Chapter 1

Sketch a quadrilateral with corners properly lettered and \(\xi \eta\) axes properly oriented if shape functions are written as \(N_{A}=\frac{1}{4}(1-\xi)(1+\eta)\), \(N_{B}=\frac{1}{4}(1+\xi)(1+\eta), N_{C}=\frac{1}{4}(1-\xi)(1-\eta), N_{D}=\frac{1}{4}(1+\xi)(1-\eta)\)

A cylindrical pipe, shown in cross section, has nominal temperatures \(T_{1}\) on the inside and \(T_{2}\) on the outside. The standard analytical solution for temperature \(T\) at arbitrary radius \(r\) in the pipe is $$ T=T_{1}+\left(T_{2}-T_{1}\right) \frac{\ln \left(r / r_{1}\right)}{\ln \left(r_{2} / r_{1}\right)} $$ However, actual circumstances may differ sufficiently from the ideal that this equation is not accurate enough, and temperature distribution must instead be determined from FEA. What are some of these circumstances?