# Calculation of Frost Depth

Some equations and variables in this section are expressed in English units only due to their source.

## Density (γd or γdry)

Typical densities for commonly encountered dry soils. Ice rich soils can have substantially smaller dry densities.

 Soil γd (lb/ft3) Gravel and sand 120 – 140 Silts and clays 90 – 100 Peat ˜ 20

Normally, dry densities are used in most calculations (gd); however, the total density of a soil (including moisture) can be calculated as follows:

For example, if a gravel has γd = 130 lb/ft3 and w = 5%, then γt = 130(1 + 5/100) = 136.5 lb/ft3

## Moisture Content (w)

Soil moisture content can be calculated as follows:

 Material γd (lb/ft3) Gravel 2 – 10 Sands 5 – 15 Silts 5 – 40 Clays 10 – 50 or more Organic (Peat) > 50

## Thermal Conductivity (k)

The thermal conductivity is the rate of heat flow through a unit area under a thermal gradient (recall that 1 BTU is the energy (heat) required to raise the temperature of 1 lb of water 1 °F):

Units: BTU/hr • ft2 • °F/ft or BTU/hr • ft • °FFurther, ksnow (loose) = 0.06, ksnow (compact) = 0.20.

In the range of water contents (5 to 10%) and dry densities (125-135 lb/ft3) commonly encountered in embankments and pavement base courses, thermal conductivity is very sensitive to moisture content and soil type. Soil thermal conductivities can be obtained from Figures 1 through 3 (from Kersten, 1949[1] and Air Force, 1966[2]):

For example, using Figures 1 through 3:

Base Course (granular)

γd = 135 lb/ft3

w = 5%

kfrozen = 1.8 BTU/hr • ft2 • °F

kunfrozen = 1.65 BTU/hr • ft2 • °F

γd = 100 lb/ft3

w = 15%

kfrozen = 0.8 BTU/hr • ft2 • °F

kunfrozen= 0.72 BTU/hr • ft2 • °F

Figure 1: Average thermal conductivity for silt and clay soils, frozen and unfrozen (redrawn from Kersten, 1949[1] and Air Force, 1966[2]).

Figure 2: Average thermal conductivity for granular soils, frozen and unfrozen (redrawn from Kersten, 1949[1] and Air Force, 1966[2]).

Figure 3: Average thermal conductivity for peat, frozen and unfrozen (redrawn from Kersten, 1949[1] and Air Force, 1966[2]).

The equations used to develop Figures 1 and 2 follow. There are separate equations for frozen (tests conducted at -4 °C) and unfrozen (tests conducted at +4 °C) conditions.

The equations for silt–clay were based on five soils and are valid for moisture contents of seven percent or higher. The equations for granular soils were based on four soils (two sands, a sandy loam, and a gravel) and are valid for moisture contents of one percent or higher. Farouki (1986[3]) noted that Kersten’s equations do not apply to dry soils or to crushed rocks. Use of the above equations to estimate k is more accurate than use of Figures 1 and 2.

## Volumetric Specific Heat (C)

The volumetric specific heat expresses the change in thermal energy in a unit volume of soil per unit change in temperature (units are in BTU/ft3 • °F).

Volumetric heat is derived from specific heat. (Recall: specific heat is the change in thermal energy per unit weight per unit change in temperature. If objects of the same weight but of different materials receive the same amount of energy (heat), they will come to equilibrium at different temperatures. Or, stated another way, how much each object’s temperature changes depends on the specific heat of the material, if the mass and energy inputs are identical).

## Latent Heat (L)

All objects have energy (heat). A portion of this thermal energy (stored heat) is released when the object cools. The sketch below represents a volume of soil with some moisture as it freezes:

As water freezes, thermal energy equal to L is released while the temperature of the soil remains nearly constant. Thus, the latent heat is the energy required to transform 1 lb of a pure substance from one phase to another at constant temperature. Further, 1 lb. of water gives off 144 BTU as it freezes. The latent heat of a soil can be represented by:

An embankment material with w = 5% and gd = 140 lb/ft3,

## Freezing and Thawing Indexes

Depth of freezing and thawing depends in part on the magnitude and duration of the temperature differential below or above freezing (32 °F) at the ground surface. The freezing or thawing index is therefore given by the summation of the degree-days for a freezing or thawing season.

### Example FI/TI Calculations

 Day Maximum Minimum Average Degree Days per Day Cumulative Degree Days 1 29 1 15 -17 -17 2 9 -11 -1 -33 -50 3 10 -8 1 -31 -81 4 15 -1 7 -25 -106 5 30 16 23 -9 -115 6 38 30 34 +2 -113 7 30 18 24 -8 -121

Notes:

1. Assume Day 1 start of freezing season. The negative sign in this case indicates freezing degree-days (normally omitted).
2. For the purpose of assessing spring load restrictions, use 29 °F in lieu of 32 °F. This accounts for the “dark” bituminous surface

## Air and Surface Indexes

Normally, data are only available for air freezing and thawing indexes (@ 1 meter in air above ground). However there is still a need to establish potential heat flow at the air-ground interface. No simple correlation exists between air and surface indexes. Differences between air and surface temperatures are influenced by:

• latitude
• cloud cover
• time of year
• wind speed
• surface characteristics
• subsurface thermal properties
• surface slope and orientation

However, designers generally use “n-factor” for purposes of correlation.

“n” increases with increases in latitude and wind speed. Snow covered surfaces reflect large portion of incoming solar radiation with a resulting larger surface freezing index.

 Surface Type “n” Snow 1.0 Pavements free of snow and ice 0.9 Sand and gravel 0.9 Turf 0.5

### n-Factor for Thawing Conditions

“n” decreases with increases in latitude and wind speed.

 Surface Type “n” Sand and gravel 2.0 Turf 1.0

### Design Freezing and Thawing Indexes

For design purposes, generally use freezing (or thawing) index based on three coldest winters (or warmest summers) in last 30 years of record. If not available, use air-freezing index for the coldest winter in last 10 years.

## Washington State Climate Data

Figure 4 and Table 6 provide an overview of Washington State mean FI data
(summarized for 1951 to 1980). Figure 5 is a contour map of Washington for design
FI data. The FI contours for both Figures 4 and 5 are only approximate. FIs
should be obtained at specific sites (projects) if possible.

Figure 4: Mean Annual Freezing Index Contour Map

 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec”’ Aberdeen 18 0 0 0 0 0 0 0 0 0 0 6 24 Anacortes 30 0 0 0 0 0 0 0 0 0 0 7 37 Battle Ground 45 0 0 0 0 0 0 0 0 0 0 8 53 Bellingham 59 6 0 0 0 0 0 0 0 0 0 18 83 Bellingham Airport 67 7 0 0 0 0 0 0 0 0 0 23 97 Bickleton 231 68 18 7 0 0 0 0 0 0 0 42 120 Blaine 68 7 0 0 0 0 0 0 0 0 0 25 100 Bremerton 20 0 0 0 0 0 0 0 0 0 0 8 28 Buckly 40 8 0 0 0 0 0 0 0 0 0 17 65 Cedar Lake 85 26 10 0 0 0 0 0 0 0 9 140 70 Centralia 29 0 0 0 0 0 0 0 0 0 0 8 37 Chelan 262 110 10 0 0 0 0 0 0 0 29 149 560 Chewelah 339 150 35 0 0 0 0 0 0 0 62 216 802 Chief Joseph Dam 293 136 17 0 0 0 0 0 0 0 34 175 655 Clearbrook 103 13 0 0 0 0 0 0 0 0 6 39 161 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Clearwater 22 0 0 0 0 0 0 0 0 0 0 6 28 Cle Elum 268 94 22 0 0 0 0 0 0 0 48 151 583 Colfax 226 45 9 0 0 0 0 0 0 0 29 99 408 Colville 321 119 29 0 0 0 0 0 0 0 75 221 765 Concrete 57 9 0 0 0 0 0 0 0 0 0 18 84 Coulee Dam 284 107 14 0 0 0 0 0 0 0 35 155 595 Coupeville 28 0 0 0 0 0 0 0 0 0 0 11 39 Dallesport Airport 179 14 0 0 0 0 0 0 0 0 8 52 253 Davenport 315 133 25 0 0 0 0 0 0 0 66 198 737 Dayton 206 30 0 0 0 0 0 0 0 0 18 61 315 Diablo Dam 126 23 9 0 0 0 0 0 0 0 10 54 222 Electron Headworks 78 20 8 0 0 0 0 0 0 0 8 40 154 Elma 22 0 0 0 0 0 0 0 0 0 0 7 29 Elwha Rngr Station 47 6 0 0 0 0 0 0 0 0 0 14 67 Ephrata Airport 285 98 5 0 0 0 0 0 0 0 41 167 596 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Everett 35 0 0 0 0 0 0 0 0 0 0 11 46 Forks 22 5 0 0 0 0 0 0 0 0 0 10 37 Glenoma 47 9 0 0 0 0 0 0 0 0 0 17 73 Grapeview 17 0 0 0 0 0 0 0 0 0 0 7 24 Hatton 266 54 0 0 0 0 0 0 0 0 43 124 487 Hoquiam 15 0 0 0 0 0 0 0 0 0 0 9 24 Kennewick 202 26 0 0 0 0 0 0 0 0 18 54 300 Kent 28 0 0 0 0 0 0 0 0 0 0 8 36 Kid Valley 43 8 0 0 0 0 0 0 0 0 0 16 67 Lacrosse 248 42 0 0 0 0 0 0 0 0 30 103 423 Landsburg 48 8 0 0 0 0 0 0 0 0 0 15 71 Laurier 350 121 35 0 0 0 0 0 0 0 82 238 826 Lind 266 62 0 0 0 0 0 0 0 0 43 126 497 Longview 36 0 0 0 0 0 0 0 0 0 0 6 42 Millin Reservoir 42 6 0 0 0 0 0 0 0 0 0 13 61 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Monroe 38 0 0 0 0 0 0 0 0 0 0 14 52 Moses Lake 292 92 0 0 0 0 0 0 0 0 50 153 587 Mt. Adams Rngr Sta. 197 48 22 0 0 0 0 0 0 0 31 90 388 Moxee City 263 63 0 0 0 0 0 0 0 0 31 134 491 Mud Mtn. Dam 56 15 5 0 0 0 0 0 0 0 0 21 97 Newhalem 88 18 6 0 0 0 0 0 0 0 7 39 158 Newport 306 112 49 0 0 0 0 0 0 0 73 199 739 Northport 275 94 19 0 0 0 0 0 0 0 46 174 608 Oakville 35 0 0 0 0 0 0 0 0 0 0 10 45 Odessa 273 82 9 0 0 0 0 0 0 0 41 144 549 Olga 29 0 0 0 0 0 0 0 0 0 0 11 40 Olympia 31 5 0 0 0 0 0 0 0 0 0 15 51 Omak 344 175 28 0 0 0 0 0 0 0 67 234 848 Othello 276 64 0 0 0 0 0 0 0 0 35 125 500 Palmer 58 14 5 0 0 0 0 0 0 0 0 25 102 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Pomeroy 201 32 7 0 0 0 0 0 0 0 22 66 328 Port Angeles 14 0 0 0 0 0 0 0 0 0 0 6 20 Prosser 240 46 0 0 0 0 0 0 0 0 22 84 392 Pullman 243 77 0 0 0 0 0 0 0 0 38 118 476 Puyallup 30 0 0 0 0 0 0 0 0 0 0 11 41 Quilcene 39 5 0 0 0 0 0 0 0 0 0 14 58 Quillayute 22 5 0 0 0 0 0 0 0 0 0 9 36 Quincy 303 106 9 0 0 0 0 0 0 0 51 189 658 Paradise 254 161 161 85 27 9 0 0 9 19 103 219 1047 Republic 408 170 73 0 0 0 0 0 0 0 117 304 1072 Richland 199 28 0 0 0 0 0 0 0 0 13 54 294 Ritzville 281 94 10 0 0 0 0 0 0 0 42 143 570 Rosalia 269 94 22 0 0 0 0 0 0 0 49 142 576 Seattle 11 0 0 0 0 0 0 0 0 0 0 6 17 Sea-Tac 24 6 0 0 0 0 0 0 0 0 0 9 39 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Sea U.W. 14 0 0 0 0 0 0 0 0 0 0 6 20 Sedro Wooley 46 6 0 0 0 0 0 0 0 0 0 15 67 Sequim 20 0 0 0 0 0 0 0 0 0 0 8 28 Shelton 21 0 0 0 0 0 0 0 0 0 0 8 29 Snqlm. Falls 44 10 0 0 0 0 0 0 0 0 0 16 70 Spokane 299 108 24 0 0 0 0 0 0 0 58 178 667 Sprague 287 94 14 0 0 0 0 0 0 0 48 148 591 Stampede Pass 283 150 116 48 13 0 0 0 0 9 105 213 937 Startup 38 7 0 0 0 0 0 0 0 0 0 13 58 Stehekin 195 70 12 0 0 0 0 0 0 0 44 127 448 Sunnyside 216 35 0 0 0 0 0 0 0 0 16 73 340 Tacoma City Hall 17 0 0 0 0 0 0 0 0 0 0 7 24 Vancouver 57 0 0 0 0 0 0 0 0 0 0 8 65 Walla-Walla Airport 192 24 0 0 0 0 0 0 0 0 18 59 293 Walla-Walla 188 20 0 0 0 0 0 0 0 0 14 50 272 Station Mean Annual Freezing Index (°F-day) Mean Annual Freezing Index (°F-days) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Wapato 214 36 0 0 0 0 0 0 0 0 18 80 348 Waterville 349 152 51 0 0 0 0 0 0 0 84 246 882 Wenatchee 233 83 0 0 0 0 0 0 0 0 22 128 466 Wilbur 306 126 20 0 0 0 0 0 0 0 53 189 694 Willapa Harbor 12 0 0 0 0 0 0 0 0 0 0 0 12 Wilson Creek 276 79 6 0 0 0 0 0 0 0 42 163 566 Winthrop 451 206 65 0 0 0 0 0 0 0 108 342 1172 Yakima 258 63 0 0 0 0 0 0 0 0 31 123 475

Source: U.S. Department of Commerce, “Degree Days to Selected Bases,” National Climatic
Center, Federal Building, Asheville, N.C., December 1982.

Figure 5: Design Annual Freezing Index Contour Map

### Depth of Freeze – Implications for Pavement Design

One of the implications of the preceding calculations, FI contour maps, etc., is that the
total depth of the pavement structure should be influenced in some way by such results.
For example, several SHAs use the rule-of-thumb that the pavement structure should
equal at least one-half of the expected depth of freeze. To this end, Figures 6 and
7 were prepared. These contour maps show the expected depths of freeze
corresponding to the design FI (refer to Figure 5) for fine-grain soil (Figure 6) and
coarse-grain soil (Figure 7). The fine-grain soil calculations assumed a γd = 100
lb/ft3 (1600 kg/m3) and water content = 20 percent. The coarse-grain soil calculations
assumed a γd = 130 lb/ft3) (2080 kg/m3)and water content = 5 percent.

Figure 6: Frost Depth Contour Map (inches) for Fine Grained Soil (dry density = 100 pcf, wc = 20%)

Figure 7: Frost Depth Contour Map (inches) for Coarse Grained Soil (dry density = 100 pcf, wc = 5%)

Figure 8 shows contours of measured depths of freeze as determined during the
extremely cold winters of 1949 and 1950 (letter correspondence from B. Tremper, State
Materials and Research Engineer to W.A. Bugge, Director of Highways, dated, October
17, 1951).

Figure 8: Frost Depth Contour Map (inches) based on Field Measurements - Winters of 1949 and 1950

The freeze depths were measured in dug holes often along the edge of the
main lanes. The freeze depths were measured during February 1949 and January and
February 1950 (a total of 401 holes). Figure 8 is, in general, similar to Figure 6
(calculated freeze depths based on Design Freezing Indices and fine-grained soil) with
the exception of the Olympic Penninsula which is closer to those results shown in Figure
7 (coarse-grained soil). Some observations made by Highway Department personnel
during the winters of 1949 and 1950:

• Greatest freeze depths were observed in sandy or gravelly soils
• Snow or ice cover substantially reduced the depth of the freeze
• Frost heaving
• Most heaving observed in coastal areas (higher availability of water)
• Heaving somewhat infrequent in Eastern Washington but more severe when it did occur (again, likely related to the availability of water (or lack of))
• Maximum differential heave of 225 mm (9 in.) noted in District 2
• Silty sands showed the largest amount of ice lenses
• District 1 (Seattle): Maximum frost depth was measured between Issaquah and North Bend 0.8 m (30 in.). On Camano Island, a 0.5 m (20 in.) frost depth was measured. Maximum differential heave was 100 mm (4 in.) (several district locations).
• District 2 (Wenatchee): Maximum depth of freeze was 0.9 m (36 in.) measured in 1949 (Wauconda Summit) and 1.3 m (51 in.) in 1950 (between Brewster and Okanogan).
• District 3 (Tumwater): Maximum depth of freeze in 1949 was 0.6 m (24 in.) and 0.4 m (17 in.) in 1950.
• District 4 (Vancouver): Maximum frost depth was 0.5 m (20 in.) in 1950.
• District 5 (Yakima): Maximum frost depth was 0.8 m (30 in.) measured in 1950 with a district-wide average of 0.6 m (24 in.) Differential heave of 150 mm (6 in.) was noted.
• District 6 (Spokane): The maximum depth of freeze was 1.1 m (43 in.) with a district average of 0.9 m (35 in.) measured in 1949. In 1950, the maximum was 1.2 m (48 in.) with a district average of 0.7 m (28 in.)

The statement about SHA frost design needs a bit of explanation. A survey conducted
during 1985 [2.23] revealed the following from several “northern” states:

 Agency Use of Frost Protection in Thickness Design Alaska DOT More than 50 percent but not full Maine DOT More than 50 percent but not full Montana DOT Frost protection not included in design North Dakota DOT Frost protection not included in design Oregon DOT More than 50 percent but not full Washington DOT Depth > 50 percent of maximum frost depth expected

Thus, SHAs such as Alaska, Maine, Oregon, and Washington use knowledge about
expected frost depths in the design process. Presumably, limiting the depth of frost into
the subgrade soils limits, adequately, the potential for frost heave and thaw weakening
for most projects/locations.

The above percentages (pavement structural section as a percentage of expected frost
depth) are further reinforced by Japanese practice. Kono et al. [2.49] reported in 1973
that on the island of Hokkaido the pavement structure is set at 70 percent of the
expected frost penetration (the pavement materials are non-frost susceptible).

### Freezing and Thawing Indices – Implications for Maintenance Operations

The calculated FI and TI can be used to estimate the depth of freeze at a specific site
(FI) and the resulting thaw (TI). The TI can be used to assess the need for seasonal load
limits by maintenance personnel. Such measures are commonly used in District 2 during
to Rutherford et al. (1985[4]) or Mahoney et al. (1986[5]).

Based on the referenced study (Rutherford et al. 1985[4], Mahoney et al. 1986[5]), the following guidelines relative to spring
highway load restrictions were developed and evaluated. These guidelines were verified
based on results from District 2 (Miller et al. 1989[6]). The guidelines can be divided into where, how
much, when, and how long to apply load restrictions.

1. Where to apply load restrictions. If pavement surface deflections are available to an agency, spring thaw deflections greater than 45 to 50 percent of summer deflections suggest a need for load restriction. Further, considerations such as depth of freezing (generally areas with air Freezing Indices of 400 °F-days or more), pavement surface thickness, moisture condition, type of subgrade, and local experience should be considered. Subgrades with Unified Soil Classifications of ML, MH, CL, and CH will result in the largest pavement weakening.
2. Amount of load reduction. The minimum load reduction level should be 20 percent. Load reductions greater than 60 percent generally are not

warranted based on potential pavement damage. A load reduction range of 40 to 50 percent should accommodate a wide range of pavement conditions.

1. When to apply load restrictions. Load restrictions should be applied after accumulating a Thawing Index (TI) of about 25 °F-days (based on an air temperature datum of 29 °F) and must be applied at a TI of about 50 °F-days (again based on an air temperature datum of 29 °F). Corresponding TI levels are less for thin pavements (e.g., two inches of asphalt concrete and six inches of aggregate base or less) in that the should TI level is 10 °F-days and the must TI level is 40 °F-days.
2. When to remove load restrictions. Two approaches are recommended, both of which are based on air temperatures. The duration of the load restriction period can be directly estimated by the following relationship which is a function of Freezing Index (FI):

Duration (days) = 25 + 0.01 (FI)
Further, the duration can be estimated by use of TI and the following relationship:
TI ~– 0.3 (FI)

Footnotes    (↵ returns to text)
1. Thermal Properties of Soils.  Engineering Experiment Station, Bulletin 28, University of Minnesota.  Minneapolis, MN.
2. Calculation Methods for Determination of Depths of Freeze and Thaw Soils — Emergency Construction.  Department of the Air Force, Manual AFM 88-40, Chapter 46, Washington, D.C.
3. Thermal Properties of Soils. Trans Tech Publications.  Clausthal-Zellerfeld, Germany.
4. Guidelines for Spring Highway Use Restrictions, Research Report WA-RD 80.1 (FHWA-RD-86-501), Washington State Department of Transportation, Olympia, Washington, August 1985.
5. Research Summary Report — Guidelines for Spring Highway Use Restrictions, Research Report WA-RD 80.2, Washington State Department of Transportation, Olympia, Washington, June 1986.
6. Implementation of Guidelines on When to Apply Seasonal Load Restrictions, Proceedings, 40th Annual Road Builders’ Clinic, Washington State University, March 1989.

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