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This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-04-079
Date: July 2006

Seasonal Variations in The Moduli of Unbound Pavement Layers

Chapter 2: Literature Review


As noted previously, this investigation is concerned with the seasonal variations exclusive of frost effects that occur in the moduli of the unbound base, subbase, and subgrade materials within pavement structures. To provide the context for this investigation, a brief discussion of the different methods of determining moduli for unbound pavement materials is followed by an overview of the means by which seasonal variations are addressed in the pavement design and evaluation process. Subsequent sections address the factors that influence the moduli of unbound pavement materials, efforts to develop relationships between those factors and moduli, the findings of field investigations of seasonal variations, related investigations, and existing environmental-effects models applicable to pavements.


The modulus of a material is a measure of stiffness, by definition, the ratio of stress to strain. The term modulus, by itself, is used here in the generic sense, and carries with it no implication as to how it was determined. In pavement engineering practice, several primary methods are used to determine the moduli of unbound pavement materials. The first is the laboratory resilient modulus, or Mr test. Another approach is to interpret nondestructive pavement deflection data through a process known as backcalculation to estimate the in situ moduli of the layer materials. The notation E (for Elastic modulus) is commonly used to refer to backcalculated moduli, and that convention will be used here, whereas the notation Mr will be used exclusively for the laboratory test result.

While Mr and E are used to characterize pavement stiffness for the same general purposes, it is important to understand that quantitative differences in magnitude may exist between these parameters. Numerical differences between E and Mr (for nominally the same materials) are well documented in the literature, as will be discussed in the next several paragraphs.

Lee, Mahoney, and Jackson compared layer moduli backcalculated using the EVERCALC program with those obtained via laboratory resilient modulus testing for 5 base layers and 16 subgrade soils at "similar stress states."[ 2 ] They reported differences in the range of 0 to 36 percent for the five base layers, with the lab moduli being consistently greater than the backcalculated moduli. Moisture content differences in the range of –0.3 to 1.0 percent (lab – backcalculated) may have contributed to the observed differences. For the subgrade layers, differences in the range of –2 to +84 percent were observed. For 11 of the 16 soils, thebackcalculated moduli were greater than the lab values. As with the base layer moduli, differences in moisture content in the range of –2.2 to 4.2 percent probably contributed to the observed differences.

Daleiden et al. report a mean ratio of laboratory resilient moduli to backcalculated subgrade modulus of 0.57. [ 3 ] The corresponding standard deviation and ranges were 0.67 and 0.01 to 10.34, respectively. The data used in their analysis were from LTPP test sections in the southern and north Atlantic regions.

The authors of the 1993 AASHTO Guide for Design of Pavement Structures (1993 guide) suggest that moduli backcalculated for fine-grained subgrade soils should be multiplied by an adjustment factor, C, not greater than 0.33 to approximate values obtained in laboratory testing.[ 4 ] They further state that the relationships between laboratory and backcalculated moduli may differ for granular materials, and that this subject requires further research.

Von Quintus and Killingsworth sought to improve upon the guidance provided in the 1993 Guide through analysis of LTPP data.[ 5 ] They reported the results presented in Table 1. The MODULUS program was used in the backcalculation for this analysis.


Table 1. Difference between laboratory and backcalculated moduli at equivalent stress states, Mr/E[ 5 ]
Layer Description Mean Standard Deviation Coefficient of Variation, %
Granular base/subbase under a PCC surface 1.32 0.978 74.1
Granular base/subbase above a stabilized material 1.43 1.14 79.9
Granular base/subbase under an asphalt concrete surface/base 0.62 0.271 43.8
Subgrade soil under a stabilized subgrade 0.75 0.095 12.7
Subgrade soil under a pavement without a granular base/subbasee 0.52 0.180 34.6
Subgrade soil under a pavement with a granular base/subbase 0.35 0.183 52.2

Overall, the literature suggests that backcalculated base and subbase layer moduli tend to be less than the corresponding laboratory values, while moduli for subgrade layers tend to be greater than the lab values, though exceptions do occur. Mr/E ratios in the range of 0.35 to 1.42 are typical, with the higher values corresponding to base layers, and the smaller end of the range representing subgrade layers.

A number of factors contribute to the observed differences between Mr and E. Whereas the laboratory resilient modulus, Mr, is appropriately termed a material property (a readily measured characteristic of a well-defined material sample), the backcalculated modulus, E, is not. The value of E depends not only on the "true" in situ characteristics of the material comprising the layer (including, but not limited to, stress state and moisture content), but also on the theoretical model used to derive (backcalculate) the value, the limitations of that model, and the details of the application of that model. Further discussion of issues related to the backcalculation process is provided in later sections of this chapter, and in Chapter 3. Other factors that may contribute to the reported differences include differences between the moisture, compaction, and confining conditions of the materials at the time of testing.


The degree to which seasonal variations in unbound pavement materials have been addressed in pavement design and evaluation, and the approaches taken to addressing them, are widely varied. Historically, the more widely known pavement design and evaluation procedures have provided for consideration of seasonal variations only indirectly. For example, early versions of the AASHTO pavement design procedure used a "regional factor" to adjust the design structural capacity of the pavement for climatic conditions more or less severe than those present at the AASHO Road Test, but did not directly address seasonal variations in the pavement structure.[ 6 ]

The 1986 AASHTO Guide for Design of Pavement Structures (1986 guide) was a watershed in relation to the treatment of environmental effects in pavement design: It was the first widely used pavement design methodology to incorporate explicit consideration of site-specific seasonal variations in the stiffness of the subgrade soil, through the effective subgrade soil resilient modulus.[ 7 ] Conceptually, the effective subgrade soil resilient modulus is a damage-weighted average. A nomographic solution is provided for the determination of the relative damage, uf. This approach was retained in the 1993 guide.

One limitation of the 1986 guide (and the 1993 guide as well) is that it makes no explicit provision for consideration of seasonal variations in the overlying pavement layers. Furthermore, incomplete knowledge of the magnitude and duration of the subgrade modulus fluctuations that occur, and the manner in which they vary as a function of location, materials, and other factors, made it difficult for highway agencies to take full advantage of this advance.

Recognition of the need for explicit consideration of seasonal variations in the structural characteristics of pavement materials has paralleled, if not arisen from, the development of mechanistically-based approaches to pavement design and evaluation. Within this context, the ultimate approach to considering seasonal variations is to divide the design period into "n" discrete periods, such that the pavement structure and loading conditions within a given period may be treated as constant. Cumulative damage concepts are applied to sum the damage caused to the pavement in each period (i.e., each combination of pavement structure and loading conditions) to obtain an estimate of the total damage induced over the design life.

Figure 1. Chart for estimating effective roadbed soil resilient modulus for flexible pavements designed using the serviceability criteria [ 4 ]

Figure 1. Chart. Chart for estimating effective roadbed soil resilient modulus for flexible pavements designed using the serviceability criteria. The figure is a table showing the results of roadbed soil modulus and relative damage tested each month. The results of the roadbed soil modulus and relative damage is for a 12-month (1 year) period. The summation of relative damage is 3.72. The average relative damage is equal to 0.31. Effective roadbed soil resilient modulus is equal to 5,000 pounds per square inch. Relative damage is equal to 1.18 times 10 to the 83.72 times effective roadbed soil resilient modulus to the negative 2.32.

With modern computing capabilities, the approach is straightforward and provides an easy mechanism to account for and explain differences in pavement performance that occur as a result of environmentally induced variations in the structural characteristics of pavement materials. This general methodology has been implemented by several researchers. (See references 8, 9, 10, 11.) The number of discrete periods considered ranges from 4 (i.e., the 4 seasons) to 12 (1 per month).

Within the ninth edition of the Asphalt Institute’s procedure for design of flexible pavements (known as MS-1), seasonal variations are considered by way of three representative temperature regimes defined by the mean annual air temperatures (MAAT) of 7 °C, 15.5 °C, and 24 °C.[ 12 ] Separate design charts are provided for each temperature regime. In developing the design charts for the 7°C and 15.5 °C temperature regimes, the subgrade modulus and the k1 coefficient in the constitutive model Mr = k1θk2 for the granular base layer were varied on a monthly basis (within a cumulative damage framework) to reflect the effects of freezing, thawing, and recovery, while the moduli for the asphalt bound layers were varied as a function of the mean monthly temperature. The monthly values used for the granular base and subgrade layers are presented in Table 2 and Table 3, respectively. Values for December are equal to the "normal" values used to define the quality of the material under consideration.

Treatment of seasonal variations in the U.S. Air Force design procedure is similar to that in MS- 1, in that design temperatures are used to determine monthly values for the asphalt concrete (AC) modulus.[ 13 ] However, only two subgrade soil conditions are considered: normal and thawed. As in the development of MS-1, the variations in modulus are considered within a cumulative damage framework.


Table 2. Subgrade moduli used in the Asphalt Institute DAMA program [ 12 ]
MAAT °C / Normal Mr Subgrade Modulus (by month), 103
Dec Jan Feb Mar Apr May June July Aug Sept Oct Nov
4.5 15.9 27.3 38.7 50.0 0.9 1.62 2.34 3.06 3.78 4.5 4.5
12.0 21.5 31.0 40.5 50.0 6.0 7.2 8.40 9.6 10.8 12.0 12.0
22.5 29.4 36.3 43.1 50.0 15.8 17.1 18.5 19.8 21.2 22.5 22.5
4.5 4.5 27.3 5.0 1.35 2.14 2.93 3.71 4.5 4.5 4.5 4.5
12.0 12.0 31.0 50.0 7.2 8.4 9.6 10.8 12.0 12.0 12.0 12.0
22.5 22.5 38.3 50.0 18.0 19.1 20.3 21.4 22.5 22.5 22.5 2.5


Table 3. Monthly granular base k1 values used in the Asphalt Institute DAMA program[ 12 ]
MAAT °C / Normal Mr Monthly Value for k 1 103, Mr = k1θk2, k2 = 0.5, Mr in kPa (psi)
Dec Jan Feb Mar Apr May June July Aug Sept Oct Nov
8.0 12.0 16.0 20.0 24.0 2.0 3.2 4.4 5.6 6.8 8.0 8.0
12.0 18.0 24.0 30.0 36.0 3.0 4.8 6.6 8.4 10.2 12.0 12.0
8.0 16.0 24.0 2.0 3.5 5.0 6.5 8.0 8.0 8.0 8.0 8.0
12.0 24.0 36.0 3.0 5.25 7.5 9.75 12.0 12.0 12.0 12.0 12.0

A common limitation in much of the work done to date, including the 1993 AASHTO guide, is the absence of significant, broadly applicable, and well supported quantitative guidance as to appropriate design values–seasonal or otherwise–to use for base, subbase, and subgrade layers. Definitive guidance in this regard has been developed, but is generally limited with respect to the geographic range over which it is valid. For example, the developers of the mechanisticempirical overlay design procedure used in the State of Washington have established a set of seasonal factors describing the relative seasonal moduli for typical Washington State base and subgrade materials for each of the two major environmental zones present in the State.[ 14, 15 ] These factors, summarized in Table 4, may be used to estimate the moduli for different seasonal conditions using the modulus for any one condition as a starting point. For example, if one knows the modulus of the material in question under dry conditions, one can estimate the modulus of that material under wet conditions by multiplying the dry value by the wet/thaw factor for the material type and environmental zone in question.


Table 4. Seasonal variations of unbound material moduli for Washington State[ 15 ]
  Base Subgrade
Region Wet/Thaw Dry/Other Wet/Thaw Dry/Other
Eastern 0.65 1.00 0.95 1.00
Western 0.80 1.00 0.90 1.00

The seasonal factor approach to considering seasonal variations presumes substantial uniformity in the materials used and the environmental conditions present within each region for which seasonal factors are identified. It does not obviate the need for basic knowledge. Rather, it represents an approach to using that knowledge, once it becomes available. Further, while the approach is "transportable," the results (seasonal adjustment factors) are not. Hence, these and other "local" solutions are insufficient to address fully the general need to characterize, quantitatively, the magnitude and timing of seasonal fluctuations in the moduli of unbound pavement materials. More broadly applicable information is key to both effective use of the most widely accepted existing pavement design procedure (the 1993 AASHTO guide) and the development and use of improved pavement design and performance prediction procedures.

The state of the art with respect to consideration of seasonal variations in pavement performance modeling is reflected in the development of the 2002 Guide for Design of New and Rehabilitated Pavement Structures, currently ongoing through National Cooperative Highway Research Program (NCHRP) project 1-37A. (See references 16, 17, 18, 19). In this work, the EICM simulation model is used to provide predictions of climatic conditions (temperature, moisture, and frost) within the pavement structure as they vary with time. For unbound materials, the moisture prediction output of the EICM is used in models relating Mr change to changes in moisture to estimate the change in Mr from the initial as-constructed condition to the equilibrium condition. The specific relationship used is seen in Equation 1:

 The log of resilient modulus divided by optimum resilient modulus equals A plus the quotient of B minus A divided by the total of 1 plus EXP of beta plus K subscript S times the degree of saturation minus the optimum degree of saturation. A is the minimum log of resilient modulus divided by optimum resilient modulus, B is the maximum log of resilient modulus divided by optimum resilient modulus, and beta is the log of negative B divided by A. (1)

In this relationship, Mropt and Sopt, are the resilient modulus and degree of saturation for the laboratory optimum moisture and density condition, while S is the degree of saturation for the moisture condition associated with Mr. This relationship is discussed further in the next section, under the subheading "Moisture Conditions."The EICM predictions of freezing and thawing are used to determine when freezing and thawing occur, so that appropriate modulus values may be assigned; predictions of the soil moisture suction are used to determine the extent of recovery, and in turn, the modulus at a given time after thaw has occurred, but before full recovery.

In summary, over the past 40 years, approaches to considering seasonal variations in the pavement design process have advanced from the use of purely empirical regional adjustment factors that do not explicitly address the issue of seasonal variations to explicit methods which relate changes in modulus to the factors that cause those changes. The latter approach is embodied in the development of the 2002 Guide for Design of New and Rehabilitated Pavement Structures.


Many factors affect the moduli of unbound pavement materials. Some are inherent to the materials themselves; others are associated with the environment in which the materials exist or the loading or stress conditions to which they are subjected. The factors found to be important in a number of laboratory investigations are summarized in Table 5. At first glance, there is considerable variation in the specific parameters considered. However, when one takes into consideration differences in the scope of the various investigations (i.e., whether the investigator was looking at a single crushed aggregate, or an assortment of fine-grained and granular materials), and the relationships between the different variables identified, there is more consensus than disagreement. Specifically, there is broad agreement that the most important factors include stress conditions, moisture conditions (most often characterized by the degree of saturation), density, and material characteristics (gradation or fines content, angularity, plasticity). Although rarely addressed in the laboratory setting, frost and thaw effects are critical when in situ conditions are considered for areas subject to freezing.

Stress Conditions

Stress conditions are generally regarded as the most important influence on resilient behavior of granular and fine-grained materials. Most often, the behavior of granular soils has been found to vary primarily as a function of the bulk stress (first stress invariant), while the applied deviator stress has been found to be more important for fine-grained soils. The stress-sensitive nature of the resilient modulus of granular materials has traditionally been characterized by Equation 2.

Resilient modulus is equal to regression constant 1 times bulk stress to the regression constant 2. (2)

where θ is the bulk stress (i.e., the sum of the principal stresses), and K1 and K2are regression constants, or a variant of Equation 2, in which the confining pressure is used in place of the bulk stress.

For the bulk stress model (Equation 2), reported values for K1 range from about 4,826 to well over 689,476 kPa (700 to well over 100,000 psi), depending on material type.[ 20 ] Most unbound pavement materials have K 1values in the range of 10,342 to 82,737 kPa (1500 to 12,000 psi). Corresponding values for K2 range from about 0.3 to 0.7, with 0.5 being a reasonable representative value.

The resilient modulus of fine–grained materials has more often been characterized by Equation 3.

Resilient modulus is equal to regression constant 1 times applied deviator stress to the negative regression constants 2. (3)

where σd is the applied deviator stress and k1 and k2 are regression constants. Note that the negative sign on the k2 coefficient in Equation 3 implies stress softening behavior (which is typical of fine–grained soils), whereas stress–hardening behavior is more often observed in granular materials.


Table 5. Parameters found to affect laboratory resilient moduli of unbound materials
Material Type Material Parameters Stress Parameter(s) Authors
Fine–grained Age, compaction method,
density, water content
Repeated stress Monismith
et al.[ 21 ]
Granular Type and gradation, void ratio,
% saturation
Confining pressure
Granular Density, % passing 200,
aggregate type, % saturation
Confining pressure or bulk stress Hicks and
[ 22 ]
Granular Density, gradation, aggregate type, % saturation, degree of crushing Confining pressure, bulk stress Monismith
et al.[ 23 ]
Subgrade % saturation, volume moisture content, plasticity index (PI),
group index, % silt, % clay,
California Bearing Ratio
(CBR), % swell, specific
gravity, % organic carbon
Deviator stress Thompson
Robnett[ 24 ]
Clays Initial and final suction,
saturation, volume moisture
content, volume soil content[ 25 ]
Deviator stress, mean stress, no. of load cycles Edris and
Lytton[ 25 ]
Granular Degree of saturation, degree of compaction, gradation Bulk stress Rada and
Witczak[ 26 ]
Granular Moisture tension, dry density,
Bulk stress or second stress invariant and octahedral shear stress Cole,Irwin,
andJohnson[ 27 ]
Base and subgrade Dry density, moisture content,
% passing 200, consolidation
2nd stress invariant and oct. shear stress (base);cyclic dev. stress and maximum shear stress (subgrade) Ishibashi,
Irwin, and
Lee[ 28 ]
Granular Soil type, moisture tension,
frozen, and total water
content, temperature, dry unit
weight, state with respect to
bulk stress or second stress invariant/octahedral shear stress Cole,
al [ 29, 30, 31, 32 ]
Subgrade Not investigated as such Saturated soil: efficiency.Confining stress anddeviation. Stress;unsaturated soil: netconfining stress, matricsuction, and deviation Stress Fredlund
Rahardjo[ 33 ]
Base and subgrade Soil type, % saturation,
grainsize distribution, density
Bulk stress due to overburden, bulk stress due to load and overburden,octahedral shear stress, anisotropic consolidation ratio Yang[ 34 ]
Granular base Gradation, material type Bulk stress Thompson
and Smith.[ 35 ]
Fine–grained Moisture Content, plasticity index, relative density, sample age Confining pressure Pezo et
al.[ 36 ]
Granular and cohesive Moisture Content, optimum Moisture Content, %
saturation, compaction,
gradation, % swell, %
shrinkage, density, CBR
Granular: bulk stress and cyclic deviation. Stress Cohesive: cyclic deviation.Stress Santha[ 37 ]
Coarse–grained Density, gradation, moisture cont., type Bulk stress Kolisoja[ 38 ]
Granular Moisture content,
temperature, dry density
Bulk stress Jin et al[ 39 ]
Fine–grained Dry density, moisture content,
soil type
Deviator stress Li and
Selig[ 40 ]
Granular and finegrained Suction, Dielectric constant,
gradation, Atterberg limits
Bulk stress and octahedral shear stress Titus–
Glover and
[ 41 ]

Equation 4 has also been used to characterize the resilient behavior of fine-grained materials.

Resilient modulus is equal to regression constant 2 plus regression constant 3 times (open first bracket) regression constant 1 minus (open first parenthesis) major principal stress minus minor principal stress (close first parenthesis) (close first bracket), for regression constant 1 is greater than (open second parenthesis) major principal stress minus minor principal stress (close second parenthesis). Resilient modulus is equal to regression constant 2 plus regression constant 4 times (open second bracket) (open third parenthesis major principal stress minus minor principal stress (close third parenthesis) minus regression constant 1 (close second bracket), for regression constant 1 is less than (open fourth parenthesis) major principal stress minus minor principal stress (close fourth parenthesis). (4)

In Equation 4, σ1 and σ3 are the major and minor principal stresses, respectively, and the Kn values are regression constants, as before.[ 42 ]

Brown and Pappin note that use of Equation 2 in pavement analysis is likely to lead to inaccurate results due to the limited range of stress paths considered in its development.[ 20 ] They further point out the need to consider effective stresses (as opposed to total stresses) in modeling the behavior of saturated or partially saturated soils, particularly those that are fine grained.

Uzan evaluated Equation 2, finding that it does not adequately describe the behavior of granular materials.[ 43 ] He further found that the relationship described by Equation (5).

Resilient modulus is equal to regression constant 1 times atmospheric pressure times (open first bracket) bulk stress divided by atmospheric pressure (close first bracket) to the regression constant 2 times (open second bracket) applied deviator stress divided by atmospheric pressure (close second bracket) to the regression constant 3. (5)

in which θ is the bulk stress and σd is the dynamic deviatoric stress, results in better agreement with observed behavior. (pa is atmospheric pressure, introduced to make the relationship independent of the system of measurements.)

Ishibashi et al. explored a number of constitutive models with regard to their ability to explain the resilient behavior of several fine-grained and granular materials.[ 28 ] An unusual aspect of their work was the consideration of anisotropic as well as isotropic consolidation. This isimportant because the anisotropic consolidation state, although rarely addressed in laboratory testing, may be a more accurate representation of the in situ condition of the pavement materials. Further, the effect of the consolidation ratio was found to be quite significant. For the four soils considered, they found that Equations 6 and 7 were well suited to explain the behavior of granular and fine-grained materials, respectively.

Resilient modulus divided by consolidation ratio to the regression constant is equal to regression constant 1 times (open bracket) repeated second stress invariant divided by octahedral shear stress (close bracket) to the regression constant 2. (6)

Resilient modulus divided by consolidation ratio to the regression constant is equal to regression constant 3 times (open bracket) repeated deviatoric stress divided by maximum shear stress (close bracket) to the regression constant 4. (7)

In these equations, J2 is the second stress invariant (σ1σ2 + σ2σ3 + σ3σ1), σd is the repeated deviatoric stress, τoct is the octahedral shear stress (1/3[(σ1-σ2)2 + (σ2-σ3)2 + (σ3-σ1)2]½), τf is the maximum shear stress, kc is the consolidation ratio (vertical consolidation stress divided by horizontal consolidation stress, or 1/k0), and K1, K2, K3, K4, and n are regression constants. Reported values for K1 and K2 were in the range of 11,032 to 64,121 kPa (1,600 to 9,300 psi) and 0.14 to .61, respectively, while K3 and K4 values of 68,948 to 147,548 kPa (10,000 to 21,400 si) and -0.25 to -0.60, respectively, were obtained. The value of n was found to be approximately 1.5 for the granular materials modeled with Equation 6, and 0.5 for Equation 7 (fine-grained soils), indicating that the consolidation ratio has a greater effect on the behavior of the granular materials than on the fine-grained materials. However, this analysis neglected the influence of several key factors.

Subsequently, Yang[ 34 ] conducted a more complete analysis of the same data set considered by Ishibashi et al.[ 28 ] In this analysis, Yang used Equation 8, which is applicable to both fine-grained and granular materials.

Elastic modulus is equal to regression constant 1 times (open first parenthesis) bulk stress due to overburden only squared plus bulk stress due to overburden and load squared (close first parenthesis) to the regression constant 2 times (open second parenthesis) 1 plus octahedral shear stress (close second parenthesis) to the regression constant 3 times consolidation ratio to the regression constant 4. (8)

In this model, θlo is the bulk stress due to overburden only; and θIp is the bulk stress due to overburden and load. Other variables are as previously defined. In contrast to the n values reported by Ishibashi et al., Yang obtained k4 values of 0.83 and 0.91 for the granular materials; 0.52 and 0.51 for the fine-grained materials; and 0.69 for the combined data set. These values for k4 are believed to more accurately reflect the true influence of the consolidation ratio than the n values obtained by Ishibashi et al., by virtue of the fact that more of the other influential factors were accounted for in the analysis. This work will be discussed in greater detail later in this chapter.

Santha conducted an investigation of the resilient moduli of 45 granular soils, in which he compared Equation 2 with Equation 5.[ 37 ] His results support Uzan’s conclusion that Equation 5 is superior to Equation 2 in describing the behavior of granular soils. For Equation 5, he reported K1 values ranging from 130 to 918, with a mean of 421; K2 values of 0.145 to 0.479 (mean 0.33) and K3 values of -0.152 to -0.574 (mean -0.37). (All coefficients are dimensionless.) Thus, the moduli decrease with increasing repeated vertical (deviator) stress (negative K3), and increase with increasing bulk stress (positive K2).

In his study of the resilient behavior of 42 cohesive soils, Santha used Equation 9.

Resilient modulus is equal to regression constant 1 times atmospheric pressure times open bracket repeated deviatoric stress divided by atmospheric pressure close bracket to the regression constant 3. (9)

Note that Equation 9 is a special case of Equation 5, in which K2, the coefficient of the bulk stress term, is taken to be zero (i.e., modulus is independent of the bulk stress). For the cohesive soils, Santha reports K1 values ranging from 188 to 1,263, with a mean of 645; and K3 values of - 0.07 to -0.60, with a mean of -0.026. (All coefficients are dimensionless.)

Witczak and Uzan[ 44 ] evaluated several constitutive relationships, including equations (8) and (10), by applying each to the laboratory test data previously developed by Rada and Witczak[ 26 ].

Resilient modulus is equal to open parenthesis regression constant 1 times atmospheric pressure close parenthesis times open parenthesis bulk stress divided by atmospheric pressure close parenthesis to the regression constant 2 times open parenthesis octahedral shear stress divided by atmospheric pressure close parenthesis to the regression constant 3. (10)

They found that equations (8) and (10) fit the observed material behavior far more closely than any other constitutive model form evaluated. The coefficients obtained for these models for the materials investigated are summarized in Table 6. As with Equations 5 and 8 (and unlike Equations 2, 3, 4, 6, 7, and 9), Equation 10 has the advantage of being applicable to both granular and fine-grained materials. The data set used in this investigation did not address the issue of anisotropic consolidation, so no evaluation of that aspect of Equation 8 was possible.


Table 6. Dimensionless constitutive model coefficients for Equations 8 (in kPa (psi)) and 10 (dimensionless)[ 44 ]
Model Ln K1 K2 K3
7 8.2 to 9.5 0.15 to 0.5 -0.4 to 0.3
9 5.5 to 6.4 0.5 to 0.95 -0.5 to 0.1

Von Quintus and Killingsworth also concluded that Equation 10 is well suited to the characterization of the stress sensitivity of laboratory resilient moduli.[ 5 ] More details regarding their work in this regard are presented under "Material Characteristics."

More recently, Andrei evaluated fourteen different constitutive model forms.[ 45 ] The models considered ranged from single–variable (θ or τ), two–parameter models to the general twovariable, five–parameter model given in Equation 11, and several special cases of the latter.

Resilient modulus is equal to regression constant 1 times atmospheric pressure times (open first bracket) bulk stress minus 3 times regression constant 6 divided by atmospheric pressure (close first bracket) to the regression constant 2 times (open second bracket) octahedral shear stress divided by atmospheric pressure plus regression constant 7 (close second bracket) to the regression constant 3. Where regression 1 and 2 is greater than or equal to 0, regression constant 3 and 6 are less than or equal to 0, and regression constant 7 is greater than or equal to 1. (11)

Both log–log and semi–log model forms were considered in Andrei’s work. He concluded that Equation 11 yielded the best overall fit of the evaluation data set, with the proviso that the regression constants must be constrained, as noted, to ensure rationality.

Other stress– or load–related parameters that have been investigated with regard to their influence on the resilient behavior of granular and/or fine–grained materials in the laboratory include the number of stress applications and the loading sequence (duration of stress application and rest periods). (See references 21, 23, 26, 28.) The effect of variations in the loading sequence has generally been found to be small in comparison to other factors.[ 26, 28 ] Similarly, as long as the number of load applications is great enough that the material being tested has reached an equilibrium state (as would be the case for moderate to high volume pavements), the effect of number of load applications is not especially significant.[ 23, 26, 28 ]

Moisture Conditions

Moisture is generally regarded as being second only to stress conditions in its influence on the moduli of unbound pavement materials, with increases in moisture content typically resulting in significant reductions in the resilient modulus of the soil. (See references 21-26, 34, 36-40) Monismith et al. note that the modulus of a fully saturated material may be as much as 50 percent lower than that of the same soil in a partially saturated condition.[ 21 ] However, Chou suggests that the general trend of decreasing modulus with increasing moisture content is much less significant when effective stress conditions rather than total confining pressures are used as the basis of comparison.[ 42 ]

For "typical" Illinois fine–grained soils, Thompson and Robnett studied the effect of degree of saturation on resilient modulus.[ 24 ] For saturation (Sr) ranging from 50 to 100 percent, and densities corresponding to 95 and 100 percent of AASHTO T–99 compaction, they obtained the relationships given in Equations 12 and 13, respectively (Mr in units of kPa (ksi)).

Resilient modulus is equal to 45.2 minus open parenthesis 0.428 times saturation close parenthesis (12)

Resilient modulus is equal to 32.9 minus open parenthesis 0.334 times saturation close parenthesis (13)

Thus, as the degree of saturation varies from 50 to 100 percent, the predicted resilient modulus decreases by roughly an order of magnitude. This work considered total (as opposed to effective) stress conditions.

Rada and Witczak note that the reduction in stiffness of granular materials with increasing moisture content is especially significant at degrees of saturation in excess of 80–85 percent, where a rapid loss of stiffness occurs with increasing saturation.[ 26 ] However, the magnitude of this effect varies from one material to another. Kolisoja’s results appear to differ somewhat, in that the resilient modulus increases with increasing saturation up to 35–45 percent, and falls off gradually thereafter.[ 38 ] However, the maximum degree of saturation investigated by Kolisojawas 77 percent. Hence, the behavior of the soils in question as they approach the fully saturated condition is not known.

Noureldin conducted an investigation of (among other things) the effect of changes in moisture content on the (backcalculated) moduli of granular base and subgrade materials for one test site in Saudi Arabia, with all other factors (for all practical purposes) held constant.[ 46 ] For this site, he found that an increase in the base course moisture content from 5 to 9 percent (4 percent increase) corresponded to a 22.4 percent reduction in the modulus. The corresponding increase in moisture content for the subgrade was from 6.8 to 13 percent (6.2 percent increase) accompanied by a 35 percent reduction in the modulus.

Ksaibati et al. looked at the effect of moisture on backcalculated moduli for highway pavement base and subgrade materials in Florida.[ 47 ] They observed modulus changes of up to 96 percent as the moisture content varied, with the magnitude of the change depending on the deflection testing device (falling weight deflectometer (FWD) or Dynaflect) used to obtain the data. However, based on the discussion provided, it appears that the potential for changes in modulus due to stress state variations arising from temperature-induced variations in the stiffness of the overlying AC layers was not considered. Thus, the author believes that some portion of the observed variation may in fact be attributable to stress sensitivity, as opposed to pure moisture effects.

Whereas most researchers have characterized moisture conditions on the basis of moisture content or degree of saturation, Edris and Lytton used soil suction, which is related to moisture content and saturation, along with the internal stress state of the soil.[ 25 ]Although there is a tremendous amount of scatter in the data presented, the general trend is for the modulus to increase with increasing suction up to a point, and then level off. Although the authors assert that suction is a more appropriate parameter than moisture content or degree of saturation for use in characterizing the effect of moisture conditions on resilient behavior, the data presented do not appear to support that assertion. While no goodness of fit statistics are presented, the graphical presentations of the data show much less scatter, and much clearer trends when either moisture content or degree of saturation, rather than suction, is used as the explanatory variable. Titus- Glover and Fernando also used suction (as well as moisture content and saturation) as an explanatory variable in their development of regression models to predict the coefficients for Equation 5.[ 41 ] The set of models selected as being best included the suction term as an explanatory variable for K1, but not for K2 or K3. This work is discussed in more detail under"Relating Resilient Moduli To Material Parameters."

Recent work by Witczak, Andrei, and Houston examined the laboratory modulus–moisture data assembled and used in a number of earlier research efforts, and found relationships of the general form presented in Equation 14,

Logarithm of open parenthesis resilient modulus divided by resilient modulus at a reference moisture start close parenthesis is equal to regression constant K sub W times open parenthesis moisture state minus reference moisture state close parenthesis (14)

with Mrref being the resilient modulus at a reference moisture state represented by mref, and m being the moisture state associated with Mr. [ 16 ] They considered both gravimetric moisture content and degree of saturation as the variables used to characterize moisture state (m), and recommend the use of degree of saturation, because data errors are more readily identified when degree of saturation is considered. They also recommend use of the laboratory optimum condition for the reference values. Witczak et al. developed the modified model presented as Equation 15 because the laboratory data set on which Equation 14 is based was composed entirely of test results within ±30 percent of optimum, whereas field data indicate that lesser degrees of saturation often occur in practice.

Logarithm of resilient modulus divided by resilient modulus at a reference moisture state is equal to A plus B minus A divided by 1 plus the exponential of (open first parenthesis) beta plus regression saturation K sub S times (open second parenthesis) saturation minus optimum saturation (close second parenthesis) (close first parenthesis). Where A is equal to the minimum (open third parenthesis) log of (open fourth parenthesis) resilient modulus divided by optimum resilient modulus (close fourth parenthesis) (close third parenthesis), B is equal to the maximum (open fifth parenthesis) log of (open sixth parenthesis) resilient modulus divided by optimum resilient modulus (close sixth parenthesis) (close fifth parenthesis), and beta is equal to natural log of (open seventh parenthesis) negative B divided by A (close seventh parenthesis). (15)

This relationship is being used in the 2002 AASHTO Guide.

Density and Soil Structure

Some measure of density has been considered in most investigations of the factors affecting the resilient behavior of granular and fine-grained materials. (See references 21-23, 26-28, 34, 36-40.) Among the parameters considered are density, degree of compaction (relative to Standard Proctor density, for example) or compaction energy, and void content. Although density affects the moduli of both granular and fine–grained materials, it has been found that the magnitude of this effect is small in comparison to those of stress and moisture conditions. This is especially true when density variations are small.

For granular soils, resilient modulus tends to increase with increasing density. Further, this general trend is relatively independent of moisture content or degree of saturation.[ 22, 23, 29 ] Hicks and Monismith found that for granular materials, the influence of changes in density decreased as the percent fines increased.[ 22 ] It has also been found that the effect of variations in density is greater for partially crushed aggregate than for crushed aggregate.[ 23 ]

For fine–grained soils, the resilient modulus may increase or decrease with increasing density, depending on the moisture conditions.[ 40 ] Wetter than optimum, the modulus tends to decrease with increasing density; whereas, dryer than optimum, the trend is for modulus to increase with increasing density. In addition, soils having a flocculated structure (typically resulting from static compaction methods) tend to have higher moduli than those having a dispersed structure (from, for example, kneading compaction).[ 28 ]

Material Characteristics

When one considers the many factors (such as gradation, mineralogy, angularity, surface roughness, and plasticity) that make one soil different from another, it is intuitive that material characteristics will have an effect on the moduli of the materials. Evidence that this is indeed the case may be found in work by Rada and Witczak [ 26 ] and Titus-Glover and Fernando,[ 41 ] among others. Key results from Rada and Witczak’s investigation of the resilient behavior of a broad array of granular materials are presented in Table 7. The coefficients given in the table are for the bulk stress model (Equation 2). Note that the mean values for the different classes vary considerably, and that even within a given aggregate class, K1 values may vary by more than an order of magnitude. While some of this variation is more than likely attributable to differences in moisture conditions and density, it is reasonable to assume that much of it is attributable to differences in other material characteristics.

The results obtained by Titus-Glover and Fernando when they applied Equation 5 to test results for a somewhat broader array of material types are summarized in Table 8.[ 41 ] The testing from which these results were derived was conducted at the optimum moisture content for each material, which ranged from 3.97 percent for the sand to 19.76 percent for the fat clay. As with the results of Rada and Witczak, there is considerable variation in the model coefficients etween the material types.

More recently, Von Quintus and Killingsworth obtained the results presented in Table 9 for laboratory test results from 125 LTPP test sections, evaluated using Equation 5.[ 5 ] They report that average R2 values of 0.85 or greater were obtained in all cases. Substantial, differences are observed between many of the mean values presented in Table 9 and those presented in Table 8. However, most of the Table 8 values fall within the ranges reported in Table 9 for comparable materials. The between-material variability reflected in the ranges and standard deviations (SD) reported in Table 9 is very high.


Table 7. Summary of K1 and K2 statistics by aggregate class[ 26 ]
Aggregate Class No. of Data Points K1Parameter (kPa (psi)) K2 Parameter
Mean SD Range Mean SD Range
Silty sands8 11,170
4,895 to 26,407 (710 to 3830) 0.62 13 0.36 to 0.80
Sand gravel37 33,646
5,929 to 88,529 (860 to 12840) 0.53 17 0.24 to 0.80
Sand aggregate blends78 29,992
12,962 to 76,325 (1880 to 11070) 0.59 13 0.23 to 0.82
Crushed stone115 49,711
11,756 to 390,726 (1705 to 56670) 0.45 23 -0.16 to 0.86
Limerock13 27,786
39,300 to 578,194 (5700to 83860) 0.40 11 0.00 to 0.54
Slag20 167,198
64,121 to 636,800 (9300 to 92360) 0.37 13 0.00 to 0.52
All data271 63,708
4,895 to 636,800 (710to 92360) 0.52 17 -0.16 to 0.86


Table 8. Summary of dimensionless K1-K3 parameters for selected materials modeled using Equation 5[ 41 ]
Material K1 K2 K3
Limestone 243 0.95 -6.5*10–5
Iron ore 75 1.01 -2.2*10-5
Sandy gravel 152 0.88 -2.9*10-4
Caliche 322 0.88 -9.8*10–5
Shellbase 318 0.80 –9.8*10–5
Sand 498 0.77 -0.01
Silt 195 0.071 –6.5*10–5
Lean clay 195 0.068 –0.19
Fat clay 122 0.19 –0.36


Table 9. Summary of average elastic coefficients and exponents for LTPP materials modeled using Equation 5 (dimensionless)[ 5 ]
Material Mean K1(Range)SD Mean K2(Range)SD Mean K3(Range)SD
Clay 594
(87 to 2039)
(-0.20 to 0.53)
(-0.55 to 0.30)
Silts 426
(136 to 838)
(-0.05 to 0.66)
(-0.57 to 0.05)
Sands 598
(103 to 3494)
(-0.33 to 0.99)
(-0.43 to 0.89)
Gravels 836
(229 to 3172)
(-0.27 to 0.59)
-0.08(-0.33 to 0.67)
Base 869
(250 to 2323)
(-01.8 to 1.07)
(-0.33 to 0.61)

Parameters describing gradation are perhaps the most widely investigated material characteristics. Several investigators have approached this issue by varying the fines content for several different aggregates. (See references 22-23, 26, 28.) The results reported typically show inconsistent trends, with the effect of increased fines content varying from one material to another[ 23, 26 ]. Although one would intuitively expect that extreme increases in the fines content will have a very marked affect on behavior, regression equations developed by Rada and Witczak indicate that the influence of fines content in the range of 3 to 17 percent is negligible in comparison to stress, moisture, and density parameters.[ 26 ]

The results of Monismith, Hicks, and Salam show an increase in modulus as the fines content increased from a coarse (2–3 percent fines) to a medium (5–6 percent fines) gradation for both crushed and partially crushed materials.[ 23 ] However a decrease in modulus occurred as the fines content was further increased from a medium gradation to a fine gradation (8–10 percent fines).

Ishibashi et al. looked at four soils. Two were gravelly silty sand base materials differing only in fines content. Two were subgrade materials, which were identical but for the substitution of a sandy silty clay material for the minus number 7 fraction of the original gravelly sandy silt in one soil.[ 28 ] For both the base and subgrade materials, the material having the higher fines content exhibited the lower modulus.

Thompson and Smith looked at the resilient modulus of seven Illinois granular materials, including two crushed stones, two crushed gravels, two gravels, and a partially (30–35 percent) crushed gravel.[ 48 ] The partially crushed gravel, and one material from each of the other types conformed to one gradation specification, while the remaining three materials conformed to another. The two gradation specifications differed only in the percentage passing the no. 200 sieve. The one with the lower fines content required 2 ±2 percent passing 200, whereas the higher fines content required 8 ±4 percent passing 200. The materials were tested at or near the maximum dry density, which varied from 1962 kg/m3 (122.5 pcf) (low fines content crushed stone) to 2300 kg/m3 (143.6 pcf) (high fines content crushed stone) and the optimum moisture content, which varied from a low of 4.0 percent (low fines content crushed stone) to a high of 9.0 percent (partially crushed stone, high fines content). At a bulk stress of 138 kPa (20 psi), their resilient modulus results ranged from a high value of 244 MPa (35.4 ksi) for the crushed stone having the higher fines content to a low of 134 MPa (19.4 ksi) for the partially crushed gravel, which also had the higher fines content. Looking at the pairs of similar material, in all cases, the material with the higher fines content had a slightly higher modulus. How much of the difference is attributable to the different fines content, and how much is attributable to different moisture and density conditions cannot be discerned from the data.

Chen et al. considered six base/subbase materials used in Oklahoma, three limestones, one sandstone, one granite, and one rhyolite, with all six materials prepared to have the same gradation.[ 49 ] Their results showed a 20–to 50–percent variation in resilient modulus with aggregate type, with the magnitude of the difference depending on the bulk stress. The samples used were compacted to 95 percent maximum dry density (relative to AASHTO T180–90D), at optimum moisture content. The ranges for the densities and optimum moisture contents were 2355 to 2403 kg/m3 (147 to 150 pcf) and 5.2 to 6.0 percent, respectively–quite narrow relative to those for the materials studied by Thompson and Smith. Hence, it is reasonable to believe that much (but not all) of the observed between material variation in modulus can be attributed to other characteristics of the materials.

One factor that contributes to between material differences in resilient behavior is the particle shape or angularity, in the sense that crushed aggregates typically exhibit higher moduli than partially crushed aggregates, due to the increasing angularity and surface roughness present in the crushed material.[ 23 ] The magnitude of this effect has been found to increase with increasing fines content.

In their investigation of Illinois subgrade soils, Thompson and Robnett found that percent clay, plasticity index, liquid limit, percent organic carbon, percent silt, and group index were all significant factors (at the 0.01 level) in explaining observed material variations in resilient behavior.[ 24 ] Interestingly, their analysis of variance results indicated that soil classification (AASHTO, Unified, or U.S. Department of Agriculture) is not a significant factor in determining the resilient behavior of the soils studied. Thus, they concluded that soil classification is not sufficient to characterize the resilient behavior of fine-grained soils. This is not surprising when one considers that, at the extremes, two soils having the same classification may be quite different, or conversely, that two soils having similar characteristics may be just different enough to fall into different classifications.

Soil State with Respect to Freezing and Thawing

Although seldom considered in laboratory investigations (due to the complexity of the required testing), the state of granular and fine-grained soils with respect to freezing and thawing can have a very great effect on their resilient behavior, and thus be an important consideration in the design and evaluation of pavements in regions subject to frost penetration. Whereas most efforts to characterize the resilient behavior of soils have addressed only nonfrozen materials, investigations at the U.S. Army Corps of Engineers’ Cold Regions Research and Engineering Laboratory (CRREL) have considered a range of soil states encompassing frozen, thawed, recovering (from thaw weakening), and fully recovered conditions. (See references 27, 29-32, 50, 51, 52). Details of the CRREL investigations are presented in the section "Relating Resilient Moduli to Material Parameters." Key findings relating to the impact of freezing and thawing on resilient modulus are as follows.

Cole, Irwin, and Johnson obtained core samples of a frozen sand base material, and conducted laboratory resilient modulus testing on the samples in frozen, thawed, and recovered states.[ 27 ] Constitutive models considered in this work included the bulk stress model (Equation 2), and Equation 16.

Regression constant 1 times open parenthesis second stress variant divided by octahedral shear stress close parenthesis to the regression constant 2. (16)

In the frozen state, the resilient modulus of the sand base material remained around 10,000 MPa (1450 ksi) (depending on the applied deviator stress) at temperatures in the range of –10 °C to about –4 °C, at which point it began to decrease rapidly with increasing temperature, reaching 1,000 MPa (145 ksi) at a temperature near 0 °C. In the thawed state, the observed resilient moduli varied (with stress and moisture tension levels) in the range of 40 to 200 MPa (5.8 to 29 ksi).

Subsequent work by Cole et al.[ 29 ] and Johnson, Bentley, and Cole[ 30 ] expanded this effort to look at additional granular materials and additional test sites in a similar fashion. They found that for frozen soils, the modulus is primarily a function of unfrozen water content, with applied stress becoming significant as the temperature approached the melting point. A significant reduction in modulus upon thawing was followed by a gradual increase as the materials drained during the recovery period.

Further extension of this work is reported by Cole et al.[ 31 ] and by Johnson et al.[ 32 ] They note that the modulus of frozen soil may be two to three orders of magnitude greater than the thawed state modulus for the same soil. The relatively fine–grained soils exhibited lower moduli in the frozen state than did the more coarse–grained soils. The difference was attributed to the greater unfrozen moisture content of the fine-grained material. In general, stress dependency in the frozen soils was found to be negligible in comparison to the temperature effects. The lone exception to this was a silty fine sand subgrade for which stress level was a significant factor in the frozen state. The Mr versus temperature relationship was found to be a strong function of the relationship between temperature and unfrozen moisturecontent for each soil. For the thawed soils, as in previous investigations, K1 was found to be primarily a function of moisture tension, with some soils being more sensitive to variations in moisture tension than others. During the recovery period, the observed increase in modulus with increasing moisture tension ranged from a factor of 1.5 for a silty fine sand to a factor of 3.5 for a silty sandy gravel.

Collectively, the CRREL investigations show that the phenomena associated with freezing,thawing, and subsequent recovery result in substantial changes in the moduli of an array of granular materials, including a crushed stone, a broad array of sands, a silty sandy gravel, anda dense graded stone. They further show that temperature and moisture conditions (the later characterized in their work by moisture tension) must be considered in any attempt to predict seasonal variations in the modulus of unbound materials. Additional details on the CRRELwork are presented under "Relating Resilient Moduli to Material Parameters."

Additional Factors Influencing Backcalculated Moduli

The influential factors discussed thus far affect the moduli of unbound pavement materials irrespective of the method by which the moduli are determined (i.e., through laboratory testing or through backcalculation from pavement deflection data). Several additional factors influence the backcalculation of pavement layer moduli, and thus the moduli derived through that process. In broad terms, the most important of those factors (assuming the input data are accurate, and the backcalculation has been done "correctly") are the extent to which the model used accurately characterizes the pavement structure and its response to load, and the extent to which measured surface deflections are sensitive to the modulus of the individual pavement layers, or between-layer differences in modulus. These factors will be discussed in the next several paragraphs.

The basic problem addressed in the backcalculation process is to identify a set of layer moduli that are theoretically consistent with input data consisting of a set pavement layer thicknesses and Poisson’s ratios, a given applied load, and a measured pavement deflection basin resulting from that load. There is no closed-form solution to this problem. In fact, for any given set of input data, more than one solution may exist. Thus, the analyst must exercise considerable judgment in evaluating the results of the backcalculation process. A number of criteria have been established to aid in this evaluation.[ 53, 54 ] Applying these criteria has proven helpful, but not infallible.

There are some situations where it is virtually impossible to derive a meaningful set of backcalculated layer moduli from pavement-deflection data. Those situations arise when the measured deflection basin is insensitive to either the modulus of one layer or to differences in modulus between two adjacent layers.[ 55 ] The first situation quite often exists for pavements having thin AC surface layers (where the definition of "thin" is typically on the order of 5–8 cm). The second situation occurs quite often when stabilized base layers are present, or where the moduli of two adjacent layers are very similar. Often, the outcome of these situations is a compensating error effect, wherein the backcalculated modulus of one layer is erroneously high, while the modulus for the adjacent layer is erroneously low.

Measurement errors and errors in layer thickness arising from spacial variability in the pavement may also result in moduli that are higher or lower than the "true" (unknown) value. As a rule, the backcalculated moduli for the upper layers of the pavement are more sensitive to errors in layer thickness than are deeper layers.[ 56 ]

Most often, the backcalculation of pavement layer moduli from deflection data utilizes a static linear layered elastic model of the pavement structure.[ 57, 58, 59 ] Key assumptions are that the pavement layers are linear (i.e., not stress dependent), elastic, homogeneous, and continuous in the horizontal plane. Real pavements violate all of these assumptions to one degree or another, and there is little doubt that the discrepancies between model and reality have an impact on the backcalculated layer moduli. For the purposes of backcalculation, the most problematic discrepancies are the particulate nature and stress dependency of the unbound materials. Some backcalculation procedures use approximate methods to address stress dependency, with mixed success.[ 60 ] Use of a finite–element model (rather than the more simplistic layered–elastic model) as the basis for backcalculation is seen as a mechanism to more correctly model the particulate nature of unbound materials. However, much more research is required to develop accurate finite–element–based backcalculation programs suitable for use in pavement engineering practice.

Despite the challenges of the backcalculation process, and the noted discrepancies between model and reality, backcalculation based on linear layered–elastic theory enjoys widespread use as the best available nondestructive technology for estimating the in situ stiffness of pavement layers. While theoretically imperfect, when used with engineering judgment, the technology yields reasonable and useful results and fills a very real need.


The CRREL work cited in preceding discussions, as well as that of Yang,[ 34 ] Santha, [ 37 ] and Titus–Glover and Fernando,[ 41 ] is of particular interest because these investigators were quite successful in their efforts to relate the coefficients in the resilient modulus constitutive models they studied to selected material characteristics. Their work and findings with regard to relationships between resilient modulus constitutive model coefficients and material parameters are discussed in this section.

In addition to the sand core samples discussed previously, Cole, Irwin, and Johnson conducted laboratory resilient modulus testing of a sandy gravel subgrade material.[ 27 ] The subgrade samples were compacted in the laboratory to a level approximating the in situ conditions, and tested only in the unfrozen state. Two stress models were considered in analysis of the test results: the bulk stress model (Equation 2) and Equation 16. Note that Equation 16 is a special case of Equation 6, in which the exponent n on the consolidation ratio is taken to be zero. It is also identical in form to the bulk stress model, with the ratio J2octreplacing the bulk stress as the stress parameter. In many cases, Equation 16 was found to yield a better fit of the data.

For the materials investigated, the CRREL investigators found that the coefficient K1 was a function of the soil moisture tension and density, whereas K2 was more or less constant for a particular soil (but varied between soils). For the thawed and nonfrozen materials, models of the general form given in Equation 17 were found to fit the data well.

Resilient modulus is equal to regression constant 1 times the (open bracket) function of psi (close bracket) to the regression constant 2 times the function of sigma to the regression constant 3 times the function of gamma sub D to the constant regression 4. Psi is equal to moisture tension in kilopascals. The function of psi is equal to 101.38 minus moisture tension divided by 1 kilopascal. The function of sigma is equal to stress function divided by 1 kilopascal. The function of gamma sub D is equal to dry unit weight divided by 1 mega gram divided by meters cubed (17)

Typical values for the constants are given in Table 10. By rearranging Equation 17 to correspond to the form of Equations 2 and 16, one can see that k1=c1f(ψ)c2f(γd)c4and k2=c3.

For the frozen soils, CRREL investigators found that a model of the general form given in Equation 18 fit the data very well in most cases.

Resilient modulus is equal to regression constant times open parenthesis unfrozen moisture content divided by total moisture content close parenthesis to the regression constant 2. (18)

Representative regression constants for Equation 18 are presented in Table 11. In one case, a silty fine sand subgrade, stress state was a significant factor for the frozen material. Models obtained for this case are given in Equations 19 and 20 (all variables as previously defined; Mr in MPa; T in °C).

Resilient modulus is equal to 2.59 times open parenthesis unfrozen moisture content divided by total moisture content close parenthesis to the negative 0.85 times open parenthesis bulk stress divided by 1 kilopascal close parenthesis to the 0.93. Unfrozen moisture content is equal to 3.14 times 10 negative squared times open parenthesis negative temperature close parenthesis to the 0.29. Total moisture content is equal to 0.29. (19)

Resilient modulus is equal to 2.66 times open parenthesis unfrozen moisture content divided by total moisture content close parenthesis to the negative 1.02 times open bracket open parenthesis second stress variant divided by octahedral shear stress close parenthesis divided by 1 kilopascal close parenthesis to the 0.78. Unfrozen moisture content is equal to 3.14 times 10 to the negative squared times open parenthesis negative temperature close parenthesis to the negative 0.29. Total moisture content is equal to 0.29. (20)


Table 10. Representative (best fit) regression constants for Equation 17 Mr in MPa[ 27, 29, 31 ]
Material State and Type/LoadPulse Stress Term c1 c2 c3 c4 n R2/SE Ref. Eq.No.
J2oct 6.68*104 –2.2948 0.414 NA 186 0.89/
29 13
J2oct 4.80 NA 0.4046 NA 36 0.87
29 16
J2oct 1.56*105 –1.76 0.136 NA 64 0.65/
29 36
J2oct 3.68*104 –2.15 0.30 3.44 222 0.84/
31 7
θ 8.00*108 –2.99 0.37 –5.55 149 0.82/
31 13
Nonfrozen Silty Fine
θ 7.73*103 –1.34 0.35 NA 262 0.78/
31 16
silty sandy
θ 1.56*106 –3.69 0.36 7.72 173 0.74/
31 20
silty fine
J2oct 3.80*106 –2.36 –3.25 –3.06 293 0.74/
31 24
silty fine
J2oct 2.49*106 2.73 0.26 2.07 278 0.82/
31 31
Standard Error referenced to ln(Mr ) γd = dry unit weight (Mg/m2)
γ0 = 1 Mg/m2), f(γd) = γd0


Table 11. Representative regression constants for Equation 18[ 27, 29, 31 ]
Material Type/Load Pulse c1 c2 n R2 Std.Error1 Ref. Eq.No.
32.14 –1.96 73 0.95 0.446 23 10
86.4 –1.32 87 0.92 0.749 23 18
Hart Bros.
40.85 –1.59 99 0.92 0.623 23 23
33.45 –2.03 69 0.95 0.617 23 32
Dense graded
82.27 –2.03 32 0.97 0.413 23 35
Sibley till/
101 –3.446 108 0.87 0.71 23 38
Crushed stone
18.9 –4.82 78 0.78 0.66 25 9
81.8 –4.02 149 0.82 0.19 25 14
Silty sandy
1.00*103 –2.63 173 0.74 0.23 25 21
  1. Standard Error referenced to ln(Mr)
  2. wu = –3*10–2(–T)–0.25, wt = 0.075, T = temperature (°C)/1°C
  3. wu = –3*10–2(–T)–0.25, wt = 0.055
  4. wu =–3*10–2(–T)–0.22, wt = 0.05

Equation 21 gives the general form of a series of regression equations developed by Yang using Equation 8 as the basic constitutive model.[ 34 ]

Resilient modulus is equal to 10 to the regression constant C sub 1 times (open first parenthesis) 1 plus degree of saturation (close first parenthesis) to the regression constant C sub 2 times (open second parenthesis) relative degree of compaction (close second parenthesis) to the regression constant C sub 3 times material constant to the regression constant C sub 4 times (open third parenthesis) bulk stress due to overburden squared plus bulk stress due to overburden and load squared (close third parenthesis) to the constant regression C sub 5 times (open fourth parenthesis) 1 minus fines content (close fourth parenthesis) times (open fifth parenthesis) 1 minus octahedral shear stress (close fifth parenthesis) to the regression constant C sub 6 times fines content times consolidation ratio to the regression constant K sub 4. (21)

In this equation, S is the degree of saturation, in percent; RC is the relative degree ofcompaction, expressed as a decimal fraction of the standard Proctor density; F is the fines content (i.e., percent passing the No. 200 sieve); and M is a material constant, as defined in Equation 22.

Material constant is equal to (open first parenthesis) 0.5 times (open second parenthesis) 10 to the fines content minus degree of saturation divided by 100 plus 10 to the degree of saturation divided by 100 minus fines constant (close second parenthesis) (close first parenthesis) times (open third parenthesis) 1 plus plasticity index (close third parenthesis) divided by the density factor. The density factor is equal to density gradation 80 divided by the square root of (open fourth parenthesis) 7.14 times (open fifth parenthesis) density gradation 80 divided by density gradation 40 plus density gradation 40 divided by density gradation 20 plus density gradation 20 divided by density gradation 10 (close fifth parenthesis) (close fourth parenthesis). (22)

By comparing Equation 8 and Equation 21, it can be seen that k1 is predicted on the basis of the relative density, gradation (specifically, the percent passing 200, D80, D40, D20, and D10), plasticity index (PI), and degree of saturation, while k2 and k3 were related to the percent passing 200. The values obtained for the regression constants are presented in Table 12. Note that the constant k4 varied within the range of 0.51 to 0.91, depending on the data set under consideration (implying that it, too, is a function of material characteristics). Higher k4 values corresponded to base materials, while the values near 0.5 were associated with subgrade soils, but no predictive relationship was developed. Moisture content and density are not statistically significant in the equation, as these parameters were held constant in the development of the data set.


Table 12. Regression constants for Equation 21[ 34 ]
Material c1 c2 c3 c4 c5 c6 kc R SE1
Base I
8.7 –2.2 9 1.2 0.37 –2.8 0.83 0.933 0.084
Base II
3.84 0 0 0 0.22 –1.8 0.91 0.924 0.075
Base I & II
9.2 –2.3 6 1.4 0.30 –1.6 0.89 0.922 0.085
Subgrade I
10.4 –3.2 17 0 0.50 –1.5 0.52 0.914 0.075
Subgrade II
4.33 0 0 0 0.18 –0.9 0.51 0.865 0.060
Subgrade I &
II (N=160)
10.2 –3.0 16 0 0.41 –1.3 0.53 0.901 0.075
All materials
6.8 –1.46 7.8 0.42 0.22 –0.34 0.69 0.863 0.105

1Standard Error referenced to log10

Likewise, Santha developed regression equations to predict the coefficients (K1–K3) for Equations 5 and 9, for granular and cohesive soils, respectively, based on moisture content (MC), optimum moisture content (MOIST), percent saturation (SATU), percent compaction (COMP), percent passing numbers 40 and 60 sieves (S40 and S60), percent clay (CLY), percent silt (SLT), percent swell (SW), percent shrinkage (SH), density (DEN), and California Bearing Ratio (CBR).[ 37 ] Santha’s approach differed from that of Yang in several respects, including: (1) the constitutive model considered; (2) the use of separate regression analyses for each of the model coefficients; and (3) the use of a much more extensive data set. The results are presented in Table 13. The R2 values reported are quite good, and the predicted moduli quite reasonable. However, one wonders whether all of the variables used are necessary, in light of the probable intercorrelation among several of them (e.g., moisture content and moisture content ratio). Santha does not address this possibility.


Table 13. Resilient modulus prediction equations developed by Santha[ 37 ]
 Equation R2
Granular materials LogK1=3.479–0.07MC+0.24 MC/MOIST+3.681COMP+
0.004(SW2/CLY+0.003(DEN2/S 40)
+0.012cbr+0.003sw2/cly –0.31(SW+SH)/CLY
Fine–grained materials LogK1 =19.813–0.045MOIST–0.131MC–9.171COMP

Titus-Glover and Fernando developed predictive models for the coefficients in Equation 10, finding that k1 could be predicted quite well with the plastic limit (PL), specific gravity of soil binder (Gsb), volumetric moisture content (θw), tangent of friction angle (tanf), percent passing number 40 (N40), suction (U, in pF (water-holding energy)), and dielectric constant (er) as variables.[ 41 ] Similarly, the predictive equation developed for k2 used the specific gravity of the soil binder, the gravimetric moisture content, and the liquid limit (LL), while that for k3 is based on the dielectric constant, liquid limit, and gravimetric moisture content. The specific equations developed are presented in Table 14. The goodness–of–fit statistics indicate that the resulting equation for k2 is not as strong as that for k1, but still acceptable. The goodness–of–fit statistics for k3 are very good. However, they are applicable primarily to fine–grained soils, as granular materials were not used in the development on the presumption that k3 is approximately zero for granular materials.


Table 14. Titus Glover and Fernando predictive models for constitutive model coefficients[ 41 ]
Equation R2 N
k1= 28659 – 417.297PL – 5706.38Gsb +12780 θw
75.44N40 – 5462.069 tan ø+58.975N40N*UN+
256.002N 40N* tan ØN–309.3217 UN* εrN
N 40N =normalized N40=n40 – 57.215
tan øn=normalized tan ø=tan ø –0.92877
UN=normalized suction=U – 2.12231
εrN=normalized dielectric constant =εr–12.5981
0.93 26
k2=0.127927+0.515759 gSB –0.084802w+0.00027N 40n2
0.009607L LNWN–0.192586L LNTAN ØN+0.003981L L2N
where WN=normalized gravimetric moisture content =W – 8.94172
L LN = normalized liquid limit = LL –24.41888
0.75 26
k3 =1.807384 – 0.181851 εr– 2.808291L LN *WN– 0.016342LLNr
LLN=normalized liquid limit = LL – 28.6667
WN=normalized gravimetric moisture content =W – 13.451.
0.92 9

Von Quintus and Killingsworth also attempted to develop regression models to predict the coefficients for Equation 5 using data from the LTPP database.[ 5 ] The equations they developed are presented in Table 15. Although the goodness-of-fit statistics reported for the transformed regressions are quite good in some cases, comparisons of the measured and predicted moduli yielded poor results. Thus, they do not recommend use of these models, by virtue of high potential for error in the predicted values.

In summary, several researchers have had reasonable success in developing models to predict Mr as a function of soil parameters. Although the specific parameters used varied among researchers, all considered some indicator of moisture conditions and soil density. Most also considered one or more parameters representing the gradation of the soil, and some measure or indicator of plasticity.


Table 15. Predictive equations for Equation 5 constitutive model coefficients as derived by Von Quintus and Killingsworth[ 5 ]
Soil Type Equation R2/Standard Error
Clay LogK1=17.622 –0.2647Wopt – 0.4430Ws + 2.6732(γdsd max)+
0.1320%slit + 0.6422LL– 0.3742PI – 0.1963γd max – 0.00087P40S
K3=3.3673 –0.01464Wopt –1.7371(Ws/Wopt)–0.1264(γdsd max)
–0.02400P40–0.03483PI+0.001779γd max
Silt LogK1=1.9823+0.01394Ws –0.5934(Ws/Wopt)+0.1500(γdsdmac)
+0.00831γd max+0.000334(γ2d max/P40
K2=6.4676 – 0.0861Wopt –0.5458(γdsd max)+0.0080S –
0.04226γd max
K35.7391+0.07929Ws–1.1778(Ws/Wopt) +0.008037%silt
+0.04549γd max
Sand LogK1=2.7602 –0.00702ws – 0.08076(Ws/Wopt)+
0.05750(γdsdmac)+ 0.000279γd max
K2=0.7386 – 0.0149W opt + 0.3916(γdsd max) – 0.00604S – 0.00157γd max 0.226/
K3=–0.04978 – 0.0092Ws+0.008377(Ws/Wopt) –0.0052%silt + 0.00487γd max 0.304/


Wopt=Optimum water content LL =Liquid limit
Ws= Water content of test specimen PI = Plasticity inex
γds=Dry density of test specimen P40=Percentage passing the No. 40 sieve
γdmax=Maximum dry unit weight of soil S=Degree of saturation
%silt=Percentage of silt  


It is well established that the in situ structural properties of pavement layers vary on a seasonal basis. The laboratory investigations discussed in the preceding section begin to explain why these variations occur, but do not, by themselves provide a complete basis for estimating the magnitude of the changes, and the duration of the different states, as is needed for design and evaluation. To address this issue fully, one must turn to field investigations. Early work in this regard, like that of Scrivner et al., was most often directed at characterizing the degree and duration of the weakened state occurring as a result of spring thaw in Northern regions to provide a basis for the application of spring load restrictions.[ 61 ] More recent investigations have addressed the need to characterize the full annual cycle of changes for purposes of pavement design and performance evaluation.

Yang investigated the seasonal variations in deflections and in situ moduli of six flexible pavements in and around Ithaca, NY, over a three–and–one–half–year period.[ 34 ] He found that significant seasonal variation occurs in all of the pavement layers. Even within this small geographic region, between–site differences in seasonal behavior (e.g., the duration of the thaw–weakened state) were observed, due to differences in the materials and drainage characteristics of the different pavements.

Newcomb et al. investigated seasonal variations at a number of sites in Washington State and Nevada.[ 62 ] They found that, for the Washington sites, seasonal variations in the moduli of the subgrade materials were much less significant than those observed in granular base materials. The data for the Nevada sites were less extensive, typically encompassing less than one year. Interestingly, they noted a tendency toward more variation from point-to-point than from season–to–season.

More recently, Uhlmeyer, Mahoney, et al. investigated seasonal variations in a broad array highway and Forest Service pavements in the State of Washington.[ 63 ] Their findings in this study supported the earlier conclusion that seasonal variations in the base layer are greater than those in the subgrade for the conditions encountered in Washington State. They selected the set of moduli ratios presented in Table 16 as being "representative of flexible pavements located in areas with modest annual freezing and thawing." They also found that the apparent magnitude of the"seasonal effect"was reduced significantly if the stress sensitivity of the pavement materials was considered, rather than treating the soils as being linearly elastic. finding is especially noteworthy because the stress-sensitive nature of pavement materials frequently neglected in field investigations.


Table 16. Representative modular ratios[ 63 ]
Month Modular Ratios
Aggregate Base Subgrade
January 0.6 0.9
February 0.6 0.8
March 0.6 0.8
April 0.6 0.8
May 0.7 0.9
June 0.8 0.9
July 0.9 0.9
August 1.0 1.0
September 1.0 1.0
October 0.8 0.8
November 0.7 0.8
December 0.6 0.9

Sebaaly et al. explored the seasonal variations in the moduli of pavements in each of three districts in Nevada over a five-year period.[ 64 ] Taking the summer moduli as the baseline condition, they developed a series of seasonal factors (multipliers) to estimate the moduli in each of four seasons for each district, as summarized in Table 17. The general approach is identical to that previously described for the State of Washington.[ 15 ] The factors applicable to the base and subgrade moduli for a given season are quite similar in magnitude, but the ranges are broader for the subgrade soils.


Table 17. Seasonal multipliers for base and subgrade soils in Nevada[ 64 ]
Layer Spring Summer Fall Winter
Base 0.68–0.70 1.00 0.93–0.98 0.87–0.95
Subgrade 0.70–0.79 1.00 0.85–1.02 0.77–0.81

Similarly, Lindly and White explored the differences between subgrade moduli backcalculated from pavement deflection data obtained in the spring and summer of a single year for 15 pavement sites in Indiana.[ 65 ] For the sites investigated, they found that the mean spring modulus was 79 to 87 percent of that obtained for the summer.


At present, the most comprehensive model addressing the effects of climate on pavements is the EICM.[ 66 ] As noted previously, this model provides the basis for consideration of seasonal variations in the 2002 Guide for Design of New and Rehabilitated Pavement Structures currently under development through NCHRP 1–37A. The EICM is an enhanced version of the Integrated Climatic Model (ICM) developed by Lytton et al. in the mid-1980s. [ 67 ]

As its name implies, the ICM integrated three separate models addressing different aspects of climatic effects on the pavement into a single comprehensive package. Those models are the Climatic-Materials-Structures (CMS) model, developed at the University of Illinois;[ 68 ] the Infiltration and Drainage (ID) model, developed at the Texas Transportation Institute;[ 69 ] and the CRREL Frost Heave and Thaw Settlement Model, developed by the U.S. Army Corps of Engineers’ CRREL.[ 70 ] Collectively, the elements of the ICM provide the capability to simulate climatic conditions at the pavement site; temperature, moisture, and freeze-thaw conditions internal to the pavement; asphalt stiffness; and base, subbase, and subgrade moduli, all as a function of time.

Solaimanian and Bolzan conducted an independent evaluation of the temperature prediction capabilities of the ICM.[ 71 ] Two types of analysis were conducted as a part of their investigation: (1) sensitivity analysis to evaluate the effects of various input parameters on the predicted pavement temperature profile; and (2) comparison of the model predictions to monitored temperature profiles. Their work did not encompass the moisture prediction capabilities of the ICM.

Factors considered in the Solaimanian and Bolzan sensitivity analysis included air temperature, percent sunshine, solar radiation, emissivity, absorptivity, and thermal conductivity. Comparisons of predicted and measured temperature profiles were conducted for five geographically dispersed sites in the United States and Canada. They reported good agreement between measured and predicted pavement surface temperatures, with the proviso that proper selection of boundary conditions, climatic parameters, and material properties is necessary to obtain reasonable results. Further evaluation of the model and enhancements to both the user interface and the manner in which the model considers the effect of windspeed were recommended.

Larson and Dempsey report that the following enhancements to the original ICM are embodied in Version 2.0 of the EICM: [ 66 ]

  • Enhancements to the user interface to simplify the creation of input files.
  • Improvements to the computational engine to provide for the use of metric units,variable length analysis periods, and use of actual daily climatic data (instead of average values)
  • Enhancements to facilitate manipulation of program output files.

In addition, they recommend further improvements to the model to (1) more accurately determine the quantity of water entering the subgrade; (2) allow use of hourly climatic data; and (3) facilitate manipulation and use of EICM output.

The following remarks summarize the state of the EICM as it existed when this study, and related work undertaken as a part of the NCHRP 1-37A, were initiated.

  • The EICM uses very sophisticated algorithms to simulate climatic conditions, both external and internal to the pavement.
  • A limited independent evaluation of the temperature-prediction capabilities found the model output to be reasonable. Similar findings relative to the moisture prediction capabilities were not reported in the literature, although the original developers of the ICM state that it provides reasonable predictions.
  • The EICM moisture and temperature prediction results are not effectively used in the simulation of seasonal variations in the moduli of unbound pavement materials because the very simple relationships used to estimate moduli neglect key factors.

The work discussed in Chapter 4 begins to address the need for independent evaluation of the EICM moisture-prediction capabilities, and the work discussed in Chapter 5 addresses the need for predictive models to use with EICM moisture output to predict backcalculatedpavement layer moduli. Work undertaken to develop the approach for considering seasonal variations in the 2002 guide (discussed previously under "Addressing Seasonal Variations in Pavement Design and Evaluation," and in the next section) has brought about substantial progress toward effective application of the EICM capabilities.


One element of the work discussed herein is an evaluation of the moisture predictive capabilities of the ICM. This work was conducted in cooperation with work by the NCHRP 1- 37A research team to develop the 2002 guide methodology for considering seasonal variations in unbound pavement layers that was summarized previously. (See references 16, 17, 18, 19.) The author’s initial findings relative to the adequacy of moisture predictionsobtained using Version 2.0 and 2.1 of the EICM resulted in the initiation of refinements to the EICM as a part of the NCHRP 1-37A research. The model improvements are summarized in Chapter 4, and documented in detail in reference 19. The author’s subsequent evaluation of the revised model is documented in Chapter 4.

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