Annex A

(normative)

# Definition and calculation of freezing index

## A.1 General

This annex gives the method of calculation of the design freezing index Fd from meteorological records of daily mean external air temperatures for the locality concerned.

A.2 defines the calculation of the freezing index, F, for one particular winter. The design data given in clauses 8 to 10 are based on Fn, the freezing index which statistically is exceeded once in n years, e.g. F10, F50, F100. These values may be obtained from a set of individual values of F calculated for several winters using the statistical treatment described in A.3.

## A.2 Calculation of freezing index for one winter

The freezing index is the 24 times sum of the difference between freezing point and the daily mean external air temperature: (A.1)

where

F
is the freezing index for one winter, in K·h;
θf
is equal to 0°C;
θd,j
is the daily mean external air temperature for day j, in °C;

and the sum includes all days in the freezing season (as defined below).

The daily mean external air temperature may be obtained as the average of several readings, or as the average of the maximum and minimum values, for the day in question.

Both positive and negative differences, within the freezing season, are included in the accumulation of equation (A.1). A negative difference (daily mean temperature above 0 °C) implies some thawing of the ground, which serves to reduce the frost penetration in the ground.

For the purposes of the summation in equation (A.1) the freezing season starts at the point from which the accumulation remains always positive throughout the winter. With reference to Figure A.1, there is initially some freezing as a result of the area marked A, followed by complete thawing as a result of the area marked B since this is greater than area A. The accumulation therefore starts after this. In Figure A.2, area A is greater than area B, so the thawing is not complete and the accumulation starts earlier as indicated on that Figure.

The freezing season ends at the point which results in the largest total accumulation for the winter. If a short thawing period is followed by a larger freezing period both are included, while if a thawing period is followed by a lesser freezing period neither is included, as illustrated in Figures A.1 and A.2. Key

• 1 Start
• 2 End
• 3 Autumn
• 4 Winter
• 5 Spring

NOTE Area B > area A, and area C > area D

Figure A.1 — Illustration of the limits of the freezing season (first example) Key

• 1 Start
• 2 End
• 3 Autumn
• 4 Winter
• 5 Spring

NOTE Area B < area A, and area C < area D

Figure A.2 — Illustration of the limits of the freezing season (second example)

NOTE 1 In the past, freezing indexes have sometimes been calculated including only positive differences in equation (A.1), i.e. ignoring the effect of thawing periods. Tables or maps of freezing indexes calculated on that basis, which give higher values of F than as defined above and so a greater margin of safety, may be used for the purposes of this standard. On the other hand an accumulation on the basis of average monthly temperatures can significantly underestimate the true freezing index and such data should not be used.

NOTE 2 An alternative, and equivalent, method of obtaining the freezing index is to plot the cumulative difference between daily mean temperature and freezing point against time for a complete 12-month period (from midsummer to midsummer). The freezing index is then the largest difference between maximum and minimum turning points on this curve.

NOTE 3 Freezing in the ground depends on the ground surface temperature. However, because air temperatures are more readily available than ground surface temperatures, this standard uses the air freezing index, i.e. the freezing index calculated from external air temperatures, as the design parameter. In most cases the use of air temperatures provides a safety margin because factors such as the presence of vegetation and snow cover, and solar radiation, result in ground surface temperatures being higher than air temperatures. However the opposite may apply for snow-free surfaces in permanent sun shadow, for which ground surface temperatures can be lower as a result of radiation to clear skies.

## A.3 Statistical determination of design freezing index

The design freezing index, Fn, is the freezing index that statistically is exceeded once in n years. This implies that the probability that the freezing index in any one winter exceeds Fn is 1/n.

NOTE 1 The appropriate value of n should be decided upon with regard to the level of safety that is required for the building in question. Parameters to consider are the expected lifetime of the structure, the sensitivity of the type of structure to frost heave, etc. For permanent buildings n is normally chosen as 50 years or 100 years.

NOTE 2 n is referred to as the return period, i.e. the average number of years between successive occurrences of freezing indexes greater than Fn.

The design freezing index for a given location is obtained from a set of freezing indexes Fi, calculated as described in A.2, of m winters at the location. Whenever possible, the value of m should not be less than 20. The use of data from m consecutive, or nearly consecutive, winters is recommended.

Use a statistical distribution that realistically reflects extreme events. The Gumbel distribution (see A.4) has been found to be suitable for many climates, and is recommended in the absence of specific information for the locality concerned.

## A.4 Use of the Gumbel distribution

Calculate the average freezing index, , using (A.2) and the standard deviation, sF, using (A.3): (A.2) (A.3)

where

i = 1,2,....., m

The design freezing index is then given by (A.4): (A.4)

where y denotes the reduced variable in the Gumbel distribution.

Obtain the appropriate values of and sy from Table A.1 corresponding to the number m of individual values of Fi used in the calculation.

Obtain the value of yn from Table A.2 corresponding to the value of n chosen for the design.

Table A.1 — Values of and sy
 m sy m sy 10 0,50 0,95 50 0,55 1,16 15 0,51 1,02 60 0,55 1,17 20 0,52 1,06 70 0,55 1,19 25 0,53 1,09 80 0,56 1,19 30 0,54 1,11 90 0,56 1,20 40 0,54 1,14 100 0,56 1,21

Table A.2 — Values of yn
 n 5 10 20 50 100 yn 1,50 2,25 2,97 3,90 4,60

NOTE For further information about the Gumbel distribution, see  and  in Bibliography.

ISO 13793-2001 Thermal performance of buildings Thermal design of foundations to avoid frost heave