Deflection of indeterminate beam system

[imcjoek]

Hello,

I am trying to find an estimate of the deflection of a pair of rolls separated by elastic material. (see attached diagram).

I am not sure what method to tackle this with. Basic indeterminate beams I can handle, but adding an elastic (viscous even) goop between two deflecting structures has me lost.

What I do not understand, is how to determine the load on the face of the rolls, as this itself is a function of the deflection of the rolls!

I suppose I could guess at a deflection, determine the distributed load, and then iterate until convergence?. How would the "squish" of the material and its modulus be related to load on the surface of the roll? Is hertzian cylinder/plane contact the way to go here?

I do not have access to FEA. Our purpose is to determine roll size and possible contour to result in uniform material thickness.

 

[GregLocock]

"How would the "squish" of the material and its modulus be related to load on the surface of the roll?"

Without knowing the material properties of the goop you are just going to have to wing it. You'll probably find that there is a time variant dependency there as well.

So can you get any tests done?

Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Wrenchbender]

Joe,
I only speak from textbook ‘experience’ on this. The buzzword is ‘viscoelasticity’ which is a characteristic of polymers (including biomaterials). Modulus is not a constant, but a strong function of strain rate, also temperature, but assume that would stay the same. In this case, the faster the rollers squish out the goop, the stiffer it becomes; the more power is needed, the higher the roller load, deflection, etc.

I would try to calculate an average strain rate imposed by the rollers’ speed and radius. Then check with the vendor of said goop for viscomechanical properties. If that is a dead end, maybe an goop-imbedded strain gauge can provide a load for the desired roller speed (before the wiring gets wrapped up).

From thenceforth, I would check a mechanics book for a closed form solution to a simply supported beam with distributed load. (I don’t have FEA capability either.)

Interesting problem. Good luck.

 

[corus]

The material passing through the rolls isn't purely elastic but elastic-plastic. In hot mills you'll need the stress strain behaviour at about 1000 deg C, together with the strain rate dependency. The roll force will depend also on the amount of reduction of the material as well as material properties. There are various papers on the subject but you'll probably have to pay for them. This site may be of use

http://www.metalpass.com/millload/

In your picture, however, you have 10,000 lbs acting at the bearings so a simple calculation would be to assume 20,000 lbs acting as a uniformly applied load across the main roll. There are other forces acting out of the paper due to friction etc. but for a simple calculation you'd have to ignore them. To calculate the stresses/deflection you'd need to consider a beam made up of two different sections (for the two diameters). I'd use a FE program like 2D!Beam, which is free.

In a full blown FE analysis it's common to model the rolls as a rigid body and then calculate the forces being applied from a transient implicit analysis as the material is pushed through the roll gap. It usually takes a few days for such an analysis to run. The forces are then applied to a seperate model of the roll using the method described. It's probably better to get a mathemtical model that has been compared to measured mill loads.

corus

 

[rb1957]

isn't the picture as drawn determinate ? ... the top roller has 20,000 lbs applied, the reaction = 20,000 lbs, the load applied to the lower roller is 20,000 lbs, the reaction to the ground is 20,000 lbs ... no ?

isn't the real question, how much force needs to be applied to the top roller to achieve the desired thickness change ?

 

[BobM3]

I think it's more the deflection shape of the rolls that the OP is interested in. You could probably do a bit better than assume a constant pressure across the length of the roller. There will be less deflection near the end supports. Maybe a linear pressure distribution (higher near the ends)? Is this a one-of design or are you making a lot of them? If it's just a one-of then play it safe and make the rollers BIG.

 

[rb1957]

maybe that's the OP's issue ... the pressure is proportional to the deflection of the rollers ... which would be close to a sine waveform.

 

[imcjoek]

Correct in that I am interested in the deflection behavior across the working face. This is a one-off-machine. As such we are leaning towards the "make it big" approach.

I have originally used a uniform pressure distribution, but was wondering if there was something better. It sounds like anything better may be more involved than we are prepared to deal with for a single machine.

Appreciate the responses.

 

[BobM3]

I think a higher pressure on the ends would be a good assumption. (smaller gap -> more squeeze on material -> higher pressure - unless of course, the material squeezes sideways!). Tough problem - I'm assuming you would want a uniform gap, not necessarily just a small deflection. The loading is material dependant, speed dependant, temperature dependant, etc. You might have to just give it an educated guess, try it, measure the deflections and iterate in - ouch!

 

[Compositepro]

The deflection will be small but perhaps important. Normal assumption is to use uniform pressure across the roll. Most deflection will be in the journals but that will not affect the flatness of the roll faces. Skewing the axis of the rolls can also compensate for deflection from flatness.

 

[chicopee]

This is what I would do and then refine:

1) Instead of estimating the max deflection of each roll, I would determine the acceptable deflection of each roll from suppliers or manufacturers if they are different in size.

2)Determine the uniform load on top of the lower roll contacting surface based on the acceptable deflection from 1). Take into account varying weight and varying moment of inertia of the nonuniform cross sectional area of roll

3)Determine the upper roll deflection using the upward uniform load calculated from 2). Again consider top roll non uniform in cross section.

A 1972 Design News by T.V. Seshadri shows the various moment and deflection formulae for statically determinate and undeterminate of non uniform beam formulae which you can apply.

 

[corus]

It's not the deflection of the rolls that is the critical factor in designing rolls but the stress at the roll neck, where it usually breaks from fatigue. Making the roll bigger isn't always the best option as the difference in roll diameters as well as the radius at the neck will determine the stress concentration there. Roark will give you the stress concentration factor to use. The total stress at the fillet radius should be less than the fatigue life of the material taking into account the surface finish.

As rb1957 says, the problem is really determined already if you're assuming that the reaction forces at the bearings are already known. Whatever method you use to determine the stresses in the 'beams' remember to multiply by the scf at the fillet radius.

corus

 

[chicopee]

No argument from me on your point corus however you need to know both. Also deflection is important if you need a consistent thickness across the width of the rolled material.