The Application of Complex Variable Theory to Reinforcing Ring Design

Abstract

The complex variable method is applied to the analysis of a traditional constant-depth reinforcing ring as used on a pressurized single discontinuous bend (Fig. 1). It is shown that there is a “cross-over” point where complex variable theory and bending strain energy theory take over one from the other. This condition occurs when ring depth divided by the cylinders’ mean radius is approximately equal to 0.3. This criterion is of special interest since it falls within the range of factors used in industrial designs, i.e., the Mackenzie and Beattie bend [1] has a ratio of 0.29. Limitations of the complex variable theory are investigated with a detailed theoretical and experimental study of an internally pressurized prismatic elliptic cylinder having a circular bore (Fig. 2). Three bore sizes are investigated to gain knowledge of the convergent/divergent characteristics of the theory. For the largest bore size, numerical results show a definite divergence.