Design of Belt Conveyors with Horizontal Curves 

Holger Lieberwirth, Germany
Courtesy : Trans Tech Publications  Bulk Solids Handling Journal
1. Introduction
Belt conveyors with horizontal curves have been successfully used for some years, as demonstrated by a variety of existing plants. The belt conveyors in the Power Station Plant Soma, Turkey, (Fig. 1) with a length of 8.5 km, and the 6 km long belt conveyor in the Optimum Coal Mine, South Africa [3], are mentioned as examples. Plans to use conventionally designed belt conveyor systems for ever narrower curves require that basic calculation principles used in the past should now be critically analysed, and that other factors, which were insufficiently considered in the past, should now be taken into account.
Belt conveyor in Soma Power Plant complex [2]
2. Present Status
The development of the theoretical principles for the design of belt conveyors over the course of time can be derived from [5]. Published research up to now has mainly investigated the influence of
a. radiaily acting force components F_{GB} from deadweight of the belt G_{B} and its troughability,
radially acting force components F_{GG} from deadweight of the material G_{G} and
radially acting components F_{R} from friction force by tilted carrying idlers on the position of the conveyor belt in the horizontal curved (Fig. 2). These forces, with an admissible offcentre belt run, are required to be in balance with the force F_{T} obtained from the belt tension force T acting radial to the inner side of the curve.
F_{T} = F_{GB} + F_{GG} + F_{R} (1)
Fig. 2: Crosssection through a belt conveyor in a horizontal curve, forces and geometry
Fig. 3 shows the basic model for calculation of the radial force resulting from the belt tension force with
F_{T} = T * I_{T} / R (2)
Fig. 3: Top view of a belt conveyor in a horizontal curve, forces and geometry
where T is the belt tension force in the relevant section of the belt, I_{T} the spacing between the carrying idlers and R the radius of the horizontal curve. Past evaluations did not consider that the radial force F_{T} at the troughed belt has to be split, analogously to the weight forces G_{B} and G_{G}, into components vertical and parallel to the carrying idlers to properly determine the effect of this force on the position of the belt.
As far as the normally used horizontal curve radii of conventionally designed belt conveyors is concerned, an approximate approach can be set up assuming the belt tension force T to be uniformly distributed over the total belt crosssection (Fig. 4). As a consequence thereof, the radial force share ΔF_{T} is in proportion to the width of the belt section in contact with the carrying idlers of the belt trough.
Fig. 4: Distribution of the belt tension force T and the radial force F_{T} in a horizontal curve over the belt crosssection
The following calculation equations are obtained for the components of the radial force F_{T1 }, F_{TM} and F_{T2} (Fig. 5)
F_{T1} S_{B1} * F_{T} (3) B F_{TM} I_{M} * F_{T} (4) B F_{T2} S_{B2} * F_{T} (5) BFig. 5: Model fro distribution of te radial force F_{T} to individual sections of the belt crosssection
where S_{B1} and S_{B2} represent the length of the outer carrying idlers in contact with the belt and I_{M} the length of the center carring idler. The components
F_{TN1} = F_{T1} * sin(λ + γ) (6)
F_{TNM} = F_{TM} * sinγ (7)
F_{TN2} = F_{T2} * sin(λ  γ) (8)are acting vertically to the shell surface of the carrying idlers and have to be considered for determination of the friction forces of tilted idlers.
The radial force components
F_{TT1} = F_{T1} * cos(λ + γ) (9)
F_{TTM} = F_{TM} * cosγ (10)
F_{TT2} = F_{T2} * cos(λ  γ) (11)act parallel to the carrying idlers and have a direct influence on the position of the belt. Thereby it was considered that the forces parallel to the carrying idlers follow the direction of the belt trough in the case of commonly used troughing angles, this being a consequence of the transverse stiffness of the belt in the transition area between two adjacent carrying idlers. This was supported by experimental investigations [1]
Thus, with F_{TT} as sum of the components of F_{T} acting parallel to the carrying idlers,
F_{TT} = F_{TT1} + F_{TTM} + F_{TT2} (12)
the following equation is valid to determine the offcentre belt run
F_{TT} = F_{GB} + F_{GG} + F_{R} (13)
Since F_{TT} < F_{T} is valid for a troughed belt, it becomes obvious that past calculation principles started from too high forces acting towards the inner curve. According to [5] the following equation is valid for the determination of F_{GB} and F_{GG}
F_{GB} = 
G_{B} * I_{T} 
[S_{B1} * sin(λ + γ) +
I_{M} * sinγ  S_{B2} * sin(λ  γ)] 
(14) 
B 
respectively.
F_{GG} = I_{T} * ρ * [A_{X1} * sin(λ + γ) + A_{XM} * sinγ  A_{X2} * sin(λ  γ)] (15)
where ρ is the bulk density of the material conveyed and A_{Xl}, A_{XM} and A_{X2} the bulk material crosssection situated above the relevant carrying idlers. The guiding force from friction between belt and tilted carrying idlers is
F_{R} = _{S1} [(k_{s} * S_{B1}/B * G_{B} + A_{X1} * I_{T} * ρ) * cos(λ + γ) + F_{TN1}] + _{M} * [(k_{M} * I_{M}/B * G_{B} + A_{XM} * I_{T} * ρ) * cosγ + F_{TNM}]  S_{2} * [(k_{s} * S_{B1}/B * G_{B} + A_{X2} * I_{T} * ρ) * cos(λ  γ)  F_{TN2}] (16)
where _{s1}, _{M} and _{s2} are the friction values between belt and the relevant carrying idlers depending on loading and degree of tilting. k_{s} and k_{m} represent experimentally determined values for consideration of the troughabiiity of the belt, the development of which is shown in [5] depending, inter alia, on belt troughing. The components F_{TN1} and F_{TN2}, hitherto not considered in the basic calculation principles, cause an increase in friction force of the inner and centre carrying idlers while that of the outer carrying idlers is decreased. In this way an additional force component directed to the outer curve is obtained for tilted carrying idlers.
Concluding Remarks
While investigations made up to now were mainly concentrated on the vertically acting components of the weight force of both belt and material conveyed as well as on the resulting force for tilted idlers, the present paper investigated the radial force of a troughed belt in more detail. It could be demonstrated that only a portion of this force contributes to a shifting of the belt towards the inner curve, while certain components are acting towards the outer curve as a function of the friction between the belt and tilted carrying idlers.
References
KESSLER, F.: Untersuchung der Fhrungskrfte quer zur Gurtlaufrichtung bei Gurtfrderern mit Horizontalkurven; Dissertation 1986, Montanuniversitt Leoben.
SAGHEER, M.: Turkey operates its first overland conveyor with horizontal curves; Mining Engineering Vol. 40 (1989) Nr. 4, pp. 234235.
BROWER, T.: GroKurvenfrderer in Sdafrika; Frdern und Heben Vol. 39 (1989) No. 1 1, pp. 914916.
KESSLER, F.: Einflu der Muldungsfhigkeit des Frdergurtes auf die Verteilung der Normakrfte zwischen Gurt und Tragrollen; Hebezeuge und Frdermittel Vol. 30 (1990) No. 17, pp. 811.
KESSLER, F., GRABNER, K., GRIMMER, K.J.: Neuer kurvengngiger Gurtfrderer mit pendelnder Aufhngung, Frdern und Heben Vol. 44 (1994) No. 12, pp. 7780.
Dr. Holger Lieberwirth, Manager Projects for Material Handling Systems,
Krupp Frdertechnik GmbH, FranzSchubertStrasse 13, D47226 Duisberg, Germany
Tel.: +49 2065 782205; Fax: +49 2065 78 3510