Plates - Mechanics of Structures: Variational and Computational Methods - page 811

The derivative of w with respect to s is

w = 3 a 3 s 2 a 1 + 2 a 2 s +

w i = At s = 0 (node i ), w = a 0 , a 1 and at s =

ij (node j ), where ij is the length of side

i

ij ij , ij

ij, The coefficients a 2 and a 3 can

be obtained from the solution of these four equations. Since the coefficients a i ,i

w = a 0 + a 1 + a 2 + a 3 w = a 1 + 2 a 2 + 3 a 3 .

j ij ,j ij

3 ,
are uniquely expressed in terms of the nodal variables which are shared by the adjacent
elements, the displacement is continuous between the elements. Substitute the expressions
for a i ,i

= 0 , ...

0 , 1 , 2 , 3 into Eq. (13.125) and let s assume the values at the nodal points. Form
the derivative of

=

w with respect to s at node k

3 1

4 w ,si +

3 1

4 w ,sj

w ,sk =−

ij w i − ij w j −

(13.126)

2 2

Because the first term of the principle of virtual work of Eq. (13.96) does not involve w ,

there is no need to define an interpolation function for w on the element. However, a cubic

variation for w on the boundary has been assumed for the sake of obtaining the expression

for w ,s . Since w varies cubically along the element boundaries, w ,s varies quadratically.

Since w ,s matches θ t at the three points along each side, i.e., let θ t =− w ,s , the form of the

quadratic function for w ,s is determined uniquely.

To obtain the relationships between the variables

θ x , θ y in Eq. (13.120) and the nodal

variables θ a , θ t , use Fig. 13.20c.

θ x
θ y

cos θ a

θ t

− w ,a

w ,s

γ ij − sin γ ij cos γ ij sin γ ij

= = (13.127)

sin γ ij cos γ ij − sin γ ij cos γ ij

and

w ,a
w ,s

− θ x

θ y

cos γ ij − sin γ ij

= (13.128)

sin γ ij − cos γ ij

where γ ij is defined in Fig. 13.20. Substitute Eqs. (13.126), (13.127), and (13.128) at each node

into Eq. (13.120) to obtain the shape functions for θ y in terms of the nodal variables.

That is, use Eqs. (13.127) and (13.128) at each node to obtain expressions for

θ x and

θ

xi and

θ

yi and

then substitute these quantities into Eq. (13.120) to find the shape functions for θ

x and

θ

y .

This results in

N x v i , N y v i

θ = θ = (13.129)

x y

Nv i , where θ

y ] T , N

[ N x N y ] T , and v i

y 3 ] T .

or θ = = [ θ θ = = [ w θ θ w θ θ w θ θ

x 1 x 1 y 1 2 x 2 y 2 3 x 3

Also, N x and N y are nine component vectors of the new shape functions for θ

x and

θ

y . These

shape functions are expressed as
N x 1 = 1 . 5 ( a 6 N 6 − a 5 N 5 ) , = N 1 − b 5 N 5 − b 6 N 6

x 2

=− − = . ( − )

N x 3 c 5 N 5 c 6 N 6 , N y 1 1 5 d 6 N 6 d 5 N 5 (13.130)

N y 2 = N x 3 , N y 3 = N 1 − e 5 N 5 − e 6 N 6

where N i ,i , 6 , are the shape functions of Eq. (13.121). The functions N x 4 , N x 5 , and

N x 6 are obtained from the above expression by replacing N 1 by N 2 and indices 6 and 5 by 4
and 6, respectively. Similarly, the functions N x 7 , N x 8 , and N x 9 are found by replacing N 1 by
N 3 and indices 6 and 5 by 5 and 4, respectively. The same procedure leads to N y 5 , N y 6 , etc.
Also

= 1 , 2 ...

2
ij

2
ij

a k =− x ij / d k =− y ij /

1
4 x ij − 2 y ij

1
4 y ij − 2 x ij

1 1

ij ij

b k = / e k = /

(13.131)

3
4 x ij y ij

2
ij

c k = /