^{1}

^{2}

^{*}

^{2}

^{3}

The generalized
constitutive model relating the resilient modulus (M_{R}) of flexible pavement layer materials to stress state,
adopted by the Mechanistic-Empirical Pavement Design Guide (MEPDG),
contains a set of constants known as k-values (k_{1}, k_{2}, and k_{3}) which are associated with the physical state of the layer materials. In Ghana, natural gravels constitute the predominant and sometimes the sole layer materials for most flexible pavements yet representative k-values of gravel materials
,
have not been determined to permit full application and implementation of the mechanistic-empirical design concept to pavements involving such materials. In this study, k-values characterising typical natural quartzitic gravels used for road construction in the country were derived by regression techniques from M_{R} values determined using laboratory repeated load triaxial test.
Using multiple linear regression technique,
correlation relationships were then developed between the k-values and the physical properties of the gravels, namely, percentages of materials passing the 9.5
mm (P_{9.5}) and 2.0
mm (P_{2.0}) sieves, liquid limit (LL), maximum dry density (ρ_{dmax}), and optimum moisture content (w_{opt}).
The regression analysis returned k_{1} values which ranged between 441 and 958 with a mean of 516; k_{2}
_{ }
which varied between 0.0636 and 0.2168 with a mean value of 0.1216;
and,
k
_{3}
values which ranged between 0.1257 and 3.1590 with a mean value of 1.723. Contrary to what is mostly reported in literature, the analysis returned positive k_{3} values for all but one gravel material, suggesting stress hardening under octahedral shear stress for those materials. While an expanded sample base is required to fully characterize the whole gamut of natural gravels used in pavement construction in the country, this study on limited quartzitic gravel samples has given a good indication of strong linear correlations between the k-values and the index properties of the gravels, to permit estimates of the constants for such gravels be made where capability and opportunity for conducting resilient modulus tests do not exist.
However, further work is recommended to fully characterise the exact nature of k_{3} values for quartzitic gravels in the country.

In the mechanistic-empirical (M-E) design/analysis of multi-layer flexible pavement structures, the resilient response of layer materials to repeated traffic loading is quantified by the resilient modulus (M_{R}) parameter. M_{R} as a material stiffness parameter was initially introduced by Seed et al. [_{R} values may be determined directly in the laboratory through repeated load triaxial testing and indirectly through correlation with empirical soil strength parameters (California Bearing Ratio, Soil Support Value, and Resistance Value) or by back-calculating from deflections measured with falling weight deflectometer. But because M_{R} is a basic mechanistic material property, its estimation through correlations with empirical soil strength parameters is not considered appropriate as the parameters do not reflect the conditions in a pavement under repeated traffic loading [

In empirical pavement design approaches, representative resilient modulus values are assigned to layer materials irrespective of position in the pavement-subgrade system. This is considered a limitation since the resilient modulus of unbound soil/aggregate layers depends on stress state and is known to increase with increasing bulk stress and confining stress [

M R = σ d ε r (1)

where,

M_{R} = resilient modulus;

σ_{d} = repeated deviator stress (σ_{1} − σ_{3});

ε_{r} = recoverable axial strain

The basic definition of M_{R} given by Equation (1) has formed the basis of several constitutive model proposals for determining resilient modulus including those of Dunlap [

M R = k 1 P a ( θ P a ) k 2 ( τ o c t P a + 1 ) k 3 (2)

where,

τ o c t = 1 3 ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 (octahedral stress)

P_{a} = atmospheric pressure

σ_{1} = major principal stress

σ_{2} = intermediate principal stress (=σ_{3} in triaxial test)

σ_{3} = minor principal stress (confining pressure in triaxial test)

θ = bulk stress (σ_{1} + σ_{2} + σ_{3})

k_{1}, k_{2} and k_{3} = constants (k-values) related to material’s physical state

The limitation inherent in empirical designs due to failure to relate M_{R} to the stress state of layer materials is addressed in the Mechanistic-Empirical Pavement Design Guide (MEPDG) [_{R} is defined by the state of stress of the layer materials in the pavement-subgrade system and the physical state of the materials through the material constants (k_{1}, k_{2}, k_{3}) known as k-values. For Level 1 analysis which has the highest design reliability in the Guide [_{R} may be estimated through correlations with index soil properties [

Unlike M_{R}, the k-values are dependent only on the physical state of the material without any stress influence. The constant k_{1} which can only be positive is related to material strength and tends to be higher for materials containing angular particles and less fines than those containing higher fines and rounded particles [_{2} should be positive as increasing bulk stress causes stiffening effect on the material and, hence, increasing modulus while k_{3} should be negative since increasing shear stress produces a softening effect on the material resulting in lower M_{R} [

For fine-grained soils, the resilient modulus decreases with increase in deviator stress at a given confining stress in a behaviour known as stress-softening whereas for granular soils, the parameter increases with increase in deviator stress (stress-hardening behaviour) [_{R} include moisture which lowers M_{R} when it is above the optimum content [_{R} when it is angular to sub-angular but lower values when rounded to sub-rounded [

The MEPDG approach to flexible pavement design has a lot of calibration protocols and performance predictions which require local input data for traffic, climate, subgrade and material properties [_{R} tests become important in the calibration process as they generate the set of data required for the determination of the k-values associated with the compacted layer materials/soils. This study forms part of preliminary efforts to establish representative k-values for natural gravels used in the country for flexible pavement construction.

The Ministry of Roads and Highway’s Standard Specification for Road and Bridge Works [

In terms of suitability as flexible pavement layer materials, natural gravels are classified into grades as G30 (CBR ≥ 30); G40 (CBR ≥ 40); G60 (CBR ≥ 60); and G80 (CBR ≥ 80) using CBR, particle gradation, Grading Modulus, Atterberg Limits, Plasticity Modulus, and 10% Fines Value. Whereas G30 and G40 grades are used for sub-base, G60 and G80 are used for base construction [

Natural gravels used for road construction are mainly of two types, lateritic and quartzitic; this study focused only on the latter. The samples were collected from gravel pits distributed in five regions in the country (

Grain size analysis was conducted on each of the gravel samples in accordance with Ghana Highway Authority (GHA) S7 Standard Method [

The M_{R }test was conducted using the Industrial Process Controls’ Universal Materials Testing Apparatus (Plate 1) equipped with a desktop computer, environmental chamber enclosing the load cell and the triaxial chamber, control and data acquisition system, air compressor, and air dryer. Gravel materials passing the 19 mm sieve were used for specimen preparation and compacted in a 100 mm

Sample Id. | Location of Depoist | Region |
---|---|---|

NNR | Ntereso | Northern |

SNR | Sankunyilli | Northern |

OAR | Ofoase | Ashanti |

AAR | Anyanso | Ashanti |

SER | Suhum | Eastern |

ECR | Ekumadze | Central |

MGR | Mobole | Greater Accra |

diameter × 200 mm high mould to the maximum dry density and optimum moisture content from the compaction characteristics of the soil. After the compaction, the specimens were removed from the mould and covered with rubber membrane for the resilient modulus test. Three specimens were prepared for each soil sample and tested for Resilient Modulus in accordance with AASHTO T307-99 test protocol [

Equation (2) which relates M_{R} to the stress state and physical state of the layer material in the repeated load triaxial test was modified to the following form;

Plate 1. Resilient modulus test set-up.

log ( M R P a ) = log k 1 + k 2 log ( θ P a ) + k 3 ( τ o c t P a + 1 ) (3)

Under laboratory triaxial loading conditions, τ_{oct} reduces to 2 3 ( σ 1 − σ 3 ) .

The logarithmic-transformed M_{R} values obtained from the repeated load triaxial test were regressed on the logarithmic-transformed test stresses to obtain the k-values in Equation (3). Correlation relationships between the k-values and the physical properties of the gravels were then developed using multiple linear regression technique. Based on review of literature [_{9.5}) and 2.0 mm (P_{2.0}) sieves, the liquid limit (LL), the maximum dry density (ρ_{dmax}) and the optimum moisture content (w_{opt}).

The natural gravels investigated were quartzitic in nature with colour ranging from light brown to yellowish brown. Visual inspection of the coarse fractions of the gravels from Ntereso (NNR), Suhum (SER), and Ekumadze (CER)revealed sub-angular particles with rough texture whereas fractions of the samples from Sankunyilli (SNR), Ofoase (OAR), Anyanso (AAR), and Mobole (MGR) were rounded in shape and smooth textured, suggesting some degree of transportation in the geologic past. The index geotechnical properties of the samples are summarised in

NNR | SNR | OAR | AAR | SER | ECR | MGR | |
---|---|---|---|---|---|---|---|

Classification | |||||||

AASHTO(Group Index) | A-2-4(0) | A-2-4(0) | A-2-4(0) | A-2-4(0) | A-2-4(0) | A-2-4(0) | A-2-4(0) |

Group Name | Clayey Sandy Gravel | Clayey Sandy Gravel | Clayey Sandy Gravel | Sandy Clayey Gravel | Clayey Sandy Gravel | Clayey Sandy Gravel | Clayey Sandy Gravel |

Grain Size Distribution | |||||||

Gravel (%) | 71 | 56 | 76 | 72 | 62 | 77 | 72 |

Sand (%) | 18 | 26 | 14 | 13 | 22 | 15 | 22 |

Silt-Clay | 11 | 18 | 10 | 15 | 16 | 8 | 6 |

Atterberg Limits | |||||||

Liquid Limit, LL (%) | 28 | 22 | 28 | 25 | 29 | 28 | 18 |

Plastic Limit, PL (%) | 19 | 15 | 19 | 16 | 19 | 19 | 11 |

Plasticity Index, PI (%) | 9 | 7 | 9 | 9 | 10 | 9 | 7 |

Compaction Characteristics | |||||||

Maximum Dry Density, ρ_{d} (kg/m^{3}) | 2152 | 2300 | 2216 | 2211 | 2038 | 2220 | 2305 |

Optimum Moisture Content, w_{opt} (%) | 4.8 | 4.2 | 5.6 | 9.0 | 9.8 | 4.9 | 5.0 |

above the A-Line on the Cassagrande plasticity chart, the fines were predominantly clay in content. Based on AASHTO Classification, all the samples classified as A-2-4(0) and were essentially clayey sandy gravel except that from Anyanso (AAR) which was a sandy clayey gravel.

On the basis of particle gradation in terms of MRH Standards [

The compaction characteristics of the gravels showed maximum dry densities ranging between 2038 kg/m^{3} and 2305 kg/m^{3} with an average of 2206 kg/m^{3}. Most samples had optimum water contents ranging between 4% and 6% with an average of 4.9%; only two samples (AAR and SER) had optimum water contents about twice the average of those of the rest of the samples. Perhaps the higher than average silt-clay contents of the two samples could partly be the reason for the relatively high optimum water contents although sight is not lost of the fact that Sample SNR with similar percentage of silt-clay content, but substantially

lower gravel fraction, belonged to the low optimum water content sample cohort.

Resilient modulus test results in the repeated load triaxial tests have been summarized in

As expected, most of the granular materials exhibited increasing modulus with increasing bulk stress for a given confining stress (typified by Figures 5-7 for samples NNR, OAR, and SNR, respectively). The only exception to this stress-hardening behaviour was the gravel from Anyanso (Sample AAR) which exhibited both stress hardening and stress softening characteristics (

In the case of variation of M_{R} with octahedral shear stress, contrary to what is touted in literature, six out of the seven gravel samples exhibited increasing modulus with increasing octahedral shear stress at a given confining stress (typified by

_{1} values ranged between 441 and 958 with a mean of 516; k_{2} varied between 0.0636 and 0.2168 with a mean value of 0.1216 while k_{3}, which is related to the octahedral shear stress, ranged between 0.1257 and 3.1590 with a mean value of 1.723.

Test Sequence | Conf. Pressure (kPa) | NNR | SNR | OAR | AAR | SER | ECR | MGR | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | Dev. Stress (kPa) | M_{R} (MPa) | ||

1 | 20.7 | 10.2 | 54.3 | 10.5 | 49.6 | 11.0 | 51.8 | 9.3 | 62.3 | 9.6 | 84.4 | 9.8 | 47.1 | 8.0 | 56.8 |

2 | 20.7 | 21.1 | 67.9 | 21.4 | 57.0 | 22.0 | 58.5 | 20.2 | 66.4 | 20.3 | 96.0 | 20.4 | 58.6 | 16.8 | 60.8 |

3 | 20.7 | 32.3 | 78.3 | 32.5 | 66.2 | 33.4 | 67.8 | 31.1 | 66.4 | 31.1 | 106.0 | 31.0 | 71.3 | 25.9 | 71.7 |

4 | 34.5 | 17.3 | 62.1 | 17.8 | 58.7 | 18.4 | 60.6 | 16.4 | 70.0 | 16.6 | 99.9 | 16.6 | 58.3 | 13.2 | 60.2 |

4 | 34.5 | 35.5 | 84.1 | 36.1 | 71.3 | 36.9 | 73.8 | 34.5 | 70.2 | 34.3 | 109.7 | 34.2 | 76.2 | 28.6 | 69.7 |

6 | 34.5 | 52.4 | 97.8 | 53.8 | 86.1 | 55.5 | 84.3 | 53.2 | 72.0 | 52.0 | 118.8 | 52.9 | 89.1 | 44.4 | 84.5 |

7 | 68.9 | 34.6 | 81.4 | 35.7 | 76.3 | 36.8 | 77.8 | 34.6 | 78.6 | 33.4 | 112.6 | 34.5 | 77.4 | 27.8 | 70.9 |

8 | 68.9 | 69.0 | 114.0 | 71.1 | 102.8 | 74.2 | 99.2 | 71.9 | 78.9 | 69.3 | 123.8 | 73.5 | 99.6 | 59.9 | 91.6 |

9 | 68.9 | 101.2 | 148.1 | 107.0 | 129.1 | 111.5 | 118.7 | 111.2 | 69.2 | 105.9 | 140.2 | 110.6 | 97.8 | 93.2 | 110.9 |

10 | 103.4 | 33.6 | 78.1 | 35.6 | 81.9 | 36.7 | 82.3 | 36.4 | 72.1 | 31.4 | 132.9 | 36.7 | 78.6 | 27.0 | 72.4 |

11 | 103.4 | 51.0 | 101.1 | 53.8 | 95.9 | 55.4 | 93.5 | 54.8 | 73.1 | 49.1 | 138.4 | 55.0 | 83.6 | 41.8 | 82.5 |

12 | 103.4 | 99.7 | 153.1 | 106.9 | 133.6 | 110.9 | 125.9 | 111.2 | 106.7 | 100.2 | 148.0 | 111.2 | 135.8 | 90.2 | 111.3 |

13 | 137.9 | 48.9 | 102.6 | 53.1 | 99.6 | 55.4 | 98.7 | 54.5 | 111.8 | 44.7 | 145.6 | 55.4 | 103.2 | 40.9 | 82.9 |

14 | 137.9 | 65.7 | 121.3 | 71.3 | 111.7 | 74.0 | 110.7 | 73.4 | 111.3 | 62.5 | 147.6 | 74.2 | 117.8 | 56.7 | 91.4 |

15 | 137.9 | 133.2 | 177.8 | 144.3 | 155.0 | 148.0 | 139.1 | 148.2 | 83.2 | 138.0 | 153.2 | - | - | 132.0 | 114.3 |

Sample ID | Average k-values | R^{2} | ||
---|---|---|---|---|

k_{1} | k_{2} | k_{3} | ||

NNR | 531 | 0.0785 | 2.3883 | 0.99 |

SNR | 503 | 0.1820 | 1.7182 | 0.99 |

OAR | 535 | 0.1843 | 1.3425 | 0.98 |

AAR | 676 | 0.1016 | -0.1257 | 0.56 |

SER | 958 | 0.2168 | 0.2772 | 0.94 |

ECR | 441 | 0.0998 | 3.1590 | 0.97 |

MGR | 568 | 0.0636 | 1.4810 | 0.94 |

Contrary to what have been reported in literature, the regression analysis returned positive k_{3} values instead of negative values for six out of the seven samples. The resilience response of a material to increase in octahedral shear stress determines the positivity or negativity of its k_{3} value. As increasing shear stress is expected to lead to the softening of materials, and hence a decreasing resilient modulus, k_{3} should be negative, but this phenomenon was not exhibited by the quartzitic gravels as they showed stress-hardening behaviour with increase in octahedral shear stress (see for example ^{2}) associated with the regressions analysis exceeded 0.90 for the investigated materials except for the sample from Anyanso (AAR) for which the value was a low 0.56 (see

The MEPDG recommends that test results and equipment should be checked for possible errors and/or test specimen disruption if R^{2} for a particular test specimen is less than 0.90 [^{2} value was rather the gravel sample for which the analysis returned negative k_{3} value in accordance with expectation based on literature. For the other samples, the higher R^{2} values from the regression analysis, therefore, appeared to rule out any test errors during the triaxial test in spite of their k_{3} values being positive.

Given the stress state in the repeated load triaxial test for which the intermediate principal stress (σ_{2}) is equal to the minor principal stress or confining pressure (σ_{3}), the generalized M_{R} constitutive model (Equation (2)) may be re-written solely as a function of deviator stress (σ_{d}) and confining pressure as follows:

M R = k 1 P a ( σ d + 3 σ 3 P a ) k 2 ( 2 σ d 3 P a + 1 ) k 3 (4)

It is reported that for granular materials, the resilient modulus increases with increasing confining pressure [_{3} parameter: 1) if k_{3} were to be positive, the observations above per references [_{3} were to be negative, the observation would also hold if and only if the increase in the value of M_{R} due to the k_{2} term (bulk stress term) more than offsets the decrease caused by the k_{3} term (octahedral shear stress term). It stands to reason, therefore, that in this case, M_{R} would only begin to exhibit decreasing values beyond the point where the decrease in its value caused by the k_{3} term overrode the increase caused by the k_{2} term. In either of the two possibilities considered above, stress-hardening behaviour of granular materials should still be anticipated only up to some threshold stress levels as confining and deviator stresses increased during the triaxial testing. This probably could be the behaviour being exhibited by Sample AAR (_{3} values can be made other than a recommendation to carry out more extensive studies and expand the gravel sample base.

The following correlation equations were the outcome of the regression analysis involving k-values from the resilient modulus test and the index physical properties, namely, percent finer than the 9.5 mm and 2.0 mm sieves, liquid limit, maximum dry density, and optimum moisture content of the samples.

k 1 = 2531.94 − 3.11 P 9.5 + 5.50 P 2.0 − 21.09 L L + 0.81 ρ d m a x + 71.30 w o p t , R^{2} = 0.95 (5)

k 2 = − 3.54 − 0.005 P 9.5 + 0.008 P 2.0 + 0.014 L L + 0.0014 ρ d m a x + 0.05 w o p t , R^{2} = 0.95 (6)

k 3 = 8.10 + 0.12 P 9.5 − 0.12 P 2.0 + 0.227 L L − 0.0055 ρ d m a x − 0.72 w o p t , R^{2} = 0.95 (7)

where,

k_{1}, k_{2} and k_{3} = k-values of gravel

P_{9.5} = percentage of gravel material passing 9.5 mm sieve

P_{2.0} = percentage of gravel material passing 2.0 mm sieve

LL = liquid limit

ρ_{dmax} = maximum dry density

w_{opt} = optimum moisture content (%)

The high R^{2} values obtained for the regression suggest a strong correlation between the k-values and the index properties of the gravel materials. Even though the recommendation is to evaluate k-values from M_{R} tests, the above correlations become especially useful when capability and opportunity for repeated load tests do not exist.

This study set out to determine the resilient modulus of quartzitic natural gravels using repeated load triaxial test and to evaluate the k-values (k_{1}, k_{2}, k_{3}) of the materials thereof. The quartzitic gravels used in the study came from limited borrow pit sources and did not cover the whole range of natural gravel types generally used for flexible pavement construction in the country which also include lateritic gravels. The k-values from the resilient modulus tests were used to develop correlation equations involving the index physical properties of the gravels to help predict resilient modulus when the materials are used as pavement layer materials. The regression analysis returned k_{1} values which ranged between 441 and 958 with a mean of 516, k_{2} values which ranged between 0.0636 and 0.2168 with a mean of 0.1216 and k_{3} values which ranged between 0.1257 and 3.1590 with a mean of 1.723. The positive nature of the constant k_{3} is in contrast to the mostly negative values reported in literature on the parameter. Nonetheless, all the k-values correlated highly with the following index properties of the gravels: the percentages of gravel material passing the 9.5 mm and 2.0 mm sieves, liquid limit, maximum dry density and optimum moisture content. The gravels exhibited stress hardening behaviour for the most part since the resilient moduli from the triaxial tests increased with increasing confining pressure, bulk stress, and octahedral shear stress. Even though the recommendation is to evaluate k-values from resilient modulus tests, the correlations developed in this study, despite the limited sample base, may permit useful estimates of the constants to be made for such gravels under any stress state and physical condition, especially when capability and opportunity for conducting resilient modulus tests do not exist. Further work, however, is recommended to cover the entire range of natural gravels deposits in the country and investigate into detail, the nature of the constant k_{3} with increasing octahedral shear stress.

The tests in this study were carried out at the Central Laboratory, Materials Division of the Ghana Highway Authority, Ministry of Roads and Highways, Accra. The support of the Directorate of the Division and the assistance of the Technical Staff of the Laboratory are gratefully acknowledged.

Obeng, E., Tuffour, Y.A., Obeng, D.A. and Koranteng-Yorke, B. (2017) Preliminary k-Values of Unbound Natural Quartzitic Gravels for Mechanistic-Empirical Pavement Design. Open Journal of Civil Engineering, 7, 509-526. https://doi.org/10.4236/ojce.2017.74034