Chapter 7

A ten-node triangular element has a node at its centroid. In area coordinates, its shape function is \(N_{10}=27 \xi_{1} \xi_{2} \xi_{3}\). Let \(\phi\) be the field variable. Imagine that, with \(\phi=0\) at all nodes on the boundary of the element, nonzero \(\phi_{10}\) produces ordinate \(\phi=\phi_{a}\) at a point midway between node 10 and a vertex of the triangle. What is the volume under the \(\phi\) surface, in terms of \(\phi_{a}\) and triangle area \(A ?\)

Let uniform pressure \(p\) act normal to one face of a quadratic tetrahedron that has straight edges and midedge nodes. What are the consistent loads at nodes on this face?