Cross Sectional Area Changes due to Plastic Bending of Prismatic Bars

Given the current trend in manufacturing to decrease part variability, and in order to increase product quality, dimensional tolerances are becoming more exacting. With this in mind, and with the decreased time allotted for components to progress from design to manufacture, it has become more critical that accurate models of the manufacturing process are developed. This paper investigates the changes in cross sectional area when a prismatic bar is plastically deformed into a ring of constant diameter. Through further processing, these rings are transformed into components that function to secure mechanical components, such as bearings, into assemblies. Failure of the ring can cause significant damage, or failure of the assembly. Typical thickness tolerances are on the order of +/−.002” (0.05 mm), but can be as small as +/−.0002” (0.005 mm). Also, a growing trend in manufacturing is for the final ring to have a specified thickness on the inner and outer edge within this tolerance band. The rings are produced in various metallic materials with different mechanical properties by continuously coiling prismatic bars to a specific diameter. An analytic model based on small strain theory was developed for the simple cross sections of rectangular and trapezoidal geometries. This model was then extended to include the effect of a hyperbolic rather than linear stress distribution through this simple section in order to relieve the constraints of small strain theory and adequately model the actual process. An empirical model was developed based on experimental observations. A numerical model was developed using the commercial finite element analysis (FEA) software Abaqus (SIMULIA, Providence, RI) to simulate the manufacturing process. This was compared to the empirical model developed from production parts for validation. Once the finite element model is validated, it could be used to explore the effects of design parameters (initial dimensions of the prismatic bar, material properties etc.) and create efficient designs for manufacturing. The empirical model can then be used in the design process. Additionally, the numerical simulation could be used to model more complex cross sectional areas which cannot be evaluated analytically. There was adequate agreement between the empirical and numerical models to the extent that the numerical model could be used for more complex cross sectional geometries. A further refinement of the analytic model to include finite strain theory should be used to expand on this.