Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 41

Fromthetheoryofthetorsionofcircularsectionbars(seeEq.(iv)inExample3.1),

τxy=29.7N/mm^2 =

Tr

J

=

T× 25

π× 504 / 32

fromwhich

T=0.7kNm

NotethatPcouldhavebeenfounddirectlyinthisparticularcasefromtheaxialstrain.Thus,from
thefirstofEqs.(1.52),

σx=Eεa= 70000 × 1000 × 10 −^6 =70N/mm^2

asbefore.

References

[1] Timoshenko,S.,andGoodier,J.N.,TheoryofElasticity,2ndedition,McGraw-Hill,1951.
[2] Wang,C.T.,AppliedElasticity,McGraw-Hill,1953.
[3] Megson,T.H.G.,StructuralandStressAnalysis,2ndedition,Elsevier,2005.

Problems..............................................................................................

P.1.1 Astructuralmembersupportsloadsthatproduce,ataparticularpoint,adirecttensilestressof80N/mm^2
andashearstressof45N/mm^2 onthesameplane.Calculatethevaluesanddirectionsoftheprincipalstressesat
thepointandalsothemaximumshearstress,statingonwhichplanesthiswillact.

Ans. σI=100.2N/mm^2 θ= 24 ◦ 11 ′

σII=−20.2N/mm^2 θ= 114 ◦ 11 ′

τmax=60.2N/mm^2 at45◦toprincipalplanes.

P.1.2 Atapointinanelasticmaterial,therearetwomutuallyperpendicularplanes,oneofwhichcarriesadirect
tensilestressof50N/mm^2 andashearstressof40N/mm^2 ,whiletheotherplaneissubjectedtoadirectcompressive
stressof35N/mm^2 andacomplementaryshearstressof40N/mm^2 .Determinetheprincipalstressesatthepoint,
thepositionoftheplanesonwhichtheyact,andthepositionoftheplanesonwhichthereisnonormalstress.

Ans. σI=65.9N/mm^2 θ= 21 ◦ 38 ′

σII=−50.9N/mm^2 θ= 111 ◦ 38 ′

Nonormalstressonplanesat70◦ 21 ′and− 27 ◦ 5 ′tovertical.

P.1.3 Listedherearevaryingcombinationsofstressesactingatapointandreferredtoaxesxandyinanelastic
material.UsingMohr’scircleofstress,determinetheprincipalstressesatthepointandtheirdirectionsforeach
combination.