Page 252 - Marine Structural Design

228 Port I1 Ultimate Strength

The objective of this chapter is to present a theoretical formulation for the modeling of strain-
rate hardening effects, and show how these effects can be implemented into three-dimensional
finite beam-column elements. The finite beam-column element is ideally suited for the impact
analysis of frames with large displacements, strain hardening, and strain-rate hardening. The
accuracy and efficiency of the element is examined by comparing the present results with
those obtained from experiments by others, rigid-plastic analyses, and from existing finite
element analysis results, see Part I1 Chapters 13 to 15. For the fundamental theory of finite
element analysis, the readers may refer to Przemieniecki (1 968), Zienkiewicz (1977), Bathe
(1987), among many other books. To understand plasticity used in the section on the plastic
node method, some basic books such as Save and Massonnet (1972), Yagawa and Miyazaki
(1985), Chen and Han (1987), Chakrabarty (1987) may be helpful. To aid the understanding of
the plastic node method, a basic theory of plasticity is presented for finite element analysis of
solids, based on Yagawa and Miyazaki (1985).
Part of the formulation presented in this Chapter appeared in Bai and Pedersen (1991) and
Fujikubo et a1 (1991). The new extension is to account for the effect of strain-rate hardening
for dynamic analysis.

12.2 Elastic Beam-Column With Large Displacements
The element has three translational displacements u,,uy , and u, and three rotational
displacements 0, ,Oy, and e,, see Figure 12.1.

aOIpt)
1.2 = END NODES OF BEAM-COLUMN ELEMENTS
3 = AUXlLlARV NODE

Figure 12.1 Three-Dimensional Beam Elements with Nodal Forces

These displacements are interpolated by using a polynomial interpolation of fimctions, which
are associated with the Timoshenko beam theory. A generalized strain vector is subsequently
established in the form: