Most steel structures require some form of fire protection in order to achieve fire resistance.
Fire resistance of protected steel can be assessed by the use of generic ratings, proprietary ratings or by calculation. Generic ratings or ‘tabulated ratings’ are those which assign a time of fire resistance to materials with no reference to individual manufacturers or to detailed specifications. Many national codes and some trade organizations provide lists of generic ratings for fire protection of structural steel members. The most common ratings are for encasement in concrete or some other generic material, with a table of the minimum thickness of material needed to provide certain ratings. Many manufacturers of passive fire protection products provide proprietary listings of approved ratings. These are similar to generic ratings in that they generally provide ratings for exposure to the standard fire, but they may be less conservative because they relate to more closely defined products. Proprietary ratings usually make no allowance for the level of load, but they often include reference to the size and shape of the member using the section factor. A list of proprietary ratings is given by the Association of Specialist Fire Protection in the UK (ASFP, 2014). This document gives the required thickness of particular proprietary spray‐on or board protection to provide fire resistance to steel beams or columns depending on their section factor F/V, where F is the exposed surface area per unit length of the beam and V is the volume per unit length of the beam.
This chapter describes calculation methods for steel structures exposed to fires. Most calculations will compare loads with load capacity in the strength domain. Using the termi- nology of the Eurocodes, there are two main types of calculation: namely, simple calculation methods; and advanced calculation methods. Simple methods are used for single members in uncomplicated structures, often using hand calculations with the equations presented below.
Advanced calculation methods require the use of computer programs for analysis of complex structures using the material properties from this chapter, and as described in Eurocode 3 Part 1.2 (CEN, 2005b).
6.2 Steel Temperature Prediction
6.2.2 Calculation Methods
The simplest hand calculation method is to use a best‐fit empirical formula (ECCS, 1985) to obtain the temperature of steel members exposed to the standard fire, assuming that the steel temperature is uniform over the cross section. A more accurate ‘lumped mass’ design method employs the step‐by‐step calculation technique in Eurocode 3 Part 1.2 (CEN, 2005b). This method assumes a lumped mass of steel at uniform temperature over the cross section, and can be used with any design fire curve as input.
More sophisticated computer‐based methods can calculate temperatures within the cross section, for any combination of materials or shapes, for exposure to any desired fire. A two‐
dimensional calculation is suitable for most situations, based on an assumption of the same temperatures at each point along the member, which is reasonable if the fire temperature is assumed to be the same throughout the fire compartment. Three‐dimensional heat transfer calculations may be useful at member junctions or other special situations. Figure 6.3 shows the temperature contours in an unprotected heavy steel section (Universal Column 356 × 406 × 634 kg/m) after exposure for 30 min to the standard fire curve, calculated by the SAFIR program (Franssen et al., 2000). It can be seen that there are temperature differences of over 100 °C within the cross section, the largest difference being between the high temper- atures in the thin web and the lower temperatures in the much thicker flanges.
6.2.3 Section Factor
The rate of temperature rise of a protected or unprotected structural steel member exposed to fire depends on the section factor, or massivity factor, which is a measure of the ratio of the heated perimeter to the area or mass of the cross section. The section factor is important because the rate of heat input is directly proportional to the area exposed to the fire environ- ment, and the subsequent rate of temperature increase is inversely proportional to the heat capacity of the member (equal to the product of the specific heat, the density and the volume of the steel segment).
The section factor can be expressed in one of four alternative ways:
• ratio of heated surface area to volume, both per unit length, F/V (m–1);
• ratio of heated perimeter to cross‐sectional area, Hp/A (m–1);
• ratio of heated surface area to mass, both per unit length, F/M (m2/t);
• effective thickness, V/F or A/Hp (m or mm).
Here F is the surface area of unit length of the member (m2), V is the volume of steel in unit length of the member (m3), Hp is the heated perimeter of the cross section (m), A is the cross‐
sectional area of the section (m2), and M is the mass per unit length of the member (t).
The first two ratios are identical and can be easily converted to the third using the density of steel (7850 kg/m3 or 7.85 t/m3). The heated surface is the actual surface area of unprotected members or members with sprayed‐on fire protection, and the area of the equivalent rectangle for box protection, with allowance for any unexposed surfaces, as shown in Figure 6.4. The fourth ratio listed above is V/F or A/Hp with units in metres (or millimetres). This ratio gives much better physical understanding because it is an effective thickness of the cross section.
Calculated in this way, the section factor for a steel plate exposed to a fire on both sides is V/F = t/2 where t is the thickness of the plate. For a hollow tube of thickness t, the section factor becomes V/F = t. For an I‐beam, the section factor V/F is one half of the average thick- ness of the different parts. Mistakes in the calculation of the section factor are much less likely when it is defined in this way.
Tables of section factors for common structural steel shapes are available from distributors of steel products. Section factors for steel members are listed by ICC (2015), ASFP (2014) and HERA (1996) for American, British and New Zealand steel sections, respectively. Some of these section factors are listed in Appendix B.
6.2.4 Thermal Properties
In order to make calculations of temperatures in fire‐exposed structures, it is necessary to know the thermal properties of the materials. The density of steel is 7850 kg/m3, remaining essentially constant with temperature. The specific heat of steel varies according to
397 397
384 371 368
397
358 384 371 410423 436 449
397 397 384 371 358 345
Figure 6.3 Temperature contours in a heavy steel section exposed to fire. Reproduced from Franssen et al. (2000) with permission of Franssen
h
c1 b c2
c1 c2
h
b
b
h
b
h
b
Contour encasement of uniform thickness
Steel perimeter Steel cross-sectional area
Hollow encasement(1) of uniform thickness
2(b + h) Steel cross-sectional area
Contour encasement of uniform thickness,
exposed to fire on three sides
Steel permieter - b Steel cross-sectional area
Hollow encasement(1) of uniform thickness,
exposed to fire on three sides
2h + b Steel cross-sectional area
(1) The clearance dimensions c1 and c2 should not normally exceed h/4
Sketch Description Section factor (F/V)
Figure 6.4 Definition of section factor in the Eurocode. Reproduced from CEN (2005b). © CEN, reproduced with permission
temperature as shown in Figure 6.5 (CEN, 2005b) where the spike results from a metallurgical change at about 735 °C. For simple calculations the specific heat cp (J/kgK) can be taken as 600 J/kgK, but it is more accurate to use the following:
cp 425 0 773T 1 69 10 T 2 22 10 T 20 T 600 666 1300
3 2 6 3
. . . C C
22 738 600 735
545 17820 731 735 900
650
/ C C
/ C C
T T
T T
9900 C T 1200 C
(6.1)
where T is the steel temperature (°C).
The thermal conductivity of steel varies according to temperature as shown in Figure 6.6, reducing linearly from 54 W/mK at 0 °C to 27.3 W/mK at 800 °C (CEN, 2005b). For simple calculations the thermal conductivity k (W/mK) can be taken as 45 W/mK but it is more accurate to use the following:
k T T
T
54 0 0333 20 800
27 3 800 1200
. .
C C
C C (6.2)
6.2.5 Temperature Calculation for Unprotected Steelwork
Unprotected steel members can heat up quickly in fires, especially if they are thin and have a large surface area exposed to the fire. The Eurocode method of calculating temperature is given below; it treats the entire steel cross section as a lumped mass. In situations where parts of the steel cross section are insulated, there will be temperature gradients within the cross section, and these temperatures cannot be calculated with a lumped mass approach, so a finite
0 200 400 600
Temperature (°C)
800 1000 1200
0
Figure 6.5 Specific heat of steel as a function of temperature. Reproduced from CEN (2005b). © CEN, reproduced with permission
element method is necessary (see Chapter 3). There are some limited cases where the lumped mass temperature calculation can be performed for different parts of the cross section, as demonstrated in Chapter 8 with a steel‐concrete composite beam.
6.2.5.1 Eurocode Method
The step‐by‐step calculation method for unprotected steelwork is based on the principle that the heat entering the steel over the exposed surface area in a small time step is equal to the heat required to raise the temperature of the steel, assuming that the steel section is a lumped mass at uniform temperature. The temperature increase over a given time step Δt in an unprotected steel member is calculated by:
T k F V
c h T T T T t
s sh
s s
c f s f s
/ 4 4
(6.3) where ρs is the density of steel (kg/m3), cs is the specific heat of steel (J/kgK), ΔTs is the change in steel temperature in the time step (°C or K), hc is the convective heat transfer coefficient (W/m2K), σ is the Stefan–Boltzmann constant (56.7 × 10‐12 kW/m2K4), ε is the resultant emissivity, Tf is the temperature in the fire environment (K), Ts is the temperature of the steel (K) and ksh is a correction factor for shadow effects. For I‐sections, ksh is calculated as 0.9 (F/V)b/(F/V), and it is (F/V)b/(F/V) for all other cross sections. (F/V) is the section factor for contour protection, while (F/V)b is the section factor for board protection, called the box value of the section factor.
Eurocode 1 Part 1.2 (CEN, 2002b) recommends a convective heat transfer coefficient of 25 W/m2K for the standard fire, 50 W/m2K for the hydrocarbon fire and 35 W/m2K for
60 50 40 30 20 10 0
Thermal conductivity (W/mK)
0 200 400 600 800 1000 1200
Temperature (°C)
Figure 6.6 Thermal conductivity of steel as a function of temperature. Reproduced from CEN (2005b).
© CEN, reproduced with permission
all parametric fires. Heat transfer in typical fires is not very sensitive to this value because radiative heat transfer dominates at typical fire temperatures (Thomas, 1997). The resultant emissivity ε is calculated as the product of the emissivity of the fire εf and the emissivity of the material εm (ε = εf εm). The Eurocodes specify values of εf =1.0 (CEN, 2002b) and εm = 0.7 for steel (CEN, 2005b).
A spreadsheet for calculating steel temperatures using this method is shown in Table 6.1.
Eurocode 3 Part 1.2 (CEN, 2005b) suggests a time step of no more than 5 s for unprotected steel, and a minimum section factor (F/V) value of 10 m‐1 (maximum effective thickness V/F of 100 mm).
6.2.6 Temperature Calculation for Protected Steelwork
Protected steel members heat up much more slowly than unprotected members because of the applied thermal insulation which protects the steel from rapid absorption of heat. The Eurocode method is described below. When using these methods with thick insulation, the section factor F/V should strictly be calculated using the fire‐exposed perimeter rather than the inside face of the insulating material, but the inside perimeter is more often used because it is published in tables such as those in Appendix B. For steel members protected with heavy insulating materials or those with temperature‐dependant thermal properties, a finite element computer program is recommended for calculating the temperatures due to the complexities involved in the calculation, even though a spreadsheet may be used.
6.2.6.1 Eurocode Method
The calculation method for protected steelwork is similar to that for unprotected steel.
The equation is slightly different and does not require heat transfer coefficients because it is assumed that the external surface of the insulation is at the same temperature as the fire gases.
It is also assumed that the internal surface of the insulation is at the same temperature as the steel. The equation is:
T k F V d c
T T
t e T T
s i
i s s
f s
f s
/ but if
1 3 10 1 0
/ / , TTf 0 (6.4)
with
i i s s
i
c
c d F V/
where ci is the specific heat of the insulation (J/kg K), ρi is the density of the insulation (kg/m3), ki is the thermal conductivity of the insulation (W/mK) and di is the thickness of the insulation (m).
The spreadsheet calculation is similar to that shown in Table 6.1 except that Equation 6.4 is used instead of Equation 6.3. Eurocode 3 suggests a maximum time step of 30 s. If the insula- tion is of low mass and specific heat such that the heat capacity of the insulation will not sig- nificantly slow the temperature increase of the steel, then Equation 6.4 can be simplified by making the ϕ term zero.
The effect of the time delay for moist materials can be incorporated into the Eurocode cal- culation method by modifying the specific heat of the insulating material to include a local increase of specific heat at 100 °C. Typical values of thermal properties of insulating materials are given in Table 6.2 (ECCS, 1995).
6.2.7 Typical Steel Temperatures
Figure 6.7(a) shows steel temperatures for a 360UB45 beam, with F/V = 210 m–1 and (F/V)b = 153 m–1, exposed to the ISO 834 standard fire. The top curve is the fire temperature and the second curve is the temperature of an unprotected steel beam. The lower two curves are for the same beam protected with insulating material, using thicknesses of 15 and 30 mm.
Figure 6.7(b) shows steel temperatures for the same beam exposed to a parametric fire, also calculated using the spreadsheet. The top curve is the fire temperature and the second curve, following the fire closely, is the temperature of the steel beam with no protection. The lower Table 6.2 Thermal properties of insulation materials
Material Density,
ρi (kg/m3)
Thermal conductivity, ki (W/mK)
Specific heat, ci (J/kgK)
Equilibrium moisture content (%)
Sprays
Sprayed mineral fibre 300 0.12 1200 1
Perlite or vermiculite plaster 350 0.12 1200 15
High density perlite or vermiculite plaster
550 0.12 1200 15
Boards Fibre‐silicate or fibre‐calcium silicate
600 0.15 1200 3
Gypsum plaster 800 0.20 1700 20
Compressed fibre boards:
Mineral wool, fibre silicate 150 0.20 1200 2
Source: ECCS (1995).
two curves are the temperatures of steel beam protected with insulating material, using thick- nesses of 15 and 30 mm as before. The kink that is observed in the steel temperature at about 750 °C is as a result of changes in specific heat capacity due to the rearrangement of its crystal structure at that temperature.
6.2.8 Temperature Calculation for External Steelwork
Steel beams and columns outside a fire compartment may be subjected to elevated tempera- tures as a result of radiation from the window opening, radiation from flames, or engulfment in flames. Methods for estimating the temperature of such exposed steel members have been
1000 800 600 400 200
00 20 40 60 80 100 120
Temperature (°C)
0
Time (min) Fire temperature
(b)
Unprotected steel
Protected steel
0 20 40 60 80 100 120
Time (min)
Figure 6.7 Typical steel temperatures for unprotected and protected steel beams exposed to: (a) the standard fire; (b) a parametric fire
developed by Law and O’Brien (1989) and are incorporated into the Eurocodes. Flame sizes, temperatures and heat transfer coefficients are given in Eurocode 1 Part 1.2 (CEN, 2002b) and methods for calculating the steel temperatures are given in Eurocode 3 Part 1.2 (CEN, 2005b).
The design method allows for conditions with or without wind creating a forced draught to influence the shape of the idealized flame.
Typical flame shapes and radiation geometries are shown in Figure 6.8 for two conditions of forced draught and no forced draught, which produce different flame shapes. The design documents show many additional shapes for conditions with cross winds, flame deflectors and other variations. Figure 6.8 shows three possible column locations which require different designs. Columns at locations A and C are exposed to radiation from the flame itself and also from the window opening behind the flame, but column C has less severe exposure. The column at location B is engulfed in the flame. The design documents mentioned above give equations for all of these situations.