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New directions and research needs in system reliability research
Structural Safety, 7 (1990) 93-100 Elsevier
93
NEW D I R E C T I O N S A N D RESEARCH NEEDS IN S Y S T E M RELIABILITY RESEARCH * Fred Moses Department of Civil Engineering, Case Western Reserve University, Cleveland, OH 44106 (U.S.A.)
(Accepted May 1989)
Key words: probability-based design; reliability; structural systems; system reliability.
ABSTRACT Structure system reliability has evolved considerably in recent years. Various theoretical studies have modelled stochastic time-varying systems for both fatigue and ultimate limit states. Implementation has been slow because of the relative emphasis on reliability oriented code member design checks. The paper surveys available system reliability methods as well as outlines for future work. Needs have arisen to include system aids in evaluating and comparing design concepts as well as providing on-line optimal strategies for allocating inspection, maintenance and rehabilitation expenditures.
1. INTRODUCTION The attention of reliability theory directed towards single component analysis had a firm historical and practical basis. Design codes, and indeed much of the attention of mechanics theory and experimentation, concerned single mode behavior, e.g. columns, connections, and pressure vessels. This reflects the activities of designers who look on design as satisfying individual component safety checks. Thus, there evolved reliability oriented prediction programs to estimate the probability of failure of single components. To apply such probabilistic results, design methods were developed (e.g. Level I code checks) so that the designers could be satisfied that a target reliability index was realized. System reliability initially arose as an extension of component failure mode analysis when it became apparent that multi-component behavior had a severe impact on the true risk of * Presented at the Workshop on Research Needs for Applications of System Reliability Concepts and Techniques in Structural Analysis, Design and Optimization, Boulder, CO, September 12-14, 1988. 0167-4730/90/$03.50
© 1990 Elsevier Science Publishers B.V.
94 structure failure. In reality, structure systems have a wide variety of different failure characteristics. In many structures, several components must fail before the structure fails. On the other hand, in statically determinate systems, a weakestqink or chain model governs so the failure of any single component is catastrophic. Clearly, most designers emphasize redundancy as being important to structure safety. But merely counting the degree of redundancy may be irrelevant for a risk evaluation. If all members of a redundant system are designed against their behavior limit and if the load uncertainty (i.e. COV) overwhelms resistance uncertainty, as is often true in seismic, ocean wave, snow, wind and similar environmental loading, then, the existence of redundant elements does not give benefits. That is, the failure mode events for each component have high correlation and if one element fails, the other elements are also likely to fail. The probability of system failure approaches that of the largest single member failure probability. The problem of predicting system reliability has been classically formulated as determining the system reliability from the component reliabilities. There are several possible techniques including (i) direct methods (e.g. Monte Carlo simulation, point estimates response surfaces) and (ii) failure mode analysis method. The direct methods, while often simpler to apply, may be time consuming and offer little further insight into the problem. However, recent advances in importance sampling and directional simulation techniques suggest significant increased efficiencies while removing some of the limitations in direct methods. The failure mode analysis method is often preferred in characterizing large structural systems. The three characteristics of a model which have been shown to significantly affect its reliability are the geometry, member behavior and statistical correlation [1]. Recent studies have investigated member behavior and considered various idealized models including elastic ductile, elastic brittle, strain hardening and semi-brittle behavior (the latter illustrates loads which drop off linearly after reaching a maximum strength value). Because of the load redistribution that occurs after component failure, the system reliability analysis must be closely connected to the structural analysis. Geometry has been a main topic of system studies with idealized series and parallel models as two extremes of system description. Such work follows similar models in electrical and mechanical systems. An important distinction for structural reliability is the large statistical correlation that may be present. This correlation in element failure events occurs either due to the common loading phenomenon that affects each component in the structure (e.g. wind, wave, seismic) or through the resistance variable in a common material, strength assessment, fabrication, inspection and testing which increases the statistical correlation of member strengths. For a series system, increasing correlation decreases the probability of failure. Conversely, for a parallel system, increasing correlation increases the probability of system failure. This influence of correlation raises serious issues regarding the definition and description of redundancy as an important ingredient in enhancing safety (see below). The material behavior, ranging from ideal ductile to brittle, has similar influences as correlation and geometry on system performance. A brittle element which carries little or no load after a component reaches its strength limit may have a counter-productive effect on a parallel system by increasing structure risk. Several studies suggest that, except for highly redundant systems, the reliability of brittle parallel systems can be approximated by a "weakest" link or series system. This effect is influenced by strength variability, correlation and overall reserve margins. Similarly for redundant systems, if all members are designed against their behavior limit, there may be little advantage in redundancy. The presumed benefits of redundancy are often only the indirect result of significant underdesign of some components due to fabrication, load combina-
95
w...0 ,
_su...s.ructur.
> ~ /
iflooding,
in elements
.
.
.
.
.
.
IIJ I IJ [1|
• / CURRENT ~If' ' "
Ill\
~
m tendons (fatigue, extreme loading)
m tendons
tendon system I
~d2nt (n per tendon )
p piles (rupture, pull-out)
Fig. 1. Tension leg platform. Failure modesand system models [2]. tion or other reasons. The enhancements in system reliability due to redundancy may be "accidental" beneficial contributions and therefore must be thoroughly checked by system reliability methods for particular applications. Examples of such direct studies include highway bridge girder systems and redundant offshore jacket frameworks. A conclusion drawn from the idealized behavior studies (e.g. series, parallel, ductile, brittle, correlated, independent) is that there is a multiplicity of possible results which are interconnected in geometry, material behavior and correlation assumptions. This suggests that detailed failure mode analyses are needed for practical systems which can be studied to define realistic reliability ranges within the parametric values likely to be encountered. An example of a relatively simple model is a tension leg platform, shown in Fig. 1. An examination of the tendon system shows each tendon is made of n cables or welded tubes in series (i.e. a weakest-link system with relatively poor reliability characteristics). By providing m tendons in parallel, some redundancy is given. For a single mode, the reliability is improved by increasing the number of components that act in parallel. However, this increase in components also increases the overall number of possible failure modes and decreases the system reliability. Some initial calculations show that the greatest marginal benefit is achieved in going to about two or three degrees of redundancy [2]. This result is of course affected by the assumptions regarding load redistribution (structure model), strength correlation and member behavior (i.e. unloading) following member failure.
2. SYSTEM ANALYSIS
After understanding the basic simple models, it is obvious that there are few cases like the tension leg illustration that can be model led by the idealized cases described above. It became
96 important several years ago, especially in modelling offshore jacket frameworks, that a general procedure was needed that could treat realistic structure geometry (combinations of series and parallel networks), a range of behaviors (some brittle, some ductile and some in between, depending on mode reached) and finally, working with basic variables, to assess correlation effects. Moses and Gorman [3] had experimented with an incremental loading method. This was extended by Moses and Stahl [4] to platform jackets. The method leads to a failure mode equation and second-order modelling (mean and variance) of structure systems appropriate for computing safety indices. For applications to large-scale realistic structures, the system analysis should be divided into two parts: (a) engineering modelling, which means identification, description and enumeration of the statistically significant collapse modes, and (b) probabilistic calculations to assess each individual model failure probability or safety index and subsequently combining these into a single system risk assessment. For simple behavior models, e.g. ductile plastic behavior, the enumeration of individual failure modes is straightforward and follows traditional plastic analysis methods. Each mode can be expressed by a reserve margin which involves the modal member resistances and active loads corresponding to that particular failure mechanism. For more general member behavior, the enumeration of significant modes is difficult and various tree search procedures are necessary. The object is to express the reserve margins, g, for each failure mode in terms of basic load and element resistance variables. This simplifies the subsequent computation of modal safety indices. The combination of these failure modes into a system reliability index is complex. Several investigations have developed bounds on the system risk using various approximations. Monte Carlo simulation offers a direct method to combine the modal descriptions into a system assessment. It is important that such simulation is done only with failure mode expressions and not involve any repeated reanalysis of the structure.
2.1. Practical system reliability analysis The basic steps of the incremental loading method are quite simple. A much greater problem is the detailed "book-keeping" that is involved in generating and keeping track of each significant possible failure alternative in a large realistically modelled system. The steps in the modelling include: (1) analyze the intact system; (2) identify a critical member which will fail as the loading is progressively increased; (3) "remove" the failed component and modify the load redistribution and identify the next component to fail; (4) repeat the reanalysis with a new failed member at every step until the system collapses. By very simple steps, it is possible to determine a failure mode equation involving the load and those resistances corresponding to the members which failed in sequence prior to collapse [1]. The simplest approach in the incremental model is to guide the selection of failure members using the mean reserve margins. This was illustrated by Moses and Stahl [4] and will usually lead to a dominant system collapse mode. The enumeration of other significant modes has been studied by several investigators. Gorman [5] used heuristic techniques and even Monte Carlo simulation to cause different failure sequences to appear. The latter were encouraged by using artificially high strength coefficients of variation. Linear programming and nonlinear programming techniques have also been used to find ductile collapse modes for frames first analyzed for
97 g=O.667R 3 + 0.555R 8 - 1.0S 1 g=l.O00R 5 + 0.555R~ - 1.0S g=0. 555R +7 + 0.555R 8
1.0S
g=0. 333R +1 + 0.500R 6
I.OS
g=l.0OOR6
+ ~.555R 8 + 0.555R10-
g=O.333R4
+ 0.277RI0-
I.OS
1.0S
g=0.555R +9 + 0. 555RI0- 1.0S g=O.667R +9 +I.000R6 etc.
+ 0.555RI0-
(See Reference
I.OS
7)
Component I0 _
Mean Strength
_
Sl
Area
Tension/ Comprn. 2
3
1-6
20/10
1.00
7-If) 5/2.5
0.25
2S 2
Fig. 2. Failure tree - enumeration strategy using incremental load model, g is failure function, R is component strength. ( - ) indicates compression, ( + ) indicates tension. Ductile components.
basic plastic mechanisms. More general approaches for systems with different member behavior have used safety index criteria to enumerate significant failure modes. Such strategies lead to failure tree searches, as illustrated in Fig. 2. Each branch may lead to a failure mode expression. It is important for large frameworks to reduce the number of structural reanalyses and several approaches have been utilized. Rashedi and Moses [6] extended the basic incremental loading model to include brittle and semi-brittle components, multiple load conditions and even partial failure modes. Applications included both evaluation of design concepts (e.g. structure topology and redundancy) and inspection criteria. The latter utilize the tree search to identify which members are critical to the system and should require additional quality review in design, construction and maintenance. The difficult task of combining the failure mode expressions into an overall system risk was also
98 studied by Rashedi [7] who used Monte Carlo simulation. Fu and Moses [8] found a new approach in importance sampling that provided both good accuracy and high efficiency. The major recent extensions of the failure tree system modelling were done by Cornell and his associates [9] supported by an oil industry project. They have developed programs for the practical modelling of offshore platform frameworks. Among the advances were improved assessments and comparisons of alternate structure geometries and topologies, the effects of redundancy, simulated damaged conditions and detailed investigation of structure redistribution capabilities, improved reanalysis, more accurate modal combination probabilities and more realistic post-failure behavior models. Further work was identified to include more detailed and realistic post-failure element models including combined loading cases and more intelligent generation of failure paths.
3. FUTURE APPLICATION OF SYSTEM MODELS Two main areas have been described which encompass the goals of system applications to date. The first is structure concept evaluation, and the second is lifetime safety strategies. In structure concept evaluation we investigate such issues as redundancy, geometry and system toughness. That is, what is the ratio of system reliability to critical member reliability? Also, what happens to the system if critical members are damaged due to original defect, fatigue, accidental loading, etc.? This requires tree searches to develop likely failure paths. Furthermore, studies are needed to show which geometries and levels of redundancy are optimal for a given loading environment including the post-failure behavior of elements and correlations. In the future, there must be more intensive aids to solve these problems, especially with regard to lifetime safety strategies. There is increasing interest in assessing safety with respect to a variety of situations other than the one idealized in design, namely a structure built as designed and subject to prescribed load combinations. What has become known as the "infrastructure crisis" has opened new decision areas in many instances much more complex than "simple" design cases. Decisions must continually be made regarding existing, structures such as highway bridges, offshore platforms, buildings, pipelines, etc. The trade-offs include restricting operation (e.g. a truck weight or traffic posted bridge), repair, rehabilitation or replacement of a structure. Further models, such as one proposed by the author, include a trade-off of safety margins with more intensive inspection, analysis or even testing all of which presumably reduce uncertainty and therefore allow for lower prescribed safety margins to attain target safety indices [10]. The applications include a revision to an existing manual for maintenance inspection of highway bridges which controls the safety of some 600,000 existing bridge structures in the United States. Increasingly, engineers with responsibility for such systems are recognizing that the conventional safety factors in design cannot be attained for existing structures except at prohibitive costs. This is due to increasing load demands occurring simultaneously with deteriorating strength because of deferred maintenance. Many engineers are relying on reliability methods to help in the decision strategy. Another example-besides the bridge structures mentioned above, is the recent AIM project being carried out by Bob Bea and his associates [11]. These efforts are aimed at improving strategies for assessing existing offshore platforms using expanded statistical data base for in-service structure performance and system models. Some of the research topics that will be needed in the future to further such development are now described.
99
3.1. More efficient failure mode Identification Cornell and his colleagues [9] have described failure mode searches that efficiently find significant failure modes. However, there are still limitations in the element behavior models that can be described especially with regard to member unloading phenomena and complex load interaction problems (e. g. axial, bending, pressure).
3.2. In general, better member descriptions are needed Our emphasis has been on ultimate strength as often characterized by the formation of a plastic failure mechanism. In reality, failure is a progressive evolution of damage which may signal "failure" to someone such as an operation or maintenance engineer that failure is imminent or has occurred. This happens at levels below conventional predictions of collapse mechanisms. There needs to be better interaction of what field people will tolerate in terms of displacement, vibration, cracking, etc. that can help in identifying and expressing true failure modes.
3.3. In addition to the item in 3.2, there is the issue of low cycle fatigue failure Often, our loads such as wave or highway do not come as a single maximum event but rather as a sequence of several, very heavy, repetitive loads. Initially, some of these load cycles weaken the structure and eventually lead to lower resistance. These phenomena also need to be included in our modelling, especially when dynamic behavior or load amplification can result in significant changes in response just prior to the failure event.
3.4. To assist in identification we will need better reanalysis and search techniques These search methods are partly functions of structural modelling and matrix methods. Directional simulation and response surface techniques may aid in better utilization and more efficient sampling data.
3.5. Modal combination techniques can be vastly improved As described above, the importance of statistical correlation can have an overwhelming effect on the solutions. A more extensive data base is needed to model such correlations and in fact we need to improve model verification. Reliability models are useful for sorting through a strategy of decision options but at some stages the relative or notional values of risk must be translated into more precise estimates for optimal resource allocation.
3.6. Finally, the author feels that we may see more application in the future of on-line sensing and control technology applied to overall system safety The technology is evolving that will permit inexpensive and widespread use within a structure of devices to sense (1) loading (e.g. pressure, forces, ground motion); (2) atmospheric conditions (e.g. temperature, moisture, chemical presence such as chloride or other corrosive conditions); (3) structure behavior (e.g. strain and acceleration); and (4) material condition (e.g. thickness losses
100 d u e to ion, loss of paint, etc.). U n l i k e t o d a y ' s situation, w h e r e d a t a are very sparce, it m a y be e x p e c t e d in the future that we can b e o v e r w h e l m e d with s t r u c t u r e i n f o r m a t i o n . T h i s d a t a m a y b e useful for assessing in-service reliability a n d p r e d i c t i n g f u r t h e r d e g r a d a t i o n a n d r e d u c t i o n in lifetime reliability. Decisions b a s e d on actual site i n f o r m a t i o n m a y b e p r a c t i c a l in o p t i m i z i n g o u r resources b e t w e e n o p t i o n s of repair, r e h a b i l i t a t i o n or r e p l a c e m e n t a n d even r e c o m m e n d i n g m o r e intensive d a t a - g a t h e r i n g o p e r a t i o n s . T h i s implies that m u c h of o u r s y s t e m m o d e l l i n g m u s t be B a y e s i a n in nature, c a p a b l e o f b e i n g u p d a t e d b y new site-specific d a t a a n d p r o v i d i n g decision strategies with g r e a t e r confidence.
ACKNOWLEDGEMENTS T h e w o r k p r e s e n t e d herein is b a s e d in p a r t on the research f u n d e d b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t E C E 85-16771 " R i s k A n a l y s i s for E v a l u a t i o n of B r i d g e s " at C a s e W e s t e r n R e s e r v e University.
REFERENCES 1 F. Moses, System reliability developments in structural engineering, Structural Safety, 1(1) (1982) 3-13. 2 M.R. Gorman, Resistance modelling, in: ASCE Short Course, Structural Reliability Analysis of Offshore Platforms, May 1985. 3 F. Moses, Further developments of the incremental loading model for computer analysis of platform reliability, Report to Amoco Production Co., September 1977. 4 F. Moses and B. Stahl, Reliability analysis format for offshore structures, Paper OTC 3046, Offshore Technology Conference, Houston, May 1978. 5 M.R. Gorman, Reliability of structural systems, Report 79.2, Department of Civil Engineering, Case Western Reserve University, May 1979. 6 F. Moses and M.R. Rashedi, The application of system reliability to structural safety, Fourth International Conference on Application of Statistics and Probability in Soil and Structural Engineering, Florence, 1983. 7 M.R. Rashedi, Studies on reliability of structural systems, Department of Civil Engineering, Case Western Reserve University, May 1983. 8 F. Moses and G. Fu, Importance sampling in structural system reliability, Fifth ASCE E M D / G T D / S T D Specialty Conference on Probabilistic Mechanics, Blacksburg, Virginia, May 1988. 9 H. Nordal, C.A. Cornell and A. Karamchandani, A structural system reliability case study of an eight-leg steel jacket offshore production platform, Marine Structural Reliability Symposium, Arlington, Virginia, October 1987. 10 F. Moses and D. Verma, Load capacity evaluation of existing bridges, NCHRP 301, Transportation Research Board, Washington, D.C., December 1987. 11 R.G. Bea and F.J. Puskar, Development of AIM (Assessment, Inspection, Maintenance) programs for fixed and mobile platforms, Paper OTC 5703, Offshore Technology Conference, May 1988.
93
NEW D I R E C T I O N S A N D RESEARCH NEEDS IN S Y S T E M RELIABILITY RESEARCH * Fred Moses Department of Civil Engineering, Case Western Reserve University, Cleveland, OH 44106 (U.S.A.)
(Accepted May 1989)
Key words: probability-based design; reliability; structural systems; system reliability.
ABSTRACT Structure system reliability has evolved considerably in recent years. Various theoretical studies have modelled stochastic time-varying systems for both fatigue and ultimate limit states. Implementation has been slow because of the relative emphasis on reliability oriented code member design checks. The paper surveys available system reliability methods as well as outlines for future work. Needs have arisen to include system aids in evaluating and comparing design concepts as well as providing on-line optimal strategies for allocating inspection, maintenance and rehabilitation expenditures.
1. INTRODUCTION The attention of reliability theory directed towards single component analysis had a firm historical and practical basis. Design codes, and indeed much of the attention of mechanics theory and experimentation, concerned single mode behavior, e.g. columns, connections, and pressure vessels. This reflects the activities of designers who look on design as satisfying individual component safety checks. Thus, there evolved reliability oriented prediction programs to estimate the probability of failure of single components. To apply such probabilistic results, design methods were developed (e.g. Level I code checks) so that the designers could be satisfied that a target reliability index was realized. System reliability initially arose as an extension of component failure mode analysis when it became apparent that multi-component behavior had a severe impact on the true risk of * Presented at the Workshop on Research Needs for Applications of System Reliability Concepts and Techniques in Structural Analysis, Design and Optimization, Boulder, CO, September 12-14, 1988. 0167-4730/90/$03.50
© 1990 Elsevier Science Publishers B.V.
94 structure failure. In reality, structure systems have a wide variety of different failure characteristics. In many structures, several components must fail before the structure fails. On the other hand, in statically determinate systems, a weakestqink or chain model governs so the failure of any single component is catastrophic. Clearly, most designers emphasize redundancy as being important to structure safety. But merely counting the degree of redundancy may be irrelevant for a risk evaluation. If all members of a redundant system are designed against their behavior limit and if the load uncertainty (i.e. COV) overwhelms resistance uncertainty, as is often true in seismic, ocean wave, snow, wind and similar environmental loading, then, the existence of redundant elements does not give benefits. That is, the failure mode events for each component have high correlation and if one element fails, the other elements are also likely to fail. The probability of system failure approaches that of the largest single member failure probability. The problem of predicting system reliability has been classically formulated as determining the system reliability from the component reliabilities. There are several possible techniques including (i) direct methods (e.g. Monte Carlo simulation, point estimates response surfaces) and (ii) failure mode analysis method. The direct methods, while often simpler to apply, may be time consuming and offer little further insight into the problem. However, recent advances in importance sampling and directional simulation techniques suggest significant increased efficiencies while removing some of the limitations in direct methods. The failure mode analysis method is often preferred in characterizing large structural systems. The three characteristics of a model which have been shown to significantly affect its reliability are the geometry, member behavior and statistical correlation [1]. Recent studies have investigated member behavior and considered various idealized models including elastic ductile, elastic brittle, strain hardening and semi-brittle behavior (the latter illustrates loads which drop off linearly after reaching a maximum strength value). Because of the load redistribution that occurs after component failure, the system reliability analysis must be closely connected to the structural analysis. Geometry has been a main topic of system studies with idealized series and parallel models as two extremes of system description. Such work follows similar models in electrical and mechanical systems. An important distinction for structural reliability is the large statistical correlation that may be present. This correlation in element failure events occurs either due to the common loading phenomenon that affects each component in the structure (e.g. wind, wave, seismic) or through the resistance variable in a common material, strength assessment, fabrication, inspection and testing which increases the statistical correlation of member strengths. For a series system, increasing correlation decreases the probability of failure. Conversely, for a parallel system, increasing correlation increases the probability of system failure. This influence of correlation raises serious issues regarding the definition and description of redundancy as an important ingredient in enhancing safety (see below). The material behavior, ranging from ideal ductile to brittle, has similar influences as correlation and geometry on system performance. A brittle element which carries little or no load after a component reaches its strength limit may have a counter-productive effect on a parallel system by increasing structure risk. Several studies suggest that, except for highly redundant systems, the reliability of brittle parallel systems can be approximated by a "weakest" link or series system. This effect is influenced by strength variability, correlation and overall reserve margins. Similarly for redundant systems, if all members are designed against their behavior limit, there may be little advantage in redundancy. The presumed benefits of redundancy are often only the indirect result of significant underdesign of some components due to fabrication, load combina-
95
w...0 ,
_su...s.ructur.
> ~ /
iflooding,
in elements
.
.
.
.
.
.
IIJ I IJ [1|
• / CURRENT ~If' ' "
Ill\
~
m tendons (fatigue, extreme loading)
m tendons
tendon system I
~d2nt (n per tendon )
p piles (rupture, pull-out)
Fig. 1. Tension leg platform. Failure modesand system models [2]. tion or other reasons. The enhancements in system reliability due to redundancy may be "accidental" beneficial contributions and therefore must be thoroughly checked by system reliability methods for particular applications. Examples of such direct studies include highway bridge girder systems and redundant offshore jacket frameworks. A conclusion drawn from the idealized behavior studies (e.g. series, parallel, ductile, brittle, correlated, independent) is that there is a multiplicity of possible results which are interconnected in geometry, material behavior and correlation assumptions. This suggests that detailed failure mode analyses are needed for practical systems which can be studied to define realistic reliability ranges within the parametric values likely to be encountered. An example of a relatively simple model is a tension leg platform, shown in Fig. 1. An examination of the tendon system shows each tendon is made of n cables or welded tubes in series (i.e. a weakest-link system with relatively poor reliability characteristics). By providing m tendons in parallel, some redundancy is given. For a single mode, the reliability is improved by increasing the number of components that act in parallel. However, this increase in components also increases the overall number of possible failure modes and decreases the system reliability. Some initial calculations show that the greatest marginal benefit is achieved in going to about two or three degrees of redundancy [2]. This result is of course affected by the assumptions regarding load redistribution (structure model), strength correlation and member behavior (i.e. unloading) following member failure.
2. SYSTEM ANALYSIS
After understanding the basic simple models, it is obvious that there are few cases like the tension leg illustration that can be model led by the idealized cases described above. It became
96 important several years ago, especially in modelling offshore jacket frameworks, that a general procedure was needed that could treat realistic structure geometry (combinations of series and parallel networks), a range of behaviors (some brittle, some ductile and some in between, depending on mode reached) and finally, working with basic variables, to assess correlation effects. Moses and Gorman [3] had experimented with an incremental loading method. This was extended by Moses and Stahl [4] to platform jackets. The method leads to a failure mode equation and second-order modelling (mean and variance) of structure systems appropriate for computing safety indices. For applications to large-scale realistic structures, the system analysis should be divided into two parts: (a) engineering modelling, which means identification, description and enumeration of the statistically significant collapse modes, and (b) probabilistic calculations to assess each individual model failure probability or safety index and subsequently combining these into a single system risk assessment. For simple behavior models, e.g. ductile plastic behavior, the enumeration of individual failure modes is straightforward and follows traditional plastic analysis methods. Each mode can be expressed by a reserve margin which involves the modal member resistances and active loads corresponding to that particular failure mechanism. For more general member behavior, the enumeration of significant modes is difficult and various tree search procedures are necessary. The object is to express the reserve margins, g, for each failure mode in terms of basic load and element resistance variables. This simplifies the subsequent computation of modal safety indices. The combination of these failure modes into a system reliability index is complex. Several investigations have developed bounds on the system risk using various approximations. Monte Carlo simulation offers a direct method to combine the modal descriptions into a system assessment. It is important that such simulation is done only with failure mode expressions and not involve any repeated reanalysis of the structure.
2.1. Practical system reliability analysis The basic steps of the incremental loading method are quite simple. A much greater problem is the detailed "book-keeping" that is involved in generating and keeping track of each significant possible failure alternative in a large realistically modelled system. The steps in the modelling include: (1) analyze the intact system; (2) identify a critical member which will fail as the loading is progressively increased; (3) "remove" the failed component and modify the load redistribution and identify the next component to fail; (4) repeat the reanalysis with a new failed member at every step until the system collapses. By very simple steps, it is possible to determine a failure mode equation involving the load and those resistances corresponding to the members which failed in sequence prior to collapse [1]. The simplest approach in the incremental model is to guide the selection of failure members using the mean reserve margins. This was illustrated by Moses and Stahl [4] and will usually lead to a dominant system collapse mode. The enumeration of other significant modes has been studied by several investigators. Gorman [5] used heuristic techniques and even Monte Carlo simulation to cause different failure sequences to appear. The latter were encouraged by using artificially high strength coefficients of variation. Linear programming and nonlinear programming techniques have also been used to find ductile collapse modes for frames first analyzed for
97 g=O.667R 3 + 0.555R 8 - 1.0S 1 g=l.O00R 5 + 0.555R~ - 1.0S g=0. 555R +7 + 0.555R 8
1.0S
g=0. 333R +1 + 0.500R 6
I.OS
g=l.0OOR6
+ ~.555R 8 + 0.555R10-
g=O.333R4
+ 0.277RI0-
I.OS
1.0S
g=0.555R +9 + 0. 555RI0- 1.0S g=O.667R +9 +I.000R6 etc.
+ 0.555RI0-
(See Reference
I.OS
7)
Component I0 _
Mean Strength
_
Sl
Area
Tension/ Comprn. 2
3
1-6
20/10
1.00
7-If) 5/2.5
0.25
2S 2
Fig. 2. Failure tree - enumeration strategy using incremental load model, g is failure function, R is component strength. ( - ) indicates compression, ( + ) indicates tension. Ductile components.
basic plastic mechanisms. More general approaches for systems with different member behavior have used safety index criteria to enumerate significant failure modes. Such strategies lead to failure tree searches, as illustrated in Fig. 2. Each branch may lead to a failure mode expression. It is important for large frameworks to reduce the number of structural reanalyses and several approaches have been utilized. Rashedi and Moses [6] extended the basic incremental loading model to include brittle and semi-brittle components, multiple load conditions and even partial failure modes. Applications included both evaluation of design concepts (e.g. structure topology and redundancy) and inspection criteria. The latter utilize the tree search to identify which members are critical to the system and should require additional quality review in design, construction and maintenance. The difficult task of combining the failure mode expressions into an overall system risk was also
98 studied by Rashedi [7] who used Monte Carlo simulation. Fu and Moses [8] found a new approach in importance sampling that provided both good accuracy and high efficiency. The major recent extensions of the failure tree system modelling were done by Cornell and his associates [9] supported by an oil industry project. They have developed programs for the practical modelling of offshore platform frameworks. Among the advances were improved assessments and comparisons of alternate structure geometries and topologies, the effects of redundancy, simulated damaged conditions and detailed investigation of structure redistribution capabilities, improved reanalysis, more accurate modal combination probabilities and more realistic post-failure behavior models. Further work was identified to include more detailed and realistic post-failure element models including combined loading cases and more intelligent generation of failure paths.
3. FUTURE APPLICATION OF SYSTEM MODELS Two main areas have been described which encompass the goals of system applications to date. The first is structure concept evaluation, and the second is lifetime safety strategies. In structure concept evaluation we investigate such issues as redundancy, geometry and system toughness. That is, what is the ratio of system reliability to critical member reliability? Also, what happens to the system if critical members are damaged due to original defect, fatigue, accidental loading, etc.? This requires tree searches to develop likely failure paths. Furthermore, studies are needed to show which geometries and levels of redundancy are optimal for a given loading environment including the post-failure behavior of elements and correlations. In the future, there must be more intensive aids to solve these problems, especially with regard to lifetime safety strategies. There is increasing interest in assessing safety with respect to a variety of situations other than the one idealized in design, namely a structure built as designed and subject to prescribed load combinations. What has become known as the "infrastructure crisis" has opened new decision areas in many instances much more complex than "simple" design cases. Decisions must continually be made regarding existing, structures such as highway bridges, offshore platforms, buildings, pipelines, etc. The trade-offs include restricting operation (e.g. a truck weight or traffic posted bridge), repair, rehabilitation or replacement of a structure. Further models, such as one proposed by the author, include a trade-off of safety margins with more intensive inspection, analysis or even testing all of which presumably reduce uncertainty and therefore allow for lower prescribed safety margins to attain target safety indices [10]. The applications include a revision to an existing manual for maintenance inspection of highway bridges which controls the safety of some 600,000 existing bridge structures in the United States. Increasingly, engineers with responsibility for such systems are recognizing that the conventional safety factors in design cannot be attained for existing structures except at prohibitive costs. This is due to increasing load demands occurring simultaneously with deteriorating strength because of deferred maintenance. Many engineers are relying on reliability methods to help in the decision strategy. Another example-besides the bridge structures mentioned above, is the recent AIM project being carried out by Bob Bea and his associates [11]. These efforts are aimed at improving strategies for assessing existing offshore platforms using expanded statistical data base for in-service structure performance and system models. Some of the research topics that will be needed in the future to further such development are now described.
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3.1. More efficient failure mode Identification Cornell and his colleagues [9] have described failure mode searches that efficiently find significant failure modes. However, there are still limitations in the element behavior models that can be described especially with regard to member unloading phenomena and complex load interaction problems (e. g. axial, bending, pressure).
3.2. In general, better member descriptions are needed Our emphasis has been on ultimate strength as often characterized by the formation of a plastic failure mechanism. In reality, failure is a progressive evolution of damage which may signal "failure" to someone such as an operation or maintenance engineer that failure is imminent or has occurred. This happens at levels below conventional predictions of collapse mechanisms. There needs to be better interaction of what field people will tolerate in terms of displacement, vibration, cracking, etc. that can help in identifying and expressing true failure modes.
3.3. In addition to the item in 3.2, there is the issue of low cycle fatigue failure Often, our loads such as wave or highway do not come as a single maximum event but rather as a sequence of several, very heavy, repetitive loads. Initially, some of these load cycles weaken the structure and eventually lead to lower resistance. These phenomena also need to be included in our modelling, especially when dynamic behavior or load amplification can result in significant changes in response just prior to the failure event.
3.4. To assist in identification we will need better reanalysis and search techniques These search methods are partly functions of structural modelling and matrix methods. Directional simulation and response surface techniques may aid in better utilization and more efficient sampling data.
3.5. Modal combination techniques can be vastly improved As described above, the importance of statistical correlation can have an overwhelming effect on the solutions. A more extensive data base is needed to model such correlations and in fact we need to improve model verification. Reliability models are useful for sorting through a strategy of decision options but at some stages the relative or notional values of risk must be translated into more precise estimates for optimal resource allocation.
3.6. Finally, the author feels that we may see more application in the future of on-line sensing and control technology applied to overall system safety The technology is evolving that will permit inexpensive and widespread use within a structure of devices to sense (1) loading (e.g. pressure, forces, ground motion); (2) atmospheric conditions (e.g. temperature, moisture, chemical presence such as chloride or other corrosive conditions); (3) structure behavior (e.g. strain and acceleration); and (4) material condition (e.g. thickness losses
100 d u e to ion, loss of paint, etc.). U n l i k e t o d a y ' s situation, w h e r e d a t a are very sparce, it m a y be e x p e c t e d in the future that we can b e o v e r w h e l m e d with s t r u c t u r e i n f o r m a t i o n . T h i s d a t a m a y b e useful for assessing in-service reliability a n d p r e d i c t i n g f u r t h e r d e g r a d a t i o n a n d r e d u c t i o n in lifetime reliability. Decisions b a s e d on actual site i n f o r m a t i o n m a y b e p r a c t i c a l in o p t i m i z i n g o u r resources b e t w e e n o p t i o n s of repair, r e h a b i l i t a t i o n or r e p l a c e m e n t a n d even r e c o m m e n d i n g m o r e intensive d a t a - g a t h e r i n g o p e r a t i o n s . T h i s implies that m u c h of o u r s y s t e m m o d e l l i n g m u s t be B a y e s i a n in nature, c a p a b l e o f b e i n g u p d a t e d b y new site-specific d a t a a n d p r o v i d i n g decision strategies with g r e a t e r confidence.
ACKNOWLEDGEMENTS T h e w o r k p r e s e n t e d herein is b a s e d in p a r t on the research f u n d e d b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t E C E 85-16771 " R i s k A n a l y s i s for E v a l u a t i o n of B r i d g e s " at C a s e W e s t e r n R e s e r v e University.
REFERENCES 1 F. Moses, System reliability developments in structural engineering, Structural Safety, 1(1) (1982) 3-13. 2 M.R. Gorman, Resistance modelling, in: ASCE Short Course, Structural Reliability Analysis of Offshore Platforms, May 1985. 3 F. Moses, Further developments of the incremental loading model for computer analysis of platform reliability, Report to Amoco Production Co., September 1977. 4 F. Moses and B. Stahl, Reliability analysis format for offshore structures, Paper OTC 3046, Offshore Technology Conference, Houston, May 1978. 5 M.R. Gorman, Reliability of structural systems, Report 79.2, Department of Civil Engineering, Case Western Reserve University, May 1979. 6 F. Moses and M.R. Rashedi, The application of system reliability to structural safety, Fourth International Conference on Application of Statistics and Probability in Soil and Structural Engineering, Florence, 1983. 7 M.R. Rashedi, Studies on reliability of structural systems, Department of Civil Engineering, Case Western Reserve University, May 1983. 8 F. Moses and G. Fu, Importance sampling in structural system reliability, Fifth ASCE E M D / G T D / S T D Specialty Conference on Probabilistic Mechanics, Blacksburg, Virginia, May 1988. 9 H. Nordal, C.A. Cornell and A. Karamchandani, A structural system reliability case study of an eight-leg steel jacket offshore production platform, Marine Structural Reliability Symposium, Arlington, Virginia, October 1987. 10 F. Moses and D. Verma, Load capacity evaluation of existing bridges, NCHRP 301, Transportation Research Board, Washington, D.C., December 1987. 11 R.G. Bea and F.J. Puskar, Development of AIM (Assessment, Inspection, Maintenance) programs for fixed and mobile platforms, Paper OTC 5703, Offshore Technology Conference, May 1988.