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Mechanistic empirical modelling of the permanent deformation

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Mechanistic empirical modelling of the permanent deformation Powered By Docstoc
					      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.