Capítulo 3 Marco Metodológico
3.3 Relato autobiográfico
3.3.8 Trayectoria musical
R.V.GOLDSTEIN
Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia
Abstract
An discrete-continual approach is developed for analysis of the problems on fireresistance estimation and improvement in hierarchical complex technical systems. The generalized characteristic of the catastrophic accidents scale. namely, the rank of the catastrophe is introduced. The conditions of the catastrophe transition from one rank to another are formulated. As an example of the approach application the model problem on improvement of fireresistance by means of protection distribution through the system structural levels is considered and the resuits of its analytical and numerical solution are given.
Keywords: Hierarchical complex technical system. fireresistance, protection distribution levels.
1 Introduction
Catastrophic accidents on complex technical systems, in particular accidents induced or accompanied by fire propagation, envelop, as a rule, many scale levels. These levels correlate with the scales of separate objects forming the complex technical system or their sets as well as with the scales inherent to appropriate physical processes.
Prediction of the catastrophic accidents and development of recommendations for improvement of resistance (e.g., fireresistance) of the complex technical systems to such catastrophes require description of conformities of their initiation and propagation in hierarchical systems taking into account the physical and/or chemical fields leading to the catastrophe and the properties of media being in contact with the complex technical system under consideration.
To analyze such problems we developed a multiscale approach which allows to model processes on various scales by discrete or continual manner [1]. The concept on the rank of the catastrophe relative to given physical field acting on the system is introduced. The rank is a generalized characteristic of the catastrophe scale. The conditions of the catastrophe transition from one rank to another are formulated.
As an example of the approach application we considered together with Dr.
D.A.Onishcenko the problem on improvement of the fireresistance of a complex multiscale technical system by appropriate distribution
Fire Engineering and Emergency Planning. Edited by R.Barham.
Published in 1996 by E & FN Spon. ISBN 0 419 20180 7.
of protective means on various scales. It is shown that the probability of the fire propagation in large scale can be reduced strongly by an optimal distribution of protective means on different scale levels of the system. Some results of the numerical modeling of the problem are obtained.
2 A discrete-continual approach
Assume that in the complex technical system one can settle out elements and/or their groups having the characteristic scales L1,…, Lk (L1<…<Lk). It is possible but not necessary that elements of one or several scales represent gemetrically identical or similar objects.
We will believe that the disturbance of the system technical state is catastrophic if it is accompanied by rise and propagation of dangerous field of action in media surrounding the system and/or being in contact with it. By the dangerous field we mean here the fields dangerous from the ecological or mechanical point of view (e.g., fire induced temperature, radiation and pressure fields).
Denote by df1,…, dfp the dangerous fields caused by a catastrophe.
If the field dfk envelops the scale Lj one will say that the catastrophe has the rank rjk
relative to the field dfk.
Let rj1,…, rjp. be the catastrophe ranks of the system under consideration relative to the fields df1,…, dfp, respectively. Then the rank, r, of the catastrophe of the system is by definition equal to maximum of the ranks rj1,…, rjp:
(1) Variation of the catastrophe rank relative to the field ield dfk, rjk, is determined by kinetics and dynamics of this field. Different fields dfm and dfk. can be conjugated and the appropriate ranks rjm. and rnk. inter related. The relation between the ranks rjm. and rnk. can be changed when the catastrophe rank increases. Increase of the catastrophe rank, relative to one field, (its scale) can be accompanied by decrease of the rank relative to other dangerous field (e.g., localization of harmful contaminants issuing at fracture caused by fire embrasing increasing scales).
We will mainly consider later on catastrophs connected with partual or total fracture of the complex technical system caused by fire. The catastrophe rank r =i will be correlated with the scale of the fire embrased region Li.
The catastrophe of i-th rank generation can be caused directly by the fire and fracture process of elements of the scale Li or can be a result of formation of catastrophe zone of the scale Li because of fire induced fracture of a certain set of elements of the scale Li−1.
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Description and modeling of the process of a definite rank catastrophe formation and extension as well as its rank increasing provide for availability of conditions which specify transition of an element of the system in the limit state, transition to the limit state of other elements of the same scale under the action of the perturbation field induced by the fire and/or fracture of an initial element as well as conditions of the next (larger) rank catastrophe formation.
The conditions of the first and second groups can be written similarly to nesessary and sufficient conditions of formation of an hierarchy of structures of fracture [2,3].
The peculiarity of the catastrophe extension process consists in its instability (dynamics). In this connection a specific role play media transmiting perturbations of different fields caused by fire and fracture of an element of the complex technical system.
Namely these processes are ascartaining for formation of the second and third group conditions.
Indeed, a perturbation propagation depends upon both the situation in the neighborhood of the fructured element and the reaction of the system in larger scales or as a whole on the arised perturbation. The system reaction is essentially determined by its capacity to retain and maintain the regime of perturbation localization as well as to provide transition to regimes of propagation of distributed perturbations.
While it is convenient to consider in frames of a discrete approach conditions of catastrophic fire induced fracture of separate elements, the analysis of the system (or its subsystem) reaction as a whole on a local perturbations can be performed using a continual approach taking into account the system structure. By the way we obtain the possibility to separate the whole problem of the catastrophe formation and extension in multiscale system onto a series of inner and outer problems.
The analysis of the process on the scale Li represents an inner problem relative to larger scales (Li+1, etc.). This problem is solved by incorporating specific mechanisms of fire and fracture in the scale Li and actions on the element under consideration from larger scales.
In turn, the problem on perturbation distribution in the complex technical system or in its subsystems of the scale Li+1 represents an outer problem relative to the scale Li. In the outer problem one considers the process of propagation of a field induced by fire and/or fracture in an element of the scale Li. Analysis of the outer problem can be performed in frames of a continual approach. The appropriate continuum can be modeled as an equivalent (generally nonlinear) composite medium (see [1]).
On can model perturbations caused by the processes in the scale Li as specific body sources.
Note, that regimes with peaking at perturbation propogation can appear in such nonlinear medium with the body sources. Similar regimes have been described in [4] in the case of the heat field.
Unstable regimes, e.g. regimes with peaking, can lead to appearence of a correlation between processes in several elements of the scale Li. This correlation, in turn, will promote increasing of the catastrophe rank.
The formulated approach allows to model the processes of catastrophe initiation and propagation in multiscale systems, It can be used also for consideration of the problems on the catastrophe protection and preventation leading to hierarchical principle of constuction of active and passive (or combined) protection.
An hierarchical approach to fire resistance improvement of complex technical systems 105
3 A model of fire protection circuit in hierarchical complex technical systems
3.1 Main assumptions of the model
We will now consider a hierarchical complex technical system of the following type:
- on the rank o the system structure consists of elements of the same type;
- the system structure of rank (j+1) is a set of blocks formed by pairwise join of j-rank blocks;
- the whole system coincides with the structure of rank n which can be represented as join of two blocks of (n−1)-rank;
Note, that the described structure corresponds to a “fractal tree”.
Assume that the intensity of an action depends on fire included structural level. Further, the system is provided with a fire protection. The protection resourses can be distributed through various structural levels of the system. The problem under consideration consists in searching for an optimal protection circuit. Here we will analyze the problem in assumption that all protection must be concentrated on a certain structural level and it is required to choice this level to achive maximum efficiency of the protection. More general case as well as detailed results of numerical calculations will be considered in separate publication with D.A.Onishcenko.
Let us formulate now the assumptions on the fire (and fracture) propagation in our hierarchical system.
The fire process can start from the structure of o-rank. Assume that a certain element of o-rank perished or failed due to fire. Then this element exerts an action of the given type on a conjugate element. Remind, that both these elements form one of the blocks of 1-rank. The conjugate element can also exert with certain probability which is determined by an appropriate distribution function and the action intensity. If this event arises one can say that a block of 1-rank is out of action.
Fracture of the block of 1-rank exerts an action on elements of a block conjugated to him. There exist two such elements for the block of 1-rank. Note, that the action caused by the failed block is constant up to fracture or perish of both of these elements forming the conjugate block. A refusal of these both elements is equivalent to a refusal of 1-rank block conjugate to failed one. Hence, two conjugate blocks of 1-rank are out of action.
This means that a block of 2-rank is failed (or perished). The fire propagation by such a way leads to refusal of (n−1)-rank block. The refusal of the whole system takes place if both (n−1)-rank blocks fall out.
Let us assume now that a certain mechanism of reduction of an action on a given block exists on each level of the system hierarchy. Denote by rj the value of the action reduction on j-rank block. Then the action on the given j-rank block, qj, can be represented as follows
qj=sj−Rj
(2) where sj is the action induced by the refusal of the conjugate j-rank block and Rj equals to the sum of the protection resources on all levels of the hierarchy from o-rank up to j-rank
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(3)
3.2 A model problem on optimal distribution of fire protection resources
Assume that the protection volume is fixed and equals to Vo. For definiteness we will believe that the protection is achieved by using of special shell. The shell thickness determines the degree of the fire action reduction. Assume for simplicity that the shell is of cylindrical type when the elements of the system are placed in a plane square region and the shell is closed one when the elements are placed in a cubic region in 3D-space.
In a general case the protection resources can be distributed partly on various levels of the hierarchical structure of the system.
We will consider the simplest approach when the whole protection is concentrated on one level only. Then the problem on protection optimization consists in searching for number of the hierarchy level where the protection efficiency has maximum. We will assume that the probability of complete fracture of the system is a measure of the protection efficiency. Hence, the maximum protection efficiency corresponds to minimum of this probability.
First let us analyze the case when all system elements are placed in a plane.
Assume that the height of the protection wall equals to h independent on the structural level. Denote by tj the thickness of the protection wall placed on j-th level.
Let us estimate the required extension of the protection wall. Suppose that it is equal to the total perimeter of all j-rank blocks. Then the protection volume Vj equals to
Vj=4 1o·r(n−j)/2·h·tj
(4) where 1o is the characteristic size of o-rank element. Taking in mind that the protection volume is given one can equate Vj=Vo in assumption that the total protection is concentrated on j-th level. Then from (4) we obtain
tj=to 2j/2
(5) Note that in the case when the system elements are placed in a cubic region similarly
tj=to 2j/3
(6) Assume that the protection degree is proportional to a certain power of the shell thickness. Then the sequence {rj} is increasing one such that
rj=ro. aj, a>1
(7) The law of the action sj increasing with the structural level growth can be sufficiently arbitrary.
An hierarchical approach to fire resistance improvement of complex technical systems 107
We will now show on numerical examples that the situations are possible when the protection distribution on the o-th level is not optimal.
Suppose that the strength distribution of o-th rank elements is described by the Weibull one
F(o) (x)=1−exp(−x2)
(8) We will analyze the two following variants:
1. the j-dependences of sj and rj are linear sj=so+j/3, rj=ro+j/3;
(9) 2. sj is the quadratic function on j and rj is the linear one
sj=so+[(j+5)2−25]/60
rj=ro+j/4 (10)
where so=0.6, ro=0.45.
The hierarchy level n equals to 10 in the both cases.
The probabi1ities of total fracture of the system were calculated according to the formulae (11)–(13) given in subsection 3.3. The results of calculation are presented below
Table
Protection Variant
1 2
with no protection 2.21. 10−1 3.02. 10−1
0 1.32. 10−3 1.27. 10−3 1 4.09. 10−4 1.98. 10−4 2 8.60. 10−4 1.49. 10−4 3 4.24. 10−3 9.08. 10−4 4 1.50. 10−2 1.18. 10−2 with protection on the level
5 3.35. 10−2 7.86. 10−2
It is seen from the Table that in Variant 1 the optimal protection is achieved if the protection is ccncentrated at o-rank elements while in Variant 2-at 2-rank blocks.
Moreover, in the last Variant the probability of total fracture of the system is almost one order of magnitude smaller with protection at 2-rank blocks than that for the protection at o-rank elements.
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3.3 Probability of refusal of the total system
In this subsection we will give a formula for calculation of the probability of refusal of the total system. This formula follows directly from some general results obtained in [5].
Omiting details we will only formulate the final result.
Denote by D(j) the probability of refusal of j-rank block having initially a refused element. In particular, D(o)=1 since the initial fire damage of o-rank element is accepted as a certain event. The value D(n) is equal to the desired probability of refusal of the total system.
Let p(j)(x) be the integral distribution function of block “strength” under the local action of the intensity x. This function depends on the distribution function of o-rank element strength, F(o)(x), the scale variation of the action, sj, and on the distribution of the protection resources through the structural levels, rj.
Under the above assumptions one can write the following formula for the value D(n) of the total system refusal
(11)
where
(12)
and
p(j)(q)=2p(j−1)(q+sj−1)−[p(j−1)(q)]2
j=1,…, (n−1) (13)
4 References
1. Goldstein R.V. (1993) About an structural-continual approach in mechanics of catastrophic fracture of complex technical systems. Reports of the Russian Academy of Sciences, 330, 45–47 (in Russian).
2. Goldstein R.V., Osipenko N.M. (1978) Fracture and structure formation. Reports of the USSR Academy of Sciences, 249, 829–832 (in Russian).
3. Goldstein R.V., Osipenko N.M. (1978) Structures of fracture (Conditions of formation. Echelons of cracks). Preprint N 110. Institute for Problems in Mechanics, USSR Academy of Sciences (in Russian).
4. Danilov V.G. (1991) Asymptotic finite solutions of degrenerate quasi1inear parabolic equations with small diffusion. Mathematical notes, 50, 77–88 (in Russian).
5. Onishcenko D.A. (1992) Refusalresistant building structures, in “Reliabi1ity, lifeteme and safety of technical systems”, SPb, 1994, PP. 102–107 (in Russian).
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