Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.7Stiffness Matrix for a Uniform Beam 189

Fig.6.7

Idealization of a beam into beam elements.

Fig.6.8

Idealization of a beam supporting a uniformly distributed load.

Fig.6.9

Idealization of beams into beam elements.

Beamssupportingdistributedloadsrequirespecialtreatmentinthatthedistributedloadisreplaced
byaseriesofstaticallyequivalentpointloadsataselectednumberofnodes.Clearly,thegreaterthe
numberofnodeschosen,themoreaccuratebutmorecomplicatedandthereforetimeconsumingwill
betheanalysis.Figure6.8showsatypicalidealizationofabeamsupportingauniformlydistributed
load.DetailsoftheanalysisofsuchbeamsmaybefoundinMartin[Ref.4].
Manysimplebeamproblemsmaybeidealizedintoacombinationoftwobeamelementsandthree
nodes.AfewexamplesofsuchbeamsareshowninFig.6.9.Ifwethereforeassembleastiffnessmatrix
forthegeneralcaseofatwobeamelementsystem,wemayuseittosolveavarietyofproblemssimply
by inserting the appropriate loading and support conditions. Consider the assemblage of two beam
elementsshowninFig.6.10.Thestiffnessmatricesforthebeamelements1–2and2–3areobtained