# On the safety prediction of deteriorating structures/Silpnejanciu konstrukciju saugos prognozavimas.

1. Introduction

The target design life of deteriorating load-carrying structures and their components must be defined in an early design stage of buildings, construction works and technological equipments. The value of this life must serve as a basis for the selections of materials and structures. The target design life is related to destruction modes of materials and structural components and failure consequences. In any case, higher durability requirements are applied to members which routine or preventive maintenance and repairs require great efforts.

Failures and collapses in load-carrying structures can be caused not only by irresponsibility of gross human errors of designers or erectors but also by some conditionalities of recommendations and directions presented in design codes and standards. The Standards EN 1990 [1] in Europe and ASCE/SEI [2] in the USA require that load-carrying structures to be designed with appropriate degrees of reliability. These Standards 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 deteriorating structures by using deterministic methods and their universal factors for loads and material properties.

The reliability degree of deteriorating structures may be objectively defined only by fully probability-based concepts and models. Only probabilistic approaches may allow us explicitly predict uncertainties of analysis models of these structures. Besides, the probabilistic analysis of deteriorating members is indispensible in order to predict their destructions or failures and to avoid of economic and psychologic losses. However, the mathematical probabilistic formats used in long-term reliability prediction of structures are based on rather complicated considerations [3-6]. Thus, the engineering modeling of survival probabilities of structures subjected to aggressive environmental actions and extreme live and climate loads are still unsolved.

The main task of this paper is to present new methodological formats on probability-based safety predictions of deteriorating members exposed to permanent loads and recurrent single or joint extreme service and climate actions.

2. Resistances and safety margins of deteriorating members

Multicriteria failure modes and safety of structural members (beams, slabs, columns, joints) may be objectively assessed and predicted only knowing survival probabilities of particular members (normal or oblique sections, connections) for which the only possible failure mode exists. Predicted durability parameters for deteriorating structures depend on chemical diagnosis and the acceptable risk of serviceability failure associated with the damage levels and losses. Besides, the predictions of safety of deteriorating members and their systems will account for all extreme action combinations. In any case, it is expedient to divide the life cycle [t.sub.n] (Fig. 1) of deteriorating structures into the initiation, [t.sub.in], and propagation, [t.sub.pr], phases [7]. The length of initiation phase is a random variable depending on a feature of degradation process, an environment aggressiveness and 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 [t.sub.in] = 0 . The propagation phase is delayed for structures protected by coats.

[FIGURE 1 OMITTED]

The resistance of particular members in the propagation phase is treated as a nonstationary random process

R(t) = [phi](t)[R.sub.in] = [phi](t)[R.sub.0] (1)

where [R.sub.0] is the initial value of member resistance, [phi](t) denotes the degradation function depending on the rate of a resistance decrease induced by an artificial ageing and degradation of materials. This function for corrosion affected particular members may be presented in the form

[phi](t) = 1 - a[(t - [t.subin]).sup.b] (2)

where b defines nonlinearity of the deterioration function and a is degradation intensity factor. A shape of the deg radation function is close to linear (b [approximately equal to] 1) and parabolic (b [approximately equal to] 2) when corresponding degradation mechanisms are steel corrosion and aggressive environmental attacks [8, 9]. However, marine corrosion of steel structures is not linear function of time [10].

Action effects of structures are caused by permanent loads g, sustained [q.sub.s] (t) and extraordinary [q.sub.e] (t)= q(t)- [q.sub.s] (t) components of live loads s(t) and wind, surf or seismic actions w(t). The annual extreme sum of sustained and extraordinary live load effects [E.sub.q] (t) caused by [q.sub.s] (t) and [q.sub.e] (t) may be modeled as a rectangular pulse renewal process described by Type I (Gumbel) distribution of extreme values with the coefficient of variation [delta]q = 0.58 and mean [E.sub.qm] = 0.47[E.sub.qk], where [E.sub.qk] is its characteristic value [11].

It is proposed to model annual extreme snow and wind action effects by a Gumbel distribution with the mean values equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [E.sub.sk] and [E.sub.wk] are the characteristic (nominal) values of action effects and [k.sub.0.98] is the characteristic fractile factor of these distributions [7, 12]. The coefficients of variation of snow and wind extreme loads depend on the feature of geographical area and are equal to [delta]s = 0.3 - 0.7 and [delta]w = 0.2 - 0.5.

The durations of extreme floor and climate actions are: [d.sub.q] = 1 -14 days for merchant and 1-3 days for other buildings, [d.sub.s] = 14 - 28 days and [d.sub.w] = 8 -12 hours. Renewal rates of annual extreme actions are equal to [lambda] = 1 / year . Therefore, the recurrence number of two joint extreme actions during the design working life of structures, tn in years, may be calculated by the formulae

[n.sub.12] = [t.sub.n] ([d.sub.1] + [d.sub.2])[[lambda].sub.1] [[lambda].sub.2] (3)

where [[lambda].sub.1] = [[lambda].sub.2] = 1 / [t.sub.[lambda]] are the renewal rates of extreme loads. Thus, the recurrence numbers of extreme concurrent live or snow and wind loads during [t.sub.n] = 50 years period are quite actual to [n.sub.qw] = 0.2 - 2.0 and [n.sub.sw] = 2.0 - 4.0. The bivariate distribution function of two independent extreme action effects may be presented as their conventional joint distribution function with the mean [E.sub.12m] = [E.sub.1m] + [E.sub.2m] and the variance [[sigma].sup.2] [E.sub.12] = [[sigma].sup.2] [E.sub.1] + [[sigma].sup.2] [E.sub.2] [13].

According to probability-based approaches (design level III), the time-dependent safety margin of deteriorating particular members exposed to extreme action effects may be defined as their random performance process and presented as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where X(t) and [theta] are the vectors of basic and additional variables, representing respectively random components (resistances and action effects) and their model uncertainties; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [E.sub.2](t) = [E.sub.s] (t) or [E.sub.2] (t) = [E.sub.w] (t). The mean values and standard deviations of additional variables of the safety margin are: [[theta].sub.Rm] = 0.99 -1.04, [sigma][[theta].sub.R] = 0.05-0.10 and [[theta].sub.Em] [approximately equal to] 1.00, [sigma][[theta].sub.E] [approximately equal to] 1.00, [sigma][[theta].sub.Em] [approximately equal to] 0.10 [11, 12].

Gaussian and lognormal distribution laws is to be used for member resistances. The permanent actions can be described by a normal distribution law [13,14]. Therefore, for the sake of simplified but quite exact probabilistic analysis of deteriorating members, it is expedient to present Eq. (4) in the form

Z(t) = [R.sub.c] (t)-E (t) (5)

where

[R.sub.c] (t) = [[theta].sub.R]R(t) - [[theta].sub.g][E.sub.g] (6)

is the conventional resistance of members the bivariate probability distribution of which may be modeled by Gaussian distribution

E(t) = [[theta].sub.1][E.sub.1](t) + [[theta].sub.2] [E.sub.2] (t) (7)

is the conventional bivariate distribution process of two stochastically independent annual extreme effects [15].

Inspite of analysis simplifications, the use of continuous stochastic processes of member resistances considerably complicates the durability analysis of deteriorating structures exposed to intermittent extreme gravity and lateral variable actions along with permanent ones. The dangerous cuts of these processes correspond to extreme loading situations of structures. Therefore, in design practice the safety margin process Eq. (5) may be modeled as a random geometric distribution and treated as finite decreasing random sequence

[Z.sub.k] = [R.sub.ck] - [E.sub.k], k = 1,2,...,n - 1,n (8)

where

[R.sub.ck] = [[phi].sub.k][[theta].sub.R][R.sub.in] - [[theta].sub.g][E.sub.g] (9)

is the conventional resistance of deteriorating members at the cut k of this sequence (Fig. 1) and n is the recurrence number of single or coincident extreme action effects, [E.sub.k], given by Eq. i.e. [E.sub.k] = [[theta].sub.1][E.sub.1k] + [[theta].sub.2][E.sub.2k].

When extreme action effects are caused by two stochastically independent variable actions, a failure of members may occur not only in the case of their coincidence but also when the value of one out of two effects is extreme. Therefore, three stochastically dependent safety margins should be considered as follows

[Z.sub.1k] = [R.sub.ck] - [E.sub.1k] k = 1,2,...,n (10)

[Z.sub.2k] = [R.sub.ck] - [E.sub.2k] k = 1,2,...,n2 (11)

[Z.sub.3k] = [R.sub.ck] - [E.sub.11k] = [R.sub.ck] - [E.sub.1k] - [R.sub.2k], k = 1,2,...,[n.sub.12] (12)

where the number of sequence cuts n12 is calculated by Eq. (3).

3. Transformed conditional probability method

For particular and structural members of deteriorating structures subjected to extreme action effects, more than one limit state situations are considered. The number of these situations is equal to recurrence numbers [n.sub.1] or [n.sub.2] and [n.sub.12] of single and coincident extreme action effects, respectively.

The statistical dependences among failure probabilities of particular members at any time [t.sub.k] or any cut k

of rank sequence and their survival probabilities at previous extreme loading situations exist. Therefore, the instantaneous failure probability of these members at sequence cut k, assuming that they were safe at cuts [1,k-1 ], may be presented in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.k] denotes the failure event of members at cut k and [S.sub.i] denotes the event of their survival at previous cut i of a sequence. Therefore, the instantaneous failure probabilities of particular members at cuts 1,2,3,...,n of their safety margin sequences are: p([Z.sub.1] [less than or equal to] 0) = p([F.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The time dependent failure probabilities of deteriorating particular and individual members as autosystems during times [t.sub.1], [t.sub.2], [t.sub.3],...,[t.sub.n] may be expressed as

P(T < [t.sub.1]) = P([Z.sub.1] [less than or equal to] 0) = P([F.sub.1]) (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Thus, according to probabilistic approaches, the prediction of time dependent survival probabilities of load-carrying particular members may be based on the analysis of decreasing sequences of random safety margins (Fig. 2), i.e. can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

When the sequence consists of two dependent cuts, the probability that either or both of two failure events of a series system occur is expressed by Eq. (17). An the evaluation of the probability of a second order intersection of failure events [F.sub.2] and [F.sub.1], i.e. p([f.sub.2] [intersection] [F.sub.1]), may be carried out by rather uncomfortable for structural engineers methods of numerical integrations or Monte Carlo simulation. It is more expedient to use in design practice the unsophisticated method of transformed conditional probabilities (TCTM). According to its approaches, the intersection probability

p([F.sub.2][intersection][F.sub.1]) = P([F.sub.2])P([F.sub.1]|[F.sub.2]) (21)

where the conditional failure probability

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The indexed correlation factor of two sequence cuts, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], characterizes an effect of their statistical dependence on the intersection probability P([F.sub.2][intersection] [F.sub.1]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 2 OMITTED]

When sequence cuts are independent, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the conditional, intersection and failure probabilities of members from Eqs. (22), (21) and (17) are defined as: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When sequence cuts are fully correlated, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], these probabilities are: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is between 0 and 1, the intersection and failure probabilities by Eqs. (21) and (17) of two cut sequences become as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Analogically to Eq. (23), the probability of an intersection of three failure events may be presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where the correlation factor [[rho].sub.3,21] [approximately equal to] 0.5([[rho].sub.32] + [[rho].sub.31]). The correlation factor and its bounded index are considered in Section 5.

4. Instantaneous survival probability

The instantaneous survival probability of particular members with respect to their single failure mode at sequence cut k, if they were safe at cuts 1-k-1 i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], can be modelled using multidimensional integral as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For design convenience, the structural safety analysis of deteriorating members may be based on the limit state criteria [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and, [R.sub.ck] -([E.sub.1k] + [E.sub.2k])> 0 where [R.sub.ck] is defined from Eq. (6). Therefore, the instantaneous survival probability may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The conventional resistance [R.sub.ck] and single extreme action effect [E.sub.k] may be treated as statistically independent variables of random safety margins.

[FIGURE 3 OMITTED]

Therefore, the instantaneous survival probability of deteriorating members can be expressed by convolution integral as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (x) is the density function of resistance and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (x) is the cumulative distribution function of their action effect (Fig. 3).

5. Long-term survival probability

Decreasing resistance of particular members must be treated as a nonstationary process. Therefore, it is rather complicated to define the failure probability of multicut sequences in easy perceptible manner. However it is fairly simple to calculate the survival probability of deteriorating members by TCPM. According to Eq. (20), the survival probabilities of these members exposed to two, three and n extreme loading situations may be expressed as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

For the sake of simplified but fairly exact probability-based analysis of deteriorating structures, the conditional survival probability of higher order [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for particular members may be defined as the probability P([S.sub.k]|[S.sub.k-1]). Therefore, the component [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of Eq. (30) my be changed by the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then,

Eq. (30) may be rewritten in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where P([S.sub.1]), ..., P([S.sub.k-1]), ..., P([S.sub.n-1]) are the instantaneous survival probabilities of members by Eq. (27). The correlation factor of dependent sequence cuts, [[rho].sub.k,1...k-1], is formed from k th row of quadratic matrix of basic coefficients of correlation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It may be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

The coefficient of correlation of rank safety margin cuts is calculated from the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where Cov([Z.sub.k], [Z.usb.l]) and [sigma][Z.sub.k], [sigma][Z.sub.l] are an auto covariance and standard deviations of safety margin values.

Then long term survival probabilities of members are calculated by Eq. (31), the bounded index, [x.sub.k], of correlation factors of random multicut sequences may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

For highly reliable load-carrying members, the instantaneous survival probability P([S.sub.k])[approximately equal to]1 and its effect on bounded indices may be ignored.

The acceptability of this index in design practice is corroborated by Fig. 4, where the position of points for decreasing sequences with two, three and four cuts is calculated by Monte Carlo simulation method. These points belong to the safety margin [Z.sub.k] = [[phi].sub.k][R.sub.0] - M, where [[phi].sub.k] is its degradation function with reference values 0.97, 0.92, 0.87 and 0.82; [R.sub.0] is the initial bending resistance and M is the bending moment. The means and variances of its independent variables are: [R.sub.0m] = 200 kNm, [[sigma].sup.2][R.sub.0] = 1600 [(kNm).sup.2] and [M.sub.m] = 60 kNm, [[sigma].sup.2] M = 36, 144, 576, 1296 [(kNm).sup.2].

[FIGURE 4 OMITTED]

When extreme action effects are caused by two independent loads and three safety margins (10), (11) and (12) are considered, the long-term survival probability of particular members as series stochastic systems may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where the ranked survival probabilities [P.sub.1] > [P.sub.2] > [P.sub.3] are calculated by Eq. (31) and the correlation factor [[rho].sub.3,21] = 0.5([[rho].sub.31] + [[rho].sub.32]).

The survival probability of members may be also introduced by the generalized reliability index

[beta] = [[PHI].sup.-1]{P(T [greater than or equal to] [t.sub.n])} (36)

where [PHI].sup.-1] (*) is the cumulative distribution function of the x standard normal distribution. The target reliability index [[beta].sub.min] of the structural members depends on their reliability classes by considering the human life, economic, social and environmental consequences of failure or malfunction [1, 15]. For persistent design situations, the values of [[beta].sub.min] are equal to 3.3, 3.8 and 4.3 for reliability classes RC1, RC2 and RC3 of structural members. The value of [[beta].sub.min] for particular members should be not less. However, for members of hyper static structures, it may be decreased to 1.64 [16].

According to TCPM, the total survival probabilities of structural members (beams, columns, plates, trusses) as series, parallel and mixed microsystems may be calculated by the equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

where [P.sub.3/par] is the greater value from the probabilities [P.sub.3] and [P.sub.par] by Eq. (38).

6. Numerical example

Consider the long-term survival probability and reliability index of deteriorating roof steel beams of a scrap metal shed exposed to atmosphere corrosion conditions induced by environmental cold, wet and dry actions (Fig. 5). The indicative design working life of beams is 25 year. The initiation degradation phase of beams [t.sub.in] = 0 and the degradation function of their bending resistance [phi](t) = 1 - 0.00375t.

[FIGURE 5 OMITTED]

The bending moments of beams [M.sub.G], [M.sub.Q] and [M.sub.S] are caused by permanent load G of steel roof structures and hanging crane crabs, variable loads Q and S of scrap metals and snow depth. The means and coefficients of variation of basic variables of a beam safety margins are: [R.sub.0m] = 363.5kNm, [delta][R.sub.0] = 0.08 ; [M.sub.gm] = 20.0 kNm, [delta][M.sub.g] = 0.10; [M.sub.qm] = 49.7 kNm, [delta][M.sub.q] = 0.20; [M.sub.sm] = 50.3 kNm, [delta][M.sub.s] = 0.5. The statistics of additional variables of beam safety margin are: [[theta].sub.Rm] = [[theta].sub.Mm] = 1.0, [[sigma].sup.2][[theta].sub.R] = 0.0025, [[sigma].sup.2][[theta].sub.M] = 0 .

The means and variances of the beam parameters are:

[([[theta].sub.R][R.sub.0]).sub.m] = 363.5 kNm, [[sigma].sup.2]([[theta].sub.R][R.sub.0]) = [(0.08x363.5).sup.2] + + 363.[5.sup.2] x 0.0025 = 1176.2 [(kNm).sup.2];

[([[theta].sub.M] [M.sub.g]).sub.m] = 20.0 kNm, [[sigma].sup.2]([[theta].sub.M][M.sub.g]) = [(0.10 x 20.0).sup.2] = = 4.0 [(kNm).sup.2];

[([[theta].sub.M][M.sub.s]).sub.m] = 49.7 kNm, [[sigma].sup.2]([[theta].sub.M][R.sub.q]) = [(0.20 x 49.7).sup.2] = = 98.8 [(kNm).sup.2];

[([[theta].sub.M][M.sub.s]).sub.m], 50.3 kNm, [[sigma].sup.2]([[theta].sub.M][M.sub.s] = [(0.50 x 50.3)2.sup.2] =

= 632.5 [(kNm).sup.2].

These parameters are described by normal (R and [M.sub.g]), lognormal ([M.sub.q]) and Gumbel ([M.sub.s]) probability distributions. The instantaneous and long-term survival probabilities are calculated by Eqs. (26) and (31) and the reliability index is defined by Eq. (36). Their decreases in time are presented in Fig. 6.

[FIGURE 6 OMITTED]

According to code recommendations [1], the minimum value for reliability index of beams is [[beta].sub.min] = 3.3. Therefore, their technical service life is equal to 17 years.

7. Conclusion

The prediction of time-dependent safety of deteriorating structures subjected to aggressive environmental conditions and recurrent extreme service and climate loads can be formulated and solved within unsophisticated probability-based approaches. It is expedient to base the analysis of survival probabilities and reliability indices of deteriorating particular members (sections, bars, connections) on the concept of random decreasing multicut sequences. The position of stochastically dependent cuts of these sequences is matched with extreme loading situations of structures.

The method of transformed conditional probabilities (TCPM) may be successfully introduced into the probability-based design of deteriorating particular and structural members in a simple and easy perceptible manner. This method help us predict the safety parameters of structural members (beams, columns, plate, trusses) as stochastical series, parallel and mixed microsystems.

A closer definition of technical service lives of deteriorating structural members allows us avoid unfounded premature replacements and unexpected damages.

The represented methodological formats on survival probability and technical service life prediction are in force for deteriorating structures subjected both to single and joint extreme loads.

Received June 18, 2009

Accepted August 07, 2009

References

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[7.] JCSS. Probabilistic Model Code: Part 1- Basis of design. -Joint Committee on Structural Safety, 2000.-65p.

[8.] Mori, Y., Nonaka, M. LRFD for assessment of deteriorating existing structures. -Structural Safety 23: 2001, p.297-313.

[9.] Zhong, W.Q. &Zhao, Y.G. Reliability bound estimation for R.C. structures under corrosive effects. -Collaboration and Harmonization in Creative Systems. -London, 2005, p.755-761.

[10.] Melchers, R.E. Probabilistic model for marine corrosion of steel for structural reliability assessment. -Journal of Structural Engineering, 2003, v.129(11), p.1484-1493.

[11.] Rosowsky, D., Ellingwood, B. Reliability of wood systems subjected to stochastic live loads. -Wood and Fiber Science, 1992, 24 (1), p.47-59.

[12.] Vrowenvelder, A. C. Developments towards full probabilistic design codes. -Structural safety, 2002, v.24,(2 4), p.417-432.

[13.] ISO 2394. General principles on reliability for structures. Switzerland, 1998.-73p.

[14.] Vaidogas, E.R., Juocevicius, V. Reliability of a timber structure exposed to fire: estimation using fragility function. -Mechanika. -Kaunas: Technologija, 2008, Nr.5(73), p.35-42.

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A. Kudzys, O. Lukoseviciene

A. Kudzys *, O. Lukoseviciene **

* Kaunas University of Technology, Tunelio 60, 44405 Kaunas, Lithuania, E-mail: asi@asi.lt

** Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: olukoseviciene@gmail.com

The target design life of deteriorating load-carrying structures and their components must be defined in an early design stage of buildings, construction works and technological equipments. The value of this life must serve as a basis for the selections of materials and structures. The target design life is related to destruction modes of materials and structural components and failure consequences. In any case, higher durability requirements are applied to members which routine or preventive maintenance and repairs require great efforts.

Failures and collapses in load-carrying structures can be caused not only by irresponsibility of gross human errors of designers or erectors but also by some conditionalities of recommendations and directions presented in design codes and standards. The Standards EN 1990 [1] in Europe and ASCE/SEI [2] in the USA require that load-carrying structures to be designed with appropriate degrees of reliability. These Standards 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 deteriorating structures by using deterministic methods and their universal factors for loads and material properties.

The reliability degree of deteriorating structures may be objectively defined only by fully probability-based concepts and models. Only probabilistic approaches may allow us explicitly predict uncertainties of analysis models of these structures. Besides, the probabilistic analysis of deteriorating members is indispensible in order to predict their destructions or failures and to avoid of economic and psychologic losses. However, the mathematical probabilistic formats used in long-term reliability prediction of structures are based on rather complicated considerations [3-6]. Thus, the engineering modeling of survival probabilities of structures subjected to aggressive environmental actions and extreme live and climate loads are still unsolved.

The main task of this paper is to present new methodological formats on probability-based safety predictions of deteriorating members exposed to permanent loads and recurrent single or joint extreme service and climate actions.

2. Resistances and safety margins of deteriorating members

Multicriteria failure modes and safety of structural members (beams, slabs, columns, joints) may be objectively assessed and predicted only knowing survival probabilities of particular members (normal or oblique sections, connections) for which the only possible failure mode exists. Predicted durability parameters for deteriorating structures depend on chemical diagnosis and the acceptable risk of serviceability failure associated with the damage levels and losses. Besides, the predictions of safety of deteriorating members and their systems will account for all extreme action combinations. In any case, it is expedient to divide the life cycle [t.sub.n] (Fig. 1) of deteriorating structures into the initiation, [t.sub.in], and propagation, [t.sub.pr], phases [7]. The length of initiation phase is a random variable depending on a feature of degradation process, an environment aggressiveness and 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 [t.sub.in] = 0 . The propagation phase is delayed for structures protected by coats.

[FIGURE 1 OMITTED]

The resistance of particular members in the propagation phase is treated as a nonstationary random process

R(t) = [phi](t)[R.sub.in] = [phi](t)[R.sub.0] (1)

where [R.sub.0] is the initial value of member resistance, [phi](t) denotes the degradation function depending on the rate of a resistance decrease induced by an artificial ageing and degradation of materials. This function for corrosion affected particular members may be presented in the form

[phi](t) = 1 - a[(t - [t.subin]).sup.b] (2)

where b defines nonlinearity of the deterioration function and a is degradation intensity factor. A shape of the deg radation function is close to linear (b [approximately equal to] 1) and parabolic (b [approximately equal to] 2) when corresponding degradation mechanisms are steel corrosion and aggressive environmental attacks [8, 9]. However, marine corrosion of steel structures is not linear function of time [10].

Action effects of structures are caused by permanent loads g, sustained [q.sub.s] (t) and extraordinary [q.sub.e] (t)= q(t)- [q.sub.s] (t) components of live loads s(t) and wind, surf or seismic actions w(t). The annual extreme sum of sustained and extraordinary live load effects [E.sub.q] (t) caused by [q.sub.s] (t) and [q.sub.e] (t) may be modeled as a rectangular pulse renewal process described by Type I (Gumbel) distribution of extreme values with the coefficient of variation [delta]q = 0.58 and mean [E.sub.qm] = 0.47[E.sub.qk], where [E.sub.qk] is its characteristic value [11].

It is proposed to model annual extreme snow and wind action effects by a Gumbel distribution with the mean values equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [E.sub.sk] and [E.sub.wk] are the characteristic (nominal) values of action effects and [k.sub.0.98] is the characteristic fractile factor of these distributions [7, 12]. The coefficients of variation of snow and wind extreme loads depend on the feature of geographical area and are equal to [delta]s = 0.3 - 0.7 and [delta]w = 0.2 - 0.5.

The durations of extreme floor and climate actions are: [d.sub.q] = 1 -14 days for merchant and 1-3 days for other buildings, [d.sub.s] = 14 - 28 days and [d.sub.w] = 8 -12 hours. Renewal rates of annual extreme actions are equal to [lambda] = 1 / year . Therefore, the recurrence number of two joint extreme actions during the design working life of structures, tn in years, may be calculated by the formulae

[n.sub.12] = [t.sub.n] ([d.sub.1] + [d.sub.2])[[lambda].sub.1] [[lambda].sub.2] (3)

where [[lambda].sub.1] = [[lambda].sub.2] = 1 / [t.sub.[lambda]] are the renewal rates of extreme loads. Thus, the recurrence numbers of extreme concurrent live or snow and wind loads during [t.sub.n] = 50 years period are quite actual to [n.sub.qw] = 0.2 - 2.0 and [n.sub.sw] = 2.0 - 4.0. The bivariate distribution function of two independent extreme action effects may be presented as their conventional joint distribution function with the mean [E.sub.12m] = [E.sub.1m] + [E.sub.2m] and the variance [[sigma].sup.2] [E.sub.12] = [[sigma].sup.2] [E.sub.1] + [[sigma].sup.2] [E.sub.2] [13].

According to probability-based approaches (design level III), the time-dependent safety margin of deteriorating particular members exposed to extreme action effects may be defined as their random performance process and presented as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where X(t) and [theta] are the vectors of basic and additional variables, representing respectively random components (resistances and action effects) and their model uncertainties; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [E.sub.2](t) = [E.sub.s] (t) or [E.sub.2] (t) = [E.sub.w] (t). The mean values and standard deviations of additional variables of the safety margin are: [[theta].sub.Rm] = 0.99 -1.04, [sigma][[theta].sub.R] = 0.05-0.10 and [[theta].sub.Em] [approximately equal to] 1.00, [sigma][[theta].sub.E] [approximately equal to] 1.00, [sigma][[theta].sub.Em] [approximately equal to] 0.10 [11, 12].

Gaussian and lognormal distribution laws is to be used for member resistances. The permanent actions can be described by a normal distribution law [13,14]. Therefore, for the sake of simplified but quite exact probabilistic analysis of deteriorating members, it is expedient to present Eq. (4) in the form

Z(t) = [R.sub.c] (t)-E (t) (5)

where

[R.sub.c] (t) = [[theta].sub.R]R(t) - [[theta].sub.g][E.sub.g] (6)

is the conventional resistance of members the bivariate probability distribution of which may be modeled by Gaussian distribution

E(t) = [[theta].sub.1][E.sub.1](t) + [[theta].sub.2] [E.sub.2] (t) (7)

is the conventional bivariate distribution process of two stochastically independent annual extreme effects [15].

Inspite of analysis simplifications, the use of continuous stochastic processes of member resistances considerably complicates the durability analysis of deteriorating structures exposed to intermittent extreme gravity and lateral variable actions along with permanent ones. The dangerous cuts of these processes correspond to extreme loading situations of structures. Therefore, in design practice the safety margin process Eq. (5) may be modeled as a random geometric distribution and treated as finite decreasing random sequence

[Z.sub.k] = [R.sub.ck] - [E.sub.k], k = 1,2,...,n - 1,n (8)

where

[R.sub.ck] = [[phi].sub.k][[theta].sub.R][R.sub.in] - [[theta].sub.g][E.sub.g] (9)

is the conventional resistance of deteriorating members at the cut k of this sequence (Fig. 1) and n is the recurrence number of single or coincident extreme action effects, [E.sub.k], given by Eq. i.e. [E.sub.k] = [[theta].sub.1][E.sub.1k] + [[theta].sub.2][E.sub.2k].

When extreme action effects are caused by two stochastically independent variable actions, a failure of members may occur not only in the case of their coincidence but also when the value of one out of two effects is extreme. Therefore, three stochastically dependent safety margins should be considered as follows

[Z.sub.1k] = [R.sub.ck] - [E.sub.1k] k = 1,2,...,n (10)

[Z.sub.2k] = [R.sub.ck] - [E.sub.2k] k = 1,2,...,n2 (11)

[Z.sub.3k] = [R.sub.ck] - [E.sub.11k] = [R.sub.ck] - [E.sub.1k] - [R.sub.2k], k = 1,2,...,[n.sub.12] (12)

where the number of sequence cuts n12 is calculated by Eq. (3).

3. Transformed conditional probability method

For particular and structural members of deteriorating structures subjected to extreme action effects, more than one limit state situations are considered. The number of these situations is equal to recurrence numbers [n.sub.1] or [n.sub.2] and [n.sub.12] of single and coincident extreme action effects, respectively.

The statistical dependences among failure probabilities of particular members at any time [t.sub.k] or any cut k

of rank sequence and their survival probabilities at previous extreme loading situations exist. Therefore, the instantaneous failure probability of these members at sequence cut k, assuming that they were safe at cuts [1,k-1 ], may be presented in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.k] denotes the failure event of members at cut k and [S.sub.i] denotes the event of their survival at previous cut i of a sequence. Therefore, the instantaneous failure probabilities of particular members at cuts 1,2,3,...,n of their safety margin sequences are: p([Z.sub.1] [less than or equal to] 0) = p([F.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The time dependent failure probabilities of deteriorating particular and individual members as autosystems during times [t.sub.1], [t.sub.2], [t.sub.3],...,[t.sub.n] may be expressed as

P(T < [t.sub.1]) = P([Z.sub.1] [less than or equal to] 0) = P([F.sub.1]) (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Thus, according to probabilistic approaches, the prediction of time dependent survival probabilities of load-carrying particular members may be based on the analysis of decreasing sequences of random safety margins (Fig. 2), i.e. can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

When the sequence consists of two dependent cuts, the probability that either or both of two failure events of a series system occur is expressed by Eq. (17). An the evaluation of the probability of a second order intersection of failure events [F.sub.2] and [F.sub.1], i.e. p([f.sub.2] [intersection] [F.sub.1]), may be carried out by rather uncomfortable for structural engineers methods of numerical integrations or Monte Carlo simulation. It is more expedient to use in design practice the unsophisticated method of transformed conditional probabilities (TCTM). According to its approaches, the intersection probability

p([F.sub.2][intersection][F.sub.1]) = P([F.sub.2])P([F.sub.1]|[F.sub.2]) (21)

where the conditional failure probability

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The indexed correlation factor of two sequence cuts, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], characterizes an effect of their statistical dependence on the intersection probability P([F.sub.2][intersection] [F.sub.1]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 2 OMITTED]

When sequence cuts are independent, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the conditional, intersection and failure probabilities of members from Eqs. (22), (21) and (17) are defined as: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When sequence cuts are fully correlated, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], these probabilities are: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is between 0 and 1, the intersection and failure probabilities by Eqs. (21) and (17) of two cut sequences become as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Analogically to Eq. (23), the probability of an intersection of three failure events may be presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where the correlation factor [[rho].sub.3,21] [approximately equal to] 0.5([[rho].sub.32] + [[rho].sub.31]). The correlation factor and its bounded index are considered in Section 5.

4. Instantaneous survival probability

The instantaneous survival probability of particular members with respect to their single failure mode at sequence cut k, if they were safe at cuts 1-k-1 i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], can be modelled using multidimensional integral as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For design convenience, the structural safety analysis of deteriorating members may be based on the limit state criteria [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and, [R.sub.ck] -([E.sub.1k] + [E.sub.2k])> 0 where [R.sub.ck] is defined from Eq. (6). Therefore, the instantaneous survival probability may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The conventional resistance [R.sub.ck] and single extreme action effect [E.sub.k] may be treated as statistically independent variables of random safety margins.

[FIGURE 3 OMITTED]

Therefore, the instantaneous survival probability of deteriorating members can be expressed by convolution integral as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (x) is the density function of resistance and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (x) is the cumulative distribution function of their action effect (Fig. 3).

5. Long-term survival probability

Decreasing resistance of particular members must be treated as a nonstationary process. Therefore, it is rather complicated to define the failure probability of multicut sequences in easy perceptible manner. However it is fairly simple to calculate the survival probability of deteriorating members by TCPM. According to Eq. (20), the survival probabilities of these members exposed to two, three and n extreme loading situations may be expressed as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

For the sake of simplified but fairly exact probability-based analysis of deteriorating structures, the conditional survival probability of higher order [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for particular members may be defined as the probability P([S.sub.k]|[S.sub.k-1]). Therefore, the component [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of Eq. (30) my be changed by the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then,

Eq. (30) may be rewritten in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where P([S.sub.1]), ..., P([S.sub.k-1]), ..., P([S.sub.n-1]) are the instantaneous survival probabilities of members by Eq. (27). The correlation factor of dependent sequence cuts, [[rho].sub.k,1...k-1], is formed from k th row of quadratic matrix of basic coefficients of correlation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It may be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

The coefficient of correlation of rank safety margin cuts is calculated from the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where Cov([Z.sub.k], [Z.usb.l]) and [sigma][Z.sub.k], [sigma][Z.sub.l] are an auto covariance and standard deviations of safety margin values.

Then long term survival probabilities of members are calculated by Eq. (31), the bounded index, [x.sub.k], of correlation factors of random multicut sequences may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

For highly reliable load-carrying members, the instantaneous survival probability P([S.sub.k])[approximately equal to]1 and its effect on bounded indices may be ignored.

The acceptability of this index in design practice is corroborated by Fig. 4, where the position of points for decreasing sequences with two, three and four cuts is calculated by Monte Carlo simulation method. These points belong to the safety margin [Z.sub.k] = [[phi].sub.k][R.sub.0] - M, where [[phi].sub.k] is its degradation function with reference values 0.97, 0.92, 0.87 and 0.82; [R.sub.0] is the initial bending resistance and M is the bending moment. The means and variances of its independent variables are: [R.sub.0m] = 200 kNm, [[sigma].sup.2][R.sub.0] = 1600 [(kNm).sup.2] and [M.sub.m] = 60 kNm, [[sigma].sup.2] M = 36, 144, 576, 1296 [(kNm).sup.2].

[FIGURE 4 OMITTED]

When extreme action effects are caused by two independent loads and three safety margins (10), (11) and (12) are considered, the long-term survival probability of particular members as series stochastic systems may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where the ranked survival probabilities [P.sub.1] > [P.sub.2] > [P.sub.3] are calculated by Eq. (31) and the correlation factor [[rho].sub.3,21] = 0.5([[rho].sub.31] + [[rho].sub.32]).

The survival probability of members may be also introduced by the generalized reliability index

[beta] = [[PHI].sup.-1]{P(T [greater than or equal to] [t.sub.n])} (36)

where [PHI].sup.-1] (*) is the cumulative distribution function of the x standard normal distribution. The target reliability index [[beta].sub.min] of the structural members depends on their reliability classes by considering the human life, economic, social and environmental consequences of failure or malfunction [1, 15]. For persistent design situations, the values of [[beta].sub.min] are equal to 3.3, 3.8 and 4.3 for reliability classes RC1, RC2 and RC3 of structural members. The value of [[beta].sub.min] for particular members should be not less. However, for members of hyper static structures, it may be decreased to 1.64 [16].

According to TCPM, the total survival probabilities of structural members (beams, columns, plates, trusses) as series, parallel and mixed microsystems may be calculated by the equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

where [P.sub.3/par] is the greater value from the probabilities [P.sub.3] and [P.sub.par] by Eq. (38).

6. Numerical example

Consider the long-term survival probability and reliability index of deteriorating roof steel beams of a scrap metal shed exposed to atmosphere corrosion conditions induced by environmental cold, wet and dry actions (Fig. 5). The indicative design working life of beams is 25 year. The initiation degradation phase of beams [t.sub.in] = 0 and the degradation function of their bending resistance [phi](t) = 1 - 0.00375t.

[FIGURE 5 OMITTED]

The bending moments of beams [M.sub.G], [M.sub.Q] and [M.sub.S] are caused by permanent load G of steel roof structures and hanging crane crabs, variable loads Q and S of scrap metals and snow depth. The means and coefficients of variation of basic variables of a beam safety margins are: [R.sub.0m] = 363.5kNm, [delta][R.sub.0] = 0.08 ; [M.sub.gm] = 20.0 kNm, [delta][M.sub.g] = 0.10; [M.sub.qm] = 49.7 kNm, [delta][M.sub.q] = 0.20; [M.sub.sm] = 50.3 kNm, [delta][M.sub.s] = 0.5. The statistics of additional variables of beam safety margin are: [[theta].sub.Rm] = [[theta].sub.Mm] = 1.0, [[sigma].sup.2][[theta].sub.R] = 0.0025, [[sigma].sup.2][[theta].sub.M] = 0 .

The means and variances of the beam parameters are:

[([[theta].sub.R][R.sub.0]).sub.m] = 363.5 kNm, [[sigma].sup.2]([[theta].sub.R][R.sub.0]) = [(0.08x363.5).sup.2] + + 363.[5.sup.2] x 0.0025 = 1176.2 [(kNm).sup.2];

[([[theta].sub.M] [M.sub.g]).sub.m] = 20.0 kNm, [[sigma].sup.2]([[theta].sub.M][M.sub.g]) = [(0.10 x 20.0).sup.2] = = 4.0 [(kNm).sup.2];

[([[theta].sub.M][M.sub.s]).sub.m] = 49.7 kNm, [[sigma].sup.2]([[theta].sub.M][R.sub.q]) = [(0.20 x 49.7).sup.2] = = 98.8 [(kNm).sup.2];

[([[theta].sub.M][M.sub.s]).sub.m], 50.3 kNm, [[sigma].sup.2]([[theta].sub.M][M.sub.s] = [(0.50 x 50.3)2.sup.2] =

= 632.5 [(kNm).sup.2].

These parameters are described by normal (R and [M.sub.g]), lognormal ([M.sub.q]) and Gumbel ([M.sub.s]) probability distributions. The instantaneous and long-term survival probabilities are calculated by Eqs. (26) and (31) and the reliability index is defined by Eq. (36). Their decreases in time are presented in Fig. 6.

[FIGURE 6 OMITTED]

According to code recommendations [1], the minimum value for reliability index of beams is [[beta].sub.min] = 3.3. Therefore, their technical service life is equal to 17 years.

7. Conclusion

The prediction of time-dependent safety of deteriorating structures subjected to aggressive environmental conditions and recurrent extreme service and climate loads can be formulated and solved within unsophisticated probability-based approaches. It is expedient to base the analysis of survival probabilities and reliability indices of deteriorating particular members (sections, bars, connections) on the concept of random decreasing multicut sequences. The position of stochastically dependent cuts of these sequences is matched with extreme loading situations of structures.

The method of transformed conditional probabilities (TCPM) may be successfully introduced into the probability-based design of deteriorating particular and structural members in a simple and easy perceptible manner. This method help us predict the safety parameters of structural members (beams, columns, plate, trusses) as stochastical series, parallel and mixed microsystems.

A closer definition of technical service lives of deteriorating structural members allows us avoid unfounded premature replacements and unexpected damages.

The represented methodological formats on survival probability and technical service life prediction are in force for deteriorating structures subjected both to single and joint extreme loads.

Received June 18, 2009

Accepted August 07, 2009

References

[1.] EN 1990. Eurocode - Basic of structural design. CEN, Brussels, 2002.-87p.

[2.] ASCE/SEI 7-05. Minimum Design Loads for Buildings and Other Structures, 2005.-388p.

[3.] Rackwitz, R. Risk acceptance and optimization of aging but maintained civil engineering infrastructures. -Safety and Reliability for Managing Risk, 2006, p.1527-1534.

[4.] Noortwijk, J.M., Kallen, M.J., Pandey, M.D. Gamma processes for time-dependent reliability of structures. -Advances in Safety and Reliability, 2005, p.14571464.

[5.] Joanni, A.E., Rackwitz, R. Stochastic dependencies in inspection, repair and failure models. -Safety and Reliability for Managing Risk, 2006, p.531-537.

[6.] Kuniewski, S.P., van Noortwijk, J.M. Sampling inspection for the evaluation of time-dependent reliability of deteriorating structures. -Risk, Reliability and Societal Safety, 2007, p.281-288.

[7.] JCSS. Probabilistic Model Code: Part 1- Basis of design. -Joint Committee on Structural Safety, 2000.-65p.

[8.] Mori, Y., Nonaka, M. LRFD for assessment of deteriorating existing structures. -Structural Safety 23: 2001, p.297-313.

[9.] Zhong, W.Q. &Zhao, Y.G. Reliability bound estimation for R.C. structures under corrosive effects. -Collaboration and Harmonization in Creative Systems. -London, 2005, p.755-761.

[10.] Melchers, R.E. Probabilistic model for marine corrosion of steel for structural reliability assessment. -Journal of Structural Engineering, 2003, v.129(11), p.1484-1493.

[11.] Rosowsky, D., Ellingwood, B. Reliability of wood systems subjected to stochastic live loads. -Wood and Fiber Science, 1992, 24 (1), p.47-59.

[12.] Vrowenvelder, A. C. Developments towards full probabilistic design codes. -Structural safety, 2002, v.24,(2 4), p.417-432.

[13.] ISO 2394. General principles on reliability for structures. Switzerland, 1998.-73p.

[14.] Vaidogas, E.R., Juocevicius, V. Reliability of a timber structure exposed to fire: estimation using fragility function. -Mechanika. -Kaunas: Technologija, 2008, Nr.5(73), p.35-42.

[15.] Kudzys, A. Survival probability of existing structures. -Mechanika. -Kaunas: Technologija, 2005, Nr.2(52), p.42-46.

[16.] Jankovski, V., Atkociunas J. Matlab implementation in direct probability design of optimal steel trusses. -Mechanika. -Kaunas: Technologija, 2008, Nr.6(74), p.30-37.

A. Kudzys, O. Lukoseviciene

A. Kudzys *, O. Lukoseviciene **

* Kaunas University of Technology, Tunelio 60, 44405 Kaunas, Lithuania, E-mail: asi@asi.lt

** Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: olukoseviciene@gmail.com

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Author: | Kudzys, A.; Lukoseviciene, O. |
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Publication: | Mechanika |

Article Type: | Report |

Geographic Code: | 4EXLT |

Date: | Jul 1, 2009 |

Words: | 4218 |

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