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 (Figure 2). Sampling locations and sample identification are detailed in Table 1.

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 [13] [14] [15] , and for proper material and physical state representation of the compacted gravels, the index properties selected as the independent variables for the correlation relationships were the percentages of materials passing the 9.5 mm (P_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 Table 2. Because the fines portions of all the samples plotted

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. Figure 3 shows the particle size distribution curves of the gravel materials superimposed on the grading envelope for G80 and G60 base-grade gravel materials [23] .

On the basis of particle gradation in terms of MRH Standards [23] , almost all the gravel materials appeared to meet the requirements for G80 gravel but failed one or more other G80 requirements such as Grading Modulus, Atterberg Limits and Plasticity Modulus and hence were all clearly G60 base materials except gravel material MGR which was of G80 grade. However, according to the AASHTO T307-99 resilient modulus test protocol [27] , all the materials classified as Type 1 unbound natural gravel since the proportions passing the 2 mm and 0.075 mm sieves were less than 70% and 20%, respectively, with plasticity indices being less than 10%.

The compaction characteristics of the gravels showed maximum dry densities ranging between 2038 kg/m³ and 2305 kg/m³ with an average of 2206 kg/m³. 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 Table 3. As shown in Figure 4, the modulus increased with confining pressure because for granular soils, confinement is known to increase interlock between the individual aggregate particles within the compacted matrix, a behaviour referred to in literature as stress-hardening.

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 (Figure 8).

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 Figure 9 and Figure 10 for samples NNR and MGR, respectively); the only exception again being Sample AAR which generally exhibited little change in modulus initially then decreasing values with further increase in shear stress (Figure 11).

Table 4 presents the summary of the k-values derived from the regression analysis of the resilient modulus values for the gravel samples. As seen in the table, k₁ values ranged between 441 and 958 with a mean of 516; k₂ varied between 0.0636 and 0.2168 with a mean value of 0.1216 while k₃, which is related to the octahedral shear stress, ranged between 0.1257 and 3.1590 with a mean value of 1.723.

Contrary to what have been reported in literature, the regression analysis returned positive k₃ 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 k₃ value. As increasing shear stress is expected to lead to the softening of materials, and hence a decreasing resilient modulus, k₃ 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 Figure 9 and Figure 10). The only exception to this behaviour was sample AAR from Anyanso which initially hardened at low shear stress levels but then softened at increased shear stress levels (Figure 11). It is not exactly clear what accounted for this unexpected behaviour but the coefficient of multiple determination (R²) 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 Table 4).

The MEPDG recommends that test results and equipment should be checked for possible errors and/or test specimen disruption if R² for a particular test specimen is less than 0.90 [9] . Because this condition did not prevail for practically all the soils except one, it is difficult to ascribe the sample behaviour under increasing shear stress to test/equipment error. And what was even more surprising is the fact that the sample which probably could be associated with test error because of a low R² value was rather the gravel sample for which the analysis returned negative k₃ value in accordance with expectation based on literature. For the other samples, the higher R² values from the regression analysis, therefore, appeared to rule out any test errors during the triaxial test in spite of their k₃ values being positive.

Given the stress state in the repeated load triaxial test for which the intermediate principal stress (σ₂) is equal to the minor principal stress or confining pressure (σ₃), 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 [28] and deviator stress [3] [29] . On the strength of those observations, and with reference to Equation (3), two possibilities could exist regarding the nature of the k₃ parameter: 1) if k₃ were to be positive, the observations above per references [3] [28] [29] would still hold but perhaps up to some stress thresholds as stress-hardening could not be perpetuated ad infinitum with increasing deviator stress and confining pressure; 2) if k₃ were to be negative, the observation would also hold if and only if the increase in the value of M_R due to the k₂ term (bulk stress term) more than offsets the decrease caused by the k₃ 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 k₃ term overrode the increase caused by the k₂ 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 (Figure 11). At this stage, however, no definite conclusion regarding the gravels’ k₃ 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 , R² = 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 , R² = 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 , R² = 0.95 (7)

where,

k₁, k₂ and k₃ = 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 R² 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.