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International Specialty Conference on ColdFormed Steel Structures
(1986) - 8th International Specialty Conference on
Cold-Formed Steel Structures
Nov 11th
On the Strength and Stability of Light Gauge Silos
John Michael Rotter
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Eighth International Specialty Conference on Cold-Formed Steel Structures
St. Louis, Missouri, U.S.A., November 11-12, 1986
ON THE STRENGTH AND STABILITY OF LIGHT GAUGE SILOS
By John Michael Rotter*
ABSTRACT
Considerable advances have been made recently in understanding the factors
which control the strength and stability of light-gauge silo structures.
This paper reviews many of these advances, makes recommendations for
procedures to be used in future designs and refers to many source documents
which can provide more complete information.
1.
INTRODUCTION
Major advances have been made in the last few years in understanding the
behaviour of light gauge silo structures, and in devising simple techniques
which allow the strength to be predicted with reasonable precision.
The
research work which has led to these advances has been stimulated by a very
high incidence of failures in silo structures, by a constant trend towards
larger silos, and by a number of attempts at innovative design to reduce
costs.
Light gauge silo structures present many challenging problems for a range of
reasons:
the patterns of loading on the silo wall are often complex and
sometimes unpredictable, the silo structure is often a stiffened shell which
displays a complex response even as a linear elastic structure, there is a
significant interaction between the displacements of the structure and the
loading to which it is subject, and the failure modes of the silo structure
are often sensitive to geometric imperfections introduced during fabrication
and erection.
These considerations lead some designers to respond by using careful
analytical techniques and conservative assumptions, whilst others prefer to
adopt the very empirical approach of basing each design closely on their own
or a competitor's existing design and rebuilding when catastrophes occur.
The latter procedure tends to restrict innovation, and is becoming
progressively less justifiable as micro-computers allow relatively thorough
strength assessments to be made quite quickly. The aim of this paper is to
direct the attention of designers who seek to upgrade their predictive
capability to research work which can be directly assimilated into the
design process.
A brief review is given of the current state of knowledge concerning the
loading on silos, the stress patterns developing in the structure, and the
modes of failure. Particular emphasis is given to information relevant to
light-gauge stiffened construction.
* Senior Lecturer in Civil Engineering, University of Sydney, N.S.W. 2006,
Australia.
543
544
2.
EIGHTH SPECIALTY CONFERENCE
PRESSURES ON SILO WALLS
A very considerable effort has been put into studying the pressures on silo
walls in the last two decades.
However, relatively few of the researchers
have a background in structural engineering, so that their ideas about the
application of the research have tended to be simplified and their awareness
of the scientific and probabilistic treatment of load specifications used
with other load types is limited.
The walls of silos are subjected to both normal pressures and frictional
shears or tractions which vary allover the wall. Slip against all internal
surfaces occurs as materials consolidate during filling, so the frictional
shear appears to be always fully developed.
In simple terms, it is to be
expected that the normal pressures will give rise to circumferential (or
hoop) tensions. and that the frictional tractions will cause cumula ti ve
axial (or vertical) compressive stresses.
However, the real loaqing is a
little more complex, and other load cases give rise to different stress
patterns in the shell, so this initial view must be modified slightly when
possible failure modes are examined.
An extensive review of the classical theories for pressures on cylindrical
silo walls was given recently (1).
This review concentrated on the
pressures in the as-filled condition in squat silos. Much larger pressures
have frequently been observed during discharge in deep silos, but very few
experiments have been reported on the pressures in squat silos during.
discharge. The common assumption, that flow pressures or over-pressures do
not occur in these structures, is consequently chiefly supported by the
evidence of many successful structures which were designed to withstand only
filling pressures.
The factor of safety for some of these silos may,
however, be smaller than is desirable.
The most widely accepted theory for wall pressures during discharge (Fig. 1)
is the minimum strain energy theory of Jenike (3, 15, 20), which preqicts
flow pressures which are in broad agreement with experimentally observed
values.
Jenike' s theory has been adopted as the principal source for
defining flow pressures in the new Australian Guidelines for the Assessment
of Loads on Bulk Solids Containers (10), but has not yet been adopted
directly in other codes.
The wide discrepancy between the flow pressures
defined by different codes is initially a source of confusion to many
designers, but the resulting silo designs in steel do not usually differ
very much. This is because normal wall pressures do not control the design
of many cylindrical walls, and the initial-filling condition is usually
critical in the design of discharge hoppers (32).
Almost all the theories and empirical approximations for silo pressures
assume perfect silo geometry with homogeneous isotropic stored solid
behaviour, and they consequently predict wall pressures which do not vary
around the circumference at a given height. The cylindrical silo structure
is well suited to carry symmetrical pressures of this kind.
By contrast,
many
experiments
on
full
scale
silos
(12,
22)
have
shown
that
unsymmetrically-distributed patches of high local pressure occur on the wall
during flow. These have been shown (37) to be potentially very serious in
steel silos (Fig. 2).
Unfortunately no current silo pressure theories
appear to deal with randomly-occurring unsymmetrical pressures and there is
insufficient experimental data to define these patches with certainty at
present.
STRENGTH AND STABILITY OF SILOS
545
In light-gauge silos constructed from horizontlllly-corrugated sheet, the
wall friction coefficient must be assessed with care. Recent experiments
(11, 21) have indicated that the nett wall friction coefficient is closely
approximated by a linear combination of the coefficients for solid sliding
on metal and for solid sliding on itself according to the geometry of the
corrugation. For most corrugation geometries this leads to wall friction
coeffi~ients only slightly less than ideally rough wall conditions for which
tan ell
(1)
and the lateral pressure ratio is given by
k
in which
Equations
the wall,
are being
(1 - sin 2e1l)/(1 + sin 2e1l)
(2)
ell is the effective angle of internal friction of the solid.
(1) and (2) are suitable for determining the frictional drag on
but lower values of ~ should be adopted when horizontal pressures
calculated.
Many failures of light-gau~e silos in service have been attributed to
eccentric discharge.
The pressures which may be expected under these
conditions have recently been studied, and the consequent failure of the
silo by buckling has been predicted (31).
In addition to defining the pressurea arising from bulk solids storage, the
new Australian Guidelines (10) provide useful information on many other
loading conditions.
Notable amongst these are loads due to differential
thermal expansion between the silo and stored solid, differential settlement
of foundations, swelling of stored agricultural products, wind loads,
seismic loads and differential thermal expansion of column supports. Loads
from the bulk solids under unusual conditions (eccentric discharge, rapid
filling with powdery solids and impact loads from filling by dumping) are
also defined.
3.
MODES OF FAILURE IN SILOS
Silos are subject to many different loading conditions, so that many
different modes of failure are possible. Nevertheless, the critical stress
conditions in the wall generally lead ultimately to one of only a few
modes. These may be briefly listed as:
For the cylindrical wall
bursting
buckling under axial (vertical) compressive stresses
buckling under circumferential (hoop) compressive stresses
buckling under membrane shear stresses
For the conical hopper
bursting of the hopper body
tear-out at the hopper/cylinder junction
buckling of the ring support
plastic collapse of the hopper/cylinder/ring junction
546
EIGHTH SPECIALTY CONFERENCE
Each of these conditions is examined briefly in the following sections,
and the loading conditions which give rise to them are defined.
Some
recently-developed design rules are also described.
More extensive but
less recent advice on the design of steel silos against these failure
modes is given in References 9, 27 and 42.
A more comprehensive list of
possible failure modes is given by Trahair (41).
4.
BURSTING OF CYLINDRICAL WALLS
The literature on over-pressures in silos (3, 14, 15, 20) suggests that very
high internal or 'switch' pressures (Fig. 1) are to be expected on limited
zones of the wall.
It is commonly recommended that the entire wall be
designed to sustain the highest pressure envelope.
However, very few
bursting failures are seen in service, and even these are generally
attri buted to poor detailing rather than excessive pressures.
There are
several reasons for this.
First, even on a simple basis, hoop tension does not usually control the
required wall thickness in smooth-walled silos except near the top where it
is overridden by the minimum acceptable thickness for fabrication purposes.
Secondly, the research which indicates that high ' swi tch' pressures may
occur has been conducted on rigid -walled silos, so that no exploration has
been undertaken of how these switch pressures decrease if the silo wall
deforms away from the solid before the wall reaches a plastic collapse
condition.
Thirdly, overpressures have been found to be very highly
correlated with inward-bulging imperfections of the wall (4, 14).
In
ductile steel silos, high internal pressures can be expected to lead to
yield at these points, so that the cause of the overpressure may disappear
before a bursting failure occurs. Finally, i t should bg. noted that 'switch'
pressures occur over a very limited range of the silo wall height (14) and
i t can be shown that very large pressures can be sustained by a ductile
steel wall provided they extend over a height less than I(Rt) in which R is
the radius and t the wall thickness.
The above considerations suggest that bursting will not be the dominant
failure mode in most silos. Light gauge bolted corrugated steel silos which
use the sheeting to sustain hoop tension and stiffeners to sustain vertical
compression should, however, have carefully designed vertical bolted
joints. Unstable catastrophic 'unzipping' failures have occurred where such
joints have begun to fail locally. Two other special circumstances in which
bursting failu,res may be expected and have been observed in the field are
swelling of the stored solid (33), and a sudden decrease in the ambient
temperature which cools the steel bin but not its contents (2, 19). Advice
on appropriate design provisions are given in Reference 10.
5.
BUCKLING UNDER AXIAL COMPRESSION
The commonest mode of failure of bins in service is probably buckling under
axial or vertical compressive stresses.
Under axisymmetric filling
conditions this is usually the controlling design consideration for most of
the bin wall. Under other loading conditions higher axial compressions may
develop over limited regions on the wall.
In particular, eccentric
discharge (31), eccentric filling (26), earthquake loading on squat bins
547
STRENGTH AND STABILITY OF SILOS
(35) and column-support forces
causes of buckling failure.
in
elevated
bins
(24)
are
all
potential
In traditional design (9, 42), a single empirical strength relation was used
to define the probable buckling strength of smooth-walled cylinders.
Such
relations generally predict buckling stresses around 0.2 of the classical
elastic critical stress
0.605 Et/R
(3)
However, it is widely recognised that the strength is extremely sensitive
to the amplitudes of initial wall imperfections (Fig. 3) (43), which are in
turn dependent on the quality of fabrication.
Unfortunately,
most
traditional design rules simply provide empirical lower bounds on laboratory
test results, and very little is known about the relevance of these to real
silos in service.
A more recent approach (30) following the lead given by the ECCS code (8),
is to allow the design strength to depend on the magnitude of expected
imperfections, by adopting Koiter' s (18) simple buckling strength relation
which can be rearranged to read
(4)
1 - q,2.t [/(1
a
+3.1!.1
) - 1]
q, 0
(5)
in which 0 is the amplitude of the largest imperfection.
The imperfection
sensitivity factor q, may be taken as 1.24 for isotropic silos under axial
compression.
As most bin designers have little idea how large the
imperfections may be in their designs, it is initially recommended that the
imperfection be assumed to be
2.. =
t
0.06
q
/(!)
t
(6)
in which q represents the quality of fabrication (q
1 for normal
cylinders, 1.5 for quality cylinders and 2.5 for very high quality
cylinders).
Equation 6 reflects the results of laboratory tests and the
'normal' quality is close to the 90 percentile lower bound of Steinhardt
and Schulz (40). An extension of this procedure to compressive stresses in
limited zones of the shell is given elsewhere (31). Further details on the
simple measurement of imperfections in full scale silos are given in the
ECCS code (8) (Fig. 4).
A comparison of different buckling strength rules
and discussion on the effects of internal pressure from the stored solid,
buckling and collapse in regions of local bending and the effects of lap
joints in bolted construction is also given in Reference 30.
Many light-gauge silos are formed using stiffened construction for which the
above relations are not applicable.
These are often proportioned using
elementary empirical rules, but if a more precise buckling strength
assessment is required, the following prediction, based on the work of
Singer et al (39), may be used
r1
d.
in which
(7)
548
EIGHTH SPECIALTY CONFERENCE
ml+(~I+CI+I+ + 2~2(CI+5+C66) + C S5 ) + C 22 + 2m2c 25
2~2(C12 + C33 )(C 22 + m2C 2S )(C 12 + m2~2c 11+)
- (~2Cll + C33)(C22 + m2C 25 ) 2
- ~2(C22 + ~2C33)(C12 + m2~2C 11+) 2
(~2Cll + C33)(C22 + ~2C33) - ~2(C12 + Cn)
+
+
C22
C
C 33
0.5(1-v)C
C25
. erEAr/(drR)
Cll
C
C12
vC
C11+
esEAs/(dsR)
CI+I+
(D
CI+5
vD/R2
C6G
[(1-v)D
Etl (1-v 2 )
D
Et 31 (12(1-v 2»)
C
~
=
EAs/d s
+
EI s /d s )/R2
C55
(D
2
EAr/d r
+
(8)
EIr/dr)/R 2
+
0.5(GJ s /d s
+
GJr/d r ) ]/R 2
1tR/(ml)
As, Ar
and (Fig. 5)
are the areas, d s , d r the separations, and e s '
er the outward centroidal eccentricities of stringer and ring stiffeners
respectively.
Equation 7 must be minimised with respect to m, the number of buckle waves
around the circumference, and with respect to 1, the half-wavelength of
buckles in the axial direction.
It is therefore best to conduct this
calculation on a microcomputer.
After solving Eq. 7, the procedure of Eqs 5 and 6 should be used. However,
test results quoted by Samuelson (38) and the theory of Hutchinson and
Amazigo (13) suggest that the imperfection sensitivity factor </! might be
taken conservatively as
(9)
The orthotropic shell theory on which the above equations are based is only
valid for shells with closely-spaced stiffeners.
The minimum number of
equally spaced stringer stiffeners is around 4m, where m is as defined
above. Further information on shells with more widely spaced stiffeners is
given in (38).
6.
BUCKLING UNDER CIRCUMFERENTIAL COMPRESSION
Both silos and tanks for liquid storage are susceptible to buckling failure
under wind when empty.
Squat ground-supported sil08 are usually the most
susceptible to wind, because they have lighter wall construction and larger
diameters than elevated silos.
The reference buckling pressure is generally taken as the classical uniform
external critical pressure.
For uniform isotropic simply-supported
cylinders this is given by (43)
(10)
STRENGTH AND STABILITY OF SILOS
549
However, many of the silos which are most susceptible to wind buckling are
light-gauge stiffened structures, so the solution developed by Baruch and
Singer (5) for eccentrically stiffened shells is of considerable value.
(11 )
in which Ai' A2 and A3 are given by Eqs 8.
Equation 11 must be minimised with respect to m, the number of buckle waves
around the circumference. It is also usually necessary to examine several
different buckling heights l in the upper part of the silo, as the critical
mode cannot normally be deduced by inspection, even though its bottom
boundary is always defined by a stiffener or change of plate thickness.
Procedures to allow for the weighted smearing of discrete stiffeners,
determination of an equivalent uniform pressure and appropriate working
stress design pressures are given by Trahair et al (42).
Further work on
the effects of non-uniform external pressures, the critical size of
stiffeners and the effects of axial restraints is given by Blackler and
Ansourian (6, 7) and by Resinger and Greiner (23). In the simplest approach
(42), these phenomena can be ignored and the wall designed to a maximum
pressure differential (unfactored) at the windward generator of
( 12)
in which Pel is determined from Eqs 10 or 11 as appropriate.
7.
BUCKLING UNDER MEMBRANE SHEAR
Silos which are subject to unbalanced horizontal shears from eccentric
filling, earthquake or mechanical handling equipment carry these loads
principally in membrane shear.
A number of failures due to buckling in
shear have been reported, but the design of silos against this mode of
failure has always been difficult because buckling predictions were not
available for cylinders with appropriate stress distributions.
Typical
distributions of membrane shear under earthquake loading (35) are shown in
Fig. 6 and it is apparent that these may be closely approximated bya.linear
variation of membrane shear.
In a recent study (17) the buckling strengths of isotropic cylindrical
shells under linearly varying membrane shear were explored. The classical
critical ritaximum shear stress for this case was found to be
~cl
=
!.) 5/ 4 (!) 1/ 2
1.41 E (R
H
(13)
in which the shear on the wall is defined by
~
cl
[1 - !.
]
H
(14)
Studies of the imperfection-sensitivity of cylindrical shells in torsion
(43) indicate that the design ultimate shear stress should probably be taken
at about
550
EIGHTH SPECIALTY CONFERENCE
(15)
in which a is given by Eq. 5, but using <I> = 0.12 for shear buckling. The
effective imperfection IS is again given by Eq. 6.
The value of a is
typically around 0.6.
8.
BURSTING IN THE HOPPER BODY
In general silo hoppers should be designed to withstand two hopper pressure
distributions:
the
'initial filling'
and the
'discharge' or
'flow'
distributions (Fig. 1).
For most silo structures, the most severe loading
condition for the body of the hopper is initial filling (32). However, in
light gauge silos using bolted meridional seams, this is not always true.
The meridional bolted seam must be designed to sustain the highest
circumferential stress to which it may be subjected at any point down its
length.
This requ:l.res a consideration of both the 'initial filling' and
'flow' pressure distributions in the hopper.
The circumferential tensile
stress in the hopper depends only on the normal pressures on the wall, and
this bolted seam is consequently more heavily stressed when the hopper wall
has a low friction coefficient, ~.
Towards the hopper outlet initial filling pressures dominate the design, but
near the hopper/cylinder
transition flow pressures
lead
to higher
circumferential tensions. Equations for these tensions are too involved to
be concisely written here, but may be found elsewhere (32).
In both bolted and welded hoppers, the location of the most highly stressed
point depends upon the relative sizes of the hopper and the cylinder above
it.
Where small hoppers are. placed below tall cylindrical silos, the
hopper/cylinder/ring junction is always the most highly stressed location
and relatively simple assumptions may be used to design the hopper.
By
contrast, where the whole structure is principally a mass-flow hopper with a
small cylinder above it, considerable care is needed to define the required
strength at any point.
9.
COLLAPSE OF THE HOPPER/RING/CYLINDER JUNCTION
Of all structural behaviour seen in silos, that in the region of the
hopper/cylinder/ring/skirt/column junction is by far the most complex. Even
when a deep skirt or many columns are used to produce an axisymmetric stress
state, failure may occur by one of two plastic collapse mechanisms or by
buckling. When a small number of column supports is used, as is often the
case in smaller silos, the prebuckling stress distributions are much more
difficult to determine and buckling is possible in the .cylinder or in the
ring, whilst the plastic collapse mechanisms cannot yet be identified with
any certainty.
In light gauge silos, two failure modes appear to cause most concern:
tear-out of the hopper body from the cylinder and ring support, and failure
modes associated with redistribution of column support forces into the body
of the shell structure.
STRENGTH AND STABILITY OF SILOS
10.
551
TEAR-OUT AT THE HOPPER/CYLINDER JUNCTION
The load on the hopper is carried by the hopper acting chiefly as a
membrane, developing meridional tensile stresses. The meridional tension at
the top of the hopper may be determined quite accurately from global
equilibrium (Fig. 7).
In light gauge silos, which often have relatively robust rings at the
hopper/cylinder junction, tear-out of the hopper is to be expected when the
meridional tension reaches the yield stress
N.p
(16)
in which th is the hopper thickness and cry the yield stress.
However, premature failure at this point may occur for several reasons.
This is the most highly stressed point in most hoppers (32), it is the site
of a critical connection which is particularly difficult to fabricate
accurately, and it is the section through which the entire hopper load must
pass if the hopper is not to fail. With the latter thought in mind, it is
vital that the total weight on the hopper should not be underestimated: the
maximum load occurs when the cylinder wall friction coefficient fL and the
cylinder lateral pressure ratio k are both at their minimum values.
A second failure mode can occur at this junction. Very high local bending
stresses develop at the structural discontinuity (9,
29),
and the
development of a fatigue crack should be considered i f the silo is to be
filled and discharged many times each day.
Where the ring at the hopper/cylinder junction is small or omitted
altogether, a plastic collapse mechanism can develop in the ring, cylinder,
hopper and skirt.
This mechanism has been studied under axisymmetric
loading conditions (29), and approximate equations developed which are
suitable for use in design.
11.
BUCKLING OF THE RING SUPPORT
The ring at the hopper/cylinder junction is subject to high circumferential
compressive stresses arising from the hopper meridional tension. Additional
compressive stresses arise in the rings of column-supported bins (Fig. 8),
where the ring is required to fulfil the role of a bowgirder beam flange
spanning between column supports.
In large bins, a beam section is usually
designed for this location, but lighter bins are often built either with
columns extending to the eaves (Fig. 8a) or with terminating engaged columns
(Fig. 8b). These three structural types act in quite different ways and are
not simple to analyse.
The
bowgirder
of
the
larger bins must
be
designed
to withstand
circumferential thrusts, bending moments about the radial axis and torques
about the circumferential axis. Statics equations defining these quantities
are given by Trahair et al (42) and extended by Rotter (29).
However, the
bowgirder ringbeam should be checked to ensure that it has adequate
stiffness to fulfil its intended role (29).
EIGHTH SPECIALTY CONFERENCE
552
The stress distribution in lighter bins with columns extending to the eaves
(Fig. 8a) can be deduced approximately using hand analyses (28), and it is
found that the eaves and transition rings are both subject to varying
circumferential thrusts and bending moments about a vertical axis.
Where
the columns terminate just above the transition ring (Fig. 8b), a more
complex stress field is set up within the bin shell (34).
In relatively
squat bins, the rings are subject to actions similar to those in bins with
extended columns.
In taller bins, only the transition ring carries these
actions.
All the above structures must be designed to resist buckling of the ring at
the transition.
Recent studies of this ring buckling (16, 36) have shown
that in-plane buckling of the ring is invariably prevented by the hopper.
However, out-of-plane buckling into a mode involving many circumferential
waves is a potential cause of failure where wide annular plate rings are
used. Discrete meridional stiffeners do not easily inhibit this mode.
In bins with terminating columns (Fig. 8b), care must be taken to ensure
that the column support forces are satisfactorily distributed into the bin
shell.
Procedures for determining when columns can be terminated are
generally based on tests, experience and in-house empirical rules (9).
However, the vertical stress distribution in the bin shell can be
approximately determined from a hand analysis (34) and compared with a
suitable buckling criterion for circumferentially-varying axial compression
(31).
10.
CONCLUSION
A brief review has been presented of an extensive recent research programme
into the behaviour, strength, stability and design of steel bins and silos.
The paper has concentrated on aspects of this programme which are especially
relevant to light gauge silo construction.
The form of the loads to which
silos are subject has been outlined, and a brief description has been given
of each of the principal failure modes of these large and complex shell
structures. Extensive references to more detailed sources have been given.
APPENDIX I
REFERENCES
1.
Abdel-Sayed, G. and Monasa, F.
'Cold-Formed Steel Farm Structures
Part I: Grain Bins', Journal of Structural Engineering, ASCE, Vol. 111,
No. 10, Oct •. 1985, pp 2065-2089.
2.
Andersen, P.
'Temperature Stresses in Steel
Civil Engineering, ASCE, Jan. 1966, pp 74-76.
3.
Arnold, P,C" McLean, A.G. and Roberts, A.W.
Flow and Handling, 2nd Ed., The University
Associates, Newcastle, Australia, 1980.
Grain-Storage
Tanks'.
'Bulk Solids: Storage,
of Newcastle Research
STRENGTH AND STABILITY OF SILOS
553
4.
Askegaard, V., Bergholdt, M. and Nielsen, J.
'Problems in Connection
with
Pressure
Cell
Measurements
in
Silos'
(in
English),
Bygningsstatiske Meddeselser, Vol. 42, No.2, 1971, pp 33-74.
5.
Baruch, M. and Singer, J.
'Effect of Eccentricity of Stiffeners on the
General Instability of Stiffened Cylindrical Shells under Hydrostatic
Pressure, Journal of Mechanical Engineering Sciences, Vol. 5, 1963,
pp 23-27.
6.
Blackler, M.J. and Ansourian, P.
'Stability of Stiffened Cylindrical
Bins', Proceedings, Ninth Australasian Conference on the Mechanics of
Structures and Materials, University of Sydney, August 1984, pp 54-60.
7.
Blackler, M.J. and Ansourian, P.
'Effect of Boundary Restraints on
Cylinder Stability.
Proceedings, Second International Conference on
Bulk Materials Storage, Handling and Transportation, Institution of
Engineers, Australia, Wollongong, July 1986.
8.
European Convention for Constructional Steelwork.
'Recommendations for
Steel Construction: Buckling of Shells', 2nd Ed., 1983.
9.
Gaylord, E.H. and Gaylord, C.N.
'Design of Steel Bins for Storage of
Bulk Solids', Prentice Hall, Englewood Cliffs, 1984.
10.
Gorenc, B.E., Hogan, T.J. and Rotter, J.M. (eds).
'Guidelines for the
Assessment of Loads on Bulk Solids Containers', Institution of
Engineers, Australia, 1986.
11.
Haaker, G. and Scott, O.
'Wall Loads on Corrugated Steel Silos',
Proceedings, Second International Conference on the Design of Silos for
Strength and Flow, Stratford-upon-Avon, Powder Advisory Centre, Sep.
1983, pp 480-503.
12.
Hartlen, J., Nielsen, J., Ljunggren, L., Martensson, G., and Wigram,
S.
'The Wall Pressure in Large Grain Silos'.
Report D2, Swedish
Council for Building Research, Stockholm, 1984.
13.
Hutchinson, J.W. and Amazigo, J.C.,
'Imperfection Sensitivity
Eccentrically Stiffened Cylindrical Shells', AIAA Journal, Vol.
No.3, March 1967, pp 392-401.
14.
Jenike, A.W., Johanson, J.R. and Carson, J.W.
'Bin Loads - Part 2:
Concepts' • Journal of Engineering for Industry, ASME, Vol. 95, Series
B, No. I, 1973, pp 1-5.
15.
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APPENDIX II
NOTATION
The following symbols are used in this paper:
E
Young's modulus
H
length of shell loaded by circumferential traction
k
horizontal to vertical stress ratio in stored solid
1
axial half-wavelength of buckling mode
m
number of complete circumferential buckle waves
classical elastic critical pressure
quality factor for cylinder buckling
cylinder radius
t
cylinder wall thickness
hopper wall thickness
z
vertical coordinate
imperfection strength reduction or knock-down factor
amplitude of imperfection
j.l
wall friction coefficient
v
Poisson's ratio
classical elastic critical stress
ultimate stress for buckling under axial compression
yield stress
classical elastical critical shear stress
imperfection sensitivity factor
STRENGTH AND STABILITY OF SILOS
Locus of pressure
peaks based on
minimum strain
energy
Active
p
-'---Switch
Passive
or
Minimum
strain
energy
I
I
Instantaneous
,.'_t--pressure
distribution
_L __ ......._ __
I
,
switch at
transition
o
p
o
P
(al Initial Pressures (bl Flow Pressures
Flu.1
TYPICAL PRESSURE DISTRIBUTIONS
557
EIGHTH SPECIALTY CONFERENCE
558
ight Ring
H/O=2
R/t=800
2a/H=0.25
b/H=0.5
00=30°
Local
Patch
~=0.4
b
Buckling
here
(a) ExamJ:!le Silo
-~
400
High stress may
cause buckling
::I:
~
QJ
Patch load
stress
300
'.....
Vl
QJ
c
200
Jenike stress
rtI
to.
~
QJ
Janssen stress
----:-- ~_-_____t_---::---
100
::I:
Or---------~r-~--_+--+_--------~
rtI
.....u'QJ
> -100
-180
o 45 90 135
-135 -90 -45
Circumferential Angle (degrees)
180
Ib) Vertical Wall Stresses near Silo Base
FIG.2
BUCKLING CONSEUUENCE OF A LOCAL PATCH OF HIGH PRESSURE
STRENGTH AND STABILITY OF SILOS
559
1. 0 r--""T"""---T-or-""'T"""-'T-or-""'T"""-"T-"-""
H = 14.5{R/t
R/t = 405
v = 0.3
I
-l-J
0.8
o
VI
VI
QJ
....
Vl
o
I-
C1
0.6
c:
~
u
::J
co
VI
:i1
198
o
0.4
~
c:
1945
o
Hutchinson and
Koiter, 1970
VI
c:
QJ
E
CJ
0.2
Experimental Observations
5
t
= Imperfection Amplitude
= Wall Thickness
o~~--~~--~~--~~--~----~
o
1.0
Dimensionless Imperfection Amplitude 6 It
FIG.3
EFFECT OF IMPERFECTION AMPLITUDE ON COLLAPSE
LOAD (after Yamaki, 1984)
560
EIGHTH SPECIALTY CONFERENCE
t
R
I
6 inward
.......,-;~
t
t
Ibl
lal
FIG.4
Icl
ECCS (1983) IMPERFECTION MEASUREMENT
R
Stringers
FIG.5
Rings
ECCENTRIC STIFFENER GEOMETRIES
561
STRENGTH AND STABILITY OF SILOS
1.0
:J:
..........
0.8
R/t=500
N
.....
Ew/Es=4x103
~
C'I
.iii
:J:
0.6
III
III
-c
cu
0
III
C
0.4
cu
.§
Cl
0.2
0
0
0.2
0.4
0.8
0.6
Dimensionless Membrane Shear Stress
FIG.6
l'mz8
1.0
t
iilR" H
HEHBRANE SHEAR STRESS IN SILO WALL (8=Jt/2)
R
FIG.7 VERTICAL EQUILIBRIUM OF ENTIRE HOPPER
562
EIGHTH SPECIALTY CONFERENCE
Transition
Ring
(a) Columns extending J.Q..
eaves
FIG.8
J..QL1ight rings and terminating
eng~ged columns
LIGHT COLUMN SUPPORTED BINS