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A reassessment of static bin pressure experiments
Powder Technology, 0 EIsevier Sequoia
(1979) 23 - 32 Lausanne -Printed
22
Sk,
23
in the Netherlands
A Reassessment of Static Bin Pressure Experiments
V. SUNDARAM and S. C. COWIN School
of Engineering,
Tulane
(Received March 6,1978;
University,
New
Orleans.
in revised form May 30, i978) c
SUMMARY The experimental studies concerned with Janssen’s formulas for bin pressures are re-
viewed and reassessed. This reassessment is motivated by a recent derivation of Janssen’s formulas under less restrictive assumptions. A number of conclusions are drawn and several recommendations formulated as a result of this study. For example, while it is found that much of the published data is reliable, doubt is cast on some of the data presented in the most extensive laboratory investigation undertaken. Recommendations for experimental procedures and procedures for the determination of bin pressures are presented.
INTRODUCTION In 1895 the German engineer H. A. Janssen [l] introduced formulas for determining the horizontal and vertical pressures a loose granular material exerted on the vertical walls and bottom of its container. In the eightythree years since Janssen’s study was published, there have been numerous experimental investigations concerning the validity of these formulas. The purpose of this paper is to reassess those experimental studies in light of the improved Janssen formulas presented recently by Cowin 123. In the following section the improved Janssen formulas are summarized and the four restrictive assumptions made by Janssen are discussedThe utilitarian value of Janssen’s formulas has been accepted for-many years, but in recent years a number of authors have questioned Janssen’s derivation. The investigations of Lvin [3], Walker [4], Hancock and Nedderman [5] and Home and
La
70128
(U.S.A.)
,,’
Nedderman [S] report studies aimed at developing a formula of the Janssen type without all of Janssen’s restrictive assumptions. This inconsistent view of Janssen’s formulas is summarized in the closing sentence of the study by Hancock and Nedderman [5] of four methods of calculating bin pressures: “Consequently the Janssen approach, though fundamentally unsound, must be recommended on grounds of simplicity and sufficient accuracy”. The derivation of the improved Janssen formulas by Cowin [Z] removes the restrictive assumptions made by Janssen and rederives the essence of Janssen’s result with a few, readily acceptable, assumptionsIn this paper the experimental literature concerned with Janssen’s formula is reassessed with the knowledge that his four restrictive assumptions are not necessary to obtain the essential features of his useful results. The experimental literature on Janssen’s formulas has been summarized twice before, first by Ketchum [7] sixty or seventy years ago, and again by Caughey et al_ [8] twenty-seven years ago. An extensive bibliography on bin loads is given by Jenike and Johanson [9]. We review that literature from the perspective detailed above. We find that there are three publications that give sufficient data to enable us to reassess the experimental results. These three works are the exceptionally clear and precise (for 1903) work of Jamieson [lo], the modem and accurate work of Huang and Savage [ll] and the extensive study by Caughey ef aZ_ [S] _ Jamieson reported only his data on wheat; Huang and Savage studied sand; but Caughey et al. experimented with six granular materials: wheat, shelled corn, soy beans, cement, sand and pea gravel. A principal conclusion of this reassessment is to endorse the reported results of Jamieson [lo]
24
for wheat, Huang and Savage Cll] for sand and of Caughey et al. [S] for wheat and cement, but to cast doubt on the reported results of Caughey et al. [S] for shelled corn, soy beans, sand and pea gravel.
THE
IMPROVED
JANSSEN
FORMULAS
The improved formulas of Janssen given in [2] prescribe lower bounds on V and H, where V is the vertical compressive stress in the granular material averaged over the crosssectional area of the container and H is the horizontal compressive stress in the granular material averaged over the perimeter of the cross-section of the container. It is assumed in 123 that these two average stresses are related by a constant K, H=KV_
(1)
K is called the pressure ratio but, more precisely, it is the ratio of the horizontal stress averaged over the perimeter to the vertical stress averaged over the crosssectional area. The improved Janssen formulas are
V>
_YA + p --.?.?) e*LKr’A PLK
(
and
where y is the weight density of the material, A is the cross-sectional area of the container, L is the iength of the perimeter of the crosssection, p is the coefficient of friction between the material and the container wall, P is the vertical surcharge stress applied at the free surface of the granular material, and z is a coordinate measured positive downward from the free surface of the material. The formulas (2) and (3) become equalities rather than inequalities if it is assumed that the full frictional force of the granular material upon the container wall is mobilized at each and every point of the container wall. The inequalities (2) and (3) reflect the fact that the full frictional force is not necessarily mobilized and therefore the actual stress is likely to be greater. In this paper we interpret K as a property of the granular material, independent of the material of the container wall and, therefore,
independent of the coefficient of friction p between the wall and the material. We view K as characterizing a mechanism of granular materials by which horizontal thrust is created by vertical compression as a result of grains being pushed into the interstices between other grains. We expect K to be a function of the porosity of the material, but we expect it to be independent of the container wall material and the associated coefficient of friction ,u_ This viewpoint is supported by the data of Jamieson [lo], who tested the same material (wheat) in containers made of different materials (wood, different steels) and found K to be constant (between 0.596 and 0.600). These data are given in Table 1. There are investigators knowledgeable in this area, who argue that K is a function of cc_ Their viewpoint is that the stress distribution in the gram&r material near the wall is influenced by cr and thereby influences K. Further experimentation is needed to clarify this point. Janssen [ 1 ] made four assumptions beyond those necessary to obtain the inequalities (2) and (3) First, he assumed that the vertical stress was uniform over a cross-section rather than using the cross-sectionally averaged vertical stress V employed in (2) and (3). To keep these ideas distinct, we denote Janssen’s uniform vertical stress by V,,. Second, he assumed that the horizontal stress was uniform over the perimeter of the cross-section. To distinguish this restrictive assumption on tine horizontal stress, we denote it by H,,_ Third, Janssen assumed that the r&c of H,, to V,-, was a constant. This constant we will denote by K,. Fourth, Janssen assumed that the full frictional force was mobilized at the conminer walls. Janssen’s formulas are special cases of the formulas (2) and (3) when these four specXizing assumptions are made: (i) V replaced by V,,, (ii) H replaced by Ho, (iii) K replaced by K. and (iv) the inequahty signs replaced by equality signs; thus Janssen’s formukis are
and
25 TABLE
1
A reassessment of Jamieson’s Bin
data on wheat (weight density
Size
12’ x 13’6”
Model Steel trough plate Flat steel sheet Wooden
1’ x 1’ x 6’6” 1’ x 1’ x 6’6” 1’ x 1'~ 6’6”
N/m3)
The values reported by Jamieson [lo] P
Full size Cribbed wood
7859
x 67’6”
K
Present analysis
P
Calculation using vert. stress data
Calculation using lateral stress data
12
fl
K
0.4410
0.5960
0.2589
0.5760
0.4495
0.4680 0.3550 0.3970
0.6000 0.5969 0.6000
0.2829 O-2148 0.2558
0.5891 0.5865 0.6089
0.4802 0.3662 0.4201
These equations of Janssen are special cases of the formulas (2) and (3), but the converse is not true. The significance of this remark is that the essential results of Janssen [l] are true under less restrictive a--umptions than have been supposed in the past. The substitution of these weaker assumptions for the stronger ones of JanTsen causes changes in the rationale of certain significant points in the experimental procedure ior the determination of K. Previous experiments have assumed the form (4) rather than (2) for Janssen’s formula. That is to say, they assumed that Janssen’s formula gives the actual vertical pressure rather than the lower bound on the vertical pressure. That previous experimenters were unaware that Janssen’s formula gave only the lower bound on the vertical pressure is graphically clear in the work of Caughey et al. [S] _ These experimenters plot, on the same graph, their experiment& determination of the vertical pressure and the vertical pressure predicted by Janssen’s formula (4), the latter based on their experimentally determined value of K. In many cases, these authors show that Janssen’s formula predicts a pressure in excess of that measured. Since Janssen’s formula specifies the lower bound on the vertical pressure, such situations are clearly not possible_ Caughey et aZ. [S] calculated the R they employed in Janssen’s formula based on all their experimental determinations of the ratio EC, not realizing that they should restrict themselves to the greatest value of R obtained. With some notable exceptions, previous experimenters also were not aware of the
weaker definition of K given by (1) as the ratio of the horizontal normal stress averaged over the perimeter H to the vertical normal stress averaged over the cross-sectional area V. It turns out, fortunately however, that most investigators did measure V. Generally, the method of measurement was to have the bottom of the bin suspended freeby so that the total force of the bulk material on the bottom could be determined by a lever balance for any depth of fill. This weight divided by the cross-sectional area of the bin gave V. On the other hand, unfortunately, the horizontal normal stress was determined at only one location and no effort to obtain a perimeter average was made by any of the previous investigators_
A SHORT
HISTORICAL
SURVEY
The history of this subject is divided here into three periods. The creative first period begins with the first experiments by the British engineer Isaac Roberts [12, 133 in 1884 and ends in 1926 with the 3rd edition of the book by Milo S. Ketchum [7] summarizing the state of the art of bin design. The work in this period is precise for its time, the problem is clearly formulated, and the picscnt&ion of results is generally done with great clarity. The middle period is from 1920 to 1965, roughly. The clarity and perception evidenced in the first period fades and misconceptions and misinformation propagate in the literature_ Janssen’s formulas are accepted as exact and their derivation is forgotten or
26
misunderstood. Somewhere before this middle period, an erroneous suggestion was made concerning the pressure ratio Ke, and its use becomes widespread in the middle period. This erroneous formula relates Ke to the angie of internal friction @ of the gm.nuIar material by K,=
1 --sin 1 +sinQ
Q
(6)
The ratio on the right-hand side of this equation is, of course, associated with Rankine active and passive stress theory. The incorrect formula appears in many standard works, for example, the Mining Engineers’ Handbook Cl43 and Spivakovsky and Dyachkov [15] _ The Mining Engineers’ Handbook also attributes Janssen’s formulas (4) and (5) to Ketchum [ 73 and gives no indication that the formulas are actually lower bounds on the stresses. The third period begins with a resurgence of criticaI anaiysis of Janssen’s formulas. Many excellent studies since 1965 have concerned themselves with the accuracy of the assumptions made by Janssen and the practitioners of the middIe period_ As an example of this critical reexamination, we cite the note by Bagster [16] pointing out the theoretical and experimental errors involved in the use of the formula (6) This third period is characterized by the clarity of the first period and the precision of modem technoIogy . The first large bins and silos were built about 1860 for the storage of grain. The first reported experiments on the pressure of grain in bins were made by Roberts [12] in 1884. Roberts discovered that the pressure on the bottom of the bin did not increase after the grain had reached a depth of more than twice the maximum width of the bin, no matter what the final depth of the grain_ The first experimental work on the determination of the pressure ratio K was done, not surprisingly, by Janssen. Janssen [l] carried out a number of experiments on wheat and corn in model wooden bins of 20, 30,40 and 60 cm square and approximately two meters in height. After he saw Janssen’s work in print, Prante [17] elected to publish some results he obtained earlier. Prante reported tests on fuI1 sized iron bins, 1.5 and 3.8 m in diameter and 19 m in height, 6IIed with wheat. Prante only measured IateraI pres-
sures, and these pressures were slightly smaller than the values calculated by Janssen’; formulas. Tine main interest in Prante’s work is the report of an increased lateral pressure of as much as four times the static pressure during the discharge of material from the bin. In 1897, Toltz [X3] in New York made a series of experiments with wheat in a wooden bin 14 ft square and 65 ft high. The main objective of his investigation was to determine pressures during flow. His tests showed a little increase in lateral pressure during discharge of the material from the bin. In two series of experiments, one in 1902 1903 and the other in 1909, Ketchum 173 studied the flow pressures in bins and concluded that they were the same as the static pressures_ He also noted that the pressure ratio was not a constant, but that it increased with depth. In 1903, Jamieson [lo] reported his tests made on a full sized bin filled with wheat. Of all the earlier investigations, Jamieson’s are reported in the greatest detail and appear to be the most reliable. He confirmed Janssen’s theory and suggested a vdue of 0.6 for the pressure ratio of wheat. Jamieson found that the pressure during discharge was only 7% greater than the static pressure. In the period 1902 - 1905, Bovey I.191 in Montreal, Lufft 1201 in Buenos ,4&s, and Pleissner [21] in Germany conducted tests that generally confirmed Janssen’s formulas. Pleissner noted that there was an increase in pressure above static pressures when the granular material was in motion. In the second period, the middle era, there was little experimental work until McCaImont 1223 reported tests on ear corn in cribs. He did not pubhsh his data, ho-wever. In 1945 Amundson [23] reported tests on the IateraI pressure of wheat and concluded that Janssen’s equations (4) and (5) were safe for design. In 1951 Caughey etal. 181 reported experiments on a model concrete bin 18 in. in diameter and 5 ft high involving six,granuIar materials: wheat, shelled corn, soy beans, cement, sand and pea gravel. These investigators found that wheat and cement followed Janssen’s lower bound formula quite consistently. However, as we mentioned in the previous section, Caughey et al. did not realize that Janssen’s formulas were lower bounds on the stresses, and these authors show the pres-
27
sums determined from Janssen’s formu& to be considerably in excess of those experimentally measured for shelled corn, soy beans, pea gravel and sand. These investigators concluded, erroneously, that shelled corn, soy beans, sand and pea gravel do not follow Janssen’s formulas. A series of tests with the cbjective of checking the new German standard DIN (Deutsche IndustrieNorm) concerning bin loads was reported by Klaus J?ieper [24] in 1969. No detaiIed data were given, but only lateral stresses were measured. Ciearly in the third period in the history of this subject are the experimental model tests reported by Deutsch and Schmidt 2251. These experiments were carefully carried out using accurate stress gauges. A large amount of scatter was observed in the lateral pressures during filling of the bin with sand. However, the Janssen formula was consistent with the mean filling pressures_ During emptying, these investigators observed over pressures as high as four times the static pressure. In the following year, 1970, Huang and Savage [ll] conducted some experiments on a model bin made of concrete pipe and containing sand. Only lateral pressures were measured. Of particular interest in this work are the two different methods the investigators used to fib the cyhnder with sand. One method was to place the sand in the bin by dumping buckets of sand from the top. The other method was to spray the sand into the bin from the top. These two methods are referred to as the “bucket-fiIIing” method and the “showerfilling” method, respectively. It was found that the bucket-filling method resulted in lesser bulk density and higher wail stresses ihan the shower-filling method, which resulted in a greater bulk density and lower waU stresses that followed more closely Janssen’s Iower bound formula. The literature on the quasi-static loading of bii-hopper combinations is surveyed by Jenike and Johanson [9] and, more recently, by Huang and Savage [11] _ It is agreed that quasistatic Ioads many times the maximum static load are possible. Jenike and Johanson’s survey [9] led them to conclude that the quasistatic load could be as much as four times the static. Templeton ]26] has recently measured a ratio of over 13.
The present work is a contribution to this reexamination of the utilitarian formulas of Janssen. THE
REASSESSMENT
OF THE
DATA
Of all the previous experimental investigations, only Jam&on [lo], Caughey et al [S] and Huang and Savage [ll] have presented the detailed data obtained in their investigations. Jamieson only reported his experiments with wheat and Huang and Savage studied only sand. Caughey et aZ_ considered six materials: wheat, soy beans, cement, sand, shelled corn and pea gravel. In the following paragraphs we present a reassessment of these data. We find that, even from the viewpoint of the improved Janssen formulas developed by Cowin [2], the investigations of Jamieson [lo] and Huang and Savage [ll J present reliable numerical data for the pressure ratio K, as does the report of Caughey et aZ_ for wheat and cement. We conclude that the pressure ratios R suggested by Caughey et al_ for soy beans, sand, shelled corn and pea gravel are not particularly reliable. We consider first the data reported by Jamieson [lo] for wheat. The method of reassessment is first to fit the lower bound specified by (2) for the vertical stress to the lowest set of vertical stress data reported in the experiment. The lowest vertical stress data correspond to the situation in which the greatest friction has been mobilized at the walk This is the physical situation that most closely approximates the conditions characterized by the lower bound (2). This curve fitting is done by least squares and yields a value of the product pK. The corresponding horizontal stress data are then curve-fitted by least-squares to the lower bound specified by (3). This yields a value of K and, since the product pK is known, p can be calculated. The results of the application of this method to Jamieson’s data are presented in Table 1, along with Jamieson’s values for p and K- The data reported in Table 1 show that the agreement of our analysis with that of Jamieson is quite good. The method of reassessment used for Jamieson’s data could not be apphed to the data of Huang and Savage [11] because the latter investigators measured only the hor-
28
TABLE
2
A reassessment of the data of Caughey, Material
Wheat Shelled corn Sand Cement Soy beans Pea gravel
Weight density U’J/m3)
Tooles
The values reported by Caughey et al.
C&l
7230 7387 14885 I2574 7230 15404
0.35 0.25 0.48 0.55 O-27 O-45
0.612 0.599 0.391 0.400 O-383 0.325
and Scheer [S] Resent
analysis
Calculation using only lateral stress data
Calculation stress data
using both vertical and lateral
P
K
CaIc. using vert. stress data @‘f
Calc. using lateral stress data K
P
0.3855 0.3594 0.6365 0.5063 0.5780 l-0701
0.6824 0.7898 0.4669 0.3436 0.4302 0.3740
0.2339
0.6372 0.7168 0.4357 0.3871 0.3951 0.3002
0.3671 0.3386 0.6125 0.5678 0.5430 0.9730
and not the vertical stress. The method used in this case was then to fit the lower bound specified by (3) for the horizontal stress to the horizontal stress data reported using the parameters cr and K to obtain the best Ieastsquares fit. Application of this method to the data of Huang and Savage yields a value of p = 0.523 and R = 0.506. Huang and Savage [ll] report a value of p = O-5317, which is in excellent agreement with this, but they employ the formula (6) for K and no corn&s& of K values can be made. This result suggests to us that the lateral stress data might be sufficient for the determination of both p and K. izontal
0.2427 0.2669 0.2166 0.2142 O-2921
Both the method of reassessment applied to Jamieson’s data and the method of reassessment applied to the data of Huang and Savage were applied to the data for the six granular materials reported by Caughey et al. [S] _ The results of this analysis are given in Table 2. The table shows that, using the same method of reassess ent that was employed for the Jamieson data, that is to say using both the vertical and horizontal stress data, our computed values of p and K agree closely with those given by Caughey ef al. [S] for wheat and cement, but not for shelled corn, sand, soy beans and pea gravel. Figures 1 through 6 are plots of the vertical stress and
WHEAT
2 STRESS
4
6 IN
IO 6 KILOPASCALS
Fig_ l_ A plot of the stress data for wheat. The measured vertical stress data reported by Caughey et al. [S] are indicated by the t’s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey eC al. is indicated by the (-) curve and the lateral stress by the (---) curve. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (---) curve.
CORN
2 4 STRESS
IN
8 6 KILOPASCAL:
Fig. 2. A plot of the stress data for shelled corn. The measured vertical stress data reported by Caughey ef al. [S] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (--) curve. The average vertical stress predicted by the present analysis t indicated by the (---) curve and the average lateral stress by the (---) curve.
0
g
SAND
0.3
g
P z -z F 4 0
SOY BEANS 0.3
P 0.6
g
0.6
z -
0.9
0.9 E Ei a
1.2
.
I-.
2
,
4
STRESS
6 IN
5
8
IO
2
KILOPASCALS
, STRESS
4
6 IN
8
IO
KILOPASCALS
6 IN
8
10
KILOPASCALS
Fig. 5. A plot of the stress data for soy beans. The measured vertical stress data reported by Caughey ef al. [S] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) cuNe and the lateral stress by the (-----) cuNe. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (---) curve. 0
CEMENT
h
4
STRESS
Fig. 3. A plot of the stress data for sand. The measured vertical stress data reported by Caughey et al_ [S] are indicated by the C’S and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et nI. is indicated by the (-) cuNe and the lateral stress by the (-----) curve. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (-- --) curve.
2
1.2
co 0.3 E w’ ZS 0.6 z g B c
o-9
1.2
2 STRESS
4
6 IN
8
10
KILOPASCALS
Fig. 4. A plot of the stress data for cement. The measured vertical stress data repormd by Caughey et cL [8] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (----) curve_ The average vertical stress predicted by the present analysis is indicated by the (----) curve and the average Lateral stress by the (---) curve.
Fig. 6. A plot of the stress data for pea graveL The measured vertical stress data reported by Caughey et al. [S] are indicated by the +‘s and the measured lateral stress data by the ok. The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (---) curve. The average vertical stress predicted by the present analysis is indicated by the (---) cuNe and the average lateral strew by the (---) curve.
horizontal stress data for the six materials studied by Caughey et al. These are not all the data for these materials, they are only the data we employed in our present analysis and they were selected according to the criteria
discussed in the reassessment of Jamieson’i data. Caughey et aZ_ did not select their data in this fashion because they were under the impression that (2) and (3) were equaiities rather than inequalities_ In Figs. 1 through 6
.
-. .
30
with both plates and they remark: “Probably these experimentally determined vertical pressure curves are the most reliable part of this work”.
the curves for the vertical and horizontal stress presented by Caughey et al. as well as the curves for the vertical and horizontal stress determined in the present analysis are plotted. Our method obviously yields curves that are a better representation of the data for shelled corn, sand, soy beans and pea gravel. It is appropriate to inquire into the sources of the discrepancy between the values of K presented by Caughey et al. and those presented here. The first source is the fact that we selected data with the lowest vertical stress to conform to the lower bound conditions of the formulas (2) and (3). The second source is the fact that we determined our coefficient of friction Z.Ifrom the bin pressure data rather than from the value reported by Caughey et aZ_ and measured independently of the bin pressure test. This suggests a third method of reassessing the data, namely, to employ the coefficient of friction y reported by Caughey et al. in conjunction with the value of z.& determined from the vertical stress data. The values of K determined in this fashion are listed in Table 3. When these R values are compared with those reported in Table 2, it appears that only for pea gravel does a more reasonable value of K result from this method. The third source of the discrepancy is the unreliability of the horizontal stress measurements reported by Caughey et al. Caughey et al. measured the horizontal stress at two different levels using different gauges, which they refer to as plate No. 1 and pIate No. 2. These investigators report that plate No. 1 behaved erratically and sometimes did not function at all_ Therefore, the reassessment reported in this paper does not employ any data from plate No. 1. Apparently Caughey et al. had considerable difficulty
TABLE
CONCLUSIONS
AND
RECOMMENDATIONS
Our reassessment of the experimental literature and associated theory leads to conclusions and recommendations presented in this section. First, Janssen’s formulas for the lower bounds on the vertical and horizontal stresses are consistent with the reliable experimental data. In fact, for many materials it is found that the actual stress is very close to the lower bound stresses predicted by Janssen. The values of K given in Table 4 for various materials are values that we believe to be accurate. Second, we concur with and endorse Bagster’s [16] criticism of the formula (6) for the pressure ratio R in terms of the angle of internal friction of the material. We recommend that its use be discontinued. Third, Janssen’s formula is only valid for the static situation. In the quasistatic situation associated with emptying of a bin-hopper, the stresses are found to be as much as 13 times the measured static stresses predicted by Janssen’s formulas. Fourth, Janssen [l] pointed out that for containers with square crosssections, the horizontal normal stress decreased toward the comers of the cross-section. It is therefore likely that the horizontal normal stress approximates its perimeter average only for circular crosssections. Fifth, there is not a single experimental study reviewed that measured the pressure ratio K in a manner consistent with improved formulas of Janssen. We
3
A reassessment of the vertical stress data of Caughey, Took Material
Wheat Shelled corn Sand Cement soy beans Pea gravel
and Scheer [8 ]
Weight density
Caughey
(N/m31
P
Calculation stress data w
7230 7387 14885 12574 7230 15404
0.3500 0.2500 O-4800 0.5500 0.2700 0.4500
0.2339 0.2427 0.2669 0.2166 9.2142 0.2921
et al_ [S]
using vert.
K
0.6682 0.9708 0.5560 0.3938 0.7933 0.6491
T-ABLE 4
ACKNOWLEDGMENT
Table of recommended
values of K
iMaterial
Weigh-t density (N/m-) (lb/ft3)
K
This work was supported by a grant from the National Science Foundation.
Wheat
7702 - 8016 (49 - 51)
0.57 - 0.64
REFERENCES
Cement
12574 - 15700 (80 - 100)
0.38 - 0.40
Sand
14146 - 16189 (90 - 103)
0.50 - 0.55
-
4
5
recommend that a standard procedure be established for the determination of K and p. Toward that end, we suggest that the following specific procedures be employed in the experimental determination of K: (1) Cyiindrical containers should be used so that the horizontal normal stresses have no geometrically induced deviations from the perimeter average. (2) The vertical normal stress averaged over the cross-sectional area, V, should be measured by dividing the entire force exerted on the bin bottom by the cross-sectional area. (3) The horizontal normal stress averaged over the perimeter, H, should be measured by taking 4 - 12 measurements at the same height around the bin and averaging the res-u&s. (4) A standardized method of measuring the coefficient of friction should be adopted. It appears that there is no ASTM standard covering the measure of the friction coefficient in the present situation. We further suggest that it might be possible to determine K from horizontal stress data only. This was done in our analysis of the Huang and Savage [ll] data; we determined both P and K by fitting (3) to the horizontal stress data. This method has an advantage in that both p and K are determined from one series of measurements. It is not possible to determine R from the verticat stress data, only the product ruK,as one can see from a scrutiny of (2). Therefore, if only vertisal stress data are available, the value of the material wail friction p must be known or determined in a separate experiment.
6
7
8
9 10 11
12 13
14 15
16
17
18 19
20
H A. Janssen, Versuche iiber Getreidedruck in Silozeiien, Z. Ver. Dtsch. Ing., 39 (1895) 1045. S. C. Cowin, The Theory of Static Loads in Bins, J. Appl. Mech., 44 (1977) 409. J. B. Lvin, Analytical Evaluation of Pressures of Granular Materials on Silo Walls, Powder Technol., 4 (1969-70) 280. D. M. Walker, An Approximate Theory for Pressures and Arching in Hoppers, Chem. Eng. Sci., 21(1966) 975. A. W. Hancock and R. M. Nedde_man, Prediction of Stresses on Vertical Bunker Walls, Trans. Inst. Chem. Engrs., 52 (1974) 170. R. M. Home and R. M. Nedderman, Analysis of the Stress Distribution in Two-Dimensional Bins by the Method of Characteristics, Powder Technol., 14 (1976) 93. M. S. Ketchurn, The Design of Walls, Bins, and Grain Elevators, McGraw-Hill, New York, 3, 1919. R. A. Caughey, C. W_ Tooles and A. C. Scheer, Lateral and Vertical Pressure of Granular Material in Deep Bins, Bull. 173, Eng. Expt. Sm., Iowa State College, 1951. A. W. Jenike and J. R. Johanson, Bin Loads, J_ Struct. Div_, ASCE, 94 (1968) 1011. J. A_ Jamieson, Gram Pressure in Deep Bins, Trans. Can. Sot. Civ. Engrs., 17 (1903) 554. J. H. S. Huang and S. B. Savage, Wall Stresses Developed by Gram&r Material in Asymmetric Bins, Dept. Civ. Eng. and Appl. Mech., McGill Univ., FML-TR 70-1, 1970. L Roberts, Pressure of Stored Grain, Eng., 34 (1882) 399. I_ Roberta, Determination of the Vertical and Lateral Pressures of Granular Substances, Rot. Roy. Sot., 36 (1884) 225. R. Peele. Mining_ Engineers’ Handbook_ John Wiley and Sons, I&, New York, 1, 19i8, 987. A. Spivakovsky and V. Dyachkov, Conveyors and Related Equipment, Peace Publishers, Moscow, 393. D. F. Bagster, A Note on the Pressure Ratio in the Janssen Equation, Powder Technol., 4 (1971) 235. N. L Prante, Messingen des Getreidedruckes Gegen Silowandungen, Z_ Ver. Dtsch. Ing., 40 (1896) 1122. M. Toltz, Discussion on Grain Pressures in Deep Bins, Trans. Can. Sot. Civ. Engrs., 17 (1903) 641. H T. Bovey, Experiments on Grain Pressures in Deep Bins and the Strengths of Wooden Bins, Eng. News, 52 (1904) 32. E. Lufft, Tests on Gram Ressure ;r~ Deep Bins at Buenos Aires, Argentina, Eng. News, 52 (1904) 531.
21 J. Pleismer, Versuche xur Ermitthmg der Boden - drucke in Getreidesilos, Z. Ver. und Se&en\ Dtsch. Ing., 50 (1906) 976. 22 J. R. McCaimont, Measuring Bin WaU Pressures Caused by Arching Materials. Eng New-Record, 120 (1938) 619. 23 L. R. Amundson, Determmation of Band Stresses and Lateral Wheat Pressure for a Cylindricai Grain Bin, Agr. Eng., 26 (1945) 321.
24 25 26
K. Pieper, Investigation of Silo Loads in Measuring Models, J. Eng. Ind., 91(1969) 365. G. P. Deutsch and L C. Schmidt, Pressure on Silo Walls, J. Eng. Ind., 91(1969) 450. J. S. Templeton III, An Experimental Study of the Mechanics of Rupture Zones in the Gravity Flow of a Granular Material, in S. C. Cowin and M. M. Carroll, The Effects of Voids on Material Deformation, Vol. 16, ASME-AMD, 1976, pp_ 71- 91_
(1979) 23 - 32 Lausanne -Printed
22
Sk,
23
in the Netherlands
A Reassessment of Static Bin Pressure Experiments
V. SUNDARAM and S. C. COWIN School
of Engineering,
Tulane
(Received March 6,1978;
University,
New
Orleans.
in revised form May 30, i978) c
SUMMARY The experimental studies concerned with Janssen’s formulas for bin pressures are re-
viewed and reassessed. This reassessment is motivated by a recent derivation of Janssen’s formulas under less restrictive assumptions. A number of conclusions are drawn and several recommendations formulated as a result of this study. For example, while it is found that much of the published data is reliable, doubt is cast on some of the data presented in the most extensive laboratory investigation undertaken. Recommendations for experimental procedures and procedures for the determination of bin pressures are presented.
INTRODUCTION In 1895 the German engineer H. A. Janssen [l] introduced formulas for determining the horizontal and vertical pressures a loose granular material exerted on the vertical walls and bottom of its container. In the eightythree years since Janssen’s study was published, there have been numerous experimental investigations concerning the validity of these formulas. The purpose of this paper is to reassess those experimental studies in light of the improved Janssen formulas presented recently by Cowin 123. In the following section the improved Janssen formulas are summarized and the four restrictive assumptions made by Janssen are discussedThe utilitarian value of Janssen’s formulas has been accepted for-many years, but in recent years a number of authors have questioned Janssen’s derivation. The investigations of Lvin [3], Walker [4], Hancock and Nedderman [5] and Home and
La
70128
(U.S.A.)
,,’
Nedderman [S] report studies aimed at developing a formula of the Janssen type without all of Janssen’s restrictive assumptions. This inconsistent view of Janssen’s formulas is summarized in the closing sentence of the study by Hancock and Nedderman [5] of four methods of calculating bin pressures: “Consequently the Janssen approach, though fundamentally unsound, must be recommended on grounds of simplicity and sufficient accuracy”. The derivation of the improved Janssen formulas by Cowin [Z] removes the restrictive assumptions made by Janssen and rederives the essence of Janssen’s result with a few, readily acceptable, assumptionsIn this paper the experimental literature concerned with Janssen’s formula is reassessed with the knowledge that his four restrictive assumptions are not necessary to obtain the essential features of his useful results. The experimental literature on Janssen’s formulas has been summarized twice before, first by Ketchum [7] sixty or seventy years ago, and again by Caughey et al_ [8] twenty-seven years ago. An extensive bibliography on bin loads is given by Jenike and Johanson [9]. We review that literature from the perspective detailed above. We find that there are three publications that give sufficient data to enable us to reassess the experimental results. These three works are the exceptionally clear and precise (for 1903) work of Jamieson [lo], the modem and accurate work of Huang and Savage [ll] and the extensive study by Caughey ef aZ_ [S] _ Jamieson reported only his data on wheat; Huang and Savage studied sand; but Caughey et al. experimented with six granular materials: wheat, shelled corn, soy beans, cement, sand and pea gravel. A principal conclusion of this reassessment is to endorse the reported results of Jamieson [lo]
24
for wheat, Huang and Savage Cll] for sand and of Caughey et al. [S] for wheat and cement, but to cast doubt on the reported results of Caughey et al. [S] for shelled corn, soy beans, sand and pea gravel.
THE
IMPROVED
JANSSEN
FORMULAS
The improved formulas of Janssen given in [2] prescribe lower bounds on V and H, where V is the vertical compressive stress in the granular material averaged over the crosssectional area of the container and H is the horizontal compressive stress in the granular material averaged over the perimeter of the cross-section of the container. It is assumed in 123 that these two average stresses are related by a constant K, H=KV_
(1)
K is called the pressure ratio but, more precisely, it is the ratio of the horizontal stress averaged over the perimeter to the vertical stress averaged over the crosssectional area. The improved Janssen formulas are
V>
_YA + p --.?.?) e*LKr’A PLK
(
and
where y is the weight density of the material, A is the cross-sectional area of the container, L is the iength of the perimeter of the crosssection, p is the coefficient of friction between the material and the container wall, P is the vertical surcharge stress applied at the free surface of the granular material, and z is a coordinate measured positive downward from the free surface of the material. The formulas (2) and (3) become equalities rather than inequalities if it is assumed that the full frictional force of the granular material upon the container wall is mobilized at each and every point of the container wall. The inequalities (2) and (3) reflect the fact that the full frictional force is not necessarily mobilized and therefore the actual stress is likely to be greater. In this paper we interpret K as a property of the granular material, independent of the material of the container wall and, therefore,
independent of the coefficient of friction p between the wall and the material. We view K as characterizing a mechanism of granular materials by which horizontal thrust is created by vertical compression as a result of grains being pushed into the interstices between other grains. We expect K to be a function of the porosity of the material, but we expect it to be independent of the container wall material and the associated coefficient of friction ,u_ This viewpoint is supported by the data of Jamieson [lo], who tested the same material (wheat) in containers made of different materials (wood, different steels) and found K to be constant (between 0.596 and 0.600). These data are given in Table 1. There are investigators knowledgeable in this area, who argue that K is a function of cc_ Their viewpoint is that the stress distribution in the gram&r material near the wall is influenced by cr and thereby influences K. Further experimentation is needed to clarify this point. Janssen [ 1 ] made four assumptions beyond those necessary to obtain the inequalities (2) and (3) First, he assumed that the vertical stress was uniform over a cross-section rather than using the cross-sectionally averaged vertical stress V employed in (2) and (3). To keep these ideas distinct, we denote Janssen’s uniform vertical stress by V,,. Second, he assumed that the horizontal stress was uniform over the perimeter of the cross-section. To distinguish this restrictive assumption on tine horizontal stress, we denote it by H,,_ Third, Janssen assumed that the r&c of H,, to V,-, was a constant. This constant we will denote by K,. Fourth, Janssen assumed that the full frictional force was mobilized at the conminer walls. Janssen’s formulas are special cases of the formulas (2) and (3) when these four specXizing assumptions are made: (i) V replaced by V,,, (ii) H replaced by Ho, (iii) K replaced by K. and (iv) the inequahty signs replaced by equality signs; thus Janssen’s formukis are
and
25 TABLE
1
A reassessment of Jamieson’s Bin
data on wheat (weight density
Size
12’ x 13’6”
Model Steel trough plate Flat steel sheet Wooden
1’ x 1’ x 6’6” 1’ x 1’ x 6’6” 1’ x 1'~ 6’6”
N/m3)
The values reported by Jamieson [lo] P
Full size Cribbed wood
7859
x 67’6”
K
Present analysis
P
Calculation using vert. stress data
Calculation using lateral stress data
12
fl
K
0.4410
0.5960
0.2589
0.5760
0.4495
0.4680 0.3550 0.3970
0.6000 0.5969 0.6000
0.2829 O-2148 0.2558
0.5891 0.5865 0.6089
0.4802 0.3662 0.4201
These equations of Janssen are special cases of the formulas (2) and (3), but the converse is not true. The significance of this remark is that the essential results of Janssen [l] are true under less restrictive a--umptions than have been supposed in the past. The substitution of these weaker assumptions for the stronger ones of JanTsen causes changes in the rationale of certain significant points in the experimental procedure ior the determination of K. Previous experiments have assumed the form (4) rather than (2) for Janssen’s formula. That is to say, they assumed that Janssen’s formula gives the actual vertical pressure rather than the lower bound on the vertical pressure. That previous experimenters were unaware that Janssen’s formula gave only the lower bound on the vertical pressure is graphically clear in the work of Caughey et al. [S] _ These experimenters plot, on the same graph, their experiment& determination of the vertical pressure and the vertical pressure predicted by Janssen’s formula (4), the latter based on their experimentally determined value of K. In many cases, these authors show that Janssen’s formula predicts a pressure in excess of that measured. Since Janssen’s formula specifies the lower bound on the vertical pressure, such situations are clearly not possible_ Caughey et aZ. [S] calculated the R they employed in Janssen’s formula based on all their experimental determinations of the ratio EC, not realizing that they should restrict themselves to the greatest value of R obtained. With some notable exceptions, previous experimenters also were not aware of the
weaker definition of K given by (1) as the ratio of the horizontal normal stress averaged over the perimeter H to the vertical normal stress averaged over the cross-sectional area V. It turns out, fortunately however, that most investigators did measure V. Generally, the method of measurement was to have the bottom of the bin suspended freeby so that the total force of the bulk material on the bottom could be determined by a lever balance for any depth of fill. This weight divided by the cross-sectional area of the bin gave V. On the other hand, unfortunately, the horizontal normal stress was determined at only one location and no effort to obtain a perimeter average was made by any of the previous investigators_
A SHORT
HISTORICAL
SURVEY
The history of this subject is divided here into three periods. The creative first period begins with the first experiments by the British engineer Isaac Roberts [12, 133 in 1884 and ends in 1926 with the 3rd edition of the book by Milo S. Ketchum [7] summarizing the state of the art of bin design. The work in this period is precise for its time, the problem is clearly formulated, and the picscnt&ion of results is generally done with great clarity. The middle period is from 1920 to 1965, roughly. The clarity and perception evidenced in the first period fades and misconceptions and misinformation propagate in the literature_ Janssen’s formulas are accepted as exact and their derivation is forgotten or
26
misunderstood. Somewhere before this middle period, an erroneous suggestion was made concerning the pressure ratio Ke, and its use becomes widespread in the middle period. This erroneous formula relates Ke to the angie of internal friction @ of the gm.nuIar material by K,=
1 --sin 1 +sinQ
Q
(6)
The ratio on the right-hand side of this equation is, of course, associated with Rankine active and passive stress theory. The incorrect formula appears in many standard works, for example, the Mining Engineers’ Handbook Cl43 and Spivakovsky and Dyachkov [15] _ The Mining Engineers’ Handbook also attributes Janssen’s formulas (4) and (5) to Ketchum [ 73 and gives no indication that the formulas are actually lower bounds on the stresses. The third period begins with a resurgence of criticaI anaiysis of Janssen’s formulas. Many excellent studies since 1965 have concerned themselves with the accuracy of the assumptions made by Janssen and the practitioners of the middIe period_ As an example of this critical reexamination, we cite the note by Bagster [16] pointing out the theoretical and experimental errors involved in the use of the formula (6) This third period is characterized by the clarity of the first period and the precision of modem technoIogy . The first large bins and silos were built about 1860 for the storage of grain. The first reported experiments on the pressure of grain in bins were made by Roberts [12] in 1884. Roberts discovered that the pressure on the bottom of the bin did not increase after the grain had reached a depth of more than twice the maximum width of the bin, no matter what the final depth of the grain_ The first experimental work on the determination of the pressure ratio K was done, not surprisingly, by Janssen. Janssen [l] carried out a number of experiments on wheat and corn in model wooden bins of 20, 30,40 and 60 cm square and approximately two meters in height. After he saw Janssen’s work in print, Prante [17] elected to publish some results he obtained earlier. Prante reported tests on fuI1 sized iron bins, 1.5 and 3.8 m in diameter and 19 m in height, 6IIed with wheat. Prante only measured IateraI pres-
sures, and these pressures were slightly smaller than the values calculated by Janssen’; formulas. Tine main interest in Prante’s work is the report of an increased lateral pressure of as much as four times the static pressure during the discharge of material from the bin. In 1897, Toltz [X3] in New York made a series of experiments with wheat in a wooden bin 14 ft square and 65 ft high. The main objective of his investigation was to determine pressures during flow. His tests showed a little increase in lateral pressure during discharge of the material from the bin. In two series of experiments, one in 1902 1903 and the other in 1909, Ketchum 173 studied the flow pressures in bins and concluded that they were the same as the static pressures_ He also noted that the pressure ratio was not a constant, but that it increased with depth. In 1903, Jamieson [lo] reported his tests made on a full sized bin filled with wheat. Of all the earlier investigations, Jamieson’s are reported in the greatest detail and appear to be the most reliable. He confirmed Janssen’s theory and suggested a vdue of 0.6 for the pressure ratio of wheat. Jamieson found that the pressure during discharge was only 7% greater than the static pressure. In the period 1902 - 1905, Bovey I.191 in Montreal, Lufft 1201 in Buenos ,4&s, and Pleissner [21] in Germany conducted tests that generally confirmed Janssen’s formulas. Pleissner noted that there was an increase in pressure above static pressures when the granular material was in motion. In the second period, the middle era, there was little experimental work until McCaImont 1223 reported tests on ear corn in cribs. He did not pubhsh his data, ho-wever. In 1945 Amundson [23] reported tests on the IateraI pressure of wheat and concluded that Janssen’s equations (4) and (5) were safe for design. In 1951 Caughey etal. 181 reported experiments on a model concrete bin 18 in. in diameter and 5 ft high involving six,granuIar materials: wheat, shelled corn, soy beans, cement, sand and pea gravel. These investigators found that wheat and cement followed Janssen’s lower bound formula quite consistently. However, as we mentioned in the previous section, Caughey et al. did not realize that Janssen’s formulas were lower bounds on the stresses, and these authors show the pres-
27
sums determined from Janssen’s formu& to be considerably in excess of those experimentally measured for shelled corn, soy beans, pea gravel and sand. These investigators concluded, erroneously, that shelled corn, soy beans, sand and pea gravel do not follow Janssen’s formulas. A series of tests with the cbjective of checking the new German standard DIN (Deutsche IndustrieNorm) concerning bin loads was reported by Klaus J?ieper [24] in 1969. No detaiIed data were given, but only lateral stresses were measured. Ciearly in the third period in the history of this subject are the experimental model tests reported by Deutsch and Schmidt 2251. These experiments were carefully carried out using accurate stress gauges. A large amount of scatter was observed in the lateral pressures during filling of the bin with sand. However, the Janssen formula was consistent with the mean filling pressures_ During emptying, these investigators observed over pressures as high as four times the static pressure. In the following year, 1970, Huang and Savage [ll] conducted some experiments on a model bin made of concrete pipe and containing sand. Only lateral pressures were measured. Of particular interest in this work are the two different methods the investigators used to fib the cyhnder with sand. One method was to place the sand in the bin by dumping buckets of sand from the top. The other method was to spray the sand into the bin from the top. These two methods are referred to as the “bucket-fiIIing” method and the “showerfilling” method, respectively. It was found that the bucket-filling method resulted in lesser bulk density and higher wail stresses ihan the shower-filling method, which resulted in a greater bulk density and lower waU stresses that followed more closely Janssen’s Iower bound formula. The literature on the quasi-static loading of bii-hopper combinations is surveyed by Jenike and Johanson [9] and, more recently, by Huang and Savage [11] _ It is agreed that quasistatic Ioads many times the maximum static load are possible. Jenike and Johanson’s survey [9] led them to conclude that the quasistatic load could be as much as four times the static. Templeton ]26] has recently measured a ratio of over 13.
The present work is a contribution to this reexamination of the utilitarian formulas of Janssen. THE
REASSESSMENT
OF THE
DATA
Of all the previous experimental investigations, only Jam&on [lo], Caughey et al [S] and Huang and Savage [ll] have presented the detailed data obtained in their investigations. Jamieson only reported his experiments with wheat and Huang and Savage studied only sand. Caughey et aZ_ considered six materials: wheat, soy beans, cement, sand, shelled corn and pea gravel. In the following paragraphs we present a reassessment of these data. We find that, even from the viewpoint of the improved Janssen formulas developed by Cowin [2], the investigations of Jamieson [lo] and Huang and Savage [ll J present reliable numerical data for the pressure ratio K, as does the report of Caughey et aZ_ for wheat and cement. We conclude that the pressure ratios R suggested by Caughey et al_ for soy beans, sand, shelled corn and pea gravel are not particularly reliable. We consider first the data reported by Jamieson [lo] for wheat. The method of reassessment is first to fit the lower bound specified by (2) for the vertical stress to the lowest set of vertical stress data reported in the experiment. The lowest vertical stress data correspond to the situation in which the greatest friction has been mobilized at the walk This is the physical situation that most closely approximates the conditions characterized by the lower bound (2). This curve fitting is done by least squares and yields a value of the product pK. The corresponding horizontal stress data are then curve-fitted by least-squares to the lower bound specified by (3). This yields a value of K and, since the product pK is known, p can be calculated. The results of the application of this method to Jamieson’s data are presented in Table 1, along with Jamieson’s values for p and K- The data reported in Table 1 show that the agreement of our analysis with that of Jamieson is quite good. The method of reassessment used for Jamieson’s data could not be apphed to the data of Huang and Savage [11] because the latter investigators measured only the hor-
28
TABLE
2
A reassessment of the data of Caughey, Material
Wheat Shelled corn Sand Cement Soy beans Pea gravel
Weight density U’J/m3)
Tooles
The values reported by Caughey et al.
C&l
7230 7387 14885 I2574 7230 15404
0.35 0.25 0.48 0.55 O-27 O-45
0.612 0.599 0.391 0.400 O-383 0.325
and Scheer [S] Resent
analysis
Calculation using only lateral stress data
Calculation stress data
using both vertical and lateral
P
K
CaIc. using vert. stress data @‘f
Calc. using lateral stress data K
P
0.3855 0.3594 0.6365 0.5063 0.5780 l-0701
0.6824 0.7898 0.4669 0.3436 0.4302 0.3740
0.2339
0.6372 0.7168 0.4357 0.3871 0.3951 0.3002
0.3671 0.3386 0.6125 0.5678 0.5430 0.9730
and not the vertical stress. The method used in this case was then to fit the lower bound specified by (3) for the horizontal stress to the horizontal stress data reported using the parameters cr and K to obtain the best Ieastsquares fit. Application of this method to the data of Huang and Savage yields a value of p = 0.523 and R = 0.506. Huang and Savage [ll] report a value of p = O-5317, which is in excellent agreement with this, but they employ the formula (6) for K and no corn&s& of K values can be made. This result suggests to us that the lateral stress data might be sufficient for the determination of both p and K. izontal
0.2427 0.2669 0.2166 0.2142 O-2921
Both the method of reassessment applied to Jamieson’s data and the method of reassessment applied to the data of Huang and Savage were applied to the data for the six granular materials reported by Caughey et al. [S] _ The results of this analysis are given in Table 2. The table shows that, using the same method of reassess ent that was employed for the Jamieson data, that is to say using both the vertical and horizontal stress data, our computed values of p and K agree closely with those given by Caughey ef al. [S] for wheat and cement, but not for shelled corn, sand, soy beans and pea gravel. Figures 1 through 6 are plots of the vertical stress and
WHEAT
2 STRESS
4
6 IN
IO 6 KILOPASCALS
Fig_ l_ A plot of the stress data for wheat. The measured vertical stress data reported by Caughey et al. [S] are indicated by the t’s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey eC al. is indicated by the (-) curve and the lateral stress by the (---) curve. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (---) curve.
CORN
2 4 STRESS
IN
8 6 KILOPASCAL:
Fig. 2. A plot of the stress data for shelled corn. The measured vertical stress data reported by Caughey ef al. [S] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (--) curve. The average vertical stress predicted by the present analysis t indicated by the (---) curve and the average lateral stress by the (---) curve.
0
g
SAND
0.3
g
P z -z F 4 0
SOY BEANS 0.3
P 0.6
g
0.6
z -
0.9
0.9 E Ei a
1.2
.
I-.
2
,
4
STRESS
6 IN
5
8
IO
2
KILOPASCALS
, STRESS
4
6 IN
8
IO
KILOPASCALS
6 IN
8
10
KILOPASCALS
Fig. 5. A plot of the stress data for soy beans. The measured vertical stress data reported by Caughey ef al. [S] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) cuNe and the lateral stress by the (-----) cuNe. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (---) curve. 0
CEMENT
h
4
STRESS
Fig. 3. A plot of the stress data for sand. The measured vertical stress data reported by Caughey et al_ [S] are indicated by the C’S and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et nI. is indicated by the (-) cuNe and the lateral stress by the (-----) curve. The average vertical stress predicted by the present analysis is indicated by the (---) curve and the average lateral stress by the (-- --) curve.
2
1.2
co 0.3 E w’ ZS 0.6 z g B c
o-9
1.2
2 STRESS
4
6 IN
8
10
KILOPASCALS
Fig. 4. A plot of the stress data for cement. The measured vertical stress data repormd by Caughey et cL [8] are indicated by the +‘s and the measured lateral stress data by the x’s_ The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (----) curve_ The average vertical stress predicted by the present analysis is indicated by the (----) curve and the average Lateral stress by the (---) curve.
Fig. 6. A plot of the stress data for pea graveL The measured vertical stress data reported by Caughey et al. [S] are indicated by the +‘s and the measured lateral stress data by the ok. The vertical stress predicted by Caughey et al. is indicated by the (-) curve and the lateral stress by the (---) curve. The average vertical stress predicted by the present analysis is indicated by the (---) cuNe and the average lateral strew by the (---) curve.
horizontal stress data for the six materials studied by Caughey et al. These are not all the data for these materials, they are only the data we employed in our present analysis and they were selected according to the criteria
discussed in the reassessment of Jamieson’i data. Caughey et aZ_ did not select their data in this fashion because they were under the impression that (2) and (3) were equaiities rather than inequalities_ In Figs. 1 through 6
.
-. .
30
with both plates and they remark: “Probably these experimentally determined vertical pressure curves are the most reliable part of this work”.
the curves for the vertical and horizontal stress presented by Caughey et al. as well as the curves for the vertical and horizontal stress determined in the present analysis are plotted. Our method obviously yields curves that are a better representation of the data for shelled corn, sand, soy beans and pea gravel. It is appropriate to inquire into the sources of the discrepancy between the values of K presented by Caughey et al. and those presented here. The first source is the fact that we selected data with the lowest vertical stress to conform to the lower bound conditions of the formulas (2) and (3). The second source is the fact that we determined our coefficient of friction Z.Ifrom the bin pressure data rather than from the value reported by Caughey et aZ_ and measured independently of the bin pressure test. This suggests a third method of reassessing the data, namely, to employ the coefficient of friction y reported by Caughey et al. in conjunction with the value of z.& determined from the vertical stress data. The values of K determined in this fashion are listed in Table 3. When these R values are compared with those reported in Table 2, it appears that only for pea gravel does a more reasonable value of K result from this method. The third source of the discrepancy is the unreliability of the horizontal stress measurements reported by Caughey et al. Caughey et al. measured the horizontal stress at two different levels using different gauges, which they refer to as plate No. 1 and pIate No. 2. These investigators report that plate No. 1 behaved erratically and sometimes did not function at all_ Therefore, the reassessment reported in this paper does not employ any data from plate No. 1. Apparently Caughey et al. had considerable difficulty
TABLE
CONCLUSIONS
AND
RECOMMENDATIONS
Our reassessment of the experimental literature and associated theory leads to conclusions and recommendations presented in this section. First, Janssen’s formulas for the lower bounds on the vertical and horizontal stresses are consistent with the reliable experimental data. In fact, for many materials it is found that the actual stress is very close to the lower bound stresses predicted by Janssen. The values of K given in Table 4 for various materials are values that we believe to be accurate. Second, we concur with and endorse Bagster’s [16] criticism of the formula (6) for the pressure ratio R in terms of the angle of internal friction of the material. We recommend that its use be discontinued. Third, Janssen’s formula is only valid for the static situation. In the quasistatic situation associated with emptying of a bin-hopper, the stresses are found to be as much as 13 times the measured static stresses predicted by Janssen’s formulas. Fourth, Janssen [l] pointed out that for containers with square crosssections, the horizontal normal stress decreased toward the comers of the cross-section. It is therefore likely that the horizontal normal stress approximates its perimeter average only for circular crosssections. Fifth, there is not a single experimental study reviewed that measured the pressure ratio K in a manner consistent with improved formulas of Janssen. We
3
A reassessment of the vertical stress data of Caughey, Took Material
Wheat Shelled corn Sand Cement soy beans Pea gravel
and Scheer [8 ]
Weight density
Caughey
(N/m31
P
Calculation stress data w
7230 7387 14885 12574 7230 15404
0.3500 0.2500 O-4800 0.5500 0.2700 0.4500
0.2339 0.2427 0.2669 0.2166 9.2142 0.2921
et al_ [S]
using vert.
K
0.6682 0.9708 0.5560 0.3938 0.7933 0.6491
T-ABLE 4
ACKNOWLEDGMENT
Table of recommended
values of K
iMaterial
Weigh-t density (N/m-) (lb/ft3)
K
This work was supported by a grant from the National Science Foundation.
Wheat
7702 - 8016 (49 - 51)
0.57 - 0.64
REFERENCES
Cement
12574 - 15700 (80 - 100)
0.38 - 0.40
Sand
14146 - 16189 (90 - 103)
0.50 - 0.55
-
4
5
recommend that a standard procedure be established for the determination of K and p. Toward that end, we suggest that the following specific procedures be employed in the experimental determination of K: (1) Cyiindrical containers should be used so that the horizontal normal stresses have no geometrically induced deviations from the perimeter average. (2) The vertical normal stress averaged over the cross-sectional area, V, should be measured by dividing the entire force exerted on the bin bottom by the cross-sectional area. (3) The horizontal normal stress averaged over the perimeter, H, should be measured by taking 4 - 12 measurements at the same height around the bin and averaging the res-u&s. (4) A standardized method of measuring the coefficient of friction should be adopted. It appears that there is no ASTM standard covering the measure of the friction coefficient in the present situation. We further suggest that it might be possible to determine K from horizontal stress data only. This was done in our analysis of the Huang and Savage [ll] data; we determined both P and K by fitting (3) to the horizontal stress data. This method has an advantage in that both p and K are determined from one series of measurements. It is not possible to determine R from the verticat stress data, only the product ruK,as one can see from a scrutiny of (2). Therefore, if only vertisal stress data are available, the value of the material wail friction p must be known or determined in a separate experiment.
6
7
8
9 10 11
12 13
14 15
16
17
18 19
20
H A. Janssen, Versuche iiber Getreidedruck in Silozeiien, Z. Ver. Dtsch. Ing., 39 (1895) 1045. S. C. Cowin, The Theory of Static Loads in Bins, J. Appl. Mech., 44 (1977) 409. J. B. Lvin, Analytical Evaluation of Pressures of Granular Materials on Silo Walls, Powder Technol., 4 (1969-70) 280. D. M. Walker, An Approximate Theory for Pressures and Arching in Hoppers, Chem. Eng. Sci., 21(1966) 975. A. W. Hancock and R. M. Nedde_man, Prediction of Stresses on Vertical Bunker Walls, Trans. Inst. Chem. Engrs., 52 (1974) 170. R. M. Home and R. M. Nedderman, Analysis of the Stress Distribution in Two-Dimensional Bins by the Method of Characteristics, Powder Technol., 14 (1976) 93. M. S. Ketchurn, The Design of Walls, Bins, and Grain Elevators, McGraw-Hill, New York, 3, 1919. R. A. Caughey, C. W_ Tooles and A. C. Scheer, Lateral and Vertical Pressure of Granular Material in Deep Bins, Bull. 173, Eng. Expt. Sm., Iowa State College, 1951. A. W. Jenike and J. R. Johanson, Bin Loads, J_ Struct. Div_, ASCE, 94 (1968) 1011. J. A_ Jamieson, Gram Pressure in Deep Bins, Trans. Can. Sot. Civ. Engrs., 17 (1903) 554. J. H. S. Huang and S. B. Savage, Wall Stresses Developed by Gram&r Material in Asymmetric Bins, Dept. Civ. Eng. and Appl. Mech., McGill Univ., FML-TR 70-1, 1970. L Roberts, Pressure of Stored Grain, Eng., 34 (1882) 399. I_ Roberta, Determination of the Vertical and Lateral Pressures of Granular Substances, Rot. Roy. Sot., 36 (1884) 225. R. Peele. Mining_ Engineers’ Handbook_ John Wiley and Sons, I&, New York, 1, 19i8, 987. A. Spivakovsky and V. Dyachkov, Conveyors and Related Equipment, Peace Publishers, Moscow, 393. D. F. Bagster, A Note on the Pressure Ratio in the Janssen Equation, Powder Technol., 4 (1971) 235. N. L Prante, Messingen des Getreidedruckes Gegen Silowandungen, Z_ Ver. Dtsch. Ing., 40 (1896) 1122. M. Toltz, Discussion on Grain Pressures in Deep Bins, Trans. Can. Sot. Civ. Engrs., 17 (1903) 641. H T. Bovey, Experiments on Grain Pressures in Deep Bins and the Strengths of Wooden Bins, Eng. News, 52 (1904) 32. E. Lufft, Tests on Gram Ressure ;r~ Deep Bins at Buenos Aires, Argentina, Eng. News, 52 (1904) 531.
21 J. Pleismer, Versuche xur Ermitthmg der Boden - drucke in Getreidesilos, Z. Ver. und Se&en\ Dtsch. Ing., 50 (1906) 976. 22 J. R. McCaimont, Measuring Bin WaU Pressures Caused by Arching Materials. Eng New-Record, 120 (1938) 619. 23 L. R. Amundson, Determmation of Band Stresses and Lateral Wheat Pressure for a Cylindricai Grain Bin, Agr. Eng., 26 (1945) 321.
24 25 26
K. Pieper, Investigation of Silo Loads in Measuring Models, J. Eng. Ind., 91(1969) 365. G. P. Deutsch and L C. Schmidt, Pressure on Silo Walls, J. Eng. Ind., 91(1969) 450. J. S. Templeton III, An Experimental Study of the Mechanics of Rupture Zones in the Gravity Flow of a Granular Material, in S. C. Cowin and M. M. Carroll, The Effects of Voids on Material Deformation, Vol. 16, ASME-AMD, 1976, pp_ 71- 91_