AP06_5.vp 1 Introduction Actions on structures are often of a time-variant nature. Special attention is in particular required when a combination of time-variant loads needs to be considered. Approaches to probabilistic structural design based on different load combi- nation models are indicated by JCSS [1]. It appears that an advanced load effect model based on renewal processes can be suitably used to describe random load fluctuations in time, enabling sufficiently accurate estimates of the reliability level in practical applications. A great number of actions on structures can be approximated by rectangular wave renewal processes with random durations between renewals, as al- ready recognized e.g. by Wen [2]. Models based on renewal processes with exponentially distributed durations between renewals and exponentially distributed durations of load pulses were also recommended for practical use by Iwankie- wicz and Rackwitz [3, 4]. When estimating the structural reliability level related to the specified observed period T, the failure probability Pf(0,T) is often assessed by the lower and upper bounds in the case of combinations of renewal processes, as indicated e.g. by Sýkora [5]. The upper bound on Pf(0,T) is of great importance for practical applications. Two basic properties of the renewal process, the expected number of renewals E[N(0,T)] and the probability of “on”-state pon(t), are needed to evaluate the up- per bound, see Sýkora [5]. The probability of „on“-state was investigated by Shino- zuka [6], considering a “sufficiently long” observed period T. Extension to an arbitrarily long period T, to the so-called non-stationary case, was then provided by Iwankiewicz and Rackwitz [4]. Formulas for the probability of „on“-state were derived considering various initial conditions. The present paper attempts to reinvestigate the formulas for pon(t) achieved by Shinozuka [6] and by Iwankiewicz and Rackwitz [3, 4]. In addition, a formula for the expected number of renewals E[N(0,T)] is verified. Both the basic properties of the renewal process are investigated under random as well as given initial conditions. Special attention is paid here to the correct definition of the random initial conditions. Initially, the expected number of renewals E[N(0,T)] is shown to be independent of the initial conditions. The initial probability of „on“-state pon(0) is then derived under random initial conditions. A two-state Markov process developed by Madsen and Ditlevsen [7] is adopted to derive the probability of „on“-state pon(t) at an arbitrary point in time. It appears that pon(t) is a time-invariant quantity under random initial conditions. In general the paper provides a comprehensive theoretical background for practical applications of advanced load models based on renewal processes. In addition to the newly derived formulas, several results already obtained by Shinozuka [6] and Iwankiewicz and Rackwitz [4] are verified. Desirable extensions for further research are outlined. 2 Basic properties of the considered renewal process It is further considered that the actual load process can suitably be approximated by the renewal process S(t) with the following properties: � the process is intermittent, i.e. the load may be “on”/pres- ent or “off”, � durations Tren between renewals are mutually independent random variables described by an exponential distribution with the rate �, � durations �Ton of “on”-state, durations of load pulses, are also mutually independent exponential variables (rate �). “On”-states are initiated when renewal occurs, � load pulses do not overlap and thus the effective load pulse duration Ton is given by T T T T T T Ton on on ren ren on ren � � � � � � � � | | (1) � load intensities S are mutually independent variables having an appropriate extreme distribution maxTren[S(t)] related to the expected duration between renewals E[Tren] = 1 / �, see e.g. Weisstein [8], 28 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 5/2006 Advanced Load Effect Model for Probabilistic Structural Design M. Sýkora In probabilistic structural design some actions on structures can be well described by renewal processes with intermittencies. The expected number of renewals for a given time interval and the probability of “on“-state at an arbitrary point in time are of a main interest when estimating the structural reliability level related to the observed period. It appears that the expected number of renewals follows the Poisson distribution. The initial probability of “on”-state is derived assuming random initial conditions. Based on a two-state Markov process, the probability of “on”-state at an arbitrary point in time then proves to be a time-invariant quantity under random initial conditions. The results are numerically verified by Monte Carlo simulations. It is anticipated that the proposed load effect model will become a useful tool in probabilistic structural design. The aims of future research are outlined in the conclusions of the paper. Keywords: rectangular wave renewal process, probability of “on”-state, expected number of renewals, action on structures. � random conditions are primarily assumed at the initial time t � 0 of the observed period T, i.e. it is of a purely ran- dom nature whether the process starts in an “on”-state or an “off”-state. Some remarks on processes with the given initial conditions are also provided in the following. The considered process S(t) is indicated in Fig. 1 where the actual load history is depicted in grey. Note that random variables are further denoted by upper-case letters X (e.g. durations between renewals are referred to as Tren) while lower-case letters x (e.g. tren) stand for their realizations/trials. 3 Random initial conditions Special attention is paid to the correct definition of ran- dom conditions at the initial state. As the process S(t) consti- tutes a sequence of intervals Tren, random initial conditions are fulfilled when the initial point is located in an interval T0 selected from a population of Tren on a purely random basis taking into account the random properties of Tren as indi- cated in Fig. 2. The interval T0 (random variable) is denoted as the first renewal. To derive a cumulative distribution function (CDF) FT0(t) of the first renewal T0, a sufficiently long sequence of a large number � �n of trials t i niren, ( )1 � � � is further considered. A trial t t j nj0 1� � � �ren, ( ) is randomly selected from the population of tren, i. Consider next that the duration of the selected trial is �, t t j0 � �ren, �. By intuition, the probability P dren,[ ( , )]T t t tj0 0� � �� � that the selected trial t0 is of duration � can be obtained as a ratio of the total duration of all t tjren, d �( , )� � over the total duration of all trials tren, i from the population. Using the probability density function (PDF) of an exponential distribution f erenT tt( ) � �� � , the central limit theorem for a sum of �n independent random variables, see Weisstein [8], and the expected value E[ ]Tren �1 �, the probability becomes � � P d P dren ren, [ ( , )] lim ( , ) T t t T t t n n j 0 0� � � � � � � � � � � � � � � � t t t n n E T i i n n j ren, ren, ren P d � � � � � � � � � � � 1 lim ( , ) [ ] � � � � � � � � � �� ��e d e d � �� � � � t t2 . (2) The cumulative distribution function of the first renewal T0 is obtained from (2), as follows F P e d eT t t t t T t t t t0 0 2 0 1 1( ) ( ) ( )� � � � � �� �� � �� � (3) and the probability density function becomes, using (3) f dF d eT T tt t t t0 0 2( ) ( ) � � �� � . (4) Note that CDF (3) can be suitably used in Monte Carlo simulations to achieve random initial conditions. The first re- newal T0 can either be randomly selected from a large popu- lation of samples of Tren, or directly simulated using CDF (3). Simulations based on CDF (3) are clearly incomparably more efficient than “the first approach” described in the beginning of this section. More details are provided by Sýkora [5]. 4 Expected number of renewals In the following, N denotes a random number of re- newals of the process S(t). The expected number of renewals E[N(0, T)] is the essential process characteristic used to esti- mate the failure probability Pf(0, T). Unlike in Section 3, consider that process S(t) is defined so that the first renewal starts at the initial time t � 0, as indicated in Fig. 3, so that the “given” initial conditions are taken into account. This as- sumption can be used e.g. for an imposed load model where the action starts to be “on” approximately at t � 0 when a new structure is put into operation. The expected number of re- newals for such a process is obtained e.g. by Weisstein [8] E N T T[ ( , )]0 � �. (5) The first renewal of the considered process S(t) is a “stan- dard” exponentially distributed duration Tren with CDF F P erenT ren tt T t( ) ( )� � � � �1 � . (6) Next consider the random initial conditions again, i.e. the first renewal T0 of the process S(t) is selected purely on a ran- dom basis and the initial time point t � 0 is randomly located in the first renewal. The process is again a sequence of expo- nentially distributed durations except for the first renewal T0. The random effective duration T0eff of T0 corresponds to the first renewal (6) of the process described above. The differ- ence between these processes is indicated in Fig. 3. Note that T0on denotes the duration of “on”-state within the first renewal and T0oneff the effective duration of T0on, i.e. the duration of the “on”-state involved in the observed period (0, T). Random properties of the durations T0on and T0oneff are discussed in the following. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 29 Acta Polytechnica Vol. 46 No. 5/2006 S t ren tren t on t on maxTren [ (S t)] t ren t Fig. 1: Rectangular wave renewal process S(t) with intermittencies t t ren,2tren,1 tren,(n�3)tren,3 tren,(n�1)tren,(n�2) tren,n t 1 t 0eff t t � 0 Fig. 2: Sequence of intervals Tren and random selection of the first renewal T0 The effective duration T0eff indicated in Fig. 3 is randomly selected from the first renewal T0 to fulfil the random initial conditions. Given T0 = t0, the effective duration T0eff (0, t0) has the rectangular distribution R(0, t0) with PDF f effT t T t t0 0 0 01( | )� � and CDF F effT t T t t t0 0 0 0( | )� � . CDF F effT t0 ( ) of the effective duration T0eff for an arbitrary T0 can be derived from the sum of probabilities of two disjoint events (t > 0): � T0 is less than t, implying T0eff < t, � T0 is greater or equal to t while T0eff < t. This can be rewritten as follows F P( P[( ( ] = P( eff eff effT t T t T t T t T T t 0 0 0 0 0 0 ( ) ) ) )� � � � � � � � ) ).� � �P( effT t T0 0 (7) The first part of the right hand side of (7) is already obtained in (3). To evaluate the second part of the right hand side of (7), consideration is initially taken that t < � < T0 < t+dt. Under this assumption, P( effT t T0 0� � ) is equal to P( deffT t t T0 0� � � �| )� � and, using CDF of T0eff for a given t0, arrives at P( deffT t t T t 0 0� � � � � �| )� � � � (8) The condition t < � implies � (t, �). Using (4) and (8), P( effT t T0 0� � ) can be obtained by the expectation P( P( d f d eff eff T t T T t t T t T t 0 0 0 0 0 � � � � � � � � � � � ) | ) ( )� � � � � � f dT t 0( ) .� � � � (9) Substitution of (4) into (9) followed by the limit passage yields � �P( e d e eeffT t T t t t t t t 0 0 2� � � � � �� � � � ��) � � � � � � �� �� � . (10) Substitution of (3) and (10) into (7) leads to CDF FT0eff(t) F e e eeffT t t tt t t0 1 1 1( ) ( )� � � � � � � � �� �� � � . (11) A comparison of CDF (6) with CDF (11) indicates that the first renewal Tren of the process with the given initial condi- tions and the effective duration of the first renewal T0eff of the process with random initial conditions are random variables with the same cumulative distribution functions. Since the following durations Tren between renewals are the same ran- dom variables, the investigated processes inevitably have the same statistical properties. The number of renewals, there- fore, remains the same for both processes. Formula (5) is thus valid for process S(t) with random initial conditions. The expected number of renewals E[N(0, T)] (5) of process S(t) as a function of T and � is numerically compared with the number of renewals predicted by the crude Monte Carlo sim- ulation method (MC) in Fig. 4. Solid lines denote the results of (5) and circles (‘o‘) denote the results of the simulations. 1000 trials of the whole processes S(t) are simulated for each considered combination of T and �, using the following procedure: � a trial t0 is randomly selected from a “sufficiently large” population of realizations t i n niren, ( , )1 � � � � � in accor- dance with the “first approach”, described in Section 3, � the effective duration t0eff is obtained as a pseudorandom number in the range from 0 to t0 (see Fig. 2), � subsequent durations between renewals t j njren, ( )2 1� � � are simulated as exponential variables with the rate �, � a realization of the number of renewals n is determined us- ing the condition t t T t tj j n j j n 0 2 0 2 1 � � � � � � � � �ren ren, , . (12) Note that n � 1 if t T t t0 0 2� � � � ren, and n � 0 if t0 > T. The number of renewals N is identified for each trial of the whole process S(t) and the expected value is then deter- mined. More details on MC verification are provided by Sýkora [5]. Fig. 4 indicates that formula (5) and MC verification lead to the same results. It is therefore concluded that formula (5) is applicable also for process S(t) with random initial conditions. 30 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 5/2006 ton S a) Process with given initial conditions trentrentrentren ton ton tren t T 0 t0on t b) Process with random initial conditions T0 t0 trentrentrentren ton t0eff ton t0oneff S Fig. 3: Investigated processes S(t) (5) 0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 T E[N(0,T)] (5) simulations � � 10 � � 1 � � 0.1 Fig. 4: Expected number of renewals E[N(0, T)] 5 Probability of “on”-state – stationary case In addition to the expected number of renewals E[N(0, T)], the probability of “on”-state pon(t) of a process S(t) needs to be known for applications of the renewal processes. The time variability of S(t) is completely described by dura- tions Tren and Ton and therefore pon(t) can be derived from the statistical properties of Tren and Ton. Note that the initial conditions in t � 0 may have an influence on the probability of “on”-state pon(t). Given that the process S(t) is “on” at t � 0, pon(t) apparently attains 1 for T�0. However, the probability of „on“-state approaches, by intuition, a “stationary” value for large T becoming independent of the initial conditions. Consider that the observed period is “sufficiently long” (conservatively T > 1/� for processes with “short” load pulses when � /� < 0.05, and approximately T > 3 /� for other pro- cesses). Note that these conditions are usually satisfied in practical applications). The probability of „on“-state then ap- proaches the stationary value obtained by Shinozuka [6] p E T E Ton on ren � ( ) ( ) , (13) where E(Ton) is the expected duration of the “on”-state and E (Tren) � 1 /� denotes the expected duration between renewals. Duration Ton is a truncated exponential variable, as de- fined in (1). If a realization of exponential duration with rate � is less than a realization tren, �ton > tren, duration Ton is truncated to ton � tren, otherwise ton � �ton . This truncation apparently influences the expected value E(Ton). CDF FTon(t) is initially derived. Two possible exclusive events can yield Ton < t: � duration Tren is less than t. This implies that duration Ton is less than t, � duration Tren is greater or equal to t. In this case, CDF of Ton in the interval (0, t) remains the “standard” cumulative distribution function of the exponential distribution and the truncation has no effect here. Probability P[Ton < t] can then be determined from CDF of the exponential distribu- tion with rate �. Using CDFs of the exponential distributions with rates � and �, FTon(t) is given for the mutually independent dura- tions Ton and Tren as follows F P P P P on on on ren ren ren T t T t T t T t T t T t ( ) ( ) ( | ) ( ) ( � � � � � � � � ) ( )[ ( )] ( ) ( )[ ( � � � � � � � � � �� � P P P e e on ren renT t T t T t t 1 1 1 1� � � � �t t t)] .( )� � � �� � �1 1e e (14) From (14), the expected value E[Ton] arrives at E on( )T � � 1 � � . (15) Substitution of (15) into (13) yields the probability of „on“-state under stationary conditions pon � � � � � (16) which is in accordance with the well-known result obtained e.g. in RCP [9]. 6 Initial probability of „on“-state Considering random initial conditions, the probability of „on“-state p0on � pon(0) at the initial time t � 0 can be defined as the probability that the sum of the effective duration of the first renewal T0eff and the initial duration of the “on”-state T0on exceeds the duration of the first renewal T0 (see the right hand side of Fig. 3) p P T T T T T0 0 0 0 0 0on on eff� � �[ ( ) ( ) ]. (17) Note that both durations T0eff and T0on are dependent on T0. If the sum is less than T0, the “on”-state is finished earlier than the observed period starts. The initial probability of “off ”-state p0off � poff(0), the complementary probability to p0on, then follows from (17) p P T T T T T0 0 0 0 0 0off on eff� � �[ ( ) ( ) ]. (18) Initially consider that T0 � �. Application of the convolu- tion integral to (18), see e.g. Weisstein [8], yields p T T T0 0 0 off 0on 0effF f d( | ) ( ) ( )� � � � � � � � � � , (19) where FT0on( ) is CDF of T0on. The upper bound for integra- tion in (19) is �, since T0eff is always less or equal to T0 (see Fig. 3). Duration T0on is defined in accordance with (1), i.e. T0on has an exponential distribution with rate � if T0on < T0 � �, otherwise it is truncated to T0on � �. Since the integration variable ��(0, �) in (19) is always positive, FT0on(�) is evaluated for values 0 � ��� � �. Duration T0on is within this interval a “standard” exponential variable with CDF FT0on(t) � 1 – e ��t. To satisfy the random initial con- ditions, the effective duration T0eff has the rectangular dis- tribution R(0, �) with PDF fT0eff(t) � 1 /�. Substitution of the aforementioned functions into (19) followed by integration leads to the probability of „off“-state conditional on T0 � � � p T0 0 0 1 1 1 1 off e d e ( | ) ( ) ( ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 1 1 1 � �� �� �� e . (20) The probability density function of T0 is provided in (4). The probability of „off “-state poff for an arbitrary T0 can be obtained by the expectation of (20), using (4) p p TT0 0 0 0 21 1 1 off offE e e � � � � � � � �� � � �� � [ ( | )]� � �� �� � ��� � �� � � � � � � �� � � � � � � � � � � d 0 2 1 . (21) Since probabilities p0on and p0off are mutually comple- mentary, the initial probability of „on“-state becomes p T T T T T p0 0 0 0 0 0 01on on eff offP� � � � � � � [ ( ) ( ) ] � � � . (22) It appears that under random initial conditions, the initial probability of “on”-state p0on (22) is equal to the “stationary” probability (16). This is an expected conclusion as, e.g., the probability of “on”-state of a wind action nearly always has a © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 31 Acta Polytechnica Vol. 46 No. 5/2006 constant value, barely dependent on the origin t � 0 of the ob- served period (0, T). the numerical study published by Sýkora [5] indicates that the result obtained in (22) is in accordance with the Monte Carlo simulations. 7 Probability of “on”-state – non-stationary case To derive the probability of “on”-state pon(t) at an arbitrary point-in-time t, consider the process S(t) with “short” pulses so that the probability of the duration of the “on”-state Ton exceeding the duration between renewals Tren is negligible, P(Ton> Tren) ~ 0. This process can be suitably modelled as the product of a non-intermittent rectangular wave renewal process S(t) and a two-state Markov process Z(t), as already proposed by Madsen and Ditlevsen [7] S t S Z tt( ) ( )( )� . (23) States of the Markov process are characterized as follows Z t( ) 0 1 � � “off” state “on” state � � � � � (24) The considered processes S(t) and Z(t) are indicated in Fig. 5. The renewal process S(t) may be “on” [Z(t) � 1] or “off ” [Z(t) � 0] in time t. Within an infinitely short time interval �t�0, the process may jump between the “on” and “off ” states or may remain in the same state. Therefore, the process S(t) may again be “on” [Z(t��t) � 1] or “off ” [Z(t��t) � 0] in t��t. It is further assumed that “on” and “off ” states form a Markov process, and states in t��t merely depend on the states in t. Consider that the process S(t) is “off ” at the time point t, i.e. Z(t) � 0. Within �t, the process may remain in the “off ”- -state [Z(t��t) � 0] or a renewal may occur and an “on”-state may be initiated [Z(t��t) � 1]. The probability P�t(N > 0) of the occurrence of at least one renewal within �t is obtained e.g. by Wen [2] P P e , � � � � � � t t t N N t o t t ( ) ( ) ( ) � � � � � � � � � 0 1 0 1 2� � � (25) where o(�t2) denotes terms with a higher order in �t for which lim ( ) � � � t o t t � � 0 2 0. This implies that the transition (transfer) probability P[Z(t��t) � 1|Z(t) � 0] of a jump from the “off”-state in t into the “on”-state in t��t given the “off”-state in t is P Z t t Z t t[ ( ) | ( ) ]� � � � �1 0 � . (26) The complementary transition probability P[Z(t��t) � 0|Z(t) � 0] that process S(t) remains “off ” becomes P P [ ( ) | ( ) ] [ ( ) | ( ) ] . Z t t Z t Z t t Z t t � � � � � � � � � � � � 0 0 1 1 0 1 � (27) Assuming an “on”-state at a time point t[Z(t) � 1], the pro- cess may remain in the “on”-state [Z(t��t) � 1] or the load pulse may be finished and an “off ”-state may be initiated [ Z(t��t) � 0]. Note that the probability of the event of an “on”-state being finished by a renewal occurrence is neglected here. This event is conditioned by Ton> Tren, but only pro- cesses with “short” load pulses are investigated in this section, and thus P[Ton> Tren] ~ 0. By analogy with (25), (26) and (27), the transition probability P[Z(t��t) � 0|Z(t) � 1] of a jump from the “on”-state in t into the “off ”-state in t��t given the “on”-state in t is P[ ( ) | ( ) ]Z t t Z t t� � � � �0 1 � (28) and the complementary transition probability P[Z(t��t) � 1|Z(t) � 1] that process S(t) remains “on” is P P [ ( ) | ( ) ] [ ( ) | ( ) ] . Z t t Z t Z t t Z t t � � � � � � � � � � � � 1 1 1 0 1 1 � (29) Fig. 6 indicates all the transition probabilities (26) to (29) and Markov states. The probability of „on“-state pon(t��t) is obtained as p t t Z t t Z t t Z t Z t on P P P ( ) [ ( ) ] [ ( ) | ( ) ] [ ( ) � � � � � � � � ! � � � � 1 1 1 1 1 0 0 1 ] [ ( ) | ( ) ] [ ( ) ] ( ) ( ) � � � � ! � � � ! � P P on Z t t Z t Z t t p t � � �� � t p t! off ( ). (30) As the probabilities of „on“-state and “off ”-state are com- plementary, pon(t)�poff(t) � 1, (30) can be rewritten as � � � � � � � � �p t t p t t p t t p ton on on on( ) ( ) ( ) ( ) ( )� � � � � �, (31) where �pon (t��t) is a derivative of the probability of „on“-state at time t. Equation (31) is in agreement with the formulas ob- tained by Iwankiewicz and Rackwitz [4]. It follows from (31) that the probability of „on“-state pon(t) at an arbitrary point in time t, the so-called non-stationary probability of „on“- -state, is p t C ton e( ) (� � � � �� � � � � , (32) where C is a constant of integration obtained from the initial conditions. Under stationary conditions, t��, the first term of the right hand side of (32) vanishes and pon(t) equals � /(���), as already given in (16). Considering the random initial conditions and an arbi- trarily long period (0, T), the initial probability of „on“-state is, in accordance with (22), pon(0) � � /(���). For t � 0, substitu- tion of (22) into (32) leads to C � 0 and the non-stationary 32 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 5/2006 S t( ) t t ren tren ton ton t ren Z t( ) t S( )t t 1 0 Fig. 5: Non-intermittent rectangular wave renewal process S(t) and two-state Markov process Z(t) � �t 0 ��t ��t “off ”“on” 1� �� t1� �� t Fig. 6: Transition probabilities and Markov states probability of „on“-state under random initial conditions becomes p ton ( ) � � � � � . (33) It appears that under random initial conditions, the prob- ability of „on“-state is time-invariant for an arbitrarily long observed period T, and process S(t) is stationary and ergodic. This is a very important conclusion, probably firstly published here. However, the initial conditions can be given in some cases. A load process S(t) can then be “on” or “off ” at t � 0. If process S(t) is assumed to be “on” at t � 0, pon(0) � 1, the non-station- ary probability of „on“-state (32) reads p t ton e( ) (� � � � � �� � � � � � � �� . (34) If the load process S(t) is “off ” at t � 0, pon(0) � 0, the non-stationary probability of „on“-state (32) writes p t ton e( ) (� � � � � �� � � � � � � �� (35) Probabilities (34) and (35) are identical with those already achieved by Iwankiewicz and Rackwitz [4]. Note that in the referenced paper the random initial conditions are approxi- mated as pon(0) � poff(0) � 0.5. Formula (32) then yields p t ton ) e( ) ( (� � � � � � �� � � � � � � � �� 2 . (36) The difference between the initial conditions applied to derive formula (33) in present paper and (36) published by Iwankiewicz and Rackwitz [4] is obvious, and needs no further comment. The probability of „on“-state pon(t) (33) is further numeri- cally evaluated using the MC simulations. For the number of trials �n � 105, “on”- and “off ”-states are identified at se- lected times t and the probability of „on“-state pon(t) is then determined. Fig. 7 indicates the probabilities pon(t) for three alternatives: � alt. A: the time variability of the considered process is defined by �A � 1 and �A � 0.1. Formula (33) predicts pon,A � 0.91. It is thus foreseen that process A is nearly always active, � alt. B: rates �B � 1 and �B � 1 and pon,B � 0.5. Process B is sometimes “on” and sometimes “off”, � alt. C: rates �C � 1 and �C � 10 and pon,C � 0.09. Process B has large intermittencies and short load pulses. Note that processes A and B hardly fulfil the condition for a process with short pulses. It can be easily shown that PA(Tren< Ton) = �A/(�A+ �A) = 0.91 and PB(Tren< Ton) = 0.5. The condition is perhaps satisfied for process C only. The dashed lines in Fig. 7 indicate the results obtained by (36), solid lines by (33) and circles ‘o’ by simulations. The MC simulations are in agreement with (33). The probability of „on“-state pon(t) proves to be time-invariant when random initial conditions are satisfied. The non-sta- tionary probability of „on“-state (36) provided in the litera- ture fails to describe alternatives A and C for lower t. It ap- pears that for a larger time (in this case approximately t > 5), the “non-stationary” effects involved in (36) vanish, and for- mula (36) corresponds well to the MC simulations. As already mentioned, formulas (33), (34) and (35) for pon(t) are derived assuming “short” load pulses, P(Ton> Tren) ~ 0. It is foreseen that deriving of the prob- ability of „on“-state for a process with general load pulses is a much more difficult task. However, the preliminary numerical results partly presented in Fig. 7 for alternatives A and B indi- cate that the derived formulas remain valid for processes with arbitrarily long pulses. 8 Concluding remarks It is indicated that the cumulative distribution function FT0(t) of the first renewal T0 obtained in the paper can suit- ably be used to simulate random initial conditions of the re- newal process. The expected number of renewals E[N(0, T)] of the renewal process considered here is shown to be inde- pendent of the initial conditions. Using the newly derived ini- tial probability of „on“-state pon(0) and a two-state Markov process, the probability of „on“-state pon(t) at an arbitrary point in time then proves to be time-invariant under correctly defined random initial conditions. It is foreseen that the investigated model based on renewal processes will become a useful tool in probabilistic structural design, particularly in applications of time-variant reliability analysis. As the formula obtained for pon(t) is derived con- sidering a process with “short” load pulses, future research should focus on deriving pon(t) for a general process with arbitrarily long pulses. Acknowledgments This research has been conducted at the Czech Technical University in Prague, Klokner Institute. Financial support has been provided by the Czech Science Foundation in the framework of GAČR research project 103/06/P237 “Probabil- istic Analysis of Time-Variant Structural Reliability”. The help given by the author’s colleague, Prof. M. Holický, is highly appreciated. References [1] JCSS (Joint Committee for Structural Safety): JCSS Probabilistic Model Code. http://www.jcss.ethz.ch\, 2001. [2] Wen, Y. K.: Structural Load Modeling and Combination for Performance and Safety Evaluation. Amsterdam: Elsevier, 1990. [3] Iwankiewicz, R., Rackwitz, R.: Coincidence Probabilities for Intermittent Pulse Load Processes with Erlang Arrivals and Durations. In: Proceeding ICOSSAR’97 – Structural Safety and Reliability (Vol. 2), (editors: © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 33 Acta Polytechnica Vol. 46 No. 5/2006 t pon( )t AA 0 0.5 1 0 2 4 6 BB CC Fig. 7: Probability of “on”-state pon(t) Shirashi, Shinozuka, Wen). Rotterdam: Balkema, 1998, p. 1105–1112. [4] Iwankiewicz, R., Rackwitz, R.: Non-Stationary and Sta- tionary Coincidence Probabilities for Intermittent Pulse Load Process. Probabilistic Engineering Mechanics, Vol. 15 (2000), No. 2, p. 155–167. [5] Sýkora, M.: Probabilistic Analysis of Time-Variant Structural Reliability. Prague: CTU Publishing House, 2005. [6] Shinozuka, M.: Notes on the Combinations of Ran- dom Loads (I) in Probability Based Load Criteria for the Design of Nuclear Structures: A Critical Review of the State-of-the-Art. BLN Report No. NUREG/CR 1979, 1981. [7] Madsen, H. O., Ditlevsen, O.: Transient Load Model- ling: Markov on-off Rectangular Pulse Process. Structural Safety, Vol. 2 (1985), p. 253–271. [8] Weisstein, E. W.: MathWorld. A Wolfram Web Resource: http://mathworld.wolfram.com/. 2006. [9] RCP Reliability Consulting Programs: Strurel (1903): A Structural Reliability Analysis Program System, Comrel & Sysrel User’s Manual. Munich: RCP Consult, 2003. Ing. Miroslav Sýkora, Ph.D. e-mail: sykora@klok.cvut.cz Department of Structural Reliability Czech Technical University in Prague Klokner Institute, Šolínova 7 166 08 Prague 6, Czech Republic 34 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 5/2006