A Probabilistic Model of Fire Spread With Time Effects

ELSEVIER

Fire Safety Journal 22 (1994) 367-398 1994 Elsevier Science Limited

Printed in Northern Ireland. 0379-7112/94/$07-00

A Probabilistic Model of Fire Spread with Time Effects

Dav id G. Pla t t

Department of Civil Engineering, Queen's Building, University of Bristol, Bristol, UK, BS8 1TR

Dav id G. E l m s & A n d r e w H. B u c h a n a n

Department of Civil Engineering, University of Canterbury, Canterbury, New Zealand

(Received 20 August 1991; revised version received 17 January 1994; accepted 19 February 1994)

ABSTRACT

The paper outlines the principal elements of a probabilistic model that analyses the spread of fire in multi-compartment buildings with respect to time. The analysis uses a graph theoretic network and an event hierarchy to determine the probability of fire spreading to different locations. The probability of fire spreading between compartments is based on a comparison of the probability density functions of the expected fire resistance and the fire severity: failure being the condition that severity exceeds resistance. The model is designed as a comparative tool to compare the performance of different fire safety strategies by calculating a 'cost index' for each design, based on the probable extent of fire damage in the building. The analysis gives attention to the compatibility of fire resistance and fire severity, and their conversion in real time parameters.

A B

AF AT Aw

N O T A T I O N

Total floor area of the building (m 2) Floor area of the fire compar tment (m 2) Internal surface area, excluding floors (m:) Ventilation area (m 2)

367

368 David G. Platt et al.

B C c,,

d

D D

Firelni

FireStartsi

FS,; g G, h hw HK

L

M, 0 Q

R S SPo. SPrel t

lo T

u,

Event fire spreads through an internal barrier Conversion factor taken as 0.07 (miniM Jim 225) Specific heat of hot gases leaving the compartment (kJ/kg K) Expected degree of damage in a compartment due to fire and sprinklers as a fraction Depth of fire compartment (m) Event fire spreads through an open doorway Time at which fire enters j from i via any path, given fire starts in i and spreads to j Probability density function of X Cumulative distribution function of X Event that during an annual period compartment i experiences a fire Event that during an annual period, a fire originates in compartment i Event fire spreads from i to j, given fire starts in i Acceleration due to gravity (9.81 m/s) Fire growth time in compartment i (s) Height of window (m) Mean weighted height of ventilation openings (m) Effective heat transfer coefficient of compartment lining (W/m 2 K) Fuel load (kg) Mass loss rate of combustible materials in ventilation controlled fire (kg/s) Mass of combustible material i (kg) Event that a door is open Rate of heat release in the fire (kW) Fire load density = ( A F ) - ' S M i . ~ ( M J / m 2) Fire resistance of a barrier (s) Duration of fully developed fire (s) Sprinkler effectiveness Sprinkler reliability Time (s) Time equivalent as given by ISO fire curve (s) Time of inception period (s) Temperature of hot gas layer in compartment (K) Ambient atmospheric temperature (K) taken as 295 K Time taken for fire to spread from i to j, given that fire does spread from i to j (s) Energy content of item i (MJ/kg)

U

W

w~ W W X

Z

Zk

o¢

13

Y

r/ O

)t

/x ~7

~p

Po

A probabilistic model of fire spread with time effects 369

Interest rate (as a fraction, not a percentage) Width of windows (m) Ventilation parameter = AF(AwATV'hw)I/2(m -'''25) Width of fire compar tment [m] Event fire spreads via external windows Horizontal projection of flame tip (m) Height of flame tip above window (m) Time taken for fire to spread along path k, given that fire does spread along path k (s)

Fire growth coefficient (kW/s z) Capital value of the sprinklers, as a ratio of the overall capital value of the building Indirect losses, as a ratio of the capital value of the building (At - a w ) / ( A w . X/hw) Ignition time for upper compar tment in window fire spread Cost of passive fire protection measures, as a ratio of the capital value of the building Discount factor = (1 - e-V~)/(e v - 1) Design life of building (years) Repair costs, as a ratio of the capital value of the building Density of ambient air (kg/m 3) Cnq,, is taken as 1.2 kJ /m 3 K

I N T R O D U C T I O N

One of the primary concerns of fire engineering is to minimize the risks caused by fire. These may be summarized as: risk to human life, risk to adjacent property and risk to the exposed property. In reducing these risks there are several strategies available to the engineer. One such strategy is the restriction of fire spread. The aim of the present work is to develop a model that can be used to assess the probable spread of fire with respect to time in mult i -compartment buildings. The model could then be used by the engineer as a tool in the comparative evaluation of different design strategies aimed at preventing fire spread. The times at which detection and suppression of the fire occur can also be included, 1 but they have not been described in this paper. The


problems associated with smoke spread are currently being addressed and have not been included in this discussion.

OBJECTIVES

In setting out to develop such a model the following objectives were identified as being important.

• The model should have a 'consistent level of crudeness '2 both internally within the analytical routines and externally with respect to the data required and results produced.

• The analysis should be based on a probabilistic rather than deterministic approach. Such an approach was felt to be more realistic given the uncertainties that exist in both the input data and the mechanisms of fire spread. For example, the mechanisms of fire spread are not confined to a single deterministic system; rather, they are impacted upon by several other systems, such as the weather conditions or the current occupancy. Therefore , any at tempt to quantify fire spread that does not consider all possible systems, will result in a probable rather than deterministic set of values.

• The analysis should use real time, rather than an equivalent or modified time such as obtained from laboratory test results using the ISO standard fire curve.

It was felt that these objectives could be met by extending the work of Elms and Buchanan, -~ who used a graph theoretic, probability approach to determine the way in which fire spreads between rooms in a mult i -compartment building. The probability of fire spreading from one compar tment to another was considered irrespective of how long it might take. In this respect their model represented a worst case scenario, and assumed that the fire would eventually burn itself out. Whilst the Elms and Buchanan model was relatively simple in its approach, it was consistent and well balanced with respect to its input data and the complexity of the analysis, and for this reason was considered a good platform from which to build a model that included time.

The nature of any model is de termined by its intended use, and also by the availability of appropriate data. The two should be consistent with one another, and both dictate the appropriate degree of approximation at each step in the model 's theoretical development. In the present case, the intention has been to use the model as a comparative


rather than an accurate predictive tool, for comparing the relative effectiveness of different fire safety strategies.

U N D E R L Y I N G CONCEPT OF THE M O D E L

Essentially the analysis is a comparison between the expected fire severity within each compartment and the probable resistance the building has to the spread of fire. In order to make such a comparison it is necessary to have a consistent unit with which both the resistance and fire severity can be measured. The unit used in the following analysis is time.

The severity of a compartment fire may be represented by the length of time the fire burns. The uncertainty of both the mechanism of fire growth and the contents of any particular compartment results in a probable rather than precise time for which the fire will burn. Associated with this value of time is a probability density function, such as that shown in Fig. l(a).

The fire resistance of the building may be dealt with in a similar way. For example, consider the fire resistance of an internal barrier. The type of barrier used will have a nominal Fire Resistance Rating (FRR) given by a testing authority. The particular barrier in a given building is unlikely to have exactly the same fire resistance, and its value can be represented by the log-normal probability density function shown in Fig. l(b). The two distributions can be combined, as shown in Fig. l(c), to evaluate the probability that the fire severity will exceed the fire resistance, i.e. the probability that fire spreads.

The expected time at which the fire will spread, given that it does spread, can be determined from the graphs by:

• evaluating the probability that at a particular point in time, say t, the fire resistance will be equal to t, and the fire severity will exceed t, so that fire will spread through the barrier at time t;

~.~ | Fi~S,v,~ity z-'~ | ~ z-~ ".~ -.~ . ~

Tune Tune

(a) (b)

it Tm'~ L-

(c)

Fig. 1. (a) Log normal probability density function for Compartment Fire Severity. (b) Log normal probability density function for Fire Resistance Rating of a barrier. (c)

Comparison of probability density functions for fire severity and fire resistance.


• repeating this for all values of time t, to give a probability distribution for the expected time at which fire will spread. From this distribution a mean value for the expected time for fire to spread can be obtained.

In essence this is the concept on which the model is built. However, there are problems of consistency in using time directly as an input parameter and consequentially the input data for the barriers have to be modified. This is done so that the barrier resistance times used in the model represent the resistance of the barriers when subject to a fire of the expected severity. The severity of the 'real' fire may be quite different from the severity of the standard ISO test fire upon which the quoted Fire Resistance Rating times are based. A further difficulty is representing the severity of a fire by the single variable time, a point that will be discussed later in the paper.

As well as barrier failure, there would normally be a number of alternative paths for the spread of fire to an adjacent compartment . Figure 2 illustrates the point: the numbers 1, 2 and 3 represent, respectively, spread of fire through an open doorway, spread via windows and spread through a barrier, which may be a wall, closed door or ceiling.

With the spread probability and time distribution obtained between adjacent compartments , the spread to any other compar tment in the building can then be found, using a graph theory approach. The final requirement for the model is to express the performance of a building as a single value. This is done through the construction of a meta-model which takes the output of the fire spread analysis and combines it with

Fig. 2. Possible fire spread paths to adjacent compartments.


economic data about the building. The result is a single performance measure expressed as an economic index. It can then be used for comparing the performance of different design strategies.

It is also possible to calculate a safety index which indicates the probability of the fire not spreading into escape routes. This was done in earlier work, 3 but is not considered appropriate in the present model because, in the absence of smoke spread, the necessary assumptions would be cruder than the rest of the model.

OUTLINE OF THE FIRE SPREAD ANALYSIS

An outline of the complete analysis is given in Fig. 3. The analysis starts with the input of physical and probability data. Expected values of the compartment fire severity, fire growth time, size of the flame projecting out of a window and a real-time barrier modification factor are calculated from these input data. A comparison is then made between the computed values. The output from these comparisons is a series of probabilities that fire will spread via each of the three possible fire path types. Combining the individual probabilities gives an overall probability of fire spreading to an adjacent compartment. Repeated for each compartment within the building, these values collectively form the Adjacency Fire Spread Matrix, and represent the probability that a fire in compartment i will spread to an adjacent compartment j.

The next stage is to determine the expected time for fire to spread to an adjacent compartment, given that fire does spread. These values form the components of the Adjacency Fire Spread Time Matrix. By combining the two matrices, the analysis computes the probability of fire spreading from an initial compartment i to any compartment j. The fire may spread along any path, but is conditional on arriving at compartment j at time t. The resulting matrix is known as the Global Fire Spread Matrix which may be considered as a three dimensional matrix with each layer being the probabilities evaluated at different values of time t.

Having formed the Global Fire Spread Matrix a 'cost index' is obtained. This is a measure of the economic performance of the building and is expressed as the ratio of the expected fire loss to the capital cost of the building. At this stage the cost index is conditional on the initial occurrence of fire in a particular compartment. It may be assumed that during the life time of a building there are certain annual probabilities of fire occurrence in each compartment. If these annual probabilities are combined with the conditional probabilities of the

Input thermal

Input Input barrier ventilation resistance data data

1 1 1

growth me barrier I modification

factor

1

l ] compute probability of fire spread via barrier failure

Input compartment

Input [ Input economic sprinkler parameters data

Output conditional cost index

Input spandrel heights properties & geometry

compete compartment fire severity

fuel loads

1

Fig. 3.

, j compute probability of fire spread via windows

It form Adjacency Fire Spread Matrix

=- calculate expected fire spread times and form Adjacency Fire Spread Time Matrix

form Global Fire Spread Matrix

/l form Fire Vectors

It Output basic cost indices

] I Input / ~ probability of / [ doorways

compute slze of name projection from window

I

I_

Outline of fire spread analysis.


compute probability of fire spread via open doorway

I

Global Fire Spread Matrix the result is a set of annual probabilities, one for each compartment, of fire burning in the compartment either due to ignition there or due to fire spreading, in a given time t, from ignition in another compartment elsewhere in the building. The complete set of these probabilities is known as the Fire Vector.

The following sections of this paper describe in greater detail a number of the different aspects of the model.


i

Flashov~

Time

Fully Developed Decay Period L r -1- q- q

Fig. 4. T i m e - t e m p e r a t u r e curve for compar tment fire.

THE C O M P A R T M E N T FIRE

It is assumed that the development of a compar tment fire follows the typical t ime- tempera ture curve outlined in Fig. 4. As can be seen from the curve the fire history can be divided into three distinct stages, the growth period, the fully developed fire and the decay period. What is required is some measure of the time duration for each of these stages.

Growth period

The growth period is described as the time from the point where fire is initiated to the stage where flashover occurs. During the growth period the average temperatures are low and the fire is localized. The growth period of a fire consists of two distinct stages; the inception period and the rapid growth period.

The inception period may last a few seconds or several hours, being primarily a function of the fuel and ignition process. As a result of uncertainties the inception time is taken as zero, thus assuming the growth period starts at the point of rapid growth. This appears justified: in the initial fire compar tment the analysis starts from the time when flames appear and the fire is detectable. For the rest of the building, any new sources of ignition are likely to be accompanied by the ' thermal momen tum ' of the already existing fire. Note that with the spread of fire through a barrier the real concern is the penetration of flames rather than insulation failure. This point is dealt with in more detail later in the paper.

The rapid growth stage of the fire is characterized by a progressive increase in the fire area and a growth in the heat output. Ramachandran 4'5 developed an exponential model of fire growth that


relates the growth time to the fire area. In moderately sized compartments it may be reasonable to assume that flashover occurs when the fire area equals the total compartment floor area. However, as yet statistical data relating to buildings other than warehouses have still to be analysed.

A more commonly adopted approach is to define the point at which flashover occurs in terms of a ceiling temperature and burning rate. Using the criterion that flashover occurs at a certain ceiling temperature it is possible to determine the rate of heat output necessary to create this condition. Having obtained a value for the necessary level of heat output the time taken to achieve this can be obtained by assuming a parabolic fire growth rate as suggested by Haskestad. 6

The analysis assumes that flashover takes place when the ceiling temperature reaches 600°C7 To calculate the rate of heat release necessary to achieve this the model uses the following relationship, ~ which expresses the change in compartment temperature as a function of the rate of heat release Q.

A T ~f O H~.A, ] (1) --~o = ] L g,/2( Cpp,,~T,,A wh ,(2 " g,/2( C,p,, )-----A-wh ~ 2 ]

This may be expressed in the form:

AT -- c . x ' , ' . (2)

L

where XI and X2 represent two dimensionless groups, the constant C and exponents m and n being determined from experimental data. McCaffrey et al. ~' analysed over 100 compartment fires, where gas temperatures did not exceed 600 °C, and offer the following values for C, It and m:

AT = 480. X~ ~-' . X~ '"' (3)

Assuming that the fire growth follows a parabolic relationship then Q the rate of heat release is given by:

Q = o~(t - t,,) 2 (4)

Values for the coefficient, a, fall in the range, c~ = 0.1, for rapid fire spread, to a = 0.01 for slowly developing fires. By rearranging eqn (4) and substituting for Q then, t, the growth period, is given by:

t = t,, + (756 ~ ~ (H~.A ,/2 ~/2 ,.,._ • . , A . h , ) ) (5)

A discussion concerning the merits of using an approach that is relatively sensitive to both the ventilation and lining characteristics of a compartment, when earlier studies have shown these parameters to be

A probabilistic model o f fire spread with time effects 377

of least significance, 1° would be of value. For this reason other alternatives have been explored by Platt e t al. 1 However, such a discussion is beyond the scope of this paper.

Fully developed fire

The heat flux generated in both the growth and decay periods is small when compared to the fully developed fire. Therefore, it seems reasonable to assume that a barrier is only at risk during the fully developed fire and not during the growth or decay periods. For this reason the term fire severity shall refer only to the fully developed stage of the compar tment fire.

The expected duration of the fully developed fire, S, is given a s "11

S = L . m 1 (6)

The mass loss rate of combustible materials, rh, depends upon whether the fire is ventilation controlled or fuel bed controlled, this being determined by the supply of oxygen. If the supply of oxygen is greater than that required to balance the stoichiometric equation then the fire is fuel bed controlled. It is usual to assume that after flashover has occurred the fire is ventilation controlled and the oxygen concentration is nominally zero. The analysis adopts a simplified approach to this problem based on a design method suggested by Law and O'Brien. 12 Theobald and Heselden 13 found that in domestic furniture fuel bed controlled fires, the fire duration is about 20 min. Taking this to represent the min imum possible duration of a fire, the duration of the fully developed fire is given by:

~L. rhf ~ where S > 1200 s

S = t1200 s where S -< 1200 s (7)

fuel is consumed in a ventilation where rh r is the rate at which controlled fire. Based on experimental test data Thomas and Heselden 14 offer the following empirical expression for rhi:

rh I = 0.18. A w h ~ v 2 ( W / D ) m . (1 - e -°'°36n) (8)

The mass loss rate mr is based on experiment using wooden cribs as the fuel. Clearly the fuel load in real compartments will be different. Currently these differences such as the volatility of the fuel and the different rates of energy output are ignored. Clearly this could have significant implications when evaluating the severity of a compar tment fire using the single parameter t ime) 5'16

The fuel load L of a compar tment is the summation of the mass of


all the individual combustible materials. It has been assumed that the distribution of these values will have a log normal density function. A comprehensive series of data tables has been published in the CIB 17 report. The report summarizes many European and American surveys of fire loads, giving average values and standard deviations for various occupancies. The important parameter is the variability, as indicated by the coefficient of variation (COW). For a large Swiss study it was found that for well defined occupancies CofV = 30-50%, and for occupancies which are rather dissimilar CofV = 50-80%.

As the mass loss rate rh s is derived from empirical data, then strictly speaking the model should treat it as a probabilistic value. However, at this stage of development it is assumed that rhy is a deterministic parameter. Therefore, the duration of the fully developed fire is a log-normal variate with an expected value and variance given by:

E{S} = rn I IE{L}; Var {S} = rhl 2 Var {L}

LIMITATION OF TIME AS A MEASURE OF SEVERITY

The direct use of the duration of the fully developed fire to represent the severity has certain limitations. For example, it gives no indication of the temperature, nor the build up of energy within the compartment. However, it does offer one major advantage, that is, the use of a common dimension, time, for both the fire severity and the barrier resistance.

Consider for example, two similar compartments with identical fuel loads but different ventilation characteristics. The time temperature curves could appear as shown in Fig. 5. Whilst the total heat released

I 0 0 0 C -

t~

O C -

F u l l y D e v e l o p e d F i r e - T i m e Pe r i o d

(a) Unrestricted Ventilation

J Co) Restricted Ventilation

TIME (0

Fig. 5. Comparison of two compartment fires.

A probabUistic model of fire spread with time effects 379

may be the same for both compartments , the duration and temperature will be quite different. Part of the complication with this stems from the fact that 40 min at 400 °C is not equivalent to 20 min at 800 °C. This is because the radiant heat flux is proportional to temperature to the fourth power. In the example above, radiant heat exposure for 40 minutes at 400 °C is equivalent to 2.5 minutes at 800°C. For an e lement of structure exposed to radiant energy, the severity experienced in the short duration high temperature fire will be greater than that of the long duration, low temperature fire. The problem is further complicated if the heat absorbed by a structure, and thus its temperature, is considered. For example, if a steel structure is clad with an insulation material then this will retard the rate at which the temperature of the steel increases. It is possible that the steel structure will reach a higher temperature in the lower temperature, longer duration fire. Of course, in a fully developed compar tment fire thermal energy will not only be transmitted by radiation but also by conduction and convection. This further adds to the difficulty of using time as the sole measure of fire severity.

The assumption in this paper, that the severity of a compar tment fire can be measured by its duration, is obviously crude. However, it represents the first stage of a pragmatic approach to quantitatively comparing the resistance of a structure to the expected fire load measured in real time. The next stage of refinement may consider means of estimating the compar tment temperature and the quantity of thermal energy reaching the structure from radiation sources and from this modifying the expected resistance. In the meant ime the above limitation of the model to realistically analyse the compar tment fire must be accepted.

D I F F E R E N T M E A S U R E S OF TIME

A further complication arises from the use of the ISO standard fire curve. An important principle in engineering design is the ability to compare the performance of different structures that have been tested against a consistent frame of reference. In this respect the ISO standard fire provides a useful f lame against which various structures such as walls and internal partitions are measured. In design it is usual to express the duration of the compar tment fire as an equivalent exposure time to the ISO standard fire, known as the ' t ime equivalent ' (re), calculated using expressions such as the one given by the CIB ~7 in eqn (9).

te = 60. wv. QI (9)


Fig. 6.

IOO0 C '

0C --

~ ~ R e a l Fire

I N . , N I S O Standard

I TIME ~[t) " te

Comparison of ISO time-temperature curve and a real compartment fire time-temperature curve.

This allows a direct comparison between the duration of the compartment fire and the fire resistance of the walls and doors, etc. The problem with this approach in modelling the actual spread of fire, is that the t ime- tempera ture relationship given by the ISO fire curve has only a limited resemblance to the relationship likely to be experienced in a real fire situation, Fig. 6.

In order to resolve, at least in part, this problem the Fire Resistance Ratings (FRRs) of the walls and barriers, as obtained by the standard fire curve, are modified to represent their expected time to failure when exposed to the real fire. This then allows a comparison between the fire severity as obtained from eqn (7) and the modified FRR of the barrier on the basis of real time.

The modified FRR, known as the 'equivalent FRR' , is obtained by multiplying the FRR, obtained from the ISO standard fire test, by the ratio of the duration of the real fire given by eqn (7) to the equivalent time given by eqn (9), giving:

t = 1"32 --~-Tl/ (1 -- e -0"036~) i ( 1 0 )

te A w . h w • W

RESISTANCE TO FIRE S P R E A D

As described earlier there are three principal ways by which fire may spread; through a door, vertically via windows or through a barrier. Each of these paths has a different resistance to the spread of fire.

A probabilistic model of fire spread with time effects

Fire spread resistance of doors

381

Where a door is closed the resistance of the door to fire spread is considered along with the barrier as outlined later. Where the door is open then it is assumed that it offers no resistance to the spread of fire. However, it is assumed that the fire will not spread beyond the compartment until flashover has occurred. The expected time therefore, for fire to spread beyond an open doorway will be given by the compartment fire growth time.

Fire spread resistance of windows

If the windows are aligned vertically and the resulting flame projection is large enough, fire may spread up the external facade of a building by 'leap frogging' from a window on the fire floor to a window above as illustrated in Fig. 7. The analysis is based on the following simplified assumption:

• Flames will emerge from the lower compartment at the time of flashover.

• Calculations of the probable height and horizontal projection of the flames from windows are based on an analysis of the mechanism of fire spread given by Law. TM The mean height of the flame tip above the window E{z} is given by:

Z = 12"8(vhf /Aw) 2/3 (11)

V///'q/~'//I///~'l///s////////~ X I

'] ~///I/////~,/s//H/////¢////l~

Fig. 7. Fire spread via windows.

11

$


whilst the horizontal projection E{x} is given by:

x = 0 . 4 5 h w ( h / 2 w ) °53 (12)

This assumes the post flashover fire is ventilated only by natural draft and ignores any forced ventilation caused by such things as internal pressure or flume effects. Furthermore, it is assumed that the ventilation area consists only of external windows and that these have a width to height ratio w / h >-2.0.

• If the height of the flame exceeds the height of the spandrel then it is assumed that fire will spread to the upper compartment.

• The smaller the horizontal projection of the flame from the window the shorter the fire spread time. The time is taken arbitrarily as 3 min if the flame tip is at the face of the building, varying linearly to 12 min if the flame tip is 1 m or more from the face of the building.

Fire spread resistance of barriers

Fire resistance tests are performed using the ISO standard time temperature curve. They are usually carried out on a 'one-off' basis due, in part, to the high cost, but mainly because sponsors only require one successful test result to obtain a FRR. As a result there is little information known about the expected variance for a particular type of barrier.

In a sur*ey of the fire tests carried out in New Zealand, Platt ~9 found that the CofV was between 5 and 12%. The larger the approved FRR the smaller the coefficient of variation. In light construction, such as timber framed walls lined with plaster-board, the most probable mode of failure in testing was found to be insulation failure. An integrity failure occurred on average some 12 min after the insulation failure was recorded. In most cases insulation failure is unlikely to result in the spread of fire. It is therefore suggested that the failure time for modelling fire spread be taken as the integrity failure time. 1~'2° For light structures this may be taken as the approved FRR plus 12 min.

In the real situation, the nominal FRR of a barrier may be substantially reduced by the presence of a weakness. These may range from electrical services and plumbing ducts to doors, etc. They may be internal or externally mounted and may or may not pass directly between the two compartments.

The analysis includes these weaknesses by reducing the expected FRR of the respective barrier. Information concerning the expected decrease in FRR due to weaknesses is scarce. Quintiere 21 reports that

A probabil&tic model of fire spread with time effects 383

experiments with electrical branch circuit fixtures lowered the fire resistance of a 1-h wood stud gypsum board by 13 to 23 min, the failure being by flame penetration at the fixture. Alan Beard (unpublished, 1988) developed a fault tree analysis with the data being collected from a Delphi group. The method developed is useful and could be readily applied once the findings of a Delphi group have been recorded.

THE PROBABILITY OF FIRE SPREADING TO AN A D J A C E N T COMPARTMENT

Fire may spread between two adjacent compartments via three possible path types as shown in Fig. 2.

• through an open doorway with probability P[D], • via external windows with probability P[W], • through an internal barrier with probability P[B],

where D, W and B represent the events that fire spreads through a doorway, a window or a barrier, respectively. Therefore, the probability of fire spreading to an adjacent compartment P[FS] is given by

P[FS] = P[D U W U B] (13)

The three types of fire path can be arranged in a hierarchy of events. The spread of fire through an open doorway is at the top of the hierarchy, that is to say, if a doorway is open fire will spread to the next compartment via this route rather than any of the other possible routes. At the bottom of thie hierarchy is the spread of fire through a barrier. The justification behind the construction of this hierarchy is as follows. If a door is open then it is very improbable that the fire path will be via an adjacent wall or window as both of these paths will involve significantly longer failure times. The spread through windows comes next. For fire to spread via a window it is necessary for the projected f ame height to exceed the spandrel height. It is assumed that at flashover the windows in a fire compartment will blow out and that the flames will then pass out reaching their maximum height. Consequently, fire spread via windows is likely to occur fairly soon after flashover. Typically if fire is going to spread in this way then it will do so within 3-12min of flashover, whereas fire spread via barriers, the lowest hierarchical route, is likely to take much longer than this. It is very

384

Fig. 8.

David G. Platt et al.

I

/Windows

_ Open Door

~Barrier

Venn diagram of fire spread event hierarchy.

unlikely that a floor will have a fire rating less than 30 min even after adjustment: remember the CofV is usually less than 12%. Therefore, if both the window and barrier mechanisms exist it is assumed that fire will spread via a window before the barrier fails. Figure 8 shows a Venn diagram of the hierarchy.

The arrangement of the three path types in a hierarchy enables them to be treated as mutually exclusive events. If fire spreads via a doorway, then the probability of fire spreading via a window is zero, or P[WID] is zero, likewise P[BID] and P[BIW] are also zero. By expanding relationship (13) and using the above results the probability of fire spreading P[FS] can be expressed as:

P[FS] = P[D] + P[W] + P[B] (14)

If the basic events for which probabilities can be calculated are;

D--f i re spreads through open doorway; / ) - - f i re does not spread through open doorway; W ]/)--f i re spreads via windows, given it can do so; /) U W--fire does not spread through an open doorway or via windows; B ](/) U 14z)--fire spreads through internal barrier given it can do so;

where a bar above means the negation of the event, then eqn (14) may be written as:

P[FS] = 1 - (1 - P[D])(1 - P[W I/5])(1 - P[B I(/) U !~,')]) (15)

Probability of fire spread through an open doorway

The probability of fire spread via an open doorway is assumed to be the probability that the door is open P[O], i.e. it is assumed that if fire is


going to spread there is a 100% chance of fire spreading through an open doorway.

Probability of fire spread via windows

The probability of fire spreading vertically up the facade of a building via windows, given that fire does not spread via a doorway, is the probability that the height of the external flames is greater than or equal to the height of the spandrel:

P [ W I/)] = (~ fname(Z) dz (16) asp andre l Ht

where fname(Z) is the density function of the flame height.

Probability of fire spread through a barrier

The probability of fire spreading through a barrier, given fire does not spread via a doorway or windows, is the probability that the fire resistance is less than the fire severity, i.e.

P[B 11D U fiT] = P[n < S] (17)

where R is the barrier resistance and S the compartment fire severity. This expression may be formulated in terms of the ratios Y = R/S. If R and S are independent log-normal variates then Y is also a log-normal variate. The case of barrier failure resulting in fire spread would be the event (Y < 1); therefore, the corresponding probability of fire spread via a barrier, is:

P[B I (/) U I,V)] = P[Y < 1] = fv(Y) dy = Fv(l'O) (18)

EXPECTED TIME TAKEN FOR FIRE TO SPREAD TO AN A D J A C E N T COMPARTMENT

Having derived an expression for the probability of fire spread to an adjacent compartment, the next stage is to determine the expected time it will take, given that fire does spread. In this section time refers to the relative time it takes for fire to spread from compartment i, to an


adjacent compar tment j, measured from the time fire first entered compar tment i.

Let the random variable Tq represent the time for fire to spread from i to j. The probability of fire spreading from compar tment i to j at time t, where t lies between t and t + dt, will be given by P[Tq = t]. Note: P[Tq = t] is used as shorthand way of writing P[t < Tq <- (t + dt)].

The Theorem of Total Probability states:

P[A] = ~ P[A I Bi]. P[Bi] all B~

provided the Bis are mutually exclusive and comprehensively exhaus- tive. Therefore, applying this theorem we have:

P[Tq = t] = ~ (P[Tq = t lfire spread condit ion] . P[fire spread condition])

(19)

However: m

P[Tq = t I fire does NOT spread] = P[Tq = t ]FS] = 0 (20)

where FS is the event that fire does not spread. Therefore, the conditional probability, that fire will spread from

compar tment i to an adjacent compar tment j at time t, given that fire does spread, is given by:

P[Tq = t l FS] P[Tq = t] (21) P[FS]

The probability of the event P[FS] is given by eqn (15), and from the Theorem of Total Probability the unconditional probability P[Tq = t] is given by the following expression:

P[Tq = t] = P[(Tq = 0 1 D ] . P[D]

+ P [ ( L = t) l W]. P[W]

+ P[(Tq = t) [ B] . P[B]

+ P[(Tq = t) IFS]. P[FS] (22)

As the last term P[(Tq = t) ]FS]. P[FS] is zero, expression (22) may be written as:

P[Tq = t] = P[(Tq = t) I O] . P[D] + P[(Tq = t) lW]

• P [ W ] + P [ ( T q = / ) I B ] . P [ B ] (23)


It is possible to further expand expression (23) by considering separately the three groups of terms.

The time taken for fire to spread given an open doorway is taken as the fire growth time for the compartment. This assumes that fire will not spread beyond a compartment until the fire has reached a fully developed state. Therefore the probability of the events, P[(T~j = 0 1 D ] . P[D], is given by:

P[D]. P[growth time for compartment i = t I open door]

= P[D].fG,.o(t) dt (24)

where fc,~o(t) represents the density function of the variate fire growth time, given an open doorway, in compartment i.

If the event 'time for fire to spread given it spreads via external windows' is taken as the fire growth time for the lower compartment i plus the ignition time Oj for the upper compartment j, then the probability of the events, P[(T~j = t) I W] . P[W], is given by:

P[W]. P[(growth time for comp i + ignition time for comp j = t) I W]

which when expanded and expressed in terms of known probabilities, becomes:

(1 - P [ D ] ) . P[W I/5]

• P[(growth time for comp i + ignition time for comp j = t) I W]

= ( 1 - P [ D ] ) . P [ W l b ] . { £ ' f o , ( O ) . f ~ , ( t - O ) d O } d t (25)

where foj(h) represents the density function of the variate ignition time of upper compartment j.

The probabilities of events (T~j = t ) [ B and P[B] are considered separately and expanded as follows. The probability P[(T~ i = t) I B] may be written as:

P[(T~j = t) l B] = P[(Rij = t) fq (S i > t)] (26)

If the barrier resistance Rij and the compartment fire severity Si are assumed independent then the above expression can be written as:

=fm(t) dt{1 - Fs,(t)} (27)

which is shown as the two areas marked on Fig. l(c). (This is not a


completely valid assumption as the value R is adjusted by the ratio tile, which is dependent in part on the variable S. Again as the dependency is likely to be small, and considering the accuracy of the other data, the assumption appears valid and consistent with the level of sophistication used throughout the rest of the analysis.)

The above expression (27) assumes that the fire starts in a fully developed state and takes no account of the growth time. In order to realistically model chronological time then the growth period of the fire needs to be included. The growth time may be handled as a random variable as described below.

Let the growth time for a fire in compartment i be represented by the random variable Gi, which has an associated probability density function fG,(t). Then the probability P[(T~j = t) lB] becomes:

P[(T~j = 01 B] = fa,(g), fR,,(t - g)[1 - Fs,(t - g)] dg dt (28)

The probability of fire spreading through a barrier P[B] can be expanded so that it contains only variables for which probabilities can be calculated.

P[B] = P[B I(D U W)]. P[D U W] + P[B I(/) U if')]. P[/5 U !¢~']

= P[B I(/) U I,V)]. P[/) U I~]

But

P[{) U 1~] = 1 - P[D U W]

= 1 - P[D] - P[W] + P[D n W]

= 1 - P [ D ] - ( 1 - P [ D ] ) . P[WI/3] + P [ W I D]. P[D]

So

P[B] = (1 - P [ D ] - (1 - P([D]). P[WI/3] ) . P[B 1(/3 U if')] (29)

Therefore, combining expressions (28) and (29) the probability P[(Tis = t) IB]. P[B] is given by:

P[(T~j = t) I B]. P[B]

= {(1 - P[D] - (1 - P[D]). P[W I/)]) . P[B I(O u w)]}

× fa,(g), fR,(t - g) . [1 - Fs,(t - g)] dg dt (30)

The substitution of eqns (24), (25) and (30) into expression (23) and the subsequent substitution of the resulting expression into eqn (21) gives


the probability that fire will spread from compartment i to j at time t, given that fire does spread, for a combination of all three paths, i.e. e[Tu = t l FS].

Correctly speaking then:

P [ T u = t I FS] =- P[t < T u <- (t + d/) I FS]

= fmFs(t) dt (31)

The expected time for fire spread, given that fire does spread, E ( T u l F S ) is then given by the first moment of area about the origin of the above density function.

The above algorithm presents a hierarchy of mutually exclusive events which combines the probability of fire spreading via each of the three path types with the expected time that each path will take. This gives a probability that fire will spread from one compartment to the next and a value for the expected time, for a combination of all three path types. In the outline shown in Fig. 3 these values are shown as the Adjacency Fire Spread Matrix and the Adjacency Fire Spread Time Matrix. The next stage is to expand the analysis to include the spread of fire between any two compartments.

FIRE SPREAD TO ANY COMPARTMENT

The spread of fire between two compartments may involve a number of intermediate compartments, forming a series of alternative paths. These alternative paths are not generally independent of each other as they may contain compartments in common. It is, therefore, far from trivial to compute the combined probability of fire spread between two compartments.

The easiest way to understand the algorithm developed is to consider the simple four compartment building shown in Fig. 9(a). Assume fire

Fig. 9.

3 1 4

2

(a)

llnk24 PATH 2 PATH I (~

~ ~llnk34

PATH 3 PATH 4

(b)

(a) Plan of four-compartment building; (b) possible paths by which fire may spread from (1) to (4).


~o~o~O~O~ .'

FNolFr~ure ~ / ~ ~ ~ ( ~ Q ~ ~nr_.._.7/.. ~ ~~"2(~ ~ - . - Pa l ®® @® USZ3~ i immure • " " Path 3 - - -- Path4

Branch only Fig . 10. Modified event tree.

starts in compartment (1) and spreads to (4). Fire may spread from (1) to (4) by any of the following four paths: (1)---~ (3)--* (4), (1)--0 (2)----~ (4), (1)--0 (2)--0 (3)--0 (4), ( 1 ) ~ ( 3 ) ~ ( 2 ) ~ (4). The simplest approach to find the probability of fire spreading from (1) to (4) is to use an event space method• This technique considers exhaustively all possible events• However, this requires vast computational effort, and is not generally feasible. Instead a modified event tree, which truncates a branch once the final outcome is inevitable, is used. A graph of this is shown in Fig. 10. The probability of failure occurring along any one of these branches is the probability that all the events on a branch occur. Consider for example the branch marked - - 0 - - 0 - - 0 - - 0 . The probability of this branch occuring is P[(01) n (12) n (13) N (32) N (24)]• If the events are assumed i n d e p e n d e n t then the branch probability becomes P[01]. P[12]. P[13]. P[32]. P(24).

Whilst the modified event tree in Fig. 10 shows seven different branches by which fire may spread to compartment (4). Figure 9(b) indicates that there are only four different paths by which the fire may reach (4). Therefore, the probability of fire spread along a path will be the summation of the branch failures that constitute that path. For example, consider path (3), (1)--0 (2)--0 (4):

P[Path(3)] = P[(branch (01), (12), (23), (34), (24))

U (branch (01), (12), (23), (24))]

As each branch is mutually exclusive then the separate branch probabilities can simply be summed• It is therefore possible to derive an expression for the probability of fire spread along each path.


TIME TAKEN FOR FIRE TO T R A V E R S E A GIVEN PATH

Given that fire does spread from i to j , then let the time taken for fire to spread along path k be represented by the random variable Zk. Then in general:

Z, = ~ Tq (32) all ij on path k

where Z, represents the conditional probability Z ~ I F S q and Tq the conditional probability Tq I FSq. In the special case where Z, = Tq, then f zK( t ) = fT, j(t) , and where Z, = Tq + T~k then:

f z~ ( t ) = fT; ,(y) fr j , ( t - y ) d y (33)

In general then, where Z k = Tq + Tj, . . . + Tpq, f zK( t ) could be solved algorithmically, using the above convolution integral at each step. Alternatively the problem of having to evaluate the above convolution integral may be avoided by applying the Central Limit Theorem to expression (32). Interpreted liberally, this states that the sum of a number of individual random variables, none of which is dominant , tends to normal distribution as the number of variables increases regardless of their initial distribution. That is, as the number of Tq terms increases Zk will tend to a normal distribution with mean E { Z k } = E E{Tq} and variance Var {Z,} = 3", 0 .2 {Tq}.

PROBABILITY OF FIRE SPREADING TO ANY COMPARTMENT IN A GIVEN TIME

Consider again path (3). The probability that fire will enter compartment (4) via path (3) at time t, is:

P[(Path (3))N (Path time to (4 )= t)]

= P[(Path time to (4) -- t) ] Path (3)]. P[Path (3)]

As all the possible paths are mutually exclusive, in general the probability that fire will enter compartment j via any path from compartment i at time t, is

P [ E T q = t l FSq] = ~ P[path time to j = t] path(k)]. P[path(k)] all paths k from i to1

(34)

392 David G. Plattet al.

where ET~j I FSgj is the entry time, that is, the time taken for fire to enter compartment , j, via any path measured from the time, t0, when fire started in compar tment i, conditional on the event fire spreads from i to ]. Substituting for Z~ eqn (34) may be rewritten as:

Thus;

P[ET~j = t l FSji] = ~ P[Z , = t]. P[path(k)] al l k

fET,~,FS,,(t) dt = ~'~ f z , ( t ) dt. P[path(k)] all k

(35)

(36)

Given that the fire starts in compar tment i, the probability that fire will enter, or will have entered, compar tment j, via any path, at or before time t, where t lies between t and t + dt is given by:

f' P[ET~j <- t l FSij] = fEr,,(t) dt

= ~ fzk(t) . P[path(k)] dt (37) all k

This expression represents the probability that compar tment j will be damaged by fire after a certain time t, measured from the time when fire first started in the building and is conditioned on the events that fire starts in compar tment i, and spreads from compar tment i to j. The time, t, can be given any reasonable value, but probably of most use are values such as the response time of the fire service and human egress times.

F IRE VECTORS

So far we have only considered the consequences of fire spread, given that fire starts in a specific compartment , and as such eqns (13)-(37) are conditional on this event. A more comprehensive measure of the performance of a building can be obtained by taking into account the probability of fire starting in different compartments.

A fire vector represents the total probability of fire occurring in a compar tment i, due either to fire starting there, or to it starting elsewhere and spreading to that compartment . It is defined as the probability of the event (Firelni) that during an annual period


compartment i experiences a fire and is conditional on the event that if fire starts elsewhere in the building the time from its point of rapid growth is less than t. If the event that during an annual period, a fire originates in compartment i is given by FireStarts~ then:

P[Fireln~] = P[FireStarts~] U ~ P[FireStart%] fq P[FSj,] N P[ETj, <- t] all . /except i

P[Fireln,] = P[FireStarts,]

U ~ P[FireStart%]. P[FSj, I FireStart%]. P[ETj~ <- t l FSj, a l l / e x c e p t i

fq FireStartsj] (38)

Where sprinklers are concerned, we can assume that if fire originates in or spreads to a compartment, and the sprinklers operate, a large proportion of the compartment will be damaged. Let such an expected degree of damage be d. With the inclusion of sprinklers the fire vector becomes:

P[FireIn,] = P[FireStarts,](1 - (1 - d)P[SPre,] . P[SPe, I SP~e,])

U {(1 - P[SPre,]. P[SPep I SP~e,]) • ~ P[FireStartsj] a l l j except i

• P [ F S j i I FireStartsj]. P[ETji <- t[ FSji ~') FireStartsj]} (39)

The sprinkler reliability (SPrel) is the probability that the system will work if a fire occurs, and the sprinkler effectiveness (SPe,) is the probability that the sprinklers, if they work, will extinguish the fire. Note that in eqn (39), the probability terms that sprinklers are reliable and effective modify the overall fire vector rather than individual elements of the path. This is tantamount to saying that if the sprinkler system fails to operate for a fire in one compartment, then it will also fail to operate in any other compartment; secondly if sprinklers fail to extinguish fire in any compartment then they will fail in all compartments. The justification for the first assumption is that a sprinkler system usually fails not because an individual head fails to operate but because of some overall factor such as the water mains being turned off. The second assumption reflects the belief that if the sprinkler heads operate in one compartment and fail to control the fire, then the fire will already have reached such a size that the operation of further


sprinkler heads will also fail to contain it; and furthermore, if a number of heads are already in operation, the water pressure will be reduced leading to a decrease in the effectiveness of the remaining heads.

P E R F O R M A N C E INDICES

An important quality of models used in engineering for comparative evaluation, is their ability to synthesize the relations between the many and varied inputs, to a manageable or preferably single index. In the example shown in Fig. 11, the inputs are synthesized to a single economic index dependent upon time. Currently the model is being extended to include the effects of smoke and this will pave the way for the development of a safety index.

The loss cost index gives the expected loss due to fire for the overall building as a percentage of the capital value of the building. It takes into account the probabilities of fire occurrence and the design life of the building. If the bulding is assumed to have a capital value of unity then the loss cost index CIL is given by:

CIL -- lO0/z (1 + A + [3)(y + q~) ~ AF,. P[Fireln,] (40) A B for all i

where /x is a discount factor to reduce the expected losses over the design life of the building to an equivalent capital sum and is given by:

- - e v T

e v - 1 (41)

The assumptions used in deriving eqn (40) are that;

• damage is confined to compartments reached by fire; • damage is proportional to floor area, unless sprinklers are

installed, in which case the expected degree of damage is given by d;

• once fire reaches a compar tment damage to that compar tment is complete;

• the total damage to a building is a linear combination of the damage to individual compartments.

Clearly, these assumptions represent only very crude approximations

A probabi l is t ic m o d e l o f f ire spread with t ime effects 395

to what happens in real fires, but as long as use of the technique is restricted to comparisons between cases, the assumptions lead to a consistent and useful measure of the relative economic performance of a building.

APPLICATIONS

The prime use of the fire spread analysis is as a tool to compare different fire spread prevention strategies. Such a comparison may, for example, consider the relative merits of automatic door closing units and sprinkler systems. The inclusion of time allows the effects of fire service intervention to be considered. The simplified approach adopted by the analysis restricts its proper use to comparisons, and as such the model is unsuited to determine the likely fire history of a building or the performance of a single building structure in a fire. The model has been used to compare different fire spread prevention strategies in small office blocks, motel units and typical apartment style dwellings. 22

As an example of typical output, consider Fig. 11 which shows the loss cost index at times up to 2 h from ignition. The building is a simple single storey building of five rooms. The expected fire severity is such that 60 min FRR on all walls prevents any fire spread. Losses are lower with sprinklers because any fire is extinguished before flashover occurs. The other curve shows the probable damage if there is a 25% or 50% probability of all the doors being open at the time of the fire, for both 30 min and 60 min FRR values. The importance of intervention by the fire service within 20 min of ignition is clearly seen. This model can be used to compare a wide range of active and passive fire protection

t60

120

¢1 ..a l,O

0 0

i i i i

- - =- 3 0 m t n FRR Ooors

~ . ~ 6 0 m i n FRR ~ ~ . . . . . 50*/* open"

/

2 / I " ~ - - 2 5 % open v v v j . 0 . . ~ - . . . . - ~

3 4 ~ ~" . . . . . . . "

c losed .~ ' i nk le r S

! I f I t I

20 z o 60 80 tOO 120

Time ( m i n u t e s /

Fig. U. Cost indices for single storey building.

I


measures in mult i-compartment buildings, provided that suitable input data are available.

LIMITATIONS

Probably the most restrictive limitation that prevents the model being able to accurately predict a real fire situation, is that the fire is assumed to develop linearly. For example, after a fire starts in a compartment , it goes through inception and growth phases before becoming fully developed. If the fully developed fire spreads to a new compartment , the model assumes that the fire growth period is the same as if the fire were first ignited in that compartment . This approach ignores the ' thermal momentum ' effect of the already developed fire, in particular, the effect of the radiated heat flux emitted by hot gases. An important aspect of fire safety is the evaluation of smoke spread. This has not been included in the model, although current developments are extending the model.

CONCLUSION

This paper outlines the development of a probabilistic model to analyse the spread of fire in mult i-compartment buildings following flashover in one or more compartments. The model is not intended to be a detailed fire growth model such as developed by others. Fire growth is included on a relatively crude basis, to allow the time and probability of fire spread to be estimated. The model allows a single damage index to be calculated for a given building. This takes into account the probability of ignition and subsequent spread within the building.

Care has been taken at each stage to maintain a consistent level of approximation, or crudeness, and to balance this against both the crudeness of available data and also the use for which the model is intended. The result is a useful tool for the comparison of different fire safety strategies.

The approach presented in this paper offers the probability modeller a means of overcoming the limitations of the more normal fault tree and event tree analyses; that they cannot allow for the time taken for events to happen.

A C K N O W L E D G E M E N T

The authors thank the Building Research Association of New Zealand for its financial support of this work.

A probabilistic model of fire spread with time effects

R E F E R E N C E S

397

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2. Elms, D. G., The principle of consistent crudeness. Proc. Workshop on Civil Engineering Application of Fuzzy Sets, Purdue University, IN, 1985.

3. Elms, D. G. & Buchanan, A. H., Fire Spread Analysis of Buildings. BRANZ Research Report R35, New Zealand, 1981.

4. Ramachandran, G., Exponential model of fire growth. Proceedings, 1st Conf. Fire Safety Science, Washington, DC, 1985.

5. Ramachandran, G., Probabilistic approach to fire risk evaluation. Fire Technology (August 1988) 205-25.

6. Haskestad, G., Engineering Relations for Fire Plumes. Society of Fire Protection Engineers, Technology Report 82-8, 1982.

7. H~igglund, B., Jansson, R. & Onnermark, B., Fire development in residential rooms after ignition from nuclear explosions. FOA Report C20016-D6 (A3), Forsvarets Forskningsanstalt, Stockholm, 1974.

8. Drysdale, D., An Introduction to Fire Dynamics. John Wiley, Chichester, 1986.

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10. Heselden, A. J. M. & Melinek, S. J., The early stages of fire growth in a compartment. A Cooperative Research Programme of the CIB (Commis- sion W14), First Phase, Fire Research Note No. 1029, 1975.

11. Heselden, A. J. M., Parameters determining the severity of fire. Paper in Joint Fire Research Organisation Symposium No. 2, Behaviour of Structu- ral Steel in Fire. HMSO, London, 1968.

12. Law, M. & O'Brien, Fire Safety of Bare External Structural Steel. Constructional Steel Research and Development Organisation, London, 1981.

13. Theobald, C. R. & Heselden, A. J. M., Fully developed fires with furniture in a compartment. Joint Research Organisation Fire Research Note 718, Borehamwood, 1968.

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15. Thomas, P. H. & Bullen, M. L., Burning of fuels in fully-developed room fires. Fire Safety Journal, 2 (1979/80) 275-81.

16. Bullen, M. L., A combined overall and surface energy balance for fully-developed ventilation-controlled liquid fuel fires in compartments. Fire Research, 1 (1977/78) 171-85.

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19. Platt, D. G., Fire-resistance of barriers in modelling fire spread. Fire Safety Journal (in press).


20. Schwartz, K. J. & Lie, T. T., Investigating the unexposed surface temperature criteria of standard ASTM El19. Fire Technology, 21(3) (August 1985) 169-80.

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22. Buchanan, A. H. & Elms, D. G., The effect of fire resistance ratings on likely fire damage in buildings. Department of Civil Engineering, Research Report 88/4, University of Canterbury, New Zealand, 1988.

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A Probabilistic Model of Fire Spread With Time Effects