Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 63

Ans. Thestressfunctionsatisfiesthebiharmonicequation.Thebeamisacantileverunderauniformlydis-
tributedloadofintensityw/unitareawithaself-equilibratingstressapplicationgivenbyσx=w(12c^3 y− 20 y^3 )/40bc^3
atx=0.Thereiszeroshearstressaty=±candx=0.Aty=+c,σy=−w/b,andaty=−c,σy=0.

P.2.6 Atwo-dimensionalisotropicsheet,havingaYoung’smodulusEandlinearcoefficientofexpansionα,is
heatednonuniformly,thetemperaturebeingT(x,y).ShowthattheAirystressfunctionφsatisfiesthedifferential
equation

∇^2 (∇^2 φ+EαT)= 0

where

∇^2 =

∂^2

∂x^2

+

∂^2

∂y^2

istheLaplaceoperator.

P.2.7 InvestigatethestateofplanestressdescribedbythefollowingAirystressfunction

φ=

3 Qxy

4 a

−

Qxy^3

4 a^3

overthesquareregionx=−atox=+a,y=−atoy=+a.Calculatethestressresultantsperunitthicknessover
eachboundaryoftheregion.

Ans. Thestressfunctionsatisfiesthebiharmonicequation.Also,

whenx=a, σx=
− 3 Qy
2 a^2

whenx=−a, σx=
3 Qy
2 a^2
and

τxy=

− 3 Q

4 a

(

1 −

y^2

a^2

)

.