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Conventional Stochastic Sequences in Reliability Assessments and Predictions of Structural Members
Available online at www.sciencedirect.com
Procedia Engineering 57 (2013) 642 – 650
11th International Conference on Modern Building Materials, Structures and Techniques, MBMST 2013
Conventional Stochastic Sequences in Reliability Assessments and Predictions of Structural Members Antanas Kudzysa, Ona Lukoševičienėb ∗ b
Department of Strength of Materials, Vilnius Gediminas Technical University, Saulėtekio st. 11, LT-10223 Vilnius, Lithuania
Abstract A simplification of probability based approaches on safety assessments and predictions of structural members is discussed. The modelling of time-dependent survival probabilities and reliability indexes of particular and single members is defined in a simple and easy perceptible manner using uniformed correlation factors of their conventional sequences. These factors of safety margin processes are analyzed and demonstrated by numerical example. © 2013 Published by Elsevier Ltd. Ltd. 2013The TheAuthors. Authors. Published by Elsevier Selection and under responsibility of theof Vilnius Gediminas TechnicalTechnical UniversityUniversity. Selection andpeer-review peer-review under responsibility the Vilnius Gediminas Keywords: reliability, safety margin, correlation coefficient, survival probability, structural member, reliability index.
1. Introduction The standards EN 1990 [1] and ASCE/SEI [2] are based on the limit state concept and, respectively, on the methods of the partial factor design and the strength or allowable stress design. However, the structural design practice shows that it is impossible to verify the safety and economy parameters of load carrying structures using deterministic methods and with their universal factors for loads and material properties. The reliability indexes of members and structural systems may be objectively defined only by fully probability-based concepts and models. According to these approaches, only probability-based concepts allow us explicitly predict basic and additional uncertainties of structural members in their analysis on the durability and long-term reliability. The probabilistic analysis of members is indispensible in order to predict their time-dependent destructions or failures. In the beginning of probabilistic structural analysis of equally correlated systems, the simplified methods of average and equivalent correlation coefficients were used [3]. The new mathematical probabilistic formats used in a long-term reliability prediction of structures are based on rather complicated considerations [4–8]. Thus, the engineering modeling of survival probabilities and reliability indexes as the counting responses of structures are still unsolved. The main task of this paper is to present for mechanical, structural, resident and management engineer’s new unsophisticated and uniform methodological formats on probability based safety assessments and predictions for nondeteriorating and deteriorating particular structural members as components of series auto systems. The second aim of this paper is to demonstrate the difference between time-variant reliability response curves of non-deteriorating and deteriorating members as auto systems.
* Corresponding author. E-mail address: [email protected]; [email protected]
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Vilnius Gediminas Technical University doi:10.1016/j.proeng.2013.04.081
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2. Safety margins of members Multicriteria failure modes and safety of structural members (beams, slabs, columns, joints) may be objectively assessed and predicted only knowing survival probabilities of their particular members (normal or oblique sections, connections). The structural reliability analysis is related with some methodological differences between non-deteriorating and deteriorating members. Therefore, it is expedient to divide the life cycle tn (Fig. 1) of deteriorating structures into the initiation, tin, and propagation, tpr, phases [9].
Fig. 1. Time-dependent model for structural reliability assessment and prediction of non-deteriorating (1) and deteriorating (2) members
A length of initiation phase is a random variable depending on a feature of degradation process, an environment aggressiveness and a quality of protective covers. The unvulnerability of structures may be characterized by the duration of this phase. When the degradation process of the members is caused by intrinsic properties of materials, the phase tin≈0. The time-dependent safety margin of particular members may be defined as their random performance process and presented as follows
Z ( t ) = g {X ( t ) , θ} = θ R R ( t ) − θ g E g − θ q Eqs ( t ) − θ q Eqe ( t ) − θ cl Ecl ( t ) ,
(1)
where X(t ) and θ are the vectors of basic and additional variables representing respectively random components (resistances and action effects) and their model uncertainties; R(t ) = ϕ (t )R0 is the resistance of particular members, where ϕ (t ) denotes their degradation function depending on an artificial ageing and degradation of materials; E g , Eq s (t ) , Eqe (t ) and Ecl (t ) are the action effects caused by permanent, sustained, extraordinary and climate (snow and wind) loads. The mean values and standard deviations of additional variables of the safety margin are: θ Rm = 0.99 − 1.04 , σθ R = 0.05 − 0.10 and θ gm ≈ θ qm ≈ θ cl , m ≈ 1.00 , σθ g ≈ σθ q ≈ σθ cl ≈ 0.10 [9–11]. Gaussian and lognormal distribution laws are to be used for resistances of reinforced concrete and steel members, respectively [10]. The permanent actions can be described by a normal distribution law [12]. Lognormal and gamma or exponential distribution laws are presented respectively for sustainable and extraordinary floor live loads [9], [10]. The duration, d, and the renewal rates, λ, of extreme floor live and climate actions are: d g = tn , λg = 1 / tn ; d qs = 5 − 10 years , λqs = 0.1 − 0.2 / years ; d qe = = 1− 14 days for merchant and 1–3 days for other buildings, λqe = 1 / years ; d s = 14 − 28 days , λs = = 1 / years ; d w = 8 − 12 hours , λw = 1 / years [9]. The recurrence number of two joint extreme actions during the design working life of structures, tn in years, may be presented by the equation
n12 = tn ( d1 + d2 ) λ1λ2 .
(2)
Thus, the recurrence numbers of extreme concurrent live loads during t n = 50 years period are: ngq = 6 − 11 , nqs qe = nqs s = 50 , nqe s = 2 − 6 nqe w = 0.2 − 2 .
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The dangerous cuts of process by Eq. (1) correspond to extreme loading situations of structures. Therefore, in design practice this safety margin process may be modeled as a random geometric distribution and treated as a finite decreasing random sequence, when the safety margin of member is
Z k = θ R Rk − θ g Eg − θ q Eqs k − θ q Eqe k − θcl Ecl ,k ,
(3)
where k = 1, 2..., n − 1, n is the number of cuts (Fig. 1). 3. Instantaneous survival probabilities of members Multi-cut sequences of random safety margins of particular members can be idealized as their series auto systems. The instantaneous probability of failure of these members in mode k is P ( Z k ≤ 0 ) . Structural members as series auto systems are safe, if each their particular member as individual component is safe. The degree of this safety expressed by probabilistic responses depends on the correlation intensity between safety margin cuts. It is safer but not relational for structural members to ignore the correlation between these cuts. The instantaneous survival probability of particular and single members with respect to their failure mode at the safety margin sequence cut k is assuming, that they were safe at sequence cuts 1, ..., k-1. Therefore, their instantaneous survival probability at cut k may be expressed as ⎛ k −1 ⎞ P ( Z k > 0 ) = P ( Sk ) = 1 − P ⎜ Fk ∩ Si ⎟ . ⎜ ⎟ ⎝ i =1 ⎠
(4)
For design convenience, the structural safety analysis of members may be based on the limit state criterion Rk − Ek > 0 , where Ek is the resultant action effect, and can be modeled using multidimentional integral as P ( Z k > 0 ) = P { Rk − Ek > 0 ∃ k ∈ [1, n ] } =
∫
g ( X k θ )> 0
fX
k θ
( X k θ ) dx ,
(5)
As it is shown, the survival probability of an single member in series systems or any particular member in series auto systems is represented by the limit state domain g Xk θ > 0 . The resistance Rk and resultant action effect Ek may be treated as statistically independent variables of random safety margins sequences. Therefore, the instantaneous survival probability of particular and structural members can be expressed by convolution multidimentional integral as
{
}
P ( Sk ) = P { Rk > Ek ∃ k ∈ [1, n ] } =
∞
∫ f Rk ( x ) FEk ( x ) dx ,
(6)
0
where f Rk (x ) is the density function of their resistance and FEk (x) is the cumulative distribution function of their action effect (Fig. 2). The value of probability by Eq. (6) may be modeled using analytical numerical integration and Monte Carlo methods or their programs.
Fig. 2. Schematic representation of an instantaneous survival probability analysis
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4. Indexed correlation factor for random sequences The correlation between resistances of particular members always has a great effect on the counting probabilistic response [13]. The time-variant safety margin of members must be treated as a non-stationary process. Therefore, it is rather complicated to define the survival probability of multicuts sequences in easy perceptible manner. However, this impediment may be removed in design practice using the indexed correlation factors of conventional random sequences. The correlation factor of a finite sequence consisted of k stochastic cuts can be formed from random k-th row, ρ ki ∃ i = 1, ..., k − 1 , or random k-th column, ρik ∃ i = 1, ..., k − 1 , of a quadratic correlation matrix by Eq. (7) of basic correlation coefficients 1
ρ12 1
ρ21 ρkl =
ρ13
...
ρ23
...
ρ2,k −1
ρ2 k
ρ1,k −1
ρ1k
ρ31
ρ32
1
...
ρ3,k −1
ρ3k
...
...
...
1
...
...
ρk −1,1 ρk1
ρk −1,2 ρk 2
ρk −1,3 ρk 3
... ...
1 ρk ,k −1
ρk −1,k 1
.
(7)
The value of correlation factors as correlation vectors ρk = ρki = ρik may be calculated from the Eqs. (8) or (9)
(
)
(8)
(
)
(9)
ρki ≈ ρk1 + ρk 2 + ρk 3 + ... + ρk ,k −1 / ( k − 1) , ρik ≈ ρ1k + ρ2k + ρ3k + ... + ρk −1,k / ( k − 1) , The basic correlation coefficients of rank safety margin cuts are calculated by the equation ρki = ρik =
Cov ( Z k , Zi ) σZ k × σZi
,
(10)
where Cov ( Z k , Zi ) and σ Z k , σ Zi are the auto covariance and standard deviations of stochastic sequence cuts. x According to our investigation data, the bounded index, xk , of indexed correlation factors, ρk k , of random sequences may be presented as
⎡ 4.5 + 4 ρkl ⎤ xk = P ( Sk ) ⎢ ⎥ ⎣1 − 0.98ρkl ⎦
υk
,
(11)
where k
υk =
∑ P ( Si )
i =2
( k − 1)
k +2
,
(12)
characterizes the effect of correlation vector size on a bounded index value. For highly reliable members, the instantaneous survival probability P(Si ) ≈ 1 and the Eq. (12) may be written as follows υk = ( k − 1) ⎡⎣( k − 1) k + 2 ⎤⎦ = 1
k +2,
(13)
For conventional sequences with equicorrelated cuts, the correlation factor ρk = ρij i.e. its value is equal to a basic correlation coefficient.
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
5. Time-dependent survival probabilities of members The methods on time-dependent structural reliability analysis were proposed by Mori Y. and his colleagues Ellingwood B.R. [14] and Nanoka M. [15], but these and other approaches are not included in design codes and standards. The total failure probability of ductile members is the union of all possible failure modes of auto systems and may be expressed as `
⎛ n ⎞ ⎛ k −1 ⎞ ⎛ n −1 ⎞ P ⎜ ∪ ( Zi ≤ 0 ) ⎟ = P ( F1 ) ∪ ...∪ P ⎜ Fk ∩ Si ⎟∪ ...∪ P ⎜ Fn ∩ Si ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠
(14)
⎛ k −1 ⎞ The transition probability P ⎜ Fk ∩ S i ⎟ denotes a statistical dependence between a failure probability of members under ⎝ i =1 ⎠ k-th loading situation and their survival probabilities under previous loading. The total survival probability of non-deteriorating members as series auto systems consisted of n random cuts of their safety margin sequences P ( S1 ) ≤ P ( S2 ) ≤ ... ≤ P ( Sn ) can be expressed as
⎛ n ⎞ ⎛ n ⎞ ⎛ n ⎞ P ⎜ ∩ Si ⎟ = P ⎜ ∩ Si ≤ S j ∃ i ∈ [1, n ] ⎟ = 1 − P ⎜ ∪ Fi > F j ∃ i ∈ [1, n ] ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠1 ⎝ i =1 ⎠ ⎝ i =1 ⎠
(
)
(
( ) × ...× P ( S
= P ( S1 ) × P S
21
k k −1,...,1
)
) × ...× P ( S
n n −1,...,1
⎛ k −1 ⎞ ⎛ n −1 ⎞ n × ... × P ⎜ Sk ∩ Si ⎟ × ... × P ⎜ Sn ∩ Si ⎟ = ∏ P ( Sk ) ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ k =1
) = P(S )× P(S ∩ S )×
⎡ x ⎢1 + ρ122 ⎣⎢
1
2
1
⎛ 1 ⎞⎤ − 1⎟⎟ ⎥ × ... ⎜⎜ P S ( ) 2 ⎝ ⎠ ⎦⎥
(15)
⎡ ⎡ ⎛ 1 ⎞⎤ ⎛ 1 ⎞⎤ xk xn × ⎢1 + ρ1,2,... − 1⎟ ⎥ × ... × ⎢1 + ρ1,2,... − 1⎟ ⎥ ⎜ k −1 k ⎜⎜ P ( S ) n n − 1 ⎟ ⎜ P(S ) ⎟ k n ⎝ ⎠ ⎦⎥ ⎝ ⎠ ⎦⎥ ⎣⎢ ⎣⎢
where the transition probability of safety margin sequence cuts k and i=1, ..., k-1 is ⎛ k −1 ⎞ P ⎜ Sk ∩ Si ⎟ = P ( Sk ) ⎜ ⎟ ⎝ i =1 ⎠
⎧⎪ x ⎨1 + ρk k ⎩⎪
⎡ 1 ⎤ ⎫⎪ − 1⎥ ⎬ , ⎢ ⎢⎣ P ( S k ) ⎥⎦ ⎭⎪
(16)
If correlation factors ρ kx k = ρ kx−k −11 ... = ρ 2x 2 = 1 transition and survival probabilities by Eqs. (16) and (15) are ⎛ k −1 ⎞ ⎛ n ⎞ P⎜⎜ S k ∩ Si ⎟⎟ = 1 and P ⎜ ∩ S i ⎟ = P (S1 ) . ⎝ i =1 ⎠ ⎝ i =1 ⎠ When the cuts of na safety margin sequence are statistically independent, i.e. ρ kxk = ρ kx−k −11 ... = ρ 2x 2 = 0 , the probability by Eq. (15) is equal to ∏ P(Si ) . i =1 For random sequences or series systems with equireliable and equicorrelated cuts or elements, the correlation factor index by Eq. (11) and transition probability by Eq. (16) may be expressed as
⎛ 4.5 + 4 ρij xk = ⎜ ⎜ 1 − 0.98 ρij ⎝
1
⎞ ⎟ ⎟ ⎠
k +2
,
⎧ ⎛ k −1 ⎞ ⎤⎫ ⎪ ⎪ x ⎡ 1 x P ⎜ Sk ∩ Si ⎟ = P ( Sk ) ⎨1 + ρk k ⎢ − 1⎥ ⎬ = P ( Sk ) + ρk k ⎡⎣1 − P ( Sk ) ⎤⎦ , ⎜ ⎟ P S ( ) ⎢ ⎥ k ⎪ ⎪ ⎣ ⎦ ⎝ i =1 ⎠ ⎩ ⎭
(17)
(18)
The survival probability of deteriorating members as series auto systems expressed by decreasing sequences, P(S1 ) > P(S 2 ) > ... > P(S n ) , may be calculated from the formulas
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
⎛ n ⎞ ⎛ n P ⎜ ∩ Si ⎟ = P ⎜ ∩ Si > S j ⎜ ⎟ ⎜ ⎝ i =1 ⎠2 ⎝ i =1
(
( )
⎞
⎛
n
⎞
⎠
⎝ i =1
⎠
) ⎟⎟ = 1 − P ⎜⎜ ∪ ( Fi < F j ) ⎟⎟ ,
⎛ n ⎞ P ⎜ ∩ Si ⎟ = P ( S1 ) × P S × ... × P S1,...,k −1 k × ... × P S1,...,n−1 n = P ( S1 ) × P ( S1 ∩ S 2 ) × ... 12 ⎜ ⎟ ⎝ i =1 ⎠ ⎛ k −1 ⎞ ⎛ n−1 ⎞ n ⎡ ⎞⎤ x ⎛ 1 × ... × P ⎜ ∩ Si Sk ⎟ × ... × P ⎜ ∩ Si Sn ⎟ = ∏ P ( Sk ) ⎢1 + ρ122 ⎜ − 1 × ... ⎜ P ( S ) ⎟⎟ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢⎣ 1 ⎝ ⎠ ⎥⎦ ⎝ i =1 ⎠ ⎝ i =1 ⎠ k =1 ⎡ ⎡ ⎛ ⎞⎤ ⎛ ⎞⎤ 1 1 xn xk − 1⎟ ⎥ , × ⎢1 + ρ1,2,... − 1⎟ ⎥ × ... × ⎢1 + ρ1,2,... ⎜⎜ n n − 1 k −1 k ⎜⎜ P ( S ⎟ ⎟ ⎢⎣ ⎢⎣ k −1 ) ⎝ P ( Sn−1 ) ⎠ ⎥⎦ ⎝ ⎠ ⎥⎦
(
)
(
(19)
)
(20)
where the transition probability of sequence cuts is
⎧ ⎛ k −1 ⎞ ⎪ x P ⎜ ∩ Si Sk ⎟ = P ( Sk ) ⎨1 + ρk k ⎜ ⎟ ⎪ ⎝ i =1 ⎠ ⎩
⎡ ⎤⎫ 1 ⎪ − 1⎥ ⎬ . ⎢ P S ⎪ ⎣⎢ ( k −1 ) ⎦⎥ ⎭
(21)
If the correlation factors of sequences are fully statistically dependent, i.e. ρ kxk = 1 , or independent, i.e. ρ kx k = 0 , the values of transition and survival probabilities, respectively, are: n ⎛n ⎞ ⎛n ⎞ ⎛ k −1 ⎞ ⎛ k −1 ⎞ P ⎜ ∩ S i S k ⎟ = P (S k ) / P (S k −1 ) and P⎜ ∩ Si ⎟ = P(S n ) or P ⎜ ∩ S i S k ⎟ = P (S k ) and P⎜ ∩ Si ⎟ = ∏ P(Si ) . ⎝ i =1 ⎠ 2 ⎝ i =1 ⎠ 2 i =1 ⎝ i =1 ⎠ ⎝ i =1 ⎠
The survival probability of structural members may be also introduces by the reliability indexes as
⎡ ⎛ n ⎞ ⎤ β1 = Φ −1 ⎢ P ⎜ ∩ Si ⎟ ⎥ , ⎢ ⎜⎝ i =1 ⎟⎠ ⎥ ⎣ 1⎦
(22)
⎡ ⎛ n ⎞ ⎤ β2 = Φ −1 ⎢ P ⎜ ∩ Si ⎟ ⎥ . ⎢ ⎜⎝ i =1 ⎟⎠ ⎥ ⎣ 2⎦
(23)
Fig. 3. Reliability indexes versus correlation factors when system consists of one, two, five and ten elements
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
The results calculated by Eqs. (15) and (22) corroborate to data presented in the papers [9, 13, 3] for equicorrelated series systems consisting of two, five and ten elements (Fig. 3). It is shown that the values of time-dependent reliability indexes based on conventional stochastic sequence approaches and expressed by Eq. (22) (curves 3, 5, 7) are close to the numerical integration data (curves 2, 4, 6). 6. Numerical example Consider the long-term survival probability and reliability index of non-deteriorating and deteriorating roof steel beams of a scrap metal shed (Fig. 4). The destroying bending moments of beams Eg, Eq and Es are caused by a permanent action G of steel roof structures and hanging crane crabs, variable loads Q of scrap metals and S of snow depth.
Fig. 4. Traversing underslung crane
The means and variances of the beam resisting and destroying parameters are:
(θ R R0 )m = 297.1 kNm, σ 2 (θ R R0 ) = 564.9
(kNm)2;
(θ E Eg )m = 18.28 kNm, σ 2 (θ E E g ) = 3.342 (kNm) ; (θ E Eq )m = 52.19 kNm, σ 2 (θ E Eq ) = 109.0 (kNm) ; 2
2
(θ E E s )m = 54.67 kNm, σ 2 (θ E Es ) = 269.0 (kNm)2.
Fig. 5. The time-dependent instantaneous and long-term reliability indexes of non-deteriorating (1, 2) and deteriorating (3, 4) members
Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
Fig. 6. The time-dependent instantaneous and long-term survival probability of non-deteriorating (1, 2) and deteriorating (3, 4) members
These parameters are described by normal (R and Eg), lognormal (Eq) and Gumbel (Es) probability distributions laws. The degradation function of deteriorating beams is ϕ (t ) = 1 − 0.00375× t . The instantaneous and long-term survival probabilities of beam are calculated by Eqs. (6) and (15), (16) and the reliability index is defined by Eqs. (22) and (23). The time-dependent instantaneous (1, 3) and long-term (2, 4) reliability indexes and survival probabilities based on conventional stochastic sequences method are presented in Figs. 5 and 6, respectively. This numerical example satisfied that the suggested method of conventional stochastic sequences is unsophisticated and may be included in structural design practice. 7. Conclusions
It is expedient to use in structural design practice the simplified approaches based on the conventional stochastic sequence and correlation factor concepts. These concepts help us to formulate the probabilistic method uniformed for an analysis of non-deteriorating and deteriorating ductile members. New approaches in applicable stochastic sequence theory help us introduce unsophisticated probability based methodological and mathematical formats for safety assessments, verifications and predictions of structural members. For practical sake of a probabilistic design, the safety margin process or time-dependent performance criterion of structural ductile members may be treated as a random sequence. Survival probabilities and reliability indexes of time-dependent particular and single structural members, respectively as normal or oblique sections and beams, columns, slabs or joints, may be defined in a simple and easy perceptible manner using uniformed correlation factors of their conventional sequences. The position of stochastically dependent cuts of conventional sequences is matched with extreme loading situations of members as components of series auto systems. The format for representation of indexed correlation factor is presented in this paper. A degradation function of member resistance may have a considerable effect not only on failure probabilities and reliability indexes of members, but also on shapes of their curves. References [1] EN 1990:2002. Eurocode - Basic of structural design. CEN, Brussels, 2002. [2] ASCE/SEI 7-05. 2005. Minimum design loads for buildings and other structures, 388. [3] Thoft-Christensen, P., Dalsgard Sorensen, J., 1982. Reliability of structural systems with correlated elements, Applied Mathematical Modelling 3(6), pp. 171-178. [4] Rackwitz, R., 2006. Risk acceptance and optimization of aging but maintained civil engineering infrastructures, Safety and Reliability for Managing Risk, pp. 1527-1534. [5] Joanni, A. E., Rackwitz, R., 2006. Stochastic dependencies in inspection, repair and failure models, Safety and Reliability for Managing Risk, pp. 531537. [6] Noortwijk, J. M., Kallen, M. J., Pandey, M. D., 2005. Gamma processes for time-dependent reliability of structures, Advances in Safety and Reliability, pp. 1457-1464.
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650 [7] Kuniewski, S. P., van Noortwijk, J. M., 2007. Sampling inspection for the evaluation of time-dependent reliability of deteriorating structures, Risk, Reliability and Societal Safety, pp. 281-288. [8] Lukoševičienė, O., Kudzys, A., 2007. “The probabilistic durability prediction of deteriorating structures”, Modern Building Materials, Structures and Techniques, Vilnius Gediminas Technical University, Vol. (3), pp. 1005-1010. [9] JCSS 2000 Probabilistic model code: Part 1. Basis of design. Joint Committee on Structural Safety, 2000. [10] ISO 2394:1998. General principles on reliability for structures. Switzerland, 1998. [11] Vrowenvelder, A. C. V. M., 2002. Developments towards full probabilistic design codes, Structural Safety 24(2-4), pp. 417-432. [12] Raizer, B. R., 1998. Theory of reliability in structural design. ACB Publishing House, Moscow (in Russian). [13] Grigoriu, M., Tursktra, C., 1979. Safety of structural systems with correlated resistances, Applied Mathematical Modelling 2(3), pp. 130-136. [14] Mori, Y., Ellingwood, B. R., 1993. Time-dependent system reliability analysis by adaptive importance sampling, Structural Safety 12(1), pp. 59-73. [15] Mori, Y., Nonaka, M., 2001. LRFD for assessment of deteriorating existing structures, Structural Safety 23, pp. 297-313. [16] Kudzys, A., Lukoševičienė, O., 2009. On the safety prediction of deteriorating structures, Mechanika 4(78), pp. 5-11. [17] Kudzys, A., 2007. “Transformed conditional probabilities in analysis of stochastic sequences”, Summer Safety and Reliability Seminars, SSARS 2007, Gdansk-Sopot, Poland, pp. 243-250.
Procedia Engineering 57 (2013) 642 – 650
11th International Conference on Modern Building Materials, Structures and Techniques, MBMST 2013
Conventional Stochastic Sequences in Reliability Assessments and Predictions of Structural Members Antanas Kudzysa, Ona Lukoševičienėb ∗ b
Department of Strength of Materials, Vilnius Gediminas Technical University, Saulėtekio st. 11, LT-10223 Vilnius, Lithuania
Abstract A simplification of probability based approaches on safety assessments and predictions of structural members is discussed. The modelling of time-dependent survival probabilities and reliability indexes of particular and single members is defined in a simple and easy perceptible manner using uniformed correlation factors of their conventional sequences. These factors of safety margin processes are analyzed and demonstrated by numerical example. © 2013 Published by Elsevier Ltd. Ltd. 2013The TheAuthors. Authors. Published by Elsevier Selection and under responsibility of theof Vilnius Gediminas TechnicalTechnical UniversityUniversity. Selection andpeer-review peer-review under responsibility the Vilnius Gediminas Keywords: reliability, safety margin, correlation coefficient, survival probability, structural member, reliability index.
1. Introduction The standards EN 1990 [1] and ASCE/SEI [2] are based on the limit state concept and, respectively, on the methods of the partial factor design and the strength or allowable stress design. However, the structural design practice shows that it is impossible to verify the safety and economy parameters of load carrying structures using deterministic methods and with their universal factors for loads and material properties. The reliability indexes of members and structural systems may be objectively defined only by fully probability-based concepts and models. According to these approaches, only probability-based concepts allow us explicitly predict basic and additional uncertainties of structural members in their analysis on the durability and long-term reliability. The probabilistic analysis of members is indispensible in order to predict their time-dependent destructions or failures. In the beginning of probabilistic structural analysis of equally correlated systems, the simplified methods of average and equivalent correlation coefficients were used [3]. The new mathematical probabilistic formats used in a long-term reliability prediction of structures are based on rather complicated considerations [4–8]. Thus, the engineering modeling of survival probabilities and reliability indexes as the counting responses of structures are still unsolved. The main task of this paper is to present for mechanical, structural, resident and management engineer’s new unsophisticated and uniform methodological formats on probability based safety assessments and predictions for nondeteriorating and deteriorating particular structural members as components of series auto systems. The second aim of this paper is to demonstrate the difference between time-variant reliability response curves of non-deteriorating and deteriorating members as auto systems.
* Corresponding author. E-mail address: [email protected]; [email protected]
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Vilnius Gediminas Technical University doi:10.1016/j.proeng.2013.04.081
643
Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
2. Safety margins of members Multicriteria failure modes and safety of structural members (beams, slabs, columns, joints) may be objectively assessed and predicted only knowing survival probabilities of their particular members (normal or oblique sections, connections). The structural reliability analysis is related with some methodological differences between non-deteriorating and deteriorating members. Therefore, it is expedient to divide the life cycle tn (Fig. 1) of deteriorating structures into the initiation, tin, and propagation, tpr, phases [9].
Fig. 1. Time-dependent model for structural reliability assessment and prediction of non-deteriorating (1) and deteriorating (2) members
A length of initiation phase is a random variable depending on a feature of degradation process, an environment aggressiveness and a quality of protective covers. The unvulnerability of structures may be characterized by the duration of this phase. When the degradation process of the members is caused by intrinsic properties of materials, the phase tin≈0. The time-dependent safety margin of particular members may be defined as their random performance process and presented as follows
Z ( t ) = g {X ( t ) , θ} = θ R R ( t ) − θ g E g − θ q Eqs ( t ) − θ q Eqe ( t ) − θ cl Ecl ( t ) ,
(1)
where X(t ) and θ are the vectors of basic and additional variables representing respectively random components (resistances and action effects) and their model uncertainties; R(t ) = ϕ (t )R0 is the resistance of particular members, where ϕ (t ) denotes their degradation function depending on an artificial ageing and degradation of materials; E g , Eq s (t ) , Eqe (t ) and Ecl (t ) are the action effects caused by permanent, sustained, extraordinary and climate (snow and wind) loads. The mean values and standard deviations of additional variables of the safety margin are: θ Rm = 0.99 − 1.04 , σθ R = 0.05 − 0.10 and θ gm ≈ θ qm ≈ θ cl , m ≈ 1.00 , σθ g ≈ σθ q ≈ σθ cl ≈ 0.10 [9–11]. Gaussian and lognormal distribution laws are to be used for resistances of reinforced concrete and steel members, respectively [10]. The permanent actions can be described by a normal distribution law [12]. Lognormal and gamma or exponential distribution laws are presented respectively for sustainable and extraordinary floor live loads [9], [10]. The duration, d, and the renewal rates, λ, of extreme floor live and climate actions are: d g = tn , λg = 1 / tn ; d qs = 5 − 10 years , λqs = 0.1 − 0.2 / years ; d qe = = 1− 14 days for merchant and 1–3 days for other buildings, λqe = 1 / years ; d s = 14 − 28 days , λs = = 1 / years ; d w = 8 − 12 hours , λw = 1 / years [9]. The recurrence number of two joint extreme actions during the design working life of structures, tn in years, may be presented by the equation
n12 = tn ( d1 + d2 ) λ1λ2 .
(2)
Thus, the recurrence numbers of extreme concurrent live loads during t n = 50 years period are: ngq = 6 − 11 , nqs qe = nqs s = 50 , nqe s = 2 − 6 nqe w = 0.2 − 2 .
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The dangerous cuts of process by Eq. (1) correspond to extreme loading situations of structures. Therefore, in design practice this safety margin process may be modeled as a random geometric distribution and treated as a finite decreasing random sequence, when the safety margin of member is
Z k = θ R Rk − θ g Eg − θ q Eqs k − θ q Eqe k − θcl Ecl ,k ,
(3)
where k = 1, 2..., n − 1, n is the number of cuts (Fig. 1). 3. Instantaneous survival probabilities of members Multi-cut sequences of random safety margins of particular members can be idealized as their series auto systems. The instantaneous probability of failure of these members in mode k is P ( Z k ≤ 0 ) . Structural members as series auto systems are safe, if each their particular member as individual component is safe. The degree of this safety expressed by probabilistic responses depends on the correlation intensity between safety margin cuts. It is safer but not relational for structural members to ignore the correlation between these cuts. The instantaneous survival probability of particular and single members with respect to their failure mode at the safety margin sequence cut k is assuming, that they were safe at sequence cuts 1, ..., k-1. Therefore, their instantaneous survival probability at cut k may be expressed as ⎛ k −1 ⎞ P ( Z k > 0 ) = P ( Sk ) = 1 − P ⎜ Fk ∩ Si ⎟ . ⎜ ⎟ ⎝ i =1 ⎠
(4)
For design convenience, the structural safety analysis of members may be based on the limit state criterion Rk − Ek > 0 , where Ek is the resultant action effect, and can be modeled using multidimentional integral as P ( Z k > 0 ) = P { Rk − Ek > 0 ∃ k ∈ [1, n ] } =
∫
g ( X k θ )> 0
fX
k θ
( X k θ ) dx ,
(5)
As it is shown, the survival probability of an single member in series systems or any particular member in series auto systems is represented by the limit state domain g Xk θ > 0 . The resistance Rk and resultant action effect Ek may be treated as statistically independent variables of random safety margins sequences. Therefore, the instantaneous survival probability of particular and structural members can be expressed by convolution multidimentional integral as
{
}
P ( Sk ) = P { Rk > Ek ∃ k ∈ [1, n ] } =
∞
∫ f Rk ( x ) FEk ( x ) dx ,
(6)
0
where f Rk (x ) is the density function of their resistance and FEk (x) is the cumulative distribution function of their action effect (Fig. 2). The value of probability by Eq. (6) may be modeled using analytical numerical integration and Monte Carlo methods or their programs.
Fig. 2. Schematic representation of an instantaneous survival probability analysis
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4. Indexed correlation factor for random sequences The correlation between resistances of particular members always has a great effect on the counting probabilistic response [13]. The time-variant safety margin of members must be treated as a non-stationary process. Therefore, it is rather complicated to define the survival probability of multicuts sequences in easy perceptible manner. However, this impediment may be removed in design practice using the indexed correlation factors of conventional random sequences. The correlation factor of a finite sequence consisted of k stochastic cuts can be formed from random k-th row, ρ ki ∃ i = 1, ..., k − 1 , or random k-th column, ρik ∃ i = 1, ..., k − 1 , of a quadratic correlation matrix by Eq. (7) of basic correlation coefficients 1
ρ12 1
ρ21 ρkl =
ρ13
...
ρ23
...
ρ2,k −1
ρ2 k
ρ1,k −1
ρ1k
ρ31
ρ32
1
...
ρ3,k −1
ρ3k
...
...
...
1
...
...
ρk −1,1 ρk1
ρk −1,2 ρk 2
ρk −1,3 ρk 3
... ...
1 ρk ,k −1
ρk −1,k 1
.
(7)
The value of correlation factors as correlation vectors ρk = ρki = ρik may be calculated from the Eqs. (8) or (9)
(
)
(8)
(
)
(9)
ρki ≈ ρk1 + ρk 2 + ρk 3 + ... + ρk ,k −1 / ( k − 1) , ρik ≈ ρ1k + ρ2k + ρ3k + ... + ρk −1,k / ( k − 1) , The basic correlation coefficients of rank safety margin cuts are calculated by the equation ρki = ρik =
Cov ( Z k , Zi ) σZ k × σZi
,
(10)
where Cov ( Z k , Zi ) and σ Z k , σ Zi are the auto covariance and standard deviations of stochastic sequence cuts. x According to our investigation data, the bounded index, xk , of indexed correlation factors, ρk k , of random sequences may be presented as
⎡ 4.5 + 4 ρkl ⎤ xk = P ( Sk ) ⎢ ⎥ ⎣1 − 0.98ρkl ⎦
υk
,
(11)
where k
υk =
∑ P ( Si )
i =2
( k − 1)
k +2
,
(12)
characterizes the effect of correlation vector size on a bounded index value. For highly reliable members, the instantaneous survival probability P(Si ) ≈ 1 and the Eq. (12) may be written as follows υk = ( k − 1) ⎡⎣( k − 1) k + 2 ⎤⎦ = 1
k +2,
(13)
For conventional sequences with equicorrelated cuts, the correlation factor ρk = ρij i.e. its value is equal to a basic correlation coefficient.
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
5. Time-dependent survival probabilities of members The methods on time-dependent structural reliability analysis were proposed by Mori Y. and his colleagues Ellingwood B.R. [14] and Nanoka M. [15], but these and other approaches are not included in design codes and standards. The total failure probability of ductile members is the union of all possible failure modes of auto systems and may be expressed as `
⎛ n ⎞ ⎛ k −1 ⎞ ⎛ n −1 ⎞ P ⎜ ∪ ( Zi ≤ 0 ) ⎟ = P ( F1 ) ∪ ...∪ P ⎜ Fk ∩ Si ⎟∪ ...∪ P ⎜ Fn ∩ Si ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠
(14)
⎛ k −1 ⎞ The transition probability P ⎜ Fk ∩ S i ⎟ denotes a statistical dependence between a failure probability of members under ⎝ i =1 ⎠ k-th loading situation and their survival probabilities under previous loading. The total survival probability of non-deteriorating members as series auto systems consisted of n random cuts of their safety margin sequences P ( S1 ) ≤ P ( S2 ) ≤ ... ≤ P ( Sn ) can be expressed as
⎛ n ⎞ ⎛ n ⎞ ⎛ n ⎞ P ⎜ ∩ Si ⎟ = P ⎜ ∩ Si ≤ S j ∃ i ∈ [1, n ] ⎟ = 1 − P ⎜ ∪ Fi > F j ∃ i ∈ [1, n ] ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠1 ⎝ i =1 ⎠ ⎝ i =1 ⎠
(
)
(
( ) × ...× P ( S
= P ( S1 ) × P S
21
k k −1,...,1
)
) × ...× P ( S
n n −1,...,1
⎛ k −1 ⎞ ⎛ n −1 ⎞ n × ... × P ⎜ Sk ∩ Si ⎟ × ... × P ⎜ Sn ∩ Si ⎟ = ∏ P ( Sk ) ⎜ ⎟ ⎜ ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ k =1
) = P(S )× P(S ∩ S )×
⎡ x ⎢1 + ρ122 ⎣⎢
1
2
1
⎛ 1 ⎞⎤ − 1⎟⎟ ⎥ × ... ⎜⎜ P S ( ) 2 ⎝ ⎠ ⎦⎥
(15)
⎡ ⎡ ⎛ 1 ⎞⎤ ⎛ 1 ⎞⎤ xk xn × ⎢1 + ρ1,2,... − 1⎟ ⎥ × ... × ⎢1 + ρ1,2,... − 1⎟ ⎥ ⎜ k −1 k ⎜⎜ P ( S ) n n − 1 ⎟ ⎜ P(S ) ⎟ k n ⎝ ⎠ ⎦⎥ ⎝ ⎠ ⎦⎥ ⎣⎢ ⎣⎢
where the transition probability of safety margin sequence cuts k and i=1, ..., k-1 is ⎛ k −1 ⎞ P ⎜ Sk ∩ Si ⎟ = P ( Sk ) ⎜ ⎟ ⎝ i =1 ⎠
⎧⎪ x ⎨1 + ρk k ⎩⎪
⎡ 1 ⎤ ⎫⎪ − 1⎥ ⎬ , ⎢ ⎢⎣ P ( S k ) ⎥⎦ ⎭⎪
(16)
If correlation factors ρ kx k = ρ kx−k −11 ... = ρ 2x 2 = 1 transition and survival probabilities by Eqs. (16) and (15) are ⎛ k −1 ⎞ ⎛ n ⎞ P⎜⎜ S k ∩ Si ⎟⎟ = 1 and P ⎜ ∩ S i ⎟ = P (S1 ) . ⎝ i =1 ⎠ ⎝ i =1 ⎠ When the cuts of na safety margin sequence are statistically independent, i.e. ρ kxk = ρ kx−k −11 ... = ρ 2x 2 = 0 , the probability by Eq. (15) is equal to ∏ P(Si ) . i =1 For random sequences or series systems with equireliable and equicorrelated cuts or elements, the correlation factor index by Eq. (11) and transition probability by Eq. (16) may be expressed as
⎛ 4.5 + 4 ρij xk = ⎜ ⎜ 1 − 0.98 ρij ⎝
1
⎞ ⎟ ⎟ ⎠
k +2
,
⎧ ⎛ k −1 ⎞ ⎤⎫ ⎪ ⎪ x ⎡ 1 x P ⎜ Sk ∩ Si ⎟ = P ( Sk ) ⎨1 + ρk k ⎢ − 1⎥ ⎬ = P ( Sk ) + ρk k ⎡⎣1 − P ( Sk ) ⎤⎦ , ⎜ ⎟ P S ( ) ⎢ ⎥ k ⎪ ⎪ ⎣ ⎦ ⎝ i =1 ⎠ ⎩ ⎭
(17)
(18)
The survival probability of deteriorating members as series auto systems expressed by decreasing sequences, P(S1 ) > P(S 2 ) > ... > P(S n ) , may be calculated from the formulas
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
⎛ n ⎞ ⎛ n P ⎜ ∩ Si ⎟ = P ⎜ ∩ Si > S j ⎜ ⎟ ⎜ ⎝ i =1 ⎠2 ⎝ i =1
(
( )
⎞
⎛
n
⎞
⎠
⎝ i =1
⎠
) ⎟⎟ = 1 − P ⎜⎜ ∪ ( Fi < F j ) ⎟⎟ ,
⎛ n ⎞ P ⎜ ∩ Si ⎟ = P ( S1 ) × P S × ... × P S1,...,k −1 k × ... × P S1,...,n−1 n = P ( S1 ) × P ( S1 ∩ S 2 ) × ... 12 ⎜ ⎟ ⎝ i =1 ⎠ ⎛ k −1 ⎞ ⎛ n−1 ⎞ n ⎡ ⎞⎤ x ⎛ 1 × ... × P ⎜ ∩ Si Sk ⎟ × ... × P ⎜ ∩ Si Sn ⎟ = ∏ P ( Sk ) ⎢1 + ρ122 ⎜ − 1 × ... ⎜ P ( S ) ⎟⎟ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢⎣ 1 ⎝ ⎠ ⎥⎦ ⎝ i =1 ⎠ ⎝ i =1 ⎠ k =1 ⎡ ⎡ ⎛ ⎞⎤ ⎛ ⎞⎤ 1 1 xn xk − 1⎟ ⎥ , × ⎢1 + ρ1,2,... − 1⎟ ⎥ × ... × ⎢1 + ρ1,2,... ⎜⎜ n n − 1 k −1 k ⎜⎜ P ( S ⎟ ⎟ ⎢⎣ ⎢⎣ k −1 ) ⎝ P ( Sn−1 ) ⎠ ⎥⎦ ⎝ ⎠ ⎥⎦
(
)
(
(19)
)
(20)
where the transition probability of sequence cuts is
⎧ ⎛ k −1 ⎞ ⎪ x P ⎜ ∩ Si Sk ⎟ = P ( Sk ) ⎨1 + ρk k ⎜ ⎟ ⎪ ⎝ i =1 ⎠ ⎩
⎡ ⎤⎫ 1 ⎪ − 1⎥ ⎬ . ⎢ P S ⎪ ⎣⎢ ( k −1 ) ⎦⎥ ⎭
(21)
If the correlation factors of sequences are fully statistically dependent, i.e. ρ kxk = 1 , or independent, i.e. ρ kx k = 0 , the values of transition and survival probabilities, respectively, are: n ⎛n ⎞ ⎛n ⎞ ⎛ k −1 ⎞ ⎛ k −1 ⎞ P ⎜ ∩ S i S k ⎟ = P (S k ) / P (S k −1 ) and P⎜ ∩ Si ⎟ = P(S n ) or P ⎜ ∩ S i S k ⎟ = P (S k ) and P⎜ ∩ Si ⎟ = ∏ P(Si ) . ⎝ i =1 ⎠ 2 ⎝ i =1 ⎠ 2 i =1 ⎝ i =1 ⎠ ⎝ i =1 ⎠
The survival probability of structural members may be also introduces by the reliability indexes as
⎡ ⎛ n ⎞ ⎤ β1 = Φ −1 ⎢ P ⎜ ∩ Si ⎟ ⎥ , ⎢ ⎜⎝ i =1 ⎟⎠ ⎥ ⎣ 1⎦
(22)
⎡ ⎛ n ⎞ ⎤ β2 = Φ −1 ⎢ P ⎜ ∩ Si ⎟ ⎥ . ⎢ ⎜⎝ i =1 ⎟⎠ ⎥ ⎣ 2⎦
(23)
Fig. 3. Reliability indexes versus correlation factors when system consists of one, two, five and ten elements
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
The results calculated by Eqs. (15) and (22) corroborate to data presented in the papers [9, 13, 3] for equicorrelated series systems consisting of two, five and ten elements (Fig. 3). It is shown that the values of time-dependent reliability indexes based on conventional stochastic sequence approaches and expressed by Eq. (22) (curves 3, 5, 7) are close to the numerical integration data (curves 2, 4, 6). 6. Numerical example Consider the long-term survival probability and reliability index of non-deteriorating and deteriorating roof steel beams of a scrap metal shed (Fig. 4). The destroying bending moments of beams Eg, Eq and Es are caused by a permanent action G of steel roof structures and hanging crane crabs, variable loads Q of scrap metals and S of snow depth.
Fig. 4. Traversing underslung crane
The means and variances of the beam resisting and destroying parameters are:
(θ R R0 )m = 297.1 kNm, σ 2 (θ R R0 ) = 564.9
(kNm)2;
(θ E Eg )m = 18.28 kNm, σ 2 (θ E E g ) = 3.342 (kNm) ; (θ E Eq )m = 52.19 kNm, σ 2 (θ E Eq ) = 109.0 (kNm) ; 2
2
(θ E E s )m = 54.67 kNm, σ 2 (θ E Es ) = 269.0 (kNm)2.
Fig. 5. The time-dependent instantaneous and long-term reliability indexes of non-deteriorating (1, 2) and deteriorating (3, 4) members
Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650
Fig. 6. The time-dependent instantaneous and long-term survival probability of non-deteriorating (1, 2) and deteriorating (3, 4) members
These parameters are described by normal (R and Eg), lognormal (Eq) and Gumbel (Es) probability distributions laws. The degradation function of deteriorating beams is ϕ (t ) = 1 − 0.00375× t . The instantaneous and long-term survival probabilities of beam are calculated by Eqs. (6) and (15), (16) and the reliability index is defined by Eqs. (22) and (23). The time-dependent instantaneous (1, 3) and long-term (2, 4) reliability indexes and survival probabilities based on conventional stochastic sequences method are presented in Figs. 5 and 6, respectively. This numerical example satisfied that the suggested method of conventional stochastic sequences is unsophisticated and may be included in structural design practice. 7. Conclusions
It is expedient to use in structural design practice the simplified approaches based on the conventional stochastic sequence and correlation factor concepts. These concepts help us to formulate the probabilistic method uniformed for an analysis of non-deteriorating and deteriorating ductile members. New approaches in applicable stochastic sequence theory help us introduce unsophisticated probability based methodological and mathematical formats for safety assessments, verifications and predictions of structural members. For practical sake of a probabilistic design, the safety margin process or time-dependent performance criterion of structural ductile members may be treated as a random sequence. Survival probabilities and reliability indexes of time-dependent particular and single structural members, respectively as normal or oblique sections and beams, columns, slabs or joints, may be defined in a simple and easy perceptible manner using uniformed correlation factors of their conventional sequences. The position of stochastically dependent cuts of conventional sequences is matched with extreme loading situations of members as components of series auto systems. The format for representation of indexed correlation factor is presented in this paper. A degradation function of member resistance may have a considerable effect not only on failure probabilities and reliability indexes of members, but also on shapes of their curves. References [1] EN 1990:2002. Eurocode - Basic of structural design. CEN, Brussels, 2002. [2] ASCE/SEI 7-05. 2005. Minimum design loads for buildings and other structures, 388. [3] Thoft-Christensen, P., Dalsgard Sorensen, J., 1982. Reliability of structural systems with correlated elements, Applied Mathematical Modelling 3(6), pp. 171-178. [4] Rackwitz, R., 2006. Risk acceptance and optimization of aging but maintained civil engineering infrastructures, Safety and Reliability for Managing Risk, pp. 1527-1534. [5] Joanni, A. E., Rackwitz, R., 2006. Stochastic dependencies in inspection, repair and failure models, Safety and Reliability for Managing Risk, pp. 531537. [6] Noortwijk, J. M., Kallen, M. J., Pandey, M. D., 2005. Gamma processes for time-dependent reliability of structures, Advances in Safety and Reliability, pp. 1457-1464.
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Antanas Kudzys and Ona Lukoševičienė / Procedia Engineering 57 (2013) 642 – 650 [7] Kuniewski, S. P., van Noortwijk, J. M., 2007. Sampling inspection for the evaluation of time-dependent reliability of deteriorating structures, Risk, Reliability and Societal Safety, pp. 281-288. [8] Lukoševičienė, O., Kudzys, A., 2007. “The probabilistic durability prediction of deteriorating structures”, Modern Building Materials, Structures and Techniques, Vilnius Gediminas Technical University, Vol. (3), pp. 1005-1010. [9] JCSS 2000 Probabilistic model code: Part 1. Basis of design. Joint Committee on Structural Safety, 2000. [10] ISO 2394:1998. General principles on reliability for structures. Switzerland, 1998. [11] Vrowenvelder, A. C. V. M., 2002. Developments towards full probabilistic design codes, Structural Safety 24(2-4), pp. 417-432. [12] Raizer, B. R., 1998. Theory of reliability in structural design. ACB Publishing House, Moscow (in Russian). [13] Grigoriu, M., Tursktra, C., 1979. Safety of structural systems with correlated resistances, Applied Mathematical Modelling 2(3), pp. 130-136. [14] Mori, Y., Ellingwood, B. R., 1993. Time-dependent system reliability analysis by adaptive importance sampling, Structural Safety 12(1), pp. 59-73. [15] Mori, Y., Nonaka, M., 2001. LRFD for assessment of deteriorating existing structures, Structural Safety 23, pp. 297-313. [16] Kudzys, A., Lukoševičienė, O., 2009. On the safety prediction of deteriorating structures, Mechanika 4(78), pp. 5-11. [17] Kudzys, A., 2007. “Transformed conditional probabilities in analysis of stochastic sequences”, Summer Safety and Reliability Seminars, SSARS 2007, Gdansk-Sopot, Poland, pp. 243-250.