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Comparison between failure sequence and failure path for brittle systems
9
Pergamon
PII: S0045-7949(94)E0282-7
COMPARISON FAILURE
Comjwters & Strwures Vol 65. No. 4, pp 625-631. 1991 1997 Pubbshed by Elsewer Science Ltd All nghts reserved Printed in Great Britain &w-7949/97 $17 00 + 0.00
BETWEEN FAILURE SEQUENCE PATH FOR BRITTLE SYSTEMS Y.
AND
IBRAHIM
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, U.S.A. (Received 11 May
1990)
classes of structures with discrete members, such as trusses and frames, system failure is generally expressed in terms of component failures. The components are presumed to be ‘two-state’
Abstract-For
members, i.e. either fail or safe. Furthermore, for such systems, it is convenient to use a failure tree to represent the various combinations of component failures that will lead to system failure. Search strategies are then developed to search the failure tree for the dominant modes of the system. As part of strategy the probability of occurrence of a failure mode is evaluated. The exact method of evaluating such probabilities is called the failure sequence formulation. Alternatively, an approximate method called the failure path formulation has been widely used. A comparison between the failure path formulation to the failure sequence formulation is considered. The implications of the differences of the two formulations on the result of search strategies are discussed. 0 1997 Published by Elsevier Science Ltd
1. INTRODUCTION
In general, the probability of failure of a timeinvariant system is given by I-J/ =
f(x) dx,
(1)
where D is the failure domain, X is the vector of basic variables and f(x) is the joint probability density function of the basic variables. A reasonably good estimate of pr can be obtained if only the dominant (usually few) modes are considered. For classes of structures with discrete members, such as trusses and frames, a number of authors [l, 2, 3,4] have studied techniques that seek the dominant modes of searching within a failure tree [5]. The members are presumed to be ‘two-state’ (failed or unfailed) members, shown in Fig. 1. These models permit the use of a failure tree. An example of a failure tree is shown in Fig. 2 (the failure tree shown is incomplete). A failure tree is a graphical layout of all possible orders of component failures. A sequence of component failures that leads to system failure is called a failure path. For example, if system failure is defined as structural instabilily, then the path 3-2 in Fig. 2 is a
failure path. The failure of component 3 followed by the failure of component 2 leads to system failure. The state of the system in which k components have failed is called damage state k. In most of the work reported in the literature, system failure is defined as structural instability. This failure condition is expressed in terms of the ordered failures of its components. For example, for the truss shown in Fig. 2, the system fails for example if components 3 and 2 fail in sequence. The probability of failure mode (3,2) occurring is given by P[failure mode (3, 2)] = P[(3 fails) n (2 fails after 3 has failed)].
Hence, the individual events in the left hand side of the above equation must be defined. System failure probability is then given by pI = P[ u f., ,(failure mode i)],
5
lx 1
2
3
Semi-Brittle
(3)
where k is the number of failure modes in the system. Thus, all the failure modes must be identified. As
L-
I&&o-Plartic
(2)
Brittle
Fig. 2. Failure tree example.
Fig. 1. Mechanical models. 625
Y. Ibrahim
626
mentioned previously, a good estimate of the pf can be obtained with only the dominant failure modes of the system. This has led to the development of a variety of search strategies. In most of the strategies, the computation of the probability of occurrence of a failure mode is based upon an approximate method called the failure path (FP) approach. In this approach, the event that a component has failed is expressed without regard to the yet unfailed components. Each failure event can then be described by a single failure function or g-function, i.e. that of the failed component. The exact method of computing the failure probability of a component, where the state of the other yet unfailed components are accounted for, is called the failure sequence (FS) approach. It is easier to estimate the probability of occurrence of a failure mode by the FP approach than the FS approach. However, there will be differences in the estimate. Clearly, if the order of the relative significance (or dominance) of the failure modes differ between the two formulations, then this will affect the results of search strategies based upon the FP formulation. The differences in the estimate of the probability of occurrences of the failure modes by the two formulations will be presented and discussed in this paper. The focus of this paper is on brittle systems. These are systems in which once a component fails, it loses both strength and stiffness completely. A comparison of the two formulations will be presented. This is followed by an example where estimates of the failure modes of the simple truss structure are obtained using both formulations. Monte Carlo simulation is used in the analysis, since it is the most accurate technique. Note that in the analysis of systems with mechanical models shown in Fig. 1. it is often implicitly assumed that a failed component does not unload after it has failed. In Fig. 1, this neglected unloading behavior would be the trajectory marked d-e. In this paper, this assumption will be used. 2. FAILURE
SEQUENCE
(FS) AND FAILURE
PATH(FP)
Consider the truss shown in Fig. 2 with brittle components and consider the failure of component 3 followed by the failure of component 2. Let the forces in the components due to a unit external load in the intact state be CI,,i = 1, 2, 3,4, 5. Consider the failure of component 3. By the failure sequence approach, this event accounts for the state of the unfailed components, i.e. component 3 is the first to fail and the other components have not failed. The probability that this event occurs is written as P[(RrcrlL)G
0 n(Rt > R)n(&
fails. The load level at which component
3 fails is
(5)
L:=$
where the superscript 1 indicates first failure and the subscript 3 is the component to fail first. The forces in components 1, 2, 4 and 5 when 3 is the first to fail are i = 1, 2,4, 5.
F,= 2 R,,
(6)
Hence, eqn (4) can be written as {RI> (ada3)R,}
P[(Rj- a4)
n{Rz > (a&~VG} n {Rs > (ah)R,} n{&
>
(a&M~}l.
(7)
The event (Rj- a,L)< 0 is a low probability event, while the other terms are relatively large probability events. The above equation is computationally very difficult. For large structures, there is usually many intersecting events and there are no computationally efficient techniques of estimating the probability of a sequence. Alternatively, the failure path (FP) formulation can be used. In this formulation, the failure of component 3 is expressed without regard to the state of the other unfailed components. Thus, the probability of failure of component 3 is just P[(R, - coL) <
01.
03)
Hence, the high probability events are dropped and the above probability can be estimated easily by, for example, the first- and second-order reliability methods (FORM/SORM) [6]. Note that the term (R,- a3L) is also called the safety margin, &. Consider now the state where component 3 has failed. Let the forces in the remaining components due to a unit external load be y,, i= 1, 2,4, 5. Assume that component 2 is the next to fail. Following the outline above it can be shown that by the FS approach, the probability that 2 fails after 3 has failed can be written as
> F2) and the FP formulation
is
n(% > F4)u (& > Fs)l,(4) where R,is the capacity of component i and F,is the force in component i at the instant when component 3
P[(Rz - yzL) G 01. By the FS formulation
(10)
the probability of occurrence
627
Failure sequence and failure path
the FP approach is therefore more difficult. Since a failure sequence is a subset of the corresponding failure path, the estimate of the system failure probability by the FP formulation is greater or at best equal to the estimate by the FS formulation. That is
of failure mode (3,2) is given by
n{& > (aJa,)&}
n (R4 > (adadR~}
n{Rs > t:aJad&}
n (Rz - YZL) S 0
n{RI > I:ah)&} n
n {R > WY~&}
{Rs > I:Y~)R~}I.
And by the FP fkmulation
p/” 2 p,‘”.
(11)
this is given by
P[(R3 - ad.) < Ofl(R2 - y&j < 01.
(12)
Clearly, eqn (12) is computationally easier. Approximate methods based upon FORM/SORM have been developed to evaluate the probability of such events [7, 1, 81. Equation (11) can be re-written as P[(R, - a31,) < 0fl(R2 - y2L) < 0 n{RI
> ~Ca~/adR~}n {Rz > (adadR~}
Thus, the FP approach can generally lead to overestimating the system failure probability. From the above formulation, it is clear that the estimate of the probability of occurrence of a failure mode by the FS formulation is smaller than that computed by the FP formulation. However, the order of the difference is not clear. The purpose of this paper is to present some results on the abovementioned differences. Estimates of the probability of occurrences of failure modes by both formulations are obtained using Monte Carlo simulation. Monte Carlo simulation provides an exact comparison, since the method is exact. Furthermore, it is difficult to evaluate the probability of occurrence of a failure mode by the Fs formulation. For completeness, the simple Monte Carlo technique is outlined [9]. Equation (1) can be recast as
n {& > (Adam)&} n {Rs > (ah)R~} P/ =
n{&
> (~hM2)
n{h
>
h/~~)R~)l.
(13)
From eqns (13) and (12), it is clear that the event described by the F% formulation is a subset of that of the FP formulation. Thus, the probability of occurrence of a failure mode estimated by the FS formulation is always smaller than that estimated by the FP formulation. An&her important observation is the fact that the FS formulation leads to events that are mutually exclusive. Consequently, the estimate of the system failure probability will be simplified. The system failure probability is given by P,=p[vr=,
occurrence of failure mode i],
(14)
where k is the number of failure modes in the system. Hence, with the FS formulation, system failure probability is the sum of the probability of occurrence of each of the failure sequences, i.e. k pr = P C occurrence of failure sequence [ ,-I
i1.
s R
n 1% > ~~Y~)Rz)
(16)
W1”04 dx,
(17)
where R is the sample space and Z(x) is an indicator function, Z(x) = 1 if x is in D and 0 otherwise. Using simple Monte Carlo simulation, an unbiased estimate for pr is computed as follows:
(18) where N is the sample size and x, is a sample from f(x). The coefficient of variation (COV) of the estimator j in eqn (18) is COV@f) =
J( > -1 -Pr
NP/
’
(19)
The estimated COV in eqn (19) with pf replaced by its estimator h. By the FS formulation, the estimator of the system failure probability is given by
(15)
Thus, if the probabilities of the failure sequences can be evaluated, the system failure probability follows easily. The system failure probability given by eqn (15) is exact. For subsequent discussion, the system failure probability by FS formulation is defined as p/‘“. In the FP formulation, the failure modes are no loqger mutually exclusive events. The determination of the system failure probability using
where Z&x,) is the indicator variable for a failure sequence. Similarly, by the FP formulation
(21) where ZFp(x,) is the indicator variable for a failure path. Hence, if for all the samples observed that lead to failure by the occurrence of failure sequences, one
Y. Ibrahim
628
or more failure paths are observed, then the two estimators will be the same. As mentioned earlier, an FS is a sub-set of an FP; the occurrence of an FS must be accompanied by the occurrence of one or several FP. However, in view of the formulations of FP and FS, there can be cases where samples lead to system failure by the occurrence of a failure path and not a failure sequence. Estimates of the probability of occurrence of failure modes by the two formulations are obtained in the same manner as above; each sample is tested for system failure by both formulations and the number of failure modes observed are counted.
3. TESTING OF SAMPLES
In this section, an outline of the procedures adopted to test each sample generated by the Monte Carlo simulation for failure by the FS and FP formulations are presented. The following discussion is for the case of a single external load. Extensions to multiple load cases are straight forward and will not be discussed here. System failure is defined as structural instability. Hence, the states of the surviving components have to be determined in order to assess structural system failure. Recall that the interest is only with brittle systems, where a component loses strength and stiffness after failure. In this study, once a component has failed, it is removed from the structure and the structure is re-analyzed from the zero load condition. 3.1. Failure sequence approach In this information, if a sample leads to system failure, then only one failure mode will occur. This is evident from the formulation given in Section 2. At each state of the system, the load is incremented from zero and the component that is the first to fail (i.e. the weakest component) in that state is obtained. The load level at which the weakest component failure occurs is checked against the sampled load. If the sample load level is smaller, then the system will not fail with that particular sample. However, if the sampled load is greater, then the weakest component will fail. The failed component is removed and the system is in a new state. This new state is checked for system failure. If the new state is a system failure state, then the structure will fail with the particular sample. If the new state is not a system failure state, then proceed to find the weakest component in the new state and repeat the above steps. The preceding is translated into steps given below. Assume that the system is in damage state k, i.e. k components have failed and are removed from the structure. 1. The structure in damage state k is analyzed with a unit applied load. The external load level to cause each component to fail is computed by dividing the sampled capacity with the respective force in the
component
due to the unit applied load, i.e. L:=%.
where u, is the force in component i due to a unit external load and R, is the sampled capacity of component i. The minimum of Lf corresponds to the weakest component, i.e., if Lf is the minimum, then component j is the weakest component in damage state k. 2. The minimum load level, i.e. L: is checked against the sampled load level, L. If L,! < L then component j will fail in damage state k; proceed to the next step. Otherwise, component j will not fail and the system also will not fail; proceed to the next sample. 3. The failed component, i.e. j, is removed from the structure. Hence, the system is in damage state k + 1. Check if the structure is stable in damage state k + 1. If it is unstable, then system failure has occurred. If the structure is stable, then it has not failed; proceed to next step. 4. Componentj is removed from the structure. The structure, now in damage state k + 1, is analyzed as in step 1. The weakest component in damage state k + 1 is obtained and steps 2 and 3 are performed to find the next damage state and check for system failure. 5. The above steps are repeated until either the system survives or fails with the particular sample. The corresponding steps in the structural analysis are shown schematically in Fig. 3. After each failure, the system becomes less stiff since the failed component is removed. This can be seen by comparing paths o-a and o-b and paths o-a, o-b and o-c in the first and second figure respectively of Fig. 3. Recall that in the analysis, after each failure, the system is loaded from zero. In the first figure, after the failure of the first component, i.e. path o-a, the force needed to fail the second component, i.e. path o-b, is higher. This implies that there was no failure during redistribution which was a consequence of the failure of the first component. However, in the second figure, L, < Lb, implying a failure during redistribution. 3.2. Failure path approach Since the events described by the FS formulation is a subset of the events described by the FP formulation, every sample that leads to failure by the FS
c
Fig. 3. Analysis of brittle systems.
629
Failure sequence and failure path
\
B
Fig. 4. Example of failure path. formulation will lead to failure by the FP formulation. The converse however is generally not true. Hence, even if a sample did not lead to failure by the FS formulation, it should be tested for failure using the FP formulatiosn. In this approach, at each damage state of the structure, the safety margins of the surviving components for the total sampled load are computed. Those components for which the safety margins are negative or zero are deemed to have failed. There could be more than one component failure. The structure is now in the next damage state. However, for that damage state, there will be several combinations of component failures leading to that damage state. For this reason, for every sample that leads to failure by the FP formulation, several failure modes can occur. In order to make the following outline clearer, reference will be made to a failure tree example shown in Fig. 4. The analysis begins at the intact state. 1. Evaluate the safety margins of the components in the intact state. The intact structure is analyzed under the full sampled load. The forces in the components are then checked against their corresponding sampled capacity, i.e. check M,= R,--E,
(23)
where M, is the safety margin for component i and F, is the force due to the sampled load, in component i. F, is given by F, = a,L,
(241
where a, is the folrce in component i due to a unit external load and L is the sampled load. 2. If none of the forces exceed the corresponding capacities, then the system survives, i.e. M, > 0 for all i. 3. If one or some of the forces exceed the corresponding capacities, then the associated components have failed in the intact state, i.e. M, < 0 for some i. Thus, one or several components, i.e. those whose
safety margin is negative, have failed. With respect to Fig. 4, the failed components would be i, j, k, 1, m. 4. Next analyse the structure in damage state 1, i.e. the structure has one failed component. Note that if there is more than one failed component, the structure is analyzed in succession with each of the failed components removed. With respect to Fig. 4, the structure is analyzed for cases where only component i is removed, only component j is removed, only component k is removed, only component 1 is removed and only component m is removed. Note that for these analyses, the structure is also checked for failure. For example, if with only component m removed, the system is unstable, then a failure mode has occurred. The safety margins for the surviving components in the various analyses cases are evaluated and checked for failure. For example, in the failure tree shown in Fig. 4, for the case where only component i is removed, components n and j failed in the subsequent analysis. However, for the case where only component j was removed, none of the components failed. Hence, with this sample, there will be no failure modes with component j as the first component to fail. 5. Repeat step 4 and analyze the structure now in damage state 2, i.e. with two components removed. For example, in Fig. 4, the cases in which the structure have to analyzed include those where the following two components are removed at a time; (i, n), (r,j), (k, i), (k, 0, (k,pl, (1, k) and U, 4). 6. Repeat steps 4 and 5 until all the possible failure combinations have been uncovered. 4. EXAMPLE
Consider the ten-bar truss structure with brittle components shown in Fig. 5. The parameters of the components are also shown in Fig. 5. The load L is normally distributed with a mean of 1.5 and a COV of 0.3. All the random variables are independent. A large scale Monte Carlo simulation was performed. For every sample observed, the structure was subjected to both the FS and FP approaches. A component fails if the force in the component due to the external load exceeds the component capacity. System failure is defined as structural instability. 4.1. Results and discussion The result of the Monte Carlo simulation is shown in Table 2. Two outcomes that lead to system failure are also produced here in Table 1 for discussion. The first sample, i.e. no. 5716, led to an outcome where a failure sequence was observed and 4 failure paths were also observed. Note that the failure mode that occurred by the FS formulation also occurred by the FP formulation, i.e. mode -4, -9. The second sample, i.e. no. 5823, is a case in which failure paths occurred but not a failure sequence. Hence, such outcomes will contribute to the differences in the estimate by the system failure probaiblity by the two
630
Y. Ibrahim
Fig. 5. Ten-bar
Table
Table 2. Failure
1. Example
truss structure.
of outcomes
probabilities
formulations. The results obtained by the FS formulation and FP formulation are shown in Table 2. Several interesting observations could be made. First, note the differences, which were expected, between the FP formulations and FS formulation for the probability of occurrence of the failure modes. However, the order of the differences is not the same for all the failure modes found. In one particular case, i.e. failure mode -4, -9, the difference is quite significant. This is a very interesting mode. In the intact state, component 9 is stressed in tension. After component 4 fails, component 9 will unload and go into compression. A more important observation is the ordering of the failure modes. This is related to the significant difference between the FS and FP formulation of failure mode -4, -9. By the FS approach failure mode -4, -9 is the third most likely failure mode. However, by the FP approach failure mode -4, -9 is the most likely failure mode. Hence, this will affect the results of the search for the dominant failure modes using the FP formulation.
by Monte
of samples
Carlo,
N = 50,000
5. CONCLUSIONS
The estimate of the probability of occurrence of a failure mode depends upon the formulation of the corresponding event. A failure sequence (FS) formulation gives rise to lower probability of occurrence than a failure path (FP) formulation. For a brittle system these differences in the estimates can also be reflected in the estimate of the system failure probability. Results of the estimates of the probability of occurrences of the failure modes of a simple truss example were obtained by the two formulations using Monte Carlo simulation. In the example chosen, the differences were quite significant for one mode. These differences have implications on search strategies that make use of the FP formulation to seek out the dominant modes of the system. The seach can be affected in cases where the differences are significant enough to cause a change in the ordering of the relative dominance of failure modes. Recall that the FS approach is the exact method. Thus, the relative
Failure sequence and failure path dominance of the failure modes as determined by the FS approach is the correct order. However, estimate of the system failure probability by the FP approach tend to be conservative. Acknowledgement-The author wishes to thank Mr Ashish Karamchandani of !3tanford University for some enlightening discussions.
REFERENCES
Y. F. Guenard, Application of system reliability analysis to offshore structures. Report No. RMS-1, Reliability of Marine Structures Program (formely Report No. 71, J’ohn A. Blume Earthquake Engineering Center), Department -of Civil Engineering, Stanford University (1984). R. E. Melchers and L. K. Tang, Dominant failure modes in stochar,tic structural systems. Srrucfural Safety 2, 127-143 (1984). F. Moses and M. R. Rashedi, The application of system reliability to structural safety. Proceedings of the 4th
631
International Conference on Applications of Statistics and Probability in Soil and Structural Engineering,
Florence, Italy (1983). 4. P. Thoft-Christensen and Y. Murotsu, Application of Structural Systems Reliability Theory. Springer, Berlin (1986). 5. P. Bjerager, A. Karamchandani and C. A. Cornell, Failure tree analysis in structural system reliability. Proceedings of the 5th International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, pp. 985-996. Vancouver,
Canada, May (1987). 6. H. 0. Madsen, S. Krenk and N. C. Lind, Methods of Strucfural Sufefy. Prentice-Hall, Englewood Cliffs, NJ (1986). 7. S. Gollwitzer and R. Rackwitz, Equivalent components in first order system reliability. Reliability Engineering 5, 99-115 (1983).
8. A. Karamchandani, Y. F. Guenard and K. Ortiz, Shasys-A software package for component and system reliability analysis. Report No. 78, John A. Blume Earthquake Engineering Center, Department of Civil Engineering, Stanford University (1986). 9. R. Y. Rubinstein, Simulation and the Monte Carlo Method. John Wiley, New York (1981).
Pergamon
PII: S0045-7949(94)E0282-7
COMPARISON FAILURE
Comjwters & Strwures Vol 65. No. 4, pp 625-631. 1991 1997 Pubbshed by Elsewer Science Ltd All nghts reserved Printed in Great Britain &w-7949/97 $17 00 + 0.00
BETWEEN FAILURE SEQUENCE PATH FOR BRITTLE SYSTEMS Y.
AND
IBRAHIM
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, U.S.A. (Received 11 May
1990)
classes of structures with discrete members, such as trusses and frames, system failure is generally expressed in terms of component failures. The components are presumed to be ‘two-state’
Abstract-For
members, i.e. either fail or safe. Furthermore, for such systems, it is convenient to use a failure tree to represent the various combinations of component failures that will lead to system failure. Search strategies are then developed to search the failure tree for the dominant modes of the system. As part of strategy the probability of occurrence of a failure mode is evaluated. The exact method of evaluating such probabilities is called the failure sequence formulation. Alternatively, an approximate method called the failure path formulation has been widely used. A comparison between the failure path formulation to the failure sequence formulation is considered. The implications of the differences of the two formulations on the result of search strategies are discussed. 0 1997 Published by Elsevier Science Ltd
1. INTRODUCTION
In general, the probability of failure of a timeinvariant system is given by I-J/ =
f(x) dx,
(1)
where D is the failure domain, X is the vector of basic variables and f(x) is the joint probability density function of the basic variables. A reasonably good estimate of pr can be obtained if only the dominant (usually few) modes are considered. For classes of structures with discrete members, such as trusses and frames, a number of authors [l, 2, 3,4] have studied techniques that seek the dominant modes of searching within a failure tree [5]. The members are presumed to be ‘two-state’ (failed or unfailed) members, shown in Fig. 1. These models permit the use of a failure tree. An example of a failure tree is shown in Fig. 2 (the failure tree shown is incomplete). A failure tree is a graphical layout of all possible orders of component failures. A sequence of component failures that leads to system failure is called a failure path. For example, if system failure is defined as structural instabilily, then the path 3-2 in Fig. 2 is a
failure path. The failure of component 3 followed by the failure of component 2 leads to system failure. The state of the system in which k components have failed is called damage state k. In most of the work reported in the literature, system failure is defined as structural instability. This failure condition is expressed in terms of the ordered failures of its components. For example, for the truss shown in Fig. 2, the system fails for example if components 3 and 2 fail in sequence. The probability of failure mode (3,2) occurring is given by P[failure mode (3, 2)] = P[(3 fails) n (2 fails after 3 has failed)].
Hence, the individual events in the left hand side of the above equation must be defined. System failure probability is then given by pI = P[ u f., ,(failure mode i)],
5
lx 1
2
3
Semi-Brittle
(3)
where k is the number of failure modes in the system. Thus, all the failure modes must be identified. As
L-
I&&o-Plartic
(2)
Brittle
Fig. 2. Failure tree example.
Fig. 1. Mechanical models. 625
Y. Ibrahim
626
mentioned previously, a good estimate of the pf can be obtained with only the dominant failure modes of the system. This has led to the development of a variety of search strategies. In most of the strategies, the computation of the probability of occurrence of a failure mode is based upon an approximate method called the failure path (FP) approach. In this approach, the event that a component has failed is expressed without regard to the yet unfailed components. Each failure event can then be described by a single failure function or g-function, i.e. that of the failed component. The exact method of computing the failure probability of a component, where the state of the other yet unfailed components are accounted for, is called the failure sequence (FS) approach. It is easier to estimate the probability of occurrence of a failure mode by the FP approach than the FS approach. However, there will be differences in the estimate. Clearly, if the order of the relative significance (or dominance) of the failure modes differ between the two formulations, then this will affect the results of search strategies based upon the FP formulation. The differences in the estimate of the probability of occurrences of the failure modes by the two formulations will be presented and discussed in this paper. The focus of this paper is on brittle systems. These are systems in which once a component fails, it loses both strength and stiffness completely. A comparison of the two formulations will be presented. This is followed by an example where estimates of the failure modes of the simple truss structure are obtained using both formulations. Monte Carlo simulation is used in the analysis, since it is the most accurate technique. Note that in the analysis of systems with mechanical models shown in Fig. 1. it is often implicitly assumed that a failed component does not unload after it has failed. In Fig. 1, this neglected unloading behavior would be the trajectory marked d-e. In this paper, this assumption will be used. 2. FAILURE
SEQUENCE
(FS) AND FAILURE
PATH(FP)
Consider the truss shown in Fig. 2 with brittle components and consider the failure of component 3 followed by the failure of component 2. Let the forces in the components due to a unit external load in the intact state be CI,,i = 1, 2, 3,4, 5. Consider the failure of component 3. By the failure sequence approach, this event accounts for the state of the unfailed components, i.e. component 3 is the first to fail and the other components have not failed. The probability that this event occurs is written as P[(RrcrlL)G
0 n(Rt > R)n(&
fails. The load level at which component
3 fails is
(5)
L:=$
where the superscript 1 indicates first failure and the subscript 3 is the component to fail first. The forces in components 1, 2, 4 and 5 when 3 is the first to fail are i = 1, 2,4, 5.
F,= 2 R,,
(6)
Hence, eqn (4) can be written as {RI> (ada3)R,}
P[(Rj- a4)
n{Rz > (a&~VG} n {Rs > (ah)R,} n{&
>
(a&M~}l.
(7)
The event (Rj- a,L)< 0 is a low probability event, while the other terms are relatively large probability events. The above equation is computationally very difficult. For large structures, there is usually many intersecting events and there are no computationally efficient techniques of estimating the probability of a sequence. Alternatively, the failure path (FP) formulation can be used. In this formulation, the failure of component 3 is expressed without regard to the state of the other unfailed components. Thus, the probability of failure of component 3 is just P[(R, - coL) <
01.
03)
Hence, the high probability events are dropped and the above probability can be estimated easily by, for example, the first- and second-order reliability methods (FORM/SORM) [6]. Note that the term (R,- a3L) is also called the safety margin, &. Consider now the state where component 3 has failed. Let the forces in the remaining components due to a unit external load be y,, i= 1, 2,4, 5. Assume that component 2 is the next to fail. Following the outline above it can be shown that by the FS approach, the probability that 2 fails after 3 has failed can be written as
> F2) and the FP formulation
is
n(% > F4)u (& > Fs)l,(4) where R,is the capacity of component i and F,is the force in component i at the instant when component 3
P[(Rz - yzL) G 01. By the FS formulation
(10)
the probability of occurrence
627
Failure sequence and failure path
the FP approach is therefore more difficult. Since a failure sequence is a subset of the corresponding failure path, the estimate of the system failure probability by the FP formulation is greater or at best equal to the estimate by the FS formulation. That is
of failure mode (3,2) is given by
n{& > (aJa,)&}
n (R4 > (adadR~}
n{Rs > t:aJad&}
n (Rz - YZL) S 0
n{RI > I:ah)&} n
n {R > WY~&}
{Rs > I:Y~)R~}I.
And by the FP fkmulation
p/” 2 p,‘”.
(11)
this is given by
P[(R3 - ad.) < Ofl(R2 - y&j < 01.
(12)
Clearly, eqn (12) is computationally easier. Approximate methods based upon FORM/SORM have been developed to evaluate the probability of such events [7, 1, 81. Equation (11) can be re-written as P[(R, - a31,) < 0fl(R2 - y2L) < 0 n{RI
> ~Ca~/adR~}n {Rz > (adadR~}
Thus, the FP approach can generally lead to overestimating the system failure probability. From the above formulation, it is clear that the estimate of the probability of occurrence of a failure mode by the FS formulation is smaller than that computed by the FP formulation. However, the order of the difference is not clear. The purpose of this paper is to present some results on the abovementioned differences. Estimates of the probability of occurrences of failure modes by both formulations are obtained using Monte Carlo simulation. Monte Carlo simulation provides an exact comparison, since the method is exact. Furthermore, it is difficult to evaluate the probability of occurrence of a failure mode by the Fs formulation. For completeness, the simple Monte Carlo technique is outlined [9]. Equation (1) can be recast as
n {& > (Adam)&} n {Rs > (ah)R~} P/ =
n{&
> (~hM2)
n{h
>
h/~~)R~)l.
(13)
From eqns (13) and (12), it is clear that the event described by the F% formulation is a subset of that of the FP formulation. Thus, the probability of occurrence of a failure mode estimated by the FS formulation is always smaller than that estimated by the FP formulation. An&her important observation is the fact that the FS formulation leads to events that are mutually exclusive. Consequently, the estimate of the system failure probability will be simplified. The system failure probability is given by P,=p[vr=,
occurrence of failure mode i],
(14)
where k is the number of failure modes in the system. Hence, with the FS formulation, system failure probability is the sum of the probability of occurrence of each of the failure sequences, i.e. k pr = P C occurrence of failure sequence [ ,-I
i1.
s R
n 1% > ~~Y~)Rz)
(16)
W1”04 dx,
(17)
where R is the sample space and Z(x) is an indicator function, Z(x) = 1 if x is in D and 0 otherwise. Using simple Monte Carlo simulation, an unbiased estimate for pr is computed as follows:
(18) where N is the sample size and x, is a sample from f(x). The coefficient of variation (COV) of the estimator j in eqn (18) is COV@f) =
J( > -1 -Pr
NP/
’
(19)
The estimated COV in eqn (19) with pf replaced by its estimator h. By the FS formulation, the estimator of the system failure probability is given by
(15)
Thus, if the probabilities of the failure sequences can be evaluated, the system failure probability follows easily. The system failure probability given by eqn (15) is exact. For subsequent discussion, the system failure probability by FS formulation is defined as p/‘“. In the FP formulation, the failure modes are no loqger mutually exclusive events. The determination of the system failure probability using
where Z&x,) is the indicator variable for a failure sequence. Similarly, by the FP formulation
(21) where ZFp(x,) is the indicator variable for a failure path. Hence, if for all the samples observed that lead to failure by the occurrence of failure sequences, one
Y. Ibrahim
628
or more failure paths are observed, then the two estimators will be the same. As mentioned earlier, an FS is a sub-set of an FP; the occurrence of an FS must be accompanied by the occurrence of one or several FP. However, in view of the formulations of FP and FS, there can be cases where samples lead to system failure by the occurrence of a failure path and not a failure sequence. Estimates of the probability of occurrence of failure modes by the two formulations are obtained in the same manner as above; each sample is tested for system failure by both formulations and the number of failure modes observed are counted.
3. TESTING OF SAMPLES
In this section, an outline of the procedures adopted to test each sample generated by the Monte Carlo simulation for failure by the FS and FP formulations are presented. The following discussion is for the case of a single external load. Extensions to multiple load cases are straight forward and will not be discussed here. System failure is defined as structural instability. Hence, the states of the surviving components have to be determined in order to assess structural system failure. Recall that the interest is only with brittle systems, where a component loses strength and stiffness after failure. In this study, once a component has failed, it is removed from the structure and the structure is re-analyzed from the zero load condition. 3.1. Failure sequence approach In this information, if a sample leads to system failure, then only one failure mode will occur. This is evident from the formulation given in Section 2. At each state of the system, the load is incremented from zero and the component that is the first to fail (i.e. the weakest component) in that state is obtained. The load level at which the weakest component failure occurs is checked against the sampled load. If the sample load level is smaller, then the system will not fail with that particular sample. However, if the sampled load is greater, then the weakest component will fail. The failed component is removed and the system is in a new state. This new state is checked for system failure. If the new state is a system failure state, then the structure will fail with the particular sample. If the new state is not a system failure state, then proceed to find the weakest component in the new state and repeat the above steps. The preceding is translated into steps given below. Assume that the system is in damage state k, i.e. k components have failed and are removed from the structure. 1. The structure in damage state k is analyzed with a unit applied load. The external load level to cause each component to fail is computed by dividing the sampled capacity with the respective force in the
component
due to the unit applied load, i.e. L:=%.
where u, is the force in component i due to a unit external load and R, is the sampled capacity of component i. The minimum of Lf corresponds to the weakest component, i.e., if Lf is the minimum, then component j is the weakest component in damage state k. 2. The minimum load level, i.e. L: is checked against the sampled load level, L. If L,! < L then component j will fail in damage state k; proceed to the next step. Otherwise, component j will not fail and the system also will not fail; proceed to the next sample. 3. The failed component, i.e. j, is removed from the structure. Hence, the system is in damage state k + 1. Check if the structure is stable in damage state k + 1. If it is unstable, then system failure has occurred. If the structure is stable, then it has not failed; proceed to next step. 4. Componentj is removed from the structure. The structure, now in damage state k + 1, is analyzed as in step 1. The weakest component in damage state k + 1 is obtained and steps 2 and 3 are performed to find the next damage state and check for system failure. 5. The above steps are repeated until either the system survives or fails with the particular sample. The corresponding steps in the structural analysis are shown schematically in Fig. 3. After each failure, the system becomes less stiff since the failed component is removed. This can be seen by comparing paths o-a and o-b and paths o-a, o-b and o-c in the first and second figure respectively of Fig. 3. Recall that in the analysis, after each failure, the system is loaded from zero. In the first figure, after the failure of the first component, i.e. path o-a, the force needed to fail the second component, i.e. path o-b, is higher. This implies that there was no failure during redistribution which was a consequence of the failure of the first component. However, in the second figure, L, < Lb, implying a failure during redistribution. 3.2. Failure path approach Since the events described by the FS formulation is a subset of the events described by the FP formulation, every sample that leads to failure by the FS
c
Fig. 3. Analysis of brittle systems.
629
Failure sequence and failure path
\
B
Fig. 4. Example of failure path. formulation will lead to failure by the FP formulation. The converse however is generally not true. Hence, even if a sample did not lead to failure by the FS formulation, it should be tested for failure using the FP formulatiosn. In this approach, at each damage state of the structure, the safety margins of the surviving components for the total sampled load are computed. Those components for which the safety margins are negative or zero are deemed to have failed. There could be more than one component failure. The structure is now in the next damage state. However, for that damage state, there will be several combinations of component failures leading to that damage state. For this reason, for every sample that leads to failure by the FP formulation, several failure modes can occur. In order to make the following outline clearer, reference will be made to a failure tree example shown in Fig. 4. The analysis begins at the intact state. 1. Evaluate the safety margins of the components in the intact state. The intact structure is analyzed under the full sampled load. The forces in the components are then checked against their corresponding sampled capacity, i.e. check M,= R,--E,
(23)
where M, is the safety margin for component i and F, is the force due to the sampled load, in component i. F, is given by F, = a,L,
(241
where a, is the folrce in component i due to a unit external load and L is the sampled load. 2. If none of the forces exceed the corresponding capacities, then the system survives, i.e. M, > 0 for all i. 3. If one or some of the forces exceed the corresponding capacities, then the associated components have failed in the intact state, i.e. M, < 0 for some i. Thus, one or several components, i.e. those whose
safety margin is negative, have failed. With respect to Fig. 4, the failed components would be i, j, k, 1, m. 4. Next analyse the structure in damage state 1, i.e. the structure has one failed component. Note that if there is more than one failed component, the structure is analyzed in succession with each of the failed components removed. With respect to Fig. 4, the structure is analyzed for cases where only component i is removed, only component j is removed, only component k is removed, only component 1 is removed and only component m is removed. Note that for these analyses, the structure is also checked for failure. For example, if with only component m removed, the system is unstable, then a failure mode has occurred. The safety margins for the surviving components in the various analyses cases are evaluated and checked for failure. For example, in the failure tree shown in Fig. 4, for the case where only component i is removed, components n and j failed in the subsequent analysis. However, for the case where only component j was removed, none of the components failed. Hence, with this sample, there will be no failure modes with component j as the first component to fail. 5. Repeat step 4 and analyze the structure now in damage state 2, i.e. with two components removed. For example, in Fig. 4, the cases in which the structure have to analyzed include those where the following two components are removed at a time; (i, n), (r,j), (k, i), (k, 0, (k,pl, (1, k) and U, 4). 6. Repeat steps 4 and 5 until all the possible failure combinations have been uncovered. 4. EXAMPLE
Consider the ten-bar truss structure with brittle components shown in Fig. 5. The parameters of the components are also shown in Fig. 5. The load L is normally distributed with a mean of 1.5 and a COV of 0.3. All the random variables are independent. A large scale Monte Carlo simulation was performed. For every sample observed, the structure was subjected to both the FS and FP approaches. A component fails if the force in the component due to the external load exceeds the component capacity. System failure is defined as structural instability. 4.1. Results and discussion The result of the Monte Carlo simulation is shown in Table 2. Two outcomes that lead to system failure are also produced here in Table 1 for discussion. The first sample, i.e. no. 5716, led to an outcome where a failure sequence was observed and 4 failure paths were also observed. Note that the failure mode that occurred by the FS formulation also occurred by the FP formulation, i.e. mode -4, -9. The second sample, i.e. no. 5823, is a case in which failure paths occurred but not a failure sequence. Hence, such outcomes will contribute to the differences in the estimate by the system failure probaiblity by the two
630
Y. Ibrahim
Fig. 5. Ten-bar
Table
Table 2. Failure
1. Example
truss structure.
of outcomes
probabilities
formulations. The results obtained by the FS formulation and FP formulation are shown in Table 2. Several interesting observations could be made. First, note the differences, which were expected, between the FP formulations and FS formulation for the probability of occurrence of the failure modes. However, the order of the differences is not the same for all the failure modes found. In one particular case, i.e. failure mode -4, -9, the difference is quite significant. This is a very interesting mode. In the intact state, component 9 is stressed in tension. After component 4 fails, component 9 will unload and go into compression. A more important observation is the ordering of the failure modes. This is related to the significant difference between the FS and FP formulation of failure mode -4, -9. By the FS approach failure mode -4, -9 is the third most likely failure mode. However, by the FP approach failure mode -4, -9 is the most likely failure mode. Hence, this will affect the results of the search for the dominant failure modes using the FP formulation.
by Monte
of samples
Carlo,
N = 50,000
5. CONCLUSIONS
The estimate of the probability of occurrence of a failure mode depends upon the formulation of the corresponding event. A failure sequence (FS) formulation gives rise to lower probability of occurrence than a failure path (FP) formulation. For a brittle system these differences in the estimates can also be reflected in the estimate of the system failure probability. Results of the estimates of the probability of occurrences of the failure modes of a simple truss example were obtained by the two formulations using Monte Carlo simulation. In the example chosen, the differences were quite significant for one mode. These differences have implications on search strategies that make use of the FP formulation to seek out the dominant modes of the system. The seach can be affected in cases where the differences are significant enough to cause a change in the ordering of the relative dominance of failure modes. Recall that the FS approach is the exact method. Thus, the relative
Failure sequence and failure path dominance of the failure modes as determined by the FS approach is the correct order. However, estimate of the system failure probability by the FP approach tend to be conservative. Acknowledgement-The author wishes to thank Mr Ashish Karamchandani of !3tanford University for some enlightening discussions.
REFERENCES
Y. F. Guenard, Application of system reliability analysis to offshore structures. Report No. RMS-1, Reliability of Marine Structures Program (formely Report No. 71, J’ohn A. Blume Earthquake Engineering Center), Department -of Civil Engineering, Stanford University (1984). R. E. Melchers and L. K. Tang, Dominant failure modes in stochar,tic structural systems. Srrucfural Safety 2, 127-143 (1984). F. Moses and M. R. Rashedi, The application of system reliability to structural safety. Proceedings of the 4th
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International Conference on Applications of Statistics and Probability in Soil and Structural Engineering,
Florence, Italy (1983). 4. P. Thoft-Christensen and Y. Murotsu, Application of Structural Systems Reliability Theory. Springer, Berlin (1986). 5. P. Bjerager, A. Karamchandani and C. A. Cornell, Failure tree analysis in structural system reliability. Proceedings of the 5th International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, pp. 985-996. Vancouver,
Canada, May (1987). 6. H. 0. Madsen, S. Krenk and N. C. Lind, Methods of Strucfural Sufefy. Prentice-Hall, Englewood Cliffs, NJ (1986). 7. S. Gollwitzer and R. Rackwitz, Equivalent components in first order system reliability. Reliability Engineering 5, 99-115 (1983).
8. A. Karamchandani, Y. F. Guenard and K. Ortiz, Shasys-A software package for component and system reliability analysis. Report No. 78, John A. Blume Earthquake Engineering Center, Department of Civil Engineering, Stanford University (1986). 9. R. Y. Rubinstein, Simulation and the Monte Carlo Method. John Wiley, New York (1981).