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Procedia Engineering 11 (2011) 179–184

The 5th Conference on Performance-based Fire and Fire Protection Engineering

Global Sensitivity Analysis of Occupant Egress Safety Model KONG De-penga, b,c, LU Shou-xianga,c,*, LO SMb,c b

.aState Key Laborotory of Fire Science, USTC, Hefei, China,230027; Department of Building and Construction, City University of Hong Kong, Hong Kong, China, 999077; c USTC-CityU Joint Advanced Research Centre, Suzhou,China, 215000

Abstract

Identification of the important factors for the occupant egress safety under building fire is important. In order to achieve this goal, a global sensitivity analysis method, Fourier Amplitude Sensitivity Test (FAST), was instead of the local sensitivity analysis to indentify the important factors for the safe evacuation level under building fire. The equation for safe egress level under building fire was determined by a series of empirical formulas. The result indicates that uncertainty of CFAST model has the dominating influence on the safety egress level while uncertainties associated with exit flow rate and population density have little influence. The relative differences between the first order sensitivity indies and the total sensitivity indies indicate that the interaction between fire growth rate and other factors has the largest influence on the safety egress level.

© 2011 Published by Elsevier Ltd. Keywords: life safety; occupant egress;global sensitivity analysis;building fire

1. Introduction In performance-based fire protection design (PFPD) and fire risk assessment, how to deal with uncertainty is an essential issue[1]. As one of the important aims for PFPD, determination of safety level of occupant egress under fire also should consider the uncertainties. Two questions should be answered for this issue[2]: (1). How we should represent uncertainty? (2). How many uncertainties we should consider? The former can be answered by the uncertainty analysis (UA) while the latter can be answered by the sensitivity analysis (SA). Due to many uncertainties involved in the occupant egress, SA seems to be more necessary for the fire engineering. SA can be categorized into local SA and global SA[3]. Local SA examines the response of the output when varying input parameters one at a time and holding other input parameters as fixed values. Local SA is flexible and easy to implement. However, it can only inspect one point at a time. Furthermore, it is practicable only when the relation between input and output is linear. Global SA can examine the response of the output by exploring a finite region. The influence of the interaction between input parameters is also can be inspected by global SA. Unfortunately, the majority of SA in fire engineering is local SA. Fu and Fan[4] studied the influence of input

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1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.04.644

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parameters for a two zone model by giving the input a small change. Using the similar SA method, Wang[5] identified the important factors for RSET. Based on Wang’s study, Wang[6] utilized single-factor sensitivity analysis to investigate the important factors for occupant evacuation under building fire. Due to the complexness of fire dynamic and evacuation, available safety egress time (ASET) and required safety egress time (RSET) are nonlinear with input parameters. In this situation, global SA is more appropriate for analyzing ASET and RSET. In this paper, we make an attempt to employ one kind of global SA, Fourier Amplitude Sensitivity Test (FAST), to analyze important factors for occupant egress under building fire. Section 2 gives a brief description of the occupant egress safety model. In Section 3, the global sensitivity analysis, FAST, is presented in details. Then we present an application of FAST to indentify the important factors for the safety level of occupant egress in a onestorey commercial building. Finally, a brief conclusion is given in Section 5. 2. The model of occupant egress safety under fire In PFPD, ASET and RSET are compared to determine the safety level of occupant egress[1]. If ASET is larger than RSET, it is considered that occupant can evacuate safely and no casualties occur. Therefore, the equation of safety level for occupant evacuation can be written as following:

G = ASET − RSET (1) Where, G is the time margin of occupant evacuation, s; ASET is available safety egress time, s; RSET is required safety egress time, s. If G>0, it indicates that occupants can evacuate successfully. If G<0, occupant cannot evacuate successfully. If G=0, Eqn.1 can be considered as the limited state equation for occupant evacuation. ASET can be determined by the zone models or filed models. RSET usually has three components, i.e. fire detection time, occupant pre-movement time and movement time. Fire detection time can be obtained by are influenced by detection actuation time models, such as DETECT-T2 model. Occupant movement time can be determined by evacuation models. Since ASET and RSET are influenced by many factors, most of which are uncertain. Therefore, Eqn. 1 can be written as following:

G = g( X )

Where, X=(X 1 , X 2 ,…, X n ) is the uncertain factors influencing occupant egress safety level.

(2)

3. The Fourier Amplitude Sensitivity Test (FAST) The Fourier Amplitude Sensitivity Test (FAST) was originally developed by Shuler et al[7-8]. It is an effective global SA when the input-output is nonlinear and non-monotonic. Considering the nonlinear relationship between inputs and G, it is appropriate to conduct sensitivity analysis using FAST. The main idea of FAST is to assign each input parameter with an integer frequency according to the Fourier transformation. The influence of a specific input parameter on the output, the sensitivity index, can be assessed by the variance contribution, which can be obtained by the characteristic integer frequency [9-10]. For the model G=g(X 1 , X 2 …X n ), every uncertain factor Xi can be transformed into a function of character frequency:

1 X i = Fi −1 ( + arcsin(sin(ωi s ))), − π ≤ s ≤ π 2 F −1

(3)

Where, i is the inverse cumulative distribution function for X i . ω i is the character frequency for X i and s is the common variable for all variables. If s varies, all X i s vary simultaneously in their own regions of variance at the rate according to ω i . Substituting Eqn.3 into G, we can present the model as a Fourier series:

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G = g ( X 1 , X 2 , X n ) ∞

= g ( s ) = A0 + ¦ { Ak cos(ks ) + Bk sin(ks )}

(4)

k =1

Where,

1 π g ( s )ds, 2π ³−π 1 π Ak = ³ g ( s ) cos(ks )ds, A0 =

Bk =

π

−π

1

π

(5)

g ( s ) sin(ks)ds, π³π −

k ∈ {1, , ∞}

Having determined A k and B k , the contribution to the total variance V i of X i can be obtained by

Vi =

1 ∞ ¦ ( Apωi + Bpωi ) 2 p =1

(6)

1 ∞ ¦ ( Ak + Bk ) 2 k =1

(7)

Vi V

(8)

And the total variance can be estimated by

V= The sensitivity index for X i can be calculated by

Si =

Similarly, the total sensitivity index for X i also can be determined. Let ω i be the frequency associated with X i , and ω -i is the set of all frequencies except ω i . The partial variance V -i then can be calculated by

1 ∞ V− i = ¦ ( A2pω + B p2ω− i ) −i 2 p =1 Correspondingly, the total sensitivity index for Xi can be determined as

(9)

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STi = 1 −

V− i V

(10) For S Ti , it can reflect the influence of both input parameter X i and its interaction with other input parameters by the variance contribution V -i. . In order to examine the influence of X i ’s interaction with other input parameters, another index, RD, is defined as follows:

RDi =

STi − Si V− i − Vi = STi V

(11)

4. Case study The object for this case study is a one-storey commercial building with area of 1100 m2 and height of 6 m. There are two exits and the total width is 7 m. Sprinklers and fire alarm systems are available in this building. In order to simplify the calculation, ASET and RSET are both calculated by empirical formulas. According to reference [11], ASET can be determined as the following empirical formula:

ASET = 0.025α −0.114 H 0.457 A1.28

(12)

The detection time, t dec , can also be calculated by the empirical formula:

tdec = 5.36α −0.478 H 0.7

(13)

For the movement time, it can be simplified as following

tmove =

q⋅ A f ⋅W

(14)

Since Eqn.11 is derived from the CFAST simulation results, the uncertainty of CFAST should be considered. The equation for occupant evacuation safety level under fire can be estimated as

G = Ms (0.025α −0.114 H 0.457 A1.28 ) − 5.36α −0.478 H 0.7 − t pre −

q⋅ A f ⋅W

(15)

The specification of related factors in Eqn.14 is described in Table 1. Table 1. The description of related factors

Factors

Denotation

Distribution

Range 2

Pre-movement time(s)

t pre

logN(4.6,0.8 )

[50,300]

Fire growth rate(kW/s2)

α

logN(-3.3,0.7)

[0.0012,0.1876]

2

2

Population density(per/m )

q

N(0.8,0.2 )

[0.2,1.2]

Exit flow rate (per/(ms))

f

N(1.3,0.35)

[0.5,2.2]

Uncertainty of CFAST model

Ms

N(1.35,0.11)

[1.0,2.0]

Employing FAST to calculate the first order sensitivity index S i and the total sensitivity index S Ti for each factor. The final results are shown in Table 2. Table 2. The description of related factors

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Factors

Si

S Ti

Relative difference

Ms

0.5184

0.5487

0.0585

t pre

0.2623

0.2771

0.0564

α

0.2072

0.2349

0.1337

f

0.0448

0.05885

0.3135

q

0.0341

0.04650

0.3638

Table 2 leads the following conclusions: (1) Both S i and S Ti indicate that we can rank the factor sensitivity as follows: the uncertainty of CFAST model (Ms), pre-movement time (t pre ), fire growth rate (α), exit flow rate (f), the population density (q). For the first three sensitive factors, Ms has the dominating influence on the model output; t pre and α have the similar influence on the output. (2) Compared with other factors, the uncertainty of CFAST model has the largest influence on the occupant egress safety level. In order to reduce the influence due to the uncertainty of CFAST model, the validation and verification of the simulated results by CFAST should be conducted. (3) The sensitivities to exit flow rate and population density are negligible because they are too small compared with sensitivities to uncertainty of CFAST model. Therefore, both of them can be considered as fixed values. Furthermore, one of the advantages for the global sensitivity analysis is that it can reflect the interaction between factors. Therefore, we can determine the interaction by the defined index, RD. The relative difference between S i and S Ti for each factor is shown in Table 2. It indicates that the fire growth rate has the largest interaction with other factors. The large relative differences for exit flow rate and population density have no useful interpretation due to their small sensitivity indices. 5. Conclusion A global sensitivity analysis method, Fourier Amplitude Sensitivity Test (FAST), is introduced to analyze the important factors for occupant egress safety under building fire. According to the result for a one-storey commercial building, the following conclusion can be drawn: (1) Factors having important influence on the occupant egress safety are: uncertainty of CFAST model, preegress time and fire growth rate. The uncertainty of CFAST model has the dominating influence while pre-egress time and fire growth rate have the similar influence. The exit flow rate and population density have little influence; therefore, their uncertainties can be ignored. (2) By comparing the first order sensitivity indices with the corresponding total sensitivity indices, the influence due to the interaction between input factors on output can be determined. The relative difference between the first order sensitivity indices and the corresponding total sensitivity indices indicate that interaction between fire growth rate and other factors has the largest influence on occupant egress safety.

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