## User's Manual for the Roller Bending Calculator

### Application

To find out how much a roller will bend under any combination of its own weight, the tension of the web and the force from a nip roller. The web and nip can act at arbitrary angles. All aspects of roller design are covered in The Web Handling Handbook by Roisum, Walker and Jones in Chapter 3, which includes an explanation of bending deflection and its effects.### Bending equation

Under a load, L, a roller of width W, modulus E and "area moment of inertia" I will bend by Delta=5LW^{4}/(384EI)

We all know that wider rollers bend more, but just a 10% increase in width gives a 50% increase in deflection, so the width effects are strong.

The "area moment of inertia" depends on the dimensions of the roller. Rollers are shells with an ID and OD. The moment of inertia depends on OD^{4}-ID^{4} so a slightly thinner shell (smaller OD or larger ID) gives more deflection - a 15% decrease in OD doubles the deflection.

There is a choice of three materials, each with its representative Density and Modulus. The "Composite" has been split into three types, Low, Med and High based on expert input on the Density and Modulus values for these materials. There is a significant cost factor across this range as the high modulus version uses super-expensive carbon fibres. Although the Medium composite has the same modulus as in earlier versions, the density has been increased by 10% so deflection under its own weight will be somewhat higher

A Dead Shaft roller uses the equation as shown. a Live Shaft roller uses a slightly more complex equation with a Width+ term which includes the bearings. For those who care just about deflection relative to the web, simply used the Dead Shaft calculation. If inputs to the calculation aren't realistic (e.g. ID>OD) the calculation provides some safe value - it's up to you to spot the fact that there is something wrong with the input.### Inset

A rather more complex roller design allows the shell to be coupled to the support shaft at a distance ("inset") from the end of the roller. As the inset moves closer to the center the deflection at the center decreases. As you move Inset away from the default ("normal") setting of 0 you will find that the central deflection decreases but after a certain point it makes no further difference. This is because beyond 22% of the distance from the edge to the center, although bending in the*centre*is reduced, bending at the

*end*increases. As this app is only concerned with central bending it would be deceptive to allow calculations beyond 22%

Note that the calculation does not work for Live and Cantilever rollers, where the Inset value will be ignored.

The formula for the Inset option was kindly provided by Timothy Walker, one of the gurus behind these web handling apps.

### Load

The Load comes from 3 effects.- The weight of the roller itself based on its calculated volume and the Density of the material. Because the volume of material depends on OD
^{2}-ID^{2}, even though a thicker shell gives a bigger load, the deflection is always less because OD^{4}is bigger than OD^{2}. - The tension from the web - this is twice the input value because the roller is being pulled by the incoming and outgoing web.
- The load from a nip roller

### Direction

Gravity always acts downwards (defined as 180°). The web and the nip can act in any direction. For example, a nip at the bottom of the roller (angle=180°) acts as a force upwards and in principle can stop the bending of the roller or, under extreme load, bend the roller upwards. The RBC calculates not only the overall bending but also its direction. The definitions of angles can be confusing, but the visual feedback and the colour coding between text and diagram should make it clear.### Results

The Deflection of the roller is calculated along with its Angle plus the Resultant of the combination of loads. Note that the Reaction shown is the Resultant divided by 2 - i.e. the load on each bearing. The ratio of Deflection/Face (multiplied by 1E6) gives an idea of how serious the deflection might be. The result is compared to a quality Class where A is the best and D is the worst. It is for you to judge if you have over- or under-engineered the roller. In Slider mode, playing with OD and ID whilst looking at the Class is a good way to see how much you can downsize the roller without sacrificing performance.### Roller Weight

Although the basic roller weight is calculated for you, you can calculate the weight of the whole roller (and get the full deflection) via the following steps which were suggested by an RBC user. Although a more sophisticated app could do this automatically, it was decided to keep things simple, hence this slightly inconvenient work-around.- Set up your roller ID, OD and material
- Set Nip and Web loads to zero
- The "React." output, i.e. the force on each end, is half the weight of the roller - so just multiply by 2
- If you want the weight of the journal, enter the total length of the journal, set Roller ID to 0 and Roller OD to the journal diameter. Multiply React. by 2
- If you want the bending from the whole weight, go back to your roller then add the journal weight as a nip load at 0°

### Mass Moment of Inertia

This is πρL/32(OD^{4}-ID

^{4}) and is included for those who are interested in the value. The calculations are based on shell only: the head, journals and other elements are considered negligible as is almost always the case. Be careful of units; especially US units.

### Critical speed

It is often said that you can calculate a*Critical Speed*at which the roller will spontaneously start to vibrate. It is also often said that this calculation isn't worthwhile. You can choose to use the output or not. There are two formulae commonly used, the one used here is the Dunkerley method. Where E is the modulus, I is the second moment of area, m is the mass and L is the roll length:

N_{c}=94.25√(EI/mL^{3})

The value is calculated both in RPM and in the peripheral velocity of the roller.

### Cantilevers

Cantilever rolls can be a dangerous option - they can bend very strongly and cause large problems with the web. They should only be considered for relatively short roller lengths. A full calculation is beyond the scope of this app, just the main deflections of the shaft itself from its own weight and that of the roll+web, amplified by the length of the roller (see the Roisum Mechanics of Rollers for a full explanation). To model deflection of a wound roll on a cantilever (by clicking the Roll option) we've used the trick of setting the nip angle to zero and allowing you to enter the weight of the roll as the nip load.### More complex cases

There is a very powerful roller bending calculator within TopWeb from RheoLogic which allows more choices of materials, multiple complex loads etc.### How To Use

Choose the units of measure as either Metric or US. As you enter the key values you get instant feedback on the key outputs in the blue boxes. Some users prefer to use Text entry (it's more precise). Others tend to prefer Slider entry (especially on smaller devices). Feel free to choose whichever is the most useful for any occasion.While you can choose either Metric or US units, you cannot specify which type of units within those systems are used. The most common usage is fixed. For example, thickness will given in µm (0.000,001 meters) and mils (0.001 inches). Pay special attention to make sure that your input/output values are in the units specified or convert to/from as needed.