SAFETY ENGINEERING OF ANTHROPOGENIC OBJECTS COMBINATIONS OF ACTIONS FOR ACCIDENTAL DESIGN SITUATION: A REVIEW, ANALYSIS, AND PROPOSITIONS

The paper discusses two main design strategies when checking for reliability and considers accidental action combinations according the various codes. If accidental actions can be identified, one of the possible design strategies is checking the “key element”. This strategy minimizes the possibility of local failure and subsequent progressive collapse. The combination of actions for accidental design situation for checking of the “key-element” resistance was proposed. In addition, the values of the combination factors for variable loads and partial factors for permanent loads in accordance with required reliability class RC for structural element and values of accidental loads was proposed. The second strategy is checking modified structural systems in accidental design situation from unidentified accidental actions. For this case, a comparison of several probabilistic models was performed, as well as a probabilistic assessment of the accidental action combinations according the various codes.

According to EN 1991EN -1-7 (2006 two groups of strategies are proposed in order to assess accidental design situations: those based on identified accidental actions and those based on limiting the extent of localized failure.
In the first group, it is proposed to design the structure to have sufficient minimum robustness, to prevent or reduce the effect of accidental action or to directly design the structural system to sustain the action.
The second group of strategies is based on limiting the extent of localized failure, either by increasing redundancy of structure or designing "key elements" to sustain notional accidental actions and applying some prescriptive rules like integrity or ductility. Ellingwood et al. (2007) proposed a following formula to assess the probability of progressive collapse:

P(C) = P(C|DH)•P(D|H)•P(H)
where P(C) is the probability of progressive collapse; In (Kokot, and Solomos, 2012) there is a good illustration of this expression (1) together with assigned appropriate terms (see Figure 1).
Considering the above Eq. (1) and Figure 1, the probability of progressive collapse can be minimized in three ways, namely by: controlling abnormal events (term P(H), controlling local element behavior (term P(D|H)) and/or controlling global system behavior (conventional probability P(C|DH)).
It is worth nothing in (Kokot, and Solomos, 2012), that controlling abnormal events by structural engineers is normally very difficult, practically impossible. However, engineer can influence the local and global system behavior, i.e. probabilities P(D|H) and P(C|DH).
Conditional probabilities presented in Eq. (1) can be obtained by a probabilistic risk analysis (PRA), in which it is possible to model uncertainties, study their propagation and the effect on the required performance of the structural systems (with damaged elements). This approach is called structural reliability analysis and failure (collapse in the case in question) is considered achieved when demand E (i.e. the effects generated by the actions) exceeds collapse resistance R. In general case, the probability of failure is equal to: where F R (x) is the CDF (cumulative distribution function) of resistance R and f E (x) is the PDF (probability density function) of E (effect of actions).
The probability of disproportionate collapse can be defined according to EN 1990EN (2009 as follows: where β is the reliability index for structural system and Φ(•) is a normal standard distribution function.
For a correct assessment of disproportionate collapse risk, it may be necessary to consider the presence of multiple hazard events and the initial stage of damage. In this case, Eq. (1) can be generalized as illustrated in the following equation (valid for independent event only): -176 - where λ H can substitute P(H) if occurrence probability is less that 10 -2 /year. Values of λ H are reported in (Ellingwood et al., 2007).
As was shown in (Ellingwood, 2005), if a performance based design approach is adopted, an acceptable value of risk tolerance has to be defined. In the case of a disproportioned collapse, which main consequence is the loss of human life, decision-makers can assume that the performance objective of safeguarding human life is achieved if the following relationship is verified: where p tag,h is the risk threshold defined as "de minimis" which in general case assumed values ranging from 10 -5 /year and 10 -7 /year. More detailed discussion presents in our publication (Tur et al., 2019).
Moreover, in particular case in which so called alternative load path method (ALP-method (Arup, 2011;Ellingwood et al., 2007)) is used in design phase, the collapse probability becomes P(C|DH), which in turn has to respect the following equation according to (Ellingwood et al., 2007): Therefore, assuming λ H equal to 10 -6 /year..10 -5 /year, the performance based target probability established by condition (6) requires that the conditional probability of collapse for the modified structural system be in the order of 10 -2 /year..10 -1 /year.
Consequently, as shown in (Ellingwood, 2005), the reference reliability index β 0 for the limit collapse state of conditioned by the occurrence of the damage will be in order of 1.5.
That is significantly lower than that assumed for ultimate limit state of new buildings for -177 -residential and office use in case of ordinary actions (i.e. β tag = 3.8, which corresponds to reference probability for structural system collapse of the order of ~10 -4 ).

Combinations of the actions according to various codes
According to (Gulvanessian, 2020;Arup, 2011), a reliability based approach can be applied to determine reasonable loading combinations for accidental design situation. The actions to be combined reflect the small probability of a joint occurrence of the accidental action and design values of imposed (or live), snow, wind loads.
Hazard events, and mainly, malicious attack are a rare events and many of them suppressed early.
Focusing on the mechanical actions, these are traditionally subdivided into permanent actions and imposed (variable) action according to (CIB, 1989). Their variability with time is an aspect of particular relevance for checking of the structural system in accidental design situation. As was shown in (Gulvanessian, 2020), in partial factor design method (PFM) for normal conditions, the load variability is considered by a characteristic or design load with a low probability of being exceeded during the service life of the structure. This ensures that the building structure are designed both safety and economically, as in setting the design requirements a balance has been sought between the cost of premature failures and the cost of additional safety investment (see ISO 2394 (2015)).  -178 - The target reliability indices derived on the basis of economic optimization might not be acceptable with regard to requirements concerning human safety, as it is stated in ISO2394 (2015). These reliability indices are denoted as β tag,h .
It is clear, the day-to-day probability of occurrence of such high (design) load value is low, just as for the day-to-day probability of occurrence of a hazard event (accidental event).
Simultaneously taking into account both events would result in very onerous design requirement for accidental design situation (in case of modified structural system robustness checking

Load combinations for accidental design situation (key-elements checking)
As shown above (Gulvanessian, 2020), in general case, hazard events can be classified in two major types: unintentional but identified (Natural and Accidental) hazards and malicious attacks. According to (Gulvanessian, 2020), the distinct, nature of two types of hazard implies that the hazard associated uncertainties, severity and frequency of occurrences are significantly different. For unintentional hazards such as earthquake, wind, scour, vessel collision, random stochastic models are typically used to represent the hazard intensity and occurrence. However, for purposely plotted malicious destruction such as explosions and -179 -intentional collisions and purposely made accidents (criminal and terrorist attacks), the ordinary random stochastic model is not longer valid.
According to EN 1990 (2009) and EN 1991-1-7 (2006) the general format of effects of actions for the accidental design situations is analogous to the general format for STR/GEO ultimate limit states. Here, the loading action is the accidental action, and the most general expression of the design value of the effects of actions is the following: which can also be expressed as: The combination for accidental design situation either involve an explicit design value of accidental action A d (e.g. impact) or refer to a situation after an accidental event (A d = 0).
The partial factors for actions for ultimate limit states in the accidental design situations are normally taken equal to 1.0, in general, not only are the reliability elements for actions modified for the partial factors for resistances.
-180 -Based on results of our own investigations (Tur, and Markovskij, 2009), we proposed to use for checking of the "key-element" resistance the following combination of actions for accidental design situation (combination comprises accidental action A d ≠ 0): where G k,j is the characteristic value of a permanent action "j"; Q k,1 is the characteristic value of the leading variable action; A d is the design value of the accidental action; γ GA,j is the combination factor applied to a permanent action "j"; ψ A,1 is the combination factor applied to the leading variable action according to Table 2.
In Table 2, we relate values of the combination factors ψ A with required reliability class RC for structural element and factor k, which is determined as ratio: where Q k,i is the characteristic value of the accompanying variable actions.  Table 1). From our point of view, this approach is not entirely correct.

Snow (S), ψ A,S
Firstly, when calibrating partial (combination) factors in combination (8)  load. Prior analysis of the expressions from Table 1 shown that values of partial coefficients applied with the characteristic loads are sufficiently different. Therefore, the total value of accidental load corresponds to various quantiles of CDF (for the total load combination K E (Q+G)) and, consequently provides a different reliability level for the same characteristic values of actions.
In an accidental design situation, when structural engineer considers malicious terrorist and criminal attacks (unidentified hazards), accounting of the climatic (show and wind) actions jointly with the imposed load in accidental load combinations, makes no sense.
The first, at the stage of the attack planning, it is very difficult and practically impossible, to foresee real point-in-time when the maximum value of the climatic actions will appear simultaneously with the design value of the imposed load.
If the wind action can have a significant influence on the high-rise building structural behavior in an accidental design situation, snow load influence is insignificant with the RCbuildings mainly (the maximum part of the snow load in total gravity load in near 15% only).
Therefore, the total combination that includes an imposed load for the assessment of the modified structural system robustness in an accidental design situation is decisive. In the general case, we should consider two types of the imposed load when accidental load combination is developed: 1) only sustained imposed load (as more realistic load value for day-to-day exploitation); 2) total value of the imposed load (sustained plus intermittent parts) for extraordinary event.
It should be noted that when the probabilistic modelling applying accidental action shall be considered as an impulse at-any-time-point. Such impulse has a very high intensity and a short period of action in comparison with permanent and sustained imposed (variable) loads.
As the occurrence of the intermittent (transient) imposed load is by its conceptualization rare, it generally does not need to be taken into account simultaneously with accident (hazard)

Imposed load modelling
With reference to the discussion in (Van Coile et   The intermittent load has normally one particular occurrence producing its maximum magnitude (see Figure 3d). The maximum total load for the combined process (see Figure 3e) might occur when the sustained load is at its maximum, when the intermittent load is at its maximum or when neither of them is at its respective maximum.

The maximum of the sustained load
If the maximum of the sustained load is of interest, it is normally sufficient to consider the marginal statistical distribution F s (x) (the index "s" means sustained), of the sequence of n independent loads (JCSS PMC, 2001; CIB, 1989). The probability distribution function for the maximum load is given by: where the number N load events can be deterministic or determined by some statistical distribution.
In the case of a deterministic value of N, N = n, the widely used Ferry Borges load model is obtained.
If the time between load changes is exponentially distributed then the number of load changes is Poisson distributed. With this assumption Eq. (12) yields where T is an appropriate reference time, for example, the anticipated life time of the building; v is the occurence rate of sustained load changes. A common procedure is to evaluate Eq. (14) at two different cumulative values in the upper tail and match a Type I extreme value distribution of these values.

The maximum of the intermittent load
In general case the maximum load which occur in a building is a combination of sustained loads and intermittent loads. As stated in (CIB, 1989), in most cases it is reasonable to assume that the sustained load and the intermittent load are mutually independent. However, a dependence may exist in special cases.
The maximum of the intermittent load during one occupancy, the duration of which is assumed to be Erlang distributed (Gamma distributed with an integer value of the shape parameter), is given by (CIB, 1989): where F p (x) is the probability distribution function of the intermittent load; ρ is the occurrence rate of intermittent loads; k is the shape factor in the Erlang distribution (k = 1, 2,…).
For example, k = 1 in the special case when the time intervals between the loads are exponentially distributed.
The maximum total load during one occupancy is obtained from the convolution integral: where f s (z) is the probability density function for the sustained load during one occupancy.

The total maximum imposed load
The total maximum imposed load during the entire reference period can be obtained, see (CIB, 1989), by considering the simultaneous distribution of completed durations.
Unfortunately, this does not lead to a closed expression. The total maximum load during the reference period T can then be expressed as: where T and v are the same as for Eq. (14).

Design total accidental action modelling
Different permanent load and imposed load models have been proposed for the structural systems checking in an accidental design situation.
-188 -It should be pointed, that these studies make limited explicit reference to the issue of time variability of the load. Most stating (directly or indirectly) that their load models correspond with arbitrary-point-in-time (APIT) permanent and imposed loads, e.g. (Van Coile et al., 2019). The study by (Ellingwood, and Culver, 1977;Ellingwood, 2005) is a notable exception, going to some depth in explaining underlying process of loading variability.
The overview of applied permanent load models is given in Table 3 (where μ is the mean value, V is the coefficient of variation (CoV), and G nom is the nominal permanent load).
-189 - The widely cited paper by (Ellingwood, and Culver, 1977;Ellingwood, 2005) does not specify a clear formulation for the total load model.
In (Van Coile et  where K E is the model uncertainty for the load effect: To assess the effect of the different load models, the total load formulations are compared. To make a direct comparison possible, the load ratio χ and total characteristic (nominal) load P k are defined through Eq. (19), where the characteristic (nominal) values Q k and G k when using the Eurocode methodology: and The variation in G and Q is thus taken into account through the stochastic variables g and q with μ and V as listed in Tables 3, 4. Let's consider the most commonly used models of the basic variables for accidental load combination and determine the reliability levels that provide accidental load combinations adopted in various codes (see Tables 5-7

Load model according to Eurocode (by Holicky and Schleich (2005))
The second family of APIT imposed load models in Table 6 (2005)).

Load model according to JCSS PMC (2001)
The The JCSS PMC (2001) further notes that one of the underlying assumptions for the equivalent uniformly distributed load model is a linear structural response.
The assumption of linearity can be omitted by considering the spatial variability of the load explicity. The latter is however considered too demanding for practical feasibility. Nonlinear behavior could be considered as part of the model uncertainty K E .

Probability analysis of combinations for checking modified (damaged) structural systems in accidental design situation
All models (see Tables 5-8) have been evaluated using 10 8 crude Monte Carlo Simulations (MCS), for load ratio χ, applying the distribution model according to Tables 5-8.
Example of obtained cumulative density functions (CDF) for the total characteristic load factor ξ according to the reviewed in Table 1 load models with different CoV values are given in Figure 4.   Table 9 shows the exceedance probabilities corresponding to the total accidental load combinations included in various codes (see Table 1), obtained from the CDF for the various reviewed load models and various category of occupancy.
As can be seen from Figure 4 values of the total accidental load calculated based on the various load combinations according to different codes (see Table 9) and corresponding exceedance probability of CDF of this total action (see Eq. (18)), varies in very wide interval (from 7% to 52%).
Moreover, Table 9 shows that for the various category of occupancy accidental load combination according one code gives various values of quantilies of CDF and reliability level.
-194 - Eurocode by (Holicky, and Schleich, 2005) 0  Table 9. The exceedance probabilities corresponding to the total accidental load combinations included in various codes.

Conclusion
Proposed combination of actions for accidental design situation Eq. (9) gives resistance of the "key-element" in accordance with required reliability class. It was found that the application of the ψ A coefficients to the accompanying (non-dominant) loads does not lead to a change in the reliability indices. Because of the significant accidental action A, the influence of all other loads decreases. Only the leading variable action have noticeable influence on the reliability indices. In this regard in Eq. (9), only dominant variable loads are taken into account in combination of actions for accidental design situation for the "key" system elements. For non-dominant variable action ψ A,i,2 = 0.
The calibrated values of the combination factors ψ A,i,1 for the key elements (see Table 2) depend on the required reliability class of structural elements and the factor k, which is