Re: Convex Curves
Hello Derek, I would suggest the the curve radii as presented in CEMA are still valid and very much in use today. We most certainly use them, as well as the Goodyear system as presented in their Red Book. I would suggest that the belt modulus is one of the more important parameters in the selection of the curve radius. Perhaps Mr Nordell would like to comment on this.
Graham Shortt ■
Re: Convex Curves
Dear Mr. Derek Bishop,
You are mentioning that you have convex curves in vertical plane. In this curve, you have also mentioned that belt center is to be checked for overstressing and edges for buckling. Your statements are opposite to the actual situation. In case of convex curves, the belt edges will tend to be overstressed and the belt center will tend to fold due to buckling.
There are systematic formulae and method to calculate the concave curvature, convex curvature and transition distance. These formulae take into consideration the belt modulus of elasticity, the safety factor, actual tension at that location, and its experimental ability to withstand the tendency to fold. It is not possible to narrate these formulae and its elaborate calculation procedure on this forum.
Regards,
Ishwar G Mulani.
Author of Book : Engineering Science and Application Design for Belt Conveyors.
Advisor / Consultant for Bulk Material Handling System & Issues.
Email : parimul@pn2.vsnl.net.in
Tel.: 0091 (0)20 25882916 ■
Re: Convex Curves
Derek
CEMA is standard. Don't forget to consider the belt mass when it is worn. ■
Re: Convex Curves
You are mentioning that you have convex curves in vertical plane. In this curve, you have also mentioned that belt center is to be checked for overstressing and edges for buckling. Your statements are opposite to the actual situation. In case of convex curves, the belt edges will tend to be overstressed and the belt center will tend to fold due to buckling.
Thanks for the correction. Not a good start. Do you have any references?
Sounds like the CEMA formulae are a good starting point then. ■
Formulae For Convex Curves
Dear Mr. Bishop,
We simple take 15 to 20 times belt width to get radius of curvature depending upon tension. Otherwise you may consult one book "Engg.Science & application design for belt conveyor" by I.G. Mulani before developing any software for the same.
Regards.
A.Banerjee ■
Re: Convex Curves
Hello again, Derek.
I would suggest that you use the formulae as noted in CEMA, taking the parameters as presented by Mr Mulani into account, namely, the belt modulus, the tension at the point of curvature, the idler wing roll angle etc. Again, look at the formulae as presented by Goodyear or CEMA, which work very well. Be careful of using a simple multiplier, as suggested by Mr Banerjee, because it can be very misleading. The traditional approach was to use exactly that - a multiplier of between 12 and 20 times the belt width, but that was always based on fabric carcass plied belting, which has a modulus much lower than steelcord belting.
Would you send me your e-mail address and I can pass some stuff on to you.
Have fun programming.
Graham Shortt
Chief Mechanical Technologist
Anglo American
Anglo Technical Division - Johannesburg Campus
email gshortt@anglotechnical.co.za ■
Curve Radius
Derek,
If you are dabbling in the grain or woodchip industries, be wary of the combination of convex curves and also 45deg x 3 roll idlers. An additional factor can come into play and I have had a couple of instances where problems have occurred with designs complying with everybody's calculated curve radius (including our own).
With these materials you have a low bulk density, non abrasive material and a requirement for high volumes so you end up with a combination of:
-wide belts
-steep idlers
-longer idler pitch
-light belt carcass (usually a ply or two lighter than the norm)
-thin belt covers (often only 1 or 2 mm)
The result is that the belt can be too light a construction to resist creasing by being pulled down into the idler nip too tightly. This would happen only where the radius is derived to limit edge tension. I suppose offset idlers would also help.
I would be very interested if anyone has derived a calculation which considers this
A combination of closer idler pitch, a good sized curve and maybe backing off the trough angle through the curve can fix.
I am not a big fan of 3xequal roll 45 deg idlers, except at the loading point.
cheers,
Ray Latchford ■
Re: Convex Curves
Regarding the reply posted by Mr Latchford:
Nice point Ray. Bear in mind that the CEMA approach (and as noted by Mr Mulani) is to consider both the maximum tensions at the belt edges and the tension at the centre, which must be greater than zero to prevent buckling. I agree with the statement regarding offset idlers, because belts running on in-line idlers can easily be drawn into the gap between the rolls, even on the horizontal. We use off-set rlls in South Africa, almost by preference, with in-line systems more the exception. My approach is to specify the gap between the rolls to be not more than half the total belt thickness, which can prevent (but not always) the belt being drawn into the gap.
With reference to the convex curves again, the idler pitch in the curve must normally be reduced, not only to provide support for the belt in the curve (bearing in mind that the "curve" is actually a segmented one - more like a lobster-back rather than a smooth radius and the resultant of the belt tension at each idler increases the load on these idlers substantially. ■
Re: Convex Curves
Interesting points. I once remember seeing some 'creasing' in the centre of a grain handling belt. I presume this was due to lack of tension on a curved section.
In most of the minerals handling I've been involved with, the idlers are in line. I take it the offset idlers help to reduce the 'pinching affect' between trough and centre idler. I think these are normally standard for smaller belt widths.
I ended up using the formula:
r = B E sin (Lambda) / 4.5 / (TR-T) - Overstress at edge
r = B E sin (Lambda) / 9.0 / (T-4500) - Understress at centre
Taken from the old Dunlop catalogue.
Where
B = Beltwidth (m)
E = Belt modulus (N)
Lambda = Trough angle (degrees)
TR = Rated belt tension. (N)
I change TR to suit the operating condition. ie run, start, braking, or coasting. Typically start/stop allowable tension is 1.5 x run allowable tension for yard conveyors. This would need to be checked to check the conveyor parameters such as type of start.
T = Calculated belt tension (N)
I also used the following to check understress at the edge of the belt for concave curves.
r = B E sin (Lambda) / 4.5 / (T-4500)
The same equation is used in the Apex belting catalogue (an Aussie belting supplier), and the old blue Goodyear catalogue has equations of a similar form to the ones quoted above. The copy of the CEMA manual I have includes a B^2 term in the numerator, along with factors which I presume relate to the sin(lamda) term.
CEMA also gives a formula r=12(b/12) to prevent buckling at centre for convex curves.
I seem to remember someone saying once that the overstress equation is not particularly good when the belt tension gets up near the rated tension of the belt. The radius then becomes excessive. ■
Re: Convex Curves
One other point. Do you set your radius to suit the calculated maximum minimum for all operating conditions ie starting/stopping/running. So say you calculated a maximum radius for the convex curve for belt centre stress under a braking condition. Would you use this to set the radius.
Seems to me a bit of discretion would need to be used here depending on the duration of the condition experienced, and the likely outcome of the event. If you got an understress on the edge of a concave curve during braking, that could result in spillage, and a possible safety hazard. ■
Re: Convex Curves
Derek, design for the worst case, especially if safety comes into the equation. Whether you are killed in 5 seconds or 5 minutes, you are equally dead. Spillage has no friends either.
Remember to check the idler loads in the curve, especially if the tensions are likely to peak under any condition.
Graham Shortt ■
Re: Convex Curves
Criteria used for vertical curves used for plant / yard belts.
Assumes conveyors always have net increase in lift.
Concave curve – lift off.
Formula: r = T / (mb+mm) g
Check start up tensions.
Belt carry side fully loaded to tangent point.
Check for empty belt case with scoop fluid coupling.
Check return side for lift off under braking if head and tail brake used.
Belt mass reduced for wear.
Maximum tension gives maximum radius.
Use average tension along the curve section. (Tstart+Tend)/2
Concave curve – under stress at edge of belt.
Formula: r = B E sin (Lambda) / 4.5 / (T-4500)
Check start up / run / stopping tensions.
Belt fully loaded.
Minimum tension gives maximum radius.
Use tension in curve section furthest from
head end of conveyor.
Concave curve – overstress at belt centre.
Formula: r = B E sin (Lambda) / 9.0 / (TR-T)
Check start up / run / stopping tensions.
Belt fully loaded.
Maximum tension gives maximum radius.
Use allowable belt tension to suit run and start conditions.
Use tension in curve section closest to head end of conveyor.
Convex curve – overstress at edge of belt
Formula: r = B E sin (Lambda) / 4.5 / (TR-T)
Check start up / run / stopping tensions.
Belt fully loaded.
Maximum belt tension gives maximum radius.
Use allowable belt tension to suit run and start conditions.
Use tension in curve section closest to head end of conveyor.
Convex curves – under stress at centre of belt
Formula: r = B E sin (Lambda) / 9.0 / (T-4500)
Check start up/ run / stopping.
Belt fully loaded.
Minimum tension gives maximum radius.
Use tension in curve section furthest from
head end of connveyor. ■
Re: Convex Curves
Remember to check the idler loads in the curve, especially if the tensions are likely to peak under any condition.
Graham,
I've seen idler spacings closed up on convex curves in books, but never actually seen it done in design. Normally we would allow for a belt deviation load on the idlers. Typical misalignment would be 3 mm. It wouldn't be hard to check the 'misalignment' of idlers on a convex curve, and include this in the idler load. If it became excessive, then would consider closing up idler spacings rather than going for a different type of idler.
How do you normally handle this situation. ■
Formulae For Convex Curves
Mr. Bishop,
The radius of curvature constraints are to constrain the max local tensile stress approximately to the manufacturer's rating and the min local stress to be a net tensile (not compressive). In this regard a troughed belt is like any other section having a neutral axis and a moment of inertia.
Curvature induces a moment according to M=(EI)/R
where:
M is moment
E is modulus of elasticity
I is moment of inertia (of troughed belt cross-section)
R is radius of vertical curvature
The compressive stresses due to moment must be offset by the belt tension while the tensile stresses due to moment, when added to those due to the belt tension must not exceed the approximate belt rating.
Starting from basics the writer has derived the radius of vertical curvature constraint equations in several published articles that can be found in the E-Library. I refer you to the latest such writing:
"Sandwich Belt High Angle Conveyors According to the Expanded Conveyor Technology", by J. A. Dos Santos
This writing derives the Goodyear equations. Previous writings derived the CEMA equations. I prefer the former because:
1.) The allowed max combined stress is 1.15 Tr, 115% of rated, reflecting that this is a local edge tensile stress not over the entire belt section.
2.) The allowed min combined stress is .05 Tr, 5% of rated. This reflects the belt's rating. By contrast CEMA's allowed min combined stress is 30 b (30 lbs per inch of belt width) regardless of the belt's tension rating.
Continuity is also important. Idler spacing must limit the idler load (material, belt and radial load=T/R) to less than the idler's rating and the breaking angle over each idler to less than approximately 3 degrees. The latter is eqivalent to limiting sag at a straight conveyor but includes that sag plus the breaking angle of the adjacent cords (straight lines between adjacent idlers).
Following these principles will lead to successful design.
Regarding thin belts, this is a separate issue. Belts must meet the load support criteria (published by the belt manufacturer) otherwise you can have creasing that will lead to fatigue and failure of the carcass along the crease. The nip between adjacent rolls aggravates this but overlapping rolls will not solve the problem if the load support criteria is badly violated.
I hope this is helpful in your basic understanding. I hope you will follow the derivation in the cited article.
Joe Dos Santos ■
Re: Convex Curves
Thankyou for your response Joe Dos Santos. I see merit in adopting the factors for maximum and minimum stresses you have indicated. The equations would then become:
Concave curve – lift off. (no change)
Formula: r = T / (mb+mm) g
Concave curve – under stress at edge of belt.
Formula: r = B E sin (Lambda) / 4.5 / (T - 0.05 TR)
Concave curve – overstress at belt centre.
Formula: r = B E sin (Lambda) / 9.0 / (1.15 TR - T)
Convex curve – overstress at edge of belt
Formula: r = B E sin (Lambda) / 4.5 / (1.15 TR - T)
Convex curves – under stress at centre of belt
Formula: r = B E sin (Lambda) / 9.0 / (T - 005 TR)
I'll attempt to derive the equations. I presume you describe the belt profile using linear equations, and then go from there to determine the position of the neutral axis and second moment of area I (m^4).
I did a search on the reference "Sandwich Belt High Angle Conveyors According to the Expanded Conveyor Technology", by J. A. Dos Santos, and could not find any further information of the derivation of the equation. ■
Formulae For Convex Curves
Derek,
The sited article can be found in the E-Library. Please follow this link: http://www.bulk-online.com/ELIB/index.php?inc=3
If you send me your Email address I'll send you a draft copy of this writing.
Joe Dos Santos ■
Equations For Curves
Derek
You can develop the classic curve equations from:
•Hookes law, stress = modulus x strain,
•Average strain using Hookes law, and the belt tension at the point.
•Additional (or less) strain = the outstand dimension from the centroid divided by the curve radius.
•Calculate the position of the centroid of the belt shape in the usual way.
Considering the vertical curve case:
The centroid is at 1/B x S^2 x sin(lambda) from the base of the trough.
The larger outstand is then (1-S/B) x S x sin(lambda), & the smaller S/B x S x sin(lambda)
B = belt width & S the belt touching the wing.
For carry side the classic equations assume S = B/3, ( This is not always the case, ie narrow or no centre roll, drift etc)
When you extend these equations you will quickly see the numbers 9 & 4.5 emerge, as per the classic equations in Goodyear, Dunlop et al.
The method applies to convex, concave & horizontal, by combining the various relevant strains at the outstand being considered. You can also do same for the drifted case.
The challenge is then the allowable safety factor. Do you use say on steel cord 6.7 / 1.15 = 5.8 or the values from say DIN 22101, February 1982 or August 2000 that can be much lower near 3 ?
The manufacturer can advise. ■
Re: Convex Curves
G'day Gary. And there is another person I've heard about, but never met. Many thanks for your comments on the topic. ■
Re: Convex Curves
Hello All,
I do not see comments on steel cord belts vertical curve analysis correction verses fabric belt. The high elastic modulus makes some steel cord vertical curves, in the lower tension region, untenable unless you consider outer cable axial compression or having a potential for buckling.
Compression stress in the outer cables is indirectly applied in the CEMA formulation. I believe they dont directly calculate the level of compression stress but state for steel cord belt apply a divider (1/2.5)of the fabric radius. (reference CEMA p244 5 ed.)
Proper treatment must include the cover geometries and reinforcements. ■
Re: Convex Curves
Larry,
Thanks for the comments. I looked in my catalogues. Some do consider this, some don't. It is not something I've considered in the past. I note the old blue Goodyear manual does give different formula for minimum edge tension for steel and fabric belts. The formula for steel belts is the same as the fabric belts, but reduced by a factor of 4500/12000 = 1/2.67. That tallies with the CEMA value of 1/2.5.
Need to check this with suppliers I guess and include it into the equations accordingly. ■
Formulae For Convex Curves
Larry, Derek,
There is no justification in mechanics for the 2.5 fold decrease in radius just because it is steel cord belting. This is an exception of convenience. Such exceptions ultimately have negative results. Indeed the belt edges will be in a buckled state not in compression, sagging locally (at the edges) between wing rolls.
I have been involved in the analysis and design of major overland and slope conveyors up to 12,000 HP and have never used this factor.
Joe Dos Santos ■
Concave Curve Radius Steel Cord Buckling Criteria
Dear Joe, Derek,
We too were skeptical of the 1/2.5 or 1/2.6 factor of CEMA and Goodyear. However, we performed a comprehensive FEM on a Hibbing iron ore steel cord belt and found it to give reasonable results.
The conveyor under study had steel cord fretting failures in the outer strands. We hoped to identify, to a reasonable certainty, the magnitude of compression, its consequential buckling magnitude, and estimation of fretting based on degree of bucking and compression.
We did have the known conditions as a starting point. We concluded the original designers errored (obviously) in the radius selection and that CEMA/Goodyear criteria would give proper life with a reasoable set of cords in compression at the set idler spacing, belt construction and a proposed expanded new curve radius.
Bottom line, we do not know what the rationale is for the steel cord reduction factor other than it mirrors half the ratio of the moduli before incoporating the cord and belt bending stiffness. This is more than the equivalent fabric belt which is moving in the right direction. If one desires to do more, I expect one would include belt thickness, cord size and construction, biaxial neutral axises analysis, etc.
HNY
Lawrence Nordell
www.conveyor-dynamics.com ■
Re: Convex Curves
Joe,
I have a desire to know more about your 12000 HP conveyor. This must have a concave curve of consequence for which you have set the curve criteria. Please share with us the parameters, if possible, since you have wetted our appetite, and inferred a strong contrary opinion.
I will do the same. Then we can compare longeviity, et al. We with compression and your more sensible, conservative, non-compression. I will state: the cost difference to a client may be substantial in civil and strucural requirements. We can claim your selection will work!
Lawrence Nordell
www.conveyor-dynamics.com ■
Re: Convex Curves
Joe,
Another note on the Hibbing:
We measured the bending properties in 4 planes - axial with top cover up, trans-axial with top cover up, axial with bottom cover up and trans-axial with bottom cover up. This is to differentiate the nonlinearity of rubber in compression verses tension to calibrate the FEM model. This was after we obtained the modulus curve over the expected range of tension-compression.
LKN ■
Re: Convex Curves
Larry,
Sounds like you have investigated this aspect thoroughly.
I have had only casual involvement with design of overland conveyors. Most of my design work has been with plant belts. I cannot think of a belt where the concave curve radius was determined by buckling criteria rather than lift off.
Joe,
Thanks for sharing your experience. Evidently some people do use it, some don't.
I don't know if this is significant, but the formula I listed above was:
r = B E sin (Lambda) / 4.5 / (T - 0.05 TR)
The Goodyear formula from the blue catalogue printed 1976 (XGP-BC-41) is effectively the same, but does not have the 0.05 TR term in the denominator (bottom line). This term is an attempt to ensure the tension at the edge of the belt doesn't reduce to zero, or worse, go into compression. Maybe this is intended to counter the same problems as the 1/2.67 factor.
I was surprised that the edge of the belt can go into compression on these curves. I presume Larry your company carried out some kind of strain gauge testing on this area of the belt, and found this to be the case. Hard to argue against a strain gauge reading that indicates the belt goes into compression. ■
Formulae For Convex Curves
Larry,
I spoke from memory and it failed me.
Up to 8000 HP, the 8000 HP system being a slope belt that you are familiar with as we subcontracted the dynamic analysis to CDI. Edmund represented CDI. I mentally extrapolated to a future max power (adding 2000 HP drive units) which I erroneously remembered to be 1200 HP. The automatic take-up is at the tail. Indeed it was a very large concave curve radius and a very high take-up tension. I will not venture to quote these from memory but I am sure you can look these up.
Regarding compression of the belt edges, this is not actually possible to any significant extent since the edges are not constrained laterally. The edges are free to sag between the idlers. In the case of the 2.5 factor the belt edges are limp and the neutral axis of the remaining belt (which is subject to net tension) is lowered to the middle part of the belt, at the bottom of the trough. A finite element analysis of this situation would be non-linear because of the non-linear behavior of wire rope and rubber but more significantly because of the large deflections. The analysis must be in stages at each stage redefining the analysis model until it is stable
The 2.5 factor is a fudge factor that allows "controlled buckling" judging that such, which is manifested in increased edge sag between wing rolls, is not detrimental to the belt and will not result in excessive spillage. There is no basis in mechanics.
Joe Dos Santos ■
Re: Convex Curves
Only the outer few cords are allowed to be in compression. THere is no observable buckling. The critical criteria is the fretting factor. In the case at Hibbing, the buckling was significant as was the 2-3 years of life in the belt as the outer cords birdcaged and failed.
Joe, you might find it of interest to do a little research on belts of significance that are designed with an apparent outer cord compression which have had a successful life.
Lawrence Nordell
wwww.conveyor-dynamics.com ■
Formulae For Convex Curves
Larry,
Thanks for your insights. Considering relative magnitude how do you distinguish edge buckling deflection from edge sag?
I would be very interested in comparing performance of edge buckled vs non-edge buckled steel cord belt conveyors. Where would one start?
Let's invite here feedback from those who have input on this matter.
Joe Dos Santos ■
Re: Convex Curves
Dear Joe,
The term "buckling" for our discussion is a misnomer. When the outer 2-3 cords go into compression there is no obvious buckling. THere may be a slight wave at the edge along the greater path of belt sag between idlers.
Extremely waviness, enough to cause internal fretting of wire strands within the cord construction, I judge to have a wave amplitude of more than 5% of the belt width. From our referenced analysis, I recall about 8-10 cables or about 200-250mm of the belts outer edge were in compression in the concave curve. THe buckling phenomenon was severe with a waviness exceeding 50 mm and repeated sinusoidal cycles between idlers.
As far as design, with concave curves that have negative edge stress, there are many. I would have to make a compliation.
I am sure I have many pics on the behavior.
Stay in touch. ■
Re: Convex Curves
Another interesting note on the wave (buckling action).
THe idler spacing is not a factor. THe buckling frequency can fluctuate between idlers. THis seems to occur as the frequency is not common between idlers. THe buckling waves are standing waves and do not move with the belt. I am sure this is obvious to you but may be less obvious to the lay engineer. ■
Re: Convex Curves
It was stated you apply the CEMA factor by dividing the radius you calculate with the fabric belt equation by 1/2.5 ie. multiply the radius by 2.5, to get the required minimum radius for the steel chord belt. This appparently needs to be larger than the radius for a fabric belt because of problems if the edge goes into compression.
I posted a note about the Goodyear formula. Essentially, the formula is:
r = B E sin (Lambda) / 4.5 / T
for fabric belt, and
r = B E sin (Lambda) / 12 / T for steel belt.
That gave the ratio 4.5/12 = 1/2.67
In this case the radius for the steel belt will be smaller, being divided by 2.67.
Does the steel belt radius need to be bigger or smaller than the fabric radius? ■
Re: Convex Curves
Here is a graph of the equations with and without the 0.05 TR term.
The equation with the 0.05 TR term gives over twice the radius when the tension gets close to the limiting tension defined by 0.05 TR, which seems to be what past experience with the steel belts suggests you need.
Further food for thought. Seems that the fabric belt radius gives the most conservative values (ie. the biggest radii). If you used this formula with the 0.05 TR term, then would that be too conservative?
Attachments
concave curve aJPEG:forum_attachments/file_container/concave curve a.jpg (BMP)
■
Re: Convex Curves
The graph was generated with values. It again highlights the affect of including the 0.05 TR term in the denominator.
Belt width 1200 mm
Belt modulus 12,000,000 N
Trough angle 35 degrees
Some of the calculated radii with the belt tension close to zero were:
Tension (N).........Radius (m)....................Radius (m)
...........................(With 0.05 TR term)......(Without 0.05 TR term)
8000...................344.1............................86.0
10000.................172.1............................68.8
12000.................114.7............................57.4
15000.................76.5..............................45.9 ■
Re: Convex Curves
I have not checked you calcs. Did you factor for the 6x higher modulus for steel cord?
Larry ■
Formulae For Convex Curves
Derek,
Since you don't state the value of Tr it is not possible to check your numbers. I suspect that the first tension value 8000 kN is close to the value .05Tr. As the tension value approaches .o5Tr the allowed radius (with min .05Tr tension) approaches infinity.
The concave radius for a steel cord belt is allowed to be smaller than the .05Tr criteria by a factor of 1/2.67 according to your formula. As I stated before this is an exception of convenience since the conforming radius is nearly 3 times as large as the exception, requiring the corresponding accomodation in the civil and structural work.
What hasn't yet been highlighted; why is the exception applied only to the steel cord belting and not to the fabric belting?
The elastic modulus of the fabric belts is low so that the required tension to offset the buckling does not cause an unreasonable uplift curve. Indeed the T2 (wrap tension) required typically governs the uplift curve and the edge buckling requires a lesser curve. This is not the case with steel cord belting whose elastic modulus is many times higher than fabric.
Joe Dos Santos ■
Re: Convex Curves
I chose numbers without reference to a given application. The intent was to demonstrate trends of the the different equations.
TR=120 000 N ■
Formulae For Convex Curves
Derek,
Thanks, indeed, .05Tr=6000 kN.
If you add to your table Tension values approaching 6000 kN , the required radius (for .05Tr min allowed) will increase dramatically untill infinity (division by zero) at 6000 kN. Below 6000 kN R will be negative which makes no sense since the intire belt tension is less than .05Tr.
Joe Dos Santos ■
Convex curves
I'd like to get some idea of the formulas typically used for say stockpile yard belts with convex vertical curves. My understanding is that the minimum radius is calculated to prevent overstress at belt centre, and minimum tension at the edge of the belt.
The ones I've seen are from suppliers such as Dunlop (old), Goodyear, Apex, and the CEMA formulas. What is the state of play with these formulas? What are the ones most commonly used now?
I would like to incorporate the formulae into a conveyor design computer program. ■