## Background theory

The lower the U-value of a construction element, the more significant is the effect of thermal bridging on the calculation of the U-value. Consequently it is usually necessary to include thermal bridging in the calculation method. The theory is based on the calculation of thermal resistances. For a single layer of material, the thermal resistance R is given by:

where d is the thickness of the layer in metres, and X is the thermal conductivity. The combined resistance of several materials depends on whether the heat flows through them sequentially, i.e. in series or in parallel, as shown in Figure 5.1.

Resistances in series

For three materials in series (Fig. 5.1a), the total combined resistance RT is given by:

Resistances in parallel

For three materials in parallel (Fig. 5.1b), the total combined resistance RT is given by:

Fig. 5.1 Resistances in series (a) and in parallel (b).

where Fi, F2 and F3 are the cross-sectional areas of each material expressed as a fraction of the total.

Once the total resistance of a structure has been found, the U-value is its reciprocal:

Construction elements with materials in series and in parallel Many practical construction elements consist of several layers through which heat passes in series, with some components embedded within them through which the heat passes in parallel (the thermal bridges). The total thermal resistance can be calculated in several ways, and the Approved Documents present one of the simplest and most direct. The method is suitable when the bridging material is timber or mortar, or some other material which is thermally similar. It is not suitable when the bridging material is metal, nor is it suitable for ground floors and basements. The method calculates an upper resistance limit for the construction element, Rupper, and a lower resistance limit, Rlower, and then finds RT by taking the average:

The upper resistance limit is found by taking each possible heat flow path separately and calculating its resistance as the sum of the resistances of each component taken in series. When all paths have been calculated, their resistances are combined in parallel, taking account of their relative areas to give Rupper. The lower resistance limit is found by first converting each thermal bridge in the construction into a single equivalent resistance by combining the elements of the bridge in parallel. When all bridges have been converted, the construction should consist of a single heat flow path whose resistance is found by summing its components in series to give Rlower. The example calculations which follow should clarify the method.