Thick-Walled Cylinder Under Uniform Boundary Pressure.General axisymmetric stress solution+ 2a, ta.(1+2nm), 0。=--4 + 2a, +a, (3+2nm)Boundary conditions-P,=0,(r) =+2a,rIap,-rp+2a20r22-rStressesr2-r: Displacements (depending on elastic constants)i +(1-x)a,r +a (r+(1 x)rlnu2G2G1+Ka,r =0Forplane strain:k =3-4vue2G11
Thick-Walled Cylinder Under Uniform Boundary Pressure • General axisymmetric stress solution • Boundary conditions ( ) ( ) ( ) 2 2 1 1 2 1 1 2 1 2 2 2 2 1 1 2 1 2 2 1 11 22 22 2 2 2 2 2 2 2 1 2 2 2 r r a r r p r a a pp r r r a rp rp pr a a r r r σ σ −= = + = − − ⇒ − −= = + = − • Displacements (depending on elastic constants) ( ) ( ( ) ) 1 2 3 1 1 1 ln 2 r a u ar a r r r G r =− + − + + − κ κ ( ) ( ) ( ) 2 2 2 2 1 2 11 22 2 2 2 1 2 2 2 1 2 1 3 1 1 1 , 2 2 1 2 r r rp rp p p r Gr r r r r u ar G θ κ κ θ − = − − +− − − + = = 0. For plane strain: 3 4 . κ ν = − • Stresses ( ) ( ) 2 2 2 2 2 2 2 2 1 2 11 22 1 2 11 22 2 2 2 1 2 22 2 2 2 1 2 22 2 1 2 1 2 1 2 1 1 1 , . r r r rp rp rr rp rp p p p p rr r rr rr r rr σ σθ − − = − + =− − + − −− − ( ) 1 2 2 3 2 1 2ln r a aa r r σ =+ + + ( ) 1 2 2 3 , 2 3 2ln a aa r r σθ =− + + + . 11
Thick-Walled Cylinder Under Internal Pressure2Internalpressureonlyr1/r2=0.51.50g/p0.5(α。)max =(r +r) / (r2 -r)p=(5 / 3)p0rrr1ao/p0.5prrr?1e0.60.70.80.50.9A-DimensionlessDistancer/rr2-r?pThin-walled tubes0,~0[r。=(ri +r)/2pr(MatchingwithStrengthof Materials Solution)t=r-r<<r112
r1 r2 p Thick-Walled Cylinder Under Internal Pressure • Internal pressure only 2 2 2 1 2 1 2 22 2 2 21 21 2 2 2 1 2 1 2 22 2 2 21 21 1 , 1 . r rr r p r rr r r rr r p r rr r r θ σ σ =− + − − = + − − 22 22 max 1 2 2 1 ( ) ( ) / ( ) (5 / 3) r r r rp p σθ =+ − = Dimensionless Distance 2 r r • Thin-walled tubes ( ) ( ) 1 2 0 2 1 0, 2 . Matching with Strength of Materials Solution r o o r rr pr trr r t θ σ σ ≈ = + ⇒ = − << ≈ 12
Hole in an Infinite Medium under Internal PressureInternal pressure only and r tends to infinity=0→80rp-rp2G2G13
• Internal pressure only and r tends to infinity Hole in an Infinite Medium under Internal Pressure 2 2 2 1 2 2 2 2 2 1 2 0, r r p r r r r σ p = →∞ = − ( ) 2 2 11 22 1 2 1 rp rp p r − − + 2 1 2 1 2 2 1 2 2 2 2 2 2 , r r r r r p r σθ − = − − ( ) 2 2 11 22 1 2 1 rp rp p r − − + 2 1 2 1 1 2 2 1 2 2 2 2 1 2 2 1 2 2 2 2 1 . , , 1 2 r r r r p r r p r r u p G r r r r θ σ σ − = − ⇒ = = − − ( ) ( ) 2 2 11 22 1 1 1 rp rp p r κ − − +− ( ) 2 1 1 2 1 2 2 2 1 . 2 r r r r u p G r r − ⇒ = 13
Hole in an Infinite Medium underBiaxial Remote TensionStress free hole in an infinite medium under remote biaxial loadingP=0,P,=-T,→00=0()=2T02G12G14
• Stress free hole in an infinite medium under remote biaxial loading 2 2 2 12 2 2 1 2 1 1 2 2 0, , r p p Tr r p r r p r σ = =− →∞ = − − ( ) 2 1 1 2 1 r p r + 2 1 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 , r r p r r p r r p r σθ − − =− − − ( ) 2 1 1 2 1 r p r + 2 1 2 1 2 2 1 2 max max 1 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 . 1 , 1 . ( ) () 2 1 2 r r r r T r r T r r T r u p r p r G r p r r θ θ θ σ σ σσ σ − − = − ⇒ = + = = = = − − − ( ) ( ) 2 1 1 1 1 r p r + −κ ( ) 2 2 2 2 2 1 2 2 1 2 2 1 2 2 r r r p r u r p r r G r κ − − − ⇒ =− − Hole in an Infinite Medium under Biaxial Remote Tension 14
Pure Bending of Curved Beams·Boundary conditions=C[α,(a) =0,(b)= 0tro(a) = tre(b) = 0oerdr=-MThis is an axisymmetric problemMMy =a +a, Inr+a,r?+a,r? inr+2a,+a,(1+2lnr),0。a+2a,+a,(3+2lnr), Tre=0:Applying the BCs4MbM2Mα2+2b Inb-2a lna), a, =-h2lna,NNNaIn-4a?b?where N =(b2-a15
0 () () 0 () () 0 b r r a b r r a dr a b a b rdr M θ θ θ θ σ σ σ τ τ σ = = = = = = − ∫ ∫ ( ) ( ) 2 2 01 2 3 1 1 2 2 2 3 2 3 ln ln 2 1 2ln , 2 3 2ln , 0 r r a a r ar ar r a a aa r aa r r r θ θ ψ σσ τ =+ + + = + + + =− + + + = Pure Bending of Curved Beams • Boundary conditions • This is an axisymmetric problem • Applying the BCs ( ) ( ) ( ) 2 2 22 2 2 2 2 1 2 3 2 2 2 2 22 4 2 ln , 2 ln 2 ln , , where 4 ln M bM M a ab a b a b b a a a b a N aN N b N b a ab a = − = − + − =− − =− − 15