Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

2.3 Inverse and Semi-Inverse Methods 51

Example 2.3
AcantileveroflengthLanddepth2hisinastateofplanestress.Thecantileverisofunitthickness,is
rigidlysupportedattheendx=L,andisloadedasshowninFig.2.3.Showthatthestressfunction

φ=Ax^2 +Bx^2 y+Cy^3 +D( 5 x^2 y^3 −y^5 )

isvalidforthebeamandevaluatetheconstantsA,B,C,andD.

ThestressfunctionmustsatisfyEq.(2.9).Fromtheexpressionforφ

∂φ

∂x

= 2 Ax+ 2 Bxy+ 10 Dxy^3

∂^2 φ

∂x^2

= 2 A+ 2 By+ 10 Dy^3 =σy

(i)

Also,

∂φ

∂y

=Bx^2 + 3 Cy^2 + 15 Dx^2 y^2 − 5 Dy^4

∂^2 φ

∂y^2

= 6 Cy+ 30 Dx^2 y− 20 Dy^3 =σx

(ii)

and

∂^2 φ

∂x∂y

= 2 Bx+ 30 Dxy^2 =−τxy (iii)

Fig.2.3

Beam of Example 2.3.