Analysis of Tandem Axle Loads by Elastic Theory

Transcription

1 7 Analysis of Tandem Axle Loads by Elastic Theory Herbert F. Southgate, Robert C. Deen, and James H. Havens, Bureau of Highays, Kentucky Department of Transportation AASHO Road Test tandem axle loads ere analyzed to determine the magnitude of the tandem axle load that causes the same damage as an 8-kN (18 OOO-lbf/in ) single axle load. The procedure is outlined and is the same as that used to analyze single axle loads. Tho essential findings ere as follos: One repetition of a 11-kN (3 OOO lbf/in ) tandem axle load appears to cause the same damage as one repetition of an 8-kN single axle load. The relationship of the log of the number of repetitions of load versus the axle load, hich is used in Kentucky, appears to be equally valid for single and tandem axle loads. The use of superposition principles and equivalency of repetitions, in combination ith terminal serviceability indexes, permits analyses of the tandemaxle-load data and comparison ith equivalent Kentucky thickness designs converted to the AASHO structural number. Single-axle-load analyses ere combined ith the tandem-axle-load analyses and superimposed by Kentucky equivalent designs. The Kentucky designs, based on elastic theory, envelop 91 percent of the AASHO Road Test data. For pavement design, the estimated fatigue for a -year design that uses the AASHO damage factors ill be reached in 1. years if the Kentucky damage factors are used. The Kentucky damage factors more closely approximate observed behavior than do the AASHO factors. Thicknesses of flexible pavement structures have been dete1'mined by seve1 a1 systems that include the Kentucky 198 and 199 curves (!J!), the AASHO interim guide(, and the Kentucky proposed 1973 design guide (). The objectives of this paper are (a) to determine tile magnitude of the tandem axle load that produces the same equivalent damage as one 8-kN C 18 -lbf (18-kip)] single axle load, (b) to develop appropriate damage factors for tandems by using the above tandem axle load as the base value, and (c) to compare the results obtained by using the above design methods. BACKGROUND The 199 Kentucky design curves ere based on empirical tests and observations of pavements that ere made in 198 and 197. They ere generally dran to separate points representing pavements that ere performing satisfactorily from those that ere not. The 19 9 curves ere an extension of the 19 8 curves to include the effects of increased traffic volumes. The 1973 Kentucky design guide is based on elastic theory; the curves (, ) are draxvn to provide structural thicknesses that have the same strain values. The limiting values of strain ere obtained by matching theoretical strain values ith field performance data. The matching as based primarily on the performance of a 8-mm (3-in) thick pavement composed of 19 mm (7.7 in) of asphalt concrete and 39 mm (1. in) of dense-graded aggregate placed on a soil having a California bearing ratio (CBR) or 7 to ithstand a field loading of 8 x 1 applications of 8-kN axle loads. A stated objective of the AASHO Road Test (7) as to determine significant relationships beteen the-numbers of repetitions of specified axle loads of different magnitudes and arrangements and the performance of different asphalt-concrete pavement structures. The emphasis as on the analysis of pavement structural fatigue and the comparison of load-damage effects. To make a direct comparison of design thicknesses obtained by the AASHO system {3) and the Kentucky system (_!) requires expressing the to systems in equivalent terms. Fatigue data for each pavement section of the test road formed the basis for the AASHO thickness design nomographs. The folloing equation as used to express the relationship beteen the number of repetitions (N), the structural numbei (SN), and the level of serviceability (pt): logn = Iog(SN +I) -.79 log(l, + L) here +.33logL + (G,/Bxl Gt= log[(. - Pt)/(. - 1.)], Bx=. + O.OS(L. + L) 3 3 /(SN + 1) 19 L 3 Lx =axle load (!tj.ps), and L = 1 for single axle loads and for tandem axle loads. Comparison of seasonally eighted data (7) ith Equation 1 as deemed appropriate and necessary:- The folloing methodology as used in Kentucky to analyze singleaxle-load data from the AASHO Road Test (_,.>: 1. Layer thicknesses for each test section ere converted to the corresponding SN by using Equation : here d1, d, and d 3 =thicknesses of surface, base, and subbase layers in the pavement structure respectively and ai, a, and aa =.,.1, and.11 respectively. (The operations of these equations require the use of U.S. customary units; therefore, SI units are not given in Figures and 7.). Graphs ere made of Pt versus N for each section of each loop. 3. Graphs at various Pts ere made for SN versus N (Figlrre 1) (7, appendix A).. The 1973 Kentucky damage factors (DFs) ere calculated by using Equation 3: OF= (1.)< -te) (3) here P = single axle load (in kips) and the constant 1. is the slope of the straight-line, single-axle load versus log repetitions relationship; the base or reference axle load as taken to be 8 kn.. For a given P,, the products of N and DF ere plotted on the graph for 8-kN data from lane 1 of loop. This procedure converts (by superposition) the N of a given axle load to an equivalent N of an 8-kN axle load. The corresponding SN is not changed.. For single axle loads, Equation 1 reduces to logw 1 = \og(SN+ l)-.79log(lx + l)+(gtfbxl () Solutions of Equation ere superimposed on the graphs of step 3. (I) ()

2 8 7. Studies had determined that the soil used at the AASHO Road Test had a CBR of. as determined by the Kentucky CBR test method. Thicknesses () corresponding to a CBR of. and an asphalt concrete-modulus of 3.31 GPa (8 lbf/in ) ere converted to SNs by Equation and superimposed on graphs from step. For Pts of. and., Equation approximated a best fit of the data( and the Kentucky design method requi'l ed thicknesses in terms of SN) greater than that required by approximately 93 percent of the Road Test single-axle-load fatigue data. For other Pt graphs, the Equation trend line and the Kentucky design curve reasonably fitted some portions, but not all, of any range of data-except for the 8-kN data at a Pt of.. 8. The relationship for single axle loads shon in Figure as used to choose the appropriate Pt versus axle load combination to be superimposed on the 8-kN base graph for a P, of. [i.e., to determine the number of repetitions of 8-kN equivalent axle loads (EALs)] (Figure 3). 9. The folloing observations ere made: (a) The 3.31-GPa asphalt-concrete-modulus Kentucky curve Figure 1. Relationship beteen number of repetitions of 8-kN (18 OOO lbf/in ) single axle loads and structural number at terminal serviceability of z Cl) f-- :r 3 <t. Pt. 8-KN (18-KIP) AXLELOAD o--r o.-q--o>-- LEQllATION 1. Note: AASHO Road Test seasonally eighted data ,o REPETITIONS Figure. Relationship beteen level of, serviceability and axle load.. KENTUCKY_ SINGLE AXLES :::; iii (,) Cl). >. IL. 1, REPETITIONS o..._._.._ AXLELOAD, KIPS Figure 3. Relationship beteen structural number and number of repetitions of 8-kN (18 OOO-lbf/in ) equivalent axle loads , i!!. 1 I I I I I ' lllll's 1. «p et _,_.- I.JI. AJCLELOAD9 AT REPETITIONS EQUIVALENT TO N REPETITIONS.. L : :a OF 8-KILONEWTON Cl8-KIPI AICLELOADS BASED UPON 1973,. Q :> KENTUCKY AllLELOAD-EQUIYALENCY FACTORS 9 1tllN['ffTON HL[CUlt z ' g z., s a a -- - e e od c a a ' REPETITIONS OF 8-klLONEWTON 118-KIP) EAL voo' ' c ' illcjj:lia. 7!13 1,. I 1.

3 9 shon in Figure 3 requires an SN greater than that required by approximately 93 percent of all AASHO singleaxle-load data points [Figure 3 also includes Kentucky moduli lines for 1.8,.1,.9, and.1 GPa (7, 3 ooo, 37, and lbf/ln )]; (b) the 8-kN single-axle-load curve for Pt =. from Equation as superimposed onto Figure 3, shoing that the SN for 9 percent of the data points as less than that from Equation, has the same general shape suggested by the data points [except for the 7-kN (-lbf) set], and is almost a direct fit of the family of asphalt-concrete elastic-moduli lines; (c) the combination of a and b confirms that the 1973 Kentucky design guide () does provide for variable terminal serviceability levels, as a function of the EAL level, ith a high probability of success in the design life; (d) elastic theory p1 ovides a sound basis for accurately describing the observed performance data of both Kentucky highays and the AASHO Road Test; (e) one level of terminal serviceability should not be used for a ide range of EALs (as s uggested by AASHO thickness design nomographs); (f) AASHO nomographs provide for an approximate -percent probability of successful pavement performance throughout the design life; and (g) the semt-loguithmic relationship beteen single axle load and repetitions that is used in the 1973 Kentucky design guide is not only a reasonable assumption, but has been confirmed by the analysis of the AASHO Road Test data (7). Kentucky had not developed its on damage factors for tandem axle loads, nor had it determined the magnitude that produces the same damage as one repetition of an 8-kN single axle load. The purpose of this investigation is to provide these ansers. ANALYSIS Single Versus Tandem Axle-Load Equivalency Analysis of the single-axle-load data (8) from the AASHO Road Test (7) proved to be rearding enough to arrant an analysis of the tandem-axle-load data (7, appendix A). AASHO (3) had determined that an 8-kN single axle load and a 1'f.kN (33 1-lbf) mean tandem axle load both produced the same damage. Hoever, it as noted that the equivalent tandem axle load varied and as a function of SN and Pt. Again, it as deemed appropriate and desirable to investigate the tandem-axle-load data by steps 1 through 3 used to analyze the single axle loads. Thus, the 1-kN (3 lbf) tandem-axle-load data ere plotted for Pts of 1.,.,., 3., and 3.; Figure ( 7, appendix A) for Pt =. is an example. The dashed line is the solution of Equation 1, and the 3.31-GPa line h; the i;ame Kenlucky solulion shon in Figure 1. The similarity beteen Figures 1 and prompted plotting of the 17-, 178-, and 1-kN (,, and 8 lbf) tandem-axle-load data for the same five values of Pt. The same trends ere evident; this is represented by the latter portion of step 8 (except that the reference axle load as 1 kn). To proceed to step, the first task as to determine the tandem axle load that produced the same equivalent damage as an 8-kN single axle load. The criteria to be satisfied ere that 1. The level of serviceability must be equal,. The same number of repetitions must be used, 3. One value of an AASHO SN must be chosen to evaluate the equivalent effects of single and tandem axle loads, and. The tandem-axle-load damage factors must be compatible ith the 1973 Kentucky () single-axle-load damage factors. - Kentucky has associated 8-kN single axle loads and a P,-value of.. After determining (8) that the 1973 Kentucky design guide () and the AASHO Road Test results (7) ere compatible, the same serviceability value of. as chosen. As shon in Figures 1 and, the AASHO Road Test data (7) at a serviceability of. had a span of repetitions from 1 to 1 ; 1 repetitions as chosen for analysis only because that point is the middle of the log scale. From Figure 1, the 3.31-GPa line intersects the 1 repetitions line at an SN value of 3.7. Thus, the criteria values ere determined. The 3.31-GPa line exceeded the SN for 93 percent of the AASHO Road Test single-axle-load data (Figure 3) and, as shon in Figure 1, one data point as above the 3.31-GPa line. The 3.31-GPa line as superimposed on Figure, and similar figures for a serviceability value of. and tandem axle loads of 17, 178, and 1 kn ere prepared. A line of equal offset to the 3.31-GPa line as constructed such that one data point as above the constructed line. The SN value at 1 repetitions as graphically determined and transferred to Figure. These four pairs of axle load versus SN values ere connected ith a smooth curve. Thus Figure shos that, at SN= 3.7, an 8-kN single axle load Figure. Relationship beteen number of , repetitions of 11-kN (3 OOO lbf/in ) tandem axle loads and structural number at 8 terminal serviceability of.. z Pt. 1-KN (3-KIP) TANDEM AXLELOAD,._ Note: AASHO Road Test Mlasonally eighted data.,,.$) --ace OOCX>.Jll- - r:fi) AASHTO EQUATION m m m REPETITIONS

4 Figure. Relationship beteen structural number and axle load. 9 B 7 Pt. ID, REPETITIONS SINGLE AXL/E ANDE!il AXLE - t SINGLE AND TANDE!il AXLELOADS EQUATED AS DA!llAGE FACTOR OF 1. AT 1, RE PETIT IONS AND AASHTO STRUCTURAL NU!llBER OF l'\l fis KILONEWTONS TANDEM AXLELOAD ould be equivalent to a 11-kN (3 lbf) tandem axle load. Tandem-Axle-Load Equivalency Factors The tandem-axle-load equivalency factors must be compatible in methodology ith the single-axle-load damage factors. Thus, the Kentucky single-axle-load relationship of axle load versus log number of r epetitions (Figure ) required the use of the same axle load ve.rsus log number of repetitions format for tandem axle loads. Also, Kentucky has associated 8 repetitions of 8-kN single axle loads as a reference pavement. Thus, a 11-kN tandem axle load must be associated ith 8 repetitions on the same pavement. Equation as used to express the single axle load versus number of repetitions relationship labeled as the 1973 Kentucky line in Figure : EAL= N(l.)<P-is) () When EAL =8 and P (single axle load) = O, N = 133. If e assume this same intercept value for a tandem axle load and use the point for a tandem axle load of 11 kn at 8 repetitions, e find that the resulting straight line in Figure is given by EAL= N(l.1)<P- 3 l () Here, the constant 1.1 is the slope of the tandem axle loads versus log number of repetitions relationship. The tandem-axle-load equivalency factors shon in Figure 7 ere derived as step from Traffic equivalency factor= N 3 /N (7) here N3 =number of repetitions for 11-kN axle load and NP = number of repetitions for any other given tandem axle load. Equation 7 can be expressed in the form of Equation 3 as DF = (l. I S)<P- 3 > (8) The traffic-equivalency-factor relationships for the 1973 Kentucky tandem axle load and the 197 AASHO interim guide tandem axle load at SN=. and P, =. are shon in Figure 7. AASHO Serviceability Ratings Figure 3 is the result of combining data for five pairs of axle-load and serviceability levels (step ) and is sh.on in Figure as the llne lapeled as single axle load. The Kentuky 3.31-GPa line (step 7) and the line from Equation (step ) are superimposed. Tandem-axle-load data ere processed in the same manner as single-axle-load data except that the reference axle load as 1i kn. In step, the data for lane of loop ere used as the base graph. In step 8, the tandem relationship shon in Figure as used. The tandem axle load versus serviceability level relationship shon in Figure as used to determine the serviceability level for the four levels of tandem axle loads used in the AASHO Road Test (step 8). Road Test data (7) ere plotted for each axle load (step 3), and the repetitions at the respective serviceability levels ere interpolated and plotted in a manner similar to Figure. By using the 197 Kentucky traffic-equivalency factors for tandem axle loads shon in Figure 7, the interpolated J.-.L.-. L.J L-. _..:-.,.1.,_L 1 C1 1-T ----J!L! U1.i.U. -.;;;-.i.;;;:.:;;-y_u«,i,.1.ju LU ;;;:ifu.i.vd.j.\;l.i.l.i.u.i. - n.l,.i. C!J C U.UUUO ao shon in Figure 8. Approximately 89 percent of the AASHO Road Test tandem data required an SN value less than the Kentucky 3.31-GPa line although only 7 percent as beneath the line obtained for tandem-axle-load solutions of Equation 1.

5 1 Figure. Relationship beteen number of repetitions of load and axle-load equ ivalencies. i: ,. 1'. ;:. ' ' : io 191' IC[NTUCK'I' TANDllll )flu.ao T'llAFC [QUl\IAlt:NCl[S lal NlllHt ' 1 1 JO O SO &O TO AXLELOAO (KIPS) loo ISO SOO SSO SO AKLELOAD (KN) &oo 'o 1 uo 1 no Figure 9 is a combination of single- and tandemaxle-load data from Figures 3 and 8. The Kentucky 3.31-GPa line is an envelope for 91 percent of the combined AASHO Road Test data; the Equation 1 solutions require an SN greater than that required by approximately percent of the data. Net Effect of Kentucky Traffic Equivalency Factors To determine the net effect of the ne, Kentucky singleand tandem-axle-load traffic-equivalency factors, data in the W- tables for 19 9 through 1973 ere analyzed. The number of axle loads of various groupings of loaded and empty trucks and truck combinations of each type eighed ere summed by axle-load groups ithin the single- and tandem-axle-load divisions. These sums ere multiplied by the corresponding traffic-equivalency factors for the 199 and 1973 Kentucky design systems and the 197 AASHO interim guide factors for SN =. and Pi =.. The total number of accumulated EALs are s hon in Figure 1 and indicate that the AASBO (3) factors ould produce a -year design that ould be reached ithin 17. years by using the 19 9 Kentucky factors and 1. years by using the 1973 Kentucky factors. Until this current analysis, there has not been a set of tandem-axle-load traffic-equivalency factors for application in Kentucky. The practice has been to eigh the individual axles of the tandem arrangement and use the single-axle-load equivalency factor for each axle load. Thus, the calculated total accumulated EALs have been higher than the 197 AASHO values but less than the 1973 Kentucky values. SUMMARY The AASHO Road Test tandem-axle-load data ere analyzed by using the procedure reported p1 eviously (8) for a single-axle-load analysis. The folloing are the major points in this report: 1. All other factors being equal, one 11-kN tandem axle load produces the same fatigue damage as one 8- kn single, rear axle load.. The logarithm of the number of repetitions versus axle load relationship used by Kentucky (, ) appears to be equally as valid for tandem axle loads -as for single axle loads. 3. A correlation beteen serviceability index and

6 axle load as developed for single and tandem axle loads.. The combination of a terminal serviceability that varies ith axle load and the equation for single and tandem axle loads permits combining the AASHO Road Test single- and tandem-axle load versus number of repetitions data (Figure 9).. The superpositioned Kentucky design curve requires SN values greater than the SN for 91 percent of the combined single- and tandem-axle-load data from the AASHO Road Test.. Pavements designed by using the 197 AASHO thickness design nomographs have been shon to reach the design fatigue level in approximately 8 percent of the expected life hen adjusted for increased volume. This discrepancy is explainable by the 1973 Kentucky axle-load damage factors. RECOMMENDATIONS The AASHO damage factors used in the Kentucky W- tables should be replaced ith the folloing factors: 1. For single axle loads-damage factor = (1. Q)(P- IS),. For tandem axle loads-damage factor = (1.1 )(P - 3 ), Figure 7. Traffic equivalency-factor relationships AASHTO SN= P 1 = IC u. 1. Cl 9 8 C!i 7.1. _ _ TANDEM AXLELOAD, KIPS 8 Figure 8. Relationship beteen structural number and number of repetitions of 11-kN (3 OOO-lbf/in ) tandem axle loads. Alll.LELOAD LEVEL OF SERVICEABILITY D D SYMBOL AXLE LOAD KILO NEWTONS D 3Z ' NUMBER Of POINTS B,. 7 i!.7 I ' ' ' ' lo' 1' REPETITIONS OF 3-KIP TANDEM AXLELOAD

7 3 Figure 9. Relationship beteen structural number and number of repetitions of 8-kN (18 OOO-lbf/in ) single or 11-kN (3 -lbf/in ) tandem axle loads. a: j ::> z _,,_ '!i u ::> a:,_ Cl> ' -[ Alll[l.QAO, KIPS - TANOOI AiJCLELOAO, ICtPS SMl[ A'lrl(l.QAOS. a JIC)llrs - S(l't >O 1.87,. Z9.,. '.. 1,. REPETITIONS OF 18- KIP SINGLE AXLELOAO OR REPETITIONS OF 3-KIP TANDEM AXLELOAO. >O 1' Figure 1. Relationship beteen accumulated total number of equivalent axle loads and time. 1 I 199 KENTUCKY 1973 KEN TIJC-, KY li97 AASHTO INTERIM GUIDE.:'.J --',_,_,_ c 1 --' ::;: u I I METHOD f, If DESIGN LIFE, YEARS 197 AASHTO INTERIM GUIDE 199 KENTUCKY KENTUCKY YEARS SINCE 198 ADDITIONAL RESEARCH IN PROGRESS Current efforts beyond the scope of this report use superposition principles, elastic theory, and strainenergy equations to analyze the AASHO Road Test loads and configurations of heel loads applied to various pavement thicknesses to separate the fatigue effects caused by front axles from those caused by rear axles. Increased use of ide tires on the front axle and of trailers having three or four axles in one group ill require additional analyses to determine the appropriate equivalent loads and associated damage factors. REFERENCES 1. R. F. Baker and W. B. Drake. Investigation of Field and Laboratory Methods for Evaluating Subgrade Support in the Design of Highay Flexible Pavements. Engineering Experiment Station, Univ. of Kentucky, Bull. No. 13, W. B. Drake and J. H. Havens. Kentucky Flexible Pavement Design Studies. Engineering Experiment Station, Univ. of Kentucky, Bull. No., AASHO Interim Guide for Design of Pavement Structures. AASHO, J. H. Havens, R. C. Deen, and H. F. Southgate. Design Guide for Bituminous Concrete Pavement Structures. Division of Research, Kentucky Bureau of Highays, R. C. Deen, H. F. Southgate, and J. H. Havens. Structural Analysis of Bituminous Concrete Pavements. HRB, Highay Research Record 7, 197, pp J. H. Havens, R. C. Deen, and H. F. Southgate. Pavement Design Schema. HRB, Special Rept. 1, 1973, pp The AASHO Road Test: Report -Pavement Research. HRB, Special Rept. 1E, H. F. Southgate, R. C. Deen, and J. H. Havens. Adaptation of AASHO Interim Guide to Fundamental Concepts. Proc., AAP T, Vol., 197, pp Publication of this paper sponsored by Committee on Theory of Pavement Design.