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SE—Structures and Environment
J. agric. Engng Res. (2001) 78 (3), 299}308 doi:10.1006/ jaer.2000.0640, available online at http://www.idealibrary.com on SE*Structures and Environment
Static and Dynamic Silo Loads using Finite Element Models F. Ayuga; M. Guaita; P. Aguado E.T.S.I. Agro& nomos, Polytechnic University of Madrid, Ciudad Universitario s/n, 28040 Madrid, Spain; e-mail of corresponding author: [email protected] E.P.S. University of Santiago de Compostela, Campus Universitario, 27002 Lugo, Spain; e-mail: [email protected] E.S.T.I. Agraria, University of Leo& n, Au, Portugal, 41.24071 Leon, Spain; e-mail: [email protected] (Received 18 October 1999; accepted in revised form 29 August 2000; published online 24 October 2000)
Di!erent "nite element models for axisymmetric silo analysis, that simultaneously considers the behaviour of the grain and the structure, are proposed in this paper. The use of commercial programmes and integration of modern theories on pressures exerted by grain are the main premises of the research. The paper presents models for both the static and dynamic conditions. In both cases, the in#uence of di!erent parameters is discussed. A new method for emptying pressure determination is also proposed. 2001 Silsoe Research Institute
1. Introduction Storing of bulk materials in silos is essential in a large number of industries, and major investments are made in their construction both in the public and private sectors. During the last 100 years, many accidents, explosions, cracking and excessive deformations have been common in silo structures due to the lack of knowledge about their structural behaviour. Such failures are costly in terms of repair or replacement cost, loss of production and injury or loss of life. Even after more than a century of research, many uncertainties still exist in various areas of silo structural behaviour. Throughout the world, there are a large number of research teams investigating silo structures, but the more advances that are made, the more complex the problem appears. Di!erent standards and rules can be used for silo design, all of which are based on theoretical and empirical methods. Almost every standard uses Janssen's (1895) theory which proposed equations based on a horizontal section of stored material that is in contact with the silo walls. In this theory, the e!ects of the stored material acting on the structure only depend on the silo hydraulic radius, the grain bulk density, the grain friction angles in contact with the silo wall and the value of k (ratio between horizontal and vertical pressures). In spite of its limitations, and a century after its appearance, this remains the theory upon which the standards of all countries 0021-8634/01/030299#10 $35.00/0
and institutions are based, including recent international standards such as Eurocode 1, Part 4 (ENV, 1995). During the silo discharge, the vertical and horizontal pressures can exceed the static results considerably. This has been known and investigated throughout the 20th century, but even today its cause, the factors that intervene and the predictable pressure values, all remain very uncertain. In order to overcome this di$culty, researchers have proposed to modify the static pressures, calculated by Janssen's theory, through the use of overpressure multiplication factors obtained in various research and veri"ed in practice. From the 1970s onwards, a large number of research teams have worked on the application of "nite element analysis to silo problems (Manbeck & Nelson, 1975; Mahmoud, 1975; Jofriet et al., 1977; Maeda & Ishiyaki, 1979). At that time, models and programmes were hampered due to the limited capacity of computers and the high cost of equipment. Furthermore, commercial programmes for these purposes had not been developed. In contrast today, it is possible to use computers whose capacity and speed is continually increasing and a large number of programs exist which not only calculate by means of "nite elements but also manipulate, analyse and present the results. A gap exists in the international bibliography concerning the use of the potent commercial programmes that are currently available for both silo research and for the development of "nite element
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models, which simultaneously work with the behaviour of the stored material and that of the silo structure. This article presents a "nite element model for the design of axisymmetric silos using the capacities of the ANSYS 5.3 (ANSYS, 1998) programme that simultaneously considers the behaviour of the grain and the structure at the same time. The main structural problem for silos occurs when the greatest pressures are produced (during their discharge). The "rst disaster caused by overpressures during discharge occurred in the 1930s when, with the intention of reducing the factor of safety, structural calculation methods were being re"ned and construction materials were being improved. Before that period it had not been easy to alter structural design, and large loads were used for the calculation. Since then, extensive research has been undertaken into the causes of overpressures during discharge, although it cannot be said that the problem has been solved yet. All current standards and suggestions for silo calculations use coe$cients of overpressures during discharge and because of this they are unable to correctly evaluate the complexity of the phenomenon. The present research assumes the validity of, and is principally based upon, the contributions of Jenike made during the 1960s (Jenike & Johanson, 1968). They clearly de"ned the di!erent types of pressures against the walls as a consequence of the change in the stress state in the grain (passing from an active to a passive state and unbalancing the load in the process). This theory has been combined with Zhang's most recent contributions (Zhang et al., 1994) relating to the decisive in#uence of the dilatancy angle of the grain during the discharge process and the consequent overpressure. As a result, this article proposes a new method of analysing silo discharge which combines both e!ects and implements them in a "nite element model. The results are compared with the previous research data and with those indicated by the various standards. Using this model, conclusions can be drawn about the di!erent in#uences of the mechanical parameters of grain on the silo design.
2. Principles of the 5nite element models The objective of this work was to produce results that could easily be incorporated into the industry and be used by silo-design engineers. A few commercial programmes that ful"l these conditions already exist, based on the theory of "nite elements. For this research, the university version of ANSYS 5.3 was used, a version that only di!ers from the commercial one in that the wavefront is limited to 800 degrees of freedom (DOF).
Sixteen "les were generated in order to represent the various problems in silo analysis (static and dynamic analysis, silos with #at bases, with hoppers, walls made of steel or concrete, etc.). With the help of these models it has been possible to confront the di!erent problems which appear during the research, such as grain be-haviour, silo wall behaviour, contact problems, the "t of the mesh, and emptying and "lling the silo. Equal number of silos with #at bases and silos with hoppers were studied. However, for discharge, silos with hoppers were chosen because of the focus of the research on the various #ow types that occur during this process, and in order to make use of the experimental data gathered by other authors, the majority of which corresponds to this type of silo. Two "nite elements were used, in accordance with ANSYS codes. A four-node element, having the possibility of axisymmetry, for ensiled material and an element of two nodes superimposed, having the possibility of taking into account friction and the loss of contact, for grain}wall and grain}grain contact.
3. Structural behaviour of the silo in a static condition In order to test the models and to study the in#uence of the principal parameters upon them, the silo was "rst studied in a static condition, that is, with the grain at rest. An example of the models which were created, a diagram of a #at-bottom silo and steel walls, can be seen in Fig. 1. The mesh is only indicative, as it was one of the parameters analysed during the research. Similar models were created using the various combinations of concrete and rigid walls ("xed nodes) and the di!erent base shapes, #at and hopper. In order to simplify the calculations, the models developed using "nite elements are axisymmetrical. Therefore, the simulated silo is cylindrical with a central discharge. Silo sizes, whether in terms of height or of diameter, could be varied as required, but for the purposes of this research two sizes were selected (Fig. 2). The material behaviour chosen is based on the Drucker-Prager criterion of plasticity, one used when considering the e!ect of expansion on #ooring materials (Drucker & Prager, 1952). The yield surface is a circular cone with the material parameters chosen such that it corresponds to the outer aspices of the hexagonal Mohr}Coulomb yield surface. The chosen material was wheat - speci"cally, Camacho wheat - since it is the most common material found in research conducted by other authors. The required characteristics of the wheat were obtained by means of laboratory tests. The values used are shown in Table 1. The
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Fig. 1. Cylindrical silo with yat base and metal sides
range of values corresponds to those used to analyse the in#uence of these parameters on the results. The entire development of these models can be consulted in Guaita (1995). The di!erent parameters that intervene were systematically analysed and the results were compared with those obtained by means of traditional calculation theories and international standards. The parameters analysed were: Young's modulus, Poisson's ratio, internal friction angle, dilatancy angle, angle of friction with the wall and bulk density. The in#uence of the element size, wall thickness and the di!erent possibilities of silo}hopper joint meshing, were also analysed. It should be taken into account that traditional theories and international standards only consider the internal friction angle, the angle of friction with the wall and the bulk density. Occasionally, the relationship between the horizontal and vertical pressures on the wall is used as a parameter although it is not one in reality. The experimental procedures to measure the material properties were: a direct shear test for the internal friction angle and the dilatancy, a direct shear test with a modi"ed box for the angle of friction with the wall, a triaxial
Fig. 2. Selected silo dimensions: (a) yat-bottom silo; and (b) silo with hopper at a slope angle (a) of 453
test for Poisson's ratio and Young's modulus from the edometric modulus using an edometer. The results from the analyses are quite varied but, for the purposes of comparison and validation, the horizontal pressures on the walls, the vertical pressures on the walls and the relation to pressure on the silo wall k were chosen. Furthermore, for the analysis of the di!erent parameters, results were obtained from the horizontal and vertical pressures of the stored material, the zones of elastic and plastic behaviour within the stored material, and the distortions and tensions within the silo wall.
4. Results of the static models 4.1. Pressures against the walls The validity of the static models, for a #at-bottom silo, becomes evident from Fig. 3(a). It can be observed that, using parameters obtained by experimental tests shown in Table 1, the curve of the horizontal pressures on the wall compares very well with those obtained by means of
Table 1 Characteristics of the wheat with the range values used to analyse their in6uence on the results and the values obtained by means of laboratory tests Material parameter Elasticity module of stored grain E, kPa Speci"c weight of grain o, kN m\ Poisson ratio of grain l Internal friction angle ,3 Wall}grain friction angle k,3 Dilatancy angle u,3
Range of values analysed 5000 7}9 0)2}0)4 22}30 14)03 0}19
Values obtained by experimental tests 5000 9 0)3 22 14)03 2)5
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Fig. 3. (a) Horizontal pressures and (b) the ratio of horizontal to vertical pressures (k) under static conditions for the silo dimensions shown in Fig. 2(a) and the material parameters shown in Table 1: , xnite element method; , Caquot and Kerisel (1956); , Janssen (1895); , Reimbert and Reimbert (1980); , Eurocode (1995); , DIN 1055-6 (1987)
traditional theories and international standards. The same occurs with the vertical pressures and with the ratio of horizontal to vertical pressures k, although this ratio remains constant in all methodologies except in that of Reimbert and Reimbert (1980) and in the "nite elements model [Fig. 3(b)].
4.2. ¹he material parameters in-uence The e!ect of the parameters in the "nite element results are obtained in the following sections. 4.2.1. >oung1s modulus As had been expected, this parameter hardly a!ected the values of the wall pressures, although it did a!ect the total grain consolidation. Its value should be carefully considered in cases where the study of the silo is undertaken when it is being progressively "lled in layers because in such cases the pressures can be a!ected. This value is determined by means of an edometric test in order to obtain the edometrical modulus E and K a triaxial test to measure Poisson's ratio l. With these values Young's modulus E is obtained by means of E"E K
1!l!2l 1!l
(1)
4.2.2. Poisson1s ratio This is a key parameter in determining the grain pressures in a static position. Variation in this coe$cient between 0)2 and 0)4 produced up to a 90% change in the pressures. This parameter is inextricably coupled to the horizontal and vertical pressures ratio k in the interior of the grain. Nevertheless, it must not be forgotten that the ratios obtained near the wall di!er from those which are obtained in the grain interior, meaning, in e!ect, a value of the ratio at rest k : k l" 1#k
(2)
Using "nite element models to simulate silos, obtaining experimental values of Poisson's ratio by means of triaxial tests becomes a necessity. The average values and ranges of variation for the various types of stored materials ought to appear in the methods of calculation. In addition, Poisson's ratio has a notable e!ect on the way in which the material behaves. With high values of Poisson's ratio, it is di$cult for the material to achieve a state of stress such that it reaches the yield surface and, as a result, the behaviour in the interior of the stored material will be elastic. Therefore, the use of elastic}plastic behaviour is no longer a necessity if the Poisson ratio is high. The values at which such e!ect may be obtained
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are di$cult to predict since it depends on the silo size and the material speci"c weight as well as on the behaviour models and their corresponding parameters. In the situation in which this research was carried out (speci"c size of silo, wheat as material and Drucker}Prager as model) with a value for l of 0)2, a complete plasti"cation of the material is produced. In such a situation, horizontal pressures can increase by up to 50% in comparison with the elastic situation and, as a result, low values of Poisson's ratio must always use an elastic}plastic behaviour model. 4.2.3. ¹he internal friction angle This is one of the required parameters for the Drucker}Prager behaviour model and one of those frequently employed in traditional theories and standards. With low values in this parameter, it is logical that plasti"cation of the stored material is produced more easily, something which implies a distinct increase in horizontal pressures. Nevertheless, the values that lead to plasti"cation - around 223*are lower than usual in granular materials - around 253. This e!ect was observed using a value for Poisson's ratio of 0)3. Using lower Poisson's ratio values, the material becomes plastic with higher internal friction angle values. 4.2.4. ¹he dilatancy angle The dilatancy angle is a new parameter that began to be considered when behaviour models for grain of an elastic}plastic type were considered in pressure calculations of stored material. Dilatancy is de"ned as the volumetric increment when a particulate material is subjected to shearing. Dilation is related to the shear displacement through the dilatancy angle u, which is the angle between the tangent of the shear plane and the horizontal shear surface (Fig. 4). Various authors have related this phenomenon to that of overpressure during discharge. Its value can vary between 03 and that of the internal friction angle, although in the bibliography (Hardin et al., 1990; Zhang, 1994) values obtained using agricultural products never exceeded 203. It can be deduced from the present research that there is practically no in#uence of this value on vertical wall pressures in the static position. The greatest static pressures were produced by an angle of 03, and with low values of Poisson's ratio (0)3) and the angle of friction (223), the pressure did not increase by more than 6%. Nonetheless, its in#uence began to appear in hopper silos where increases of up to 16% were produced in the horizontal pressures on the silo}hopper joint. In this case, the increase was produced wherever any dilatancy angle di!erent from 03 was considered and varied little between the di!erent values that were tested.
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Fig. 4. Saw-tooth model for describing dilatancy angle in particulate materials
4.2.5. ¹he wall friction coe.cient The wall friction coe$cient k is one of the parameters employed in traditional theories and in international standards. This analysis has con"rmed previous "ndings that the greater the friction coe$cient, greater the horizontal pressure. There is a 30% increase when moving from a coe$cient with a value of 0)25 to that of 0)5. Increasing the values of wall friction coe$cient, increases the level of plasti"cation and, as a result, accentuates the di!erence in pressure by varying other parameters such as the Poisson ratio or the internal friction angle. 4.2.6. Bulk density No di!erences were found to exist between the results of this research and those produced using traditional theories. The static pressures were proportional to the bulk density. However, the bulk density of the grain could be varied during a simulated loading by using the "nite element method. 4.2.7. Mesh size and distribution It has been possible to demonstrate that with large mesh sizes (of about 1 m) good results are obtained, producing little di!erence compared to the "ner mesh sizes. However, the need for a "ner mesh in the joint between the silo and the hopper has been observed. Di!erences in pressure of up to 100%, using an element of 1 m and another of 0)08 m, were discovered. In such cases, the use of a progressive mesh is recommended in the silo}hopper joint. Furthermore, it is essential to establish that the friction at this point follows the direction of the hopper, and not that of the silo wall, otherwise exaggerated distortion phenomena are produced (Fig. 5). 4.2.8. =all thickness The results of the present research show that for the purpose of pressure calculations, walls of reinforced concrete behave exactly the same as would a perfectly rigid wall. Therefore, it is valid (in the application of the "nite element method) to consider this assumption for the calculation of such pressures. It is not the same with metal walls. A smaller wall thickness causes the stored
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Fig. 5. Pressures with diwerent mesh sizes: , Eurocode; , xnite element method with regular mesh and size of 0)5 m; , progressive meshing
material to exert less pressure, due to radial de#ection. Di!erences in pressure of up to 25%, using a #at-bottom silo model with an internal friction angle of 303 and a wall thickness of 1 mm, were discovered. However, if the material behaviour is elastic}plastic and total plasti"cation of the material is produced in the zone adjacent to the wall pressures are not a!ected by wall material or wall thickness.
5. Structural behaviour of the silo during discharge It is known that the main structural problem of silos is the increase in horizontal pressures when discharge starts. In earlier silo designs overpressures were not considered, even though as early as 1950s, it was fairly well recognized that overpressures occur during emptying (Caquot & Kerisel, 1956). This pressure increase has been measured both in model and full-scale silos. Nevertheless, disagreement still exists between the magnitude of these overpressures and their causes. Traditionally, the increase in pressures has been attributed to a change in the state of equilibrium of the
granular material. It was thought that the full silo in a static position would respond according to Rankine's active state of equilibrium and that once the grain began to move, it would pass to a passive state of equilibrium (Jenike & Johanson, 1968). Some authors later modi"ed this simpli"cation and they took into account the grain}wall friction and also the types of #ow during discharge (Rotter et al., 1997; Aguado et al., 1997). More recently, this overpressure has been attributed to the dilatancy phenomenon which is seen as the cause of the grain not displacing parallel to the rupture line that provokes an increase in volume and, as a consequence, an increase in the lateral pressures (Zhang et al., 1994). On the other hand, Jenike and Johanson attributed the overpressures to the types of #ow during discharge (Jenike & Johanson, 1968). Attaching strain gauges demon strated that in the case of mass #ow the pressure peak occurred at the silo}hopper joint the same as occurred in the case of a static silo. However, in the case of funnel #ow, the pressure peak was produced exactly where the funnel for grain #ow started. No variation was observed below this point owing to the fact that the grain against the wall remained at rest in this area. This line of reasoning has not led to a method of predicting overpressures during discharge. The results of this research show that all these phenomena are intimately related to overpressure during discharge. The prime objective of this research was to analyse the phenomenon of silo discharge using the "nite element method as a tool. Models were developed based on those previously created for static stored material incorporating a series of modi"cations derived from the following three basic hypotheses. (1) Maximum pressures Maximum pressures are produced during the initial moments of discharge (Drescher, 1978; Sugita, 1972). Discharge is simulated in the present analysis by a small displacement of the grain in the silo outlet. At the same time, a vertical cylindrical surface of contact elements with "xed grain}grain friction was included in the model in order to simulate the movement of granular material within the silo (Fig. 6). (2) ¹he existence of di+erent -ow types (mass or funnel) This implies that the rupture lines within the granular material can take di!erent directions. Thus, in models developed for mass #ow, the material descends creating friction against the silo wall. In models for funnel #ow, a new line of rupture is formed creating an angle, called b, with the horizontal axis (Fig. 6), and which, due to the axisymmetric character of the model, produces a funnel of stored material.
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is produced. The uncertainty about the slide line position in the latter case is due to the scarcity of research and the still limited development of discrete element methods (Rong, 1994; Rotter et al., 1997; Jofriet et al., 1997). By this method the behaviour of the material is simulated grain by grain and, as a result, it is not necessary to force its movement. Owing to this uncertainty about the slide line position, it was agreed to retain this ambiguity, developing a model which would solve the problem of di!erent angles of inclination along the rupture line in the grain b, record the data and, "nally, obtain a curve of the maximum pressures reached at each sectional height of the silo wall.
6. Results obtained using discharge models 6.1. Pressures against the walls
Fig. 6. Schematic discharge model; b, rupture angle
(3) ¹he dilatancy e+ect This is simulated by means of the change in the friction direction (depending on the dilatancy angle) in the contact element. It is assumed that the material is separated following the line of rupture thus provoking an increase in the lateral pressures. Depending on the geometric characteristics of the silo and the type of stored material, either mass or funnel #ow
For mass #ow, the pressure peak coincided with silo}hopper joint [Fig. 7(a)]. Pressure distribution obtained using funnel #ow models conformed with the experimental observations by Jenike and Johanson (1968) that the pressure peak coincided with the junction of the rupture line in the grain and the vertical wall of the silo [Fig. 7(b)]. A pressure envelope curve was developed which represents the maximum values of the di!erent curves obtained in the di!erent #ow conditions (Fig. 8). Di!erences in pressure of up to 100%, using an element of 1 m (regular mesh) and another of 0)08 m (progressive mesh) in the silo}hopper joint, were also discovered (Fig. 8).
Fig. 7. (a) Mass yow (regular mesh) pressures and (b) funnel yow pressures for a rupture angle (b) of 553, an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , static model; , discharging model
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Fig. 8. Final discharge model for an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , static model; , discharge for b"453; , discharge for b"503; , discharge for b"553; , discharge for b"603; , discharge for b"653; , discharge for b"703; , discharge for b"753; , enveloping curve for regular mesh; , enveloping curve for progressive mesh
The values obtained using a progressive mesh agree with those obtained by means of Eurocode 1, Part 4, except in the hopper wall where they were lower (Fig. 9). The ratio of horizontal to vertical pressures k was 12% greater than in the static example, and was similar (except in the hopper wall) to that of the reduced value proposed by Eurocode 1, Part 4. All of this justi"es the focus of the research and the validity of the models used.
6.2. ¹he material parameters in-uence It is interesting to examine the e!ect on wall pressures of Poisson's ratio and of the dilatancy angle with regard to the discharge models. The Poisson ratio a!ects the lateral pressures, although notably less than in the static example. An increase from 0)2 to 0)4 increase overpressures by 30% on the hopper wall, 20% in the silo}hopper joint and 10% on the vertical wall [Fig. 10(a)]. Unlike the static example, in the discharge model, an increase in the dilatancy angle from 5 to 193 produces an increase of 30% in the lateral pressures. As a consequence, there is an increase in the coe$cient k [Fig. 10(b)].
Fig. 9. Eurocode and proposed model pressures comparison for an internal friction angle of 223, a dilatancy angle of 193 and , proposed model using progressa bulk density of 9 kN m!3: ive meshing; , Eurocode for mass yow; , Eurocode for funnel yow
7. Conclusions Finite element models capable of simulating the behaviour of silo-stored granular material were developed, both for the static example and for that of centralized discharge. The in#uence of the type of wall and the geometrical structure of the silo, as well as the di!erent parameters of silo-stored granular material, were analysed with regard to their e!ect on pressures, not only in the case of material at rest but also while being discharged. By means of these model values for pressure against the silo walls were obtained in mass #ow and funnel #ow. Values obtained in mass #ow were similar to those proposed by other authors and calculations using the Eurocode. Nevertheless, by using these models it is possible to get a more detailed analysis and better adaptation to particular situations. The models also allow an explanation of the pressure distribution in funnel #ow, phenomena observed in the experimental trials of other authors. In these models, parameters such as that of dilatancy angle or the Poisson ratio, which exert a great in#uence on lateral pressures, are taken into account. Dilatancy angle acts during the discharge while the Poisson ratio
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Fig. 10. (a) Poisson+s ratio inyuence on pressures for regular mesh, an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , Poisson+s ratio"0)2; , Poisson+s ratio"0)3; , Poisson+s ratio"0)4: (b) Dilatancy angle inyuence on pressures for regular mesh, an internal friction angle of 223, a Poisson+s ratio of 0)3 and a bulk density of 9 kN m!3: , dilatancy"53; , dilatancy"103; , dilatancy"153; , dilatancy"193
does so both during the discharge and in the static state. In spite of their great in#uence, these two parameters are not taken into account in current guidelines. This is due to the fact that they were simply not considered in traditional methods of calculation and that, furthermore, very few experimentally obtained values exist for agricultural materials. Acknowledgements The authors are grateful to the CICYT (Spanish Research and Technology Commission) for funding this project (AGF97-1141). References Aguado P J; Ayuga F; Guaita M (1997). Comparative evaluation of numerical methods for predicting #ow and stress "elds in silos. Filling of a Silo. EPSRC DEMFEM International Collaboration CA-SILO Collaborative Action: WG5. Problem 1. Universidad PoliteH cnica de Madrid Group Contribution ANSYS (1998). User's Manuals for Revision 5.3. Vols. I, II, III, IV and V. Swanson Analysis Systems Inc., Houston, USA
Caquot A; Kerisel J (1956). Traite Mecanique des Sols. [Treatise on soil mechanics.] Gauthier-Villars EditeurImprimeur-Librairie. Paris DIN 1055 Part 6 (1987). Design loads for buildings. Loads in silo bins Drescher A (1978). Kinematics of the mass #ow of granular material through a plane hopper Geo. 28(1), 27}42 Drucker D C; Prager W (1952). Soil mechanics and plastic analysis on limit design. Quarterly Applied Mathematics, 10(2), 157}165 Eurocode 1, ENV 1991-4. (1995). Basis of design and actions on structures. Part 4: Actions on silos and tanks Guaita M (1995). CreacioH n de Modelos para la SimulacioH n de Silos por el MeH todo de los Elementos Finitos y AnaH lisis de los Empujes EstaH ticos del Material Almacenado. [Model development for silo simulation using the "nite element method and analysis of the static pressures of the stored materials.] Tesis Doctoral. Universidad PoliteH cnica de Madrid Hardin B O; Hardin K O; Ross I J; Schwab C V (1990). Triaxial compression, simple shear, and strength of wheat en masse. Transactions of the ASAE, 33(3), 933}943 Janssen H A (1895). Versuch uK ber Getreidedruck in Sillozellen. [Experiments on grain loads in silo cells.] Zeischrift des Vesein Deutscher Ingenieure, 39, 1045}1049 Jenike A W; Johanson J R (1968). Bins loads. Journal of Structure Division, Proceeding of ASCE, 94(ST4), 1011-1041 Jofriet J C; Lelievre B; Fwa T F (1977). Friction model for "nite element analyses of silos. Transactions of the ASAE, 20(4), 735}744
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Jofriet J C; Negi S C; Lu Z (1997). Parametric study of bin loads using a hybrid numerical model. ASAE Meeting Presentation. Paper no 97-4102. Minneapolis Maeda Y; Ishizaki S (1979). Analysis of cylindrical shells for design of steel silos. Journal Civil Engineering Design, 1(4), 325}354 Mahmoud M H (1975). Silage}silo interaction using material characterization and "nite element analysis. PhD Thesis. Department of Civil Engineering, Ohio State University, Columbus, OH Manbeck H B; Nelson G L (1975). Three dimensional constitutive equations for wheat en masse. Transactions of the ASAE, 18(6), 1122}1127 Reimbert M; Reimbert A (1980). Pressures and overpressures in vertical and horizontal silos. International Conference on Design of Silos for Strength and Flow, Powder Advisory Centre, London
Rong G (1994). Discrete element modelling for #ow of particulate materials in bins PhD Thesis. University of Guelph, Canada Rotter M; Ooi J; Holst M; Zhong Z (1997). Comparative evaluation of numerical methods for predicting #ow and stress "elds in silos. EPSRC DEMFEM International Collaboration CA-SILO Collaborative Action: WG5. http://oats.civ.ed.ac.uk/research/silo/demfem/ Sugita M (1972). Flow and pressures of non-cohesive granular materials in funnel #ow bin. ASME No. 72-MH-20, pp 1}8 Zhang Q; Britton M G; Xu S (1994). Using dilatancy angle to predict dynamic loads during discharge in bulk solid storage structures. Proceeding of Powder and Bulk Solids'94, Rosemont, IL 383-390 Zhang Y (1994). Mechanical properties of soyabean and corn. MS Dissertation, University of Guelph
Static and Dynamic Silo Loads using Finite Element Models F. Ayuga; M. Guaita; P. Aguado E.T.S.I. Agro& nomos, Polytechnic University of Madrid, Ciudad Universitario s/n, 28040 Madrid, Spain; e-mail of corresponding author: [email protected] E.P.S. University of Santiago de Compostela, Campus Universitario, 27002 Lugo, Spain; e-mail: [email protected] E.S.T.I. Agraria, University of Leo& n, Au, Portugal, 41.24071 Leon, Spain; e-mail: [email protected] (Received 18 October 1999; accepted in revised form 29 August 2000; published online 24 October 2000)
Di!erent "nite element models for axisymmetric silo analysis, that simultaneously considers the behaviour of the grain and the structure, are proposed in this paper. The use of commercial programmes and integration of modern theories on pressures exerted by grain are the main premises of the research. The paper presents models for both the static and dynamic conditions. In both cases, the in#uence of di!erent parameters is discussed. A new method for emptying pressure determination is also proposed. 2001 Silsoe Research Institute
1. Introduction Storing of bulk materials in silos is essential in a large number of industries, and major investments are made in their construction both in the public and private sectors. During the last 100 years, many accidents, explosions, cracking and excessive deformations have been common in silo structures due to the lack of knowledge about their structural behaviour. Such failures are costly in terms of repair or replacement cost, loss of production and injury or loss of life. Even after more than a century of research, many uncertainties still exist in various areas of silo structural behaviour. Throughout the world, there are a large number of research teams investigating silo structures, but the more advances that are made, the more complex the problem appears. Di!erent standards and rules can be used for silo design, all of which are based on theoretical and empirical methods. Almost every standard uses Janssen's (1895) theory which proposed equations based on a horizontal section of stored material that is in contact with the silo walls. In this theory, the e!ects of the stored material acting on the structure only depend on the silo hydraulic radius, the grain bulk density, the grain friction angles in contact with the silo wall and the value of k (ratio between horizontal and vertical pressures). In spite of its limitations, and a century after its appearance, this remains the theory upon which the standards of all countries 0021-8634/01/030299#10 $35.00/0
and institutions are based, including recent international standards such as Eurocode 1, Part 4 (ENV, 1995). During the silo discharge, the vertical and horizontal pressures can exceed the static results considerably. This has been known and investigated throughout the 20th century, but even today its cause, the factors that intervene and the predictable pressure values, all remain very uncertain. In order to overcome this di$culty, researchers have proposed to modify the static pressures, calculated by Janssen's theory, through the use of overpressure multiplication factors obtained in various research and veri"ed in practice. From the 1970s onwards, a large number of research teams have worked on the application of "nite element analysis to silo problems (Manbeck & Nelson, 1975; Mahmoud, 1975; Jofriet et al., 1977; Maeda & Ishiyaki, 1979). At that time, models and programmes were hampered due to the limited capacity of computers and the high cost of equipment. Furthermore, commercial programmes for these purposes had not been developed. In contrast today, it is possible to use computers whose capacity and speed is continually increasing and a large number of programs exist which not only calculate by means of "nite elements but also manipulate, analyse and present the results. A gap exists in the international bibliography concerning the use of the potent commercial programmes that are currently available for both silo research and for the development of "nite element
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models, which simultaneously work with the behaviour of the stored material and that of the silo structure. This article presents a "nite element model for the design of axisymmetric silos using the capacities of the ANSYS 5.3 (ANSYS, 1998) programme that simultaneously considers the behaviour of the grain and the structure at the same time. The main structural problem for silos occurs when the greatest pressures are produced (during their discharge). The "rst disaster caused by overpressures during discharge occurred in the 1930s when, with the intention of reducing the factor of safety, structural calculation methods were being re"ned and construction materials were being improved. Before that period it had not been easy to alter structural design, and large loads were used for the calculation. Since then, extensive research has been undertaken into the causes of overpressures during discharge, although it cannot be said that the problem has been solved yet. All current standards and suggestions for silo calculations use coe$cients of overpressures during discharge and because of this they are unable to correctly evaluate the complexity of the phenomenon. The present research assumes the validity of, and is principally based upon, the contributions of Jenike made during the 1960s (Jenike & Johanson, 1968). They clearly de"ned the di!erent types of pressures against the walls as a consequence of the change in the stress state in the grain (passing from an active to a passive state and unbalancing the load in the process). This theory has been combined with Zhang's most recent contributions (Zhang et al., 1994) relating to the decisive in#uence of the dilatancy angle of the grain during the discharge process and the consequent overpressure. As a result, this article proposes a new method of analysing silo discharge which combines both e!ects and implements them in a "nite element model. The results are compared with the previous research data and with those indicated by the various standards. Using this model, conclusions can be drawn about the di!erent in#uences of the mechanical parameters of grain on the silo design.
2. Principles of the 5nite element models The objective of this work was to produce results that could easily be incorporated into the industry and be used by silo-design engineers. A few commercial programmes that ful"l these conditions already exist, based on the theory of "nite elements. For this research, the university version of ANSYS 5.3 was used, a version that only di!ers from the commercial one in that the wavefront is limited to 800 degrees of freedom (DOF).
Sixteen "les were generated in order to represent the various problems in silo analysis (static and dynamic analysis, silos with #at bases, with hoppers, walls made of steel or concrete, etc.). With the help of these models it has been possible to confront the di!erent problems which appear during the research, such as grain be-haviour, silo wall behaviour, contact problems, the "t of the mesh, and emptying and "lling the silo. Equal number of silos with #at bases and silos with hoppers were studied. However, for discharge, silos with hoppers were chosen because of the focus of the research on the various #ow types that occur during this process, and in order to make use of the experimental data gathered by other authors, the majority of which corresponds to this type of silo. Two "nite elements were used, in accordance with ANSYS codes. A four-node element, having the possibility of axisymmetry, for ensiled material and an element of two nodes superimposed, having the possibility of taking into account friction and the loss of contact, for grain}wall and grain}grain contact.
3. Structural behaviour of the silo in a static condition In order to test the models and to study the in#uence of the principal parameters upon them, the silo was "rst studied in a static condition, that is, with the grain at rest. An example of the models which were created, a diagram of a #at-bottom silo and steel walls, can be seen in Fig. 1. The mesh is only indicative, as it was one of the parameters analysed during the research. Similar models were created using the various combinations of concrete and rigid walls ("xed nodes) and the di!erent base shapes, #at and hopper. In order to simplify the calculations, the models developed using "nite elements are axisymmetrical. Therefore, the simulated silo is cylindrical with a central discharge. Silo sizes, whether in terms of height or of diameter, could be varied as required, but for the purposes of this research two sizes were selected (Fig. 2). The material behaviour chosen is based on the Drucker-Prager criterion of plasticity, one used when considering the e!ect of expansion on #ooring materials (Drucker & Prager, 1952). The yield surface is a circular cone with the material parameters chosen such that it corresponds to the outer aspices of the hexagonal Mohr}Coulomb yield surface. The chosen material was wheat - speci"cally, Camacho wheat - since it is the most common material found in research conducted by other authors. The required characteristics of the wheat were obtained by means of laboratory tests. The values used are shown in Table 1. The
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Fig. 1. Cylindrical silo with yat base and metal sides
range of values corresponds to those used to analyse the in#uence of these parameters on the results. The entire development of these models can be consulted in Guaita (1995). The di!erent parameters that intervene were systematically analysed and the results were compared with those obtained by means of traditional calculation theories and international standards. The parameters analysed were: Young's modulus, Poisson's ratio, internal friction angle, dilatancy angle, angle of friction with the wall and bulk density. The in#uence of the element size, wall thickness and the di!erent possibilities of silo}hopper joint meshing, were also analysed. It should be taken into account that traditional theories and international standards only consider the internal friction angle, the angle of friction with the wall and the bulk density. Occasionally, the relationship between the horizontal and vertical pressures on the wall is used as a parameter although it is not one in reality. The experimental procedures to measure the material properties were: a direct shear test for the internal friction angle and the dilatancy, a direct shear test with a modi"ed box for the angle of friction with the wall, a triaxial
Fig. 2. Selected silo dimensions: (a) yat-bottom silo; and (b) silo with hopper at a slope angle (a) of 453
test for Poisson's ratio and Young's modulus from the edometric modulus using an edometer. The results from the analyses are quite varied but, for the purposes of comparison and validation, the horizontal pressures on the walls, the vertical pressures on the walls and the relation to pressure on the silo wall k were chosen. Furthermore, for the analysis of the di!erent parameters, results were obtained from the horizontal and vertical pressures of the stored material, the zones of elastic and plastic behaviour within the stored material, and the distortions and tensions within the silo wall.
4. Results of the static models 4.1. Pressures against the walls The validity of the static models, for a #at-bottom silo, becomes evident from Fig. 3(a). It can be observed that, using parameters obtained by experimental tests shown in Table 1, the curve of the horizontal pressures on the wall compares very well with those obtained by means of
Table 1 Characteristics of the wheat with the range values used to analyse their in6uence on the results and the values obtained by means of laboratory tests Material parameter Elasticity module of stored grain E, kPa Speci"c weight of grain o, kN m\ Poisson ratio of grain l Internal friction angle ,3 Wall}grain friction angle k,3 Dilatancy angle u,3
Range of values analysed 5000 7}9 0)2}0)4 22}30 14)03 0}19
Values obtained by experimental tests 5000 9 0)3 22 14)03 2)5
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Fig. 3. (a) Horizontal pressures and (b) the ratio of horizontal to vertical pressures (k) under static conditions for the silo dimensions shown in Fig. 2(a) and the material parameters shown in Table 1: , xnite element method; , Caquot and Kerisel (1956); , Janssen (1895); , Reimbert and Reimbert (1980); , Eurocode (1995); , DIN 1055-6 (1987)
traditional theories and international standards. The same occurs with the vertical pressures and with the ratio of horizontal to vertical pressures k, although this ratio remains constant in all methodologies except in that of Reimbert and Reimbert (1980) and in the "nite elements model [Fig. 3(b)].
4.2. ¹he material parameters in-uence The e!ect of the parameters in the "nite element results are obtained in the following sections. 4.2.1. >oung1s modulus As had been expected, this parameter hardly a!ected the values of the wall pressures, although it did a!ect the total grain consolidation. Its value should be carefully considered in cases where the study of the silo is undertaken when it is being progressively "lled in layers because in such cases the pressures can be a!ected. This value is determined by means of an edometric test in order to obtain the edometrical modulus E and K a triaxial test to measure Poisson's ratio l. With these values Young's modulus E is obtained by means of E"E K
1!l!2l 1!l
(1)
4.2.2. Poisson1s ratio This is a key parameter in determining the grain pressures in a static position. Variation in this coe$cient between 0)2 and 0)4 produced up to a 90% change in the pressures. This parameter is inextricably coupled to the horizontal and vertical pressures ratio k in the interior of the grain. Nevertheless, it must not be forgotten that the ratios obtained near the wall di!er from those which are obtained in the grain interior, meaning, in e!ect, a value of the ratio at rest k : k l" 1#k
(2)
Using "nite element models to simulate silos, obtaining experimental values of Poisson's ratio by means of triaxial tests becomes a necessity. The average values and ranges of variation for the various types of stored materials ought to appear in the methods of calculation. In addition, Poisson's ratio has a notable e!ect on the way in which the material behaves. With high values of Poisson's ratio, it is di$cult for the material to achieve a state of stress such that it reaches the yield surface and, as a result, the behaviour in the interior of the stored material will be elastic. Therefore, the use of elastic}plastic behaviour is no longer a necessity if the Poisson ratio is high. The values at which such e!ect may be obtained
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are di$cult to predict since it depends on the silo size and the material speci"c weight as well as on the behaviour models and their corresponding parameters. In the situation in which this research was carried out (speci"c size of silo, wheat as material and Drucker}Prager as model) with a value for l of 0)2, a complete plasti"cation of the material is produced. In such a situation, horizontal pressures can increase by up to 50% in comparison with the elastic situation and, as a result, low values of Poisson's ratio must always use an elastic}plastic behaviour model. 4.2.3. ¹he internal friction angle This is one of the required parameters for the Drucker}Prager behaviour model and one of those frequently employed in traditional theories and standards. With low values in this parameter, it is logical that plasti"cation of the stored material is produced more easily, something which implies a distinct increase in horizontal pressures. Nevertheless, the values that lead to plasti"cation - around 223*are lower than usual in granular materials - around 253. This e!ect was observed using a value for Poisson's ratio of 0)3. Using lower Poisson's ratio values, the material becomes plastic with higher internal friction angle values. 4.2.4. ¹he dilatancy angle The dilatancy angle is a new parameter that began to be considered when behaviour models for grain of an elastic}plastic type were considered in pressure calculations of stored material. Dilatancy is de"ned as the volumetric increment when a particulate material is subjected to shearing. Dilation is related to the shear displacement through the dilatancy angle u, which is the angle between the tangent of the shear plane and the horizontal shear surface (Fig. 4). Various authors have related this phenomenon to that of overpressure during discharge. Its value can vary between 03 and that of the internal friction angle, although in the bibliography (Hardin et al., 1990; Zhang, 1994) values obtained using agricultural products never exceeded 203. It can be deduced from the present research that there is practically no in#uence of this value on vertical wall pressures in the static position. The greatest static pressures were produced by an angle of 03, and with low values of Poisson's ratio (0)3) and the angle of friction (223), the pressure did not increase by more than 6%. Nonetheless, its in#uence began to appear in hopper silos where increases of up to 16% were produced in the horizontal pressures on the silo}hopper joint. In this case, the increase was produced wherever any dilatancy angle di!erent from 03 was considered and varied little between the di!erent values that were tested.
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Fig. 4. Saw-tooth model for describing dilatancy angle in particulate materials
4.2.5. ¹he wall friction coe.cient The wall friction coe$cient k is one of the parameters employed in traditional theories and in international standards. This analysis has con"rmed previous "ndings that the greater the friction coe$cient, greater the horizontal pressure. There is a 30% increase when moving from a coe$cient with a value of 0)25 to that of 0)5. Increasing the values of wall friction coe$cient, increases the level of plasti"cation and, as a result, accentuates the di!erence in pressure by varying other parameters such as the Poisson ratio or the internal friction angle. 4.2.6. Bulk density No di!erences were found to exist between the results of this research and those produced using traditional theories. The static pressures were proportional to the bulk density. However, the bulk density of the grain could be varied during a simulated loading by using the "nite element method. 4.2.7. Mesh size and distribution It has been possible to demonstrate that with large mesh sizes (of about 1 m) good results are obtained, producing little di!erence compared to the "ner mesh sizes. However, the need for a "ner mesh in the joint between the silo and the hopper has been observed. Di!erences in pressure of up to 100%, using an element of 1 m and another of 0)08 m, were discovered. In such cases, the use of a progressive mesh is recommended in the silo}hopper joint. Furthermore, it is essential to establish that the friction at this point follows the direction of the hopper, and not that of the silo wall, otherwise exaggerated distortion phenomena are produced (Fig. 5). 4.2.8. =all thickness The results of the present research show that for the purpose of pressure calculations, walls of reinforced concrete behave exactly the same as would a perfectly rigid wall. Therefore, it is valid (in the application of the "nite element method) to consider this assumption for the calculation of such pressures. It is not the same with metal walls. A smaller wall thickness causes the stored
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Fig. 5. Pressures with diwerent mesh sizes: , Eurocode; , xnite element method with regular mesh and size of 0)5 m; , progressive meshing
material to exert less pressure, due to radial de#ection. Di!erences in pressure of up to 25%, using a #at-bottom silo model with an internal friction angle of 303 and a wall thickness of 1 mm, were discovered. However, if the material behaviour is elastic}plastic and total plasti"cation of the material is produced in the zone adjacent to the wall pressures are not a!ected by wall material or wall thickness.
5. Structural behaviour of the silo during discharge It is known that the main structural problem of silos is the increase in horizontal pressures when discharge starts. In earlier silo designs overpressures were not considered, even though as early as 1950s, it was fairly well recognized that overpressures occur during emptying (Caquot & Kerisel, 1956). This pressure increase has been measured both in model and full-scale silos. Nevertheless, disagreement still exists between the magnitude of these overpressures and their causes. Traditionally, the increase in pressures has been attributed to a change in the state of equilibrium of the
granular material. It was thought that the full silo in a static position would respond according to Rankine's active state of equilibrium and that once the grain began to move, it would pass to a passive state of equilibrium (Jenike & Johanson, 1968). Some authors later modi"ed this simpli"cation and they took into account the grain}wall friction and also the types of #ow during discharge (Rotter et al., 1997; Aguado et al., 1997). More recently, this overpressure has been attributed to the dilatancy phenomenon which is seen as the cause of the grain not displacing parallel to the rupture line that provokes an increase in volume and, as a consequence, an increase in the lateral pressures (Zhang et al., 1994). On the other hand, Jenike and Johanson attributed the overpressures to the types of #ow during discharge (Jenike & Johanson, 1968). Attaching strain gauges demon strated that in the case of mass #ow the pressure peak occurred at the silo}hopper joint the same as occurred in the case of a static silo. However, in the case of funnel #ow, the pressure peak was produced exactly where the funnel for grain #ow started. No variation was observed below this point owing to the fact that the grain against the wall remained at rest in this area. This line of reasoning has not led to a method of predicting overpressures during discharge. The results of this research show that all these phenomena are intimately related to overpressure during discharge. The prime objective of this research was to analyse the phenomenon of silo discharge using the "nite element method as a tool. Models were developed based on those previously created for static stored material incorporating a series of modi"cations derived from the following three basic hypotheses. (1) Maximum pressures Maximum pressures are produced during the initial moments of discharge (Drescher, 1978; Sugita, 1972). Discharge is simulated in the present analysis by a small displacement of the grain in the silo outlet. At the same time, a vertical cylindrical surface of contact elements with "xed grain}grain friction was included in the model in order to simulate the movement of granular material within the silo (Fig. 6). (2) ¹he existence of di+erent -ow types (mass or funnel) This implies that the rupture lines within the granular material can take di!erent directions. Thus, in models developed for mass #ow, the material descends creating friction against the silo wall. In models for funnel #ow, a new line of rupture is formed creating an angle, called b, with the horizontal axis (Fig. 6), and which, due to the axisymmetric character of the model, produces a funnel of stored material.
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is produced. The uncertainty about the slide line position in the latter case is due to the scarcity of research and the still limited development of discrete element methods (Rong, 1994; Rotter et al., 1997; Jofriet et al., 1997). By this method the behaviour of the material is simulated grain by grain and, as a result, it is not necessary to force its movement. Owing to this uncertainty about the slide line position, it was agreed to retain this ambiguity, developing a model which would solve the problem of di!erent angles of inclination along the rupture line in the grain b, record the data and, "nally, obtain a curve of the maximum pressures reached at each sectional height of the silo wall.
6. Results obtained using discharge models 6.1. Pressures against the walls
Fig. 6. Schematic discharge model; b, rupture angle
(3) ¹he dilatancy e+ect This is simulated by means of the change in the friction direction (depending on the dilatancy angle) in the contact element. It is assumed that the material is separated following the line of rupture thus provoking an increase in the lateral pressures. Depending on the geometric characteristics of the silo and the type of stored material, either mass or funnel #ow
For mass #ow, the pressure peak coincided with silo}hopper joint [Fig. 7(a)]. Pressure distribution obtained using funnel #ow models conformed with the experimental observations by Jenike and Johanson (1968) that the pressure peak coincided with the junction of the rupture line in the grain and the vertical wall of the silo [Fig. 7(b)]. A pressure envelope curve was developed which represents the maximum values of the di!erent curves obtained in the di!erent #ow conditions (Fig. 8). Di!erences in pressure of up to 100%, using an element of 1 m (regular mesh) and another of 0)08 m (progressive mesh) in the silo}hopper joint, were also discovered (Fig. 8).
Fig. 7. (a) Mass yow (regular mesh) pressures and (b) funnel yow pressures for a rupture angle (b) of 553, an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , static model; , discharging model
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Fig. 8. Final discharge model for an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , static model; , discharge for b"453; , discharge for b"503; , discharge for b"553; , discharge for b"603; , discharge for b"653; , discharge for b"703; , discharge for b"753; , enveloping curve for regular mesh; , enveloping curve for progressive mesh
The values obtained using a progressive mesh agree with those obtained by means of Eurocode 1, Part 4, except in the hopper wall where they were lower (Fig. 9). The ratio of horizontal to vertical pressures k was 12% greater than in the static example, and was similar (except in the hopper wall) to that of the reduced value proposed by Eurocode 1, Part 4. All of this justi"es the focus of the research and the validity of the models used.
6.2. ¹he material parameters in-uence It is interesting to examine the e!ect on wall pressures of Poisson's ratio and of the dilatancy angle with regard to the discharge models. The Poisson ratio a!ects the lateral pressures, although notably less than in the static example. An increase from 0)2 to 0)4 increase overpressures by 30% on the hopper wall, 20% in the silo}hopper joint and 10% on the vertical wall [Fig. 10(a)]. Unlike the static example, in the discharge model, an increase in the dilatancy angle from 5 to 193 produces an increase of 30% in the lateral pressures. As a consequence, there is an increase in the coe$cient k [Fig. 10(b)].
Fig. 9. Eurocode and proposed model pressures comparison for an internal friction angle of 223, a dilatancy angle of 193 and , proposed model using progressa bulk density of 9 kN m!3: ive meshing; , Eurocode for mass yow; , Eurocode for funnel yow
7. Conclusions Finite element models capable of simulating the behaviour of silo-stored granular material were developed, both for the static example and for that of centralized discharge. The in#uence of the type of wall and the geometrical structure of the silo, as well as the di!erent parameters of silo-stored granular material, were analysed with regard to their e!ect on pressures, not only in the case of material at rest but also while being discharged. By means of these model values for pressure against the silo walls were obtained in mass #ow and funnel #ow. Values obtained in mass #ow were similar to those proposed by other authors and calculations using the Eurocode. Nevertheless, by using these models it is possible to get a more detailed analysis and better adaptation to particular situations. The models also allow an explanation of the pressure distribution in funnel #ow, phenomena observed in the experimental trials of other authors. In these models, parameters such as that of dilatancy angle or the Poisson ratio, which exert a great in#uence on lateral pressures, are taken into account. Dilatancy angle acts during the discharge while the Poisson ratio
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Fig. 10. (a) Poisson+s ratio inyuence on pressures for regular mesh, an internal friction angle of 223, a dilatancy angle of 193 and a bulk density of 9 kN m!3: , Poisson+s ratio"0)2; , Poisson+s ratio"0)3; , Poisson+s ratio"0)4: (b) Dilatancy angle inyuence on pressures for regular mesh, an internal friction angle of 223, a Poisson+s ratio of 0)3 and a bulk density of 9 kN m!3: , dilatancy"53; , dilatancy"103; , dilatancy"153; , dilatancy"193
does so both during the discharge and in the static state. In spite of their great in#uence, these two parameters are not taken into account in current guidelines. This is due to the fact that they were simply not considered in traditional methods of calculation and that, furthermore, very few experimentally obtained values exist for agricultural materials. Acknowledgements The authors are grateful to the CICYT (Spanish Research and Technology Commission) for funding this project (AGF97-1141). References Aguado P J; Ayuga F; Guaita M (1997). Comparative evaluation of numerical methods for predicting #ow and stress "elds in silos. Filling of a Silo. EPSRC DEMFEM International Collaboration CA-SILO Collaborative Action: WG5. Problem 1. Universidad PoliteH cnica de Madrid Group Contribution ANSYS (1998). User's Manuals for Revision 5.3. Vols. I, II, III, IV and V. Swanson Analysis Systems Inc., Houston, USA
Caquot A; Kerisel J (1956). Traite Mecanique des Sols. [Treatise on soil mechanics.] Gauthier-Villars EditeurImprimeur-Librairie. Paris DIN 1055 Part 6 (1987). Design loads for buildings. Loads in silo bins Drescher A (1978). Kinematics of the mass #ow of granular material through a plane hopper Geo. 28(1), 27}42 Drucker D C; Prager W (1952). Soil mechanics and plastic analysis on limit design. Quarterly Applied Mathematics, 10(2), 157}165 Eurocode 1, ENV 1991-4. (1995). Basis of design and actions on structures. Part 4: Actions on silos and tanks Guaita M (1995). CreacioH n de Modelos para la SimulacioH n de Silos por el MeH todo de los Elementos Finitos y AnaH lisis de los Empujes EstaH ticos del Material Almacenado. [Model development for silo simulation using the "nite element method and analysis of the static pressures of the stored materials.] Tesis Doctoral. Universidad PoliteH cnica de Madrid Hardin B O; Hardin K O; Ross I J; Schwab C V (1990). Triaxial compression, simple shear, and strength of wheat en masse. Transactions of the ASAE, 33(3), 933}943 Janssen H A (1895). Versuch uK ber Getreidedruck in Sillozellen. [Experiments on grain loads in silo cells.] Zeischrift des Vesein Deutscher Ingenieure, 39, 1045}1049 Jenike A W; Johanson J R (1968). Bins loads. Journal of Structure Division, Proceeding of ASCE, 94(ST4), 1011-1041 Jofriet J C; Lelievre B; Fwa T F (1977). Friction model for "nite element analyses of silos. Transactions of the ASAE, 20(4), 735}744
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Jofriet J C; Negi S C; Lu Z (1997). Parametric study of bin loads using a hybrid numerical model. ASAE Meeting Presentation. Paper no 97-4102. Minneapolis Maeda Y; Ishizaki S (1979). Analysis of cylindrical shells for design of steel silos. Journal Civil Engineering Design, 1(4), 325}354 Mahmoud M H (1975). Silage}silo interaction using material characterization and "nite element analysis. PhD Thesis. Department of Civil Engineering, Ohio State University, Columbus, OH Manbeck H B; Nelson G L (1975). Three dimensional constitutive equations for wheat en masse. Transactions of the ASAE, 18(6), 1122}1127 Reimbert M; Reimbert A (1980). Pressures and overpressures in vertical and horizontal silos. International Conference on Design of Silos for Strength and Flow, Powder Advisory Centre, London
Rong G (1994). Discrete element modelling for #ow of particulate materials in bins PhD Thesis. University of Guelph, Canada Rotter M; Ooi J; Holst M; Zhong Z (1997). Comparative evaluation of numerical methods for predicting #ow and stress "elds in silos. EPSRC DEMFEM International Collaboration CA-SILO Collaborative Action: WG5. http://oats.civ.ed.ac.uk/research/silo/demfem/ Sugita M (1972). Flow and pressures of non-cohesive granular materials in funnel #ow bin. ASME No. 72-MH-20, pp 1}8 Zhang Q; Britton M G; Xu S (1994). Using dilatancy angle to predict dynamic loads during discharge in bulk solid storage structures. Proceeding of Powder and Bulk Solids'94, Rosemont, IL 383-390 Zhang Y (1994). Mechanical properties of soyabean and corn. MS Dissertation, University of Guelph