Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

2.3 Inverse and Semi-Inverse Methods 49

Fig.2.1

Required loading conditions on rectangular sheet in Example 2.1.

Example 2.2
Amorecomplexpolynomialforthestressfunctionis

φ=

Ax^3

6

+

Bx^2 y

2

+

Cxy^2

2

+

Dy^3

6

Asbefore

∂^4 φ

∂x^4

=

∂^4 φ

∂x^2 ∂y^2

=

∂^4 φ

∂y^4

= 0

sothatthecompatibilityequation(2.9)isidenticallysatisfied.Thestressesaregivenby

σx=

∂^2 φ

∂y^2

=Cx+Dy

σy=

∂^2 φ

∂x^2

=Ax+By

τxy=−

∂^2 φ

∂x∂y

=−Bx−Cy

WemaychooseanynumberofvaluesforthecoefficientsA,B,C,andDtoproduceavarietyofloading
conditionsonarectangularplate.Forexample,ifweassumeA=B=C=0,thenσx=Dy,σy=0,and
τxy=0,sothatforaxesreferredtoanoriginatthemid-pointofaverticalsideoftheplate,weobtain