Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

14 CHAPTER 1 Basic Elasticity

Fig.1.11

Stress system on two-dimensional element of the beam of Example 1.2.

Thestresssystemactingonatwo-dimensionalrectangularelementatthepointisshowninFig.1.11.
Notethatsincetheelementispositionedatthebottomofthebeam,theshearstressduetothetorqueis
inthedirectionshownandisnegative(seeFig.1.8).
AgainσnandτmaybefoundfromfirstprinciplesorbydirectsubstitutioninEqs.(1.8)and(1.9).
Notethatθ= 30 ◦,σy=0,andτxy=−28.3N/mm^2 ,thenegativesignisarisingfromthefactthatitisin
theoppositedirectiontoτxyasinFig.1.8.
Then,

σn=−21.2cos^230 ◦−28.3sin60◦=−40.4N/mm^2 (compression)

τ=(−21.2/ 2 )sin60◦+28.3cos60◦=5.0N/mm^2 (actinginthedirectionAB)

DifferentanswerswouldhavebeenobtainediftheplaneABhadbeenchosenontheoppositeside
ofAC.

1.7 PrincipalStresses....................................................................................

Forgivenvaluesofσx,σy,andτxy,inotherwordsgivenloadingconditions,σnvarieswiththeangleθ
andattainsamaximumorminimumvaluewhendσn/dθ=0.FromEq.(1.8),

dσn

dθ

=− 2 σxcosθsinθ+ 2 σysinθcosθ+ 2 τxycos2θ= 0

Hence,

−(σx−σy)sin2θ+ 2 τxycos2θ= 0

or

tan2θ=

2 τxy

σx−σy

(1.10)