Two-Dimensional Problemsin Polar Coordinates
Two-Dimensional Problems in Polar Coordinates
Outline.Polar Coordinate FormulationAxisymmetric Solutions toBiharmonicEquationsCylinders under Boundary PressuresHole in Infinite MediaPure Bending of Curved BeamsRotating Disk/Cylinder ProblemGeneral Solutions to Biharmonic equationStress Concentration around a HoleTransverse Bending of Curved BeamsWedge ProblemsQuarter-Plane ProblemsHalf-Plane Problems2
Outline • Polar Coordinate Formulation • Axisymmetric Solutions to Biharmonic Equations • Cylinders under Boundary Pressures • Hole in Infinite Media • Pure Bending of Curved Beams • Rotating Disk/Cylinder Problem • General Solutions to Biharmonic equation • Stress Concentration around a Hole • Transverse Bending of Curved Beams • Wedge Problems • Quarter-Plane Problems • Half-Plane Problems 2
Polar Coordinate Formulation - ReviewStrain-Displacementougou1 u.ue2a0a02OrarrrHooke's Law3-K13-Kd0ap = 2GSGapOOapaβαβaβry2G42(1-x)(1+x)(1 + x)13- x13- x1oO60Gr000Oro2G2G2G441+ K1+KGG+x)。 +(3-)c,),Tro= 2G8ro)8, +(3-x)8(1-3-VFor plane strain: K = 3- 4v; For plane stress: 1+v3
Polar Coordinate Formulation – Review • Strain-Displacement • Hooke’s Law 1 1 1 , . 2 r r r rr u u u u u u r r r rr θ θ θ θ θ εε ε θ θ ∂ ∂ ∂ ∂ = = + = −+ ∂ ∂ ∂∂ 3 3 For plane strain: 3 4 ; For plane stress: . 1 ν κ ν κ ν − = − = + ( ) ( ) ( ) ( ) (( ) ( ) ) ( ) (( ) ( ) ) 1 3 3 2 2 4 2 1 13 13 1 1 1 , , . 24 1 24 1 2 1 3 , 1 3 , 2. 1 1 r r rr r r r rr r G G GG G G G G αβ αβ γγ αβ αβ αβ γγ αβ θ θ θ θ θ θ θ θ θ θ κ κ ε σ σδ σ ε εδ κ κ κ κ κ ε σ σε σ σε τ κ κ σ κε κε σ κε κε τ ε κ κ − − = − =− − + + − − = −= − = + + =− + + − =− + + − = − −
Polar Coordinate Formulation - ReviewEquilibrium equations00,+10tre +,-0e+F, = 0atre100.2tre + E= 0Orr a0Orr a0rrBeltrami-Michell equation4aFF1 aFV?(, +0。)r 00Or1+K2Navier's equation2GV(V·u)+F = 0.GV?u1-Ka'u.1 'u,22Ga1ou,OueOurur10ueu,-F=0r2Or2002r2r200rrOr001-x rrr1 0"ueo'ue2a10ueour2G 1our1ougUgur=0002r2r2Or?r2a0Orara0r1-xr001r4
Polar Coordinate Formulation – Review 1 1 2 0, 0. r rr r r F F r rr r rr r θ θ θ θθ θ σ τ σσ τ σ τ θ θ ∂∂ − ∂ ∂ + + += + + += ∂ ∂ ∂ ∂ ( ) 2 4 1 . 1 r r r F F F r rr θ σ σθ κ θ ∂ ∂ ∇ + =− + + +∂ ∂ 4 • Equilibrium equations • Beltrami-Michell equation ( ) 2 2 2 2 22 2 2 2 2 2 22 2 2 2 0. 1 11 2 2 1 0, 1 1 1 2 21 1 0. 1 r r r r r r r r r r G G u u u u uu u u G G F r rr r r r r r r r uu u u u G u u u G F r rr r r r r r r r θ θ θθ θ θ θ θ κ θθ κ θ θ θ κθ θ ∇ − ∇ ∇⋅ + = − ∂∂∂ ∂ ∂ ∂ ∂ + + − − − ++ += ∂ ∂ ∂ ∂ −∂ ∂ ∂ ⇒ ∂∂∂ ∂ ∂ ∂ ∂ + + + − − ++ += ∂ ∂ ∂ ∂ − ∂∂ ∂ u uF • Navier’s equation
Polar Coordinate Formulation - Review?Airy Stress Function representationa2022a211 a12(1-x)a21 01aOr200200212Or2r.2aer orr Or1+xOra'ya'y1a1ay1ayaOOOr?P6a02r2Orr orr0. Traction boundary conditionsCRf,(r,0) = T(") =0,n, + Tronefo(r,0)=T(") =Tron, +Ogne. Without body forces, the plane problem is then reducedto a single governing biharmonic equation5
( ) 2 2 2 2 2 22 2 22 2 2 2 2 2 2 2 22 2 11 11 1 1 1 , . 2 1 1 1 . 1 r r r rr r r rr r r r r r r r V r rr r V V θ θ ψ θ θ ψψ ψ ψ σ σσ θ θ κ κ θ − ∂∂∂ + + ∂ ∂ ∂∂ ∂ ∂ ++ ++ = ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = +∂ ∂ ∂ + == − ∂ ∂ ∂ ∂∂ + + R S x y θ r • Airy Stress Function representation ( ) ( ) (, ) (, ) r r rr r r r fr T n n fr T n n θ θ θ θ θ θθ θ στ θ τσ = = + = = + n n • Traction boundary conditions • Without body forces, the plane problem is then reduced to a single governing biharmonic equation. 5 Polar Coordinate Formulation – Review