Chapter 9

The following question is strictly mathematical, and serves as a review of the Lagrange multiplier method. What is the area of the largest rectangle that can be inscribed in the ellipse \((x / a)^{2}+(y / b)^{2}-1=0\) ?

A bar of axial stiffness \(k=A E / L\) lies along the \(x\) axis and is allowed only axial displacements \(u\). Its right end carries a force \(P=3\) in the \(+x\) direction. Its left end (node 1\()\) is to be displaced two units to the right. Impose displacement \(u_{1}=2\) and solve for \(u_{2}\) by use of a Lagrange multiplier.

A rectangular bilinear element (four nodes) is subjected to the constraint \(\iint\left(\epsilon_{x}+\epsilon_{y}\right) d x d y=0 .\) Show that the same constraint is produced by setting \(\epsilon_{x}+\epsilon_{y}=0\) in a one-point Gauss quadrature rule.