Three-Dimensional Finite Element Analysis of Wall Pressure on Large ...
Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao*, Luan Maotian, Yang Qing, Nian Tingkai State Key Laboratory of Coastal and Offshore Engineering and Institute of Geotechnical Engineering, School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian, 116085, China [email protected], [email protected], [email protected], [email protected] doi: 10.4156/jcit.vol5.issue7.15
Abstract With the increasing volume demand of silos, silo diameters are bigger and bigger. However, present wall pressure computation methods are mostly based on small diameter silos. There is little system research on large diameter silos. To solve this problem, systematical research by three-dimensional finite element method on large diameter silos was carried out in this paper. Static wall pressure and wall pressure at the end of filling were analyzed. And the influence of mechanical parameters of bulk materials to wall pressure was studied. Two lamination ways to simulate filling process was studied. The result shows that scale of silo take important effect to wall pressure. Wall pressures at the end of filling of large diameter silos are much larger than static wall pressures near silos bottom. Wall pressure of filling should be considered in design of large diameter silos. Study in this paper has heavy theoretical significance and provide an important basis for the large diameter silo design.
Keywords: Large Diameter Silos, Three-Dimensional Finite Element Method, Wall Pressure, Influence of Bulk Materials Parameters.
1. Introduction Silos are structures that have always concerned mankind due to their capacity to store bulk materials, such as grain, coal, cement and so on. With the increasing production demand, the demand of silo volume increases continuously. Diameters of silos are bigger and bigger. The maximum diameter of present silos exceeds 100 meters [1]. With the passing of time, knowledge about silos has been further developed. However, this knowledge is mostly based on small silos whose diameter is smaller than 20 meters. There is little system research on large diameter silos whose diameter is larger than 20 meters. Scale effect is considered to particularly important to silos [2]. Therefore, it is meaningful to study on larger diameter silos. Wall pressure is a key parameter to silos’ design. It has an important effect on the safety and efficiency of silos. Research methods about wall pressure include three kinds: theoretical analysis, experiment method and numerical method. Many scholars studied on silo pressure with theoretical analysis, such as Janssen (1895), Airy (1897), Reimbert M and Reimbert A [3], Walker, Walkers, Coulomb, Rankine, Jenike, Liu, Yuan etc [1, 4]. Most of this research applied to small diameter silo. Only little system research directed to large diameter silos. Yuan (2004) studied static wall pressure on large diameter silos used limit equilibrium method in her PhD thesis. Part of her research was adopted in China Code for design of reinforced concrete silos (GB50077-2003) [1]. As an intuitional method, experiments were often carried out to study wall pressure on model silos or real silos. In early times, many scholars did experiments on small silos (Tachtamishev’s (the former Soviet Union, 1938-1940), Japanese academics (1962), Reimbert (France, 1941-1943), Kim (USSR, 1948-1953), Pieper and Wenzel (Germany, 1963), Deutsch and Clyde (Australia, 1967-1968), Sugden (South Africa) etc.) [1]. Li [8], Yuan [1], Chen [4] carried out experiments on large diameter silos of national grain reservoir in Henan province, China. They obtained several groups of valuable data about static wall pressures, one group of which were used as example in this paper. Numerical method includes two kinds: finite element method and discrete element method.
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Journal of Convergence Information Technology Volume 5, Number 7, September 2010
Much research has been done on wall pressure of silos with finite element method (Mahmoud (1975), Jofriet (1977), Bishara, Liu [5, 6], Jenike [7], Ooi [9, 10], Zhang [11], Negi [13], Zeng [14], Wieckowski [15], Rombach [16], Karlsson [17], Briassoulis [18], Nilaward [19], Ayuga F and Guaita M etc. [20], Vidal and Couto etc. [26]). Most of this research was about small diameter silos and used two-dimensional finite element methods. Many scholars used discrete element method to study wall pressure of silos, such as Langston [21], Zhou [22], Gavrilov and Vinogradov [23], Yu and Xing [24], Hirshfeld and Rapaport [25]. Limited to computation speed, large silos are difficult to simulate by discrete element methods. Sanad and Ooi etc. compared between finite element method and discrete element method. The result shows that discrete element models give good qualitative predictions of flow patterns, and the finite-element models give good quantitative predictions of pressure regimes [1]. Therefore, the writer used finite element method to analyze wall pressure of large diameter silos in this paper. In previous studies, numerical analysis of large diameter silos is very little. As far as the writers are concerned, this is the first research that uses a 3D finite-element model to analyze wall pressures of large diameter silos.
2. Sizes Objectives The following aims are established in this work: • Develop a 3D model of large diameter flat bottom silos with rigid walls; • Compared two lamination ways to simulate central vertical filling process; • Analyze static wall pressures and wall pressures at the end of filling on large diameter silos; • Study influence of bulk material parameters on wall pressures;
3. Static Wall Pressure Predictions To study wall pressures of large diameter silos, a real silo experiment made by Chen was taken as example in this paper. Chen carried out several groups of experiments on large diameter flat bottom silos of national grain reservoir in Henan province, China [4]. Several groups of static wall pressures were obtained. The experiment silo is reinforcement concrete silo with an inside diameter 28 meters. The height of bulk solid in the silo is 13.5 meters. The angle between the free surface of bulk solid and level is the angle of repose of the bulk solid. Bulk material in the silo is wheat. Mechanical parameters of the bulk material are shown in table 1. Table 1. Mechanical Properties for bulk material Parameter Values Density, ρ (kg m-3) Young’s modulus, E (kPa) Poisson’s ratio, ν Grain-wall friction coefficient, μ Cohesion, c (kPa) Internal friction angle, φ (degrees) Angle of repose, φr(degrees)
788 20,000 0.3 0.4 0 25 23.5
A 3D finite-element model shown in figure 1 was developed to study pressures on the large diameter silo wall used FEM software ADINA. The element used to represent the stored material was 3D solid element, a cubic element defined by eight nodes with three degrees of freedom and a nodal displacement at x, y, and z. This structural element is compatible with surface-to-surface contact elements that admit different plasticity models and laws of behavior of the bulk material. This study simulates silos filled with wheat, a granular material that can be reasonably considered isotropic, particularly when it is randomly packed. Because wheat is a granular material, a law of behavior of the stored material must be used that reproduces the behavior of wheat grains with low cohesion. In this study, the bulk solid was simulated by Mohr-Coulomb model which was based on a non-associated flow rule, a perfectly-plastic Mohr-Coulomb yield behavior and tension cut-off. The Mohr-Coulomb yield equation can be written as
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
f = α I1 +
(1)
J2 − k
Where α and k are stress dependent: α=
θ=
1 3
cos
−1
2 sin φ 3(1 − sin φ ) sin θ +
3 ( 3 + sin φ ) cos θ
, k=
6c sin φ 3(1 − sin φ ) sin θ +
3 ( 3 + sin φ ) cos θ
,
⎛ 3 3 J3 ⎞ , I1 is the first stress invariant at time t, J2 is the second deviatoric stress ⎜ 3 2 ⎟ ⎝ 2 J2 ⎠
invariant at time t, J3 is the third deviatoric stress invariant at time t. φ is the friction angle (a material constant), c is the cohesion (a material constant). Although more complex models exist, the Mohr-Coulomb model is sufficiently accurate and easy to use with numerical models.
Figure 1. FEM model A surface-to-surface contact model was used between the bulk solid and silo wall. Two surfaces that contactor surface and target surface made up a contact pair. The contactor surface was the out surface of bulk solid and the target surface was silo wall. Because the Young’s modulus of reinforcement concrete is about 20 GPa, which is much larger than the Young’s modulus of stored material, the silo wall was considered to be rigid for good convergence in this paper.
3.1 Static Wall Pressure Distribution
Vertical coordinate h (m)
14
FEM method Experiment Rankine method
12 10 8 6 4 2 0 0
10
20
30
40
50
Wall pressure p (kPa)
Figure 2. Static wall pressure distribution along wall height Variable h in the figure 2 is vertical coordinate. The zero point is at the bottom of the silo. After the calculation was performed, static wall pressures distribution along the wall height was obtained, as shown in Figure 2. Rankine method equation which is often used to predict the wall pressure of bunkers can be written as [2, 4]:
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P = Kγ s
(2)
Where P is the pressure intensity; K is the wall pressure coefficient and K = tan (45 − φ 2) ; φ is internal friction angle of material, γ is the unit weight of material; s is the depth whose origin is at the free surface of material in silos when the free surface is level or at the geometry center of gravity of the top cone when the free surface is conical surface. Many scholars support to compute wall pressure of silos with Rankine method. China code use Rankine method to compute wall pressure of bunkers. Figure 2 shows three kinds of wall pressure distribution along wall height. Values of static wall pressures by FEM method are similar to the experiment results. Compared to the wall pressure distribution of Rankine method, FEM result is closer to the experiment result at the bottom of silo wall. 2
3.2 Influence of silo scale
Vertical coordinate system h (m)
Can compute wall pressure of large diameter silos with methods based on small diameter silos? To study the influence of silo scale to wall pressure, following analysis was made. Establish another model whose scale was only a quarter of the original silo. Other parameters were the same. Then wall pressures of the small silo were enlarged 4 times and compared with wall pressures of the large silo. These two groups of wall pressures were shown in Figure 3. 14 12
D=28m D=7m
10 8 6 4 2 0 -2 -10
0
10
20
30
40
50
Wall pressure P (kPa)
Figure 3. Wall pressure of silos with different scales Figure 3 shows that wall pressures are different between large diameter silo and small diameter silo. Specifically near the bottom of silos, the difference is obvious. Therefore, silo scale takes heavy effect to silo wall pressure. It is not appropriate to predict wall pressure of large diameter silos by small silo computation methods. Research on large diameter silos is very important.
3.3 Influence of Material Parameters To study the influence of material parameters to wall pressure, following analysis was made. Change one parameter shown in table 1 and others unchanged. Wall pressure distributions with a mutative parameter were obtained. Figure 4 shows three groups of wall pressure distribution with different densities. Wall pressure shows an increase with density. Figure 5 shows that wall pressure increases with Young’s modulus E near the bottom of silo. At the upside and the middle of the silo wall, the value change of Young’s modulus has little influence on static wall pressure. Figure6 shows that wall pressure increases with Poisson’s ratio v and the increase range also increases with Poisson’s ratio. Figure 7 shows wall pressure distributions with different grain-wall frictional coefficients. Static wall pressure shows an increase with grain-wall frictional coefficient, especially near the bottom. Figure 8 shows that the value change of bulk solid cohesion has little influence on static wall pressure.
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
14
ρ = 600kN/m3
12
ρ = 700kN/m3
14
ρ = 788kN/m3
10 8 6 4 2 0 -2 -10 -5
E=5M Pa E=20M Pa E=68.9M Pa
12
Vertical coordinate h (m)
Vertical coordinate h (m)
Figure 9 shows that the value change of dilatation angle δ of bulk solid has no effect on static wall pressure.
10 8 6 4 2 0 -2
0
5
10 15 20 25 30 35 40 45
0
10
Wall pressure P (kPa)
20
30
40
50
W all pressure P (kPa)
Figure 4. Influence of density
Figure 5. Influence of Young’s modulus E 14
ν=0.4 ν=0.3 ν=0.2
12 10
Vertical coordinate h (m)
Vertical coordinate h (m)
14
8 6 4 2 0
12
μ=0 μ = 0.2 μ = 0.4
10 8 6 4 2 0
-2 0
10
20
30
40
50
0
60
Figure 6. Influence of Poisson’s ratio, v
20
30
40
10
Vertical coordinate h (m)
c=0 c=2kPa c=10kPa
12
8 6 4 2 0 -2 0
10
20
30
50
Figure 7. Influence of grain-wall frictional coefficient, μ δ =0
14
14
Vertical coordinate h (m)
10
Wall pressure P (kPa)
Wall pressure P (kPa)
δ =5ο
12
δ =15ο
10
δ =25ο
8 6 4 2 0 -2
40
0
Wall pressure p (kPa)
10
20
30
40
Wall pressure p (kPa)
Figure 8. Influence of cohesion, c
Figure 9. Influence of dilatation angle, δ
The angle φr between the free surface of bulk solid and horizontal plane is generally taken as the angle of repose of the bulk material. The angle of repose and the grain-wall friction coefficient μ are concerned to φ. Therefore, to study the influence of φ on wall pressure, φr =0 and μ̃=0 were set. That is main the free surface of bulk solid is horizontal plane and the grain-wall friction is neglected when the influence of internal friction angle was analyzed. Figure10 shows that wall pressure increases while φ reduce when φ < 25 degrees but value change of the internal friction angle has no influence on static lateral wall pressure when φ > 25 degrees.
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From figure 4 to figure 10, it is shown that Yong’s modulus, Poisson’s ratio, grain-wall frictional coefficient and internal friction angle have heavy effect on static lateral wall pressure while dilatation angle and cohesion have little influence on static wall pressure.
φ =15ο φ =20ο φ =25ο φ =35ο φ =40ο
Vertical coordinate h (m)
14 12 10 8 6 4 2 0 -2 0
10
20
30
40
50
60
70
Wall pressure p (kPa)
Figure 10. Influence of internal friction angle φ
4. Wall Pressure at the End of Filling For the FEM analyses, the true filling process is not easy to model. In the calculation, a central vertical filling process was specified. Progressive filling with incremental layers was suggested. There are two lamination ways as shown in figure 11. In the first way shown in figure 11(a), the first layer is a cone with a diameter 28 meters and a height 6 meters, then every layer’s thickness is 1 m and the top layer’s thickness is 1.5 m. In the other way shown in figure 11(b), layers are a cylinder with a diameter 28 meters and a height 1 meter and the top layer is a combination of a cylinder with a height 1 meter and a cone with a height 6 meters. To simulate the time effect of filling process, element birth and death function was used in the calculation. Element birth and death function in ADINA is able to model processes during which material is added to and/or removed. If the element birth option is used, the element is added to the total system of finite elements at the time of birth and all times thereafter. If the element death option is used, the element is taken out of the total system of finite elements at times larger than the time of death. If the element birth-then-death option is used, the element is added to the total system of finite elements at the time of birth and remains active until the time of death. The time of death must be greater than the time of birth. The element is taken out of the total system of finite elements at all times larger than the time of death. Layers of stored material were simulated by different elements whose material and death time were the same but birth time was different. The birth time of the first layer is zero, and then the birth time of layers enlarges uniform.
(a)
(b)
Figure 11. Filling process
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
Vertical coordinate h (m)
14 12
the first way the second way
10 8
static prssure
6 4 2 0 0
10
20
30
40
50
60
70
Wall pressure P (kPa)
Figure 12. Wall pressure at the end of filling Wall pressure distributions at the end of filling in this paper are shown in figure 12. There is big different between wall pressure distributions of two lamination ways at the bottom of silo wall. From the physical sense, the second way shown in figure 11(b) better indicate the progressive filling process. Therefore, this paper applied the way shown in figure 11(b) to study wall pressure of filling finally. In figure 12, it is shown that wall pressure much bigger than static wall pressure near the silo bottom and basically equal to static wall pressures at the middle and upside of the silo wall. Near the silo bottom, wall pressure at the end of filling enlarged suddenly. This phenomenon doesn’t agree with the general view of academia. Academia generally accounts that wall pressure at the end of filling is not bigger than static wall pressure (Japanese academics, 1962 [1]). Considering previous study on wall pressure of filling all based on small diameter silos, following calculation was made. The value of silo diameter was changed and other parameters unchanged.
Vertical coordinate h (m)
14
D = 5m D = 14m D = 28m
12 10 8 6 4 2 0 -2 0
10
20
30
40
50
60
70
Wall pressure P (kPa)
Figure 13. Wall pressure at the end of filling with different diameters Figure 13 shows the Wall pressure distributions at the end of filling with different silo diameters. When silo diameter is small, such as 5 meters, wall pressure at the end of filling has no sudden enlargement. With the increasing of silo diameter, the sudden enlargement of wall pressure near the silo bottom is more and more obvious. It can be conclude that when silo diameter is small, wall pressure at the end of filling is similar to static wall pressure. There is no need to consider wall pressure of filling in silo design. When silo diameter is large, wall pressure at the end of filling is much larger than static wall pressure near silo bottom. Therefore wall pressure of filling should be considered in the design of large diameter silo.
5. Conclusion
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In this paper, a large diameter silo was established by three-dimensional finite element. Static wall pressures and wall pressure at the end of filling were studied. Two lamination ways to simulate filling process was studied. And the influence of mechanical parameters of bulk solid to wall pressure was studied. The following conclusions can be drawn from the results of the generated finite-element model: • Silo scale has heavy influence on wall pressure. It is not feasible to compute wall pressure of large diameter silos with methods based on small diameter silos. It is important to study on large diameter silos. • Yong’s modulus, Poisson’s ratio, grain-wall frictional coefficient and internal friction angle have heavy effect on static wall pressure while dilatation angle and cohesion have little influence to static wall pressure. The value change of Yong’s modulus has no effect to wall pressure at the end of filling. • Wall pressures of large diameter silos at the end of filling were much larger than static wall pressures near silos bottom. Wall pressure of filling should be considered in design of large diameter silos. There is little system research on large diameter silo currently. Study in this paper has heavy theoretical significance and provide an important basis for the large diameter silo design.
6. References [1] Yuan Fang, Analysis of bulk-solid pressures on curve walls and its engineering application, PhD thesis, Dalian university of technology, China, 2004. [2] Sanad A M, Ooi J Y, Holst F G, Rotter J M, “Computations of Granular Flow and Pressures in a Flat-bottomed Silo”, Journal of Engineering Mechanics, vol. 127, no. 10, pp. 1033-1043, 2001. [3] Reimbert M, Reimbert A, Silos: theory and practice. Trans Tech Publications, Clausthal, 1976. [4] Chen Changbing, Research on lateral bulk-solid pressures on silo’s wall, PhD thesis, Hefei university of technology, China, 2006. [5] Liu Dinghua, Hao Jiping, “Research of lateral pressure upon wall of reinforced concrete silo”, Journal of Building Structures, vol. 16, no. (5), pp. 57-63, 1995. (in China) [6] Liu Dinghua, Wei Yihua, “Calculation and Testing of Lateral Pressure in a Reinforced Concrete Silo”, Architectural Science, vol. 14, no. 4, pp. 14-18, 1998. (in China) [7] Jenike A W, “A theory of flow of particulate solids on converging and diverging channels based on a conical yield function”, Powder technology, vol. 50, no. 3, pp. 229-236, 1987. [8] Li Xingzhao, Testing and Analysis on wall pressure of larger diameter grain squat silos, phD thesis, Zhengzhou Institute of Technology, China, 2002. [9] Ooi J Y, Bulk solids behavior and silo wall pressure, PhD thesis, University of Sydney, 1990. [10] Ooi J Y, Pham L, Rotter J M, “Systematic and random features of measured pressures on fullscale silo walls”, Engineering Structures, vol. 12, no. 2, pp. 74-87, 1990. [11] Zhu Ruizhen, Zhang Ou, “Elastic-plastic finite element analysis of static wall pressure on silos”, Northeast Water Resources and Hydropower, no. 11, pp. 11-15, 1995. (in China) [12] Hao Jiping, Xiao Linshan, “Dynamic pressure on circular silo wall”, Engineering Mechanics, vol. 9, no. 3, pp. 95-100, 1992. (in China) [13] Negi S C, Lu Z, Jofriet J C, “A numerical model for flow of granular materials in silos, Part 2: Model validation”, Journal of Agricultural Engineering Research, no. 68, pp. 231-236, 1997. [14] Zeng Ding, Hao Baohong, Huang Wenbin, “Finite element analysis of static wall pressure on silos”, China Powder Technology, vol. 6, no. 5, pp. 6-11, 2000. [15] Wieckowski Z, “Finite Deformation Analysis of Motion of Bulk Materials in silo”, in Proceeding of Application of Finite Element Method Research Report in Lulea University of Technology on Division of Structural Mechanics, 1994. [16] Rombach G, Eibl J, “Granular Flow of Materials in silos: Numerical results”, Bulk Solid Handing, no. 1, pp. 65-70, 1995. [17] Karlsson T, Klisinski M, Runesson K, “Finite Element simulation of Granular Material Flow in Plane Silos with Complicated Geometry”, Powder Technology, pp. 29-39, 1998. [18] Briassoulis D, “Finite element analysis of a cylindrical silo shell under unsymmetrical pressure distributions”, Computer and Structures, pp. 271-281, 2000.
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
[19] Nilaward T, Analysis of Bulk-solid Pressure in Silos by Explicit Finite Element Method, PhD thesis, Purdue University, the USA. 2001. [20] Ayuga F, Guaita M, Aguado P J, Couto A, “Discharge and the Eccentricity of the Hopper Influence on the Silo Wall Pressures”, Journal of Engineering Mechanics, vol. 127, no. 10, pp. 1067-1075, 2001. [21] Langston P A, Tuzun U, “Continuous Potential Discrete Particle Simulations of Stress and Velocity Fields in Hopper: Transition from Fluid to Granular Flow”, Chemical Engineering, vol. 49, no. 8, pp. 1259-1275, 1994. [22] Zhou Deyi etc, “Discrete Element Simulation for Arch Flowing of Agricultural Particle Material in Outlet”, Journal of Agricultural Engineering, vol. 12, no. 2, pp. 186-190, 1996. (in China) [23] Gavrilov D, Vinogradov O G, “Micro Instabilities in a System of Particles in Silos during Filling Process”, Computational Mechanics, no. 24, pp. 166-174, 1999. [24] Yu Liangqun, Xing Jibo, “Discrete Element Method Simulation of Forces and Flow Fields during Filling and Discharging Materials in Silo”, Journal of Agricultural Engineering, vol. 16, no. 4, pp. 15-19, 2000. (in China) [25] Hirshfeld D, Rapaporta D C, “Granular flow from a silo: Discrete-particle simulations in three dimensions”, Eur. Phys. J. E, no. 4, pp.193-199, 2001. [26] Vidal P, Couto A, Ayuga F, Guaita M, “Influence of Hopper Eccentricity on Discharge of Cylindrical Mass Flow Silos with Rigid Walls”, Journal of Engineering Mechanics, vol. 132, no. 9, pp. 1026-1033, 2006.
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao*, Luan Maotian, Yang Qing, Nian Tingkai State Key Laboratory of Coastal and Offshore Engineering and Institute of Geotechnical Engineering, School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian, 116085, China [email protected], [email protected], [email protected], [email protected] doi: 10.4156/jcit.vol5.issue7.15
Abstract With the increasing volume demand of silos, silo diameters are bigger and bigger. However, present wall pressure computation methods are mostly based on small diameter silos. There is little system research on large diameter silos. To solve this problem, systematical research by three-dimensional finite element method on large diameter silos was carried out in this paper. Static wall pressure and wall pressure at the end of filling were analyzed. And the influence of mechanical parameters of bulk materials to wall pressure was studied. Two lamination ways to simulate filling process was studied. The result shows that scale of silo take important effect to wall pressure. Wall pressures at the end of filling of large diameter silos are much larger than static wall pressures near silos bottom. Wall pressure of filling should be considered in design of large diameter silos. Study in this paper has heavy theoretical significance and provide an important basis for the large diameter silo design.
Keywords: Large Diameter Silos, Three-Dimensional Finite Element Method, Wall Pressure, Influence of Bulk Materials Parameters.
1. Introduction Silos are structures that have always concerned mankind due to their capacity to store bulk materials, such as grain, coal, cement and so on. With the increasing production demand, the demand of silo volume increases continuously. Diameters of silos are bigger and bigger. The maximum diameter of present silos exceeds 100 meters [1]. With the passing of time, knowledge about silos has been further developed. However, this knowledge is mostly based on small silos whose diameter is smaller than 20 meters. There is little system research on large diameter silos whose diameter is larger than 20 meters. Scale effect is considered to particularly important to silos [2]. Therefore, it is meaningful to study on larger diameter silos. Wall pressure is a key parameter to silos’ design. It has an important effect on the safety and efficiency of silos. Research methods about wall pressure include three kinds: theoretical analysis, experiment method and numerical method. Many scholars studied on silo pressure with theoretical analysis, such as Janssen (1895), Airy (1897), Reimbert M and Reimbert A [3], Walker, Walkers, Coulomb, Rankine, Jenike, Liu, Yuan etc [1, 4]. Most of this research applied to small diameter silo. Only little system research directed to large diameter silos. Yuan (2004) studied static wall pressure on large diameter silos used limit equilibrium method in her PhD thesis. Part of her research was adopted in China Code for design of reinforced concrete silos (GB50077-2003) [1]. As an intuitional method, experiments were often carried out to study wall pressure on model silos or real silos. In early times, many scholars did experiments on small silos (Tachtamishev’s (the former Soviet Union, 1938-1940), Japanese academics (1962), Reimbert (France, 1941-1943), Kim (USSR, 1948-1953), Pieper and Wenzel (Germany, 1963), Deutsch and Clyde (Australia, 1967-1968), Sugden (South Africa) etc.) [1]. Li [8], Yuan [1], Chen [4] carried out experiments on large diameter silos of national grain reservoir in Henan province, China. They obtained several groups of valuable data about static wall pressures, one group of which were used as example in this paper. Numerical method includes two kinds: finite element method and discrete element method.
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Journal of Convergence Information Technology Volume 5, Number 7, September 2010
Much research has been done on wall pressure of silos with finite element method (Mahmoud (1975), Jofriet (1977), Bishara, Liu [5, 6], Jenike [7], Ooi [9, 10], Zhang [11], Negi [13], Zeng [14], Wieckowski [15], Rombach [16], Karlsson [17], Briassoulis [18], Nilaward [19], Ayuga F and Guaita M etc. [20], Vidal and Couto etc. [26]). Most of this research was about small diameter silos and used two-dimensional finite element methods. Many scholars used discrete element method to study wall pressure of silos, such as Langston [21], Zhou [22], Gavrilov and Vinogradov [23], Yu and Xing [24], Hirshfeld and Rapaport [25]. Limited to computation speed, large silos are difficult to simulate by discrete element methods. Sanad and Ooi etc. compared between finite element method and discrete element method. The result shows that discrete element models give good qualitative predictions of flow patterns, and the finite-element models give good quantitative predictions of pressure regimes [1]. Therefore, the writer used finite element method to analyze wall pressure of large diameter silos in this paper. In previous studies, numerical analysis of large diameter silos is very little. As far as the writers are concerned, this is the first research that uses a 3D finite-element model to analyze wall pressures of large diameter silos.
2. Sizes Objectives The following aims are established in this work: • Develop a 3D model of large diameter flat bottom silos with rigid walls; • Compared two lamination ways to simulate central vertical filling process; • Analyze static wall pressures and wall pressures at the end of filling on large diameter silos; • Study influence of bulk material parameters on wall pressures;
3. Static Wall Pressure Predictions To study wall pressures of large diameter silos, a real silo experiment made by Chen was taken as example in this paper. Chen carried out several groups of experiments on large diameter flat bottom silos of national grain reservoir in Henan province, China [4]. Several groups of static wall pressures were obtained. The experiment silo is reinforcement concrete silo with an inside diameter 28 meters. The height of bulk solid in the silo is 13.5 meters. The angle between the free surface of bulk solid and level is the angle of repose of the bulk solid. Bulk material in the silo is wheat. Mechanical parameters of the bulk material are shown in table 1. Table 1. Mechanical Properties for bulk material Parameter Values Density, ρ (kg m-3) Young’s modulus, E (kPa) Poisson’s ratio, ν Grain-wall friction coefficient, μ Cohesion, c (kPa) Internal friction angle, φ (degrees) Angle of repose, φr(degrees)
788 20,000 0.3 0.4 0 25 23.5
A 3D finite-element model shown in figure 1 was developed to study pressures on the large diameter silo wall used FEM software ADINA. The element used to represent the stored material was 3D solid element, a cubic element defined by eight nodes with three degrees of freedom and a nodal displacement at x, y, and z. This structural element is compatible with surface-to-surface contact elements that admit different plasticity models and laws of behavior of the bulk material. This study simulates silos filled with wheat, a granular material that can be reasonably considered isotropic, particularly when it is randomly packed. Because wheat is a granular material, a law of behavior of the stored material must be used that reproduces the behavior of wheat grains with low cohesion. In this study, the bulk solid was simulated by Mohr-Coulomb model which was based on a non-associated flow rule, a perfectly-plastic Mohr-Coulomb yield behavior and tension cut-off. The Mohr-Coulomb yield equation can be written as
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
f = α I1 +
(1)
J2 − k
Where α and k are stress dependent: α=
θ=
1 3
cos
−1
2 sin φ 3(1 − sin φ ) sin θ +
3 ( 3 + sin φ ) cos θ
, k=
6c sin φ 3(1 − sin φ ) sin θ +
3 ( 3 + sin φ ) cos θ
,
⎛ 3 3 J3 ⎞ , I1 is the first stress invariant at time t, J2 is the second deviatoric stress ⎜ 3 2 ⎟ ⎝ 2 J2 ⎠
invariant at time t, J3 is the third deviatoric stress invariant at time t. φ is the friction angle (a material constant), c is the cohesion (a material constant). Although more complex models exist, the Mohr-Coulomb model is sufficiently accurate and easy to use with numerical models.
Figure 1. FEM model A surface-to-surface contact model was used between the bulk solid and silo wall. Two surfaces that contactor surface and target surface made up a contact pair. The contactor surface was the out surface of bulk solid and the target surface was silo wall. Because the Young’s modulus of reinforcement concrete is about 20 GPa, which is much larger than the Young’s modulus of stored material, the silo wall was considered to be rigid for good convergence in this paper.
3.1 Static Wall Pressure Distribution
Vertical coordinate h (m)
14
FEM method Experiment Rankine method
12 10 8 6 4 2 0 0
10
20
30
40
50
Wall pressure p (kPa)
Figure 2. Static wall pressure distribution along wall height Variable h in the figure 2 is vertical coordinate. The zero point is at the bottom of the silo. After the calculation was performed, static wall pressures distribution along the wall height was obtained, as shown in Figure 2. Rankine method equation which is often used to predict the wall pressure of bunkers can be written as [2, 4]:
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P = Kγ s
(2)
Where P is the pressure intensity; K is the wall pressure coefficient and K = tan (45 − φ 2) ; φ is internal friction angle of material, γ is the unit weight of material; s is the depth whose origin is at the free surface of material in silos when the free surface is level or at the geometry center of gravity of the top cone when the free surface is conical surface. Many scholars support to compute wall pressure of silos with Rankine method. China code use Rankine method to compute wall pressure of bunkers. Figure 2 shows three kinds of wall pressure distribution along wall height. Values of static wall pressures by FEM method are similar to the experiment results. Compared to the wall pressure distribution of Rankine method, FEM result is closer to the experiment result at the bottom of silo wall. 2
3.2 Influence of silo scale
Vertical coordinate system h (m)
Can compute wall pressure of large diameter silos with methods based on small diameter silos? To study the influence of silo scale to wall pressure, following analysis was made. Establish another model whose scale was only a quarter of the original silo. Other parameters were the same. Then wall pressures of the small silo were enlarged 4 times and compared with wall pressures of the large silo. These two groups of wall pressures were shown in Figure 3. 14 12
D=28m D=7m
10 8 6 4 2 0 -2 -10
0
10
20
30
40
50
Wall pressure P (kPa)
Figure 3. Wall pressure of silos with different scales Figure 3 shows that wall pressures are different between large diameter silo and small diameter silo. Specifically near the bottom of silos, the difference is obvious. Therefore, silo scale takes heavy effect to silo wall pressure. It is not appropriate to predict wall pressure of large diameter silos by small silo computation methods. Research on large diameter silos is very important.
3.3 Influence of Material Parameters To study the influence of material parameters to wall pressure, following analysis was made. Change one parameter shown in table 1 and others unchanged. Wall pressure distributions with a mutative parameter were obtained. Figure 4 shows three groups of wall pressure distribution with different densities. Wall pressure shows an increase with density. Figure 5 shows that wall pressure increases with Young’s modulus E near the bottom of silo. At the upside and the middle of the silo wall, the value change of Young’s modulus has little influence on static wall pressure. Figure6 shows that wall pressure increases with Poisson’s ratio v and the increase range also increases with Poisson’s ratio. Figure 7 shows wall pressure distributions with different grain-wall frictional coefficients. Static wall pressure shows an increase with grain-wall frictional coefficient, especially near the bottom. Figure 8 shows that the value change of bulk solid cohesion has little influence on static wall pressure.
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
14
ρ = 600kN/m3
12
ρ = 700kN/m3
14
ρ = 788kN/m3
10 8 6 4 2 0 -2 -10 -5
E=5M Pa E=20M Pa E=68.9M Pa
12
Vertical coordinate h (m)
Vertical coordinate h (m)
Figure 9 shows that the value change of dilatation angle δ of bulk solid has no effect on static wall pressure.
10 8 6 4 2 0 -2
0
5
10 15 20 25 30 35 40 45
0
10
Wall pressure P (kPa)
20
30
40
50
W all pressure P (kPa)
Figure 4. Influence of density
Figure 5. Influence of Young’s modulus E 14
ν=0.4 ν=0.3 ν=0.2
12 10
Vertical coordinate h (m)
Vertical coordinate h (m)
14
8 6 4 2 0
12
μ=0 μ = 0.2 μ = 0.4
10 8 6 4 2 0
-2 0
10
20
30
40
50
0
60
Figure 6. Influence of Poisson’s ratio, v
20
30
40
10
Vertical coordinate h (m)
c=0 c=2kPa c=10kPa
12
8 6 4 2 0 -2 0
10
20
30
50
Figure 7. Influence of grain-wall frictional coefficient, μ δ =0
14
14
Vertical coordinate h (m)
10
Wall pressure P (kPa)
Wall pressure P (kPa)
δ =5ο
12
δ =15ο
10
δ =25ο
8 6 4 2 0 -2
40
0
Wall pressure p (kPa)
10
20
30
40
Wall pressure p (kPa)
Figure 8. Influence of cohesion, c
Figure 9. Influence of dilatation angle, δ
The angle φr between the free surface of bulk solid and horizontal plane is generally taken as the angle of repose of the bulk material. The angle of repose and the grain-wall friction coefficient μ are concerned to φ. Therefore, to study the influence of φ on wall pressure, φr =0 and μ̃=0 were set. That is main the free surface of bulk solid is horizontal plane and the grain-wall friction is neglected when the influence of internal friction angle was analyzed. Figure10 shows that wall pressure increases while φ reduce when φ < 25 degrees but value change of the internal friction angle has no influence on static lateral wall pressure when φ > 25 degrees.
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From figure 4 to figure 10, it is shown that Yong’s modulus, Poisson’s ratio, grain-wall frictional coefficient and internal friction angle have heavy effect on static lateral wall pressure while dilatation angle and cohesion have little influence on static wall pressure.
φ =15ο φ =20ο φ =25ο φ =35ο φ =40ο
Vertical coordinate h (m)
14 12 10 8 6 4 2 0 -2 0
10
20
30
40
50
60
70
Wall pressure p (kPa)
Figure 10. Influence of internal friction angle φ
4. Wall Pressure at the End of Filling For the FEM analyses, the true filling process is not easy to model. In the calculation, a central vertical filling process was specified. Progressive filling with incremental layers was suggested. There are two lamination ways as shown in figure 11. In the first way shown in figure 11(a), the first layer is a cone with a diameter 28 meters and a height 6 meters, then every layer’s thickness is 1 m and the top layer’s thickness is 1.5 m. In the other way shown in figure 11(b), layers are a cylinder with a diameter 28 meters and a height 1 meter and the top layer is a combination of a cylinder with a height 1 meter and a cone with a height 6 meters. To simulate the time effect of filling process, element birth and death function was used in the calculation. Element birth and death function in ADINA is able to model processes during which material is added to and/or removed. If the element birth option is used, the element is added to the total system of finite elements at the time of birth and all times thereafter. If the element death option is used, the element is taken out of the total system of finite elements at times larger than the time of death. If the element birth-then-death option is used, the element is added to the total system of finite elements at the time of birth and remains active until the time of death. The time of death must be greater than the time of birth. The element is taken out of the total system of finite elements at all times larger than the time of death. Layers of stored material were simulated by different elements whose material and death time were the same but birth time was different. The birth time of the first layer is zero, and then the birth time of layers enlarges uniform.
(a)
(b)
Figure 11. Filling process
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Three-Dimensional Finite Element Analysis of Wall Pressure on Large Diameter Silos Fu Jianbao, Luan Maotian, Yang Qing, Nian Tingkai
Vertical coordinate h (m)
14 12
the first way the second way
10 8
static prssure
6 4 2 0 0
10
20
30
40
50
60
70
Wall pressure P (kPa)
Figure 12. Wall pressure at the end of filling Wall pressure distributions at the end of filling in this paper are shown in figure 12. There is big different between wall pressure distributions of two lamination ways at the bottom of silo wall. From the physical sense, the second way shown in figure 11(b) better indicate the progressive filling process. Therefore, this paper applied the way shown in figure 11(b) to study wall pressure of filling finally. In figure 12, it is shown that wall pressure much bigger than static wall pressure near the silo bottom and basically equal to static wall pressures at the middle and upside of the silo wall. Near the silo bottom, wall pressure at the end of filling enlarged suddenly. This phenomenon doesn’t agree with the general view of academia. Academia generally accounts that wall pressure at the end of filling is not bigger than static wall pressure (Japanese academics, 1962 [1]). Considering previous study on wall pressure of filling all based on small diameter silos, following calculation was made. The value of silo diameter was changed and other parameters unchanged.
Vertical coordinate h (m)
14
D = 5m D = 14m D = 28m
12 10 8 6 4 2 0 -2 0
10
20
30
40
50
60
70
Wall pressure P (kPa)
Figure 13. Wall pressure at the end of filling with different diameters Figure 13 shows the Wall pressure distributions at the end of filling with different silo diameters. When silo diameter is small, such as 5 meters, wall pressure at the end of filling has no sudden enlargement. With the increasing of silo diameter, the sudden enlargement of wall pressure near the silo bottom is more and more obvious. It can be conclude that when silo diameter is small, wall pressure at the end of filling is similar to static wall pressure. There is no need to consider wall pressure of filling in silo design. When silo diameter is large, wall pressure at the end of filling is much larger than static wall pressure near silo bottom. Therefore wall pressure of filling should be considered in the design of large diameter silo.
5. Conclusion
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In this paper, a large diameter silo was established by three-dimensional finite element. Static wall pressures and wall pressure at the end of filling were studied. Two lamination ways to simulate filling process was studied. And the influence of mechanical parameters of bulk solid to wall pressure was studied. The following conclusions can be drawn from the results of the generated finite-element model: • Silo scale has heavy influence on wall pressure. It is not feasible to compute wall pressure of large diameter silos with methods based on small diameter silos. It is important to study on large diameter silos. • Yong’s modulus, Poisson’s ratio, grain-wall frictional coefficient and internal friction angle have heavy effect on static wall pressure while dilatation angle and cohesion have little influence to static wall pressure. The value change of Yong’s modulus has no effect to wall pressure at the end of filling. • Wall pressures of large diameter silos at the end of filling were much larger than static wall pressures near silos bottom. Wall pressure of filling should be considered in design of large diameter silos. There is little system research on large diameter silo currently. Study in this paper has heavy theoretical significance and provide an important basis for the large diameter silo design.
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