Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

12 CHAPTER 1 Basic Elasticity

Example 1.1
Acylindricalpressurevesselhasaninternaldiameterof2mandisfabricatedfromplates20mmthick.
Ifthepressureinsidethevesselis1.5N/mm^2 and,inaddition,thevesselissubjectedtoanaxialtensile
loadof2500kN,calculatethedirectandshearstressesonaplaneinclinedatanangleof60◦totheaxis
ofthevessel.Calculatealsothemaximumshearstress.

Theexpressionsforthelongitudinalandcircumferentialstressesproducedbytheinternalpressure
maybefoundinanytextonstressanalysisandare

Longitudinalstress(σx)=

pd

4 t

=1.5× 2 × 103 / 4 × 20 =37.5N/mm^2

Circumferentialstress(σy)=

pd

2 t

=1.5× 2 × 103 / 2 × 20 =75N/mm^2

Thedirectstressduetotheaxialloadcontributestoσxandisgivenby

σx(axialload)= 2500 × 103 /π× 2 × 103 × 20 =19.9N/mm^2

Arectangularelementinthewallofthepressurevesselisthensubjectedtothestresssystemshownin
Fig.1.9.Notethattherearenoshearstressesactingonthexandyplanes;inthiscase,σxandσythen
formabiaxialstresssystem.
Thedirectstress,σn,andshearstress,τ,ontheplaneABthatmakesanangleof60◦withtheaxisof
thevesselmaybefoundfromfirstprinciplesbyconsideringtheequilibriumofthetriangularelement
ABCorbydirectsubstitutioninEqs.(1.8)and(1.9).Notethatinthelattercase,θ= 30 ◦andτxy=0.
Then,

σn=57.4cos^230 ◦+75sin^230 ◦=61.8N/mm^2

τ=(57.4− 75 )(sin( 2 × 30 ◦))/ 2 =−7.6N/mm^2

Fig.1.9

Element of Example 1.1.