Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.1Prandtl Stress Function Solution 67

where∇^2 isthetwo-dimensionalLaplacianoperator

(

∂^2

∂x^2

+

∂^2

∂y^2

)

Therefore,theparameter∇^2 φisconstantatanysectionofthebarsothatthefunctionφmustsatisfy
theequation

∂^2 φ

∂x^2

+

∂^2 φ

∂y^2

=constant=F(say) (3.4)

atallpointswithinthebar.
Finally, we must ensure thatφfulfills the boundary conditions specified by Eqs. (1.7). On the
cylindricalsurfaceofthebar,therearenoexternallyappliedforcessothatX=Y=Z=0.Thedirection
cosinenisalsozero,andthereforethefirsttwoequationsofEqs.(1.7)areidenticallysatisfied,leaving
thethirdequationastheboundarycondition;thatis,

τyzm+τxzl= 0 (3.5)

ThedirectioncosineslandmofthenormalNtoanypointonthesurfaceofthebarare,byreference
toFig.3.2,

l=

dy

ds

m=−

dx

ds

(3.6)

SubstitutingEqs.(3.2)and(3.6)intoEq.(3.5),wehave

∂φ

∂x

dx

ds

+

∂φ

∂y

dy

ds

= 0

or

∂φ

ds

= 0

Fig.3.2

Formation of the direction cosineslandmof the normal to the surface of the bar.