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DYNAMIC ANALYSIS FOR BALL MILL FOUNDATION

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DYNAMIC ANALYSIS FOR BALL MILL FOUNDATION

Y. C. Han1 and F. Guevara2

1 Fluor, 1075 W. Georgia Street, Vancouver BC, V6E 4M7 Canada E-mail: yingcai.han@fluor.com 2 Fluor, 1075 W. Georgia Street, Vancouver BC, V6E 4M7 Canada E-mail: fernando.guevara@fluor.com Keywords: soil-structure interaction, stiffness and damping of soil, structural dynamics, soil

dynamics, response spectrum analysis.

ABSTRACT

The dynamic analysis of ball mill foundation is a typical problem of soil-structure interaction, and the sub-structure method is used to estimate the structural vibration. In this study a practical case of ball mill foundation is investigated to illustrate the approach and the dynamic behaviour of structure. The concrete mat foundation and piers are modelled by FEM model, and the stiffness and damping of soil (rock) are generated by a computer program. Then the stiffness and damping input to the FEM model as the base boundary condition. A series of dynamic experiments had been done in the field to validate the values of radiation damping which can be generated from the program. Different design options are compared to obtain the better solution. The large mill foundation is an irregular structure and located in a severe seismic zone. The response spectrum analysis is used to determine the earthquake forces and seismic response.

1. INTRODUCTION

The foundation acts as a rigid body assumed normally in classical empirical methods for dynamic analysis, such as Barkan model (1962) [3]. However, the structure of mill foundation and piers with large dimension is flexible rather than a rigid body. Numerical methods such as the general finite element method are also difficult to apply, as the direct simulation of radiation damping is not possible. Radiation damping is the dominant energy dissipation mechanism in most dynamically loaded foundation systems. The dynamic analysis is challenging for the flexible mill foundations using the standard analytical or numerical methods, and it is a typical problem of soil-structure interaction. Another challenge is that the dynamic loads come from not only the unbalanced forces by mill charge rotation, but also from the unbalanced magnetic pull force in high frequency domain when a gearless mill drive (GMD) is used to drive the grinding mill.

The diameter of ball grinding mills may be much large in mining industry. A practical case of ball mill foundation is examined herein. The diameter of mill is 8.2 m with length of 15.2 m, operating at 12 rpm. The height of mill shaft is 18.4 m above ground. The weight of mill and charge (ore and grinding media) is 3,000 tons, and GMD motor weight is 310 tons. The sub-structure method is used for dynamic analysis of the ball mill foundation, that is, the structure and soil are considered as two parts separately. The structure (mat foundation and piers) are modelled by FEM model. The impedance of soil (stiffness and damping) are generated by a computer program, and then input to the FEM model as the base boundary condition. A series of dynamic experiments had been done in the field to verify the values of radiation damping, and it can be generated by the program DynaN [5]. 2. FOUNDATION OF BALL MILL WITH GMD There are a number of Ball mills in operation around the world with diameter up to 8 m. Aspect ratio L/D varies for ball mills, L/D >1, typically 1.5 to 2.5 factor. Installed power of ball mills is close to 22 MW. Mining operations continually invest in new technologies to improve their energy efficiency and capacity in their grinding circuit. No doubt that mills size will continue increasing. Grinding mills are designed to break mineral ore into smaller pieces by the action of attrition and impact using grinding media. Ball mills are basically a horizontal rotating cylinder partially filled with steel balls as grinding media. Bearing pads are located at the end of the mills. One bearing has no axial float, while the second bearing has sufficient float to accommodate the thermal expansion of the mill.

Figure 1. Ball mill shell supported design

In mining industry, ball mills normally operate with an approximate ball charge of 30% with a rotational speed close to 11 rpm. The mill is fed at one end of the cylinder and the discharge is at the other. Ball mills can be driving by medium voltage motors or wrap around motors, knows as Gearless Mill Drives (GMD). The grinding mills are manufacture using steel plate and some casting parts. Mill design can be trunnion supported or shell supported. In shell supported design, the mill shell supports the weight at the circumference through T-shaped fabricated riding rings and slipper pad bearings (see Figure 1). The load of the mill body, lining, and charge is transferred directly from the sliding ring to the bearing shoes and then to the foundations. The motion of charge, rocks and balls, in grinding mills is performed by metal liners installed in the mills shells. The purpose of installing liners in grinding mills is to protect the mill shell from wear and efficiently transfer the energy to the grinding media. Liners lift the charge producing a cascade motion of the charge inside the mill. The frequency of the cascade motion, and then the charge impact frequency, is a function of the number of lifters and the rotational speed of the mill. The GMD motor design does not have shaft neither bearings. The mill is used as a direct rotor, moved by poles which are divided into a number of segments. The poles are mounted directly on the mill shell through a flange motor carrier ring. One option is to mount this flange by using bolting connection on to the mill shell into the fixed mill bearing side. This is the location with least axial movement of the mill due to thermal expansion. Also the mill deflection due to bending after shut down is smallest at this location. The motor air gap, distance between poles and stator frame, is defined together with the mill manufacturer and need to be kept in the range of 16 mm. The stator frame is designed as a self-supporting ring construction. Usually, the stator is split in four sections to allow easy transport. It is mounted on a motor foundation with integral stator bedplates. During installation and overhaul the stator can be moved on these bedplates.

3. SOIL-STRUCTURE INTERACTION AND RADIATION DAMPING

Many authors have made contributions to the subject of soil-structure interaction, such as Dobry & Gazetas (1988) [4], Gazetas & Makris (1991) [6], Benerjee & Sen (1987)[1] and Wolf (1988) [12]. Different approaches are available to account for dynamic soil-structure interaction but they are usually based on the assumptions that the soil behaviour is governed by the law of linear elasticity or visco-elasticity, and that the soil is perfectly bonded to the footing. In practice, however, the bonding between the soil and the foundation is rarely perfect, and slippage or even separation often occurs in the contact area. Furthermore, the soil region immediately adjacent to the foundation can undergo a large degree of straining, which would cause the soil-structure system to behave in a nonlinear manner. A lot of efforts have been made to model the soil-structure interaction using the 3D Finite Element Method (FEM). However, it is too complex and costly. Several problems of soil-structure interaction are concerned for dynamic analysis in practice. It is a consideration how to account for the nonlinear properties of soil. As an approximate analysis, a procedure is developed using a combination of the analytical solution and the numerical solution, rather than using the general FEM applied to the entire system composed by soil, foundation, mill and motor. The relationship between the foundation vibration and the resistance of the side soil layers is derived using elastic theory by Baranov (1967) [2]. A model for the boundary zone with a non-reflective interface was proposed for nonlinear

properties of soil by Han and Sabin (1995) [9]. The effects of soil-pile-structure interaction on dynamic response were discussed by Han, (2008) [8]. The radiation damping is a very important subject to the dynamic response of foundation. The elastic-wave energy is dissipated from foundation vibration in three dimensions to form the radiation damping. The radiation damping is the dominant energy dissipation mechanism in most dynamically loaded foundation systems. The formula of radiation damping is derived based on elastic half-space theory in which the soil is assumed to be a homogeneous isotropic medium. As a matter of fact, however, the soil is not a perfect linear elastic medium as assumed. A series of dynamic experiments have been done and indicated that the damping is overestimated in the elastic half-space theory, see Han and Novak (1988) [10]. The values of radiation damping are modified and reduced in the program based on the measurements carried out in the field. It is also an important subject for the coupled horizontal and rocking vibration of an embedded foundation. As for the approximate analysis, the plain strain method is considered as an efficient technique for solving the problem of coupled horizontal and rocking vibration of an embedded foundation (Luco 1982) [11]. The relationship between the foundation vibration and the resistance of soil layers was derived using the elastic theory. Then, the solutions of coupled horizontal and rocking vibration of embedded footings were formulated. Six vibration parameters horizontal stiffness Kx, damping Cx, rocking stiffness Kand damping C, and cross coupled stiffness Kx and damping Cx are included in the displacement expression. However, the foundation embedment conditions are very complex practically. An inverse problem is often required in experimental research: all the parameters of the embedded foundation need to be determined, while the dynamic response is given from measurements. It is not convenient to back-calculate for all of the six parameters in the displacement expressions. A simplified mathematical model of the coupled horizontal and rocking vibration of an embedded foundation is proposed by Han, (1989) [7]. Vibration tests of the foundation with different embedment were conducted and compared with different methods. Four parameters are required in the displacement expression based on this method, but six parameters are required in the traditional method. The four parameters Kx, Cx, K and C can be back-calculated from the dynamic response of the foundation.

4. STIFFNESS AND DAMPING OF SOIL

Based on the geotechnical report, the top 3.8 m depth is rigid soil or extremely weak sedimentary rock, then weak to medium strong intrusive rock. In construction, the top soil cut off, and the foundation installed on the intrusive rock. The average shear wave velocity of soil at site is 900 m/sec. The unit weight of soil is 22 kN/m3, Poisson’s ratio is 0.3 and material damping ratio is 0.05.

Stiffness Damping

Frequency

(Hz)

Kx

(kN/m)

Kz

(kN/m)

K

(kN.m/ra)

Cx

(kN/m/s)

Cz

(kN/m/s)

C

(kN.m/rad/s)

0.2 1.12 x 10 8 1.32 x 10 8 3.45 x 10 10 2.22 x 10 6 2.68 x 10 6 5.73 x 10 8

12.8 1.12 x 10 8 1.10 x 10 8 2.88 x 10 10 0.92 x 10 6 1.36 x 10 6 2.30 x 10 8

Table 1. Stiffness and Damping of Soil for Ball Mill Foundation

The concrete mat foundation is 28 m x 21 m for each unit, with thickness of 2.0 m. The impedance of soil is frequency dependent. The frequency of mill operation is 0.2 Hz and the unbalanced force from GMD motor is 12.8 Hz. The impedances of soil are generated from the program for different frequencies, as shown in Table 1, where Kx, Kz and K are stiffness in horizontal, vertical and rocking direction, and Cx,, Cz and C are damping constants in horizontal, vertical and rocking direction.

5. DYNAMIC ANALYSIS

Two methods can be used to carry out the dynamic analysis for ball mill foundations, the free vibration analysis and the forced vibration analysis. 5.1. Free Vibration Analysis For small ball mill, mill diameter less than 3.6 m, with small dynamic loads, the method of free vibration analysis can be used. This is also call modal analysis. The natural frequencies of foundation and piers can be calculated using the free vibration analysis to avoid the resonance. The natural frequency should be less than 0.7 f n or larger than 1.4 f n, where f n is the operation frequency of the machine. 5.2. Forced Vibration Analysis For large ball mill or mill with GMD, the method based on forced vibration analysis is recommended. The vibration amplitudes should be calculated to meet the requirement of allowable vibration limit.

Dynamics analysis is difficult for the flexible mill foundations using standard analytical or numerical methods. Classical empirical methods assume that the foundation acts as a rigid body. However, the structure of mill foundation and piers with large dimension is flexible rather than a rigid body. Numerical methods such as the general finite element method are also difficult to apply, as the direct simulation of radiation damping is not possible. Radiation damping is the dominant energy dissipation mechanism in most dynamically loaded foundation systems. The sub-structure method is used for dynamic analysis of ball mill foundation, that is, the structure and soil are considered separately. The structure part (mat foundation and piers) are modelled by FEM model. The impedance of soil (stiffness and damping) are generated by the computer program, and then input to the FEM model as the base boundary condition. So the reasonable values of radiation damping can be used with the help of program. Different design options are compared to get the better solution in this case. In this case, the diameter of mill is 8.2 m with length of 15.2 m, operating at 12 rpm. The height of mill shaft is 18.4 m above ground. The weight of mill and charge (ores) is 3,000 tons and GMD motor weight is 310 tons. The mat foundation is 28 m x 21 m with thickness of 2.0 m, and the height of piers is 14.5 m above ground. The plan view and section view of mill foundation are shown in Fig. 2. The concrete volume of bearing piers and mat foundation is 2,500 m3 for each unit. The ratio of foundation mass and mill with charge mass is 2.0. In general, the ratio of mass of foundation with machine (including charge) should be 1.5 to 2.5, depending on the soil properties and the foundation structure. To meet large plant capacity is required four to six ball mills and the amount of concrete is huge for mill foundation construction. If the foundation design is over conservative, it may lead to higher costs.

Figure 2. Ball mill foundation plan and section view The operating speed of the mill is relatively slow, usually in the range of 12 to 30 rpm (0.2 to 0.5 Hz), the vibration amplitudes calculated are less than the allowable vibration limit in general. The motor driving the mill operates at different speed, synchronous low speed motor in the range of 90 to 200 rpm, and induction motor in 1800 rpm. The motors are typically well balanced before they leave the factory. Any residual imbalance normally does not give rise to significant excitation forces. It is desirable to tune the supporting piers so that their lowest natural frequency is at least 33% above the operating speed of the synchronous motor. However, with the applications of large gearless mill drive (GMD), the significant dynamic forces are caused (such as taken as 5 % of static loads), and in a higher frequency domain than that for mill operation. So, the challenge rises up on dynamic analysis. Two dynamic loads should be considered, from the mill charge rotation and unbalanced magnetic pull force. The unbalanced forces from mill charge rotation is shown in Fig. 3, and the shaded area represents the charge. Area of segment (shaded), As = r 2cos -1 (m/r) – m (r2 - m2 )1/2 (1) Ratio of area of segment to area of circle, As / A0

As / A0 = 1/ [cos -1 (m/r) – m /r ( 1 – (m/r)2 )1/2] (2) Location of centroid of segment from centre of circle, C C/r = 2/3 [ sin3sin cos

Figure 3. Unbalanced force from mill charge rotation

Unbalanced force F = M e2

The charge unit weight= 5,400 kg/ m3. For ball mills usually m / r = 0.25, and As / Ao = 35 %. The thickness of liner is 0.13 m, so r = 4.1 – 0.13 = 3.97 m. The mass of charge, M = 0.35 x 3.14 x 3.97 2 x 15.16 x 5,400 = 1,418,000 kg. Eccentricity e = C = 0.56 r = 2.22 m. Mill rotational maximum speed 12.0 rpm. f = 12 / 60 = 0.2 Hz, = 2 x 3.14 x 0.2 = 1.256 rad/ sec. Unbalanced force, F = 1,418,000 x 2.22 x 1.256 2 = 4,966 kN. This maximum load value can be reduced up to 50% in the steady state vibration, considering the cascading effect of charge in operation.So the unbalanced forces from the rotation of mill-charge may be F = 2,483 kN. The dynamic loads come from the unbalanced magnetic pull force is shown in Fig. 4. The magnetic pull forces are unbalanced around the machine shaft centre-line. For a mill speed of N rpm, the air-gap variation at a point on the stator will pass through N p cycles per minute, where p is the number of poles in the GMD. The frequency of dynamic force in hertz, f = N p / 60 In this case, N = 12 rpm, p = 64, then frequency f = 12.8 Hz。 Per vendor’s information, the values of harmonic force are taken as 5% of static loads. Base on the data, horizontal dynamic load Fx = -31.3 kN at pier 1 and Fx = 156.2 kN at pier 2, vertical dynamic load Fy = 150.7 kN at pier 1 and Fy = 118.2 kN at pier 2. The concrete mill foundation is modelled using solid element by SAP 2000 program, and the dimension of element is 1m x 1m x 1m as shown in Fig. 5. The mat foundation is 28 m x 21 m, with thickness of 2.0 m. The height of piers is 14.5 m with thickness of 2.5 m for piers supporting motor. The thickness of piers supporting mill is 1.7 m at fixed bearing end and 1.45 m at free bearing end. To examine the dynamic behaviour of soil and concrete foundation, the mill and motor machines are assumed as a sole rigid body and modelled by rigid link element.

Figure 4. Ball mill motor (GMD) foundation

The stiffness of soil generated from the program is distributed at each base node of mat foundation as the value of springs in six directions. The damping constants generated by the program input into the model by link element of damper. The time history analysis is carried out to get the vibration amplitudes. The sine function is used for harmonic loads, and time step is taken as 1/20 of period T. Thus, time step is 0.25 second for mill rotation and 0.0039 second for GMD motor. The amplitude calculated is 68 m = 2.7 mils at top of mill pier under the unbalanced forces of mill rotation. The amplitude calculated is 17.0 m = 0.67 mils at top of motor pier under the unbalanced forces of GMD motor. It is noticed that the amplitude is calculated to be 30.6 m = 1.2 mils if no damping is accounted for. The effect of radiation damping is increased with frequency. So, the effect of radiation damping is small for mill rotation at low frequency but significant for the unbalanced magnetic pull force in high frequency domain. The amplitudes calculated meet the requirement of allowable vibration limit, since the properties of soil (rock) are strong in this case. If soft soil is used for the mill’s foundation, the dynamic response of mill foundation may be challenged.

Figure 5. FEM model of ball mill foundation with solid elements

6. SEISMIC RESPONSE SPECTRUM ANALYSIS

The project is located in a severe earthquake area, and the structural design of bearing piers and mat foundation may be governed by the seismic load. The spectral response acceleration is SS (0.2s) = 1.9 g, and S1 (1.0s) = 0.64 g, that is very severe earthquake.

The method of equivalent static load can be applied for a regular structure, with mass and stiffness distributed uniformly. However, due to the big mass of mill supported on top of foundation, this is an irregular structure. The height of mill top is larger than 20 m, and the spectrum analysis has to be used to determine the earthquake forces. The design response spectrum is formed based the local data per ASCE / 7. The spectral response adjusted for site class effect as defined Sms = Fa * Ss = 1.9 g and Sm1 = Fv * S1 = 0.64 g. Where Fa = 1.0 and Fv =1.0 is the site coefficient. The design spectral response acceleration SDS = 2/3 Sms = 1.26 g and SD1 = 2/3 Sm1 = 0.42 g. T0 = 0.2 SD1/ SDS = 0.07 sec, TS = SD1/ SDS = 0.33 sec, long period transition period TL = 4.0 sec. Based on these parameters, the response spectrum is formed as shown in Fig. 6. Per ASCE, the seismic response coefficient Cs can be determined, based on the important factor I = 1.0, and the response modification coefficient R = 2.5.

Cs = SDS / (R/I) = 0.50 for Maximum Cs = SDS / T (R/I) for T < TL (6) Cs = SDS / T

2(R/I) for T > TL From Equation (6), the curve of seismic response coefficient is formed as shown in Fig. 7.

Figure 6 Design response spectrum

Figure 7. Seismic response coefficient

The response spectrum analysis can be done using FEM model by RISA-3D program. The most difficult part of the entire procedure is calculating the scaling factor. The elastic base shear was calculated using the program. The spectra were normalized using modal participation. In the calculation for the scale factor, 20 vibration modes are calculated making the modal participation to be over 90%. With the local design response spectrum and the spectrum analysis, the earthquake forces and seismic response are calculated. The foundation structure is modelled with plate element, and the dimension of element is 1m x 1m as shown in Fig. 8. The values of springs and damping come from the program as the same as the model of solid elements. The mill and motor machines are modelled by rigid link elements. The frequencies and modes participation are calculated by the program and fill into Table 2 to calculate the scaling factor. The seismic response coefficient Cs (x) = 0.48 and Cs (z) = 0.50 are obtained from the analysis. The horizontal seismic force is F = Cs * W, where W is the weight of foundation structure.

Figure 8 FEM model of ball mill foundation with plate elements Table 2 Normalizing Sa using the model participation for each of the different models

Modal Participation Seismic RSA Contribution Mode Frequency Period

X Spectra Z Spectra Response, Sa X dir'n Z dir'n

(Hz) (s) (%) (%)

1 5.938 0.168 51.459 1.2600 0.64838

2 8.568 0.117 0.526 1.2600 0.00663

3 9.816 0.102 8.187 1.2600 0.10316

4 9.965 0.1 60.734 1.2600 0.76525

5 11.728 0.085 12.438 1.2600 0.15672

6 13.362 0.075 0.104 1.2600 0.00131

7 13.721 0.073 2.892 1.2600 0.03644

8 15.086 0.066 1.2168

9 15.323 0.065 2.289 1.2060 0.02761

10 15.898 0.063 7.468 1.1844 0.08845

11 17.515 0.057 1.479 1.1196 0.01656

12 19.798 0.051 0.293 1.0548 0.00309

13 21.482 0.047 1.011 1.0116 0.01023

14 21.708 0.046 11.59 1.0008 0.11599

15 22.379 0.045 8.462 0.9900 0.08377

16 23.006 0.043 2.537 0.9684 0.02457

17 23.275 0.043 4.958 0.9684 0.04801

18 24.333 0.041 2.455 0.9468 0.02324

19 24.834 0.04 1.55 0.9360 0.01451

20 25.534 0.039 1.114 0.9252 0.01031

Totals: 91.863 89.683 1.09151 1.09272

7. CONCLUSIONS

The soil-structure interaction is investigated based on the practical case of dynamic analysis for ball mill foundation. The dynamic response depends on both parts of soil and concrete foundation structure (mat and piers), and the sub-structure method is efficient to solve the problem. The following results are concluded from this study. (1) The stiffness and damping of soil (rock) can be generated by the computer program, and

the values of radiation damping have been validated by many dynamic tests. The analysis of mat foundation and supporting piers can be done using FEM models.

(2) The amplitudes calculated at the frequency of unbalanced magnetic pull force (12.8 Hz)

are different significantly with the damping and without the damping. The radiation damping increases with the frequency.

(3) The ball mill foundation is an irregular structure with a big mass (mill and charge) on the

top of piers. The response spectrum analysis has to be used to determine the earthquake forces and seismic response.

8. REFERENCES

[1] Banerjee, P.K. and Sen, R. Dynamic behavior of axially and laterally loaded piles and pile groups. Chapter 3 in Dynamic Behavior of Foundations and Buried Structures, Elsevier App. Sc., London, 95-133, 1987.

[2] Baranov, V.A., On the calculation of excited vibrations of an embedded foundation. Voprosy Dynamiki Prochnocti, No.14, 195-209, (in Russian), 1967.

[3] Barkan, D.D. Dynamics of bases and foundations. McGraw-Hill Book Co. New York, 1962.

[4] Dobry, R. and Gazetas, G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechnique, Vol.38, No.4, 557- 574, 1988.

[5] DynaN 2.0 for Windows, Dynamic analysis of shallow and deep foundations, Ensoft. 2003. www.ensoftinc.com.

[6] Gazetas, G. and Makris, N. Dynamic pile-soil-pile interaction. I: Analysis of Axial Vibration. J. Earthq. Eng. and Struct. Dyn. Vol. 20, No.2, 1991.

[7] Han, Y.C. Coupled vibration of embedded foundation, Journal of Geotechnical Engineering, ASCE, 115(9), 1227-1238, 1989.

[8] Han, Y.C., Study of vibrating foundation considering soil-pile-structure interaction for practical applications. J. of Earthquake Engineering and Engineering Vibration, Vol.7, No.3, 321-327, 2008.

[9] Han, Y.C. and Sabin, G. Impedances for radially inhomogeneous soil media with a non- reflective boundary. J. of Engineering Mechanics, ASCE, 121(9), 939-947, 1995.

[10] Han, Y.C. and Novak, M. Dynamic behavior of single piles under strong harmonic excitation. Canadian Geotechnical Journal, 25(3), 523-534, 1988.

[11] Luco, J.E. Linear soil – structure interaction: A Review. Applied Mech. Div., Vol.53, ASME, 41-57, 1982.

[12] Wolf, J.P. Soil – structure interaction analysis in time domain. Englewood Cliffs, NJ: Printice - Hall, 446p, 1988.

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