Annex B

(normative)

# Numerical calculations

## B.1 General

The general case of frost penetration into the ground adjacent to buildings or structures is a three- dimensional, time-dependent, non-linear heat transfer problem, which can be modelled using suitable numerical techniques (for example finite differences or finite elements).

The design procedures given in this standard are based on such numerical calculations for buildings on homogeneous ground consisting of frost-susceptible soil with properties as given in 5.1, and with other conditions as described in B.2.

The procedures described in clauses 8 to 10 will give adequate frost protection of foundations in most cases. If, however, the soil properties differ substantially from those given in 5.1 (in particular if the dry density of the soil is outside the range 1100 kg/m^{3} to 1600 kg/m^{3} or if the water saturation is less than 80 %), numerical calculations according to B.2 shall be undertaken.

NOTE The calculated soil temperatures adjacent to the building are increasingly sensitive to the precise values of the soil properties as the freezing index increases, as the internal temperature decreases, and as the floor insulation increases.

Numerical calculations which conform with B.2 may be used as an alternative to the tables and graphs given in this standard.

## B.2 Conditions for numerical calculations

### B.2.1 Subdivision of the geometrical model

The geometrical model of the ground is subdivided in such a way that the subdivisions are smallest near to the edge of the floor, and gradually increasing in size to much larger subdivisions near the truncation planes. The criteria given in ISO 10211-1 for judging whether sufficient subdivisions have been used (related to the calculation of heat flows and surface temperatures) are recommended.

### B.2.2 Dimensions of the ground

The following minimum dimensions of the ground define the truncation planes in the geometrical model:

- in the horizontal direction inside the building: 0,5
*B*; - in the horizontal direction outside the building: 2,5
*B*; - in the vertical direction below ground level: 2,5
*B*;

where *B* is the width (smaller dimension) of the floor.

### B.2.3 Three- or two-dimensional calculations

If the smaller dimension of the floor does not exceed 4 m, three-dimensional calculations shall be used. For other cases, the frost conditions along the walls can be judged from two-dimensional calculations with the building width set equal to the smaller dimension of the floor. The frost conditions at corners should then be judged from three-dimensional calculations or by using the appropriate tables and graphs in the standard.

### B.2.4 Boundary conditions

For two-dimensional calculations, there is a vertical symmetry plane mid-way across the floor, which is taken as an adiabatic boundary (so that one half of the building is modelled). For three-dimensional calculations on a rectangular building, there are two vertical symmetry planes mid-way across the floor in each direction, which are taken as adiabatic boundaries (so that one quarter of the building is modelled).

Outside the building, the vertical truncation plane is taken as an adiabatic boundary. The horizontal truncation plane in the ground is taken as an adiabatic boundary.

Surface resistances as specified in ISO 6946 apply at the inside floor surface and at the outside ground surface.

### B.2.5 Thermal properties

For the thermal properties of the ground:

- a) if known, use values for the actual location, allowing for the normal moisture content;
- b) otherwise, use the values specified in 5.1.

When water in the soil freezes or melts, there is a change in the heat capacity per volume and in the thermal conductivity of the soil, and the latent heat of the water in the soil is released during freezing. Numerical calculations should allow for these effects.

The latent heat of water in the soil may be treated as an apparent increase in the heat capacity of the soil over a temperature interval of 1 K below 0 °C. Soil at a temperature of –1 °C or below is considered as fully frozen.

For materials other than the ground, use values according to 5.2.

### B.2.6 Design external temperature

Use a sinusoidal variation of external temperature given by (B.1):

where

*t*, in °C;

*t*

_{p}

*F*

_{d}(see 6.1).

In order to start the calculation of the design year with an appropriate temperature distribution in the ground:

- the initial conditions should be the annual average external air temperature throughout the ground;
- the calculation period should extend over two consecutive design years, with the results being taken from the second year.

### B.2.7 Design criterion

The foundation design is considered to be protected against frost heave when no fully frozen soil occurs below the foundation during the design winter, i.e. the temperature remains above –1 °C under the whole of the base of the foundation. This can be done by examining the maximum penetration of the –1 °C isotherm towards the base of the foundation. An example of such an isothermal plot is shown in Figure B.1.