MICH-PA VE: A Nonlinear Finite Element Program for Analysis of Flexible Pavements

Transcription

1 TRANSPORTATION RESEARCH RECORD MICH-PA VE: A Nonlinear Finite Element Program for Analysis of Flexible Pavements RONALD s. HARICHANDRAN, MING-SHAN YEH, AND GILBERT Y. BALADI A nonlinear mehanisti finite element program alled MICHp A VE has been developed for use on personal omputers to aid in the analysis and design of flexible pavements. The program has three major features. First, it utilizes a newly developed flexible boundary onept for pavement analysis. Seond, it uses performane models for the predition of fatigue life and rut depth, utilizing results of the mehanisti analysis (stresses, strains, and other pavement variables). Third, it is "user-friendly." The program was developed for the Mihigan Department of Transportation and is in the publi domain. The methodology used, and the features of the program, are desribed. More and more in reent years pavement design is being based on mehanisti analysis. The migration from empirial methods to mehanisti analysis has been failitated by the availability of relatively inexpensive miroomputers that an be used in daily pratie. Early omputer programs for mehanisti analysis (suh as CHEV5L, BISAR, ELSYM5, et.) modeled pavements as being omposed of linear elasti layers and omputed defletions, stresses, and strains within a pavement arising from a wheel load. In these programs eah pavement layer is assumed to extend infinitely in the horizontal diretions, allowing the three-dimensional problem to be redued to an axisymmetri two-dimensional problem. Beause of the linear elasti assumption, multiple wheel loads an be analyzed by superposing the results from single wheel loads. The main drawbaks of these linear elasti layer programs are that They annot model the nonlinear resilient behavior of granular and ohesive soils; They normally assume weightless pavement material; They may yield tensile stresses in granular material, whih annot physially our; and They do not aount for "loked-in" stresses from ompation during onstrution. To overome these shortomings, nonlinear analysis programs based on the finite element method have been developed (e.g., ILLI-PAVE). However, beause of the large memory and omputational effort requirements, these programs primarily have been implemented on mainframe omputers. Fur- R. S. Harihandran and G. Y. Baladi, Department of Civil and Environmental Engineering, Mihigan State University, East Lansing, Mih M-S. Yeh, Engineering Offie of Taipei Railway Underground Projet, 3 Pei-Ping West Road, 3rd Floor, Taipei, Taiwan 1, Republi of China. ther, the interation of the user and these programs, in terms of data input and interpretation of the output, is not "friendly," making their use in daily pratie undesirable. For most state highway agenies, a "user-friendly" flexible pavement program that an be used for the design or rehabilitation of flexible pavements in daily pratie is desired. This program should onsider all the major fators affeting the design or rehabilitation of pavements. To ahieve this, a researh study was sponsored by Mihigan Department of Transportation (MDOT). The main goal of this researh was to review existing analysis and design methods and then to develop a user-friendly nonlinear finite element program that an be used on personal omputers in daily pratie. Sine urrent personal omputers have limited memory apaities, the traditional finite element method, whih requires a large amount of memory, annot be suitably implemented on them unless auray is sarified. To overome this shortoming, Harihandran and Yeh (1) proposed a new tehnique of plaing a relatively shallow finite element mesh on a flexible boundary. This tehnique substantially redues the memory and omputational requirements of the nonlinear finite element method without sarifiing auray. In this researh, this tehnique is implemented with user-friendly input and output features, to develop a nonlinear finite element program for the analysis and design of flexible pavements. The program has been named MICH-PAVE. MATERIAL NONLINEARITY There are many nonlinear material models that may be used in mehanisti analysis. Four suitable models-the hyperboli, the resilient modulus, the shear and volumetri stressstrain (also alled the ontour model), and the third-order hyperelasti models-were reviewed (2-7). Of these, the resilient modulus model was hosen as the most suitable at the present time. This hoie is based on the appliability of the model to repeated loading patterns experiened by pavements, and on the relative ease of determining the model parameters by state highway agenies. The resilient modulus model haraterizes the resilient stress-strain properties of soils through a stress-dependent modulus and a onstant Poisson ratio. For granular soils, the resilient modulus is haraterized as (1)

2 124 TRANSPORTATION RESEARCH RECORD 1286 Mr \ Log Bulk Stress (Log 9) FIGURE 1 Resilient modulus model for granular material. "I I K1 I Deviator Stress (a - a ) 1 3 FIGURE 2 Resilient modulus model for ohesive material. where M, = resilient modulus (psi), = u, + rr 2 + rr 3 = bulk stress (psi), and Ki and K 2 are material onstants. This relationship is illustrated in Figure 1. For ohesive soils, the resilient modulus is expressed through the bilinear relationship for K i > (rr 1 for K i :5 (rr 1 where (rri - rr 3 ) = deviator stress (psi), and K,, K 2, K 3, and K 4 are material onstants. This relationship is illustrated in Figure 2. If the finite element method is used with only the resilient modulus model, it will onverge extremely slowly. Therefore, Raad and Figueroa (8) applied the resilient modulus model with the Mohr-Coulomb failure riterion. The Mohr-Coulomb failure riterion is used to modify the prinipal stresses of eah element in the granular layers and roadbed soil after eah iteration so as not to exeed the Mohr-Coulomb failure envelope. Thompson (9) used this algorithm in the ILLI PA VE program. A similar algorithm was developed for MICH PAVE. FEATURES OF THE MICH-PAVE PROGRAM The MICH-PA VE program is apable (depending on the user's hoie) of performing either linear or nonlinear finite element analysis of flexible pavements. It assumes axisymmetri loading onditions arising from a single irular wheel load on pavement layers of infinite horizontal extent. The various features of the program are desribed in this setion. Tehnial details and results from various sensitivity analyses an be found elsewhere (1,11). Mesh Generation and Flexible Boundary The finite element method needs to satisfy some basi requirements for the mesh, suh as the loation of the side and bottom boundaries, the size and shape of the elements, and the distribution of the elements in the various regions. To establish riteria for loating the boundaries in a finite element mesh, (2) Dunan et al. (12) showed omparisons between displaements and stresses omputed using the finite element method and those omputed using elasti half-spae and layered system analysis. For an elasti half-spae subjeted to a uniform irular load, displaements and stresses omputed by the finite element method ompare well with those determined from the Boussinesq solution, when the bottom boundary in the finite element mesh is fixed at a depth of about 18 radii of the loaded area; and the vertial side boundary is loated at a distane of about 12 radii from the enter and onstrained from moving radially. For a three-layered system, a reasonable omparison between the two proedures an be obtained if the bottom boundary in the finite element mesh is moved to a depth of about 5 radii, while maintaining the side boundary at 12 radii. Previous experiene has shown thal stresses based on quadrilateral elements will be aurate provided that the lengthto-width ratio for the elements does not exeed five to one. Furthermore, smaller elements should be used lose to the loaded area, and progressively larger elements may be used in the regions away from the loaded region. Based on the above onsiderations and experiene, the following mesh generation rules were used. In the radial direliun, a mesh wilh a Lota! width of 1 radii was divided into four zones. The first zone, between and 1 radius, is equally divided into four elements; the seond zone, between 1 radius and 3 radii, is equally divided into four elements; the third zone, between 3 radii and 6 radii, is equally divided into three elements; and the fourth zone, between 6 radii and 1 radii, is equally divided into two elements. The MICH-PAVE program will automatially generate the default values of the finite element mesh along the radial and vertial diretion. A typial mesh is shown in Figure 3. In the MICH-PA VE program, a flexible boundary is used instead of a fixed bottom boundary. Therefore, a mesh that is 5 radii deep is not required. Normally when a mesh depth of about 1 radii is used, suffiiently aurate results will be obtained. In general, if the flexible boundary is loated too lose to the top of the roadbed soil, the displaements may still be aurate, but the stresses may not be. If the boundary is plaed too deep, the primary advantages of using the flexible boundary are lost. The flexible boundary is usually plaed

3 Harihandran et al. 125 Asphalt l 11 Depth." 1." Granular 3." Roadbed Soil a 3a 6a 1a5." Radial Distane in Radii (Radius of loaded area, a= 5.35 in) FIGURE 3 Typial finite element mesh. at about 12 in. below the upper surfae of the roadbed soil, or at a depth of 5 in., whihever is greater. In the MICH PA VE program, the depth at whih the flexible boundary is plaed is speified by the user, by inputting the depth of the roadbed soil to whih stress and strain alulations are required. If the boundary is plaed at a depth of Jess than about 5 in., then the stresses still will be aurate, but the vertial defletions may be overestimated. where K = 1 - sin <!> for ohesionless soil and gravel; K = sin<!> for ohesive soil; and <!> = angle of internal frition. To approximately aount for "loked-in" stresses aused by ompation, the user an input a value for K higher than the oeffiient of earth pressure at rest. Modulus of Half-Spae Below Flexible Boundary The half-spae below the flexible boundary is assumed to be homogeneous and linear elasti. The boundary is therefore always plaed within the last layer (roadbed soil). Although the modulus of the roadbed soil below the boundary may in reality be stress dependent and vary from one loation to another, in most pavement. setions the stresses from wheel loads are substantially diminished at the level of the roadbed soil. Thus, the use of a onstant modulus below the boundary has a negligible effet on the stresses and displaements above the boundary. In MICH-PAVE, the modulus used for the half-spae below the boundary is the average moduli of the finite elements immediately above the boundary. To avoid undesirable "edge effets," the elements losest to the right vertial boundary are not used in omputing this average. Gravity and Lateral Stresses The MICH-PA VE program inludes the effet of gravity and lateral stresses arising from the weight of the materials. At any loation within the pavement, the vertial gravity stress (r 8 ) is omputed as the aumulation of the layer thiknesses multiplied by the appropriate unit weights. The lateral stress (rh) is alulated from the oeffiient of earth pressure at rest (K ) and vertial gravity stress, as (3) Iterative Solution and Convergene Criterion MICH-PA VE performs nonlinear analysis by the following steps: 1. Initially, the wheel load at the surfae is assumed to spread over a 2:1 region (i.e., the radius of the loaded area is assumed to inrease by 1 in. for every 2 in. of depth). The stresses arising from this assumed distribution of the wheel load are ombined with gravity stresses, to ompute the initial resilient moduli for eah of the finite elements in granular and ohesive materials. 2. A linear analysis is performed, and a more aurate stress distribution is omputed. 3. For those elements that exeed the Mohr-Coulomb failure riterion, the omputed stresses are adjusted aording to the proedure outlined by Raad and Figueroa (8). 4. A new value of the resilient modulus is omputed for eah element based on the latest stresses. 5. Convergene in the resilient moduli is heked by omputing the relative error, e = l (M,,, - M,,,_ 1) 2 /l M;.,_,, where M,,, are the urrent resilient moduli, M,,,_ 1 are the previous resilient moduli, and the summations are taken over all nonlinear elements. 6. Steps 2 through 5 are repeated until the relative error e is less than.1.

4 126 TRANSPORTATION RESEARCH RECORD Displaements, strains, and stresses are then omputed at all speified loations, and the fatigue life and rut depth of the pavement are estimated. Interpolation and Extrapolation of Stresses and Strains at Layer Boundaries For a pavement setion in whih various layers are fully bonded (i.e., when no slip ours at layer boundaries), quantities suh as the vertial and shear stresses and the radial and tangential strains should be ontinuous aross layer interfaes. However, beause of the low-order interpolation funtions hosen in the finite element approah, these quantities are not ontinuous aross element boundaries. Thus, if these quantities are estimated by finite element approah at two adjaent points aross an interfae, the results will show an apparent disontinuity that is an artifat arising from the error in the finite element method. This result is undesirable and an be overome by using linear interpolation to estimate these quantities at an interfae from those at the middle of the adjaent elements. For example, if a 1 is the stress at the enter of an element immediately above the interfae and a 2 is the stress at the enter of an element immediately below the interfae, then the stress at the interfae obtained by linear interpolation is where z 1 and z 2 are the depths of the points at whih a 1 and a 2 are evaluated, and z is the depth of the interfae. Smee the finite element approah gives aurate estimates of stresses and strains at the enter of elements, but an yield signifiant error at element edges, even those stresses and strains at the interfaes that are disontinuous aross the interfae an be in signifiant error. For quantities suh as the radial and tangential stresses that are disontinuous aross an interfae, it is possible to estimate their values at one side of the interfae by linear extrapolation of the values at the enter of two elements on that side of the interfae. For example, if a 1 and a 2 are stresses at the enter of two onseutive elements below (or above) an interfae, then the linearly extrapolated stress at the lower (or upper) side of the interfae is also given by Equation 4, where z 1 and z 2 are the depths of the points at whih a 1 and a 2 are estimated, and z is the depth of the interfae. (4) Finally, based on prior knowledge of the solution (vertial and shear stresses must be zero at the surfae, exept for the vertial stress below the wheel load, whih must be idential to the tire pressure), the surfae stresses are arbitrarily set to their proper values in MICH-PA VE. The improvement in auray obtained through interpolation and extrapolation is illustrated by onsidering a typial pavement setion for linear analysis using the MICH-PAVE and CHEVSL programs. (Results from CHEVSL are aurate for linear analysis and are used as the benhmarks.) In this example, a wheel load of 9, lb and a tire pressure of 1 psi were used, with material having the following material properties: Layer 1-AC, E = 3, psi, v =.4; thikness = 8 in.; Layer 2-Base, E = 2, psi, v =.38, thikness = 12 in. ; and Layer 3-Roadbed soil, E = 8, psi, v =.45, semi-infinite depth. A omparison of stresses omputed by CHEV5L, and by MICH-PAVE before and after extrapolation, are given in Table 1. Sine the materials are assumed to be weightless, gravity stresses are negleted in both programs. Similar omparisons for stresses omputed before and after interpolation, and adjustment of surfae stresses, are given in Table 2. The plus and minus symbols used for depth indiate points just above and below interfaes. It is lear that interpolation and extrapolation improve the auray of these stresses. Similar improvements are obtained for strains. Equivalent Resilient Moduli for Linear Analysis In some ases it may be desirable to ompare the results of nonlinear finite element analysis with those from a linear analysis or those of elasti layer programs suh as CHEV5L, or to utilize the speed or multiple-wheel load apabilities of existing linear analysis programs. In suh ases it would be desirable to input to the linear program an equivalent resilient modulus for eah layer that reflets the true stress-dependent variation of the modulus within the layer. The equivalent moduli an be estimated from a nonlinear finite element analysis, using the various resilient moduli in eah finite element. MICH-PA VE omputes an equivalent resilient modulus for eah pavement layer that is obtained as the average of the moduli of the finite elements in the layer that lie within an assumed 2:1 load distribution zone, as shown in Figure 4. The average moduli of those elements within the regions ABGH, BCFG and CDEF are taken as the equivalent moduli for layers 1, 2 and 3, respetively. TABLE 1 COMPARISON OF RADIAL AND TANGENTIAL STRESSES AT 67 IN. FROM THE CENTER OF THE LOADED AREA Depth Radial stress (psi) Tangential stress (psi) (in) Before After CHEV5L Before After CHEV5L (+) ' ( - ) 1 84, (+) (.)

5 Harihandran et al. 127 TABLE 2 COMPARISON OF VERTICAL AND SHEAR STRESSES AT 67 IN. FROM THE CENTER OF THE LOADED AREA Depth Vertial stress (psi) Shear stress (psi) (in) Before After GHEV5L Before After GHEV5L , 8(+) (.) (+) (.) , , , , 7, 7, 11 7 The use of equivalent resilient moduli obtained through nonlinear analysis utilizing the MICH-PAVE program, in a linear program (CHEV5L), is investigated through typial examples. Tables 3 and 4 show the material properties of a 12-in. fulldepth asphalt onrete (AC) setion, and a three-layer setion with 3 in. of AC and 12 in. of granular material on roadbed soil (RS). The material properties shown in the tables, inluding the modulus, Poisson ratio (v), lateral earth pressure oeffiient (K ), material onstants (K 1, K 2, K 3, and K 4 ) and density have been obtained previously (5). A ohesion of 6.45 psi and an angle of frition of degrees were used for the roadbed soil in the full-depth setion. For the three-layer setion, = psi and l> = 45 degrees were used for the granular material and = 6.45 psi and l> = degrees were used for the roadbed soil. For the granular material and roadbed soil, the equivalent resilient moduli are also shown in the tables. In the MICH-PAVE analysis, the flexible boundary was plaed beneath the roadbed soil, at depths of 52 in. and 6 in. for the full-depth asphalt and three-layer setions, respetively. The setions were analyzed for a wheel load of 9, lb and a tire pressure of 1 psi. Figures 5 and 6 show omparisons of surfae defletions for nonlinear analysis using MICHp A VE, and the linear analysis using CHEV5L with the equiv- ' i :. I I! I, 1,.j j : _...! i 1 i 2! I I I! I D I : I I! Layer1 Layer 2!! i 1 Layer 3.i j... --~ i _.._......_, Flexible Boundary FIGURE 4 Elements used to ompute equivalent resilient moduli. alent moduli. Figures 7 and 8 show omparisons of the vertial and radial stresses at.5 in. from the enter of the loaded area. Defletions and stresses omputed by the nonlinear finite element program ILLI-P A VE are also shown in these figures. The results indiate that the use of equivalent resilient moduli obtained from nonlinear analysis, in a subsequent linear analysis, gives reasonably similar displaements. Notable differenes are obtained in the stresses, beause the linear analysis using CHEV5L neglets the weight of the material. In partiular, linear analysis an give tensile stresses in granular soils. The stresses omputed using MICH-PAVE and ILLI-PAVE are almost idential. However, ILLI-PAVE tends to give lower surfae defletions than MICH-PAVE. Linear analysis, for whih aurate results an be obtained through CHEV5L, has shown that displaements obtained by using a flexible boundary (as in MICH-PAVE) are slightly greater and more aurate than those obtained by using a deep fixed boundary (as in ILLI-PA VE) (1,11). Using a deep finite element mesh tends to produe a "stiffening" effet, resulting in smaller displaements. For the three-layer pavement, the variation of modulus within the granular layer and roadbed soil is shown in Table 5 for the first eight elements from the left boundary. The depth and radial distane to the middle of the elements are tabulated. The finite element mesh was similar to that shown in Figure 3, exept that the thiknesses of the layers are as given in Table 4. It is apparent that the modulus dereases with depth and radial distane within the granular layer. This is beause the bulk stress () dereases with depth and radial distane; hene, aording to Equation 1, the resilient modulus follows the same pattern. Within the roadbed soil, however, the modulus inreases with depth and radial distane. This inrease is beause the deviatori stress ( u 1 - u 3 ) dereases with distane away from the wheel load; hene, aording to Equation 2, the modulus must also do the same. In MICHp A VE, the modulus of the half-spae below the flexible boundary is taken to be the average modulus of the elements immediately above the boundary. Care must be taken in MICHp A VE to use a thikness for the roadbed soil that is suffiiently deep; otherwise the modulus of the half-spae below the flexible boundary may turn out to be too small, yielding large displaements. It is reommended that the depth from the pavement surfae to the flexible boundary (whih is determined by the user inputs of layer thiknesses inluding that of the roadbed soil) be about 5 in. Larger depths have no appreiable effet on displaements.

6 128 TRANSPORTATION RESEARCH RECORD 1286 TABLE 3 MATERIAL PROPERTIES FOR THE FULL-DEPTH AC SECTION LAyer Thir:-l<'nr:><;;; Md11l115 " Ko Kl K, K, K, npn.:; i ry Type (inhes) (psi) (pf) AC 12 5, ' RS 4 8,753 * * Equivalent resilient modulus TABLE 4 MATERIAL PROPERTIES FOR THREE-LA YER SECTION Layer Thikness Modulus v Type (inhes) (psi) AC 3 5,. 4 Granular 12 22' 543 *. 38 RS 45 7,435* 45 Ko K, K, Ka Ka Density (pf) 15 5' ' * Equivalent resilient moduli Fatigue and Rut-Depth Preditions Results from the nonlinear mehanisti analysis, together with other parameters, are used as input to two performane models (fatigue life and rut depth) derived on the basis of field data (13), to predit the fatigue life and rut depth of flexible pavements. The models relate the fatigue life and rut depth to the number of equivalent 18-kip single-axle loads, surfae defletion, moduli and thiknesses of the layers, perent air voids in the asphalt, tensile strain at the bottom of the asphalt layer, average ompressive strain in the asphalt layer, kinemati visosity of the asphalt binder, and average annual air temperature. Fatigue life and rut depth are useful design parameters on whih to base the seletion of a suitable pavement setion. ifiation of existing data, analysis, plotting of results, and subtasks within these. The spreadsheet-like data-entry forms allow input data to be entered or edited easily with no format requirements. When possible, suitable data for typial pavements are suggested on the sreen. New or hanged data are immediately heked for errors, and the user is prompted for orretions. Output from the analysis, suh as stresses, strains, and displaements at speified loations, is saved in a file and may also be plotted on the sreen. These features make the program easy to use in daily pratie. The program has been strutured and optimized for speed, within the memory limitations of present personal omputers (64 KB RAM). The total analysis time for typial three- and four-layer pavements is about 2 and 3.5 minutes, respetively, on an IBM AT-ompatible omputer with an 8287 math oproessor. User-Friendly Features MICH-PA VE is designed with menus, data-entry forms, and on-sreen plots, to failitate easy input of data and interpretation of results. It also has extensive error-trapping features. Menus failitate the seletion of various stages of a typial pavement analysis, suh as the speifiation of new data, mod- PROGRAM AVAILABILITY The MICH-PA VE program is in the publi domain and may be obtained on request from Larry Heinig, Mihigan Department of Transportation, Division of Testing and Researh, P.O. Box 349, Lansing, Mih A user's manual is also available (14).

7 ,.-.., 5 1.._, * 15 '..::; u 2 ' u '+-- L 3 t.--6 ILLl-PAVE :J (/) 13- -a MICH-PAVE,.. _ {fj:ioo,l,,, 11 ' """1 """"'I'"""" I" 1111 "' I"''''"' I''"""' Radial Distane from Center of Loaded Area (in) FIGURE S Comparison of surfae defletion (12-in. full-depth AC). -rr 'I''' * t;. '+- u '+ I.. :J (/) t.--6 ILLl-PAVE t -a MICH-PAVE 45-t-rTG--..,.,.,.,...-rp-.-nrn-ITT,..,.,.'rnTnrrt",.,..,...,.,.,.,.,..,..,.,..,..,.,...,..,..,.,.,,...,...,.,.TTTTnTnOTTTTTTJ"ITnrTj Radial Distane from Center of Loaded Area (in) FIGURE 6 Comparison of surfae defletion (3-in. AC and 12-in. base). 8

8 ]'""""I""'''" I'""'" ' I'""'" 'I"'"""I' ""rrrfl'tttti. l ~~-~-----,....._,, _..., t I \ 6 l>--t. ILLl-PAVE &-8 MICH-PAVE 7.J.n,.,.,.,.-rrnrri...-r.;;...,.,~~~~~~~~~~~~~~~~~...i Vertial Stress (psi) FIGURE 7 Comparison of vertial stress (12-in. full-depth AC). 1,....._,, _ +-' &-D 5 ILLl-PAVE MICH-PAVE G--- CHEV5L l I Radial Stress (psi) FIGURE 8 Comparison of radial stress (12-in. full-depth AC).

9 Harihandran et al. 131 TABLE 5 VARIATION OF MODULUS (PSI) IN GRANULAR AND ROADBED SOIL LAYERS Layer Depth Radial Distane from Center of Wheel Load (inhes) (in.) , , BOO 35 '4 32 ' , 6 3' 5 28' 6 Gran, '8 25 '2 24, ,4 21,1 2' ' 2 18,1 17' ,64 5,66 5,69 RS ,34 7,35 7' B, 25 8 '25 8,26 27' 8 22, 1 15,3 12, 1' 3 26' 1 2,4 16' 12,7 1' 9 22 '7 2,3 15,5 13' 2 11, 5 19,7 18,5 16,1 13 ' 11, , , 12,1 5,75 5,86 6,4 6,29 6,58 7,38 7 '43 7,52 7,63 7,75 B, 26 B, 27 8,3 B,34 8,38 CONCLUSIONS The features and methodology used by a state-of-the-art nonlinear finite element program for the analysis of flexible pavements, MICH-PA VE, are desribed. The analysis aounts for material nonlinearity, the unbound nature of granular soils, and " loked-in" lateral stresses arising from ompation. By utilizing a flexible bottom boundary in the finite element model, the memory and omputational effort required by nonlinear analysis are drastially redued, allowing the program to be used on personal omputers. The program is designed to be user friendly, making it suitable for use in daily pratie. Results from the MICH-PA VE program have been ompared with those from CHEV5L and ILLI-PA VE. For linear analysis, MICH-PA VE and CHEV5L give similar strains and displaements. The stresses omputed by MICH-PA VE, however, aount for the weight of pavement materials that are negleted by CHEV5L. For nonlinear analysis, MICH-PAVE and ILLI-PAVE give very similar stresses, but displaements omputed by ILLI-PAVE are smaller than those omputed by MICH-PAVE. It is believed that the use of a flexible boundary in MICH-PA VE alleviates the "stiffening" effet that results when a deep finite element mesh with a fixed boundary is used. A method of obtaining a onstant equivalent modulus for a pavement layer with a stress-dependent resilient modulus is outlined. The equivalent modulus, estimated from the results of nonlinear analysis, may be used with linear analysis programs to yield surfae defletions that would be similar to those omputed by nonlinear analysis. Results of the mehanisti analysis are used as input to two pavement performane models developed using field data. The models employ omputed stresses and strains, as well as the average annual temperature and the kinemati visosity and air voids of the asphalt binder. ACKNOWLEDGMENTS T he authors would like to expres their gratitude to Mihigan D epartment of Tran portation (MOOT) for providfog the finanial support. The onlributio n of MOOT personnel and the suggestions made by partiipants of the 4-week NH.I ourse held at Mihigan State University in February 1989, are greatly appreia ted. REFERENCES 1. R. S. Harihandran and M-. Yeh. Flexible Boundary ia Finite Elrnout Annlysis of Pwrnent. In Tra 11 portatio11 Nesearh Re ord 127, TRB, National Re enrh Counil, Washington..., 1988, pp J. M. Dunan and C. Y. Chang. Nonlinear Analy i of tr s and train in Soils. Joumal of Soil Melumis 11d Fo 1111da1io11s, ASCE l97, pp. L629- l W. F. hen and A. F. Saleb. 11sti1ute q11"'io11s for E11si 11eeri11g Mmerials. John Wiley, nd S n. In., New York, 1982, pp S. F. Brown and J. W. Pappin. Analy is of Pav mnt with Gran ular Bases. Jn TrallSporwtio11 Researh Reord 81. TRB, ational Researh Counil, Washington, D.C., pp E. J. Yoder and M. W. Witzak. Pri11iples of Pa11eme11t Design, John Wiley and ns, In., New York, H. J. Haynes and E. J. Yoder. Effets of Repemed Loading n Gravel and Crushed tone Base our.;e Mat rial U d in the AASHO Road Test. Highway Re. ear/r Reor<l 39, T R13, National Re.searh uni l, Wn hington, D.. '1963, pp. 82-' H. B. Seed,. K. han, and. E. Lee. Resiliene Char11terisli of Subgrnde Soils and Their Relation to Fatigue Failures in Asphalt Pavement. Pro.,./st lnum1a1io11a/ 11fere 11e 11 the Stm111ra/ Design of M plwl1 Pave111e11is, Ann Arbor, Mih., 1962, pp L. Raad and J. L. Figueroa. Load Response of Tran ponntion Supp rt Sy tern. Journal of Tm11Sportatio 11 E11gineeri11g, AS E, Vol. 16, 19, pp. 1.l'l M. R. Thompson. JLLl-PAVE, User's Manual. Tran portation Failities Group, Department or Civil Engineerin g. University or Illinois, Urbana hampaign, M. S. Yeh. Nonlinear Finile Element Analysis a11d De ign of Flexible Pavemellfs. Ph.D. dissertation. Mihigan tale Unive r sity, East Lansing, R. S. Harihand ran, G. Y. Baladi, and M-. Yeh. Develop t of a Compuler Program for De.sign of Pa veme111 System on sistillg of Layers of Bound and U11bow1d Materials. Report FHWA MI-RD Mihigan epanment oftransportntion Lansing J. M. Dunan, C. L. Monismitb. and E. L. Wilson. Finite Element Analysi of Pavements. ln Highway Researh Reord 228, TRB, Naliooal Researh o"unil, Wa hington, D.., 1968, pp G. 'Baladi. Fa 1.igue Life and l'ermanent Deformation harater istis of Asphalt onret Mixe. Tra11spor1111io11 Researh Reord 1227 TRB, ationalresarh ounil Washingt n D R. S. 1-larihandran, G. Y. Baladi, and M-S. Yeh. MICH-PAVE User's Ma1111al. Report FHWA-Ml RD Mihiga n Department of Tran portation, Lansing, Publiation of this paper sponsored by Committee on Flexible Pavement Design.