FPE eXTRA Issue 59, November 2020

Proposed Framework for a Reliability-Based Method for Structural Design Fires

By: Kevin LaMalva, PE, Panagiotis Kotsovinos PhD, and Colleen Wade, PhD

Background

Uncontrolled fire within a building is a low likelihood but extraordinary event that can have severe consequences. Fire sprinkler systems significantly reduce the probability of this occurrence, but these systems are generally not effective against very large fires. Accordingly, structural fire resistance is intended to serve as a secondary safety measure in the unlikely event that a fire becomes uncontrolled and grows to become fully developed.

Heating of structural systems from fire causes thermal load effects that are not considered in conventional structural engineering design, such as reduced material strength, thermal strains and/or stresses induced by restrained thermal expansion. Under these conditions, it is critical that structural systems remain stable to protect occupants and the operation of fire-fighters and satisfy other required performance objectives (such as supporting a fire rated façade for external fire spread etc.).

Structural fire engineers can learn from existing guidance and aspects of conventional structural engineering, which has a well-developed reliability-based framework. This article describes a framework for a reliability-based method to calculate structural design fires that is under consideration by the SFPE Standards Making Committee on Thermal Exposures to Structures.

RELIABILITY THEORY IN ENGINEERING

Each parameter in an engineering design is a random (stochastic) variable with a probability distribution, which can be idealized; a lognormal distribution is commonly used in structural engineering where the coefficient of variation is a measure of a parameter’s scatter and is defined as the ratio of the standard deviation to the mean of the random variable. In addition to parameter variability, there also exists uncertainty in the ability of engineering equations and models to produce a certain result, which may be due to simplifications (generally known as “model” uncertainty).

Reliability-based engineering design methods are typically based on the establishment of a target reliability, which limits the probability of design exceedance or failure to an ‘As Low as Reasonably Practicable’ (ALARP) level. Due to economic practicalities, this probability must be finite and somewhat appreciable. The target reliability index [β] relates to the probability of exceedance (failure) in accordance with the following relation :

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The relation between target reliability and probability of exceedance is plotted in Figure 1 below:

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RELIABILITY-BASED FRAMEWORK IN STRUCTURAL ENGINEERING

Load and Resistance Factored Design (LRFD) methods employ a statistical-based approach for predicting loads and material strength and use criteria to reduce the probability of the load effect exceeding a capacity to an acceptable level (ALARP). The target reliability can be expressed in terms of mean and standard deviation values of the load effect and the capacity to withstand the load effect as follows:

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This expression can be represented graphically as shown in Figure 2 below. The area under the two curves where they intersect represents the exceedance of a limit state (failure).

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Accordingly, this method results in members that are sized to withstand all considered load effects during the design life of the structural system, with an appropriate level of reliability for each relevant limit state. For instance, live loads (e.g., bookcases) may exceed design levels or actual material mechanical strength may be less than assumed under extremely rare circumstances. Specific limit states that have a higher consequence of failure (e.g., brittle failure modes) typically have higher target reliability.

The load effect combinations used for conventional structural engineering design pertain to those resulting from dead (i.e., self-weight), live (i.e., movable weights), snow, rain, wind, and seismic load effects. Nominal loads are frequently defined with reference to a probability level (e.g., 50-year snow load). The additional load effect combination for uncontrolled fire exposure may also be included, for example ASCE/SEI 7 specifies the following :

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This load effect combination recognizes the small probability of uncontrolled fire occurring during a peak live or snow load event by applying associated load factors that are less than unity. The nominal loads are typically based on a combination of measured data and engineering judgment. Thus, there is a small but finite probability that the nominal load will be exceeded each year. Other standards such as the Eurocodes adopt different safety factors for dead and live loads and for example snow loads are not considered in the fire limit state.

RELEVANT FIRE EXPOSURE PARAMETERS AND DATA

Enclosure fires are typically assumed to be ventilation-controlled for small and medium sized compartments. The calculation of the time-temperature history of a ventilation-controlled fire commonly includes the following parameters:

  • Distributed fuel load [MJ/m^2]
  • Total area of enclosure boundaries [m^2]
  • Total area of ventilation openings [m^2]
  • Height of ventilation openings [m]
  • Density of enclosure boundaries [kg/m^3]
  • Thermal conductivity of enclosure boundaries [W/mK]
  • Specific heat of enclosure boundaries [J/kgK]

The height/area of ventilation openings can be highly uncertain given the randomness of window breakage which usually necessitates sensitivity studies.

RELIABILITY-BASED METHOD FOR DETERMINING FUEL LOADS

The NFPA 557 standard provides a reliability-based method for calculating distributed fuel loads that involves calculating a fuel load risk factor reflecting the likelihood of an uncontrolled fire occurring with a target β-value of approximately 4.8 (P_f=1.0 ×10^(-6)) on an annual basis, which is a function of the following:

  • Occupancy type
  • Construction characteristics
  • Presence or absence of active fire protection systems
  • Level of inherent and applied fire protection present

Based on specific fuel load surveys and studies, this standard specifies average and standard deviation values of the distributed fuel load for a very limited number of occupancies types. These values reflect a 99 percent upper confidence bound. The design distributed fuel load is calculated as a function of these statistical values and the fuel load risk factor. In contrast, Eurocode 1 treats the nominal fuel load as a variable parameter with a Gumbel distribution, and suggests the use of an 80 percent upper confidence interval. Also, risk factors are determined considering the β-value to be approximately 4.7 (P_f=〖1.3 ×10〗^(-6)) on an annual basis and reliability class 2.

VENTILATION AREA

To account for the randomness of window breakage, the probabilistic model code of the Joint Committee on Structural Safety (JCSS) provides an expression for a truncated log-normally distributed variable, which is used as a modifier for the maximum opening factor as follows:

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PROPOSED RELIABILITY-BASED FRAMEWORK FOR FIRE

The parametric equations in Annex A of the Eurocode serve as the appropriate “base model” for the proposed framework. This “base model” is supplemented with provisions for reliability-based treatment of certain input parameters. In this context, it is proposed that the following parameters be treated as direct inputs:

  • Total enclosure area of a single floor surrounded by exterior walls and fire resistance rated construction [m^2].
  • Adjusted area of ventilation openings [m^2].
  • Weighted average height of ventilation openings [m].
  • Ambient density of enclosure boundaries [kg/m^3].
  • Ambient thermal conductivity of enclosure boundaries [W/mK].
  • Ambient specific heat of enclosure boundaries [J/kgK].

Furthermore, it is proposed that the following parameters are treated as probabilistic inputs:

  • Distributed fuel load per NFPA 557 reduced to an 80 percent upper confidence interval [MJ/m^2].
  • Temperature correlation uncertainty factor to be multiplied by the nominal temperature at each time increment [-].
  • Time correlation uncertainty factor to be multiplied by the nominal fire duration [-].

It is assumed that temperature and time correlation uncertainty factors implicitly capture the uncertainty of the direct input parameters. Accordingly, the SFPE is in the process of expanding and synthesizing its database of fully developed fire tests to be utilized for the development of this proposed method. The method would be applicable to compartments with characteristics where a typical post-flashover fire has been shown to be applicable. Further research is ongoing to adequately characterize fires in large and well-ventilated compartments (also known as travelling fires).

There exist certain pitfalls for extending ambient reliability targets to structural fire engineering (e.g., the effect of fire exposure on both the load and capacity, which differs from ambient design). Considering this, the temperature and time uncertainty factors should be derived and provided as a function of the target reliability index. For structural fire engineering, it would seem reasonable that the resultant target reliability index should be not less than about 4.2 (P_f=10^(-5)) on an annual basis. The resultant target reliability could be adjusted downward or upward for lower consequences or higher consequences as deemed appropriate. Subsequent editions of ASCE/SEI 7 could address such risk-based modifiers.

Kevin LaMalva, PE is with Warringtonfire, Panagiotis Kotsovinos PhD is with Arup and Colleen Wade, PhD is with Fire Research Group.

References

  1. Victorsson, Victor, K., ‘The Reliability of Capacity-Designed Components in Seismic Resistant Systems,’ PhD Dissertation, Stanford University, 2011
  2. Van Coile, R., Reliability-Based Decision Making for Concrete Elements Exposed to Fire, PhD Dissertation, University of Ghent, Belgium, 2015.
  3. ASCE/SEI 7: Minimum Design Loads and Other Criteria for Buildings and Other Structures, American Society of Civil Engineers: Structural Engineering Institute, 2016
  4. NFPA 557: Standard for Determination of Fire Loads for Use in Structural Fire Protection Design, National Fire Protection Association, Quincy, MA, 2016
  5. JCSS: Probabilistic Model Code, Joint Committee on Structural Safety, 2001
  6. EN 1991-1-2 Actions on structures – Part 1-2: General actions – Actions on structures exposed to fire Annex A, 2002.