Appendix A
Prediction of HMA Permanent Deformation
(Prepared by: A. Abu Abdo and F. Bayomy)
This appendix describes further literature review of permanent deformation models for asphalt
pavements, which is performed under Task A1 – Review of previous studies and available data.
Prediction of HMA Permanent Deformation
Permanent deformation or rutting is one of the major stresses in flexible pavements, which is caused by
the densification and movement of materials under repeated loads, and also might results from lateral
plastic flow under the wheel track (NCAT 2000). When the new Superpave binder grade system was
developed under SHRP (Strategic Highway Research Program), it was suggested that choosing the right
upper binder grade could extend the life of flexible pavements with regards to permanent deformation;
unfortunately that is not the case.
Many approaches have been used to predict permanent deformation in HMA. Most of the widely used
approaches are based on the classic power relationship between the permanent strain and number of
load cycles as shown in Eq. 1.
𝜀𝑝 = 𝑎𝑁 𝑏
(Eq. 1)
where,
εp: Accumulated Permanent Strain at N cycle,
N: Number of Load Repetitions, and
a,b: Non-linear Regression Coefficients.
This classic power equation is derived from the secondary stage in a typical behavior of HMA tested
sample under repetitive loads as shown in Figure 1, where “a” is the intercept at N = 1 cycle and “b” is
the slope of the line.
The widely used approaches to predict permanent deformation in flexible pavements includes; Layered
Vertical Permanent Strain Approach, Plastic-Elastic Vertical Strain Ratio Approach, Permanent Strain
Rate Approach, and Rutting Rate Approach.
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 1
𝜀𝑝 = 𝑎𝑁 𝑏
Figure 1 Typical Permanent Deformation Behavior for HMA under Repetitive Loading (After NCHRP 1-37A)
Layered Vertical Permanent Strain Approach
Layered Vertical Permanent Strain Approach is based on determining the permanent strain in each layer
as a function of repeated load applications. Then the total deformation is computed by summing up
permanent deformation in each layer. Two famous models used this approach; Allen and Dean Models
(Allen and Dean 1986) and Asphalt Institute Model (May and Witczak 1992).
Allen and Dean Models were developed for all pavement layers. These models are incorporated in
software called PAVRUT. In addition to permanent deformation models, traffic and temperatures
models were added to the software (Allen and Dean 1986). This software can be used to estimate the
permanent deformation for any flexible pavement. The permanent deformation model for HMA is as
follow,
𝐿𝑜𝑔𝜀𝑝 = 𝐶𝑜 + 𝐶1 [𝐿𝑜𝑔(𝑁)] + 𝐶2 [𝐿𝑜𝑔(𝑁)]2 + 𝐶3 [𝐿𝑜𝑔(𝑁)]3
(Eq. 2)
where,
εp: Permanent Axial Strain,
N: Number of Stress or Wheel Load Repetitions,
Co = -0.000663.T2 + 0.1521.T – 13.304 + (1.46 – 0.00572.T).Log (σ1),
σ1: Deviator Stress, psi,
T: Temperature, °F,
C1 = 0.63974,
C2 = -0.10392, and
C3 = 0.00938.
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 2
Based on triaxial tests on 251 samples, the Asphalt Institute has developed a model that determines the
permanent deformation of HMA by including the effects of mix design properties. Thus, Eq. 3 was
developed. Users need to take into consideration that this model cannot be used for mixes with less
than 3% air voids or for deviator stresses larger than 90 psi (May and Witczak 1992).
𝐿𝑜𝑔𝜀𝑝 = −14.97 + 1.4808𝐿𝑜𝑔(𝑁) + 6.865𝐿𝑜𝑔(𝑇) + 1.107 log(𝑆𝑑 ) ⋯
−0.117𝐿𝑜𝑔(𝑉) + 1.908𝐿𝑜𝑔(𝑃𝑒𝑓𝑓 ) + 0.971(𝑉𝑣 )
(Eq. 3)
where,
εp: Permanent Axial Strain,
N: Number of load repetitions to failure,
T: Temperature, °F,
Sd: Deviator Stress, psi,
V: Viscosity at 70 °F, Ps x106,
Peff: Percent by volume of effective asphalt content, and
Vv: Percent by volume of air voids.
Plastic-Elastic Vertical Strain Ratio Approach
Plastic-Elastic Vertical Strain Ratio Approach was based on the constitutive relationship developed from
the statistical analysis of repeated load permanent deformation lab tests. This model is in a form of the
classical power model, but determines the ratio of the permanent strain to the elastic (resilient) strain
as in Eq. 4,
𝜀𝑝
𝜀𝑟
= 𝛽𝑟 (𝑎𝑁 𝑏 )
(Eq. 4)
where,
εp: Accumulated Permanent Strain at N cycle,
εr: Resilient Strain as a function of asphalt mixes properties,
N: Number of load repetitions,
a,b: Non-linear regression coefficients, and
𝛽r: Field adjustment factor.
Many studies have been conducted to determine the regression factors by relating these factors to
different mixes properties. Leahy (1989) argued that temperature was the most important factor.
Leahy’s model was less sensitive to loading condition and mix properties as seen in Eq. 5.
𝜀𝑝
𝐿𝑜𝑔 ( 𝜀 ) = −6.631 + 0.435𝐿𝑜𝑔(𝑁) + 2.767𝐿𝑜𝑔(𝑇) + 0.110 log(𝜎𝑑 ) ⋯
𝑟
−0.118𝐿𝑜𝑔(𝜂) + 0.930𝐿𝑜𝑔(𝑉𝑏𝑒𝑓𝑓 ) + 0.5011(𝑉𝑎 )
(Eq. 5)
where,
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 3
εp: Accumulated Permanent Strain at N cycle,
εr: Resilient Strain as a function of asphalt mixes properties,
N: Number of load repetitions,
T: Temperature, °F,
σd: Deviator stress, psi,
η: Viscosity at 70 °F, 106 poise,
Vbeff: Percent by volume of effective asphalt content, and
Va: Air voids, percent.
Similar to Leahy’s approach, Kaloush and Witczak (1999) developed two models that have fewer
parameters, but with approximately the same accuracy. These models are,
𝜀𝑝
𝐿𝑜𝑔 ( 𝜀 ) = −3.74938 + 2.02755𝐿𝑜𝑔(𝑇) + 0.4262𝐿𝑜𝑔(𝑁)
𝑟
(Eq. 6)
and,
𝜀𝑝
𝐿𝑜𝑔 ( 𝜀 ) = 0.1981 + 0.404𝐿𝑜𝑔(𝑁)
𝑟
(Eq. 7)
where,
εp: Accumulated Permanent Strain at N cycle,
εr: Resilient Strain as a function of asphalt mixes properties,
N: Number of load repetitions, and
T: Temperature, °F,
These models tend to under predict the εp/εr ratio at higher number of load repetitions, mainly due to
determination of the regression coefficients from steady-state zone of creep or repeated load tests. To
obtain better results, Kaloush and Witczak (1999) presented another model that predict the number of
cycle at failure based on the asphalt mix volumetric properties, binder viscosity, temperature and stress
level. The model is expressed as follow,
−3.6444 −0.9421
𝑁𝑓 = (1.00788𝐸5)𝑇 −1.6801 𝑆 −0.1502 𝜂 0.2109 𝑉𝑏𝑒𝑓𝑓
𝑉𝑎
(Eq. 8)
where,
Nf: Number of load repetitions to failure,
T: Temperature, °F,
S: Deviator stress, psi,
η: Viscosity at 70 °F, 106 poise,
Vbeff: Percent by volume of effective asphalt content, and
Va: Air voids, percent.
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 4
Permanent Strain Rate Approach
Permanent Strain Rate Approach is based on modifying the classical power model (Eq. 1) and
determining the rate of permanent strain with changes per cycle as shown in Eq. 9,
𝜀𝑝𝑛 =
𝜕𝜀𝑝
𝜕𝑁
=
𝜕(𝑎𝑁 𝑏 )
𝜕𝑁
= 𝑎𝑏𝑁 𝑏−1
(Eq. 9)
Divide by the resilient strain (εr),
𝜀𝑝𝑛
𝜀𝑟
=
𝑎𝑏 𝑏−1
𝑁
𝜀𝑟
By letting, 𝜇 =
(Eq. 10)
𝑎𝑏
𝜀𝑟
and 𝛼 = 1 − 𝑏. The final model form is similar to the classic power form as shown in
Eq. 11.
𝜀𝑝𝑛
𝜀𝑟
= 𝜇𝑁 −𝛼
(Eq. 11)
The term “μ” is the permanent deformation parameter representing the constant proportionality
between εp and εr. The term “𝛼” is the permanent deformation parameter that indicates the decrease in
permanent deformation as the number of cycles increases.
Rutting Rate Approach
The most famous model under Rutting Rate Approach is the Ohio State University Model (Majidzadeh et
al. 1981); the model describes the progression of rutting in all pavement layers. The permanent strain in
any layer is determined as follows,
𝜀𝑝
𝑁
= 𝐴(𝑁)𝑚
(Eq. 12)
where,
εp: Permanent Strain,
N: Number of allowable load applications, and
A, m: Experimental constants.
It was found (Barenberg and Thompson 1990) that values of “m” does not vary significantly (0.83 –
0.86). However, “A” varies widely (12.4x10-4 – 138x10-4) as per the material type, the stress level and the
environment.
Role of E* in the Prediction of Permanent Deformation in the 2002 AASHTO
MEPDG (NCHRP 1-37A)
MEPDG utilizes the Layered Vertical Permanent Strain Approach to determine the permanent
deformation for HMA layers. Where the general form is,
𝜀𝑝
𝜀𝑟
= 𝑎1 𝑇 𝑎2 𝑁 𝑎3
(Eq. 13)
where,
εp: Accumulated Permanent Strain at N cycle,
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 5
εr: Resilient Strain as a function of asphalt mixes properties,
N: Number of load repetitions,
T: Temperature, °F, and
ai : Non-linear regression coefficients.
While this relation provides adequate results, field shift factors (𝛽ri) were introduced to provide more
accurate predictions of this model, as shown in Eq. 14,
𝜀𝑝
𝜀𝑟
= 𝛽1 𝑎1 𝑇𝛽2 𝑎2 𝑁𝛽3 𝑎3
(Eq. 14)
Based on 88 Long Term Pavement Performance (LTPP) sites located in 28 states, Eq. 2.29 was calibrated
and modified using 387 observations to obtain better predictions than the original model. Later, a depth
parameter “k1” was introduced based on MnRoad test site (Stroup-Gardiner and Newcomb 1997), to
increase the accuracy of this model as follows,
𝜀𝑝
𝜀𝑟
= 𝑘1 × 10−3.4488 𝑇 1.5606 𝑁 0.479244
(Eq. 15)
where,
𝑘1 = 0.328196𝑑𝑒𝑝𝑡ℎ (𝐶1 + 𝐶2 . 𝑑𝑒𝑝𝑡ℎ) ,
depth: Depth to computational point, in,
2
𝐶1 = −0.1039ℎ𝑎𝑐
+ 2.4868ℎ𝑎𝑐 − 17.342 ,
2
𝐶2 = 0.0172ℎ𝑎𝑐
− 1.7331ℎ𝑎𝑐 + 27.428 , and
hac: Total Asphalt Layer thickness, in.
The resilient (elastic) strain is determined using the general Hooke’s Law (Eq. 16), where the asphalt
dynamic modulus is incorporated. Therefore, the determined resilient modulus is a function of materials
properties, temperature and load frequency.
1
𝜀𝑟𝑧 = 𝐸 ∗ (𝜎𝑧 − 𝜈𝜎𝑥 − 𝜈𝜎𝑦 )
(Eq. 16)
where,
εrz: Resilient Strain in the vertical direction,
E*: Dynamic Modulus of HMA,
σz, σx, and σy: Stresses in the z, x, and y directions, and
ν: Poison’s Ratio of HMA.
Finally, the determined permanent strain in any layer is used to determine the permanent deformation
by,
𝑅𝐷 = ∑𝑛𝑖=1(𝜀𝑝𝑖 . Δℎ𝑖 )
(Eq. 17)
where,
QR4_Appendix A: Prediction of HMA Permanent Deformation - Page 6
RD: Total Permanent Deformation,
i: Layer number,
n: Total number of layers,
εpi: Permanent Strain in layer i, and
∆hi: Thickness of layer i.
References
Allen, D.L. and R.C. Deen. A Computerized Analysis of Rutting Behavior of Flexible Pavement.
Transportation Research Record 1095, Washington D.C., 2005.
ARA, Inc., ERES Consultants Division. Guide for Mechanistic-Empirical Design of New and
Rehabilitated Pavement Structures. NCHRP Final Report 1-37A, Illinois.
Barenberg, E.J. and M. R. Thompson. Calibrated Mechanistic Structural Analysis Procedures for
Pavements, Phase 2 of NCHRP Project 1-26. National Cooperative Highway Research
Program. TRB. Washington, DC., 1990.
Kaloush, K., and Witczak, M.W.. Development of Permanent to Elastic Strain Ratio Model for
Asphalt Mixtures. Development of the 2002 Guide for the Design of New and
Rehabilitated Pavement Structure, University of Maryland, College Park, Maryland,
1999.
Leahy, R.B.. Permanent Deformation Characteristics of Asphalt Concrete. PhD Dissertation,
University of Maryland, College Park, Maryland, 1989.
Majidzadeh, K., et al. Implementation of a Pavement Design System - Volumes 1 and 2. Final
Report, Research Project EES 579, Ohio State University. 1981.
May, R.W. and M.W. Witczak. An Automated Asphalt Concrete Mix Analysis System. Proceeding
of the Association of Asphalt Paving Technologists, Volume 61. South Carolina, 1992.
NCAT. Hot Mix Asphalt Materials, Mixture Design and Construction. National Center for Asphalt
Technology. NAPA Research and Education Foundation, 2nd Edition, Maryland, 2000.
Stroup-Gardiner, M. and D. Newcomb. Investigation of Hot Mix Asphalt Mixtures at Mn/Road.
Minnesota Department of Transportation, Office of Research Administration, Final
Report, Minnesota, 1997.
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