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E cient computational algorithms for forward and backward analysis of a dynamic pavement systemRobert Y.Liang *,J.X.ZhuDepartment of Civil Engineering,Center for Infrastructure Materials and Rehabilitation,The University of Akron,Akron,OH 44325-3905,U.S.A.Received 11December 1996;accepted 21August 1997AbstractA simple dynamic analysis algorithm is presented in this paper for both forward and backward calculations of a pavement system consisting of an asphalt concrete layer,underlain by a uniform subgrade to a depth H wherein the bedrock is located.The subgrade soil is represented by a higher-order continuum modelÐthe modi®ed Vlasov model.The asphalt concrete layer is represented by a three-parameter complex compliance function in a frequency domain.The governing equations of the dynamic pavement system,along with the solution algorithms for both forward and backward computations,are presented in detail.A numerical example is provided to illustrate the importance of considering dynamic e ect in predicting pavement response under dynamic load.In addition,numerical examples are given to demonstrate the use of nondestructive testing data to back calculate the material properties,such as the modulus,damping,creep compliance and fatigue cracking speed for an asphalt concrete layer and the modulus damping for the subgrade layer.#1998Elsevier Science Ltd.All rights reserved.Keywords:Dynamic;Pavement;Forward;Backward;Numerical algorithms1.IntroductionIn recent years,the dynamic response of a pavement system subjected to moving tra c loads or dynamic loads from nondestructive testing (e.g.Falling Weight De¯ectometer and Dyna¯ect)has been the subject of intensive studies.Earlier,Kausel and Roesset [1]devel-oped the solutions for Green's functions of a layered medium on a rigid ter,Roesset and Shao [2]used these Green's functions to analyze the test results of Falling Weight De¯ectometer (FWD)and Dyna¯ect tests.Cebon [3]and Hardy and Cebon [4]used the nu-merical convolution theory for calculating the response of a pavement subjected to moving dynamic loads.The e ects of vehicle speed,frequency of loading on the pavement response were considered.Trochanis et al .[5]developed a uni®ed procedure based on the fastFourier transform (FFT)technique for analyzing the steady-state response of an in®nite long beam sup-ported on a linear elastic,damped foundation and sub-jected to moving loads.Luco and Wong [6]developed an identi®cation tech-nique that can be used to determine dynamic soil prop-erties such as wave velocities and material damping ratios from the knowledge of experimentally deter-mined impedance functions.In particular,it is interest-ing to note that the soil properties can be identi®ed by use of just one impedance function (say the vertical impedance function).Uzan [7]presented a dynamic lin-ear back calculation procedure for estimating pave-ment material properties.It was found that the parameters of the generalized power law of the creep compliance of the asphalt concrete from back calcu-lation and laboratory results do not compare well.He suggested that the sample tested in the laboratory may not be representative of the asphalt concrete layer.Computers and Structures 69(1998)255±2630045-7949/98/$-see front matter #1998Elsevier Science Ltd.All rights reserved.PII:S 0045-7949(97)00107-7PERGAMON*Corresponding author.There are various approaches for representing sub-grade support in pavement analysis.The simplest approach of all is the use of a Winkler model.In fact,some of most commonly used computer programs have been developed based on the ®nite element method for pavements on Winkler's foundation.These include ILLISLAB of the University of Illinois [8],KENSLAB of the University of Kentucky [9]and DYNASLAB of University of California at Berkeley [10].However,the Winkler assumption is insu cient to represent a continuous medium.Realizing the inadequacy of the Winkler model,Vlasov and Leont'ev [11]developed a two-parameter model for a beam on an elastic subgrade.The so-called Vlasov model requires an estimate of a parameter,g ,that controls the decay of the stresses in the conti-nuum.Vlasov did not give a precise value for g ,instead he recommended arbitrary values for g .Vallabhan and Das [12,13]developed a unique itera-tive technique to determine a consistent value of the g parameter.Their model is called a modi®ed Vlasov model.In this paper,the modi®ed Vlasov model previously developed for the static conditions is further expanded for dynamic conditions involving an asphalt concrete layer underlain by Vlasov subgrade.The proposed sub-grade model is considered to be more realistic and accurate compared to the traditional Winkler model.A back calculation algorithm based on a dynamic com-pliance function with a high rate of convergence is also developed.The methodology for predicting moduli and damping of asphalt and subgrade layers and creep compliance,fatigue cracking growth and fatigue life of the asphalt concrete layer in the ®eld using the backcalculation procedure in conjunction with nondestruc-tive testing data is developed as well.2.Problem statementThe pavement system is idealized as shown in Fig.1.An in®nite beam with width b and thickness h is rested on a uniform viscoelastic soil subgrade of thickness H ,which,in turn,is underlain by a bedrock.A plane strain condition is assumed,hence the beam width b is considered as unity.The pavement layer is assumed to be an asphalt concrete material and is subjected to a tra c load q (t )traveling at a constant speed V and oscillating vertically with circular frequency o .A con-venient approach for deriving the governing equations of motion for such a complex dynamic boundary value problem is to use Hamilton's principle,which is given by:d t 1t 0& I ÀI r r d "w d t 2d x b 2 I ÀIH 0r s d w d t 2d z d xÀ12I ÀIE p I p d 2"wd x 2 2d xÀb 2 I ÀIH 0r s s x e x s z e z g xz t xz 2v d z d x 'd t t 1t 0 IÀIq x ,t d "w d x d t 0, 1where w (x ,z ,t )="w(x ,t )f (z ,t )is the vertical displace-ment and f (0)=1and f (H )=0;s x ,s z and t xz are components of stress at a point in the soil;E x ,E z and g xz are corresponding strains in the soil;E p =E *p (1+i 2D p ),where E *p is Young's modulusofFig.1.Dynamic pavement±subgrade interaction model.R.Y.Liang,J.X.Zhu /Computers and Structures 69(1998)255±263256Table1Material properties of asphalt concrete pavement and soilParameter Symbol ValueParameters of AC pavement D00.5Â10À8psfÀ1(0.1044GpaÀ1)D10.5Â10À7psfÀ1(1.044GpaÀ1)m0.4Mass density of pavement r p 4.46lb s2/ft4(2300kg/m3) Poisson's ratio of pavement u s0.35Width of pavement b1ft(0.3048m)Thickness of pavement h1ft(0.3048m)Young's modulus of soil E s0.432Â107psf(0.2069GPa) Mass density of soil r s 3.98lb s2/ft4(2052kg/m3) Poisson's ratio of soil u s0.4Damping ratio of soil D s0.05Depth of soil H25ft(7.62m)the pavement layer,D p is the damping ratio and I p is the moment of inertia,E*p t and D p(t)will vary with time;E s=E*s(1+i2D s),where E*s is Young's mod-ulus and D s is damping ratio of the soil;r p and r s are mass densities of the pavement and soil,respectively. The material properties of asphalt concrete pavement and soil are summarized in Table1.3.Representation of asphalt concrete pavementA generalized time-domain power law representation of asphalt concrete layer,a three-parameter represen-tation,is adopted as follows:D t D0 D1t m, 2 where D(t)is the compliance,t is time,D0is the elastic compliance,D1is the intercept at t=1s and m is the slope of the log creep compliance vs log time. Because of the reciprocal relationship between com-pliances and moduli,the®rst and second compliances in Eq.(2)can be written as follows:D0 EÀ10 3a andD1 EÀ11, 3b where E0is the elastic modulus and E1is the viscoelas-tic modulus at t=1s.Expressing Eq.(2)in terms of the modulus in Eqs.(3a)±(b),one getsD t 1a E0 t m a E1 4 This representation is equivalent to two springs inseries when evaluated at t=1s.The time-domain creep compliance functions can be transformed into the frequency domain for use in the dynamic analysis.The frequency-domain represen-tation is called the complex compliance because it can be expressed as a complex number having a real part and an imaginary part.Performing a Fourier integral transform on Eq.(2),one obtains the following three-parameter complex compliances[14]:D o D0 D1G 1 m oÀm cos m p a2 Ài sin m p a2 ,5 where G is the gamma function.The complex modulus is directly related to the com-plex compliance by:E p o E*p o 1 i2D p o1D o, 6 where7aD p o 12D1G 1 m oÀm sin m p a2D0 D1G 1 m oÀm cos m p a2X 7bThe Young's modulus E*p and the material dammingratio D p of the asphalt concrete will vary with fre-quency,o and can be determined frequency by fre-quency using Eqs.(7a)±(b).R.Y.Liang,J.X.Zhu/Computers and Structures69(1998)255±263257erning equationsBy taking variations of the energy and virtual work functions,Eq.(1),with respect to w and f ,the follow-ing dynamic equation for the pavement system is obtained:E p I pd 4"w d x 4Àt s d 2"w d x 2 k s "w m s m p d 2"wd t2 q x ,t for ÀI `x ` I ,8where t s G s bHF 2d z ,9k s l s 2G s b H 0d F d x 2d z ,10m s r s b HF 2d z X11Introducing a moving coordinate X traveling with the load,i.e.X =x ÀVt ,where x is the spatial coordi-nate along the tra c moving direction,as shown in Fig.1and concentrating on steady state solutions ofthe form "w(X ,t )=W (X )e i o t for the pavement de¯ec-tion and q (X ,t )=Q (X )i o t for the moving load,the explicit time dependence in Eq.(8)can be eliminated.An ordinary di erential equation for W (X )is written as:E p I p d 4W d X 4Àt sd 2W d X 2 k s W m s m p  V 2d 2W d X 2Ài 2o Vd Wd X Ào 2W Q d X for ÀI `X ` I X12The displacement under the load is obtained for a con-stant load Q with a constant speed V (o =0):W movingQ8E p I p k s 4E p I p s k s 4E p I pst s À m s m p V 2 4E p I p v u u t X 13For a stationary harmonic load Q e i o t (V =0):W dynamicQ8E p I pk s À m s m p o 2 4E p I p sk s À m s m p o 24E p I p s t s4E p I pv u u t X 14For the static load Q (V =0and o =0):W staticQ8E p I pk s 4E p I p s k s 4E p I p s t s 4E p I pv u u t X 155.Fatigue of asphalt concrete pavementFatigue is the phenomenon of fracture e ected byrepeated application of stresses less than the strength of the material.Under tra c loading,the asphalt con-crete pavement is subjected to ¯exural action in which ¯exural strain is dependent on the wheel loading con-ditions,the thickness of the pavement and the proper-ties of the pavement material and subgrade soil.Fatigue transfer functions are used to relate the num-ber of load repetitions to reach certain pavement cracking failure conditions (i.e.crack initiation,10%area cracking,etc.)to the maximum tensile strain in the asphalt concrete pavement.The strain-based fatigue relationship [15±17]has been the most commonly used phenomenologically-based fatigue equation:f K 1a E n ,16where N f is the number of load repetitions to failure,Eis the maximum tensile strain,and K and n are re-gression constants.The constants K and n are derived from regression analysis of laboratory test data.A large variability of these constants for various asphalt concrete mixtures is noted.There is no unique asphalt concrete pavement fati-gue algorithm that can be utilized in a mechanistic empirical pavement design procedure.However,the strain-based fatigue algorithm is utilized in The Transportation and Road Research Laboratory [18],Nottingham University [19],the Mobil Pavement Design Guide [20],the National Road Directorate of Denmark [21],The Belgian Road Research Center [22]and Illinois DOT/University of Illinois [23,24].In general,the strain-based approach has provided a reasonably simple procedure that has been widely adopted.However,it bears the limitation that it can-R.Y.Liang,J.X.Zhu /Computers and Structures 69(1998)255±263258not take into account both crack initiation and propa-gation.Such a di erentiation may be quite important in estimating fatigue life.If a pavement material is brittle,the time required to initiate the crack will con-stitute the major portion of its fatigue life and crack propagation will be rather rapid.On the other hand, as the pavement material becomes more ductile,the time needed to propagate the crack to failure will con-stitute an increasing portion of the fatigue life of the pavement.A fracture mechanics approach for predicting the fatigue life of the asphalt concrete has been developed by Majidzadeh et al.[25]and Little and Mahboub[17]. Both studies indicated that in the fracture mechanics approach,the rate of crack propagation in asphalt concrete can be predicted by using an empirical power law relationship developed by Paris:d c a d N A D K n, 17 where N is the number of load cycles,D K is the ampli-tude of stress intensity factor,and A and n are fracture parameters of the material.The Paris equation relates the rate of crack growth (d c/d N)to the induced stress intensity factor(K)in the form of a power law.With the knowledge of material properties A and n,Eq.(17)can be integrated over a given range of crack length to calculate fatigue life(N f): N f cfc0d c a A K n , 18where c0is initial crack length,c f is®nal crack length and K is the stress intensity factor for cyclic zero-to-tension loading.At a constant loading amplitude(s),K is de®ned as: K s p C1a2 X 19 Combining Eq.(18)and(19)and performing the inte-gration,one obtains:N f 2a 2Àn A s n p n a2 c 1Àn a2f Àc 1Àn a220Jayawickrama and Lytton[26]in their study have shown that the relationship:n 2a m 21 can be used to predict n value with reasonable accu-racy,where m is the slope of the log creep compliance vs the log time relationship.They indicated that a lin-ear relationship exists between n and log A.For tra c-associated loading,the following relationship was pro-posed by Lytton and Shanmugan[27]:n À1X558À0X401log A X 22 6.Back calculation algorithmIn the back calculation procedure,the falling weight drops to a pavement surface,generating both body waves and surface waves.The geophone sensors pick up the vertical velocity of the pavement surface and a single analog integration of the signal produces the de¯ection vs time ually these signals are used to extract the maximum force and the maximum de¯ection from each geophone and to print them out for analysis by static methods.But there is much more information in these signals than simply their maxima. One method of tapping this additional information is to perform a fast Fourier transform on the force±time impulse and on each de¯ection±time response. The transform breaks up a signal into its component frequencies and produces a complex number for each frequency.If the transform of the de¯ection signal is divided,frequency by frequency,by the transform of the load impulse,the result is a transfer function, which is also a complex number and a function of fre-quency.The magnitude is the de¯ection per unit of force at each frequency and the phase angle represents the time lag of the response behind the impulse at each frequency.The complex frequency response function(dynamic compliance function)of the pavement±soil system from Eq.(14)is given as F(o)=W dynamic/Q.The com-plex frequency response function contains information about the mechanical properties of the pavement sys-tem.It is convenient to split the complex frequency re-sponse function into its real part and imaginary part in the form:F oH3E p I pf R o iH3E p I pf I o , 23where f R(o)and f I(o}are the real and imaginary parts of the nondimensional dynamic compliance function, respectively.Supposing that these functions derived from exper-imental data are represented by"f R o and"fIo XDenote m=[D0,D1,m,E s,D s]as the vector contain-ing the unknown properties of the pavement±soil sys-tem.For the inversion process,a set of frequencies o r(r=1,N l)is selected and it is required that the model m minimize the sum of squares:E mN lr 1f f R o r,m À"f R o r 2 f I o r,m À"f I o I 2g X24 The conditions for an extreme corresponding tod Ed m i0(i=1,2,3,4,5)are given by:R.Y.Liang,J.X.Zhu/Computers and Structures69(1998)255±263259N i r 1& f R o r ,m À"f Ro r d f R d m if I o r ,m "f Io I d f I d m i' 0,25which are implicit equations for the minimizing model m .In the following numerical examples,®ve frequen-cies are used;i.e.,0.1,0.5,1,5and 10Hz.7.Numerical examples and discussions 7.1.Example of forward calculationTo demonstrate the calculatoin results of a forward analysis,the following properties are assumed for as-phalt concrete:D 0=0.5Â10À8psf À1(0.1044GPa À1),D 1=0.5Â10À7psf À1(1.044Gpa À1)and m =0.4,the mass density of pavement r p =4.46lb s 2/ft 4(2300kg/m 3),width of pavement b =1ft (0.3048m),thickness of pavement h =1ft (0.3048m),speed of vehicle V =0,Young's modulus of soilE s =0.432Â107psf (0.2069GPa),mass density of soil r s =3.98lb s 2/ft 4(2052kg/m 3),Poisson's ratio of soil u s =0.4,damping ratio of soil D s =0.05,depth of subgrade H =25ft (7.62m)and 50ft (15.24m).The calculated complex frequency response functions are shown in Fig.2.As can be seen,the displacements computed using a dynamic analysis can be quite di erent from those computed using a static analysis (Young's modulus of asphalt concrete pavement E p =2.0Â108psf).The di erences in displacement for the case of thick sub-grade (H =50ft)are much smaller than those for the case of thin subgrade (H =25ft).A signi®cant ampli-®cation is observed near the fundamental frequencyofFig.2.(a)E ect of depth of subgrade on dyamic displacement;(b)e ect of depth of subgrade on dynamic displacement;(c)e ect of depth of subgrade on dynamic displacement;and (d)e ect of depth of subgrade on dynamic displacement.R.Y.Liang,J.X.Zhu /Computers and Structures 69(1998)255±263260the asphalt concrete pavement±subgrade soil system.This numerical example clearly shows the importance of considering dynamic e ects in analyzing pavement responses.7.2.Example of back calculationThe approach used in the back calculation pro-cedure is basically an iterative scheme in which the parameters of asphalt concrete pavementÐD 0,D 1,m and Young's moduli E s and damping D s ratios of the subgrade soilÐare assumed to be unknown and are determined by minimizing the di erences between the calculated and the measured frequency response func-tions.When the absolute sum of the di erences in the frequency response functions is less than a certain tol-erance,then D 0,D 1,m ,E s and D s used in the fre-quency response computations are considered to be representative of these materials in the ®eld.In Table 2,the initial guess of the material parameters is shown inthe ®rst row.The forward calculation results of pre-vious examples are used as measured dynamic compli-ance functions.This is a back calculation example,so that material properties can be identi®ed.In this nu-merical example,it is assumed that the mass densities,Poisson's ratios and thicknesses of the asphalt concrete layer and the subgrade soil layer are known.The sum of the square of the errors corresponding to the iter-ation number is shown in Fig.3.The results clearly in-dicate that it is possible to deduce the parameters of the asphalt concrete layer and the Young's modulus and damping ratio of the subgrade soil,from the measured complex frequency response functions from FWD test.It is noted that the initial estimate of nu-merical values of unknown variables does not signi®-cantly a ect the ®nal converged results.However,to ensure convergence,the initial estimate needs to be reasonable.7.3.Prediction of fatigue life of AC layer from nondestructive testingIn this numerical example,the parameters A ,n ,c 0,c f in the Paris law will be determined on the basis of nondestructive test results.The constant c 0is related to size and statistical variability in the initial ¯aws in the asphalt concrete material.Majidzadeh et al .[28]reported that c 0varies between 0.025and 0.1in.for sand±asphalt and asphalt concrete mixtures.The con-stant c f is determined based on the failure conditions,where the critical size may be regarded as the pave-ment depth or some fraction of it.The constants A ,n ,Eqs.(21)and (22)are determined from the back calcu-lation technique,which is depicted in Fig.4in a ¯ow-chart format.s is the state of stress and is determined from the forward calculation.The relationship of the crack growth rate d c /d N vs stress intensity factor K relationships for the asphalt concrete pavement±subgrade soil system are shown in Fig.5for m =0.4,0.5,0.6and n =5.0,4.0,3.33.As can be seen,these crack growth reactions are quiteTable 2Back calculation of parameters for asphalt concrete pavement and soil Iteration Errors D 0/108psf À1D 1/107psf À1m E s Â107psf D s Initial Ð0.2000.2000.2000.2000.020014606000000.2400.2400.2400.2400.023821259000000.2880.2880.2830.2880.02853293600000.3450.3450.3250.3450.0342454850000.3770.4140.3630.4090.0411********.4290.4870.3910.4350.0477634220.5010.4990.4000.4320.04997 3.440.4990.5000.3990.4310.050080.040.5000.5000.4000.4320.0500ActualÐ0.5000.5000.4000.4320.0500Fig.3.Backcalculation of parameters for asphalt concrete pavement and subgrade soil.R.Y.Liang,J.X.Zhu /Computers and Structures 69(1998)255±263261di erent for various values of the parameter m .Majidzadeh et al .[28]pointed out that at low tempera-ture,A and n can be considered as material constants.However,at high temperature,due to pronounced creep e ect in asphalt concrete,parameters A and n can no longer be considered as constants.It is possible to predict the crack growth and fatigue life of the as-phalt concrete layer in the ®eld using the nondestruc-tive test data,as demonstrated in this example.8.Conclusions(1)A new dynamic analysis procedure for the pave-ment layer±subgrade soil interaction,cast in both for-ward and back calculations,is developed.The soil is represented by a high-order continuum-based on modi-®ed Vlasov model.The proposed dynamic analysis method provides a more realistic and accurate rep-resentation of subgrade soil compared to the tra-ditional Winkler model.Further,the calculation is fairly e cient,compared to other techniques such as ®nite element methods.(2)A back calculation algorithm with a high rate of convergence is also developed.The back calculation method can be used to predict moduli,damping,creep compliance and fatigue cracking properties of the as-phalt concrete layer and the modulus and damping of the subgrade soil in the ®eld,using NDT data.(3)In this paper,the parameters in the Paris fatigue equation of asphalt concrete layer are predicted based on NDT data.The slope of log creep compliance vs log time,m ,is actually determined from the back cal-culation.Then,the corresponding constants in the Paris fatigue equation,A and n ,are calculated from the correlations established by Lytton and his associ-ates.It is possible to predict the crack growth and fati-gue life of the asphalt concrete layer in the ®eld immediately from the nondestructive testing data.References[1]Kausel E and Roesset JM.Sti ness matrices for layeredsoils.Bulletin of Seismological Society of America.1981;71(6):1743±61.[2]Roesset JM and Shao KY.A note on de¯ections ofplates on a viscoelastic foundation.J Applied Mechanics 1985;25:144±5.[3]Cebon,D.An investigation of the dynamic interactionbetween wheeled vehicles and road surfaces.Ph.D.thesis,University of Cambridge,Cambridge,UK,1985.[4]Hardy MSA,Cebon D.Response of continuous pave-ments to moving dynamic loads.Journal of Engineering Mechanics,ASCE 1993;119(9):1762±80.[5]Trochanis AM,Chelliah R,Bielak J.Uni®ed approachfor beams on elastic foundation under moving loads.Journal of Geotechnical Enineering,ASCE 1987;113(8):879±95.[6]Luco JE,Wong HL.Identi®cation of soil propertiesfrom foundation impedance functions.Journal of Geotechnical Engineering,ASCE 1992;118(5):780±95.[7]Uzan J.Dynamic linear back calculation of pavementmaterial parameters.Journal of Transport Engineering,ASCE 1994;120(1):109±25.[8]Tabatabaie AM and Barenberg EJ.Finite element analy-sis of jointed or cracked concrete pavements.Transportation Research Record No.671,Transp.Res.Board,Washington,DC1978;11±19.Fig.4.Back calculation algorithm of AC pavement±soil sys-tem.Fig.5.Crack growth rate vs stress intensity factor for AC pavement±subgrade soil system.R.Y.Liang,J.X.Zhu /Computers and Structures 69(1998)255±263262[9]Huang YH.Pavement analysis and design.Prentice Hall,Englewood Cli s,NJ,1993.[10]Chatti K,Lysmer J and Monismith CL.Dynamic modelfor analysis of concrete pavements.Proc Struct Congress XII.Baker NC,Goodno BJ,editors.ASCE,New York, N.Y.1994;984±9.[11]Vlasov VZ and Leont'ev,NN.Beams,plates and shellson elastic foundations.Israel Program for Scienti®c Translations,Jerusalem,Israel,1966.[12]Vallabhan CVG,Das YC.Parametric study of beams onelastic foundations.Journal of Engineering Mechanics, ASCE1988;114(12):2072±82.[13]Vallabhan CVG,Das YC.Modi®ed Vlasov model forbeams on elastic foundations.Journal of Geotechnical Engineering,ASCE1991;117(6):956±66.[14]Lytton RL.Backcalculation of pavement layer proper-ties.Nondestructive testing of pavements and backcalcu-lation of moduli.ASTM STP1989;1026:7±38.[15]Monismith CL,Secor KE,Blackmer EW.Asphalt mix-ture behavior in repeated¯exure.Proceedings of the American Association of Paving Technologists 1961;30:188±222.[16]Pell PS,McCarthy PF.Rheological Acta1962;1(2):174±9.[17]Little DN,Mahboub K.Engineering properties of®rstgeneration plasticized sulfur binders and low temperature fracture evaluation of plasticized sulfur paving mixtures.Transportation Research Record1985;1034:103±11. [18]Powell WD et al.,The Structural Design of BituminousPavements.TRRL Laboratory Report1132,Transport and Road Research Laboratory,UK,1984.[19]Brunton JM,Brown SF,Pell PS.Developments to theNottingham analytical design method for asphalt pave-ments.Proc6th Int Conf on Structural Design of asphalt pavements.Ann Arbor,MI,1987.[20]Asphalt Pavement Design Manual for the U.K.MobilOil Company Ltd,London,1985.[21]Shell Pavement Design Manual,Shell InternationalPetroleum Co.,Ltd.,London.1978.[22]Verstraeteu J,Romain JE,Veverka V.The Belgian RoadResearch Center'overall approach to asphalt pavement structural design.Proc4th Int Conf on Structural Design of Asphalt Pavements.Vol.1,August,1977.[23]Thompson MR,Cation K.A proposed full-depth asphaltconcrete thickness design procedure.Civil Engrg studies.Transportation Engineering Series No.45,University of Illinois at Urbana-Champaign,1986.[24]Thompson MR,Elliott RP.ILLI-PAVE based responsealgorithms for design of conventional¯exible pavements.Record No.1043.Transportation Research Board, 1985;30±40.[25]Majidzadeh K,Buranrom C,Karakouzian M.Applications of Fracture Mechanics for Improved Design of Bituminous Concrete.FHWA Report,1976, pp.76±91.[26]P.W.Jayawickrama and R.L.Lytton,Methodology forpredicting asphalt concrete overlay life against re¯ection cracking.Proceedings of the Sixth International Conference on Structural Design of Asphalt Pavements, Ann Arbor,MI,1987.[27]Lytton RL,Shanmugham U.Fabric Reinforced Overlaysto Retard Re¯ection Cracking.Report RF-3424-5,Texas A&M Research Foundation,TX,1983.[28]Majidzadeh K,Kau mann EM,Saraf CL.Analysis offatigue of paving mixtures from the fracture mechanics viewpoint.Fatigue of Compacted Bituminous Aggregate Mixtures.ASTM STP508,1972,pp.67±83.R.Y.Liang,J.X.Zhu/Computers and Structures69(1998)255±263263。