 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  (c) A.F. Bower, 2008
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