SIMULATION-BASED FORECASTING EFFECTS OF AN ACCIDENTAL EXPLOSION ON THE ROAD. PART I: METHODOLOGICAL FRAMEWORK

<jats:p>Forecasting mechanical actions induced by accidental explosions on the road is of crucial importance to assessing potential damage to structures and non‐structural property exposed to them. A logical result of such forecasting may be expressed in the form of probabilistic models. They should quantify likelihood of occurrence and physical characteristics of accidental explosions. Generally the models are to be selected under the conditions of sparse statistical information on intensities and likelihood of explosive actions. The first part of the present paper proposes a simulation‐based procedure intended for selection of the probabilistic models in the absence of direct statistical data on the explosive actions. The proposed procedure is formulated in the context of the classical Bayesian approach to risk assessment. The main idea of the procedure is that statistical samples necessary for fitting the probabilistic models can be acquired from a stochastic simulation of an accident involving an explosion on the road. The proposed simulation‐based procedure can be used for damage assessment and risk studies within the methodological framework provided by the above‐mentioned approach. A case study illustrating an application of the proposed procedure is given in the second part of the paper.</jats:p>

section_content: Man-made accidents occurring during transportation of hazardous substances by road and rail include such adverse physical phenomena as explosions, fires, and releases of dangerous materials [13]. Mechanical, thermal, and sometimes chemical actions (effects) induced by these phenomena can cause considerable damage to structural and non-structural property. Highly probable targets of explosions and fires are structures built in vicinity of roads. In terms of structural engineering, an action induced during a manmade accident is called an accidental action. The present paper will be focussed on one type of accidental actions, namely, actions which can be generated by an accidental explosion (AE 1 ) on the road. 

A complete prevention of accidents capable of inducing AEs is not always possible or economically feasible. What is possible it is a limitation of potential damage which can be caused by AEs. Predicting characteristics of AEs is indispensable for assessing this damage. Characteristics of a blast wave generated by an AE can be predicted either by experiment (direct measuring) or by means of mathematical modelling. . ull-scale and scaled experiments imitating AE and allowing a direct measurement of blast wave characteristics can be expensive and cover only a part of values of the random factors which determine these characteristics (e. g. [4] ). 

In some cases, predicting effects of AEs is possible by combining mathematical modelling and data from direct measurements of AEs [5, 6]. However, the available knowledge can sometimes allow predicting accidental actions merely by mathematical modelling. It can be carried out by applying a stochastic (Monte Carlo) simulation of accidents inducing accidental actions [7]. Such modelling is fully applicable to accidents involving an AE on the road. 

The stochastic simulation allows coping with considerable uncertainties related to AEs. . ormally, the simulation can be used for propagating these uncertainties through mathematical models of AEs. The uncertainties themselves can be expressed by means of a quantitative risk assessment (QRA). QRA is well suited to dealing with sparseness of statistical data on man-made accidents and so accidents involving AEs on the road [8, 9]. This paper proposes an approach to a simulationbased forecasting effects of an AE which may take place on the road. The result of this forecasting is a probabilistic model describing possible effects which can be induced by this AE. The model can be applied to assessing potential damage to exposed structures in the framework of QRA or, alternatively, it can be used for specifying traditional, deterministic intensities of the effects for the conventional structural analysis. 

The paper consists of two parts. The first part develops a procedure for simulating the accident involving AE on the road and selecting the probabilistic model suitable to describe mechanical effects of this AE. The second part presents a case study which illustrates an application of the proposed procedure [10].
section_name: Introduction

section_content: AEs and similar dangerous phenomena occurring on the road are, by their very nature, a matter for QRA [1113]. Mathematical tools of QRA allow dealing with accidents involving AEs in a formalised way. However, many of the QRAs, which consider AEs occurring in transportation, do not approach accidents of this type from the standpoint of structural engineering. The following examination seeks to stress the structural aspect. What is more, the author of the present paper strongly believes that dealing with AEs on the road by the traditional, deterministic means prevailing in the structural engineering does not hold much promise. 

Logical models of QRA can be applied to trace causes and consequences of an accident involving AE. The causes of AE can vary substantially. Therefore this paper will be focussed on an accident which can be initiated by a collision of two vehicles. One of them carries explosive substances. Thus AE can be a consequence of vehicular impact. . ig 1 shows an event tree developed for the initiating event E 0 = collision of vehicles. Branches of this tree diagram end with the consequences C 1 to C 4. The first two, C 1 and C 2, are determined mainly by AE. 

The standard result of QRA applied to the event tree shown in. ig 1 is the so-called risk profile {(. r(C k ), S k ), k = 1, 2, 3, 4}, where. r(C k ) and S k is the frequency and significance of C k, respectively (e. g. [8, 9] ). The significances of C 1 and C 2 are determined in essence by characteristics of AE. . r(C 1 ) and. r(C 2 ) coincide with the frequency of AE related to respective event tree branch. This frequency will be denoted next by. r(AE), where AE is the random event (re) of AE. 

Potential damage to structures from a particular AE and, more generally, the significance S 1 and S 2 can be assessed by means of a probabilistic model describing characteristics of this AE (probabilistic action model, pam). When selected pam can serve as a tool for predicting characteristics of AE and estimating probabilities of specific damage to structures. 

The problem considered in this paper is how to select pam. r(m) which expresses the frequency (annual probability) of exceeding the magnitude m of a mechanical effect generated by AE. This model can be schematically defined as. r(AE) × P(m | AE), where P(m | AE) is the conditional probability of exceeding m given AE. In the interests of brevity the selection of. r(m) will be considered only for one type of accident, namely, the accident represented by the sequence of res E 0, E 1, and E 2 (. ig 1). 

The pam. r(m) breaks down the modelling problem into smaller problems of estimating the likelihood of AE and modelling characteristics of AE. A more precise definition of the pam. r(m) and thus peculiarities of its selection depend on an interpretation of the frequency. r(AE) and probability P(m | AE). Estimating. r(AE) and fitting P(m | AE) solely on the basis of the data gained from occurrences of an AE will more often than not be impossible. Data on AEs are usually sparse or unsuitable to a particular situation of exposure to AE, or, what is not uncommon, unavailable at all. 

The situation of sparseness or absence of data can be alleviated by mixing hard data (relevant experience data) with subjective information (expert opinions and judgements of analysts) as it is done in QRA [14]. A methodological framework for the combination of hard data and subjective information is provided by a general approach to QRA known as the classical Bayesian approach (CBA) [14, 15]. Principles of CBA will be followed in this paper. In view of the selection of. r(m),

. ig 1. Event tree diagram with the initiating event collision of vehicles and tracing the consequences involving an accidental explosion on the road CBA may be defined as a tool for estimating. r(AE) and P(m | AE).
section_name: Looking at the road accident from the standpoint of quantitative risk analysis

section_content: In line with CBA, the frequency. r(AE) and probability P(m | AE) are treated as true, even if unobservable and unknown, quantities. Both. r(AE) and P(m | AE) express the aleatory (stochastic) uncertainty in the res AE and exceedance of m given AE. Another type of uncertainty distinguished in CBA is the epistemic (state-of-knowledge) uncertainty in the true values of. r(AE) and P(m | AE). Thus a pam based on CBA should incorporate both types of uncertainty. 

In terms of CBA, a model of an accidental action can be expressed as a family of probability distributions (pds) quantifying both aleatory and epistemic uncertainty [7]. As applied to AE, CBA yields the following form of the pam. r(m):

where x is the vector of action characteristics; X is the random vector which models an aleatory uncertainty in x; p a is the frequency of AE; P a is the random variable (rv) which expresses an epistemic uncertainty in p a ; ) | ( A simplified example of such a pam is the family of hazard curves of ground motion acceleration used in a seismic risk assessment [16]. 

The definition (1) follows the principles of CBA which require keeping apart the aleatory and epistemic uncertainties in the final form of risk analysis [15]. It follows from the definition (1) that a selection of the pam

as well as assignment of the respective weights p i.
section_name: Mathematical model used to predict mechanical effects of explosion on the road

section_content: In most cases the analyst will face a complete lack of direct data allowing a direct fitting the cdfs ) | (

In principle, some data are almost always available, say, from back (postmortem) investigations of one or more analogous accidents. However, it is usually impossible to group the data in samples which are large enough for fitting these cdfs. 

The situation of the virtual lack of direct data will be usually encountered when the analyst has to do with unique situations of exposure to AE. However, this situation can not be considered hopeless if the analyst can employ probabilistic and deterministic models describing physical phenomena behind the events E 0, E 1, and E 2. The selection of cdfs ) | (

θ x X can be based on these models. Models describing E 0, E 1, and E 2 are normally developed independently of each other. Therefore it is hardly probable that the entire set of them will be present in a ready-to-use form. In authors experience, extra effort will usually be required to make these models compatible with each other. 

The
section_name: Selecting the model of accidental explosion on the road 4.1. How to deal with virtual lack of direct data on explosion effects?

section_content: The uncertainty propagation via a stochastic simulation of the accident represented by E 0, E 1, and E 2 is visualised by. ig 2. It shows the event tree path containing two branching points. A nested loop simulation procedure can be applied to the propagation [7, 17]. An application of the two loops stems from the necessity to distinguish between the aleatory and epistemic uncertainties. They can be expressed by the model sets = : The rule used to decide at the two branching points represented by the complementary events E k and E k (k = 1, 2) can be expressed as follows:

where k ω is the output domain defining an occurrence of E k ;

) | ( x is defined as l x = ( jl x, j = 1, 2,, n al ). These samples can serve for fitting cdfs ) | (

as well as assigning the weights p i.
section_name: Procedure for simulating the accident involving an explosion on the road

section_content: A need for expert judgement arises from the fact that AEs are typically rare events backed by sparce hard data. The pam ) ( x X Fr defined by ( 1 ) reflects uncertainties expressed by the sets of lower-level models 1

Ξ and 2 Ξ. Given these sets, the selection of

amounts to uncertainty propagation by means of the above simulation-based procedure. 

According to the model classification used in QRA, the models

can be considered to be process

Energy of collision exceeds tolerable value, above which explosion can be triggered in vehicle A (event event E 1 )

Calculate characteristics of AE:

Update frequency of AE:

Store x j and n al in computer memory Entire accident leading to AE is simulated,

Current accident simulation is terminated and the next one started anew
section_name: Expert judgement in forecasting effects of accidental explosions on the road

section_content: (index l, n p repetitions) Nested loop

belong to stochastic models [14, 18]. The structure of 1 Ξ and 2 Ξ depends mainly on the available knowledge about the physical phenomena behind the res E 0, E 1, and E 2. A part of this knowledge should be the process models

which are a prerequisite for an application of the stochastic accident simulation. Another part of the knowledge should be information which allows assigning and updating the stochastic models included in 1

Ξ and 2 Ξ. This information may consist of available hard data and expert judgements. In some instances expert judgements may be the only source of information for specifying elements of 1 Ξ and 2 Ξ. 

Expert judgement can be incorporated in modelling AEs by applying organisational and mathematical methods developed in the field of QRA. These methods determine choice of experts, formal elicitation of expert opinions as well as quantification and combination of these opinions [14, 18]. 

. ormal means applied to QRA in combining expert judgement with available hard data are provided by the Bayesian statistical theory [8, 18]. In CBA, elements of this theory are used for updating expert opinions expressed as prior (posterior) distributions of epistemic uncertainty when new information becomes available. These distributions are related to input parameters of such QRA models as event trees and fault trees. Bayesian updating schemes can be used for combining probabilities and p. d. s subjectively assessed by experts [19]. 

A number of methods for constructing prior distributions have been proposed in the Bayesian statistical theory (see e. g. [20, 21] ). Certain of them can be used for constructing prior distributions. ormal methods for eliciting probabilities and p. d. s from experts have been extensively developed in the field of QRA [19, 22, 23]. Many practical situations covered by the above-mentioned methods can be encountered when simulating accidents involving AEs on the road. These methods can be applied to specifying the epistemic uncertainty measures included in the sets 1 Ξ and 2 Ξ. Incorporation of expert judgement in modelling physical phenomena, which can escalate into AE, is to a large degree a problem that can be solved by standard means of QRA. They were developed for a model uncertainty analysis and are closely linked to an expert opinion use [24]. The meaning of model uncertainty and the role of expert judgement in quantifying this uncertainty under the CBA format are discussed in [25]. 

The practical application of the simulation-based procedure proposed in Sec. 4. 2 can be complicated by the fact that the process models

, which embody the state-of-knowledge about the physical phenomena related to the events E 1, E 2, and AE, may be deterministic and imperfect (not error free). This can require supplementing deterministic constituents

with subjective uncertainty distributions which could allow applying these models in the Bayesian framework. A candidate approach to such a probabilistic upgrading of the models

is the method of adjustment factors [26]. The application of this method to the model

will be illustrated in the second part of the present paper [10]. 

Specifying the stochastic models belonging to 1 Ξ and 2

Ξ may be a highly case-specific task. It is determined mainly by available knowledge on E k s (general information on the physical phenomena related to E k s, relevant hard data on E k s, and expert judgements). Therefore, the case study described in the second part of the paper gives methodological (howto-do and where-to-find) descriptions of the procedures which can be applied to assigning and updating elements of 1 Ξ and 2 Ξ which are used to quantify expert judgement [10].
section_name: Outer loop

section_content: This paper proposed a computational procedure intended for selecting mathematical models for actions induced during accidental explosions (AEs) on the road. It was suggested to choose mathematical models for AEs (action models) in the framework of the quantitative risk assessment (QRA). It was shown that methods of QRA allow considering the specific nature of the rare and difficult-to-predict physical phenomena occurring as AEs. The proposed procedure is underpinned by the classical Bayesian approach to QRA. This approach is well suited to quantifying aleatory (stochastic) and epistemic (state-of-knowledge) uncertainties related to actions induced by AEs on the road. 

The procedure implements a stochastic simulation of an accident initiated by a vehicular collision. It relates the aleatory and epistemic uncertainties in characteristics of AE to the uncertainties in the physical phenomena, the sequence of which may end up in AE. Thus the procedure serves the purpose of propagating uncertainties. The action model selected by applying the procedure expresses both aleatory and epistemic uncertainty related to AE. 

The practical application of the proposed procedure requires establishing mathematical models describing the physical phenomena which precede AE. They determine initial information (hard data and expert judgements) which must be collected and transformed via uncertainty propagation into an action model. The simulation of the accident involving AE is not an end in itself. A preparation of this simulation indicates what initial information is to be gained to allow using the available models. This is an important, albeit indirect result of the analysis of an accident involving AE. 

Clearly, results of the simulation will be conditioned on the models being applied. 

The classical Bayesian approach to QRA was considered in this paper as a means of combining hard data and expert judgements within the problem of predicting AE. This approach allows compensating sparce experience data on AE by subjective information. The number of recipes given in QRA and Bayesian statistical theory for a formal use of subjective information is relatively large, albeit not sufficiently systemised for practical use. 

The simulation-based procedure proposed in the paper can be used for design of future structures (assessment of existing structures) which are built in vicinity of roads used for transportation of explosive substances and goods. The second part of the present paper contains a case study which illustrates an application of the proposed procedure [10].
section_name: Conclusions

section_name: Appendix. Abbreviations used in the paper