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CHAPTER 3
RESILIENT MODULUS EVALUATION OF AGGREGATE BASE
AND SIJBGRADE MATERIALS
INTRODUCTION
Resilient modulus, when used In a layer system analysis, is a very important variable in predicting
the resilient (i.e., recoverables stress, strains and deflections in a flexible pavement. In an unstabilized
layer, the stress or strain can be related to permanent deformation provided the permanent deformation
properties of the material are known. Resilient moduli are also now used in the AASHTO design method
even though it is statistically based rather than based on mechanistic design principles. By the year 2002
a mechanistic based AASHTO design memos should be available. The widespread use of resilient moduli
in pavement thickness design indicates the importance of obtaining reliable values of this design variable.
A primary objective of this study is to develop laboratory resilient modulus testing procedures
suitable for use by a state transportation agency. To help achieve this goal, the emphasis of the study was
placed on evaluating the effects on resilient modulus of laboratory testing details such as equipment
calibration and testing conditions. A limited amount of effort was also devoted to the permanent
deformation characterization of base and subgrade materials. Permanent deformation characterization
of all layers is as important, or even more important, than resilient modulus characterization.
Permanent deformation can be measured as an extension of the resilient modulus test with little extra
effort.
Based on a Borough review of the literature, He repeated load, biaxial test was selected to
characterize in this study the resilient modulus of unstabilized base and subgrade soils.
RESILIENT MODULUS CONCEPI
The resilient modulus is equal to the peak applied axial repeated stress (Figure 3a) divided by He
recoverable axial strain occurring within the specimen. The resilient axial strain is equal to the recoverable
deformation (bounce) which the specimen undergoes when subjected to one pulse, divided by the axial
distance over which the bounce is measured. The resilient modulus (MR) IS calculated as
MR ((~1 ~ 031/~Ir (5)
where: MR =
(] 1~3
~_
'or
resilient modulus
maximum repeated axial stress (deviator stress)
maximum resilient axial strain
97
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In addition to being called MR, the resilient modulus is also sometimes called Er or just the modulus of
elasticity (E). In the repeated load test, and also other dynamic tests, the applied peak stress occurs before
the peak strain develops and hence stress and strain are said to be out of phase with each other. Because
stress and strain are out of phase with each other, the resilient modulus concept is an approximation.
The resilient Poisson's ratio (ur) is determined by
_e
~2
fir
~r
where: v. =
~_
Or
~1
resilient Poisson's ratio
resilient radial strain
resilient axial strain
(6,
Once the resilient modulus and resilient Poisson's ratio have been evaluated, other dependent elastic
constants can be calculated. For example, resilient bulk modulus (K) and resilient shear modulus (G) can,
if desired, be calculated using the following equations:
MR
G =
2(1 +V )
MR
K

3( 1 2vr)
The above equations assume linear elastic response.
RESILIENT MODULUS LABORATORY TESTING METHODS
(7)
(8,
The resilient modulus of pavement materials can be evaluated in the laboratory using the following
different types of repeated load and cyclic laboratory testing equipment: (~) biaxial cell, (2) simple shear
device, (3) resonant column/torsional apparatus, (4) hollow cylinder, and (5) true biaxial cell. These tests
all have potential, under certain conditions, for characterizing both stabilized and unstabilized pavement
materials. A brief general discussion of this equipment is therefore presented including advantages and
disadvantages with regard to resilient modulus testing.
Triaxial Cell
Advantages. The biaxial test offers three very important advantages in resilient modulus testing:
98
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2
Stress State. In the biaxial test known principal stresses a, and 03 are applied to the
specimen In known directions as illustrated in Figure 46. As a result, the stress conditions
within the specimen on any plane are also defined throughout the test. The stress
conditions applied are, in fact, those which occur when an isolated wheel loading is
applied to the pavement directly above We element of material simulated in the test.
Spec~mea;;)rainage, Use of Me biaxial test allows relatively simple, controlled drainage
of the specimen in He axial and/or radial directions. If desired, lateral drainage occurring
in, for example, a granular base can be simulated reasonably easy in the biaxial test. Pore
pressures can also be easily measured at the ends of the specimen or, with more difficulty,
within the specimen.
Strain Measurement. Axial, radial and volumetric strains can all be measured relatively
easily in the biaxial test.
In addition to the above advantages, undisturbed tube samples of the subgrade obtained from the
field can be extruded and tested with a minimum amount of specimen preparation. Finally, the biaxial
cell used for the repeated load biaxial test can also be employed in static testing.
Disadvantages. The most important disadvantage of the biaxial cell is its limited ability to handle (~)
rotation of the principal stress axes and (2) shear stress reversal. Only a fixed orthogonal rotation of
principal stress axes are possible in this test. Both of these limitations of the test come into play when a
wheel load moves across the pavement. Also, the intermediate principal stress applied to a specimen
cannot be controlled in the biaxial test. Careful equipment calibration is required, which is especially
critical when applied strain is measured outside the cell.
Use. The repeated load biaxial compression test has for about the last 35 years been used as the basic
testing apparatus to evaluate the resilient modulus of cohesive and granular materials for pavement design
applications [361. Early repeated load tests on stabilized materials were also conducted using the biaxial
test [37, 381. The repeated load biaxial compression test is relatively straightforward to perform compared
to most other alternatives, and it reasonably closely simulates the conditions encountered in the field.
Triaxial Cell Design. Conventional Cell  The conventional biaxial cell, which has external tie
rods to hold the cell together, has traditionally been used in resilient modulus testing. In this type cell the
tie rods are located on the outside of the biaxial chamber (Figure 471. To form a seal in the conventional
cell, circular flat gaskets or rubber "0" rings are located between the chamber and the top and bottom of
the biaxial cell. If flat gaskets are used, the nuts on the tie rods should be tightened evenly using a torque
wrench. Uniform tensioning of the rods insures Me load remains properly aligned. If "0" rings are used
the seal should be designed to allow the top and bosom plates to contact the chamber.
Triaxial Cell wig Internal Tie Bars  An alternative to the conventional biaxial cell design has
the tie bars located on He inside of the chamber. In a cell having internal tie bars, the top of the cell is
supported by the tie bars through machined, steel to steed contact. Both the conventional and internal tie
bar type cells require careful machine work to insure accurate alignment. No big advantage appears to
exist for either cell.
BishopWesley Type Cell  The BishopWesley type biaxial cell, developed for static stress path
testing, is completely self~ontained and does not require a separate load frame [39]. This biaxial cell uses
99
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Total Axial Stress, cr'
(major principle stress)
a'  Al = Repeated (Cyclic)
Deviator Stress
Shear Stresses ~ = 0
tS3= Confining Pressure
(minor principle stress)

Note: () = a1 + Cr2 + o3 = C7d + 3
external tie bars, acne load is applied to the specimen hydraulically through the base of the cell. The
toadying ram is guided by a linear bearing. Two rolling diaphragm seals separate the hydraulic loading
fluid from the chamber fluid.
on   or  cow   r ~ ~     a ~o
In the BishopWesley type cell, the axial stress (ok and confining pressure (~3) are controlled
ind~ernend~ently of each over. Independent control in the conventional triaxiad cell can also be accomplished
by making the load rain diameter He saline as the specimen and using a rolling diaphragm to form the seam.
Self~ontainment, together with the ability to nan stress paw tests, appear to be ache advantages of this type
cell for routine resilient modulus testing. This cell has not been used for rapid cyclic loading, and some
problems have been encountered in slow loading due to the diaphragm pulling out.
Torsional Shear/Resonant Column Test
Torsional Shear. The torsional shear test subjects a cylindrical specimen of material to a cyclic torsional
shear loading (Figure 48~. The bottom of Me specimen is considered fixed while the top of the specimen
is connected to a drive system that generates torsional motion. In the torsional shear test, the frequency
of loading is generally less than 10 Hz. The modulus is determined from the shear stressshear strain
hysteresis loop.
The torsional shear test can be used to determine moduli at shear strains up to about 103 in/in.
For moderately thick pavement sections and/or weak materials, the shear strains developed under typical
heavy wheel loadings can be greater than the capability of the test apparatus. The torsional shear apparatus
is, however, relatively easy to calibrate, and the test straightforward to perform. The drainage condition
of the specimen is not as well defined as in the biaxial test. The applied strain level beneath a pavement
is smaller in the subgrade than the base. Hence, the test is most attractive for testing subgrade soils.
Furler consideration of adapting this test for subgrade testing appears to be desirable.
Resonant Column. The resonant column test [401, which is somewhat similar to the torsional shear test,
subjects the specimen to cyclic axial strains considerably smaller than applied by heavy wheel loadings
near the surface. Hence the resonant column test is not suitable for testing highway materials unless
empirical corrections for strain level are applied to the measured moduli.
Simple Shear
In the simple shear test, shear stress is applied to the top and bottom of either a square block or
disk shaped specimen as shown in Figure 49. In the repeated load simple shear test, shear stress is applied
alternately in each direction. This loading condition results In a fixed O to 90° principal stress axis rotation
and accompanying reversal of shear stress. Since shear stress reversal occurs in the field, the simple shear
test is more realistic than the repeated load test with regard to permanent deformation. The stress paths
occurring in the field due to application of a moving wheel load, however, are not faithfully reproduced
in He simple shear test. The general effects of principal stress axis rotation and shear stress reversal can
be studied using the simple shear test [4l, 421.
Resilient shear moduli determined on sand using the simple shear apparatus and the repeated load
biaxial compression test have been found to be similar at comparable stress levels [411. In ache simple
shear test, however, permanent strain continued to logarithmically accumulate for 100,000 repetitions.
101
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Torsion ~
cy3
~3
.
~ ~17~ 
l
 c73= Confining Pressure
Figure 48 . Stresses Applied in a Cyclic Torsional Shear Test
\^ is
~ ~ ~ ~1~" ~ ~
/~
(a) Shear to leD
= Applied normal stress
~ = Applied shear stress
y = Shearing strain
Shear ModulUsG=7ly
'a r
OTT r 7 '
~ ~ ~ '. ~ ~ ~ ~ ~ ~
~ ~ Speclmen' ~ ~ ~ ~
\\r\
(b) Shear to nght
Figure 49. Stress Applied in Cyclic Simple Shear Test
102
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Very little change In permanent strain occurred in the repeated load biaxial test after 10,000 to 20,000 load
repetitions which is not true in a pavement. Thus, the permanent strain when stress reversal occurs is
greater, and the behavior quite different than In the repeated load biaxial compression test. This difference
in permanent strain behavior is explained by stressrotation induced "anisotropy". Arther, et al. [43] have
developed a "directional shear cell" for studying stress induced anisotropy. The directional shear cell,
which is more complex, apparently overcomes the limitation of the conventional simple shear apparatus
, , ~ , , ,
which applies only a fixed rotation to the specimen.
Conceptually, He simple shear test is easy to perform. In practice, however, applying a uniform
shear stress distribution to the top of the specimen and Inducing uniform deformation has frequently caused
problems [421. Also, the complex stress state to which the specimen is subjected is not easy to evaluate
[411. The simple shear test has been used on a limited basis for a long time, but because of these
disadvantages has not become well accepted for either static, cyclic or repeated load testing. The simple
shear test does offer an attractive alternative for evaluating permanent deformation under more realistic
stress reversal conditions than the repeated load biaxial test. The test, however, is not considered to be
suitable for routine resilient modulus testing of pavement materials.
Hollow Cylinder Test
The hollow cylinder test can more closely simulate the complex stress conditions, including
principal stress axis rotation, to which pavement materials are subjected than either He biaxial test or the
simple shear test [441. In this test a moderately ~ickwalled, hollow cylindrical specimen is used as
shown in Figure 50. Because of the limited wall Sickness, material particle size is limited to about a
coarse sand size. Both the inside and outside of the specimen is enclosed by a membrane. In addition to
being able to vary the axial and torsional stress, the pressure on the inside of He hollow cylinder can be
varied relative to He outside cell pressure, thus changing the tangential stress within the specimen. The
hollow cylinder test, because of its great flexibility In applying different stress paths, offers a valuable tool
for advanced research into pavement material characterization. The hollow cylinder apparatus appears
particularly suited to investigate permanent deformation under stress states simulating moving wheel loads.
Because of the complicated test apparatus required, extensive instrumentation and complicated specimen
preparation, the hollow cylinder is not suitable for routine resilient modulus testing.
True (Cubical) Triaxial Device
The true biaxial device tests a cubeshaped specimen and is capable of independent control of each
of the three orthogonal normal stresses applied to the specimen. Only two of the normal stresses are
controlled in tests using standard biaxial, simple shear, and plane strain devices. True biaxial apparatuses
have been built wig either rigid platens or flexible membrane boundaries and have been used on both sand
and clay materials. The cubical device can be used to investigate the influence of intermediate principal
stress, initial stress anisotropy, strength anisotropy, and stress invariants [451. Because of its complexity,
however, the true triaixal device is not suitable for routine repeated load biaxial testing.
103
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Hydraulic
Pressure
Supply
l
r ~
~ By ~
_ L A
AL _ _
LiT
I 1 1=__ ~1
. ....
ll l

: ~
~ .
. . ~
. I.D.
_ _
O.D.
_f
500 mm
(19.7 in.)
Steel
Frame
Serv  Controlled
Hydraulic Actuator
Slip Coupling
 Smoothing Bush
Load Cell
Top Plate
Top Platen (toothed)
Hollow Cylinder
Specimen
Bottom Platen (toothed)
I. De = Intemal Diameter
224 mm (8.8 in.)
O. D. = Outer Diameter
280 mm (! I.0 in.)
Figure 50. Stresses and apparatus used for hollow cylinder
test (after Reference 44)
104
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GRANULAR MATERIAL RESILIENT MODULUS MODELS
The resilient modulus of unstabilized granular base and subgrade soils is highly dependent upon
the stress state to which the material is subjected within the pavement in addition to other variables. As
a result, constitutive models must be used to present laboratory resilient modulus test results, including
the effect of stress state, in a form suitable for use in pavement design. This section summarizes resilient
modulus models presently used for granular materials. The accuracy of selected granular material resilient
modulus models is examined later in this chapter.
KO Mode}
The K8 model, summarized in Table I8, in the past has been overwhelmingly the most popular
nonlinear resilient modulus model. This mode! expresses resilient modulus as a function of He bulk stress
(~) to which the specimen is subjected. The bulk stress (~) equals Be sum of the principal stresses
(~+~2+~) acting on Be specimen. Me experimentally determined relationship between resilient
modulus (MR) and ~ is a straight line on a logIog plot. As a result, the constants K, and K2 used in Be
KO mode] can be readily obtained using a linear regression analysis of the log MR ~ log ~ data. May and
Witczak [46] and later Uzan [4] have pointed out the most serious drawback of the KO mode! is neglecting
He important effects of shear stress (Figure 51a). Other modifications of the K8 mode} have used the
minor principal stress (~3) rather than ~ and also a stress ratio term to consider behavior at stress levels
above the static failure stress [41.
Simple Models that Consider Shear Stress Effects
Uzan Model. Uzan [41 developed a mode] that considers both bulk stress (~) and He deviator
stress (od) The deviator stress is directly related to He maximum shear stress (Im) applied to the specimen
(i.e., Tm = O`' /2~. The Uzan mode} (Table ~ 8) therefore overcomes the most serious deficiency of the K0
model: not considering shear stress effect. The Tree material constants K3, K4 and Ks given in Table IS
must be evaluated by multiple regression analysis from a sequence of repeated load resilient modulus tests.
The stress sequences specified in either the AASHTO T292, AASHTO T294, or SHRP P46 test methods
can, for example, be used to evaluate the constants in the KO and Uzan models.
The relatively good agreement between laboratory resilient moduli and the Uzan mode! is shown
in Figure Sib. The Uzan mode} also exhibits quite good agreement with the considerably more
complicated contour mode} of Brown and Pappin [481.
Octahedral Shear Stress. Witczak and Uzan [491 modified the original Uzan mode! by replacing
the deviator stress wig octahedral shear stress which is a more fundamental parameter. The bulk stress
and shear stress were also normalized using atmospheric pressure. Since deviator stress and octahedral
shear stress are proportional, the two models should have the same accuracy.
UTAustin Model. The University of TexasAustin mode} [50] uses confining pressure instead
of the bulk stress employed in He Uzan mode] (Table Age. Me KD mode! is not statistically sound since
the resilient modulus is related to the deviator stress (oaf) which is hidden in the prediction variable bulk
stress (O = O] +~2+~3 = O~ +3CT31. The UTAustin mode! overcomes this problem by using as response
the axial strain (ea) instead of the resilient modulus, and by using as independent predictors the deviator
stress (~: and the confining pressure (~3~.
105
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Table IS. Resilient modulus models used in the experimental data analysis
MODEL  MODEL  LINEAR REGRESSION MODEL PLOT
EXPRESSION USED
K8 I M,` = KI8K2 I Log Me= LogK, + K:Log' Log MR VS
.
UZAN Ml = K3yc4(a~)K ~Log MR= L~gK3 + K.Log~ ~ K5Logo ~Log MR VS
M,, N6a~N7a3N' Log M,' VS
N. = K6 = 10. Log al
UTAUSllN N1 = 1 K? LogC. = a + K] Legal + K,LO8~3 Or
N. = K, Log M,, VS
{TrEP M' = K' SKIS, Kll LOB Me = l~gK' + KloLog' ~* Log M,` vs
l  K.' Log`, Log' l
106
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6
O '
·_
C

:'

o
._

._
Cat
Is
3
2
1
O . .,

lt .tl~ll  . ''tt't'
620pt~> if/
lo.
7~ $; ,~C, 10
5 ~ 7.5
I  C~a,~
Legend
· · Test Results
~ ~ ~ KD Equation
~ ~ tatted ~ ~ ~ ~ Al
. 1os 10 ~ 103
Axial Vertical Strain
(a) K8 mode}
B
_ STATE ~ T L l · 1~
DGALS2~) ~
~ 5 ~ \: ~
P: ~3p~' I__
2  Legend
. . Experimental Results
~   ~ Uzan's Equation
1 I , ~ t · ~ l ~ ~ t"'J
os 10 ~103
Axial Vertical Strain
(b) Uzan model
Figure 51. Test results and predicted behavior using K8
and Uzan models for a densegraded aggregate
(af ter ref erence 4)
10/
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it:
·m
co
c
a.
q At
Ed
·
~ 
o ~
·  4 
~ I:
 · 
~ 
·
o ~
v
a)
D
Ig 1~ 1~ 1: ~ ~ ~
:~m
.
~ I ~ ~   0  0 0  0 0  0 0
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1~ T ~m
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j_~0. ~0.
7~
1 ~ 1 10 1= ~ 1~ ~ 1 ~
Lath.
W~ ~
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240
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 a
Z 0
0 ~
Zip
O
~ ~ ~ 0
no ~ ~
Cal ~0
e.
4 
C~
c:
~ ^
0
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~ V
Cal
Cd
0 ~
V ~
0 ~
._
3
o ~
.
En
AM
Y . If. }. ~1  18 1 1~1 §~ § ~ t ?
o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ 3 3 3 3 3 3 ~ ~ 3 3 3 3 3 3 3 3 3 3 .
a ~ 0 ~ ~ ~ Ifs I °l   2~ I ~~ ~ ILIAC ~
~L: 1211 Lit 121 ~ ~ ~ ~ ~1

~ :~3~ ~
~ ~ ~ ~ ~ ~ o _ =~ =~ ~! ~ ^~ m~ _ ~ ~ ! =! ~! =~ ~! s~ m1 ol 1 ol 1 ~
ILL I I ~ ~ 1 ~: 1i ili~ili ilili i ili
1 ~ i44~T
1 1  1   1 ~ 1 X _ 1  1  ~ 1 1 ~ 1 1 ~ 1 1 ~ 1 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 
!1:L ~ ~ ~ ~ ~ ~ ~ ~ 1 1 1 ~1 ~1 =1 1 =1 1 1 1 ~
. 1 ~ ~ ~ t] ~ ~
241
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a
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Example. The Li and Selig approach can be best illustrated by the following practical example. ORen
the design moisture content (w) and dry density (Yd) is different from the reference state. For example,
to simulate a weak wet season subgrade condition, He resilient modulus to be usM in one portion of a
pavement design might be at a moisture content above the optimum value wig the corresponding dry
_ _ _
A"na;~r 1~o than V ao ;ll'~c!  q.~1;  ll;r  ~" 1~^ Tl~^ '.~  ^~q^~;r~ ^~.~r~ ;~ ~;~^ In ~
U~ll~lt' I`;~ Ul=1 rd(malc) ~ 1ll"~"at`;U Ill rl'&Ult; 14U 111O Uppt;l ~Ulili)~LlUll SHIV\; lit (l't;Ult; l4U Was
Aetemuned by performing an AASHTO T99 compaction test. This curve establishes the reference state
defined by w,,p~ and Y`' Oman' (point ~ in He figured. The resilient modulus is determined for this reference
state by performing a laboratory test. Point 4 represents the point of higher moisture content and lower
density at which the design resilient modulus is desired. The curve upon which point 4 lies corresponds
to a compaction curve that could be obtained using a lower constant energy than for the reference upper
curve.
The resilient modulus must now be determined at the different physical state (i.e., different density
and moisture content) than He reference value. Visualize a change in resilient modulus occurring in three
increments while moving in Figure 120 from point ~ to 4 as shown on the figure. Point 4 represents the
correct resilient modulus at the corresponding desired physical state. In moving along the curves,
however, the following two rules must always be followed to obtain the correct end point resilient
modulus: (~) movement must be along a compaction curve which represents a paw of constant compaction
energy or else (2) along a horizontal line which represents a path of constant dry density. Movement must
also either begin or end at He reference point (i.e., the point on the compaction curve corresponding to
wape art Ya tmax')
In Figure 120, the path which satisfies these requirements is from ~ to 2, then from 2 to 3 and finally
from 3 to 4. To solve this problem, the resilient modulus is calculated at the end point of each increment
of movement.
Step I. Now apply this incremental movement approach to He problem summarized and solved in
Table 57. Steps given in the table correspond to Hose given here. As the first step in determining He
required resilient modulus, move down the constant energy compaction curve from ~ to 2 and calculate
the resilient modulus at point 2. When movement is from the reference point down a constant energy
compaction curve, He new resilient modulus is determined from He following general expression
developed by Li and Selig:
where
MRC = [0.96  0.~S (W  WOP, ~ + 0.0067 (WW0Pt ~ ~ · MR(OPL)
MRC =
MR (~)
resilient modulus at point defined by w on a compaction curve
reference resilient modulus at w<,p, and Yd~ma,`' for He compaction curve
(14)
The above expression is valid, as a good approximation, for all fine "rained soils and its derivation is
discussed subsequently.
. · ~. ~a 
Step 2. Now move from 2 to 3 along a constant density line and calculate MR((,Pl) for the lower
compaction curve from He following general expression given by Li and Selig for constant density:
MR = [0.98  0.28 (ww~,p) + 0.029 (wwOp~ ~ ~ MR(OP~
242
~.~.~ap ~
.
(1~
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110~0
108.0
106.0
A; 1 04.0

.2 ~102.0
c
100.0
98.0
9660
94.0 _
92.0  1
15.0 17.0
. _
. _
(I) Initial Reference State: MR(OPt) Known
(Wopt 18%, Ed = 107 pep
_
(I Reference Case
~_ _

AASI ITO T99
Compaction Curve

Lowl)ensitY \~ 3) Desired:
Constant Energy
Compaction Curve (W = 22C%o, ad = 97 pcf)
1 1 1 1
19.0 21.0 23.0 25.0
Water Content, w(°/O)

Figure 120. Illustrative example problem  Li and Selig resilient modulus model
243
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Table 57. Numerical example problem for determining resilient modulus
finegrained subgrade: Li and Selig Method [57, Il0]
Required: Determine the resilient modulus of a clayey silt at a moisture content of 22% and a dry
density of 97 pcf (Point 4, Figure 120~.
Solution: Perform an AASHTO T9 compaction test on the clayey silt to determine the reference
compaction characteristics as shown on We upper curve in Figure 120 (We, = IS%, ym"~ =
107 pc0. The soil has a T99 compaction reference resilient modulus of 22,000 psi at We design
deviator stress. Ihe reference resilient modulus at W,~,p and York can be determined from design
relationships such as equations, graphs or a catalog of soil types and corresponding reference
resilient moduli. The resilient moduli selected should be based on the results of laboratory
resilient modulus tests.
Next, knowing the moisture content (22%) and dry density (97 pcf) at which We design resilient
modulus is desired, sketch in the lower compaction curve on a scale plat. The corresponding
optimum moisture content is about 19.596 for a maximum dry density of 101 pcf. In drawing
in We curve remember the zero air voids curve is approximately parallel to the line of optimum
moisture contents for different energy levels.
Now perform Steps 1 to 3 calculations as described in the text:
Step 1: Points 1 to 2 using Equation (14) Point 2: Yd = 101 pcf, w = 23%)
MRC = [0.96  0.18 (ww,,pt) + 0.0067 (WW(,Pt) ] · MR(q~t)
= [0.96  0.18 (2318) + 0.0067 (2318)2] . (22,000 psi)
= 5,005 psi at Point 2
Step 2: Points 2 to 3 using Equation (15) (Point 3: Yd = 1Ol pcf, w = 19.5%)
MR(OPt) = MR / [0~98  0.28 (ww~ + 0.029 (ww~,p~ ]
5,005 psi / [0.98~.28 (2319.5) + 0.029 (2319.5)2]
= 14,089 psi at Point 3
Step 3: Points 3 to 4 using Equation (14) (Point 4: Ye = 97 pcf, w = 22%)
MRC = [0~96~18 (ww.,p) + 0.0067 (WW{,PL) ] MR(opt)
[0.96~.18 (2219.5) + 0.0067 (2219.5)2] . (14,089 psi)
7,775 PS; at Point 4, which is the required design resilient modulus
1 1
244
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where
MRD = Resilient modulus at moisture content w; on constant density line (point 2, Figure
120)
MR(OlPl)
reference resilient modulus at was and Y`yma~c, on constant density path
Equation (15) is solved for MR(OPl) for the lower compaction curve (refer to Step 2 in Table 57~.
Step 3. The desired resilient modulus is then calculated by moving down the lower compaction
curve from point 3, the reference point for that curve, to point 4. Calculate MRC from equation (14) using
the known resilient modulus MR(OP,) previously calculated in Step 2.
For most design problems, the physical state at which the design resilient modulus is desired will
probably lie on the AASHTO T99 (or T180) compaction curve. For this condition, only Step
calculations are required.
Equation Development. Equations (143 and (15) were developed by correlating for a wide range of
cohesive soils the resilient modulus ratios MRC/MR`op`, and MR6/MR`op~with the corresponding difference
in moisture content (ww<,p~) as shown in Figures 121 and 122. Equation (14), which is for constant
compactive effort has a correlation coefficient r2 = 0.83, and is based upon data from 26 repeated load
tests on 10 finegrain soils. Equation (15), which is for constant density, has a correlation coefficient
r2 = 0.76 and was developed from 27 repeated load tests on ~ ~ finegrained soils. The repeated load test
data was taken from the literature from 8 different sources for each correlation.
Breakpoint MR Estimation. The bilinear resilient modulus mode} for f~negrain~ soils has a distinct
breakpoint as shown in Figure 53. The resilient modulus at the breakpoint can be estimated using the
following expression developed by Thompson and LaGrow [85,1111:
MA= 4.46 + 0.098 (% clay) + 0.~19 (Pl)
where
(16)
MR((,Pt) = Breakpoint resilient modulus at optimum moisture content and 95% of AASHTO
T99 maximum dry density
% clay = % particles finer than the 2 micron size
PI = Atterburg plasticity index
Equation (16) is for cohesive soils compacted to 95% of AASHTO T99 maximum dry density at the
optimum water content. Equation 16, although useful, has the important disadvantage that the resilient
modulus is at only He breakpoint. The breakpoint is often, but not always, at or close to He minimum
value of the resilient modulus. Thick pavement sections apply low deviator stress to the subgrade. As
a result, He breakpoint resilient modulus is lilcely to be too low for strong sections resulting in unnecessary
additional thickness.
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R  '
5
3
~ ~ 2
.
~/M, (  i) = 0.~  0.18 (W  W  i)
+ 0.0067 (w  we,,,) 2
~ = 0.83
1.
6 5 ~ 3 2  1 0 1 2 3 4 5 6
(w wqpt\%)
Figure 121. Correlation of resilient modulus ratio MC~/M~,p~ with moisture content
change constant compactive effort (after reference 57 and ~ 103
Ret,
5
~3
\ ^^
dS I\
~
_~
6 5~ ~2  1 0 ~ ~3 ~5
(W Wollty%)
M!~/M, (  ,) = 0.98  0.~ (w  w~,)
+ 0.~29 (wwept) 2
= 0.76
. I . I . I . I .A ~ 1

Figure 122. Correlatmn of resilient modulus ratio MD,'/M~.p`) moisture content
change constant density (after reference 57 and 1 10)
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SUMMARY AND CONCLUSIONS
General Conclusions  Granular and Cohesive Materials
The repeated load biaxial test is proposed as the most appropriate method at this time for evaluating
in the laboratory the resilient modulus of aggregate base and granular subgrade materials. An unconfined
repeated load test is proposed for cohesive subgrade soils. Great caution, however? must be exercised in
performing these tests. The proposed test procedures are given in Appendix E. General important
findings relating to laboratory testing details are as follows:
I. Testing. A large number of resilient modulus tests were performed on both aggregate base
subgrade and synthetic specunens. A closedIoop, electrohydraulic testing system was used to apply
a 0. ~ sec. haversine shaped load pulse. The tests were performed on a very carefully calibrated
testing system. Resilient moduli were determined using (~) EVDTs mounted externally to the
biaxial cell and also using EVDTs located on clamps attached directly on the specimen. A limited
number of tests were also performed with EVDTs mounted between the top and bottom platens.
2. System Compliance, Calibration and Specimen Misalignment. To obtain reliable, accurate
results from the repeated load biaxial test, compliance in the testing system must be minimized and
the system carefully calibrated, including testing synthetic specimens. Follow the procedures given
in Appendix D. During testing, to insure good load alignment, the ratio of the maximum to
minimum displacements obtained from two independent transducers should initially be less than
1.10. Complete system calibratiorl, carefully performed is absolutely assert if axial
defonnation is measured outside the biaxial cell.
With proper, tedious, equipment calibration, both clamp mounted L~VDTs and external EVDTs gave
satisfactory results for resilient moduli of aggregate bases less than about 60,000 to 70,000 psi. The
use of external L`VDTs should not be attempted, even under He best of conditions, when Be
resilient modulus is greater than about 70,ED0 psi. The calibration procedure required when external
EVDTs are used is very tedious, timeconsuming and may be subject to erratic, unknown errors in
He resilient modulus. Exterrud [VDTs are considered not suitable for routine laboratory testing
of ei~herg~r or cohesive materials if reliable resilient moduli are required. This conclusion
was supported by a limited multilaboratory study on aggregate base material given in Appendix H.
3. ~L=. Measurement of axial dLefonnati~n and system calibrafio It
are the most important factors in obtaining accurate, rehinge resilient modulus test rests.
Preconditioning, pulse shape, pulse time, type testing equipment, etc. all have a relatively minor
effect on resilient modulus compared to these two factors. Adequate sensor sensitivity, for specimen
loading conditions giving small deformations, is also an important consideration often overlooked.
Measurement of axial Spacemen outside of the triadic cell cannot be Reperked upon to give
reliable values of resilient mod~di. This finding is true for most laboratories even if He calibration
procedures given In Appendix D are follows including correcting measured deformations for system
compliance. To obtain accede resilient modFuli, awl displacement of grander materials shard
be measured Erectly on Me speezmen. Axial displacement can be measured using either an optical
extensometer, clamp mounted L`VDTs or proximity gages. In general, as sensitive as possible
displacement measurement gage should be used. Maximum recommended excitation voltages should
be used; for small displacements even consider exceeding these voltages by 10 to perhaps 20%.
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4. Modern Optical Extensometer. A modern optical extensometer offers at this time the most
practical, reliable memos of axial displacement measurement. The optical extensometer, however,
costs a minimum of about $30,000 to $40,000. Use of this measurement system requires a special
biaxial cell or else a vacuum test has to be performed.
Aggregate Base and Granular Subgrade Materials
I. Vacuum Shear Triaxial Test. Most aggregate base and granular subgrade resilient modulus tests
are performed at optimum moisture content and maximum dry density. For this condition, a
repeated load can be applied to a granular specimen subjected to a vacuum inside the specimen rawer
than the conventional external confining pressure. This approach eliminates the need for using a
biaxial cell. As a result, axial deformation can be readily measured using a modern optical
extensometer. The vacuum type biaxial test is limited to a maximum confining pressure of about
10 to 12 psi which is adequate for granular base and subgrade materials. Use of a vacuum type
repeated load test with axial strain measured with an optical extensometer offers a very practical and
accurate test procedure.
2. QuasiStatic Modulus Test. For some laboratories, the quasistatic modulus test offers a
realistic, simple test which does not require either an electrohydraulic testing system or a data
acquisition system. The quasistatic test, in summary, consists of first conditioning Me specimen
and then applying a single pulse to the specimen for a short time (2 to 5 min. was used in the tests
but shorter times can be used) and then removing the load. The test, if desired, can readily be
performed using an inexpensive, pneumatic loading system and simple instrumentation consisting
of clamp mounted EVDTs and a voltmeter to read the output.
3. Moisture Sensitivity. The resilient modulus moisture sensitivity of a material indicates how
much detrimental effect an increase in moisture has on the resilient modulus of a particular class
material. Resilient modulus tests should be performed to evaluate moisture sensitivities of pavement
materials.
4. I. For the good quality aggregate base materials tested in this study a
reduction in resilient modulus was not observed due to bedding errors (i.e., irregular particle contact
with the end platens). As a result, 200 repetitions of a conditioning loading was found to be
adequate for good quality aggregate base. A realistic principal stress ratio (o,/a2) for an aggregate
base is about 3.5. The experimental findings, however, indicate using a principal stress ratio of 2.0
during conditioning gives similar resilient moduli during subsequent testing. Therefore, to avoid
causing excessive permanent deformations in weak specimens, a principal stress ratio of 2.0 was
used in the recommended test procedure. After vibratory compaction, the top platen was placed on
the specimen and vibrated for about 10 sec. To obtain good contact.
5. Repeatability of Ma. In the present study the coefficient of variation (CV) of
resilient modulus measurements using one specimen was 9.3 using external transducers and ~ I. ~ %
using EVDTs mounted on clamps. Using two specimens, the coeff'cientof variation decreased
sign~ficandy to 6.6% and 7.~% for external and clamp measurements, respectively. The coefficient
of variation of course varies wig Be equipment used, skill and experience of Be technician
performing Be test and materials being tested. Nevertheless, this 30% reducfion in variation gives
a great deal of justification for using a duplicate set of specimens as a routine MR test pro cedFure.
Even more support is given to using duplicate specimens considering all the problems that can
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develop during a resilient modulus test involving sample preparation, instrumentation and even
human error (particularly if the test is not automated).
6. Resilient Modulus Models. The popular KD resilient modulus, for duplicate specimens, was
found to have an average r2 value of 0.89 with a mean square error of 0.00389. Improved models
had average r2 values between 0.96 and 0.98 and mean square errors 409to or more smaller Han the
KD model. Perhaps more way, the KB mode! ~ not have the required accuracy to
flush one muted from another at the 95% confidence [eve!. The Uzan, U.T.Austin, and
UTEP models did have this ability. These three models are superior to the K8 model. Either the
Uzan or the U.T.Aust~n models are considered suitable for use in practice. The Uzan, UTEP and
U.T.Austin models (also the related SHRP P46 Qune, 199~ model) all work with materials having
important cohesive properties such as I'meflyash stabilized base materials; the KO is not applicable
for such materials and should not be used.
7 LimeFlyash Stabilization. The addition of limeflyash to an aggregate base can increase its
.
resilient modulus by a factor of about 10 provided the limeflyash specimens are permitted to cure.
. ~ ~ ,
. . ~a ~ · , _ ~1 _ ~ ~ _
The repeated load test can still be used to measure me resilient modulus. Axial Deformations,
however. are extremely small. As a result, external axial displacement measurement is unreliable
. .. . . ~ ... . . . ~ , . . ~ ~ ~ A. ~ ^~ ~ ~ A _
even under the best of conditions, and deformation must be measured Erectly on the specimen. An
optical extensometer, which has a very high sensitivity, is well suited for such stiff materials. If
conventional EVDT or proximity transducers are used to obtain reliable results, instrumentation
should be selected having as high sensitivity as available and practical.
S. Cement Stabilization. Resilient modulus tests on cement stabilized specimens were not
performed during this study. The high stiffness of cement stabilized specimens gives small axial
displacements which require a very high precision measurement system. To accurately measure
resilient moduli of cement stabilized materials (~) use an optical extensometer having high precision
and (2) use cylindrical specimens either 4 in. or 6 in. in diameter depending upon the precision of
the extensometer and size of aggregate present. An alternative approach would be to use the
diametral test and optical extensometer.
9. Permanent Deformation. For most miseries and inserrice conditions, the measurement of
pennanent defo~n is mom import than the measurement of resilient modulus. Permanent
deformation repeat load biaxial tests should be performed for all classes of base/subbase materials,
particularly for marginal materials. To perform this test, 50,000 or more load cycles should be
applied to He specimen to evaluate their potential rutting characteristics. A pneumatic loading
system can be used wig permanent deformations being measured using dial indicators.
Cohesive Subgrade Soils
I. Unconfined Compression Test. An unconfined resilient modulus test is proposedfor cohesive
sods. The unconfined test does not require a biaxial cell, has a fewer number of stress sequences
and allows the easy use of an optical extensometer to measure axial `deformation. The effect of
eliminating in He test the small confining pressure present in He upper part of He subgrade is only
slightly conservative for cohesive soils. The proposed unconfined resilient modulus test procedure
is given in Appendix E. S`milady to aggregate base minerals, equipment calibration and the
correct measurement of resilient al specimen defonnaiion are considered the most ~mpo~a~
factors in obtaining accurate resilient moduli test results.
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2. Axial Deformation. Axial deformation of unconfined cohesive subgrade specimens should be
either measured directly on We specimen or else between solid end platens. If deformation is
measured between solid end platens, either the specimen ends should be grouted, or else the resilient
moduli obtained should be corrected empirically for end effects. EM effects due to irregular
particle contact are not eliminated by con~ionzag am become zmpoffant for resilient moduli
greater than about lO,OOO psi. The solid end platens proposed for the test prevent drainage and
hence give a conservative resilient modulus.
3. Moisture Sensitivity. The reduction in resilient modulus due to an increase in moisture content
is an important factor that should be considered in developing a laboratory testing program and
subsequently in design.
4. SDecimen Conditioning. Special conditioning has been considered in both the SHRP and
AASHTO test procedure to eliminate end effects and specimen aging effects. Recent carefully
conducted research, however, shows conditioning does not eliminate either of these problems
which both must be considered in resilient modulus testing.
5. Strength Gain with Time. After compaction, cohesive soil specimens undergo up to 30% or
more strength gain wig time. Contrary to early results obtained by Seed am his associates,
conditioning compacted specimens does not hale the strength gain with time effect on the
resilient Mobius. Because of this strength gain, compacted specimens should be tested at the same
age (2 days is recommended). For design, the laboratory resilient moduli should preferably be
empirically corrected for longterm strength gain.
6. Specimen Preparation Me~od. The resilient modulus of cohesive soils is dependent upon the
method used to compact the specimen. This variation in resilient modulus caused by different
compaction melons results from differences in soil structure. Therefore, laboratory compacted
specimens of cohesive soils should be prepared by different compaction methods depending upon
He ascompacted and anticipated future moisture content.
7. practical MR Models for Design.
models were developed for compacted cohesive soils.
Fundamental resilient modulus and permanent deformation
To apply this approach, generalized
normalized relationships are first developed for a specific AASHTO soil class using He repeated
load tria~cial test. Specimens are tested at only He optimum moisture content and dry density. After
once developing the generalized relationships, resilient moduli for site specific conditions can then
be determined by performing a conventional static shear strength test and several over routine tests.
Another fundamental resilient modulus mode! proposed by hi and Selig is also presented. Both of
these models offer a practical approach for design involving the performance of a minimum
number of resilient modulus tests when the approach is first developed.
250