The Structural Design of Light Gauge Silo Hoppers 2

8/18/2019 The Structural Design of Light Gauge Silo Hoppers 2

1/27

Missouri University of Science and Technology

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Te structural design of light gauge silo hoppers John Michael Roer

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2/27

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


3/27

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


4/27

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


5/27

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


6/27


7/27


8/27

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


9/27

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


10/27


11/27


12/27

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


13/27

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.


14/27

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,


15/27

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.


16/27

54

Roof

Ring

Cylinder

Ring

kirt

Conical

Hopper

FIG 1

TYPICAL COLD FORMED ELEVATED SILO


17/27

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


18/27


19/27


20/27

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


21/27


22/27

o

o

L.f l

(Y l

o

o

L.f l

,

3500

o=9kN/m


23/27


24/27

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


25/27

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


26/27

55

a) Collapse at Top

of

Hopper

b)

Collapse

Mode

of Junction

FIG 11 PL STIC COLL PSE MODES

OF

HOPPER

ND

TR NSITION


27/27

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