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Problems
for Chapter 10
Approximate
Theories for Solids with Special Shapes:
Rods,
Beams, Membranes, Plates and Shells
10.6. Simplified
Versions of General Shell Theory
10.6.1. Consider a shell that is so thin that the internal
moments  all vanish.
Find the simplified equations of motion for the internal forces and the transverse force in terms of relevant geometric parameters.
10.6.2. The figure shows a thin circular plate with
thickness h, mass density ,
Young’s modulus E and Poisson’s
ratio that is simply supported at its edge and is
subjected to a pressure distribution acting perpendicular to its
surface. The goal of this problem is
to derive the equations governing the transverse deflection of the plate in
terms of the cylindrical-polar coordinate system shown in the figure.
10.6.2.1.
Write down the
position vector of a point on the mid-plane of the undeformed plate in terms
of ,
expressing your answer as components in the basis.
10.6.2.2.
Calculate the
basis vectors and ,
expressing your answer as components in the basis shown in the figure.
10.6.2.3.
Find the
components of the Christoffel symbol for the
undeformed plate;
10.6.2.4.
Calculate the
contravariant components of the metric tensor
10.6.2.5.
Find the basis
vectors for the deformed plate, neglecting terms of
order ,
etc
10.6.2.6.
Show that the
curvature tensor has components
10.6.2.7.
Express the
internal moments in the plate in terms of and and its derivatives.
10.6.2.8.
Write down the
equations of motion for the plate in terms of and
10.6.2.9.
Hence, show
that the transverse displacement must satisfy the following governing
equation
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