WOW! Why ask questions of such complexity that cannot be answered in this forum?

You can go to our website, listed below, to get references on the treatment of analyzing the stress limiting factors in designing pulley assemblies. A number of PhD papers have been published such as:

1. Helmut Lange 1963 - known by many to be the father of the modern triaxial stress analysis approach for hub, end disk and shell wrt turbine end disk shapes

2. Schmoltzi about 1974 on the stress mechanics of the Rf locking device with its methods and its advantages

3. Qiu, et al on the "Modified Transfer Matrix Method" in defining a classical approach to the stress/strain field transfer between end disk and shell connection published by Bulk Solids Handling for Conveyor Dynamics, Inc. as noted on our website. THis is not a PhD paper. However, it shows that a classical approach can come very close to FEM and is much easier to use. CDI has supplied this code PSTRESS V3.0 to a number of well respected North American pulley manufacturers.

4. Dr. Dietrich (from the old PWH firm) PhD on stub shaft designs

5. Many others - South Africa, England, Australia, Germany

Go do your appropriate literature search! ■

Dear Mr. Singh,

Disc thickness can be found out by Roark's formulae for stress & as given:

dr= pxb(L-X )x X / 40000xDx txtx I(1/at+X/20000xI)

Where

dr= radial stress in N/mm Sq.

p= total resultant belt pull on pulley in N

D=disc dia. in mm

a= disc constant

b=disc constant

L=bearing centre in mm

X =disc centre in mm

t= disc thickness in mm

I=moment of inertia of shaft in cm to the power 4

We have develop one programe for it. Mr. Singh it is very difficult for me to type the relation properly & to furnish table also.I think Mr. Nordell is very correct in this regard.Have a nice day

Regards.

A.Banerjee ■

Dear Mr.Singh,

RIM THICKNESS

Av. belt pull= (T1+T2)/2 in N

Bending Moment= Fn x F/8

Where Fn= 2T Sin A/2

F= Pulley face in mm

t= Rim thk. in mm

A= Distorted angle=20 deg.

Zb= Bending moment/fb

fb= 100N/mm sq.

t= Sq. root of (6Zb/F)

T= Av. belt pull. in N

Regards.

A.Banerjee ■

Dear Mr. Singh,

Conveyor pulley shells have traditionally been designed using simple formulae, similar to as has been supplied to you in a previous reply. However, with the latest high-speed computers, the use of computational intensive formulae can easily and quickly be incorporated in a typical pulley design. As a world leader in the design and manufacture of conveyor pulleys Prok uses a technique where a force and moment equilibrium is analysed by considering the stress distribution over an infinitely small element of the pulley shell. Axial, radial and tangential external belt loads, shearing moments, bending moments, normal stress and shearing stress are considered in this equilibrium analysis. Biaxial stress and strain equations are applied to the small element to determine the deformation resulting from the stress levels induced. Calculus is used to replicate the solution for this infinitely small element over the entire conveyor pulley shell. Boundary conditions due to the restraint of the end discs are applied to this solution. The linear differential equations resulting from the aforementioned can be solved using Fourier series and matrix techniques. A minimum application requires the formulation of a 15 term Fourier series representing the axial, radial and tangential loads due to the belt, the generation and solution of the 30 3 x3 matrices describing the geometry of the pulley shell, and back substitution into equilibrium equations are used to determine the stress level at any particular point. This process must be repeated several times following the adjustment of the shell thickness before final arriving at the final solution. Using modern computing power this can all be achieved within a fraction of a second.

The technique used, and described above, has been verified by us against modern finite element analysis techniques using Von Mises stress criterion.

Paul Attiwell

Group Engineering Manager

Sandvik Materials Handling ■

Dear Mr. Singh,

You can find the derivation in various publications of Timoshenko and others for treatment as noted by Mr. Banerjee.

I have not validated his expression.

This method is not used today because of the many simplifications and omissions such as: no influences of locking pressure, hub strain, end disk to shell connection elasticity and shear sensitivity. No treatment of weldments. No metal triaxial fatigue criteria.

References on Timoshenko are:

---------------------------------------------------------------------------------

Simplified Methods: do not define principal and combined (radial, tangential, bending and shear) stress components without fatigue and proper yield (von Mises) stresses

1. "Theory of Plates and Shells" Timoshenko & Woinowsky-Krieger

Sect 63 pgs 288-289 1959 Original work 1929-1930 and citations back to 1911. Shell theory, Chapter 16, is also given but is very difficult reading. This work is expanded by H. Lange below.

2. "Formulas for Stress & Strain" Roark pgs 189-210 Second Edition 1943 with tables for ratio of inner to outer diameter and clamped or free outer edge -- this is based on Timoshenko and

is the citation by Mr. Banerjee.

------------------------------------------------------------------------------

Comprehensive Triaxial Stress & Fatigue Methods

3. "Investigations of Stresses in Belt Conveyor Pulleys" doctoral thesis by Helmut Lange 1963 based on Timoshenko Plates & Shells with the formulations noted by Paul Atttiwell

4. "The Design of Conveyor Belt Pulleys with Continuous Shafts" Doctoral Thesis by W. Schmoltzi 1974 -- expanded but erroneous treatment of locking mechanism carrying on from H. Lange

5. "A New Pulley Stress Analysis Method Based on Modified Transfer Matrix Method" by X. Qiu and V. Sethi BSH Vol. 13 No. 4 Nov 1993 -- this is the most comprehensive treatment of end disk and shell design except for FEA. This is difficult reading. Many of the North America Pulley design and manufacturing companies have used this prior to using FEA. The work is also based on Timoshenko and expanded beyond Helmut Lange/ W. Schmoltzi. This method covers the details noted by Mr. Atiwell. The big difference is this method allows full triaxial stress mechanics to establish magnification of critical principal stress and combined Von Mises stresses for yield and fatigue life criteria in native metal and in weldments.

The problems with Mr. Banerjee's citation are:

1. does not properly address the boundary conditions such as end disk to shell coupling. The only treatment that does, in a classical continuum mechanics form, is Dr. X. Qiu's work.

2. does not discuss the big stress gradient between the end disk and shell connection where many pulleys fail due to the transfer of tangential(hoop) and axial bending to a shear stress into the end disk with no treatment of the weldment at this connection. This is the reason for the T type end connection and turbine end disk shapes.

3. does not treat the end disk loading by the locking device expansion pressure



4. does not treat the end disk to hub connection -- welded from one side or both sides and the hub strain sink influence

5. ignores the stress limiting criteria which is weld type, location, and surface treatment sensitive

The concept offered by Mr. Banerjee has been is use since the 1930's. By 1960, many pulleys suffered failures and the end disk and shell. This precipitated the 1963-1974 Ph. D. works of H. Lange and W. Schmoltzi. CDI developed and expanded procedure with 3-D metal fatigue and yield criteria due to many high tension pulley failures. WE found the work of Lange needed refinement due to the limited end disk and shell Fourier expansion terms produced errors beyond 7%. The locking mechanism was not properly analyzed. This was due to the lack of sufficient computer capacity at the time.

The concept offered by Mr. Banerjee has been is use since the 1930's. By 1960, many pulleys suffered failures at the end disk and shell connections. This precipitated the 1963-1974 Ph. D. works of H. Lange and W. Schmoltzi.

CDI developed and expanded these procedures with 3-D metal fatigue and yield criteria due to many high tension pulley failures. We found, as others have, the work of Lange needed refinement due to the limited end disk and shell Fourier expansion terms. The CDI criteria can be found in the SME publication Chapter 14 1993 entitled: "Modern Pulley Design Techniques and Failure Analysis Methods" by V. Sethi and L Nordell.

The big problem with all classical mechanics methods based on Lange, except Dr. Qiu and modern FEA codes, is that there is no solution for transmission and compatability of the strain and bending moment around the corner of the end disk to shell. This produced errors beyond 7%. The locking mechanism was not properly analyzed. This was due to the lack of sufficient computer capacity at the time. ■

I also want to thank you for such an interesting reading.

Best Regards,

Mike ■

Dear Mr. Witt,

I am not sure why you addressed me in particular since I agree with your thoughts and stated so in my thread second reply, points 5 and 5 regarding the treatment of weldment stress endurance limits and other factors which need consideration.

CDI does have computer codes that provide the triaxial stresses in shaft, locking device, hub, end disk of many forms, and shell. The triaxial stresses are evaluated for Von Mises yield limiting criteria as well as Goodman (Schmitz) mean and alternating stress fatigue criteria for native metal and welded zones. The code does allow for optimization of the end disk, hub and locking deive optimization.

In welded zones ,we apply the British and American weld standards for the various weld types (stress flow) and the altered weld limits by type (fillet; butt weld) and surface treatment (machined and non-machined to differing surface classes).

I commend you on your vigor for also developing such code. Hopefully, it is also well verified when applying it to such pulleys are Los Pelambres. HAve you published your findings?

CDI did also make the audit of Los Pelambres pulley designs in association with Krupp.

THe thread starter was asking about a formulae for the calculations of the above. THis is too difficult to provide herein. ■

In the FORUM I cannot naked myself. But for first hand information which I expressed is good enough for further development. In the second reply, I expressed the difficulties & before taunting any body, one should be very careful.

Mr. MIKE, I appreciate your reply.

Regards.

A.Banerjee ■

Dear Kjshelat,

The a,b constants were developed by Timoshenko. Read his purpose. These values were conceived to correct for end disk fixture or flexity. Should the design require a fixed connection treatment between end disk and shell or should the design treat the connection as a knife edge restrained in the end disk transverse to the end disk plane ie. fixed in the shell axial plane. Read from an earlier post:

Simplified Methods: do not define principal and combined (radial, tangential, bending and shear) stress components without fatigue and proper yield (von Mises) stresses

1. "Theory of Plates and Shells" Timoshenko & Woinowsky-Krieger

Sect 63 pgs 288-289 1959 Original work 1929-1930 and citations back to 1911. Shell theory, Chapter 16, is also given but is very difficult reading. This work is expanded by H. Lange below.

2. "Formulas for Stress & Strain" Roark pgs 189-210 Second Edition 1943 with tables for ratio of inner to outer diameter and clamped or free outer edge -- this is based on Timoshenko and

is the citation by Mr. Banerjee. ■

Originally Posted by nordell

href="showthread.php?p=13886#post13886" rel="nofollow">

WOW! Why ask questions of such complexity that cannot be answered in this forum?

You can go to our website, listed below, to get references on the treatment of analyzing the stress limiting factors in designing pulley assemblies. A number of PhD papers have been published such as:

1. Helmut Lange 1963 - known by many to be the father of the modern triaxial stress analysis approach for hub, end disk and shell wrt turbine end disk shapes

2. Schmoltzi about 1974 on the stress mechanics of the Rf locking device with its methods and its advantages

3. Qiu, et al on the "Modified Transfer Matrix Method" in defining a classical approach to the stress/strain field transfer between end disk and shell connection published by Bulk Solids Handling for Conveyor Dynamics, Inc. as noted on our website. THis is not a PhD paper. However, it shows that a classical approach can come very close to FEM and is much easier to use. CDI has supplied this code PSTRESS V3.0 to a number of well respected North American pulley manufacturers.

4. Dr. Dietrich (from the old PWH firm) PhD on stub shaft designs

5. Many others - South Africa, England, Australia, Germany

Go do your appropriate literature search!

Dear sirs

I have come across the intested topic in context of belt conveyor and is a challangeble issue . plea to respond me . Attached is the clarification

Attachments

clarification0 (DOC)

■

And what has this latest post got to do with the design of pulleys??? ■

Could you please tell me where i can get way to determinate a and b values named disc constant

Originally Posted by A Banerjee

Dear Mr. Singh,

Disc thickness can be found out by Roark's formulae for stress & as given:

dr= pxb(L-X )x X / 40000xDx txtx I(1/at+X/20000xI)

Where

dr= radial stress in N/mm Sq.

p= total resultant belt pull on pulley in N

D=disc dia. in mm

a= disc constant

b=disc constant

L=bearing centre in mm

X =disc centre in mm

t= disc thickness in mm

I=moment of inertia of shaft in cm to the power 4

We have develop one programe for it. Mr. Singh it is very difficult for me to type the relation properly & to furnish table also.I think Mr. Nordell is very correct in this regard.Have a nice day

Regards.

A.Banerjee

■

Dear designer,

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I thank you all sincerely for your cooperation.

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Administrator ■

Originally Posted by A.Banerjee

Dear Mr.Singh,

RIM THICKNESS

Av. belt pull= (T1+T2)/2 in N

Bending Moment= Fn x F/8

Where Fn= 2T Sin A/2

F= Pulley face in mm

t= Rim thk. in mm

A= Distorted angle=20 deg.

Zb= Bending moment/fb

fb= 100N/mm sq.

t= Sq. root of (6Zb/F)

T= Av. belt pull. in N

Regards.

A.Banerjee

Dear Mr. Banerjee,

I am trying to understand the equation you have provided for calculating the required shell thickness.

Do you have a reference you could provide where this is derived? The same equation is published in "Recommended Practice for Trough Belt Conveyors" by MHEA in 1986 but they also do not provide a reference or derivation.

Looking at the calculation:

a) Fn= 2T Sin A/2 = 2 T Sin (20 deg / 2) : This is the total force acting on a 20 deg section of the shell where the belt either begins or finishes contact with the pulley. Not clear why you would use the average tension (T1 + T2) / 2 as the maximum would occur where the high tension belt makes contact with the pulley.

b) BM = Fn x F/8 : This is the maximum bending moment, in the longitudinal direction, assuming a beam of length F simple supported at its ends, with a uniformly distributed load of Fn / F. BM = Fn / 2 * F / 2 * 1 /2 = Fn * F / 8. This maximum occurs in the center of the beam.

c) I = F t^3 / 12 : The moment of inertia for the piece of the shell in the circumferential direction

d) fb= 100N/mm sq. : The maximum allowed stress. (MHEA suggest 93 N/mm^2)

e) stress = Mc / I, with c = t/2 , fb = BM * 6 / (F t^2)

f) solving for t: t = sqrt( BM * 6 / (fb * F) ) = sqrt ( 6 * Zb / F ) where Zb = BM/ fb which is in agreement with your equation above.

The 2 things that I am unclear on if the above math is correct:

1. The maximum bending moment is calculated in the longitudinal direction but used to calculate the bending stresses in the circumferential direction.

2. The maximum bending moment, as calculated above, only occurs at the center of the beam, but is assumed to be at this same level across the entire face width when calculating the bending stresses.

Curious if there is more to this old equation than what I have shown above?

Best regards,

Andrew ■

Hi Andrew,

The old equation mentioned in the MHEA Blue Book. MHEA publications bear a strong resemblance to CEMA and were long considered as a conveniently metricated and simplified version. Perhaps more light could be gained from examining early CEMA.

In the earlier MHEA Grey Book, which I mislaid, there was a shorter equation which was considered safe to use in those days. When improved performance was demanded at eg Selby coalfield, the early work was found wanting. I was instructed to replace 147 pulleys due to cracking issues. The new pulleys also cracked up and were eventually replaced by equipment designed using pioneering FEA techniques. The first and second vendors had probably decided that FEA was a NASA thing and anyway they had been paid, up to retention, by a very generous Owner. MHEA published their 1986 revision just after the 1984 experience described. ■

Originally Posted by johngateley

Hi Andrew,

The old equation mentioned in the MHEA Blue Book. MHEA publications bear a strong resemblance to CEMA and were long considered as a conveniently metricated and simplified version. Perhaps more light could be gained from examining early CEMA.

In the earlier MHEA Grey Book, which I mislaid, there was a shorter equation which was considered safe to use in those days. When improved performance was demanded at eg Selby coalfield, the early work was found wanting. I was instructed to replace 147 pulleys due to cracking issues. The new pulleys also cracked up and were eventually replaced by equipment designed using pioneering FEA techniques. The first and second vendors had probably decided that FEA was a NASA thing and anyway they had been paid, up to retention, by a very generous Owner. MHEA published their 1986 revision just after the 1984 experience described.

Hi John,

Appreciate your insight. I don't have a copy of the earlier MHEA book. Is this the one published in 1977 or ???. If you do find your copy I'd be interested in the section on pulleys.

I've looked at the early CEMA books. They don't address the shell thickness in any way. This was left up to the manufacturers.

Best regards,

Andrew ■

Hi Andrew,

I'm quite thoroughly gobsmacked that MHEA had deviated from CEMA and done something off their own bat regarding shell thickness. I never paid much attention because once the metric version is there the old avoirdupois version never gets a second glance. I had just taken CEMA's original work to have covered the issue.

Your 1977 copy is the earlier 'Grey Book'. The "Blue Book" came out in 1986. ■