tailieunhanh - Ebook A first course in the finite element: Part 2

(BQ) Part 2 book "A first course in the finite element" has contents: Isoparametric formulation, three dimensional stress analysis, plate bending element, heat transfer and mass transport, fluid flow, thermal stress. | 10 CHAPTER Isoparametric Formulation Introduction In this chapter, we introduce the isoparametric formulation of the element sti¤ness matrices. After considering the linear-strain triangular element in Chapter 8, we can see that the development of element matrices and equations expressed in terms of a global coordinate system becomes an enormously di‰cult task (if even possible) except for the simplest of elements such as the constant-strain triangle of Chapter 6. Hence, the isoparametric formulation was developed [1]. The isoparametric method may appear somewhat tedious (and confusing initially), but it will lead to a simple computer program formulation, and it is generally applicable for two- and threedimensional stress analysis and for nonstructural problems. The isoparametric formulation allows elements to be created that are nonrectangular and have curved sides. Furthermore, numerous commercial computer programs (as described in Chapter 1) have adapted this formulation for their various libraries of elements. We first illustrate the isoparametric formulation to develop the simple bar element sti¤ness matrix. Use of the bar element makes it relatively easy to understand the method because simple expressions result. We then consider the development of the rectangular plane stress element sti¤ness matrix in terms of a global-coordinate system that will be convenient for use with the element. These concepts will be useful in understanding some of the procedures used with the isoparametric formulation of the simple quadrilateral element sti¤ness matrix, which we will develop subsequently. Next, we will introduce numerical integration methods for evaluating the quadrilateral element sti¤ness matrix and illustrate the adaptability of the isoparametric formulation to common numerical integration methods. Finally, we will consider some higher-order elements and their associated shape functions. 443 444 d d 10 Isoparametric Formulation d Isoparametric .