Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.8Mohr’s Circle of Stress 19

to200N/mm^2 (tension).Calculatetheallowablevalueofshearstressatthepointonthegivenplanes.
Determinealsothevalueoftheotherprincipalstressandthemaximumvalueofshearstressatthepoint.
VerifyyouranswerusingMohr’scircle.

ThestresssystematthepointinthematerialmayberepresentedasshowninFig.1.13byconsidering
the stresses to act uniformly over the sides of a triangular element ABC of unit thickness. Suppose
thatthedirectstressontheprincipalplaneABisσ.Forhorizontalequilibriumoftheelement,

σABcosθ=σxBC+τxyAC

whichsimplifiesto

τxytanθ=σ−σx (i)

Consideringverticalequilibriumgives

σABsinθ=σyAC+τxyBC

or

τxycotθ=σ−σy (ii)

Hence,fromtheproductofEqs.(i)and(ii),

τxy^2 =(σ−σx)(σ−σy)

Nowsubstitutingthevaluesσx=160N/mm^2 ,σy=−120N/mm^2 ,andσ=σ 1 =200N/mm^2 ,wehave

τxy=±113N/mm^2

ReplacingcotθinEq.(ii)by1/tanθfromEq.(i)yieldsaquadraticequationinσ

σ^2 −σ(σx−σy)+σxσy−τxy^2 =0(iii)

Fig.1.13

Stress system for Example 1.3.