic backcalculation scheme.
Westover, T.M. and Guzina, B.B. 3
INTRODUCTION
Degradation of pavements over time is a crucial concern for pavement engineers. To decide on a proper corrective measure, one must be able to characterize and quantify the condition of the pavement and its substructure in an efficient and nonintrusive manner. With the increasing need for economical nondestructive testing methods, the Falling Weight Deflectometer (FWD) test has emerged as a viable way to estimate the pavement structural adequacy while leaving the pavement section intact and traffic largely unaffected.
Due in part to its simplicity, elastostatic backcalculation is the most common form of FWD data interpretation. Dynamic analyses are significantly more complex and often require a nearly prohibitive computational effort in the context of field applications. Though some simplified elastodynamic models [1,2] and artificial neural networks [3, 4, 5] have been employed to circumvent these difficulties, elastostatic backcalculation remains the dominant method of analysis. Consequently, the traditional methods of backcalculation assume static deformation while interpreting the dynamic (i.e. peak) deflections of the pavement system. In particular, since the FWD test is decidedly dynamic in nature, phenomena such as wave reflection and refraction as well as the viscoelastic response of asphalt concrete are unaccounted for in the elastostatic backcalculation. Recently, a simple and efficient preprocessing procedure invoking the concepts of the Fourier transform and frequency response functions was proposed [6] that can potentially elevate the FWD backcalculation of pavement elastic layers within the framework of conventional elastostatic analyses, especially in the presence of a stiff layer. In the approach, the (dynamic) peak values of the pavement response are replaced, as an input to backcalculation, by their zerofrequency (i.e. static) counterparts stemming from the Fourier analysis of transient deflection records.
Despite its potential for improving the pavement diagnosis from FWD records, the proposed approach necessitates the use of the complete deflection time histories for the preprocessing procedure. In this regard, the Fourierbased extraction of static deflections faces two key challenges caused by measurement errors, namely i) problem of the socalled baseline offset, and ii) poor signaltonoise ratio characterizing the pavement response at low frequencies. A systematic treatment of these issues, together with its implementation in a userfriendly environment, is the main focus of this study.
Baseline offset. In the FWD test, the response of the pavement to the imparted load is measured by an array of vertical velocity transducers known as geophones. As a result of measurement noise and possible sensor drift, integration of the temporal velocity records to obtain pavement deflections can result in an accumulating offset that pollutes the deflection records and compromises the accuracy of backcalculation. Such
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accumulating error, manifest via a nonzero drift at the end of the deflection record, is known as the baseline offset [7,8]. In a conventional (peakbased) elastostatic backcalculation such error rarely accumulates to a significant level by the time the pavement deflection reaches its maximum value, typically occurring early in the temporal record. In the context of the featured Fourier analysis that utilizes the entire deflection record this error becomes significant and must be addressed. In this study, a polynomialtype correction of the baseline offset is implemented and shown, through field examples, to significantly smoothen the frequency content of the pavement response and enhance the extraction of the zerofrequency (i.e. static) pavement deflections.
Lowfrequency response. Due to physical limitations of a geophone, the lowfrequency response of the captured velocity records are characterized by poor signaltonoise ratios, manifest in the erratic behavior of the pavement response function in the range of 010 Hz. This very part of the response is however critical for the preprocessing procedure and must be dealt with appropriately. In this study, the deficiencies of the lowfrequency pavement response are overcome by extrapolating the intermediate frequency (1020 Hz) data through the noisepolluted region. This is achieved by implementing a singledegreeoffreedom extrapolation procedure that is both stable and capable of closely mimicking the physical characteristics of the lowfrequency pavement response for a variety of multilayer systems.
Graphical user interface. The above developments are implemented as a comprehensive preprocessing procedure for FWD backcalculation in a graphical application, GopherCalc. The interface is equipped with a number of options, allowing a user to i) perform automated averaging of the devicestored deflection records, ii) visualize the temporal and frequencydomain signatures of pavement deflections, iii) select the type of baseline correction, iv) view the lowfrequency extrapolation, and v) compare the peakbased and extrapolatedstatic deflection basins. In the future, this preprocessing tool will be integrated with a variety of established elastostatic backcalculation procedures. Performance of the integrated preprocessing procedure is examined using both (noisepolluted) synthetic and field records.
EXTRACTION OF STATIC DEFLECTIONS FROM FWD RECORDS
To extract the static deflections from the transient pavement response due to imparted FWD loading, it is necessary to perform a frequency domain analysis. This is accomplished by the application of a Fourier transform to the temporal deflection and force records. The Fourier transform is commonly used and widely implemented as an integral transform in signal processing which decomposes e.g. a function of time, into a
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series of harmonics. Since any FWD record, g(t), is inherently digitized and of finite duration, a discrete Fourier transform
( ) ( ) ( )2
0, 0,1, 2,...,m j
Mi f t
m jj
G f t g t e m Mp-
=
= D =
must be applied where g(tj) is the value of g at the sampling point tj=j∆t (j=0,1,2…M) and fm = m ∆f=m/(M∆t). There are several efficient algorithms for implementing the discrete Fourier transform, commonly referred to as Fast Fourier Transforms (FFT), [9] For the application of (1) to FWD records with e.g. N geophones, let Q(fm) and Wk(fm) denote respectively the Fourier transforms of the fallingweight force, q(t), and pavement deflection at the kth sensor, wk(t). In this setting, the frequency response function of the pavement system at the kth sensor (k=1,2..N) is formally defined as FRFk(fm) = Wk(fm)/Q(fm).
When multiple realizations (i = 1,2,…T) of the FWD test are available, the featured deflectionperunitload ratio in the frequency domain is computed more robustly using the crossspectral Sqk and autospectral Sqq density functions [10] where
( ) ( ) ( )*
1
1 , 0,1, 2,...,T
qk m m k m iii
S f Q f W f m MT =
= =
( ) ( ) ( )*
1
1 , 0,1, 2,...,T
qq m m m iii
S f Q f Q f m MT =
= =
and ‘*’ denotes the complex conjugate. The frequency response functions FRFk
characterizing the pavement system are then computed as
( ) ( )( )
, 1, 2,..., , 0,1, 2,...,qk mk m
qq m
S fFRF f k N m M
S f= = =
Note that FRFk are inherently complex valued, with their real and imaginary parts representing the in and outofphase components of the pavement response at the kth
station. In the context of (4), it is important to note as in [11] that an application of the Fourier transform to FWD data requires the use of the “long” (1200ms) records to avoid unnecessary systematic errors associated with premature signal truncation.
BASELINE OFFSET
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During an FWD test the pavement velocity records at geophone locations are inevitably polluted with noise due e.g. to ambient vibrations. When integrated to estimate the pavement deflections, such measurement noise often accumulates, resulting in a nonzero deflection value at the end of the temporal record. Such phenomenon is illustrated in Figure 1a where the deflections exhibit a noticeable drift at times >0.1s when one would expect the pavement system to return to the undeformed configuration. While the effect of baseline offset may not be critical in the context of peak deflections, its importance can not be neglected when computing the frequency response functions FRFk as they depend on the entire temporal records. Figure 1b plots FRFk computed from the baselineoffsetpolluted records in Figure 1a. As can be seen from the display, the frequency records are highly oscillatory despite computing FRFk using (2)(4) from six repetitions of the FWD test. Clearly, such oscillations are unacceptable as they often exceed the respective mean values and must be dealt with before a meaningful frequencydomain analysis of FWD records can be applied.
FIGURE 1 FWD field data: a) timehistories wk(t) and b) frequency response functions FRFk(f), k = 1,2…9 computed from 6 repeated drops at the Mn/ROAD testing facility, Test Section 33, May 22nd,
2001. No baseline correction applied.
In seismology, the presence of even small amounts of baseline offset in ground acceleration data has been shown to generate large errors in displacement calculations
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[7,8]. To deal with the problem in the context of FWD records, a baseline correction is applied to the temporal deflection records such as those in Figure 1a in the form of
( ) ( ) ( ) ( )( )
( )0
0
, 0,1, 2,...nk M kbc
k j k j jnM
w t w tw t w t t j M
t t-
= - =-
where n is the order of the polynomial baseline correction. It is useful to note that in the case of linear baseline correction (n = 1), transformation (5) amounts to a ‘rotation’ of the temporal record to achieve the zero offset wk
bc(tM) = 0.With reference to Figure 1, the baselinecorrected FWD records (with n = 1) and
the associated frequency response functions are plotted respectively in Figures 2a and 2b. From the display, it is evident that the baseline correction removes the artificial oscillations while maintaining the overall character of the pavement response in the frequency domain.
FIGURE 2 FWD field data: a) timehistories wk(t) and b) frequency response functions FRFk(f), k = 1,2…9 computed from 6 repeated drops at the Mn/ROAD testing facility, Test Section 33, May 22nd,
2001. Linear baseline correction applied (n = 1).
LOWFREQUENCY RESPONSE
By definition the frequency response function FRFk(fm) = Wk(fm)/Q(fm), when evaluated at a zero frequency f0 = 0, yields the pavement deflection at the kth sensor due to a static force component Q(f0). As a result, once the frequency response functions are computed,
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the static pavement response is given by their readings at zero frequency. Unfortunately, due to physical limitations of geophone construction [11] the lowfrequency portion of each FRFk, including its static value, is inherently associated with inadequate signaltonoise ratios and is thus of limited practical value. As an illustration, the phenomenon is highlighted in Figure 3. The FRFs begin to experience questionable fluctuations in the lowfrequency regime (below 510 Hz), exemplified by the unreasonable crossing and reversal of FRF8 and FRF9 at approximately 2.5 Hz. Note however that the integrated geophone records in the intermediate regime (1025 Hz) are typically characterized by strong signaltonoise ratios and are thus deemed reliable. For these reasons, it is proposed that the static values be obtained by a robust lowfrequency extrapolation scheme, anchored in the trusted (intermediate) frequency range.
FIGURE 3 Lowfrequency response of a baseline corrected FWD test. Mn/ROAD testing facility, Test Section 33, May 22nd, 2001.
In deciding the trusted frequency range, it is further noted as in Guzina and Nintcheu [12] that the centroid of the Fourier spectrum of a typical FWD load pulse is located slightly above 20 Hz. Additionally, the energy transferred to the pavement is primarily limited to frequencies less than 40 Hz. Owing to the necessity to fit a curve to the reliable data and extrapolate through the unusable region, the fit range is chosen as 1020 Hz.
Depending on the mechanical properties and physical characteristics of the pavement system, its response to FWD loading in the range of 1020 Hz can vary widely, from being smooth and wellbehaved to featuring a sharp resonant peak [11]. Since the ability to mimic the mechanics of lowfrequency response is critical for zerofrequency extrapolation, a suitable curvefitting model must be developed that is able to capture the variety of pavement behaviors while remaining stable as the frequency tends toward zero.
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It is therefore proposed that a SingleDegreeofFreedom (SDOF) model, expressed in the context of dynamic stiffness with units of displacementperunitforce, be implemented for fitting and extrapolation. The key advantages of this model are i) its inherent stability manifest in the zeroslope at zero frequency, and ii) its ability to represent a wide range of mechanical behaviors ranging from a monotonic deflectionload variation to that exhibiting a sharp peak. The frequency response function (representing the deflectionperunitload ratio in the frequency domain) for a SDOF system with mass m, spring constant s, and damping ratio can be written as
( ) 022 2
0 0
1
,
1 2
s sSDOFFRF
mk k
w w
w wx
w w
= =
- +
where =2f denotes the circular frequency. To extrapolate the experimental FWD data safely through the lowfrequency region, is fitted to each FRFk, using nonlinear minimization, in the featured intermediate range (1020 Hz). Through numerical simulations, it was found that a proper choice of initial parameters in terms of s, m, and
is critical for the success of the minimization procedure and the quality of the SDOF fit in general.
Initial Values for SDOF Fitting
In general, it is reasonable to assume that the layered pavement system is laterally homogeneous over the FWD sensor array (typically less than 2 meters). Consequently, geophone records stemming from a single FWD test will typically have similar features. For example, a resonant peak will be visible in all geophone records, though much less pronounced for geophones farther away from the loading plate. Nonetheless, each geophone record will have its own individual features (Figure 3) and it is necessary to fit the SDOF model to each FRFk separately as a means to ensure accurate extrapolation towards the static pavement response.
Initial Spring Stiffness
With reference to , one may note that the denominator of FRFSDOF approaches unity as the circular frequency 0 so that the static value of FRFSDOF equals 1/ s. By virtue of this behavior, selection of the initial value for the spring constant, s
initial as the value of
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FRFk at the lowfrequency cutoff (10 Hz) provides a reasonable approximation while avoiding the noisedominated behavior of the lower frequencies. An expression for the initial spring constant then becomes:
( )1 .10 Hz
initials
kFRFk =
FIGURE 4 Effect of damping ratio on the frequency response function of a SDOF system.
Initial Damping
From numerical simulations it was found that the ability of the SDOF system to capture the lowfrequency peak (if any) in the pavement response is primarily dependent upon its damping ratio 0 ≤ ≤ 1. As can be seen in Figure 4, variations in result in a wide variety of shapes. The variation in shape can be broken down into three primary categories for the purpose of determining initial values, namely i) pronounced peak, ii) gentle peak, and iii) no discernable peak. In the case where the peak is pronounced, the halfpower method [13] provides a convenient means to determine the proper value. In the context of this study, the halfpower method can be adapted to the selection of an initial damping ratio as follows. Let
max
2reduced k
kFRFFRF =
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where FRFkmax is the peak value of the fitted function FRFk, located at fmax in frequency
range of interest. Next, let fL and fR be the frequencies to the left and right of fmax whose value is FRFk
reduced. The amount damping can then be approximated as:
max
.2
initial R Lf ff
x-
=
The above described halfpower approach was found to be most effective for geophone records where the values of FRFk
reduced can be reasonably determined on both sides of fmax. In situations where FRFk
max occurs within the frequency interval of interest (1020 Hz) but is not pronounced enough to yield reasonable values for fL and fR, experience dictates that taking the initial damping ratio of intial= 0.3 is sufficient to ensure convergence. Further, if FRFk varies monotonically over the featured frequency interval, experience suggests that intial= 0.8 provides a robust initial guess. Here it is important to note that the above described procedure for choosing intial is implemented in a fully automated algorithm within GopherCalc.
Once the initial values of the spring and damping parameters have been chosen, the initial value of the SDOF mass can be calculate as
( )( )2
max
1 2, 2
2
initial initials
initial initialmf
k xx
p
-= <
which ensures that the locations of the peaks of the SDOF and fitted FRFk curve coincide. For intial>√2, the SDOF curve does not have a peak and experience suggests an initial value of minitial 0 is sufficient to provide reasonable convergence. ≈
Figure 5 illustrates the performance of the MATLAB fitting procedure (fminsearch) with the initial values of the SDOF parameters selected as described above. As can be seen, the SDOF system is capable of accurately fitting all geophone records in a way that provides stable and consistent extrapolation through the noisepolluted lowfrequency range.
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FIGURE 5 Lowfrequency response of the pavement system fitted with SDOF model and extrapolated toward zero frequency.
IMPLEMENTATION: GopherCalc
An FWD test typically consists of many repeated drops at various heights and locations, thus automation of the above frequencydomain and fitting analyses was necessary to expedite computations. A graphical application, GopherCalc was developed using MATLAB. The user interface is equipped with a number of options, allowing i) automated averaging of multiple FWD records (by station and drop height), ii) visualization of the temporal and frequencydomain pavement response, iii) selection of the order of polynomial baseline correction, iv) visualization of the (automated) SDOF fit and lowfrequency extrapolation, and v) visual comparison of the peakbased and extrapolatedstatic deflection basins. In the future, this preprocessing tool will be integrated with a variety of established elastostatic backcalculation procedures. As an illustration, a screenshot of GopherCalc is shown in Figure 6 which shows some of the available options.
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FIGURE 6 GopherCalc graphical user interface.
RESULTS AND DISCUSSION
To validate the preprocessing methodology presented thus far, analysis was performed on both syntheticallygenerated data and field data from the Mn/ROAD facility. In what follows, synthetic FWD waveforms are generated using the elastodynamic FWD model [12] that makes use of the method of propagator matrices, the Fourier transform, and the Hankel integral transform [14].
Westover, T.M. and Guzina, B.B. 14
Parametric Study with Synthetic Data
With reference to Table 1, six test profiles consisting of either three or four elastic layers were used to examine the effectiveness of the proposed preprocessing procedure. Using the noisepolluted FWD time histories generated by the forward model, composite deflection basins are computed using both i) the peaktopeak (P2P) method, and ii) the proposed frequencyresponsefunction (FRF) technique. A sample comparison of the deflection basins is shown in Figure 7 where the peak method recovers decidedly larger values at most geophone locations.
TABLE 1 Information on Synthetic Layer Profiles
CaseNumber AC Base Subbase Stiff AC Base Subbase Stiff
FIGURE 7 Comparison of static (FRF) and dynamic (P2P) deflection basins, Case 4.
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The P2P and FRF deflection basins generated for each test layered profile were used as an input to the elastostatic backcalculation program EVERCALC [15] together with i) the information concerning layer thicknesses and Poisson’s ratios (Table 1), and ii) the seed modulus values shown in Table 2. The results of the elastostatic backcalculation are shown in Figure 8ad where the ‘true’ modulus values (Table 1) are also listed to provide a point of reference. As shown in the figures, the effect of the dynamic portion of the pavement response, inherently embedded in the P2P analysis, is evident in the backcalculated moduli. In particular, one of the wellknown shortcomings of the peaktopeak method is its sensitivity to shallow stiff layers [16]. This is confirmed by the P2P backcalculation results for Cases 3, 4, and 5 where the modulus of the AC layer is overestimated (Figure 8a) while the stiff layer remains virtually undetected (Figure 8d). In contrast, the elastostatic backcalculation, when applied to the FRF deflection basin, yields modulus values that are consistent with their ‘true’ counterparts. With reference to Figure 8ad, the only situation where the FRFbased backcalculation fails to identify the pavement modulus correctly deals with the stiff layer of Case 4. This anomaly, however, may be caused by a limitation of the FWD testing configuration itself rather than the data interpretation methodology. Indeed, Meier and Rix [4] suggest that when the stiff layer is located at depths greater than 3 meters (Case 4), the FWD deflection basin is largely unaffected by its presence.
FIGURE 8 Results of EVERCALC backcalculation for the a) AC layer, b) base layer, c) subgrade layer, d) stiff layer (if present).
Initial Study with Field Data
Mn/ROAD Section 31 (low volume road). To highlight the differences in elastostatic backcalculation when applied to P2P and (static) FRF deflection basins in the context of field applications, a comparison similar to that presented earlier is performed using FWD data from Section 31 (low volume road) at the Mn/ROAD testing facility. In the first example, the FWD testing was performed on a fourlayer pavement system with base and subbase but no stiff layer, see Table 3. The results of the EVERCALC backcalculation are shown in Figure 9. Despite the apparent differences between the Mn/ROAD section profile and those used in synthetic examples, the P2P backcalculation appears to overestimate the AC modulus while underestimating the moduli of deeper layers; a trend that is similar to those observed in the context of synthetic data, see Figure 8. For completeness, the comparison is also performed using an alternative backcalculation routine, ELMOD [17]. Notwithstanding the apparent differences in the estimated moduli, the relative (FRF versus P2P) trend again mimics that in Figure 8.
FIGURE 9 Backcalculation results for Mn/ROAD LowVolume Road (Test section 31, March 16th, 2001).
TABLE 3 Layer Information for Test Section 31 (Mn/ROAD testing facility )
Class5 Special Base 10.2 0.35Class3 Special Base 30.5 0.4
Clay Subgrade ∞ 0.45
Mn/ROAD Section 33 (phantom subbase layer). In the second example, the performance of the P2P and FRFbased backcalculation is compared using the FWD data taken at Section 33 at the Mn/ROAD facility. This section was constructed by placing a 0.1m asphalt layer directly on top of the 1.2mthick layer of Class6 base material that was used to construct the embankment [3]. The soil underlying the embankment was insitu clayey material. For backcalculation purposes, the BH =1.2m
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thick layer of Class6 material was artificially divided, as 1 2B B BH H H= + , into two
“phantom” layers of thickness 1BH and 2
BH given by
1 2[m] 0.2 0.1 , [m] 1.0 0.1 , =0,1,...4B BH k H k k= + = -
Although the exact mechanical characteristics of the Class6 material were unavailable, one would expect a moderate increase of the modulus with depth within the embankment layer due to insitu stresses and a pressuredependent behavior of granular materials. Accordingly, the backcalculation analysis is expected to yield somewhat higher modulus for the lower sublayer (thickness 2
BH ) compared to that of the upper sublayer (thickness
1BH ). On performing the FRFbased backcalculation with variable thicknesses 1
BH and 2BH of the base and subbase according to (11), their respective moduli were obtained as
B1 =83±17 MPaE and B
2 =207±29 MPa.E On the other hand, the P2Pbased method produced the moduli values as B
1 =36±13 MPaE and B2 =946±498 MPaE , which are far
less reasonable. On the basis of these initial field comparisons, it appears that the FRFbased backcalculation produces more consistent results when the subsurface conditions (i.e. layer thicknesses) are not precisely known.
CONCLUSION
In this study, an automated preprocessing procedure was developed to extract the static deflections from the FWD test. Using a simple preprocessing modification of the FWD time histories, the key inconsistency associated with an elastostatic backcalculation of the peak (i.e. dynamic) FWD data is removed by means of a frequency domain analysis of the FWD deflection records. Through the use of a Fourier transform and frequency response functions, the static pavement response is extracted from the transient FWD data records, thus eliminating the dynamic effects that undermine the conventional elastostatic backcalculation. This alternative method of analysis can be applied to a conventional FWD test, although it requires the use of the full, 1200 ms temporal record to avoid truncation errors associated with shorter time histories [11]. When combined with a proper treatment of the experimental errors in FWD data, namely i) application of a suitable baseline correction, and ii) extrapolation of the frequency response functions from the (stable) intermediate frequency range through the noise polluted lowfrequency region, the proposed preprocessing procedure is shown, through numerical and field examples, to elevate the performance of the conventional elastostatic backanalyses. For
Westover, T.M. and Guzina, B.B. 19
pavement engineering applications, the developments are implemented in a userfriendly graphical environment GopherCalc.
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