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corner nodes and four rotational (slope) DOF at the midpoints of the sides. Use the
displacement function
4 x 2
6 y 2
7 x 3
8 y 3
N u
w =
w
= w
+ w
2 x
+ w
3 y
+ w
+ w
5 xy
+ w
+ w
+ w
1
13.36 Set up a triangular element analogous to the rectangular element of Problem 13.35.
Use six DOF.
13.37 Formulate the stiffness matrix of a four node quadrilateral element with
w
θ
,
x , and
θ
y as independent variables at the nodes. Use linear interpolation polynomials for all
of these variables. Use the Reissner-Mindlin plate theory.
Hint:
Follow the development of the rectangular element in Section 13.5.1.
13.38 Derive the stiffness matrix for a 12 DOF rectangular element based on Kirchhoff
theory. The DOF at each node are
y .
13.39 Calculate the deflection of the plate shown in Fig. P13.16 using the finite element
method and compare your results with those of Problem 13.16.
13.40 Use a finite element solution to see how close you can approximate the circular plate
results of Problem 13.21 (Fig. P13.21).
13.41 Based on rectangular elements, find the maximum deflection in a simply supported
rectangular plate that is twice as long
w
,
θ x =− ∂w/∂
x,
θ y =− ∂w/∂
. The load is uniformly
distributed of magnitude p 0 (force/area). Use numerical values to compare your so-
lution to the answer given for Problem 13.11.
13.42 Use the finite element method to find the natural frequencies for a simply supported
rectangular plate. For particular numerical values, compare your frequencies with
those of Problem 13.17.
13.43 Find the natural frequencies of a circular plate with no center hole if the outer rim
is simply supported. For particular numerical values, compare with the frequencies
given in Problem 13.26.
13.44 Derive the expressions for the shape functions in Eq. (13.130).
13.45 Obtain the formulas for the rotations of Eq. (13.129) for a four-node quadrilateral
DKT element.
13.46 Derive L ij of Eq. (13.154) by using the relationships in Eqs. (13.128), (13.129), (13.152),
and (13.153).
13.47 Give the explicit form of B of Eq. (13.146) and B 1 of Eq. (13.158) for a homogeneous
isotropic plate.
13.48 Calculate C ij of Eq. (13.155).
13.49 Give the explicit form of the matrix C of Eq. (13.157).
(
2 L
)
as it is wide
(
L
)
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