Chapter 12

Imagine that a garden hose has an elliptical cross section. Explain why internal pressure causes bending stresses to appear, and sketch their approximate variation over the outside perimeter of the cross section. Also explain why, if internal pressure is to be carried by membrane stresses alone, the cross section must first become circular,

Write an expression for strain \(\epsilon_{s}\) if the arch radius \(R\) is a function of \(s\).

The conical shell shown is fixed at its base and loaded by torque \(T\) at the top. Use mechanics of materials concepts, not finite elements, to answer the following questions. (a) What is shear stress \(\tau_{s \theta}\), in terms of \(T, r_{1}, t, \phi\), and \(s ?\) (b) What is the angle of rotation of the top of the truncated cone relative to the bottom, in terms of \(T, r_{1}, t, \phi, L\), and shear modulus \(G ?\)

Consider a thin cylindrical shell of radius \(R\) whose midsurface axial strain \(\epsilon_{m}\) is unrestrained. By considering the energy associated with membrane strains \(\epsilon_{m s}\) and \(\epsilon_{m g}\), show that displacement \(w\) is in effect resisted by an elastic foundation of modulus \(E t / R^{2}\) (as well as being resisted by bending stiffness).

Consider a cylindrical shell, thin-walled and symmetrically loaded, but without axial loads. Thus \(N_{s}=0, \epsilon_{m s}=-\nu \epsilon_{m \theta}\), and displacement \(u\) need not be considered. Generate the 4 by 4 element stiffness matrix. Use a cubic \(w\) field.

(a) Define terms in the Jacobian matrix [J] in terms of \(\zeta, N_{l}, N_{i, k}, N_{i, m}\) nodal coordinates, and components of \(V_{3 i}\) (b) Specialize your result for the case of an element that is flat, of constant thickness, and whose midsurface coincides with the \(x y\) plane.