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Discussion on ‘non-invariant failure probabilities with invariant reliability index’
Reliabifity Engineering and System Safety 33 ( 1991 ) 3 ! 5-321
Letters to the Editor
DISCUSSION ON 'NON-INVARIANT FAILURE PROBABILITIES WITH INVARIANT RELIABILITY INDEX' The aims of the paper by Van Uu Nguyen t appear to show (i) that safety indices are not necessarily good estimators of the probability of failure and (ii) that the so-called 'invariant' reliability index of Hasofer and Lind is not invariant for some limit state formulations. The first part need not be discussed, since it is well known (e.g. see Ref. 2). The argument for the second part is flawed because the assumptions used are not carefully examined prior to drawing conclusions. However, the author raises an interesting point in developing his arguments. Some preliminary remarks about the Cornell index are in order. It is known that this index is not invariant and that it only corresponds to the Hasofer-Lind index under the very specific conditions of second-moment description of the basic variables and a linear limit state function. As argued by Ditlevsen, 3 the second-moment representation is essentially equivalent to that of an independent component multi-dimensional normal probability density function. There is, therefore, little point in examining the Cornell index for invariability. Further, many other indices can be developed-Cornelrs index enjoys no particularly theoretical advantage (cf. Ref. 4). The Hasofer-Lind index is invariant only under the conditions spelt out in their paper--unfortunately, the author's cases do not fit into these limitations. However, he has exposed, unknowingly it seems, a possibility not foreseen specifically by Hasofer and Lind, namely that the limit state equation can have multiple roots. Of course, the arguments by Hasofer and 315
Reliability Engineering and System Safety (33) (1991 ~ Ltd, England. Printed in Great Britain
1991 Elsevier Science Publishers
316
Letters to the editor
Lind (and others subsequently) were based on linear or near-linear limit state functions, for which the possibility of multiple roots does not arise. Nevertheless, multiple roots have been noted previously (e.g. by DolinskyS). F r o m a mathematical point of view, the Hasofer-Lind index should hold for all descriptions of the limit state equation Z = 0 in terms of the load L and resistance R. Consider the case Z = R 2 - L 2. In the paper this is shown in Fig. 4 as two linear limit state equations in the (R, L) space, with a different fl for each. The union of the two failure regions describes the total failure region, so that it is clear that neither of the fl values is necessarily a good description of the actual failure probability. There is no argument with this. However, how can the limit state function Z = R: - L 2 lead to this situation, apparently so much at variance with the ideas of Hasofer and Lind? It should have been recognized at once that Z has two roots: Zx = R + L = 0 and Z 2 = R - L = 0. Thus there are actually two limit state equations and the problem should have been considered as a problem in system reliability from the outset. Similar remarks apply to the case Z = (R - L ) 3. This has three roots, which are, however, identical, so that the fl values are the same. This explains why it is sufficient to obtain fl only for the linear form Zt = R - L. (It is difficult to understand why the results are claimed to be 'poor' in relation to a Monte Carlo simulation--the error is less than 1%, a result usually considered excellent, given the nature of Monte Carlo methods.) Turning now to the case Z = (R/L) - 1 = 0, and still looking only at the mathematics, it is easily seen that this has two roots, represented by the equations Z I = R - L = 0 and Zz = 1/L = 0. If this had been recognized at the outset, a system interpretation could have been applied and the probability calculated would be
p, = P[(R < L) u (L < 0)] ¢ P[R < L] This is not, however, the interpretation given in Fig. 3, since, under the present formulation, the lower 'safe' (triangular) region would be a 'failure" region in addition to zones B and D. The lower triangular region in Fig. 3 is associated with highly negative values of R and lower negative values o f L, such that R/L is positive and more than 1 and such that Z > 0. Thus it would appear that the triangular region should be a 'safe' zone, as shown in Fig. 3. The contradiction can be explained only by examining the physical meaning of negative R and L. The meaning of the factor o f safety F in this context must be examined also. Consider an elementary reliability problem: a bar of tensile strength R loaded in tension by a load L. Conventionally, F = R/L. If L is negative in value, the load is applied so as to put the bar in compression. The safety
Letters to the editor
317
margin Z = R - ( - L ) is now positive definite, which means 'safe" and suggesting that a negative load adds to the strength of the structure, particularly if there was a further load that might be applied in the direction to cause tension in the bar. This accords with convention. However, the factor of safety F = R/(-L) is negative and therefore less than 1, which means 'failure'. Which of these two alternatives is correct depends on the physics of the problem, in particular on the compressive strength of the bar. In probabilistic terms, the problem strictly should have been formulated (i) to specifically disallow negative loading by appropriate formulation of the probabilistic model which describes the loading, (ii) to specify the compressive strength and hence the probability density function that goes with it, or (iii) to accept that negative loading can occur, being then a resistance, and that the strength can have corresponding 'negative values' also. The latter is the convention adopted in structural reliability analysis when simplified probabilistic models of loading are used, such as the secondmoment description used in the example. The error in so doing is often ignored. Thus, in using a normal distribution for loads or resistances the probability density function (p.d.f.) tail may enter the negative region; this error is usually ignored. Note that if R is negative, the 'failure' condition is indicated since Z will be negative and F < 1, which agrees. In this case, the negative resistance can be interpreted as either a load (which is the usual convention) or as strength in the opposite direction. It is clear from the above that if the factor of safety F is to be used in a reliability formulation, it must be further defined, in particular when either or both R and L are negative. One way in which the reliability problem could have been specified to avoid the above problem is (a) Pt = P[(R < L)I(L > O, R > 0)] = P[R < L]/{P[R > O] •P[L > 0]} This corresponds to the upper right-hand corner of Fig. 3. If the condition on R is removed, that is if negative R is also allowed (corresponding to compressive strength in the above example), the region to the right o f / = - 6 in Fig. 3 applies. The region to the left is simply not part of the failure region of interest. Hence (b)
Pt = P[(R < L) IL > O] = P[R > L]/P[L > O]
The corresponding probability P[R > L] is then 0.007 65 - 0.000 024 3 = 0.007626, which corresponds closely with the probability of 0.00765 obtained by simply ignoring the existence of the left region, i.e. ignoring the condition L > 0 in the lower part of Fig. 3. The probability P[L > 0] is given
318
Letters to the editor
by 1 - 0 ( - 6 ) (where 6 = 2-0) or 0.9773. Hence P f = 0.007 626/0-9773 = 0-007 803. However, the author interprets the region L < 0 (except for the small lower triangular region) as structural failure, and so arrives at the m u c h larger failure probability of 0-0302. It should be clear that the latter result is entirely a consequence of i m p r o p e r problem definition associated with the use of a ratio (the factor o f safety) to express structural adequacy. A similar problem with using a ratio has been noted in the field of costbenefit analysis. There the trend is to use the 'net benefit' (equivalent to the safety margin concept) in order to obviate the problems associated with the interchangeability of costs and benefits (e.g. see Ref. 6). I f a ratio such as F m u s t be used (and to date the reasons advanced for this do not seem to be compelling) it is necessary to define t h e probabilistic properties of R and L such that the p.d.f.s for (R > 0 and R < 0) and (L > 0 and L < 0) are all separately specified. As noted above, this would also be desirable for conventional reliability formulations if there is a need for detailed and complete accuracy. A final point is that F becomes difficult to manage if there are a n u m b e r of loads and resistances to be considered (see, for example, Ref. 2).
REFERENCES 1. Nguyen, Van Uu, Non-invariant failure probabilities with invariant reliability index. Reliability Engng and System Safety, 28 (1990) 99-110. 2. Melchers, R. E., Structural Reliability: Analysis and Prediction. Wiley, 1987. 3. Ditlevsen, O., Generalised second moment reliability index. J. Structural Mechanics, 7(4) (1979) 435-51. 4. Turkstra, C. J. & Duly, M. J., Two moment structural safety analysis. Can. J. Ciril Engng, 5 (1978) 414-26. 5. Dolinsky, K., First order second moment approximation in reliability of structural systems: critical review and alternative approach. Structural Safeo', 1(3) (1983) 211-31. 6. de Neufville, R. & Stafford, J. H., Systems Analysisfor Engineers and Managers. McGraw-Hill, New York, 1971. R. E. Melehers
Department of Civil Engineering and Surveying, The University of Newcastle, New South Wales 2308, Australia (Received 5 May 1990)
Letters to the Editor
DISCUSSION ON 'NON-INVARIANT FAILURE PROBABILITIES WITH INVARIANT RELIABILITY INDEX' The aims of the paper by Van Uu Nguyen t appear to show (i) that safety indices are not necessarily good estimators of the probability of failure and (ii) that the so-called 'invariant' reliability index of Hasofer and Lind is not invariant for some limit state formulations. The first part need not be discussed, since it is well known (e.g. see Ref. 2). The argument for the second part is flawed because the assumptions used are not carefully examined prior to drawing conclusions. However, the author raises an interesting point in developing his arguments. Some preliminary remarks about the Cornell index are in order. It is known that this index is not invariant and that it only corresponds to the Hasofer-Lind index under the very specific conditions of second-moment description of the basic variables and a linear limit state function. As argued by Ditlevsen, 3 the second-moment representation is essentially equivalent to that of an independent component multi-dimensional normal probability density function. There is, therefore, little point in examining the Cornell index for invariability. Further, many other indices can be developed-Cornelrs index enjoys no particularly theoretical advantage (cf. Ref. 4). The Hasofer-Lind index is invariant only under the conditions spelt out in their paper--unfortunately, the author's cases do not fit into these limitations. However, he has exposed, unknowingly it seems, a possibility not foreseen specifically by Hasofer and Lind, namely that the limit state equation can have multiple roots. Of course, the arguments by Hasofer and 315
Reliability Engineering and System Safety (33) (1991 ~ Ltd, England. Printed in Great Britain
1991 Elsevier Science Publishers
316
Letters to the editor
Lind (and others subsequently) were based on linear or near-linear limit state functions, for which the possibility of multiple roots does not arise. Nevertheless, multiple roots have been noted previously (e.g. by DolinskyS). F r o m a mathematical point of view, the Hasofer-Lind index should hold for all descriptions of the limit state equation Z = 0 in terms of the load L and resistance R. Consider the case Z = R 2 - L 2. In the paper this is shown in Fig. 4 as two linear limit state equations in the (R, L) space, with a different fl for each. The union of the two failure regions describes the total failure region, so that it is clear that neither of the fl values is necessarily a good description of the actual failure probability. There is no argument with this. However, how can the limit state function Z = R: - L 2 lead to this situation, apparently so much at variance with the ideas of Hasofer and Lind? It should have been recognized at once that Z has two roots: Zx = R + L = 0 and Z 2 = R - L = 0. Thus there are actually two limit state equations and the problem should have been considered as a problem in system reliability from the outset. Similar remarks apply to the case Z = (R - L ) 3. This has three roots, which are, however, identical, so that the fl values are the same. This explains why it is sufficient to obtain fl only for the linear form Zt = R - L. (It is difficult to understand why the results are claimed to be 'poor' in relation to a Monte Carlo simulation--the error is less than 1%, a result usually considered excellent, given the nature of Monte Carlo methods.) Turning now to the case Z = (R/L) - 1 = 0, and still looking only at the mathematics, it is easily seen that this has two roots, represented by the equations Z I = R - L = 0 and Zz = 1/L = 0. If this had been recognized at the outset, a system interpretation could have been applied and the probability calculated would be
p, = P[(R < L) u (L < 0)] ¢ P[R < L] This is not, however, the interpretation given in Fig. 3, since, under the present formulation, the lower 'safe' (triangular) region would be a 'failure" region in addition to zones B and D. The lower triangular region in Fig. 3 is associated with highly negative values of R and lower negative values o f L, such that R/L is positive and more than 1 and such that Z > 0. Thus it would appear that the triangular region should be a 'safe' zone, as shown in Fig. 3. The contradiction can be explained only by examining the physical meaning of negative R and L. The meaning of the factor o f safety F in this context must be examined also. Consider an elementary reliability problem: a bar of tensile strength R loaded in tension by a load L. Conventionally, F = R/L. If L is negative in value, the load is applied so as to put the bar in compression. The safety
Letters to the editor
317
margin Z = R - ( - L ) is now positive definite, which means 'safe" and suggesting that a negative load adds to the strength of the structure, particularly if there was a further load that might be applied in the direction to cause tension in the bar. This accords with convention. However, the factor of safety F = R/(-L) is negative and therefore less than 1, which means 'failure'. Which of these two alternatives is correct depends on the physics of the problem, in particular on the compressive strength of the bar. In probabilistic terms, the problem strictly should have been formulated (i) to specifically disallow negative loading by appropriate formulation of the probabilistic model which describes the loading, (ii) to specify the compressive strength and hence the probability density function that goes with it, or (iii) to accept that negative loading can occur, being then a resistance, and that the strength can have corresponding 'negative values' also. The latter is the convention adopted in structural reliability analysis when simplified probabilistic models of loading are used, such as the secondmoment description used in the example. The error in so doing is often ignored. Thus, in using a normal distribution for loads or resistances the probability density function (p.d.f.) tail may enter the negative region; this error is usually ignored. Note that if R is negative, the 'failure' condition is indicated since Z will be negative and F < 1, which agrees. In this case, the negative resistance can be interpreted as either a load (which is the usual convention) or as strength in the opposite direction. It is clear from the above that if the factor of safety F is to be used in a reliability formulation, it must be further defined, in particular when either or both R and L are negative. One way in which the reliability problem could have been specified to avoid the above problem is (a) Pt = P[(R < L)I(L > O, R > 0)] = P[R < L]/{P[R > O] •P[L > 0]} This corresponds to the upper right-hand corner of Fig. 3. If the condition on R is removed, that is if negative R is also allowed (corresponding to compressive strength in the above example), the region to the right o f / = - 6 in Fig. 3 applies. The region to the left is simply not part of the failure region of interest. Hence (b)
Pt = P[(R < L) IL > O] = P[R > L]/P[L > O]
The corresponding probability P[R > L] is then 0.007 65 - 0.000 024 3 = 0.007626, which corresponds closely with the probability of 0.00765 obtained by simply ignoring the existence of the left region, i.e. ignoring the condition L > 0 in the lower part of Fig. 3. The probability P[L > 0] is given
318
Letters to the editor
by 1 - 0 ( - 6 ) (where 6 = 2-0) or 0.9773. Hence P f = 0.007 626/0-9773 = 0-007 803. However, the author interprets the region L < 0 (except for the small lower triangular region) as structural failure, and so arrives at the m u c h larger failure probability of 0-0302. It should be clear that the latter result is entirely a consequence of i m p r o p e r problem definition associated with the use of a ratio (the factor o f safety) to express structural adequacy. A similar problem with using a ratio has been noted in the field of costbenefit analysis. There the trend is to use the 'net benefit' (equivalent to the safety margin concept) in order to obviate the problems associated with the interchangeability of costs and benefits (e.g. see Ref. 6). I f a ratio such as F m u s t be used (and to date the reasons advanced for this do not seem to be compelling) it is necessary to define t h e probabilistic properties of R and L such that the p.d.f.s for (R > 0 and R < 0) and (L > 0 and L < 0) are all separately specified. As noted above, this would also be desirable for conventional reliability formulations if there is a need for detailed and complete accuracy. A final point is that F becomes difficult to manage if there are a n u m b e r of loads and resistances to be considered (see, for example, Ref. 2).
REFERENCES 1. Nguyen, Van Uu, Non-invariant failure probabilities with invariant reliability index. Reliability Engng and System Safety, 28 (1990) 99-110. 2. Melchers, R. E., Structural Reliability: Analysis and Prediction. Wiley, 1987. 3. Ditlevsen, O., Generalised second moment reliability index. J. Structural Mechanics, 7(4) (1979) 435-51. 4. Turkstra, C. J. & Duly, M. J., Two moment structural safety analysis. Can. J. Ciril Engng, 5 (1978) 414-26. 5. Dolinsky, K., First order second moment approximation in reliability of structural systems: critical review and alternative approach. Structural Safeo', 1(3) (1983) 211-31. 6. de Neufville, R. & Stafford, J. H., Systems Analysisfor Engineers and Managers. McGraw-Hill, New York, 1971. R. E. Melehers
Department of Civil Engineering and Surveying, The University of Newcastle, New South Wales 2308, Australia (Received 5 May 1990)