Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

210 CHAPTER 6 Matrix Methods

Fig.6.15

Tetrahedron and rectangular prism finite elements for three-dimensional problems.

thatis,

τxy=−115.4N/mm^2

Theapplicationofthefiniteelementmethodtothree-dimensionalsolidbodiesisastraightforward
extensionoftheanalysisoftwo-dimensionalstructures.Thebasicthree-dimensionalelementsarethe
tetrahedronandtherectangularprism,bothshowninFig.6.15.Thetetrahedronhasfournodeseach
possessing three degrees of freedom, a total of 12 for the element, while the prism has 8 nodes and
thereforeatotalof24degreesoffreedom.Displacementfunctionsforeachelementrequirepolynomials
inx,y,andz; for the tetrahedron, the displacement function is of the first degree with 12 constant
coefficients, while that for the prism may be of a higher order to accommodate the 24 degrees of
freedom.
Adevelopmentinthesolutionofthree-dimensionalproblemshasbeentheintroductionofcurvilinear
coordinates.Thisenablesthetetrahedronandprismtobedistortedintoarbitraryshapesthatarebetter
suitedforfittingactualboundaries.Formoredetaileddiscussionsofthefiniteelementmethod,reference
shouldbemadetotheworkofJenkins[Ref.5],ZienkiewiczandCheung[Ref.6],andthemanyresearch
paperspublishedonthemethod.
Newelementsandnewapplicationsofthefiniteelementmethodarestillbeingdeveloped,someof
whichlieoutsidethefieldofstructuralanalysis.Thesefieldsincludesoilmechanics,heattransfer,fluid
andseepageflow,magnetism,andelectricity.

References

[1] Argyris,J.H.,andKelsey,S.,EnergyTheoremsandStructuralAnalysis,ButterworthScientificPublications,
1960.
[2] Clough,R.W.,Turner,M.J.,Martin,H.C.,andTopp,L.J.,Stiffnessanddeflectionanalysisofcomplexstructures,
J.Aero.Sciences, 23 (9),1956.
[3] Megson,T.H.G.,StructuralandStressAnalysis,2ndedition,Elsevier,2005.
[4] Martin,H.C.,IntroductiontoMatrixMethodsofStructuralAnalysis,McGraw-Hill,1966.