91Ó°ÊÓ

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

(a) Define terms in Jacobian matrix \([\mathrm{J}]\) in terms of \(\zeta, N_{i}, N_{i, \xi}, N_{i, \eta}\), nodal coordinates, and components of \(\mathbf{V}_{3 i}\) (b) Specialize the results of part (a) for an element that is flat, of uniform thickness, and whose midsurface coincides with the \(x y\) plane.

Consider a cylindrical shell, thin-walled and symmetrically loaded, but without axial loads. Thus \(N_{s}=0\), membrane strains have the relation \(\varepsilon_{m u}=-v \varepsilon_{m \theta}\), and neridional displacement \(u\) need not be considered. Use a cubic \(w\) field, and deternine the 4 by 4 element stiffness matrix that operates on nodal d.o.f. \(w_{i}\) and \(\psi_{p}\).

Model a complete circular ring by four straight elements of equal length. The FE model is therefore a square, as shown. Calculate the relative separation of loads \(P\), accounting for deformation due to bending only. Take the element length as (a) chord length \(L=\sqrt{2} R\), and (b) arc length \(L=\pi R / 2\). The exact result is \(0.1488 P R^{3} / E I\). Suggestion: Apply elementary beam theory.