2.094 — Finite Element Analysis of Solids and Fluids
Fall ‘08
Lecture 21 - Plates and shells
Prof. K.J. Bathe
MIT OpenCourseWare
Timoshenko beam theory, and Reissner-Mindlin plate theory
For plates, and shells, w, βx , and βy as independent variables.
w = displacement of mid-surface, w(x, y)
A = area of mid-surface
p = load per unit area on mid-surface
w = w(x, y)
(21.1)
w(x, y, z) = w(x, y)
(21.2)
The material particles at “any z” move in the z-direction as the mid-surface.
u(x, y, z) = −βx z = −βx (x, y)z
v(x, y, z) = −βy z = −βy (x, y)z
(21.3)
(21.4)
∂βx
∂u
= −z
∂x
∂x
∂βy
∂v
= −z
=
∂y
∂y
�
�
∂u ∂v
∂βx
∂βy
=
+
= −z
+
∂y
∂x
∂y
∂x
�xx =
(21.5)
�yy
(21.6)
γxy
∂βx
∂x
⎛
⎞
⎜
�xx
⎜
⎝ �yy ⎠ = −z ⎜
⎜
⎜
γxy
⎝
⎞
⎛
�
∂βy
∂y
∂βx
∂y
+
��
κ
(21.7)
∂βy
∂x
⎟
⎟
⎟
⎟
⎟
⎠
(21.8)
�
90
MIT 2.094
21. Plates and shells
∂w ∂u
∂w
− βx
+
=
∂x
∂z
∂x
∂w ∂v
∂w
− βy
=
+
=
∂y
∂z
∂y
γxz =
γyz
⎛
⎞
⎡
1
τxx
E
⎝ τyy ⎠ =
⎣ ν
1 − ν2
τxy
0
ν
1
0
(21.9)
(21.10)
0
0
1−ν
2
⎤⎛
⎞
�xx
⎦ ⎝ �yy ⎠ = C · �
γxy
(21.11)
(plane stress)
�
τxz
τyz
�
E
=
2(1 + ν)
�
�
γxz
γyz
= Gγ
(21.12)
Principle of virtual work for the plate:
� �
A
+ 2t
⎡
�
�xx
�yy
γ xy
− 2t
� �
+ 2t
k
A
− 2t
�
�
1
E ⎣
ν
1 − ν2
0
γ xz
γ yz
⎞
�xx
⎦ ⎝ �yy ⎠dzdA+
1−ν
γxy
2
�
��
�
�
�
1 0
γxz
G
dzdA =
wpdA
0 1
γyz
A
ν
1
0
0
0
⎤⎛
(21.13)
Consider a flat element:
⎛
⎜
⎜
⇒ Kb ⎜
⎝
w1
θx1
θy1
.
..
⎞
⎛
⎟
⎟
⎟ , also Kpl.
⎠
where Kb is 12x12 and Kpl.
str.
⎞
u1
⎜ v ⎟
str. ⎝ 1 ⎠
...
(21.14)
is 8x8.
91
MIT 2.094
21. Plates and shells
For a flat element:
⎛
�
⇒
Kb
0
0
Kpl. str.
⎜
⎜
⎜
⎜
⎜
�⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
w1
θx1
θy1
..
.
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
4 ⎟
θy ⎟ = · · ·
u1 ⎟
⎟
v1 ⎟
⎟
.. ⎟
. ⎠
v4
(21.15)
u=
�
hi u i
(21.16)
v=
�
hi vi
(21.17)
w=
�
hi wi
�
βx = −
hi θyi
�
βy =
hi θxi
(21.18)
(21.19)
(21.20)
From (21.13)
⎡
1
3
Et
⎣ ν
κT
12 (1 − ν 2 )
A
0
�
ν
1
0
0
0
1−ν
2
⎤
�
⎦ κdA +
γ T Gt · k
A
�
1
0
0
1
�
γdA
(21.21)
where k is the shear correction factor.
Next, evaluate
∂w ∂w ∂βx
∂x , ∂y , ∂x ,
. . . etc. ⇒ This element, as it is, locks!
This displacement-based element “locks in shear”. We need to change the transverse shear interpola­
tions.
γyz =
1
1
A
C
(1 − r)γyz
+ (1 + r)γyz
2
2
(21.22)
where
A
γyz
=
�
��
�
∂w
− βy ��
∂y
evaluated at A
(21.23)
from the w, βy displacement interpolations.
1
1
B
D
(1 − s)γxz
+ (1 + s)γxz
(21.24)
2
2
with this mixed interpolation, the element works. Called MITC interpolation (for mixed interpolatedtensional components)
γxz =
92
MIT 2.094
21. Plates and shells
Aside: Why not just neglect transverse shears, as in Kirchhoff plate theory?
βx = 0 ⇒ βx = ∂w
∂x
� 2
�
• Therefore we have ∂∂xw2 , · · · in strains, so we need continuity also for
� ∂w
�
∂x
· · ·
• If we do, γxz =
∂w
∂x −
•
w=
�
�
1
1
r(1 + r)w1 − r(1 − r)w2 + 1 − r2 w3
2
2
w2 and w3 never affect w1 (∴ w|r=1 = w1 ).
But,
∂w
1
1
= (1 + 2r)w1 − (1 − 2r)w2 − 2rw3
∂r 2
2
�
�
w2 and w3 affect ∂w
∂r 1 . ⇒ This results in difficulties to develop a good
element based on Kirchhoff theory.
With Reissner-Mindlin theory, we independently interpolate rotations such that this problem does
not arise.
For flat structures, we can superimpose the plate bending and plane stress element stiffness. For
shells, curved structures, we need to develop/use curved elements, see references.
References
[1] E. Dvorkin and K.J. Bathe. “A Continuum Mechanics Based Four-Node Shell Element for General
Nonlinear Analysis.” Engineering Computations, 1:77–88, 1984.
[2] K.J. Bathe and E. Dvorkin. “A Four-Node Plate Bending Element Based on Mindlin/Reissner Plate
Theory and a Mixed Interpolation.” International Journal for Numerical Methods in Engineering,
21:367–383, 1985.
[3] K.J. Bathe, A. Iosilevich and D. Chapelle. “An Evaluation of the MITC Shell Elements.” Computers
& Structures, 75:1–30, 2000.
[4] D. Chapelle and K.J. Bathe. The Finite Element Analysis of Shells – Fundamentals. Springer,
2003.
93
MIT OpenCourseWare
http://ocw.mit.edu
2.094 Finite Element Analysis of Solids and Fluids II
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.