Sensitivity and Uncertainty Analysis of a Fire Spread Model with Correlated Inputs

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Procedia Engineering 00 (2017) 000–000 Procedia Engineering Engineering00 211 (2018) 403–414 (2017) 000–000 Procedia Engineering 00 (2017) 000–000 Procedia Engineering 00 (2017) 000–000 Procedia Procedia Engineering Engineering 00 00 (2017) (2017) 000–000 000–000 Procedia Engineering 00 (2017) 000–000

Sensitivity and Uncertainty Analysis of a Fire Spread Model with Sensitivity and Uncertainty Analysis of a Fire Spread Model with Correlated Inputs Sensitivity and Uncertainty Analysis of and a Fire SpreadEngineering Model with 2017 8th International Conference on Fire Science Fire Protection Correlated Inputs Sensitivity and Uncertainty Analysis of a Fire Spread with a, b a Xiao LI *, George HADJISOPHOCLEOUS , Xiao-qian SUN (onUncertainty the Development of Performance-based Fire Code) Model Sensitivity and Analysis of a Fire Spread Model with Correlated Inputs Sensitivity and Uncertainty Analysis of aa Fire Spread Model with a, b a Xiao LI *, George HADJISOPHOCLEOUS , Xiao-qian SUN Correlated Inputs Sensitivity and Uncertainty Analysis of Fire Spread Model with International Consultant (Shanghai) Ltd, 37F 1045 Huaihai Road, Shanghai, 200031, China, Correlated Inputs Sensitivity Arup and Uncertainty Analysis of a Fire Spread Model with a, a Carleton University, Ottawa, Canada, 1125 Colonel By Drive, bOttawa, ON K1S 5B6 Correlated Inputs Xiao LIa,*, George HADJISOPHOCLEOUS Xiao-qian Arup International Consultant (Shanghai) Ltd, 37F 1045 Huaihai Road,, Shanghai, 200031,SUN China, Correlated Inputs b a Xiao LI *, George HADJISOPHOCLEOUS , Xiao-qian Carleton University, Ottawa, Canada, 1125 Colonel By Drive, bOttawa, ON K1S 5B6SUNa Correlated Inputs Xiao LIa,*, George Xiao-qian Arup International ConsultantHADJISOPHOCLEOUS (Shanghai) Ltd, 37F 1045 Huaihai Road,, Shanghai, 200031,SUN China, a a

b b

a

Abstract

a, b a Xiao LI *, George HADJISOPHOCLEOUS ,, Shanghai, Xiao-qian SUN a, b Arup International Consultant (Shanghai) Ltd,1125 37F Colonel 1045 Huaihai Road, 200031, China,aa a,*, George bOttawa, Carleton University, Ottawa, Canada, By Drive, ON K1S 5B6SUN Xiao LI HADJISOPHOCLEOUS Xiao-qian Xiao LI *, George HADJISOPHOCLEOUS , Xiao-qian SUN a, b Arup International Consultant (Shanghai) Ltd,1125 37F Colonel 1045 Huaihai Road, 200031, China,a Carleton University, Ottawa, Canada, By Drive, Ottawa, ON K1S 5B6SUN Xiao LI *, George HADJISOPHOCLEOUS , Shanghai, Xiao-qian Arup International Consultant (Shanghai) Ltd, 37F 1045 Huaihai Road, Shanghai, 200031, China, a a a a aArup aArup

b b

b Abstract Carleton University, Ottawa, Canada, 1125 Colonel By Drive, Ottawa, ON K1S 5B6 China, International Consultant (Shanghai) Ltd, 37F 1045 Huaihai Road, Shanghai, 200031, b Sensitivity and uncertainty analysis is a very important toolCanada, to identify model uncertainties in quantitative International Consultant (Shanghai) Ltd,1125 37Fand 1045treat Huaihai Road, Shanghai, 200031, Carleton University, Ottawa, Colonel By Drive, Ottawa, ON K1S 5B6 China, fire risk analysis. An b Arup International Consultant (Shanghai) Ltd, 37F 1045 Huaihai Road, Shanghai, 200031, China,and selected input Carleton University, Ottawa, Canada, 1125 Colonel By Drive, Ottawa, ON K1S 5B6 b existing Fire Spread model with correlated input variables are presented for sampling-based sensitivity analysis, Abstract Carleton University, Ottawa, Canada, 1125 Colonel By Drive, Ottawa, ON K1S 5B6 b Sensitivity and uncertainty analysis is a very important toolCanada, to identify treatBy model in quantitative fire risk analysis. An Carleton University, Ottawa, 1125and Colonel Drive,uncertainties Ottawa, ON K1S 5B6 Abstract variables include fire growth rate, fire resistance rating and its standard deviation, fire load density and its standard deviation. A sampling existing AbstractFire Spread model with correlated input variables are presented for sampling-based sensitivity analysis, and selected input approach isand proposed to dealanalysis with structure ofits input variables, which introduces a noise and can deviation. transform correlated Sensitivity uncertainty is correlated aresistance very important tool to identify and treat model uncertainties interm quantitative fire risk analysis. An Abstract include variables fire growth rate, the fire rating and standard deviation, fire load density and its standard A sampling Abstract Sensitivity and uncertainty analysis is a veryinput important toolare to presented identify and treat model uncertainties in fire quantitative fire risk analysis. An input variable structure into an independent one. Furthermore, sensitivity analysis of input variables of spread model is performed and existing Fire Spread model with correlated variables for sampling-based sensitivity analysis, and selected input Abstract approach isand proposed to dealanalysis with the structure ofto input variables, which introduces a noiseinterm and can fire transform correlated Sensitivity uncertainty is correlated a veryinput important tool identify and treat model uncertainties quantitative risk analysis. An Abstract existing Fire Spread model with correlated variables are presented for sampling-based sensitivity analysis, selected input an order of variable is given. Results show that fire resistance rating and itsinput standard deviation twoand very important inputAn variables include firesensitivity growth rate, fire resistance rating and its standard deviation, fire load density and itsare standard deviation. A sampling Sensitivity and uncertainty analysis is a very important tool to identify and treat model uncertainties in quantitative fire risk analysis. input variable structure into an independent one. Furthermore, sensitivity analysis of variables of fire spread model is performed and existing Fire Spread with input variables are presented fortreat sampling-based sensitivity analysis, anddeviation. selected input Sensitivity and uncertainty analysis isfire very important tool to identify and model uncertainties interm quantitative fire riskthe analysis. An variables include firemodel growth rate,correlated fire rating its standard deviation, fire load density and its are standard A sampling while standard deviation ofis load density isand the least sensitive parameter. Further discussions provided on approach isand proposed to deal with the correlated structure of input variables, which introduces a deviation noise and can transform correlated Sensitivity uncertainty analysis aaresistance very important tool to identify and treat model uncertainties in quantitative fire risk analysis. An existing Fire Spread model with correlated input variables are presented for sampling-based sensitivity analysis, and selected input an order of variable sensitivity is given. Results show that fire resistance rating and its standard are two very important input variables include fire growth rate, fire resistance rating and its standard deviation, fire load density and its standard deviation. A sampling Sensitivity uncertainty is correlated a very important tool to identify and treat model uncertainties in quantitative fire risk analysis. An existing Fire Spread model with correlated input variables are presented for sampling-based sensitivity analysis, and selected input approach isand proposed tointo dealanalysis with the structure of input variables, which introduces a noise and can transform correlated effectiveness of the technique and the use theisand results of sensitive the analysis. input variable structure an independent one. Furthermore, sensitivity analysis of input ofterm fire spread model performed and existing Fire Spread model with correlated input variables presented for sampling-based sensitivity analysis, and selected input variables include firesampling growth rate, fire resistance rating its standard deviation, fire load variables density and its are standard deviation. A sampling while standard deviation of fire load density theare least parameter. Further discussions onisthe approach is proposed tointo deal with the structure input variables, whichof introduces a noise term andprovided can transform correlated existing Fire Spread model with correlated input variables are presented for sampling-based sensitivity analysis, and selected input variables include firesensitivity growth rate, fire correlated resistance rating andof its standard deviation, fire load density and itsare standard deviation. A sampling sampling input variable structure an independent one. Furthermore, sensitivity analysis input variables of fire spread model is performed and an order of variable is given. Results show that fire resistance rating and its standard deviation two very important input variables include fire growth rate, fire resistance rating and its standard deviation, fire load density and its standard deviation. A approach is proposed to deal with the correlated structure of input variables, which introduces a noise term and can transform correlated effectiveness of the sampling technique and the use the results of the analysis. input variable structure into an independent one.show Furthermore, sensitivity analysis input variables of fire spread model is performed variables include firesensitivity growth rate, fire correlated resistance rating andof its standard deviation, fire load density and itsare standard deviation. A sampling approach is proposed to deal with the structure input variables, which introduces aa deviation noise and can transform correlated an order of is given. Results fire resistance rating andof its standard two important input and variables while standard deviation ofby fire load density is the least sensitive parameter. Further discussions are provided onisthe © 2017 The Authors. Published Elsevier Ltd. that Selection and peer-review under responsibility of the Academic Committee approach is variable proposed tointo deal with the correlated structure of input variables, which introduces noiseofterm term and canvery transform correlated input variable structure an independent one. Furthermore, sensitivity analysis of input variables fire spread model performed an order of sensitivity is given. Results show that fire resistance rating andofintroduces itsinput standard are two important input and approach is variable proposed tointo deal with the correlated structure of input variables, which a deviation noiseofterm and canvery transform correlated input variable structure an independent one. Furthermore, sensitivity analysis variables spread model performed and variables while standard deviation of fire load isresults the least sensitive parameter. Further discussions are provided onis effectiveness of the sampling and thedensity use thethat of and the analysis. input variable structure into antechnique independent one. Furthermore, sensitivity analysis ofits input variables of fire fireare spread model isthe performed of 2017 ICFSFPE 2017. an order of variable sensitivity is given. Results show fire resistance rating and standard deviation two very important input and © The Authors. Published by Elsevier Ltd. Selection peer-review under responsibility of the Academic Committee variables while standard deviation of fireand load density isresults thefire least sensitive parameter. Further discussions are provided onisthe input variable structure into antechnique independent one. Furthermore, sensitivity analysis variables of fireare spread model performed an order of variable sensitivity is given. Results show that resistance rating andofits itsinput standard deviation two very important input and effectiveness of the sampling the use the of the analysis. an order of variable sensitivity is given. Results show that fire resistance rating and standard deviation are two very important input variables while standard deviation of fireand load isresults thefire least parameter. are two provided on the input of order ICFSFPE 2017. effectiveness of the sampling technique thedensity use thethat of sensitive the analysis. an of variable sensitivity is given. Results show resistance rating and itsFurther standarddiscussions deviation are very important variables while standard deviation of fire load density is the sensitive parameter. Further discussions on variables while standard deviation ofby fire load density the least least sensitive parameter.under Further discussions are are provided on the the Committee Keywords: risk of assessment; modeling; fire growth, statistics © 2017 The Authors. Published Elsevier Ltd. Selection peer-review responsibility of provided the Academic effectiveness the sampling technique and the use theis results of and the analysis. variables while standard deviation of fire load density is the least sensitive parameter. Further discussions are provided on the Committee © 2018 The Authors. Published by Elsevier Ltd. effectiveness of the sampling technique and the use the results of the analysis. © 2017 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Academic effectiveness of the sampling technique and the use the results of the analysis. of ICFSFPE 2017. Keywords: risk assessment; modeling; fire growth, statistics Peer-review responsibility ofbythe organizing committee 2017.under responsibility of the Academic Committee effectiveness of the sampling technique and the use the results of of theICFSFPE analysis. © 2017 Theunder Authors. Published Elsevier Ltd. Selection and peer-review

of ICFSFPE 2017. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Academic Committee © 2017 The Authors. of ICFSFPE 2017. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Academic Committee © 2017 The Authors. Nomenclature © 2017 The Authors. bygrowth, Elsevier Ltd. Selection and peer-review under responsibility of the Academic Committee Keywords: risk assessment; modeling; fire statistics of ICFSFPE 2017. Published © 2017 The Authors. Published bygrowth, Elsevier Ltd. Selection and peer-review under responsibility of the Academic Committee of ICFSFPE 2017. Keywords: risk assessment; modeling; fire statistics of ICFSFPE 2017. 2 Nomenclature ) growth, statistics α fire growth rate (kW/sfire Keywords: risk assessment; modeling; of ICFSFPE 2017. Keywords: number risk assessment; modeling; fire growth, statistics 2 growth, statistics nS of samples Keywords: risk modeling; ) growth, statistics α fire growth rate (kW/sfire Nomenclature Keywords: risk assessment; assessment; modeling; fire Keywords: risk assessment; modeling; fire fire spread time (minute) tnS sp Nomenclature number of samples 2 growth, statistics Nomenclature α fire spread growth rate (kW/s ) (kg/m2) FLDμ mean of Fire Load Density fire time (minute) tNomenclature sp 2 ) α fire growth rate (kW/s nS number samples 2 FRRμ mean of of Fire Resistance (minute) 2 Rating Nomenclature ) FLDμ mean of Fire Load Density Nomenclature ) (kg/m α fire growth rate (kW/s 2 nS number of samples Nomenclature tFRRμ fire spread time (minute) FLDσ standard deviation of Fire Load Density (kg/m2) sp ) α growth rate (kW/s 2 mean of Fire Resistance Rating (minute) nS number of samples 2 ) α fire growth rate (kW/s 2 spread time (minute) tα sp ) fire growth rate (kW/s FLDμ mean of Fire Load Density (kg/m ) 2 FRRσ standard deviation of Fire Resistance Rating (minute) 2 nS number of samples ) FLDσ standard deviation of Fire Density (kg/m ) Load αsp growth rate (kW/s fire spread time (minute) tnS 2 number of samples ) FLDμ mean of Fire Load Density (kg/m nS number of samples FRRμ mean of Fire Resistance Rating (minute) noise Factor of Fire Load Density NF fire spread time (minute) tFLDμ 2 FLD sp FRRσ standard deviation of Fire Resistance Rating (minute) nS number of samples ) mean of Fire Load Density (kg/m fire spread time (minute) sp 2 mean of Fire Fire Resistance (minute) fire time (minute) ttFRRμ sp FRR FLDσ standard deviation of FireRating Load Density (kg/m2) noisespread Factor of Fire Resistance Rating NF ) FLDμ mean of Load Density (kg/m 2 noise Factor of Fire Load Density NF fire spread time (minute) tFRRμ mean of Fire Fire Resistance Rating (minute) sp FLD 2 2) FLDμ mean of Load Density (kg/m ) FLDσ standard deviation of Fire Load Density (kg/m ) FLDμ mean of Fire Load Density (kg/m FRRσ standard deviation of Fire Resistance Rating (minute) 2 FRRμ mean of Fire Fire Resistance Rating (minute) 2 noise Factor of Fireof Resistance Rating NF ) FLDμ mean of Load Density (kg/m FRR ) FLDσ standard deviation Fire Load Density (kg/m FRRμ mean of Fire Resistance Rating (minute) 2 FRRσ standard deviation of Fire Resistance Rating (minute) FRRμ mean of Fire Resistance Rating (minute) NF noise Factor of Fire Load Density FLD ) FLDσ standard deviation of Fire Load Density (kg/m 2 FRRμ mean of Fire Resistance Rating (minute) FRRσ standard deviation of Fire Resistance Rating (minute) 2) FLDσ standard deviation of Fire Load Density (kg/m noise Factor of Fire Load Density NF FLD FLDσ standard deviation ofResistance Fire Resistance Load Rating Density (kg/m NF noise Factor of Fire 2) FRR FRRσ standard deviation of Fire Rating (minute) 1. Introduction ) FLDσ standard standard deviation ofLoad Fire Density Load Density (kg/m noise Factor of Fire Fireof NF FRRσ deviation Fire Resistance Rating (minute) noise Factor of Resistance RatingRating NFFLD FRR FRRσ standard deviation Fire Resistance (minute) noise Factor of Fire Fireof Load NF FLD FRRσ standard deviation ofResistance Fire Density Resistance noise Factor of RatingRating (minute) NF 1. Introduction FRR noise Factor of Fire Load Density NF FLD noise Factor of Fire Fire Load Density NF FLD noise Factor of Resistance Rating NFQuantitative FRR fire risk assessment models have been frequently proposed to predict fire risks, especially with the shift noise Factor of Fire Fire Load Density NF FLD noise Factor of Resistance Rating NF FRR noise Factor of Fire Resistance Rating NF FRR 1. Introduction from torisk performance-based fire have safetybeen design. Generally those to firepredict models usedespecially to estimate results and noise Factor of Fire Resistance Rating NF FRRprescriptive Quantitative fire assessment models frequently proposed firearerisks, with the shift 1. Introduction consequences of assigned fire scenarios with certaindesign. amountGenerally of input variables outputs, presuming that the modeland is 1. Introduction from prescriptive to performance-based fire safety those fireand models are used to estimate results Quantitative fire riskproblems. assessment models the haveuncertainty been frequently proposed to predict fire risks, especially with themakes shift 1. Introduction valid to solve the given However, of the input variables propagates through the model and consequences of assigned fire scenarios with certain amount of input variables and outputs, presuming that the model is 1. Introduction Introduction Quantitative fire assessment models have frequently proposed to with the shift 1. from prescriptive torisk performance-based fire safetybeen design. Generally those firepredict models arerisks, usedespecially to estimate results and the uncertainty of the results unpredictable, specifically when thethe model isvariables a submodel offire large system. 1. Introduction Quantitative fire risk assessment models have been frequently proposed to predict fire risks, especially with themakes shift valid to solve the given problems. However, the uncertainty of input propagates through the model and from prescriptive to performance-based fire safety design. Generally those fire models are used to estimate results and consequences of assigned fire scenarios with certain amount of input variables and outputs, presuming that the model is Quantitative fire risk assessment models have been frequently proposed to predict fire risks, especially with the shift from prescriptive torisk performance-based fire safety design. Generally those firepredict models arerisks, used to estimate results and the uncertainty of the results unpredictable, specifically when the model is a submodel offire large system. Quantitative fire assessment models have been frequently proposed to especially with the shift consequences of assigned fire scenarios with certain amount of input variables and outputs, presuming that the model is Quantitative fire risk assessment models have been frequently proposed to predict fire risks, especially with the shift valid to solve the given problems. However, the uncertainty of the input variables propagates through the model and makes from prescriptive to performance-based fire safety design. Generally those fire models are used to estimate results and Quantitative fire risk assessment models have been frequently proposed to predict fire risks, especially with the shift consequences of assigned fire scenarios with certain amountof of input outputs, presuming that the model is from prescriptive to fire safety design. Generally those fire models are used to estimate results and valid to solve the given problems. However, the uncertainty the inputvariables variables propagates model and makes from prescriptive to performance-based performance-based fire safety design. Generally those fireand models are through used to the estimate results and the uncertainty of the results unpredictable, specifically when the model is a submodel of large system. consequences of assigned fire scenarios with certain amount of input variables and outputs, presuming that the model is from prescriptive to performance-based fire the safety design. Generally those fireand models are through used to the estimate results and valid to solve the given problems. However, uncertainty of the input variables propagates model and makes consequences of assigned fire scenarios with certain amount of input variables outputs, presuming that the model is the uncertainty of the results unpredictable, specifically when the model is a submodel of large system. consequences of assigned fire scenarios with certain amount of input variables and outputs, presuming that the model is valid to solve the given problems. However, the uncertainty of the input variables propagates through the model and makes *the Corresponding author. Tel.: +86-21-3118-8561; consequences ofofassigned fire scenarios with certain amount of input variables and outputs, presuming that theand model is uncertainty the results unpredictable, specifically when the model is a submodel of large system. valid to solve the given problems. However, the uncertainty of the input variables propagates through the model makes valid toaddress: solve the given problems. However,specifically the uncertainty ofthethe inputisvariables propagates through the model and makes the uncertainty of the results unpredictable, when model a submodel of large system. E-mail [email protected] valid to solve the given problems. However, the uncertainty of the input variables propagates through the model and makes *the Corresponding author. Tel.: +86-21-3118-8561; uncertainty of of the the results results unpredictable, unpredictable, specifically specifically when when the the model model is is aa submodel submodel of of large large system. system. the uncertainty the uncertainty of the results unpredictable, specifically when the model is a submodel of large system. E-mail address: [email protected] *1877-7058 Corresponding author. Tel.: +86-21-3118-8561; © 2018 The Authors. Published by Elsevier Ltd. *Peer-review Corresponding author. Tel.: +86-21-3118-8561; E-mail address: [email protected] under responsibility of the organizing committee of ICFSFPE 2017 * Corresponding Tel.: +86-21-3118-8561; E-mail address:author. [email protected] *10.1016/j.proeng.2017.12.029 Corresponding author. Tel.: +86-21-3118-8561; E-mail address: [email protected] * Tel.: * Corresponding Corresponding author. Tel.: +86-21-3118-8561; +86-21-3118-8561; E-mail address:author. [email protected] * Corresponding author. Tel.: +86-21-3118-8561; E-mail E-mail address: address: [email protected] [email protected] E-mail address: [email protected]

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Sensitivity and uncertainty analysis are becoming increasingly widespread in many fields of engineering and science. Hamby [1] explained some reasons for sensitivity and uncertainty analysis, including determining which parameters require additional research for strengthening the knowledge base, which parameters are insignificant and which inputs contribute most to output variability. Cacuci et al. [2] stated that the methods of sensitivity and uncertainty analysis are based on deterministic and statistical methods, which can be further categorized as follows: sampling-based methods (simple random sampling, stratified importance sampling, and Latin Hypercube sampling), first and second-order reliability algorithms, variance-based methods, and screening design methods. The most popular uncertainty analysis technique is sampling based methods (also called Monte Carlo method) [3,4]. Such methods involve the generation and exploration of a mapping from uncertain analysis inputs to uncertain analysis results. Among the sampling based methods, the simple random sampling approach is the simplest but it needs a large amount of samples to effectively cover the distribution range, and it becomes non-feasible to conduct uncertainty analysis when the inputs are multivariate and the system takes time to generate outputs. Most designs use Stratified Sampling to improve the rate at which estimated quantities converge to the true quantities [5]. The principle behind stratified sampling with a single variate is to partition the sample space into non-overlapping regions and to guarantee sampling from each region. The purpose for Stratified Sampling is to ensure that all parts of the sample space are represented, for improved (i.e. less uncertain) mean and variance estimates. The idea of fully covering the range of each parameter is further extended in the Latin Hypercube sampling procedure [6], which is a frequently used method and many applications exist in the fire safety engineering area [7-10]. For Latin hypercube sampling, the range of all the input variables are exhaustively divided into same number of disjoint intervals of equal probability and one set of input values is selected at random from each interval. This requires that the investigated input variables should have clear and continual distributions, such as normal distribution, but practically these are sometimes hard to define. Furthermore, if a correlation structure exists among the input variables, but the actual sampling takes place as if the input variables were independent, the theoretical properties of the statistics formed from the output may no longer be valid [11]. A detailed illustration of differences of the Random Sampling, Stratified Sampling and Latin hypercube sampling can be found in [12]. Until now, only a few studies were done to deal with the sensitivity analysis of correlated variables [11,13-16]. A method for inducing a desired rank correlation matrix on multivariate input vectors for simulation studies was developed by Iman and Conover [11,13]. The primary intention of this procedure is to produce correlated input variables for use with computer models. Xu [16] used a regression-based method to quantitatively decompose the variances in the model output into partial variances contributed by the correlated and uncorrelated variations of parameters. Mara et.al [15] created a procedure that generalizes Xu’s [16] approach to the case of conditionally dependent inputs by an ANOVA decomposition of the original model. Jacques et.al [14] proposed a variance-based application of the multidimensional generalization of classical sensitivity indices, and applied group sensitivities for models with correlated inputs. Sebastien [17] considered a method based on local polynomial approximations for conditional moments to deal with correlated inputs. Xu [18] extended Fourier amplitude sensitivity test (FAST) to models with correlated parameters. All the above studies except the work of Iman were variance-based methods and actually do not involved with sampling of input variables. In this study, uncertainty and sensitivity analysis of a fire spread model with dependent input variables and discrete distributions is investigated. A sampling method adapted to this specific model is proposed and parameter sensitivity analysis is performed. 2. Description of Fire Spread Model The Fire Spread model is a probabilistic model using Bayesian Network approach and probability theory [19,20]. It is one of the submodels of CUrisk, which is a Fire Risk Analysis computer model being developed at Carleton University. This sub-model is used to calculate the probability of fire spread across rooms throughout a building at different simulation times. The results reflect the combination of the fire development process and the boundary failure process. The probability of fire spread from room B to room A, P(a|b), is written as, P(a | b)  P(a|a') Pi g (a'|b)

And,

Pig (a ' | b)  P(a'|b) P(b)

(1)

(2) If a room has several adjacent fire rooms, heat could be transferred to this room simultaneously from all adjacent fire rooms; Assuming the room A has two adjacent fire rooms B and C, the probability of fire spread to room A due to fire rooms B and C is, P(a | b, c)  P(a | a ') Pig (a ' | b, c) (3)



Xiao LI et al. / Procedia Engineering 211 (2018) 403–414 Xiao LI et al / Procedia Engineering 00 (2017) 000–000

And,

405

Pig (a ' | b, c)  P(a ' | b) P(b)  P(a'|c) P(c)  P(a ' | b)P(a ' | c) P(b) P(c)

(4) Where, P(a|b) is the probability of fire spread from room B to room A, P(a|a’) is the probability of fire growth from ignition to a fully-developed fire in room A; Pig(a’|b) is the probability of fire ignition in room A due to the heat transfer from room B; P(a’|b) is the probability of barrier failure indicating the probability that heat is transferred from the fire room B to the adjacent room A and ignites the combustible materials in room A; P(b) is the probability of a fully-developed fire occurring in room B; Pig(a’|b,c) is the probability of fire ignition in room A due to heat transfer from room B and room C. Both the probability of barrier failure and the probability of fire growth to a fully-developed fire are assumed to follow the properties of a normal distribution (µ, σ). The probability of boundary failure is calculated based on the equivalent failure time in a real fire to the Fire Resistance Rating of the building components, and failure probability of a building component is actually the cumulative probability from the time of flashover in this room. The mean and standard deviation of fire resistance of a building component exposed to a fully developed fire are, FRR (i ) bf ,i  (  / 230)3/ 2 (5) FRR (i)  bf ,i  (  / 230)3/ 2 (6) Where, FRRµ(i) is the Fire Resistance Rating (FRR) of a building component i, which indicates the mean of the failure time of a building component (for example, a wall) when subjected to the standard ISO 834 fire; FRRσ(i) is standard deviation of the failure time under the standard ISO 834 fire for i; µbf,i is the mean of the failure time of building component i when subjected to a fully developed compartment fire; σbf,i is the standard deviation of the failure time of building component i subjected to a fully developed compartment fire; and β is a parameter of the Japanese parametric model for compartment boundaries, (K∙s-1/6), which is given by the expression, 1/3

 A Ho    3.0T0  o   A k c   T 

AT k  c   i AT ,i ki i ci

Ao H o   j Ao, j H o, j

(7) (8)

(9) Where, T0 is ambient temperature; ki is the thermal conductivity of boundary material i, kW/(m∙k); ρi is the density of the boundary material i, kg/m3; ci is the specific heat of the boundary material i, kJ/kg∙K; Ao,j is the area of the jth opening (door or window), and Ho,j is the height of the jth opening. The probability of fire growth to a fully-developed fire depends on several factors including the fire load density, the geometry of the compartment and its ventilation conditions, and the availability of fire suppression systems. More details are given in [19]. Input parameters of the Fire Spread model include simulation time and step size, ambient temperature, type of each room (fire load density and its standard deviation), fire growth rate of each room, Fire Resistance Rating and its standard deviation and also materials of each building component (wall, floor/ceiling, door), size of each room and its openings, linking type between every two room on the same floor and across floors. The outputs of the Fire Spread model are probabilities of fire spread (0 to 1) at each time step for each room, and also the ignition and flashover time of each room. As a submodel of a large fire risk analysis model, the outputs of the Fire Spread model will be read by other models of CUrisk in order to predict life risk and property damages for the fire scenario being considered. 3. Fire Scenario A fire scenario is designed to occur in a two-storey building shown in Fig. 1. The 12 rooms in the building are identical and measure 4 meters wide by 5 meters deep by 3 meters high. Each room has a window (1 meter high by 1 meter wide) and a door measured 1 meter wide by 2 meters high. A corridor of 2 meter width separates the rooms, and also connects the 6 rooms through the doorways.

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1m

1m

Room 3

Room 5

Room 6

5m

Room 4

Room 2

2m

Room 1

4m

Fig. 1. Two-storey building plan of Fire scenario.

It is assumed that fires in all the rooms in the building would have the same fire growth rate, and same fire load density and its standard deviation. Similarly, all the rooms are constructed with same type of building components (including wall, floor/ceiling, door and window), and each component has a specific Fire Resistance Rating. Ambient temperature is assumed to be 20°C. The fire origin room is assigned to be Room 1 on the first floor, and fire destination room is set to be Room 6 on the second floor. Fire suppression measures such as sprinklers and fire department are not considered since the fire spread process is the focus of this study. The maximum fire spread time used for the simulation is 600 minutes, and the time step size is 1 minute. A completed fire spread process denotes the time duration from ignition of Room 1 to 100% probability of fire spread in Room 6, and this time duration is named as the fire spread time (tsp), and is chosen as the single model response in the sensitivity analysis later. tsp can well represent the level or severity of fire spread for the given model inputs. In performance-based design, parameters such as fire growth rate (α), Fire Resistance Rating (FRRμ) and Fire Load Density (FLDμ) are always important and their values significantly affect fire consequences. Similarly, those parameters are also key parameters in the Fire Spread model. Therefore, input parameters considered in this uncertainty and sensitivity study include, the fire growth rate (α), Fire Resistance Rating (FRRμ) and its standard deviation (FRRσ), and Fire Load Density (FLDμ) and its standard deviation (FLDσ). Please note the presence of two pairs of dependent input variables, (FRRμ, FRRσ) and (FLDμ, FLDσ). 4. Sampling Technique And Sensitivity Analysis Approaches 4.1. Sampling technique There are several studies on uncertainty and sensitivities analysis with correlated variables as introduced earlier, and only Iman’s method [11,13] deals with the sampling technique. To use this method, however, the existing correlation structure among the input variables should be specified as accurately as possible. Iman’s method works with well-defined uncertainty of variables and only linear models with correlations are supported, which is hard to be applied to models with complex input vectors and non-linear output, such as the model discussed later in this study. For the dependent input variables, such as the two pairs of dependent variables (FRRμ, FRRσ) and (FLDμ, FLDσ) in the fire spread model, Saltelli [5] stated that dependent input samples are more laborious to generate and, even worse, the sample size needed to compute sensitivity measures for dependent samples is much higher than in the case of uncorrelated samples. For this reason it is advised to work on uncorrelated samples as much as possible. Meanwhile he suggested that dependent variables can be treated as explicit relationships with a noise term, and in this way dependent variables can be treated as independent ones. Inspired by this suggestion, a sampling method for dependent variable pairs is proposed below, which considers a combination of independent variables to achieve a desired correlation structure: For dependent input variable pair (X1, X2), and if X2 is an dependent variable of X1, X2 will be considered as noise of X1; that is, instead of sampling X1 and X2, X1 and X3 will be sampled, with variable X3 being a factor describing noise of X1, and consider X2 as a function of X1 and X3. An example is given by assuming a simple linear combination of X1, X2 and X3, as



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following: given a sample x1 from sample space of variable X1, and by defining X3 as a variable of noise factor with uniform distribution U(0,1) and x3 is a sample from it, a sample x2 from variable X2 can be obtained by x2 = function(x1) = x3 ∙x1. Let’s name variable X3 as Noise Factor, which will be assumed to be a uniform distribution U(0,1) in this study. The dependent relationship of X2 and X1 can be shown as below, X X1  X 3 2 (10) By doing this, instead of dealing with dependent variable pairs (X1, X2), independent variables (X1, X3) are sampled. As mentioned in the Introduction section of this paper, random sampling is not a choice of sampling technique in this study due to its requirement of large amount of samples, and the resulting extremely long computer running time. Also discussed is Latin hypercube sampling method which is not suitable for this study because of the presence of a discrete distribution of a variable presented later. In this study, the Stratified Sampling technique is used to sample each input variable and Noise Factors when the variable has a continuous probability distribution, such as normal or uniform distribution. This will be illustrated in detail later. 4.2. Input uncertainty analysis Fire growth rate (α) In the Fire Spread model, the input fire is a t-square fire, and four different fire growth rates are used as input data, which denote slow, medium, fast and ultrafast fires, with fire growth rates of 0.00293, 0.01172, 0.0469 and 0.1876 kW/s 2, respectively [21,22]. One of these fire growth rates is assigned to each room. According to a study of Holborn [23], the distributions of fire growth parameter values observed were reasonably well approximated by the log-normal distribution, based on the Greater London Area data from real fire incidents. Similar applications are also employed at [7,10]. In order to study the impact of the fire growth rates on the model output, the fire growth rate is characterized by a log-normal distribution, logN (-6.5, 22). The fire growth rates are sampled from the range (0.003, 0.19). Fire Resistance Rating (FRRμ) and its standard deviation (FRRσ) The Fire Resistance Rating (FRRμ) and its deviation (FRRσ) are two very important input parameters of the Fire Spread model, and are therefore given great emphasis in terms of sampling and analysis. FRRμ and FRRσ are assigned to 4 building components, i.e., wall, floor, door and window in the Fire Spread model. In practice, minimum Fire Resistance Ratings (FRRμ) of building components are required by building codes, and those ratings are in the form of certain discrete values (in minutes) based on occupancy type, building height, etc. Therefore, in this paper FRRμ is assumed as a variable with discrete distribution of equal probability. Five sets of FRRμ combination samples are assigned as shown in Table 1, covering the Fire Resistance Rating requirements in the NBCC (National building code of Canada 2010) [24]. For instance, sample #2 is chosen in line with residential buildings up to 2 storeys. The Fire Resistance Ratings of windows are based on experimental findings [25]. Table 1. Fire Resistance Rating (in minutes) samples and probability. Components

Wall

Door

Ceiling/floor

Window

Samples #

(FRRWμ, min)

(FRRDμ, min)

(FRRFμ, min)

(FRRWDμ, min)

Probability

1

30

10

30

3

0.2

2

45

20

45

4

0.2

3

60

30

60

5

0.2

4

90

40

90

8

0.2

5

120

50

120

10

0.2

The standard deviation of the Fire Resistance Rating (FRRσ) is a dependent variable of the Fire Resistance Rating (FRRμ), and both of the two input parameters are processed to estimate the probability of boundary failure of different building components. In the Fire Spread model, FRRσ is actually a parameter used to adjust the model output values to be reasonable. Later it was found that this parameter is very useful to represent the differences of various construction materials. For instance, failure time of walls of the same Fire Resistance Rating but different construction materials may be different if exposed to the same realistic fire. This point will be discussed later.

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The sampling technique for dependent variable pairs detailed in the previous section is employed to sample FRRσ, according to which Noise Factor will follow U(0,1). Noise factor of FRRμ is named as NFFRR, and thus FRRσ = FRRμ × NFFRR. All the components of each sample share the same values of Noise Factor. Fire load density (FLDμ) and its standard deviation (FLDσ) In the Fire Spread model, the fire load density FLDμ of each room is an input parameter and it is used to calculate the duration of the fully-developed fire phase if the room flashovers or the maximum fire growth time if the fire cannot reach flashover. Meanwhile, the standard deviation (FLDσ) is used to calculate the maximum fire growth time in cases when flashover does not occur in the room, details of data manipulation can be found at [19,20]. Seven input options of occupancy types were originally given in the fire spread model, corresponding to different combinations of (FLDμ, FLDσ) [26], which include dwelling (30.1,4.4), office (24.8,8.6), school (17.5,5.1), hospital (25.1,7.8), hotel (14.6,4.2), stairwell estimated (2,0) and elevator estimated (0,0). Au et al [27] presumed a normal distribution of fire load density with a mean value of 950 MJ/m2 (about 47.5 kg/ m2) and a standard deviation of 95, which is 10% of the mean value, in the so-called advanced Monte Carlo simulation. Hasofer and Qu [28] used a uniform distribution of fuel density with an interval of 20 - 60 kg/m2. For simplicity, and based on the above analysis, FLDμ is assumed to have a uniform distribution U(14, 60). FLDσ is sampled using the proposed sampling technique by assuming Noise Factor a Uniform distribution U(0,1). Noise factor of FLDμ is named as NFFLD, and thus FLDσ = FLDμ × NFFLD. Please note that by assuming a Uniform distribution for the Noise Factor, the variables FRRσ and FLDσ actually follow the same type of distribution. And the so-called 3σ-rule is ignored here in order to study the model mathematically and cover the maximum possible sampling space. 4.3. Generation of samples A summary of uncertain variables and their distributions is shown in Table 2. The number of samples of each variable is of great importance because it controls the total sample number of Monte Carlo simulation, which should be balanced between precision and computing time cost. The sample size of each variable is determined by the importance of the variable. NFFRR (or FRRσ) is the focus of this study, so a size of 50 is given to it; size of 50 is also applied to α in order to effectively cover the given sampling space. Sample size of 10 is used for both FLDμ and NFFLD because it can represent the changes of those two variables while at the same time produce a total sample size of acceptable computing time. Sample sizes of the input variables are listed in Table 2. Table 2. Uncertainty variables and their distributions. Variable

Variable name

Unit

Distribution

Range

Sample size

α

Fire growth rate

kW/s

LogNormal(-6.5, 22)

(0.003, 0.19)

50

FLDμ

Fire load density

kg/m2

NFFLD

Uniform(14, 60)

(14, 60)

10

Noise factor of FLDμ

-

Uniform(0,1)

(0,1)

10

FRRμ

Fire Resistance Rating

minutes

Discrete

See Table 1.

5

NFFRR

Noise factor of FRRμ

-

Uniform(0,1)

(0,1)

50

2

A computer program was written to facilitate the sampling process, which flows as in Fig. 2, Stratified Sampling technique is adopted to generate samples for variable α, FLDμ, NFFLD and NFFRR

Paired samples are generated for (FLDμ , NFFLD) (10×10) and (FRRμ, NFFRR) (5×50).

All the samples of the 5 input variables are grouped together producing input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) with sample size nS = 1.25 million.



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Transform input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) into model readable input vector X’ = (α, FLDμ, FLDσ, FRRWμ, FRRWσ, FRRDμ, FRRDσ, FRRFμ, FRRFσ, FRRWDμ, FRRWDσ)

Monte Carlo run of input vector X’ with sample size nS = 1.25 million, and produce output Y = tsp with same sample size.

Fig. 2. Flow chart illustrating the sampling process.

The Stratified Sampling technique is adopted to generate samples for variable α, FLDμ, NFFLD and NFFRR, separately, according to Table 2. Paired samples are generated for (FLDμ , NFFLD)(10×10) and (FRRμ, NFFRR)(5×50). Sample pairs of (FRRμ, NFFLD) are then grouped with pairs (FRRμ, NFFRR) and then grouped with 50 samples of α, producing the final input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) with a sample size nS = 1.25 million. The samples of input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) cannot be read directly by the fire spread model, and they have to be transformed into readable vector X’ = (α, FLDμ, FLDσ, FRRWμ, FRRWσ, FRRDμ, FRRDσ, FRRFμ, FRRFσ, FRRWDμ, FRRWDσ). FRRWμ, FRRWσ, FRRDμ, FRRDσ, FRRFμ, FRRFσ, FRRWDμ, FRRWDσ means Fire Resistance Ratings for wall, door, floor and window (see Table 1 for meaning of these notations) and their corresponding standard deviations. The standard deviations of Fire Resistance Rating can be simply obtained by (FRRμ × NFFRR), and the FLDσ is calculated by (FLDμ × NFFLD), for each sample. Furthermore fire spread models input those input samples of X’ and generates model output samples of Y = tsp . Samples of input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) and output Y = tsp will be used for sensitivity analysis. As mentioned earlier, the reason of using the input vector X = (α, FLDμ, NFFLD, FRRμ, NFFRR) is to keep uncorrelated structure among input variables, since most of the sensitivity analysis approaches are not appropriate to be applied to correlated input variables because they will likely generate misleading and even false results. 4.4. Sensitivity analysis approaches As reviewed at the beginning, the purpose of a sensitivity analyses is to determine which of the input parameters exerts the most influence on model outputs. And thereafter this information can be used to eliminate unimportant parameters and provides direction for further research to reduce parameter uncertainties and increase model accuracy [1]. Regarding to sampling based sensitivity analysis, there are many analysis techniques. Helton et.al [3,4] provided a comprehensive review of those techniques in conjunction with sampling-based methods. The sensitivity analysis techniques and procedures considered in this study will include correlation coefficients (CCs), partial correlation coefficients (PCCs), standardized regression coefficients (SRCs), rank correlation coefficients (RCCs), partial rank correlation coefficients (PRCCs), standardized rank regression coefficients (SRRCs), and SRCs and SRRCs are obtained by stepwise regression analysis. For the detailed information of those approaches please refer to [3,4]. 5. Results 5.1. Uncertainties in model output After 1.25 million Monte Carlo runs of the Fire spread model using the sample data, same number of model responses are obtained. The single response variable is defined as the fire spread time (tsp) from Room 1 on 1st floor to Room 6 on 2nd floor; see Fig. 1 for building layout.

410

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Fig. 3. Histogram (lower part) and Cumulative Probability (upper part) distribution of Fire Spread time tsp.

A frequency distribution (histogram) of the output samples (nS = 1.25 million) is presented at the lower part of Fig. 3. This probability distribution can be approximately simulated by a lognormal distribution, but the exact distribution type and parameters are not the subject this study, and the dashed curve in the graph is just for illustration purpose. It should be noted that the amount of samples with size nS’ = 41555 (about 3.3% nS) were outside the distribution. For these a fire spread time of 600 minutes (same as model simulation time) was assigned. Those samples (nS’) were exacted and recalculated by changing the simulation time to 1200 minutes, however, only 8% of the results showed successful fire spread (tsp <1200). These results actually indicate that the nS’ = 41,555 input samples produced extremely slow fire spread or no fire spread. These points are excluded from the descriptive statistics in Fig. 3, in order to avoid misleading presentation. Unfortunately, real-life data of multi-room and multi-floor fire spread are not found in the literature to compare with the model output. Nevertheless, the probability density curve of tsp shows a good shape in terms of the distribution of fire spread times, see Fig. 3. Only 8.6% of tsp data are under 30 minutes, and 25% of data are under 56 minutes. Furthermore, 50% of the data lie in the range between 56 minutes to 98 minutes, and data over 150 minutes only account for 5.7%. In the above analysis, the two Noise Factors NFFLD and NFFRR had a uniform distribution (0, 1), which means that FLDσ and FRRσ are actually sampled from a uniform distribution (0, FLDμ) and (0, FRRμ), respectively. Practically, according to the 3σ rule, it is almost impossible that mathematical expectation is more than three times the standard deviation; therefore, the lowest possible value should be around (μ - 3σ) > 0, as in practice failure times (or Fire Resistance Ratings) are always greater than 0. In this study, if the 3σ rule is applied, the following condition should apply, NFFLD < 1/3 and NFFRR < 1/3. Input and output samples that fit this condition are analyzed and shown as in Fig. 4.



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Fig. 4. Histogram and Cumulative Probability distribution of tsp after applying the 3σ rule.

The cumulative probability and frequency distribution of samples after applying the 3σ rule are shown in Fig. 4. The number of samples became nS = 120,000 after filtering samples with Noise Factors (both NFFLD and NFFRR) less than 1/3. Furthermore, to avoid misleading statistical results, results of unsuccessful fire spread (tsp > 600) (nS’= 12,438) are also removed from the analysis, leading to a total sample size nS = 107,562. It was found that the mean value of tsp in this case is about 42 minutes higher than the earlier case; and the 1st quartile, median and 3rd quartile are also about 35 to 50 higher than the earlier case. Even though greater variations are showed in Fig.3 than that in Fig. 4 (see skewness and coefficient of variation on figures), results in Fig. 4 are more reasonable. 5.2. Sensitivity analysis of input variables CC, also called Pearson product-moment correlation coefficient, provides a measure of the strength of the linear relationship between single input variable xj and output y, and PCC provides a representation of the linear relationship between two variables after a correction has been made to remove the linear effects of all other variables in the analysis. CCs, and PCCs have been frequently used as measures of the relationship between uncertain variables and analysis results. CCs and PCCs between each input variable and model output tsp and the order of input sensitivity are shown in Table 3. Table 3. Correlation coefficients with raw data and rank-transformed dataa . Raw data

Rank-transformed data

Variable

CC

Orderb

PCC

Order

RCC

Order

PRCC

Order

α

-0.120

4

-0.150

4

-0.524

2

-0.790

2

FLDμ

-0.161

3

-0.200

3

-0.014

4

-0.034

4

NFFLD

0.000

5

0.000

5

0.000

5

0.000

5

FRRμ

0.282

2

0.335

2

0.245

3

0.515

3

NFFRR

-0.502

1

-0.536

1

-0.706

1

-0.866

1

Model response is tsp Order number of variable sensitivity It is worth to note that CCs and PCCs are based on determining linear relationships and typically perform poorly when the underlying relationships are nonlinear [3]. When these relationships are nonlinear, but still monotonic, the rank transformation can be used to linearize the underlying relationships between sampled and calculated variables. This situation applies to the Fire spread model. The model inputs and outputs relationship cannot be described explicitly, for example, as a linear one; however, it does show monotonic relationships between uncertain input variables and model responses. Therefore, a rank transformation was performed and RCCs and PRCCs were calculated and shown in Table 3. From Table 3, it can be seen that the sensitivity rankings of five input variables through 4 different coefficients are somewhat different, but all of them presented NFFRR as the most significant variable and no sensitivity of NFFLD. The only a

b

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difference is the ranking of the fire growth rate α. All the values of CCs and RCCs are in the range of -1 to 1, with 0 no linear relationship, -1 perfect negative linear, 1 perfect positive linear and somewhat linear in between. Range of absolute values of CC (maximum 0.502) in raw data shows weak linear relationship among model inputs and response, however, the value range of RCCs indicate that the model inputs and output can be described as monotonic, thus monotonic non-linear. Furthermore, the negative coefficients of variable NFFRR, α, and FLDμ mean that when they increase, the model response tsp changes in the opposite direction, while the single positive coefficients of FRRμ indicate that when Fire Resistance Rating increases, the fire spread time tsp also increases. Table 4. Stepwise regression analysis with raw data and rank-transformed dataa . Stepb

Raw data

Rank-transformed data

Variable

SRC

R

Variable

SRRC

Cumulative R2

1

NFFRR

-0.502

2

FRRμ

0.282

0.252

NFFRR

-0.706

0.499

0.332

α

-0.524

0.774

3

FLDμ

4

α

-0.161

0.358

FRRμ

0.245

0.833

-0.120

0.372

FLDμ

-0.014

0.834

2

Model response is tsp Steps in stepwise regression analysis Regression techniques allow evaluation of sensitivity of individual model inputs, taking into account the simultaneous impact of other model inputs on the result. Rank regression can capture any monotonic relationship between an input and the output, even if the relationship is nonlinear. Stepwise regression analysis was performed to the raw data and ranktransformed data between model inputs and output of the Fire spread model, as show in Table 4. Variable sensitivity is indicated by the steps in which variables are selected in the stepwise process, the absolute value of the SRCs or SRRCs can be used to provide a measure of variable importance. Table 4 also shows the cumulative (coefficient of determination) R2 ∈ [0, 1]. A large value of R2 indicates the regression model accounts well for most of the uncertainties in model response tsp , otherwise it may denote that the regression analysis is used inappropriately. The order of sensitivity level of input variables in Table 4 are exactly the same as the orders given by the four correlation coefficients in Table 3, for both raw data and rank transformed data. Moreover, the relatively lower value of cumulative R2 (0.372) from raw data and higher value from rank-transformed data (0.834) reinforce the judgment that the model is nonlinear but monotonic. Meanwhile, it can be said that regression analysis is successful in the rank-transformed data. Therefore, sensitivity analysis should be mostly relied on analysis of rank-transformed data. Combining the results in Table 3 and Table 4, an order of variable sensitivity can be given as NFFRR, α, FRRμ, FLDμ, NFFLD. Most importantly, it is found that NFFRR is the most sensitive input variable, and accounts for about 49.9% of observed model variation, indicated by the R2 value in Step 1 in Table 4. The least important input variable is NFFLD, which actually shows zero sensitivity to model output (see Table 3), and is excluded from the stepwise regression analysis because it accounts for no model uncertainty. Regarding variable NFFLD, a reinvestigation of the inner mechanisms of fire spread model uncovered the reason why variation of NFFLD results in no model responses. FLDσ, which is determined by (FRRμ× NFFLD), is used to calculate the maximum fire growth time in cases when flashover does not occur in a room; however, flashover criteria is satisfied in all the rooms in this scenario. The lower bound of FRRμ to cause flashover is 6.76 kg/m2 in this fire scenario, which is out of the sampling space of FRRμ. Analysis after applying the 3σ rule After applying the 3σ rule, input samples conforming to (NFFLD < 1/3 and NFFRR <1/3) are analyzed (nS = 120000). RCCs, PRCCs and SRRCs are calculated and the ranking of input variable sensitivity are given as FRRμ, NFFRR, α, FLDμ, NFFLD. Compared to the previous results, FRRμ becomes the most sensitive parameter, while NFFRR turns into the second place instead of being the most sensitive as in the earlier case. The two least sensitive variables are still FLDμ and NFFLD, with zero sensitivity of NFFLD. Validity of sampling technique As mentioned earlier in this paper, sensitivity analyses based on the above coefficients can give very misleading results when correlations exist among the input elements. The introduction of the Noise Factor approach in this study is a way to serve this purpose during the sampling process. The original input vector is X’ = (α, FLDμ, FLDσ, FRRWμ, FRRWσ, FRRDμ, FRRDσ, FRRFμ, FRRFσ, FRRWDμ, FRRWDσ), which has readable inputs by fire spread model; however, strong correlations exist among variables. CCs are calculated to illustrate the existence of correlation structure, for notational simplification, CC(x1,x2) means correlation coefficient between variable x1 and x2. After calculation, CC(FLDμ, FLDσ) = 0.514; CC(FRRWμ, FRRWσ) = CC(FRRDμ, FRRDσ) = CC(FRRFμ, FRRFσ) = CC(FRRWDμ, FRRWDσ) =0.6; and also a

b



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correlations between other variables, for example CC(FRRWμ, FRRDσ) ≈0.6. Those results indicate strong correlation structure among the input variables. CCs between input variables and model output tsp were also obtained. The correlation coefficient between FLDσ and tsp shows misleading results CC(FLDσ, tsp) = -0.087, which however was proven to be 0 by using the Noise Factor NFFLD, i.e., CC(NFFLD, tsp) = 0. 6. Discussion and Conclusion Nowadays comprehensive models with massive amount of input parameters are used in every field including fire engineering. This is not only in the academic world but also in the industries. For instance, CFD computer models are used to model fire and smoke behaviour in buildings, and complex evacuation models are also used frequently in the fire protection consulting jobs. Many of the applications and models are taken for granted and presumed as valid before used in real world to solve safety problems. While two sides of model application shall be understood: one is whether the model is good at handle all the input parameters appropriately, and it is always good to know the range of each input parameter that the model could handle well; the other side is that given a model is well verified against experiments, one would like to know which input parameter affect the results the mostly, and then could be used positively to improve the real building design, for instance. In short, sensitive analysis of model inputs and outputs could help identify the model inadequacies and also could help us use the model in a better way. Normal sensitivity analysis approaches such as CC, RCC, and regression analysis perform well when dealing with independent variables but produce misleading results if applied to correlated input variables. The introduction of Noise Factor can transform the correlated model input structure into independent variables, and this technique can be used to overcome this problem. It should be noted that the proposed sampling technique is a specific method to solve the given problem in this paper. One important condition is that the correlated variable can be expressed by a function of noise factor and one or more other variables, while at the same the noise factor and those variables are independent with each other. For example, in this study, FLDμ, FLDσ can be expressed as a linear relationship FLDσ =FLDμ× NFFLD, while FRRDμ and NFFLD are independent with each other. Mapping of CCs, RCCs, SRCs, and SRRCs (from stepwise regression analysis) presented a non-linear but monotonic relationship between input variable and output variable, especially for variables with higher ranking of sensitivity. A sensitivity ranking of different input variables was given and showed NFFRR (corresponding to FRRσ) as the most sensitive and accounting for almost 50% of model uncertainty, as discussed earlier. However, after considering the 3σ rule, which filtered the portion of samples when Noise Factors are less than 1/3, the sensitivity ranking changed, showing that Fire Resistance Ratings (FRRμ) as the most sensitive and NFFRR the second sensitive. This clearly indicates that the selection of sampling space boundaries is very important and could significantly affect the sensitivity analysis results. One of the expectations of this study is to find out sensitivity of input parameter FRRσ, and intend to utilize this parameter to take into consideration the effects of different building characteristics on the fire spread process, for instance, performance of wood framed wall assembly and steel framed wall assembly with the same Fire Resistance Rating might fail at different times under the same realistic fire. It is good to know that the standard deviation of the Fire Resistance Ratings (FRRσ) has a high variable sensitivity. But further work is needed to better understand the effect of changes in FRRσ on the model outputs. Sensitivity analysis also uncovered that the variable FLDσ accounted to no model response uncertainties. Information such as those will definitely help to better understand the model and handle model uncertainties. Acknowledgements The authors would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC), FPInnovations and all other sponsors of NEWBuildS network for their funding support. Thanks are also given to Qimiao Xie for her help with sampling of fire growth rates from Lognormal distribution. Thanks to Arup Shanghai for funding the conference trip. References [1] [2] [3]

Hamby, D. M. (1994). "A review of techniques for parameter sensitivity analysis of environmental models." Environmental Monitoring and Assessment, 32(2), 135-154. Cacuci, D., Ionescu-Bujor, M., and Navon, M., Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-Scale Systems, CRC Press, USA, 2005, p.2. Helton, J.C., and Davis, F.J., (2002) Illustration of Sampling-Based Methods for Uncertainty and Sensitivity Analysis, Risk Analysis 22: 591-622, DOI: 10.1111/0272-4332.00041.

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