Case Study for Reliability Analysis for Individual Members
Cai and Fu et al. (2020) developed a new approach for postfire reliability analysis of concrete beams retrofitted with CFRP sheet in bending using the Monte Carlo method. It is introduced here.
6.5.1.3.1 Limit state function
The limit state function of a beam in bending in ambient temperature can be written as follows:
where g(X) is the failure function, X,, X_{2}, ..., X„ are n mutually independent random variables, R is the resistance of the structure, and S is the action effect of the structure. Values of Z greater than 0, less than 0, or equal to 0 indicate that the structure is under a reliable status, a failure status, or a limit status, respectively.
The flexural capacity of RC beams at ambient temperature was (GB, 2010) as follows:
where M_{c} is the flexural capacity of RC beams at normal temperature. With a random variable y„, that represents the uncertainty coefficient of the resistance calculation, the limit state function of RC beams at normal temperature is as follows:
where M_{Gm} is the mean value of M_{G} and Mq„, is the mean of M_{Q}.
The limit state function of RC beams after fire is as follows:
Since the flexural capacity of RC beams was deteriorated after fire exposure, CFRPs can be used to reinforce the bottom of postfire RC beams. If the bonding between CFRP and concrete is assumed to be perfect (GB, 2013), the flexural capacity of postfire CFRPreinforced RC beams can be computed as follows:
where
M_{D} is the flexural capacity of postfire RC beams strengthened with CFRPs,
f_{fu s} is the mean tensile strength of CFRPs,
A_{fe} is the valid sectional area of CFRPs,
ЧС
is the strength use coefficient of CFRPs.
The limit state function of postfire RC beams strengthened with CFRPs was determined as follows:
6.5.1.3.2 Statistical parameters of the variables
The statistical parameters are selected based on the Chinese Code (GB, 2010), EN 199212(2004)), and the research of Cai (2016) and Coile et al. (2014) (Table 6.2).
6.5.1.3.3 Monte Carlo simulation
In the Monte Carlo simulation, the random variables for the limit state function were repeatedly simulated using the program coded in MATLAB, and the reliability can be calculated. The specific procedures are listed as follows:
 • The random variables of the limit state function were integrated with their probability distributions.
 • Random values were simulated repeatedly using the Monte Carlo method with the probability distribution of these random variables.
 • g(X) was calculated using the simulated values.
 • When the number of repetitions reached the preset value, the simulations were terminated.
 • Calculate the reliability based on the simulation results.
Case study for reliability analysis for a whole building
To assess the probability of a whole building failure under fire is more complicated. Van Coile et al. (2014) developed a method considering the following probability:
p_{ig} is the probability of a fire to develop, pf _{u} is the probability of early intervention by the occupants, pf _{s} is the probability active measures, such as sprinklers, pff_{b} is the probability of the fire brigade,
Pf fi is the probability of the structure damage,
Pf, 1 is the probability of the structure fails.
Table 6.2 Statistical parameters of the variables used in the limit function
Symbol 
Variable 
Distribution 
Units 
Bias^{a} (mean) 
CoV'fsttf) 
Nominal value 
Source 
C30 compressive strength 
Lognormal 
MPa 
1.395 
0.15 
20.1 
GB (2010) 

HRB335 steel yield stress 
Lognormal 
MPa 
1.139 
0.07 
335 
GB (2010) 

Live load 
Extreme type 1 
kNm 
0.859 
0.233 
^{_} 
GB (2012) 

Dead load 
Normal 
kNm 
1.060 
0.070 
 
GB (2012) 

Effective depth of section 
Normal 
mm 
1.000 
0.030 
565 
CAI et al. (2016) 

Beam width 
Normal 
mm 
1.000 
0.010 
250 
CAI et al. (2016) 

CFRP crosssectional area 
Normal 
mm^{2} 
1.00 
0.02 
 
CAI et al. (2016) 

CFRP tensile strength 
Weibull 
MPa 
1.152 
0.08 
3,100 
CAI et al. (2016) 

Total model uncertainty 
Normal 
 
1 
0.025 
 
Coile et al. (2014) 

CFRP strip thickness 
Lognormal 
mm 
1.00 
0.010 
0.167 
CAI et al. (2016) 

T (°C) concrete compressive strength reduction factor 
Beta 
(Temperaturedependent, conforms to EN 199212) 
(Temperature dependent) 
^{} 
EN 199212:2004 

T (°C) reinforcement yield stress reduction factor 
Beta 
(Temperaturedependent, conforms to EN 19921 2) 
(Temperature dependent) 
" 
EN 199212:2004 
* Bias: mean value/nominal value. ^{b} CoV: coefficient of variation. ^{c} std: standard deviation.
This analysis process proposed by Van Coile et al. (2014) is shown in Figure 6.14. It comprises two domains: the ‘event instigation’ and ‘response’ domains.
Based on this framework, Hopkin et al. (2017) made a reliability analysis of the probability of failure of a tall building in fire. The Monte Carlo simulation is used to sample different variables, such as fire load density and temperature spread rate; therefore, different fire scenarios can be simulated and the probability of failure of the building can be assessed.
Fire Fragility Functions
Fragility function has been recently used by researchers to characterize the probabilistic vulnerability of buildings to fire. In earthquake engineering, fragility functions are widely used to assess the likelihood of structural damage due to an earthquake. A fragility function provides the probability of exceeding a damage state for a given intensity of earthquake load. Similarly, fire fragility functions can be developed to measure the probability of exceeding a damage state (e.g. column failure, excessive beam deflection) for a given intensity of fire load.
This method is quite new, and the influence of the different uncertain parameters on the functions has not been systematically studied. Uncertainties in fire, heat transfer and structural models, fire load intensity, compartment geometry and openings, the thickness and thermal conductivity of fire protection, and the material degradation, all generate significant variability in the fire fragility. The prevailing parameters in constructing fire fragility functions for steel frame buildings identified through sensitivity analyses are conducted using the Monte Carlo simulations and a
Figure 6.14 Stochastic factors leading to a fireinduced structural failure (Coile et al., 2014).
variancebased method. One important parameter in defining a fragility function is intensity measure for fire load. Gernay et al. (2019) used average fire load (in MJ/m^{2} of floor area) as the intensity measure.
CompartmentLevel Fragility Function
Khorasan et al. (2016) and, Gernay et al. (2019) developed the below equation:
where
PfiHfi is the probability of reaching a damage state condition to the occurrence of a fire Нр,
F_{D}_{Hfi} (.) is the cumulative distribution function of the demand relative to the fire Hp,
f_{c}() is the probabilistic distribution function of capacity.
is obtained through repeated structural fire analysis under fire load densities (q values) in the same compartment. The analysis needs to be repeated sufficient times to be able to work out P_{F}Hfi Based on P_{F}j_{Hfi}, the fragility function is built by fitting a function to the obtained points, assuming a lognormal distribution:
where
q is the fire load (MJ/m^{2}),
Ф[»] is the standardized lognormal distribution function.
There are two parameters c and £: c is the mean of lognormal distribution,
C is the standard deviation of the lognormal distribution. c and C are determined by the best fit to the data points from structural fire analysis.
BuildingLevel Fragility Function
Fire fragility functions should first be developed for each compartment under different fire scenarios and then combined to derive a fire fragility function for the entire building. The combined fragility function is also a lognormal function, the same as in Equation 6.35. Khorasan et al. (2016) and Gernay et al. (2019) developed the following equations to work out the two lognormal parameters:
where
q_{c} is the mean of combined lognormal distribution,
C_{c} is the standard deviation of the combined lognormal distribution, Other parameters are as follows:
n is the number of ‘nominally identical but statistically different’ fragility curves,
Cj is the median associated with each individual fragility curve, p, is the conditional probability for a fire in compartment, i, a fire occurs in the building,
P is the vector of the probabilities p„
Z is the vector of the variances,
C;^{2} is associated with each individual fragility function,
A is the vector of the expected values (In c,), and
Based on the above approaches, fire fragility curve can be derived. Figure 6.15 shows typical fire fragility curves for different types of buildings.
Other Probabilistic Approaches in Fire Safety Design
Fu (2020) used the Monte Carlo simulation to simulate a probability distribution for different variables for the random inputs for fireinduced tall building collapse using machine learning. This will be introduced in detail in Chapter 9.
Figure 6.15 Fire fragility curves for different types of buildings.