91Ó°ÊÓ

If a problem is to be mathematically two-dimensional, \(\theta\) independence is required of all dependent variables. Explain by example why this requires that \(\theta\) be a principal direction of an orthotropic material. Suggestion: Consider axial load on a cylinder.

The sketch shows the cross section of a flat element shaped like a metal washer. D.o.f. are radial displacements \(u_{1}\) and \(u_{2}\) at nodal circles 1 and \(2 .\) The material is isotropic. (a) Formulate matrices [N] and [B]. (b) Let \(\nu=0\), and generate \([k]\) for a one-radian segment by explicit integration. (c) Let \(L=r_{2}-r_{1}\) and \(r_{m}=\left(r_{1}+r_{2}\right) / 2\). Simplify integration by assuming that \(r=r_{m} .\) Hence, determine [k] (for a nonzero Poisson's ratio). For what geometry is this [k] a good approximation? (d) For \(\nu=0\), show that the \([k]\) 's of parts (b) and (c) agree for \(r_{m}>>L\) (e) From part (b), obtain \([\mathrm{k}]\) for the special case \(\nu=r_{1}=u_{1}=0 .\) (f) From part (c), obtain \([\mathrm{k}]\) for the special case \(\nu=r_{1}=u_{1}=0\).

The sketch represents three nodes on a \(z=\) constant face of an axisymmetric quadratic element. Node 7 is at midside. Determine the consistent nodal load vector for these three nodes if \(z\)-direction surface traction \(\Phi_{z}\) is applied as follows. (a) \(\Phi_{z}\) is the constant value \(p\) over the face. (b) \(\Phi_{z}=\left(\xi^{2}-\xi\right) p_{4} / 2+\left(1-\xi^{2}\right) p_{7}+\left(\xi^{2}+\xi\right) p_{3} / 2\), which is a parabolic variation based on nodal values \(p_{4}, p_{7}\), and \(p_{3}\).

Show that \(\epsilon_{r}=\epsilon_{\theta}\) at \(r=0\) in an axially symmetric problem.