Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

42 CHAPTER 1 Basic Elasticity

σx(N/mm^2 ) σy(N/mm^2 ) τxy(N/mm^2 )

(i) + 54 + 30 + 5

(ii) + 30 + 54 − 5

(iii) − 60 − 36 + 5

(iv) + 30 − 50 + 30

Ans. (i) σI=+55N/mm^2 σII=+29N/mm^2 σIat11.5◦toxaxis.

(ii) σI=+55N/mm^2 σII=+29N/mm^2 σIIat11.5◦toxaxis.

(iii) σI=−34.5N/mm^2 σII=−61N/mm^2 σIat79.5◦toxaxis.

(iv) σI=+40N/mm^2 σII=−60N/mm^2 σIat18.5◦toxaxis.

Fig. P.1.4

P.1.4 Thestateofstressatapointiscausedbythreeseparateactions,eachofwhichproducesapure,unidirec-
tionaltensionof10N/mm^2 individuallybutinthreedifferentdirections,asshowninFig.P.1.4.Bytransforming
the individual stresses to a common set of axes (x,y), determine the principal stresses at the point and their
directions.

Ans. σI=σII=15N/mm^2 .Alldirectionsareprincipaldirections.

P.1.5 Ashearstressτxyactsinatwo-dimensionalfieldinwhichthemaximumallowableshearstressisdenoted
byτmaxandthemajorprincipalstressbyσI.
Derive, using the geometry of Mohr’s circle of stress, expressions for the maximum values of direct stress
whichmaybeappliedtothexandyplanesintermsofthethreeparametersjustgiven.

Ans. σx=σI−τmax+

√

τmax^2 −τxy^2

σy=σI−τmax−

√

τmax^2 −τxy^2.

P.1.6 Asolidshaftofcircularcrosssectionsupportsatorqueof50kNmandabendingmomentof25kNm.If
thediameteroftheshaftis150mm,calculatethevaluesoftheprincipalstressesandtheirdirectionsatapointon
thesurfaceoftheshaft.

Ans. σI=121.4N/mm^2 θ= 31 ◦ 43 ′

σII=−46.4N/mm^2 θ= 121 ◦ 43 ′.