coupled dem and fem simulations for the analysis of conveyor belt deflection

BulkSolids Europe 2012

1

Coupled DEM and FEM simulations for the analysis of conveyor belt

deflection

M. Dratt1, P. Schartner

2, A. Katterfeld

1, C. Wheeler

2, C. Wensrich

2

1Institute of Logistics and Material Handling Systems, University of Magdeburg, Germany

2School of Engineering, University of Newcastle, Australia

[email protected], [email protected]

Abstract

The application of numerical analysis and simulation methods is a main part of the design process of todays bulk solids

handling facilities. The Finite Element Method (FEM) is mainly used for static and dynamic structural analysis when

designing and optimising components. Meanwhile the Discrete Element Method (DEM) is used to simulate and optimise the

transport and material flow processes. The advantage of these two methods is proven by the rapidly growing interest in the

use of these techniques in both industry and research.

The Finite Element Method is a very useful tool to analyse the deformation behaviour of components. The main

disadvantage of this technique is that all the load assumptions that accrue from the interaction of the components and the

bulk material can not be found easily so some simplifications are needed. This is the point where the Discrete Element

Method becomes useful. It provides the opportunity to calculate the contact forces between the bulk material and

components of the bulk handling equipment that is being simulated. The main disadvantage of this method is that the

components can only be represented by rigid walls.

The coupling of these two methods can help to overcome the disadvantages occurring from the use of the individual

methods. This paper will explain the theoretical background of the coupling of the two methods and how they can be used

for analysing conveyor belt deflection. It will introduce the use of ANSYSTM

Classic to cover the Finite Element Method

and the use of PFC3D

for the Discrete Element section.

1 INTRODUCTION

The Finite Element Method (FEM) is widely used in Engineering for design and analysis of structural components. It solves

problems by dividing a large object into small sections and solving each individually. When this technique is used to

investigate conveyor belt deflection the bulk material load assumptions have traditionally been based on theoretical

continuum models. For a more accurate prediction of the belt deflection a more accurate bulk material load model is

required.

The Discrete Element Method (DEM) is a fairly new technology but widely accepted and used in industry to analyse bulk

material flow. It models the bulk material as spheres with simulated contact properties that represent the real bulk material

properties. The method calculates the contact forces within the particles as well as between the particles and the surrounding

walls. The weakness of this technique is that it only allows for the use of rigid walls to represent the boundaries of the

enclosure.


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To combine the strength of these two methods a coupling of the techniques is needed. This paper will explain the theoretical

background of a coupling technique using two commercial software packages. It will introduce the use of ANSYSTM

Classic

to cover the Finite Element Method and the use of PFC3D

for the Discrete Element Modelling. The application to be

discussed will cover the analysis of static and dynamic deflection of conveyor belts. To verify the coupled simulations

experiments have been undertaken under a range of loading conditions. While restricted to a conventional three roll

troughing idler set in the present analysis, further work is planned to expand the technique to more complex belt geometries

like pipe and pouch conveyors.

2 THEORETICAL APPROACH

As mentioned before, FEM breaks problems down to small elements that are connected at nodes and solves them

individually. In applications that model thin walled components in three-dimensions the use of shell elements is preferred.

This special type of element unites the membrane stiffness of membrane elements as well as the bending stiffness of plate

elements [1] [2]. Typical examples of these elements are three-dimensional, square, 6-node, triangle and 8-node rectangle

elements. The term square relates to the polynomial grade of the shape function of the element type that describes the

deformation of those elements. Compared to linear elements this type has extra nodes in the middle of its sides that allows

adapting the edges to a quadric function. The increase in the degrees of freedom results in higher accuracy of the results

while keeping the number of elements low [3].

Figure 1 Left side: Basic principal of the DEM-FEM coupling by using a 8-node rectangle square element. Right side: Structure of a 6-node

master triangle and a 8-node master square


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To develop a realistic coupling approach it is important to identify the corresponding elements in the finite and discrete

element simulations. To verify this, the finite element mesh with its element nodes I to L is transferred into an stl-format. By

breaking the 8-node rectangle elements into a pair of triangle areas it is guaranteed that the derived areas are planar, which

is needed by the DEM contact laws. The same happens for 6-node triangle elements that are transferred into a triangular

area. If the FEM mesh is mixed and contains rectangle and triangle elements together, it is important to verify that one pair

of triangle stl-areas is corresponding to one rectangle element and a single triangle element to only one triangle stl-area.

This is ensured by exporting the element number as additional information in the stl-file.

Figure 1 shows the basic principal of the DEM-FEM coupling in the case of the ith

8-node rectangle element and its related

pair of derived triangle areas. The vectors resulting from the contact forces of particles and walls and is

summed up to the load vector . This is not needed for the corresponding area of a triangle element; its load vector results

directly from the contact force. As published by Dratt in 2010 [4], the contribution of the global x, y and z-components of

the load vector is assigned to equivalent node forces. The required weighting factors are related to the type of finite

elements used (line, surface, or volume elements) and the degree of polynomial of their shape functions.

Figure 2 Assignment of the resulting nodal forces for corner, side and shared nodes. Left side: Regular arranged FE-mesh; Right side: mixed FE-

mesh

The node displacements of a 8-node rectangle element are described by the following eight shape functions (2.1) to (2.3)

used with dimensionless, natural coordinates and in a range of , = 1. These equations correspond to the unit

displacement functions of the 8-node master square as shown on the left side of Figure 1 on which every 8-node rectangle

element is mapped with Cartesian coordinates. This helps to avoid changing shape functions used for mutable element

geometry in the Cartesian coordinate system [3].


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(2.1)

(2.2)

(2.3)

The shape function can be summed as a matrix as shown in equation 2.4:

[

] (2.4)

The elastic potential of a general finite element is given by [5]:

(2.5)

In equation (2.5) v is the node displacement vector, f is the element force vector and K is the element stiffness matrix. The

contribution of the elements weight is the force vector resulting from the volume load fp.

(2.6)

According to Bathe [5] the load p is interpolated using the shape function approach and results in the vector for the element

node loads pk:

(2.7)

Deriving from the volume loads the general form of the force vector is:

(2.8)

The boundaries and differential of the volume integral (2.8) is positioned in the Cartesian coordinate system while the shape

function approach (2.1) and (2.3) is in natural -coordinates. The coordinates transformation is completed using the

reduced two-dimensional Jacobi-matrix:

[

]

[

]

(2.9)

The differential dz corresponds to the constant imaginary element thickness telem. The transformation relation results

according to Betten [3] as the determinant of the reduced Jacobi-matrix:

(2.10)

The force vector fp resulting from the volume load of the weight of the element results in natural -coordinates from the

relationship:


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(2.11)

By solving the integral equation (2.11) a constant value of -1/12 results for the corner nodes I to K and 1/3 for the side nodes

M to P as shown in Figure 1on the right side. These weight factors can be applied for elements in the shape of a square, a

rectangle and a parallelogram and can be adapted directly for three dimensions. The same approach can be used to calculate

the weight factors for the 6-node triangle elements, only the integration boundaries have to be changed according to Figure

1 right side. The factors are 1/3 for the side nodes M, N, P and 0 for the corner nodes I, J and K. The extension of the

calculations over the whole model takes the neighbouring relations of nearby elements with shared nodes into consideration.

If a node belongs to several elements then as a first step the x-,y-, and z- load vectors of the element are calculated and

multiplied by the weight factors. The equivalent node forces are then summed and applied to the nodes. This is shown in

Figure 2.

Based on the preceding analysis we now describe how to find the weight factors for distorted, planar 8-node rectangle

elements as shown in Figure 3. A prerequisite of plane element surfaces, requires all Nodes, IP to be in the xy-plane, and

the mid side nodes MP are on a direct connection line of the corner nodes IL. Using an imaginary element thickness of

tElem and a constant material density Elem the determinate of the Jacobi-transformation matrix will not result in a constant

value, but rather in a polynomial in relation to that has to be included in the integration of the element areas, as show in

(2.11). The weight factors are always referenced to the xy-coordinates so they cannot be directly adapted for three

dimensions. For this reason a calculation of the weight factor is needed that is independent of the shape function.

Figure 3 Context of the areas Ai(k) in comparison to the centre of gravity of an element Si(IP) of a planar, distorted 8-node square element

If such an element is loaded with a volume force fp, having a constant element thickness tElem and a density Elem and is

mounted in the centre of gravity it has to be in balance. In that case the potential i for the element is calculated following

the principal of the elastic potential at an extreme, in this case the minimum, according to Mueller [1]. In the current

situation the boundary conditions for the centre of gravity of the area Si(IP) in relation to the individual areas is given by the

following geometrical relation:


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(2.12)

(2.13)

Based on the preceding geometrical relations, if the sum of opposite individual areas Ai(k) is half of the area Ai of the planar,

distorted 8-node element, then the sum of the weight factors has to be -2/12 for the corner nodes IL and 2/3 for the side

nodes MP as in the master square to ensure equilibrium.

This statement allows the determination of the weight factors Wi(j,k) for the planar, distorted 8-node element using the

relation of the single areas Ai(k). Wi(j,k) is the weight factor on the ith

element and the jth

component, x, y or z, at the node

position k=I..P.

{[

]

}

(

) (2.14)

{[

]

}

(

) (2.15)

The weight factors in equation (2.14) and (2.15) left for the nodes I and M result from the relation of the results per node

using equation (2.11) and the total element area Ai and match with the results calculated using the sub areas Ai(k) on the right.

If using planar, distorted 6-node triangle elements the weight factors of the master triangle stay the same since the relation

between the individual areas and the element area is consistent. When an element is wrapped or has curved sides the sub

area relations are used automatically as an approximation. The error that occurs from this approximation is correlated to the

element size and can be pushed under 1% using a well-conditioned FE-mesh.

3 NUMERICAL VALIDATION

To validate the coupled simulation using a mixed FE-mesh (shown in Fig 4 (a)) the deflection of a beam, supported on both

sides and loaded with a constant area load is compared with the results just using FE-analysis. In the coupled simulation the

areal load is represented by particles simulating an equivalent stress. This is shown in Figure 4.

Figure 4 a) mixed FE-mesh; b) derived stl-geometry model of the calculation area; c) DEM-simulation with randomly generated particles; d)

FEM-analysis of the beam deflection;

c) d) b) a)

DEM-simulation FEM-analyses FE-mesh STL


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The error that occurs when comparing the two different solutions to the problem is less than 1% and results from the

irregular distribution of the particle bed. Further investigations have shown that by using a regular FE-mesh and using

particles that have the same diameter as the element length the error can be reduced further.

3.1 ANALYSIS OF CONVEYOR BELT LOADS AND DEFLECTION

The accurate analysis of the loads and deflection of a conveyor belt can serve to provide valuable information in regards to

induced belt stress, idler loads, and energy consumption due to belt and bulk material flexure. Loads and corresponding

deflections are dependent on the properties of the conveyor belt, idler configuration and pitch, as well as the properties of

the bulk material. The application of coupled FEM and DEM simulations to this area provides the opportunity to gain much

greater detail than presently obtainable via analytical approaches.

While the simulation of rigid bulk material handling plant such as bins and transfer chutes typically requires one-way

coupling of the FEM and DEM, the relatively large scale deflection of the conveyor belt between successive idler sets

requires two-way coupling. This requires the load data as well as the deformed shape of the belt to be transferred from

ANSYSTM

Classic to PFC3D

.

Initial modelling of a static belt conveyor was undertaken by Dratt et al [4]. This approach analysed a fabric reinforced

conveyor belt using a linear orthotropic material model and resulted in good correlation with measured maximum belt

deflections. In the current work, as in the static case, the belt is tensioned and loaded with particles. After the particles settle

the dynamic simulation is started. Once a constant load condition is reached in the particle bed the loads calculated by

PFC3D

are exported to ANSYSTM

and the particle positions and their rotational and transversal velocities are saved. After

applying the loads to the belt, a new belt geometry is generated and exported as an stl-file that can be imported back into

PFC3D

. This procedure is repeated until a steady belt deflection is reached.

Figure 5 shows a belt section with three idler stations with a spacing of 1m and a belt width of 0.8m modelling a test rig

used by Hettler [6]. Using a belt speed of 2 m/s and a bulk material density of b = 1,4 t/m and a theoretical cross section of

the material bed of Ath = 0,07 m this results in a belt capacity of around IM = 700 t/h. The theoretical approach to calculate

the bearing loads in the idler rolls is derived from the theory of Krause and Hettler [7], and is summarised in Table 1.

Transport direction

Figure 5 Particle bed running over two idler stations (v=2m/s) after a simulation period of about 12.5 sec. Colours due to particle velocity,

Bottom left: Definition of idler roll support


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Position Left idler Centre idler Right idler

Bearing load [N] radial axial radial radial axial

Fr1a Fr1b Fa1 Fr2a Fr2b Fr3a Fr3b Fa3

Krause/Hettler 62 244 -26 352 352 62 244 26

FEM-DEM 83 203 -20 379 378 81 199 21

Table 1 Comparison of the analytical results following the theory of Krause and Hettler and the coupled FEM-DEM Simulations after a

simulation period of about 12.5s

To calculate the bearing loads over the simulation time additional nodes are positioned at the idler roll supports. The outer

nodes of the wing idler roll are modelled as a floating support and the inner ones as well as nodes on the centre idler roll are

modelled as fixed nodes. As shown in Table 1 there is a good correlation between the theoretical approach and the coupled

simulations.

Further verification of the coupled FEM and DEM simulations are being undertaken to compare belt deflection profiles.

This work involves directly measuring the 3-dimensional profile of a loaded conveyor belt. A section of steel cord

conveyor belt (ST2500) is mounted on two end frames in the shape of a three-roll idler set. These two end frames are fitted

to a base frame, with one secured and the other free to slide along the base frame to allow the belt to be pre-tensioned. The

tensioning frame is mounted on two bearings to minimise friction between the end frame and the base frame. Figure 6

shows details of the test facility.

Figure 6 Conveyor belt deflection test rig

The conveyor belt is bolted onto the idler frames and a Perspex screen allows material to be loaded all the way to the end.

The belt is pre-tensioned using threaded bars that are instrumented with 2000 kg S-type load cells to measure the applied


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tension. The bulk material used for the tests was gravel with a particle size range from 16 to 25 mm. The belt was loaded in

a cross section according to CEMA with an edge clearance of 80 mm based on a surcharge angle for gravel of 25 degrees.

Tests have been conducted for both the empty and fully loaded cases for a range of belt pre-tensions from 5 to 25 kN, in

steps of 5 kN. At each load step the profile of the belt was measured using a 3D Laser. The point cloud detected by the laser

is processed and the data points later transferred into CAD software for further processing. Figure 7 shows a typical point

cloud from the 3D laser scan prior to further processing.

Figure 7 Setup of the 3-D laser and resulting measured point cloud

Figure 8 shows an example of a conveyor belt deflection profile that was extracted from the measured point cloud and

imported into AutoCAD for analysis. From the CAD image it is then possible to extract two-dimensional belt cross-

sectional profiles that can be directly compared to the results obtained from the coupled simulations. This experimental

work is ongoing and will provide valuable data to verify the coupled simulations under a range of loading conditions.

Figure 8 AutoCAD surface plot and cross-sectional view obtained from the measured point cloud

Cross section I

Cross section II

Cross section I

Cross section II


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4 FURTHER EXPERIMENTAL WORK

To verify the coupled simulations for the dynamic case a series of experiments is planed using a full scale conveyor belt

shown in Figure 9. In these tests the belt deflection will be measured using 3-dimensional photo imaging techniques and

will then be compared with the simulation results. The test will be conducted for various belt speeds and idler pitches.

Figure 9 TUNRA Bulk Solids belt conveyors and instrumented idler roll set

During the testing program it is also planned to measure the loads on the idler rolls. This will be achieved using the

instrumented idler roll set shown in Figure 9. The rig contains a support frame that can be mounted to the conveyor

structure. The idler rolls are supported on knife edges and instrumented with various load cells. Experiments will be

conducted for different belt speeds and idler spacings. The results from these experiments will be compared with the results

of the coupled simulations and the theoretical approach by Krause and Hettler [7].

5 CONCLUSIONS

The Finite Element Method is widely used for the design and dimensioning of structural components, while the Discrete

Element Method is a very popular technique to investigate bulk material flow. The coupling of DEM and FEM simulations

provides a useful tool to examine and design material handling operations. The current paper presents an application of the

coupled DEM and FEM simulation technique to model the deflection of a conveyor belt. The analysis of the loads and

deflection of a conveyor belt will lead to useful design information to calculate induced belt stresses, idler loads and energy

consumption. Traditionally this has proven to be a difficult problem due to the relatively large scale belt deflection, however

the application of coupled FEM and DEM simulations to this area provides the opportunity to gain much greater insight

than presently obtainable via analytical approaches.


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6 REFERENCES

[1] Mller, G.; Groth, C.: FEM fr Praktiker Band 1: Grundlagen. 8. Auflage. Renningen: Expert Verlag, 2007.

[2] Klein, B.: Grundlagen und Anwendungen der Finite Elemente Methode im Maschienen- und Fahrzeugbau. 7.

Auflage. Wiesbaden: Vieweg Verlag, 2007.

[3] Betten, Josef: Finite Elemente fr Ingenieure 1&2. Heidelberg: Springer Verlag, 1997.

[4] Dratt, M.; Katterfeld, A.; Wheeler, C. A.: Prediction of belt deflection by coupling of FEM and DEM simulations.

In: Bulk solids handling. Wrzburg: Vogel Business Media, Bd. 30.2010, 7, S. 380-384.

[5] Bathe, K.-J.; Finite-Elemente-Methoden. 2.Auflage. Berlin, Heidelberg: Springer Verlag, 2009.

[6] Hettler, W.: Beitrag zur Berechnung der Bewegungswiederstnde von Gurtbandfrderern. Magdeburg, Technische

Hochschule Otto von Guericke, 1976.

[7] Krause, F.; Hettler, W.: Die Belastung der Tragrollen von Gurtbandfrderern mit dreiteiligen Tragrollenstationen

infolge Frdergut unter Beachtung des Frdervorgangs und der Schttguteigenschaften. Wissenschaftliche

Zeitschrift der Technischen Hochschule Otto von Guericke, Magdeburg, 18 Heft 6/7. pp 667-674, 1974.

coupled dem and fem simulations for the analysis of conveyor belt deflection

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