ARTICLE IN PRESS

Fire Safety Journal 44 (2009) 901–908

Contents lists available at ScienceDirect

Fire Safety Journal

0379-71

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/firesaf

The modeling of fire spread in buildings by Bayesian network

Hao Cheng �, George V. Hadjisophocleous

Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Canada ON K1S 5B6

a r t i c l e i n f o

Article history:

Received 31 August 2007

Received in revised form

7 May 2009

Accepted 22 May 2009Available online 17 June 2009

Keywords:

Fire spread

Bayesian network

Building

Static model

12/$ - see front matter & 2009 Elsevier Ltd. A

016/j.firesaf.2009.05.005

esponding author.

ail address: haocheng5@gmail.com (H. Cheng

a b s t r a c t

Fire spread modeling is very important to fire safety engineering and to insurance industries involved in

fire risk–cost analysis of buildings. In this paper, the Bayesian network is introduced. The directed

acyclic graph of a fire spread model is presented. When the fire ignition location is known, the fire

spread model based on the Bayesian network from the compartment of fire origin to another

compartment can be built, and the probability of fire spread can be calculated by making use of the joint

probability distribution of the Bayesian network. A specific application for an office building is

presented for a case without sprinkler and one with sprinkler installed.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Fires in buildings pose a significant risk to building occupantsand cause property damage. A lot of research has been conductedover the last decades aiming to understand the mechanism of fireignition and growth as well as smoke movement to adjacentcompartments. This body of research resulted in computer modelsthat predict fire growth and smoke spread through a building,which can be used to design effective strategies to control firegrowth and spread in a building to improve life safety and reduceproperty damage [1].

Mathematical models to simulate fire spread between com-partments are particularly important for fire risk assessment oflarge buildings. There are two kinds of approaches that can beused to simulate fire spread, the deterministic and probabilisticmethods. Deterministic models such as WALL2D [2,3] can beused to predict the time of failure of a wall when subjected to afire attack. The results of these models can also used in aMonte-Carlo simulation to predict the probability of failure atdifferent times.

Ramachanandran [4,5] summarized the studies of probabilisticapproach model done over last decades. In the earlier studies, theepidemic theory [6,7], random walk theory [8,9], Markov process[10–12], percolation process [13,14] and probabilistic network[15,16] were used to model the fire spread. These models couldsuccessfully describe the fire spread process in building in somerespects. But there are some disadvantages to simulate fire spread

ll rights reserved.

).

process using these models. The epidemic theory can not explainthe fire spread to adjacent combustible materials or compart-ments, which can not be reached by the burning flame or the firespread due to radiation. The random walk theory [8,9] andpercolation process [13,14] can simulate the fire spread from a firecompartment to one of its adjacent compartments, and then fromthis fire compartment to another adjacent compartment. But theyare not good at simulating the scenarios that fire may spread froma fire compartment to multiple adjacent compartments or firespreads from multiple fire compartments to their adjacentcompartments. The transition probability in Markov process[10–12] is not the probability of fire spread from fire compartmentto the adjacent compartment. It only presents the probability offire will spread from fire compartment to a compartmentcomparing to the other compartments, i.e. there are two similarcompartments at the each side of a fire compartment, thetransition probability of each compartment will always equal to50%, no matter how long the fire lasts. In addition, the fire spreadprocess from one compartment to multiple compartment ormultiple compartments to adjacent compartments at the sametime can not be described by Markov process.

Ling and Williamson [15] first presented a probabilisticnetwork approach to study room-to-room fire spread and anetwork of fire spread in a building floor was presented. Thismodel did not consider the barrier breach because of radiationand the network is complicated. If the fire initial change, a newnetwork had to be developed. Platt et al. [16] developed a simpleand clear model in which event tree was used to determine theprobability of fire spread form fire initial compartment to othercompartments. This model is very good to express the fire spreadprocess for small buildings. But it is hard to develop a tree for

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Nomenclature

A variables describing whether a fully developed fireoccurs in compartment A or not

a a fully developed fire occurs in compartment Aa a fully developed fire does not occur in compartment AA0 variables describing whether an ignition occurs in

compartment A or not

P(a|b) the probability of fire spread from compartment B tocompartment A.

P(a0|b) the barrier failure probability indicating that heattransfers from fire compartment B to adjacentcompartment A and ignites the combustible materialsin compartment A.

P(a|a0) the probability of fire grows from ignition to fullydeveloped fire.

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908902

large buildings. If the initial fire compartment changed, a new treehas to be developed even for same buildings which make thismodel difficult to be programmed. The digraph (directed graph)approach was used for the fire spread sub-model of the fire riskevaluation and cost assessment model (FiRECAM) [17]. To simplifythe problem, all compartments of the same type such as rooms,corridors, stairwells in one floor are combined as a node of thenetwork of buildings. The developed algorithm searches allpossible pathways for fire to spread from one node to another.

In this paper, the Bayesian network (BN) [18–20] is used tosimulate the fire spread process. Bayesian network had been usedin the fire risk assessment [21]. Bayesian network can overcomethe disadvantages mentioned in previous models. But Bayesiannetwork can not directly describe fire spread process. To build thefire spread model, a general fire spread network has to firstly bebuilt according to the floor plan of a building. Once the fire initialcompartment is known, a detail fire spread model using a directedacyclic graph (DAG) of Bayesian network to express the fire spreadprocess from the fire initial compartment to any destinationcompartment in the floor can be constructed and the probabilityof fire spread from the initial compartment to the destinationcompartment can be calculated by marginalizing the jointprobability distribution of the Bayesian network.

A B

DC

E

7.0)( =aP 7.0)( =bP

2.0)~(

7.0)(

=

=

acP

acP1.0)

~,~(,3.0),~(

5.0)~

,(,8.0),(

==

==

badPbadP

badPbadP

1.0)~

,~(,3.0),~(

3.0))~

,(,6.0),(

==

==

dcePdceP

dcePdceP

Fig. 1. An example of a Bayesian network.

2. Fundamentals of Bayesian network

The Bayesian network model is a tool to manage uncertaintyusing probability. A Bayesian network is a graphical model thatcombines graph theory and Bayesian probability theory. Bayesianprobability theory deals with the problem of reasoning underuncertainty.

2.1. The fundamentals of probability

If A is an event, P(a) represents the probability that event A istrue, and PðaÞ denotes the probability that event A is not true.Some basic axioms can be expressed as follows:

0 � PðAÞ � 1 (1)

PðaÞ þ PðaÞ ¼ 1 (2)

If event A and event B are mutually exclusive, the probability ofthe union of events A and B is

PðA [ BÞ ¼ PðAÞ þ PðBÞ (3)

If events A and B are not exclusive

PðA [ BÞ ¼ PðAÞ þ PðBÞ � PðA \ BÞ (4)

where P(A\B) is called the joint probability of events A and B.Usually P(A\B) is shorten as P(A, B).

The joint probability P(A, B) can be derived by

PðA;BÞ ¼ PðBjAÞ � PðAÞ (5)

where P(B|A) is called the conditional probability, which isthe probability that event B occurs given that event A has alreadyoccurred.

P(A, B) can also be written as

PðA;BÞ ¼ PðAjBÞ � PðBÞ ¼ PðBjAÞ � PðAÞ (6)

Rearranging above equation leads to the famous Bayes theorem

PðAjBÞ ¼PðBjAÞ � PðAÞ

PðBÞ(7)

2.2. The basics of Bayesian network

2.2.1. Definitions

The Bayesian network is based on a fundamental assump-tion—the probability distributions in BN are subjected to theMarkov condition. A Bayesian network or Bayesian belief networkconsists of two components:

(1)

A graphical structure, called directed acyclic graph G (DAG).G ¼ (V, E) where V are the set of nodes representing randomvariables on which the Bayesian network is defined and E arethe set of directed edges representing relations among thevariables. Fig. 1 is an example of a DAG.In DAG, the family notation is often used to express therelationships between variables. For AAV, the parents of A, orpa(A), are the set of variables from which there is an arrowgoing to Node A. The children of A are the set of variableswhich are reached by an arrow from Node A. The ancestors ofA are the set of variables who are the parents of A, its parent’sparents and so on. The descents of A are the set of variableswho are the children of A, its child’s children and so on. Thenodes without parents are called root nodes. The nodeswithout children are called leaf nodes.In Fig. 1, Nodes A and B are root nodes. Node E is a leaf node.The nodes C, D are the children of Node A, and Nodes A, B arecalled the parents of Node D. Nodes A, B, C, D are the ancestorsof Node E, and Node E is called the descendant of Nodes A, B, C,D. Node C is a non-descendent of Node B, or ND(B).

ARTICLE IN PRESS

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908 903

(2)

Fig.dive

a set of probability P, each of which is associated with a nodeof the DAG as shown in Fig. 1. Each root nodes possesses aprior probability distribution table. Each of other nodespossesses a Conditional Probability Table (CPT). In this paper,we assume all variables to be binary, that is, we use v todenote V ¼ true and v to denote V ¼ false.

2.2.2. Independence and d-separation

Suppose that a conditional probability P(A|B, C) has theproperty that it is always equal to the conditional probabilityP(A|B) as C varies, therefore A is conditional independent of C

given B, written as A‘

C|B.A Bayesian network may have a complicated DAG. No matter

how complicated, any Bayesian network can be consideredto consist of the following three types of basic connections(Fig. 2).

(1)

The serial connection (Fig. 2(a)): There is a directed path fromthe start node to the end node A-B-C. In Fig. 2(a), Nodes A

and C are separated by Node B. If the state B is known, A and C

become independent on each other according to the Markovcondition, therefore A and C are referred to be d-separated by B.

(2)

The diverging connection (Fig. 2(b)): In the diverging connec-tion in Fig. 2(b), Node B effectively d-separates Nodes A and C

from each other, making them conditional independent.

(3) The converging connection (Fig. 2(c)): In the converging

connection in Fig. 2(c), if the state of B is known, A and C

become conditional dependent. This is known as d-connection.

2.2.3. Representation of the joint probability distribution

If a BN contains n nodes V ¼ {X1, X2,y, Xn}, a particular valuein the joint probability distribution can be represented byP(X1 ¼ x1, X2 ¼ x2,y, Xn ¼ xn), or P(x1, x2,y, xn).

The chain rule in the classical probability theory can be used tofactorize it as follows:

Pðx1; x2; . . . ; xnÞ ¼ Pðxnjx1; x2; . . . ; xn�1ÞPðx1; x2; . . . ; xn�1Þ

¼ Pðx1ÞPðx2jx1ÞPðx3jx1; x2Þ . . . Pðxnjx1; x2; . . . xn�1Þ (8)

Assume that the variables in V are ordered ancestrally, or in atopological ordering. That is for every node XiAV; the index of itsancestor Xj has the property jo1. According to the Markovcondition property, we know that any variable Xi is conditionallyindependent to its nondescents ND(Xi)) given its parents pa(Xi),which can be stated as Xi

‘ND(Xi)|pa(Xi). Thus the conditional

probability distribution of any variable Xi may be expressed asP(xi|x1, x2y, xi�1) ¼ P(xi|pa(xi)).

A

B

C

A B C

A

B

C

2. Three basic connections in Bayesian network: (a) serial connection; (b)

rging connection; and (c) converging connection.

Therefore the joint probability distribution over n variables X1,X2,y, Xn can be defined as

Pðx1; x2; . . . ; xnÞ ¼ Pðx1Þ � Pðx2jpaðx2ÞÞ

� Pðx3jpaðx3ÞÞ . . . PðxnjpaðxnÞÞ ¼Yn

i¼1

PðxijpaðxiÞÞ (9)

2.2.4. Inference and marginalization in Bayesian network

Bayes’ rule is one of the fundamental theorems in Bayesiannetwork. Based on the observed evidence of some variables,Bayes’ rule can predict the outcome of other variables byproviding the posterior probability distribution when thesevariables are linked in the form of a network. For the randomvariables X1, X2,y, Xn, the impact of an observed variable Xj ¼ xj

on another variable Xi ¼ xi can be stated as

PðxijxjÞ ¼Pðxi; xjÞ

PðxjÞ(10)

The marginal probability P(xi, xj) is computed by summing thejoint probability distribution P(x1, x2,y, xn) over all instantiationsof the variables except Xi and Xj. The marginal probabilities can beexpressed as

Pðxi; xjÞ ¼X

X=fxi[xjg

Pðx1; x2; . . . ; xnÞ (11)

PðxjÞ ¼X

xi

Pðxi; xjÞ (12)

3. A fire spread model for an office building usingBayesian network

A floor of a building may consist of rooms, corridors, stairwells,elevator shafts, and ducts. Due to the uses and fuel loads, some ofthe compartments may have higher probabilities for fire to beinitiated in them than other compartments. If a fire occurs in onecompartment, the fire may spread to other compartments in thebuilding. In order to build a fire spread model in a building basedon Bayesian network, the first step is to transform the buildingplan into a DAG. To simplify the problem, in this paper the firespread model is focused on the case where fire only spreadshorizontally on a single floor.

3.1. Transforming the compartment floor into the DAG of

Bayesian network

Fig. 3 shows the floor plan of the office building considered.The floor has 11 rooms, 2 stairwells, and one elevator shaft. Wecan transform this floor plan into a DAG of Bayesian network(Fig. 4). In the DAG of Fig. 4, there are 15 nodes, each ofthem representing one compartment such as room, stairwell,elevator shaft, or duct, and edges connecting the 15 nodes. Thearrows represent the possible paths and directions of firespread. The DAG can be considered as the general fire spreadmodel that includes all possible fire spread paths from anycompartment of fire origin to any destination compartment inthe building. Once the compartment of fire origin is known, thegeneral fire spread network can be immediately convertedinto a specific fire spread Bayesian network for fire spread fromthe compartment of fire origin to a specific destinationcompartment, and calculate the probability of fire spread fromthe fire initial source to the destination compartment usingBayesian network theory.

ARTICLE IN PRESS

R1

S2

R10R8

S1

R11R9

E

R6R4

R5

R2

R3

C

R7

Fig. 3. The floor plan of an office building.

S2

R1 R3 R5 R8 R10

R2 R4 R6 E R9

S1 C

R7

R11

Fig. 4. The DAG of fire spread model based on Bayesian network.

R5

R6

R8

R9

R7

E R11

R10

S2

R3

R4

R1

R2

S1

Fig. 5. The simplified general fire spread network.

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908904

To further simplify the problem, it is assumed that although itis possible for a corridor fire to occur, a corridor fire will notspread to other compartments because usually there is notenough fuel in the corridor. With this assumption, the fire spreadnetwork of Fig. 4 can be simplified resulting in the fire spreadnetwork shown in Fig. 5.

3.2. The probability of fire spread

Fire spread from fire compartment to an adjacent compart-ment includes two processes: (1) Heat overcomes the fireresistance of the barrier between the two compartments andtransfers to the adjacent compartment, igniting combustiblematerial located in the adjacent compartment. (2) Followingignition, fire grows into a fully developed fire in the adjacentcompartment.

The main reason a fire spreads from the fire compartment to anadjacent compartment is that the barrier between these twocompartments fails to contain the fire. The failure depends on threefactors—the fire severity and duration in the fire compartment andthe fire resistance of the barrier between the two compartments.Fire penetrates a barrier and spreads to another compartmentmainly via the weakest part of the barrier such as a door, a window,etc. After ignition, the fire may grow and reach the fully developedstage. The duration of the fire depends on the amount and type offuel in the compartment and its ventilation condition.

The probability of fire spread from compartment B tocompartment A is written as P(a|b). Therefore

PðajbÞ ¼ Pðaja0Þ � Pða0jbÞ (13)

where P(a0|b) is the barrier failure probability indicating that heattransfers from the fire compartment B to adjacent compartment Aigniting the combustible materials in compartment B.

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H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908 905

P(a|a0) is the probability of fire growth to fully developedfire indicating the fire in A grows from ignition to a fullydeveloped fire.

3.2.1. The barrier failure probability

A barrier failure probability is the probability that heat in thefire compartment overcomes the fire resistance of the barrier andpenetrates to an adjacent compartment igniting the combustiblematerial in the adjacent compartment. The barrier failureprobability depends on the fire severity, amount and types ofcombustible materials, the material and thickness of the barrier,the position of doors or windows and the material and size of thedoor or window, etc. Fire always spreads into adjacent compart-ments through the weakest part of its barrier, such as door orwindow. Whether the door or window is open or closed also has agreat effect on the speed of fire spread.

Fire may overcome the fire resistance of a building elementand spread to its adjacent compartments through the followingmechanisms:

(1)

Conduction heat transfer through walls, ceiling/floor, or aclosed door separating two compartments, causing anincrease of temperature on the unexposed side and ignitingcombustible materials in the adjacent compartment.

(2)

Convection heat transfer: hot gas flows through openings suchas open doors or windows, igniting combustible materials inother compartments. Hot gases travel quickly through acorridor and a stair shaft to areas remote from the fire.

(3)

Radiation heat transfer from flames or hot gases to thecombustible materials in adjacent compartments.

Values of the probability of barrier failure are difficult toestimate due to the complexity of the processes involved.A computer model is being developed to calculate the probabilityof failure of timber frame walls exposed to fire [22] using aMonte-Carlo simulation in the WALL2DN computer model [2,3].

3.2.2. Probability of fire growth to fully developed fire

Once ignition occurs in a compartment, the fire may or maynot grow from ignition to a fully developed fire. The probability offire growth to fully developed fire is the probability that givenignition fire grows to the fully developed fire state. Thisprobability depends on the following factors:

�

The fuel load: fuel amount and fuel types in the compartment; � The geometry of the compartment and its ventilation condi-

tion;

� The presence of a fire suppression system.

In this paper, the following assumptions are made for thecalculation of the probability of fire spread:

�

If a fire burns in a compartment and spreads to its adjacentcompartments, the fire cannot return to this compartment. � When two rooms are separated by a wall, the barrier failure

probabilities for fire to spread in both directions through thewall are the same.

� When two similar compartments (such as rooms) are sepa-

rated by a corridor, the barrier failure probabilities for fire tospread in both directions through the corridor from the firecompartment to the opposite compartment are the same.

� When two similar compartments (such as rooms) are sepa-

rated only by a door, the barrier failure probabilities for fire tospread in both directions through the door from the firecompartment to the adjacent compartment are the same.

�

The barrier failure probability of fire spread to upper floorcompartments (upward direction) is greater than that in thedownward direction. � The barrier failure probability for fire in the stairwell, elevator

shaft or duct to spread in the upward (vertical) direction isgreater than that in the horizontal or downward direction.

� Though a corridor fire may occur, the probability of barrier

failure for fire to spread from the corridor to its adjacentcompartments can be ignored since there is not enough fuel inthe corridor to support the fire to spread.

� Though the probability of barrier failure from a corridor to a

room is assumed to be zero, heat can spread from one room toa room separated by a corridor due to radiation and convectiveheat transfer.

� The compartments of same type have the same probability of

fire growth to fully developed fire.

� The probabilities of fire growth to fully developed fire in

stairwells, elevator shafts or ducts are lower than that in acompartment (room).

4. Example of a fire spread model using Bayesian network

To calculate the probabilities of fire spread in a building, ageneral fire spread network has to be built based on the buildingfloor plan. Following that, a Bayesian network can be constructedto calculate the probability of fire spread from the fire compart-ment to a destination compartment.

As an example, a Bayesian network model for the buildingshown in Fig. 3 will be developed using the principles describedabove for calculating the probability of fire spread from a fire thatstarts in room 1 to room 6, both of which are located on the samefloor. Once the model is developed the fire spread probability fromroom 1 to room 6 will be computed for the following two cases:

�

Case 1: There is no suppression system in the building orintervention during the fire spread process. � Case 2: There is a sprinkler system installed in the rooms of the

building.

The steps to calculate the probability of fire spread from room1to room 6 using the fire spread model based on Bayesiannetwork are listed below:

�

Step 1. Build a general fire spread network according to thebuilding floor plan. This network is shown in Fig. 5. � Step 2. Find all the possible pathways, through which fire could

spread from room 1 to room 6. The possible pathways areshown in Fig. 6, which represents the directed acyclic graph ofthe Bayesian network.

From Fig. 6 it can be seen that fire can spread from R1 to S1 andR2; from S1 to R2; from R2 to R4; from R4 to R3 and R6; etc. Usingthis information the joint probability for the Bayesian network forfire spread from room 1 to room 6 can be expressed as

PðR1;R2; . . . ;R11; S1; S2; EÞ

¼ PðR1Þ � PðS1jR1Þ � PðR2jR1; S1Þ

� PðR3jR1;R4Þ � PðR4jR1;R3Þ � PðR5jR3Þ

� PðR6jR4;R5; EÞ � PðR7jR5Þ

� PðEjR7;R9Þ � PðR8jR7Þ � PðR9jR8;R11Þ

� PðR10jR8Þ � PðR11jR10; S2Þ � PðS2jR10Þ (14)

The terms of Eq. (14) can be re-written to explicitly considerboth the probability of barrier failure and the probability of firegrowth to fully developed fires in the adjacent compartments,

ARTICLE IN PRESS

Table 1Barrier failure probability distribution tables.

Node Barrier failure probability

R1 P(r1) ¼ 1, Pðr1Þ ¼ 0

S1 P(s01|r1) ¼ 0.57, Pðs01jr1Þ ¼ 0

R2 Pðr02jr1; s1Þ ¼ 0:84; Pðr02jr1 ; s1Þ ¼ 0:60; Pðr02jr1; s1Þ ¼ 0:936; Pðr02jr1 ; s1Þ ¼ 0

R3 Pðr03jr1; r4Þ ¼ 0:81;Pðr03jr1; r4Þ ¼ 0:84; Pðr03jr1; r4Þ ¼ 0:9696; Pðr03jr1 ; r4Þ ¼ 0

R4 Pðr04jr2; r3Þ ¼ 0:81;Pðr04jr2; r3Þ ¼ 0:84; Pðr04jr2; r3Þ ¼ 0:9696; Pðr04jr2 ; r3Þ ¼ 0

R5 Pðr05jr3Þ ¼ 0:81; Pðr05jr3Þ ¼ 0

R6 Pðr06jr4; r5 ; eÞ ¼ 0:81;Pðr06jr4; r5; eÞ ¼ 0:84; Pðr06jr4; r5; eÞ ¼ 0:60; Pðr06jr4; r5; eÞ ¼ 0:9696; Pðr06jr4; r5; eÞ ¼ 0:924;

Pðr06jr4 ; r5 ; eÞ ¼ 0:936;Pðr06jr4 ; r5 ; eÞ ¼ 0:98784;Pðr06jr4; r5 ; eÞ ¼ 0

R7 Pðr07jr5Þ ¼ 0:81; Pðr07jr5Þ ¼ 0

E Pðe0 jr7 ; r9Þ ¼ 0:28; Pðe0 jr7 ; r9Þ ¼ 0:57; Pðe0 jr7; r9Þ ¼ 0:6904; Pðe0 jr7; r9Þ ¼ 0

R8 Pðr08jr7Þ ¼ 0:81; Pðr08jr7Þ ¼ 0

R9 Pðr09jr8; r11Þ ¼ 0:84; Pðr09jr8 ; r11Þ ¼ 0:81;Pðr09jr8 ; r11Þ ¼ 0:9696;Pðr09jr8; r11Þ ¼ 0

R10 Pðr010jr8Þ ¼ 0:81; Pðr010jr8Þ ¼ 0

R11 Pðr011jr10 ; s2Þ ¼ 0:84; Pðr011jr10; s2Þ ¼ 0:60; Pðr011jr10; s2Þ ¼ 0:936; Pðr011jr10; s2Þ ¼ 0

S2 Pðs02jr10Þ ¼ 0:57;Pðs02jr10Þ ¼ 0

R5

R6

R8

R9

R7

E R11

R10

S2

R3

R4

R1

R2

S1

Fig. 6. The DAG of the Bayesian network for fire to spread from room 1 to room 6.

Table 2Fire spread probability distribution tables without intervention.

Node Fire spread probability

R1 P(r1) ¼ 1, Pðr1Þ ¼ 0

S1 P(s1|r1) ¼ 0.0285, Pðs1jr1Þ ¼ 0

R2 Pðr2jr1; s1Þ ¼ 0:20328;Pðr2jr1; s1Þ ¼ 0:1452; Pðr2jr1; s1Þ ¼ 0:226512;Pðr2jr1; s1Þ ¼ 0

R3 Pðr3jr1; r4Þ ¼ 0:19602; Pðr3jr1 ; r4Þ ¼ 0:20328; Pðr3jr1 ; r4Þ ¼ 0:2346432;Pðr3jr1; r4Þ ¼ 0

R4 Pðr4jr2; r3Þ ¼ 0:19602; Pðr4jr2 ; r3Þ ¼ 0:20328; Pðr4jr2 ; r3Þ ¼ 0:2346432;Pðr4jr2; r3Þ ¼ 0

R5 Pðr5jr3Þ ¼ 0:19602; Pðr5jr3Þ ¼ 0

R6 Pðr6jr4; r5 ; eÞ ¼ 0:19602; Pðr6jr4; r5; eÞ ¼ 0:20328;Pðr6jr4; r5 ; eÞ ¼ 0:1452;Pðr6jr4 ; r5 ; eÞ ¼ 0:2346432; Pðr6jr4; r5; eÞ ¼ 0:223608;

Pðr6jr4 ; r5 ; eÞ ¼ 0:226512; Pðr6jr4; r5; eÞ ¼ 0:23905728;Pðr6jr4; r5 ; eÞ ¼ 0

R7 Pðr7jr5Þ ¼ 0:19602; Pðr7jr5Þ ¼ 0

E Pðejr7 ; r9Þ ¼ 0:014; Pðejr7 ; r9Þ ¼ 0:0285;Pðejr7; r9Þ ¼ 0:03452;Pðejr7 ; r9Þ ¼ 0

R8 Pðr8jr7Þ ¼ 0:19602; Pðr8jr7Þ ¼ 0

R9 Pðr9jr8; r11Þ ¼ 0:20328;Pðr9jr8; r11Þ ¼ 0:19602; Pðr9jr8; r11Þ ¼ 0:2346432; Pðr9jr8; r11Þ ¼ 0

R10 Pðr10jr8Þ ¼ 0:19602; Pðr10jr8Þ ¼ 0

R11 Pðr11jr10 ; s2Þ ¼ 0:20328; Pðr11jr10; s2Þ ¼ 0:1452; Pðr11jr10; s2Þ ¼ 0:226512; Pðr11jr10 ; s2Þ ¼ 0

S2 Pðs2jr10Þ ¼ 0:0285;Pðs2jr10Þ ¼ 0

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908906

which is a condition of spreading from a fire compartment to theadjacent compartments, as per Eq. (13).

As described in Eqs. (1) and (2), each event (fire spread) inEq. (14) can be true or not true, both of which have to beconsidered in the calculations. Table 1 shows the probabilities ofsuccess and failure of all events that lead to barrier failure and fireignition in the different rooms of the building. Table 2 showsthe probabilities of success and failure of all events that givenignition result in fully developed fires in compartments. UsingEq. (11) the fire spread probability from room 1 and room 6 canbe computed by summing the joint probability distributionP(R2,y, R11, S1, S2, E) over all instantiations of the variablesexcept R1 and R6 as follows:

Pðr6=r1Þ ¼X

X=R6

PðR1;R2; . . . ;R11; S1; S2; EÞ (15)

A computer model was developed to perform the summationsof Eq. (15). The model was used to compute the probability of firespread from room 1 to room 6 for two cases: a case withoutsprinklers in the building, and a case with sprinklers installed inevery room in the building. A discussion of the two cases is givenbelow. The probabilities of barrier failure and probability of firegrowth to fully developed fires are estimates obtained from Refs.[17,23]. They are used to demonstrate the ability of the model tocompute the probability of fire spread from compartment of fireorigin to other compartments in the building.

4.1. Case 1: when there is no intervention of the floor

during fire spread

In Case 1, the building has no sprinklers and it is assumed thatthere is no manual fire suppression intervention. The values of the

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Table 3Fire spread probability distribution tables with sprinklers installed.

Node Fire spread probability

R1 P(r1) ¼ 1, Pðr1Þ ¼ 0

S1 P(s1|r1) ¼ 0.00285, Pðs1jr1Þ ¼ 0

R2 Pðr2jr1; s1Þ ¼ 0:04284;Pðr2jr1; s1Þ ¼ 0:0306; Pðr2jr1 ; s1Þ ¼ 0:047736; Pðr2jr1 ; s1Þ ¼ 0

R3 Pðr3jr1; r4Þ ¼ 0:04131; Pðr3jr1 ; r4Þ ¼ 0:04284; Pðr3jr1 ; r4Þ ¼ 0:0494496;Pðr3jr1; r4Þ ¼ 0

R4 Pðr4jr2; r3Þ ¼ 0:04131; Pðr4jr2 ; r3Þ ¼ 0:04284; Pðr4jr2 ; r3Þ ¼ 0:0494496;Pðr4jr2; r3Þ ¼ 0

R5 Pðr5jr3Þ ¼ 0:04131; Pðr5jr3Þ ¼ 0

R6 Pðr6jr4; r5 ; eÞ ¼ 0:04131; Pðr6jr4 ; r5 ; eÞ ¼ 0:04284;Pðr6jr4; r5 ; eÞ ¼ 0:0306; Pðr6jr4; r5; eÞ ¼ 0:00494496;

Pðr6jr4; r5 ; eÞ ¼ 0:047124; Pðr6jr4 ; r5 ; eÞ ¼ 0:047736;Pðr6jr4 ; r5 ; eÞ ¼ 0:05037984; Pðr6jr4 ; r5; eÞ ¼ 0

R7 Pðr7jr5Þ ¼ 0:4131; Pðr7jr5Þ ¼ 0

E Pðejr7 ; r9Þ ¼ 0:0014; Pðejr7 ; r9Þ ¼ 0:00285;Pðejr7; r9Þ ¼ 0:003452;Pðejr7 ; r9Þ ¼ 0

R8 Pðr8jr7Þ ¼ 0:04131; Pðr8jr7Þ ¼ 0

R9 Pðr9jr8; r11Þ ¼ 0:04284; Pðr9jr8; r11Þ ¼ 0:04131; Pðr9jr8; r11Þ ¼ 0:0494496;Pðr9jr8; r11Þ ¼ 0

R10 Pðr10jr8Þ ¼ 0:04131; Pðr10jr8Þ ¼ 0

R11 Pðr11jr10 ; s2Þ ¼ 0:04284; Pðr11jr10 ; s2Þ ¼ 0:0306; Pðr11jr10 ; s2Þ ¼ 0:047736; Pðr11jr10; s2Þ ¼ 0

S2 Pðs2jr10Þ ¼ 0:00285; Pðs2jr10Þ ¼ 0

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 44 (2009) 901–908 907

probabilities of barrier failure shown below are used in thecalculations. These values have been obtained from [17].

(1)

The probabilities of barrier failure between the variousbuilding elements [17]� room to room (separated by wall) 0.81;� room to room (separated by corridor) 0.84;� room to stairwell, elevator shaft or duct (separated by wall)

0.57;� room to stairwell, elevator shaft or duct (separated by

corridor) 0.28;� stairwell, elevator shaft or duct to room (separated by wall)

0.60.Using the above data, the barrier failure probabilitydistributions between the adjacent compartments ornodes in Bayesian network in Fig. 6 were calculated andare shown in Table 1.

(2)

The probabilities of fire growth to fully developed fire in thecompartments without a suppression system and intervention� room 0.242 [23]� stairwell, elevator shaft or duct 0.05.

Using the barrier failure probability distributions in Table 1 andabove probabilities of fire growth to fully developed fire in thecompartments without a suppression system and intervention,the probabilities of fire spread were calculated and are shown inTable 2. Using these values, the model computed the probabilityof fire spread from room 1 to room 6 to be 0.022.

4.2. Case 2: building with sprinklers

In Case 2 the building has a sprinkler system installed in the allrooms.

When a sprinkler system is installed in the building, it isassumed that the effect of sprinkler on the probabilities of barrierfailure between the various building elements is limited and theprobabilities of barrier failure have the same values as in Case 1.But the values of probabilities of fire growth to fully developed firein compartments have great change as below:

�

room 0.051 [23] � stairwell, elevator shaft or duct 0.005.

Using the barrier failure probability distributions in Table 1 andabove probabilities of fire growth to fully developed fire in thecompartments with the intervention of sprinkler system, then theprobability of fire spread can be calculated (shown in Table 3).

Using these probabilities, the model computed the probability offire spread from room 1 to room 6 to be 0.00022.

The probabilities of failure used in the case studies and theresults shown are for the purpose of demonstrating the concept.The impact of the sprinkler system on fire spread probabilities isheavily dependent on the values used. In cases where theprobability of failure is low the effect of sprinklers will be lowas well, hence to use the model for real case studies it is importantto have valid values for these probabilities. With valid values ofprobabilities of failure the model is beneficial as it can be used tocompare the impact of different systems on fire spread and allowthe estimation of property damages from fires.

One of the limitations of the model is that it does not providetime-based calculation of the probability of fire spread so that itcannot be used to determine the impact of fires on occupantevacuation and life safety calculations. For this, a transient modelis needed, which will require as input transient probabilities offailure.

5. Conclusions

A fire spread model of a building is a key factor of a fire riskanalysis of big buildings used for fire safety designs. Theprobability of fire spread from the compartment of fire origin toother compartments in the building in conjunction with smokeconditions in the building are required to calculate the expectedrisk to life and expected losses in a building during a fire. Theresults of a fire spread model can also be used to determine thefire prevention strategies for buildings.

This paper describes a model developed to calculate the firespread probability from the fire compartment to other compart-ments using a Bayesian network. To demonstrate the use of themodel to calculate fire spread from the compartment of fire originto a remote compartment in the building two cases wereconsidered: one without a sprinkler system and one with asprinkler system installed in the building. The results show thatsprinklers reduce significantly fire spread in the building. It isimportant to note, however, that the probabilities of barrier failureand probability of fire growth to fully developed fire used for thecase studies are for demonstration only. The relative impact ofsprinkler is expected to be dependant on and probability of firegrowth to fully developed fire.

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