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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.