Computer Aided Engineering Design

310 COMPUTER AIDED ENGINEERING DESIGN

time taken will be more. Thus, there is a trade off involved between the average element size and
solution time taken which a designer should keep in mind when performing mesh generation which,
by itself, is a very vast and active field of research. Appendix 1 is provided to familiarize the reader
with some preliminary methods and algorithms on discretization, mostly in two dimensions. As this
chapter deals with discrete (truss, beam and frame) and continuum (triangular and quadrilateral)
elements in two dimensions, some methods pertaining to only the abovementioned elements are
discussed in the appendix noting that most methods may be extended for use in three dimensions. We
may realize at this stage that discrete representation of solids is another approach in solid modeling
wherein a solid’s volume may be regarded as the sum total of the volumes of constituting tetrahedral
or hexahedral elements. To create a discrete representation using mesh generation would, however,
require the B-rep information of the solid.
With regard to the finite element analysis, there are many texts available for an in-depth study.
This chapter, however, introduces preliminary concepts to the reader by presenting linear elastic
analysis using some widely used basic elements. The finite element method as known today was
investigated in the papers of Turner, Clough, Martin and Topp, Argyris and Kelsey and many others.
The name finite element was coined by Clough. The advent of digital computers in the 1960s and
1970s provided a rapid means of performing intricate calculations involved in the analysis that made
the method practically viable. With the development of high speed digital computers, the application
of the finite element method also progressed at a very impressive rate. Zienkiewicz and Cheung
presented a broad interpretation of the method and its applicability to any general field problem. As
a result, the finite element equations could also be derived using general methods like the weighted
residual (Galerkin) method. This led to a widespread interest among other researchers working with
generic nonlinear differential equations.

11.2 Springs and Finite Element Analysis

Preliminary concepts of the finite element analysis are presented here using linear springs. Consider
a spring of stiffness kp shown in Figure 11.1(a). The nodal displacements are allowed along the
horizontal direction which makes the spring a two degree-of-freedom system, one at each node. Let
the displacements at nodes i and j be ui and uj and the external forces acting along the axis be fi and
fj, respectively. Considering the equilibrium at nodes i and j using Newtonian mechanics, we have

fi = kp(ui– uj)

fj = kp(uj – ui) (11.1)

(a) (b)

(c)

uj ul uk

fi fj fl fk

Fi

Fj Fk

ui uj u

k

Figure 11.1 Springs: (a) and (b) with stiffness kp and kq and (c) assembled in series