A tentative method of estimating k and u for Janssens bin pressure equations
by Harry F Steeves
A THESIS Submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree
of Master of Science in Civil Engineering
Montana State University
© Copyright by Harry F Steeves (1961)
Abstract:
In this thesis empirical equations are developed for estimating k (the ratio of lateral to vertical pressure)
and µ'(the effective coefficient of wall friction) for use in Janssen's bin pressure equations. Also, ah
attempt is made to approximate values of "a” (the ratio of vertical pressure at the bin wall to the
average vertical pressure) for use in Scheer's theoretical equations for obtaining values of k.
Values of k, µ', and "a” were determined from lateral and vertical pressure measurements in a test bin,
with and without sandpaper glued to the walls, for five fill materials varying from glass marbles to very
rough angular aggregate. Three simple laboratory friction tests were used to estimate &mu' for each of
the materials.
Empirical equations which were developed to correlate the results of the laboratory friction tests with k
and µ' give results that estimate observed values of k and µ' with a maximum error of 22&. Although
Scheer's equations proved valid for estimating k, the empirical equations developed gave a more
accurate value of it by a faster method.
Incidental to the aforementioned studies, pressures were also measured with grain in motion, and with
the bin subjected to severe vibrations. With fill material flowing, the maximum lateral pressure
observed was 53.7% above the static lateral pressure. The largest observed lateral pressure after
vibration was 134% higher than the static lateral pressure. A TENTATIVE METHOD OF ESTIMATING k A N D y
FOR JANSSEN'S BIN PRESSURE EQUATIONS
by
Harry F . S+eeves
A THESI S
Submitted to the Graduate Faculty
in
p a rtia l f u lfillm e n t of the requirements
for the degree of
Master o f Science In G lv il Engineering
at
Montana S tate College
Approved:
Head, Major Department
Ghairaa
Dean, Graduate D ivision
Bozemap, Montana
August, 1961
//3 7 ?
S ir 3 17
i2
ACKNOWLEDGEMENTS
The author takes th is opportunity to express his g ra titu d e to
Professor A lfred C . Scheer and the thesis committee for t h e ir guidance
in preparing th is thesis as well as in the research which preceded i t .
Also, the author wishes to thank his w ife , Jackie, for her typing and
the patience th a t went with i t .
150577
3
TABLE OF CONTENTS
Abstract
,
’
'
4
Introduction
5
Bi n TermI nol ogy
-7
Review o f L ite ra tu re
g
Object!ves
'
“
'
24
Experimental Equipment and M aterial
Test Bin
26
'
26
Bin Cal I b r a ti on
Laboratory Apparatus
_ 39
fo r Estim ating 0 and 0*
31
M a te ria ls
33
Experimental Procedure
35
Experimental Results
39
Analysis of Results
53
'
Discussion
Recommendations
'
6
5
50
Conclusions
69
L ite ra tu re C ited
70
Appendix
73
I - D erivatio n of Janssen’ sEquations
I I - D erivatio n o f Scheer’ sEquations
III
- Experimental Data
75
75
80
4
ABSTRACT
In th is th esis em pirical equations are developed for e s tim a te
ing k (th e r a tio o f la te ra l to v e rtic a l ,pressure) and j j , - (th e e ffe c ­
t iv e c o e ffic ie n t of wall f r ic t io n ) for use in Janssen's bin pressure
equations. A lso, ah attempt is made to approximate values o f "a”
(th e r a tio o f v e rtic a l pressure a t the bin wall to the average v e r t i­
cal pressure) fo r use in Scheer 1S th e o re tic a l equations fo r obtaining
values o f k .
Values o f k , yu', and "a” were determined from la te ra l and
v e rtic a l pressure measurements in a te s t b in , with and without sand­
paper glued to the w a lls , for fiv e f i l l ' m aterials varying from glass
marbles to very rough angular aggregate. Three simple laboratory
fr ic tio n te s ts were used to estim ate ^i1 for each o f the m a te ria ls .
Empirical equations which were developed to c o rre la te the
re su lts of the laboratory f r ic t io n te s ts with k and p ’ give re su lts
th a t estim ate observed values o f k and p' with a maximum e rro r of
22$. Although Scheer's equations proved v a lid fo r estim ating k , the
em pirical equations developed gave a more accurate value o f I t by a
fa s te r method. ,
Incidental to the aforementioned s tu d ies, pressures were also
measured with grain in motion, and with the bin subjected to severe
vibration s*. With f i l l m aterial flo w in g, the maximum la te ra l pres­
sure observed was 53.7# above the s ta tic la te ra l pressure. The
larg est observed la te ra l pressure a fte r v ib ra tio n was 134# higher
than the s ta tic la te ra l pressure.
INTRODUCTION
Since the Introduction of Janssen's equations in 1995 invest!
gators have conducted experiments to evaluate the v a lid it y o f his
solution and also to determine the constants necessary for the sol­
ution of his equations.
B r ie fly s ta te d , Janssen's equations are?
(See D erivation o f Janssen's, Equations - Appendix I )
■
V = M
/j'k
and
L = kV =_Rw ( ,
/i';
R
•
t i l
K
}
,
where, using ty p ic a l English u n its ,
V = average v e rtic a l u n it pressure a t depth y, I Ip. /s q . f t .
I
'
L = ayerage la te ra l u n it pressure on bin wall a t ! depth y,
I b.,/sq. f t .
R = hydraulic radius (area of bin gross section divided by
perim eter o f b in ), f t .
W = uni+ weight of f i l l m a te ria l, I b ./c u . f t ,
y = depth from surface of f i l l
m a te ria l, f t .
^i'= e ffe c tiv e c o e ffic ie n t of f r ic t io n between f i l l
m aterial
and bin w a ll, dimensionless.
k = L/V = r a tio o f average la te ra l u n it pressure to average
v e rtic a l u n it pressure, dimensionless.
Al I factors In Janssen's equations are re a d ily obtainable
with the exception o f k and ya'.
In th is th esis an attempt is made
to develop a general, method for estim ating k and ya' by simple labor­
atory fr ic tio n te s ts .
Although p rio r in vestig ato rs have reported
6
values o f Janssen's constants for m aterials they have in vestig ated,
to th is author's knowledge only one attempt (See Scheer's Theory Appendix Il) ' has been made to estim ate k from simple laboratory f r i c ­
tio n te s ts .
• Since considerable mention Is made in the Review o f L ite ra tu re
o f the constants k and yu', a short discussion of bin terminology w ill
be given before proceeding to th a t section.
7
BIN TERMINOLOGY
In addition to the terms in Janssen’ s equations which are d is­
cussed in the in tro d u ctio n , several other factors need to be defined
and discussed.
These are 0* 0,
p, "a " , and smooth and rough bin
conditions.
For use in Janssen’ s equation yu’ was defined as the e ffe c tiv e
c o e ffic ie n t of fr ic t io n between f i l l m aterial and bin w a ll.
I t is
also defined as tan 0 ’ where 0 ' is the e ffe c tiv e angle o f fr ic tio n
o f the f i l l
m aterial on the bin w a ll.
As distinguished from yu ’ , yu is the e ffe c tiv e c o e ffic ie n t of
in tern al fr ic t io n o f a f i l l m aterial in the b in .
yj is also defined
as the tan 0 where 0 is the e ffe c tiv e angle of in ternal fr ic t io n of
the m aterial
in the b in .
Some p rio r in vestig ato rs have indiscrim in­
a te ly equated 0 with the angle of repose o f a m a te ria l; however,
because o f the r e la tiv e confinement and other fa c to rs , the angle
o f repose is not necessarily equal to 0 as w ill be subsequently shown.
Before s ta tin g Scheer's equations fo r estim ating k , a d e fin i­
tio n o f smooth and rough bin conditions w ill be made.
bin condition yj’ is less than yj, so th a t, when the f i l l
consolidates and s e ttle s v e r t ic a lly , f i l l
on the bin w a ll.
For a smooth
m aterial
m aterial p a rtic le s s lid e
In a rough bin condition s lid in g occurs w ithirr.the
f i l l m aterial adjacent to the wall instead o f d ire c tly on the wall
surface.
In th is case
yj’
Is equal to
y j.
8
Scheer, fo r his equations, defines a term "a" as the r a tio of
v e rtic a l u n it pressure a t the bin wall to the average v e rtic a l u n it
pressure a t a given depth.
Scheer ha& shown t h a t , th e o r e tic a lly , for a rough bin condition:
k = ak' = a
and fo r a smooth bin condition:
k == ak' = I
(
where(X s a tis fie s the equation,
cos©< + tan 0 ' sinCX = tan 0 '
sin 0
Three laboratory fr ib tio n te s t methods were used to obtain
values to estim ate
equations.
ji, ^i1,
0 and
0 ' fo r use in Scheer1s and Janssen's
To distingu ish these as fr ic t io n te s t re su lts the follow ­
ing nomenclature w ill be used:
'
/Ir =
tan 0r =stan of angle from angle of repose te s ts
JUg =
tan 0S ==c o e ffic ie n t o f f r ic t io n
Jtif == tan 0 f = c o e ffic ie n t of. fr ic t io n
from d ire c t shear te s ts
from d ire c t fr ic tio n ■
te s ts
The Review o f L ite r a tu r e , in addition to general discussion
o f p rio r in v e s tig a to rs , w ill emphasize the values they reported for
k and Jti1 as well as methods used, and values obtained in estim ating
/ i ' by te s ts separate from t h e ir bin studies.
9
REVIEW OF LITERATURE
MHo S. Ketchum ( 1 5 )* gave a chronological discussion o f bin
in vestig ations including his own, with a good discussion o f some of
them in regard to s ta tic and dynamic pressures and the values of k
for wheat and some other grains.
This review draws on Ketchum1s discussion o f the works of
Janssen, Prante, and P le iss n e r, because t h e ir papers are in German.
The works o f the other authors were consulted firs t-h a n d .
Isaac Roberts (22) made the f i r s t recorded observations of
pressure in deep bins.
He used model bins as well as a fu ll sized
bin to measure pressures o f wheat.
The model bins were
I in . to
20<75 in . in diameter and the fu ll sized bin was 6.75 f t . by 6 f t .
in cross section and 52.17 f t . deep.
His apparatus fo r measuring
pressure was not too s a tis fa c to ry , being plates in the side and
bottom "connected by levers to scales s im ila r to a weighing machine"
H is conclusion th a t " a ll
Increase o f pressure on the bottom ceases
before the bins are f i l l e d two diameters" was appreciated by most
In vestig ato rs follow ing him.
H. A, Janssen (14) developed formulas (See Appendix I ) for
determining la te ra l and v e rtic a l pressures in deep bins
i
* Number In pare n th e se s and subsequent numbers In pare n th e se s r e f e r
t o ite m o f L it e r a t u r e GI te d , page 7 0 .
IO
and in 1895 made experiments in model bins to v e rify his equations.
fcHs te s ts o f dry sand, wheat, corn, and other grains were run with
'
square model. bins, 200 cm. deep and having sides of 20, 30, 40, and
60 cm.
The bases o f his bins were supported by a scale giving base
load d ir e c t ly .
for wheat.
Ketchum states Janssen found k = 0.67 and
p 1 = 0 .3
Ketchum does not say how Janssen determined these con­
stants but Janssen apparently found"what he assumed to be a tru e yo'
by some fr ic t io n te s t and then selected a k which made his v e rtic a l
pressure equation f i t his Experimental v e rtic a l pressure curve.
Prante (2 0 ), in 1896, made studies on fu ll size iron bins,
1.5 m. and 3 .8 m. in diameter and 19 m. deep, containing wheat.
<
I
Prante's experiments fo r wheat a t re s t gave pressures in f a i r agree­
ment with Janssen's equation.
However, for wheat in motion, he
found pressure in excess of 4 times the s ta tic pressure.
Both
Jamieson (13) and Ketchum (1 5 ) questioned the accuracy o f P rante's
re su lts as did Prante him self.
In 1897, to check his own design o f a grain e le v a to r. Max
T o ltz (30) made experiments on a large wood bin 14 f t . square and
65 f t . deep,
Using wheat, T o ltz found a maximum la te ra l pressure
fo r th is bin o f only 3 psi a t the bottom o f the w a ll, by means of
a 1*5 f t . by 3 f t . steel p la te held r ig id a t two sides across an
opening of th is s ize In the b in .
He determined his pressure by ■
c a lib ra tin g the d e flectio n s o f his p la te using known weights.
A iry ( I ) , also In 1897, determined experimental I y for several
grains the angle o f repose and c o e ffic ie n t o f fric tio n 'o n wood, iron,
and concrete.
The values were determined fo r use in a formula which
he developed.
Although Ketchum compared A iry 's solution for a long
rectangular bin with Janssen's and found close agreement, A iry 's
'
solution requires m odification fo r shallow bins and does not take
bin shape in to account as a facto r th a t a ffe c ts pressure.
As stated above A iry found angle of repose and c o e ffic ie n t of
fr ic tio n values for several grains.
He found the tan of the angle
o f repose of his wheat to be 0.466 and his c o e ffic ie n ts o f fr ic tio n
o f wheat on rough wood, on smooth wood, on Iro n , and on concrete
were 0.412, 0 .3 6 1 , 0 .414, and 0.4 4 4 , re s p e c tiv e ly .
The angle of
repose was determined by measuring the slopes o f grain in p ile s and
he assumed th a t th is was equal to 0»
His fr ic t io n c o e ffic ie n ts of
wheat on wood, Iro n , and concrete were determined by finding the
slope a t which a piece o f the m aterial would s lid e down the grain
surface.
A iry assumed, the angle of th is slope to be equal to the
e ffe c tiv e angle o f wall f r ic t io n , 0 ' .
In 1902 and 1903, under the d ire c tio n o f M ilo S . Ketchum (1 5 ),
experiments were made with a model bin a t the U niversity o f I l l i n o i s .
Pressures fo r wheat were determined by two p la te s , one serving as the
base o f the bin restin g d ire c tly on a platform scale and the other
located in one side near the bottom connected by levers to a pI a t -
12
form s c ale .
A k of 0 .4 fo r wheat was reported, being the r a tio of
la te ra l to v e rtic a l pressure as determined in the bin experiment.
No pressure increase was found for wheat in motion over pressure of
wheat a t r e s t.
Tests were also run on flow of wheat from an o r if ic e .
Ketchum states th a t flow is independent of head and varies d ire c tly
as the cube of the width o f a square o r if i c e ,
M r. J . A. Jamieson (13) made extensive te s ts of bin pressures,
both in model and f u ll s iz e bins, for wheat, peas, corn, and fI axseed.
Jamieson used hydraulic c e lls which were placed in the walls
and base o f b in .
The side of the c e l l , which was set flush with
the in te r io r wall o f b ln , consisted o f a rubber diaphram which was
acted upon by the fill;!p re s s u re .
The c e ll was connected through
the wall to a v e rtic a l glass tube which indicated the pressure on
the diaphram.
G lycerin was used in the c e ll and a mercury column
was used in the glass tube to reduce it s h eig h t.
Full scale bin experiments, under Jamieson’ s d ire c tio n , were
sta rte d in 1900 on a wood bin in New Brunswick, Canada.
The bin
was 67.5 f t . deep and had a 12 f t , by 13.5 f t . rectangular cross
s e c tio n .
The bin was f i l l e d and emptied in 3.75 f t . increments.
Pressure readings were taken a f t e r each increment was added or
removed.
Curing f i l l i n g , each increment was allowed to stand 18 hours
before the next increment was placed.
Jamieson reported no change
13
in pressure during th is tim e, whereas L u ff t (16) la te r reported a
d e fin ite decrease In la te ra l pressure when the f i l l m aterial was
allowed to stand s ta tic fo r a few hours between increments.
Testing pressures of flowing g ra in , Jamjeson reported a
maximum
A% increase over s ta tic pressure during emptying.
Closing
the emptying gate during emptying gave a s lig h t decrease.
For wheat in th is wood bin Jamieson reported k = 0 ,6 , tan
0* = 0,441 and tan 0 = 0.532*
Jamieson's k is the r a tio o f the
la te ra l and v e rtic a l pressures he found from his experiments.
His
reported 0 was determined by t i l t i n g a box f u ll of wheat u n til the
surface m aterial sta rte d to s lid e .
The angle through which the box
was t i l t e d was taken as 0 . ' For bis 0' a bottomI ess box was placed
on wood surface and f i l l e d with wheat,
th e angle through which the
wood surface was t i l t e d (u n til the box o f wheat started to move)
was accepted as 0 ' .
In his model bin studies Jamieson used bins of both round
and square cross section and 6,5 f t . high*
The round bins were
6 in . and 12 in . in diameter and the square bins were 6 in . and
12 in . in w idth.
f u ll sized b in .
The same type o f diaphrams were used as with the
His bins were constructed of wood, smooth s te e l,
and corrugated steel (th e corrugation being h o riz o n ta l).
Jamieson emptied his bins from both the bottom and the side.
He obtained a maximum v e r t ic a l. pressure increase o f 7.3$ when f i l l
m aterial was emptying from the bottom.
During side emptying the
14
la te ra l pressure decreased on the side of the opening and increased
on the opposite s id e .
Both L u fft and Ketchum reported th is same
s itu a tio n fo r side emptying.
In addition Jamieson experimented with a surcharge on the
f i l l e d b in .
He placed two 50 lb . weights on top the f i l l
observed only a s lig h t Increase in v e rtic a l pressure.
and
Upon removal
o f the weights the pressure returned to normal in dicating granular
m aterial behaves as an e la s tic s o lid when confined*
Jamieson made a very in te re s tin g v ib ra tio n te s t by tapping
the bin with a hammer a fte r I t was f i l l e d .
He observed a decrease
in v e rtic a l pressure on the base by tapping on the side near the
bottom of the b in .
As he continued tapping upward on the wall the
pressure returned to normal and then increased to a maximum when
he tapped on the side a t the top o f the b in .
This tapping a t the
top also gave a settlem ent o f f i l l m aterial o f 2 to 3 in .
This
author observed the same re su lts from v ib ra tio n .
When he placed t i e rods, o f 24 gage sheet steel 0 .5 in . wide,
horizontal Iy in the bin Jamieson found no increase in pressure but
only a decrease in v e lo c ity of emptying g ra in .
These t i e rods were
placed.w ith the 0*5 in . dimension in a v e rtic a l plane.
Jamieson reported values for wheat from his c o e ffic ie n t of
fr ic tio n te s ts o f wheat on wheat
bin o f
0,424 to 0 .4 5 0 .
0.532;, wheat on cribbed wood
Jamieson also reported he found no s ig n if i-
15
cant d iffe re n c e between his model bin te s t re su lts and his f u ll size
bin te s t re s u lts .
Jamieson designed a grain e le v a to r in M ontreal, Canada, using
his te s ts re s u lts and Janssen's equations.
Nis design was condemned
by a panel o f experts hired by the Board o f Harbor Commissioners on
the basis i t was not strong enough to support the pressure of a
flu id having a density equal to th a t o f the g ra in .
This prompted the Board of. Harbor Commissioners to re ta in
Henry T . Bovey to run experiments o f fu ll sized grain elevato rs to
check Jamieson's design.
Using a te s tin g apparatus s im ila r to Jamieson's, Bovey ( 6)
ran te s ts on an e le v a to r a t M ontreal> Canada, and one a t Quebec,
Canada.
The Montreal bin was constructed o f wood with a 12 f t . by
14 f t . rectangular cross section, and the Quebec bin was of cribbed
wood construction with a 13 .4 f t . by 12.35 f t . rectangular cross
section*
Bovey' s re su lts v e rifie d Jamieson's fin ding s.
In addition to checking Jamieson's re s u lts , Bovey checked
th e diaphram method by using diaphrams o f d iffe r e n t s ize s , the
smal le s t 26 sq. in . and the largest I IO sq^ in .
diaphram s ize had l i t t l e
He found th a t
i f any e ffe c t on pressure determ inations.
Bovey: also placed four d Iaphrams in the base of the Quebec
b in , equally spaced from center to w a ll, to determine v e rtic a l
pressure d is trib u tio n s .
Ketchum re p lo tte d these s ta tin g "the
16
grain mass producing bottom pressure might be represented by a
portion o f an e llip s o id o f re v o lu tio n , with major axis o f the
el I ipse v e r t ic a l" .
In Buenos A ire s, Argentina, in 1902 and 1903, Eckhardt L u fft
(1 6 ) made experiments of fu ll-s iz e d c ir c u la r bins of concrete con­
s tru c tio n .
Two bins were te s te d , one of 23.83 f t . diam eter, the
other 11,25 f t .
Both were 54.8 f t . high.
Pressure c e lls s im ila r
to those of Jamieson and Bovey were used.
For an o u tle t a t the
s id e , la te ra l pressure increased on a c e ll adjacent to the o u tle t
when the gate was slowly opened, and decreased when the gate was
ra p id ly opened.
L u fft found no increase in la te ra l pressure when
the grain was flowing*
L u fft found th a t when a few hours passed between f i l l
ments, there was a d e fin ite decrease in la te ra l pressure.
Incre­
He gave
no re s u lts for v e rtic a l pressure fo r th is condition,
Using bins of d iffe r e n t m a te ria ls , J* P leissner (19) in
1902 through 1905 made bin experiments in Dresden-Plauen, Germany.
H is bins were o f wood and also of concrete.
He used cribbed timber
construction and also v e rtic a l plank construction in his wood bins.
Several grains including wheat were tested In these bins
which were from 9 to 18 meters high and from 1,5 m. square, to 2.5 m.
by 3.15 m. of rectangular cross s e ctio n .
Pleissner determined bis
pressure from the d e fle c tio n of simply supported pitch pine planks.
17
Later c a lib ra tio n o f these deflectio n s gave him his pressures.
P ie is s n e r's k values' were determined from the r a tio o f la te r ­
al to v e rtic a l pressure as measured in the bins*
He found k to vary
with the depth and to be 0 .30 to 0 .3 5 , and 0.34 to 0 .4 6 , for wheat
in a concrete bin and wood plank bin re s p e c tiv e ly .
concrete bin.and wood plank bin he reported
re s p e c tiv e ly .
For wheat in a
p 1 to be 0.71 and 0.25
I t was not stated how p* was .determined.
For wheat in motion Pleissner found la te ra l pressures as
much as two times as great as s ta tic pressures.
These pressures
were on the opposite wall from a bottom wall gate.
In Ketchum1S discussion of the preceding in v e s tig a to r’ s
re su lts he states the follow ing conclusions.
1.
The la te ra l pressure of grain on bin walls is less than
the v e rtic a l pressure and increases very l i t t l e a fte r a depth of
2.5 to 3 times the width or diameter o f the bin is reached.
2.
The r a tio o f la te ra l to v e rtic a l, pressure, k> is not a
constant but varies with d iffe r e n t grains and d iffe r e n t bins.
The
value o f k can only be determined by experiment,
3.
The la te ra l pressure of moving grain Is only s lig h tly
greater than the pressure o f grain a t re s t (maximum v a ria tio n for
r
ordinary conditions is.probably 10# ) *
4.
Discharge gates in bins should be located a t or near the
center o f the bin base.
18
In 1928, W .W . Hay ( I I ) gave recommendations fo r design of deep
c ir c u la r bins.
Using Janssen's theory he expressed equations for max­
imum la te ra l and v e rtic a l pressures.
fo r c o e ffic ie n t o f f r ic t io n
May recommended s p e c ific values
(p' ) for d iffe r e n t f i l l m aterials but made
no e f f o r t to specify p f what m aterial the bin was constructed fo r de­
term ination of ^ i'.
In a drawing of a ty p ic a l b in , however, he shows
concrete construction.
In 1943, Marcel M. Reimbert ( 8 ) made some bin studies in
France.
His studies were made on steel s ilo s 13.5 f t , square and.
33 f t . deep.
E le c tr ic s tra in gages were used on peripheral straps
around the b in .
pressure.
This peripheral s tra in was converted to la te ra l
From his f i r s t te s ts he found la te ra l emptying pressures
up to I .45 times the s ta tic pressure, fo r bin opening a t center of
bottom.
He reduced th is emptying pressure In his second te s ts by
placing a perforated tube down the center of the bin to the empty­
ing gate.
These te s ts were run on an octagonal si Io 71.5 f t . deep
with sides of 6 .4 f t .
On- these te s ts he found no Increase of
emptying pressure over s ta tic pressure thereby recommending a per­
forated tube fo r emptying bins.
L . R. Amudson ( 3 ) , in 1944, ran te s ts on bins belonging to
the U. S. Department of A g ricu ltu re a t Jamestown, North Dakota.
These bins were IO f t . deep and had a 9.5 f t . radius base,
He used
19
SR s tra in gages on bands around periphery to determine la te ra l
pressures.
Me found Janssen's solution to be safe for wheat using w =
49 p c f, yu1 = 0.400 and k = 0*50.
Mis methods fo r determining k
and y i' are not given, but presumably he determined k from bin te s t
pressures and
p ' from a f it t e d curve using Janssen's equations.
Robert A. Saul (24) conducted te s ts on a bin a t Iowa Stgte
C o lleg e.
This bin was 12 f t . by 32 f t . in rectangular cross sec­
tio n and 10 f t . deep.
I t was constructed of 2 in . by 6 in . studs
lined with car s id in g .
By making his studs r ig id he could determine the s tra in on
peripheral steel rods connecting them by means o f e le c tr ic s tra in
gages.
These s tra in s he converted to la te ra l pressures.
Mis conclusions from his te s ts were:
1.
Method o f f i l l i n g w ill
influence the d is trib u tio n of
grain pressure on w alls and flo o r ,
2.
F l e x i b il it y of wall Influences the pressure d is tr ib u tio n .
The more r ig id the wall the greater load i t w ill c a rry .
A lfred Ci Scheer (25) and Calvin W. Tooles (3 1 ), in 1950,
under the d ire c tio n o f Robert A, Caughey ( 7 ) , made investigations
o f the la te ra l and v e rtic a l pressures of d iffe r e n t m aterials in ­
cluding wheat, shelled corn, soy beans, cement, sand, and pea
g ra v e l.
The bin used was 5 f t , deep and 1.5 f t . in inside diam eter.
20
The base o f the bln was supported d ire c tly .b y a platform s c ale .
By means o f steel plates in holes in the bin wall connected.by rods
to a peripheral s ta in le s s steel s tr ip the la te ra l pressures were
determined*
The rods were also connected to a c a n tile v e r bar ex-
tended from bottom Sf b in .
This arrangement held the p lates in
po sition without contact with the holes.
Tension in peripheral
s trip s was determined by 8R4 s tra in gages*
By c a lib ra tin g the
tension In the band by means o f a spring balance the la te ra l
pressure was determined.
Rising a value of k equal to the average r a tio o f la te ra l to
v e rtic a l pressure and c o e ffic ie n ts o f .f r ic t io n determined from
separate te s ts , Tooles (7 ) concluded;
I*
Shelled corn# soy beans# sand# and pea gravel do not
follow Janssen's theory# but the values of k from experimental
re s u lts y ie ld re s u lts which are on the safe side*
(Tooles' 0 '
values fo r corn, soy beans* sand, and pea gravel appear to be in
e r ro r.
With c o rre c t values of 0 ' he may well have concluded th a t
these m aterials do follow Janssen's th e o ry ).
2.
There is a s lig h t increase in la te ra l pressure on the
side of a grain bin opposite a side emptying hole when the grain
s ta rts to flow .
A fte r the Missouri River had flooded several grain elevators
which subsequently bu rst, a very in te re s tin g experiment was c arried
out by A. G* Dale and R. N. Robinson (9 ) in 1954.
Rising a bin
21
5 f t . deep of 14 gage sheet steel ro lle d Into an 18 In . diameter
c y lin d e r they found pressures caused by corn sw elling in a b in .
Using pressure c e lls s im ila r to Jamieson they f i l l e d the
bin with corn with a I# moisture content.
By pumping humid a ir
for 96 hours they Increased the moisture content to 4#. I n i t i a l l y
was
the load, on the baseA286 lb s . and a fte r 96 hours of pumping i t had
increased to 430 lb s.
(th e to ta l weight of the c o rn .)
This
occurred on t h e ir f i r s t te s t when the base and w alls were supported
s e p a ra te ly . .
For t h e ir second te s t the base and w alls were fastened so l­
id ly and the bln was f i l l e d with corn.
The corn was then flooded
with water fo r 10 minutes and then drained.
The moisture content
was Increased approximately 10$ and pressures were increased by a
fa c to r of 10.
John K. Rudd (2 3 ), in 1954, made a series o f te s ts to d e te r­
mine how m aterials flow out o f bins*
P relim inary te s ts were run to
determine la te ra l and v e rtic a l pressure*in d iffe r e n t shaped hoppers
Janssen's equation was in good agreement with s tra ig h t sided hop­
pers but was found inadequate for hoppers with battered sides.
Rudd thought a m odification o f Janssen's solution could be
used fo r battered hoppers to determine pressures and flow r a te .
For use in th is equation he determined k fo r d iffe r e n t m aterials
using a confined compressive te s t c y lin d e r,. He gave no values for
these k determ in atio n s.-
22
For his te s ts on flow Rudd used a bin th a t was cut v e r tic a lly
down the middle, and placed "a piece o f glass over th is exposed c u t.
Then, placing d iffe r e n t colored layers o f m aterial In the bin and
takin g moving pictures o f th is m aterial flow ing, he could determine
the flow p a tte rn ,
I
From his t e s t re s u lts Rudd stated the follow ing conclusions:
1.
A central column of flow e x is ts above the discharge
opening and is as wide as the opening.
Jf m aterial was being put
in to the bin a t the same ra te i t was being withdrawn, m aterial out­
side the area of the central column consequently would remain s t a t ic .
2.
M a terial flows in to the central column area as soon as
the central column is discharged.
In I960, a t Montana'State C ollege, Wesley Boulanger (5 )
conducted experiments to determine the influence o f base move­
ment, r e la tiv e s iz e of p a rtic le s o f f i l l
m a te ria l, and shape of
bin on v e rtic a l pressure.
Using model wood bins with base supported on a scale he
concluded:
1.
A movement o f the base of _+ 0,0020 in . w ill re s u lt in a
v a ria tio n in the base load o f less than + 10$ .
2.
Varying the r a tio of the sides o f a rectangular bin ,
hydraulic radius remaining constant, does not cause any s ig n ific a n t
v a ria tio n in the average base pressure o f wheat.
(This agrees with
23
Janssen’ s theory, fo r with the same f i l l m aterial and bin construc­
tio n , a t large depths, pressure varies d ir e c tly with the hydraulic
r a d iu s ).
The values o f .k , 0 , and 0 ’ given by the various investigators
w ill be compared with the re su lts o f th is in vestig ation in a discus­
sion of re s u lts .
)
)
24
OBJECTIVES
The o b jectives o f th is thesis can be summarized as an attempt
to answer the follow ing questions.
1,
What is the re la tio n s h ip between k and
p' for any f i l l
m aterial confined in a bin?
2,
Can yu' for a f i l l m aterial confined in a bln be accurate­
ly estimated by simple, laboratory fr ic t io n te s ts , and I f so, what
are the mathematical relationships?
3,
What e ffe c t does v ib ra tio n o f the b in , and m aterial
flowing from the bin have on the la te ra l and v e rtic a l pressures
e x is tin g w ithin the bin?
Most p rio r in vestig ato rs have concentrated on obtaining k
and ^ur for s p e c ific m aterials in speci f ic bins and have reported
very l i t t l e data for an approach to the problem o f how to estimate
k and ya' and, thereby, la te ra l and v e rtic a l pressures in a bin,
w ith o u t're s o rtin g to bin pressure studies in a te s t b in .
In th is in vestig atio n a te s t bin was used to determine, as
accurately as possible, values of L and V from which co rrec t values
o f k and ^u1 were obtained,
As one approach in determining the re la tio n s h ip between k and
yu', Scheer’ s th e o re tic a l equations for estim ating k were In v e s ti­
gated.
To estim ate "a"* (V ’ /V ) , in his equation, the bin base was
composed of two square concentric p la te s , the peripheral p la te to
25
measure the approximate v e rtic a l u n it pressure a t the bin wall (V 1)
and both plates to measure the average v e rtic a l u n it pressure (V ).
In th is manner i t was hoped to obtain representative values
o f "a" to use in Scheer's equations for obtaining k more a ccu rately.
As a second approach to the re la tio n s h ip between'^Rir^lid ^li1,
t h e ir values for d iffe r e n t f i l l m aterials were plotted.on a graph
o f k versus yn'.
A f it t e d curve was then passed through these
points to empirical ly determine the re la tio n s h ip between k and yii*.
Three d iffe r e n t laboratory fr ic t io n testswere used to e s t i­
mate yu and y j*.
and ^ i*.
These were compared with the bin te s t values of
In th is way te n ta tiv e empirical relatio n sh ip s were develop­
ed to i nterre I ate the la b o ra to ry fr ic t io n te s t values and
I f these te n ta tiv e em pirical re la tio n s h ip s prove v a lid in
the fu tu re , when tested against a wider range of bins and f i l l
m aterials# the problem o f determining la te ra l and v e rtic a l pres­
sure for a m aterial in a bin w ill only involve making simple labor­
atory fr ic t io n te s ts on the m aterial ra th e r than extensive bin
pressure te s ts in a te s t b in .
For determining the pressure e ffe c ts when m aterial was
flowing from the b in , and when the bin was v ib ra te d , la te ra l and
v e rtic a l pressures were measured when the bin was emptying, and
a fte r the f i l l e d bin had been vibrated by tapping with a rubber
m alle t ,
26
EXPERIMENTAL EQUIPMENT AND MATERIAL
/
Test Bin
The bin and supporting frame (shown in Figs. I and 2) used
in th is experimentation were constructed of selected pine.
The-main body of the bin was 47 in . deep and consisted of
two nominal I in . by 8 in . boards between two I in , by 12 in .
boards, which formed an inside cross section approximately 7 ,5 in .
square.
The overlap of the 12 in . boards formed lip s which were
used to attach the bin to a supporting fram e.' Pressure measuring
plates were placed on the bottom and one side o f the b in .
A" 4 in ,
by 6 in . emptying gate was placed a t the bottom of one bin wall ^
The emptying gate was in the opposite wall from the side pressure
p la te .
The f i r s t series o f te s ts were run In th e ’ bin using the
natural surface fo r a smooth bin co n d itio n .
To obtain a rough bin
condition fo r the second series of te s ts 2/ 0* sandpaper was glued
to the inside of the bin and p la te s ,
. The supporting frame was constructed of 2 in . by 6 in ,
boards, forming a long horizontal rectangular opening, the short
side of which ju s t accepted the width of the b in .
The long sides
o f the frame gave s t a b ili t y fo r the bin and a securing place for
the long I ever arms attached to the base p la te s .
The bin was
secured in th is opening by screws and s ta b iliz e d with braces from
759 IN. SQUARE.
7.46 IN. SQUARE.
SECTION 5 - 6
SIDE PLATE
LEVER ARMS
SIDE
PLATE
SECTION A-A
STRAIN
GAGE
ALUM
STRl P —r
HINGE
HANGERS
LEVER
ARMS FOR
BASE ffiS
F I GU R E
I
T E S T BIN
SECTION C-C
28
FIG. 2
DETAILS OF TEST BIN
29
the top of the bin to the ends of the supporting frame. The base o f th e bin consisted o f two concentric square
p la te s .
A clearance o f 0.04 in . was maintained between outside
periphery of the center p la te and the inside periphery o f the
edge p la te .
The edge p la te had a clearance of 0.06 in . with the
bottom edges o f the bin w a lls .
The center and edge p la te were
supported s e p a ra te ly , the edge p la te by two I ever arms and the :
center p la te by one.
These I ever arms were secured a t one end by
hinges on the bin frame and on the other end by aluminum s trip s
secured to the b in .
SR4 e le c tric a l s tra in gages were attached
to the aluminum s trip s to determine pressure on the base p la te s .
The cross sections shown in F ig . I give the d e ta ils o f th is con­
s tru c tio n .
The base was constructed of these two plates to ex­
perimental Iy determine approximate values of "a” .
The side p la te was fastened to two I ever arms which were
hinged near the top of the b in .
The bottom side of the. p la te was
fastened to the bin by a th in s tr ip o f aluminum to which was
attached an SR4 s tra in gage.
A 0.04 in . clearance was maintained
between the side p la te and the b in .
gangers were attached to the bottom o f the base p lates for
use in c a lib ra tin g the s tra in gages and for applying s ta b iliz in g
preloads.
A hanger was attached to the side p la te fo r the same
purposes by means ©f a p u lley attached to the frame.
These
30
hangers and p u lle y are shown in Figs. 1, 2a and 2b>j. Al I three of
the aluminum s trip s were subjected to a s ta b iliz in g i n i t i a l tension
before c a lib ra tin g or loading the b in .
C a lib ra tio n curves were
drawn to r e la te the change in te n s ile s tra in to the pressure on
the p la te s .
j
Bin C a lib ra tio n — To c a lib ra te the bin fo r pressure measurements,
known loads, in addition to the s ta b iliz in g preloads, were applied
to the hangers and the corresponding s tra in s of the aluminum s trip s .
recorded.
The loads were applied in 2 lb . increments, up to a to ta l
load of 20 lb s. on each p la te .
The p lo t o f load versus s tra in was
then used as a c a lib ra tio n curve to determine load from known s tra in
during bin te s ts .
These c a lib ra tio n curves are shown in F ig . 21
through F ig . 23, Appendix I I I .
The hangers were symmetrically attached to the edge and base
p la te s , so th a t the applied c a lib ra tio n loads were acting through
the center o f the bin base area.
This allowed a determination of
average v e rtic a l pressures fo r bin te s ts by dividin g the load,
determined from the c a lib ra tio n curves, by the area over which i t
acted.
See F ig . I for d e ta ils of these hangers.
The v e rtic a l
load used to c a lib ra te the side pressure plate
was applied across a p u lley to an extension of the aluminum s trip
holding the side p la te to the b in .
The d e ta ils of th is arrange­
ment are shown in sectionG-0 of F ig . I and F ig . 2a.
The p u lley
31
used in c a lib ra tin g was found to be 96.5# e f f ic ie n t , due to pulley
f r ic t io n , so only th is percentage o f the applied load was used to
p lo t the c a lib ra tio n curve.
To c a lc u la te la te ra l pressure a f t e r te s tin g , a summation of
moments was taken about the hinges on the side p late I ever arms.
The hinges were In lin e with the inside surface o f bin.and side
plates so th a t the fr ic t io n force acting on the side p la te caused
no moment about the hinges.
The tension in the aluminum s tr ip was
known from the c a lib ra tio n curve, and the la te ra l pressure was then
determined from the summation o f moments.
Laboratory Apparatus for Estim ating 0 and 0 ’
For determining 0 (angle of repose) fo r the f i l l
a box, open a t the top and one end, was used.
The inside dimen­
sions o f th is box were 8.75 in . by 10,25 in . by 24 in .
is shown in F ig . 3a.
m a te ria l,
This box
A hinged p la te was attached to the open end
which, could be fastened securely during f i l l i n g o f the box.
p la te could also be removed without ja r r in g the box*
This
This would
al l ow t he f i l l m aterial to flow from the end of the box.
Also fo r estim ating 0 and 0 * , a combination d ire c t shear
and d ire c t fr ic t io n device was b u i l t .
te s tin g are given as 0g and 0f
The re su lts obtained during
re s p e c tiv e ly .
The d ire c t shear device consisted o f two boxes.
The top
box was open, top and bottom, and attached to a s trin g to which
32
a - ANGLE OF REPOSE BOX
FIGURE 3 A P P A R A T U S FOR FRICTION TESTS
33
a load could be a p p lie d .
/
The bottom box was open a t the top only
and was secured to a s o lid mounting plank.
Using a clearance be
tween the two boxes to maintain shear in the f i l l m a te ria l, the
device was f i l l e d , and a horizontal
load was applied through the
attached s trin g u n til s lid in g occurred.
F ig . 3b shows th is device.
The d ire c t shear device was a lte re d to make the d ire c t
fr ic tio n device by replacing the bottom box with a piece o f a
board used in construction of the b in .
A fte r fr ic t io n te s ts were
run on the p la in board, 2 /0 sandpaper was.glued to i t and f r ic tio n
te s ts were run on t h i s .
F ig . 3c shows th is d ire c t fr ic t io n device.
The boxes used in both the d ire c t shear te s t and d ire c t
fr ic tio n te s t contained b a ffle s to minimize progressi ve .y ie ld in g .
M a te ria ls
Five m aterials were te s te d , four In the bin without sand­
paper, and a ll
fiv e In the bin with sandpaper.
They consisted of
glass marbles, two types of wheat, a coarse angular sand, and ah
angular I i ghtweig ht coarse aggregate.
The glass marbles were selected to represent m aterials with
a very low c o e ffic ie n t of f r ic t io n .
The marbles were 0.5 in . in
diam eter, o f c le a r glass, and very uniform in shape.
The two types o f wheat used w ill be designated as old wheat
and new wheat.
The old wheat had been used for previous bin
studies and was d ir ty and contained many broken p a r tic le s .
The new
/
34
wheat was very hard anql clean with sharp beards„
These two types were used to see what v a ria tio n might e x is t
in bin pressures not only fo r two d iffe r e n t types of wheat, but
also fo r fresh , newly reaped wheat and o ld e r, much handled wheat.
These two wheats have f r ic t io n angles which f a ll between those of
the marbles and the aggregate.
Both th e sand and the lig h t aggregate were sharp, machine
crushed m a te ria ls .
They d iffe re d mainly in weight, however, the
lig h t aggregate was composed o f larger p a r tic le s .
The sand was
obtained from the concrete laboratory aggregate bins a t Montana
S tate Gol lege.
a No. 14 s ie v e .
The sand passed a No, 4 sieve and was retained on
The lig h t aggregate was a commercial cinder
m aterial and passed a sieve o f 0.525 in . opening and was retained
on a No. IO s ie v e .
Both m aterials were q u ite d ir ty causing consid­
erab le dust when emptying from the b in .
The bulk u n it weights ( l b . per cu. f t . ) of these m aterials
were 9 1 .5 , 5 2 .3 , 5 1 .0 , 8 3 .0 , and 38*8 re s p e c tiv e ly , fo r glass
marbles, old wheat, new wheat, sand, and lig h t aggregate.
35
EXPERIMENTAL PROCEDURE
Al I m aterials were placed in to the bin without impact by means
o f a No. IO can with a hole in the bottom.
The can had a four foot
wooden handle by which i t was lowered in to the b in .
When the proper
depth fo r depositing was reached, a piece o f clo th covering the hoik
was pulled away by means o f a f le x ib le w ire , and the m aterial gently
deposited I t s e l f .
The bin was f i l l e d in increments fo r each m a te ria l.
The f i r s t
increment was 8 in . In depth, th is being the distance from the base
to the bottom o f the side p la te . Each successive increment was equal
■H
in height to the width o f the b in , 7,59 in . in the smooth bin condi­
tio n and 7.48 in . in the.rough bin co n d itio n ,
including the f i r s t
8 In . increment a to ta l o f 6 Increments gave a to ta l depth o f 45.95
in , and 45.40 in . re s p e c tiv e ly , fo r the smooth bin condition and
rough bin c o n d itio n .
Enough marbles were a v a ila b le fo r only four
increments.
Readings o f a ll three s tra in gages were taken with the bin
empty and a f t e r each increment was placed.
A fte r the bin f i l l i n g was
completed the bin was tapped with a hammer to simulate v ib ra tio n ,
u n til a maximum s tra in reading occurred.
Readings were taken also
when the side emptying gate was opened and when the m aterial was flow­
in g ,
The s tra in s from v ib ra tio n and emptying were not always taken as
two people were re q u ire d .
(Two people were not always a v a ila b le ).
36
Spot checks were usual I y made a fte r every two or three runs to
catch any deviation from the o rig in a l c a lib r a tio n .
With both sand and
lig h t aggregate, as weI I as occasionally with wheat, p a rtic le s were
wedged in to the clearance spaces o f the plates a fte r the te s t was run
and the bin emptied.
A fte r cleaning the bin , the c a lib ra tio n was
checked and i f much deviation from the o rig in a l c a lib ra tio n curve was
present the gages were re c a lib ra te d and new c a lib ra tio n curves were
drawn fo r the next t e s t .
The procedure fo r determining the angle o f repose was qu ite
sim ple.
The angle o f repose box was f i l l e d with the m aterial to be
tested with the hinged end secured in place.
The m aterial was placed
loosely with a scoop so th a t i t had e s s e n tia lly the same density in
the box as i t had in the b in .
When the hinged end was removed a wedge
o f m aterial flowed out leaving a smooth plane, in most cases.
The
angle which i t made with the horizontal was taken as the angle of
repose.
In the te s ts with marbles a d e fin ite plane did not occur.
The surface remaining a f t e r the end of the box was removed was a con­
cave surface.
The value of 0 was taken as the slope o f a lin e drawn
from the lower to the upper end of th is surface*
The sides o f the angle o f repose box had no apparent e ffe c t on
the slope o f the m a te ria !.
The surface o f m aterial lay in a continuous
plane from one side o f the box to the o th e r.
When using the d ire c t shear device, the top box and bottom box
37
were separated by shims o f 0.04 thickness Iri rear and 0 ,10 thickness
in fr o n t.
This d if fe r e n t ia l clearance compensated for a tendency of
the top box to drop down a t the fro n t during a shearing t e s t .
A fte r the ghear boxes were set up with proper clearance, they
were gently f i l l e d and leveled a t the top *
The shims were removed
and force was applied by slowly f i l l i n g a bucket, hung from the
p u lle y , with water.
(See F ig . 3 b ).
The p a rtic le s of m aterials would
read just themselves by y ie ld in g a very small amount as an Increasing
force was a p p lie d .
The maximum value o f fr ic t io n was determined
when the m aterial yielded to permit the top box to s lid e fr e e ly .
The
c o e ffic ie n t o f f r ic t io n tyas) was taken as the r a tio o f the to ta l load
required to shear the m aterial divided by the to ta l weight acting on
the shearing plane,
.
Sheet metal b a ffle s were placed in both upper and lower boxes
to minimize progressive y ie ld in g .
Without the b a ffle s the m a te ria l,
when i t sheared, would hump up in the back o f the top box and slump
down In the fr o n t.
An increase in values of 0S of approximately one
degree occurred in the te s ts a fte r b a ffle s were added, in d icatin g
additional fr ic tio n a l resistance due to added confinement.
The 0S
values reported in th is th e s is were from te s ts with b a ffle s in the ,
boxes*. The clearance used In the shear box was varied from 0,04 to
0.15 and indicated no apparent v a ria tio n in the values of 0S obtained.
38
T e s ts -fn the d ire c t fr ic tio n device were run using a method
id en tic a l with the d ire c t shear te s ts except the lower box was replaced
with a piece of the 12 In . board used in constructing the b in .
F ig . 3 c ).
/
(See
39
EXPERIMENTAL RESULTS
The re su lts of the bin pressure te s ts are shown in F ig . 4
through F ig . 12.
From the c a lib ra tio n curves the loads on edge and
center plates were determined fo r the s tra in s recorded a f t e r each Incre­
ment was placed.
The edge p la te load when divided by the edge p late
area gave the average v e rtic a l edge pressure ( V ) fo r a depth of f i l l
m aterial equal to the to ta l depth of Increments placed.
S im ila rly
the average v e rtic a l u n it pressure (V) was determined by dividin g the
sum o f edge and center p la te loads by to ta l bin base a rea .
age r a tio of V / V
The aver­
fo r each m aterial is recorded in Table I for both
smooth and rough bin conditions, as a ' , an approximation of Seheer11s
"a".
A pressure d is trib u tio n diagram shape Had to be assumed for
the la te ra l
placed.
load on the side pressure p la te a fte r each increment was
The logical .diagram would be o f the same shape as the v e r t i­
cal pressure diagram.,. A fte r s ta rtin g the calcu latio n s fo r la te ra l
pressures I t was found th a t, assuming a tria n g u la r d is trib u tio n for
the f i r s t increment and one o f rectangular shape fo r subsequent in crements, The re s u ltin g deviatio ns, from pressures determined using a
d is trib u tio n s im ila r in shape to the v e rtic a l pressure d is trib u tio n ,
were n e g lig ib le a t depths greater than two bin widths.
The re s u lta n t o f the side p la te pressure was computed from a /
summation, o f moments about the hinges, using the distance from the
hinges to it s point o f ap p lic a tio n as it s moment arm.
The only other
•
40
D E P T H OF F IL L
( lN .)
—©— E X P E B I M E N T A L V E B T I C A L P E E S S U B E
—<2>— E X P E R I M E N T A L L A T E R A L P E E S S U B E
--------- J A N S S E N F I T T I N G
P R E S S U R E (L B /S Q .F T )
FIG U R E 4 - P R E S S U R E OF OLD WHEAT IN SMOOTH BIN
41
- O - E X P E R I M E N T A L V E R T IC A L PRESSURE
-9E X P E R IM E N TA L LATERAL PRESSURE
--------J -A M S S E M F I T T I N G
IO —
IO
70
PRESSURE
FIG U R E
30
( L B /S Q .F T .)
40
5 - P R E S S U R E OF N E W WHEAT IN S M O O T H
BIN
42
E X P E R IM E N T A L VERTICAL P R E S S U R E
- O - E X P E R IM E N T A L LA TER AL PRESSURE
-------- J -A N S S E N F I T T I N G
P R E S S U R E (L B ./ SQ.FT)
FIG U R E
Q3
- P R E S S U R E OF G L A S S M A R B L E S IN SM O OT H B I N
43
- O - E X P E R IM E N T A L V E R T IC A L P R E S S U R E
- 0 - E X P E R I M E N T A L L A T ER. A L P R E S S U R E
------- J A N S S E N F I T T I N G
P R E S S U R E ( L B ./SQ. FT.)
FIG U R E 7 - P R E S S U R E
OF S A N D IN
S M O O T H e>l N
44
-O E X P E R IM E N T A L VERTICAL P R E S S U R E
-O E X P E R I M E N T A L L A T E R A L Pl?E S S U R E
------- J A N S S E N F I T T I N G
P R E S S U R E (L E )/S Q .F T )
FIGURE S
- P R E S S U R E . OF O L D W H E A T IN ROUGH B lN
45
— G>- E X P E R I M E N T A L V E R T I C A L P R E S S U R E
-O E X P E R IM E N TA L L A T E R A L P R E S S U R E
-------- J A N S S E N F \ T T \ N G
PRESSURE
(L E / SQ FT)
F I G U R E 9 - P R E S S U R E OF N E W W H E A T IN ROUGH B I N
46
—O - E X P E R I M E N T A L V E R T I C A L P R E S S U R E
- 0 - E X P E R IM E N T A L LATERAL P R ES SU R E
-------- J A N S S E N
FITTING
20
40
GO
P R E S S U R E ( L B . / SG). F T )
F I G U R E IO - P R E S S U R E
OF M A R B L E S
80
INI R O U G H B I N
47
—O— e x p e r i m e n t a l v e r t i c
-O e x p e r im e n t a l
l a t e r
---------J A N S S E N F I T T I N G
a l
p r e s s u r e
a l
p r e s s u r e
IO —
IO
FIGURE
20
PRESSURE
I I - PRESSURE
30
(L B ./S Q .F T .)
OF S A N D
40
IN R O U G H B l N
48
-G —O --------
E X P E R I M E N T A L V E R T I C A L PRE SSUEE
E X P E R IM E N T A L LA TER AL PR ESSU R E
TANSSEN FITTING
PRESSURE
(L B ./S Q .F T .)
FI GU RE 12 - PRESSURE OF LI GH T AGGREGATE
I N RO UG H © I N
49
moment needed fo r the c a lc u la tio n was caused by the tension change in
the aluminum s tr ip which was obtained from the side p la te c a lib ra tio n
curve using the recorded s tra in s .
D ividing the re s u lta n t load by the area of the side p la te gave
the average la te ra l u n it pressure acting on the p la te .
With n e g lig ib le
e rro r th is was assumed equal to the actual u n it pressure a t the center
o f the p la te .
A fte r p lo ttin g the la te ra l and v e rtic a l pressures from te s t/
re s u lts , a curve, obtained from a t r i a l and e rro r solution of Janssen’ s
v e rtic a l pressure equation, was! f it t e d to the experimental v e rtic a l
pressure data.
This procedure determined the value o f kp' necessary
to f i t Jannsen’ s equation to the experimental data and gave an excel­
len t f i t with the observed pressure curves, e s p e cia lly a t depths
'
greater than two bin w idths. The value o f k was taken as the. average
r a tio of la te ra l pressure to v e rtic a l pressure from the bin te s t pres­
sure curves, taken a t three depths.
The"maximum deviation from the
average k a t any o f these depths was + 2 .5 $ ,
Knowing tyi’ from the fitted'Janssen curve, and k from the ra tio ,
of la te ra l to v e rtic a l pressure from bin te s ts ,
was computed for each
m aterial in both smooth and rough bin conditions.
A fte r having determined
l . e . tan 0 ”, for each m aterial in
both smooth and rough bin conditions, k ' was calculated from Scheer's
equations.
For use in these equations 0', as determined from bin te s ts ,
was taken as the co rrec t 0 *in a smooth bin cond itio n, and as tbd correct
50
0 I r a rough bin con ditio n (by d e f i n i t i o n o f a rough bin condition
0 = 0*).
T h e . r a t i o o f k to k* was taken as th e c o r r e c t value o f Scheer6S
" a " .'
Values o f tyi6, k ,
6, 0 ’ , a 6, a , and k * are recorded in Table I .
Also recorded in Tab le I are values o f 0r , 0S„ add 0^.»
The values o f 0 r are slope angles o f f i l l
found by angle o f repose t e s t s .
m aterial surfaces
The values o f tan 0g and tan 0^ are
th e r a t io s o f the load necessary t o shear the. m aterial t o the to ta l
normal
load a c tin g on the shearing plane.
The normal
loads app lied during th e d i r e c t shear and d i r e c t f r l c ■ %
t i on t e s t s were used t o give normal pressures as close as possible to
the actual
la te ra l
pressures occurring in the bi n.
In Table 11 percent increases o f s t a t i c
pressures are given fo r m aterial
h o le , and m aterial v ib r a t e d .
sures a t large depth,
la te r a l and v e r t ic a l
in motion, flowing from sid e emptying
Al I pressure increases given are fo r pres­
TABLE I - RESULTS 0F BIN & FRICTION TESTS
SMOOTN BIN CONDITION
Marb Ies 0 . Wheat N,Wheat
Sand
a n Cf?
0.173
0.530
0,326
18.0
8 3.0
0.193
0.410
0.471
25.2
" 91 .5
0.107
0.580
0.185
10,5
W
k/V
k
K
W
52.3
. 0.168
0.470
0.358
19.7
0=0«
0f
.yuf
0S
/is
0r
Pr
(D ire c t F ric tio n Test)
= :.tan 0 f ,
(D ire c t Shear Test)
* .tan 0S
(Angle o f Repose Test)
» tan 0r
.
14.3
0.255
19.7
0.358
28.0
17.1
0.308 . 0.531
-
__________ ROUGH BIN CONDITION
Marbles 0 . Wheat, . N.Wheat , .Sand
L . Aggr.
91,5
0 .1 60
0.500
0.320
52.3
0.256
0.430
0.593
SJ .0
0.232
0.456
0.508
8 3 .0
0.326
0.370
0.880
38.8
0.297
0.410
0.725
17.7
30.7
26.9
41 .4
36.0
17.3
0.312
35.4
0.717
- 12.5
0.222
28.0
0.531
28,5
0.542
28.0
0.53I
24.6
0.457
24.5
0.455
25.0
0.466
35.5
0.714
37.4
0.764
37.3
0.761
40,1
0.841
40.8
0.861
40.1 .
0.841
k 1 (From Bin 0 & 0 ' )
0.844
0.576
0,624
0.428
0,830
0.587
0.660
0.393
0.486
a = k/k* ( from bin)
a" (From Base P la te s )
0.687
0.847
0.816
0.835
0.848
0.829
0.958
0,935
.0 .6 0 3
0.870
0.733
0.700
0.691
0.742
0.942
0.881
0.843
0.940
-
.
TABLE I I - PRESSURE INCREASES FROM MATERIAL FLOWING & VIBRATED
SMOOTH BIN CONDITION
_____ Marbles Q.. Wheat N. Wheat
Sand
'
Marbles
ROUGH BIN CONDITION
O .Wheat N.Wheat Sand
S ta tic
L
Pressure ( Ib ./s q f t ) V
21 .7
48.5
226:6
67.5
38.0
87.8
13.6
31 .7
F Iowing
Pressure ( I b/sq f t )
L
V
22.8
44.1
32.1
69.8
49.6
19.1
22.6
% Increase
FJowing .
L
V
+20,7
+ 3 .4
+30.5
+40.4
+53.7
Vibrated
Pressure ( I b/sq f t )
L
V
33.5
82.3
9 0.2
148.0
22.0
36.9
% Increase
L
V
+25.9
+22.0
+ 13 7 .0
+ 68.6
+61 .7
+ 16.4
V ibrated
+5.1
-9.1
.
15.6
34.1
24.9
41 .8
14.7
39.8
34.4
55.7
' +59.6
+ 134
+22.6 \ + 40
L.Aqgr.
53
ANALYSIS OF RESULTS
F ig . 13 shows a graph, fo r smooth bin conditions, o f ^i1 versus
k.
These values are from Table I .
The s tra ig h t lin e A (and a ll subse­
quent s tra ig h t lin e f it t in g s in th is analysis) was f it t e d to the plotted
points by the s ta tis t ic a l theory of le a s t squares.
The maximum deviation from lin e A (k = 0.701-G .S I6
showing good c o rre la tio n between k and
be assumed to hold for O J 6 0 ^ '^ 0 .5 0 0 .
p ’ ) is 6$,
p ' . This c o rre la tio n can only
As
p 1 approaches zero, k must
approach 1.0 (th is is the lim itin g case o f a flu id pressure condition;
I . e . , V = L and k = L/V = I .0 ) .
In F ig . 13 In addition to the k values obtained from bin te $ ts ,
k estimates obtained from the pptiduqt o f k ! and a* from Table I were
p lo tte d againstJti1.
These p lo tted points show th a t Scheer1s equation
c o rre la te s well with lin e A fo r values o f
p 1 greater than. 0 .3 .
The
large deviation of a ’ k ’ fo r marbles w ill be explained follow ing the
'
analysis o f the c o rre la tio n s fo r the rough bin condton.
Line C
(p' = 0.923 U f) of F ig . 14 shows the c o rre la tio n of
d ire c t fr ic t io n te s t re s u lts with
and Uf would be zero*
p 1.
For a liq u id f i l l
m aterial
p'
For th is reason the lin e © was forced to pass
through the point ( 0 ,0 ) .
Estimated values o f k and
p' fo r the smooth bin condition are
compared with bln te s t values o f k and
in Table I I I .
Values o f
p'
were estimated from the equation o f lin e C using Table I values of
These values of
p' were then used in the equation of lin e A to estimate k ,
54
O
O
Ic FEOM TABLE I
k = a' k' FkOM TABLE I
LINEA
0 ,7 0 1-0 ,6 1 6 u
_
0, IGO Z h 1 ^ 0 ,5 0 0
/
FIGURE IB - SMOOTH B I N COERE L A T I O N OF k W I T H y u '
55
LIN E C -y u = 0 .9 2 3
0.4 —
Zj f
F I G U R E 14 - SMOOTH B lN C O e ^ E L A T lO N O FyU j W I T H yU'
56
TABLE I 11
k MB p' FOR SMOOTH BIN ESTIMATED FROM /Jf
Pi
P'
E s t. p 1
# e rro r
k._
E s t. k
t e rro r
Ol d Wheat
New Wheat
Marb Ies
0.358
0,358
0.330
-7 .8
0.47©..
0.490':
4 4.3
0.308
0.326
0.285
-1 2 .6
0.530
0.523
- I .3
0,255
0.185
0.238
428,6
0.580
0.550.
- 5 .2
: Sand
L . Aaar.
0.531
0.471 '
■0.490
44.0
...0.410.
0v395
-3 .7
TABLE IV
k AND p 1 FOR ROUGH BIN ESTIMATED FROM Uf
i"
.; '
Sand •
,Old Wheat New Wheat Marbles
L . Aaar.
/Jf
■
E s t. p'
X e rro r
k. ........... ..
E s t. k
% e rro r
0.531
0.593
0.565
- 4 .7 ;
• 0 .4 3 0 .
0.44,0
4 2.3
•• ■
»
Pr
Pi
E s t. p'
% e rro r
k
E s t. k
% e rro r
>,
0.457
0.508
0.485
-4i. 5
0.456
0.460
40 *9
0.312
0.320
0.330
43.1 :
0.500
0.495
- I .0
TABLE V
k AND"u' FOR ROUGH BIN ESTIMATED FROM
•• ' /. 1
- -i-.:
Old, Wheat New Wheat M arbles
Sand '
,
0.531
0.593
0.570
-3 .9
0.430
0.442
42.8
0.841
0,725
0.890
422.1
0.410
0,370
-9 .8
0.714
0.880
0.760
-1 3.‘6 !.>f
. .0 .3 7 0 ...
0.400
48. 1
0.466
0.508
0.500
- I .6
0.456
0.458
40,4
0.220
0.320
0.240
-2 5 .0
0.500
0,520
4 4 .0
0.761
0.880
0.820
- 6 .8
0.370
0.385
4 4 .1
Ur
'
L . Aqqr.
0.841
0.725
0.900
424.2
0.410
0.368
-9 .8
57
F ig . 15 shows a graph of yj1 versus k fo r m aterials in a rough
bin c o n d itio n .
Line D (k = 0.572^0.229 yu1) shows an excel le n t c o rre la ­
tio n between k and
the maximum deviation being 1.2$.
tio n can only be re lie d upon for 0.300
0 .900.
This c o rre la ­
The extension' of
lin e D is assumed to approach 1.0"as yu* approaches zero because th is
would be the c o rrect condition fo r a liq u id f i l l
m a te ria l.
F ig . 15, as In F ig . 13 fo r a smooth bin co nd itio n, shows k = a ’ k*
p lo tte d against yu1.
Again, a 'k * estimates k qu ite well fo r a ll the
m aterials except marbles.
The large deviation of a,'k' fo r marbles is-
believed to be caused by a large discrepancy in a ’ caused by It s large
p a r t ie le s iz e .
The lines F (yu* = I .058 yj
and G Cyu1 = I .070 yur > o f F ig . 16
show the c o rre la tio n o f yu* with yj^. and yj^, re sp ec tiv e ly , fo r a rough
bin c o n d itio n .
The lin e s were both forced through the point (0 ,0 ) be­
cause th is would be the condition fo r a liq u id f i l l m a te ria l.
Tables
IV
and
V
show estimates o f k and
yj8
compared with bin
te s t values, determined from values o f yj^ and yir re s p e c tiv e ly .
Values
o f yw8 were estimated using the equations o f lin es F and G resp ectively
with Table I values of ytif and yur .
Values of k were then estimated using
the equation of lin e D with estimated values o f y j 8.
The values o f yag were not co rrelate d with yi8, fo r , as can be
seen from Table I , yug values are very s im ila r to values.of yur with the
exception o f marbles.
Excluding marbles the maximum d iffe re n c e in
these values is .le s s than 3$.
"
58
O k FROM T A B L E I
<$> I c = Q 1Ic1 FROM T A B L E I
L IN E D
k = 0 , 5 1 1 - 0 / 2 "29
0.3 0 0 ^ ec' ^ OlSOOy
F I G U R E 15 - R O U G H 5 I N C O R R E L A T I O N OF k _ W lT H y u '
59
FKOM T A B L E I
FROM T A B L E I
O O
LINE G
-
1 .0 7 0
LINE F y u = 1.058yU^
y A f A N D yU -V
FI GU RE IG - ROUGH B I N CORRELATION OFyU.' W I T H A N C y A r
60
For marbles,
ps Is much higher than yj’ because to cause shear In
a m aterial with large p a r tic le size a large force is necessary to over­
come in terlo ck in g as well as s lid in g f r ic t io n .
The marbles must ro ll
and s lid e over one another to have fa ilu r e and must ris e as much as
almost h a lf t h e ir diameter v e r t ic a lly in order to pass by .one another.
In the angle o f repose te s t the marbles gave a concavfe fa ilu r e
slope ra th e r than a smooth plane slope.
Placing a h a lf inch deep s trip
of wood a t the fro n t edge o f the bottom of the angle of repose box,
an average slope angle o f 22.9 degrees was obtained.
(See Angle of
Repose Tests - Appendix I I I ) Without the s tr ip an average angle of
!.
12.5 degrees was obtained. N either o f these values agree with the
bin value of 17 .'7 degrees.
The
f&f obtained fo r marbles was the best c o rre la tio n with ^u1
because the method o f fa ilu r e was the same in the d ire c t fr ic t io n te s t
as in the b in .
The marbles in the d ire c t fr ic t io n te s t fa ile d by a
combination of r o llin g on the wall and s lid in g on each o th e r.
This
was indicated by no v e rtic a l movement o f the marbles to enable them
to r o lI and s lid e over one another.
Also, there were no scratches on
the marbles from s lid in g on the sandpaper.
(Previous te s ts by Boulanger
( 5 ) , with marbles glued to a block o f wood and dragged over the sand­
paper showed deep scratches in th e m arbles.)
The marbles used in the
bin te s ts showed no scratches.
The I ines H (a = 0.508 + 0.952
p') and I (a = 0.388 + 0 .6 1 8 / i 1)
c o rre la tin g "a" and yj* in F ig . 17 and 18, re s p e c tiv e ly , show th a t "a"
61
O
O
a
a'
FROM T A B L E I
FROM T A B L E I
0 ,4 —
F I G U R E 17 - S M O O T H B I N
C O R R E L A T I O N O F yu.'W I T H CL AN D CL*
62
1.0
—
0.6
—
AMD
O.'
G
a FROM TABLE I
O
a' FEOM T A B L E I
LINE
(X- 0 . 5 8 8 +
F I G U R E 16 ~ ROUGH B l N CORRELATION O F y u ' W I T H a A N D CL*
'63
is not constant but v a rie s as a function of p 1.
F urther evidence
given t o t h i s by the a d d itio n a l p lo ts o f a* against jj 1 in F i g .
18.
is
17 and
With the exception o f marbles, (as s tated before, the extreme
d e v ia tio n o f a* from "a" fo r marbles is f e l t t o be due t o large p a r t i c l e
s i z e ) the p l o t in d ic a te s t h a t a* does estim ate "a” and also shows t h a t
"a" is not con sta nt.
To show comparisons with ScheerVs equation fo r a rough bin
c o n d itio n ,
line I , Fig.
18, was used t o estim ate "a” fo r values o f p '
determined from l i n e F, F i g .
16.
The computed values o f k , determined
from these estimated values o f "a" and
v a l ues o f k. in T a b le V I .
jj ' ,
are compared with bin t e s t
64
TABLE VI.
k AND /*' FOR ROUGH BIN ESTIMATED FROM /Jf
and Scheer' s Equation
Old Wheat
Fi
E s t. /j ’
a
sin 0'
E s t. k
k
X' e rro r
0.531
0.565
0.738
0.492
0.450
0.43O
+4.7
New Wheat
.0 .457
0.485
0.690
0.436
0.470
0.456
+ 3 .1
Marb Ies
Sand
L . Aqqr.
0.312
0.330
0.595
0.314
0.488
0.500
-2 .4
0.714
0.760
0.858
0.605
0.398
0.370
+ 7.6
0.841
0.890
0.940
0.665
0.362
'0.410
- I I .7
65
DISCUSSION
Although the experimentation done for th is thesis was carried
out in a square model t e s t bin, the resu lts o f previous investigators
Indicate there is l i t t l e reason to believe th a t the correlations of k
and yii' should not be applicable to fu ll size bins of various cross sec­
tions ,
In regard to differences in k and
p ' due to bin s iz e , Jamieson
stated none for te s ts in model bins compared to fu ll sized bins.
Roberts also made te s ts on both model bins and f u ll- s iz e d bins and
because he did not make any statement o f e ffe c ts of bin size his s i ­
lence can be taken as an indication he observed no s ig n ific a n t d i f f e r ­
ences worthy of reporting .
Boulanger found no s ig n ific a n t change in base pressure due to
change in b--in cross, geetiioru
in addition he reported no s ig n ific a n t
change, in k for d if f e r e n t shapes o f cross sections.
Using the c o rre la tio n "a" with
t a i n l y be used to estimate k,
' , Scheer1s equation can c e r-
.
However, because "a" is not constant
but varies as a function of ^i1, there is no reason to use th is longer
procedure when k varies d ir e c t ly with ^u1 and may be estimated d ir e c t ly .
In c orrelations of
1 with it s estimators yjf and ^ur the corre­
la tin g lines indicate confinement is a greater factor in rough bins
than in smooth bins.
Is greater than
is less than
p1
In the correlations in the rough bin case
pf and pr, however in the smooth bin correlatio n
This is logical because In a rough bin
equals
66
ji so t h a t confinement does a f f e c t fi' whereas, in a smooth bin confine­
ment would have l i t t l e or no e f f e c t on yu1.
A comparison with p r io r investigations Inspires confidence in
k, Jii, and ji values determined in t h is in ve s tig atio n .
ported
Jamieson re­
jir si 0.532 for wheat which compares with values 0.531 and 0.466
determined in t h is Investigation for old wheat and new wheat respect­
iv e ly .
Airy reported a value of
Jir = 0,466 for wheat,
value fo r wheat of 0.300 compares with
Janssen's
Jit
ji' of 0.326 for new wheat.
Values of k for wheat in a smooth bin of 0.600 and 0,400 were reported
by Jamieson and Ketchum re s p ec tiv e ly .
Values o f 0.470 and 0,530 were
determined in t h is investigation for k i n a smooth bin fo r old wheat '
and new wheat resp ec tiv e ly .
The very close agreement of the correlations of yjf and
jir with
ji1 for rough bin conditions indicates th a t the simple angle of repose
t e s t is as good an estimator of
f r ic t io n t e s t .
Jii as the more complicated d ire c t,
For smooth bin conditions, however, the d ire c t f r ic t io n
was the only applicable t e s t .
Table 11 shows th a t fo r both emptying and v ib ra tin g conditions,
the pressure:, Increases over s t a t ic pressure are greater in a rough
bin condition than in a smooth bin c o n d itio n .
This is not surprising
because the s t a t ic pressures for any material
in a rough bin are much
less than those in a smooth bin, and Vibration tends to destroy the
I.
effectiveness of wall roughness as a pressure reducing agent.
67
Other investigators found large increases in la te ra l pressure
when the material was flowing from the bin.
Prante, Pleissner, and
Reimbert reported la te ral pressure increases of 300$, 100$,' and 45$
re sp ec tiv e ly .
The accuracy of Prante1s results is highly questionable.
Prante1S and P leis s n e r’ s results were obtained with a side emptying
gate, whereas Reim berf s results were for a bottom emptying gate.
Jamieson reported an increase in la te ral pressure of less than 10$ for
material emptying from the side'.of a bin.
In comparison, the maximum
la te ra l pressure increases,, found from th is investigation for material
flowing from a side gate, were 5$ and 54$ for a smooth bin and rough J
bin, re s p ec tiv e ly .
68
■RECOMMENDATIONS
From the analysis and discussion of the results of t h is in v e s ti­
gation, the following recommendations are presented.
I.
Investigations should be made on large bins to v e r if y the
te n ta tiv e correlations found in th is thesis and to determine what, i f
any, modifications should be made for large bins.
■2 .
Investigations to observe the e ffe c ts of gradation, of mater­
ia ls such as sand and aggregate, on pressures should be made.
3.
Modifications of the f r ic t io n t e s t methods should be in v e s ti­
gated to determine i f b e tte r correlatio n with ju1 could be obtained.
One modification is v ib ra tin g the material before te s tin g to obtain, as
closely as possible, the same density th a t exists in the bin.
4.
Investigations should be made to determine i f a relationship
exists between the la te ral pressure increase due to emptying and the
ra ti os.of'emptyIng gate area to bin cross section area.
5,.
In the absence of more accurate determinations of k and ju",
design engineers may use the empirical formulas found in t h is In v e s ti­
gation for bin design problems, but should make allowance for the fact
th a t these formulas are approximate and also th a t vibration and empty­
ing operation may increase pressures.
6.
Investigations should be made to determine the shape of the
v e rtic a l pressure d is trib u tio n on the base of a bin, and the shape of
the lateral pressure d is trib u tio n on the walls of a non-cylin d r ical
bin.
69
CONCLUSI ONS
On the basis of the results for the bin and m aterials in v e s ti­
gated, the following conclusions are made.
1.
p 1 may. be estimated from the f r ic t io n te s t estimators
and
p r by the following equations:
p 1 = 0.923 p^i smooth bin condition
i
'
p ' =s I .058 p^} rough bin condition
p 1 == I .070 pr', rough bin condition
2.
k may be d ir e c t ly estimated from
k = 0.701-0.616
0.160
3.
p * > smooth bi n condition where
^ 0.500
k = 0.572-0.229
0.300
p 1 by the equations;
p 1; rough bin condition where
Z. 0.900
For use in Scheer's equation k = ak1, "a" may be estimated
by the equations:
a =x 0.508 + 0.952 y j'; smooth bin condition
a = 0.388 + 0.618
4.
p '; rough bin condition
M aterials emptying from the side of a bin cause a s ig n ific a n t
increase in la te ral pressure.
The maximum increase recorded in th is
investigation was 54#.
5.
Vibration of m aterials in a bin causes a large increase in
lateral and v e rtic a l pressure.
Maximum increases over s t a t i c pressures
of 137# and 69#, resp ec tiv e ly , were observed in th is in vestig atio n .
6.
Vibration causes greater pressure increases in rough bins
than in smooth bins.
70
LITERATURE CITED
1.
A iry , W ilfre d . The Pressure of Grain,
of the In s t it u t io n of C iv il Engineers.
Minutes of Proceedings
1315347-358, 1898.
2.
American Society of C iv il Engineers, Proc. V77, Separate n82.
August 1951, P. 13, Pressures in Shallow Rectangular Bins.
3.
Amundson, L . R. Determination of Band Stresses and Lateral Wheat
Pressure fo r a C ylind rical Grain Bin. A gricultural Engineering.
26:321 -324 1945.'
4.
Beton-u Stahlbetonbau V48 n8 , August 1953,' P l92-4
nung in Einem S i l o t r i c h t e r .
5.
Boulanger, Wesley P. An Investigation of Some Factors Affecting
Pressures in Bins. Unpublished M. S . Thesis, Montana State
College L ib ra ry , Bozeman, Montana. I960.
6.
Bovey, Henry T . Experiments on Grain Pressures in Deep Bins .and
the Strength of Wooden Bins. Engineering News. 52:32-34. 1904.
7.
Caughey, R. A ., Tooles, C.W. and Scheer, A. C. Lateral and
V ertical Pressure o f Granular Material in Deep Bins. Iowa
Engineering Experiment Station B u lle tin No. 172, Ames, Iowa.
1951.
8.
Concrete and Constr. Engr., V50 n4, April
Design of S ilo s .
9.
Dale, A. G ., and Robinson, R. N. Pressure in Deep Grain Storage
Structures. A gricultural Engineering 35:570-573 8 August 1954.
Druckberich-
1955, P . 170-172.
10.
Ford.ham, A. A. The D ire ct Measurement of Lateral Pressure on
Walls and Bins. Engineering. 143; 561-562 1937.
11.
Hay, W. W. Design of Deep C irc u la r Bins.
43-44 1928.
12.
Henry, G. E . J . Bulk Grain Storage with P a rtic u la r Reference to
Design on V ertical S ilo s . South African In s t, of C iv il Engineers
Transactions 7:225-242 July 1957.
13.
Jamieson, H. A. Grain Pressures in Deep Bi,ns
5 1:236-243 10 March 1904.
Concrete.
32 (No 6):
Engineering News
71
14.
Janssen, H. . A. Versuehe uber G e tr e idedruck in Si Iozet Iem. Z e i t s c h r if t des Vereines Deufscher lngenieure. 39:1045-1049 1895.
15.
Ketchum, M ilo S. The Design o f Walls, Bins, and Grain Elevators
3rd Ed. .323-354, McGraw M i l l , New York 1919.
16.
L u f f t , Eckhardt. Tests of Grain Pressure in Deep Bins a t Buenos
A ires, Argentina. Engineering News 52:531-532 1904.
' /.
McGaImont, J . R. Measuring Bin Wpll Pressures Caused by Arching
M a te r ia ls . Engineering News-Record 120:619-620 1938.
17.
18.
McOaImont, J . R. and Ashby, Wallace. .Pressures and Loads of
Ear Corn in Gribs. A gricultural Engineering 15:123-125,128 1934
19.
Pleissmer, J. Versuche zur Ermettlung der Boden und Seitenwanddrucke in G e tr e idesiI os. Z e i t s c h r i f t des Vereines Deutscher
lngenieure. 50 (p a rt I ) : 976-986 1906.
20.
Prante. Messungen des Getreidedruckes gegen S i lowandungen.
Z e i t s c h r i f t des Vereines Deutscher lngenieure 30:1122-1125
1896
21.
Roberts, Isaac. Determination of the Vertical and Lateral
Pressures o f Granular Substances. Proceedings of the Royal
Society of London 36:225-240 1884.
22.
Roberts, Isaac.
34:399 1882.
23.
Rudd, John K.
Mow M aterials Flow from a Bin
26:72-74 February 1954
24.
Saul, Robert A. Measurement of Grain Pressure on Bin Walls
and Floors. A gricultural Engineering 34:231-234 4 April 1953.
25.
Scheer, Alfred C. Measurements of Horizontal Pressures Exerted
by Confined Granular M a te ria ls . M, S . Thesis, Iowa State College
L ib ra ry , Ames, Iowa, 1950.
26.
Scheer, Alfred C. Bin Theory and Soil Mechanics, Unpublished
Paper, Montana State College I960.
27.
Spangler, M. G. Underground Conduits - An Appraisal of Modern
Research Transactions. American Society of C iv il Engineers
113:316-374.
1948.
The Pressure of Stored Grain
Engineering
Food Engineering
72
■28.
S ta h l, Benton M.
G irc u la r Mo. 835
Grain Bin Requirements.
1950..
EJ. S . Dept, of AgrI .
29.
TerzaghI, Karl S t a b i l i t y and S tiffn e s s of C e llu la r Cofferdams
(and Discussions) Transactions, American Society of C iv il
Engineers
I 10 : 1083-1202 1945.
30*
To lt z , Max
Discussion of Grain Pressures In Deep Bins
Transactions of Canadian Society of C iv il Engineers 17 (p a rt I) ;
641-644 1903.
31.
Tooles, Calvin W.
Experimental Determinations of Lateral Press­
ures in Deep Grain Bins. U n p u b l i s h e d M. S . Thesis, Iowa State
College L ib ra ry , Ames, Iowa 1950.
32.
Tschebotanio f f , Gregory P,
Earth Structures.
1st Ed.
Soil Mechanics, Foundations, and
265-267 1951.
73
APPENDIX
I
DERIVATION OF JANSSEN'S EQUATION
V
y
L
L
V+dV
V = average v e rtic a l un it pressure a t depth y , I b . / s q . f t .
L = average lateral u n it pressure at depth y, I b . / s q . f t .
R = hydraulic radius (area of horizontal cross section of bin divided
by the inside perimeter of b in ), f t .
w= un it weight of stored m a te r ia l, I b . / c u . f t .
^j '= e f f e c t iv e c o e f f ic ie n t of f r ic t io n between stored material and bin
w a ll, dimensionless
y = depth from top of stored material to point under consideration, f t .
k = r a t io of average la te ra l to average v e rtic a l pressure, dimension I ess
A = area of horizontal cross section of bin, sq. f t .
U = inside perimeter of bin, f t .
IV = 0;
(VtdV)A t yu l LUdy - V A - wAdy = 0
VA t AdV t yj'LUdy - V A - wAdy = 0
74
AdV = wAdy - yj’ LAdy, where
R = Ai, thus U =
U
73|>
AdV = wAdy - yj'Ltidy
dV = wdy - /J11Ldy
R
dy = . dV
=
w-/J 1L
/ R
dV
W
- U
1kV
R
I n+egrating
Y = -_B_
^ lk
log (w - /Jt KV) + G
y R
when y = 0, V ~ 0
0
log w + C
f'l
6 =
R log w
yu ’ k
y = - _R_
yj’ k
log ( W - ju'kV) + Fl_
R
^j1k
log
M u ltip ly in g by - u'k „
R* ‘
-U 1Ky = log Cw -U 1KV) - log w = log
R
R
e -yilkx
R
w
Z wItiJM. X
w - yj 'k V
S o lv ing for V:
we -yj':kv
kV
= w - yu'kV
w - we
R
u'kV = Rw CI- e "irlJ.K.y.
r
R
- U t Ky
V = RW ( I -e '^ p
R
)
,
f ’k
and
L = kV =iT T CI -e
75
APPENDIX I I
DERIVATION OF SCHEER' S EQUATIONS
Rough Bin
Figure 1.9 shows a rough c y lin d ric a l bin and the stresses acting
on a smalI element of f i l l
adjacent to the bin walI .
V' is the
v e rtic a l pressure a t the wall and V is the average v e rtic a l pressure.
V' w ill generally be somewhat less than V.
k' is defined as L /V 1.
From the d e fin itio n of the rough bin condition, 0 ’ = 0, the shearing
stresses on four faces of the element must then have a magnitude of
L tan 0 because the v e rtic a l
faces are f a ilu r e surfaces, and from
s ta tic s the shear stresses oh the horizontal faces must be the same
as those on the v e rtic a l
faces.
From the r ig h t t r ia n g le Inscribed in the Mohr c i r c l e of Fig.
19c, i t is evident th a t
V1 - L
~
2 L tan 0
= tan 0
This reduces to
_L_ _
V*
■
I
......
I + 2 tan%0
= 1 - Sin^B = k'
I t sin^0
k' would.be equal to k only i f V1 = V .
CD
In general, V* w ill be
somewhat smaller than V, as indicated in F ig . 19a; th e re fo re , k w ill
be somewhat smalIe r than k * .
k may be regarded as the upper lim it of
k ’ j th a t is , k ^ k ' .
Now le t V1 = aV where a is a p o sitiv e number somewhat less than
76
(a)
COUGH WAiLED BIN
t
(C)
MOHR CIRCLE FOR STRESSES AT A
F IG . 19 - STRESSES IN FILL IN COUGH WALLED B IN
77 '
u n ity .
When t h is is substituted in equation ( I ) the re s u lt is:
L/V = k = ak'
'
Assuming a constant
(2)
0, and s ubstituting ( I ) into ( 2 ) , one gets:
k = a / I - s i 0 A for a rough b in .
\ I t s i In'2 0 /
(3)
Smooth Bin
For a smooth bin, where 0 * / 0 , the correct Mohr c i r c l e for an
element such as the one of F ig . 20a, is not uniquely defined.
This is
il lu s t r a t e d in F ig . 20b, where point D represents the stress on the
v e rtic a l plane of s lid in g .
Any Mohr c i r c l e which passes through point
D, and does not in ters e ct the 0 envelope, represents a possible solu­
tio n .
For instance, points F, F 1, and F" give possible values for V1.
Let point F on the large c i r c l e be the maximum value of V1, and the
corresponding value of k * , which w ill be designated k 1^, be the
smallest value of k * .
From the r ig h t t r ia n g le inscribed in the large
Mohr c i r c l e i t may be shown th a t
V
k'
A
=
?■"
1_______ ___
I + 2 tanqttan 0'
(4)
The angle(X may be obtained from the equation:
tan 0 '
cos<x + tan 0 ' s in # = ■s j n g
The small Mohr c i r c l e of F ig . 20b, which is the tangent to the
0’ envelope lin e s , gives what Scheer considers to be the probable upperlim it f o r k * .
This p a r t ic u la r k' value w ill be designated k 'g .
The
geometry of the small c i r c l e Is s im ila r to the geometry of the Mohr
78
WALL
FIG . 2 0 - STATE OF STRESS ADJACENT TO WALL OF SMOOTH B IN
79
c i r c l e of F ig . 19c.
k'
B
Therefore
I_- sin2 0*
I + Sin^ 0 1
(5)
For +he smooth bin case to get an admittedly approximate equa­
tio n fo r k 1, k' lie s midway between k '^ and k ’g.
k' = 1
This gives
Ck'A + k' ) .
(6)
Again le t t in g k = ak% and using the results from equations (4)
and (5 ) in equation ( 6) , the resu lting equation for k is
a / ___ I
2 V I + 2 tancX+an 0*
+
I -s in ^ 0 *
I + sin^0* / fo r a smooth bin
80
APPENDIX I I I
EXPERIMENTAL DATA
81
o - CALIBeATIO
A U BBATIONI CURVE A l
ri-T-r
-—»—
—
—
j--L—
——
—
•
4' —
4--I—f- --4—f—
-4
.........
T fr rr
GURE 21 -
82
83
84
Increment
Run #1 Side Pl
S tra in Edge Pl
Gen. Pl
Run #2 Side Pl
S t r a i n Edge Pl
Gen. Pl
Run #3 Side Pl
S tra in Edge Pl
Cen. Pl
8 i n.
Depths
7.59 in .
.
.
.
.
.
.
.
,
.
Weight:
13.3 lb .
3
4
0 '
SI
62
28
73
94
61
90
I 14
81
93
125
90
100
132
98
100
138
103
93
I 18
0
55
58
25
76
93
61
80
113:
90
94
124'
93
97
134
100
102
137
I 10
I 10
98
0
55
58
24
75
91
64
88
III
82
' 94
125
95
97
127
99
97
130
N:R.
0
54
59
62
26
75
86
93 . I 13
84
94
125
93
98
131
99
100
135
106
102
108
2 .0 4 .8 6 .4 7.1 7.5
5.5 6 .3 7 .0 7.3 7,5
8 .7 10.2 I I ,1 I I .6 I I .9
7.9
7.7
9.9
21 ,7
40.3
48.5
,832
22.8
0
5,.5
2.1 .5 29,6
25.0 : 35.5
.860 .833
13.8
33.9
41 .3
,822
. 5
18.4 20.5
37.7 39.3
4 5 .2 47.3
.833, .832
a
F I QW2
V ib . 3
n . r .!
»
It
ft
Tl
H
H
It
It
I!
1,1
< EQ O
Side Pl .
0
Edge PI . 4 .0
Gen. PI . 6 .0
L (p sf)
V1 "
V
Ii
a' = V '/V
14.,0 lb.
Weight:
2
J
Avg.
Side Pl .
S tra in Edge Pl ,
Gen, Pl .
Avg.
Load
(lb .)
Depths
OLD WHEAT
. '
44.1
Il
In c r. 2-6
M A T E R IA L :
<
I
SMOOTH
3»
In c r .
B IN :
CD
B IN T E S T #1
.835
! ,...S t r a in in a ll te s ts was mi oroi nches; per inch,
2,
Maximum values a t large depth for material .flowing from bin.
S.
Maximum values a t large depth fo r bln vibrated,
4.
C alib ra tio n Curve used to determine average load from average
s tra in .
5,
Not run.
i /
85
B IN T E S T # 2
B IN :
SMOOTH : : M A TE R I'A L : NEW WHEAT
In c r . I
Depth:
8 in .
Weights
In c r . 2”6
Depth; ■7.559 in .
13.7 lb.
WeIght:
13.0 lb.
I
2
3
4
5
6
Flow
O
56
58
29
76
97
72
89
I 14
94
93
122
98
96
126
I IO
96
126
N;R.
'V
Run 02 Side Pl .
S t r a i n Edge Pl .
52
•Gen. P I. ' 61
30
72
93
71
84
I 14'
96
87
120
100
91
127-
106
92
128
U
Il
I?
tl
Vl
Avg*
Side P l .
S t r a i n Edge Pl .
Gen, P l .
0
54
60
30
74
95
72
86
114
95
90
121
99 108
94 • 94
126 127
M
H
Il
Vt
11
Tl
Avg*
Side P I.
Load
Edge P l .
( l b . ) . Gen. P l .
0
4 .0
6.0
1ncrement
Run 01 Side P l .
S t r a i n Edge P l .
Cen. P l ,
it
I!
2.4 5 .6 7 .2 7.5 8 .2
5 .4 6 .3 6.7 7 .0 7.1
8.8 10.3 10.8 I I .2 I I .3
0 6.6 16.0
21 .5 29.0 33.9
25.0 35.5 41 .5
. .860 .817 .817
20.8
36.0
43.8
.822
21 .7
37.6
45.5
.827
23.7
3 8.2
46.0
.830
< co o
L (p s f)
y in
V
"
a' = V' / V
'.I!
Vi b*
Avg. =:
*829
829
86
B IN T E S T # 3
B IN :
■ I nor, I
I nor. 2-6
Increment
SMOOTH :. M ATERI A L :;;'. MARBLES
Depth:
8 In .
Depth:
7.591 in .
I
2
Weight:
3
4
Run m Side P l .
S t r a i n Edge P l .
Cenv P I .
0 105
1,17 165
151 234
191
201
289
249
219
321
Run #2 Side P l .
S train Edge Pl .
Gen. P I .
0 103 193
I 16 171 ' 196
147 233 288
249
219
327
Avg.
Side P l .
S t r a i n Edge P l .
Gen. Pl .
0 104
I 16 168
149 233
249
219
324
Weight;
23.3 I b ,
6
Flow.
Vib
N.R.
M
N.R
5
H
M
U
11
11
U
JT
M
Tl
L (ps f)
V* "
V
"
a' = V '/V
0
47.1
54.5
.865
2J .7
69.7
81 .8
.852
39.8
82.7
98.8
.838
51 .2
91 .6
I I 0 .0
.832
CD
>
Side Pl .
'0 7.9 13*9 17.8
Edge Pl . 8 .8 13.0 15.4 17.0
Gen. P l . 13.0' 19.8 24.2 27.0
O
Avg.
Load
( Ib )
192
198
288
24.5 Ib..
Avg. = .847
87
B IN T E S T # 4
B IN ;
In c r . I
SMOOTH
Depth;
8 in .
In c r. 2-6 Depth:
Increment
I
M A T E R IA L ;
SAND
Weight:
7.59 in .
22.2 lb.
Weight;
21.0 lb .
2
3
4
5
6
Flow.
Vlb
Run #1 Side Pl .
S t r a i n Edge Pl .
Cen. P l .
0
94
95
41
129
140
91
143
158
III
147
171
122
152
175
127
I 54
177
164
169
169
N.R
U
it
Run #2 Side PL.
S t r a i n Edge Pl .
Cen. P l .
0
91
95
36
128
140
83 105
142 146
164 . 176
I 16
149
179
120
150
182
138
137
205
158
145
259
Avg.
Side P l .
S t r a i n Edge P l .
Cen. P I.
0
92
95
38
128
140
87
142
161
108
146
174
119
150
177
124
152
180
164
169
. 169
158
145
259
Side P l .
Edge P l .
Gen. P I.
L (psf)
V1 "
V
"
a' = V '/V
0 3 .2 6.6 8 .2 8 .9 9 .2
6 .8 9 ,8 10.8 I I .2 I I .5 I I .6
8 .8 12.3 14.0 14,9 15.2 15.4 0 8 .8 18.9 23.6
56.6 52.7 58.1 60.3
39.0 5 5.2 62.0 65.2
.938 .954 .937 .927
25.7
61 .8
66.8
.927
26.6
62.4
67.5
.925
12.0
13.0
14.7
I I .6
I 1.1
21.8
32.1
33.5
82.3
69.8
Avg. = .935
<c C O o
Avg.
Load
(lb .)
88
BIN TEST
m
Blltk
ROUGH
MATERIAL:
I ncr.
I
Depth:
8 i n.
I ncr.
2-6
Depth:
7.48 I n .
OLD WHEAT
Weight:
13 .6 lb.
Weight;
12.75 lb.
Increment
I
Run #1 Side Pl .
S t r a i n Cen. P I .
Edge Pl .
13
31
39
39
O
39
68
Discarded because of interaction
between center and ed^e
.i
2
3
4
6
5
Run #2 Side Pl .
0
S t r a i n Gen. P I . . 68
Edge Pl . 40
II
89
50
27
93
56
32
94
56
34
97
56
34
97
56
0
Run #3 Side Pl
S t r a i n Gen. P I .' 68
Edge Pl . 41
14
93
49
30
IOI
54
37
106
54
45
108
54
49
120
54
Run #4 Side Pl .
S t r a i n Cen. P I .
Edge Pl .
0
67
40
IO
89
45
25
IOI
50
34
103
50
38
105
50
38
105
50
Run #5 Side Pl
S t r a i n Cen. Pl .
Edge Pl .
0
69
39
18
93
49
34
107
52
42
I IO
52
49
I 14
58
50
I 15
58
Run #6 Side Pl
S t r a I n Gen. P I .
Edge Pl .
0
71
39
IO
90
51
27
100
53
34
103
53
34
107
53
' 34
107
53 .
Run #7 Side Pl
S t r a i n Gen. Pl .
Edge Pl .
0
63 '
40
IO
89
51
25
97
■51
30
100
52
35
102
52
35
103
52
12
90
49
I .5
7 .3
3.4
4.1
19.6
27.6
.707
28
100
53
3.6
8 .0
3.6
10.3
20.7'
29.9
.692
35
103
53
4 .2
8 .3
3.6
12. 1'
21 .3
30.9
.690'
39
106
54
4 .6
8.5
3 .7
13.3
21 .6
31 .4
.688
40
108
54
4 .7
8 .6
3.7
13.6
21 .7
31 .7
.685
0
Avg.
Side Pl .
S tra in Gen. Pl ■. 68
Edge Pl . 40
Avg.
Side Pl .
0
Load
Gen. Pl . 5.5
( l b . ) Edge Pl . 2.7
0
L (psf)
V »
"
15.5
V
"
21 . 1
.735
a' = V '/V
'
Flow *
V ib r e
N.R.
J59
N.R
it
N.R*
Tl
It
n
IT
Tl
Tl
IT
It
Tl
Il
Il
M
Il
IT
It
Tl
Tl
Tl
Tl
U
83
121
60
Tl
70
129
65
Il
II
11
64
6 * 6
19*1
Avg .
.
76
125
62
7 .6
10.0
4 .3
22.0
36.9
.700
=
Al
BI
Cl
89
B IN T E S T # 6
ln c r.
B IN :
I:
I n c r . 2-6:
ROliIGH
M A T E R IA L :
NEW WHEAT
Depth: 8 ' in .
Weight:
13.3 lb.
Depth: 7.48 in .
Weight:
12.45 lb.
2
3
4
5
6
Flow.'
Vi b r .
Run #1 Side Pl . 0
S t r a i n Cen. Pl . 68
Edge Pl . 42
15
91
58
37
102
61
45
I 10
62
47
I 10
62
52
I 10
62
N.R.
it
it
N.R.
ti
it
Run #2 Side Pl . 0
S t r a i n Cen. Pl . 67
JEdge P. 45
10
95
57
33
41
104 108
60 ' 63
45
I 10
64
47
I 13
64
H
ii
n
ii
ii
ii
Run #3 Side Pl . 0
S tra in Cen. Pl . 67
Edge Pl . 41
13
91
53
36
102
55,
41
106
56
43
107
56
45
107
56
it
IT
It
82
I30
67
Run #4 Side Pl . 0
Strain Cen. Pl . 65
Edge Pl . 44
16
92
57
38
104
63
48
107
65
53
107
65
55
107
66
M
11
II
96
140
83
Avg.
Side Pl . 0
S t r a i n Gen. Pl . 67
Edge Pl . 43
14
92
56
36
103
60
44
108
62
47
108
62
50
109.2
63.0
4 .33 4 .9
8 .3 8.7
4.1 4 .2
4 .2
8 .8
4 .2
5 .4
8 .9
4.-3
8 .6
10.7
5.5
0
5 .0 12.4 14.1
16.7 21.8 23.6 24.2
21.4 28.9 32.0 33.3
.780 .754 .738 .727
15.0
24,5
33.8
.726
15.6
24.7
34.1
.725
24.9
Increment
41 .8
Avg, = .742
150877
O
L (ps f)
V* »
V
"
a' = V'/V
I .8
7 .4
3 .8
CO
Side Pl . 0
Gen. Pl * 5.4
Edge Pl . 2.9
89
135
80
<
Avg.
Load
(lb .)
I
90
B IN T E S T # 7
B IN !
ROUGH
M A T E R IA L :
MARBLES
I n o r, I
Depth! 8 i n .
Weight; 2 3 ,8 lb .
I n c r . 2-6
Depth: 7 .4 8 i n . Weight;
22.35 lb .
Fl ow.
Vibr.-
178
291
190
N.R.
M
Il
N.R.
it
it
151
287
173
Il
.ill
Il
H
Ii
H
62
234
160
I 19 150
265. 285
188 188
Il
Il
Il
ii
n
H
Run #4 Side Pl . 0
S train Cen. Pl . 170
Edge Pl . 108
65
245
148
121
279
166
148
300
174
206
. N.R.
Il
Il
Il
Run #5 Side Pl . 0
S t r a i n Cen. Pl . 160
Edge Pl . I 16
67
231
149
124
267
166
155
281
170
Il
Il
Il
420
483
308
Avg.
Side Pl . 0
S train Cen. Pl . 161
Edge Pl . I 16
66 122
235 276
152 . 172
151
288
176
206
420
483
308
6.7 I I .0 13.2
Side Pl .. 0
Gen. Pl .1 I .8-1 7 . 2 19.8 21.2
Edge Pl . 8 .2 10.9 12.5 12.8
' 17.2
31 .3
34.0
23.3
49,6
90.2
2
3
4
Run #1 Side Pl . 0
Strain Cen. Pl . 164
Edge Pl . 124
80
230
155
151
287
192
Run #2 Side Pl . 0
S t r a I n Cen. Pl . 160
Edge Pl . 120
68
229
150
125
269
167
Run #3 Side Pl . 0
S t r a i n Cen, P I , 155
Edge Pl . I 20
Increment
Avg.
Load
(lb .)
L ( p s f .)
V1 "
V
"
a' = V '/V
I
0
47.1
SI .6
.913
18.4
62.7
72.5
.866
31 .0 38.0
71 .9 75.6
8 3 .3 '8 7 .8
.863 .839
5
6
-
148.0
Avg. = .870
C.G
Al
B2
Cl
91
B IN T E S T # 8
B IN :
ROUGH
M A T E R IA L :
I ncre
Depth:
8 .i n . '
I n c r e . 2-6
Depth:
7.48 in
Increment
I ■
.
. SAND
W eight:
21.6 lb .
Weig h t :
20.15 lb .
2
3
4
5
6
Flow .
Vi b r .
Run #1 S i d e Pl .
S t r a i n Gen. P I .
Ed g e P l .
O
95
69
13
I 15
78
32
122
81
38
122
81
38
122
81
41
I 24
81
78
N. R.
I!
N.R.
Tl
11
Run # 2 S i d e P l .
Gen. P I .
Edge Pl .
0
91
70
12
I 12
78
32
121
79
39
122
80
41
124
80
'4 5
126
82
78
N.R.
M
11
11
11
Run # 3 S i d e P I .
Gen. P l .
Ed g e Pl .
0
I 15
74
15
135
86
38
143
89
44
146
90
47
148
91
49
149
91
'.'1.18
N.R.
it
130
253
81
Run 0 4 S i d e P l .
Cen.' P l .
Ed g e P l .
0
I 10
62
9
126
71
30
133
73
37
135
73
39
138
73
41
138
74
106
N.R.
M
137
253
75
12
33
39
41
43
78
134
Avg.
Side
Run I & 2
Gen.
Run 3 & 4
Gen.
S t r a i n Edge
.P l.
0
P I.
93
I 14
122
122
123
125
P l.
P l.
1.12
69
130
78
138
80
141
81
143
81
143
82
I .5
8 .4
8 .0
5 .3
4 .0
8 .9
8 .5
5 .5
4 .5
9.1
8 .7
5 .6
4 .7
9 .2
8 .8
5 .6
4 .8
9 .3
8 .9
5 .7
0
Avg.
Side P I .
" I & 2 Gem. P I . 6 . 9
6.8
" 3&4
"
"
Load
Edge P l . 4 . 7
253
78I
7 .8
I 1.9
16.3
5 .3
Al
B2
83
Cl
\
92
B IN T E S T # 8 A
(C o n tin u a tio n
o f Test 8)
Inorement '
I
2
3
4
5
6
Flow.
Vi b r .
Run #5 Side Pl .
Cen. Pl
Edge Pl .
0
75
69
18
94
82
40
102
.86
48
106
89
50
108
90
■5 I
108
91
N.R.
rr
H
N.R.
H
Il
M•
11
Il
Il
N.R.
Ii
Ii
N.R.
Run
#6 Side Pl .
Gen. P I .
Edge Pl .
49
15
39
51
0
98 107 I 13 I 17
78
Damaged s tra in gage
51 1
I 17
13
109
85"
34
121
88
42
123
90
45.
124
91
51
125
95
Avg.
Side Pl .
S t r a i n Cen . Pl .
Edge Pl .
0
81
71
15
100
84
38
I 10
87
46
I 14
90
49
I 16
91
51
I 17
92
Avg. ' Side Pl .
O
Load ' Cen . Pl . 7 .0
( l b . ) Edge Pl . 4 .8
2 .2
8.5
5.7
4 .5
9.1
6 .0
5.1
9.5
6 .2
5.4
9.7
6 .2
5.5
9.7
6.3
Avg.
Side Pl .
0
Load
Cen. Pl . 6.9
8 & GA Edge Pl . 4 . 8
I .8
8 .4
5.5
4 .2
8.9
5 .8
4 .8
9 .2
5.9
5 .0
9 .3
6 .0
5.1
9.4
6 .0
7.8
I I .9
16.3
5 .3
0
27.6
30.2
.915
5 .0
31 .6
35.9
.881
12.1
33.4
38.0
.879
13.8
33.9
39.0
.870
14.4
34.5
39.5
.874
14.7
34.5
39.8
.868
22.6
34.4
L (ps f)
"
v
"
a' = V '/V
V1
ii
0
89
73
>
<
CO
Run #7 Side Pl .
Cen. Pl .
Edge Pl .
ii
ii
55.7
.881
93
B I N TEST # 9
B IN :
ROUGH
MATERI A L :
L IG H T AGGEG.
I ncr.. I
Depth: B i n ,
Weight:
IO.IO i b .
I n e r . 2-6
Depth: 7.48 in .
Weight:
9.48 lb.
Increment
I
Run #1 Side Pl .
S t r a i n Gen. Pl .
Edge Pl
IO
17
0
39
32
41
38 . 45 . 46
Run #2 Side Pl .
S t r a i n Gen. Pl .
Edge Pl .
0
31
34
I I:
39
42
20
19
40 ■ 41
43
42
Side Pl .
Avg.
S t r a i n Cen. Pl
• Edge Pl ,
0
32
36
IO
■ 39
18
40
44.
Avg. • Side Pl *
0
Gen. PI . 3 .4
Load
Edge Pl . 2.7
(lb .)
L (psf)
V' "
V
"
a * = V '/V
2
43
I .8
4 .2
3.1
3
2.7
4 .2
3 .2
4
5
18
40
46
18 '
41
45
21
42
42
6
Flow.'
V ib r.
N.R.
M
M
N.R.
11
n
H
11
11
Ii
it
it
C.G.
19 . 20 '
41
42
44
44
2.8
4 .3
3 .3
A3
B5
G2
2.8
4 .3
3 .3
15.5., 18.4 18.7 19.0 19.0
16.2 19.7 20.0 20.2 20.3
.957 .933 .935 .938 .936
Avg = ,940
94
ANGLE OF REPOSE TESTS
Rurtl ■#
I
2
3
V e rt. *
18.2 I7i5. I 8 .0
H oriz . ** 22.8 23.6 23.5
tan 0
0.798 0.742 0.767
38.6 36.5 37.5
^r
V e rt. *
17.7 17.7 17.9
Woriz , * * ' 32.9 33.3 33.9
tan 0r
0.538 0.532 0.528
28.3 28.0 27.9
«r
V e rt. *
18.1
18.2 18.2
M oriz. * * 1 39.4 38.7 3 8.2
0.460 0.470 0.477
tan 024.7 25.2 25,5
0r
8
4
5
6
7
. SAND
17.6 18.1
17.4 18.1
17.6
23.6 23.8 23.1 23.7 23.2
0.747 0.761 0.754 0.763 0.758,.
36.7 37.3 37.0 37.3 3 7 .2
Avg. = 37.3°
OLD WHEAT
17.6 17.8
32.7 32.7
0.538 0.544
28.3 28.5
Avg. = 28.2°
NEW WHEAT
18.1
39.7 .
0.456
24.5
Avg. = 25 .0°
MARBLES
9
IQ-
12.9 13.3 12.8
V e rt. *
Hori z . **■ '5 9 .2 59.2 59.2
0.218 0.225 0.217
tan 0r
12.3 12.7 12.2
0r
O
I
S
S
O
CO
O
Avg. = 12.4°
MARBLES (1 /2 " wood.strip a t fro n t of bottom edge)
16.0 |6 .5
V e rt. *
16.2 15.9 16.2 15.6 16.0 15.7 16.1
15.1:
Horiz . * * <38.8 38.1 39.4 36.9 :3 8 .0 38.1 36.9 37.5 37.6 36.2.
tan 0_
0.418 0.418 0.408 0.423 0.422 0.412 0.437 0.427 0.438 0.417
22.7 22.7 22.2 22.9 22.9 22,4 23.6 23.1 23.6 22 . 6 0r
Avg. = 22.9
LIGHT AGGREGATE
22.7 23.1
V e r t. *
24,5 24.1 24.9 ' 24.5
(
H oriz . * * • 18.9 19.9 20.1 20.7 20.2 20.9
0.833 0.862 0.820 0.861 0.812 0.854
tan 0r
39.6 40.8 39.4 40,7 39.1 40.5
^r
Avg. = 40.0
LIGHT AGGREGATE
17.3 16.6 16.9
15.7 "15.9 16.4 17.1
“ 15.7 16.4 16.1
V e rt. *
19.3 18.8 18.8 18.9 19.3 19.0 20.7 19.4 20.2 19.9
H oriz . * *
0.891 0.822 0.850
tan 0 _
0.813 0.872 0.857
3
9.2
41.1
40.6
39.8
39.5
40,8
39.6
41.7 Ag.4 40.4
0r
Avg • = 40 *2
•^Vertical trac e of slope of repose angle
■^Horizontal trace of slope of repose angle
95
DIRECT FRietIQIU TESTS GKI SMOOTH BOARD
•Run #
1
Total Normal Load
Total Shear Load '
tan 0 f
0f
2
3
4
OLD WHEAT
5
6
12.23 12.23 12.23 12.23 I7i38 I'7.'38
4.11
4.54 4.54
4.30 6.20 6.20
0.336 0.371 0.371 0.352 0.357 0.357
18.6 20.4 20.4
19.4 19.6 19.6^
Avg. = 19.7
NEW WHEAT
Total Normal Load
Total Shear Load
tan 0 f
Of
11.92 11.92 17.07 17.07 21.07 21.07
' 3.77
3.94 5.05- 5.38 6.32 6.32
0.316 0.330 0.296 0.316 0.301 0.301
17.6 18.3 16.5
17.6 16.8 16.8 Av g. =
17.1
GLASS MARBLES
Total Normal
Total Shear
tan 0 f
0f
Load
Load
19.19 19.19 28.34 28.34
4.68 4.68 7.60 7.45
0.245 0.245 0.268 0.262
13.8 13.8 15.0 14.7
Avg. =14.3
SAND
Total Normal
Total Shear
tan 0 f
0f
Load
Load
18.06 18.06 18.06 26.21
9.23 9.80 9.70 .13.60
0.511 0.543 0.538 0.519
27.2 28.5 28.3 .2 7 .4
26.21
14.20
0.542
28.5
Avg. = 28.0
96
DIRECT FRICTION TESTS ON SANDPAPER COVERED BOARD
2'
Run #
3
4
OLD WHEAT
Normal Load
Shear Load
tan
0f
12.23
6.55.
0.535
28.2.
12.23
6.39
0,522
27.6
17.38
9.27
0.534
28.2'
17.38
9.15
0.527
27.8
Avg; = 28.0
NEW WHEAT
Normal Load
Shear Load
tan '04
0f
11.92 I I .92
5.27 5.51
0.443 0.462
23.9 24.8
17.07
17.07
8.08 7.67
0.473 0.450
25.3 24.2
Avg. = 24.6
GLASS MARBLES
Normal Load
Shear Load
tan 0 f
0f
19.19 19.19,
5.93 6.14
0.309 0.320
17.2 17.8
28.34 28.34
8.90 9.07
0.296 0.320
16.5 17.8
Avg. = 17.3
HEAVY AGGREGATE
Normal Load
Shear Load
tan 0 f
0f
18.06 18.06
12.57 12.60
0.696 0,698
34.9 35.0
27.21 27.21
19.70 19.90
0.724 0.732
35.9 36.2
Avg. = 35.5
LIGHT AGGREGATE
Normal Load
Shear Load
tan 0 f
0f
9.43 C9.43
8.37 7.53
8.887 0.798
41.6 38.6
14.58 14.58
12.33 12.18
.847 0.836
4 0 .3 39.9
Avg. = 40.1
97
DIRECT SHEAR TESTS
Run #
2
3
4
5 - 6
OLD WHEAT
Normal Load.
Shear Load tan 0S
0s
12.23 12.23 17.38 17.38 21.38 21 .38
6.47 6.55 9.85 9 .92 I I .30 I I .20
0.528 0.535 0.567 0.572 0.528 0.524
27.9 28.2 29.6 29.8 27.9 27.7
Avg. = 28.5
NEW WHEAT
Normal Load
Shear Load
tan 0_
0 S
I I .92 I I .92 17.07 17.07 21.07 21 .07
5.51 5.54 7.46 7.44 9.59 9.92
0.462 0.465 0.538 0.437 0.455 0.471
24.8 25.0 23.7 23.6 24.5 25.2
Avg. « :24.5
GLASS MARBLES
Normal Load
Shear Load
tan 0«.
%
19.19 19.19 28.34 28.34
13.10 13.56 20.02 20.80
0.682 0.707 0.707 0.733
34.3 35.5 35.5 3 6.3
,
'
■
■
' Avg. = 35.4
HEAVY AGGREGATE
Normal Load
Shear Load
tan 0c
0 S
18.06 18.06 27.21 27.21 27.21
13.85 13.68 19.60 21.85 20.95
0.768 0.758 0.721 0.803 0.770
37.6 3 7.2 35.8 38.8 37.6
Avg. = 37.4
LIGHT AGGREGATE
Normal Load
Shear Load
tan 0S
0
S
.9 .4 3 9.43 14.58 14.58
7.80 8.50 12.70 12.43
0.826 0.900 0.871 0.854
39.6 42.0 41.1 40.5
Avg. = 40.8
MONTANA
N378
SC317
cop.2
150877
150877