FREEZING AND THAWING OF FROST-SUSCEPTIBLE SOILS (DEVELOPMENT OF A
RELIABLE PREDICTIVE MODEL)
By
Pegah Rajaei

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of requirements
for the degree of
Civil Engineering- Doctor of Philosophy
2017

ABSTRACT
FREEZING AND THAWING OF FROST-SUSCEPTIBLE SOILS (DEVELOPMENT OF A
RELIABLE PREDICTIVE MODEL)
By
Pegah Rajaei
Frost depth is an important factor that affects the design of various transportation
infrastructures including pavements, retaining structures, bridge foundations, utility lines, and so
forth. Soil freezing can lead to frost heave and heave pressure, which may cause serious stability
issues. On the other hand, at the beginning of spring season, the ice starts to thaw from the top
down and to a lesser extend from the bottom up. The melted water below the pavement surface is
trapped (setting on impermeable frozen materials). It saturates the top part of the upper pavement
layer. Consequently, the stiffness of the saturated layer decreases causing substantial decrease in
its load bearing capacity and high deformations, which lead to premature and localized failure.
To decrease the spring thaw damage, Spring Load Restrictions (SLR) signs are usually placed
along the roads. The objectives of this study are to develop accurate and reliable frost and thaw
depth and frost heave prediction models, estimate heave pressure and develop a reliable SLR
policy.
After extensive literature review, various existing frost depth models were identified and
tested. These include the finite difference UNSAT-H, the Stefan, the Modified Berggren, and the
Chisholm and Phang models. Unfortunately, some of these models require substantial input data
that are not available and all models yielded inaccurate results. Therefore, statistical frost depth
models were developed using frost depth and air temperature data collected by Michigan
Department of Transportation (MDOT); one for clayey soils and one for sandy soil. The two

models were then combined using the measured thermal conductivity of clayey and sandy soils.
The combined statistical model was then verified using frost depth and air temperature data
collected by Minnesota Department of Transportation (MnDOT).
Additionally, The Gilpin’s mechanistic-empirical model was employed to predict frost
heave. The model produced inaccurate and counterintuitive results in some cases. Therefore, the
model was modified and the empirical frost depth model developed in this study was
incorporated into the model. The resulting model was then simplified to replace some of the
required of input data that are not available. The modified model accuracy was assessed using
the frost heave data measured at 5 sites in Oakland County, Michigan. Further, the relationship
between frost heave and heave pressures were established for four soil types.
Moreover, a new statistical model was developed for calculating the cumulative thaw
degree-day (CTDD) using pavement surface temperature and air temperate data collected by
MDOT. Then, the thaw depth data measured in the state of Michigan were used to assess Nixon
and McRoberts thaw depth predictions model. Since the model did not produce accurate and
acceptable results, statistical thaw depth models were developed using the calculated CTDD
values and thaw depth data collected by MDOT and MnDOT; one for clayey soils and one for
sandy soils. The models were then verified using the calculated CTDD values and thaw depth
data collected by MnDOT. Finally, based on the results of thaw depth model a new SLR policy
was proposed.

In dedication to my beloved parents,
Without whom none of my success would be possible.

iv

ACKNOWLEDGEMENTS

I would like to express my special appreciation and thanks to my advisor, Dr. Gilbert
Baladi, for providing me the opportunity to complete this work, encouraging my research and for
allowing me to grow as a research scientist. His advice on both research as well as on my career
have been priceless. I would also like to thank my committee members, Dr. Karim Chatti, Dr.
Emin Kutay and Dr. Neil Wright for serving as members of my PhD committee, and for their
professional comments and suggestions.
I would also like to thank the Michigan Department of Transportation (MDOT) for the
financial support and the MDOT staff for their cooperation and for providing soil samples and
the soil temperature data. In addition, I gratefully acknowledge the Minnesota Department of
Transportation (MNDOT) for providing the soil temperature data and for its assistance.
At the end, a special thanks to my family. Words cannot express how grateful I am to my
sister, brother, mother, and father for all of the sacrifices that they have made on my behalf. I
would also like to thank all of my friends, particularly Nan Hu, Andisheh Ranjbari and Jenifer
Banks who supported me in writing, and incented me to strive towards my goal.

v

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................................................... viii
LIST OF FIGURES ...................................................................................................................... xii
CHAPTER 1 ................................................................................................................................... 1
INTRODUCTION & RESEARCH PLAN .................................................................................... 1
1.1
Problem Statement ........................................................................................................... 1
1.2
Study Objectives .............................................................................................................. 3
1.3
Research Plan ................................................................................................................... 3
1.3.1
Task 1 - Conduct Comprehensive Literature Review ............................................... 3
1.3.2
Task 2 - Development of Heat Transfer Predictive Model ....................................... 4
1.3.3
Task 3 – Development of Coupled Heat and Mass Transfer Models for Prediction
of Frost Heave and Frost Pressure ........................................................................................... 4
1.3.4
Task 4 – Development of SLR Policy ...................................................................... 5
1.4
Dissertation Layout .......................................................................................................... 5
CHAPTER 2 ................................................................................................................................... 7
LITERATURE REVIEW ............................................................................................................... 7
2.1
Frost Depth ....................................................................................................................... 7
Numerical Models ..................................................................................................... 7
2.2.2.1 UNSAT-H Modeling ........................................................................................... 14
Mechanistic Empirical Models ............................................................................... 15
Empirical Models .................................................................................................... 21
2.2
Frost Heave .................................................................................................................... 24
Frost Heave Mitigation ........................................................................................... 24
2.2.2.1 Capillary Theory ................................................................................................. 28
2.2.2.1 Frozen Fringe Theory .......................................................................................... 32
Frost Pressure .......................................................................................................... 51
2.3
Thaw Depth and Seasonal Load Restrictions ................................................................. 54
Thaw Depth ............................................................................................................. 54
Seasonal Load Restriction....................................................................................... 56
CHAPTER 3 ................................................................................................................................. 72
DATA MINING ........................................................................................................................... 72
3.1
Data Base........................................................................................................................ 72
3.2
Frost Depth Data ............................................................................................................ 73
3.2.1
The State of Michigan ............................................................................................. 73
3.2.2
The State of Minnesota ........................................................................................... 79
3.3
Soil Properties Data ........................................................................................................ 79
3.4
Frost Heave Data ............................................................................................................ 82
3.5
Pavement Surface Temperature and Thaw Depth Data ................................................. 83

vi

CHAPTER 4 ................................................................................................................................. 85
DATA ANALYSIS & DISCUSSION .......................................................................................... 85
Introduction and Research Objectives............................................................................ 85
Hypotheses ..................................................................................................................... 85
Frost Depth ..................................................................................................................... 87
4.3.1
UNSAT-H Model.................................................................................................... 88
4.3.2
Freezing Index and Freezing Degree Day Calculation ........................................... 90
4.3.2.1 Minnesota Cumulative Freezing Degree Day ..................................................... 90
4.3.2.2 Boyd Cumulative Freezing Degree Day ............................................................. 91
4.3.3
Existing Frost Depth Prediction Models ................................................................. 94
4.3.3.1 Stefan’s Equation ................................................................................................ 94
4.3.3.2 Modified Berggren’s Equation ............................................................................ 97
4.3.3.3 Chisholm’ and Phang’s Equation ...................................................................... 100
4.3.4
Frost Depth Empirical Models .............................................................................. 104
Frost Heave .................................................................................................................. 121
4.4.1
Insulation Effect on the Frost Depth ..................................................................... 126
4.4.1.1 Example............................................................................................................. 129
4.4.2
Gilpin’s Frost Heave Model.................................................................................. 130
4.4.2.1 Basic Assumptions ............................................................................................ 130
4.4.2.2 Heat and Mass Balance Equations .................................................................... 132
4.4.2.3 Ice Pressure Distribution in the Frozen Fringe Zone ........................................ 134
4.4.3
Revised Frost Heave Model .................................................................................. 136
4.4.4
Discussion of the Results of the Revised Frost Heave Model .............................. 138
4.4.5
Heave Pressure ...................................................................................................... 146
Thaw Depth .................................................................................................................. 150
4.5.1
Calculation of Cumulative Thawing Degree day (CTDD) ................................... 150
4.5.1.1 Pavement Surface Modeling ............................................................................. 151
4.5.1.2 Model Verification ............................................................................................ 155
4.5.2
Nixon and McRoberts Equation............................................................................ 160
4.5.3
Thaw Depth Empirical Models ............................................................................. 161
Spring Load Restrictions .............................................................................................. 169
CHAPTER 5 ............................................................................................................................... 172
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .............................................. 172
5.1
Summary ...................................................................................................................... 172
5.1.1
Frost Depth Modeling ........................................................................................... 172
5.2
Frost Heave Model ....................................................................................................... 173
5.2.1
Thaw Depth Model ............................................................................................... 173
5.3
Conclusions .................................................................................................................. 174
5.4
Recommendation .......................................................................................................... 176
APPENDICES………………………………...………………………………………………..177
APPENDIX A: Frost And Thaw Depth Analysis ……………………………………....…178
APPENDIX B: Frost Heave Stations Profile………………………………………………....198
REFERENCES …………………………………………………………………………….…..205
vii

LIST OF TABLES

Table 2.1 Thermal resistance values (R-values) at different mean temperature........................... 26
Table 2.2 Design values for FPSF insulation materials based on ACSE 32-01 ........................... 27
Table 2.3 Yearly variation in the resilient modulus of roadbed soils and the associated damage
(AASHTO, 1993) .......................................................................................................................... 58
Table 2.4 CTDD threshold for posting SLR (Mahoney et.al., 1986) ........................................... 60
Table 2.5 Complementary guild for SLR .................................................................................... 62
Table 2.6 Reference temperature .................................................................................................. 64
Table 3.1 RWIS stations in the State of Michigan........................................................................ 75
Table 3.2 Data availability in each database ................................................................................. 78
Table 3.3 Sensor use guide (Decagon Devices Inc., 2008). ......................................................... 81
Table 3.4 Measured thermal properties of different types of soil using KD2 Pro ........................ 83
Table 3.5 Measured total heave and frost depths in different soil types, I75, Oakland County,
Michigan. ...................................................................................................................................... 84
Table 4.1 Hydraulic and thermal properties for different layers in UNSAT-H model ................. 89
Table 4.2 Cumulative freezing degree day calculation, Waters station, Lower Peninsula ........... 91
Table 4.3 Maximum frost depth predicted by Stefan’s equation for RWIS stations .................... 98
Table 4.4 Frost depth calculation using the modified Berggren’s equation, Benzonia, Lower
Peninsula, Michigan .................................................................................................................... 101
Table 4.5 Maximum frost depth predicted by Modified Berggren’s equation for RWIS stations
..................................................................................................................................................... 103
Table 4.6 Performance metrics of Equation 4.8 and Equation 4.9 for frost depth estimations in
the State of Michigan and Minnesota ......................................................................................... 108
Table 4.7 Performance metrics of Equation 4.8 and Equation 4.9 for frost depth estimations in
the State of Michigan and Minnesota ......................................................................................... 115
Table 4.8 Performance metrics of Equation 4.12 for frost depth estimations in the State of
Michigan and Minnesota............................................................................................................. 119
viii

Table 4.9 Frost susceptibility classification (COE, 1984) .......................................................... 124
Table 4.10 Different input values for each site, I75, Oakland County, Michigan. ..................... 145
Table 4.11 Different input values for each soil type ................................................................... 147
Table 4.12 Statistical coefficients in Equation 4-30 for each soil type. ..................................... 150
Table 4.13 Performance metrics for the model ........................................................................... 153
Table 4.14 Performance metrics of Equation 4.42, MNDOT and WSDOT models for the average
daily pavement surface temperature for 12 RWIS stations in the LP and 9 RWIS stations in the
UP in 2012, 2014, and 2015........................................................................................................ 159
Table 4.15 Performance metrics of Equation 4.44, MnDOT and WSDOT models for the CTDD
calculation for 12 RWIS stations in the LP and 9 RWIS stations in the UP in 2012, 2014, and
2015............................................................................................................................................. 159
Table 4.16 Maximum thaw depth predicted by Nixon and McRoberts equation for RWIS stations
..................................................................................................................................................... 161
Table 4.17 Performance metrics of Equation 4.45 for thaw depth estimations in the State of
Michigan and Minnesota............................................................................................................. 165
Table 4.18 Performance metrics of Equation 4.46 for thaw depth estimations in the State of
Minnesota.................................................................................................................................... 169
Table 4.19 Time gap between the measured and calculated thaw depth for sandy and clayey soil
in the state of Minnesota……………………………………………………………………..…170
Table A.1 Frost depth calculation using Stefan equation, Benzonia, LP ....................................180
Table A.2 Frost depth calculation using Modified Berggren equation, Benzonia, LP ................180
Table A.3 Thaw depth calculation using Nixon and McRoberts equation, Benzonia, LP ..........180
Table A.4 Frost depth calculation using Stefan equation, Cadillac, LP ......................................181
Table A.5 Frost depth calculation using Modified Berggren equation, Cadillac, LP ..................181
Table A.6 Thaw depth calculation using Nixon and McRoberts equation, Cadillac, LP ............181
Table A.7 Frost depth calculation using Stefan equation, Grayling, LP .....................................182
Table A.8 Frost depth calculation using Modified Berggren equation, Grayling, LP .................182
Table A.9 Thaw depth calculation using Nixon and McRoberts equation, Grayling, LP ...........182
ix

Table A.10 Frost depth calculation using Stefan equation, Houghton Lake, LP.........................183
Table A.11 Frost depth calculation using Modified Berggren equation, Houghton Lake, LP ....183
Table A.12 Thaw depth calculation using Nixon and McRoberts equation, Houghton Lake, LP183
Table A.13 Frost depth calculation using Stefan equation, Ludington, LP .................................184
Table A.14 Frost depth calculation using Modified Berggren equation, Ludington, LP ............184
Table A.15 Frost depth calculation using Stefan equation, Reed City, LP .................................185
Table A.16 Frost depth calculation using Modified Berggren equation, Reed City, LP .............185
Table A.17 Thaw depth calculation using Nixon and McRoberts equation, Reed City, LP .......185
Table A.18 Frost depth calculation using Stefan equation, Waters, LP ......................................186
Table A.19 Frost depth calculation using Modified Berggren equation, Waters, LP ..................186
Table A.20 Thaw depth calculation using Nixon and McRoberts equation, Waters, LP ............186
Table A.21 Frost depth calculation using Stefan equation, Williamsburg, LP............................187
Table A.22 Frost depth calculation using Modified Berggren equation, Williamsburg, LP .......187
Table A.23 Frost depth calculation using Stefan equation, Au Train, UP...................................188
Table A.24 Frost depth calculation using Modified Berggren equation, Au Train, UP ..............188
Table A.25 Thaw depth calculation using Nixon and McRoberts equation, Au Train, UP ........188
Table A.26 Frost depth calculation using Stefan equation, Brevort, UP .....................................189
Table A.27 Frost depth calculation using Modified Berggren equation, Brevort, UP ................189
Table A.28 Thaw depth calculation using Nixon and McRoberts equation, Brevort, UP ...........189
Table A.29 Frost depth calculation using Stefan equation, Cooks, UP .......................................190
Table A.30 Frost depth calculation using Modified Berggren equation, Cooks, UP ..................190
Table A.31 Frost depth calculation using Stefan equation, Engadine, UP ..................................191
Table A.32 Frost depth calculation using Modified Berggren equation, Engadine, UP .............191

x

Table A.33 Frost depth calculation using Stefan equation, Golden Lake, UP ............................192
Table A.34 Frost depth calculation using Modified Berggren equation, Golden Lake, UP ........192
Table A.35 Thaw depth calculation using Nixon and McRoberts equation, Golden Lake, UP ..192
Table A.36 Frost depth calculation using Stefan equation, Harvey, UP .....................................193
Table A.37 Frost depth calculation using Modified Berggren equation, Harvey, UP .................193
Table A.38 Thaw depth calculation using Nixon and McRoberts equation, Harvey, UP ...........193
Table A.39 Frost depth calculation using Stefan equation, Michigamme, UP ............................194
Table A.40 Frost depth calculation using Modified Berggren equation, Michigamme, UP .......194
Table A.41 Frost depth calculation using Stefan equation, Seney, UP ......................................195
Table A.42 Frost depth calculation using Modified Berggren equation, Seney, UP ...................195
Table A.43 Thaw depth calculation using Nixon and McRoberts equation, Seney, UP .............195
Table A.44 Frost depth calculation using Stefan equation, St. Ignace, UP .................................196
Table A.45 Frost depth calculation using Modified Berggren equation, St. Ignace, UP .............196
Table A.46 Frost depth calculation using Stefan equation, Twin Lakes, UP .............................197
Table A.47 Frost depth calculation using Modified Berggren equation, Twin Lakes, UP ..........197
Table A.48 Thaw depth calculation using Nixon and McRoberts equation, Twin Lakes, UP ....197

xi

LIST OF FIGURES

Figure 2.1 Heat transfer between pavement surface and air temperature (After Dempsey and Pur,
1990) ............................................................................................................................................. 11
Figure 2.2 A schematic representation of two phase heat conduction .......................................... 16
Figure 2.3 Fusion parameter (Îź) versus correction factor (Îť) ....................................................... 20
Figure 2.4 a. Equilibrium between ice and water, b. simple ice-water model .............................. 30
Figure 2.5 Schematic diagrams for the frost heave process (Peppin and Style, 2012) ................. 32
Figure 2.6 Schematic diagram of a freezing soil with frozen fringe (After Peppin and Style,
2012) ............................................................................................................................................. 33
Figure 2.7 A schematic representations of equilibrium conditions for ice and water near a
substrate (Gilpin, 1980) ................................................................................................................ 36
Figure 2.8 the frost heave simulation model (Gilpin, 1980) ......................................................... 38
Figure 2.9 a zone of frozen soil, a freezing fringe, and an underlying zone of unfrozen soil
(Nixon ,1991) ................................................................................................................................ 41
Figure 2.10 Numerical verification of linear temperature profile assumption (Nixon 1991) ...... 43
Figure 2.11 Predicted and observed heave for Konrad test No. 4 (Nixon, 1991) ......................... 43
Figure 2.12 Frost action under foundation causing uplift pressure and behind retaining structure
causing horizontal pressure ........................................................................................................... 53
Figure 2.13 Frost heave and thaw consolidation process (Dore, 2004) ........................................ 57
Figure 2.14 MnDOT CDTT calculation flowchart. ..................................................................... 63
Figure 2.15 Sinusoidal temperature variation (After Berg et al) .................................................. 66
Figure 3.1 RWIS station locations, Michigan ............................................................................... 74
Figure 3.2 A soil profile at a typical RWIS station showing the depths of the temperature sensors.
....................................................................................................................................................... 76
Figure 3.3 Maximum frost depth contours in a typical year in the State of Michigan. ................ 77
Figure 3.4 MNDOT stations location, Minnesota. ....................................................................... 79
Figure 3.5 Thermal conductivity measurement using KD2 pro. .................................................. 81
xii

Figure 3.6 Five locations where the soil thermal properties were measured in each soil sample
using KD2 pro. .............................................................................................................................. 82
Figure 3.7 MDOT frost heave station locations, Oakland County, Michigan. ............................. 84
Figure 4.1 Flow chart of the research plan ................................................................................... 86
Figure 4.2 Schematic of the cross section of the two modeled pavement sites ............................ 88
Figure 4.3 Calculated frost depth using UNSAT-H model and measured frost depth versus time
in Waters station, Lower Peninsula, Michigan. ............................................................................ 89
Figure 4.4 Calculated frost depth using UNSAT-H model and measured frost depth versus time
in Cadillac station, Lower Peninsula, Michigan. .......................................................................... 90
Figure 4.5 Calculation of freezing index using cumulative freezing degree day ......................... 92
Figure 4.6 Comparison of calculated CFDD using Boyd (Boyd 1976) and Minnesota (MnDOT
2009) methods. .............................................................................................................................. 94
Figure 4.7 soil Thermal conductivity of different types of soil based on water content and dry
density obtained by US army cold region and engineering laboratory (CRREL) (Edgar, 2014) . 96
Figure 4.8 The maximum frost depths predicted by Stefan equation versus the measured
maximum frost depths in Michigan. ............................................................................................. 97
Figure 4.9 Measured maximum frost depths in Michigan versus the maximum calculated ones
using the modified Berggren’s equation ..................................................................................... 102
Figure 4.10 Measured maximum frost depths versus calculated ones using Chisholm’ and
Phang’s equation. ........................................................................................................................ 104
Figure 4.11 Frost depths versus cumulative freezing degree day for clayey and sandy soils in the
State of Michigan. ....................................................................................................................... 106
Figure 4.12 Measured frost depths in State of Michigan versus the calculated ones using
Equation 4.8 ................................................................................................................................ 107
Figure 4.13 Measured frost depth in State of Minnesota versus the calculated ones using
Equation 4.8 ................................................................................................................................ 107
Figure 4.14 Measured frost depth in State of Michigan versus the calculated ones using Equation
4.9................................................................................................................................................ 109
Figure 4.15 Measured frost depth in State of Minnesota versus the calculated ones using
Equation 4.9 ................................................................................................................................ 109
Figure 4.16 Measured frost depths in Michigan versus cumulative freezing degree day for clayey
soil showing the best fit and the U.S. Corps of Engineers equations. ........................................ 111
xiii

Figure 4.17 Measured versus calculated frost depth data in clayey soils in Michigan (Equation
4.10). ........................................................................................................................................... 111
Figure 4.18 Frost depths versus cumulative freezing degree day for sandy soil showing the best
fit and the U.S. Corps of Engineers equations in the State of Michigan. ................................... 112
Figure 4.19 Measured versus calculated frost depths in sandy soils in Michigan (Equation 4.11).
..................................................................................................................................................... 112
Figure 4.20 Measured frost depths in clayey soil in the state of Minnesota versus the frost depth
values calculated using Equation 4.10. ....................................................................................... 114
Figure 4.21 Measured frost depths in sandy soil in the state of Minnesota versus the frost depth
values calculated using Equation 4.10. ....................................................................................... 114
Figure 4.22 Correlation between the statistical power coefficient (b) of Equation 4.10 and
Equation 4.11 and the corresponding average thermal conductivity of the soil ......................... 117
Figure 4.23 Correlation between the statistical coefficient (a) of Equation 4.10 and Equation 4.11
and the corresponding average thermal conductivity of the soil ................................................ 117
Figure 4.24 Calculated frost depths using Equation 4.12 versus the measured frost depth in
clayey and sandy soil in the State of Michigan........................................................................... 118
Figure 4.25 Calculated frost depths using Equation 4.12 versus the measured frost depth in
clayey and sandy soil in the State of Minnesota ......................................................................... 119
Figure 4.26 Frost depths versus cumulative freezing degree-day for clayey soil showing the best
fit statistical model and the proposed model (Equation 4.12) in Minnesota. ............................ 120
Figure 4.27 Frost depths versus cumulative freezing degree-day for sandy soil showing the best
fit and proposed model (Equation 4.12) in the State of Minnesota ........................................... 121
Figure 4.28 Effect of capillary and permeability on frost susceptibility (ACPA, 2008). ........... 122
Figure 4.29 Heaving Rate in laboratory test on different disturbed soil types (COE, 1984)...... 123
Figure 4.30 Frost susceptibility criteria, Canadian Department of Transportation (Edgar, 2014).
..................................................................................................................................................... 125
Figure 4. 31 AASHTO four environmental regions (ACPA, 2008) ........................................... 127
Figure 4.32 The schematic of frost heave model (Gilpin, 1980) ................................................ 131
Figure 4.33 Ice pressure along frozen fringe zone (Gilpin, 1980) .............................................. 135
Figure 4.34 Particle separation pressure ..................................................................................... 136

xiv

Figure 4.35 Calculated frost heave for three soil types when GWL= 9 m, TTOP= -3 oC for 100
days, POB = 150 kPa. ................................................................................................................... 139
Figure 4.36 Calculated frozen fringe for three soil types when GWL= 9 m, TTOP= -3 oC for 100
days, POB = 150 kPa. ................................................................................................................... 140
Figure 4.37 Calculated total heave versus overburden pressure for clayey silt in different ground
water table depths when TTOP= -3 oC in 100 days. ..................................................................... 141
Figure 4.38 Calculated total heave versus overburden pressure for sandy clayey silt in different
ground water table depths when TTOP= -3 oC in 100 days. ......................................................... 141
Figure 4.39 Calculated total heave versus overburden pressure for fine sand and silt with pebbles
in different ground water table depths when TTOP= -3 oC in 100 days. ...................................... 142
Figure 4.40 Calculated total heave versus overburden pressure for clayey, silty, gravely, sand in
different ground water table depths when TTOP= -3 oC in 100 days. .......................................... 142
Figure 4.41 Total heave versus overburden pressure for clayey silt when TTOP is fixed at TTOP= -3
o
C and when TTOP is decreasing with a rate of -.057 per day in 100 days. ................................. 143
Figure 4.42 Measured versus calculated frost heave under the shoulder and pavement in 5 sites,
Oakland County, Michigan. ........................................................................................................ 145
Figure 4.43 heave pressure versus calculated total heave for clayey silt in different ground water
table depths when TTOP= -3 oC in 100 days. ............................................................................... 148
Figure 4.44 Heave pressure versus calculated total heave in four soil types when TTOP= -3 oC
over 100 days. ............................................................................................................................. 149
Figure 4.45 Calculated versus measured surface temperature in 2011 using Equation 4.41 and
Equation 4.42 for 12 RWIS stations in the first 120 days of the year. The data are shown in 10day interval.................................................................................................................................. 154
Figure 4.46 Calculated versus measured surface temperature in 2012, 2014 and 2015 for 12
RWIS stations in the LP of the State of Michigan. ..................................................................... 156
Figure 4.47 Calculated versus measured surface temperature in 2012, 2014 and 2015 for 9
RWIS stations in the UP of the State of Michigan. .................................................................... 156
Figure 4.48 CTDD of the calculated surface temperature versus CTDD of the measured surface
temperature in 2012, 2014, and 2015 for 12 RWIS stations in the LP of the State of Michigan.
..................................................................................................................................................... 158
Figure 4.49 CTDD of the calculated surface temperature versus CTDD of the measured surface
temperature in 2012, 2014, and 2015 for 9 RWIS stations in the UP of the State of Michigan. 158

xv

Figure 4.50 Maximum thaw depths predicted by Nixon and McRoberts equation versus the
measured maximum thaw depths in Michigan. .......................................................................... 160
Figure 4.51 Measured thaw depths versus cumulative thawing degree-day for sandy soils in the
State of Michigan ........................................................................................................................ 163
Figure 4.52 Measured thaw depth in sandy soils in the state of Michigan versus the calculated
values using Equation 4.45 ......................................................................................................... 163
Figure 4.53 Measured thaw depths in sandy soil in the state of Minnesota versus the thaw depth
values calculated using Equation 4.45. ....................................................................................... 164
Figure 4.54 thaw depths versus cumulative thawing degree-day for sandy soil showing the best
fit and proposed model (Equation 4.45) in the State of Minnesota. ........................................... 165
Figure 4.55 Thaw depths versus cumulative thawing degree-day for clayey soils in the state of
Minnesota.................................................................................................................................... 167
Figure 4.56 Measured thaw depths in clayey soil in the state of Minnesota (Ada, Marshal,
Starbucks, and Rochester) versus the thaw depth values calculated using Equation 4.46. ........ 167
Figure 4.57 Measured thaw depths in clayey soil in Gatzke station versus the thaw depth values
calculated using Equation 4.46. .................................................................................................. 168
Figure B.1 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 528+88.
..................................................................................................................................................... 200
Figure B.2 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 652+00.
..................................................................................................................................................... 201
Figure B.3 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 719+00.
..................................................................................................................................................... 202
Figure B.4 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 724+00.
..................................................................................................................................................... 203
Figure B.5 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 474+00.
..................................................................................................................................................... 204

xvi

CHAPTER 1
INTRODUCTION & RESEARCH PLAN
1.1

Problem Statement
In cold regions such as Michigan, where air temperature drops below 0 C for extended

periods of time, frost depth is an important factor that affects the design of infrastructures
including pavements, building and bridge foundations and/or utility lines. As the winter begins
the pavement starts to freeze from the top down. empirical and localized models for predicting
frost depth in soils are fairly well established and not complex for a single layered system.
However, it is a challenging task for multi-layered systems subjected to various surface boundary
conditions where the air temperature fluctuates. Perhaps complex and sophisticated models to
predict the propagation of freezing and thawing fronts can be developed. However, such models
require a large amount of data that are expensive to obtain and hence, they cannot be easily
implemented. Therefore many State Highway Agencies (SHAs) tend to use simplified analytical
or empirical models for estimating frost and thaw depth. The problem with such models is that
they require calibration from one state to another or even within a state from one region to
another.. Therefore, the focus of this study is to develop and customize accurate semi-empirical
models to predict freezing and thawing depths in soils. The input data for these models must be
readily available or can be obtained at a minimum cost.
The other complex aspect of freeze-thaw cycles is the estimation of the frost heave of
multi layered system due to ground freezing. During freezing, wet soils undergo heave due to the
formation and growth of ice lenses. Frost heave is a function of many variables including soil
type, its water holding capacity, and its thermal conductivity, air temperature, and frost depth.
The heave could result in significant vertical and lateral stresses and movements which could lift

1

foundations or apply substantial additional stresses to exposed retaining structures. In addition,
frost heave is typically followed by thaw consolidation and settlement. Therefore, spread
footings located on soils subjected to freeze-thaw cycles would experience up and down vertical
movements. For bridges, such movements may create unsafe driving conditions at the boundaries
between the bridge and the adjacent pavement structure.
On the other hand, as the spring begins the pavement starts to thaw from the top down
and to a lesser extend from the bottom up (Marquis, 2008). The extend of frost and the time in
which the pavement starts to thaw is a function of the material types, their thermal properties,
water content, and climatic condition, such as temperature, wind speed, precipitation and solar
radiation. At the beginning of the spring season , the pavement is in a critical condition where
the upper and lower layers are thawed but the layer in between, which is still frozen, acts as an
impermeable layer and trap water between the pavement surface and the undelaying materials
causing saturation. As a results, the stiffness of the saturated layer and its bearing capacity
decrease considerably leadingto substantial pavement deformations (Chapin et.al, 2012). Studies
have shown that up to 90% of pavement damage occurs at this critical state (Tighe et.al, 2006).
This phenomenon occurs particularly in low volume roads, these roads are often built by tight
budgets and therefore they have minimal subbase and surface treatments. (Tighe et.al, 2006;
Chapin et.al, 2012). Spring load restrictions (SLR) signs are usually placed along the roads as
preservation strategies. The accuracy of the SLR implementation is critical in avoiding pavement
damage. Even few days could lead to substantial damages. Further, the trucking industry should
receive sufficient advance notice (at least 7-day notice) prior to posting the SLR in order to be
prepared to follow the weight restriction. It is estimated that accurate posting and removing the
SLR increase the life of low volume asphalt road’s by about 10 percent, which leads to a

2

potential saving of about $10,000,000 annually (Ovik et.al, 2000). Therefore an SLR policy
which is simple, effective and accurate is essential. In this study, a simplified methodology was
developed and verified to accurately estimate when to enforce SLR.
1.2

Study Objectives
The objectives of this study are:

1. Review the advantages and shortcomings of some of the exsiting frost and thaw depth
models.
2. Develop accurate and reliable models for predicting the frost depth during freezing period in
Michigan.
3. Develop a model to predict heave and the resulting pressure under the pavement or behind
existing retaining structures due to freezing of frost-susceptible soils in Michigan.
4. Develop accurate and reliable model for predicting the thaw depth under the pavement in
spring season in Michigan.
5. Investigate changes in pavement bearing capacity in the cycles of freeze and thaw.
6. Develop a model to estimate when to post and remove SLR signs.
1.3

Research Plan
To accomplish the objectives of the study, a research plan consisting of 4 tasks was

drawn . The 4 tasks are summarized in the next few subsections.
1.3.1 Task 1 - Conduct Comprehensive Literature Review
The literature review includes:
1.

The state of the art of modeling of freeze and thaw in soils and their applicability to this
study.

2.

The state of art of modeling of frost heave and the resulting pressure behind retaining walls.

3

3.

The state of the practice of SHAs for forecasting frost and thaw depth and the time at which
to post and remove SLR .

1.3.2 Task 2 - Development of Heat Transfer Predictive Model
After reviewing available models to predict the propagation of the freezing and thawing
front in multilayered soils; the following steps will be taken:
1. The existing frost depth models that simulate the Michigan Department of Transportation
(MODT) data the most will be further scrutinized and modified.
2. The thawing front will be modeled using a modified version of Nixon and McRoberts (1973)
equation and other empirical models to fit the MDOT measured field data.
3. The models will be validated using Minnesota Department of Transportation (MNDOT) data.
1.3.3 Task 3 – Development of Coupled Heat and Mass Transfer Models for Prediction of
Frost Heave and Frost Pressure
The initiation and growth of ice lenses in a soil deposit in cold environment exert uplift
pressure against the foundation and lateral pressures against a retaining structure and the soil
behind it. Both pressures can be estimated using existing theory of coupled mass and heat
transfer for estimating the rate of ice growth.The efforts in this task consist of the four steps
listed below:
1. Estimate the freezing depth (Task 2).
2. Estimate the rate of flow of water to the frozen depth from a water supply (ground water
table, surface water source, etc.) to calculate the rate of growth of ice lenses during the
critical time period and at the most critical location of the site. The estimation of the rate of
flow of water to the frozen depth could be based on several parameters including sub-zero

4

temperature, existence of frost susceptible soil, and the depth to the water table or the
distance to the closest free supply of water.
3. Estimate the amount of heave pressure based on the frost heave model results.
4. Evaluate the accuracy of frost heave model using MDOT heave data.
1.3.4 Task 4 – Development of SLR Policy
As ice in the pavement melts, the base and subgrade become saturated. Melting water is
trapped in the upper subgrade, could not be drained through the frozen zone and consequently
these layers lose strength. Since the frost-susceptible soils have relatively low hydraulic
conductivity, it takes a long time for water to drain from the saturated layers. Therefore
pavement weakness could continue for weeks after it is completely thawed. It is critical to
implement and remove the load restrictions accurately. Therefore, the efforts in this task consist
of the steps listed below:
1. The beginning of SLR period will be estimated using the surface temperature and thaw depth
model.
2. The result of thaw depth model will be used to use to develop recommendations for
removing the SLR signs.
1.4

Dissertation Layout
This dissertation organized in 5 chapters and appendices. The contents of each chapter

are detailed in the table of contents. The title of each chapter is listed below.
Chapter 1 – Introduction & Research Plan
Chapter 2 – Literature Review
Chapter 3 – Data Mining
Chapter 4 – Data Analyses & Discussion

5

Chapter 5 – Summary, Conclusions & Recommendations
Appendices – Additional data, figures and drawings are presented in the appendices.

6

CHAPTER 2
LITERATURE REVIEW
2.1

Frost Depth
One of the most important aspects of infrastructure design such as pavement,

foundations, or utility line is frost depth prediction. Frost depth is a function of the material type,
soil thermal properties, soil water content, and climatic conditions such as temperature, wind
speed, precipitation, and solar radiation. In order to neutralize the effects of frost, foundations are
usually built below the frost line. For pavements, most State Highway Agencies (SHAs) use nonfrost susceptible soils (granular materials). However, over time, fine aggregates migrate from the
lower soil layers and soil becomes frost susceptible. In general, any soil might be considered
frost susceptible when the percent fine (passing sieve number 200) exceeds about seven percent.
Since silt has high water holding capacity and relatively low permeability, it is the most frost
susceptible soil. Depending on the availability of the input data and the required accuracy, frost
depth can be estimated by numerical, empirical, and/or mechanistic-empirical models.
Numerical Models
Different numerical techniques (finite element and finite difference) have been used for
modeling complex transient heat flow in pavement layers. Hsieh et al (Hsieh et al., 1989)
developed a three-dimensional finite difference computer program for predicting temperature
profile in concrete pavements and rainfall infiltration into the layered system. The program
inputs consist of typical meteorological year (TMY) data and typical physic al soil and concrete
properties. They reported that their results were in a good agreement with the test results
provided by the Florida DOT.

7

Dempsey (Dempsey 1985) developed a transient one-dimensional finite difference heat
transfer model (Climatic-Materials-Structural (CMS)) to predict the temperature in asphalt layer.
The required inputs for this model are thermal properties of materials, air temperature, solar
radiation, and wind velocity. In this model, for temperature prediction the Fourier’s law was used
as follows:
𝜕 2𝑇
𝜕𝑇
=
2
𝜕𝑧
𝛼𝜕𝑡

Equation 2.1

Where T = temperature (oC);
z = depth (m);
t = time (s); and
Îą = thermal diffusivity (m2/s).
The model has two boundaries, the surface temperature, and the temperature at the top of
the base layer (McCartney et al., 2010). As can be seen in Figure 2.1, for estimating the surface
temperature, the model considers heat convention and heat radiation in the energy balance
equation as follows (ARA Inc., 2004) :
𝑄𝑖 − 𝑄𝑟 + 𝑄𝑎 − 𝑄𝑒 ± 𝑄𝑐 ± 𝑄ℎ ± 𝑄𝑔 = 0
𝑄𝑠 = 𝑄𝑖 − 𝑄𝑟 = 𝑎𝑠 𝑅 ∗ [𝐴 + 𝐵

𝑆𝑐
]
100

Equation 2.2
Equation 2.3

9
0.28
𝑁𝑊
𝑄𝑎 = 𝜎𝑠𝑏 (𝑇𝑎𝑖𝑟 + 273.15) ∗ (0.77 − 0.074𝑝 ) (1 −
)
5
10
100

Equation 2.4

9
𝑁𝑊
𝑄𝑎 = 𝜎𝑠𝑏 𝜀((𝑇𝑠 + 273.15) ∗ )4 (1 −
)
5
100

Equation 2.5

𝑄𝑐 = ℎ (𝑇𝑎𝑖𝑟 − 𝑇𝑠 )

Equation 2. 6

ℎ𝑐 = 122.93 [0.00144 (𝑇𝑚 + 273.15)0.3 𝑈 0.7 − 0.00097(𝑇𝑠 − 𝑇𝑎𝑖𝑟 )0.3 ]

Equation 2.7

8

Where Qi = incoming short wave radiation (W/m2);
Qr = reflected short wave radiation (W/m2);
Qa = incoming long wave radiation (W/m2);
Qe = outgoing long wave radiation (W/m2);
Qc = energy transferred to or from the body as a result of convection (W/m2);
Qh = effect of transpiration, condensation, evaporation and sublimation (W/m2);
Qg= energy absorbed by the ground (W/m2);
Qs = net short wave radiation (W/m2);
Ql = net long wave radiation (W/m2);
as = surface short wave absorptivity of pavement surface;
R* = extraterrestrial radiation incident on a horizontal surface at the outer atmosphere;
A,B = constants that account for diffuse scattering and adsorption by the atmosphere;
Sc = sunshine percentage;
N = cloud base factor;
W = average cloud cover during the day or night;
Tair = air temperature (oC);
σsb = Stefan-Boltzmann constant, 0.98 *10-8 (W/(m2.oC));
p = vapor pressure of the air (1 to 10mm Hg);

Îľ = emissivity of the pavement;
Ts = surface temperature (oC);
hc = convection heat transfer coefficient;
Tm = average of surface and air temperature (oC); and
U = average daily wind speed (m/s).

9

Furthermore, this model was implemented by the Federal Highway Agency (FHWA)
Integrated Climate Model (ICM) to investigate the environmental effect on the pavement. The
ICM integrates the Infiltration and Drainage model (ID model), developed at Texas A&M
University, the CMS model, the Frost Heave and Thaw Settlement model (CRREL model),
developed at the United States Army Cold Region Research and Engineering Laboratory
(CRREL) (Solaimanian and Bolzan, 1993, Johanneck, 2011).
The ICM model considers that thermal properties of pavement layer do not change over
time but for unbound layer the thermal properties vary due to the change in water and ice
contents. Therefore, the pavement temperature predictions are coupled with the moisture
estimations. As state before, the CMS is used to estimate the temperature within the asphalt
layer. The boundary conditions for CRREL model are the ID and CMS outputs. Also, the
required input variables are soil thermal conductivity, soil specific capacity, soil hydraulic
conductivity and the soil-water characteristic curve (SWCC). The CRREL model is used to
estimate the temperature within the subbase and subgrade layer. The model applies finite element
solution to the governing water and temperature equations. The following equations are used for
predicting the distribution of total hydraulic head and temperature, respectively (McCartney et
al., 2010):
𝑑
𝑑𝐻
𝑑𝐻
(𝑘𝑢
)−𝑆( ) = 0
𝑑𝑧
𝑑𝑧
𝑑𝑡

Equation 2.8

𝑑
𝑑𝑇
𝑑
𝑑𝑇
(𝐶𝑤 𝑇) − 𝐶𝑠 ( ) = 0
(𝑘ℎ ) −
𝑑𝑧
𝑑𝑧
𝑑𝑧
𝑑𝑡

Equation 2.9

Where Ku = unsaturated permeability (m/s);
H = total hydraulic head (m);
S = slope of the soil water retention;

10

Kh = soil thermal conductivity (W/(m.oC));
Cw = water heat capacity (J/(Kg.oC));
Cs = soil heat capacity (J/(Kg.oC)); and
All other parameters are the same as before.

Figure 2.1 Heat transfer between pavement surface and air temperature (After Dempsey and Pur,
1990)

11

Solaimanian and Bolzan investigated the capability of the model in predicting the
pavement temperature profiles accurately (Solaimanian and Bolzan, 1993). They performed a
sensitivity analysis in order to evaluate the effect of different input parameters in the prediction
results. The sensitivity analysis showed the following results:
1.

While the air temperature significantly affects the pavement temperature predictions, the
difference between air temperature and pavement temperature can be as low as 5.5 to 8 oC or
as high as 22 to 28 oC depends on the solar radiation and percent sunshine.

2.

In the same solar radiation and percent sunshine, air temperature and surface temperature has
a linear relationship.

3.

An increase in solar radiation from 395 to 526 (W/(m2)) leads to an increase of 4.5-5.5 oC in
pavement temperature.

4.

An increase in percent sunshine from 45% to 90% increases the pavement temperature by
4.5-5.5 oC in pavement temperature.

5.

An increase in absorptivity from 0.7 to 0.8 or from 0.8 to 0.9 yields a 3 oC drop in pavement
temperature at any depth.

6.

An increase in emissivity from 0.7 to 0.8 or from 0.8 to 0.9 results in an increase of 3 oC in
pavement temperature at any depth.

7.

An increase in thermal conductivity from 1.7 to 3.4 and from 3.4 to 5.11 (W/(m.oC)) leads to
a reduction of 2 oC and 1 oC in pavement surface temperature, respectively. However, the
effect of thermal conductivity changes is found to be greater in larger depths.
Solaimanian and Bolzan results showed that if the proper input variables were chosen the

pavement temperature predictions were within Âą1 oC of the measured surface temperature.

12

However, they recommended modification in user interface of the model in order to reduce the
large number of required inputs variables (Solaimanian and Bolzan, 1993).
Furthermore, in a collaboration study between the University of Illinois and Applied
Research Associates (ARA), the ICM was modified and its moisture prediction capability was
improved. The modified model was called Enhanced Integrated Climate Model (EICM) (Zapata
and Houston, 2008). Since then, the accuracy of the results of EICM was investigated by
different researchers. Liang (Liang, 2006) used EICM to estimate the temperature and moisture
profile in six pavement section in the State of Ohio. The results showed that the predicted
temperature profiles did not coincide with the measured data but they could be considered within
an acceptable range. They also found that there is a good correlation between frost depth
predictions and measured ones in sections with unbound base materials but not in sections with
bounded base material. Heydinger (Heydinger, 2003) evaluated the EICM temperature prediction
in two sites in the State of Ohio for the year 2000. He used the default input data in the model
and his results showed that the EICM consistently over predicted pavement temperature. Ahmed
et al (Ahmed et al, 2005) used site specific input values and compared the EICM temperature
and moisture predictions with the measured values in different site in New Jersey. Their results
showed that while the predicted temperatures follow the same trend as the measured ones, their
difference can be significant. However, other researchers’ results indicated that after local
calibration, EICM temperature predictions are relatively accurate (Khazanovich, 2013; Chung,
and Shin, 2015).
Yavuzturk et al (Yavuzturk et al., 2005) proposed a transient two-dimensional finite
difference model to assess the thermal behavior and temperature distribution in asphalt
pavement. TMY weather data were used and sensitivity analyses were conducted to determine

13

the influence of different thermal properties of the materials on the predicted asphalt
temperature. They reported that the temperature predictions were most effected by variation of
the absorptivity, volumetric heat capacity, emissivity, and thermal conductivity of the materials.
Chapin et al (Chapin et al., 2012) utilized finite element program TEMP/W (GEO-SLOPE 2007)
to simulate freezing and thawing front in the pavement. They applied the program to two sites in
northern Ontario with considerably different pavement structures. First, a steady state analysis
was conducted to establish the initial conditions within the model and second, a transient
analysis was conducted. By using adiabatic1 conditions on the lateral boundaries they induced
one-dimensional heat flow. They reported that the predicted frost front was several days behind
the measured frost front.
2.2.2.1 UNSAT-H Modeling
UNSAT-H is a one dimensional, finite difference computer program developed at the
Pacific Northwest Laboratory (Fayer and Jones, 2000). UNSAT-H can simulate the water and
heat balance in a layered cross section simultaneously. The input properties for the models are
listed below:
1. Hydraulic Properties - To solve the water balance equations, relationships for both water
content and hydraulic conductivity as a function of suction head are required.
To describe soil water retention from measured data the van Genuchten function has been
used:

1

An ADIABATIC process is the changing temperature of air due to its movement. Rising air will cool adiabatically, whereas sinking air
warms adiabatically. The DIABATIC process, on the other hand, is any change in air temperature not associated with adiabatic vertical
displacement of air. The prime source of heating in the DIABATIC process is the sun, while the main cause of cooling is evaporation and
the emission of long wave energy from the ground surface.

14

𝜃 = 𝜃𝑟 + (𝜃𝑠 − 𝜃𝑟 )[1 + (𝛽ℎ)𝑛 ] −𝑚

Equation 2.10

Where 𝜃𝑟 = residual water content;
𝜃𝑠 = saturated water content;
h = suction; and 𝛽, n, m = fitting parameters
2. Thermal Properties - UNSAT-H model use Cass et al. equation to express thermal
conductivity (𝑘ℎ ) as a function of water content (Stormont and Zhou, 2001):
𝜃
𝜃
−[𝐶 ]
𝑘ℎ = 𝐴 + 𝐵 − (𝐴 − 𝐷) exp 𝜃𝑠
𝜃𝑠

𝐸

Equation 2.11

Where 𝑘ℎ = thermal conductivity (W/(m.oC);
𝜃 = the water content corresponding to the measured 𝑘ℎ ;
A, B, C ,D, E = the fitting parameters; and
All other parameters are the same as before.
Mechanistic Empirical Models
Neumann proposed the first solution to the heat transfer phase-change problem in his
lectures in the 1860’s; he then published his work in 1912 (Jiji, 2009). In his solution, onedimensional heat transfer in a semi-infinite region was assumed. The above freezing initial
surface temperature (Ti) drops to T0 (a temperature below the freezing point) and freezing starts
to propagate through the liquid phase as shown in Figure 2.2 (Jiji, 2009).
The governing heat conduction equations for solid and liquid phases are stated in
Equation 2.12 and Equation 2.13, respectively.
𝜕 2 𝑇𝑓
1 𝜕𝑇𝑓
=
2
𝜕𝑥
𝛼𝑓 𝜕𝑡

15

0<𝑥<𝑃

Equation 2.12

𝜕 2 𝑇𝑢
1 𝜕𝑇𝑢
=
𝜕𝑥 2
𝛼𝑢 𝜕𝑡

𝑥>𝑃

Equation 2.13

Where the subscripts u and f refer to unfrozen and frozen, respectively;
t = time since the freezing starts (s);
𝑥𝑖 = frost depth (m);
T = temperature (oC); and
𝛼 = thermal diffusivity (m2/s) calculated using Equation 2.14.
𝛼=

𝑘
𝜌𝑐𝑝

Equation 2.14

Where 𝑘 = thermal conductivity of the soil (W/(m.oC));
𝑐𝑝 = specific heat at constant pressure (J/(Kg.oC));
𝜌 = density (Kg/m3).

Moving
Interface

Frozen
zone

Unfrozen
zone

xTf (x,t)
P

Figure 2.2 A schematic representation of two-phase heat conduction
The interface energy equation is stated in Equation 2.15:
𝜕 2 𝑇𝑓 (𝑥𝑖 , 𝑡)
𝜕 2 𝑇𝑢 (𝑥𝑖 , 𝑡)
𝑑𝑥𝑖
𝑘𝑓
− 𝑘𝑢
= 𝜌𝑓 𝑙
2
2
𝜕𝑥
𝜕𝑥
𝑑𝑡
The boundary conditions are
𝑇𝑓 (0, 𝑡) = 𝑇0
16

Equation 2.15

𝑇𝑓 (𝑥𝑖 , 𝑡) = 𝑇𝑚
𝑇𝑢 (𝑥𝑖 , 𝑡) = 𝑇𝑚
𝑇𝑢 (∞, 𝑡) = 𝑇𝑖
And the initial conditions are
𝑇𝑢 (𝑥, 𝑡) = 𝑇𝑖
𝑥𝑖 (0) = 0
Where the subscripts u and f refer to unfrozen and frozen, respectively
l = latent heat of fusion (J/Kg);
𝑇𝑚 = bulk freezing temperature (oC); and
All other parameters are the same as before.
The frost depth can be estimated using Equation 2.16:
𝑃 = 𝜇 √4𝛼𝑓 𝑡

Equation 2.16

Where P = frost depth (m);
𝜇 = constant obtained from; and
All other parameters are the same as before.
The parameters 𝜇 can be calculated using Equation 2.17:

𝛼𝑓 𝑘𝑢 𝑇𝑖 − 𝑇𝑚
exp(−𝜇2 )
−√
𝑒𝑟𝑓𝜇
𝛼𝑢 𝑘𝑓 𝑇𝑚 − 𝑇0

𝜇 2 𝛼𝑓
exp (− 𝛼 )
𝑢
𝛼𝑓
1 − erf (√𝛼 𝜇)
𝑢

=

√𝜋𝜇𝑙
𝑐𝑝𝑓 (𝑇𝑚 − 𝑇0 )

Where the subscripts u and f refer to unfrozen and frozen, respectively;
erf = Gauss error function;
Ti = initial surface temperature (oC);
T0 = surface temperature at t≠0 (oC); and
17

Equation 2.17

All other parameters are the same as before.
Further, Stefan solved Neumann’s equation for a special case of no heat transfer in liquid
layer in 1891, (Jiji, 2009) as follow:

𝑃=√

2𝑘𝑓 (𝑇𝑚 − 𝑇0 )
𝑡
𝜌𝑙

Equation 2.18

Where all parameters are the same as before.
Stefan assumed that the applied constant surface temperature (𝑇𝑚 − 𝑇0 ) multiplied by the
time (t) is equivalent to the freezing index (FI) at that time. He further introduced a
dimensionless multiplication parameter (n) to converts air temperature to surface temperature.
then , Equation 2.19 became:
172.8 𝑘𝑓 ∗ 𝑛 ∗ 𝐹𝐼
𝑃=√
𝐿

Equation 2.19

𝐿 = 334𝑤𝛾𝑑

Equation 2.20

Where P = depth of freeze or thaw (m);
𝑘𝑓 = thermal conductivity of soil (W/(m.oC));
n = dimensionless parameter which converts air index to surface index;
FI = freezing index (oC-day); note that the freezing index in Stefan equation is similar to
the cumulative degree-day at time t, it is not the conventionally defined freezing
index for a winter season.
L = volumetric latent heat of fusion (KJ/m3);
w = water content and; and
𝛾𝑑 = dry density (Kg/m3).

18

Since Stefan’s equation does not consider the volumetric heat capacity of the soil and
water the accuracy of the results are debatable. Consequently, several studies have been
conducted to improve the prediction of frost depth, including the modified Berggren’s equation
(Aldrich et al., 1953). Berggren’s Equation is very much similar to the early work of Neumann.
Therefore, it is not explained here. Aldrich et al. applied a correction factor to Berggren’s
equation, which is a function of two dimensionless parameters, the thermal ratio (Îą), and the
fusion parameter (Îź) (see Figure 2.3). In this figure V0 is the initial temperature differential
(mean annual temperature -0 0C), Vs is the average temperature differential (nFI/t), C is the
average volumetric heat capacity, and L is the volumetric heat of fusion.
These parameters take the effect of temperature changes in the soil mass into account and
depend on the freezing index, the annual average temperature in the site and the thermal
properties of the soil (USACE, 1988). The modified Berggren’s Equation can be written as
follows:

𝑃 = 𝜆√

172.8 𝑘 ∗ 𝑛 ∗ 𝐹𝐼
𝐿

Equation 2.21

Where 𝜆 = correction factor; and
All other parameters are the same as before.
A multilayer solution to the modified Berggren’s equation can be applied to
nonhomogeneous soils by calculating the required cumulative freezing degree-day (CFDD) for
frost to penetrate each layer. The maximum summation of the CFDDs must be equal to or less
than the regional and seasonal freezing index. The frost depth can be estimated as the sum of the
thicknesses of all the frozen layers (USACE, 1988). The CFDD required to penetrate the nth layer
is defined as:

19

𝐶𝐹𝐷𝐷𝑛 =

𝑛−1
𝐿𝑛 𝑑𝑛
𝑅𝑛
[(∑
𝑅𝑛 ) + ]
2
86.4𝜆𝑛
2
1

Equation 2.22

Where Ln = volumetric latent heat of fusion of the nth layer (KJ/m3);
Rn= thermal diffusivity of the nth layer= dn/kn ((m2.oC)/W);
dn = depth of the nth layer (m);
𝜆𝑛 = correction factor of the nth layer;
kn = thermal conductivity of the nth layer (W/(m.oC)); and
CFDDn = cumulative freezing degree day required for frost to penetrate the nth layer (°Cdays).

Figure 2.3 Fusion parameter (Îź) versus correction factor (Îť)
20

The Pavement-Transportation Computer Assisted Structural Engineering (PCASE)
software provided a more accurate numerical solution of the Modified Berggren’s equation,
(Bianchini et al. 2012).
Berg (Berg, 1996) applied Modified Berggren’s equation to 40 sites in the state of
Minnesota for 3 years to assess the accuracy of the results. He reported that predicted frost
depths were within Âą15 percent of the measured frost depth. He also conducted different
sensitivity analysis to assess the dependence of the predicted frost depths to the n-factor (defined
on page 2-5), water content, dry density, thermal conductivity, and each layer thickness. Berg
concluded that small variation in thickness, water content and dry density of each layer would
have a small effect on the predicted frost depths. On the other hand he found that increases in the
n-factor values would result in deeper frost depths prediction. Whereas increasing the measured
thermal conductivity by 25 percent would lead to better frost depths prediction. Stated
differently, Berg found that the modified Berggren’s equation produced more accurate estimates
of the frost depth when the measured thermal conductivity was artificially increased by 25
percent.
Empirical Models
Chisholm and Phang used the data from different stations throughout Ontario and
developed an empirical equation to correlate the calculated cumulative freezing degree day
(CFDD) and the measured frost depths (Chisholm and Phang, 1983).
𝑃 = 0.0578 √𝐶𝐹𝐷𝐷 − 0.328
Where P = depth of freeze or thaw (m); and
All other parameters are the same as before.

21

Equation 2.23

Many State Highway Agencies (SHAs) used similar approach to generate their own
equations or simply calibrated Equation 2.23 using local frost depth data and CFDD.
Dore (Tighe et al, 2007) conducted a research to develop an empirical model for frost
depth in Quebec, Canada. First, he developed Equation 2.24 to estimate pavement surface
temperatures (PST) based on the measured air temperatures. Second, he calculated the
cumulative freezing degree day (CFDD) based on the estimated pavement surface temperature
(PST of Equation 2.24) and estimated the frost depth using Equation 2.25. Third, he correlated
the estimated frost depths from Equation 2.25 to the measured frost depth and obtained statistical
Equation 2.26.
𝑃𝑆𝑇 = 𝑇𝑀𝐸𝐴𝑁 + [0.178(𝑇𝑀𝐴𝑋 − 𝑇𝑀𝐼𝑁 )] + 1.628

Equation 2.24

𝑃 = 𝐶 √𝐶𝐹𝐷𝐷

Equation 2.25
2 0.5

𝑃𝑐𝑜𝑟𝑟

(√𝐶𝐹𝐷𝐷 − 𝑋𝑀𝐸𝐴𝑁 )
1
= 𝑃 + [𝐶𝐼(𝑆𝑒 ) (1 +
)+(
)
∑(𝑋𝑖 − 𝑋𝑀𝐸𝐴𝑁 )2
398

]

Equation 2.26

Where; TMEAN = (TMAX+TMIN)/2;
TMAX = maximum daily air temperature (oC);
TMIN = minimum daily air temperature (oC) and; and
PST = estimated pavement surface temperature (oC).
P = frost depth (cm);
C = regression constant; and
CFDD = cumulative freezing degree days based on the estimated pavement surface
temperature (PST) (oC-day).
Pcorr = corrected frost depth;

22

CI = confidence interval for a population mean, a function of significance level, alpha =
0.4, one standard deviation and a sample size of one;
Se = sum of squared errors;
Xi = measured frost depth (cm); and
XMEAN = Average measured frost depth (cm).
Tighe et.al (Tighe et.al, 2007) used data from one study site along Highway 569 in
Northern Ontario and calibrated the Chisholm and Phang model. Furthermore, they used CFDD
and cumulative thawing degree day (CTDD) and developed a modified model for estimating the
frost depths as follow:
For 0 ≤ 𝑖 ≤ 𝑖0
For i ≥ 𝑖0

𝑃𝑖 = 𝑎 + 𝑏√𝐶𝐹𝐷𝐷𝑖 + 𝑐√𝐶𝑇𝐷𝐷𝑖

Equation 2.27

𝑃𝑖 = 𝑑 + 𝑒√𝐶𝐹𝐷𝐷𝑖 + 𝑓√𝐶𝑇𝐷𝐷𝑖

Equation 2.28

Where i = number of days after the day indexed as day i=0 (i= 0 day on which air temperature
first falls below 0 oC);
i0 = day after which the CTDD consistently increases;
Pi = depth of frost on day i;
CFDDi = cumulative freezing degree day on day i (oC-days);
CTDDi = cumulative thawing degree day on day i (oC-days); and
a,b,c,d,e,f = calibration coefficients.
Moreover, they used Road Weather Information Systems (RWIS) in three sites close to
the study site to estimate the frost depths and compare them with the study site data. They
estimated the calibration coefficients and calculated the frost depth. Although, the coefficient of
determination was 91%, the reliability of the model is questionable since only one year of data
was used.
23

2.2

Frost Heave
In seasonally frozen regions, soil freezing causes frost heave, which may cause extensive

damage to various civil engineering structures, such as pavements and utility lines (Liu et.al,
2013).
Frost heave refers to the uplifting of ground surface caused by freezing of water within
the layers of soil. Taber (Taber, 1930) was the first one that demonstrated experimentally the
features of frost heave. Before Taber frost heaving was explained based on experiments with
closed systems. Taber showed that under normal conditions, the freezing occurs in an open
system. Therefore in the freezing process water migrates through the soil voids below the
freezing zone, causes excessive heaving by creating segregated ice layers. Tendency of a soil to
heave under the freezing conditions is known to be influenced by parameters such as soil type,
freezing rate, availability of water and the applied load or overburden pressure
Frost Heave Mitigation
The effects of frost heave on various structures vary from one structure to the next.
Typically, structural foundations are constructed below the expected frost depths and hence, they
are not affected by frost heave. Frost susceptible soils or free standing water behind bridge
abutments and/or behind exposed retaining structures (such as retaining structures along
depressed highways), are subjected to frost and frost heave causing active pressure against the
structures. Basement retaining walls are rarely affected by frost due to heat loss from the
basement interior that keeps the soil in the vicinity of the wall in relatively warm conditions.
Pavement structures are frost heave susceptible especially if the roadbed soils are not protected
from frost action or if the granular base and subbase are subjected to saturation due to lack of
proper drainage. Given the potential damage due to frost heave, different techniques have been

24

proposed to mitigate frost heave damage especially in the pavement. The most common
techniques are:
1.

Cutting off the Water Source - The source of water can be cut off in many different ways.
One common technique is to install a barrier between the water source and the frost zone
(Edgar, 2014). The barrier reduces the capillary action and consequently reduces frost heave.
A blanket or a layer of gravel and crushed stone under the pavement or wrapping the roadbed
soil by a geo-membrane layer could be effective in decreasing access to water (Wallace,
1987; Edgar, 2014). Another technique is to remove water using a proper drainage system. In
pavements, drain tile, edge drain, and/or open side ditches can be built to remove the water.
In retaining wall, weep holes can be installed at the foot of the wall which is exposed to frost
(Wallace, 1987).

2.

Removing Frost Susceptible Soil - As stated in the previous section, some soils are more
frost susceptible than others. Such soils can be replaced by non-susceptible soils if the cost is
not prohibitive. In a typical scenario, the various frost heave mitigation options are assessed
against their costs. The most cost effective option is typically chosen.

3.

Reducing Freezing Depth – Although different approaches can be used to prevent frost
penetration, two of these approaches are insulation and chemical additives to lower the water
freezing temperature. Since insulation is the most common method, it is detailed further
below.

4.

Insulation Method – This method could be used in many different structures to decrease heat
loss from the soil to the atmosphere. In pavements, an insulation layer is typically placed
above the roadbed soils to protect the soils from freezing. Rigid polystyrene foams (RPF) are
commonly used for frost protection under different building foundation and infrastructures.

25

Two types of polystyrene have been used; expanded polystyrene (EPS) and extruded
polystyrene (XPS).
The insulation materials are usually known by their thermal resistivity (R-value). R-value
is an indication of material resistance to heat flow. It has inverse relationship with thermal
conductivity of the material (Edgar, 2014). Table 2.1 shows the R-value of different RPF
according to ASTM C578. It should be noted that the nominal R-value varies depending on
moisture exposures condition. Moisture condition could vary from one site to another and it
depends on the drainage system and on the direction along which the insulation is installed
(vertical or horizontal). Therefore in the design process, the effective R-values are calculated or
estimated and used.
Table 2.1 Thermal resistance values (R-values) at different mean temperature
Classification XI
I
VII
II
IX XIV
Minimum
Density
11.2 14.4 18.4 21.6 28.8 38.4
(kg/m3)
Mean
Temperature
-3.9 Âą 1 oC
4.4 Âą 1 oC
43.3 Âą 1 oC

XII

X

IV

VI

VII

V

19.2

20.8

23.2

28.8

35.2

48.1

Thermal Resistance of 2.54 Centimeters Thickness Minimum ((m2.oC)/W)
1.97 2.39 2.51 2.62 2.74
1.88 2.28 2.39 2.51 2.62
1.65 1.85 1.97 2.08 2.19

2.74
2.85
2.19

2.96
3.08
2.45

3.19
3.08
2.65

3.19
3.08
2.65

3.19
3.08
2.65

3.19
3.08
2.65

3.19
3.08
2.65

Another important property of the PRF is the minimum thickness. Non-uniform
distribution of moisture in RPF leads to edge effects and as the insulation thickness decreases it
impacts the thermal performance of the RPF. It should be noted that the effect of the thickness
varies depending on the insulation type and moisture conditions (Crandell, 2010). Table 2.2
shows the Design values for frost protected shallow foundation (FPSF) RPF based on ASCE 3201.
26

Various researches investigated the effect of temperature and moisture conditions on the
RPF properties. Ojanen and Kokko (1997) used data from different highway projects to evaluate
the EPS performance. They found that the thermal conductivity measured at -5 oC is the most
relevant to the highway conditions. Their data showed that with proper drainage, the long term
moisture contents in EPS under highways are in the range of 0.5 to 2.5 %. Sandberg (1986) did
research on RPF performance under highways. He found that moisture content distribution is
highly non-uniform in XPS which reduces the influence of moisture content on R-value in
comparison to EPS (Crandell, 2010). Nevertheless, it is apparent that XPS performs consistently
under different conditions. But EPS performance could vary based on the moisture content,
density and manufacturing process (Crandell, 2010).
Table 2.2 Design values for FPSF insulation materials based on ACSE 32-01

Insulation
Type

Minimum
Density
(kg/m3)

Effective Resistivity
(R/cm.)
Vertical

Horizontal

Nominal
Resistivity
(R/cm.)

Allowable
Bearing
Capacity
(kg/m2)

Minimum Insulation
Thickness (cm.)
Vertical

Horizontal

ESP
II
IX

21.6
28.8

0.7
0.8

0.6
0.6

0.9
0.9

N/A
5859

5.1
3.8

7.6
5.1

1.1
1.1
1.1
1.1
1.1

N/A
5859
9374
14061
23436

3.8
2.5
2.5
2.5
2.5

5.1
3.8
2.5
2.5
2.5

XPS
X
IV
VI
VII
V

21.6
25.6
28.8
35.2
48.1

1.0
1.0
1.0
1.0
1.0

0.9
0.9
0.9
0.9
0.9

The magnitude and the rate of frost heave can be predicted in terms of certain
characteristics of the freezing system and some boundary conditions by use of a practical theory
explaining the frost heave of a specific soil (Konrad and Morgenstren, 1980). In general the
theories toward this matter can be classified into two categories, capillary theory and frozenfringe theory.
27

2.2.2.1 Capillary Theory
Capillary theory, also known as primary frost heave theory, is characterized by a frozen
and an unfrozen zone within the soil strata. Consider pure water to be at equilibrium with ice,
when a differential amount of water freezes at constant temperature and pressure:

For two phases of ice and water;

𝑑𝐺 = 𝑉𝑑𝑃 − 𝑆𝑑𝑇 = 0

Equation 2.29

𝑑𝐺𝑖 = 𝑑𝐺𝑤

Equation 2.30

𝑉𝑖 𝑑𝑃 − 𝑆𝑖 𝑑𝑇 = 𝑉𝑤 𝑑𝑃 − 𝑆𝑤 𝑑𝑇

Equation 2.31

By rearrangement the equation becomes
𝑑𝑃 𝑆𝑖 − 𝑆𝑤 ∆𝑆𝑤𝑖
=
=
𝑑𝑇 𝑉𝑖 − 𝑉𝑤 ∆𝑉𝑤𝑖

Equation 2.32

Where the subscripts “i” and “w” stand for ice and water, respectively (Takagi, 1978).
G = Gibbs free energy (J);
S = entropy (J/oC);
V = volume (m3);
P = pressure (Pa); and
T= temperature (oC).
The entropy change ∆𝑆𝑤𝑖 and the volume change ∆𝑉𝑤𝑖 are the changes, which occur
when a unit amount of water is transferred from phase w to phase i at the equilibrium
temperature and pressure.
Clapeyron substituted the latent heat of phase transition as ∆𝐻𝑤𝑖 = 𝑇𝑚 ∆𝑆𝑤𝑖 in Equation
2.32 and obtained Equation 2.33 (Smith et al. 2001):
𝑑𝑃
∆𝐻𝑤𝑖
=
𝑑𝑇 (𝑇𝑚 + 273)∆𝑉𝑤𝑖

28

Equation 2.33

Where ∆𝐻𝑤𝑖 = the enthalpy change when a unit amount of water is transferred from water to ice;
Tm = the bulk freezing temperature, (oC); and
All other parameters are the same as before.
Equation 2.33 can be rewrite as (Peppin and Style, 2012):
𝑃𝑖 − 𝑃𝑤 =

𝜌𝑤 𝐿
(𝑇 − 𝑇)
𝑇𝑚 𝑚

Equation 2.34

Where T = the thermodynamic equilibrium temperature of the system (oC);
L = latent heat of fusion (J/kg);
𝑃𝑖 = ice pressure (Pa);
𝑃𝑤 = water pressure (Pa); and
𝜌𝑤 = density of water (kg/m3).
The Clapeyron equation explains thermodynamically why lowering the temperature
below the freezing temperature causes water to move (be sucked) toward the ice.
Black (Black, 1995) solved Clapeyron equation for different scenarios.
1. If the pressure difference in ice and water are the same, by increasing the confining pressure
of 1 MPa the melting temperature decreases by 0.074 oC.
2.

If the change in confining pressure in water is 1.09 times greater than the change in ice
pressure then the melting temperature remains constant.

3. If water pressure is constant by increasing the ice confining pressure of 1 MPa the melting
temperature decreases by 0.893 oC.
4. If ice pressure is constant by decreasing the water confining pressure of 1 MPa the melting
temperature decreases by 0.810 oC.
Everret (Everret, 1960) constructed a simple model for explaining the capillary theory. He
considered two cylinders each closed up by a piston and joined by a capillary tube as shown in
29

Figure 2.4. By lowering the temperature, water starts to freeze in the upper cylinder. When the
upper cylinder is completely filled up with ice; further decreases in temperature result in water
flow from the lower cylinder to the upper one.
According to Laplace equation if the radius of the capillary tube is r (m), ice can only
penetrate to the capillary tube when Equation 2.35 is satisfied.
𝑃𝑖 − 𝑃𝑤 =

2𝜎𝑖𝑤
𝑟

Equation 2.35

Where 𝜎𝑖𝑤 = ice-water surface energy (J/m2);
All other parameters are the same as before.

Ice

Ice

r=Radius of
capillary tube

Ice

Pi

Capillary
tube

r
Pw

p

Figure 2.4 (a)

water

Figure 2.4 (b)

Figure 2.4 a. Equilibrium between ice and water, b. simple ice-water model
Since the capillary tube represents the soil pores, it implies that segregated ice forms
when
𝑃𝑖 < 𝑃𝑤 +
And pore ice forms when
30

2𝜎𝑖𝑤
𝑟

Equation 2.36

𝑃𝑖 > 𝑃𝑤 +

2𝜎𝑖𝑤
𝑟

Equation 2.37

This implies that the growth of ice lenses will stop as the ice invades the soil at the
maximum heaving pressure given in Equation 2.38 (Loch and Miller, 1975):
𝑝𝑚𝑎𝑥 = 𝑃𝑤 +

2𝜎𝑖𝑤
𝑟

Equation 2.38

The temperature Tl at which ice invades the pores can be found by combining Equation
2.34 and Equation 2.35 into Equation 2.39 (Peppin and Style, 2012):
𝑇𝑙 = 𝑇𝑚 (1 −

2𝜎𝑖𝑤
)
𝑟𝜌𝑤 𝐿

Equation 2.39

Where 𝑇𝑙 = The temperature at which ice invades the pores (oC), and
All other parameters are the same as before.
The capillary theory has various limitations including:
1.

Predictions of the maximum frost-heave pressure works well with idealized soils composed
of particles with one size. But, in soils with different particle sizes the heaving pressures are
considerably larger (Peppin and Style, 2013).

2.

Capillary theory can be used to predict the flow rate towards the ice lenses in the frozen
region. By assuming that the porous medium is incompressible, Darcy’s law can be used to
determine the flow rate of water towards the lenses by Equation 2.40. But the equation tends
to over predict the measured values of flow rate (Peppin and Style, 2012)
𝑉=

𝑘 𝑃𝑅 − 𝑃𝑓
𝜇 𝑍ℎ

Where V =flow rate (m/s);
k = permeability of the soil (m/s);
Îź = dynamic viscosity of water (Pa.s);

31

Equation 2.40

Zh = distance between the ice lens and the water reservoir (m);
PR = ground water pressure (Pa); and
Pf = pressure of the water directly below the warmest lens (Pa).
1. No mechanism for initiation of new lenses has been explained by this method.
2.2.2.1 Frozen Fringe Theory
Since the capillary theory has limitations, some researchers explained the propagation of
frost heave phenomenon by another theory, Frozen Fringe theory. Frozen Fringe theory, also
termed as secondary frost heave, is characterized by three zones: a frozen zone, a partially frozen
zone and unfrozen zone. According to this theory, frost heave can continue to occur at ice-lens
temperatures above 𝑇𝑙 (the temperature at the bottom of the frozen zone) when a frozen fringe is
shaped by formation of ice in the soil pores, see Figure 2.5.

Figure 2.5 Schematic diagrams for the frost heave process (Peppin and Style, 2012)
Stated differently, if the rate of extracting heat is too large or the soil column is too tall,
or too impervious to prevent ice entry the frozen fringe is created beneath ice lenses at the top.
32

Therefore ice pressure could rise above the maximum (Pmax).This process is called secondary
heaving (Loch and Miller, 1972).
At the interface of ice lens and soil particles, there are repulsive intermolecular forces
(surface tension). These forces act like a disjointing pressure that separate ice and soil particles
and initiating a microscopically thin layer of water between the ice lenses and the soil particles
below the freezing temperature, Tm (Dash et al. 2006), see Figure 2.6. Because of the repulsive
forces between the ice lenses and the soil particles, the pressure in the thin water film is reduced
causing suction and upward water movement toward the growing ice lenses (Peppin and Style,
2012).

Figure 2.6 Schematic diagram of a freezing soil with frozen fringe (After Peppin and Style,
2012)
Secondary frost heave can be affected by the suction pressure. The specific
characteristics of the soil determine the practical relation between the suction and the unfrozen
water content. As the ice-water interface curvature is increasing, the unfrozen water content
decreases which consequently yields an increase in suction.

33

According to the Clapeyron formula, an increase in load results into a decrease in the
amount of unfrozen water, which consequently increases the suction. However, increase in load
makes the onset of new ice lens formation more difficult. Therefore, higher suction is required to
separate the soil grains. These dual effects of increase in load make the ice lenses initiation in
clays more easily. The secondary frost heave occurs mostly under any load condition in clays,
while rarely takes place in sands (Fowler and Krantz, 1994).
When the freezing front penetrates into the soil, it absorbs the moisture in the soil, which
stands for a process of both heat and mass transfer (Harlen1973). The complexity in the frost
heave theory arose from this coupled effect of heat and mass transfer. The first model, which
considers heat and mass flow in the soil, was proposed by Harlan (Harlan, 1973).
He proposed that the generalized one-dimensional mass flow for steady or unsteady flow
in a saturated or partially saturated soil media can be modeled by Equation 2.41 and the onedimensional transient heat transfer can be modeled by Equation 2.42
𝜕
𝜕𝐻
𝜕(𝜌𝑤 𝜃𝑙 )
[𝜌𝑤 𝐾(𝑥, 𝑇, 𝜓) ] =
+ ∆𝑀
𝜕𝑥
𝜕𝑥
𝜕𝑡

Equation 2.41

𝜕
𝜕𝑇
𝜕(𝜗𝑥 𝑇) 𝜕(𝐶𝑇)
[𝑘(𝑥, 𝑇, 𝑡)
− 𝑐𝑙 𝜌𝑤
=
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑡

Equation 2.42

Where t = time (min);

ρw = density of water fraction (gr/cm3);
θl = volumetric water content = volume of water/ total volume (cm3/cm3) ;
K = hydraulic conductivity (cm/ min);
T = temperature (Co);
H = total head (cm);
𝜓 = capillary pressure head (cm);

34

∆𝑀 = change in mass of ice per unit volume, unit time (gr/(cm3. min));
cl = bulk specific heat of water (cal /gm /Co);
𝜗𝑥 = water flow velocity in x direction (cm/ min);
C = 'apparent' volumetric specific heat (cal /cm2/Co); and
All other parameters are the same as before.
Gilpin (Gilpin 1979, 1980a, 1980b) studied water flow towards the ice layer and
proposed a physical model for prediction of ice lensing and heave rate, he suggested that frost
heave is a function of basic soil properties and boundary conditions. He assumed that the free
energy of water in the pores is lowered by the surface effect of the solid. Figure 2.7 shows the
pressures in the water near the solid soil surface in the case of the existence of tension between
the water meniscus and the soil. The effect of the tensile surface force on free energy could be
described as follow:
𝐺𝑤 = 𝐺𝑤𝑂 + 𝜈𝑤 𝑃𝑤 − 𝑆𝑤 𝑇𝑤 − 𝑔(𝑦)

Equation 2.43

Where the subscripts w stand for water;
g(y) = a dummy variable expressing the effect of the particle surfaces on the free energy
of water (KJ/mol), g(y) is estimated using Equation 2.44 by setting y equal to h (the
distance between the soil particle surface and the ice lenses);
𝐺𝑤 = the free energy of water near the surface (KJ/mol);
𝐺𝑤𝑂 = the free energy at bulk conditions T0 and P0 (KJ/mol) ;
νw = specific volumes of water (m3/Kg); and
All other parameters are the same as before.

35

In the case of thermodynamic equilibrium, Gw should be constant in the water layer and
also the temperature could be considered constant because the layer is thin. Therefore the effect
of surface can be obtained as:
𝑃𝑤𝑦 =

1
𝑔(𝑦)
𝜈𝑤

Equation 2.44

Where 𝑃𝑤𝑦 = pressure at the distance y from the surface; and
All other parameters are the same as before.

Figure 2.7 A schematic representations of equilibrium conditions for ice and water near a
substrate (Gilpin, 1980)
On the other hand, the free energy in the ice (Gi) can be obtained as:
𝐺𝑖 = 𝐺𝑖𝑂 + 𝜈𝑖 𝑃𝑖 − 𝑆𝑖 𝑇𝑖
Where the subscripts i stand for ice
36

Equation 2.45

𝐺𝑖 = the free energy of ice (KJ/mol);
𝐺𝑖𝑜 = the free energy at bulk conditions T0 and P0 (KJ/mol);
νi = specific volumes of ice (m3/Kg); and
All other parameters are the same as before.
The temperature cannot be different across the phase boundary, but the pressure
difference can be gained as:
̅
𝑃𝑖 − 𝑃𝑤ℎ = 𝜎𝑖𝑤 𝐾

Equation 2.46

̅ = the mean curvature of the interface,
Where 𝐾
𝑃𝑤ℎ = pressure at the distance h from the surface, and
All other parameters are the same as before
Equating Equation 2.43 and Equation 2.45 and using Equation 2.46, g(h) can be
calculated as
̅−
𝑔(ℎ) = −Δ𝜈𝑃𝑤ℎ − 𝜈𝑖 𝜎𝑖𝑤 𝐾

𝐿𝑇
𝑇𝑚

Equation 2.47

Using Equation 2.46 and Equation 2.47, the pressure gradient can be obtained as
𝑑𝑃𝑤𝑦
𝜈𝑖 𝑑
𝐿𝑇
𝜈𝑖 𝑑
𝐿𝑇
̅+
=
[𝑃𝑤ℎ + 𝜎𝑖𝑤 𝐾
]=
[𝑃𝑖 +
]
𝑑𝑥
𝜈𝑤 𝑑𝑥
𝜈𝑖 𝑇𝑚
𝜈𝑤 𝑑𝑥
𝜈𝑖 𝑇𝑚

Equation 2.48

Therefore the flow rate through water layer (q) can be calculated as
𝑞 = −𝑘

𝜈𝑖 𝑑
𝐿𝑇
[𝑃𝑖 +
]
𝜈𝑤 𝑑𝑥
𝜈𝑖 𝑇𝑚

Equation 2.49

The above equation can be modified to obtain the water velocity in the frozen fringe as
𝑉𝑓𝑓 = 𝐾𝑓

𝜈𝑖 𝑑
𝐿𝑇
[𝑃𝑖 +
]
𝑔 𝑑𝑥
𝜈𝑖 𝑇𝑚

Where 𝐾𝑓 = the permeability in the frozen fringe (m/s),
𝑉𝑓𝑓 = the velocity of water flow in the frozen fringe (m/s); and
37

Equation 2.50

All other parameters are the same as before.
Figure 2.8 illustrates schematically the frost heave simulation. A linear temperature
profile is assumed in each layer and the heat balance equation can be written as:

Figure 2.8 the frost heave simulation model (Gilpin, 1980)
−

𝑘𝑓 (𝑇𝑇𝑂𝑃 − 𝑇𝑙 ) 𝑘𝑓𝑓 (𝑇𝑓𝑓 − 𝑇𝑙 ) 𝐿
−
= 𝑉𝐻
𝐻
𝑎
𝜈𝑖

𝑘𝑓𝑓 (𝑇𝑓𝑓 − 𝑇𝑙 ) 𝑘𝑢𝑓 (𝑇𝐵𝑂𝑇 − 𝑇𝑓 )
𝑑𝑧
−
= 𝜌𝑠𝑖 𝐿
𝑎
𝑍
𝑑𝑡

Where a = thickness of the frozen fringe (m);
H = thickness of frozen zone (m);
kf = thermal conductivity of the frozen zone (W/(oC.m));

38

Equation 2.51

Equation 2.52

kff = thermal conductivity of the partially frozen zone (W/(oC.m));
kuf = thermal conductivity of the unfrozen zone (W/(oC.m));
L = latent heat of fusion of water (J/Kg);
TTOP, TBOT = temperatures at the top and bottom of the soil column (oC);
VH = frost heave rate (m/s);
Z = distance between bottom of soil column and position of ice penetration (m);
Tl = temperature at the base of the active ice lens (oC);
Tff = temperature at the base of the frozen fringe (oC);
𝑑𝑧
𝑑𝑡

= frost depth propagation rate (m/s); and

𝜌𝑠𝑖 = mass of ice per unit volume of soil (kg/m3).
The water pressure at the Tf boundary is:
𝑃𝑤𝑓 = −𝑔

𝑃𝑤𝑓

𝑉𝑢𝑓
𝑍
(1 +
)
𝜈𝑖
𝐾𝑢𝑓

𝑑𝑧
𝑧
𝜈𝑤 (𝑉𝐻 + 𝜌𝑠𝑖 Δ𝜈 𝑑𝑡 )
= −𝑔 (1 +
)
𝜈𝑤
𝜈𝑖
𝐾𝑢𝑓

Equation 2.53

Equation 2.54

And finally the velocity in the active frozen zone can be obtained as:
𝑉𝐻 =

𝜈𝑖2
𝑔𝜈𝑤

𝐿(−𝑇𝑙 )
[
− 𝑃𝑂𝐵 + 𝑃𝐿𝑓 ]
𝑎𝐼𝑓𝑙
𝜈𝑤 𝑇𝑚
1
[𝑇 − 𝑇 ] + (𝐾 )
𝑓𝑓
𝑙
𝐿
1

Equation 2.55

𝑇𝑙

1
𝑑𝑇
𝑇𝑓𝑓 𝐾𝑓𝑓

𝐼𝑓𝑙 = ∫

Where 𝑔 = acceleration of gravity (m/s2);
𝑃𝑤𝑓 = water pressure at the edge of the frozen fringe (kPa);
Δ𝜈 = specific volume difference (νi − 𝜈𝑤 );
POB = overburden pressure (kPa);
39

Equation 2.56

KL = permeability of ice lenses (m/s);
𝐾𝑢𝑓 = permeability of unfrozen zone (m/s); and
All other parameters are the same as before.
Using different boundary conditions and solving Equation 2.51, Equation 2.52, Equation
2.54 and Equation 2.55 simultaneously, the water pressure at the bottom of the frozen fringe and
the heave rate can be obtained. In addition, Gilpin proposed an approximate analytical solution.
Nixon (Nixon, 1991) modified the approximate analytical solution of Gilpin. In this
approach, a relationship between the frozen hydraulic conductivity and temperature is needed to
predict the distinct location of each ice lens within the frozen zone. As shown in Figure 2.9, a
linear temperature distribution (see Equation 2.57) and permeability distribution (see Equation
2.58) across the frozen fringe were assumed.
𝑥
𝑇 = (𝑇𝑙 − 𝑇𝑓𝑓 ) (1 − ) + 𝑇𝑓𝑓
𝑎
𝑘=

Equation 2.57

𝑘𝑢𝑓
𝛼
𝑥
[−(𝑇𝑙 − 𝑇𝑓𝑓 ) (1 − 𝑎 ) − 𝑇𝑓𝑓 ]

Equation 2.58

Where x = the depth from the face of the active ice lens (cm);
All other parameters are as before.
If the assumptions of no pore-water phase expansion and incompressible soil are made, the
continuity of water flow indicates that
𝑑
=0
𝑑𝑃𝑤
𝑑𝑥 {
}
𝑑𝑥

Equation 2.59

So the velocity of water flow should be constant in the frozen fringe. At any temperature
the unfrozen water content can be characterized by

40

𝑤𝑢
𝐴(−𝑇)𝐵
𝑊𝑢 =
=
𝑤𝑡𝑜𝑡
𝑤𝑡𝑜𝑡

Equation 2.60

Where T = the temperature (oC);
𝑤𝑢 = the gravimetric unfrozen water content;
𝑊𝑢 = the fraction of the unfrozen water content;
𝑤𝑡𝑜𝑡 = the total gravimetric moisture content; and
A , B = constants.

Figure 2.9 a zone of frozen soil, a freezing fringe, and an underlying zone of unfrozen soil
(Nixon ,1991)
The frost heave can then be calculated as
41

𝑎

𝐻𝑓 = 𝑛 ∫ (1 − 𝑊𝑢 )𝑑𝑥

Equation 2.61

0

Where n = porosity of soil;
Hf = frost heave (cm); and
All other parameters are as before.
The unfrozen water content parameter is redefined as follow: Al = A/𝑤𝑡𝑜𝑡 , and the
distribution of 𝑊𝑢 with depth in the frozen fringe x is
𝑊𝑢 = 𝐴(−𝑇)𝐵 = 𝐴[−(𝑇𝑙 − 𝑇𝑓𝑓 )(1 − 𝑥/𝑎) − 𝑇𝑓 ]𝐵

Equation 2.62

After integration, the frost heave can be calculated using Equation 2.63
1+𝐵

1 + 𝐴{(−𝑇𝑓 )
− (−𝑇𝑙 )1+𝐵 }
𝐻𝑓 = 𝑛 𝑎
(1 + 𝐵)(𝑇𝑓𝑓 − 𝑇𝑙 )

Equation 2.63

The frost heave rate can be obtained from Equation 2.64
𝑑𝐻𝑓
𝐿𝑑𝐻𝑓
𝑑𝐻𝑓
𝑑𝑎 𝑑𝑇𝑙 𝑑𝑇𝑙
= 𝐿(
𝑑𝑎
+
) = (𝑄𝑓𝑓 − 𝑄𝑢 )
𝑑𝑡
𝑑𝑎
𝑑𝑡
𝑑𝑡

Equation 2.64

Where Qff = heat flux through frozen fringe;
Qu = heat flux through unfrozen zones; and
All other parameters are the same.
By comparison with a numerical solution, the assumptions of linearity of the temperature
profile can be checked. The results of the finite difference calculation and the comparison with
the approximate analytical solution are displayed in Figure 2.10. Also the model was used for
different kinds of soil and the comparison between the predicted and observed laboratory results
was made as shown in Figure 2.11 for one of the cases.

42

Figure 2.10 Numerical verification of linear temperature profile assumption. T = + 2.7 oC at
sample base, cooling rate = 0.84 oC /day, sample height = 10 cm, initial lens temperature = 0.14"C, A = 0.05, B = 0.5, W = 20%, and initial freezing point = - 0.04 oC, (Nixon 1991)

Figure 2.11 Predicted and observed heave for Konrad test No. 4 (Nixon, 1991)

43

Fowler and Krantz (Fowler and Krantz ,1994) developed a generalized model for the
secondary frost heave. In order to simplify the governing model’s equations, dimensional
analysis techniques (i.e. normalization and scaling) were used and a new dimensionless
parameter was introduced.
The model could predict the thickness of ice lenses. It was also shown that the thickness
of the frozen fringe initially starts to increase then it reaches a steady state. The results were in
good agreement with experimental data based on a step freezing process. The model is also
capable of being extended to incorporate the solute effects on the freezing temperature and the
unsaturated soils effects in the secondary frost heave.
Konrad and Morgenstern (Konrad and Morgenstern, 1980) proposed a semi empirical
model to solve the mass and heat transfer. The model is based on two assumptions. The first is
zero overburden pressure, which implies that the weight of the overlying soil can be ignored and
the second, Clapeyron equation at the base of the ice lens is valid.
𝑃𝑤 =

In term of total head;

Neglecting the elevation head yields;

𝜌𝑤 𝐿
(𝑇 − 𝑇𝑚 ) = 𝑀𝑇𝑙∗
𝑇𝑚
𝑃𝑤
𝐻𝑤 = ( ) + ℎ𝑒
𝛾𝑤
𝐻𝑤 = (

𝑃𝑤
𝑀
) = ( ) 𝑇𝑙
𝛾𝑤
𝛾𝑤

Equation 2.65

Equation 2.66

Equation 2.67

Where 𝑇𝑙∗ = 𝑇 − 𝑇𝑚 (oC);
𝑀= constant;
ℎ𝑒 = elevation head (cm);
𝐻𝑤 = total head (cm);
Tl = the temperature at the bottom of the frozen zone (the top of the frozen fringe) which
depends on the soil type (oC); and
44

All other parameters are as before.
Assuming Darcy law is valid and considering a two layered system consisting of
unfrozen soil of thickness lu(t) having hydraulic conductivity ku and a frozen fringe thickness d(t)
with the overall hydraulic conductivity ̅̅̅
𝐾𝑓 (𝑡), the velocity of water movement can be attained
using Equation 2.68.
𝑣(𝑡) =

|𝐻𝑤 (𝑡)|
𝑑(𝑡)
𝑙𝑢
+ ̅̅̅
𝐾𝑢 𝐾𝑓 (𝑡)

Equation 2.68

Where lu(t) = thickness of unfrozen soil (cm);
ku = hydraulic conductivity in unfrozen zone (cm/s);
̅̅̅
𝐾𝑓 (𝑡) = overall hydraulic conductivity of frozen fringe (cm/s);
d(t) = frozen fringe thickness (cm);
𝑣(𝑡)= water flow velocity (cm/s); and
All other parameters are the same as before.
At last, the integration of the heave rate over the duration of freezing yields the total
heave hs(t) as stated in Equation 2.69.
𝑡

ℎ𝑠 (𝑡) = ∫
0

𝑡
𝑑ℎ𝑠
𝑑𝑡 = 1.09 ∫ 𝑣(𝑡)𝑑𝑡
𝑑𝑡
0

Equation 2.69

For one-dimensional heat flow the above Fourier equation can be written as
𝜕
𝑑𝑇
𝜕𝑇
(𝑘 ) + 𝑄 = 𝐶
𝜕𝑧 𝑑𝑧
𝜕𝑡

Equation 2.70

where C = volumetric heat capacity (J/(kg.oC));
k = thermal heat conductivity (W/(m.oC));
Q = internal heat generation term per unit area and per unit time (J/(s.kg)); and
All other parameters are the same as before.
45

By solving the coupled heat and mass flow equations, heave can be calculated. The 𝑇𝑙
and ̅̅̅
𝐾𝑓 (𝑡) are physical parameters of the soil that can be determined in the laboratory and must
be known in order to use equations Equation 2.65 to Equation 2.70.
In another development, Konrad and Morgenstern introduced the concept of segregation
potential (SP). They conducted a simple linear analysis based on the following three assumptions
1. Clapeyron equation is valid at the ice lens base.
2. Water flows continuously with an overall hydraulic conductivity in frozen fringe.
3. The temperature (Tl) at the top of the frozen fringe measured in the laboratory for certain soil
type is the same as that in the field. This temperature is called segregation temperature.
The results of their analysis indicate that when the temperature of the warm-side of a soil
sample is held constant and the other side freezes under various freezing temperatures, the water
intake flow toward the ice lenses (heave rate) increases linearly as the temperature gradient
increases. The slope of such linear line is called segregation potential (SP) which can be
expressed as:
𝑉𝐻 = 𝑆𝑃 ∆𝑇

Equation 2.71

Where ∆T = the temperature gradient in the frozen fringe (oC/cm);
SP = segregation potential (cm2/(day. oC)); and
all other parameters are the same as before.
They also conducted laboratory test in order to evaluate the theory. Their laboratory
results were consistent with the theory (Konrad and Morgenstern, 1981).
Nixon 1991 stated that the segregation potential theory published by Konrad and
Morgenstern (Konrad and Morgenstern, 1981) addressed the velocity of the migrating water
toward the freezing front and temperature gradient in the frozen fringe. He stated that “the
46

velocity of water arriving at an advancing frost front is related to the temperature gradient in the
frozen soil just behind the frost front”.
Since this theory is empirical, laboratory tests must be conducted to find the segregation
potential in each soil type under different field conditions. Also, since laboratory and field
conditions are not necessarily based on the same physical conditions, the predictions might not
be fully reliable (Gilpin, 1982).
Additionally, Konrad and Morgenstern investigated the effect of the overburden pressure
on the frost heave rate. They concluded that
1.

As the overburden pressure increases, the segregation temperature decreases.

2.

Increasing the overburden pressure causes decrease in the unfrozen water content and
consequently decrease in permeability. They reported that 400kPa overburden pressure
causes 25 percent decrease in the permeability relative to zero overburden pressure.

3.

Decrease in segregation temperature leads to lager frozen fringe.

4.

Increasing overburden pressure causes decrease in the heave rate. The reason is that increases
in the overburden pressure cause decrease in the overall permeability and decrease in the
suction pressure to move water toward the frozen front.
They also, investigated the concept of “shut-off pressure” at which no water will flow

into or out of the soil. This concept is controversial, some researchers showed that such pressure
can be found in different soil in laboratory conditions and water is drawn to the freezing front in
pressures less than the shut-off pressure and expelled from the frost front as the pressure
exceeded the shut-off pressure. Others believe that given a sufficient freezing time, the water
expulsion could be followed by water intake again.

47

Konrad and Morgenstern also pointed out that because frozen fringe is relatively large in
the field and frost penetration rate is small it is reasonable to assume that the Tf/dt is zero.
Applying this condition in laboratory, they developed an equation for SP and for different
overburden pressure (POB) as follows
𝑆𝑃 = 𝑎 exp(−𝑏𝑃𝑂𝐵 )

Equation 2.72

Where a and b = statistical constants that can be obtained by modeling the data obtained from
laboratory tests; and
All other parameters are the same as before.
Gilpin (Gilpin, 1982) tried to relate the SP approach to his model (Gilpin, 1980). This
model gave a physical foundation to the empirical model. He introduced a dimensionless
segregation potential (DSP) as follows:
𝐷𝑆𝑃 =

𝐿 𝑉𝐻
𝜗𝑖 𝐾𝑓𝑓 (𝑇𝑙 − 𝑇𝑓 )

Equation 2.73

Where DSP = dimensionless segregation potential; and
All other parameters are the same as before
Then, he used Equation 2.73 and the laboratory data to obtain the constant values in the
following two equation forms
𝐷𝑆𝑃 = 𝐶1 (𝑃𝑂𝐵 + 𝑃𝑠𝑒𝑝 − 𝑃𝐿𝑓 )−𝛼

Equation 2.74

−(𝑃𝑂𝐵 − 𝑃𝐿𝑓 )
𝐷𝑆𝑃 = 𝐶2 exp [
]
𝑃𝑠𝑒𝑝

Equation 2.75

Where all parameters are the same as before.
The results indicated that the correlation coefficients are approximately the same (0.97)
for both forms, so either one of them could be used for the calculation of DSP.

48

The applicability of segregation potential was investigated by some researchers (Nixon,
1982; Hayhoe and Balchin, 1990). Nixon installed two circular frost heave test plates at Foothills
Pipe Lines test facility in Calgary, Canada and compared the measured data to the calculated
ones. The SP values were obtained from laboratory data. The water intake at the bottom of ice
lens was calculated using the following equation
𝑉𝑓𝑓 = 𝑆𝑃 𝑔𝑟𝑎𝑑 𝑇

Equation 2.76

Where Vff = the velocity of water flow in the frozen fringe (cm/day);
SP = segregation potential (cm2/(day.oC)) and;
grad T= the temperature gradient in the frozen fringe (oC/cm).
The total heave was then estimated the following three equations
∆ℎ𝑡𝑜𝑡𝑎𝑙 = ∆ℎ𝑠 + ∆ℎ𝑖

Equation 2.77

∆ℎ𝑠 = 1.09 ∗ 𝑉𝑓𝑓 ∗ ∆𝑡

Equation 2.78

∆ℎ𝑖 = 0.09 ∗ 𝑛 ∗ 𝐻

Equation 2.79

Where ∆ℎ𝑡𝑜𝑡𝑎𝑙 = total frost heave (cm);
∆ℎ𝑠 = frost heave due to water intake (cm);
∆ℎ𝑠 = frost heave due to in-situ pore water freezing (cm);
∆𝑡 = time interval (day);
N = soil porosity and; and
H = frost depth (cm).
The results were in relatively good agreement with the measured data in the field (Nixon,
1982). Other researchers used field data in Ottawa, Canada and the segregation potential
approach to calculate the frost heave (Hayhoe and Balchin, 1990). Their data showed that SP

49

could change due to the change in soil properties with depth. However assuming a fixed value
for SP, the estimated heave values were in relatively good agreements with the field data.
Han and Goodings (Han and Goodings, 2006) investigated the differences between clay
and silt freezing behavior due to lower hydraulic conductivity and higher water content of the
saturated clay. They used a geotechnical centrifuge in order to observe the behavior of the soil.
They found that in the unfrozen zone, consolidation occurs due to water migration toward the
frozen zone. This consolidation reduced the total heave. The results of their tests indicated that,
as for other soil type, the heave in clay decreases with increasing overburden pressure. They also
found that due to the low hydraulic conductivity of the clay, the water content effect heave more
than the ground water level (GWL). In other words, due to low hydraulic conductivity, it requires
long time for water to migrate from the GWL toward the frozen zone. Therefore freezing in clay
appears to be a close system and the immediate accessible supply of water has more effect than
the GWL. Further, they developed a simple analytical model (see Equation 2.80) based on using
consolidation concept for 100% saturated soils.
0.09𝑒𝑓
∆𝐻 = (
) ∗ 𝐻𝑓
1 + 1.09𝑒𝑓

Equation 2.80

Where ∆𝐻= heave (cm);
𝐻𝑓 = frost depth including heave (cm); and
𝑒𝑓 = final void ratio in the frozen fringe.
For the case where saturation falls below 100% during the freezing the heave can be
estimated as follows
1.09𝑒𝑓 − 𝑆𝑟𝑓 𝑒𝑖
𝑒𝑓 − 𝑒𝑖
∆𝐻 = [(
)−(
)] ∗ 𝐻𝑓
𝑆𝑟𝑓 + 1.09𝑒𝑓
1 + 1.09𝑒𝑓
Where 𝑆𝑟𝑓 = degree of saturation in the frozen zone,
50

Equation 2.81

𝑒𝑖 = initial void ratio.
Frost Pressure
The magnitudes of heave pressure and heave caused by frost action vary from one
scenario to another and they are function of water availability, soil type, overburden pressure,
below freezing temperatures and the duration of cold season (Loch and Miller, 1972; Nixon,
1991; Konrad and Morgenstern, 1981). The magnitude of frost pressure highly depends on the
particle sizes. In fine sand, the pressure is low, whereas it is intermediate for silt and high for
clay. In fact, the pressure could vary between 420 psf in sand to 6300 psf in clay (Hoekstra,
1969).
For certain design scenario where the frost potential cannot be eliminated, the heave
pressure and heave should be accounted for in the design phase of the structure. Therefore,
accurate estimation of the magnitudes of frost pressure and frost heave are essential part of the
design process. The direction of frost heave and frost pressure is parallel to the heat flow
direction (Penner and Irwin, 1969). For example, under the pavement and bridge foundations, the
frozen front advances vertically downward whereas for retaining wall, frost heave progresses
vertically and horizontally behind the wall (Andersland and Anderson, 1978). In order to
eliminate or decrease frost heave potential, building foundations are typically placed below the
frost line. Otherwise, frost will cause upward pressure against the foundation. When the upward
pressure becomes higher than the downward pressure (due to the foundation weight and applied
load), the foundation will move upward as shown in Figure 2.12a. Behind most retaining
structures, the frost pressure is oriented horizontally against the wall as shown in Figure 2.12b.
The combination of frost and active earth pressures may cause the wall to slide horizontally
along its foundation.

51

The behavior of frozen soil was of interest in the past century, particularly after the
ground freezing techniques were developed. Vialov (Vialove 1965) was the first one who
investigated the viscoelastic behavior of frozen soil comprehensively. The viscoelastic behavior
is especially important in the estimation of creep in artificial ground freezing (Lackner et al.,
2008; Klein, 1981). Estimation of frost pressures and relaxation is of interest in cold-regions
tunnel design. Klein (1981) developed a finite element time dependent model for investigating
the behavior of temporary frozen earth support system in tunneling. Lai et al. (2000) used
numerical inversion of Laplace transform for calculating forces and lining stress in tunnels. They
assumed elastic-viscoelastic behavior for the frozen rock and used Poyting-Thomson model for
illustrating the viscoelastic behavior in their model. Yuanming et al. (2005) assumed that the
frozen soil is nonlinear elastic- plastic isotropic body and developed a finite element model for
estimating the frost heaving pressure along a pile of a land bridge in china. Unfortunately, none
of the mentioned researches conducted laboratory or field testing to evaluate their results.
In the field, there are different complications in estimating frost pressure. The pressure
could be different when the amount of free water is different or when the level of homogeneity is
different. Sometimes the ice penetrates along the cracks and fissure instead of through the pores
which results in lower frost pressures (Penner and Irwin, 1969). Different researchers reported
measurements of frost pressure in different conditions and for different soil types (Penner 1969;
Penner and Gold 1971; Kinosita 1967). Kinosita measured frost pressure both in field and
laboratory conditions. His results showed decrease in frost pressure when the frost depth
penetration stops or slows down in the cycles of freeze and thaw. He assumed that the decrease is
because of viscoelastic behavior of frozen soil; i.e. stress relaxation.

52

(a)

(b)
Figure 2.12 Frost action under foundation causing uplift pressure and behind retaining structure
causing horizontal pressure
The results of field testing showed the following equation for decrease in frost force after
a stop in frost penetration
𝐹 = 𝐹0 (𝑎𝑡 + 1)−𝑛

53

Equation 2.82

Where F0 = the force at the time of stoppage (Pa);
t = time from frost penetration stoppage (hour);
a and n = constants that can be obtained from experiments ; for temperature of -4 oC, a
and n are 35 hour-1 and 1/6, respectively.
By assuming a viscoelastic relationship between frost heave rate and force, he also
suggested the following equation for estimation of frost heave force
𝐹(𝑡) =

𝐶𝑏
𝑎𝑡(2𝑛 − 1) + 1
{1 −
}
(𝑎𝑡 + 1)𝑛
𝑎(1 − 𝑛)

Equation 2.83

Where b = constant frost heave rate;
C = constant; and
All other parameters are the same as before.
His laboratory results also showed that frost heave and frost pressure have very similar
trend.
2.3

Thaw Depth and Seasonal Load Restrictions
This section consists of review of the literature regarding the thaw depth during spring

season and the load restriction models.
Thaw Depth
With the assumptions of no heat transfer in the frozen zone and a linear temperature
distribution in the thawed zone, to estimate the thaw depths over time, Stefan simplified
Neumann’s equation as follow (Jiji, 2009)

𝑋=√

2𝑘𝑢 𝑇𝑠
𝑡
𝜌𝑙

Where X = thaw depth (cm);
t = time since the thawing starts (s);
54

Equation 2.84

ku = thermal conductivity of the unfrozen soil (W/(m.oC));
𝑇𝑠 = applied constant surface temperature (oC);
𝑙 = latent heat of fusion (J/Kg); and
𝜌 = density (Kg/m3).
Nixon and McRoberts (Nixon and McRoberts, 1973) modified the Stefan solution for
multi layered systems. They considered a two layered system, the first layer with the depth of H
and thermal conductivity of k1 and volumetric latent heat of L1 overlying the second layer of soil
with thermal conductivity of k2 and volumetric latent heat of L2. They calculated the time to
thaw the upper layer completely using Equation 2.85. Re-arranging Equation 2.85 yields
Equation 2.86. The product 𝑡0 𝑇𝑠 in the last equation is defined as the cumulative thawing degree
day (CTDD) that is required to completely thaw the first layer. Further, they estimated the thaw
depth for the second layer using Equation 2.87
𝐻 2 𝐿1
2𝑘1 𝑇𝑠

𝑡0 =

𝑡0 𝑇𝑠 = 𝐶𝑇𝐷𝐷 =

𝑋 = [(

Equation 2.85
𝐻 2 𝐿1
2𝑘1

𝑘2 2
𝑘2
𝑘2
𝐻) + 2 𝑇𝑠(𝑡 − 𝑡0 )]1/2 − ( − 1) 𝐻
𝑘1
𝐿2
𝑘1

Where the subscript 1 refers to the first layer;
t0 = time to thaw the overlying layer (s);
H = thickness of the first layer (m);
L = volumetric latent heat of fusion (J/m3);
k = thermal conductivity (W/(m.oC));
Ts = the mean surface temperature during the thawing period (C);

55

Equation 2.86

Equation 2.87

X= the thaw penetration depth (m);
the subscripts 1 and 2 refer to first and second layer, respectively; and
All other parameters are the same as before.
Thus, the estimation of thaw depths using Equation 2.85 through Equation 2.87 requires
knowledge of the thermal conductivity and latent heat of fusion of the soil. Since these inputs are
not always available or expensive to collect, average values are typically used in the calculations
of frost depths.
Seasonal Load Restriction
A typical pavement structure consists of two or three layer system depending on the
pavement class. The thicknesses of these layers vary substantially from one road class to another.
For all Interstate and primary roads, the thicknesses of the layers are designed and constructed to
provide adequate protection of the roadbed soil against freezing. Such protection is not provided
for the majority of the secondary roads including most county roads, city streets and farm-tomarket roads. During the winter season, available water in the pavement structure freezes
creating ice lenses and causing increases in the stiffness of the various pavement layers. Over the
winter months, the ice lenses grow in volume due to migration of water from the ground water
table toward the freezing front. The growth of ice lenses causes the pavement to heave. During
spring season, the frozen ice lenses melt from the top due to warmer temperature and from the
bottom due to the internal heat of the earth. Melted water at the top of the ice lenses cannot drain
by gravity because of the impermeable ice. Hence, the water from the melted ice saturate the
pavement layers especially the upper portion of the roadbed soil. This causes substantial
softening of the roadbed soil. Figure 2.13 shows the pavement deformations due to frost heave
and thaw consolidation. At the beginning of the winter season, suction is build up at the frost

56

front due to frost action. This suction can cause volume reduction in the subgrade soil (See path
“a”). On the other hand, as the frost front progresses, the suction causes upward water flow to the
frost front which leads to formation of excessive ice lenses. As a result of ice lenses formation,
the subgrade soil volume increases as shown in path “b”. When spring thaw occurs, the ice
lenses start to melt from top down making the subgrade soil saturated and resulting in bearing
capacity reduction (See path “c”). The extent of effective stress reduction depends on the degree
of saturation, rate of thaw and drainage system efficiency. On the other hand, as thaw begin,
excess pore pressure start to dissipate gradually resulting in thaw consolidation. Also, as the
water drains out, depends on the subgrade soil permeability and effective hydraulic gradient the
bearing capacity recovers slowly (See path “d”). It is also possible that freeze and thaw action
results in negative of positive residual volume change at the end (Dore, 2004).

∆h

h

∆𝝈’

𝝈’
Figure 2.13 Frost heave and thaw consolidation process (Dore, 2004)

57

It is at this critical time, the damage delivered by the traffic load increases substantially
causing the well- known spring break-up of the pavement structures. Various studies have shown
that up to ninety percent of the pavement damage occurs during the yearly spring thaw period
(Tighe et.al, 2007; Janoo, 2002). Table 2.3 depicts the changes in the MR and the resulting
damage in a typical year (AASHTO, 1993). It can be seen from the table that the resilient
modulus of the roadbed soils decreases during the thawing period and the relative damage
increases substantially.
Table 2.3 Yearly variation in the resilient modulus of roadbed soils and the associated damage
(AASHTO, 1993)

Month

Roadbed Soil Relative
Modulus, MR Damage
(Kpa)
uf

January

137895

0.01

February

137895

0.01

March

17237

1.51

April

27579

0.51

May

48263

0.13

June

48263

0.13

July

48263

0.13

August

48263

0.13

September

48263

0.13

October

48263

0.13

November

27579

0.51

Decemeber

137895

0.01

To minimize this damage and to extend the life of the pavement structures, most highway
authorities post seasonal load restriction (SLR) signs. A study in the State of Minnesota showed
that 20% and 50% weight reduction are expected to increase the pavement service life by 62%
and 95%, respectively (Huen et.al, 2006). The problem stems from the fact that the time for
posting SLR varies substantially from one year to the next and it is a function of the environment
58

and the water flow regime in the area. Such practice is exercised by most European Countries,
Canadian Provinces, and road owners in USA. Some agencies restrict the maximum load that can
be carried by certain axles, others restrict load and speed. Since the severity of winter varies
substantially from one year to the next, most existing practices unintentionally lead to some
pavement damage. On the other hand, The SLR causes hardship to the trucking industry and
increases the number of trucks on the road. Thus, accurate knowledge of the time when the SLR
signs should be posted and removed is crucial to the road owners and the road users. Such timely
posting and removing the SLR signs cannot be had unless accurate prediction of frost and thaw
depths as a function of time can be accomplished. In addition, accurate prediction of freeze-thaw
cycles is critical to an effective load restriction approach (Ovik et al., 2000).
Various practices are used by road owners to establish the value of the maximum loads
and the dates for posting and removing the SLR signs. These practices include:
1. Engineering judgment and visual observations.
2. Fixed dates for posting and removing the SLR signs.
3. Upper bound or critical values of pavement deflections measured using either a falling
weight deflectometer (FWD) or a portable falling weight deflectometer (PFWD). Such
practice is expensive and time consuming and requires repeated FWD testing (Kestler et.al,
2007, Tighe et.al, 2007).
4. Quantitative approaches based on location and observed severity of the winter season (Ovik
et.al, 2000; Miller et.al, 2013, Kestler et.al, 2007, Tighe et.al, 2007).
Kestler et.al (Kestler et.al, 2007) recommended using practice 3, 4 or combination of
both along with complementary guidance stated in Table 2.4. Washington Department of
Transportation (WSDOT) proposed a SLR guideline, which has been adapted and modified in

59

several State Highway Agencies (SHAs). In this policy, the CTDD is calculated using Equation
2.89 (WisDOT, 2003)
𝑛

𝐶𝑇𝐷𝐷𝑛 = ∑ 𝑇ℎ𝑎𝑤𝑖𝑛𝑔 𝑑𝑒𝑔𝑟𝑒𝑒 𝑑𝑎𝑦

Equation 2.88

𝑖=1
𝑇𝑚𝑎𝑥 +𝑇𝑚𝑖𝑛

𝑇ℎ𝑎𝑤𝑖𝑛𝑔 𝑑𝑒𝑔𝑟𝑒𝑒 𝑑𝑎𝑦 = (

2

− 𝑇𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 )

Equation 2.89

Where Tmax = maximum daily air temperature (°C);
Tmin = minimum daily air temperature (°C); and
Treference = -1.7 °C.
Since various temperature measurements indicated that during the thawing period, the
asphalt pavement surface temperature is 0°C when the air temperature is about -1.7°C, the
reference temperature was set at -1.7°C (Mahoney et.al., 1986).
Table 2.4 provides a list of Washington’s seasonal load restriction policy for thin and
thick pavements. The posting of the SLR signs are based on the two levels listed in Table 2.4;
should and must is placed when the 5-day weather forecast shows that the CTDD will reach the
“should be posted or the must be posted” levels. The CTDD value for each level and pavement
type is listed in Table 2.4 (Mahoney et.al., 1986).
Table 2.4 CTDD threshold for posting SLR (Mahoney et.al., 1986)
Pavement Structure

CTDD
“Should” Level

“Must” Level

THIN Asphalt 2” or less
Base course 6” or less

6°C- degree day

22°C-degree days

THICK Asphalt more than 2”
Base course more than 6"

15°C- degree days

28°C- degree days

60

They also developed a regression equation between freezing index (FI), CTDD and thaw
duration (D) based on the results of a heat flow simulation model on fine-grained subgrades.
They recommended to remove the SLR as soon as one of these conditions were reached
(Mahoney et.al., 1986)
𝐷 = 25 + 0.006 ∗ 𝐹𝐼

Equation 2.90

𝐶𝑇𝐷𝐷 = 0.17 ∗ (𝐹𝐼)

Equation 2.91

Where D = thaw duration (days);
FI = freezing Index oC-days; and
All other parameters are the same as before.
The proposed thaw duration was consisting of the thaw duration and the base recovery
time after thaw ends. Yesiller et al (Yesiller et al, 1996) showed that the WSDOT equations
predicts the thaw duration for fine-grained better than granular materials. A study in Minnesota
indicated that WSDOT equation does not accurately predict the thaw duration. Since WSDOT
used numerical model for developing the equation, the model may not reflect the field conditions
precisely and this might be a possible reason for deviation of the prediction from the actual
duration (Ovik et.al, 2000).
The Minnesota Department of Transportation (MnDOT) developed SLR policy based on
the following criteria (Ovik et.al, 2000)
1. Thaw had penetrated to the depth of 15 cm.
2. The weather forecast shows continuous thaw.
3. Different district within the same zone should reach to the same thaw depth (Minnesota was
divided into 5 zones).

61

It was also recommended that deflection measurement should be collected and SLR
should be removed three weeks after the maximum deflection. The deflection measurements
showed that maximum deflection occurred when the thaw depth was about 102 centimeters.
Table 2.5 Complementary guild for SLR
Which pavements need SLR?
1- Pavements with frost susceptible soils in base and subgrade.
2- Pavements with fine materials in subgrade such as ML,MH,CL and CH
3- Pavements with poor drainage system or high groundwater table.
4- Pavements in which distresses like fatigue cracking or rutting has been detected
5- Pavements in which surface deflection is 45% to 50% higher in spring than summer
When SLR should be placed?
1- When pavement surface deflection increases up to 45-50% higher than summer.
2- When the thaw depths reaches the subgrade (This could be determined which a thaw
index method or temperature measurements with frost tubes or electric resistance gauge)
What load restrictions should be placed?
1- Allow load levels that reduce the pavement deflections to those occur in summer.
2- Allow load levels that lead to desired proliferation in service life.
3- Allow load levels based on guidelines developed by different researchers
When SLR should be removed?
1-When pavements recover and the measured surface deflections reduce to the summer
values.
Henceforth, MNDOT revised the policy. They modified the thawing index calculation
approach and adapted a variable reference temperature instead of a fixed one and also
implemented SLR when thaw reaches 30.5 centimeters. The revised SLR policy is based on the
cumulative thawing degree-days (CTDD) and the cumulative freezing degree days (CFDD)
(MnDOT, 2009). The MnDOT procedure for the calculation of the CFDD and CTDD is
summarized in a flowchart format shown in Figure 2.14.
The decision to post and remove the SLR signs is based on the results of the calculation
of the CFDD and CTDD and the corresponding reference temperatures are listed in Table 2.6.
The reference temperature accounts for the effect of the duration and intensity of the sun

62

radiation on the pavement thawing. As stated before, the MnDOT’s SLR policy divides the state
into five zones. The SLR is posted when the 3-day weather forecast shows that the CTDD for a
given zone is more than 15 oC-degree day and the continued warmth is predicted for longer time
period.

CTDD0=0 and n starts at 1

For day n
Taverage-T reference > 0 oC
YES

NO

Thawing degree day = T average-T reference
o

CTDDn-1> (0.5*(0 C-T average))
NO

Freezing degree day = 0 oC- day
YES

Thawing degree day = 0 oC- day

Thawing degree day = 0 oC- day

Freezing degree day = 0 oC- day

Freezing degree day = 0 oC-T average

n=n+1

CTDDn=CTDDn-1+(Thawing degree day- 0.5*Freezing degree day)

Figure 2.14 MnDOT CDTT calculation flowchart. In this flowchart CTDDn is the cumulative
thawing degree day calculated over ‘n’ days , CTDDn-1 is the cumulative thawing degree day
calculated over ‘n-1’ days, Taverage is the daily average air temperature ((Tmax-Tmin)/2) and the
Treference is the reference temperatures listed in Table 2.6

63

Table 2.6 Reference temperature
Date

Reference Temperature
(oC)

January 1 – January 31

0

February 1 – February 7

-1.5

February 8 – February 14

-2

February 15 – February 21

-2.5

February 22 – February 28

-3

March 1 – March 7

-3.5

March 8 – March 14

-4

March 15 – March 21

-4.5

March 22 – March 28

-5

March 29 – April 4

-5.5

April 5 – April 11

-6

April 12 – April 18

-6.5

April 19 – April 25

-7

April 26 – May 2

-7.5

May 3 – May 9

-8

May 10 – May 16

-8.5

May 17 – May 23

-9

May 24 – May 30

-9.5

June 1 – December 3

0

They also revised the thaw duration relationship as follow:
𝐷 = 0.15 + 0.010 ∗ 𝐹𝐼 − 19.1𝑃 − 12090 ∗

𝑃
𝐹𝐼

Equation 2.92

Where P=frost depth (m); and
All other parameters are the same as before.
It is noteworthy that the standard error of estimate for this equation is 8 days and
although the R2 is about 0.5, the equation predicts the thaw duration more accurately than
WSDOT equations. However, because of the low correlation MnDOT, the length of SLR policy
was considered to be 8 weeks. They also verified the 8 weeks length by using the rate of strength
64

recovery in back calculation moduli. It is of interest that their results showed that at two weeks
past the end of thaw the pavement strength recovers between 50 to 100 percent depends on the
soil type. The increase in fines in the base material leads to longer recovery period.
Leong et al. (Leong et al, 2005) used Long Term Pavement Performance (LTTP)
database in 6 stations in Ontario to develop a thaw index base method for predicting SLR. It
should be noted that the frequency of the data in LTTP varies based on the data type, for instance
the FWD and surface temperature data were collected once or twice every two years. They
plotted the average daily temperature against the asphalt temperature and adapted the horizontal
intercept of the graph as their reference temperature for the thaw index calculation. They
analyzed the data for the first 100 oC-days of the year and obtained thawing index of 13 oC-days
as the threshold. This threshold is very similar to MNDOT threshold thawing index of 15 oCdays.
Berg et al (Berg et al, 2006) developed an alternative method for SLR, which considers
the effect of pavement temperature. They assumed that the air temperature can be fitted into the
following equation (See Figure 2.15)
2𝜋
𝑇𝑡 = 𝑀𝐴𝑇 + 𝐴𝑚𝑝 ∗ sin(( ) ∗ (𝑡 − 𝐿𝑎𝑔))
𝑃

Equation 2.93

Where Tt = sinusoidal temperature on Julian day t;
P = period of sinusoidal variation (365 days);
MAT = mean annual temperature in 20 years;
Amp= amplitude of temperature sinusoid and;
Lag= the time that it takes for temperature sinusoid to reach the MAT (110 days Lag was
used)

65

Surface Temp (oC)
Air Temp (oC)

Temperature (oC)

Mean Annual Surface Temp (oC)
Mean Annual Air Temp (oC)

Freeze (oC)

Julian Date (Days)

Figure 2.15 Sinusoidal temperature variation (After Berg et al)
They also estimated the surface temperature using the n-factor of 0.5 and 1.7,
respectively. Then, they calculated the difference between air and surface temperature for each
day. This difference was added to measured air temperature to estimate the surface temperature.
The estimated surface temperature was used to calculate the surface thawing index (CTI) as
follows
𝑛

𝐶𝑇𝐼 = ∑ 0 − (𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑡𝑒𝑚𝑝𝑎𝑟𝑎𝑡𝑢𝑟𝑒)

Equation 2.94

𝑖=1

They recommended to place the SLR with CTI reaches 17 oC-days but they did not
provide a protocol for removing the SLR. It should be noted that this method does not work well
in the mountain area where the closes weather station is not relatively close (Kestler et al, 2011).
Bradley et al (Bradley et al. 2012) developed a SLR policy in Manitoba by using the
FWD data, moisture content, and temperature profile in pavement layers. They calculated
thawing index as follows
66

𝑇𝑚𝑎𝑥 + 𝑇𝑚𝑖𝑛
𝑇𝑚𝑎𝑥 + 𝑇𝑚𝑖𝑛
≥ 0−→ 𝐶𝑇𝐼 = ∑(𝑇𝑟𝑒𝑓 +
)
2
2

Equation 2.95

𝑇𝑚𝑎𝑥 + 𝑇𝑚𝑖𝑛
𝑇𝑚𝑎𝑥 + 𝑇𝑚𝑖𝑛
< 0 −→ 𝐶𝑇𝐼 = ∑(𝑇𝑟𝑒𝑓 +
)
2
4

Equation 2.96

𝑖𝑓
𝑖𝑓

Where Tref = 1.7 oC at March 1 and increases by 0.055 oC per day until May 31 (CTI cannot be
negative, i.e. it resets when CTI values drops below 0), and
All other parameters are the same as before.
They recommended a CTI threshold of 16 oC-days for posting the SLR. Based on their
FWD data they found that the pavement recovers noticeably when thaw depth reaches to 119 cm.
They suggested removing the SLR in 8 weeks or when CTI reaches 350 oC-days, whichever
occurs first.
Miller al (Miller et al, 2013) used Bradley et al, WSDOT and MnDOT methods and
compared their results with measured data in New Hampshire. They used FWD data and
temperature logger data in their analysis. Since Manitoba is closer in latitude to Minnesota, they
calculated thawing index based on MnDOT method. Their results indicated that in posting SLR
the MnDOT and Bradley et al methods are slightly conservative, whereas WSDOT method dates
for applying SLR are consistently late. On the other hand, in most cases Equation 2.91 and
Bradley et al threshold were able to predict the time of SLR removal accurately. However, both
of these methods can be non-conservative in some cases. Generally speaking, the 8 weeks
duration seems to yield fairly conservative results for SLR removal.
Manitoba department of transportation developed the following Equation for thawing
index calculation:
𝑇𝐴𝑖𝑟
2

𝑖𝑓 𝑇𝑀𝑒𝑎𝑛 < 0

𝑇𝑀𝑂𝐷 =

𝑖𝑓 𝑇𝑀𝑒𝑎𝑛 ≥ 0

𝑇𝑀𝑂𝐷 = 𝑇𝑀𝑒𝑎𝑛

67

Equation 2.97
Equation 2.98

𝑇𝐼𝑖 = 𝑇𝐼𝑖−1 + 𝑇𝑀𝑂𝐷 + 𝑇𝑅𝑒𝑓

Equation 2.99

𝑇𝐼𝑅𝑒𝑓 = 1.7 + 0.06𝑖

Equation 2.100

Where TMOD = Modified Temperature (oC);
TAir = Air temperature (oC);
TRef = Reference Air temperature (oC) – 1.7 (oC);
TIi = Thaw index for day i (oC-day);
TIi-1 = Thaw index for day i-1 (oC-day); and
i = number of days since March 1.
They recommended posting SLR when thawing index reaches 15 (oC-day);
Kestler et al (Kestler et al, 2007) investigated the applicability and accuracy of three
methods for posting and removing SLR and reached the following conclusions
1.

Using subsurface temperature and moisture sensors: Their results showed that the
temperature-moisture system can be reliable method to determine when to post and remove
SLR signs. SLR can be commenced when the surface temperature nears 0 oC and can be
ended when the moisture dissipates. It should be noted that although this method is fairly
accurate, it is site specific.

2.

Using the lightweight or potable FWD (PFWD): Since the FWD initial cost is beyond the
some agencies budgets, the use of PFWD for placement and removal of SLR was
investigated. Their results showed that the trend in composite moduli obtained from PFWD
and the moduli back calculated using FWD data is the same. On the other hand, as the
asphalt thickness decreases the correlation between the PFWD and PFW modulus increases.
They concluded that PFWD can be used as a complementary approach for posting and
removing SLR in low volume road with gravel or thin asphalt surface.
68

3.

Using thaw index methods: the thawing index approach proposed by MnDOT showed
promising results but required different reference temperatures that might vary by time and
location. On the other hand, Since the Berg et al approach does not require a reference
temperature; the method can be used in any site without calibration. However, their results
indicated that although the method works well for posting the SLR it is unable to accurately
predict the SLR removal. Furthermore, Kestler el at ( Kestler el at, 2011) developed a toolkit
for implementing the SLR using the following three methods

1. Everseries suite: Everseries suite developed in the State of Washington was used to post
SLR. Firstly, pavement moduli were back-calculated using Evercalc. Secondly, the stress and
strain and damage factor were estimated using Everpave. Damage factor is the ratio of the
load cycle required to reach failure in normal condition to the load cycle required to reach
failure under thaw condition. They results showed the highest damage factor when thaw
reaches 46 centimeters. Also, the damaged factor evened out after 5 weeks of implementing
the SLR.
2. Enhanced Integrated Climatic Model (EICM): EICM was used as a tool to develop the thaw
predictor. EICM was modified to predict the thermal and mass transfer between pavement
layers. They indicated that since the output of the model is frost and thaw depth, therefore the
model can be used as a tool to help the DOTs in the SLR decision making. They
recommended using another climatic database along with National Climatic Data Center
(NCDC). NCDC is consisting of 851 stations across the United States but most of these
stations are at the airports or cities therefore there are not good representatives of remote site
climate conditions. The recommended database is consisting of 2000 remote automated
weather stations (RAWS) which are frequently at isolated areas. Also, since the low volume

69

road could be covered by few centimeters of snow, they also recommended considering the
effect of snow cover in the model. Their results indicated that the EICM model can predict
the beginning of the thaw season fairly well but still needs additional modifications to predict
the end of the thaw as well.
3. The Lightweight Deflectometer (LWD): The LWD estimates the in-situ stiffness modulus of
the soil layers. They showed that the LWD followed the seasonal trend in stiffness changes
and has a good correlation with FWD-derived moduli. For posting the SLR, they
recommended to take different tests in order to obtain modulus in the normal condition and
then take multiple readings during spring thaw season. It was recommended to implement the
SLR when composite modulus is less than 20% of the modulus in normal condition and end
the SLR when the composite gain the 80% of modulus value.
Chapin et al (Chapin et al., 2012) utilized finite element program TEMP/W (GEOSLOPE 2007) to simulate freezing and thawing front in the pavement. They applied the program
to two sites in northern Ontario with considerably different pavement structures. First, a steady
state analysis was conducted to establish the initial conditions within the model and second, a
transient analysis was conducted. For the upper boundary conditions, they investigated two
different scenarios. First scenario was converting the average daily air temperature to pavement
surface temperature by using n-factor. They indicated that n-factor can be between 1.4 to 2.3 and
between 0.29 to 1 for thaw and frost depth prediction, respectively. The second scenario
involved using increasing weekly reference temperatures similar to MnDOT method. Their
results showed that the first scenario yielded the most accurate predictions. But the n-factor can
be different in different locations and even in different years. In Ontario n factor of 2.3 and 0.6
yielded the most accurate thaw and frost depth predictions, respectively. Their results indicated

70

that TEMP/W is able to accurately simulate the thaw depth and rate of thawing at the beginning
of thaw, which is the critical time for posting SLR. But as the thaw progresses the accuracy of
the simulation can be decreased. Therefore, they concluded that TEMP/W simulation results can
be used along with the thawing index approach for posting the SLR but not for removing it.
Moreover, they recommended that thawing index threshold for applying SLR should be refined
actively as more data become available and also, considering the climate difference as well as
pavement structures, different CTI threshold in different regions should be used.

71

CHAPTER 3
DATA MINING
3.1

Database
The evaluation of the accuracy of existing frost depths and heave models or the

development of new accurate and representative ones require field data that represent the
environment and the various pavement structures (Tighe et.al, 2007). Fortunately, such data were
available and obtained from the Michigan Department of Transportation (MDOT) and the
Minnesota department of Transportation (MnDOT). The following databases and their sources
were used in this study:
1.

Road Weather Information System (RWIS) for frost depth and weather data measured in the
state of Michigan. It should be noted that the RWIS subsurface sensors do not measure the
frost depth directly, they measure the subsurface temperature. In the analyses, it was assumed
that the ground water freezes at 0oC.

2.

National Oceanic and Atmospheric Administration (NOAA) weather data in the states of
Michigan and Minnesota.

3.

Minnesota Department of Transportation (MnDOT) database for frost depth measured in the
state of Minnesota.

4.

Michigan Department of Transportation (MDOT) database for frost heave measured in the
state of Michigan.

5.

Michigan State University Enviro-weather (MSU-EW) for weather data in the state of
Michigan.

72

3.2

Frost Depth Data

3.2.1 The State of Michigan
RWIS uses different technologies that collect, transmit and publish weather and road
condition information. The weather data is collected by the environmental sensor station (ESS).
In these stations the sensors collect and transmit weather and pavement data (US DOT, 2002). In
general RWIS may encompass:
1.

Meteorological sensors for measuring atmospheric pressure, temperature, relative humidity,
visibility, wind speed and direction, and precipitation (amount and type).

2.

Pavement sensors for measuring pavement temperature and condition (wet, dry, snow),
subsurface pavement temperature, the amount and type of deicing chemical used on the
pavement surface.

3.

Pavement temperature and weather condition forecast based on the site (Boselly et al.,
1993).

In this study, the RWIS database that was provided by MDOT was used for subsurface pavement
temperature data (RWIS, 2012). RWIS consist of 25 stations located throughout the State of
Michigan. However, in this study, only 18 stations were used (MDOT 2008, MDOT 2009a, and
b) due to partially missing data in seven stations. Figure 3.1shows the stations’ location in the
state of Michigan.
Table 3.1 shows the RWIS stations ID, latitude, longitude and soil type. The detailed soil
log for each station was provided by MDOT. In all stations, one year data (2010-2011) were
available and used except for Au Train, Harvey and Brevort Stations in the UP where two years
of data (2009-2010) were available and used.

73

Figure 3.1 RWIS station locations, Michigan
As mentioned before the RWIS collect meteorological as well as surface and subsurface
pavement temperature data. The data were collected approximately every 10 minutes. The RWIS
data were provided by MDOT and processed in order to be used in this study. Figure 3.2 depicts
a soil profile showing the locations of temperature sensors at a typical RWIS station. The data
were then used to develop a GIS contour map of maximum frost depth in a typical year in
Michigan as shown in Figure 3.3.

74

Table 3.1 RWIS stations in the State of Michigan
Region

Upper
Peninsula

Lower
Peninsula

Station

Latitude

Longitude

Au Train

46.43

-86.84

Brevort

46.01

-85.01

Cooks
Engadine

45.91
46.10

-86.48
-85.62

Golden Lake

46.16

-88.88

Harvey

46.49

-87.23

Michigamme

46.54

-88.13

Seney
St. Ignace

46.35
45.90

-86.04
-84.74

Twin Lakes

46.88

-88.86

Benzonia

44.59

-86.10

Cadillac

44.25

-85.37

Grayling

43.89

-85.53

Houghton
Lake

44.77

-85.40

Ludington

45.36

-85.18

Reed City

44.61

-84.71

Waters

44.33

-84.81

Williamsburg

45.76

-84.73

75

Subgrade Soil (up to 3.05 m)
Upper 0.9 meter of Sand with
Gravel and Silt, below Which
Loose Moist Fine Sand
Loose To Moderately
Compact Moist Fine Sand
Silty Clay with Sand
Plastic Moist Sandy Clay
Moderately Compact Moist
Fine Sand with Gravel
Loose to Moderately Compact
Moist Fine to Medium Sand
Upper 1.5 meter Clayey Sand,
1.5 meter and below Wet Find
to Medium Sand with Silt
Loose Moist to Wet Sand
Silty Clay with Sand
Loose Moist Fine to Medium
Silty Clayey Sand
Very Loose Fine to Medium
Sand
Medium Compact Fine Sand
Fine to Coarse Sand with
Gravel
Loose to Medium Compact
Fine Sand
Very Loose Silty Sand with
Trace Clay
Loose to Medium Fine Sand
with Trace Gravel
Medium Compact to Loose
Fine Sand with Trace Gravel
Medium Compact Fine Sand
with 0.5 meter of Silty Clay

Unfortunately, the meteorological data were not available for the whole 2010-2011
winter. Therefore the NOAA and/or the MSU-EW weather data were used in the analyses
(MSU-EW, 2012; NOAA, 2012). The selected NOAA and/or MSU-EW stations were within 16
kilometers of an RWIS station, otherwise the RWIS data were not used in the study. Table 3.2
shows the data availability in each database.

Surface

0 cm
7.6 cm

Base

13.2 cm
22.9 cm
38.1 cm
53.3 cm
71.1 cm

Sub-base
86.4 cm
101.6 cm
116.8 cm
132.1 cm
Roadbed
Soil

147.3 cm
157.5 cm
172.7 cm
182.9 cm

Figure 3.2 A soil profile at a typical RWIS station showing the depths of the temperature sensors

76

146.8
132.1
132.1
132.1

146.8
146.8

147.3

172.7
101.6

157.5
101.6

86.4

132.1

101.6

91.4
86.4
91.4
91.4

76.2

89.0- 93.0 cm
96.5- 101.6 cm
104.0- 106.7 cm
109.2- 111.8 cm
114.3- 119.4 cm
121.9- 124.5 cm
127.0- 129.5 cm
132.1- 137.2 cm
139.7- 182.9 cm

Figure 3.3 Maximum frost depth contours in a typical year in the State of Michigan

77

Table 3.2 Data availability in each database

Location

Lower
Peninsula

Upper
Peninsula

1.
2.
3.
4.

RWIS
Station
Name

Database

Station
Name

Pr1

Temp2

WS3

SR4

Database

Station Name

Pr1

Temp2

WS3

Station Name

Temp2

Pr1.

Benzonia

Benzonia

✓

✓

✓

✓

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Cadillac

McBain

✓

✓

✓

✓

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Grayling

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Houghton
Lake
Ludington

MSUEW

NOAA
Ludington

✓

N/A

✓

Roben- Hood
Airport

Williamsburg

Traverse
City

✓

✓

✓

✓

N/A

Waters

Gaylord

✓

✓

✓

✓

Gaylord-9 SW

Michigamme

N/A

N/A

N/A

N/A

N/A

N/A

Harvey

N/A

N/A

N/A

N/A

N/A

Sawyer Int.
Airport

Au Train

Chatham

Brevort

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Seney

MSUEW

McMillan

✓

✓

✓

N/A

✓

✓

N/A

✓

N/A

✓

✓

Reed City

Engadine

N/A

✓

Graying Army
Airfied
Roscommon
Co. Airport

✓

✓

N/A

✓

✓

✓
N/A
✓
N/A
✓

✓

USC00203391

✓

✓

N/A

✓

✓

N/A

✓

✓

✓

Big Rapids
Waterwork

✓

✓

N/A

N/A
N/A
✓
N/A
N/A

N/A
✓
N/A
✓

✓
N/A

✓
N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Marquette

✓

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

USC00205591

✓

✓

Engadine MDOT

✓

✓

Naubinway

✓

N/A

✓

N/A

✓
N/A

NOAA
N/A
Schoolcraft Co.
Airport

N/A
✓

N/A

✓

N/A

Cooks

N/A

N/A

N/A

N/A

N/A

Twin Lakes

N/A

N/A

N/A

N/A

N/A

N/A

✓

N/A

✓

Stambaugh 2 SSE

✓

✓

✓

N/A

Golden Lake

N/A

N/A

N/A

N/A

N/A

Kings Land O'
Lakes Airport

St. Ignace

N/A

N/A

N/A

N/A

N/A

St. Ignace

Precipitation
Temperature
Wind Speed
Solar Radiation

78

N/A

N/A

N/A

N/A

N/A

Garden Corners
N/A

N/A
✓
N/A
✓
N/A

N/A
✓
N/A
✓
N/A

3.2.2 The State of Minnesota
In order to evaluate and verify the frost and thaw depth model which was developed
using the measured frost depth data in the state of Michigan, the Minnesota frost depth data were
requested, received and used in this study. The MnDOT data consisted of 9 years of data (2003
to 2012) collected at 8 stations located throughout the State of Minnesota as shown in Figure 3.4.
Similar to the Michigan case, the meteorological database from the nearest NOAA station was
obtained and used in the study.

Figure 3.4 MNDOT stations location, Minnesota
3.3

Soil Properties Data
Disturbed soil samples from different RWIS stations in the State of Michigan were

provided by MDOT. The thermal properties of the soil were then measured in the laboratory at
79

Michigan State University. Since only disturbed soil samples were received, the insitu dry
densities and water contents were unknown. Nevertheless, seven representative soil types were
selected and their thermal properties were measured in the laboratory using KD2 pro thermal
properties analyzer. The KD2 pro is a small and portable device with the capability of measuring
different thermal properties of almost any material. The device has three sensors 6 cm single
needle (KS-1), 10 cm single needle (TR-1), and 3 cm dual-needle (SH-1). Each sensor could be
used depending on the thermal properties being measured and the material type. The KS-1 sensor
was used to measure the thermal conductivity and thermal resistivity of the soil. The sensor is
most accurate in liquid samples and insulating materials. In liquid samples free convection could
be a source of error. Since the sensor applies small amount of heat to needles, free convection
could be prevented making KS-1sensor a good choice for liquid samples. On the other hand, in
granular samples like soil or powders, contact resistance could be a source of error. Size of the
sensor and short heating time could maximize this error making the KS-1sensor a poor choice for
these types of materials (Decagon Devices Inc., 2008).
The TR-1 sensor was also used to measure the thermal conductivity and thermal
resistivity. TR-1 is a hollow needle and comes with drill bits. The size and relatively longer
heating time could minimize the contact resistance error, which make this sensor the primary
choice for soil and granular materials. The sensor fully complies with ASTM D5334-08
specifications and standard test procedure for measuring the thermal conductivity of soils
(Decagon Devices, Inc., 2008).
The SH-1 sensor measures volumetric heat capacity, thermal diffusivity as well as
thermal conductivity and thermal resistivity. This sensor could be used in most solid and

80

granular material but not in liquid samples. Table 3.3 summarizes the applicability of each sensor
for different materials (Decagon Devices, Inc., 2008).
Table 3.3 Sensor use guide (Decagon Devices Inc., 2008)
Sample Material
Low viscosity liquids (Water)
High viscosity liquids (glycerol, oil)
Insulation and insulating materials
Moist soil
Dry soil, powders, and granular materials
Concrete and rock
Other solids

KS-1
Best
Best
Best
N/A
N/A
N/A
N/A

TR-1
N/A
OK
N/A
Best
Best
Best
Best

SH-1
N/A
N/A
N/A
OK
OK
OK
OK

As stated before, seven types of soil were tested to obtain the required thermal properties.
TR-1 and SH-1 sensors were used for measurements (Figure 3.5). In order to minimize the error
five measurements were done on each sample as shown in Figure 3.6. The thermal properties of
the samples were considered to be the average of these readings. Table 3.4 depicts the measured
thermal properties of seven different types of soil in saturated condition. It is noteworthy that all
soil samples were disturbed and were not compacted in the laboratory.

Figure 3.5 Thermal conductivity measurement using KD2 pro
81

As mentioned before, detailed soil log for each RWIS station were available in the State
of Michigan. On the other hand, the soils at the State of Minnesota stations were categorized as
clayey and sandy soils. Therefore, in the model the average values of Table 3.4 for clayey and
sandy soils were used for both states.
3.4

Frost Heave Data
Data from 5 sites in Oakland County, Michigan were provided by MDOT (Novak, 1968).

The data was collected in the winter of 1962-63 in a six-mile section (EBI 63172, CR5H) along
I-75; Figure 3.7 depicts the location of the sites. Table 3.5 shows soil type, the maximum
measured frost heave and frost depth under pavement and shoulder and at different stations. The
detailed measurements with time can be found in Appendix B.
Side of
the mold

Probe hole

Figure 3.6 Five locations where the soil thermal properties were measured in each soil sample
using KD2 pro

82

Table 3.4 Measured thermal properties of different types of soil using KD2 Pro

Station Name

Moisture
Condition

Material

Williamsburg

Silty Fine Sand with Trace of
Gravel
Fine Sand
Fine Sand with Trace of
Gravel
Fine Sand
Soft Clayey Sandy , Some
Silt & Some Gravel
Silty Clay

Rudyard

Silty Clay

Houghton
Lake

Wolverine

Saturated

Thermal
Conductivity
(W/m.K)

Heat
Capacity
(MJ/m3.K)

2.57

2.67

2.55

2.84

2.49

2.69

2.42

2.69

1.74

2.98

1.51

3.1

1.12

3.2

It should be noted that for station/528+88 the difference between the measured frost
heave under the pavement and under the shoulder is approximately 1 centimeter which is much
higher than the other sites. At this site, an undercut of approximately 30.5 centimeter was made
while constructing the pavement for frost protection.
3.5

Pavement Surface Temperature and Thaw Depth Data
In order to develop an effective spring load restriction policy, accurate models for

calculating Cumulative Thawing Degree Day (CTDD) and estimating thaw depth are required.
For developing a CTDD model, air temperature, pavement surface temperature data from the 18
RWIS stations in 2011, 2012, 2014, and 2015 were used (See Table 3.1). For developing a thaw
depth model, data from 14 RWIS station in 2011, 2014, and 2015 were used. Unfortunately, the
subsurface temperature data were not available for the whole 2012 spring. Therefore, that year
was not included in the study. In addition, among the 18 RWIS stations in Table 3.1, the data
from Cooks, St. Ignace, Twin Lake, and Williamsburg were excluded from the study due to
partially missing data in 2011, 2014 and 2015.
83

Figure 3.7 MDOT frost heave station locations, Oakland County, Michigan.
Table 3.5 Measured total heave and frost depths in different soil types, I75, Oakland County,
Michigan
Station
Name
Sta/724+00
Sta/719+00

Sta/652+00

Soil Type
Fine Sand and Silt
with Pebbles
Fine Sand with Silt
Pockets with
Pebbles
Insitu Sub Soil
Clayey, Silty,
Gravely, Sand

Frost
Depth
(cm)

Max Heave in
Shoulder (cm)

Max Heave
in Pavement
(cm)

Duration
(days)

60.96

2.54

1.91

65

71.12

2.16

1.91

40

86.36

2.29

2.16

60

Sta/528+88

Sandy Clayey Silt

76.20

1.91

1.02

70

Sta/474+00

Clayey Silt

63.50

2.54

2.29

55

84

CHAPTER 4
DATA ANALYSIS & DISCUSSION
Introduction and Research Objectives
In this chapter, the analysis methods that were used in this study and the results of this
analysis are presented and discussed. The analyses are based on the objectives and research plan
presented in Chapter 1. For convenience, the specific objectives of the study were included
below as well.
•

Review the advantages and shortcomings of some of the existing frost and thaw depth
models.

•

Develop accurate and reliable models for predicting the frost depth during freezing period in
Michigan.

•

Develop a model to predict heave and the resulting pressure under the pavement or behind
existing retaining structures due to freezing of frost-susceptible soils in Michigan.

•

Develop accurate and reliable model for predicting the thaw depth under the pavement in
spring season in Michigan.

•

Investigate changes in pavement bearing capacity in the cycles of freeze and thaw; and

•

Develop a model to estimate when to begin and end SLR.

To accomplish these objectives a comprehensive research plan was drawn as presented in
Chapter 1 and depicted in Figure 4.1.
Hypotheses
Previous studies indicate that frost and thaw depths are a function of many
variables including intensity and duration of the freezing period, water availability, soil
permeability and capillarity, grain size and grain size distribution, and the soil thermal

85

conductivity. Hence, it was hypothesized that these variables are a function of the soil type
such as clayey and sandy soils.
Furthermore, it was hypothesized that the various missing soil parameters (such as
insitu density, water content, grain size, grain size distribution, soil permeability and
capillarity) can be expressed by one related property; the saturated thermal conductivity of
the soil.
Task 1
Conduct comprehensive literature review which includes the state of the
art of modeling of freeze and thaw in soils and their applicability to this
study, the state of art of modeling of frost heave and the resulting
pressure behind retaining walls and the state of the practice of SHAs for
forecasting frost and thaw depth and the time at which to post and
remove SLR .

Task 2
Develop heat transfer predictive model
by firstly, scrutinize the existing
models for frost, and thaw depth
prediction using the MDOT database.
Secondly, develop new statistical
models for frost and thaw depth
predictions using MDOT database.
Thirdly, verify the accuracy of the
developed model using MnDOT
database.

Task 3

Develop coupled heat and mass
transfer models for prediction of
frost heave and frost pressure by
firstly estimate the freezing depth.
Secondly, estimate the rate of flow of
water to the frozen depth from a
water supply. Thirdly, estimate the
amount of heave pressure based on
the frost heave model results. Finally,
evaluate the accuracy of frost heave
model using MDOT heave data.

Task 4
Develop SLR policy based on different
research on resilient modulus in literature
and by using the results of thaw depth
model

Figure 4.1 Flow chart of the research plan
86

Frost Depth
Existing frost depth prediction models can be classified into numerical, analytical, semiempirical, and empirical models. Some models require as inputs various thermal and hydraulic
properties of soil and different meteorological data. Others require only the freezing index (FI) or
the cumulative freezing degree day (CFDD). Any of these models can be used depending on the
availability of the input data and the required accuracy. As mentioned in chapter 2, different
numerical models were developed in the past three decades. In this chapter, UNSAT-H model
was used to predict the frost depth. The results were then evaluated with the field data in
Michigan. Furthermore, the accuracy of different analytical and semi-empirical frost depth
prediction models including Stefan model (Jiji, 2009), Modified Berggren’s model (Aldrich et
al., 1953) and Chisholm and Phang empirical model (Chisholm and Phang, 1983) were evaluated
using the RWIS soil temperature data measured in the state of Michigan. Since none of the
models yielded accurate results, revised empirical models that require only cumulative freezing
degree day as input were developed. First, the data in the State of Michigan was used to develop
an empirical model regardless of the soil types. Further, for model validation the 2003 to 2012
frost depth data from 8 stations in the State of Minnesota were used. Third, by considering the
soil types two empirical models were developed for clayey and sandy soils. The two models
were also evaluated using the Minnesota frost depth data. Finally, using the thermal conductivity
data of each soil type, the two models were combined and one general model was developed
which required CFDD and soil thermal conductivity. The accuracy of the general model was also
checked using the frost depth data measured in the state of Minnesota.

87

4.3.1 UNSAT-H Model
As stated in chapter 2, UNSAT-H is a one dimensional finite difference heat and mass
balance model. The model inputs are meteorological data including air temperature,
precipitation, solar radiation, wind speed, cloud cover, dew point, soil hydraulic, and thermal
properties data. The soil can be modeled in different layers in UNSAT-H.
For evaluating the UNSAT-H model, two of the RWIS sites located in the Lower
Peninsula of Michigan were chosen. The pavement profile corresponding to both sites is
illustrated in Figure 4.2.
Hot mix asphalt (12.7 cm.)
Gravel (30.5 cm.)
Compacted Sand (61 cm.)

Loose Sand
(76 cm.)

Figure 4.2 Schematic of the cross section of the two modeled pavement sites
Table 4.1 shows the hydraulic and thermal properties that were used for each layer. Using
these properties and the weather data as the boundary conditions, soil temperature profile was
estimated. Figure 4.3 and Figure 4.4 show the estimated and the measured freezing depths as a
function of time. It can be seen from the figures that the model predicted the freezing front up to
three weeks earlier than the measured front. Although timing is not an issue in freezing period, it
certainly is problematic during the thaw season. Also, in Cadillac station the model over
predicted the frost depth by approximately 50 centimeters. One possible reason for the
unfavorable results could be the fact that UNSAT-H does not consider latent heat of fusion in the
88

heat balance analysis. Therefore, while the model can predict temperatures above the freezing
temperature quite accurately, the accuracy of temperature predictions below the freezing
temperature is questionable. In addition, the use of the estimated input properties (such as
thermal conductivity, volumetric specific heat, and hydraulic conductivity) contributed to the
discrepancy between the estimated and the measured data.
Table 4.1 Hydraulic and thermal properties for different layers in UNSAT-H model

Asphalt

0.070 0.360 0.0050

5.60E-08

1.090

0.08257

3.9

Volumetric
Specific
Heat
(kJ/(m3.oC)
2

Gravel

0.005 0.420 1.0000 1.00E+01

2.190

0.54338

1.25

1.36

0.020 0.375 0.0431

3.100

0.67742

1.5

2.39

Material

Frost depth (cm)

Sand

θr

200
180
160
140
120
100
80
60
40
20
0

θs

Îą

Ks
(cm/sec)

n

4.63E-01

Thermal
m (1-1/n) Conductivity
(W/(m.oC))

Calculated frost depth (cm)
Measured frost depth (cm)

Date

Figure 4.3 Calculated frost depth using UNSAT-H model and measured frost depth versus time
in Waters station, Lower Peninsula, Michigan
89

Calculated frost depth (cm)
Measured frost depth (cm)

200
180

Frost depth (cm)

160
140

120
100
80
60
40
20
0

Date
Figure 4.4 Calculated frost depth using UNSAT-H model and measured frost depth versus time
in Cadillac station, Lower Peninsula, Michigan
4.3.2 Freezing Index and Freezing Degree Day Calculation
One of the common inputs to most analytical and semi-empirical models is the freezing
index or the cumulative freezing degree day. Two different methods have been considered for
calculating the cumulative freezing degree day; the Minnesota method (MnDOT, 2009) and
Boyd method (Boyd, 1976).
4.3.2.1 Minnesota Cumulative Freezing Degree Day
The cumulative freezing degree-day (CFDD) was calculated using Equation 4.1
(MnDOT, 2009):
𝑛

𝐶𝐹𝐷𝐷𝑛 = ∑ 𝐹𝑟𝑒𝑒𝑧𝑖𝑛𝑔 𝐷𝑒𝑔𝑟𝑒𝑒 𝐷𝑎𝑦 ≤ 0
𝑖=1

90

Equation 4.1

Freezing Degree day = (

Tmax + Tmin
− 0o C)
2

Equation 4.2

Where 𝑇𝑚𝑎𝑥 = Maximum daily air temperature (°C); and
𝑇𝑚𝑖𝑛 = Minimum daily air temperature (°C).
It should be noted that in the Minnesota method, the cumulative freezing degree day is
reset on July 1 of each year and the freezing index is the maximum CFDD at the end of the
winter season. Table 4.2 lists an example calculation of the cumulative freezing degree day.
Table 4.2 Cumulative freezing degree day calculation, Waters station, Lower Peninsula

Date

Freezing
Average Air Degree day
Temperature
(FDD)
o
o
( C)
( C-day)

Cumulative
freezing degree
day (CFDD)
(oC-day)

Absolute
Cumulative
freezing degree
day (CFDD)
(oC-day)

11/16/2010

5.8

5.8

0

0

11/17/2010

3.2

3.2

0

0

11/18/2010

-0.95

-0.95

-0.95

0.95

11/19/2010

-0.75

-0.75

-1.7

1.7

11/20/2010

-1.25

-1.25

-2.95

2.95

11/21/2010

1.75

1.75

-1.2

1.2

11/22/2010

10.55

10.55

0

0

11/23/2010

4.15

4.15

0

0

11/24/2010

-2.3

-2.3

-2.3

2.3

11/25/2010

-2.35

-2.35

-4.65

4.65

11/26/2010

-6.4

-6.4

-11.05

11.05

11/27/2010

-1.95

-1.95

-13

13

11/28/2010

-1.15

-1.15

-14.15

14.15

11/29/2010

1.15

1.15

-13

13

4.3.2.2 Boyd Cumulative Freezing Degree Day
If the CFDD is calculated and plotted as a function of time as shown in Figure 4.5, the
graph will have a minimum value in the fall and a maximum value in spring. The Freezing Index

91

(FI) for that winter is estimated as the difference between the maximum and minimum
cumulative degree days as shown in the Figure 4.5 (Boyd, 1976).
Any spring or fall month that includes a seasonal maximum or a seasonal minimum
degree days is called a “changeover” month. Boyd (1976) proposed Equation 4.3 that can be
used for calculating the cumulative freezing degree days in the change-over month:
𝑌2 + 𝑁 ∗ 𝑋 ∗ 𝑌 = 𝑁2 𝑘2

Equation 4.3

where k = 2.5 constant;
N= number of days in the month;
𝑋 = (𝑇 − 0℃ );
T= the average temperature in the change-over month of N days; and
Y= Cumulative degree day of the change-over month.
1200
1000

CFDD (OC)

800

Freezing Index

600
400
200
0

Figure 4.5 Calculation of freezing index using cumulative freezing degree day

92

The solution of this equation yields two values for Y; a positive and a negative value. For
the changeover month, the cumulative freezing degree days (CFDD) for the month can be
calculated using Equation 4.4. Whereas, for all other months, the CFDD is calculated using
Equation 4.5.
𝐶𝐹𝐷𝐷 = |𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑌|
𝐶𝐹𝐷𝐷 = 𝑁𝑋

Equation 4.4
Equation 4.5

Where all parameters are the same as before.
CFDD values were calculated using both the Minnesota (Equation 4.1) and the Boyd
methods and the data from different RWIS stations in Michigan. The results are plotted in Figure
4.6. It can be seen from the figure that the CFDD values calculated using Boyd equation differ
from the CFDD values calculated using the Minnesota equation by more than 20 percent. This
difference could be attributed to various reasons including:
1. The k value in Equation 4.3 is an empirical value based on the 10 years data collected at 22
stations across Canada. This value could change from one year to the next and from one
region to another.
2. By using the monthly average temperature in the Boyd equation instead of the daily average
temperature in calculating CFDD, the daily variations in the degree days are disregarded,
which may lead to errors.
Because of the above reasons, in this study, the CFDD values were calculated using the
Minnesota equation (Equation 4.1).

93

Percent differnce with Boyd CFDD %

160
140
120
100
80
60
40
20
0
0

200

400
600
800
Minnesota CFDD (oC)

1000

1200

Figure 4.6 Comparison of calculated CFDD using Boyd (Boyd 1976) and Minnesota (MnDOT
2009) methods
4.3.3 Existing Frost Depth Prediction Models
There are several frost prediction models that were developed. Some of these models are
empirical in nature, some others are semi-empirical, and still others are mechanistically based.
Some of these models are enumerated and discussed below.
4.3.3.1

Stefan’s Equation
As stated before, Stefan solved the heat transfer phase-change problem for the special

case of no heat transfer in the unfrozen zone (Jiji, 2009) and estimated the frost depths. His
solution is one of the first frost depth prediction models (see Equation 2-19 of Chapter 2) and
modified versions of his solution are still being used by some State Highway Agencies (SHAs).

94

In order to evaluate the accuracy of Stefan Equation relative to the measured data in
Michigan, measured frost depths at different RWIS stations in the state of Michigan were used.
Unfortunately the in-situ water content and dry density data of the soils were not available.
Therefore, the soil water content and dry density were estimated using the graphs developed by
the U.S Army Corps of Engineers (USACE, 1998) and shown in Figure 4.7. The various curves
in the figure relate thermal conductivity to dry density and moisture content in the frozen and
unfrozen conditions of various soil types. For each soil type, the measured thermal properties
were used and its dry density and water content were estimated from the graphs. Next, the
volumetric latent heat of fusion (L) was calculated using Equation 2-20; as it was expected, the
calculated values of L decreased as the water content decreased. The values of the freezing index
for the years 2010 and 2011 were calculated using the NOAA data obtained from the appropriate
weather stations. Finally, Equation 2-19 was used to calculate the frost depths as a function of
time for the two years. The details of frost depth calculation for each RWIS station were
presented in appendix A. Figure 4.8 and Table 4.1 depicts the maximum calculated versus the
maximum measured frost depth data for saturated condition. In Figure 4.8 the straight line is the
line of equality between the measured and the calculated frost depth data. It can be seen from the
figure that, for all soil types, the calculated maximum frost depths in saturated condition are
much higher (more than 63.5 centimeters) than the measured values. The discrepancy between
the measured and calculated data could be related to:
1.

The volumetric heat capacity of the soil and water, which were not considered in Stefan’s

Equation (Equation 2-19).
2.

Errors in estimating the in-situ water content, dry density using the soil thermal

conductivity and the Corps of Engineers graphs.

95

20

20

Dry Unit Weight (kN/m3)

19

17

16

1 2 .3 3 .4 4.5 5.7 6.8 8

9.1

19
7.9

Dry Unit Weight (kN/m3)

K=(W/(m. C))

14

K=(W/(m.oC))

1.1
2.3
3.4
4.5
5.7
6.8
8
9.1
10
11
12
13

o

6.8

17
5.7

16

5.1

1.1
1.7
2.3
2.8
3.4
4
4.5
5.1
5.7
6.2
6.8
7.4
7.9
8.5
9.1

4.5

14

9.1 10

4.0
2.3 3.4

13

13
0

10

20
30
Water content (%)

0

40

10

20
30
Water content (%)

Figure 4.7(a) Frozen sand and Gravel

Figure 4.7(b) Unfrozen sand and Gravel

20

20
1.1

o

K=(W/(m. C))

K=(W/(m.oC))

2.3

2.3

19

2.8

Dry Unit Weight (kN/m3)

4.5
5.7
6.8

17

16

14
1

2 .3

3 .4

4.5

5.7

1.1
1.7

3.4

19

Dry Unit Weight (kN/m3)

40

3.4
4
4.5

17

5.1
5.7

16

14

6.8

8

13

13
0

10
20
30
Water content (%)

40

Figure 4.7(c) Frozen silt and clay

1.1

0

10

20
30
Water content (%)

40

Figure 4.7(d) Unfrozen silt and clay

Figure 4.7 soil Thermal conductivity of different types of soil based on water content and
dry density obtained by US army cold region and engineering laboratory (CRREL) (Edgar, 2014)

96

Line of equality between the measured and calculated data

Maximum calculated frost depth (cm)

400
350
300
250
200
150
100
50
0
0

50

100
150
200
250
300
Maximum measured frost depth (cm)

350

400

Figure 4.8 The maximum frost depths predicted by Stefan equation versus the measured
maximum frost depths in Michigan
Given the significant differences between the measured frost depth data and the
calculated ones using Equation 2-19, Stefan’s equation was abandoned and the modified
Berggren’s equation was studied. The results are presented and discussed below.
4.3.3.2

Modified Berggren’s Equation
Aldrich et al., (1953) made the two assumptions listed below and modified the original

Berggren’s equation. Equation 2-21 was the resulting equation.
1.

Heat transfer is one-dimensional problem and the soil is at its mean annual temperature
before freezing begins (USACE, 1998).

97

Table 4.3 Maximum frost depth predicted by Stefan’s equation for RWIS stations

Location

Station
Name
Benzonia
Cadillac
Grayling
Houghton
Lake

Lower
Peninsula

Ludington
Reed City
Waters
Williamsburg

Au Train

Type of soil

Year

Loose Sand
Dense Sand
Dense Sand

2010-2011
2010-2011
2010-2011

Maximum
Measured
Frost
Depth
(cm)
86.4
116.8
157.5

Dense Sand

2010-2011

132.1

254.0

2010-2011

86.4

167.6

2010-2011

101.6

236.2

2010-2011

172.7

312.4

2010-2011

101.6

182.9

2009-2010

132.1

264.2

2010-2011

132.1

363.2

2009-2010
2010-2011

116.8
147.3

231.1
264.2

2009-2010

147.3

248.9

2010-2011

172.7

279.4

2010-2011

132.1

309.9

2010-2011
2010-2011
2010-2011
2010-2011
2010-2011
2010-2011

132.1
116.8
132.1
116.8
116.8
116.8

251.5
271.8
264.2
180.3
205.7
193.0

Loose Sand with
clay
Compacted Sand
Loose Sand
Compacted Sand
Loose Sand
Dense Sand
Silty Clay
Sand with Gravel
and Silt
Loose Sand

Upper
Peninsula

Brevort

Loose Sand

Harvey

Sand with Gravel
and Silt

Golden Lake
Seney
Cooks
Michigamme
St.Ignace
Twin Lakes
Engadine

Dense Sand
Dense Sand with
Gravel
Loose Sand
Clayey Sand
Clayey Sand
Silty Clayey Sand
Silty Clayey Sand
Silty Clayey Sand

98

Maximum
Calculated
Stefan Eq.
(cm)
221.0
271.8
256.5

2.

At the beginning of the freezing season, the surface temperature decreases in a step-function
manner from the mean annual temperature to some degrees below the freezing point and
remains at this temperature (steady state) during the entire freezing season (Bianchini et al.,
2012).
In this study, the maximum frost depths were calculated using the modified Berggren’s

equation for multilayered system (Equation 2-22).Table 4.4 shows an example of the step by step
frost depth calculation using Equation 2-22.
In the calculations, in order to obtain the correction factor (Îť) from Figure 2.3, two
dimensionless parameters (the thermal ratio (Îą) and the fusion parameter (Îź)) must be calculated.
As stated in Chapter 2, fusion parameter (Îź) depends on the volumetric heat capacity and
latent heat of fusion of every layer and was calculated using Equation 4.6. The thermal ratio (Îą)
is a fixed number for all layers and depends on the FI and annual average temperature; it was
calculated using Equation 4.7.
𝜇=

𝐶
∗𝑣
𝐿 𝑠

𝛼=

𝑣0
𝑣𝑠

Equation 4.6
Equation 4.7

Where C=volumetric heat capacity (KJ/m3);
L= latent heat of fusion (KJ/m3);
vs= average temperature differential= n(FI)/t;
t= duration of winter period (used in calculation of vs);
v0= initial temperature differential = annual average temperature -0 oC; and
All other parameters are as before.
After calculating Îą and Îź, the correction factor Îť was obtained from Figure 2-3 and the
maximum frost depth was calculated using Equation 2-22 (See Table 4.4).
99

Using the Modified Berggren’s equation, the maximum frost depths were calculated for
the RWIS stations in Michigan. The details of frost depth calculation for each station were
presented in appendix A. The maximum calculated frost depths were compared to the maximum
measured frost depth data in the State of Michigan in Figure 4.9 and Table 4.5. It can be seen
from the figure that the Modified Berggren’s equation leads to more accurate results than the
Stefan equation. However, the differences between the calculated and measured values in some
cases are more than 51 centimeters. The discrepancy between the measured and calculated data
could be related to:
1. Equation 2-22 does not account for the water movement in the soil.
2. Potential errors in estimating the thermal conductivity, water content and dry density.
Given the substantial differences between the measured frost depth data and the
calculated ones using Equation 2-22, the Modified Berggren’s equation was also abandoned and
the Chisholm and Phang equation was studied. The results are presented and discussed in the
next section.
4.3.3.3

Chisholm’ and Phang’s Equation
One of the first empirical equations, which relate CFDD and frost depth, was developed

by Chisholm and Phang in 1980 to predict frost depths under asphalt pavements in Ontario,
Canada (Equation 2-23). It should be noted that since the local daily air temperature data were
not available on a daily basis, the CFDD values were calculated using Boyd approach (Chisholm
and Phang, 1980). Their results indicated that the frost depth predictions were within 30.5
centimeters of the measured values.

100

Table 4.4 Frost depth calculation using the modified Berggren’s equation, Benzonia, Lower Peninsula, Michigan
Column
No.

1

2

3

4

5

Material

Îłd

w

d

C

k

HMA

21.7
20.4

0

0.13 894
0.075 0.30 1118

1.49
2.60

6

7

Äš=
∑Ld/∑d
14792
0
50300 35508
L

8

9

Ĉ=
∑Cd/∑d
894
1043

Îź=
Ĉ / Ě *vs
0

10

11

12

13

14

15

Îť

R

∑R

∑R+R/2

FI

∑FI

0

0.3
0.4

0.0
0.3

0.1
0.5

0
82

0
82

Gravel
0.17
0.56
Loose
0.16
0.58 0.8 0.7
19.6 0.09 0.58 1565
2.42
57938 48362
1341
1.0
372 457
Sand
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m); C= volumetric heat capacity(KJ/m3); k= thermal conductivity (W/(
m.oC ); L= latent heat of fusion (KJ/m3); Îź= fusion parameter; Îť= correction factor; R= thermal diffusivity ((m2.oC)/W);
FI=freezing Index (oC-day)
Step by step calculation
1. k values were obtained from the laboratory measurements (Table 3.4). Îłd and w values were obtained from Figure 4.7 using k values. d
values were obtained from the pavement profile of RWIS station, and C values are assumed based on the soil type (Columns 1-5).
2. L and Îź were calculated using Equation 2.20 and 4.6, respectively (Columns 6 and 9).
3. Îą can be calculated by Equation 4.7 as follow:
FI=445 (oC-day) ; vs=0.9(445)/t = 3.1 ; v0= 9.2-0=9.2  ι=v0/vs=2.97
4. Using Îą and Îź values, Îť can be obtained from Figure 2.3 for each layer (Columns 10)
5. R values were calculated as R= d/K for each layer (Columns 11)
6. Freezing index required for each layer to freeze were calculated using Equation 2.22 (Columns 14)
7. The summation of freezing indexes in column 14 should be approximately equal to the seasonal freezing index (FI=445)

101

Line of equality between the measured and calculated data

Maximum calculated frost depth (cm)

400
350
300

250
200
150
100
50
0
0

50

100
150
200
250
300
Maximum measured frost depth (cm)

350

400

Figure 4.9 Measured maximum frost depths in Michigan versus the maximum calculated ones
using the modified Berggren’s equation
Equation 2-23 was used to predict the maximum monthly measured frost depths at
different RWIS stations in the state of Michigan. Figure 4.10 shows the results. It can be seen
that in most cases, Equation 2-23 underestimates the maximum monthly frost depths. In fact, in
some cases for small values of CFDD, the calculated frost depths could be negative. The
differences between the predicted and the measured frost depths could be as high as 76
centimeters. Based on the fact that the empirical equation was developed using the measured
frost depth data in Ontario, it can be concluded that the equation is regional and calibration is
required for using it in other regions.

102

Table 4.5 Maximum frost depth predicted by Modified Berggren’s equation for RWIS stations

Location

Station
Name
Benzonia
Cadillac
Grayling
Houghton
Lake

Lower
Peninsula

Ludington
Reed City
Waters
Williamsburg

Au Train

Type of soil

Year

Loose Sand
Dense Sand
Dense Sand

2010-2011
2010-2011
2010-2011

Maximum
Measured
Frost
Depth (cm)
86.4
116.8
157.5

Dense Sand

2010-2011

132.1

2010-2011

86.4

2010-2011

101.6

2010-2011

172.7

2010-2011

101.6

2009-2010

132.1

2010-2011

132.1

2009-2010
2010-2011

116.8
147.3

2009-2010

147.3

2010-2011

172.7

2010-2011

132.1

2010-2011
2010-2011
2010-2011
2010-2011
2010-2011
2010-2011

132.1
116.8
132.1
116.8
116.8
116.8

Loose Sand with
clay
Compacted Sand
Loose Sand
Compacted Sand
Loose Sand
Dense Sand
Silty Clay
Sand with Gravel
and Silt
Loose Sand

Upper
Peninsula

Brevort

Loose Sand

Harvey

Sand with Gravel
and Silt

Golden Lake
Seney
Cooks
Michigamme
St.Ignace
Twin Lakes
Engadine

Dense Sand
Dense Sand with
Gravel
Loose Sand
Clayey Sand
Clayey Sand
Silty Clayey Sand
Silty Clayey Sand
Silty Clayey Sand

103

Maximum
Calculated
Stefan Eq.
(cm)
101.6
152.4
162.6
160.0
96.5
139.7
170.2
109.2
149.9
213.4
154.9
175.3
132.1
154.9
215.9
162.6
177.8
193.0
129.5
148.9
132.1

Line of equality between the measured and calculated data

Maximum Calculated Frost depth (cm)

400
350
300
250
200
150
100
50
0
0

50

100

150

200

250

300

350

400

-50
Maximum Measured Frost Depth (cm)
Figure 4.10 Measured maximum frost depths versus calculated ones using Chisholm’ and
Phang’s equation
4.3.4 Frost Depth Empirical Models
Since none of the existing models yielded accurate frost depth results, new empirical
models were developed in this study using the RWIS data in the State of Michigan. These new
models are presented below.
Among the 25 RWIS stations located in the Upper and Lower Peninsulas of Michigan,
the data from 18 stations were used for developing empirical models. Seven stations were not
considered due to incomplete data (some of the data are missing). For air temperature, the
databases from the nearest NOAA stations were obtained and used due to higher consistency and
accuracy relative to the RWIS database. Sites not situated close to NOAA stations are not
considered in this study. Due to data availability, only data from the winter of 2011 was used,

104

except for Au Train, Brevort and Harvey sites where data from the 2010 winter were also
available.
First, the air temperature data were used to calculate the CFDD for each RWIS station
location. Second, the measured frost depth data in all RWIS stations and the calculated CFDD
values were used to develop one simple statistical prediction equation regardless of the soil type.
This resulted in Equation 4.8.
𝑃 = 4.759 (𝐶𝐹𝐷𝐷)0.5339

Equation 4.8

Equation 4.8 is more or less similar to Equation 4.9 , which was developed by the U.S.
Corps of Engineers (Yoder, 1975)
𝑃 = 4.210 (1.8 𝐶𝐹𝐷𝐷)0.478

Equation 4.9

Where P = frost depth (cm); and
CFDD = cumulative freezing degree day (oC – day).
The results of both equations are depicted in Figure 4.11. The data in the figure indicate
that Equation 4.9 under predicts the majority of the data. On the other hand, Equation 4.8
represents the measured frost depths more accurately. In fact, the calculated coefficient of
determination (R2) is 0.91. Figure 4.12 show the calculated frost depths using Equation 4.8
versus the measured ones. The solid straight line is the locus of equality between the measured
and calculated data. It can be seen that, the majority of the calculated frost depth data are within
a few centimeters from the measured values and the maximum difference is 25-centimeter.
As stated in Chapter 3, frost depth data measured from 2003 to 2012 in 8 stations in the
State of Minnesota were requested and received from MnDOT. Equation 4.8 was then used to
calculate the frost depth data at all 8 stations and for the ten year period. The measured frost
depth data and the calculated ones are depicted in Figure 4.13. The straight line in the figure

105

indicates the line of equality between the measured and the calculated values. Examination of the
data shown in the figure indicates that Equation 4.8 does not predict the measured frost depth
data in Minnesota accurately. In fact, Equation 4.8 over predicts the Minnesota frost depth data
by as much as 102 centimeters. Moreover, the calculated coefficient of determination (R2) for the
Minnesota data is 0.77, which is much lower than the calculated R2 of 0.91 for the Michigan
data. Other performance metrics of Equation 4.8 and Equation 4.9 are shown in Table 4.6. In this
table, standard error of the estimate (SEE), mean absolute error (MAE) and mean absolute
percentage error (MAPE) are shown for each model. The SEE is proportional to the width of the
confidence interval so smaller SEE is an indication of a better fit. As can be seen in, Table 4.6
for the Michigan, the SEE value of Equation 4.9 is 60% larger than SEE of Equation 4.8. On the
other hand, Equation 4.9 performs better in the State of Minnesota. In general, all of the
performance metrics indicate that Equation 4.8 does not predict frost depth data in the State of
Minnesota accurately.

U.S.Corp of Engineers
200

Clayey soil

P = 4.7591CFDD0.5339
R² = 0.9135
n= 158

160

Frost depth (cm)

Sandy soil

120
80
40
0
0

200
400
Cumulative freezing degree days (oC-day)

600

Figure 4.11 Frost depths versus cumulative freezing degree day for clayey and sandy soils in the
State of Michigan
106

Line of equality between the measured and calculated data
Clayey soil-Michigan
Sandy soil-Michigan

Calculated frost depth (cm)

200
n= 158

160

120
80
40
0
0

40

80
120
Measured frost depth (cm)

160

200

Figure 4.12 Measured frost depths in State of Michigan versus the calculated ones using
Equation 4.8

Calculated frost depth (cm)

Line of equality between the measured and calculated data
Calyey soil-Minnesota
Sandy soil-Minnesota
240
n= 621

200
160
120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

240

Figure 4.13 Measured frost depth in State of Minnesota versus the calculated ones using
Equation 4.8
107

Table 4.6 Performance metrics of Equation 4.8 and Equation 4.9 for frost depth estimations in
the State of Michigan and Minnesota

Location

Michigan
Minnesota

Model

Standard Error
of the Estimate
(SEE)

Mean Absolute
Error (MAE)

Equation 4.8

15.0

11.2

Mean Absolute
Percentage
Error (MAPE)
0.18

Equation 4.9

19.1

12.7

0.18

Equation 4.8

24.4

18.3

0.29

Equation 4.9

17.5

13.2

0.22

Figure 4.14 and Figure 4.15 show the calculated frost depths using Equation 4.9 versus
the measured ones in the state of Michigan and Minnesota, respectively. The solid straight line in
the figures is the locus of equality between the measured and the calculated data. It can be seen
that Equation 4.9 predicts frost depths in clayey soils better than in sandy soils in both states. For
the latter soils, the differences between the measured and calculated values could be as high as
63.5-centimeter. Please note that, like Equation 4.8, Equation 4.9 does not separate sandy from
clayey soils.
The results shown in Figure 4.11 were further scrutinized to improve the accuracy of
Equation 4.8. Previous studies indicate that frost depths are a function of many variables
including intensity and duration of the freezing period, water availability, soil permeability and
capillarity, grain size and grain size distribution, and the soil thermal conductivity. Hence, it was
hypothesized that these variables are a function of the soil type such as clayey and sandy soils.
Therefore, the frost depth data measured at various RWIS stations in the state of Michigan was
divided into two groups according to soil type at the stations; clayey and sandy soils. It should be
noted that dividing the data into two groups of clayey and sandy soil was based on the soil log
provided by MDOT for each station.

108

Line of equality between the measured and calculated data
Clayey soil-Michigan
Sandy soil-Michigan

Calculated frost depth (cm)

200
160

n=158

120
80
40
0
0

40

80
120
Measured frost depth (cm)

160

200

Figure 4.14 Measured frost depth in State of Michigan versus the calculated ones using Equation
4.9

Calculated frost depth (cm)

Line of equality between the measured and calculated data
Calyey soil-Minnesota
Sandy soil-Minnesota
240

200

n= 621

160
120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

240

Figure 4.15 Measured frost depth in State of Minnesota versus the calculated ones using
Equation 4.9

109

After dividing the data per soil type, a mathematical power function was used to model
each group of frost depth data as a function of the calculated CFDD. This resulted in Equation
4.10 Equation 4.11 for clayey and sandy soils, respectively.
For Clayey Soils

𝑃 = 5.3858 (𝐶𝐹𝐷𝐷)0.4896

Equation 4.10

For Sandy Soils

𝑃 = 4.6473 (𝐶𝐹𝐷𝐷)0.5423

Equation 4.11

Figure 4.16 and Figure 4.17 depict the measured frost depth data in clayey soils in
Michigan and the calculate frost depth data using Equation 4.10. Figure 4.16 also show the U.S.
Corps of Engineers equation (Equation 4.9). It can be seen from the figure that Equation 4.10
and Equation 4.9 fit the data very well. The coefficient of determination for both equations is
0.94. Further, Figure 4.17 depicts the measured frost depth data in clayey soils in Michigan
versus the frost depth data calculated using Equation 4.10. The solid line in the figure is the line
of equality between the measured and the calculated data. The results in the figure indicate that
Equation 4.10 predict the frost depth data in clayey soils in Michigan more accurately.
Figure 4.18 Figure 4.19 depict the measured frost depth data in sandy soils in Michigan
and the calculate frost depth data using Equation 4.11. Figure 4.18 also show the U.S. Corps of
Engineers equation (Equation 4.9). It can be seen from the figure that Equation 4.11 represents
the measured frost depth data much better than Equation 4.9. Indeed, the coefficient of
determination is 0.91 and 0.76 for Equation 4.11 and Equation 4.9, respectively.
Further, Figure 4.19 depicts the measured frost depth data in clayey soils in Michigan
versus the calculated values using Equation 4.11. The solid line in the figure is the line of
equality between the measured and the calculated data. The results in the figure indicate that
Equation 4.11 provides similar prediction of the measured frost depth data to Equation 4.8. Since

110

the majority of the data were from sandy soil, the similarity between the predictions of both
equations was expected.
U.S.Corp of Engineers

Frost depth (cm)

200

P = 5.3858(CFDD)0.4896
R² = 0.94
n=29

160
120
80
40
0

0

200
400
Cumulative freezing degree days (oC- day)

600

Figure 4.16 Measured frost depths in Michigan versus cumulative freezing degree day for clayey
soil showing the best fit and the U.S. Corps of Engineers equations

Calculated frost depth (cm)

Line of equality between measured and calculated data
Clayey soil-Michigan
200
160

n= 29

120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

Figure 4.17 Measured versus calculated frost depth data in clayey soils in Michigan (Equation
4.10)

111

U.S.Corp of Engineers
200

P = 4.6473(CFDD)0.5423
R² = 0.9119
n= 129

Frost depth (cm)

160
120
80
40
0
0

200
400
600
Cumulative freezing degree days (oC-day)

Figure 4.18 Frost depths versus cumulative freezing degree day for sandy soil showing the best
fit and the U.S. Corps of Engineers equations in the State of Michigan
Line of equality between the measured and calculated data

Calculated frost depth (cm)

Sandy soil-Michigan
200
160

n= 129

120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

Figure 4.19 Measured versus calculated frost depths in sandy soils in Michigan (Equation 4.11)
It should be noted that for frost depth more than 127-centimeter in sandy soils in both
states, Equation 4.11 underestimates the measured data by as much as 25-centimeter.
112

Examination of the results depicted in Figure 4.16 through Figure 4.19 indicates that Equation
4.10 predicts the frost depth data in clayey soils better than Equation 4.11 in sandy soils. The
main reason is that the variability of the measured frost depth data in sandy soils is higher than
that in clayey soil. Such variability is a function of the grain size and grain size distribution,
which impact the distribution of water and the hydraulic conductivity of the soils. Unfortunately,
such data are not available at this time to improve Equation 4.11. Nevertheless, the equation does
predict the frost depth data in sandy soils relatively accurately. One important point should be
noted is the number of measured data points in clayey soils is much less than in sandy soils. A
total of 29 data points are available for clayey soils, whereas 129 data points are available in
sandy soils.
Once again, to evaluate the accuracy of Equation 4.10 and Equation 4.11 , they were used
to predict the measured frost depths in clayey and sandy soils in the State of Minnesota. Figure
4.20 and Figure 4.21 depict the results. It should be noted that for clayey soil (Figure 4.20) the
number of measured data points is 374 while for sandy soil (Figure 4.21) it is 247.
Examinations of Figure 4.20 and Figure 4.21 indicates that the prediction of frost depth
data in clayey and sandy soils in Minnesota using Equation 4.10 and Equation 4.11 is better and
more accurate than the prediction using Equation 4.8 (see Figure 4.13, Figure 4.20, and Figure
4.21). In fact, for clayey and sandy soil the calculated coefficients of determination (R2) are 0.88
and 0.9, respectively. The relatively high values of R2 indicate that Equation 4.10 and Equation
4.11 predict the frost depth data in clayey and sandy soils in the states of Michigan and
Minnesota relatively accurately. In order to further evaluate the performance of Equation 4.8 to
Equation 4.11, the performance metrics of all four equations in clayey and sandy soils for both
states listed in Table 4.7 were examined.

113

Line of equality between the measured and calculated data
Clayey soil- Minnesota

Calculated frost depth (cm)

240
n= 374

200
160
120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

240

Figure 4.20 Measured frost depths in clayey soil in the state of Minnesota versus the frost depth
values calculated using Equation 4.10

Line of equality between the measured and calculated data
Sandy soil- Minnesota

Calculated frost depth (cm)

240
n= 247

200
160
120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

240

Figure 4.21 Measured frost depths in sandy soil in the state of Minnesota versus the frost depth
values calculated using Equation 4.10
114

As expected, the performance of Equation 4.9 and 4.10 are almost the same in both
states. On the other hand, while performance of Equation 4.10 in Minnesota is not as good as in
Michigan, it is substantially better than Equation 4.8 in both states. In addition, as expected,
since the majority of the data in Michigan were from sandy soil, Equation 4.11 and Equation 4.8
performance are similar in both states.
Table 4.7 Performance metrics of Equation 4.8 and Equation 4.9 for frost depth estimations in
the State of Michigan and Minnesota

Location

Soil type

Clayey Soil
Michigan
Sandy Soil

Clayey Soil
Minnesota
Sandy Soil

Model

Standard
Error of the
Estimate
(SEE)

Mean
Absolute
Error
(MAE)

Equation 4.8

11.9

8.4

Mean
Absolute
Percentage
Error
(MAPE)
0.15

Equation 4.9

7.4

5.1

0.11

Equation 4.10

7.1

4.6

0.11

Equation 4.8

15.5

11.7

0.18

Equation 4.9

20.8

14.2

0.19

Equation 4.11

15.2

11.7

0.18

Equation 4.8

28.4

21.8

0.35

Equation 4.9

13.7

11.2

0.23

Equation 4.10

15.0

11.9

0.25

Equation 4.8

15.7

12.4

0.17

Equation 4.9

22.1

16.5

0.18

Equation 4.11

17.3

13.0

0.18

The thermal conductivity of any soil type (see Chapter 2) depends upon its water content,
dry density, void distribution, and grain size and grain size distribution. These physical
properties vary substantially from one soil type to another. Therefore, the disturbed clayey and
sandy soil samples that were obtained by MDOT from various RWIS stations were saturated and
the thermal conductivity of each soil type was measured in the laboratory at Michigan State
115

University using the KD2 pro. The results are listed in table 3-4 of Chapter 3. To consolidate
Equation 4.10 and Equation 4.11 into one equation, it was hypothesized that the various missing
soil parameters (such as insitu density, water content, grain size, grain size distribution, soil
permeability and capillarity) can be expressed by one related property; the saturated thermal
conductivity of the soil.
Based on the hypothesis, the statistical parameters of Equation 4.10 and Equation 4.11
were correlated to the average thermal conductivity of each soil type. The statistical parameters
of the two equations were then replaced by the resulting correlation equation, which yielded
Equation 4.12 for both clayey and sandy soils. Figure 4.22 and Figure 4.23 show the correlation
between the statistical parameters and the average thermal conductivity of each soil type.
Unfortunately, only two data points (two soil types) were available, hence the best correlation
between the statistical parameters and the average thermal conductivity is a straight line as
shown in the figures. It should be noted that such straight line correlations may not be accurate
and may result in errors in the resulting frost prediction equations. To produce more accurate
nonlinear equations (power, exponential or logarithmic function), data from three or more soil
types must be available. Unfortunately, this was not the case and the straight line equations are
the best scenario for the given data. Nevertheless, the equations for the two straight lines in
Figure 4.22 and Figure 4.23 were used to replace the statistical constants of Equation 4.10 and
Equation 4.11. Equation 4.12 is the resulting equation for both types of soils clayey and sandy.
𝑃 = (−0.7392 𝑘 + 6.4408) ∗ (𝐶𝐹𝐷𝐷)(0.0035𝑘+0.4846)
Where k= the average thermal conductivity of the soil (W/(m.oC)); and
All other parameters are the same as before.

116

Equation 4.12

0.55
0.54
b = 0.0035k + 0.4846

0.53

b

0.52
0.51
0.5
0.49

0.48
0

0.5
1
1.5
2
2.5
o
Thermal conductivity (W/(m. C))

3

Figure 4.22 Correlation between the statistical power coefficient (b) of Equation 4.10 and

a

Equation 4.11 and the corresponding average thermal conductivity of the soil
5.5
5.4
5.3
5.2
5.1
5
4.9
4.8
4.7
4.6

a = -0.7392k + 6.4408

0

1
2
Thermal conductivity (W/(m. oC))

3

Figure 4.23 Correlation between the statistical coefficient (a) of Equation 4.10 and Equation 4.11
and the corresponding average thermal conductivity of the soil
Equation 4.12 was then used to calculate the frost depths in clayey soils in the States of
Michigan and Minnesota. The inputs to the equation consisted of the calculated CFDD for each
state and the average measured thermal conductivity of the soil samples obtained from MDOT.
117

Figure 4.24 and Figure 4.25 depict the calculated and the measured frost depths in Michigan and
in Minnesota, respectively. Comparing the results shown in the two figures and those shown in
Figure 4.17 and Figure 4.20 using Equation 4.10 indicate that the two equations produce similar
results for clayey soils. Similarly, for sandy soils in Michigan and Minnesota, Equation 4.11 and
Equation 4.12 produced almost the same results. These results can be seen in Figure 4.19 and
Figure 4.21 for Equation 4.11 and in Figure 4.24 and Figure 4.50 for Equation

4.12.

Performance metrics of Equation 4.12 in clayey and sandy soils for both states are listed in
Table 4.8. As expected, all performance metrics indicate the similarity between the performance
of Equation 4.10, 4.11, and 4.12.
Line of equality between the measured and calculated data
Clayey soil- Michigan
Sandy soil-Michigan

Calculated frost depth (cm)

200
n = 158

160
120
80
40
0
0

40

80
120
Measured frost depth (cm)

160

200

Figure 4.24 Calculated frost depths using Equation 4.12 versus the measured frost depth in
clayey and sandy soil in the State of Michigan

118

Line of equality between the measured and calculated data
Clayey soil-Minnesota
Sandy soil-Minnesota

Calculated frost depth (cm)

240
n= 621

200
160
120
80
40
0
0

40

80
120
160
Measured frost depth (cm)

200

240

Figure 4.25 Calculated frost depths using Equation 4.12 versus the measured frost depth in
clayey and sandy soil in the State of Minnesota
Table 4.8 Performance metrics of Equation 4.12 for frost depth estimations in the State of
Michigan and Minnesota

Location

Soil type

Model

Michigan

Clayey Soil
Sandy Soil

Equation 4.12

Minnesota

Clayey Soil
Sandy Soil

Equation 4.12

Standard
Error of the
Estimate
(SEE)

Mean
Absolute
Error
(MAE)

7.112

4.826

Mean
Absolute
Percentage
Error
(MAPE)
0.11

17.272

11.938

0.18

14.986

11.938

0.25

16.002

12.446

0.16

Recall that the average thermal conductivity values obtained from seven different soil
samples (two soil types) were used in the prediction of the frost depths in Michigan and
Minnesota. This implies that:
119

1. If thermal conductivity data of more soil types are available, the prediction of frost depth
could improve.
2. Equation 4.12 could perhaps be used at the regional level to estimate the frost depth data.
To further evaluate the validity of Equation 4.12, a statistical model was developed using
the measured frost depth and the calculated CFDD for both soil types in Minnesota. The results
are shown in Figure 4.26 and Figure 4.27, respectively.
The dashed and solid curves in Figure 4.26 and Figure 4.27 represent the statistical model
and Equation 4.12, respectively. Examination of Figure 4.26 indicates that Equation 4.12 and
the statistical model produce almost the same results for clayey soils in Minnesota. On the other
hand, the data in Figure 4.27 indicate that for sandy soil the differences between the results of the
two models are less than 12.5 centimeters. This implies that Equation 4.12 could be used in
Minnesota without calibration.

Statistical power function
Clayey soil-Minnesota
Proposed model (Eq.4-12)

240

Frost depth (cm)

200
160

n= 374

120
80

40
0
0

400
800
1200
1600
o
Cumulative freezing degree days ( C-day)

Figure 4.26 Frost depths versus cumulative freezing degree-day for clayey soil showing the best
fit statistical model and the proposed model (Equation 4.12) in Minnesota

120

Statistical power function
Sandy soil-Minnesota
Poposed model (Eq.4-12)

280

Frost Depth (cm)

240
200

n= 247

160
120
80
40
0
0

400
800
1200
o
Cumulative freezing degree days ( C-day)

Figure 4.27 Frost depths versus cumulative freezing degree-day for sandy soil showing the best
fit and proposed model (Equation 4.12) in the State of Minnesota
Frost Heave
Frost heave refers to the uplifting of the ground surface caused by freezing of water
within the soil layers. In cold regions, frost heave could cause uplifting of the pavement
structure, shoulders, and even unprotected foundations of bridges and trusses supporting
highway signs and utility lines (Liu et al., 2012).
Frost heave can be influenced by various conditions including:
1.

Frost Susceptibility – In general, the frost susceptibility of a soil is a function of its grain size
and grain size distribution, which affect its capillarity and hydraulic conductivity (ACPA,
2008). There are various methods and criteria for the determination of soil frost
susceptibility. In general, frost susceptibility could be affected by soil type. In coarse material
such as gravel and coarse sand hydraulic conductivity is high and capillary potential is low,
whereas clay has low hydraulic conductivity and high capillary potential. Only fine sand and
121

silt seem to have a balance between hydraulic conductivity and capillary potential. Figure
4.28 illustrates the dual effect of hydraulic conductivity and capillary potential on frost
susceptibility. One of the most common criteria regarding frost susceptibility is based on the
grain size distribution and the percent passing sieve number 200. Figure 4.29 and Table 4.9
show the susceptibility criteria developed by the U.S Corp of Engineers (COE). The
Canadian Department of Transportation developed another soil frost susceptibility criterion
that also based on soil grain sized distribution as shown in Figure 4.30.

Figure 4.28 Effect of capillary and permeability on frost susceptibility (ACPA, 2008)

122

Figure 4.29 Heaving Rate in laboratory test on different disturbed soil types (COE, 1984)

123

Table 4.9 Frost susceptibility classification (COE, 1984)
Typical Soil
Types Under
Unified Soil
Classification
System

Frost Group

Soil

Percentage
Finer Than
0.02 mm by
Weight

Non-frost
susceptible

Gravel
Crushed stone
Crushed rock

0- 1.5

GW, GP

Sands

0- 3

SW,SP

Gravel
Crushed stone
Crushed rock

1.5- 3

GW,GP

3- 10

SW,SP

Possibly frost
susceptible,
requires lab tests

Sands

S1

Gravely soils

3- 6

S2

Sandy soils

3- 6

F1

Gravely soils

6- 10

Gravely soils

10- 20

Sands

6- 15

F2

F3

F4

GW, GP, GWGM, GP-GM
SW, SP, SW-SM,
SP-SM
GM , GW-GM,
GP-GM
GM , GW-GM,
GP-GM
SM , SW-SM,
SP-SM

Gravely soils

Over 20

GM, GC

Sands, except very fine
silty sands

Over 15

SM, SC

Clays, PI>12

---

CL, CH

Silts

---

ML, MH

Very fine, silty sand

Over 15

SM

Clays, PI< 12

---

CL, CL-ML

---

CL, ML and SM,
CL, CH and ML,
CL, CH, ML and
SM

Varved clays and other
fine-grained, banded
sediments

124

Figure 4.30 Frost susceptibility criteria, Canadian Department of Transportation (Edgar, 2014)
2.

Below Freezing Temperature - As stated in Chapter 2, freezing point depression occurs in
pore water because of different reason such as intermolecular forces between water and soil
(soil water surface tension) and salt solution. Therefore, pore water starts to freeze when the
air temperatures and consequently the ground surface temperature drops below the freezing
temperature of 0oC. The rate of water freezing is a function of the actual temperature below
freezing and its duration. Colder and more sustainable below freezing temperatures
accelerate the freezing rate and increases the depth of frost penetration and consequently
increases ground heave. Snow cover acts like insulator reducing frost depth substantially
unless the air temperature and consequently the soil surface temperature drop significantly
below the freezing temperature. However, for safety reasons, snow is typically removed from
125

the pavement surface and accumulated near the shoulder as soon as possible. This causes
higher frost depth and higher frost heave under the pavements relative to other areas covered
by snow (Yoder, 1975). Further, salt and other deicing chemicals (typically used on roads
during winter season) decrease the temperature at which water starts to freeze and causes
decreases in frost depth and frost heave.
3.

Availability of Water Source – If no free water is available, no water frost action will take
place, hence, a source of water should be available under the pavement to start the free water
freezing process. The water source could be as deep as 6 meters (Edgar, 2014). If the ground
water level is shallow, frost heave can be observed even in course material (COE, 1984).
Figure 4.31 shows the ASSHTO four different environmental regions in the United States.
Only two regions, wet-freeze, and dry freeze are subjected to water freezing under the
pavements. The wet freeze region is considered to be the most frost susceptible region
(ACPA, 2008). As can be seen, the state of Michigan is located in the most frost susceptible
region, the wet-freeze region. Hence, the estimation of frost depths and frost heave are two
important factors that are typically considered in the design of pavement and bridge and other
structural foundations.

4.

The Presence or Absence of Insulation- See chapter 2 for frost heave mitigation methods and
the impacts of insulation on frost depth.

4.4.1 Insulation Effect on the Frost Depth
As stated before, insulation is typically used to reduce heat flow and prevent heat loss in
temperatures below 0 oC. Since the thermal conductivity of the insulation material is low, the
heat loss decreases across the soil layer and the temperature remains above freezing point.
Equation 4.13 that is based on conservation of energy governs the Temperature variation in a soil

126

layer (Jiji, 2009). Since no energy is generated in the freezing process, the equation can be
rewritten as Equation 4.14. Using Fourier’s law, the heat flux can then be calculated using
Equation 4.15.

Figure 4.31 AASHTO four environmental regions (ACPA, 2008)
Equation 4.13
𝐸̇𝑖𝑛 + 𝐸̇𝑔 − 𝐸̇𝑜𝑢𝑡 = 𝐸̇
𝐸̇𝑖𝑛 − 𝐸̇𝑜𝑢𝑡 = 𝐸̇
𝐸̇𝑖𝑛 − 𝐸̇𝑜𝑢𝑡 = −𝑘
Where

𝜕 2𝑇
𝜕𝑧 2

𝐸̇ = rate of energy change within the region (J/s);
𝐸̇𝑖𝑛 = rate of energy added (J/s);
𝐸̇𝑔 = rate of energy generated (J/s);
𝐸̇𝑜𝑢𝑡 = rate of energy removed (J/s);
T= temperature in the soil layer (oC);
k= Thermal conductivity of the soil layer (W/(m.oC)); and
127

Equation 4.14
Equation 4.15

z= depth from the ground surface (m).
The rate of energy change within the region can be calculated using Equation 4.16. Thus,
Equation 4.14 can be rewritten as Equation 4.17 (Edgar, 2014).
Ė = 𝐶

∂T
∂t

𝜕𝑇
𝜕 2𝑇
𝐶
= −𝑘 2
𝜕𝑡
𝜕𝑧

Equation 4.16

Equation 4.17

It should be noted that Equation 4.17 does not consider the phase change effect in the
soil layer but since in RPF latent heat of fusion is negligible, this equation can be used for
modeling an insulation layer. By assuming that surface temperature varies in a sinusoidal
manner, solution to Equation 4.17 can be obtained using Equation 4.18 (Edgar, 2014).
−𝑧
𝑧
𝑇(𝑧, 𝑡) = 𝑇𝑎𝑣𝑒 + 𝐴0 ∗ 𝑒 𝑑 ∗ sin (𝜔𝑡 − )
𝑑

Equation 4.18

Where 𝑇(𝑧, 𝑡)= temperature variation at depth for each time interval (oC);
Tave= the average temperature in soil layer (oC);
A0= amplitude of the sine wave which relates to surface temperature fluctuation;
d= depth that relates the reduction in temperature fluctuation A0 to depth (m);
ω = time frequency; and
All other parameters are the same as before.
As stated above, the parameter “d” in Equation 4.18 is the characteristic depth that
relates the reduction in surface temperature fluctuation to depth and can be calculated using
Equation 4.19 (Edgar, 2014).
𝑑=(

2𝑘 1
)2
𝐶𝜔

Where all parameters are the same as before.

128

Equation 4.19

According to Equation 4.19, adding a low thermal conductivity layer could significantly
influence the temperature pattern in the soil. In fact, adding an RPF layer has the same effect as
adding additional soil to the layer.
Therefore, the thickness of the RPF can be modeled as

𝑡𝑅𝑃𝐹

𝑘𝑅𝑃𝐹
= 𝑑 𝑠𝑜𝑖𝑙 ∗ √
𝑘𝑠𝑜𝑖𝑙

Equation 4.20

Where tRPF = insulation thickness (cm);
dRPF = depth of the soil layer (cm);
𝑘𝑠𝑜𝑖𝑙 = thermal conductivity of soil (W/(m.oC)); and
𝑘𝑅𝑃𝐹 = the effective thermal conductivity of insulation layer (W/(m.oC)).
The thickness of the RPF can be calculated using Equation 4.21

𝑡𝑅𝑃𝐹 = (((−0.7392 𝑘 + 6.4408) ∗ (𝐶𝐹𝐷𝐷)

(.0035𝑘+0.4846)

𝑘𝑅𝑃𝐹
) − 𝑑𝑅𝑃𝐹 ) ∗ √
𝑘𝑠𝑜𝑖𝑙

Equation 4.21

Where dRPF = depth of insulation (cm);
CFDD= cumulative freezing degree day in design year (oC-day); and
All other parameters are the same as before.
4.4.1.1 Example
Calculate the thickness of the insulation for the given data.
1. CFDD in design year = 800 oC-day;
2. dRPF = 36 cm;
3. 𝑘𝑠𝑜𝑖𝑙 = 2.42 (W/(m.oC));
4. The effective insulation R-value=1.0  k RPF =1/(1*100)= 0.01 (W/(m.oC));
The thickness of the insulation layer can be calculated using Equation 4.21.

129

0.01
𝑡𝑅𝑃𝐹 = (((−0.7392 ∗ 2.42 + 6.4408) ∗ (800)(.0035∗2.42+0.4846) ) − 36) ∗ √
= 5.7 𝑐𝑚
2.42
4.4.2 Gilpin’s Frost Heave Model
Different theories and models for modeling frost heave are reviewed in Chapter 2 of this
report. In this study, the Gilpin’s model, which is based on frozen fringe theory, was used to
predict the frost heave under field conditions. As stated before the original Gilpin model is a
mechanistic-empirical model based on heat and mass balance equations and laboratory data. The
Gilpin model is a laboratory based model; applying it to field conditions having different
boundary values led to some errors in the results. Further, the required input data to the model
are not available and are expensive to obtain. Therefore, in this study, the model was simplified
to include the empirical frost depth prediction model developed in this study. The resulting
model was verified by comparing the predicted frost heave under pavements and shoulders to the
measured values at 5 different sites in Oakland County, Michigan.
4.4.2.1

Basic Assumptions
Assume that a saturated and salt-free soil column was subjected to a constant overburden

pressure (POB) as shown in Figure 4.32. The top of the column was subjected to a fixed subfreezing temperature (TTOP), whereas the bottom of the column (at the ground water table
elevation) was at a fixed above freezing temperature (TBOT). The soil column was further
assumed to consist of three zones; frozen zone at the top followed by a frozen fringe zone and
then by an unfrozen zone. The top of the unfrozen zone begins at a point where water and ice can
exist in the pore spaces of the soil at below freezing temperature (Tf). In this model, frost
penetration and frost heave were predicted using analytical iteration solution. In each iteration, it
was assumed that

130

1.

The temperature variation in each zone is linear.

2.

The thermal conductivity in each zone is constant.

3.

The water content and permeability in the unfrozen zone is constant.

4.

Steady state water flows through the frozen fringe and unfrozen zones.
POB
X
TTOP
Frozen Zone

VH

H

Bottom of ‘Active’ Ice Lens
Tl
Frozen Fringe

a
Tff

Further Advance of Pre Ice

Unfrozen Zone

Z

TBOT

PL=0
Figure 4.32 The schematic of frost heave model (Gilpin, 1980)

131

4.4.2.2 Heat and Mass Balance Equations
As stated in Chapter 2, for simulating the heat transfer in his model, Gilpin used the phasechange heat transfer equations. After imposing the boundary conditions in each zone, Equation
2-42 and 2-43 were obtained for heat transfer between the frozen and the frozen fringe zones and
between the frozen fringe and the unfrozen zones, respectively. For convenience, these equations
were converted to English system as follow

𝑘𝑓 (𝑇𝑇𝑂𝑃 − 𝑇𝑙 ) 𝑘𝑓𝑓 (𝑇𝑓𝑓 − 𝑇𝑙 )
𝐿
−
= 𝑉𝐻
𝐻
𝑎
𝜈𝑖

Equation 4.22

𝑘𝑓𝑓 (𝑇𝑓𝑓 − 𝑇𝑙 ) 𝑘𝑢𝑓 (𝑇𝐵𝑂𝑇 − 𝑇𝑓 )
𝑑𝑧
−
= 𝜌𝑠𝑖 𝐿
𝑎
𝑍
𝑑𝑡

Equation 4.23

−

Where a = thickness of the frozen fringe (m);
H = thickness of frozen zone (m);
kf = thermal conductivity of the frozen zone (W/(oC.m));
kff = thermal conductivity of the partially frozen zone (W/(oC.m));
kuf = thermal conductivity of the unfrozen zone (W/(oC.m));
L = latent heat of fusion of water (J/Kg);
TTOP, TBOT = temperatures at the top and bottom of the soil column (oC);
VH = frost heave rate (m/s);
Z = distance between bottom of soil column and position of ice penetration (m);
Tl = temperature at the base of the active ice lens (oC);
Tff = temperature at the base of the frozen fringe (oC);
νi = specific volumes of ice (m3/kg);
𝑑𝑧
𝑑𝑡

= frost depth propagation rate (m/s); and

132

𝜌𝑠𝑖 = mass of ice per unit volume of soil (kg/m3).
Using the mass balance equation and imposing boundary conditions, Gilpin proposed
Equation 2-45 for calculating the water pressure at the bottom of the frozen fringe zone. Finally,
Equation 2-46 was obtained for frost heave calculation. For convenience, the equations were
converted to English system as follows:

𝑃𝑤𝑓

𝑉𝐻 =

𝑑𝑧
𝑍
𝜈𝑤 (𝑉𝐻 + 𝜌𝑠𝑖 Δ𝜈 𝑑𝑡 )
= −𝑔 (1 +
)
𝜈𝑤
𝜈𝑖
𝐾𝑢𝑓

𝜈𝑖2
𝑔𝜈𝑤

𝐿(−𝑇𝑙 )
[
− 𝑃𝑂𝐵 + 𝑃𝐿𝑓 ]
𝑎𝐼𝑓𝑙
𝜈𝑤 𝑇𝑚
1
[𝑇 − 𝑇 ] + (𝐾 )
𝑓𝑓
𝑙
𝐿
1

Equation 4.24

Equation 4.25

𝑇𝑙

1
𝑑𝑇
𝑇𝑓𝑓 𝐾𝑓𝑓

𝐼𝑓𝑙 = ∫

Equation 4.26

Where 𝑔 = acceleration of gravity (m/s2);
𝑃𝑤𝑓 = water pressure at the edge of the frozen fringe (Pa);
νw = specific volumes of water (m3/kg);
Δ𝜈 = specific volume difference (νi − 𝜈𝑤 );
POB = overburden pressure (Pa);
KL = permeability of ice lenses (m/s);
𝐾𝑢𝑓 = permeability of unfrozen zone (m/s); and
All other parameters are the same as before.
It should be noted that Gilpin proposed semi-empirical models for estimating the
hydraulic conductivity of frozen fringe and the temperature at the bottom of the frozen fringe
zone, Tf (Gilpin 1980).

133

4.4.2.3

Ice Pressure Distribution in the Frozen Fringe Zone
Gilpin calculated the ice pressure distribution in the frozen fringe zone in order to model

the initiation of new ice lenses. He assumed that the initiation of new ice lenses takes place
where the ice pressure in the frozen fringe zone exceeds the critical pressure, which is also
known as the separation pressure. This pressure is a function of the overburden pressure and
water-ice curvature. Figure 4.33 illustrates the ice pressure distribution. Based on Clapeyron
equation, at zero flow rate in the frozen fringe zone, the ice pressure increases along the solid
line (L(-T)/(vsTa)). Further, at non-zero flow rate the ice pressure increases along the Ps line so
that it becomes equal to the overburden pressure at the top of the frozen fringe zone. Ice pressure
in the frozen fringe zone could be estimated using Equation 4.27.
𝑉𝐻 ∗

𝑣𝑤
𝜈𝑖 𝑑
𝐿𝑇
= 𝐾𝑓𝑓
[𝑃𝑖 +
]
𝑣𝑖
𝑔 𝑑𝑥
𝜈𝑖 (𝑇𝑚 + 273)

Equation 4.27

Where 𝐾𝑓𝑓 = the permeability in the frozen fringe (m/s);
Pi = pressure in ice (Pa);
Tm = bulk freezing temperature (oC);
T= temperature along the frozen fringe (oC.); and
All other parameters are the same as before.
For modeling the separation pressure, a pair of spherical soil particles was considered as
shown in Figure 4.34. In the absence of ice, the overburden pressure is acting on the interface of
the particles. This pressure could be transmitted from one particle to the other. However, at a
critical ice pressure, the pressure at the contact point drops to zero allowing the ice to separate
the particle from each other. This critical ice pressure was assumed to be the separation pressure.
Where ice pressure in frozen fringe zone exceeds the separation pressure, new ice lenses are

134

formed. Estimation of the separation pressure was a matter of debate between researchers; Gilpin
suggested the following equation for separation pressure:

Figure 4.33 Ice pressure along frozen fringe zone (Gilpin, 1980)
𝑃𝑠𝑒𝑝 = 𝑃𝑂𝐵 +

2 ∗ 𝜎𝑖𝑤
𝐷10

Where 𝑃𝑠𝑒𝑝 = separation pressure (Pa);
D10= = particle size at 10 percent passing (m);
𝜎𝑖𝑤 = ice-water surface energy (N/m); and
𝑃𝑂𝐵 = overburden pressure (Pa).
135

Equation 4.28

Other equations were developed and are available in the literature. However, in this study, the
Gilpin equation was revised to simplify the required inputs and used to predict the frost heave
potential.

R

Ice

Pore Water
Particle

Figure 4.34 Particle separation pressure
4.4.3 Revised Frost Heave Model
In the Gilpin model at the beginning of the solution, initial non-zero values were chosen
for a and H parameters of Equation 4.22 and Equation 4.23 in order to avoid the infinite
temperature gradient (See Figure 4.32). At each iteration (Δt), the systems of four equations
(Equation 4.22 to Equation 4.25) were solved to calculate the four unknowns, i.e. VH, Pwf, Tl and
dz/dt. It should be noted that since Equation 4.25 is nonlinear the accuracy of the results are
highly related to the nonlinear solution. After calculating VH, ice pressure in the frozen fringe
was calculated using Equation 4.27. If the ice pressure did not exceed the critical pressure then H
was increased by VH*Δt; a was increased by dz/dt* Δt and the equations were solved for the next
iteration. Otherwise a new ice lens was assumed to initiate where ice pressure exceeded the

136

critical pressure, H was increased and a was decreased accordingly. Then VH, Pwf, Tl and dz/dt
were calculated again for the same time step (Gilpin, 1980).
In Gilpin model, the hydraulic conductivity of the frozen fringe zone (Kf) was estimated
based on the laboratory data. Since laboratory conditions were not necessarily correlated well
with the field conditions and field data were not available for calibration, an overall permeability
(Kf) was assumed for the frozen fringe zone in order to avoid nonlinear solution. Further, due to a
large frozen zone thickness in the field, the frozen zone was assumed to be impermeable.
Tff (temperature at the bottom of frozen fringe) was calculated using the following
empirical equation (Gilpin 1980).
𝑇𝑓𝑓 = −

8𝜎𝑖𝑤 𝜈𝑤 𝑇𝑚
𝐷10 ∗ 𝐿

Equation 4.29

Where all parameters are the same as before.
In the revised model, instead of using Equation 4.23 for calculating the frost depth
propagation, the empirical frost depth model that was developed based on the measured frost
depths data in Michigan was used (Equation

4.12). The analyses were conducted using

analytical iterative solution. In each iteration, the frost depth propagation rate (

𝑑𝑧
𝑑𝑡

) was

calculated using Equation 4.30 and Equation 4.31 and the Tl was estimated using Equation 4.32.
𝑑𝑧
= (−0.7392 𝑘𝑢𝑓 + 6.4408) 𝛽
𝑑𝑡

Equation 4.30

𝛽 = (𝐶𝐹𝐷𝐷𝑛 (0.035𝑘𝑢𝑓+0.4846) − 𝐶𝐹𝐷𝐷𝑛−1 (0.0035𝑘𝑢𝑓+0.4846) )

Equation 4.31

𝑇𝑙 = 𝑇𝑓𝑓 −

𝑎
𝑑𝑧 𝑘𝑢𝑓 (𝑇𝐵𝑂𝑇 − 𝑇𝑓 )
∗ (𝜌𝑠𝑖 𝐿 +
)
𝑘𝑓𝑓
𝑑𝑡
𝑍

Where CFDD= Cumulative freezing degree day (oC- day); and
All parameters are the same as before

137

Equation 4.32

Furthermore, Equation 4.25 was revised into Equation 4.33 and by solving Equation 4.24
and Equation 4.33; the values of VH and Pwf were calculated.
𝜈𝑖2 𝐾𝑓𝑓
𝐿(−𝑇𝑙 )
𝑉𝐻 =
[
− 𝑃𝑂𝐵 + 𝑃𝑤𝑓 ]
𝑔𝜈𝑤 𝑎 𝜈𝑤 (𝑇𝑚 + 273)

Equation 4.33

Where 𝐾𝑓𝑓 = over all permeability of frozen fringe (m/s); and
All other parameters are the same as before.
Finally, the ice pressure variation in the frozen fringe zone was calculated using Equation
4.27. New ice lens formation was assumed where the pressure value is higher than the separation
pressure. Therefore, the thicknesses of the frozen fringe and the frozen zone were changed
accordingly and consequently, the calculations of VH, Tl, Pwf were repeated.
The total frost heave was then estimated using the following equation
∆ℎ𝑡𝑜𝑡𝑎𝑙 = ∆ℎ𝑢 + ∆ℎ𝑖 = 𝑉𝐻 ∗ ∆𝑡 + 0.09 ∗ 𝑛 ∗ 𝐻

Equation 4.34

Where ∆ℎ𝑡𝑜𝑡𝑎𝑙 = total frost heave (m);
∆hu = frost heave due to water uptake (m);
∆hi = heave due to freezing of in-situ pore water (m);
∆t = time interval (s);
n = soil porosity, and 0.09 is the ratio of volumetric expansion of water in phase change
(Nixon 1982); and
all other parameters are the same as before.
4.4.4 Discussion of the Results of the Revised Frost Heave Model
At the beginning of the frost, the heave rate is high therefore the ice pressure in the frozen
fringe zone could surpass the separation pressure and the boundary of the frozen zone keeps
moving downward in the soil column. As frost progresses, the heave rate decreases and
consequently the ice pressure decreases and it does not exceed the separation pressure anymore.
138

This causes growth in the frozen fringe zone. Since the hydraulic conductivity of the frozen
fringe zone is less than that of the unfrozen zone the larger frozen fringe zone leads to lower
heave. The extent of the frozen fringe zone depends on the types of the soil and overburden
pressure.
Soil Type - The hydraulic conductivity of fine sand is higher than the clayey silt so the flow
rate is greater in fine sand. However, as mentioned before, capillary pressure is mainly
responsible for the frost heave phenomenon. Due to the aggregate size, suction is smaller in
fine sand than in clayey silt, therefore larger frost heave rates are expected in clayey silt.
Further, in fine sands relative to clayey silts a larger frozen fringe zone is observed. Figure
4.35 and Figure 4.36 depicts the calculated frost heave and frozen fringe thickness in three
different types of soil versus time, respectively.
Clayey silt

Sandy clayey silt

Fine sand

35
30

Frost heave (mm)

1.

25
20
15
10
5
0
10

30

50
Time (Day)

70

90

Figure 4.35 Calculated frost heave for three soil types when GWL= 10 m, TTOP= -3 oC for 100
days, POB = 150 kPa

139

Clayey silt

120

Sandy clayey silt

Fine sand

Frozen fringe(mm)

100
80
60
40
20
0
10

30

50
Time (Day)

70

90

Figure 4.36 Calculated frozen fringe for three soil types when Z= 10 m, TTOP=- 3 oC for 100
days, POB = 150 kPa
2.

Overburden Pressure - Various overburden pressures were used to assess the impact of
overburden pressure on the frost heave and frozen fringe zone. Konrad and Morgenstern
(1982) found out that the overall permeability of the frozen fringe zone decreases
approximately by 25% as the overburden pressure increases up to 400kPa. Accordingly, in
the revised model, the frozen fringe zone permeability was reduced as the overburden
pressure was increased.
Figure 4.37 to Figure 4.40 show the model results for total heave in different
overburden pressures for different Z values when the TTOP was fixed at -3 oC for 100 days in
different soil types. As can be seen in the figures, when the ground water table is deep, for
the same freezing time period, less amount of water can reach the frozen zone and therefore
the frost heave decreases.

140

6

z=6 m

z= 7.5 m

z=9 m

Total heave (cm)

5
4
3
2
1
0
0

50

100
150
200
Overburden pressure (kPa)

250

300

Figure 4.37 Calculated total heave versus overburden pressure for clayey silt in different ground
water table depths when TTOP=-3 oC in 100 days

5

z=6 m

z= 7.5 m

z=9 m

Total heave (cm)

4
3
2
1
0
0

50
100
150
Overburden pressure (kPa)

200

Figure 4.38 Calculated total heave versus overburden pressure for sandy clayey silt in different
ground water table depths when TTOP=-3 oC in 100 days
141

4.5

z=6 m

z= 7.5 m

z=9 m

4

Total heave (cm)

3.5
3
2.5
2

1.5
1
0.5
0
0

50
100
150
Overburden Pressure (kPa)

200

Figure 4.39 Calculated total heave versus overburden pressure for fine sand and silt with pebbles
in different ground water table depths when TTOP=-3 oC in 100 days

4.5

z=6 m

z= 7.5 m

z=9 m

4

Total heave (cm)

3.5
3
2.5
2
1.5
1
0.5
0
0

50
100
Overburden Pressure (kPa)

150

Figure 4.40 Calculated total heave versus overburden pressure for clayey, silty, gravely, sand in
different ground water table depths when TTOP=-3 oC in 100 days
142

At higher overburden pressures, the separation pressure is larger. Therefore, it is
expected that the frozen fringe zone thickness increases as the overburden pressure increases.
This leads to lower frost heave values in higher overburden pressures.
In order to assess the impact of surface temperature (i.e. TTOP), the model was
assessed with a fixed TTOP and with a changing TTOP but a fixed rate of cooling. In both
scenarios, the freezing index was the same. The results are depicted in Figure 4.41. It can be
seen from the figure that the results are almost the same in both assessments, therefore using
a fixed TTOP based on the cumulative freezing index and length of frost period is a good
assumption.

7

Variable Ttop

Fixed Ttop

Total heav (cm)

6
5
4
3

2
1
0
0

100

200
300
Overburden pressure (Kpa)

400

Figure 4.41 Total heave versus overburden pressure for clayey silt when TTOP is fixed at -3 oC
and when TTOP is decreasing with a rate of -.057 per day in 100 days
Furthermore, the revised model was used to predict the frost heave under shoulder and
pavement. Data from 5 sites located in Oakland County, Michigan were provided by the

143

Michigan Department of Transportation (See Chapter 3-4). In order to validate the revised
model, this data was used. According to each soil description and size distribution, hydraulic
conductivity and D10 values were chosen. Thermal conductivity values were chosen according to
the soil type from the measured thermal conductivity values (See Table 3-4).
Furthermore, by using the measured frost depth at maximum heave and Equation 4.12,
the CFDD values and the TTOP for each site were calculated. The inputs for each soil type are
shown in Table 4.10.
As stated before, the frost heave occurs when water migrates from the water table to the
frozen layer. Therefore, the controlling layer is the natural soil. It can be assumed that frost
heave could be calculated by a single layer model consisting of the natural soil under the
pavement layers. The weight of the asphalt, base, and subbase layers can be considered as the
overburden pressure.
The frost heave was estimated under the shoulder and pavement in Oakland County sites.
The results are shown in Figure 4.42. The only difference between the two models is the
overburden pressure. The overburden pressure was modeled by a 51-centimeter thick soil layer
having density of 19.6 (kN/m3) for the shoulder and a 77.5-centimeter thick soil layer with the
same density for the pavement. It should be noted that the ground water table was set at the
average measured ground water table level in Oakland County, which was 9 meters.
Figure 4.42 indicates that in both cases the calculated frost heave values are within 0.25
centimeters of the measured ones. It can be concluded that different simplifications and
modifications, which were applied to the Gilpin model, did not affect the accuracy of the model
significantly, indeed, it produced better results.

144

Table 4.10 Different input values for each site, I75, Oakland County, Michigan.

Station Name

Duration
(days)

Ttop
(oC)

Sta/724+00

65

-1.94

Hydraulic
Conductivity
(m/s)
1.0*10^-6

Sta/719+00

40

-3.06

Sta/652+00

60

Sta/528+88
Sta/474+00

9

D10
(mm)
0.01

1.0*10^-6

9

0.01

-3.89

5.0*10^-7

9

0.02

70

-1.94

1.0*10^-7

9

0.002

55

-1.94

5.0*10^-8

9

0.001

Line of equality between calculated and measured data
Frost heave under shoulder
Frost heave under pavement

3

Calculated heave (cm)

GWT
(m)

2.5
2
1.5
1
0.5
0.5

1

1.5
2
Measured heave (cm)

2.5

3

Figure 4.42 Measured versus calculated frost heave under the shoulder and pavement in 5 sites,
Oakland County, Michigan

145

It is noteworthy that for station 528+88 (See Table 4.10); the measured frost heave under
the pavement is approximately 0.9 centimeters less than that under the shoulder (See Table 3-5).
This difference is higher than those at the other sites (about 0.3-centimeter). At station 528+88,
an undercut of approximately 30.5 centimeters was made for frost protection of the roadbed soil.
Therefore, the frost penetration in the clayey silt roadbed soil decreased by 30.5 centimeters and
hence, as it was expected, the frost heave decreased. In the analyses, the undercut was modeled
as a part of the overburden pressure against the surface of the clayey silt roadbed soil. The results
are also shown in Figure 4.42.
4.4.5 Heave Pressure
Heave pressure can cause real stability issues in different structures such as retaining
walls, utility poles, and shallow foundations.
In the revised frost heave model presented in the previous section, frost heave can be
calculated as a function of overburden pressure. If the overburden pressure is equal to or greater
than the heave pressure, no heave will occur. That is equilibrium scenario is reached. Otherwise,
frost heave will take place due to the net pressure against the structure in question. The heave
pressure can be calculated as follows:
𝑃𝐹𝐻 = 𝑃𝐸 − 𝑃𝑂𝐵

Equation 4.35

Where PFH = pressure due to heave (kPa);
PE = the equilibrium overburden pressure (see Figure 4.37), (kPa);
POB= the actual overburden pressure (kPa);
In order to develop a model for estimating the frost heave pressure, the following three
steps were used:

146

1. For four soil types, the revised heave model (Equation 4.30 to Equation 4.34) was used to
calculate the amount of heave as a function of the overburden pressure and the depth to the
ground water table (see Figure 4.37 through Figure 4.39). Table 4.11 shows the different
input values for each soil type. For each scenario, the corresponding equilibrium pressure
was also calculated.
Table 4.11 Different input values for each soil type

Soil Type

Hydraulic
D10
Conductivity
(mm)
(Kuf, (m/s)

Thermal
Conductivity
(W/(m.oC))

Dry unit
Weight
(Îłd, kN/m3)

Water
Content
(w%)

POB
(kPa)

Clayey,
silty,
1.0*10^-6
0.02
2.57
10
19.6
gravely,
sand
Fine sand
variable
5.0*10^-7
0.01
2.42
15
18.9
and silt
Sandy
1.0*10^-7
0.002
1.75
20
18.1
clayey silt
clayey silt
5.0*10^-8
0.001
1.52
25
15.7
1- Soil type: can be obtained from the boring log on site (known)
2- Duration: the time period that CFDD (cumulative freezing degree day) is calculated over
(known or assumed)
3- Ttop: temperature at the ground surface= CFDD/Duration (known or assumed)
4- Hydraulic conductivity: can be measured on site or assumed based on the soil type
5- GWTD: ground water table depth (known)
6- D10: the effective size of the soil; can be obtained from the soil distribution curve or
assumed based on the soil type (known or assumed)
7- Thermal conductivity: measured at MSU soil laboratory (known)
8- Tbottom: temperature at the ground water table level; assumed based on GWTD (assumed)
9- Dry unit weight of soil: can be measured on site or obtained from the CRREL graphs
based on the thermal conductivity values (known or assumed)
10- Water content: can be measured on site or obtained from the CRREL graphs based on the
thermal conductivity values (known or assumed)
11- Void ratio: can be measured on laboratory or assumed based on the soil type and its
density= 0.5 (known or assumed)
12- POB= overburden pressure

147

2. The data in Figure 4.37 through Figure 4.39 were used to estimate (for each amount of heave
and ground water depth), the corresponding overburden pressure.
3. The estimated overburden pressure and the equilibrium pressure were used as inputs to
Equation 4.35 to estimate the heave pressure. Figure 4.43 shows the results for clayey silt. As
can be seen the heave pressure is almost the same in different ground water table depth.
Therefore, heave pressure can be estimated regardless of the ground water table depth as a
function of frost heave. Figure 4.44 shows the heave pressure versus frost heave in four soil
types.

6

z=6 m

z=7.5 m

z=9 m

Total heave (cm)

5

4

3

2

1

0
0

50

100

150

200

250

300

Frost pressure (kPa)

Figure 4.43 heave pressure versus calculated total heave for clayey silt in different ground water
table depths when TTOP= -3 oC in 100 days

148

The results showed that in the same winter duration the heave pressure has a unique
polynomial relationship with frost heave as follows
2
𝑃𝐹𝐻 = 𝑎 ∗ ∆ℎ𝑡𝑜𝑡𝑎𝑙
+ 𝑏 ∗ ∆ℎ𝑡𝑜𝑡𝑎𝑙

Equation 4.36

Where a,b,c = constant values which are different in each soil type; and
all other parameters are the same.
It should be noted that in Equation 4.36 the soils were considered to be saturated. Also,
the effect of void ratio was not considered in the model (since the void ratio is a function of soil
density) Table 4.12 shows the statistical parameters of Equation 4.36 for each soil type.
Clayey Silt
Sandy Clayey Silt
Clayey, Silty, Gravely Sand
Fine Sand with Silt

250

Heave pressure (kPa)

200

150

100

50

0
0

1

2

3
Total heave (cm)

4

5

6

Figure 4.44 Heave pressure versus calculated total heave in four soil types when TTOP= -3 oC
over 100 days

149

Table 4.12 Statistical coefficients in Equation 4-30 for each soil type.
Soil Type

Equation 4.36

clayey silt

PFH = -2.86∆htotal2 + 60.2∆htotal

Sandy clayey silt

PFH = -2.64∆htotal 2 + 49.1∆htotal

Fine sand and silt

PFH = -2.63∆htotal 2 + 45.4∆htotal

clayey, silty, gravely, sand

PFH = -2.17∆htotal 2 + 37.1∆htotal

It should be noted that field data for evaluating the accuracy of the results were not
available. Therefore, the model should be validated as data become available.
Thaw Depth
At the end of the freezing season, the soils start to thaw. The prediction of frost and thaw
depths are crucial for estimating the amount of heave due to frost action and to estimate the
proper time to post and remove seasonal load restriction signs. The calculation of thaw depth is
presented and discussed below.
4.5.1 Calculation of Cumulative Thawing Degree day (CTDD)
Calculating the CTDD accurately is the first step in developing an accurate thaw depth
model and consequently an effective and reliable SLR policy. In this study, the average daily air
and the pavement surface temperatures in 2011(which were available at the early stage of the
study) from 12 Road Weather Information System (RWIS) stations in Lower Peninsula (LP) of
the State of Michigan were used to develop a new CTDD approach. First, surface and air
temperature data and the calculated solar radiation were used to develop a model for estimating
the average daily pavement surface temperature. Second, the estimated surface temperatures
were used to calculate the CTDD. Later on, when more data became available, the 2012, 2014,
and 2015 data from the same RWIS stations were used to check the accuracy of the predicted
surface temperatures for different years with different winter severity. Third, the accuracy of the
model was further evaluated using the air and the average daily pavement surface temperatures
150

in 2012, 2014, and 2015 from 9 RWIS stations in the Upper Peninsula (UP) of the State of
Michigan. The results of the model were also compared with WSDOT and MnDOT predictions
(See Chapter 2, Section 2.3.2).
4.5.1.1 Pavement Surface Modeling
For modeling the daily pavement surface temperature, the air-pavement system can be
considered as a thermodynamic system and the energy balance (equilibrium) at the pavement
surface can be calculated using Equation 2.2.
As mentioned in Chapter 2, Solaimanian and Bolzan proposed a mechanistic empirical
nonlinear model for solving Equation 2.2 (Solaimanian and Bolzan, 1993). The model requires
various inputs such as air temperature, solar radiation, wind speed, percent of sunshine,
pavement emissivity, pavement absorptivity, and so forth. Since most of these input values are
not available and/or expensive to collect, most highway agencies only consider air temperature
as an input in their calculations of the pavement surface models and CTDD. However,
Solaimanian and Bolzan results showed that while the air temperature significantly affects the
pavement surface temperature predictions, the difference between air temperature and pavement
temperature can be as low as 5.5 to 8 oC or as high as 22 to 28 oC depending on the solar
radiation and percent sunshine. The solar radiation can change based on the latitude and time of
the year. In fact, different locations of earth receive different amount of solar radiation due to
solar inclination, i.e. the tilt of the earth north-south axis with respect to the orbital plane. On the
other hand, during the year the distance between the earth and the sun changes, which causes a
daily variation in the amount of solar radiation (Diefenderfer et al, 2006). Therefore, in this
study, in addition to air temperature, calculated solar radiation was also considered as an input of
the developed model. At any geographical location, the daily amount of solar radiation was

151

calculated using the latitude and the day of the year as follows (Diefenderfer et al, 2006; Iqbal,
2012)
𝜔𝑠 ∗ 𝜋
− tan 𝜔𝑠 )
180𝑜

𝐷𝐶𝑆𝑅 = (76.39) ∗ 𝐼𝑠𝑐 ∗ 𝐸0 ∗ sin(𝜑) ∗ sin(𝛿) ∗ (

Equation 4.37

𝑑𝑛
]
365

Equation 4.38

𝐸0 = 1 + 0.033 ∗ cos [2𝜋
𝛿 = 23.45𝑜 ∗ sin [

360𝑜
∗ (𝑑𝑛 + 284)]
365

𝜔𝑠 = cos−1 (− tan(𝜑) ∗ tan(𝛿))

Equation 4.39

Equation 4.40

Where DCSR = daily calculated solar radiation ((kJ/(m2day));
Isc = solar constant= 4,871 (kJ/(m2.hr))
E0 = daily eccentricity factor;
φ = latitude (deg);
δ = solar declination (deg);
ωs= sunrise hour angle which is the angle between the sun’s highest point each day,
which is zero, and the location of the sun at sunrise in that day(deg); and
dn = day of the year starting at 1 for January first; for example, February 4 is the 35th day
of the year, hence, the corresponding dn is 35.
In order to develop a prediction model for pavement surface temperature, the latitude of
each RWIS station and the day of the year (dn) were used as inputs to Equation 4.37 to calculate
the solar radiation for each day where the air and pavement surface temperatures were measured.
Two statistical pavement surface temperature models were then developed; linear and nonlinear
as stated in Equation 4.41 and Equation 4.42. For both models, the measured pavement surface
temperature was treated as a dependent variable and the output of Equation 4.37 (the solar

152

radiation), and the average measured daily air temperature as the independent variables. The
metrics of the two models are depicted in Table 4.13
𝑇𝑠𝑢𝑟𝑓 = 0.687𝑇𝑎𝑖𝑟 + 4.377 ∗ 10−4 ∗ 𝐷𝐶𝑆𝑅 − 6.4

Equation 4.41

𝑇𝑠𝑢𝑟𝑓 = 0.674𝑇𝑎𝑖𝑟 + 1.61 ∗ 10−4 ∗ 𝐷𝐶𝑆𝑅1.09 − 5.8

Equation 4.42

Where Tsurf = average daily pavement surface temperature (oC);
Tair = average daily air temperature (oC); and
all other parameters are the same as before.
Table 4.13 Performance metrics for the model
Model

R2

Standard Error of
the Estimate (SEE)

Mean Absolute
Error(MAE)

Equation 4.41

0.89

3.6

2.9

Equation 4.42

0.91

2.8

2.2

Figure 4.45 depicts the measured and the calculated pavement surface temperatures using
Equation 4.41 and Equation 4.42 for 12 RWIS stations in 2011. It should be noted that since in
most years the pavement is completely thawed in the first 120 days of the year, in this figure data
from 12 RWIS stations of the first 120 days of the year in 10-day interval are shown. The solid
straight line in Figure 4.45 indicates the locus of equality between the measured and the
calculated pavement surface temperatures. It can be seen from the figure that Equation 4.42
predicts the pavement surface temperature slightly better than Equation 4.41 especially for
pavement surface temperature higher than 15.5oC. Therefore, it is recommended to use Equation
4.41 for more accurate prediction of the pavement surface temperature.

153

Equation 4.41

Equation 4.42

Line of Equality

Calculated surface temperature (oC)

25
20
15
10
5
0
-15

-10

-5

0

5

10

15

20

25

-5
-10
-15
Measured surface temperature (oC)

Figure 4.45 Calculated versus measured surface temperature in 2011 using Equation 4.41 and
Equation 4.42 for 12 RWIS stations in the first 120 days of the year. The data are shown in 10day interval
After developing the statistical model for predicting the pavement surface temperature,
Equation 4.42 was used to calculate, for each day, the thawing degree-day using Equation 4.43.
Finally, the cumulative thawing index was calculated using Equation 4.44.
𝑇ℎ𝑎𝑤𝑖𝑛𝑔 𝐷𝑒𝑔𝑟𝑒𝑒 𝑑𝑎𝑦 = (𝑇𝑠𝑢𝑟𝑓 − 0𝑜 𝐶)
𝑛

Equation 4.43
Equation 4.44

𝐶𝑇𝐷𝐷𝑛 = ∑ 𝑇ℎ𝑎𝑤𝑖𝑛𝑔 𝐷𝑒𝑔𝑟𝑒𝑒 𝐷𝑎𝑦 ≥ 0
𝑖=1

Where 𝐶𝑇𝐷𝐷𝑛 = Cumulative thawing Index inform January first to the nth day of the year (°Cday); and
All other parameters are the same as before.

154

4.5.1.2 Model Verification
In order to evaluate the accuracy of the statistical model, Equation 4.42 was used to
calculate the pavement surface temperatures for three years other than the year 2011 because the
2011 data were used to develop the statistical model. The three years (2012, 2014, and 2015)
have substantially different winter characteristics and severity. For example, the 2015 winter was
an average winter, 2012 winter was a relatively warm one, and 2014 winter was a particularly
cold winter (Based on the National Oceanic and Atmospheric Administration (NOAA) weather
data, the Normal freezing index for LP and UP is 360 and 555 degree-days, respectively.
Therefore, the year with approximately the same freezing index counted as average, the year
with the lower freezing index counted as warm and the year with the higher freezing index
counted as cold). Nevertheless, Equation 4.42 was used to calculate the average daily pavement
surface temperature for the three years and for the 12 RWIS stations in the LP of the State of
Michigan. The latitudes of these stations are approximately in the range of 43.9o to 45.8o. The
results are depicted in Figure 4.46. In the figure, the number of points (n) and the coefficient of
determination (R2) are shown. The data in the figure indicates that Equation 4.42 accurately
predicts the measured pavement surface temperature for all three years and for the 12 RWIS in
the Lower Peninsula of the State of Michigan. Indeed, the coefficient of determination (R2) of
Equation 4.42 is slightly higher than the value of R2 for the 2011 data, which were used in
developing Equation 4.42. In addition, Equation 4.42 was also used to predict the measured
pavement surface temperatures at 9 additional RWIS stations located in the Upper Peninsula
(UP) of the State of Michigan. The latitudes of these stations are approximately in the range of
45.9o to 46.8o. The results are shown in Figure 4.47. It can be seen from the figure that Equation
4.42 accurately predicted the daily pavement surface temperatures for most days.

155

Line of Equality

Calculated surface temperature(oC)

30

-20

R2=.94
n=3294

20
10
0

-10

0

10

20

30

-10
-20
Measured surface temperature(oC)

Figure 4.46 Calculated versus measured surface temperature in 2012, 2014 and 2015 for 12
RWIS stations in the LP of the State of Michigan

Calculated Surface Temperature(oC)

Line of Equality

-20

30
R2=0.91
n=1691

20
10
0

-10

0

10

20

30

-10
-20
Measured Surface Temperature(oC)

Figure 4.47 Calculated versus measured surface temperature in 2012, 2014 and 2015 for 9
RWIS stations in the UP of the State of Michigan
156

Furthermore, Equation 4.44 was used to calculate the CTDD for the measured and the
calculated pavement surface temperature. Figure 4.48 depicts the results for the first 120 days of
the year in 2012, 2014 and 2015 for 12 RWIS stations in the LP of the state of Michigan. In this
figure, the number of data points (n) and the mean absolute percentage error (MAPE) are shown.
In addition, Equation 4.44 was also used to calculate the CTDD for the measured and the
calculated pavement surface temperature in 2012, 2014 and 2015 for 9 RWIS stations in the UP
of the State of Michigan. The results are shown in Figure 4.49. It can be seen that Equation 4.44
predicts the CTDD less accurately for the UP stations. In fact, the MAPE of Equation 4.44 is
18% for the LP stations and 23% for the UP stations.
Additionally, the WsDOT and MnDOT methods were used to calculate the average daily
pavement surface temperature in 2012, 2014, and 2015 for 12 RWIS stations in the LP and 9
RWIS stations in the UP of the state of Michigan. Table 4.14 shows the different performance
metrics for each model. In this table, the standard error of the estimate (SEE), the mean absolute
error (MAE) and the mean absolute percentage error (MAPE) are shown for each model. As can
be seen in Table 4.14, for the LP stations, the SEE values of MnDOT and WSDOT models are
about 28% and 60% larger than SEE of Equation 4.42, respectively. This difference is even
larger for the UP stations. This indicates that Equation 4.42 predicts the average pavement
surface temperature better than the other two methods. In general, all of the performance metrics
indicate that Equation 4.42 predicts the average pavement surface temperature with higher
accuracy.
Further, the accuracy of the calculated CTDD was also compared to the accuracy of the
WSDOT and MnDOT methods. The CTDD values for the years 2012, 2014, and 2015 and for

157

each of the 12 RWIS stations in the LP and 9 RWIS stations in the UP were calculated using the
three methods. Table 4.15 shows various performance metrics for each model.
Line of Equality

CTDD of calculated surface
temperature (oC-day)

700
MAPE= 18%
n-= 1677

600
500
400

300
200
100
0
0

100
200
300
400
500
600
o
CTDD of measured temperature( C-day)

700

Figure 4.48 CTDD of the calculated surface temperature versus CTDD of the measured surface
temperature in 2012, 2014, and 2015 for 12 RWIS stations in the LP of the State of Michigan

CTDD of calculated surface
temperature (oC-day)

Line of Equality
600

MAPE= 23%
n= 718

500

400
300
200
100
0

0

100
200
300
400
500
600
o
CTDD of measured surface temperature ( C-day)

Figure 4.49 CTDD of the calculated surface temperature versus CTDD of the measured surface
temperature in 2012, 2014, and 2015 for 9 RWIS stations in the UP of the State of Michigan
158

Table 4.14 Performance metrics of Equation 4.42, MNDOT and WSDOT models for the average
daily pavement surface temperature for 12 RWIS stations in the LP and 9 RWIS stations in the
UP in 2012, 2014, and 2015

Location

Model

Equation 4.42
Lower
MnDOT Model
Peninsula
WSDOT Model
Equation 4.42
Upper
MnDOT Model
Peninsula
WSDOT Model

Standard Error
of the Estimate
(SEE)

Mean Absolute
Error (MAE)

2.3
2.7
3.6
2.9
3.8
4.7

1.8
2.2
2.5
2.3
2.8
3.7

Mean Absolute
Percentage
Error (MAPE)
0.13
0.17
0.19
0.10
0.29
0.35

As can be seen, for all of the UP and LP stations, the standard error of the estimate (SEE)
values of MnDOT and WSDOT models are, respectively, about 15% and 135% higher than the
SEE of Equation 4.44. Such differences are significant and adversely affect the timely posting
and removing the SLR signs. Such timing could be off by few days to about a week when the
pavement is in critical condition states. Accurate prediction of posting and removing the SLR
signs saves the road owners and the trucking industries unnecessary expenses.
Table 4.15 Performance metrics of Equation 4.44, MnDOT and WSDOT models for the CTDD
calculation for 12 RWIS stations in the LP and 9 RWIS stations in the UP in 2012, 2014, and
2015

Location
Lower
Peninsula
Upper
Peninsula

Model

Standard Error
of the Estimate
(SEE)

Mean Absolute
Error (MAE)

Equation 4.44
MnDOT Model
WSDOT Model
Equation 4.44
MnDOT Model
WSDOT Model

19
22
44
22
26
51

11
13
24
11
13
25

159

Mean Absolute
Percentage
Error (MAPE)
0.17
0.26
0.34
0.23
0.32
0.48

4.5.2 Nixon and McRoberts Equation
After calculating the CTDD, Nixon and McRoberts equation (Equation 2-87) was used to
estimate the depth of thaw at the various RWIS stations in the state of Michigan. Figure 4.50 and
Table 4.16 depict the results. It can be seen from the figure that the results are not satisfactory. In
fact, Equation 2-87 under predicts thaw depth by as much as 76 centimeters in some stations.
The error could be related to the simplifying assumptions made in the equation, the lack of exact
input data, or error in calculating the thaw index.
Since Nixon and McRoberts Equation did not yield accurate thaw depth results, new
empirical models were developed in this study using the RWIS data in the State of Michigan.

Line of equality between the measured and calculated data

Maximum calculated thaw depth (cm)

160
140
120
100
80
60
40
20
0
0

20

40
60
80
100
120
140
Maximum measured thaw depth (cm)

160

Figure 4.50 Maximum thaw depths predicted by Nixon and McRoberts equation versus the
measured maximum thaw depths in Michigan.
160

Table 4.16 Maximum thaw depth predicted by Nixon and McRoberts equation for RWIS stations

Location

Lower
Peninsula

Name of the
Station

Benzonia
Cadillac
Grayling
Houghton
Lake
Reed City
Waters
Au Train
Brevort

Upper
Peninsula

Harvey
Golden Lake
Seney
Twin Lakes

Type of Soil

Year

Maximum
Measured
Thaw Depth
(cm)

Loose Sand
Dense Sand
Dense Sand

2010-2011
2010-2011
2010-2011

53
53
117

26
25
50

Dense Sand

2010-2011

117

62

2010-2011

71

24

2010-2011

53

34

2010-2011

117

48

2010-2011

117

53

2010-2011

117

56

2010-2011

157

61

2010-2011
2010-2011

102
102

58
48

Compacted Sand
Loose Sand
Compacted Sand
Loose Sand
Sand with Gravel
and Silt
Loose Sand
Loose Sand
Sand with Gravel
and Silt
Dense Sand
Dense Sand with
Gravel
Loose Sand
Silty Clayey Sand

Nixon
Maximum
Calculated
(cm)

4.5.3 Thaw Depth Empirical Models
Since Nixon and McRoberts Equation did not yield accurate thaw depth results, new
empirical models were developed in this study using the RWIS data in the State of Michigan and
Minnesota. These new models are presented below.
Among the 25 RWIS stations located in the Upper and Lower Peninsulas of Michigan,
only 12 stations were used for developing the empirical models. Thirteen stations were not
considered due to incomplete data (some of the data are missing). For air temperature data the
161

nearest NOAA station database was used due to higher consistency and accuracy with respect to
the RWIS database. Sites not situated close to NOAA stations are not considered in this study.
Due to data availability, data from the spring of 2011, 2014, and 2015 were used. Unfortunately,
all of the 12 stations contain sandy soils.
First, the air temperature data were used to calculate the CTDD for each RWIS station
location. Second, the measured thaw depth data in all RWIS stations and the calculated CTDD
values were used to develop a statistical prediction equation for sandy soil. This resulted in
Equation 4.45
𝑇 = −0.0028 ∗ (𝐶𝑇𝐷𝐷)2 + 1.6999 ∗ 𝐶𝑇𝐷𝐷

Equation 4.45

Where T = thaw depth (cm); and
CTDD = cumulative thawing degree day (oC – day).
The results of Equation 4.45 are depicted in Figure 4.51. The calculated coefficient of
determination (R2) is 0.85. As expected, the accuracy of the thaw depth model is lower than the
frost depth model. The main reason is that the variability of the measured thaw depth data is
higher than that of the frost depth data. Such variability is a function of the measured solar
radiation, percent sunshine, absorptivity, emissivity, and so forth. Unfortunately, such data are
not available at this time to improve Equation 4.45. Nevertheless, the equation does predict the
thaw depth data relatively accurately.
Figure 4.52 shows the calculated thaw depths using Equation 4.45 versus the measured
ones. The solid straight line in the figure is the locus of equality between the measured and
calculated data. As can be seen, the majority of the calculated thaw depth data are within a few
centimeters from the measured values.

162

220

T = -0.0028CTDD2 + 1.6999CTDD
R² = 0.8519
n= 143

200

Thaw depth (cm)

180
160
140
120
100
80
60

40
20
0
0

20

40
60
80
100
120
o
Cumulative thawing degree day ( C-day)

140

160

Figure 4.51 Measured thaw depths versus cumulative thawing degree-day for sandy soils in the
State of Michigan

Calculated thaw depth (cm)

Line of equality between the measured and calculated data
Sandy soil-Michigan
200
180
160
140
120
100
80
60
40
20
0

n= 143

0

20

40

60
80 100 120 140
Measured thaw depth (cm)

160

180

200

Figure 4.52 Measured thaw depth in sandy soils in the state of Michigan versus the calculated
values using Equation 4.45

163

As stated in Chapter 3, thaw depth data measured from 2003 to 2012 in 8 stations in the
State of Minnesota were requested and received from MnDOT. Equation 4.45 was then used to
calculate the thaw depth data in sandy stations and for the ten-year period. The measured thaw
depth data and the calculated ones are depicted in Figure 4.53 . The straight line in the figure
indicates the line of equality between the measured and the calculated values.

Calculated thaw depth (cm)

Line of equality between the measured and calculated data
Sandy soil- Minnesota

220
200
180
160
140
120
100
80
60
40
20
0

n= 124

0

20

40

60

80 100 120 140 160 180 200 220
Measured thaw depth (cm)

Figure 4.53 Measured thaw depths in sandy soil in the state of Minnesota versus the thaw depth
values calculated using Equation 4.45.
It should be noted that for sandy soil (Figure 4.53) the number of measured data points is
130. The calculated coefficient of determination (R2) for Minnesota data is 0.82 which is slightly
lower than R2 for Michigan data. The relatively high values of R2 indicate that Equation 4.45
predict the thaw depth data in sandy soils in the states of Michigan and Minnesota relatively
accurately. In order to further evaluate the performance of Equation 4.45, various performance
metrics for both states are shown in Table 4.17.
164

Table 4.17 Performance metrics of Equation 4.45 for thaw depth estimations in the State of
Michigan and Minnesota

Location

Standard Error
of the Estimate
(SEE)

Mean Absolute
Error (MAE)

Michigan

20.3

16.0

Mean Absolute
Percentage
Error (MAPE)
0.30

Minnesota
0.29
20.6
16.3
In this table, the standard error of the estimate (SEE), the mean absolute error (MAE) and
the mean absolute percentage error (MAPE) are shown for each model. The SEE is proportional
to the width of the confidence interval so smaller SEE is an indication of a better fit. As can be
seen in, Table 4.17 the model performance is almost identical in both states.
To further evaluate the validity of Equation 4.45, a statistical model was developed using
the measured thaw depth and the calculated CTDD for sandy soils in Minnesota. The results are

Thaw depth (cm)

shown in Figure 4.54.
Statistical polynomial function
Poposed model (Eq.4-45)

220
200
180
160
140
120
100
80
60
40
20
0

n=124

0

20

40 60 80 100 120 140 160 180 200 220
Cumulative thawing degree days (oC-day)

Figure 4.54 thaw depths versus cumulative thawing degree-day for sandy soil showing the best
fit and proposed model (Equation 4.45) in the State of Minnesota.

165

The dashed and solid curves in Figure 4.54 represent the statistical model and Equation
4.45, respectively. Examination of Figure 4.54 indicates that up to 150 centimeters of thaw
depth, Equation 4.45 and the statistical model produce almost the same results for sandy soils in
Minnesota. After 150 centimeters, the differences between the results of the two models are less
than 20 centimeters. In any events, the number of measured data points for more than 150centimeter thaw depth is very much limited. Nevertheless, the results could be interpreted as
Equation 4.45 could be used in Minnesota without calibration.
As mentioned before, there are no data available for clayey soils in Michigan. However,
since all of the developed frost and thaw depths model indicate that a statistical model developed
based on the data in one state can be used in another state without calibration, data from
Minnesota were used to develop a statistical model for clayey soils. As mentioned in Chapter 3,
data from 5 stations were available in ten-year period. Therefore, the data from four of those
stations (Ada, Marshal, Starbucks, and Rochester) were used for developing the model and data
from Gatzke were used for validation. The statistical analysis is resulted in Equation 4.46
𝑇 = 8 ∗ 10−6 ∗ (𝐶𝑇𝐷𝐷)3 − 0.0051 ∗ (𝐶𝑇𝐷𝐷)2 + 1.15 ∗ 𝐶𝑇𝐷𝐷

Equation 4.46

Where all other parameters are the same as before.
The results of Equation 4.46 are depicted in Figure 4.55. The calculated coefficient of
determination (R2) is 0.85. Figure 4.56 shows the calculated thaw depths using Equation 4.46
versus the measured ones. The solid straight line in the figure is the locus of equality between the
measured and calculated data. As can be seen, the majority of the calculated thaw depth data are
within a few centimeters from the measured values.

166

Clayey soil
180
T = 8E-06CTDD3 - 0.0051CTDD2 + 1.15CTDD
R² = 0.8566
n= 288

160

Thaw depth

140
120

100
80
60
40
20
0
0

50

100
150
200
250
300
o
Cumulative thawing degree day ( C-day)

350

400

Figure 4.55 Thaw depths versus cumulative thawing degree-day for clayey soils in the state of
Minnesota.
Line of equality between the measured and calculated data
Clayey soil-Minnesota

Calculated thaw depth (cm)

160
140

n= 288

120
100
80
60

40
20
0
0

20

40

60
80
100
120
Measured thaw depth (cm)

140

160

Figure 4.56 Measured thaw depths in clayey soil in the state of Minnesota (Ada, Marshal,
Starbucks, and Rochester) versus the thaw depth values calculated using Equation 4.46.

167

As mentioned before, to evaluate the accuracy of Equation 4.46, this equation was used
to predict the measured thaw depths in Gatzke stations in the State of Minnesota. Figure 4.57
depicts the results. It should be noted that the number of measured data points in this figure is 61.
Examinations of Figure 4.57 indicates that the prediction of thaw depth data in Gatzke
station using Equation 4.46 is even more accurate than in the other four stations. In fact, for the
calculated coefficient of determination (R2) is 0.86.

Line of equality between the measured and calculated data
Clayey soil-Minnesota

Calculated thaw depth (cm)

160
140

n= 60

120
100
80
60
40

20
0
0

20

40

60
80
100
120
Measured thaw depth (cm)

140

160

Figure 4.57 Measured thaw depths in clayey soil in Gatzke station versus the thaw depth values
calculated using Equation 4.46.
In order to further evaluate the performance of Equation 4.46, performance metrics are
shown in Table 4.18 for both databases. As expected, performance of Equation 4.46 indicates
that the equation predicts thaw depth for both database telatively accurately.

168

Spring Load Restrictions
As stated before, the accuracy of the SLR posting and removing times is critical in
avoiding pavement damage. Even few days could lead to substantial damages. Most studies so
far, used thawing index critical threshold to post the SLR and another critical threshold or a fixed
period of 8-weeks for removing the SLR (See Chapter 2, Section 2.3.2). However, since the
severity of winter varies substantially from one year to the next, using thaw depth threshold
instead of thawing index threshold or a fixed period for posting and removing SLR could lead to
more accurate SLR policy.
Table 4.18 Performance metrics of Equation 4.46 for thaw depth estimations in the State of
Minnesota

Location
Minnesota (Ada, Marshal,
Starbucks, and Rochester)
Minnesota (Gatzke)

Standard Error
of the Estimate
(SEE)

Mean
Absolute
Error (MAE)

Mean Absolute
Percentage Error
(MAPE)

5.4

4.0

0.25

2.7

4.7

0.22

The Minnesota Department of Transportation (MnDOT) study showed that pavement is
in the most critical state when thaw is between 15 to 30 centimeters (Ovik et.al, 2000). Their
study on the rate of strength recovery and back calculation of resilient moduli also showed that at
two weeks past the end of thaw the pavement strength recovers between 50 to 100 percent
depends on the soil type and the increase in the fine contents in the base material led to longer
recovery period (Ovik et.al, 2000).
Therefore, in this study, it is recommended to post the SLR when thaw depth is at 15
centimeters and remove the SLR two weeks after the thaw completion in sandy soils and three

169

weeks after the thaw completion in clayey soils. It should be noted that these recommendations
are based on literature and there was no data available to verify them.
As stated before, the accuracy of the SLR posting and removing times is critical in
avoiding pavement damage. On the other hand, early posting and late removal of SLR increases
the trucking industry cost. Therefore, it is important to evaluate the accuracy of the thaw depth
model relative to time. In other word, it is of interest to investigate the time gap between the
calculated thaw depth and the measured ones.
Therefore, since data in Michigan were used to develop the thaw depth model for sandy
soil, data from Minnesota were used to investigate the time gap. Also, for the same reason data
from Gtzake station were used to evaluate the accuracy of the thaw depth model relative to time
in clayey soils. Table 4.19 shows the average time gap between the calculated and measured
thaw depths in these stations.
Table 4.19 Time gap between the measured and calculated thaw depth for sandy and
clayey soil in the state of Minnesota

Location

Minnesota Sandy Soils
Minnesota Clayey Soils
(Gatzke)

Average
Absolute Time
gap for 6” thaw
depth (day)
Âą2

Average Absolute
Time gap for
thaw completion
(day)
Âą3.8

Average
Absolute Time
gap for all thaw
depth (day)
Âą3

Âą3.6

Âą8.5

Âą8

As can be seen in Table 4.19, while the thaw depth model (Equation 4.46) predictions for
clayey soils were more accurate than thaw depth model (Equation 4.45) for sandy soils, Equation
4.45 predictions lead to much lower time gap between the measured and calculated thaw depth.
However, both models predict the beginning of the thaw with less than 4-day time gap.

170

Therefore, the accuracy of Equation 4.46 in predicting the removal of SLR should be further
evaluated as more data become available.

171

CHAPTER 5
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
5.1

Summary

5.1.1 Frost Depth Modeling
Frost depth is an important factor that affects the design of all infrastructures including
pavements, retaining structures, building and bridge foundations and/or utility lines. Hence,
accurate estimation of frost depth data plays significant roles in estimating heave pressure and
heave of various structures including pavements. In this study, existing frost depth prediction
models were scrutinized. These include the finite element based model (UNSAT-H) and various
semi-empirical models (Stefan, Modified Berggren and Chisholm and Phang). Unfortunately,
none of the model predicted the measured frost depth with and reasonable degree of accuracy.
Further, some of the required input data to the models are not available and/or expensive to
obtain by State Highway Agencies. Therefore, during the study, new statistical models were
developed based on available and easily measured data to predict frost depths.
First, the measured frost depth data in the State of Michigan were divided into two groups
according to the soil types; clayey and sandy soils. For each soil type, a statistical model was
developed relating the frost depth to the calculated cumulative freezing degree day (CFDD). The
two statistical models were then validated using the measured frost depth data in the State of
Minnesota. Both models produced reasonable estimates of the measured frost depth data. The
two statistical models were then combined into one statistical model based on the average
laboratory measured thermal conductivity of saturated clayey and sandy soil samples obtained
from MDOT. The accuracy of the combined statistical model was then assessed using the frost

172

depth data measured in the states of Michigan and Minnesota. The calculated frost depth data
were reasonable and closely represent the measured frost depth data in both states.
5.2

Frost Heave Model
As soils freeze, water migrates through the soil voids below the freezing zone toward the

freezing front, coats existing ice lenses causing them to grow and producing excessive frost
heave. Frost have can be mitigated by removing and replacing the frost susceptible soil by
drainable materials, stopping water flow by intercepting its path using drainage lines, cutting off
the source of water, and/or reducing the frost depth by installing insulation. In this study, frost
mitigation semi-empirical model was developed based on heat balance in the soil layer and on
the newly developed statistical frost depth model. The frost mitigation model estimates the
required insulation thickness to reduce or eliminate frost depth.
In addition, the Gilpin’s mechanistic- empirical model was used to predict frost heave.
The model yielded unreasonable results and did not simulate the frost depth data measured by
MDOT. Consequently, the model was modified by replacing the heat balance equation for
calculating the frost depths by the newly developed statistical frost depth model. The modified
Gilpin’s model yielded relatively accurate results that represent the frost heave data measured at
5 sites in Oakland County, Michigan. Lastly, results of the frost heave model was used and heave
pressure models were developed to estimate heave pressure for four soil types.
5.2.1 Thaw Depth Model
During the thawing period, pavement structures, in general, are in critical conditions. The
melted water at the top of the frozen area saturates the top part of the upper pavement layer
causing substantial decreases in their stiffness and their load bearing capacity leading to
premature and localized failure. In general, spring thaw damages are observed along city streets,

173

county roads and some state roads. These roads are designed and constructed on relatively low
permeability roadbed soil and have relatively thin base and/or subbase layers. Stated differently,
because of limited budget, the roadbed soils of these roads are not properly protected from frost
heave. To decrease the spring thaw damage, Spring Load Restrictions (SLR) signs are usually
placed along the roads. The problem stems from the fact that the time for placing and removing
the SLR significantly increases the cost of pavement preservation and user costs especially the
trucking industry. Hence, accurate prediction of the dates of posting and removing the SLR
becomes very important. In this study, the Nixon and McRoberts thaw depth prediction model
was evaluated relative to the thaw depth data measured by MDOT. The results were not
satisfactory. Therefore, the cumulative thaw degree-day (CTDD) were calculated using the
pavement surface and air temperatures data collected by MDOT and accurate thaw depth
prediction models were developed for clayey and for sandy soils. The two models were then
verified using the calculated CTDD values and thaw depth data collected by MnDOT. Finally,
based on the results of the thaw depth prediction models a new policy for posting and removing
SLR signs was proposed.
5.3

Conclusions
Based on the results of the analyses, the following conclusions were drawn:

1. The UNSAT-H numerical finite element model for frost depth prediction requires various
meteorological and soil properties data that were not available and/or expensive to collect.
2. Existing mechanistic and empirical-mechanistic models for predicting frost depths do not
accurately predict the measured frost depth data in the States of Michigan. The models
assume that volumetric heat capacity and water movement can be neglected. Existing
statistical models are not reliable and require substantial calibration for each region.

174

3.

The newly developed two statistical models for clayey and sandy soils predicted the
measured frost depth data in Minnesota relatively accurately.

4. The single statistical model developed based on the average thermal conductivity of saturated
clayey and sandy soils produced accurate results for both soils in the states of Michigan and
Minnesota.
5.

The Gilpin frost heave model was modified. The modified model yielded frost heave data
that are representative to the measured data under the shoulder and under the pavement in
Michigan.

6.

Heave pressure model was developed based on the result of the frost heave model. However,
since heave pressure data were not available, the accuracy of the model was not evaluated.

7. A new model for estimating the cumulative thawing degree day (CTDD) was developed.
Based on the measured pavement surface and air temperatures in Michigan.
8. The existing mechanistic model for predicting thaw depths did not produce acceptable
results.
9. Two statistical models were developed; one for clayey and one for sandy soils using the
measured thaw depth data in Michigan and Minnesota. The two models predicted the
measured frost depth data in Minnesota relatively accurately.
10. The two models for predicting thaw depths were used to develop a new SLR policy, the
results showed that for clayey soil the model leads to average time gap of Âą3 days between
the measured and calculated thaw depth and for sandy soils the models leads to average time
gap of Âą8 days.

175

5.4

Recommendation
Based on the results of this study and the enumerated conclusions, the following

recommendations were made:

1. Undisturbed soil samples be collected from various soil types in Michigan and their thermal
conductivity be measured. The resulting data be used to improve the accuracy of the
statistical frost and thaw depth models developed in this study.
2.

The developed frost model be implemented to calculate frost depth data in those areas where
no temperature sensors are installed with depth.

3. The Mechanistic-Empirical Pavement Design Guide (MEPDG) or FWD data be used to
estimate the range and variability of the resilient modulus of the roadbed soil and determine
the sensitivity of the newly proposed SLR policy.

176

APPENDICES

177

APPENDIX A

Frost and Thaw Depth Analysis

178

This appendix houses the details of frost depth calculations for the winter of 2010-2011 using Stefani
equation and Modified Berggren equation for all RWIS stations in Michigan. Also provides the details of
thaw depth calculations using Nixon and McRoberts equation for all of the stations. It should be noted that
LP stands for Lower Peninsula and UP stands for Upper Peninsula.

179

Table A.1 Frost depth calculation using Stefan equation, Benzonia, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08 0.3
2.51
50288 11
12
Sand
20
0.09 1.8
2.34
57924 433 444
3
Îłd = unit weight (Kn/m ); w = water content (%); d = layer depth
(m); k= thermal conductivity (W/(m.o C); L= latent heat of fusion
(kJ/3m); FI=freezing Index (oC-day)

Table A.2 Frost depth calculation using Modified Berggren equation, Benzonia, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
894
1.49
14788
0
894
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
35499
1043
0.17
0.56 0.4
91.9
Sand
20
0.09
0.6
1565
2.42
57924
48351
1341
0.16
0.58 1.6
418.8
3
o
d = layer depth (m); C= volumetric heat capacity(kJ/ m); k= thermal conductivity (W/(m. C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
82
457

0
82
457

Îą=v0/vs

2.9

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

9.1

vs= n(CFI)/t

3.1

FI

445

Table A.3 Thaw depth calculation using Nixon and McRoberts equation, Benzonia, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20
0.08 0.3
o
FI=freezing Index ( C-day)

k
1.44
2.51

180

L
0
35201

TI
0
6

∑TI
0
7

Table A.4 Frost depth calculation using Stefan equation, Cadillac, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08 0.3
2.51 50288 11
11
Sand
20
0.09 2.3
2.34 57924 726 737
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer
depth (m); k= thermal conductivity (W/(m.o C); L= latent heat
of fusion (kJ/3m); FI=freezing Index (oC-day)

Table A.5 Frost depth calculation using Modified Berggren equation, Cadillac, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
35499
1229
0.17
0.56 0.4
66.3
Sand
19
0.09
1.1
1565
2.42
57924
51591
1453
0.16
0.58 1.6
746.3
3
o
d = layer depth (m); C= volumetric heat capacity(kJ/ m); k= thermal conductivity (W/(m. C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
59
722

0
59
722

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

6.3

vs= n(CFI)/t

4.7

FI

729

Table A.6 Thaw depth calculation using Nixon and McRoberts equation, Cadillac, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20
0.08 0.3
TI=Thawing Index (oC-day)

181

k
1.44
2.51

L
0
35201

TI
0
6

∑TI
0
6

Îą=v0/vs

1.35

Table A.7 Frost depth calculation using Stefan equation, Grayling, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 55.64
0
0
0
Gravel
20
0.08 0.3 189.00 668
10
11
Sand
19
0.09 2.1 217.73 623
633 644
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer
depth (m); k= thermal conductivity (W/(m.o C); L= latent heat
of fusion (kJ/3m); FI=freezing Index (oC-day)

Table A.8 Frost depth calculation using Modified Berggren equation, Grayling, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
14805
0
1490
0.00
0.00 0.3
7
Gravel
20
0.08
0.3
1118
2.60
50288
39851
1227
0.17
0.56 0.4
49.8
Sand
19
0.09
1.2
1565
2.42
57931
53114
1475
0.16
0.58 1.7
677.7
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
50
653

0
50
653

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

4.5

vs= n(CFI)/t

5.2

FI

637

Table A.9 Thaw depth calculation using Nixon and McRoberts equation, Grayling, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
19 1.08 0.1
TI=Thawing Index (oC-day)

182

k
1.44
2.51
2.34

L
0
35201
40565

TI
0
6
20

∑TI
0
6
29

Îą=v0/vs

0.86

Table A.10 Frost depth calculation using Stefan equation, Houghton Lake, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08 0.3
2.51 50288 11
12
Loose
Sand
20
0.09 2.1
2.34 57924 618 629
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer
depth (m); k= thermal conductivity (W/(m.o C); L= latent heat
of fusion (kJ/3m); FI=freezing Index (oC-day)

Table A.11 Frost depth calculation using Modified Berggren equation, Houghton Lake, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
39858
1229
0.17
0.56 0.4
50.0
Sand
20
0.09
1.2
1565
2.42
57924
53007
1453
0.16
0.58 1.7
634.4
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
52
616

0
52
616

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

4.5

vs= n(CFI)/t

5.2

FI

629

Table A.12 Thaw depth calculation using Nixon and McRoberts equation, Houghton Lake, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Loose
Sand
20 1.08 0.1
TI=Thawing Index (oC-day)

183

k
1.44
2.51

L
0
35201

TI
0
18

∑TI
0
19

2.34

40565

35

53

Îą=v0/vs

0.84

Table A.13 Frost depth calculation using Stefan equation, Ludington, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 1.44
0
0
0
Gravel
20
0.08
0.3 2.51 50288 11
12
Sand with
Clay
18
0.09
1.2 2.34 92529 344
356
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth
(m); k= thermal conductivity (W/(m.o C); L= latent heat of fusion
(kJ/3m); FI=freezing Index (oC-day)

Table A.14 Frost depth calculation using Modified Berggren equation, Ludington, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
39858
1229
0.17
0.56 0.4
58.1
Sand
18
0.09
0.5
1565
2.42
92529
68950
1416
0.16
0.58 0.8
357.5
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
62
369

0
62
369

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

5.1

184

vs= n(CFI)/t

3.1

FI

361

Îą=v0/vs

1.63

Table A.15 Frost depth calculation using Stefan equation, Reed City, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 1.44
0
0
0
Gravel
20
0.08
0.3 2.51 52299 11
11
Sand
20
0.09
1.9 2.34 60345 526
536
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth
(m); k= thermal conductivity (W/(m.o C); L= latent heat of fusion
(kJ/3m); FI=freezing Index (oC-day)

Table A.16 Frost depth calculation using Modified Berggren equation, Reed City, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
52299
41273
1229
0.17
0.56 0.4
53.1
Sand
20
0.09
1.0
1565
2.42
60345
54497
1453
0.16
0.58 1.4
550.6
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
56
545

0
56
545

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

4.7

vs= n(CFI)/t

4.6

FI

533

Table A.17 Thaw depth calculation using Nixon and McRoberts equation, Reed City, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
TI=Thawing Index (oC-day)

185

k
1.44
2.51

L
0
35201

TI
0
16

∑TI
0
17

Îą=v0/vs

1.03

Table A.18 Frost depth calculation using Stefan equation, Waters, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 50288 11
11
Sand
20
0.09
0.6
2.51 50288 42
52
Loose Sand
with Gravel
19
1.09
2.1
2.34 57924 603
655
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.19 Frost depth calculation using Modified Berggren equation, Waters, LP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
39858
45967

Ĉ=
∑Cd/∑d
1490
1229
1155

Îź=
Ĉ / Ě *vs
0.00
0.17
0.16

Îť

R

∑R+R/2

FI

∑FI

0
51
233

0
51
233

648

648

Îą=v0/vs

0.77

HMA
22
0
0
1490 1.49
0
0.00 0.3
0
Gravel
20 0.08 0.3 1118 2.60 50288
0.56 0.4
48.8
Sand
20 0.09 0.6 1118 2.60 50288
0.58 0.8
205.0
Loose Sand with
19 1.09 0.7 1565 2.42 57924
50660
1304
0.16
0.58 1.0
466.9
Gravel
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

4.1

5.3

vs= n(CFI)/t

FI

656

Table A.20 Thaw depth calculation using Nixon and McRoberts equation, Waters, LP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
20 1.08 0.1
TI=Thawing Index (oC-day)

186

k
1.44
2.51
2.51

L
0
36617
35201

TI
0
14
6

∑TI
0
14
19

Table A.21 Frost depth calculation using Stefan equation, Williamsburg, LP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 50288
10
11
Sand
20
0.09
0.3
2.34 57924
10
21
Silty Clay
18
1.09
1.1
1.83 101767 388
409
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.22 Frost depth calculation using Modified Berggren equation, Williamsburg, LP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20 0.08
0.4
1118
2.60
50288
35499
1229
0.17
0.56 0.4
55.6
Sand
20 0.09
0.7
1453
2.42
57924
44216
1341
0.16
0.58 0.4
100.6
Silty Clay 18 1.09
1.1
1565
1.90
101767
64890
1527
0.16
0.58 0.7
286.3
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
49
139
393

0
49
139
393

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

5.1

187

vs= n(CFI)/t

3.4

FI

408

Îą=v0/vs

1.45

Table A.23 Frost depth calculation using Stefan equation, Au Train, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 52299
11
11
Sand
20
0.09
0.5
2.34 60345
34
46
Loose Sand
19
1.09
2.7
2.34 57924 1027
1072
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.24 Frost depth calculation using Modified Berggren equation, Au Train, UP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
36915
49356

Ĉ=
∑Cd/∑d
1490
1229
1416

Îź=
Ĉ / Ě *vs
0.00
0.17
0.16

Îť

R

∑R+R/2

HMA
22
0
0
1490
1.49
0
0.00 0.3
0
Gravel
20 0.08
0.3
1118
2.60
52299
0.56 0.4
49.4
Sand
20 0.09
0.5
1565
2.42
60345
0.58 0.7
186.3
Loose
19 1.09
1.2
1565
2.42
57924
54236
1490
0.16
0.58 1.8
966.3
Sand
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;

FI

∑FI

0
44
209

0
44
209

1068

1068

Îą=v0/vs

0.44

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.8

vs= n(CFI)/t

8.8

FI

1072

Table A.25 Thaw depth calculation using Nixon and McRoberts equation, Au Train, UP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
19 1.08 0.1
TI=Thawing Index (oC-day)

188

k
1.44
2.51
2.34

L
0
36617
42242

TI
0
15
3

∑TI
0
14
18

Table A.26 Frost depth calculation using Stefan equation, Brevort, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 52299
11
11
Sand
20
0.09
2.2
2.34 60345
707
718
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.27 Frost depth calculation using Modified Berggren equation, Brevort, UP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
52299
36915
1229
0.17
0.56 0.4
41.3
Sand
20
0.09
1.3
1565
2.42
60345
54534
1490
0.16
0.58 1.9
761.9
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
37
714

0
37
714

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.7

vs= n(CFI)/t

5.3

FI

711

Table A.28 Thaw depth calculation using Nixon and McRoberts equation, Brevort, UP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
20 1.08 0.1
TI=Thawing Index (oC-day)

189

k
1.44
2.51
2.34

L
0
35201
40565

TI
0
18
3

∑TI
0
18
21

Îą=v0/vs

0.7

Table A.29Frost depth calculation using Stefan equation, Cooks, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 52299
11
11
Sand
20
0.09
1.9
2.34 60345
525
536
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.30 Frost depth calculation using Modified Berggren equation, Cooks, UP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
35499
1229
0.17
0.56 0.4
46.9
Sand
20
0.09
1.1
1565
2.42
57924
51480
1416
0.16
0.58 1.6
545.6
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
42
526

0
42
526

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.7

190

vs= n(CFI)/t

3.7

FI

533

Îą=v0/vs

1.03

Table A.31 Frost depth calculation using Stefan equation, Engadine, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 52299
11
11
Sand
20
0.09
0.6
2.34 60345
58
69
Silty Clay
18
1.09
1.4
1.83 101767 551
620
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.32 Frost depth calculation using Modified Berggren equation, Engadine, UP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20 0.08
0.4
1118
2.60
52299
36915
1229
0.17
0.56 0.4
43.1
Sand
20 0.09
0.7
1453
2.42
60345
50623
1341
0.16
0.58 0.9
228.8
Silty Clay 18 1.09
1.1
1565
1.90
101767
70105
1490
0.16
0.58 1.0
415.0
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
38
242
611

0
38
242
611

Îą=v0/vs

0.66

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.8

191

vs= n(CFI)/t

5.7

FI

617

Table A.33 Frost depth calculation using Stefan equation, Golden Lake, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 50288
10
11
Sand
20
0.09
2.7
2.34 57924
989
999
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.34 Frost depth calculation using Modified Berggren equation, Golden Lake, UP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
50288
35499
1229
0.17
0.56 0.4
43.8
Sand
20
0.09
1.8
1453
2.42
57924
53454
1490
0.16
0.58 2.5
1068.1
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
39
988

0
39
988

Îą=v0/vs

0.39

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

2.9

vs= n(CFI)/t

7.4

FI

1002

Table A.35 Thaw depth calculation using Nixon and McRoberts equation, Golden Lake, UP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
20 1.08 0.1
TI=Thawing Index (oC-day)

192

k
1.44
2.51
2.34

L
0
36617
42242

TI
0
19
3

∑TI
0
19
22

Table A.36 Frost depth calculation using Stefan equation, Harvey, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 55.64
0
0
0
Gravel
20
0.08 0.3 196.56
668
11
12
Sand with Gravel and Silt
20
0.09 0.3 226.80
623
18
29
Sand
20
1.09 2.0 189.00
668
453
483
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m); k= thermal
conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m); FI=freezing Index (oCday)

Table A.37 Frost depth calculation using Modified Berggren equation, Harvey, UP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
36915

Ĉ=
∑Cd/∑d
1490
1229

Îź=
Ĉ / Ě *vs
0.00
0.17

Îť

R

∑R+R/2

HMA
22
0
0
1490
1.49
0
0.00 0.3
0
Gravel
20 0.08
0.3
1118
2.60
52299
0.56 0.4
45.6
Sand with
Silt
19 0.09
0.5
1453
2.42
57924
48053
1416
0.16
0.58 0.7
166.3
Sand
20 1.09
0.6
1118
2.60
50288
48984
1304
0.16
0.58 0.9
320.6
3
o
d = layer depth (m); C= volumetric heat capacity(kJ/ m); k= thermal conductivity (W/(m. C); Îź= fusion parameter;

FI

∑FI

0
41

0
41

188
473

188
473

Îą=v0/vs

0.74

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

2.9

4.0

vs= n(CFI)/t

FI

480

Table A.38 Thaw depth calculation using Nixon and McRoberts equation, Harvey, UP
Material
Îłd
w
HMA
22
0
Gravel
20
0.08
Sand with Silt
19
1.08
o
TI=Thawing Index ( C-day)

d
0.1
0.3
0.1

193

k
1.44
2.51
2.34

L
0
40565
35201

TI
0
7
3

∑TI
0
7
10

Table A.39 Frost depth calculation using Stefan equation, Michigamme, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 1.44
0
0
0
Gravel
20
0.08
0.3 2.51 52299
11
11
Sand
20
0.09
0.6 2.34 60345
54
64
Clayey Sand
19
1.09
2.3 1.83 57924
904
969
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m); k=
thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.40 Frost depth calculation using Modified Berggren equation, Michigamme, UP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
36915
50623

Ĉ=
∑Cd/∑d
1490
1229
1416

Îź=
Ĉ / Ě *vs
0.00
0.17
0.16

Îť

R

∑R+R/2

HMA
22
0
0
1490
1.49
0
0.00 0.3
0
Gravel
20 0.08
0.3
1118
2.60
52299
0.56 0.4
45.6
Sand
20 0.09
0.6
1453
2.42
60345
0.58 0.9
240.0
Clayey
19 1.09
1.3
1118
1.90
57924
54646
1267
0.16
0.58 2.4
783.8
Sand
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;

FI

∑FI

0
41
254

0
41
254

951

951

Îą=v0/vs

0.5

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.2

194

vs= n(CFI)/t

6.5

FI

964

Table A.41 Frost depth calculation using Stefan equation, Seney, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1
1.44
0
0
0
Gravel
20
0.08
0.3
2.51 52299
11
11
Sand
20
0.09
2.1
2.34 60345
624
619
3
Îłd = unit weight (kN/m ); w = water content (%); d = layer depth (m);
k= thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.42 Frost depth calculation using Modified Berggren equation, Seney, UP
Äš=
Ĉ=
Îź=
Îť
R
∑R+R/2
∑Ld/∑d ∑Cd/∑d Ĉ / Ĺ *vs
HMA
22
0
0
1490
1.49
0
0
1490
0.00
0.00 0.3
0
Gravel
20
0.08
0.3
1118
2.60
52299
36915
1229
0.17
0.56 0.4
51.9
Sand
20
0.09
1.2
1565
2.42
60345
53975
1490
0.16
0.58 1.7
658.1
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;
Material

Îłd

w

d

C

k

L

FI

∑FI

0
46
631

0
46
631

Îą=v0/vs

0.78

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

4.1

vs= n(CFI)/t

5.2

FI

618

Table A.43 Thaw depth calculation using Nixon and McRoberts equation, Seney, UP
Material
Îłd
w
d
HMA
22
0
0.1
Gravel
20 0.08 0.3
Sand
20 1.08 0.2
TI=Thawing Index (oC-day)

195

k
1.44
2.51
2.34

L
0
36617
42242

TI
0
9
11

∑TI
0
9
20

Table A.44 Frost depth calculation using Stefan equation, St. Ignace, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 1.44
0
0
0
Gravel
20
0.08 0.3 2.51 52299
11
11
Sand
20
0.09 0.6 2.34 60345
54
64
Silty Clayey Sand
18
1.09 1.3 1.83 101767
542
607
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m); k=
thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.45 Frost depth calculation using Modified Berggren equation, St. Ignace, UP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
41273
52448

Ĉ=
∑Cd/∑d
1490
1229
1341

Îź=
Ĉ / Ě *vs
0.00
0.17
0.16

Îť

R

∑R+R/2

HMA
22
0
0
1490 1.49
0
0.00 0.3
0
Gravel
20 0.08
0.3
1118 2.60 52299
0.56 0.4
41.3
Sand
20 0.09
0.6
1453 2.42 60345
0.58 0.9
213.1
Silty Clayey
10176
18 1.09
0.6
1788 1.90
71222
1527
0.16
0.58 1.2
406.3
Sand
7
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;

FI

∑FI

0
37
226

0
37
226

587

587

Îą=v0/vs

0.4

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

2.2

196

vs= n(CFI)/t

5.2

FI

600

Table A.46 Frost depth calculation using Stefan equation, Twin Lakes, UP
Material
Îłd
w
d
k
L
FI
∑FI
HMA
22
0
0.1 1.44
0
0
0
Gravel
20
0.08
0.3 2.51 52299
11
11
Sand
20
0.09
0.6 2.34 60345
54
64
Silty Clayey Sand
18
1.09
1.6 1.83 101767
796
861
Îłd = unit weight (kN/m3); w = water content (%); d = layer depth (m); k=
thermal conductivity (W/(m.o C); L= latent heat of fusion (kJ/3m);
FI=freezing Index (oC-day)

Table A.47 Frost depth calculation using Modified Berggren equation, Twin Lakes, UP
Material

Îłd

w

d

C

k

L

Äš=
∑Ld/∑d
0
36915
50623

Ĉ=
∑Cd/∑d
1490
1229
1341

Îź=
Ĉ / Ě *vs
0.00
0.17
0.16

Îť

R

∑R+R/2

HMA
22
0
0
1490
1.49
0
0.00 0.3
0
Gravel
20 0.08
0.3
1118
2.60
52299
0.56 0.4
41.3
Sand
20 0.09
0.6
1453
2.42
60345
0.58 0.9
218.1
Silty Clayey
18 1.09
0.9
1788
1.90 101767
74128
1565
0.16
0.58 1.7
688.8
Sand
d = layer depth (m); C= volumetric heat capacity(kJ/3m); k= thermal conductivity (W/(m.o C); Îź= fusion parameter;

FI

∑FI

0
37
231

0
37
231

843

843

Îą=v0/vs

0.5

Îť= correction factor; R= thermal diffusivity (m2.oC/W); FI=freezing Index (oC-day)
v0= Annual Temp average-0

3.1

6.3

vs= n(CFI)/t

FI

881

Table A.48 Thaw depth calculation using Nixon and McRoberts equation, Twin Lakes, UP
Material
Îłd
w
HMA
22
0
Gravel
20
0.08
Sand
20
1.08
o
TI=Thawing Index ( C-day)

d
0.1
0.3
0.1

197

k
1.44
2.51
2.34

L
0
35201
42242

TI
0
9
3

∑TI
0
9
12

APPENDIX B

Frost Heave Stations Profile

198

This appendix houses the details pavement profile of frost heave stations in Michigan. In the
figures both measured frost heave and frost depth were shown for each station (Novak, 1968).

199

1963

1964

Heave Rate at Pavement
and Shoulder Edge, cm

1.3
2.5
1.3
2.5

Frozen Zone Below Pavement and Shoulder, cm

0
13
25
38
51
64
76
89

PAVEMENT UNDERCUT 30 CM FOR FROST
PROTECTION

102

'

Figure B.1 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 528+88.

200

1963

1964

Heave Rate at Pavement
and Shoulder Edge, cm

1.3
2.5
1.3
2.5

Frozen Zone Below Pavement and Shoulder, cm

0
13
25
38
51
64
76
89
102

Figure B.2 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 652+00.

201

1963

1964

Heave Rate at Pavement
and Shoulder Edge, cm

1.3
2.5
0
1.3
2.5

Frozen Zone Below Pavement and Shoulder, cm

0
13
25
38
51
64
76
89
102

Figure B.3 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 719+00.

202

1964

Heave Rate at Pavement
and Shoulder Edge, cm

1963

1.3
2.5
1.3
2.5

Frozen Zone Below Pavement and Shoulder, cm

0
13
25
38

51
64
76
89
102

Figure B.4 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 724+00.

203

1963

1964

1.3

Heave Rate at Pavement
and Shoulder Edge, cm

2.5
0
1.3
2.5

Frozen Zone Below Pavement and Shoulder, cm

0
13
25
38

51
64
76
89
102

Figure B.5 Frost depth and corresponding frost heave of shoulder and pavement, Sta. 474+00.

204

REFERENCES

205

REFERENCES

1. American Concrete Pavement Association, “Frost-Susceptible Soils”, 2008.
2. Aldrich Jr, P. Harl, and H. M. Paynter, “Analytical Studies of Freezing and Thawing of
Soils”, Technical Report No. 42, Arctic Construction and Frost Effects Laboratory, New
England Division U.S. Army Corps of Engineers, Boston, MA, U.S. Arctic Construction and
Frost Effects Lab Boston Ma, 1953.
3. Andersland, O. B., and D. M Anderson, “Geotechnical Engineering for Cold Regions”,
McGraw-Hill Inc, 1978.
4. ASTM C578-14a., “Standard Specification for Rigid, Cellular Polystyrene Thermal
Insulation”, ASTM International, West Conshohocken, PA, 2014, www.astm.org.
5. Berg, R. L., “Calculating Maximum Frost Depths At Mn/Road: Winters 1993-94, 1994-95
And 1995-96”, No. MN/RC-97/21, 1997.
6. Bianchini, A., and R.G. Carlos., “Pavement-Transportation Computer Assisted Structural
Engineering (PCASE) Implementation of the Modified Berggren (ModBerg) Equation for
Computing the Frost Penetration Depth within Pavement Structures”, Publication No.
ERDC/GSL-TR-12-15. Engineer Research and Development Center Vicksburg Ms
Geotechnical and Structures Lab, 2012.
7. Black, P. B., “Applications of the Clapeyron Equation to Water and Ice in Porous Media”,
No. CRREL-95-6. Cold Regions Research and Engineering Lab Hanover, 1995.
8. Boselly III, S. E., G. S. Doore, J. E. Thornes, C. Ulberg, and D. Ernst, “Road Weather
Information Systems Volume 1”, Research report. Strategic Highway Research Program
Publication-SHRP-H-350, National Research Council, Washington, DC, 1993
9. Boyd, D. W., “Normal Freezing and Thawing Degree-Days from Normal Monthly
Temperatures” Canadian Geotechnical Journal, Volume 13.2, pp.176-180, 1976.
10. Bradley, H., M. A. Ahammed, S. Hilderman, and S. Kass, “Responding to Climate Change
with Rational Approaches for Managing Seasonal Weight Programs in Manitoba.” Cold
Regions Engineering, pp. 391-401, 2012.
11. Chapin, J., B. H. Kjartanson, and J. Pernia, “Comparison of Two Methods for Applying
Spring Load Restrictions on Low Volume Roads”, Canadian Journal of Civil Engineering,
Vol. 39, Issue 6, pp.599-609, 2012.
12. Chisholm, R. A., and W. A. Phang, “Measurement and Prediction of Frost Penetration in
Highway”, Transportation Research Record: Journal of Transportation Research Board, No
205

918, Transportation Research Board of the National Academies, Washington D.C., pp. 1-10,
1983.
13. Crandell, J. H., "Below-Ground Performance of Rigid Polystyrene Foam Insulation: Review
of Effective Thermal Resistivity Values Used in ASCE Standard 32-01—Design and
Construction of Frost-Protected Shallow Foundations”, Journal of Cold Regions Engineering,
Volume 24.2, pp. 35-53, 2009.
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