American Institute of Aeronautics and Astronautics
1
A State Transition Approach to Reliability based Design and Optimization
S. Vittal* and P. Hajela** Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute, Troy, NY 12180
Abstract In this paper, a new approach in probabilistic
design and optimization is presented based on predicting operational risk inherent in parts that transition through multiple probabilistic mission environments and are subjected to multiple failure modes. The proposed method is unique in that it can compute the probability of a part existing in different states (rather than failure modes) and is a function of the parts operating history as well as relationships between multiple failure modes. This makes it possible to develop more accurate risk models and manage the residual life of the part in a manner that maximizes usage and minimizes risk. Conventional “competing risk” methods are focused on rolling up various failure modes and treat all modes with equal severity. The proposed effectiveness-based approach is different in that the focus is on “part states” and the consequences of a part transitioning from one state to another. By changing the probabilistic design philosophy from evaluating failure mode probabilities to computing different state probabilities, a “true” composite estimate of part or system risk can be developed. This forms the underlying philosophy behind the new state-transition based probabilistic design algorithms being proposed. Case studies drawn from industry are used to illustrate the proposed method and explain the development of the algorithm. They include a turbine disk operating in different environments and subject to different failure modes; and the optimal design of a two-bar truss subjected to multiple failure models of varying severity.
Introduction It is generally recognized that most parts and
systems have more than one single mode of failure, each of which can cause the part to fail. Traditionally, mechanical reliability has been
based extensively on the use of the “Competing Risk Model” to evaluate the probability of failure. For the competing risk model to work, the following assumptions need to be strictly satisfied
1) Each failure mechanism (mode) is statistically independent of the others, at least till the occurrence of the first failure.
2) The component is said to have failed when the first of all of these modes reaches a failure state – i.e. all failure modes are assigned the same degree of severity, and any of them could fail the part completely. Mathematically,
[ ]1min , ,fail kT T T= … (1)
Here Tfail is the time to first fail the part, and [T1,…,Tk] are the first times to failure in each of the ‘k failure modes. A distribution of the expected part life is obtained by randomly simulating [T1…Tk] from their parent distributions and computing Tfail.
3) Each of the ‘k’ failure modes can be described completely by a known failure distribution, Fi(t) . The distribution is usually computed using standard life analysis techniques like rank regression or Maximum Likelihood techniques. The part life distribution takes the form,
( ) ( )∏=
=k
iiC tRtR
1
(2)
( ) ( )( )1
1k
C ii
F t F t=
= −∏ (3)
( ) ( )∑=
=k
iiC thth
1
(4)
Here Rc(t) is the reliability of the part at time ‘t’ using the competing risk model, Fc(t) is the probability of failure and hc(t) is the hazard rate of the failure distribution. The probability of failure obtained from the competing risk model can subsequently be written in a more compact form as shown below.
( ) ( )10
1 expt k
c ii
F t h t dt=
= − −
∑∫ (5)
*Graduate Student, **Professor, Fellow AIAA Copyright 2003 by P. Hajela . Published by theAmerican Institute of Aeronautics and Astronautics, Inc. with permission
44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia
American Institute of Aeronautics and Astronautics
2
While the method of competing risks is popular owing to it’s simplicity, it has a few serious drawbacks. From a mechanical reliability perspective, the biggest drawback is that it assigns equal severity to all failure modes. For example, excessive deflection would be treated at the same level of severity as catastrophic fracture. In addition, several other variables like the stochastic nature of the operating environment and mode of operation are not taken into account while computing the probability of failure.
On the other hand, the concept of systems effectiveness has been in use for some time in the field of operations research. “Effectiveness” is a dynamic measure indicating the expected level of system performance and seeks to quantify the degree of accomplishing a specific mission [Pukite, 1998]. The basic model takes the form shown in (6).
E = A x D x C (6)
Effectiveness, as defined in (6) is a measure of operational risk, and takes into account part availability, dependability and capability. I.e.,
Effectiveness = Availability x Dependability x Capability
In matrix form this can be written as,
{ }
11 12 13 1 11
21 22 23 2 21
31 32 33 311 2
1 2 3 1
..
..
.. ... ... .. .. .. .. ..
..
j
j
i
i i i ij j
D D D D CD D D D CD D D CE A A A
D D D D C
= ∗ ∗
(7)
This can also be written as follows.
∑∑= =
=n
j
n
ijiji CDAE
1 1
(8)
E = System effectiveness, a measure of the extent to which a system will perform it’s mission and is function of it’s probability to survive various states as well as the consequences of arriving in a particular state (condition).
A = Availability, a measure of the system condition at the start of a mission. Ai is the probability of starting its mission in state ‘i’.
D = Transition matrix, populated by elements dij, where dij is the probability of a part starting its mission in state ‘i’ and ending up in state ‘j’ at the end of that mission sequence.
C = Capability, a measure of the system’s ability to achieve a mission objective. Cj represents the ability (usually scored between 0
and 1) of the system to the part to complete its mission if it terminates in state ‘j’.
The competing risk model is too conservative, and often results in discarding parts with useful residual life. The proposed effectiveness-based method is more risk-driven rather than reliability-driven, and strikes a balance between the benefits of extending the useful life of a part versus the risks of it failing in service
Generalized problem formulation The following definitions are used in the state
transition approach. A part is assumed to have several “failure modes” which can result in its performance being degraded. This degradation can have different levels of severity, ranging from catastrophic to benign. Each failure mode is uniquely described by its probability density function. A part can exist in several “states”, where a state is a condition that can have several failure modes. For example, a part can transition to a state where it is subjected to two failure modes F(2) and F(3) and it’s probability density function is a joint function of the distributions Pf1 and Pf2.
The proposed state transition approach is based on the following assumptions.
1) A part is subjected to several failure modes, F = (1,..,k) which are assumed to be statistically independent. This restriction can be removed using any standard statistical approach like computing the variance-covariance matrix for the problem and using that to decompose the variables into components that are statistically independent [Haldar, 2000]
2) The part can begin its mission in any of ‘i’ starting states, and transition to any of ‘j’ final states at the end of its mission life. For a multi-mission calculation, the ending states ‘j’ at the (n-1)th mission are usually the starting states for the nth mission. The probability that a part is in a particular state ‘j’ at the end of the (n-1)th mission will usually be the availability (i.e. the probability of starting out in a particular state) in the next nth mission.
3) The failure modes can be ranked in order of criticality, and are assigned to states based on the condition of the part. It is possible for a part to transition from one state ( a combination of particular failure modes) to another with some rules. For example, it may not be physically possible for a part to transition from a damaged state an undamaged one, or to undergo transitions
American Institute of Aeronautics and Astronautics
3
that are not physically possible. The introduction of transition rules ensures that the solutions are physically realizable and that some of the physics of failure are captured.
4) A part can continue to provide useful, though degraded service even if damaged. The degree of useful service depends on it’s state and can be explicitly computed
5) Each failure mode can be described either with a single probability of failure (time independent) or with a time-dependent probabilistic density function. Again note that the restriction on parametric distributions is to keep the initial formulation tractable.
Algorithm description Step 1: The first step is to list all possible
failure modes, F=(1,..k) in increasing order of severity. i.e. F(i) is less severe than F(i +1).
Step 2: Determine, based on engineering judgment, all possible states that the part can exist in. This is laid out in a table as shown in the turbine blade case study. The absence of a failure mode ‘i’ is written as NO F(i) and its presence is written as YES F(i)
Step 3: The next step is to compute, based on the physics of the problem, the allowed state transitions in the problem. From the failure mode list, it is clear that a part in mode ‘k’ is in the worst possible condition and cannot be degraded further regardless of the number of modes above it. The rules state that if a part is in state ‘k’, then it cannot transition to any state above it. However, subject to physical constraints, any part in a higher failure mode can transition to one lower.
Step 4: From the probability density functions for each failure mode, compute a matrix of transition probabilities where each term of the matrix D(p,q) , p ¶ q, is the probability of a part in state p transitioning to state q. It is quite possible for some of the terms in the upper diagonal to be zero, and this indicates that the probability of that transition is zero, i.e. that transition is not allowed
Step 5: Populate the D matrix. Here, one needs to be careful while computing the probabilities. Based on the initial assumption of statistical independence of failure modes, the joint probability density function for two or more failure modes usually reduces to the product of the individual failure modes. It is also useful to start with the worst failure mode and work up to the most benign mode. At each step, construct a
decision tree and use that to compute the transition probabilities. Note that the sum of all terms in each row equals unity, i.e.,
1
1q
ijj
D=
=∑
Step 6: Compute a starting state probability vector, A = [A1, …,Aq]. This is a (1 x q) vector with each column, j, consisting of the probability of a part beginning it’s service in state ‘j’. For a brand new part, it is assumed that there is no prior damage and the vector becomes A = [1 0 0 0…0]. However, if the part has already been in usage, then there is a probability that it could already exist in the other states, which changes the composition of the starting vector.
Step 7: Compute a mission completion vector, C = [C1,…,Cj] which is a metric that computes the utility of a part that has ended in a particular state. It is a useful measure of the consequences of landing in a particular state. For example, if a part ended in a state Sj that had no damage of any kind, it’s corresponding Cj = 1.0. A part that is in a completely damaged state will have a Cj = 0.0. If the objective of a part is to complete it’s intended mission, then the elements of C will be probabilities of mission completion of the part if it ends up in a particular state.
Step 8: Once the vectors A and C, and the D matrix have been computed in closed form, it is possible to compute the “starting state probability”, A1(t) defined as follows.
( )( )
( )
( ) ( )
( ) ( ) ( )( ) ( )
( ) ( )
1 11 12 1
2 11 22
1
1
0 0 . . 00 0 . . .
.. . . . . .0 .. .. .. 0
.. . . . . .0 . . . 0
j
i
j i ij
A t d t d t d tA t d t d t
A A
A t d t d t
− − − − − = − −
(9)
Step 9 : Depending on the number of mission cycles, Step 8 is repeated using the rule,
( ) ( ) ( )∑∑= =
−− −=n
j
n
innijnjni ttDtAtA
1 111
(10)
( ) ( ) ( ) j
n
j
n
innijnin CttDtAtE ∑∑
= =−− −=
1 111
(11)
This method is now illustrated with the help of two case studies. In the first case study, the method is applied to get a better estimate of a turbine blade failure. In the second case study, the effectiveness-based approach is incorporated within a multiobjective optimization framework.
American Institute of Aeronautics and Astronautics
4
Case Study 1: Turbine blade Consider a following simple example in
which a turbine blade is subjected to two mission profiles. Each mission environment lasts for a specified period of time that may be expressed in hours. For a given profile, the environment variables like speed, temperature and pressure follow known statistical distributions. As the blade goes through different missions, it accumulates damage that may be in the form of tip wear, fatigue crack initiation, spalling, metal oxidation, etc. The objective of this case study is to compute the reliability of the system using the conventional competing risk model as well as the proposed part effectiveness method.
Three typical life-limiting failure modes of a turbine blade considered in this case study are High Cycle Fatigue (HCF), creep and oxidation (spalling). It is assumed that probability density functions have been developed that predict the probability of a turbine blade failing in any of these modes. The density functions follow the well-known two-parameter Weibull model with a characteristics life, h and a slope b, as the distribution parameters. To protect proprietary data, all parameters have been scaled. This does not in any way affect the validity of the technique or the accuracy of the results presented in this section.
Table 1 shows the cumulative density function for each failure mode for one particular mode of operation (part load), called mission type “A”. The distribution function for a more severe operating cycle (full load, mission type “B”) is described in Table 2. HCF, Creep & Oxidation are considered to be failure modes ‘1’, ‘2’, and ‘3’ respectively. For example, PA1 is the probability of failure in HCF of a part operating in Mission type A for duration of (t2 – t1) hours.
Table 1: Turbine failure distributions, Type A
Gas turbines are usually expected to last 24,000 hours before a major inspection or
overhaul. This total duration is divided into three operating phases, which take place in sequence. It is assumed that a typical turbine blade has the following exposure. It operates 30% of its time in Mission type 1 (7200 hrs, Partial Load) , then 50 % in Mission type 2 (12,000 hours, Full load) and then 20 % in mission type 1 ( 4,800 hours at partial load)
Table 2: Turbine failure distributions, Type B
Competing Risk Solution: The probability of failure for a typical blade subjected to the operating history described in the previous paragraph is computed using the competing risk method, and detailed in Table 3 as shown.
Table 3: Results of competing risk model for turbine blade
The results indicate that there is approximately a 1.79 % chance of a blade having failed at the end of 24,000 hours. However, it is unclear if the failure is due to oxidation, creep, HCF or a combination of the above. Another vital piece of information that is missing is the fraction of blades that have failed in a particular mode. This is important as some degraded parts can be rebuilt, e.g. a burnt blade can sometimes be rebuilt and put back in service. It is, therefore, important to recognize that several “degrees” of failure exist – for example; HCF failure implies destruction of the part. However, a blade can continue to work with degraded performance if it is partially burnt,
Failure Mode Distribution , for mission time t1 ..t2 Remarks
High Cycle FatiguehA1 = 284200
bA1 = 2.0Creep
hA2 = 145670
bA2 = 2.2
Oxidation
hA3 = 113345
bA3 = 2.5
Worst failure mode
Can transition to
HCF
Can transition to
creep
−
−−=
11
1
1
1
21 exp1
AA
AAA
ttP
ββ
ηη
−
−−=
22
2
1
2
22 exp1
AA
AAA
ttP
ββ
ηη
−
−−=
33
3
1
3
23 exp1
AA
AAA
ttPββ
ηη
Failure Mode Distribution , for mission time t1 ..t2 Remarks
High Cycle FatiguehB1 = 241570bB1 = 2.0
CreephB2 = 123800bB2 = 2.2
Oxidation
hB3 = 96300bB3 = 2.5
Worst failure mode
Can transition to
HCF
Can transition to
creep
−
−−=
11
1
1
1
21 exp1
BB
BBB
ttP
ββ
ηη
−
−−=
22
2
1
2
22 exp1
BB
BBB
ttP
ββ
ηη
−
−−=
33
3
1
3
23 exp1
BB
BBB
ttPββ
ηη
Reliability (Per phase)
Phase 1, Partial Load (7200 hrs)
0 - 7200 0.997006848
Phase 2, Full Load (12000 hrs)
7200 - 19200 0.964347891
Phase 3, Partial Load (4800 hrs)
19200 - 24000 0.981443952
Completed (24000 hrs)
0 - 24000 0.943620527
Operating Phase Duration (hours)
American Institute of Aeronautics and Astronautics
5
or if limited creep is present. Again, the traditional method of evaluating reliability does not take this factor into account.
Effectiveness Solution: A blade can exist in several states – one state is the absence of defects (a “perfect” blade). Another state could be the presence of creep only, and another could be a burnt (oxidized) blade, etc. For the purposes of this discussion, we assume that the failure modes can be ranked in an increasing order of criticality. At the top, creep deflection is the most “benign” failure mode. This is followed by oxidation damage and finally HCF is at the bottom (it is the most critical). A part at the top of the list can transition to one below it (for example a blade that is oxidized can also develop HCF cracks) but a part in a critical failure mode cannot move up the list ( a part with HCF cracks cannot transition to a state of creep as it is assumed destroyed in the HCF stage itself). In addition, a part in any state (say, creep) can stay in that state till the end of its mission life. A part can begin its mission life in five states shown in the diagram below.
Figure 1: Turbine blade transition states
The diagram shows that a turbine blade, during the course of operation, can transition to the worst possible state (HCF, State 5) with a probability PA. The chance of it escaping State 5 is (1-PA). It then has the option of failing by oxidization with a probability PB or escaping that with a probability (1- PB). There is also a probability of it failing in creep and that can happen regardless of oxidation, and this is reflected in the tree diagram. Based on these three failure modes, the part can exist in any of the following five states described in Table 4. The transition diagram is shown in Figure 2.
State Description 1 No HCF, No Oxidation and No Creep 2 No HCF, No Oxidation, some Creep 3 No HCF, Some oxidation, no Creep 4 No HCF, Some Oxidation, Creep 5 HCF Table 4: Turbine Blade State Description
State 1
2
3
4
5 Figure 2: State transition rules diagram for
turbine blade case study
The first step is to obtain the starting state probability vector A = [A(1),..A(5)] where each element A(i) is the probability of the part starting it’s mission life in state (i). At time, t = 0 it is assumed that the part is free from any defects. The starting vector reduces to the form
A(0) = [ 1 0 0 0 0 ] (12)
The transition matrix “D” is as shown in (13)
=
55
4544
353433
252422
1514131211
0000000
0000
dddddddddddddd
D (13)
The best way to compute the terms in the state transition matrix would be to start at the bottom. A part that starts in State 5 has essentially failed, and it stays failed in State 5 for the duration of its mission life, i.e. it cannot be degraded any further. It’s probability, therefore is 1.0.
A part in State 4 has two choices – it can either go to State 5 with a probability of Pf,A(t) or remain in its own State 4 with a probability (1- Pf,A(t)). Thus, d45 equals Pf,A(t) and d44 is (1- Pf,A(t)).
A part in State 3 follows the following path. In it’s worst case, it can go directly to State 5 with probability Pf,A(t) , i.e. d35 = Pf,A(t) . It moves on
Part
PAHCF
(1- PA)No HCF
Oxidation
(1- PB)No Oxidation
PBOxidation
Creep
(1- PC)No Creep
PCCreep
Creep
(1- PC)No Creep
PCCreep
State 1
State 2
State 3
State 4
State 5
Part
PAHCF
(1- PA)No HCF
Oxidation
(1- PB)No Oxidation
PBOxidation
Creep
(1- PC)No Creep
PCCreep
Creep
(1- PC)No Creep
PCCreep
State 1
State 2
State 3
State 4
State 5
American Institute of Aeronautics and Astronautics
6
to its remaining states with a probability (1- Pf,A(t)). Having survived state 5 with probability (1- Pf,A(t)), it can move to State 4 with probability Pf,C(t) or remain in its original state with a probability of (1-Pf,C(t)). Thus the final probability of a part starting in State 3 and ending up in State 4, d34, is (1- Pf,A(t))*Pf,C(t). The final probability that remains in it’s starting configuration, i.e. in State 3, d33, is (1- Pf,A(t))*(1-Pf,C(t)).
The probability calculations for a part starting in State 2 follow a similar line of reasoning. Stating in State 2, it can fail and go to it’s condition which is State 5 with a probability Pf,A(t) , i.e. d25 = Pf,A(t) . It survives this with probability (1- Pf,A(t)) and can then fail and move to State 4 with probability Pf,C(t) or remain in it’s original configuration with probability (1-Pf,C(t)). Thus the probability of a part stating in State 2 and ending up in State 4, d24, is (1- Pf,A(t))* Pf,B(t). The probability of starting in State 2 and remaining in State 2, d22, is (1- Pf,A(t))*(1- Pf,B(t)).
The calculations for a part starting brand new, i.e. in State 1 follow the same logic as detailed in previous paragraphs but require a few more calculations. This process is best visualized with the aid of Figure 3 shown below.
Figure 3: Example of state transitions
The various “options” for a part starting in State 1 can be easily computed as shown. For example, the probability of going directly into State 5 is d15, is Pf,A(t). The part moves on to the rest of its possible conditions with a survival probability of (1-Pf,A(t)). It can then transition to State 4 with a probability, d14, of (1-Pf,A(t))* Pf,B(t)* Pf,C(t) or move to State 3 with a final probability, d13, of (1-Pf,A(t))* Pf,B(t)* (1-Pf,C(t)). Alternately, having survived both modes ‘A’ and ‘B’ it could fail in mode ‘C’ by transitioning to State 2 with a final probability of d12, of (1-Pf,A(t))* (1-Pf,B(t))* Pf,C(t). Or, if could survive all these failure modes and return to its original undamaged state at the end of mission interval ‘t’ with the final probability, d11, of (1-Pf,A(t))* (1-Pf,B(t))* (1-Pf,C(t)).
A part is said to have completed its mission life successfully if it has performed its primary function (transferring power to the rotor) and this is a function of its survivability. If the part has survived the mission without any degradation, the probability of completing the mission is 1.0. Depending on the degree of damage this number can vary from 0 to 1.0. For the turbine blade example, the probabilities are in Table 5. Note that this number is currently based on expert opinion and should be determined based on experience and prior field data. This could also be replaced by mission completion probabilities based on Bayesian methods.
State Description Cj 1 No HCF, No Oxidation, No Creep 1.0 2 No HCF, No Oxidation, Creep 0.85 3 No HCF, oxidation, No Creep 0.70 4 No HCF, Oxidation, Creep 0.60 5 HCF 0.0
Table 5: Mission Completion Vector
The final step is to compute the “part effectiveness” at the end of a given mission (or at any time during a mission), which is a product of the starting reliability vector, state transition matrix and mission completion vector. In addition, the “starting reliability” for the next time period is the product of A and D. The results are summarized in Table 6. The results from the two approaches are compared in Table 7.
Table 6: Effectiveness Results – Turbine Blade Case Study
It should be noted that the reliability estimate obtained from the competing risk model is equivalent to the probability of a part ending up in State 1 in the part effectiveness method. Thus, using the proposed method it is possible to extract the competing risk model results as well.
From these results, a number of interesting inferences can be drawn
1) The conventional method of computing reliability cannot predict the fraction defective in various failure modes; the proposed method can compute this.
2) The traditional method provides overtly conservative estimates indicating that 1.787% of the parts will have to be discarded, and the risk associated with the part is 0.01787*Cost of failure. The proposed method indicates that 1.787 %of the parts will not be in pristine condition, and it is quite possible to salvage parts that exist in other states. The risk associated with this part can be computed only for those failure modes that have high consequences of failure, rather than penalizing all failure modes equally.
3) The probability of a part ending in State 1 is numerically equal to the traditional reliability of the part. In addition, the effectiveness-based approach provides a more realistic estimate of the risk associated with the part.
4) The proposed method also tracks the degradation of the part at various times – the starting reliability vector shows the gradual increase in probability of a part ending up in various failure modes. This can also be used to identify the severity of various failure modes so that corrective action can be taken at the design stage itself.
Case Study 2: Two-bar truss The preceding example demonstrated the use
of the part effectiveness method in modeling the risk of structural systems during their operating lives. This technique can be extended to include probabilistic design and optimization as well, as demonstrated by the following case study. The two-bar truss optimization problem layout is shown in Figure 5.
The objective of the design exercise is to minimize the weight of the design, f1, as well as its deflection at node 3, f2 . The reliability of the
truss structure, f3 and effectiveness ‘E’ is computed using the method described in this paper. The structure is assumed to fail catastrophically if the tensile stress exceeds the yield stress of either of the two members, i.e the systems is a “series-reliability system” for this primary failure mode. The truss is also assumed to fail (secondary failure mode) if the displacement at node 3 exceeds a predefined limit.
Figure 4: Two-Bar Truss Layout
The weight of the design to be minimized is,
21 2 12 1f hx xρ= + (14)
and the displacement at node 3 to be minimized is as follows.
( ) ( )1.5 0.52 41 1
2 21 2
1 1
2 2
Ph x xf
Ex x
+ += (15)
The reliability of the structure, f3 is derived from the limit state equations for the problem. The deterministic stress constraints are as shown in (23) and (24).
( )( )0.521 1
1 01 2
1 10
2 2
P x xg
x xσ
+ +≡ − ≤ (16)
( )( )0.521 1
2 01 2
1 10
2 2
P x xg
x xσ
− +≡ − ≤ (17)
The side bounds on the design variables are applied in context to the type of problem being solved. Typical values are shown below.
American Institute of Aeronautics and Astronautics
8
1
2
0.1 2.2290.5 2.498
xx
≤ ≤
(18)
The design variables are defined as,
1
wxh
= , 2
min
AxA
= (19),(20)
The nominal values for the design and problem variables are, Load P = 10000 lbs, Yield Stress ‘s0’ = 20000 lbs/in2, Density ‘r’=0.283 lbs/in3, Young’s Modulus ‘E’ = 30 x 106 lbs/in2, Cross section Amin = 1.0 in2. The maximum allowed displacement is 0.08”.
Where b1 and b2 are the reliability safety indices for truss members 1 and 2 respectively. The maximum displacement of the truss,d, can be written as,
( ) ( )2
21
41
232
1
2211
xExxxPh ++
=δ (21)
Equation (21) is converted into the third limit state in the problem as follows.
( ) ( )3
2 421 1
3 max21 2
1 1
2 2
Ph x xg
Ex xδ
+ += − (22)
The partial derivatives of the two limit states with respect to their probabilistic design variables
1x , 2x and P can be obtained analytically for this simple problem. The reliability safety indices for stress failures in members 1, 2 and displacement are shown in (23), (24) and (25) respectively.
( ) ( )
( ) ( ) ( )
12 2
1 10
1 2
1 2 2 21 1 1
1 21 2
1 1
2 2
P x x
x x
g g gVar x Var x Var Px x P
σ
β
+ + − =
∂ ∂ ∂ + + ∂ ∂ ∂
(23)
( )( )
( ) ( ) ( )
12 2
1 10
1 2
2 2 2 22 2 2
1 21 2
1 1
2 2
P x x
x x
g g gVar x Var x Var Px x P
σ
β
− + − =
∂ ∂ ∂ + + ∂ ∂ ∂
(24)
( ) ( )
( ) ( ) ( )PVarPg
xVarxg
xVarxg
FxEx
xxPh
23
2
2
2
31
2
1
3
max2
21
41
232
1
3
2211
∂∂
+
∂∂
+
∂∂
−
++−
=β
(25)
Using the competing risk model, the reliability of the truss is written as,
( )( )3
11truss j
jR β
=
= −Φ −∑ (26)
The probability of failing due to excessive stress and excessive displacements are (27) and (28) respectively.
( )( ) ( )( )1 21 1 1SP β β= − −Φ − ∗ −Φ − (27)
( )31DP β= −Φ − (28)
Using the effectiveness-based approach, the following “state-transition” diagram can be constructed as shown in Figure 5. It assumed that a structure undergoes partial failure due to excessive displacement, and complete failure under excessive stress.
Figure 5: Truss state-transition diagram
The various states are defined as follows.
State 1: No excessive displacement or stress
State 2: Benign failure, truss has excessive displacement but can still function partially.
State 3: Catastrophic failure, excessive stresses and truss cannot function at all.
The transition matrix D is as shown in (29)
( )( ) ( )( )
1 1 10 10 0 1
S D S D S
S S
P P P P PD P P
− − − = −
(29)
Equation (29) can be derived using the same logic used in the turbine blade case study. If a truss were to start its mission in State 3, it will obviously remain in that state; hence the probability d33 equals 1.0. A truss that starts its mission in State 2 (i.e. already has excessive displacement) can either fail in stress and transition to State 3 with a probability d23 equal to PS, or it can remain in it’s own displaced state, d22, with a probability (1-PS). A brand new truss, in State 1 can fail catastrophically and transition
3
2
1
State
American Institute of Aeronautics and Astronautics
9
directly to State 3 with a probability d23 equal to PS or it can survive that and fail in displacement, State 2, with a total probability d12 equal to ((1-PS) PD). It can also survive both failure modes and remain undamaged with a probability, d11 equal to ((1-PS) (1-PD)).
The starting availability vector, A(0) = [1 0 0] assumes the truss starts operation with no defects. Using the part effectiveness approach, the effectiveness of each design, E is computed as shown in (30). C’ = [1.0 0.5 0] is the mission-completion vector, which contains the probabilities of successfully completing the mission by transitioning to a particular state.
[ ]
=
3
2
1
33
2322
131211
*00
0*001CCC
dddddd
E (30)
The effectiveness-based model is introduced into probabilistic optimization using a three-step approach. In the first step, the two bar truss optimization case study is set up as a deterministic multiobjective optimization problem subject to trade offs between the truss weight and displacement, both of which have to be minimized. A classical weighted-objective approach is used for formulating the multiobjective optimization problem. Both truss reliability and effectiveness are computed for each of the optimal solutions obtained from the Pareto front. This helps establish the case for effectiveness, E, as a viable metric in probabilistic optimization. The deterministic optimization problem formulation is as shown in (31).
{ } { } { }
1 21 2* *
1 2
1
2
. .00
L U
f fMin f W Wf f
s tgg
X X X
= +
≤≤
≤ ≤
(31)
The optimal solutions obtained from (31) are shown in Table 8.
The reliability-based optimization problem is formulated as shown in (32). The target system reliability is set to 0.90.
{ } { } { }
1 21 2* *
1 2
min
. .
truss
L U
f fMin f W Wf f
s tR R
X X X
= +
≥
≤ ≤
(32)
Optimal designs obtained for various combinations of the weighting functions are documented in Table 9.
The optimization problem with effectiveness-based constraints is formulated as shown in (33). The minimum effectiveness was set to 0.9. The results are shown in Table 10.
{ } { } { }UL XXX
EEts
ff
Wff
WfMin
≤≤
≥
+
=
min
*2
22*
1
11
.. (33)
The optimal solutions obtained from the three formulations are shown in Figure 6.
It is important to note that the designs generated from (31) have not factored in structural risk. Designs generated from (32) and (33) have risk factored in different ways – one based on traditional reliability, and the other based on part effectiveness.
Table 8: Results from deterministic optimization (Two bar truss case study)
American Institute of Aeronautics and Astronautics
10
Table 9: Results from Reliability-based optimization, Two-bar case study
Table 10: Effectiveness-based optimization results, two bar case study
Table 11: Comparison of effectiveness & reliability-based designs – two bar truss
Some interesting observations can be made from Figure 6. As expected, the Pareto front for the deterministic optimization case spanned the largest region of feasible, optimal solutions. This
is due to the fact that risk and variability were not considered in the problem formulation. The optimal front for the effectiveness-based formulation was larger than that obtained for the case with reliability-based constraints.
Figure 6: Pareto solutions from deterministic, reliability & effectiveness-based optimization
This implies that the effectiveness-based approach produced a larger range of optimal designs without any increase in operational risk. In fact, the proposed approach produced designs that were 5% lighter than that those produced by purely reliability-based methods. A comparison of weight savings from the two methods (effectiveness and reliability based optimization) is shown in Table 11.
Summary & Discussion
In this paper, a new approach to formulating probabilistic design problems is proposed, and is based on the concept of “part effectiveness” rather than part reliability. This problem was motivated by problems currently faced in the gas turbine industry, where expensive turbine blades are often scrapped even though they have a lot of useful residual life left in them. By reformulating the problem using the proposed effectiveness-based metric, a realistic level of risk is calculated. The state transition probabilities can be obtained analytically using probabilistic design algorithms, or obtained empirically from field data. Both approaches are discussed in the paper.
The proposed effectiveness based method has several advantages over the reliability-based method in both areas - risk assessment as well as design optimization. The proposed technique can handle multiple failure modes and their interactions with the appropriate level of risk
American Institute of Aeronautics and Astronautics
11
associated with each failure mode. It takes into account the consequences of failing in different modes, as the fact that a part can degrade into several states until it is deemed unfit for use.
From the case studies, it is seen that the proposed approach provides optimal designs that are superior to those produced by reliability-based methods for the same level of operational risk. The effectiveness metric can be used either as a constraint or as an additional objective to be maximized. The only disadvantage in the proposed method is the extra effort required to populate the availability vector, the state transition matrix and mission completion vector. However, considering the benefits of the technique, the extra effort appears to be justified. Hence, it is hoped that the proposed approach will be an additional tool for the probabilistic design community.
References
[1] Cornell, C.A., “A probability based structural code”, Journal of the American Concrete Institute, Vol. 66, No. 12, pp. 974-985, 1969
[2] Frangopol D. and Iizuka, M., “Probability based structural system design using multicriteria optimization”, Proc. of AIAA/USAF/NASA/OAI Symposium on Multidisciplinary Design and Optimization, Sept. 21-23, 1992, Technical Papers. Pt. 2 (A93-20301-06-66), pp. 794-798
[3] Ghiocel, D.M. and Roemer, M.J., “Probabilistic integration of relevant technologies for a risk-based life prediction of GTE components”, Technical Paper No. AIAA-99-1591, American Institute of Aeronautics and Astronautics, 1999
[4] Hahn, G and Shapiro, S., Statistical Methods in Engineering, Wiley Classics Library Edition, 1994
[5] Haldar, A and Mahadevan S., Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, 2000
[6] Haldar, A and Mahadevan S., Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, 2000
[7] Madsen, H.O., Krenk, S. and Lind, N.C., Methods of Structural Safety, Prentice Hall Inc., NJ, 1986
[8] Pukite, J. and Pukite, P., Modeling for Reliability Analysis: Markov modeling for Reliability, Maintainability, Safety & Supportability Analysis of Complex Systems, IEEE Publishers, June 1998
[9] Rackwitz, R. and Fiessler, B., “Structural reliability under combined random sequences”, Computers and Structures, Vol. 9, No. 5, pp. 484-494, 1978
[10] Rackwitz, R., “Practical probabilistic approaches to design”, Bulletin No. 112, Comite Eurpoean du Beton, Paris, France 1976
![Page 1: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/1.jpg)
![Page 2: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/2.jpg)
![Page 3: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/3.jpg)
![Page 4: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/4.jpg)
![Page 5: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/5.jpg)
![Page 6: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/6.jpg)
![Page 7: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/7.jpg)
![Page 8: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/8.jpg)
![Page 9: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/9.jpg)
![Page 10: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/10.jpg)
![Page 11: [American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC](https://reader035.documents.pub/reader035/viewer/2022080406/575095251a28abbf6bbf4c01/html5/thumbnails/11.jpg)
