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Plate and Shell Analyses 197

z y z y

4 3 4 7 3
8 6
x x

1 2 1
t t 5 2

FIGURE 6.14
Four- and eight-node quadrilateral plate elements.

6.4.3 Discrete Kirchhoff Elements

This is a triangular element with only bending capabilities. First, start with a six-node
triangular element (Figure 6.15). There are 5 DOFs at each corner node (,w ∂ w ∂ , x ∂ w/ , y ∂
/
θ x , θ y ) 2 DOFs (,θ x θ y ) at each mid-node, and a total of 21 DOFs for the six-node element.
Then, impose conditions γ = γ = 0, and so on, at selected nodes to reduce the DOFs
yz
xz
(using relations in Equation 6.18), to obtain the following discrete Kirchhoff triangular
(DKT) element (Figure 6.16).
For the three-node DKT element shown above, there are 3 DOFs at each node, that is,
(
(
w, θ= ∂w/ ∂ ) x , and θ= ∂w/ ∂ ) y , and a total of 9 DOFs for the element. Note that w(x, y)
y
x
is incompatible for DKT elements [7]; however, its convergence is faster (w is cubic along
each edge) and it is efficient.
6.4.4 Flat Shell Elements
A flat shell element can be developed by superimposing a plane stress element to a plate
element (Figure 6.17).

y
z 3

4 6
2
1 x
5

FIGURE 6.15
A six-node triangular element with 5 DOFs at each corner node and 2 DOFs at each mid-node.

y
z 3

1 2 x

FIGURE 6.16
Discrete Kirchhoff triangular element with 3 DOFs at each node.