Plastic deformation behaviour of pavement granular materials under low traffic loading

Transcription

1 Granular Matter (2010) 12:57 68 DOI /s Plastic deformation behaviour of pavement granular materials under low traffic loading I. Pérez L. Medina J. Gallego Received: 3 February 2009 / Published online: 14 October 2009 Springer-Verlag 2009 Abstract This paper analyses the permanent deformation performance of an unbound granular material for base layers of low traffic roads. The material has been subjected to repeated triaxial loads. Three models were fitted to express the cumulative permanent strain as a function of the number of load cycles. In general, the predictions of two models previously studied by other researchers proved to be good but in the long-term, they tended to underestimate the measured values. In contrast, a third new model-the sum of two well known models-offered excellent predictions, which in the long term did not tend to either underestimate or overestimate the measured values. The granular material did not give satisfactory results to be used in low traffic volume road pavements. Keywords Unbound granular material Permanent deformation Rutting Low-traffic roads 1 Introduction Low-traffic roads are usually made with flexible pavements in which unbound granular materials (UGM) provide the most important structural element. Rutting is one of the main causes of damage in low-traffic flexible pavement roads and it is the result of an accumulation of permanent Communicated by: S. Luding. I. Pérez (B) L. Medina Universidade da Coruña. E. T. S. I. Caminos, Campus de Elviña, A Coruña, Spain iperez@udc.es J. Gallego Universidad Politécnica de Madrid. E. T. S. I. Caminos, Profesor Aranguren, s/n, Madrid, Spain deformation in the various pavement layers, both bound and unbound. Accordingly, the structural design of these pavements requires the mathematical description of the permanent deformation behaviour of UGM. The goal is to predict the development of rutting in granular base courses. The repeated load triaxial (RLT) permanent deformation test has been widely used to determine permanent deformation characteristics of granular materials. In the RLT test, under repeated load cycles the accumulation of permanent strain is a gradual process where each load application contributes to the accumulation of strain by a small increment. Therefore, the number of load cycles is an important factor to consider in the analysis of the long term behaviour of granular materials. In this sense, Pérez et al. [13] carried out several analyses related to permanent deformation. In their conclusion they pointed out that mathematical models that predicted permanent deformation as a function of load cycles presented certain deficiencies. To be precise, in the long term these models underestimated or overestimated the observed values. In this paper additional work was done to improve the predictions. The objective was to examine the applicability of a non-linear model depending on the number of load cycles to predict the development of rutting in granular base courses of low traffic roads. The model is not found in the technical literature and consists of a combination of two well known models: the Sweere model and the Wolff model. 2 Background 2.1 UGM response ranges A pavement is liable to show progressive accumulation of permanent strains (evident as rutting) under traffic loading if the magnitude of the applied loads exceeds the limit value.

2 58 I. Pérez et al. If the applied traffic loads are lower than this limit, after any post-compaction stabilisation, the permanent strains will level off and the pavement will reach a shakedown state at which time it starts to undergo only resilient deformation under additional traffic loading. This implies that the pavement has adapted to the loading. This could be due to a change in material response (e.g. from compaction), a change in stress state (e.g. from the development of lockedin stresses) or a combination of both effects. However, the ideal concept does not relate in a straight-forward manner to what was observed in laboratory testing of UGM. Multilayer systems such as pavements are exposed to repeated traffic loads of various magnitudes and numbers of load cycles. For performance prediction, it is of great importance to know whether a given UGM will experience progressive accumulation of permanent deformation leading to a state of incremental collapse or if the increase will cease, resulting in a stable response (shakedown state). In this sense, the ideal shakedown concept maintains that there are four regimes of behaviour under repeated loading [4,7]: Purely elastic; Elastic shakedown; Plastic shakedown (cyclic plasticity) and Incremental collapse (ratchetting). Nevertheless, Werkmeister et al. [19 21] have proposed only three regimes of behaviour of UGM to the applications of load cycles (N) in highway pavements [19 21]: Range A (Plastic Shakedown). The material response shows high cumulative permanent strain rates (ε 1p /N) for a finite number of load repetitions during the initial compaction period. After the compaction period ε 1p /N decreases until it reaches a state of equilibrium. The deformation is entirely resilient and does not produce permanent deformations. The material does not reach failure. This range occurs at low stress levels. Range B (Plastic Creep). Initially, during the compaction period the response is the same as range A. After that period ε 1p /N is either decreasing, constant or slightly increasing. Although the deformation is not entirely resilient, for the period of the RLT test the permanent deformation is acceptable. For a great number of load cycles the material can reach range A or range C. Range C (Incremental Collapse). An initial compaction period is observed and after that period ε 1p /N increases or remains constant, failure occurs with a relatively small number of load cycles. Werkmeister et al. [19 21] assumed that a pavement section is well designed when the UGM response is under stable conditions, i.e. it should be within range A (Plastic shakedown). However, range A is difficult to find in practical pavement engineering. It is extremely complicated to attain stable equilibrium behaviour in response to the loading, since accumulation of permanent deformation (Rutting) is almost always produced in flexible pavements because of heavy vehicle load repetitions. Over the passage of time it takes place even in pavements that are well designed, well constructed and with excellent maintenance [5]. Range B (Plastic creep) is a more realistic and practical range. It is very appropriate for low-traffic roads provided that a suitable rut depth has been produced during the pavement design life. Range C should not be allowed to occur in a well designed pavement. 2.2 Permanent deformation models as a function of number of load cycles Sweere [15] modelled the relationship between the logarithm of the cumulative permanent axial strain and the logarithm of load repetitions (Eq. 6 in Table 1). In this model, ε 1p (%) is the cumulative permanent axial strain after N load cycles are attained by means of RLT tests, and A and B are non-linear regression parameters. Intercept A represents the permanent strain at N=1 on a log-log scale. Slope B corresponds to the rate of change of the ε 1p (%) logarithm as a function of the change in the logarithm of N. As can be seen, Eq. 6 predicts an infinite deformation for an infinite number of load cycles. Sweere [15] found a satisfactory linear relationship between the two factors on a log-log scale. However, Pérez et al. [13] confirmed a satisfactory fit but observed that Eq. 6 tended to underestimate the measured values. Wolff and Visser [22] measured the accumulated permanent deformation produced in granular road bases using Heavy Vehicle Simulator (HVS) tests. They verified that for a large number of cycles (N >10 6 ) the values predicted for Eq. 6 differed from the real values. Therefore, Wolff et al. [22] suggested an improved model (Eq. 2 in Table 1) with which they obtained a good fit. This model was also fitted by Theyse [16] with good results. Equation 2 predicts an infinite deformation for an infinite number of cycles, as well. Coefficients m, A and B are non-linear regression parameters and e is the base of the natural logarithm. Equation 2 consists of a linear and exponential component. The exponential component rapidly decays with an increasing number of load cycles. Hence, the permanent deformation tends to form a straight line (m N + A) at high numbers of load cycles. Parameters m and A are the slope and the intercept of the straight line (asymptote) respectively; B is a constant controlling the bend of the curve. With regard to this model, the fit found by Pérez et al. [13] may be considered quite satisfactory but it was observed that Eq. 2 tended to overestimate the measured values. Francken and Clauwaert [3] and Kaloush and Witczak [8] used a non-linear model composed of two components (Eq. 3 in Table 1) to study the permanent deformation behaviour of asphalt mixtures subjected to RLT tests. The same model was

3 Plastic deformation behaviour of pavement granular materials under low traffic loading 59 Table 1 Models linking permanent deformation to the number of load cycles Equation Model Reference Regression parameters 1 ε 1p (%) = A N B Sweere [15] A, B 2 ε 1p (%) = (m N + A) ( 1 e B N ) Wolff et al. [22] A, B, m 3 ε 1p (%) = A 1.N B1 + (A 2 ) ( e B2 N 1 ) Francken et al. [3], Kaloush et al. [8], A 1, B 1, A 2, B 2 Huurman [6], Werkmeister [19], Arnold [1] 4 ε 1p (%) = (m N) + ( A 2 ) ( 1 e B2 N ) Theyse [17,18] A 2, B 2, m 5 ε 1p (%) = A 1.N B1 + (m N + A 2 ) ( 1 e B2 N ) A 1, B 1, m, A 2, B 2 also proposed by Huurman [6]. Later, Werkmeister [19] and Arnold [1] selected this model for the practical application of the shakedown concept to model the permanent deformation of UGM. In Eq. 3, the first component is the same power-law proposed by Sweere. According to Werkmeister [19], it is able to express the material phase in behavioural range A (Plastic shakedown). Parameters A 1 and B 1 are similar to parameters A and B of Eq. 6. The second component is a function which represents an exponential increase of ε 1p (%) with N on the same log ε 1p (%) log N. The second summand describes the material phases in range B (Plastic creep) and range C (Incremental Collapse) behaviour [19]. Werkmeister [19] made the model stress dependent as long as A 2 and B 2 were equal to zero in range A stable behaviour. Theyse [17,18] also measured the accumulated permanent deformation produced in granular road bases using HVS tests. He proposed several non-linear functions [17,18] to model the accumulated permanent deformation of granular materials, obtaining good results. Table 1 presents one of these models (Eq. 4) used by this researcher for materials under stable conditions. The model in question comprises two phases. First, an initial exponential deformation phase and second, a long-term linear increase rate in the permanent deformation. The model has an initial slope equal to the product of the two coefficients A 2 and B 2, a curvature determined by the value of coefficient B 2, an eventual linear slope equal to coefficient m, and, finally, an intercept with the Y -axis represented by coefficient A 2. 3 Materials and methods The UGM tested was crushed granitic stone, frequently used as a base for road pavements in the region of Galicia (Spain). Its grading curve falls within the limits corresponding to the unbound granular material designated as ZA25, defined in the General technical specifications for works on roads and bridges in Spain [12]. Figure 1 provides the grading curve of the material and Table 2 reflects the results of other % passing Granular Material upper limit ZA25 lower limit ZA ,01 0, Sieve Size (mm) Fig. 1 Grading curve of the material characterization tests. It fulfils the conditions laid down in the above-mentioned specifications, with the exception of the Los Angeles Coefficient, which, with a value of 37%, is slightly higher than the 35% specified as the maximum. The laboratory sample was prepared using a special cylindrical aluminium mould consisting of two pieces which allowed it to be opened lengthwise for easy removal of the compacted sample. In the first part of the test, four laboratory samples of granular material were compacted in the mould (without a membrane), and their dry densities and optimum moistures were obtained. Afterwards, the specimens that were to be subjected to the triaxial test were fabricated. Before compacting the material, the first membrane was attached to the wall of the container by applying a vacuum between the mould and the membrane. Compaction was carried out in three layers with a vibrating hammer 20 s for each layer. The laboratory samples were compacted at the optimum moisture content (6.4%) to reach the maximum dry density (2.3g/cm 3 ). After compaction was completed and before implementing the actual dynamic tests, the specimens were held for 24 hours in a humid chamber at a temperature of 20 C with a relative humidity of 95%. Repeated load triaxial Constant Confining Pressure tests (CCP) were carried out with a dynamic apparatus. The tests were carried out with a dynamic triaxial apparatus, consisting

4 60 I. Pérez et al. Table 2 Characterization tests Test Parameter Result Modified proctor test Dry density (g/cm 3 ) 2.3 Optimum moisture content (%) 6.4 Atterberg limits Non plastic Non plastic Sand equivalent Sand equivalent 51 Los Angeles Abrasion Coefficient (%) 37 Organic material content Proportion (%) 0.19 Surface cleanness Coefficient (%) 1.3 Elongation index Elongation index (%) 3.0 Percentage of fractured particles Proportion (%) 100 Fig. 2 Triaxial Dynamic equipment essentially of a main unit containing the axial load system generator and a removable chamber (Fig. 2). Three controllers are used to generate the chamber and back pressure, as well as the air pressure, in the partially saturated samples. The most commonly used chamber fluid is deaerated water, although with laboratory samples of granular material, oil can also be used. The analogical signals sent from the transducers and load cell are received by a module where they are transformed into digital signals. The system has a converter for the digital signals sent from the computer. This system is located in the main unit and facilitates the transmission of the user s orders to the motor controller. The whole system is controlled by a PC which is equipped with the appropriate software allowing for easy and accurate communication between the user and the triaxial apparatus. This software makes it possible to select the type of test to be performed as well as all the parameters, stress paths, data to be stored, etc. This apparatus is limited to laboratory samples with a maximum diameter of 100 mm and a height of 200 mm. Moreover, the apparatus allows the laboratory sample to be subjected to cyclic axial deviatoric stresses, but it is not feasible to vary the confining radial stresses at the same time. The principal stresses (σ 1 and σ 3 ) were calculated in the pavement section shown in Fig. 3. This section was built with 100 mm of hot mix asphalt and 200 mm of UGM resting upon a sub-grade of cement stabilised soil with a CBR greater than 20 [11]. The analysis employed a linear, elastic multi-layer model [5]. A single axle load of 130 kn was assumed using single tires having a tire pressure of 900 kpa and a contact area of cm in the radius (Fig. 3). The principal stresses were found at 11 different points in the UGM layer. The results were expressed in terms of the stress invariants: deviator stress q = σ 1 σ 3 and mean normal stress p = (σ 1 + 2σ 3 )/3 (Fig. 4; Table 3), where σ 1 is the main principal axial stress (kpa) and σ 3 is the minor principal

5 Plastic deformation behaviour of pavement granular materials under low traffic loading 61 D=303 mm. Asphalt mixture E=3000 MPa Unbound granular material E=300 MPa Cement Stabilised soil Fig. 3 Pavement section =0.35 υ = KPa. 100 mm. 200 mm. 4 Proposed permanent deformation model As expressed by Pérez et al. [13], Eq. 6 tends to underestimate the measured values while Eq. 2 tends to overestimate them, resulting in biased predictions. Therefore, for the purpose of improving the predictions, it was decided TO investigate the possibility of fitting a new model. In this way, since Eq. 2 predicts underestimated values while Eq. 3 results in an overestimation, the objective was to join the two models in order to predict unbiased estimated values. The applicability of this model will be limited to low-traffic roads. The model in question (Eq. 5 in Table 1) also comprises two summands: The first summand is the Sweere model; the second is the Wolff model. As a result, it is made up of five parameters (Fig. 5). However, as discussed earlier, the first term of Eq. 5 produces a linear increase of ε 1p (%) in relation to N on a log(ε 1p ) log(n) scale. It has a slope equal to at N = 0 and equal to the product of the two coefficients A 1 and B 1 at N = 1. After a certain number of load cycles, the second summand of Eq. 5 reproduces a linear increase of ε 1p (%) with N on a ε 1p N scale. This increase is asymptotic to (m N + A) at very high N values. It has a slope equal to the product of the two coefficients A 2 and B 2 at N = 0. The model has a curvature determined by the value of coefficients B 1 and B 2 and an eventual linear slope equal to coefficient m. 5 Load cycles during pavement design life q (kpa) failure line pavement stresses stress paths P11 P10 P9 P8 P7 P1 P2 P3 P4 P5 P6 To design a highway pavement it is necessary to predict the number of repetitions of equivalent axe loads that will occur during the design life. Under the Spanish pavements regulations [11] it is foreseen that 50 heavy vehicles per day will be supported in the pavement service lane shown in Fig. 3. On the basis of this information, the heavy traffic flow was considered in order to estimate (by means of Eq. 6) the number of repetitions of 130 kn axles in the service traffic lane during the design life period [5]: p (kpa) Fig. 4 Stress Paths radial stress (kpa). Eleven stress paths (Fig. 4; Table 3) were selected according to the stress produced in the pavement section. Dynamic tests of load cycles were carried out with a sinusoidal wave frequency of 1 Hz. ESAL = AADT HV TGF EF 365 (6) Where ESAL is the number of repetitions of the equivalent single axle loads of 130 kn during the design life; AADT HV is the annual average daily heavy traffic in the service lane during the design life; TGF is a traffic growth factor and EF is an equivalence factor of a heavy vehicle on standard axles. Estimating an annual traffic growth of 2% for a design life of 20 years results in a TGF equal to On the other hand, EF is equal to 0.70 [11]. Based on these premises during the design life ESAL is equal to load cycles. Therefore, this number of load cycles will be utilized to estimate the rutting produced in the UGM during its design life.

6 62 I. Pérez et al. Table 3 Stresses for permanent deformation tests Stress path q(kpa) p(kpa) Confining Stress σ 3 (kpa) P P P P P P P P P P P Fig. 5 Equation 5 parameters ε 1p (%) Initial deformation rate second summand = A 2 B 2 m = Eventual deformation rate 1 (mn+a 2 )(1- e (-B 2 N) ) A 1 N B1 A 1 N B1 (mn+a 2 )(1- e (-B 2 N) ) 1 m = Eventual Deformation rate A 2 N 6 Results 6.1 Material response In Fig. 6, ε 1p (%)/N versus ε 1p (%) is represented for the eleven stress conditions under which the dynamic triaxial tests were carried out. This figure shows that in the curves with stress paths P1, P2, P3, P6 and P11 the aforementioned rate decreases rapidly and the cumulative permanent axial strain undergoes a relatively minor increase. The lines are nearly vertical and parallel and might almost be located in range A. However, as can be observed, a complete stabilization of permanent deformation is not produced. Possibly, a further increase in permanent deformation would take place with a greater number of load cycles. For this reason and for the practical proposal, the material in these cases is considered to be in range B (Plastic Shakedown). Consequently, rutting is produced in the material. On the other hand, in the line that represents stress path P5, at first the increase in ε 1p (%) is even more accentuated. Curiously, it would seem that a slight stabilisation was produced in the central zone (between ε 1p =1.4% and ε 1p =1.6%), with ε 1p (%) increasing considerably at the end of the curve. Here, the material is in range C (Incremental Collapse), especially at the end of the line. Finally, for the five stress paths P4, P10, P7, P9 and P8 the material behaves between range B and range C. However, when N increases it gives rise to a sharp increment in ε 1p (%), clearly passing over to range C, where rutting will be produced. 6.2 Models goodness-of-fit Equations 3, 4 and 5 were selected for fitting the data in order to compare the functioning of the different models. The parameters were obtained by means of the Levenberg- Marquardt method [10]. The Levenberg-Marquardt algorithm interpolates between the Gauss-Newton algorithm and the method of gradient descent. It is more robust than the

7 Plastic deformation behaviour of pavement granular materials under low traffic loading 63 Fig. 6 Cumulative permanent strain rate versus cumulative permanent strain 1,E-01 ε 1p (%)/Ν 1,E-02 1,E-03 1,E-04 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 1,E-05 1,E-06 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 ε 1p (%) Gauss-Newton algorithm, which means that in many cases, it finds a solution even if it starts far from the final minimum. Table 4 shows all the values for the parameters of the eleven stress paths as well the determination coefficients R 2 for Eqs. 3, 4 and 5. Figure 7 reflects the values predicted for Eq. 3 and the measured data. The experimental data are very close to the continuous lines of the model. The adjustment is satisfactory, since the determination coefficients (R 2 ) yield a result of over 97.50% in all cases. Figure 8 presents the values predicted for Eq. 4 and the measured data. The experimental data are not as close to the continuous lines of the model as in Eq. 3. The fit is not as good, since the R 2 coefficients are only over 79.40% (Table 4). Figure 9 shows the values provided by Eq. 5 as well as the measured data. This fit is considered excellent, given that the R 2 coefficients yield a result over 99.50% in all cases (Table 4). 6.3 Rutting produced during the design life To estimate the rut depth, the highest values of the permanent deformation will be taken into account; that is to say, those that correspond to stress paths P8. Such a depth would simply be the product of the predicted accumulative permanent strain ε 1p and the thickness of the granular material (200 mm) where ε 1p = ε 1p (%)/100 has been obtained with Eqs. 3, 4, and 5 and where N = ESAL = Thus, during the pavement design life, in UGM, rut depths of 4.98, 4.73, and 5.55 mm are predicted with Eqs. 3, 4 and 5, respectively. 7 Discussion of results 7.1 Response From the results of the repeated load triaxial tests it can be deduced that the increase in permanent deformation does not behave in the same way under all load conditions (Fig. 6). In the curves with stress paths P1, P2, P3 and P6 avery slow increase of ε 1p (%) is produced, giving rise almost to stabilization ( range B ). In the curve with stress path P11 the increase of ε 1p (%) is a little more accentuated ( range B ). On the other hand, in the curves with stress paths P4, P5, P7, P8, P9 and P10 the increase is much more accentuated and the stabilization of the material is not observed ( range C ). This poor result is due to the fact that when the values of the deviator stresses approach the failure envelope of the material, a clear acceleration in the increase of ε 1p (%) is produced. 7.2 Goodness-of-fit The graphical results of these three models for the particular stress paths P8 (maximum permanent deformation) are reflected in Fig. 10. The fit of E3 (Eq. 3) is quite satisfactory, although some of the measured data do not match the model completely. It can be seen that 1PE3 (First part Eq. 3) is an increasing line with a positive curvature, showing predicted values which are much higher than the measured

8 64 I. Pérez et al. Table 4 Model parameters Stress path A1 B1 m A2 B2 R2 Equation Equation Equation Equation Equation Equation P P P P P P P P P P P data. However, 2PE3 (Second part Eq. 3) is a decreasing line with a negative curvature until it stabilizes at N = 4 10 load cycles, predicting from this point a ε 1p constant value equal to 0.49%. In other words, 2PE3 predicts negative values. It is evident that that E3 (Eq. 3) does not provide a very satisfactory fit to the measured data. In Fig. 10 it is also possible to observe that the values predicted by 1PE4 (First part Eq. 4) and 2PE4 (Second part Eq. 4) are lower than the measured values. Moreover, 1PE4 is a straight line that intercepts the Y -axis in zero, while 2PE4 has an initial curvature until it stabilizes, predicting a ε 1p (%) constant value equal to 1.81 % starting at approximately 100 load cycles. It is evident that that E4 (Eq. 4) does not provide a very satisfactory fit to the measured data. Figure 10 highlights a very close fit between the predicted values with E5 (Eq. 5) and the measured values. Undoubtedly, this model adjustment is much better than the fits of Eqs. 3 and 4. This figure also displays the values predicted for 1PE5 (First part Eq. 5) and 2PE5 (Second part Eq. 5), separately. Both predictions are below the measured values. It can be seen that starting at approximately load cycles, the values predicted for the asymptote are the same as those predicted for 2PE5. InFig.10 it is also possible to see that the values predicted for 1PE5 are higher than those predicted for 2PE5. E5 clearly provides an excellent fit as the measured values match the model almost perfectly. It is only natural that the fit to the data would improve when the number of parameters in the model is increased. Actually, E5 includes Sweere s and Wolff s models as particular cases. However, the resulting accuracy proved to be remarkable even when this effect is taken into account, and it is well worth having to manipulate a few more parameters. By examining each term s contribution to the fitting curves, it becomes apparent that Sweere s power-like term governs the behaviour of the solution for relatively small values of N, while Wolff s perturbed linear term gradually acquires relevance as the number of load cycles increases, eventually giving the curve an asymptotic slope. 7.3 Rutting prediction On the other hand, in Fig. 11 the rut predictions (in mm.) estimated by means of Eqs. 3 and 4 are not close to the values calculated as the product of the measured strain and 200 mm. It is also demonstrated that predictions of Eqs. 3 and 4 tend to underestimate or overestimate the values, whereas Eq. 5 does not. However, they are quite similar up to a number of cycles equal to (Fig. 11). From here the predictions of Eqs. 4 and 5 increase very rapidly, and are higher in Eq. 5. The prediction of Eq. 3 increases more slowly and tends to stabilize. In Fig. 11 it can be verified that during the design life of the pavement, rut depths of 4.98, 4.73 and 5.55 mm are

9 Plastic deformation behaviour of pavement granular materials under low traffic loading 65 Fig. 7 Permanent deformation versus load cycles (with Eq. 3) Standard error bars represent the 99% confidence interval of a mean P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P ε 1p (%) E E E E E E+06 N Fig. 8 Permanent deformation versus load cycles (with Eq. 4) Standard error bars represent the 99% confidence interval of a mean P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P ε1p(%) E E E E E E+06 N

10 66 I. Pérez et al. Fig. 9 Permanent deformation versus load cycles (with Eq. 5) Standard error bars represent the 99% confidence interval of a mean P1 P2 P3 P4 P5 P6 P7 P8 P9 2.0 P10 P ε1p (%) E E E E E E+06 N predicted with Eqs. 3, 4 and 5 respectively. A criterion proposed by the Transportation Research Laboratory (TRL) of the United Kingdom to determine the end of the service life of a flexible pavement is to consider a 10 mm rut in the surface of pavement [14]. This criterion marks the point known as the Critical Life when the pavement should be strengthened to retain its structural integrity. It is evident that by adopting the failure criterion of the Transportation Research Laboratory (rut depth of 10 mm in the surface of the pavement), the predictions offered by Eqs. 3, 4 and 5 do not exceed the maximum permanent deformation that would be considered admissible. Nearly half of the permanent deformation allowed is produced in the UGM. Nevertheless, to be able to compare this with the aforementioned limit, it would be necessary to determine the permanent deformation in each layer (asphalt mixture layer, granular material layer and sub-grade) and sum up the results. It would be interesting to compare the results of these models with a TRL lineal rut model determined from the analysis of the performance of full-scale experimental pavements [9]. The rut model in question is: RD = N; and it predicts the rut depth (RD) on the surface of a granular base pavement. In Fig. 11 shows that when N = 0, the initial rut depth is 2.66 mm (assumed for new surfaces at the time of construction). In the design life, when ESAL is , rutting is equal to 2.92 mm, but this depth includes the rutting of various materials (not only the base course). Also, it can be compared with a modification to the Asphalt Institute sub-grade strain criterion utilized by Deacon et al. [2]. According to these authors, when the compressive strain (ε v ) at the top of the sub-grade is equal to ,therut depth contributed by unbound layers is assumed to accumulate as follows: RD=0.052 N 0.372, where the coefficients are experimentally determined from least squares analyses for the WesTrack data (USA). Rutting in the latter model is somewhat accentuated, thus in the design life it is 5.72 mm (Fig. 11). It can be seen that the long-term predictions provided by the TRL equation are lower than the predictions of Eqs. 3, 4 and 5. The predictions resulting from the WesTrack equation are lower than those of Eqs. 3, 4 and 5, until load cycles. From until load cycles the predictions of Eq. 5 and WestTrack are very close. According to these results, it would also seem that in the long-term, Eqs. 4, 5, and WestTrack are conservative (they err on the safe side and therefore it can be deemed that these laws tend to reject materials that could be structurally valid), while

11 Plastic deformation behaviour of pavement granular materials under low traffic loading 67 Fig. 10 Permanent deformation versus load cycles (stress path P8) Measured 1PE3 2PE3 E3 1PE4 2PE4 E4 1PE5 2PE5 E ε1p (%) E E E E E E E E E E E+05 N E3=Equation 3; 1PE3= First part equation 3; 2PE3=Second part equation 3 E4=Equation 4; 1PE4= First part equation 4; 2PE4=Second part equation 4 E5=Equation 5; 1PE5= First part equation 5; 2PE5=Second part equation 5 Eq. 3 is non-conservative (predict deformations lower than the laboratory ones and consequently could accept improper materials from a structural point of view). It is important to note, however, that these rut models are based on pavement materials, designs and construction standards practiced in the US and United Kingdom. Finally, in view of the above, it would seem advisable to take, as the main failure criterion, the fact that the granular material is located in range C owing to the number of axle loads and stress level that have to be supported by the pavement. Therefore, the use of this material under this pavement section (Fig. 3) would have to be rejected because of the inadmissible plastic ruts it would produce. 8 Conclusions From the results of this research the following conclusions follow: The model proposed (Eq. 5) offers unbiased predictions since it neither underestimates nor overestimates the measured values. Hence, Eqs. 1 and 2 are models that work much better as a single unit than separately. It has been proven that this model offers an excellent description of the material behaviour in range B (Plastic Creep) and range C (Incremental collapse). The granular material has been shown to be inadequate for use under a low traffic road pavement. This is so because under the stress levels that it is required to support, it would fall within range C (Incremental collapse), indicating that during the design period plastic ruts would be produced. Finally, it is important to clarify that this paper merely presents a preliminary approach to the problem. Hence, future research will require systematic procedures using a higher number of stress paths. Also to be taken into account is the fact that these results are based on repeated load triaxial (RLT) permanent deformation tests and not on tests with actual road pavements.

12 68 I. Pérez et al. Fig. 11 Prediction of rut depths Standard error bars represent the 99% confidence interval of a mean 10,00 9,00 8,00 7,00 Measured Equation 3 Equation 4 Equation 5 TRL WestTrack 6,00 Rut depth (mm) 5,00 4,00 3,00 2,00 1,00 0,00 1,0E+01 1,0E+02 1,0E+03 1,0E+04 1,0E+05 1,0E+06 1,0E+07 N References 1. Arnold, G.: Rutting of Granular Pavement. Ph.D. Thesis. University Of Nottingham, UK (2004) 2. Deacon, J.A, Harvey, J.T, Guada, I, Popescu, L, Monismith, C.L.: An Analytical-Based Approach to Rutting Prediction. In: Annual meeting of the Transportation Research Board (TRB), Washington, DC, USA (2002) 3. Francken, L., Clauwaert, C.: Characterization and structural assessment of bound materials for flexible road structures. In: Proceedings of the 6th International Conference on the Structural Design of Asphalt Pavements. Ann Arbor, Michigan, pp (1987) 4. García-rojo, R., Herrmnan, H.J.: Shakedown of unbound granular material. Granul. Matter 7, (2005) 5. Huang, Y.H.: Pavement Analysis and Design. Pearson Prentice Hall, 2nd edn, USA (2004) 6. Huurman, M.: Rut development in concrete block pavements due to permanent strain in the substructure. In: First International Conference on Concrete Block Paving, pp , Israel (1996) 7. Johnson, K.L.: Plastic flow, residual stress and shakedown in rolling contact. In: Proceedings of the Second International Symposium on Contact Mechanics and Wear of Rail/Wheel Systems. University of Waterloo Press, Canada, pp (1986) 8. Kaloush, K., Witczak, M.: Tertiary flow characteristics of asphalt mixtures. J. Assoc. Asphalt. Paving Technol. 71, (2002) 9. Kannemeyer, L.: Modelling road deterioration and maintenance effects in HDM-4. Chapter 5 Rutting of Paved Surfaces. ISO- HDM. RETA REG Highway Development and Management Research. Final Report. Prepared for Asian Development Bank (1995) 10. Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, (1963) 11. Ministerio de Fomento. Secciones de firme de la instrucción de carreteras 6.1 IC-IC. Madrid (2002) 12. Ministerio de Fomento. Anexo actualización PG-3. Capas estructurales de firmes. Anexo del pliego de prescripciones técnicas generales para obras de carreteras y puentes. Madrid (2002) 13. Pérez, I., Medina, L., Romana, M.G.: Permanent deformation models for a granular material used in road pavements. Constr. Build. Mater. 20, (2006) 14. Powell, W.D., Potter, J.F., Mayhew, H.C., Nunn, M.E.: The Structural Design of Bituminous Roads, Laboratory Report 1132, Transport and Road Research Laboratory, Department of Transport, UK (1984) 15. Sweere, G.T.H.: Unbound granular bases for roads. Ph.D. Thesis, University of Delft, Holland (1990) 16. Theyse, H.L.: Mechanistic-empirical modelling of the permanent deformation of unbound pavement layers. In: ISAP 8th International Conference on Asphalt Pavement. Seattle (USA) (1997) 17. Theyse H.L.: The development of mechanistic-empirical permanent deformation design models for unbound pavement materials from laboratory accelerated pavement data. In: Proceedings of the fifth international symposium on unbound aggregates in roads, pp , Nottingham, UK (2000) 18. Theyse, H.L.: Stiffness, Strength, and Performance of Unbound Aggregate Material: Application of South African HVS and Laboratory Results to California Flexible Pavements. University of California Pavement Research Center, pp (2002) 19. Werkmeister, S.: Permanent deformation behaviour of unbound granular materials in pavement construction. Ph.D. Thesis. Dresden University of Technology, Germany (2003) 20. Werkmeister, S., Dawson, A., Wellner, F.: Permanent deformation behaviour of granular materials and the shakedown concept. Transp. Res. Record 1757, (2001) 21. Werkmeister, S., Dawson, A., Wellner, F.: Pavement design model for unbound granular materials. J. Transp. Eng.-ASCE 130, (2004) 22. Wolff, H., Visser, A.T.: Incorporating elasto-plasticity in granular layer pavement design. In P. I. Civil Eng.-Transp. 105, (1994)