MECHANISTIC-EMPIRICAL MODELLING OF THE PERMANENT DEFORMATION OF UNBOUND PAVEMENT LAYERS H L Theyse Division of Roads and Transport Technology CSIR P O Box 395 Pretoria 0001, South Africa Abstract. This paper describes recent research the aim layers and the roadbed. The method is based on a critical of which was to develop permanent deformation design layer approach whereby the shortest layer life of the transfer functions for unbound pavement layers from individual pavement layers determines the pavement life. Heavy Vehicle Simulator (HVS) test data. Two types of This approach may be suited to the fatigue failure data generated during an HVS test are used to develop of bound layers, but does not allow for each of the the permanent deformation models on which the design pavement layers to contribute to the total surface rut. transfer functions are based. These are the in-depth Current research is therefore aimed at developing deflection and permanent deformation data obtained permanent deformation models for individual pavement from the Multi-Depth Deflectometer (MDD) layers, to enable the designer to predict each layer’s measurements taken at regular intervals during an HVS contribution to the total permanent deformation (rut) of test. Test data from a number of HVS tests, selected the pavement system. from the moderate and wet regions in South Africa, were This paper describes the process followed during the used for the development of the permanent deformation development of such permanent deformation transfer models. functions for pavement foundation layers. The same A multi-dimensional, conceptual model for process was also followed for granular, structural permanent deformation was developed and calibrated pavement layers and examples of permanent with HVS test data for pavement foundation and deformation transfer functions for both the structural structural layers of different material qualities. These and foundation layers are illustrated. The use of these models provide permanent deformation design transfer transfer functions is illustrated by a number of design functions at different expected performance reliabilities examples. The ultimate aim is to develop similar for unbound pavement layers in South Africa. transfer functions for all road-building materials, The use of these design transfer functions is including asphalt and cemented material. illustrated by a number of examples. The design Accelerated pavement test data from selected Heavy approach allows each of the pavement structural layers Vehicle Simulator (HVS) tests done in South Africa and the pavement foundation to contribute to the total during the past decade proved to be invaluable in deformation or surface rut of the pavement structure. developing these permanent deformation transfer Keywords. Heavy Vehicle Simulator test data, Granular functions. The whole process centres around resilient material, Permanent deformation, Design transfer pavement response and permanent deformation data functions. collected at various depths in a pavement structure during HVS testing. A brief discussion on specific HVS INTRODUCTION instrumentation and the data collected is therefore essential. The South African Mechanistic Design Method (SAMDM) has been used in South Africa for a number HVS INSTRUMENTATION, PAVEMENT of years (Theyse et al. 1996). This method is a RESILIENT RESPONSE AND PERMANENT mechanistic-empirical design method which includes DEFORMATION fatigue transfer functions for asphalt surfacing, asphalt base and lightly cemented layers as well as permanent Various types of data are collected during an HVS deformation transfer functions for unbound structural test. The most important data from the viewpoint of developing permanent deformation models are the data obtained from the Multi-Depth Deflectometer (MDD) system. The MDD system (De Beer et al. 1989) is basically a stack of Linear Variable Displacement Transducers (LVDTs) referred to as MDD modules, installed at predetermined depths in the pavement structure with a reference point at the anchor, normally at 3 m depth. Installation is done after pavement construction and the MDD modules are placed at the layer interfaces and near the road surface, unless the layer thicknesses dictate otherwise. A minimum clearance of about 150 mm is required between two successive layer interfaces to be able to fit an MDD module at each interface. Two kinds of output are obtained from the MDD FIGURE 2. TYPICAL IN-DEPTH DEFLECTION stack. Firstly, the resilient deflection of each MDD PROFILES AT VARIOUS STAGES OF AN HVS module relative to the reference point at the anchor is TEST measured under a slow moving wheel load. A total of 256 points are sampled for each MDD module resulting in a smooth, well defined deflection bowl at each depth By doing a back-calculation from the peak deflection where an MDD module is installed. Figure 1 shows a profiles, the elastic moduli are obtained for the pavement plot of the in-depth deflection bowls at 9 points selected layers, allowing the stresses, strains and any stress from the total of 256 points. invariant to be calculated at any position in the pavement structure. The second type of data obtained from the MDD stack, is the permanent movement of each MDD module relative to the reference point at the anchor for the duration of the HVS test. An example of this type of data is illustrated in Figure 3. FIGURE 1. TYPICAL IN-DEPTH DEFLECTION BOWLS FROM AN MDD STACK By selecting the peak deflection values for each of the in-depth deflection bowls and plotting these values against the depth at which the bowl was measured, a deflection profile is obtained at various stages of the HVS FIGURE 3. TYPICAL IN-DEPTH PERMANENT test as illustrated in Figure 2. DEFORMATION DATA FOR THE DURATION OF AN HVS TEST A number of concepts may de defined, based on the data shown in Figure 3. As already mentioned, the data in Figure 3 represent the permanent movement of each MDD module relative to the anchor. The permanent deformation or plastic strain for a specific layer is obtained from the difference between the permanent movement of the two MDD modules on either side of the layer. The permanent movement of an MDD module in the pavement foundation or roadbed (consisting of the in- • In-depth permanent deformation data as illustrated situ and imported, selected material) represents the total in Figure 3 had to be available. permanent settlement of the pavement foundation from • The material for each of the pavement layers had to the depth of that particular MDD module downwards. be classified according to the material classification The term “permanent deformation” will, however, be system used in South Africa (CSRA, 1985). used in general to refer to the plastic strain of a The HVS tests selected are listed in Table 1. The structural layer as well as the permanent settlement of locations of the HVS sites for these tests are shown on the pavement foundation. the map in Figure 4. An abbreviated specification for the In terms of modelling the permanent deformation of material codes used in Table 1 is given in Table 2 the layered pavement system, the permanent deformation (modified from CSRA, 1996) is calculated for the structural layers and added to the permanent settlement of the foundation layers. The total deformation of the pavement structure therefore consists of the permanent settlement of the pavement foundation from a specific depth downwards, plus the permanent deformation of each of the structural layers. By combining the stresses, strains and stress invariants calculated from the in-depth deflection profiles with the permanent deformation and settlement data for a specific HVS test, permanent deformation design transfer functions can be developed. SELECTED HVS SITES AND TESTS HVS tests have been done in South Africa for about two decades. Permanent deformation measurements have, however, only been collected since about the mid eighties. HVS tests selected for developing permanent deformation models therefore had to fulfil the following prerequisites: • In-depth deflection profiles had to be available at a number of stages during the test, preferably measured under a number of different wheel loads. TABLE 1. HVS TESTS FROM WHICH RESILIENT RESPONSE AND PERMANENT DEFORMATION DATA WERE OBTAINED Region District, climatic region and Construction site number as per Figure 4 Gauteng Bronkhorstspruit, moderate, Asphalt overlay (various thicknesses) site no 1 200 mm G5 natural gravel base G8/G9 gravel/soil foundation Bultfontein, moderate, site no 110 mm C3 lightly cemented base 2 200 mm C4 lightly cemented subbase G4 natural gravel foundation Eastern Cape Port Elizabeth, moderate, 60 mm asphalt surfacing layer site no 3 140 mm G2 crushed stone base G5 natural gravel foundation 200 mm asphalt and Bitumen Treated Base (BTB) 175 mm G6 G6 foundation Macleantown, wet, site no 4 150 mm G1 150 mm G4/G5 150 mm C4 150 mm C4 G9 foundation 150 mm G2 150 mm C4 150 mm C4 150 mm C4 G9 foundation KwaZulu-Natal Umkomaas, wet, site no 5 65 mm asphalt 150 mm G1 250 mm C3/C4 G5 foundation 65 mm asphalt 150 mm WM1 250 mm C3/C4 G5 foundation Amanzimtoti, wet, site no 6 40 mm asphalt 100 mm Dense Bitumen Macadam (DBM) 150 mm C3 220 mm C4 G5 foundation FIGURE 4. MAP OF SOUTH AFRICA WITH LOCATIONS OF SELECTED HVS TEST SITES INDICATED TABLE 2. ABBREVIATED SPECIFICATION FOR THE UNBOUND AND LIGHTLY CEMENTED MATERIAL CLASSIFICATION USED IN TABLE 1 (MODIFIED FROM CSRA, 1996) Material Abbreviated material specification code G1 Dense-graded unweathered crushed stone; maximum aggregate size 37,5 mm; 86 - 88% bulk relative density; soil fines PI<4 G2 Dense-graded crushed stone; maximum aggregate size 37,5 mm; 100 - 102% Mod. AASHTO density or 85% bulk relative density; soil fines PI<6 G3 Dense-graded stone with soil binder; maximum aggregate size 37,5 mm; 98 - 100% Mod AASHTO density; soil fines PI<6 G4 CBR 80% @ 98% Mod. AASHTO density; maximum aggregate size 37,5 mm; 98 - 100% Mod. AASHTO density; soil fines PI<6; maximum swell 0,2% @ 100% Mod. AASHTO density. G5 CBR 45% @ 95% Mod. AASHTO density; maximum aggregate size 63 mm or 2/3 of layer thickness; density as prescribed for layer type; soil fines PI<10; maximum swell 0,5% @ 100% Mod. AASHTO density. G6 CBR 25% @ 95% Mod. AASHTO density; maximum aggregate size 63 mm or 2/3 of layer thickness; density as prescribed for layer type; soil fines PI<12; maximum swell 1% @ 100% Mod. AASHTO density. G7 CBR 15% @ 93% Mod. AASHTO density; maximum aggregate size 2/3 of layer thickness; density as prescribed for layer type; soil fines PI<12; maximum swell 1,5% @ 100% Mod. AASHTO density. G8 CBR 10% @ 93% Mod. AASHTO density; maximum aggregate size 2/3 of layer thickness; density as prescribed for layer type; soil fines PI<12; maximum swell 1,5% @ 100% Mod. AASHTO density. G9 CBR 7% @ 93% Mod. AASHTO density; maximum aggregate size 2/3 of layer thickness; density as prescribed for layer type; soil fines PI<12; maximum swell 1,5% @ 100% Mod. AASHTO density. G10 CBR 3% @ 93% Mod. AASHTO density; maximum aggregate size 2/3 of layer thickness; density as prescribed for layer type. C3 UCS: 1 to 3,5 Mpa @ 100 % Mod.AASHTO; ITS 250 kPa @ 95 to 97% Mod. AASHTO; maximum aggregate size 63 mm; PI 6 after stabilization; maximum fine loss 20%. C4 UCS: 0,75 to 1,5 Mpa @ 100 % Mod.AASHTO; ITS 200 kPa @ 95 to 97% Mod. AASHTO; maximum aggregate size 63 mm; PI 6 after stabilization; maximum fine loss 30%. BASIC PERMANENT DEFORMATION repetitions” as used in this document, therefore MODEL actually implies “number of stress repetitions”. The independent variables may be grouped as Before the actual permanent deformation models primary and secondary independent variables. The could be developed from HVS test data, the nature and two primary independent variables are defined as the general form of these models had to be considered. A stress or strain level and the number of stress conceptual model of permanent deformation was repetitions. Without either one of these two variables, therefore developed first. there would not be any permanent deformation. The The model assumes that the permanent remaining independent variables, such as the material deformation of a pavement layer depends on a number type (or material shear strength) moisture content and of variables and cannot be controlled specifically asphalt temperature, are referred to as secondary during an experiment. The permanent deformation is independent variables and will not cause any therefore defined as the dependent variable of the permanent deformation by themself, but will control model. The variables determining the rate and value the rate of permanent deformation. of permanent deformation are referred to as the By only considering the relationship between the independent variables and are controlled to some dependent variable (permanent deformation) and the extent, or at least measured during an experiment. two primary independent variables (stress or strain These may include variables such as the stress or strain level and the number of stress repetitions), a three level, the number of load repetitions, the moisture dimensional, non-linear regression model such as that content and initial density of unbound materials, the illustrated in Figure 5 may be formulated. If an operating temperature of asphalt material and the experiment is repeatedly done at any combination of inherent resistance of the particular material to the two primary independent variables, the outcome of deformation, as quantified by its shear strength the experiment will exhibit a variation in the value of parameters C and . the dependent variable (permanent deformation). Although the number of load repetitions is Some of this variation may be attributed to the controlled during an HVS test and is therefore referred influence of the secondary independent variables and to as an independent variable, the material layers in some to pure experimental error. the pavement actually experience a stress condition for If all the significant secondary independent each load repetition, determined by the load magnitude variables are included in the regression model, then and by the way in which the load is distributed the variation of the dependent variable at any given throughout the pavement. The term “number of load combination of independent variables, will only include the pure experimental error. The influence of result that the permanent deformation tends towards a the secondary independent variables was not included straight line at large numbers of load repetitions. The in the permanent deformation models reported in this initial exponential increase, followed by a linear paper. increase in permanent deformation as measured during HVS testing, is clearly illustrated by the data in Figure 3. The function listed in Equation 1was fitted to the permanent deformation data from each MDD module in the pavement foundation layers of each of the HVS tests listed in Table 1. Most of these test sections were instrumented with two or more MDD stacks, often with more than one MDD module in the pavement foundation, resulting in a large number of permanent deformation data sets. Figure 6 illustrates the function listed in Equation 1 fitted to the data of Figure 3. In this case, the three deepest MDD modules are FIGURE 5. BASIC THREE DIMENSIONAL effectively in the pavement foundation, at depths of PERMANENT DEFORMATION MODEL 440 mm, 660 mm and 900 mm. The regression and correlation coefficients for Equation 1 obtained for each data set, from each HVS DEVELOPMENT OF PERMANENT test listed in Table 1, are listed in Table 3. Figure 6 DEFORMATION MODELS FROM HVS TEST and the correlation coefficients listed in Table 3 clearly DATA illustrate that Equation 1 provides an accurate regression model for permanent deformation as a Once the conceptual permanent deformation function of load repetitions. Again it should be model had been developed, this model had to be emphasized that each HVS load repetition on the road expressed as a mathematical function. The shape of surface corresponds to a stress repetition in the the model was, however, unknown except along the pavement system. two axes where the value of the primary independent variables are zero and the permanent deformation is therefore also zero. By developing regression functions describing the basic model in the two perpendicular directions of the primary independent variable axes, it is possible to establish the mathematical function describing the total surface. Permanent deformation as a function of load repetitions. Wolff (1992) suggested the use of the function listed in Equation 1 to describe the increase in permanent deformation with increasing load repetitions during an HVS test, with high accuracy. bN PD (mN a)(1 e ) (1) FIGURE 6. REGRESSION FUNCTION FITTED TO IN-DEPTH PERMANENT DEFORMATION DATA Where PD = permanent deformation (mm) Permanent deformation as a function of stress or N = number of load repetitions strain level. As mentioned previously, only HVS tests m,a,b = regression coefficients for which in-depth deflection profiles were available e = base of the natural logarithm were selected for analysis. Elastic moduli were back- Equation 1 consists of a linear and exponential calculated from the peak deflection values at the layer component. The exponential component rapidly interfaces for the HVS test sections under investigation decays with increasing load applications, with the as opposed to the back-calculation of layer moduli from the deflection bowl. The magnitude of the test repetitions, plotted against the corresponding vertical loads varied from 40 to 100 kN dual wheel loads with strain and vertical stress from Table 3. Each data tyre pressures ranging from 520 to 700 kPa. The peak point on these plots is assumed to represent the deflections of the two deepest MDD modules were permanent deformation of a semi-infinite half-space of extrapolated to estimate the depth to zero deflection more or less homogenous material, subjected to the and the back-calculation was done with a linear elastic corresponding value of the critical parameter. multi-layer program. The stresses and strains at any position in the pavement structure could therefore be calculated from the elastic material parameters, the layer geometry and the loading condition for a particular HVS test. Two parameters were considered for use as the critical parameter relating the stress/strain condition to the development of permanent deformation. These were the vertical stress and vertical strain on top of the subgrade. Vertical stress is a continuous function over the interface between two layers with different elastic properties, due to equilibrium conditions, and was therefore calculated at the exact depth of the interface above the pavement foundation. Vertical strain is, however, a discontinuous function at such an interface and was therefore calculated just below the upper interface of the pavement foundation. The values of vertical stress and vertical strain corresponding with each set of permanent deformation data are also listed in Table 3. The HVS tests listed in Table 1 were not all trafficked to the same number of load repetitions. The regression coefficients listed in Table 3 were therefore used to extrapolate the permanent deformation at specific load repetition values to produce plots of the permanent deformation against the value of the critical parameter at these load repetition values. In some cases the permanent deformation had to be extrapolated to load repetition values far beyond the duration of the test. It is therefore implicitly assumed that Equation 1 will remain valid and give accurate estimates of permanent deformation for load repetition values higher than the duration of the HVS test. This assumption is believed to be valid, as the accuracy of Equation 1 is illustrated by the high correlation coefficients obtained (Table 3) with 87% of the values above 0.900 and 5% below 0.700. It is also believed that once the permanent deformation for a particular test has settled down to a constant rate of increase at high numbers of load repetitions, as quantified by the linear component of Equation 1, there will not be any drastic deviation in permanent deformation from the basic trend unless there is a significant change in either or both the load condition and the pavement moisture content. Figure 7 shows the plots of permanent deformation at one million and ten million load TABLE 3. REGRESSION AND CORRELATION COEFFICIENTS FOR EQUATION 1 FITTED TO PERMANENT DEFORMATION DATA SETS FROM HVS TESTS Material Weinert Depth Critical Parameter Regression Coefficients (Eq. 1) r2 Type** Region* (mm) v (µ ) v (kPa) a b m G4 2 310 970 155.5 2.730 3.00e-06 2.50e-06 0.9991 G4 2 310 901 172.8 1.250 1.60e-05 3.10e-06 0.9989 G4 2 310 1171 182.1 1.750 1.20e-05 5.00e-06 0.9978 G4 2 350 311 131.1 0.580 2.35e-04 2.10e-06 0.9874 G4 2 310 584 105.4 2.950 1.00e-06 1.10e-06 0.9970 g5 2 800 661 18.9 0.023 2.10e-05 1.50e-07 0.9999 g5 2 800 392 18.7 0.010 3.41e-04 2.00e-07 0.8648 G5 1 615 1274 54.1 0.410 1.50e-05 4.90e-07 0.9165 G5 2 630 441 21.1 0.008 1.00e-05 4.90e-07 0.9258 G5 1 430 1364 84.5 0.350 3.20e-05 1.90e-06 0.9919 G5 2 480 552 69.4 0.580 2.60e-05 2.40e-06 0.9985 G5 2 630 512 30.7 0.300 5.00e-06 7.80e-07 0.9379 G5 2 480 335 63.6 3.120 1.00e-06 9.30e-07 0.9986 G5 1 510 535 119.3 2.500 7.00e-06 6.40e-06 0.9931 G5 1 650 1318 68.2 1.500 7.00e-06 5.40e-06 0.9764 G5 2 600 446 27.0 0.160 5.00e-06 6.70e-07 0.9840 G5 1 650 881 77.3 2.100 7.00e-06 5.00e-06 0.9784 G5 2 480 326 39.7 0.800 4.00e-06 1.00e-06 0.9927 G5 1 765 1422 45.9 0.340 1.41e-04 1.50e-07 0.4755 G5 1 430 1132 79.6 0.400 2.30e-05 1.20e-06 0.9158 G5 2 480 399 81.7 0.890 1.50e-05 3.50e-06 0.9985 G5 2 630 718 38.7 0.160 1.28e-04 1.50e-06 0.9925 G5 1 765 1211 47.5 0.200 1.35e-04 2.00e-07 0.9468 G5 2 375 703 38.9 0.070 3.10e-05 4.00e-07 0.9581 G5 2 375 798 48.0 0.077 3.01e-04 7.00e-07 0.8929 G5 2 630 606 42.6 0.190 1.20e-04 2.30e-06 0.9941 G5 2 550 322 31.4 0.013 3.61e-04 4.50e-07 0.7338 G5 1 765 1831 42.0 0.220 1.00e-06 2.80e-07 0.3138 G5 1 430 1063 63.6 0.220 3.40e-05 1.80e-06 0.9841 G5 2 550 638 26.2 0.070 2.10e-05 2.70e-07 0.8144 G5 1 615 1705 47.4 0.300 9.20e-06 5.50e-07 0.9854 G5 1 510 1018 95.6 2.100 7.00e-06 7.40e-06 0.9957 G5 1 615 1056 62.6 0.200 3.22e-05 8.90e-07 0.9907 G5 1 615 1163 52.7 0.190 2.23e-05 1.00e-06 0.9458 G5 1 430 1580 60.7 1.880 2.21e-05 1.75e-06 0.9955 g6 2 900 201 18.3 0.030 1.90e-06 7.00e-08 0.7731 g6 2 660 525 21.6 0.120 2.40e-06 7.00e-08 0.9317 g6 2 660 232 33.3 0.400 4.80e-06 3.20e-07 0.9924 g6 2 900 340 22.6 0.090 6.00e-06 1.00e-07 0.9233 g6 2 550 571 62.0 1.253 2.00e-05 2.00e-06 0.9559 g6 2 900 254 21.1 0.080 1.10e-06 1.00e-09 0.9450 * Weinert climatic “n” value (Weinert, 1980). 2 indicates moderate regions and 1 indicates wet regions. ** South African road building material classification (Table 2 modified from CSRA, 1996), lower case indicates uncertain material classification. TABLE 3 (CONTINUED). REGRESSION AND CORRELATION COEFFICIENTS FOR EQUATION 1 FITTED TO PERMANENT DEFORMATION DATA SETS FROM HVS TESTS Material Weinert Depth Critical Parameter Regression Coefficients (Eq. 1) r2 Type** Region* v (µ ) v (kPa) a b m g6 2 550 461 30.5 1.178 1.50e-05 1.00e-06 0.9780 g6 2 660 238 28.4 0.310 2.10e-06 2.80e-08 0.9844 G6 2 440 877 48.0 0.760 3.70e-06 3.80e-07 0.9926 G6 2 440 140 35.1 0.550 1.60e-06 6.00e-08 0.9368 G6 2 375 893 90.2 2.413 2.00e-05 4.00e-06 0.9055 G6 2 375 718 36.7 1.797 2.00e-05 1.70e-06 0.9825 G6 2 440 609 40.0 0.350 3.10e-06 7.00e-08 0.9892 G8 2 350 3041 96.1 0.509 2.50e-04 8.10e-06 0.9936 G8 2 350 2814 79.7 0.114 1.35e-04 4.22e-06 0.9974 G8 2 295 2453 55.9 0.322 6.61e-05 3.50e-06 0.9891 G8 2 295 3326 50.6 0.312 7.61e-05 3.00e-06 0.9944 G8 2 350 2292 83.7 0.200 1.70e-04 6.30e-06 0.9976 G8 2 325 2314 49.2 0.153 1.51e-04 1.00e-06 0.9409 G8 2 325 1856 45.0 0.034 1.48e-03 4.00e-07 0.9675 G8 2 350 3055 81.4 0.510 1.12e-04 7.90e-06 0.9963 G9 2 720 441 45.1 0.111 7.76e-04 6.50e-07 0.9883 G9 2 1000 965 16.9 0.033 3.11e-04 3.00e-07 0.9188 G9 2 445 1574 37.2 0.294 6.01e-05 2.50e-06 0.9967 G9 2 665 1025 25.0 0.039 5.91e-05 5.00e-07 0.9665 G9 2 500 1622 62.2 0.100 1.60e-03 4.20e-06 0.9959 G9 2 720 502 43.4 0.057 2.50e-04 1.10e-06 0.9821 G9 2 720 630 46.1 0.071 2.80e-04 1.60e-06 0.9582 G9 2 500 1286 70.6 0.363 2.00e-04 5.70e-06 0.9906 G9 2 475 1272 33.6 0.010 2.07e-03 4.00e-07 0.9111 G9 2 500 1259 62.7 0.099 7.86e-05 2.21e-06 0.9917 G9 2 445 1974 38.2 0.193 5.51e-05 2.50e-06 0.9802 G9 2 665 919 25.4 0.043 4.01e-05 5.00e-07 0.9876 G9 2 500 2427 60.0 0.474 1.03e-04 5.00e-06 0.9968 G9 2 720 329 26.0 0.096 2.11e-04 1.00e-07 0.8791 G9 2 720 429 22.6 0.007 2.31e-04 2.00e-07 0.7883 G9 2 1000 481 15.6 0.007 2.04e-03 1.00e-07 0.5902 G9 1 630 2219 80.3 0.890 1.22e-05 3.76e-06 0.9971 G9 2 1000 379 17.5 0.038 2.41e-04 1.00e-07 0.5682 G9 1 630 2248 84.2 1.110 1.47e-05 4.11e-06 0.9737 G9 2 720 751 42.8 0.221 1.15e-04 2.20e-06 0.9911 G9 1 600 659 52.9 0.020 3.68e-05 8.90e-07 0.9097 G9 2 475 1219 38.2 0.116 2.01e-04 1.00e-06 0.9410 * Weinert climatic “n” value (Weinert, 1980). 2 indicates moderate regions and 1 indicates wet regions. ** South African road building material classification (Table 2modified from CSRA, 1996), lower case indicates uncertain material classification. (a) PERMANENT DEFORMATION PLOTTED AGAINST (c) PERMANENT DEFORMATION PLOTTED AGAINST VERTICAL STRAIN AT 1 MILLION LOAD REPETITIONS VERTICAL STRESS AT 1 MILLION LOAD REPETITIONS 100.0 80.0 60.0 40.0 20.0 0.0 (b) PERMANENT DEFORMATION PLOTTED AGAINST (d) PERMANENT DEFORMATION PLOTTED AGAINST VERTICAL STRAIN AT 10 MILLION LOAD REPETITIONS VERTICAL STRESS AT 10 MILLION LOAD REPETITIONS FIGURE 7. PERMANENT DEFORMATION OF THE PAVEMENT FOUNDATION PLOTTED AGAINST VERTICAL STRAIN AND VERTICAL STRESS The plots of permanent deformation against at such an interface. The vertical stress therefore also vertical strain show no clear correlation between corresponds to the trend of decreasing permanent these two parameters. The plots of permanent deformation with increasing depth, as illustrated by deformation against vertical stress indicate a better Figures 3 and 6. correlation between the applied stress and the It was therefore decided to develop permanent resulting permanent deformation. This contradicts deformation transfer functions for the pavement current design practice where the permanent foundation with vertical stress as the critical deformation of the pavement foundation is usually parameter. linked to the vertical strain calculated at the top of the At a zero vertical stress value on Figures 7(c) and foundation layers. (d), the permanent deformation should also be zero In addition to this, as mentioned previously, because if there is no applied stress condition, there vertical strain is a discontinuous function at the should not be any permanent deformation. As the interface of two materials with different stiffness value of the vertical stress increases, there seems to an moduli and the vertical stress is a continuous function exponential increase in the permanent deformation as plotted on Figures 7(c) and (d). The results of the regression analysis are A regression function as listed in Equation 2 was summarised in Table 4. Figures 8 and 9 show the therefore fitted to the data as plotted in Figures 7(c) regression coefficients A and B plotted on a log-log and (d) at a number of load repetition values for the G4 and a log-linear scale respectively against the number material quality data, the G5 and G6 material quality of load repetitions N, for the different material quality data combined and the G8 and G9 material quality groups. data combined. These material groups seem to The regression coefficient A clearly exhibits a represent the actual material quality of pavement linear correlation with the number of load repetitions foundations for the HVS test sections listed in Table 1. N, on a log-log scale (Figure 8). Furthermore, the same relationship seems to exist between A and the B PD A(e v 1) (2) number of load repetitions N, regardless of material type. On the other hand, there does not seem to be any correlation between the regression coefficient B and Where PD = permanent the number of load repetitions N (Figure 9). The value deformation (mm) = vertical stress on top of pavement of B does however, seem to depend on material quality, v foundation (kPa) with the value of B decreasing with increasing material A,B = regression constants quality. e = base of the natural logarithm TABLE 4. REGRESSION COEFFICIENTS A AND B OF EQUATION 2 FOR DIFFERENT MATERIAL GROUPS Number of load or G4 material G5 and G6 material G8 and G9 material stress repetitions, N A B r2 A B r2 A B r2 1 000 0.004 0.0100 0.813 0.005 0.0140 0.810 0.008 0.0250 0.326 3 000 0.011 0.0100 0.812 0.013 0.0148 0.817 0.017 0.0255 0.454 10 000 0.030 0.0100 0.786 0.014 0.0151 0.811 0.040 0.0250 0.534 30 000 0.075 0.0100 0.728 0.123 0.0149 0.725 0.069 0.0255 0.609 100 000 0.200 0.0098 0.812 0.266 0.0161 0.723 0.140 0.0255 0.752 300 000 0.497 0.0100 0.966 0.576 0.0181 0.701 0.300 0.0255 0.809 1 000 000 1.171 0.0100 0.912 1.297 0.0181 0.683 0.890 0.0250 0.802 3 000 000 2.793 0.0100 0.924 2.953 0.0187 0.686 2.420 0.0255 0.789 10 000 000 8.302 0.0100 0.917 8.320 0.0194 0.679 7.990 0.0255 0.783 30 000 000 24.027 0.0100 0.907 24.280 0.0194 0.676 24.050 0.0255 0.781 100 000 000 78.105 0.0101 0.904 73.050 0.0203 0.675 79.000 0.0255 0.781 c = -10.919 s = 0.813 r2 = 0.993 Equation (5) then becomes: B PD 18×10 6 N 0,813 [e v 1] (6) The value of B was assumed to be constant for each material group and taken as the average of the B- values listed in Table 4 per material group. This gives the following B-values: FIGURE 8. REGRESSION COEFFICIENT “A” B = 0,01 for G4 material OF EQUATION 2 PLOTTED AGAINST THE = 0,117 for G5/G6 material and NUMBER OF LOAD REPETITIONS, N = 0,025 for G8/G9 material Equation 6, combined with the B-values listed above, provides the regression function governing the basic, 3-dimensional permanent deformation model for a semi-infinite pavement foundation of G4, G5/G6 and G8/G9 material quality. PERMANENT DEFORMATION DESIGN TRANSFER FUNCTIONS The mathematical permanent deformation models developed in the previous section have the shape of the surface illustrated in Figure 5. Unfortunately, it is difficult to illustrate these models graphically and to read data from the graphical representation of these FIGURE 9. REGRESSION COEFFICIENT “B” models for design purposes. The permissible vertical OF EQUATION 2 PLOTTED AGAINST THE stress may, however, be calculated from these models NUMBER OF LOAD REPETITIONS, N at specific permanent deformation values for a range The relationship between A and N may be of load repetition values. By doing this, the ordinates expressed by Equation 3. of contour lines on the 3-dimensional permanent lnA s ln N c (3) deformation model are actually calculated. These contour lines may be used as permanent deformation design transfer functions. or The permanent deformation models developed in the previous section represent the best fit model, fitted A e cN s (4) to the permanent deformation data. As illustrated in Figure 5, the permanent deformation data will vary around this best fit model. By calculating upper Where s is the slope and c is the intersect of the prediction intervals at increasing probability values straight line relationship between A and N, on a log- and using these as the permanent deformation models, log scale. Equation 2 then becomes: rather than the best fit model, more and more of the data points will be included below the permanent B PD e cN s (e v 1) (5) deformation model. By doing this, the probability of a single permanent deformation occurrence exceeding the value predicted from the permanent deformation A linear regression analysis was done for all the model is reduced. The permanent deformation models data, regardless of material type, plotted on Figure 8. developed in such a way are used to limit the The following values were obtained from this analysis: probability of the actual permanent deformation exceeding the predicted permanent deformation for different road categories. Table 5 provides a summary of the approximate design reliability values required for different road categories in South Africa (CSRA, 1996). A 95% design reliability for an A category road implies that the pavement should be designed so as to limit the probability of actual permanent deformation exceeding the predicted permanent deformation to 5%. This requires the use of the permanent deformation transfer function obtained from a 95% probability upper prediction limit. Contour lines on different upper prediction limit permanent deformation models (corresponding to the approximate design reliabilities for the different road categories) may therefore be used as design transfer functions for the different road categories. Figure 10 illustrates the G8/G9 material quality, pavement foundation permanent deformation design transfer functions for the road categories listed in Table 5. By following a similar approach for the pavement structural layers, transfer functions of the type illustrated in Figure 11 were developed for all road categories for G2, G4 and G5/G6 material quality groups. The critical parameter selected for the structural layers is the bulk stress (the sum of the principal stresses calculated at the mid-depth of the structural layer). The other main difference between the transfer functions for the pavement foundation and structural layers is the units in which permanent deformation is expressed. The transfer functions for the foundation layers express the permanent deformation in mm and represent the settlement of the pavement foundation. The transfer functions for the structural layers express the permanent deformation as a permanent strain percentage. The permanent strain obtained for a structural layer at a specific combination of and N must therefore bemultiplied by the thickness of the layer to obtain the permanent deformation in mm. TABLE 5. APPROXIMATE DESIGN RELIABILITY VALUES USED IN SOUTH AFRICA FOR DIFFERENT ROAD CATEGORIES (CSRA, 1996) Road category Description of road category and typical examples Approximate Allowable probability of design actual distress exceeding reliability predicted distress A Very important with a very high level of service. Major interurban 95 % 5% freeways and roads B Important with a high level of service. Interurban collectors and 90 % 10 % major rural roads. C Less important with a moderate level of service. Lightly trafficked 80 % 20 % rural roads and strategic roads. D Least important with a moderate to low level of service. Rural 50 % 50 % access roads. (a) 50% design reliability, road category D (c) 90% design reliability, road category B (b) 80% design reliability, road category C (d) 95% design reliability, road category A FIGURE 10. G8/G9 MATERIAL QUALITY PAVEMENT FOUNDATION PERMANENT DEFORMATION TRANSFER FUNCTIONS 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 (a) 50% design reliability, road category D (c) 90% design reliability, road category B 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 (b) 80% design reliability, road category C (d) 95% design reliability, road category A FIGURE 11. G5/G6 MATERIAL QUALITY PAVEMENT LAYER PERMANENT DEFORMATION TRANSFER FUNCTIONS EXAMPLES OF PERMANENT between two of these contour lines. DEFORMATION CALCULATIONS FOR PAVEMENT STRUCTURES Once the road category has been selected and the design bearing capacity has been determined in terms of the number of standard axles for which the facility will be designed, potential pavement designs are analysed to determine the bulk stress at the mid-depth of the structural layers and the vertical stress on top of the foundation layers. The calculated value of the critical parameter is entered on the vertical axis of the transfer function and a horizontal line is extended across the transfer function. The design bearing capacity in terms of the number of standard axle repetitions is entered on the horizontal axis and a vertical line is extended upwards until it intersects the horizontal line. The point of intersection will lie on one of the contour lines indicated on the transfer function or, more often, If the point lies between two contour lines, the value of the permanent deformation should be interpolated along the horizontal line between the two contour lines. The total pavement permanent deformation is calculated as the sum of the permanent deformation of the individual structural layers and the permanent settlement of the foundation layers. Table 6 shows a number of design examples for category C and D roads for different design bearing capacity values. The permanent deformation is calculated for each of the structural layers based on the value of the bulk stress and in the case of the foundation layers, it is based on the value of the vertical stress calculated on top of the layer. The structures shown in Table 6 were selected from the pavement design catalogue used in South Africa (CSRA, 1996). These structures were all designed originally with the South African Mechanistic Design Method. TABLE 6. PERMANENT DEFORMATION CALCULATION FOR A NUMBER OF PAVEMENT STRUCTURES Road Number of Pavement Layer critical Layer and total Category, Load Structure parameter permanent Design Repetitions deformation reliability (ESA*) C, 80% 100 000 = 505 kPa (G4) 4 mm = 185 kPa (G6) 2 mm v = 101 kPa (Top of G7) 4 mm 10 mm 300 000 = 510 kPa (G4) 12 mm = 181 kPa (G6) 3 mm v = 88 kPa (Top of G7) 5 mm 20 mm D, 50% 100 000 = 550 kPa (G4) 3 mm = 223 kPa (G6) 1 mm v = 99 kPa (Top of G9) 3 mm 7 mm 300 000 = 505 kPa (G4) 7 mm = 185 kPa (G6) 1 mm v = 85 kPa (Top of G9) 4 mm 12 mm * ESA = Equivalent Standard Axles Although the pavement structures shown in Table should include: 6 for category D roads generally have thinner structural layers and a lower quality upper selected • In-depth deflection bowls and peak deflections at layer than those shown for category C roads, the a range of wheel loads permanent deformation predicted for the category D • In-depth permanent deformation roads is less than that of the category C roads at the • A proper material classification for the full same number of load repetitions. This is because the pavement depth, including the pavement transfer functions for the category C roads allow a foundation or roadbed smaller probability of the actual permanent • All other relevant variables, such as the moisture deformation exceeding the predicted permanent content of unbound layers and the temperature of deformation. asphalt layers. The data listed above should be collected for the CONCLUSIONS AND RECOMMENDATIONS duration of the test. A basic, conceptual model has been developed for Permanent deformation transfer functions were the permanent deformation of pavement layers. developed for the following unbound structural Permanent deformation design transfer functions were pavement layer material quality groups: developed for a number of unbound material quality groups, following the principles of the basic model. • G2 dense graded crushed stone The main (practically the only) source of data for • G4 base quality natural gravel developing these design models was Heavy Vehicle • G5/G6 subbase quality natural gravel Simulator test data collected over a long period of time in South Africa. and for the following pavement foundation material The success of utilizing accelerated pavement quality groups: testing data for the development of such design models depends largely on having a centralized data storage • G4 natural gravel system where all the data collected during accelerated • G5/G6 natural gravel pavement testing may be stored. Data which should be • G8/G9 gravel/soil captured and stored on a centralised system during accelerated pavement testing for the purpose of These material groups only represent a small developing permanent deformation design models selection of the materials normally used for road construction in South Africa. In order to develop a There are also quite a number of less conventional comprehensive permanent deformation component for materials being considered for road building in South a mechanistic design method, similar transfer Africa. Transfer functions for these materials are functions will have to be developed for materials such lacking, thereby reducing the confidence in using these as asphalt and lightly cemented material. materials in pavement design. Because of the time involved in large scale accelerated pavement testing, it is believed that a combination of laboratory testing and accelerated pavement testing would yield the quickest results. It should be possible to generate the type of data needed to develop transfer functions for these materials from laboratory test methods such as the dynamic triaxial test. The models developed from such data may then be verified by means of a limited number of large scale, accelerated pavement tests. The permanent deformation of a number of pavement structures was calculated from the transfer functions for the material quality groups listed above. The advantage of these transfer functions and of the way they are applied is that they allow for each of the pavement layers to contribute to the total pavement deformation. The design approach has therefore shifted from a critical layer approach to a pavement system approach. These transfer functions do not, however, provide for the influence of secondary independent variables such as moisture content and initial density on the development of permanent deformation of unbound layers. By including the influence of these variables in the transfer functions, the amount of scatter in the data should decrease and the accuracy of the models should increase. The influence of these variables on the development of permanent deformation could be quantified by a detailed laboratory test programme or by recording these variables during accelerated pavement testing. A further improvement to the models may be achieved by more advanced stress/strain analysis techniques and by a further investigation into the appropriate critical parameters to be used. The development of these models has, however, only started recently and, although there is a lot of scope for improvement, the basic approach has been established and illustrated. REFERENCES Committee of State Road Authorities (CSRA), TRH 14: 1985, Guidelines for Road Construction Materials. Department of Transport, South Africa, 1985. Committee of State Road Authorities (CSRA), DRAFT TRH 4: 1996, Structural Design of Flexible Pavements for Interurban and Rural Roads. Department of Transport, South Africa, 1996. De Beer, M., Horak, E., Visser, A.T., “The Multi- Depth Deflectometer (MDD) System for Determining the Effective Elastic Moduli of Pavement Layers.”, Nondestructive Testing of Pavements and Backcalculation of Moduli, ASTM STP 1026, A.J. Bush III and G. Y. Baladi, Eds., ASTM, Philadelphia, 1989. Theyse, H.L., De Beer, M., Rust, F.C., “Overview of the South African Mechanistic Design Method.”, Paper presented at the 75th annual Transportation Research Board meeting, Washington, 1996. Weinert, H.H., “The Natural Road Construction Materials of South Africa”, H & R Academica, Cape Town, 1980. Wolff, H., “Elasto-Plastic Modelling of Granular Layers.”, Research Report RR92/312, Department of Transport, South Africa, 1992.