SYSTEMS
D.A.ONISHCENKO
VNIPIMORNEFTEGAS, Moscow, Russia Abstract
This paper presents a model describing progressive fire propagation in a discrete multicomponent system with combustible elements. Each element of the system can catch fire with some probability which depends on fire intensity and element’s fire resistance, where the last is supposed to be random.
The propagation of fire over the system is characterized by means of a stepwise procedure. The ignition of the next element or elements results in load increase for elements that are not yet in flames. If the intensity of fire load exceeds the corresponding value of fire resistance for some elements, these ones are considered to be ignited at this procedure step. With a view of obtaining conservative results, the conjecture of instantaneous ignition is adopted. If at any step none of the elements catches fire, this means the fire localization; otherwise, the progressive fire covers all the system.
The principal characteristic of fire emergency for the system is the probability of progressive fire propagation. It is shown that an analytical recurrence relation for the required probability can be found. Previously, similar approaches were used in the progressive collapse analysis of redundant engineering structures. As an example, the results of calculations for model systems are presented. Some directions for future research are also proposed.
Keywords: analytical estimation, complex system, fire path, probabilistic approach, progressive fire propagation.
Fire Engineering and Emergency Planning. Edited by R.Barham.
Published in 1996 by E & FN Spon. ISBN 0 419 20180 7.
1 Introduction
The process of fire propagation in a complex system composed of many elements (objects) is often characterized by successive ignition of the constituents. For systems in which the origin of fire is a negative phenomenon, it is desirable to have a localization property, e.g. fire stop following the ignitions of some elements. When analyzing
complex systems, the probabilistic approach is used as adequate one. In this case the fire resistance of a system is characterized by the probability of fire localization, while its fire emergency is characterized by a complementary probability, with respect to 1, of the beginning of global fire covering all the system.
Note that the probabilistic nature of fire propagation process can be caused by various reasons, including random character of loads, scattering of element fire resistance, the presence of randomness in an ignition criterion and so on. Besides, if the system consists of similar elements, the probabilistic setting of the problem is one of the approaches used for proper description of multivariant progressive failure process in complex systems [6].
In many cases, the process of fire propagation and, more general, the process of progressive failure has a hierarchical feature. It is caused by natural and artificial hierarchy of system’s structure as well as internal properties of the process [7]. As a result, some intricate and interesting problems arise. An example of the study of such a problem may be found in [5], where one model system is analysing. The present paper proposes a probabilistic model which describes a fire propagation process in complex system with combustible elements under the condition that all elements are of the same hierarchy level. The parameter of element fire resistance is supposed to be random. A recursion relation for the calculation of the desired probability is analytically found.
Some results of model system analysis are also presented.
2 The principal features of the model
Consider a system being a collection of n elements arbitrary enumerated. Let us assume that the elements of the system are combustible and that after their ignition they influence on non burning elements by means of fire loading. We assume that the loads have prescribed values which may differ for different elements. When some elements are in flames, the corresponding loads are summarized.
Let us suppose further that every element has the property to withstand fire loading.
The corresponding quantitative measure is characterized by fire resistance that we will designate r. Under some load, the ignition of non burning element can only occur if the load intensity q exceeds the fire resistance r. So, the ignition criterion takes the form of
r<q.
(1) Now describe the process of fire propagation over the system. Let us assume that the system, in its initial state, is subjected to any external influence which may cause the ignition of the elements with some definite probability. Note that in the capacity of such influence we can also consider the action from the element that accidentally caught fire.
The process of fire propagation will be analyzed with the help of step by step techniques. At the first step we select all those elements whose parameters of fire resistance are smaller then the intensity of the load on them. According to the criterion (1), such elements must catch fire. At the next step we determine loads on non burning elements. Their values equal the sum of the initial load and the loads caused by inflamed elements. It is clear that the new values of the loads are, at any case, no less then those at
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the preceding step. We find again those non burning elements that are overload in accordance with (1), and then they are regarded to be in fire. And so on.
The stepwise procedure described may come to its close in two ways. In the first variant, the consecutive ignition of all elements will occur after a number of steps. In another one, the step will be found where the criterion (1) will not be satisfied for any non burning element, i.e. the process of fire propagation will stop and the fire will be localized. Such states we will call the stationary ones.
Let us make an important remark. For the sake of simplicity, we will consider the case when the ignition criterion takes into account the intensity of loading only, but not its duration. In this paper the subject of inquiry is the systems with instantaneous ignition of elements, that is the limiting case. The analysis of such systems has usually resulted in conservative estimations when one investigate the question of fire resistance of complex systems.
For the convenience of further presentation, refer a sequence of the elements ignitions as a fire path. Then introduce the notion of a critical state of the system. First of all, the ignition of all elements of a system is, obviously, a critical state. Besides, there may exist such states that, by virtue of some reasons, can be considered as identical to the full system collapse. The critical states can be characterized by the fact of the ignition of the most important elements or by the event in which the ignition of preassigned, generally speaking, great enough number of elements take place. Below, we suppose the collection of critical states to be defined in advance.
We say now that the fire resistance of a system is ensured under given external load and for given values of elements fire resistance, if the relevant fire path will not lead to any critical state. Otherwise, the fire resistance is considered to be not ensured.
The above presented model is not of any interest, when all parameters of the system and the loading are determinate quantity. On the other hand, if at least some of the parameters are random, then the problem of the system fire resistance estimation complicates significantly.
We will not treat the question of random loading and will regard the loads as determinate parameters. As to fire resistance parameters, let us suppose that they are random and are characterized by given distribution functions.
In this case, a fire path is a random entity. Indeed, at any step of the procedure describing the process of fire propagation every non-burning element may catch fire—
with some probability. Hence, the consequence of elements ignitions is random. It means that the fire path can reach any critical states with some probability only. Within this approach, system fire resistance may be quantitatively characterized by the probability of the event that the fire path will come to a stationary state.
Note that the described statement of a question is similar to a considerable extent to that widely used in the study of engineering structures reliability [1,3]. The common feature of these two problems is the presence of ‘redundancy factor’: ignition (failure) of one or several elements does not immediately lead to system collapse, since the system continues to operate in a damaged state. The outlined probabilistic model of fire propagation is similar, in its common features, to the models that were earlier applied by the author to the analysis of carrying capacity and reliability of complex technical systems [8,9].
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It may be shown that, within the scope of the model presented, the required probability can be found with the help of analytical methods in a form of a recursion relation (See Appendix A). This relation defining the quantity complementary with respect to 1, i.e. the probability of global fire emergence, may be written as
q(n)=L(q(n−1), q(n−2),…, q(1)),
(2) where L is a linear function, and q(n−j) is the probability of global fire emergence for a subsystem in the initial state of which j elements are already in flames. The coefficients in (2) depend only on the probability of elements ignition when the system is in its initial undamaged state. The formula (2) is recurrent, and the probabilities q(n−j) for all j are determined by appropriate initial states.
It should be emphasized that despite a relatively simplicity of the deciding relation (2), the execution of the relevant calculations will entail great difficulties, a part of which are, however, typical, when one programmers recurrent formulae. First of all, as n increases, the number of treatment of the formula (2) is enlarged abruptly. For example, the number of different system states considered as initial is of order 2n, that raises the required computer time up to the inaccessible level when a straightforward algorithm is used even on n=20−30. The other obstacle is associated with the dangerous of loss of significant figures. This is caused by the next reason. The terms in the right side of (2) have alternating signs and large absolute values. At the same time their sum, being a probability, should be of order 1 and below. Because register length and, hence, number precision in computers are bounded (for instance, the latter equals 20 for Turbo Pascal), some significant digits may be lost even on moderate values of n.
The elaboration of appropriate stable and effective algorithms for n large enough demands an additional investigation. Nevertheless, some recommendations can be stated just now. Firstly, the cases are often met when if the ignition of several elements occurs, the load on the other elements abruptly increases. The process of fire propagation becomes of avalanche type. So, one may assume that if the fire covers more then some definite number of elements (for example, 10 or more from total 50), then all the probabilities q(m) with m<40 may be taken at once equal to 1 instead of their direct computation. The necessary enumeration of various system states will be principally reduced.
Secondly, the qualitative analyses of possible fire paths in a given system may lead to a conclusion that some of them can’t be realized at all, or may be realized with very little probability in comparison with others. In this case there are good reasons to put the corresponding quantities q(m) equal to 0, since their full accounting would give a little correction only. Note that such states may be numerous. The similar approach has been earlier applied repeatedly in analyses of structural reliability [4,10].
3 The results of a model system analysis
As an illustration, we will present some results concerning the systems composed of uniform elements. The uniformity we will treat here as an identity of all determinate
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parameters of the elements. As to random characteristic of fire resistance, we assume that for all elements it obeys to the same probability distribution.
Determine some parameters of the system analyzed. Let the system contains n elements. We define loads on the elements in the following way: 1) the initial load has the intensity s0; 2) when an arbitrary element catches fire, the load on each of the remaining (non-burning) elements increases by the value s1. Thus, if m elements are in flame, the remaining n−m elements are influenced by fire load with the intensity s0+ms1. Let F(x) with 0≤x<∞, be a distribution function of elements fire resistance which is a random parameter.
In such a situation the common relation (2) may be rewritten in a considerably more short form:
(3)
where q(k)(xk) is the probability of global fire emergence in a system consisting of k elements under initial load xk; are the binomial coefficients equal to k!/[i!(k−i)!], and xk equals s0+(n−k)s1.
Note an interesting analogy with the model describing the classical problem on the strength of fiber bundles resolved by H.Daniels [2]. It was found [11] that the relation, derived previously by Daniels, may be written in another way, similar to (3). In Daniels model the quantities q(k) equal the probability of rupture of a bundle consisting of n fibers; the function F(x) describes the strength distribution for individual fiber, and xk are defined slightly differently:
xk=(n/k)s0,
where s0 is a specific load (per fiber) on a bundle. The total load is equal then to ns0. The main qualitative difference between the two models is that as fire (failure) propagates, the load on remaining elements increases linearly in our model, while in Daniels model—in accordance with the hyperbolic law. Below, some numerical results in comparison for both models are presented.
Let s1=s0 in the fire model; this corresponds to the case when the initial external influence is caused by some burning element. In this case, the initial loads in both models are the same. As a distribution of random parameter of fire resistance (strength) of elements we take the Weibull one:
F(x)=1−exp[−(x/x*]m,
(4) where x* and m are some constants. Let for definiteness x*=1, m=2. In Table 1 and corresponding Fig.1 some relevant results of the calculations are given.
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