M.Sc Thesis
M.Sc Student Peysin Alon Finite Element with Drilling Degrees of Freedom - Static and Dynamic Analysis Department of Civil and Environmental Engineering Professor Moshe Eisenberger

Abstract

In the finite element method, the analysis for the solution of linear plane problems often uses elements with 2 degrees of freedom in each node - vertical and horizontal movement. There are many different elements. The primary differences are in the shape functions that are used to describe the displacement field within the element as a function of the displacements at the element nodes, or the different ways to derive the stiffness matrix. There are triangular or quad elements. One often uses Cartesian coordinates, or other type of coordinate - such as local or area coordinates.

In the present work, new elements are developed, which belong to the family of elements with 3 degree of freedom in every node, and the additional degrees of freedom are drilling or vertex degrees of freedom. Elements of this family were developed originally in 1984. This type of elements has a great advantage over elements with two degrees of freedom leading to higher precision of the solution and faster convergence for the same meshes of element.  Additional advantage is in the ability to connect correctly between two plate elements in space and to connect between plate elements and one-dimensional beam elements.

The new element is obtained by starting with a high order element with many nodes on the edges and inside the element. For building a new triangle element, high order triangular element with 10 nodes are used with 2 degree of freedom in each node. For building the new quad element, high order quad element with 12 edge nodes is taken, with two degree of freedom in each node. The new shape functions are expressed by relating between the drilling degrees of freedom and the displacements of the corner nodes and internal nodes of the element. The new triangular element has three nodes in the corners of the element with 3 degree of freedom in each of them. The new quad element has four nodes in the corners of the element with 3 degree of freedom in each of them.

In the present work, we solved many static and dynamic known examples and compared them to the best-known result. The results of the new element in some of examples are the same as those of other elements, and more accurate in some of the examples. The new elements have fast rate of convergence and good accuracy even for coarse meshes.