The document summarizes a resilient modulus model developed for unbound pavement layers that accounts for the effects of moisture, stress state, and freezing/thawing. Key aspects include: - A "universal" resilient modulus model relates MR to confining pressure, deviator stress, and moisture. - MR decreases nonlinearly with increased moisture content according to sigmoid curves developed for coarse-grained and fine-grained materials. - Freezing/thawing is modeled using adjustment factors based on material type and season to account for very high or reduced modulus. - The model is implemented in the Mechanistic-Empirical Pavement Design Guide to predict seasonal variations in MR at the node and layer

1. Resilient Modulus Model
2009 TRB Annual Meeting Workshop
Environmental Effects in the ME-PDG
January 11, 2009
Dragos Andrei, Ph.D., P.E.

2. Stress Dependent MR Model
A harmonized resilient modulus test
method was developed at the University of
Maryland, under NCHRP project 1-28A
30 MR tests were performed on 6 materials
from different sources: FHWA-ALF,
MnRoad, USACE-CRREL.
Data was analyzed with 14 models
including the classical “k1-k2”models, the
“Universal” model and the “SHRP-
Superpave” MR model
1999

3. “Universal” MR Model
Applicable to both coarse-grained and
fine-grained materials
Includes both the volumetric and shear
components of stress
Normalization by pa makes ki parameters
dimensionless
k2 k3
⎛ θ ⎞ ⎛ τ oct ⎞
M R = k1 ⋅ pa ⋅ ⎜ ⎟ ⋅ ⎜
⎜p ⎟ ⎜ p ⎟ ⎟
⎝ a⎠ ⎝ a ⎠

4. Variations of the “Universal”
Model
Use semi-log instead of log-log form
Replace θ with (θ – 3k6) or σ3
Replace τoct/pa with (τoct/pa + 1) or
(σcyc/pa+1)
Replace τoct/pa with (τoct/pa + k7), where
k7>1
k2 k3
⎛ θ − 3k6 ⎞ ⎛ τ oct ⎞
M R = k1 ⋅ pa ⋅ ⎜
⎜ p ⎟ ⎜ p ⎟ ⋅⎜ + k7 ⎟
⎟
⎝ a ⎠ ⎝ a ⎠

5. Key Findings of the 1-28A
Study
Models including both θ and τoct were
clearly superior to the classical k1-k2
models
Log-log models were more accurate than
the corresponding semi-log models
Models using θ and τoct were generally
more accurate than those using σ3 and
σcyc
The higher the number of ki parameters –
the better the goodness of fit

6. Model Selection for
Implementation in ME-PDG
Goodness of fit
Computational stability
Implementable in the general framework of
the ME-PDG
k2 k3
⎛ θ ⎞ ⎛ τ oct ⎞
M R = k1 ⋅ pa ⋅ ⎜ ⎟ ⋅ ⎜
⎜p ⎟ ⎜ p + 1⎟
⎟
⎝ a⎠ ⎝ a ⎠

7. MR - Moisture Effects
Literature review performed at Arizona
State University in an effort to quantify the
effect of changes in moisture and density
on MR
Data retrieved from published papers:
Li and Selig, Drumm et al, Jin et al, Jones
and Witczak, Rada and Witczak, Santha,
CRREL, Muhanna et al.
2000

8. Key Findings of ASU Study
MR reduces with increased moisture; the
reduction in modulus is greater for fine
grained materials
Regardless of the model used, a linear
relationship is observed when plotting:
log(MR) versus moisture
Some researchers used S while others
preferred w
The compactive energy (standard or
modified) was not always specified

9. Analysis
Use approach from Li and Selig paper to
normalize MR, w and S with respect to
values at optimum and to plot change in
MR versus change in moisture
Use the literature models to create MR-
moisture data points
Divide materials into:
Coarse-Grained and Fine-Grained
Use sigmoid model form to fit the “data”

10. M R - M oisture M odel for Coarse-Grained M aterials
2.5
2
1.5
MR/MRopt
1
Literature Data
0.5
Predicted
0
-70 -60 -50 -40 -30 -20 -10 0 10 20 30
(S - S opt)%

11. M R - M oisture M odel for Fine-Grained M aterials
2.5
2.0
1.5
MR/MRopt
1.0
Literature Data
0.5
Predicted
0.0
-70 -60 -50 -40 -30 -20 -10 0 10 20 30
(S - S opt)%

12. MR – Moisture Model
b−a
a+
( (
1+ EXP β + k m ⋅ S − Sopt ))
M R = 10 ⋅ M Ropt
MOISTURE
ADJUSTMENT MR = FU*MRopt
FACTOR (FU)
MR = Resilient Modulus at S
MRopt = Resilient modulus at Sopt
a, b, km = Regression parameters
β = lne(-b/a) from condition of (0,1) intercept

13. a, b, km Values for ME-PDG
Coarse-Grained:
a = -0.3123
b = 0.3 (maximum MR/MRopt ratio of 2)
km = 6.8157
Fine-Grained:
a = -0.5934
b = 0.4 (maximum MR/MRopt ratio of 2.5)
km = 6.1324

14. MRopt Estimates in the ME-PDG
Several options available:
USCS Classification
AASHTO Classification
CBR
R-Value
AASHTO Structural Layer Coefficient
Gradation and Atterberg Limits

15. Combined Effects of Moisture
and Stress in ME-PDG
b−a k2 k3
a+
( (
1+ EXP β + k m ⋅ S − Sopt )) ⎛ θ ⎞ ⎛ τ oct ⎞
M R = 10 ⋅ k1 ⋅ pa ⋅ ⎜ ⎟ ⋅ ⎜
⎜ p ⎟ ⎜ p + 1⎟⎟
⎝ a⎠ ⎝ a ⎠
MOISTURE STRESS
ADJUSTMENT DEPENDENT
FACTOR (FU) MR MODEL
This form was implemented in the ME-PDG
for “unfrozen” unbound materials
Calibration/validation of the model with
laboratory test data was desired

16. Moisture Variation in Unbound
Pavement Layers
Compaction – optimum moisture content
FU With time – equilibrium moisture content
Seasonal – variations around equilibrium
Freezing – soil becomes very stiff
?
Thawing – temporary softening below
equilibrium stiffness

17. Freeze-Thaw Effects: Freezing
From Literature:
MR = 2,500,000 psi for non-plastic materials
MR = 1,000,000 psi for plastic materials
Model Form:
MR = FF*MRopt
FF = Adjustment factor for frozen materials
2001

18. Freeze-Thaw Effects: Thawing
Modulus Reduction Factor
0.40 … 0.85 as a function of plasticity index and
% fines (wPI)
Recovery Period
90 … 150 days as a function of wPI
Model Form:
MR = FR*MRopt
FR = Adjustment factor for thawing
(recovering) materials

19. Example
M innesota
100
FROZEN
10
Fenv
OPTIMUM
TR
1 EQUILIBRIUM
EQUILIBRIUM
RECOVERY
0.1
08/23/96 12/01/96 03/11/97 06/19/97 09/27/97
Tim e

20. From NODE to LAYER …
Tim e (days)
Nodes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 SPRING
1 AC ANALOGY
2
3 FF FF FF FF FF FF FF FF FR FR FR FR FR FR BASE
4 FF FF FF FF FF FF FF FF FR FR FR FR FR FR
5 FF FF FF FF FF FF FF FR FR FR FR FR FR FR
6 FF FF FF FF FF FF FF FR FR FR FR FR FR FR
7 FF FF FF FF FF FF FF FR FR FR FR FR FR FR
8 FF FF FF FF FF FF FF FR FR FR FR FR FR FR
9 FF FF FF FF FF FF FF FR FR FR FR FR FR FR SUBBASE
10 FF FF FF FF FF FF FF FR FR FR FR FR FR FR
11 FF FF FF FF FF FF FR FR FR FR FR FR FR FR
12 FF FF FR FR FR FR FR FR FR FR FR FR FR FR
13 FF FR FR FR FR FR FR FR FR FR FR FR FR FR
14 FR FR FR FR FR FR FR FR FR FR FR FU FU FU
15 FR FR FR FR FR FR FR FR FR FR FU FU FU FU
16 FR FR FR FR FR FR FR FR FU FU FU FU FU FU
17 FR FR FR FR FR FU FU FU FU FU FU FU FU FU SUBGRADE
18 FR FR FU FU FU FU FU FU FU FU FU FU FU FU
19 FU FU FU FU FU FU FU FU FU FU FU FU FU FU
20 FU FU FU FU FU FU FU FU FU FU FU FU FU FU
21 FU FU FU FU FU FU FU FU FU FU FU FU FU FU LEGEND:
22 FU FU FU FU FU FU FU FU FU FU FU FU FU FU FROZEN
23 FU FU FU FU FU FU FU FU FU FU FU FU FU FU RECOVERING
24 FU FU FU FU FU FU FU FU FU FU FU FU FU FU UNFROZEN

21. Fenv = Layer Adjustment Factor
Principle: Find Fenv corresponding to an equivalent (composite) modulus that produces
the same average displacement over the total thickness of the layer/sublayer for the
considered analysis period (1 month or 2 weeks).
t total ⋅ htotal
Fenv =
⎛ n ⎛ hnode
t total
⎞⎞
∑ ⎜ node =1 ⎜ F
⎜ ∑ ⎜ ⎟⎟
⎟⎟
t =1 ⎝ ⎝ node ,time ⎠⎠
hnode = Length between mid-point nodes
htotal = Total height of the considered layer/sublayer
ttotal = The desired time period (either a two-week period or a
month period)
Fnode,t = Adjustment factor at a given node and time increment
which could be FF , FR , or FU

22. Fenv Calculation Example
Tim e (days)
Nodes 1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 50 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 BASE
4 50 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 F env = 1.45
5 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 0.7
6 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 0.7
7 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 0.7
8 50 50 50 50 50 50 50 0.7 0.7 0.7 0.7 0.7 0.7 0.7
9 75 75 75 75 75 75 75 0.6 0.6 0.6 0.6 0.6 0.6 0.6 SUBBASE
10 75 75 75 75 75 75 75 0.6 0.6 0.6 0.6 0.6 0.6 0.6 F env = 0.92
11 75 75 75 75 75 75 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7
12 75 75 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7
13 75 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.7 LEGEND:
14 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 1 1 FRO ZEN
15 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 1 1 1 RECOVERING
16 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 1 1 1 1 1 UNFRO ZEN
MR (layer, analysis period) = Fenv*MRopt
All calculations done in EICM !

23. ADOT MR-Moisture Lab Study
Arizona DOT Materials
4 base materials
4 subgrade soils
Each material tested at:
3 moisture contents (optimum, soaked and dried)
2 compactive efforts (standard and modified)
2 replicates (minimum)
Total: 96 tests performed using the
NCHRP 1-28A test protocol
2002

24. Key Findings
Density strongly affects the MR-S
relationship and should be added as a
predictor to the model based on S
When gravimetric moisture content was
used instead, the effect of density was
greatly minimized
MR – Moisture models including stress
dependency (like the one in the ME-PDG)
were successfully used to fit the measured
lab test data

25. Effect of Density (Compactive
Energy)
Phoenix Valley Subgrade (A-2-4, SC), Hot Conditions
1,000,000
Resilient Modulus (psi)
100,000
Standard Measured
Standard Sigmoid
10,000
Modified Measured
Modified Sigmoid
1,000
0.0 20.0 40.0 60.0 80.0 100.0
Degree of Saturation (%)

26. Using Moisture Content
Phoenix Valley Subgrade (A-2-4, SC), Hot Conditions
1,000,000
Standard
Modified
Predicted
Resilient Modulus (psi)
100,000
10,000
1,000
0 2 4 6 8 10 12 14 16 18
Moisture Content (%)

27. Goodness of Fit – Phoenix
Valley Subgrade
PVSG (A-2-4, SC) - MR(w-w opt , θ, τoct) Model
2
n =142, Se/Sy =0.15, R = 0.98
1,000,000
100,000
10,000
MR Predicted
Line of Equality
1,000
1,000 10,000 100,000 1,000,000
Measured Resilient Modulus (psi)

28. Goodness of Fit – Gray
Mountain Base GMAB2 (A-1-a, GW) - MR(w-w opt , θ, τoct) Model
2
n = 254, R = 0.90, Se/Sy = 0.32
1,000,000
100,000
10,000
MR Predicted
Line of Equality
1,000
1,000 10,000 100,000 1,000,000
Measured Resilient Modulus (psi)

29. Fu for ADOT Base Materials
Grey Mountain Base (A-1-a, GW)
100
10
MR/MRopt
1
0.1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
wi - wopt (%)

30. Fu for ADOT A-2/SC Subgrade
Soils
All A-2 Subgrades, M R - Moisture Model
2
n = 36, R = 0.96, Se/Sy = 0.20
100
PVSG (A-2-4), PI=9.9, p200=21.6
FCSG (A-2-6), PI=17.2, p200=31.5
SCSG (A-2-4), PI=12.1, p200=25
Predicted
10
1
0.1
-12 -10 -8 -6 -4 -2 0 2 4 6
wi - wopt (%)

31. ADOT Database of MR Model
Parameters
Material ID AASHTO USCS a b kw β k1 k2 k3 w opt std
%
Phoenix Valley Subgrade A-2-4 SC 0.24 41.88 67.255 0.974 467 0.358 -0.686 11.3
Yuma Area Subgrade A-1-a GP 1.00 94.01 82.757 8.714 1,468 0.838 -0.888 11.0
Flagstaff Area Subgrade A-2-6 SC 0.31 10.93 74.489 0.722 634 0.187 -0.855 19.0
Sun City Subgrade A-2-6 SC 0.13 19.22 53.166 0.360 747 0.224 -0.104 11.3
Grey Mountain Base A-1-a GW 0.00 2096.40 2.559 -0.539 1,423 0.758 -0.288 6.7
Salt River Base A-1-a SP 0.59 2096.41 22.401 2.666 1,170 0.919 -0.572 6.9
Globe Area Base A-1-a SP-SM 0.68 2096.44 35.787 2.981 1,032 0.830 -0.307 6.7
Precott Area Base A-1-a SP-SM 1.00 2096.45 144.223 8.711 1,092 0.784 -0.236 6.3
ADOT A-1-a AB2 Base Materials A-1-a SP-SM 0.60 2096.65 24.221 2.721 1,075 0.841 -0.305 6.7
ADOT A-2 Subgrade Materials A-2 SC 0.22 21.79 58.965 0.699 - - - -

32. Final Remarks
Moisture, density and state of stress all
affect MR and should be included in a M-E
predictive methodology
Changes in moisture will trigger
significant changes in MR, especially for
fine-grained materials
Coarse-grained materials are especially
affected by changes in the state of stress

33. Final Remarks (Cont’d)
The MR-Moisture material models
implemented in the ME-PDG were verified
through a limited laboratory testing study
performed at ASU
Agencies could engage in similar studies
to develop a database of material
properties for typical unbound pavement
materials used on highway construction
projects.