48 CHAPTER 2 Two-Dimensional Problems in Elasticity
TheEnglishmathematicianAiryproposedastressfunctionφdefinedbytheequationsσx=∂^2 φ
∂y^2σy=∂^2 φ
∂x^2τxy=−∂^2 φ
∂x∂y(2.8)
Clearly,substitutionofEqs.(2.8)intoEqs.(2.6)verifiesthattheequationsofequilibriumaresatisfied
by this particular stress–stress function relationship. Further substitution into Eq. (2.7) restricts the
possibleformsofthestressfunctiontothosesatisfyingthebiharmonicequation
∂^4 φ
∂x^4+ 2
∂^4 φ
∂x^2 ∂y^2+
∂^4 φ
∂y^4= 0 (2.9)
The final form of the stress function is then determined by the boundary conditions relating to the
actualproblem.Therefore,atwo-dimensionalprobleminelasticitywithzerobodyforcesreducestothe
determinationofafunctionφofxandy,whichsatisfiesEq.(2.9)atallpointsinthebodyandEqs.(1.7)
reducedtotwodimensionsatallpointsontheboundaryofthebody.
2.3 InverseandSemi-InverseMethods................................................................
The task of finding a stress function satisfying the preceding conditions is extremely difficult in the
majorityofelasticityproblems,althoughsomeimportantclassicalsolutionshavebeenobtainedinthis
way. An alternative approach, known as theinverse method, is to specify a form of the functionφ
satisfyingEq.(2.9),assumeanarbitraryboundary,andthentodeterminetheloadingconditionswhich
fittheassumedstressfunctionandchosenboundary.Obvioussolutionsariseinwhichφisexpressedas
apolynomial.TimoshenkoandGoodier[Ref.1]consideravarietyofpolynomialsforφanddetermine
the associated loading conditions for a variety of rectangular sheets. Some of these cases are quoted
here.
Example 2.1
Considerthestressfunction
φ=Ax^2 +Bxy+Cy^2whereA,B,andCareconstants.Equation(2.9)isidenticallysatisfied,sinceeachtermbecomeszero
onsubstitutingforφ.Thestressesfollowfrom
σx=∂^2 φ
∂y^2= 2 C
σy=∂^2 φ
∂x^2= 2 A
τxy=−∂^2 φ
∂x∂y=−B
Toproducethesestressesatanypointinarectangularsheet,werequireloadingconditionsproviding
theboundarystressesshowninFig.2.1.