Plates - Mechanics of Structures: Variational and Computational Methods - page 818

Substitute Eq. (13.158) into Eq. (13.157) to find

1
2 v T C T

1
2 v T C T B − 1 Cv

Π ∗ H =− B − 1

T BB − 1 Cv

v T C T B − 1

T Cv

( ) + ( ) =

The second derivative of this Π ∗ H
with respect to v shows that (Chapter 3, Example 3.3) the stiffness matrix is given by

B − 1

T

B T ) − 1 B − 1

where, because B is symmetric, ( ) = ( = .

C T B − 1 C k HSM = (13.159)

In evaluating the stiffness matrix k HSM , the matrix B − 1 has to be calculated. This matrix

can be expressed as

e 11 Λ e 12 Λ e 13 Λ
e 21 Λ e 22 Λ e 23 Λ
e 31 Λ e 32 Λ e 33 Λ

B − 1

= (13.160)

where e ij are the elements in E B and

I xy I xx I yy − 0 0

1
A I xx I yy −

φ − 1

I xy

Λ = = 0 AI yy − AI xy (13.161)

0 − AI xy AI xx

After the nodal displacements are found, the stress parameters can be obtained from Eq.
(13.158) and the bending moments in the plate can be calculated using Eq. (13.144).

EXAMPLE 13.7 Square Plate under Concentrated Load with Clamped and
Simply Supported Edges
Examine the accuracy of the DKT and HSM elements through the analysis of a square plate.
Consider the square plate of sides 2 a , with either simply supported or clamped edges,
shown in Fig. 13.21. A concentrated load P is applied at the center of the plate. Owing to
the symmetry of the plate, only one-quarter of the plate is modeled. Two different mesh
orientations (1 and 2 of Fig. 13.21) are used in the analysis. Four different sizes of meshes are
considered: 2, 8, 32, and 128 triangular elements are used to form the meshes corresponding
to N 1 , 2 , 4 , and 8, respectively, where N is the number of rows and columns of elements

in the mesh. In all cases, simply supported as well as clamped boundary conditions are
considered.
The results of the computations are shown in Fig. 13.22 to Fig. 13.25. Figures 13.22 and
13.23 show that the DKT and HSM elements are quite effective. However, mesh 2 does
not appear to be very effective in modeling the clamped plate problem, since all DOF of
the corner element vanish. The influence of mesh orientation on the displacement is more
severe for the DKT element than for the HSM element, as it is seen that the error curves of
the DKT are further apart than for the HSM element. It is apparent that the mesh orientation
has a significant effect on the accuracy.
The stress resultant calculations are shown in Figs. 13.24 and 13.25. The results for the
moment reaction at the center of the side of the plate are given in Fig. 13.24, where it can be
seen that all of the results converge. This means that the boundary condition m n

=

0 can be
satisfied. The moment reactions of the corner points are shown in Fig. 13.25 and very good
convergence is observed.

=

In addition to the three-node triangular elements discussed in the previous sections,
higher order elements are also available. One example is a 21-DOF element shown in