Mechanical Engineering ETDs

Publication Date



An eigenvalue-eigenvector approach is taken to study the details of the causes of and remedies for the overly stiff behavior exhibited by this element. The exact functional variation of the eigenpairs with element aspect ratio and material properties are derived for rectangular elements. These analytic expressions for the eigenpairs have not previously appeared in the literature. It is then shown that only the eigenvalues of the two flexure modes are decreased by the lower integration rules that are often applied, thereby causing a more flexible response. Knowing the variation in the eigenvalues with the various integrations then allows the element to be formulated in terms of the stabilization matrix. With this formulation, finite element simulations of a cantilever beam were run in which several methods of setting the flexure mode eigenvalues were used. The ability of each method to provide more flexible displacements for three aspect ratios and two Poisson ratios were compared.

The results show that decreasing the flexure eigenvalues works well for systems whose response is dominated by the flexure modes. This is the case when the beam thickness is modelled by only a few elements. However, care must be taken to ensure that monotonic convergence is not lost. But as the number of elements used through the beam thickness increases, the contribution of the flexure modes to the overall system response decreases and so does the effect of changing the element's flexure mode eigenvalues.

Degree Name

Mechanical Engineering

Level of Degree


Department Name

Mechanical Engineering

First Committee Member (Chair)

Howard Linn Schreyer

Second Committee Member

C. Randall Truman

Third Committee Member

Samuel W. Key

Document Type