6.8 Finite Element Method for Continuum Structures 1936.8 FiniteElementMethodforContinuumStructures...............................................
Intheprevioussections,wehavediscussedthematrixmethodofsolutionofstructurescomposedof
elementsconnectedonlyatnodalpoints.Forskeletalstructuresconsistingofarrangementsofbeams,
thesenodalpointsfallnaturallyatjointsandatpositionsofconcentratedloading.Continuumstructures,
suchasflatplates,aircraftskins,shells,andsoon,donotpossesssuchnaturalsubdivisionsandmust
thereforebeartificiallyidealizedintoanumberofelementsbeforematrixmethodscanbeused.These
finiteelements,astheyareknown,maybetwo-orthree-dimensional,butthemostcommonlyusedare
two-dimensionaltriangularandquadrilateralshapedelements.Theidealizationmaybecarriedoutin
any number of different ways depending on such factors as the type of problem, the accuracy of the
solution required, and the time and money available. For example, acoarseidealization involving a
small number of largeelements would provideacomparatively rapid but very approximatesolution,
whileafineidealizationofsmallelementswouldproducemoreaccurateresultsbutwouldtakelonger
andconsequentlycostmore.Frequently,gradedmeshesareusedinwhichsmallelementsareplaced
in regions where high stress concentrations are expected—for example, around cut-outs and loading
points.TheprincipleisillustratedinFig.6.12whereagradedsystemoftriangularelementsisusedto
examinethestressconcentrationaroundacircularholeinaflatplate.
Althoughtheelementsareconnectedataninfinitenumberofpointsaroundtheirboundaries,itis
assumedthattheyareonlyinterconnectedattheircornersornodes.Thus,compatibilityofdisplacement
is only ensured at the nodal points. However, in the finite element method, a displacement pattern is
chosenforeachelementwhichmaysatisfysome,ifnotall,ofthecompatibilityrequirementsalongthe
sidesofadjacentelements.
Sinceweareusingmatrixmethodsofsolution,weareconcernedinitiallywiththedetermination
ofnodalforcesanddisplacements.Thus,thesystemofloadsonthestructuremustbereplacedbyan
equivalentsystemofnodalforces.Wheretheseloadsareconcentrated,theelementsarechosensuch
that a node occurs at the point of application of the load. In the case of distributed loads, equivalent
nodalconcentratedloadsmustbecalculated[Ref.4].
Fig.6.12
Finite element idealization of a flat plate with a central hole.