Figure 5.3(a) shows schematically that load U and resistance R are both probabilistic quan- tities, with a distribution of values about a mean. There is a small probability of failure which can be calculated from the area of overlap between the two curves if their distributions are known. The characteristic value of member resistance usually represents the lower 5th percen- tile tail of the strength distribution, and the design load represents a high percentile of likely loads for a given return period.
When considering Figure 5.3(a) for fire design, the load and resistance curves can be quan- tified in the time domain, the temperature domain or the strength domain (as shown in Chapter 4). If Figure 5.3(a) represents load and resistance at room temperatures, both the curves will shift to the left under fire conditions because the expected loads are less and the strength decreases due to elevated temperatures.
The ultimate limit state representing failure occurs if R < U, so the likelihood of failure is related to the difference R − U. Figure 5.3(b) shows the frequency distribution of R − U. The probability R − U < 0 is given by the shaded area under the distribution. Limit state design codes are usually calibrated to give a certain reliability index β, which is the number of stan- dard deviations of the mean value of R − U above zero, as shown in Figure 5.3(b). For given distributions, the strength reduction factor Φ is derived by code writers to give a target reli- ability index β, in the range between 2 and 3 (roughly equivalent to a probability of failure between 10–2 and 10–3).
The above discussion shows that although there is a probabilistic framework behind the ultimate strength code formats, day‐to‐day design is a deterministic process.
Structural design for fire safety has far more uncertainly than structural design for normal temperature conditions. This book considers structural fire safety in a deterministic frame- work, rather than a probabilistic framework. The science of structural reliability is rapidly developing, but applications to structural fire safety are still in their infancy despite pioneering work many years ago by Magnusson (1972) and Schleich (1999).
The above factors may be different for different materials. For example, Figure 5.4(a) shows how failure of a simply supported steel beam occurs when the yield strength of the material drops so low that it is exceeded by the actual stress in the member at the time of the fire. The stress in the member does not change during the fire because the loads are constant and the section properties do not change. In contrast, Figure 5.4(b) shows a similar situation for a timber beam, where the stresses increase steadily (under constant load) due to loss of cross section by charring. The material strength only decreases very slightly due to elevated temper- atures within the beam. As before, failure occurs when the stress in the member exceeds the material strength.
5.3.1 Design Equation
Verification of design for strength during fire requires that the applied loads are less than the load capacity of the structure, for the duration of the fire design time. This requires satisfying the design equation given by:
Ufire* fireRfire (5.8)
Material strength
Failure
Time
Steel Timber
(a) (b)
Time Stress in member
Material strength
Failure Stress in member
Stress Stress
Figure 5.4 Member failure in fire, due to internal stresses exceeding material strength
where U*fire is the design action from the applied load at the time of the fire, Rfire is the nominal load capacity at the time of the fire and Φfire is the strength reduction factor for fire design.
The design force U*fire may be axial force N*fire, bending moment M*fire or shear force V*fire occurring singly or in combination, with the load capacity calculated accordingly.
The strength reduction factor Φfire accounts for uncertainty in the estimates of material strength and section size. Fire design is based on the most likely expected strength, so most national and international codes specify a strength reduction factor of Φfire = 1.0. In the Eurocodes, the partial safety factor γM is also equal to 1.0 for fire design. In both the North American and European formats the design equation for fire conditions now becomes:
*
fire fire
U R (5.9)
This is the equation that will be used in the following chapters for design of steel, concrete and timber structures.
5.3.2 Loads for Fire Design
5.3.2.1 Load Combinations
In the ‘accidental’ event of a fire, the most likely applied loads are much lower than the maximum design loads specified for normal temperature conditions. Most codes refer to an
‘arbitrary point‐in‐time load’ to be used for the fire design condition, often known as the fire limit state loads. Fire limit state load combinations from the Eurocode (CEN, 2002a), the US Standard (ASCE, 2010) and the Australia/New Zealand Standard (SA, 2002), are given in Equation 5.10a, Equation 5.10b and Equation 5.10c respectively.
Lf Gk 0 5. Qk or Lf Gk 0 9. Qk (5.10a)
Lf 1 2. Gk 0 5. Qk (5.10b)
Lf Gk 0 4. Qk or Lf Gk 0 6. Qk (5.10c) where Lf is the factored load combination for fire, Gk is the characteristic dead load and Qk is the characteristic live load.
Where two equations are given, the second is the combination for storage occupancies with semi‐permanent live loads, and the first equation is for all other occupancies. It can be seen that the loads under fire conditions are much less than in normal temperature conditions. This is especially true for members which have been designed for load combinations including wind, snow or earthquake, or for members over‐sized for deflection control or architectural reasons.
National standards should be consulted for more detail. For example, ASCE (2010) requires that 1.2Gk in Equation 5.10b should become 0.9Gk if the dead load has a stabilizing effect.
Some codes including the Eurocode and US codes have additional load combinations to con- sider the possible effects of snow or wind occurring at the same time as a fire.
The fire itself may induce forces in a structure, and these must also be included in the design. These are most likely as a result of restraint from the surrounding structure preventing
rload Ufire* /Rcold (5.11) Most buildings have a load ratio of 0.5 or less, at most times, so that the strength of any member could drop by half or more before collapse would be expected. The load ratio is far less than 0.5 for buildings or parts of buildings designed to resist extreme events such as rare snowstorms, hurricanes or earthquakes. The lower the load ratio, the greater the fire resis- tance, because of the large loss in load‐carrying capacity which can occur before failure would occur in a fire. This is a most important concept for structural fire design.
5.3.2.3 Working Stress Design
Most modern loading codes, which specify load combinations for fire design, are in limit states (LRFD) format with loads similar to those described above. Loading codes which are in working stress format do not usually include load combinations for fire design, so designers have to use the normal temperature design load combination of Lw G Q for fire design, which is very conservative because it does not recognize the likely reduction of loads at the time of an unexpected fire.
5.3.3 Structural Analysis for Fire Design
Structural analysis for fire design is essentially the same process as structural analysis for normal temperature design, but it is complicated by the effects of elevated temperatures on the internal forces and the properties of materials.
For many simple structural elements exposed to fire, load carrying capacity can be calcu- lated with simple hand calculation methods, using the same techniques as for cold conditions.
Examples for steel, concrete and timber structures are given in later chapters. The major changes from cold conditions are the use of lower fire limit state loads and temperature‐
reduced material properties. For some materials such as wood, an alternative approach is to use reduced section properties with no change in the material properties.
Hand calculations are most appropriate for single elements with simple supports, especially where internal temperatures are uniform or where the temperature of one part of the member is critical. Structural analysis must consider the possibility of instability failures as well as
strength failures. Many tools are available for calculating the structural fire performance of load‐bearing construction, ranging from hand calculations and design charts to a variety of computer programs discussed in Chapter 11.