Chapter 13

Plane triangles (a) and (b) shown have homogeneous material, uniform thickness, straight sides, and uniformly spaced side nodes. Their stiffness matrices can be exactly integrated, by a six-point rule for triangle (a) and by a ten-point rule for triangle (b). In a refined mesh, for an incompressible material, what constraint ratios do the respective elements provide?

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

Model a uniform cantilever beam by a single conventional beam element, fixed at its left end. Neglect transverse shear deformation. For the following constraint conditions, use the Lagrange multiplier method to determine the deflection of a transverse load \(P\) applied at the right end. (a) The right end is to remain tangent to a straight line between the two ends. (The line rotates as the beam deforms.) (b) The right end is to rotate half as much as the midpoint of the beam, but in the. opposite direction.

Consider the equations \(8 u_{1}-4 u_{2}=-20,-4 u_{1}+8 u_{2}=4\). (a) To what simple arrangement of identical springs does this equation correspond? (b) Impose the constraint \(u_{1}=0\) by the Lagrange multiplier method. Solve for \(u_{2}\) and \(\lambda\) and interpret the meaning of \(\lambda\). (c) Impose the constraint \(u_{1}=-1\) by the Lagrange multiplier method. Solve for \(u_{2}\) and \(\lambda\), and interpret the meaning of \(\lambda\).