Chapter 3
Finite Element Formulation of Beam Elements
3.1
What Is a Finite Element?
To illustrate the concept of the finite element method (FEM), let us go back to the era of 250 BC, when the famous ancient Greek mathematician, physicist, engineer, inventor, and astronomer Archimedes studied rectification of a circle. The problem is “to find a relationship between the perimeter P and the diameter D of a circle.” The answer to this problem is to find a numerical value for π from the following equation: P¼π D P n where, π ¼ i¼1 sin ð180=nÞ ¼ n sin ð180=nÞ
ð3:1Þ
The perimeter P of a circle with diameter D as shown in Fig. 3.1a will be approximated by n polygons with equal side lengths (Fig. 3.1b) that can be calculated from a simple trigonometry relationship (Fig. 3.1c). Values of π calculated from (3.1) are shown in Table 3.1. Since the perimeter P of the circle can be approximated by summing the length of n-polygon sides, we need to increase the value of n more than one million divisions to reach the exact value of π to 12 decimal places. This example has been presented in most of the FEM textbooks and literature to teach the finite element concept (Bathe 1982; Cook 1981; Fung 1969; Shames and Dym 1985; Washizu 1982).
3.2
Beam Element in the Real World
Most of the structural components in man-made engineering structures are composed of beams. A beam is a long-block-like structural member whose primarily function is to support transverse loading and distribute it to the supports (Fig. 3.2). © Springer International Publishing AG 2018 B.S. Gan, An Isogeometric Approach to Beam Structures, DOI 10.1007/978-3-319-56493-7_3
[email protected]
61
