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Missouri University of Science and Technology
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Ninth International Specialty Conference on Cold-Formed Steel Structures
St. Louis, Missouri, U.S.A., November S-9, 1988
THE STRUCTURAL DESIGN OF LIGHT GAUGE SILO HOPPERS
by
J.
Michael Rotter l
Synopsis
Elevated light
gauge
silos usually have
a
conical discharge hopper
a t the
bottom. Although
this
hopper often carries much of the total weight of the
stored
solids within
the silo,
i t can often
be
cold-formed
from
thin
steel
sheet because
the s t ructural form
is
very efficient. However, guidance on
the design of
light
gauge
hopper
structures
is rare. The
term light
gauge is used here to describe the class of cold-formed silo
s t ructure
which
is
not
restr icted
by
a
nominal
minimum
plate
thickness
requirement
(eg 1/4 or 6 mm .
This
paper addresses several
aspects of the
design
of
light
gauge
hoppers.
Current proposals concerning hopper loads are
discussed first ,
and
recommendations
are
made. Appropriate s t ructural analysis
is then
presented.
The
potential failure modes
of
the hopper are identified, and
corresponding
s t rength
checks
described.
1. INTRODUCTION
Silos made from
cold-formed
steel
sheets are
widely
used
in
agriculture
throughout the
world.
Failures in silos
are common,
and improvements in
silo
technology
are clearly desirable.
This paper is concerned with the design of l ight gauge metal
silo
hoppers,
and those aspects of
the r ings and column support
conditions
which are
intimately
related
to
the
hopper (Fig. 1). t relates only to silos
of
circular
planform.
Conical
discharge hoppers are
generally subjected
to only symmetrical stored
solids loading. However, the pressures on hoppers are
less
well understood
than those on
vertical silo
walls;
the
structural action
is
a little more
complex, and little attention has
been
paid to hopper design in the past .
These
factors
underlie the present review paper.
Cold-formed
steel
silo hoppers are
susceptible
to more modes of failure
than
the
larger
industr ial
silo hoppers, because they
often
have
bolted joints of
limited strength. The
design
of these
joints
requires
a
more careful
assessment
of
hopper
loading patterns, so
a
significant
part of
this paper is
devoted
to the
definition of hopper
loads.
Farm silos
often differ considerably from industr ial
silos,
and
much
of the
available design
advice
is concerned either with industrial
and
mmmg
applications
(Wozniak,
1979;
Trahair
et
aI,
1983;
Gaylord
and
Gaylord,
1984;
Rotter, 1985 or
with
the cylindrical walls
of light
gauge
silos (Trahair et
aI, 1983; Abdel-Sayed
et
aI, 1985; Rotter, 1986b).
Senior
Lecturer
in
Civil Engineering,
University of
Sydney,
New South
Wales, Australia.
5 9
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530
2. LOADS ON HOPPER WALLS
2.1
Introduction
The
chief loading
on
conical
discharge hoppers der ives
from
symmetrically
placed stored solids. However,
the
assessment
of these
loads involves both
the cylinder
and
the hopper. A detailed argument concerning
appropriate
pressures
for
l ight gauge hoppers is presented
elsewhere
(Rotter, 1988), and
only the recommendations are given here.
The
most commonly used theories for pressures in hoppers
are
those
of
Walker (1966), Walters (1973) and Jenike
et
al (1973).
J vIost
codes
of
pract ice
(American
Concrete
Insti tute, 1977;
DIN
1055,
1986;
Gorenc et aI,
1986;
BJ vIHB, 1987) specify ei ther
a
constant pressure within the hopper or
a
l inearly varying pressure. J vIost
include a
local high s,,,itch pressure
near
the
hopper/cylinder junction (the transition) to account for flow conditions,
but the
details
of
the pressure distr ibution in the
body
of the hopper are
often thought to be relatively unimportant. However, l ight-gauge bolted
steel hoppers
require
a
more careful assessment
of
pressure distributions.
2.2
Defining
the Total Load on the Hopper
In elevated silos, the hopper suppor ts the
majority
of the total weight
of
stored material. The
total load
on
the hopper is defined by the
hopper
volume and the mean vertical str ss in the s tored material a t the transition
(hopper/cylinder junction)
(Fig.
2a). The la t ter depends
on
the height of
stored material
above
this
point, and
the proportion which is supported
by
friction
on
the
cylindrical
walls.
Cylindrical
wall
pressures are
therefore
needed to
define
the loading on the hopper. The
pressures on the
walls
of
the
cylinder
p, wall
frictional
tractions v,
and vertical s t r ss
in the s tored
solid
q (Fig. 3a)
are most
easily
assessed using Janssen's
equation
(1895)
p
= Po (1
-
e-Y/Yo)
( 1 )
v =
/ Lp
2 )
3
)
in
which Po
=
yR/2/.L
= yR/2/.Lk, Zo = R/2/.Lk, R is the
silo radius,
y is
the
distance
below
the
effective
surface
of
the
solid,
y
is
the
bulk
solid
density, / L
is the wall friction coefficient, and k is
the
lateral pressure ratio
(ratio
between
horizontal
and mean vertical stresses in
the
stored solid).
2.3
J vIaximising
the Loading
due to
Bulk Solids
J vIost silos are used to store
a
range
of
materials, so
that
the properties
may vary significantly from time
to time. Other
changes may
occur
as the
silo becomes polished or roughened by stored solids.
The silo
should
therefore
be designed for a
variety
of different values of Y, k and
/ L
in
Eqs
1-3.
All
loads
are
maximised
when
the
value
of
y
is
maximised.
The
largest
values of wall pressure occur when k is a t
i ts
maximum value and
/ L
at
i ts
minimum
value. The maximum vertical
loads
on hoppers
occur when both k
and
/ L
take their
minimum
values. The smallest
possible
value of
k
is given
by the simple Rankine pressure ratio
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k
=
-
s i n
+ s i n
53
4 )
in
which
is
the effective angle
of internal
frict ion
of
the stored
solid
(usually in the
range 28-33
0
for grains) . More
realist ic values
of
k
can
be
derived from
the
relation
f irs t
advanced by Walker
(1966)
and since adopted
by
many
others
k
=
1 +
s in
2
< >
- 2 / s i n
2
-g2cos
2
< »
4g2
+ co s
2
< >
5 )
However, the
values
obtained from Eqs 4
and
5 differ only
slightly
unless
the
wall is very
rough.
For
steel
silos, i t
therefore
is arguable tha t
Eq.
4
should be
used
to determine k
for
the
cyl inder
when
the
total
hopper
load
is being determined.
Corrugated
silos provide the one possible exception to
th is
proposal (Rotter, 1988).
2.4
Initial
Filling Loads
on
Hopper
Walls
The
simplest
theory of
hopper
filling pressures is tha t of Walker (1966). I t
assumes
tha t
the stored
material
in the
hopper
carr ies
no shear
stresses.
The maximum pressure occurs at the outlet (Fig. 4b).
This
pat tern is often
the
worst
pressure distr ibut ion for welded hoppers without s t rong
transit ional
r ing suppor t (Rotter, 1986a)
as i t
places the
maximum
load
as
far
from the suppor t
as
possible. However,
i t
may be
unduly conservat ive.
Walker
filling theory is
used
in
some codes (Gorenc e t
aI, 1986;
BMHB, 1987),
but not in
others DIN 1055, 1986; American
Concrete
Inst i tute,
1977).
t
should be noted
tha t the
frictional
t ract ion v has sometimes
been omitted in
draft ing
code
rules
based on
Walker
theory
BMHB,
1987)
leading
to
an
unsafe
definition,
s ince the hopper
is
deemed
to
carry less than
the total
load on it .
Walker
(1966) also presented a
theory for
discharge conditions. This
theory
involves the
general
solution of
the hopper
equilibrium equation subjec t to
the
condition
(similar to the Janssen
assumption)
tha t the ratio
of
wall
pressure to
mean
vertical
st ress in the
solid is invariant with depth
(Figs
4b
and 4c). Walker
also
proposed
one
method
of determining
th is
ratio.
Many writers
have
made
modifications to
th is
basic theory by determining
the ratio in different ways. McLean s (1985) modification of the Walker
theory
is supported
by
some experimental evidence (Motzkus, 1974;
Hofmeyr,
1986)
and
the
finite
element
calculations of
Ooi
and Rotter
(1987).
The
general Walker theory
leads
to
the
pressure
distr ibut ion (Figs 4b
and
4c)
p =
F
q
6
v
g p 7 )
8
in which p is the normal wall pressure , v is the frictional
traction,
q is the
vertical
s t ress in
the s tored
solid at height z above the apex, qt is the
value of
q at
the transi t ion
and
H
is
the height
of
the
hopper.
Equilibrium
of the complete
hopper
requires
tha t
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53
n
=
2 Fg c o t ~ + F
1 )
9
)
McLean 1985) suggests that F 1.0 leads
to reasonable estimates
of
hopper
initial filling
pressures ,
so
Eq.
9
reduces
to
n
= 2g
c o t ~
10
This theory
is recommended
here
to define hopper
i ~ t i l
filling
pressures.
2.5 Mass Flow
and
Funnel Flow Pressures
The pat tern of solids flow
from a
silo is known to affect
both the pattern
and the magnitude of the
pressures .
Two simple forms of flow pat tern have
been widely accepted (Jenike
e t
aI,
1973)
and are known as the mass flow
and funnel flow
modes (Fig. 3b). The hopper pressures are normally
defined with one or other of these
flow
modes in mind.
Most published
theories are concerned with the pressures dur ing mass flow.
I t is widely recognised tha t
the
pressures a t the
outlet
decrease
dur ing
discharge,
as
only then can
flow
of the solids
occur.
A
local high pressure
also
develops a t the t ransi t ion (Fig. 4c), and
most
design guides recommend
that
th is switch pressure
be considered.
The magnitude of the t ransi t ion
switch
pressure only
becomes
really large when
a
very steep hopper is
used. Most cold-formed
silos
have quite
shallow
hoppers (-45
0
) ,
so th is
switch
pressure is not
a very significant item.
In cold-formed bolted
hoppers,
the
crit ical point
is
usually
a short
distance
from the t ransi t ion, and this distance is defined by the hopper st ruc tura l
behaviour.
In these
circumstances,
i t is difficult to decide between the
different
theories and
codified rules , but
the original Walker
(1966)
discharge
theory may provide reasonable values. I t is certainly preferable
to
uniform
pressures
when
cold-formed hoppers
with
bolted joints
are
being
designed. I t is
given
by Eqs 6-9 with
F =
1 + sin
cosE:
11)
1 - sin c o s 2 ~ - E : )
.
1 [ /
[ g2
] ]
Sl.n
sin l+g2
12
)
in which g
is the hopper
wall
frict ion coefficient
and \>
is
the
effective
angle
of internal friction.
Funnel flow pressures
are less well understood. Gaylord and
Gaylord (1984)
and Rotter
(1988)
have presented
s t rong
criticisms of
the
commonly assumed
pressure distr ibut ions. The best current t reatment is probably to assume
tha t all hoppers are
subject to
mass flow hopper pressures
i r respective
of
their flow
pat terns
(Eqs
6-9,
11-12).
3.
STRESSES
IN
HOPPER
WALLS
3.1
Introduction
When
the silo is
continuously
suppor ted (not supported
on
columns),
the
st resses developing in the
grea ter part
of
a
steel hopper
can be
determined
using the
membrane
theory
of shells. Bending st resses
in the
hopper body
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535
design were studied:
one
with a cylinder of
simple rolled
sheet steel Design
A ,
and
the other made from corrugated sheeting Design
B).
The
choice
of
cylinder
wall
type will be seen to
have
a bearing
on
the design of the
hopper.
The
st ruc ture
was
analysed
f i rst
as
if it were
continuously
supported all
around the
circumference. This
illustrates
the
pat tern
of
st resses
ar is ing
in
and
near the
r ing as
a
consequence
of the
r ing
compression.
The
resul ts a re virtual ly independent of
the form
of
the
cylindrical wall. The
circumferential
membrane
st resses
are shown in Fig.
8a. The
compression
varies
considerably
through
the r ing, being only sensibly
constant
in the
annular plate element of the channel. Hand methods of analysis are chiefly
aimed at
determining
the value in
this
element alone. t is also clear tha t
circumferential compressive
st resses
ar ise in the hopper and cylindrical
wall,
that
these differ in magnitude from the
value
in the r ing, and that they
decline in a non-linear but
rapid
manner away
from
the
ring. Thus,
a
calculation based on
an effective section
must be interpreted with care,
as
the
s t ress
distr ibut ion is
very different
from those found in steel frame
members
which
do
not
distort.
The meridional membrane st resses near the top of the hopper are
shown
in
Fig. 8b. When
very
small
r ings
are used and the
hopper
is relatively
thick,
the meridional
tension
falls
below
the value
defined
by Eq. 21,
because some of
the
hopper
load
is suppor ted by
t ransverse shear ing
in
the hopper Rotter,
1987c). The reduced tension leads to
a
sl ight ly reduced
compression
in
the r ing,
but very
high bending st resses occur at
the
top
of
the hopper.
The meridional bending
st resses are
shown in Fig.
8c,
where
the very
high
very
local
maximum
at the
t ransit ion junction
is clearly
seen. Alternative
hand methods
of determining these
bending
st resses
were
advanced
by Barthelmes 1977),
Fuchssteiner and
Olsen
1979), Gaylord and
Gaylord 1984)
and
Rotter 1985). Some
authors
e.g. Gaylord
and Gaylord,
1984)
have argued tha t the
bending
discontinuity) st resses should
be
determined and allowed for in the design. However,
unless
the silo is
to be
loaded and unloaded so many times
that
a
fatigue failure
is possible Rotter,
1985), the bending st resses are not direct ly implicated in a definable
failure
mode, as
noted below. The
tedious
calculation
to determine
these
st resses
is
therefore
not normally warranted.
3.4
Stresses in Hoppers of Column-Supported Silos
The top of
the
hopper
in a column-supported silo
is subjec t to
a s t ress
state
which
is
closely
related
to
tha t
of
the
t ransit ion r ing or
junction.
The hopper must therefore be
designed
with the r ing and
support
condition
in mind.
Column-supported
silos present
a much more difficult problem
than
continuously suppor ted silos. They were f i rst described by Ketchum 1907),
and
the procedure
which
he suggested, al though
often
in serious error was
not
re-examined until the 1980s. Column-supported silos have been the
subject of a number of recent s tudies see Rotter,
1988)
and
several
hand
methods for est imating the s t resses in both the t ransi t ion r ing and the
cylindrical shell have
been
advanced. Many of
these
hand
methods
are
both
complicated and give
ra ther inaccurate
results .
More reliable st resses
are
obtained
from
finite element calculatio n Rotter,
1982).
The problem of the column-supported silo is essentially three-fold: f i rst the
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5 6
r ing a t
the
transition must be examined for bending or combined bending
and torsion: second, non-uniform
axial forces develop in the cylindrical
wall
above the r ing, and these
may
lead
to
buckling of the cylinder: and
thirdly, the
hopper is
in
non-uniform
meridional
tension,
and
this
non-uniformity may
lead to
rupture of the hopper. This paper deals only
with the last of these three.
The meridional membrane
s t resses
in
the
hopper of Design A are shown in
Fig. 9a.
The
non-uniformity of
hopper meridional
tensions extends
approximately
half-way
down the hopper.
The
meridian of the column
support is much more
highly
st ressed than that of
the
midspan.
The corresponding st resses for Design B are shown in Fig. 9b. These
st resses are significantly higher than those for Design A. This is
because
the cylindrical
wall
of
Design
A
plays an
important role
in
redistr ibuting
the forces from
the
columns,
but the corrugated wall of
Design
B
is
very
flexible in
vert ical
deformation, so it cannot fulfil
this
role. As a result the
r ing
in Design B
is more
highly stressed, sustains larger deformations
and
leads
to greater
non-uniformity
in the hopper meridional stresses. The
column-support
condition
affects
the
circumferential s t resses
in
only
a small
zone
a t
the top of the hopper,
and most
of the hopper is
stressed as
defined by Eq.
15.
The circumferential variation of
meridional membrane
st resses
in the
top of
the
hopper
is shown
in Fig. lOa for the
three
conditions of continuous
support C),
Design A and
Design
B. The corresponding
variation
of
circumferential
st resses
is shown
in
Fig.
lOb.
The column
support
causes
a
major change in hopper meridional
tension, and
a
significant
change in
circumferential
s t ress at the
top
of the
hopper.
However, the difference
between
a rolled
steel and
corrugated
cylinder wall causes
a
less easily
anticipated
but large
difference in
meridional tension
(40 ). Safe
hopper
design
clearly
depends on more than the loads acting on the hopper itself.
4. CRITICAL STRENGTH
ASSESSMENTS
FOR THE HOPPER
4.1
Introduction
The conical hopper of a silo is susceptible to several different
failure
modes,
including plastic collapse, meridional
seam
rupture ,
and
transition
joint
rupture . The transition
junction,
which
is intimately related
to
the
hopper, is susceptible to
plastic
collapse
and buckling,
and may
init iate
either hopper rupture or
cylinder
compression
buckling.
The
transition
junction is referred to here only insofar as it
affects
the hopper
design.
4.2 Plastic Collapse of Hoppers
Because the hopper is in biaxial
tension,
i ts
res is tance
exceeds
the
value
determined
by
simply restr ict ing the
effective
membrane
s t ress to
the yield
stress,
provided the hopper seams are s t rong
enough.
A clear
distinction
must therefore be made between fully welded
hoppers and
bolted hoppers.
The st rengths of fully welded hoppers have
been
the subject of a recent
study Teng and Rotter,
1988),
in which i t was shown that
the
st rength
estimated
using the
results of membrane theory underestimates
the
real
st rength
by about
10 . A
typical
collapse mode
at the top of the hopper is
shown
in Fig.
l la
The plastic collapse design st rength N
evm
for a
hopper
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539
junction,
t
c
t s and
th
are
hopper respectively,
and
the
may be assessed as lec
0.975/ (Rth/cosf3).
the thicknesses of
effecth-e
lengths
of
0.975/(Rt
c
)' les
=
the
cylinder,
sl t ir t
and
adjacent shel l segments
0.975/
(Rt
s
) and leh
=
For continuously supported silos, tradit ional simple t ransi t ion junct ion des ign
techniques
(Wozniak,
1979;
Trahair
e t
ai,
1983;
Gaylord
and
Gaylord,
19841
are general ly conservath-e, but
sometimes
they are
e17
conservative.
Useful savings
may thel 'efore be
made
by designing each component to i ts
real s t rength. Nevertheless, i t should be realised tha t
the t ransi t ion r ing
is
ineffective
if
placed only
a
shor t
distance above the hopper/cyl inder
intersection
(Rotter,
1985).
4.7
The
Column-Supported Silo and
i ts
Transi t ion
Ring
One of the
most
difficult
tasks in silo
design
is
to achieve
an economic
solution to the
problem of
an elevated
silo on
columns. The
column
suppor ts
int roduce high local
\-ertical compressive
forces into the
shell.
These lead to high
vertical
compressions
in the cyl inder ,
and
high
local
meridional tensions in the hopper. In
addition,
the t ransi t ion r ing
is
subjected to axial compression, bending about two axes
and torsion.
The
problem
of
st ress analysis of
these
components was mentioned above, but
very l i t t le work has been under taken to
establ ish
rational failure cri ter ia
for
any of
them.
t has been shown above tha t the meridional
higher in the hoppers
of
column-supported
cont inuously-supported
silos. No previous
s tudy
failure
cri ter ia for the hopper when the
silo
is
part icular ,
no
r igorous calculations
are
known.
s t resses are
locally
much
silos
than in those
of
appears
to have
discussed
supported on columns. In
Three
cri ter ia are
therefore proposed here:
one
for
r ing
plastic collapse,
one for
hopper
plastic
collapse
and
the third
for
hopper l 'upture . t
is
proposed tha t
the r ing plastic collapse s t r eng th should be assessed as the
same as
tha t
of
the
continuously supported st ruc ture (Eq. 24). This
proposal is based
on the observat ion
tha t
the junct ion
collapse mode
involves
large bending deformations
of
the shell
elements meeting
at the
tl 'ansition,
and tha t
signif icant redistr ibut ion may therefore be possible.
t is
also
proposed
tha t
hopper plastic collapse should be assessed as i f the
hopper were
continuously
supported
(Eq. 23). The
reasons
are similar to
those
for
junction
collapse,
but
the addit ional observat ions
may
be
made
tha t
the
hopper plastic collapse occurs within
the hopper ,
away
from the
region
most
inf luenced
by the column supports .
Fur ther
the
non-uniformity
of
the
meridional membrane st ress extends
fur ther into
the hopper than
tha t
of the
circumferential
membrane
stress .
The
s t r eng th of
most des igns
(shallo\-:,
smooth walled hoppers)
should not
be affected
very
much
by
locally elevated meridional membrane s t resses because
the
shape of
the
biaxial yield cri ter ion indicates insensi t ivi ty to th is parameter.
By
cont ras t
with
the two
above cri ter ia,
i t is proposed that hopper
t ransi t ion joint rupture and c i r c u m f ~ r e n t i l joint rup ture should be expected
when the maximum
meridional membrane
st ress
(Fig.
lOa) at tains
the
yield
st ress or the
s t r eng th
of the
joint. This proposal
recognises
tha t
local
rup ture
of the hopper near the
column could
lead
to
complete
rup ture of
the hopper. t is also based on the observat ion tha t
there
is
l i t t le
scope
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540
for redistrib:, tion e c ~ u s e the support
condition (ring)
is
in
bending, whilst
the
h.opper IS stret?hmg.
The
scope for redistr ibution of
the
high
hopper
merIdIonal
stresses
IS thus very limited.
A joint
of
high strength
is
most
easily
achieyed
by
using
thicker
steel
sheeting
for
the hopper.
Unfor·tunately, no
simple
hand
method of predicting the
local
high meridional
tensions in
the hopper
is
known,
so
a finite
element
analysis
is
currently
required
hen designing this
joint
to the proposed
criteria.
The rings on
column-supported silos
require separate
and careful
analysis
(Rotter, 1982,
1984, 1985),
which is beyond the
scope of
the
present
paper.
A
number
of practical
matters, relat ing to the use
of
steep hoppers, column
braeing
and ground-supported
skirts are discussed elsewhere (Rotter, 1988).
5.
SUJ vlJ vlARY
In this paper ,
a
review
of
design advice for
l ight-gauge
cold-formed steel
silos has been presented. The review has made specific recommendations on
the
pressures which
should be used in design and has described
appropriate
stress analysis
of
the
s t ructure .
The
failure
modes which
control
the design have been defined, and simple rules
for
some of these
have been presented.
For
column-supported
silos,
it
has been
shown that
the hopper
must be
thicker
than
it is for continuously
supported
silos. t has also been shown
that
the
form
of the
cylindrical silo wall can affect hopper s t resses
markedly. Design criteria for the
hoppers
of
column-supported
silos have
been proposed.
No
comparable existing criteria are known. J vlore detailed
information is
given in a
longer report (Rotter,
1988).
6.
ACKNOWLEDG EJ vlENTS
This
paper forms par t
of
a
major
research
program
into the
loading,
behaviour,
analysis
and design of
silo
and
tank
structures being
undertaken
a t the University
of
Sydney. Support
for
this program from the Australian
Research
Grants Scheme, the Universi ty
of
Sydney and cooperating
commercial
organisations is gratefully
acknowledged.
APPENDIX.- REFERENCES
Abdel-Sayed,
G., J'vlonasa, F.
and Siddall, W. (1985) Cold-Formed Steel Farm
Structures
Part I: Grain Bins' ' ' , Jnl of Structural Engineering,
ASCE, Vol.
111,
No.
STlO, Oct.,
pp
2065-2089.
American
Concrete
Institute
(1977)
Recommended Practice for Design and
Construction of
Concrete Bins, Silos
and
Bunkers
for
Storing Granular
J'vlaterials , ACI 313-77,
Detroit (revised 1983).
Barthelmes, W. (1977) Ermittlung del' Schnittafte
in
kreiszylindrischen Silos
mit kegelrmigem Boden , Bauingenieur,
Vol.
52,
pp 423-435.
BJ vlHB
(1987) Silos:
Draft
Design Code for Silos,
Bins,
Bunkers and
Hoppers ,
British Materials
Handling Board and
British
Standards Institution, London.
-
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541
DIN
1055 (1986) Design
Loads for
Buildings; Loads
on Silos ,
German
Standard,
Sheet
6, September.
Fisher, J.W. and Struik, J.H.A.
(1974)
Guide to Design Criteria for Bolted
and Rivetted Joints , Wiley, New York.
Fuchssteiner , W
and
Olsen,
O.W.
(1979)
Ein Problem
der
Stahlblechsilos ,
Bauingenieur,
Vol. 54, pp 17-21.
Gaylord, E.H. and
Gaylord,
C.N.
(1984) Design
of Steel Bins for Storage of
Bulk
Solids,
Prentice
Hall Englewood Cliffs, New Jersey.
Gorenc, B.E.,
Hogan,
T.J.
and Rotter,
J.M. (eds) (1986) Guidelines
for
the
Assessment
of Loads on Bulk Solids Containers , Insti tution
of
Engineers,
Australia.
Hofmeyr, A.G.S. (1986) Pressures in Bins ,
MSc(Eng) Thesis,
University
of
Witwatersrand, Johannesburg.
Janssen, H.A.
(1895)
Versuche uber Getreidedruck in
Silozellen ,
Zeitschrift
des Vereines Deutscher
Ingenieure,
Vol. 39, No. 35, pp 1045-1049.
Jenike, A.W. Johanson,
J.R.
and Carson, J.W.
(1973) Bin
Loads- Part 2:
Concepts; Part
3: Mass Flow
Bins , Jnl
of
Engng
for
Industry,
ASME Vol.
95,
Series B No.1 ,
Feb.,
pp 1-12.
Ketchum,
M.S.
(1907) Design
of
Walls,
Bins and
Grain
Elevators ,
1st
edition,
McGraw-Hill,
New York
(2nd edn 1911,
3rd edn
1919).
McLean,
A.G.
(1985)
Initial
Stress
Fields
in Converging
Channels ,
Bulk
Solids
Handling,
Vol 5, No 2, April.
Motzkus, u.
(1974)
Belastung von
Siloboden
und Auslauftr ichtern durch
kornige
Schuttguter , Dr.-Ing
Dissertation,
Technical Universi ty of
Braunschweig.
Ooi J.Y.
and
Rotter, J.M.
(1987) Elastic
Predictions of Pressures in
Conical
Silo Hoppers , Research Report
R555,
School of Civil and
Mining
Engng,
Univ. of Sydney, Oct.
Rotter,
J.M. (1982) Analysis
of
Ringbeams
in Column-Supported
Bins ,
Eighth
Australasian
Conference on
the
Mechanics of
Structures
and
Materials,
University
of
Newcastle, Aug.
Rotter, J.M. (1983) Effective Cross-sect ions of
Ringbeams
and Stif feners for
Bins ,
Proc., International
Conference on Bulk
Materials
Storage
Handling and
Transportation, Insti tution of Engineers, Australia, Newcastle, Aug., pp
329-334.
Rotter, J.M. (1984) Elastic Behaviour of
Isolated
Column-Supported
Ringbeams ,
Journal
of Constructional Steel Research,
Vol 4 1984, pp 235-
252.
Rotter,
J.M. (1985)
Analysis
and
Design of Ringbeams ,
in
Design of Steel
Bins
for the Storage of Bulk
Solids,
edited by J.M. Rotter, Univ. Sydney,
-
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542
March, pp
164-183.
Rotter,
J.M.
(1986a) On
the
Significance
of Switch Pressures
a t
the
Transition
in
Elevated Steel Bins ,
Proc.,
Second International Conference on
Bulk
Materials Storage
Handling and
Transportation,
Institution
of
Engineers,
Australia,
Wollongong, July, pp
82-88.
Rotter,
J.M. (1986b)
Recent
Studies
of
the
Stability of
Light Gauge
Steel
Silo Structures , Proc.,
Eighth International Specialty
Conference on
Cold-Formed Steel
Structures, St.
Louis, Missouri,
Nov.,
pp
543-562.
Rotter, J.M.
(1987a)
The
Buckling
and
Plastic
Collapse
of
Ring Stiffeners
at
Cone/Cylinder
Junctions , Proc., International
Colloquium on
Stability
of Plate
and Shell
Structures, Ghent,
April, pp 449-456.
Rotter,
J.M.
(1987b) Membrane Theory of Shells for
Bins
and
Silos ,
Transactions
of
Mechanical
Engineering, Insti tution
of
Engineers, Australia,
Vol. ME12 No.3 Sept., pp
135-147.
Rotter, J.M.
(1987c) Bending
Theory
of Shells
for Bins and and Silos ,
Transactions
of Mechanical Engineering,
Institution
of Engineers, Australia,
Vol. ME12 No.3
Sept.,
pp 148-159.
Rotter,
J.M. (1988) The
Structural
Design
of
Light Gauge
Research
Report,
School
of Civil
and Mining
Engineering,
Sydney,
May.
Silo
Hoppers ,
University
of
Teng, J.G.
and
Rotter,
J.M. (1988)
Plastic Collapse of Steel Silo Hoppers ,
Research Report
R568, School of Civil
and Mining Engineering, University of
Sydney,
April.
Trahair,
N.S.,
Abel,
A.,
Ansourian, P., Irvine,
H.M.
and Rotter,
J.M. (1983)
Structural
Design
of
Steel Bins
for
Bulk Solids, Australian Inst i tute of Steel
Construction, Nov.
Walker, D.M.
(1966)
An Approximate
Theory for Pressure and Arching
in
Hoppers ,
Chern.
Eng.
Sci., Vol. 21, pp
975-997.
Walters,
J.K. (1973)
A
Theoretical
Analysis of
Stresses
in Axially-Symmetric
Hoppers
and Bunkers , Chern.
Engng
Sci., Vol.
28, No.3 ,
March, pp
779-89.
Wozniak, R.S. (1979) Steel Tanks
in
Structural Engineering
Handbook,
2nd
edn, Section
23,
Eds.
E.H.
and
C.N.
Gaylord, McGraw-Hill.
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54
Roof
Ring
Cylinder
Ring
kirt
Conical
Hopper
FIG 1
TYPICAL COLD FORMED ELEVATED SILO
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S
n
M
e
d
o
T
o
\
R
a
E
b
u
m
a
N
a
o
e
z
c
b
s
m
m
e
c
S
e
R
a
s
F
G
2
H
P
G
E
O
M
E
R
L
O
A
N
A
N
T
O
N
c
n
-
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-
8/18/2019 The Structural Design of Light Gauge Silo Hoppers 2
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-
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I
1
0
-
N
.
;
0
8
-
<
t
>
=
3
.
.
:
.
.
0
6
'
'
.
/
,
/
0
.
4
,
~
_
c
o
0
2
Q
E
o
o
0
2
0
4
0
6
0
8
1
0
W
a
P
e
e
p
o
R
a
P
e
e
D
s
b
o
0
1
0
2
0
3
0
4
C
r
c
m
f
e
e
a
S
R
N
e
o
b
C
rc
m
f
e
e
a
M
m
a
S
R
,
/
,
,
/
/
/
0
5
0
2
0
4
0
6
0
8
1
0
1
2
M
e
d
o
S
R
c
M
e
d
o
M
m
a
S
R
F
G
5
T
C
S
R
R
A
D
S
R
B
O
H
O
N
L
Y
C
l
1
-
l
-
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-
8/18/2019 The Structural Design of Light Gauge Silo Hoppers 2
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o
o
L.f l
(Y l
o
o
L.f l
,
3500
o=9kN/m
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55
Support
Position
200MPa
L ............
Support
Position
200MPa
T
a) Design
b) Design B
FIG.9
MERIDIONAL
MEMBRANE STRESSES
IN
HOPPERS
O COLUMN-SUPPORTED SILOS
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3
2
L
V
I
V
I
Q
J
V
1
o
:
0
I
5
0
,
,
-
-
~
-
-
-
-
r
-
-
-
-
-
-
~
-
-
-
-
r
-
-
-
,
T
o
o
\
I
r
o
a
L
V
5
0
V
I
Q
J
-
V
-
1
C
m
p
e
o
D
e
g
B
5
0
I
-
1
.
L
L
I
_
_
-
1
-
1
5
0
5
1
1
-
1
C
r
c
m
f
e
e
a
C
d
n
e
d
e
a
M
e
d
o
M
e
m
b
a
S
e
-
1
-
5
o
5
1
C
r
c
m
f
e
e
a
C
d
n
e
d
e
b
C
r
c
m
f
e
e
a
M
e
m
b
a
S
e
F
G
1
C
R
M
F
A
V
A
O
N
O
S
R
A
T
O
P
O
H
1
0
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55
a) Collapse at Top
of
Hopper
b)
Collapse
Mode
of Junction
FIG 11 PL STIC COLL PSE MODES
OF
HOPPER
ND
TR NSITION
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