THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS · equipment. A steel model silo with an anti-dynamic tube was set up in the laboratory to measure the friction on the tube during material

THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

A THESIS SUBMITTED TO THE UNIVERSITY OF WESTERN AUSTRALIA FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL AND RESOURCE ENGINEERING

DESIREE NORTJE

2002

ABSTRACT

Storage of granular solids in silos has been the practice for many years. Engineers

have been faced with the problem of making the silos empty more efficiently and

minimising the forces acting on the walls of the silo during material discharge. To

this end the anti-dynamic tube was invented. The tube has a smaller diameter than

the silo and consists of several portholes along its height and around its

circumference. When the discharge gate of the silo is opened the granular material

enters the tube through the portholes, flows down the inside of the tube and exits

the silo through the discharge gate. Most tubes have been installed such that there

was sufficient space between the base of the tube and silo bottom for the granular

material to flow simultaneously through the discharge gate.

The flowing material causes a down drag on the tube from the friction of the

granular material on the walls of the tube. Previous research has underestimated

the magnitude of these frictional forces resulting in catastrophic buckling failure of

the tubes, blocking the discharge gate of the silo. A blockage of the discharge gate

requires top emptying of the silo resulting in financial losses and down time of

equipment. A steel model silo with an anti-dynamic tube was set up in the

laboratory to measure the friction on the tube during material flow. From the

results of these experiments, an equation has been derived to estimate the

magnitude of the down-drag force. Furthermore, an empirical expression was

developed for the effects of the speed of the flowing material on the magnitude of

the down-drag force.

To keep construction costs down, it is necessary to optimise the wall thickness of

the tube. There is currently no theory for the buckling capacity of a thin walled

cylindrical shell with multiple perforations around its height and circumference.

Therefore additional experiments were undertaken on a cylindrical shell with

multiple perforations subjected to a combination of an axial as well as an external

lateral pressure. Following on from the experiments, finite element analyses were

undertaken to compare with the experimental results. For each finite element

analysis an out-of-roundness was introduced as an initial wall imperfection. From

these analyses and the cylinder experiments, a method of producing interaction

curves for tubes with varying ratios of open area has been developed.

DISCLAIMER

No portion of the work presented in this thesis has been submitted in support of an

application for another degree or qualification from this, or any other, university or

institute of learning.

Désirée Nortje, August 2002.

ACKNOWLEDGEMENTS

My sincere thanks are given to my supervisor, Dr Ken Kavanagh, for his

encouragement, insight, informality, approachability, understanding and extreme

patience. Without his motivation and kindness I would have given up in the early

stages of the project. I would also like to thank Dr Kavanagh for supporting the

application for my research scholarship provided by the Department of Civil and

Resource Engineering.

Thanks are also due to the workshop staff who assisted me in setting up both my

model silo and cylinder experiments. Special thanks goes to Jim Carrol, Jim Waters,

and Neil McIntosh for their assistance with the laboratory machinery. Without their

help and guidance the experiments would have been cumbersome, difficult to

execute and probably never would have happened. Thanks are also due to

Wladyslaw Bzdyl and Sun Nichersen for maintaining the electrical equipment and

their endless patience in explaining my queries.

ACI Glass, Penrith Plant, kindly allowed me to print and bind my thesis at their

offices. I am very grateful for their generosity in making space available for me and

providing invaluable assistance for binding.

Finally, there are not enough words which can describe my gratitude to my

husband, Richard, and my two daughters, Stephanie and Jennifer, for their patience

while I undertook this PhD research. Many of our decisions revolved around this

project and it’s eventual completion date. We can now finally start making our

plans become a reality.

Thank you to all.

Desiree Nortje, August 2002

LIST OF SYMBOLS

hopper half angle

a constant

stress ratio

c strain in the circumferential direction

r strain in the r direction

strain in the q direction

angle in the circumferential direction in the hopper

m internal angle of friction of the granular material

t anti-dynamic tube wall friction angle

w wall friction angle

material bulk density

angle in the material. w < <m

angle subtended by the hopper axis and centered at the hopper vertex

c vertical stress on the silo axis

h horizontal stress

J Janssen static vertical pressure

JH Janssen static horizontal pressure

v vertical pressure

vw vertical stress adjacent to the hopper wall

shearing stress

v shear stress in the vertical direction at an arbitrary distance from the silo

axis

vw shear stress in the vertical direction adjacent to the hopper wall

angle of inclination of the major principal stress

A silo cross sectional area

surface area of a cylindrical shell

Ac plan area of a cut out in the wall of a cylindrical shell

AP projected plan area of an object

B factor derived from the Mohr circle in the circumferential section of the silo

Cc Reimberts characteristic constant of the silo

constant

dt diameter of the anti-dynamic tube

dg diameter of the silo discharge gate

D silo diameter

flexural rigidity of a thin walled shell

E Youngs modulus

factor derived from the Mohr circle in the hopper

a constant

F vertical pressure distribution factor in the cylindrical section of the silo

a constant

FH vertical pressure distribution factor in the hopper

G a constant

h height of cone of material above the top of the silo

h step size used in the Runge-Kutta method of numerical integration

H total height of the silo

Hh height of hopper

ht height of the anti-dynamic tube

iN ratio of zN to hopper height =zN/ Hh

K stress ratio

K0 at rest stress ratio

Ka active stress ratio

Knn factor used in the Runge-Kutta method of numerical integration

Kp passive stress ratio

L length of a cylindrical shell

m coefficient of friction

M bending moment

N axial force

Nx axial load at buckling of a cylindrical shell with cut outs

N0 axial load at buckling of a cylindrical shell with no cut outs

P silo perimeter

Q stress ratio

q shear stress

R hydraulic radius = A/P

r radius of a thin walled cylinder

rc radius of a cut out in the wall of a cylindrical shell

r0 radius of the hopper at the level of the transition

rav average radius

rb radius of the bottom of an element in the hopper

rt radius of the top of an element in the hopper

sN pressure normal to the hopper wall

Sv dimensionless pressure ratio = v/D

t wall thickness of a cylinder

u,v,w displacements in the cartesian plane

Uc strain energy in the circumferential direction

UB bending energy

x,y,z cartesian co-ordinates

z depth co-ordinate

Z dimensionless depth ratio = z/H

z element thickness

zN depth of the maximum pressure in the hopper

CONTENTS

ABSTRACT

DISCLAIMER

ACKNOWLEDGEMENTS

LIST OF SYMBOLS

CHAPTERS

1 INTRODUCTION 1.1 Introduction 1.1

1.2 Silo Types and Flow Patterns 1.6

1.3 Silo Inserts 1.7

1.4 Introduction to Wall Pressures 1.10

2 CLASSIC WALL PRESSURE THEORIES

2.1 STATIC WALL PRESSURES

2.1.1 CYLINDRICAL SECTION 2.1.1.1 Janssen 2.1

2.1.1.2 Reimbert 2.4

2.1.1.3 Janssen vs Reimbert’s theory 2.7

2.1.2 HOPPER SECTION

2.1.2.1 Walker 2.8

2.1.2.2 Jenike Radial Pressures 2.11

2.1.2.2.1 Linear Normal Wall Pressure 2.11

2.1.2.2.2 Radial Pressure Field in the Solid 2.16

2.1.2.2.3 Position of the Maximum Pressure in the Hopper 2.21

2.1.2.3 Walters Static Hopper Pressures 2.23

2.2 DYNAMIC WALL PRESSURES

2.2.1 CYLINDRICAL SECTION

2.2.1.1 Walters 2.25


2.2.2.1 Flow/Slip in the Hopper by Equilibrium of a Slice 2.33

2.2.2.2 Walters Pressures in Converging Channels 2.39

2.2.2.3 Jenike Radial Stress Field 2.49

2.3 SWITCH PRESSURES


2.3.1.1 Jenike Upper Bound Pressures 2.62

2.3.1.2 Walters Switch Pressure in the Cylinder 2.68


2.3.2.1 Jenike Switch Pressure in the Hopper 2.74

2.3.2.2 Walters Switch Pressure in the Hopper 2.78

3 WALL PRESSURE MEASUREMENTS

3.1 LITERATURE SURVEY

3.1.1 STATIC PRESSURES

3.1.1.1 Cylindrical Section 3.1

3.1.1.2 Hopper Section 3.3

3.1.2 DYNAMIC PRESSURES

3.1.2.1 Cylindrical Section 3.5

3.1.3 STRESS RATIOS 3.8

3.2 EXPERIMENTAL SET-UP

3.2.1 Steel Model 3.11

3.2.2 Bulk Solid material 3.12

3.2.3 Data Acquisition 3.12

3.2.4 Strain Gauge Bridges 3.14

3.2.5 Floating Pressure Cells 3.16

3.2.5.1 Ball type pressure cell 3.17

3.2.5.2 Tube type pressure cell 3.17

3.2.5.3 Plate type pressure cell 3.18

3.2.6 Pressure Cell Calibration 3.19

3.2.7 Multi-turn potential meters 3.21

3.2.8 Gate Switches 3.22

3.3 EXPERIMENTAL RESULTS

3.3.1 Description 3.23

3.3.2 Static Tests 3.24

3.3.3 Dynamic Tests 3.27

3.3.3.1 Switch Pressure 3.30

3.3.4 Stress Ratios 3.32

4 ANTI-DYNAMIC TUBE THEORY


4.1.1 Pieper 4.1

4.1.2 Reimbert 4.2

4.1.3 Ravenet 4.2

4.1.4 McLean 4.3

4.1.5 Ooms and Roberts 4.4

4.1.6 Kaminski and Zubrzycki 4.7

4.1.7 Schwedes and Schulze 4.10


4.2.1 Anti-Dynamic Tube Model 4.12

4.3 EXPERIMENTAL RESULTS 4.15

4.4 MATHEMATICAL MODEL 4.23

4.4.1 Tube Parameters 4.25

4.4.2 Variable Vertical Pressure across a Slice 4.26

5 BUCKLING OF THIN CYLINDRICAL SHELLS

5.1 ELASTIC SHELL BUCKLING THEORY

5.1.1 Cylinder subjected to uniform external lateral pressure 5.1

5.1.2 Cylinder subjected to axial pressure

5.1.2.1 Special Case 5.10

5.1.2.2 General Case 5.19

5.1.3 Cylinder subjected to combined axial and lateral pressure 5.24

6 PERFORATED CYLINDRICAL SHELLS


6.1.1 Tennyson 6.1

6.1.2 Almroth and Holmes 6.3

6.1.3 Starnes Jr 6.5

6.1.4 Scutella 6.7

6.2 DISCUSSION 6.10

6.3 PERFORATED CYLINDER EXPERIMENTS

6.3.1 Experimental set-up 6.14

6.3.2 Experimental results 6.19

6.4 FINITE ELEMENT ANALYSIS

6.4.1 Description 6.26

6.4.2 Cylinder with 16.5% open area 6.30

6.4.3 Cylinder with 36.6% open area 6.35

6.4.4 Solid Cylinder 6.39

6.5 COMPARISON WITH LABORATORY TESTS

6.5.1 Cylinder with 16.5% Open Area 6.43

6.5.2 Cylinder with 36.6% Open Area 6.44

6.5.3 Interaction Plots for Cylinders with Multiple Perforations 6.45

7 CONCLUSIONS

7.1 SILO WALL PRESSURES

7.1.1 Static pressures 7.1

7.1.2 Dynamic pressures 7.2

7.1.3 Switch pressures 7.3

7.1.4 Stress Ratios 7.4

7.2 ANTI-DYNAMIC TUBE FRICTIONAL DRAG 7.5

7.3 PERFORATED CYLINDERS 7.8

7.4 INTERACTION CURVES 7.8

8 APPENDICES

8.1 APPENDIX A: IMPLEMENTATION OF THE RUNGE-KUTTA

METHOD

8.1.1 The Runge-Kutta Equations A.1

8.1.2 Equilibrium Slice Method A.1

8.2 APPENDIX B: CALIBRATION CONSTANTS

8.2.1 Pressure Cell Calibration B.1

8.2.2 Anti-Dynamic Tube Support Calibration B.3

8.3 APPENDIX C: CHECK LISTS

8.3.1 Pre-Static Test Check List C.1

8.3.2 Pre-Dynamic Test Check List C.2

8.4 APPENDIX D: MODEL SILO WALL PRESSURES TESTS

8.4.1 Static Test Results D.1

8.4.2 Dynamic Test Results D.5

8.5 APPENDIX E: ANTI-DYNAMIC TUBE TESTS

Frictional Drag Test Results E.1

8.6 APPENDIX F: SHELL THEORY

8.6.1 Uniformly Compressed Circular Ring F.1

8.6.2 Flexural Rigidity of a Shell F.1

8.7 APPENDIX G: PERFORATED CYLINDER TEST RESULTS

8.7.1 Cylinders with 16.5% Open Area G.2

8.7.2 Cylinders with 36.6% Open Area G.4

8.7.3 Lateral Pressure Tests G.9

8.8 APPENDIX H: EIGENVALUE BUCKLING MODE SHAPES

8.8.1 Cylinders with 16.5% Open Area H.1

8.8.2 Cylinders with 36.6% Open Area H.4

8.8.3 Solid Shell H.7

9 REFERENCES

INTRODUCTION 1.1

CHAPTER 1

INTRODUCTION

1.1 INTRODUCTION

A silo is a structure well known to most people. It’s use is for the storage of any

bulk material which is of a granular nature such as grain, wheat, lupins, salt, sugar,

cement, coal, etc… While there are many silos in existence, this does not imply that

all knowledge about silos has been determined and that very little is left to still be

discovered. Silos in all sizes are being constructed all around the world, some of

which operate very successfully, and others which do not.

Silo discharge is classified into two main groups, either concentric or eccentric

discharge. In eccentric discharge the gate is off-centre with respect to the centre

line of the silo. Due to this eccentricity, flowing material causes large bending

stresses on the walls of the silo. These bending stresses are erratic in nature and

difficult to predict due to the erratic nature of eccentric flow. In the case of

concentric discharge, the centre of the gate aligns with the centre line of the silo, or

the centre of the group of gates aligns with the centre line of the silo. The research

covered in this thesis focuses only on concentrically discharging silos.

Generally speaking a silo consists of two main sections, namely a hopper and an

upper bin. The joint between the upper bin and the hopper is referred to as the

transition. A flat bottom silo has an effective transition which is formed within the

stored material which does not exit the silo during discharge. This remaining

material is referred to as the “dead material” . Figure 1.2b shows a flat bottom silo

with dead material forming an effective transition with the walls.

Silos vary in shape from circular to square and rectangular. Depending on the

shape of the silo, there may be one discharge gate in the hopper as for the circular

case, or several discharge gates as for the rectangular silo. Figure 1.1 a,b,c shows

some of the typical silos in use. In figure 1.1d, a bank of silos has been shown

where the interstitial areas between the silos have also been used for the storage of

material, shown by the shaded area. Some silos also have their cones inverted as

shown in figure 1.2. This type of silo is used mainly for storing and blending bulk

materials which are in powder form, such as raw meal, cement and lime. This type

of silo has not been considered in this thesis.

1.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

Some of the problems associated with the storage of bulk materials are the

segregation of particles of varying sizes, excessive wear on the walls due to the

flowing material and cracked walls from the flowing material. The wall pressures

are generally referred to as the static and dynamic pressures, ie the filling and

emptying pressures, respectively.

One of the methods proposed to alleviate these problems is a silo insert called the

anti-dynamic tube. Other names for the anti-dynamic tube are the tremmie tube,

decompression tube, discharge tube and static-flow pipe. This method consists of

placing a tube centrally inside the silo, which has a smaller diameter than the silo.

The tube may extend the full height of the silo with multiple perforations around its

circumference and along its length. The material then flows into the tube through

(b) square or

doubly symmetrical

silo

(c) rectangular silo with a group of

outlet gates

(d) a bank of

circular silos

Figure 1.1: Typical silo geometries.

Figure 2.1a: Inverted cone silo

Effective transition

Dead material

Figure 2.1b: Flat bottom silo.

INTRODUCTION 1.3

the holes, and down the tube to the discharge gate. This type of tube causes the

silo to empty in successive layers resulting in a first-in-last-out situation. For

materials which degrade with time (either biologically or mechanically) this is highly

undesirable. An alternative arrangement is a shorter tube which extends only a

portion of the silo height. The optimum length of the tube is determined from the

internal friction angle of the material. In this arrangement the silo empties in two or

three stages only. In some instances, port holes are accommodated at the base of

the tube allowing material discharge to occur simultaneously from the bottom of

the silo as well as through the tube at higher levels.

This PhD thesis considers anti-dynamic tubes placed in mass flow silos storing free

flowing granular materials, such as sand, grain, lupins etc. This introductory

chapter gives a background to the various silo types and flow patterns. A brief

background of the different types of silo inserts used to overcome material flow and

wall pressure problems has been given, as well as a general description of the wall

pressures acting on the silo during filling and material discharge.

Chapter two consists of a study of the classic theories for static, dynamic and

switch pressures acting on the wall of the silos. This includes the Janssen, Jenike,

Walker and Walters theories for the pressures in a silo. From these pressure

theories it has been established that one of the main factors affecting the

determination of the horizontal wall pressures is the assumption of a suitable stress

ratio. Thus a section of the literature survey has been dedicated to the stress ratio

as recommended by other researchers.

Chapter three gives an overview of the wall pressures as measured by researchers

world wide in either model silos or full scale silos. This has been categorised into

the static, and dynamic pressures for the cylindrical and hopper section of the silo.

There is not much data available for the measurement of the stress ratio in silos

and consequently the literature survey covering stress ratios is relatively short.

Following the literature survey is a description of the steel model silo set up in the

structures laboratory. To enable the measurement of the pressures during material

flow, three novel types of floating pressure cells were developed. These pressure

cells were inexpensive, easy to construct, easy to calibrate and were found to be

very responsive to the instantaneous pressures found in the flowing material. The

results of the test are discussed in section three of this chapter, while a full set of

the data has been given in Appendix D in the form of graphs. Throughout this


thesis, reference is made to the appendices for further detailed results.

Chapter four is dedicated to the anti-dynamic tube. The first section consists of the

limited research work by others that has been done on tubes. In many cases the

reports are merely descriptive, rather than relating to the actual frictional drag on

the tube. There has been much verbal discussion with Professor Roberts, from the

Centre for Bulk Solids and Particulate Technologies in Newcastle, Australia, about

the difficulty of determining the friction force acting on the tube during material

flow. In one such a discussion, recount was given of a tube installed in a full scale

silo which started punching through the base of a flat bottomed silo when the

discharge gate was opened. It was mentioned that several attempts were made to

support the tube from the walls and the roof of the silo to prevent excessive loading

on the silo bottom from the tube. This discussion gives a good appreciation of the

extent of the drag forces on the tube during material flow.

The second section of chapter four gives an in-depth description of the tube

experiments undertaken for this PhD research, and the method of measuring the

force on the tube during material flow. The experimental results of the drag force

measurements with a full set of graphs have been given in Appendix E. From the

literature survey on the pressures in the material, a mathematical model of the

pressures exerted on the anti-dynamic tube has been presented in the last section

of chapter four.

Since the tube is a shell structure, chapter five of this thesis has been dedicated to

shell theory, in particular, cylindrical thin shells. The topic of a thin shell is

appropriate to the anti-dynamic tube, as the wall thickness of the tube would need

to be a optimised for financial reasons. Furthermore, during material flow there is

wear on the walls of the tube which results in thinning of the walls over time. There

is no theory for the structural stability and strength capacity of a thin shell with

multiple perforations around it’s circumference. Consequently the classic theories of

shells subject to axial, lateral and a combination of both pressures have been

studied.

There has been limited work from previous researchers who have conducted tests

on thin cylindrical shells with either one or two cut outs, placed at the mid height of

the shell. The shells considered had a varying ratio of radius to wall thickness as

well as the diameter of the cut out in the shell wall. These shells were subjected to

an axial load only and the results from this work has been presented in the

INTRODUCTION 1.5

literature survey of chapter six. Included in this survey is the work from an honours

thesis, L Scutella, University of Western Australia, 1998, which described tests on

thin shells with multiple perforations subjected to an axial load only. In Scutella’s

work, four different percentages of open area were considered. The open area is

defined as the ratio of the area of the cut outs to the surface area of the shell.

Scutella’s research presented a good basis for comparison with the experiments

conducted in this PhD research on similar shells subjected to both axial and

external lateral pressures. However only two percentages of open area were

considered, 16.5% (a cut out radius of 51mm) and 36.6% (a cut out radius of

76mm). These tests consisted of subjecting the shells to a combination of a varying

axial load and a constant external lateral pressure.

The final section of chapter six describes the large deflection, non-linear finite

element analyses undertaken on a solid shell and shells with the same open areas

as the laboratory experiments. In this thesis the term “solid shell” has been

employed to describe a thin walled cylindrical shell with no holes in the wall, hence

solid. As there is no shell which has perfectly curved walls, the finite element

analyses included initial geometric imperfections which were imposed on the walls.

As there are an infinite variety of geometric imperfections which could be imposed

on the walls in such analyses, it was considered reasonable to use the mode shapes

from elastic eigenvalue buckling analyses. However, there is no way of predicting

which mode shape as an imposed wall imperfection will result in the lowest failure

load in the large deflection, non-linear buckling analyses. Consequently, the first

thirty mode shapes from the eigenvalue buckling analyses were expanded and

imposed as imperfections in an axial load analysis and an external lateral pressure

analysis for each shell. From these analyses the final mode shapes were chosen as

the required wall imperfection in the non-linear buckling analyses of the shells

subjected to simultaneous axial loads and lateral pressures. The degree of wall

imperfection imposed on each shell was varied till the finite element results

reasonably matched the results from the shell experiments in the laboratory. Finally

from these analyses, interaction curves have been presented as a design tool for

shells with multiple perforations of varying open area ratios.

Finite element analysis is a topic undergoing much research on a continuous basis.

It is therefore necessary to stress that in this PhD research, finite element analysis

has been used as a means to an end, similar to the use of pressure cells, strain

gauges and other equipment in the laboratory tests. Consequently no attempt has


been made to give an in depth theory of finite element analysis.

The final chapter in this thesis presents the overall conclusion of this PhD research

including the work on silo wall pressures, the resulting frictional drag on an anti-

dynamic tube during material flow and the recommended interaction curves for thin

cylindrical shells with multiple perforations.

1.2 SILO FLOW PATTERNS

Silos are defined according to the resulting flow pattern within the silo. A W Jenike

(1964) defined types of flow pattern, mass flow and funnel flow, as shown in figure

1.3a and b respectively. Figure 1.3c shows a combination of the two types of silos.

1.2.1 MASS FLOW SILOS

Mass flow silos are characterised by steep hopper half angles and smooth wall

surfaces enabling all the material to flow when the discharge gate is opened. This is

most desirable for materials which degrade or consolidate with time, as mass flow

guarantees complete discharge of the material. If the material becomes segregated

during filling, re-mixing during discharge can be ensured. Mass flow silos are

classified according to the hopper shape; axi-symmetric silos have conical hoppers

and plane flow silos have wedge shaped hoppers with long slotted openings or a

group of discharge gates.

1.2.2 FUNNEL FLOW SILOS

In funnel flow silos the material forms a funnel within itself above the hopper

outlet, causing a last-in-first-out situation. The silo does not completely empty

when flow has stopped and an area of dead material remains inside the silo. This is

Dcr

Figure 1.3: Basic silo types: a) Mass flow ; b) Funnel flow ; c) Expanded flow

(a) (b)

Funnel flow cylinder

(c)

Mass flow hopper

INTRODUCTION 1.7

undesirable for most materials, but has the advantage of protecting the walls

against excessive wear. Funnel flow silos have shallow half angles or are flat

bottomed and cause segregation problems and erratic discharge.

1.2.3 EXPANDED FLOW SILOS

A third type of silo is the expanded flow bin which combines the two types of flow.

The critical pipe diameter Dcr in the funnel flow cylindrical section determines the

minimum dimension for the mass flow hopper below.

1.3 SILO INSERTS

Since the anti-dynamic tube is a silo insert a brief discussion of types of inserts has

been given.

Most silo inserts are discharge devices used as a solution for a poorly flowing

hopper. Ideally, the silo should first be designed with a gravity flow hopper that

would discharge the material satisfactorily, as gravity flow is the cheapest and most

reliable method of discharge. If a gravity flow hopper cannot be installed then an

appropriate discharge aid should be considered. Reed and Duffell (1983) have

categorised discharge aids into three types:

1) Pneumatic: those which rely on the application of air to the material to

initiate flow

2) Vibrational: those which rely on the application of high frequency low

amplitude vibrations to the hopper wall

3) Mechanical: those which rely on mechanical means to discharge/extract the

material from the hopper.

1.3.1 PNEUMATIC DEVICES

These devices rely on controlled quantities of air at low pressure being applied to

the material thereby reducing its strength and improving the flow characteristics.

By introducing the air at the wall of the hopper, the wall friction is reduced and the

material in the region of the wall becomes “liquid”. Reed and Duffell (1983) state

that this method works best with dry (or very low moisture content) materials less

than 300microns in size.

1.3.2 VIBRATIONAL DEVICES

Vibrational devices rely on the ability of the material to transmit vibrations thereby

reducing the strength of the bulk solid. This method should not be operated when


the discharge gate is closed as the vibrations cause densification of the material

and a higher strength material than initially loaded results. Their effectiveness in

handling sticky, flaky and fibrous materials is doubtful as they rely on the ability of

the material to transmit the vibrations.

1.3.3 MECHANICAL DEVICES

The simplest mechanical device consists of chains suspended in the silo. If the

material arches, an upward pull of the chain breaks the arch and the material starts

flowing. Alternatively, paddles within the material which rotate about a horizontal

or vertical axis maintain the material in a state of continuous motion. For hard

coarse materials, wear on the chain and paddles can be significant. Screw feeders

are a commonly used mechanical method of discharging and controlling flow rate of

the bulk solid, but are also subject to high wear rates.

1.3.3.1 Binsert

The binsert approximates the concept of the anti-dynamic tube except that it has

sloping sides, not parallel like the anti-dynamic tube.

The binsert is a patented device invented by JR

Johanson (1982) consisting of a cone-in-cone insert

as shown in figure 1.4. The smaller hopper inside the

silo hopper must be designed in accordance with

mass flow principles. He states that the location of

the insert relative to the outer hopper is such that the

included angle, satisfies mass flow criteria.

Johanson states that the outer hopper half angle,

must be twice the angle required for mass flow.

1.3.3.2 Anti-dynamic tube

The Reimberts (1976) claim to be the inventors of anti-dynamic tube in France in

the 1950’s. Figure 1.5c is a photograph taken from the Reimbert’s book (1976)

illustrating the use of their tubes. The main purpose of the tube was to alleviate the

pressures on the walls during flow of the material. The Reimbert tube is a small

diameter tube with multiple perforations along it’s length, which is placed centrally

in the silo, as shown in figure 1.5a. This design caused the silo to operate on a last-

in-first-out basis. A modification to the Reimbert tube is reported in Ooms and

Roberts (1985) and was installed in full scale silos in Port Adelaide. The

Figure 1.4: Binsert

INTRODUCTION 1.9

modification consisted of reducing the overall height of the tube and omitting all the

holes such that the silo empties in two stages only, as shown in figure 1.5b.

Figure 1.5: a) Reimbert tube; b) Roberts tube

(a) (b)

Material Flow

Figure 1.5c: Photograph taken from Reimbert(1976) showing the use of anti-dynamic tubes.


1.4 INTRODUCTION TO WALL PRESSURES

The two pressures which are of interest to researchers are the vertical pressures

acting on the silo bottom, and the horizontal (or normal) pressures acting on the

silo walls. Depending on which type of pressure exists in the silo, the vertical

pressure is either larger than, or smaller than, the horizontal pressure.

The pressures exerted by the stored material in the silo can be divided into three

main types. The first type of pressures are those which develop during loading of

the solid into the empty silo, and are generally referred to as the static pressures.

Since the silo consists of two main sections, ie the cylindrical and the hopper

sections, the static pressure distribution in each section is different. When the

material is loaded into the silo, an active pressure field develops and the lines of

major principal stress are nearly vertical. The associated minimum stress ratio

possible is the active stress ratio, Ka which is less than one. The static vertical

pressures are greater than the pressures acting normal to the silo walls. The static

normal pressures are the minimum pressures which can be expected in a silo.

The second type of pressures are those which develop during discharge of the

material from the silo, and are generally referred to as the dynamic pressures.

Again, the distribution of the dynamic pressures in the cylindrical section are

different to those found in the hopper. When the discharge gate is opened the

relative motion of the solid with respect to the wall is the same as the wall moving

in towards the solids, and hence the pressure field changes from an active to a

passive state. The stress ratio now has a value greater than one, and the lines of

major principal stress are approximately horizontal. The dynamic pressures acting

normal to the silo walls are greater than the dynamic vertical pressures, as well as

the static normal pressures, but are not necessarily the largest that can be

expected.

Based on the above discussion, a passive stress state cannot theoretically develop

in the cylindrical section of the silo, since the walls are parallel and there is no

relative motion in towards the material. However, it is doubtful whether a perfectly

parallel wall can be achieved in practice, and consequently many researchers allow

for the possibility of a small convergence in the walls of the cylindrical section. Thus

in the theories which have been presented in this chapter, a passive stress state

has been assumed in the cylindrical section, to determine the possible distribution

of flow pressures.

INTRODUCTION 1.11

The third pressure on the walls of the silo also occurs during discharge of the

material. This is the switch pressure, which is a transient pressure exerted over a

small area of the walls for a small period of time. The switch pressure is caused by

the change-over from a static to a dynamic stress state and travels quickly up the

hopper the instant the gate is opened. The switch pressure may extend the full

height of the silo, or it may become trapped at the transition from the cylindrical to

hopper section. In both the hopper and the cylindrical section, the switch pressure

is the largest expected pressure acting on the walls of the silo. Most researchers ,

and silo design codes, give the switch pressure as a multiple of the static pressure,

the actual ratio being dependant on the theory developed.

The direction of the major principal stresses have been shown by the lines in figure

1.6 for the three types of pressures found in the silo. Figure 1.6a shows the near

vertical lines of the major principal stress during filling of the silo. Figure 1.6b

shows the lines of the major principal stress as near horizontal during emptying of

the silo, with the assumption that the passive stress state develops in the cylinder.

Figure 1.6c shows the location of the switch at an instant in time when the switch

has travelled up in to the cylinder and the stress changes from an active to a

passive state.

(a) (c) (b)

Figure 1.6: Schematic representation of the lines of major principal stress.

CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES 2.1

CHAPTER 2

CLASSIC WALL PRESSURE THEORIES

2.1 STATIC WALL PRESSURES


2.1.1.1 The Janssen Theory

Prior to Janssen’s experiments and his paper of 1895, it was assumed that the

material in a silo exerted a triangular hydrostatic pressure distribution down the

height of the wall. Now the Janssen formula is used in engineering standards world

wide for computing the initial filling pressures in the vertical section of the silo.

Roberts(1995) reports on Janssen’s experiments which were performed on wooden

model silos filled with wheat. The silos being of square cross-section with sides

measuring approximately 200, 300, 400 and 600mm. The models were mounted on

adjustable screws, while the bottoms were formed by close fitting movable boards

connected to a weigh bridge. In this way the pressure on the bottom of the silo

could be measured.

The Janssen theory was developed by considering the equilibrium of the vertical

forces acting on a horizontal elemental slice in the cylindrical section of the silo, as

shown in figure 2.1.

s = length of the side

A = silo cross sectional area

P = silo perimeter

z = depth from the top

dz = element thickness

v = average vertical pressure

dv /dz = change in vertical pressure

through the element thickness.

= shearing stress between the material

and the silo wall

h = horizontal pressure

= material density

= wall friction angle Figure 2.1: Forces acting on a horizontal elemental slice

v

v + dv dz dz

z

dz h

s


Equilibrium of forces on the elemental slice is given by :

dzAAdzPAdzdz

dv

vv

(2.1)

Simplifying the above equation gives :

AP

dzd v

(2.2)

In equation (2.2) P/A is the perimeter divided by the cross-sectional area and for a

circular silo P/A = D/(D2/4) = 4/D (2.3)

The shearing stress between the material and the wall is given by :

hh Tan (2.4)

Janssen assumed a constant stress ratio of K= h /v =Ka , through the depth of

the silo.

Substituting equations (2.3) and (2.4), and substituting for h = K/ v into

equation (2.2) gives the following expression for the vertical pressure:

D/K4ze1

K4Ddz

z

0Dh4

v (2.5)

The derivation of Janssen’s equation was based on the following assumptions:

1) the vertical pressure is constant across a horizontal slice

2) the stress ratio K = h /v is assumed constant at all depths in the silo

3) the bulk density, does not vary with depth

4) the wall friction is fully mobilised and the material is on the point of slip

Roberts(1995) reports that Janssen showed from his experimental results that the

vertical pressure distribution across the cross-section was not uniform. He showed

that the pressure was higher in the centre, at 1.15 times the average value, and

lower in the corners of his models, at 0.8 times the average value. He concluded

that the wall pressures could be estimated with sufficient accuracy by assuming a

constant pressure distribution across a horizontal slice.

In this thesis the horizontal pressure on the silo wall is of interest, which is given by

D/K4zVh e1

4DK

(2.6)


The graph in figure 2.2 shows the distribution of the horizontal pressure as a

function of the height to diameter ratio of the silo. The values used in the graph

are: =16.8kN/m3, D=0.96m, =Tan220.404. The stress ratio used for the

purposes of this graph is the “at rest” ratio given by: K=0.29=1-Sin45 material

friction angle of 45).

From equation 2.6, as the depth z, tends to infinity, the exponential term tends to

zero, and the pressure tends towards a maximum given by the asymptote:

4D .

Therefore, in a very tall silo, there is no increase in the vertical pressure at the silo

bottom when more material is loaded on top. The additional material weight is

carried by the walls of the silo.

This asymptote is inversely

proportional to the wall friction

angle and therefore as the wall

friction angle increases the

horizontal pressure acting on the

wall decreases, as shown by the

curve labelled 2 in figure 2.2.

The horizontal pressure is directly

related to the stress ratio since

the horizontal pressure is

calculated from the vertical

pressure by multiplying by the

stress ratio. As can be seen in

figure 2.2, increasing the stress

ratio by a factor of two has the

greatest influence in the upper

regions of the silo.

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

0 3 6 9

2

2K H/D

Rat

io

Horizontal Pressure (kPa)

Figure 2.2: Graph of Janssen horizontal pressure distribution


2.1.1.2 The Reimbert Theory

Marcel and Andre Reimbert (1976) developed a theory for the static pressures in

the cylindrical section of the silo. They approached the problem by taking a

horizontal cut through the silo and considering the vertical forces acting on the free

body diagram of the upper portion, as shown in figure 2.3a. The depth of the silo,

z, is measured from the top of the silo, while the cone of material on top has a

height of, h. The Reimberts state that if there were no friction on the walls of the

silo, the graph of vertical pressure would be a straight line as shown in figure 2.3b.

The cone of material above the silo is shown by the offset, h/3 in figure2.3b.

However, since there is friction on the walls of the silo, the equation of vertical

equilibrium, can be written as:

0321 PzhAPAAv (2.7)

where the cross-sectional area of the silo is given by A, the circumference is given

by P, and the volume of the cone of material above the silo is given by Ah/3. In

equation 2.7, v is the vertical reaction from the material below acting on the upper

portion, is the shear stress acting on the sides of the walls and is the bulk

density of the material.

From their experimental results, the Reimberts state that the shape of the curve of

the shear stress on the walls of the silo is as shown schematically in figure 2.4.

1

2

v

w

h

z

Figure 2.3a: Free Body Diagram

Depth(z)

Stress h/3

Figure 2.3b: Graph of vertical stress

z + h/3


The Reimberts state that with increasing depth the curve of the shear stress

approaches an asymptote, which is parallel to the line of the hydraulic stress, and

has been shown dotted in figure 2.4. Therefore, as the depth of the silo increases

the equation of vertical equilibrium becomes:

0321 z

APhPmaxv (2.8)

The shear stress has the following expression, which was derived from their

experiments:

cCzz)z(

2

(2.9)

The Reimberts define the characteristic constant, Cc, as follows:

3/hC maxv

c (2.10)

Now the unknown term, vmax , in the expression for the friction function is

contained in the expression for the characteristic constant. As the silo is being

filled, v reaches a maximum limit and any additional elemental slice of material of

thickness dz, loaded into the silo is carried by friction on the walls. This can be

expressed as:

Pdz = We = Adz (2.11)

But the friction term can also be expressed as : = hmax (2.12)

Equating equation 2.11 and 2.12 gives:

hmax = (A)/(P) = (/) (2.13)

This expression is a constant since the vertical pressure in the silo becomes

constant as the depth increases and therefore the horizontal stress also becomes

constant.

Figure 2.4: Graph of shear stress

Depth(z)

Stress

(z) v max

Line of hydraulic stress in the silo for zero friction

z+h/3


The maximum vertical stress can be found from the maximum horizontal stress by

using an appropriate stress ratio. The stress ratio they propose is the active stress

ratio. Therefore, substituting for vmax into the expression for the characteristic

constant gives:

Cc = (/(Ka) - h/3) (2.14)

And hence, the friction function is fully described as:

= (z2) / (z+/(Ka)-h/3) (2.15)

The Reimbert theory then considers the general expression for friction as given by

equation 2.12, and state that since the horizontal stress varies with depth, the

expression for the friction can be written as:

(z) = h dz (2.16)

Therefore, the derivative of the friction function as given in equation 2.15 would

give an expression for the horizontal pressure. The derivative of equation 2.15 is:

2

c

2c

2c

2c

Cz

C1

Cz

)z()z2(Czdzd (2.17)

Therefore, the final expression for the horizontal stress can be written as:

2

c

2c

hCz

C1 (2.18)

From figure 2.4, the total weight of the material in the silo, z+h/3, equals the

shear stress on the walls of the silo plus maximum vertical stress acting on the

bottom. This is expressed as:

vmax + =z+h/3 (2.19)

The final expression for the vertical stress on the bottom of the silo, can be found

since the expression for the friction function is given in equation 2.15. Therefore,

the expression for the vertical stress is:

3h1

Czz

1

cv (2.20)


2.1.1.3 Discussion of Janssen vs Reimbert Theory

Briassoulis(1991) states that both the Janssen and Reimbert’s theories are

“unconditionally applicable for any silo geometry and stored material”.

For the silo model and material used in the laboratory, a comparison between the

two theories is shown in figure (2.5). The Janssen theory gives a greater vertical

stress as the depth increases, while the Reimbert theory gives a greater horizontal

stress in the upper parts of the silo. Both theories tend to the same value of

horizontal pressure as the depth of the silo increases.

Re-writing Janssen’s expression for the horizontal pressure, equation (2.1.6) as:

DzK4

h e1K1

4D (2.21)

Noting that for a cylindrical silo A/P = D/4; Reimbert’s expression for the horizontal

pressure, equation (2.20) can be re-written as:

2

ch 1

Cz1

4D (2.22)

Therefore, only the terms in square brackets need be considered when comparing

equation (2.21) and (2.22).

For a silo with no cone of material on top, the expression for the characteristic

abscissa in Reimbert’s theory can be re-written as Cc = D/(Ka 4) and substituting

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

0 5 10 15 20Vertical Pressure

Janssen

Reimbert

Dep

th b

elow

sur

face

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

0 2 4 6 8Horizontal Pressure

Janssen

ReimbertD

epth

bel

ow s

urfa

ce

Figure 2.5: Comparison between Janssen and Reimbert theories


this in equation (2.22) gives the term in square brackets as: 1-(4Kz/D + 1)-2

The graphs in figure 6 show the influence of the wall friction angle and the stress

ratio, K, on the terms in square brackets for both equations of horizontal pressure.

The Janssen term has been shown with a blue line and the Reimbert term in a red

line. The graphs show that as the wall friction angle increases, the Janssen and

Reimbert expressions tend to coincide, while a smaller stress ratio causes them to

diverge.


The theory for the wall pressure in a convergent hopper was presented by Walker in

1960. The theory is included in this section for completeness only, since the results

appear to be at a variance with physical intuition and with the results of Janssen

theory in the cylindrical section

2.1.2.1 Walker Theory

Walker (1960) considered a smooth walled hopper with a hopper half angle (,

filled with a nearly incompressible material. His initial premise was that the

principal stress planes in the hopper were vertical and horizontal, so that the shear

stress on a vertical plane is zero. Under the assumption of zero shear, the normal

stresses must increase hydrostatically with depth.

Figure 2.6: Comparison of terms in Janssen and Reimbert equations: (a): Wall Friction ; (b) Stress ratio

(b): Stress ratio varies from 0.5 to 1

0

1

2

3

4

5

0.0 0.4 0.8 1.2 1.6

1

0.5

JanssenReimbertD

epth

bel

ow s

urfa

ce (

m)

Horizontal pressure (kPa)

0

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.0

10o 50o

JanssenReimbert

(a) Wall friction angle varies from 10 to 50 degrees

Dep

th b

elow

sur

face

(m

)

Horizontal pressure (kPa)


Therefore: hV zzg (2.23)

Therefore, plotting a graph of the

static pressure in the silo using

the Janssen theory for the

cylindrical section, and the Walker

theory for the hopper section,

gives the diagram shown in Figure

2.7 for the model silo in the

laboratory. This graph illustrates

the vertical pressure distribution

in the silo.

If the material in the hopper lies within the yield surface, and only the wall is

assumed to slip, Walker gives the Mohr circle for an element on the wall as shown

figure 2.8. Point P on the circle represents the stresses at the wall which is

tangential to the wall yield locus. The wall friction angle is given by w and the

material friction angle is given by m. Walker assumes the horizontal stress to be

equal to the minor principal stress, 3, on the circle.

Figure 2.7: Graph of static vertical pressure in a model silo

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40Vertical pressure (kPa)

Dep

th b

elow

top

of si

lo (

m)

Cyl

indr

ical

sec

tion

Hop

per

Janssen

Walker

Figure 2.8: Mohr’s circle for stresses at the hopper

3 N 1

hopper wall yield locus

material effective yield locus

W 2C 0

P m


The angle which the normal stress, N, acting on the hopper wall makes with the

horizontal stress, is given by 2 in Figure 2.8. The value of N is obtained from the

geometry of the Mohr circle.

11ww

wN C

Sin2Sin

Cos2Sin

(2.24)

From equation 2.24, it can be seen that Walker gives the normal stress at the wall

as a function of the hopper half angle and the wall friction angle. The material

friction angle does not influence the value of N. Figure 2.9 shows the variation of

the ratio given by C in equation 2.24 as a function of the wall friction angle. The

four curves shown in figure 2.9 are for a hopper angle of 5 increasing to 20 in 5

increments.

As can be seen, the shallower the hopper slope, the higher will be the normal wall

stress according for the same wall friction. The lowest normal wall stress will be

achieved using a shallow hopper half angle, ie 5, and a higher wall friction angle.

However too high a value for the wall friction angle will inhibit mass flow and result

in funnel flow.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70

=5

=20

Wall friction angle (w

Con

stan

t C

Figure 2.9: Variation of C as a function of the wall friction angle

CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER 2.11

2.1.2.2 Jenike Theory

Jenike has written several papers on the pressures exerted on the hopper walls as

well as the pressures within the solid stored in the hopper. The theories developed

were covered in several papers. To fully understand Jenike’s derivations, a brief

summary, in chronological order, describing the outline of each theory is given

below. A similar outline has been given in chapter 2.2 for Jenike’s theories of

pressures due to the flowing material in the silo.

1) Radial pressure field in the solid stored in a hopper.

In this theory Jenike (1961) considers an element of material, at an arbitrary

location in the stored solid, and evaluates the forces acting on this element by

considering equilibrium of all forces. The equations which result after extensive

algebraic manipulation, are lengthy differential equations which are solved using

numerical methods. Jenike shows from the results of these equations that the

pressures in the solid increase linearly from zero at the vertex of the hopper. Jenike

defines this as a radial stress field.

2) Linear normal pressure exerted on the hopper wall .

In this analysis, Jenike (1968) states that the weight of the material in a hopper

without a surcharge, is carried by the vertical components of the shear and normal

wall stresses. By simple equilibrium of forces, Jenike derives an expression for the

pressure normal to the hopper wall. This analysis was extended by Jenike (1973) to

include normal pressures acting on the walls of hoppers with a surcharge.

3) Position of the maximum pressure in the hopper.

Having found the expressions for the normal wall pressure and the radial pressure

field, Jenike(1968) equates the two expressions to determine the location, iN, of the

maximum normal wall pressure in the hopper.

Jenike’s theory for the linear normal pressures on the hopper walls has been dealt

with first in this thesis, followed by the radial pressure field and the location of the

maximum static pressure.

2.1.2.2.1) Linear normal pressure exerted on the hopper wall

Jenike(1968) states that the pressures in a hopper have been shown

experimentally to decrease to zero at the vertex of the hopper. Therefore, Jenike

assumes a linear pressure distribution acting normal to the wall during filling of a


hopper with no surcharge as shown in figure 2.10(b) below. This pressure field

cannot extend all the way to the free surface in a hopper without a surcharge, but

will be topped off by a compatible pressure field, with pressures decreasing towards

zero at the free surface. The maximum pressure in the hopper occurs at the

interface of the two pressure fields. The distance of the maximum pressure from

the vertex has been defined by Jenike using the ratio iN, as shown in figure 2.10(c).

Jenike states that the total weight of the material must be supported by the walls

and ignores any support gained from the material below. Therefore, the sum of the

vertical components of the wall shear and normal pressure, and N, acting on an

element of thickness dz, equals the weight of the element in the hopper.

The vertical components of the wall support, acting over an area of 2rdz / Cos.

are given by:

Cos/dzr2CosTanSin )( wNN (2.25)

The radius of the hopper is given by: r = z Tan (2.26)

The volume of the element is approximately given by:

V= z2 Tan2 dz (2.27)

Therefore, the weight of the element is given by:

dzzTan 22W (2.28)

By equating equation 2.25 and 2.28, re-arranging and integrating over the hopper

wall from 0 to Hh gives the following, (where Hh is the height of the hopper):

r

N

(a)

iN=zN/Hh

i=1

(c)

Figure 2.10: (a) Hopper; (b) Pressure distribution; (c) Ratio i

Hh

zN

z

N max

(b)

Dep

th

Pressure


hH

0nw

3h zdz)TanTan(6HTan (2.29)

In Figure 2.10(c), for 0< z <zN : N= z N max /zN

And for zN< z <Hh : N = (Hh-zN) N max / (Hh-z )

Substituting for N in equation 2.29, and integrating gives the following expression:

3

z

3

H

2

zH

2

H

zH

1z3

z)TanTan(6HTan

3N

3h

2Nh

3h

NhN

3N

Nw3h (2.30)

Solving for N gives the expression for the normal pressure on the hopper wall as:

N=D/[2(Tan+Tanw)(1+zN /Hh)] (2.31)

Figure 2.11 shows the

maximum normal wall

pressure in the hopper as

given by equation 2.31,

for various values of zN/Hh

The graph was drawn for

values of =18 kN/m3,

hopper diameter D=1m,

hopper half angle =15,

and a wall friction angle

w = 22. From equation

2.31, the material bulk

density and diameter of

the hopper have the

greatest effect on the wall

pressure. As can be seen,

from figure 2.11 the maximum pressure decreases, as the position of the maximum

increases within the hopper. The locus of the maxima has been shown in figure

2.11. Since the maximum cannot be at the free surface, the locus has been

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

Rat

io z

N/H

h

Normal wall pressure (kPa)

zN/Hh = 0.25 zN/Hh = 0.5 zN/Hh = 0.75 Locus of Maxima

Figure 2.11: Normal Pressures on the Hopper Wall for various values of the Ratio iN


arbitrarily terminated at a value of iN=0.875

The shape of the locus can be closely approximated by a cubic polynomial of the

general form: iN = AN3+ BN

2+ CN+D. The constants in the equation are different

for each hopper being analysed.

Having derived an expression for the normal wall pressure in a hopper with no

surcharge, Jenike(1973) adopts the same approach in deriving an expression for a

hopper with a surcharge. This surcharge, J, calculated from the Janssen equation

at the level of the transition, increases the pressure at the top of the hopper, from

zero to some value t at the transition. Jenike states that this surcharge is

supported by the walls above the location of the maximum normal pressure, as

shown by the shaded area in Figure 2.12 (b).

By equilibrium, the additional pressure is given by:

hH

Nz12

wJ zdzpp

TanTanTan2

(2.32)

where p1 is the pressure at some level, z, for no surcharge and p2 is the additional

pressure at the same level, z, for a surcharge acting on the hopper

and (p2 - p1) = t (z –zN)/(Hh-zN) (2.33)

The ratio i is defined by Jenike as: i = z /Hh

Substituting equation 2.33 into equation 2.32, and substituting for iN=zN/Hh gives:

Pressure distribution in hopper with no surcharge

Additional pressure due to material in the cylindrical section

t

Hh

zN z

N max

(b) Pressure

p2 p1 r

Figure 2.12: (a) Hopper ; (b) Pressure distribution with surcharge

N

(a)

J


hH

NzN

N

twJ d)(

1TanTanTan2

iiiii

(2.34)

Integrating and solving for t gives:

wNNJt

TanTan21Tan3

ii (2.35)

Figure 2.13 shows the normal wall pressure acting on a hopper with a surcharge.

The Janssen pressure, J, for the hopper surcharge was calculated for a cylinder

height of six times the hopper diameter, since the vertical pressure in a cylinder

does not increase substantially beyond this height to diameter ratio. The graph has

been drawn for values of =18 kN/m3, a hopper diameter D=1m, a hopper half

angle =15, and a wall friction angle w = 22. As the position of the maximum

pressure on the hopper walls increases, so the pressure at the transition also

increases.

The dotted lines in figure 2.13 indicate the lines of pressure for no surcharge in the

hopper, as given in figure 2.11.

Figure 2.13: Normal Pressures on the Hopper Wall for various values of Ratio iN for a hopper with a surcharge

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Rat

io i N

Normal wall pressure (kPa)

zN/Hh=0 zN/Hh=0.25 zN/Hh=0.5 zN/Hh=0.75


2.1.2.2.2) Radial pressure field in the solid stored in a hopper.

Jenike [1961] defines a radial stress field as a field in which all the stresses

increase linearly with the radius, r, for the initial loading of material in the hopper.

Youngs modulus for the material is eliminated during the process of derivation and

Poisson’s ratio is assumed constant. The material is assumed to be non-linearly

elastic and is assumed to slip at the walls. In his derivation Jenike considers an

element in the hopper with a set of spherical co-ordinates (r,,) with origin at the

vertex of the hopper. Jenike does not give a complete diagram of the element in his

papers. The element shown in figure 2.14 has been drawn to fully understand the

shape of the element and the directions of the forces acting on it. The areas of each

face of the element have been shown in figure 2.15, with the expressions for the

length of each side and the area of each face.

Considering equilibrium in the r-direction (refer figure 2.14):

2dCosAASindddrr2Adr

r rightrbottrbottr

r

………………...

+ rightrightr

r A2d

2dCosCosdddrrAd

…………..…….

ddrrSin2d2CosdddrrA

2dd cright …..…….

+ 0Sindddrr2dCos 2

(2.36)

Divide throughout by:( r2drddSin )

02dCos

r1

r1

r1Cot

r1

rr2 c

rr

rr

(2.37)

Re-arranging gives the following final equation for equilibrium in the r-direction:

02dCosCot2

r1

r1

r rcrrr

(2.38)

This expression differs from Jenike’s expression by the term Cos(+d/2) instead of

Cos(). By small angle approximations, this difference in the equations has little

effect on the final solution and the term d/2 has been ignored.

The expression for equilibrium in the direction is given after figures 2.14 and 2.15

+


Figure 2.14: Element in the stored material in a conical hopper

Plan on hopper

A A

B

B

Element in the stored solid

Section A-A

Section B-B Plan View

d/2

d

r

drr

rr

r

r

dr

r

drrr

r

d

d

d/2 c

c


The position of the element

within the hopper is defined as

follows:

r = radius from hopper vertex

dr = small change in r

= angle of rotation of r

d = small change in

d = small change in the

horizontal angle

The lengths of the element

sides are defined as follows:

S1 = r d

S2 = (r+dr) d

S3 = r Sin(+d) d

S4 = (r+dr) Sin(+d) d

S5 = r Sin d

S6 = (r+dr) Sin d

The areas of the element faces

can be estimated as follows:

Area bottom = S1 (S3+S5)/2

=r2 d d Sin

Area top = S2 (S4+S3)/2

r2 d d Sin +

2r dr d d Sin

= Abott + 2r dr d d Sin

Area right = dr (S5+S6)/2

r dr d Sin

Area left = dr (S3+S4)/2

r dr d Sin +

r dr d d Cos

=Aright + r dr d d Cos

Area front = Area back

= dr (S1+S2)/2 r dr d

The volume of the element is given by: V = dr (Atop + Abott)/2 r2 dr d d Sin

Figure 2.15: Element in the Stored Material in a Conical Hopper

S4 S3 S6 S5 d

r Sin(+d)

r Sin

(r+dr) Sin(+d)

(r+dr) Sin

r

d

dr S1

S2

Area top

Area left

Area right

Area front

Area back

Area bottom


Considering equilibrium in the -direction :

CosdddrrA2dCosdA

2dCos rightright …………………….….…

-

CosdddrrA

2ddA

2d

rightr

rrightr ………………………………..

+ bott2 ArSindddrr

2dSin

………………..………………………………………………

- 02dCos2

2dddrrcSindddrr2Adr

rr

r bott

(2.39)

Dividing throughout by:( r2 dr d d Sin ) , collecting terms and re-arranging gives :

02dSinCot3

r1

rcr

r

(2.40)

This expression differs from Jenike’s expression by the term Sin(+d/2) instead of

Sin(). By small angle approximations, this differences has little effect on the final

solution, and has therefore been ignored.

To complete the solutions of equations 2.38 and 2.40, Jenike(1961) first finds

expressions for , r, c and r . As the material is loaded in the hopper of the silo,

it contracts both vertically and horizontally. Therefore, the material does not reach

the limiting state of stress but is in an elastic-active state of pressure.

The elastic stress strain relations in the hopper are therefore given by:

211E cr

r1

211

1E cr (2.41 a, b, c)

211

1E rc

c

The strains given in equations 2.41 a,b,c can be written as functions of the radial

displacement as follows:

r = -u/r

= c = -u/r

+

(2.42 a,b)


From equation 2.42b it follows that = c

Jenike defines the stress ratio in the hopper as: k = /r (2.43)

This is similar to the definition used by Janssen for the stress ratio in the cylindrical

section of the silo.

In a Mohr circle for the stresses on the element, the average stress is given by:

= (r+)/2 (2.44)

The general equation for the radial stress is given by Jenike as: = rS (2.45)

where S is the radial stress field and is a function of the co-ordinates r and

Therefore, from equations 2.43, 2.44 and 2.45:

r = k /(1+k) = k rS /(1+k) (2.46)

= c = 2k /(1+k) = 2k rS /(1+k) (2.47)

The shear stress on the wall is given by:

r= Tanw= 2k rS Tanw /(1+k) (2.48)

The derivatives of r , and r are as follows:

drdSrS

k12

rrrr (2.49)

ddSr

k1k2

r (2.50)

dd

wCosk1

rSk2ddSrTan

k1k2

dd w

2ww

w

rrr (2.51)

drd

wCosk1

rSk2drdSrSTan

k1k2

rdrd

rw

2ww

w

rrr

(2.52)

Now the derivatives of r , and r given by equations 2.49, 2.50, 2.51 and 2.52

can be substituted into the equations of equilibrium given by equations 2.38 and

2.40. In his analysis, Jenike has assumed dS/dr and dw/dr to be zero. Therefore,

collecting terms in dS/d and dw/d and dividing throughout by (1+k)/(2k) gives

the following:

0Cos

k2k1CotTan2

k2S

dd

Cos

SddSTan w

w

w2w

(2.53)

0

kSink1STan4

ddS

w

(2.54)


Substitution of ds/d from equation 2.54 into equation 2.53 gives the following for

dw/d:

w2

www2w CosCos

Sk2k1

CosSinCotSinSk2k1

Cosk

k234

dd

(2.55)

Assigning a constant value to k, the stress ratio in the hopper, equations 2.53 and

2.55 can be solved by numerical integration. The value of S, for the given value of

k, is then substituted into equation 2.51 to determine the value of the stress

normal to the wall, , which is a linear function of the hopper radius, r. From

equation 2.51, the stress field, S, must be unitless since the density, , has units of

kN/m3 and radius, r, has units of m. This results in the units of kPa for the stress,

, normal to the wall.

2.1.2.2.3) Position of the Maximum Pressure in the Hopper

The locus of maximum pressures is unique for a given hopper and stored solid, as

shown in figure 2.11. This locus has been drawn by assuming values for the ratio of

zN/Hh. Jenike states that the position, zN, of the peak pressure is determined from

the intersection of the locus of maximum pressures with the linear (radial) pressure

field shown in figure 2.10. This has been shown schematically in figure 2.16.

The normal wall pressure acting on the hopper due to the radial pressure field, has

been given by equation 2.47, and is repeated as follows : = 2krS/(1+k).

Referring to figure 2.15, the radius, r, in equation 2.47 is measured from the vertex

of the hopper, and does not denote the radius of the hopper. Therefore, at the level

of the maximum pressure, let r=rN, which can be written in terms of iN as follows:

Locus of maxima

Radial pressure field zN

Peak pressure

Hopper

D

Figure 2.16: Position of Peak Pressure in a Hopper


CosHi

Cosz

r hNNN

Then equation 2.47 becomes: k1CosHiSk2 hN

(2.56)

The equation for the maximum pressure has been given by equation 2.31 and is

repeated as follows:N=D/[2(Tan+Tanw)(1+zN /Hh)]

In equation 2.31, D is the diameter of the hopper as shown in figure 2.16, and can

be re-written as :

D=2r=2HhTan

And by definition iN=zN/Hh

Therefore, equation 2.31 becomes: Nw

hN i1TanTan2

TanH2

(2.57)

By letting equation 2.56 equal equation 2.57, and re-arranging, an equation for the

position of the peak pressure, ratio iN, is given as follows:

w

NN TanTanSk2Sink1

i1i

(2.58)

where S is the radial pressure field in the solid and is determined from equations

2.54 and 2.55.

The position of the maximum pressure is directly related to the stress ratio, k, and

the hopper half angle, , and is indirectly related to the stress field, S, and the

hopper wall friction angle, w.

Re-arranging equation 2.58 in terms of w and gives the following:

Tan

kS2i1iSink1

TanNN

w (2.59)

A graph of equation 2.59 for k=0.8, S=0.4 and varying the hopper half angle has

been plotted in figure 2.17 below. Unlike the graph given in Jenike (1968), the

contours of iN in figure 2.17, decrease from iN=0.8 closest to the horizontal axis to

iN=0.1 closest to the vertical axis. This is the reverse of the graph given by Jenike.

Increasing the values of both the stress ratio, k, and the stress field, S, from 0.3 to

0.9 has the effect of decreasing the required wall friction angle for a given hopper

half angle. Since these graphs represent filling conditions in the hopper, the stress


ratio, k, cannot have a value greater than one. Increasing the stress field, S,

beyond a value of 1 gives a negative value for the wall friction angle, which is not

possible.

2.1.2.3 Walters Static Hopper Pressures.

In his paper Walters(1972) gives the same analysis for the static vertical pressure

in the hopper as for the dynamic vertical pressure in the hopper. The full analysis of

the equilibrium on a horizontal elemental slice in the hopper section, by Walters,

has been given in chapter 2.2.1.1 of this thesis. The difference between the static

and the dynamic case, is that in the static case, the shear stress at the wall is the

minimum value. This minimum value is determined from the intersection of the wall

yield locus and the Mohr circle representing the stresses at the wall, as shown

graphically in figure 2.18.

Walters analysis of a horizontal elemental slice in a hopper results in the same

differential equation (equation 2.132) for both the static and dynamic conditions,

except that the constants E and FH are as follows:

Dm

Dm

22CosSin1

22SinSinE

(2.105 repeated)

0

10

20

30

40

50

0 5 10 15 20 25Hopper half angle

Wal

l frict

ion

angl

e w

iN=0.1 iN=0.2 iN=0.4 iN=0.6 iN=0.8

Figure 2.17: Contours of iN for a conical hopper for k=0.8 and S=0.4


mm2

2m

2m

2

HySin2Sin1Cos

SinSin2CosSin1F (2.128 repeated)

where the +ve sign refers to static conditions and the –ve sign refers to dynamic

conditions.

Static shear value

Dynamic shear value

Wall yield locus

Material yield locus

+

+

Figure 2.18 General Mohr circle for stresses in the material adjacent to the wall.

CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: CYLINDRICAL SECTION 2.25

2.2 DYNAMIC WALL PRESSURES


2.2.1.1 The Walters Theory

Walters (1973a) follows the same assumption made by Jenike, that the lines of

major principal stress are approximately vertical during initial filling of the solids

into the silo, as shown in figure 2.19(a). When the discharge gate is opened and

the material starts flowing, the lines of major principal stress become nearly

horizontal, as shown in figure 2.19(b). In both cases, the angle which the major

principal stress makes with the wall is S and D, for the static and dynamic cases.

In the same manner as Janssen, Walters solves for the equilibrium of vertical forces

acting on an elemental slice of thickness, dz, in the cylindrical section of the silo.

The result is stated again as follows:

vwv

D4

dzd

(2.60)

where v is the average vertical stress acting across the elemental slice and vw is

the shear stress on the silo wall. To solve equation 2.60, the shear stress at the

wall, vw must be related to the average vertical stress, v acting across the slice.

First vw is related to v and then vw is related to vw .

Since the vertical stress acting across a horizontal elemental slice is not constant,

Walters assumes the average vertical stress, v, is related to the vertical stress

adjacent to the wall, vw, by a distribution factor, as follows:

vw = F v (2.61)

S

(a)

D

(b)

Figure 2.19: Lines of major principal stress (a) Static,(b) Dynamic; (c) Force balance on an elemental slice

(c)

dz H

v vw

dzzv

v


The distribution factor has been determined by Walters from the mohr circle given

in figure 2.20, which has been greatly enlarged in this thesis for clarity. The

distribution factor has been given the symbol F and not D (as in Walters), to avoid

confusion with D used for the silo diameter. The calculations have been continued

after figure 2.20

Figure 2.20: Mohr Circle for Stresses at the Wall and in the Material for Flow Conditions

H

vw

v

w

m

C

v

vw

Wal

l yie

ld lo

cus

Mat

eria

l yie

ld lo

cus

Str

esse

s at

th

e w

all

Ave

rag

e st

ress

es

Str

esse

s at

th

e ce

ntr

e

x


The green circle in figure 2.20, represents the stresses at the centre of the silo, the

blue circle represents the average stresses, and the red circle represents the

stresses at the wall. All three circles are tangential to the material effective yield

locus. The vertical stress, vw, and the shear stress, vw, at the wall, are determied

from the intersection of the wall yield locus with the circle for wall stresses. Walters

assumes that the horizontal stress, H, remains constant across the slice as shown

in figure 2.20.

The circle for average stresses has a radius, x, and the centre of the circle is given

by (H+v)/2

The centre of the circle is related to the internal angle of friction of the material as

follows:

m

vHSin

x2

which can be re-written as: Hm

v Sinx2

(2.62)

The radius of the Mohr circle, x, for average stresses is given by:

2Hv

H2

v2

2x

(2.63)

Substituting equation 2.62 into equation 2.63, solving for x and re-arranging gives

the following expression for the radius of the circle:

m

22vwm

22HH

m

m CosSinCosTan

x (2.64)

The shear stress at the centre of the silo is

zero, while the shear stress is a maximum at

the silo wall. Walters assumes that the shear

stress varies linearly as shown in figure 2.21.

The shear stress at an arbitrary distance, r,

from the centre of the silo is given by v, and

is related to the shear stress at the wall, vw,

as follows:

v =vw r / (D/2) (2.65)

Substituting for x from equation 2.64, and from equation 2.65 into equation 2.63

gives the following expression for the distribution factor, F given in equation 2.61,

0

vw=max

Silo centre line

Silo wall

v

r

D/2

Figure 2.21: Shear stress variation across the silo


as follows:vw = F v where F is given by:

2/3

m2

w2

w2

mm2

m2

w

w2

m2

m2

w

Tan

Tan11

Tan3

SinTan4Sin1Cos

SinSin2Sin1CosF (2.66)

The graph of the distribution factor, F, as a function of the material friction angle,

m, and the wall friction angle, w, has been shown in figure 2.22 below.

The curves of the wall friction angle in figure 2.22 have been given in increments of

10. From equation 2.66 for F, a wall friction angle of 90 has no meaning as this

would require a division by zero, therefore a maximum value of w= 89 has been

shown in figure 2.22. From the figure it can be seen that for all values of material

friction angle and wall friction angle, the vertical stress at the wall of the cylinder,

vw, is greater than the vertical stress, v, at the centre of the cylinder.

Now that vw has been defined as a function of v the next step in the solution of

equation 2.60, is to define the shear stress at the wall, vw as a function of vw.

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0 10 20 30 40 50 60 70 80 90Material Friction Angle (m)

Dis

trib

utio

n Fa

ctor

(F)

w=10

w=40

Figure 2.22: Distribution factor as a function of both material friction angle (m) and wall friction angle (w)

w=89


To find an expression of vw as a function of vw the Mohr stress circle for the

stresses at the wall, as given in figure 2.20, has been repeated in figure 2.23

below. From the Mohr circle Walters derives an expression for the ratio of vw/vw.

Let the radius of the circle be denoted by ‘a’ = PW = NW

Let B= vw/vw = PH / (OW – Wvw )

And PH = a Sin2D ; OW = a / Sinm ; Wvw= WH = a Cos2D (2.67a,b,c)

The subscript D in D refers to the angle shown in figure 2.19b for the case of

material discharge ie the dynamic case.

Therefore, B = Sin2D Sinm / (1 - Sinm Cos2D ) (2.68)

From triangle MWO: + 2D = /2 + w (2.69)

Therefore, 2D = /2 + w - (2.70)

From triangle PMW: = Arc Cos (MW/a) (2.71)

And MW = OW Sinw (2.72)

Substituting equation 2.67b for OW into 2.72 gives:

MW = a Sinw / Sinm (2.73)

Substituting equation 2.73 into equation 2.71 and re-arranging gives :

2D/2 + w - Arc Cos( Sinw / Sinm ) (2.74)

H vw w m

vw Wall yield locus


Stresses at the wall

P

W O

2D

M

N

Figure 2.23: Mohr circle for stresses at the wall


Substituting equation 2.74 into equation 2.68 and re-arranging gives the following

expression for B:

w2

m2

m2

w

m2

w

SinSin2Sin1Cos

CosSinB

(2.75)

Therefore, vw = B vw (2.76)

The graph of B as a function of the material friction angle has been given in figure

2.24 below. Walters gives the plot of the factor B as a function of the wall friction

angle, therefore the shape of the graph in figure 2.24 does not correspond to that

given by Walters. The curves of varying wall friction angle have been given in

increments of 10. The first two curves for wall friction angles of 5 and 10 have

been plotted in dotted lines for clarity only.

Substituting equation 2.61 into equation 2.76, and then substituting this into the

equation of equilibrium of forces (equation 2.60) gives the following:

vv

DBF4

dzd

(2.77)

where F is given by equation 2.66 and B is given in equation 2.75.

Plotting the product of the distribution factor, F, and the ratio, B, gives a set of

curves similar in shape to the curves in figure 2.24. This product has been shown in

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90

Rat

io (

B)

Material Friction Angle (m)

w=5 w=10

w=20

w=80

w=40

Figure 2.24: Ratio B as a function of material friction angle.


figure 2.25 below. The curves for wall friction angles of 5, 10 and 20 have been

plotted in dotted lines for clarity only. For most silo problems the wall and material

friction angles lie within the range shown by the red dotted line in figure 2.25.

Before Walters gives the solution to equation 2.77, he first puts the variables in

dimensionless form by dividing throughout by D and letting SV equal v/d and

Z=z/D as follows:

1BFS4dZ

dSV

V (2.78)

The solution to equation 2.78 for the vertical pressure in a silo has the form :

BFZ4V e1

BF41S (2.79)

Walters gives the horizontal pressures acting normal to the silo wall from the

relationship given below:

SH = SV*B*F/m (2.80)

As Walters points out these equations are of similar form to the classical Janssen

equation, with the factor BF in place of Janssen’s Ka , where Tanw and Ka is the

active stress ratio.

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90

Prod

uct

BF

Material Friction Angle (m)

w=10 w=20

w=80

w=40

Figure 2.25: Product of distribution factor, F, and ratio, B.


The vertical pressures acting across a slice have been given in figure 2.26a, and the

horizontal pressures acting normal to the silo wall have been shown in figure 2.26b.

The static pressures acting in the silo have been calculated using the Janssen

equations as given by equation 2.5 for the vertical pressures, and equation 2.6 for

the horizontal pressures. The vertical dynamic pressures have been calculated

using Walters equation 2.79, and the horizontal pressures have been calculated

using equation 2.80. To keep the equations consistent, the Janssen equation has

been divided throughout by D to present it in dimensionless form.

From figure 2.26 it can be seen that the vertical dynamic pressures are

approximately 3 times less than the static pressures for values of Z greater than 5.

However, the dynamic horizontal pressures are greater than the static pressures by

a factor of approximately 2.6. This excludes the effect of a switch pressure which

has been discussed in chapter 2.3. The dynamic pressures approach a constant

value at a height to diameter ratio of approximately 2, whereas in the case of static

pressures, the asymptote is only reached at height to depth ratios of approximately

5.

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

Vertical pressure Horizontal pressure

Dep

th w

ithin

silo

Z=

z/D

Janssen static

Walters dynamic

Figure 2.26: Static and Dynamic Pressures: (a) Vertical pressure (b) Horizontal pressures

(a) (b)

Janssen static

Walters dynamic



2.2.2.1 Flow/Slip in the Hopper by Equilibrium of a Slice

In order to examine the variation of stress within the hopper, a slice equilibrium

model similar to the Janssen model has been investigated, as shown in figure 2.27.

The model assumes that the wall slip and the material flow occur simultaneously,

so that the Mohr circle is tangential to the yield surface and the wall stress is

defined by the friction angle w. A circular hopper with a half angle (, a radius of

r0 at the transition has been assumed. The depth (z) is measured from the

transition, and the stress at the transition is given by Janssen theory. The vertical

stress v is assumed to act uniformly over the slice, and the pressures normal and

tangential to the wall are given by N and respectively. The material in the

hopper has a bulk density of .

It will be shown that the slice model leads to a first order differential equation with

non-constant coefficients, and that a solution can be obtained numerically.

Importantly, the first order differential equation allows only one boundary condition

for stress at the top of the hopper. The variation of stress with depth and the stress

at the hopper bottom are dependent on the material properties and the hopper half

angle, and are obtained from the numerical equation solution.

The radius of the top of the element is

given by:

Tanzrr 0t (2.81)

The radius of the bottom of the element

is given by:

Tan)dzz(rr 0b (2.82)

And the average radius of the element is

given by:

Tan)2/dzz(rr 0av (2.83)

The shear stress acting on the wall of

the hopper is given by:

wv

wN TanQ

Tan

(2.84)

The pressure normal to the hopper wall N, has been substituted by v / Q, where

the expression for Q will be derived later.

Figure 2.27: Elemental slice of material in the hopper

dz

z

r0

rt

rb N

V


By taking the sum of the vertical forces equal to zero, and considering the

downwards direction as positive, gives the following equation for equilibrium:

0dzrCosdzr2Cos

Cosdzr2Sinrdz

zr 2

avavavN2

bv

v2

tv

(2.85)

Substituting equations 2.81, 2.82, 2.83 and 2.84 for rt, rb, rav and N in equation

2.85, and re-arranging, collecting and canceling common terms, gives the equation

below.

v0

w02

0

v zTanrQ

TanQ11zTan1

Q1rTan

zTanr

2z

(2.86)

The boundary value problem given in equation 2.86 is a first order differential

equation and can be re-written as:

v

v cz

(2.87)

where c is the term given in the square brackets in equation 2.86.

The term (r0-zTan)2 in the denominator of c, was found to make the solution of

equation 2.87 unstable. Therefore, the differential equation was re-written as:

20vv2

0 zTanrCz

zTanr

(2.88)

where

zTanr

QTan

Q11zTan1

Q1rTan2C 0

w0 (2.89)

To find the expression for Q in equations 2.84

and 2.86, the stresses acting on an element of

material, adjacent and parallel to the hopper

wall, has been shown in Figure 2.28. The

corresponding Mohr circle has been shown in

Figure 2.29. The sign convention used is that of

Gere and Timoshenko (1996), where

compressive stresses and shear stresses acting

in a clockwise direction are negative, and shear

stresses acting in a counter clockwise direction

are positive. A positive or negative symbol has

been placed next to the arrowhead of each

-

N

V D

+

+ -

-

-

-

Hopper wall

Figure 2.28: Stresses acting on an element


stress in Figure 2.28. The hopper wall has a half angle inclined to the vertical,

and a wall friction angle of w. The stress acting normal to the wall is N and the

stress acting at right angles to N is D. The vertical stress is V and acts on a

horizontal plane as shown in Figure 2.28.

In Figure 2.29 below, the arrowheads on the axes indicate the positive directions of

normal and shear stresses. The circle has been drawn tangential to the material

yield locus since the material has not yet yielded. The maximum shear stress which

can occur between the element and the wall, is determined by the point of

intersection of the wall yield locus and the circle. This point locates the normal

stress N on the circle. Using the method of origin of planes, the plane of the

hopper wall has been plotted at an angle, to the vertical and has been shown by

the dotted line through N in Figure 2.29. The intersection of this line with the circle

locates the origin of planes labeled OP. Since the vertical stress acts on a horizontal

plane, a line has been drawn through OP to locate the point of the vertical stress on

the circle, labeled V . Thus the conjugate pairs of stress D and H are located,

since the other points have been established.

The radius of the circle can be given by:

2W

2N

2DN

2r

where W=TanW . (2.90)

Figure 2.29: Mohr circle for an element adjacent to the hopper wall.

V

C

N

D

H

W

2

Wall yield locus

O

+

+

OP

M



Let = D /N which is less than 1 since D < N on the Mohr circle.

Then the radius of the circle can be re-written as:

2w

2N 412

r (2.91)

The vertical stress can be given by:

2Cosr

2DN

v (2.92)

where

1

Tan2ATN

2

TanATN w

DN

wN (2.93)

Substituting equation 2.91 and 2.93 into 2.92, and re-arranging gives the following

expression for the vertical pressure acting in the hopper under filling conditions:

Nv Q (2.94)

where

21Tan2

ATNCos41121Q w2

w2 (2.95)

The only unknown in the expression for Q is the stress ratio given by D/N .

To solve the boundary value problem given by equation 2.88, the Runge-Kutta

method of numerical mathematics has been used. This is stated as follows:

4n3n2n1nn1n

KK2K2K6h

(2.96)

where: Kn1=ƒ(zn;Nn)

Kn2=ƒ(zn+h/2 ; Nn+Kn1 h/2)

Kn3=ƒ(zn+h/2 ; Nn+Kn2 h/2)

Kn4=ƒ(zn+h ; Nn+Kn3 h)

The solution of equation 2.88 using equation 2.96 and 2.97a,b,c and d, has been

done using an excel spreadsheet. The full set of calculations showing the

implementation of the Runge-Kutta method on the differential equation 2.88, has

been given appendix A.

(2.97 a,b,c & d)


The solution of equation 2.88 gives a

curve as shown in Figure 2.30. The

shape of this curve is in contrast to

that given by Walker, refer Figure 2.7.

According to the theory given in this

thesis, the maximum vertical pressure

is at the transition from cylinder to

hopper, and not at the base of the

hopper, as suggested by Walker.

Furthermore, the pressure decreases

rapidly with depth in the hopper.

To determine the effect of changing the variables on the solution of equation 2.88,

a sample silo was analysed. In this example, the silo has a height of 7m, a

diameter of 2m, a hopper depth of 2.6m and hopper half angle of 15, a wall

friction angle of 20 and a bulk density of 17kN/m3. The value of the stress ratio, K,

used to determine the vertical pressure at the level of the transition was 0.4. These

results have been shown respectively in Figures 2.31 a,b,c,d and e, for varying

hopper radius, material bulk density, wall friction angle, hopper half angle and the

stress ratio, .

Figure 2.30: Typical vertical pressure curve in a hopper

Vertical Pressure

Dep

th o

f ho

pper

0

2

4

6

8

10

0 20 40 60

r=1r=2r=3

Dep

th (

m)

Vertical Pressure (kPa)

0

1

2

3

4

5

6

7

0 20 40 60 80

=10=15=20

Dep

th (

m)


Figure 2.31 (a & b): Effect of variables on vertical pressure in the hopper.

(a): Changing silo radius (b): Changing material bulk density


In Figure 2.31(a), the silo radius has been varied from 1m to 3m. It can be seen

that at depths equal to one silo diameter, there is an increase in the vertical

pressure. For a radius of 1m, the vertical pressure at a depth of 2m is 3.34kPa,

while for a radius of 3m the vertical pressure at a depth of 6m is4.76kPa. This is an

increase of 42.5%. In Figure 2.31(b), the material bulk density has been increased

from 10kN/m3 to 20kN/m3. At a depth of 1m, there is an increase in the vertical

pressure of 9.27kPa to 18.54kPa for an increase in bulk density from 10kN/m3 to

20kN/m3. Thus when the bulk density is doubled, so too is the vertical pressure.

Since the bulk density of a material can vary with time, moisture content and

source, a range of densities needs to be specified when designing a silo.

In Figures 2.31 c & d below, the effects of changing the stress ratio and the wall

friction angle have been shown. From Figure 2.31 (c) it can be seen that at a depth

of 1m, the vertical pressure is 15.76kPa for a stress ratio of 0.3, while for a stress

ratio of 0.9 the vertical pressure increases to 54.44 kPa. This is an increase of

approximately 245%. Therefore, using the correct stress ratio has a significant

effect on the value of the calculated vertical pressure. From Figure 2.31(d) at a

depth of 1m below the transition, it can be seen that changing the wall friction

angle from 10 to 20 has the effect of decreasing the vertical pressure from

22.53kPa to 15.11kPa. This is a decrease of approximately 33%.

0

1

2

3

4

5

6

7

0 20 40 60 80

=10=15=20

Dep

th (

m)

Vertical pressure (kPa)

0

1

2

3

4

5

6

7

0 20 40 60 80

=0.3=0.6=0.9

Dep

th (

m)


Figure 2.31 (c & d): Effect of variables on vertical pressure in the hopper.

(c): Changing hopper stress ratio (d): Changing wall friction angle


0

1

2

3

4

5

6

7

0 20 40 60 80

=10=20=30


Dep

th (

m)

(e): Changing the hopper half angle

Figure 2.31 (e): Effect of variables

on the vertical pressure in the hopper

From Figure 2.31 (e) it can be

seen that increasing the hopper

half angle from 10 to 30 has

the effect of increasing the

vertical pressure, up to a depth

of 0.75r below the transition.

Beyond this point, the vertical

pressure decreases with

increasing hopper half angle.

2.2.2.2 Walters Pressures in Converging Channels

Walters (1972b) developed his theory for pressures acting on the walls of axially

symmetric hoppers in the same manner as his theory for pressures in vertically

sided silos. Figure 2.32a below shows the lines of the major principle stress in the

hopper when the material starts flowing. The stresses acting on a horizontal

elemental slice have been shown in figure 2.32b.

Figure 2.32: (a) Lines of major principle stresses (b) Elemental slice

D

(a)

D

Area A

Area (A+dA)

v

v + dv

w

w dz

z

(b)


Walters gives the results of the vertical force balance on the elemental slice in the

hopper as follows:

gTanAP

dzdA

A1

dzd

wwvv

(2.99)

where A is the cross sectional area of the element at a depth z from the top of the

hopper, P is the perimeter at that depth, v is the uniform average vertical stress

acting across the slice, w is the shear stress acting along the wall, w is the stress

normal to the hopper wall and is the hopper half angle.

To solve equation 2.99, Walters finds a relationship between the third term in

equation 2.99, (w+wTan), and the vertical stress at the hopper wall, vw, from

the geometry of the Mohr circle given in figure 2.33. As before, the green circle

represents the stresses along the centreline of the hopper and the red circle

represents the stresses at the hopper wall. The stresses at a distance r from the

centre of the hopper have been shown in figure 2.33 by the blue circle. Since

Walters makes the assumption that the horizontal stress, H, remains constant

across the slice, the points P and N can be located on the Mohr circles.

Walters further defines the angle as that angle which the line through point P

makes with the horizontal axis of the graph in figure 2.33.

Let the radius of the circle for the stresses at the wall be denoted by a, then:

w = aSin2D (2.100)

Similarly w = OC + Cw = a/Sinm + aCos2D (2.101)

Therefore, substituting equations 2.101 and 2.100 into the third term of equation

2.99 and rearranging gives the following:

w + w Tan = a ( Sinm Sin2D + ( 1+Sinm Cos2D )Tan ) / Sinm (2.102)

To eliminate the radius, a, from the expression in equation 2.102, Walters finds a

relationship between the vertical shear stress at the wall and the vertical stress at

the wall, as follows:

Let vw = E vw (2.103)

CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION 2.41

where both vw and vw are found from the Mohr circle to be:

vw= a Sin (2 + 2D) and vw= a /Sinm – a Cos(2 +2D) (2.104a,b)

Mat

eria

l yie

ld lo

cus

vw

v

w

m

vr

vw

Wal

l yie

ld lo

cus

Str

esse

s at

th

e ce

ntr

e

x

P

N

H

w

vc

O

2 D

2

w

Str

esse

s at

th

e w

all

Figure 2.33: Mohr circle for stresses in the hopper during discharge

C

a

Str

esse

s at

dis

tan

ce r

fr

om h

op

per

cen

trel

ine

2.42 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS

Substituting 2.104a,b into 2.103 and rearranging gives the expression for E as

follows:

Dm

Dm22CosSin1

22SinSinE

(2.105)

The expression for angle 2D in equation 2.105 is found from the Mohr circle shown

in figure 2.34 as follows:

In triangle MCO, MC = OC Sin (2.106)

In triangle NCO, OC = a / Sinm (2.107)

Therefore, MC = a Sin / Sinm (2.108)

In triangle PCM, = ArcCos(MC/a) (2.109)

Substituting for MC from equation 2.108: = ArcCos (Sin / Sinm) (2.110)

From triangle MCO, +2+2D= /2 + (2.111)

Therefore, 2 + 2D = /2 + -/2 + - ArcCos (Sin / Sinm) (2.112)

The expression for 2 + 2D in equation 2.112 is similar to the expression derived

for 2D in equation 2.92, except w is replaced by . It can be seen from figure 2.34

that as the hopper half angle tends to zero, ie a vertically walled silo, the angle

tends to the wall friction angle w. The method for determining the angles w and

has been given at the end of this section.


vw w

m

vw Wall yield locus

P

H O

2D

2

w

C

a

Figure 2.34: Mohr circle for stresses at the wall

M

N


Now the radius, a, can be eliminated from the expression in equation 2.102 by

dividing (w + w Tan ) by vw to give the following:

Dm

DmDm

vw

ww22CosSin1

Tan2CosSin12SinSinTan

(2.113)

To simplify equation 2.113, Walters adds and subtracts TanSinmCos(2+2D)

from the numerator on the RHS, which after rearranging gives the following:

TanETan22CosSin1

22SinSinTan

Dm

Dm

vw

ww (2.114)

Substituting equation 2.114 into equation 2.99 gives:

gTanEAP

dzdA

A1

dzd

vwvv

(2.115)

Equation 2.115 still cannot be solved due to the unknown term of vw, therefore

Walters defines the relationship between the average vertical stress, v, and the

vertical stress at the wall, vw, as before (refer chapter 2.2.1.1):

vw = Fhv (2.79 repeated)

where Fh is the distribution factor associated with the hopper stresses. To

determine the expression for the distribution factor Fh, Walters defines the

relationship between the average vertical stress, v , and the vertical stress at a

distance, r, from the centreline of the hopper, vr , as shown in figure 2.33.

Figure 2.33 shows a typical

horizontal elemental slice in a

hopper, with the line of average

vertical stress acting on the

element. The line of the real

vertical stress has been shown

arbitrarily by the curved line in

figure 2.33. The vertical stress at

a distance, r, has been shown by

the dotted vertical line.

In Walters paper, the relationship between v and vr has the general form:

r

C L Line of average vertical stress, v

Horizontal elemental slice

Vertical stress,vr at distance r

Figure 2.33. Average and general stress distribution across an elemental slice in the hopper.


wr

0vr2v dzr2

r

1 (2.116)

Walters assumes that the vertical shear stress,vr at a distance r from the centre of

the hopper is a linear function of the vertical shear stress at the wall of the hopper,

vw , as before, so that:

vr/vw = r/rw vr = vw r/rw (2.117)

where r is an arbitrary distance from the centre, and rw is the radius of the hopper

at the wall. Let the radius of the circle through point N be x, then:

x2= vr

2 + (x / Sinm-r)

2 (2.118)

And hence:

m

22vrm

22HH

m2

m CosSinCos

Sinx (2.119)

In the Mohr circle for stresses at the wall, 2+2D < /2 + m. If the hopper half

angle is greater than this limit, mass flow in the hopper can not occur. From the

geometry of the Mohr circle:

v = 2x / Sinm- H (2.120)

Substituting for x from equation 2.119, and for vr from equation 2.117 into

equation 2.120, and re-arranging, gives the following:

2

w2

mm2

m2H

v r/cr1Sin2Sin1Cos

(2.121)

where c=(Tan / Tanm)2

Substituting equation 2.121 for v into equation 2.1116, the average stress across

the slice can be integrated to give the following:

2/3

mm2

m2H

v c11c3

2Sin2sin1

Cos

mm2

m2H Siny2sin1

Cos

(2.122)


where y = 2[1-(1-c)3/2]/3c =

2/3

m2

2

2m

2

Tan

Tan11

Tan3

Tan2

(2.123)

From the Mohr circle in figure 2.32, H=vw/Tan (2.124)

Substituting for vw from equation 2.103 into equation 2.126 gives:

H = E vw/Tan where E is given in equation 2.105 (2.125)

Substituting for H from equation 2.125 into equation 2.122 gives the following:

m2

mm2

vwvCosTan

Siny2Sin1E

(2.126)

Now substituting for E from equation 2.105 into equation 2.126, and re-arranging

gives the following:

h

vwvw

2m

2m

2

mm2

v FSinSin2Sin1Cos

Siny2Sin1Cos

(2.127)

where

mm2

2m

2m

2

HSiny2Sin1Cos

SinSin2Sin1CosF

(2.128)

Substituting for vw from equation 2.127 into equation 2.115 gives the following:

gFTanEAP

dzdA

A1

dzd

vHvv

(2.129)

For a conical hopper:

zTan2DTan4

dzdA

A1 (2.130)

and

zTan2D4

AP (2.131)

Substituting equation 2.130 and 2.131 into equation 2.129, and re-arranging gives

the following:

g1FTanFEzTan2D

4dz

dHH

vv

(2.132)

The equation for the vertical force balance on the elemental slice given in equation

2.132 can now be solved. As before Walters puts this equation into dimensionless

form;

Let Sv=v/(gD) , Z=z/D , Z0= z0/D (z0 = 0 at the top of the hopper)

and let 1FTan

FE2M H

H

(2.133)


Then equation 2.132 can be re-written as:

1MTan21

SMTan2dZ

dS vv

(2.134)

The solution to the linear first order differential equation, 2.134 is:

1M

0v TanZ21

ZTan211

1MTan2ZTan21

S (2.135)

To find the vertical pressure in the hopper, the unknowns in the expressions for E

and M need to be determined.

From equation 2.112 for 2 + 2D , the graph in figure 2.34 has been plotted for

determining the value of 2D and . The curves of material friction angle varying

from 10 to 80 has been shown in increments of 10. To demonstrate the use of

figure 2.34, a curve has been plotted for a silo with a wall friction angle of 22 a

material friction angle of 45 and a hopper half angle of 25. The value of 22 for

the wall friction is entered along the x-axis. The intersection of this vertical line with

the curved line gives a value of 2D ,for this example, on the y-axis as 54. To this

value of 2D a value of 2 is added 2 = 50 which gives 2 +2D as 104.

Figure 2.34: 2D + 2 as a function of w and

(w) or ()

2 D

or

(2 D

+ 2)

0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50 60 70 80

m=10

40

70


The intersection of this horizontal line with the curve gives the value of as 39

along the x axis of the graph.

Having thus found the value of , this can be used in equation 2.128 to evaluate FH

Substituting equation 2.123 for y into equation 2.128 shows that equation 2.128 is

the same as equation 2.84 for F, except that w is replaced by . The graph for FH

is therefore, the same as figure 2.26 for various curves of angle .

All the unknowns in the expression for the average vertical pressure in the hopper

given by equation 2.135 have been defined and the stress distribution in a hopper

without a surcharge can be plotted, as shown by the graphs in figure 2.36. For all

three graphs the scales of the vertical pressure, on the horizontal axis, and the

depth, on the vertical axis, are the same. Figure 2.36a was calculated for a material

friction angle of 50, wall friction angle of 22 and the hopper half angle varied from

5 to 15 in increments of 5.

0

20

40

60

80

100

120

140

0 4 8 12 16 20 24 28 32 36 40 44

2D=54

=39 w=22

+2

2D+2 =104

m=45

2D o

r (2 D

+ 2)

(w) or ()

Figure 2.35: 2D + 2 as a function of w and for a material friction angle, m =45


Similarly figure 2.36b was calculated for a wall friction angle of 22, a hopper half

angle of 10 and the material friction angle varied from 25 to 35 in increments of

5. In figure 2.36c the material friction angle used was 50, the hopper half angle

was 10 and the wall friction angle varied from 15 to 25 in increments of 5.

From figures 2.36a and b it can be seen that a larger hopper half angle and higher

material friction angle both have the effect of lowering the vertical pressure acting

in the hopper, while from figure 2.36c it can be seen that the wall friction angle has

a negligible effect on the vertical pressure in the hopper.

For a hopper with a surcharge pressure due to the cylindrical section above the

solution to equation 2.134 would be as follows:

M

00v

1M

0v TanZ21

ZTan21S

TanZ21ZTan21

11MTan2

ZTan21S

(2.136)

where all variables are as previously, Z0 is the dimensionless depth ratio at the

transition, Sv0 is the vertical pressure acting at the transition due to the material in

Figure 2.36: Average vertical pressure in conical hopper : (a) Effect of changing hopper half angle , (b) Effect of changing material friction angle (c) Effect of changing wall friction angle

Dimensionless Average vertical pressure Sv

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0 0.1 0.2 0.3

=5 =10 =15

m=25 m=30 m=35

w=15 w=20 w=25

Z=

z/D

(a) (b) (c)


the cylindrical section of the silo. The value of Sv0 can be calculated from equation

2.97 (the Walters expression for the vertical pressure in a cylinder).

Figure 2.37 shows the

dynamic average vertical

stress acting in a silo filled

with material with a

friction angle of 50, a wall

friction angle of 22 and a

hopper half angle of 10.

The dotted line in figure

2.37, shows the pressure

distribution in the hopper

with no surcharge acting in

the cylindrical section.

2.2.2.3 The Jenike Radial Stress Field

Jenike (1968) defines the condition of flow as a particular case of failure, which

occurs when pressures are such that shear occurs without destroying the isotropy

of the material. During flow, the bulk density of the material is a function of the

pressures. When the pressures are constant, the solid shears at a constant density.

When the pressures increase, the solid compacts and the density increases, and

when the pressures decrease, the solid expands and the density decreases. Thus,

flow can proceed indefinitely. In the hopper the mass of solid contracts laterally and

expands vertically which implies horizontal, or nearly horizontal, major principal

stresses and a plastic-passive state of pressure exists in the hopper. This state may

extend to the top of the bin. Because the solid slides along the walls as it flows, the

vertical pressure at the wall, vw is accompanied by a frictional stress at the wall w.

There is a change in the wall pressure at the transition from the cylindrical section

Hop

per

0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2

Dimensionless average vertical pressure, Sv

Hei

ght

to d

iam

eter

rat

io,

Z

Cyl

inde

r

Figure 2.37: Dynamic average vertical stress acting in a silo


to the hopper section, with the pressures decreasing to zero at the vertex of the

hopper. The speed of the flowing material is generally sufficiently slow and close to

the steady state condition for the inertia forces to be negligible. Therefore, the

conditions of equilibrium are satisfied and the vertical force supported by the walls

is equal to the weight of the stored material. The element in the hopper is the same

as that shown in figure 2.18 and figure 2.19. The symbols used in this chapter are

the same as those used in chapter 2.1.2.3.2 resulting in the same equations of

equilibrium as previously. These equations have been repeated here for continuity.

Considering equilibrium in the r-direction

02d

CosCot2r1

r1

r rcrrr

(2.56 repeated)

Considering equilibrium in the -direction

02dSinCot3

r1

rcr

r

(2.58 repeated)

In Bulletin 108, Jenike (1961) gives the following relationships from the Mohr circle

as follows:

2CosSin1 m (2.137)

2CosSin1 mr (2.138)

mc Sin1 (2.139)

from the assumption that in axial symmetry the circumferential stress is equal to

either the major or minor principal stress of the median plane.

r = SinmSin2 (2.140)

In Bulletin 108, Jenike(1961) gives the following relationship for the average

vertical stress in the hopper:

v=rs (2.141)

where is the density and is assumed a function of r and as follows: = (r,)

s is the radial stress field and is also function of r and as follows: s=s(r,)

The angle between the major principal stress and the ‘r’ co-ordinate in figure 2.18

is given as ; = (r,)

+

+


The derivative of the shear stress given in equation 2.140 therefore becomes:

rrrrrr

r2CosSin2s

rrs

rsr2SinSin mm

(2.142)

The derivative of the average vertical stress with respect to is as follows:

2SinSin2rssr2CosSin1 mm (2.143)

The derivative of the average vertical stress with respect to r is as follows:

rrrrrr

r

2SinSin2srsr2CosSin1 mm

(2.144)

Substituting equations 2.142 and 2.143 into the equation of equilibrium in the

direction given by equation 2.58 gives the following:

.............r

2CosSin2sr

rsrsr2SinSin mm

.......2SinSin32SinSin2rss

r2CosSin1r1

...... mmm

02dSinSin2CosSinCot

r1...... mm

(2.145)

Now substituting for =rs into equation 2.145 and collecting terms in s/r, s/ ,

s and the remaining terms gives the following:

0DCss

Brs

A

(2.146)

where A = r SinmSin2 (2.147)

B = (1-SinmCos2 (2.148)

.......r

2rCosSin22CosSin1r

r2SinSinC mmm


2Cos1CotSin2SinSin32SinSin2...... mmm (2.149)

D = -Sin (2.150)

Solving for s/r in equation 2.146 gives :

s

AB

sAC

AD

rs (2.151)

Similarly in the r direction, substituting equations 2.142 and 2.144 into the

equation of equilibrium given by equation 2.56, and collecting terms in s/r, s/ ,

s and the remaining terms gives the following:

0HGss

Frs

E

(2.152)

where: 2CosSin1rE m (2.153)

2SinSinF m (2.154)

.......2CosSin2r

r22SinSinr

r2CosSin1G mmm

Cot2SinSinSin12CosSin12CosSin12..... mmmm

(2.155)

CosH (2.156)

Substituting for s/r in equation 2.151 into equation 2.152 and collecting terms in

s/ , s and the remaining terms, gives the following:

0A

EDHsAECGs

AEBF

(2.157)

Due to the lengthy size of equation 2.157, each term has been evaluated

separately. To maintain an overview of the solution, equation 2.157 has been

repeated several times during the following calculations.

The first term of equation 2.157 is as follows:

2SinSinr2CosSin1

2CosSin1r2SinSinAEB

Fm

mmm

2SinSinCos

2SinSin1Sin

.............m

m2

m

m2

(2.158)


Equation 2.157 repeated: 0A

EDHsAECGs

AEBF

..........r

r22SinSinr

r2CosSin1AECG mm

......Sin12CosSin12CosSin122CosSin2...... mmmm

.......

2SinSin2CosSin1

rr2CosSin1Cot2SinSin......

m

2m

2

mm

....2CosSin132CosSin12r

2Cos2Sin

2CosSin1r2.... mm

m

2Sin

Cot12Cos2CosSin1.... m (2.159)

Now multiplying out all the terms , replacing Cos22 with(1-Sin

22), cancelling out,

and after many lengthy algebraic manipulations, the second term of equation 2.157

can be simplified to:

......Sin2Cosr2Sin

r22

2SinSinCos

3SinAECG m

m

m2

m

1Sin12Cos2Sin

Cot...... m

(2.160)

Equation 2.157 repeated: 0A

EDHsAECGs

AEBF

2SinSinSin2CosSin1

CosA

EDH

m

m (2.161)

Solving for s/ in equation 2.157 is as follows:

0

AEBF

AEDH

s

AEBF

AECG

s

(2.162)

As described previously, solving the terms in equation 2.162 is a lengthy process,

equation 2.162 has again been repeated throughout the calculations to maintain an

overview of the solution progress.

To evaluate the first term in equation 2.162, substitute equations 2.160 and 2.158

to give the following:


..........

Cos

2SinSin21

Cos

3Sin2SinSin

AEBF

AECG

m2

m

m2m

m

1Sin12CosCos

SinCotr

Sin2CosCos

rSin2...... m

m2

mm

m2

m

This can be simplified to give the following:

........

rCos

SinSin2Cosr2

Cos

2SinSin21

A

EBF

A

ECG

m2

mm

m2

m

2Cos2Cos1Sin1Cot3Sin2Sin

Cos

Sin..... mm

m2

m (2.163)

This is the same as the f(r,) term given in Jenike’s Bulletin 108 (1961).

Equation 2.162 repeated: 0

AEBF

AEDH

s

AEBF

AECG

s

To solve the second term in equation 2.162, substitute equations 2.161 and 2.158

to give the following:

2SinSin

2CosSin12SinSin

2SinSin

Sin2CosSin1Cos

A

EBF

A

EDH

m

2m

2

m

m

m

which can be simplified to give:

m

2m

2m

Cos

Sin2Sin

Cos

Sin

A

EBF

A

EDH

(2.164)

This is the same as the g(r,) term given in Jenike’s Bulletin 108 (1961)


Thus, equation 2.162 has the form:

0),r(gs),r(fs

Therefore, (s/ f(r,)s - g(r,) (2.165)

where f(r,) is given by equation 2.163 and g(r,) is given by equation 2.164

Having solved for s/, equation 2.165 can be substituted into equation 2.151 to

solve for s/r as follows:

D

EBFAA

EDHABsC

EBFAA

ECGAB

r

s (2.166)

As previously, the evaluation of both terms in equation 2.166 is a lengthy process

and equation 2.166 has been repeated throughout the following calculations to

maintain an overview of the solution progress.

Evaluating the first term in equation 2.166 :

.........

Cos

2SinSin21

2rSin

2Cos

2SinSinr

1C

EBFAA

ECGAB

m2

m

m

......3Sin2SinCos

Sin

rCos

SinSin2Cosr2........ m

m2

m

m2

mm

......

rr2SinSin2Cos2Cos1Sin1Cot

Cos

Sin....... mm

m2

m

......2SinSin3r

2rCosSin22CosSin1........ mmm

2Cos1CotSin2SinSin2...... mm

This can be simplified by multiplying out all the terms and then collecting terms in

/r, /, /r, / and the remaining terms. After considerable mathematical

manipulation the result is as follows:


......r

2SinCos

Sin2Sin2Cos

Cos

Sin1

r

2

r

1C

EBFAA

ECGAB

m2

mm

m2

m

12Cos2SinCotSin1Cosr

Sin

r

1..... m

m2m

(2.167)

Equation 2.166 repeated:

D

EBFAA

EDHABsC

EBFAA

ECGAB

r

s

To evaluate the second term in equation 2.166, substitute equations 2.147, 2.148,

2.150, and equation 2.163 as follows:

Sin

Cos

Sin2SinmCos

Sin

2SinSinr

2CosSin1D

EBFAA

EDHAB

m22

m

m

m

Multiplying out, cancelling and collecting terms, the above equation can be further

simplified to give:

m2

m2

m

Cos

Cos2Cos

Cos

Sin

r

1D

EBFAA

EDHAB (2.168)

Both equations 2.167 and 2.168, can be multiplied throughout by the radius, r, of

the hopper. This then gives the final expressions as found in Bulletin 108.

Finally the two differential equations have now been defined as follows:

0),r(gs),r(fs

(2.169)

0),r(js),r(hr

sr

(2.170)

where f(r,) is given by equation 2.163, g(r,) is given by equation 2.164.

CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER 2.57

......r

2SinCos

Sinr2Sin2Cos

Cos

Sin121),r(h

m2

mm

m2

m

12Cos2SinCotSin1Cos

Sin

r

r..... m

m2

m

(2.171)

m

2m

2m

Cos

Cos2Cos

Cos

Sin),r(j

(2.172)

Jenike further simplifies his calculations by making the assumption that is only a

function of and that the density g is a constant. Then the terms /r, / /r

become zero and equations 2.169 and 2.170 become:

0)(gs)(fs

(2.173)

0)(js)(hr

sr

(2.174)

Both equations 2.173 and 2.174 are first order linear partial differential equations.

where .........3Sin2SinCos

Sin

Cos

2SinSin2)(f m

m2

m

m2

m

2Cos2Cos1Sin1Cot

Cos

Sin....... m

m2

m (2.175)

m

2m

2m

Cos

Sin2Sin

Cos

Sin)(g

( as before ) (2.164 repeated)

12Cos2SinCotSin1Sin2CosCos

Sin121)(h mm

m2

m

(2.176)

m

2m

2m

Cos

Cos2Cos

Cos

Sin)(j

( as before ) (2.172 repeated)

In Bulleting 108, Jenike(1961) gives the solutions to the differential equations given

in equations 2.173 and 2.174. Then Jenike assumes s/r to be equal to zero. This

simplifies the solution process considerably to give only two unknowns in equations

2.174 and 2.175. These unknowns are / and s/ .


The unknowns / and s/ can easily be solved for. Equation 2.174 now

becomes:

h()s = -j() (2.177)

Multiplying equation 2.177 throughout by Cos2m/Sinm and dividing throughout by

2s, re-arranging and collecting terms ,gives the following expression for /as

follows:

.............

Sin2CosSins2

Coss2CosSinCos

mm

m2

m

1

Sin2CossSin2

12cos2SinCotSin1sSin...........

mm

mm

(2.178)

Equation 2.178 represents the

variation of with respect to along a

given ray of radius, r as shown in

figure 2.38.

On the axis of the hopper, equals

zero and equals 90. At some

arbitrary distance in the hopper, when

varies from 1 to 2, varies from 1

to 2 on the same ray. Thus a family of

solutions can be plotted for equation

2.178 for a range of material friction

angles, m, and stress field values of s.

Substituting for / from equation 2.178 into equation 2.173 and solving for s/

gives the following:

m

m

Sin2Cos

2Sin2Cos1CotSins2Sin2Sinss

(2.179)

Equations 2.178 and 2.179 can be solved using numerical methods and applying

the physical boundary conditions of = 0 on the axis of the hopper, and = at

the wall.

1

2

1

2

r

Figure 2.38: Variation of with respect to within the hopper.


The solution to equation 2.179 is substituted into the expression for the average

vertical stress given by equation 2.141, to give a radial stress field as follows

v = r s() (2.180)

In Bulletin 108, Jenike(1961) considers the

general case of a hopper as shown in figure

2.39, where the walls of the hopper have

different friction angles, the slopes of the

hopper walls are not the same on either side of

the axis and the wall friction angle approaches

the material friction angle, ie rough walls. In

this thesis only the solution of an axially

symmetric hopper (ie one hopper half angle)

and a wall friction angle less than the material

friction angle has been considered.

Since the density in equation 2.180 does not equal zero, the radius does not equal

zero and the stress field s() does not equal zero, the radial stress field given by

equation 2.180 cannot extend upwards to a free surface. Therefore, the stress field

in a hopper without a surcharge deviates significantly from a radial stress field in

the upper part of the hopper.

The solution of equation 2.178 for can be plotted on a graph with as the vertical

axis and as the horizontal axis. Jenike (1961) states that the boundary conditions

of equation 2.178 are not mathematically uniquely defined, but can be determined

from the physical boundary

conditions. There are two

physical boundary conditions

imposed on the equation for

/which are symmetrically

located as shown in figure 2.40.

On the axis of the hopper, =

/2 and = 0; and at the wall

= . Since there is no direct

way of finding a solution which

connects two boundary points,

Jenike has computed several

solutions and interpolated the

required functions.

Figure 2.39: General Hopper geometry

1 2

w2w1

(0,/2)

(0,)

(0,0)

Figure 2.40: General shape of the function / in (,) co-ordinates

(,)

(-,)


Having thus solved for the radial stress field, Jenike derives the normal wall

pressure acting on the hopper during flow from equation 2.137, which has been

repeated below:

2CosSin1 m (2.137 repeated)

In the derivation of N, the normal pressure

on the hopper wall, Jenike uses the (x,y) co-

ordinates as shown in figure 2.41 below.

The hopper has a diameter, D, at the top, a

width of B=2y at some distance –x above

the vertex, and a hopper half angle of .

The polar co-ordinates used in the derivation

of the radial stress can be expressed in

terms of (x,y) as follows:

r = Sin(B/2) at the wall. (2.181)

Substituting equation 2.181 into equation 2.180 and in turn substituting this into

equation 2.137 gives the following expression for the normal stress acting on the

wall:

2CosSin12BSins mN )(

which Jenike has written in the following form:

2CosSin1

2Sins

B mN )( (2.182)

Putting equation 2.182 as a function of the depth of the hopper gives the following

equation for the normal stress acting on the hopper wall:

2CosSin12

SinHxs mh

N D)( (2.183)

where B has been replaced with Dx/Hh . In equation 2.183 above, the term x/Hh is

the depth within the hopper as a ratio of the hopper height. From equation 2.183 it

can be seen that N is a linear function of the radial stress s(), the bulk density of

the material, , the diameter of the hopper at the transition, D, and the depth ratio,

x/Hh. The graph in figure 2.42 shows the hopper normal wall stress, and has been

plotted for s() = D = =1.

x

y

D

B

N -x

Figure 2.41: Co-ordinate system in the hopper

Hh


Figure 2.42(a) shows the effect on the value of N , of varying the hopper half angle

from 25 to 15, for a material friction angle of 45and an angle of 60. Using the

same scale, figure 2.42(b) shows the effect of changing the material friction angle

from 45 to 25 (=60 as in figure (a) and =15). The third curve in figure

2.42(b) shows the effect of changing the angle from 60 to 45 (m=45 as in

figure (a) and =15).

From equation 2.183, the greatest effect on the hopper normal wall pressure is due

to the radial stress field s(), the material bulk density, , and the diameter of the

hopper at the transition, D. The graphs in figure 2.42(a,b) show that the hopper

half angle has a greater effect on the normal wall pressure compared to the

material friction angle and the angle of the major principal stress, .

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 0.2 0.3

Dep

th r

atio

(x/

Hh)

=15

=25 =25 m=25

Normal wall pressure N (kPa) Normal wall pressure N (kPa)

Figure 2.42: Normal wall stress N, acting in the hopper during flow

(a) (b)


2.3 SWITCH PRESSURES


2.3.1.1 Jenike Upper Bound Pressures

The pressures exerted by a solid stored in a silo are affected by the wall

imperfections of the silo (ie deviations from cylindrical) and the boundary layers of

material which form at the walls. Therefore, Jenike only considers the bounds on

wall pressures, the minimum being described by Janssen (equation 2.5) and the

critical upper bound which Jenike has based on the considerations of strain energy

of the stored material..

During flow of a mass solid, Jenike(1973b) states that the energy is lost at the

maximum rate possible. Therefore, the recoverable strain energy tends towards a

minimum, which is approached as closely as the wall imperfections will allow. Using

Janssen’s assumption of the stresses being independent of the horizontal co-

ordinates, Jenike solves for the strain energy in the cylinder in one dimension. He

neglects the strain energy due to shear stresses and assumes the cylinder walls to

be rigid. Hence in his analysis, the vertical co-ordinate, z, is the independent

variable. Figure 2.43 shows a typical horizontal element in a cylinder, of cross

sectional area A.

In his derivation, Jenike assumes the vertical

pressure,V , to be the major principal stress,

the horizontal pressureH, is the minor principle

stress and the circumferential pressure,C, is

the intermediate stress. Therefore:

1 = V = RS (2.184)

2 = 3 = C = H = KRS (2.185)

where K=H/V is the stress ratio.

Since Jenike(1973b) does not give a definition of the symbol S, it has been

assumed in this thesis that S has been used to denote a stress field. Furthermore,

while equation 2.184 appears to be similar to equation 2.63, the symbol R denotes

the hydraulic radius of the cylindrical section and not the radial co-ordinate as

previously. There is no explanation for this change of symbols in either Bulletin 108

(Jenike 1961) or in the paper relating to the switch pressure, Jenike (1973b)

dz

v

℄

z

Figure 2.43: Horizontal Element in the cylinder

CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION 2.63

The recoverable strain energy in an element of thickness dz is given by:

0

V

0

H

HHVV d2ddzAU (2.186)

The recoverable parts of the energy expression given in equation 2.185 are as

follows:

E

d

E

d1d,

E

d2

E

dd VH

HHv

V

(2.187a,b)

In equation 2.187, Young’s Modulus of the stored solid, E, and Poison’s ratio, , are

assumed constant.

Substituting equations 2.187a,b into 2.186 gives the following:

z

0

z

0

dzRSRSK1E1RSK2dzRSK2RS

E1RSAU

Collecting terms gives:

dzSK12K41E

RAU

2z

0

22

(2.188)

As the switch proceeds up into the cylindrical section, it is located at some arbitrary

level, z0, as shown in figure 2.44. Above this level the stress ratio K is equal to the

Janssen stress ratio, and below this level the stress ratio varies and therefore is a

function of S.

Vertical Pressure

Dep

th

Janssen static pressure z0

Switch pressure

Cyl

indr

ical

silo

Figure 2.44: Location of the switch pressure during flow

Static pressures: K=KJanssen

Dynamic pressures: K varies


Jenike (1973b) assumes that the switch propagates slow enough so that

acceleration terms are negligible, and therefore the equation of equilibrium of a

horizontal elemental slice as shown in figure 2.43 is as follows:

01R

Kdz

d1 vv

(2.189)

The derivative of equation 2.184 is as follows:

dzdSR

dz

d v

(2.190)

And re-arranging equation 2.184 in terms of S gives: S = v/R (2.191)

Substituting equations 2.190 and 2.191 into equation 2.189 gives the following

differential equation in terms of the stress field S:

R1S

RK

dzdS

(2.192)

Equation 2.192 integrates to: K

e1SR/Kz

(2.193)

Substituting the expression for S given by equation 2.193, into the equation for the

strain energy, given by equation 2.188, gives the following;

......X......K12K412

K1U

2

3

R/0Kz2R/0Kz0 e121e12

R

zK........ (2.194)

Re-arranging equation 2.192 gives K as a function of the stress field S, as follows :

SdzdSR1

K

(2.195)

By applying variational calculus to equation 2.188 for strain energy, the minimum

energy can be obtained by letting U = 0.

dzSKSK14SKSSSK4SS2ERA

U

h

o

2}{

(2.196)

Both S, (S+S), and SK, (SK+SK) must satisfy equation 2.195, which is the

equilibrium equation re-arranged. Substituting into equation 2.195 gives:


RdS/dz + SK = Rd(S+S)/dz + (SK+SK) (2.197)

Cancelling out SK and RdS/dz and re-arranging equation 2.197 gives SK as follows:

dz)S(dRSK

(2.198)

Substituting equation 2.198 into 2.196 and integrating by parts gives the following:

.............SKS14S4RERA

Uh

0z

2

dzKS14S4dzdRK21S2S..........

h

0z

(2.199)

There is a boundary condition on S such that at z=z0, S=SJ (the Janssen static

pressure) and therefore no variation in S is admissible.

At z=h, any value of S is possible to force ΔU to be zero for any value of S.

Therefore: KS14S4 at z = h must always hold. (2.200)

Thus Jenike(1973) has set the second term in the integral sign of equation 2.199

equal to zero. Jenike re-arranges equation 2.200 to give an expression for the

stress ratio in the material as follows:

K = /(1-) (2.201) Jenike then states that the integrand in equation 2.199 must vanish to zero for any

admissible value of S , and since S is arbitrary, this requires that:

KS14S4dzdRKS4S2

in the range z0< z < h (2.202)

Substituting for K from equation 2.195 into equation 2.202 gives the following:

SR114S4dzdRSR14S2

Cancelling terms and dividing throughout by 2 gives:

SR122S2

2

(2.203)

Jenike now introduces a new co-ordinate system such that:

12R

zzx 0 (2.204)


Then a general solution to equation 2.203 is as follows:

S= Aex + Be

-x + 2/ (2.205)

where A and B are constants determined from boundary conditions.

From the boundary condition at z=z0 ; S=S0 it follows that x =0 and substituting

this into equation 2.205 gives:

A + B = S0 – 2/ (2.206)

At the bottom of the silo: z=h and x=X; and from equation 2.201: K=/(1-) gives

the following solution for A:

xx

1x

0

e1KMe1KM

KNMeNS1KMA

(2.207)

where M = √(2(1-)) and N = 2/ (2.208a,b)

Substituting equation 2.207 for A into equation 2.206 gives the expression for the

constant B.

Three curves of the switch pressure envelope, as given by equation 2.205, have

been shown in dotted lines in figure 2.45 for Poisson’s ratio, 0.3. The solid

curves are the Janssen static pressures in the silo for the same H/D ratios and wall

friction angles. The first two curves were calculated for a H/D ratio of 10(D=1) and

5(D=2), both with a wall friction angle of 20. The third curve was calculated for a

H/D ratio of 10, but varying the wall friction angle to 15. Also shown in figure 2.45

is the Janssen horizontal static pressure distribution in dimensionless form, ie

H/D, for a static stress ratio of K=0.4. The values of the switch pressure at three

different levels in the silo (ie 1.5, 3.5, 5, and 7m) have been given as a multiple of

the static horizontal pressure next to the graph for all three curves.

The graph of Jenike’s upper bound of switch pressures shows that the envelope of

switch pressures tends towards an asymptote, and the greatest changes in

pressures occur in the top half of the cylindrical section. It can also be seen that the

switch pressure varies from approximately two to four times the static pressure

value. The greatest horizontal pressures acting on the wall of a silo during flow

would therefore occur on a very tall silo with a low wall friction angle .


As the switch reaches the top of the cylindrical section the stress field, S, tends to a

minimum. From equation 2.195 this implies that the stress ratio, K, approaches

infinity, which makes Jenike’s analysis less reliable for positions close to the top of

the silo. Jenike states that the switch pressure stops at a height approximately one

silo diameter below the top, which Jenike states has been observed experimentally.

Although the expression for the switch pressure does not give an area of influence,

Jenike states that the maximum pressure acts over an area of approximately one

third the diameter of the silo, (D/3). Jenike emphasises that the curves given in

figure 2.45 are only an upper bound to the wall pressures which can be expected

during flow of the material. Small deviations in the shape of the cylinder as well as

thin boundary layers of material on the wall, will reduce the maximum pressure.

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2Dimensionless horizontal pressure

Dep

th H/D=10; w=20

H/D=5; w=20

H/D=10; w=15

S=3.10SJH S=4.01SJH S=3.53SJH




Figure 2.45: Jenike switch pressure envelope


2.3.1.2 Walters Switch Pressure

Walters(1973) considers an instant in time during the discharge of the silo when

the switch pressure is at a height z below the top of the silo as shown in figure

2.46.

Walters assumes that above the switch, at depth h, the stress field is undisturbed

and therefore the static pressure field applies. Below the switch, dynamic pressures

exist with a surcharge pressure equal to the static pressure at that point. Below the

switch the dynamic pressures are given by the differential equation as follows:

vv

DBF4

dzd

(2.95 repeated)

where F is given by equation 2.84 and B is given by equation 2.93.

With a uniform surcharge pressure acting above the level of the switch, the limits of

integration for equation 2.95 become =J (the Janssen static pressure) at z=z0.

The solution to equation 2.95 then becomes:

D/BFz4

J

D/BFz4

Vee1

BF4D

(2.209)

In dimensionless form: BFZ4

J

BFZ4

VeSe1

BF41S

(2.210)

where S=/D and Z=z/D as before

2

1

Figure 2.46: Location of the Switch Pressure according to Walker

Dep

th

Horizontal wall pressure

h

Janssen static pressure

Walters dynamic pressure

Switch pressure


Therefore, below the switch, at depth Z h, the pressures are as follows:

hZBF4

J

hZBF4

VeSe1

BF41S

(2.211)

The normal wall pressures at the level of the switch are determined as follows:

SH = BF SV/Tanw = BF SV/ for dynamic pressures (2.212)

and SJH=K SJ= (1-e-4 K z / D)/(4) for static pressures (2.213)

Multiplying equation 2.211 by equation 2.212 and substituting equation 2.213 for SJ

gives the following expression for the horizontal wall pressures:

hZBF4

JH

hZBF4

HeS

KBFe1

41S

(2.214)

At the level of the switch, Z=h, and the first term in equation 2.214 becomes zero

Therefore, at the level of the switch, Walters gives the horizontal pressures as:

JHHS

KBFS

(2.215)

which is simply the ratio (BF/K) multiplied by the static horizontal pressures.

Figure 2.47 shows the ratio (BF)/(K) for a Janssen stress ratio of 0.4, for material

friction angles varying in increments of 20.

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Figure 2.47: Walters Switch Ratio for horizontal wall pressures: for material friction angle varying from 20 to 80

m=80

Wall friction angle w)

Rat

io (

BF)

/(K

)

m=60


Equation 2.214 gives the height over which the switch pressure acts from the term

Z-h in the exponent. By changing the co-ordinate system as shown in figure 2.48

below, the distance over which the switch acts can be determined. The horizontal

axis is at the level of the switch and the vertical axis lies on the dynamic pressure

graph.

For the new co-ordinate system, as SH tends to zero the value of x can be

determined as follows:

xBF4

JH

xBF4eS

KBFe1

410

(2.216)

Multiplying equation 2.216 by 4 and re-arranging gives the following:

A=e-4BFx (2.217)

where 1

JHS

KBF1A

(2.218)

The solution to equation 2.217 is n (A) = -4BFx and therefore x is as follows:

x = n (A) / (-4BF) (2.219)

Equation 2.219 depends on the factors B and F, which in turn, are functions of the

material friction angle and the wall friction angle. Therefore, the depth over which

the switch pressure acts varies for each silo.

Figure 2.48: New co-ordinate system to determine height of switch

Dep

th

Horizontal wall pressure

x

Janssen static pressure

Walters dynamic pressure

Switch pressure

0


Figure 2.49 shows the switch pressure at various levels in a tall silo with a wall

friction angle of 20 and a material friction angle of 40. The silo height to diameter

ratio is 10:1. The pressures have been plotted in dimensionless form according to

equation 2.97 and 2.98 for the dynamic pressures and equation 2.6 divided by D

for the Janssen static pressure. The switch pressures have been plotted using

equations 2.214 and 2.215. The calculations were done for the switch at level 1.5,

3.5, 5 and 7m below the top of the cylinder. As can be seen from figure 2.49, the

maximum value of the switch pressure decreases from 5.2SJH to 3.2SJH as the

switch moves up the cylinder. However, the area over which it acts remains

constant, in this example the area of influence of the switch is approximately 1.0

silo diameter.

Figure 2.50 shows the switch pressure in a silo with a height to diameter ratio of 5.

The wall friction angle, the material friction angle and the levels of the switch are

the same as in the previous example . From figure 2.50 it can be seen that the area

of influence of the switch is still approximately 1 silo diameter. However, the

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6Dimensionless horizontal pressures

Dep

th

Static pressures (SJH)

Dynamic pressures

Switch pressures (SH)

Locus of switch

Figure 2.49: Walters Switch pressure at various levels in a silo of H/D=10

SH = 5.2 SJH

SH = 5 SJH

SH = 4.6 SJH

SH = 3.1 SJH


maximum values of the switch pressure have decreased by 11.5%=(5.2-4.6)/5.2 at

the lowest level to 39%=(3.08-1.87)/3.08 at the highest level.

Figures 2.51 and 2.52 show the switch in a silo with a H/D ratio of 10, as in figure

2.49. However, the material friction angle has been varied from 40 in figure 2.49,

to 25 in figure 2.51, while the wall friction angle has been varied from 20 in figure

2.49 to 15 in figure 2.52.

By reducing the material friction angle by 37.5%, the magnitude of the switch

pressures is decreased by approximately 44%. A reduction in the wall friction angle

of 25% has the effect of increasing the switch pressure by 179% at the lowest level

to 136% at the highest level in the silo.

Therefore, using the Walters equation for determining the switch pressures during

flow, the worst case pressures would occur in a silo with a high H/D (height to

diameter) ratio combined with a high material friction angle and a low wall friction

angle. A “short” silo with a rough wall would experience lower switch pressures, but

consequently may not undergo mass flow of the material.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5

Static pressures (SJH)

Dynamic pressures

Switch pressures (SH)

Locus of switch

SH = 4.6 SJH

SH = 4.1 SJH

SH = 3.38 SJH

SH = 1.87 SJH

Dimensionless horizontal pressures

Dep

th

Figure 2.50: Walters Switch pressure at various levels in a silo of H/D=5


0

1

2

3

4

5

6

7

8

9

10

0 1 2 3

Static pressures (SJH) Dynamic pressures Switch pressures (SH) Locus of switch

SH = 2.89 SJH

SH = 2.78 SJH

SH = 2.56 SJH

SH = 1.71 SJH


Dep

th

Figure 2.51: Walters Switch pressures in a silo of H/D=10, m=30, w=20

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8

Static pressures (SJH) Dynamic pressures Switch pressures (SH) Locus of switch

SH = 8.08 SJH

SH = 7.51 SJH

SH = 6.61 SJH

SH = 4.03 SJH


Dep

th

Figure 2.52: Walters Switch pressures in a silo of H/D=10, m=40, w=15



2.3.2.1 Jenike Switch Pressures in the Hopper

The material starts to flow at the instant the discharge gate of the silo is opened.

vertical support of the solids has been removed and the material above the outlet

starts to expand vertically downwards. This reduces the vertical pressure within

the material and causes a change from a static to a dynamic stress field. The major

principal stresses now arch across the outlet of the silo. As more material expands

the region of flow extends upwards into the hopper and the switch travels upwards.

Jenike(1969) states that this change in stress fields results in a deficiency in the

wall support during flow. Figure 2.53 shows an instant when the switch is at level z

above the vertex of the hopper. Above the switch the material is still in the static

state while below the switch the material is in the passive state of stress. The

shaded volume of solid between the two stress fields does not belong to either, but

is in transition from an active to a passive state. The area under the dynamic

pressure curve represents the total weight of the solid which has not changed

significantly compared to the curve for the static pressures. Therefore, there is a

deficiency in the wall support as shown by the shaded area between the static and

dynamic pressure curves. Equilibrium is maintained by a switch pressure which is

exerted on the walls of the hopper and travels upwards from the discharge gate to

the transition, where it can become locked in position, or move up into the

cylindrical section of the silo. The force of the switch pressure is equal to the

shaded area under the pressure curve.

Static pressures Dynamic pressures Switch pressure

Deficiency in wall support

Dep

th

Pressure

z

Figure 2.53: Deficient wall support in the hopper during flow

t N

J

CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: HOPPER SECTION 2.75

Since both the static and dynamic curves are known, Jenike (1968) states that the

magnitude of the concentrated force needs to be superimposed over the dynamic

pressures to obtain the envelope of the design pressures for flow conditions.

Jenike(1969) assumes that the switch pressure has a triangular distribution acting

normal to the hopper wall over a depth of 0.3D parallel to the hopper wall as shown

in figure 2.54. During flow, the mass of the material in the hopper remains constant

for both static and dynamic conditions. When the switch is located at the transition,

the deficiency in wall support is due to the difference between the static hopper

pressures, t , and the dynamic hopper pressures, N. This difference in pressure

(t-N) is also equal to the difference between the initial surcharge acting on the

hopper given by the Janssen pressure, J , and the radial flow pressure, , as


The vertical components of the switch pressure, SZ and the shear stress, SZ at

level zs from the vertex of the hopper, acting on an elemental slice of thickness dz,

given in figure 2.54, are as follows:

SinSZ

and CosSZ (2.220)

In figure 2.54 the change in pressure during flow varies from 0 at level zb to a

value of S , at level h. Therefore, SZ at a level, zs, can be given by:

b

bsSSZ zh

zz

(2.221)

Figure 2.54: Switch pressures acting on the hopper wall.

h zs zb

SZ

SZ 0.3D

J

S

Pressure

Dep

th

dz

t


Therefore, the additional force due to the switch acting normal to the wall can be

given by:

h

bz

SZ

h

bz

SZhJ CosdzPCos

CosdzPSinA (2.222)

where Ah is the area of the hopper at the level of the transition and is given by:

Ah = r2= h

2 Tan

2 (2.223)

and P is the perimeter of the elemental slice given by:

P = 2r = 2zTan (2.224)

Note that in the integration of the area under the curve, Jenike has ignored the

error as shown by the red shaded area in figure 2.5, due to the approximation of

the area as a right angle triangle.

The shear stress along the wall is given by SZ = SZTanw (2.225)

Substituting for SZ from equation 2.221, SZ from equation 2.225, and for Ah and

P given by equations 2.223 and 2.224, into equation 2.222 gives the following:

h

bz

bs2

sb

sw

22J dzzzz

zhTanTanTan2Tanh (2.226)

In equation 2.226, h-zb= 0.3D Cos . Therefore, integrating equation 2.226 and

cancelling out Tan, gives:

h

bzb

2s

3sSw

2J zz

21z

31

CosD3.01TanTan2Tanh

3

h

zzh2h

61

CosD3.0TanTan2

2

2b

b2s

w (2.227)

Equation 2.227 can be simplified to give:

3

h

zzh2

CosD9.0TanTanTan

2

2b

bs

wJ (2.228)

Solving for the additional pressure at the transition due to the switch, S , gives:


3h

zzh2TanTan

TanCosD9.0

2

2b

bw

Js (2.229)

where h is the height of the hopper from the vertex (not the gate), and zb is the

bottom of the area of influence of the switch, refer figure 2.54, and zb =h-0.3DCos

Therefore, all the variables in equation 2.229 are known and S can be calculated.

Jenike then adds this additional pressure at the transition to the dynamic static

pressure in the hopper at the level of the transition as follows:

= t+S (2.230)

where t is given by equation 2.53 in chapter 2 2.1.2.3.1 and S is given by

equation 2.229 above.

The shape of the curve for the switch pressure acting at the transition between the

cylindrical and hopper sections, is as is shown in figure 2.54.


2.3.2.2 Walters Switch Pressures in the Hopper.

In his derivation of the switch pressures in the hopper, Walters(1972) adopts the

same approach as for the cylindrical section given in chapter 2.3.1.2. Figure 2.55

shows the lines of the major principle stresses above the switch and below the

switch during flow of the material when the discharge gate of the silo is opened.

Therefore, below the switch, the dynamic pressures are given by equation 2.136 for

a hopper with a cylindrical section above, resulting in a surcharge pressure acting

at the level of the transition.

M

00

1M

0v TanZ21

ZTan21STanZ21

ZTan211

1MTan2ZTan21

S

(2.136 repeated)

where 1FTan

FE2M H

H

(2.133 repeated)

mm

2

2m

2m

2

HSiny2Sin1Cos

SinSin2Sin1CosF

(2.128 repeated)

Dm

Dm22CosSin1

22SinSinE

(2.105 repeated)

In equation 2.136, Z0=Zs the level of the switch, and the variable Z now starts from

below the switch. (Note Z=z/D). S0 in equation 2.136 is the Janssen vertical

pressure in dimensionless form, ie SJ=v/D. However, to determine the dynamic

vertical pressure below the switch, at depth Zs, the value of S0 should be replaced

with the static value of the vertical pressure in the hopper, Sv.

Dynamic pressures

Static pressures

Switch

zs

Figure 2.55: Switch pressure at depth z from the transition.


Walters(1972) gives the expression for the dimensionless normal wall stress as a

function of the dimensionless average wall stress as follows:

SN=EFHSv/Tanw (2.231)

This was determined from the general expression (in dimensionless form):

T=TanwSN , where T is the dimensionless wall shear stress = w/D

Therefore, SN= vw/D = T/Tanw (2.232)

In chapter 2.2.2.1 it was shown that Walters(1972b) relates the wall shear stress

vw to the vertical pressure at the wall,vw, by equation 2.103. Substituting

equation 2.103, in dimensionless form, into equation 2.232 gives:

SN= ESvw/Tanw (2.233)

where Svw is the dimensionless form of the vertical wall pressure acting at the wall.

Walters then relates vw to the average vertical pressure acting across the hopper

slice by equation 2.127. Substituting the dimensionless form of equation 2.127 into

equation 2.233 results in equation 2.231

Substituting equation 2.231 into equation 2.136 gives the dimensionless normal

pressure during flow of the material as follows:

M

0v

w

H1M

0v

w

HN TanZ21

ZTan21STan

EF

TanZ21ZTan211S

1MTanTan2

)ZTan21(EFS

(2.234)

The envelope of the switch pressure is found by letting Z=Z0 in equation 2.234.

Since the term (1-2ZTan)/(1-2Z0Tan) becomes equal to 1, the first term equals

zero, and equation 2.234 becomes:

SN=EFHSv/Tanw (2.235)

where Sv is the dimensionless form of the static vertical pressure in the hopper.

Substituting the dimensionless form of the Walters equation for the static normal

wall pressure, into equation 2.235 results in the following expression:

NSSH

DHN S

)EF(

)EF(S (2.236)


where (EFH)D is the dynamic value of the constants E and FH ; and (EFH)S is the

static value of the constants E and FH. This ratio is as follows:

Since the variables in equation 2.237 are all constant for a given hopper geometry,

the ratio (EFH)D/(EFH)S for the envelope of the switch pressures is also a constant.

The shape of the switch pressure envelope is therefore dependant on the shape of

the static pressure curve.

In his paper, Walters substitutes the following expression for 2+2 in equation

2.237:

2+2 = /2 + ArcCos(Sin/Sinm) (2.112 repeated)

where the +ve sign refers to static conditions and the –ve sign refers to static

conditions.

The value of can be determined as shown in figure 2.35 of chapter 2.2.1.1.

Walters states that the ratio given in equation 2.237 gives a value of 3.23 for a

material friction angle, m=50, a wall friction angle, w=25 and a hopper half

angle, =4. However, this was checked on a spreadsheet, where the value of

was solved by trial and error, and found to be 28.279 for dynamic conditions and

1.582 for static conditions. These values of in equation 2.237 give the following

numerical values:

(FH)D=1.620 ; (E)D=1.012 and (FH)S=1.000 ; (E)S=0.004

Substituting these values in equation 2.237 gives a switch pressure ratio of 447.8,

which is excessively large. The spreadsheet was checked several times by

comparison with hand calculations and found to be correct. It is therefore

suggested in this thesis that the switch pressure ratio as given by Walters in

equation 2.237 is not reasonable.

(2.237)

mm2

2m

2m

2

Dm

Dm

ySin2Sin1Cos

SinSin2CosSin1

22CosSin1

22SinSin

mm2

2m

2m

2

sm

sm

ySin2Sin1Cos

SinSin2CosSin1

22CosSin1

22SinSin


WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3.1

CHAPTER 3

WALL PRESSURE MEASUREMENTS



3.1.1.1 Cylindrical Section

Bishara et al (1981) undertook finite element (FE) analyses of the pressures in the

cylindrical section of a silo of 7.3m internal diameter and 24.4m tall (H/D=3.3). The

material used in their simulations was a granular cohesionless sand. They state that

the FE horizontal pressures were shown to be 10% larger than the calculated

Janssen horizontal pressure. However, on closer inspection of their graphs for the

horizontal pressure, the amount by which the FE solution is larger than the Janssen

formula, varies from 28% at 1.2 diameters from the top, to 11% at 2.6 diameters

from the top. This is higher than they have reported.

From the graphs of their results for the vertical pressures, the finite element

solution gives values higher than Janssen for a depth of 0 to 1.4 diameters, while

below this level the Janssen formula gives higher values. At full depth, the Janssen

formula gives a 26% higher vertical pressure than the finite element solution while

at a depth of 0.9 diameters, the Janssen formula gives a 14.6% lower vertical

pressure than the FE solution. They reported that the distribution of the vertical

pressure across a horizontal plane was about 50% higher in the centre of the silo

than in the vicinity of the walls.

The stress ratio Bishara et al (1981) used in their calculations was not given, so it

has been interpolated from their graphs for the purposes of this thesis. The stress

ratio used in their calculation of the Janssen horizontal pressure, was found to be

approximately 0,4 taken from three points: (9m depth: 47/120=0.39, 14m depth:

43/106=0.41, 19m depth: 33/84=0.39). The graphs from their finite element

results suggest a stress ratio of approximately 0.51 taken from the same three

points. (53/103=0.51, 50/97=0.51, 46/87=0.53)

Suzuki et al (1985) conducted tests on small and medium sized model silos. The

dimensions were 0.3m internal diameter by 1.7m tall for the small model

(H/D=5.7), and 1.4m internal diameter by 6.4m tall for the medium sized model

(H/D=4.6). The test material used in the small model was Milo(™) and in the

medium model tests were done using Milo(™), maize, soybean meal and alfalfa


meal pellets were used. They report that their test results from both models

compare favourably with the Janssen horizontal pressure, particularly at depths of

less than half the model diameter (ie in the upper part of the silo). From their

graphs of the measured horizontal pressure in the smaller model, their plotted test

results are 6% , 25% and 37% less than the Janssen equation.

As part of their experiments, Suzuki et al also measured the vertical pressure at six

points across the bottom of the model for various levels of fill. Their graph has been

reproduced in this thesis by scaling the points off their report and replotting them,

as shown in figure 3.1. The average values of the vertical pressure are shown in

dotted lines in figure 3.1.

As can be seen from this graph,

Suzuki et al show that the central

vertical pressure is the highest, and

rapidly diminishes to a minimum

within one third of the radius from

the centre. The central pressure is

approximately 26% greater than

the average, while the minimum is

approximately 15% less than the

average.

Their measured value of the stress

ratio which they calculated from the

measured average vertical stress

and the measured horizontal stress

was K= 0.51.

Blight et al (1989) conducted a set of tests on two identical full-scale silos

containing cement. The silos were 20m internal diameter and 65m overall height

and were strain gauged across the height with six temperature compensated

gauges. For the calculation of emptying pressures, a stress ratio of 1 was used to

calculate the horizontal pressures. Their internal angle of friction was measured at

42, (which is higher than the design values of 28). They measured the stress ratio

as being 0.35 to 0.37 while the calculated value of the at-rest stress ratio (given by

Ko=1-Sin) was 0.29. They showed that the measured pressures were within the

envelope of Janssen’s pressures using a Ko ratio of 0,35. However, the test results

also showed the measured pressures in areas of low overburden were larger than

the calculated values. This is equivalent to areas close to the top of the silo.

0

1

2

3

4

5

0 0.25 0.5 0.75 1

Distance from centre to wall: r/ro

Ver

tical

pre

ssur

e on

mod

el b

otto

m

Figure 3.1: Radial Distribution of Vertical pressure taken from Suzuki et

20kg fill

40kg fill

60kg fill

80kg fill

2.1

2.9

3.4 3.8


Molenda et al (1993) have conducted experiments to determine the effects of filling

method on the wall loads. They tested concentric, eccentric and uniform sprinkle

filling methods using soft and hard wheats. From their experiments they found that

the grains aligned themselves parallel to the free surface during filling. For the

concentric filling methods, the bulk density of the material after filling was

approximately 6% lower than for the cases where the grains were uniformly

sprinkled in the silo.

It is well established that the Janssen equation for the static vertical pressure in the

silo, gives a good initial estimate of the minimum loads to be expected in the

cylindrical section of the silo. The vertical pressure in the centre of the cylindrical

section has been shown to be greater than the average value by approximately

26% to 50%, while Janssen only found it to be 15% greater. For the purposes of

this thesis, a value of 30% greater than average will be used for the vertical

pressure in the centre of the silo (where 30% is the average of the three values

quoted). The shape of the individual particles affects the bulk density of the

material in the silo due to the filling method employed. This effect does not show

up in the finite element analyses, as can be seen by the average vertical pressures

being higher than those calculated from the Janssen equation.

3.1.1.2 Hopper section

Blair-Fish and Bransby (1973) conducted tests on a model silo 0.15m square and

0.45m in height, with a hopper half angle which can vary between 30 and 20,

filled with dry sand. From their report, the normal pressure on the hopper wall is

approximately constant throughout the depth of the hopper.

Van Zanten and Mooij (1977) conducted tests on a model silo, 1.5m in diameter

and 6m tall, fitted with a hopper half angle of 15. Two types of fill materials were

used, viz PVC powder and sand. The graph of their results shows a large scatter of

data, which has the average minimum value in the lower portion of the hopper and

the maximum occuring at approximately half the hopper height. At the transition,

the pressure in the hopper is shown to be approximately three times greater than

the pressure in the cylinder. In their graph, Van Zanten and Mooij also show the

calculated line of pressure in the cone according to Jenike. Of their nearly 50 data

points for normal wall pressure in the hopper, only the four maximum points fall

outside the limit given by Jenike.


Suzuki et al (1985) measured the wall pressures in the hoppers of two mass flow

silos filled with milo. Their results show the pressures in the hopper to be the

greatest at the base decreasing towards the transition between hopper and

cylindrical section. Their results show the static pressure in the hopper at the

transition is approximately 2.3 times greater than in the cylindrical section. A

diagrammatic representation of the pressure distribution in the hopper has been

taken from the graph of their experimental results and shown in figure 3.2. In this

thesis, the vertical axis is the ratio below the transition, z, to the total depth of the

hopper,zh. The horizontal axis is given as the ratio of the pressure, P, to the

maximum pressure, Pmax , at the base of the hopper when the silo is fully loaded.

From figure 3.2 it can be seen that the pressure distribution varies from nearly

linear for no surcharge in the hopper, to a curved distribution for a fully loaded silo,

with a minimum at approximately one third the depth.

Kmita (1991) gives a very different pressure distribution for the normal pressure on

the hopper wall. The tests were conducted on a plane flow silo 0.8m wide and

3.6m overall height, filled with rinsed grit of particle diameter ranging between

3mm and 5mm. The silo has a hopper half angle of 15 and hopper height of 1.2m.

Kmita conducted tests for the case of the silo being filled from empty as well as

partially emptied and then re-filled to the same height. The pattern of the pressure

distribution is shown to be the opposite of that given by Suzuki et al in figure 3.2,

with a maximum occurring at approximately one fifth the depth of the hopper below

the transition. There is no difference in the pressure distribution for the partial

emptying and re-filling case compared to filling from completely empty. Kmita

shows the maximum to be five times greater than the maximum wall pressure in

the vertical section of the model.

It is clear from the varied results for the pressures in the hopper, that a large

variety of factors influence these pressures and the exact pressure distribution

00.250.5

0.751

0 0.25 0.5 0.75 1

Dim

ensi

onle

ss

para

met

er z

/zh

Dimensionless parameter P/Pmax

Figure 3.2: Distribution of normal pressure on hopper wall (a) no surcharge, (b) silo fully loaded

a b

position of pressure cells


cannot be predicted. However, all the research reports studied indicate the static

pressures generally do not exceed those given by Jenike, which is therefore a good

initial assessment of the maximum static pressures to be expected in the hopper.


3.1.2.1 Cylindrical Section

Pieper (1969) conducted tests on three model silos filled with a quartz sand. The

two cylindrical models were 0.6m in diameter and 3m tall, and 0.8m in diameter

and 6m tall. The third model was a square silo of 0.7m cross section and 5m tall.

This gives H/D ratios of 5.1, 7.5 and 7.1 respectively. In the graph of flow results,

Pieper shows the flow pressures in all three models to be approximately 1.3 times

the static value.

Blair-Fish and Bransby (1973) conducted tests on a sand filled mass flow silo,

150mm square cross section and 375mm tall (H/D=2.5). with a 30 hopper half

angle. They presented their results for the measured pressures in the form of a bar

graph at the point measured and the flow results were presented for each

increment of emptying

Their flow results have been

scaled and given as a ratio of

the static pressure at the point,

as shown in figure 3.3. This

graph shows their flow results

only reached a maximum of 3

times the static pressures in the

cylinder, while in the hopper,

the maximum did not exced 2.5

times the measured static

value.

Richards ( 1977) conducted experiments on a model silo 0.6m in diameter and 1m

in height with a 15 hopper half angle, filled with sand (wet sand to study minimum

opening dimensions and dry sand to study flow pressures). Richards reports that as

soon as the gate was opened only slightly, the normal wall pressures just below the

0 1 2 3

Figure 3.3: Test results scaled from Blair-Fish and Bransby

Ratio of flow/static pressure


transition increased substantially above the static pressure, although the maximum

pressure did not always occur at the start of flow when the silo was full. The flow

rate was varied by a factor of 6 in these experiments and had no effect on the

measured wall pressures. When the flow was stopped, the overpressures remained

at the points where they were recorded.

In the tests conducted by Van Zanten and Mooij (1977) on model silos filled with

PVC powder and sand, they considered flow in a “perfect” silo as well as flow in a

silo with irregularities on the walls. The following discussion does not include the

measured pressures at the points of irregularities in their model. The geometric

aspect ratio of their model silo is shown in figure 3.4 below. First they measured

the circumferential distribution of the overpressure at four points, 90 apart, at the

transition during flow. They report that the distribution was highly assymmetrical

and “simultaneous peak pressures at two points occurred only occasionally”.

For the discussion in this thesis, their test results of

the vertical distribution of the overpressures for sand,

have been divided into two sections. Those between

the transition and 1.5D above the transition, and

those at levels greater than 1.5D above the transition.

In the lower section of the cylinder, the flow pressures

are greater than the pressure envelope given by

Jenike Strain Energy. Their test results show these

pressures to be 5.8 times the maximum Janssen static

value and twice the maximum Jenike Strain Energy

value. In the upper portion of the cylinder, their test

results are 4.3 times the Janssen static pressure, but

are within the envelope given by Jenike Strain Energy.

At the transition, the maximum pressures are 15% less than the value given by

hydrostatic pressure at a depth of 4D, and 39% less than the Jenike Peak Pressure.

At a depth of 0.7D below the transition, the maximum test results are 1.4 times

greater than the Jenike static value in the hopper. In the hopper, nearly all the test

results lie outside the line of the Jenike peak Pressure. (Note: In their report, the

Jenike peak pressures are smaller than the Jenike static value in the hopper)

For their tests with PVC powder, the same division in their results has been made in

this thesis. In the lower section of the cylinder, the pressures are approximately 3.7

times the maximum Janssen static value and twice the value given by Jenike Strain

1.5

D

1.9

D

4 D

Figure 3.4: Geometric aspect of the model silo


Energy. In the upper portion, the results are twice the Janssen static value, and are

also within the envelope given by Jenike strain energy. At the level of the

transition, nearly half the test results lie beyond the maximum value given by

hydrostatic pressure. In the hopper, the results again lie beyond the Jenike peak

pressure envelope. At a depth of 0.7D below the transition the test results are 1.2

times the Jenike static value in the hopper.

Nielsen and Andersen (1982) conducted tests on full scale silos, 7m in diameter and

46m tall (H/D=6.6), filled with barley. They conducted several tests for various

arrangements of filling and emptying but only their tests for central emptying have

been considered in this thesis. The

pressure cells were placed at four points,

90 apart, around the circumference, and

at seven different vertical levels, giving a

total of 28 pressure cells. From the

results of their filling and emptying tests,

the ratios of flow vs static pressures have

been plotted in figure 3.5. These results

show that in the upper half of the silo, the

flow pressures are only 1.5 times the

static pressures whereas in lower portion,

the ratio is approximately 1,5 to 3.

Rombach and Eibl (1995), conducted finite element tests on material flow in the

hopper. The hopper half angle in their model was 20, and the wall friction angle

was 21.8. Their results show the dynamic pressure, 0.2 seconds after flow was

initiated, was 1.4 times the static pressure.

0 1 2 3

Dep

th b

elow

sur

face

11

18

25

32

39

46m

7m

Flow/Static

Figure 3.5: Ratio of flow pressures to static pressures taken from Nielsen and Andersen’s test results


3.1.3 STRESS RATIOS

In his derivation of the vertical pressure in a silo, Janssen assumes a constant value

of 0.4 for the stress ratio. Jenike and Johanson(1969) recommend a constant value,

assuming Poisson’s ratio for the material does not vary and the material remains

isotropic. They recommend a minimum value of 0.4 in the cylinder and 0.8 in the

hopper.

In their finite element analysis of the storage and flow pressures in silos, Bishara et

al (1981) give the following expressions for the horizontal and vertical pressures on

the walls, immediately after filling:

25.025.0

75.0h

HD46.0 and 33.0

62.055.0v HD82.1

Since the stress ratio is defined as : K=h/v : an equation for K can be found by

substituting in their expressions for h and v to give the following:

58.0

08.038.02.0 HD253.0K (3.1)

Equation (3.1) implies that the stress ratio is independent of the internal friction

angle of the material. It would be reasonable to expect the stress ratio to have a

term for the internal friction angle in the expression.

Furthermore, substituting for (D, H, and ) the values 1m, 3m, 0.4, and 16kN/m3

respectively in equation (3.1), gives a value of K = 0.071. Irrespective of the

material used (whether cohesive or free flowing), this value is too low to give

reasonable results.

Ravenet (1983) reports of tests done by the

Reimbert brothers in 1943 on full scale, flat

bottomed silos in France which were exhibiting

signs of being overstressed. According to Ravenet,

the Reimbert brother’s strain gauged these silos to

measure the horizontal and vertical pressures on

the walls. From the graph of Reimbert’s results,

given in Ravenet’s report, the stress ratio at various

5m: K=0.67

10m: K=0.81

15m: K=0.89

1m: K=0.4

Figure 3.6: Stress ratios in silo from Reimbert’s tests


points along the height of the silo has been determined by scaling off the graph.

These values are shown in figure 3.6. These results show the stress ratio increases

with increasing depth in the silo.

Briassoulis (1991) derives an expression for the stress ratio in the Reimbert

formula from the ratio of the horizontal stress to the vertical stress, and gives this

as follows:

p/y1

p/y2K)y(K a

(3.2)

In the above expression, Ka is the active stress ratio, p=r/(2Ka), and y is the

depth below the surface of fill. Briassoulis states that the Reimberts assume the

stress ratio, K, to decrease with depth, which is shown in equation 3.2. However,

the results given in figure 3.6 from Ravenet’s (1983) report of the Reimberts’ tests,

show the stress ratio to increase with depth.

Referring to the Mohr circle for stresses at the hopper wall, as shown in figure 3.7,

an expression for the stress ratio at the hopper wall under static conditions can be

derived as follows:

OC=OH+HC (3.3)

W

Op

13 C

P

Wall yield locus

plane on which the vertical stress acts plane on which the

horizontal stress acts

2

O

VH

Figure 3.7: Mohr circle for stresses at the hopper wall under static conditions.


But OC=r/SinW and HC=rSin(W+2) (3.4a&b)

Substituting these values into equation (3.3) gives the expression for the horizontal

stress as follows:

W

WWH

Sin

Sin2Sin1r (3.5)

Similarly, the expression for the vertical stress is given as follows:

W

WWV

Sin

Sin2Sin1r (3.6)

Therefor the expression for the stress ratio in the hopper becomes:

WW

WWh

Sin2Sin1

Sin2Sin1K

(3.7)

As the hopper half angle varies to a minimum, in the limit, this would give the

expression for the stress ratio at the wall in the cylindrical section of the silo as:

K=(1-Sin2w)/(1+Sin2w) (3.8)

These expressions (3.7 and 3.8) imply the stress ratio is a constant value and not

dependant on the level of overburden material in the silo.

WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP 3.11


3.2.1 STEEL MODEL SILO

Nielsen and Askegaard (1977) studied the effects of model scale on the results of

pressure measurements. Their experiments were done in a centrifuge, using a

sample of 40mm in diameter and 150mm in height (H/D ratio of 3.75), filled with a

cohesive material (silica gel) and dry sand. They concluded that test results on

models filled with a cohesionless material can be transferred to a geometrically

similar full scale silo, provided the model is “not too small”.They do not give a

definition of what is considered not too small. However, they stated that it was not

necessary to test cohesive materials in a centrifuge, if the model diameter was at

least 12 times larger than their model. This value of 480mm (=12x40mm) was

therefore used as an indication of their description “not too small”, for the purposes

of this this research.

The available sections in the laboratory used to set up the model, were just more

than twice the minimum requirement, defining it as a large model. Therefore, it has

been assumed in this thesis that there should be no scaling errors applicable to the

test results.

A steel model silo of height 3.21m,

0.98m in diameter (H/D=3.28) and

outlet opening of 0.18m in diameter, was

set up in the laboratory as shown in

figure 3.8. The cylindrical section of the

model was made up of four equal semi-

circular sections, 1.2m in length. The

sections were bolted together through

the outside flanges, so that there were

no obstructions to the material flow. The

hopper was made up in two halves, and

fitted to the cylindrical section through

matching flanges at the transition. A

hopper half angle of 25 was chosen from

Jenike (1967) as the maximum angle

which would still cause mass flow of the

material. The silo was filled by means of

a bucket elevator.

980

180

25

2400

810 B

UCKET

EL

EVATO

R

Figure 3.8 : Steel Model Silo


3.2.2 BULK SOLID MATERIAL

The pressures exerted on the silo walls during flow are affected by obvious criteria

such as wall friction angle, hopper slope, shape and the number of outlets in the

hopper. The less obvious influences are the material properties such as

cohesiveness, the ability to segregate and the degree of segregation due to the

filling technique. Arnold (1991) states that particle segregation influences the flow

pattern in the silo and hence the wall pressures during flow as well as during

storage. Coarser materials have a better degree of flowability than finer material,

as well as a lower coefficient of wall friction. Based on this, a uniformly graded, dry

cohesionless sand of particle diameter between 0.8mm and 1.6mm was chosen as

the fill material in the experiments. The internal friction angle of the material was

determined in a triaxial cell and was found to be 45. The wall friction angle

between the material and the silo (also determined in a triaxial cell), was found to

be 22. The material density was determined in the laboratory by allowing a sample

of the material to fall from a height of 1m into a container of known volume and

self weight. Two different methods of filling the container were used to simulate the

filling operations of the model. The material was allowed to rain into the container

and also to fall in a constant stream. There was very little difference between the

density as determined from the two different filling methods and an average

density of 16 kN/m3 was used.

3.2.3 DATA ACQUISITION

Data acquisition was obtained from a 16 channel AD card (Metrobyte Dash16)

driven by a fortran computer code (K.Kavanagh 1986). The original code was

written for a 2 cycle internal timing clock, which had to be hand calibrated in these

experiments for the a/d card with a single cycle clock. The program was structured

around background filling and emptying of 512 word buffers. Calculations were

post-processed from data transferred to hard disc. Maximum data rates of 25kHz

were obtainable without data interuption due to disc transfers.

The input file has the four entries: number of channels, number of combinations,

number of buffers and the input frequency. Therefore, if eight different types of

pressure cells were used, the number of channels would be eight. As there were no

combinations of pressure cells, this entry was entered as zero. The number of

buffers and the number of channels entered in the input file, affected the time


taken for data acquisition and size of the output data file. The output data file had

the form of columns and rows, where the columns represented the individual

channels and the rows represented the next data point for each channel.

As the number of data points in the output file was used as a time reference, the

exact sampling frequency of the single cycle clock in the computer, had to be

determined. This hand calibration was done by timing a series of blank runs for an

input file of 1 channel, 0 combinations, 5

buffers and input frequency. The input

frequency was varied from 500 to 8000 Hz

in steps of 500. The number of 5 buffers was

chosen so that at higher input frequencies,

the time could be measured with a stop

watch. If too few buffers were used, the test

duration was too short to enable timing on a

stop watch. For each input frequency, ten

tests were run and the average was taken as

the sampling time for that input frequency.

The results of these tests are shown in figure

3.9. The slope of this graph gives a factor of

0.095369 applied to the sampling frequency

in the input file.

The total time, in seconds, taken for the computer to record data, was required to

ensure data acquisition was not cut short before the end of a test, resulting in a

loss of results. The computer’s total sampling time was found to be given by:

Sampling Duration=(512NB)/(Fi 0.095369) (3.9)

The time between rows of data, tr , was determined from the following equation:

tr = NC/( Fi 0.095369) (3.10)

where NB is the number of buffers, NC is the number of channels and Fi is the input

frequency.

The size of the output data file was determined from:

Number of data rows = 512 NB/NC (3.11)

Figure 3.9: Actual frequency of computer data acquisition

0

200

400

600

800

0

2000

4000

6000

8000

Act

ual f

requ

ency

Input frequency


3.2.4 STRAIN GAUGE BRIDGES

From the literature on wall pressures in silos it was recognised that the pressures at

the transiton during flow, were likely to be the greatest. Therefore, the first attempt

at measuring the pressures was concentrated in the area just above and below the

transition, in three groups of four points around the circumference of the model, as

shown in figure 3.10. The concept was to make small holes in the silo walls and to

place an instrumented bridge across each opening. The strain in each bridge was

recorded as a measurement of the applied pressure during flow of the material.

The holes were made in the silo walls by cutting 50mm disks at each point shown in

figure 3.10. These disks were then fixed to the middle leg of the bridge as shown in

figure 3.11. Due to the fact that the disks were curved in one direction only, the

bridges were placed on the silo wall so that the curvatures of the disks and silo

were aligned. The holes were covered with a clear plastic sheeting on the inside of

the silo wall. The dimensions of the aluminium bridge is shown in figure 3.11, with

two strain gauges glued to the top and bottom of the bridge. The strain gauges

used were Kyowa type KFG-5-120-C1-11, with a temperature compenstion for steel

and a gauge length of 5mm.

After the gauges were connected in series to give an average reading of deflection,

they were tested for specified resistance to ensure no gauges had become damaged

in the soldering process. Tape was placed over the gauges to reduce the risk of

damage during handling.

Figure 3.10: Location of Strain Gauge Bridges


Each bridge was loaded and unloaded three times to ensure the gauges had

adhered to the bridge before they were calibrated. The gauges were then connected

to the amplifier, and the bridges loaded up to 5 kilograms in increments of 1

kilogram. Readings were taken after each increment. The bridges were then

unloaded to 3 kilograms, a reading taken, and allowed to stand for 30 minutes after

which another reading was taken. This was done to check for drift in the gauges.

The results for each bridge were plotted on a graph of load versus voltage output

and the slope of the graph gave the calibration factor for each bridge.

Static readings were taken after the silo had been completely filled. The silo was

emptied in stages into drums which were placed under the hopper. Emptying

stopped when the drum was full and reloaded into the silo. Readings were taken for

50mm disk cut out of silo wall

Strain gauges

Figure 3.11: Aluminium strain gauge bridge

10mm

10mm

10mm

40mm

40mm 20mm 2mm

Aluminium bridge


each stage of emptying. The results from the bridges did not record any strain for

the static pressures or for the first stage of emptying, the output being attributed to

electrical noise. The likely cause was that the bridges were too flexible relative to

the model wall so that the material arched over the bridges. One bridge became

dislodged during testing, and the plastic sheeting managed to hold the material in

place without rupturing. This lockup of material is a strong indication that arching

will occur over small diameters, where the opening is softer than the bin wall. This

phenomenon may be more critical, the smaller the diameter of the opening.

Due to the failure of these experiments, a radically different pressure measuring

cell was developed. Rather than being mounted to the silo wall, the cells were

placed in the material during filling, and the cells travel with the material during

flow.

3.2.5 PRESSURE CELLS

Richards (1977) reports of vibrations being felt and heard during discharge from a

mass flow model. The frequencies of the individual fluctuations were reported to be

in the range of 15Hz to 85Hz, the frequency increasing with flow rate and being

independent of particle size. Therefore, a responsive measurement and fast

recording system was needed to investigate the material pressures during flow.

Three types of material pressure cells, as shown in figure 3.12, were developed in

the laboratory to allow continuous measurement of the pressures during filling and

emptying of the silo contents. The concept of the cells was to place a standard

pressure sensor in a small hollow object with a flexible wall, which could then be

filled with an incompressible, low viscosity oil. The most important criteria was to

ensure that there were no air bubbles retained in the oil or pressure sensors to

affect the incompressibilty.

X

Y

Z

Y

Z

X

Figure 3.12: Floating Pressure Cells: a) Ball type, b) Tube type, c) Plate type

(a) (b)

(c)


3.2.5.1 The Ball Type Pressure Cell

The first type of cell was the ball cell, where a pressure sensor (psi absolute) of

capacity 15psi, was modified to fit inside a hollow thin walled rubber sphere, as

shown in figure 3.13. The front ports and the side connection holes were clipped

off, leaving only the shell of the sensor which was then fitted in the sphere. A large

container filled with silicon oil was placed inside a vaccuum chamber to remove any

air bubbles in the oil. The sphere and sensor were then placed in the vaccuum

chamber and the air bubbles removed. The opening in the sphere, through which

the sensor was inserted, was sealed off with a glue suitable for elastomeric

materials. Removing the air bubbles from the inside of the sensor proved to be very

difficult, and in some ball cells the prescence of an air bubble showed up during

calibration.

3.2.5.2 The Tube Type Pressure Cell

The second type of pressure cell was the tube type cell which consisted of a 14mm

diameter hollow Tygon(™) tube, 100mm long, as shown in figure 3.14. The

pressure sensor used in these cells was a standard 5psi differential pressure sensor,

which was chosen above the 15psi sensor, as the sensor’s capacity produced a

larger signal-to-noise ratio. The smaller capacity sensors were not available in the

absolute form, so that one port was blocked off from the atmosphere with a plug,

leaving the remaining port as the active port. A solid end cap to close off the end of

the tygon tube, and a plug with a central hole, were fabricated in the laboratory.

The separate components for this cell were filled with oil by placing them in the

container of oil in the vacuum chamber. Once the air bubbles were removed, the

cell was then assembled while submerged in the oil, taking care not to introduce

new air bubbles while handling the tube and components.

Absolute port

Pressure port

Connection holes

Electrical pins and cables

Thin walled rubber sphere

Pressure sensor shell

Figure 3.13: Pressure sensor modified to fit inside rubber sphere.


The sensor and the plug were connected using a flexible plastic tubing which was

pushed over both the port and the plug and fixed on with a plastic tie. The plug and

end cap were pushed into the tube and also fixed in place with a plastic tie. This

system proved very successful as the air bubbles from the active port of the sensor

were easily removed. Since the Tygon(™) tube was translucent, a visual check of

the existence of air bubbles was easily made. The tube cell was left to stand

overnight in a vertical position resting on the sensor, and any trapped air bubbles

would then float up into the tygon tubing.

3.2.5.3 The Plate Type Pressure Cell

As shown in figure 3.13, the ball cell measured the average pressure acting in three

dimensions, where as the tube cell measured the average pressure acting in a

plane. The third type of pressure cell was therefore developed to measure the

average pressure acting in one direction only. This was the plate type cell which

consisted of an aluminium plate with a slot cut out the middle connected to the

pressure sensor as shown in figure 3.15.

14mm diameter tygon

End cap plug with centre hole

Pressure sensor

Electric pins and cables

open port blocked off

flexible tubing pushed over port and plug, and fastened with a tie

Figure 3.14: Tube type pressure cell

standard pipe connection threaded end cap

100mmx30mm aluminium plate covered with rubber membrane on both faces

10mm wide slot

rigid plastic tubing

Electric pins and cables

pres

sure

sens

or

port blocked off

Figure 3.15: Plate type pressure cell


Again, a standard 5psi differential pressure sensor was used which had one port

plugged off. Two threaded holes were made at either end of the aluminium plate

extending into the slot. Both faces of the plate were sealed by a rubber membrane

which was glued on to the plate surface. One hole was closed off with a threaded

end cap, and a standard pipe fitting was used to screw into the other hole. A short

length of stiff plastic tubing was pushed into the pipe fitting and then heated up in

boiling water to fit over the active port of the sensor. The individual components of

the pressure cell were placed in the container of low viscosity oil and a vacuum

applied to remove all the air bubbles. Once the air bubbles were removed the cell

was assembled while submerged in the oil.

3.2.6 PRESSURE CELL CALIBRATION

The cells were placed in an air tight vessel with an absolute pressure sensor, first to

determine their responses to an instantaneous pressure and then to calibrate each

cell. The inlet and outlet to the vessel were sealed off with silicon sealant and

allowed to stand for five hours before applying air. Compressed air was supplied to

the vessel by means of a valve and regulator, which gave a digital read out of the

pressure supplied to the vessel. The set up is shown in figure 3.16.

Figure 3.16a: Calibration of pressure cells (i) Ball cell, (ii) Tube cell, (iii) Plate cell, (iv) Plain sensor

valve and digital readout unit

amplifier

To computer

compressed air supply

air tight container

inlet and outlet sealed with silicon

cables from pressure cells

i

ii iii

iv


Before calibration, the cells were tested to ensure there were no air bubbles

trapped inside and also to determine their responsiveness to an instantaneous

pressure. The instantaneous pressure was applied to the cells by first regulating the

pressure to a known value with the valve closed and once the pressure had been

reached the valve was suddenly opened. The cells were then subjected to a series

of instantaneous pressures and the output recorded on the computer. When the

cells contained trapped air bubbles, the output of the cell compared to the plain

pressure sensor exhibited a lag in the peak reponse, as well as a damped response

in the decay curve of the graph. As shown in figure 3.17, the dotted shows there is

no lag in the peak response of the cells.

Vol

tage

ou

tput

Vol

tage

ou

tput

Data point Data point

Ball cell

Plain sensor

No lag in peak response

Cells very responsive to instantaneous pressures

Figure 3.17: Typical response of a ball type cell

Figure 3.16b: Photograph of pressure Cell Calibration


Once the responsiveness of each cell had been determined, they were calibrated by

applying a series of pressures, in increments of one, up to thirty two kilopascals.

Before the cell output was recorded, the air pressure in the vessel was first set at a

constant for each increment, thereby ensuring good data readings. For each

increment of pressure, 170 data points were recorded. The average voltage output

for each pressure increment was plotted as a point on the graph, and a linear least

squares regression analysis applied to the data. The graph for each cell has been

shown in Appendix B, with the calibration constant.

The pressure cells were connected to the data logger by means of 6m long cables

which enabled the cells to flow freely during material discharge. The cells were

placed at various depths in the silo during the filling operation, and their positions

measured.

3.2.7 MULTI-TURN POTENTIAL METERS

To determine the vertical position of the

pressure cells during material flow, two

multi-turn potentiometers were each

connected to a pulley, supported at the

top of the silo. Two flat plates, 100mm in

diameter each, were fixed to an

inextensible wire which connected the

plates to the pulleys, as shown in figures

3.18a and 3.18b.

The multi-turn potentiometers were

connected to the data logger and

calibrated by recording the voltage

output for each quarter, or half turn of

the pulley. The pulleys each had a

circumference of 685mm, giving a

reading for every 171mm travelled. The

graph of results and the calibration

constants for each multi-turn potential

meter has been shown in Appendix B.

Figure 3.18a: Multi-turn potential meters

100mm flat plate

inextensible wire

pulley

multi-turn potential meter


The plates were placed in the material during filling for each test, and their

positions from the top of the silo measured. When the material started flowing, the

position of the pressure cells relative to the multi-turn potential meters was

therefore easily determined.

3.2.8 GATE SWITCHES

To determine the exact time or data point when the gate was opened, two on/off

switches were positioned on the gate. One switch was triggered by the closed gate

and the second switch was triggered by a leverarm which had been placed on the

sliding gate. When the gate was opened the voltage output changed from zero volts

to five volts, and when the gate was fully opened the leverarm triggered the second

switch, which changed the voltage output from five volts back to zero volts. This

also gave the time taken and hence the number of data rows, to open the gate.

Figure 3.18b: Photograph of Multi-Turn Potential Meter connected to the Pulleys on top of the Model Silo

WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.23


3.3.1 DESCRIPTION

A total of twenty four tests were performed on the silo, of which four tests gave no

useable results due to problems with the data logger. The first five tests were

performed on the silo without an anti-dyanmic tube installed, to measure the static

and dynamic pressures in the material. The remaining fifteen tests were undertaken

to determine the frictional drag on tubes of varying lengths and diameters. In the

drag tests, the pressure cells were placed in the silo to gain additional data about

the speed with which the swich pressure travels up the silo, as well as to measure

the effect of the tube on the wall pressures. The discussion of the drag tests on the

anti-dynamic tube is given in chapter four.

Owing to the large amount of effort involved in filling the silo for each test, a check

list was made to ensure correct preparation was carried out. An example of the

check lists for both the static and dynamic tests has been given in Appendix C. The

static test check list included checking the electrical signals from all the cells, the

multi-turn potential meters and the switches on the gate. The positions of the

pressure cells from the top of the silo were measured and recorded, as well as the

cell orientation relative to the silo wall. The check list also noted the calibration

constants, the input data file and the method of filling the silo for later reference.

During the filling stage a stopper was placed in the pulleys of the multi-turn

potential meters to avoid the falling material causing the pulleys to turn and reach

their full rotation before the dynamic tests were performed.

The dynamic test check list included the input data file, calibration constants and

checking electrical signals from the data loggers. At the end of each test, the cone

of material at the top was levelled off and the silo was filled to it’s capacity. The

sampling frequency for each test was set approximately seven seconds longer to

ensure the start and finish of each test would be recorded. Each test was timed

with a stop watch to compare the test duration with the output from the multi-turn

potential meters. The cables from the pressure cells were laid out individually next

to the silo to ensure they would not become entangled. It was essential to leave

enough spare cable for the pressure cells to flow freely down the silo. The stoppers

from the pulleys were removed before the start of each dynamic test. However, due

to the slip-stick nature of the material flow, the pulleys overshot their turning

giving exaggerated flow rates. The friction of the pulleys was increased to avoid this

problem.


A complete list of the tests performed is given in table 3.1.

3.3.2 STATIC TESTS

The silo was filled by means of a bucket elevator without any attachments at the

outlet, for the first five tests. From figure 3.19 it can be seen that the position of

the cone of material varied with the filling process, making it a combination of

eccentric and central filling. To determine the effect of this on the density of the

material several samples were taken from various positions in the silo by placing a

TestNumber

-1 1 Ball1 1 Ball 12 4 tubes 13 3 tubes 14 3 tubes 15 2 tubes 3 plates 16to9

Test Tube Tube HopperNumber Diameter Length Gate

diameter(mm) (mm) (mm)

10 140 2610 180 Tube support broke11 140 2610 18012 140 2610 18013 140 2610 18014 140 3 sections 180 Middle section broke off ten

895,840,845 seconds after flow started.15 140 3 sections 180

895,840,84516 140 3 sections 180

895,840,84517 140 3 sections 140

895,840,845 Smaller gate opening18 140 3 sections 140 resulted in a funnel

895,840,845 flow pattern. 19 140 2 sections 140

895,84020 120 1800 18021 120 1800 180 Flow down the inside22 120 1800 180 and outside of the tube23 120 1800 180

No useable results:Electrical problems with

the data loggersAnti-Dynamic Tube Drag Force

TABLE OF SILO TESTS Static and Dynamic pressures

Pressure cells withMtpmgood test results

Table 3.1: Static and Dynamic test list


small bucket of known weight and volume in

the material during filling. Each sample was

then weighed to determine the density at that

level. The bucket elevator was also equipped

with a flexible hose which was positioned in the

silo to cause a central filling situation. Again

samples were taken from various depths in the

silo and the density calculated. Table 3.2

shows the densities at various levels for both

filling methods. The difference in the material

density between the two methods was 2.7%

which was not considered large enough to have

a noticeable effect on the results. Therefore, an

average value of 16kN/m3 as determined in the

laboratory has been used throughout this

thesis.

Test No Depth (1) Mass Density Test No Depth (1) Mass Density (m) (g) kN / m3 (m) (g) kN / m3

Free fall from bucket elevator Flexible hosing fixed to bucket elevator 7 2.62 2299.7 16.13 6 2.62 2270 15.92

1.89 2353.7 16.53 1.66 2306.8 16.18 0.98 2320.7 16.29 0.67 2196.8 15.38

8 2.8 2335.5 16.39 14 2.84 2325.9 16.32 - - - 2.2 2338.3 16.41

1.53 2244.6 15.73 1.43 2263.5 15.87 0.47 2286.9 16.04 0.93 2072.5 14.47

9 2.8 2334.1 16.38 15 2.71 2340.4 16.43 2.02 2327.2 16.33 1.87 2247.2 15.75 1.35 2306.9 16.18 1.13 2313.1 16.23

- - - 0.74 2098.5 14.66 Average 16.22 Average 15.78

OVERALL AVERAGE = 16,0 kN/m3

(1) Depth of container from top of silo

(2) Container self weight: 87.92g and volume: 0.001345m3

The results from the first five static tests have been shown in table 3.3 and plotted

in figure 3.20, with the equivalent Janssen horizontal pressure. The Janssen

pressure was calculated using a stress ratio of 0.4. The pressure cells used in these

Buc

ket

elev

ator

Cone of material changes position during filling

Figure 3.19: Position of the cone of material

Table 3.2: Density measurements during filling of the silo


tests were tube type cells, which were all placed vertically in the material

approximately 100mm from the wall. The tube cells measured the average of the

circumferential and horizontal pressures acting in the plane, and is given by:

2HC

tube

From Jenike (1964), the horizontal and circumferential pressure acting in the

horizontal plane of the axially symmetric cylindrical section are equal in magnitude.

Therefore, the horizontal pressure acting normal to the wall is given by:

tubeH

The pressure cells were placed in the silo at various depths. The exact orientation

and depths of the cells from the top of the silo were recorded on a drawing which

formed part of the check list, as shown in Appendix C. The filling process was

therefore interrupted to place the pressure cells in the material. The total time

taken to fill the silo, place the cells and note the depth and orientation, was

approximately three to four hours. Thus, there was not enough time between

successive layers for the material to be affected by the process, and filling has been

considered as a continuous operation.

Figure 3.20: Static test results

2

3

1

0 D

epth

bel

ow s

urfa

ce (

m)

Horizontal Pressure (kPa)

Hop

per

Cyl

inde

r

0 2 4 6 8

Test Depth Pressure Janssenbelow cell (kPa) horizontal

surface (H+C )/2 pressure1 1.1 5.57 5.252 0.6 3.15 3.35

0.93 4.73 4.671.61 6.65 6.612.11 8.06 7.552.58 3.1

3 0.69 4.04 3.741.11 6.5 5.281.39 6.64 6.082.28 7.932.54 3.88

4 0.28 2.57 1.741.04 4.53 5.052.2 5.33 7.69

5 1.26 5.66 5.732.17 8.29 7.65

Table 3.3: Static Test Results


Although most of the static test results are slightly greater than those shown by the

line of the Janssen horizontal pressure, these results are deemed to be in good

agreement with the Janssen theory. Therefore, the static results will be used to

compare the ratio of the dynamic to the static pressures. This ratio gives an

indication of the factors applied to wall pressures, and hence to the pressures

exerted on the anti-dynamic tube in the material. From the literature survey, the

vertical pressure in the centre of the silo has been shown to be approximately 30%

greater than the average value. Therefore, the vertical pressure exerted on the

anti-dynamic tube can be estimated from the ratio of dynamic to static results by

multiplying the ratio by a factor of 1.3.

3.3.3 DYNAMIC TESTS

The dynamic tests all followed directly after the static tests with no waiting time

between tests in which the material had time to settle or de-aerate.

At the end of each static test, the computer program was shut down and a new

input data file was entered for the dynamic test. Zero offsets were recorded with

the static pressure acting on the cells. Hence, the dynamic tests show a negative

pressure when the cells passed through the gate of the silo with the moving

material. This negative value is equivalent to the static pressure at the depth the

cell was placed during filling. Figure 3.21 is a typical sample output showing the

general trend of the dynamic tests.

Figure 3.21: Sample output of a typical dynamic test

0

40 30 20 -4

4

8

12

50 60

Pres

sure

(kP

a)

Time (sec) 70

initial static pressure

First peak

Second peak

zero offsets

gate

ope

ned


Nearly all the tests showed a first peak pressure at the instant the discharge gate

was opened and the material started moving. The pressure then reduced to

approximately the static value (zero on the graph), although in some cases the

pressure was considerably higher. The second pressure peak shown on the graph

occurred when the cell passed through the transition and entered the hopper. As

the cell left the silo the pressure reduced to a negative value. When the cell was

placed at, or just above, the transition, the first and second peak curves in the

output graph, merged to form a single curve. However, when the cells where

placed below the transition only a single peak appeared in the dynamic output

curve, followed by a rapid decrease in pressure as the cell moved down the hopper.

A list of all the tests performed on the silo, as well as a complete set of the output

results for each test, is given in appendix D.

A table of the results for the first five dynamic tests is given in table 3.4, showing

the values of the first and second measured peak pressures. The total pressures

acting on the silo wall during flow are equal to the dynamic value plus the static

value, which have also been given in table 3.4. The values plotted in figure 3.22 are

the total pressures acting on the silo walls.

Table 3.4: Dynamic Test Results

Test Depth StaticNo from top pressure First Second First Second

of silo (m) peak peak peak peak-1 1.1 3.64 4.83 8.89 8.47 12.531 1.1 5.57 3.73 5.89 9.3 11.462 0.6 3.15 4.13 6.67 7.28 9.82

0.93 4.73 2.13 11.49 6.86 16.221.61 6.65 - 12.26 18.912.11 8.06 - 11.93 19.99

3 0.69 4.04 5.18 14.67 9.22 18.711.11 6.5 1.66 22.18 8.16 28.681.39 6.64 2.35 20.13 8.99 26.772.28 7.93 - 12.36 20.292.54 3.88 - 3.88

4 0.28 2.57 0.53 3.26 3.1 5.831.04 4.53 2.46 4.24 6.99 8.772.2 5.33 10.4 21.41 15.73 26.74

5 1.26 5.66 7.89 12.86 13.55 18.522.17 8.29 5.24 11.25 13.53 19.54

Dynamic pressures Total pressure


The ratios of the measured dynamic to static pressures have been shown in figure

3.22(b). This ratio was calculated by dividing the value of the first peak pressure by

the measured static pressure at the depth the cell was placed. To calculate the ratio

of the dynamic to static pressure at the transition, the value of the second peak in

the output curve was used for the dynamic pressure, and an average value of 7.10

kPa was used for the static pressure at the transition. This average value has been

calculated from the results of six static tests.

Shown in figure 3.22(c) is the ratio of the measured dynamic pressure to the

calculated Janssen static pressure. For test numbers three to five, tube type

pressure cells were used. These tubes were placed horizontally parallel to the silo

wall and thus measured the average of the pressures acting in the meridian plane.

Therefore an equivalent average static pressure has been calculated as follows:

Average pressure at the point: AV = (H+V)/2 , and stress ratio: K=H/V

Therefore: AV=V(K+1)/2 where V is the Janssen vertical pressure given in

equation 2.1.5, using =16.8kN/m3, D=0.98m, =0.404 and K varies.

A varying stress ratio has been used in the above calculation of the Janssen

pressure. For a depth of fill of zero to one diameter K=0.5; from one to two

diameters fill level, K=0.3; and for two to three diameters fill, K = 0.2. These

values have been determined from the static tests and have been explained in

chapter 3.3.4.1.

Figure 3.22: Dynamic test results: (a) Measured horizontal pressures (kPa) ; (b) Ratio of measured dynamic to measured static pressure ; (c) Ratio of measured dynamic pressures to Janssen static. In all three graphs: * First peak ; Second peak

0 1 2 3 4

0

1

2

30 10 20 30 0 1 2

Dep

th b

elow

sur

face

(m

)

(a) (b) (c)


The experimental results show the dynamic pressures to be between one and two

times greater than the Janssen static pressures, and one to four times greater than

the experimental static pressures. This is smaller than the values taken from the

tests by Blair-Fish and Bransby (1973), as well as Van Zanten and Mooij (1977).

3.3.3.1 Speed of the Switch Pressure

The height of the pressure cells above the gate and the sampling rate of the

computer, give an estimate of the speed with which the switch pressure travels up

the silo. The calculation of the time between columns of data and the time between

rows of data has been given in equations 3.9 and 3.10, and has been used to

determine the time when the cells registered a change in pressure. A sample output

data file has been given in table 3.5.

For this test the sampling frequency was 5300 Hz, therefore the time between

columns and rows of data was 0.00198 and 0.0237 seconds, respectively. It is

reasonable to assume that the pressure wave may have passed over a pressure cell

before the computer was able to record the data change.

In the spreadsheet the cell number represents both the channel number of the data

logger and the column number in the data output. Tube cell 2 and plate cell 6,

which were placed together at a height of 0.87m above the discharge gate,

registered a pressure change at 0.0810 seconds after the gate was opened. This

gives the pressure wave speed was 10.74m/s for this sample output. By the time

the pressure wave reached tube cell 4 and plate cell 7, which were both placed at

Seconds Tube 2 Tube 4 Tube 5 Plate 6 Plate 7 Plate 8 Switches

-0.0948 -0.012 -0.018 0.184 -0.024 -0.009 0.032 -0.002-0.0711 -0.021 0.001 0.157 -0.014 -0.029 0.042 -0.002-0.0474 -0.104 0.011 0.157 -0.097 -0.1 0.022 4.972-0.0237 -0.067 -0.018 0.157 -0.014 0.001 0.052 4.982

0.0000 -0.058 0.001 0.221 -0.033 -0.07 -0.018 4.9770.0237 -0.095 0.001 0.231 -0.088 -0.019 0.012 4.9770.0474 -0.169 -0.018 0.231 0.004 -0.019 0.032 4.9790.0711 0.007 -0.027 0.24 -0.005 0.001 0.042 4.9750.0948 0.888 0.02 0.231 1.028 -0.321 0.192 0.0010.1185 4.281 0.523 0.695 2.816 0.624 0.9 0.0010.1422 5.236 1.755 1.456 2.881 0.995 0.69 0.0080.1659 4.986 1.3 1.196 2.816 0.794 0.7 0.0030.1896 4.921 1.167 1.112 2.844 0.804 0.71 -0.0020.2133 5.217 1.044 1.103 3.037 0.754 0.631 0.0030.2370 5.625 1.082 1.149 3.406 0.784 0.631 -0.004

Table 3.5: Sample Output Data File


the same level higher in the silo, the computer had already recorded channel 4,

with no pressure change. However since plate cell 7 registered a pressure change,

the time taken for the wave to reach these cells has been taken from channel 5.

This is reasonable, since tube cell 5 was placed together near the top of the silo and

therefore registered no pressure change at the instant in time when the pressure

wave reached plate cell 7.

Tube cell 5 and plate cell 8 were placed together at a height of 2.51m above the

discharge gate. The pressure wave passed over these cells at a time of 0.1106

seconds after the gate was opened. This calculation gives the speed of the switch

as 22.69m/s.

Due to the possibility of large errors being introduced when trying to determine

when the pressure wave passed over the cells, only those cells which were placed

in the upper sections of the silo have been used. This gives the average speed of

the switch pressure travelling up the silo. Table 3.6 gives the average speed from

13 tests using only those cells which were placed in the upper third of the silo.

The average speed from all the tests has been shown in table 3.6 and is found to

be 20.31m/s. The results from test numbers 17, 18, 20 and 21 have been excluded

from the calculation of the average due to the large scatter of their results

Table 3.6: Switch Pressure Speed

Test Sampling Height of Switchnumber frequency cell above pressure

(Hz) gate (m) speed (m/s)5 401 2.88 23.056 477 2.54 16.8210 477 2.78 21.8011 429 1.99 23.2412 496 2.18 18.6513 496 2.61 19.4714 505 2.51 22.6915 668 2.47 22.4916 906 2.04 17.8417 906 2.33 12.918 291 2.36 13.2519 305 2.29 17.0920 944 2.64 52.2821 954 2.63 58.02

Average speed: 20.31m/s


3.3.4 STRESS RATIO

By placing two types of cells at the same level

in close proximity to each other, the stress ratio

can be determined for filling as well as during

flow as shown in figure 3.22. When a tube cell

is placed horizontally and parallel to the silo

wall the average of V and H is measured.

Therefore by placing a plate cell to measure V

or H, the stress ratios can be determined.

3.3.4.1 Static Stress Ratio

The results of the static stress ratio from test number 11 to 14 is given in table 3.7.

The data from test numbers 12 and 13 did not give reasonable results and have

therefore been ignored. Thus, the overall average of the measured stress ratio is

0.32. This is approximately equal to the stress ratio for the material at rest namely:

Ko = 1 – Sin = 1 – Sin 45 = 0.293

V

C

H

Figure 3.22: Directions of the stresses in the cylinder

0

1

2

30.0 0.4 0.8

Dep

th f

rom

top

(m

)

Figure 3.23: Graph of static stress ratios

Stress Ratio

Ka

Ko

K

0 to

1D

1

to 2

D

Hop

per

Table 3.7: Static Stress Ratios

Test Depth Average StressNo. from Pressure Ratio

top(m) (kPa) V H H V

1.47 2.861.47 1.240.71 0.140.71 0.082.8 2.582.8 0.831.77 2.541.77 0.901.22 2.281.22 1.371.43 1.361.43 0.101.24 1.421.24 4.392.34 3.722.34 1.181.17 3.841.17 0.960.7 1.670.7 1.42

0.96 0.14

1.92 1.42 0.74

13 4.39 -1.6 -0.35

14

6.26 1.18 0.19

6.72

12 0.10 2.62 26.47

11

4.33 0.83 0.19

4.18 0.90 0.22

3.19 1.37 0.43

0.28

0.2 0.08 0.40

Measuredpressure

104.48 1.24


The active stress ratio given by : Ka = (1 – Sin )/(1 + Sin ) = 0.17, is

considerably less than the overall average. However from the graph in figure 3.23,

it can be seen that the stress ratio decreases with the depth of material in the silo.

It is proposed in this thesis, to divide the silo into three sections to calculate the

stress ratio. At a depth of zero to one diameter below the top of the silo the ratio

can be approximated from: K = 1–Sin2 = 0.5.

From one diameter to two diameters below the top, the ratio approaches the value

for the material at rest: Ko = 0.29.

At depths greater than two diameters and in the hopper, the stress ratio is

approximately equal to the active stress ratio, Ka = 0.17.

From Briassoulis (1991) the Reimberts give the stress ratio as decreasing with

depth, which is in agreement with these test results.

3.3.4.2 Dynamic Stress Ratio

For each pair of pressure cells placed at the same level, two ratios have been

determined; one at the start of flow and one when the cells passed through the

transition. These results are shown in table 3.8 and plotted in figure 3.24. There is

a large scatter in the measured stress ratios.

Test Depth StressNo. below Ratio

surface H V

(m)11 1.22 1.89

2.42 1.951.77 0.462.3 1.73

12 1.43 3.22.4 0.58

13 2.4 1.5214 2.34 0.4

1.17 1.772.3 1.690.7 1.712.45 1.22

15 1.41 0.512.4 0.530.74 0.912.41 0.31

16 1.8 2.8

Table 3.8: Dynamic Stress Ratios

0

1

2

30 1 2 3

Dep

th b

elow

sur

face

(m

)

Stress Ratio

Figure 3.24: Graph of Dynamic stress ratios

Hop

per

Cyl

inde

r

ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4.1

CHAPTER FOUR

THE ANTI-DYNAMIC TUBE


Other names for anti-dynamic tubes are decompression tubes, discharge tubes,

static flow pipes and tremmie tubes.

4.1.1 PIEPER

Pieper (1969) conducted tests which measured the force exerted on a horizontal

bar during flow. A 10mm diameter bar was placed in the material, 1.5m above the

outlet of a 0.8m square silo filled with quartz sand of 15kN/m3 bulk density. The

bar was supported by a frame hanging from the ceiling, as shown in figure 4.1. The

graph shown in figure 4.2, has been taken from the results by Pieper, as the

average of the two values for the left hand and right hand support. In the paper,

Pieper does not say if the deflection of the bar itself was taken into account in the

output. Furthermore, no description is given of the method to take into account any

downward movement of the bar supports, which may have caused it to derive

additional support from the walls of the model. It must therefore be assumed that

these results may have an experimental error in the value of the force recorded.

Pieper states that the output of the vertical force on the bar during filling is

1.5m

3.

35

0.8m

10mm bar

Figure 4.1: Horizontal tie placed in silo during flow. (Tests by Pieper)

Pb

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

Pres

sure

(kP

a)

Time (min)

65.1

filling emptying

Figure 4.2: Average values taken from Pieper’s test results


approximately the same as the Janssen static value for that depth.

As can be seen from figure 4.1, the force suddenly increases fourfold when the gate

is opened, decreasing slowly with the decrease in the height of the material. The

maximum pressure on the bar recorded by Pieper was 65.1 kPa approximately one

and a half minutes after the gate was opened.

4.1.2 REIMBERT

The Reimbert brothers (1976) claim to be the inventors of the anti-dynamic tube as

a means to produce a homogeneous outflow from a silo. Initially, this consisted of

perforated tubes fitted to the side walls of the silo, connected to inclined tubes in

the hopper, as shown in figure 4.3(a), or alternatively placed centrally as shown in

figure 4.3(b). The Reimbert’s state that the centrally placed tube caused the

material to flow down the tube in successive layers, resulting in a first-in-last-out

situation. In both cases, material flow along the walls is eliminated, thereby

reducing the wall pressures as well as wear on the walls. The Reimberts claim to

have successfully installed the tubes as a retro-fit in many existing concrete silos

which were cracked due to excessive wall pressures.

However, according to Ravenet’s report (1983), the Swedish specialist Bergau,

records that anti-dynamic tubes were first used at the beginning of the 1900’s by

Miersch in the Frankfurt/Main Silos; Duhle used them in the Alexander Dock Silo, in

Liverpool; and Huart and Kvapil also made use of them in the early 1900’s.

4.1.3 RAVENET

Ravenet (1983) conducted tests with an anti-dynamic tube in a transparent model

to emulate flow in layers as reported by the Reimbert brothers. Ravenet states that

Figure 4.3: The Reimbert’s tube: (a) Section and Top View of Tube down the side walls of the silo; (b) Centrally placed tube.

(a)

(b)


the tube failed to discharge the material as the holes frequently became blocked.

After several attempts, layered flow was achieved. He states that when the tube

operated succesfully, the dynamic to static pressure ratio was approximately 1.35.

4.1.4 MCLEAN

McLean (1985) reports on general arrangements of the anti-dynamic tube being

successfully installed in silos to allow safe side discharge. These arrangements

alleviate the bending stresses in the wall associated with an eccentric outlet. The

diagrams shown in figure 4.4 have been taken from McLean’s report by scaling his

drawings. While it is uncertain if McLeans’ diagrams were drawn to scale, the ratio

of the tube to silo diameter in his diagrams is 0.2. This is approximately twice the

ratio given by other researchers.

The material is drawn

from two or three levels

in the case of the silo

with multiple outlets.

McLean reports on the

importance of adequate

support given to the

side discharge chutes

which protrude into the

path of the flowing

material.

McLean (1985) gives the following equations to determine the vertical forces acting

on objects placed in the flowing material:

For a tall silo:

Je1

K4DA5.2F pv (4.1)

where sHz4DKJ

(4.2)

and Ap is the projected plan area of the object and Hs is the height of surcharge

above the silo. If there is no cone of material above the top of the silo, equation 4.1

reduces to a modification factor of 2.5 times the Janssen equation multiplied by the

projected area of the object.

Both Pieper and McLean report that the pressure on an object submerged in the

Figure 4.4: Side Discharge Outlets: (a) Single Outlet (b) Multiple Outlets

(a)

Single outlet

(b)

Multiple outlets


material is approximately equal to the Janssen static pressure during filling of the

silo. However McLean’s formula (equation 4.1) gives the force during flow of the

material as 2.5 times the static value, whereas Pieper’s test results show the flow

force to be nearly four times the static value. Thus Pieper’s results give values that

are approximately 1.6 times larger than predicted by McLean.

4.1.5 OOMS AND ROBERTS

Ooms and Roberts (1985) conducted tests on a flat bottomed acrylic model silo,

3.8m tall and 1.2m in diameter, fitted with an anti-dynamic tube and filled with

wheat. Their model and tube arrangement has been shown in figure 4.5. Unlike the

Reimbert’s tube, theirs did not extend the full height of the silo and was open at the

top, with port holes only at the base of the tube. The purpose of their tests was to

determine the effectiveness of this tube arrangement in controlling flow patterns

and wall pressures, before installing tubes in full scale silos. After installation of the

tube, the model silo emptied in two stages as shown in figure 4.5, thus operating

as two short silos in series. They report that with no tube installed, the flow

pressures at the effective transition were over three times the static measured

value. However, the flow pressures were nearly equal to the measured static

pressures after the tube was installed.

Ooms and Roberts state that the minimum height of the tube is determined by the

angle , such that no effective transition is formed between the material and the

silo wall during the first stage of discharge. Therefore, from the geometry of the

model silo and tube:

Stage 1

Stage 2

Dead material

hmin

hmax

Effective transition

Figure 4.5: Ooms and Roberts Tube: (a) Discharge Sequence; (b) Tube geometry

(a) (b)


hmin = H–D/(2Tan) (4.3)

Similarly, the maximum height of the tube is determined such that no effective

transition intersects the wall during the second stage of discharge. Therefor

hmax =D/(2Tan) – (D/2) Tan (4.4)

The angle () of the effective transition to the vertical is given by Jenike et al

(1973c) and Arnold et al (1989) to be a function of the internal friction angle of the

material, whereas Hasra and Bazur (1980) give as a function of both the wall and

internal friction angles.

Ooms and Roberts derived an expression for the static vertical pressure in a silo

with a tube installed, by considering the equilibrium of a horizontal element.

The forces acting on a horizontal slice

of thickness dz in the silo at the level of

the anti-dynamic tube have been

shown in figure 4.6. The silo and tube

diameters are given by D and d

respectively. At the level of z=zo the

vertical pressure, zo is equal to the

static pressure given by the Janssen

equation (2.1)

B/B/ BzoBzzov ee (4.5)

and B=4(DKoo+dKii)/(D2-d2) (4.6)

In their derivation, Ooms and Roberts assumed the stress ratios on the inside and

outside of the tube to be different. In equation 4.5, Ko is the stress ratio on the

outside of the tube and Ki is the stress ratio on the inside of the tube.

Assuming fully mobilised flow along the outside of the tube wall, Ooms and Roberts

have given the vertical drag as a direct function of the internal friction angle and

stress ratio, as shown in equation 4.7.

B11e

BBH

dKF tBHzo

tiiv (4.7)

However, as there is no direct relationship between the vertical drag force on the

v

t w dz

zo z

v+(v/z)dz

Figure 4.6: Forces acting on a horizontal slice


outside of the tube and the internal properties of the tube, equation 4.7 has been

re-written as shown in equation 4.8. Therefore, Ko and o have been substituted for

Ki and i, and the drag down force becomes:

B11e

BBH

dKF tBHzo

toov (4.8)

The effect of the tube diameter on the vertical pressure and the vertical drag down

force as given in equation 4.5, has been shown in figure 4.7 (a) and (b)

respectively. These graphs were calculated for the material propeties and silo

geometry of the model used in the laboratory. The tube height used in the

calculations is 2.6m at a depth of 0.6m below the top of the silo. By increasing the

tube to silo diameter ratio from one quarter to a half, the vertical pressure as

calculated by equation 4.5 decreases from 84% of the Janssen pressure to 61% of

the Janssen pressure. Thus, the vertical static pressure in a silo, and hence the

horizontal static pressures on the walls, can be reduced by introducing a larger tube

to silo diameter ratio.

The function B in the vertical drag down force given by equation 4.6, was calculated

assuming equal internal and external stress ratios, and equal internal and external

wall friction angles. The vertical drag force increases with increasing height of the

tube as expected, as this is directly related to the tube surface area. The graph in

figure 4.7(b) shows the relation to be approximately linear at depths of 1.4m and

greater.

Figure 4.7: Effect of tube to silo diameter ratio of 0.25 and 0.5 on: (a) Vertical pressure ; and (b) Vertical drag on outside of tube

Dep

th b

elow

sur

face

(m

)


(a)

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.20 10 20 30

0.5

Janssen

0.25

Hei

ght

of t

ube

(m)

Vertical drag (kN)

(b)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 2 4 6 8

0.25

0.5


The vertical drag force on the inside of the tube during flow has been given in the

report by Ooms and Roberts, as:

4

Hde1

K4d

4dF t

2d/tHiKi4

iizo

2vi

(4.9)

This expression contains only the properties relating to the inside of the tube, as

would be expected. The same material and silo properties as used in the above

calculations, was used in the calculation of the vertical drag inside the tube. For a

tube of length 2.6m the vertical drag force is 2.07kN, which is 45% of the vertical

drag on the outside of the tube. This implies that the drag should be increased by a

factor of 1.5 when the material flows inside and outside the tube at the same time.

4.1.6 KAMINSKI AND ZUBRZYCKI

Kaminski and Zubrzycki (1985) conducted experiments on a concrete model silo

1.25m in diameter by 3.78m tall, fitted with anti-dynamic tubes and filled with

wheat. They report that the wheat had a bulk density 8 t/m3 , which is too heavy

for wheat and should probably be 8kN/m3. In their report, they do not give the

diameters of the tubes nor do they give an indication of the number of tubes

installed. A sketch of their model arrangement is shown in figure 4.8.

The purpose of their tests was to determine

the vertical forces acting on perforated and

non-perforated tubes during material flow.

The tubes were suspended by steel rods

from a supporting structure above the

model silo. Spring dynamometers, with a

maximum capacity of 2kN, were fixed to

each steel rod to measure the loads on the

tubes as shown in the diagram. The results

of their tests showed that the drag on the

non-perforated tubes increased by a factor

of 6 times the filling vertical force, whereas

for a perforated tube there was no increase

in the force during discharge of the material.

Kaminski and Zubrzykci state that the flexibility of the tube supports has a large

influence on the measured value of the vertical force. They conducted experiments

by varying the flexibility of the supports and then measuring the deflection of the

bottom of the tube, dS, during material flow.The results of their tests have been

Figure 4.8: Kaminski and Zubrzycki model silo

spring dynamometer

steel rods

tube

support

dS


reproduced in figure 4.9. The vertical axis in their graph shows the ratio of the

measured vertical deflection, dS , to the hydraulic radius of the tube, rh. Thus, they

define the flexibility parameter as the ratio of the deflection, ds, to the hydraulic

radius of the tube, rh. This seems inappropiate as the flexibility can not be a factor

of the tube geometry. The horizontal axis is the ratio of the measured vertical

force, FV , to the calculated Reimbert value, FR. They state that an increase in the

flexibility parameter beyond 25.6, does not give a further decrease in the measured

vertical force on the tube. The lowest measured vertical force was 4% of the

calculated Reimbert value. For an inflexible support (a value of 3.3x10-3), the

measured vertical force was 70% of the calculated Reimbert value. They propose

correction factors to be applied to the calculated Reimbert value, as shown in the

graph by the stepped solid line, which represents an envelope of their test results.

They state that their results were compared with measurements on full scale silos.

However, there is no reference or description of the extent of the full scale tests.

In their report they state that the optimal geometric parameters for an anti-

dynamic tube were previously derived in a dissertation in 1977. Unfortunately this

reference is not available in English, and hence the recommendations from their

report have been included in this thesis for completeness only. The parameters are

as follows:

1) The diameter of the tube should fall within the following range:

0.32 dt / H 0.64 (4.10)

where H is the hydraulic radius of the silo and dt is the tube diameter.

0

3.2

6.4

9.6

12.8

16

19.2

22.4

25.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d s /

rh

(x

10-3

)

FV / FR

Figure 4.9: Kaminski and Zubrzycki test results for varying support flexibility


In this relationship, the tube diameter is a linear function of the silo diameter.

2) The total area of the tube perforations should fall within the following limits:

0.08 A 0.2 (4.11)

and A = Ah/At = Ah/ dt Ht (4.12)

where Ah is the total area of the perforations in the tube wall, and At is the surface

area of a solid tube. This parameter gives the tube perforations as a linear function

of the tube surface area.

3) Flow characteristics of the tube perforations

0.25 0.8 (4.13)

and = A * dt3 /(dG * D) (4.14)

where dG is the silo discharge gate diameter, and D is the silo diameter.

For a cylindrical or square silo, H =D/4, equation (4.10) can be re-written as:

0.32 D/4 dt 0.64 D/4 (4.15)

Re-writing the second tube parameter given in equation (4.11), in terms of the

tube diameter gives:

0.08 dt Lt Ah 0.2 dt Lt (4.16)

Re-writing the third tube parameter gives the discharge gate diameter as a function

of the area of the tube perforations:

D8.0dA

dD25.0

dA 2th

g

2th

(4.17)

The maximum and minimum limits as given by equations (4.15), (4.16) and (4.17)

have been plotted in figure 4.10.

Therefore, for a tube diameter of 2m, the acceptable range for the total area of the

tube perforations must fall within 0.5m2 upto 1.25m2 per meter length of tube,

while the silo discharge gate must fall within 0.2m up to 1.25m in diameter.


4.1.7 SCHWEDES AND SCHULZE

Schwedes and Schulze (1991) report on a discharge tube being succesfully installed

in two cement clinker silos of 16m diameter and 7m tall, with a funnel flow hopper.

The tube diameter was 1.8m, with a wall thickness of 25mm. The ratio of the tube

diameter to the silo diameter was 0.11. The vertical hole spacing was set at 3.5m c/c with four rectangular holes at each level. The holes were orientated vertically,

with an open area of 0.6m2 (0.6x1.0m). In their report the maximum vertical stress

inside the tube of diameter dt was calculated from:

v=dt/(4K) (4.18)

This is the Janssen equation with the exponential term approximately equal to one.

Using the above formula given by Schwedes and Schulze, the pressure on the tube

during flow of the material, for the silo and tube model used in this research, would

be: v = ( 160.14) / (40.4Tan19) = 4.07 kPa

For the tube length of 2.61m and diameter of 140mm, this gives a total drag down

force of:

Fv=4.07 * 2.61* *0.14 = 4.66 kN = 475.5 kg

This is 2.1 times larger than the value determined using the equation

recommended by Ooms and Roberts.

Silo diameter 5 15 10 20

1

2

3

Tube

Dis

char

ge

gate

1 2

Tube perforation area 2

0.4

0.3

0.2

0.1

Figure 4.10: Maximum and minimum limits for tube parameters

THE ANTI-DYNAMIC TUBE: EXPERIMENTAL SET-UP 4.11


4.2.1 ANTI-DYNAMIC TUBE MODEL

The model silo set up in the laboratory has been described in chapter 3.2.1 of this

thesis. The same cohesionless material as described in chapter 3.2.2, was used for

the experiments of the frictional drag on the anti-dynamic tube. Anti-dynamic tubes

of 0.14m and 0.12m in diameter, and 0.4mm wall thickness, were placed centrally

in the model silo and suspended from a support at the top of the model silo, as

shown in figure 4.11. The angle of wall friction (t) between the tube and the

material was measured in a standard shear box test and was found to be 19.

The support frame was made up from rectangular hollow sections (RHS) of 65mm x

35mm, welded to two short struts with base plates which were clamped to the top

flange of the model silo, as shown in figure 4.12. The deflection of the support

frame is directly related to the frictional drag acting on the side of the tube during

flow of the material. The beam in the support frame was deliberately orientated

across its weaker axis to ensure adequate deflection, as well as provide sufficient

space for fixing of the strain gauges. The chain connecting the tube to the support

Figure 4.11 : Model Silo and Tube

strain gauged support beam

inextensible chain

anti-dynamic tube: 2.61m total length

locating rods

980

180

25

2400

810 B

UCKET

EL

EVATO

R


beam was an inextensible chain, thus avoiding excessive vertical movement of the

tube during material flow. The chain was located across the centre of the support

beam.

The chain was connected to the top of the tube by a strip of plate which was bent

to fit across the tube, as shown in figure 4.12. Two holes were drilled through the

side of the plate, which was then connected to the top of the tube. A hole was

drilled through the centre of the strip to accommodate a threaded bolt and eye with

two nuts, top and bottom of the bent strip. The chain was passed through the eye

of the threaded bolt and fixed over the top of the support beam. The cover to the

top of the tube was fixed around the threaded bolt to prevent material flowing

down the inside of the tube. For three tests this cover was removed giving the drag

down force for material flow on the inside as well as the outside of the tube.

The model tube lengths varied from 1.8m to 2.6m and were made up from three

sections of galvanized pipes which were riveted together. To ensure the tubes

Figure 4.12: Support frame and tube connection

average strain readings from top and bottom of beam

65 x 35mm RHS

Base plates clamped on to top flange of model silo

inextensible chain

threaded rod and eye

bent plate

pop-rivets nut, top and bottom

cover over top of tube

anti-dynamic tube

THE ANTI-DYNAMIC TUBE: EXPERIMENTAL SET-UP 4.13

remained in a central position during filling of the silo, six locating rods, three near

the base and three at mid height of the tube, were used. The locating rods were

removed when the fill level reached the level of the rods. The three sections of the

tube were connected by means of disks and 5mm threaded rods, placed inside the

tubes, as shown in figures 4.13a and4.13b. These 5mm rods were strain gauged

at three points around their circumference to give the average strain reading.

The support beam at the top of the silo, and the 5mm rods were calibrated by

hanging a series of weights at the bottom of the tube and recording the voltage

output. The weights were added in 10kg increments up to a maximum of 160kg.

The calibration constants of the strain gauges, with units of kg/V, were determined

from the slopes of the data output which plotted as a straight lines as shown in the

Appendix B.2.

To determine the rate of flow of the material during discharge, the multi-turn

potential meters (mtpm) and 100mm plates, as described in chapter 3.2.7, were

used. The filling process was interrupted to place the plates in the material and

record their positions. The total time taken to empty the silo was also recorded and

compared to the flow rate given by the multi turn potential meters.

A photograph of the inside of the silo with the tube installed has been shown in

figure 4.14. The photograph shows the group of experiments with the tube divided

into three sections.

Anti-dynamic tube

Disks bolted to side of tube

Strain gauged 5mm diameter threaded rods

Skirt around the gap to prevent entry of material (not shown)

Figure 4.13b: Detail of Disks and Rods

Figure 4.13a: Anti-Dynamic Tube Sections

disk inside the pipe

strain gauged rods

inextensible chain

cover over top of pipe

skirt around gap (not shown)



A total of 14 tests were conducted with the anti-dynamic tube installed in the model

silo. The tube lengths, diameters and the diameter of the silo discharge gate were

varied. A matrix of tube tests performed is shown in table 4.1. The first group of

tests, Group A, were done with a silo discharge gate diameter of 160mm, while the

second group of tests, Group B, had a discharge gate diameter of 140mm.

Three tests were performed with a tube of 2610 mm long, followed by another

three tests with the tube separated into three sections as shown in figure 4.13a.

These six tests had a silo discharge gate diameter of 160mm and a tube diameter

of 140mm.

Figure 4.14: View of the Anti-Dynamic tube installed in the steel model silo.

Table 4.1: Matrix of tube tests

Silo Solid Tube Solid Flow downdischarge tube split in 3 tube inside &

gate 2610 mm sections 1800 mm outside ofdiameter long long solid tube

Group A: 160 mm 3 3 2 3

Group B: 140 mm 3

Number of tests performed

Tube diameter (140mm)

Tube diameter (120mm)

THE ANTI-DYNAMIC TUBE: EXPERIMENTAL RESULTS 4.15

The opening size of the silo discharge gate was then varied from 160mm to 140mm

by placing a wooden disk in the base of the hopper, and another three tests

performed with the tube split into three sections. The tube diameter was then

changed to 120 mm and the length reduced to 1800 mm. Two tests were then

performed with a silo discharge gate diameter of 160mm. Finally, the last three

tests were performed with material flowing down the inside and outside of the tube,

with a silo discharge gate of 160mm in diameter. These variables on the tube were

considered to be of greatest influence on the magnitude of the drag down force.

The complete set of tube test results has been given in Appendix E. In nearly all the

tests there was an initial peak force which only lasted a matter of seconds, followed

by a second peak force of a longer duration. The maximum value of the initial peak

varied considerably between tests, as can be seen in the two typical test results

shown in figure 4.15 and 4.16 below. In figure 4.15, the initial peak value was

lower than the second peak, where as in figure 4.16, the initial peak value is

greater than the second peak value. The second peak represents established flow in

the cylindrical section of the silo. Once the level of material drops below the top of

the tube, there is a non-linear decrease in the drag force, which can be seen in

figures 4.15 and 4.16. In figure 4.16, the drag force on the top and middle portion

of the tube is decreasing, while on the bottom portion of the tube there is a

constant drag force acting, until the material level drops below the bottom portion

of the tube.

0

60

240

120

180

10 20 30 40 50 60 70 80

157.1

245.6

Figure 4.15: Test number 12:Drag down force on solid tube 2.61m long

Time (seconds)

Dra

g D

own

Forc

e (k

g)


The first test performed to determine the drag down force, was test number 10. As

can be seen from the output of this test in Appendix E, the pop rivets fixing the

plate to the tube broke approximately 16 seconds after the discharge gate was

opened. The tube moved down the silo with the flowing material and blocked the

silo outlet. This test had to be abandoned and the silo top emptied before testing

could resume. The plate was re-fixed to the top of the tube using larger diameter

pop rivets, 5mm, to ensure this did not occur again.

The results from Group A

Tests for the solid tubes,

have been given in table 4.2

The shear stress on the tube

has been determined from

the drag down force divided

by the surface area of the

tube. The column of

constant drag in the table

refers to established flow in

the silo, which occurs after

the switch pressure has

Dra

g D

own

Forc

e (k

g)

0

40

80

120

160

200

240

280

320

10 20 30 40 50 60 70 Time (seconds)

Figure 4.16: Test number 16: Drag Down Force on tube split into three sections

358.0

212.5

52.6

245.5

211.6

66.5

Test Tube Initial Shear Constant ShearNo. length Drag Stress drag Stress

(kg) (kPa) (kg) (kPa)

11 2610 183.1 1.56 252.7 2.1612 2610 157.1 1.34 245.6 2.1013 2610 193.8 1.66 278.4 2.38

20 1800 74.8 1.08 - -21 1800 177.5 2.57 134.3 1.94

22 1800 118.3 0.86 127.9 0.9223 1800 110.9 0.80 104.8 0.7624 1800 - 103.4 0.75

Tube Diameter (120 mm)

Tube Diameter (120 mm)Flow down inside and outside of tube

Silo Discharge Gate Diameter (160 mm)


Table 4.2: Results from Group A Tests


travelled up the silo. In some graphs of the output, the established flow is clearly

visible, as in test numbers 13, 14, 16, 17 and 18. Except for test number 13, these

tests correspond to the tube being split into three sections. In test number 13, the

section of constant flow is very short due to the level of material going below the

top of the tube, as is the case for the remaining tests using a solid tube.

The results for the Group A tests with the tube split into three sections have been

given in table 4.3 below. For this group of tests, there was an initial peak drag

force followed by a second peak representing established flow in the cylindrical

section of the silo.

The test results for Group A tests have

been plotted in graphs as shown in

figures 4.17 and 4.18. These test

results have been divided into two sub-

groups showing the initial peak drag

value, shown in figure 4.17, and the

peak drag value for established flow,


Test Tube Shear ShearNo. length Top Middle Bottom Stress Stress

(kPa) Top Middle Bottom (kPa)

840 254.4 2.20 296.6 2.5714 840 254.4 2.31 414.3 2.79

895 74.6 1.86 197.4 4.92840 287.6 2.28 287.6 2.28

15 840 201.6 3.00 201.6 3.00895 88.7 2.21 88.7 2.21840 358 3.86 245.5 0.90

16 840 212.5 4.25 211.6 3.85895 52.6 1.31 66.5 1.66

Drag Force (kg)


Silo Discharge gate Diameter (160mm)Initial Drag Force (kg) Established Flow

Table 4.3: Group A Test Results for the tube split into three sections

0

0.5

1

1.5

2

2.5

3

0.0 0.5 1.0 1.5 2.0 2.5

Gate = 160 mm: Initial peak

Dra

g Fo

rce

(kN

)

Tube Surface Area (m2)

Figure 4.17: Group A Tests: Initial Peak Drag Force


The drag force on the vertical axis, has been given in units of kiloNewtons, while

the tube surface area on the

horizontal axis has been given

in units of meters squared.

From figure 4.17 and 4.18 it

can be seen that there was a

wide scatter of results, with

the maximum initial drag on

the tube reaching

approximately 2.6 kN while

the maximum drag value for

established flow reached

approximately 2.8 kN.

The results from the Group B tests have been given in table 4.4. For these tests the

tube was split into three sections and the discharge gate diameter was changed to

140mm diameter. The data output showed no initial peak value in the drag force on

the tube when the discharge gate was opened, as can be seen in the graphs given

in Appendix E.

The values given in table 4.4 represent the maximum values of the drag force on

the tube. For each section of the tube, the shear stress has been calculated by

dividing the drag force by the surface area of the tube. The drag for each section

has been determined by

subtracting the value of the

drag force for the lower

sections. Thus for test number

17, the drag force acting on

this section is equivalent to

85.9kg subtract 46.3kg, which

gives 39.6kg. This value was

then divided by the surface

area over the length of 0.84m.

Test Tube ShearNo. length Top Middle Bottom Stress

(kPa)840 91.9 0.16

17 840 85.9 1.05895 46.3 1.15840 108.3 0.26

18 840 98.5 1.21895 52.8 1.32840 96.2 0.86

19 840 -895 64 1.59

Drag Force (kg)

Silo Discharge Gate Diameter (140mm)Tube Diameter (140 mm)

Table 4.4: Group B Test results

0

0.5

1

1.5

2

2.5

3

0.0 0.5 1.0 1.5 2.0 2.5

Peak Drag Values for Established Flow

Dra

g Fo

rce

(kN

)


Figure 4.18: Group A Tests: Established Flow Drag Force


The results for Group B

tests have been shown in

figure 4.19 for established

flow only, as there was no

initial peak drag for these

tests. From figure 4.19 it

can be seen that the

maximum drag force

reached a value of

approximately 0.64 kN.

The time taken to completely empty the silo was determined using a stop watch.

This time gives the average flow rate of the material, compared to the results from

the multi turn potential meters, which give the variation of the flow rate during

material flow. These average flow rates are given in units of meters per second,

which represents the rate of travel of the top surface of material. Therefore, the

total height of the silo was divided by the time measured on the stop watch. While

it is accepted that this is not an accurate measurement of the flow rate of a silo, it

gives an indication of the variability of the flow rate for varying sizes of discharge

gate diameter. The results are given in table 4.5 and 4.6 below.

Test Time from FlowNumber stop watch Rate

(seconds) (m/s)3 88 0.03655 76.72 0.04186 87.2 0.03688 80.72 0.03989 80.28 0.040011 81 0.039612 77 0.041713 94 0.034115 74.84 0.042916 75.25 0.042720 82.07 0.039122 79.34 0.040523 81 0.0396

Average 81.34 0.0396

Silo Gate Diameter 160 mm

Table 4.5: Average Flow Rates of silo with gate diameter of 160mm

Test Time from Flownumber stop watch Rate

(seconds) (m/s)17 355.09 0.009018 378.87 0.008519 364.96 0.0088

Average 366.31 0.00877

Silo gate diameter 140 mm

Table 4.6: Average Flow Rates of silo with gate diameter of 140mm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.36 0.38 0.40

Peak Drag Values for 140mm Gate

Dra

g Fo

rce

(kN

)


Figure 4.19: Group B Tests: Established Flow Drag Force


From table 4.5, it can be seen that the average time taken to empty the silo was

one minute and thirty four seconds. For the total silo height of 3.21 meters, this

gives an average flow rate of 0.0396 m/s. This is faster than the rate given by the

multi turn potential meters, since the material undergoes free fall in the hopper

section of the silo. The average flow rates for the silo with a discharge gate

diameter of 140mm are given in table 4.6. The discharge gate of 160mm has a plan

area of 0.0201 m2 while the gate of diameter 140mm has a plan area of 0.01539

m2. The gate of diameter 140mm is approximately 23.4 percent smaller than for

the gate of diameter 160mm. The comparison between the flow rates shows that

the 140mm gate gives an average flow rate of 4.5 times slower than the 160mm

gate.

The average value of the initial shear stress from table 4.2 and 4.3 for Group A

tests is 2.25 kPa. This frictional stress corresponds to a silo with a gate diameter of

160mm, which has a plan area of 0.0201m2. The average shear stress for

established flow is 2.52 kPa for the same silo discharge gate . These values exclude

the results for material flow down the inside as well as the outside of the tube. The

average value of the shear stress for established flow for Group B tests, is 0.95

kPa, taken from table 4.4. This corresponds to a silo gate diameter of 140mm

which has a plan area of 0.01539m2 . These results have been shown in graphical

form in figure 4.20.

Figure 4.20: Shear Stress vs Discharge Gate Area

0

4

8

12

16

20

0 0.01 0.02 0.03 0.04

She

ar S

tres

s (k

Pa)

Silo Gate Plan Area (m2)

Exponential Trend Line

Power TrendLine

Linear Trend Line


Only three trend lines could be fitted through the two data points. These are an

exponential curve, a power curve and a linear trend line. The equation for each of

the trend lines are:

Exponential Curve:

= 0.0392 e207.12x

Power Curve:

= 4x106 A

3.6537

Linear Trend Line:

= 101.85 A

where is the shear stress acting on the walls of the tube, and A is the plan area of

the gate.

To determine which of these lines would be most appropriate, the drag force on a

full scale silo which is five times larger than the model silo has been calculated. The

full scale silo would have a diameter of 4.8m and a height of 16m, while the

discharge gate would be 0.8m in diameter. The corresponding tube size would be

0.7m in diameter and 13.05m tall. For a discharge gate of 0.8m in diameter, the

plan area is 0.503m2 . This would give the following shear stresses:

Exponential Curve: = 6.897x1043

kPa.

Power Curve: = 3.248x106 kPa.

Linear trend Line: = 51,23 kPa.

The full scale anti-dynamic tube would have a surface area of approximately

28.7m2. Thus the corresponding total friction force acting on the full scale tube, as

determined from each of the trend lines, would give 1.979x1045

kN, 93.22x106 kN,

and 1470 kN, respectively. Clearly the calculated value from the exponential curve

is far too large and therefore not practical. The value determined from the power

curve is 9.5x106 tonnes of force, which is equally impossible for the scope of the

assumed model. When the silo discharge gate is opened the material suddenly

changes from a stationary state to a state of constant motion under the


acceleration of gravity. Therefore, the impact force as the material starts moving

can be calculated as follows:

F = mg = 16 (4.8216/4 – 0.7

213.05/4) = 4552.1 kN (4.19)

If the value of the friction force as determined from the linear trend line was

greater than the calculated impact force, it would imply that the material had an

acceleration greater than gravity. Since the value of the friction force calculated

from the linear trend line, 1470 kN, is less than the force calculated for impact, as

given in equation 4.19, the linear trend line can be accepted as a reasonable curve

applied to the two data points, as shown in figure 4.20.

Therefore, it is proposed in this thesis that an equation for estimating the effect of

the speed of the flowing material on the magnitude of the friction stress acting on

the sides of a tube is as follows:

= 101.85 A (4.20)

where is the shear on the walls of the tube and A is the plan area of the discharge

gate.

This equation is an empirical equation and it is not proposed that this is an

acceptable expression for all silo and tube arrangements. This expression is limited

to the scope of this thesis only.

THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL 4.23

4.4 MATHEMATICAL MODEL

A mathematical model of the friction force on an insert tube has been developed by

considering the equilibrium of vertical forces acting on an elemental slice of

material as shown in figure 4.11. The figure represents an element of material of

thickness dz, from a silo fitted with an anti-dynamic tube. The lengths of the

element sides are S1, S2, and S3 where:

S1 = Rd S2 = rd S3 = (R- r) (4.21)

The top and bottom surface areas of the element are equal, and are given by:

A (S1+S2) S3/2 d (R2-r2)/2 (4.22)

The volume of the element is given by: V = Adz = d (R2-r2)dz/2 (4.23)

The areas of the element in contact with the silo wall and tube wall are given

respectively by:

A silo = S1dz = Rd dz and A tube = S2dz = r d dz (4.24)

By considering downward forces on the element as positive, equilibrium of the

vertical forces acting on the elemental slice gives:

0Adzz

AAVA vvtubetsilosv

(4.25)

Noting that = KvTanw , and substituting equations 4.22, 4.23, and 4.24 in

equation 4.25 gives the following differential equation for the vertical pressure

acting on the element :

S2

R

r

S1

S3

d

dz

dzz

vv

v

s

t

A silo

A tube

A top

A bottom

Figure 4.11: Forces acting on an elemental slice of material in a silo fitted with an anti-dynamic tube


vv G

dzd (4.26)

where : G = [2 (Ks Tansw R + Kt Tantw r)]/[ (R2-r2)] (4.27)

This is the same expression as derived by Ooms and Roberts (1985). To solve equation 4.26, multiply throughout by a function (z) and add and subtract

(z)v from the left hand side of the equation. Re-arranging the terms gives:

)z()z(G)z()z()z( vvv (4.28) Noting that the first term of equation (4.28) is the derivative of the product of

(z) and v, and if the second term equals zero then:

zz v (4.29)

From the second term in equation (4.28), v 0 , therefor let: G = (z)/(z) , and

G)z(Lnz

Therefore, GzCGz Aeez (4.30)

Substitute equation 4.30 into 4.29 and solving gives:

zG

v eCG1

(4.31)

To find the constant of integration: at z = 0, v = 0, and C= -1/G , Then equation

4.31 becomes:

zGv e1

G

(4.32)

O’Neil (1995) gives this general method for solving linear first order differential

equations.

Substitute equation (4.32) into the expression for t to give:

zGtttvtt e1

GTanKTanK

(4.33)

The total friction force acting on the tube can be found by multiplying equation

4.33 by the circumference of the tube and integrating over the length of the tube:

Ce

G1hr2Tan

GKdzr2F hG

t

h

0

ttF (4.34)

From the boundary conditions at h=0, FF = 0 , and therefore the constant of

integration can be found:


G1C (4.35)

Substituting equation 4.35 in equation 4.34 gives the expression for the total

friction force acting on the tube.

1er2TanG

KF hGt2tF

(4.36)

where G is given in equation 4.27

4.4.1 TUBE PARAMETERS

From the first geometric parameter defined by Kaminski and Zubrzycki 1985, the

diameter of the tube divided by the hydraulic radius of the model silo used in this

research gives a value of 0.5833 which lies within their recommended optimum

range. Therefore, this model tube and silo is considered a good geometric

arrangement. As mentioned previously in this paper, the second and third

parameters have no meaning for a tube with zero perforations and have not been

considered.

As can be seen from equation 4.36 the friction force on the tube is directly related

to the stress ratio within the material at the tube wall, the material density and

the tube radius. Substituting for =16kN/m3, t=19, s=22 ,r=140mm,

R=960mm, h=2.61m and using various values of the stress ratio, the variation of

the friction force has been shown in figure 4.12. The stress ratios used in figure

4.12 have been discussed on the following page. From this figure it can be seen

that the correct value of stress ratio must be used to determine the friction force.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Friction Force FF

Leng

th o

f tu

be (

m)

Figure 4.12: Graph of Friction force for various stress ratios

4 3 2 1

1: Active Stress ratio 2: At Rest Stress ratio 3: Minimum given in AS 3774 4: Blight’s Upper limit Experimental results


The active stress ratio is given by: )Sin1/()Sin1(Ka = 0.172, for the tube

and silo wall. Using this value in equation 4.36 gives the first curve in figure 4.12.

The stress ratio for the “at rest” condition is given by )Sin1(Ko =0.2929 for

the tube and silo wall, which gives the second curve in figure 4.12

The stress ratio given in the Australian Code [10] is :

22

2222

Cos4

CosSin2Sin1K (4.37)

where is the coefficient of wall friction and is the material friction angle

For the silo and tube model this gives: Kt = 0.1768 and Ks = 0.1790.

The minimum value recommended by AS 3774 is 0.35, which is given in figure 4.12

by the third curve.

The fourth curve of the friction force in figure 4.12 has been calculated using the

upper limit recommendation of the stress ratio for granular materials from Blight

(1993). This upper limit is given by: K = 1+Sin = 1.7071

Substituting the values for the model into equation 4.36 and using the Blight upper

limit stress ratio gives a total friction force of 293.1 kg. Substituting a stress ratio

of 1.7071 into the expression given by Ooms and Roberts, equation 4.7, the total

friction force acting on the tube is:

3012.361.2161e

3012.316

3012.314.019Tan7071.1F 61.23012.3

0zF

kg64.294kN8904.2FF

In the above expression, the values of the wall friction and stress ratio inside the

tube have been taken equal to the outside of the tube, and the v0 has been

assumed zero. In both equations 4.9 and 4.36 for the friction force, the variables G

and B have the same numerical value because of the assumption that the internal

tube wall has the same surface as the external tube wall.

4.4.2 VARIABLE VERTICAL PRESSURE ACROSS A SLICE

From the literature survey of the pressure measurements in the silo, the vertical

pressure in the centre of the silo is approximately 15% to 50% greater than the

average vertical pressure. In this thesis an average value of 30% is recommended.

Suzuki et al (1985) give an approximation of the vertical pressure distribution


across a horizontal element in the silo. This distribution has been given in chapter

3.1.1.1 and has been shown schematically in figure 4.13

This pressure distribution

has been adapted to a silo

fitted with an anti-dynamic

tube. Since the vertical

pressure near the silo wall

has been shown to be less

than average, this has

been assumed at the wall

of the tube. Similarly,

since the vertical pressure

at the centre is greater

than average, it has been

assumed in this thesis that the vertical pressure at the midspan between the tube

and silo wall is greater than average. This results in the pressure distribution shown

in figure 4.14.

The shape was chosen such that the area of the curve below the average pressure

line equals the area above the average pressure line. Two distributions were

considered where the maximum and the minimum varied from 10% above and

below the average line to 30% above and below the average pressure line.

The equations for the two distributions are :

x2Cos1.01v for the 10% variation (4.38)

Silo center line

Silo wall

Average vertical stress

1.26

0.8

1.1

1/3 R 3/4 R

Figure 4.13: Vertical Pressure distribution according to Suzuki et al.

1.1 to 1.30

Tube wall

Silo center line

Silo wall

Average vertical stress

0.90 to 0.70 1/4 3/4 1/2

0.90 to 0.70

Figure 4.14: Assumed vertical pressure distribution in a silo fitted with an anti-dynamic tube

F1 F3

F2


and

x2Cos3.01v for the 30% variation (4.39)

Thus the element has been divided into four segments as shown in figure 4.15. The

vertical forces acting over the element segments, F1, F2 and F3, have been shown in

figure 4.14. These forces were determined by integrating the expression for the

vertical pressure distribution across the element and multiplying by the average

length of the segment.

The lengths of the segments S4, S5 and S6 are given by the following:

S4 = (r+(R-r)/4) d = (3r+R) d /4

S6 = (r+(R-r)3/4) d = (r+3R) d /4 (4.40a,b)

For the distribution of a 10% variation above and below the average pressure line,

the vertical force, F1 is given by:

2

SSdxx2Cos1.0F 42

vv1 (4.41)

where (S2+S4)/2 is the average length of the segment and is given by:

(S2+S4)/2 = (7r+R) d /4 (4.42)

Integrating equation 4.41 from 0 to 0.25(R-r) and substituting equation 4.42 in

equation 4.41 gives:

8R1.0

16R

8r7.0

16r7

d4

Rr721.0

4F vvvvvv1

Figure 4.15: Segments of the elemental slice

S2

R

r

S1

S3

d

dz S4

S6

S5

1/4 1/2

3/4


which can be simplified to give:

d1.0

21

8R

8r7

F vv1 (4.43)

Similarly the expression for the vertical force acting over the middle segment is

given by:

5vv2 Sdxx2Cos1.0F

(4.44)

and the length of the segment S5 is given by S5 = (r+R) d /2 (4.45)

Integrating equation 4.44 and substituting equation 4.45 gives the expression for

F2, after simplification, as follows:

d1.0

21

2R

2r

F vv2 (4.46)

And the expression for the vertical force, F3, acting over the third segment of the

element is given by:

2

SSdxx2Cos1.0F 61

vv3 (4.47)

where (S1+S6)/2 is the average length of the segment and is given by:

(S1+S6)/2 = (r+7R) d /4 (4.48)

Again integrating equation 4.47 and substituting equation 4.48 into 4.47, after

simplification, gives the following expression for the force F3:

d1.0

21

8R7

8r

F vv3 (4.49)

Therefore, there are three forces acting upwards on the element from the material

below. These three forces are:

(∂F1/∂z)dz , (∂F2/∂z)dz , and (∂F1/∂z)dz (4.50a,b,c)

These three forces have an expression which can be found in a similar manner to

the definition of the vertical forces acting downwards. The final expression for the

vertical forces acting upwards on the element are:

d1.0

21dz

z8R

8r7F v

v1 (4.51)


d1.0

21dz

z2R

2rF v

v2 (4.52)

d1.0

21dz

z8R7

8rF v

v3 (4.53)

The terms containing σv in the expressions of the vertical forces acting downwards

cancel out with the σv terms in the expressions of the vertical forces acting upwards

on the element. Therefore, only terms containing (∂σv/∂z)dz will appear in the

expression for vertical equilibrium of the forces acting on the element, which is as

follows:

ddzz16

Rr7dzdrTandzdRTandzd2

rR vtHsH

22. . . . . . . . . .

ddzz2

Rr1.0ddz

z4Rrddz

z8R1.0r7.0 vvv . . . . . . . . . . . . .

0ddzz8

R7.0r1.0ddz

z16R7r vv

(4.54)

Dividing equation 4.54 throughout by dzdθ and cancelling terms gives the following

expression:

22vttss

v rRrKRK2z

Rr1.023

(4.55)

which can be written as:

JCz

vv

(4.56)

where

Rr1.023

rKRK2C ttss

(4.57)

and

1.023

rR

Rr1.023

rRJ22

(4.58)

The expression for C in equation 4.57 differs from the expression for G given in

equation 4.27 by the term: π/((3π/2-0.1)(R-r)), which is inversely proportional to

the difference between the silo radius and the tube radius.


The expression for J in equation 4.58 differs from the right hand term in equation

4.26 by the term: π(R-r)/(3π/2-0.1), which is linearly related to the difference

between the silo and tube radius.

Solving equation 4.56 by the same method as given for equation 4.29 gives the

solution for σv as:

Czv e1

CJ (4.59)

where J and C have been defined in equations 4.57 and 4.58.

If the silo wall and the tube wall have the same coefficient of friction and assuming

Ks=Kt=K, then the expression for J/C becomes:

K)rR(

)rR(K)rR(

CJ 22

(4.61)

The total drag force on the tube can be determined from:

h

0

tF dzr2F (4.62)

The shear stress on the walls of the tube is determined from:

t=hTant = Ktvt (4.63)

Therefore, the friction force is given by:

h

0

vttF dzKr2F (4.64)

Substituting the expression for v from equation 4.59 into the equation 4.64, and

integrating over the height of the tube gives the following expression for the friction

force acting on the tube walls:

1e

ChJKr2F Ch

ttF (4.65)

where h is the total height of the tube, Kt is the stress ratio at the wall of the tube,

t is the coefficient of friction of the tube wall, r is the radius of the tube and C and

J are constants defined in equations 4.57 and 4.58.


Thus the expression for the friction force assuming a vertical pressure distribution

across the element, as given by equation 4.38, has been determined. Substituting

in the values for the model, where =16kN/m3, t=Tan19, s=Tan22, and

assuming the Blight (1993) upper limit on the stress ratio of K=1+sin, is

applicable gives the total friction force on the tube of 230.43kg.

The second expression for the vertical pressure distribution, as given by equation

4.39, can be solved in a similar manner. For this pressure distribution the final

expression for the friction force is as given in equation 4.65. However the constants

J and C are as follows:

rR

41

23

rRJ22

(4.66)

rR

41

23

rKRK2C ttss

(4.67)

Substituting the model constants for the silo and tube walls into equations 4.66 and

4.67, and solving equation 4.65 gives a total friction force acting on the tube of

327.60kg.

These equations give values of the friction force in accordance with the values

measured in the laboratory, refer table 4.3.

BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY 5.1

CHAPTER 5

INTRODUCTION TO CYLINDRICAL THIN SHELL

BUCKLING THEORY

Shell structures are defined as thin by their radius to thickness ratio, r/t. The

definition of small and large in terms of r/t is very ambiguous in most texts. The

minimum limit for the radius to thickness ratio is approximately, 150 to 200

whereas the maximum limit is solely dependent on the practical aspects of

constructing such a thin shell. As the r/t ratio gets larger, so the walls of the shell

deviate from their initial curved shape. Thus there are “imperfections” in the shape

of the shell. These imperfections greatly influence the maximum load carrying

capacity of the shell. The early theories dealing with shell analysis assume a

perfectly curved wall and therefore the load capacities are much higher than the

results obtained from experimental research. These imperfections are most

pronounced in the axial load capacity of the shell.

The middle surface of an element of the cylindrical shell wall, shown in figure 5.1,

is synonymous to the neutral axis of a beam. The shell element has a radius r, a

wall thickness t and sides of length dx and rd. The x, y, and z-axes are as shown

in figure 5.1, where the x-axis is parallel to the length of the shell, the y axis is

tangential to the shell wall and the z-axis is positive towards the centre of the shell.

The corresponding displacements along the x, y and z axes are u, v, and w

respectively.

d

dx

x

y

z

dz

t/2

-t/2

rd

middle surface

Figure 5.1: Element of a cylindrical shell wall


5.1 UNIFORM EXTERNAL LATERAL PRESSURE

In the consideration of a thin cylindrical shell subjected to a uniform external lateral

pressure, the simple equation, as given in appendix F, for a uniformly compressed

circular ring can be used as the theoretical buckling load, if the ratio of length to

shell diameter is larger than fifty. For shorter cylinders, Timoshenko and Gere

(1963) use the general equations for the deformation of a cylindrical shell, to

determine the critical buckling load due to a uniform external pressure. This allows

the effects of end restraints to be taken into account, which introduces local

bending for shells with rigid end supports. The analysis of a thin shell requires the

determination of the equilibrium of forces acting on a small element of the shell

wall. Figure 5.2 shows a cylinder subjected to a uniform external lateral pressure,

q, around it’s circumference and along it’s length. The internal membrane forces

acting on the deformed shell element have been shown in Figure 5.3.

x

y

z

Figure 5.2: External lateral pressure on a Cylindrical shell

q

y

d

x

z

rd

r dx

Nx

Nxy

Ny

dxx

NN xx

d

NN y

y

Qx

dxx

QQ xx

Nyx

d

NN yx

yx

dxx

NN xy

xy

Figure 5.3: Enlarged Deformed Element showing the internal forces


The general equations of equilibrium of membrane forces for deformation of a

cylinder due to uniform external lateral pressure are given below:

In the x-direction:

0xw

xvN

N

xN

r2

yyxx

(5.1)

In the y-direction:

0Qx

Nr

Ny

xyy

(5.2)

In the z-direction:

0rqr

wr

v1NQ

xQ

r2

2y

yx

(5.3)

In equations 5.1, 5.2 and 5.3, Timoshenko and Gere assumed that the resultant

forces, except Ny , are small. They neglected all terms containing products of these

resultants with the derivatives of the displacements u, v and w.

Figure 5.4 shows the internal moments acting on the sides of the deformed

element.

Figure 5.4: Deformed Element showing the internal moments

y

d

x

z

rd

r

dx

Mx

Mxy

My

Mx

Myx

Myx

My

Mxy


Equilibrium of moments with respect to the x, y, and z axes given in figure 5.4 are

as given below. Again Timoshenko and Gere assumed that the bending and twisting

moments are small and therefore neglected the products of these moments with

the derivatives of the displacements u, v, and w.

x

MM

r1Q0Qr

M

x

Mr xyy

yyyxy

(5.4)

x

MM

r1Qx0Qr

x

Mr

Mxxy

xxyx

(5.5)

where Qx and Qy are the shear forces acting perpendicular to the element sides as


Substituting for Qy and Qx from equations 5.4 and 5.5 into equations 5.1, 5.2, and

5.3 gives the following:

In the x-direction:

0xw

xvN

N

x

Nr

2

yyxx

(5.6)

In the y-direction:

0x

MM

r1

x

Nr

N xyyxyy

(5.7)

In the z-direction:

0qrwr1v

r11N

M

r1

x

Mr

2

2

y2y

2

2x

2

(5.8)

Timoshenko and Gere give the following expressions for Nx, Nxy and Ny:

wrr

vxu

1

EtdzN2

2/t

2/t

xx (5.9)

xvu

r1

12EtdzN

2/t

2/t

xyxy (5.10)

Ny=-qr (5.11)


Substituting equation 5.11 and the derivatives of equations 5.9 and 5.10 and into

equation 5.6 for the forces in the axial direction, and multiplying this result

throughout by r(1-2)/Et gives the following:

In the x-direction:

0

xw

xvr

xv

2r1u

21

xwr

x

ur22

2

2

2

22

(5.12)

where =qr(1-2)/Et (5.13)

The equation for equilibrium in the x-direction, given by Timoshenko and Gere

(1963) has been repeated below:

0

xw

xvr

xv

2r1u

21

xwr

x

ur22

2

2

2

22

(5.14)

The difference between the two equations (5.12 & 5.14) lies in the second term

in brackets, which comes from the substitution for Nxy. Thus there is a printing error

in equation 5.14 given in Timoshenko and Gere. Working back from this equation,

and integrating this second term in brackets with respect to , and multiplying by

Et/r(1-2) gives the following substitution for Nxy :

xv

11u

r1

12EtNxy (5.15)

Unless =0 , which for steel is not true, equation 5.15 does not equal equation

5.10, and therefore equation 5.14 is incorrect.

Only small deflections from the uniformly compressed form of equilibrium have

been considered, and therefore Ny differs by a small amount from the value of –qr:

Ny = -qr+Ny’ (5.16)

where Ny’ is the small change in the value of Ny due to the deformation of the shell.

Also taking into account stretching of the middle surface during buckling gives the

following expressions for Ny and the external pressure q:

Ny=Ny(1+1) and q=q(1+1)(1+2) (5.17a,b)


where the axial and circumferential strains are given by:

1=u/x and 2=(v/-w)/r (5.18a,b)

The bending moments in the x and y directions are given by:

2

2

222

2

xx

w

r

v

rxwDM (5.19)

2

2

2

2

22yx

w

x

w

r

1v

r

1DM (5.20)

The twisting moment is given by:

xwv1

rDM

2

xy (5.21)

Substituting the derivatives of equations 5.10, 5.16, 5.20 and 5.21 into equation

5.7 for the forces in the y-direction gives the following:

In the y-direction:

.............................wv

x

v2

1rx

u2

1r2

2

2

22

2

0x

v1rx

wrwv................2

22

2

32

3

3

2

2

(5.22)

where =t2/(12r2) (5.23)

Similarly substituting the derivatives of equations 5.16, 5.19, and 5.20 into

equation 5.8 for the forces in the z-direction gives the following:

In the z-direction:

..........w

x

wrx

vr2vwvxur

4

4

4

44

2

32

3

3

2

2

22

42 ww

x

wr2.......... (5.24)

The displacements in the x, y, and z-directions, for the buckled shape, are given in

Timoshenko and Gere(1963) by the following equations, respectively :

u = Asin(n)sin(x/L) , v = Bcos(n)cos(x/L) , w = Csin(n)cos(x/L) (5.25a,b,c)


where 2n is the number of half waves around the circumference and A, B and C are

arbitrary constants.

The derivatives of these displacement equations with respect to x is:

LxCosnSin

LA

xu and u

Lx

u2

2

2

2

(5.26a,b)

LxSinnCos

LB

xv and v

Lx

v2

2

2

2

(5.27a,b)

LxSinnSin

LC

xw and w

Lx

w2

2

2

2

and wLx

w4

4

4

4

(5.28a,b,c)

Similarly the derivatives of equations 5.25a,b,c with respect to is:

LxSin)n(CosAnu and unu 2

2

2

(5.29a,b)

LxCosnBnSinv and vnv 2

2

2

(5.30a,b)

LxCos)n(CnCosw and wnw 2

2

2

and wnw 44

4

(5.31,a,b,c)

Substituting the derivatives given in equations 5.26 to 5.31 into equations 5.12,

5.22 and 5.24, after considerable manipulation, gives the following three

simultaneous equations with three unknown constants A, B, and C, in Timoshenko

and Gere as follows:

In the x-direction:

0Cnn2

1Bn2

1A 22

(5.32)

In the y-direction:

0nnnC1nn2

1Bn2

1A 232222

(5.33)

In the z-direction:

0n1n2n1Cn2nnBA 2224423 (5.34)

where

Et1qr 2

and 2

rt

121

and

Lr (5.35a,b,c)


These three equations can be written in matrix form as follows:

C

B

A

11nn1n2n

1nn1n211n

21

21nn

21

0

0

0

22222

2222

22

(5.36)

Buckling of the cylinder is only possible if the constants A, B, and C in equations

5.32, 5.33 and 5.34, are not equal to zero. Therefore, the determinant of the above

matrix, equation 5.36, must be equal to zero.

The equation for the critical lateral pressure can be simplified by further neglecting

small terms in the determinant to give the following final expression for the critical

lateral pressure acting on the cylinder:

2

22

222

2

22

22

222

2

2cr

rLn11n

1

rLn1

1n21nr12

t

1r

Etq (5.37)

The expression as given by equation 5.37 has been plotted in graphical form in

figure 5.5, for a steel shell with a Young’s Modulus of 200x103Mpa, a Poisson’s ratio

of 0.3 and a length to diameter ratio of 4. A group of curves has been plotted for

n=2 to n=8 and the ratio of thickness to radius was varied from 0.00025 to 0.005.

For a thickness to radius ratio of 0.0035, the shell buckles at a pressure of 0.0166

MPa with 8 half waves around the circumference, as shown by point A in figure5.4.

However, a similar shell will buckle at a higher pressure with only 6 half waves, as

shown by point B, and with 4 half waves as shown by point C in figure 5.5.

Therefore, as the lateral pressure acting on a shell increases, so the number of half

waves into which the shell buckles decreases.


Figure 5.6 shows the pressure on a cylinder as given by equation 5.37 for a steel

cylinder with the number of half waves around the circumference kept constant at

n=3. A group of curves has been plotted for length to diameter ratios of 5, 6 and

50. As can be seen from figure 5.6, the critical pressure on the shell decreases as

the length to diameter ratio increases, for a given thickness to radius ratio.

Figure 5.7 shows the critical lateral pressure on the same cylinder, however the

number of half waves around the circumference has been increased to 8. As can be

seen for any given thickness to radius ratio of the shell, there is very little change

in the value of the critical pressure for a cylinder of length 5 times the diameter

compared to a cylinder of length 50 times the diameter.

Therefore, the statement made by Timoshenko and Gere that the equation for a

uniformly compressed circular ring can be used to calculate the critical pressure for

a cylinder if the length to diameter ratio is large, is only partially true. Figures 5.6

and 5.7 show that as the number of half waves around the circumference increases,

so the critical pressure tends towards the same value, regardless of the cylinder’s

length.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5100t/r

Exte

rnal

late

ral p

ress

ure,

qcr (

MPa

) n=2

n=4

n=6

n=8

Figure 5.5: Critical External Lateral Pressure acting on a Thin Cylindrical Shell

B

A

C


5.2 SYMMETRICAL BUCKLING DUE TO A UNIFORM

AXIAL PRESSURE

The stiffness of a shell around it’s circumference (the membrane stiffness), is much

larger than the bending stiffness of the shell, ie along its axis. Therefore, a great

deal of membrane energy can be absorbed without deforming too much. However

to absorb the same amount of strain energy in bending, the shell has to deform

much more. When the shell is loaded such that most of it’s strain energy is in the

form of membrane compression, the shell will fail dramatically in buckling if this

stored up membrane energy is converted to bending energy.

Two types of buckling exist, non-linear collapse and bifurcation buckling. Non-linear

collapse is predicted by means of a non-linear stress analysis, while the onset of

bifurcation buckling is predicted by means of an eigenvalue analysis. An idealised

load-deflection curve has been given in figure 5.8 for an axially compressed

cylinder, showing the general shape of the various types of buckling paths. For all

paths of buckling, as the load approaches the maximum load, the load-deflection

curve has a nearly zero slope. If the load is maintained, snap through buckling

occurs resulting in dramatic and instantaneous failure.

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Figure 5.7: Critical Lateral pressure for n=8

100t/r

Critic

al L

ater

al P

ress

ure,

qcr,

(Mpa

)

L=5D

L=50D

0.00

0.01

0.02

0.03

0.04

0 0.1 0.2 0.3 0.4 0.5100t/r

Critic

al L

ater

al P

ress

ure,

qcr,

(Mpa

)

Figure 5.6: Critical Lateral pressure for n=3

L=5D

L=50D

L=6D

BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5.11

In figure 5.8, the cylinder deforms approximately axi-symmetrically along the line

OA, until the limit load PL is reached at point “A”. At point A, snap through buckling

occurs and the load-deflection curve follows the line AC. The path OAC is called

fundamental or primary buckling and is associated with axi-symmetric buckling.

The bifurcation buckling point, B, lies between O and A, as shown in figure 5.8. At

this buckling load, the deformations begin to grow in a pattern which is different

from the pre-buckled pattern. Failure of this new deflection mode occurs if the post-

bifurcation load-deflection curve has a negative slope and the applied load is

independent of the deformation amplitude. The post bifurcation buckling path BD is

called the secondary or post buckling path and along this path the cylinder buckles

non-symmetrically.

In both load paths of collapse and bifurcation buckling, the maximum limit occurs at

loads for which some or all of the material is stressed beyond its elastic limit point.

In the case of real structures which contain imperfections there is no such thing as

true bifurcation buckling. The loading path of a real shell follows the curve OEF,

with the failure corresponding to snap through at point E. The collapse load ,Ps, at

point E involves significant non-symmetric displacements.

A

B C

D E

F

PS

PC

PL

Load

“P”

Deflection “”

Limit load of a perfect shell

Limit of an imperfect shell

Bifurcation

Post-Buckling Curve

Figure 5.8: General P- curve for a non-linear analysis


Although true bifurcation buckling is fictitious, a bifurcation buckling analysis is

valid as it gives a good approximation of the actual failure load and mode. The

difference between the bifurcation load PC of the perfect shell, and the actual

collapse load PS of the imperfect shell, depends on the magnitude of the initial

imperfections. A plot of PC/PS versus deflection characterises the sensitivity of the

maximum load, PS, to initial geometric imperfections. This corresponds to the term

of an “imperfection sensitive” shell.

Figure 5.9 shows a plot of test data from Brush and Almroth of the normalised

buckling stress, Pc/Ps, for various r/t ratios of cylindrical shells. The curve shows

that the greater the r/t ratio (ie smaller wall thicknesses) the lower is the Pc/Ps

ratio. Therefore, shells with very thin walls are highly sensitive to initial

imperfections in the walls.

Figure 5.9: Test data for cylinders subjected to axial

compression. Taken from Brush and Almroth

0

0.2

1.0

0.8

0.6

0.4

500

1000

1500

2000

2500

3000

3500

Radius/Thickness

Nor

mal

ised

buc

klin

g st

ress

Theory

A design recommendation

Practical range


5.2.1 BUCKLING DUE TO A UNIFORM AXIAL PRESSURE ( SPECIAL CASE )

Timoshenko and Gere (1963) give a theoretical

method for determining the critical axial buckling

load acting on a thin shell by considering the

potential energy of the shell during deformation.

The shell in figure 5.10 shows a typical thin shell

subjected to a uniform axial pressure, Ncr per unit

circumference length, which Timoshenko and

Gere assume remains constant during buckling.

The shell has a radius r, a wall thickness t, and a

length of L. They define the middle surface of the

shell as the surface which is located in the centre

of the wall thickness. The associated

displacements for the axes system, x,y,z , shown

in figure 5.10 is u,v and w, respectively.

In this theory they have used energy methods to determine the critical axial force

by equating the strain energy to the work done by the external forces. The strain

energy of the shell during buckling consists of the strain due to axial compression

as well as strain of the middle surface in the circumferential direction and bending

of the middle surface.

The strain energy term is made up of bending energy and the strain energy due to

stretching of the middle surface.

U = UC+UB (5.38)

UC is the circumferential strain energy and is given by:

L

0

2

0

ca2

ca2C dxdy12)1(2

hEU (5.39)

where a is the axial strain and c is the circumferential strain.

The circumferential strain energy of the middle surface has been given in

Timoshenko and Gere, using the following expressions for the axial and

circumferential strains. These strains have been shown schematically in figure 5.11.

Figure 5.10: Axial buckling of a thin cylindrical shell

x

y

z

Ncr/unit circumference


o is the axial strain before buckling and is given by:

o = Ncr/Et (5.40)

a is the axial strain and is given by:

a = o-w/r (5.41)

c is the circumferential strain and is given by

c =w/r - o (5.42)

The function for the axial displacements has been

given by Timoshenko and Gere as:

w= A Sin(mx/L) (5.43)

Substituting equation 5.40, 5.41, 5.42 and 5.43 into equation 5.39 gives the

following expression for the change in circumferential strain energy:

dxdrL

xmSinrA

LxmSin

rA12

12

Et

dxdrL

xmSinrA

LxmSin

rA

12

EtU

oo

L

0

2

02

L

0

2

0

2

oo2c

(5.44)

In equation 5.38, UB is the bending energy term and is given in Timoshenko and

Gere as:

dxdy2DU 2

L

0

2

0

xB

(5.45)

where D is the flexural rigidity, (the derivation of D is given in Appendix F):

2

3

112

tED

(5.46)

In the expression for the change in bending strain energy given by equation 5.45

x is the curvature of the shell along its length and is given by:

x = 2w/x2 (5.47)

Figure 5.11: Schematic diagram of strains due to axial pressure

a

c

o

Ncr


Substituting the second derivative of the displacement function given by equation

5.43 into equation 5.47, gives the curvature of the shell along it’s axis as:

x = A(mx/L)2 Sin(mx/L) (5.48)

Substituting equation 5.48 into 5.45 gives the following expression for the change

in bending strain energy:

dxlxmSinr2

lmA

2DU

l

0

24

2B

(5.49)

Substituting equation 5.49 and 546 into equation 5.38 gives the final expression for

the increase in strain energy in the shell during buckling by Timoshenko and Gere

as:

DLrL2

mAr2

LtEAdx

LxmSinAEt2U

4

4422L

0

o

(5.50)

The work done by the compressive forces, W, is equal to force times distance. The

distance through which the axial force travels is given by:

X = (a - 0)+L (5.51)

where L is the shortening due to bending effects. In general terms L is

determined as given in figure 5.12, which shows the wall of the shell buckling in the

vertical plane.

The distance through which the axial force travels is

given by L=L0-L2 , where L0 is the original length,

L2 is the length after buckling, z is the radial axis,

and dz is the change in radius due to buckling.

By small angles :

dz/dx

dxz21L

dx.......dx2

dz1dxCosL

L

0

20

L

0

L

0

2

(5.52)

dxz21LLL

L

0

220

dx

dz

L2

L0

Shell wall

Figure 5.12: Shell wall during buckling


Therefore, the work done is given by: W = Ncr (a - c) + Ncr (L) (5.53)

The radial displacements in the z-direction are given in equation 5.42. Substituting

equation 5.40, 5.41 and 5.42 into 5.53, and multiplying around the circumference

of the shell gives:

L

0

L

0

2

cr dxL

xmCosL

Am21dx

LxmSin

rANr2W (5.54)

Timoshenko and Gere equate the strain energy in the shell to the work done by the

axial compressive force, to give the following expression for the critical axial stress

acting on the shell:

22

2

22

2cr

crBr

EtBD

BrDE

tBD

tS

N

(5.55)

where B= m/L, S is the shell circumference and m is the number of half waves

along the length of the shell. The shell in figure 5.10 has been shown with 5 half

waves along it’s length. Since D, E, S, r and t are all constants, the minimum axial

force required to cause the shell to buckle is given by:

32mincrcr

Br

StE2SDB2NtS

dB

d

(5.56)

The expression for the critical load given by equation 5.55, has been plotted in

figure 5.13 for a steel shell with a Youngs modulus of 200x103 MPa, a Poisons ratio

of 0.3 and a wall thickness of 0.8mm. The graphs in figure 5.13 have been plotted

as a function of the number of half waves along the length of the shell. Four curves

have been shown for a cylinder with a length to diameter (2r) ratio varying from 2

to 8, in increments of 2.

Figure 5.13 shows that the minimum axial load required to cause buckling remains

constant, which in this example is 486.75kN. For a constant shell length at this

minimum buckling load, the shell diameter increases as the number of half waves

along the shell decreases, as shown by points A and B in figure 5.13. Alternatively,

for the same number of half waves along the length of the shell, the axial load

increases as the diameter of the shell decreases, as shown by points C and D.


Since the number of lobes around the circumference of the shell is zero, this

analysis given by Timoshenko and Gere, represents a special case of buckling of a

thin cylindrical shell. Figure 5.14 graphically demonstrates this special case of a

cylinder buckling due to an axial pressure, as given by equation 5.56 with the

number of longitudinal half waves increasing from m=1 to m=36.

450

500

550

600

650

700

750

6 10 14 18 22 26 30

Ncr(k

N)

Length/number of half waves , L/m (mm)

Figure 5.13: Variation of the Critical Axial Load as a function of the Cylinder Length to Number of Half Waves ratio

Min = 486.75 kN

L/2r=2 L/2r=4 6 8

A B

C

D

m=1 m=3 m=9 m=36

Figure 5.14: Special case of axial buckling with number of half waves increasing from 1 to 36.


From equation 5.56 the length of waves into which the shell buckles can be

determined as follows:

3222cr

Br

E2tBD2

B

1dBd

r

EtBD20tS

dB

d

(5.57)

Therefore, L

m

Dr

EtB 42

(5.58)

Solving for L/m gives:

4 2

22

112

trmL

(5.59)

For steel =0.3, therefore equation 5.59 can be approximated by:

tr72.1m/L (5.60)

Equation 5.60 can be re-arranged to give the number of half waves, m, into which

a shell of known length, radius and thickness will buckle, as follows:

m = L / (1.72 rt) (5.61)

The number of half waves as a function of the t/r ratio as given by equation 5.61

has been plotted in figure 5.15 for shell lengths varying from 10r to 30r. The graph

shows that as the length of shell increases so does the number of half waves into

which it will buckle. It can also be seen that thicker shells buckle into fewer half

waves than do shells with a smaller t/r ratio, ie thinner shells.

Num

ber

of h

alf

wav

es,

m.

Shell thickness to radius ratio, 100t/r

100

200

300

400

500

600

700

800

900

1000

0.02 0.06 0.10 0.14 0.18

Figure 5.15: Number of half waves as a function of t/r

L=10r

L=20r

L=30r


5.2.2 GENERAL CASE OF BUCKLING DUE TO AN AXIAL

PRESSURE

The analysis given in section 5.2.1 gives the result for the special case of buckling

when the deformed shape only has half waves along it’s longitudinal axis. There are

cases when the cylinders will buckle with m half waves in the longitudinal direction

as well as 2n half waves around it’s circumference.

Timoshenko and Gere have approached the general case of axial buckling in the

same way as the analysis for buckling due to a uniform external lateral pressure.

Referring to figure 5.3 given in section 5.1, the resultant internal forces due to an

axial pressure give the following three equations of equilibrium in the x, y and z-

directions. Assuming that all the forces except Nx are small, and neglecting the

products of these forces with the derivatives of the displacements u,v and w:

In the x-direction:

0N

x

Nr yxx

(5.62)

In the y-direction:

0Qx

vNrx

Nr

Ny2

2

xxyy

(5.63)

In the z-direction:

0Nx

wNrQ

x

Qr y2

2

xyx

(5.64)

Similarly, referring to figure 5.4, neglecting products of moments with the

derivatives of the displacements, the equations of moment equilibrium are the

same as given in section 5.1. These equations have been repeated below:

In the x-direction

x

MM

r1Q0Qr

M

x

Mr xyy

yyyxy

(5.4 repeated)

In the y-direction

x

MM

r1Qx0Qr

x

Mr

Mxxy

xxyx

(5.5 repeated)

In the z-direction:

0M0)NN(rM yxyxxyyx (5.65)


Substituting for Qx and Qy from equations 5.5 and 5.4, respectively, into equations

5.63 and 5.64 gives the following:

Equilibrium of forces in the x-direction remains unchanged:

0N

x

Nr yxx

(5.57 repeated)

In the y-direction:

0x

MM

r1

x

vNrx

Nr

N xyy2

2

xxyy

(5.66)

In the z-direction:

0NxwrN

M

r1

x

Mr y

2

2

x2y

2

2x

2

(5.67)

The definitions of Nx, Nxy, Mx, My and Mxy are as given by equations 5.9, 5.10, 5.19,

5.20 and 5.21 respectively. Substituting for Nx and Nxy into equation 5.57 gives:

In the x-direction:

0u

r2

1xw

rxv

r21

x

u2

2

2

2

2

2

(5.68)

Timoshenko and Gere give the following definition for Ny:

xu

rwv

r1

1

EtdzN

2/t

2/t2yy (5.69)

Substituting for Ny from equation 5.69, and for My and Mxy into equation 5.66 gives

the following:

In the y-direction:

.............................wv

x

v2

1rx

u2

1r2

2

2

22

2

0x

vrx

v1rx

wrwv...........2

222

222

323

3

2

2

(5.70)


Note that equation 5.70 differs from equation 5.22 only by the last term: ie

equation 5.70 has the extra term of r2v/x2.

Finally substituting for Mx, My, Nx and Ny into equation 5.68 gives the following

equation for the equilibrium of forces:

In the z-direction:

0x

wr2wr1

x

wrx

v2rvr1

22

4

4

4

4

432

3

3

3

(5.71)

The expressions for the displacements have been given in equations 5.25a,b,c in

section 5.1. Substituting the derivatives of these displacements into the above

three equations, gives the following three simultaneous equations, in matrix form:

C

B

A

n12n1n

n1n1n15.12

1n

21nn

21

0

0

0

222222

2222

22

(5.72)

where, as before, =t2/(12r2) and = mr/L

The solution to equation 5.72 can be found by setting the determinant equal to zero

and solving for . Timoshenko and Gere have ignored small quantities of higher

order containing terms in 2 and 2. The solution for then becomes:

GF

Et1N

2

x (5.73)

where F and G are as follows:

..........n32n1F 2442242

64224224 n2n3n7n12.......... (5.74)

...........1n2n12

1n12G 2222222

12n..........

222222 (5.75)


Both the numerator and denominator to equation 5.73 are unitless. Therefore, the

total axial force carried by a shell is given by:

GF

1

Etr2N2

TOTALx

(5.76)

Timoshenko and Gere state that since 2 is a large number, equation 5.76 can be

approximated as follows:

222

2

22

2223

TOTALx

n1

n

r12Et2N (5.77)

Both equations 5.76 and equation 5.77 for the total critical axial load have been

plotted in figure 5.16, as a function of the longitudinal half waves, m. These curves

were calculated for a shell with a Young’s Modulus of 200x103MPa, a Poisson’s ratio

of 0.3, a radius of 300mm, a thickness of 0.6mm, (t/r=0.002), a length of 30 times

the radius (L=9000mm) and the number of half waves around the circumference

2n=4, ie n=2.

0

3000

6000

9000

12000

15000

0 10 20 30 40 50 60 70 80

Number of Longitudinal half waves, m.

Tota

l Axi

al L

oad

Cap

acity

, N

xTO

T (k

N)

Figure 5.16: Total axial load as a function of the number of longitudinal half waves.

Equation 5.76

Equation 5.77

14 138 kN

11 535 kN


From figure 5.15 it can be seen that the approximate equation 5.77 predicts a

higher axial load capacity than given by the full solution, equation 5.76 for m=19.

The difference for this particular cylinder is 22.5% (=100(14138-11535)/11535).

From figure 5.15 it was shown that shells with smaller thickness to radius ratios

buckle into a larger number of half waves than do shells with larger ratios.

Furthermore as the length of the shell increases, so does the number of longitudinal

half waves. For the shell under discussion the approximate number of half waves

into which the shell will buckle, at the minimum axial load capacity has been

calculated from equation 5.61 and found to be 390 waves. Thus the area of the

graph in figure 5.16 applicable to this shell lies beyond the graph where the two

equations coincide.

As the number of half waves around the circumference increases, so the two curves

approach each other. The case of n=2 results in the largest difference between the

approximate curve and the exact curve. For n=3 this difference is reduces to 9.7%

and for n=4 the difference is only 5%.

Therefore, in figure 5.16, it is not possible to be working within the range of fewer

longitudinal half waves, m, where there is a noticeable difference between the two

curves. Therefore, the approximation made by Timoshenko and Gere is

reasonable.

5.3 COMBINED AXIAL AND UNIFORM EXTERNAL

LATERAL PRESSURE

The problem of a thin cylindrical shell subjected to a combination of an axial and

uniform external lateral pressure is approached in the same way as before.

Equilibrium equations for the internal forces and moments acting in the x, y and z-

directions due to the applied loads are first assembled. It is to be expected that

these equations are a combination of the equilibrium equations for the case of

lateral pressure and the case of axial loads. They can be written as follows:

In the x-direction (this is the same as the equation for lateral pressure only):

0xw

xvN

N

xN

r2

yyxx

(5.1 repeated)


In the y-direction (this is the same as for axial pressure only)

0Qx

vNrx

Nr

Ny2

2

xxyy

(5.63 repeated)

In the z-direction (this is a combination of both cases)

0rqx

wNrr

wr

v1NQ

x

Qr

2

2

x2

2

yyx

(5.78)

Similarly the equations of internal moment equilibrium are as follows:

In the x-direction (this is the same as the case of lateral pressure):

x

MM

r1Q0Qr

M

x

Mr xyy

yyyxy

(5.4 repeated)

In the y-direction (this is the same as the case of lateral pressure):

x

MM

r1Qx0Qr

x

Mr

Mxxy

xxyx

(5.5 repeated)

Substituting equations 5.4 and 5.5 into equations 5.1, 5.63 and 5.78 gives the

following:

In the x-direction:

0xw

xvN

N

x

Nr

2

yyxx

(5.6 repeated)

In the y-direction:

0x

MM

r1

x

vNrx

Nr

N xyy2

2

xxyy

(5.66 repeated)

In the z-direction:

0qrxwrNw

r1v

r11N

M

r1

x

Mr

2

2

x2

2

y2y

2

2x

2

(5.77)


The definitions for Nx, Ny, Nxy, Mx, My, and Mxy have been given previously in

equations 5.9, 5.69, 5.10, 5.19, 5.20 and 5.21 respectively. Substituting these

equations into equations 5.6, 5.66, and 5.77 gives the following:

In the x-direction:

0

xw

xvr

xv

2r1u

21

xwr

x

ur2

q

2

2

2

2

22

(5.12 repeated)

In the y-direction:

.............................wv

x

v2

1r

xu

21r

2

2

2

222

0x

vrx

v1rx

wrwv...........2

2

n2

2

222

323

3

2

2

(5.70 repeated)

In the z-direction:

..........w

x

wrx

vr2vwvxur

4

4

4

44

2

32

3

3

2

22n2

2

q22

42

x

wrwwx

wr2..........

(5.78)

where q is due to the external lateral pressure and is given by:

q=qr(1-v2)/(Et) (5.79)

and n is due to the axial pressure and is given by:

n=Nx(1-v2)/(Et) (5.80)

and is as before: =t2/12r2

Using the definitions for the displacements as follows:

u=A Sin(n) Cos(mx/L)

v=B Cos(n) Sin(mx/L) (5.81a,b,c)

w=C Sin(n) Sin(mx/L)


Substituting the derivatives of the displacements given in equations 5.81,a,b,c in

equations 5.12, 5.70 and 5.78 gives the matrix given in equation 5.82.

Note that in the x-direction the term Sin(n)Cos(mx/L) cancels out; in the y-

direction the term Cos(n)Sin(mx/L) cancels out and in the z-direction the term

Sin(n)Sin(mx/L) cancels out.

C

B

A

M

0

0

0

and =mx/L as before (5.82)

where [M]=

n22

2q

22

2222

22

2

22

222

2n22

qq222

r

n1

r

nr12

r

n1nr

r

rn1n1n15.1

r2

1nr

r2

1rnn

21

r

(5.83)

Furthermore, Timoshenko and Gere give the solution to the matrix in equation 5.82

by setting the determinant equal to zero. This results in the following solution:

H + R = Kq + Pn (5.84)

where:

H=(1-2)4 (5.85)

R=(2+n2)4 - 2(6 + 34n2 + (4-)2n4 + n6) + 2(2-)2n2 + n4 (5.86)

K = n2(2+n2)2 – (32n2+n4) (5.87)

P = 2(2+n2)2 + 2n2 (5.88)


Equation 5.84 shows that the lateral pressure and the axial pressure are linearly

related. This equation has been plotted in figure 5.17 for a cylinder with a Young’s

modulus of 200x103MPa, a Poisson’s ratio of 0.3, a wall thickness of 0.6mm, a

radius of 150mm (t/r=0.0004), a length of 7500mm (=50r) and 4 half waves

around the circumference ie n=2.

Three curves have been plotted for the number of longitudinal half waves,

increasing from m=15 to m=35 in increments of 10. Figure 5.17 shows that the

axial load capacity for a cylinder buckling into 35 longitudinal waves is less than the

load capacity for the cylinder to buckle in 25 longitudinal waves. Furthermore from

the difference in the slope of the three lines in figure 5.17, it can be seen that the

external lateral pressure has a greater influence on the axial pressure for a lower

value of m than a higher value.

Figure 5.18 shows the interaction curve, as given by equation 5.84, for a thin

cylindrical steel shell with a length to radius ratio of 4 and a thickness to radius

ratio of 0.0025. The number of half waves along the length and around the

External Lateral Pressure, qcr (MPa)

Tota

l App

lied

Axi

al L

oad,

Nx

(kN

)

Figure 5.17: Total Axial Load, (Nx), as a function of External Lateral Pressure, (qcr).

0

1000

2000

3000

4000

5000

6000

7000

0 1 2 3 4 5 6

m=15

m=25

m=35


circumference have been varied to give the lowest possible failure combination of

the axial and lateral pressures.

0

200

400

600

800

0.00 0.02 0.04 0.06 0.08 0.10 0.12

m=7,n=14

m=3,n=13

m=3,n=14

m=3,n=15

m=3,n=16

Tota

l axi

al lo

ad (

kN)

Lateral Compressive Pressure (MPa)

Figure 5.18: Interaction Graph for a Thin Shell with L=4r.

CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6.1

CHAPTER SIX

CYLINDRICAL THIN SHELL EXPERIMENTS


6.1.1 TENNYSON

Tennyson(1968) conducted tests on thin cylindrical shells with cut outs. The shells

were made from photo-elastic plastic which were spun cast by pouring liquid epoxy

plastic into a cylindrical mould and rotated at high angular speed until the epoxy

plastic had cured. Tennyson conducted tests on 16 cylinders, each with only one

circular cut out located at mid height of the cylinder. The shells were fitted with end

plates to provide clamped constraints. A diagram drawn to

scale of a shell with the largest cut out has been given in

figure 6.1. From a photograph of the shells given in

Tennyson, the shells appear to have a height to diameter

ratio of 2. As can be seen, the holes in Tennyson’s tests

were small compared to the radius of the cylinder. The

shells were tested axially till elastic buckling occurred.

Each cylinder was loaded axially before the holes were cut

out, to determine the reference buckling load. The results

from Tennyson’s tests have been given in table 6.1 along

with the dimensions of the cylinders and holes.

The radius of the cylinders is r, while the radius of the cut out is rc. In Table 6.1,

the reference failure load for cylinders without holes is N0 , and the failure load for

cylinders with holes is Nx. The ratio of shell thickness to radius (t/r) has been given

in the third column of the table, with the inverse of this ratio, ie r/t, in brackets.

The last column of the table gives the ratio of the buckling load of the cylinders

with cut outs to that of the reference buckling load for no cut outs.

A graph of the results from Tennyson has been given in figure 6.2 for the ratio of

Nx/No to the ratio of hole radius to cylinder radius. As Tennyson points out there is

a unique correlation between the test data which appears to be independent of the

cylinder thickness to radius ratio, t/r.

Figure 6.1: Diagram of thin shell with cutout.

2r

2rc


r (mm) t (mm) t/r (r/t) rc(mm) rc /r Nx (kN) Nx/N0

140.97 0.43 0.00302

(331)

0 0 1.557 1 12.7 0.09 0.868 0.597 15.88 0.113 0.801 0.514

140.21 0.48 0.00342

(292)

0 0 1.958 1 5.82 0.042 1.686 0.861 8.33 0.059 1.357 0.693 10.24 0.073 1.210 0.618 11.71 0.084 1.139 0.582 15.86 0.113 0.997 0.509 20.90 0.149 0.899 0.459 23.39 0.167 0.916 0.468

67.31 0.42 0.00617

(162)

0 0 1.842 1 3.18 0.047 1.406 4.78 0.071 1.166 0.632 6.35 0.094 1.005 0.546 12.7 0.189 0.828 0.449

Table 6.1: Test data from tests by Tennyson (1968)

As can be seen from the graph the larger the cut out the lower is the buckling load

of the cylinder. Tennyson gives a trendline through the data which gives an

apparent minimum for the ratio of Nx/No. This does not seem to be a reasonable

conclusion as it is doubtful whether a hole to shell radius of 0.5 gives the same

buckling load as a ratio of 0.2, and therefore the trend line has been omitted in

figure 6.2.

Figure 6.2: Results from Tennyson(1968)

Nx/

No

rc/r 0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2


Tennyson states that the reason for the drastic reduction of buckling load for a

cylinder with a cut out is due to the growing imperfection in the shape of the

cylinder in the region of the cut out.

6.1.2 ALMROTH AND HOLMES

Almroth and Holmes (1972) conducted tests on 11 thin walled aluminium cylinders,

four with unreinforced cutouts and five with reinforced cutouts. In all cases two

rectangular cutouts were made in the cylinder at mid height and 180 apart on the

circumference. Before the rectangular holes were cut, the cylinders were subjected

to axial pressure to determine a reference buckling load No. Table 6.2 gives the test

data from Almroth and Holmes, converted to SI units. The length of the rectangular

cutouts has been given in Almroth and Holmes as a 30 or 45 arc in plan, as shown

in figure 6.3 drawn to scale. The segment length of the arc, S, has been calculated

and added to table 6.2. Their cylinders were 9 inches high with an outer diameter

of 12.75 inches, and a varying wall thickness. The average radius of each cylinder,

rav, has been given to the middle surface of the cylinder.

The first four rows of data in table 6.2 are for cylinders with unreinforced cut outs

while the remaining seven rows of data are for cylinderrs with reinforced cut outs.

From table 6.2 it can be seen that the cut outs were all of approximately the same

dimensions, with only one being smaller than the rest. Consequently little

information can be gained from plotting their results as a function of the hole

dimensions. However, the wall thicknesses of the cylinders tested do vary, and

therefore the ratio of the buckling load to the reference buckling load, Nx/No, has

45

S

2r

2r

S

76.2

228.

6

Plan Elevation

Figure 6.3: Diagram of cylinders with cut outs tested by Almroth and Holmes

Cut outs


been plotted against the ratio of the wall thickness to the radius, t/r, for each

cylinder. The graph of their test results has been shown in figure 6.4. The data

points marked with a dot are for unreinforced cut outs, while the data points

marked with a star are for reinforced cut outs.

rav

(mm)

t

(mm) A (mm2)

S

(mm) Ac (mm2) t/r Nx (kN) Nx/No

161.83 0.38 232444.1 84.74 6456.8 0.0083 12.19 0.68 161.83 0.37 232444.8 127.10 9685.2 0.0067 11.30 0.55 161.84 0.33 232444.9 127.11 9685.9 0.0059 9.12 0.48 161.86 0.25 232444.1 127.13 9687.1 0.0044 3.59 0.64 161.83 0.37 232445.2 127.10 9685.2 0.0067 14.19 0.85 161.84 0.34 232457.7 127.11 9685.7 0.0061 12.68 0.77 161.84 0.35 232454.1 127.11 9685.6 0.0063 11.39 0.76 161.84 0.35 232453.5 127.11 9685.6 0.0063 11.57 0.85 161.86 0.24 232492.1 127.13 9687.2 0.0043 5.67 0.89 161.86 0.24 232491.8 127.13 9687.2 0.0044 4.58 0.75 161.86 0.24 232491.8 127.13 9687.2 0.0044 4.69 0.68

Table 6.2: Test data from Almroth and Holmes (1972)

The data from Almroth and Holmes shows that cut outs which are reinforced around

the edges fail at approximately 68% to 88% of the reference buckling load,

whereas unreinforced cut outs fail at approximately 48% to 68% of the reference

buckling load.

0.4

0.6

0.8

1.0

0.000 0.003 0.006 0.009

Wall thickness to cylinder radius ,t/r.

Rat

io o

f bu

cklin

g lo

ad t

o re

fere

nce

buck

ling

load

, N

x/N

o

Figure 6.4: Test data from Almroth and Holmes (1972)

Reinforced cut outs Unreinforced cut outs


In some instances, reinforcing the cut out edges can almost double the axial load

capacity of a cylinder with perforations. From figure 6.4 it can be seen that the

buckling load of cylinders with cut outs, reinforced or unreinforced, depends on the

ratio of wall thickness to cylinder radius, which is in contrast with the test results

from Tennyson. One possible explanation is the difference in the shape of the cut

outs and the different materials of the cylinders. Tennyson used plastic cylinders

whereas Almrtoh and Holmes used aluminium cylinders.

6.1.3 STARNES JR

Starnes Jr (1972) conducted tests on thin walled cylindrical shells made of Mylar™

polyester film and made of copper, with a single circular hole at mid height of the

cylinder. Mylar was chosen by this researcher for it’s ability to buckle elastically as

well as not being sensitive to handling, thereby allowing the same shell to be tested

several times with an increasing hole size. These shells were made from

rectangular sheets of Mylar™ which were lap jointed with a flexible adhesive.

Starnes Jr does not state in the report where the location of the hole was relative to

the lap joint, and it is assumed in this thesis that the hole was placed diametrically

opposite to the joint. The copper shells were manufactured by an electroforming

process. A fly cutter was used to cut the hole in the shell and subsequent surface

measurements showed no apparent bending around the hole due to cutting. The

Mylar™ shells were 203.2mm in diameter and 254mm long, while the copper

cylinders had a length and diameter of 203.2mm. The results from Starnes Jr tests

on copper shells have been given in table 6.3 and the results from the Mylar shells

have been given in table 6.4 in SI units. Starnes Jr does not give the value of Nx

and No for the copper shells however, the ratios of the stress Sx due to the applied

load to the reference buckling stress So have been given.

rc (mm) r (mm) rc/r Sx/So

4.06 101.63 0.0399 0.433 5.03 101.63 0.0495 0.395 6.10 101.63 0.0600 0.391 4.06 101.63 0.0399 0.507 3.05 101.68 0.0299 0.531 10.16 101.68 0.0999 0.398

Table 6.3: Test data for copper shells from Starnes Jr (1972)


rc (mm) rc/r Nx (kN) Nx/No

0 0 0.994 1 0.64 0.00625 0.994 1 1.27 0.0125 0.994 1 1.91 0.01875 0.994 1 2.54 0.025 0.983 0.989 3.18 0.03125 0.894 0.9 4.06 0.04 0.705 0.709 5.08 0.05 0.583 0.586 6.35 0.0625 0.505 0.508 7.62 0.075 0.516 0.519 8.26 0.08125 0.449 0.452 8.89 0.0875 0.444 0.446 10.16 0.1 0.438 0.441 15.88 0.1563 0.416 0.418 21.34 0.21 0.394 0.396 26.04 0.2563 0.366 0.368 31.12 0.3063 0.338 0.34 40.64 0.4 0.283 0.284 51.44 0.5063 0.242 0.244

Table 6.4: Test data for Mylar shells from Starnes Jr (1972)

A graph of the test results on the copper shells have been given in figure 6.5 with

the vertical axis in terms of Sx/So plotted as a function of the ratio of the cut out

radius to the cylinder radius. The results from the Mylar shells have been given in

figure 6.6 with the vertical axis in terms of Nx/No also plotted as a function of the

ratio of the cut out radius to the cylinder radius, rc/r. As can be seen from figure

6.5 and 6.6, the results lie on a similar shaped curve to the results from Tennyson.

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 6.5: Test results for copper shells from Starnes Jr (1972)

Ratio of cut out radius to cylinder radius, rc/r

Rat

io o

f bu

cklin

g st

ress

to

refe

renc

e bu

cklin

g st

ress

, N

x/N

o


6.1.4 SCUTELLA

L. Scutella (1998) conducted eighteen tests on thin walled cylindrical steel shells

with multiple perforations around the circumference and along the length of the

shell. The cylinders had a wall thickness of 0.8mm, a radius of 150mm and a length

of 600mm. The cylinders were made from cold formed steel sheets which were pre-

punched and then rolled and butt welded to form a cylinder. The cylinders were

fitted with end caps which were clamped on, giving constrained edge conditions.

The cylinders were then subjected to a uniform axial compression in a displacement

controlled test. A diagram of the layout of the holes on the flat sheet prior to rolling

has been given in figure 6.7.

The ratio of the hole area to the cylinder surface area (here after called cut out

area) was varied by increasing the hole sizes from an 8mm diameter hole to a

102mm diameter hole. The same ratio of cut out area was also achieved by

increasing the centre to centre spacing of the holes. The cut out area has been

calculated by the ratio of one hole area to the area of one diamond square. These

concepts have been diagrammatically shown in figure 6.8 (a) and (b).

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6

Rat

io o

f bu

cklin

g lo

ad t

o re

fere

nce

buck

ling

load

, N

x/N

o

Ratio of cut out radius to cylinder radius, rc/r

Figure 6.5: Test results for Mylar™ shells from Starnes Jr (1972)


600m

m

942mm

45

Figure 6.7: Hole layout for flat sheet rolled into cylinders

Figure 6.8 (a): Increasing the cut out area by increasing the hole sizes and maintaining the centre to centre spacing of the holes.

Cut out area =5.3%

Cut out area =16.5%

Cut out area =36.6%

Cut out area =65.9%

Figure 6.8 (b): Maintaining the cut out area at 5.3% by increasing the hole sizes and increasing the centre to centre spacing of the holes.

Cut out diameter =29mm




Three sets of tests were done on a cylinder with no holes to determine a reference

buckling load, No , which was found to give an average value of 113.46kN. Table

6.5 gives the average results of the data for three tests of each cut out area and

hole arrangement, by Scutella. From the data for a cut out area of 0.05% there

appears to be little difference between a hole size of 8mm and 13mm, however

when the spacing and hole size was increased to 29mm there appears to be a drop

in the buckling capacity of the cylinder by approximately 37%.

rc (mm) Ac (mm2) Ac/A Nx (kN) Nx/No

8 50.3 0.05 96.42 0.85 12 113.1 0.112 77.72 0.68 13 132.7 0.05 96.46 0.85 29 660.5 0.05 60.04 0.53 51 2942.8 0.16 31.4 0.28 76 4536.5 0.37 13.04 0.12 102 8171.3 0.66 2.37 0.02

Table 6.5: Test data from Scutella (1998)

A graph of the test results from Suctella has been shown in figure 6.9. As can be

seen from figure 6.9, the data points tends to lie on a curve similar to those from

Tennyson and Starnes Jr. There is a very rapid decrease in the buckling capacity of

a cylinder with multiple perforations as the ratio of cut out area is increased from

0.05 to approximately 0.25.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Ratio of cut out area to cylinder surface area, Ac/A

Rat

io o

f bu

cklin

g lo

ad t

o re

fere

nce

buck

ling

load

, N

x/N

o

Figure 6.9: Test results from Scutella (1998)


Figure 6.10 shows the data points for a cylinder with a cut out area of 5% with an

increasing hole size and an increasing centre to centre hole spacing. Since the data

points appear to give a linear trend, a line has been drawn through these points

with the intercept on the y-axis set to 1 which represents the data point for a

cylinder with no cut outs.

From the results by Scutella, given in figure 6.9 and 6.10, the buckling capacity of

a cylinder with multiple perforations appears to be a function of not only the cutout

area but also the centre to centre hole spacing. The trendline through the data

points in figure 6.10 has the following equation:

Nx/No = 1 - 0.004 s

where s is the centre to centre spacing of the holes.

6.2 DISCUSSION

In both the papers by Tennyson(1972) and Starnes Jr (1972), reference is made to

a parameter which the researchers suggest governs the buckling capacity of the

cylinder with cut outs. In Tennyson this parameter is as follows:

tr8

r112 c4 2

(6.1)

0.4

0.6

0.8

1

0 40 80 120

Centre to centre hole spacing (mm)

Rat

io o

f bu

cklin

g lo

ad t

o re

fere

nce

buck

ling

load

, N

x/N

o.

Figure 6.10: Test results from Scutella (1998) for a cylinder with a cut out area of 0.05%


where is Poissons ratio, rc is the cut out radius, r is the cylinder radius and t is the

cylinder thickness.

Since 8=22, equation 6.1 can be reduced to the following expression:

tr2

rK

tr2

rK

2cc (6.2)

where K = 0.5 [12(1-v2)]1/4

(6.3)

A similar parameter is found in Starnes Jr as follows:

tr

r112

21

2c4 2

(6.4)

which can be reduced as follows:

tr

rK

2c (6.5)

where K is as before and all other symbols are as defined previously.

The only difference between equation 6.2 and 6.5 is the value of 2 in the

denominator of equation 6.2. Multiplying the numerator and denominator in both

equations by gives the following:

tr2

rK

2c

for Tennyson’s expression (6.6)

tr

rK

2c

for Starnes Jr’s expression (6.7)

Both equations 6.6 and 6.7 are expressions in terms of a constant multiplied by the

square root of the ratio of the cut out area to the area of cylinder material in plan.

However, in equation 6.7 the ratio has only half the cylinder material in the

denominator. Both these expression suggest that the buckling capacity is a function

of how much material has been removed from the cylinder by the cut out.


If both Tennysons data and Starnes Jr’s data are plotted as a function of the ratio

of the cut out area to the cylinder surface area ( viz (rc2/2rL), the results plot as


For both sets of data a power series trendline has been plotted through the test

results. As can be seen there is a very good agreement between the two sets of

test results when using the ratio of cut out area to the cylinder surface area rather

than the ratio of cut out area to the cylinder plan area. This similarity between the

results may be due to the fact that both Tennyson and Starnes Jr conducted their

tests on plastic cylinders which could be manufactured to a very high standard.

In a similar manner plotting the test results of Almroth and Holmes (1972) and

Scutella(1998), as a function of the ratio of the cut out area to the cylinder surface

area, produces the graphs shown in figure 6.12.

As can be seen these curves also result in a power series plot, however the

agreement between the results is not very good. While both tests were conducted

on steel cylinders, the difference between the results may be due to a different

degree of wall imperfection in each case.

Nx/No = 0.1584(Ac/A)-0.1825

0

0.2

0.4

0.6

0.8

1

0.000 0.001 0.002 0.003 0.004

Tennyson(1968)

Nx/No = 0.1574(Ac/A)-0.1789

0

0.2

0.4

0.6

0.8

1

0.000 0.005 0.010 0.015 0.020

Starnes Jr (1972)

Ratio of cutout area to cylinder surface area, Ac/A

Ratio of cutout area to cylinder surface area, Ac/A

Nx/

No

Nx/

No

Figure 6.11: Test results from Tennyson (1968) and Starnes Jr (1972) plotted as a function of the cut out area to the cylinder surface area.

PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL SET-UP 6.13

The buckling capacity of a shell with perforations is therefore a function of the

amount of material that has been removed from the walls of the shell, and is not a

function of the plan area of the cylinder as initially suggested by Tennyson and

Starnes Jr. Furthermore, from figure 6.10 it can be seen that the reduction in

buckling carrying capacity of a cylinder with multiple perforations is also a function

of the centre to centre spacing of the cut outs.

From the test results given in figure 6.11 and 6.12, it is can be seen that the

reduction in buckling capacity of a cylinder with perforations, either multiple or

single, subjected to a uniform axial pressure plots as a power series given by

equation 6.8 as follows:

Nx/No = C (Ac/A)-B (6.8)

where Nx is the buckling capacity of a shell with perforations

No is the reference buckling load of a shell without perforations

C, and B are empirically determined constants

Ac is the total area of the cut outs

and A is the surface area of the cylinder.

Nx/No = 0.023(Ac/A)-1.2306

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

Scutella (1998)

Nx/No = 0.1562(Ac/A)-0.5089

0

0.2

0.4

0.6

0.8

1

0.00 0.02 0.04 0.06 0.08 0.10

Almroth and Holmes(1972)

Nx/

No

Nx/

No

Ac/A Ac/A Figure 6.12: Test results from Almroth and Holmes(1972) and Scutella(1998)


6.3 PERFORATED CYLINDER EXPERIMENTS

6.3.1 EXPERIMENTAL SET-UP

Following on from the experiments done by L Scutella, tests were done in the

laboratory on thin shells with multiple perforations subjected to a combined uniform

axial and lateral pressure. The shells were formed from thin cold-formed steel

sheets with a thickness of 0.8mm. The holes were pre-punched in the sheets prior

to rolling into a cylindrical shell. Figure 6.14 and 6.15 shows photographs of the

punching of the cut outs in the sheets and the rolled sheet once punching was

complete. The setting out of the holes started from the centre of the sheet to

minimise the effects of accumulated measurement errors. The centre to centre hole

spacing was kept constant at 157.5mm and the diameters of the cut outs were

51mm and 76mm. This gave a cut out open area of 16.5% and 36.6% respectively.

To ensure that no holes were located on the seam, the setting out was such that

the centreline of the sheet aligned with the three-quarter line of the hole group.

The dimensions of the sheet and hole centre to centre spacing have been drawn to

scale in figure 6.16.

Figure 6.14 : Photo of cut out punching process

Figure 6.15 : Photo of punched sheet rolled into cylindrical shape prior to welding the seam


Figure 6.16a shows the sheet with the 51mm diameter hole and figure 6.16b shows

the sheet with the 76mm diameter holes. These figures give a good visual

appreciation of the difference between cut out open area of the two groups of

cylinders investigated.

157.

5mm

157.5mm

45

942mm

600m

m

Figure 6.16a: 0.8mm thick sheet with 16.5% cut out open area (51mm diameter pre-punched holes)

Figure 6.16b: 0.8mm thick sheet with 36.6% cut out open area (76mm diameter pre-punched holes). No holes were located on the shell seam


After the holes were punched, the sheet was rolled into a cylindrical shape and the

seam was butt welded. To weld the seam, the rolled sheet was clamped into the

two end caps which had a diameter of 300mm. A diagram of the end clamps, which

were also used during the tests to ensure constraint end conditions of the cylinder,

is shown in figure 6.17.

The circular clamps and end caps were ridged on the inside to ensure a good grip

on the cylinder wall and thereby stop any possibility of slippage. The end caps were

closed with a plate welded to the sides of the cross beams and the cap. To apply

the axial pressure to the cylinder, one end cap was fitted with a solid half steel ball

and notched for seating of the loading rod, as shown in figures 6.18a and b. The

bottom end cap was also notched for seating on a small round stainless steel ball.

Top view

25mmx45mm deep cross beams for carrying axial load

circular end caps

cylinder fitted over end caps

circular clamps (not shown) hold cylinder in place Side view

Figure 6.17: Top View and Side view of cylinder end caps

Spaces between cross beams closed off with plates welded to the beams and cap

Loading sphere

Figure 6.18b: Photo of Loading ball on end cap

Loading ball

Loading rod connected to Instron

Top of end cap

Figure 6.18a : Loading ball and rod


A dummy PVC tube of 250mm diameter (50mm smaller than the perforated

cylinders) was placed inside the perforated cylinders. Polystyrene filler plugs were

then fixed with double sided tape to the inner dummy tube through the cut outs. A

plastic sheath was fitted over the perforated cylinder and clamped to the end caps

with the cylinders, to ensure that the whole assembly was water tight. This

assembly has been shown in diagrammatically figure 6.19 below.

To ensure the dummy PVC tube did not carry any of the applied axial load, the top

of the tube was notched by 30mm to match the alignment of the cross beams of

the end caps. This allowed the end caps on the perforated cylinders to move the

required vertical distance of 25mm as well as maintaining the dummy tube in place

while lifting the assembly inside the loading bin. This has been shown

diagrammatically in figure 6.20.

Figure 6.19: Perforated cylinder assembled with plastic sheath, dummy tube, plugs and end caps

Dummy PVC tube

Polystyrene filler plugs

plastic sheath

Perforated cylinder

End cap

End cap

Dummy PVC tube with notched top edge

End cap and cross beams

Cross beams fit inside notches

Figure 6.20: Notched dummy PVC tube to accommodate end cap.


To apply the lateral compressive pressure to the outside of the cylinder, the whole

assembly as shown in figure 6.19 was placed inside a large container which had a

removable top lid. This container, the loading bin, was fitted with a small stainless

steel ball in the centre of its base, thereby ensuring the cylinder assembly could be

placed centrally. The loading bin was first placed in the Instron machine and then

the cylinder assembly was lifted into place so that the notched bottom end cap

fitted the stainless steel ball at the base of the bin. The loading bin was then filled

with water up to a short distance from the top, leaving an air gap. The lid of the

loading bin was fitted with a rubber O-ring, and bolted to the main body of the

container. Through the lid were two valves, to which an air supply and pressure

gauge were fitted. Thus the top air gap inside the loading bin was pressurised from

the compressed air supply, and the amount of pressure applied was read off the

gauge. In this manner a constant lateral pressure was applied to the cylinder

assembly during the testing process. A photograph of the loading bin and top

valves has been shown in figure 6.21. The lid also had an opening with a rubber O-

ring through which the loading rod from the Instron machine was placed.

Opening for loading rod

Valve for compressed air supply

Valve for air pressure gauge

Top lid bolted to container

Valve for draining water after completed test

Construction joint

Figure 6.21: Photograph of container for cylinder assembly

PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL RESULTS 6.19

6.3.2 EXPERIMENTAL RESULTS

Once the perforated cylinder, dummy PVC tube, polystyrene fillers, plastic sheath

and end caps had been assembled and placed in the loading bin, the perforated

cylinders were subjected to axial displacement controlled tests in the Instron. For

each test a constant lateral pressure was applied first and then the displacement

controlled test started. For each group of cylinders, ie 51mm diameter cut outs and

76mm diameter cut outs, the lateral pressure was varied from 10kPa to 40kPa in

increments of ten. The data from Scutella(1998) has been used as the reference

buckling capacity for perforated cylinders with only applied axial pressure. Table 6.6

gives a summary of the results from the combined axial and lateral pressure tests.

The results from Scutella(1998) have been included in Table 6.6 for the case of

zero lateral pressure. The full set of test results have been plotted graphically in

Appendix G.1 for the 16.5% cut out area and in Appendix G.2 for the 36.6% cut

out area.

Applied Lateral Pressure (kPa)

Maximum Applied Total Axial Load (kN)

51mm Cutouts 76mm Cutouts

0 31.4 (Scutella) 13.04 (Scutella) 10 26.27 11.51 20 32.25 9.36 30 17.13 12.78 40 18.55 6.21

Table 6.6: Test results for cylinders with multiple perforations

As a reference point, the capacity of the perforated cylinders with a 16.5% and

36.6% cut out area due to a lateral pressure only was also determined. This was

done using the assembled system in the loading bin and applying a lateral pressure

in increments of 5kPa up to a maximum of 60kPa. Inside the perforated cylinders

were placed four displacement transducers around the circumference, which

measured the inwards movement of the cylinder walls. The applied lateral pressure

was plotted as a function of the inward wall movement as shown in appendix G.3

for the 16.5% and 36.6% cut out areas. Figure 22 shows a photograph of the

cylinder with a 16.5% cut out area after failure due to a lateral pressure only.


The failure load for the cylinders with 16.5% and 36.6% cut out areas were 42kPa

and 47kPa, respectively.

The results from the perforated cylinder tests have been plotted in figure 6.23 for

the perforated cylinders with 16.5% cut out area and in figure 6.24 for the

cylinders with 36.6% cut out area.

Figure 6.22: Buckled perforated cylinder due to lateral pressure only (16.5% cut out area)

0

5

10

15

20

25

30

35

0 10 20 30 40 50

Tota

l Axi

al L

oad

(kN

)

Figure 6.23: Test results for combined pressure on perforated cylinders with 16.5% cut out area


0

2

4

6

8

10

12

14

0 10 20 30 40 50

Tota

l Axi

al L

oad

(kN

)


Figure 6.24: Test results for combined pressure on perforated cylinders with 36.6% cut out area


From the theory of thin walled shells subjected to a combined axial and lateral

pressure, given in chapter 5.3, it was shown that the relationship between the axial

and lateral pressure is linear. Furthermore, from the finite element analyses given

in section 6.4 of this chapter, it was shown that the relationship between the axial

and lateral pressure for a cylinder with multiple perforations consists of two straight

lines as shown in figure 6.23 and 6.24.

On closer inspection of the test results for the 16.5% cut out area, the data point

for an applied lateral pressure of 20kPa would appear to be too high. Therefore,

ignoring this point, a best fit straight line can be drawn through the remaining data

points for an applied lateral pressure up to 40kPa. The equation of this line is given

as follows:

Nx16.5 = 31.4 - 0.382 q (6.9)

where Nx16.5 is the total applied axial load in kN acting on a cylinder with cut out

area of 16.5%, q is the applied lateral pressure in kPa and 31.4 is the total axial

load capacity of a cylinder with 16.5% cut out area with zero lateral pressure.

The second best fit line has been drawn through the data point for an applied

lateral pressure of 40kPa and the reference pure lateral pressure, 47kPa. This line is

given by the following equation:

Nx16.5 = 124.55 - 2.65q (6.10)

Similarly from the results for the cylinder with 36.6% cut out area, the data point

for an applied lateral pressure of 30kPa appears too high and has therefore also

been ignored. The best fit straight lines through the remaining points have been

shown in figure 6.24. For the data points up to an applied lateral pressure of 40kPa,

the equation of the straight line is as follows:

Nx36.6 = 13.04 - 0.172 q (6.11)

where Nx36.6 is the total applied axial load in kN acting on a cylinder with cut out

area of 36.6%, q is the applied lateral pressure in kPa and 13.04 is the total axial

load capacity of a cylinder with 36.6% cut out area with zero lateral pressure.

The second straight line through the data point for an applied lateral pressure of

40kPa and the reference buckling pressure of 42kPa for an applied lateral pressure

only is as follows:


Nx36.6 = 130.41 – 3.105 q (6.12)

Using these equations, interaction curves can be drawn for the capacity of a

cylinder with multiple perforations subjected to a combination of an applied lateral

pressure and total applied axial load as shown in figure 6.25. The areas under the

curves represent the load combinations which can safely be sustained by cylinders

with a cut out area of 16.5 and 36.6% .

From figure 6.25 it can be seen that the influence of the lateral pressure is to

reduce the applied axial load up to a point when the lateral pressure is close to the

value of the cylinders lateral failure pressure. At this point the applied axial load is

approximately half the value of the axial load for zero lateral pressure acting.

Beyond this point the lateral pressure has a stronger influence on the axial load by

rapidly reducing it’s value to zero.

From equation 6.9, the axial load which a cylinder with 16.5% cut out area can

carry when subjected to a lateral pressure of 40kPa is as follows:

Nx16.5 = 31.4 - 0.382*40 = 16.12 kN

Similarly for a cylinder with a cut out area of 36.6%, the total axial load which can

be carried with a combined lateral pressure of 40kPa is as follows:

0

5

10

15

20

25

30

35

0 10 20 30 40 50

Tota

l axi

al lo

ad N

x ,

(kN

)

Applied lateral pressure q, (kPa)

Safe combined loads for cut out area of 16.5%

Safe combined loads for cut out area of 36.6%

Figure 6.25: Interaction graphs for cylinders with multiple perforations (cut out areas of 16.5% and 36.6%)


Nx36.6 = 13.04 - 0.172*40 = 6.16 kN

Plotting the ratio of these values to the reference buckling loads (31.4 for the

16.5% cut out area and 13.04 kN for the 36.6% cut out area) as a function of the

cut out area of the cylinder, has been shown in figure 6.26.

From the work by Tennyson(1968), Starnes Jr (1972), Almroth and Holmes (1972)

and Scutella (1998) it was shown that the curve for the axial load on cylinders with

perforations has the shape of a power series. It has been assumed in this thesis

that this same shape curve will be applicable to a cylinder with multiple

perforations, which has a combination of an applied axial and lateral pressure.

Therefore, plotting a power series trendline through the points in figure 6.26 gives

the curve of the total axial load on a cylinder with multiple perforations and a

constant lateral pressure of 40kPa as a function of the cut out area. An additional

set of curves have also been plotted for a constant lateral pressure of 20kPa and

5kpa.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4

Constant lateral pressure:

5kPa

20kPa

40kPa

Ratio of cut out area, Ac/A

Rat

io o

f ax

ial l

oad

to r

efer

ence

buc

klin

g, N

x/ N

o

Figure 6.26: Axial load as a function of cut out area for a constant lateral pressure

0.165 0.366


Table 6.7 gives the data points used to plot the curves in figure 6.26. The equations

of the resulting power series trendlines have also been given in table 6.7 for each

curve of constant lateral pressure.

Lateral Pressure

(kPa)

Cut out area

Equation Number

Calculated Nx

(kN)

Nx/No Equation of power series trendline

Nx/ No =

5 0.165 6.9 29.15 0.257 0.0357 q (-1.0947) 0.366 6.11 12.14 0.107

20 0.165 6.9 23.37 0.206 0.0274 q (-1.1194) 0.366 6.11 9.64 0.085

40 0.165 6.9 15.77 0.139 0.0165 q (-1.1824) 0.366 6.11 6.13 0.054

Table 6.7 : Data points for curves in figure 6.25

Photographs of the buckled cylinders with 16.5% and 36.6% cut out area subjected

to a combined total axial load and external lateral pressure have been shown in

figures 6.27 and 6.28 respectively.

Figure 6.27a: Perforated Cylinder with 16.5% cut out area subjected to a total axial load of 26.27kN and 10kPa lateral pressure

Figure 6.27b: Perforated Cylinder with 16.5% cut out area subjected to a total axial load of 17.13kN and 30kPa lateral pressure


Figure 6.27c: Perforated Cylinder with 16.5% cut out area subjected to a total axial load of 18.55kN and 40kPa lateral pressure

Figure 6.28a: Perforated Cylinder with 36.6% cut out area subjected to a total axial load of 11.51kN and 10kPa lateral pressure

Figure 6.28b: Perforated Cylinder with 36.6% cut out area subjected to a total axial load of 9.36kN and 20kPa lateral pressure


6.4 FINITE ELEMENT ANALYSES

6.4.1 DESCRIPTION

Since there is no existing theory for the buckling capacity of cylinders with multiple

perforations with which to compare the test results presented in chapter 6.3.2,

finite element analyses were performed on cylinders with similar ratios of cut out

area. The thin walled shells considered in the finite element analyses were cylinders

with 50mm diameter cut outs and cylinders with 76mm diameter cut outs. The

arrangement and spacing of the holes was as given in figure 6.16 of chapter 6.3.1.

In addition to cylinders with multiple perforations, a solid shell with no cut outs was

also analysed and the results used as a reference for cylinders with multiple

perforations.

The finite element (FE) software package used for these analyses was the student

version of Ansys 5.6/Mechanics which includes structural and thermal non-linear

capabilities.

For all the cylinders analysed, the most suitable element chosen from the FE

software package was the quadrilateral element “shell63” which has both bending

and membrane capabilities. Loads on this element can be in-plane node or edge

loads, or positive surface loads acting into the element. Stress stiffening and large

deflection capabilities are also included. The element is defined by four nodes, four

thicknesses specified at each node and orthotropic material properties. The

geometry, node locations and local co-ordinate system of this element have been


L

K

J

I

x

z

y

I

J

K,L

Triangular option of ‘shell63’

Figure 6.29: Geometry and node location of element ‘shell63’

PERFORATED CYLINDERS : FINITE ELEMENT ANALYSIS 6.27

The external lateral pressures were applied as equivalent element loads at the

nodes as this produced more accurate stress resultants on the flat elements

representing the curved surface of the cylinders.

The easiest method of entering the model geometry into the FE package was to use

the solid modelling option. In this method a repeatable segment of the cylinder was

modelled with the cut outs of the required diameter. The element mesh density was

then generated and the segment copied 360 around the circumference of the

cylinder. The resulting portion of the cylinder was then copied several times to

obtain the required cylinder length. Figure 6.30 shows this procedure.

Once the FE cylinder models had been generated, the cylinders were analysed for

non-linear buckling with initial wall imperfections, to simulate a more realistic

model. To obtain the initial imperfections, an eigenvalue buckling analysis was first

performed on each FE cylinder model and several modes from each analysis were

expanded. The eigenvalue buckling analysis represents the theoretical buckling

strength of an ideal linear elastic structure. The wall displacements of each mode

shape were entered separately into the model data base as an updated geometry of

the cylinder, and the model analysed first for axial loads and then lateral pressures.

Each mode shape had to be entered as an initial imperfection, since it was not

possible to state with any certainty which mode shape would result in the lowest

buckling load.

Once the geometry of the FE models had been updated, both axial and lateral non-

linear buckling analyses were performed separately on each updated FE cylinder

XY

ZRepeatable segment with cut outs and element mesh generated

Segment copied around the circumference to make up a portion of the cylinder Portion of meshed

cylinder copied to make up the required cylinder length. Figure 6.30: Procedure for element mesh generation

of cylinder model with multiple perforations.


model for all the eigenvalue modes expanded. Since the yield stress of the steel

sheets used for making the cylinders was given as 180MPa by BHP, the loads

corresponding to a nodal von Mises stress of 180MPa was chosen as the buckling

failure load of the cylinder. From these non-linear analyses, a graph was plotted of

the buckling failure load for each eigenvalue mode shape representing the initial

imperfection. The mode resulting in the lowest buckling load was then used as the

‘correct’ mode shape in the final non-linear buckling analysis of a cylinder with

multiple perforations subjected to a combined axial and lateral pressure. An

example of the buckling failure loads corresponding to the mode shapes has been

shown diagrammatically in figure 6.31, and does not represent any real analysis.

From these analyses two different mode shapes resulted in the lowest buckling load

of the cylinders, one for an axial load and one for a lateral pressure. Therefore, a

non-linear analysis with a combined axial and lateral pressure was performed on

the FE cylinder models for both mode shapes as an initial wall imperfection. The

method of load application in the FE analyses was similar to that of the perforated

cylinder experiments in the laboratory. The external lateral pressures were applied

first and kept constant while the axial load was applied incrementally until the nodal

Axial Loads only Lateral pressures only

0

20

40

60

80

100

0 5 10 15 20 250

20

40

60

80

100

120

0 5 10 15 20 25

Buc

klin

g Lo

ad (

kN)

Buc

klin

g Lo

ad (

kN)

Eigenvalue Mode Shape Eigenvalue Mode Shape

lowest buckling load lowest buckling load

Figure 6.31: Diagrammatic plot of Eigenvalue mode corresponding to Lowest Buckling Stress.


von Mises stress reached 180MPa. This resulted in an interactive curve of the axial

load vs the lateral pressure for each cylinder.

In the non-linear analyses, the loads were applied gradually using ramped substeps

so that an accurate solution could be obtained. The difference between ramped and

stepped loads has been shown in figure 6.32. The lateral pressures were applied on

the cylinder first, in three substeps and the subsequent axial loads were applied in

10 substeps. To ensure convergence, 5 equilibrium iterations were calculated at

each substep.

To obtain the load-displacement graph, three nodes were monitored for each

analysis and the output from these nodes was transferred to an excel spreadsheet.

For each node the variable to be monitored was specified before the analysis was

started. In the case of an axial load, the circumferential nodes were monitored for

vertical displacements while the nodes on the body of the cylinder were monitored

for the von Mises stresses. For lateral pressures the nodes were monitored for an

inwards displacement.

(a) Stepped Loads (b) Ramped Loads

Full load applied at substep 1

Load

Loadstep 1 Substep 3

Load



Time Time

Figure 6.32: Application of load increments: (a) Stepped Loads, (b) Ramped Loads


6.4.2 CYLINDER WITH 16.5% OPEN AREA

Figure 6.33 shows the final arrangement of the FE cylinder model with 50mm

diameter cut outs.

The cylinder was modelled with 96 Nodes

around the circumference. A total of 5274

nodes were used with 4392 elements to

analyse the cylinder. The global axes

have been shown at the base of the

cylinder. To simulate the capped end

conditions of the laboratory experiments,

the base nodes were fully fixed in all

directions and top nodes were free to

move down vertically.

The axial load was applied at each

circumferential node and a ramped load

controlled analysis undertaken. Once the

analysis had reached the total load, the

nodal von Mises stresses were checked

and the load corresponding to a 180MPa

stress was accepted as the failure

buckling load. For the cases when the

stresses were not exactly 180MPa, a

linear interpolation was undertaken.

Before an eigenvalue bucking analysis could be done, a static analysis was first

performed on the cylinder with the option of prestress set to on. The subsequent

eigenvalue buckling analysis performed was the subspace buckling analysis. A total

of 26 modes were extracted with a subspace working size of 5 modes. This resulted

in the least computer time at no cost of accuracy. The extracted mode shapes have

been shown in Appendix H1, from an end view of the cylinder.

To determine the failure load of each mode shape the applied axial load was

multiplied by the number of circumferential nodes and by the load factor for each

mode. For the FE cylinder with 50mm diameter cutouts, the graph of Axial Non-

Linear Buckling loads for each mode shape has been shown in figure 6.34. Similarly

the graph of Lateral Non-Linear Buckling Pressure for each mode shape has been

XY

Z

X

Z

Figure 6.33: Finite Element Model of cylinder with 50mm diameter cut outs



X

Y

ZMode 4

40

45

50

55

60

65

70

75

85 90 95 100 105 110 115 120

Non

Lin

ear

Failu

re L

oad

(kN

)

Eigenvalue Buckling Load (kN)

1

18

12

26

16

5&62&3

7&8

4

2521

23&24

22

Figure 6.34: Graph of Non-linear Axial Buckling Load for each Eigenvalue buckling mode shapes.

Lowest Buckling Load from mode shape 4. Non-Linear Buckling Load = 44.25kN Eigenvalue Load = 95.11kN

X

Y

ZMode 23 & 24

80

85

90

95

100

105

110

115

85 90 95 100 105 110 115 120

Non

Lin

ear

Buc

klin

g (k

Pa)

Eigenvalue Buckling (MPa)

122

23&24

26&27

4

8 2513

2&3

5&6

18

21

Lowest Buckling Load from mode shape 24. Non-Linear Buckling Load = 82.85kPa Eigenvalue Load = 113.29kN

Figure 6.35: Graph of Non-linear Lateral Buckling Load for each Eigenvalue buckling mode shape.


The maximum displacement of the wall in applying the initial geometric

imperfection was set equal to one wall thickness, which was 0.8mm. From figures

6.34 and 6.35 it can be seen that different mode shapes result in the lowest

buckling load for an axial load compared to a lateral pressure application. While it is

not possible to predict which modes shape produce the lowest buckling failure,

there is a general trend that the first few mode shapes produce lower buckling

loads in an axial load application. Similarly the higher mode shapes produce the

lower failure loads for a buckling analysis due to a lateral pressure. In the case of

the cylinder with 50mm diameter holes, mode shape 4 produced the lowest failure

load in an axial buckling analysis while mode shape 23 and 24 produced the lowest

failure load in a lateral pressure buckling analysis.

Once these modes were established, the non-linear buckling analyses due to a

combined axial and lateral pressure were performed for both mode shapes 4 and

24. Note that mode 23 has the same shape as mode 24 and therefore there was no

difference in which shape was applied as the initial imperfection. The resulting

interaction curve for a cylinder with 50mm diameter holes has been shown in figure

6.36.

From figure 6.36 it can be seen that analyses on both mode shapes are needed to

produce the final interaction curve. The graph can be divided into two distinct

0

10

20

30

40

50

60

0 20 40 60 80 100

Tota

l Axi

al L

oad

(kN

)

Lateral Pressure (kPa)

Figure 6.36: Combined Axial and Lateral Load for mode shapes 4 and 24

Mode shape 4

Mode shape 24

Axial load dominates

Lateral pressure dominates


sections, the first being dominated by the axial load and the second section being

dominated by the lateral pressure. Since mode shape 4 produced the lowest failure

buckling load for an axial load only, it is reasonable that this mode shape as an

initial geometric imperfection gives the lower results in the first section of the

interaction curve given in figure 6.36. The second section of the interaction plot

comes from an analysis of Mode shape 24 as an initial geometric imperfection since

this mode gives the lower values of a combined loading.

Since the results from the FE analysis are higher than the values produced from the

laboratory tests, another set of analyses was performed on the FE model using the

same mode shapes but with a wall displacement of 1.6mm as the maximum initial

imperfection scale. The interaction curve for this imperfection scale has been shown

in figure 6.37. However since it has already been shown that mode shape 4

dominates the axial loading and mode shape 24 dominate the lateral loading, only

these final two curves have been shown.

From figure 6.37 it can be seen that the axial capacity of the cylinder has dropped

while there has been very little change in the capacity of the cylinder in carrying a

lateral pressure.

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70 80 90

Tota

l Axi

al L

oad

(kN

)


Mode shape 4

Mode shape 24

Figure 6.37: Interaction curve for FE model with 50mm diameter holes and a wall displacement of 1.6mm


The combined graphs of the two scales of wall imperfections has been shown in

figure 6.38.

As can be seen from figure 6.38, doubling the scale of wall imperfections has the

effect of reducing the axial capacity of the cylinder while there is very little effect on

the lateral pressure of the cylinder.

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70 80 90

Tota

l Axi

al L

oad

(kN

)


Figure 6.38: Interaction plot for cylinder with 50mm diameter cut outs and wall imperfection scales of 1 and 2.

Imperfection scale = 1 wall thickness

Imperfection scale = 2 wall thicknesses

FINITE ELEMENT ANALYSIS: 76mm DIAMETER CUT OUTS 6.35


Figure 6.39 shows the final arrangement of the FE cylinder model with 76mm

diameter cut outs.

The cylinder was modelled with 70 nodes



analyse the cylinder. The global axes have

been shown at the base of the cylinder. To

simulate the capped end conditions of the

laboratory experiments, the base nodes

were fully fixed in all directions and top

nodes were free to move down vertically.




analysis had reached the total load, the

nodal von Mises stresses were checked and

the load corresponding to a 180MPa stress

was accepted as the failure buckling load.

For the cases when the stresses were not

exactly 180MPa, a linear interpolation was

undertaken.









mode. For the FE cylinder with 50mm diameter cutouts, the graph of Axial Non-

Linear Buckling loads for each mode shape has been shown in figure 6.40. Similarly

XY

Z

X

Z

Figure 6.39: Finite Element Model of cylinder with 76mm diameter cut outs


the graph of Lateral Non-Linear Buckling Pressure for each mode shape has been


X

Y

Z

Mode 18&19 Loadfactor

7.347

Lowest Buckling Load from mode shape 18. Non-Linear buckling Load = 66.73kPa Eigenvalue Load = 45.178kN

60

65

70

75

80

85

30 35 40 45 50

Non

Lin

ear

Buc

klin

g (k

Pa)


1

18

13

22

20

5

3

8

2

25


20

22

24

26

28

30

30 35 40 45 50

Non

Lin

ear

Failu

re L

oad

(kN

)


1

18

13

22

20

5

3

6

2

X

Y

Z

Mode 1 Loadfactor

5.396

Lowest Buckling Load from mode shape 1. Non-Linear buckling Load = 21.98kN Eigenvalue Load = 33.61kN

Figure 6.40: Graph of Non-linear Axial Buckling Load for each Eigenvalue buckling mode shapes.


The maximum displacement of the wall in applying the initial geometric

imperfection was set equal to one wall thickness, which was 0.8mm. From figures

6.40 and 6.41 it can be seen that different mode shapes result in the lowest

buckling load for an axial load compared to a lateral pressure application. While it is

not possible to predict which modes shape produce the lowest buckling failure,

there is a general trend that the first few mode shapes produce lower buckling

loads in an axial load application. Similarly the higher mode shapes produce the

lower failure loads for a buckling analysis due to a lateral pressure. In the case of

the cylinder with 76mm diameter holes, mode shape 1 produced the lowest failure

load in an axial buckling analysis while mode shape 18 and 19 produced the lowest

failure load in a lateral pressure buckling analysis.



18. Note that mode 19 has the same shape as mode 18 and therefore there was no

difference in which shape was applied as the initial imperfection. The resulting

interaction curve for a cylinder with 50mm diameter holes has been shown in figure

6.42.

The shape of this plot is similar to the plot for the cylinder with 51mm diameter cut

outs. There are two distinct sections to the curves, the first being dominated by the

axial load and the second dominated by the lateral pressure.

Figure 6.42: Combined Axial and Lateral Load for mode shapes 1 and 18

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80

Tota

l Axi

al L

oad

(kN

)

External Lateral Pressure (kPa)

Mode shape 18

Mode shape 1




Since this plot produced results greater than those from the laboratory tests, the

analysis was redone for an imperfection of twice the wall thickness. the results of

these analyses have been shown in figure 6.43 for mode shape 1 and mode shape

18.

The combined graphs of the two scales of wall imperfections has been shown in

figure 6.44.

0

2

4

6

8

10

0 10 20 30 40 50 60 70

Tota

l Axi

al L

oad

(kN

)


Mode shape 1

Mode shape 18

Figure 6.43: Interaction curve for FE model with 76mm diameter cut outs and a wall displacement of 1.6mm

0

5

10

15

20

25

0 10 20 30 40 50 60 70Lateral Pressure (kPa)

Tota

l Axi

al L

oad

(kN

)

Figure 6.44: Interaction plot for cylinder with 76mm diameter cut outs and wall imperfection scales of 1 and 2




6.4.4 SOLID CYLINDER

Figure 6.45 shows the final arrangement of the FE cylinder model with no cut outs.

The cylinder was modelled with 72 Nodes



analyse the cylinder. The global axes have

been shown at the base of the cylinder. To

simulate the capped end conditions of the

laboratory experiments, the base nodes were

fully fixed in all directions and top nodes were

free to move down vertically.




analysis had reached the total load, the nodal

von Mises stresses were checked and the load

corresponding to a 180MPa stress was

accepted as the failure load. For the cases

when the stresses were not exactly 180MPa, a

linear interpolation was undertaken.









mode. The graph of Axial Non-Linear Buckling loads for each mode shape has been

shown in figure 6.46. Similarly the graph of Lateral Non-Linear Buckling Pressure

for each mode shape has been shown in figure 6.47.

XY

Z

X

Z

Figure 6.45: Finite element model of cylinder with no holes


100

110

120

130

140

150

160

550 570 590 610 630

Non

Lin

ear

Failu

re L

oad

(kN

)


1&2

3

106

11

18

25

19

22

17

5

9X

Y

ZMode shape 18 Factor=80,2200

Lowest Buckling Load from mode shape 18. Non-Linear buckling Load = 106.96kN Eigenvalue Load = 577.58kN

Figure 6.46: Graph of Non-linear Axial Buckling Load for each Eigenvalue buckling mode shape.

80

120

160

200

240

280

320

550 570 590 610 630

Non

Lin

ear

Buc

klin

g (k

Pa)


1

3

10

511

1825

622

2016

X

Y


Lowest Buckling Load from mode shape 5. Non-Linear buckling Load = 85.42kPa Eigenvalue Load = 557.3kN



As in the analyses of the 51mm and 76mm diameter cut out areas, the maximum

displacement of the wall in applying the initial geometric imperfection was set equal

to one wall thickness, which was 0.8mm. Again, different mode shapes result in the

lowest buckling load for an axial load compared to a lateral pressure application.

Unlike the analyses for the perforated cylinders, the lower mode shapes produce

the lower failure loads for a buckling analysis due to a lateral pressure. In the case

of the solid cylinder, mode shape 18 produced the lowest failure load in an axial

buckling analysis while mode shape 5 produced the lowest failure load in a lateral

pressure buckling analysis.



5. The resulting interaction curve for the solid cylinder has been shown in figure

6.48.

The shape of this plot is similar to that for the perforated cylinder analyses with two

distinct sections to the graph, ie axially dominated and laterally dominated.

Although no tests were done on solid cylinders, the finite element analyses on solid

cylinders was undertaken to form a reference for the perforated cylinders.

Therefore the analysis was repeated for an imperfection scale of two wall

thicknesses, ie 1.6mm. These results have been plotted in figure 6.49.

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

Tota

l Axi

al L

oad

(kN

)




Mode shape 18

Mode shape 5

Figure 6.48: Combined Axial and lateral Load for mode shapes 18 and 5


The combined graphs of the two imperfection scales have been shown in figure

6.50.

0

50

100

150

200

250

0 20 40 60 80 100 120 140

Tota

l Axi

al L

oad

(kN

)


Figure 6.49: Interaction curve for a solid cylinder and a wall displacement of 1.6mm

Mode shape 5

Mode shape 18

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120

Tota

l Axi

al L

oad

(kN

)

External lateral pressure (kPa)

Figure 6.50: Interaction plot for a solid cylinder subjected to a combined axial and external lateral pressure for wall imperfection scales of 1 and 2.



COMPARISON BETWEEN FINITE ELEMENT ANALYSES AND LABORATORY TESTS 6.43

6.5 COMPARISON WITH LABORATORY TESTS


The interaction curve from the laboratory tests on the cylinder with 51mm diameter

cut outs has been shown in figure 6.51. Included in this graph are the curves from

the finite element analyses for a geometric imperfection of 1 and 2 wall

thicknesses.

From figure 6.51 it can be seen that the wall imperfection of the cylinders tested in

the laboratory lies in between 1 and 2 times the wall thickness as predicted by the

finite element analyses. This is only true for the section of the graph where the

axial load dominates. For the region of the graph where the lateral pressure

dominates, the wall imperfection scale of the cylinders in the laboratory tests lies

below the region predicted from the finite element analyses. The lateral pressure

from the laboratory test results is approximately 50% of that predicted by the finite

element analyses. It is therefore reasonable to assume the safe working loads of

the cylinder, as shaded in figure 6.51, lies in the area bounded by the laboratory

tests and the finite element analyses for a geometric imperfection of 2 wall

thicknesses.

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100

Figure 6.51: Interaction curves from finite element analyses and laboratory tests on cylinder with 16.5% open area.


Tota

l Axi

al L

oad

(kN

)

FE analysis: WI = 0.8mm

FE analysis: WI = 1.6mm

Laboratory test results

6.44 DES NORTJE; PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

From the test results the equation of the line for the region of the graph dominated

by the lateral pressure has been given in chapter 6.3.2 and has been repeated as

follows:

Nx16.5 = 124.55 - 2.65q (6.10 repeated)

where Nx16.5 is the total axial load in kN applied on a cylinder with 16.5% open area,

and q is the external lateral pressure in kPa.

The equation of the straight line for the region of the interaction dominated by the

total applied axial load is as follows:

Nx16.5 = 17.29 + 0.0017q (6.13)

where the symbols are as before.


Similarly, the interaction curves from the laboratory tests on the cylinder with

76mm diameter cut outs has been shown in figure 6.52. Also included in this graph

are the curves from the finite element analyses for a geometric imperfection of 1

and 2 wall thicknesses.

0

5

10

15

20

25

0 10 20 30 40 50 60 70


Tota

l Axi

al L

oad

(kN

)

Figure 6.52: Interaction curves from finite element analyses and laboratory tests on cylinder with 36.6% open area.

FE Analysis WI = 0.8mm

FE Analysis WI = 1.6mm

Laboratory test results


Figure 6.52 is similar to figure 6.51 since the laboratory test results on the cylinder

with 76mm diameter cut outs also lies between the two curves from the finite

element analyses. From the laboratory test results, the region of the graph

dominated by the lateral pressure appears to be approximately 64% lower than the

results from the finite element analyses. Since this result is similar to the case of

the cylinder with 51mm diameter cut outs it is again reasonable to assume the safe

working loads of the cylinder lie within the area bounded by the laboratory test

results and the finite element analyses for a geometric imperfection of 2 wall

thicknesses. This area has been shaded in figure 6.52.

Equation 6.12 given in chapter 6.3.2 describes the straight line of the graph for the

region dominated by the lateral pressure in figure 6.52. This equation has been

repeated below:

Nx36.6 = 130.41 - 3.105q (6.12 repeated)

where Nx36.6 is the total axial load in kN applied on a cylinder with 36.6% open

area, and q is the external lateral pressure in kPa.

The equation for the straight line in figure 6.52 dominated by the total axial load

applied is given as follows:

Nx36.6 = 9.08 – 0.0135q (6.14)

6.5.3 INTERACTION PLOTS FOR CYLINDERS WITH

MULTIPLE PERFORATIONS.

Using equations 6.10, 6.12, 6.13 and 6.14, the interaction plots of the safe working

loads for cylinders with 16.5% open area (51mm diameter cut outs) and 36.6%

open area (76mm diameter cut outs) have been shown in figure 6.53. Included in

figure 6.53 is the interaction plot for a cylinder with no cut outs and a geometric

imperfection of two wall thicknesses.

From figure 6.53 it can be seen that a cylinder with an open area of 16.5% can

sustain only approximately 16% of the total axial load of a solid cylinder. Similarly,

a cylinder with 36.6% open area can carry only approximately 8% the total axial

load of a cylinder with no cut outs.


For a cylinder with no cut outs, the region of the graph dominated by the lateral

pressure has the following equation:

Nx = 306.45 – 3.496q (6.15)

where Nx is the total axial load in kN, applied on a cylinder with no cut outs, and q

is the external lateral pressure in kPa.

Closer inspection of figures 6.51, 6.52 and 6.53, suggests that the portion of the

graphs dominated by the axial load could be estimated by a horizontal line. For the

cylinders with 16.5% and 36.6% open area these lines have been drawn horizontal

are as shown in figure 6.53. It is therefore reasonable to assume these lines will

also be horizontal for cylinders with other ratios of open areas.

Furthermore, from figures 6.51, 6.52 and 6.53, the slopes of the lines representing

the region of the curve dominated by the lateral pressure, appear to be very

similar. From equations 6.10, 6.12 and 6.15 the slopes of these lines are all

negative and are 2.65, 3.105 and 3.496 respectively. Taking an average of the

three slopes results in a slope of approximately –3.08.

External lateral pressure (kPa)

Tota

l Axi

al L

oad

(kN

)

Figure 6.53: Interaction curves for cylinders with 16.5% and 36.6% Open Area: Geometric Imperfection = 2 x wall thickness

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

Solid Cylinder

16.5% Open Area

36.6% Open Area


Since the general shape of the interaction curves have now been established, the

interaction curves of cylinders with multiple perforations subjected to a combination

of an axial load and an external lateral pressure can easily be plotted. To produce

these graphs, only two points are needed: one is the capacity of the cylinder with

an axial load only, and the other point is the capacity of the cylinder with an

external lateral pressure only.

The interaction curves for cylinders with, say, 5% and 10% open area will be drawn

using the method described above.

From the literature survey it was shown that the shape of the curve for the total

applied axial load as a function of the percentage open area of the cylinder, is a

power series. Therefore, using the value for a cylinder with no cut outs, and the

points for cylinders with 16.5% and 36.6% open area, the power series curve has

been shown in figure 6.54 for the assumption of a geometric imperfection of 2

times the wall thickness. Since it was shown in section 6.5.1 and 6.5.2 that the real

cylinder values lie approximately within the range of an imperfection of 2 times the

wall thickness, this would be a reasonable value to use in general.

0

20

40

60

80

100

120

0 10 20 30 40Percent Open Area

Tota

l App

lied

Axi

al L

oad

(kN

)

Figure 6.54: Capacity of cylinders with a geometric imperfection of 2 x wall thickness, subjected to an axial load

48.86kN

27.05kN


The results from the work by Starnes Jr (1972) have been given in figure 6.5 of the

literature survey, chapter 6.1. From this figure it can be seen that a cylinder with a

cut out value of rc/r (the ratio of the cut out radius to the cylinder radius) of 0.025

(2.5%) can carry the same axial load as a cylinder with no cut outs. Therefore in

figure 6.54, it has been assumed that a cylinder with 2% open area can sustain the

same load as a cylinder with no cut outs.

The equation of the power series given in figure 6.54 is as follows:

A = 192.72 * C(-0.8527) (6.16)

where A is the total axial load in kN and C is the percentage open area of the

cylinder as defined by figure 6.7a in chapter 6.1.

Using equation 6.16, the total axial load which cylinders with 5% and 10% open

area can sustain are 48.86kN and 27.05kN respectively.

The capacities of the cylinders with no cut outs, a 16.5% open area and a 36.6%

open area due to an external lateral pressure only, has been shown in figure 6.55.

It has been assumed that a similar relationship exists for the lateral pressure as for

the axial load and therefore a power series trendline has been shown through these

points. The equation of this curve is as follows:

q = 104.18 * C(-0.2558) (6.17)

where q is the external lateral pressure in kPa and C is the percentage open area of

the cylinder.

Using equation 6.17, the external lateral pressure which cylinders with 5% and

10% open area can sustain are 69.02kPa and 57.81kPa respectively.

Now the interaction curves for cylinders with 5% and 10% open area can be

plotted, as all the points needed for the curves have been determined. The results

have been shown in figure 6.56 with the curves for cylinders with 16.5% and

36.6% open areas. The plot for a cylinder with no cut outs has also been shown.


30

40

50

60

70

80

90

0 10 20 30 40Percent Open Area

Exte

rnal

Lat

eral

Pre

ssur

e (k

Pa)

Figure 6.55: Capacity of cylinders with a geometric imperfection of 2 x wall thickness, subjected to an external lateral pressure only.

69.02kPa

57.81kPa

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

5% open area

10% open area

Tota

l Axi

al L

oad

(kN

)


Figure 6.56: Interaction curves for cylinders with 5%, 10%, 16.5% and 36.6% Open Area: Geometric Imperfection = 2 x wall thickness.

48.86

27.05

69.02 57.81

Slope = -3.08

CONCLUSIONS 7.1

CHAPTER SEVEN

CONCLUSIONS

7.1 SILO WALL PRESSURES


Until Janssen’s paper in 1895, engineers calculated the vertical pressure on the

bottom of a silo in accordance with a hydrostatic pressure distribution. However

Janssen proved that the vertical pressure distribution in a silo varies from zero at

the top increasing exponentially towards a maximum asymptote as the depth

increases. The Reimberts(1976) conducted experiments, which demonstrated the

vertical pressure distribution varies hyperbolically from zero at the top to a

maximum asymptote at great depths in the silo. Both the Janssen and Reimbert

theories for the horizontal pressure acting normal to the walls of the silo tend

towards the same maximum asymptote. However, the Reimbert theory gives higher

values of horizontal pressures in the upper regions of the silo, and consequently

also in squat silos. In the case of a squat silo or in the upper regions of the silo, the

horizontal pressure distribution is best described by the Reimbert theory.

Conversely, the Janssen theory for the vertical pressures in the silo tends towards a

larger value than the Reimbert theory at the same depth. The Janssen vertical

pressure distribution gives the highest vertical pressure acting on the bottom of the

silo. This difference between the two theories has been shown in figure 2.5 in

chapter 2.

The prediction of the static vertical pressure in the hopper has been made by

several researchers and the results of their research varies greatly. Walker(1960)

assumed the principal stress planes were vertical and horizontal, consequently

Walker states that there can be no shear on the vertical planes. Hence Walker gives

the vertical pressure distribution in the hopper as a hydrostatic variation increasing

linearly from the value of the vertical pressure at the transition to a maximum

value at the discharge gate.

The Walker static normal wall pressure in the hopper is given as a constant times

the vertical pressure, where this constant is a function of only the wall friction angle

and the material friction angle, and is not influenced by the hopper half angle.


In direct contrast to Walker, Jenike(1968) states that the normal wall pressures in

the hopper have been shown experimentally to increase linearly from zero at the

vertex to an intermediate value at some height in the hopper. Beyond this

intermediate value the Jenike normal wall pressure distribution increases linearly to

a maximum value, equal to a constant times the Janssen vertical pressure at the

transition. This constant is a function of the hopper half angle and the wall friction

angle and does not depend on the material friction angle. This has been shown in

figure 2.17 of chapter 2.

Conversely, it has been shown in this thesis by the equilibrium of forces acting on a

horizontal slice of material in the hopper that the resulting differential equation of

the vertical pressure is a boundary value problem. This boundary value problem

was solved using the Runge-Kutta method of numerical integration as given in

Appendix A. The resulting distribution of the vertical pressure increases from a

minimum at the discharge gate to a maximum at the transition. The shape of this

curve has been shown in figure 2.12. This pressure distribution is in agreement

with the Jenike distribution at the discharge gate only, while giving larger values at

the transition, as shown in figure 2.22.

The equilibrium of a slice method shows that the hopper half angle and wall friction

angle have a small effect on the vertical pressure. The assumption of a stress ratio

increasing from 0.3 to 0.9 affects the pressure within the hopper, while not

affecting the value at the gate or transition. This has been shown in figure 2.13c

and 2.13d. It is believed in this thesis that the equilibrium slice method gives a

better indication of the static vertical pressure distribution in a converging channel.


Walters(1973a) solves the equilibrium of forces on a horizontal slice in the

cylindrical section of the silo during material flow. Walters assumed the average

vertical stress across the elemental slice was related to the vertical stress at the

wall by a distribution factor F. This distribution factor was determined from the

Mohr circle given in figure 2.24 and is a function of the material friction angle and

the wall friction angle. The Walters distribution factor varies from 1 up to a

maximum value of 3, and shows that the vertical pressure at the wall is higher than

at the centre of the cylindrical section. The Walters theory for the dynamic vertical

pressures in the cylindrical section of the silo predicts values which are

approximately 3 times smaller than the Janssen static vertical pressures. However

CONCLUSIONS 7.3

the Walters pressures normal to the wall during material flow are approximately 2.6

times larger than the Janssen static horizontal pressures.

The Walters (1972b) theory for the vertical pressure distribution in the hopper

during material flow is based on the equilibrium of forces on a horizontal elemental

slice of material. The resulting pressure distribution varies linearly from some non-

zero value at the discharge gate increasing to a maximum value at the transition.

The Jenike(1961) radial stress theory for vertical pressures in the hopper is an

exceedingly lengthy calculation requiring extensive algebraic manipulation.

Ultimately Jenike arrives at a linear pressure distribution increasing from zero at

the discharge gate to a maximum value at the transition. The normal pressures

acting on the hopper wall during material flow are given by Jenike(1961) is a

constant times the stress in the material. This constant is a function of the material

friction angle, the hopper half angle and the inclination of the major principal stress

to the vertical.

7.1.3 SWITCH PRESSURES

All researchers conclude there is a third pressure, the switch pressure, which

travels up through the silo. This switch occurs the instant the discharge gate is

opened and the stress state changes from an active state to a passive state. Switch

pressures are generally quoted as a constant times the Janssen static horizontal

pressures in the cylinder.

The Jenike(1973b) switch pressure varies from approximately 2 times the Janssen

pressure in the lower regions of the cylinder to approximately 3 times the Janssen

pressure in the upper regions of the cylinder. Jenike showed that for the same H/D

ratio of the silo, a smaller wall friction angle resulted in a higher switch pressure in

the cylinder. However, keeping the wall friction angle constant and decreasing the

H/D ratio tends to decrease the switch pressure in the cylinder. This has been

shown in figure 2.45 of chapter 2.

Walters(1973) theory for switch pressures predicts the switch to vary from

approximately 5.2 times the Janssen static pressure in the lower regions of the

cylinder up to approximately 3.1 times the Janssen pressure in the upper regions of

the cylinder.


Decreasing the H/D ratio of the silo by a factor of 2, for the same friction angles

(wall and material), has the effect of decreasing switch pressure by 11.5% in the

lower regions to 39% in the higher regions of the cylinder. By reducing the material

friction angle by 37.5% (keeping the H/D ratio and the wall friction angle constant)

the magnitude of the switch pressures is decreased by approximately 44%.

Conversely a reduction in the wall friction angle of 25% has the effect of increasing

the switch pressure by 179% at the lowest level to 136% at the highest level in the

silo.

Both the Jenike and Walters theories for determining the switch pressures during

flow, predict the worst case would occur in a tall silo (a high H/D ratio) with smooth

walls (a low wall friction angle). A “short” silo with a rough wall would experience

lower switch pressures, but consequently may not undergo mass flow of the

material. Furthermore, the Walters theory predicts that a material with a high

internal angle of friction will result in higher switch pressures on the wall.

Therefore, the magnitude of the switch pressures on the walls of a steel silo

increases over a period of time since the walls become smoother from the wear of

the material. However, in a concrete silo the walls become rougher due to the wear

of the flowing material and therefore the magnitude of the switch pressures

decreases.

The area of influence of the switch is given as approximately one silo diameter in

both the Jenike and Walters theories. Therefore, these high pressures are localised

as well as being transient effects on the walls. However, the magnitudes vary

greatly ranging from approximately 2 to 8 times the Janssen static pressures.

From the experimental results in this thesis it was shown that the switch pressure

travels up the silo at a speed of approximately 22.69m/s (81.6km/h). Therefore in

a 30m tall silo it takes approximately 1.3 seconds for the switch to reach the top.

Assuming the silo has a diameter of 10m, the effects of the switch will be felt on

the walls for approximately 0.44 seconds. The experimental results show the switch

pressures to be between one to four times greater than the experimental static

pressures. This is smaller than the results from the tests by Blair-Fish and Bransby

(1973) as well as van Zanten and Mooij (1977).

CONCLUSIONS 7.5

7.1.4 STRESS RATIOS

In all the theories, the greatest influence on the magnitude of the static and

dynamic wall pressures for a given bulk material comes from the assumption of a

suitable stress ratio. There are no theories predicting which stress ratio should be

used in calculating the static and dynamic pressures. The recommended minimum

and maximum limits are however given as the active and passive stress ratios,

respectively.

Both Janssen and Jenike recommend a value of between 0.3 and 0.4 for static

filling conditions. Any value less than 0.3 is unrealistically low and should not be

used in determining the static pressures for design purposes.

It was shown from the experimental results in this thesis that the stress ratio is not

constant throughout the depth of the silo. For static conditions in the upper regions

of the silo the stress ratio is the highest and can be approximated by K = 1–

Sin2m. The ratio decreases to a minimum value in the lower regions of the silo,

and should not be less than the stress ratio for active pressures.

However for dynamic conditions, no pattern to the varying values of measured

stress ratios was determined. The stress ratios measured varied from a minimum

value of 0.31 up to a maximum value of 3.2. This confirms the recommendation to

use the passive stress ratio as the maximum limit in the calculations of the dynamic

pressures in the silo.

7.2 ANTI-DYNAMIC TUBE FRICTION DRAG

Both Pieper(1969) and McLean(1985) report that the pressure on an object

submerged in the material is approximately equal to the Janssen static vertical

pressure at the same depth during filling of the silo. However McLean’s formula

(equation 4.1) gives the force during flow of the material as 2.5 times the static

value, whereas Pieper’s test results show the flow force to be nearly four times the

static value. Thus Pieper’s results give values that are approximately 1.6 times

larger than predicted by McLean.

Ooms and Roberts (1985) showed the friction force on the anti-dynamic tube to be

directly related to the stress ratio in the material, the wall friction angle of the tube,

the tube diameter and the tube height. They derived an expression for the static


vertical pressure in a silo with a tube installed, by considering the equilibrium of an

elemental slice. By increasing the tube to silo diameter ratio, dt/D from one quarter

to a half, the vertical pressure in the silo decreases from 84% to 61% of the

Janssen static pressure. Thus, the static vertical pressure in a silo, and hence the

horizontal static pressures on the walls, can be reduced by introducing a larger tube

to silo diameter ratio. This has been shown in figure 4.7 of chapter 4.

Ooms and Roberts(1985) also derived an equation to predict the frictional drag on

the outside of the tube during material flow. In their analysis of the equilibrium of

forces on a horizontal elemental slice, they assumed a constant vertical pressure

acting across the slice. They showed that the frictional drag force increases with

increasing height of the tube, as this is directly related to the tube surface area.

The relationship between the height of the tube and the vertical drag force was

shown to be approximately linear at depths greater than 1.4 times the silo

diameter.

Furthermore, from their theory it was shown that the frictional drag force should be

increased by a factor of 1.5 when the material flows inside and outside the tube at

the same time. This situation occurs when portholes have been provided at the

base of the tube for material flow to take place.

Kaminski and Zubrzycki(1985) state that the flexibility of the tube supports has a

large influence on the measured value of the vertical drag force. Thus when the

tube is allowed to move with the flowing material, the magnitude of the frictional

drag force on the tube will be decreased.

For optimum tube performance and material flow through the holes in the walls of

the tube, Kaminski and Zubrzycki derived three relationships. The first relationship

gives the minimum and maximum limits for the ratio of tube diameter to the

hydraulic radius of the silo. Their second relationship defines the limits for the total

area of the tube perforations as a function of the tube surface area. The third

relationship defines the flow characteristics of the tube perforations as a function of

the silo and discharge gate diameters. These have been shown in figure 4.10 of

chapter 4.

Schwedes and Schulze (1991) derived an expression for calculating the vertical

stress inside the tube during flow. This expression is the same as the Janssen

equation with the exponential term approximately equal to one. Using their

CONCLUSIONS 7.7

expression multiplied by the internal surface area of the tube, results in a friction

force approximately 2 times larger than given by Ooms and Roberts.

In this thesis an expression was derived for the frictional drag on the tube walls

during material flow. It was shown that the drag force is directly related to the

stress ratio in the material. From the experimental results of the drag on the tube,

the upper limit for the stress ratio as given by Blight(1993) should be used. This

limit is stated as follows:

K = 1+Sinm . (7.1)

A similar approach to Ooms and Roberts in deriving the expression for the drag

force was adopted, however the distribution of the vertical pressure across the

elemental slice was not assumed constant. The shape of the variation has been

shown in figure 4.14 and is given by the following equation:

x2Cos3.01v (7.2)

where x is the distance from the tube wall, is the vertical pressure at distance x

from the tube wall and v is the average Janssen static vertical pressure acting on

the slice. The expression derived in this thesis for the calculation of the total drag

force on the tube is as follows:

1e

ChJKr2F Ch

ttF (7.3)

where the constants J and C are as follows:

rR

41

23

rRJ22

(7.4)

rR

41

23

rKRK2C ttss

(7.5)

Furthermore, it was shown in this thesis that the rate of discharge of the material

influences the magnitude of the drag force acting on the tube. It was shown

experimentally that the relationship between the shear on the walls of the tube and

the plan area of the discharge gate for the experimental model in this research is as

follows:


= 101.85 A (7.6)

where is the shear on the tube walls and A is the plan area of the discharge gate.

7.3 PERFORATED CYLINDERS

From the test results by Tennyson(1968), Starnes Jr(1972) and Scutella(1998) it

was shown in this thesis that the buckling capacity of a thin walled cylindrical shell

with perforations subjected to an axial load can be estimated by a power series as

follows:

Nx/No = C (Ac/A)-B (7.7)

where Nx is the buckling capacity of a shell with perforations

No is the reference buckling load of a shell without perforations

C, and B are empirically determined constants

Ac is the total area of the cut outs

and A is the surface area of the cylinder.

The buckling capacity of a shell with perforations is therefore a function of the

amount of material that has been removed from the walls of the shell, and is not a

function of the plan area of the cylinder as initially suggested by Tennyson(1968)

and Starnes Jr(1972).

Using the test results from Scutella(1998) it was shown in this thesis that the

reduction in buckling capacity due to an axial load of a cylinder with multiple

perforations is a function of the centre to centre spacing of the cut outs. The

relationship between the buckling capacity of the cylinder and the centre to centre

spacing of the holes for a constant open area, was shown to be linear. This was

shown in figure 6.10 of chapter 6.1.

7.4 INTERACTION CURVES

The finite element analyses were undertaken to enable a comparison with the test

results of the cylinders in the laboratory. Finite element models were set up with

the same open areas as the cylinders used in the laboratory experiments. However

the method of applying an initial wall imperfection

CONCLUSIONS 7.9

From the finite element analyses it was shown that a cylinder with an open area of

16.5% can sustain only approximately 16% of the total axial load of a solid

cylinder. Similarly, a cylinder with 36.6% open area can carry only approximately

8% the total axial load of a cylinder with no cut outs. Similarly it was shown that

cylinders with 16,5% and 36.6% open area can sustain approximately 54% and

46%, respectively, of the external lateral capacity of a cylinder with no cut outs.

Furthermore, from the finite element analyses it was shown that the interaction

curves for thin walled cylindrical shells with multiple perforations subjected to a

combination of axial and lateral pressures consists of two straight lines. Thus the

graph is divided into two regions, the first region being dominated by the axial load.

As the lateral pressure is increased from zero there is very little discernible

reduction in the axial capacity of the cylinder. The second, smaller region of the

graph is dominated by the external lateral pressure. As the external lateral pressure

approaches the value of the lateral capacity of the cylinder, the axial load carrying

capacity of the cylinder is reduced. This has been shown in figure 6.53.

The straight line describing the region of the interaction curve dominated by the

external lateral pressure was shown to have an average negative slope of –3.08,

and passing through the value of the lateral capacity for no axial load applied.

Since the general shape of the interaction curves have been established in this

thesis, the interaction curves of cylinders with multiple perforations subjected to a

combination of an axial load and an external lateral pressure can easily be plotted.

To produce these graphs, only two points are needed: one is the capacity of the

cylinder with an axial load only, and the other point is the capacity of the cylinder

with an external lateral pressure only.

APPENDIX A: IMPLEMENTATION OF THE RUNGE-KUTTA METHOD A.1

CHAPTER 8

APPENDIX A

8.1 IMPLEMENTATION OF THE RUNGE-KUTTA

METHOD

8.1.1 THE RUNGE-KUTTA EQUATIONS

The Runge-Kutte method was chosen to solve to the differential equation 2.31 as

derived in chapter 2 since it has a local truncation error that is proportional to the

step size raised to the power of five, ie h5. This is a smaller error than in the Taylor

and improved Euler methods, which have a truncation error of h3, and the simple

Euler method, which has a truncation error of h2.

The Runge-Kutte method involves a weighted average of the values of ƒ(z,) taken

at different points over the interval zn z zn+1, and is given by:

4n3n2n1nn1n

KK2K2K6h

where Kn1=ƒ(zn;Vn)

Kn2=ƒ(zn+h/2 ; Vn+Kn1 h/2)

Kn3=ƒ(zn+h/2 ; Vn+Kn2 h/2)

Kn4=ƒ(zn+h ; Vn+Kn3 h)

8.1.2 EQUILIBRIUM SLICE METHOD

The initial value of the stress at the level of the transition, ie z=0, has been

calculated from the Janssen equation. The following values were taken from a

fictitious silo, for the calculations of this example.

Silo diameter = 2m

Depth to transition = 7m

Hopper depth = 2.6m

Hopper half angle = 15 and Tan 15 = 0.267949

Wall fricition angle = 20 and Tan 20 = 0.363970

(A.1 a,b,c & d)

A.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

Material bulk density = 17 kN/m3

Static stress ratio = 0.4

Step size = 2.6/10 = 0.26. The choice of suitable step sizes has been discussed at

a later stage in this chapter.

The equation 2.32, given in chapter 2, has been re-written as follows:

v2

0v2

0 CzTanrz

zTanr

(A.2)

The first calculation was done at the step z=0, as follows:

K01 = ƒ(z0;V0) = (r0-z0Tan2- C01 V0 .

The constant C01 is dependent on the depth within the hopper and is calculated as

follows:

20Tanz1

4.020Tan1

4.0120Tanz120Tan2C 0001 =1.495417

Therefore K01 = (r0-zTan2- C01 V0 = 17(1)2-1.495417*50.77889 = -58.935624

The calculation of the second constant K02 was done at step z=0+h/2=0.13 as

follows:

K02 = ƒ(z0+h/2 ; V0+K01 h/2) = (r0-(z0+h/2)Tan2- C02 (V0+0.13K01)

The constant C02 is calculated as follows:

20Tan13.01

4.020Tan1

4.0120Tan13.0120Tan2C02 =1.524836

Therefore: K02 =(r0-(z+h/2)Tan2- C02 (V0+0.13K01)

=17(1-0.13Tan20)2-1.524836*(50.77889-0.13*58.935624) = -48.746751

Since K03 is also evaluated at the step interval z0+h/2, the value of C03 will be equal

to C02 and the constant K03 becomes:

K03 = ƒ(z0+h/2 ; V0+K02 h/2) = (r0-(z0+h/2)Tan2- C03 (V0+0.13K02)

K03=17(1-0.13Tan20)2-1.524836*(50.77889-0.13*48.747651) = -50.766478


The last constant K04 was calculated at the step interval, z0+h=0.26, as follows:

K04=ƒ(z0+h ; V0+K03 h)= (r0-(z0+h)Tan2- C04 (V0+0.26K03)

where

20Tan26.01

4.020Tan1

4.0120Tan26.0120Tan2C04

Therefore C04=1.554255 , and

K04=17(1-0.26Tan20)2-1.554255*(50.77889-0.26*50.766478) = -41.408287

The vertical pressure at the next step level z1 can now be calculated from:

040302010V1V2

00 KK2K2K6hTanzr

=50.77889 - 0.26(58.935624 + 2*48.746751 + 2*50.766478 + 41.408287)/6

V1 = 37.806173 kPa.

The calculation of the vertical pressure at the next step level, ie z=0.26, is carried

out in the same way as before, with z=0.26 and z+h/2=0.26+0.13=0.39.

Thus after calculation the following values were found:

C11 = 1.554255 K11 = -44.739010

C12 = 1.583674 K12 = -36.640465

C13 = C12 K13 = -38.307774

C14 = 1.613093 K14 = -30.897003

Now the vertical pressure at the third step level can be calculated as before:

141312111V2V2

10 KK2K2K6hTanzr

V2 = 28.033098 kPa

The full set of calculations was set up on a spreadsheet, and the results of the

vertical pressure in the hopper were plotted on a graph as shown in figure A.1:

A.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

To determine the most suitable step size to minimise the error in the calculation of the

Runge-Kutta method, four step sizes were considered. These have been tabulated below

along with the results for each step size.

Depth (m)

Vertical Pressure in the hopper (kPa)

h=0.1300 h=0.01300 h=0.00325 h=0.00162

0 50.77888 50.77888 50.77888 50.77888

0.13 43.73370 43.58646 43.57504 43.57315

0.26 37.57858 37.39880 37.38471 37.38238

0.39 32.29345 32.08983 32.07378 32.07112

0.52 27.75899 27.53880 27.52138 27.51849

0.65 23.87033 23.63958 23.62128 23.61824

0.78 20.53873 20.29936 20.28059 20.27747

0.91 17.67536 17.43740 17.41849 17.41535

1.04 15.22011 14.98380 14.96501 14.96190

1.17 13.11039 12.87833 12.85988 12.85681

1.3 11.29507 11.06930 11.05135 11.04837

1.43 9.73046 9.51256 9.49523 9.49236

1.56 8.37938 8.17054 8.15394 8.15118

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50


Dep

th b

elow

tra

nsiti

on (

m)

Figure A.1: Static vertical pressure in a hopper

Table A.1: Effect of step size on accuracy


From Table A.1 it can be seen that there is convergence to two decimal paces for

step size 0.00325 and step size 0.00162. However, for engineering purposes a step

size of 0.00325 is considered acceptable as the error is only:

(43.57504 - 43.57315) / 43.57315 = 0.004 %

Therefore in the subsequent calculations of the equilibrium slice method, a step size

of 0.00325m has been used.


A.2 JENIKE RADIAL STRESS THEORY FOR DYNAMIC

PRESSURES IN THE HOPPER

The equation for the angle of the major principle stress, with respect to the angle

, is given by equation 2.178 as follows:

.............

Sin2CosSins2

Coss2CosSinCos

mm

m2

m

1

Sin2CossSin2

12cos2SinCotSin1sSin...........

mm

mm

(2.178 repeated)

Equation 2.178 has been solved for a silo with a material friction angle, m, of 20,

a stress value of s=10, and an initial value of =0.01 (ie an infinitesimally small

angle off the axis of the hopper). Equation 2.178 cannot be solved for a value of

= 0, since Cotangent 0 is undefined.

APPENDIX B: CALIBRATION CONSTANTS B.1

APPENDIX B: CALIBRATION CONSTANTS

8.2.1 PRESSURE CELL CALIBRATION

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35

TUBE 1

Vol

tage

out

put

Applied pressure (kPa)

1/0.2725= 3.669 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35

Vol

tage

out

put


TUBE 2

1/0.2635= 3.795 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

TUBE 4

1/0.2578= 3.879 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

TUBE 5

1/0.2633= 3.798 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

PLATE 7

1/0.2432= 4.112 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

PLATE 6

1/0.2651= 3.774 kPa/V

B.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

PLATE 8

1/0.2449= 4.083 kPa/V

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35


Vol

tage

out

put

PLATE 12

1/0.2581= 3.874 kPa/V

0

1

2

3

4

5

0 1 2 3 4 5 6 7

Distance (m)

Vol

tage

out

put

MTPM 10

1/0.703= 1.422 m/V

0

1

2

3

4

5

0 1 2 3 4 5 6

Vol

tage

Out

put

Distance (m)

MTPM 3

1/0.7272= 1.375 m/V

0

0.1

0.2

0.3

0.4

0 3 6 9 12 15 18

Ball Cell 1

Vol

tage

out

put


1/0.0236= 42.37 kPa/V

APPENDIX B: CALIBRATION CONSTANTS B.3

8.2.2 ANTI-DYNAMIC TUBE SUPPORT CALIBRATION

0

20

40

60

80

100

120

140

160

0 0.1 0.2 0.3 0.4

393.06 kg/V

Mas

s (k

g)

Votlage Output (V)

Support Beam

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1

163.01 kg/V

Votlage Output (V)

Mas

s (k

g)

Middle Rod

0

20

40

60

80

100

120

140

160

0 0.1 0.2 0.3 0.4 0.5

Votlage Output (V)

Mas

s (k

g)

Bottom Rod

295.26 kg/V

APPENDIC C: CHECK LISTS C.1

APPENDIX C: CHECKLISTS

Tube: 120mm Static23Flow down inside of tube 08 Oct 99

1) Check all data acquisition cables and connections

2) Check signals from sensors, switches, strain gges & mtpm

3) Fully wind up multi-turn potential meter pulleysEnsure pulley ropes are secured during filling

4) Measure height of tube from top of silo

5) Edit input file for static test.Chnnls Combin Buffers Input freq { Max input freq = 500 000 }

11 0 1 1000Test Duration = No of buffers * 512/(input freq * 0.095369)

= 5.3686 secs = 0.001491 hrs =No of data rows = 46 Times no of runs: if using "KENHO"

6) Note calibration constantsChnnl 1 Chnnl 2 Chnnl 3 Chnnl 4 Chnnl 5 Chnnl 6Str Gge Tube Mtpm Tube Tube Plate96.154 3.795 1.375 3.879 3.798 3.774 Cal cnstnt

Chnnl 7 Chnnl 8 Chnnl 9 Chnnl 10 Chnnl 11 Chnnl 12Plate Plate Switches Mtpm Str Gge Str Gge4.112 4.083 1 1.422 78.74 95.238 Cal cnstnt

7) Note positions from top of siloChnnl 1 Chnnl 2 Chnnl 3 Chnnl 4 Chnnl 5 Chnnl 6Str Gge Tube Mtpm Tube Tube Plate

Chnnl 7 Chnnl 8 Chnnl 9 Chnnl 10 Chnnl 11 Chnnl 12Plate Plate Switches Mtpm Str Gge Str Gge

8) Note method of filling silo Flexible hose

9) Density measurements: Bucket self weight =87.92g & Vol = 0.001345m3

DepthWeight

Pre Static Test Check List

Gate: 180mm

C.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

Tube: 120mm Dynmic23Flow down inside of tube 08 Oct 99

1) Level off top surface of sand

2) Measure height of sand from top of silo

3) Remove stopper from pulley and set to zero positionSlow pulley down so it cannot overshoot itself.

4) Ensure cables can move freely

5) Check switches are working

6) Edit input fileChnnls Combin Buffers Input freq { Max input freq = 500 000 }

11 0 185 10000Test Duration = No of buffers * 512/(input freq * 0.095369)

= 99.31949 secs = 1 min 39 secs

No of data rows = 8510 Use "SILO" program

7) Note calibration constantsChnnl 1 Chnnl 2 Chnnl 3 Chnnl 4 Chnnl 5 Chnnl 6Str Gge Tube Mtpm Tube Tube Plate96.154 3.795 1.375 3.879 3.798 3.774 Cal cnstnt

Chnnl 7 Chnnl 8 Chnnl 9 Chnnl 10 Chnnl 11 Chnnl 12Plate Plate Switches Mtpm Str Gge Str Gge4.112 4.083 1 1.422 78.74 95.238 Cal cnstnt

8) Time to empty silo using stop watch

Pre Dynamic Test Check List

Gate: 180mm

APPENDIC C: CHECK LISTS C.3

Pressure Cell Orientation Test 2308 Oct 99

support beam

a-d tube

bucketelevator

top ofsilo

180mm diameter opening

OUTPUT FROM DYNAMIC TESTSTEST -1 TEST 1

TEST 2TEST 1

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80Time (seconds)

Test -1: Ball cell placed 1.1m from top

Pres

sure

(kP

a)

Staticpressure

=3.64 kPa

4.83

8.89

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

0 20 40 60 80

Test 2: Cell 2 placed 2.11m from top

Pres

sure

(kP

a)

Time (seconds)

11.93

Staticpressure

=8.56 kPa

-8

-6

-4

-2

0

2

4

6

8

10

12

14

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

12.26

Static pressure

=6.69 kPa

-8

-6

-4

-2

0

2

4

6

8

0 20 40 60 80

Test 1: Ball cell placed 1.1m from top

Pres

sure

(kP

a)

Time (seconds)

3.73

5.89

Staticpressure

=5.89 kPa

0

1

2

3

0 10 20 30 40 50 60 70

Test 1: Mtpm 3 placed 1.1m from top

Dis

tanc

e (m

)

Time (seconds)

Constant flowrate in thecylinder

Acceleratedflow rate inthe hopper

Cylinder Hopper

-6

-4

-2

0

2

4

6

8

10

12

0 20 40 60 80 100

Pres

sure

(kP

a)

Time (seconds)

2.13

11.49Test 2: Cell 4 placed 0.93m from top

Staticpressure

=4.79 kPa

TEST 3TEST 2

-4

-2

0

2

4

6

8

0 20 40 60 80 100Time (seconds)


Pres

sure

(kP

a)

4.13

6.67

Static pressure

=2.94 kPa

-12

-9

-6

-3

0

3

6

9

12

15

-10 0 10 20 30 40


12.36

Staticpressure

=10.95 kPa

Pres

sure

(kP

a)

Time (seconds)

0

1

2

-10 0 10 20 30 40 50


Cylinder

Hopper

Dis

tanc

e (m

)

Time (seconds)-8

-4

0

4

8

12

16

20

24

-15 0 15 30 45 60 75 90


2.35

20.13

Pres

sure

(kP

a)

Time (seconds)

Staticpressure

=7.19 kPa

-10

-5

0

5

10

15

20

25

-15 0 15 30 45 60 75


Time (seconds)

Pres

sure

(kP

a)

1.66

22.18

Staticpressure

=7.81 kPa

-5

0

5

10

15

-10 10 30 50 70 90


Pres

sure

(kP

a)

14.67

5.18

Staticpressure

=4.33 kPa

Time (seconds)

TEST 5

TEST 4

-20

-15

-10

-5

0

5

10

15

20

25

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

21.41

Staticpressure

=16.8 kPa

10.4

-1

1

2

3

0 10 20 30 40 50


Dis

tanc

e (m

)

Time (seconds)

Cylinder

Hopper

Change of flow ratein cylinder due

to side discharge

-9

-6

-3

0

3

6

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

2.46

4.24

Staticpressure

=7.53 kPa

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

Staticpressure

=5.75 kPa

3.26

0.53

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30


Pres

sure

(kP

a)

Time (seconds)

11.25

-2

0

2

4

6

8

10

12

14

0 10 20 30 40


Pres

sure

(kP

a)

Time (seconds)

11.25

Cablescut

5.24

TEST 5

TEST 6

-2

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60


Pres

sure

(kP

a)

Time (seconds)

12.86

7.89

Cablescut

-8

-4

0

4

8

12

16

0 10 20 30 40 50 60


Pres

sure

(kP

a)

Time (seconds)

13.90

Staticpressure

=5.92 kPa

6.58

-2

-1

0

1

2

3

4

5

6

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

5.33

electricalinterference

2.93

0.0

1.0

2.0

3.0

0 20 40


Dis

tanc

e (m

)

Cylinder

Hopper

Change inflow rate

-8

-4

0

4

8

12

0 20 40 60 80


Pres

sure

(kP

a)

Time (seconds)

Cablescut

11.31

7.42

-4

-2

0

2

4

6

8

-10 0 10 20 30 40 50 60 70 80


Pres

sure

(kP

a)

4.49

Cell passesthrough thetransition

Time (seconds)

2.28

APPENDIX D: DYNAMIC TEST RESULTS D.1

APPENDIX D:

DYNAMIC TEST RESULTS

A table of the results for the first five dynamic tests is shown in table D.1, showing

the values of the first and second measured peak pressures. The total pressures

acting on the silo wall during flow are equal to the dynamic value plus the static

value, which have also been given in table D.1.

To calculate the ratio of the dynamic to static pressure at the transition, the value

of the second peak in the output curve was used for the dynamic pressure, and an

average value of 7.10 kPa was used for the static pressure. This average value has

been calculated from the results of six static tests.

Table D.1: Dynamic Test Results

Test Depth StaticNo from top pressure First Second First Second

of silo (m) peak peak peak peak-1 1.1 3.64 4.83 8.89 8.47 12.531 1.1 5.57 3.73 5.89 9.3 11.462 0.6 3.15 4.13 6.67 7.28 9.82

0.93 4.73 2.13 11.49 6.86 16.221.61 6.65 - 12.26 18.912.11 8.06 - 11.93 19.99

3 0.69 4.04 5.18 14.67 9.22 18.711.11 6.5 1.66 22.18 8.16 28.681.39 6.64 2.35 20.13 8.99 26.772.28 7.93 - 12.36 20.292.54 3.88 - 3.88

4 0.28 2.57 0.53 3.26 3.1 5.831.04 4.53 2.46 4.24 6.99 8.772.2 5.33 10.4 21.41 15.73 26.74

5 1.26 5.66 7.89 12.86 13.55 18.522.17 8.29 5.24 11.25 13.53 19.54

Dynamic pressures Total pressure

D.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN SILOS

TEST -1 TEST 1

TEST 2

-8

-6

-4

-2

0

2

4

6

8

0 20 40 60 80

Test 1: Ball cell placed 1.1m from top

Pres

sure

(kP

a)

Time (seconds)

3.73

5.89

Staticpressure

=5.89 kPa

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80Time (seconds)

Test -1: Ball cell placed 1.1m from top

Pres

sure

(kP

a)

Staticpressure

=3.64 kPa

4.83

8.89

0

1

2

3

0 10 20 30 40 50 60 70


Dis

tanc

e (m

)

Time (seconds)

Constant flowrate in thecylinder Accelerated

flow rate inthe hopper

Cylinder Hopper

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

0 20 40 60 80


Pres

sure

(kP

a)

Time (seconds)

11.93

Staticpressure

=8.56 kPa

-6

-4

-2

0

2

4

6

8

10

12

0 20 40 60 80 100

Pres

sure

(kP

a)

Time (seconds)

2.13

11.49Test 2: Cell 4 placed 0.93m from top

Staticpressure

=4.79 kPa-8

-6

-4

-2

0

2

4

6

8

10

12

14

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

12.26

Static pressure

=6.69 kPa


TEST 2 TEST 3

-4

-2

0

2

4

6

8

0 20 40 60 80 100Time (seconds)

Test 2: Cell 6 placed 0.6m from topPr

essu

re (

kPa)

4.13

6.67

Static pressure

=2.94 kPa

-12

-9

-6

-3

0

3

6

9

12

15

-10 0 10 20 30 40


12.36

Staticpressure

=10.95 kPa

Pres

sure

(kP

a)

Time (seconds)

0

1

2

-10 0 10 20 30 40 50


Cylinder

Hopper

Dis

tanc

e (m

)

Time (seconds)-8

-4

0

4

8

12

16

20

24

-15 0 15 30 45 60 75 90


2.35

20.13

Pres

sure

(kP

a)

Time (seconds)

Staticpressure

=7.19 kPa

-10

-5

0

5

10

15

20

25

-15 0 15 30 45 60 75


Time (seconds)

Pres

sure

(kP

a)

1.66

22.18

Staticpressure

=7.81 kPa-5

0

5

10

15

-10 10 30 50 70 90


Pres

sure

(kP

a)

14.67

5.18

Staticpressure

=4.33 kPa

Time (seconds)


TEST 4

TEST 5

-20

-15

-10

-5

0

5

10

15

20

25

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

21.41

Staticpressure

=16.8 kPa

10.4

-1

1

2

3

0 10 20 30 40 50


Dis

tanc

e (m

)

Time (seconds)

Cylinder

Hopper

Change of flow ratein cylinder due

to side discharge

-9

-6

-3

0

3

6

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

2.46

4.24

Staticpressure

=7.53 kPa

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

Staticpressure

=5.75 kPa

3.26

0.53

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30


Pres

sure

(kP

a)

Time (seconds)

11.25

-2

0

2

4

6

8

10

12

14

0 10 20 30 40


Pres

sure

(kP

a)

Time (seconds)

11.25

Cablescut

5.24


TEST 5

TEST 6

-2

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60


Pres

sure

(kP

a)

Time (seconds)

12.86

7.89

Cablescut

-8

-4

0

4

8

12

16

0 10 20 30 40 50 60


Pres

sure

(kP

a)

Time (seconds)

13.90

Staticpressure

=5.92 kPa

6.58

-2

-1

0

1

2

3

4

5

6

0 20 40 60 80 100


Pres

sure

(kP

a)

Time (seconds)

5.33

electricalinterference

2.93

0.0

1.0

2.0

3.0

0 20 40

Test 5: Mtpm 10 placed 0.76m from topD

ista

nce

(m)

Cylinder

Hopper

Change inflow rate

-8

-4

0

4

8

12

0 20 40 60 80


Pres

sure

(kP

a)

Time (seconds)

Cablescut

11.31

7.42

-4

-2

0

2

4

6

8

-10 0 10 20 30 40 50 60 70 80


Pres

sure

(kP

a)

4.49

Cell passesthrough thetransition

Time (seconds)

2.28


TEST 15

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

-2 0 2 4 6 8 10 12

Test 15:Tube 2 placed 2.34m from top

Time (seconds)

Pres

sure

(kP

a)

4.3

0.5

-3

-2

-1

0

1

2

3

4

5

6

-2 0 2 4 6 8 10 12

Test 15: Plate 6 placed 2.34m from top

Time (seconds)

Pres

sure

(kP

a)

4.3

-6

-3

0

3

6

9

12

15

18

-5 0 5 10 15 20 25 30 35 40

Test 15: Tube 4 placed 1.41m from top

Time (seconds)

2.4

16.33

-2

0

2

4

6

8

10

12

14

-5 0 5 10 15 20 25 30 35 40


Time (seconds)

2.4

12.2

-4

-2

0

2

4

6

8

10

12

14

16

18

20

-10 0 10 20 30 40 50 60 70


Time (seconds)

Pres

sure

(kP

a)

2.1

17.5

-4

-2

0

2

4

6

8

10

-10 0 10 20 30 40 50 60 70


Pres

sure

(kP

a)

2.1

7.2

Time (seconds)


TEST 17

TEST 18

-2

-1

0

1

2

-10 10 30 50 70 90 110


cells flow down the tube

1.9

Time (seconds)

Pres

sure

(kP

a)

-2

-1

0

1

2

-10 10 30 50 70 90


cells flow down the tube

1.4

Time (seconds)

Pres

sure

(kP

a)

0

1

2

3

-20 0 20 40 60 80 100 120


Time (seconds)

Pres

sure

(kP

a)

2.3

0

1

2

3

0 20 40 60 80 100 120

Dis

tanc

e (m

)

Time (seconds)


0.0365m/sAverage flowin the tube

-6

-4

-2

0

2

4

6

8

10

12

-50 50 150 250 350


10.2

Time (seconds)

Pres

sure

(kP

a)

-6

-4

-2

0

2

4

6

8

-50 50 150 250 350


1.3

Time (seconds)

4.8

Pres

sure

(kP

a)


TEST 18

TEST 19

-10

-8

-6

-4

-2

0

2

4

6

8

10

-50 0 50 100 150 200 250

Test 18: Tube 5 placed 0.85m from topPr

essu

re (

kPa)

Time (seconds)

8.6

-4

-3

-2

-1

0

1

2

3

4

-50 0 50 100 150 200 250


Pres

sure

(kP

a)

Time (seconds)

1.7

0

1

2

-50 50 150 250


Time (seconds)

Pres

sure

(kP

a)

1.49

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

-50 50 150 250 350 450


Pres

sure

(kP

a)

Time (seconds)

-10

0

10

20

30

-20 20 60 100 140


Pres

sure

(kP

a)

Time (seconds)

26.7

-1

0

1

2

-50 50 150 250 350Time (seconds)


Pres

sure

(kP

a) 1.65


TEST 20

-4

-3

-2

-1

0

1

2

3

4

-5 5 15 25 35 45 55 65 75


Time (seconds)

Pres

sure

(kP

a)

2.31.8

-10

-8

-6

-4

-2

0

2

4

6

0 10 20 30 40 50


Pres

sure

(kP

a)

Time (seconds)

4.8

-3

-2

-1

0

1

2

3

4

5

6

-5 5 15 25 35 45 55


Time (seconds)

4.7

Pres

sure

(kP

a)

-4

-3

-2

-1

0

1

2

0 10 20 30 40 50


Pres

sure

(kP

a)

Time (seconds)

0.7


TEST 10

-4

-2

0

2

4

6

8

10

12

14

-5 0 5 10 15 20 25 30

Test 10: Tube 2 placed 2.37m from top 13.3

Time (seconds)

1.28

-1

0

1

2

3

4

5

6

7

8

-5 0 5 10 15 20Time (seconds)

Pres

sure

(kP

a)


6.1

-2

0

2

4

6

8

10

12

-10 10 30 50 70 90


Time (seconds)

Pres

sure

(kP

a) 11.4

-4

-2

0

2

4

6

8

10

-2 0 2 4 6 8 10 12 14

Pres

sure

(kP

a)


8.4

Time (seconds)

3.22

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

Pres

sure

(kP

a)

Time (seconds)


5.4

0.6

-2

-1

0

1

2

3

4

-2 0 2 4 6 8 10

Pres

sure

(kP

a)


Time (seconds)

2.7

TEST 10

-30

0

30

60

90

120

150

180

210

240

-4 0 4 8 12 16 20

219.5

Test 10: SG 11 & 12: Drag force on tube

Forc

e (k

g)

Time (seconds)

121.2

00.20.40.60.8

11.21.41.61.8

22.2 Multi-turn potential meter : Channel 10

Data point number

Dis

tanc

e tr

avel

led

(m)


TEST 16

-0.5

0.5

1.5

2.5

3.5

-10 10 30 50 70


Flow rate=0.037 m/s

Cylinder

Hopper

Time (seconds)-9

-6

-3

0

3

6

9

12

15

18

-5 5 15 25 35


15.2

Pres

sure

(kP

a)

Time (seconds)

-8

-4

0

4

8

12

16

20

-10 0 10 20 30 40 50 60


Time (seconds)

Pres

sure

(kP

a) 19.8

2.7

-8

-4

0

4

8

12

16

20

24

28

-10 0 10 20 30 40 50 60 70


23.6

2.5

Time (seconds)

Pres

sure

(kP

a)

-3

0

3

6

9

12

-10 0 10 20 30 40 50 60


Time (seconds)

Pres

sure

(kP

a)

9.9

-4

0

4

8

12

16

20

24

-10 0 10 20 30 40 50 60 70


1.7

18.9

Time (seconds)

Pres

sure

(kP

a)

APPENDIX E: TEST RESULTS: ANTI-DYNAMIC TUBE FRICTIONAL DRAG E:1

8.5 APPENDIX E: TEST RESULTS

ANTI-DYNAMIC TUBE FRICTIONAL DRAG

-30

0

30

60

90

120

150

180

210

240

-4 -2 0 2 4 6 8 10 12 14 16 18 20

219.5Fo

rce

(kg)

Time (seconds)

121.2

Test number 10

0

30

60

90

120

150

180

210

240

270

-10 0 10 20 30 40 50 60 70 80 90

252.7

183.1

Time (Seconds)

Test number 11

0

30

60

90

120

150

180

210

240

270

0 10 20 30 40 50 60 70 80 90

245.6

157.1

Forc

e (k

g)

Time (seconds)

Test number 12

E:2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

0

50

100

150

200

250

300

-10 0 10 20 30 40 50 60 70 80 90 100

278.4

193.8

Time (seconds)

Forc

e (k

g)

Test number 13

-150-100-50

050

100150200250300350400450

-4 -2 0 2 4 6 8 10 12 14 16

296.6: Top

414.3: Middlesection broke off

254.4

74.6

197.4: Bottom

Forc

e (k

g)

Time (Seconds)

Test number 14

0

50

100

150

200

250

300

-10 0 10 20 30 40 50 60 70 80

287.6: Top

88.7: Bottom

Forc

e (k

g)

Time (seconds)

201.6: Middle

Test number 15


0

40

80

120

160

200

240

280

320

360

-10 0 10 20 30 40 50 60 70 80

358.0

211.6: Middle

52.6

212.5

245.5: Top

66.5: Bottom

Time (seconds)

Forc

e (k

g)

Test number 16

0

10

20

30

40

50

60

70

80

90

100

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

Bottom

Middle

Top85.9

46.3

91.9

Time (seconds)

Forc

e (k

g)

Test number 17

0

20

40

60

80

100

120

-30 0 30 60 90 120 150 180 210 240 270 300 330 360

Top

Middle

Bottom

52.8

98.5

108.3

93.7

38.5

Test number 18

E:4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

0102030405060708090

100110

-25 25 75 125 175 225 275 325

Top

Bottom

Time (seconds)

96.2

64

Forc

e (k

g)Test number 19

0

10

20

30

40

50

60

70

80

-10 0 10 20 30 40 50 60 70

74.8

Time (seconds)

Forc

e (k

g)

Test number 20

0

40

80

120

160

200

-10 0 10 20 30 40 50 60 70 80

134.3

177.5

Forc

e (k

g)

Time (seconds)

Test number 21


0

20

40

60

80

100

120

-10 0 10 20 30 40 50 60 70 80

Forc

e (k

g)

Time (seconds)

103.4

Test number 24

0

20

40

60

80

100

120

140

-10 0 10 20 30 40 50 60 70 80

127.9

Forc

e (k

g)

Time (seconds)

Test number 22

118.3

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

104.8

Time (seconds)

Test number 23 110.9

APPENDIX F: SHELL THEORY: FLEXURAL RIGIDITY OF A SHELL F.1

8.6 APPENDIX F: SHELL THEORY

8.6.1 UNIFORMLY COMPRESSED CIRCULAR RING

For thin cylindrical shells which have a length to diameter ratio greater than fifty

the following equation can be used as the theoretical buckling load due to a uniform

external lateral pressure:

3crr

IE3q (F.1)

8.6.2 FLEXURAL RIGIDITY OF A SHELL

To determine the expression for the flexural rigidity of a shell, the beam equation is

used as a starting point,

RE

IM

z

(F.2)

In this equation, is the stress across the face of an element, z is the distance

from the neutral axis to the most extreme fibre on the element, M is the bending

moment in the structure, I is the second moment of area, E is Youngs modulus of

elasticity and R is the radius of curvature of the neutral axis of the element in

bending.

A cross section through a simple element of thickness t, is shown in figure F.1 .

qcr

x

y

z

Figure F.1: External lateral pressure on a Long Cylindrical shell

F.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

Since the shell is a thin structure, the strain of the element can be considered as a

two dimensional problem. Therefore the strains of the element in the x and y

directions are given by:

yxx

xE1

Rz

and xyy

yE1

Rz

(F.3)

These equations can be re-arranged in terms of the stresses in the x and y

directions as follows:

yx2x1

zE

and xy2y

1zE

(F.4)

where is the curvature of the shell and is equal to 1/R

2/t

2/t

xx dzdyzdyM (F.5)

Substituting equation F.4 into F.5 gives:

24t

24t

1Edzz

1zEM

33yx2

2/t

2/t

2yx2x

And therefore:

yxyx2

3x D

112tEM

(F.6)

where D is the flexural rigidity of the shell

M x x

Stress across the face of the element

Strain across the face of the element

Cross section through an element

Figure F.1: Stress distribution across an element

z

x

y

APPENDIX G: PERFORATED CYLINDER TEST RESULTS: 51mm CUTOUTS G.1

8.7 APPENDIX G

PERFORATED CYLINDER TEST RESULTS

The results of the perforated cylinder tests have been given on the following four

pages. These have been grouped according to the diameter of the cut outs, ie first

the 51mm diameter cut outs (16.5% open area) followed by the 76mm diameter

cut outs (36.6% open area). The graphs have been deliberately placed on the next

pages so that the reader can obtain an overall view of the tests undertaken. Each

group of cylinders was subjected to a lateral pressure of 10kPa increasing to 40kPa

in increments of 10, which constitutes four tests per cut out size. Therefore pages

G.2 and G.3 shows the test results of the cylinder with 51mm diameter cut outs,

while pages G.4 and G.5 show the results for the 76mm diameter cut outs.

G.2 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

8.7.1 CYLINDERS WITH 16.5% OPEN AREA

0

5

10

15

20

25

30

35

0 5 10 15 20 25

32.25 kN Lateral pressure 20 kPa

Tota

l Axi

al L

oad

(kN

)

Vertical displacement (mm)

0

5

10

15

20

25

30

0 5 10 15 20 25

Lateral pressure 10 kPa

26.27kN


Tota

l Axi

al L

oad

(kN

)

APPENDIX G: PERFORATED CYLINDER TEST RESULTS: 51mm CUTOUTS G.3

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

17.13 kN

Tota

l Axi

al L

oad

(kN

)

Vertical Displacement (mm)

Lateral pressure 30kPa

0

4

8

12

16

20

0 2 4 6 8 10 12 14 16

Tota

l Axi

al L

oad

(kN

)

18.55kN


Lateral pressure 40kPa

G.4 DES NORTJE: PhD THESIS : THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS

8.7.2 CYLINDERS WITH 36.6% OPEN AREA

0

2

4

6

8

10

12

0 4 8 12 16 20

Tota

l Axi

al L

oad

(kN

)


Lateral pressure 10kPa11.51 kN

0

2

4

6

8

10

0 5 10 15 20 25

Tota

l Axi

al L

oad

(kN

)

9.36 kN Lateral pressure 20 kPa


APPENDIX G: PERFORATED CYLINDER TESTS: 76mm DIAMETER CUTOUTS G.5

0

2

4

6

8

10

12

14

0 4 8 12 16 20

Tota

l Axi

al L

oad

(kN

)


12.78 kNLateral Pressure 30 kPa

0

1

2

3

4

5

6

7

0 4 8 12 16 20

6.21 kN

Lateral pressure 40 kPa

Tota

l Axi

al L

oad

(kN

)


G.6 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS

8.7.3 LATERAL PRESSURE ONLY

CYLINDER WITH 16.5% OPEN AREA

CYLINDER WITH 36.6% OPEN AREA

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8Inwards wall displacement (mm)

Late

ral P

ress

ure

(kPa

)

47kPa

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8Inwards wall displacement (mm)

Late

ral P

ress

ure

(kPa

)

42kPa

APPENDIX H: FINITE ELEMENT ANALYSIS: EIGENVALUE BUCKLING: 50mm CUT OUTS H.1

8.8 APPENDIX H FINITE ELEMENT ANALYSIS:

EIGENVALUE BUCKLING MODE SHAPES


X

Y

ZMode 1

factor=12.4647 X

Y

ZMode 2 & 3

factor=12.7623

X

Y

ZMode 4

factor=13.2104 X

Y

ZMode 5 & 6 factor=13.5373

X

Y


X

Y


H.2 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS

X

Y

ZMode 11

factor=14.4791 X

Y

ZMode 12 factor=14.5045

X

Y

ZMode 13 & 14

factor=14.8181 X

Y

ZMode 15 & 16 factor=14.8464

X

Y

Z

Mode 17 factor=14.9217

X

Y

ZMode 18

factor=15.4534

X

Y

ZMode 20 & 21 factor=15.4953

X

Y

ZMode 23 & 24 factor=15.7345

k


X

Y

ZMode 26 & 27

factor=16.1719

Eigenvalue Buckling Mode 4: 3-D view


H.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS


X

Y

Z

Mode 7&8 Loadfactor

6.528

X

Y

Z

Mode 5&6 Loadfactor

6.390

X

Y

Z

Mode 1 Loadfactor

5.396 X

Y

Z

Mode 2&3 Loadfactor

5.726

X

Y

Z

Mode 4 Loadfactor

6.077

X

Y

Z


6.970


X

Y

Z

Mode 24 Loadfactor

7.959 X

Y

Z


7.951

X

Y

Z


7.567

X

Y

Z


7.347

X

Y

Z


7.208

X

Y

Z


7.197 X

Y

Z


7.115

X

Y

ZMode 17

Loadfactor 7.317


X

Y

Z

Mode 25 Loadfactor

7.963


Y

Z


APPENDIX H: FINITE ELEMENT ANALYSIS: EIGENVALUE BUCKLING: SOLID SHELL H.7

8.8.3 SOLID SHELL

X

Y


X

Y

ZMode shape 3&4 Factor=77,1401

X

Y

ZMode shape 1&2 Factor=77,1074

X

Y


X

Y


X

Y



X

Y

ZMode shape 11 factor=79,6821

X

Y

Z

Mode shape 13 factor=79,9256

X

Y


X

Y


X

Y


X

Y


X

Y


X

Y


APPENDIX H: FINITE ELEMENT ANALYSIS: EIGENVALUE BUCKLING: SOLID SHELL H.9

X

Y


X

Y


X

Y

ZMode shape 27 factor=96,1378

X

Y


REFERENCES 9.1

CHAPTER NINE

REFERENCES

Almroth BO and Holmes AMC (1972): “Buckling of Shells with Cutouts, Experiment

and Analysis”, International Journal of Solids Structures, Vol 8, 1972, pg

1057-1071

Arnold PC (1991): “On the Influence of Segregation on the Flow Pattern in Silos”,

Bulk Solids Handling, Vol 11, Number 2, may 1991, pg 447-449.

Arnold PC, McClean AG and Roberts AW (1985): “The Design of Storage Bins for

Bulk Solids Handling”, The Best of Bulk Solids Handling, Selected Articles

1981-1985, VolA/86, pg 7

Arnold PC, McClean AG and Roberts AW (1989): “Bulk Solids: Storage, Flow and

Handling”, TUNRA Bulk Solids Handling Research Associates, University of

Newcastle, Australia, Second edition, December 1989.

Australian Standards (1990): “AS 3774 – 1990 : Loads on Bulk Solids

Containers”; First edition; Standards Association of Australia, Standards

House, 80 Arthur Street, North Sydney, NSW, ISBN 0 7262 6159 9

Bishara AG, El-Azazy SS and Huang T (1981): “Practical Analysis of Cylindrical

Farm Silos Based on Finite Element Solutions”; ACI Journal, November/

Decmeber 1981, pg 456-462.

Blair-Fish PM and Bransby PL (1973): “Flow patterns and Wall Stresses in a Mass

Flow Bunker”; Journal of Engineering for Industry, Transactions of the ASME,

February 1973, pg 17-27

Blight GE, Fliss L and Schaffner RH (1989); “Design and Performance of Two Types

of Cement Storage Silos”; Journal of the Civil Engineer in South Africa, Vol 31,

number 1, January 1989, pg 11-21

Blight, G E (1993): “Loading Applied to Silos by Retained Granular Materials and

Powders”; GDE Lecture Notes, The University of the Witwatersrand, South

Africa, 1993


Briassoulis D (1991): “Limitations in the Range of Applicability of the Classic Silo

Theories”; ACI Structural Journal, July-August 1991, pg437-444

Brown RL and Richards JC (1970): “Principles of Powder Mechanics”; First Edition,

1970, Pergamon Press

Drescher A, Cousens TW and Bransby PL (1978): “Kinematics of the Mass Flow of

Granualr material through a Plane Hopper”; Geotechnique, Vol 28, No 1,

1978, pg 27-42

Gere JM and Timoshenko (1996): “Mechanics of Materials”; Third SI Edition,

Chapman & Hall, London, 1996, ISBN 0-412-36880-3

Hasra and Bazur (1980): “Considerations in the Design of Silos for Process

Industry”; Proceedings of the International Conference on Design of Silos for

Strength and Flow, University of Lancaster, U.K, 2-4 Sept, 1980

Jenike AW (1961): “Gravity Flow of Bulk Solids”; Bulletin 108, Utah Engineering

Experiment Station, University of Utah, Salt Lake City, October 1961

Jenike AW (1964): “Storage and Flow of Solids”; Bulletin 123, Utah Engineering

Experiment Station, University of Utah, Salt Lake City, November 1964

Jenike AW (1967): “Quantitative Design of mass Flow Bins”; Journal for Powder

Technology, Vol 1, 1967, pg 237-244

Jenike AW (1968): “Bins Loads”; Journal of the Structural Division, Proceedings of t

the American Society of Civil Engineers, April 1968, pg 1011-1040

Jenike AW and Johanson JR (1969): “On the Theory of Bin Loads”; Journal of

Engineering for Industry, Transactions of the ASME, May 1969, pg 339-344

Jenike AW, Johanson JR and Carson JW (1973a): Bin Loads – Part 2: Concepts”,

Journal of Engineering for Industry, Transactions of the ASME, February 1973,

pg 1-5

Jenike AW (1973b): “Bin Loads – Part 3: Mass-Flow Bins”; Journal of Engineering

for Industry, Transactions of the ASME, February 1973, pg 6-12

REFERENCES 9.3

Jenike AW, Johanson JR and Carson JW (1973c): Bin Loads – Part 4: Funnel Flow

Bins”; Journal of Engineering for Industry, Transactions of the ASME, Series

B, Vol 95, Number 1, February 1973, pg 13-16

Johanson JR (1982): “Controlling Flow Patterns by use of an Insert”; Bulk Solids

Handling, Vol 2, No 3, Sept 1982.

Kmita J (1991): “An Experimental Analysis of Internal Silo Loads”; Bulk Solids

Handling, Vol 11, Number 2, May 1991, pg 459-467

Lansgton PA, Tuzun U and Heyes UK (1995): “Computer Simulations of Hopper

Flows - A Design Tool or a Research and Teaching Facility”; Bulk Solids

Handling, Vol 15, No 3, July/September 1995, pg 385-389

Luong MP (1993): “Flow Characteristics of Granular Bulk Materials”; Journal of

Particle-Particle-System Characteristics, Vol 10, 1993, pg 79-85

McLean AG (1985): “The Design of Silo Side Discharge Outlets for Safe and Reliable

Operation”; Bulk Solids Handling, Vol 5, Number 1, February 1985, pg 185-

190

Molenda M, Horabik J and Ross IJ (1993): “Loads in Model Grain Bins as Affected by

Filling Methods”; American Society of Agricultural Engineers, Vol 36, number

3, May/June 1993, pg 915-919.

Nielsen J and Andersen E Y (1982): “Loads in Grain Silos”; Bygningsstatiske

Meddelelser, Danish Society for Structural Science and Engineering, Vol 53,

number 4, 1982, pg 123-135. ( In English).

Nielsen J and Askegaard V (1977): “Scale Errors in Model Tests on Granular Media

with Special Reference to Silo Models”; Powder Technology, Volume 16, 1977,

pg 123-130

O’Neil, P V: “Advanced Engineering Mathematics”; Fourth Edition, PWS Publishing,

Boston, Massechusets, 1995, ISBN 0-534-94320-9

Ooms M and Roberts AW (1985): “The Reduction and Control of Flow Pressures in

Cracked Grain Silos”; Bulk Solids handling, Vol 5, No 5, October 1985.


Pieper K (1969): “Investigation of Silo Loads in Measuring Models”; Journal of

Engineering for Industry, Transactions of the ASME, May1969, pg 365-372

Ravenet J (1983): “The Development of Industrial Silos Throughout the World

During the Last 100 years”; Bulk Solids Handling, Vol 3, number 1, March

1983, pg 127-140

Reed AR and Duffell CH (1983): “ A Review of Hopper Discharge Aids”; Bulk Solids

Handling, vol 3, No 1, March 1983

Reimbert M and A (1976): “Silos Theory and Practice”; First edition, Trans Tech

Publications, Germany, 1976, ISBN 0-442-22684-5

Richards PC (1977): “Bunker Design - Part 1: Bunker Outlet Design and Initial

Measurements of Wall Pressures”; Journal of Engineering for Indistry,

Transactions of the ASME, November 1977, pg 809-813

Roberts AW (1995): “100 Years of Janssen”; Bulk Solids Handling, Vol 15, No 3,

July/September 1995.

Rombach G and Eibl J (1995): “Granular Flow of Materials in Silos: Numerical

Results”; Bulk Solids Handling,Vol 15, number 1,January/March 1995,pg 65-

70

Schulze D (1996): “Silos- Design Variants and Special Types”; Bulk Solids Handling,

Vol 16, Number 2, April/June 1996

Schwedes J and Schulze D (1991): “Examples of Modern Silo Design”; Bulk Solids

Handling, Vol 11, number 1, March 1991, pg 47-52

Scutella L (1998): “Local Buckling of Perforated Cylindrical Shells under Uniform

Axial Compression”; Honours Thesis, University of Western Australia,

Department of Civil Engineering, October 1998

Starnes JH Jr(1972): “Effect of a Circular Hole on the Buckling of Cylindrical Shells

Loaded by Axial Compression”; AIAA Journal, Vol 10, Number 11, November

1972, pg 1466-1472

REFERENCES 9.5

Suzuki M, Akashi T and Matsumoto K (1985): “Flow Behaviour and Stress

Conditions in Small and Medium Silos”; Bulk Solids Handling, Vol 5, number 3,

June 1985, pg 63-72.

Tennyson RC (1968): “The Effects of Unreinforced Circular Cutouts on the Buckling

of Circular Cylindrical Shells under Axial Compression”; Journal of Engineering

for Industry, Transactions of the ASME, November 1968, pg 541-546

Timoshenko SP and Gere JM (1963): “Theory of Elastic Stability”; Second edition,

McGraw-Hill Book Compnay, Singapore, 1963, ISBN 0-07-085821-7

Van Zanten DC and Mooij A (1977): “Bunker Design-Part 2: Wall Pressures in Mass

Flow”; Journal of Engineering for Industry, Transactions of the ASME,

November 1977, pg 814-818

Walker DM (1966): “An Approximate Theory for Pressures and Arching in Hoppers”;

Chemical Engineering Science, 1966, Vol 21, pg 975-997.

Walters JK (1973a): “A Theoretical Analysis of Stresses in Silos with vertical walls”;

Chemical Engineering Science, 1973, Vol 28, pg 13-21.

Walters JK (1973b): “A Theoretical Analysis of Stresses in Axially-Symmetric

Hoppers and Bunkers”; Chemical Engineering Science, 1973, Vol 28, pg 779-

789.

Documents

THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS · equipment. A steel model silo with an anti-dynamic tube was set up in the laboratory to measure the friction on the tube during material