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