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6
[p j ,P 2,P 3 ] = [p 2,P 3 ]
P j +P 2 +P 3 = 1
0 0.357 0.643 0.8 0 0.2 0.385 0.615 0
we get
pj = 0.374, p2 = 0.321, p3 = 0.305,
then the limit values of the transient probabilities pb (t) at the operation states zb, according to (9), are given by
px = 0.684, p2 = 0.183, p3 = 0.133.
(16)
The subsystem S4 consists of three chain conveyors. Two of these are composed of 162 components and the remaining one is composed of 242 components. Thus it is a non-regular series-parallel system. In order to make it a regular system we conventionally complete two first conveyors having 162 components with 80 components that do not fail. After this supplement subsystem S4 consists of kn =3 conveyors, each composed of ln = 242 components. In two of them there are:
- two driving wheels with reliability functions R(u)(t) = exp[-0.0798t],
- 160 links with reliability functions R°,2)(0 = exp[-0.124t],
- 80 components with "reliability functions" R(1,3)(t) = exp[ -l (1) t], where \ (1) = 0.
The third conveyer is composed of:
- two driving wheels with reliability functions R(21)(t) = exp[-0.167t]
- 240 links with reliability functions
R(2'2)(t) = exp[-0.208t].
At the operation state zx the subsystem S4 becomes a non-homogeneous regular series-parallel system with parameters
kn = 3, ln = 242, a = 2, qx = 2/3, q2 = 1/3,
e = 3, e2 = 2,
pii = 2/242, pi2 = 160/242, pB = 80/242,
P21 = 2/242, p22 = 240/242,
and from (10)-(11) the reliability function of this system is given by
[R3,242 (t)](1)
= 1 - [1 - exp[-19.9892t]]2[1 - exp[-50.2628t]] = 2exp[-19.9892t] - 2exp[-70.252t] + exp[-50.2628t ] + exp[-90.2412t ]
- exp[-39.9784t] for t > 0.
(17)
According to (14)-(15), the subsystem lifetime mean value and the standard deviation are
Mx @ 0.078, a j @ 0.054.
(18)
At the operation state z2 the subsystem S4 becomes a non-homogeneous regular series-parallel system with parameters
kn = 2, ln = 162, a = 1, qj = 1, ej = 2,
pii = 2/162, p12 = 160/162.
and from (10)-(11) the reliability function of this system is given by
[R2,i62 (t)](2)= 1 - [1 - exp[-20.007t]]2
= 2exp[-20.007t] - exp[-40.014t] for t > 0. (19)
According to (14)-(15), the subsystem lifetime mean value and the standard deviation are
M2 @ 0.075, a 2 @ 0.056.
(20)
At the operation state z3 the subsystem S4 becomes a
non-homogeneous regular series-parallel (series) system with parameters
kn = 1, ln = 162, qi = 1, ei = 3,
pii = 2/162, pi2 = 160/162,
and from (10)-(11) the reliability function of this system is given by
[fl (t)](3) = exp[-19.999t] for t > 0. (21)
According to (14)-(15), the system lifetime mean value and the standard deviation are
M3 @ 0.050, c 3 @ 0.050.
(22)
Finally, considering (12), the subsystem S4 unconditional reliability is given by
R(t) @ 0.684 • [^3,242 (t)](1) + 0.183 • [^2,162(t)](2)
+ 0.133 • [ *16a(t )](3),
(23)
where [R3,242(t)](1), fl2,162(t)](2), [*U(a (t)](3), are given by (17), (19), (21).
Hence, applying (16) and (18), (20), (22), we get the mean values and standard deviations of the subsystem unconditional lifetimes given by
M @ 0.684• 0.078 + 0.183• 0.075
+ 0.133-0.050 @ 0.074,
c (1) @ 0.054. References
(24)
(25)
[7] Kwiatuszewska-Sarnecka, B. (2003). Analiza niezawodnosciowa efektywnosci rezerwowania w systemach szeregowych. PhD Thesis. Maritime Academy, Gdynia - Systems Research Institute, Polish Academy of Sciences, Warsaw.
[8] Kwiatuszewska-Sarnecka B. (2006). Reliability improvement of large multi-state parallel-series systems. International Journal of Automation & Computing. 2, 157-164.
[9] Kwiatuszewska-Sarnecka, B. (2006). Reliability improvement of large parallel-series systems. International Journal of Materials & Structural Reliability. Vol. 4, No 2.
[10] Milczek, B. (2004). Niezawodnosc duzych systemow szeregowo progowych k z n. PhD Thesis. Maritime Academy, Gdynia - Systems Research Institute, Polish Academy of Sciences, Warsaw.
[11] Soszynska, J. (2006). Reliability of large seriesparallel system in variable operation conditions. International Journal of Automation & Computing. 2, 3, 199-206.
[12] Soszynska, J. (2006). Reliability evaluation of a port oil transportation system in variable operation conditions. International Journal of Pressure Vessels and Piping. 4, 83, 304-310.
[13] Soszynska, J. (2007). Analiza niezawodnosciowa systemow w zmiennych warunkach eksploatacyjnych. PhD Thesis. Maritime Academy, Gdynia - Systems Research Institute, Polish Academy of Sciences, Warsaw.
[1] Grabski, F. (2002). Semi-Markov Models of Systems Reliability and Operations. Warsaw: Systems Research Institute, Polish Academy of Sciences.
[2] Kolowrocki, K. (2004). Reliability of Large Systems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.
[3] Kolowrocki, K. & Soszynska, J. (2005). Reliability and availability of complex systems. Quality & Reliability Engineering International. 1, 22, J. Wiley & Sons Ltd., 79-99.
[4] Kolowrocki, K. (2006). Reliability and risk evaluation of complex systems in their operation processes. International Journal of Materials & Structural Reliability. Vol. 4, No 2.
[5] Kolowrocki, K. (2007). Reliability and safety modelling of complex systems - Part 1. Proc. Safety and Reliability Seminars - SSARS 2007. Sopot.
[6] Kolowrocki, K. (2007). Reliability of Large Systems, in Encyclopedia of Quantitative Risk Assessment. Everrit, R. & Melnick, E. (eds.), John Wiley & Sons, Ltd., (to appear).
Kudzys Antanas
KTU Institute of Architecture and Construction, Kaunas, Lithuania
Transformed conditional probabilities in the analysis of stochastic sequences
Keywords
safety margin, stochastic sequence, conditional probability, system safety Abstract
The need to use unsophisticated probability-based approaches and models in the structural safety analysis of the structures subjected to annual extreme service, snow and wind actions is discussed. Statistical parameters of single and coincident two extreme variable actions and their effects are analysed. Monotone and decreasing random sequences of safety margins of not deteriorating and deteriorating members are treated, respectively, as ordinary and generalized geometric distributions representing highly-correlated series systems. An analytical analysis of the failure or survival probabilities of members and their systems is based on the concepts of transformed conditional probabilities of safety margin sequences whose statistically dependent cuts coincide with extreme loading situations of structures. The probability-based design of members exposed to coincident extreme actions is illustrated by a numerical example.
1. Introduction
The stochastic systems and their subsystems consist of some particular members representing the only possible failure mode. To particular members belong cross and oblique sections of tension, compression, flexural and torsional structures. The structural members (beams, slabs, columns, walls) of buildings consist of two or three design particular members and may be treated as auto systems representing multicriteria failure modes. An overloading of members during severe service and climate actions may provoke a failure of structures. Therefore, the requirements of design codes should be satisfied at all sections along structural members. Structural failures and collapses in buildings and construction works can be caused not only by irresponsibility and gross human errors of designers, builders or erectors but also by some conditionalities of recommendations and directions presented in design codes and standards. A possibility to ensure objectively the safety degree of structures subjected to extreme service loads, wind gust and snow pressures or wave surfs is hardly translated into reality using the traditional deterministic design methods of partial safety factors in Europe or load and resistance factors in the USA. It is understandable that probabilistic design approaches are inevitable for the calibration of partial factors. However, it should be more expedient to
analyse the structural safety of particular members and their systems by probability-based methods. Regardless of efforts to improve and modify deterministic design approaches, it is inconceivable to fix a real reliability index of structures a failure domain of which changes with time. The time-dependent safety assessment and prediction of deteriorating members and systems using unsophisticated methods is a significant concern of researchers. Despite of fairly developed up-to-date concepts of reliability, hazard and risk theories, including the general principles on reliability for structures [6], [7], [15], it is difficult to apply probability-based approaches in structural safety analysis. These approaches may be acceptable to designers and building engineers only under the indispensable condition that the safety performance of members and their systems may be considered in a simple and easy perceptible manner. In other words, probabilistic methods may be implanted into structural design practice only using unsophisticated mathematical models helping us to assess all uncertainties due to the features of resistances and action effects of structures. This paper deals with probability-based safety analysis of deteriorating and not deteriorating members and their systems under extreme gravity and lateral (horizontal) actions using unsophisticated but fairly exact design models.
2. Time dependent safety margin
According to probability-based approaches (design level III), the time-dependent safety margin as the performance of deteriorating particular members may be presented as follows:
Z(t) = g[h,X(t)] = 0RR(t) - 0 gSg - 0_ Sq. (t) -
-0 q2 Sq2(t )-0 wSw (t ) ,
(1)
where h is the vector of additional variables characterizing uncertainties of models which give the values of resistance R, permanent Sg , sustained Sqi
and extraordinary Sq^ service and extreme wind Sw
action effects of members (Figure 1, a). This vector may represent also the uncertainties of probability distributions of basic variables.
According to Rosowsky and Ellingwood [11], the annual extreme sum of sustained and extraordinary occupancy live action effects Sq (t) = S (t) + Sq^ (t)
can be modelled as an intermittent process and described by a Type 1 (Gumbel) distribution with the coefficient of variation %Sq = 0.58, characteristic Sqk
and mean Sqm = 0.47Sqk values. Latter on Ellingwood
and Tekie [4] recommended modelling extreme values of this sum during a 50 years period by a Type 1 distribution with the coefficient of variation
aSq = 0.25 and mean value Sqm = S
It is proposed to model the annual extreme climate (wind and snow) action effects by Gumbel distribution law with the mean values equal to
Swm = Swkl(l + k0.98^Sw ) and Ssm = Ssk/(l+k0.98^Ss )
[3, 6, 7, 13, 15]. According to meteorological data, the strong wind conditions are characterized by a small wind extreme velocity variation, i.e. rv » 0.1. On the contrary, a large variation is characteristic of strong snow loading. Therefore, the coefficients of variation of wind and snow loads depending on the feature of a geographical area are equal to aw = 0.2 - 0.4 and as = 0.3 - 0.7.
Probability distributions of material properties are close to a Gaussian distribution [3], [6], [9], [12]. Therefore, a normal distribution or a log-normal distribution may be convenient in resistance analysis models [5], [6], [7]. The permanent action effect Sg
can be described by a normal distribution law [4], [5], [6], [10], [12]. Thus, for the sake of design simplifications, it is expedient to present the expression (1) in the form:
Z (t) = Rc (t) - S (t), where the component process Rc(t) = 0RR(t) -0gSg ,
(2)
(3)
>qk ■
may be considered as the conventional resistance of members which may be modelled by a normal distribution;
a)
S, R
di
JLjl
t t
■2t
R(t)
11
.J
UL n II H
b)
Figure 1. Real (a) and conventional (b) models for safety analysis of particular members (sections) of deteriorating structures
S (t) = 0 qSq (t ) + [0 WSW (t )], (4) S (t) = 0 sSs (t ) + [0 WSW (t )], (5)
S
q
S
g
S
g
5555555555555555555555555555555
are the joint processes of two annual extreme action effects when floor and roof structures, respectively, are under consideration. The components in square brackets belonging to the wind action effect are used in design analysis of wind-resistant members and systems. The action effect Ss (t) in Equation (5) is caused by extreme snow loads.
3. Safety margin sequences with independent cuts
The data presented in Section 2 allow us to model extreme service and climate action effects as intermittent rectangular pulse renewal processes. These time-variant intermittent action effects belong to persistent design situations in spite of the short period of extreme events being much shorter than the design working life of structures. When variable action effects may be treated as rectangular pulse processes, the time-dependent safety margin (2) may be expressed as the finite rank random sequence and written as:
Zk - Rck Sk ■
There
k - 1, 2, 3,
n -1, n .
Rck RRk -qgSg-
(6)
(7)
instantaneous survival probabilities of members Pf 1 < Pf 2 < ... < Pfk < ...Pf,n-1 < Pn calculated by Equation (10).
P
s1
-f^- Z2 —Zn-1— Zn
Pf
-2
Pf2
P
fk
Ps, n- > P 1 sn
Zn-1 Zn
Pf, n 1 Pfn
Figure 2. The scheme of series systems
Failure probabilities of structures should always be defined for some reference period tn or as a number of extreme events n during this period. The scheme of series systems representing the safety margin sequences is given in Figure 2. When the cuts of rank random sequences are statistically independent, the cumulative distribution function and similarly a failure probability of members during their service life [0, tn ] with n extreme situations may be presented as follows:
Pf - Fn (n) - P{N < n}= Ë (1 - Psk )HPs
k-1
Sk = qqSqk wSwk or Sk = qsSsk + qwS
w wk
(8)
are the components of this non-stationary sequence; n - 1tn is the number of sequence cuts as critical events (situations) during design working life tn of members (Figure 1, b), where l - 1/1x is a mean renewal rate of these events per unit time when their return period is tl .
Usually the components Rck and Sk are stochastically independent. The instantaneous survival probability of a member at k-th extreme situation (assuming that it was safe at the situations 1, 2, ..., k -1) is:
k-1
- 1 - Ps1 +(1 - Ps2 P + .. + (1 - Psk )n Ps
i-1
, ,n-1 n
... + (1 - Psn )n Pi - 1 -n Psk .
i-1
k-1
(11)
When the resistance R(t) is a time-invariant function and treated as a stationary process, the instantaneous survival probability Psk by (9) is characterized by the same value for all cuts of the monotone sequence. In this case, Equation (11) becomes a cumulative distribution function of an ordinary geometric distribution as follows:
¥
Psk - P{Rck > Sk } - J fRck (x)Fsk (x)dx,
(9)
Pf - Fn (n) - P{N < n}- 1 -(1 - Pfk ) n .
(12)
where fR
(x) and FS ( x) are the density and
Rck^' aiiyx £'Sk
distribution functions of a conventional resistance Rck by (7) and an extreme action effect Sk by (8). In this case, the instantaneous failure probability of members may be presented as:
k-1
Pfk =(1 - Psk )n. (1°)
i=1
Thus, the random sequence of safety margins may be treated as a geometric distribution with ranked
The failure probability of members may be approximated by Equations (11) and (12) only for
situations in which a variance of the action effect y 2 S is much larger than the value y2 Rc for their conventional resistance by (7). 4. Safety margin sequences with dependent cuts
In design practice, only recurrent extreme action effects caused by extraordinary service and climate loads may be treated as stochastically independent variables. Usually, random sequence cuts of the safety margin (6) are dependent. The value of a coefficient of
autocorrelation pkl of sequence cuts depends on uncertainties of material properties and dimensions of members. This coefficient may be defined as:
Pki = p(zk,zi) = Cov(zk,zi)/(yZk xyzi), (13)
where Cov(Zi-,Z/) and yZ^yZ, are an autocovariance and standard deviations of the random safety margins Zk and Z,.
The finite random sequence of member safety margins may be treated as a series stochastic system. The survival probability of highly correlated series systems consisting of two dependent elements can be expressed as follows:
p{zi > 0 • Z2 > 0}= Psl X i>{z2 > 0|Zj > o}
(14)
where a«4.5/(l - 0.98p12) is the bond index of survival probabilities of second-order series systems. The data calculated by (14) and computed by the complex numerical integration method presented by Ahammed and Melchers [1] are very close. Thus, a conditional probability p\z2 > 0\Zl > o} may be
transformed to a probability P
s 2
1 + p" --1
si
Therefore, Equation (14) may be presented in the form: P{ZX>Q- Z2>0}«PilxPi2
i I „ö
Pl2
V^l
-1
(15)
For not deteriorating structures, a member resistance is a time-invariant fixed random function the numerical values of which are random only at the beginning of a process. Therefore, the coefficient of correlation (13) of monotone sequence cuts may be expressed as:
(16)
When the monotone rank sequence of safety margins consists of n dependent elements, a failure probability of members is:
Pf =P{N< n} = P\ Zk < 0 = 1 -Pi Zk > 0
A=1
A=1
1 P.
sk
i + Pi
p
V sk
— 1
(17)
When a ratio of variances y2Sk/y2Rc>\, the coefficient p":/ «0 and the failure probability (17) becomes Pf = 1 - (l - P^ ) " as it is expressed by Equation (12).
A long-term survival probability of not deteriorating members is:
Ps=\-Pf=n
■P M
P
V sk
-1
n-1
(18)
The decreasing rank sequence of safety margins of deteriorating members may be treated as a generalized geometric distribution. Similar to Equation (17), the failure probability of these members as series systems may be calculated by the formula:
J^PIa^h}*!-]!^
k=1
sk
l + p
H...1
1
p
V .VJ? 1
-1
l+p
A-...1
P
V s,k-\
-1
1 I CI
P21
V^i
-1
(19)
where the transformed rank coefficient of correlation is P*...i = (p/U-1 +Pk,k-2 +- + Pi-2 +Pki)/(k-l) (2°)
The long-term survival probability of deteriorating members P =\-Pf, where the probability Pf is given in (19).
The presented method of transformed conditional probabilities may also be successfully used in the reliability analysis of random systems consisting of individual components and characterizing different failure modes of structures. In this case, it is expedient to base the structural safety analysis of systems on the ranked survival probabilities of their members as: >P2 ... ... >PStn_x >Psn {Figure 2). A
rank correlation matrix of systems is constructed taking into account this analysis rule.
4. The system of safety margin sequences
Due to the complexity of mathematical models, it is rather difficult to assess and predict a failure probability of structures subjected to two and more coincident recurrent and different by nature extraordinary actions. The methods based on the Markov-chain model and Turkstra's rule [14] may be quite unacceptable in a probabilistic analysis of not only deteriorating but also not deteriorating members and their systems. The Markov-chain model may be quite inaccurate for reliability analysis of members exposed to multiple combination of action-effect processes [12]. The Turkstra's rule may be assumed only in the case when the principal extreme load is strongly dominant [10]. Failure probabilities of members may be computed by modified numerical integration methods. It is suggested to use the theoretical expression of the cumulative distribution function of the maximum intensity of two load processes [10], the load overlap method [12] and the improved upper bounding techniques [13]. It leads to sufficiently accurate values but it is hard to realize these recommendations in engineering practice. The need to simplify a reliability analysis of deteriorating structures is especially urgent. In any analysis case, it must be taken into account that a member failure caused by two statistically independent extreme action effects 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 finite random sequences of safety margins should be considered:
M 1k = Rck - S1k, k = 1 2 ..., «1 =
M2k = Rck - S2k , k = 1 2= ...= «2
M3k = Rck - S3k , k = 1 2= ...= «3.
(21)
(22)
(23)
There S3k = S1k + S2k is the joint action effect, the recurrence number of which during the period of time [0, tn ] may be calculated by the equation:
n3 = tn (d 1 + d 2 ) I1I
2! 12 *
(24)
where d1 , d2 and l1 , l 2 are durations and renewal rates of extreme actions [8].
Mostly, the duration dq of annual extreme gravity
service loads is from 1 to 3 days. The durations of annual extreme snow and wind loads, respectively, are: ds = 1428 days and dw = 8-12 hours. The renewal rates of these actions are: lq = ls = lw = 1/year. Therefore, for 50
years reference period, the recurrence numbers of extreme actions are: nqw = 0.2-0.5 and nsw = 2-4.
When probability distributions of random variables X and Y obey a Gumbel distribution law, the bivariate density function of the random variable Z = X + Y may be presented in the form:
¥
fz (z) = J fx (z - y, Xm - 0.45yX)
— ¥
X fy (y,Ym - 0.45yY)dy, (25)
where Xm, Ym and yX, y Y are means and standard deviations of these variables.
Taking into account that y 2Z = y 2X + y 2Y is the variance of bivariate probability distribution, the joint density function may be expressed as:
fZ (z) » fZ az ),
(26)
Xm + Ym -0.19(yX + yY)-0.45(y2X + y2y)1/2.
fz (z) 0.6 0.5 0.4 0.3 0.2 0.1 0
5 6 7 8 9 10 11 12 13 14 15 16 z=x+y
Figure 3. Bivariate density functions calculated by Equations (25) - 1 and (26) - 2: the coefficients of correlation nX = nY = 0.10 (a) and 0.224 (b)
The probability density curves of joint extreme variable Z = X + Y are given in Figure 3. It is not difficult to ascertain that the difference between the values computed by Equations (25) and (26) is fairly small. Besides, the upper tails of both density curves coincides. Therefore, in design practice it is expedient to use the conventional bivariate distribution function of two independent extreme action effects with the mean S3k ,m = S1k ,m + S 2k ,m and the variance
y 2 S3k = y 2 S1k + y 2 S2k .
6. Numerical example
The knee-joints of not deteriorating concrete frames of reliability class RC2 are under exposure of shear forces during 50 years period (Figure 4). The shear resistance of knee-joints is expressed as: R = 0.068bhfc. The
az =
characteristic, design and mean values of the concrete compressive strength and shear resistance of knee-joints are:
fk = 30 MPa, fcd = 20 MPa, fCm = 38 MPa;
Rk = 306 kN, Rd = 204 kN, Rm = 387.6 kN.
The variance of shear resistance of knee-joints is: y 2 R = (0.128 x 387.6)2 = 2461.4 (kN)2.
Vg
Vs Vw
1 - 1
b = 0.3 m
f = °'5 m|
Figure 4. The knee-joint of concrete frames
The characteristic and design values of shear forces caused by permanent, snow and wind loads are:
Vgk = 77.72 kN,
Vsk = Vwk = 38.86 kN;
Vgd = 77.72 x 1.35 = 104.92 kN,
Vsd = 38.86 x 1.5 = 58.29 kN,
Vwd = 38.86 x 0.7 x 1.5 = 40.8 kN.
Thus, the joint design shear force
Vd = Vgd + Vsd + Vwd = 204 kN = Rd .
Therefore, according to deterministic calculation data, the frame knee-joints are reliable. The coefficients of variation, means and variances of these extreme shear forces are:
aVg = 0.1,
y 2Vg = 60.4 (kN)2; aVs = 0.6,
Vsm = Vsk/(1 + k 0.98flVs )= 15.21 kN, y 2VS = 83.25 (kN)2; V = 0.3,
Vwm = Vwk/ (1 + k 0.98flVw ) = 21.86 kN;
y 2Vw = 43.0 (kN)2.
The parameters of additional variables are:
q Rm = 1.0, = 0.1;
q Vm = 1.0,
g = s = w = 0.1,
^e sw = 0.15.
Thus, the variances of revised shear forces are:
y2 (e gVg )=120.8 (kN)2,
y 2 (e SVS ) = 85.56 (kN)2,
y2 (e wVw )=47.8 (kN)2,
y 2 (e swVsw )= 157.17 (kN)2.
Vgm = Vgk = 77.72 kN,
The parameters of conventional shear resistance (3) are:
Rcm = 387.6 - 77.72 =309.9 kN, y 2Rc = 1.0 x 2461.4 + 387.62 x 0.01
+ 120.8 = 4084.6 (kN)2.
According to (16), the coefficients of autocorrelation of the safety margins Zw = Rc - Vw, Zs = Rc - Vs and Zsw = Rc - Vs - Vw of considered knee-joints are:
p w,k/ = 0.9884,
1
