DevelopmentofEmpiricalFragilityCurvesinEarthquake...

Research ArticleDevelopment of Empirical Fragility Curves in EarthquakeEngineering considering Nonspecific Damage Information

Jung J Kim

Professor Department of Civil Engineering Kyungnam University Changwon-si 51767 Republic of Korea

Correspondence should be addressed to Jung J Kim jungkimkyungnamackr

Received 3 September 2018 Revised 4 November 2018 Accepted 11 November 2018 Published 13 December 2018

Guest Editor Tiago Ferreira

Copyright copy 2018 Jung J Kimis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a function of fragility curves in earthquake engineering the assessment of the probability of exceeding a specific damage stateaccording to the magnitude of earthquake can be considered Considering that the damage states for fragility curves are generallynested to each other the possibility theory a special form of the evidence theory for nested intervals is applied to generate fragilityinformation from seismic damage data While the lognormal distributions are conventionally used to generate fragility curves dueto their simplicity and applicability themethodology to use the possibility theory does not require the assumption of distributionsSeismic damage data classified by four damage levels were used for a case study e resulted possibility-based fragility in-formation expressed by two monotone measures ldquopossibilityrdquo and ldquocertaintyrdquo are compared with the conventional fragilitycurves based on probability e results showed that the conventional fragility curves provide a conservative estimation at therelatively high earthquake magnitude compared with the possibility-based fragility information

1 Introduction

In earthquake engineering fragility curves have been used toestimate damages of infrastructures according to the mag-nitude of earthquake While fragility curves can providereasonable estimation of earthquake damages with damagelevels those might neglect the possible slight damage oc-currence at the relatively low earthquake magnitude due tothe nature of probability density functions consists of twoparameters expected mean value and dispersion

Traditionally probability theory has been used to modeluncertainties in structural engineering especially whenaddressing reliability for structural safety [1 2] However thetypes of uncertainties considered in probability theory arerandom chance and likelihood and there are limitations tomodel other types of uncertainties such as nonspecificityfuzziness and strife using probability theory [3 4] Randomuncertainty known as aleatory uncertainty is from inherentrandomness and therefore is irreducible However othertypes of uncertainties known as epistemic uncertainties arisefrom lack of knowledge and therefore are reducible andsubjective Research on generalized information theory (GIT)

[5 6] showed that three types of epistemic uncertainties dueto lack of knowledge andor variability thrive when modelingcomplex environments [7] While nonspecificity representsthe difficulty to choose from many modeling alternativesfuzziness represents the uncertainty due to lack of sharpness(imprecise boundaries) of the modeling parameter Strifeexpresses the uncertainty due to conflict among alternativesGiven are there various types of uncertainties appropriatemodeling of uncertainty has been an interesting and chal-lenging topic in many areas during the last few decades[8ndash10] A number of theories to model uncertainties ade-quately have been introduced evidence theory [11 12]possibility theory [3 13] and fuzzy set theory [14 15]

In earthquake engineering empirical fragility curveswere generally presented in the form of lognormal cumu-lative distribution function (CDF) with respect to peakground acceleration (PGA) representing the ground motionintensity due to earthquake [16] To generate empiricalfragility curve for a damage state the damage reports byexperts are used e damage reports usually present thedamage states in linguistic ways such as ldquono damagerdquo ldquoslightdamagerdquo ldquomoderate damagerdquo ldquoextensive damagerdquo and

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 6209137 13 pageshttpsdoiorg10115520186209137

ldquocollapserdquo for a structure experiencing earthquake of a PGAAsthe fragility curve of a damage state represents the fragility ofldquoat leastrdquo of the damage level the evidence for a damage stateincludes possible higher damage states For example the ev-idence for ldquomoderate damagerdquo of a structure by an earthquakemeans that the structure is damaged at least moderately and itmight be possible for the structure to be damaged extensivelyor collapsed As there exist ambiguous boundaries between thedamage states fuzzy logic and possibility theory were appliedto resolve the ambiguity [17 18]

Recently extensive earthquake damage data are used togenerate fragility curves [19ndash25] Postearthquake surveys ofapproximately 340000 reinforced concrete structures wereused to derive fragility curves for a European seismic riskassessment scenario [19] A database of 7597 reinforcedconcrete buildings located in the city and the province ofLrsquoAquila in Italy was used in order to derive fragility curves[20] e observed damage to 9500 of low-rise residentialbuildings from earthquakes in South Iceland was studied bytypological fragility curves [21] Moreover fragility curveswere developed frommillions of data on the basis of 665515building damage cases by earthquake in Nepal [22 23] InItaly the postearthquake damage surveys of approximately90000 buildings in order to derive fragility curves wereconsidered [24 25] Even with the increase of damage data togenerate fragility curves there is still an uncertainty ofnonspecificity the difficulty to choose from many modelingfunctions of fragility curves such as lognormal extreme typeI extreme type II functions and so on

In this study the evidence of damage state is dealt withpossibility theory It is noticeable that the fragility curvesfrom possibility distribution representing the certainty ofdamage state and those are generated without any as-sumption of distributions erefore there is no uncertaintyof nonspecificity to choose functions of fragility curves

2 Possibility Information in Fragility Curves

eories for modeling uncertainties present different types ofuncertainty assignment and monotone measures As un-certainty assignment terms the degree of belief probabilitydistribution and possibility distribution are used for evidencetheory probability theory and possibility theory respectivelyTo quantify the assigned uncertainties monotone measuresare used such as dual monotone measures of plausibility andbelief dual monotone measures of possibility and certaintyand single monotone measure of probability for evidencetheory probability theory and possibility theory respectively[26ndash28] Considering the relationship between uncertaintyassignment terms and monotone measures used for eachtheory it can be known as probability theory and possibilitytheory are special forms of evidence theory [6] Consider adiscrete universe D that consists of a set of damage levels

D dN dS dM dE dC1113864 1113865 (1)

where dN dS dM dE and dC represent no damage slightdamage moderate damage extensive damage and collapseof a structure due to a seismic force level respectively

In evidence theory which is also known asDempsterndashShafer theory [11 12] the degree of belief mbased on evidence is assigned to all countable subsets A(eg Oslash dN dN dS dN dC) with the constraint of

1113944AsubD

m(A) 1 (2)

Dual monotone measures belief bel (A) and plausibilitypl (A) for a subset A are calculated as

bel(A) 1113944BisinA

m(B) (3)

pl(A) 1113944BcapAneempty

m(B) (4)

While belief measure represents the degree of evidencefor a subset A plausibility measure is defined as ldquoCom-plement of the belief of the complement of a subset Ardquo as

pl(A) 1minus bel(A) (5)

As belief measure is based on the degree of belief with itsevidence belief measure of ldquoComplement of a subset Ardquo alsoneeds its evidence erefore if there is no evidence forldquoComplement of a subset Ardquo one cannot determine thebelief of ldquoComplement of a subset Ardquo as 1-bel (A) edifference between these two measures can represent ourignorance (lack of knowledge) of a subset A (denoted ign) as

ign(A) pl(A)minus bel(A) 1minus[bel(A) + bel(A)] (6)

In probability theory probability distribution which isequivalent to the degree of belief m in evidence theory isassigned to a single variable (eg dN dC) on universe Dsuch as

1113944diisinD

p di( 1113857 1(7)

where di denotes the damage state dN dS dM dE and dCOnly one monotone measure probability prob(A) for asubset A is defined as

prob(A) 1113944diisinA

p di( 1113857(8)

and probability measure of ldquoComplement of a subset Ardquo isdefined as

prob(A) 1minus prob(A) (9)

with the excluded middle axioms [4] Unlike evidence theoryldquoComplement of a subset Ardquo can be determined as1 ndash prob (A) erefore our lack of knowledge measured byign (A) in equation (6) cannot be measured in probabilitytheory

In possibility theory possibility distribution π is assignedto a single damage level in possibility theory such as

π di( 1113857 1113944diisinA

m(A)

max π di( 11138571113864 1113865 1

(10)

2 Advances in Civil Engineering

e relationship between the uncertainty assignment πand the degree of belief m in equation (10) indicates thatpossibility theory is a special form of evidence theory whenthe collective body of evidence is consonant [26ndash28] (seeFigure 1)

Dual monotone measures certainty cert (A) and possi-bility pos (A) for a subset A are determined as

pos(A) sup π di( ) xi isin A (11)

cert(A) 1minus pos(A) (12)

is relationship can be converted to a single measurethat represents the degree of conshyrmation C (A) of a subsetA whose range is from minus1 to 1 [5]

C(A) cert(A) + pos(A)minus 1 (13)

Negative value of the degree of conshyrmation expressesthe degree of disconshyrmation of a subset A It is noticeablethat the assignment of possibility distribution of 1 to a singlevariable means that the occurrence of the variable is possibly1 but certainly ldquono evidencerdquo while the assignment of degreeof possibility of zero conshyrms that the occurrence of thevariable is possibly zero and certainly zero For a consonantbody of evidence the following relationship for two dierentsubsets A and B can be proven [6]

pos(AcupB) max[pos(A) pos(B)] (14)

cert(AcapB) min[cert(A) cert(B)] (15)

When the ignorance in equation (6) is zero the excludedmiddle axioms are satisshyed and the evidence can be describedusing one monotone measure called the probability measureMoreover when body of evidence is consonant monotonemeasures in evidence theory belief and plausibility can berepresented as those in possibility theory certainty andpossibility respectively (see Appendix for examples)

As the fragility curve of a damage state represents thefragility of ldquoat leastrdquo of the damage level probable damageinformation for higher damage state is included in a damagestate For example the evidence for ldquono damagerdquo means that astructure seems undamaged but there might be possibledamages for the structure slight moderate extensive orcollapsed damages Considering the characteristics of evi-dence ldquoat leastrdquo we can deshyne subsetsN SM E andC for ldquoatleast no damagerdquo ldquoat least slight damagerdquo ldquoat least moderatedamagerdquo ldquoat least extensive damagerdquo and ldquoat least collapserdquo asshown in equation (16) It can be recognized that a damagesubset includes higher than and equal to the damage levels

N dN dS dM dE dC S dS dM dE dC M dM dE dC E dE dC C dC

(16)

It is seen that the subsets in equation (16) are nested toeach other such as C sub E subM sub S sub N Based on the degree

of belief m is assigned to a consonant set of damage levelspossibility distributions for predetermined PGA region aregenerated using the assignment of the degree of belief esequence of monotone measures of each damage level withrespect to PGA is presented and compared with conven-tional fragility curves generated by maximum likelihoodestimation It is noticeable that the evidence for at least nodamage N is considered as any damage level

3 Case Study

A schematic representation of the proposed framework togenerate fragility curves using possibility distributions isshown in Figure 2 At the shyrst step the empirical damagedata are rearranged in the shape of binomial damage data fordamage states In this step the damage data are rearrangedin the ascending order with respect to PGA As this pro-cedure is out of scope of this study the arranged data byother researchers [16] are used for case study Some part ofthe arranged data is presented in Table 1 Using the arrangeddata set PGA intervals are determined based on the in-clusion of damage levels at the second step At the third stepthe degree of belief m is assigned to the sets in equation (16)based on the occurrence of damage levels in each PGAinterval At the fourth step possibility distribution is gen-erated based on the degree of belief At the shynal stepmonotone measures for PGA intervals are calculated and thesequence of monotone measures with respect to PGA isdetermined

For the case study the damage data of bridges from the1994 Northridge earthquake are used to develop empiricalfragility curves [16] e PGA value at the location ofbridges is interpolated and extrapolated from the PGA data[16] e binomial damage information of damage statesldquono damagerdquo ldquoat least minorrdquo ldquoat least moderaterdquo ldquoat leastmajorrdquo and ldquocollapserdquo in original reference [16] are used togenerate possibility-based empirical fragility curves of ldquonodamage fragility curverdquo ldquoslight damage fragility curverdquoldquomoderate damage fragility curverdquo ldquoextensive damagefragility curverdquo and ldquocollapse fragility curverdquo respectivelyin this study

For the comparison the binomial information for thedamage state at PGA is used to generate empirical fragilitycurves by maximum likelihood estimation with the as-sumption of lognormal and extreme type I and II distri-butionse results of the four families of fragility curves arepresented in Figure 3

a b cm1

m2

m3

Figure 1 Consonant body of evidence

Advances in Civil Engineering 3

4 Results and Discussions

From the binomial information of damage state data PGAintervals are determined based on the inclusion of damagelevels and presented in Table 2 For the last PGA interval it isdivided into two regions as the first collapse evidence at PGAof 0385 g seems possible but the next evidence is far fromthe PGA such as 0682 g During the 0385 g to 0682 g asthere is no evidence for collapse the PGA interval is dividedfor engineering sense

Based on the occurrence of damage levels in each PGAinterval the degree of belief m is assigned to the sets inequation (16) and presented in Table 3 e possibilitydistributions for each PGA intervals are generated as shownin Figure 4 e possibility distribution π values for each

damage level are calculated based on the degree of beliefand presented in Table 4 For example π for dM for theinterval [0385 0680] is calculated as the summation of thedegree of belief for N S M in the respective interval ofTable 3 It follows the definition of π as presented inequation (10)

Using the generated possibility distributions in Figure 4monotone measures for N S M E and C can be measuredAn example to measure the dual monotone measures inpossibility theory certainty cert(M) and possibility pos(M)for M of the interval [0323 0384] is presented in Figure 5e maximum possibility distribution π inside of the in-terval will be the measure of pos as 100 as defined inequation (11) and the complement of the maximum pos-sibility distribution π outside of the interval will be themeasure of cert as 8 as defined in equation (12) e degreeof confirmation is calculated using the dual monotonemeasures as 8 as shown in equation (13) e certainty andpossibility measures are presented in Tables 5 and 6 re-spectively It is noticeable that the belief and plausibilitymeasures for the sets in equation (16) are same with thecertainty and possibility measures respectively as possibilitytheory is a special form of evidence theory e belief andplausibility measures in evidence theory are presented inTables 7 and 8

As the possibility measures for all intervals are 100 aspresented in Table 6 the degree of confirmation is going tobe the same with the certainty measure in Table 5 eempirical fragility curves can be constructed by plottingcertainty or the degree of confirmation with respect to thecorresponding intervals as shown in Figure 6

Comparisons with the conventional fragility curves thatgenerated by assuming a distribution are presented inFigures 7ndash10 It is noticeable that the caption for y-axis in thefigures ldquofragilityrdquo represents ldquoprobabilityrdquo in probabilitytheory and ldquocertaintyrdquo in possibility theory For the com-parison of ldquoSlightrdquo damage state as shown in Figure 7 theconventional fragility curves underestimate the failureprobability at the PGA interval between 0323 g and 05 gFor the comparison of ldquoModeraterdquo and ldquoExtensiverdquo damagestates as shown in Figures 8 and 9 respectively the con-ventional fragility curves underestimate the failure proba-bility at the PGA interval between 0385 g and 05 g Finally

Empirical damage data Binomial damage data for damage states

Determine PGA intervals

Generate possibility distribution for each PGA interval

Assign the degree of belief m for damage state subsets in equation (16) at each PGA interval

Generate fragility curves by measuring certainty measure for each PGA interval

Figure 2 Procedure to generate empirical fragility curves using possibility distributions

Table 1 Some part of the arranged binomial data to generateempirical fragility curves for the case study [16]

No PGA None geMin geMod geMaj geCol Firstappearance

1 0069 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮57 0079 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮62 0080 1 1 0 0 0 Minor damage⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮397 0137 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

415 0138 1 1 1 0 0 Moderatedamage

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1303 0322 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1307 0323 1 1 1 1 0 Major damage⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1555 0384 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1561 0385 1 1 1 1 1 Collapse⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1859 0680 1 1 1 1 01860 0682 1 1 1 0 01861 0682 1 1 1 1 1 Collapse⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮1998 0889 1 1 1 1 0


for the comparison of ldquoCollapserdquo damage state as shown inFigure 10 the conventional fragility curves underestimatethe failure probability at the PGA interval between 0682 gand 08 g While there is no evidence for damage over thePGA of 0889 g the conventional fragility curves for alldamage states clearly overestimate the failure probabilityover the PGA of 0889 g ese results are obvious as the

conventional fragility curves are generated by assuming thedistributions with mathematical formulations such as log-normal and extreme type I and type II distributions

With extensive earthquake damage database the uni-form PGA interval so called as ldquoPGA binrdquo can be usedinstead of the PGA intervals determined by experts in Ta-ble 2 In this case the shape of fragility curve also can be

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(a)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(b)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(c)

Figure 3 Empirical fragility curves determined by maximum likelihood estimation using different distribution functions At each figurecurves from top to bottom are fragility curves for damage state subsets of SM E and C respectively (a) Lognormal CDF (b) Extreme type I(Gumbel) CDF (c) Extreme type II (Frechet) CDF

Table 2 Number of damage information at each damage state in PGA intervals

PGA interval (g) [θ1 θ2] None (N) Slight (S) Moderate (M) Extensive (E) Collapse (C) No of evidence[0069 0079] 56 0 0 0 0 56[0080 0137] 350 8 0 0 0 358[0138 0322] 862 14 16 0 0 892[0323 0384] 205 24 14 6 0 249[0385 0680] 221 23 39 20 1 304[0682 0889] 74 14 25 21 5 139Sum 1768 83 94 47 6 1998


Table 3 Degree of belief m based on the fraction of each damage state in PGA intervals

PGA interval (g) [θ1 θ2] None (N) Slight (S) Moderate (M) Extensive (E) Collapse (C) Sum[0069 0079] 1000 0000 0000 0000 0000 1000[0080 0137] 0978 0022 0000 0000 0000 1000[0138 0322] 0966 0016 0018 0000 0000 1000[0323 0384] 0823 0096 0056 0024 0000 1000[0385 0680] 0727 0076 0128 0066 0003 1000[0682 0889] 0532 0101 0180 0151 0036 1000

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )

Figure 4 Possibility distributions of damage states for the corresponding PGA (g) intervals (a) [0069 0079]g (b) [0080 0137]g(c) [0138 0322]g (d) [0323 0384]g (e) [0385 0680]g (f ) [0682 0889]g


Table 4 Assigning π based on the degree of belief m in Table 3

PGA interval (g) [θ1 θ2] dN () dS () dM () dE () dC () Max ()[0069 0079] 1000 1000 1000 1000 1000 100[0080 0137] 978 1000 1000 1000 1000 100[0138 0322] 966 982 1000 1000 1000 100[0323 0384] 823 920 976 1000 1000 100[0385 0680] 727 803 931 997 1000 100[0682 0889] 532 633 813 964 1000 100

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92

Figure 5 Calculated certaintymeasure cert 8 and possibilitymeasure pos 100 of the setM dM dE dC for the PGA (g) interval of[0323 0384]g

Table 5 Certainty measure of damage sets in equation (16) using equation (12) based on the possibility distribution in Figure 4

PGA interval (g) [θ1 θ2] None (N) () Slight (S) () Moderate (M) () Extensive (E) () Collapse (C) ()[0069 0079] 1000 000 00 00 00[0080 0137] 1000 223 00 00 00[0138 0322] 1000 336 18 00 00[0323 0384] 1000 1767 80 24 00[0385 0680] 1000 2730 197 69 03[0682 0889] 1000 4676 367 187 36

Table 6 Possibility measure of damage sets in equation (16) using equation (11) based on the possibility distribution in Figure 4


Table 7 Belief measure of damage subsets in equation (16) usingequation (3) based on the degree of belief m in Table 3

PGA interval (g) [θ1 θ2] N () S () M () E () C ()[0069 0079] 1000 000 00 00 00[0080 0137] 1000 223 00 00 00[0138 0322] 1000 336 18 00 00[0323 0384] 1000 1767 80 24 00[0385 0680] 1000 2730 197 69 03[0682 0889] 1000 4676 367 187 36

Table 8 Plausibility measure of damage subsets in equation (16)using equation (4) based on the degree of belief m in Table 3



formulated by connecting the necessity measure at thecenter of bin instead of using step function that has thesame value in a PGA interval (bin) An example usingldquoPGA binrdquo of 01 g is presented in Figures 11ndash14 for therespective damage state of ldquoSlightrdquo ldquoModeraterdquo ldquoExten-siverdquo and ldquoCollapserdquo In the shygures the proposed fragilitycurves using expertrsquos interval are compared with thoseusing the uniform interval e alternative formulation ofthe proposed fragility curves by connecting the certaintymeasure at the center of PGA interval is also presented inFigures 11ndash14e comparisons in the shygures showed thatthe uniform interval would be used to generate theproposed fragility curves in this study with a large numberof damage data

5 Conclusions

A framework to consider the ldquononspecishycityrdquo of damagedata is presented Considering that the damage states forfragility curves are generally nested to each other thepossibility theory a special form of the evidence theoryfor nested intervals is applied to generate fragility in-formation from seismic damage data Seismic damagedata classishyed by four damage levels were used for a casestudy Based on the damage state evidence the degree ofbelief is assigned to a consonant set of damage levelsPossibility distributions for predetermined PGA regionare generated using the assignment of the degree of beliefe sequence of monotone measures of each damage level

0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse

Figure 6 Empirical fragility curves of damage state subsets determined by measuring certainty of possibility distributions for PGA (g)intervals

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)

MLE-lognormalMLE-gumbel

MLE-FrechetProposed empirical fragility curve

Figure 7 Comparison of empirical fragility curves of damage state with Slight


with respect to PGA is presented and compared withconventional fragility curves generated by maximumlikelihood estimation It was shown that the generatedsequence of certainty measure could be used as fragilitycurves in alternative perspective Noticeably the fragilitycurves from possibility distribution represent the cer-tainty of damage state and those are generated withoutany assumption of distributions e empirical fragilitycurves using possibility theory were compared with thosegenerated by maximum likelihood estimation e resultsshowed that the conventional fragility curves generatedby assuming the distributions overestimate the failureprobability at the relatively high PGA while those un-derestimate the failure probability at the medium range ofPGA erefore the proposed empirical fragility curves

using possibility theory can be used as an alternativemethodology for earthquake engineering

Appendix

All possible subsets of a set constitute a special set deshyned aspower set

X a b c P(X) [ a b c a b b c a c a b c

(17)

Example 1 Evidence theory

Consider a discrete universe X 1 2 3 e degree ofbelief (evidence) m is assigned to the corresponding power set

0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)



Figure 8 Comparison of empirical fragility curves of damage state with Moderate



0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)

Figure 9 Comparison of empirical fragility curves of damage state with Extensive


with the summation of m as unity as shown in equation (2)such as

P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)

e monotone measures for a subset are calculated byequation (3) for belief and equation (4) for plausibility aspresented in Table 9

Example 2 Special case of evidence theory (same withprobability theory)

For the same universe X in example 2 the degree ofbelief (evidence)m is assigned to the corresponding powerset such as

P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)

e monotone measures belief and plausibility arecalculated as presented in Table 10

As shown in results belief and plausibility measures havethe same values for all subsets and ignorance measures are



0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)

Figure 10 Comparison of empirical fragility curves of damage state with Collapse

0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

Slight (uniform interval)Slight (uniform interval center)

Slight (expertrsquos interval)Slight (expertrsquos interval center)

Figure 11 Comparison of the proposed fragility curves for Slight about the dierent interval selections with alternative formulation of theproposed fragility curves by using step function and connecting the necessity measure at the center of PGA interval


going to be zero erefore belief and plausibility measurescan be represented as one monotone measure such asprobability

Example 3 Special case of evidence theory (same withpossibility theory)

As shown in Figure 1 if a body of evidence is nestedwithin each other for a discrete universe a b c this body ofevidence is called ldquoconsonant body of evidencerdquo For aconsonant body of evidence belief and plausibility are equalto necessity and possibility respectively

For the same universeX in example 2 when the degree ofbelief (evidence) m is assigned to the corresponding powerset such as

P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)

e monotone measures belief and plausibility arecalculated as presented in Table 11 Moreover possibilitydistributions can be generated using equation (10) as pre-sented in Table 12 Using these possibility distributions in

Moderate (uniform interval)Moderate (uniform interval center)

Moderate (expertrsquos interval)Moderate (expertrsquos interval center)

0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

Figure 12 Comparison of the proposed fragility curves for Moderate about the different interval selections with alternative formulation ofthe proposed fragility curves by using step function and connecting the necessity measure at the center of PGA interval

0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

Extensive (uniform interval)Extensive (uniform interval center)Extensive (expertrsquos interval)

Figure 13 Comparison of the proposed fragility curves for Extensive about the different interval selections with alternative formulation ofthe proposed fragility curves by using step function and connecting the necessity measure at the center of PGA interval


Table 12 certainty and possibility measures in possibilitytheory are calculated using equations (11) and (12) as pre-sented in Table 13 As expected the belief and plausibilitymeasures in Table 11 are equal to the certainty and possi-bility measures in Table 13 respectively

0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

Collapse (uniform interval)Collapse (uniform interval center)

Collapse (expertrsquos interval)Collapse (expertrsquos interval center)

Figure 14 Comparison of the proposed fragility curves for Collapse about the different interval selections with alternative formulation ofthe proposed fragility curves by using step function and connecting the necessity measure at the center of PGA interval

Table 9 e calculated monotone measures in evidence theory

Subset Monotone measures in evidence theory

1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)

01 + 02 + 01 + 03 07ign(1) pl(1) minus bel(1) 07 minus 01 0

1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +

m(1 3) + m(1 2 3) 1ign(1 2) pl(1 2) minus bel(1 2) 1 - 04 06

1 2 3bel(1 2 3) all 1pl(1 2 3) all 1

ign(1 2 3) pl(1 2 3) minus bel(1 2 3) 1 minus 1 0

Table 10 e calculated monotone measures in a special case ofevidence theory which is the same with probability theory


1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06

pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3)+m(1 3) +m(1 2 3) 03 + 03 + 0 + 0 + 0 + 0 06ign(1 2) pl(1 2) minus bel(1 2) 06 minus 06 0

1 2 3bel(1 2 3) all 1pl(1 2 3) all 1


Table 11 e calculated monotone measures in a special case ofevidence theory which is the same with possibility theory


1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1


Table 12 e possibility distributions


1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04

Table 13 e calculated monotone measures in possibility theory

Subset Monotone measures in possibility theory

1cert(1) 1 ndash sup [π(2) π(3)] 1 ndash sup [07 04]

03pos(1) π(1) 1

1 2 cert(1 2) 1 minus π(3) 1 minus 04 06pos(1 2) sup [π(1) π(2)] sup [1 07] 1

1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability

e data used for the case study of this study can be found inReference [16]

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is work was supported by Kyungnam University Foun-dation Grant 2016

References

[1] C O Li and R E Melchers ldquoTime-dependent reliabilityanalysis of corrosion-induced concrete crackingrdquo ACIStructural Journal vol 102 no 4 pp 543ndash549 2005

[2] J Seo G Hatfield and J-H Kimn ldquoProbabilistic structuralintegrity evaluation of a highway steel bridge under unknowntrucksrdquo Journal of Structural Integrity and Maintenancevol 1 no 2 pp 65ndash72 2016

[3] D Dubois ldquoPossibility theory and statistical reasoningrdquoComputational Statistics and Data Analysis vol 51 no 1pp 47ndash69 2006

[4] T J Ross Fuzzy Logic with Engineering Applications JohnWiley amp Sons Hoboken NJ USA 3rd edition 2010

[5] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic 6eory andApplications Prentice-Hall Upper Saddle River NJ USA1995

[6] G J Klir Uncertainty and Information John Wiley amp SonsHoboken NJ USA 2006

[7] O F Hoffman and J S Hammonds ldquoPropagation of un-certainty in risk assessment the need to distinguish betweenuncertainty due to lack of knowledge and uncertainty due tovariabilityrdquo Risk Analysis vol 14 no 5 pp 707ndash712 1994

[8] J C Helton and D E Burmaster ldquoGuest editorial treatmentof aleatory and epistemic uncertainty in performance as-sessments for complex systemsrdquo Reliability Engineering andSystem Safety vol 54 no 2-3 pp 91ndash94 1996

[9] H G Matthies C E Brenner C G Bucher and C GuedesSoares ldquoUncertainties in probabilistic numerical analysis ofstructures and solids-stochastic finite elementsrdquo StructuralSafety vol 19 no 3 pp 283ndash336 1997

[10] W L Oberkampf S M DeLand B M RutherfordK V Diegert and K F Alvin ldquoError and uncertainty inmodeling and simulationsrdquo Engineering and System Safetyvol 75 no 3 pp 333ndash357 2002

[11] A P Dempster ldquoUpper and lower probabilities inducedby a multivalued mappingrdquo Annals of MathematicalStatistics vol 38 no 2 pp 325ndash339 1967

[12] G Shafer Mathematical 6eory of Evidence Princeton Uni-versity Press Princeton NJ USA 1976

[13] D Dubois and H Prade Possibility 6eory An Approach toComputerized Processing of Uncertainty Plenum Press NewYork NY USA 1988

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[15] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[16] M Shinozuka M O Feng H Kim T Uzawa and T UedaStatistical Analysis of Fragility Curves Technical reportMCEER Federal Highway Administration Washington DCUSA 2001

[17] F Colangelo ldquoA simple model to include fuzziness in theseismic fragility curve and relevant effect compared withrandomnessrdquo Earthquake Engineering and StructuralDynamics vol 41 no 5 pp 969ndash986 2012

[18] J Rohmer and C Baudrit ldquoe use of the possibility theory toinvestigate the epistemic uncertainties within scenario-basedearthquake risk assessmentsrdquo Natural Hazards vol 56 no 3pp 613ndash632 2010

[19] T Rossetto and A Elnashai ldquoDerivation of vulnerabilityfunctions for European-type RC structures based on obser-vational datardquo Engineering Structures vol 25 no 10pp 1241ndash1263 2003

[20] C Del Gaudio G De Martino M De Ludovico et alldquoEmpirical fragility curves from the damage data on RCbuildings after the 2009 LrsquoAquila earthquakerdquo Bulletin ofEarthquake Engineering vol 15 no 4 pp 1425ndash14502017

[21] B Bessason and J O Bjarnason ldquoSeismic vulnerability of low-rise residential buildings based on damage data from threeearthquakes (MW 65 65 and 63)rdquo Engineering Structuresvol 111 pp 64ndash79 2016

[22] D Gautam G Fabbrocino and F Santucci deMagistris ldquoDeriveempirical fragility functions for Nepali residential buildingsrdquoEngineering Structures vol 171 pp 612ndash628 2018

[23] D Gautam ldquoObservational fragility functions for residentialstone masonry buildings in Nepalrdquo Bulletin of EarthquakeEngineering vol 16 no 10 pp 4661ndash4673 2018

[24] M Rota A Penna and C L Strobbia ldquoProcessing Italiandamage data to derive typological fragility curvesrdquo SoilDynamics and Earthquake Engineering vol 28 no 10ndash11pp 933ndash947 2008

[25] M Colombi B Borzi H Crowley M Onida F Meroni andR Pinho ldquoDeriving vulnerability curves using Italian earth-quake damage datardquo Bulletin of Earthquake Engineeringvol 6 no 3 pp 485ndash504 2006

[26] J J Kim ldquoUncertainty quantification in serviceability ofreinforced concrete structuresrdquo PhDDissertation in Universityof New Mexico USA 2009

[27] J J Kim M M Reda Taha and T J Ross ldquoEstablishingconcrete cracking strength interval using possibilitytheory with an application to predict the possible rein-forced concrete deflection intervalrdquo Engineering Struc-tures vol 32 no 11 pp 3592ndash3600 2010

[28] J J Kim M M Reda Taha and T J Ross ldquoBinary damageclassification in SHM using possibility distributionsrdquo inProceedings of the Second International Conference onVulnerability and Risk Analysis and Management (ICV-RAM) and the Sixth International Symposium on Un-certainty Modeling and Analysis (ISUMA) Liverpool UKJuly 2014


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

ldquocollapserdquo for a structure experiencing earthquake of a PGAAsthe fragility curve of a damage state represents the fragility ofldquoat leastrdquo of the damage level the evidence for a damage stateincludes possible higher damage states For example the ev-idence for ldquomoderate damagerdquo of a structure by an earthquakemeans that the structure is damaged at least moderately and itmight be possible for the structure to be damaged extensivelyor collapsed As there exist ambiguous boundaries between thedamage states fuzzy logic and possibility theory were appliedto resolve the ambiguity [17 18]

Recently extensive earthquake damage data are used togenerate fragility curves [19ndash25] Postearthquake surveys ofapproximately 340000 reinforced concrete structures wereused to derive fragility curves for a European seismic riskassessment scenario [19] A database of 7597 reinforcedconcrete buildings located in the city and the province ofLrsquoAquila in Italy was used in order to derive fragility curves[20] e observed damage to 9500 of low-rise residentialbuildings from earthquakes in South Iceland was studied bytypological fragility curves [21] Moreover fragility curveswere developed frommillions of data on the basis of 665515building damage cases by earthquake in Nepal [22 23] InItaly the postearthquake damage surveys of approximately90000 buildings in order to derive fragility curves wereconsidered [24 25] Even with the increase of damage data togenerate fragility curves there is still an uncertainty ofnonspecificity the difficulty to choose from many modelingfunctions of fragility curves such as lognormal extreme typeI extreme type II functions and so on

In this study the evidence of damage state is dealt withpossibility theory It is noticeable that the fragility curvesfrom possibility distribution representing the certainty ofdamage state and those are generated without any as-sumption of distributions erefore there is no uncertaintyof nonspecificity to choose functions of fragility curves

2 Possibility Information in Fragility Curves

eories for modeling uncertainties present different types ofuncertainty assignment and monotone measures As un-certainty assignment terms the degree of belief probabilitydistribution and possibility distribution are used for evidencetheory probability theory and possibility theory respectivelyTo quantify the assigned uncertainties monotone measuresare used such as dual monotone measures of plausibility andbelief dual monotone measures of possibility and certaintyand single monotone measure of probability for evidencetheory probability theory and possibility theory respectively[26ndash28] Considering the relationship between uncertaintyassignment terms and monotone measures used for eachtheory it can be known as probability theory and possibilitytheory are special forms of evidence theory [6] Consider adiscrete universe D that consists of a set of damage levels

D dN dS dM dE dC1113864 1113865 (1)

where dN dS dM dE and dC represent no damage slightdamage moderate damage extensive damage and collapseof a structure due to a seismic force level respectively

In evidence theory which is also known asDempsterndashShafer theory [11 12] the degree of belief mbased on evidence is assigned to all countable subsets A(eg Oslash dN dN dS dN dC) with the constraint of

1113944AsubD

m(A) 1 (2)

Dual monotone measures belief bel (A) and plausibilitypl (A) for a subset A are calculated as

bel(A) 1113944BisinA

m(B) (3)

pl(A) 1113944BcapAneempty

m(B) (4)

While belief measure represents the degree of evidencefor a subset A plausibility measure is defined as ldquoCom-plement of the belief of the complement of a subset Ardquo as

pl(A) 1minus bel(A) (5)

As belief measure is based on the degree of belief with itsevidence belief measure of ldquoComplement of a subset Ardquo alsoneeds its evidence erefore if there is no evidence forldquoComplement of a subset Ardquo one cannot determine thebelief of ldquoComplement of a subset Ardquo as 1-bel (A) edifference between these two measures can represent ourignorance (lack of knowledge) of a subset A (denoted ign) as

ign(A) pl(A)minus bel(A) 1minus[bel(A) + bel(A)] (6)

In probability theory probability distribution which isequivalent to the degree of belief m in evidence theory isassigned to a single variable (eg dN dC) on universe Dsuch as

1113944diisinD

p di( 1113857 1(7)

where di denotes the damage state dN dS dM dE and dCOnly one monotone measure probability prob(A) for asubset A is defined as

prob(A) 1113944diisinA

p di( 1113857(8)

and probability measure of ldquoComplement of a subset Ardquo isdefined as

prob(A) 1minus prob(A) (9)

with the excluded middle axioms [4] Unlike evidence theoryldquoComplement of a subset Ardquo can be determined as1 ndash prob (A) erefore our lack of knowledge measured byign (A) in equation (6) cannot be measured in probabilitytheory

In possibility theory possibility distribution π is assignedto a single damage level in possibility theory such as

π di( 1113857 1113944diisinA

m(A)

max π di( 11138571113864 1113865 1

(10)














(16)



3 Case Study




a b cm1

m2

m3


















1 0069 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮57 0079 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮62 0080 1 1 0 0 0 Minor damage⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮397 0137 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮







0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(a)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(b)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(c)







00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )





00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














(16)



3 Case Study




a b cm1

m2

m3


















1 0069 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮57 0079 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮62 0080 1 1 0 0 0 Minor damage⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮397 0137 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮







0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(a)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(b)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(c)







00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )





00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

















1 0069 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮57 0079 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮62 0080 1 1 0 0 0 Minor damage⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮397 0137 1 0 0 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮







0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(a)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(b)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(c)







00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )





00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(a)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(b)

0

02

04

06

0 01 02 03 04 05 06 07 08

Prob

abili

ty o

f exc

eedi

nga d

amag

e sta

te

PGA (g)

(c)







00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )





00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(a)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(b)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(c)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(d)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(e)

00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

(f )





00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




00

200

400

600

800

1000

dN dS dM dE dC

π

Damage states

M = dM dE dC

pos = 100

cert= 100ndash92= 8

92












5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



5 Conclusions


0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)

SlightModerate

ExtensiveCollapse


0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)







Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Appendix



(17)



0

01

02

03

04

05

06

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)






0

005

01

015

02

025

03

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 01 01 01 02 01 01 03]

(18)




P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3 ⟶ m [0 03 03 04 0 0 0 0]

(19)





0

001

002

003

004

005

006

0 01 02 03 04 05 06 07 08

Frag

ility

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)









P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






P(X) [ 1 2 3 1 2 2 3 1 3 1 2 3

⟶ m [0 03 0 0 03 0 0 04]

(20)




0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)


0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)





0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

10

20

30

40

50

60

0 01 02 03 04 05 06 07 08

Certa

inty

mea

sure

of e

xcee

ding

a da

mag

e sta

te

PGA (g)






1

bel(1) m(1) 0pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 04pl(1 2) m(1)+ m(2) + m(1 2) + m(2 3) +


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) +m(2) +m(1 2) 03 + 03 +0 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1

bel(1) m(1) 03pl(1) m(1) + m(1 2) + m(1 3) + m(1 2 3)


1 2

bel(1 2) m(1) + m(2) + m(1 2) 03 + 0 +03 06


1 2 3bel(1 2 3) all 1pl(1 2 3) all 1




1 π(1) m(1) +m(1 2) +m(1 3) +m(1 2 3)

03 + 03 + 0 + 04 1

2 π(2) m(2) +m(1 2) +m(2 3) +m(1 2 3)

0 + 03 + 0 + 04 07

3 π(3) m(3) +m(1 3) +m(2 3) +m(1 2 3)

0 + 0 + 0 + 04 04




03pos(1) π(1) 1


1 2 3cert (1 2 3) 1

pos(1 2 3) sup [π(1) π(2) π(3)] sup [107 04] 1


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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