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.

"Ye cannot go amiss"

In Doctrine and Covenants section 80, the Lord calls a member of the church on a mission to preach the Gospel. In verse three the Lord says, “Wherefore, go ye and preach my gospel, whether to the north or to the south, to the east or to the west, it mattereth not, for ye cannot go amiss.” In class we discussed how this scripture can be referred to missionaries around the world who wonder if they are where they are supposed to be. I also think that this scripture can refer to other aspects of life in a very important way.

I have been thinking hard about what exactly I want to do with my education when I graduate from college. As an engineer, there are so many options, and it is important to me to find something that I can enjoy and really get good at. While growing up, I noticed that I can be good at a variety of skills, and because of this realization I did everything I could. Even while I was in high school I noticed that I didn’t really have any focus in life other than the idea that I liked the sciences, and yet I didn’t focus on those studies much more than anything else. Now, after focusing more on studying the sciences and learning more about them, I am making the decision how I should even more specify my focus. Because I wanted to bring the Lord into my decision, I started studying the matter of choosing a direction, and this is what I’ve found:

“Wherefore, go ye…whether to the north or to the south, to the east or to the west, it mattereth not, for ye cannot go amiss.” In studying this subject, I found that Psalm 25 can be all about bringing the Lord into decisions like this. “The meek will he guide in judgment: and the meek will he teach his way. All the paths of the Lord are mercy and truth unto such as keep his covenant and his testimonies….What man is he that feareth the Lord? Him shall he teach in the way that he shall choose. His soul shall dwell at ease; and his seed shall inherit the earth” (Psalm 25:9-13). As we see in Revelation 3:15, the Lord wishes that we be useful in one way or another, and not lukewarm: “I know thy works, that thou art neither cold nor hot: I would that thou wert cold or hot.” And because He wants us to be useful in one way or another, he will guide us in those ways, and teach us in the ways that we shall choose.

I would like to quote from Elder M. Russell Ballard’s April 2009 Conference talk on this subject. This talk is titled, “Learning the Lessons of the Past.” “Look to your parents and others who have gone before you, [and] you will find examples of faith, commitment, hard work, dedication, and sacrifice that you should strive to duplicate….It’s hard to imagine a scenario in which it would not be worthwhile to consider and learn from the experience of other. Many professions require internships, during which aspiring professionals shadow seasoned veterans to learn from their years of experience and accumulated wisdom….Of course, there are times when we have no choice but to venture out on our own and do the best we can at figuring things out as we go along….It is my message and testimony to you today, my young friends, that for the most important questions of your eternal lives, there are answers in the scriptures and in the words and testimonies of apostles and prophets.”

My original idea about making decisions about what to do with my life was that I would continue studying diligently what I had been studying, and that somehow the Lord would help me see my mind open in one direction or another. From this study I realized that the Lord expects me to study what I am studying, apply myself in the direction that I am most inclined to go, and then He will support me and teach me in that way as long as I keep my covenants and testimony. He is waiting for me to make my decision and then He will tell me that it is the right decision for me.

Helpless

I never thought I would be in sunny Hawaii and wish I were home in Virginia in February. I timed this trip as poorly as any I have ever been on. It is a helpless feeling to be here and not be able to do anything or even help make decisions.
You have all been hearing about the big storm that has dumped nearly 30 inches of snow on Northern Virginia. I left on Thursday morning and we were already starting to worry a little, but not too much. After all, we had been snowed-in a couple of times this winter already, and it just meant that we would be about to stay home and enjoy the day. Who knew that it would knock out the power, and that now, two days later, the power would still be off with no estimated for when it would be back on. By Friday when I called Mom from Hilo the snow had started to pile up and the power went off, and we were starting to wonder if she should get out while she still could and stay with friends. The estimate from Dominion Power was that the power would be back on by 11 pm. The house was not any colder that it would have been on a normal night with the heat turned down, so Joyce decided to stay and to decide in the morning. It snowed all day on Saturday, and by then it was too late. The snow was too deep to drive through it. Joyce went out and shoveled a path for the car down the driveway, but the snow on the road was too deep, and it was still snowing. Sunday morning it had stopped snowing, but the temperature in the house was down to 45 degrees and there was still no estimate from the power company when it would be back on. Joyce finally got a neighbor with a four-wheel-drive to take her over to our friends, the Latimers, to stay until the power came back on.
I may have gotten some of the details wrong in what was going on at home – and there is the topic of this essay. What do you do when you want to help but are completely powerless to do anything except cheer from the sidelines? My feelings here ranged from real anxiety on Friday as we were trying to decide whether Mom should go or stay and not knowing exactly what the conditions were or, more importantly, what they were going to be, to envy and regret on Saturday that I couldn’t be there to enjoy being snowed in together, to worry again on Sunday when there still wasn’t heat or any way to drive anywhere and the temperature was dropping in the house. How often we have watched teenage or young adult children with the same emotions – anxiety not being able to help with a decision and not knowing the future, regret that we can’t be there to share an experience, and worry when things aren’t going as we expected and maybe even appearing to be getting worse. How often we have watched new converts, home teaching and visiting teaching families, and others we are working with in the Church in the same way.
It is good to know that if things get really bad there are good people around who can help. In this case we had neighbors who shared hot meals and helped when Joyce decided to leave. We had ward members who were concerned. Whenever I called we would be interrupted by other phone calls, usually from concerned ward members. It was good to have Jon’s home for a backup if Joyce could have gotten there. Jared and Daniel called to offer encouragement (thanks!). When you get teenagers you realize how valuable that support is. The genius of the Young Men/Young Women program is that you have someone there you can trust to teach correct principles to your children when you can’t be there yourself. The genius of the Church organization is that when all are doing their duty, the support is there for each one of us. The catch, of course, is that not all “concerned” helpers are faithful Church members teaching correct principles. Sometimes these voices are the hardest to counter and it at times becomes counterproductive to even suggest that they might be off the mark.
So this comes down to the final resort. Sometimes you are completely helpless and all that is left is prayer and faith. Can a person change the elements, get power restored, influence thoughts and actions of another, by praying and having faith? Will Dominion Power get our power back on sooner if I pray really hard about it tonight? Can I influence people I don’t even know who in turn influence the lives of a loved one? I know that it is not me doing the influencing, but will the Lord change things just because I ask him, no matter how sincere I am? The answer has to be “yes” or the scriptures would not tell us to do it.
This must be where faith comes in. Maybe it is not so much that we ask in faith as that we have faith, after we have asked, that the right thing will happen. Not necessarily that the thing that we want will happen, but that whatever happens the Lord will make it right. That is perhaps the hardest aspect of faith. How hard is it to see a situation develop that we don’t like or don’t agree with, and let it happen. When is it right to have faith that God is in charge, and when is it right to step in and try to change it? Should we always try to change things, right up to the last minute that we think we can? Do we have a moral obligation to do everything we can to make things right?
I don’t know the answer to these questions. It is perhaps the hardest tightrope there is to walk in our relationships. One difference between the Law of Christ and the Law of Moses is that Christ tells us that it is always our duty to do good without worrying about reward or consequences. On the other hand, I don’t recall any instance of Christ doing good for someone who didn’t want it.
Anyway, here I am forced to cheer from the sidelines as Joyce does battle with the elements. They have already issued another winter storm warning for Tuesday. At least she is warm and has good company for now.

Interviews

Few parenting ideas – no, it would be more accurate to say that NO OTHER parenting idea has ever come to me with more force than the idea of doing regular interviews with my kids. This is a rather recent idea (within the last year) but I feel that if I weren’t to do it, God would hold me personally responsible. I plan on being consistent with interviews until each of my kids leaves the house. Maybe longer…

I do an interview with two of my kids one Sunday, and then my other two kids on the next Sunday. I remember by writing their names in my planner with a box around them. I have their boxed names written in my calendar on each Sunday for months ahead of time.

I don’t have a time of day scheduled, but they usually happen in the evening. If Sunday doesn’t work out, then they happen on Monday night after FHE.

Perhaps one reason that this idea hit me with such force, is that I never challenged my assumption that interviews were only a male/priesthood assignment. I guess that I needed a clonk on the head to notice the fact that I can give interviews, too.

Mike likes the idea of interviews, and whenever he’s home, he does them with me. I love having him do interviews with me because he brings the Spirit into the conversation in such a strong way. But, I’ve done interviews many times on my own when he’s gone, and, to be frank, even when we do them together, I’m the one who remembers to makes interviews happen. I think this is just fine. He has other amazing parenting ideas and inspirations – things that he remembers to make happen.

When we do interviews, we each sit on one side of our kid, snuggled on our bedroom couch. The door is closed and everyone, including baby, are outside. Sometimes we pray first, and sometimes not. And then what happens? We simply talk. Unstructured.

There are, however, two things that I always bring up at some point in the conversation: 1) personal scripture study, and 2) personal prayers. I plan on being consistent with these two questions for as long as I’m giving interviews. I ask a few questions about both of these topics: When are you reading the scriptures on your own? What are you reading? Tell me about it. Where are you saying your prayers? Are you interrupted by the other kids coming and going from the room? What kinds of things to you like to pray about in the morning?

This is partially prompted by a study on juvenile delinquency done at BYU, quoted also in an excellent book by Gene Cook called “Raising Up a Family to the Lord,” about how no other no other actions besides personal prayer and study can be firmly linked to teenage behavior – not regular church attendance, not seminary attendance, not even family home evening or family prayers. Personal worship is really what makes the difference.

Other than personal worship, I like to ask about friends, about sibling relationships, about anything the child would like to change in their life or at home, about job assignments, about goals. I like to remind my kids at every interview that they can always talk to me about anything or any question, and that they don’t have to wait for an interview.

I bring up one of those “uncomfortable” growing-up topics at almost every interview – what to expect during puberty, sex ed, standards, addictions, gender differences. For example, I went through a list of bad words with Danny – we put words into either a “rude words” or “swear words” category. I told Noah about menstruation and advised him about how to act when/if he hears girls talking about it or if he hears boys talking about it.

I also like to praise the kid up one side and down the other, and tell them specific things that I’ve seen them doing that please me. I think this is important for any kid to hear, but especially kids from large families. Also, I think it’s safer to give effusive praise a child in private, and not in front of the other kids, so interviews provide the chance.

How long to my interviews last? Anywhere from 5 – 30 minutes. 15-20 minutes would be average.

I’ve already had several specific insights into my kids that I know I would have missed otherwise. I already wrote about Oak’s desire to have his own scriptures and go to scouts. Another time I was talking with Danny about how to react to Noah’s teasing. I thought this was a serious issue and a new phase in Noah and Danny’s relationship. Danny, however, shrugged it off, saying, “I especially love it when Noah teases me when Daddy’s there, because when I yell ‘Nooo-ahhh’ and Daddy hears me, then Noah gets in big trouble!” The little scamp - I had no idea!

As the kids get older and start dating, I plan to regularly ask them questions about worthiness and help them set standards. I recently heard one grown man talk about how much easier it was for him to resist temptation as a youth because he knew that his parents would be asking him specific questions at his next interview on Sunday. He knew he didn’t want to lie to them, and he knew they’d be asking.

I also remember talking to Patricia Martin in Germany about how she got involved with a good, LDS boy, and when things started going too far, she wanted so desperately to tell her parents, but they never asked, and she could not bring herself to start the conversation. When she came home at night, she said that she’d hang around an extra long time with her mom, just hoping against hope that her mom would ask some hard questions, but instead her mom would just go on about what a great young man he was, etc, etc.

There are some outstanding ideas about interviews in the book about adolescents that I recommended recently, “Parenting Today’s Adolescents” by the Raineys. I plan on re-reading that book and jotting down some ideas, maybe make a pot of questions and scenarios and pull one or two out at every interview, or use them for FHE.

A friend named Julie Sessions told me just yesterday that she does interviews with her kids, too. She always does them without her husband. She calls it M&M time, because she eats M&M’s with each kid when they come in. She keeps a folder on each kid (scouts, duty to God, personal goals) and uses the time to help kids set their own goals for life. I’m sure her interviews are a great thing for her kids, even though the style is different than mine.

I can easily see that interviews are not for everyone. Some mothers are more natural talkers, or are less “busy” during a regular day, or have private time with each kid in the normal course of life, or whatever it may be. I think that some people may accidentally turn interviews into grilling sessions that make the kid want to be rebellious. I hope that I won’t!

I’m sure that my interviewing style will morph over the years. Who knows where it will end up! Perhaps this essay will seem quaint when I read it again in ten or fifteen years, after raising most of my kids through their teenage years. However, I do know that I feel good and I feel true after doing interviews with my kids. I know that it is right for me and right for my kids, and I feel better about it than almost anything else in my life. I feel ecstatic when I finish interviews! I feel the Spirit confirm to me that I’m doing the right thing. I’m so thankful for the inspiration.