Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home   |   Pavements Home
REPORT
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-12-072
Date: May 2013

 

Smart Pavement Monitoring System

CHAPTER 6. LABORATORY FATIGUE TESTING AND DEVELOPMENT OF SENSOR-SPECIFIC DAMAGE PROGNOSIS ALGORITHMS

This chapter focuses on evaluating the performance of the sensor for fatigue testing. The main objective is to develop a robust data interpretation algorithm that is able to use the diminished data provided by the sensor to achieve reasonable predictive capabilities comparable to what is obtained using conventional measuring techniques (conventional strain gauges). The time compressed cumulative data provided by the sensor result in a loss of information. A set of laboratory tests were designed to quantify and characterize the effect of these losses. The objective is to recreate the damage index variation curves using only the cumulative information tracked by the sensor.

6.1 DATA INTERPRETATION ALGORITHMS

Measured peak strain distributions in a pavement are approximated by Gaussian distributions for all of the considered cases. The variation of the strain amplitude over time is due to the increase of the compliance (i.e., induction of fatigue damage in the specimen). Figure 168 shows the strain amplitude variation of a concrete beam over time under cyclic loading at a constant amplitude. The strain amplitude is shown to increase, which explains the loss in the elastic modulus of the beam. This increase in amplitude causes the increase of the output voltage amplitude that is recorded by the sensor.

The considered hypothesis is that a shift in the measured strain distributions toward higher strains over time is indicative of damage accumulation. Thus, monitoring the mean and the standard deviation of the distribution over time and tracking the mean variations allow researchers to determine the levels of strains that are induced in the specimen between reading periods. By determining these response levels, the compliance can be evaluated. A simplified model, described in this section, shows how these measurements are obtained and related to damage.

This graph shows strain amplitude variation of a concrete beam under cyclic load with constant amplitude. The x-axis shows the number of cycles, and the y-axis shows the strain amplitude. The line on the graph beings at a strain of 0.0009E-6 and zero cycles and increases to a strain of 0.0011E-6 at 140,000 cycles.
Figure 168 Graph. Strain amplitude variation of a concrete beam under cyclic load with constant amplitude.

The strain cumulative density function (CDF) is characterized by figure 169 as follows:

F open parenthesis epsilon closed parenthesis equals beta divided by two times open bracket 1 minus erf times open parenthesis epsilon minus mu divided by sigma times the square root of 2 closed parenthesis closed bracket.
Figure 169 . Equation. Strain cumulative density.

Where:
μ= Mean of the strain distribution.
σ= Standard deviation reflecting the width of the normal distribution.
β= Total cumulative time of applied strain.

The sensor output data are defined by these three parameters, which are obtained by fitting the sensor's output distributions collected from all the memory cells.

Figure 170 shows the measured strain CDF from the sensor at different life stages of the beam. The amplitude is expressed in voltage, which is directly related to the event's cumulative durations.

This graph shows cumulative distribution of strain expressed in voltage. The x-axis shows strain, and the y-axis shows output voltage. There are 12 lines shown on the graph. All start horizontally from a strain of 675E-6 until they reach a strain of 950E-6, where all of the lines decrease to an output voltage of zero. The line representing 140,000 is the highest line and begins at an output voltage of 4.8 V. the next highest is 120,000 at an output voltage of 4.3 V, followed by 100,000 at an output voltage of 4.6 V, 90,000 at 3.3 V, 80,000 at 3 V, 70,000 at 2.5 V, 60,000 at 2.2 V, 50,000 at 1.9 V, 40,000 at 1.5 V, 30,000 at 1.1 V, 20,000 at 0.7 V, and 10,000 at 1.3 V.
Figure 170. Graph. Cumulative distribution of strain expressed in voltage.

The shift of the mean due to the strain amplitude variation cannot be directly obtained from the cumulative distributions. Figure 171 shows the normalized density distribution reconstructed from the measured CDF at different life stages. The mean of the distributions is equal to the average induced strains amplitude, thus proving the consistency of the assumptions.

To determine the final design of the sensor, the number of gates per sensor was evaluated using a sensitivity analysis. Figure 172 shows the variation of the relative error per gate versus the sensor strain level for different specimens at different life stages. Starting from eight gates per sensor, the relative error is less than 1 percent.

This graph shows the normalized density distribution expressed as normalized voltage. The x-axis shows strain, and the y-axis shows voltage. There are three bell-shaped lines shown on the graph. The first line is for 10,000 cycles, and its mean is at strain of 900E-6 at a voltage of 0.9 V. The next line represents 20,000 cycles, and its mean is at a strain of 1000E-6 at a voltage of 5 V. The last line represents 140,000 cycles and its mean strain value of 1100E-6 and a voltage of 0.8 V.
Figure 171. Graph. Normalized density distribution expressed as normalized voltage.

This graph shows the relative error of fitting per gate for specimens at different life stages versus the number of gates per sensor. The x-axis represents the number of gates per sensor, and the y-axis represents the relative error per gate. There are 12 lines shown on the graph. They all begin at errors in the range of 1.2 to 3.8 percent at 2 gates per sensor and decrease to an error of 1 percent at 12 gates per sensor.
Figure 172. Graph. Relative error of fitting per gate for specimens at different life stages versus number of gates per sensor.

Due to the missing load data and the fact that the output response is collected periodically (e.g., once every year), the damage index cannot be evaluated as a deterministic value. In this project, the damage index is considered to be the ratio of the elastic moduli of the beam at any time, t, with respect to a predefined initial condition (baseline). The sensor output, as defined in figure 169, is a cumulative distribution of multiple normally distributed strain histograms defined by the equation in figure 173.

h subscript epsilon times open parenthesis epsilon closed parenthesis equals alpha divided by the square root of 2 times pi times sigma squared subscript i end square root times by e raised to the power of negative open parenthesis epsilon minus mu closed parenthesis squared divided by 2 times sigma squared subscript i, which equals alpha divided by the square root of 2 times pi times sigma squared end square root all times e raised to the power of negative open parenthesis epsilon minus mu closed parenthesis squared divided by 2 times sigma squared.
Figure 173. Equation. Cumulative distribution.

The standard deviation, the cumulative loading time, and the mean of the cumulative strain are evaluated from the parameters of the strain loading distributions shown in figure 174 through figure 176.

Alpha equals the sum of alpha subscript i.
Figure 174. Equation. Cumulative loading time.

E times open bracket epsilon closed bracket equals the sum of alpha subscript i divided by alpha times mu subscript i.
Figure 175 . Equation. Mean of the cumulative strain.

Var times open bracket epsilon closed bracket equals the sum of alpha subscript i divided by alpha times sigma subscript squared i.
Figure 176. Equation. Standard deviation.

Using figure 175 and figure 176, the mean and the standard deviation of the applied strain amplitude at a time t can be evaluated using two consecutive readings, as expressed by the following equations in figure 177 and figure 178.

Mu subscript t equals delta times open parenthesis mu times alpha closed parenthesis divided by delta times alpha.
Figure 177. Equation. Mean of the applied strain amplitude at time t.

Sigma subscript t equals open parenthesis delta times open parenthesis sigma squared times alpha closed parenthesis divided by delta times alpha closed parenthesis raised to the one-half power
Figure 178. Equation. Standard deviation of the applied strain amplitude at time t.

Once the mean and the standard deviation of the strain distributions are evaluated, Taylor series with exact derivation are used to derive the mean and the variance of the damage coefficient, which are given by figure 179 and figure 180.

E times open bracket D closed bracket equals mu subscript 0 divided by mu subscript N.
Figure 179. Equation. Mean of the damage coefficient.

Var times open bracket D closed bracket equals sigma squared subscript 0 divided by mu squared subscript N all plus mu squared subscript 0 times sigma squared subscript N divided by mu raised to the fourth power subscript N.
Figure 180. Equation. Variance of the damage coefficient.

The reliability index, considered with respect to a damage coefficient equal to 0, is then evaluated as follows:

Beta equals mu subscript 0 divided by the square root of sigma squared subscript 0 plus mu squared subscript 0 times sigma squared subscript N all divided by sigma squared subscript N end square root.
Figure 181. Equation. Reliability index.

The probability of failure, which is defined as the probability of the damage coefficient being less than 0, is then given by the following equation:

P times open parenthesis failure closed parenthesis equals one-half times open bracket 1 plus erf times open parenthesis negative
Figure 182. Equation. Probability of failure.

Expressing the failure of the structure in terms of probability of failure is more meaningful, given that the damage coefficient at failure is not a predefined value, and it varies from one specimen to another.

6.2 ALGORITHM EVALUATION USING CONCRETE BEAM FLEXURAL BENDING FATIGUE TESTS

A total of 23 plain PCC three-point single-edge notched beam specimens were tested under constant and variable amplitude fatigue loading. Two beam sizes were considered: the large-sized beams had a span of 16 inches (406.4 mm), a depth of 4 inches (101.6 mm), and a width of 4 inches (101.6 mm). The small-sized beams had a span of 8 inches (203.2 mm), a depth of 2 inches (50.8 mm), and a width of 2 inches (50.8 mm). The notch to depth ratio for each specimen was 0.35. A COD gauge was used to measure the crack mouth opening and was attached to a pair of knife edges, which were mounted to the bottom face of the beam by a fast-drying epoxy resin, as recommended by Shah et al.(6) Each specimen was subjected to a 2-Hz cyclical load. Ten specimens were subjected to constant amplitude loading using a stress ratio (max load/peak load) of 0.85 and 0.95. The other specimens were subjected to variable loading in which both the R ratio (minimum/maximum load) and the stress ratio were varied at several stages throughout the test.

The concrete mix used in this research consisted of ASTM C-150 type I cement, a natural sand, and a limestone coarse aggregate (nominal maximum size of 1 inch (25.4 mm)). The water-to-cement ratio was 0.45, and the air content was 6.5 percent. The unit weight was 142 lb/ft3 (22.7 kg/m3).

The average 28-day modulus of rupture and the split tensile strength, f't, were 760 and 419 psi (5,236.4 and 2,886.91 kPa), respectively. The 28-day compressive strength was 3,626 psi (24,983.14 kPa). The specimens were cured for 1 year inside of a humidity room and then placed in ambient temperature for one more month to ensure minimal strength gain during fatigue testing.

The full strain-time history output from the COD gauge was used as an input into the damage algorithm proposed. The measured peak strain distributions monitored by the COD gauge over the entire life of the specimens under constant and variable loading can be approximated by Gaussian distributions as shown in figure 183 through figure 185. The figures show the strain distribution at different life stages of a specimen subject to variable loading. The shift of the strain amplitude over time is due to the variation of material stiffness, which happens because the material is damaged.

The same observations remain valid for strain distributions at different life stages of a specimen under constant loading. However, the standard deviation is higher under variable loading, which is expected because there is an additional strain bandwidth caused by the variation in loading amplitude (and not damage).

This graph shows the strain distribution histogram at different life stages of the beam at 100 cycles. The x-axis shows strain amplitude, and the y-axis shows frequency. The mean value is at a strain of 3E-4 and a frequency of 16.
Figure 183. Graph. Strain distribution histogram at different life stages of the beam at 100 cycles.

This graph shows the strain distribution histogram at different life stages of the beam at 25,000 cycles. The x-axis shows strain amplitude, and the y-axis shows frequency. The mean is a strain of 4.2E-4 and a frequency of 350.
Figure 184. Graph. Strain distribution histogram at different life stages of the beam at 25,000 cycles.

This graph shows the strain distribution histogram at different life stages of the beam at 40,500 cycles. The x-axis shows strain amplitude, and the y-axis shows frequency. The mean strain amplitude is at 4.8E-4 and a frequency of 600.
Figure 185. Graph. Strain distribution histogram at different life stages of the beam at 40,500 cycles.

Using eight gates per sensor, the cumulative strain-time distributions are fitted (see figure 186 through figure 188). Using the mean and the amplitude of the distribution, the actual induced strains distribution can be evaluated using figure 177 and figure 178. The initial mean strain (o) is evaluated at the initial stage of specimen life (less than 100 cycles). Approximation of the extent of damage can thus be obtained.

This graph shows fitting the sensor's output at different life stages of the specimen at 100 cycles. The x-axis represents the strain level, and the y-axis represents cumulative time. At a strain of 7E-4, the cumulative time is 0 s. When the strain level decreases to zero, the cumulative time is 1,000 s.
Figure 186. Graph. Fitting the sensor's output at different life stages of the specimen at 100 cycles.

This graph shows fitting the sensor's output at different life stages of the specimen at 25,000 cycles. The x-axis represents the strain level, and the y-axis represents cumulative time. At a strain of 8E-4, the cumulative time is 0 s. When the strain level is zero, the cumulative time is 4,000 s.
Figure 187. Graph. Fitting the sensor's output at different life stages of the specimen at 25,000 cycles.

This graph shows fitting the sensor's output at different life stages of the specimen at 40,500 cycles. The x-axis represents the strain level, and the y-axis represents cumulative time. At a strain level of 8E-4, the cumulative time is just above 0 s. When the strain level becomes zero, the cumulative time is 8,500 s.
Figure 188. Graph. Fitting the sensor's output at different life stages of the specimen at 40,500 cycles.

Figure 189 shows the variation of the damage coefficient distribution versus the number of applied load cycles. The accumulation of damage is shown as a decrease of the damage coefficient mean value and a flattering of the distribution, explained by the increase of the uncertainty. As shown in figure 189, the mean damage index is decreasing over time, which is inversely proportional to the strain amplitude variation. However, the variance of the distribution is almost constant over the lifetime, with a fast change at the failure stage of the beam explained by important variability of the induced stain during failure. Once the standard deviation and the mean are evaluated, the reliability index and the probability of failure can be calculated using figure 181 and figure 182.

This graph shows the probability distribution of the damage coefficient versus the number of cycles of loading. The x-axis shows the damage index, the y-axis shows the density, and the z-axis shows the number of cycles. The graph shows a peak at a damage index in the range of 0.5 to 0.8 and at density of 1 for all of the cycles.
Figure 189. Graph. Probability distribution of the damage coefficient versus the number of cycles of loading.

Figure 190 through figure 193 show the variation of the reliability index of the damage coefficient as well as the probability of failure versus the number of cycles.

This graph shows the variance damage coefficient distribution. The number of cycles is on the x-axis, and damage mean is on the y-axis. The line begins at just above 0.8 for the damage mean and at 2,000 cycles. It quickly decreases to a damage mean of 0.55 at 60,000 cycles.
Figure 190. Graph. Variance damage coefficient distribution.

This graph shows variation of the mean. The number of cycles is on the x-axis, and variance is on the y-axis. The line begins at a variance around 0.2 at 2,000 cycles and remains at a steady variance until about 50,000 cycles when it increases to a variance of 0.95 at around 63,000 cycles.
Figure 191. Graph. Variation of the mean.

This graph shows the probability of failure of one of the samples versus the number of load cycles. Number of cycles is on the x-axis, and probability of failure is on the y-axis. The line begins at a probability of 0.025 during the first 4,000 cycles and increases to a probability of failure of 0.3 at 60,000 cycles.
Figure 192. Graph. Probability of failure of one of the samples versus the number of load cycles.

This graph shows the reliability index of one of the samples versus the number of load cycles. The number of cycles is on the x-axis, and reliability index is on the y-axis. The line begins at a reliability index of 2 and decreases to a reliability index of 0.5 at 60,000 cycles.
Figure 193. Graph. Reliability index of one of the samples versus the number of load cycles.

6.3 ESTIMATION OF REMAINING LIFE-PRELIMINARY RESULTS

Researchers estimated remaining life of the host structure using only the compressed data from the sensor and the models discussed in section 6.2. The evaluation of the deterministic values of the damage coefficient based only on the mean value has proven to be an unreliable indicator of remaining life. This is due to the high variability of the coefficient around failure.

Table 5 shows the reliability index, the damage coefficient, and the probability of failure just before failure. Due to the high variability of the damage index at failure, the probability of failure just before failure and the reliability index are not consistent. For a better remaining life estimation, the damage index variability at failure needs to be accounted for.

Table 5. Reliability index, probability of failure, and damage coefficient at failure for different specimens.


Sample

Reliability Index at Failure

Probability of Failure Just Before Failure

Damage Coefficient at Failure

1

0.54

0.29

0.3

2

0.77

0.25

0.44

3

0.72

0.31

0.56

4

0.51

0.31

0.46

6.3.1 Mechanistic-Empirical Approach

Figure 194 and figure 195 show the linear damage accumulation rule that is used in mechanistic-empirical (M-E) models. The coefficients,βi, are calibrated for every specimen using the sensor damage reading at the damage inflection point. Subramaniam et al. observed that under constant amplitude loading, the inflection point between the deceleration and the acceleration cracking region occurs at approximately 40 to 50 percent of the total life of the specimen.(7) Thus, the coefficients pertaining to the first half of the specimen's life should be similar to the second half. Once the coefficients are known, a remaining life prediction can be made.

D equals the sum of 1 divided by N subscript f.
Figure 194. Equation. Linear damage accumulation rule.

The log of open parenthesis N subscript f closed parenthesis equals beta subscript 0 times open parenthesis 1 divided by SR closed parenthesis raised to the power of beta subscript 1 plus beta subscript 2.
Figure 195 . Equation. Remaining life.

Table 6 shows the predicted remaining life using the described method based on the sensor output (models described in section 6.2) and based on the calibrated coefficient of the linear damage accumulation rule for different tested specimens. The loading of the specimens was stopped, and the remaining life was estimated using the different methods and based on the evaluated damage at that stage. The tests were then continued until failure in order to record the actual remaining life. As observed in the results, for the considered cases, the predictions evaluated using the localized sensor data are closer to reality than linear damage accumulation predictions based on averaged values.

Table 6. Predicted remaining life cycles using M-E calibrated coefficients and using the updated sensor output.


Exact Remaining Life (Loading Cycles)

Predicted Remaining Life Using Linear Damage Accumulation (Loading Cycles)

Predicted Remaining Life Using the Sensor (Loading Cycles)

391

709

325

20,527

716

5,873

420

835

425

9,350

902

7,125

7,022

922

11,048

10,980

990

23,011

6.3.2 Probabilistic Approach

Reliability engineering and survival analysis mostly deal with a positive random variable called "lifetime." The lifetime is manifested by a failure or other end event. In this case, failure is defined by the total break of the beam, and the lifetime variable is the time, T, at which the failure occurs with a cumulative distribution function F(T), defined by the probability of the damage index at time T being higher than the damage index at failure, as seen in figure 196.

F times open parenthesis T closed parenthesis equals P subscript r times open parenthesis D subscript f is less than D closed parenthesis.
Figure 196. Equation. Cumulative distribution function.

Figure 197 shows the density function of the damage coefficient at failure. A total of 63 specimens have been tested, and the index has been measured using COD. The fitted distribution is a logit-normal distribution.

This graph shows the probability density function of the damage index at failure. The graph shows a line in the shape of a bell curve and a bar graph that shows the results that the curve follows. The bar graph beings at a damage index at failure of 0.2 at a density 0.75, increases to a peak density of 3.25 at a damage index of failure of 0.4, and decreases to a density just below 0.25 at a damage index of 0.9.
Figure 197. Graph. Probability density function of the damage index at failure.

Researchers evaluated the survival probability function of the specimens based on the evaluated damage index obtained using the sensor and also evaluated the probability density function of the index at failure.

The remaining life CDF is defined using the law of conditional probability, the condition being that the beam did not fail at time t = x.

F subscript x times open parenthesis T closed parenthesis equals P subscript r open parenthesis, when T is greater than x but less than x plus t closed parenthesis divided by P subscript r, when T is less than x closed parenthesis. This is equal to F open parenthesis x plus t closed parenthesis minus F times open parenthesis x closed parenthesis divided by the average F times open parenthesis x closed parenthesis.
Figure 198. Equation. Remaining life CDF.

The corresponding survival probability function of the beam is given by the following equation:

F average subscript x times open parenthesis T closed parenthesis equals F average times open parenthesis x plus t closed parenthesis divided by the average F times open parenthesis x closed parenthesis.
Figure 199. Equation. Survival probability function of the beam.

Remaining life is then estimated to be the expectation of the survival probability function:

Rem equals E times open parenthesis T subscript t closed parenthesis, which also equals the integral from t to infinity of F average as a function of open parenthesis u closed parenthesis times du divided by F average times open parenthesis t closed parenthesis.
Figure 200. Equation. Expectation of the survival probability function.

However, the life probability function is not defined.

Remaining life should be expressed as a function of the damage index probability function (the only information that the sensor can provide). Using a change of variable, figure 199 can be expressed as a function of the damage index.

Rem equals E times open parenthesis T subscript t, which also equals the integral from t to infinity of F average subscript d times open parenthesis D closed parenthesis times the derivative of D divided by the derivative of D with respect to t divided by F average subscript d times open parenthesis D closed parenthesis.
Figure 201. Equation. Function of the damage index.

Where dD/dt is the variation of the damage index with respect to time evaluated by fitting a shape function to the discrete values evaluated using the sensor. The assumed shape functions are linear, exponential, and arcsine.

Figure 202 shows the normalized predicted remaining life (see figure 201) derived using the developed methodology based solely on the sensor's output. The associated probability (from figure 198) is shown in figure 203. As the number of applied cycles increases, more readings are incorporated into the adaptive models, which are used as fitting points. This implies that as the specimen gets closer to failure, the prediction accuracy improves, which is shown by the higher probability (reliability) of the estimated remaining life.

As discussed, the probability of the remaining life is a good indicator of predictions reliability. As shown in table 7, for a probability higher than 0.6, the relative error of the predicted remaining life is less than 50 percent.

This graph shows the normalized estimate remaining life versus the normalized specimen's lifetime using three fitting shape functions. The normalized specimen's lifetime is on the x-axis, and the normalized estimated remaining life is on the y-axis. The graph is a scatter plot, and there are three variables: linear fitting in blue, exponential fitting in green, and arcsine fitting in red. The arcsine fitting flows a linear path beginning around 0.45 on the y-axis and 0.25 on the x-axis and decreases to 0.25 on the y-axis and 1 on the x-axis. The exponential fitting begins following a linear path at around 0.35 on the y-axis and 0.18 on the x-axis and decreases to 0.3 on y-axis and 1 on the x-axis. Linear fitting begins at 0.2 on the y-axis and around 0.19 on the x-axis and runs almost horizontally to 0.15 on the y-axis at 1 on the x-axis.
Figure 202. Graph. Normalized estimated remaining life versus the normalized specimen's lifetime using three fitting shape functions.

This graph shows the remaining life probability versus normalized specimen's remaining lifetime using three fitting shape functions. The normalized specimen's remaining life is on the x-axis, and the remaining life probability is of the y-axis. The graph is a scatter plot, and there are three variables: linear fitting in blue, exponential fitting in green, and arcsine fitting in red. The arcsine fitting flows an increasing path beginning at the origin and ending at 0.8 on the y-axis and at 1 on the x-axis. The exponential fitting begins following a linear path at around 0.5 on the y-axis and zero on the x-axis and increases to 0.8 on y-axis and 1 on the x-axis. Linear fitting begins on 0.4 on the y-axis and increases to around 0.7 at 0.1 on the x-axis and increases linearly to 0.95 on the y-axis and 1 on the x-axis.
Figure 203. Graph. Remaining life probability versus normalized specimen's lifetime using three fitting shape functions.

Table 7. Estimated remaining life using the different fitting shape function.


Exact Remaining Life

Exponential Shape Function

Linear Shape Function

Arcsine Shape Function

Remaining Life

Prob.

Error (Percent)

Remaining Life

Prob.

Error (Percent)

Remaining Life

Prob.

Error (Percent)

0.9

0.1809

0.6406

71.91

0.3003

0.6071

59.97

0.4064

0.2133

49.36

0.8

0.2014

0.6565

59.86

0.3435

0.6149

45.65

0.4347

0.3459

36.53

0.7

0.2026

0.6913

49.74

0.3539

0.6309

34.61

0.4207

0.4346

27.93

0.6

0.2116

0.7688

38.84

0.3703

0.6746

22.97

0.4221

0.5455

17.79

0.5

0.2031

0.807

29.69

0.3629

0.6959

13.71

0.3881

0.5988

11.19

0.4

0.1919

0.8444

20.81

0.3496

0.717

5.04

0.3481

0.6436

5.19

0.3

0.1796

0.8751

12.04

0.3368

0.735

3.68

0.3115

0.6784

1.15

0.2

0.1719

0.9068

2.81

0.3287

0.7575

12.87

0.2836

0.7218

8.36

0.1

0.1617

0.9315

6.17

0.3171

0.7748

21.71

0.2525

0.757

15.25

6.4 DATA IMPUTATION-MISSING FULL-FIELD DATA GENERATION

Several researchers studied the effects of traffic wander on pavement performance.(8–10) It was shown that the wheel wander is critical because it determines the distribution of load location on the pavement as well as the frequency at which a point is loaded. Measurements from a field study conducted by Timm and Priest showed that wheel wander tends to be normally distributed with a standard deviation ranging from 8 to 24 inches (203.2 to 609.6 mm).(11) Given that only a limited number of sensors can be implemented on a pavement section, it is critical to generate full field data and incorporate the effect of traffic wander in the fatigue prediction algorithms that use the data collected from the implemented sensors. In addition, the obtained full field data can be used to backcalculate the traffic wander distribution at a given location by tracking strain peaks in the generated data in a pavement cross section. These problems could be considered as missing data problems. Several techniques have been used to generate missing data from a set of measurements at a set of predetermined locations. In this report, the Kriging technique was implemented to test its applicability to the specific data collected from the developed sensors.

The novel self-powered microsensors presented in chapter 2 of this report are capable of continuously monitoring local strains of the host structure. They implement the level-crossing counting algorithm. A network of N implemented sensors will generate N random variables
(X1,...,XN)that describe the response of the system at a given state.

Examples of strain output distributions are shown in figure 204. The shown data are obtained from the model of piezo-powered sensors attached on a simply supported beam with dimensions 12 × 1 × 1 inch (304.8 × 25.4 × 25.4 mm) and subject to a random applied loading at its center, inducing a mid-span deformation of amplitude varying between 0.04 and 0.12 inches (1.02 and 3.04 mm). PZT piezoelectric generators with dimensions of 0.4 × 0.2 × 0.004 inches (10.16 × 5.08 × 0.10 mm) were used in the simulation. The generators' output were obtained at three different positions (x = 2.9, 3.7, and 4.9 inches (73.66, 93.98, and 124.46 mm)), where x is the position of the piezo-strip center measured from the left end of the beam. The cumulative strain data at each sensor node can be fit to a variation of the exponential discrete probability distribution of the form f(y)= θ1eθ2y2+θ3y, where y = (1,2,...,7) is the memory cell number also associated with a strain amplitude level, and θT = (θ1, θ2, θ3)represents the parameter of the strain distribution at a given location. The vectors θT are specific to a location and a system's condition; they are the parameters to estimate at the missing locations.

This graph shows an example of data from distributed sensors on a simply supported beam under random loading. The x-axis represents memory cells, and the y-axis represents probability mass function. There are three lines displayed on the graph, the first is for the probability data for sensor 1. It begins at a probability mass function of 0.31 at one memory cell and lineally decreases to a probability mass function of 0.02 and seven memory cells. The next line represents the probability data for sensor 2. The line begins at a probability mass function of 0.25 at one memory cell and lineally decreases to a probability mass function of 0.05 and seven memory cells. The third line represents the probability data for sensor 3. It begins at a probability mass function of 0.2 at one memory cell and lineally decreases to a probability mass function of 0.09 and seven memory cells.
Figure 204. Graph. Example of data from distributed sensors on a simply supported beam under random loading.

In its simplest form, the basic goal of the Ordinary Kriging (OK) is to estimate the attribute value at an unobserved location by interpolating the observed values in the neighborhood locations. If (ui, i = 1…, n) at n locations in a region of interest R where the field data X has been observed, and u denotes a specified but arbitrary unobserved location in the region R, then the value to estimate X* (u) at location u is given by the following equation:

X times open parenthesis mu closed parenthesis equals the sum of alpha subscript i times open parenthesis mu closed parenthesis times X times open parenthesis mu subscript i closed parenthesis plus the absolute value of 1 minus the sum of alpha subscript i times open parenthesis mu closed parenthesis end absolute value times m times open parenthesis mu closed parenthesis.
Figure 205. Equation. Estimate of X as a function of mu.

Where m(u) is the mean of X(u) and αi(u) are the Kriging weights that can be determined for the case of an OK formulation by solving the following system of equations:

The sum of alpha subscript j times open parenthesis mu closed parenthesis times gamma times open parenthesis mu subscript i minus mu subscript j closed parenthesis plus l times open parenthesis mu closed parenthesis times open parenthesis the summation of alpha subscript j times open parenthesis mu closed parenthesis closed parenthesis equals gamma times open parenthesis mu subscript j minus mu closed parenthesis. The second equations states that the sum of alpha subscript j times open parenthesis mu closed parenthesis equals 1.
Figure 206 . Equation. System of equations to solve for the case of an OK formulation.

Where:
(uiuj) = Distance between location i and location j.
(uiu) = Distance between location i and the location to be estimated.
l(u) = Lagrange parameter.
y = Predefined semi-variance property that expresses the degree of spatial dependence
between points.

A simply supported beam setup was initially used to test and tune the developed techniques under simplified strain distributions; the load configuration consisted of a moving concentrated load for generating linear strains along the beam and a distributed load for quadratic strains. Simulated strain data were generated using the piezoelectric generator and the sensor models described in chapters 2 and 3. The frequency distribution of strain levels at a selected reference location (x = 3.7 inches (93.98 mm)) of the beam was estimated for different loading and known data combinations. One million cycles of random amplitude for each loading combination were simulated. Values of y in figure 206 were extracted using linear and quadratic variograms. It was determined that higher order variograms do not improve the obtained estimations for the considered strains. It was also observed that a better estimation is obtained when the missing data point is located between the known points. For example, the cumulative relative error when the known couple is located at 3.34 and 4.1 inches (84.84 and 104.14 mm) is 0.0193, while it is equal to 0.0567 for the case of a known couple at 4.1 and 5.3 inches (104.14 and 134.62 mm). The error decreases to 0.0433 for 3.34 and 5.3 inches (84.84 and 134.62 mm) even though the relative distances to the point at which an estimate is desired are the same.

An example of estimated probabilities obtained using the Kriging technique is shown in figure 207. The probability mass function of the strain levels at the location x = 3.7 inches (93.98 mm) if a sensor was to be installed at that position is also shown. Second order polynomial variograms were used for this case. The computed errors are 0.0415, 0.0694, and 0.072, respectively, for the cases 3.34, 4.1, and 4.5 inches (84.84, 104.14, and 114.3 mm), 3.34, 5.3, and 5.7 inches (84.84, 134.62, and 144.78 mm), and 4.1, 5.3, and 5.7 inches (104.14, 134.62, and 144.78 mm).

This graph shows the theoretical and estimated strain probability distributions at 3.74 inches (94.99 mm) using data from groups of three sensors at different locations. The x-axis shows the number of memory cells, and the y-axis shows the probability mass function. The scatter plot shows the estimated data for three different sets of locations the overall highest results begin for locations at 4.13, 5.31, and 5.71 inches (10.5, 13.5 and 14.5 cm). The next highest points are for the locations of 3.35, 4.13, and 4.53 inches (8.5, 10.5, and 11.5 cm). The lowest results are for the locations at 3.35, 5.31, and 5.71 inches (8.5, 13.5, and 14.5 cm). All the data follow the same decreasing trend, beginning at 0.37 at one memory cell and decreasing to a probability mass function of zero at seven memory cells.
1 inch = 2.54 cm
Figure 207 . Graph. Theoretical and estimated strain probability distributions at 3.74 inches (94.99 mm) using data from groups of three sensors at different locations.

The refined algorithms were then tested under more realistic conditions with complicated strain profiles. The objective was to prove the validity of the methods for pavement structures. The three-dimensional response of a layered system under a moving load with static and dynamic components has been modeled. The properties of the used layered system were obtained from Chabot et al.(12) The three-layered system consists of a top viscoelastic layer modeled through the Huet-Sayegh model, a road base (3.14-inch (79.76-mm) thickness), and a subbase layer (16.5-inch (419.1-mm) thickness) both assumed to be elastic and dependent on thermal and moisture characteristics.(13–16) Detailed properties are given in Chabot et al. and Nilsson et al.(12,16)

Traffic distributions were generated and applied as input loading to the pavement structure. Four different types of trucks were considered in the analysis: classes 9, 11, 5, and 16.(17) To simulate traffic wander, five different possible positions within the wheel path were selected. In total, 1,000 passing truck events were simulated. Each event corresponds to a randomly selected truck type passing at a randomly selected position within the wheel path. The loading per axle as applied in this analysis was as follows: 15,400 lb (6,991.6 kg) for steering axle, 18,000 lb (8,172 kg) for single axle, 16,000 lb (7,264 kg) for tandem axle, and 13,000 lb (5,902 kg) for tridem and higher axles. Figure 208 and figure 209 show an example of generated longitudinal strain response for the class 9 trucks at the bottom of the viscoelastic layer. The results indicate a significant interaction between the axles and relatively large tensile strains (compressive strains at the surface of the pavement). At each node location in figure 208 and figure 209, the cumulative strain data induced by randomly generated truck traffic distributions were compressed into probability mass function histograms using the piezoelectric generator and the sensor models described above. The objective of this exercise is to recreate the probability mass functions of cumulative strains induced by all the loading events at all locations within the pavement section using only a finite number of sensors implemented at determined node locations.

This illustration shows an example of a truck used for strain response data generation. A side view of a semi-truck is shown. There are five tires with the distances marked between its tires shown. The distance from the first tire to the second is 12 ft, the distance from the second to third is 52 inches, the distance from the second to the third is 34 ft, and the distance from the third to the fourth is 48 inches.
Figure 208. Illustration. Example of a class 9 truck used for strain response data generation.

This figures shows a three-dimensional graph of a longitudinal strain profile evaluated at the bottom of the hot mix asphalt (HMA) layer for a moving load induced by a class 9 truck. The x-axis represents time, the y-axis represents strain, and the z-axis represents the traverse position. There are 10 major peaks in strain all reaching almost 80E-6 at transverse positions of -3.28 and 3.28 ft (-1 and 1 m).
1 ft = 0.305 m
Figure 209. Graph. Example of longitudinal strain profile evaluated at the bottom of the HMA layer for a moving load induced by a class 9 truck.

The frequency distribution of strain levels at a selected location 9 inches (228.6 mm) away from the center of the wheel path under the moving load is shown in figure 210. Estimated probability mass functions using two sensors placed on each side of the unknown data node are also shown. Results from known nodes at 3.93- and 7.87-inch (99.82- and 199.89-mm) spacings (1.96 and 3.93 inches (49.78 and 99.82 mm) from the location to estimate) are shown. Values of y in figure 206 were extracted using cubic variograms. It was determined that higher order variograms do not improve the obtained estimations for the considered strains.

This graph shows the theoretical and estimated strain probability distributions at a selected transverse location using data from two sensors at different spacing distances. The x-axis represents the number of memory cells, and the y-axis represents the probability mass function. Two sets points are shown on the graph all following the same linear tread. The two sets of points represent the estimated data using distances of 3.93 and 7.87 inches (10 and 20 cm), respectively. The lines follow the linear path of the line representing the probability mass function. The data begin at a probability mass function between 0.16 and 0.165 at one memory cell and decrease to a probability mass function 0.12 at seven memory cells.
Figure 210. Graph. Theoretical and estimated strain probability distributions at a selected transverse location using data from two sensors at different spacing distances.

Figure 211 shows the computed relative error obtained from estimated probability distributions at all the nodes in the pavement section for different known nodes spacing distances varying from 4 to 40 inches (101.6 to 1,016 mm) (the nodes' spacing resolution is 1.96 inches(49.78 mm)). The averaged and maximum observed values are shown for each case. It can be seen that in order to achieve full reconstruction of the data in all the field points with an average error less than 10 percent, the maximum spacing between placed sensors has to be less than 7.87 inches (199.89 mm).

This graph shows the maximum observed relative error and average relative error from generated data at all field points using known nodes at different spacing distances. The x-axis represents the distance between two known points, and the y-axis represents the relative error. There are two sets of data points shown on the scatter plot: the maximum observed error and the average error. Both sets of data vary. If a trend line was added, the maximum observed error would begin at a relative present error between 70 and 80 percent at 0.98 ft (0.3 m) and decrease to between 50 and 60 percent and 3.28 ft (1 m). The trend line for the average error would begin around 10 percent at an estimated distance of 0.65 ft (0.2 m) and increase to a relative error between 30 and 40 percent at 3.28 ft (1 m).
1 ft = 0.305 m
Figure 211. Graph. Maximum observed relative error and average relative error from generated data at all field points using known nodes at different spacing distances.

6.5 CONCLUSION

In this chapter, a sensor-specific data interpretation algorithm for predicting remaining fatigue life of a pavement structure was developed using cumulative limited compressed strain data stored in the sensor memory chip. The algorithm was verified using actual laboratory fatigue test results of a notched concrete beam under constant, variable, and random loading histories.


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration