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On the criteria for mass flow in hoppers
Powder Technology, 73 (1992) 251-260
2.51
On the criteria for mass flow in hoppers A. Drescher” University of Glasgow, Department of Civil Engineering, Glasgow GI2 8LT (UK) (Received February 24, 1992)
Abstract The criteria which define the transition state from mass flow to funnel or plug flow in hoppers are considered. These criteria are expressed as functions of the hopper angle, granular material friction angle, and dry friction angle at the walls. The existing criteria based on exact and approximate radial stress fields are reassessed, and a new criterion for cohesionless materials is derived based on the flow pattern at the hopper outlet. Next, the dependence of the angle of internal friction on the state of stress is analyzed, and the results are implemented in the criteria for mass flow.
Introduction During the discharge of granular materials from bin/ hopper structures two basic types of flow pattern are observed [la]. In the first, all the material is in motion and the flow is called mass flow. In the second, stagnant zones develop outside the moving mass, and the flow is called funnel flow. An extreme case of the latter is plug flow, where only a central, nearly vertical zone undergoes flow. Sometimes, with the discharge progressing, one flow pattern switches to another when the upper surface of the material reaches a critical height [5]. This paper deals only with flow in hoppers where, in general, steep or smooth walls lead to mass flow, whereas less inclined or rough walls enforce either funnel or plug flow. The type of flow also depends on the geometry of the hopper; steeper walls are required for mass flow in conical hoppers [2]. A relation between the wall inclination, wall roughness, and granular material properties, which describes the transition state from mass flow to funnel or plug flow, can be termed the criterion for mass flow. The objective of this paper is frrst to reassess some of the existing criteria for mass flow, and to present a new criterion for cohesionless materials based on the flow pattern in the vicinity of the hopper outlet. Next, to investigate the dependence of the granular material friction angle on the states of stress in plane and conical hoppers. Several yield conditions available in the lit*On sabbatical leave from the University of Minnesota, Department of Civil and Mineral Engineering, 500 Pillsbury Dr. S. E., Minneapolis, MN 55455 (USA).
0032-5910/92/$5.00
erature are discussed, and the corresponding relations between the friction angle in triaxial compression, extension, and plane strain are derived. These are implemented subsequently in some of the criteria for mass flow. Emphasis is placed on assumptions, limitations, and overall predictions of the criteria rather than on experimental verification based on a specific set of tests. Criteria for mass flow based on radial stress field
The derivation of the criteria for mass flow originates from the works of Jenike [l, 2, 61 and Johanson [7]. These authors assumed that, because of the large shearing deformation developing during discharge, the granular material can be regarded as incompressible rigid-perfectly plastic, with the yield condition (yield function) containing one parameter: the effective (critical state) angle of internal friction, 6. Jenike [l, 2, 61, and Johanson [7], presented exact solutions for a particular plastic stress field in hoppers called a radial stress field. In the radial stress field, all the stress components are linear functions of the distance r measured from the hopper vertex, and can be expressed as functions of the mean stress a,, and the angle 17 between the principal stress O, and the r-direction. The mean stress is related to r by U, = ‘)‘K?-
(1)
K is a function of the hopper angular @-coordinate. With the help of the local equilibrium equations, and the Mohr-Coulomb yield condition
where y is the bulk unit weight, and
F=a,-a,-(a,+~+) s i n S=O (aI>u33.cra)
(2)
0 1992 - Elsevier Sequoia. All rights reserved
252
function and angle n can be found from a set of ordinary differential equations. These equations can be integrated numerically utilizing the boundary conditions at the symmetry axis, O= 0, and at the walls, O= O,, where the stress vector is inclined at the wall friction angle &,. It turns out that the solution to these equations can only be obtained for a limited range of 8, &,, and @,-values. Within the admissible range of these angles, it is possible to construct a steady, incompressible radial velocity field. Jenike [l, 2, 61 asserted that the corresponding stresses and velocities can be regarded as those developing in mass flow. On the other hand, beyond the limits for 6, &, and O,, flow other then mass-type should take place. The solid curves in Fig. l(a) show bounds of regions where the radial stress field solutions exist in conical hoppers, and they represent graphically the criterion for mass flow. The broken horizontal lines corresponding to plane hoppers indicate that limits to the solution result from the requirement that angle & should not be greater than S. In other words, in plane hoppers solutions exist for any angle 0, as long as & < 8. Jenike [2] suggested lower empirical limits for mass flow in plane hoppers. In a subsequent paper, Jenike [S] considered another yield condition, originally suggested by Drucker and Prager [9], K
___________________---.~._.________.-~ 30”
6, 0
30”
60”
90”
(4
0
30”
60”
%v
90”
Fig. 1. Bounding curves for mass flow based on exact radial stress fields: (a) yield condition (2)-Jenike [2, 61; (b) yield condition (3)- Jenike [S].
=0
(3)
where k, is a constant whose value can be selected to match the Mohr-Coulomb yield condition at a given stress state. In his analysis, Jenike [8] selected k, to match yield condition (2) at the state of stress corresponding to incompressible plane strain which leads, as we will show later, to k, = 5 sin26 In addition, Jenike assumed that the stresses in the radial stress field are related to the strain-increments by a law which proportionally relates the stress deviator components to the strain-increment deviator components, and the velocities result from a steady, incompressible radial flow. Again, solution for conical hoppers with non-zero velocities can only be obtained for a limited range of angles 8, &, and O,, and the bounding curves are presented in Fig. l(b). For plane hoppers, however, there is no difference in the limits of solution for yield conditions (2) and (3). Although paper [S] aimed at predicting the slope of the funnel flow zone in hoppers, the curves in Fig. l(b) can be regarded as bounds for mass flow for materials described by yield condition (3). The comparison between Figs. l(a) and l(b) indicates that condition (3) leads to a greater range of angles & and 0, for mass flow. Approximate analytic expressions for the mean stress at the walls in a radial stress field can be obtained by means of the differential slice method operating with global rather than local equilibrium equations. Various types of differential slices can be selected, and various additional assumptions can be made to relate the average stresses acting on the slice to those at the wall [4]. For example, Arnold and McLean [lo] used the slices and assumptions originally postulated by Walker [ll] and generalized by Walters [12] and Clague [13]. Other slices and assumptions such as those due to Enstad [14, 151 and Benink [5] can also be used. In all cases the mean stress at the wall can be expressed as
where A and B are coefficients related to S, 4,,,, and O,, whose explicit forms are given in the Appendix. Note that in all solutions the Mohr-Coulomb yield condition has been assumed. Extending Jenike’s idea, we can arrive at the criterion for mass flow by seeking a critical combination of S, &,, and O,, which gives
253
physically unrealistic stresses, i.e. when a,,., becomes zero, infinite, or negative (tension). Figures 2(a), 2(b), and 2(c) present the results for slices and assumptions due to Walters [12] and Clague [13], Benink [5], and Enstad [14, 151, r e s p e c t i v e l y . Following the Walters-Clague analysis, the bounding curves for conical hoppers are located slightly outside the curves for plane hoppers. Beniuk’s slices and assumptions lead to practically identical curves for plane and conical hoppers, whereas those of Enstad to horizontal lines. This implies that mass flow should occur at any angle O,, and the limits are due to the requirement that &,
mw
0
30"
60"
%z
900
Fig. 3. Comparison of bounding curves for mass flow in conical hoppers, 6 = SO”.
means higher values of &, or 0, beyond which mass flow should no longer occur. Furthermore, the Walters-Clague and Benink solutions give bounds for plane hoppers that depend on angle 0, which is not the case for the Jenike and Enstad solutions. These bounds, however, are inside or coincide with those for conical hoppers and this is not supported experimentally.
Criterion for mass flow based on the flow pattern at the hopper outlet
0
1%
30"
60
91 I”
30"
60"
%I 90"
(4 60"
0
hu
Fig. 2. Bounding curves for mass flow based on approximate radial stress fields and yield condition (2): (a) Walters [12] and Clague [13]; (b) Benink [S]; (c) Enstad [14, 151.
In the criteria for mass flow based on the exact radial stress field, the velocities are assumed to be continuous functions of their location. No account is taken of the velocities in criteria derived from approximate radial stress fields. Experiments indicate that, even in mass flow, the velocities undergo rapid changes across zones of dilation, termed rupture surfaces or shear bands [16-191. Figure 4 shows radiographs which illustrate the formation of dilated zones in a plane hopper during discharge of sand; the apparent parallel upper sides in the radiographs merely indicate the width of the Xray film. In Fig. 4(a) the hopper walls are smooth, and mass flow occurs. With progressing discharge, the shear bands propagate upwards, reflect from the walls, cease to dilate, and move towards the outlet as zones of looser material. In Fig. 4(b) the walls are rough, and plug flow develops. The boundary between the flowing and stationary regions moves laterally, and eventually disappears with progressing discharge. The mechanism of shear band formation shown in Fig. 4 is complicated, and was analyzed in detail by Michalowski [20-221. Here, we concentrate on the orientation of shear bands shortly after the onset of flow. It is evident that in mass flow the shear bands are not vertical but oriented towards the opposite wall. In contrast, in plug flow, the orientation of one of the shear bands next to the walls is vertical. This observation suggests that the initial type of flow in cohesionless materials can be related to the orientation of shear
(4
(b)
dun
Fig. 5. (a) Stress circle representing the stress state at the wall; (b) incremental displacements across shear band.
* = arcsin
@I Fig. 4. Radiographs of mass and plug flow in a hopper.
bands in the vicinity of the outlet: mass flow terminates when one of the shear bands becomes vertical. The orientation of shear bands can be determined with the help of the Mohr stress circle representing the passive plastic state of stress at the wall (Fig. 5(a)). Point A corresponds to the end-point of the stress vector OA inclined at angle & to the wall normal, and point P represents the location of the pole of planes. The shear bands are parallel to the straight lines joining pole P and points B and C, which represent the end-points of the stress vectors acting at shear bands. The location of points B and C depends on the ratio of the normal, du,, and tangential, du,, components of the displacement-increment vector du across the shear band (Fig. 5(b)). This ratio defines the dilatancy angle + du I)= arctan n
du,
which, in terms of the principal strain-increment components, dei (i= 1, 2, 3), can be written as [23]
- (de, + dc) de, -de2
(7)
For the Mohr-Coulomb yield condition (2), with 6 replaced by the instantaneous (peak state) friction angle 4, the dilatancy angle may vary from $=O for incompressible flow to +I= 4 for a fully dilatant flow. If the principal axes of stress and strain-increments coincide, as is often postulated, the location of point B varies from B, for $=O, to B, for $= 4; a similar, reflected range of locations applies to point C. From Fig. 5(a) it then follows that the shear bands parallel to PB and PC are inclined to the vertical at angles
(8) The shear band parallel to PB becomes vertical when %B=O, i.e. when tan &=
sin 4 cos(2f3,- $) 1 + sin d, sin(26, - I+%)
and eqn. (9) is the mathematical form of the resulting criterion for mass flow. Figures 6(a) and 6(b) show the bounding curves for += 4 and q?= 213 4. Note that the curve for $= 4=30” is identical with the curve for S = 30” obtained by Jenike [2, 61 for the yield condition (2). The criterion for mass flow (9) has been derived on the basis of results of model tests on plane hoppers. Some evidence exists [2, 5, 241 that dilation zones also form in cylindrical and conical vessels, where the stress
255
60”
4..
0
30”
60”
6w 900
(4 60”,
$w
triaxial compression (a1 > a, = a3) or triaxial extension (a1 = a,> aZ), is sufficient to determine 6, or 4, and its value should be the same. Experiments performed with general stress states indicate, however, that neither S nor 4 is constant, and yield conditions that incorporate the effect of the intermediate principal stress have been proposed in the literature. In the following analysis of the various yield conditions we will use only one symbol S for friction angle, which can be replaced by 4 if the instantaneous angle of internal friction for cohesionless materials is being considered. A yield condition that is a formal extension of the von Mises yield condition for metals on granular media, rather than being derived from experimental data, is the yield condition (3) considered by Jenike [8]. Based on the results of tests performed on Monterey No. 0 sand in truly triaxial states of stress (a, # a, it: s), Lade and Duncan [25] suggested the following yield condition F = (a, + a, + u3)3 - k,ulu2u3 = 0
0
30”
6 0
c
Fig. 6. Bounding curves for mass flow based on the flow pattern at the outlet: (a) $=4; (b) JI=2+/3.
state at the wall may differ from plane strain. This will not affect the derivation of criterion (9) and, when plotted in Fig. 6, will give identical bounds if the same value for the instantaneous friction angle is selected. The curve for $= 4=50” is drawn in Fig. 3 as the broken line. As with the criteria based on eqn. (6), criterion (9) does not account for the experimentally observed differences in limits for 4,,, and 0, in plane and conical hoppers. Influence of the yield condition
As demonstrated for conical hoppers by Jenike [2, 6, 81, the form of the yield condition has a profound effect on the criterion for mass flow identified by means of bounds to the exact radial stress field solutions. It may be expected that it also affects the criteria based on eqn. (5) and criterion (9) and, in particular, is responsible for the location of bounds for plane hoppers at or below those for conical hoppers. In the following, we will discuss several yield conditions and their relation to the Mohr-Coulomb one. The Mohr-Coulomb yield condition was originally derived from failure tests on sand in plane strain, with results being described as a relation between the shear and normal stress on a failure plane. The form (2) is a generalization to an arbitrary state of stress. Since yield condition (2) is independent of the intermediate principal stress, any type of test, e.g. plane strain (Ed = 0),
(10)
The concept of a shear plane in a general stress state led Matsuoka [26] to a yield condition of the form F=(ul+u,+u3)(ulu2+u,u3+ulu3)-k3ulu,u3=0 (11)
whose adequacy in describing test results on Toyoura sand was shown by Matsuoka and Nakai [27]. In the expressions above, k2 and k, are constants similar to k,, and can be related to the friction angle S, or 4, in condition (2) for a given state of stress. To compare conditions (2), (3), (lo), and (ll), it is convenient to make use of their geometrical representations as surfaces in the space of the principal stresses a,, u2, and a,. The Mohr-Coulomb yield condition is represented by a hexagonal pyramid, the Drucker-Prager by a circular cone, and the remaining two conditions by the non-circular cones. Figure 7(a) shows the comparisons of their cross-sections in a constant mean stress plane (r-plane), drawn so that all the traces coincide at the state of triaxial compression (points A’, A”, and A”‘). Then,
(3-sin S)3 k2 = (1 - sin S)‘( 1 + sin S)
(12)
k,=9+8 tan*S
For triaxial extension (points B’, B”, and B”‘) the expressions for kl and k2 are different, and the expression for k3 remains unchanged. It is evident from Fig. 7(a) that all the traces possess at least a six-fold symmetry. The traces are drawn in Fig. 7 for S= 45” in triaxial compression.
256
Application to criteria for mass flow
Until the exact radial stress solutions for the yield conditions (10) and (11) become available, it is not possible to assess conclusively their effect on the resulting criteria for mass flow. It may be expected, however, that the corresponding bounding curves for conical hoppers would be located close to those for yield condition (2), and the curves for plane hoppers would not change. This conjecture is derived from the fact that the changes in angle 6 for yield conditions (10) and (11) are small with respect to a=const for yield condition (2), see Figs. 8(b) and 8(c). Consider now criteria for mass flow based on eqn. (5) and criterion (9). Both types of criteria are ap-
I
0 trmxial compression
I
I
,*.
30"
R
60 triaxial extension
(4
Fig. 7. (a) Yield conditions and strain-increment vectors in the r-plane after [25-271; (b) location of a plane strain point.
For every point of a trace we may determine the angle of friction S which, for a, 2 a, > a,, is defined as 6 = arcsin z 1
2
(13)
The angle of friction S= const o n l y f o r t h e Mohr-Coulomb yield condition. This can be shown directly from (2). All other conditions give values of 6 that vary with the location of the point. This is shown in Figs. 8(a), 8(b), and 8(c), where angle 6 is plotted (solid lines) as a function of angle R measured as in Fig. 7(a). The Drucker-Prager yield condition predicts angle S in triaxial extension signiticantly greater than in triaxial compression. Furthermore, angle 6 may reach the unrealistically high value of 90” for some values of CR, if in triaxial compression 6 2 arcsin(3/5) = 37”. These features of yield condition (3) severely hamper its application for describing plastic flow of real granular materials. The remaining two yield conditions predict much smaller variations of 6 with IR. The Lade-Duncan yield condition predicts a slightly greater friction angle in triaxial compression than in extension, whereas the Matsuoka yield condition gives identical values.
*. .: 0 iriaxial compression
30"
I
R
60"
triaxial extension
(4 Fig. 8. Variation of 6 with R: (a) yield condition (3); (b) yield condition (10); (c) yield condition (11).
257
proximate in that only stresses at the wall contribute to the derivation. In plane hoppers, it is reasonable to assume that the stresses correspond to those for plane strain. Then, the angle of internal friction should be taken as S,, or &, where the subscript ‘p’ indicates plane strain. The type of stress state that exists in conical hoppers is more difficult to assess; however, an admissible engineering approximation would be an axisymmetric passive state, and the friction angle can then be taken as that at triaxial extension, S,, or &, where the subscript ‘e’ stands for extension. To determine the friction angle S,, or &,, for yield conditions (3), (lo), and (ll), it is necessary to make use of the relationship between the stresses and strainincrements during plastic flow. In terms of the principal stresses aj and strain-increments &, this relationship is usually expressed [9, 231 as dei=dh g I
(14)
where dh is a positive multiplier, and G = 0 is a function of 0;:. This function can be represented in the principal stress space as a surface to which the strain-increments are orthogonal when plotted as vectors. Knowing the function G = 0, calculating de3 from eqn. (14), setting de3 =0, utilizing the function F=O, and selecting an arbitrary mean stress, we may determine the state of stress and, from eqn. (13), the friction angle in plane strain. The procedure above requires the knowledge of the function G = 0, whose form can be deduced from tests at various stress states by plotting the strain-increments as vectors in stress space. The experiments described by Lade and Duncan [25], and by Matsuoka and Nakai [27], show that G #F, however, the shape of G =0 is such that in the r-plane the strain-increment vectors are nearly orthogonal to the trace of the appropriate yield condition (Fig. 7(a)). With this assumption, it is then possible to apply another procedure which only makes use of the dilatancy angle for plane strain defined by eqns. (6) and (7). For the Mohr-Coulomb yield condition (2) and G = F, the dilatancy angle determined from eqns. (7) and (14) is *=S
smooth trace on the r-plane and, hence, also to yield conditions (3), (lo), and (11). By means of this procedure, the results of calculations can be represented in Figs. 8(a), 8(b), and 8(c) as the broken lines at which +=const. The intersection point of a broken line and a solid line gives the value of the angle S,, or &,, and R, for a given I,+. For the Drucker-Prager yield condition the points for plane strain and 0 Q $< S, are located at fi> 30”, and for yield conditions (10) and (11) at R Q 30”. In particular, for the Drucker-Prager yield condition and $=O as assumed by Jenike [8], angle CR = 30” where a, = (a1 + &/ 2. This information allows to determine such a value of k, in eqn. (3) that the Drucker-Prager yield condition matches the Mohr-Coulomb yield condition in plane strain, i.e. the traces of both yield conditions in the rrplane intersect at R = 30”. The resulting expression for k, is given by eqn. (4). With the help of Figs. (8a), 8(b), and 8(c), it is possible to construct diagrams which relate friction angles in triaxial compression, S,, and extension, S,, to S, (or & and +e to &), for different values of the dilatancy angle I(I . This is shown in Figs. 9(a), 9(b), and 9(c) for $= 0 and $= S,. For small values of $, the yield conditions (3) and (10) predict S, > S,, with their difference decreasing with increasing I,% at higher values of $, yield condition (10) gives S,< S,. On the other hand, S, < S, and their difference increases with $. The yield condition (11) predicts S, = S, < 6, which decreases with increasing I/J. Except for the Drucker-Prager yield condition, which predicts an unrealistically large difference between friction angles in triaxial compression and extension, the remaining two conditions account for lower friction angles in triaxial compression and extension than in plane strain. For S,, or $J,,, in the range of 30-50”, the difference is about 3-5”. Since the angles S, and & are smaller than S, and +p, respectively, the bounding curves for conical hoppers in Fig. 2, and also in Fig. 6, will shift inside those for plane hoppers, and their locations will correspond, at least qualitatively, with experimental results. This is demonstrated for criterion (9) in Fig. 10, where #+, and & were located according to yield condition (11) and *= 4,.
(15)
This means that a point on the r-plane trace of a yield condition other than (2), which is common with a circumscribed Mohr-Coulomb hexagon with S = S’, will correspond to plane strain with $= S’ (Fig. 7(b)). Thus, selecting a value of I+%, constructing the circumscribed Mohr-Coulomb hexagon with S’=JI, we can find the location of the point corresponding to plane strain. This procedure applies to any yield condition with a
Closing remarks In the analysis presented above, no account was taken of a possible change from mass flow to other types of flow with progressing discharge and lowering of the upper surface of the material in hoppers. This effect
A ,’
30” -
,,
,,
,’
,,
,_-’
,’ ,/
,’
,,’
,‘,’
,,’ *+
~=O~.._.---
__--
___---
‘v=sp
,_*--
triadal compresson
,
triaxial extension 60”
triaxial compression
0
30”
60”
6, 9 0
30”
60”
SP 900
(b) go0
SC?
se
60” -
0
(4 Fig. 9. Relation between SC, S,, and Sp for $=O and $=Sp: (a) yield condition (3); (b) yield condition (10); (c) yield condition (11).
was considered in [19], and a criterion for mass flow incorporating the height of the material in plane hoppers was suggested there. Also, the presence of a vertical bin section in a bin/hopper structure was disregarded in the analysis. Experiments by Benink [5] show that, with the lowering of the material in the cylindrical bin section, initial mass-type flow may turn into funneltype. Utilizing the method of differential slices, Benink
0
30”
60”
S
Fig. 10. Modified bounding curves for mass flow criterion (9) and yield condition (11).
proposed a criterion for this mixed-type flow, which gives bounds that are located below those shown in Fig. 2(b). It is striking that realistic bounds on mass flow derived from the exact radial stress fields are only obtained for conical hoppers [Z, 6,8]. This limitation is alleviated in some criteria that are based on approximate radial stress fields, and in criterion (9), though bounds for plane hoppers and constant friction angle are incorrectly located if yield condition (2) is assumed; contrary to predictions, experiments show greater hopper angles for mass flow in plane hoppers than in conical ones. It should be remembered, however, that the method of differential slices suffers from inherent freedom in the selection of the geometry of the slices and additional assumptions [4]. This causes widely scattered bounds for mass flow for different slices, on the one hand, and allows for adjusting the slices and assumptions to match experimental results in a selected type of hoppers, on the other. The criteria for mass flow derived from the radial stress field assume large shearing deformation of the material, and apply to both cohesionless and cohesive materials characterized by the effective friction angle S. Criterion (9) was derived for cohesionless materials, and makes use of the instantaneous friction angle 4 and the dilation angle @, in most cohesionless materials, angle 4 is slightly greater than or equal to S. In spite of these limitations, the bounds for mass flow are surprisingly close to those for some other criteria if 4 is replaced by 6. In particular, identical bounds are obtained for criterion (9) and for Jenike’s [2,6] criterion, if 4=S=30”. The dilatancy angle $ is readily available from the direct shear test if the increments du, and du, in eqn. (6) are measured in the shear cell. The thus determined angle + is usually significantly smaller than &. It should be stressed, however, that the normal stresses at the outlet level may be lower than those induced in the direct shear test. Furthermore, the mechanism of shear band formation in hoppers, and the accompanying
259
dilation, may differ from that in the direct shear test, where strong geometric constraints on the deformation field are enforced by the rigid walls and the cap of the shear cell. The material near the hopper outlet may dilate more, as the opening of the gate allows the material to deform freely. Accordingly, the dilation angle may be close to the instantaneous friction angle, though no simple experimental technique has been developed to validate this statement. The analysis of the influence of stress state on friction angles of granular materials indicates the way in which corrections can be made to some mass flow criteria. Similar corrections can be made in predicting loads in storage structures. Besides yield conditions (10) and (ll), other approximations to experimental data have been proposed in the literature [28, 291. Even though they all deviate from the Mohr-Coulomb yield condition they are far away from yield condition (3). The significance of properly assessing the magnitude of the friction angle in various stress states has already penetrated geotechnics, where the difference of about 3-5” between soil friction angles in plane strain and axisymmetric states has been accepted [30]. However, when dealing with plastic flow phenomena, it is the displacement-increment (velocity) field that plays a more important role than the stresses alone. This is particularly important in non-steady flow, which often develops during discharge of granular materials from storage structures. The mass flow criterion (9) presented here is merely the outcome of one possible approach, and should by no means be regarded as the ultimate solution. A more sophisticated approach has recently been investigated by Tejchman and Gudehus [31].
5 E. J. Benink, Ph.D. Thesic, Univ. of Twente, 1989. 6 A. W. Jenike, J. Appl. Mech., Trans. ASME, 31 (i964) 5.
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
J. R. Johanson, J. Appl. Mech., Trans. ASME, 31 (1964) 499. A. W. Jenike, Powder Technol., 50 (1987) 229. D. C. Drucker and W. Prager, Q. Appl. Math., 10 (1952) 157. P. C. Arnold and A. G. McLean, Powder Technol., 13 (1976) 255. D. M. Walker, Chem. Eng. Sci., 21 (1966) 975. J. K. Walters, Chem. Eng. Sci, 28 (1973) 779. K. Clague, Ph.D. Thesis, Univ. of Nottingham, 1973. G. Enstad, Chem. Eng. Sci., 30 (1975) 1273. G. Enstad, Chem. Eng. Sci, 32 (1977) 339. W. G. Pariseau, Powder Technol., 3 (1970) 218. P. M. Blair-Fish and P. L. Bransby, J. Eng. Ind., Trans. ASME, Ser. B, 95 (1973) 17. J. Lee, S. C. Cowin and J. S. Templeton III, Trans. Sot. Bheol., 18 (1974) 247. A. Drescher, T. W. Cousens and P. L. Bransby, Geotechnique, 28 (1978) 27.. R. L. Michalowski, Powder Technol., 39 (1984) 29. R. L. Michalowski, Chem. Eng. Sci., 42 (1987) 2587. R. L. Michalowski, Geotechnique, 40 (1990) 389. D. Muir Wood, Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press, 1990. J. S. Giunta, J. Eng. Ind., Trans. ASME, Ser. B, 91 (1969) 406. P. Lade and J. M. Duncan, J. Geotech. Eng. Div., ASCE, IO1 (1975) 1037. H. Matsuoka, Soils Found., 24 (1974) 47. H. Matsuoka and T. Nakai, in P. Venneer and J. Luger (eds.), Proc. IUTAM Symposium on Deformation and Failure and Granular Materials, Delft, 1982, p. 253. G. Gudehus, Zng. Arch., 42 (1973) 45. 0. C. Zienkiewicz and G. N. Pande, in G. Gudehus (ed.), Finite Elements in Geomechanics, 1977, p. 171. C. P. Wroth, Geotechnique, 34 (1984) 449. J. Tejchman and G. Gudehus, in J. Eibel (ed.), Silos-Forschung und Praxis, Tagung ‘88 in Karlsruhe, 1988, p. 211.
Appendix Acknowledgments
The author gratefully acknowledges financial support provided by the Carnegie Trust and the Royal Society during his stay at the University of Glasgow. Special thanks for discussions and review of the paper are directed to Cormack Professor David Muir Wood. Discussions with Professor J. Michael Rotter of the University of Edinburgh also are appreciated.
References 1 A. W. Jenike, Utah Univ. Eng. E&n. St., BulL, 108 (1961). 2 A. W. Jenike, Utah Univ. Eng. Exp. Stat., Bull., 1.23 (1964). 3 P. C. Arnold, A. G. McLean and A. W. Roberts, Bulk Solids: Storage, Flow and Handling, TUNRA, 1982. 4 A. Drescher,Analyncaf Methods in Bin-LoadAnalysis, Elsevier, Amsterdam, 1991.
Coefficients A and B in eqn. (5) Walters [12] and Clague [13] D A = 1 -sin 6 cos(28, + 2p)
B=(l+m) $& + D - 1 1 ( sin S sin[2( 0, + p)] ’ = 1 - sin 6 cos[2( 0, + p)] D=
cos T( 1 + sin’@ - 2Jsin26 - sin277 cos q( 1 + sin26 + 2E sin S)
sin S sin[2( 0, + $I)] ’ =arctan 1 + sin S c0s[2(ew + p)]
260
E=
($ )m[l-(l-F)1.5)]m l- -m xi[C )I FF+
arcsin YF
@
2
Enstad [14, 151 sin p sin’+“@ + &J A= (l-sin 8) sin’+“(p+&,) + 2”[ 1 - cm@ + ew)yyp+ 0,)’ --m sin e, (l-sin 8) sin’+“(p+e,)
Benink [5]
1
A= l+C tan 0, tan(&+p) l-sin S
B= (l+m) sin 6 l+ sin(2p+e,) l-sin 6 [ sin e,
B= 2(1+m) sin S cos p sin(&+p) (l-sin 6) sin 0,
In the equations above m = 0 for plane hoppers, m = 1 for conical hoppers, and angle p is given by
c=
2+m
1+n+m
1 sin & p= - 4,+arcsin2 ( sin 6 )
2.51
On the criteria for mass flow in hoppers A. Drescher” University of Glasgow, Department of Civil Engineering, Glasgow GI2 8LT (UK) (Received February 24, 1992)
Abstract The criteria which define the transition state from mass flow to funnel or plug flow in hoppers are considered. These criteria are expressed as functions of the hopper angle, granular material friction angle, and dry friction angle at the walls. The existing criteria based on exact and approximate radial stress fields are reassessed, and a new criterion for cohesionless materials is derived based on the flow pattern at the hopper outlet. Next, the dependence of the angle of internal friction on the state of stress is analyzed, and the results are implemented in the criteria for mass flow.
Introduction During the discharge of granular materials from bin/ hopper structures two basic types of flow pattern are observed [la]. In the first, all the material is in motion and the flow is called mass flow. In the second, stagnant zones develop outside the moving mass, and the flow is called funnel flow. An extreme case of the latter is plug flow, where only a central, nearly vertical zone undergoes flow. Sometimes, with the discharge progressing, one flow pattern switches to another when the upper surface of the material reaches a critical height [5]. This paper deals only with flow in hoppers where, in general, steep or smooth walls lead to mass flow, whereas less inclined or rough walls enforce either funnel or plug flow. The type of flow also depends on the geometry of the hopper; steeper walls are required for mass flow in conical hoppers [2]. A relation between the wall inclination, wall roughness, and granular material properties, which describes the transition state from mass flow to funnel or plug flow, can be termed the criterion for mass flow. The objective of this paper is frrst to reassess some of the existing criteria for mass flow, and to present a new criterion for cohesionless materials based on the flow pattern in the vicinity of the hopper outlet. Next, to investigate the dependence of the granular material friction angle on the states of stress in plane and conical hoppers. Several yield conditions available in the lit*On sabbatical leave from the University of Minnesota, Department of Civil and Mineral Engineering, 500 Pillsbury Dr. S. E., Minneapolis, MN 55455 (USA).
0032-5910/92/$5.00
erature are discussed, and the corresponding relations between the friction angle in triaxial compression, extension, and plane strain are derived. These are implemented subsequently in some of the criteria for mass flow. Emphasis is placed on assumptions, limitations, and overall predictions of the criteria rather than on experimental verification based on a specific set of tests. Criteria for mass flow based on radial stress field
The derivation of the criteria for mass flow originates from the works of Jenike [l, 2, 61 and Johanson [7]. These authors assumed that, because of the large shearing deformation developing during discharge, the granular material can be regarded as incompressible rigid-perfectly plastic, with the yield condition (yield function) containing one parameter: the effective (critical state) angle of internal friction, 6. Jenike [l, 2, 61, and Johanson [7], presented exact solutions for a particular plastic stress field in hoppers called a radial stress field. In the radial stress field, all the stress components are linear functions of the distance r measured from the hopper vertex, and can be expressed as functions of the mean stress a,, and the angle 17 between the principal stress O, and the r-direction. The mean stress is related to r by U, = ‘)‘K?-
(1)
K is a function of the hopper angular @-coordinate. With the help of the local equilibrium equations, and the Mohr-Coulomb yield condition
where y is the bulk unit weight, and
F=a,-a,-(a,+~+) s i n S=O (aI>u33.cra)
(2)
0 1992 - Elsevier Sequoia. All rights reserved
252
function and angle n can be found from a set of ordinary differential equations. These equations can be integrated numerically utilizing the boundary conditions at the symmetry axis, O= 0, and at the walls, O= O,, where the stress vector is inclined at the wall friction angle &,. It turns out that the solution to these equations can only be obtained for a limited range of 8, &,, and @,-values. Within the admissible range of these angles, it is possible to construct a steady, incompressible radial velocity field. Jenike [l, 2, 61 asserted that the corresponding stresses and velocities can be regarded as those developing in mass flow. On the other hand, beyond the limits for 6, &, and O,, flow other then mass-type should take place. The solid curves in Fig. l(a) show bounds of regions where the radial stress field solutions exist in conical hoppers, and they represent graphically the criterion for mass flow. The broken horizontal lines corresponding to plane hoppers indicate that limits to the solution result from the requirement that angle & should not be greater than S. In other words, in plane hoppers solutions exist for any angle 0, as long as & < 8. Jenike [2] suggested lower empirical limits for mass flow in plane hoppers. In a subsequent paper, Jenike [S] considered another yield condition, originally suggested by Drucker and Prager [9], K
___________________---.~._.________.-~ 30”
6, 0
30”
60”
90”
(4
0
30”
60”
%v
90”
Fig. 1. Bounding curves for mass flow based on exact radial stress fields: (a) yield condition (2)-Jenike [2, 61; (b) yield condition (3)- Jenike [S].
=0
(3)
where k, is a constant whose value can be selected to match the Mohr-Coulomb yield condition at a given stress state. In his analysis, Jenike [8] selected k, to match yield condition (2) at the state of stress corresponding to incompressible plane strain which leads, as we will show later, to k, = 5 sin26 In addition, Jenike assumed that the stresses in the radial stress field are related to the strain-increments by a law which proportionally relates the stress deviator components to the strain-increment deviator components, and the velocities result from a steady, incompressible radial flow. Again, solution for conical hoppers with non-zero velocities can only be obtained for a limited range of angles 8, &, and O,, and the bounding curves are presented in Fig. l(b). For plane hoppers, however, there is no difference in the limits of solution for yield conditions (2) and (3). Although paper [S] aimed at predicting the slope of the funnel flow zone in hoppers, the curves in Fig. l(b) can be regarded as bounds for mass flow for materials described by yield condition (3). The comparison between Figs. l(a) and l(b) indicates that condition (3) leads to a greater range of angles & and 0, for mass flow. Approximate analytic expressions for the mean stress at the walls in a radial stress field can be obtained by means of the differential slice method operating with global rather than local equilibrium equations. Various types of differential slices can be selected, and various additional assumptions can be made to relate the average stresses acting on the slice to those at the wall [4]. For example, Arnold and McLean [lo] used the slices and assumptions originally postulated by Walker [ll] and generalized by Walters [12] and Clague [13]. Other slices and assumptions such as those due to Enstad [14, 151 and Benink [5] can also be used. In all cases the mean stress at the wall can be expressed as
where A and B are coefficients related to S, 4,,,, and O,, whose explicit forms are given in the Appendix. Note that in all solutions the Mohr-Coulomb yield condition has been assumed. Extending Jenike’s idea, we can arrive at the criterion for mass flow by seeking a critical combination of S, &,, and O,, which gives
253
physically unrealistic stresses, i.e. when a,,., becomes zero, infinite, or negative (tension). Figures 2(a), 2(b), and 2(c) present the results for slices and assumptions due to Walters [12] and Clague [13], Benink [5], and Enstad [14, 151, r e s p e c t i v e l y . Following the Walters-Clague analysis, the bounding curves for conical hoppers are located slightly outside the curves for plane hoppers. Beniuk’s slices and assumptions lead to practically identical curves for plane and conical hoppers, whereas those of Enstad to horizontal lines. This implies that mass flow should occur at any angle O,, and the limits are due to the requirement that &,
mw
0
30"
60"
%z
900
Fig. 3. Comparison of bounding curves for mass flow in conical hoppers, 6 = SO”.
means higher values of &, or 0, beyond which mass flow should no longer occur. Furthermore, the Walters-Clague and Benink solutions give bounds for plane hoppers that depend on angle 0, which is not the case for the Jenike and Enstad solutions. These bounds, however, are inside or coincide with those for conical hoppers and this is not supported experimentally.
Criterion for mass flow based on the flow pattern at the hopper outlet
0
1%
30"
60
91 I”
30"
60"
%I 90"
(4 60"
0
hu
Fig. 2. Bounding curves for mass flow based on approximate radial stress fields and yield condition (2): (a) Walters [12] and Clague [13]; (b) Benink [S]; (c) Enstad [14, 151.
In the criteria for mass flow based on the exact radial stress field, the velocities are assumed to be continuous functions of their location. No account is taken of the velocities in criteria derived from approximate radial stress fields. Experiments indicate that, even in mass flow, the velocities undergo rapid changes across zones of dilation, termed rupture surfaces or shear bands [16-191. Figure 4 shows radiographs which illustrate the formation of dilated zones in a plane hopper during discharge of sand; the apparent parallel upper sides in the radiographs merely indicate the width of the Xray film. In Fig. 4(a) the hopper walls are smooth, and mass flow occurs. With progressing discharge, the shear bands propagate upwards, reflect from the walls, cease to dilate, and move towards the outlet as zones of looser material. In Fig. 4(b) the walls are rough, and plug flow develops. The boundary between the flowing and stationary regions moves laterally, and eventually disappears with progressing discharge. The mechanism of shear band formation shown in Fig. 4 is complicated, and was analyzed in detail by Michalowski [20-221. Here, we concentrate on the orientation of shear bands shortly after the onset of flow. It is evident that in mass flow the shear bands are not vertical but oriented towards the opposite wall. In contrast, in plug flow, the orientation of one of the shear bands next to the walls is vertical. This observation suggests that the initial type of flow in cohesionless materials can be related to the orientation of shear
(4
(b)
dun
Fig. 5. (a) Stress circle representing the stress state at the wall; (b) incremental displacements across shear band.
* = arcsin
@I Fig. 4. Radiographs of mass and plug flow in a hopper.
bands in the vicinity of the outlet: mass flow terminates when one of the shear bands becomes vertical. The orientation of shear bands can be determined with the help of the Mohr stress circle representing the passive plastic state of stress at the wall (Fig. 5(a)). Point A corresponds to the end-point of the stress vector OA inclined at angle & to the wall normal, and point P represents the location of the pole of planes. The shear bands are parallel to the straight lines joining pole P and points B and C, which represent the end-points of the stress vectors acting at shear bands. The location of points B and C depends on the ratio of the normal, du,, and tangential, du,, components of the displacement-increment vector du across the shear band (Fig. 5(b)). This ratio defines the dilatancy angle + du I)= arctan n
du,
which, in terms of the principal strain-increment components, dei (i= 1, 2, 3), can be written as [23]
- (de, + dc) de, -de2
(7)
For the Mohr-Coulomb yield condition (2), with 6 replaced by the instantaneous (peak state) friction angle 4, the dilatancy angle may vary from $=O for incompressible flow to +I= 4 for a fully dilatant flow. If the principal axes of stress and strain-increments coincide, as is often postulated, the location of point B varies from B, for $=O, to B, for $= 4; a similar, reflected range of locations applies to point C. From Fig. 5(a) it then follows that the shear bands parallel to PB and PC are inclined to the vertical at angles
(8) The shear band parallel to PB becomes vertical when %B=O, i.e. when tan &=
sin 4 cos(2f3,- $) 1 + sin d, sin(26, - I+%)
and eqn. (9) is the mathematical form of the resulting criterion for mass flow. Figures 6(a) and 6(b) show the bounding curves for += 4 and q?= 213 4. Note that the curve for $= 4=30” is identical with the curve for S = 30” obtained by Jenike [2, 61 for the yield condition (2). The criterion for mass flow (9) has been derived on the basis of results of model tests on plane hoppers. Some evidence exists [2, 5, 241 that dilation zones also form in cylindrical and conical vessels, where the stress
255
60”
4..
0
30”
60”
6w 900
(4 60”,
$w
triaxial compression (a1 > a, = a3) or triaxial extension (a1 = a,> aZ), is sufficient to determine 6, or 4, and its value should be the same. Experiments performed with general stress states indicate, however, that neither S nor 4 is constant, and yield conditions that incorporate the effect of the intermediate principal stress have been proposed in the literature. In the following analysis of the various yield conditions we will use only one symbol S for friction angle, which can be replaced by 4 if the instantaneous angle of internal friction for cohesionless materials is being considered. A yield condition that is a formal extension of the von Mises yield condition for metals on granular media, rather than being derived from experimental data, is the yield condition (3) considered by Jenike [8]. Based on the results of tests performed on Monterey No. 0 sand in truly triaxial states of stress (a, # a, it: s), Lade and Duncan [25] suggested the following yield condition F = (a, + a, + u3)3 - k,ulu2u3 = 0
0
30”
6 0
c
Fig. 6. Bounding curves for mass flow based on the flow pattern at the outlet: (a) $=4; (b) JI=2+/3.
state at the wall may differ from plane strain. This will not affect the derivation of criterion (9) and, when plotted in Fig. 6, will give identical bounds if the same value for the instantaneous friction angle is selected. The curve for $= 4=50” is drawn in Fig. 3 as the broken line. As with the criteria based on eqn. (6), criterion (9) does not account for the experimentally observed differences in limits for 4,,, and 0, in plane and conical hoppers. Influence of the yield condition
As demonstrated for conical hoppers by Jenike [2, 6, 81, the form of the yield condition has a profound effect on the criterion for mass flow identified by means of bounds to the exact radial stress field solutions. It may be expected that it also affects the criteria based on eqn. (5) and criterion (9) and, in particular, is responsible for the location of bounds for plane hoppers at or below those for conical hoppers. In the following, we will discuss several yield conditions and their relation to the Mohr-Coulomb one. The Mohr-Coulomb yield condition was originally derived from failure tests on sand in plane strain, with results being described as a relation between the shear and normal stress on a failure plane. The form (2) is a generalization to an arbitrary state of stress. Since yield condition (2) is independent of the intermediate principal stress, any type of test, e.g. plane strain (Ed = 0),
(10)
The concept of a shear plane in a general stress state led Matsuoka [26] to a yield condition of the form F=(ul+u,+u3)(ulu2+u,u3+ulu3)-k3ulu,u3=0 (11)
whose adequacy in describing test results on Toyoura sand was shown by Matsuoka and Nakai [27]. In the expressions above, k2 and k, are constants similar to k,, and can be related to the friction angle S, or 4, in condition (2) for a given state of stress. To compare conditions (2), (3), (lo), and (ll), it is convenient to make use of their geometrical representations as surfaces in the space of the principal stresses a,, u2, and a,. The Mohr-Coulomb yield condition is represented by a hexagonal pyramid, the Drucker-Prager by a circular cone, and the remaining two conditions by the non-circular cones. Figure 7(a) shows the comparisons of their cross-sections in a constant mean stress plane (r-plane), drawn so that all the traces coincide at the state of triaxial compression (points A’, A”, and A”‘). Then,
(3-sin S)3 k2 = (1 - sin S)‘( 1 + sin S)
(12)
k,=9+8 tan*S
For triaxial extension (points B’, B”, and B”‘) the expressions for kl and k2 are different, and the expression for k3 remains unchanged. It is evident from Fig. 7(a) that all the traces possess at least a six-fold symmetry. The traces are drawn in Fig. 7 for S= 45” in triaxial compression.
256
Application to criteria for mass flow
Until the exact radial stress solutions for the yield conditions (10) and (11) become available, it is not possible to assess conclusively their effect on the resulting criteria for mass flow. It may be expected, however, that the corresponding bounding curves for conical hoppers would be located close to those for yield condition (2), and the curves for plane hoppers would not change. This conjecture is derived from the fact that the changes in angle 6 for yield conditions (10) and (11) are small with respect to a=const for yield condition (2), see Figs. 8(b) and 8(c). Consider now criteria for mass flow based on eqn. (5) and criterion (9). Both types of criteria are ap-
I
0 trmxial compression
I
I
,*.
30"
R
60 triaxial extension
(4
Fig. 7. (a) Yield conditions and strain-increment vectors in the r-plane after [25-271; (b) location of a plane strain point.
For every point of a trace we may determine the angle of friction S which, for a, 2 a, > a,, is defined as 6 = arcsin z 1
2
(13)
The angle of friction S= const o n l y f o r t h e Mohr-Coulomb yield condition. This can be shown directly from (2). All other conditions give values of 6 that vary with the location of the point. This is shown in Figs. 8(a), 8(b), and 8(c), where angle 6 is plotted (solid lines) as a function of angle R measured as in Fig. 7(a). The Drucker-Prager yield condition predicts angle S in triaxial extension signiticantly greater than in triaxial compression. Furthermore, angle 6 may reach the unrealistically high value of 90” for some values of CR, if in triaxial compression 6 2 arcsin(3/5) = 37”. These features of yield condition (3) severely hamper its application for describing plastic flow of real granular materials. The remaining two yield conditions predict much smaller variations of 6 with IR. The Lade-Duncan yield condition predicts a slightly greater friction angle in triaxial compression than in extension, whereas the Matsuoka yield condition gives identical values.
*. .: 0 iriaxial compression
30"
I
R
60"
triaxial extension
(4 Fig. 8. Variation of 6 with R: (a) yield condition (3); (b) yield condition (10); (c) yield condition (11).
257
proximate in that only stresses at the wall contribute to the derivation. In plane hoppers, it is reasonable to assume that the stresses correspond to those for plane strain. Then, the angle of internal friction should be taken as S,, or &, where the subscript ‘p’ indicates plane strain. The type of stress state that exists in conical hoppers is more difficult to assess; however, an admissible engineering approximation would be an axisymmetric passive state, and the friction angle can then be taken as that at triaxial extension, S,, or &, where the subscript ‘e’ stands for extension. To determine the friction angle S,, or &,, for yield conditions (3), (lo), and (ll), it is necessary to make use of the relationship between the stresses and strainincrements during plastic flow. In terms of the principal stresses aj and strain-increments &, this relationship is usually expressed [9, 231 as dei=dh g I
(14)
where dh is a positive multiplier, and G = 0 is a function of 0;:. This function can be represented in the principal stress space as a surface to which the strain-increments are orthogonal when plotted as vectors. Knowing the function G = 0, calculating de3 from eqn. (14), setting de3 =0, utilizing the function F=O, and selecting an arbitrary mean stress, we may determine the state of stress and, from eqn. (13), the friction angle in plane strain. The procedure above requires the knowledge of the function G = 0, whose form can be deduced from tests at various stress states by plotting the strain-increments as vectors in stress space. The experiments described by Lade and Duncan [25], and by Matsuoka and Nakai [27], show that G #F, however, the shape of G =0 is such that in the r-plane the strain-increment vectors are nearly orthogonal to the trace of the appropriate yield condition (Fig. 7(a)). With this assumption, it is then possible to apply another procedure which only makes use of the dilatancy angle for plane strain defined by eqns. (6) and (7). For the Mohr-Coulomb yield condition (2) and G = F, the dilatancy angle determined from eqns. (7) and (14) is *=S
smooth trace on the r-plane and, hence, also to yield conditions (3), (lo), and (11). By means of this procedure, the results of calculations can be represented in Figs. 8(a), 8(b), and 8(c) as the broken lines at which +=const. The intersection point of a broken line and a solid line gives the value of the angle S,, or &,, and R, for a given I,+. For the Drucker-Prager yield condition the points for plane strain and 0 Q $< S, are located at fi> 30”, and for yield conditions (10) and (11) at R Q 30”. In particular, for the Drucker-Prager yield condition and $=O as assumed by Jenike [8], angle CR = 30” where a, = (a1 + &/ 2. This information allows to determine such a value of k, in eqn. (3) that the Drucker-Prager yield condition matches the Mohr-Coulomb yield condition in plane strain, i.e. the traces of both yield conditions in the rrplane intersect at R = 30”. The resulting expression for k, is given by eqn. (4). With the help of Figs. (8a), 8(b), and 8(c), it is possible to construct diagrams which relate friction angles in triaxial compression, S,, and extension, S,, to S, (or & and +e to &), for different values of the dilatancy angle I(I . This is shown in Figs. 9(a), 9(b), and 9(c) for $= 0 and $= S,. For small values of $, the yield conditions (3) and (10) predict S, > S,, with their difference decreasing with increasing I,% at higher values of $, yield condition (10) gives S,< S,. On the other hand, S, < S, and their difference increases with $. The yield condition (11) predicts S, = S, < 6, which decreases with increasing I/J. Except for the Drucker-Prager yield condition, which predicts an unrealistically large difference between friction angles in triaxial compression and extension, the remaining two conditions account for lower friction angles in triaxial compression and extension than in plane strain. For S,, or $J,,, in the range of 30-50”, the difference is about 3-5”. Since the angles S, and & are smaller than S, and +p, respectively, the bounding curves for conical hoppers in Fig. 2, and also in Fig. 6, will shift inside those for plane hoppers, and their locations will correspond, at least qualitatively, with experimental results. This is demonstrated for criterion (9) in Fig. 10, where #+, and & were located according to yield condition (11) and *= 4,.
(15)
This means that a point on the r-plane trace of a yield condition other than (2), which is common with a circumscribed Mohr-Coulomb hexagon with S = S’, will correspond to plane strain with $= S’ (Fig. 7(b)). Thus, selecting a value of I+%, constructing the circumscribed Mohr-Coulomb hexagon with S’=JI, we can find the location of the point corresponding to plane strain. This procedure applies to any yield condition with a
Closing remarks In the analysis presented above, no account was taken of a possible change from mass flow to other types of flow with progressing discharge and lowering of the upper surface of the material in hoppers. This effect
A ,’
30” -
,,
,,
,’
,,
,_-’
,’ ,/
,’
,,’
,‘,’
,,’ *+
~=O~.._.---
__--
___---
‘v=sp
,_*--
triadal compresson
,
triaxial extension 60”
triaxial compression
0
30”
60”
6, 9 0
30”
60”
SP 900
(b) go0
SC?
se
60” -
0
(4 Fig. 9. Relation between SC, S,, and Sp for $=O and $=Sp: (a) yield condition (3); (b) yield condition (10); (c) yield condition (11).
was considered in [19], and a criterion for mass flow incorporating the height of the material in plane hoppers was suggested there. Also, the presence of a vertical bin section in a bin/hopper structure was disregarded in the analysis. Experiments by Benink [5] show that, with the lowering of the material in the cylindrical bin section, initial mass-type flow may turn into funneltype. Utilizing the method of differential slices, Benink
0
30”
60”
S
Fig. 10. Modified bounding curves for mass flow criterion (9) and yield condition (11).
proposed a criterion for this mixed-type flow, which gives bounds that are located below those shown in Fig. 2(b). It is striking that realistic bounds on mass flow derived from the exact radial stress fields are only obtained for conical hoppers [Z, 6,8]. This limitation is alleviated in some criteria that are based on approximate radial stress fields, and in criterion (9), though bounds for plane hoppers and constant friction angle are incorrectly located if yield condition (2) is assumed; contrary to predictions, experiments show greater hopper angles for mass flow in plane hoppers than in conical ones. It should be remembered, however, that the method of differential slices suffers from inherent freedom in the selection of the geometry of the slices and additional assumptions [4]. This causes widely scattered bounds for mass flow for different slices, on the one hand, and allows for adjusting the slices and assumptions to match experimental results in a selected type of hoppers, on the other. The criteria for mass flow derived from the radial stress field assume large shearing deformation of the material, and apply to both cohesionless and cohesive materials characterized by the effective friction angle S. Criterion (9) was derived for cohesionless materials, and makes use of the instantaneous friction angle 4 and the dilation angle @, in most cohesionless materials, angle 4 is slightly greater than or equal to S. In spite of these limitations, the bounds for mass flow are surprisingly close to those for some other criteria if 4 is replaced by 6. In particular, identical bounds are obtained for criterion (9) and for Jenike’s [2,6] criterion, if 4=S=30”. The dilatancy angle $ is readily available from the direct shear test if the increments du, and du, in eqn. (6) are measured in the shear cell. The thus determined angle + is usually significantly smaller than &. It should be stressed, however, that the normal stresses at the outlet level may be lower than those induced in the direct shear test. Furthermore, the mechanism of shear band formation in hoppers, and the accompanying
259
dilation, may differ from that in the direct shear test, where strong geometric constraints on the deformation field are enforced by the rigid walls and the cap of the shear cell. The material near the hopper outlet may dilate more, as the opening of the gate allows the material to deform freely. Accordingly, the dilation angle may be close to the instantaneous friction angle, though no simple experimental technique has been developed to validate this statement. The analysis of the influence of stress state on friction angles of granular materials indicates the way in which corrections can be made to some mass flow criteria. Similar corrections can be made in predicting loads in storage structures. Besides yield conditions (10) and (ll), other approximations to experimental data have been proposed in the literature [28, 291. Even though they all deviate from the Mohr-Coulomb yield condition they are far away from yield condition (3). The significance of properly assessing the magnitude of the friction angle in various stress states has already penetrated geotechnics, where the difference of about 3-5” between soil friction angles in plane strain and axisymmetric states has been accepted [30]. However, when dealing with plastic flow phenomena, it is the displacement-increment (velocity) field that plays a more important role than the stresses alone. This is particularly important in non-steady flow, which often develops during discharge of granular materials from storage structures. The mass flow criterion (9) presented here is merely the outcome of one possible approach, and should by no means be regarded as the ultimate solution. A more sophisticated approach has recently been investigated by Tejchman and Gudehus [31].
5 E. J. Benink, Ph.D. Thesic, Univ. of Twente, 1989. 6 A. W. Jenike, J. Appl. Mech., Trans. ASME, 31 (i964) 5.
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
J. R. Johanson, J. Appl. Mech., Trans. ASME, 31 (1964) 499. A. W. Jenike, Powder Technol., 50 (1987) 229. D. C. Drucker and W. Prager, Q. Appl. Math., 10 (1952) 157. P. C. Arnold and A. G. McLean, Powder Technol., 13 (1976) 255. D. M. Walker, Chem. Eng. Sci., 21 (1966) 975. J. K. Walters, Chem. Eng. Sci, 28 (1973) 779. K. Clague, Ph.D. Thesis, Univ. of Nottingham, 1973. G. Enstad, Chem. Eng. Sci., 30 (1975) 1273. G. Enstad, Chem. Eng. Sci, 32 (1977) 339. W. G. Pariseau, Powder Technol., 3 (1970) 218. P. M. Blair-Fish and P. L. Bransby, J. Eng. Ind., Trans. ASME, Ser. B, 95 (1973) 17. J. Lee, S. C. Cowin and J. S. Templeton III, Trans. Sot. Bheol., 18 (1974) 247. A. Drescher, T. W. Cousens and P. L. Bransby, Geotechnique, 28 (1978) 27.. R. L. Michalowski, Powder Technol., 39 (1984) 29. R. L. Michalowski, Chem. Eng. Sci., 42 (1987) 2587. R. L. Michalowski, Geotechnique, 40 (1990) 389. D. Muir Wood, Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press, 1990. J. S. Giunta, J. Eng. Ind., Trans. ASME, Ser. B, 91 (1969) 406. P. Lade and J. M. Duncan, J. Geotech. Eng. Div., ASCE, IO1 (1975) 1037. H. Matsuoka, Soils Found., 24 (1974) 47. H. Matsuoka and T. Nakai, in P. Venneer and J. Luger (eds.), Proc. IUTAM Symposium on Deformation and Failure and Granular Materials, Delft, 1982, p. 253. G. Gudehus, Zng. Arch., 42 (1973) 45. 0. C. Zienkiewicz and G. N. Pande, in G. Gudehus (ed.), Finite Elements in Geomechanics, 1977, p. 171. C. P. Wroth, Geotechnique, 34 (1984) 449. J. Tejchman and G. Gudehus, in J. Eibel (ed.), Silos-Forschung und Praxis, Tagung ‘88 in Karlsruhe, 1988, p. 211.
Appendix Acknowledgments
The author gratefully acknowledges financial support provided by the Carnegie Trust and the Royal Society during his stay at the University of Glasgow. Special thanks for discussions and review of the paper are directed to Cormack Professor David Muir Wood. Discussions with Professor J. Michael Rotter of the University of Edinburgh also are appreciated.
References 1 A. W. Jenike, Utah Univ. Eng. E&n. St., BulL, 108 (1961). 2 A. W. Jenike, Utah Univ. Eng. Exp. Stat., Bull., 1.23 (1964). 3 P. C. Arnold, A. G. McLean and A. W. Roberts, Bulk Solids: Storage, Flow and Handling, TUNRA, 1982. 4 A. Drescher,Analyncaf Methods in Bin-LoadAnalysis, Elsevier, Amsterdam, 1991.
Coefficients A and B in eqn. (5) Walters [12] and Clague [13] D A = 1 -sin 6 cos(28, + 2p)
B=(l+m) $& + D - 1 1 ( sin S sin[2( 0, + p)] ’ = 1 - sin 6 cos[2( 0, + p)] D=
cos T( 1 + sin’@ - 2Jsin26 - sin277 cos q( 1 + sin26 + 2E sin S)
sin S sin[2( 0, + $I)] ’ =arctan 1 + sin S c0s[2(ew + p)]
260
E=
($ )m[l-(l-F)1.5)]m l- -m xi[C )I FF+
arcsin YF
@
2
Enstad [14, 151 sin p sin’+“@ + &J A= (l-sin 8) sin’+“(p+&,) + 2”[ 1 - cm@ + ew)yyp+ 0,)’ --m sin e, (l-sin 8) sin’+“(p+e,)
Benink [5]
1
A= l+C tan 0, tan(&+p) l-sin S
B= (l+m) sin 6 l+ sin(2p+e,) l-sin 6 [ sin e,
B= 2(1+m) sin S cos p sin(&+p) (l-sin 6) sin 0,
In the equations above m = 0 for plane hoppers, m = 1 for conical hoppers, and angle p is given by
c=
2+m
1+n+m
1 sin & p= - 4,+arcsin2 ( sin 6 )