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12
T.Derda
inloadscarriedbyindividualcomponents.Beforetheloadingprocessthesystem
isintacti.e.allthepillarsareworking.
Inthefirststepofthequasi-staticloading,theexternalloadFisappliedonthe
systemjusttodestroytheweakestcomponentofthesystem.Afterthedestruction
oftheweakestpillaritsloadissharedbyremainingcomponents,accordingto
agivenloadtransferrule.Aftertheloadtransfer,thesubsequentfailuresmayoccur
followedbynextloadtransfers.Iftheloadtransferdoesnotprovokeanyfailure
astablestateisachieved.InthissituationtheexternalloadFisincreasedby
anamountδFjusttodestroyonecomponentandtheabovedescribedprocedureof
loadtransfersisrepeated.TheprocedureofincreasingloadFiscontinueduntilall
thepillarsaredestroyed.Theprocessofloadingisinstantaneous,andthefailure
developsinaformofavalanchesofconcurrentlydestroyedcomponents.
Indistinctiontothegradualloadingcase,theprocedureofsuddenloadingis
realisedbyapplicationoffiniteexternalloadF,thatiskeptconstantthroughthe
entireprocess.TheloadFisapplieduniformlyontheinitiallyunloadedsystem,
consequentlytheloadperpillarinitiallyis
σ±
FN
/.
Hence,allthecomponents
withstrength-thresholdssmallerthanσareinstantlydestroyed.Then,theload
redistributionsmaycausenextfailures.Thesuddenloadingprocedureleadstoone
ofthethreestablestates.Assumingthattheinitialbreakingsoccur,thesystemwill
bepartiallyorfullydestroyed.Thethirdpossiblestablestateoccurswhenthe
initialloadσisnotgreaterthanstrength-thresholdoftheweakestcomponent.
3.Resultsofcomputersimulations
Basedonthemodeldescribedintheprevioussection,wehavedeveloped
computercodesinWolframMathematicatosimulatefailureprocessesinthepillar
arrayswithFréchetdistributedstrength-thresholds.Usingthesecodes,wehave
performedintensivenumericalsimulations.Ouranalysisisrestrictedtothecases
withfinitemeanandfinitevarianceofthestrength-thresholddistributions(
m>
2).
Withoutlossofgenerality,thescaleparameterYisassumedtobe1.
3.1.Quasi-staticloading
Inthecaseofquasi-staticloading,thefailureprocessproceedsinanavalanche-
likemanner.EachincreaseofexternalloadFisfollowedbyavalancheofdestroyed
pillars.Thelast,catastrophicavalancheisinitiatedbyexceedingoftotalcritical
load
FThus,thevalueof
c
.
Fcanbeinterpretedasthemaximalloadsupportedby
c
thesystem.Inordertocomparetheresultsfordifferentsystemsizes,itisreasona-
bletoscalevalueof
Fbythesystemsize,thusgivingcriticalloadperpillar
c
σ±
c
FN
c
/.
FortheGLSsystems,thevaluesofcriticalloadsinthecaseofuniformand
Weibulldistributionsareknownfromtheliterature[12-14].Criticalloadof
