chapter  3
20 Pages

- Mechanics Aspects of Lattice Models

A continuum can be discretized in a lattice of truss or beam elements. This has been known for a long time: in 1941 Hrennikoff showed how the element parameters should be set to obtain the same elastic properties in a truss lattice and a shell element loaded in plane stress. A lattice can consist of different types of elements. Linear elements are always used, but the connectivity may vary. Truss elements (Figure 3.1a) have two degrees of freedom (dof) in each node (displacements ui and vi); beam elements also include rotations (φi) and have 3dof per node (Figure 3.1b). Thus for the beam element of Figure 3.1b the displacement vector is

where the subscript i and j refer to the two nodes. For the simple truss element of Figure 3.1a the two rotations φi and φj are missing. The hyphen in v indicates that displacements are considered in the local coordinate system. This is the situation for simple linear elements deforming in a two-dimensional plane. We start by brie¸y describing the set of equations needed in a 2D truss or beam-lattice. Generalization to three dimensions is quite elaborate but straightforward.