10 - Structural mechanics problems in one dimension — rods

Abstract

The aim of this chapter is to discuss structural mechanics problems in one dimension—rods. Bending of rods is generally associated with a beam theory, such as the classical Euler-Bernoulli theory studied in introductory strength of materials. If one attempts to model a rod with a standard three-dimensional finite element model, two aspects give difficulty. One is purely numerical and associated with large round-off errors when attempting to solve the simultaneous equations. The other is a new form of locking in interactions among bending, shear, and axial behavior when low-order elements are used. Often, a much more economical solution is to use a structural mechanics approach in which the problem is formulated as a one-dimensional problem along the axis of the rod. Using this approach and appropriate interpolation forms one can avoid numerical difficulties associated with round-off and locking. This chapter presents the approximation for two classical rod theories. The first combines the Euler-Bernoulli theory of bending with axial and torsion theories. The second form presents the Timoshenko theory of bending together with the axial and torsion theories.