SPECTRAL ANALYSIS OF HIGHWAY

PAVEMENT ROUGHNESS

By Jorge Marcondes,' Gary J. Burgess,2 R. Harichandran,3 Associate Member, ASCE, and Mark B. Snyder,4 Associate Member, ASCE

(Reviewed by the Highway Division)

ABSTRACT: This paper reports new findings in the spectral analysis of pavement roughness. Elevation profiles of federal and interstate highways were measured with a profilometer, and spectral analysis was used to compute power spectral densities for the longitudinal elevation of pavements. The longitudinal profiles ob­tained were also used to compute the International Roughness Index (IRI) using a quarter-car simulation method. An extensive review of the existing techniques for pavement spectral analysis was done from which it was found that previously de­veloped models do not explain pavement spectral properties very well, especially for long wavelengths in the spectral decomposition. A new equation that fits the power spectral density curves for several pavement types was developed. A set of models was also developed to predict the power spectral density as a function of pavement type and age, and the IRI.

INTRODUCTION

Pavement roughness analysis has been the subject of considerable research for many years (Gillespie 1985). A satisfactory mathematical model that ex­plains the spectral properties of pavement sections has not been developed to date. It has been found that most pavement profiles have very similar power spectral densities (PSD) and that a single parameter known as the roughness index shifts the power density levels up or down, depending on the overall level of roughness (Gillespie 1985). Some authors have even reduced the PSD curves to only two families, one for rigid pavements and one for flexible pavements (Gillespie 1985). Since pavement roughness has become increasingly important in the prediction of truck bed acceleration levels in relation to transportation hazards (shock and vibration), it has be­come necessary to develop better PSD models.

Spectral techniques are necessary for analyzing pavement roughness be­cause pavement profiles generally have many of the statistical properties of random signals. Several studies of the spectral characteristics of various pavement profiles showed many similarities (Gillespie et al. 1980; Gillespie and Sayers 1981; Gillespie 1985). Eq. (1) was developed to predict the PSD levels for average road properties (Gillespie 1985)

Gz(v) = G0[l + (v20v-2)] • ( 2 ™ r 2 (1)

where G2(v) = power density value, cu in./cycle (m3/cycle); v = spatial frequency, cycle/in. (cycle/m); G0 = roughness magnitude parameter (the

'Lect., Packaging Tech., Massey Univ., Palmerston North, New Zealand. 2Assoc. Prof., School of Packaging, Michigan State Univ., East Lansing, MI 48824-

1223. 3Asst. Prof., Civ. and Envir. Engrg., Michigan State Univ., East Lansing, MI. 4Asst. Prof., Civ. and Envir. Engrg., Michigan State Univ., East Lansing, MI. Note. Discussion open until February 1, 1992. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on May 30, 1990. This paper is part of the Journal of Transportation Engineering, Vol. 117, No. 5, September/October, 1991. ©ASCE, ISSN 0733-947X/91/0005-0540/ $1.00 + $.15 per page. Paper No. 26135.

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parameter that differentiates rough from smooth roads), cu in./cycle (m3/ cycle); and v0 = spatial frequency cutoff (the lowest spatial frequency to be used in the model), cycles/in. (cycles/m).

Eq. (1), used in combination with random number sequences provides a method for generating pavement-elevation profiles of random roughness with the spectral qualities of typical roads.

Early studies (Houbolt 1962; Van Deusen and McCarron 1967) of lon­gitudinal profiles of airport runways and roads exhibited PSDs that could be expressed by

G2{v) = A(2TTV)-2 (2)

where A = roughness magnitude parameter with the units of cu in./cycle (m3/cycle).

Sayers (1985) demonstrated that (2) fails for profiles with a large range of spatial frequencies and suggested a more generalized PSD

Gz{v) = Ava (3)

where a = an experimentally determined exponent. Eq. (2) is a special case of (3) with a = - 2 [the 2TT factor of (2) is

incorporated in A of (3)]. When (3) cannot be used throughout the entire range of spatial frequencies, a piecewise fit is suggested to cover different ranges (Dodds 1974)

GUv) = GM)(^j . v s v0 (4c)

Gz{v) = GJivJ^-J , v>v0 (4b)

Characteristic power spectral density functions for pavement roughness were also studied and developed by Sayers (1985) who used measured pave­ment profiles to develop an empirical model for random road inputs to ve­hicles. The model is based on a summation of white-noise sources. The following equations were suggested for road elevations, road slopes, and road accelerations, respectively

Gz(v) = Ga(2irw)~4 + GS(2TTV)'2 + Ge (5)

G,'(v) = Ga(2-nvy2 + GS + Ge(2-nv)+2 (6)

G2"(v) = Ga + GS(2TTW)+2 + Ge(2-uv)+A (7)

where Ga = white-noise acceleration, in."'/cycle (m~'/cycle); Gs = white-noise slope, inch/cycle (m/cycle); and Ge = white-noise elevation, in.3/cycle (nrVcycle).

According to Sayers et al. (1985), these models fit the data obtained from the international road roughness experiment conducted in 1982 involving several countries and from the 1984 Ann Arbor profilometer meeting (a val­idation meeting for profilometers conducted at the University of Michigan, Ann Arbor, Mich.) as well as PSD functions for European and Texas roads. It is important to point out that (5), (6), and (7) were developed for profiles with no peculiar characteristics such as "concentrated roughness" (e.g. rail­road tracks, potholes, and other inhomogeneities).

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EXPERIMENTAL DESIGN

Twenty different pavement sections approximately 0.19 mi (306 m) in length [as recommended for roughness studies (Sayers et al. 1986b)] were surveyed using a Michigan Department of Transportation (MDOT) profilom­eter to measure the pavement profile.

The pavement sections used in this experiment were selected with several considerations in mind.

1. The sections should be typical and represent common routes for tractor-semitrailer trucks.

2. The pavement types and conditions should be uniform along the entire length of each section but different between sections.

3. Each site should have enough distance between the point of access and the beginning of the section to allow the profilometer to develop the required test speed [minimum of 25 mph (40.2 km/hr)].

4. Each section should be of nearly constant longitudinal slope. 5. Each section should be located with easily identifiable starting and ending

points.

The sections that met these requirements and were used in the experiment are located on 1-69, 1-94, and US-127, in Michigan near Lansing.

Profilometer readings were taken at 3 in. (0.0762 m) intervals (as rec­ommended by MDOT and other highway agencies) and stored in a digital computer. This sampling frequency was at least twice as large as the max­imum frequency to be considered in the PSD plot and therefore avoided the problem of aliasing (Newland 1984). At this sampling rate, 4,000 elevation readings were taken over each 0.19 mi. (306 m) section.

The profilometer was run over each test section four times. The resulting measurements were very precise, with a coefficient of variation (standard deviation/mean) of no more than 5%. The pavement elevation PSDs were computed for a maximum spatial frequency of 0.06 cycle/in. (2.36 cycle/ m), which corresponds to an input of approximately 50 Hz at 45 mph (72.4 km/hr), the upper limit of frequencies that have been shown to cause damage to most truck shipments (Tevelow 1983; Marcondes et al. 1988; Antle 1989). Contact between the truck tire and the pavement smooths out high-frequency oscillations in the pavement profile, thereby limiting the maximum spatial frequency that can be studied.

DATA AND RESULTS

The PSD plots and root mean square (RMS) elevation values were com­puted using the measured elevation profile as input to a locally written com­puter program. The program estimates the PSD by computing a suitably windowed autocorrelation function followed by a Fourier transform (Cran-dall and Mark 1963; Bendat and Piersol 1971; Newland 1984). The RMS elevations for the pavement sections varied between 0.07 in. and 0.16 in. (0.18 cm and 0.41 cm). Since the RMS elevation is the square root of the variance about the mean, and the mean is zero, the variance is numerically equal to the area under the PSD curve between zero and 0.063 cycles/in. (2.48 cycle/m). The RMS elevation is a useful intensity parameter in com-

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U- , , , , , I 0.000 0.005 0.010 0.015

Spatial Frequency (cycle/in.)

FIG. 1. Graph of (8)

paring PSD curves of the same shape. An equation was fitted to the pavement-elevation power-spectrum-density

data to facilitate its estimation at any point. Several of the equations pro­posed in the literature were tried first, including ( l ) - (5) . Because the section profiles exhibited some peculiarities in the form of concentrated roughness, none of these equations fit the data very well. Thus, it was necessary to develop new equations for these sections,

PDpe(v) = A, exp (-kvp), v<Vl (8a)

PDpe(v) = A2(v - v0)\ v>vt (fib)

where PDpe (v) = power density value [cu in./cycle (m3/cycle)] for the pave­ment elevation as a function of spatial frequency v; v{ = discontinuity fre­quency, cycles/in. (cycles/m); v0 = asymptote frequency, cycles/in. (cycles/ m); and Au A2, k, p, and q = constants.

A graph of (8) is shown in Fig. 1. The fundamental difference between ( l ) - (5) , and (8) lies in the predicted PSD values for v < vx (large wave­lengths), which are important in predicting truck-trailer response. Large wavelengths are responsible for the low-frequency inputs to the vehicle.

The value of vx was set to 0.005208 cycles/in. (0.205 cycles/m) by in­spection since this is the maximum value for v that fits the data well for the first part of (8).

TABLE 1. Ranges of Variable Values for (8) (v s vs)

Category (1) OC NC AC

Aj

(2)

1.3-7.2 1.5-3.4 1.8-5.7

k (3)

7,000-67,000 24,000-83,000 63,000-240,000

P (4)

1.6-2.0 1.8-2.0 2.0-2.2

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TABLE 2. Ranges of Variable Values for (8) fr > vt)

Category d) OC NC AC

A2

(2)

5 .9E-7 -4 .2E-5 6 .0E-7 -6 .0E-5 1 .4E-4-7 .7E-4

Wo

(3)

0 -3 .9E-3 2 . 5 E - 3 - 4 . 9 E - 3 4 . 6 E - 3 - 5 . 2 E - 3

1 (4)

- 2 . 6 — 1 . 5 - 2 . 2 — 1 . 1 - 1 . 1 — 0 . 5

Tables 1 and 2 present the values of the remaining parameters for (8) that were found to fit the data obtained for each pavement section using a least-squares best-fit approach.

Eq. 8 was found to produce PSD curves for each of the pavement test sections. These curves were plotted and compared. It was found by inspec­tion that the pavement sections studied in this experiment could be divided into three categories differentiated according to pavement type and age.

1. Category OC—old portland cement concrete (PCC) pavement. 2. Category NC—new portland cement concrete pavement. 3. Category AC—asphalt concrete-surfaced pavement flexible pavements and

overlay of PCC pavements.

One sample PSD and fitted equation is shown for each category in Figs. 2-4. The International Roughness Index (IRI) was computed for each sec­tion using the quarter-car simulation technique described by Sayers et al. (1986a). The average IRI computed for each test section varied between 46 and 280 in./mi (0.73 and 4.42 m/km).

Fig. 5 shows the relationship between IRI and RMS pavement elevation

10-]

v o o

" c 1E-01-

'w c

Q 1E-02-k_ 4) 5 o °- IE-03-

1E-04-I , , r - r - i , , r - r - , , . r - r - , 1E-04 1E-03 1E-02 1E-01

Spatial Frequency (cycle/in.)

FIG. 2. Elevation PSD and Fitted Equation for One Section of Pavement Cate­gory OC

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o >> o

" c 1E-01

& <u * o

Q.

1E-02-

1E-03-

1E-04-

Fitted curve

Elevation PSD

1E-04 T r - j 1 1 i i—l

1E-03 1E-02

Spatial Frequency (cycle/in.)

1E-01

FIG. 3. Elevation PSD and Fitted Equation for One Section of Pavement Cate­gory NC

for the sections studied. Each point on the graph represents one of the four replicates of the profilometer readings for each section. Correlations between RMS and IRI were somewhat weak [best-fit curves for each category using regression analysis produced correlation coefficients (r2) of less than 0.7].

Fitted curve

Elevation PSD

1E-04 1E-04 1E-03 1E-02

Spatial Frequency (cycle/in.)

1E-01

FIG. 4. Elevation PSD and Fitted Equation for One Section of Pavement Cate­gory AC

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0.20-,

0.16-

0.12-

0.08-

0.04-

„ . -D ocP

A Category AC p Category NC o Category PC

100 200

IRI (inch/mile)

FIG. 5. RMS of Pavement Elevation versus IRI

300

Comparing the IRI values for all of the sections studied with the param­eters in (8) leads to the following conclusions.

1. The value of vx = 0.005208 cycle/in. (0.205 cycle/m) corresponds to 4.1 Hz at 45 mph (72.4 km/hr), 4.8 Hz at 52.5 mph (84.5 km/hr) and 5.5 Hz at 60 mph (96.5 km/hr). These frequencies of truck bed vibrations, mainly produced by the pavement roughness, usually match the resonant frequencies of many truckload shipments.

2. In (8), parameters Al and k do not correlate with the IRI. The parameter p, which represents the sharpness of the exponential curve in (8), ranges from 1.8 to 2 for PCC pavements and between 1.9 and 2.3 for asphalt concrete-surfaced pavements.

3. Parameters A2, v0, and q (8), vary with pavement type and age, but do not correlate strongly with the IRI.

The lack of correlation between the IRI and the parameters of (8) is ex­plained by the fact that these parameters describe a section of PSD pavement elevation that is well below the resonant frequency of the quarter-car model (Marcondes 1990). It was found that for spatial frequencies between 0.002 cycle/in. (0.0787 cycle/m) and 0.015 cycle/in. (0.591 cycle/m), a good cor­relation exists between the IRI and the PSD values for the pavement ele­vation. The following equations for this relationship were developed for sev­eral frequency ranges and each of the three pavement categories described previously.

1. OC, 0.002-0.005 cycle/in. (0.0787-0.1969 cycle/m)

PDpe = -20.26 + 0.2583 • IRI + 3,806 • v - 48.32 • v • IRI

-5.814E-4-IRI2 + 0 .1107-vIRI 2 (9)

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with R2 = 0.697; and sum of the square of errors = 0.6361.

2. NC and AC, 0.002-0.005 cycle/in. (0.0787-0.1969 cycle/m)

PDpe = 5 . 1 0 9 4 - 6 . 9 9 7 E - 2 - I R I - 1,132.4-v + 16.64-w-IRI

+ 5.473E-4-IRI2-0.1179-w-IRI2 (10)

with R2 = 0.869; and sum of the square of errors = 0.3096.

3. OC, NC, and AC, 0.005-0.007 cycle/in. (0.1969-0.2756 cycle/m)

PDpe = - 0.072887 + 0.01146 • IRI - 1.1756 • v • IRI

+ 1.1637E-3-W-IRI2 (11)

with R2 = 0.938; and sum of the square of errors = 0.0657.

4. OC, NC, and AC, 0.007-0.01 cycle/in. (0.2756-0.3937 cycle/m)

PDpe = - 0.015746 + 2.8309E-5 • IRI2 - 2.3897E-3 • v • IRI2 (12)

with R2 = 0.984; and sum of the square of errors = 0.0161.

5. OC, NC, and AC, 0.01-0.013 cycle/in. (0.3937-0.5118 cycle/m)

PDpe = - 7.3642E-4 + 1.0968E-5 • IRI2 - 6.805E-4 • v • IRI2 (13)

with R2 = 0.969; and sum of the square of errors = 0.0089. 6. OC, NC, and AC, 0.013-0.015 cycle/in. (0.5118-0.5906 cycle/m)

PDpe = 2.5276E-3 - 7.1989E-3 • v • IRI + 5.6971E-6 • IRI2

- 2.565 • v • IRI2 (14)

with R2 = 0.934; and sum of the square of errors = 0.0077.

Eqs. 9-14 were obtained using multiple linear regression analysis with 95% confidence levels. These equations are valid only for the IRI in in./mi and PDpe in cu in./cycle.

High levels of correlation were also observed between the IRI and the PSD values for pavement elevation for spatial frequencies that correspond to input frequencies above approximately 5 Hz for vehicle speeds between 45 mph (72.4 km/hr) and 60 mph (96.5 km/hr). This is explained by the fact that the quarter-car simulation model used to compute the IRI is much more sensitive to pavement oscillations in this range (Marcondes 1990).

Relationships between IRI and PSDpe above 12 Hz were not studied be­cause previous research has shown that frequencies in the 3—7 Hz range cause the greatest truck-trailer response and produce the greatest amount of product damage during transportation (Marcondes et al. 1988; Tevelow 1983).

CONCLUSIONS

Power spectral density functions of pavement elevation can be modeled for a broader range of spatial frequencies using (8). This is very advanta­geous for situations that require accurate PSD values at low spatial frequen­cies. Such is the case when predicting the truck-trailer response.

Eqs. (9)-(14) strongly support the conclusion that the IRI and pavement categorization can be used to predict PSDs of pavement elevations.

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Applications to Civil Engineering Practice The results of this study are directly applicable in the spectral analysis of

pavements, since the spectral densities of longitudinal profiles can be pre­dicted using the International Roughness Index, which has become fre­quently used, after the international road roughness experiment, conducted in 1982 (Sayers et al. 1985).

Findings of the present study can be immediately applied to predict the road inputs to vehicles and therefore relate to the effects of traffic on pave­ments such as damage potential and so forth. These findings are also pow­erful tools to predict the dynamic response of trucks once the transmissibility function of the vehicle is known. Testing the dynamic performance of loads for truck shipments is another broad application in the transportation indus­try.

ACKNOWLEDGMENTS

The writers would like to thank the Michigan Department of Transpor­tation for providing a road profilometer to collect the data for this study.

APPENDIX I. REFERENCES

Antle, J. R. (1989). "Measurement of lateral and longitudinal vibration in commer­cial truck shipments," thesis presented to Michigan State University, at East Lan­sing, Mich., in partial fulfillment of the requirements for the degree of Master of Science.

Bendat, J. S., and Piersol, A. G. (1971). Random data: analysis and measurement procedures. Wiley Interscience, John Wiley and Sons, New York, N.Y.

Crandall, S. H., and Mark, W. D. (1963). Random vibration in mechanical systems. Academic Press, New York, N.Y.

Dodds, C. J. (1974). "The laboratory simulation of vehicle service stress." J. Engrg. Industry: Trans. ASME, 96(3), 391-398.

Gillespie, T. D. (1985). Heavy truck ride: SP-607, Society of Automotive Engineers, Warrendale, Pa.

Gillespie, T. D., and Sayers, M. W. (1981). "Better method for measuring pavement roughness with road meters." Transp. Res. Record 836, Transportation Research Board, Washington, D.C.

Gillespie, T. D., Sayers, M. W., and Segel, L. (1980). "Calibration of response-type road roughness measuring systems." NCHRP Report 228, Transportation Re­search Board, Washington, D.C.

Houbolt, J. C. (1962). "Runway roughness studies in the aeronautical fields." Paper N3364, ASCE Trans., ASCE, New York, N.Y.

Marcondes, J. (1990). "Modeling vertical acceleration in truck shipments using the International Roughness Index for pavements," thesis presented to Michigan State University, East Lansing, Mich., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Marcondes, J., Singh, S. P., and Burgess, G. J. (1988). "Dynamic analysis of a less than truck load shipment." Paper 88-WA/EEP-17, ASME, New York, N.Y.

Newland, D. E. (1984). An introduction to random vibration and spectral analysis. Longman, New York, N.Y.

Sayers, M. W. (1985). "Characteristic power spectral density functions for vertical and roll components of road roughness." Proc. Symp. on Simulation Control of Ground Vehicles and Transp. Systems, ASME, New York, N. Y.

Sayers, M. W., Gillespie, T. D. and Paterson, D. O. (1986). "Guidelines for con­ducting and calibrating road roughness measurements." World Bank Tech. Paper No. 46, World Bank, Washington, D.C.

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Sayers, M. W., Gillespie, T. D. and Queiroz, C. A. V., (1986). "The international road roughness experiment—establishing correlation and calibration standard for measurements." World Bank Tech. Paper No. 45, World Bank, Washington, D.C.

Tevelow, F. L. (1983). "The military logistical transportation vibration environment: Its characterization and relevance to MIL-STD fuze vibration testing." U.S. Army Electronics Research and Development Command, Harry Diamond Laboratories, Adelphi, Md.

Van Deusen, B. D., and McCarron, G. E. (1967). "A new technique for classifying random surface roughness." Trans. SAE 670032, Society of Automotive Engi­neers, New York, N.Y.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A = roughness magnitude parameter (cu in./cycle); AuA2,k,p,q = constants;

Ga = white-noise acceleration (in. - ' /cycle); Ge = white-noise elevation (cu in./cycle); Gs = white-noise slope (in./cycle);

Gz(v) = power spectral density value (cu in./cycle); G0 = roughness magnitude parameter (the parameter that dif­

ferentiates rough from smooth roads) (cu in./cycle); PDpe(u) = power density value (cu in./cycle) for the pavement el­

evation as function of spatial frequency v; v = spatial frequency (cycle/in.);

w0 = asymptote frequency (spatial) (cycles/in.); vl = discontinuity frequency, cycles/in.; and a = experimentally determined exponent.

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