Arborless Bending of Steel Tubing (Prospectus)

Introduction

Conventional bending of round tubing requires a form, called an arbor, around which the tube is bent, by means of a forming tool called a shoe. Both the arbor and shoe are shaped to enclose the tube, which assures that the tube retains its circular cross section throughout the bending process, preventing cross section ovalization and buckling.

However, the arbor creates undesirable constraints on the process, which prevent the conversion to modern, quick-changeover, flexible manufacturing. The arbor tooling and fixtures must be changed each time there is a change of tube size or bend radius. This is a major task, which shuts production down for an extended time. Large production runs are required in order to spread the cost of tooling changes over a large number of parts.

In addition, specialty bends, such as refrigeration coils, do not give access to the tubing, due to the closely spaced fins around each tube. The arbor is therefore a smooth cylinder of the proper radius. Although the fins provide some support to the tube cross section, an arbor is still required to achieve the desired bend radius of the coil.

In an arborless bend, major tool changes would be eliminated. Ideally, a new program would be downloaded to change the bend radius, bend angle and tubing size for an automated machine. Lot size could be as small as a single part. Or, several different bends could be applied in sequence to a single part. Overbend requirements could be built into the controls to account for springback.

The objective of this thesis is to define the boundaries and necessary conditions to allow an arborless bend with a pure bending moment load to provide the desired geometry, while avoiding unnecessary ovalization and without allowing buckling to occur. A boundary will be established by a defined critical bend radius and bend angle combination where buckling will occur within the 90° final bend radius limit. A variety of tube diameters and wall thicknesses will be considered.

In order to define this boundary a laboratory system has been assembled which places a pure bending load on a tube. With this system and the associated controls, we are able to produce a pure bending load that will create the desired geometry. By bending the tube to a radius that buckles the tube, we can see what factors are associated with that buckling. Using this data, the equations put forth by previous researchers, which relate the buckling moment and associated curvature will be verified and a failure boundary for the machine will be defined.

Background

Based on the basics of the mechanics of a material in pure bending set forth by Datsko, a better understanding of the maximum strain to failure can be understood. Also from Datsko, it is established that the ultimate engineering strain (εu) is equal to the exponent m in the true stress-strain equation. This allows us to find the maximum bend radius based on failure due to the ultimate strain. Datsko also dictates a more descriptive notation for the stresses and strains in a system where strain in all three dimensions occur. (Datsko 1966)

Calladine’s additions to the subject include establishing the equations for the critical buckling moment, curvature, and strain for a part considering both the bifurcation point splitting the tube that fails at one main buckling point versus corrugated buckling at multiple points. (Calladine 1974)

Brazier observed that buckling at a single location on a tube corresponds to a certain point on the moment-curvature plot. This behavior is called the Brazier Buckling phenomenon. He also established equations to predict the ovalization displacement along the cross section and the process of finding the maximum point on the moment versus curvature plot. This point defines the point where buckling occurs. (Brazier 1927)

Korol showed empirically that Brazier’s work was valid by using a moment-curvature plot of Korol’s data to show where buckling occurred. (Korol 1978)

K. Pan and K. A. Stelson cite Von Karman’s research in establishing ovalization occurring because of a continual change of direction of the forces parallel to the center line or neutral axis. (Pan and Stelson 1995)

Finally, P. Cheng of MIT published a thesis on pure bending and compression bending of tubes containing support structures within the outer shell of the tube. Her discussion of pure bending is invaluable for this project, as it not only brings much of the previous research together on tube bending and buckling, but she also corrects some of the ovalization equations set forth previously. (Cheng 1996)

Although pure bending of hollow structures has been covered theoretically by Cheng and others, none of the research described above has been applied to an arborless bender designed for production. Empirical data must be established in order to know where the moment-curvature buckling condition occurs in relation to the arborless machine setup. Ovalization of the tube in this configuration needs to be understood along with moment-curvature values in relation to the bend angle. Through empirical data, as well as mathematical models, these relationships will be investigated and a theoretical and empirical failure boundary for arborless bending will be established.

Research objectives

The objectives of this research are to establish the following:
• An empirically based buckling failure boundary for arborless bending processes, based on the desired bend radius, desired bend angle, tube diameter and wall thickness.
• A theoretical model of the ovalization and buckling of the metal tubes as pure bending is applied. This will be a model of the stresses and strains occurring within the tube at the time of buckling as well as throughout the bend.
• A theoretical model of the buckling failure boundary similar to the empirical boundary above. The model will include the parameters of the bend process itself and show what conditions will occur within the machine at the failure of the tube.

Proposed Research

The approach to finding a valid buckling failure theory will be based on both empirical data and theoretical models. The empirical model will be established by testing a number of different bend angles and radii. As a tube is bent to the desired geometry, the buckling mode will be noted as well as the bend angle at which the buckling occurred. This bend angle will be the boundary for this specific radius. Repeating this process with many tubes and bend angle verses radius combinations will produce an empirical failure boundary.

Another factor that will be studied is the effect of the desired final bend angle to the system. The distance the clamps are positioned apart through the bend depends on both the desired bend angle and the desired bend radius. These will affect the buckling as they will change the clamp separation.

The theoretical model of ovalization and buckling as a result of a pure bending moment has been mostly established by Calladine, Brazier, and Cheng. These models need to be applied to the arborless forming process at hand, which will require a study of the models to extract the terms that apply specifically to loads caused by pure bending. This process will lead to a theoretical model designed specifically for the arborless bending machine. Once this model has been established, we can apply the instantaneous curvature at each angle to this theoretical model to determine a theoretical failure boundary of the system.

The steps to merging the theoretical and empirical data are to first follow the guide of Datsko’s strain equations to understand the strain in the critical region and establish an empirical criterion for buckling using this method. Then, bend tube at decreasing radii until buckling occurs. At that point find what angle buckling occurs for each bend radius. Follow this process with different tube diameters and wall thicknesses. Finally, compare this data with the information gathered from the strain equations and show the compatibility.

Equipment

An arborless tube bending machine has been provided by the BYU capstone program and their sponsor, Burr Oak Tool Inc. The control system was built by Kevin Cole. The machine is controlled by a feedback control loop using a compact RIO controller, which is run with LabView and a desktop computer. The system controls three motors. In the center of the system, an AC gear motor drives the angular displacement of the bend. Two linear actuators clamped to the ends of the tubing make up the rest of the control. One of these actuators is fixed to the table and the other is on an arm whose angle is controlled by the motor. As the arm moves through the bend angle, the control system changes the linear actuators displacement in order to keep the neutral axis of the tube at its original length. The continuous clamp adjustments keep axial pressure from occurring and allow the system to apply only a pure bending moment to the tube.

Readings from the machine will be taken by the computer in two forms. One form will be purely visual. A web camera is attached to record the visual angle at which the buckling occurred. The data acquisition system records the current to the bend motor, which will provide a discontinuity when buckling occurs, and which will be tied to a time and angle. Using geometry, the angle can be related to a curvature and the critical bending moment will be detected when the current becomes discontinuous. The data will be gathered into the LabView control system set up by Kevin Cole and will provide the necessary information to compare the empirical data to the theoretical model to see if the model is valid.

Collaborated Efforts

This project was started in 2007 as a BYU Capstone project with an industrial sponsor. Kevin Cole has put many hours into the control system to get the computer to read the correct information, control the motors, and keep the neutral axis load free. Dr. Ken Chase has developed the equations for keeping the neutral axis at a constant length as the angle changes. Complete results will be shared with Burr Oak and the operational arborless bender will be shipped to their site, except for the computer and other components belonging to BYU.

References

Brazier, L. G."The flexure of thin cylindrical shells and other ‘thin’ sections." Proc.
Royal Society Series A no. 116 (1927): 104-114.

Calladine, C. R. "Limit analysis of curved tubes." Journal of Mechanical
Engineering Science 16 (1974): 85-87.

Cheng, Phoebe. Weight Optimization of Cylindrical Shells with Cellular Cores.
Cambridge, MA: MIT, 1996.

Datsko, Joseph. Materials in Design and Manufacturing. Ann Arbor, MI: Malloy Inc.,
1977.

Datsko, Joseph. Material Properties and Manufacturing Processes. New York, NY:
John Wiley and Sons Inc., 1966.

Korol, RM. "Buckling of circular tubes in bending." Journal of Engineering Mechanics
ASCE 104 no. 4 (1978): 289-290, 939+.

Pan, K., K. A. Stelson. "On the Plastic Deformation of a Tube during Bending."
Transactions of the ASME 117 (1995): 494-500.