## Mechanical Engineering ETDs

#### Title

A Finite Element for the Three-Dimensional Deformation of a Circular Ring of Arbitrary Cross Section

#### Publication Date

12-22-1982

#### Abstract

A two-node finite element for relatively compact, solid curved beams is derived. A node is located at the centroid of each end of the beam, and each node has six degrees of freedom, consisting of three translations and three rotations in mutually perpendicular directions. The element stiffness matrix has dimensions 12x12. A cylindrical polar coordinate system is used to express the components of the associated displacement and load vectors. As a consequence of the assumption of a nonzero cross-section product of inertia, deflections in the plane and out of the plane of beam curvature are coupled. Hence in-plane loads can lead to out-of-plane deflections, and out-of-plane loads to in-plane deflections. The stiffness matrix is determined by applying specially chosen sets of boundary conditions to a curved beam displacement general solution to determine the values of its unknown constants. These constants are substituted into a related set of curved beam stress resultant equations to give the stiffness matrix coefficients. Primarily for checking purposes this approach is also applied to the simpler case having a zero cross-section product of inertia, which results in the uncoupling of the in-plane and out-of plane deflections; two separate 6x6 stiffness matrices are obtained. A multi-element stiffness equation for a complete ring is assembled.

Either coupled, in-plane, or out-of-plane stiffness matrices can be used in this assembly. A modified stiffness equation that allows more compact computer storage is given. Each of the three types of elements is checked for internal consistency, and tested against independent results.

#### Degree Name

Mechanical Engineering

#### Level of Degree

Masters

#### Department Name

Mechanical Engineering

#### First Committee Member (Chair)

Alan O. Lebeck

#### Second Committee Member

Howard L. Schreyer

#### Third Committee Member

Illegible

#### Sponsors

Office of Naval Research

#### Document Type

Thesis

#### Language

English

#### Recommended Citation

Knowlton, Jeffrey Stuart. "A Finite Element for the Three-Dimensional Deformation of a Circular Ring of Arbitrary Cross Section." (1982). https://digitalrepository.unm.edu/me_etds/179