Application of AASHO Road Test Equations To Mixed Traffic F. H. SCRIVNER AND H. C. DuzAN, respectively, Research Engineer, Texas Transportation Institute, and Chief, Differential Cost Studies Group, Highway Cost Allocation Study, Bureau of Public Roads The general pavement performance equations developed at the AASHO Road Test describe the behavior of pavements when subjected to repeated applications of an axle load of a given type and weight. To apply the equations to the problems of highway design it is necessary to transform the single-load equations to multiload form. The "mixed traffic theory" produces a multiload equation in the form of an integral that can not be readily evaluated, but the "equivalent applications approach" yields a manageable equation that approximates the results obtained by mechanical integration of the equation derived by the mixed traffic theory. The difference in the results obtained by the two methods is negligible for all except relatively thin flexible pavements. AASHO Road Test Report 5 (HRB Special Report 61E) presents two pavement performance equations, one applying to flexible and the other to rigid pavements. Either may be expressed in the following general form. At any instant during the life of a pavement of a given design, in which log W = logp + (1) W = total number of axle loads of a given type and weight which have passed over the pavement, no other loads having been applied; P = number of axle load applications of the given type and weight required to reduce the serviceability index, p, of the pavement from an initial value, Co, to a value of 1.5 (co is 4.5 for rigid and 4.2 for flexible pavements) ; and G = l o g f - ^ ) (2) \ Co - 1.5 / Log p and p are given for each pavement type as functions of the design and load variables defined as follows: 387 Variable A In Flexible Equations for log p and p Thicness of asph. cone, (m.) D, Thicness of base (in.) D, Thicness of subbase (in.) L, Weight of axle (ips) U Type of axle (= 1 for single, 2 for tandem axle) Meaning In Rigid Equations for log p and p Not used Thicness of concrete (in.) Not used Same as for flex. pvt. Same as for flex. pvt. It will be seen from the definitions of W and p that neither of the pavement performance equations directly yields information as to the behavior of pavements acted on by mixed traffic (that is, normal highway traffic composed of both single and tandem axles of a variety of weights). Thus, it is necessary to transform the Road Test equations (referred to hereafter as "single-load equations") to multiload form in order to permit their application directly to problems of highway pavement design. Two methods of accomplishing the transformation one nown as the "mixed traffic theory" (1)
388 CONFERENCE ON THE AASHO ROAD TEST and the other as the "equivalent applications theory" (2) are described herein. MIXED TRAFFIC THEORY An existing highway is considered with given design parameters, Di, Dj, and D3, and design, materials and environment similar to those at the AASHO Road Test. The highway is subjected to mixed traffic of nown composition. It is desired to find an equation, analogous to Eq. 1, connecting the serviceability index, p, of the pavement with design and load variables. It is first necessary to characterize the traffic. For this purpose let N be the number of axles of all weights and types which have passed a fixed point in a selected lane of the highway since it was opened to traffic. Let N be divided into categories, each containing axles of the same type (single or tandem) and weight. Let N, represent the number of axles in the category, and L. the corresponding weight in ips. Let g{p) be any function, g, of p, continuous in the interval 0<p<Co. (Such a function is G, defined previously.) Since fl' is a function of p, and p has been shown to be a function of pavement design, axle load, and axle applications in the special ind of traffic used at the Road Test, it is assumed that in the more general case of mixed traffic, p (and therefore f) is a function of pavement design and traffic. That is g = a function of Ni, N2,. -.Nr,..., Nri Li, L2,..., L...Li; Di, D2, D3 (3) It is further assumed that a representative sample of the traffic passes a fixed point in the pavement within a period of time so short, when compared to the life of the pavement, that this period may be treated as a differential quantity without excessive error. Thus, the particular order in which the various types and weights of loads pass over the pavement is neglected, and the quantities Lj in Eq. 2 are considered to be independent of time (constant). Since Di, and D3 are also constant, a differential increase in g, occurring as a result of a representative sample of traffic passing over the pavement, is given by = -^iv, + +. -aiv,. + dat* (4) If C, is the proportion of JV occurring in the i'" load category (i.e., C. = N,/N or N, = C,N), dn, = C^dN and da da da = C^^-dN + C^-^N +...+ a dn 9Ni dn^ (5) or dft (6) For application to the Road Test equations let Co P (7) Co 1.5 Then Eq. 1 leads to the following relation which holds for a Road Test section of the same design as the highway, but subjected to repeated applications of loads in the i"' category only: W, = p,g,'/p. (8) in which W, = N, only if A; = 1. Differentiating and inverting gives dw, _ /8.fli(i-v3.) Pi (9) (In Eqs. 8 and 9 the subscript, i, is taen to mean that the axle load representing the i'" category, as well as the highway design parameters Di, D. and D have been used to determine p,, p, and Wt from Road Test equations.) If the Road Test section and the highway are in the same condition {g is the same for both), dn, dw^ Thus, according to Eqs. 9 and 10, _ i8,ff'^-vp.' dn, ~ p, Use of Eqs. 11 and 3 gives dn~ i ^ l p. By separation of variables. N = 9i I.2 I = 1 Pt (10) (11) (12) (13) in which g is the terminal value of g for which N is to be computed. Eq. 13 is the multiload equation according to the mixed traffic theory. Because the exponent of g varies from term to term in the denominator, the integral cannot be readily evaluated, and to date solutions for particular cases have
USE OP PAVEMENT RESEARCH FINDINGS 389 been found only by resort to numerical methods of approximate integration. As is shown later, however, the equivalent applications theory requires no integration and yields results which are, for practical cases, nearly the same as those found by numerical integration of Eq, 13. The value of can be obtained from the Road Test equations by substituting a and i in the load terms of p and p and dividing; that is, i ^ ^ P ^ (14) EQUIVALENT APPLICATIONS APPROACH To use the Road Test equations in the incremental solution made as a part of the Highway Cost Allocation Study it is necessary to have a means of woring the combined effect of the variety of axle loads found in mixed traffic into the single-load Road Test equations. Because the equations contain terms for both axle weight and number of axle applications it is natural to try to introduce mixed traffic into the equations by converting the number of applications of each axle load (or each axleload group) found in mixed traffic into an equivalent number of applications of a selected axle load so that the equations may be handled as single-load equations. This approach is by no means unique to the Highway Cost Allocation study. The AASHO Design Committee, and perhaps others struggling with the problem of applying the equations to mixed traffic, also hit upon what may be called the "equivalent applications approach." This approach (it can hardly be called a theory) is based on the following assumptions: 1. The effect of any axle load can be expressed in terms of an equivalent number of applications of any selected axle load. For convenience, the selected axle load is designated axle load a. 2. The combined effect of all axle loads in mixed traffic can be expressed in terms of the combined number of equivalent applications of axle load a. 3. The effect of mixed traffic can be introduced into a single-load equation for load a by means of the combined number of equivalent applications of axle load a. These assumptions mae it possible to write an equation for mixed traffic in terms of the single-load equations for axle load a. As will be seen, however, the first assumption is a little too broad for the form of the Road Test equations. Except as noted, the succeeding description of the equivalent applications approach employs the terminology established in describing the mixed traffic theory. If /?, denotes the ratio of equivalence (or equivalent application factor) that will convert each application of axle load i to an equivalent number of applications of axle load a, by definition Ra=l. If W denotes the total number of axle applications in mixed traffic and C, the proportion of axle applications of weight i, the number of applications of axle load i is C^W. The number of applications of each axle load is converted to an equivalent number of applications of axle load a by means of the appropriate ratio of equivalence. The sum of the products is the equivalent number of applications of axle load a in mixed traffic. This summation, which is represented by 2 C,R i& subi = 1 stituted for W in the Road Test equation for axle load a to obtain the equivalent applications version of a multiload equation: or, solving for W, i = 1 W = ^C,R. (15) (16) Eq. 16 can be written in a different form by substituting for R, the expression by which it is obtained in Eq. 14; that is, W = which simplifies to W = - 4 r P"^'''' (17) (18) It will be seen that W is not the equivalent number of applications of axle load a, but simply the total number of axle applications in mixed traffic, without regard to weight. It corresponds to the N used in the mixed traffic theory. Of the two forms, Eq. 16, in which R appears, is the easier to use. Tables or curves giving the value of R appears, is the easier to use. Tables or curves giving the value of R, for each axle load for each value of pavement design (that is, D or Di) and for values of Pt can be pre-
390 CONFERENCE ON THE AASHO ROAD TEST pared. Table 1 gives equivalent 18-ip application ratios for rigid pavement for a value of Pt = 2.0 for selected axle loads and pavement thicnesses; Table 2 gives ratios for flexible pavement. The values given in these two tables are taen from the AASHO Interim Guides for the Design of Rigid and Flexible (2). To solve the equation for W it is simply a matter of substituting in p and 8 the value of D or D2 and axle load a (18 ips for the equivalence ratios in Tables 1 and 2) and computing 2 C, from the axle load composition of traffic and the equivalent application ratios. Solving for the D (or D,) required to provide the desired number of applications to a selected value of Pt is more difficult. The design term appears in both p and p and cannot be solved for directly. The tas can be made easier, however, by the use of curves of D versus equivalent applications or by means of nomographs such as those used by the AASHO Design Committee in the Interim Design Guides (2). Source TABLE 1 EQUIVALENT 18-KIP APPLICATION FACTORS Axle FOR RIGID PATOMENT; P, = 2.0 Factors for Values of A (ips) 7 9 11 2 S 0.0002 0.0002 0.0002 10 S 0.08 0.08 0.08 18 S 1.00 1.00 1.00 24 S 3.34 3.47 3.53 10 T 0.01 0.01 0.01 26 T 0.64 0.62 0.62 32 T 1.45 1.50 1.51 44 T 5.59 5.82 6.03 AASHO Interim Guide for the Design of Rigid Structaies COMPARISON OF THE TWO METHODS To test for equivalence of the two theories, the derivative, /dw, may be found from Eq. 18 and compared with Eq. 12, as follows: Eq. 18 may be written, which differentiates to or or dw dw dw W = ^-i/p. (19) i?i^.p. r.2 1 p«eq. 12 may be written dn y i = 1 ^.p.1 = 1 _i = l P' i = 1 AP. i = l (20a) (20&) (20c) (21) According to Eq. 20c and 21, /dw and /dn are equivalent if Source TABLE 2 EQUIVALENT 18-KIP APPLICATION FACTORS Axle Load (ips) FOR FLEXIBLE PAVEMENT; P, = 2.0 Factors for Values of D 2 S 0.0002 0.0002 0.0002 10 S 0.08 0.08 0.08 18 S 1.00 1.00 1.00 24 S 3.62 3.33 3.51 10 T 0.01 0.01 0.01 26 T 0.34 0.35 0.33 32 T 0.82 0.84 0.82 44 T 3.36 3.18 3.31 AASHO Intel im Guide for the Design of Flexible Structures _i = 1 P» J Li = 1 P* C, 0-^/3." 2 (22) The right-hand member of Eq. 22 can be made identical with the left by letting /?, = P = constant, and subsequently dividing numerator and denominator by p. Thus, it is concluded that the two methods yield exactly the same result only if /? does not vary with axle load. But, according to the Road Test equations, js does vary with the type of axle as well as the weight, and it is concluded that the two methods must give different results. As will be shown, the difference, at least in practical cases, is small.
USE OF PAVEMENT RESEARCH FINDINGS 391 The results obtained by the two methods were compared for rigid pavements 7 in. and 10 in. thic and for flexible pavements having a D of 2, 4, and 7 (the value of D is given by the Road Test expression a^di + azdi + a-ids). The two different compositions of traffic given in Table 3 were used. In the traffic composition designated A, about 79 percent of the axle loads were less than 3 ips and nearly 4 percent were 20 ips or more. In traffic composition B nearly 96 percent of the axle loads were under 3 ips and less than 0.1 percent were 16 ips or more. The equivalent applications method almost always overestimated pavement life in comparison with the results obtained by the mixed traffic theory, and by about the same amount for both traffic compositions. Table 4 shows that the overestimate is not enough to be of real concern, however, except for the thin bituminous pavements with a D = 2. For both traffic compositions the equivalent applications approach gave a life (to a present serviceability index of 3 or less) a little more than double that found by the mixed traffic theory. For bituminous pavements with Z) = 4 the overestimate is in the neighborhood of 6 percent and for those with D = 7 it is hardly 1 percent. For 7-in. rigid pavements the overestimate is in the neighborhood of 6 percent and for 10-in. rigid pavements the difference is less than 1 percent. For the incremental solution and for design purposes the difference in the design required to provide a given life is of more significance than the difference in life provided by a given design. Accordingly, the number of applications computed by the mixed traffic theory was used to compute the required design by the equivalent applications method. The difference in design was, of course, found to be greatest for the thin bituminous pavements. As shown in Table 5, the equivalent applications approach indicated that a flexible pavement with a Z? of about 1.75 would provide the TABLE 3 TRAFFIC COMPOSITIONS USED IN COMPARING RESULTS OBTAINED BY MIXED TRAFFIC THEORY AND EQUIVALENT APPLICATIONS APPROACH Single-Axle Load (ips) 3 7 12 16 20 22 Total Axle-Load Distribution Assumed for Traffic Composition Designated A 0.7877 0.0811 0.0659 0.0263 0.0257 0.0133 1.0000 B 0.9587 0.0338 0.0056 0.0011 0.0005 0.0003 1.0000 same pavement life to Pt = 2.5 as was found for D = 2.0 by the mixed traffic theory. This 121/2 percent reduction in D is significant percentagewise. In terms of pavement thicness the difference represents a little less than 0.6 in, of surface, or 1.8 in. of base, or 2.3 in. of subbase. For the thicer bituminous pavements and for the rigid pavements the difference in D (or D2) was negligible. For example, the equivalent applications method yielded a flex- TABLE 4 PERCENTAGE BY WHICH NUMBER OF AXLE APPLICATIONS COMPUTED BY EQUIVALENT APPLICATIONS METHOD DIFFERS FROM NUMBER COMPUTED BY MIXED TRAFFIC THEORY Type and Design Present Serviceability Index Diiference in Number of Axle Applications (%) TrafRc Comp. A Traffic Comp. B Flexible 4.0 -M13.5 +8.8 D = 2 3.5 -f 138.4 + 60.6 3.0 + 131.5 +94.1 2.5 + 121.0 +115.4 2.0 + 114.0 + 126.5 1.5 + 108.6 +131.5 Flexible 4.0 + 5.0-3.8 ZJ = 4 3.5 +4.3 + 5.6 3.0 +5.9 + 7.1 2.5 +5.8 +6.7 2.0 +4.2 + 7.0 1.5 +3.2 + 6.0 Flexible 4.0 +9.4 +8.1 D = 7 3.5-2.6-1.7 3.0 +0.3-0.8 2.5-0.1 0.0 2.0 + 1.3 +1.5 1.5-0.6 +0.3 Rigid 4.0 + 6.8 +3.6 Z?2 = 7 3.5 + 6.5 +5.6 3.0 +6.6 +7.3 2.5 + 6.1 +6.5 2.0 +6.0 +5.8 1.5 + 6.3 +6.6 Rigid 4.0 +1.4 + 1.4 D2 = 10 3.5 + 1.0-0.4 3.0-0.8 + 0.3 2.5-0.2 +0.3 2.0-0.2 +0.2 1.5-0.4 +0.5 TABLE 5 COMPARISON OF REQUIRED DESIGN COMPUTED BY MIXED TRAFFIC THEORY AND EQUIVALENT APPLICATIONS METHOD FOR P, = 2.5 Type Flexible Flexible Rigid Required Design Computed by Mixed Traffic Theory 2.00 4.00 7.00 Equivalent Applications Method 1.75 3.96 6.94
392 CONFERENCE ON THE AASHO ROAD TEST ible pavement D of about 3.96 versus 4.00 by the mixed traffic theory, a difference representing less than 0.1 in. of surface or about 0.3 in. of base. Similarly, the rigid pavement thicness found by the equivalent applications approach was 6.94 in comparison with the 7.00 in. by the mixed traffic theory, a difference of only 0.06 in. of Portland cement concrete pavement. Similar comparisons were not made for the thicer flexible and rigid pavements because it was obvious that the difference in Z) (or ^2) would be even less. REFERENCES 1. SCRIVNER, F. H., "A Theory for Transforming AASHO Road Test Performance Equations to Equations Involving Mixed Traffic." HRB Special Report 66 (1961). 2. AASHO Committee on Design "AASHO Interim Guide for the Design of Flexible Structures," and "AASHO Interim Guide for the Design of Rigid Structures." (Not yet released).