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An Unbiased Method for Probabilistic Fire Safety Engineering, Requiring a Limited Number of Model Evaluations

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Abstract

The rise of Performance Based Design methodologies for fire safety engineering has increased the interest of the fire safety community in the concepts of risk and reliability. Practical applications have however been severely hampered by the lack of an efficient unbiased calculation methodology. This is because on the one hand, the distribution types of model output variables in fire safety engineering are not known and traditional distribution types as for example the normal and lognormal distribution may result in unsafe approximations. Therefore unbiased methods must be applied which make no (implicit) assumptions on the PDF type. Traditionally these unbiased methods are based on Monte Carlo simulations. On the other hand, Monte Carlo simulations require a large number of model evaluations and are therefore too computationally expensive when large and nonlinear calculation models are applied, as is common in fire safety engineering. The methodology presented in this paper avoids this deadlock by making an unbiased estimate of the PDF based on only a very limited number of model evaluations. The methodology is known as the Maximum Entropy Multiplicative Dimensional Reduction Method (ME-MDRM) and results in a mathematical formula for the probability density function (PDF) describing the uncertain output variable. The method can be applied with existing models and calculation tools and allows for a parallelization of model evaluations. The example applications given in the paper stem from the field of structural fire safety and illustrate the excellent performance of the method for probabilistic structural fire safety engineering. The ME-MDRM can however be considered applicable to other types of engineering models as well.

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Abbreviations

E[.]:

Expected value operator

F x :

Cumulative density function for the variable X

\( F_{X}^{ - 1} \) :

Inverse cumulative density function for the variable X

f xl :

Probability density function describing the lth stochastic input variable

f y :

Probability density function describing Y

\( \hat{f}_{y} \) :

ME-MDRM estimate for f y

h(.):

Model response indicator

h l (.):

Unidimensional cut function for h(.) where all stochastic input variables expect the lth are evaluated by their median value

h 0 :

Model response y when all stochastic variables are given by their median values

L :

Number of Gauss integration points for the Gaussian interpolation

\( M_{Y}^{{\alpha_{i} }} \) :

α th i sample moment of Y

m :

Order of the Maximum Entropy estimate

n :

Number of stochastic variables

P f :

Probability of failure

P f,fi :

Probability of failure conditional on the occurrence of a ‘significant’ fire

w j :

Gauss weight for the jth Gauss integration point

x :

Vector of stochastic input variables x l

Y :

The stochastic model output

y :

Realization of the model output

y j,l :

Model realization where the lth stochastic variable is defined by the jth Gauss integration point, and all other variables are evaluated by their median value

z j :

jth Gauss integration point

α i :

Exponent i for the ME-MDRM estimate of the PDF

λ i :

Coefficient i for the ME-MDRM estimate of the PDF

λ 0 :

Normalization coefficient for the ME-MDRM estimate of the PDF

μ X :

Mean value of stochastic variable X

\( \hat{\mu }_{X} \) :

Estimated mean value of stochastic variable X

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }_{X} \) :

Median value of stochastic variable X

σ X :

Standard deviation of stochastic variable X

Φ :

Standard normal cumulative distribution function

φ:

Standard normal probability density function

LHS:

Latin Hypercube Sampling

MCS:

Monte Carlo simulations

ME-MDRM:

Maximum Entropy Multiplicative Dimensional Reduction Method

MDRM–G:

Parameter estimation based on Multiplicative Dimensional Reduction Method and Gaussian interpolation

PDF:

Probability density function

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Authors and Affiliations

  1. BRE Centre for Fire Safety Engineering, School of Engineering, The University of Edinburgh, Edinburgh, UK

    Ruben Van Coile

  2. Department of Structural Engineering, Ghent University, Ghent, Belgium

    Ruben Van Coile & Robby Caspeele

  3. Department of Civil and Environmental Engineering, Rice University, Houston, Texas, USA

    Georgios P. Balomenos

  4. Department of Civil and Environmental Engineering, Waterloo University, Waterloo, Canada

    Mahesh D. Pandey

Authors
  1. Ruben Van Coile
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  2. Georgios P. Balomenos
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  3. Mahesh D. Pandey
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  4. Robby Caspeele
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Correspondence to Ruben Van Coile.

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Van Coile, R., Balomenos, G.P., Pandey, M.D. et al. An Unbiased Method for Probabilistic Fire Safety Engineering, Requiring a Limited Number of Model Evaluations. Fire Technol 53, 1705–1744 (2017). https://doi.org/10.1007/s10694-017-0660-4

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