Formulas: Temperatures in Structural Systems Under Fire Conditions

Formula 1

\Delta T_s = \frac{a}{c_s \left( \frac{W}{D} \right)} (T_F - T_s) \Delta t

Formula 2

c_{s} W / D>d_{p} \rho_{p} c_{p} / 72

Equation C-A-4-6

then, Equation C-A-4-6 can be applied to determine the temperature rise in the steel:

\Delta T_{s}=\frac{k_{p}}{c_{s} d_{p}\left(\frac{W}{D}\right)}\left(T_{F}-T_{s}\right) \Delta t

Equation C-A-4-7

If the thermal capacity of the insulation needs to be considered (such that the inequality in Equation C-A-4-5 is not satisfied), then Equation C-A-4-7 should be applied:

\Delta T_{s}=\frac{k_{p}}{d_{p}} \frac{T_{F}-T_{s}}{c_{s}\left(\frac{W}{D}\right)+\frac{c_{p} \rho_{p} d_{p}}{288}} \Delta t

where

$c_{p}=$ specific heat of the fire-protection material, Btu/lb-°F (J/kg-°C)

$d_{p}=$ thickness of the fire-protection material, in. (m)

$k_{p}=$ thermal conductivity of the fire-protection material, Btu/ft-sec-°F (W/m-°C)

$\rho_{p}=$ density of the fire-protection material, $\mathrm{lb} / \mathrm{ft}^{3}\left(\mathrm{~kg} / \mathrm{m}^{3}\right)$

Equation C-A-4-8

External Steelwork. Temperature rise can be determined by applying the following equation:

\Delta T_{s}=\frac{q^{\prime \prime}}{c_{s}\left(\frac{W}{D}\right)} \Delta t