Chapter 16

The three-node bar shown is uniform. Temperature \(T_{1}\) is prescribed, \(Q=\) 0, node 3 is insulated \(\left(q_{3}=0\right)\), and heat is transferred across the lateral surface by convection. In the units used in Section 16.1, let \(k=180, h=\) \(12, A=0.1, p=1\), and \(L=2\) (a) Use two two-node elements to determine \(T_{2}\) and \(T_{3}\) in terms of \(T_{1}\) and \(T_{f}\). (b) Repeat part (a), but use a single three-node element of length \(2 L\). Note that \(\left[h_{i s}\right.\) ] has the form of a mass matrix, and use an "optimally lumped" form (see Eq. 13.3-9).

The three-node triangle shown is to be used for heat conduction analysis. The body in question is homogeneous, plane, isotropic, and of unit thickness. (a) Evaluate \([\mathrm{k}]\) in terms of \(k\) and nodal coordinates. (b) Evaluate \([\mathrm{h}]\) in terms of \(h\) and nodal coordinates if only side \(1-3\) transfers heat by convection. (c) Write the "lumped" forms of \([\mathrm{h}]\) and \([\mathrm{c}]\). (d) Write \(\left\\{\mathbf{r}_{Q}\right\\}\) in terms of \(Q\) and nodal coordinates if \(Q\) is constant over the element.