Construction and Building Materials 159 (2018) 440–450

Contents lists available at ScienceDirect

Construction and Building Materials

journal homepage: www.elsevier.com/locate/conbuildmat

Correlation between resilient modulus (MR) and constrained modulus

(MC) values of granular materials
Muhammad Arshad
Department of Geological Engineering, University of Engineering & Technology, G.T Road, Lahore, Pakistan

h i g h l i g h t s

A new regression model for the estimation of resilient modulus value for granular materials.

Potentials use of reclaimed asphalt pavement.

Statistical analysis for the strength and significance of the correlation.

a r t i c l e i n f o a b s t r a c t

Article history: The resilient modulus (MR) value of unbound aggregates has been widely accepted as the principal
Received 2 July 2017 mechanical property required in the mechanistic-empirical design/analysis of pavement structures.
Received in revised form 15 September Resilient modulus measured through repeated triaxial load test is highly desirable to characterise gran-
2017
ular materials. However, because of the complexities encountered with this test, other laboratory tests
Accepted 8 October 2017
would be desirable if a reliable correlation could be established.
Reclaimed asphalt pavement (RAP) and recycled concrete aggregate (RCA) have potential to be used as
base and subbase material in substantial percentage to achieve economy and to minimise the undesirable
Keywords:
Pavement structure
environmental effects linked to the use of virgin aggregates. This paper presents a correlation developed
Recycled material between MR value and constrained modulus (MC) value on the basis of experimental results obtained for
Material properties 32 reconstituted granular samples containing virgin aggregates, RAP and RCA materials. Proposed model
Empirical correlation considers the effect of stress state through the experimentally determined MC value while model param-
Laboratory experimentation eters are correlated to the percentage of RAP materials in the reconstituted blends. Statistical analysis of
the suggested correlation by t-test demonstrates encouraging results in terms of its strength and
significance.
Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Furthermore, experimental investigations in recent years have

demonstrated the influence of the metric section (w) on the MR
The reliability of pavement design depends on the success in value of various pavement materials [5–13]. However, an inclusive
determining the material properties which are used in the perfor- state-of-the-art review of the equations for interpreting and pre-
mance prediction models. Resilient modulus (MR) which is defined dicting the variation of the MR value with respect to the metric sec-
as the ratio of cyclic deviator stress (rd) to resilient strain (er ) is the tion for pavement materials remains outstanding and hence an
most critical material property, among others that are used in the open question [14].
mechanistic pavement design and evaluation [1]. Depending upon the level of confidence, three different
Factors affecting the MR value for base/subbase layer of the approaches have been used to figure out the MR value for base/sub-
pavement, primarily include the soil type, physical properties of base layer of the pavements. More specifically: Level 1, conducting
the soil such as gradation characteristics, moisture contents, den- repeated-load triaxial (RLT) laboratory tests; Level 2, estimation
sity and stress conditions in addition to stress level [2–4]. A sum- using correlations developed on the basis of strength/stiffness/-
mary of such influencing factors is given in Table 1. physical properties of soils and/or back-calculation from the
in situ test measurements; and Level 3, using the representative
MR value from the existing database prepared on the basis of past
experience and soil type or classification [15–17]. AASHTO (Amer-
E-mail address: arshadm@uet.edu.pk ican Association of State Highway and Transport Officials) provides

https://doi.org/10.1016/j.conbuildmat.2017.10.047
0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.
 M. Arshad / Construction and Building Materials 159 (2018) 440–450 441

Table 1 As for as RAP is concerned, it consists of coarse and fine aggre-

Summary of the factors influencing the resilient modulus value. gates and bitumen refer to a collection of loose materials from
Factor Resilient modulus Importance asphalt pavements. All over the world major application of RAP
Shear stress level " ; High material is in the hot mix asphalt. For instance, in Europe, 47% of
Bulk stress " " High the available RAP was used in hot mix asphalt applications as per
Deviatoric stress " no effect Low data provided by European Asphalt Pavement Association (EAPA)
Number of load repetition " "; Low/high [30]. While for the USA, statistics published by National Asphalt
Density of sample " " Low
Fine aggregate content " not clear Medium
Pavement Association (NAPA) and documented by Zaumanis
Medium aggregate content " " Medium et al., [31] approximates a total of 70 million tonnes of RAP pro-
Coarse aggregate content " " Medium duced in 2011 and 84% of which were used in asphalt applications.
RAP has been used in the surface layer of the hot mix asphalt pave-
ments in varying percentages with the fresh material that reached
in some cases up to 80% [32]. Nevertheless, range of 20 to 50% has
standard test procedures for the determination of resilient modu- been suggested in most studies [33–35]. This situation mandate for
lus using the repeated load triaxial test, which include AASHTO T the RAP/RCA to be used as granular base/subbase materials in addi-
292, AASHTO T 294 and AASHTO T 307. There were some compli- tion to using them as a replacement for fresh granular material in
cations and concerns associated with some procedures, which hot mix asphalt applications.
were improved with time. The current protocol for determination In recent years, many researchers have made efforts through
of resilient modulus of soils and aggregate materials is the AASHTO commendable studies to incorporate RAP into pavement base or
T 307 which in fact has been evolved from the Long-Term Pave- subbase applications [18–21,36–41]. The application of RAP and
ment Performance (LTPP) protocol P46. Although the determina- RCA in pavement base/subbase as an aggregate has a long history.
tion of MR value through laboratory methods is highly desirable However, their use has been limited due to the lack of reported lab-
but this requires a higher level of technical expertise and more oratory and field-testing results [42]. More specifically, structural
sophisticated equipment hence leading to higher cost and more behaviour of the blended samples containing RAP/RCA materials
time for the design/construction of pavement project. Conse- under traffic loading is yet need to be explored, further.
quently, under this situation use of prediction models for the esti-
mation of MR value may prove to be a reliable alternative.
The objective of this paper is to develop a resilient modulus pre- 3. Existing correlations for the estimation of MR-Value
diction model for the materials containing RAP/RCA as major con-
stituent. For the proposed prediction model, the constitutive From the existing literature, the correlations for the estimation
behaviour of the remoulded samples under different stress condi- of MR value for granular material can be categorically divided into
tion was captured through one dimensional cyclic constrained four types as discussed below.
modulus tests which were conducted through a specially designed
apparatus capable of accommodating soil particle up to 50 mm in
size. Regression constants of the model were correlated to the per- 3.1. Regression equations for resilient modulus values based on single
centage of RAP contents in the blended samples. Statistical analysis soil parameter
reveals that the proposed regression model has moderate to strong
prediction reliability for the MR values. To the author’s best knowl- These prediction models have been developed using strength or
edge, this is the first study of its kind about the development of stiffness characteristics of the unbound pavement material
correlation for the estimation of MR value on the basis of MC value including CBR values [43–48]; resistance, resistance value (R-value)
and the percentage of RAP contents in the blended samples. [49–53]; unconfined compressive strength [54] and undrained shear
Additionally, a review on the use of RAP/RCA material in strength [55].
unbound pavement layer and various existing correlations/models In addition to the laboratory-based correlations, there are many
for the estimation of MR value has been presented in the next correlations (for the estimation of MR value) which are based on
sections. the data obtained during in-situ field testing. Most commonly used
instrument/devices for this purpose includes cone penetrometer
[56], dynamic cone penetrometer [57–59], light falling weight
2. Use of RAP/RCA material in different pavement layers
deflectometer [60], dilatometer, geogauge [61].
The fundamental problem with empirical relationships devel-
RAP [18–20] and RCA [21–23] are amongst the most frequently
oped to correlate resilient modulus to soil strength or stiffness
used recycled materials in different layers of the flexible pave-
characteristics such as CBR or R-value is that those tests them-
ments. RCA resulting from renovation or demolition of structures
selves are pretty much empirical in nature. Whereas, resilient
such as commercial buildings, individual residences or civil engi-
modulus is a mechanistic parameter and depends on a host of soil
neering structures waste is one of the waste products that have
index properties and stress state.
become more common to use in road applications during the last
two decade [24–26]. Studies conducted by Nataatmadja and Tan
[27], Jitsangiam et al. [28] concluded that the resilient modulus 3.2. Regression equations for resilient modulus based on soil properties
of RCA was comparable, if not better, than that of typical virgin and stress state
road aggregates. Therefore, RCA products were believed to have
suitable applications as either a subbase or base course, provided This approach utilises the various soil index properties and
appropriate product quality standards were met. Leek and Siri- strength characteristics or different in situ strength test-related
pun’s report [29] found that RCA was suitable for high stress appli- parameters for the estimation of resilient moduli values. For exam-
cations and should be considered for use as a premium base course ple, there are many correlations for the evaluation of MR values
product. Studies documented by Grab and Cameron [23] suggests which are based on bulk stress and index properties of the soil
that the RCA material performed better than the quartzite aggre- [62]; unconfined compressive strength and index properties of
gate regarding resilient modulus and accumulation of permanent the soil [63]; R-value and index properties of the soil [64].
deformation during repeated load triaxial test. Statistical regression tools are usually adopted for this exercise.
442 M. Arshad / Construction and Building Materials 159 (2018) 440–450

Correlations obtained through this approach are of varying com- blend (0, 50, and 75% RAP) were stored in wrapped plastic bags
plexity and acceptability [54,65–67]. until they were required for different types of testing in the
laboratory.
3.3. Resilient modulus constitutive models It should be noted that procedures for classification of RAP
materials are quite varied across the world. The only characteriza-
In this approach, MR values for a particular soil is obtained by tion that is common to the various standards and procedures is the
taking into account the various types of stress invariant, for exam- particle size distribution curves (gradation curves). However
ple bulk stress [53]; bulk stress and atmospheric pressure [68]; Tebaldi et al., [76,77] have proposed that a detailed description
bulk stress, atmospheric pressure and deviator stress [69]; confin- would require a series of testing including but not limited to:
ing pressure and deviator stress [70]; atmospheric pressure, octa- determination of particle sizes (as RAP and of recovered aggre-
hedral normal stress and octahedral shear stress [56]; bulk gates); determination of active binder content in RAP; shape
stress, atmospheric pressure and octahedral shear stress [71]. MR description of recovered aggregates; cleanliness of RAP and
values obtained through this approach are more reliable when homogeneity.
compared with those obtained through the first two approaches
however, this approach requires extensive efforts with higher eco- 4.1. Grain size distribution characteristics
nomic cost.
The grain size distribution characteristics were obtained
3.4. Resilient modulus based on constitutive equation according to AASHTO Designation T27-99 [78] for all materials/
blends. Fig. 1 shows the gradation curves for the virgin aggregates,
Most of the State Highway Agencies around the world and par- RCA and RAP materials. Each curve corresponds to the average of
ticularly in the USA do not routinely measure MR value in the lab- two independent tests. Based on the gradation curves, virgin aggre-
oratory but estimate the design MR either from experience or other gate A is finer than other virgin aggregate materials (F & W) while
material properties. The potential benefit of estimating MR value the virgin W is the most coarser among them. Similarly, RAP(3) is
from soil physical properties is that the seasonal variations in the coarser when compared with other RAP materials and RAP(4)
MR value can be determined from seasonal changes in the mate- being the most finer among them. RCA was derived from plain con-
rial’s properties; however, the effect of stress sensitivity is not cap- crete aggregate designed for 25 MPa compressive strength after 28
tured. In order to capture the effects of stress sensitivity and days having limestone as crushed aggregate, ordinary Portland
physical properties on design MR, Von Quintos and Killingsworth cement and coarse sand.
[72], Dai and Zollars [73], Santha [74] and Mohammad et al. [75],
among others, have developed prediction equations for MR value 4.2. Physical properties of recovered bitumen binder
by regressing the coefficients of selected constitutive equations
relating them to soil physical properties. The RAP samples were washed with tap water to eradicate for-
eign materials, and the samples were put in the oven at a temper-
4. Material characterization ature of 140 °C to lose the aggregate from the binder agent.
Recycled binder contents for different RAP types were extracted
For this research work index properties tests, shear strength from the RAP samples by solvent extraction method using a rota-
tests, resilient modulus tests, one-dimensional constrained modu- tional viscometer apparatus. The recovered bitumen contents
lus tests (1D) were conducted to evaluate the influence of RAP con- ranging from 4.3 to 3.3% were characterised in terms of penetra-
tent on the resilient properties of the various blends prepared by tion (ASTM D5-06), viscosity (ASTM D4402-06) and softening
mixing RAP contents (designated as ‘RAP(1)’, ’RAP(2)’, ‘RAP(3)’ & points (ASTM D36-06).
‘RAP(4)’) with virgin aggregates (granular) samples (designated
as ’A’, ‘F’ & ‘W’) and RCA material. Each of the virgin aggregate sam- 4.3. Compaction characteristics
ples, as well as RAP materials, were containing crushed limestone
of angular to sub-angular particles in terms of shape while flat and Results of the Modified Proctor Compaction tests (AASHTO
elongated particles in the samples/materials were limited to 6% as T180) [79] were used to get the compaction characteristics of all
per ASTM D 4791. Table 2 shows the matrix of the testing program the tested materials. The optimum moisture contents (wopt) of
designed for resilient modulus test and constrained modulus test.
Virgin aggregates (A, F and W), RAC and RAP samples were 100
transported to the laboratory, air-dried at ambient laboratory tem-
perature (typically 23 ± 2 °C), then carefully mixed before being 90
divided into equal parts by using a standard rifle box. After propor-
Material passing by mass (%)

80
tioning and blending the materials, suitable quantities of each 70

60
Table 2
50
Matrix of testing program.
40
Virgin aggregate/RCA RAP Minimum number Minimum number virgin (A)
of resilient of constrained 30 virgin (F)
virgin (W)
modulus tests modulus tests RCA
20 RAP(1)
Virgin aggregate A, F, W – 3 3 RAP(4)
RCA – 1 1 10 RAP(2)
RAP(3)
Virgin aggregate A, F, W 4  2* 3  2  4 = 24 3  2  4 = 24
0
RCA 4  2* 24=8 24=8 0.01 0.1 1 10 100
Total 36 36 Diameter (mm)
*
Blended samples were prepared by mixing 50% and 75% (by weight) of each RAP
type with the virgin aggregates and RCA. Fig. 1. Particle size distribution curves for virgin aggregates (A, F, W), RCA and RAP
materials.
 M. Arshad / Construction and Building Materials 159 (2018) 440–450 443

Fig. 2. Typical shape of applied repeated load cycles and the generated accumulative strain curve.

that includes a 0.1s load duration and a 0.9 s rest period. The
Table 3
Compaction characteristics of the tested materials. repeated axial load is applied on top of a cylindrical specimen
under specified confining pressure. The total recoverable axial
Material Maximum dry Optimum moisture
deformation response of the specimen is measured and used to cal-
density content
culate the resilient modulus. AASHTO T 307 [83] involves the use of
kN/m3 (%)
a load cell and deformation devices mounted outside the triaxial
virgin (A) 21.9 7.1
virgin (F) 22.5 5.7 chamber. Air is used as the confining fluid, and the specimen size
virgin (W) 23.3 5.5 is required to have a minimum diameter to length ratio of 1:2.
RCA 20.7 7.5 For the current study a closed-loop, servo-controlled electro-
RAP(1) 21.4 6.9 hydraulic MTS testing machine was used in conjunction with a
RAP(2) 20.6 6.4
RAP(3) 20.9 6.5
function generator that is capable of applying repeated cycles of
RAP(4) 19.7 9.1 haversine-shapes load pulse following a load history supplied by
the microcomputer control software. A typical variation of applied

granular and RCA were found in the range of 5.5% and 7.5%. The
maximum dry unit weight varied from 20.7 to 23.3 kN/m3, with
granular W having the highest maximum dry unit weight and
RCA the lowest. The values of wopt for the RAP were in the range
of 6.4–9.1% while the maximum dry unit weight remained
between 19.7 and 21.4 kN/m3 as shown in Table 3. Even though
the proctor compaction tests for blends of virgin aggregates with
RAP were not performed, it was assumed that the wopt values of
the blends with different RAP contents would be in the range of
5.5–7.5%.

4.4. Shear strength parameters

To have an idea about the shear strength parameters (especially

angle of internal friction) of the virgin aggregates, 100% RAP and
RCA, quick shear tests (triaxial) were performed using the appara-
tus used for resilient modulus test as per AASHTO T 307 [80–82].
All the samples were initially compacted to a density of at least
95% of compaction test (AASHTO T180) at optimum moisture con-
tent. Shearing load (in the vertical direction) was applied at a strain
rate of 1 mm/min after applying the confining pressure. The Angle
of internal friction and cohesion of the tested materials was found
in the range of 36°–43° and 20–30 kPa, respectively. These values
of shear strength parameters were obtained for three distinct
levels of confining pressure values of 20, 35 and 50 kPa.

4.5. Resilient modulus test

Resilient modulus tests were performed as per AASHTO T 307 Fig. 3. Apparatus for the constrained modulus test. Key: 1, loading frame; 2, load
[83] which requires a haversine-shaped loading waveform. The measuring gauge; 3, deflection gauge; 4, mould containing test specimen; 5,
load cycle duration, when using a hydraulic loading device, is 1 s hydraulic pump.
444 M. Arshad / Construction and Building Materials 159 (2018) 440–450

load impulse and the accumulated strain with the passage of time distinct levels of axial stress (ra) at which unloading of the sample
is shown in Fig. 2. was performed. The maximum vertical pressure in the condition-
AASHTO T 307 protocol completes the MR testing with 15 load- ing stage is selected as 205 kPa, which is, in fact, equal to the axial
ing series consisting of different combinations of confining stress, stress in the conditioning stage of the resilient modulus testing.
cyclic axial stress and contact stress. For the first series, the maxi- Ten number of cycles of loading-unloading were applied to the
mum axial stress was set at 3 psi (21.0 kPa) and the confining pres- specimen by the manual control hydraulic piston at the average
sure was 3 psi (21.0 kPa). One hundred repetitions of load cycles loading rate in the range of 100–150 kPa/min before the data
were applied during each series of the loading. The readings of ofloading and settlement were recorded. Fig. 4 shows the typical
each LVDT and the load cell for the last five cycles of each loading loading-unloading cycles corresponding to four different levels of
sequence were considered to calculate the corresponding resilient axial stress. The slope of the average trend line drawn for each loop
moduli. Under a unique stress condition, 100 cycles of loads were gives a calculated value of the constrained modulus at the particu-
applied while conditioning stage was completed with 750 number lar stress level [84].
of load cycles. Further details about the sample preparation, sam-
ple installation, loading sequence and data acquisition system 5. Development of correlation between MR and MC
can be found in the related AASHTO guide and same has been doc-
umented by Arshad and Ahmed [84].
5.1. General form of correlation
4.6. Constrained modulus test
To develop a correlation between MR and Mc values the test
4.6.1. General description results of 36 samples were analysed. For each sample, four values
A series of one-dimensional (1D) constrained compression/- of the constrained modulus Mc were calculated at four selected
modulus tests were carried out using a 30 cm diameter and 30 levels of axial stress. For all these samples the angle of internal fric-
cm high mould. This mould can accumulate materials with a max- tion ð/Þ was assumed to be 40° to estimate the coefficient of earth
imum particle size of approximately 40–50 mm. The constrained pressure at rest (Ko) using Jacky’s equation:
modulus, which reflects the compressibility and hence the stiffness
K 0 ¼ 1  sinð/Þ ð1Þ
of a material can be obtained from these tests.
The bulk stress (rbulk = h = r1 + r2 + r3) in the 1D compression
4.6.2. Loading system and data acquisition test can be determined at the selected axial stress (ra) levels, with
Fig. 3 shows the test setup of 1D cyclic compression test for the the results being summarised in Table 5.
constrained modulus. The deformation of the sample is measured Fig. 5 shows the variation of resilient modulus and constrained
using three dial gages fixed at 120 relative to each other. Appar- modulus value with bulk stress. From this figure, it can be inter-
ently, such an arrangement of the dial gauges provides ‘‘best” esti- preted that both of the resilient modulus and constrained modulus
mate for the average deformation even when the loading plate has values increase with the increase in bulk stress level (h) and the
a slight tilt. A load cell is used to measure the load that is controlled percentage of RAP material in the blends. In addition to that MR
manually using a hydraulic pump.
During the 1D compression test, the sample was unloaded at
Table 5
different stress levels to obtain the unloading characteristics and Estimation of bulk stress corresponding to the axial stress
the constrained modulus of the material. Table 4 shows the four during constrained modulus test.

ra (kPa) rbulk = (1 + 2K 0 ) ra (kPa)

Table 4
Axial stress for different stages of unloading. 205 348.5
275 467.5
Stages Axial stress (kPa) 310 523.6
1st 205 410 689.7
2nd 275
3rd 310
4th 410

450
y = 9.7243x0.5713
Virgin (W)
Resilient/constrained modulus (MPa)

400 R² = 0.9625
450 25%virgin(W)+75%RAP(1)
350
400 50%virgin(W)+50%RAP(1) y = 10.097x0.5498
300 R² = 0.9773
350

300 250
Axial stress (kPa)

y = 5.1683x0.6369
250 200
R² = 0.9923
200 150
MR
Mc
150 100 vs θ
vs θ
Cycle 1
100 Cycle 2 50
50 Cycle 3
Cycle 4 0
0 0 100 200 300 400 500 600 700 800
0 0.002 0.004 0.006 0.008 0.01 Bulk stress (Kpa)
Strain (%)
Fig. 5. Variation of resilient modulus and constrained modulus value with bulk
Fig. 4. Typical stress-strain loop for the estimation of the constrained modulus. stress.
 M. Arshad / Construction and Building Materials 159 (2018) 440–450 445

600 and Mc values can be approximated in terms of bulk stress using a

power function and linear function, respectively, having higher
500 values of coefficient of correlation.
Fig. 6 shows the relationship between MR and Mc values at four
400 distinct levels of bulk stress for the 36 number of samples tested.
From this figure, it can be observed that there exists a certain trend
Mc (MPa)

300 between these two quantities, although the data points are very
scattered. As the first order approximation, MR can be estimated
as approximately 2–4 times of the value of Mc. In addition, it can
200
R² = 0.1624 be concluded that both MR and Mc value depend on not only the
stress state but also on the nature of the virgin aggregates and
100
the RAP/RCA.
One of the most widely utilised relationships for the estimation
0 of resilient modulus value for an unbound granular material is the
0 100 200 300 400 500 600
MR (MPa) one proposed by Hicks and Monismith [18] as follows:

Fig. 6. Relationship between MR and Mc. MR ¼ aðhÞb ð2Þ

Table 6
Evaluation of regression model parameters (a, b) used in Eq. (3).

Material MR (MPa) MR(estimated) = a(Mc)b Mc (MPa) a b R2

Virgin(A) 215 227 118 0.1 1.62 0.98
250 263 129 0.1 1.62
310 300 140 0.1 1.62
Virgin(F) 180 180 75.8 0.12 1.69 0.99
211 216 84.4 0.12 1.69
225 226 86.6 0.12 1.69
263 270 96.2 0.12 1.69
Virgin(W) 220 189 77 0.128 1.68 0.97
264 258 87 0.128 1.68
284 285 98.4 0.128 1.68
342 371 115 0.128 1.68
RCA 337 317 102 0.122 1.7 0.95
405 418 120 0.122 1.7
436 424 121 0.122 1.7
523 498 133 0.122 1.7
50%Virgin(A) + 50%RAP(1) 181.24 193 65 0.13 1.75 0.94
212.52 220 70 0.13 1.75
227.24 225 71 0.13 1.75
266.8 290 82 0.13 1.75
50%Virgin(A) + 50%RAP(2) 230 248 94 0.11 1.7 0.97
276 286 102 0.11 1.7
299 325 110 0.11 1.7
368 371 119 0.11 1.7
50%Virgin(A) + 50%RAP(3) 252.08 242 95 0.11 1.69 0.99
270.48 259 99 0.11 1.69
320.16 334 115 0.11 1.69
50%Virgin(A) + 50%RAP(4) 219.88 244 83 0.1 1.75 0.95
255.76 268 90 0.1 1.75
272.32 292 97 0.1 1.75
50%Virgin(F) + 50%RAP(1) 197 202 109 0.111 1.6 0.94
231 211 112 0.111 1.6
247 223 116 0.111 1.6
290 255 126 0.111 1.6
50%Virgin(F) + 50%RAP(2) 250 211 102 0.085 1.69 0.98
300 252 113.4 0.085 1.69
325 258 115 0.085 1.69
400 315 129.3 0.085 1.69
50%Virgin(F) + 50%RAP(3) 231 221 92.6 0.11 1.68 0.98
274 273 104.9 0.11 1.68
294 284 107.4 0.11 1.68
348 361 123.8 0.11 1.68
50%Virgin(F) + 50%RAP(4) 239 254 72.1 0.12 1.79 0.974
278 273 75 0.12 1.79
296 296 78.5 0.12 1.79
345 339 84.7 0.12 1.79

(continued on next page)

446 M. Arshad / Construction and Building Materials 159 (2018) 440–450

Table 6 (continued)

Material MR (MPa) MR(estimated) = a(Mc)b Mc (MPa) a b R2

50%Virgin(W)+50%RAP(1) 188 179 76.8 0.09 1.75 0.94

221 208 83.7 0.09 1.75
249 229 88.4 0.09 1.75
276 288 100.7 0.09 1.75
50%Virgin(W)+50%RAP(2) 200 203 81.5 0.125 1.68 0.98
235 234 88.6 0.125 1.68
265 263 95 0.125 1.68
293 304 103.6 0.125 1.68
50%Virgin(W) + 50%RAP(3) 190 214 85 0.09 1.75 0.98
206 228 88.1 0.09 1.75
233 258 94.5 0.09 1.75
260 281 99.2 0.09 1.75
50%Virgin(W) + 50%RAP(4) 190 187 84 0.1 1.7 0.99
223 230 95 0.1 1.7
251 255 101 0.1 1.7
278 282 107 0.1 1.7
50% RCA + 50%RAP(1) 337 317 80 0.13 1.78 0.99
370 361 86 0.13 1.78
383 395 90.5 0.13 1.78
420 455 98 0.13 1.78
50% RCA + 50%RAP(2) 227 224 85 0.09 1.76 0.99
262 254 91.4 0.09 1.76
277 272 95 0.09 1.76
320 325 105 0.09 1.76
50% RCA + 50%RAP(3) 306 298 82.9 0.125 1.76 0.99
362 351 91 0.125 1.76
387 394 97.2 0.125 1.76
458 474 108 0.125 1.76
50% RCA + 50%RAP(4) 283 274 79 0.12 1.77 0.99
325 316 85.6 0.12 1.77
345 341 89.3 0.12 1.77
396 401 98 0.12 1.77
25%Virgin(F) + 75%RAP(1) 268 273 96.8 0.1 1.73 0.93
309 285 99.3 0.1 1.73
329 324 107 0.1 1.73
380 385 118.2 0.1 1.73
25%Virgin(F) + 75%RAP(2) 258 264 90 0.11 1.73 0.98
300 296 96 0.11 1.73
318 328 102 0.11 1.73
368 373 109.8 0.11 1.73
25%Virgin(F) + 75%RAP(3) 300 297 84.8 0.12 1.76 0.98
355 343 92 0.12 1.76
380 373 96.4 0.12 1.76
450 471 110.2 0.12 1.76
25%Virgin(F) + 75%RAP(4) 254 254 60 0.12 1.87 0.9
290 296 65.1 0.12 1.87
307 333 69.4 0.12 1.87
25%Virgin(W) + 75%RAP(1) 281 298 78 0.058 1.95 0.98
332 328 82 0.058 1.95
355 352 87.1 0.058 1.95
418 411 98 0.058 1.95
25%Virgin(W) + 75%RAP(2) 255 244 78.4 0.07 1.87 0.99
294 284 85 0.07 1.87
311 313 89.6 0.07 1.87
359 363 97 0.07 1.87
25%Virgin(W) + 75%RAP(3) 295 302 90.1 0.08 1.83 0.96
340 341 96.2 0.08 1.83
360 355 98.4 0.08 1.83
416 446 111.4 0.08 1.83
25%Virgin(W) + 75%RAP(4) 280 274 81 0.12 1.76 0.96
322 299 85.1 0.12 1.76
341 330 90 0.12 1.76
392 395 99.6 0.12 1.76
25%RCA + 75%RAP(1) 252 248 85 0.07 1.84 0.99
292 284 91.4 0.07 1.84
310 311 96 0.07 1.84
360 363 104.5 0.07 1.84
 M. Arshad / Construction and Building Materials 159 (2018) 440–450 447

Table 6 (continued)

Material MR (MPa) MR(estimated) = a(Mc)b Mc (MPa) a b R2

25% RCA + 75%RAP(2) 303 277 72.7 0.065 1.95 0.97
350 342 81 0.065 1.95
372 402 88 0.065 1.95
430 467 95 0.065 1.95
25% RCA + 75%RAP(3) 314 301 83.6 0.08 1.86 0.99
367 382 95 0.08 1.86
391 397 97 0.08 1.86
456 476 107 0.08 1.86
25% RCA + 75%RAP(4) 300 289 80 0.07 1.9 0.96
344 327 85.3 0.07 1.9
364 362 90 0.07 1.9
418 459 102 0.07 1.9
25%Virgin(A) + 75%RAP(1) 236 248 81 0.07 1.86 0.99
295 295 88.8 0.07 1.86
344 334 95 0.07 1.86
25%Virgin(A) + 75%RAP(2) 240 253 88 0.07 1.83 0.99
300 294 95.4 0.07 1.83
350 320 100 0.07 1.83
25%Virgin(A) + 75%RAP(3) 363 392 94 0.08 1.87 0.94
466 471 103.8 0.08 1.87
555 499 107 0.08 1.87
25%Virgin(A) + 75%RAP(4) 317 325 76.5 0.072 1.94 0.99
342 341 78.5 0.072 1.94
363 355 80.1 0.072 1.94

The values of regression constants, a and b depend on the mate- ficient of correlation remained in the range of 0.95 to 0.99, which
rial properties and laboratory test procedure such as AASHTO in return reveals that 95–99% values of the dependent variable
T-307 [83]. Seed et al. [85] have also suggested a similar K–h model (MR) are dependent on the employed independent variable
(relation) for the estimation of resilient modulus in terms of bulk (Mc for the current study) [86]. Fig. 7 compares the measured MR
stress. values (experimentally) to those calculated (estimated) on the
For the study under consideration, a functional relationship basis of Eq. (3), using the specific regression model parameters
between MR and Mc values of the selected materials/blends was for each sample tested. From this figure, it can be observed that
demonstrated through the power relation as shown in Eq. (3): all the data points remain in a narrow band about the line of unity
with an approximate variation of ±25 MPa.
M Rðcorrelated:Þ ¼ aðM c Þb ð3Þ

where a and b are the model parameters. 5.2. Generalised values of the model parameters (a,b) for the
Table 6 summarises the values of these model parameters estimation of MR values
obtained using the computer software package ‘‘SOLVER” for each
sample tested in this study. Four distinct levels of bulk stresses, as Virgin aggregates and the RAP materials cover a wide range of
given in Table 6, were used in the above regression analysis. The gradation characteristics as shown in Figure 1 and moreover, there
coefficient of determination (R2), that is in fact square of the coef- is no unique way to characterise these materials, so it is difficult to
attach a single value of regression coefficients for all the tested
samples (virgin aggregates and blended materials). A careful anal-
ysis of the experimental data reveals that tested samples can be
categorised by the percentage of RAP content such that:

A. Blended samples containing 75% RAP content and 25% virgin

aggregates/RCA
B. Blended samples containing 50% RAP content and 50% virgin
aggregates/RCA
C. Samples containing 100% virgin aggregates/RCA

Fig. 8 shows the performance of the Eq. (3) when three separate
sets of regression constants (each one for a particular category) are
used which have been developed using a computer software tool
‘SOLVER’. Furthermore, the regression constant ‘a’ has been gener-
alised in term of percentage of RAP content as shown in Table 7
such that, ‘a’ can be treated as dependent variables while the per-
Fig. 7. Comparison between measured and estimated values of MR. centage of RAP content can pretend as an independent variable.
448 M. Arshad / Construction and Building Materials 159 (2018) 440–450

Pn



600 i¼1 ðui  ui Þðt i  t i Þ
R² = 0.593 R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ð4Þ
Pn
½ðu i  u
i Þ2
ðt i 
ti Þ2 
500 i¼1
MR(estimated) (MPa)

400
where ui and t i are the experimental and predicted/estimated val-
ues; respectively; u
i or
t i is the average of the respective values
300
and n is the size of sample.
200
The numerical value of R may range from 1 to +1, whereas
positive value represents an uphill correlation and negative value
100
corresponds to a downhill correlation. Additionally, the strength
(a) of the linear relationship can be categorised as very strong, moder-
0
ately strong and fair for R values in the range 0.8–1.0, 0.6–0.8 and
0 100 200 300 400 500 600 0.3–0.5, respectively [86]. However, if R = 0.0, doesn’t mean that
MR(measured) (MPa) the two variables are unrelated. Hence it is often mandatory to per-
form a statistical test of significance to talk about the strength of a
500 relationship indicated by the correlation coefficient [88]. For this
R² = 0.5912 study, three different types of statistical tests were performed to
evaluate the significance of the developed correlation presented
400
in Eq. (3) and Table 7.
MR(estimated) (MPa)

300 Case-1: T-test for evaluation of Rat a particular degree of free-

dom and level of significance [89].

200 To test the correlation coefficient for statistical significance, it is

necessary to define the true correlation coefficient (q) that would
100 be observed if all population values were obtained. The null
hypothesis is that there is no relationship between the two vari-
(b)
ables X and Y (in our case (MR)measured and (MR)estimated).
0
0 100 200 300 400 500
MR(measured) (MPa) H0 : q ¼ 0
600
R² = 0.8091 The alternative hypothesis could be
500
HA : q – 0
MR(estimated) (MPa)

400
Whereas, the standardised t statistic on the basis of R can be written
300 as:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
200 N2
t¼R ð5Þ
1  R2
100
(c)
where N is the number of paired observations in the given
0
0 100 200 300 400 500 600 population.
The null hypothesis is evaluated by comparing t statistic of Eq. (5)
MR(measured) (MPa)
with t-critical obtained from t distribution having N  2 degree of
Fig. 8. Comparison between measured and correlated values of MR for: (a) blended freedom.
samples containing 75% RAP; (b) blended samples containing 50% RAP and (c)
samples containing virgin aggregates/RCA. Case 2: Direct comparison of R-value with R-critical value at a
particular degree of freedom and level of significance [90].
5.3. Statistical analysis of the proposed model Case 3: To perform a standard t-test for paired samples at a par-
ticular degree of freedom and level of significance [90,91].
To evaluate the performance of the proposed models, Pearson
correlation coefficient ‘R’ as given by Eq. (4), which is a measure Table 8 summarise the statistical analysis of the above three
of the strength of the relationship between or among variables is cases for the three categories of tested samples (group) investi-
used [1,84,87]. gated in this study.

Table 7
Generalised model parameters on the basis of percentage of RAP content.

Sample/Cluster Type ‘a’ ‘b’

A. Blended samples containing 75% RAP content and 25% virgin aggregate/RCA 1 + %RAP (in fraction) = 1.75 1.16
B. Blended samples containing 50% RAP content and 50% virgin aggregate/RCA 1 + %RAP (in fraction) = 1.5 1.16
C. Samples containing 100% virgin aggregate/RCA 1 + %RAP (in fraction) = 1.0 1.24
 M. Arshad / Construction and Building Materials 159 (2018) 440–450 449

Table 8
Summary of the statistical analysis for the validation of proposed model for the estimation MR value.

Soil Correlation statistical a- R or t-critical R or t-statistic Comments

group coefficient R tests value
A 0.77 Case 1 0.01 2.66 9.1 Since for all the values of a (significance level) t-statistic > t-critical
0.02 2.39 which implies that value of the correlation coefficient not due to
0.05 2.00 sampling error.
B 0.75 0.01 2.69 7.7 Therefore, null hypothesis is rejected and it is conclude that there
0.02 2.41 is a significant correlation between
0.05 2.01 (MR)measured and (MR)estimated in the population
C 0.90 0.01 3.01 7.4
0.02 2.65
0.05 2.16
A 0.77 Case 2 0.01 0.32 0.77 Since for all the values of a (significance level) R > R-critical which
0.02 0.29 implies that null hypothesis is rejected.
0.05 0.25 It is quite realistic to accept the ‘‘alternative hypothesis”, that is
B 0.75 0.01 0.35 0.75 the value of R that we have obtained from our sample represents a
0.02 0.32 real relationship between (MR)measured and (MR)estimated in the
0.05 0.27 population
C 0.90 0.01 0.64 0.90
0.02 0.59
0.05 0.51
A 0.77 Case 3 0.01 2.66 0.09 Since for all the values of a (significance level) t-statistic < t-critical
0.02 2.39 which implies that null hypothesis is accepted.
0.05 2.00 Therefore, it is conclude that there is insignificant difference
B 0.75 0.01 2.66 0.03 between (MR)estimated and (MR)measured values in the population
0.02 2.39
0.05 2.00
C 0.90 0.01 2.94 0.76
0.02 2.60
0.05 2.13

6. Conclusions and recommendations Author also acknowledges the valuable comments and suggestions
of the reviewers for the improvement of this article.
The RAP/RCA materials have substantial potential to be used in
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