PSI - Issue 20

D. Reznikov / Procedia Structural Integrity 20 (2019) 17–23 D. Reznikov / Structural Integrity Procedia 00 (2019) 000–000

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6

Table 2. Values of societal criteria factor χ . Type of TS Places of public assembly, dams Domestic, offshore, trade and industry



0,005

0,05

Bridges

0,5

Towers, masts, offshore structures

5

Complex systems may fail due to multiple failure mechanisms i =1, 2, …, n . Bearing this in mind a vector of failure probabilities { P f ( t )}={ P f1 ( t ), P f1 ( t ) …. P fn ( t )} may be introduced. The components of this vector can be defined as follows:   1 1 1 ( ) ( ) ( ) f P t P c t d t   ,   2 2 2 ( ) ( ) ( ) f P t P c t d t   , (8) . . .   ( ) ( ) ( ) f n n n P t P c t d t   . And the condition for insuring structural integrity in a probabilistic formulation may be written as follows:   [0; ], 1, 2, , ( ) ( ) ( ) [ ] d f i i i f t T i k P t P c t d t P         . (9) Using intuitive considerations one can assume that at least in some cases a decreasing functional dependence P f = P f ( n ) between failure probability and safety factor exists (Fig. 2). If such function does exist we can determine the condition of equivalence between deterministic and probabilistic approaches to securing structural integrity, Ching (2009). Then using the deterministic approach we can define the acceptable failure probability [ P f ] n that is equivalent to the specified normative safety factor [ n ] and vice versa in case of probabilistic approach determine the minimum allowable safety factor [ n ] P that is equivalent to the normative failure probability [ P f ]. Unfortunately such dependence exists only in some specific cases. Let’s consider the relationship between safety factor n and failure probability P f for a structural component that is subjected to static loading (demand) L and has load carrying capacity R . If both L and R are considered as random variables that are distributed according to normal laws F L ( x ) = N ( E { L }, S { L }) and F R ( x ) = N ( E { R }, S { R }) (where E { L }, S { L }, E { R }, and S { R } are mathematical expectations and mean square deviations of the load L and capacity R ) then the following relationship holds, Elishakoff (2004), Makhutov and Reznikov (2011), Makhutov and Reznikov (2014): 4. The relationship between deterministic and probabilistic criteria of structural integrity

  

2 2 2 1 n

    ,  

P

   

(10)

f

L R v v n

where Φ (·) is the Laplace function, n 1 = E { R }/ E { L } is the central safety factor under a single static loading; ν L = S { L }/ E { L } and ν R = S { R }/ E { R } are coefficients of variation of the load and loading capacity. It means that the functional relationship between the failure probability and the safety factor exists when coefficients of variation of the demand and capacity parameters are constant values.