Models of calculating flexure resistance of conveyor belt and solid material

(1)

Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

This report consists of 44 pages and 0 appendices. It may only be reproduced literally and as a whole. For commercial purposes only with written authorization of Delft University of Technology. Requests for consult are only taken into consideration under the condition that the applicant denies all legal rights on liabilities concerning the contents of the advice.

Specialization: Transport Engineering and Logistics Report number: 2017.TEL.1608

Title: Models of calculating flexure

resistance of conveyor belt and solid material

Author: H.C. Kalff

4087518

Title (in Dutch) Modellen voor het berekenen van buigingsweerstand van een transportband en zijn bulk materiaal

Assignment: literature Confidential: no

Initiator (university): Dr.ir. Y Pang Initiator (company): -

Supervisor: Dr.ir. Y Pang Date: October 31, 2017

(2)

TUDelft

F A C U L T Y O F M E C H A N I C A L , M A R I T I M E AND M A T E R I A L S E N G I N E E R I N G

Delft University of Technology Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone + 3 1 (0)15-2782889 Fax + 3 1 (0)15-2781397 www.mtt.tudelft.nl Student: H.C. Kalff Supervisor; Dr. ir. Y. Pang Specialization: TEL Creditpoints (EC): 10

Assignment type: Literature Report number: 2017.TEL.1608 Confidential: No

Subject: Models of calculating flexure r e s i s t a n c e of c o n v e y o r belt a n d solid m a t e r i a l

Belt conveyors are extensively applied in bulk solid handling systems and as i m p l i e d by the standard D I N 2 2 1 0 1 , the main resistances contribute more than 80% o f the total motional resistances o f long horizontal belt conveyors. The main resistances consist o f the indentation resistances o f belt on idlers, the flexure resistances o f belt and solid material, and the rotating resistances o f idlers. I n the past decades, different models have been built to calculate the flexure resistances. Hence i n this assignment, the calculation models are expected to be reviewed and compared. The main tasks f o r this literature assignment include:

o to overview o f the motional resistances o f belt conveyors, especially the main resistances

• to interpret the flexure resistances o f the belt and the carried solid material

o to review the existing models o f calculating flexure resistances o f the belt and the solid material,

respectively

9 to compare the models and indicate their the pros and cons

This report should be ananged in such a way that all data is structurally presented i n graphs, tables, and lists w i t h belonging descriptions and explanations i n text.

The report should comply w i t h the guidelines o f the section. Details can be f o u n d on the website.

(3)

Abstract

Belt conveyors are a frequently used form of transportation for bulk material. To make more efficient belt conveyors and decrease the costs and energy consumption, the influence of all the motional resistances working on the belt need to be researched. The second contributor to the main resistance, the flexure resistance, is not widely studied, but the body of work concerning the flexure resistance can not be neglected.

Motional resistances

DIN22101 describes the total motional resistance as the sum of the primary, secondary, gradient and special resistances. Zooming in to the primary resistance to determine the friction coefficient, the following resistances can be distinguished: idler rotational resistance, sliding resistance, indentation resistance, flexure re-sistance of a belt and flexure rere-sistance of bulk material. These all contribute differently to the total resistance under different circumstances. However, it can be concluded that the indentation resistance has the biggest share when running an average belt conveyor. The flexure resistances can also be of great influence depending on the circumstances such as the type of bulk material or the belt tension.

Flexure resistance

Focusing on the two types of flexure resistance, a few main researches are surveyed. The different mathematical models are based on the energy balance and the equilibrium of forces on the belt. It is also important to calculate the active and passive pressure factors the bulk material experiences due to the opening in closing of the belt, in order to calculate the transition zone of the stress states. The mathematical models, verified by laboratory set-ups and simulations, show that the flexure resistances of the belt and bulk material are influenced by the belt sag, the mass and height of the bulk material, the type of bulk material, the tensile force of the belt and the type of belt.

Implementation

When the different parameters of the flexure resistances are known, the design tools can be improved. Better simulations can be made to predict the behaviour of the belt conveyor and the bulk material even more accurately. This will lead to more efficient designs of belt conveyors and thereby saving capital costs and energy consumption.

Conclusion

The information gained of the flexure resistance can be used when large belt conveyor systems in the future will have intelligent control systems that will keep track of all the dynamic parameters using sensors. This results in belt conveyors with an increased overall system safety and reliability. However, still some research needs to be done on the dynamic effect of the flexure resistances and the other motional resistances. This can result in improved simulation techniques and more efficient belt conveyor designs

(4)

List of symbols

a Idler spacing

C Factor for secondary resistance d Thickness of layer bulk material Enb Energy dissipation

Enf Energy dissipation

EI Flexural rigidity of the belt EJ Flexural rigidity of the belt f Coefficient of friction

fb Coefficient of the flexure resistance of a belt

fd Dynamic factor

fi Coefficient of the indentation resistance

fs Coefficient of the flexure resistance of bulk material

F Motional resistance

Fb Flexure resistance of a belt

FH Primary resistance

Flong_f lex Longitudinal flexure resistance force

FN Secondary resistance

Fs Flexure resistance of bulk material

Fs Special resistance

Fsn Normal force per unit length

Fst Gradient resistance

Ftr Resistance due to the troughing shape

Ftrans_f lex Transverse flexure resistance force

Fv Vertical force due to the weight of the belt and bulk solid

Fvs Normal force due to the weight of the bulk material

g Acceleration of gravity h Height of bulk solid

hbs Height of bulk solid above the centre idler roll

h0_bs Height of bulk solid bounded by the failure plane H Material lifting height

Ka Active pressure factor

Kla Longitudinal active pressure factor

Klp Longitudinal passive pressure factor

Kp Passive pressure factor

Kta Transverse active pressure factor

Ktp Transverse passive pressure factor

(5)

iii

lg Spacing between consecutive idlers

lz Contact length

L Length of the belt conveyor

Lss Length of bulk solid in contact with the inclined side of the conveyor belt

m0_B Mass of the belt m0_L Mass of the load m0_r Mass of the idler rolls ∆M Loss of moment M Moment

Pz_bs Pressure distribution due to bulk solid

q Homogeneous load distribution belt and material s Indentation depth

T Belt tension w Belt deflection

Wb Flexure resistance of a belt

Wf Flexure resistance of bulk material

y(x) Bending lines of a belt β Troughing angle

β Conveyor surcharge angle of bulk material δ Inclination angle of the belt conveyor ∆1, ∆2,

∆3, ∆4,

Angles in Mohr circle η Angular deflection θa, θp

Rotations of the major principal stresses in active and passive stress states

λ Conveyor surcharge angle λa Active stress coefficient

λav Active stress coefficient in a vertical plane

λpv Passive stress coefficient in a vertical plane

λp Passive stress coefficient

ρ Radius of curvature of the belt ρ Bulk density

φ Angle

φi Bulk solid internal friction angle

φw Angle of friction between belt and bulk

Ψb Damping factor of energy loss due to bending

(6)

1

Introduction

1.1. General introduction

Belt conveyors are economical, efficient and are extensively applied in bulk solid handling systems. As implied by the standard DIN22101 the main resistances contribute more than 80% of the total motional resistances of long horizontal belt conveyors. The material on the belt is an important issue which directly and indirectly affects the load on idlers, belt tension, belt life and overall performance of the conveyor [1]. While the indentation resistance is the main contributor to the motional resistances and therefore widely researched, the flexure resistance is the next in line. Unfortunately the flexure resistance is therefore a lot less researched. However it is important to also consider the body of work concerning the other resistances besides the indentation resistance, because they all contribute to the main resistance. With more knowledge of all the contributors of the main resistance, designers can design better and more efficient belt conveyors, which benefits the costs and energy consumption.

In this report the results of a literature survey are presented, giving insight in the theoretical analysis of the physical behaviour of the running conveyor belt to identify the impact of all the important parameters on the flexure resistance. The report is split into three main chapters. In chapter 2 the different contributors to the main motion resistance are explored. Chapter 3 zooms in to the flexure resistance of the belt and its bulk material. It focuses on advanced calculation methods by multiple reseachers which are tested and verified by laboratory tests and in-situ measurements. The last chapter, chapter 4, will describe the possible implementations of the knowledge gained in the previous chapters.

(8)

2

Motional resistances

2.1. Introduction to belt conveyors

In order to understand the resistances of motion working on a belt conveyor, one must first understand the composition of a belt conveyor. The basic concept of a standard belt conveyor is displayed in figure 2.1. The conveyor shown is a simple design shown in longitudinal sections. It consists of a frame, idlers, the head and tail pulleys, the take-up system, loading chutes and, of course, the belt. Idlers (figure 2.2) are used to carry the belt. They consist of a frame an rolls and can be used to create a trough that increases the load carrying capacity of the belt in case of belt conveyors that transport bulk solid materials [2]. At the tail end bulk solids are charged via a loading chute on the belt continuously and accelerated by the conveyor belt to belt speed. Over the conveyor length the bulk solids are transported on the upper strand of the conveyor to the discharge point at the head of the conveyor.

Belt conveyors are in operation throughout the world for the transport of large amounts of bulk solids over great distances. Capacities of up to 40,000 t/h are achieved. The average conveyor length is about 1,000 m, but head to tail distances of greater than 3,000 m are no exception [3]. Besides the troughed belt conveyor, there are also special belt conveyors to transport bulk under extraordinary circumstances, e.g. inclination or declination. In that case the material needs to be transported by a force or form to prevent an uneven load due to gravity. The four typical belt conveyors to use in this case are: pouch conveyors, pipe conveyors, high angle conveyors and pocket conveyors. These are however exceptional conveyors for unusual circumstances, so are not considered in this report. The belt conveyor with a conventional troughed conveyor belt is of

Figure 2.1: A typical belt conveyor [2].

(9)

2.2. Motional resistances 3

Figure 2.2: Idlers of different angles in a transition zone to get a troughed belt [2].

considerable importance for the handling of bulk materials and is therefore taken as standard in this report.

2.2. Motional resistances

As the length of belt conveying systems increases it becomes even more important to accurately calculate the motional resistances at the design stage, with the view of minimizing these resistances to improve the efficiency of the installation. When a belt is in motion the conveyor has to deal with a lot of resistances. These resistances have a physical effect on the running belt conveyor and therefore it is vital to take them into account when designing a conveyor. Many authors have provided primary motional resistance models. However, only the primary motional resistance model delivered by DIN22101 is complete and of analytical nature [4]. This design norm is used for determining all the specifications of belt conveyors e.g. dimensioning and selecting components. In the DIN22101 standard the total motion resistance is defined as:

F = FH + FN + FSt+ FS (2.1) With: F Motional resistance FH Primary resistance FN Secondary resistance Fst Gradient resistance Fs Special resistance

As can be seen in equation 2.1 the motional resistance is the sum of all the individual resistances. These resistances according to DIN22101 are described below.

(10)

2.2. Motional resistances 4

Primary resistance FH

The primary resistance is the sum of all friction-related resistances along the belt. This is the running resistance of the idlers (idler rotational resistance and the sliding resistance of a belt on idlers) and the running resistance of the belt (indentation resistance, flexure resistance of a belt and the flexure resistance of bulk material). In this resistance, the material load on the belt is of major influence.

Secondary resistance FN

The secondary resistances are caused by friction and inertia. These are the feed resistance of transported bulk material, friction between the bulk material and the loading chute, friction of the belt cleaners and the deflection resistance of the belt at the pulleys. Because these resistances only occur at certain parts of the belt conveyors, the magnitude of these resistances is independent of the belt conveyor length. Therefore the significance of the secondary resistance decreases for longer belt conveyors. The secondary resistance is usually taken into account as a percentage of the primary resistance. Under normal circumstances it is about 78% of the primary resistance for belt conveyors of 100 m, and drops to only 9% for belt conveyors longer than 1000 m [4].

Gradient resistance FSt

The gradient resistance is the force the belt conveyor needs to overcome the difference in altitude between the loading and unloading section of the belt conveyor. It is dependent only on the actual material flow and the material lifting height.

Special resistance FS

The special resistance describes resistances that do not occur for all belt conveyors, for example the tilt resistance of idlers and the resistances of equipment used for feeding the belt conveyor alongside the conveying track. These resistances generally have a relatively small influence on the total motional resistance (1%) [4].

When looking at those four resistances, the primary resistance is the most impor-tant resistance. For long belt conveyors the primary resistance is even larger due to the decreasing influence of the secondary resistance. Therefore, we are going to focus on this resistance in the following text. Many authors have provided primary motional resistance models. A full complete model for the calculation of the main resistance on the basis of the physical properties of the belt and bulk material and the geometry of the belt conveyor was first published by Spaans in 1978 , but soon other researches followed. However, only the model delivered by DIN22101 is complete and of analytical nature [4]. DIN22101 defines the motional resistance as in equation 2.2.

(11)

2.3. Idler rotational resistance 5

With:

C Factor for secondary resistance f Coefficient of friction

L Length of the belt conveyor g Acceleration of gravity m0_r Mass of the idler rolls m0_B Mass of the belt m0_L Mass of the load

δ Inclination angle of the belt conveyor H Material lifting height

As can be seen, the motional resistance depends of the sum of the moving masses, the length, and a fixed friction coefficient. This coefficient is the most important of this research. The inclination angle and the material lifting height can be neglected, because we are only looking at a long horizontal conveyor belt without the secondary, gradient and special resistances. The value of f is based on the experience of the belt conveyor engineer or on measurements on a real belt conveyor. It depends on the belt conveyor and the operating conditions and therefore it is not entirely generic. It is generally between 0.016 and 0.027 [4]. In order to determine the friction coefficient, we have to look further into the main primary resistances. Those are:

• Idler rotational resistance

• Sliding resistance of a belt on idlers • Indentation resistance

• Flexure resistance of a belt

• Flexure resistance of bulk material

In this chapter all the different contributors to the main motional resistances as listed above are briefly reviewed.

2.3. Idler rotational resistance

As Gladysiewicz [5] stated, the idler rotational resistance is a sum of values of a rotational resistance of each roller. However, it is difficult to verify this due to a great variety in results in the laboratory tests. Therefore, the resistance is only calculated for single rollers and used as a approximation of the rotational resistance. The rotating resistance occurs due tot the friction of the rolling elements in the bearings, the viscous drag of the lubricant and the friction of the contact lip seals, as stated by Lodewijks [6].

(12)

2.4. Sliding resistance of a belt on idlers 6

2.4. Sliding resistance of a belt on idlers

Gladysiewicz [5] also stated, the sliding resistance of a belt on idlers consists of two components. The first is caused by forward tilt of side rollers while the second is a result of belt misalignment and its transverse movements on idlers. The sliding resistance of a belt on idlers can be combined from friction forces on rollers which are calculated from their radial reactions and kinetic friction factor between rollers and the belt. The assessment of the impact of belt misalignment is more difficult as it depends on the random operating conditions of a conveyor like lateral and vertical tilt, lateral offset and horizontal skew of idlers or poor belt or bulk stream tracking. However, these parameters cannot be measured or predicted precisely so the belt misalignment can only be assessed.

The sliding resistance of a belt on a given idler caused by belt misalignment can be calculated with the use of actual exact numbers (values that are individually generated from fuzzy numbers with regard to their membership functions).

2.5. Indentation resistance

The indentation rolling resistance occurs due to the viscoelastic nature of the bottom cover of the belt [6]. When moving over a roll a belt will be indented by the roll in the contact area, and as this deformation is not free of hysteresis, it will cause resistance [7]. The indentation is of very short duration and occurs periodically.

The indentation resistance depends on different parameters, namely the belt parameters (damping factor and elementary flexural rigidity of a belt), roller diameter and radial force on a roller [5].

The indentation resistance of the whole idler set is a sum of values of an indentation resistance of each roller. Just like the idler rotational resistance, the indentation resistance is constant, while the other three resistances - the flexure resistance of a belt and bulk material and the sliding resistance of a belt - depend on the belt tension. [5].

This resistance is thought to be the main contributor to the motional resistances, so the most research is done to the indentation resistance.

2.6. Flexure resistance of the belt

While the indentation resistance is extensively researched, due to the fact that it is the highest contributor to the total motional resistance, there is a lot less known about the flexure resistance. Only in recent years researchers started truly exploring the physical effects of the flexure resistances. The first to publish about the flexure resistance was Spaans in 1979 [8]. He divided the flexure resistance in two parts, the flexure resistance of bulk material and the flexure resistance of a

(13)

2.7. Flexure resistance of the bulk material 7

belt. Gladysiewicz [5] caught up the physical model of the flexure resistance and since then a lot of researches on the matter have been done.

When a belt moves from one idler to another, the belt bends a little bit. This results in the flexure resistance of the belt. While it is further explored in chapter 3.1 the flexure resistance of the belt depends of a damping factor, the distance between two idler sets and the energy dissipation [5].

2.7. Flexure resistance of the bulk material

The bulk material also has an effect on the belt and therefore the flexure resistance. The volume flow of bulk material experiences a cyclical deformation between two idler sets, both lengthwise and clockwise, due to the sag of the belt and its limited flexural rigidity. The relative movement results in energy losses due to the internal friction of the bulk solid [9]. The belt and bulk solid properties influence the bulk solid flexure resistance to varying degrees, as does the belt speed, belt sag and idler spacing, thereby offering potential reduction with informed design. The flexure resistance of the bulk material is further explored in chapter 3.2.

2.8. Total motional resistances

All the motional resistances together have a big influence on the conveyor belt. In order to achieve optimal usage of the belt conveyor we need to get more insight in the contribution of all the individual motional resistances to the total resistance. The original method of calculation of belt conveyor motional resistance has been developed on the basis of theoretical analysis of the energy dissipation processes in a conveyor belt and in the material load stream, as well as the analysis of the interaction between the belt and idlers. With multiple laboratory tests and in situ-measurements supported by verified theoretical calculation methods all the different resistances can be identified from the total resistance. The main resistance force is prevailing for conveyors longer than 80 m. Therefore the accuracy of calculations of the main resistance force components is vital for the proper choice of a belt conveyor design. Taking into account the energy dissipation processes in a conveyor belt and in the material load stream the components of the main resistance forces are considered as described in this chapter.

Length

At first the length of the conveyor belt heavily influences the contribution of the secondary resistances. As is clearly apparent, these resistances are independent of the length of the conveyor, are constant in decrease in importance with the primary resistances, which are distributed throughout the length of the conveyor, and grow as head-to-tail distances increase [3]. The pie charts of figure 2.3 show the distribution of the individual resistance fractions for three different conveyor

(14)

2.8. Total motional resistances 8

Figure 2.3: Percental distribution of the individual motion resistances of bulk conveyors [3].

types. The large chart shows the distribution for long heavy horizontal conveyors. The first notable value is the 61% share of the indentation rolling resistance. While both the flexure resistance are relatively small (5% and 18%), they can be influenced by the belt characteristics. They are, just as the secondary resistances and the extraordinary (special) resistances, not negligible as to amount, but represent minor factors with respect to their effect on total energy requirement. The other two pie charts also show a large influence of the indentation rolling resistance with respect tot the total resistance. Those two charts however have a big secondary resistance resp. gradient resistance due to their special condition as described (a short conveyor and an ascending conveyor). The improvement in conveyor belts with respect to their indentation resistance will also have positive effects on the flexure resistance of the belt.

Belt tension

The impact of belt tension on the value of the main resistance to motion becomes more important for longer conveyors where the belt tension changes within a wide range. As can be seen in figure 2.4 the idler rotational resistance and the indentation resistance are constant, while the flexure resistance of a belt and bulk material and the sliding resistance of a belt on idlers depend on the belt tension. The result from Gladysiewicz match the measurement figures within satisfactory accuracy especially for bigger actual capacity where the standard DIN22101 method has given poorer required drivepower.

(15)

Figure 2.4: Main resistances to motion along the top belt of a typical belt conveyor from a lignite open pit [5].

Table 2.1: Specifications for figure 2.4

Transported material Overburden Type belt Steel cord belt

Idler type Garland, well maintained Density ρ 1600 kg/m3 Strength 3150 kN/m Length L 1100 m Lift H 10 m Belt width B 2.25 m Speed vt 5.24 m/s

Spacing on carrying side lg 1.0 m

Troughing angle λ 45◦ Ambient temperatue Tc 0◦C

Drive pulleys at the head station 2

Drive units 4

(16)

Figure 2.5: Main resistances versus belt speed for gravel [10].

Figure 2.6: Main resistances versus belt speed for bauxite.[10]

Table 2.2: Specifications for figures 2.5 and 2.6

Transported material Gravel (fig. 2.5) and bauxite (fig. 2.6)

Length L 60 m

Belt width B 600 mm

Idler spacing 1250 mm

Density gravel ρg 1400 kg/m3

Density bauxite ρb 1400 kg/m3

Internal friction angle gravel φig 45◦

Internal friction angle bauxite φib 34◦

Friction angle between belt and gravel φwg 28◦

Friction angle between belt and bauxite φwb 25◦

Bulk material

Figures 2.5 and 2.6 show the contributions of each of the components of the main resistance for gravel and bauxite respectively. As can be seen, there is a huge difference between those two materials. The idler rotational resistance and the indentation rolling resistance are approximately the same for both bulk solids, but the bulk solid flexure resistance differs a lot. This is primarily due to the higher internal friction of the gravel. These two figures show that the bulk solid flexure resistance can indeed be a significant factor in the calculation of the total motional resistance, depending on the internal friction angle φi of the bulk

(17)

3

Flexure resistance

In the previous chapters the different motional resistances are discussed. Knowing what influences those resistances is important to a high degree, because it can affect the working of the conveyor belt. As already said, the indentation resistance is the main contributor to the total motion resistance. This is also the reason there is a lot of research done on this particular resistance. The flexure resistance of the belt and its bulk material is the next resistance in line, but the diversity of the researches of this resistance is limited in comparison with the indentation resistance. In this chapter the main researches about flexure resistances are analyzed. The flexure resistance is divided in two parts, the flexure resistance of the belt and the flexure resistance of the bulk material, therefore those two are also separately discussed in the following chapter. Furthermore, of each resistance the mathematical model is first examined and at the end, some laboratory tests are looked into.

3.1. Flexure resistance of a belt

When a belt moves from one idler to another, the belt bends a little bit. This results in a flexure resistance of the belt. Main researchers of the flexure resistance of the belt are Spaans and Gladysiewicz.

3.1.1. Spaans

According to Spaans [7], [8] there are two different situations in which a flexure resistance of the belt can be calculated. At first we consider the mathematical model made by Spaans of the flexure resistance a flat belt and after that we consider that of a curved belt.

Flat belt

If we consider a flat belt moving over flat rolls and no flexure resistance occurs, then the belt should deform symmetrically. However, this happens hardly ever and therefore most belts adhere asymmetrically to the roll. This is due to the difference in flexural rigidity when moving over the rolls, even when the belt has been loaded evenly.

In figure 3.1 the forces on the roll and the curved belt are showed. The belt has to be in equilibrium, otherwise it will slide of the idler rolls. This results in

(18)

3.1. Flexure resistance of a belt 12

Figure 3.1: Part of belt between the "area of contact of roll and belt" and the "vertical symmetry line of the belt" [7].

a relationship between the difference in the different moments and the flexure resistance. This follows in the formula Fv∗ ρ ∗ y0 = ∆M . Also, the forces have

to be in balance, so therefore Fb/Fv = y0. From these two formulas, equation 3.1

can be derived.

Fb = ∆M/ρ (3.1)

With:

Fb Flexure resistance of a belt

∆M Loss of moment

ρ Radius of curvature of the belt

Curved belt

When a belt is curved, it is supported by two rolls with a spacing between them. Spaans [8] made a mathematical model of the curved belt based on the fact that the belt again has to be balanced, see figure 3.2. Taken into account that the moment in the belt causes resistance, although very small compared to the moment above the roll, and the radius of curvature is very large at the lowermost point, equation 3.2 is derived.

(19)

3.1. Flexure resistance of a belt 13

Figure 3.2: Mathematical model of a belt [7].

Fb = Ψb· EI ( Fv 2ωEI − q T 2 +h−q T i2 ) (3.2) With:

EI Flexural rigidity of the belt

q Homogeneous load distribution belt and material T Belt tension

In this equation it is assumed that the belt consists of isotropic and homogeneous material and that the flat cross-section does not deform. With belts with rub-ber covers and fabric carcass this is certainly not the case, but more detailed calculations are very time consuming and hardly contribute to the improvement of the qualitative insight. q/T can be neglected because naturally the degree of curvature of the belt between two trough idlers is negligible with respect to the curvature above the rolls. The formula can thus be deduced to:

Fb =

Ψb· Fv2

4T (3.3)

When the main resistance is calculated, it is important to know the share of the flexure resistance. This is expressed in the coefficient of the flexure resistance. This coefficient will be:

(20)

3.1. Flexure resistance of a belt 14 fb = Fb Fv = Ψb· Fv 4T (3.4) With:

fb Coefficient of the flexure resistance of a belt

It is also important to know the belt sag. Usually the sag s will not become greater than 1% of the idler spacing. As Fv/T is equal to eight times the relative sag of

the belt, and this sag usually does not amount more than 0.01, the coefficient for the flexure resistance will normally not be grater than 0.012. In that case the damping factor of energy loss due to the bending of the belt sag stands at 0.6. The sag of a throughed belt is calculated by equation 3.5 according to DIN21101 and therefore it appears that the flexure resistance of the belt contributes relatively little to the main resistance.

s a = Fv 8T (3.5) With: s Indentation depth a Idler spacing

3.1.2. Gladysiewicz

While Spaans approaches the model with an equilibrium of forces, Gladysiewicz [5] derives an equation from the energy balance. The force of flexure resistance of a belt performs a work of deformation of a belt on the spacing between consecutive idlers which balanced the energy dissipation.

Wb· lg = Ψb · Enb (3.6)

With:

Wb Flexure resistance of a belt

lg Spacing between consecutive idlers

Enb Energy dissipation

The strain energy of bending of a belt is treated as a linear elastic beam supported by idlers with a constant flexural rigidity over its length is calculated from the integral: Enb = q · lg 2 · EJ · Z lg 0 M2(x) · dx (3.7)

(21)

3.2. Flexure resistance of bulk material 15

Figure 3.3: Scheme of bending of a belt between two consecutive supporting idlers [5].

With:

q Homogeneous load distribution belt and material EJ Flexural rigidity of the belt

M Moment

The strain energy of bending is calculated with regard to transverse flexural rigidity of a belt resulting from idler troughing angle. If any misalignment of a belt occurs then the cross section of a belt on idlers is unsymmetrical which influences on the flexure resistance of a belt.

The damping factor has to be calculated with the use of supplementary empirical formulas developed for textile and steel-cord belts.

The equation for the flexure resistance of a belt from Gladysiewicz is therefore:

Wb = q · Ψb 2 · EJ · Z lg 0 M2(x) · dx (3.8)

3.2. Flexure resistance of bulk material

The volume flow of bulk material experiences a cyclical deformation, both length-wise and clocklength-wise, due to the sag of the belt and its limited flexural rigidity. This results in a flexure resistance of the bulk material.

Limited research has been conducted in the area of bulk solid flexure resistance primarily due to the perception that this resistance contributes little to the total resistance. Additionally, the bulk solid properties are generally considered a constraint rather than a variable parameter by the conveyor designer. While the conveyor designer has little control over the properties of the bulk solid being conveyed, the influence of these properties on the main resistances in some instances are significant and should not be overlooked. The bulk solid flexure resistance is consistently the second largest of the main resistances, as noted by

(22)

Hager and Hintz [3], and may even exceed the indentation rolling resistance in the case of wide conveyor belts, as discussed by Spaans [7]. The belt and bulk solid properties influence the bulk solid flexure resistance to varying degrees, as does the belt speed, belt sag and idler spacing, thereby offering potential reduction with informed design.

Bulk solid flexure resistance occurs between successive idler sets as the bulk solid undergoes transverse and longitudinal displacement due to belt sag. As the belt progresses from one idler set to the next the bulk solid undergoes cyclic expansion and contraction in the transverse direction, in addition to variation in height in the longitudinal direction, as shown in figure 3.7. The relative movement results in energy losses due to the internal friction of the bulk solid. [9].

The flexure resistance of bulk material results from cyclic deformations of material stream. These deformations are strictly connected with the bending line of a belt between idlers and therefore in previous methods of calculations of the main resistances of motion both the flexure resistances (bulk and belt) used to be identified as single components from an approximated formula. Assuming that transported bulk material is a kind of grainy medium, two zones of movement of a belt with material can be identified:

• active zone, where the material stream causes deformations of a belt (bending and an increase of an actual throughing angle).

• passive zone, where deformations of belt cause deformations of a material stream.

The length of each zone equals half of the distance between idlers.

3.2.1. Spaans

Spaans [7] was the first to provide an analytical model to calculate the flex-ure resistance of the bulk solid due to the cyclic transverse and longitudinal deformation.

In the research of Spaans [8] he neglects the resistance due to the opening and closing of the belt, because the other components of the resistance, the volume flow and the internal friction of the bulk material, are dominant. In the case of very wide belts with very thick layers of bulk material the flexure resistance contributes considerably to the main resistance.

Just like the calculation of the flexure resistance of the belt, Spaans considered a flat belt and a curved belt apart from each other.

Flat belt

As a result of its own weight and the weight of the bulk material, the belt is always a little bit curved and is thus cyclically deformed when the belt moves

(23)

Figure 3.4: Deformation pattern of volume flow [7].

the bulk material. From figures 3.4 and 3.5 Spaans made an mathematical model which resulted in the formula of the flexure resistance Fs over spacing a:

Fs = Fvs· Fv √ T · EI · (λpv− λav) · d2 12 ∗ a (3.9) in which: (λpv− λav) = 1 + sin φ 1 − sin φ− 1 − sin φ 1 + sin φ = 4 · sin φ cos φ2 (3.10) With:

Fs Flexure resistance of bulk material

Fvs Normal force due to the weight of the bulk material

T Belt tension

EI Flexural rigidity of the belt

λpv Passive stress coefficient in a vertical plane

λav Active stress coefficient in a vertical plane

d Thickness of layer bulk material a Idler spacing

(24)

Figure 3.5: Deformation of a volume part

a. the volume part; b. the pressure distribution on the normal plane [7].

According to Spaans, this equation is only valid if the deformations of the belt are slight. Because the opening and closing of the belt’s section is neglected, the equation is also only valid for a qualitative consideration.

Curved belt

When the neutral line of a curved belt and its bulk material lies on the same place as in case of the flat belt, namely in the belts lowest point, equation 3.9 can be used. However, this is not always the case. Because of the belt’s curvature and troughed shape, due to the weight of the bulk material, the tensions will be higher than with a flat belt. The following is assumed:

• At the place of the greatest sag the flexural rigidity of the troughed belt is equal to the flexural rigidity of the flat belt.

• At the place of a trough idler the belt is forced to take on the troughed shape.

The flexural rigidity of the troughed belt is very large there, as a consequence of which the radius of curvature of the belt and the bulk material is high as well. The flexure resistance therefore needs a correction of e−ωx, with ω is pT /EI, which results in equation 3.11.

Fs = Fvs· Fv √ T · EI · (λpv− λav) · d2 12 ∗ a · e −ωx (3.11)

The troughed shape opens and closes when going from the first idler roll to the second. Due to this deformation of the troughed shape the flexure resistance Ftr

(25)

Figure 3.6: Model of the opening and closing of the troughed shape [7].

is calculated by Spaans as follows, with lz referring to the section of bulk material

and not to the length of the side-roll as shown in figure 3.6.

Ftr = 1 6∆β · ρ · g · l 3 z · (λp − λa) · cos φw (3.12) With:

Ftr Resistance due to the troughing shape

β Troughing angle ρ Bulk density

g Acceleration of gravity lz Contact length

λp Passive stress coefficient

λa Active stress coefficient

φw Angle of friction between belt and bulk material

Just like the flexure resistance of the belt, the flexure resistance of the bulk material also has a coefficient fs to make is easier to calculate with all the other

main resistances. fs = Fvs √ T · EI · (λpv− λav) · d2 12 ∗ a · e −ωx (3.13) With:

fs Coefficient of the flexure resistance of bulk material

3.2.2. Wheeler

Wheeler only looks at the flexure resistances of bulk solids. In his papers [9] and [10] he clarifies his vision on the flexure resistance of bulk solid material. In

(26)

Figure 3.7: Induced active and passive stress states for a loaded conveyor belt [9].

his analysis he considers that the flexure resistance consists of both transverse an longitudinal components. Figure 3.7 shows the expansion and contraction of the bulk solid as the belt progresses from one idler set (position A) to the other (position E). When leaving the idler set and moving to position B, the troughed belt opens up under the action of gravity. This results in an active stress state in the transverse direction due to the bulk solid that can relax. Longitudinally, however, the bulk solid is undergoing compressive stresses due to the contraction of the bulk solid arising from the longitudinal sag of the belt. When the bulk reaches roughly half the idler spacing (position C) the stress states theoretically reverse. Here the conveyor has to close again and therefore a passive stress state is induced in the transverse direction resulting from the compressive stresses due to the narrowing profile of the belt. Meanwhile the bulk solid in the longitudinal direction widens generating an active stress state as it moves away from the point of maximum sag. This repeats between each idler set creating an cyclic transverse and longitudinal flexure of the bulk solid which results in flexure losses due to internal friction and friction at the belt and bulk solid interface.

Wheeler [10] adopts a similar approach to that of Spaans [7] by calculating the transverse and longitudinal components of the bulk solid flexure resistance individually. He uses in his analysis orthotropic plate mechanics to calculate the belt deflection to provide a means of predicting the flexure resistance due to the relative movement of the bulk solid. Rather than calculating the resulting normal force acting on the conveyor belt due to the induced stress states, the analysis calculates the pressure distribution over the surface area of the conveyor belt, using also the pressure factors given by equations 3.19 and 3.20. The basis for the transverse model is pictured in figure 3.8, while figure 3.9 shows the model of the longitudinal flexural component. The total bulk solid flexure resistance for the idler spacing is calculated from the sum of the two equations.

(27)

Figure 3.8: Transverse active and passive stress states that are formed within the bulk solid as the belt opens and closes between successive idler sets [9].

Figure 3.9: Longitudinal active and passive stress states that are formed within the bulk solid as the belt deflects longitudinally between successive idler sets [9].

(28)

3.2. Flexure resistance of bulk material 22 Ftrans_f lex = 2 s nmax X n=njunct+1 ×    Pmmax m=mtrans 1

2∆xy.ρ(a)m.n[sin β(h)n(n − njunct)] 2

Ktpcos φw(w)m,n

−Pmtrans−1

m1

1

2∆xy.ρ(a)m.n[sin β(h)n(n − njunct)] 2 Ktacos φw(w)m,n    (3.14) Flong_f lex= 2 s njunct X n=1 ×        Pmmax m=mtrans(Pz_bs)m,nKlp∆xy(η)m,n(h0bs)2n h 1 2 − (h0 bs) 2 n 3(hbs)n i −Pmtrans−1 m=1 (Pz_bs)m,nKla∆xy(η)m,n(h0bs) 2 n h 1 2 − (h0 bs)2n 3(hbs)n i        (3.15) With:

Ftrans_f lex Transverse flexure resistance force

Flong_f lex Longitudinal flexure resistance force

ρ Bulk density β Troughing angle h Height of bulk solid

hbs Height of bulk solid above the centre idler roll

h0_bs Height of bulk solid bounded by the failure plane Ktp Transverse passive pressure factor

Kta Transverse active pressure factor

w Belt deflection

Pz_bs Pressure distribution due to bulk solid

η Angular deflection

Both transverse (equation 3.14) and longitudinal (equation 3.15) flexure resis-tances are calculated from the difference between the work done in deflecting the bulk solid during the active and passive stress states. A significant benefit of the finite difference model described by Wheeler [10] is the ability to accurately predict the transition between the opening and closing belt, which is calculated from the point of maximum sag along the inclined side of the conveyor belt. However, in a later study with Ilic and Ausling [11] Wheeler preferred the theory with the normal force acting on the side idler roll as described next.

The transverse flexure is modeled by calculating the difference between the work done during the opening and closing of the belt, as the belt moves between

(29)

Figure 3.10: Force analysis for active stress case [10].

consecutive idler sets. The normal forces acting on the side idler rolls are calculated using a method developed by Krause and Hettler [12], who provide an analysis of the total force acting on the idler rolls due to the formation of active and passive stress states within the cross-section of bulk solid, as shown in figure 3.10 and 3.11. They applied a modified version of Coulomb’s earth pressure theory to calculate the normal forces acting on the side idler rolls of a three-roll idler set. The normal force per unit length Fsn, acting on the side idler roll due

to the bulk solid is then approximated as

Fsn= 1 2ρgL 2 ss Kta+ Ktp 2 cos φw (3.16) With:

Fsn Normal force per unit length

ρ Bulk density

g Acceleration of gravity

Lss Length of bulk solid in contact with the inclined side of the conveyor belt

The most difficult part in calculating the flexure resistance of bulk material is determining the place of transition between the passive and active stress zone. After further researching and testing the theory of the stress states in the transition zone ([11], [13]) it is stated that the transition zone is at 2/3 of the idler spacing.

Pressure factors

The formulas described in equations 3.14, 3.15 and 3.16 all use pressure factors. As already explained, when moving from one idler to the other, active and passive

(30)

Figure 3.11: Pressure distribution acting on inclined side of conveyor belt [10].

stress states are induced in the bulk solid due to the constant opening and closing of the belt. This leads to a transit zone where the bulk material experiences a lot of stresses. These stresses need to be predicted accurately in order to make a satisfying assumption of the pressure factors. There are multiple researches done in order to calculate these pressure factors.

As stated by Ilic, Wheeler and Ausling [11], Krause and Hettler provided an analysis on the total force acting on the idler rolls as a result of the formation of active and passive stress states within the cross-section of the bulk solid material. The longitudinal passive and active pressure factors are approximated by:

Klp = 1 + sin φi 1 − sin φi (3.17) Kla= 1 − sin φi 1 + sin φi (3.18)

The transverse active pressure factor for the opening conveyor belt was derived in terms of the troughing angle, bulk solid internal friction angle, belt and bulk solid friction angle and the conveyor surcharge angle, and is given by

(31)

3.2. Flexure resistance of bulk material 25 Kta =   sin (β + φi)/ sin β psin (β − φi) + q sin (φi+φw) sin (φi−λ) sin (β+λ)   2 (3.19)

Similarly, the transverse passive pressure factor, for the closing conveyor belt is

Ktp =   sin (β − φi)/ sin β psin (β + φi) − q sin (φi+φw) sin (φi+λ) sin (β+λ)   2 (3.20) With:

Ktp Transverse passive pressure factor

β Troughing angle

λ Conveyor surcharge angle

This model is widely accepted though it is realized that it overestimates the forces on the inclined sides of the conveyor belt compared to experimental results [11]. Liu [14], [15] presented another analytical approach based on the stress discontinuity method. This model incorporates a new hypothesis of bulk movement on the belt and rigorous stress field analysis. These new calculations of the coefficients are:

Ka =

1 − sinφicos∆2+ φw

1 + sinφicos∆1+ β

fd· cos2βe2θatanφi (3.21)

Kp =

1 + sinφicos∆4+ φw

1 + sinφicos∆3+ β

fd· cos2βe2θptanφi (3.22)

With:

Ka Active pressure factor

Kp Passive pressure factor

β Conveyor surcharge angle of bulk material ∆1, ∆2,

∆3, ∆4,

Angles in Mohr circle fd Dynamic factor

θa, θp

Rotations of the major principal stresses in active and passive stress states

(32)

Figure 3.12: Scheme of identification of the flexure resistance of bulk material [5].

These equations are used in an experimental model and are verified by some tests. This model can predict the pressure distribution on the belt based on active and passive stress states of the bulk material.

3.2.3. Gladysiewicz

Just like Wheeler, Gladysiewicz [5] considers the two different zones, the active and the passive stress zone, where the bulk material behaves differently. As with the flexure resistance of the belt, Gladysiewicz derives an equation from the energy balance.

Elementary forces that cause the deformations of a belt together with transported material stream are linear loads, q, uniformly distributed over a single distance between idlers. Within the passive zone the work of loads on the deformations equals: Enf = q · Z 0.5·lg 0 y(x) · dx (3.23) With: Enf Energy dissipation

q Homogeneous load distribution belt and material lg Spacing between consecutive idlers

y(x) Bending lines of a belt

The bending lines of a belt y(x) assumed for calculations the flexure resistance of bulk material and the flexure resistance of a belt are the same. Assuming that the two separate zones are proportional to pressure between the belt and the transported material, following the grainy medium mechanics with regard to energy dissipation the flexure resistance of bulk material is:

Wf = Ψf ·

Enf

lg

(33)

3.3. Laboratory tests 27

Table 3.1: Specifications for figures 3.13 and 3.14

Conveyor number 1 2 3 4 5 6 7 8

Mass flow t/h 1,000 15,900 750 2,500 15,900 5,000 6.000 6,000 Density kg/m3 ₇₄₀ _1,000 _1,580 _2,500 _1,650 _1,736 ₈₁₁ _2,765

Speed m/s 2.5 6.5 2.9 2.6 5.2 4 6 3

Filling ratio φv 1 1 0.65 0.7 0.8 0.95 1 0.62

Idler spacing (top) m 1.2 1.88 1.25 1.2 1.25 1.24 1.25 1.25

Idler spacing (bottom) m 3.0 7.5 4 3 3.75 3 5 5

Diameter roll (top) m 0.133 0.219 0.108 1.133 0.194 0.133 0.159 0.159 Diameter roll (bottom) m 0.108 0.194 0.108 0.108 0.108 0.089 0.108 0.108

Troughing angle β ◦ 30 37 35 30 41 30 40 40

Mass idlers kg/m 27.1 64.8 10.6 42.8 79.1 27.9 38.8 38.8

Mass belt kg/m 23 99 23.4 39.4 99 45.6 55.8 30.2

Belt tension (top) kN 905 570 100 120 260 120 100 80 Belt tension (bottom) kN 51.7 530 40 73.2 227 73.9 69 46

Belt width m 1.2 2.2 0.8 1.2 2.2 1.4 1.6 1.6 Belt type st 800 st 2500 st 1800 st 1600 st 2500 st 1600 st 2000 D 1850 With Ψf = 1 − tan2₍π 4 − φw 2 ) tan2₍π 4 + φw 2 ) (3.25) With:

Wf Flexure resistance of bulk material

Ψf Damping factor of material

φw Angle of friction between belt and bulk material

3.3. Laboratory tests

The mathematical models described in the previous sections also need to be verified. This can be done by simulating the belt conditions using a test rig or making a DEM (Discrete Element Modelling) simulation.

As can be seen in equation 3.4 the coefficient for the flexure resistance of the belt is a function of the ratio between the vertical load on the idler roll and the belt tension. Spaans used data from Limberg [16] to verify that. Limberg used multiple steel cord belts in his experimental research and measured the resistance per idler and further the coefficient of the main resistance according to the calculation method of DIN21101. Figure 3.13 indicates this linear relation as well as the position of the separate conveyors (numbers 1 to 8 in the figure) are all lined out. Using the same data, Spaans also made a graph of the coefficients of different resistances as can be seen in figure 3.14. This gives some insight in the share of these resistances under different circumstances.

An important aspect of the analysis is the allowance for the influence of belt speed. As indicated previously the transition between the stress states is assumed to occur at 50% to 60% of the idler spacing as indicated in position C in figure 3.7.

(34)

Figure 3.13: Coefficient fb for the flexure resistance of the belt as a function of Fv/T [7].

Figure 3.14: Calculated coefficients of indentation resistance, fi, flexure resistance of the belt,

fb, and flexure resistance of bulk material, fs, arranged according to the coefficient of the

(35)

The exact location of the transition is heavily dependant on the belt speed since as the belt speed increases the transition, and therefore the point of maximum sag moves further away from the midpoint of the idler spacing. Typically at high belt speeds the transition will occur at 55% to 60% of the idler spacing.

Since the bulk solid flexure resistance is calculated from the difference between work done during each stress state, increasing belt speed has the effect of increasing bulk solid flexure resistance. Wheeler used Discrete Element Modelling to simulate the movement of bulk solid as it is conveyed from one idler set to the next. In his research he examined the influence of the different parameters on the bulk solid flexure resistance.

While the belt conveyor designer typically has little control over the properties of the bulk solid being conveyed, it is still worth noting the influence of the internal friction angle on the bulk solid flexure resistance. Figure 3.15 shows the internal friction angle versus the bulk solid flexure resistance for a range of idler spacings. As the internal friction angle increases the ratio between the passive and active stress factors also increases. This has the effect of increasing both longitudinal and lateral components of the bulk solid flexure resistance, which are calculated from the difference between the work done during the passive and active stress states.

Also of interest is the reduction in the bulk solid flexure resistance coefficient with increasing idler spacing. This occurs since the magnitude of the flexure resistance per idler set only increases marginally with idler spacing since in the present example the sag ratio is maintained at 2%. Consequently, the flexure resistance force per unit length decreases with increasing idler spacing providing belt tension is increased accordingly to maintain 2% sag.

Figure 3.16 shows the influence of keeping the idler spacing constant and varying the sag ratio. As expected increasing sag results in higher bulk solid flexure resistance since the belt and therefore the bulk solid undergoes greater relative movement that results in higher frictional losses. Furthermore, as the internal friction of the bulk solid becomes greater there is a relative increase in bulk solid flexure resistance.

Figure 3.17 shows the influence of belt speed on the bulk solid flexure resistance. As the belt speed increases the bulk solid flexure resistance also increases since the transition between the active and passive stress states takes place at a location greater than 50% of the idler spacing. The increasing flexure resistance occurs for each of the sag ratios shown and is slightly more pronounced with higher sag ratios.

Given the influence of particular conveyor parameters on the bulk solid flexure resistance it is clear that the conveyor designer has control over many of there variables at the design stage. The selection of these variables should be made with consideration to the other components of the main resistance in addition to the total life cycle cost of the conveyor installation. Additionally, while the present discussion has been limited to static conveyor design in practice this particular

(36)

Figure 3.15: Calculated bulk solid flexure resistance coefficient versus kinematic internal friction angle for a range of idler spacings [9].

Belt speed vb 5 m/s

Belt width B 1.2 mm

Sag ratio s 2%

Density ρ 1000 kg/m3

Angle of friction between belt and bulk φw 30◦

Belt speed vb 5 m/s

Belt width B 1.2 mm

Idler spacing a 2 m

Angle of friction between belt and bulk φw 30◦

Belt width B 1.2 mm

Idler spacing a 2 m

(37)

Figure 3.16: Calculated bulk solid flexure resistance coefficient versus kinematic internal friction angle for a range of sag ratios [9].

Figure 3.17: Calculated bulk solid flexure resistance coefficient versus belt speed for a range of sag ratios [9].

(38)

Table 3.5: Normal force distribution for 24 mm belt deflection (2% sag) - DEM results [11]

Bulk Solid Normal Force on Each side (N) Normal Force on Centre (N)

Coal 64 207

Gravel 118 397

Magnetite 190 534

River Sand 132 428

Table 3.6: Normal force distribution for 32 mm belt deflection (2.7% sag) - DEM results [11]

Bulk Solid Normal Force on Each side (N) Normal Force on Centre (N)

Coal 66 204

Gravel 119 406

Magnetite 194 533

River Sand 140 414

methodology may also be incorporated into a dynamic analysis program where the bulk solid flexure resistance is expressed as a function of belt tension and belt speed.

To experimentally verify the theoretical model Wheeler developed an instrumented idler set to measure the total motion resistance at a given point in a belt conveyor. To determine the contribution of the bulk solid flexure resistance the other main resistance components were measured and then subtracted from the total resistance. After some tests the results showed an increase in flexure resistance with the internal friction angle which correlated well with the theoretical analysis. Ilic [11] tested the influence of belt sag using a DEM model and the equations as described by Wheeler. When comparing the results of table 3.5 and 3.6 there is very little difference in the normal forces exerted on the belt. To investigate the influence of the sag further, a comparison of the normal force variation during an opening and closing cycle was also undertake, focussing on coal and magnetite, since their relative particle densities are smallest and largest respectively. The results are shown in figures 3.18 and 3.19 As can be seen, the amplitude of the force between the open and closed belt positions is greater with the increased belt sag of 32 mm. This implies a greater amount of work resulting in more energy.

(39)

Figure 3.18: Coal - normal force as a function of time [11].

(40)

4

Implementation

With all the information acquired in the previous chapters a lot of research can be done. First and most important is that with the formulas and insight in the flexure resistance following research will be al lot easier. While today two-dimensional models are still state-of-the-art, the full three-dimensional finite element models of belt conveyor systems are getting more and more integrated in research. Besides Wheeler [9], who used DEM to simulate the movement of bulk solid as it is conveyed from one idler set to the next, and Ilic [11], who used DEM for simulating the pressure factors, Mustoe [17] used DEM to calculate the influence of belt sag on the forces and therefore the energy losses. However, three-dimensional models are much more complicated and still need to be verified by laboratory set-ups and known researches. On the other hand, when the simulation models are working correctly it is much easier and cheaper to run a simulation instead of building a set-up to test an hypothesis. Each time we go longer, higher, wider or faster, we stretch the limits of our analytical tools to predict system performance. And because each conveyor is unique, the only way we have to predict performance is our numerical analysis and simulation tools. Therefore it is imperative we continue to improve our design tools as our goals get bigger [18]. Now that we explored the main researches about the flexure resistance of the belt and its bulk material, we can look at what variables influence the flexure resistance. When designing a conveyor belt the capital costs of the system is a restriction and when the designer knows how to reduce motional resistances and thus the energy consumption of the belt, the costs can also be reduced. While reducing energy is an important consideration, the design of the conveyor should also be made with consideration to maintenance and capital costs of the system. As a result the economic performance of a belt conveyor system is suited to evaluation using life cycle cost analysis. Roberts [19] made a very detailed economic analysis of belt conveyor systems. When looking at figure 4.1, made by Lodewijks using the data given by Roberts, it is quite clear that the structure has the biggest influence on the costs. However, the effect of the indentation resistance and both the flexure resistances should not be overlooked. Therefore, it is important to look more into the options to reduce the flexure resistances to safe energy and therefore costs.

(41)

35

Figure 4.1: Annual equivalent costs for a horizontal steel cord conveyor belt versus idler spacing [6].

Throughput 1000 t/hr Length 1000 m Belt speed 3 m/s Belt width 1 m

(42)

36

From chapter 3.2 it follows that the flexure resistance of the bulk material in the main resistance is larger if:

• The mass of the bulk material per meter is larger • The thickness of the layer of bulk material is larger • The internal friction of the bulk material is greater • The tensile force of the belt is smaller

• The elasticity modulus of the belt is smaller

• The flexural rigidity of both the belt itself and the troughed belt is smaller • The belt sag is larger

It also shows that for a certain geometry the thickness of the layer of bulk material has a great influence on the flexure resistance of the bulk material. This means that the volume flow and the height-width ratio of that volume flow can, to a large extent, influence the flexure resistance. A high tensile force T of the belt causes a considerable decrease of the resistance. For a certain volume flow d2_/√_T

is determining for the magnitude of the flexure resistance. The flexural rigidity of the belt and the internal friction of the bulk material is thick and the tensile force in the belt is small. A large flexure resistance of the belt always consides with a large flexure resistance of the bulk material.

Under normal conditions the share of flexure resistance of the belt in the main resistance is very small. In the case of large belt conveyors the flexure resistance of the bulk material flow can be considerable and will even exceed the indentation resistance. While the share of the flexure resistance of the belt is relatively small to the main resistance, it is still important to take it into account. This resistance is larger if:

• The mass of the material per meter is larger • The tensile force of the belt is smaller • The elasticity modulus of the belt is smaller

• The flexural rigidity of both the belt itself and the troughed belt is smaller Observing those two lists we find that they are almost similar. It is important to reduce both flexure resistances, while not increasing the other resistances. From the model described above, it can be derived how the flexure resistance in those cases can be limited. However, most of the time the designer can’t choose which type or the amount of material is transported by the belt conveyor. This leads to limited flexibility in designing. On the other hand, when the designers knows the type of material and the throughput (and therefore the mass of the material per meter, the thickness of the layer of bulk material and the internal friction force) he can anticipate by changing the tensile force, the elasticity modulus or the flexural rigidity of the belt in order to decrease the flexure resistance.

(43)

37

Figure 4.2: Typical tension distribution in a driven belt conveyor [2].

For example by increasing the tensile force. In the research of Lodewijks and Pang [20] it is given that since a conveyor belt is not a flywheel but an elastic band, it also behaves like an elastic body. This means that it takes time for a tension wave to travel around the conveyor. The belt tension decreases with an increase in the magnitude of the dip. This can be a problem with starting after an emergency stop, but also under normal circumstances it belt tension must not be to small. A solution for increasing the belt tension is to increase the belt pre-tension [20]. If the belt’s pre-tension T2 (see figure 4.2) is seriously increased

then the belt tension in the dip may be acceptable. However, if this is the only measure taken then the pre-tension may have to be raised to unpractical levels. The impracticality comes from the required raise in belt rating, which would be quite expensive. Other solutions given by Lodewijks and Pang is to counteract the decrease of the belt tension by locally increasing the belt tension or decrease the idler pitch in the dip. This last solution does not help to prevent low belt tensions, but it will limit the belt sag and therefore the flexure resistance. Furthermore the belt tension is the main factor that influences longitudinal vibration [21]. This indicates that more attention should be paid to controlling tensile loading in belt conveyor design.

Both Hiltermann et.al [4] and Lodewijks et.al [22] researched the methods to save power of a conveyor belt by speed control. The speed can be controlled by using different methods like a varying load or material. Narrow belts running at speeds up to 15 m/s are technically feasible and are more cost effective than wide, slow belts [23]. While the indentation resistance is still the biggest resistance, the mathematical models of the flexure resistance can be used to predict speed control savings even further. Furthermore, a review of both fabric and steel cord belts and variable belt widths and troughing profiles should also be undertaken.

(44)

38

Summarily, there are a lot of ways to research the option to reduce the flexure resistance. But it is also important to keep looking at all the other resistances, cause reducing the flexure resistance may lead to increasing another one.

The most important effect of all the research done on the different aspects of a belt conveyor is that better models can be made and dynamic simulation techniques can be applied in their control systems. This development will affect the future of belt conveyor systems [1]. Nowadays engineers are faced with the fact that they have to design control procedures that work more or less under all operational circumstances. These procedures however are far from optimal and put the conveyor system in most operational conditions under a relative high strain just to make sure that they also work in the worst case conditions. In the future, however, large belt conveyor systems will have intelligent control systems that will keep track of all the dynamic parameters using sensors. This results on one hand that safety factors on the different components can be decreased and on the other hand that the overall system safety and reliability will increase.

(45)

5

Conclusion

In this report a number of researches about the flexure resistance are presented. The research used to create this report date back more than 40 years. With the results of this literature survey, more insight is given in the theoretical analysis of the physical behaviour of the running conveyor belt. When looking at the main resistances working on the belt, the flexure resistance is mainly the second contributor after the indentation resistance. While the indentation resistance is extensively research, the analysis of the flexure resistances lags behind due to the fact that its contribution is much smaller.

Luckily some researchers like Spaans, Wheeler and Gladysiewicz did do some research to the flexure resistance. It is split into the flexure resistance of the belt and the flexure resistance of the bulk material. While Gladysiewicz derives his equations from the energy balance, Spaans and Wheeler use the principle of the equilibrium of forces and Coulomb’s earth pressure theory. The bulk material also experiences stresses due to the opening and closing of the belt. In order to calculate the influence of this cyclic deformations, the transit zone has to be calculated using pressure factors.

When looking at the equations and the laboratory tests, both the flexure resis-tances are influenced by the belt sag, the mass and height of the bulk material, the type of bulk material and belt, the tensile force of the belt, the elasticity modulus of the belt and the flexural rigidity of the belt. When the impact of all this parameters is known, the engineer can design more efficient belt conveyors. With the information gained a better analysis of the whole belt conveyor can be made. However, some research still needs to be done concerning the flexure resistances and the other resistances working on the belt and the bulk material. Also the simulation models still need to be tested thoroughly in order to correctly predict the dynamics of the bulk material. In the future the models, laboratory set-ups as well as dynamic simulation techniques, can be improved and more intelligent control systems can be implemented in the design.

Models of calculating flexure resistance of conveyor belt and solid material

TUDelft

Abstract

List of symbols

Contents

1

Introduction

1.1. General introduction

2

Motional resistances

2.1. Introduction to belt conveyors

2.2. Motional resistances

2.3. Idler rotational resistance

2.4. Sliding resistance of a belt on idlers

2.5. Indentation resistance

2.6. Flexure resistance of the belt

2.7. Flexure resistance of the bulk material

2.8. Total motional resistances

Length

Belt tension

Bulk material

3

Flexure resistance

3.1. Flexure resistance of a belt

3.1.1. Spaans

Flat belt

Curved belt

3.1.2. Gladysiewicz

3.2. Flexure resistance of bulk material

3.2.1. Spaans

Flat belt

Curved belt

3.2.2. Wheeler

Pressure factors

3.2.3. Gladysiewicz

3.3. Laboratory tests

4

Implementation

5

Conclusion