cp4: Examples CP4, CP6, and CP7, adapted from PRobability Of...

In crackR: Probabilistic damage tolerance analysis for fatigue cracking of metallic aerospace structures

Description

Examples CP4, CP6 and CP7 are adapted from the documentation of the software PRobability Of Fracture (PROF) v3.0. The data has been modifed from the documentation in the following ways: crack growth is indexed to number of flights instead of flight hours; probability of inspection is set to 1.0; and the PROF parameter 'number of similar locations' is not part of this routine (effectively always set equal to 1). For example CP7ext, the crack growth and geometry curves have been extrapolated to a larger final crack size. The objects cp4, cp6, cp7, and cp7ext are of analysis type 'single'; i.e., there is a single damage tolerance analysis (DTA) associated with each. cp7ext.Mult.simple and cp7ext.Mult.complex are of the 'multiple' type; i.e., there are two or three DTAs associated, respectively. Finally, cp7.CD is of the continuing damage variety. Any one of these can be used as a starting point for creating a new probabilistic damage tolerance analysis problem, though it is recommended to use the appropriate type (single, multiple, or continuing damage) due to differences regarding the structure of each list. All parameters are discussed in detail below along with an indication of whether they apply to Sequential Importance Sampling (SIS) or Explicit Monte Carlo (EMC). This information is repeated from crackRinit.

Usage

data(cp4)
data(cp6)
data(cp7)
data(cp7ext)
data(cp7ext.Mult.simple)
data(cp7ext.Mult.complex)
data(cp7.CD)

Format

List (class Sing, Mult, or CD).

Details

• Np – integer – SIS/EMC – number of Monte Carlo samples to utilize

  • For SIS (e.g., using analyze) this is the number of particles in the set. In this case the complete set of particles is carried through the analysis, so Np > several million may use up available memory. If many millions of particles are needed, the runs can be split and re-combined later either by using analyzeParallel or by using the add=TRUE option in analyze.
    

    For EMC run (using crackRmc) this is the number of trials. Detailed results of each trial are not maintained, thus many millions is fine from a memory standpoint. crackRmc is NOT fast, so testing a run with Np around 1000 is recommended (see system.time).

• analysis.type – character – SIS/EMC – the type of probabilistic damage tolerance analysis being performed, be it a single-type at a single crack ('single'), multiple-type at a single crack ('multiple'), or a continuing damage analysis involving two cracks ('CD')

  • Must be either 'single', 'multiple', or 'CD'. For an SIS run, this determines the class of the object created by crackRinit (either Sing, Mult, or CD). For an EMC run, the appropriate method of crackRmc is selected according to this parameter.

• survival.updating – character – SIS – whether to proceed flight-by-flight ('fbf') or in intervals ('int')

  • Must be either 'fbf' or 'int' for an SIS run (specifically, in calcInterval). This parameter is not used in an EMC run (crackRmc).

• flt.calc.interval – integer – SIS/EMC – frequency of output of SFPOF calculations in number of flights for each sub-interval

  • Regardless of the value of survival.updating, SFPOF is calculated and output at the beginning and end of sub-intervals of length flt.calc.interval. The actual sub-interval lengths will often vary slightly since sub-intervals are forced to end at a scheduled inspection. The sub-intervals within an inspection interval are rounded to integers which minimize the variation from flt.calc.interval. A value of 1 may be used, but for long service lives the results object may become large and calculation speeds may suffer.

• sfpof.threshold – numeric – SIS/EMC – if specified, a horizontal line will appear on results plots at this value.

  • This parameter is not currently used for anything important.

• sfpof.min – numeric – SIS/EMC – minimum value of SFPOF to include in results

  • If > 0, SFPOF calculated below sfpof.min will be recorded as sfpof.min. If = 0, SFPOF will be allowed to be zero in results. In this case, when plotting SFPOF on the log scale, warnings will be issued and the plot will not appear at the zero values.

• bootstrap.sfpof – logical – SIS – whether or not to perform obtain bootstrap confidence intervals of SFPOF during a run

  • If bootstrap confidence intervals are desired, set this parameter to TRUE. The number of bootstrap samples to generate at each calculation of SFPOF is set by bootstrap.samples and the desired quantiles to estimate by bootstrap.quantiles. Results will be in additional column(s) with SFPOF in the SFPOF data.frame of the crackRresults object. Note that there are many SFPOF calculations and using a large number of bootstrap samples may significantly increase the run time.

• bootstrap.pcd – logical – SIS – whether or not to perform obtain bootstrap confidence intervals of PCD during a scheduled inspection

  • If bootstrap confidence intervals are desired, set this parameter to TRUE. The number of bootstrap samples to generate at each calculation of PCD is set by bootstrap.samples and the desired quantiles to estimate by bootstrap.quantiles. Results will be in additional column(s) with PCD in the PCD data.frame of the crackRresults object. Inspections occur less frequently than SFPOF calculations, but large values of bootstrap.samples may significantly increase the run time.

• bootstrap.samples – integer – SIS – number of bootstrap samples to take for either SFPOF or PCD

  • More samples = better bootstrap estimates of the quantiles of the sampling distribution(s), with more samples taking longer to run. Note the bootstrap run time is also a function of the number of particles (Np).

• bootstrap.quantiles – numeric – SIS – quantile(s) at which to estimate the sampling distribution of either SFPOF or PCD

  • this is simply the probs argument to the quantile function.

• ms.gumbel – numeric – SIS/EMC – vector of length two specifying the location (loc) and scale parameters for the Gumbel distribution which describes the maximum applied stress per flight

  • For example: cp4$ms.gumbel <- c(loc=31.079, scale=0.832)

  • The names "loc" and "scale" are required!

  • crackR is currently hard-coded to utilize the Gumbel distribution for max stress per flight. Expanding to a generic distribution is not difficult; it just hasn't been done yet. Mostly this is because max stress per flight is almost always Gumbel distributed in the literature, so I haven't bothered.

• inspections – data.frame – SIS/EMC – specification of inspection intervals (in number of flights) and the inspection type to occur at the end of the interval

  • For example: cp4$inspections <- data.frame(flt.interval = c(4615,2308,2308), type = c(1,2,1))

  • The names "flt.interval" and "type" are required!

  • The above example specifies three inspection intervals over the 4615+2308+2308=9231 flights. The three inspections are of types 1, 2, and 1, respectively, which refers to parameter pod.func.

• pod.func – list – SIS/EMC – list of POD functions

  • For example: cp7ext$pod.func <- list(); cp7ext$pod.func[[1]] <- function(a) lognormalPOD(a, 0.03, 1); cp7ext$pod.func[[2]] <- function(a) lognormalPOD(a, 0.01, 1)

  • The above example specifies two POD curves. At inspection time the parameter inspections specifies which of these to use. If the inspection type calls for a POD curve that is not specified here, errors will occur. See lognormalPOD and inspection.

• pod.threshold – numeric – SIS/EMC – either a matrix or a vector specifying the crack sizes at which different severities of repair will occur (see below for details)

  • For a run of analysis.type=="single" or analysis.type=="CD", this is a weakly increasing vector of thresholds at which to partition the PCD results. For example, if = 0, then no partitioning will occur; if = c(0.01, 0.05), then PCD will be split into three: a < 0.01, 0.01 < 0.05, and a > 0.05.

  • For analysis.type=="multiple", there is more than one type of repair that is possible and the behavior of each is different. Because of this, pod.threshold is an upper triangular matrix of dimension c(dta.types, dta.types), where each row is the POD threshold vector for that type. Note each row must be weakly increasing from left to right. It is upper triangular since it is assumed that a repair can only get more severe...unless the part is replaced because the crack grew to exceed the largest threshold value. This is admittedly confusing, so here's an example. Suppose there are two DTA types describing the possible condition at a single fastener hole, where the first represents the as-manufactured condition and the second represents an oversized fastener installation. Suppose pod.threshold==matrix(c(0.005,NA,0.25,0.005), nrow=2, ncol=2). For a type 1 particle, the POD threshold values used are taken from the first row of this matrix: c(0.005, 0.25). If a crack is found smaller than 0.005, it can be repaired to as-manufactured condition. If 0.005 < a < 0.25, the oversized fastener is installed and the type switches to type 2 as a result of the inspection being performed. If a > 0.25, the part is replaced (back to type 1). If instead when heading into inspection the repair was of type 2, then pod.threshold==0.005. If the found crack is smaller than 0.005, it can be repaired back to the oversized fastener condition (but not back to type 1, a smaller fastener). If a > 0.005, the part must be replaced (since the oversized fastener repair has already been utilized and no other repair is possible). Note that zero and Inf values may be used. Also the NA values can optionally be zero as they are not used by the code.

• dta.types – integer – SIS/EMC – number of Damage Tolerance Analysis (DTA) data sets present in the analysis

  • This is used only for analysis.type=="multiple" in the methods corresponding to a crackR object of class Mult.

• ismc.bool – logical – EMC – whether or not to utilize importance sampling to set the initial state in an EMC run

  • If FALSE standard Monte Carlo sampling is used and the initial crack size will be generated using, for example, ifs.ractual.

  • If TRUE importance sampling is used and the initial size will be generated using ifs.rsamp instead. Subsequently ifs.dsamp and ifs.dactual are used to determine the importance weights.

  • See parameter dta for more information regarding specification of initial flaw size distributions

• cg.cc – numeric – SIS/EMC – critical crack length for a run of either analysis.type=="single" or analysis.type=="multiple"

  • The crack length at which failure is assumed to occur with certainty. Note that if cg.cc is larger than the largest values in the DTA data item cg, linear extrapolation will occur until reaching cg.cc. If cg.cc is smaller than the largest values of said table, failure will occur earlier.

  • Generally cg.cc should be set to the last value of crack length in the DTA data item cg.

• cg.cc.pc/cg.cc.ph/cg.cc.sc/cg.cc.sh – numeric – SIS/EMC – critical crack length for a run of analysis.type=="CD"

  • For a continuing damage analysis, there are two cracks involved (primary and secondary), each of which can be in either a 'cold' state or a 'hot' state, depending on whether the other crack has previously reached critical size. Thus there are four sets of DTA data: primary cold (dta.pc), primary hot (dta.ph), secondary cold (dta.sc), and secondary hot (dta.sh). See below for more details.

  • The various critical crack sizes associated with a continuing damage analysis are as follows: cg.cc.pc, the length at which the primary crack goes critical (secondary goes hot); cg.cc.sc, the length at which the secondary crack goes critical (primary goes hot); cg.cc.ph, the length at which a primary hot crack fails; and cg.cc.sh, the length at which a secondary hot crack fails.

• dta – list – SIS/EMC – deterministic damage tolerance analysis input data for analysis.type=="single" or analysis.type=="multiple". There are several components to this list, each discussed in turn below. Note that for analysis.type=="single", this is a list of the following parameters. For analysis.type=="multiple", dta is a "list of lists", where each item of the list (e.g., dta[[1]]) is a complete dta listing representing that type of part condition.

  • cg – data.frame – crack growth data with columns flight and crack; note this differs from some other probabilistic damage tolerance analysis codes which are indexed to flight hours instead of flight numbers. Linear extrapolation will occur using the last two rows of this table if cg.cc is larger than the last crack size in the table.

  • geo – data.frame – geometry data relating crack length to normalized stress intensity K/sigma with columns crack and ksig. The data in this table is not extrapolated; if a K/sigma value is needed for a crack size beyone the end of this table, the final value of K/sigma will be used.

  • kc.rsamp – function – generates values from fracture toughness distribution

  • ifs.rsamp – function – generates values from initial flaw size sampling distribution

  • ifs.dsamp – function – density function for initial flaw size sampling distribution

  • ifs.ractual – function – generates values from initial flaw size actual distribution

  • ifs.dactual – function – density function for initial flaw size actual distribution

  • rfs.rsamp – function – generates values from repair flaw size sampling distribution

  • rfs.dsamp – function – density function for repair flaw size sampling distribution

  • rfs.ractual – function – generates values from repair flaw size actual distribution

  • rfs.dactual – function – density function for repair flaw size actual distribution

• dta.pc/dta.ph/dta.sc/dta.sh – list – SIS/EMC – deterministic damage tolerance analysis input data for a continuing damage run

  • For a continuing damage run there are four distinct sets of DTA data, primary cold (dta.pc), primary hot (dta.ph), secondary cold (dta.sc), and secondary hot (dta.sh). The cracks become hot at cg.cc.pc or cg.cc.sc, and fail at cg.cc.ph or cg.cc.sh.

  • At manufacture and following a repair, cracks are always 'cold'. Hence dta.pc and dta.sc need to have the various distributions for fracture toughness and initial flaw size, but dta.ph and dta.sh don't require them.

Source

PRoF Ver. 3.0 PRobability of Fracture, Software Package, University of Dayton Research Institute, Dayton, OH, 2005

References

Halbert, K. "Estimation of Probability of Failure for Damage-Tolerant Aerospace Structures" PhD Thesis, Temple University Department of Statistics, Philadelphia, PA, Apr 2014

Examples

data(cp7)
head(cp7)

crackR documentation built on May 29, 2017, 8:45 p.m.