ABSTRACT
InauniformlystressedtensilespecimenofvolumeVtheprobabilityof failureatastressuorlessisgivenby
P(u,V)=1-exp[-pV(uju0)m]
andtheprobabilitydensityfunctionisgivenby
dP(u)mpV(0")m-1mp(u,V)=-d-=- - exp[-pV(uju0)] uuouo
Themeanstrengthofthetensilespecimenisgivenby
jj=Joeupdu=O"o1/r(1+1/m) 0(pV)m
(6.5)
(6.6)
(6.7)
Thereislittledifferencebetweenthemeanstrengthandthemedianstrength, O"m,givenby
( 0.693) 1/m
O"m=O"opV(6.8)
form>10.Thecoefficientofvariationofthestrengthoftheuniformtensile specimensfLisgivenapproximatelyby
fL=1.283/m(6.9)
form;:::10(Coleman,1958). Underthree-pointbendingthestress,inabeamofrectangularcross-
sectionwithaspanSanddepthW,isgivenbytheengineers'theoryof bendingas
(6.10)
whereO"maxisthemaximumstressattheloadpointand(x,y)arethecoordinatesmeasuredfromthecentreofoneendofthebeam.Ifitisassumedthat failureonlyoccursonthetensionsideofthespecimen,theprobabilityof failureatamaximumstressofO"maxorlessisfoundbyintegrationofeqn 6.4tobegivenby
P(umax•V)=1-exp[- pV2(umax/uo)m](6.11)2(1+m)
Equation6.11canberefinedbyincludingtheSeewald-Karmancorrection forthestressneartheloadpoint(DiazandKittl,1988).Ifthemedian strengthofathree-pointbendspecimeniscomparedtothatforatensile specimenofthesamevolumetheratioinstrengthsisgivenby
O'bend/O'tensile=[2(1+m) 2]1fm(6.12) Thisratioispredictedtobeindependentofthesizeofthespecimen.Other assumptionsofriskofrupturedoshowadecreaseintheratiowithan increaseinthesizeasdoesthetheoryfortwo-phasebrittlematerials(Hu eta/.,1985).However,themajorcontributiontosizeeffectincementitious beamscomesfromthecrackgrowthresistanceofthematerialratherthan fromastatisticaldistributionofthestrength(seesection4.6).Thereisno deterministicsizeeffectundertensionincementitiousmaterialssinceonce thematerialreachesitsmaximumstrength,attheonsetofstrain-softening, atensilespecimenmustfail.Hence,foradeterministiccementitiousmaterial theratioofbendingtotensilestrengthmustdecreaseasthesizeincreases, tendingtounityforverylargebeams.
