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Angle of repose and internal friction
Int. J. Rock Mech. Min. Sci.Vol. 3, pp. 155--161. PergamonPressLtd. 1966. Printed in Great Britain
ANGLE OF REPOSE AND INTERNAL FRICTION J. R. METCALF Postgraduate School of Mining, Sheffield University ( R e c e i v e d 30 S e p t e m b e r 1965)
Abstract--The relation between angle of repose and angle of internal friction in a fragmented mass is discussed, and experiments with crushed rocks are reported. It is concluded that the angle of repose is not generally the angle of internal friction of the material in the pile, but of the same material in a more closely packed condition. It approximates to the angle of solid friction of the material upon itself. Implications and limitations of the concept of angle of repose are discussed. INTRODUCTION
THERE appears to exist considerable scope for the application of theory to the bulk handling and storage of fragmented material. In this a knowledge of internal friction within the fragmented mass is fundamental. It governs not only shearing and flow, but the transmission of force within the mass. Angle of internal friction is difficult to measure; angle of repose easy. If there were a relationship between the two, then investigation might be greatly facilitated. Angle of repose is obviously related to friction in some way. Several reputable works on Soil Mechanics make statements to the effect that angle of repose is equal to the angle of internal friction at the loosest condition of packing. Such a statement, although it may appear self-evident, has been found most misleading. A short study has therefore been made of the mechanical principles that appear to govern angle of repose, and the results of this form the following paper. The angles of repose of a number of materials are listed in some engineering handbooks, but no record of original readings, of any theoretical treatment of the subject, could be traced. The study had therefore to begin with the generally-accepted view that a dry, noncohesive, fragmented material possesses a characteristic physical property, known as its angle of repose. L O O S E PACKING IN A FRAGMENTED MASS
It is not advisable to consider here whether there exists an ultimate condition of "loosest packing" of any fragmented material. But the loosest packing that could be obtained during the present study resulted from the following procedure. Crushed material (most of this work was done with coal, but the same effects were seen with other materials) was fed gently into a flat-bottomed basin, beginning at one point on the circumference, until the pile reached the rim. The pile was then extended across the basin by feeding at the level of the rim until the basin was full. The surface was then levelled with minimum disturbance. Weighing indicated packing at about 50 per cent void space. By contrast, if the basin was filled by sprinkling material gently all over the bottom and allowing it to build up as a level surface, then the basin was found to hold 10 per cent more by weight. This would be equivalent to 45 per cent voids. 155
156
J.R. METCALF
The explanation of the difference in effect of the two filling procedures appeared to be as follows. In the first case it was noted, as is commonly seen when a heap is formed, that the material came to rest near the top of the slope. The fragments evidently settled in the gaps between those already in place. Only the largest sometimes reached the foot of the slope. In this way the slope steepened, until a considerable mass appeared to shear off, and, moving as an entity, to slide down and come gently to rest. Further feeding would then again build up the upper part of the slope. The angle of repose appeared to be that of the steepest slope that could be formed without the risk of shearing taking place. Now for a mass of fragments to flow down a slope, it would need to have low internal friction. This would be possible if the fragments could roll freely on one another. If the mass came gently to rest it would retain much of the open structure this implies. Hence material in a heap formed by gentle feeding at its summit would tend to have a loose packing. By contrast, if the material were scattered on a level surface each fragment would settle in a space between others and a denser packing would result. This loose packing condition is stable in that the material will remain in place indefinitely and will support a considerable distributed load if this is applied without shock. On the other hand vibration or shearing causes consolidation which proceeds very rapidly at first and then relatively slowly.
CONSOLIDATION OF A LOOSE PACKING It was found that if a basin was filled with crushed coal by the first (loose packing) procedure, and then subject to a standard vibration for 2 rain, the coal decreased in volume by about 12 per cent. During 30 rain of further vibration the volume continued to decrease at the rate of only 1 per cent/5 min. In a basin filled by the second procedure (scattering on a level surface) the shrinkage was 2 per cent in the first minute and I per cent/5 rain thereafter. A shear box was filled by the loose packing procedure and the coefficient of internal fric.tion recorded as shearing proceeded. This increased rapidly at first; then more slowly. A definite change-point could not however be detected. It is well known that in a shear box consolidation deflects the original shear plane so that consolidation and increasing friction spread gradually through the sample under test. Loosely packed material in a basin was sheared by stabbing it with a ½-in. blade on a regular pattern of vertical strokes. A pattern of two stabs/in 2 was found to consolidate the mass by 10-12 per cent. Additional stabbing, up to six stabs/in 2 had no detectable effect. There appears, therefore, to be a condition of packing where shearing or vibration produces further consolidation at a slow rate only. It may be suggested that the significance of this condition is as follows. The loosest condition of packing involves an open structure in which the fragments can roll on one another. Shearing causes a complex pattern of rotation of the fragments, extending well away from the shear plane. In this changing condition actual frictional movement of fragment upon fragment may be smaller than linear movement on the shear plane. Internal friction is therefore low. But this rotation allows consolidation under the effect of gravity. Hence shearing will cause consolidation, unless, as when a mass of fragments is sliding downhill, gravitational energy is otherwise employed. Consolidation will first cause the fragments to interlock so that they cannot easily rotate. Shearing then becomes a matter of the sliding of fragment upon fragment. Thereafter consolidation will proceed more slowly. High internal friction will develop as interlocking proceeds to the stage where shearing involves appreciable fracture.
ANGLE OF REPOSE AND INTERNAL FRICTION
157
CONDITIONS AT ANGLE OF REPOSE It is now possible to consider the idea of angle of repose theoretically. If, on the side of a heap, the angle of repose is exceeded then the side of the heap will shear off. The onset of shearing will consolidate the material at the shear plane, if this is loosely packed, and will produce a characteristic condition of packing that allows sliding but not rolling. The angle of the shear plane to the horizontal will then be the angle of internal friction of the material at that particular condition of packing. This must be the angle of repose of the material. As movement is essentially a matter of sliding, then the angle of repose of the material should approximate to the angle of solid friction of the material upon itself. It might be objected that the foregoing discussion is devious and unnecessary, and that it is only necessary to consider the case of a single fragment lying on a slope of the same material. This would slide down so long as the angle of the slope exceeded the angle of friction of the material upon itself, and so the angle of repose should equal that of solid friction. The answer to this objection is that a heap does not form by the addition of single fragments. An isolated fragment will either incorporate itself with the heap, or sometimes it may start to roll. Only in special cases will it slide. SMALL-SCALE EXPERIMENTS A series of experiments were performed to test the conclusions of the foregoing discussion. Samples of a number of materials, typical of those handled or stored in bulk, and covering a wide range of frictional properties were crushed until about 95 per cent of the sample passed a L-in. aperture. These were tested for angle of repose, angle of internal friction after forming into a pile, and the same after the pile had been consolidated. The angle of solid friction of the material on itself was measured on uncrushed pieces. The measurement of angle of repose, in piles a few inches high, presented no difficulty except in the case of rocksalt and shale. Both proved to be cohesive and liable to form steep irregular slopes. The figures eventually adopted were obtained, reproducibly, as follows. In both cases a steep slope was undercut by removing the toe, so that material ran down freely. As this might give too low an angle owing to kinetic energy in the moving fragments, the readings were confirmed, in the case of the rock-salt, by drying at boiling point and testing in the usual way, while the sample was still warm; and in the case of the shale by sieving out some of the fine material (20 mesh). The measurement of internal friction in loosely packed material is difficult, as the act of measurement consolidates the material. The following procedure was used. The apparatus comprised a wooden base supporting two vertical glass plates, parallel and 4½ in. apart. A wooden block, with a vertical face could travel between the glass plates. Initially the block was at one end of the plates, forming a box. The material to be tested was fed gently into this, beginning at the block and building up a pile to a depth of about 1 in. This pile was gradually extended through the box, in the same way as in the basin-filling procedure that gave loose packing. When the block was moved inwards slightly a shear plane was formed on which the wedge in front of the block slid upwards (Fig. 1). The intersection of this shear plane with the upper surface of the material was noted and by aligning a straight-edge on the outside of the glass, the inclination of the shear plane to the horizontal was read. This inclination would be, by accepted theory, (45 °-~/2). From this, q~,the angle of internal friction was obtained. It was not possible to obtain a second reading without refilling the box, as the initial shearing evidently caused consolidation. Before refilling, however, the material was
158
J.R. METCALF /
.Gloss plates
h i
f
//
IIf/
%
/*/
so-qs/p, F~(;. . Measurement of internal friction.
consolidated by the standard procedure of two vertical stabs with a ~-in. blade/in 2 of surface. Readings of internal friction in this condition were then obtained. It will be noted that the theory of this method ignores friction of the sliding wedge on the glass and the block. It was found that, by restricting the height of the pile to 1 in., this effect ~as rendered unimportant. Angle of solid friction of each material on itself was measured by mounting a lump so that a flat surface, either a natural break or cut with a diamond saw, was horizontal. A fragment of the material was pushed over this surface by gripping it in tweezers, on which was mounted a clinometer. A rod passing through the upper end of the tweezers was gripped between thumb and finger so that thrust could only be applied axially (Fig. 2). By raising and lowering the hand, the angle of running friction could be determined. It is well known that friction is liable to vary with pressure. The device used was intended to give the lightest practicable loading. Slight rocking action of wrist
../..: X. Selected horizoflfol surface
FIG. 2. Measurement of angle of solid friclion.
ANGLE OF REPOSE AND INTERNAL FRICTION
159
All the foregoing measurements were made a number of times, this number being varied so as to give a mean with a standard deviation of about ½°. The results are summarized in Table 1. Blank entries were for various reasons unobtainable. The results show clearly that the angle of repose is not equal to the angle of internal friction at loose packing. They indicate approximate agreement between angle of repose and the angle of internal friction after consolidation by a standard procedure, and also with the angle of friction of the material on itself. TABLE
Angle of repose
Material Rock-salt Shale (coal measure, weathered) Coal, dry steam Coal, coking Coal, hard, high volatile Anhydrite Sand (washed, quartz) Crushed sandstone, (hard siliceous) Coke
1
Angle of internal friction Loose packing
30
Consolidated
Angle of solid friction
30
3O
33 38 35½
30 18 23
32 35
33 37½ 37
35~ 37 37
27~ 31 32
35 36½ 36
36 37 37*
39 40
36 32
40 39.~
39
i ! [
I
* This reading was taken using massive vein quartz. DISCUSSION OF RESULTS The three coals, the anhydrite and the quartz sand have very similar angles of repose and coefficients of friction upon themselves. This covers a wide range of hardness, the dry steam coal being particularly friable. Lower values are however obtained with two soft materials (weathered shale and rock salt). Higher values are shown by coke and crushed standstone, both of which form very irregular fragments. It may therefore be tentatively suggested that angle of repose increases with the hardness of the material but that this is complicated both by variations in shape of fragment (at least where this is extreme) and by the tendency of some soft materials to show a relatively high coefficient of friction when tested upon themselves. The effect of these factors together is that although the extreme range of angle of repose of the materials met in mining and quarrying is perhaps 30°-40 °, these normally tend to have a value of approximately 37 °. MIXTURES
It is to be expected, from the theoretical viewpoint that the angle of repose of a mixture will be that of the material having the lower value, unless this is present in only a small proportion. This was confirmed experimentally. Mixtures of equal parts of several pairs of the materials used in the previous experiments showed, in each case, the lower angle of repose. Then the shale and the hard coal were mixed in varying proportions. The angle of repose was that of the shale (33 °) provided there was 25 per cent of shale in the mixture. With 10 or 20 per cent of shale the angle was slightly affected (33½°). Reducing the shale to 5 per cent gave a reading of 35 °. With less than 5 per cent the angle of repose was that of the coal (35~°).
160
J.R. METCALF
THE QUESTION OF SCALE The idea of angle of repose implies that it is unaffected by the size of the fragments and of the pile. Exceptions might occur with materials that change their fragments shape or frictional character with crushing, as is certainly the case with some ores. It was considered important to test the general validity of this idea because the experiments mentioned were carried out on a very small scale and angle of repose is of practical application in such cases as stock-piles, involving large tonnages. Direct experiments were made with a batch of the hard coal. This was found to keep the same angle of repose from L-in. size through successive crushings to a fine powder. A slightly steeper angle was found with l-in. or ½-in. material. It appears common for a steeper angle to be obtained when the height of the pile is not great in relation to the coarsest fragments. A casual examination of mining literature shows many photographs of mine waste dumps. On these it is possible to measure angle of repose provided the photograph includes a feature allowing it to be orientated, or else shows both sides of a dump in suitable profile, in which case the mean value may be taken. The results of such a casual examination were : Witswatersrand: Development rock dumps (presumably predominantly quartzite) 37' Witswatersrand: Sand dumps (quartzite) 35 3 7 British colliery washery refuse (predominantly shale) 33 ' Anhydrite stock-piles (20 ft high) 37 ~ Published figures of angle of repose presumably refer to full-sized piles. These include rock salt and coke at the same values as in Table 1. Published figures for coal and sand are very variable. OTHER IMPLICATIONS Discussion on size in the previous section assumed the material to comprise a wide range of sizes characteristic of the result of mining or other process involving comminution. If material is sized, the angle of repose should, by theory be unaltered. Experiments with sieve fractions of some of the materials tested showed this to be so. It should be pointed out, however, that the term "sized" material often embraces quite a wide range of size, typically the difference between two apertures, one double the other, and neither screen doing its work perfectly. The shape of the pile has theoretically no effect. Small-scale experiments showed no difference between conical piles, fiat-topped piles of various shapes, and piles banked against walls. Except that the reading for a conical pile may be steeper than normal, this being due to local steepening at the point of the cone. The point of the cone itself may be considered as a cone that is small in relation to fragment size, and so liable to steepen. The inclination and frictional character of the surface on which the pile is built have theoretically no effect, and experiments confirmed this. The method of formation of the pile, whether by feeding from the top or by uniform sprinkling onto a level surface, would appear to have no effect, because conditions at the face of the pile will remain the same. A single experiment supported this view. LIMITATIONS The idea of angle of repose becomes meaningless: (a) When the size of the pile is not large compared to the largest fragments. This applies also when the pile is confined by closely spaced vertical surfaces, as in a narrow hopper.
A N G L E OF REPOSE A N D I N T E R N A L F R I C T I O N
161
(b) When the pile is so high that it crushes under its own weight. (c) With powders that entrain air as they flow. (d) With wet materials. Very wet material cannot come to rest at angle of repose. Damp materials may eventually do so if they drain easily, but cohesive materials cannot have a constant angle of repose. (e) With rounded fragments, fairly closely sized. This was observed to be the case in one sample of washed sand, though another sand had a lower angle of repose than quartz owing to the low friction impurities present. (f) When any material transiently forms a slope steeper than the angle of repose. This is a phenomenon analagous to static friction in sliding. A small addition of energy will generally cause this slope to collapse. Coal for example will generally do this if left for a few seconds. On the other hand mine waste dumps can sometimes be quarried with quite a steep face. Attempts to measure a "static angle of repose" during the present study gave results which were neither reproducible nor capable of interpretation. CONCLUSIONS
The phenomenon of angle of repose has been studied in relation to materials of the type produced by mining or quarrying or by the comminution of these mine products. Rounded, accurately sized, cohesive or wet materials have not been considered. 1. It has been confirmed that angle of repose of such a material is a real physical property, independent of fragment size, size range, pile size, shape of pile, its method of formation and the surface on which it rests. 2. The formation of a pile by feeding gently to the top results in a loose condition of packing. 3. The angle of repose is not the angle of internal friction at this loose packing. 4. If loosely packed material is consolidated it contracts rapidly at first to a well-defined condition and then very slowly. 5. The angle of repose is the angle of internal friction at this condition. 6. This angle also approximates to the angle of solid friction of the material upon itself. 7. The tendency of some soft materials to have a relatively high coefficient of friction when tested upon themselves results in a wide range of materials presenting only a narrow range of angle of repose. 8. The angle of repose of a mixture is that of the component having the lower value.
L
ANGLE OF REPOSE AND INTERNAL FRICTION J. R. METCALF Postgraduate School of Mining, Sheffield University ( R e c e i v e d 30 S e p t e m b e r 1965)
Abstract--The relation between angle of repose and angle of internal friction in a fragmented mass is discussed, and experiments with crushed rocks are reported. It is concluded that the angle of repose is not generally the angle of internal friction of the material in the pile, but of the same material in a more closely packed condition. It approximates to the angle of solid friction of the material upon itself. Implications and limitations of the concept of angle of repose are discussed. INTRODUCTION
THERE appears to exist considerable scope for the application of theory to the bulk handling and storage of fragmented material. In this a knowledge of internal friction within the fragmented mass is fundamental. It governs not only shearing and flow, but the transmission of force within the mass. Angle of internal friction is difficult to measure; angle of repose easy. If there were a relationship between the two, then investigation might be greatly facilitated. Angle of repose is obviously related to friction in some way. Several reputable works on Soil Mechanics make statements to the effect that angle of repose is equal to the angle of internal friction at the loosest condition of packing. Such a statement, although it may appear self-evident, has been found most misleading. A short study has therefore been made of the mechanical principles that appear to govern angle of repose, and the results of this form the following paper. The angles of repose of a number of materials are listed in some engineering handbooks, but no record of original readings, of any theoretical treatment of the subject, could be traced. The study had therefore to begin with the generally-accepted view that a dry, noncohesive, fragmented material possesses a characteristic physical property, known as its angle of repose. L O O S E PACKING IN A FRAGMENTED MASS
It is not advisable to consider here whether there exists an ultimate condition of "loosest packing" of any fragmented material. But the loosest packing that could be obtained during the present study resulted from the following procedure. Crushed material (most of this work was done with coal, but the same effects were seen with other materials) was fed gently into a flat-bottomed basin, beginning at one point on the circumference, until the pile reached the rim. The pile was then extended across the basin by feeding at the level of the rim until the basin was full. The surface was then levelled with minimum disturbance. Weighing indicated packing at about 50 per cent void space. By contrast, if the basin was filled by sprinkling material gently all over the bottom and allowing it to build up as a level surface, then the basin was found to hold 10 per cent more by weight. This would be equivalent to 45 per cent voids. 155
156
J.R. METCALF
The explanation of the difference in effect of the two filling procedures appeared to be as follows. In the first case it was noted, as is commonly seen when a heap is formed, that the material came to rest near the top of the slope. The fragments evidently settled in the gaps between those already in place. Only the largest sometimes reached the foot of the slope. In this way the slope steepened, until a considerable mass appeared to shear off, and, moving as an entity, to slide down and come gently to rest. Further feeding would then again build up the upper part of the slope. The angle of repose appeared to be that of the steepest slope that could be formed without the risk of shearing taking place. Now for a mass of fragments to flow down a slope, it would need to have low internal friction. This would be possible if the fragments could roll freely on one another. If the mass came gently to rest it would retain much of the open structure this implies. Hence material in a heap formed by gentle feeding at its summit would tend to have a loose packing. By contrast, if the material were scattered on a level surface each fragment would settle in a space between others and a denser packing would result. This loose packing condition is stable in that the material will remain in place indefinitely and will support a considerable distributed load if this is applied without shock. On the other hand vibration or shearing causes consolidation which proceeds very rapidly at first and then relatively slowly.
CONSOLIDATION OF A LOOSE PACKING It was found that if a basin was filled with crushed coal by the first (loose packing) procedure, and then subject to a standard vibration for 2 rain, the coal decreased in volume by about 12 per cent. During 30 rain of further vibration the volume continued to decrease at the rate of only 1 per cent/5 min. In a basin filled by the second procedure (scattering on a level surface) the shrinkage was 2 per cent in the first minute and I per cent/5 rain thereafter. A shear box was filled by the loose packing procedure and the coefficient of internal fric.tion recorded as shearing proceeded. This increased rapidly at first; then more slowly. A definite change-point could not however be detected. It is well known that in a shear box consolidation deflects the original shear plane so that consolidation and increasing friction spread gradually through the sample under test. Loosely packed material in a basin was sheared by stabbing it with a ½-in. blade on a regular pattern of vertical strokes. A pattern of two stabs/in 2 was found to consolidate the mass by 10-12 per cent. Additional stabbing, up to six stabs/in 2 had no detectable effect. There appears, therefore, to be a condition of packing where shearing or vibration produces further consolidation at a slow rate only. It may be suggested that the significance of this condition is as follows. The loosest condition of packing involves an open structure in which the fragments can roll on one another. Shearing causes a complex pattern of rotation of the fragments, extending well away from the shear plane. In this changing condition actual frictional movement of fragment upon fragment may be smaller than linear movement on the shear plane. Internal friction is therefore low. But this rotation allows consolidation under the effect of gravity. Hence shearing will cause consolidation, unless, as when a mass of fragments is sliding downhill, gravitational energy is otherwise employed. Consolidation will first cause the fragments to interlock so that they cannot easily rotate. Shearing then becomes a matter of the sliding of fragment upon fragment. Thereafter consolidation will proceed more slowly. High internal friction will develop as interlocking proceeds to the stage where shearing involves appreciable fracture.
ANGLE OF REPOSE AND INTERNAL FRICTION
157
CONDITIONS AT ANGLE OF REPOSE It is now possible to consider the idea of angle of repose theoretically. If, on the side of a heap, the angle of repose is exceeded then the side of the heap will shear off. The onset of shearing will consolidate the material at the shear plane, if this is loosely packed, and will produce a characteristic condition of packing that allows sliding but not rolling. The angle of the shear plane to the horizontal will then be the angle of internal friction of the material at that particular condition of packing. This must be the angle of repose of the material. As movement is essentially a matter of sliding, then the angle of repose of the material should approximate to the angle of solid friction of the material upon itself. It might be objected that the foregoing discussion is devious and unnecessary, and that it is only necessary to consider the case of a single fragment lying on a slope of the same material. This would slide down so long as the angle of the slope exceeded the angle of friction of the material upon itself, and so the angle of repose should equal that of solid friction. The answer to this objection is that a heap does not form by the addition of single fragments. An isolated fragment will either incorporate itself with the heap, or sometimes it may start to roll. Only in special cases will it slide. SMALL-SCALE EXPERIMENTS A series of experiments were performed to test the conclusions of the foregoing discussion. Samples of a number of materials, typical of those handled or stored in bulk, and covering a wide range of frictional properties were crushed until about 95 per cent of the sample passed a L-in. aperture. These were tested for angle of repose, angle of internal friction after forming into a pile, and the same after the pile had been consolidated. The angle of solid friction of the material on itself was measured on uncrushed pieces. The measurement of angle of repose, in piles a few inches high, presented no difficulty except in the case of rocksalt and shale. Both proved to be cohesive and liable to form steep irregular slopes. The figures eventually adopted were obtained, reproducibly, as follows. In both cases a steep slope was undercut by removing the toe, so that material ran down freely. As this might give too low an angle owing to kinetic energy in the moving fragments, the readings were confirmed, in the case of the rock-salt, by drying at boiling point and testing in the usual way, while the sample was still warm; and in the case of the shale by sieving out some of the fine material (20 mesh). The measurement of internal friction in loosely packed material is difficult, as the act of measurement consolidates the material. The following procedure was used. The apparatus comprised a wooden base supporting two vertical glass plates, parallel and 4½ in. apart. A wooden block, with a vertical face could travel between the glass plates. Initially the block was at one end of the plates, forming a box. The material to be tested was fed gently into this, beginning at the block and building up a pile to a depth of about 1 in. This pile was gradually extended through the box, in the same way as in the basin-filling procedure that gave loose packing. When the block was moved inwards slightly a shear plane was formed on which the wedge in front of the block slid upwards (Fig. 1). The intersection of this shear plane with the upper surface of the material was noted and by aligning a straight-edge on the outside of the glass, the inclination of the shear plane to the horizontal was read. This inclination would be, by accepted theory, (45 °-~/2). From this, q~,the angle of internal friction was obtained. It was not possible to obtain a second reading without refilling the box, as the initial shearing evidently caused consolidation. Before refilling, however, the material was
158
J.R. METCALF /
.Gloss plates
h i
f
//
IIf/
%
/*/
so-qs/p, F~(;. . Measurement of internal friction.
consolidated by the standard procedure of two vertical stabs with a ~-in. blade/in 2 of surface. Readings of internal friction in this condition were then obtained. It will be noted that the theory of this method ignores friction of the sliding wedge on the glass and the block. It was found that, by restricting the height of the pile to 1 in., this effect ~as rendered unimportant. Angle of solid friction of each material on itself was measured by mounting a lump so that a flat surface, either a natural break or cut with a diamond saw, was horizontal. A fragment of the material was pushed over this surface by gripping it in tweezers, on which was mounted a clinometer. A rod passing through the upper end of the tweezers was gripped between thumb and finger so that thrust could only be applied axially (Fig. 2). By raising and lowering the hand, the angle of running friction could be determined. It is well known that friction is liable to vary with pressure. The device used was intended to give the lightest practicable loading. Slight rocking action of wrist
../..: X. Selected horizoflfol surface
FIG. 2. Measurement of angle of solid friclion.
ANGLE OF REPOSE AND INTERNAL FRICTION
159
All the foregoing measurements were made a number of times, this number being varied so as to give a mean with a standard deviation of about ½°. The results are summarized in Table 1. Blank entries were for various reasons unobtainable. The results show clearly that the angle of repose is not equal to the angle of internal friction at loose packing. They indicate approximate agreement between angle of repose and the angle of internal friction after consolidation by a standard procedure, and also with the angle of friction of the material on itself. TABLE
Angle of repose
Material Rock-salt Shale (coal measure, weathered) Coal, dry steam Coal, coking Coal, hard, high volatile Anhydrite Sand (washed, quartz) Crushed sandstone, (hard siliceous) Coke
1
Angle of internal friction Loose packing
30
Consolidated
Angle of solid friction
30
3O
33 38 35½
30 18 23
32 35
33 37½ 37
35~ 37 37
27~ 31 32
35 36½ 36
36 37 37*
39 40
36 32
40 39.~
39
i ! [
I
* This reading was taken using massive vein quartz. DISCUSSION OF RESULTS The three coals, the anhydrite and the quartz sand have very similar angles of repose and coefficients of friction upon themselves. This covers a wide range of hardness, the dry steam coal being particularly friable. Lower values are however obtained with two soft materials (weathered shale and rock salt). Higher values are shown by coke and crushed standstone, both of which form very irregular fragments. It may therefore be tentatively suggested that angle of repose increases with the hardness of the material but that this is complicated both by variations in shape of fragment (at least where this is extreme) and by the tendency of some soft materials to show a relatively high coefficient of friction when tested upon themselves. The effect of these factors together is that although the extreme range of angle of repose of the materials met in mining and quarrying is perhaps 30°-40 °, these normally tend to have a value of approximately 37 °. MIXTURES
It is to be expected, from the theoretical viewpoint that the angle of repose of a mixture will be that of the material having the lower value, unless this is present in only a small proportion. This was confirmed experimentally. Mixtures of equal parts of several pairs of the materials used in the previous experiments showed, in each case, the lower angle of repose. Then the shale and the hard coal were mixed in varying proportions. The angle of repose was that of the shale (33 °) provided there was 25 per cent of shale in the mixture. With 10 or 20 per cent of shale the angle was slightly affected (33½°). Reducing the shale to 5 per cent gave a reading of 35 °. With less than 5 per cent the angle of repose was that of the coal (35~°).
160
J.R. METCALF
THE QUESTION OF SCALE The idea of angle of repose implies that it is unaffected by the size of the fragments and of the pile. Exceptions might occur with materials that change their fragments shape or frictional character with crushing, as is certainly the case with some ores. It was considered important to test the general validity of this idea because the experiments mentioned were carried out on a very small scale and angle of repose is of practical application in such cases as stock-piles, involving large tonnages. Direct experiments were made with a batch of the hard coal. This was found to keep the same angle of repose from L-in. size through successive crushings to a fine powder. A slightly steeper angle was found with l-in. or ½-in. material. It appears common for a steeper angle to be obtained when the height of the pile is not great in relation to the coarsest fragments. A casual examination of mining literature shows many photographs of mine waste dumps. On these it is possible to measure angle of repose provided the photograph includes a feature allowing it to be orientated, or else shows both sides of a dump in suitable profile, in which case the mean value may be taken. The results of such a casual examination were : Witswatersrand: Development rock dumps (presumably predominantly quartzite) 37' Witswatersrand: Sand dumps (quartzite) 35 3 7 British colliery washery refuse (predominantly shale) 33 ' Anhydrite stock-piles (20 ft high) 37 ~ Published figures of angle of repose presumably refer to full-sized piles. These include rock salt and coke at the same values as in Table 1. Published figures for coal and sand are very variable. OTHER IMPLICATIONS Discussion on size in the previous section assumed the material to comprise a wide range of sizes characteristic of the result of mining or other process involving comminution. If material is sized, the angle of repose should, by theory be unaltered. Experiments with sieve fractions of some of the materials tested showed this to be so. It should be pointed out, however, that the term "sized" material often embraces quite a wide range of size, typically the difference between two apertures, one double the other, and neither screen doing its work perfectly. The shape of the pile has theoretically no effect. Small-scale experiments showed no difference between conical piles, fiat-topped piles of various shapes, and piles banked against walls. Except that the reading for a conical pile may be steeper than normal, this being due to local steepening at the point of the cone. The point of the cone itself may be considered as a cone that is small in relation to fragment size, and so liable to steepen. The inclination and frictional character of the surface on which the pile is built have theoretically no effect, and experiments confirmed this. The method of formation of the pile, whether by feeding from the top or by uniform sprinkling onto a level surface, would appear to have no effect, because conditions at the face of the pile will remain the same. A single experiment supported this view. LIMITATIONS The idea of angle of repose becomes meaningless: (a) When the size of the pile is not large compared to the largest fragments. This applies also when the pile is confined by closely spaced vertical surfaces, as in a narrow hopper.
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(b) When the pile is so high that it crushes under its own weight. (c) With powders that entrain air as they flow. (d) With wet materials. Very wet material cannot come to rest at angle of repose. Damp materials may eventually do so if they drain easily, but cohesive materials cannot have a constant angle of repose. (e) With rounded fragments, fairly closely sized. This was observed to be the case in one sample of washed sand, though another sand had a lower angle of repose than quartz owing to the low friction impurities present. (f) When any material transiently forms a slope steeper than the angle of repose. This is a phenomenon analagous to static friction in sliding. A small addition of energy will generally cause this slope to collapse. Coal for example will generally do this if left for a few seconds. On the other hand mine waste dumps can sometimes be quarried with quite a steep face. Attempts to measure a "static angle of repose" during the present study gave results which were neither reproducible nor capable of interpretation. CONCLUSIONS
The phenomenon of angle of repose has been studied in relation to materials of the type produced by mining or quarrying or by the comminution of these mine products. Rounded, accurately sized, cohesive or wet materials have not been considered. 1. It has been confirmed that angle of repose of such a material is a real physical property, independent of fragment size, size range, pile size, shape of pile, its method of formation and the surface on which it rests. 2. The formation of a pile by feeding gently to the top results in a loose condition of packing. 3. The angle of repose is not the angle of internal friction at this loose packing. 4. If loosely packed material is consolidated it contracts rapidly at first to a well-defined condition and then very slowly. 5. The angle of repose is the angle of internal friction at this condition. 6. This angle also approximates to the angle of solid friction of the material upon itself. 7. The tendency of some soft materials to have a relatively high coefficient of friction when tested upon themselves results in a wide range of materials presenting only a narrow range of angle of repose. 8. The angle of repose of a mixture is that of the component having the lower value.
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