Polar CoordinatesAxi-symmetric problems3-node Triangular ElementThin PlateKirchhoff Plate TheoryMindlin Plate TheoryThin Shell ElementsThree Dimensional ElementsTetrahedronHexahedron (Brick) ElementIso-parametric:M. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D1Polar CoordinatesThe small strain-displacement equations in polar coordinates are:εrrr∂∂ur= εθθurr----1r---θ∂∂vθ+= εzzz∂∂w=γrθ1r---θ∂∂urr∂∂vθvθr-----–+= γrzr∂∂wz∂∂ur+= γzθ1r---θ∂∂wz∂∂vθ+=The Generalized Hooke’s Law can be written as:σrr2G12ν–()--------------------1ν–()εrrνεθθνεzz++[]= γrθτrθG------=σθθ2G12ν–()--------------------1ν–()εθθνεrrνεzz++[]= γrzτrzG------=σzz2G12ν–()--------------------1ν–()εzzνεθθνεrr++[]= γzθτzθG-------=GE21 ν+()--------------------=Axi-symmetric problemsFor a problem to be axi-symmetric the following requirements must be met:1. The geometry must be symmetric about an axis of revolution.2. The material properties must be symmetric about the axis of revolution.3. The loading and boundary conditions must be symmetric about the axis of revolution.Implications: Displacements and stresses must be independent of angular location (θ) and there can be no twist (vθ must be zero).εrrr∂∂ur= εθθurr----= εzzz∂∂w= γrθ0= γrzr∂∂wz∂∂ur+= γzθ0=• Note radial displacement causes tangential normal strain. zrθσrrσzzσθθτrzPressurizedrzSmoothSurfaceM. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D23-node Triangular Element• Displacements are linear in r and z directionsura0a1ra2z++=wb0b1rb2z++=132zrurrz,() Nirz,()uie()i1=3∑=wrz,() Nirz,()vie()i1=3∑=de(){}ur1e()w1e()ur2e()w2e()ur3e()w3e()⎩⎭⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎫=ur1e()ur2e()ur3e()w1e()w3e()w2e()εrrx∂∂ura1== εzzz∂∂wb2== γrzz∂∂urr∂∂w+a2b1+==εθθurr----a0r-----a1a2zr--------++==Same as CSTTangential normal strain is not constant• You can use any 2-D element, but will need to post-process the results of displacements and strains to get εθθ, σrr, σθθ, σzz.M. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D3Thin PlateA thin two-dimensional structural element that is subjected to bending loads.• Plane stress in z-directiontyxzMid-surfaceis neutral surfacepz(x,y)• Mid-plane is initially flat• Plane sections before deformation remain plane after deformation. (displacements u and v are linear in z, i.e., through the thickness.)Kirchhoff Plate Theory• Plane sections initially perpendicular to the mid-surface remains perpendicular after deformation ( γxz0≈γyz0≈ ) --Shearing action is small)uzx∂∂w–=vzy∂∂w–=w is the displacement in the z-direction and is only a function of x and y. u and v are displacements in x and y direction.For small strain:εxxx∂∂uzx22∂∂ w–== εyyy∂∂vzy22∂∂ w–== γxyy∂∂ux∂∂v+zx∂y2∂∂ w–==Stresses in plane stress:σxxEεxxνεyy+[]1 ν2–()-----------------------------Ez–1 ν2–()-------------------x22∂∂ wνy22∂∂ w+⎝⎠⎜⎟⎛⎞==σyyEεyyνεxx+[]1 ν2–()-----------------------------Ez–1 ν2–()-------------------y22∂∂ wνx22∂∂ w+⎝⎠⎜⎟⎛⎞==τxyE21 ν+()-------------------- γxyE–z21 ν+()--------------------x∂y2∂∂ w==M. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D4Internal Forces and Moments:Mxzσxxzdt2⁄–t2⁄∫=Myzσyyzdt2⁄–t2⁄∫=Mxyzτxyzdt2⁄–t2⁄∫= qxτxzzdt2⁄–t2⁄∫=qyτyzzdt2⁄–t2⁄∫=The moments and shear forces have units of moments and forces per unit length.Moment Curvature Formulas: MxDx22∂∂ wνy22∂∂ w+–=MyDy22∂∂ wνx22∂∂ w+–=MxyD1 ν–()x∂y2∂∂ w–=where, DEt312 1 ν2–()-------------------------= is called the plate rigidity.Differential Equation: Bi-harmonic Equationx44∂∂ wx2∂y24∂∂ wy44∂∂ w++pzxy,()= or ∇4w ∇2∇2wpzxy,()==where, ∇2x22∂∂y22∂∂+= is the harmonic operator.• A kinematically admissible deflection w requires continuity of wx∂∂wy∂∂w,, at all points. • At a corner the requirement that x∂y2∂∂ wy∂x2∂∂ w= results in the condition that x∂y2∂∂ w be continuous at the corner. • Rectangular element: Each node has four degrees of freedom (dof) per node: wx∂∂wy∂∂wx∂y2∂∂ w,,, . Can be used only with rectangular sides parallel to x and y axis.xy16 dof21 dof Hermite polynomials are used for interpolation functions.M. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D5• To ensure x∂y2∂∂ wy∂x2∂∂ w= at any orientation, requires all second derivatives to be continuous at nodes. • Triangular element: Each corner node has six degrees of freedom per node wx∂∂wy∂∂wx22∂∂ wy22∂∂ wx∂y2∂∂ w,,,,, and the middle node on each side has one degree of freedom n∂∂w where the n direction is the normal direction to the side.• The continuity of second derivatives implies that moments must be continuous. If there is a line load of moment then this will leads to problems.• Non-conforming elements do not satisfy all continuity requirements. Non-conforming elements are used in plate analysis.Mindlin Plate Theory• Mindlin plate theory differs from Kirchhoff plate theory in the same way as Timoshenko’s beam theory differs from classical beam theory. • The assumption of plane sections initially perpendicular to the mid-surface remains perpendicular after deformation is dropped and transverse shear is accounted. Displacements:uzθy=vzθx–=where, θx and θy are the rotation about x and y axis, respectively, of a line that was initially perpendicular to the mid surface. Strainsεxxx∂∂uzx∂∂θy== εyyy∂∂vzy∂∂θx–== γxyy∂∂ux∂∂v+zy∂∂θyx∂∂θx–⎝⎠⎛⎞==γxzz∂∂ux∂∂w+x∂∂wθy+⎝⎠⎛⎞== γyzz∂∂ux∂∂w+y∂∂wθx–⎝⎠⎛⎞==• Note θyx∂∂w⎝⎠⎛⎞–= and θxy∂∂w= reduces Mindlin’s theory to Kirchhoff’s theory.• Kinematically admissibility requires that w θxθy,, must be continuous. Can use Lagrange polynomial for interpolation functions.M. Vable Notes for finite element method: Axi-symmetric, plates and shells, 3-D6Thin Shell Elements• Curved plate: Combination of membrane (2-D in-plane) and plate bending.•