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Evaluation of Resilient Modulus and Modulus of Subgrade

Reaction for Florida Pavement Subgrades

W. Virgil Ping1, Biqing Sheng

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1 Corresponding Author, Professor, Department of Civil and Environmental Engineering, FAMU-FSU College of

Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA; ping@eng.fsu.edu, 1-

850-410-6129 (phone), 1-850-410-6142 (FAX) 2 Graduate Research Assistant, Department of Civil and Environmental Engineering, FAMU-FSU College of

Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA; shengbi@eng.fsu.edu,

1-850-410-6131 (phone), 1-850-410-6142 (FAX)

ABSTRACT: The performance of subgrade generally depends on its load bearing capacity,

which can be determined by the resilient modulus and the modulus of subgrade reaction.

AASHTO pavement design guide (1993) suggested a theoretical relationship between modulus

of subgrade reaction and resilient modulus of subgrade based on the assumption that the

subgrade material is linear elastic, which was not evaluated by experimental work. This paper

presents an experimental study to evaluate the load-deformation and resilient modulus

characteristics of the granular subgrade soils using field and laboratory tests. An extensive field

static plate bearing load testing program was conducted to evaluate the in-situ bearing

characteristics of typical Florida pavement subgrade soils. In addition, laboratory cyclic triaxial

tests were performed to evaluate the resilient modulus characteristics of the subgrade materials.

Based on the experimental results, correlation relationships were developed between the

subgrade soil resilient modulus and the modulus of subgrade reaction to calibrate the AASHTO

theoretical relationship. It was found that the calibrated relationship was close to the AASHTO

theoretical relationship with a difference around 10%.

INTRODUCTION

Subgrade materials are typically characterized by their resistance to deformation under load,

which can be either a measure of their strength or stiffness. A basic subgrade stiffness/strength

characterization is resilient modulus (MR). Resilient modulus is a measurement of the elastic

property of soil recognizing certain nonlinear characteristics, and is defined as the ratio of the

axial deviator stress to the recoverable axial strain. Both the AASHTO 1993 Guide for Design of

Pavement Structures (AASHTO, 1993) and the mechanistic based design methods (AASHTO,

2008) use the resilient modulus of each layer in the design process. The modulus of subgrade

reaction (k) is a required parameter for the design of rigid pavements. It estimates the support of

the layers below a rigid pavement surface course. The modulus of subgrade reaction is

determined from field plate bearing load tests (Huang, 1993). However, the field plate bearing

load test is elaborate and time-consuming. Recently, resilient modulus has been commonly

applied for both flexible and rigid pavement in the design guide (AASHTO, 1993). Therefore, it

is necessary to develop a relationship between the modulus of subgrade reaction (k) and the

subgrade soil resilient modulus (MR). This allows the designer to treat the seasonal variation of

the subgrade soil k-value by simply converting the same seasonal resilient modulus that would

be used for flexible pavement design.

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In Florida, several major research studies in the past years have been conducted to evaluate the

resilient modulus characteristics of Florida pavement soils (Ping et al., 2000; Ping et al., 2001;

Ping et al., 2001; Ping et al., 2002; Ping et al., 2008). An extensive field static plate bearing load

testing program was performed on selected field pavement sites to evaluate the bearing

characteristics of pavement subgrade soils (Ping et al., 2002). A laboratory triaxial testing

program was carried out to evaluate the resilient modulus of subgrade materials. Recently,

comparative studies were conducted between the resilient modulus and modulus of subgrade

reaction for Florida subgrade soils (Ping and Sheng, 2011). Calibrated correlation relationships

were developed from experimental results.

This paper presents a series of characterization effort of the subgrade modulus from the

laboratory cyclic triaxial test and field experimental studies such as field plate bearing load test.

The subgrade soil resilient modulus and the modulus of subgrade reaction (k) using field

measured experimental results were evaluated. The details could be found elsewhere (Ping et al.,

2000; Ping et al., 2001; Ping et al., 2008). The experimental programs are described briefly as

follows.

EXPERIMENTAL PROGRAMS

An extensive field static plate bearing load testing program was carried out to evaluate the in-situ

bearing characteristics of pavement base, subbase, and subgrade soils (Ping et al., 2001; Ping et

al., 2002). Typical granular subgrade soils were excavated from the field test sites and obtained

for laboratory resilient modulus evaluation. In conjunction with the field experimental programs,

a laboratory triaxial testing program was performed to evaluate the resilient modulus

characteristics of the subgrade materials.

Field Static Plate Bearing Load Test

A series of tests were conducted on selected field flexible pavement sites around Florida (Ping et

al., 2000; Ping et al., 2002). The sites were evenly scattered within the state to better represent

different soil conditions in Florida. Granular materials (A-3 and A-2-4 soils) were most

commonly encountered as roadbed in Florida. Thus, only the granular soils were analyzed in the

field study.

The plate bearing load test procedures employed may vary somewhat, depending on the adoptive

agencies, but the method is generally in close agreement with ASTM D 1196 (ASTM, 2004). In

Florida, the plate load test is designated as FM 5-527 in the Manual of Florida Sampling and

Testing Methods (FDOT, 2000). At each site, the asphalt concrete structural layer was cut and

removed. For each layer of the pavement beneath the asphalt concrete, including the base,

subbase (stabilized subgrade), and subgrade (embankment), the in-situ moisture content and

density were measured using a nuclear gauge device. Representative bag samples of each layer

were taken for future testing of resilient modulus in the laboratory. The plate bearing load test

was conducted on the subgrade (embankment) layer. A 305 mm (12 in.) diameter circular steel

plate was used for applying the load (ACI, 2006). A schematic illustration of the test setup is

shown in Figure 1. After completion of the plate bearing load test program, the subgrade soil

layer was excavated up to more than 1 m below the tested stratum to check the layer

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homogeneity. All soil materials were reconstituted in the laboratory to the in-situ moisture and

density conditions for the resilient modulus test.

Figure 1. Field Plate Bearing Load Test

Laboratory Triaxial Test

The triaxial test setup is shown in Figure 2. Both T292-91I (AASHTO, 1991) and T307-99 test

(AASHTO, 2003) methods were adopted for preparing and testing untreated subgrade materials

for the determination of resilient modulus. At least two duplicate resilient modulus tests were

conducted on each type of soil. The differences between most of the two replicate tests were

within about 5 percent (Ping and Ge, 1997). Thus, the resilient modulus test was repeatable.

Figure 2. Schematic of Triaxial Cell for Resilient Modulus Measurement

ANALYSIS OF EXPERIMENTAL RESULTS

Field Experimental Results

The field experimental program was conducted to evaluate the supporting characteristics of in

situ pavement layers. The plate bearing load test results were calibrated by using secant modulus

concept (Ping et al., 2000; Ping et al., 2002). In the field testing program the number of load

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applications, the angle of internal friction, and the geometry of the bearing plate are constant.

Based on this information, after analyzing the data obtained from the field plate load test, a two-

constant hyperbolic model was proposed to represent the relation of the load-deformation as

follows (Ping et al., 2002):

baP (1)

where P = load,

Δ = deflection, and

a, b = constants.

The representation of the load-deflection curve is illustrated in Figure 3. It was found that the

hyperbolic models have good agreements with the experimental curve.

Deformation, Δ

Lo

ad

, P

Pult

asymptote

Pult/21/b

Ei Esec(at P = Pult/2)

tanθ = 1/a

P = Δ/(a+bΔ)

Figure 3. Rectangular Hyperbolic Representation of Load-Deflection Curve

Modulus of subgrade reaction k is defined as the following equation:

k 0

(2)

where k = modulus of subgrade reaction,

σ0 = pressure applied to the surface of the plate, and

Δ = deflection of the plate.

The k values were calculated with σ0 = 10 psi (68.9 kPa) on 12 in. diameter plate. The detailed

calculation may be found elsewhere (Ping and Sheng, 2011).

Laboratory Triaxial Test Results

The resilient modulus (Mr or MR) was calculated from the load and deformation using the

following equation:

R

drM

(3)

where d = axial deviator stress, and

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R = axial resilient strain.

For granular soils, the resilient modulus Mr is commonly expressed by the following regression

models to show the variation of the Mr versus the bulk stress (θ) and confining pressure (3):

k1r

2k=M (4) 4

3

k

3r k=M (5)

where = sum of the principal stresses, (1 + 2 + 3),

3 = confining pressure, and

k1, k2, k3, k4 = regression constants.

In actual field conditions, the confining pressure at subgrade layers was found to be

approximately 13.8 kPa (2.0 psi). Because the laboratory resilient modulus is stress dependent, a

constant stress level has to be determined in selecting the resilient modulus of roadbed soils for

pavement design. In a laboratory resilient modulus test, the resilient modulus value obtained at a

deviator stress of 34.5 kPa (5.0 psi) under the confining pressure 13.8 kPa (2.0 psi) was

considered representative of the in-situ subgrade modulus (Ping et al., 2001). The subgrade

resilient modulus was then obtained from the bulk stress of 75.8 kPa (11.0 psi) at different

moisture conditions.

DISCUSSIONS ON RESILIENT MODULUS AND MODULUS OF SUBGRADE

REACTION

A theoretical relationship between the k-value and resilient modulus was developed in the

Appendix HH of the AASHTO design guide (AASHTO, 1993), which is as follows:

4.19

)psi()pci( RM

k (6)

It should be noted that this theoretical relationship was developed based on the assumption that

the roadbed material is linear elastic. The elastic layer theory and equation provide the basis for

establishing the relationship. Some other relationships based on the Long Term Pavement

Performance (LTPP) database were established in the Mechanistic-Empirical Pavement Design

Guide (M-EPDG) (AASHTO, 2008) and elsewhere (Setiadji and Fwa, 2009; Khazanovich et al.,

2001).

It is usually impractical to conduct plate bearing load tests in the field on representative subgrade

soils for design projects. Thus, it is necessary to develop a relationship between the modulus of

subgrade reaction (k) and the roadbed soil resilient modulus (MR). This allows the designer to

obtain the k value by simply converting the soil resilient modulus. By changing the units of MR

and k to MPa and MPa/m, Equation (6) becomes the following:

Rmeasured Mk 028.2 (7)

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Equation (7) was based on the definition of k using a 30 in. (762 mm) diameter plate. The

deflection Δ of a plate on a solid foundation can be determined by the following equation (Huang,

1993):

RM

qav

2

)1( 2

(8)

where q = applied pressure, 10 psi;

v = Poisson’s ratio;

a = radius of the plate; and

MR = resilient modulus.

The modulus of subgrade reaction, which is defined as the ratio between an applied pressure q

and the deflection Δ, can be expressed as:

av

Mqk R

)1(

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(9)

It can be found that the modulus of subgrade reaction k is inversely proportional to the diameter

of the plate (Huang, 1993)). If v = 0.45 and a = 15 in. (381 mm), then Equation (9) becomes:

8.18

(psi)(pci) RM

k (10)

Due to the rigid plate restriction of the computer programs, AASHTO re-defined the equation of

modulus of subgrade reaction, which is as follows:

V

Pk (11)

where P is the magnitude of the load (in pounds) applied to the 30 in. plate and V is the volume

(in cubic inches) of soil (directly beneath the plate) that is displaced by the load. This is

considered a valid re-definition and allows the rigid loading plate constraint to be relaxed.

Without the rigid plate restriction, an elastic layer computer program was used to predict the

deflected shapes, displaced volumes and k-values under a 30 in. plate for a range of roadbed soil

resilient moduli. Then, Equation (10) becomes Equation (6). However, it is well known that

granular materials and subgrade soils are nonlinear with an elastic modulus varying with the

level of stresses. Therefore, this theoretical relationship needs to be calibrated in order to be

accommodated in the pavement design.

It should be noted that the field experiments were conducted using a plate with diameter of 305

mm (12 in.) in the plate bearing load test. Since the modulus of subgrade reaction k is inversely

proportional to the diameter of the plate, the k-values need to be converted to the values that

resulted when the plate diameter in the plate bearing load test was 762 mm (30 in.). The

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comparisons between laboratory resilient modulus and modulus of subgrade reaction for

subgrade soils were made and the relationship is shown as follows:

)MPa(25.2)MPa/m( RMk (12)

or 5.17/)psi()pci( RMk (13)

%difference 2.25 2.03

2.0311% (14)

As shown in Equation (14), it appears that the correlation relationship obtained from the

experimental results is close to the theoretical relationship with a difference about 11 percent,

which is reasonable. This calibrated correlation relationship could be utilized in the Florida

pavement design guide for obtaining realistic resilient modulus values of Florida subgrade soils

from laboratory measured resilient modulus values.

CONCLUSIONS

Several major field and laboratory experimental studies were conducted in Florida to evaluate

the resilient modulus and load-deformation characteristics of Florida subgrade soils. The resilient

modulus measured in laboratory was compared to the modulus of subgrade reaction (k)

measured from field test to evaluate the AASHTO theoretical relationship. A calibrated linear

relationship was developed to correlate resilient modulus and modulus of subgrade reaction (k).

It was found that the calibrated relationship based on the experimental results was close to the

AASHTO theoretical relationship.

Conducting the soil resilient modulus test in laboratory and selecting an appropriate resilient

modulus value for pavement design are very complex processes. The processes are even more

time-consuming, labor intensive, and costly on conducting in-situ field plate bearing load test

and obtaining field measured k-values. Therefore, the calibrated relationship between the

resilient modulus and modulus of subgrade reaction (k) could be utilized in the Florida pavement

design guide for obtaining realistic subgrade resilient modulus design values from laboratory

resilient modulus measurements.

ACKNOWLEDGEMENTS

Funding for this research was provided by Florida Department of Transportation (FDOT) and

Federal Highway Administration (FHWA) through the Research Center of the FDOT. The strong

support from the FDOT managers, Bruce Dietrich, David Horhota, Bill Miley, Robert Ho, and

Emmanual Uwaibi, are gratefully acknowledged. Harold Godwin, Rick Venick, and Ron Lewis

with the FDOT State Materials Office conducted the filed plate load tests. Zenghai Yang, Ginger

Ling, Chaohan Zhang, Haitao Liu, and Jian Lan, all former research assistants, performed most

of the laboratory resilient modulus tests. The opinions, findings, and conclusions expressed in

this paper are those of the authors and not necessarily those of the sponsors.

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