Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.6 Determination of Stresses on Inclined Planes 11

Fig.1.

(a) Stresses on a two-dimensional element; (b) stresses on an inclined plane at the point.

stateofstressonotherplanesonwhichthedirectandshearstressesmaybegreater.Weshallrestrict
theanalysistothetwo-dimensionalsystemofplanestressdefinedinSection1.4.
Figure 1.8(a) shows a complex stress system at a point in a body referred to axes Ox,Oy.All
stressesarepositiveasdefinedinSection1.2.Theshearstressesτxyandτyxwereshowntobeequalin
Section1.3.Wenow,therefore,designatethembothτxy.Theelementofsideδx,δyandofunitthickness
issmall,sostressdistributionsoverthesidesoftheelementmaybeassumedtobeuniform.Bodyforces
areignored,sincetheircontributionisasecond-orderterm.
SupposethatwewanttofindthestateofstressonaplaneABinclinedatanangleθtothevertical.
ThetriangularelementEDCformedbytheplaneandtheverticalthroughEisinequilibriumunderthe
actionoftheforcescorrespondingtothestressesshowninFig.1.8(b),whereσnandτarethedirect
andshearcomponentsoftheresultantstressonAB.Then,resolvingforcesinadirectionperpendicular
toED,wehave

σnED=σxECcosθ+σyCDsinθ+τxyECsinθ+τxyCDcosθ

DividingbyEDandsimplifying

σn=σxcos^2 θ+σysin^2 θ+τxysin2θ (1.8)

NowresolvingforcesparalleltoED,

τED=σxECsinθ−σyCDcosθ−τxyECcosθ+τxyCDsinθ

AgaindividingbyEDandsimplifying,

τ=

(σx−σy)

2

sin2θ−τxycos2θ (1.9)