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### Design Formulae - Concrete Members

PD7974-3 (2003) adopts the simple calculation method proposed by Wickström (1986) for calculating the temperatures in concrete members exposed to the standard fire in accordance with BS476 or real fire conditions. It should be noted that this method does not take into account of possible spalling of concrete.

The fire-exposed surface temperature Ts of a concrete member at a time t is first given by:

 (1)
 with
where
 ns is the ratio between gas and surface temperatures of concrete member [°C]; Tg is the gas atmosphere temperatures [°C]; ts is the scaled time as given by Eq.(2) [hr].

The scaled time ts , accounting for the variation in thermal properties between the concrete being considered and a nominal standard mix for normal weight concrete, is given by:

 (2)
 with
where
 ρc is the density of concrete at elevated temperatures [kg/m3]; At is the total internal area, including openings, of the enclosure [m2]; Aw is the area of the openings [m2]; hw is the opening height [m]; cc is the specific heat capacity of concrete at elevated temperatures [J/kg L]; kc is the thermal conductivity of concrete at elevated temperatures which is assumed to reduce linearly from approximately 1.25 W/m K to 0.5 W/m K between 100 °C and 1200 °C [W/m K]; t is the time [hr].

It is worth noting that when predicting the response of normal weight concrete exposed to the standard fire, the scaling of time is unnecessary and ts may be set to equal t.

For uniaxial heat flow condition, such as in a slab, the temperature rise Tx at any depth x [m] beneath the fire-exposed surface of the member is a factor nx of the surface temperature Ts with nx given by:

 (3)
 with

where Kc is the thermal diffusivity of concrete [m2/s].

Hence, the temperature rise Tx is given by:

 (4)

The method can be applied to concrete members heated on parallel faces simultaneously, in which nx is simply the superimposed total of the nx values calculated with respect to each face.

The method can also be used for corners of beams where there is accommodated heat flow from two directions, through superimposition of the contributions from the orthogonal faces nx and ny as follows:

 (5)

where ny is calculated in the same way as nx .

In case of normal weight concrete exposed to the standard fire in accordance with BS476, the scaling of time is unnecessary and ts in Eq.(2) may be set to equal t. Hence, Eq.(4) can be greatly simplified as follows:

 (6)