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# Polar Coordinates

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• pg 1
```									M. Vable                Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Polar Coordinates
The small strain-displacement equations in polar coordinates are:
∂u r                 u r 1 ∂v θ                        ∂w
ε rr =                  ε θθ = --- + --
- -                    ε zz =
∂r                    r r ∂θ                           ∂z
1 ∂u r ∂v θ v θ                               ∂w ∂u r                  1 ∂w ∂v θ
γ rθ = --
-    +    – ----
-                  γ rz =       +               γ zθ = --
-   +
r ∂θ ∂r        r                              ∂r ∂z                     r ∂θ ∂z
The Generalized Hooke’s Law can be written as:
2G                                                           τ rθ
σ rr = ------------------- [ ( 1 – ν )ε rr + νε θθ + νε zz ]
-                                        γ rθ = ------
( 1 – 2ν )                                                          G
2G                                                            τ rz                       E
σ θθ = ------------------- [ ( 1 – ν )ε θθ + νε rr + νε zz ]
-                                         γ rz = -----
-                                -
G = -------------------
( 1 – 2ν )                                                           G                2( 1 + ν)
2G                                                            τ zθ
σ zz = ------------------- [ ( 1 – ν )ε zz + νε θθ + νε rr ]
-                                        γ zθ = -------
( 1 – 2ν )                                                           G
Axi-symmetric problems
For 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.
revolution.
Implications: Displacements and stresses must be independent of angular location
(θ) and there can be no twist (vθ must be zero).
∂u r                ur                        ∂w                                       ∂w ∂u r
ε rr =                ε θθ = ---
-             ε zz =                  γ rθ = 0          γ rz =     +                  γ zθ = 0
∂r                   r                        ∂z                                       ∂r ∂z

• Note radial displacement causes tangential normal strain.
σzz
τrz
σθθ                                        σrr

z

r         θ

1
M. Vable              Notes for finite element method: Axi-symmetric, plates and shells, 3-D

z
r
Smooth
Pressurized                       Surface

3-node Triangular Element
• Displacements are linear in r and z directions
ur = a0 + a1 r + a2 z            w = b0 + b1 r + b2 z
(e)
z            (e)
3               ⎧ u r1 ⎫
w3                    (e)                                      (e)      ⎪        ⎪
(e)
u r3
w2            u r ( r, z ) = ∑ N i ( r, z )u i        ⎪ (e) ⎪
3                                               i=1                      ⎪ w1 ⎪
(e)                         2             (e)
⎪        ⎪
w1                                       u r2                                      ⎪u  (e) ⎪
(e)  ⎪ r2 ⎪
1         (e)                                         3                {d } = ⎨        ⎬
u r1                   r
(e)        ⎪ w( e ) ⎪
w ( r, z ) = ∑ N i ( r, z )v i           ⎪ 2 ⎪
⎪ (e) ⎪
i=1                       ⎪ u r3 ⎪
⎪        ⎪
⎪ (e) ⎪
⎩ w3 ⎭

∂u r                        ∂w                              ∂u r ∂w
ε rr =       = a1          ε zz =      = b2                γ rz =       +   = a2 + b1      Same as CST
∂x                          ∂z                              ∂z ∂r
ur    a0           a2 z
ε θθ   = --- = ---- + a 1 + -------
-      -               -                        Tangential normal strain is not constant
r      r             r

• You can use any 2-D element, but will need to post-process the results of
displacements and strains to get ε θθ , σ rr , σ θθ , σ zz .

2
M. Vable           Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Thin Plate
A thin two-dimensional structural element that is subjected to bending loads.
• Plane stress in z-direction
z

t               y
pz(x,y)
x                                                              Mid-surface
is neutral surface

• 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 ( γ xz ≈ 0                   γ yz ≈ 0 ) --Shearing action is small)

∂w                       ∂w
u = –z                  v = –z
∂x                       ∂y
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:
2                                      2                                            2
∂u     ∂w                           ∂v     ∂w                             ∂u ∂v      ∂w
ε xx   =    = –z 2                 ε yy    =    = –z 2                    γ xy   =   +   = –z
∂x     ∂x                           ∂y     ∂y                             ∂y ∂x      ∂ x∂y

Stresses in plane stress:
2           2
[ ε xx + νε yy ]                     – Ez ⎛ ∂ w           ∂ w⎞
σ xx   = E ----------------------------- = ------------------ ⎜ 2 + ν 2 ⎟
-
(1 – ν )⎝∂x               ∂y ⎠
2                          2
(1 – ν )
2           2
[ ε yy + νε xx ]                     – Ez ⎛ ∂ w           ∂ w⎞
σ yy   = E ----------------------------- = ------------------ ⎜ 2 + ν 2 ⎟
-
(1 – ν )⎝∂y               ∂x ⎠
2                          2
(1 – ν )
2
E                    –E z ∂ w
τ xy   = ------------------- xy = -------------------
-γ                       -
2(1 + ν)                 2 ( 1 + ν ) ∂ x∂y

3
M. Vable                               Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Internal Forces and Moments:
t⁄2                                                        t⁄2                                            t⁄2

Mx =              ∫       zσ xx dz                               My =         ∫      zσ yy dz                  M xy =        ∫      zτ xy dz
–t ⁄ 2                                                      –t ⁄ 2                                         –t ⁄ 2
t⁄2                                                 t⁄2

qx =             ∫      τ xz dz                             qy =     ∫      τ yz dz
–t ⁄ 2                                              –t ⁄ 2
The moments and shear forces have units of moments and forces per unit length.
Moment Curvature Formulas:
2                    2                                         2                2                                      2
∂w                ∂w                                        ∂w               ∂w                                         ∂w
Mx = –D                                +ν                            My = –D                          +ν                     M xy      = –D ( 1 – ν )
∂x
2
∂y
2
∂y
2
∂x
2                                  ∂ x∂y
3
Et
where, D = ------------------------- is called the plate rigidity.
2
12 ( 1 – ν )
Differential Equation: Bi-harmonic Equation
4              4                         4
∂w             ∂w                       ∂w                                            4               2    2
4
+          2        2
+            4
= p z ( x, y ) or ∇ w = ∇ ∇ w = p z ( x, y )
∂x             ∂ x ∂y                   ∂y
2                2
2            ∂                ∂
where, ∇ =                                   2
+            2
is the harmonic operator.
∂x           ∂y

∂w ∂w
• A kinematically admissible deflection w requires continuity of w,                                                                                     ,
∂x ∂y
at all
points.
2                2                                                  2
•   At a corner the requirement that ∂ w = ∂ w results in the condition that ∂ w
∂ x∂y ∂ y∂x                             ∂ x∂y
be continuous at the corner.
• Rectangular element: Each node has four degrees of freedom (dof) per node:
2
∂w ∂w ∂ w
w,    ,   ,      . Can be used only with rectangular sides parallel to x and y
∂ x ∂ y ∂ x∂y
axis. Hermite polynomials are used for interpolation functions.
y

21 dof
16 dof
x

4
M. Vable            Notes for finite element method: Axi-symmetric, plates and shells, 3-D

2          2
•   To ensure ∂ w = ∂ w at any orientation, requires all second derivatives to be
∂ x∂y ∂ y∂x
continuous at nodes.
• Triangular element: Each corner node has six degrees of freedom per node
2        2     2
∂w ∂w    ∂w ∂w ∂w
w,     ,    ,      ,      ,       and the middle node on each side has one degree
∂x ∂y    ∂x
2
∂y
2   ∂ x∂y

of freedom ∂w where the n direction is the normal direction to the side.
∂n
• 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:
u = zθ y        v = – zθ x
where, θ x and θ y are the rotation about x and y axis, respectively, of a line that was
initially perpendicular to the mid surface.
Strains
∂u     ∂θ y                ∂v      ∂θ x                   ∂u ∂v      ∂θ y ∂θ x⎞
ε xx =      = z            ε yy =      = –z               γ xy =     +   = z⎛
∂x     ∂x                  ∂y      ∂y                     ∂y ∂x    ⎝∂y – ∂x ⎠
∂u ∂w     ∂w                              ∂u ∂w     ∂w
γ xz =     +   = ⎛    + θ y⎞              γ yz =     +   = ⎛    – θ x⎞
∂z ∂x   ⎝∂x       ⎠                       ∂z ∂x   ⎝∂y       ⎠

∂w           ∂w
• Note θy      = – ⎛ ⎞ and θ x =    reduces Mindlin’s theory to Kirchhoff’s theory.
⎝∂x ⎠         ∂y
• Kinematically admissibility requires that w,                  θ x, θ y must be continuous. Can use
Lagrange polynomial for interpolation functions.

5
M. Vable          Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Thin Shell Elements
• Curved plate: Combination of membrane (2-D in-plane) and plate bending.
• The elements are similar to plate elements but requires definition of curved
geometry.
• FEM codes usually have shallow thin shell elements which can be used to also
simulate plate elements.

6
M. Vable            Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Three Dimensional Elements
Tetrahedron
• Displacements are linear in x and y, resulting in constant strains.
Constant Strain
u = a0 + a1 x + a2 y + a3 z
v = b0 + b1 x + b2 y + b3 z
w = c0 + c1 x + c2 y + c3 z

1

z
z2
x                     y
xz               yz                z3
2                xy                      2                 z3
x                                        y
z3

x3              x2 y                  xy2                y3

x4           x3y                x2y2                xy3          y4

x4y           x3y2                             xy4            y5
5
x                                           x2y3

7
M. Vable                  Notes for finite element method: Axi-symmetric, plates and shells, 3-D

Hexahedron (Brick) Element
Trilinear
ζ
5                               7                            1
N 1 = -- ( 1 – ξ ) ( 1 – η ) ( 1 – ζ )
-
8
6                                                                               1
8                                   N 4 = -- ( 1 + ξ ) ( 1 + η ) ( 1 – ζ )
-
η                    8

3
ξ        1
4
2

ζ

η

ξ

Iso-parametric:
(e)
u =        ∑ Ni ( ξ, η, ζ )ui                                         x =   ∑ Ni ( ξ, η, ζ )xi
i=1                                                              i=1
n                                                                n
(e)
v =        ∑ Ni ( ξ, η, ζ )vi                                         y =   ∑ Ni ( ξ, η, ζ )yi
i=1                                                              i=1
n                                                                n
(e)
w =        ∑ Ni ( ξ, η, ζ )wi                                         z =   ∑ Ni ( ξ, η, ζ )zi
1 1 1
(e)                                                                            T
∫ ∫ ∫ [B]                                            ∫ ∫ ∫ [B]
T
[K         ] =                   [ E ] [ B ] ( dx ) ( dy ) ( dz ) =          ˜          [ E ] [ B ] J ( dξ ) ( dη ) ( dζ )
˜
–1 –1 –1

8

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