 # Probabilistic Method for Fire Safety Design

The probabilistic method has been increasingly used for fire safety design. However, it is based on the repeated evaluation of the structural behavior under fire loading, which is computationally expensive even for simple structural models, thereby severely hindering probabilistically.

## Reliability-Based Structural Fire Design and Analysis

Reliability-based method has been used for design codes to determine partial load factors and material safety factors. They are derived from First-Order Reliability Methods (FORM) with the intention of ensuring that structural elements or sub-frame assemblies have an appropriately low probability of failure. EN 1990:2002+A1 (2005) specifies that all design solutions should Figure 6.13 Vertical deformation distribution of WTCI in fire. (Abaqus® screenshot reprinted with permission from Dassault Systemes.)

achieve a reliability index (/J) of 3.8 in a building’s conceptual design life (50 years).

### The Basic Reliability Design Principles

In structural fire, design reliability method to assess the fire resistance (capacity) of a building element or structure is sufficient to withstand the fire severity (demand). Due to the complex nature of this problem, the available capacity and demand may be modeled as random variables. The reliability of either a building system or a structural member can be measured in terms of probability. The probability of failure Pf is thus expressed as follows: where

R=fire resistance (capacity),

S=fire severity (demand),

M=safety margin defined as M=R - S.

The equation M=R-S is also known as the limit state function.

The objective of a reliability-based design is to ensure that given the outbreak of any fire, the event (M>0) or G(R,S)>0. This assurance is only possible in terms of the probability P(M> 0). This probability therefore represents a realistic measure of the reliability of the system or a structural member in fire.

If the probability density functions fR(r) and fs(s) of R and S are available or can be approximated, and if R and S are statistically independent, the probability of failure P, may can be expressed as follows: The reliability-based structural fire design and analysis are to work out Pf shown in Equation 6.26. The procedure for reliability-based design and analysis will be introduced in the following sections.

### Reliability-Based Design and Analysis Procedure

6.5.1.2.1 Determination of limit state function

The limit state function M=R-S needs to be determined first. This is based on the type of design problems (whether to design the flexural capacity of a beam under fire or the overall stability of a building in fire).

6.5.1.2.2 Monte Carlo simulation method

The Monte Carlo technique is applicable for either stochastic or probabilistic problem. The process is computational and involves selecting input values at random for use in engineering calculations. Monte Carlo simulation is a choice of probability distributions for the random inputs. It uses the randomness to solve problems that might be deterministic in principle. Monte Carlo methods can be used to sample using a known probability distribution.

It first needs to select a probability distribution for each individual variable. It is also essential to determine the dependencies between simulation inputs. Ideally, input data to a simulation should reflect what is known about dependence among the real quantities being modeled. Probability distributions for each variable need to be created from statistical data or information taken from real-life observations or experimental data.

In the fire safety design, there are some key parameters that will affect the design values. For example, when determining atmosphere temperature, opening factor and fire load density are the two key parameters to affect its value, and they are mutually independent to each other. The probability distribution or the range of these parameters is readily known from design guidelines such as Eurocode and other research as shown in Table 6.1. Therefore, using the available distributions and key statistic index such as mean and standard deviation obtained from Eurocodes design practice (see Table 6.1), the random value of opening factor and fire load density can be generated. Subsequently, the corresponding atmosphere temperature can be calculated based on design formula. The random variables used are determined from large-scale data analysis and tests.

6.5.1.2.3 Determining statistical parameters of the variables

The statistical parameters of design are given by Eurocode 1 (2002) and Eurocode 3 (2005). Using these statistic parameters or the range of the design parameters, the values of the variable can be generated using Monte Carlo simulation. Table 6.1 shows examples of statistical parameters of the variables used in the Monte Carlo simulation.

Table 6.1 Examples of statistical parameters of the variables used in the limit function

 Variable Distribution Units Mean Standard deviation Range Source Opening factor Normal N/A N/A N/A 0.02-0.2 Eurocode 1; Part 1.2 (2002) Fire load density Imposed load Gumbel Extreme type 1 MJ/m2 KN/m2 420 126 1-5 Eurocode 3 (2005) Eurocode 1; Part 1.2 (2002) Yield strength of steel Log-normal MPa 280 28 275-355 Eurocode 1; Part 1.2 (2005) Partial safety factors Normal 1.5-2 Eurocode 1; Part 1.2 (2002)