60 CHAPTER 2 Two-Dimensional Problems in Elasticity
Fig.2.8
(a) Distortion of cross section due to shear; (b) effect on distortion of rotation due to shear.
Therefore, if allowed, the cross section would take the shape of the shallow reversedSshown in
Fig. 2.8(a). Eq. (xv) does not include the previously discussed effect of rotation of the neutral plane
causedbyshear.However,itmerelyrotatesthebeamsectionasindicatedinFig.2.8(b).
Thedistortionofthecrosssectionisproducedbythevariationofshearstressoverthedepthofthe
beam.Thus,thebasicassumptionofsimplebeamtheorythatplanesectionsremainplaneisnotvalid
whenshearloadsarepresent,althoughforlong,slenderbeamsthebendingstressesaremuchgreater
thanshearstressesandtheeffectmaybeignored.
ItwillbeobservedfromFig.2.8thatanadditionaldirectstresssystemwillbeimposedonthebeam
atthesupportwherethesectionisconstrainedtoremainplane.Formostengineeringstructures,this
effectissmallbut,asmentionedpreviously,maybesignificantinthin-walledsections.
Reference
[1] Timoshenko,S.,andGoodier,J.N.,TheoryofElasticity,2ndedition,McGraw-Hill,1951.
Problems..............................................................................................
P.2.1 AmetalplatehasrectangularaxesOx,Oymarkedonitssurface.ThepointOandthedirectionofOxare
fixedinspaceandtheplateissubjectedtothefollowinguniformstresses:
compressive,3p,paralleltoOx
tensile,2p,paralleltoOy
shearing,4p,inplanesparalleltoOxandOy
inasensetendingtodecreasetheanglexOy
Determinethedirectioninwhichacertainpointontheplatewillbedisplaced;thecoordinatesofthepointare
(2,3)beforestraining.Poisson’sratiois0.25.
Ans. 19.73◦toOx.P.2.2 What do you understand by an Airy stress function in two dimensions? A beam of lengthl, with a thin
rectangularcrosssection,isbuilt-inattheendx=0andloadedatthetipbyaverticalforceP(Fig.P.2.2).Show
thatthestressdistribution,ascalculatedbysimplebeamtheory,canberepresentedbytheexpression
φ=Ay^3 +By^3 x+Cyx