safety factor and inverse reliability

8/3/2019 Safety Factor and Inverse Reliability

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I. Introduction1Inverse reliability measures are becoming popular in reliability community. Various2

inverse reliability measures have been developed. This paper focuses on discussing the3

interconnection of various inverse reliability measures, examining methods to compute4

inverse reliability measures and the advantages of using them over direct reliability measures.5

Traditionally, structural safety was defined in terms of safety factors, which were used6

to compensate for uncertainties in loading and material properties, and for inaccuracies in7

geometry and theory. Safety factors permit design optimization using computationally8

inexpensive deterministic methods. In addition, it is relatively easy to estimate the change in9

structural weight of over or under designed structures needed to satisfy a target safety factor10

requirement (Qu and Haftka 2003, 2004).11

Probabilistic approaches in design optimization allow incorporation of available12

uncertainty data and thus provide more accurate measures of safety. Structural safety is13

measured in terms of probability of failure to satisfy some performance criterion. The14

probability of failure is often expressed in terms of a reliability index. This reliability index is15

the ratio of the mean to the standard deviation of the safety margin distribution, which is the16

difference between the capacity and the resistance of the system.17

Optimization using probabilistic approaches called reliability-based design18

optimization (RBDO) enables to gauge the structural safety better but is computationally19

significantly more expensive compared to deterministic approaches. In addition, the20

difference between the computed probability of failure or reliability index and their target21

values does not provide the designer with easy estimates of the change in the design cost22

needed to achieve these target values.23

Safety factors are defined as the ratio of the capacity to resistance of a system. In24

deterministic approaches, safety factors are determinate, typically calculated for mean values25


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inverse measure as percentile performance. Traditionally, design for robustness involves1

minimizing the mean and standard deviation of the performance. Here, Du et al. (2003)2

proposed to replace the standard deviation by percentile performance difference, which is the3

difference between the percentile performance corresponding to the left tail of a CDF and the4

right tail of that CDF. They demonstrated increased computational efficiency and more5

accurate evaluation of the variation of the objective performance. In an effort to address6

reliability-based designs when both random and interval variables are present, Du and7

Sudijianto (2003) proposed the use of percentile performance with worst-case combination of8

the interval variables for efficient RBDO solutions.9

Du and Chen (2004) developed the sequential optimization and reliability assessment10

(SORA) to improve the efficiency of the probabilistic optimization. The method is a serial11

single loop strategy, which employs percentile performance and the key is to establish12

equivalent deterministic constraints from probabilistic constraints. This method is based on13

evaluating the constraint at the most probable point of the inverse measure in Section IV14

below) based on the reliability information from the previous cycle. This is referred to as15

design shift (Chiralaksanakul and Mahadevan 2004; Youn et al. 2004). They show that the16

design quickly improves in each cycle and is computationally efficient. The sequential17

optimization and reliability assessment, however, is not guaranteed to lead to an optimal18

design. Single-level (or unilevel) techniques that are equivalent to the standard RBDO19

formulation are based on replacing the RIA or PMA inner loop by the corresponding Karush-20

Kuhn-Tucker conditions. Here again, Agarwal et al. (2004) showed that the PMA approach is21

more efficient than the unilevel RIA approach due to Kuschel and Rackwitz (2000).22

The several inverse measures discussed above are all based on the common idea of23

using the inverse of the cumulative distribution function. The numerous names for the inverse24

measures contemplate that they were developed by different researchers for different25


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applications. Since these inverse measures come under various names, it is easy to fail to1

notice the commonality among them. The main purpose of this paper is to highlight the2

relationship between the inverse measures. The objectives of this work include:3

(i) To discuss the relationship (or minor differences) between the various inverse4

measures5

(ii) To discuss methods available for calculating these inverse measures6

(iii) To explore the advantages of using inverse measures instead of direct measures.7

Section II describes inverse reliability measures. Calculation of inverse measures by MCS is8

discussed in Section III. Section IV describes calculation of inverse measures using moment-9

based techniques, followed by discussion of using inverse measures in RBDO in Section V.10

Section VI demonstrates the concepts with the help of a beam design example, and Section11

VII provides concluding remarks.12

13

II. Inverse Reliability Measures14Birger Safety Factor15

The safety factor, S is defined as the ratio of the capacity of the system cG (e.g.,16

allowable strength) to the response rG with a safe design satisfying r cG G . To account for17

uncertainties, the design safety factor is greater than one. For example, a load safety factor of18

1.5 is mandated by FAA in aircraft applications. To address the probabilistic interpretation of19

the safety factor, Birger (1970) proposed to consider its cumulative distribution function20

(CDF) SF :21

( ) Prob( )cSr

GF s s

G= (1)22

Note that unlike the deterministic safety factor, which is normally calculated for the23


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mean value of the random variables, c rG G in Eq. (1) is a random function. Given a target1

probability, ftargetP , Birger suggested a safety factor s* (which we call here the Birger safety2

factor) defined in the following equation3

Prob( ) Prob( )cS ftarget r

GF (s*) s* S s* P

G= = = (2)4

That is, the Birger safety factor is found by setting the cumulative distribution function5

(CDF) of the safety factor equal to the target probability. That is, we seek to find the value of6

the safety factor that makes the cumulative distribution function of the safety factor equal to7

the target failure probability. This requires the inverse of the CDF, hence the terminology of8

inverse measure.9

10

Probabilistic Sufficiency Factor11

Qu and Haftka (2003, 2004) developed a similar measure to the Birger safety factor,12

calling it first the probabilistic safety factor and then the probabilistic sufficiency factor13

(PSF). They obtained the PSF by Monte Carlo simulation and found that the response surface14

for PSF was more accurate than the response surface fitted to failure probability. Later, they15

found the reference to Birgers work in Elishakoffs review (2001) of safety factors and their16

relations to probabilities. It is desirable to avoid the term safety factor for this entity because17

the common use of the term is mostly deterministic and independent of the target safety level.18

Therefore, while noting the identity of the Birger safety factor and the probabilistic19

sufficiency factor, we will use the latter term in the following.20

Failure happens when the actual safety factor S is less than one. The basic design21

condition that the probability of failure should be smaller than the target probability for a safe22

design may then be written as:23

Prob( 1) (1)f S ftarget P S F P= = (3)24


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Using inverse transformation, Eq. (3) can be expressed as:1

11 ( )S ftarget F P s*

= (4)2

The PSF concept is illustrated in Fig. 1. The design requirement ftargetP is known and the3

corresponding area under the probability4

density function of the safety factor is the5

shaded region in Fig. 1. The upper bound of6

the abscissa s* is the value of the PSF. The7

region to the left of the vertical line 1S = 8

represents failure. To satisfy the basic9

design condition s* should be larger than or10

equal to one. In order to achieve this, it is11

possible to either increase cG or12

decrease rG . The PSF s* , represents the factor that has to multiply the response rG or divide13

the capacity cG , so that the safety factor be raised to 1.14

For example, a PSF of 0.8 means that rG has to be multiplied by 0.8 or cG be15

divided by 0.8 so that the safety factor ratio increases to one. In other words, this means that16

rG has to be decreased by 20 % (1-0.8) or cG has to be increased by 25% ((1/0.8)-1) in order17

to achieve the target failure probability. It can be observed that PSF is a safety factor with18

respect to the target failure probability and is automatically normalized in the course of its19

formulation.20

PSF is useful in estimating the resources needed to achieve the required target21

probability of failure. For example, in a stress-dominated linear problem, if the target22

probability of failure is 10-5

and a current design yields a probability of failure of 10-3

, one23

cannot easily estimate the change in the weight required to achieve the target failure24

Fig. 1 Schematic probability density of

the safety factor S . The PSF is the value

of the safety factor corresponding to the

target probability of failure


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can also be used to approximate the structural response in order to reduce the computational1

cost. Qu and Haftka (2004b) employ PSF with MCS based on RSA to design stiffened panels2

under system reliability constraint.3

4

IV. Inverse Measure Calculation by Moment-Based Methods5Moment-based methods provide for less expensive calculation of the probability of6

failure compared to MCS, although they are limited to a single failure mode. These methods7

require a search for the most probable point (MPP) on the failure surface in the standard8

normal space. The First Order Reliability Method (FORM) is the most widely used moment-9

based technique. FORM is based on the idea of the linear approximation of the limit state10

function and is accurate as long as the curvature of the limit state function is not too high.11

When the limit state has a significant curvature, second order methods can be used. The12

Second Order Reliability Method (SORM) approximates the measure of reliability more13

accurately by considering the effect of the curvature of the limit state function (Melchers,14

1999, pp 127-130). All the random variables are to be transformed to the standard normal15

variables with zero mean and unit variance.16

Moment-based methods are employed to calculate the reliability index, which is17

denoted by and related to the probability of failure as:18

( )fP = (13)19

where is the standard normal cumulative distribution function. Respective target values of20

and failure probabilities are also related in the same manner. In FORM, can be21

calculated as = U , where U is the vector of standard normal variates (variables with22

normal distribution of zero mean and unit variance). The standard normal variates are23

obtained from a transformation of the basic random variables X, which could be non-normal24


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and dependent. In the standard normal space the point on the first order limit state function at1

which the distance from the origin is minimum is the MPP.2

Figure 4 illustrates the concept of the reliability index and MPP search for a two3

variable case in the standard normal space. In reliability analysis, concerns are first focused4

on the ( ) 0G =U curve. Next, among the various values possible (denoted by 1 , 2 , 3 ),5

the minimum is sought. The corresponding point is the MPP. This process can be6

mathematically expressed as:7

To find* = u ,8

MinimizeT

U U 9

Subject to: ( ) 0G =U (14)10

where u* is the MPP. The calculation of the failure probability is based on linearization of11

the limit function at the MPP.12

Inverse reliability measures can also be computed through moment-based methods.13

Figure 5 illustrates the concept of inverse reliability analysis and MPP search. The circles14

represent the curves with the target curve represented by a dashed circle. Here, among15

the different values of limit state functions that pass through the target curve, the one with16

minimum value is sought. The value of this minimal limit state function is the PPM as shown17

by Tu et al. 1999. The point on the target circle with the minimal limit state function is18

sought. This point is also an MPP and in order to avoid confusion between the usual MPP and19

MPP in inverse reliability analysis, Du et al. (2003) coined the term most probable point of20

inverse reliability (MPPIR) and Lee et al. (2002) called it the minimum performance target21

point (MPTP). Du et al. (2003) developed the sequential optimization and reliability analysis22

method in which they show that evaluating the probabilistic constraint at the design point is23


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1equivalent to evaluating the deterministic constraint at the most probable point of the inverse2

reliability. This facilitates in converting the probabilistic constraint to an equivalent3

deterministic constraint. That is, the deterministic optimization is performed using a4

constraint limit which is determined based on the inverse MPP obtained in the previous5

iteration. Kiureghian et al. (1994) proposed an extension of the Hasofer-Lind-Rackwitz-6

Fiessler algorithm that uses a merit function and search direction to find the MPTP. In Fig. 5,7

the value of the minimal limit state function or the PPM is -0.2. This process can be8

expressed as:9

Minimize: ( )G U 10

Subjected to: T target= =U U U (15)11

In reliability analysis the MPP is on the ( ) 0G =U failure surface. In inverse reliability12

analysis, the MPP search is on the targetcurve.13

14

15

16

Fig. 5 Inverse reliability analysis and MPP

for target probability of failure of 0.00135

( = 3)

Fig. 4 Reliability analysis and MPP


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Table 4 Comparison of inverse measures for

w=2.4526 and t=3.8884 (stress constraint)

p

1f fP ( P )

N

= (19)1

In this case, the failure probability of 0.0013 calculated from 100,000 samples has a standard2

3

Minimize objective functionA=wtsuch that 3 4

Inverse Reliability AnalysisMethod Reliability

Analysis

FORM

(RIA)

FORM

(PMA)

MCS (PSF)

(Qu and Haftka,2003)

Exact

Optimum

(Wu et al.

2001)

w 2.4460 2.4460 2.4526 2.4484Optima

t 3.8922 3.8920 3.8884 3.8884

Objective Function 9.5202 9.5202 9.5367 9.5204

Reliability Index 3.00 3.00 3.0162 3.00

Failure Probability 0.00135 0.00135 0.00128 0.00135

5

deviation of 1.1410-4

. It is seen from Table 3 that the designs obtained from RIA, PMA and6

PSF match well. Since the stress Eq. (17) is a linear function of random variables, the RIA7

and PMA are exact. The more conservative design from PSF is due to limited sampling of8

MCS.9

10

Comparison of Inverse Measures11

The relation between PSF and PPM in Eq. (10) is only approximate when PPM is12

calculated by FORM and PSF by MCS. PSF suffers from sampling error, and PPM from13

error due to linearization. For the linear stress constraint, and with a large Monte Carlo14

sample, the difference is small, as seen in Table 4. It may be expected that the MPTP16

(minimum performance target point)18

should also be close to the point used19

to calculate PSF. This result may be20

useful, because when a response21

surface is used for an approximation to the response, it is useful to center it near the MPTP.22

Method FORM MCS (1107samples)

Pf 0.001238 0.001241

Inverse

Measure PPM: 0.00258 PSF: 1.002619

Table 3 Comparison of optimum designs for the stress constraint


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one. The structure can be made safer by scaling both w and tby the same factor c. This will1

change the stress and displacement expressed in Eq. (17) and Eq. (18) by a factor ofc3

and c4,

2

respectively, and the area by a factor ofc2. If stress is most critical, it will scale as c

3and the3

PSF will vary with the area, A as:4

1 5.

*0

0

As* s

A

=

(20)5

Equation (20) indicates that a one percent increase in area will increase the PSF by 1.56

percent. Since non-uniform scaling of width and thickness may be more efficient than7

uniform scaling, this is a conservative estimate. Thus, for example, considering a design with8

a PSF of 0.97, the safety factor deficiency is 3% and the structure can be made safer with a9

weight increase of less than two percent, as shown by Qu and Haftka (2003). For a critical10

displacement state, s* will be proportional to 2A and a 3% deficit in PSF can be corrected in11

under 1.5% weight increase. While for more complex structures we do not have analytical12

expressions for the dependence of the displacements or the stresses on the design variables,13

designers can usually estimate the weight needed to reduce stresses or displacements by a14

given amount.15

16

Design for System Reliability by MCS and PSF17

Monte Carlo simulation is a good method to use for system reliability analysis with18

multiple failure modes. The allowable deflection was chosen to be 2.25" in order to have19

competing constraints (Wu et al., 2001). The results are presented in Table 6. It can be20

observed that the contribution of the stress mode to failure probability dominates the21

contribution due to displacement and the interaction between the modes. The details of the22

design process are provided in Qu and Haftka (2004). They demonstrated the advantages of23

using PSF as an inverse safety measure over probability of failure or the safety index as a24


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normal safety measure. They showed that the design response surface (DRS) of the PSF was1

much more accurate than the DRS for the probability of failure. For a set of test points the2

error in the probability of failure was 39.11% from the DRS to the PSF, 96.49% for the DRS3

to the safety index, and 334.78% for the DRS to the probability of failure4

5

6

7

**100,000 samples, Pf1, Pf2, Pf1 Pf2 - Failure probabilities due to stress constraint,8

displacement constraint, and intersection between the modes, respectively.9

10

Design for System Reliability by Moment-Based Method and PPM11

Moment-based methods are restricted to address single mode failures. A common way12

of addressing multi-mode failures using moment-based methods is to prescribe reliability13

indices for each failure modes. Tu et al. (1999) adopted this procedure to solve system14

reliability problems using PMA. They showed that the PPM helps accelerate the15

convergence. The system reliability for the beam design example with separate reliability16

indices for the stress mode and displacement mode are performed by RIA and PMA and the17

results are presented in Table 7 for RIA and Table 8 for PMA. The convergence of the18

objective function is shown in Fig. 7. Comparing Tables 7 and 8, it is clear that the optimal19

values converge more quickly in PMA compared to RIA. Figure 7 illustrates that there is a20

smoother convergence in PMA compared to RIA.21

22

23

24

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Optima Objective

FunctionPf Safety

Index

Pf1 Pf2 Pf1 Pf2 PSF

*s

w = 2.6041t= 3.6746

9.5691 0.001289

3.01379 0.001133 0.000208 0.000052 1.0006

Table 6 Design for System Reliability by PSF, Qu and Haftka (2004)**


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Fig. 7 Convergence of objective function in (a) RIA and (b) PMA

(a) (b)

1

V. Concluding Remarks2The paper described various inverse measures and their usage for reliability based3

design optimization (RBDO). In particular, the relationship between two inverse safety4

measures, PPM and PSF was established. The computation of inverse measure by Monte5

Carlo simulation (MCS) and moment-based techniques was discussed. Several advantages of6

inverse measures were illustrated. They can accelerate convergence in RBDO, increase the7

accuracy of design response surfaces and maintain accuracy with MCS even when the failure8

probability is very low. Moreover, inverse measures can be employed to estimate the9

additional cost required to achieve the target reliability. These features of inverse measure10

make it a valuable resource in RBDO. A simple beam example was used to demonstrate some11

of the benefits.12

Acknowledgments13

This work has been funded in part through NASA Cooperative Agreement NCC3-994, the14

"Institute for Future Space Transport" University Research, Engineering and Technology15

Institute.16


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safety factor and inverse reliability

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