Lesson 6 - Strain Gradient Representation of Discrete Structures

Abstract

The purpose of this chapter is to present procedures for identifying and utilizing linearly independent sets of strain gradient quantities that a two-dimensional truss is capable of representing as a way to highlight the role of individual strain states as generalized coordinates. The chapter uses strain gradient notation to provide an alternative set of independent coordinates for describing altered configurations of the truss. The nodal movements are expressed in terms of rigid body motions and various types of deformations or strain states. The strain gradient quantities are considered as generalized coordinates and are used to extract equivalent continuum parameters from truss structures. Because the joints of a truss correspond to the nodes of a finite element, the capabilities developed in the chapter apply directly to the formulation of finite element stiffness matrices. Before beginning the formal development, the relationship between the strain gradient coefficients and the deformations of a square, four-node truss is presented to illustrate the type of results produced in this chapter. A four-node configuration is capable of representing the five deformations. These deformations correspond to the five strain states. The result is used to formulate transformation matrices from nodal displacements to strain gradient coordinates. The [Ф] matrices for the seven configurations are used to develop finite element stiffness matrices. These transformations are applied to compute the equivalent continuum parameters of a truss. These examples are designed to highlight the generalized coordinate nature of the strain gradient quantities. The equations that compute the strain energy in the discrete structures and its specializations, hint at an alternative procedure for formulating finite element stiffness matrices.