Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TU-Delf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary in ime. This means ha he probabilisic analysis does no include random variables only, as in he sandard case, bu also random funcions of ime, usually referred o as random processes. As a resul he failure probabiliy is no longer jus a single number, bu also a funcion of he ime (see figure 1). In general one should always keep in mind ha jus menioning a value for he failure probabiliy does no make any sense wihou specifying he period of ime for which is was derived. For he sake of compleeness we menion ha a complee probabilisic analysis also includes variabiliy in space. These random funcions of spaial coordinaes are usually referred o as random fields. These, however, will no be considered in his paper. The wo main classes of random processes are he saionary and insaionary processes. The basic feaure of saionary processes is ha heir saisical properies (mean, sandard deviaion and so on) do no change wih ime. Saionary processes may be subdivided ino ergodic and nonergodic ones. Ergodic means ha all saisical properies can be inferred from a single observaion, supposing i is long enough. For a more exensive reamen of hese imporan noions, reference is made o he relevan ex books. R R S S ime ime P F P F ime ime Figure 1: Time varian an ime invarian reliabiliy problems
The core of ime independen reliabiliy analysis is he limi sae formulaion (see he paper by Michael Faber). The same holds for he ime dependen case. In secion 2 of his paper we shall discuss he formulaions for he failure probabiliy on he basis of limi sae funcions for various ypes of ime dependen problems. Firs load and resisance ime dependen models will be reaed separaely. Afer ha we will consider he case ha resisance and loads boh are changing wih ime, bu ineracion is absen. Finally also he case of ineracion beween load and resisance will be considered. In chaper 3 we will discuss he echniques of how o calculae he failure probabiliies as defined in secion 2. We will also consider relaed noions like life ime disribuion and failure raes. Finally in secion 4 he effecs of inspecion and repair will be shorly reviewed. The models and echniques in his paper can be considered as a summary of he models and echniques in he JCSS Probabilisic Model Code. The reader is advised o have a look a his documen for more deailed informaion. The documen is available on he inerne sie www.jcss.ehz.ch 2. Failure Probabiliy formulaions for ime varian load and resisance 2.1 Load variabiliy The sandard case in reliabiliy heory is he case where we have only simple (no ime dependen) random variables and ime iself is absen in he limi sae funcion. Such a case (a leas in principle) can be presened by wo horizonal lines in a load-resisance ime-diagram as already indicaed in figure 1. The failure probabiliy for (whaever) considered period of ime T is hen given by; P F (T)= P(Z<0) (1) where Z is he limi sae margin which may be wrien as: Z = g(r,s) = R S (2) g = limi sae funcion R = random variable represening he resisance S = random variable represening he corresponding load or acion (effec) Of course, R and S may be funcions of oher variables, deerminisic as well as random. The firs sep ino a more general ime dependen presenaion is o keep he resisance independen of ime and o consider he acions as flucuaing funcions of ime. Wih respec o hese variaions he following classificaion is usually made: - permanen acions: (self weigh, earh pressure, presressing, imposed deformaion, emperaure effecs, shrinkage, selemens; - variable acions: live loads, wind, snow - excepional acions: impac, fire, explosion, avalanches. In he case of permanen loads here is of course no need for ime dependen modelling. I can well be modelled by a se of deerminisic and random variables. For variable and excepional acions we need, however, ime dependen models by definiion. I should be kep in mind ha hese models may also conain a number of ime independen random variables as well.
The appropriae ime dependen models for acion parameers may vary very much depending on he naure of he load. Some ypical and useful process models are (see figure 2): a) Coninuous and differeniable process b) Random sequence c) Poin pulse process wih random inervals d) Recangular wave process wih random inervals e) Recangular wave process wih equidisan inervals If he load inensiies in subsequen ime inervals of model (e) are independen, he model is referred o as a FBC model (Ferry Borges Casanhea model). In many applicaions a combinaion of models is used, e.g. for wind he long erm average is ofen modelled as an FBC model while he shor erm gus process is a coninuous Gaussian process. Such models are referred o as hierarchical models. Each erm in such a model describes a specific and independen par of he ime variabiliy. a c b d Figure 2: Various ypes of load models A se of load models based on hese saring poins may be found in he JCSS model code menioned before. As in his case only he load is flucuaing in ime, he limi sae funcion can be formulaed as: g() = R S() (3) g() = ime dependen limi sae funcion R = random variable represening he resisance S() = random process represening he load or acion (effec)
Failure in his case will occur if he limi sae funcion is negaive a one or more poins of ime in he inerval 0 < < T. In oher words, failure will occur if he minimum of g() is negaive (see figure 3), and so we may wrie he failure probabiliy as: wih P F (T) = P( Z T < 0 ) (4) Z T = min g() (for 0<<T) (5) As in his case only S is ime dependen we may also wrie: Z T = R S max (for 0<<T) (6) If S is a simple sequence of n independen and saionary spikes or blocks (like in figure 2 b or d), he probabiliy disribuion of S max can easily be found from: P(S max < s) = P(S 1 <s and S 2 <s and... and S n <s) = P(S i <s) n (7) And so he disribuion funcion for S max is given by: F Smax (s) = F Si (s) n (8) This way he ime dependen problem has been reduced o a ime independen formulaion. This procedure may also be applied in he case of one consan load and one varying load. In he case of more randomly flucuaing load parameers a more advanced calculaion procedure for a more advanced echnique is needed (see secion 3) R g() = R S() S ime ime Figure 3: Failure occurs if S max > R or min[g()] < 0. Noe ha one may also calculae P(g() < 0). This, however, is only he probabiliy of being in a failed sae a some chosen poin in ime. If S() is a saionary process he resul is he same for every. In ha case we call P(g()<0) he failure probabiliy for he arbirary poin in ime. This probabiliy, however, does no have much meaning. In general i is only an inermediae resul.
2.2. Time dependency of he resisance. In his secion we will discuss he limi sae formulaion: Z T = R() S (9) So in his case he load is consan in ime bu he resisance is a (ofen decreasing) funcion of ime. We will assume in his secion ha he resisance R is no physically influenced by he load S. So we disinguish more or less arificially beween he acions ha reduce he resisance and he acions ha lead o collapse of he weakened srucure. Examples of hose resisance reducing processes are abrasion, corrosion or roing of wood. Similar o he previous case we may wrie: wih P F (T) = P(Z T < 0 ) (10) Z T = min g() for 0<<T (11) As in his case S is ime independen we may also wrie: Z T = R min - S (for 0<<T) (12) If (as in many cases) R is a simply monoonically decreasing funcion of ime, he minimum of g() is always reached a he end of he period for =T, so: R min = R(T) (13) Again we have been able o reduce he ime dependen problem a ime-independen formulaion. Consider as an example he abrasion and/or corrosion mechanism for a recangular beam wih can be modelled by: g() = d() w f S (14) d() w f = he decreasing hickness of he bar due o corrosion [mm] = widh of he bar [mm] = he maerial rupure srengh [N/mm2] The process of hickness reducion may be wrien as (see figure 4): d() = d o d = d o - a u b (15) d o d a b u = he original maerial hickness [mm] = loss of hickness due o corrosison = he average corrosion / abrasion speed [mm/year] = measure for he scaer in he process = random process describing deviaions from he average speed
d µ( d) = a limi mean life iem Figure 4: Corrosion as funcion of ime I is convenien now o divide by wf and o wrie Z T = R(T) S o as: Z T = g(t) = d o a T - u b T S/wf (16) Assume for a numerical example: d o = 15 mm, T = 3 years, a = 1 mm/year, b = 0.5 mm/ year, S/wf = 10 mm and u = a zero mean Gaussian variable wih uni sandard deviaion. In ha case: µ(z) = d o at S/wf = 15 1* 3 10 = 2.00 mm (17) σ(z) = b T = 0.5 3 = 0.86 mm (18) As he variable u has a Gaussian disribuion so has Z. From here we find P F (T) = P(Z<0) = 0.011 2.3 Load and resisance ime dependen, no ineracion If boh he resisance R and he load S are ime dependen we have: Z T = min g() wih g()= R() S() (19) In his secion we sill mainain he assumpion ha here is no ineracion beween R and S. So he acions in S do no belong o he acions ha cause he degradaion of R. In his case here is no simple reducion o a ime independen case. However, someimes we may use he approximaion: Z T = min R() max S() = R(T) S max (20) This, of course, is an upper bound approximaion. The maximum load and he minimum value of R need no o occur a he same poin in ime. This safe side approximaion and can be used as an inermediae sep o decide wheher more advanced echniques are necessary.
2.4 Load and resisance ime dependen, ineracion In he general case he resisance a ime may have been affeced by he loading S in he period from he sar up o. This for insance is wha happens for mechanisms like faigue and load of duraion rupure. The following formulaion is hen adequae: P F (T) = P(Z T <0) (21) Z T = min g() for 0<<T (22) g() = R(S(τ),, x) S() wih 0<τ< (23) The funcion R is ofen presened as a produc of he resisance R o a =0 and a degradaion facor based on a degradaion model. Consider a simple duraion of load example, where he load is consan (S() = S o ), as in many experimens. If in addiion he degradaion is a simple linear funcion of he load, we have: g() = R o (1- a S o ) S o (24) Here (1- as o ) is he degradaion facor depending on he load S o and a is a (random) degradaion parameer. In his case one may easily ransform he limi sae funcion ino (see figure 5): R 0 g () = 1 + a R o - S o (25) This way we have again reduced he limi sae funcion o a formulaion ha can be used by he sandard mehod. In he general case (see he paper by JanWillem van der Kuilen) his is no possible and we may need more general soluion mehods. R,S R 0 R (τ) failure S ime o failure Figure 5: Load duraion dependen srengh under consan load
3. Failure probabiliy calculaion / life ime disribuion and failure rae The probabiliy ha a srucure fails wihin a period (disregarding repair) is idenical o he probabiliy ha he life ime is less han. From here i follows ha he failure probabiliy for some period of ime is equivalen o he disribuion of he life ime L: F L () = P F () (26) Differeniaing his funcion wih respec o gives he probabiliy densiy of he life ime L: f L() = dp F () / d (27) See also figure 6. Z f Z f L µ( L ) Figuur 6: Relaion beween he disribuion of Z and he disribuion of he life ime. The life ime densiy f L () is he so called (uncondiional) probabiliy of failure per uni of ime. Anoher relaed quaniy is he condiional failure rae λ(), expressing he probabiliy of failure per uni ime, given ha no failure has occurred ye: λ() d = P( failure in, +d no failure in 0,) (28) To find λ() one should divide f() by he probabiliy of no failure: λ() = f() /(1-P F ()) (29) One may also derive P F (T) from λ by inegraing (31) on boh sides: T 0 λ() d = ln (1-P F (T)) (30) Or:
T P F (T) = P F (0) + { 1- exp ( - 0 λ() d ) } (31) where P F (0) is he failure probabiliy a he sar. This formulaion is ofen used as a basis o find an approximaion for P F (T) by: λ() d = P( g(+d) < 0 g(τ) > 0 for all 0<τ<) P( g(+d) < 0 g() > 0 ) (32) or: λ() = P[g() > 0 g( + ) < 0] (33) This approximaion is called he oucrossing mehod and λ is in ha case referred o as he oucrossing rae. The mehod is no very accurae if ime-invarian variables (yield sress a ime zero, self weigh) have a grea influence. The mehod can hen be approved by using he oal probabiliy heorem: T P F (T) =... [ P F(0) {1- exp ( - 0 λ( X=x ) d ) } ] f X (x) dx (34) Here X is he vecor conaining all ime independen random variables. The oucrossing rae λ() has now o be found condiional upon X. The failure probabiliy according o (36) can sill be found wih ime saving echniques like FORM alhough a so called nesed mehod needs o be used. Addiionally (36) may even furher be refined ( see he JCSS model code) 5 Effec of inspecion In he case of deerioraing processes i may be uneconomic o design a srucure in such a way ha he reliabiliy is sufficien for a normal design life of 30 or 50 years. In hose cases a more economical soluion can be obained by he definiion of an inspecion scheme. In hose cases failure will no occur if he inspecion reveals some predefined deerioraion crierion and he srucure is repaired adequaely. The sequence of evens can be represened in an even ree as indicaed in figure 7 Le he firs inspecion I i be planned a ime i. In ha case we may have hree possibiliies. 1) a failure occurs before i (branche F) 2) he inspecion deecs a serious defec and repair is necessary (branche R) 3) no serious defec is deeced and a nex inspecion a = 2 is planned If he srucure is repaired, one may usually assume ha all variables are rese o he iniial siuaion. From every even R hen a new even ree of he same ype as he one in figure 7 is sared. For reasons of simpliciy we will sar by having one inspecion only. Using he oal probabiliy heorem, he probabiliy of failure for a period may hen formally be wrien as: where P F () = P[ F Z i > 0 ] P(Z i > 0) + P[ F Z i < 0 ] P(Z i < 0) (35)
F Z i = failure = inspecion resul of inspecion a ime i (negaive values correspond o he deecion of damage) If we assume ha in he case of a serious damage revealed a he inspecion (ha is Z<0) he srucure will be repaired adequaely, (37) may be reduced o (replacing F by min τ g (τ) < 0, where g( ) is he limi sae funcion and 0 < τ< ): or simply: P F () = P[ min τ g(τ ) < 0 Z i > 0 ] P(Z i > 0) ] (36) P F () = P[ min τ g(τ ) < 0 Z i > 0 ] (37) If more inspecions in fixed inervals are presen we arrive a: P F () = P [ min τ g(x(τ);τ) < 0 { Z i (x( i ); i ) } > 0 for 0 < τ < ] (38) i = ime of inspecion; only inspecions wih i < τ are relevan Noe: wheher or no an inspecion is planned, of course, is a maer of economy. a a cri a lim i i + i Figure 7: Failure in he inerval i, i + i wih a(i) < a lim a he beginning of he inerval. 6. Closure This paper has presened a number of possible formulaions o deal wih ime dependency loads and resisance. Is has hopefully been made clear ha: ime dependen load models and adequae calculaion mehods are available. As hings look now one may sar calculaing ime dependen failure probabiliies for every imber duraion model. The nex sep hen should be o find how hese advanced calculaions can be cooked down o give relaively easy bu reasonably accurae rules in for insance he nex Eurocode.