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with t he differential stiffness operator k D
u D T ED u
The quantities p V and p contain p z
and p s , the applied transverse loading and the loading on the edges of the plate, respectively.
Since u
=
.
=
[
w
] and
2
x
2
y
D u =
(13.83)
2
x
y
the virtual work becomes
∂(∂ xx + ν∂ yy ) +
xx
w
δ
W
=−
A δw
K
2
(
1
ν)∂ xy +
p z
dA
+
S p δw
p s
ds
=
0
(13.84a)
yx
∂(∂ yy + ν∂ xx )
yy
so that
k D
=
K [ xx
∂(∂
+ ν∂
) +
2
(
1
ν)∂
+
∂(∂
+ ν∂
)
]
(13.84b)
xx
yy
yx
xy
yy
yy
xx
13.4
Shear Deformation Effects
The history of the attempts to develop analytical models to represent the bending of a
plate can be traced to the early 1800s and the work of Sophie Germain 4 , Lagrange, and
Poisson [Todhunter and Pearson, 1886]. These efforts resulted in what we now refer to
as Kirchhoff or classical plate theory, with transverse shear strains
yz neglected.
Poisson (1829) addressed the question of boundary conditions and contended that three
boundary conditions should be prescribed on each edge. As mentioned in Section 13.3.1,
Kirchhoff (1850) reasoned that two edge conditions are more suitable than three and defined
a special shear force to reduce the number of forces on the boundary from three to two. As
mentioned previously, Thomas and Tait (1883) supplemented an energy-related explanation
of the equivalent shear force with a physical clarification.
The fundamentals of the plate theory became clearer when Eric Reissner (1945) intro-
duced the effect of transverse shear strains. Mindlin 5 (1951) formulated the plate bending
problem with the influence of rotary inertia and shear deformation, which is consistent
with Reissner's formulation. Plate theory, including transverse shear deformation effects,
which is sometimes called the Reissner-Mindlin plate theory, will be considered briefly here.
γ
xz and
γ
4 Sophie Germain (1776-1831) was France's most significant female mathematician during her time. She was self-
educated, using the library of her well-to-do father. She decided to become a mathematician, over the objections of
her parents, when, at thirteen, she read of Archimedes being killed by a Roman soldier. As a teenager she obtained
some lecture notes from the Ecole Polytechnique, including some of those of Lagrange. Using the pseudonym Le
Blanc, she submitted a major homework project to Lagrange. He was so impressed that he discovered her real
name and began to assist her. She initiated correspondence with scholars such as Legendre. She returned to the
use of the name Le Blanc in corresponding with Gauss. He discovered her real name after she inquired of a French
general as to the safety of Gauss when French troops were occupying parts of Germany.
5 Raymond David Mindlin (1906-1987), an American civil engineer, was born in New York City and attended
Columbia University for all of his degrees. He remained as a faculty member at Columbia, where he had a
distinguished career including significant developments in plate theory. His 1936 PhD dissertation presented the
elasticity solution of the “Mindlin Problem,” the effect of a concentrated force in the interior of a semi-infinite
solid.
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