2017 3rd International Conference on Applied Mechanics and Mechanical Automation (AMMA 2017) ISBN: 978-1-60595-479-0

Optimization and Experimental Research on Vertical Section of Pipe Belt Conveyor

Tong-bing ZHU1,2, Kun HU1,2,* and Shuang WANG1,2

1Anhui Mine Electromechanical Equipment Cooperative Innovation Center, Anhui University of Science and Technology, Huainan, 232001, China;

2School of Mechanical Engineering, Anhui University of Science and Technology, Huainan 232001, China

*Corresponding author

Keywords: Pipe belt conveyor, Vertical section, Bending stress, Roller spacing.

Abstract. Due to the material extrusion of the vertical section and the supporting force of roller

groups of pipe belt conveyor, the periodic wave deformation of conveyor belt was undergoing

between the supporting rollers in the vertical lifting process, with the complex change of belt

bending stress. In this paper, the bending stress of vertical section of pipe belt conveyor was studied.

And the calculation formula of the relationship between the belt bending strain and the roller

spacing in vertical section was deduced from the vertical pipe belt bending model, which

considered the wave deformation condition in actually. Moreover a testing apparatus with the

adjustable rollers spacing was designed for the verification experiment. And the results showed that

the theoretical calculation was consistent with the experiments. Therefore, under the certain design

conditions, the layout of supporting roller groups of pipe conveyor could be optimized and

determined by calculating the appropriate rollers spacing ratio C, to improve the belt stress of

vertical section, the service life and reliability of the pipe belt conveyor.

Introduction

Pipe belt conveyor is one special belt conveyor which forces to roll the conveyor belt to tubular

shape with external force and it is widely applied to bulk material transportation in industries like

coal, building material and chemistry. Due to its special tubular structure, the tubular belt conveyor

boasts the advantages of closed space, green and environmental-friendly and small space[1-3];

besides, its maximum transportation inclination is greatly better than normal trough-type belt

conveyor. In particular, the Chinese tubular belt type conveyor improved by professor Zhang Yue,

the famous belt conveyor expert in China, increases substantially the encapsulation and clamping

force of the material, which improves greatly the transportation inclination and even vertically, and

its industrial application has been realized in many regions in China[4].

As for the vertical lifting section of the tubular belt conveyor, bulk material particles are filled

inside the tube. The conveyor belt is subject to action of many kinds of force like gravity, friction

force, side material force, tensile force and positive force of cradle roller, so the force situation is

more complicated than normal transportation section; particularly, the side material pressure makes

the conveyor belt produce the external expansion, while the uniform cradle roller group supports

inwardly so that the conveyor belt has to withstand the periodic stress change during the continuous

transportation process, causing the maximum stress to increase and the reliability and service life of

the conveyor belt to increase greatly. Since vertical lifting theory occurs late, the design of vertical

lifting section of the tubular belt conveyor focuses usually on the experience and there is littler

literature report about relevant study[5]. In order to safeguard the reliability of the vertical lifting of

the tubular belt conveyor, it is necessary to conduct deer study.

74

Force Analysis of the Vertical Lifting Section

Side Pressure of the Vertical Lifting Section

When the bulk material enters the vertical lifting section of the tubular belt conveyor, it is just like

the material enters one thin and long cylindrical bin. The cylindrical shape at this time is relatively

stable. Assume the material is static relative to the conveyor belt, then static bin model can be used

for analysis[6-7]. Balance analysis of force is conducted by selection of small portion of vertical

section in height of dh, as shown in Figure 1.

Figure 1. Force analysis of vertical section.

As shown if Figure1, the material is subject to the self-weight of the material and the extrusion

force on the upper layer of material by the lower one in the vertical direction inside the tubular

conveyor belt. The stress in the vertical direction transfers to the side according to proportion,

forming the side pressure to the conveyor belt by the material, while greater bending moment will

be produced when the side pressure acts on the conveyor belt between two cradle rollers which

increases the stress of the conveyor belt[8].

The equation as follows can be obtained from infinitesimal force balancing:

0)( =+−−+ AdppdfgAdhAp vvv ρ (1)

ldhpdf sµ= (2)

vs pp λ= (3)

Among which: pv—vertical pressure, Pa;

ps—side pressure produced by the material inside the vertical section, Pa;

ρ—material density, kg/m3;

l—circumference of the conveyor belt tube, l=πD;

µ—coefficient of friction between material and conveyor belt;

A—cross section area of the material inside the conveyor belt, A=πD2 /4;

λ—side pressure coefficient, i.e. the ratio between the side pressure and the vertical pressure;

h—the distance between the cross section surface to the top surface.

Substitute formula (2) to formula (1) to simplify the differential equation in the vertical direction,

then we get:

A

lpg

dh

dp vv µλρ −= (4)

Perform variable separation and integration for formula (4), and use the boundary conditions h=0

and pv=0 at the same time to get the vertical pressure produced by the material in the vertical

section.

)1( A

h

v el

gAp

µλ

µλ

ρ −

−= (5)

Then the side pressure on the conveyor belt produced by the material is:

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)1( A

h

s el

gAp

µλ

µ

ρ −

−= (6)

Bending Model of Conveyor Belt in Vertical Section

During vertical lifting transportation of tubular belt conveyor, the vertical section of the conveyor

belt has lateral deformation and expansion due to the side material pressure and it is also subject to

the reverse supporting action of the cradle roller, so there is wave shape deformation on the

conveyor belt; the law of load distribution in the vertical section can be represented by formula (6).

In order to calculate conveniently, the tensile force influence on the horizontal section is not

considered in this paper and linear approximation is conducted for load in different sections

between adjacent cradle rollers, i.e. the stress in each section is assumed to change linearly and then

bending model of the vertical section of the conveyor belt is obtained, as shown in Figure2.

Figure 2. Belt bending model of vertical section. Figure 3. Bending moment analysis diagram.

Now the bending moment is analyzed for the conveyor belt inside the vertical section. Since the

bending model for the conveyor belt in the vertical section is statically indeterminate issue, now the

supporting point of the cradle roller is cut to be replaced with middle ream, which is equivalent to

remove the rotating inner restraint. One pair of bending moment M is added to make redundant

constraint. Now conveyor belt of unit width is taken for analysis, as shown in Figure 3.

Equation for Bending Moment of Conveyor Belt in Vertical Lifting Section

The equation set of bending moment within the vertical lifting section can be obtained from the

bending model of conveyor belt above.

+−=+++

+−=+++

+−=+++

+−=+++

+−=+++

+−=+++

−

−−−−−

−

−−

−

−−−−−−−−

+

++++++

−

−−+−−−

)0(60)0(2

)(6)(2

......

)(6)(2

)(6)(2

......

)(6)(2

)0(6020

1

11111

1

11

2

22112122

1

111211

1

111111

2

11

1

112321211

1

111211

n

nnnnnn

n

nn

n

nnnnnnnnn

i

ii

i

iiiiiiiii

i

ii

i

iiiiiiiii

L

awLMLM

L

bw

L

awLMLLMLM

L

bw

L

awLMLLMLM

L

bw

L

awLMLLMLM

L

bw

L

awLMLLMLM

L

bwLMLM ）（

(7)

76

Change the above equation to the matrix form A M = B, then A and B are indicated as follows:

A=

020......00

0.....00

.....

00...0....00

00...........00

......

00....0

00....002

1

1)(2

1)(2

)(2

)(2

1

122

1

11

2211

11

−

−

+

−

−+−−

++

+−−

+

nL

nL

iL

iL

n

nLnLn

iLiLi

iLiLi

LL

L

L

L

L

LL

LL

(8)

B=

1

11

1

11

2

22

1

11

1

11

2

22

1

11

1

11

......

....

6-

−

−−

−

−−

−

−−

+

++

−

−−

+

+

+

+

n

nn

n

nn

n

nn

i

ii

i

ii

i

ii

i

ii

L

aw

L

bw

L

aw

L

bw

L

aw

L

bw

L

aw

L

bw

L

aw

L

bw

(9)

Based on the linear matrix equation above, the constraining moment Mn within the rotation of

each supporting point can be solved.

Among which, Ln—the distance between No. n and No. n+1 cradle rollers.

wn—area of bending moment diagram due to the load on span Ln and Ln+1.

an—the distance from the graph wn to the No.n supporting area.

bn—the distance from the graph wn to the No.n+1 supporting area.

Then the bending moment at any point in each section can be obtained.

∫+= dxxpMxM sn)( (10)

Among which, x is the distance from any point in each section after cutting to the upper

supporting point.

From material mechanics, we know

εσ E= (11)

zI

yxM ')(=σ (12)

Among which, E—elastic modulus of the conveyor belt;

Iz—the inertia moment of the cross section of the conveyor belt for the neutral shaft;

y'—the distance from the point on the cross section to the neural shaft.

From formulas (10), (11) and (12), we get:

z

sn

EI

ydxxpM ')( ∫+=ε (13)

As suggested above, the strain due to the bending at any point within the vertical lifting section of

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the tubular belt conveyor is related to the distance of cradle rollers and the distance to the cradle

rollers. Therefore, we can further improve the strain situation of the conveyor belt by optimizing the

distance of cradles so as to improve the service life and reliability of the conveyor belt. The

calculation is explained by one example as follows[9].

Calculation Analysis

Example

Assume the length of the vertical lifting section of the tubular belt conveyor is 2m and the diameter

is d=250 mm. There are five cradle roller groups in total and the transported material is yellow sand

with the density of ρ=1.3×103

kg/m3. The coefficient between the material and the conveyor belt is

µ=0.46. The top and bottom ends of the conveyor belt are fixed. The diagram of bending moment

analysis in this example is indicated in Figure4.

Figure 4. Bending moment analysis of calculation example.

The equation set can be obtained based on the bending moment analysis:

+−=+++

+−=+++

+−=+++

+−=+++

+−=+++

)0(60)0(2

)(6)(2

)(6)(2

)(6)(2

)0(6)0(20

4

444544

4

44

3

33

4543433

3

33

2

22

3432322

2

22

1

11

2321211

1

11

1211

L

awLMLM

L

bw

L

awLMLLMLM

L

bw

L

awLMLLMLM

L

bw

L

awLMLLMLM

L

bwLMLM

(14)

In the design of belt conveyor, the distance between cradles can be selected with constant

proportion to improve the force of conveyor belt. Take the relation of distance between cradles

Li=CLi-1. ∑Li=2 m in this example. Set

LCL

LCL

CLL

LL

3

4

2

3

2

1

=

=

=

=

(15)

When C=1, we can get from the formula above:

M1= 6 N•m; M2= -22 N•m; M3= -21 N•m; M4= -722 N•m; M5= -124 N•m;

Then the bending moment of the conveyor belt within each section can be obtained

78

−−+

+−+

+−+

+

=

6312342619

38776532442

4824617392

7206

)(

23

23

23

3

xxx

xxx

xxx

x

xM

)0.2,5.1(

)5.1,0.1(

)0.1,5.0(

)5.0,0(

∈

∈

∈

∈

x

x

x

x

(16)

Substitute formula (16) into the formula (13). Take the spring modulus approximate to the rubber

material for E, E=4.676 Mpa, then we can get the bending strain of each section of the conveyor

belt.

Calculation Result and Analysis

Similarly, the strain in each section of the conveyor belt can be obtained when the distance between

cradle rollers is arranged in accordance with C=0.8 and C=0.7. After fitting, the bending strain

change diagram of the vertical lifting section of the conveyor belt can be drawn, as shown in Figure

5[10-11].

Figure 5. Strain variation.

In the figure, C=1, εmax=0.00308; C=0.8, εmax=0.00246; C=0.7, εmax=0.00296. The further

comparison suggests that when C=0.8, the maximum bending strain of the conveyor belt in the

vertical lifting section is small and the strain change is relatively uniform.

Experiment Test

The Truss Structure of the Distance of Cradle Rollers Can Be Adjusted

In order to better verify the theoretical calculation, the task group designed independently one kind

of truss structure of tubular belt conveyor whose cradle roller distance can be adjusted steplessly, as

shown in Figure6. Its main principle is that: the hexagon cradle roller group is installed on the end

plate which is connected with the truss through resilient clip where there are bolt holes. When the

cradle roller distance has to be adjusted, the bolts of the resilient clip on four corners of the end

plate have to be loosened and the end plate can move along the truss at this time; when it moves to

the set position, the bolts shall be tightened and fixed with the friction between the resilient clip and

truss.

Figure 6. Conveyor truss with adjustable spacing rollers. Figure 7. Testing circuit design.

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Test System Design

The bending strain value and theoretical calculation value of each section are tested by experiment

to make comparison. The design of multichannel data acquisition circuit of the test system is shown

in Figure7. One quarter bridge is adopted for the test circuit. The power voltage is 5V, the resistor of

bridge is 120Ω gold film resistor, the strain foil is 120Ω high accuracy resistance strain foil, the

sampling mode is continuous sampling and the sampling rate is 1 KHz. Differential signal control is

added in the test circuit so as to show separately the tested four-line signals; in order to prevent the

intervention of other signals, the low-pass cut-off frequency of the filter is set to be 20 Hz[12-15].

Test Result and Analysis

The cradle roller distance has to be adjusted during the test. The SMT test is conducted separately

for different distances. Firstly the upper, middle and lower positions of any section of the conveyor

belt are tested. The comparison shows that the maximum strain is at the lower end of each section,

so it is in line with the theoretical calculation; then SMT test is conducted at the maximum strain

area in each section. See Figure 8 for the test device and Figure 9 for the test result (partially).

The strain test results are shown in Table 2.

Figure 8. Experimental apparatus.

Figure 9. Test results(C=0.8).

80

The test results and theoretical calculation for each section of the vertical lifting section are

shown in Figure 10.

Table 2. Maximum strain of experiment.

C=1.0 C=0.8 C=0.7

L1 0.00156 0.00238 0.00259

L2 0.00287 0.00266 0.00293

L3 0.00255 0.00240 0.00313

L4 0.00349 0.00280 0.00347

Maximum 0.00349 0.00280 0.00347

(a) C=1.0 (b) C=0.8

(c) C=0.7

Figure 10. Experimental results vs. theoretical calculation.

The comparison results suggest that the test results of the bending in vertical section of the

tubular belt conveyor are basically the same as the theoretical calculation results, which proves that

the calculation with formula (11) with certain design conditions can determine the proper cradle

roller distance ratio C so as to optimize the cradle roller arrangement, improve the stress and strain

of the conveyor belt and enhance the service life and reliability of the lifting section of the tubular

conveyor belt.

Conclusions

The study of bending strain of the vertical section of the tubular belt conveyor leads to the

conclusions as follows: (1) the bending model of conveyor belt in the vertical lifting section of the

tubular belt conveyor is established, the wave shape deformation conditions that actually happen are

considered, and the calculation formula for the relation between the bending strain and cradle roller

distance of the tubular conveyor belt in the vertical section is derived; (2) the test device which can

adjust steplessly the cradle roller distance is designed and the verification test is conducted, the

results of which suggest that the theoretical calculation is in line with the test results. (3) with the

design conditions fixed, proper cradle roller distance ratio C could be determined by calculation to

81

optimize the arrangement of cradle roller group, improve the force of conveyor belt in the vertical

section and enhance the service life and reliability of the tubular conveyor belt.

Acknowledgement

This research work was supported by National Natural Science Fund Project (Grant No. 51641501),

the First-Class General Financial Grant from the China Postdoctoral Science Foundation (Grant No.

2013M540506) and Doctoral Fund of Ministry of Education of China(Grant No. 20133415110003).

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