MODELING INDIRECT STATISTICS OF SURFACE ...

MODELING INDIRECT STATISTICS

OF

SURFACE ROUGHNESS

By Lu Sun,1 Zhanming Zhang,2 and Jessica Ruth3 (Reviewed by the Highway Division) ABSTRACT: Quarter-car model-based indirect statistics of road surface roughness are analyzed in this paper. The stochastic process theory is used to establish the relationship between the international roughness index (IRI ) statistic of a quarter-car response and the power spectral density (PSD) of road random excitation. Under the circumstance of a linear vehicle model and homogeneous roughness with a zero-mean Gaussian distribution, we prove that the average of the absolute response of the quarter-car model is directly proportional to the standard deviation of that response quantity. This result enables us to further correlate the indirect statistics with PSD roughness. A theoretical relation between the IRI and the PSD roughness is established. Based on the road classification proposed by the International Standards Organization (ISO), it is found that a linear correlation exists between the IRI and the standard deviation of roughness. Furthermore, corresponding to the ISO PSDbased road classification, an IRI-based road classification is provided to categorize pavement into five classes, ranging from very good surface condition to very poor surface condition.

INTRODUCTION Background The concept of pavement management has been proposed for over two decades. The methodology and procedure developed in the process of pavement management have been successfully applied to bridge management, railway management, and airport management. They have also been widely accepted and adopted by transportation agencies to facilitate more generalized transportation infrastructure management. As a key component of pavement mangement systems, pavement performance is considered the most critical quantity that serves as the overall objective of the optimization for allocating funds and selecting maintenance strategies. Pavement performance is scaled by the present serviceability index (PSI) in the early stage of pavement design and management. Carey and Irick (1960) are the first who proposed this concept of serviceability during the American Association of State Highway Officials (AASHO) road test. The PSI is given as the regressed panel rating of pavement serviceability, which is obtained while a panel is driving a test vehicle and traveling over a pavement section. As a combined index, the PSI is a function of pavement surface roughness, cracking, and rutting. The fundamental relationship found from the AASHO road test shows that approximately 95% of the PSI is contributed by roughness (Haas et al. 1994). Since the PSI is, in fact, the human perception or subjective judgment of ride quality, neither cracking nor rutting has a significant contribution to the human feeling. It is then realized that it is not very rational to evaluate PSI in terms of three different factors (i.e., roughness, cracking, and rutting) (Janoff et al. 1985; Janoff 1988; Haas et al. 1994). In fact, today many state highway departments and transportation agencies measure only road roughness for estimating road serviceability (Gillespie et al. 1980). As a result, further studies have also been conducted in order 1 PhD Candidate, Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712. E-mail: [email protected] 2 Asst. Prof., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX. E-mail: [email protected] 3 Grad. Student, Dept. of Civ. Engrg., Univ. of Texas, Austin, TX. E-mail: [email protected] Note. Discussion open until September 1, 2001. 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 December 22, 1999; revised September 22, 2000. This paper is part of the Journal of Transportation Engineering, Vol. 127, No. 2, March/April, 2001. 䉷ASCE, ISSN 0733-947X/01/00020105–0111/$8.00 ⫹ $.50 per page. Paper No. 22223.

to quantify pavement performance. For example, the concept of ride number has been proposed in a National Cooperative Highway Research Program (NCHRP) report to assess ride quality, in which only roughness is accounted for (Janoff et al. 1985; Janoff 1988). In the highway industry, many statistics have been developed for evaluating pavement surface roughness since the 1960s (Gillespie et al. 1980; Hudson 1985; Haas et al. 1994). One of the most popular statistics is the international roughness index (IRI ), developed and recommended by the World Bank to evaluate road roughness (Gillespie et al. 1980, 1993; Sayers 1985; Marcondes et al. 1992). Today the IRI has been a widely used indicator to evaluate the ride quality of roads in the highway industry all over the world. It basically measures the output of a specified test vehicle as it traverses a tested pavement section at a constant speed. The test vehicle is designed as a quarter car and is calibrated before conducting the test. The quarter car actually acts as a response-type device that reflects pavement surface roughness in terms of the vehicle’s response, such as displacement, velocity, and acceleration. The IRI is defined as the overall relative velocity between the axle and sprung mass of the quarter car (Gillespie et al. 1980; Sayers 1985). Clearly, the IRI is an indirect statistic of roughness, since it measures the response of the test vehicle to rough pavement rather than surface roughness itself. Ride quality is also a concern for the vehicle industry, where the term ride quality is sometimes called ride comfort, and it is usually evaluated in terms of suspension acceleration (Dodds 1974; Robson 1979; Wei 1988; Marcondes et al. 1992; Sun 1998). To provide better vehicle performance, it is important to evaluate the ride quality of a certain type of vehicle. The process of evaluating ride quality of a specific vehicle involves simulation of both dynamic vehicle models and pavement surface roughness. Since it is convenient to conduct the analysis of vehicle dynamics in the frequency domain (this is true especially when linear vehicle models are considered), it is preferred in the vehicle industry to employ the power spectral density (PSD) of pavement surface roughness as a direct indicator of road profile (Dodds 1974; Zhu 1992; Sun 1998). In addition, PSD roughness plays an important role in the research community when issues of dynamic pavement loading, vehicle safety, energy consumption, and suspension optimization and control are treated (Gillespie et al. 1980; Sun and Deng 1998). Motivations and Objectives The PSD roughness is different from the IRI in that the IRI is an indirect statistic of roughness and the PSD roughness is

JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001 / 105

a direct statistic of roughness. The PSD roughness has been routinely adopted by the vehicle manufacturing industry for automobile design for many years, while the IRI is more favored in the community of highway agencies because of its summary nature. Certainly it will be of benefit to both the highway industry and the vehicle industry if a relationship can be established between the IRI and PSD roughness. For instance a pavement engineer will anticipate knowing the value of IRI if the PSD roughness of a pavement is given. It is evident that pavement dynamic loads are significantly influenced by surface roughness (Sweatman 1983; Heath and Good 1985; Hedric and Markow 1985; Woodrooffe and LeBlanc 1986; Monismith et al. 1988). Previous studies by one of the writers (Sun and Deng 1997, 1998; Sun 1998) have established a quantitative relation between the PSD loads and PSD roughness. Thus, if we can make the IRI related to the PSD roughness, most likely we will further be able to correlate the PSD loads with the IRI. This will help us to better understand the effect of the IRI on pavement dynamic loading. Another motivation prompting us to address this problem arises from the theoretical perspective. It is known that since both the IRI and the PSD roughness are statistics of a random field, it is of interest to examine whether these two statistics are independent—and if not, how they are correlated. This paper describes a theoretical investigation on this subject. We find that it is possible to establish the relationship between the IRI and PSD roughness under certain circumstances. DESCRIPTIONS OF ROAD PROFILE Numerous measurements indicate that roughness can be modeled as a zero mean Gaussian isotropic random field in the spatial domain and becomes a normal stationary ergodic random process in the time domain if it is measured by a device moving at a constant speed (Newland 1984; Zhu 1992). The height ␰ of the surface profile, representing pavement roughness, is a function of spatial distance x along the road. Low-frequency components correspond to long wavelength roughness, while high-frequency components correspond to short wavelength roughness. From the Wiener-Khintchine theory, the following forms constitute a Fourier-transform pair (Zhu 1992):

冕 冕

⬁

1 S␰(⍀) = 2␲

R␰(X )e⫺i⍀X dX

S␰(␻) =

R␰(X ) = E [␰(x)␰(x ⫹ X )]

(2)

R␰(␶)e⫺i␻␶ d␶

(5a)

⫺⬁

S␰(␻)ei␻␶ d␻

(5b)

where f and ␻ = natural and angular frequences, respectively; ␶ represents the time lag; and R␰(␶) = temporal autocorrelation function. Assume that a vehicle moves along a pavement of length X at a constant speed v. The relation of length and speed gives X = v ␶, where ␶ is the time that the vehicle takes to traverse the pavement section. Substituting this relation into (1a), we have 1 S␰(⍀) = 2␲

冕

⬁ ⫺i(␻/v)v␶

R␰(v ␶)e

⫺⬁

v d(v ␶) = 2␲

冕

⬁

R␰(␶)e⫺i␻␶ d␶

⫺⬁

= vS␰(␻)

(6)

Also, since ␻ = 2␲ f, based on (5a), it is straightforward to see that S␰(␻) =

1 2␲

冕

⬁

R␰(␶)e⫺i 2␲f␶ d␶ =

⫺⬁

1 S␰( f ) 2␲

(7)

Comparing (6) and (7), we obtain a general relationship among the PSD roughness expressed by three different frequencies; i.e., 2␲S␰(⍀) = 2␲vS␰(␻) = vS␰( f )

(8)

␻ = 2␲ f = v⍀

(9)

Here

In practice, the single-sided PSD roughnesses G␰(⍀) and G␰( f ) are often employed (Zhu 1992), and are defined by G␰( f ) =

G␰(⍀) =

where X represents the distance between two points along the longitudinal direction of the road; S␰(⍀) = PSD roughness in terms of a wave number ⍀, which represents the spatial frequency; and R␰(X ) = spatial autocorrelation function, defined by

⬁

⫺⬁

and

(1b)

(4b)

⬁

⫺⬁

S␰(⍀)ei⍀X d⍀

冕 冕

1 2␲

R␰(␶) =

(1a)

⫺⬁

S␰( f )ei 2␲f ␶ df

⫺⬁

⬁

R␰(X ) =

冕

R␰(␶) =

⬁

再

2S␰( f ) for f ⱖ 0 0 for f < 0

(10a)

再

2S␰(⍀) for ⍀ ⱖ 0 0 for ⍀ < 0

(10b)

Thus, (8) can also be written in the form 2␲G␰(⍀) = 4␲vS␰( f ) = vG␰(␻)

(11)

QUARTER-CAR MODEL A quarter-car model with two degrees of freedom on a rough pavement is shown in Fig. 1. The motion equations governing this suspension system are given by ordinary differential equations

where E[ ⭈ ] represents the expectation of a random process, which can be estimated from E[␰(x)] = lim X→⬁

1 X

冕

X

␰(x) dx

(3)

0

According to the stochastic process theory (Zhu 1992), the PSD of a random process, S␰( f ) and S␰(␻), are respectively described by

冕

⬁

S␰( f ) =

⫺⬁

R␰(␶)e⫺i 2␲f ␶ d␶

(4a)

FIG. 1. Quarter-Car Model Used for the International Roughness Index

106 / JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001

TABLE 1.

Quarter-Car Parameters

Vehicle (1)

kt /ms (s⫺2) (2)

ks /ms (s⫺2) (3)

mt /ms (4)

cs /ms (s⫺1) (5)

ct /ms (s⫺1) (6)

HSRI BPR

653.0 643.0

62.3 128.7

0.15 0.162

6.0 3.86

0 0

˙ ⫹ ks( yt ⫺ ys) mt y¨i ⫹ cs( y˙t ⫺ y˙s) ⫹ ct ( y˙t ⫺ ␰) ⫹ kt ( yt ⫺ ␰) = 0

(12a)

ms y¨s ⫺ cs( y˙t ⫺ y˙s) ⫺ ks( yt ⫺ ys) = 0

(12b)

where the road profile is modeled as a one-dimensional Gaussian random field with a zero mean and is represented by the height of pavement surface irregularities ␰; and ys and yt = absolute displacements of the sprung mass and unsprung mass, respectively. One or two dots above the foregoing symbols are the first and second derivative processes, respectively, i.e., the velocity process dy/dt = y˙ and the acceleration process d 2y/dt 2 = y¨. Let zt(t) = ␰ ⫺ yt(t);

zs(t) = yt(t) ⫺ ys(t)

(13a,b)

where zt(t) and zs(t) = relative displacements of the sprung mass and unsprung mass, respectively. Taking (13) into (12) gives mt ␰¨ ⫹ mt z¨t = ct z˙t ⫹ kt zt ⫺ cs z˙s ⫺ ks zs

(14a)

ms ␰¨ ⫹ ms z¨t ⫹ ms z¨z = cs z˙s ⫹ ks zs

(14b)

Historically, the Bureau of Public Roads (BPR) roughness profilometer is one of the first devices developed to measure road roughness at normal highway speeds (Hudson 1985). Since that time, mathematical models that simulate the response of a quarter car ‘‘traveling’’ over a pavement profile have been developed and used as roughness summary statistics for calibrating a response-type road roughness system (RTRRMS). One of the most popular RTRRMS devices in the United States during the 1970s and 1980s is the Mays Ride Meter (MRM), developed for the Texas Highway Department in the 1960s (Phillips and Swift 1969; Haas et al. 1994). The IRI is defined as the response of the MRM at a speed of 80 km/h; the vehicle parameters are recommended by the Highway Safety Research Institute (HSRI). Quarter-car parameters of both the BPR and the HSRI quarter-car models are provided in Table 1 (Gillespie et al. 1980; Hudson 1985; Sayers 1985; ‘‘Section’’ 1997).

FIG. 2. FRF Corresponding to Quarter-Car Model with HSRI and BPR Parameters

are shown in Fig. 2. It can be seen that the BPR quarter-car model has a larger response at almost all frequencies of [0, 25] Hz than does the HSRI quarter-car model. According to the stochastic process theory, if the input of a linear time-invariable system is a stationary random process, then its output is also a stationary random process. In most cases, pavement roughness could be modeled as a zero mean Gaussian ergodic random process (Newland 1984; Zhu 1992). Hence, the response of the quarter-car system is also a zero mean Gaussian stationary random process. The relationship between the PSD of the system response and the PSD of the system excitation is SZ t(␻) = 兩Ht(␻)兩2S␰(␻);

SZs(␻) = 兩Hs(␻)兩2S␰(␻)

(17a,b)

where SZ t(␻) and SZ t(␻) = PSDs of displacement responses of the unsprung mass and sprung mass, respectively. The meansquare functions of the displacements, ␴2Z t and ␴2Zs, are then expressed by

冕 冕

⬁

␴2Z t = E [Z 2t ] =

SZ t(␻) d␻

(18a)

SZs(␻) d␻

(18b)

⫺⬁ ⬁

␴2Zs = E [Z 2s ] =

⫺⬁

The mean-square function is equal to the variance if a random process is a zero mean process. The PSDs of the velocity response, SZ˙ t(␻) and SZ˙ t(␻), are given by SZ˙ t(␻) = ␻2SZ t(␻);

SZ˙ s(␻) = ␻2SZs(␻)

(19)

and the mean-square response functions of the velocities, ␴2Z˙ t and ␴2Z˙ s, are as follows:

冕 冕

⬁

␴2Z˙ t = E [Z˙ 2t ] =

VEHICLE RESPONSE CHARACTERISTICS Frequency Response Function (FRF)

Hs(␻) = zs(␻)/␰

(15a,b)

where Ht(␻) and Hs(␻) = FRF of the unsprung mass and sprung mass, respectively. To obtain the FRF, we take the Fourier transform on both sides of (14). It gives Ht(␻) = Dt(␻)/D(␻);

Hs(␻) = Ds(␻)/D(␻) 3

2

␴2Z˙ s = E [Z˙ 2s ] =

SZ˙ s(␻) d␻

(20b)

⫺⬁

The PSDs of acceleration response, SZ¨ t(␻) and SZ¨ t(␻), are given by SZ¨ t(␻) = ␻4SZ t(␻);

3

SZ¨ s(␻) = ␻6SZs(␻)

(21a,b)

and the mean-square response functions of accelerations, ␺ 2Z¨ t and ␺ 2Z¨ s, are as follows:

(16a,b)

where Dt = ␻ ⫺ i␣s(1 ⫹ ប)␻ ⫺ ␤s(1 ⫹ ប)␻ ; Ds = ⫺i␣t␻ ⫺ ␤t␻2; and D = ␻4 ⫺ i(␣t ⫹ ប␣s ⫹ ␣s)␻3 ⫺ (␣t␣s ⫹ ␤t ⫹ ␤s ⫹ ប␤s)␻2 ⫹ i(␣t␤s ⫹ ␣s␤t)␻ ⫹ ␤t␤s. Also, in (16) ប = ms /mt, ␣s = cs /ms, ␣t = ct /mt, ␤s = ks /ms, and ␤t = kt /mt. The FRFs of both the HSRI and the BPR quarter-car models 4

(20a)

⬁

The FRF of the system controlled by (14) is defined as the response of zt and zs as the input ␰ = ei␻t. In other words, we have Ht(␻) = zt(␻)/␰;

SZ˙ t(␻) d␻

⫺⬁

冕 冕

⬁

␴2Z¨ t = E [Z¨ 2t ] =

SZ¨ t(␻) d␻

(22a,)

SZ¨ s(␻) d␻

(22b)

⫺⬁ ⬁

␴2Z¨ s = E [Z¨ 2s ] =

⫺⬁

JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001 / 107

The PSDs of jerk response, S ᠮZ t (␻) and S ᠮZ t (␻), are given by S ᠮZ t (␻) = ␻ SZt(␻);

S ᠮZ t (␻) = ␻ SZs(␻)

6

6

(23a,b)

and the mean-square response functions of the jerks, ␺ ᠮZ t and 2 ␺ ᠮZ s, are as follows: 2

冕 冕

⬁

2 ᠮ 2t ] = ␴ ᠮZ t = E [Z

⫺⬁

S ᠮZ t (␻) d␻

(24a)

As was mentioned in the previous section, any response of the quarter-car model is a normally distributed stationary process with a zero mean because of the nature of linearity. In other words, we have z (s j)(t) ⬃ N [0, ␴2Z (s j )]

The probability density function (PDF) of z (s j)(t) is given by

ᠮ ]= ␴ ᠮ s = E [Z 2 s

1

fZ (s j )[z (s j)] =

⬁

2 Z

⫺⬁

S ᠮZ s (␻) d␻

(24b)

(31)

兹2␲␴Z (s j )

exp

再

冎

[z (s j)]2 ⫺ 2 ; 2␴Z (s j )

⫺⬁ < z (s j) < ⬁

(32)

The following theory from Casella and Berger (1990) is used here to derive the PDF of x (s j)(t).

INDIRECT STATISTICS OF ROUGHNESS Definitions

Theorem

So far we have discussed the quarter-car model and the associated vehicle response properties. Now we may have adequate premises to construct statistics for representing roughness. It is clear that any response of the quarter car to a rough pavement is not a direct indicator of roughness. Consequently, any statistic constructed based on the response of the quartercar model is indirect statistics (IS) of roughness, in which the vehicle dynamic properties are also involved. Let T be the time that a vehicle takes to traverse a pavement section of length L at a constant speed v. Construct indexes j) ( j) and IS abs,L as the average of the absolute value of IS (abs,T z (s j )(t) over time T and length L, respectively, i.e.,

Let Z have a PDF of fz(z), and let X = g(Z ). Suppose there exists a portion, A0, A1, . . . , Ak, such that P (x 僆 A0) and fZ(z) is continuous on each Ai. Further, there exist monotone functions g1(Z ), g2(Z ), . . . , gk(Z ) on Ai (i = 1, 2, . . . , k) and g ⫺1 i (X ) has a continuous derivative on X for each i = 1, 2, . . . , k. Then

j) IS (abs,T =

IS

( j) abs,L

1 T

1 = L

冕 冕

T

兩z (s j)(t)兩 dt

(25)

再

冘

fZ [g ⫺1 i (X )]

i=1

x僆X (33)

For the current problem, g[z (s j)] = 兩z s( j)兩 and g[z s( j)] is monotone on (⫺⬁, 0) and on (0, ⬁). The set X (s j) = (0, ⬁). Applying this theorem, we take A0 = {0}

兩z (t)兩 dt ( j) s

( j) ( j) A1 = (⫺⬁, 0), g1[z ] = x (s j) = 兩z (s j)兩, g ⫺1 1 [x s ] = x s ( j) s

(26)

0

IS (stdj) = ␴Z (s j ) = 兹E{[z (s j)(t)]2}

A2 = (0, ⬁), g2[z s( j)] = x s( j) = 兩z (s j)兩, g 2⫺1[x s( j)] = ⫺x s( j)

The PDF of x (s j)(t) then takes the form

(27) ⫹

where E{[z (t)] } can be estimated by 1 T

冕

1

( j)

fX (s j )[x s ] =

2

T→⬁

冏

d ⫺1 g i (X ) ; dx

otherwise

T

E{[z (s j)(t)]2} = lim

冏

0;

0

where j = 0, 1, 2, . . . ; and superscript represents the nth time derivative of the specified quantity. Note that L = vT, and we j) ( j) have IS (abs,L = v ⭈ IS abs,T . Also, we may define index IS (stdj) , the ( standard deviation ␴Z s j ) of z s( j)(t), as an indirect statistic of roughness, i.e., ( j) s

fX (x) =

k

兹2␲␴Z

1

兹2␲␴Z

(28)

=

0

( j) s

In theory, ␴Z (s j ) can also be given by (18), (20), (22), and (24) for j = 0, 1, 2, 3, respectively. Therefore, these formulas can also be used to calculate the value of ␴Z s( j ). Apparently, since each statistic contains road roughness in a different form, it will be appropriate if any statistic from definitions (25), (26), and (27) is selected as an indirect statistic of roughness.

1

兹2␲␴Z

( j) s

再 再

exp ( j) s

再

冎

( j)

[⫺x s ]2 ⫺ 2␴Z2 (s j ) ( j)

冎 冎

[⫺x s ]2 ⫺ 2␴2Z (s j )

exp

T

[z (s j)(t)]2 dt

exp

( j)

[⫺x s ]2 ⫺ ; 2␴2Z (s j )

兩⫺1兩

兩1兩 0 < xs < ⬁ ( j)

(34) ( j) s

According to the definition of expectation, E [兩z (t)兩] can also be expressed in terms of the PSD of 兩z (s j)(t)兩; i.e.,

冕

⬁

E [兩z (s j)(t)兩] = E [x (s j)(t)] =

x (s j)fX (s j )[x (s j)] dx (s j)

0

Substituting (34) into the expectation above, we obtain

Properties j) j) It is of interest to examine if IS (abs,T and IS (abs,L are statistically independent. According to the stochastic process theory (Zhu 1992), the expectation of a random process 兩z (s j)(t)兩 can be estimated as follows:

E [兩z (s j)(t)兩] = lim T→⬁

1 T

冕

T j) 兩z (s j)(t)兩 dt = lim IS (abs,T

0

(29)

T→⬁

j) Thus, indirect statistic IS (abs,T might be thought of as an average of a sample from the random process 兩z (s j)(t)兩 over time. Eq. (29) simply implies that, as long as the sample duration is j) sufficiently long, the sample average IS (abs,T will be approach( j) ing the time average E[兩z s (t)兩] with a probability of one. Define a transformation

x (s j)(t) = 兩z (s j)(t)兩

(30)

( j) s

E [兩z (t)兩] =

2

兹2␲␴Z

exp ( j) s

再

冎冏

[x (s j)]2 ⫺ 2 2␴Z s( j )

⬁

= 兹2/␲␴Z (s j )

(35)

0

Here, the integration by parts is used in the derivation of (35). Comparing (25)–(27), (30), and (35), we find the relationship j) j) between IS (abs,T and IS (std,T when sufficiently long time is taken into account j) j) IS (abs,T = v ⭈ IS (abs,L = 兹2/␲IS (stdj)

(36)

j) Eq. (36) is a remarkable result, indicating that neither IS (abs,L ( j) nor IS abs,T is statistically independent with respect to the indirect statistic IS (stdj). Moreover, a direct proportion relationship j) exists between IS (abs,T and IS (stdj). In particular, if a constant vej) hicle speed is considered, IS (abs,L is also directly proportional ( j) to IS std .

108 / JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001

RELATIONS OF INDIRECT STATISTICS AND PSD ROUGHNESS

IRI BASED ROAD CLASSIFICATION

We may now examine the correlation of some indirect statistics of roughness proposed by previous researchers with the direct statistic—the PSD roughness. Statistics IS (1) abs,T and IS (1) abs,L are both proposed in an NCHRP report by Gillespie et (1) al. (1980), in which IS (1) abs,T and IS abs,L are named the average rectified velocity (ARV ) and average rectified slope (ARS ), respectively. In a later study (Sayers 1985), the World Bank recommends the use of the indirect statistic ARS at a constant speed v = 80 km/h as the IRI. Similar to (36), the following relations hold for ARV, ARS, and IRI; ˙ ARV = v ⭈ ARS = 兹2␲IS (1) std = 兹2/␲␴Z s ⫺1

IRI = v 兹2/␲IS with

(37)

⫺1

= v 兹2/␲␴Z˙ s

(1) std

v = 80 km/h = 22.22 m/s

(38)

Obviously, the IRI is related to ␴Z˙ s /v through a linear relationship. In this sense, the IRI is actually dimensionless. However, in practice, the unit of the IRI is often chosen as meters per kilometer or inches per mile. Now we take this relationship into consideration. Replacing ␴Z˙ s in (37) and (38) by (20), we see the connection among these statistics

冋冕

册

⬁

ARV = v ⭈ ARS = 兹2/␲

冋冕

1/2

␻2兩Hs(␻)兩2S␰(␻) d␻

⫺⬁

with

(39)

册

⬁

IRI = v ⫺1兹2/␲

1/2

␻2兩Hs(␻)兩2S␰(␻) d␻

⫺⬁

v = 22.22 m/s

(40)

Here, (17) and (19) are both used for the derivation of (39) and (40). Substituting (8) into (39) and (40), we obtain ARV, ARS, and IRI in terms of PSD roughness expressed in the form of a wave number ARV = v ⭈ ARS =

IRI =

2

兹␲ v 3

冕

2

兹␲v

冕

So far, we have established the relationship between the IRI and PSD roughness. In this section, we will address the issue of road classification. The International Standards Organization (ISO) has proposed a road classification based on different levels of PSD roughness (International 1972). From the connection of (42), it is of interest to classify roads in terms of the IRI. Various forms of the PSD roughness have been suggested over the past years (Dodds and Robson 1973; Gillespie et al. 1980, 1993). These wave-number spectrums are regressed from the actually measured PSD roughness. A commonly used PSD roughness takes the form (Dodds 1974) S␰(⍀) = S0(⍀ /⍀ 0)⫺m

(43)

where S0 = roughness coefficient; and reference frequency ⍀ 0 usually is set as 1/(2␲) cycles/m. Substitution of (43) into (41) and (42) gives a more specified result ARV = v ⭈ ARS = g(m, v) ⭈ v ⭈ S 1/2 0 ⬁ 0

(44)

where g(m, v) = 2/兹␲ 兰 ⍀ ⍀ 兩Hs(v ⍀)兩 d⍀] . Figs. 3 and 4 show the curves of g(v, m) as a combined function of speed v and power m. In view of (43), one may realize that power m in the PSD roughness virtually acts as an indicator that reflects the component of spatial frequency or the wavelength of surface roughness. The greater the m value, the greater the low spatial frequency (corresponding to the long wavelengths) components in roughness. As was observed from Figs. 3 and 4, for smaller m values, say m < 2.0, function g(v, m) does not have significant variations with the changing speed. Nevertheless, for larger m values, say m > 2.0, function g(v, m) increases significantly as the vehicle speed increases. This phenomenon implies that for those pavement surfaces in which high frequency components or short wavelengths play a dominating role (i.e., m < 2.0), the vehicle speed has no significant effect 2⫺m

m 0

2

1/2

⬁

␻2兩Hs(␻)兩2S␰(⍀) d␻

(41)

0

⬁

␻2兩Hs(␻)兩2S␰(⍀) d␻

with

v = 22.22 m/s

0

(42)

where spatial frequency ⍀ is given by (8). Eqs. (40) and (41) are universal results and are suitable for many types of pavement surface conditions. It should be noted that, here, we do not require an analytical representation of the PSD roughness S␰(⍀). As a result, if the PSD roughness takes the numerical form, which is often the case, as real profile data are collected and the fast Fourier transform is used to compute S␰(⍀), as long as an appropriate method of numerical integration is chosen, (41) and (42) will apply. Eqs. (41) and (42) also provide a great advantage, in that whether or not the time domain data of the road profile are available, one can figure out the ARV, ARS, and IRI with only a given PSD roughness. From the expression of ARV, ARS, and IRI in (41) and (42), we can further recognize that all of these indirect statistics are actually the area surrounded by a certain kind of weighted PSD roughness curve and the frequency axis. Since the standard deviation ␴␰ is also defined as the area embraced by the PSD roughness curve and the frequency axis, we believe that the PSD roughness is a more essential statistic than indirect statistics.

FIG. 3. Model)

Eq. (43) Based Function g(v, m) against Speed (HSRI

FIG. 4. Model)

Eq. (43) Based Function g(v, m) against Speed (BPR

JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001 / 109

on the indirect statistic ARS. As far as pavements with m > 2.0 are concerned, the higher the speed, the greater the indirect statistics ARS and ARV. In addition, since the IRI is defined as ARS given the vehicle speed v = 22.22 m/s, i.e., IRI = g(m, v22.22) ⭈ S 1/2 0

再

Csp ⍀ ⫺2 for 0 ⱕ ⍀ ⱕ ⍀ 1 1 Csp ⍀⫺2 for ⍀ 1 ⱕ ⍀ ⱕ ⍀ 2 0 for ⍀ 2 < ⍀

(46)

where Csp, named the roughness coefficient = positive constant and reflects the intensity of surface roughness. The ISO suggests that ⍀ 1 = 0.1 cycles/m and the cutoff frequency ⍀ 2 = 2.0 cycles/m (Wei 1988). Based on the PSD roughness of the form of (46), the ISO classifies pavement into five classes, depending on different surface conditions. Table 2 gives specified values for this ISO classification (Wei 1988). Also, in

FIG. 5. IRI as a Function of Roughness Coefficient 兹S0 against Power m

冋冕

册 冋 冉

⬁

␴␰ =

1/2

S␰(⍀) d⍀

⫺⬁

=

冊册

2 1 ⫺ 2 ⍀1 ⍀

2Csp

1/2

(47)

Taking (46) into (44) and (45), we have

(45)

it is not difficult to see that the IRI is a function of the unique variable 兹S0. Fig. 5 illustrates the influence of power m on the IRI. Apparently, the IRI increases with an increase of the m value. Also, the relationship between them is nonlinear, and the IRI will dramatically get higher as the m value becomes greater. Another commonly used formulation for representing the PSD roughness is proposed by the ISO (International 1972; Wei 1988), in which the following standard expression is adopted to describe the pavement roughness PSD: S␰(⍀) =

Table 2, the standard deviation ␴␰ of roughness is calculated from

ARV = v ⭈ ARS = G(m, v) ⭈ v ⭈ C 1/2 sp IRI = G(m, v22.22) ⭈ C

where G(m, v) =

2

兹␲

冋冕 冉 冊 ⍀1

0

⍀ ⍀1

(48)

1/2 sp

2

兩Hs(v ⍀)兩 d⍀ ⫹ 2

(49)

冕

⍀2

册

1/2

兩Hs(v ⍀)兩 d⍀ 2

⍀1

The IRI is then calculated using (49) and the ISO road classification. The IRI values corresponding to five pavement classes are provided in Table 2, where both HSRI and BPR quarter-car models are considered. A linear relation between the IRI and the standard deviation of roughness is found and is plotted in Fig. 6, where the slopes for linear correlation are 0.300 for the HSRI model and 0.342 for the BPR model. Since the ISO-specified PSD roughness serves as the basis for the derivation of the IRI-based road classification, it implies that (46) should be adopted to characterize pavement surface roughness if the IRI values provided in Table 2 are employed for any purpose. In other words, this IRI-based classification assumes that (46) with exponential term ⫺2 can be suitably used to fit the real PSD roughness curves, which usually are derived by measuring the road profile. Clearly, this requirement virtually means that all pavement surfaces obey the power law of the form of (46), and the magnitude of a pavement surface fluctuation is only affected by the roughness coefficient Csp. Therefore, if the PSD roughness in the form of (46) is not an optimal fitness of the real surface spectrum and other forms of PSD roughness functions are used for the regression analysis, one must be aware that the utilization and interpretation of Table 2 will not always be as valid as when (46) is suitable. The IRI-based road classification can be conveniently used in practice by highway agencies for the purpose of routine management. However, it is also noteworthy to point out that the ISO road classification requires that the cutoff frequency of PSD roughness ⍀ 2 = 2.0 cycles/m (Wei 1988). It simply means that the surface fluctuations with wavelengths less than ␲ m will be filtered as representing the PSD roughness in the form of (46). Clearly, this range of cutoff frequencies does not cover these small-scale fluctuations caused by surface defects such as cracks, potholes, and corrugations. DISCUSSION

FIG. 6. Relationship between the IRI and Standard Deviation of Roughness ␴␰ TABLE 2.

There have been a number of statistics proposed for describing pavement surface roughness. From a theoretical perspective, these statistics can be characterized into two categories. One category can be called direct statistics, and another category can be called indirect statistics. Direct statistics mean that people abstract statistical information directly from roughness

ISO Proposed Road Classification and Corresponding IRI Level

Pavement classes (1)

Csp (10⫺7 m2 cycles/m) (2)

␴␰ (10⫺3 m) (3)

IRIHSRI (m/km) (4)

IRIHSRI (in./mi) (5)

IRIBPR (m/km) (6)

IRIBPR (in./mi) (7)

Very good Good Average Poor Very poor

1 4 16 64 256

2 4 8 16 32

0.60 1.20 2.41 4.81 9.62

37.87 75.74 152.12 303.61 607.21

0.68 1.37 2.73 5.47 10.94

42.92 86.47 172.32 345.27 690.53

110 / JOURNAL OF TRANSPORTATION ENGINEERING / MARCH/APRIL 2001

itself. For instance, the average and standard deviation of roughness, PSD roughness, the autocorrelation function, and the so-called root-mean square vertical acceleration (Hudson 1985) all belong to the direct statistics category. In theory, direct statistics are the most essential statistical summaries of roughness. However, it is sometimes difficult to get the real road profile data with some devices at the early stage of pavement research and development. In such a case, it may not be possible to analyze the direct statistics by processing raw roughness data. Nevertheless, it should be recognized that quarter-car based indirect statistics are, in fact, the summary of the weighted PSD roughness. From the analysis of the paper, it is clear that direct statistics such as the PSD roughness are the essential summaries that can completely reflect the statistical characteristics of roughness; they also can be very useful in other kinds of studies; such as studies on dynamic pavement loading, energy consumption, and vehicle fatigue. In the meanwhile, many laserbased noncontact profilometers have been available since the late 1980s. We believe that direct statistics, in general, are better than indirect statistics, although the latter can also serve very well for some specific purposes. In other words, direct sampling of pavement roughness will enable us to maximize the utilization of the collected time domain data of roughness, and therefore generate more benefits for research and application. It is also worth noting that a fundamental relationship, i.e., (36), is proved under the hypothesis that surface roughness can be well modeled by a homogeneous Gaussian random field with a zero mean. This hypothesis is a reasonable assumption, and has been widely accepted in the research community and vehicle industry. However, there do exist pavements whose surfaces cannot be well described by a Gaussian random field. In such cases, it is helpful to be aware that (36), (41), and (42) will no longer hold. CONCLUSIONS This paper proved that indirect statistics such as the ARS are directly proportional to the standard deviation of the relative vertical velocity between the axle and vehicle body. Both the ARV and the IRI are correlated to the PSD roughness using the relationship between the standard deviation and PSD of a stationary random process. Two kinds of commonly used PSD roughnesses are employed to reveal the variation of the ARV and IRI caused by the PSD roughness and quarter-car model parameters. Corresponding to the ISO PSD-based road classification, an IRI-based road classification is provided, which allows one to convert a PSD roughness of a pavement to the IRI of that pavement. APPENDIX.

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