Help suporting a shaft with tail stock center

I posted a while ago about turning 1-1/2" square rod round on the ends. When chucking the rod in a 4 Jaw chuck and using an indicator as best I could to get the rod concentric a few inches before the square was to be turned round. Problem I had was stress would bow the shaft so that the round was not concentric with the square by quite a bit. Suggest cure was to make a "bushing" to be able to place the square in a steady rest. I managed to buy quite a few bearings from HGR that look new. 3" bore or larger (about size used simple measurement as they are probably metric) Some are Angular contact, Roller and another is Double Row being at least 3" wide. Was thinking of making a bushing from 4" turned to fit bearing then press on a hub on the other side. Drill and tap both sides with 4 adjustment screws and possibly mount the bearing in a steady rest type fixture. Would need at least a 2-1/8" hole thru the bushing.
The round shaft to be turned is 27" long so I will need a live center for support.
That special Steady bearing rest will hold the square as concentric as possible but if I use the standard live center there will be a major bow in the shaft when I remove the center. I'm thinking maybe it would be best to make a live "center" using a 4 jaw chuck rather than a 60° center. Take a few cuts, then readjust the 4 jaw "center". When near finished size (1.250")about1-5/16" center drill the end then finish the shaft between centers (other side being only 4" long) so that both sides are concentric.
Any better ideas? Suggestions??

If the shaft is bending as you machine it, you need to stress relieve the ends. Heat them with a torch to dull red and cool in oil dry should do it.

GBent is on the right track with stress relieving. My question is: what material is the shaft being machined from ? Cold rolled steel is notorious for warping once machined, as it releases 'locked in stresses'. Hot rolled square bar is a better choice, but even that will have some locked in stresses. What you are doing when you machine that much off a square bar is to release a lot of the locked in stress created by the processes when the bar was rolled at the mill. Even hot rolled bar is going to have some locked in stress as it was never made with the idea it would be machined as you require. Starting with stress-relieved bar stock would go some ways to curing the warping problems you are experiencing.

"Stress Proof"- a tradename= steel is formulated and processed to avoid this sort of warpage as a result of machining. Stressproof uses a combination of formulation of the elements in the steel, along with a 'drawing' process to take the steel to finished dimensions. I do not know if 'Stressproof' is available in square cross sections.

Assuming you are using hot rolled square bar of some common steel (such as A-36 or 1018), I'd start by having the bars stress relieved. I also do not know how accurate the square section must be, and what kind of surface finish your job requires. As-rolled hot rolled steel will have a thin layer of mill scale on it. Do-it-yourself heat treating methods over the length of the bar are going to produce more scaling. Sending the blanks out to a heat treating shop with an inert atmosphere heat treating oven is the 'Cadillac' way.

As for supporting a square bar in the lathe, what is sometimes known as a 'cathead' is used to provide a journal to run in a steady rest. The cathead is a piece of round tube (or round bar with a hole bored thru it sufficiently large to pass the work thru). Four (4) jaw screws are tapped thru the wall of the tube at 90 degrees to each other, and fitted with check or jam nuts. For heavier jobs, the cathead is made long enough to have two rows of jaw screws with space between them for a journal to run in the steady rest.

As for putting a live chuck on the tailstock, while this will provide more support,it will allow you to support the square cross section quite rigidly. However, it will not address the warpage issues. As you machine the shaft, if it is restrained at both ends in chucks, the machining will still want to slightly bow the work away from the tool. The live chuck on the tailstock will certainly provide more support for the work. While the live chuck on the tailstock is something that is often done, but for your job, I'd do it in conjunction with a cathead and steady rest. My sense of it is that if you support the work in a chuck at each end, when you remove it from the chucks, it is going to really warp on you. A complete stress relieving job up front is going to help a lot.

For centering, a thought would be to do the centering by some means other than in the lathe. If you have access to a horizontal milling machine, or to a vertical mill with horizontal head, I would do the centering in the mill. Namely: set the job up in the mill, indicate the work so it is square/parallel with the table and spindle, mike the actual dimensions of the stock, and locate the centers. This gives you a center that is based on the as-found square cross section.

We had a job years ago that required putting a new center in the ends of journals on hydro turbine wicket gates. Each gate weighed 5500 lbs. and looked somewhat like a ship's rudder. The gates were to have their stem journals turned and new journal sleeves shrunk on. We set each journal up on the table of a planer mill with a right-angle head, indicated around the journal to find center, and put the centers in. The gates then went to the lathe. On smaller odd-cross section or odd-shaped jobs, I've put centers in using milling machines prior to doing any turning.

Square Rod looks to be Hot Rolled SS, yes it is SS and probably 304 supplied by the customer for food conveyor shaft. Square section fits plastic conveyor belting cogs, round end for bearing on both sides and hollow output gearbox on the other side. Also being production line repair there is little time. I have heated the square shaft red and allowed to cool slow but in most cases didn't help. In the past I would chuck the square and indicate the area where the end cog was located. Then center drilled the end. Of course the end was quite a bit out because the hot rolled square rod is not straight. After rough turning I then completed the round ends between centers. That worked reasonably well until the last shaft I made especially since the 1.250" round end was 27" long.
That being said I believe holding the square in a "Steady Rest" will work, at least it will keep the round end meeting the square in a reasonable center of the square and the round will have a bow. To compensate moving the center will work and simplest way seems to be using a 4 jaw chuck. Then after the round is near size, remove the chuck, drill center hole and finish shaft between centers.
Just asking if there might be a simpler way.

With that application I would consider drilling then boring the ends and pressing in rounds. Pinning them if necessary. I have done quite a few jobs like that

Boring the end and inserting a shaft is not a good option. Being the Gear box has a 1.250 opening rather than a shaft still has the same torque requiring 1.250" shaft size. Boring 1-1/2" square will leave 1/8" wall thickness. There is quite a bit of load on the conveyor. Yes the shaft is not space craft and the slight bow in the square section is acceptable. But the round ends must be inline that is why I complete the turning of both sides between centers. The shaft is supported via pillow block or flange mount bearings. The bearing pivots to adjust for any mounting alignment but if the shaft wobbles the bearing wobbles in it's mount and soon is badly worn.
Simply put if I rough the shaft between centers the stress causes the entire shaft to bow, the greatest amount is at the junction of the square and round being it is near the middle of the length protruding out of the chuck. If I were to provide a support (Steady Rest) at the square then the bow will be in the round shaft some place near the middle of the 27". I can remove the shaft and straighten it to limit the amount of the bow but will have a center hole off center. I guess I can make the shaft longer and cut the end off and recenter. But if I was to make a "4 Jaw chuck live center" I can adjust for the stress bow each time I take a cut.

Could you supply your own material and use stress relieved 303 square bar? That would probably be the easiest.

First I don't have a lathe with 2-1/8" bore plus 2-1/8" would not be large enough given the size tolerance and the bow in the bar. I did buy a W&S 1A turret lathe (3-1/16" bore) at HGR but don't have it yet. However that would be the same as having a steady rest as I want to make using the bearing. Thought the 1A is not a ram turret lathe I can put a center in the rear carriage but I'm back to the same situation in the large bore lathe as I am using a steady rest! The shaft will bow from stress and when removed the center hole will not be correct! I'm back to making a live center using a 4 jaw chuck! However my shop is small, I'm retired and make extra cash in my shop. However to me the shop is small, I have 6 lathes from 24" to 12" rated though they swing larger. 2 Vert Mills, 1 Horz. Mill, 2 Moore Jig borers, HBM, 2 Surface grinders and other machinery but now my floor space is limited! Not sure where I can put that 1A though it's the shortest centers machine. On the other hand a special steady rest can be stored anyplace even in outside storage shed!
This is maintenance item, I have no idea when shaft will be needed again. When required it's ASAP! I've told the company to do it plus I said get a square shaft and put it outside in the sun and let it age. I've also asked they give me a length and I'll do it but haven't gotten anything yet.
I have made many smaller shafts that require large diameter shafting turned to a much smaller diameter. Something like 1-3/4" turned to fit 16mm bearings that is 6" long. Without a center each cut will remove stress movement of previous cut, as finished size is approached and less material removed the movement is less and eventually the shaft is straight. I can then finish the other side to 42mm or what ever.
Furthermore I have in the past machined material that was stress relieved, it's better but still has stress that will bow the shaft. Plus if I were to machine a shaft in a lathe with large spindle bore and a center the shaft will be bowed when the center, flipping it over and doing the other side event though the length is only 4" long the 2 shafts will not be concentric! To be honest other shops have done that though the one I just did is the longest round shaft extension (27") because it fit's a bearing, a sprocket to power another conveyor and a transducer to transmit rotational data. Those other shafts have destroyed bearing housings! I stopped that type shafting as I was the Engineer at the company!

Thanks Greg! I'm getting some precision roller bearings and plan to put a 4 jaw chuck on #4 Morse taper. I'll have it for when I get the turret lathe, if not I will make the Bearing "Steady Rest" since checking the Bearing I got from HGR are Precision Roller 7316. Might be a bit of work to mount 2 angular contact bearings that big. I do have another 80mmish beaning from HGR that is probably double row. I'll check the number. Simply put I see no other way. Chucking the bar in a large spindle bore lathe or using a Steady Rest is the same. Square bar stress relived will not be straight so turning between centers will be required. I'm thinking because of the stresses in a square bar they all will not be eliminated by stress relieving. I may have to straighten the turned round 27" before finish machining between centers. I will post the results. However having a live center using a 4 jaw chuck will probably come in handy for other jobs.