Plates - Mechanics of Structures: Variational and Computational Methods - page 809

is one of a number of nonconforming but viable plate bending elements. The element
displacement field properly represents the rigid body motion and constant strain states.
Furthermore, the element passes the patch test [Bazeley, et al., 1965].
The plate elements based on the Kirchhoff plate theory, which use w , θ =− ∂w/∂ x, and

x

θ y as nodal DOF, are usually nonconforming. An exception is a rectangular

element which has

=− ∂w/∂

y

,xy as nodal unknowns. If desired, there are means
for avoiding nonconforming elements. The first approach is to use the plate theory that
takes shear deformation effects into consideration. Such an element was considered in
the previous section, and another will be treated in the next section. For these elements,
independent shape functions are used for

w , w

,x ,

w

,y and

w

so the shape functions for these variables
are continuous at the element boundary. The second approach is to use more DOF at the
nodes to make the slope continuous. For example, if two more DOF related to the slope of
the deflection are added to the nodes of the triangular element of this section, the slope can
be completely defined and the discontinuity can disappear. This type of triangular element
is discussed later.

w and θ

Discrete Kirchhoff Theory (DKT)
It is apparent from the formulation of the triangular plate bending element in the previ-
ous section that it is difficult to formulate a compatible triangular element with nine DOF
using a single polynomial approximation for . A viable approach [Batoz, et al., 1980] for

formulating a compatible triangular Kirchhoff plate element is to start as though Reissner-
Mindlin plate theory applies, so that the nodal variables for the deflection and the rotations
are independent of each other. Then the shape functions for these quantities in a triangular
element can be made continuous at the inter-element boundaries, forming a compatible C 0
element. Since the plate is very thin, the terms in the principle of virtual work correspond-
ing to shear deformation effects are assumed to be negligible. The Kirchhoff plate theory
assumptions are introduced at discrete points along the boundary of the element. The shape
functions are designed to maintain compatibility, so the element is still conforming. The
resulting element is called the discrete Kirchhoff theory element, or simply the DKT element.
The DKT element uses the same nodal variables as those in Fig. 13.18. The formulation
of the stiffness matrix and the loading vector of this element starts from the principle of
virtual work of Eq. (13.96). The first term of Eq. (13.96) corresponds to the bending of the
plate and the second term to the effects of shear deformation, which is to be neglected. For
the bending response, only the rotations appear in the expression for the principle of virtual
work of Eq. (13.96), so that only the shape functions for rotations are needed in developing
an element. For an element with six nodes (Fig. 13.20) the rotations are approximated by

w

6 6

θ x = N i θ xi θ y = N i θ yi (13.120)

i = 1 i = 1

in which

yi are the rotations at the nodes, and nodes 4, 5, 6 are midside nodes be-
tween the corner nodes 1, 2, 3. The shape functions N i , formed with natural coordinates as
quadratic polynomials, are expressed as

θ

xi ,

θ

N 1 = L 1 ( 2 L 1 − 1 ) = ( 1 − L 2 − L 3 )( 1 − 2 L 2 − 2 L 3 ) , = L 2 ( 2 L 2 − 1 )

2

(13.121)

N 3 = L 3 ( 2 L 3 − 1 ) , = 4 L 2 L 3 ,

4

N 5 = 4 L 1 L 3 = 4 ( 1 − L 2 − L 3 ) L 3 , = 4 L 1 L 2 = 4 ( 1 − L 2 − L 3 ) L 2

6

where L i ,i 1 , 2 , 3 , are the area coordinates defined in Chapter 6. The shape functions

can be obtained from Eqs. (6.76) and (6.77). For a discrete Kirchhoff theory plate element

=