An investigation of some factors affecting pressures in bins
by Wesley P Boulanger
A THESIS Submitted to the Graduate Faculty i n partial fulfillment of the requirements for the degree
of Master of Science in Civil Engineering at Montana State College
Montana State University
© Copyright by Wesley P Boulanger (1960)
Abstract:
Model deep bins were used to determine the effect of several fundamental variables on the vertical
pressure in bins. These variables were (I) the effect of relative movement between the walls and the
base on vertical pressure; (2) the effect of particle size on vertical pressure; (3) the effect of shape of
bin cross-section on vertical pressure; and (4) the effect of changing wall roughness on vertical
pressure. Observed values of K are compared with values predicted by the method suggested by
Professor Alfred C. Scheer in his paper, "Bin Theory and Soil Mechanics." Base pressure of wheat was
plotted against relative movement to illustrate the effect of base movement on vertical pressure. The
variation in base pressure was small over a large range of base movement. The data obtained were
inconclusive as to the effect of particle size on base pressure. The result's on the effect of bin shape
verified Janssen’s theoretical approach where the only reference to bin shape is the hydraulic radius.
An increase in wall roughness caused an Increase in the value of K for wheat.
Very little work has been done where these variables have been controlled in order to determine their
individual effects upon bin pressures, it is believed that a knowledge of their- effects will add greatly to
the overall understanding of the problem of vertical pressure in deep bins, and will assist in the
interpretation of results of future experiments. AM INVESTIGATION OF SOME FACTORS
AFFECTING PRESSURES IN BINS
by
WESLEY P„ BOULANGER
A TMESIS
S u b m itte d t o t h e G raduate F a c u lt y
in
p a rtia l
f u l f i l l m e n t o f t h e req u ire m e n ts
f o r t h e degree o f
M aste r o f S cience in C i v i l E n g in e e rin g
at
Montana S t a t e C o lle g e
Approved;
i
Head, M a jo r Department
C h a jz ^ a n P B Kami ni ngi Committee
6
Dean, G raduate D i v i s i o n
Bozeman, Montana
December, I960
r.:': ,. ■,
-2 -
TABLE OF CONTENTS
Page
Acknowledgements ........................................................................................
3
A b s t r a c t .........................................................................................................
4
In tro d u c tio n
5
.................................................................................................
Review o f L i t e r a t u r e
...............................................................................
6
E xp erim e ntal E q u i p m e n t ...........................................................................14
M a t e r ia l
U s e d .............................................................................................. 15
E xp e rim e n ta l P r o c e d u r e ...........................................................................19
E xp e rim e n ta l R e s u lts
1) E f f e c t o f R e la t iv e Movement ................................................
26
2) E f f e c t o f P a r t i c l e S i z e .............................................................. 29
3) E f f e c t o f B in S h a p e ....................................................................... 34
4) E f f e c t o f Wal I R o u g h n e s s .......................................................... 39
P r e d i c t i o n o f K V a l u e s ................................................................................41
D is c u s s io n and Recommendations .........................................................
47
C o n c l u s i o n s ........................................................................................................ 51
Append!x
D e r i v a t i o n o f Ja n sse n 's E q u a t io n ......................................................53
O u t l i n e o f A i r y ' s S o lu t io n ...........................................................
56
E xp erim e ntal D a t a .................................................................................... 60
M icrobeads S iz e D i s t r i b u t i o n .............................................................. 75
B i b l i ography .................................................................................................
76
146215
-3 -
ACKNOWLEDGEMENTS
The w r i t e r wishes t o express h i s s in c e r e g r a t i t u d e t o P ro fe s s o r
A l f r e d C. S ch eer, w it h o u t whose guidance and a s s is ta n c e t h i s t h e s i s
would n o t have been accom plishe d, and h i s a p p r e c ia t io n t o P r o fe s s o r
Theodore T . W illia m s and A s s i s t a n t Dean George Herman f o r t h e i r t i m e l y
guidance and s u g g e s t io n s .
Thanks a re due a ls o t o M rs . G e t t i e Tschache f o r h e lp in g com pile
and ty p e t h i s t h e s i s .
-4 -
ABSTRAOT
Model
deep b in s were used t o determ ine t h e e f f e c t o f .several
mental v a r ia b le s on t h e v e r t i c a l
p re s s u re in b i n s .
funda-
These v a r ia b le s were
C l) th e e f f e c t o f r e l a t i v e movement between t h e w a lls and th e base on
v e rtic a l
p re s s u re ;
(2 ) t h e e f f e c t o f p a r t i c l e s iz e on v e r t i c a l
( 3 ) t h e e f f e c t o f shape o f b in c r o s s - s e c t i o n on v e r t i c a l
<4) t h e e f f e c t o f changing w a ll
roughness on v e r t i c a l
p re s s u re ;
p re s s u re ; and
pre ssu re .
Observed
v a lu e s o f K !are compared w ith v a lu e s p r e d i c t e d by t h e method suggested by
P r o fe s s o r A l f r e d G . Scheer in h is p ap er, " B in Theory and S o il M e chan ics."
Base p re s s u re o f wheat was p l o t t e d a g a in s t r e l a t i v e movement t o
i I I u s f r a t e t h e e f f e c t o f base movement on v e r t i c a l
pressure.
The v a r i ­
a t io n Tn base p re s s u re was small o ver a l a r g e range o f base movements
The
d a t a . o b t a in e d were in c o n c lu s iv e as t o t h e e f f e c t o f p a r t i c l e s iz e oh” base
p ressure.
The r e s u lt 's on t h e e f f e c t o f b in shape v e r i f i e d Janssen’ s 1th e o ­
r e t i c §1 'approach where t h e o n ly r e fe re n c e t o b in shape is t h e h y d r a u lic
ra d iu B ';
fAn in c re a s e ' in w a ll
roughness caused an Incre a se in th e value o f
K f o r 'wheat.
■-
•••»
Very I f t t l e work has been done where th e s e v a r ia b le s have been con­
tro lle d
It
in o r d e r t o dete rm in e t h e i r in d i v i d u a l
e f f e c t s upon b in p r e s s u r e s .
is - b e l i e v e d t h a t a knowledge o f t h e i r - e f f e c t s w i l l
add g r e a t l y t o th e
overaM' u n d e rs ta n d in g o f t h e problem -of v e r t i c a l ..pressure In deep b in s ,
and w r i t a s s i s t In t h e i n t e r p r e t a t i o n o f r e s u l t s o f f u t u r e e x p e rim e n ts .
r
-5 I MTRQDUCTI PM
L ittle ,
i t any, re search has been perform ed t o determ ine th e se p a ra te
e f f e c t s o f b a s ic f a c t o r s which a f f e c t t h e base p re ssu re in deep b in s .
o f t h e in v e s t i g a t i o n s
Most
in t h e f i e l d o f b in p ressures have been t o g a th e r data
f o r s p e c i f i c m a t e r i a ls in s p e c i f i c b i n s , w i t h o u t a tte m p ts t o g e n e r a liz e .
m ajor purpose o f t h i s t h e s i s
A
is t o in v e s t i g a t e th e e f f e c t s o f base move­
ment, p a r t i c l e s i z e , b in shape, and w a ll
a knowledge o f th e s e f a c t o r s w i l l
rough ness.
|t
i s b e lie v e d t h a t
enhance t h e u n d e rs ta n d in g o f b in p ressures
I t was decided t o weigh base 'Idads d i r e c t l y on a p l a t f o r m s c a le .
Several
p r i o r i n v e s t i g a t i o n s have used t h i s method, b u t w i t h o u t c o n s id e r ­
ing t h e e f f e c t which base movement may have had on t h e measured base lo a d s .
I t was a ls o decided t o use wood model b in s because o f t h e i r
low c o s t
and t h e t im e which c o u ld be saved in t h e numerous lo a d in g and u n lo a d in g
o p e ra tio n s .
F u rth e rm o re , t h e l i t e r a t u r e c i t e s cases where t h e r e s u l t s from
model b in s checked w e ll w it h r e s u l t s from f u l l
s iz e b in s .
—6*"
REVIEW QF LITERATURE
Isaac R o b e r t s , i n
1882, perform ed t h e f i r s t reco rd e d experim ents
on t h e p re s s u re o f wheat in b i n s .
lin d r ic a l
H is t e s t s were perform ed on model c y ­
b in s 7 t o 2 0 -3 /4 inches In d ia m e te r.
led him t o th e o b s e r v a tio n t h a t " a l l
The r e s u l t s o f th e se t e s t s
In c re a s e o f p re s s u re on t h e bottom
ceases b e fo r e th e c e l l s are f i l l e d two d ia m e te r s . "
In 1884, Roberts p e r ­
formed a second s e r ie s o f exp e rim e n ts u s in g a r e c t a n g u la r b in 6 ' 9" by
6 ' 0" in c ro s s s e c t io n and 5 2 ' 2" h ig h .
Levers arranged l i k e a weighing
machine were used t o dete rm in e th e pressure,.
R e a liz in g t h a t any movement
o f t h e le v e r s would a f f e c t h is measurement, Roberts added an excess o f
w e ig h t t o t h e s c a l e 's beam p r i o r t o
A fte r f i l l i n g
lo a d in g t o e l im in a t e t h i s movement.
t h e b i n , he g r a d u a ll y reduced th e w e ig h t and made h is re a d ­
ings a t t h e i n i t i a l
movement o f th e beam.
These e xpe rim e nts v e r i f i e d th e
r e s u l t s he o b ta in e d from h i s f i r s t s e r ie s o f t e s t s .
Janssen,
Jtn 1895, perform ed e xpe rim e nts t o dete rm in e th e p re s s u re
o f g r a in on b in w a l l s .
H is t e s t s were perform ed on square model b in s u s in g
wheat, c o r n , and o t h e r g r a i n s .
The bases o f h is b in s were placed d i r e c t l y
on t h e s c a l e ' s p l a t f o r m and t h e s id e s were supported above th e base by
ja c k s .
s till
These expe rim e n ts led t o th e development o f h i s fo rm u la s which are
in use t o d a y .
The development o f th e s e form ulas a re presented in th e
appendix o f t h i s r e p o r t .
In 1896, t h e r e s u l t s o f a s e r ie s o f t e s t s perform ed in Germany by
P r a n te ^ 24) were p u b lis h e d .
Two c y l i n d r i c a l
ir o n b in s were used In these
^N u m ber i n p a r e n t h e s e s r e f e r s t o B i b l i o g r a p h y
i t e m . Page
76 .
. -7 -
exper I m erits,
h ig h .
They were 1.5 and 2 .8 m eters in d ia m e te r, and were 19 meters
S id e p re ssu re s were recorded f o r t h e c o n d it io n s o f wheat a t r e s t and
m oving.
The r e s u l t s o f h is t e s t are ,not c o n s id e re d r e l i a b l e because o f t h e
Inadequacies o f h i s measuring apparatus,,
t h e in c re a s e in l a t e r a l
W ilfre d A i r y ^ ^ ,
However, h is e xp e rim e n t d id re v e a l
p re s s u re d u r in g t h e d is c h a rg e o f g r a in from a b i n ,
in 1897, made t h e n e x t m ajor c o n t r i b u t i o n t o t h i s
in a paper t o t h e I n s t i t u t e o f G iv M E n g in e e rs .
fie ld
H is paper c o n ta in e d th e
development o f e q u a tio n s f o r g r a in p re s s u re on w a lls due t o a wedge o f g r a in
between t h e w a ll and t h e plane o f r u p t u r e .
He a ls o r e p o r te d on a s e r ie s
o f t e s t s perform ed in England t o dete rm in e t h e va lu e s f o r a n g le o f repose
o f g r a in s and th e c o e f f i c i e n t s o f f r i c t i o n o f g r a in s on b in w a l l s .
( 17 )
A s e r ie s o f t e s t s on a f u l l - s i z e d b in were performed by J .A. Jamieson
in 1900.
The b i n , owned by t h e Canadian P a c i f i c Railway E l e v a t o r , West S t .
John, til,B . , was o f t im b e r c r i b c o n s t r u c t i o n
s e c t i o n , and 67' 6" h ig h .
12' 0" by 13 ' 6" In cross
M anitoba wheat was used f o r th e s e t e s t s .
Hy­
d r a u l i c diaphragms w ith w a ter column gages were used t o measure both v e r t i c a l and l a t e r a l
pressure.
T h is method o f measuring p ressure was found t o be
both s e n s i t i v e and a c c u r a t e , and was used on a l a t e r s e r ie s o f t e s t s Jamieson
perform ed on model b i n s .
r e s t and d u r in g e m p tyin g .
P ressure r e a d in g s were made w ith t h e g r a in a t
■
The r e s u l t s showed c lo s e agreement w ith those
o b ta in e d from J a n sse n 's form u la s u s in g a v a lu e f o r K o f a p p ro x im a te ly Q„6.
The maximum in c re a s e in p re s s u re d u r in g em ptying o ver t h e s t a t i c c o n d it io n
was f o u r p e r c e n t, a number c o n s id e r a b ly le s s than, t h a t r e p o r te d by P r a n t e ,
J a m i e s o n ^ l a t e r made e xpe rim e nts on 12 inch and 6 inch square model
b in s 6 ' 6" h ig h , and on c y l i n d r i c a l model b in s 12 inches and 6 inches In
—8~
d ia m e te r and 6* 6" h ig h .
T e s ts were made u sin g wheat, peas, c o r n , and f l a x ■
se e d . ,When th e g r a in was being em ptied from t h e c e n t e r o f t h e bottom , a
maximum in c re a s e in p re s s u re o f 7 . 3 p e r c e n t was o b se rve d.
made w ith t i e
bars in th e b i n .
T e s ts were a ls o
I t was observed t h a t th e y would decrease t h e
f lo w o f g r a i n , b u t d id n o t a f f e c t t h e p re s s u re due t o g r a i n .
About t h i s same tim e> a s e r ie s o f t e s t s on g r a in p re s s u re s were p e r ­
formed a t t h e U n i v e r s i t y o f I l l i n o i s
by A l b e r t G . Varnes and M.S. Ketc&um!
The t e s t s were perform ed on a square model b ln I ' 0" by I ' 0" .and 8 ‘ 6"
h ig h .
The bottom o f t h e b in was f r e e from t h e s id e s and was placed d i r e c t l y
on a p l a t f o r m s c a le t o r e c o rd v e r t i c a l
pressure.
S ide p re s s u re s on a one
f o o t square diaphragm were t r a n s m i t t e d by le v e rs t o a p l a t f o r m s c a le f o r
o b s e r v a t i o n . ■Measurements o f th e s id e and v e r t i c a l
when t h e g r a in was a t r e s t and being e m p tie d .
p re s s u re s were made
No d i f f e r e n c e in th e
p re s s u re s w ith t h e g r a in a t r e s t o r in m otion was o b se rve d .
la te ra l to v e r tic a l
The r a t i o o f
p re s s u re was a p p ro x im a te ly 0 .4 f o r t h i s s e r ie s o f t e s t s .
To check t h e r e s u l t s o f Jamieson, P r o fe s s o r Henry T . B o v e y ^ , M c G ill
U n i v e r s i t y , M o n t r e a l, perform ed f i v e s e r ie s o f t e s t s on f u l l - s i z e d b i n s .
Four s e r ie s o f t e s t s were perform ed on a b in 12' by 14'
and 4 4 '
in c ro s s s e c tio n
10" h ig h , a t t h e Canadian P a c i f i c R a ilw ay E le v a t o r a t M o n t r e a l.
The maximum l a t e r a l
was f i l l i n g ,
p re s s u re f o r b in e m p ty in g , compared t o t h a t when t h e b in
showed an in c re a s e o f . 9 .7 p e r c e n t.
s e r ie s was a t im b e r c r i b b in 13 .4 ' by 1 2 .3 5 '
N o rth e rn R a ilr o a d E l e v a t o r , Quebec.
The b in f o r th e f i f t h
in cro ss s e c t io n a t th e G re a t
H is re a d in g s agreed v e ry c l o s e I y w ith
th o s e r e p o r te d by Jamieson.
In 1904, E c k h a rd t L u f f t r e p o r t e d
on a s e r ie s o f t e s t s made on .
-9 -
fu I I - s i zed b in s a t Buenos A i r e s , A r g e n t in a .
c i r c u l a r b in s 5 4 .8 f e e t deep, one 2 3 '
These t e s t s were made on two
10" and th e o t h e r I I ' 3 " in d ia m e te r.
The In n e r s u r fa c e s o f the se b in s were coated w ith a cement m o r t a r . Pressures
were measured by a ru b b e r diaphragm w ith a mercury column.
w ith most t e s t s perform ed about t h i s t im e , v e r y l i t t l e
In agreement
in c re a s e in p ressures
was observed when t h e g r a in was moving, compared t o th e p re s s u re s when th e
g r a in was a t r e s t „
In 1906, M r . J . P l e i ssner^23) r e p o r te d on a s e r ie s o f t e s t s performed
in Germany.
The t e s t s were perform ed on model b in s o f t im b e r and c o n c r e t e .
The p re s s u re s were determ ined by measuring t h e d e f l e c t i o n s from a s e r ie s
o f w a te r t e s t s ,
Values o f K f o r wheat o f from 0 . 3 f o r c o n c r e te b in s t o
0 .5 f o r t im b e r b in s were o b t a in e d .
M i l o S . KetchumC19), in 1919, p u b lis h e d a book, "The Design o f W a lls ,
B in s and G ra in E l e v a t o r s , "
H is p u b l i c a t i o n re p o r te d on t h e work done t o
t h a t date on t h e p re ssu re s in g r a in b i n s , and c o n ta in e d a d is c u s s io n o f
some e x p e rim e n ts he had perform ed a t th e U n i v e r s i t y o f C o lo ra d o w ith P ro ­
fe s s o r C . C. W il li a m s .
8 ' h ig h .
These e xpe rim e nts were on model b in s 15" square and
The base p re ssu re s were recorded by p la c in g t h e bottom o f th e b in
on a k n i f e edge c a r r i e d a t t h e m id d le o f a I e v e r .
One end o f th e I ever was
r e s t i n g on a p l a t f o r m s c a le where measurements were made.
L a te ra l
p ressures
were determ ined by measuring th e v a r i a t i o n
in an e l e c t r i c c u r r e n t passing
th ro u g h carbon p l a t e s .
i t was observed t h a t th e v a lu e K
For g r a in a t r e s t ,
was n o t a c o n s t a n t , b u t in cre a se d w ith t h e depth o f g r a i n .
The la t e r a l
p re s s u re in c re a s e d from two t o f o u r p e r c e n t when th e g r a in was being e m p tie d .
L . R. Amundson^J p u b lis h e d in 1945 t h e r e s u l t s o f t e s t s run on a
- 1Q-
Gyl rn d ri.ca l
g r a in b in a t t h e 0 . S . Department o f A g r i c u l t u r e ’ s b in s i t e a t
Jamestown, N o rth D a kota .
The b in was b u i l t o f I n s u l i t e G r a y l i t e Board, and
was 10’ high and 9 ’ 6 - 1 / 2 " in d ia m e te r.
was wheat.
la te r a l
The m a te r ia l used in the se t e s t s
Several bands were placed around t h e b in t o determ ine th e
pressure.
s t r a i n gages.
The s t r a i n s
in th e s e bands were measured by WhIttemore
The re a d in g s from th e s e mechanical s t r a i n gages were then
c o n v e rte d t o s t r e s s , and t h e l a t e r a l
p re s s u re s were then o b t a i n e d .
M r.
Amundson concluded from h i s r e s u l t s t h a t Janssen’ s e q u a tio n s are adequate
f o r s a fe d e s ig n .
A v a lu e f o r K o f a p p ro x im a te ly 0.458 was o b t a in e d .
Most o f t h e p r e v io u s e xpe rim e nts
p r i o r t o 1950 were
perform ed on w h e a t.
To supplement t h e la ck of, in f o r m a t io n
on o t h e r m a t e r i a l s ,
a s e r ie s .of t e s t s
were run on s i x g r a n u la r m a t e r i a ls a t
Iowa S t a t e C o lle g e by A l f r e d C. Scheer
and C a lv in W. Tool
v e r t i c a l p ressures were measured f o r
L a t e r a l and
wheat, s h e lle d c o r n , soybeans, cement, sand, and pea g r a v e l ,
A c o n c re te
b in f i v e f e e t h ig h and e ig h te e n inches in dia m e te r was u s e d . The movable
i
■
'
.v .
■■ .
bottom o f th e b in was supp orted d i r e c t l y on a p la t f o r m s c a l e , making i t
p o s s ib le t o measure v e r t i c a l
load d i r e c t l y .
S ix s te e l p re s s u re p la te s used
t o measure l a t e r a l p re s s u re were supp orted by c a n t i l e v e r bars and s te e l
:
.
!
; - ' ■ ■ ■ ■ ' r" I
■;
bands wrapped around t h e b i n . The a d d i t i o n a l s t r a i n in t h e bands caused
:
.
f
*
by t h e l a t e r a l p re s s u re was measured by 8R-4 Type A - I 2 e i e c t r l c a l - r e s i s t :
.
't:
■■
.
:
Values o f K were o b ta in e d f o r a I I m a t e r i a ls in th e co n ■
■
•
' ■-■'■
■
ance s t r a i n gpges.
’
c re te b in .
R. A.
S a u l
( 2 ) r e p o r te d on a d d it io n a l
Iowa S t a t e C o lle g e in 1953.
work done on g r a in pressures a t
T e s ts were run on a r e c t a n g u la r b in 12' by
3 2 ’ in c ro s s s e c t i o n , and 10’ h ig h .
Panels which were small
areas o f th e
f l o o r o r w a ll s were used t o measure p r e s s u r e s .
These pan els were supported
by w e igh ing d e vice s which c o u ld weigh both t h e normal and p a r a l l e l
th e panel
beams„
fa c e „
The w e igh ing d evices were s t e e l
The s t r a i n s
load on
rods s u p p o rte d as s im p le
In th e s e rods were o b ta in e d by SR-4 Type A~7 e l e c t r i c a l -
r e s i sta n ce s t r a i n gages.
Three methods o f f i l l i n g
th e b in were used.
one case, th e g r a in 'w a s k e p t a p p ro x im a te ly le v e l d u r in g f i l l i n g
t h e spou t around w i t h i n th e b l n .
In
by moving
T h is method gave a more u n ifo rm p re ssu re
d i s t r i b u t i o n on t h e base, and a ls o gave s m a lle r values o f l a t e r a l
The r e s u l t s showed a r a p id decrease in t h e l a t e r a l
p re s s u re .
p re s s u re j u s t above t h e
b ase.
In 1943, M. Marcel R e lm b e r t^ O ) prese n te d a form u la f o r c a l c u l a t i n g
h o r iz o n t a l
p re s s u re on t h e w a ll s o f s i l o s .
R e s u lts o b ta in e d from t e s t s p e r ­
formed' in 1954 showed c lo s e agreement w ith h i s c a l c u la t e d v a lu e s .
R eim bert s t a t e d t h a t t h e la t e r a l
L =
p re s s u re a t a depth y equ als;
w R
ta n 0 '
where G Is a c o n s ta n t depending on th e shape o f . t h e s i l o
and t h e p r o p e r t i e s o f t h e c o n ta in e d m a t e r i a l .
® =
•
R
_______ - R ta n 0
ta n 0 ’ t a n ^ f r t
3
U
2)
R = H y d r a u lic Radius
.An i n t e r e s t i n g p o i n t which he d iscu sse s i s t h e v a r i a t i o n o f l a t e r a l
p re ssu re s on t h e long and s h o r t s id e s o f a r e c ta n g u la r b i n .
th e la te r a l
He s ta te d t h a t
p re s s u re on t h e s h o r t s id e would be th e same as t h a t on th e w a ll
o f a square b in w ith
l i k e dim en sion s, w h il e t h e l a t e r a l p re s s u re on th e long
-I 2-
s id e may be assumed t o b e - t h a t ora th e w a ll s o f a square s i l o w ith le ngth
o f 2 ab-a , where a i s th e le n g th o f t h e s h o r t s id e , and b i s th e le n g th o f
b
t h e long s i d e .
P r o fe s s o r A l f r e d G . S c h e e r(^ G ), j n 1960, presen ted a method o f p r e d i c t ­
in g v a lu e s f o r t h e r a t i o o f l a t e r a l ' t o v e r t i c a l
p re s s u re .
H is paper a ls o
discussed t h e a p p l i c a t i o n o f Ja n sse n's e q u a tio n s t o problems in pressures on
r e t a i n i n g w a l l s , tr e n c h s h e e tin g , c e l l u l a r c o ffe rd a m s , underground ,c o n d u its ,
c o n s o lidom eter f r i c t i o n
and f r i c t i o n a l
drag on p i l e s .
v a lu e s ( t h e r a t i o o f l a t e r a l t o v e r t i c a l
The p r e d i c t i o n o f K
p r e s s u re ) i s d iv id e d i n t o two
cases, rough b in s and smooth b i n s .
Rough b in s a re b in s where 0 , g i s g r e a t e r than 0, and t h e r e f o r e 0 ' = 0 .
To p r e d i c t K:
K = a K'
K' = I - si m2 0
I + s i n2 0
0=
A ngle o f in t e r n a l
fric tio n o f f i l l
0 ' b = A ngle o f f r i c t i o n o f f i l l
b lo c k , on b in w a l l .
m a te ria l.
m a t e r i a l , glu e d t o a
0 ' = E f f e c t i v e a n g le o f f r i c t i o n o f f i l l
bi n wa l I .
m a te r ia l on
Smooth b in s a re b in s where 0 is g r e a t e r than 0 ' g .
K = a K'
,
0 ' = 0'b
K 'm in
= _________ I __________
I t 2 ta n
ta n 0 '
cos e<
+ ta n 0 ' si n
= ta n 0 '
s in 0
To p r e d i c t K:
-1 3 -
K' max
I - s i 0'
1 + s in 2 0 '
K' = -L (K'm i n + K'max)
2
P r o fe s s o r Scheer s t a t e s t h a t a v a lu e o f 0 .7 f o r " a " w i l l
o f K> which w i l l
a p p ro xim a te expe rim e n ta l
o u t t h a t e x p e rim e n ta l
v a lu e s o f K.
y i e l d a v a lu e
Eowever, he p o in t s
va lu e s o f K should be used whenever a p p r o p r ia t e v a lu e s
a re o b t a i nab I e .
The l i t e r a t u r e search d id n o t reveal any re c o rd s o f experim ents in
which s y s te m a tic e f f o r t s were made t o i s o l a t e th e in d i v i d u a l
movement, p a r t i c l e s i z e , b in shape, and w a ll
e f f e c t s o f base
roughness on base p r e s s u r e . In
t h i s t h e s i s p r o j e c t , model wood b in s were used t o s tu d y each o f th e s e v a r i a b Ie s .
-1 4 -
EXPERI MEMTAL EQUIPMENT
S e le c te d f i r
was used in t h e c o n s t r u c t i o n o f a l l
b in s .
Al I boards
chosen were f r e e o f k n o t s , .had t h e g r a in ru n n in g l o n g i t u d i n a l , and had
a p p ro x im a te ly t h e same roug h n e ss.
S in ce in Jansse n's e q u a tio n s , th e o n ly
r e fe r e n c e t o th e b i n ' s c ro s s s e c t io n i s t h e h y d r a u lic r a d iu s , a l l
b ins
were c o n s t r u c te d so t h a t th e y would have equal h y d r a u lic r a d i i .
T hree r e c t a n g u l a r b in s were b u i l t halving h y d r a u lic r a d i i equal t o
th a t o f a s ix -in c h c i r c l e ,
Thddr dTiti'e'nsions were I I .48 by 4 .0 7 In ch e s,
7 .5 7 by 4 .9 8 In ch e s, and 6 . do' by. 6 .0 0 in c h e s .
each b in was 0.125 f e e t .
The h y d r a u lic r a d iu s f o r
Thdv h e ig h t o f each b in was a p p ro x im a te ly two
f e e t , which gave a depth o f from two t o f o u r tim e s t h e l a r g e s t d im e n s io n .
To a p p ro xim a te a c i r c u l a r b l n , u s in g m a te r ia l o f th e same roughness,
a b in was b u i l t h a ving f i f t e e n s id e s , w it h a minimum in s id e dimension o f
s i x in c h e s .
The j o i n t s o f t h i s b in were q u i t e i r r e g u l a r , a f a c t o r which
must n o t be o v e rlo o k e d In t h e c o n s id e r a t io n o f a n y . r e s u l t s o b ta in e d from
t e s t s ' perform ed on t h i s b i n .
In p e r fo rm in g t h e v a r io u s t e s t s t o determ ine th e e f f e c t o f base move­
ment, p a r t i c l e s i z e , and w a ll
roughness, t h e r e c t a n g u la r b in s were used.
The c i r c u l a r b in was used o n ly in th e e x p e rim e n ta l
d e te r m in a tio n o f th e
e f f e c t o f b ln shape.
A F a irb a n ks-M o rse p l a t f o r m s c a le , which .could be read t o 0.01
was used d u r in g t h i s
in v e s tig a tio n .
a p p ro x im a te ly t h r e e hundred pounds.
The c a p a c it y o f t h i s s c a le was
lb s .,
- 15 -
MATERtAL USED
In t h i s
i n v e s t i g a t i o n , wheat and f o u r s iz e s o f g la s s spheres were
u sed,
S iz e and Shape C h a r a c t e r i s t i c s
The wheat used was o f an undeterm ined v a r i e t y ,
No a tte m p t a t d e t e r ­
m in in g g r a in s iz e o r shape c h a r a c t e r i s t i c s was made.
The la r g e s t s iz e o f g la s s spheres used was 1/2 inch in d ia m e te r.
The
n e x t la r g e s t s i z e , r e f e r r e d t o as s iz e AP, had 95 t o 100 p e rc e n t passing
t h e number 20 s ie v e , and 0 t o 15 p e rc e n t passin g t h e number 50 s ie v e .
S iz e PM had 100 p e rc e n t passin g t h e number 70 s ie v e , and 0 t o 10 p e rc e n t
p a s sin g t h e number 230 s ie v e .
The s m a lle s t s i z e , MS-ML, had 90 t o 100
p e r c e n t p a ssin g t h e number 140 s ie v e , and 0 t o
number 270 s ie v e .
10 p e rc e n t passing th e
A more d e t a i l e d c l a s s i f i c a t i o n
i s diagrammed in th e
a p p e n d ix .
F r ic tio n a l C h a ra c te ris tic s
S in ce both t h e c o e f f i c i e n t o f i n t e r n a l
and t h e c o e f f i c i e n t o f f r i c t i o n
f r i c t i o n o f the f i l l
between t h e f i l l
m a te ria l
m a te r ia l and th e bin
w a ll s were im p o r ta n t f a c t o r s , c o n s id e r a b le e f f o r t was made t o determ ine
-
th e s e v a lu e s .
To dete rm in e t h e c o e f f i c i e n t o f in t e r n a l
f r i c t i o n o f th e m a te ria l,
t h e em ptying ang le o f repose method was used, s in c e p r e v io u s i n v e s t i ­
g a t io n s r e p o r t t h i s as being a c o n s i s t e n t method.
23" box was f i l l e d
and t h e m a te r ia l
then measured.
An 8 - 3 / 4 " x 9 - 1 / 2 " x
w ith t h e m a t e r i a l ; th e n one end o f t h e box was removed
was a llo w e d t o f lo w o u t . . The angle o f repose (0 ) was
There were no p r o j e c t i o n s on t h e open end o f t h e box t o
-.1 6 -
prevemt t h e f lo w o f t h e m a te r ia l
near t h e base.
T h is method o f measuring
t h e a ng le o f repose was fo llo w e d f o r a l l m a t e r i a l s .
However, In th e case
o f t h e 1 /2 inch d ia m e te r m a rb le s, t h e s u r fa c e o f t h e m a te r ia l- w a s concave,
and t h e a n g le recorded was t h e s lo p e o f t h e c e n te r l i n e
th e s u r fa c e to . t h e to p o f t h e s u r f a c e .
from th e base o f
The ta n g e n t o f t h e a n g le o b ta in e d
i s assumed equal t o t h e c o e f f i c i e n t o f in t e r n a l
fric tio n
{ JL{ ) w i t h i n th e
m a te r ia l .
Several methods were t r i e d t o d e te rm in e th e c o e f f i c i e n t o f f r i c t i o n
between t h e m a te r ia l and t h e b i n .
A b lo c k o f wood, 2 x .4 x 7 Inches,
was prepared by g l u in g t h e m a te r ia l on one s u r f a c e .
The b lo c k was then
dragged o v e r t h e s u r f a c e o f t h e b in by a sp rin g - b a la n c e .
T h is method was
found t o be in ade q u a te , s in c e th e re a d in g s were q u i t e I n c o n s i s t e n t .
The
c o e f f i c i e n t o f f r i c t i o n was then o b ta in e d from t h e ang le o f f r i c t i o n .
The b in was placed a t a g ive n s lo p e and t h e b lo c k was placed on th e
in n e r s u r f a c e .
The a ng le o f f r i c t i o n was t h a t ang le a t which th e b lo c k ,
i f pushed up t h e p la n e , more o f t e n than n o t , would s l i d e back when r e ­
le a s e d .
The ta n g e n t o f t h i s a ng le was assumed equal t o
' g , th e co­
e f f i c i e n t o f f r i c t i o n o f t h e m a te r ia l on t h e b i n .
For t h e rough bin case, whenever 0 'g was g r e a t e r than 0 , t h e e f f e c t ­
iv e c o e f f i c i e n t o f f r i c t i o n
(
' ) was assumed equal t o t h e ta n g e n t o f 0.
In t h e case o f t h e smooth b i n , whenever 0 was g r e a t e r than 0 ' g , th e
e ffe c tiv e c o e ffic ie n t o f f r ic t io n
(
' ) was assumed equal t o th e ta n g e n t
of 0'b .
■The f r i c t i o n a l
c h a r a c t e r i s t i c s o f t h e m a t e r i a ls are q u i t e v a r i a b l e ,
depending upon t h e method used t o dete rm in e them.
T h is f a c t o r must n o t
-1 7 -
be o v e rlo o k e d in t h e c o n s id e r a t io n o f t h e e x p e rim e n ta l
r e s u lts .
F r i c t i o n v a lu e s used in t h i s t h e s i s a re shown in T a b le I I .
These
v a lu e s were taken from u n p u b lis h e d data in P r o fe s s o r S c h e e r"s f i l e s
gathered as p a r t o f an M.S.G. E n g in e e rin g Experim ent S t a t i o n research
p ro je c t.
—18 -
TABLE I I
F r i c t i o n Angles f o r M a t e r i a l s Used
Wheat
1/2 D ia . M arbles '
AP
PM
MS-ML
0
28.0°
1.2.5°
25.0°
25.9°
27.0°
0 'b *
21 .7°
19.25°
25.9°
25.2°
24.8°
0 'B * *
Al I v a lu e s g r e a t e r than 0 .
*
For wood "bln.
**
For sandpaper b i n .
- 19 -
EXPERiMEMTAL PROCEDURE
In v e s tig a tio n
i n t o t h e e f f e c t s o f f o u r f a c t o r s on t h e base pressure
in b in s was p e rfo rm e d .
These f a c t o r s were r e l a t i v e movement between th e
w a ll s and t h e base, p a r t i c l e s i z e , b in shape, and w all
A to ta l
e ig h t ru n s .
o f 17 t e s t s was made.
ro u g h n e ss.
Each te s t, c o n s is te d o f fro m f o u r t o
Each run re p re s e n te d one f i l l i n g o f th e b i n .
Al I 17 t e s t s
a re sum m arized.in t h e a p p e n d ix.
The f i r s t n in e t e s t s were conducted p r i m a r i l y t o determ ine th e
e f f e c t o f r e l a t i v e movement between t h e w a ll s and th e base.
T e s ts IO
th ro u g h 13 were made t o i n v e s t i g a t e th e e f f e c t o f p a r t i c l e s i z e .
T e s ts
2, 1,4, 15, and 16 fu r n is h e d th e d ata t o determ ine th e e f f e c t o f bin shape.
Data from t e s t 2 and t e s t 17 show t h e e f f e c t o f changing w a ll
roughness.
E f f e c t o f R e la t iv e Movement
The f i r s t
I n v e s t i g a t i o n was t o dete rm in e th e e f f e c t o f r e l a t i v e
movement between t h e w a ll s and th e base on t h e v e r t i c a l
p re s s u re in bins-.
■
Al I t e s t s w ere-perform ed on t h e 4 .9 8 x 7 .5 7 inch b i n , u s in g w h eat.
I n d i c a t o r s were placed a t t h r e e o f t h e lower c o r n e r s .
D ia l
They were a tta ch e d
t o t h e s id e s o f t h e b in and recorded t h e r e l a t i v e d e f l e c t i o n between th e
w a ll s and t h e base.
Two o f th e d ia l
in g measurements t o 0.0001
o p p o s it e c o r n e r s .
in c h e s .
The t h i r d d i a l
i n d i c a t o r s were ca p a b le o f o b t a i n ­
These two in d i c a t o r s were placed oh
i n d i c a t o r , which c o u ld be read t o 0.001
in c h e s , was p laced a t one o f t h e two re m a in in g c o r n e r s .
T h e r e f o r e , any
i r r e g u l a r d e f l e c t i o n o f t h e base o r s id e s c o u ld be ob se rve d .
The base
o f t h e b in was placed on t h e p l a t f o r m o f t h e s c a le a long w ith a p r e lo a d ,
and t h e s id e s were r i g i d l y suspended above i t by a r e c t a n g u la r frame, as
/
-2 0 -
shown in F ig u r e I . '
i n t o th e b i n .
A c o n ic a l m e t a l l i c funnel
So t h a t a l l
m a te r ia l
was used t o
load th e m a te ria l
would be placed i n t o th e b in in a p p r o x i­
m a te ly th e same f a s h io n , th e f u n n e l , s u p p o rte d by hand, was s e t l i g h t l y
on t h e m a te r ia l
in th e b in ( o r on t h e base o f th e b in ) and was then f i l l e d
w ith t h e m a t e r i a l .
t o f lo w o u t „
The funnel was th e n s lo w l y r a is e d , a llo w in g th e m a te ria l
As th e m a te r ia l
was f lo w in g o u t , th e funnel was moved about
w i t h i n t h e b i n , th e r e b y keeping th e s u r fa c e o f th e m a te r ia l a p p ro x im a te ly
le v e l „
Two ty p e s o f runs were made.
c o m p le te ly f i l l e d
b in was f i l l e d
In one s e t o f ru n s , t h e b in was
b e fo re any re a d in g s were made.
in f i v e
in cre m e n ts.
Four equal
In t h e o t h e r ru n s , t h e
■
increm ents o f f i v e inches
o f m a te r ia l were placed in th e b i n , and t h e f i f t h
Increm ent then f i l l e d
t h e b in t o t h e same le ve l, as I t was in t h e f i r s t s e t o f r u n s .
Readings o f
t h e d ia l
A f t e r ru n n in g
I n d i c a t o r s were made a f t e r each increm ent was added»
th e f i r s t t h r e e t e s t s , t h e in n e r s u r fa c e o f t h e b in appeared t o have become
sm oother• than i t had been a t t h e b e g in n in g o f these t e s t s .
To v e r i f y t h i s ,
wheat was g lued on one s u r fa c e o f a p ie c e o f wood and dragged o ver th e
In n e r and o u t e r s u r fa c e o f th e b i n .
Readings were made by a s p r in g b a la n c e .
A lth o u g h t h e method o f measurement d id n o t g iv e a c c u ra te f r i c t i o n v a lu e s ,
th e y d id show t h a t In n e r s u r fa c e was smoother than o u t e r s u r f a c e .
T a b le I
.fs howsI t h e s p r in g balance re a d in g s o b ta in e d by dragg ing t h e b lo c k o f wood
o v e r t h e in n e r and o u t e r s u r f a c e s .
The averages are shown f o r com pariso n.
-2 1 -
Frame
P re lo a d
Base
Pl a t form S e a le
SET-UP FOR BASE ON SCALES
F ig u re
I
-2 2 TABLE I
Force Required t o Drag B lo c k Over S u rfa ce s
! finer
S u rfa c e
(0 z )
O uter
S u rfa ce
(Oz)
,
12.4
13.4
9 ,9
1.2.0
I I .6
13.6
■ 12.7
14.1
Ave = 11.9
Ave = I 3 .0
To o b t a i n r e s u l t s under th e same f r i c t i o n a l
r e s is t a n c e , t h r e e more t e s t s
were c o n d u c te d .
In re a d in g t h e p l a t f o r m s c a le lo a d , when i n v e s t i g a t i n g r e l a t i v e move­
ment, two d i f f e r e n t methods were u s e d .
Readings were ta ke n w ith th e s c a l e ' s
beam in i t s extreme upper and i t s extreme lower p o s i t i o n .* When readings
were taken w ith t h e beam in i t s
excess w e ig h t p r i o r t o f i l l i n g
lower p o s i t i o n , th e beam was loaded w ith an
the b in .
A f t e r th e b in had been f i l l e d
t h e m a t e r i a l , t h e w e ig h ts on t h e s c a l e 's beam were g r a d u a ll y reduced.
w ith
The
re a d in g was recorded when i t was j u s t p o s s ib le t o see l i g h t between th e
■scale's beam and i t s
re s t.
The maximum movement o f th e s c a l e ' s p la t f o r m
d u r in g w e ig h in g never exceeded 0.0002 in c h e s .
When w e igh ing was done w ith
t h e beam in i t s upper p o s i t i o n , no w e ig h t was added t o t h e s c a l e 's beam p r i o r
t o lo a d in g .
o f th e s c a le .
A fte r f i l l i n g
th e b i n , w e ig h ts were g r a d u a ll y added t o th e beam
The re a d in g was recorded under t h e same c o n d i t i o n as when
t h e beam was in i t s
lower p o s i t i o n .
P o s i t i v e va lu e s o f d e f l e c t i o n d e s ig n a te a movement o f t h e base away
from t h e s id e s , and n e g a tiv e va lu e s d e s ig n a te movement o f t h e base towards
-2 3 -
+he s id e s .
Movements o f t h e base away from t h e s id e s o f a p p ro x im a te ly 0.0035
inches were o b t a in e d .
By in c r e a s in g t h e p re lo a d on th e p l a t f o r m o f th e
s c a l e , t h e movement was reduced t o a p p ro x im a te ly 0.0008 in c h e s .
To ob­
t a i n a r e l a t i v e d e f l e c t i o n o f t h e base tow ards th e s id e s , t h e frame
s u p p o r tin g t h e s id e s wa's placed on t h e p l a t f o r m o f t h e s c a l e .
The base
was then s u p p o rte d on two b lo c k s on o p p o s it e s id e s o f t h e s c a l e , as shown
in F ig u r e 2.
Thus, t h e base was now f r e e o f both th e s id e s o f th e bin and
th e p la tfo rm o f th e s c a le .
lower p o s i t i o n .
Readings were taken w ith th e s c a l e ’ s beam in i t s
In t h e f i r s t t e s t run w i t h t h i s arrangement (T e s t # 6 ) , t h e
b lo c k s s u p p o r tin g t h e base were placed a t t h e ends o f t h e base, and p o s i t i v e
movements o f a p p ro x im a te ly 0.0008 inches were re c o rd e d .
When t h e b lo c k s
' .
were moved in alo n g t h e base u n t il , th e y were n e xt t o t h e s c a l e , d e f l e c t i o n
o f t h e base tow ard t h e s id e s was o b t a in e d .
E f f e c t o f P a r t i c l e S iz e
The second f a c t o r t e s t e d was t h e e f f e c t o f p a r t i c l e s iz e on base lo a d .
Al I t e s t s were perform ed on t h e 4 .9 8 by 7 .5 7 inch b i n .
Al I f o u r s iz e s o f
g la s s spheres were used.
The base o f t h e b in was placed on t h e p la t f o r m o f t h e s c a le , along
w i t h a p re lo a d o f a p p ro x im a te ly 130 pounds.
The s id e s o f t h e b in were sus­
pended- above t h e base by r e c t a n g u la r fra m e s .. T h e T o a d in g procedure was t h e
same as f o r t h e f i r s t s e r ie s o f t e s t s .
The d i a l
s in c e t h e i r r e a d in g s were no lo n g e r d e s ir e d .
i n d i c a t o r s were removed,
The end ope nin g in th e funnel
was r e a d ju s te d f o r each o f t h e p a r t i c l e s iz e s so-as t o keep a more u n ifo rm
f lo w .
However, in t h e case o f t h e 1 /2 inch diam e te r m a rb le s , i t was
-2 4 -
- Bin
Frame
P re lo a d
Base
Base
Support
i— Pl a tfo rm S e a le
SET-UP FOR BIN ON SCALES
F ig u re 2
Base S u pport
-2 5 -
necessary a t . t i m e s t o s l i g h t l y shake t h e funnel t o keep them f lo w in g .
E f f e c t o f B in Shape
The n e x t f a c t o r t e s t e d was t h e e f f e c t o f b in shape on t h e v e r t i c a l .
p re s s u re in b i n s .
Three r e c t a n g u l a r b in s anti th e c i r c u l a r b in were used
f o r t h i s s e r ie s o f t e s t s .
by 7 .57 in c h e s ,
T h e i r dimensions were 6 .00 by 6 .0 0 in ch e s, 4 .9 8
I I .48 by 4 .0 7 in c h e s , and 6 .0 0 inches in d ia m e te r.
The
Vodding proced ure and setup f o r th e s e t e s t s were s i m i l a r t o th o s e used f o r
th e in v e s tig a tio n o f p a r t i c l e s iz e .
Wheat was used f o r t h i s s e r ie s o f t e s t s
E f f e c t o f Wal I Roughness
The f i n a l
f a c t o r t e s t e d was th e e f f e c t o f v a r y in g roughness o f th e b in
s id e s on t h e v e r t i c a l
th e s e t e s ts . ,
p ressure.
The 4 .9 8 by 7 .57 inch b in was used f o r
J t was take n a p a r t , s iz e 2 /0 emery paper was glued t o th e
in n e r s u r fa c e t o o b t a in a rough t e x t u r e , and th e n th e b in was reassembled.
The dimensions were s l i g h t l y changed, g iv in g '7 .5 4 by 4 .9 3 in c h e s .
a f i n a l c ro s s s e c t io n o f
Wheat was used f o r t h i s s e r ie s o f t e s t s .
The setup
and lo a d in g procedure was th e same as t h a t used f o r th e p re c e d in g t e s t s .
The c o e f f i c i e n t o f f r i c t i o n o f t h e m a te r ia l on t h e b in w a l l , and th e
a ng le o f repose o f th e m a te r ia l were determ ined by th e Montana S ta te G o lle g e
E n g in e e rin g Exp erim e nt S t a t i o n .
~26-
EXPERI MENTAL RESULTS
E f f e c t o f R e la t iv e Movement
The e f f e c t o f movement between base and s id e s , on t h e base load o f
wheat, i s
i l l u s t r a t e d by F ig u r e 3 ,
When t h e base o f t h e b in was s u p p o rte d on th e s c a l e ' s p l a t f o r m , and
t h e beam in' th e upper p o s i t i o n , th e p l a t f o r m had a s l i g h t movement toward
t h e g r a in d u r in g w e ig h in g .
T h is movement caused a com paction o f th e g r a i n s ,
th u s g i v i n g a h ig h e r v a lu e o f base lo a d .
W ith t h e beam in t h e lower
p o s i t i o n , th e movement was down away from t h e g r a i n , g i v i n g t h e o p p o s ite
e ffe c t.
When t h e b in w a ll s are supp orted on t h e s c a l e 's p l a t f o r m , th e
movements w i l l
be j u s t re ve rse d In each case, g i v in g t h e o p p o s it e e f f e c t
on base lo a d .
These movements o f t h e base and s id e s , d u r in g w e ig h in g , never
exceeded 0 .0 0 0 2 in c h , and t h e r e f o r e had a m inor e f f e c t on v e r t i c a l
p re s s u re
a c c o rd in g t o F ig u r e 3 ;
-
%
,
The in n e r s u r fa c e o f t h e b in became smoother d u r in g t h e 4 i r s t few t e s t s ,
which may account f o r t h e base loads from t h e f i r s t t h r e e t e s t s being le ss
than v a lu e s from subsequent t e s t s w it h
lik e d e fle c tio n s .
The d e f l e c t i o n s , th a t were o b ta in e d f o r l a t e r t e s t s were between 0 and
a p p ro x im a te ly 0.0025 o f .an in c h .
O b s e rv a tio n from F ig u r e 3 would i n d ic a t e
t h a t th e base loads o b ta in e d from th e s e l a t e r t e s t s were s l i g h t l y
le ss than
would have been o b ta in e d i f t h e r e had been no base movement, w ith a maximum •
e r r o r o f perhaps 100, u n le s s t h e o t h e r m a t e r i a ls behave d r a s t i c a l l y
d i f f e r e n t from w h e a t.
T h is i s
a f a c t o r w o rth y o f f u r t h e r i n v e s t i g a t i o n .
The o t h e r m a t e r i a ls were n o t s i m i l a r l y t e s t e d , because t h e t e s t s were v e ry
tim e -c o n s u m in g , and i t was d e s ire d t o in v e s t i g a t e o t h e r f a c t o r s besides
'
I
-
-27■ : ...............................
•
base movement.
Inal!
subsequent t e s t s t o dete rm in e t h e e f f e c t o f p a r t i c l e s i z e , b in
shape, a n d -w a ll
roughness on base p r e s s u r e , t h e base was su pp orted on th e
s c a l e 's p l a t f o r m , and' re a d in g s o f load were taken w ith t h e beam in i t s
lower p o s i t i o n .
NO
340
20
D lE T Z G E N
GRAPH
2 0 X 2 0 PER IN C H
PAPER
|f
EUGENE
M •;' e
D'ETZd**.
U. S
6
11— j —4-.
i— r
*
CO
-2 9 E ffe c t o f P a rtic le
S iz e '
F ig u re s 4 t o 7 are p l o t s o f th e p re s s u re d i s t r i b u t i o n s
in th e 7.57 x
4 .9 8 inch b in f o r th e d i f f e r e n t m a t e r i a ls t e s t e d t o determ ine th e e f f e c t o f
p a r t ic le s iz e .
Values o f / ^ t K were s e le c te d t h a t would g iv e curves u s in g Janssen's
e q u a tio n , which would appro xim ate th e e x p e rim e n ta l c u r v e s .
The m a t e r i a ls used in o r d e r o f d im in is h in g p a r t i c l e s iz e s are I / 2 "
d ia m e te r m a r b l e s , AP m icrobea ds, PM m icrobea ds, and MS-ML m icrobeads.
•
340
2 0 D iE T Z G E N
GRAPH
2 0 X 2 0
PE R IN C H
2>
V E K T iL A - .
P k g ^ -J H fg
ON
O A & e j-( P * fj)
r iaufctan ItMiwwAlJ
PAPER
E U G E N E D lE T Z G E N
MADE i n u . S A.
CO.
NO
340
2 0 O lE T Z G E N G R A P H
2 0 X 2 0 PE R IN C H
PAPER
EUGENE
MAD
4 J k N f e ^ N vSI S
NO. 3 4 0
2 0 D lE T Z G E N
GRAPH
2 0 X 2 0
P E R IN C H
PAPER
MACE
EUGENE
D iE T Z G E N
IN u . S. A .
CO.
C KPEFttM O VTM CUftvC
j w v e k c ^ ' s E Q b K rm
6<»7 r<5M
or IUN
NO
340
-2 0 D IE T Z G E N G R A P H
2 0 X 2 0 PER IN C H
PAPER
E U G E N E D IE T ZG E N
MADE IN U. 6. A.
7 I : I : : : t.
E IP E NirtertTKw «iumv€g
-34 ~
■E f f e c t o f B in Shape
,
Four b in s were t e s t e d t o determ ine t h e e f f e c t o f b in cro s s s e c tio n
on base p r e s s u r e .
8 to I I .
T h e i r p re s s u re d i s t r i b u t i o n curves are shown in F ig u re s
Wheat was used as t h e f i l l
m a te ria l.
As discu sse d under t h e s e c t io n on Design o f B in s , due t o i t s con­
s t r u c t i o n , t h e r e s u l t s from t h e c i r c u l a r b in may n o t be v e ry r e l i a b l e .
Values o f yljj'K were s e le c te d such t h a t Janssen’ s e q u a tio n would g iv e
a th e o re tic a l
curve t h a t would a pp ro xim ate t h e exp e rim e n ta l c u r v e s .
t h e o r e t i c a l cu rv e s are superimposed upon t h e e xpe rim e nta l c u r v e s .
These
NO
340
- 2 0 D IE T ZG E N G R A P H
20 X 2 0
PER IN C H
PAPER
O IE T Z G E N
U. S. A.
M A D t IN
EUGENE
CO.
WHCMf
— tit
ftfMw
- J A k d S fCN': c k *
NO
340
- 2 0 D IE T Z G E N
GRAPH
2 0 X 2 0 PER IN C H
PAPER
E U G E N E D IE T Z G E N
MADE I N u . S . A.
CO.
NO. 3 4 0
* 2 0 D IE T ZG E N G RA P H
2 0 X 2 0 PER IN C H
PAPER
EUGENE
MADE
DIETZGEN
IN U. S A .
CO.
NO
340
2 0 D iE T Z G E N
GRAPH
2 0 X 2 0 PEP IN C H
PAPER
MADE
EUGENE
IN U. S. A .
D IE T Z G E N
CO.
JkN 4± i£*V ex aUAT Mjru
>
C^rioM dr I Htst
-3 9 E f f e c t o f Wal I R oughness
F ig u r e 12 shows t h e p re s s u re d i s t r i b u t i o n curve f o r t h e b in which had
Its
in n e r s u r fa c e covered w ith sandpaper t o
Incre ase th e roughness.
The dimensions o f t h i s b in were 7 .5 4 x 4 .9 3 in c h e s , which is v e r y •
s i m i l a r in c ro s s s e c t io n t o t h e 4 .9 8 x 7 .5 7 inch b i n .
v a lu e o f
Elsing wheat, a
,yLf 'K o f a p p ro x im a te ly 0 .2 4 0 y i e l d s a curve u s in g Jansse n's
e q u a tio n , which shows c lo s e agreement w ith t h e exp e rim e n ta l c u r v e .
4 .9 8 x 7 .5 7 inch b i n , which had no c o v e r in g on i t s
The
in n e r s u r f a c e , r e q u ir e d
a v a lu e o f / 4 'K o f a p p ro x im a te ly 0.146 in Ja n sse n's e q u a tio n f o r th e ex­
p e rim e n ta l and t h e o r e t i c a l
curves t o be in agreem ent.
For t h i s b i n , 0 < 0 ' g ; t h e r e f o r e ,
b i n , and th e e f f e c t i v e
i t would be c l a s s i f i e d as a rough
/4 ' v a lu e equ als ta n 0.
NO
340
2 0 O IE T Z G E N G HAF-K
2 0 X 2 0 PER IN C H
PAPEH
MACE
EUGENE
D IE T ZG E N
IN u. S. A .
CO
- 4 1—
PREDICTION OF K VALUES
P r e d i c t i o n o f K va lu e s were made by t h e method suggested by P r o fe s s o r
A l f r e d Soheer in h i s p ap er, " B in T heory and S o il Meehanic s
Wheat
0*B C 0
0 = 28.0°
0 'B = 21 .7°
0 ’ = 0 'B
0' s in of
Cos
+ ta n
Cos of
+ 0.398 s in o f
= ta n 0'
si n 0
0.398
0.469
o< = 5 9 .5 '
I
I + 2 ta n e< ta n 0 '
K 'a =
I ■
I + 2 ta n 59.5°
<
it:
0.425
Ii
CD
k'b
Il
K\
I .- si Fi^ 0 1
I + s i n^ 0 ’
= I - s i n 2 21 .7°,
I + s in 2 21 .7°
Ke = 1 ( K 'a + K ^ )
2 '
K'
= 0 .5 9 2
^ B i b l i o g r a p h y R e f e r e n c e Number Page 77.
Smooth Bin
•
-4 2 _
172 Dia . M arbles
0 = I 2.5°
0 /„ Rough Bi n
0 'B >
U
CO
0'
19.25°
=
0
K' = I - s i n^ 0 ' "
I + s i n^ 0
K' = I - s i n^ 12.5°
I + s i n^ 12.5°
0.910
AP M icrobeads
0 = 25 . 0°
0'B > 0 .» Rough B in
0.'
0 ' b = 25.9°
=
0
K' = I - s i n ^ 0
I + s in ^ 0
K' = I - s i n^ 25.0°
I + s i n 2 25.0°
0.699
PM M icrobeads
0. = 25.9°
01B ^ 1S», Smooth B in
B = 2 5 .2 °
0' = 0 B
Cos
+ ta n 25.20 si n ©< = ta n 25.2°
s in 25.9°
o< = 38° .
K';
I + 2 ta n 38° ta n 25.2'
-4 3 K 1a = 0.576
K ' b = I - s i n 2 25.2°
I + s ln 2 25,2°
K 'b
= 0.692
K'
= 1/2 (0 .5 7 6 + 0 .6 9 2 )
K'
a 0.634 ■
MS-ML M icrobeads
0 = 27 . 0°
0 l B ^ 0 /o Smooth Bin
0 ' b = 2 4 .8 °
0' = 0’b
Cos erf + ta n 24.8° s in erf
= ta n 24.8°
s in 27.0°
cd = 4 7 °
K 1a = __________ I______________
' I + 2 ta n 47° ta n 24.8°
K 1a = 0.425
K ' b = I - s i n 2 24.8°
I + s i n 2 24.8°
-
K 1B = 0.700
K'
= 1/2 (0 .4 2 5 + 0 .7 0 0 )
K'
= 0 .5 6 2
-4 4 -
Samd Paper Lin e d B in
-
M a te ria l:
Wheat
B in :
0'g ^ 0
Rough B in
0 ' = 0,
K' = I - s i n 2 28°
I + s i n 2 28°
K' = 0.640
The c a l c u l a t i o n s a re summarized in T a b le I I I .
4 .9 3 x 7.54
-4 5 -
TABLE I 11
Wheat
( M 'K ) F i t t e d
Bi n
>4'
Kmh =i
K 'c a l
a = k /K 1
6 .0 0 x 6 .0 0
0.146
.398
.366
0 .5 9 2
.619
I I .48 x 4.07
0.146
.398
.366
0 .5 9 2
.619
4 .9 8 x 7.57
0.146
.3 9 8
.366
0 .5 9 2
.619
0 .2 4 Q
.469*
.512
0 .6 4 0
.800
M 1
Kmhc
KV=,
0.105
. 222*
.473
,910
.520
0 .1 5 7
.466*
.337
.699
.482
0.204
.471
.433
.634
.684
0:167
.460
,363
.562
.646
4 .9 3 x 7 .54
Csandpaper li n e d )
B in :
4 .9 8 x 7.57
M a te ria l
I / 2 M a rb les
AP
.
PM
MS - ML
( X 'K )
F itte d
*Rough b in (Al I o t h e r s smooth b in ) .
0 < 0 'g ; t h e r e f o r e
= ta n 0
a = k /K '
-4 6 -
P r o fe s s o r Scheer in h i s paper, " B in Theory and S o il M e ch a n ics,"
suggests a v a lu e o f 0 .7 f o r " a " .
The p r e d ic t e d va lu e s o f K f o r " a " o f
0 .7 are compared w ith t h e observed va lu e s o f K in T a b le IV .
TABLE IV
Comparison o f Observed W ith P r e d ic te d Values o f K
M a t e r ia l
B in
Obs K
Wheat
4 .9 8 x 7.57
0.366
0.414
Wheat
6 .0 0 x 6 .0 0
0 .3 6 6
0.414
Wheat .
4 .0 7 x I I .48
0.366
0.414
Wheat
4 .9 3 x 7.54
(sandpaper lin e d )
0 .5 1 2
' 0.448
Wheat
6 .0 0 i n . d i a .
(c ir c u Ia r)
0 .3 0 2
0.414
I / 2 M a rb les
4 .9 8 x 7.57
- 0.47.3
0 .6 3 6
AP
4 .9 8 x 7.57
0.337
0.488
PM
4 .9 8 x. 7.57
0.433
0.443
MS-ML
4 .9 8 x 7.57
0.363
0.393
^
P r e d ic te d K
:
-A l-
DI SGUSS I ON AND RECOMMENDATIONS
C o n s id e ra b le c a u tio n must be used in r e l a t i n g model b in r e s u l t s t o f u l l
s iz e b i n s .
The g r e a t e r depths in f u l l
s iz e b in s causes much la r g e r p re s s u re s
which r e s u l t in g r e a t e r c o n s o li d a t io n o f t h e f i l l
m a te ria l.
c o n s i d e r a t io n s , p o s s ib ly a f l u f f y c o m p re s s ib le m a te r ia l
From s i m i l i t u d e
l i k e mica c h ip s o r
ru b b e r c h ip s shou ld be used in model b in s t o g e t e q u iv a le n t c o n s o l i d a t i o n .
However, l i t e r a t u r e r e p o r t s , though meager, g e n e r a ll y f u r n is h evidence
t h a t K v a lu e s f o r model b in s may be a p p lie d t o f u l l
dence.
s iz e b in s w ith c o n f i ­
z
Al I t e s t s , w ith a l l
m a t e r i a l , e xcep t f o r t e s t s numbered 3 , 4 , 6 , 7, and
8 , were conducted w ith t h e beam in t h e lower p o s i t i o n and t h e base o f th e
b in r e s t i n g on t h e s c a l e ' s p l a t f o r m .
T h is method is b e lie v e d t o g iv e r e ­
s u l t s t h a t a re more n e a r ly r e p r e s e n t a t iv e o f c o n d it io n s in f u l l - s i z e d b in s
where c o n s o li d a t io n
Is more pronounced than i t
i s in model b i n s .
In f u l l
s iz e b i n s , c o n s o li d a t io n causes c o n s id e r a b le downward movement o f th e f i l l
r e l a t i v e t o t h e b in w a l l s .
When th e beam i s balanced under t h e above
c o n d i t i o n s , t h e base has a s l i g h t d e f l e c t i o n away from t h e f i l l
T h is d e f l e c t i o n
in c re a s e s th e movement o f t h e f i l l
E x c e lle n t c o r r e la t io n ,
m a te ria l.
r e l a t i v e t o th e b in w a l l s .
r e l a t i v e t o b in shape, was o b ta in e d f o r th e
t h r e e r e c t a n g u la r b in s t e s t e d .
When a v a lu e o f
o f 0.146 i s used in
Ja n sse n 's e q u a tio n , th e p re s s u re d i s t r i b u t i o n curve o b ta in e d c l o s e l y a p p r o x i­
mated th e e x p e rim e n ta l curves f o r each o f t h e r e c t a n g u la r b i n s .
In
Ja n sse n 's e q u a t i o n , t h e s o le geo m e tric p r o p e r t y o f th e b i n ' s cross s e c t io n
i s th e h y d r a u lic r a d iu s .
th e o ry .
The r e s u l t s o b ta in e d s tro n g Iy s u p p o r t Janssen's
Even t h e c i r c u l a r b i n , when viewed in th e l i g h t o f i t s crude w ork­
-48~
manship, which y ie ld e d a
'K v a lu e o f 0 . 120, showed f a i r l y good c o r r e ­
l a t i o n w ith th e o t h e r b i n s .
However, t o o b t a in a more com plete undei—
s ta n d in g o f t h e e f f e c t o f b in c ro s s s e c t io n on base p r e s s u r e .
I t would
be necessary t o p e rfo rm a s e r ie s o f t e s t s on c i r c u l a r b i n s , r e c ta n g u la r
b in s , and b in s o f o t h e r shapes, perhaps i n c lu d in g an e q u i l a t e r a l t r i a n g l e ,
a lI
b u i I t o f t h e same m a te r ia l .
i t was im p o s s ib le t o a r r i v e a t a c o n c lu s io n r e g a r d in g t h e e f f e c t o f
p a r t i c l e s iz e on base p r e s s u r e .
11 was o r i g i n a l l y hoped t h a t th e use o f
t h e g la s s m a rb le s , and t h e v a r io u s s iz e s o f g la s s m icro b e a d s, would
e ffe c tiv e ly
i s o l a t e p a r t i c l e s iz e as t h e p rim a ry p h y s ic a l
e v e r , th e s e m a t e r i a ls had d i f f e r e n t f r i c t i o n a l
v a ria b le .
How­
c h a r a c t e r i s t i c s which were
u n d o u b te d ly a t t r i b u t a b l e , p a r t l y a t l e a s t , t o f a c t o r s o t h e r than p a r t i c l e
s iz e .
The percentage o f t r u e spheres was v a r i a b l e , and t h e microbeads were
n o t u n ifo r m in s i z e .
T h is l a t t e r f a c t o r c o u ld have been p a r t i a l l y
c o r r e c te d th ro u g h s e l e c t i v e s i e v i n g , b u t i t was not f e l t t o be w arranted
in view o f t h e u n c e r t a i n t i e s , which would s t i l l
rem ain.
The la rg e s iz e
gap, between t h e m arbles and th e l a r g e s t beads, was a ls o a s e r io u s
o b s ta c le .
It
Is suggested t h a t f u t u r e re s e a rc h e rs s e r i o u s l y c o n s id e r using a
s i n g l e s iz e m a te r ia l
in b in s o f d i f f e r e n t s i z e , t o dete rm in e whether th e
r a t i o o f p a r t i c l e s iz e t o b in d iam e te r a f f e c t s p re s s u re .
I f d iffe re n t, f i l l
m a t e r i a ls are used, th e y shou ld be more c a r e f u l l y c o n t r o l l e d , as t o s iz e
and shape, than was done h e re .
Except f o r an o u t l i n e o f A i r y ' s s o l u t i o n in th e a p p e n d ix , v e ry l i t t l e
e ls e has been mentioned o f I t
in t h i s t h e s i s .
Of th e two mathematical
-4 9 -
appro a c h e s , J a n sse n 's is +he more r a t i o n a l
and th e more w id e ly used by
e n g in e e rs , and t h e r e f o r e was used t o a id in a n a ly z in g t h e data th ro u g h o u t
th is
in v e s tig a tio n ..
A iry ,
in h i s s o l u t i o n , ta k e s a one f o o t wide s e c t io n
o u t o f t h e b in and f a i l s c o m p le te ly t o r e l a t e bin shape t o p re s s u re .
F u rth e rm o re , -he in c lu d e s t h e fo rc e s on o n l y one s id e o f t h i s s e c t io n ,
where i t bears a g a in s t t h e w a l l , and n e g le c ts t h e fo rc e s on t h e o p p o s ite
s id e .
In g e n e r a l, t h e p re s s u re d i s t r i b u t i o n curves d id n o t f l a t t e n o u t as
r a p i d l y as t h e t h e o r e t i c a l
Ja n sse n's c u rve s do.
T h is t r e n d appears t o be
d e f i n i t e enough and im p o r ta n t enough t o w a rra n t f u r t h e r i n v e s t i g a t i o n .
It
i s p o s s ib le t h a t s l i g h t l y d i f f e r e n t v a lu e s o f yLf'K would have been s e le c te d
t o f i t t h e e x p e rim e n ta l curves i f th e b in s had been deeper.
I t is
recommended t h a t f u t u r e t e s t s be perform ed on r e l a t i v e l y deeper b ins t o
f i n d t h e depth a t which t h e base p re s s u re does become e s s e n t i a l l y c o n s t a n t .
The observed v a lu e o f K f o r wheat in t h e r e c t a n g u la r b in s was 0 .3 6 6 .
M i l o Ketchum re p o r te d a v a lu e o f K f o r a wood r e c t a n g u la r b in o f 0 , 4 , w h ile
P l e is s n e r observed va lu e s from s i m i l a r b in s o f 0 .3 t o 0 , 5 , using a f i l l
m a te r ia l o f wheat.
Janssen and Jamieson, u s in g wheat in wood blns^ r e ­
p o rte d h ig h e r v a lu e s f o r K; t h e i r v a lu e s were 0.67 and 0 .5 9 6 , r e s p e c t­
iv e ly .
C o n s id e ra b le v a r i a t i o n
pected because o f t h e v a r i a t i o n
in t h e r e p o r te d values o f K c o u ld be ex­
in t h e f r i c t i o n a l
wheat and t h e d i f f e r e n t f r i c t i o n a l
c h a r a c te r is tic s o f
c h a r a c t e r i s t i c s o f b in s o f th e same t y p e .
When t h e b in w a ll s were covered by emery paper, t h e a n g le o f f r i c t i o n
0 ' b o f t h e wheat on th e w a ll
o f repose , 0 ,
was in c re a se d u n t i l
i t had exceeded th e ang le
The e f f e c t i v e c o e f f i c i e n t o f f r i c t i o n
was th e n equal t o th e
-SO™
ta n g e n t o f 0»
T h is in c re a s e in. roughness r e s u lt e d in an in c re a s e in K
f o r wheat o f from 0.366 t o 0 .5 1 2 .
P r e d i c t i o n o f K va lu e s were then made by th e method presen ted by P ro ­
f e s s o r A l f r e d 0 . S c h e e r.
The p r e d ic te d v a lu e s shown in T a b le IV , Page 46,
gave a f a i r e s tim a te o f th e observed K v a lu e s .
H is use o f a = 0 .7 i s , o f
c o u rs e , n o th in g more th a n an e s t im a t e , because " a " c o u ld not p o s s ib ly be
a u n iv e r s a l
co n s ta n t.
The r e s u l t s do g iv e hope, however, t h a t K may
u l t i m a t e l y be a c c u r a t e l y p r e d ic te d i f a r e f i n e d method o f e s t im a t in g " a "
i s d e v is e d .
- 5 1CONCLUSIONS
I,
From F ig u r e 3,
± 0.0020 inches w i l l
it
i s concluded t h a t a movement o f t h e base o f
r e s u l t in a v a r i a t i o n
in th e base load o f less than
± 10# f o r wheat in a b in o f t h i s s iz e and t y p e .
T h is v a r i a t i o n
is
•
r e l a t i v e l y s m a ll, and t h e r e f o r e i n s p ir e s c o n fid e n c e in t h e p la t f o r m s c a le
method o f measuring base lo a d s .
' 2.
The near i d e n t i c a l - n a t u r e o f F ig u re s 9 , 10, and I I ,
c o n c l u s i v e ly
dem onstrates t h a t v a r y in g t h e r a t i o o f t h e s id e s o f a r e c t a n g u la r b in d id
n o t cause any s i g n i f i c a n t v a r i a t i o n
3.
In t h e average base p re s s u re o f w h e a t.
In I n v e s t i g a t i n g t h e e f f e c t o f p a r t i c l e s i z e , extreme Iy ^ lo s e
c o n t r o l must be e x e r c is e d o ver th e s iz e and shape o f t h e p a r t i c l e s , I f a
s i g n i f i c a n t c o n c lu s io n i s t o be reache d.
The c o n t r o l was i n s u f f i c i e n t in
th is p r o je c t.
4.
jt
i s concluded t h a t a s i g n i f i c a n t v a r i a t i o n
in w all
roughness,
which causes a change from th e smooth b in case t o th e rough b in case, may
be expected t o cause a s i g n i f i c a n t change in K«
When emery paper was glued
t o t h e w a lls o f a r e c t a n g u la r wood b i n , t h e observed v a lu e o f K f o r wheat
in t h i s b ln in cre a se d from 0.366 t o 0 .5 1 2 ;
5.
To o b t a i n t h e observed K ' s , u s in g Seheer’ s K p r e d i c t i o n e q u a tio n s ,
i t would be necessary t o use va lu e s o f " a " t h a t l i e between 0 .5 and 0 . 8 .
-5 2 -
APPENDI X
-5 3 -
DERIVATION OF JANSSEN'S EQUATION
T
i
L
A 1M
- M U
T
dY
t
V
Ul I I
'I
'r
"
I
'L
4
L
t 1 1 1I 'TT - T T
V + dV
NOMENCLATURE
0 ' = Angle o f f r i c t i o n o f th e g r a in on t h e b in w a ll s .
JH' = Tan 0 ' = C o e f f i c i e n t o f f r i c t i o n o f g r a in on th e b in w a l l s ,
w = Weight o f g r a in in lb s . per c u . f t .
V
= V e rtic a l
p re s s u re in l b s . per s q . f t .
L
= L a te ra l
A
= Area o f b in in s q . f t .
U
= C irc u m fe re n c e o f b in in f t .
R
=A=
p re s s u re in lb s . per s q . f t .
H y d r a u lic r a d iu s in f t .
U
Y
= Depth from to p o f b in t o p o i n t where p re ssu re is d e s ire d in f t .
K
= R a tio o f l a t e r a l t o v e r t i c a l
pressure.
-54~
% V = O
(V + dV) A + ^ '
LWdY - V A -
wAdY = O
VA + AdV +
j q ' LUdY - V A - wAdY = O
dV .= wdY -
jy 'L W dY
M
A
R = A
U
W= A
R
T h e r e fo r e
M t L dY
R
dV = wdY -
S o lv in g f o r dY
dY =
dV
M 'L
R
w -
L
= KV
dY =
L a te ra l
p re s s u re i s equal t o v e r t i c a l
dV
W - M tK V
‘ 1R " ' - .
I n t e g r a t i n g we g e t
Y = ~
At Y = Q
R
/f 'K
log (w -
,
V=O
S o I v I nq f o r Q g i ves
0. =
R
lo g w
M *K V) + c o n s ta n t
R
p re s s u re tim e s a c o n s ta n t
Then:
Y = -
lo g (w - M ' K V) +
R
R
JLi i K
-
/ Y 'K Y S lo g (w R
,
-
vW'K
Y=
R
1
lo g
R
log w
JH'K
/Y 'K V) - log w
R
Zw - / C t K v \
I - W
- /C t K Y
e
R
= w- -
M 't V
S o lv in g f o r V
- / f tK Y
we
^
V=
R
’= w - yC t K V
- / C tK T
R
I w - w e
- V ^ tK f
V =
R w
/f'K
11 - e
V e rtic a l
p re ssu re a t depth Y.
L = KV
T h e re fo r e
L = Rw
I - e
/Ct K r
R‘
L a t e r a l p re s s u re a t depth Y.
—5 6 —
AIRY'S SOLUTION
W = Weight o f AEDCA one f o o t t h i c k .
P
= P re ssu re a g a in s t s id e AC.
= C o e ffic ie n t o f f r i c t i o n
between t h e g r a in and th e w a l l .
= C o e f f i c i e n t o f f r i c t i o n o f g r a in on g r a i n ,
d
= Bread th o f th e b i n .
h
= The depth o f th e g r a i n ,
w = Weight o f g r a in l b s . per c u . f t .
-5 7 -
Deep B in
Gne in which t h e pla n e o f r u p t u r e AE meets t h e o p p o s ite s id e
o f t h e b in w i t h i n t h e mass o f g r a i n /
Summation o f Forces ?n D i r e c t i o n o f AE
/-fR + Pcos x t yt^'Rsj ri x - ^ s i n x = ®
/ f R + Pcos x = CW - j y ’ P) si.n x
Summation o f Forces P e rp e n d ic u I a r t o AE
P + Zlf Peos x " P sin x - ^cos x = ®
R - P s in x = CW - > f ' P )
cos x
' - '
S o lv in q f o r P
= (W - M 'P ) s i n x - P Cosx
CW -
"P) Gosx + P s in x
CW ~ /4 t P) cosx t P sin x -
• J / (W - / / ’ P) t Pta n x
,
CW -
**P) s i n x . - Pcos x
= CW " / f 'P ) ta n x - P
P Cyl/ ta n x t yv" ta n x -
t
I ) = W Ctan x - / / )
P = W _________ ta n x - M _________
' - Z fZ f' + (/4
ta n *
However
W = w (Area AEDCA) = w (GD.)
(AC t DE
2
= w ji
'
2
(2h - d tan x)
.
-5 8 T h e re fo re
P = w _d (2h - d ta n x )
2
ta n x - M
1 ~
' + cJV + / / '> +an x
or
J
2h ________ ta n x - /-4__________
1 - M M '
I
+
- .d C ta n ^ x ~ / / t a n x
+an x
V
To f i n d x which makes Pmaximum s e t dP = O
dx
dP
dx
= W d /
2h
I
(I
2 j
Cos^ x
V
L
I
/-.^r-2
L
2 ta n x
COS^ x
Al I d i v i ded by
So t h a t
{
( - M i + A i ' * +an x
co s^ x
+M '*
( / ^ t / / ' ) ' ta n x + ___ I__
o n e ;2 x
..
d
-M M "* +
~M— 4
cos^ >
1 -M n '
' + ( M + M ' * "*"an
+ ( - M + M '*
2
Which g iv e s
ta n x = N
I 2h
\]
( I * M h T 7\ + A i h ( I - M M ' S - I - M M 1
d M +M '
M +M '
\M+M' J
M +Ai '
x
-5 9 T h e re fo re
P = J_ wd^
2
N
i- k ( / / + / / ' )
+ 1 ~m
m
..
To d e te r m in a te u n i t p r e s s u r e L a t depth Y
D i f f e r e n t i a t e P w ith re s p e c t t o depth ,
L =
wd
// + //'
V e rtic a l
V = L
K
p re s s u re a t any p o i n t
"
\Ji + / / 2
—6Q~
EXPERIMENTAL DATA
T e s t sffl
B in :
4 .9 8 x 7.57
Method o f L o a d inq;
M a te ri a'I:
Wheat
P re lo a d on base , arm in I ower p o s i t i o n .
Run
I increment
Base P ressure
W t. o f G r a i n
I
I
4 0 .4 2 p s f
27.26 lb s
2
I
4 4 .66
27.20
3
I
40.46
2 7 .4 2 .
7.25
4
I
41 .03
27.47
7 .0
5
I
40.65
27.41
I I .0
6
I
38.93
27.02
9 .25
7
37.41
27.40
I I .25
8
41 .41
27.43
10.0
D e fl a c t io n
5.2 5 /1 0 0 0 0
3 3 .0
-61 —
T e s t #2
g in :
4 .9 8 x 7 .57
Method o f Loading;
M a te ria l ;
Wheat
P re lo a d on base, arm Irn lower p o s i t i o n .
Run
Increm ent
Base P ressure '
I
I
2
3 ’
4
5
17.94 p s f
2 7 . 1Q
33.78
3 9 .3 2
41 .41
5 .7 3 lbs
I I .46
17.19
22.92
27.47
I
2
3
4
5
17.75
25.57
31 .87
3 7 .2 2
41 .99
5 .7 3
I I .46
17.19
22.92
27.3©
16.79
26.53
33.01
38.93
■ 42.18
5 .7 3
I I .46
17.19
22.92
27.35
6.75
7 .0
5.75
4 .2 5
2.5
I
2
3
4
5
18.70
26.15
32,64
3 7 .0 2
42.18
5 .7 3
I I .46
17.19
22.92
27.31
8 .0
■ 13.0
14.25
15.0
15.5
I
2
3
4
5
15.08
25.38
29.58
34,93
40.65
5 .7 3
I I .46
17.19
22.92
27.41
7 .0
9 .0
I I .5
10.5
8 .5
2
3
I
2 ■•
■3
4
4
5
'
W t. o f G ra in
■'
D e f l e c t i on
8 . 0/10000
5 .0
0 .0
- 3 .0
- 5 .25 ’
9.25 ■
'14.0
15.5
16.0 .
16.5
-6 2 T e s t #3
4 .9 8 x 7.57
M a te ria l:
Wheat
Method o f LoadInq':
P re lp a d on base, arm in upper p o s i t i o n .
Run
Inerememt
Base P ressure
I
I
2
Wt * o f G r a i n
D e f l e c t io n
4 3 .9 0 p s f
27.37 lbs
0 . 0/10000
I
42.94
27.44
14.5
I
43*32
27.37
14.25
4
I
2
3
4
5
■ 17.18
26,15
34.16
40*08
44.28
5 .7 3
I I .46
17.19
2 2 .9 2
27.43
12.5
19.0
23.0
26.0
27.0
5
I
2
3
4
5
17.17'
26.34
34.35
4 0 .8 4
44.85
5 .7 3
I I .46
17.19
22.92
2 7 .4 2
8.75
14.0
17.0
2 0 .0
20.25
3
'
-63'
Test M
B ill:
4 .9 8 x 7.57
M a te ria l :
Wheat
Method o f L o a d inq;
Mo p re lo a d on bQse j arm in upper p o s i t i o n .
Run.
I increment
Base Pressure
I
I
4 3 .7 0 p s f
27.39 lb s
28.0/100 00
W t. o f O r a i n
Def l a c t io n
2,
'I
46.95
27.36
36.5
3
'I
4 3 .3 2
27.31
3 6.5
4
I
2
3
4
5
18.89
27,29
34.54
41 .80
43.70
5 .7 3
I I .46
17.19
22.92
27.26
18.0
23.75
31 .75
3 8 .0
3 7 .0
5
I
2
3.
4
5
17.37
27.10
36.26
4 2 .5 6 ' ■
46.76
5 .7 3
I I .46
17.19
22.92
27.34
I 2.5
19.0
2 5 .0
2 9.0
3 2 .0
-6 4 T e s t ^5
B in :
4 .9 8 x 7 .5 7
Wheat
J
4-
No p re lo a d on base. arm in I ower
!
Method o f L o a d inq:
M a te ri a I ;
Run
I increment
Base Pressure
I
I
39.51 p s f
27.46 lbs
31 .5/10000
2
I
41 .41
27.53
3 3.5
3
I
41 .80
27.35
34 .'75
4
I
42.75 _
27.38
4 3 .5
5
I
41 .80
27.40
3 4 .0
2
3
4
5
17.94
26.34
32.25
39.51
41 .03
5 .7 3
I I .46
17.19
22.92
27.38
15.5
23 „5
.27.25
32.25
3 3 .0
I
2
3
4
5
' 18.32
27.29
33.78
38.93
4 3 ; 13
5 .7 3
I I .46
17.19
22.92
27.36
21 .25
28.25
3 2 .0
3 3 .0
35.25
6
7
•
Wt . o f G rai n
‘
D efl ectiora
'
-6 5 —
. T e s t #6
B in :
4 .9 8 x 7 .5 7
M a te ria l:
Wheat
Method o f L o a d in g ; No p r e lo a d , arm in lo wer p o s i t i o n „ Sides
su pp orted on p la t f o r m o f scale-, b lo c k s s u p p o r tin g base a t th e
extreme ends o f b ase.
Run
Increment
Base Pressure
I
.1
46.07 p s f
27.37 lbs
2 '
I
42.64
27.37
3 '
I
47.64
27.43
8.75
4
I
2
3
4
I
19.39
29.05
37.37
43.78
4 6 .3 0
.5.73
I I .46
17.19
22.92
27.18
9,75
9 .5
7 .5
6.5
4 .0
5
I
2
3
4
I
18.63
29.24
36.41
41.11
47.98
5 ,7 3
I I .46
17.19
. 2 2 .9 2
27.27
11.5
17.75
17.75
16.75
14.0
'
W t. o f G r a i n
D e f l e c t io n
9 .75/100 00
'
4 .0
T e s t #7
B in :
M a te r ia l :
4 .98 x 7.57
Wheat
No p r e lo a d , arm in lower p o s i t i o n . S ides supported
Method o f L o a d inq;
on p la t f o r m o f s c a le , b lo c k s s u p p o r tin g base moved in a l o n g • base nearer
t o s c a le .
Run
I ncrement
I
Base P ressure
W t. o f G rai n
Def I e c t io n
4 9 .7 0 p s f
27.37 lbs
- 18 .0 /1 0 0 0 0
2
I
47.71
27.40
- 2 7:0
3
I
52.48
27.45
- 18.5 -
4
I
51 .07
27.53
- 24.0
'5
I
48.86
■ 27.45
- 18.5
-6 6 T e s t #8
B in s
4 .9 8 x 7 ,5 7
M a te ri a I :
Wheat
Method o f L o a d in g s Base o f b in f ix e d in p o s i t i o n , s id e s supported
on p l a t f o r m o f s c a le s , p re lo a d on s c a le , arm in lower p o s i t i o n .
Run
Increm ent
Base P ressure
W t. o f G r a i n
I
I
4 6 .8 3 p s f
27.42 lbs
- 3 .0 /1 0 0 0 0
2
I
4 7 .0 3
27.42
- 0.75
3
I
4 8 .0 6 '
27.34
- 2.5
4
I
48.17
27.47
- 3 .0
D e f l e c t io n
T e s t #9
Bi ns
4 .9 8 x 7.57
Method o f Loadings
Run
I ncrement
I
M a te ri a I s • Wheat
P re lo a d on base, arm in lower p o s i t i o n .
. Base P ressure
W t. o f G r a i n
4 5 .2 3 p s f
27.39 lbs
I 3 .5 /1 0 0 0 0
D e f l e c t io n
2
I
43.51
27.38
13.5
3
I
44.09
27.44
14.25
I
4 7 .3 3
27.40
I 1.5
46.19
27 31
13.5
4
5
.
-6 7 T e s t #10
B in :
4 . 9 8 x 7 .5 7
Method o f L o a d inq:
Run'
Increm ent
M a te ria l :
Gl ass M a rb les ( I / 2 " Di a
P re lo a d on pI a t form , arm in I ower p o s i t i o n
Base Pressure
W t. o f M a rb les
I*
I
84.74 p s f
47.37 lbs
2
I
SI .87
47.81
3
I
79.01
47.90
4
I
76.91
4 7 .7 7
I
3 4 .7 5
9 .9 8
2
3 '
4
59.35
68.13
81.11
87.60
19.96
5
I
6
34.35 .
57.06
68.71
I
2
3
4
I
7 9 .2 0
-
82.07
^Funnel was n o t p r o p e r l y a d j u s t e d .
2 9 .9 4
39.92
47.54
9.98
9.98
9.98
39.92
37.57
'- 6 8 -
T e s t #1 I
B ln :
M a te ria l:
No. AP C la ss I l
4.98 i x 7 .5 7
Method o f L o a d inq:
P re lo a d on p l a t f o r m , arm in lower p o s i t ! o r
W t. o f M a te ri at
Run
I ncremen.t
Base Pressure
I
I
67.75 p s f
50.01
lb s
2
I
68.71
50.00
\
3
I
69 .85
50.07
4
I
69.66
50.00
5
I
2
3
4
I
2 6 .7 2
4 8 .2 9
10.50
, 21.00
31 .50
42.00
4 9 .6 2
I
2
3
4
I
27.10
45.61
58.97
67.18
68.52
6
57.64
66.61
71.57 _
'
10.50
21 .00
31.50 '
42.00
49.70
-6 9 T e s t #1 2
B in :
4 .9 8 , x ,7.57
Method o f L o a d in g :
Run
M a te ri al :
P re lo a d on p l a t f o r m , arm in lower p o s i t i o n i
Increm ent
I
I
2
I
3
No. PM C lass I I
W t. o f M a te r ia l
53.25 p s f
' 47.44 lb s
54.58
47.39
I
52.10
47.51
4
I
51.34
47.41
5
I
2
3
4
• I
,26.15
38.55
48.86
52.10
55.92
9.91
19.82
29.73
39.64
47.29
25.76
38.93
49.43
51 .15
54.96
9.91
19,82
29.73
39.64
47.45
6
I
2
3'
4
I
-
Base Pressure
-
, ( i ; 1' '
~70~
T e s t #13
B ln :
4 .9 8 x 1 3 1
Method o f Loa d in g :
Run
M a te ria l:
No. MS-ML Gl ass I I I
P re lo a d on pI a t fo r m , arm in lower p o s i t i o n
Increm ent
Base P ressure
Wt . o f M a t e r ia l
I .
I
61 .64 p s f
45.54 lbs
2
I
60.88
45.26
3
I
55.35.
45.10
4
I
65.27
45.16
5
I
2
3
4
I
25.00
37.22
46.38
56.49
59.16
9.46
18.92
28.38
37.84
45.76
I
2
3
4
I ■
24,24
37.98
50.96
57.83
60.50
9 .46
18,92
28,38
37.84
45.70
6
. .
■
-7 1 T e s t #14
B ln ;
6 .0 0 x 6 .0 0
Method o f Load!nq:
Run
I ncremeiiit
M a te ria l;
Wheat
P re lo a d on p l a t f o r m . arm in lower p o s i t i o n
Base P ressure
Weiq h t o f Wheat
36.40 p s f
25.65 lb s
I
I
2
I
' 41.80
25.62
3
I
39.40
25.73
4 ■
I
40.40
25.73
16.40
5 .38
10.76
16.14
21 .52
25.75
5
6
I
2
3
'4
3 6 .2 0
I
41 .80
I
2
3
. 4
I
I 6.80
28.60
2 8 .0 0
32.40 _
5.38
3 3 .8 0
I Q ,7 6
1 6 .1 4
39.80
42.40
21 .52
25.71
-7 2 Test #15
B in :
I I .48 x 4 .0 7
Method o f L o a d inq:
Run
M a te ria l:
P re lo a d on base, arm in
. Increm ent
Wheat
lower p o s i t i o n .
Base P ressure
Weiqht o f Wheat
I
I
36.83 p s f
33.00 lb s
2
I ,
3 7 .6 0 .
3 2 .9 6 ■
38.06 .
33.02
3
- I
4
I
38.83
33.1 I
I
2
3
■4
I
16.49
25.12
'32.42
38.68
4 1 .9 2
6 .85
13.70
20.55
27.40
33.13
5
6
.
I
2
3
4
I
-
15.41
.23.73
32.67
38.68
40.68
•
6.85
13,7®
20.55
27.40
■ 33.05
-7 3 T e s t #16
B ln :
C i r c u l a r , 6 in . D Ia .
Method o f L o a d Inq;
M a te ria l:
Wheat
P re lo a d on p l a t f o r m . arm in lower p o s i t i o n
Run
Increm ent
Base Pressure
Weight o f Wheat
I
I
4 2 .0 2 p s f
20.97 lbs.
.2
I
45.84
20.98
3
I
43.29
21 .01
4
I
57.55
20.95
'5
I
2
3
4
5
16.81
27.50
35 . 65
43.29
49.66 .
4 .3 6
8 .7 2
13.08
17.44
20.55
15.79
28.27
35.91
4 2 .0 2
45.58
4.36
8 ,7 2
13,08
17.44
20.67
6
I
2
3
4
I
■
.
-7 4 T e s t S a n d p a p e r-L in e d B in
!L
r:
4 . 9 3 x 7.54
Method o f L o a d inqs
M a te ria l:
P re lo a d on base, arm in
Wheat
lower p o s i t i o n .
I increment
Base Pressure
Weiq h t o f G ra in
I
I
26,74 p s f
27.17 lbs
2
I
25.58
27.14
I
26.16
27,15
4
I
25.58
27.15
5
I •
2
3
' 4
5
17.05
21 .71,
2 4 .4 2
25.58
27.13
5 .7 3
I I .46
17,19
22.92
27.29
I
15.12
20.54
23.64
24.81
25,58
I I .46
17.19
22.92
M
l
3
6
■
2
3
4
5
5 .7 3
2 7 .31
-
-7 5 -
TYPIOAL SIZE DI STRI B O T iI PM
>
C la ss I I
Minimum o f 13% t r u e spheres
P e rce n t Passing U „ S . Sieves
C a ta lo g '
Number
AP
20
30
50
95-100
70. '
80
IOG
140
230
325
0-15
100
PM
85-1.06,
15-55
0-10
\■
______ I
TYPICAL SIZE DISTRIBUTION
C lass I 11
Minimum o f 85# t r u e spheres
P e rc e n t Passing Id. S . Sieve's
C a ta lo g
Number
MS-ML
60
70
100
140
90-100
200
270
325
0-10
,
400
I
J
-7 6 BI BLt QGRAPHY
I.
A g ri c . Eng. V 34, 4 Apr I 953, P . 231 -4,, Measurement o f G ra in
P re ssu re on B in W a lls and F l o o r s .
2.
A g r i o . Eng. V 35, 8 Aug 1954, P . 5 7 0 -3 , P ressure in Deep G ra in
S to ra g e S t r u c t u r e s .
3.
A i r y , W i l f r i d . The P re ssu re o f G r a i n .
o f t h e I n s t i t u t i o n o f G i v i l E n g in e e rs .
4.
American S o c ie t y o f C i v i l E n g in e e rs , P r o c . V 77, S e parate n 82,
Aug. 1951, P . 13, P ressures in S h a llo w R e c ta n g u la r B in s .
5.
Amundson, L . R. D e te r m in a tio n o f Band S tre s s e s and L a t e r a l Wheat ,
P re s s u re f o r " a C y l i n d r i c a l G ra in B i n . A g r i c u l t u r a l E n g in e e r in g .
2 6 :3 2 1 -3 2 4 .
1945.
6.
B a lla g h , T . J . S . , C o ncre te Si I o s , I n s t n . o f C i v . Engrs.- o f I r e l a n d ,
Trans V 84 n 4 , Mar 1940, P . I 2 3-46 .
7.
B e to n -u S ta h lb e to n b a u V 48 n 8 , Aug 1953, P . 19 2 -4 , Druckberechnung
In Einem Si I o t r i ob ite r.
8.
Bovey, Henry T . E xp erim e nts on G ra in Pressures in Deep B in s and
t h e S t r e n g th o f Wooden B i n s . E n g in e e rin g News. 5 2 :3 2 - 3 4 .
1904.
9.
Caughey, R o b e rt A . , T o o le s , C a lv in W., and S ch eer, A l f r e d G.
L a t e r a l 'a n d V e r t i c a l P ressure o f G r a n u la r M a te r ia l in Deep B in s .
Iowa E n g in e e rin g E xperim ent S t a t i o n B u l l e t i n 172, 1951.
M inutes o f P roceedings
I 3 1 :3 4 7 -3 8 5 .
1898.
10,
C o ncre te & G o n s t r . E n g r . , V 50 n 4 , Apr 1955, P . 17 0 -2 .
o f S ilo s .
11,
E n g in e e r in g .
12,
Food Eng. V 26 n 2, F eb, 1954, P . 7 2 - 4 .
A B in .
13,
Fordham, A. A. The D i r e c t Measurement o f L a te r a l P re s s u re on W alls
and B i n s . E n g in e e r in g .
143:561-562.
1937.
14,
Hay, W, W. Design o f Deep C j r c u l a r B i n s .
4 3 -4 4 .
1928.
15,
Haugh, R . R . , C a l c u l a t i n g S tr e s s P re s s u re on W a lls o f R ig id .C o n ­
t a i n e r s , Package Eng. V 2 n 5 , May 1957, P , 4 0 -2 .
3 4 ;3 9 9 ,
1882.
Design
The P re s s u re o f S tored G r a in .
How M a t e r ia l
C o n c r e te .
Flows From
32 (n o . 6 ) ;
)
16 „
.................................. ~ '
-77Henry, G . E . J . , B u lk G ra in S to re s With P a r t i c u l a r R e t. To Design o f
V e r t i c a l Si I o s , S . A f r ic a n I n s t n . , G iv . E n g rs . T ra n s , V 7 n 7 , J u ly
1957, R, 225-42.
17.
Jamieson, J . A. G ra in P ressures in Deep B i n s . T r a n s a c tio n s o f th e
Canadian S o c ie t y o f C i v i l E n g in e e rs .
17:554-607.
1903.
18.
Janssen, H . A. Versuche uber G e tr e id e d r u c k in Si I o z e l le n . Z e i t s c h r f i t
des V e re in e s Deutscher I n g e n ie u r e . 3 9 ; 1045-1049.
1895.
19 .
Ketchum, M i l o S . , The Design o f W a lls , B in s and G ra in E l e v a t o r s .
Ed. 323-354.
20»
L u f f t , E c k h a r d t . T e s ts o f G ra in P ressure In Deep B in s a t Buenos
A i r e s , A r g e n t in a . E n g in e e rin g News. 5 2 :5 3 1 -5 3 2 .
1904.
21.
McGaImont, J . R. M easuring B in Wall P ressures Caused by A rc h in g
M a t e r i a l s , E n g in e e rin g News-Record.
12 0 :6 1 9 -6 2 0 .
1938.
22.
P e tr o v , B .A . Eksp erim e ntal 1nue O p re d e le n ie D a v le n iy a Tsementa Na
S tenki Zhelezobetonnykh S is o s o v , Tsemeht, V 24 n 2 Mar Apr 1958,
P. 2 1 -5 .
' -
23.
P l e i s s n e r , J . Versuche z u r E r m i t t l u n g der- Boden und Seitenw anddrucke
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50 P a r t I : 976-986.
1906.
.
3 rd
24.
P r a n t e . Messungen des G e tre Id e d ru c k e s gegen S i I owandungen. Z e i t s c h r i f t des V e re in e s D e utsch er I n g e n ie u r e . , 3 0 ; I I 22-1 I 25. 1896.
25.
R o b e r ts , Is a a c . D e te r m in a tio n o f t h e V e r t i c a l and L a t e r a l P ressures
o f G r a n u la r Substances. P roceedings o f the Royal S o c ie t y o f London.
3 6 :2 2 5 -2 4 0 .
1884.
26.
Sch eer, A l f r e d C . B in T heory and S o il M echanics, U n published Paper,
I960, Montana S t a t e C o lle g e .
27.
S c h e e r, A l f r e d 0 . Measurements o f H o r iz o n t a l P ressu re s E xe rte d by
C o nfined G ra n u la r M a t e r i a l s . M aste r o f S cience T h e s is , Iowa S ta te
Col Ieg e , I 950.
28.
S t a h l , Benton M.
No. 835.
1950.
29.
Tol t z , M . D is c u s s io n on G ra in P ressures in Deep Bins.. T r a n s a c tio n s
o f Canadian S o c ie t y o f C i v i l E n g in e e rs .
(17 ( p a r t I ) ; 641-644.
1903.
G ra in B in R equirem ents.
U. S . D e p t. A g r . C i r c u l a r
MONTANA STATE UNIVERSITY
762 1001 28 9 2 3