Plates - Mechanics of Structures: Variational and Computational Methods - page 810

FIGURE 13.20
The geometry of a triangular element.

the Kirchhoff assumptions are to be imposed at particular points. For the element shown
in Fig. 13.20, the assumptions imposed are

θ

+ w

x ,x

γ = = 0 (13.122)

θ + w

y ,y

at the corner nodes and

θ tk =− w ,sk (13.123)

at the middle nodes where k = 4 , 5 , 6 . Here θ t denotes the rotation about the normal to

the boundary of the element at node k and with respect to the

s direction (Fig. 13.20), which coincides with the boundary. It is assumed that the rotation
about the s direction of Fig. 13.20 at the middle nodes

w ,sk is the derivative of w

( k = 4 , 5 , 6 ) is the average value of

the rotations of the end nodes, i.e.,

1
2 (θ

θ = + θ )

(13.124)

ak ai aj

where the corner nodes are ij 23 , 31 , and 12. As will be seen later, Eqs. (13.123) and

(13.124) will lead to relationships between the variables at the in-span nodes and the corner
nodes so that the stiffness matrix will involve corner variables only.
Let the variation of

=

w along the element boundaries be cubic. Thus, along a boundary of

length

ij with nodes i and j at the ends (Fig. 13.20), the displacement

w takes the form

a 2 s 2 a 3 s 3

w = + + +

a 0 a 1 s (13.125)