302 CHAPTER 9 Thin Plates
9.6 FailureStressinPlatesandStiffenedPanels......................................................
Thepreviousdiscussiononplatesandstiffenedpanelsinvestigatedthepredictionofbucklingstresses.
However,aswehaveseen,platesretainsomeoftheircapacitytocarryloadeventhoughaportionof
theplatehasbuckled.Infact,theultimateloadisnotreacheduntilthestressinthemajorityoftheplate
exceedstheelasticlimit.Thetheoreticalcalculationoftheultimatestressisdifficult,sincenonlinearity
resultsfrombothlargedeflectionsandtheinelasticstress–strainrelationship.
Gerard[Ref.1]proposesasemiempiricalsolutionforflatplatessupportedonallfouredges.After
elasticbucklingoccurs,theoryandexperimentindicatethattheaveragecompressivestress,σ ̄a,inthe
plateandtheunloadededgestress,σe,arerelatedbythefollowingexpression:
σ ̄a
σCR=α 1(
σe
σCR)n
(9.8)where
σCR=kπ^2 E
12 ( 1 −ν^2 )(
t
b) 2
andα 1 is some unknown constant. Theoretical work by Stowell [Ref. 7] and Mayers and Budiansky
[Ref.8]showsthatfailureoccurswhenthestressalongtheunloadededgeisapproximatelyequalto
the compressive yield strength,σcy, of the material. Hence, substitutingσcyforσein Eq. (9.8) and
rearranginggive
σ ̄f
σcy=α 1(
σCR
σcy) 1 −n
(9.9)wheretheaveragecompressivestressintheplatehasbecometheaveragestressatfailureσ ̄f.Substituting
forσCRinEq.(9.9)andputting
α 1 π^2 (^1 −n)
[12( 1 −ν^2 )]^1 −n=αyield
σ ̄f
σcy=αk^1 −n[
t
b(
E
σcy)^12 ]^2 (^1 −n)
(9.10)or,inasimplifiedform,
σ ̄f
σcy=β[
t
b(
E
σcy)^12 ]m
(9.11)whereβ=αkm/^2 .TheconstantsβandmaredeterminedbythebestfitofEq.(9.11)totestdata.
Experimentsonsimplysupportedflatplatesandsquaretubesofvariousaluminumandmagnesium
alloys and steel show thatβ=1.42 andm=0.85fit the results within±10 percent up to the yield
strength.Correspondingvaluesforlong,clamped,flatplatesareβ=1.80,m=0.85.