Pure Bending of Curved Beams Stress field6a?b2ba'b?4M4MQ1InIrONNbNbaa1.2Displacement fieldb/a= 4ssessoe0.8Theoryof ElasticityStrengthof Materials0.6)a,10.442G0.2(r+(1-x)rlnr)+a, (+u。 cosO+v, sin?0.20.41+ Ka,ro-u,sin0+v,cos0+o.re0.6531.522.53.52GDimensionlessDistance,rlaRotation of a polar differential elementoue1u1+0212Or002G16
( ) ( ( ) ) 1 2 3 3 1 1 2 1 ln cos sin 1 sin cos 2 r o o oo o a a r u r G ar r r u v u ar u v r G θ κ κ θ θ κ θ θ θω + − = − + +− + + + = −++ 2 2 2 2 2 2 2 2 22 2 2 4 4 ln ln ln , ln ln ln r M ab b r a M ab b r a b a b a ba Nr a b r N r a b r σ σθ =− + + =− − + + + − • Stress field • Displacement field 21 3 11 1 2 2 r o u u u a r r G θ θ κ ω θ ω θ ∂∂ + = − −= + ∂ ∂ • Rotation of a polar differential element Pure Bending of Curved Beams 16
Rotating Disk/Cvlinder Problem: Load: centrifugal force due to constant rotationF =po?rEquilibrium equations0aa1oy,-0e+F=0araxrGe + po'r= 0(ro,)+ por2Propose a special stress function for this casedy+por2ro,=y, OdrThe equilibrium condition is automatically satisfied17
1 r r r r θ τ θ ∂σ ∂ ∂ + ∂ ( ) ( ) 2 2 2 0, 1 0 r r r r F r d r r r dr r d r r dr θ θ θ σ σ σ σ ρω σ σ ρω − + += ⇒ −+ = ⇒= + Rotating Disk/Cylinder Problem • Load: centrifugal force due to constant rotation 2 F r r = ρω • Equilibrium equations • Propose a special stress function for this case 2 2 , r d r r dr θ ψ σ ψ σ ρω = = + • The equilibrium condition is automatically satisfied. 17
Rotating Disk/Cylinder ProblemBeltrami-Michell equationaF84H72g.+o001+xar1+1ao02a1 12dya+po°r2Or2r ora0drdrdi7*84(3+KC1+1+Kddt3+1-24(1+K3+KW0T24(1+ K2dy5-K1Cdr2224(1+xThe C, term leads to multivalued displacement behavior, and is notfoundfollowingthedisplacementformulationapproachC, =00, (0,0), 0。(0,0) must be finite3+KStress fieldpo'a0, (a,0)=02(1+18
( ) 2 4 1 1 r r r F F r r F r θ θ σ θ σ κ ∂ + ∂ ∇ + =− + + ∂ ∂ ( ) 2 2 2 2 2 2 2 2 1 1 8 1 r r r r r σ σ ρω θ κ θ ⇒ ∇ + =− + ∂ ∂ ∇ + ∂ ∂ + ∂ = ∂ ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 3 3 1 2 2 2 1 1 , 1 1 8 11 4 3 , 1 1 3 1 11 ln 4 1 2 2 2 3 4 1 r r d d d d r rrr r dr dr r dr r dr dd d dd d r rr r r r dr dr r dr r dr dr r dr C r Cr r Cr r r r θ ψ ψ σ σ ρω ψ ρω κ ψ ρω ρω ψ ρω κ κ κ ψ ρω κ ψ κ σ ρω κ = + =+ + = + + ⇒ + =− ⇒ = − + + + ⇒ =− + − + + + + = = − + ⇒ ( ) 3 1 2 2 2 2 2 2 3 1 2 2 1 11 ln 2 22 5 1 11 ln 4 1 2 2 2 C Cr C r d C r r Cr C dr r θ ψ κ σ ρω ρω κ + −+ + − = + =− + + + − + Rotating Disk/Cylinder Problem • Beltrami-Michell equation 18 • The C1 term leads to multivalued displacement behavior, and is not found following the displacement formulation approach. • Stress field ( ) ( ) ( ) ( ) 2 2 2 3 0 0, , 0, must be fini 0 te. 3 , 2 1 r r a C C a θ κ σ ρω σ θσ θ θ κ ⇒ + = = + =