Modelling the Moisture Dependent Permanent
Deformation Behavior of Unbound Granular Materials
Mohammad Shafiqur Rahman
1*and Sigurdur Erlingsson
1, 21 Swedish National Road and Transport Research Institute (VTI), Linköping, 581 95, Sweden 2University of Iceland, Reykjavik, 107, Iceland
[email protected], [email protected]
Abstract
The impact of moisture on the permanent deformation (PD) behavior of unbound granular materials (UGMs) was investigated based on multistage (MS) repeated load triaxial (RLT) tests. Two UGMs with different particle size distributions were tested for a range of moisture contents and the accumulation of permanent strains for the different moisture contents were modelled using a simple predictive model. Moisture was found to increase the accumulation of PD in the materials. Analyses of the variation of the parameters of the model with respect to moisture showed that it is possible to capture the moisture dependent PD behavior of the materials assuming a simple linear relationship between one of the parameters of the model and the moisture content.
Keywords: unbound granular materials, permanent deformation, moisture, repeated-load triaxial test, model
1
Introduction
The unbound granular materials (UGMs), used in the base and sub-base layers of flexible pavements experience cyclic stresses from the moving traffic load (Lekarp, 1999). The induced deformation in the UGM in response to each cycle of loading consists of two parts. The major part is resilient or recoverable deformation (RD) and a small part is plastic or permanent deformation (PD) (Lekarp, 1999). Albeit small for each cycle of loading, PD in UGMs may accumulate to significantly large value with repeated loading, leading to rutting and failure of the structure (Hornych and El Abd, 2004; Erlingsson, 2012). For structural design of pavements, it is essential to predict the performance of UGMs in selecting appropriate materials (based on availability and economic considerations) to control rutting. However, the performance of a certain UGM in a pavement structure is largely influenced by its moisture content among several factors (Lekarp, 1999; Erlingsson, 2010; Cary and Zapata, 2011). Thus the mathematical models used in the design process should be able to predict the performance of the UGMs considering the impact of moisture content that naturally varies with climatic conditions.
* Corresponding author
Volume 143, 2016, Pages 921–928
Advances in Transportation Geotechnics 3 . The 3rd International Conference on Transportation Geotechnics
(ICTG 2016)
The objective of this study was to model the influence of moisture on the PD behavior of UGMs based on multistage (MS) repeated-load triaxial (RLT) tests. Two crushed rock aggregates commonly used in pavement construction in Sweden were tested for a range of moisture contents. The results were modelled using a simple PD model. The sensitivities of the material parameters of the model to moisture were analyzed to capture the impact of moisture on PD.
2
Permanent Deformation Properties of UGMs
PD in UGMs accumulates with the number of load applications. The accumulation of PD in UGMs is dependent on stress levels, stress history, number of load cycles, degree of compaction, particle size distribution, moisture content and aggregate type (Lekarp, 1999). Based on the shakedown theory, Dawson and Wellner (1999) and Werkmeister et al. (2001) have identified that the accumulation of the PD in UGMs falls within the three shakedown ranges depending on stress levels (Figure 1). Range A occurs for relatively low stress levels when permanent strain accumulates up to a finite number of load applications after which the response becomes entirely resilient with no further permanent strain. For stress levels higher than this, Range B occurs where the accumulation of permanent strain continues at a constant rate (per cycle). When stress levels are even higher, Range C behavior is observed where the permanent strain accumulates at an increasing rate that may eventually lead to failure.
Figure 1: Different types of PD behavior, depending on stress level
For RLT tests, the boundaries of the different shakedown ranges may be defined using the following criteria (Werkmeister 2003, CEN 2004a):
3 3000 5000
10
045
.
0
)
ˆ
ˆ
(
H
pH
pu
Range A: 3 3000 5000 3 (ˆ ˆ ) 0.4 10 10 045 . 0 uH
pH
p u Range B: 3 3000 5000 ˆ ) 0.4 10 ˆ (H
pH
p ! u Range C: (1)where
H
ˆp3000 and Hˆp5000are accumulated permanent strains at 3000th and 5000th load cycles, respectively, in the RLT test.Several models are available in the literature to characterize the PD behavior of UGMs. The following model proposed by Rahman and Erlingsson (2015) was used for this study:
Number of load cycles, N Range C Range B Range AA Accum u lated perm anent str ain, εˆp
bS f p N aN fS
H
ˆ (2)where Hˆp
N is the accumulated permanent strain, N is the total number of load cycles, a and b are regression parameters associated with the material and the term Sf takes into account the effect of stress state in permanent deformation accumulation given as:
D ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § a a f p p p q S (3)
where p is the hydrostatic stress (one third of the sum of the principal stresses, θ), q is the deviator stress,
pa is the reference stress here taken equal to the atmospheric pressure 100 kPa and α is a parameter
obtained from regression analysis. The advantages of this model are that a) it contains relatively fewer numbers of material parameters, b) unlike some other models, it does not require the shear strength parameters and c) it directly utilizes applied stress level as a predicting variable. These simplify its application and it is convenient to use for tests with a series of moisture contents.
This model is used for MS RLT tests by applying the time hardening formulation proposed by Erlingsson and Rahman (2013). The basic principle of this approach is the calculation of Nieqwhich is the equivalent number of load cycles that would be required for any single stress path (i) of the MS RLT test to attain the same amount of strain that was accumulated in the specimen due to all the previous stress paths [from the first to the (i-1)th stress path] applied to it. Then the term N in Equation (2) is replaced with the expression (NNi1Nieq), where Ni1 is the total number of load cycles at the end of the (i-1)th stress path (Figure 2). If the calculated Nieq approaches infinity, it infers that no further permanent strain will occur for this stress path.
Figure 2: Demonstration of the time hardening approach
Thus, for MS RLT tests the accumulation of permanent strain during any stress path (i) is calculated using the modified form of Equation (2) as follows:
Ni-1 N2 N1 N N2 0 N N3 0 N N i0 N2eq N3eq
…
Nieq 11, 2, 3,...i = Stress paths
2
3
i
1 Number of load cycles, N
Accumulated per m an en t a xial strain , ε ˆp 0 εˆp1 εˆp2 εˆp3 εˆpi
eq bS f i i i pi N a(N N N ) f i(S ) ˆ ( ) 1
H
(4)where the suffix i refers to the ith stress path. eq i N is calculated as:
1 1 1 ) ( ˆ » » ¼ º « « ¬ ª f i i S b i f p eq i S a N H (5) where 1 ˆ i p
H is the accumulated permanent strain at the end of (i-1)th stress path. This modified version
of the model when calibrated using MS RLT tests yields values of the material parameters that are applicable for a large range of stress situations with better reliability while at the same time takes into account the effect of stress history.
3
Experimental Investigation
The RLT test was adopted for this study since it is a widely used and convenient approach. Despite some limitations it reveals a great deal of information regarding the material behavior. The RLT tests were carried out in accordance with the European standard EN-13286-7 (CEN, 2004a). The MS loading approach was preferred since it allows for a more comprehensive study of the PD behavior including the effect of stress history using a single test specimen with reduced effort. Cylindrical specimens of 150 mm in diameter and 300 mm in height with a range of moisture contents (w) were tested. The ranges of w were selected assuming the possible variation in the field. The tests were carried out applying a set of different stress paths according to the standard, referred to as ‘high stress level’ as presented in Table 1. Each of the stress path was applied for 10,000 cycles with a frequency of 10Hz with no rest period. The total number of load cycles applied during the tests were 280,000 (28 stress paths). Lesser number of load cycles had to be applied when the specimen experienced excessively large deformation. This resulted in the missing data in some of the latter figures. Further details of the RLT test setup can be found in Rahman (2015).
Sequence 1 Sequence 2 Sequence 3 Sequence 4 Sequence 5 Confining stress, σ3 kPa Deviator stress, σd kPa Confining stress, σ3 kPa Deviator stress, σd kPa Confining stress, σ3 kPa Deviator stress, σd kPa Confining stress, σ3 kPa Deviator stress, σd kPa Confining stress, σ3 kPa Deviator stress, σd kPa constant min max constant min max constant min max constant min max constant min max
20 0 50 45 0 100 70 0 120 100 0 200 150 0 200 20 0 80 45 0 180 70 0 240 100 0 300 150 0 300 20 0 110 45 0 240 70 0 320 100 0 400 150 0 400 20 0 140 45 0 300 70 0 400 100 0 500 150 0 500 20 0 170 45 0 360 70 0 480 100 0 600 150 0 600 20 0 200 45 0 420 70 0 560
Table 1: Stress levels used for the MS RLT tests (HSL from the European standard)
The UGMs used for this study were crushed rock aggregates (granite) obtained from two quarries in Skärlunda and Hallinden in Sweden. The grain size distributions are provided in Figure 3. The optimum moisture content (wopt = 6.5% for Skärlunda and 5.5% for Hallinden) and the maximum dry density (2220 kg/m3 for Skärlunda and 2075 kg/m3 for Hallinden) were determined using the modified Proctor
of the specimens were 97% of the modified Proctor dry density, prepared using a vibrocompactor. The tests were performed on identical specimens of each material with varying ws and all of the tests were replicated once again.
Figure 3: Particle size distributions
4
Results and Modelling
The accumulated permanent strains with N in the two UGMs with different ws during the MS RLT tests are shown in Figure 4. It can be seen that the permanent strain increased with the increase in w. The accumulation of permanent strain was modelled for each case using Equation 4 by optimizing the material parameters using a least square curve fitting method. For this study the following approach was adopted for the optimizations:
x Since during the MS RLT test, the value of Sf changes for each stress path, the calibration of the model was performed considering all of the stress paths at once that yielded average best fit values of the parameters.
x It was observed during several trials with free optimization of all the parameters that an average value of α = 0.75 (α varied from 0.72 to 0.78) yielded good results for all cases. x The value of b was found to be material specific and was not much affected by moisture.
Hence it was restricted to 0.063 (varied from 0.55 to 0.70) for Skärlunda and 0.065 (ranged from 0.52 to 0.73) for Hallinden.
x Then the only parameter that was freely optimized and was sensitive to w was a. The values of a obtained for the different ws of the two UGMs are plotted in Figure 5.
The reason for restricting the parameters was to minimize scatter and to look for some trend in variation of the parameters with respect to moisture. Even though the qualities of fits for the model were somewhat sacrificed this way, Figure 4 shows that these were within reasonable limits provided the scatter usually experienced with RLT tests on UGMs. The shakedown ranges for the permanent deformation curves for different stress paths were calculated using Equation (1). It was found that all the three shakedown ranges occurred during the tests and the model fitted quite well to all of these ranges. Examining Figure 5, the parameter a was assumed as a linear function of w and the best fit linear equations are shown in this figure. Thus it appears that the variability of accumulated permanent strain with respect to w may
0 10 20 30 40 50 60 70 80 90 100 0.0625 0.125 0.25 0.5 1 2 4 8 16 32 P as si ng [% b y w ei ght ] Sieve size [mm] Skärlunda Hallinden
be modelled satisfactorily by simply expressing the parameter a as a linear function of w in Equation 4 within the range of w used in this study. In general form this may be expressed as:
>
@
f i S b eq i i pi N a a w (N N N ) f i(S ) ˆ ( ) 1 2 1H
(6)where a1 and a2 are regression parameters evaluated by performing a few RLT tests with different
moisture contents. Here, the values of a1 and a2 are -0.004 and 0.0047 respectively for the Skärlunda
aggregates and 0.0009 and 0.0004 respectively for the Hallinden aggregates.
It should be noted in Figure 4 that the UGM from Skärlunda with higher amount of fines were more affected by moisture compared to the UGM from Hallinden which has relatively low amount of fines. Thus the a versus w plot in Figure 5 has a steeper slope (higher a2 value) for the Skärlunda aggregates.
Figure 4: Accumulated permanent strain for different ws (measured and modelled): (a) Skärlunda, (b)
Hallinden 0 0.02 0.04 0.06 0.08 0.1 0.12 0 60000 120000 180000 240000 300000 A ccu m ul at ed pe rm an en t s tr ai n, ε ˆp
Number of load cycles, N
w = 1% w = 2% w = 3.5% w = 5% w = 6% Model Sequence 1 Sequence 2 Sequence 3 Sequence 4 Sequence 5
(a) Skärlunda 0 0.02 0.04 0.06 0.08 0.1 0.12 0 60000 120000 180000 240000 300000 A ccu m ul at ed pe rm an en t st ra in , εˆp
Number of load cycles, N
w = 1% w = 4% w = 5.5% w = 6.5 Model
Sequence 1 Sequence 2 Sequence 3 Sequence 4 Sequence 5
Figure 5: Parameter a as a function of w
5
Conclusions
The accumulation of permanent deformation in UGMs in pavement structures is significantly affected by the seasonal variation of moisture. The objective of this work was to mathematically characterize this behavior which may enhance the design and maintenance of pavements by controlling the contribution of UGMs to rutting. The results obtained by conducting MS RLT tests in the laboratory corroborated the general consensus that moisture increases the accumulation of PD in UGMs (Lekarp, 1999; Erlingsson, 2010). The model used in this study successfully captured the PD behavior of the two crushed rock UGMs with a range of moisture contents. The MS RLT tests allowed for a comprehensive study of the material behavior for a large range of stress conditions including the effect of stress history. The model was convenient to use since for its adoption, it is not necessary to conduct separate static failure tests to determine the shear strength properties of the specimen and the number of regression parameters in the model are minimal. From the sensitivity study of the parameters, it appeared that the moisture dependent PD behavior of the UGMs may be modelled reasonably well with this model if the parameter a is expressed as a linear function of w (within the range of w used in this study). However, this should be further validated by performing tests on identical specimens with a series of moisture contents but applying different combination of stress levels (for example the ‘low stress level’ from the European standard) other than that used for the calibrations. Additionally, the optimization approach should be further explored with free optimization of other parameters (for example α) including several combinations of the values of the parameters which may yield satisfactory quality of fit for the model.
This study was limited to two crushed rock granite aggregates. Hence it is necessary to extend the study further to different types of UGMs. Replicating the tests are recommended to reduce the uncertainties involved in RLT tests on UGMs. Besides moisture content and stress conditions, the PD behavior of UGMs is influenced by factors such as the amount of fines and the initial degree of compaction. Thus the model should be improved to take these factors into account as well. For field application it may require further modification and validation of the model that was developed based on RLT tests which involves some simplifications. Thus, with further verification and modification, the model may be used for mechanistic-empirical design of pavements.
a = 0.0047w - 0.004 a = 0.0004w + 0.0009 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0 1 2 3 4 5 6 7 8 9 10 Pa ra m et er , a [-] Moisture content, w [%] Skärlunda Hallinden
Acknowledgements
This work was sponsored by the Swedish Transport Administration (Trafikverket).
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