CHAPTER 2 Two-Dimensional Problems in Elasticity..............................................
Theoretically, we are now in a position to solve any three-dimensional problem in elasticity, having
derived three equilibrium conditions, Eqs. (1.5); six strain-displacement equations, Eqs. (1.18) and
(1.20);andsixstress–strainrelationships,Eqs.(1.42)and(1.46).Theseequationsaresufficient,when
supplemented by appropriate boundary conditions, to obtain unique solutions for the six stress, six
strain,andthreedisplacementfunctions.Ithasbeenfound,however,thatexactsolutionsareobtainable
onlyforsomesimpleproblems.Forbodiesofarbitraryshapeandloading,approximatesolutionsmaybe
foundbynumericalmethods(e.g.,finitedifferences)orbytheRayleigh–Ritzmethodbasedonenergy
principles(Chapter7).
Twoapproachesarepossibleinthesolutionofelasticityproblems.Wemaysolveinitiallyeitherfor
the three unknown displacements or for the six unknown stresses. In the former method the equilib-
riumequationsarewrittenintermsofstrainbyexpressingthesixstressesasfunctionsofstrain(see
Problem P.1.7). The strain–displacement relationships are then used to form three equations involv-
ing the three displacementsu,v,andw. The boundary conditions for this method of solution must
be specified as displacements. Determination ofu,v,andwenables the six strains to be computed
fromEqs.(1.18)and(1.20);thesixunknownstressesfollowfromtheequations,expressingstressas
functionsofstrain.Itshouldbenotedherethatnousehasbeenmadeofthecompatibilityequations.
Thefactthatu,v,andwaredetermineddirectlyensuresthattheyaresingle-valuedfunctions,thereby
satisfyingtherequirementofcompatibility.
Inmoststructuralproblems,theobjectisusuallytofindthedistributionofstressinanelasticbody
producedbyanexternalloadingsystem.Itis,therefore,moreconvenientinthiscasetodeterminethe
six stresses before calculating any required strains or displacements. This is accomplished by using
Eqs. (1.42) and (1.46) to rewrite the six equations of compatibility in terms of stress. The resulting
equations,inturn,aresimplifiedbymakinguseofthestressrelationshipsdevelopedintheequations
ofequilibrium.Thesolutionoftheseequationsautomaticallysatisfiestheconditionsofcompatibility
andequilibriumthroughoutthebody.
2.1 Two-DimensionalProblems........................................................................
For the reasons discussed in Chapter 1 we shall confine our actual analysis to the two-dimensional
casesofplanestressandplanestrain.Theappropriateequilibriumconditionsforplanestressaregiven
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00002-6 45