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Load combination analysis by ‘Directional simulation in the load space’
Probabilistic Engineering Mechanics 21 (2006) 159–170 www.elsevier.com/locate/probengmech
Load combination analysis by ‘Directional simulation in the load space’ William A. Gray, Robert E. Melchers * The University of Newcastle, Callaghan, NSW, 2308, Australia Received 23 August 2004; received in revised form 8 August 2005; accepted 10 August 2005 Available online 14 October 2005
Abstract The reliability of structures subjected to multiple time-varying random loads is considered herein. It is well-known that the reliability of such systems may be evaluated by considering outcrossings of the load process vector out of a safe domain, and the contribution of individual loads to structural failure may be evaluated by considering outcrossings caused by combinations of one or more loads. In this paper the ‘Directional Simulation in the Load Space’ approach to reliability analysis is developed to consider explicitly outcrossings caused by all possible combinations of loads, during analysis of systems comprising stationary continuous Gaussian loads. To do this, the direction of the load process vector is ‘fixed’ at each point of outcrossing (to physically represent the particular combination of loads causing the outcrossing), and, by considering each possible load combination, all loads not causing an outcrossing are then held constant during radial integration (to model correctly that they do not contribute to each outcrossing). A numerical example demonstrating the validity of the proposed formulation is presented. q 2005 Elsevier Ltd. All rights reserved. Keywords: Directional simulation in the load space; Time-dependent reliability; Outcrossings; Load combinations; Load modelling
1. Introduction One approach to handling systems involving time-varying loads is to reduce the time-variant problem to a time-invariant one. For problems with multiple loads, this then requires employing a suitable (but approximate) load combination rule (e.g. ‘Turkstra’s Rule’ (Turkstra [14])), that, typically, involves analysing combinations of (time-invariant) extreme value and ‘average’ loading to allow for the fact that all loads might not be at their extremes at the time of failure. However, this procedure does not accurately account for the time-variant behaviour of individual loads, or combinations of loads. It is well-known that by modelling the loads as (timevariant) random processes (implying that high-level loads are applied more than once in a given lifetime), the failure probability may be evaluated by considering the outcrossing of a vector process out of a safe domain. In general, the probability of failure occurring in the time interval 0Kt (i.e. pf(t)) is given by the well-known upper bound (c.f. Veneziano et al. [15]) pf ðtÞ% pf ð0Þ C ð1Kpf ð0ÞÞð1KeKnt Þ
(1)
* Corresponding author. Tel.: C61 2 4921 6044, fax: C61 2 4921 6991. E-mail address: [email protected] (R.E. Melchers).
0266-8920/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2005.08.001
where pf(0) is the probability the structure will fail on first loading (i.e. at tZ0) and is obtained by time-invariant reliability analysis, and n (denoted herein as nC D ) is the mean outcrossing rate of a vector process out of a deterministic safe domain D (Belyaev [1]). However, it is possible—at the time of failure—for only one or a few loads to be actively changing in value (and therefore causing failure). To account for this the treatment of vector-outcrossings may need to allow for the possibility of outcrossings being caused by individual loads, as distinct from combinations of all loads. The procedure used to analyse combinations of timevarying loads depends on the stochastic process model used to represent the loads. Two well-known load models that have been presented in the literature are herein referred to as the ‘onoff’ model (e.g. Wen [19]) and the ‘standard’ model (Fig. 1). The ‘on-off’ model assumes loads are either ‘on’ (e.g. their value is non-zero and constantly changing) or ‘off’ (e.g. their value is strictly zero). They can contribute to failure only when they are ‘on’. To implement this model in load combination analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’. In contrast, the ‘standard’ model (which is more commonly used, and is considered herein) effectively assumes each load is always ‘on’. The possibility of one or more loads not causing failure (i.e. such loads are effectively ‘off’ at the time of failure) needs to be allowed for. A procedure to enable this, during analysis by the time-variant
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Fig. 1. The ‘On-Off’ Model.
analysis method ‘Directional Simulation in the Load Space (DS-LS)’ (Melchers [9]) will be described herein. In DS–LS, the loads are separated from all other variables and analysis is performed in the load space. This is convenient for time-variant analysis as it enables the loads to be represented as random processes, and to be modelled properly. The technique has so far been shown to work well for timevariant reliability analysis of relatively simple structural systems (e.g. Melchers [10]), and simple combinations of loads represented by the ‘on-off’ model (e.g. Melchers [11]). It has also been used to examine vector-outcrossings in systems comprising loads represented by the ‘standard’ model (e.g. Melchers [9]); however, it appears not to have considered outcrossings caused by explicit combinations of only one or a few such loads. Developments to DS–LS to achieve this are proposed in this paper. In Section 2 a review of DS–LS and stochastic process models used to represent loads will be given. Developments to DS–LS will be derived in Sections 3 and 4. Equivalent Monte Carlo expressions will then be discussed briefly in Section 5, for comparison in the numerical example that will be presented in Section 6. 2. Review of time-variant structural reliability analysis
Fig. 2. Directional simulation in 2D Q space.
occurs if any system of loading QZq is in the ‘failure’ region. Since, for each value QZq, the realisation g(q,x)Z0 exists as a ‘probabilistic’ boundary, the location of the safe and failure regions may be defined only probabilistically. It follows that nC D denotes the mean rate at which the load process vector QZ q(t) crosses out of the ‘safe’ region, and is obtained by integrating over all outcrossing rates that are evaluated conditional to each realisation G(Q,X)Zg(q,x)Z0. A time-variant DS–LS expression will now be reviewed. The expression may be used specifically for analysis of systems comprising stationary continuous gaussian loads, as it is only this type of load that is considered by the numerical example presented herein. 2.1.1. Time-variant DS–LS expression The ‘usual’ generalised DS-LS expression for nC D is given by (Gray [6], modified from [9]) ð nC Z nC D D ðaÞfA ðaÞda A
2.1. Directional simulation in the load space (DS–LS) In DS-LS the loads are separated from all other variables and ‘Directional Simulation’ (e.g. Bjerager [2]) is performed in the load space. This is often convenient because (i) there are usually only a small number of loads compared with other variables (therefore the overall dimension of space is reduced), and (ii) DS–LS enables the loads to be modelled properly. Let the system being considered be represented by the vectors Q (comprising a total of nQ random loads) and X (comprising a total of nX variables-all variables other than loads). It follows that the system structural resistance is represented by a nQ-dimension vector of random resistance RZR(X), comprising individual resistances Ri (with Ri corresponding to each load Qi in Q). The load capacity of the structure is reached when qZr (where q and r are realisations of Q and R, respectively), and qRr represents failure. The system limit state function G(Q,X), which is an expression that relates Q to X, is constructed piece-wise, from individual limit state functions Gi(Q,X). The realisation G(Q, X)Zg(q,x)Z0, which will be referred to herein as the ‘limit state’, separates the load space into a ‘safe’ (i.e. g(q,x)O0) and ‘failure’ (i.e. g(q,x)%0) region, where q and x represent realisations of Q and X, respectively (Fig. 2). System failure
2N ð
4 Z SnQ A EA
3 E½Q_ n ðsjaÞCfSjA ðsjaÞfQ ðsa C cÞsnQK1 ds 5 janðsjaÞj
(2)
0
where fA is the probability density function (pdf) of the C direction variable A, nC D (a) is the value of nD evaluated in the nQ direction AZa, SA is the surface area of the nQ-dimension unit (hyper-) sphere, EA is an expectation operator, SZsR0 is a random variable used to allow for uncertainty in the limit state (in the direction AZa) due to variables in X, fSjA is the pdf _ vQðtÞ=vt is the load of ‘S’ in the direction AZa and QZ process time derivative vector. Also, the term E[ ]C—which will be described later-is the mean outcrossing rate at the point (SZsjAZa), CZc is the origin of DS–LS (and is related to R and S by the relationship RZS$ACC), fQ is the system load pdf and n(sja) is the limit state unit outward normal vector at the point ðsjaÞ. Here, E[ ]C is the mean outcrossing rate, given that the limit state function is positioned at the point ðSZ sjAZ aÞ and the load process vector is at the point QZ qðSZ sjAZ aÞ. This term is weighted by the ‘probability’ fSjA that the limit state function is at the point (SZsjAZa) and the ‘probability’ fQ(s$aCc) that the load process vector is at the point QZ qðSZ sjAZ aÞ. Also, division by the term jan(sja)j
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161
converts the elemental surface area DSeZsnQK1da (which is normal to AZa (Fig. 2)) into an equivalent area DSD on the limit state surface (through which individual outcrossings occur); the absolute value j j ensures the surface area is positive if the point CZc is positioned in the failure region (Gray [6]). Some important features of (2) will now be discussed. 2.1.2. Deterministic ‘s’ and random ‘S’ The variable ‘S’ may exist as either a deterministic (i.e. ‘s’) or random variable (i.e. ‘S’) quantity in a given direction AZa, depending on the statistical properties of the quantities in X contributing to G(Q,X). If all such quantities are deterministic, then ‘s’ is also deterministic (i.e. the failure boundary can be defined deterministically at one or more points in the direction AZa), and fSjA is given by fSjA ðsjaÞ Z 1 if gðsjaÞ Z 0 Z0
otherwise
(3)
Conversely, ‘S’ is a random variable (i.e. the failure boundary exists as a random function in the direction AZa) if one or more such quantities exist as random variables. Procedures to evaluate fSjA for such situations have been discussed in the literature (e.g. Guan and Melchers [7]). Although defined for the DS–LS approach, the terms Deterministic ‘s’ and Random ‘S’ will be used in a generic sense throughout this paper, to describe the uncertainty in each limit state due to variables other than loads. 2.1.3. General assumptions The following two general, but important, assumptions are made by (2): (i) Load Path Independence, and (ii) ‘Star-shaped’ Integration Region. These assumptions are also made by the developments proposed herein, and are discussed briefly as follows. Load path independence: This is the assumption that ‘an outcrossing does NOT depend on the path travelled by the load process vector up to the point of outcrossing’ (e.g. Ditlevsen and Madsen [5], Melchers [12]). Herein, this means an outcrossing can occur at any point (sja), irrespective of how the load process vector reaches that point. Note that the concept of Load Path Independence is independent of the load type (i.e. continuous or discrete), and its practical implications are given elsewhere in the literature. Star-shaped integration: Due to the considerable complexity involved with a limit state being intercepted at multiple points in a given direction AZa (cf Bjerager [2], Fig. 3b), herein it is assumed the limit state is intercepted only once in a given direction. This means the integration region is ‘starshaped’ (Bjerager [2]) with respect to the origin of DS–LS. 2.1.4. Outcrossing of load process vector The term E½Q_ n ðsjaÞC in (2) is described as follows. Here, each component ni(sja) of n(sja) and Q_ i Z vQi =vt of Q_ corresponds to each component Qi of Q. Also, the term _ ni ðsjaÞ. Q_ i represents the component of Q_ i that is Hi Z QZ normal to the limit state, with hi Z q_ ni representing its
Fig. 3. Possible directions of an outcrossing in the 2D load space.
particular value. Clearly, the particular value hZ q_n ðsjaÞ of the scalar quantity H (which is given by the dot product _ represents the sum of all particular H Z Q_ n ðsjaÞZ nðsjaÞQ) values hi (for 1%i%nQ), and therefore the rate at which Q exits (or enters) the safe region at the point (sja). An outcrossing at the point (sja) will occur only if hO0; it follows that E½HCZ E½Q_ n ðsjaÞC is the mean outcrossing rate at the point (sja), and may be evaluated from the expression (e.g. Melchers [9]) N ð
sH E½H Z hfH ðhÞdh Z pffiffiffiffiffiffi 2p C
(4)
0
where fH and sH are the pdf and standard deviation respectively of H. Eq. (4) does not consider explicitly combinations of loads causing an outcrossing at each point (sja)—any load combination can implicitly cause an outcrossing if the condition E[ ]CO0 is satisfied. This implies that the direction of the load process vector at any point (sja) is a random variable, such that the load process vector can ‘cross out’ in any outward direction. To consider explicitly specific load combinations causing each outcrossing, the direction of the load process vector needs to be ‘fixed’ at each point of outcrossing, according to the particular load combination being considered. For example, Fig. 3 shows the possible directions in which an outcrossing at any point (sja) in the 2D load process space can be caused by either one load only or two loads. The concept of ‘fixing’ the direction of the load process vector is herein termed ‘Development 1 for Load Combination Analysis by DS–LS’, and will be derived below. Before proceeding however, two stochastic process models used to represent individual loads will be reviewed briefly, and the implementation herein of one such model in load combination analysis performed by DS–LS will be described. 2.2. Stochastic process models used for load combination analysis In a given time interval, time-varying loads can be idealised as being either ‘active’ (i.e. their value changes constantly) or ‘inactive’ (i.e. their value is strictly zero, or constant at some non-zero value). The procedure used to analyse combinations of time-varying loads depends on the stochastic process model used to represent these state changes (i.e. without loss of
a
4
3
2
b
ni is the mean upcrossing rate of Qi (Rice [13]). Prob( ) denotes probability.
Qj
C Evaluate nC Dj , nDij
Repeat steps 1-3, to consider all possible load combinations
ð b Z nC Di jqj fQj ðqj Þdqj
nC Di jqj : Probðqj Þ
nC Di Z ni qi pj qj
X
a C Evaluate nC D for Qi acting alone (i.e. nD Z ni )
Evaluate nC Di jqj
nC Di z
Set QjZqjZ0 Identify Qj
Proposed DS–LS
Identify Qi 1
Step
Table 1 Procedure to evaluate load combinations
Assume a combination of loads that will cause failure (i.e. the ‘active’ loads) Identify all loads not causing failure (i.e. the ‘inactive’ loads) ‘Fix’ the value of all ‘inactive’ loads, by holding their value constant at the time of failure Conditional to the ‘fixed’ values, evaluate the system failure for the condition of failure caused by all ‘active’ loads Integrate over all possible ‘fixed’ values of the ‘inactive’ loads, to allow for the probability that the loads are ‘inactive’
Typical calculation of nC Di
Set QjZqjZany value
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Equivalent ‘On-Off’
162
generality, the pdf fQi of load Qi). Two well-known load models that have been presented in the literature are referred to herein as the ‘on-off’ model (e.g. Wen [19]) and the ‘standard’ model. They will be described below, along with the implementation herein of the ‘standard’ model in load combination analysis performed by DS–LS. ‘On-off’ model: The ‘on-off’ model assumes each load is either ‘on’ (e.g. its value is non-zero, and constantly changing) with probability pi, or ‘off’ (e.g. its value is strictly zero) with probability qiZ1Kpi (Fig. 1). It typically assumes loads are non-negative-its pdf might be obtained, for example, by truncating the load ‘average value’ or ‘extreme value’ pdf to allow positive values only. Loads can contribute to failure only when they are ‘on’; to implement this model in load combination analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’, using combinations of pi and qi. ‘Standard’ Model: This is more commonly used, and is considered herein. Unlike the ‘on-off’ model, it can allow each load to be negative; its pdf is typically equal to the load ‘average value’ or ‘extreme value’ pdf. It effectively assumes each load is always ‘on’ (i.e. piZ0); to apply this model in load combination analysis, the possibility of load Qi not causing failure (i.e. the ‘equivalent’ of piO0) must be allowed for, as follows. Implementation of ‘Standard’ Model: The system load pdf fQ essentially describes the probability that Q is ‘moving around’ in the load space; it does not allow for the probability of one or more loads Qi not causing failure (i.e. the ‘equivalent’ of piO0). However, this probability can be derived from fQ and fQi (and subsequently evaluated by DS–LS) as follows. C Denoting nC Di , nDij as the mean rate at which outcrossings are caused by load Qi only, and both loads Qi and Qj only respectively, it is proposed herein that each of the terms nC Di , nC etc be evaluated by DS–LS, using the procedure described Dij in Table 1. It shows how the value of each load not causing failure is held ‘constant’ at the time of failure, while the value of all remaining loads is allowed to change. For clarity, Table 1 also demonstrates typical steps taken during DS–LS to evaluate nC Di for a system of two loads Qi and Qj, and the ‘equivalent’ steps that would be taken to examine loads represented by the ‘On-Off’ model. In doing so, it demonstrates how the probability pjO0 (of load Qj) is ‘built-in’ to the DS–LS calculations (using the term Prob(qj)ZfQj(qj)dqj); further details of this will be given later in this paper. To consider all possible combinations of loads that can cause each vector outcrossing, the following expression for nC D is proposed:
nC D Z
nQ X iZ1
nC Di C
nQ X nQ X
nC Dij C.
(5)
iZ1 jZ1
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} isj
and in what follows, the developments to the ‘usual’ DS–LS C formulation that enable the terms nC Di , nDij , etc. to be evaluated
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(specifically for application to systems comprising stationary continuous gaussian loads) will be derived.
E½Q_ n jQ_ nij ðsjaÞC C ðN
3. Development 1 for load combination analysis by DS–LS
Z
2CN ð .4 Iij ½hfH
KN
To evaluate the mean rate E[ ]C at which outcrossings at any point (sja) are caused by a specific load combination, the conditions enabling outcrossings to be caused by the specific load combination need to be considered, as follows. For clarity, outcrossings caused by either one load only (Qi(t), say) or two loads only (Qi(t) and Qj(t), say, with isj) will be considered, and time ‘(t)’ will be omitted. Consider first the situation with outcrossings caused by Qi only. An outcrossing will occur at the point (sja) only if hO0; an outcrossing will be caused by Qi only if hiO0 and hj%0 (for all 1%j%nQ, with jsi). Hence, the mean rate E½Q_ n jQ_ ni ðsjaÞC at which outcrossings at the point (sja) are caused by Qi only may be expressed as the integral 2CN 3 CN ð ð E½Q_ n jQ_ ni ðsjaÞC Z .4 Ii ½hfH1 .HnQ ðh1 ; .; hnQ ÞdhnQ 5dh1 KN
KN
(6)
with ‘h’ given by the sum hZ
nQ X iZ1
hi Z q_n Z
nQ X
q_ ni
(7)
iZ1
where fH1 .HnQ is the joint pdf of the independent variables H1,.,HnQ and Ii[ ] is an indicator function given by the expression Ii ½ Z 1
if ½ðhO 0Þ and ðhi O 0Þ and ðhj % 0Þ c1%j%nQ ðj siÞ
(8)
Z 0 otherwise Similarly, an outcrossing will be caused by both Qi and Qj only if hO0, hiO0, hjO0 and hk%0 (for all 1%k%nQ, with ksjsi), so that the mean rate E½Q_ n jQ_ nij ðsjaÞC at which outcrossings at the point (sja) are caused by both Qi and Qj only becomes the integral
163
3 1 .HnQ
ðh1 ; .; hnQ ÞdhnQ 5dh1
(9)
KN
where Iij[ ] is an indicator function given by the expression Iij ½ Z 1 if ½ðhO0Þ and ðhi O0Þ and ðhj O0Þ and ðhk %0Þ c1%j%nQ ðjsiÞ;c1%k%nQ ðksjsiÞ Z0 otherwise (10)
Expressions for E[ ]C for other load combinations can be obtained similar to (6) and (9). For systems comprising only two loads (Q1 and Q2, say), the regions of integration for (6) and (9) are shown in Fig. 4(a). Herein, expressions for E[ ]C are evaluated by directional simulation in H-space, using a procedure that will be described below. A discussion on the use of these expressions to evaluate nC Di , nC Dij etc, using a procedure similar to (2) (i.e. simulating directions on the entire surface of the nQ-dimension unit hypersphere), and the need for an additional development to DS–LS to model the loads properly during load combination analysis will follow this. 3.1. Evaluation of E[ ]C by directional simulation Consider a typical direction QZq generated from an origin of directional simulation EZe (Fig. 4(b))). Evidently, the scalar variable H and a vector variable U-with particular value UZuZ[h1 . hnQ]-are related to E, Q and the radius PZr by the relationships UðPjQÞZ P:QC E and HZ hðr$qC eÞZ sumfug, where sum{ } is the sum of the elements in the vector described by { }. The term E½ZQ_ n jQ_ nij ðsjaÞC, for example, may be obtained by integrating over all values PZr
Fig. 4. Integration in H-space.
164
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and all directions QZq, according to the expression ð E½C Z E½qCfQ ðqÞdq Q
2N 3 ð 4 Ii ½rq C ehðr:q C eÞfH .H ðrq C eÞrnQK1 dr5 Z SnQ A EQ 1 nQ 0
(11) where E[jq]C is the value of E[ ]C in the direction QZq, fQ is the pdf of Q, EQ is an expectation operator, fH1 .HnQ ðrqC eÞrnQK1 dr is the hypervolume under fH1 .HnQ at the point (rjq), and details regarding the numerical evaluation of (11) are given in Section 6.1. C 3.2. Evaluation of nC Di and nDij and load modelling implications C Using ‘Development 1’, the terms nC Di , nDij etc can be C expressed in a form similar to (2). For example, nC Di and nDij may be expressed as ð C nC Di Z nDi ðaÞfA ðaÞda A
2N 3 ð C nQK1 _ _ 4 E½Qn jQni ðsjaÞ fSjA ðsjaÞfQ ðsa C cÞs ds5 Z SnQ A EA janðsjaÞj 0
(12) and
ð Z nC nC Dij Dij ðaÞfA ðaÞda
model, in which all loading is governed by one variable (i.e. ‘S’). In (12) and (13), this model implies all loads change ‘in proportion’, and therefore all simultaneously cause each outcrossing. Although this implication is generally strictly true only when all loads are fully dependent (i.e. the correlation coefficient ‘r’ between all loads is the limit rZ1.0), it means that when the loads are not fully dependent, (14) should only be used to evaluate the mean rate at which outcrossings are caused by all loads (i.e. nDi.nCQ ). Furthermore, the ‘one-parameter’ model implies all loads constantly change value in time (and therefore at the time of outcrossing), such that the probability of an outcrossing being caused by only one or a few loads is approximately zero. Unless the loads are fully dependent, there is actually a finite non-zero probability that one or more loads may remain ‘approximately’ constant at the time of outcrossing, so that the ‘one-parameter’ model does not necessarily model the loads realistically. For example, for extreme wind load Qi acting on a building and causing an outcrossing while the building occupancy load Qj remains approximately constant at the time of outcrossing, the loading is more appropriately modelled by the expression " # qi Z sai C ci Q Z qðsjaÞ Z sa C c Z (15) qj Z constan t Hence, to model properly the time-dependent behaviour of each load during DS–LS, it is necessary to perform a further development-to complement ‘Development 1’-‘fixing’ the particular values of loads not causing each outcrossing, prior to radial integration. This development is herein termed ‘Development 2 for Load Combination Analysis by DS–LS’, and will now be derived.
A
2N 3 ð C nQK1 _ _ E½ Q j Q ðsjaÞ f ðsjaÞf ðsa C cÞs n nij SjA Q 4 ds5 Z SnQ A EA janðsjaÞj 0
(13) respectively. Although (12) and (13) are analytically exact, they do not necessarily model properly the time-variant behaviour of the loads, for reasons that will be discussed below. Load Modelling Implications: Radial integration in (12) and (13) is performed according to the random structural strength variable ‘S’, to allow for any uncertainty in the limit state in each direction AZa (due to random variables other than loads). In doing so, the particular value q of the system of loading Q at any point (sja) is given by the expression 2 3 sai C ci 6 7 Q Z qðsjaÞ Z sa C c Z 4 saj C cj 5 (14) « where, without loss of generality, ai and ci are the ith component of AZa and CZc, respectively. Although (12) and (13) assume load path independence, (14) is a ‘one-parameter’
4. Development 2 for load combination analysis by DS–LS ‘Development 2’ involves ‘fixing’ the value of one or more loads prior to radial integration. Based on the above discussions, this procedure-in combination with the existing radial integration, involving the variable ‘S’-provides implicitly a practical means of modelling the time-dependent behaviour of the system of loading (i.e. the ‘exact’ load path travelled by the system load process vector is not defined explicitly). The procedure to perform ‘Development 2’ may be described as follows. First, depending on the particular load combination being investigated, all loads not contributing to each individual outcrossing are assigned particular values (i.e. held constant) prior to radial integration. Radial integration is then performed conditional to these particular values. During radial integration, individual outcrossings at each point (sja) are idealised as being caused by all of the loads not being held constant. Using the formulation presented for ‘Development 1’, the direction of the load process vector at each point of outcrossing is then ‘fixed’, to represent the combination of loads causing an outcrossing. Finally, integration over all possible values of the load(s) being held constant accounts for
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
the uncertainty in the particular load(s). This procedure implies some loads remain ‘fixed’ at the point of outcrossing, and therefore do not cause the outcrossing. To facilitate the integration requirements of ‘Development 2’, some directions AZa will be ‘fixed’ (i.e. instead of being random, as in ‘usual’ DS–LS); as will be shown, these directions are used solely to analyse situations with outcrossings caused by one load only (Qi, say), and hence evaluation of the term nC Di . The ‘complete’ formulation proposed herein to enable load combination analysis by DS–LS will now be derived. This will involve derivations of the formulation for ‘Development 2’, together with the ‘Development 1’ formulation derived earlier. The formulations will consist of those involving ‘fixed’ directions (for analysis of systems comprising two loads) and, for discussion, those involving random directions (for analysis of systems comprising three loads). The formulation for analysis of more than three loads is similar. Finally, the evaluation of all formulations by Monte Carlo and Importance Sampling will be derived.
To illustrate, the formulation for nC Di will be derived in the space of the loads Qi and Qj (with isj). Here, nC Di can be expressed in the form nC Di Z
Fig. 5. Typical ‘fixed’ direction in the 2D load space.
(cf Eq. (15))
qi Z sai C ci QððsjaÞjqj Þ Z sa Cc Z qj
nC Di ðqj ÞfQj ðqj Þdqj
(16)
C _ _ nC Di ððsjaÞjqj Þ Z E½Qn jQni ðsjaÞ fQi jQj ðsa CcÞ
(17)
nC Di
where is the value of evaluated conditional to the value QjZqj, fQj is the marginal pdf of Qj and the term fQj(qj) dqj—which is the ‘probability’ Qj will have the particular value qj—is the ‘equivalent’ of the probability pjO0 (for load Qj) in the ‘on-off’ model (Section 2.2), and is ‘built-in’ to the DS–LS calculations. Eq. (16) enables load Qi to cause each outcrossing, load Qj to be ‘held constant’ during radial integration and the uncertainty in load Qj to be accounted for. Details regarding its evaluation will now be discussed. 4.2. Fixed directions-characteristics of AZa and SZs Only one direction (AZa*, say) can exist in the region defined by QjZqj (Fig. 5); hence, this direction is a certain event, with ‘probability’ fA(a*)Z1. Also, the random variable S is ‘completely continuous’ (e.g. Walpole and Myers [16]), with SZsZ0 corresponding to the point CZc. Note that a variable ‘M’ is completely continuous when all particular values MZm change continuously within the range K N%m%N. 4.3. Fixed directions-evaluation of nC Di ðqj Þ The coordinates of the load process vector Q at the point SZs along the direction AZa* are given by the expression
(18)
where fQijQj is the pdf of Qi conditional to Qj. Hence, by integrating over all values SZs, the (mean) value of nC Di in the direction AZa* can be obtained from
N
nC Di ðqj Þ
where ai is the ith component of a*. Accordingly, the ‘local’ value of nC Di at the point (sja*) can be expressed as
4.1. Fixed directions-concept
C ðN
165
nC Di ða
jqj Þ Z
C ðN
nC Di ððsjaÞjqj ÞfSjA ðsjaÞds
KN C ðN
Z
E½Q_ n jQ_ ni ðsjaÞCfQi jQj ðsa CcÞfSjA ðsjaÞds
(19)
KN
where the limit state element surface DSD at each point (sja*) is constant. Since only the direction AZa* exists in the region defined by QjZqj, the (mean) value of nC Di evaluated conditional to QjZqj is given by the expression nC Di ðqj Þ Z
ð
C nC Di ðajqj ÞfA ðaÞda Z EAjQj ½nDi ðajqj Þ
AjQj
Z nC Di ða jqj Þ
(20)
where AjQj refers to all directions that can exist in the region defined by QjZqj, EAjQj is an expectation operator, nC Di ðajqj ÞZ C nC ða jq Þ is the value of n evaluated in the direction AZa j Di Di and the limit state element surface DSD is again constant. By integrating over all values of QjZqj (according to (16)) and evaluating the limit state element surface DSD at each point (sja*)—by dividing by the term ja $nðsjaÞj (Fig. 5)—the unconditional (mean) value of nC Di can be determined from
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the expression C ðN
nC Di
Z
nC Di ðqj ÞfQj ðqj Þdqj
N
2CN 3 ð _ n jQ_ ni ðsjaÞCfQijQj ðsa CcÞfSjA ðsjaÞ E½ Q Z EQj 4 ds5 ja nðsjaÞj KN
(21) where EQj is an expectation operator and the term fQijQj( ) is the ‘probability’ Qi will have the particular value qi(sja*) when Qj is ‘fixed’ with particular value qj. As expected, (21) is analogous to DS–LS Eq. (2). Limit State Hyperplanes: The numerical example presented herein involves a limit state function represented by a single (linear) hyperplane of the form (for example, for deterministic ‘s’) (Ditlevsen [3]) GðQÞ Z a0 C
nQ X
ai Q i
(22)
iZ1
where ai are constants (with a0O0). It follows that the term E[ ]C is independent of each point (sja) on the limit state, thereby reducing (21) to 2CN 3 ð f ðsa CcÞf ðsjaÞ Qi jQj SjA _ _ C 4 ds5 (23) nC Di Z E½Qn jQni EQj ja nðsjaÞj KN
4.4. Random directions-formulation To illustrate, the formulation for nC Dij will be derived in the space of the loads Qi, Qj and Qk (with isjsk). Using similar derivations to (18)–(21), the term nC Dij can be expressed in the form (Gray [6]) C ðN
nC Dij
Z
Z N
2
ð
6 4
4.5. Monte Carlo sampling (MC)
3 7 nC Dij ðajqk ÞfA ðaÞda5fQk ðqk Þdqk
All equations proposed in Sections 4.1–4.4 may be evaluated using well-known MC. For example, using a total of ‘N’ samples, the MC estimate of (16) becomes
AjQk
Z EQk EAjQk ½nC Dij ðajqk Þ
(24)
C where nC Dij ðqkÞ is the value of nDij evaluated conditional to the C value QkZqk and nDij ðajqk Þ is the value of nC Dij ðqk Þ evaluated in the direction AZa, and is given by the expression
nC Dij ðajqk Þ
surface (Fig. 6). As expected, (24) is analogous to DS–LS Eq. (2). Eq. (24) enables both loads Qi and Qj to cause each outcrossing, load Qk to be ‘held constant’ during radial integration and the uncertainty in load Qk to be accounted for. An important aspect of its evaluation will now be discussed. Characteristics of AZa and SZs: In ‘usual’ DS–LS analysis of a system comprising nQ loads (with nQO1), random directions AZa are generated on the surface of a nQdimension unit hypersphere. In contrast, they are now generated on the surface of a nA-dimension unit hypersphere, with 1!nA!nQ. For example, Fig. 6 shows a typical direction that is generated from a pdf fA which is assumed to be uniformly distributed on a (2D) circle and exists within the Qi– Qj plane (i.e. nAZ2); at each point (sja) the elemental surface area is DSe Z snAK1 da dqk Z sda dqk . Similar to ‘usual’ DS–LS analysis, the random variable ‘S’ is ‘positive continuous’, with SZsZ0 corresponding to the point CZc. Note that the term ‘positive continuous’ is used here to describe a variable ‘M’ when all particular values MZ m change continuously within the range 0%m%N.
nC Dij ðqk ÞfQk ðqk Þdqk
N C ðN
Fig. 6. Typical ‘random’ direction in the 3D load space.
N ð
Z
E½Q_ n jQ_ nij ðsjaÞCfQi Qj jQk ðsa C cÞfSjA ðsjaÞ janðsjaÞj
nC Di z
N 1 X nC ðq Þ N mZ1 Di jm
(26)
where qjm is the mth sample of qj drawn from fQj.
ds
0
(25) where division by the term ja$nðsjaÞj converts the element surface DSe into the equivalent surface DSD on the limit state
4.6. Importance sampling (MC-IS) The proposed equations may also be evaluated using well-known MC-IS. For example, the MC-IS estimate of
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
(16) becomes C ðN
nC Di
Z
nC Di ðqj Þ
N N X
! mZ1
fQj ðqj Þ hQj ðqj Þ
hQj ðqj Þdqj z
"
fQj ðqjm Þ nC Di ðqjm Þ hQj ðqjm Þ
1 N
# (27)
where hQj is an importance sampling pdf—details for its construction herein are given by Gray [6]. It is assumed formulation similar to (27) can be derived to evaluate the term nC Di ðqj Þ by MC-IS; however, this is not considered herein. 5. Monte carlo integration By performing analysis in original ‘X’ space (which C comprises all variables), the terms nC Di , nDij , etc. can be evaluated by well-known MC and MC-IS, using the formulations given by Gray [6]. This is considered herein because there are no presumptions in its formulation (for example, unlike the assumption that the pdf fA in (24) is uniformly distributed), and therefore it may be used to obtain unbiased results to compare results obtained using all DS–LS formulation proposed herein.
where mQ1 Z mQ1 Z 20 and sQ1 Z sQ1 Z 2; QC2 describes the situation where mQ1Z20, sQ1Z2, mQ2Z5 and sQ2Z0.5; QC3 describes the situation where mQ1Z5, sQ1Z0.5, mQ2Z20 and sQ2Z2. Clearly, load cases QC2 and QC3 represent illproportioned load spaces. Two load correlation cases (RC1 and RC2) were examined, as follows: RC1 describes the situation where rQijZ0.0 (i.e. the loads are fully independent) and RC2 describes the situation where rQijZ0.5. Individual directions AZa (during DS–LS) and QZq (during directional simulation in H-Space) were generated either systematically (e.g. Melchers [12]), by ‘The Hyperspace Division Method (HDM)’ (Katsuki and Frangopol [8]), or randomly, by Monte Carlo (MC). Convergence of the terms nC Di and nC Dij was examined by considering the convergence of both: (1) the mean outcrossing rate E[ ]C at any point (sja) on the limit state surface; and (2) the mean outcrossing rate nC D . Some critical aspects of this will now be discussed. 6.1. Convergence of E[ ]C and nC D Convergence of E[ ]C: Numerical estimates of E[HjHi]C and E[HjHij]C were used to evaluate E[ ]C, using the wellknown expression (cf (5)) E½HC Z
6. Numerical example
nQ X
E½HjHij C
iZ1
Because systems seldom comprise more than 2 or 3 loads, C and because the pdf fSjA (and hence nC Di , nDij , etc.) is, in general, difficult (and computationally demanding) to evaluate for even simple structural systems, a numerical example demonstrating support for proposed Eqs. (16) and (13) (i.e. a system comprising 2 loads only) will now be presented. Further examples (involving 2 or more loads), and a detailed discussion of all results presented herein, are given by Gray [6]. C The terms nC Di and nDij were evaluated for a single linear limit state function G(Q1,Q2), which is a function of two (stationary) continuous gaussian loads Q1 and Q2 (which may be correlated) and involves deterministic ‘s’ (as defined for DS–LS). The following two limit states were analysed separately: G1 Z G1ðQ1 ; Q2 Þ Z a0 K3Q1 K3Q2 G2 Z G2ðQ1 ; Q2 Þ Z a0 K3Q1
167
(28)
where a0O0 is a constant and QiZQi(t) is a time-variant load. Clearly, both loads can cause failure of G1, whereas only Q1 can cause failure of G2. The term ‘Load Case (QC)’ was used herein to describe the values of the first and second moments of the loads in a given system (i.e. the mean mQi and standard deviation sQi of each load Qi) and the term ‘Load Correlation Case (RC)’ was used to describe the corresponding correlation between the loads (i.e. the correlation coefficient rQij between two loads Qi and Qj, with isj). Three load cases (QC1, QC2 and QC3) were examined, as follows: QC1 describes the situation
nQ X nQ X
C iZ1 jZ1
sH E½HjHi C C/Z pffiffiffiffiffiffi 2p
(29)
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} isj
with vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u nQ uX n2i COVQ_ ði; iÞ sH Z t iZ1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ iÞ Z r 00 ð0ÞCOVQ ði; iÞ COVQði;
(30) r 00 ð0Þ Z
vrðsKtÞ vsvt
where COVQ is the covariance matrix of Q and r(sKt) is a correlation function of time points ‘s’ and ‘t’. The value r 00 (0)Z0.5 is used herein. Using the point EZeZ0 in (11), a total of 40,000 sample directions was required. This ‘large’ value was needed to adequately sample the pdf fH1.HnQ because the magnitude of the statistical moments of Q1 and Q2 caused the pdf to be relatively ‘tall and thin’. Similar examples involving three loads show that the number of sample directions required may increase significantly with the number of loads (Gray [6]). C C Convergence of nC D : Numerical estimates of nDi and nDij C were used to evaluate nD (using (5)); this was compared with the analytical value of nC D , which was obtained as follows. Since each limit state is expressed in the (linear) form given by (22) and all variables are normal distributed, the analytical expression for the mean outcrossing rate nC D is
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Table 2 Notation for methods Notation
Method a
DS–LS LC1 (HDM)
DS–LS LC1
DS–LS LC2 (SR) DS–LS LC2 (MC) DS–LS LC2 (MC–IS)
DS–LS LC2b
MCc MC–IS
Monte Carlo Importance sampling
a b c
Comments on integration/simulation
Outcrossing rates evaluated
Simulation by ‘Hyperspace Division Method’ Integration of load pdf’s by ‘Simpsons Rule’ Integration of load pdf’s by ‘Monte Carlo’ Integration of load pdf’s by ‘Importance Sampling’ – –
C nC Di , nDij C nC Di , nDij
C nC Di , nDij C C nDi , nDij
‘Usual’ DS–LS, but with ‘Development 1’ (Section 3) performed at each time of outcrossing. ‘Development 1’ (Section 3) used in combination with ‘Development 2’ (Section 4). The notation ‘MC-i’ is used in some figures; this means ‘sampling from fQi’.
given by the well-known expression (Ditlevsen at al [4]) sffiffiffiffiffiffiffiffiffiffiffi nQ P a2i ð m iZ1 G C nC fQ ðqÞdq Z E½Cf (31) D Z E½ sG sG
Similar examples involving three loads show that the number C of ‘samples’ required (and hence CPU time) to evaluate nC D , nDij C and nDijk , etc. may increase significantly with the number of loads (Gray [6])
with
Typical plots of results are given in Figs. 7 and 8. For C clarity, only mean results are shown (i.e. E½nC Di and E½nDij ). The majority of results were found to compare well. Fig. 7 shows that all results obtained by methods using importance sampling were computationally superior (in terms of convergence) to results obtained by methods using Monte Carlo sampling. The ill-proportioning effect of the load space was found to have little effect on the accuracy of all results. An important aspect of the results will now be discussed.
SD
mG Z a0 C a1 mQ1 C a2 mQ2 s2G
vGðQÞ vGðQÞ T Z COVQ vQ vQ
(32)
where mG and sG are the mean and standard deviation, respectively, of G, vG(Q)/vQ is a vector containing derivatives of G with respect to Q and f is the Standard Normal Probability Density Function. Calculations were performed by DS–LS, MC and MC–IS, with notation given in Table 2. A total of: 700 sample directions was needed by DS–LS LC1(HDM); 6E5 samples was needed by DS–LS LC2(MC) and MC; 8E3 samples was needed by DS–LS LC2(MC–IS) and MC–IS. The majority of results obtained required what may be considered negligible computation (CPU) time. DS–LS LC1 (HDM) was found to be computationally superior, whereas calculations involving Monte Carlo sampling were found to require the most CPU time. In all results, the minority of the computational effort was required for the calculation of the term E[ ]C.
Fig. 7. Results for the combination G2, QC1, RC2.
6.2. Numerical results
C 6.2.1. Relationship between nC Di and nDij For situations with outcrossings caused by any combination of Q1 and Q2 (i.e. GZ GðQ1 ; Q2 Þ), the relationship nC D12 R C maxfnC ; n g was displayed by all results (e.g. Fig. 8), with D1 D2 max{ } representing the maximum value of the arguments in { }. This result was due to the relationship between the terms E[HjHi]C and E[HjHij]C, as follows. Because each load Qi(t) is stationary, continuous and Gaussian, it’s time derivative Q_ i ðtÞ—and therefore each variable Hi Z Q_ ni Z nðsjaÞQ_ i ðtÞ—is also stationary, continuous
Fig. 8. Results for the combination G1, QC1, RC2.
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
and Gaussian (but with zero mean). Hence, as shown by Fig. 4(a), the region in the 2D H-space where the conditions for which an outcrossing at any point (sja) in the load space are caused by both load processes are satisfied (‘region ij’, say) comprises larger values of ‘h’ and hfH1 .HnQ ðh1 ; .; hnQ Þ than the regions where the conditions for which an outcrossing at the same point are caused by only one load process are satisfied (‘region i’, say). It follows immediately that individual values of the product hfH1 .HnQ ðh1 ; .; hnQ Þ are greater in magnitude in ‘region ij’ than in ‘region i’. Hence, based on (6) and (9), the relationship E½HjHij CR maxfE½HjHi C; E½HjHj Cg holds, so that the above-mentioned relationship nC D2 R maxf$g is to be expected. Evidently, these relationships can be physically interpreted from Fig. 3; it shows that outcrossings in the load space that are caused by either Q1 only or Q2 only can occur only along one of the two exclusive directions shown, whereas outcrossings that are caused by Q1 and Q2 outcrossing simultaneously can occur over the much broader range of directions shown. Hence, one would expect an outcrossing caused by both Q1 and Q2 to provide a greater contribution to nC D than an outcrossing caused by only one load. 6.2.2. Implications of results The results obtained herein suggest that individual outcrossings are more likely to be caused by combinations of C C loads, than individual loads (i.e. nC Dij R maxfnDi ; nDj g). This implication appears at first to contradict what has been published previously in the literature. For example, results obtained by Wen [18] using the ‘Load Coincidence Method’ (which requires the loads to be represented by the ‘on-off’ model (Section 2.2) typically indicated that failure was most likely to be caused by individual loads Qi acting ‘alone’ (i.e. they have qiZ1, and all loads Qj not acting have pjZ1). This implication is assumed to have occurred for the following two reasons: 1. The ‘standard’ model (Section 2.2) was used to represent each load, thereby implying each load Qi is effectively always ‘on’ (i.e. qi/1). It follows that all loads are likely to be constantly changing and therefore all loads are more likely to contribute to failure than only one or a few loads. In contrast, Wen [18] used the ‘on-off’ model, and each load Qi was assumed to be ‘off’ most of the time (i.e. pi/ 1), so that it was less likely for multiple loads to be ‘on’ simultaneously (and therefore less likely they would all contribute to failure), and 2. Each load was modelled as being (completely) ‘continuous’, so that it’s value changes continuously (though not necessarily constantly) in time. In contrast, Wen [18] modelled each load as being ‘discrete’, so that it’s value changes either only at discrete points in time, or only within discrete periods of time. Clearly a vector process comprising continuous loads is more likely to be ‘moving around’ in the load space (and hence outcrossing due to the combined behaviour of the loads) than one comprising discrete loads.
169
This reasoning appears to agree with results that have been published in the literature. For example, Wen and Chen [17] examined two poisson pulse processes (i.e. processes represented by a random occurrence time, length of duration and non-negative intensity (Ditlevsen and Madsen [5])). When both processes were ‘sparse’ (i.e. pi/1), it was found that, ‘.only 7.4% of the total failure can be attributed to the simultaneous action of the two loads because of their transient nature.’ (pp. 816). However, when the mean load duration was increased by a factor of 10 (i.e. qi was increased), it was found that, ‘.the load coincidence accounts for 45%).’ (pp. 816). The reason for this trend is that the ‘probability of occurrence’ of combinations of loads decreases with the ‘sparseness’ of each load C (Wen [19]), so that nC Di OOnDij . This reasoning (as well as the discussed published results) suggests that the trends displayed C by the terms nC Di , nDij etc may be influenced significantly by the way in which the loads are modelled.
7. Conclusions In this paper the ‘Directional Simulation in the Load Space (DS–LS)’ approach to reliability analysis was developed to consider explicitly outcrossings caused by all possible combinations of loads, during analysis of systems comprising stationary continuous Gaussian loads. A relatively simple numerical example comprising two loads and one linear limit state function was presented to demonstrate the validity of the proposed formulation. The results obtained by DS–LS were compared with equivalent Monte Carlo results derived from well-established theory. The majority of the results were found to compare well. For the example considered herein, it was found—for situations with outcrossings caused by any combination of loads (i.e. there was no restriction to failure being caused by only one load)—that failure was more likely to be caused by combinations of loads than individual loads. It was argued that this finding was achieved because of the way in which the loads were modelled—for example, due to the stochastic process model used to represent the loads. This reasoning appears to agree with results that have been published previously in the literature. Acknowledgements The funding and support from The University of Newcastle for this research project is gratefully acknowledged. References [1] Belyaev YK. On the number of exits across the boundary of a region by a vector stochastic process. Theory Probab Appl 1968;13(2):320–4. [2] Bjerager P. Probability integration by directional simulation. J Eng Mech 1988;114(8):1285–302. [3] Ditlevsen O. Gaussian outcrossings from safe convex polyhedrons. J Eng Mech 1983;109(1):127–48. [4] Ditlevsen O, Bjerager P, Olesen R, Hasofer AM. Directional simulation in gaussian processes. Probab Eng Mech 1988;3(4):207–17.
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[5] Ditlevsen O, Madsen HO. Structural reliability methods. Chichester: Wiley; 1996. [6] Gray WA. Generalisation of the directional simulation in the load space approach to structural reliability analysis. PhD Thesis. The University of Newcastle, Australia: 2004. http://www.newcastle.edu.au/services/liberary/ adt/public/adt-NNCU20050728.134322/index.html [7] Guan XL, Melchers RE. A load space formulation for probabilistic finite element analysis of structural reliability. Prob Eng Mech 1998;14:73–81. [8] Katsuki S, Frangopol DM. Hyperspace division method for structural reliability. J Eng Mech 1994;20(11):2405–27. [9] Melchers RE. Load space formulation of time dependent structural reliability. J Eng Mech ASCE 1992;118(5):853–70. [10] Melchers RE. Structural system reliability assessment using directional simulation. Struct Saf 1994;16(1 and 2):23–37. [11] Melchers RE. Load space reliability formulation for poisson pulse processes. J Eng Mech ASCE 1995;121(7):779–84. [12] Melchers RE. Structural reliability analysis and prediction. 2nd ed. Chichester: Wiley; 1999.
[13] Rice SO. Mathematical analysis of random noise. Bell Syst Technol J 1944;23:282–332 [1945; 24:46–156]. [14] Turkstra CJ, Madsen HO. Load combination in codified structural design. J Struct Div ASCE 1980;106(ST12):2527–43. [15] Veneziano D, Grigoriu M, Cornell CA. Vector-process models for system reliability. J Eng Mech Div ASCE 1977;103(EM3): 441–60. [16] Walpole RE, Myers RH. Probability and statistics for engineers and scientists. 2nd ed. New York: Macmillan; 1978. [17] Wen YK, Chen H-C. System reliability under time-varying loads I. J Eng Mech 1989;115(4):808–23. [18] Wen YK. Statistical combination of extreme loads. J Struct Div ASCE 1977;103(ST5):1079–93. [19] Wen YK. Structural load modelling and combination for performance and safety evaluation. Developments in civil engineering, vol. 31, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA. Amsterdam: Elsevier; 1990.
Load combination analysis by ‘Directional simulation in the load space’ William A. Gray, Robert E. Melchers * The University of Newcastle, Callaghan, NSW, 2308, Australia Received 23 August 2004; received in revised form 8 August 2005; accepted 10 August 2005 Available online 14 October 2005
Abstract The reliability of structures subjected to multiple time-varying random loads is considered herein. It is well-known that the reliability of such systems may be evaluated by considering outcrossings of the load process vector out of a safe domain, and the contribution of individual loads to structural failure may be evaluated by considering outcrossings caused by combinations of one or more loads. In this paper the ‘Directional Simulation in the Load Space’ approach to reliability analysis is developed to consider explicitly outcrossings caused by all possible combinations of loads, during analysis of systems comprising stationary continuous Gaussian loads. To do this, the direction of the load process vector is ‘fixed’ at each point of outcrossing (to physically represent the particular combination of loads causing the outcrossing), and, by considering each possible load combination, all loads not causing an outcrossing are then held constant during radial integration (to model correctly that they do not contribute to each outcrossing). A numerical example demonstrating the validity of the proposed formulation is presented. q 2005 Elsevier Ltd. All rights reserved. Keywords: Directional simulation in the load space; Time-dependent reliability; Outcrossings; Load combinations; Load modelling
1. Introduction One approach to handling systems involving time-varying loads is to reduce the time-variant problem to a time-invariant one. For problems with multiple loads, this then requires employing a suitable (but approximate) load combination rule (e.g. ‘Turkstra’s Rule’ (Turkstra [14])), that, typically, involves analysing combinations of (time-invariant) extreme value and ‘average’ loading to allow for the fact that all loads might not be at their extremes at the time of failure. However, this procedure does not accurately account for the time-variant behaviour of individual loads, or combinations of loads. It is well-known that by modelling the loads as (timevariant) random processes (implying that high-level loads are applied more than once in a given lifetime), the failure probability may be evaluated by considering the outcrossing of a vector process out of a safe domain. In general, the probability of failure occurring in the time interval 0Kt (i.e. pf(t)) is given by the well-known upper bound (c.f. Veneziano et al. [15]) pf ðtÞ% pf ð0Þ C ð1Kpf ð0ÞÞð1KeKnt Þ
(1)
* Corresponding author. Tel.: C61 2 4921 6044, fax: C61 2 4921 6991. E-mail address: [email protected] (R.E. Melchers).
0266-8920/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2005.08.001
where pf(0) is the probability the structure will fail on first loading (i.e. at tZ0) and is obtained by time-invariant reliability analysis, and n (denoted herein as nC D ) is the mean outcrossing rate of a vector process out of a deterministic safe domain D (Belyaev [1]). However, it is possible—at the time of failure—for only one or a few loads to be actively changing in value (and therefore causing failure). To account for this the treatment of vector-outcrossings may need to allow for the possibility of outcrossings being caused by individual loads, as distinct from combinations of all loads. The procedure used to analyse combinations of timevarying loads depends on the stochastic process model used to represent the loads. Two well-known load models that have been presented in the literature are herein referred to as the ‘onoff’ model (e.g. Wen [19]) and the ‘standard’ model (Fig. 1). The ‘on-off’ model assumes loads are either ‘on’ (e.g. their value is non-zero and constantly changing) or ‘off’ (e.g. their value is strictly zero). They can contribute to failure only when they are ‘on’. To implement this model in load combination analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’. In contrast, the ‘standard’ model (which is more commonly used, and is considered herein) effectively assumes each load is always ‘on’. The possibility of one or more loads not causing failure (i.e. such loads are effectively ‘off’ at the time of failure) needs to be allowed for. A procedure to enable this, during analysis by the time-variant
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Fig. 1. The ‘On-Off’ Model.
analysis method ‘Directional Simulation in the Load Space (DS-LS)’ (Melchers [9]) will be described herein. In DS–LS, the loads are separated from all other variables and analysis is performed in the load space. This is convenient for time-variant analysis as it enables the loads to be represented as random processes, and to be modelled properly. The technique has so far been shown to work well for timevariant reliability analysis of relatively simple structural systems (e.g. Melchers [10]), and simple combinations of loads represented by the ‘on-off’ model (e.g. Melchers [11]). It has also been used to examine vector-outcrossings in systems comprising loads represented by the ‘standard’ model (e.g. Melchers [9]); however, it appears not to have considered outcrossings caused by explicit combinations of only one or a few such loads. Developments to DS–LS to achieve this are proposed in this paper. In Section 2 a review of DS–LS and stochastic process models used to represent loads will be given. Developments to DS–LS will be derived in Sections 3 and 4. Equivalent Monte Carlo expressions will then be discussed briefly in Section 5, for comparison in the numerical example that will be presented in Section 6. 2. Review of time-variant structural reliability analysis
Fig. 2. Directional simulation in 2D Q space.
occurs if any system of loading QZq is in the ‘failure’ region. Since, for each value QZq, the realisation g(q,x)Z0 exists as a ‘probabilistic’ boundary, the location of the safe and failure regions may be defined only probabilistically. It follows that nC D denotes the mean rate at which the load process vector QZ q(t) crosses out of the ‘safe’ region, and is obtained by integrating over all outcrossing rates that are evaluated conditional to each realisation G(Q,X)Zg(q,x)Z0. A time-variant DS–LS expression will now be reviewed. The expression may be used specifically for analysis of systems comprising stationary continuous gaussian loads, as it is only this type of load that is considered by the numerical example presented herein. 2.1.1. Time-variant DS–LS expression The ‘usual’ generalised DS-LS expression for nC D is given by (Gray [6], modified from [9]) ð nC Z nC D D ðaÞfA ðaÞda A
2.1. Directional simulation in the load space (DS–LS) In DS-LS the loads are separated from all other variables and ‘Directional Simulation’ (e.g. Bjerager [2]) is performed in the load space. This is often convenient because (i) there are usually only a small number of loads compared with other variables (therefore the overall dimension of space is reduced), and (ii) DS–LS enables the loads to be modelled properly. Let the system being considered be represented by the vectors Q (comprising a total of nQ random loads) and X (comprising a total of nX variables-all variables other than loads). It follows that the system structural resistance is represented by a nQ-dimension vector of random resistance RZR(X), comprising individual resistances Ri (with Ri corresponding to each load Qi in Q). The load capacity of the structure is reached when qZr (where q and r are realisations of Q and R, respectively), and qRr represents failure. The system limit state function G(Q,X), which is an expression that relates Q to X, is constructed piece-wise, from individual limit state functions Gi(Q,X). The realisation G(Q, X)Zg(q,x)Z0, which will be referred to herein as the ‘limit state’, separates the load space into a ‘safe’ (i.e. g(q,x)O0) and ‘failure’ (i.e. g(q,x)%0) region, where q and x represent realisations of Q and X, respectively (Fig. 2). System failure
2N ð
4 Z SnQ A EA
3 E½Q_ n ðsjaÞCfSjA ðsjaÞfQ ðsa C cÞsnQK1 ds 5 janðsjaÞj
(2)
0
where fA is the probability density function (pdf) of the C direction variable A, nC D (a) is the value of nD evaluated in the nQ direction AZa, SA is the surface area of the nQ-dimension unit (hyper-) sphere, EA is an expectation operator, SZsR0 is a random variable used to allow for uncertainty in the limit state (in the direction AZa) due to variables in X, fSjA is the pdf _ vQðtÞ=vt is the load of ‘S’ in the direction AZa and QZ process time derivative vector. Also, the term E[ ]C—which will be described later-is the mean outcrossing rate at the point (SZsjAZa), CZc is the origin of DS–LS (and is related to R and S by the relationship RZS$ACC), fQ is the system load pdf and n(sja) is the limit state unit outward normal vector at the point ðsjaÞ. Here, E[ ]C is the mean outcrossing rate, given that the limit state function is positioned at the point ðSZ sjAZ aÞ and the load process vector is at the point QZ qðSZ sjAZ aÞ. This term is weighted by the ‘probability’ fSjA that the limit state function is at the point (SZsjAZa) and the ‘probability’ fQ(s$aCc) that the load process vector is at the point QZ qðSZ sjAZ aÞ. Also, division by the term jan(sja)j
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161
converts the elemental surface area DSeZsnQK1da (which is normal to AZa (Fig. 2)) into an equivalent area DSD on the limit state surface (through which individual outcrossings occur); the absolute value j j ensures the surface area is positive if the point CZc is positioned in the failure region (Gray [6]). Some important features of (2) will now be discussed. 2.1.2. Deterministic ‘s’ and random ‘S’ The variable ‘S’ may exist as either a deterministic (i.e. ‘s’) or random variable (i.e. ‘S’) quantity in a given direction AZa, depending on the statistical properties of the quantities in X contributing to G(Q,X). If all such quantities are deterministic, then ‘s’ is also deterministic (i.e. the failure boundary can be defined deterministically at one or more points in the direction AZa), and fSjA is given by fSjA ðsjaÞ Z 1 if gðsjaÞ Z 0 Z0
otherwise
(3)
Conversely, ‘S’ is a random variable (i.e. the failure boundary exists as a random function in the direction AZa) if one or more such quantities exist as random variables. Procedures to evaluate fSjA for such situations have been discussed in the literature (e.g. Guan and Melchers [7]). Although defined for the DS–LS approach, the terms Deterministic ‘s’ and Random ‘S’ will be used in a generic sense throughout this paper, to describe the uncertainty in each limit state due to variables other than loads. 2.1.3. General assumptions The following two general, but important, assumptions are made by (2): (i) Load Path Independence, and (ii) ‘Star-shaped’ Integration Region. These assumptions are also made by the developments proposed herein, and are discussed briefly as follows. Load path independence: This is the assumption that ‘an outcrossing does NOT depend on the path travelled by the load process vector up to the point of outcrossing’ (e.g. Ditlevsen and Madsen [5], Melchers [12]). Herein, this means an outcrossing can occur at any point (sja), irrespective of how the load process vector reaches that point. Note that the concept of Load Path Independence is independent of the load type (i.e. continuous or discrete), and its practical implications are given elsewhere in the literature. Star-shaped integration: Due to the considerable complexity involved with a limit state being intercepted at multiple points in a given direction AZa (cf Bjerager [2], Fig. 3b), herein it is assumed the limit state is intercepted only once in a given direction. This means the integration region is ‘starshaped’ (Bjerager [2]) with respect to the origin of DS–LS. 2.1.4. Outcrossing of load process vector The term E½Q_ n ðsjaÞC in (2) is described as follows. Here, each component ni(sja) of n(sja) and Q_ i Z vQi =vt of Q_ corresponds to each component Qi of Q. Also, the term _ ni ðsjaÞ. Q_ i represents the component of Q_ i that is Hi Z QZ normal to the limit state, with hi Z q_ ni representing its
Fig. 3. Possible directions of an outcrossing in the 2D load space.
particular value. Clearly, the particular value hZ q_n ðsjaÞ of the scalar quantity H (which is given by the dot product _ represents the sum of all particular H Z Q_ n ðsjaÞZ nðsjaÞQ) values hi (for 1%i%nQ), and therefore the rate at which Q exits (or enters) the safe region at the point (sja). An outcrossing at the point (sja) will occur only if hO0; it follows that E½HCZ E½Q_ n ðsjaÞC is the mean outcrossing rate at the point (sja), and may be evaluated from the expression (e.g. Melchers [9]) N ð
sH E½H Z hfH ðhÞdh Z pffiffiffiffiffiffi 2p C
(4)
0
where fH and sH are the pdf and standard deviation respectively of H. Eq. (4) does not consider explicitly combinations of loads causing an outcrossing at each point (sja)—any load combination can implicitly cause an outcrossing if the condition E[ ]CO0 is satisfied. This implies that the direction of the load process vector at any point (sja) is a random variable, such that the load process vector can ‘cross out’ in any outward direction. To consider explicitly specific load combinations causing each outcrossing, the direction of the load process vector needs to be ‘fixed’ at each point of outcrossing, according to the particular load combination being considered. For example, Fig. 3 shows the possible directions in which an outcrossing at any point (sja) in the 2D load process space can be caused by either one load only or two loads. The concept of ‘fixing’ the direction of the load process vector is herein termed ‘Development 1 for Load Combination Analysis by DS–LS’, and will be derived below. Before proceeding however, two stochastic process models used to represent individual loads will be reviewed briefly, and the implementation herein of one such model in load combination analysis performed by DS–LS will be described. 2.2. Stochastic process models used for load combination analysis In a given time interval, time-varying loads can be idealised as being either ‘active’ (i.e. their value changes constantly) or ‘inactive’ (i.e. their value is strictly zero, or constant at some non-zero value). The procedure used to analyse combinations of time-varying loads depends on the stochastic process model used to represent these state changes (i.e. without loss of
a
4
3
2
b
ni is the mean upcrossing rate of Qi (Rice [13]). Prob( ) denotes probability.
Qj
C Evaluate nC Dj , nDij
Repeat steps 1-3, to consider all possible load combinations
ð b Z nC Di jqj fQj ðqj Þdqj
nC Di jqj : Probðqj Þ
nC Di Z ni qi pj qj
X
a C Evaluate nC D for Qi acting alone (i.e. nD Z ni )
Evaluate nC Di jqj
nC Di z
Set QjZqjZ0 Identify Qj
Proposed DS–LS
Identify Qi 1
Step
Table 1 Procedure to evaluate load combinations
Assume a combination of loads that will cause failure (i.e. the ‘active’ loads) Identify all loads not causing failure (i.e. the ‘inactive’ loads) ‘Fix’ the value of all ‘inactive’ loads, by holding their value constant at the time of failure Conditional to the ‘fixed’ values, evaluate the system failure for the condition of failure caused by all ‘active’ loads Integrate over all possible ‘fixed’ values of the ‘inactive’ loads, to allow for the probability that the loads are ‘inactive’
Typical calculation of nC Di
Set QjZqjZany value
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Equivalent ‘On-Off’
162
generality, the pdf fQi of load Qi). Two well-known load models that have been presented in the literature are referred to herein as the ‘on-off’ model (e.g. Wen [19]) and the ‘standard’ model. They will be described below, along with the implementation herein of the ‘standard’ model in load combination analysis performed by DS–LS. ‘On-off’ model: The ‘on-off’ model assumes each load is either ‘on’ (e.g. its value is non-zero, and constantly changing) with probability pi, or ‘off’ (e.g. its value is strictly zero) with probability qiZ1Kpi (Fig. 1). It typically assumes loads are non-negative-its pdf might be obtained, for example, by truncating the load ‘average value’ or ‘extreme value’ pdf to allow positive values only. Loads can contribute to failure only when they are ‘on’; to implement this model in load combination analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’, using combinations of pi and qi. ‘Standard’ Model: This is more commonly used, and is considered herein. Unlike the ‘on-off’ model, it can allow each load to be negative; its pdf is typically equal to the load ‘average value’ or ‘extreme value’ pdf. It effectively assumes each load is always ‘on’ (i.e. piZ0); to apply this model in load combination analysis, the possibility of load Qi not causing failure (i.e. the ‘equivalent’ of piO0) must be allowed for, as follows. Implementation of ‘Standard’ Model: The system load pdf fQ essentially describes the probability that Q is ‘moving around’ in the load space; it does not allow for the probability of one or more loads Qi not causing failure (i.e. the ‘equivalent’ of piO0). However, this probability can be derived from fQ and fQi (and subsequently evaluated by DS–LS) as follows. C Denoting nC Di , nDij as the mean rate at which outcrossings are caused by load Qi only, and both loads Qi and Qj only respectively, it is proposed herein that each of the terms nC Di , nC etc be evaluated by DS–LS, using the procedure described Dij in Table 1. It shows how the value of each load not causing failure is held ‘constant’ at the time of failure, while the value of all remaining loads is allowed to change. For clarity, Table 1 also demonstrates typical steps taken during DS–LS to evaluate nC Di for a system of two loads Qi and Qj, and the ‘equivalent’ steps that would be taken to examine loads represented by the ‘On-Off’ model. In doing so, it demonstrates how the probability pjO0 (of load Qj) is ‘built-in’ to the DS–LS calculations (using the term Prob(qj)ZfQj(qj)dqj); further details of this will be given later in this paper. To consider all possible combinations of loads that can cause each vector outcrossing, the following expression for nC D is proposed:
nC D Z
nQ X iZ1
nC Di C
nQ X nQ X
nC Dij C.
(5)
iZ1 jZ1
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} isj
and in what follows, the developments to the ‘usual’ DS–LS C formulation that enable the terms nC Di , nDij , etc. to be evaluated
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(specifically for application to systems comprising stationary continuous gaussian loads) will be derived.
E½Q_ n jQ_ nij ðsjaÞC C ðN
3. Development 1 for load combination analysis by DS–LS
Z
2CN ð .4 Iij ½hfH
KN
To evaluate the mean rate E[ ]C at which outcrossings at any point (sja) are caused by a specific load combination, the conditions enabling outcrossings to be caused by the specific load combination need to be considered, as follows. For clarity, outcrossings caused by either one load only (Qi(t), say) or two loads only (Qi(t) and Qj(t), say, with isj) will be considered, and time ‘(t)’ will be omitted. Consider first the situation with outcrossings caused by Qi only. An outcrossing will occur at the point (sja) only if hO0; an outcrossing will be caused by Qi only if hiO0 and hj%0 (for all 1%j%nQ, with jsi). Hence, the mean rate E½Q_ n jQ_ ni ðsjaÞC at which outcrossings at the point (sja) are caused by Qi only may be expressed as the integral 2CN 3 CN ð ð E½Q_ n jQ_ ni ðsjaÞC Z .4 Ii ½hfH1 .HnQ ðh1 ; .; hnQ ÞdhnQ 5dh1 KN
KN
(6)
with ‘h’ given by the sum hZ
nQ X iZ1
hi Z q_n Z
nQ X
q_ ni
(7)
iZ1
where fH1 .HnQ is the joint pdf of the independent variables H1,.,HnQ and Ii[ ] is an indicator function given by the expression Ii ½ Z 1
if ½ðhO 0Þ and ðhi O 0Þ and ðhj % 0Þ c1%j%nQ ðj siÞ
(8)
Z 0 otherwise Similarly, an outcrossing will be caused by both Qi and Qj only if hO0, hiO0, hjO0 and hk%0 (for all 1%k%nQ, with ksjsi), so that the mean rate E½Q_ n jQ_ nij ðsjaÞC at which outcrossings at the point (sja) are caused by both Qi and Qj only becomes the integral
163
3 1 .HnQ
ðh1 ; .; hnQ ÞdhnQ 5dh1
(9)
KN
where Iij[ ] is an indicator function given by the expression Iij ½ Z 1 if ½ðhO0Þ and ðhi O0Þ and ðhj O0Þ and ðhk %0Þ c1%j%nQ ðjsiÞ;c1%k%nQ ðksjsiÞ Z0 otherwise (10)
Expressions for E[ ]C for other load combinations can be obtained similar to (6) and (9). For systems comprising only two loads (Q1 and Q2, say), the regions of integration for (6) and (9) are shown in Fig. 4(a). Herein, expressions for E[ ]C are evaluated by directional simulation in H-space, using a procedure that will be described below. A discussion on the use of these expressions to evaluate nC Di , nC Dij etc, using a procedure similar to (2) (i.e. simulating directions on the entire surface of the nQ-dimension unit hypersphere), and the need for an additional development to DS–LS to model the loads properly during load combination analysis will follow this. 3.1. Evaluation of E[ ]C by directional simulation Consider a typical direction QZq generated from an origin of directional simulation EZe (Fig. 4(b))). Evidently, the scalar variable H and a vector variable U-with particular value UZuZ[h1 . hnQ]-are related to E, Q and the radius PZr by the relationships UðPjQÞZ P:QC E and HZ hðr$qC eÞZ sumfug, where sum{ } is the sum of the elements in the vector described by { }. The term E½ZQ_ n jQ_ nij ðsjaÞC, for example, may be obtained by integrating over all values PZr
Fig. 4. Integration in H-space.
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and all directions QZq, according to the expression ð E½C Z E½qCfQ ðqÞdq Q
2N 3 ð 4 Ii ½rq C ehðr:q C eÞfH .H ðrq C eÞrnQK1 dr5 Z SnQ A EQ 1 nQ 0
(11) where E[jq]C is the value of E[ ]C in the direction QZq, fQ is the pdf of Q, EQ is an expectation operator, fH1 .HnQ ðrqC eÞrnQK1 dr is the hypervolume under fH1 .HnQ at the point (rjq), and details regarding the numerical evaluation of (11) are given in Section 6.1. C 3.2. Evaluation of nC Di and nDij and load modelling implications C Using ‘Development 1’, the terms nC Di , nDij etc can be C expressed in a form similar to (2). For example, nC Di and nDij may be expressed as ð C nC Di Z nDi ðaÞfA ðaÞda A
2N 3 ð C nQK1 _ _ 4 E½Qn jQni ðsjaÞ fSjA ðsjaÞfQ ðsa C cÞs ds5 Z SnQ A EA janðsjaÞj 0
(12) and
ð Z nC nC Dij Dij ðaÞfA ðaÞda
model, in which all loading is governed by one variable (i.e. ‘S’). In (12) and (13), this model implies all loads change ‘in proportion’, and therefore all simultaneously cause each outcrossing. Although this implication is generally strictly true only when all loads are fully dependent (i.e. the correlation coefficient ‘r’ between all loads is the limit rZ1.0), it means that when the loads are not fully dependent, (14) should only be used to evaluate the mean rate at which outcrossings are caused by all loads (i.e. nDi.nCQ ). Furthermore, the ‘one-parameter’ model implies all loads constantly change value in time (and therefore at the time of outcrossing), such that the probability of an outcrossing being caused by only one or a few loads is approximately zero. Unless the loads are fully dependent, there is actually a finite non-zero probability that one or more loads may remain ‘approximately’ constant at the time of outcrossing, so that the ‘one-parameter’ model does not necessarily model the loads realistically. For example, for extreme wind load Qi acting on a building and causing an outcrossing while the building occupancy load Qj remains approximately constant at the time of outcrossing, the loading is more appropriately modelled by the expression " # qi Z sai C ci Q Z qðsjaÞ Z sa C c Z (15) qj Z constan t Hence, to model properly the time-dependent behaviour of each load during DS–LS, it is necessary to perform a further development-to complement ‘Development 1’-‘fixing’ the particular values of loads not causing each outcrossing, prior to radial integration. This development is herein termed ‘Development 2 for Load Combination Analysis by DS–LS’, and will now be derived.
A
2N 3 ð C nQK1 _ _ E½ Q j Q ðsjaÞ f ðsjaÞf ðsa C cÞs n nij SjA Q 4 ds5 Z SnQ A EA janðsjaÞj 0
(13) respectively. Although (12) and (13) are analytically exact, they do not necessarily model properly the time-variant behaviour of the loads, for reasons that will be discussed below. Load Modelling Implications: Radial integration in (12) and (13) is performed according to the random structural strength variable ‘S’, to allow for any uncertainty in the limit state in each direction AZa (due to random variables other than loads). In doing so, the particular value q of the system of loading Q at any point (sja) is given by the expression 2 3 sai C ci 6 7 Q Z qðsjaÞ Z sa C c Z 4 saj C cj 5 (14) « where, without loss of generality, ai and ci are the ith component of AZa and CZc, respectively. Although (12) and (13) assume load path independence, (14) is a ‘one-parameter’
4. Development 2 for load combination analysis by DS–LS ‘Development 2’ involves ‘fixing’ the value of one or more loads prior to radial integration. Based on the above discussions, this procedure-in combination with the existing radial integration, involving the variable ‘S’-provides implicitly a practical means of modelling the time-dependent behaviour of the system of loading (i.e. the ‘exact’ load path travelled by the system load process vector is not defined explicitly). The procedure to perform ‘Development 2’ may be described as follows. First, depending on the particular load combination being investigated, all loads not contributing to each individual outcrossing are assigned particular values (i.e. held constant) prior to radial integration. Radial integration is then performed conditional to these particular values. During radial integration, individual outcrossings at each point (sja) are idealised as being caused by all of the loads not being held constant. Using the formulation presented for ‘Development 1’, the direction of the load process vector at each point of outcrossing is then ‘fixed’, to represent the combination of loads causing an outcrossing. Finally, integration over all possible values of the load(s) being held constant accounts for
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
the uncertainty in the particular load(s). This procedure implies some loads remain ‘fixed’ at the point of outcrossing, and therefore do not cause the outcrossing. To facilitate the integration requirements of ‘Development 2’, some directions AZa will be ‘fixed’ (i.e. instead of being random, as in ‘usual’ DS–LS); as will be shown, these directions are used solely to analyse situations with outcrossings caused by one load only (Qi, say), and hence evaluation of the term nC Di . The ‘complete’ formulation proposed herein to enable load combination analysis by DS–LS will now be derived. This will involve derivations of the formulation for ‘Development 2’, together with the ‘Development 1’ formulation derived earlier. The formulations will consist of those involving ‘fixed’ directions (for analysis of systems comprising two loads) and, for discussion, those involving random directions (for analysis of systems comprising three loads). The formulation for analysis of more than three loads is similar. Finally, the evaluation of all formulations by Monte Carlo and Importance Sampling will be derived.
To illustrate, the formulation for nC Di will be derived in the space of the loads Qi and Qj (with isj). Here, nC Di can be expressed in the form nC Di Z
Fig. 5. Typical ‘fixed’ direction in the 2D load space.
(cf Eq. (15))
qi Z sai C ci QððsjaÞjqj Þ Z sa Cc Z qj
nC Di ðqj ÞfQj ðqj Þdqj
(16)
C _ _ nC Di ððsjaÞjqj Þ Z E½Qn jQni ðsjaÞ fQi jQj ðsa CcÞ
(17)
nC Di
where is the value of evaluated conditional to the value QjZqj, fQj is the marginal pdf of Qj and the term fQj(qj) dqj—which is the ‘probability’ Qj will have the particular value qj—is the ‘equivalent’ of the probability pjO0 (for load Qj) in the ‘on-off’ model (Section 2.2), and is ‘built-in’ to the DS–LS calculations. Eq. (16) enables load Qi to cause each outcrossing, load Qj to be ‘held constant’ during radial integration and the uncertainty in load Qj to be accounted for. Details regarding its evaluation will now be discussed. 4.2. Fixed directions-characteristics of AZa and SZs Only one direction (AZa*, say) can exist in the region defined by QjZqj (Fig. 5); hence, this direction is a certain event, with ‘probability’ fA(a*)Z1. Also, the random variable S is ‘completely continuous’ (e.g. Walpole and Myers [16]), with SZsZ0 corresponding to the point CZc. Note that a variable ‘M’ is completely continuous when all particular values MZm change continuously within the range K N%m%N. 4.3. Fixed directions-evaluation of nC Di ðqj Þ The coordinates of the load process vector Q at the point SZs along the direction AZa* are given by the expression
(18)
where fQijQj is the pdf of Qi conditional to Qj. Hence, by integrating over all values SZs, the (mean) value of nC Di in the direction AZa* can be obtained from
N
nC Di ðqj Þ
where ai is the ith component of a*. Accordingly, the ‘local’ value of nC Di at the point (sja*) can be expressed as
4.1. Fixed directions-concept
C ðN
165
nC Di ða
jqj Þ Z
C ðN
nC Di ððsjaÞjqj ÞfSjA ðsjaÞds
KN C ðN
Z
E½Q_ n jQ_ ni ðsjaÞCfQi jQj ðsa CcÞfSjA ðsjaÞds
(19)
KN
where the limit state element surface DSD at each point (sja*) is constant. Since only the direction AZa* exists in the region defined by QjZqj, the (mean) value of nC Di evaluated conditional to QjZqj is given by the expression nC Di ðqj Þ Z
ð
C nC Di ðajqj ÞfA ðaÞda Z EAjQj ½nDi ðajqj Þ
AjQj
Z nC Di ða jqj Þ
(20)
where AjQj refers to all directions that can exist in the region defined by QjZqj, EAjQj is an expectation operator, nC Di ðajqj ÞZ C nC ða jq Þ is the value of n evaluated in the direction AZa j Di Di and the limit state element surface DSD is again constant. By integrating over all values of QjZqj (according to (16)) and evaluating the limit state element surface DSD at each point (sja*)—by dividing by the term ja $nðsjaÞj (Fig. 5)—the unconditional (mean) value of nC Di can be determined from
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W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
the expression C ðN
nC Di
Z
nC Di ðqj ÞfQj ðqj Þdqj
N
2CN 3 ð _ n jQ_ ni ðsjaÞCfQijQj ðsa CcÞfSjA ðsjaÞ E½ Q Z EQj 4 ds5 ja nðsjaÞj KN
(21) where EQj is an expectation operator and the term fQijQj( ) is the ‘probability’ Qi will have the particular value qi(sja*) when Qj is ‘fixed’ with particular value qj. As expected, (21) is analogous to DS–LS Eq. (2). Limit State Hyperplanes: The numerical example presented herein involves a limit state function represented by a single (linear) hyperplane of the form (for example, for deterministic ‘s’) (Ditlevsen [3]) GðQÞ Z a0 C
nQ X
ai Q i
(22)
iZ1
where ai are constants (with a0O0). It follows that the term E[ ]C is independent of each point (sja) on the limit state, thereby reducing (21) to 2CN 3 ð f ðsa CcÞf ðsjaÞ Qi jQj SjA _ _ C 4 ds5 (23) nC Di Z E½Qn jQni EQj ja nðsjaÞj KN
4.4. Random directions-formulation To illustrate, the formulation for nC Dij will be derived in the space of the loads Qi, Qj and Qk (with isjsk). Using similar derivations to (18)–(21), the term nC Dij can be expressed in the form (Gray [6]) C ðN
nC Dij
Z
Z N
2
ð
6 4
4.5. Monte Carlo sampling (MC)
3 7 nC Dij ðajqk ÞfA ðaÞda5fQk ðqk Þdqk
All equations proposed in Sections 4.1–4.4 may be evaluated using well-known MC. For example, using a total of ‘N’ samples, the MC estimate of (16) becomes
AjQk
Z EQk EAjQk ½nC Dij ðajqk Þ
(24)
C where nC Dij ðqkÞ is the value of nDij evaluated conditional to the C value QkZqk and nDij ðajqk Þ is the value of nC Dij ðqk Þ evaluated in the direction AZa, and is given by the expression
nC Dij ðajqk Þ
surface (Fig. 6). As expected, (24) is analogous to DS–LS Eq. (2). Eq. (24) enables both loads Qi and Qj to cause each outcrossing, load Qk to be ‘held constant’ during radial integration and the uncertainty in load Qk to be accounted for. An important aspect of its evaluation will now be discussed. Characteristics of AZa and SZs: In ‘usual’ DS–LS analysis of a system comprising nQ loads (with nQO1), random directions AZa are generated on the surface of a nQdimension unit hypersphere. In contrast, they are now generated on the surface of a nA-dimension unit hypersphere, with 1!nA!nQ. For example, Fig. 6 shows a typical direction that is generated from a pdf fA which is assumed to be uniformly distributed on a (2D) circle and exists within the Qi– Qj plane (i.e. nAZ2); at each point (sja) the elemental surface area is DSe Z snAK1 da dqk Z sda dqk . Similar to ‘usual’ DS–LS analysis, the random variable ‘S’ is ‘positive continuous’, with SZsZ0 corresponding to the point CZc. Note that the term ‘positive continuous’ is used here to describe a variable ‘M’ when all particular values MZ m change continuously within the range 0%m%N.
nC Dij ðqk ÞfQk ðqk Þdqk
N C ðN
Fig. 6. Typical ‘random’ direction in the 3D load space.
N ð
Z
E½Q_ n jQ_ nij ðsjaÞCfQi Qj jQk ðsa C cÞfSjA ðsjaÞ janðsjaÞj
nC Di z
N 1 X nC ðq Þ N mZ1 Di jm
(26)
where qjm is the mth sample of qj drawn from fQj.
ds
0
(25) where division by the term ja$nðsjaÞj converts the element surface DSe into the equivalent surface DSD on the limit state
4.6. Importance sampling (MC-IS) The proposed equations may also be evaluated using well-known MC-IS. For example, the MC-IS estimate of
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
(16) becomes C ðN
nC Di
Z
nC Di ðqj Þ
N N X
! mZ1
fQj ðqj Þ hQj ðqj Þ
hQj ðqj Þdqj z
"
fQj ðqjm Þ nC Di ðqjm Þ hQj ðqjm Þ
1 N
# (27)
where hQj is an importance sampling pdf—details for its construction herein are given by Gray [6]. It is assumed formulation similar to (27) can be derived to evaluate the term nC Di ðqj Þ by MC-IS; however, this is not considered herein. 5. Monte carlo integration By performing analysis in original ‘X’ space (which C comprises all variables), the terms nC Di , nDij , etc. can be evaluated by well-known MC and MC-IS, using the formulations given by Gray [6]. This is considered herein because there are no presumptions in its formulation (for example, unlike the assumption that the pdf fA in (24) is uniformly distributed), and therefore it may be used to obtain unbiased results to compare results obtained using all DS–LS formulation proposed herein.
where mQ1 Z mQ1 Z 20 and sQ1 Z sQ1 Z 2; QC2 describes the situation where mQ1Z20, sQ1Z2, mQ2Z5 and sQ2Z0.5; QC3 describes the situation where mQ1Z5, sQ1Z0.5, mQ2Z20 and sQ2Z2. Clearly, load cases QC2 and QC3 represent illproportioned load spaces. Two load correlation cases (RC1 and RC2) were examined, as follows: RC1 describes the situation where rQijZ0.0 (i.e. the loads are fully independent) and RC2 describes the situation where rQijZ0.5. Individual directions AZa (during DS–LS) and QZq (during directional simulation in H-Space) were generated either systematically (e.g. Melchers [12]), by ‘The Hyperspace Division Method (HDM)’ (Katsuki and Frangopol [8]), or randomly, by Monte Carlo (MC). Convergence of the terms nC Di and nC Dij was examined by considering the convergence of both: (1) the mean outcrossing rate E[ ]C at any point (sja) on the limit state surface; and (2) the mean outcrossing rate nC D . Some critical aspects of this will now be discussed. 6.1. Convergence of E[ ]C and nC D Convergence of E[ ]C: Numerical estimates of E[HjHi]C and E[HjHij]C were used to evaluate E[ ]C, using the wellknown expression (cf (5)) E½HC Z
6. Numerical example
nQ X
E½HjHij C
iZ1
Because systems seldom comprise more than 2 or 3 loads, C and because the pdf fSjA (and hence nC Di , nDij , etc.) is, in general, difficult (and computationally demanding) to evaluate for even simple structural systems, a numerical example demonstrating support for proposed Eqs. (16) and (13) (i.e. a system comprising 2 loads only) will now be presented. Further examples (involving 2 or more loads), and a detailed discussion of all results presented herein, are given by Gray [6]. C The terms nC Di and nDij were evaluated for a single linear limit state function G(Q1,Q2), which is a function of two (stationary) continuous gaussian loads Q1 and Q2 (which may be correlated) and involves deterministic ‘s’ (as defined for DS–LS). The following two limit states were analysed separately: G1 Z G1ðQ1 ; Q2 Þ Z a0 K3Q1 K3Q2 G2 Z G2ðQ1 ; Q2 Þ Z a0 K3Q1
167
(28)
where a0O0 is a constant and QiZQi(t) is a time-variant load. Clearly, both loads can cause failure of G1, whereas only Q1 can cause failure of G2. The term ‘Load Case (QC)’ was used herein to describe the values of the first and second moments of the loads in a given system (i.e. the mean mQi and standard deviation sQi of each load Qi) and the term ‘Load Correlation Case (RC)’ was used to describe the corresponding correlation between the loads (i.e. the correlation coefficient rQij between two loads Qi and Qj, with isj). Three load cases (QC1, QC2 and QC3) were examined, as follows: QC1 describes the situation
nQ X nQ X
C iZ1 jZ1
sH E½HjHi C C/Z pffiffiffiffiffiffi 2p
(29)
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} isj
with vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u nQ uX n2i COVQ_ ði; iÞ sH Z t iZ1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ iÞ Z r 00 ð0ÞCOVQ ði; iÞ COVQði;
(30) r 00 ð0Þ Z
vrðsKtÞ vsvt
where COVQ is the covariance matrix of Q and r(sKt) is a correlation function of time points ‘s’ and ‘t’. The value r 00 (0)Z0.5 is used herein. Using the point EZeZ0 in (11), a total of 40,000 sample directions was required. This ‘large’ value was needed to adequately sample the pdf fH1.HnQ because the magnitude of the statistical moments of Q1 and Q2 caused the pdf to be relatively ‘tall and thin’. Similar examples involving three loads show that the number of sample directions required may increase significantly with the number of loads (Gray [6]). C C Convergence of nC D : Numerical estimates of nDi and nDij C were used to evaluate nD (using (5)); this was compared with the analytical value of nC D , which was obtained as follows. Since each limit state is expressed in the (linear) form given by (22) and all variables are normal distributed, the analytical expression for the mean outcrossing rate nC D is
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Table 2 Notation for methods Notation
Method a
DS–LS LC1 (HDM)
DS–LS LC1
DS–LS LC2 (SR) DS–LS LC2 (MC) DS–LS LC2 (MC–IS)
DS–LS LC2b
MCc MC–IS
Monte Carlo Importance sampling
a b c
Comments on integration/simulation
Outcrossing rates evaluated
Simulation by ‘Hyperspace Division Method’ Integration of load pdf’s by ‘Simpsons Rule’ Integration of load pdf’s by ‘Monte Carlo’ Integration of load pdf’s by ‘Importance Sampling’ – –
C nC Di , nDij C nC Di , nDij
C nC Di , nDij C C nDi , nDij
‘Usual’ DS–LS, but with ‘Development 1’ (Section 3) performed at each time of outcrossing. ‘Development 1’ (Section 3) used in combination with ‘Development 2’ (Section 4). The notation ‘MC-i’ is used in some figures; this means ‘sampling from fQi’.
given by the well-known expression (Ditlevsen at al [4]) sffiffiffiffiffiffiffiffiffiffiffi nQ P a2i ð m iZ1 G C nC fQ ðqÞdq Z E½Cf (31) D Z E½ sG sG
Similar examples involving three loads show that the number C of ‘samples’ required (and hence CPU time) to evaluate nC D , nDij C and nDijk , etc. may increase significantly with the number of loads (Gray [6])
with
Typical plots of results are given in Figs. 7 and 8. For C clarity, only mean results are shown (i.e. E½nC Di and E½nDij ). The majority of results were found to compare well. Fig. 7 shows that all results obtained by methods using importance sampling were computationally superior (in terms of convergence) to results obtained by methods using Monte Carlo sampling. The ill-proportioning effect of the load space was found to have little effect on the accuracy of all results. An important aspect of the results will now be discussed.
SD
mG Z a0 C a1 mQ1 C a2 mQ2 s2G
vGðQÞ vGðQÞ T Z COVQ vQ vQ
(32)
where mG and sG are the mean and standard deviation, respectively, of G, vG(Q)/vQ is a vector containing derivatives of G with respect to Q and f is the Standard Normal Probability Density Function. Calculations were performed by DS–LS, MC and MC–IS, with notation given in Table 2. A total of: 700 sample directions was needed by DS–LS LC1(HDM); 6E5 samples was needed by DS–LS LC2(MC) and MC; 8E3 samples was needed by DS–LS LC2(MC–IS) and MC–IS. The majority of results obtained required what may be considered negligible computation (CPU) time. DS–LS LC1 (HDM) was found to be computationally superior, whereas calculations involving Monte Carlo sampling were found to require the most CPU time. In all results, the minority of the computational effort was required for the calculation of the term E[ ]C.
Fig. 7. Results for the combination G2, QC1, RC2.
6.2. Numerical results
C 6.2.1. Relationship between nC Di and nDij For situations with outcrossings caused by any combination of Q1 and Q2 (i.e. GZ GðQ1 ; Q2 Þ), the relationship nC D12 R C maxfnC ; n g was displayed by all results (e.g. Fig. 8), with D1 D2 max{ } representing the maximum value of the arguments in { }. This result was due to the relationship between the terms E[HjHi]C and E[HjHij]C, as follows. Because each load Qi(t) is stationary, continuous and Gaussian, it’s time derivative Q_ i ðtÞ—and therefore each variable Hi Z Q_ ni Z nðsjaÞQ_ i ðtÞ—is also stationary, continuous
Fig. 8. Results for the combination G1, QC1, RC2.
W.A. Gray, R.E. Melchers / Probabilistic Engineering Mechanics 21 (2006) 159–170
and Gaussian (but with zero mean). Hence, as shown by Fig. 4(a), the region in the 2D H-space where the conditions for which an outcrossing at any point (sja) in the load space are caused by both load processes are satisfied (‘region ij’, say) comprises larger values of ‘h’ and hfH1 .HnQ ðh1 ; .; hnQ Þ than the regions where the conditions for which an outcrossing at the same point are caused by only one load process are satisfied (‘region i’, say). It follows immediately that individual values of the product hfH1 .HnQ ðh1 ; .; hnQ Þ are greater in magnitude in ‘region ij’ than in ‘region i’. Hence, based on (6) and (9), the relationship E½HjHij CR maxfE½HjHi C; E½HjHj Cg holds, so that the above-mentioned relationship nC D2 R maxf$g is to be expected. Evidently, these relationships can be physically interpreted from Fig. 3; it shows that outcrossings in the load space that are caused by either Q1 only or Q2 only can occur only along one of the two exclusive directions shown, whereas outcrossings that are caused by Q1 and Q2 outcrossing simultaneously can occur over the much broader range of directions shown. Hence, one would expect an outcrossing caused by both Q1 and Q2 to provide a greater contribution to nC D than an outcrossing caused by only one load. 6.2.2. Implications of results The results obtained herein suggest that individual outcrossings are more likely to be caused by combinations of C C loads, than individual loads (i.e. nC Dij R maxfnDi ; nDj g). This implication appears at first to contradict what has been published previously in the literature. For example, results obtained by Wen [18] using the ‘Load Coincidence Method’ (which requires the loads to be represented by the ‘on-off’ model (Section 2.2) typically indicated that failure was most likely to be caused by individual loads Qi acting ‘alone’ (i.e. they have qiZ1, and all loads Qj not acting have pjZ1). This implication is assumed to have occurred for the following two reasons: 1. The ‘standard’ model (Section 2.2) was used to represent each load, thereby implying each load Qi is effectively always ‘on’ (i.e. qi/1). It follows that all loads are likely to be constantly changing and therefore all loads are more likely to contribute to failure than only one or a few loads. In contrast, Wen [18] used the ‘on-off’ model, and each load Qi was assumed to be ‘off’ most of the time (i.e. pi/ 1), so that it was less likely for multiple loads to be ‘on’ simultaneously (and therefore less likely they would all contribute to failure), and 2. Each load was modelled as being (completely) ‘continuous’, so that it’s value changes continuously (though not necessarily constantly) in time. In contrast, Wen [18] modelled each load as being ‘discrete’, so that it’s value changes either only at discrete points in time, or only within discrete periods of time. Clearly a vector process comprising continuous loads is more likely to be ‘moving around’ in the load space (and hence outcrossing due to the combined behaviour of the loads) than one comprising discrete loads.
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This reasoning appears to agree with results that have been published in the literature. For example, Wen and Chen [17] examined two poisson pulse processes (i.e. processes represented by a random occurrence time, length of duration and non-negative intensity (Ditlevsen and Madsen [5])). When both processes were ‘sparse’ (i.e. pi/1), it was found that, ‘.only 7.4% of the total failure can be attributed to the simultaneous action of the two loads because of their transient nature.’ (pp. 816). However, when the mean load duration was increased by a factor of 10 (i.e. qi was increased), it was found that, ‘.the load coincidence accounts for 45%).’ (pp. 816). The reason for this trend is that the ‘probability of occurrence’ of combinations of loads decreases with the ‘sparseness’ of each load C (Wen [19]), so that nC Di OOnDij . This reasoning (as well as the discussed published results) suggests that the trends displayed C by the terms nC Di , nDij etc may be influenced significantly by the way in which the loads are modelled.
7. Conclusions In this paper the ‘Directional Simulation in the Load Space (DS–LS)’ approach to reliability analysis was developed to consider explicitly outcrossings caused by all possible combinations of loads, during analysis of systems comprising stationary continuous Gaussian loads. A relatively simple numerical example comprising two loads and one linear limit state function was presented to demonstrate the validity of the proposed formulation. The results obtained by DS–LS were compared with equivalent Monte Carlo results derived from well-established theory. The majority of the results were found to compare well. For the example considered herein, it was found—for situations with outcrossings caused by any combination of loads (i.e. there was no restriction to failure being caused by only one load)—that failure was more likely to be caused by combinations of loads than individual loads. It was argued that this finding was achieved because of the way in which the loads were modelled—for example, due to the stochastic process model used to represent the loads. This reasoning appears to agree with results that have been published previously in the literature. Acknowledgements The funding and support from The University of Newcastle for this research project is gratefully acknowledged. References [1] Belyaev YK. On the number of exits across the boundary of a region by a vector stochastic process. Theory Probab Appl 1968;13(2):320–4. [2] Bjerager P. Probability integration by directional simulation. J Eng Mech 1988;114(8):1285–302. [3] Ditlevsen O. Gaussian outcrossings from safe convex polyhedrons. J Eng Mech 1983;109(1):127–48. [4] Ditlevsen O, Bjerager P, Olesen R, Hasofer AM. Directional simulation in gaussian processes. Probab Eng Mech 1988;3(4):207–17.
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