Plates - Mechanics of Structures: Variational and Computational Methods - page 812

where k = 4 , 5 , and 6 correspond to the sides ij = 23 , 31 , and 12, respectively. The quantities

x ij and y ij are defined in Fig. 13.20.
The stiffness matrix of the DKT element is formulated from the principle of virtual work
of Eq. (13.96). Since the shear deformation effect is neglected, the shape functions for

θ

x and

θ

y are substituted into the first part of Eq. (13.96), giving

v iT θ T D T N T D T E B D κ N dA v i

δ W i =

A δ

E B D κ θ dA = δ

κ κ

A

v iT v iT kv i B T E B B dA v i = δ = δ (13.132)

A

with

B T E B B dA

k = (13.133)

A

and

y 31 N x, L 2 + y 12 N x, L 3

1
2 A

x 31 N y, L 2 − x 12 N y, L 3 B = − (13.134)

−

x 31 N x, L 2

−

x 12 N x, L 3 + y 31 N y, L 2

+

y 12 N y, L 3

where A is the area of the element. The matrices N x , N x, L 2 , N y, L 2 , N x, L 3 , and N y, L 3 are given

by

P 6 ( 1 − 2 L 2 ) + ( P 5 − P 6 ) L 3

− 4 + 6 ( L 2 + L 3 ) + r 6 ( 1 − 2 L 2 ) − L 3 ( r 5 + r 6 )

− q 6 ( 1 − 2 L 2 ) + ( q 5 + q 6 ) L 3

− P 6 ( 1 − 2 L 2 ) + L 3 ( P 4 + P 6 )

N x, L 2 = − 2 + 6 L 2 + r 6 ( 1 − 2 L 2 ) + L 3 ( r 4 − r 6 ) (13.135a)

− q 6 ( 1 − 2 L 1 ) + L 3 ( q 6 − q 4 )

− L 3 ( P 5 + P 4 )

− L 3 ( r 5 − r 4 )

− L 3 ( q 4 − q 5 )

t 6 ( 1 − 2 L 2 ) + ( t 5 − t 6 ) L 3

− q 6 ( 1 − 2 L 2 ) + L 3 ( q 5 + q 6 )

− − ( − ) + ( + )

1 r 6 1 2 L 2 r 5 r 6 L 3

− ( − ) + ( + )

t 6 1 2 L 2 L 3 t 4 t 6

N y, L 2 =

− q 6 ( 1 − 2 L 2 ) − L 3 ( q 4 − q 6 )

(13.135b)

1 − r 6 ( 1 − 2 L 2 ) − L 3 ( r 4 − r 6 )

− L 3 ( t 5 + t 4 )

− L 3 ( q 4 − q 5 )

L 3 ( r 5 − r 4 )