11AnexampleforphysicalapproachThecomponentunderconsideration isa connecting rod oflengthLand diameterDthatissubjectedtoanaxial compressiveforceY.ThestrengthofthecomponentistheEulerbuckling loadoftheconnectingrod,givenby元3ED4X64L2Themeandiameterμp is tobe selectedto ensurea reliabilityof o.99againstbucklingunderthestressY.Wemakethefollowingassumptions:Yisnormallydistributedwithmeanμy=2000Ibsandstandarddeviationoy=200Ibs;Lisnormallydistributedwithmeanμy=20 inand standarddeviationo,=0.5 inEisassumedtobedeterministicwithavalue30×106(lb/in2)Disalsonormallydistributedwithmeanμp and standarddeviationop=0.1μpNTNU-TrondheimNorwegianUniversityofScienceandTechnologywww.ntnu.edu
11 An example for physical approach The component under consideration is a connecting rod of length L and diameter D that is subjected to an axial compressive force Y. The strength of the component is the Euler buckling load of the connecting rod, given by ܺ ൌ ସܦܧଷߨ 64ܮଶ The mean diameter ߤ is to be selected to ensure a reliability of 0.99 against buckling under the stress Y. We make the following assumptions: • Y is normally distributed with mean ߤ ൌ 2000 lbs and standard deviation ߪ ൌ 200 lbs; • L is normally distributed with mean ߤ ൌ 20 in and standard deviation ߪ ൌ 0.5 in • Ε is assumed to be deterministic with a value 30 ൈ 10 (lb/inଶ) • D is also normally distributed with mean ߤ and standard deviation ߪ ൌ 0.1ߤ
12Anexampleforphysicalapproach3EuD元3(30×106)μ=36331μμx64×20264μ2And2axaE哆o2axaLaDNotethataxax24xxand1aDalMLμDSo,Ox=15856μb(X-Y) canbeapproximated as a normaldistribution.The specified reliability ofo.99correspondstoalowerlimitof-2.3265forthestandard normal distribution36331μ-20000.99Pr(X -Y ≥ 0) =1- Φ/(15856μb）2+2002And then,we cangetNTNU-TrondheimONorwegian University ofμD=0.7451Science and Technologywww.ntnu.edu
12 An example for physical approach ൌ ߤ ସߤܧଷߨ 64ߤଶ ൌ ߨଷ 30 ൈ 10 ߤସ 64 ൈ 20ଶ ൌ 36331ߤସ And ଶߪ ଶ ܦ߲ܧ߲ ଶߪ ଶ ܮ߲߲ܺ ൌ ଶߪ Note that ߲ܺ ߤߤ 4 ൌ ܦ߲߲ܺ and ߤߤ െ2 ൌ ܮ߲ So, ߪ ൌ 15856ߤସ ሺܺ െ ܻሻ can be approximated as a normal distribution. The specified reliability of 0.99 corresponds to a lower limit of ‐2.3265 for the standard normal distribution Pr ܺെܻ0 ൌ1െΦ 36331ߤସ െ 2000 15856ߤସ ଶ 200ଶ ൌ 0.99 And then, we can get ߤ ൌ 0.7451
13TimedependentphysicalapproachInPractices,boththestrengthSandtheloadLvarywiththetimetStrength, S(t)FailureTimetofailure,TTimetThetimetofailureTisT=min(t;S(t)<L(t))AndthereliabilityR(t)maynowbedefinedasR(t) = Pr(T >t)NTNU-TrondheimNorwegianLniversityof梦Science and Technologywww.ntnu.edu
13 Time dependent physical approach In Practices, both the strength ܵ and the load ܮ vary with the time ݐ The time to failure ܶ is ሻݐሺܮ ൏ ݐ ܵ ;ݐ min ൌܶ And the reliability ܴሺݐሻ may now be defined as ܴ ݐ ൌ Pr ሺܶ ݐሻ
14Challengesof physical approachThestrengthismulti-dimensionalandmustbeexpressedasavectorSTheloadismulti-dimensionalandmustbeexpressedasavectorLThestrengthandtheloadmaybedependent randomvectorsThe strengthat timet isgenerallydepending on the load historyH,uptotimetNTNU-TrondheimNorwegianUniversityofScienceandTechnologywww.ntnu.edu
14 Challenges of physical approach • The strength is multi‐dimensional and must be expressed as a vector S • The load is multi‐dimensional and must be expressed as a vector L • The strength and the load may be dependent random vectors. • The strength at time t is generally depending on the load history Ht up to time t
15ActuariaapproachThetimetofailureTisconsidered asa randomvariablewitha probabilitydistributionF(t),withparametersthatareestimatedbasedonfielddata.Thedistribution may be decided based onphysical interpretation of thepropertiesof thedistribution,e.g.,bystudyingthefailureratefunction.Thisisthemaintopicofthecurrentcourse.NTNU-TrondheimNorwegian University of梦Science and Technologywww.ntnu.edu
15 Actuaria approach The time to failure ܶ is considered as a random variable with a probability distribution ܨሺݐሻ, with parameters that are estimated based on field data. The distribution may be decided based on physical interpretation of the properties of the distribution, e.g., by studying the failure rate function. This is the main topic of the current course