Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 43

P.1.7 Anelementofanelasticbodyissubjectedtoathree-dimensionalstresssystemσx,σy,andσz.Showthat
ifthedirectstrainsinthedirectionsx,y,andzareεx,εy,andεz,then

σx=λe+ 2 Gεx σy=λe+ 2 Gεy σz=λe+ 2 Gεz

where

λ=

νE

( 1 +ν)( 1 − 2 ν)

and e=εx+εy+εz

thevolumetricstrain.

P.1.8 Showthatthecompatibilityequationforthecaseofplanestrainsuchthat

∂^2 γxy

∂x∂y

=

∂^2 εy

∂x^2

+

∂^2 εx

∂y^2

maybeexpressedintermsofdirectstressesσxandσyintheform

(

∂^2

∂x^2

+

∂^2

∂y^2

)

(σx+σy)= 0

P.1.9 Abarofmildsteelhasadiameterof75mmandisplacedinsideahollowaluminumcylinderofinternal
diameter75mmandexternaldiameter100mm;bothbarandcylinderarethesamelength.Theresultingcomposite
bar is subjected to an axial compressive load of 1000kN. If the bar and cylinder contract by the same amount,
calculatethestressineach.
Thetemperatureofthecompressedcompositebaristhenreducedby150◦C,butnochangeinlengthispermitted.
CalculatethefinalstressinthebarandinthecylinderifE(steel)=200000N/mm^2 ,E(aluminum)=80000N/mm^2 ,
α(steel)=0.000012/◦C,andα(aluminum)=0.000005/◦C.

Ans. Duetoload:σ(steel)=172.6N/mm^2 (compression)

σ(aluminum)=69.1N/mm^2 (compression).

Finalstress:σ(steel)=187.4N/mm^2 (tension)

σ(aluminum)=9.1N/mm^2 (compression).

P.1.10 InFig.P.1.10,thedirectstrainsinthedirectionsa,b,care−0.002,−0.002,and+0.002,respectively.IfI
andIIdenoteprincipaldirections,findεI,εII,andθ.

Ans. εI=+0.00283 εII=−0.00283 θ=−22.5◦or+67.5◦.

Fig. P.1.10