..MASTER OF SCIENCE IN CIVIL ENGINEERING

- - .- -

1111111111111111111111111111111111#88056#

NOVEMBER, 1994

MD. ALA UDDIN

A THESIS BY

Submitted in partial fulfilment of the requirements for the

degree of

DEPARTMENT OF CIVIL ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY

A DESIGN RATIONALE FOR CIRCULARSILOS BASED ON FINITE ELEMENT

ANALYSIS

R624-15)(~q9~ A DESIGN RATIONALE FOR CIRCULARilL SILOS BASED ON FINITE"ELEMENT

ANALYSIS

••11,

Member

Member(External)

Member

Chairman(Supervisor)

~Dr. M. Shamim Z. BosuniaProfessorDepartment of Civil EngineeringBUET, Dhaka

~~~Dr. Md. Wahha$UddinProfessorDepartment of Mechanical EngineeringBUET, Dhaka

A .N. tv\ .ct J.9....L-Dr. A. M. M. SafiullahProfessor and HeadDepartment of Civil EngineeringBUET, Dhaka

ii

£~94-Dr. Sohrabuddin AhmadProfessorDepartment of Civil EngineeringBUET, Dhaka

A THESIS BY

MD. ALAUDDIN

Approved as to the style and content on November 29, 1994 by

TO

MY

TEACHERS

iii

DECLARATION

Declared that, except where specific references are made to otherinvestigators, the work embodied in this thesis is the result of investigation carriedout by the author under the supervision of Dr. Sohrabuddin Ahmad, Professor of

Civil Engineering, BUET.

Neither this thesis nor any part of it has been submitted or is beingconcurrently submitted in candidature for any degree at any other institution.

Author

r

ACKNOWLEDGEMENT

The author expresses his indebtedness to Dr. Sobrabuddin Ahmad,Professor of Civil Engineering, BUET, for his continuous guidance, invaluablesuggestions and affectionate encouragement at all phases of this work.

Profound gratitude is expressed to Dr. Sobrabuddin Ahmad for providingthe author the valuable Finite Element Program that was the basic tool in this

research.

The author is grateful to Dr. M. Shamim Z. Bosunia, Professor of CivilEngineering, BUET, for his constructive and valuable suggestions at various stagesof this work. .

The author expresses his deep gratitude to Mr. Md. Golam Mohiuddin,Associate Professor of Industrial and Production Engineering, BUET, for providingvaluable reports and information, as well as for facilitating a visit to Narayanganjsilo site, which were very useful in this research.

Heartiest thanks are expressed to Mr. M.A. Malek of Civil EngineeringDepartment for typing this thesis with extreme care.

Particular appreciation is expressed by the author to all his friends andcolleagues for their encouragement and support

vi

Chapter 2: LITERATURE REVIEW

2.1 Introduction 82.2 Types and Defmition 82.3 Effect of Stored Material and other Factors on

Silo Analysis and Design 92.4 Pressures in Silo 11

2.4.1 The Janssen Method 112.4.2 The Reimbert Method 142.4.3 The Airy Method 152.4.4 Pressure Normal to Inclined Surfaces 16

2.5 Flow Patterns 172.6 Total Pressure - Static plus Overpressure 182.7 Effect of Eccentric Discharge and Non-symmetrical Flow 19

2.7.1 ACI 313-77 Approach 202.7.2 Safarians Method 212.7.3 A more nearly Rational Procedure 21

2.8 Loads in Silos 232.8.1 Dead Loads 232.8.2 Live Loads 232.8.3 Wind Loads 232.8.4 Seismic Loadings 242.8.5 Thermal Effects 25

13467

IV

V

IX

X

CONTENTS

Chapter 1: INTRODUCTION

1.1 General1.2 History and Past Research1.3 Objective and Justification of the Research1.4 Scope of the Study1.5 General Remarks

DeclarationAcknowledgementAbstractNotations

vii

Chapter 4: PARAMETRIC STUDY

4.1lntrodeuction 534.2 Silo Parameters 534.3 Stress Resultants in Silos 544.4 Effect of Variation of Parameters

4.4.1 Effect of Height of Vertical wall 554.4.2 Effect oflntemal Diameter of Silo 594.4.3 Effect of Inclination of Conical Hopper with Horizontal 624.4.4 Effect of Bottom Thickness of Vertical Wall 634.4.5 Effect of Conical Hopper Thickness at Top 654.4.6 Effect of Depth of Bottom Ring Beam 664.4.7 Effect of Unit Weight of Stored Materials 67

252626

2.8.6 Other Loads2.8.7 Load Combinations

2.9 Silo Roofs

Chapter 3: ANALYSIS OF SILOS

3.1 Conventional Method of Analysis and Design3.1.1 General 283.1.2 Design Steps - Conventionally Reinforced Concrete Silos 283.1.3 Analysis and Design of Circular Silos 293.1.4 Analysis of Prototype Silo 37

3.2 Finite Element Analysis3.2.1 General 393.2.2 The Finite Element Program 393.2.3 Finite Element Idealisation of Silo 413.2.4 Determination of Forces and Moments at a Section 433.2.5 Capability of the Program at the Present Stage 443.2.6 Analysis and Presentation of the Results 46

3.3 Comparative Study3.3.1 Mode of Comparison 473.3.2 Forces Obtained from Both the Methods of Analysis 483.3.3 Moments Obtained from Finite Element Analysis 503.3.4 Hoop Force due to Wind Load Predicted by Finite

Element Analysis 513.3.5 Remarks 51

4.4.8 Effect of Angle ofIntemal Friction of StoredMaterial 694.4.9 Effect ofCo~efficient of Wall Friction 70. 4.4.10 Effect of Wind Pressure Intensity 724.4.11 Effect of Height of Hopper Bottom Above Floor Level 73

4.5 Remarks 74

Chapter 5: A DESIGN RATIONALE

5.1 General 755.2 Basis of the Proposal 765.3 Proposed Design Rationale 76

5.3.1 Maximum Values of Stress Resultants 775.3.2 Variation of Stress Resultants in Vertical Direction 98

5.4 Comparison of Results Obtained from Propoesd DesignRationale with those of Finite Element Analysis 99

105106108109

List of References

viii

Chapter 6: CONCLUSIONS

6.1 General6.2 Findings from the Investigation6.3 The Design Rationale6.4 Scope for Future Research

Appendix

ABSTRACT

The current practice of silo analysis is based on several assumptions andidealisation. Analysis of silo by Finite Element method and comparison of theresults with corresponding values obtained from conventional method reveals thatthe. conventional method cannot predict all the stress resultants (forces andmoments) required for silo design. Again the functions predicted by theconventional method deviate largely from the actual values for a region near thering beam of a silo. Conventional method can analyse a silo for axisymmetricloading only and cannot evaluate various types of moments which may be ofconsiderable magnitude. Since silo is an elevated structure it may be subjected to aconsiderable amount. of wind load which is non-axisymmetric in nature.Earthquake loading haS significant effect on silo behaviour which is also non-axisymmetric. Finite Element approach can analyse a silo for axisymmetric as wellas non-axisymmetric loadings easily. It was, therefore, felt that the application ofFinite Element will lead to a realistic analysis and to a more rational designprocedure for silos.

With this objective an extensive investigation was carried out on thebehaviour of silos of different types under various loading conditions using theAxisymmetric thick shell Finite. Element program by Alunad. A number ofparameters influencing silo behaviour were selected and a detail parametric studyhas been carried out to reveal the sensitivity of stress resultants with respect to aparticular parameter. From this study it became obvious that the effect of restraintprovided by the ring beam at the bottom of vertical wall can not be ignored.Actually, the moments developed in a silo are due to the restraint provided by thethickened ring beam at the bottom of vertical wall (top of conical hopper). Thebehaviour of silos under non-axisymmetric loading, such as wind load, has beenstudied elaborately. It was observed that wind load produces considerablemeridional force and circumferential moment in the vertical wall which must beconsidered in the design of a silo.

A detail study has also been carried out to know the effect of temperaturedifference between inside and outside of a silo. The conventional equations for thecomputation of meridional moment and circumferential moment have beenmodified to take into account the effect of restraint provided by the ring beam.

Finally, on the basis of the study a design rationale has been presented.Using this rationale the stress resultants required for silo design can easily becomputed.

ix

A

. ArAsA'sAsvCCdCi

C2, C3DE

FFmGHH'

M'

NOTATIONS

= Area; Interstice dimension= Area of ring-beam cross section= Area of tensile reinforcement per unit width= Area of compression reinforcement per unit width= Area of reinforcement in vertical direction per unit width= Reimbert's characteristic abscissa= Overpressure coefficient= Impact factor= Factors for ring-beam analysis= Diameter; Dead load= Modulus of elasticity= Force= Meridional force per unit width= Shear modulus of elasticity= Horizontal force; height of storage zone= Total height of vertical wall (Type-3)= Moment of inertia= Torsion factor= Load factors for live and dead load= Ccoefficient for wall temperature gradient= Factor for ring beam analysis= Length of column; Length of conical hopper= Moment; Meridional moment; Circumferential moment= Moment applied to ring-beam by column= Nominal (theoretical) ultimate strength of wall per unit width= Ultimate load= Hydraulic radius= Temperature=Temepratureof&medmmerial= Atmospheric temperature .

x

xi

Uf

.'f'

aa'

beff

d

d'

=Cross-section perimeter=Depth of stored material above point in question; upwarddistance from bottom of vertical wall (Type-I, Type-2)

=Upward distance from bottom of vertical wall (Type-3)= Opening width; width of rectangular or polygonal silo= Fictitious length for side of rectangular silo= Effective width=Effective depth of flexural section, from compression face ofconcrete to centroid of tensile reinforcing,

=Distance from compression face of concrete to centroid ofcompression reinforcing bars

e = Eccentricityecc = Subscript meaning" eccentric" or "eccentricity"eff = Subscript indicating" effective"f =Actual or computed stressfc =Compressive stress; compressive stress in concretef~ =Unit compressive strength of concreteis =Computed tensile stress in reinforcing steelf; =Ultimate tensile strength of concretej; = Specified yield strength of steelg =Acceleration due to gravity; subscript meaning "gravity"h =Wall thickness; subscript indicating" hopper"; depth of vertical

wall below pressure zoneh' = Height of hopper bottom above floor levelI =Distance measured downward from top of conical hopperm = Concrete shrinkage coefficient; subscript meaning "meridional"max =Maximum; subscript meaning "maximum"min = Subscript meaning "minimum"a = Subscript indicating" initial"p = Lateral pressure due to stored materialq =Vertical pressure due to stored material; intensity of wind

pressurer = Radius; subscript meaning "ring-beam"s = Subscript meaning "static" or "steel"t = Thickness; subscript meaning" total", "top"u = Subscript indicating "ultimate"

v = Subscript meaning "vertical"Wcr =Width of crackw}. W2, W3 =Width of crack due to various loadingsx = Subscript for "x-direction"y = Subscript for "y-direction"X, y = Co-ordinates of centroidL1 =Displacement (linear); deflectionL1T = Temperature difference, outside and inside wall facesL = Sum of reinforcing bar perimeters per unit width of walla =Angle of hopper slope; subscript for forces or pressures on

sloping surfaceat = Linear coefficient of thermal expansionfJ = factor for crack-width computatione =Angle around perimeter; subscript meaning" tangential"

directionA, = Factor for ring-beam analysisf.l =Angle of wall friction (storied material against wall or hopper)f.l' =Coefficient of wall friction (tan f.l)v = Poisson's ratiop =Angle of internal friction for stored material(J" = StressrjJ = Strength-reduction factor; subscriptfor "meridional" directionIf/ 1' If/ 2' If/ 3 = Factors for crack-width computation

xii

CHAPTER!

INTRODUCTION

1.1GENERALThe custom of storing grain in upright containers is centuries old. Not until

the mid 1800's, however, relatively large storage containers were built forcoinmercial purposes. Since then silos and bunkers have come into extensive usenot for storing grain alone but for storing a wide variety of granular materials. Inagriculture and industry improved production methods and mechanisation ofhandling have opened the way for large storage complexes, with sophisticatedfilling, unloading and handling systems.

Bunker or Bin is the tenn applied commonly to a structure in which drygranular materials are stored. Such structures, generally elevated above the ground,may be rectangular or circular in plan and may comprise one or morecompartments.

The tenn 'Silo' includes both deep bins and shallow bins, the latter usuallyreferred to as bunkers. However the tenn 'Bin', 'Silo' and 'Bunker' have differentmeaning in different. parts of the world. Actually the tenn 'silo' represents deepbins.

As stated above concrete silos and bunkers may be single or multiple and ofvarious plans. (Fig.1-1). The most common shape is circular, since under unifonnlateral pressure around the circumference the circular wall is under tension with nobending moment. For this reason, circular silos are built with diameters farexceeding practical lateral dimensions for rectangular or square silos.Unfortunately, large diameter circular silos usually have several dischargeopenings. All of them or all but one are eccentric so that a moment free condition is

(e)

(g)

(e)POCKET BIN

INTERSTICE

INTERSTICE WrTH_FLAT INSIDE WALLS

-L~~.

, I I

(a)

(f)

(b)

(d)

Fig. 1-1. Typical Silo and Bunker groups

seldom realised. Again, non-symmetric loadings, such as wind load or earthquakeload, may produce meridional or circumferential bending moment in the silo wall.From the architectural point of view, circular silos are also more pleasant looking.Although circular form costs more but requires less construction material comparedto other shapes.

Concrete IS the material most frequently used for Silo or Bunkerconstruction. Usually it is cast in place, but occasionally, it is precast. It may beconventionally reirIforced or prestressed, unlined or lined with protective material.

Concrete may be used for the complete structure - foundation, walls, roof,and flat bottom or hopper; or certain components (such as the hopper bottomsupports or roof) may be of steel while the remainder is concrete.

Wheather steel or concrete will prove more economical for a particularapplication depends. on many factors including cost, size, complexity of thestructure, locations of silos, and problems of delivering construction materials tothe site.

A circular silo essentially consists of a number of axisymmetric structuralelements, namely the roof, cylindrical vertical wall and the bottom. The top roofmay be of concrete, doweled to the walls by providing full or partial continuity ofwalls or it may be supported in a marmer permitting free expansion and contractionand slight movement due to lateral forces. The vertical wall may be of uniform orvarying thickness. Flat bottom may create problem in the removal of material. Onthe other hand conical hopper, either of uniform or varying thickness, is selfcleaning.

Vertical wall and conical hopper of a silo may be monolithicallyconstructed and supported on columns or continuous circular vertical wall. Againvertical wall and conical hopper may be supported separately. Various types ofsupports and foundation are shown in the Fig.1-2. Since silos are elevatedstructures, overtuming moment due to lateral load must be considered in the designof foundation

2

(g)

(c)

(f)

( b)

(e)

~tV 0~u ~C tV

:r: 0 <J) -u 0

:r: <J) E-- - ~0

~ J: ::>:r: c

0~CJ>

(0)

(d)

:r:

k\. /: .c:.c

Fig. 1-2. Typical Vertical cross section of silos. (a) Silo walls on continuous footing, silo bottom consistingof tunnel and fill around and on top of tunnel. (b) Silo on raft foundation, independent hopper resting onpilasters attached to wall. (c) Silo with wall footing and independent bonom slab supported on fill. (d) Silowith hopper-forming fill and bottom slab supported on thickened lower wall. (e) Silo with multipledischarge openings and hopper-forming fill resting on bottom slab, all supported by columns; raftfoundation with stiffening ribs on top surface. (f) Silo on raft foundation, with hopper independentlysupported by a ring beam and column system. (g) Silo walls on continuous footing; bottom is a slab ongrade.

1.2 HISTORY AND PAST RESEARCH

The fIrst large silos were constructed over 120 years ago for storing grain.Today, silos are increasing rapidly in height, diameter and storage capacity. Thevariety of stored materials are also on the increase. Designers and builders areemploying higher strength steels and concrete, prestressing, and a great variety ofwithdrawal systems having high throughput.

Before 1860, designers assumed that granular material behaved as aquasiliquid and exerted pressures similar to hydrostatic pressure. Theseassumptions lead to overdesign for static horizontal loads and bottom loads, butfailed to account for the vertical material friction load on the walls. Early in the1880's, Roberts [35] discovered that the pressure on the bottom of a grain silo didnot increase after the material depth reached twice the silo width. The frictionbetween the granular material and the wall transferred weight to the wall. In 1885,Janssen [17] derived formulas for granular material pressures on the silo walls andbottom. The fIrst period of growth in silo design, starting with Roberts [35] in 1882and ending with Ketchum [24] in 1909, was characterised - for that time - byexperimental precision and great clarity of formulation. In the middle period,Reimbert [32] introduced the 'antidynarnic tube', and modifIed Janssen's formulabased on a hyperbolic function. Reimbert's formula is used in many codes aroundthe world. Few other contributions were made until 1965 as investigators primarilytried to relate flow pressures to Janssen's static pressure.

Since 1965, many excellent studies have appeared in the literature. Theexperimentation reflects the precision of modem technology, and the formulationsreflect the clarity of the early investigators.

Several investigators questioned and investigated many of Janssen's basicassumptions of a constant ratio of vertical to horizontal pressure, and constantdensity. Cowin and Sundaram [12] derived a modifIed Janssen formula reflecting.linear variations in both variables.

One of the most important silo developments in the last two decades is theincreased understanding of granular flow. Jenike [18] defmed mass and funnel flowand derived differential equations for mass flow. Johanson [22] used the method ofcharacteristics to determine the stresses in converging .flow charmels. Walker [40],followed by Walters [41], developed the most practical and simplest approaches for

3

calculations of mass-flow pressures. Clague and Wright [10] and Bransby et al [8].have experimentally measured the pressures created by mass flow. Johanson [23]and Williams [42] developed formulas for computing discharge rates from mass-

flow conical hoppers.

1.3 OBJECTIVE AND JUSTIFICATION OF THE RESEARCH

While earlier silos were only for more or less sedentary storage, the silo oftoday often plays an active role in the manufacturing and distribution process.Mixing, blending, proportioning - all are done using the silo as a vital part of the

process system.

In Bangladesh, silos for granular materials are constructed mainly to fulfilstorage requirement. The volume of annual food-grain import is considerably highin our country. In contrast of normal storage facilities such as godown, silo facilityrequires lesser time for the withdrawal of granular material from ship. In case ofnormal storage facility the unloading time for a large ship may be as high as two orthree months. On the other hand, pneumatic unloading of grain reduces theunloading time, greatly. Due to this reduction of unloading time, shipping chargebecomes lower. At the time of national emergency supply of food grain should beat a higher rate. Only silos, using mechanical withdrawl system, can provide thisfacility. Again silos require less plan area than normal storage facility for the samecapacity.

Recently, the desire to withdraw stored material faster has led to a demandfor larger capacity silos, having either greater height or greater diameter or both.Each new trend brings new challenges to silo designers and builders.

Remarkable progress has already been made in understanding the behaviourof granular material in silos. This progress has resulted largely from years ofexperiments conducted in many parts of the world to study the pressure of storedgranular materials against the walls and bottom of silos.

A number of structural analysts have attempted to develop a procedure forsilo analysis and design. The methods followed for silo analysis and design,however, are more or less the same. These methods are based on severalassumptions and idealisation of silos, and much remains to be leamed in thisrespect.

4

A designer is interested to know the exact state of stress in a loaded bodyeven though he may use simple formulas in his design process. In conventionalmethods exact analysis of stress is not possible and designers prefer short andapproximate methods. In such cases the approximation must be rational and thedesigner should have a clear idea about the degree of approximation. The extent oferror of an approximate method can be assessed only when the exact state of stressis known. But the exact analysis of a complicated structure like silo is not possiblebecause of the limitation of analytical formulation of the problem. In that case wecan use a suitable numerical technique like Finite Element approach.

Again the conventional method stated above, can deal with the design of asilo for only axisymmetric loading dile to gravity and stored materials. But a silo isan elevated structure which may be subjected to tremendous lateral loads due towind and earthquake. The conventional methods can not incorporate the effect oflateral loads in their design procedures effectively. Ahmad, S. [3, 4, 5] developed ageneral Finite Element program (1969) for the analysis of axisymmetric shellstructures for symmetric as well as non-symmetric loadings. Initial work carried outby the author with this program in his undergraduate research [7] has revealed thepotentials for developing a design rationale for silos based on the use of thisprogram in the detailed analysis for axisymmetric as well as non-symmetricloadings. Observing the drawbacks of conventional methods and versatility ofFinite Element method, a study was undertaken with the following objectives:

i) To investigate the state of stress in silos using Finite Element methodof analysis.

ii) To compare the design values of various stress resultants obtainedfrom conventional method and Finite Element method.

iii) To investigate the effects of various parameters on the values of stressresultants required for silo design.

iv) To develop a simple and straight forward design rationale forcircular silos.

5

1.4 SCOPE OF THE STUDY

For developing a design rationale for circular silos, conventional methodsare studied in details to find out its drawbacks in respect of approximations and

stress conditions.

A full-scale model or prototype is selected for comparison of conventionalanalysis with Finite Element analysis. For conventional analysis a FORTRANprogram is developed which is also capable of computing temperature stress andmeridional force due to wind. Design values of stress resultants obtained from theconventional method are compared with those obtained from Finite Elementanalysis. Stress resultants such as meridional moment and circumferential moment,which can not be obtained from conventional method, are also studied with theirdistribution along a vertical section.

Various forces and moments required for silo design such as meridionalforce, meridional moment, hoop force and circumferential moment vary with thechanges of a number of parameters. These parameters include both the geometricdimensions and the properties of the stored materials. In order to know theinfluence of various parameters on the overall behaviour of a silo, a sensitivityanalysis or parametric study is carried out. The results of the parametric study areessential in visualising the structural response of silo and in establishing the relativeimportance of different parameters. For parametric study the same model isselected. In every case of analysis only one parameter is varied and the others arekept constant. Finally, on the basis of the parametric studies, a design rationale isdeveloped.

For the analysis of ring beam of a silo a conventional method is suggestedby Safarian and Harris [36]. Using this method a FORTRAN program isdeveloped, a detailed investigation is made, and a design recommendation ispresented for ring beam.

In this study a systematic way is followed in presentation. Some generalaspects of silos and various methods of pressure computation are presented inChapter 2. Different types of loadings which should be considered in silo designare also discussed in Chapter 2. Chapter 3 deals with the review of theconventional methods of analysis, various aspects of the Finite Element analysisused in this research and a comparative study of both the methods. Details of the

6

parametric study is presented in Chapter 4. Subsequently, on the basis of thefindings in Chapter 4, a design rationale is presented in Chapter 5. Finally, inChapter 6 findings of the study are discussed.

1.5 GENERAL REMARKS

The investigations into the behaviour of silos revealed that the conventionalmethod cannot predict moments developed in silo, while the Finite Elementmethod can predict them easily. The effect of restraint provided by the ring beamor roof slab can also be assessed by Finite Element method while conventionalmethod can not do that.

Parametric study is done to reveal the SenSItiVItyof various stressresultants with respect to different parameters. This acted as a guide for thedevelopment of a design rationale. The suggested design rationale can dealeffectively with the analysis and design of silos having wide range of parameters.

***

7

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

Reinforced concrete silos have almost replaced the normal concrete andsteel storage structures for storing coal, cement, food grains and other granularmaterials because of their ease of construction, greater capacity, economy inhandling of material and superior architectural qualities. In modem world there area number of silos in evry country. The codes and standards of silo design vary fromcountry to country. These include DINlO55 silo code in Germany, CH302 silocode in the Soviet Union, the French silo code, and the ACI 313 standardrecommended practice in the United States. In this research work all the codes havebeen studied and ACI 313 standard recommended practice is followed for analysisand design. However, for the computation of pressure and frictional force inconical.hopper the German silo code is followed.

2.2 TYPES AND DEFINITION

The proportions of a bin, especially the ratio of material depth to leastlateral dimension, affect the behaviour of stored materials both at rest and duringdischarge. Since bin geometry affects pressures, it is classified either as a silo (deepbin) or a bunker (shallow bin). Proper classification (which should also considerthe flow condition) may soon be feasible, but presently the following methods arewidely used in practice:

(a) Empirical approximations - preferred by many engineers. Two suchapproximations are:

I. By Dishinger [2]H > 1.5A

2. By the Soviet Code [39]H > 1.5]) forcircular silosH > 1.5a for rectangular silos

Where H = height of storage zoneA = interstice dimensionD = diameter of circular silosa = width of wall of rectangular silos

If the storage structure in question satisfies either of the above, it is considered asilo. If it satisfies neither rule, it is considered to be a bunker.

(b) An approximation based on the position of the plane of rupture. Fig.2-Ishows bins of two different depths. The plane of rupture is determined by theCoulomb theory. Neglecting friction against the wall, for the case of a vertical wall,and horizontal top surface, the Coulomb plane of rupture is midway between theangle of internal friction ( p) and the vertical wall. (For other wall positions orsurface slopes, the plane of rupture can be located analytically or graphically - byCulmann's method for example). According to A Reimbert [33], the angle ofrupture should be given by (Jr/4 - P /3) rather than by the classic definition (Jr/4 - P/2), both shown in Fig.2-1. If the rupture plane intersects the top surface of thestored material, the bin is a bunker (Fig.2-Ia), otherwise it is a silo (Fig.2-Ib).

However, engineers do not agree on the location of the plane of rupture.Some would start the plane at the bottom of the hopper, point C of Fig.2-Ib, whileothers would pass it through point D, at the bottom of the vertical wall. Thus, byone interpretation the bin would be a silo; by the other, a bunker.

2.3 EFFECT OF STORED MATERIAL AND OTHER FACTORSON SILO ANALYSIS AND DESIGN

The physical properties of materials stored in silos and bunkers influencethe flowability of the material and the forces which the material applies to the silowalls and bottom. Obviously those properties will vary from one material toanother, but they may also vary within a supposedly uniform material. In materialsof this latter type (coal is a good example) large variations of properties occurbetween materials from different sources or even in materials from a commonsource. The physical properties may also vary with age of the material, degree ofcompaction, and changes of environment.

9

Fig.2-1 Classification of bins using plane of rupture

-lPlane ofrupture

Top of materialin Bunker -,

I-----,----

By Classic approach(90 -P)/2

:." ::

: Top of material: in8i1°i: L

(a) Bunker

(b) Silo

o

Plane ofrupture

By Reimbert approach(45 -P/3)

By Classic approach

(90-P)/2 I

By Reimbert approach(45 -P/3)

For pressure computation, the properties considered most important are unitweight ( y), angle of internal friction ( p, approximately the same as angle ofrepose), and the coefficient of friction ( f.I.') between the stored material and the binwall.

Other properties that may influence flowability or pressure include particlesize and gradation (which affect moisture content), physical strength (which affectsdegree of compaction), cohesiveness, and shrinkage or swelling characteristics.

Unit weight, 1, may vary with depth below the surface of the storedmaterial, the lower material being compacted by pressure from that above. Unitweight may vary also with time in storage, and with the method of filling.

Wherever the unit weight is expected to vary, it is prudent to obtain materialproperties from laboratory tests covering the full range of materials to beencountered. The tests should include 1, p, and f.I. ~ and may also consider thevariation of these properties with pressure from the material above.

The coefficient of friction, f.I.' between stored material and the bin wall may.also vary with age of the bin. Whether the wall is metal or concrete, it willprobably become smoother with age, from abrasion by the sliding material.Powdery materials (cement, for example) may adhere to the walls causing thecoefficient of friction ( f.I.') to approach the coefficient of internal friction (tanp).

Materials containing oils or waxes (soya beans, for example) may lubricate thewall, thus reducing the friction coefficient, and increasing lateral pressures on the

walls.

Table 2-1 [CommentaIy of ACI 313-77] gives approximate values of unitweight, angle of internal friction, and coefficient of friction against steel andconcrete for various materials. The Table shows values typical of what might begiven by tests, but cannot possibly give precise values or the range of values forany given material. Caution must be exercised in using these values. •.._",

"Some materials are hot when stored. Cement, cement clinker, and fly ash

are examples. Large volumes of hot stored material can cause serious thermalstresses on walls, bottom, and roof of the bin structure. It is important to learn inadvance the temperature of the material to be stored. Abnormally cold materialscould conceivably be as troublesome as hot materials.

10

2.4 PRESSURES IN SILO

Early silo designers did not recognise the vertical friction between the storedmaterial and the vertical wall, assumed lateral pressures to vary hydrostatically.Subsequently analytical methods have been developed that consider wall friction.These methods provide means for computing (I) Pressure of the stored materialagainst vertical wall, sloping surfaces, and flat bottom; (2) friction forces and wallcompression forces; and (3) vertical pressures at various depths in the storedmaterial itself.

Analytical methods normally give the static pressures (pressures whenmaterial is at rest) only. The structural designers need to know the final totalpressure, or "Design Pressure". This design pressure can be estimated by modifyingthe computed static pressure to account for material movement, eccentricdischarge, and other pressure-affecting conditions or by using analytical methodsintended to give design pressures directly.

The analytical methods are based on equilibrium of the stored material in astatic condition. Elastic interaction with the bin structure is not considered, nor isstrain energy in either the stored material or the structure. In this study at first thestatic pressure is computed which is converted to design pressure from multiplyingthe static pressure by overpressure factor which is discussed in Article 2-6. Threewidely used methods for computing static pressures in silo are briefly discussedhere.

2.4.1 The Janssen Method

The mltior breakthrough in computation of stored material pressures came in1895 when H.A. Janssen developed equations for computing lateral and verticalpressures of granular material in deep bins [17].

Janssen's method is based on equilibrium of a thin horizontal layer of storedmaterial, as shown in Fig. 2-2. Equating the vertical forces to zero gives

qA + r Ady = A[ q + dY~;] + j..l'p(Udy)

Where q = static vertical pressure at depth Y below surface of stored materialA = area of horizontal cross section through the siloU = perimeter of horizontal cross sectionp = pressure of stored material against walls at depth Y

II

Hopper

Upward and downward pressures are assumed uniformover entire area

t-I P

(b) Horozontallamina

q

(a) Silo

Fig. 2-2 Horizontal lamina for derivation of Janssen'sequations

p = Horozontal pressure between wall and storedmaterial, assumed uniform around perimeter U

Y Ady. = Lamina weight

J.r'p = Friction force per unit area of wall in contactwith lamina

12

Table 2-1. Physical Properties of Granular Materials [2].

fl' k--qR

dq-=ydy

The solution to this differential equation is the Janssen formula for vertical pressureat depth Y and is given by

Substituting kq for p, and "hydraulic radius" R for A/V and rearranging, thedifferential equation of equilibrium becomes

Material Unit Wcigh~ y Angle of Rep- Coefficient of wall Friction. j.l

per kg/m' ose. Deg. = p Against Concret, Against Stee

Cement Clinker 88 1,410 33 0.60 0.30

Cement Portland 84 - 100 1,344 - 1,600 24 to 30 0.36 - 0.45 0.30

Clay 106-138 1.810-2.210 15 to 40 t1.20 -0.50 0.36 - 0.70

Coal, Bituminous 50 - 65 800 - 1,040 32 to 44 0.50 - 0.60 0.30

Coal, Anthracite 60 -70 960 - 1,120 24lo 30 0.45 - 0.50 0.30

Coke 38 600 40 0.80 0.50

Flour 38 600 40 0.30 l1.30

Gravel 100 - 125 1,600 - 2,000 25 to 35 0.40 -0.45 -

Grain (small): Whea~ Com,

Barley, Beans (navy, kidney) 44 - 62 736 - 990 23 to 37 0.29-0.47 0.26-0.42

Oats, Rice. Rye

Gypsum in Lumps, Limeston 100 1,600 40 0.50 0.30

Iron ore 165 2,640 40 0.50 0.36

Lime, Burned (pebbles) 50 - 60 800 - 960 35 to 55 0.50 - 0.60 0.30

Lime, Burned, Fine 57 910 35 0.50 0.30

Lime, Burned, Coarse 75 12,00 35 0.50 0.30

Lime, Powder 44 700 35 0.50 0.30

Manganese ore 125 2,000 40 - -

Sand 100 - 125 16,00 - 2,000 25 to 40 0.40 - 0.70 0.35 - 0.50

Soy Beans, Peas 50-60 800 - 960 23 0.25 0.20

Sugar Granular 63 1.000 35 0.43 -

in which D is the inside diameter.

(2-4)

(2-3)

(2-2)

(2-1)

13

or simply k = tan' (45° - pi 2 )

V=R(yY -q)

k=(I-sinp)(I+sinp)

q = Y R [I_e-P'kYIR]j./ k

The silo wall designer needs to know the total vertical force applied to thewall by friction from the stored material. This force, from materials above anydepth Y, is equal to the weight of those materials minus the upward force fromvertical pressure q. The friction force, per unit length of wall, from above is

The above derivation makes no assumption as to shape of the silo cross section. Ifthe cross section is circular, then the hydraulic radius is

JrD'/4R = area 1perimeter = --- = D 14

JrD

The wall friction force is p'p per unit area of wall at depth Y. Verticalfriction forces cause vertical force in the wall: compression if the wall is supportedfrom below, tension if suspended from above. Integrating from the top of thestored material to depth Y, the vertical force in the wall (per unit of wall perimeter)

at depth Y is given by

V = p'Jpdy = y R[Y-~(I- e-"'kYIR]Y p'k

which is the Rankine coefficient for active earth pressure - the ratio of horizontalpressure to vertical, Hence, to compute the horizontal pressure p, Eq. (2-1) ismultiplied by k. Thus, the Janssen equation for horizontal pressure is

Koenen improved Janssen's method by introducing the term

(2-6)

(2-5)

C= L h47<j.1.'k 3

yRPmax =

j.1.'

C=~-"--4j.1.'k 3

yDPmM = 4j.1.'

The Reirnbert equations for static pressure are as follows:

C=_G__ h7<j.1.'k 3

For rectangular silos, on the short wall of width G:

14

Vertical pressure at depth Y below stored material surface is

For polygonal silos of more than four sides:

For circular silos the terms Pmax and C (characteristic abscissa) in the aboveequations are:

and lateral static pressure at depth Y is

In 1953 and 1954, Marcel and Andre Reimbert [34] presented their methodfor computing static pressure due to stored material. Their derivation recognisesthat at large depths Y, the curve of lateral pressure becomes asymptotic to thevertical axis.

2.4.2 The Reimbert Method

15

(2-7)

q = P (2-8)k

Bunkers: Fig. 2-3a shows dimensions used in Airy's equation for bunkers.Using symbols of this thesis, Airy's equations for bunkers are as follows

Where a' = (2ab - a')/ b

a' hC=----Jrj.lk 3

2.4.3 The Airy Method

Developed in 1897, the method of Wilfred Airy [6] presents separatesolutions for bunkers and silos. Airy's equations were derived considering staticequilibrium of wedge shaped portions of stored material above the plane of rupture.

For the longer wall of width b:

In the equations above, R and k are defined as for the Janssen method. Frictionforce is determined in the same manner as for Janssen's method, using Eq.(2-4)

Lateral pressure at depth Y is

Where fJ = tan p

Silos: Fig. 2-3b shows similar dimensions for silos. Airy's derivation for asilo leads to:

and vertical pressure at depth Y is

h

D

D-.

E

/

D

Plane ofrupture l,,' E

(b) Silo

D

(a) Bunker

Surface of grain

Surface of grain

fJ'p

A"I.

c

c

Fig. 2-3 Dimensions for use in Airy's equations.

16

(2-9)

(2-10)

(2-11)

(2-12)

(2-13)

qa =2.4yR(sin'a)/.J17

1+ fl'2Y( ).-. fl + fl' + 1- flfl'D

yDp= 1-

fl + fl'

qa = psin2a+qcos2a

Friction per unit area V =~a 2

Normal pressure

Lateral pressure at depth Y is

and vertical pressure at depth Y is

2.4.4 Pressure Normal to Inclined Surfaces

This equation can be derived from equilibrium of a triangular element of materialas shown by Fig.2-4.

Pressure on an inclined swface, such as the sloping wall of a hopper, isusually computed as:

The German Silo Code: German Code considers the sloping wall, wherea > 20

0

, to be. subject to both normal pressure, qa' and friction force per unitarea, Va . If angle ex between the hopper wall and the horizontal is greater than 200,the code suggests a simple method in which the effects of material in the hopperand. of material above the hopper are considered separately, and then both areadded together.

From-material in the hopper (Fig. 2-5a):

This loading is applicable also to hopper bunkers.

From material above the hopper (Fig. 2-5b): Pressures pf and qf at the topof the hopper are computed first for the filling condition. Then normal pressures arecomputed for the upper end and lower end of the sloping wall, as follows:

cosa

Fig. 2-4 Pressure on inclined surface ( ACI Code)

q cosa2

~qCOSa

I

Fig. 2-5 Hopper pressure diagrams ( German Silo Code)

o

n

D

q

D

( a ) Material in the hopper

( b) Material above the hopper

Upper

17

Since loads and stresses are related to flow pattern, the structural engineer isrequired to consider the effect of flow pattern in designing silos or bunkers.

(2-14)

(2-15)

(2-16)

(, , )(I+Sin2aJUpperend, qa = q,cos-a+Pjsiwa 4,u'

Lower end, qo = qf cos' a

Va = 0.5 xaverage qa

Friction force per unit area of inclined wall is

2.5 FLOW PATTERNS

In mass-flow, the hopper is sufficiently steep and smooth to cause flow ofall the solids, without stagnant regions, whenever any solid is withdrawn. Mass-flow silos are usually recommended for cohesive materials (coal, for example),materials that degrade with time, powders (unless means of withdrawal such asaeration are used), and materials in which segregation needs to be minimized.

Flow of stored material from silos is of two main patterns, funnel flow (coreflow) and mass flow. In mass flow, all of the stored material is in motion duringdischarge. In funnel flow, movement occurs only in a channel within the storedmaterial, and this channel is surrounded by nonflowing material. The two types are.illustrated by Fig. 2-6.

Funnel flow occurs when the hopper is not sufficiently steep and smooth toforce material to slide along the walls, or when the outlet of a mass-flow bin is notfully effective. In a funnel-flow silo, solid flows toward the outlet through achannel that forms within stagnant material. Usually, funnel-flow bins are suitableonly for coarse, free-flowing or slightly cohesive, nondegrading solids in whichsegregation is unimportant.

Fig. 2-7 shows charts by Jenike [19] that may be used to predict (for twoshapes of hopper) whether mass flow or funnel flow will occur. The regionsmarked "uncertain" indicate conditions under which flow type may changeabruptly. These conditions should preferably be avoided, since they may lead tononsymmetric flow patterns and frequent vibration and shock loads, which canseriously affect the silo.

Effectivetransition

.<=oo

~ E'" ~0.0.-0o CI 0

0.

'"'"~

(a) Mass flaw

(bl Funnel flow(or core flow)

Fig. 2-6 Mass flow and funnel flow

Stagnant--solid

40.

Funnel- flow30•

. jJ.' 20°

Moss flow10.

O. 10. 20. 30. 40. 50.Cone ec

(Q)

50.

Funnel- flow40.

jl. 30.

1 20.

IMass - flowJ\- 7ep 10.

ec0° 10. 20. 30. 40. 50. 60.

Slotted opening ep(b)

Fig. 2-7 Mass-flow / funnel-flow bounds (after Jenike-Johanson)

18

2.6 TOTAL PRESSURES - STATIC PLUS OVERPRESSURE

(2-17)v = (rY - 0.8q)R

There are two general approaches to determine total pressures. One is tomodify the computed static pressure using "overpressure factors"; the second is tocompute total pressures directly. At the present stage of development, neitherapproach is completely satisfactory.

The total pressures may be called "operation", or "flow" pressures. Totalpressures, both lateral and vertical, can exceed computed static pressures by a widemargin. In earlier silo designs overpressures were not considered, even though asearly as in 1950s, it was fairly well recognised that overpressures occur duringemptying. The result of ignoring overpressure is to reduce the overall factor ofsafety. A marginal structure is produced, with increased probability of bulgingwalls, damaging cracks, or even collapse.

Overpressures are due to various causes, including arching of the storedmaterial; collapse of material arches; sudden change of flow channels, velocities,and directions; and changes between fimnel flow and mass flow.

In 1977, the American Concrete Institute published its standard _"Recommended Practice for Design and Construction of Concrete Bins, Silos andBunkers for Storing Granular Materials (AC! 313-77) and Commentary". Arevised edition of this standard appeared in 1983.

Table 2-2 and Table 2-3 show the minimum values recommended by ACI313 for overpressure and impact factors.

Material design pressure against walls and bottoms are determined asfollows [2]:

I. Static pressures are obtained using either Janssen's or Reimbert's method.When calculating vertical frictional forces on silo walls, ACI 313 uses a slightlymodified form of Janssen's equation for frictional force.

By either Janssen's or Reimbert's method, static unit pressure normal toa surfaceinclined at angle a to the horizontal and at depth Y below the surface of storedmaterial is computed as

Table 2-2 Values of overpressure factor, Cd

.Cd values given In rhis table are bwdeqllare for the higher loads associated Witll mass flow.NOTES:J. Cd factor for lateral pressures is givell for the bottom of each height ZOlle sllow1l.2. III the regioll of a flow-correcting Illsert (e.g., Buhler.Nase) lateral pressures may be many times greater t1IQIIstatic, and

the Cd values abol'e are 1l0t suflicient.3. Silo bottom pressures /Iced f10t be considered larger rhall tile pressure caused by J 00% of weight of silo contents.4. If H] < H "21-f l' use the seco/ld Cd value from the top for the ellflre silo height H.5. Vollies of factor Cd for HID between those gil'e/l I" Table 2-2s1,01lId be determined by linear interpolation.6, Values of Cd [actor given IfI Table 2.2[or calculating design bottom pressures sllall be IIIultiplied by 0_ 75 for noncolJeslve

material except for homogenizing silos in whiell pllellmatic withdrawal is used.7. The Cd [actors showlI In Table 2.2are minimum recommended vahles. However, lower Cd factors may be used, but only

for particular cases for which the designer call demonstrate rh~t Slid, lower Cd factors are satisfactory_

OVERPRESSURE FACTOR Cd

FOR POWDERY-COHESIVEN '" •. LIKE CEMENT OR FLOUR I

VI II AI WHEN EMPTYING IS- - - DONE PNEUMATICALLYIlo Ilo Ilo •. "'5 5 5 " AI~ - - rio xlo"Io Ilo Ilo 2 3TOP OF MATERIAL AND SILO --. Ilo Ilo

\z I- Z

l- I- I- Z I-~~

'" '" z '" z '" '"wi!! w w w w w w w w~2~ '" '" <D '" <D '" <D '" <D'" " '" " '" " '" ;!i '" "

• O~ ZW z W z W z z

WE

"" "" "" ~ II! """ " , '" , '" , '" , '"° xI." "'''' "' 0 "'o£ '" !:? 0 '" ill 0 '"

00

'" ..OJ '" N ~ ~ '"~ ~ ~ - - ~ - ~..

~

"-

'"-0 '" 0 ° '" 0 0 "' 0I .. "! '" '" ~ '" " .. " '"

, - ~ ~ - ~ - - ~ -I-.. \"-

'" '" 0-

'" '" '" 0 ° 0 "'I of '" .. '" '" " '" <D " '" "~ ~ ~ ~ ~ - - - ~ ~I-

.. \"-

'" '" '" 0- '" '" '" g 0 0I '" '" " " <D "l '" 0 0LATERAL DESIGN ,./ ~ - ~ - - ~ ~ oj ojI PRESS. CURVE -

LATERAL STATIC..PRESSURE CURVE"-- BY JANSSEN'S OR

'" '" '" '" '" '" ~0 0I REJMBERT'S ,~ '" '" " " ., <D 0 0 0'"\ ~ - ~ ~ - ~ ~ oJ ojE THEORY ___

" USE SAME PRESSURE WITHIN HOPPER HEIGHT OR, IF"'~ 0 I DESIRED, REDuCE PRESSURES IN ACCORDANCE WITH.. lr ...J.J

I I HYDRAULI C RADIUS CHANGE.I a.. ..J ilimr-o...Ji;: <l:Q.:x:::::! ~...J / I~ ~ \LlJ...ZVl~ 0-

oc°C) lr:fBOTTOM OF HOPPER I I IF DESIRED, PRESSURES MAY BE REDUCED fROM

WI~IOOQ..t-71-a..t-FLAT SLAB OR FILl IT : TOP OF FILL TO TOP OF FLAT SLAB AS SHOWNg,~~~~~::tou-ott')

OVERPRESSURE FACTOR Cd FOR CONCRETE '" 0 '" 0 '" 0 '" 0 '" 0'" '" '" '" ~ ~ '" "' '" '"USE IN CALCULATING DESIGN

BOTTOM -' ~ ~ ~ - - ~ ~ ~BOTTOM PRESSURES IN SILOSSTEEL 0(SEE NOTE 6 ) "' 0 '" 0 '" 0 '" 0 ~~ " ~ " '" " '" ~ '" ~BOTTOM - ~ ~ - - ~ -

19

Pressure increase due to eccentric discharge must also be considered.

(2-18)

(2-19)

1.0

1.25

1:6 and less1:5

I.I

1.35

1:4

1.2

1.5

1.3

1:3

Ratio of volrnne drnnped in one load to total Silo capacity

1:2

• 2 ,qa = psm a +qcos. a

Factor Steel bottom 1.75 1.60

Impact Concrete bottom 1.4

Table 2-3. Recommended Minimum Values ofImpact Factor Ci

2. Design pressures Pd~' qd~. qa,d~ are obtained by simply multiplyingthe static pressure by the appropriate overpressure correction factor Cd (or impactfactor C;, which applies to bunkers only). The overpressure factor does not apply tofrictional force; thus:

Pd~ =Cdpqd~ = Cdqv,,~= V

2.7 EFFECT OF ECCENTRIC DISCHARGE ANDNON-SYMMETRICAL FLOW

Experimental studies show that withdrawal of granular and powderymaterials through eccentric openings causes lateral pressure changes much differentfrom the pressure change occurring during withdrawal through a concentricopenmg.

Early tests showed that the pressure increase due to eccentric discharge,compared with that of concentric discharge, occurs on the side opposite to thedischarge opening eccentricity, and that the pressure decreases on the side nearer tothe opening.

More recent tests were made by Pieper [30, 31] and his associates onlaboratory scale model silos and bunkers, using different arrangements of dischargeopenings. These tests revealed that pressures due to eccentric discharge are erratic,that they may increase or even decrease against any side at different levels and atdifferent storage depths above the opening. This irregularity causes horizontal andvertical bending moments in silo and bunker walls.

All these nonsymmetrical flow conditions cause non-uniform pressures on

the silo walls. The nonuniform pressures cause horizontal and vertical bending

moments, which should be considered in design of the concrete silos. In circular

steel silos, however, because of the flexible character of the thin circular walls,

horizontal bending moment is usually not considered, the walls merely change in

shape so that horizontal bending moment practically does not occur. Nevertheless,

if there is possibility of a pressure decrease, which could cause denting of the wall

towards the inside, such a condition should be checked and prevented. There are

several methods for considering the effect of eccentric discharge. Some silo codes,

standards, and other methods for computing material pressures consider, in varying

degrees, the effect of eccentric withdrawal.

The following discussion introduces several approximate methods for

computing additional lateral pressures P <cc due to eccentric discharge. Such

computed values are usually added to calculated lateral pressures computed for the

concentric discharge condition to obtain design pressures P d" .

2.7.1 ACI 313-77 Approach

This approach is given by the Commentary to ACI 313 [2] for lateralpressure increase:

Assume the increase of design lateral pressure to be at least 25 percent of

the static pressure at the bottom of the silo when an opening is next to the silo wall.

If the eccentricity (e) of the opening from the center of the silo is less than radius

(r), consider the increase to be at least 25% of elr. Assume this design pressure

increase to be constant from the top of the hopper (or hopper forming fill ) to a

height equal to D (or a or b) and to reduce linearly from that point to the top of the

silo. The increase need not be multiplied by Cd. Expressed as an equation, the

design pressure at any depth Y below the surface of the stored material would be

Since the pressure variation around the wall (perimeter) is unknown, it is

common to assume the increased pressure all around.

Recent experiments show that the lateral pressures computed by this method

are inadequate, and therefore it is suggested to increase the computed values of

Pe<e by 50%.

20

21

(2-20)

Eccentric discharge may be considered by adding a correction, P,ee' to thelateral design pressure, P des' computed at depth H by either the Janssen orReimbert formula; P ,ee is assumed to be constant from the top of the hopper (orhopper forming fill) to a height equal to D (or a or b) and to reduce linearly fromthat point to the top of the silo. Within height H - D, the lateral design pressure atdepth Y is then:

2.7.2 Safarian's Method [37, 13]

The correction P,ee at depth His:

where PH = static pressure at depth H.

Pressure P; is the lateral static pressure at depth H in an imaginary silo, asshown by Fig. 2-8 and Fig. 2-9. For rectangular silos, the imaginary silo isdetermined as shown by Fig. 2-8. When the opening is displaced toward side a,correction P,ce for sides a is computed using an imaginary silo measuring(a+2ea)x b. (If ea is larger than a, the imaginary silo should measure 3ax b)Similarly, if the opening is eccentric toward side b, the imaginary silo will measure(b+2eb)X a. If both eccentricities occur, each correction is computed separately,using the first described imaginary silo to determine P 'ce for sides a and thesecond for sides b.

The imaginary circular silo of Fig. 2-9 is centered on the dischargeopening and has a radius equal to that of the actual silo plus the eccentricity.

Where multiple discharge openings occur, even though the group IS

centrally located, eccentric discharge is always possible and should be considered.

2.7.3 A More Nearly Rational Procedure

Many design engineers believe that the methods given above for .determining additional lateral pressures due to eccentric withdrawal do not offer asatisfactory solution to nonsymmetrical withdrawal conditions, and that these arenot reliable and do not account for the actual flow pattern.

[(Pecc)~Q = ~Pi)~a-(PH)a

(decreose lConsider only when checkingsleel woll 'for denting

'--~Pecc 'HJa = ~Pi)~a -(PH'a(increose)

For both II all walls

--,__-1

a

Silo

------ ------,

VIShe~r force~ vi Outline of imaginary(Friction) I ' 'I "" ( ..I 510 a for eccentricity

, I Joward shorl wall)

j II

, Silo I~t~iI ----I 0' h ._ _ __ _ _ _ .J - ISC arge opemng

IIIIII

r--L.. __

Fig. 2-8 Pressure change due to eccenuicity of discharge opening inrectangular silo

N"-

.D

N"-.D

.D

.----- Use for whole circumference

Probable variationof pressure changesdue tue eccentric discharge

(Peee lH = (Pi lH- PH( increase)

(PeeelH =(PilH- PH(decrease 1

Consider only when checkingsleel silo wall for denting

r-- Outline of imaginary silo

I

//

Fig. 2-9 Pressure change due to eccentric discharge opening incircular silo(after Safarian)

\\

"- "- ./'.••.•...• .---

and the maximum static pressure p, in the silo (away from the flow channel) is

If force F is assumed acting as a concentrated load, then the approximate maximum

positive and negative horizontal bending moments may be expressed as follows:

(2-21)

(2-22)

(2-23)

(2-24)

(2-25)

(2-26)

22

P, =- 2y D, /4p'

M m~ = - 0.125FDM min = + 0.090FD

P, =-yD/ 4p'

Where r = turit weight of stored material

D I = diameter of flow channel

D = silo diameter

p' = coefficient of wall friction

F = MD, = (yD' / 4p')(2D,' / D' - D, / D)

!1p = p, - P, = (y D / 4p')(2D, / D -I)

Then the pressure differential,

and the eccentric wall loading is expressed as

Colijn and Peschl [11, 27, 28, 29] have suggested an approach for

computing overpressures in silos due to nonsymmetrical flow or eccentric

discharge of material. Their approach assumes that the maximum horizontal

discharge pressure (PI) in the circular flow channel (Fig. 2-10) is twice that ofstatic pressure in the same channel:

From the above equation,

F is maximum when D, = 0.25,D,

Dand F = 0 when -' = 0.5.

D

yD'Hence, Fm~ = - 0.125--

4p'

diD

F

I.---t-...../' I "'

/' I. "11\11'

0.25 05

Moment

F.D/2

o

D

0.16

+

d

DeformationF

Flow channel

Fig. 2-10 Forces and moments on silo wall (after Colijn and PescW)

The horizontal reinforcement needed to resist these bending momentsshould be added to the reinforcement computed for pure tension and temperatureeffects.

2.8 LOADS IN SILOS

The principal loads for silo and bunker design come from action of thestored material. Other loads includes dead loads, equipment loads, wind, floor androof live loads, seismic loads, forces from thermal effects and forces applied byrestraint of attached items.

2.8.1 Dead Loads

Dead loads include the weight of the walls, roof, ring-beams, hopper, plusthe weight of items supported by the silo. The suppported items include inside andoutside stairways and service platforms, equipment on the silo roof (such as dustcollectors and conveyors), buildings supported by the roof, overhead gallery, etc.

2.8.2 Live Loads

Pressures due to stored materials are considered in Art. 2.4. Under strengthdesign methods, these pressures are treated as live load. Live loads on platforms,roof, and floors should be as required by applicable building codes, or larger. Insome cases a buildup of dust (cement dust, for example) may cause significant liveload, perhaps much more than the code-specified floor or roof load.

2.8.3 Wind Loads

All silo and bunker structures should be designed to resist the overturningeffects caused by wind or earthquake forces. Wind and earthquake loads should notbe assumed to act simultaneously. Whichever of these loads is more serious shouldbe used in design.

As for buildings, wind loads for silos or bunkers can be in any lateraldirection, and generally should be considered as positive (inward) pressures on thewind-ward side, acting simultaneously with suction (negative pressure) on the lee-ward side. Wind pressure distribution should preferably take into account adjacentstructures.

23

Circumferential bending due to wind on the empty silo or bunker should beconsidered. Wind may affect the stability of empty silos, and of all narrow silos orsilo groups, particularly those made of steel, wood, or fiberglass. Foundationpressures and colmnn stresses, however, may be worse with wind acting on the fullsilo.

In conventional method the silo is. considered as vertical cantilever beamwith complete fixity at the level of ring beam for wind load analysis (Fig. 2-11).The wind pressure is assumed 36.9 psf (equivalent to a wind velocity of 120 mph)on a surface normal to the directions of wind all over the depth and a reductionfactor of 0.6 is used to take into account the effect of circular geometry.

For Finite Element analysis it is assumed that the pressure distributionaround the circumference depends on the Renolds number. The distributionassumed here is taken from Reference [3] by Ahmad and is shown in Fig. 2-12.

It has been found that about seven Fourier harmonics represent the abovedistribution quite accurately. The Fourier coefficients used are show in Table 2- 4.In case of a different distribution around the circumference, the Fourier co-efficients will have to be recalculated.

Table 2-4. Fourier Coefficients for the Pressure Dstribution of Fig. 2.11

Harmonics Coefficients

0 0.24706

1 0.31387

2 0.58763

3 0.42213

4 0.02466

5 -0.11481

6 -0.00451

2.8.4 Seismic Loading

Earthquake loads may affect both stability and strength of silos and bunkers.Walls and colmnns supporting silos and bunkers may be particularly vulnerable to.earthquake forces. The foundation, especially if supported on piles or on caissons,may also be affected.

24

H

ASilo

Hopper

( b) Section A - A

( a) Vertical distribution

A

Fig. 2-11 Wind pressure distribution in conventional method.

Fig. 2-12 Variation of Wind Pressure along the circumferentialdirection on a horizontal plane

18015060 90 120

Angular Distance a (Degree)30

r

~

'" ,

\

\\

- \ /"r \ /

"- /

2,00

1,75

~ 1.50CD~ 1,25Q.

~ 1.00::;]lJl18 0.75a:"C 0,50c:~ 0,25

'5-' °c:Q)'13 -0,25;e~ -0,50o() -0,75

-1.00

°

In computing stability against overturning due to earthquake, the entireweight of the stored material may be used. However, if the silo has anindependently supported bottom, the material weight must be divided between silowalls (fnction) and the bottom structure.

2.8.5 Thermal Effects

Two types of thermal effect may need to be considered. The first is athrough-the-wall temperature gradient, important in concrete walls, caused bystoring material that is much hotter than the air temperature around the silo.

The second is the daily temperature changes due to intense sunlight whichmay cause expansion and contraction of silo groups. Stresses due to this action canbe large enough to cause wall concrete to crack. Seasonal temperature change canalso have a similar effect.

2.8.6 Other Loads

(i) Loads from External Restraint: A silo is a flexible membrane. The wallof an isolated circular silo under uniform internal pressure around its circumferenceexpands radially. Such a wall has a high horizontal membrane tensile stress, but nohorizontal bending moment. Vertically it will have compression and a small(usually not computed) vertical bending moment. .

However, if at any point the silo wall is attached to something that restrainsits radial movement, the wall is "dented", and significant horizontal and verticalbending moments occur. These bending moments, when their effect is added to thehoop tension and vertical compression, could cause wall failure.

Anything that, in effect, causes a "hard spot" in the wall can generate thisproblem. It could be a platform connected to two separated silos or a structuralmember or rigid duct connected to each.

(ii) Equipment Loads: In addition to dead load, equipment items may applysevere live load on the silo Structure. Theoretically, equipment manUfacturersshould be able to predict the load their equipment will impose; but if thisequipment is vibrating, it may bring about changes in other loading. For example,the stored material may become compacted, acquiring higher density and alteredflow characteristics with resultant changes in lateral and vertical pressures.

25

26

2.9 SILO ROOFS

Table 2-5. Suggested Load Combinations for Silo or Bunker Design.

Certainly all reasonable load combinations should be considered by thedesigner. Combinations that are extremely improbable are, of course, usuallyneglected (combinations including both wind and earthquake, for example).

Belt conveyors and their structural supports can bring large live and deadloads to a silo structure. Often, the end of a conveyor bridge will be supported on asilo roof. This bridge will transfer lateral wind load to the silo, as well as verticaldead and live loads.

2.8.7 Load Combinations

What load combinations should be considered is a matter of experience andjudgement. Table 2-5 shows a matrix ofload types and suggested combinations forsilo or bunker design. Combinations not shown would ordinarily be ignored. Thisshould not be taken as a fast rule, however, the designer must be alert to specialcircumstances that warrant considering other combinations than those shown.

::!:: = "consider or neglect, whichever is more severe"• = for eccentric flow, flow-improving device, etc.•• = consider also material piled outsideX = considerXX = consider, but reduce if required by code.

Designers are divided on the subject of attachment of concrete roof slabs tosilo walls. Some believe that the slab should be supported only vertically at thewalls (on elastomeric material or heavy tar paper) so as to be free to contract or

Types of Load CombinationsA B C D . E

Dead Load X X XX XX XXFloor and Roof Live Load X :t XMaterial Pressure •• Static X X X

Over Pressure X X X• Modification X X X

Thermal Loads X X XEquipment Loads X X XWind X XEar1hquake

X XRestraint Loads :t :t :t

(a)

This portion af roof is.doweled into walls

\'Shear pin

( b)

(c)

Elevalor tower ~

Fig. 2-13 Roof expansion-controlling devices.

/This portion of roof is

-8@00@QWOI'SSh.arpin . 00000 Shearpin

Fig. 2-14 Typical silo roof-to-wall details

(b)

(d)

• •

[(;;::.Ir .=. =.=. r. ---j

~~---""'- Dowels

Inside verticalreinforcingextends intoroof slab

(e)

•

••

.

Dowel

.

. .

(e)

(a)

•r.\

Drip groove/

expand with temperature changes and to move slightly during earthquake. On theother hand, attaching the roof slab to the walls stiffens tall silos against wind andearthquake loads and reduces lateral deformations (Fig. 2-13 and Fig. 2-14).

Roof systems should be designed for dead and live loads, including allexpected equipment loads. Live load deflections should preferably not exceed1/360 of the span.

Roof structures usually consist of a reinforced concrete slab on steel beams.For very small units a slab without beams may be sufficient. Cast-in-place concretebeams are of limited use. The most commonly used system - steel beams withconcrete slab - is particularly well adapted to slipforming because the steel beamscan be incorporated in the slipform work deck.

Ample bearing on the concrete wall should be provided at the ends of thesteel beams, and the concrete below and to each side of the beam should bereinforced to prevent undue cracking or even a concrete fallout after some years ofsefVlce

The roof slab should be at least 4 inch (100 mm) thick. Either deformedreinforcing bars or welded wire mesh reinforcing may be used. When the slab iskeyed or doweled to the wall it is subjected to combined bending and tension. If thetension is significant, it must be considered in design.

Extra reinforcing bars should be added around roof slab openings. Foropenings larger than 2 ft. square, diagonal corner bars should be provided.

Heavy loads acting on the roof between supporting beams and wallpreferably should be transferred to the supports by providing additional beams.Light and moderate concentrated loads, however, may be supported by the roofslab merely by adding extra reinforcing bars beneath the loads.

***

27

CHAPTER 3

ANALYSIS OF SILOS

3.1 CONVENTIONAL METHOD OF ANALYSIS AND DESIGN

3.1.1 General

A munber of authors have attempted to present design methods for silos intheir papers and textbooks. But the methods followed for silo design have beenmore or less the same and these are based on several assumptions and idealisationof silos.

3.1.2 Design Steps - Conventionally Reinforced Concrete Silo

The whole design process can be divided into the following steps:i) Determination of silo geometry. The shape and size must be chosen to

provide the desired capacity, also considering how the silo is to be filled,how material will be withdrawn, and required elevation of roof and silobottom to align properly with other plant structures or process systems.

ii) Determination of the necessaI)' properties of the material to be stored,including the temperature it will have when placed in the silo. The design.engineer must determine the values of y, p and f.J. ~

iii) Determination of what, if any, other structures or equipment will beattached to the silo walls or roof and to determine how much force each ofthese items will apply.

(iv). Selection of the method of computation of internal pressure andoverpressure.

v) .Computation of static and design values of lateral pressure on the walls atvarious depths below the surface of the stored material and to adjust thedesign pressure for eccentric discharge

vi) Design of the silo walls.

vii) Design of the bottom. This includes bottom slabs, bottom supports, steel orconcrete hoppers, and ring beam, depending on the type of bottom systemselected.

viii) Design of the silo roof and roof supports.ix) Design of the foundation.

3.1.3 Analysis and Design of Circular Silos

Whether isolated or connected in groups, circular silos are usually fIrst ()designed as single silos. Interaction among silos of a silo group is then consideredand necessary modifIcations are made where necessary.

In the conventional analysis and design of circular silos certain assumptionsare necessary. Some of these are listed below:

i) The radial pressure from the stored material is uniform around thecircumference at a particular elevation.

ii) Silo is a thin-walled cylinder stressed in circumferential tension only due tolateral pressure and there is no bending moment or shear.

.iii) If the ring beam is monolithic with the vertical wall or conical hopper thereis no effect of the restraint provided by the ring beam either on the verticalwall or on the conical Hopper. ~••..

iv) The vertical wall can expand freely at the bottom of pressure zone.v) The conical Hopper can expand freely at its junctions with the bottom of

ring beam.

The analysis and design of circular silo can be performed following thesteps below:

(aJ Selection of minimum wall thickness

Minimum wall thickness depends on material strength, permissible crackwidth and details of reinforcements. According to the assumption of the horizontaltension without bending moment may create regularly spaced vertical cracks, andthe entire resistance would be provided by the hoop steel. Except for its influenceon the width of those cracks, the thickness of concrete would be unimportant,

29

m = concrete shrinkage co-efficient and assumed to be equal to0.0003

Regardless of the thickness suggested by the above equation, thicknesses less than6 inches should not be used.

(3-1)

(3-3)

(3-4)

(3-2)

( WSD method)

( USD method)

[mE +f -n+ ]h. = ' s !Ic."" D / 2

mm ff Pd"S c,len .

[mE +f -n+ ]h . = s s !I e.te" D / 2

mm 100Islcoten Pdes

F=PdcsD/2

F" = k,PdcsD / 2

30

where is = allowable (WSD) stress, generally 40 to 45% of hie t<" = the allowable stress for concrete in tension (PCA suggests 0.1 fc)n = modular ratio, Es

Ee

But, actually, the lateral pressure distribution is not unifonn and the ring

beam (if exists) provides some restraint due to which there may be some momentin

the vertical wall. This moment cannot be computed by conventional method. To

resist the accidental bending moment, a minimum wall thickness is to be selected.

Conventionally walls having two layers of reinforcement should be greater than 8inches in thickness.

According to portland cement Association [9] a lower limit of wallthickness is given by

provided it is sufficient to (i) protect the steel (ii) ensure proper bond strength at lapsplices and (iii) resist the vertical force.

The metric equivalent of the above equation is

(b) Determination of required horizontal (hoop) reinforcing

The horizontal tensile force per unit height of cylindrical wall is

where k, is live load factor (ACl. 313-17 recommends 1.7).

The hoop steel required per unit height is:

Walls that will notfail by buckling: The Commentary to ACI 313-77 offerssome suggestions for considering bucking of circular walls, as follows: a1. For circular walls with uniform radial pressure that restrains buckling:

(a) For walls that are continuous around the entire circumference (noopenings) use the permissible axial load strength of Eq.3-6; that is,buckling does not need to be considered.

. . •.

(3-5)

(3-6)

(WSD)(USD)

A, = F! f,A, = F;, !((bfy)

where (b is the strength reduction factor (0.9 for tension as per ACI 318).

3\

Fw = 1.7 x (Vertical friction force + roof and other vertical live load)+ 1.4 x (wall self weight above + roof and other Dead load)

All terms in parenthesis are per unit length of wall.

(c) Determination of vertical load in the wall

After getting the minimum waIl thickness in 'step-a' the thickness should bechecked for adequacy insupporting the vertical load. Vertical force per unit lengthof wall is the largest at the bottom of the silo, but vertical stress may be higher inthe thinner wall at the elevation where waIl thickt)ess changes.

Sources of vertical force (meridional force) include: (i) friction from thestored material above the height in question; (ii) roof live load and dead loadincluding equipment and structures mounted on the silo; (3) weight of the waIlitself above the height in question; (4) for lower walls load imposed by the silobottom system; and (5) overturning moment due to wind or seismic action.

Total factored (ultimate) meridional force Fw per unit width of the waIl isgiven by (not considering wind)

This computed ultimate vertical load should not be greater than

(3-7)

(3-8)

(3-9)

(3-10)

(3-11 )

(3-12)

(3-13)

2. For circular walls not subject to uniform radial pressure (including walls belowthe pressure zone):

(a) For walls continuous throughout their entire circumference (noopenings) the limiting stress is

(b) Adjacent to openings that have no stiffening members at their sides,the limiting stress is

(b) For walls with openings but no stiffening member at the openingedge;

in which ho is the height of the opening.

(d) Checking the adequacy of selectedwallfor lateralload~

Lateral load due to earthquake or wind (whichever is worse) should now be

considered in combination with the vertical loads. Different load factors are used,

of course. According to ACI 318 [1], the factored combinations are

0.75( lAD + 1.7L + l.lE)

With wind:

32

0.9D + 1.43E

0.9D + l.3W

0.75 (lAD + l.7L + 1.7W)or

whichever is more severe.

With earthquake:

or

whichever is more severe.

(3-14)

(3-15)

(3-16)

(3-17)

(3-18)

81' = [(1') - l' JKI des 0 t

In Eq.3-10 and Eq.3-12, live load L should have its full value or zero, whicheverleads to larger required strength per unit length.

In the case of hot cement, for example, 8 in. of cement adjacent to the insideface of the silo is commonly assumed to act as an insulating material withtemperature varying linearly across the strip. <l--. .

It has been observed that the temperature of hot granular materials in silos isnot uniform but drops appreciably near the wall.

(in-Ib per ft when E~is psi and h is in.)

or

33

(e) Determination of steel neededjor thermal stresses

In building design, certain levels of temperature difference. are commonlyignored. Similarly, for walls of silos having hot stored materials, a difference of upto 80

0F (44.S°C) is often ignored. When this is done, the design temperature of the

hot material whose actual internal temperature is T, becomes

Fig. 3-lb shows the temperature variation through the 8 in. (20 cm) ofcement and the silo wall. The temperature difference 81' between inside andoutside face of the wall is

Temperature drop, 81', within the wall is a portion of the total designtemperature difference, corresponding to the ratio Kt of the thermal resistance ofthe wall alone to that of cement, wall and outside air combined. Values of 81' maybe obtained from

where K, is determined by heat transfer analysis. For cement, it is shown in Fig.3-la. Assuming shape and/or restraint to prevent warping (curling), the thermalultimate bending moment per unit of wall height is

Fig. 3-1 Computation of liT for wall of cement storage silo

2422

Outside face

of silo wall

R3To

20

1.8 inch.:.h .1Ti,des

T,

(b) Temperature drop across silo wallliT = [,T;,d.' - To] xKt

T2

8 10 12 14 16 18

Wall thickness, h ( inch)

(a) Determination of K t for use in computing liT for wallof cement storage silo ..

6

- I

- //VI/V

I.

/

V ./

./

/V

0.05

T; = Temperature of stored cement ( of)

Tj,des= (T - 80 of) Design temperatureof stored cement

To = Design winter dry-bulb temperature (oF)h ~ Silo walilhickness

Above curve is based on the following assumptions

1. Rres;stence of 8 inch ( 203 mm ) cement ( considered to

act as insulating material) = 3.92

2. Resistance of 1 inch (25 mm) thick concrete ~ 0.083. Resistance of outer surface film = 0.17

0.1

03

0.35

••••••••• O. 15II

•••••••••0.25.r=g:J

.r= 0g:J + 0.2o ~

<i

. 34

(g) Checking of crack width

(3-21)

(3-20)

. (3-19)

MA = x'.U for horizontal reinforcement'.X /y(d - d')

MA = yt.u for vertical reinforcement'.Y /y(d - d')

M t = M, = 1.25£ h2a,t:>.TKgX,u y,u c

Crack width due to pure tension in a circular bin wall is estimated by the

method shown by Lipnitski and Abramovitsch [26], which considers the effects of

both short-term and long-term horizontal (hoop) tension .

Although very fine, well distributed and harmless shrinkage cracks are

always present in conventionally reinforced concrete, significant silo wall cracks

due to silo loading can be prevented or minimised in width so as not to be harmful.

Cracks that are too wide may: (1) admit moisture, damaging the reinforcing steel or

the stored material; (2) allow fine stored material to escape; or (3) be objectionable

in appearance, Cracks also reduce the capability of the wall to resist horizontal

bending moment without appreciable distortion of the wall.

. The designer must decide what crack width is acceptable. A limit of 0.008 .

ill. 1S often used for silos exposed to the weather, storing moisture-sensitive

materials such as grain, cement, fly ash, etc.

(j) Selection of tentative hoo~bar sizes and spacings

The pressure varies continuously with depth and so does the required area

of hoop reinforcing. A practical solution, however, would vary the bar spacings (or

size or both) by groups.

In each case, the thermal reinforcing should be added to the layer closer to the cold

side.

The required additional steel area is approximately:

The factor of 1.4 in the above equation is the load factor, Kg.Using poisson's ratio, v = 0.2 above equation can also be written as

(3-22)

(3-24)

(3-23)

(3-26)

(3-25)

35

If/, = 1- 0.7[0.8AJ,'J but not less than 0.31;ot

If/, = 1- O.7[O.8AJ,'J but not less than 0.31'"

In this method, it is asswned that the walls are in pw'e h0l1zontal tension

and that the hoop reinforcing is centrally located. The total crack width Wcr for a

vertical crack is given by

Values of WI, W2 and W3 are each estimated using the formula

where WI = crack width that would be caused by a short-term occurrence of 7~o/'

the horizontal tension force per unit height due to the unfactored sum

of static pressure plus overpressure

W2 = that portion of WI due to force Tst, the horizontal tension per unit

height due to unfactored static pressure alone, acting during the

presence of force TtotW3 = the crack width due to long-term application of horizontal tensile .

force, Tst

in which f, is the actual steel stress under theunfactored tensile force Tst or Ttot(I, = T / A,), sC, is the crack spacing, and 1//" is a constant.

The estimated crack spacing is given by

where A is the gross concrete area per unit height, /3= 0.7 or 1.0 for deformed and

plain hoop bars, respectively, and Io is the sum of the hoop-bar perimeters per unitheight of wall.

The constants If/J, 1f/2, and 1f/3, used to compute WI, W2, and W3, respectively, are:

36

According to USD;

(3-27)[08A 1',]

11/.= 1- 0.35 . c. , but not less than 0.65'1', T

"

Often a designer will compute We, at only that elevation where the lateralpressure is maximum. This is a mistake; since this method may indicate larger

crack widths at higher elevations.

(h) Design of conical hopper

The conical concrete hopper may be rigidly attached to the silo wall, as in abunker, but ordinarily it is supported by a concrete ring-beam around the upperperimeter of the hopper. The bottom of the hopper preferably should not berestrained or supported. Fig. 3-2 shows dimensions to be used in computing verticalpressures and pressures normal to the walls of a conical hopper.

If the computed We, is larger than its acceptable limiting value, crack widthmay be reduced by modifying wall thickness, by increasing the hoop steel area As,or by increasing the sum of perimeters Lo. Changing wall thickness should be thelast resort, since the other two parameters can be modified more easily. It should benoted that while factored (ultimate) pressures are used to compute the steel areaneeded for strength, service pressure (unfactored) are used to compute crack width.

The conical hopper shell is subject to two tensile membrane forces. Themeridional force, Fm, is parallel to the generator line of the cone. The tangentialforce Ft , is in the plane of the shell and horizontal. The meridional force per unit. width at depth Y is computed from equilibrium of the loads on the cone below thatdepth. These loads, shown in Fig. 3-2 , are the resultant of vertical pressures, qdes

(at depth Y), and W, the combined weights of WL, (material in the hopper belowdepth Y),and Wg(hopper plus equipment supported by the hopper below depth Y).

ACI 313 suggests that 4.5JJj psi be used for the tensile strength of concrete inthe above equations for If. (In metric units ft' = 1.194,fJ: kglcm

2)

Since tensile forces Tst and Ttob As, and Lo all vary with depth below thesurface of the stored material, it is better to calculate crack width at the bottom ofeach group of hoop bars having a common size and spacing.

a at depth YSilo bottomand hopperbase

\ ",. -;>-"tJ'

Qdes at depth Yb'

~ Fm

Conical hopper

Fig. 3-2 Forces In conical hopper in conventional method of analysis

37

(3-31)

(3-30)

(3-29)

(3-28)

(horizontally)

(meridionally)A, reqd=F"", /(~fy)A, reqd = F",/ (Ny)

[D W] <[ W ]F = 1.7 qd,., + L + 1.4 '

m" 4sina nDsina . nDsina

In this study a prototype silo is analysed using both conventional method

and Finite Element method and a comparison is made thereafter. For this purpose a

prototype of standard dimensions with common properties is used. The geometric

parameters of the prototype are shown in Fig. 3-3. Data used for prototype silo isgiven below.

The minimwn acceptable thickness for the hopper is determined considering

the acceptable crack width, as for a circular silo wall. It is preferable, however, that

the shell thickness is never less than 5 in. The required reinforcement area per unit

width of shell is

Both forces are maximum at the upper edge of the hopper, and approach

zero at the lower edge.

3.1.4 Analysis of a Prototype Silo

The ring-beam and upper edge of a conical hopper supported at isolated

points along its boundary by columns, pilasters, or wall pockets (Type-l and

Type-3) can be designed in the manner shown in Appendix- 1.

(i) Design of the ring beam

Simply supported ring beams are usually designed for the horizontal

component of Fmu only. If the hopper wall is eccentric to the centroid of the ring-

beam, the beam will also receive uniform bending moment. The monolithically cast

ring-beam and conical shell is very stiff, however, and this moment is usually

neglected in design of ring-beams with all-around support. The area of longitudinal

steel in such ring-beams is arbitrary, but should not be less than 0.5% of the cross-

sectional area of the beam.

Ttop

•

H

1

« h

/ ~JJj]:t;;"1

o

Tbottom

[Type.' ]

: Type.2t top

Fig. 3-3 Diagram showing various dimensions of a silo

........fl rf.>~om

I

38

To avoid duplication, the results of analysis using both conventional method

and Finite Element method are presented in article 3-3 and a comparative study is

made using both the results.

50.0 Ib/cft

30.0 degree

0.45

42.0 inch

-12.0 inch

30.42 inch

6.0 inch

9.0 inch

9.0 inch

5.0 inch.

55.0 Degree

180.0 ft

160.0 ft.30.0 ft.

= 3.Ox106 pSI= 150.0 Ib/cft= 3000.0 pSI= 0.2= 29.Ox106 psi= 60000.0 pSI.= 36.9 psf

Grain.

Janssen's Method

WSD

=

=

=

=

=

=

=

=

=

=

=

=

=

=

Dimensiom:Height of vertical wall, HDiameter of the silo(internal), D

Overall depth of silo (From bottom of hopper to top of

vertical wall)

Size of Bottom Ring beam

(A) Depth of bottom ring beam, d(B) Width of bottom ring beam at top, b(C) Width of bottom ring beam at bottom

Thickness of vertical wall at top, TtopThickness of vertical wall at bottom, Tbollom

Thickness of hopper at top, ttop

Thickness of hopper at bottom, tbollom

Angle of conical hopper with horizontal

Properties of stored material:Unit weight of material, yAngle of internal friction, pCo-efficient of wall friction, f.l'

Other data:Stored material

Method of pressure computation

Design Method

Properties of construction material:Modulus of elasticity of concrete, EcUnit weight of concrete, YcUltimate strength of concrete, f~Poisson's Ratio of concrete, v

Modulus of Elasticity of steel, EsUltimate strength of steel, J;Air pressure due to wind

39

3.2.1 General

(3-32){R) {'LR cosn(}}

. Z = 'LZ: cosn(}T 'Ll: sinn(}

The cubk'type of element (Fig. 3.4) has four nodal normals, each havingfive degrees of freedom - axial, radial, tangential and two inplane rotations asshown in the figure .

.The program at present can handle isotropic elastic material. The materialproperties are defined for every element, thus allowing the program to deal withmaterials varying from element to element.

Finite Element approach is a powerful and versatile numerical tool for theanalysis of complex structures. In this technique a high speed digital computer isessential. Analysis of silos using this method is discussed in the following articles.

3.2.2 The Finite Element Program

3.2 FINITE ELEMENT METHOD OF ANALYSIS

(a) Features ofthe computer program

Ahmad [3, 4, 5] developed a very general Finite Element computer programfor analysis of axisymmetric shell structures. This computer program is a thickshell Finite Element program capable of analysing axisymmetric shells loadedsymmetrically as well as non-symmetrically. It can also deal adequately with thinshells.

Using Fourier analysis the non-symmetric load acting on an axisymmetricshell is replaced by a set of harmonics and each harmonic is treated separately andthe loads are presented circumferentially by

where R, Z and T are in radial, axial and circumferential directions respectively.

The displacements and stresses are calculated independently for eachharmonic and the results are automatically super imposed to give the final effects ofthe loading at every node. If these vary circumferentially, the fmal results are .calculated at a specified number of sections along the circumferential direction andthe results are printed for every point indicated by its angular distance from the

• r

r', u

a

(b) Displacement components

z", v

s,w

~

(a) Axisymmetric Element

Fig. 3-4 Axisymmetric thick shell element (cubic).

z

reference diameter ( e = 0° ). As the results are symmetric about this diameter onlyone half of the shell needs to be taken into consideration.

This Finite Element program has been adapted and used for the analysis ofsilo for the following load cases.

i) Gravity (Self weight), considered axisymmetricii) Stored material pressure, considered axisymmetriciii) Temperature difference, considered axisymmetriciv) Wind load considered, asymmetric

(b) Assumptions and limitations of the program

The axisymmetric thick shell element computer program developed byAhmad is based on some assumptions such as the material within an element isisotropic, elastic and obeys Hooke's law. This program cannot deal with branchingas would be encountered in certain axisymmetric shell problems. The performanceof the element suffers with sharp kinks. Moderate kinks can however be treated bytaking smaller elements near the kinks.

(c) Modifications of the program

The author has adapted the above mentioned program. In order to analysisthe silo, considerable modifications and additions are made specially in respect ofinput of data and output of results. The modifications are discussed below:i) As stated earlier, due to its highly general nature, the program needed large

volume of input Data, the preparation of which was time consuming. Asubroutine is, therefore, written to generate the necessary data for silo withminimum input.

ii) In the original program the output was stresses in global co-ordinates whichcould not be used conveniently for design. Huda [15] modified the programto obtain the stress resultant N 1» N(),M~and Me directly from the computerfor the analysis of Intze tank. The same modifications are adapted for siloanalysis.

iii) The original program treated axisymmetric loads such as gravity load,temperature load, and the stored material pressure (requiring one harmoniconly) and non-axisymmetric loads such as wind, earthquake (requiring anumber of harmonics) separately. Considering this fact the program wasmodified in such a manner that the analysis is carried out in a single

40

hannonic for gravity, temperature difference and stored material pressurewhile that for wind load is carried out with as many hannonies as desired.The flexibility of the original program had to be sacrificed to some extent to

attain this specific goal.

The output of the original program is the nodal displacement in theascending order of the nodes for every right hand side (load case). Huda[15] has modified the program to offer choice between nodal points andGauss integrating points for stress calculation. The same procedure is

followed for silo analysis.

In silo analysis, three different types are considered. In each type thegeometric dimensions, the loading conditions and the support conditions aredifferent as shown in Fig. 3-5. In order to handle the three types in a singleFORTRAN program necessary modifications were also made.

The program, in its present form, can provide the stress resultants directly

necessary for silo design.

3.2.3 Finite Element Idealisation of Silo

For Finite Element analysis, the silo is represented by a chain of unbracedaxisymmetric shell elements placed end to end. Since the program can not dealwith branching the actual structure needs idealisation. these idealisation is neededmainly near joints. In this case the following assumptions are made.

Type-l and Type-2

i) The mid surface of vertical wall meets with the mid surface of

the bottom ring beam.ii) The bottom supporting ring beam is divided in to two parts. One part

(one-third) is associated with the vertical wall as a part of shellwith greater thickness. Another part (two-thirds) is associatedwith the conical hopper as a part of shell of greater thickness.

iii) The bottom of the vertical wall is assumed to be supported bycolumn for Type-I and by continuous wall for Type-2. The wholestructure is supported at the junction of vertical wall and conicalhopper in such a way that there is no vertical displacements but theremay be radial movements. Idealisation of Type-I and Type-2 is

shown in Fig. 3-6.

41

(c) Type - 3

Column as asupport forC6nicai Hopper

Ring Beam

Continuous Wallas a support forVertical Wall andConical Hoppar

(b) Type - 2

Conical Hopper ,l[.B

Ring Beam

Column as asupport forVertical Wall andConical Hopper

'iij ., .,3: 3: 3:., ., .,0 00'E 'E 'E" "" > >>

(a) Type - 1

Fig. 3,5 Various Types of silos depending on ring beam support(a) Ring Beam Supported by column(b) Ring Beam Supported by Monolithic Continuous Wall(c) Ring Beam Supported by Separate Column

B

Conical Hopper li.

Fig, 3-6 Idealisation and element numbering of silos

2

3

4

5

6

7

8

9

10

11

12

13

14

2 15161718

6 1978

9 201011

12 2113

22

(b) Type-3 2325 24

3

4

2

5

6

7

8

9

10

1113 1'2

15 1417 16

182022

25

(a) Type-1, Type-2

Type-3

In Type-3 it is considered that the vertical wall and the conicalhopper with ring beam are completely separate components of silo.These are independent structures. In this case the whole analysis isperfonned separately and the element numbering of the conical partis independent of that of the vertical wall. The idealisation of Type-3 is shown in Fig. 3-6. Owing analysis the vertical wall is consideredas problem-I with the stored material pressure in the pressure zoneonly and the wind load all over the depth. The conical part isconsidered as problem-2 and subjected to symmetric load only withno wind load acting on the conical hopper. In Table 3-1 the lengthof different elements are shown.

In Type-3 the vertical wall is considered to be supported onfoundation such that the foundation acts as fixed support at thebottom. The conical hopper is supported by ring beam on separatecolumns. In this case only one displacement is considered to be zeroi.e. the columns support the ring beam only vertically. The centre of ~column section passes through the node which is closest to the centreof graVityof the ring beam section.

The actual Structures are shown in Fig. 3-5 . Fig 3.6 shows the schemefollowed in diVidingthe silo into 38 elements in Type-I and in Type-2. In Type-3the vertical wall is diVidedinto 25 elements and the conical part is diVided into 15elements. Length of different elements are shown in Table 3-1.

Near the junctions the element shapes become somewhat odd due to lack ofcontinuity of slopes of the middle surface of the two elements on the two sides ofthe junction. Such elements near the junctions can be kept smaller compared toother nonnal elements in order to limit the shortcomings of the odd elements.

42

Table 3-1. Length of Elements in Finite Element analysis.

Vertical Wall COIucal HopperSilo Type Element No. Length of Element Element No. Length of Element

ItolO 80%ofH 26,27 d/3TYPE-! II to 21 (18.011I) % of H 28,37 Ll64& 22 to 23 1.0%ofH 29,36 7L/64

TYPE-2 24 to 25 d/6 30 to 35 Ll8

38 12"I to 14 {H - 0.025(H+h)}/14 I to 2 b/215 15 %of (H + h) I 3 to 4 bj/216 1.0 % of (H + h) 5, 14 Ll64

TYPE-3 17 to 18 d/2 6,13 7L/64. 19 to 22 {h - d. 0.04 (H + h)}/4 7 to 12 Ll8

23 to 24 15 %of (H + h) 15 12"25 1.0%of (H+h)

Alternatively, a technique applied by Huda [15] and illustrated in Fig. 3.7 asapplied to joint 'A' and joint 'B' may be used to eliminate the shortcomings. Thismakes the nodal normals of each element perpendicular to its middle smface at thenode. This technique consists of removing small quantity of material from one sideof the middle smface and adding it to other side so that the odd shaped elementnow assumes a normal shape. Shifting a small quantity of material from tensionside to compression side or vice versa does not change the total quantity of strainenergy so long as the behaviour of the material is linearly elastic. Since the FiniteElement formulation is based on the minimisation of strain energy the aboveidealisation does not affect the stiffness term, though it ensures a gentle behaviourof the element.

3.2.4 Determination of Forces and Moments at a Section

The forces and moments that act at a section are shown in Fig. 3-8. Theseare the Meridional force N(>. Hoop force No. Meridional bending momentM(> andCircumferential bending moment Mo. In silo analysis Meridional force and Hoopforce are positive for tension and negative for compression. Moments producinginside tension is negative and outside tension is positive. In some places in this

43

.-

(b)-iii. Joint B

b ~I

(b)-ii. Joint A, Type-3

(b)-i. Joint A, Type-1, Type-2

,--Vertical Wall

_L~CentroidalI Distance

ColumnSupport

(a)-iii. Joint B

Fig, 3-7 Idealisation of joints(a) Actual Joint(b) Idealised Joint

:~; Col~mn

Vertical Wall

z

(a)-ii. Joint A, Type-3

Ylr-~

(a)-i. Joint A, Type-1, Type-2

Td

Support

d

44

where" t " is the nodal thickness of the shell.

(O'~/+ O'~b)xl12

(0'01 + O'Ob) x II 2(O'~I- O'~b)xl 2112(0'01 - O'Ob) x 12112

Np

No =

M~=Mo =

study the melidional bending moment is denoted by only meridional moment andcircwnferential bending moment by circumferential moment.

3.2.5 Capability of the Program at the Present Stage

These quantities are required for design and are to be calculated from thelocal nodal stresses. The procedure described below demonstrates how to calculatethese forces from stresses. Fig. 3-8. shows the stresses acting at top and bottom of anodal normal. These stresses have two subscripts. The first subscripts indicates thedirection and the second subscripts indicates wheather it is at top or bottom of thenodal normal. Now the forces and moments acting at the node can be calculated asfollows:

The Finite Element program developed in this work for the analysis of silocan provide a designer all the necessary stress resultants (Forces and moments)and their locations required for a silo design. With this program a designer has thefollowing options:

I) Silo analysis using FPS or MKS method.2) Silo analysis using WSD or USD method.

3) Silo analysis using Janssen's or Reimbert's method of pressurecomputation.

4) Silo analysis of anyone of the three major types considered in thisstudy.

5) Silo analysis for Grain or Cement (cohesive) material.

The load combinations considered are:a) Self weight

b) Self weight + Stored material pressurec) Self weight + Stored material pressure + windd) Self weight + wind.

Fig. 3-8 Forces and Moments with nodal stresses acting ata node of an axisymmetric shell .

OOb

N~

M~

(c) Forces and Moments

(a) Nodal Stresses

r'

a

(b) Local Axes

z'

If USD method is used then ACI load factor is considered for analysis. Forthe above load combinations following load factors are used.

a) 1.4xSelfweight

b) 1.4xSeif weight + I.7xStored material pressure

c) O.75x(1.4xSelfweight + 1.7xStored material pressure + l.7xWindpressure).

d) O.9xSelfweight + I.3xWind pressure.

The present program provides the following stress resultants:

i) The absolute maximum values of various forces and moments eitherin vertical wall or in conical hopper along with the maximumpositive values and maximum negative values. These are required forthe thickness selection of silo components.

ii) Distances from a reference point for all the stress resultants stated in(i)

iii) Values of various forces and moments at each node for individualeffect (load case).

iv) The values of moments developed due to temperature difference ateach node.

v) The program considers three types of load at a time - gravity load,stored material pressure and wind load. For wind load it calculatesforces and moments at an angle of 15° interval along thecircumference. For each node it considers the four loadcombinations mentioned above and for combination (c) and (d) at 13separilte points along the circumference. After considering all thepossibilities it finds out the maximum forces and moments at a node.Finally for all the nodes either in vertical wall or in conical hopper, itprovides design values of each force or moment (positive andnegative) in the ascending order of node number. It also gives theload combination number from which the maximum values arecalculated.

45

(3-19)

(3-33)

(3-34)

(Ii - Ih/Ji)

(fi - Ihlji)

(a) Analysis for self weight, material pressure and wind load

As stated earlier a prototype silo with data given in Art.3.1.3 is analysedusing' both conventional method and Finite Element method and the results arepresented in Art. 3.3 for comparison.

(b) Analysis for temperature difference

In Finite Element analysis an elaborate study has been made to know thebehaviour of meridional bending moment and tangential bending moment due totemperature difference across the silo wall. In this analysis a broad and practicalrange of various geometric dimensions and temperature difference has beenstudied. Temperature difference of inside and outside of silo for all the nodes hasbeen taken the same in some cases. On the other hand in some other casestemperature difference, LIT, varied from node to node. Geometric parameters arealso varied in this investigation. Vertical wall below pressure zone in Type-3 isconsidered separately. In this research, temperature difference is considered onlyin the pressure zone. Below the pressure zone (Type-3) inside and outsidetemperature are the same. But due to temperature difference in pressure zoneconsiderable moments also develop in this portion. Fig. 3-18 and Fig. 3-19 showthe variation of ratio of moment at any depth "y" to the moment at the bottom ofpressure zone in percent, "R", with "(y/h)xIOO".

3.2.6 Analysis and Presentation of the Results

According to conventional method, (Eq.3-19) moments due to temperaturedifference are given by

where MTtu = Ultimate circumferential momentMyt,u = Ultimate meridional moment.

46

This equation can also be written as

or

where at . = Coefficient of thermal expansion of concreteL).T =Temperature difference across the wall

47

(3-35)

(3-36)

Moment at depth y = (R x Moment at bottom of pressure zone)/] 00

In the above equations

Cx' = Temperature coefficient for Circumferential moment in (ft - lb/ft)Cy, = Temperature coefficient for Meridional moment in (ft - lb/ft)

C, =A constant and in this study it is expressed as 'TemperatureCoefficient' .

According to the conventional method C, is constant for all the depth of

vertical wall and corncal hopper and the same for circumferential moment and

meridional moment. Finite Element analysis. predicts values of C, for the verticalwall and conical hopper and it is observed that beyond a certain distance from the

ring beam this values are constant. Further, C, is not the same (0.1042) for

circumferential moment and meridional moment. Values of C, may increase 50%.or more near the ring beam where restraint exists. Actual variation of C, formeridional moment and circumferential moment. for various types of silos are

shown in Fig. 3-10 to Fig. 3-19. The symbols (H, 1"; L, l) used in this presentation

are shown in the Fig. 3-9. On the basis of this study a set of Design Curves are

recommended for the determination of C,. These curves are presented in Fig. 3-20

to Fig. 3-24. Using these curves, equations adapted for meridional orcircumferential moments are as follows:

For vertical wall below pressure zone in Type-3 the Moments (Meridionalor Circumferential) at any depth "y" can be obtained as follows:

Values of "R" can be obtained from Fig. 3-22.

3.3 COMPARATIVE STUDY

3.3.1 Mode of Comparison

In orderto reveal the merits of Finite Element method of analysis in relation

to conventional method of analysis of silo, a comparative study is made. Results

obtained from both conventional method and Finite Element method are presented

in the same figure. At first, the stress resultants which can be obtained from both

conventional and Finite Element method are presented. A number of forces and

Fig. 3-9 Various Types of silo depending on ring beam supportwith various symbols.

(a) Ring Beam Supported by column(b) Ring Beam Supported by Monolithic Continuous Wall(c) Ring Beam Supported by Separate Column

Y'

H

--n-II

Y W

i

(c) Type - 3

Y

H

T

(b) Type - 2

H

Y

(a) Type - 1

Fig. 3-10 Temperature coefficient for meridional moment in vertical wall( Type-1, Type-2 )

. 16144 6 8 10 12

Temperature Coefficient x 100

2

90

o

80

70

60

100

40

~ .. . . .--.: ..

1'2. ....~0

0

C-

o •..

0

o 0.0

.~ 0t-

o

.

c-~~ 0f-

.0 .c .

.~

0

'.~... .. ..o -" , " " " '" " "

3

20

xso:r:-->-

Fig. 3-11 Temperature coefficient for circumferential moment in verticalwall (Type-t, Type-2 )

14138 9 10 11 12

Temperature Coefficient x 100

7

.....~.

.

...•....••• ..'F-e . -t- 1,-

f..

.. -.... r-.. -u

.. .I- .. ...

. .

. ..

. .....

~ ... .. .. .. .

~ 1,f- ;

;i

.,, , , ,," , , iU

90

100

70

80

60

o6

40

30

20

10

aax~ 50:r:-..>-

14138 9 10 11 12

Temperature Coefficient x 100

7

Fig, 3-12 Temperature coefficient for meridional moment in verticalwall ( Type-3 )

~ .~ . . .~ . . .. " .....~ ... .

....~•...••.~••• •••I••

~ •e- .,.•.'".,.•• Ii-•• .••

. •:.

~ ."~ .~~~ ..~;~ .. "... .- , .

90

80

o6

70

60

40

30

100

20

10

><~ 50I-->-

Fig. 3-13 Temperature coefficient for circumferential moment in verticalwall ( Type-3 )

14138 9 10 11 12

Temperature Coefficient x 100

7

......~,.).: I

, r::-., , -.. , 1-,,

.. -.

.. ......... -~ ..

~ ..' ..~ ..' .~ .'~ .'.~~ .' •

"~ . •.. •.', •" ...-

-•~

.~, •••..•

~ •~ -.~" .••, .}"

I I I I I • I I

80

90

70

o6

60

50

40

100

30

20

10

x8

I-->-

Fig. 3-14 Temperature coefficient for meridional moment in conicalhopper ( Type-1, Type-2 )

14138 9 10 11 12Temperature Coefficient x 100

7

, - - ! II - - 1i\-- -

.)

-I--=

i _ n

"..- .-

-• 0

t-o

-..-.-

r ....t-

..-

\~

-

i:>.:. .""" , " " " '" I""", - .. . ,.-, ," ,

90

70

80

100

o6

30

20

10

6000~x

50-J.--•••

40

. Fig. 3-15 Temperature coefficient for circumferential moment in conicalhopper ( Type-1, Type-2 )

14138 9 10 11 12Temperature Coefficient x 100

7

..-\.•. . ...

.. ...f- . ..

0 0 - .

... .....

I ..= .. ..~ .

-o _ -

ooo

~ . ..10-

o.

-f--...

.

J= -'"'''''' """ '" " "1\' " " " "

90

60

80

70

100

50

o6

40

30

20

10

x..J--•••

't,

14124 6 8 10

Temperature Coefficient x 1002

Fig, 3-16 Temperature coefficient for meridional moment in conicalhopper ( Type-3 )

:~ ..l-..t--.

.

.

.

,

.t.\-. -

~~ ...~~ / . .f-F . .

~.--;--:-.

, ,

70

80

60

40

30

10

20

100

, 90

x50

-l~•••

Fig. 3-17 Temperature coefficient for circumferential moment in conicalhopper ( Type-3 )

.. \~

......-.

C- .-~~ ..~~ . .~

.

~E ...~~~ I-~ ... .

.-.-

. ....

.. ..

J./ .. .

14124 6 8 10

Temperature Coefficient x 1002

10

30

20

40

60

70

90

80

100

x50-'-•••

100 120 140 1608060R

4020o

R = (moment at depth y)/(moment at bottom ofof pressure zone)x1 00

-20

Fig. 3-18 R for meridional moment in vertical wall below pressurezone ( Type-3 )

r~~~~

tEE~FF

/

~ I~r •~

~tl-

., , , ,

70

80

90

60

40

30

20

10

100

x1:: 50

160140120100804020o-20 60R

Fig. 3-19 R for circumferential moment in vertical wallbelow pressurezone ( Type-3 )

R = (moment at depth y)/(moment at bottom ofpressure zone)x1 00

90

~f-

~f-

0

0-

~

f-

~

~r

l-, , , , , . , , . , , , ,

80

70

100

60

50

o-40

40

30

20

10

x

16

16

15

15

10 11 12 13 14Temperature Coefficient x 100

10 11 12 13 14Temperature Coefficient x 100

9

9

~II Meridional Moment--I Circumferential MomentI ---,IIIIII

E,IE I

III .

I;II

('-

Fig. 3-20 Temperature coefficient for vertical wall ( Type-1, Type-2)(A) Full height( B ) Enlarged Lower portion of ( A )

~. I

IMeridional MomentI --

I Circumferential Moment---I

:I

II

. II

~ II

~-

~

"""" r---....\, r----f- ,"

o8

08

100

90

80

a 70a~x 60I-- 50>-

40

30

20

10

20

18

16

14aa 12~xI 10-->- 8

6

4

2

14

14

13

13

8 9 10 11 12Temperature Coefficient x 100

8 9 10 11 12

Temperature Coefficient x 100

7

7

0 II Meridional Moment-; Circumferential Moment---I

--lIIII

~ ;IIII

:,..........•---::::.I

~

. ;~

I Meridional Moment-Circumferential MomentI ---II

IE :\r

:\I

1\I

\\\_.

t= ,)t=

JI ./

~..-

"".;,"

4

Fig. 3-21 Temperature coefficient for vertical wall (Type _3 )( A ) Full height( 8 ) Enlarged Lower portion of ( A )

14

2

10

x

100

12

90

80

a 70a~~ 60I-..>- 50

40

30

20

10

aa~ 8

I):6

Fig. 3-22 Ratio of moment at any depth y to the moment at thebottom of pressure zone ( Type - 3, below pressure zone)

~ Meridional Moment--Circumferential Moment---

I

/i/ I

rJ

"- I-..:, ---~ -~ -

100 120 140 160

100 120 140 160

80

80

60R

60R

40

40

20

20

o

o

R = (Moment at depth y ) I (Moment at bottomof pressure zone)x1 00

( A ) Full depth( B ) Enlarged lower potion of ( A )

-20

-20

~Meridiooal Moment--

Circumferential Moment---

//:•...•.•.. \

o-40

o-40

40

35

30

2500~)( 20J::->- 15

10

5

100

90

80

7000~ 60)(

J:: 50->-40

30

20

10

Fig. 3-23 Temperature coefficient for conical hopper ( Type-1, Type-2)( A) Full depth( B ) Enlarged lower portion of ( A )

14

14

13

13

\\\\\

10 11 12Temperature Coefficient x 100

8 9 10 11 12Temperature Coefficient x 100

9

7

~~\

\ Meridional Moment-.Circumferential Moment

\ ---\

\\

I. ,~ ,~

~E

~~" -..r---, r--.. ~

o6

5

o8

100

90

80

0 700~x 60~...J--"" 50~

40

30

20

10

25

~Meridional Moment

Circumferential Moment

20I,I0

0 15~X~...J--"" 10

14

14

12

12

4 6 8 10Temperature Coefficient x 100

2

2

0 \Meridional Moment\-Circumferential Moment \---\

i,IE II

I

~1

l-iU)

-"--- --oo

4 6 8 10Temperature Coefficient x 100

Fig. 3-24 Temperature coefficient for conical hopper ( Type _3 )( A) Full depth( B ) Enlarged lower portion of ( A)

~ ~ Meridional Moment :/-~ CirCUmferential Moment---

~I

~ II i

E

I \I

i

~~ =-- ~/~

/- /- /-"

~oo

5

15

10

100

90

80

0 700~x 60...J-- 50•••

40

30

20

10

50

45

40

3500 30~X

...J 25--••• 20

bending moments, which can be obtained only from Finite Element analysis andconventional method provides no means for the computation of such functions, arepresented later. In case of hoop forces, meridional moments and circumferentialmoments due to wind load, the maximum values are shown considering 13equidistant points (at intervals of ISo) along the circumference at each node.

3.3.2 Forces Obtained from Both the Methods of Analysis

(a) Meridional force

Vertical wall: In the vertical wall meridional force due to self weight andstored material pressure are more or less the same both for conventional methodand Finite Element method, and it is negative all over the depth (Fig. 3-2Sa andFig. 3-26a).

Various stress resultants such as meridional force, hoop force, meridionalmoment and circumferential moment due to wind load VlUY circumferentially as aresult of non-symmetric distribution of wind pressure in circumferential direction(Art. 2.8.3). Fig. 3-27 to Fig. 3-29 show the circumferential variations of variousstress resultants on a horizontal plane at different levels for different types of silos.From these diagrams, the locations of maximum wind effect for different stressresultants are obvious. These diagrams also indicate that the locations of maximumeffe~t may change depending on the level of the horizontal plane.

Both positive and negative meridional force develop for wind load.Investigation shows that maximum positive meridional forces occur at e = 0° forall types of silos and maximum negative meridional forces, away from the bottomsupports, occur at e = 180°.Near the bottom support maximum negative meridionalforce occur between e = lOSo and e = 120°. At e = 0° the wind direction isperpendicular to the surface and the diametraI line is parallel to the wind direction.Fig. 3-30a and Fig. 30b show positive meridional forces at e = 180° for all typesand negative meridional forces at e = lOSofor Type~1 and Type-2 and at e = 1200for Type-3. Finite Element analysis and conventional analysis are in closeagreement in respect of positive meridional force for the upper part of the verticalwall (Fig. 3-30a). But for lower portion of the vertical wall the conventionalmethod and Finite Element method predict different values. The negativemeridional force due to wind predicted by conventional method is always greaterthan that of Finite Element method (Fig. 3-30b).

48

,

cO

o

Finite Element, Type 1,2

Finite Element, Type 3

Conventional,Type 1,2,3

................

.......•.••••

......'..'..'

..'..'

Finite Element,Type 1,2

.......'

Finite Element, Type 3

....

Conventional,Type 1,2,3

.'..'

.'

200 400 600 800 1000 1200 1400

Meridional Force ( Ib / ft )

(b) Conical Hopper, £ = Distance from bottom of ring beam

Fig. 3-25 Meridional Force due to self weight along vertical section

o

1500

2500

-500

2000

13.5 1000

-1000 0 -5000 -10000 -15000 -20000

Meridional Force ( Ib / ft )

(a) Vertical Wall, Y = Distance from bottom of pressure zone( upward positive)

>-8 500c:'"ini5

50

250

300 0

13 100.5

•••8 150c:~i5 200

Fig. 3-26 Meridional Force due to stored material pressure along verticalsection

35000

-50000

30000

-40000

.';.,',.,' ,,,- ,

.,' ,,'-'

Conventional,Type 1,2,3

Finite Element,Type 1,2

Finite Element, Type 3

Finite Element, Type 1,2

Conventional,Type 1,2,3

-30000-20000-10000

5000

Finite Element, Type 3

10000 15000 20000 25000

Meridional Force ( Ib / ft )

(b) Conical Hopper, £ = Distance from bottom of ring beam

Meridional Force ( Ib / ft )

(a) Vertical wall, Y = Distance from bottom of pressure zone( upward positive)

a

300 a

-1000 a

-500

50

2500

a

2000

1500

250

"" 1502lc:'"~ 200

:2100()

.S

.<::()

-"' 1000>-2l 500c:~o

Fig. 3-27 Forces and Moments along circumferential direction on ahorizontal section due to wind pressure (Type-1, Type-2 )

.,-.,£

EQ)

E0 ~:>a;~~J11E:J~G-.,-£Q)

~0l>-e.00I

~.£e.8I

.,-£

105 120 135 150 165 1809075

(A)AtY= 1455 inch.

(B )At Y = 843 inch.

Angular Distance 0 ( Degree)

60 75 90 105 120 135 150 165 180Angular Distance 0 ( Degree)

6045

45

30

30

15

15

Meridional Force Meridional moment-- .. -_ ..Hoop Force Circumferential moment

.••. - -, ........ - -/ ........,

I,,-r-- I ,- .:..:.:: - .•- -- -- ..- '.....

.........- _::-< ..,....., -- •.... ...... ............ .'....... -- ..' ..... ,- --- -..--- '-..!.... . ' ---- --- ..---- I '. ...... .".....-- ........ -- ....."I

........

/ - -//

I- ,/

F-I-rr

---- .............. Meridional Force Meridional moment-- - - --

""Hoop Force Circumferential moment........ --

rf- ""~ -- •...r ><- ,

,/ '" ...../ ..../ ~- "-/ ..--- ---- --- .. "-- .. _-- ......... ........ .........

~~.~~;.:: :.....~.-;~-- .--.--),.......... ...•.-.::.. --- .. ---- -_ .. -................... .........

"",

I- '. --f- / --f- ,/ ""-r ,/ -~-F' ......

t ......•

"--r

o

Q)~ -1000.£a;c:.Q -2000-0"iij:>-3000

0

"EQ)

g 2000:>a;c:o'6 1000'5i:>-., 0

_ 4000.,-.,i! 3000

3000.,.,£ 2000

EQ)

E0 1000:>a;c:0:2iii 0:>-.,-~ -1000Q)

~0I>-a; -2000c:0'6.~:> -3000

Fig. 3-28 Forces and Moments along circumferential direction on ahorizontal section due to wind pressure ( Type-1, Type-2)

o

<=-f!~~a.gI

180

( A ) At Y = 383 inch.

( B ) At Y = 226 inch.

60 75 90 105 120 135 150 165Angular Distance 0 ( Degree)

"~~"-gI

60 75 90 105 120 135 150 165 180

Angular Distance 0 ( Degree)

45

45

30

30

15

15

Meridional Force Meridional moment-- - - --•.•.•.••

"'-Hoop Force Circumferential moment........ --

•• ""~

I". I.

'\"\

...... -~--:;;.~..... ......... '\"I\. .

""i'- L.-- j....--

••~••

:--........ Meridional Force Meridional moment-- - - --~ "'- HOOp Force Circumferential moment

~........ --

"E '"~ '\tt

~t -:::: - --- --t.e. e.e..c - - ......... .......... ....... . .... . . ....e: -- .:,:.; .•.•...

'"••t "- ""-

"= -

"~ -3000

0;is -6000'C.<:

"::;; -90000

_15000<=-<=,B 12000

E9000"E

0::;;0; 6000<:0:g" 3000::;;-<=- 0f!

"~ -2000~0;~ -4000-0.~::;; -6000

o

-10000<=-<=

8000f!

E6000"E

0::;;0; 4000<:0'C.~

2000::;;-<=- 0f!

0- ,,.

/, .•.. .•..

/ ,/

,-,"-; ....-- --- - ---- ... .........- -- ..- .' ..... ............... ........ ."./ ..

.... ----- ------.-' ... -.--_ .. - - ..... ...... ... .•...... ........ ...... .../ .•.. ,

/ --// Meridional Force Meridional moment

/ -- ...... -"" Hoop Force Circumferential moment........ - -

"-""

""~gI

180

180

165

185

165

150

150

150

135

135

1201059075

Angular Distance 0 ( Degree)60

60 75 90 105 120 135

Angular Distance 0 ( Degree)

45

45

45

30

30

30

15

15

15

~-........... "- "," --- ,

c .•..

~ / "- .•../

~.•..,

~... _. ---- .. --- -"- .. -'. .•.. .......... .... .......... .....~_.-- ..... -/-. ...~ .......... ..... ., - .. _- ----.--- -_ .. -.......... ....... .. ....... . .•.....•.... .•..~ /

,'.

~ / i'-.... --.~

/ Meridional Force Meridional moment

"-.'" -- ...... -F Hoop Force Circumferential moment i'---c- ........ --~

60 75 90 105 120Angular Distance 0 ( Degree)

Fig. 3-29 Forces and Moments along circumferential direction on ahorizontal section due to wind pressure at (A) Y = 1517inch, ( B ) Y = 986 inch, ( C ) Y = 8 inch ( Type-3 )

o

o

~ @] Meridional Force Meridional moment~ -- ...... -

~ ~Hoop Force Circumferential moment........ --

~ .......~ .•...•...•...•. I-~~

- 2~ --F.' ..." .:,:.1. ••••••• ....... .•...•...•..c .•..•......

I---o

" 2000

""E 1000•Ea"0;c 0gu"•"~" -1000

"g~ -20000;cII"•" -3000

" 3000

""- 2000E•Ea" 1000..cgu"• 0

"~"" -1000

~ol'0; -2000cII~

-3000"

16000

""" 12000

E•E 8000a

"0;cg 4000u"•"~ 0~"~ -4000a~0;c -<lOOOg

~" -12000

30000

.14000-12000

25000

.....

Finite Element,Type 3

20000

...•.... .

.........•.•

15000

Conventional,Type 1,2,3

Finite Element, Type 1,2

Finite Element, Type 3

--.". -.-

Finite Element, Type 1,2

Conventional,Type 1,2,3

..... .... ...... ........ . .

........ ....

" .' .." .

-'.

..............

10000Meridional Force ( Ib / ft )

(a) Positive (Tensile) Meridional Force

-4000 -6000 -8000 -10000Meridional Force ( Ib / ft )

(b) Negative (Compressive) Meridional Force

"1":-.".........•

5000

-2000

Fig. 3-30 Meridional Force in vertical wall due to wind pressure.

'" ..•., .., ...•.•'~~.~

-500a

-sao a

1500

a

2000

J:::UoS 1000~>-Q)uctl 500is

2000,,,,

1500

~J:::U 1000.s>-Q)uc 500en-II)is

a

Conical hopper: For conical hopper the meridional forces due to self weightin Finite Element analysis of Type-! and Type-2 are almost identical to thoseobtained from conventional method (Fig. 3-25b). However, this is considerablysmaller in Type-3. On the other hand for material pressure, conventional methodalways underestimates the value of meridional force in conical hopper in relatiOiltoFinite Element analysis (Fig. 3-26b). Again for Finite Element analysis, in Type-3lesser meridional force develope than those in Type-l and Type-2.

(b) Hoop force

Vertical wall: Both in conventional method and in Finite Element methodthe Hoop force due to self weight is zero for most part of the vertical wall (Fig. 3-3!). But Finite Element method provide some positive and negative hoop forcenear the ring beam. The positive hoop force is negligible but negative hoop forcedue to self weight in Finite Element method is appreciable.

Due to grain load, hoop force predicted by conventional method and FiniteElement method are almost identical for the upper part of the vertical wall(Fig. 3-32) . Near the ring beam some discrepancy is observed. In conventionalmethod there is no negative hoop force for the vertical wall. But in Finite Elementmethod there exists considerable negative hoop force near the ring beam in Type-land Type-2 and near the foundation in Type-3.

From Fig. 3-33 it is obvious that hoop force due to self weight is verysmall in comparison to that of stored material pressure and this can be neglected fordesign.

Conical hopper: Hoop force in conical hopper due to self weight and storedmaterial pressure are in close agreement for conventional method and Finite.Element method in Type-! and Type-2 beyond certain distance from the ringbeam (Fig. 3-34a and Fig. 3-34b). Finite Element analysis predicts much smallerhoop force near the ring beam than that of conventional method. Fig. 3-35compares the hoop forces for grain load and self weight. From this the relativeimportance of these two functions is revealed. In conventional method themaximum hoop force occurs at the junction of conical hopper and ring beam but inFinite Element method the maximum value of hoop force is found at a considerabledistance from the ring beam. For Type-! and Type-2 the distance of maximumhoop force is about 25% of the length of the conical hopper and for Type-3, it isabout 14% of the length of conical hopper. The maximum value of hoop force in

49

.G\...

1000o-1000-2000

Hoop Force ( Ib / fl )

-3000

Fig. 3-31 Hoop Force in vertical wall due to self weight.

Conventional,Type 1,2,3

- Finite Element,Type 1,2------

Finite Element, Type 3............

~

~

..-- .----------- --- .----.----

'...'::.......

...... ................ ....... .................. ........ ................ .......

~, , , , , , , , , , , , , , , , , , , , , , , ,

.250

o

250

500

-500-4000

2000

1250

1750

1500

>-~c 750tlCi

131000.5'

Fig. 3-32 Hoop Force in vertical wall due to stored material pressure.

•

n

3000020000o 10000Hoop Force ( Ib / ft )

.10000

~ Conventional,Type 1,2,3

Finite Element,Type 1,2-----_.

~

Finite Element, Type 3............." ,'.,'.,,,f\\

. '\.

\

I

,

- - -- ------ - ,:;.- - -- ----------- 1----------- ............ ..'..................

.

....... .............

-

", , , ,

250

o

.750.20000

.500

.250

1750

1500

2000

2250

1250

~J::() 1000.!:~>-'"() 750c'"-enis

500

Fig. 3-33 Hoop Force in vertical wall due to self weight and storedmaterial pressure.

3000020000o 10000Hoop Force ( Ib - fI / fI )

-10000

"Fin~e Element Analysis

"Type-1, Type-2

" Self weight

\,Material pressure ------

Type-<!Self weIght ............Material pressure ---

1\\'.

\\\\\\\\\\\\I

\\

t.,

,.------..:.-:: .-\;.-----_ .../ ------------ .-'---- .---\

--~..~-,"", ....

, " , , , , , " " ,

-250

-500-20000

o

500

250

1500

1750

1250

2000

~ 1000.I:.o.5

>-~ 750cCll"liiis

1400

50000

",..'

1200

Finite Element, Type 3

Conventional,Type 1,2,3

Finite Element,Type 1,2

40000

Finite Element, Type 3

Finite Element,Type 1,2

Conventional,Type 1,2,3

1000

30000

800600Hoop Force ( Ib 1ft)

Hoop Force ( Ib 1ft)

(a) Due to self weight

20000

400

--. -'-

(b) Due to stored material pressure

....

Fig. 3-34 Hoop Force in conical hopper.

.................. . .--. ......--- --. '-, -,- -'- -.

...•.....•.....

10000

'. '-- ".

.......

"

200

....•....

a

50

~ 100;:::()

.~••• 150Q)()c'"- 200'"is

250

300 a

a

50

100;:::()

.~~ 150•••Q)()c'" 2007iiis

250

300 a

500004000020000 30000Hoop Force ( Ib 1ft)

10000

Fig. 3-35 Hoop Force in conical hopper due to self weight and storedmaterial pressure.

.", - - ---", ". ". - --."- ~'-.-', J'. , --.• -. ----- -:~-

---- -.-..- ...............-.- ..- .-- ...-- ../ --- .. Finne Element Analysis

/' .... Type-1, Type-2-- ./' .. 8elfwelght..- ... Material pressure ------.... Type.3.

~

- Setfwelght ............Material pressure ---

,

o

50

250

300 o

•••~ 150

~is 200

-5 100.5

conventional method is about 21% greater than that of Finite Element value inType-I and Type-2. But the maximum value of hoop force in Type-3 of FiniteElement analysis is about 10% greater than that of conventional method.

3.3.3 Moments Obtained from Finite Element Analysis

Conventional method of analysis of silo can not predict any value of eitherMeridional moment or Tangential moment due to any loading. But Finite Elementmethod can predict those easily. The variation of meridional moment andCircumferential moment predicted by Finite Element method for different load

cases are discussed in the subsequent articles.

(a) Meridional moment

Vertical wall: The value of meridional moment due to grain load isconsiderable near the ring beam (Fig. 3-36). Both the negative and positivemeridional moments in Type-I and Type-2 are much grater than that of Type-3. InType-3 the maximum negative bending moment occurs at bottom of vertical wall.In Type-I and Type-2 the maximum negative value occurs at the junction ofvertical wall with the ring beam and maximum positive value occurs at a smalldistance above this.

Analysis of vertical wall in Type-I and Type-2 are similar for self weightand grain load (Fig. 3.36) but for wind load analysis there is slight difference inloading. In Type-2, due to wall support of the ring beam, the conical hopper is notsubjected to the wind pressure. But in Type-I, through the opening of column,wind may create pressure on the conical hopper. For this reason wind load analysisis shown for three different cases, separately. Fig. 3-37a and Fig. 3-37b shows thebending moment due to wind load. From this figure it is clear that the differencebetween Type-I and Type-2 is negligible.

Conical hopper: Conventional method predicts no value of meridionalmoment in conical hopper for any of the stated load cases. But Finite Elementanalysis predicts considerable amount of meridional moment in conical hopper( Fig. 3-38). For Type-I and Type-2, at the junction of conical hopper and ringbeam, negative bending moment develops due to grain pressure. In Type-3, at thesame point positive meridional moment is predicted. For all the types themeridional moment reduces sharply as the distance increases from the junction ofhopper and ring beam.

50

Fig. 3-36 Meridional moment in vertical wall due to self weight andstored material pressure.

2000o-2000

Meridional moment ( Ib - ft / ft )

-4000

Fin~e Element AnalysisType-1, Type-2

setfweight

- Material pressure ------Type-3

.self weight .............Material pressure -- _.

:."., - .. _- .•- .....----------- -------------- -_.:.._---------- --~....•-------,.-- ,

f-

f- - .•..•.. ,.- ... ?)-- -,

-500-6000

-250

o

500

250

1250

1500

2000

1750

>-~ 750ctli:5

~ 100013c

.-Finite Element,Type 1--Finite Element Type 2

- - --: Finite Element, Type 3

f-........

f-

f- J!C-f-f-

~/../f- . .....:' .

r ./f- ...•.

~(................... ........ ............................. ........................... . ............

....

<; Finite Element Type 1•.. --Finite Element, Type 2

- ---/) Finite Element, Type 3

~ ........c- /c / ....

..

~ ........---- .....

J ........................ .......-

,.::.r:~."C- .....r ............ ................. ................ ................ ............... ............

-700

2000

-600

1500

-500-400-300

Meridional moment ( Ib - ft 1ft)

1000Meridional moment ( Ib - ft 1ft)

(b) Negative Meridional Moment

(a) Positive Meridional Moment

-200

500

-100

Fig. 3-37 Meridional moment in vertical wall due to wind pressure.

-500 a

-500 a

2000

2000

1500~uc

1000>-CDUC

'" 500-<Jli5

a

1500

~u.!: 1000~>-CDUC

'" 500-<Jli5a

Fig. 3-38 Meridional moment in conical hopper due to self weight andstored material pressure.

80006000-2000 0 2000 4000

Meridional moment ( Ib - ft 1ft)-4000

--- .. - .... -------} - l- .---.,.. .....-'. ----..:,--,- ...

/ ..I, ...~I,

Fin~e Element AnalysisType-1, Type-2

Self weight

Material pressure ------~ Type-3

~ Self weight ............

~ Material pressure ---~

50

o

300-6000

250

•••'" 150oc~15 200

."5 100.!:

Due to wind load meridional moments also develop but these areinsignificant in comparison to the moment developed by stored material pressure(Fig. 3-39a and Fig. 3-39b).

For all the load cases the meridional bending moment is significant near thering beam because of the ring beam restraint.

(b) Circumferential moment.

Vertical wall: Circumferential moment, developed due to grain pressure isinsignificant over all the depth of the vertical wall except near the ring beam (Fig.3-40). In Type-3 the tangential bending moment due to stored material pressure inthe vertical wall is insignificant .

Tangential bending moments, both positive and negative, develop due towind load on the vertical wall (Fig. 3-41a and Fig. 3-41b). The maximum value oftangential bending moment exist in the top portion of the vertical wall and itreduces to zero at the bottom of vertical wall.

Conical hopper: Tangential bending moment in conical hopper predicted byFinite Element method forall the types due to wind load is negligible (Fig. 3-42aand Fig 3-42b). But these values for the grain pressure are somewhat higher forType-3 than for Type-l and Type-2 (Fig. 3-43).

3.3.4 Hoop Force due to Wind Load Predicted by Finite Element Analysis

Conventional method provides no means of prediction of the hoop forcesdue to wind load neither in vertical wall nor in conical hopper. But Finite Elementmethod can predict them. Fig. 3-44 through Fig. 3-45 show the variations ofmaximum hoop forces in the prototype. But these values are insignificant except atthe bottom of vertical wall.

3.3.5 Remarks

Investigation of the prototype silo reveals that the conventional method cannot predict the values of all the stress resultants required for silo design accurately.The conventional method is completely unable to compute the values of meridionalmoment and circumferential moment. Finite Element method, on the other hand,analyses silo to obtain meridional force, hoop force and moments and computethese values with acceptable accuracy. In practice Finite Element analysis of a silo

51

Fig. 3-39 Meridional Moment in conical hopper due to wind pressure.

... -------- 1--------.. --- ..... -- .

--.7;;...'...0.•,,,,

, I,,•••,

••,,,,

(•,,• Finite Element,Type 1•. ,

Finite Element, Type 2------

-300

-700600

-250-200-150-100Meridional moment ( Ib - ft / ft )

(b) Negative Meridional Moment

(a) Positive Meridional Moment

200 300 400 500Meridional moment ( Ib - ft / ft )

-50

100

---- .----.----- -_.-.~, \•

~,

~:

Finite Element, Type 1\

~ "' Rnite Element,Type 2~ ------

t

3000

300 0

0

50

.r:;0 100.£:•••OJ 1500c:CIl-'"is 200

250

0

50

.r:; 1000.£:

••• 150OJ0c:CIl-'" 200is

250

Fig. 3-40 Circumferential Moment in vertical wall due to self weight andstored material pressure.

500-1500 -1000 -500, 0Circumferential moment ( Ib - ft / ft )

Finne Element AnalysisType-1, Type-2

8elfwelght

f-' Material pressure -----_.Type-3

I- Setfwelght ............Material pressure --- I

I

IIII

IIII

III,,,,,,,IIII!I

- "l,- .". '.----------- ----------- a..,.--_ ....-

'- 0

....•. ;~

- r:"I:

.--.~.'";J

r

, " , , , , ,

o

250

-500-2000

.250

1250

1500

1750

2000

1000J::C,)

.5

>-'"

750C,)c:C1l-'"is

500

2000

-2500

....././

..'..'.. '

(b) Negative Circumferential Moment

-500 -1000 -1500 -2000

Circumferential moment ( Ib - ft 1ft)

Finite Element, Type 2

Finite Element,Type 3

SOO 1000 1500

Circumferential moment ( Ib - ft 1ft)

Finite Element, Type 1

(a) Positive Circumferential Moment

...............

Finite Element, Type 2

Finite Element,Type 1

Finite Element, Type 3

...-.................::.::::: •...•:::.. ' ...

..........':::.

-500 o

1000

1000

1500

o

-500 o

o

Fig.3-41 Circumferential Moment in vertical wall due to wind pressure.

2000,

2000

1500

.l:<.>c:

>-~c:ttlen 500is

>-~c:-m 500is

.l:<.>,",

. -200

200

Finite Element, Type 2

FInite Element, Type 1

Finite Element, Type 2

Finite Element, Type 1

-150-100

.Circumferential moment ( Ib - ft / ft )

(b) Negative Circumferential Moment

(a) Positive Circumferential Moment

50 100 150

Circumferential moment ( Ib - ft / ft )

-50

,,.,,

_ .... -

,,,

•••,•,•,

•,,•,•

.'

••,••.•

••.•••

,,,.,

.'"

,,,,.•

"

.'r

Fig. 3-42 Circumferential Moment in conical hopper due to wind pressure.

250

300 0

50

o

•••~ 150c:.~(f)

is 200

13 100.S

0

50

.c 100<.>.S

••• 150(])<.>c:«l-(f)is 200

250

300 0

Fig. 3-43 Circumferential Moment in conical hopper due to self weight andstored material pressure.

t .._--- .._-~ ----..- .. - .. ----.-- ---~ _-f" -~ - ,t - -,- .

r ---t \ ,,,

~ ~t ',I"\I Fin~e Element Analysis\ Type-1, Type-2

\ Self weight

, Material pressure ------~ ,

Type-3,

t Setfwelght ............Material pressure ---

~2000o 500 1000 1500

Circumferential moment ( Ib - It / It )

.500300

-1000

0

50

.<:: 1000.s~••• .150Q)0cOl-.!!1 200

250

5000

-2500

4000

-2000

3000

-1500

2000

-1000Hoop Force ( Ib / ft )

Hoop Force ( Ib / ft )

(a) Positive (Tensile) Hoop Force

-SOD

(b) Negative (Compressive) Hoop Force

1000

...... ..................,,::: ::::::::::::: ':',::':::": ........... .

Fig. 3-44 Hoop Force in vertical wall due to wind pressure.

"'= 5" Finite Element, Type 1--Finite Element, Type 2

--- -Finite Element,Type 3

........

/1~~ .

\........,d .................. ....................... ............... ........ . ....................... ................. ....

o

o

-500 o

-500

2000

1500

.s::<.lc: 1000~>-Ql<.lc:Cll 500'li5i5

0

2000Finite Element,Type 1

1500 Finite Element, Type 2

Finite Element, Type 3~.s::<.l.5: 1000>-Ql<.lc:Cll 500-'"i5

.•.

1600

-1600

1400

-1400

1200

-1200

1000

-1000

800

-800

600

-600

Hoop Force ( Ib 1ft)

(a) Positive (Tensile) Hoop Force

400

Hoop Force ( Ib 1ft)

(b) Negative (Compressive) Hoop Force

-400

200

-200

Fig. 3-45 Hoop Force in conical hopper due to wind pressure.

<::....:..- ------ - -', ---.-',

...--- ,• Finite Element, Type 1- -'--- --'- -- Finite Element, Type 2-.---- ----~ - -

"~"

•000

•.0,•

0

~~f ---,

"- --.I-

~I-

._-~ --,,,• Finite Element, Type 1,

,, --.. ----- Finite Element, Type 2,,.

')- ---,,

••• /,,

/••1-' ,,• V1-'

l1-'•I- •I- ~

c- 00

~C- oI-

,,I-

,,- - --- .-,I-~l-I-

300 0

300 0

0

50

.r::. 100t.l

-'=••• 150'"t.lc:'"-lJ)is 200

250

0

50

~ 100.r::.'-'.!:

••• 150'"t.lc:'"-lJ) 200is

250

may not be possible. Because a high speed digital computer and a suitablecomputer program may not always be available. Hence an attempt will be made todevelop a simple but rational analysis procedure based on this extensive study forquick calculation of forces and moments required for design. Again there are anumber of parameters (geometric parameters, material properties) affecting thevalues of both forces and moments. For this purpose it is necessary to examine theeffect of various parameters on the overall behaviour of silo. Therefore aparametric study applying Finite Element approach is made. Results of this study ispresented in the following chapter.

***

52

CHAPTER 4

PARAMETRIC STUDY

4.1 INTRODUCTION

In the previous chapter the conventional and Finite Element methods are

applied for the analysis of a model silo and a comparative study is made. From this

study some drawbacks of the conventional approach is revealed. On the other hand

the potentials and melits of Finite Element analysis have become apparent.

However, the analysis and study of a model silo is not enough for the

understanding of the overall behaviour of silo. The development of a design

rationale for circular silos requires an extensive study of the effect of variations of

parameters on different stress resultants. )I

4.2 SILO PARAMETERS

Behaviour of silos can be influenced by a number of parameters, which

include geometric dimensions of silo, the properties of the stored materials and

wind velocity. These are listed below along with their ranges of study.

Table 4-1. Geometric parameters (Fig. 3-7) of a Silo

Name oflhe Parameters Range

Height of vertical wall, H 40 ft. 10 280 ft.

Internal diameter of silo, D 10 ft. to 100 ft.

Thickness of vertical wall at bottom, Tbn"om 6 inch to 13 inch

Thickness of conical hopper at top, ftop 6 inch to 13 inch

Depth of ring beam, d 24 inch to 96 inch

Inclination of conical hopper with horizontal, a . 40° to 75°

Height.ofhopper bottom (opening) above floor level, h' 8 ft. to 25 ft.

Table 4-2. Material Properties

Nameof the Parameter Range

Unitweight,r 35 lb/cft. to 160Ib/cft.

Angleof internalfiiction,p 150to 500

Coefficientofwallfiiction,JI 0.2 to 0.7

Other parameter

Effect of variations of wind pressure on stress resultants are also studied.The range of wind pressure on silo was 9.2 psf to 103 psf which are equivalent to awind velocity of 60 to 200 miles per hour.

4.3 STRESS RESULTANTS IN SILOS

Conventionally designing a silo requires only two forces - meridional forceand hoop force, for various components of the structure. But Finite Elementanalysis has revealed that there may be considerable meridional andcircumferential moments under different loading conditions. So in this study fourstress resultants, shown below, are considered:

a) Meridional force, N<p

b) Hoop force, Noc) Meridional moment, M<pd) Circumferential moment,Mo

Since silo is an elevated structure the value of above stress resultants varyalong the height for the vertical wall or conical hopper considerably. To arrive at aneconomic design these variations must be considered. Evidently the value of a forceor moment at any location is a function of its maXimum value. In this chapter,therefore, only the sensitivity of maximum values for various functions with respectto various parameters are shown.

In this study, as mentioned in Chapter-3, four loading cases are considered.Parametric study, however, is made on three load cases - self weight, storedmaterial pressure and wind pressure. Temperature effects have already beendiscussed in Chapter-3 in details. According to code of practice (ACI) temperaturesteel is provided in addition to what is required from the analysis for other loads.

54

In the design of any concrete structure it is mandatory to know the criticallocations where maximum values of forces and moments occur. As for example, arational design for moment requires both the maximum value of moment and itslocation with reference to a fixed point. In this study, as mentioned earlier, thevariation of maximum values of various stress resultants with parameters arepresented graphically. The location for these maximum values may change alsowith the change of various parameters. A comprehensive study also has been madeon the location of maximum forces and moments. These are not presentedgraphically but discussed in the subsequent article in details for each of theparameter separately. For the vertical wall these locations are expressed by thedistances from the bottom of vertical wall. For conical hopper the similar approachis followed and the distances are measured from the junction of ring beam andconical hopper.

In Fig. 4-1 to Fig. 4-10 the effect of variation of height of vertical wall, H onvarious stress resultants are shown. The variation of H from 40 ft. to 280 ft. isshown on the horizontal axis of each figure. The forces and moments are plotted asordinates. In a similar way the effects of other parameters are shown in Fig. 4-11 toFig. 4-72. The results are discussed in details in the following section.

4.4 EFFECTS OF VARlATiONOFPARAMETERS

4.4.1 Effect of Height of Vertical Wall, H

Fig. 4-1 through Fig. 4-10 show the effect of variation ofH on various stressresultants (maximum values) for various parts of silo.

(a) Vertical wall

(i) Meridional force: For all of the Type-I, Type-2 and Type-3 maximummeridional forces in the vertical wall due to stored material pressure vary linearlywith changes of H (Fig. 4.Ia and Fig. 4.2a). Maximum meridional force due to selfweight also shows linear variations. In this case the meridional forces are alwayscompressive. Due to wind load both tensile and compressive meridional forcesdevelop and the variations are not linear. The higher the vertical wall, more is therate of increase in meridional forces. For all the silo types and for all the load casesthese maximum values occur at the bottom of veltical wall.

55

iI _'-"

- ---.-----r-_. .-..-.-._- .--

I-->--_.-

~~'!:--~..~..~~.::.._..r-~---L----+ I............. .................. _ .._ ..- ..- ..1.

~- - ..............I....~.:- ..'. - 0

0

~

0o •

1

....0

00

0

~......... ". Il- ';.

~ I....

I-- - - ----- -~-,--I

7.~. ,=- - - -"r"' --.._ .. .. .. .. .. .. .. f' .._ .._ .... _ .. ..-

300

300

250

250200150

( a ) Vertical Wall

( b ) Conical Hopper

100

100 150 200Height of Vertical Wall, H ( It )

Height of Vertical Wall, H ( It )

Self Weight Material Pressure Wind PressurePositive -- Positive - -- Positive -----Negative Negative ----- Negative

_ .._ ..-

50

Self Weight Material Pressure Wind PressurePositive -- Posrtive -- - Positive _._ ..Negative ......... Negative ----- Negative _ .._ ..-

50

Fig. 4-1 Change in maximum Meridional Force due to change in H(Type-1, Type-2)

a

::. 25000.c--.; 20000uo 15000LL

coc 10000o'0.~ 5000::2

-5000o

a

35000

60000

40000

-80000

30000

-100000 a

""~ 20000.c

Q)uoLL -20000co5 -40000'0.;:Q) -60000::2

---'--------'---- f--

--_.----- -_._.-'--.._ .._ .._ ..- ._ .._ .._ ..-..- - ....~..:~..:~::~:..- - - .... -. -- 1--.__ ._---... . -- "":":', ..•..._ .. :.•.•.:-.....--. ....-.--.- .- -...-.- - ..-.

f-

Fig. 4-2 Change in maximum Meridional Force due to change in H(Type-3)

300

300

250

250

( b ) Conical Hopper

-( a ) Vertical Wall

100 150 200Height of Vertical Wall, H (It )

100 150 200Height of Vertical Wall, H ( It )

Self Weight Material PressurePositive PositiveNegative . . . . . . . . . Negative

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._--Negative ......... Negative ----- Negative _ .._.'-

~

- ---------/

o

5000

o o

~ 20000olLOJ 15000c:o:g 10000'":2

-100000o

35000

40000

<l:'£ 25000

30000

20000

<l:'--f1~ -20000olLOJ -40000c:o:g -60000'":2

-80000

Fig. 4-5a shows the variation of meridional forces due to change in H for allthe load cases for Type-3 below pressure zone. The trend of variation is similar tothe variation in pressure zone as explained above.

(ii) Hoop Force: Conventional method can not predict any negative hoopforce for stored material in vertical wall. But Fig. 4-3a shows that in Type-l andType-2 considerable negative hoop force develop and increases nearly linearly withthe increase of H. Positive hoop force, due to grain develop forall types and thevalues are much greater than those of other loadings. For pressure zone in all thetypes the positive hoop force due to grain is the most predominant. It increases .initially parabolically with decreasing rate up to a: value of H about 165.0 ft.Beyond this the hoop force in pressure zone due to material pressure remainsconstant with the increase ofH for all types.

For Type-3 there exists no negative hoop force in the pressure zone due tomaterial pressure. For other loads such as self weight and wind load some hoopforces develop. But these are very small in comparison to that due to materials(Fig. 4-3a and Fig. 4-4a). Below the pressure zone in Type-3, (Fig. 4.5b)considerable hoop forces of both signs exist due to wind load and materialpressure. For material pressure the trend of variation is similar to that of verticalwall in pressure zone. But due to wind load there exist considerable positive andnegative hoop force below pressure zone ofType-3 .

For all the types and for all the load cases the maximum negative hoopforces occur at the bottom of vertical wall. In Type-l and Type-2 the location ofmaximum positive hoop force is more or less fixed and it occurs at a distance of7.75 ft. to 8.5 ft. from the bottom of vertical wall. For Type-3 the maximumpositive hoop force exists at the bottom of pressure zone (top of ring beam).

(iii) Meridional Moment: Considerable positive and negative meridionalmoments occur in Type-l and Type-2 due to material pressure (Fig. 4-6a). In thiscase maximum positive and negative meridional moment may be as high as 1835lb-ft./ft. and 7058 lb-ft./ft. respectively. For other loads meridional moments are sosmall that their effect may be neglected in design. For Type-3 negligible negativemeridional moment develop in the pressure zone due to material pressure (Fig. 4-7a) and the highest positive meridional moment is only about 750. lb-ft./ft. Invertical wall below pressure zone in Type-3 the negative meridional moment at thebottom of vertical wall (foundation level) due to material pressure is considerable

56

------------- -------.••..••."

------_.-- _._.-._-- _._-_.._'.. '._ .. .. .._ .._ .._ .. .. .. .. .. .. .. "-"-"- ..-.. -"- ..-......... .... .. ... ... ..... ....... ... .. ....". .. ...... .... " ... .. ....

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

~~~ - - --~ - ..---- f-- - ----~ --~ , .••.~ .••.~~~~I-

- .-_ .._ .._ .. .._ .._. .. .. .. .. .. .. .. .. .._.' .. .._ .. .. ..-

300

300

250

250

( a ) Vertical Wall

100 150 200Height of Vertical Wall, H (It )

100 150 200Height of Vertical Wall, H (It )

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ ..-..-

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative

_ .._ ..-

( b ) Conical Hopper

Fig. 4-3 Change in maximum Hoop Force due to change in H(Type-1, Type-2)

-10000o

o

30000

20000

-30000o

40000

30000~

""--,Q 20000

'"(J~0~ 100000-00I

0

~o~ -100000-ooI -20000

10000~

f"'.- ...

300

300

250

250

( b ) Conical Hopper

( a ) Vertical Wall

100 150 200Height of Vertical Wall, H (It )

100 150 200Height of Vertical Wall, H (It )Se~Weight Meteriel PressurePositive PositiveNegative . . . . . . . . . Negative

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Fig. 4-4 Change in maximum Hoop Force due to change in H (Type-3)

-- ---- -- --------~"'"

'"

------ -- ------"-,.-"'"

'"'"

//

/

- ------- - ~- --_ ..f- .._ .._ .._ ..- ._ .._ .._ .._ .. .._ .. .. .' ..>-.. .. .. .. ..-

50000

00

20000

40000

25000

-50000

o

'"~~ 20000a.aaI 10000

f1 30000

:::.15000f1

~ 10000au..a.g 5000I

300

300

250

250200150100

100 150 200

Height of Vertical Wall, H ( ft )

Height of Vertical Wall, H (ft ) _

( b ) Vertical Wall below pressure zone

( a )Vertical Wall below pressure zone

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive -_. Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Fig_ 4-5 Change in maximum Meridional Force and Hoop Force due to

change in H (Type-3)

- - -- -- _.-'-- - _.-.-'-'-1-- _.-.--'1-'-'-'-'- _.-._._'-~.,..f.', _::::--:~~-------------- -..-.._ .._ ..-................... .... .... .... .-:::..~..-..-..---. --... ....,_ ..--.-.-... ..- - ..-. .......- .-....

.

~ -'-~ -,-'-'-'- _.-.-.-'

.-_._' .-'-'~.•..1"«'::;-:-:-:""-:::-:: -------------- _ .._ ..-.._ ..-.................... --- -..-..- - .... --- --- .........--:..:':':" .

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o

5000

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-80000

300

300

250

250

( a ) Vertical Wall

100 150 200Height of Vertical Wall, H (ft )

100 150 200

Height of Vertical Wall, H (ft )

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _.---Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..•.

50

50

( b ) Conical Hopper

Fig. 4-6 Change in maximum Meridional Moment due to change in H(Type-1, Type-2)

-- 1--- --- -- - - - -- - - - 1--- -- -- - ---- - '-- ----------- 1----------- ----------- --:

.. ,::-::::.:::.::::: ::::.- .:::':'::;;::':"::.:.:.::. ...::.:. ...•.•..._ .._. _ •••••••• I'T'O ••••••••••••••••• ~......•.._ ..::

: - -- --.: -..'.~ -..-.--.: - .. _- --- -. -- .. _-: - .._- ". -. -~~

,..- -- - I-

- ,..---- ,..--------- ._____ ._0 ---------- f-------1-----------~,•.•...:.;.:.::.:.:.::.:.:.: ..j:-: ••••~ •••-- •••••••••••••.••••• " .•.•.•,_.,_ ...-•... .•.••.•...•.•.•..••.•......•.•....... ".,",':".:" '.:.

- -- - - - - .-- -. - - .- .- -- -.. .- ..- .- - - - --..- - . ..- - - --I-

-6000 0

-10000o

2000

4000

-= 2000---= 0,Q- -2000c::Q)

E0E -4000(ijc::0 -6000'0-;::Q)

~ -8000

=- 1000---= 0,,Q~-1000_.c::Q)E -2000o.!: -3000'"c::_2 -400032(;:;~ -5000

-- -- - -- - - - - -- ----..-..-..-..- f-' .-_.----.....-.::~

- - - - - - - - - _. - - - - - - - - - .- - - - - - - - - - -- - - - - - - - - - .-~~

,'.

.....•........-.."-"-" " ,. .. .._ .. '. .. "-"--'-"-" .._'._ ..

300

300

250

250200150100

( b ) Conical Hopper

( a ) Vertical Wall

100 150 200Height of Vertical Wall, H ( It )Se~Weight MaterialPressurePositive PositiveNegative . . . . . . . . . Negative

Height of Vertical Wall, H ( It )

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._0-Negative ......... Negative ---_. Negative _ .._ ..-

Fig. 4-7 Change in maximum Meridional Moment due to change in H(Type-3)

~---- -- --- -- -------..-~-- .-

- - --------- --------- --.------ --------- ----_.

~~

1000

-2000 a

8000

.600 a

~ 6000

*',f!':;:::'"4000cQ)

EoE 20001i'ico'5.~ 0:2

'EQ)

E 200oE1i'i ac~ .200'CQ)

:2 -400

*' 600,

~ 800

*'.0:::. 400

and may be as high as 2790 lb-ft./ft. forH = 280 ft. (Fig. 4-l0a). Due to wind loadmaximum positive meridional moment of2424Ib-ft./ft. may develop in the verticalwall below pressure zone. Both positive and negative meridional moments varies ina way similar to meridional force or hoop force variations. For other cases notmentioned above negligible moments may exist. Maximum negative meridionalmoment always occur near the restraint i.e at the bottom of vertical wall. Maximumpositive meridional moment for Type-l and Type-2 occur at a distance of 4.0 ft. to4.5 ft. from the bottom of vertical walL In Type-3 below pressure zone bothmaximum positive and negative meridional moment due to wind load occur at thefoundation leveL The positive and negative moments occur at differentcircumferential locations.

(iv) Circumferential moment: Maximum circumferential moment in verticalwall is predominant due to wind load for all types and among all load cases(Fig.4.8a,Fig. 4-9a and Fig. 4-l0b). Considerable negative circumferential momentmay also be developed due to material pressure (Fig. 4-8a) in Type-l and Type-2.For other load cases circumferential moments are negligible for design. For windload, positive and negative circumferential moment may be as high as 1893 lb-ft./ft.and 2203 lb-ft./ft. respectively for Type-l and Type-2. For Type-3 these values are1890.0 lb-ft./ft. and 2196.0 lb-ft./ft. as shown in Fig. 4-9a. The variations ofcircumferential moments due to wind pressure in all the cases follow the samepattern initially increasing and then remaining constant with change of H above150 ft. Due to material pressure, negative circumferential moment developed inType-l and Type-2, remain more or less constant. It varies from -1127 lb-ft./ft. to-1406 lb-ft./ft. for H varying from 60 ft. to 280 ft. (Fig. 4.8a). Maximumcircumferential moments due to wind load in pressure wne occur within top 15%to 30% of the vertical wall depending on the value of H. Maximum negativecircumferential moment due to stored material pressure occur at the bottom ofvertical wall for Type-l and Type-2. In Type-3 and below pressure zone themaximum positive and negative circumferential moments occur at the level of topof ring beam and they remain constant when H varies from 70.0 ft. to 280.0 ft.(Fig.4.8b). Here the maximum positive and negative circumferential moments are 844.0lb-ft./ft. and 958Ib.ft./ft. respectively.

(b) Conical Hopper

(i)Meridional force: Variations of H affects only the maximum meridionalforce due to stored material pressure as shown in Fig. 4-1b and 4-2b. The patterns

57

~~~ -" ._' _. - _. -_.-'-'--'.-_/

•.•..."•.•...-- -- - -- --- ~-- -- -...,-- -- -- -

.... ................... ................... ................... .................. ...........

..•.....•...- - - ...•.•...- - - - -,",,:: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

.•.••.•..••.•.."- ._ .. .. " "- ._ .._ .._ .._ .._. _ .._ .._.

1--- - ------- ..- .-' --' -

...... ........ .......... .. ..... .... ...... ..... .. ..... .... ..... .... .. .....

-" .._ .. .. ,,- " " ..-.' .._ .._ .._ .._ .. .._ .._ .._ .._ . _ .._ ..-

- - --- - - - - - - - - - -- - - -= - - - - - - - - - - - - -~

- - - - - - -~

-

300

300

250

250

( b ) Conical Hopper

( a ) Vertical Wall

100 150 200

Height of Vertical Wall, H (f1 )

100 150 200

Height of Vertical Wall, H ( fI )

Se~Weight Material Pressure Wind PressurePos_ -- Positive --- Pes_ _._--Negative ......... Negative ----- Negative _ .._ ..-

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Fig.4-8 Change in maximum Circumferential Moment due to changein H (Type-1, Type-2)

a

-1000a

600~<I:' 400--<I:'

f1 200

~- acQ)

E0 -200EOi -400:;::cQ)~Q) -600-E:Ju -BOO~G

3000

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EoEOi:;::as -1000

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<I:'-- 2000<I:'

eI-

~--.-'~.-'-'-- - ~- - ,..- -

-', ..,,0

~

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

•.........•.~.

••..................._ ..-.. "- ._.. "-"-"- ._.._ .._ .._ ..- ,,-"-"-

- ---- ---,..-

-

~-- --------- e--- ______--------- --------- -----

300

300

250

250

( b ) Conical Hopper

( a ) Vertical Wall

100 150 200

Height of Vertical Wall, H (ft )

SenWeigh! Material PressurePositive PositiveNegative . . . . • . . • . Negative

100 150 200Height of Vertical Wall, H ( ft)

50

SenWeight Material Pressure Wind PressurePositive -- Positive --- Positive _._ ..Negative. ......... Negative ----- Negative _ .._ ..•

50

Fig.4-9 Change in maximum Circumferential Moment due to changein H (Type-3)

2500

"" 2000-""f1 1500-c:'"E 10000E0;:;:; 500c:'"~'"~E 0:::J~G

-500o

3000

""- 2000""f1~ 10001::'"Eo 0E0;E -1000'"$!5 -2000~G

-3000 o

300

300250

250

200150100Height of Vertical Wall, H ( ft )

100 150 200Height of Vertical Wall, H ( ft)

( b ) Vertical Wall below pressure zone

( a ) Vertical Wall below pressure zone

50

50

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative .......... Negative ----- Negative

_ .._ ...

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative

_ .._ ..•

Fig.4-10 Change in maximum Meridional and Circumferential Momentdue to change in H (Type-3)

_.-'-'-._' "-'-,-,-, .. . ._. .-

-- -- -"';..' -- . ........ ... ..........-. .. .... .. .... ........- .. ..... ..... ......... ..... .. .. ..- ... .. .. ......, - -.- .,...-. - - - - -

~ '" 1-.•. - •• _ .• .._ ..-.'-"-" .._ .._ .._ .._ .. ,,-"-".._ .. .. .._ ..-.._ ..

E -_.-.- .-.-'-~ .---'-'---'---,-,-'-' ~-_.--_.-.

-- - - --- ---f-. --- -"':': ':"':'.::.:'" ........ ................... 1--•• -._,,~".,.,; ••7

.. .._ ..~.~.:.~:::.:-..-..-.._ .. :'.":::'.7: '::-:-"".,._ ..•..•.........- " .. "-- .. .. .. .. .. .. ,-, .., , - .. ,-'- .. - , - - .. - .. -. -

-10000

-30000

1000

"" 750--"",Q 500

- 250cQ)

E0 0:2:(ij .250""cQ)~ -500Q)-E:> -750u~0

3000

""2000--

"",Q 1000~-cQ) 0E0:2:(ij -1000c0'5'C -2000Q)

:2:

of variation for Type-I, Type-2 and Type-3 are the same. Initially the slope of thecurve is relatively steep. As the depth of the vertical wall increases the rate ofincrease of meridional force in the conical hopper due to material pressuredecreases and at H = 200 ft. and above the variation becomes asymptotic to a valueof 30000 lb/ft. while other variables of the model remained fIxed. Meridionalforces in conical hopper due to other loads are negligibly small. Due to self weightthe meridional force is only about 4.5% of that of material pressure. For all theabove mentioned cases maximum meridional forces occur at the top point ofconical hopper i.e. at the junction of ring beam and conical hopper.

(ii) Hoop Force: Maximum hoop force in conical hopper due to materialpressure also varies in a way similar to that of meridional force as mentioned inprevious paragraph (Fig.4-3b and Fig. 4-4b). In conventional method the maximumhoop force occurs at the top point of conical hopper but in Finite Element analysisit occurs at a distance of 6 ft. to 6.75 ft. from the junction of conical hopper and

ring beam.

(iii) Meridional Moment: Meridional moment of both signs in conicalhopper due to material pressure are predominant (Fig.4.6b). The maximum positiveand negative meridional moment at H = 180.0 ft. may be as high as 1510 lb-ft./ft.and 4810 lb-ft./ft. respectively for Type-l and Type-2. In this case the positivemeridional moment is almost independent of the variation of H, but negativemeridional moment varies considerably with the change of H. Initial variation isparabolic and beyond a depth ofH of about 125.0 ft. the variation becomes almostlinear. For Type-l and Type-2 positive meridional moment due to wind load isrelatively higher and the highest value of this is about 925 lb-ft./ft. for H = 200 ft.In this case the variation is approximately linear. In Type-3 only the maximumpositive meridional moment is considerable (Fig. 4.Th). It increases with a higherrate for lower value of H and the rate of increase decreases gradually as the heightof vertical wall increases and fInally it becomes independent of the variation of H.For other load cases meridional moments remain constant at a negligible value. InType-l and Type-2 the maximum positive meridional moment occur at a distanceof 3 ft. to 4 ft. from the junction of conical hopper and ring beam and the negativemeridional moment exists at the junction. For Type-3 the maximum positivemeridional moment develop at the junction of ring beam and conical hopper.

(iv) Circumferential moment: Circumferential moments in conical hopperare very small for all the load cases except due to material pressure. For Type-l

58

and Type-2 maximmn negative circumferential moment due to material pressure isonly worth consideration (Fig. 4.8b). Initially it increases with decreasing rate andabove a height of 150 ft. its variation follows a straight line pattern. For Type-3only the positive circumferential moment due to stored material pressure isconsiderable (Fig.4.lOb). For the model data and at H of 280 ft. the maximmncircumferential moment is 1967 Ib-ft./ft.. It increase with a decreasing rate as Hincreases and finally becomes asymptotic to a horizontal line. For all the cases themaximmn circumferential moments occur at the junction of ring beam and conicalhopper.

4.4.2 Effect of Internal Diameter of Silo,D

(a) Vertical wall

(i)Meridional force: Variation of internal diameter of silo has considerableinfluence on the maximmn meridional force in the vertical wall due to materialpressure and wind load for Type-I, Type-2 and Type-3. For all the types thepattern of variations are same (Fig. 4-11a, Fig. 4-12a and Fig. 4-15a). Maximmnnegative meridional force due to stored material pressure increases with decreasingrate and when the diameter of the silo exceeds about 45ft., it increases linearly.Both positive and negative meridional force due to wind load initially decreaseswith the increase of D. When D exceeds about 27.5 ft. then the maximmnmeridional force increases in a decreasing rate and finally it becomes more or lessconstant with the variation of D. From this study it is found that the maximmnmeridional force whether negative or positive occur at the bottom of vertical wall.

(ii) Hoop Force: Variation of Diameter D has significant influence on thepositive and negative hoop forces due to material pressure for Type-I and Type-2(Fig. 2.13 a). In this case the negative hoop force increases at a faster rate than thepositive hoop force. For Type-3 there exists only positive hoop force in the verticalwall due to material pressure and the variation is similar to that in Type-I. Forother load cases, not mentioned above, maximmn hoop forces are negligibly smallin comparison to that of material pressure.

Positive hoop force below pressure zone in Type-3 due to material pressureshow similar variation as that in pressure zone but in this case the rate of increase issimilar. Variation of negative hoop force below pressure zone is nearly linear.Considerable positive hoop force develop below pressure zone due to wind load(Fig. 4.15b).

59

._,-'-'-'-'- .-'-'-'-'-'--'---.- .------.-'-"

.. ..-.,_ ..-..-..- :;':" ...................~.• ..' ... ... .. ' . .. .....' .. ... ... .. ' . .. ........ .. ..' .. ..".. ..,..... - "-"-- .-.._._-.- - -. _ .._ .._ .._ ..-- - .. .. .._ .. ..- - - - - - --,- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

100

100

80

80

( b ) Conical Hopper

( a) Vertical Wall

40 60Internal diameter of silo, D ( ft )

40 60Internal diameter of silo, D ( ft )

20

20

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-

Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._--Negative ......... Negative ----- Negative _ .._ ..-

Fig. 4-11 Change in maximum Meridional Force due to change in D

(Type-1, Type-2)

//

/

~/

"/

~ "~ ""~ . ""t ""I- ..-t ..-..-I- ..-~ - -~ ----- .. .._ .._ .. .. .._", .. .. .. ..

50000

100000

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a

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Cll 150000"o

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100

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80

80

( b ) Conical Hopper

40 60Internal diameter of silo, 0 ( It )

Se~Weight Material PressurePositive PositiveNegative _.. Negative

40 60Internal diameter of silo, 0 ( It)

( a ) Vertical Wall, pressure zone

20

20

Self Weight Material Pressure Wind Pressure .

Positive -- Positive --- Positive _._'~Negative ......... Negative ----- Negative

_ .._ ..-

Fig. 4-12 Change in Maximum Meridional Force due to change in 0

(Type-3).

//

,,/,;

,;

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-----. -'- --: -I- --c ----- -- .--- --, .•......•- --,- -'.--. , , .- '...,..!=

....~~

100

100

80

80

( a ) Vertical Wall

( b ) Conical Hopper

40 60Internal diameter of silo, D ( It )

40 60Internal diameter of silo, D (It )

20

20

Se~Weight Material Pressure Wind PressurePositive -- Posjtive --- Positive _._--Negative ......... Negative ----- Negative _ .._.'-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Fig. 4-13 Change in maximum Hoop Force due to change in D(Type-1, Type-2)

-------- -- .- --- --- --- ---- - - - - - - .. .. .... ......... ..........., ,

" , - - - .'- ...... .........-.. ........ .....

400000

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100000

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~~ 0a-g -100000I

=- 200000

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~ 0

100

100

80

80

( b ) Conical Hopper

40 60Internal diameter of silo, 0 ( ft )

40 60Internal diameter of silo, 0 ( ft )

SelfWeight MaterialPressurePositive PositiveNegative Negative

( a ) Vertical Wall, pressure zone

20

20

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive

_._ ..Negative ......... Negative ---- . Negative _ .._.'.

Fig. 4.14 Change in maximum Hoop Force due to change in 0 (Type-3)

~ --'--"~

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o

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40 60Internal diameter of silo, D ( ft )

( b ) Vertical Wall below pressure zone

( a ) Vertical Wall below pressure zone

20 40 60Internal diameter of silo, D ( ft )

20

Self Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._ ..Negative ......... NegatIve ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

".~.~ --.-"'-._.-._'

::.>< -,~:::,::::-:..::-::.::-::..,-"<:..: ..~..~.~. .. .. . .•..•.~...•... - .-..-..-..

Fig. 4-15 Change in maximum Meridional Force and Hoop Force due

to change in D (Type-3)

ccf- "f- /"r- ""r- ~"r- "r- "..-f- ..-f- ..-~..-f- ..-f- --f- ..- -.-.-.-.-'-r- - ..- _.-.-.--_.-r- - _.-._-.--.--'-r- -.-.;::::- c.:::._._.-'---I- ::'.~i:"'::'_ ..~.::-:-..:....:::-:-: ...:-.... -'-""":::"::::"::::"::. ...................... ......................I- - .... _--- .. _-- ..I- , - - ..- .._ .._ .._ .._ ..

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100000

~J: 0CIlCo'6'C'" -50000:2 -

Maximum negative hoop force due to stored material pressure for Type-land Type-2 occur at the bottom of the vertical wall and maximum positive hoopforce occur at a distance of 7ft. to 15ft. above the bottom. For Type-3 themaximum positive hoop force in pressure zone occur within a distance of 3 ft. to11.5 ft. from the bottom of pressure zone depending on the value of diameter.Maximum positive hoop force below pressure zone in Type-3 occurs at the top ofthat portion (top of ring beam) and negative value occurs at the bottom of verticalwall.

(iii) Meridional Moment: Both positive and negative meridional bendingmoments are predominant for material pressure for all types in pressure zone andshow a sharp sensitivity with the changes of diameter (Fig 4.16a). Positivemeridional moment due to material pressure increases with a increasing rate and itmay be as high as 22160 Ib-ft./ft. when D is 100 ft.. Maximum negative meridionalmoment also follow the same pattern of variation but the rate of increase is muchgreater and the maximum value may reach 127000 Ib-ft./ft. for a Diameter of 100ft. For Type-3 the maximum positive meridional moment increases almost linearly(Fig. 4-17a). In this case relatively greater meridional moment of both signs occurdue to wind load and it becomes constant with the changes of D when D exceedsabout 50 ft. Below pressure zone of Type-3 a relatively higher meridional momentof both signs develop due to wind load and stored material pressure.

Maximum negative meridional moment always occurs at the bottom ofvertical wall. But the location of positive meridional moment is sensitive to thechange of diameter D. For Type-1 and Type-2 this occurs within a distance of 2 ft.to 8 ft. for the range of diameter considered. This distance vary from 1 ft. to 3 ft.for Type-3 from the bottom of pressure zone.

(iv) Circumferential moment: Negative circumferential moment due tomaterial pressure is sensitive to the change of diameter for all the range consideredfor Type-1 and Type-2 (Fig. 4-18a) and it increases with an increasing rate. In thiscase considerable maximum circumferential moment of both sign develop due towind load. For Type-3 considerable circumferential moment of both sign occuronly due to wind load (Fig. 4-19a). In this case both positive and negativecircumferential moment becomes approximately constant when diameter exceedsabout 50.0 ft. For diameter below 50.0 ft. the variation can be approximated aslinear. Circumferential moment below pressure zone is predominant only for wind

60

f-f-

---- --- --- --- ------ - - - - ., __ .0_ .. .- .. -- - - - - - -. .......-... ........... .. ......... ....

100

100

80

80

( b ) Conical Hopper

( a ) Vertical Wall

40 60Internal diameter of silo, D ( ft )

40 60Internal diameter of silo, D ( ft )

20

20

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-

Negative ......... Negative ----- Negative _ .._ .. -

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._0-Negative ......... Negative ----- Negative _ .._ ..-

Fig. 4-16 Change in maximum Meridional Moment due to change in D

(Type-1, Type-2)

-- ------- --- --- - --- ....---

f- -'- -'. '- ..~

.' ....~

....~

..'...

~...

~....

~ ....

~....

~...

~~

40000

'*' 20000--'*'f1 a-c'"E -200000:2(ij -40000c.2"0'C

'" -60000:2

-800000

-1500000

25000

50000

f1 -25000"E~ -50000o:2 -75000(ijcoii -100000

'C'":2 -125000

Fig. 4-17 Change in maximum Meridional Moment due to changein D (Type-3)

100

100

80

( b ) Conical Hopper

40 60 80Internal diameter of silo, D ( It )

40 60Internal diameter of silo, D ( It )

Se~Weight Material PressurePositive PositiveNegative . . . . . . . . . Negative

( a ) Vertical Wall, pressure zone

20

SelfWeight Material Pressure Wind PressurePositive -- Positive -- - Positive _._--Negative ......... Negative ----- Negative _ .._ ..-

20

c~~E ,t ,/

C ,/

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~ ,/

~ ,/

t ,/

C •...~ •...~

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- - _.-,_.--_.--- _.-.-'::::-_.:::::. ~.-'--_ .._ .._~.~~.~~.~- - - - - - - - - - - - - - - - - - - - - - - _C - - - - - - - --"-"-" - - -

'. "- .._ .. .. ..

200000

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::;;

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_._.-.-.-.---- ------------ .=.=.=.~._.~-::::_=-=-:::_::---- - -

'.:"':'.":'.::"7.:..,:,.::..,,:,.:..~._ ..••

'. ....-.._ .._ .._ .._ .. .._ .._ .._ .._ .._ ..

";':-..,:':;-"-"-"-"- - - - - - - - - - - -...- - - -. .-..-.- ...- -.100

100

80

80

( b ) Conical Hopper

( a ) Vertical Wall

40 60Internal diameter of silo, D ( It )

40 60Internal diameter of silo, D ( It )

20

20

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative

_ .._ ..•

Se~Weight Material Pressure Wind PressurePositive -- Positive --- PoSitive --_.-Negative ......... Negative ----- Negative

_ .._ ...

Fig. 4-18 Change in Maximum Circumferential Moment due to changein D (Type-1, Type-2)

--- - ------ - --I-- --- - - --- ---_.-- - - - - -.- ',_ .. .. .._ .._ .. .. .. .' '. .. .. ";":::"::::":.: .. '.::':':::':'::~'~ .- -. - -..--.

...- - - - - - - - - -... - ---..- ..-- - -~-15000 a

cCD aEo::;;;]i -5000cCD-~E .10000::J(J-G

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$ -20000E::Je -25000G

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f1

!:. 5000;;::

f1 a

100

10080

80

( b ) Conical Hopper

40 60Internal diameter of silo, D (ft )

40 60Internal diameter of silo, D ( ft )

Se~Weight Material PressurePositive PositiveNegative Negative

( a ) Vertical Wall, pressure zone

20

20

Sel/Weight Material Pressure Wind PressurePositive -- Positive - -- Positive _0_--Negative ......... Negative ----- Negative _ .._ ...

Fig. 4-19 Change in maximum Circumferential Moment due to changein D (Type-3)

/- /

"- "E ",,'

" .

"- ""~/

: ~~

~ - I--~--- - - ------~----- ---------- -~

- - -----------~

~.-._-_.-.-.

~.-.-.- .•••.....

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..•.....••....'. ,

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load (Fig. 4-20b). The pattem of variation is similar to that of pressure zone of

Type-3.

In Type-1 and Type-2 maximum positive circumferential moment occurs

within a distance of about 3 ft. to 8 ft. from the bottom of vertical wall and

maximum negative circumferential moment develop at the bottom of vertical wall

for material pressure. Circumferential moments of both signs due to wind load for

all types occur at the top of the vertical wall.

(b) Conical Hopper

(i) Meridional force: Meridional force in conical hopper due to material

pressure only is predominant among those of all other loads cases for all types of

silos (Fig. 4-11 b, Fig. 4-12b). For all types the maximum meridional force

increases with a increasing rate upto a diameter of about 50 ft. and beyond that the

variation follows a linear path.

For all of the three types the location of maximum meridional force vary

with the change of diameter. In Type-1 and Type-2 the maximum meridional force

occur within a distance of 0 to 7 ft. from the junction of ring beam and conical

hopper. For Type-3 this range is 2 ft. to 8 ft ..

(ii) Hoop Force: Maximum positive and negative hoop force due to stored

material pressure is the most predominant in all types of silos. Negative hoop force

exist only when the diameter exceeds about 50 ft. for Type-1 and Type-2 (Fig. 4-

13b) and 55 ft. for Type-3 (Fig. 4-14b). In each type maximum hoop force due to

material pressure increases initially with increasing rate and when D exceeds 40 ft.

it varies linearly.

Maximum positive hoop force occurs within a distance of 2 ft. to 15.5 ft.

from the top of the conical hopper (Junction of hopper and ring beam) for Type-I

and Type-2. For Type-3 this range is about 2 ft. to 5.5 ft .. Maximum negative hoop

force always occurs at the junction of ring beam with conical hopper.

(iii) Meridional Moment: Maximum meridional moment in conical hopper

of both negative and positive signs are predominant due to material pressure for

Type-land Type-2 (Fig. 4-16b). For Type-3, only positive meridional moment due

to material pressure is predominant and increases with an increasing rate (Fig. 4-

17b).

61

100

100

80

8040 60

Internal diameter of silo, D (ft )

( b ) Vertical Wall below pressure zone

( a ) Vertical Wall below pressure zone

20 40 60

Internal diameter of silo, D (ft )

20

Self Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

Fig. 4-20 Change in Maximum Meridional and Circumferential Momentdue to change in D (Type-3)

~ ---------~ -'-'---- _.-._---_.-~ .1-'--

E-'_.-'- ---'--

E--"':'::,::,: ...................... ....................... ....................... ......................-.,- ..-.. ..- - - - - - - - - - - - - - - - - - - - - - - - - - - - -"- '- - - - - - -"- - - - - -

~"- .. ..-.._ ..-.._ ..-

"-. -..-"- --._ .. ..

_.-.-",-,-'-'-'- .-.--

---'-'-'••...-.'-- _._.-.-'

----- - - ------------- :~:::.~:~~:.~''''''''-''''~:.:....:'' . ... ..... .......... .. ..... .... .. ... .... .... "... ... ... ...- - - -..- - - "-- ..- ..- -..- - - - -.-- - :::""."::- --,~:o:.::.:':~::..:..

':':':.:.~ .••...-.

.=".::

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3000

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6000

== 4000-==, .f1 2000-CQl

E 00:2:c;; -2000c0'5'C

-4000Ql:2:

Negative melidional moment develop near the ring beron. Positive

meridional moment in Type-3 also occur near the ring beatn. Maximum positive

meridional moment exist within a distance of 2 ft. to 6 ft. from the junction of ring

beatn and conical hopper in Type-l and Type-2.

(iv) Circumferential moment: In Type-l and Type-2 the maximum positive

and negative circumferential moments due to material pressure are dominant (Fig.

4-l8b). For Type-3, only maximum positive circumferential moment due to

material pressure is considerable (Fig. 4-19b).

Maximum negative circumferential moment occur at the top of hopper and

maximum positive circumferential moment exists within a distance of 1.5 ft. to 6 ft.

Ji-om the junction of ring beatn and hopper for Type-1 and Type-2. For Type-3 the

maximum positive circumferential moment develop near the ring beatn.

4.4.3 Effect of Inclination of Conical hopper with horizontal, a

(a) Vertical wall

(i)Meridionalforce: Meridional force in vertical wall IS independent of

variation of a (Fig. 4.21a).

(it) Hoop Force: Hoop force due to stored material pressure changes

considerably with the change of a and it decreases in a linear fashion with increase

of a (Fig. 4.22a). But for the maximum negative hoop force the rate of decrease is

much more pronounced. The location of both maximum positive and negative hoop

force do not change with change of a.

(iii)Meridional Moment: Maximum meridional moment in vertical wall due

to material pressure changes considerably with the change of a (Fig. 4.23a) and itdecreases in a linear fashion with the increase of a. The rate of decrease for

maximum negative melidional moment is much more pronounced. The locations of

maximum meridional moment does not change with the change of a.

(il) Circumferential moment: Maximum negative circumferential moment

only decreases considerably with the increase of a (Fig. 4.24a). The location of

maximum circumferential moment remains fixed due to change of a.

62

~ •.. -~ ...~ ~ ...~ -- ...- -F

.. - ,,~ ~.: .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. '.

75

75

( b ) Conical Hopper

( a ) Vertical Wall

00 ~ w • roInclination of conical hopper, a ( degree)

50 55 60 65 70Inclination of conical hopper, a ( degree)

SelfWeighf Material Pressure Wind PressurePositive -- Positive --- Positive _A_'.Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..•

45

45

Fig. 4-21 Change in maximum Meridional Force due to change in a(Type-1, Type-2)

.

.._ .._ .._ .. .._ .._ .._ ..- .._ .._ .._ ..- ._ .._ .._ .._. _ .._ .._ .._ .. .._ .._ .._ .. .._ .._ .._ ..-.. ... .. ..... .. .. .. .. .. .... .__ ...

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

.

-1000040

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40000

=- 30000--fl'" 20000t.lou..Cii<:: 10000o'C.;::

'"::;; 0

=- 20000--fl'" 0t.l~ou..Cii<:: .20000.Q-0.;::

'"::;; -40000

--- -- -- ------ -- --- --- - - --

'_. -_.' . .. ,. .. .._ .._ .. ..- .. '._ .. .. .. .. .. ..- .. .. .._ .. _ .._.' .._ .. .. .. .. ..

'"" -- - - --- - --- -- --- -------- -- --

e-.-._.-. '-'_0_. -_. ..- - - -1::-" '. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .... .. ... ...... .... .. .. ' ...... .. .... .. ... ..- - - - -- - - - -- - - - -- - - -- - -.- - -.- --.- -.- -..-- - - - -.-- - - - -

75

75

( b ) Conical Hopper

( a ) Vertical Wall

50 55 60 65 70Inclination of conical hopper, a ( degree)

50 55 60 65 70Inclination of conical hopper, a ( degree)

Se~Weight Material Pressure Wind PressurePositive -- Posttive --- Positive _.- ..Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

45

45

Fig. 4-22 Change in maximum Hoop Force due to change in a(Type-1, Type-2)

o

40000

-1000040

30000

-20000

20000

~ 10000.0

-3000040

30000

~ 0olLC-O -10000oI

4=--~ 20000

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:. - ----- -- --- -- - --- -------_. f-.-._-_.- _._-_._. _._._._-- -------- _._._--- .=.=._.-.,•.••..,..•...••.•....• .••.•."' ..•...••.•...• ...•.••..••.•...,•.••.._ ••... ••••••_ •••••••' ••••••H ••••••• _.'._ .._ .._ ..~_.._ .._ ..- .._ .._ .._ ..-

- - - - - -.- - - - -- -- - - - -- - - -- - - - -.- - - - -.- - - -- - - - -- .--- - - -

75

75

70

70

( a ) Vertical Wall

( b ) Conical Hopper

50 55 60 65Inclination of conical hopper, a ( degree)

50 55 60 65Inclination of conical hopper, a ( degree)

Se~Weight Material Pressure Wind PressurePositive -- Positive. --- Positive -----Negative ......... Negative ----- Negative -'._ ..-

Self Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._--Negative ......... Negative ----- Negative _ .._ ..-

45

45

Fig. 4-23 Change in maximum Meridional Moment due to change in a(Type-1, Type-2)

-- --- ---- ---- ~-- -----._-_. ~._._--.-_._._.-. --_._-_.- -----_.- -'-'-'-' '-.-._.-_ .._.__ .._- ~-_._-----.- .'- _ •••••••••••••••• wo' ••• _ •• wo •••••••••• ..:.: •••. ..............,......c :." ':." ':." ':.:';..:

.........-- -..-.--........-- ..- -.-.- .--..-.-.- - --

-800040

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== 2000-.==£2 a

4000

2000==-.== a£2

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:;; -10000

75

7570

( a ) Vertical Wall

( b ) Conical Hopper

50 55 60 65Inclination of conical hopper, a ( degree)

50 55 60 65 70Inclination of conical hopper, a ( degree)

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive --_.-Negative ......... Negative ----- Negative -,,_ ..•

Self Weight I Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

45

45

Fig. 4-24 Change in maximum Circumferential Moment due to changein a (Type-1, Type-2)

••

--- --- - --~ --- --- - - .,.. ---1----- --- -I--- --- - -

••

... .. .. .. .. .. --. -- . .... -- -- .-- . .. -- -- .... -- .1=-- .._ .._. "-"-"-"- ._ .._ .._ .._ ..1- .. _ .._ .. - .. "-"-"_ .._. - .._ .. -._.,

'-"

- - - - - -- - - - - -.- - - ..-.- - - - - - - - .- -- - - - - - - - - - -.-.- -1>-- - - - -

~-------- ._._._-- _._._--- 1--------- --- -_.

~-- -. - ------- >-- - -- - - - -- - - - -- --... .. .. .. .. .. .. .. .. ... .. .... ...

.- - - - .-- - ..- - - - -- - - -- - -.- -- - - -- - - - - -- - - -.-- -!:"",.-."l'.- .. .. .. .. .. .. +..... .. .. .. .. .._ .._ .. .. "-"-"- ._ .._ .._ .._ ..1-.. .. ..

-300040

1000

;I:: 750-;I::500

f1- 250c:Q)

E0 0:2:c;;:;:;-250c:~

J!! -500E"~ -750o

-100040

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.01000-c:Q)

E 00:2:c;;:;:; -1000c:Q)~Q)-E -2000""~0

(b) Conical Hopper

(i) Meridional force: Maximum positive meridional force due to storedmaterial pressure increases noticeably due to change in a only when a exceeds 65°for all the types (Fig. 4.21b and Fig. 4.25a). For a below 65° it is more or lessconstant with the variation of a.

(ii) Hoop Force: Maximum positive hoop force due to material pressureonly decreases almost linearly with the increase of a for all the types (Fig. 4-22b,Fig. 4-25b). The locations of maximum hoop forces are more or less fixed.

(iii)Meridional Moment: Maximum meridional moments of both signs dueto stored material change in Type-l and Type-2 due to change in a. Both thesecases the variation of meridional moment may be approximated by a straight-line(Fig.4.23b).

For Type-3 the maximum Positive meridional moment due to materialpressure (Fig. 4.26a) changes with changes in a . Initially it decreases slightly, thenit increases.

(iv) Circumferential moment: Maximum circumferential moment, bothpositive and negative, in conical hopper decreases due to increase of a in a linearfashion (Fig. 4.24b) for Type-l and Type-2 due to material pressure. For Type-3the maximum positive circumferential moment due to material pressure initiallydecreases and then it increases (Fig. 4-26b).

4.4.4 Effect of Bottom Thickness of Vertical Wall, Tbottom

(a) Vertical Wall

(i) Meridional force: The variation of bottom thickness of vertical wallslightly influence the maximum positive and negative meridional force due to windload (Fig. 4-27a) and the variations are linearly decreasing. It has no effect on themeridional forces due to material pressure. Only the maximum negative meridionalforce due to self-weight increases due to the increase in weight of concrete with

higher Thol/om .

In Type-3, changes in Thol/om has no effect on maximum meridional force inpressure zone due to wind or material pressure. (Fig. 4.29a). Due to self weightonly the maximum negative meridional force increases linearly with the increase of

ThOI/Om.

63

--- I- • - -- ---- -- -- -- ----- - --

l-I-

~--- --- f-- -- ---- -I. --- ----

.

, "

75

75

Material PressurePositiveNegative

( a ) Conical Hopper

( b ) Conical Hopper

Se~WeightPositiveNegative

~ ~ ~ ffi roInclination of conical hopper, a ( degree)

Se~Weight MaterialPressurePositive PositiveNegative Negative

50 55 60 65 70Inclination of conical hopper, a ( degree)

45

45

Fig. 4-25 Change in maximum Meridional Force and Hoop Force dueto change in a (Type-3)

40000

Q)

E& 20000a.gI

10000

o40

50000

35000

30000

5000

=-,Q 30000

=- 25000,Q

~ 20000~oLLCii15000co:g 10000Q)

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" .•.. .•...•.. .•.. .•.. .•..-~- - +-- -- ---- ----

- - - - - - - - - - - - - -- - - - - - - - - - - - - -- - - - - - - - - -- - - - -- - -- - - - - - - -75

75

( b ) Conical Hopper

( a ) Conical Hopper

~ U 00 • roinclination of conical hopper, a ( degree)

Se~Weight MaterialPressurePositlve PositiveNegative Negative

~ U 00 • roInclination of conical hopper, a( degree)

Se~Weight MaterialPressurePositive PositiveNegative Negative

45

45

Fig. 4-26 Change in maximum Meridional and Circumferential Momentdue to change in a (Type-3)

-------- --~ -- --- -- ------~--- --

- - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -- - - - - - - - - - - - - - - -- - - - - -.

-50040

.0::::. 1500

-200040

8000

4:'-. 20004:',

2500

4:'-. 60004:'

f1:;:- 4000c'"Eo:2 2000c;;co'5"i:: a'":2

f- . - ----

-f'-c. .. .. .. .. .. .. .. .. .._ .. ,.~.. .. .. .. .. .. .. _ .. .. .. .. .. .. ..

-'-'-._. -'-'-'-,- ----'-'- -----'- -'-'-'-' -------- ------'-.

.

..-.._ .._'.- .._ .._ .._ ..- ._ .._ .._._ . _ .._ ..~.._ .. .._ .._ .._-- ,,-"-"-'-_ .._ .._ .._ .... .......... ...... .. .. ...... ..... .. .. .. ..... ........ ... ..... ....

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

13

13

Se~Weight MaterialPressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

( b ) Conical Hopper

( a ) Vertical Wall

7 8 9 10 11 12Bottom thickness of vertical wall, Tbottom (inch)

7 8 9 10 11 12Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-27 Change in maximum Meridional Force due to change inTbottom (Type-1, Type-2)

35000

-50006

30000

o

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30000

15000

::. 25000,9~20000~~ 15000OJ5 10000'6"ij3 5000:2

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13

13

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Pos_ _._--Negative ......... Negative ----- Negatlve _ .._ ..-

Se~Weight Material Pressure Wind PressurePos_ -- Positive --- Positive _._.~Negative ......... Nega1lve ----- Negative _ .._ ..-

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

( a ) Vertical Wall

( b ) Conical Hopper

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-28 Change in maximum Hoop Force due to change in Tbottom

(Type-1, Type-2)

"-- -- f-- --- - ------- -- --- -- ---- - -

~c-c- o

c-f-f-,c- o

c-c-,,-. - . - -p .. .. .. .. .. .._ .. .. .. .. .. .._ .. .. .. .. .. .. .. .. .. .. .._ .. .. .. ..c-f-f-

~r- -- -- -- -------- -- --- ---- - ---

~._--._._. ._. -.. .._ .. .. .._ .. .. .. .._,. .. .. .. .. .. .. .. .. .. .. "_ .. .. .. .. ...... .. .. ...... ....... .. ..... ..... .. ... ..... .. .. ... ... .. ..

- - - - - - - - - - - - - -- - - - - - - - - -- - -- - - - - - - -- - - - - -- - - - -- - - - - --

30000

a

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13

13

SeWWeight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

SeWWeight Material Pressure Wind Pressure.Positive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

( b ) Vertical Wall

7 8 9 10 11 12Bottom thickness of vertical wall, Tbottom (inch)

( a ) Vertical Wall

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-29 Change in maximum Meridional Force and Hoop Force dueto change in Tbottom (Type-3)

tl. - - .1.... - -tt.._ .._ .._ .. .._ .._ .._ ..- "._"._ .._ ..- -"-"_.'_ .. .._ .._ .._ .. .._ .._ .._ ..- .._ .._ .._ ..-.. .. .. .. .... '. ' ........ ...... ...... .. .. .. ...... .. .. ....". ' . .. .. .. ".

- - - - - - - -......- ....- -. .....- - - - - - - - - - - - - - - - - - - - - - - - - -

~--- f-- --. --- ----- --- -- -- - - - -

----~._.1-._._. .- . - -~.._ .._ .._ .. .._ .._ .._ .. ,'-"-"-"- .._ .._-._ ..- ._ .._.'-"-' _ .._ .._ .._ .. .. .. .. ..

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-500006

20000

~ -10000~ou.(ij -20000co:2 -30000~'"~ -40000

Maximum positive and negative meridional force due to wind load belowpressure zone in Type-3 decreases linearly with increase of Tboffom (Fig. 4-30a). Forpositive values the rate of decrease is more pronounced.

(ii) Hoop Force: The effect of change in Tbol/om in Type-l and Type-2 onmaximum hoop force is limited to these functions due to matereial pressure only(Fig. 4.28a). In this case both negative and positive maximum values decreaseslinearly with the increase of T bol/o';'. The rate of decrease for negative values ismuch more pronounced. In Type-3 only the maximum positive hoop force inpressure zone due to stored material pressure changes slightly and linearly due tochange in Tbollom (Fig. 4-29b). Below pressure zone of Type-3, only maximumpositive and negative hoop forces due to wind load decreases linearly with changein T boffom but the rate of decrease is relatively smaller (Fig. 4.30b).

(iii) Meridional Moment: In Type-l and Type-2 the maximum positive andnegative meridional moment due to stored material pressure increases with adecreasing rate (Fig. 4.31a). For Type-3 and in pressure zone only the positivemaximum meridional moment increases linearly with change in Tboffom (Fig. 4.33a).In this type below pressure zone, maximum positive meridional moment due towind and maximum negative meridional moment due to material pressure increaseslinearly with the increase in Tbol/om(Fig, 4-34a).

(iv) Circumferential moment: For Type-l and Type-2 the maxunumcircumferential moment due to wind and material pressure changes with change inTbollom (Fig. 4.32a). But this change is not significant for design purpose. In Type-3the varation in maximum circumferential moment due to increase in T boffom isnegligible (Fig. 4-33b) for the pressure zone. In this type, below pressure zone, themaximum positive and negative circumferential moment due to wind load increasewith increase in TbOl/om (Fig. 4-34b). But the maximum values are so small that thechanges can be ignored

(b) Conical Hopper

(i) Meridional force: Changes in Tbol/om in Type-l and Type-2 has littleeffect on the maximum meridional force for all the load cases (Fig. 4-27b).

(ii) Hoop Force: Hoop force variation in conical hopper due to change inTbol/om is again negligible for all the load cases (Fig. 4-28b).

64

13

13

Seij Weigh! Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

( a ) Vertical Wall below pressure zone

( b ) Vertical Wall below pressure zone

7 8 9 10 11 12Bottom thickness of vertical wall, Tbottom (inch)

SeijWeigh! Material Pressure Wind PressurePositive -- PoslIIve --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

7 8 9 10 11 - 12Bottom thickness of vertical wall, Tbottom (inch)

Fig_4-30 Change in maximum Meridional Force and Hoop Force due tochange in Tbottom (Type-3)

r

- -- - ---- -- -- - - -

_.-- -.

,._ .._ .._ .. .._ .._ .._ .. .._ .._ .._ ..- ._ .._-._ .._ . ________ c ___ ------------ -------------- .... --- ......... --- -- --- -- -- ....... .... -- -- -- -- -- .... ..... -- -- --- -- -- ........... -- ......

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - -

,...-.-._. --_.-._- '-'-._-- -.-._-_. -._----- -_._. - -

.._ .._ .._ .. --_ .._------ .._ .._ .._ .._. _ .._ .._ .._ .. ------------ ------------ .._ .._ .._ ..-....... -- -- ..... -- -- -- -- --- -- .... -- -- --- -- .... -- -- -- -- -- -- --- ---

- -- - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

15000

10000

""-... 5000f1~

'"U~0 0LLc-o0I -5000

40000

-100006

-600006

~ 20000-...f1

'" 0u~oLL

~ -20000_Qu-;::

'":a: -40000

F-- -- ---. -- -- -- -- -- - - -- ----~ --F-------- ------_. ._._0_0- -------- --_._--. ----_. __ . ---_.---................ .":.:.:,::.:.:.::.:.:,::.' '.''''.'.'''''.=.'~.'=.'.=.'=.'- ,•.•.•......... _-- _ .•..•...--_ ...... _._~--_...._ .._ .._ ..

... -..- --. - -- - - -..- ..- - --. -......- ..-.... ..- ..- ..

13

13

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative

_ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _0_'-Negative ......... Nega1ive ----- Negative _ .._ ..-

( a ) Vertical Wall

( b ) Conical Hopper

7 8 9 10 11 12Bottom thickness of vertical wall, Tbottom (inch)

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-31 Change in maximum Meridional Moment due to change inTbottom (Type-1, Type-2).

-- ---- _.F-"--- - -f-'---'-- .-._._- ---_._ .. ._------ ._._---- _._-_.- -'-'-'-~..."'...=..."'., • •:.:.:.••"' ••••••H•••••\ '.._",_.'-- -_.._ .._ ..- •..•.•.•..•...•..•...•..•.. •.•.•.••.•.•.•_ •.•.•. .....••.•...•.......••.••..

-. .'. - '. - -..'-.....--. .... - .-...-.-. - - .- - . - - - ..- --10000

6

2000

-60006

~ 1000

""-"" 0.0:::..-10001:IIIE '2000o~ -3000c:~ -4000-'"III:2 -5000

4000

""2000-

"" 0f!- -2000c:IIIE0 -4000:2c;;c: -6000015-'"III -8000:2

13

13

SeijWeight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Seij Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

( a ) Vertical Wall

( b ) Conical Hopper

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-32 Change in maximum Circumferential Moment due to

change in Tbottom (Type-1, Type-2).

...

_. - .._. ..- - . --- - -_.- ._._---- _._--._........ .. .. ....... " ...." ..... .... ...... " ... .... .." ... .. .... .. " ...... ...... ................_ .. .. .. .._ .. .. .. ._ .. '._ .. +-.. .. .._ .. .._ ..-"-"- ._.._ .._ .._ . _ .._ .._ .._ ..- - ..- ..- -... ...- - -.- - - - - -.....- -.- - - - - - .- - - - - - .- - - - - -

::.. .- '- - - +-. ,.. - ,-- . - -.

-- - -- -- -- - --- -- -- "- --- - - - -----................ ," ....... ...... .... '" " " ,." ............. " ....... .... " " .

- - - - - - . .-- - - -.- - - - - - -- - - - - - - - - - -.- - - - - - - - - -.- - - -- - - - - - -._ .._ ..- ._ .._" .. .. .. .. .. .. .. .. .. .. .._ .._ .._ .-.._ ..-.._ ..

~,,-,,-"-" ..-'.3000

6

(ij'E -400~~ -600:J~ .800(3

.10006

600~

'*' 400--'*'fl 200-c: 0CDE0 -200:2

3000

'*'--'*' 2000

fl- 1000c:CDE0 0:2(ij:;:;c: -1000CD~CD-E:J -2000()~(3

Fig. 4-33 Change in Meridional and Circumferential Moment in verticalwall due to change in Tbottom (Type-3).

'-. -- . _ .. - . - - '-'

- .. - - - - - - - - - - - - - - - - - - - .. - - .. - - - - - --- - - - - - - - - - - - - - - - -- - -

~~c ..-..-..-..- .._ .._ .._ .._. -"-"_.'_ .. .._ .._ .._ .. .._ .._ .._ ..- .-.._ .._ ..-_ ..-..-.._ ..=

13

13

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._--Neglrtive ......... Negative ----- Negative _ .._ ..-

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

( a ) Meridional Moment

( b ) Circumferential Moment

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

--------f--- ------- -

;;..- --. - - - . .-

.. .. ..- .. - - - - - - - - - - - - - - - - .. - - - - - - - - .. - - - - .. .. .. - - .. - - - - - - - - - .. - -.

..-.._ .._" .. .._.' .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..;;..._ .._ ..-..

.

a

3000

-6006

:: 1000c:IIIEo:2(ij••ai -1000~" -2000~i:3

-30006

'*'-'*' 2000

1200

~ 1000

'*'-'*' 800

fa 600-c: 400IIIE0 200:2(ij ac:0'i5 -200.;::III:2 -400

13

13

Sen Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

SenWeighl Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negatlve ......... Negative ----- Negative _ .._ ..-

( b ) Vertical Wall below pressure zone

( a ) Vertical Wall below pressure zone

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

7 8 9 10 11 12

Bottom thickness of vertical wall, Tbottom (inch)

Fig. 4-34 Change in maximum Meridional and Circumferential momentdue to change in Tbottom (Type-3)

.-' - - -_.--_.-- --------.----_.- ,-'-'---f-.-.----~~rrf .... ......... ...... ..... .. ........ ...... .........~- .... .. ...... .. ..... ...... .. ..- - - -.. - - - - .- ........ ..~

- - - - - -.- - - - - - - - .- - - - - ...- - - - - ...- - -- - - - -~~._..-..-.. "-"- ..-..-r .-.,,_ .._ ..-. _ .._ ..-..- -.. .. .._ ..~

"-'''- ._ .._ .._ ..-.~

'-'---- _.--_.-'--------- --_.----- -'-'----_.-.----

~ --- - -------~ .- - - -r1".:;;':.:;;.::.;;.::: ""::'::.::.:,::.:.:,::.:.:.. •.;.:.. .... :,;...•..•...,-..-

-- •••• ',.,..••f.~ ,,": ':":":::':':." ::::":::-. '-"-"-"-" -.._ ..-.._ ..r - - -. .... ........ ........- - .- -.- - - '- - - .... - - - .- - - - - - .- - .. ..- -. -~

- .- -.-

1000

iI:'-iI:' 500fa~-c: 0'"E0:2

3000

Cii -500:;::;c:~'"E -1000

'"u~(3 -1500

6

-30006

1:'"E 0o:2Cii -1000c:o'6.~ -2000:2

iI:' 2000-iI:'fa 1000

(iii) Meridional Moment: Variation of Thol/omhas considerable effect on themaximum negative meridional moment due to grain (Fig. 4.3!b).

(iv) Circumferential moment: Circumferential moments developed inconical hopper due to various load cases are relatively smaller. Only the maximumnegative circumferential moment due to material pressure in Type-! and Type-2varies noticeably (Fig. 4-32b).

4.4.5 Effect of Conical Hopper Thickness at Top, (top

Since the change in hopper thickness has no effect on the vertical wall forType-3, vertical wall mentioned in the subsequent paragraphs refers to verticalwall in Type-! and Type-2 only.

(a) Vertical Wall

Variation of top.thickness of conical hopper, ttop,has no effect on any of thestress resultants e.g. meridional force, hoop force, meridional moment andcircumferential moment (Fig. 4-35, Fig. 4-36a, Fig. 4-38a, Fig. 4-39a).

(b) Conical Hopper

(i) Meridional force: Variation of top thickness of conical hopper, ttop, haslittle effect on the meridional force in any of the types (Fig. 4-35b, Fig. 4-37a).

(ii) Hoop Force: Variation of top thickness of conical hopper, ttop,has alsolittle effect on the hoop force in any of the types (Fig. 4-36b, Fig. 4-37b).

(iii) Meridional Moment: Maximum meridional moment of both signs dueto material pressure and maximum positive meridional moment due to wind loadvaries with the variation of ttop for Type-! and Type-2 (Fig. 4-38b). For Type-3 themaximum positive meridional moment shows considerable sensitivity with thevariation of ttop(Fig. 4.40a).

(iv) Circumferential moment: In Type-! and Type-2 the maximumcircumferential moment due to material pressure varies with the variation of ttop(Fig. 4-39b) but magnitudes are small. For Type-3 maximum positivecircumferential moment due to material pressure increases considerably with theincrease of tlop(Fig. 4.40b).

65

13

13

1211

ttop ( inch)

( b ) Conical Hopper

8 9 10

Top thickness of conical hopper,

( a ) Vertical Wall

Se~Weight Material Pressure Wind PressurePosltive -- Positive --- Posltl ••••e _._.-Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

7

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

Fig. 4-35 Change in maximum Meridional Force due to change in ttop

(Type-l, Type-2)

.. .. .. .. .. .. .. .. .. .. .. ..- .._ .._., _ .._ .. .. .. .. .. .. .. .. .. .. .... .. ........... ........... ...... ...... ..... .. ....... ...... .... .. ... .... ... .. ....... .....

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - -

~F -- .- - -- -- - - -~e~~~~t~~~~

- . ... .. .. .. .._ .._'._ .. .. .. .. .. ._ .. .. .. .. .. .. .. .. .._ .. .. .. .. .. ...

o

20000

30000

o

-500006

-40000

-50006

35000

30000

~ou. -10000(ijco -20000'C

~ -30000

.;::--. 10000£!

::. 25000£!i 20000

~ 15000(ijB 10000'C0;::'" 5000:2

13

13

1211

ttop ( inch)

( b ) Conical Hopper

( a ) Vertical Wall

8 9 10

Top thickness of conical hopper,

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive

_._ ..Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight MaterielPressure Wind PressurePositive -- Positive --- Positive

_._ ..Negative ......... Negative ----- Negative _ .._ ..-

7

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

Fig. 4-36 Change in maximum Hoop Force due to change in ttop(Type-1, Type-2)

-- -- -- -- ---- ---- - --- -- -- ---

--- ._._._- ._._._- ._._._- ._._---- -------- --------.. .. .. .. .. .. .. .. .. .._ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .... ......... ...... .......... ........ ........ ...... .... .. .... .... ... ... ....... ... ..

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --

---- -- --- ---- -------- -- -- -- --

~- - -~.. .. .. .. .._ .. .. .. .._ .. .._ .. ._ .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

o

o

'"2~ 10000a.ooI

40000

30000

-200006

-=--f1 20000

30000

-100006

20000

'"2olLa.ooI

-10000

-=--f1 10000

-~rF r .- -- ---- ---- f----~~~

.

.

~~~~

~

~

Fig. 4-37 Change in maximum Meridional Force and Hoop Force dueto change in ttop (Type-3)

( b ) Conical Hopper

13

13

Material PressurePositiveNegative

Material PressurePosIIIveNegative

( a ) Conical Hopper

SeijWeightPositiveNegative .

SeijWeightPositiveNegative .

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

'-- -- -- -- ---- ---- ----40000

-100006

o

50000

35000

5000

o6

30000

""£" 25000

CDE 20000olL

~ 15000o:g rooooCD::;;;

"" 30000--.f2~ 20000EolLa. 10000gI

13

13

1211

ttop ( inch)

( a ) Vertical Wall

( b ) Conical Hopper

8 9 10

Top thickness of conical hopper,

Se~Weight Material Pressure Wind PressurePosttlve -- Positive --- Positive _._ ..Negative ......... Negative ----- Negative _ .._ ..-

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

7

Fig. 4-38 Change in maximum Meridional Moment due to changein ttop (Type-1, Type-2)

t~1--- - 1--- - - -- - - - 1---~._._._.1--------. ----_._. -------- -------- ._._"_.- -------..... :.:.. ....:.:...~.•..•... :f"'.'-"-"-', ,....•.•..._.,-., "-"-"-'- ._ ..__ ._-- --.._._ ..- ._.._ .._ ..-

.

- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- -------- --- ---- -- - -- -- - -----_.--- -------- __._._0- ----:..-------

~., ••••••• loI_'., ••••••• ...:.•.•.,:.:.:" ..•.•.:..•• :....:.: ...:.:.: ..•':.:.....:.:. :~'.:.:.:,::.:.:.::.:.:.: .:.:.::.:.:: ..•:,:...:.:....: ".::':':.::.:.:.:':':':: ::.:..:.::.:..:.::.:..:.::

.

.

- - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - -

3000

2000

-50006

-10000 6

o"E'"E -1000a~ .-2000<:.2 -3000"tl.;::

'"~ -4000

;f;'-;f;' 1000.0

4000

2000;f;'-;f;'

0,Q- -2000<:'"Ea -4000~(ij<: -6000a'6.;::'" -8000~

Fig. 4-39 Change in maximum Circumferential Moment due to changein ttop(Type-1, Type-2)

13

13

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._ ..Negative ......... Negative ----- Negative _ .._.'.

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -.-.-Negative ......... Negative ----- Negative _ .._ ..-

( a ) Vertical Wall

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

( b ) Conical Hopper

7 8 9 10 11 12

Top thickness of conical hopper, t,op ( inch)

--.----------.-- --- --_. _._. ._. -_. ._.- ---_._.- --_._-_. _._- -

............... .......... .. ...... .. .... ... .. ............. .... ..... ..... .. .. .. .. .. .._ .. .._ .. '._ .. .. .. .. .. .. .. .. .. .. .. ..

- - -- - -- - --- - -- --- - - -- - - - - -- -- - - - - - - - - - - - - - - - - - - - -

f-- - f-- - - ._- . _. _. . .~

,...- - - f-- - -- ---- -------- -- -- -- --.......... ..... .. .... ..... .. .... .. ...... ....f-f-f-

f-t--- - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -f-

b. .. .. .. .. .. .._ .. .._'._" .. .. .._ .._. _ .. .. .. .._ .. .. .. .. .. .. .. ..f-f-

800

= 600--=,Q 400

- 200c'"E0 0:;;0; -200:;::;c'"~'" -400-E"~ -600

<:5

1000C'"Eo 0:;;0;:;::;~ -1000

~" -2000~

<:5-3000

6

3000

-8006

=;;- 2000

,Q

----- -1----------------" ----- - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

.~I ."

13

13

( a ) Conical Hopper

7 8 9 10 11 12

Top thickness of conical hopper, ttop ( inch)

SenWeight MaterialPressurePositive PositiveNegative . . . . . . . . . Negative

( b ) Conical Hopper

7 8 9 10 11 12Top thickness of conical hopper, ttop ( inch)

SenWeighl MalerialPressurePositive PositiveNegative . . . . . . . . . Negative

Fig. 4-40 Change in maximum Meridional and Circumferential momentdue to change in ttop (Type-3)

~

.

E- --- -----~--t ----~---

.

------- -------. -------- -------- ------- .. -------- -------

4000

-20006

"EQ) 2000Eo:i!:0; 1000:g~~E 0:Jf::G

-10006

10000

""8000-

""f1 6000~-cQ)E 40000:i!:0; 2000c0'5";:Q) 0:i!:

""-~ 3000f1

4.4.6 Effect of Depth of Bottom Ring Beam, d

Since depth of bottom ring beam has no effect on the analysis of verticalwall in Type-3, the vertical wall mentioned in the subsequent paragraphs refers tothe vertical wall ofType-l and Type-2 only.

(a) Vertical Wall

(i)Meridional force: variation of depth of ring beam has no effect on themeridional force due to various load cases in the vertical wall (Fig. 4.4la).

(ii) Hoop Force: Due to increase in the depth of ring beam d, onlymaximum negative hoop force due to stored material pressure show sensitivity.(Fig. 4-42a). In this case the maximum negative hoop forces due to grain loaddecrease approximately linearly with increase in d.

(iii) Meridional Moment: Maximum positive and negative meridionalmoment due to grain in Type-l and Type-2 decreases with increases of d (Fig. 4-44a). But the rate of decrease of the maximum negative meridional moment ismuch higher.

(iv) Circumferential moment: Variation of d has influence on the maximumnegative circumferential moment in the vertical wall due to stored material only.(Fig. 4.45a). In this case the maximum negative circumferential moment decreaseslinearly with the increase in d.

(b) Conical Hopper

(i) Meridional force: In Type-l and Type-2 maximum positive meridionalforce due to stored material only decreases with increase in the depth of ring beam(Fig. 4-4lb). In Type-3 the maximum positive meridional force due to materialpressure only decreases linearly (Fig. 4.43a).

(ii) Hoop Force: The maximum positive hoop force due to material pressureonly decreases (Fig. 4-42b). For Type-3 the maximum positive hoop force due tostored material pressure only decreases with the increase in the depth of ring beam(Fig. 4-43b).

(iii) Meridional Moment: Maximum negative and posluve meridionalmoment due to material pressure in Type-l and Type-2 decrease with the increaseof d (Fig. 4.44b). For Type-3 the maximum positive meridional moment due to

66

~- --- f...- ---- .

~-

~

. . . . . . ... .. .._ .. .._. _ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...

100

100

90

90

807060

( a ) Vertical Wall

( b ) Conical Hopper

50Depth of ring beam, d ( inch)

40 50 60 70 80Depth of ring beam, d ( inch)

40

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _.---Negative ......... Negative ----- Negative _ .._ ..-

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._0-Negative ......... Negative ----- Negative _ .._ ..-

30

30

Fig.4-41 Change in maximum Meridional Force due to change in d(Type-1, Type-2).

---- f-._._.- --_.-. __ ._0_- _._._.- ._-_._0 ._._.- _0_0

_ .._ .. .._ .._ .._. _ .._ .._ .. .._ .._ .._ ..-"-"-"- ._.._ .._ .. .._ .._ .._. _ .._ ..

...... .... .. ...... ... .. .. .. .. ........ ... .... ... ..... ...... ...........

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

35000

.6000020

.500020

30000

o

;I:: 25000--.c;::. 20000

~o 15000U-n;c 10000o'5'iii 5000::;;

30000

15000;I::--f! 0

'"u~0 .15000U-n;c0'5 -30000'C

'"::;;-45000

Fig. 4-42 Change in maximum Hoop Force due to change in d(Type-1, Type-2).

100

10090

90

( b ) Conical Hopper

( a ) Vertical Wall

40 50 60 70 80Depth of ring beam, d ( inch)

40 50 60 70 80Depth of ring beam, d ( inch)

Sel/Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

SeWWeight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

30

30

~ -- 1-- - ---- --- -- -- --- 1-- ----

._._0_. ------- ._-_._ . ._._.- ----.. .. .. .._ ..- .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..-...... ...... .. .. .. ....... .. .. ... .. ...... .. .. ........ ....... .. ..

- - - -- - - - - - ..- - - - -- - - - - - -- .- - - -- - - - -- - -- - - -

t -- 1------ -. - - ---~ --~ --- --~~~~~

~

-.. .. .. .._ .._. .. .._ .. .. .. .._ .. .. .. ..- .. .. .. .._ .._ .. .. ..~~~

o

20000

30000

-20000

-1000~0

-3000020

40000

30000

'l:'--:Q 20000

Ql()~0u. 10000Q.00I

0

~oU.Q.

g -10000I

'l:'-- 10000:Q

30 40 50 60 70 80 90 100Depth of ring beam, d ( inch)

SenWeighl Material PressurePositive PositiveNegative ......... Negative

c

.... '.... .... .... .... .... .... -- -- -- ---- --- --

1009080706050

( b ) Hoop Force

( a ) Meridional Force

Depth of ring beam, d ( inch)SenWeight Malerial PressurePositive PositiveNegative . . . . . . . . . Negative

4030

- ------- -- --- --- .. -- --

--

:

Fig. 4-43 Change in maximum Meridional Force and Hoop force inconical hopper due to change in d (Type-3)

60000

50000

;I:- 40000,Q

'" 30000()~0LLa. 2000000I

10000

30000

35000

5000

;I:

~ 25000

~ 20000oLL

0; 15000c:o:g 10000'":2

100

100

90

90

( b ) Conical 'Hopper

40 50 60 70 80

Depth of ring beam, d ( inch)

40 50 60 70 80Depth of ring beam, d ( inch)

.( a ) Vertical Wall

Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive -.-.-Negative ......... Negative ----- Negative _ .._ ..-

Se~Wei9ht Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negatlve ----- Negative _ .._ ..-

30

30

-- ---- -- --- -- -----. -- --- .---- _._._.- -'-'-" -_._._- -'---'- ------ ._-_._. _._.--- _. ._ .._. .....:.•J.:: •.. ':." .......:.. '.:' ':: .... .._ . .. .. .. .. .. .. .. .. ..-

..

0 .. .. 0 0 - .. 0 0 0 - .. 0 0 0 .- - - - - - _0 ,~-, 0

, -. .'.. - 0 . .

.. ..-

.. - --.

..

..

..

r

~ - - - --. --- ---- f:.=.=.= :::::1._._._._.- ._._--- ---'-'- ._._.-.r ---- ,..._._.-~ ..•...".••.., ..,•........•..,•.•. .••....... ,._ .. .•. ....,....•......•... "-,,.•.•.,~.•.. ...•••......•._ ..• ,.._._ .._ . _ .._ ..~

--- ~.-.-_.- 00'

0"0-0

---- ----.-----00

0 ,--

Fig. 4-44 Change in maximum Merodional Moment due to changein d (Type-1, Type-2)

-800020

-1000020

4000

""2000--

"".0 0~-CQl -2000E0::;;(ij -4000c.Q""0.", -6000Ql

::;;

4000

~ 2000""--"" 0,g~- -2000cQl

E0 -4000::;;(ijc -60000'5.",Ql

-8000::;;

Fig. 4-45 Change in maximum Circumferential Moment due to changein d (Type-1, Type-2)

100

100

90

90

( a ) Vertical Wall

( b ) Conical Hopper

40 50 60 70 80Depth of ring beam, d ( inch)

40 50 60 70 80Depth of ring beam, d ( inch)

SeW Weight . Material Pressure Wind PressurePosltive -- Positive --- Positive

_._ ..Negative ......... Negative ----- Negative

_ .._ ...

SeW Weight Material Pressure Wind PressurePositive -- Positive --- PositiveNegative ......... Negative ----- Negative _ .._ ..•

30

30

_._. _._._.- _._._ . ._-_._- ------- ------- ------

- - -- - ---- - - - -- -- - -- -- ----.- .... ... ...... .... .. ...... ..... .... .. ........... ......... ..

-. - - - - - ..- - - -- - - - .. .- - -- - - - .- --.-.- - - .- -...... - -.. .. .. .._ .. _ .. .. ..- .._.' .. .. .. .. .._ .. .. .. .. .. .. .. ..

-- ---. --- -- ------ --- ,...- ._._- --_._-- _._--_. ._---_. _._---- --_._ .. -_._--- '--_ .......... .. .. .... .... ...... ........

-"-" .._ .._ .._. _ .._ .._ ..• .._ .._.'-" .._ .._ .._ ._ .._ .._ ..• .._ .._ .._ .. f- .._ ..'

- - - - - -..-.- - - - .- .- - - -. .- - - - - - - - - -...-..--- --- -.

o

1000

>l:'->l:' 500

.£-c 0CDE0

::!E

3000

1ti -500~~~E -1000:::l~(3

-150020

_ 1000cCDEo

::!E1ti:;::;ai -1000~:::l -2000~(3

>l:'->l:' 2000.£

material pressure decreases sharply for the initial variation of d and then the rate ofdecrease slows down (Fig. 4-46a).

(iv) Circumferential moment: Maximwn circumferential moment of bothsign due to stored material pressure in Type-l and Type-2 decreases withdecreasing rate with respect to the increase in the depth of ring beam (Fig. 4.45b).For Type-3 the maximwn positive circumferential moment due to material pressuredecreases sharply for the initial variation of d and then the rate of decrease slowsdown (Fig. 4-46b).

4.4.7 Effect of Unit Weight of Stored Materials, r(a) Vertical Wall

(i) Meridional force: Meridional force due to stored material pressure isalways negative. It increases linearly with the increase of unit weight of storedmaterial (Fig. 4.47a). For all the three types of silos maximwn meridional forcesare approximately the same. For Type-3 and below pressure zone maximwnnegative meridional force also increases linearly with the increase in unit weightof stored material (Fig. 4.49a).

(ii) Hoop Force: For Type-l and Type-2 both positive and negative hoopforce exists and increases linearly (Fig. 4.48a). But in Type-3 only positive hoopforce exists and it increases linearly with the increase of r. Below pressure 'zone inType-3 both positive and negative hoop force increase linearly as unit weightincreases (Fig. 4-49b). Variation of r has no effect on the location ofmaximwnhoop forces for all the three types of silos.

(iii) Meridional Moment: Maximwn meridional moment, both positive andnegative, increases linearly for all the three types of silos (Fig. 4.50a). In Type-3 nonegative meridional moment exists and the maximwn positive meridional momentincreases at a rate of 14.1 lb/ft. per unit increase of y. Maximwn positive andnegative meridional moment below pressure zone in Type-3 also increase linearlywith respect to the increase of unit weight (Fig. 4.52a). For Type-l and Type-2 themaximwn negative meridional moment occur at the bottom of vertical wall and themaximwn positive meridional moment occurs at a distance of 4.5 ft. from thebottom of vertical wall. For Type-3 and in pressure wne the maximwn positivemeridional moment occurs at a distance of 1.71 ft. from the bottom of pressure

67

100

10090

9080

80

70

70

60

60

50

50

( a ) Meridional Moment

( b ) Circumferential Moment

Depth of ring beam, d ( inch)Sen Weight Malerial PressurePositive PositiveNegative • • • . . . . . . Negative

Depth of ring beam, d ( inch)Sen Weight Malerial PressurePos~ PosltiwNegative Negattve

40

40

30

30

Fig. 4-46 Change in maximum Meridional and Circumferential momentin conicl hopper due to change in d (Type-3) .

,",

" ," ,,

' ..•. ~--- -- 1---- 1-- ---- -------- ------- ------- ------ - - - -- - - ........----

.

""" '.'

"",..•. ..•. ..•.

--- 1---- 1---- -- -- -------- ------ ------- ------- ------- .. -_ .......... -------

o

2000

14000

.0

':E 3000

'"E~ 2000

"iii~ 1000e~:J ,0u~

<:5-1000

20

<I:';;- 4000

5000

-200020

4000"iiico'i5'C

'"::;;;

.0:::::. 8000"E~ 6000o

::;;;

~12000<I:'-<I:'10000,

( b ) Conical Hopper

Fig, 4-47 Change inmaximum Meridional Force due to change in unitweight of stored material.

160

160

1401201008060

.( a ) Vertical Wall

60 80 100 120140

Unit weight of stored material ( Ib / CU.ft)Type-1,Type-2 Type-3Postttve PositiveNegative . . . . . . . . . Negative

Unit weight of stored material ( Ib / CU.ft)Type-1,Type-2 Type-3Positive PositiveNegative . . . . . . . . . Negative

40

40

" •...•...•.",

" .•...•..

t ".",.0,f- " '.,f- •••• .c,..

",t .•..•...~;"..t ...•...;,.

'''',f- .....: ...: ..•~.t

.... ;...;. .•..;..;. ..:" ..;. ..

f-

f-

t

100000

-16000020

80000<I:'-.0Q) 60000<.>~0LLOJ

40000c:0'0';::Q)

:2 20000

0

-20000

<I:' -40000-:fa -60000~Q)<.>~0 -80000LLOJc: -1000000'0';::Q)

-120000:2

-140000

.--:: ~

~,-::::--

-"""~- ------= ....~ ..... .........". ..... .... " .r . .....~

...... ..... ..... ........ ....'"

'" .....f- ....

" ......r ".r

160

160

( a ) Vertical Wall

60 80 100 120 140Unit weight of stored material ( Ib / cU.ft )

Type-1,Type-2 Type-3Positive PositiveNegative . . . . . . . . • Negative

40 60 80 100 120 140Unit weight of stored material ( Ib / CU.ft)

Type-1,Type-2 Type-3Positive PositiveNegative . . . . . . . . . Negative

.40

~

~,,'

""f- "~ """~

" ~"- ... " .............--~

,,- ----. ",,;,- --" ~"" ;.------

( b ) Conical H.opper

Fig. 4-48 Change in maximum Hoop Force due to change in unit weightof stored material.

20000

140000

160000

-6000020

Q)u~ 800000LLC-O 600000I

40000

~ 120000--:: 100000

100000

80000

60000

~ 40000--£!Q) 20000u~0LL 0C-O0I -20000

-40000

Fig. 4-49 Change in maximum Meridional Force and Hoop Force due tochange in unit weight of stored material (Type-3)

160

160

40 60 80 100 .120 140

Unit weight of stored material ( Ib / CU.ft)

IMaterial Pressure I

Positive -- Negative - - - - -

40 60 80 100 120 140

Unit weight of stored material ( Ib / CU.ft)

I. Material Pressure I

Positive -- Negative - - - - - .

( b ) Hoop Force in Vertical Wall below pressure zone

( a ) Meridional Force in Vertical Wall below pressure zone

~ ----~

~

-----~------" ", ".

". ". ",". ". , ", ,

'" '" ", ," ",

"

~rr

,, ,,,,.,., ., .,,

'", ,,, .

,,,. ,.,,,, ,,,,., .,

, ,,,,,,

-3000020

-16000020

-140000

40000

30000

*' 20000-f1., 10000"~0lL.c- o00I -10000

-20000

o

-20000

*'- -40000

e -60000o~ -80000co:g .100000<;;::l: -120000

160

1601401201008060

( a ) Vertical Wall

Unit weight of stored material ( Ib I cu,ft )

Type-1, Type-2 Type-3Positive -- PositiveNegative . . . . . . . . . Negative

40 60 80 100 120 140

Unit weight of stored material ( Ib I cu,ft )

Type-1, Type-2 Type-3Positive PositiveNegative . . . . . . . . . Negative

40

( b ) Conical Hopper

Fig, 4-50 Change in maximum Meridional Moment due to changein unit weight of stored material.

,.:

~~ .,~ --~ --~ - 1--~ 1----~ ---~ ---~ ---~ -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -.... ........... ..... ..... .... .... .. .... .... ...... .. ........ -..... ......... ....... ...... ....

.' - - - - -::- - - - - - - -

"" "

" "," ",

",

"",

" "," " ....

" ",

"," ",

"", ..

" "-

" ","

30000

-2000020

-2500020

== 20000-==,£- 10000c:'"E0~ 00;c:015';:

'" -10000~

10000

5000

==-== 0,£~- -5000c:'"E0~ -100000;c:0 -1500015';:

'"~ -20000

( b ) Conical Hopper

Fig, 4-51 Change in maximum Circumferential Moment due to changein unit weight of stored material.

160

160

( a ) Vertical Wall

60 80 100 120 140

Unit weight of stored material ( Ib / cu,ft )Type-1,Type-2 Type-3Positive PositiveNegative Negative

40 60 80 100 120 140

Unit weight of stored material ( Ib / cu,ft )Type-1,Type-2 Type-3Positive -- PositiveNegative . . . . . . . . . Negative

40

,--- - - - - -- - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - - - - - - - -." '" " ", " ", "

",

" ",",

"""'"" ,

"'" " ....

"" " '" "

'" ",",

"",",

l

---~-------------. -j... --

- - - - - - - -.- - - - -.- - -" - - - - - - - - - - - - - - - - - - - -.......'" - - - - - - - - - -", " " ....... ..... ..... ", " " ..... ...... ..,

'" ...... .., ..

-600020

8000

;:: 6000-;::,,Q 4000-c.,E 20000:2c;;:;::;c 0.,~.,-E::l -2000u~B

2000

1000;::-;::0

,Q~- -1000c.,E0 -2000:2c;;:;::; -3000c.,~.,-E -4000::lU~B -5000

160

160

1401201008060Unit weight of stored material ( Ib I cU.ft )

I . Material Pressure IPcs_ -- Negative - - - - -

40 60 80 100 120 140

Unit weight of stored material ( Ib I CU.ft)

IMaterial Pressure I

,POSitive -- Negative - - - - - ,

40

( a ) Meridional Moment in Vertical Wall below pressure zone

( b ) Circumferential Moment in Vertical Wall below pressure zone

Fig. 4-52 Change in maximum Meridional and Circumferential momentdue to change in unit weight of stored material (Type"3)

tt~

~

~

- - -..- - - - - -t - --- ---t - - - - -~ - - - -t -.-- - -~ - -t .....-t - - - -- -

~t .

~tt~t~t -- -- -t -- -- - -~ -- --t - - - - --~ - .t - - -- -t -- - - .~ - - -t - - - -~ -- - -tt

-100020

-600020

200

=- 0=,Q

-200-cQ)E0 -400::i[;iii:;::;c -600Q)~J!!E:J -600u~U

2000

1000=-= 0,,Q~ -1000-cQ)E -20000::i[;iii -3000c0'6 -4000'CQ)

::i[;-5000

zone and below pressure zone the maximum positive and negative values occur at adistance of 4.5 ft. from bottom of vertical wall and at the bottom respectively.

(iv) Circumftrential moment: Circumferential moments in vertical wall inpressure zone for different types of silos also increase linearly with the increase inunit weight (Fig. 4.5la). For Type-3, in the pressure zone the circumferentialmoments due to material pressure is negligible. Below pressure zone in Type-3 themaximum circumferential moments, both positive and negative, are also small andthese increase linearly with the increase of y. (Fig. 4.52b).

(b) Conical Hopper

(i) Meridional force:' Meridional force in conical hopper due to materialpressure is always positive. For all types of silos maximum meridional forces dueto material pressure increases linearly with increase in the unit weight (Fig. 4-47b)and variations are the same. Maximum meridional force always exist at the top ofhopper.

(ii) Hoop Force: Hoop force in conical hopper due to material pressure isalways positive. Maximum hoop forces increase linearly with increase in unitweight and the values of maximum hoop force for Type-3 is always greater thanthat of Type-l and Type-2 (Fig. 4.48b). The location of maximum hoop force is ata distance of 5.04 ft. from the junction of ring beam and hopper in Type-l andType-2. ForType-3 this distance is 2.54 ft..

(iii) Meridional Moment: Maximum meridional moment of both signs forall types of silos increase linearly with the increase of unit weight (Fig. 4.50b). Thelocation of maximum positive and negative meridional moments are at a distance of4 ft. from the junction of ring beam and hopper and at the junction respectively forType-l and Type-2. For Type-3 the maximum positive meridional moment existsat the junction and negative maximum values occur at a distance of 7.11 ft. fromthe top of hopper.

(iv) Circumferential moment: Maximum circumferential moment alsoincreases linearly for all types of silos with the increase of unit weight (Fig. 4.51b).The variation of unit weight has no effect on the locations of maximum values ofcircumferential moments. Maximum negative circumferential moments for Type-land Type-2 and maximum positive circumferential moments in Type-3 occur at thejunction of ring beam and hopper. Maximum positive circumferential moments in

68

Type-l and Type-2 exist at a distance of 2.97 ft. from the top of ring beam andmaximum negative circwnferential moment in Type-3 occurs at a distance of 0.6 ft.from the top of hopper (junction of hopper and ring beam).

4.4.8 Effect of Angle of Internal Friction of Stored Material, p

(a) Vertical Wall

(i) Meridional force: Maximum negative meridional force for Type-l andType-2 are always slightly greater than those of Type-3. In this case the maximumvalues decreases as the angle of internal friction increases and the variation may beapproximated by a straight line (Fig. 4-53a). Maximum negative meridional forcein vertical wall below pressure zone in Type-3 decreases with increasing rate withincrease of angle of internal friction (Fig. 4.55a). In pressure zone maximummeridional force always occur at the bottom of pressure zone. But for Type-3 andbelow pressure zone the maximum negative meridional force occur at a distance of4.5 ft. from the bottom of vertical wall.

(ii) Hoop Force: Maximum positive hoop force for all types of silosdecrease with the increase of angle of internal friction (Fig. 4.54a). But maximumnegative meridional force in Type-l and Type-2 increases as the angle ofintemalfriction increases. For maximum positive hoop force, Type-l and Type-2 alwaysgive the greater value than that of Type-3. In this case for lower value of p the rateof decrease is very small and as the value of p increases the rate of decrease alsoincreases. The variation of maximum negative hoop force in Type-l and Type-2can be approximated by a straight line. Below pressure zone in Type-3 bothpositive and negative values of maximum hoop force decrease as the angle ofinternal friction increases (Fig. 4.55b). For the smaller values of p the rate ofdecrease is very small and as the value of pincreases the rate of decreasegradually increases. Variations of angle of internal friction do not affect thelocation of maximum hoop force.

(iii) Meridional Moment: Only negative meridional moment in Type-l andType-2 increases considerably with the increase of angle of internal friction and thevariation may be approximated by a straight line (Fig. 4.56a). Below pressure zonein Type-3 the maximum negative meridional moment decreases appreciably withthe increase of p (Fig. 4.58a). Change in p has no effects on the location ofmiOOmummeridional moments.

69

( b ) Conical Hopper

Fig. 4-53 Change in maximum Meridional Force due to change in angleof internal friction of stored rnaterial.

-'--;.::::.:- l.-------.-:::.-~

----------

50

50

45

45

( a ) Vertical Wall

25 30 35 40

Angle of internal friction ( Degree)Type-1,Type-2 Type-3PosItiVe PositiveNegative Negative

25 30 35 40

Angle of internal friction ( Degree)Type-1,Type-2 Type-3PositiVe PosttlveNegative . . . . . . . . . Negative

20

20

~~..•::.::..-:.':::';..;..;..;..;. ..;.

~..:"..'

~ :...:..;. .... :.. ;..:..•..;..;. .;. •••• ;..,,: •.;.-.;10:

."' ..•.;..;. ..; ...::-..::.: ~'J.';":'-50000

-6000015

50000

~ 30000~ou.(ij5 20000'C.;:CD

::iE10000

=-40000-f1

-10000

o

CDU

~ -30000(ijco'C -40000.;:CD::iE

=-f1 -20000

( b ) Conical Hopper

Fig. 4-54 Change in maximum Hoop Force due to change in angle ofrepose of stored material.

50

50

45

45

25 30 35 40

Angle of repose ( Degree)Type-1,Type-2 Type-3Positive PositiveNegative Negative

( a ) Vertical Wall

25 30 35 40

Angle of repose ( Degree)Type-1, Type-2 Type-3Positive Pos~veNegative . . . . . . . . . Negative

20

20

~ - - ----

.... .... ..... .... ... ........ .... ......... .. ... ... ... .. ... .... .... .. .. .... .. ...

- -."-"-." "'-- ."-'1--- - --- ----- ---- -

o

-3000015

20000

60000

70000

10000

-20000

30000

20000

<I:' 50000--.0-;- 40000~o~ 30000ooI

Q)

~oLL

g- -10000oI

<I:' 10000--,Q

- - - - - - -- - - - - - - -- - - - - - -~ - - - - - - - -- - - - - -- - - - - - - - - - -50

50

45

4525 30 35 40

Angle of internal friction ( Degree)I Material Pressure IPos_ -- Negative - - - - -

25 30 35 40

Angle of internal friction ( Degree)

I Malerial Pressure IPositive -- Negative - - - - -

20

20

( a ) Meridional Force in Vertical Wall below pressure zone

( b ) Hoop Force in Vertical Wall below pressure zone

Fig_4-55 Change in rnaximum Meridional Force and Hoop Force due tochange in angle of internal friction of stored material (Type-3)

- - -- - - -- - - --- - - --- - - --- - - --- - -- - - -- - - -- - -- - -- - - --

15000

-1000015

10000

==--,Q~ 5000'"E,fg- aoI

-5000

-10000

-50000

a

~,f -30000(ijco'i5 -40000-'"'":2

==--,Q -20000

----1-------- --- -1-- -~---

I- - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -'-t'-~

........... ................ ................. ..............

'-50

5045

45403530

( a ) Vertical Wall

25Angle of internal friction ( Degree)Type-1,Type-2 Type-3Positive PositiveNegative Negative

25 30 35 40Angle of internal friction (Degree)Type-1,Type-2 Type-3Positive PositiveNegative Negative

20

20

( b ) Conical Hopper

Fig. 4-56 Change in rnaximum Meridional Moment due to changein angle of internal friction of stored material.

,""- -- ---- - ------ - -- - -- - - -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

f-I-

'-

.... .... '. '" ' ........ .... '. '. '" ' .. '.'" .." .' ..... '. '. '...... '. ". ". '. '.

'" '. '. '.

-600015

.1000015

12000

10000.;=- 8000.;=:Q 6000-c 4000Q)

E0 2000:2iiic 00'C'C -2000Q)

:2-4000

2000

0.;=-.;=:Q

-2000

-CQ)-4000E

0:2iiiC -60000'C'CQ)

:2 .8000

50

50

45

45

403530

( a ) Vertical Wall

25

25 30 35 40

Angle of internal friction ( Degree)Type-1,Type-2 Type-3Positive PositiveNegative Negative

Angle of internal friction ( Degree)Type-1,Type-2 Type-3Positive PositiveNegative Negative

20

20

( b ) Conical Hopper

Fig. 4-57 Change in maximum Circumferential Moment due to changein angle of internal friction of stored material.

--------- --- 1------- ----... - - -

,c- o - - - 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - -, - 0 - - 0 - - - - - - - - - - - - -r .... .... ........ ... .. ... ........ ............. .. .. .... .. ......~.. ........ ..... ..

c

~--- --- -- -- - - -- --- ------ - - -,- - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - 0 - - - - - - -,rCrr

.

c .. ...... .., - ..... ........ ..... .. ....... ..... ........ ..... .. .... ............. ... .. .....

-100015

-200015

4000

<l:'- 3000<l:',a- 2000c:Q)

E0~(ij 1000••c:Q)~.l!1E 0::JU~U

500

<l:'- 0<l:'

a-c: -500Q)

E0~(ij -1000••c:Q)~.l!1E::J -1500u~U

50

50

45

45

25 30 35 40

Angle of internal friction ( Degree)

IMaterial Pressure I

~o61t1ve -- Negative - - - - -

25 30 35 40

Angle of internal friction ( Degree)

I. - Material Pressure I

Pos""" -- Negative - - - - -

20

20

( b ) Circumferential Moment in Vertical Wall below pressure zone

( a ) Meridional Moment in Vertical Wall below pressure zone

Fig. 4.58 Change in maximum Meridional and Circumferential momentdue to change in angle of internal friction of stored material(Type-3)

- --.--- - - -..-- .-.- - - ..-- - - - - .- --- - - - -. - .-.-.-...- ..-.-

- - - --f-- - -- - -- - - -- - - - -- -.- - - -- - .-..-- . - --- - - - -.-.- -...-

-ai -100Eo~iii -200~~.l!!E -300:J~(3

-40015

-200015

500

100

a~ -500-cQl

Eo~-1000iiico'0.5i-1500~

~ 0-~,

(iv) Circumferential moment: Maximum negative circumferential momentin Type-l and Type-2 increases approximately linearly with the increase of p (Fig.4.57a). Below pressure zone in Type-3the values of maximum circumferentialmoments are so small that it can be neglected in design (Fig. 4.58b).

(b) Conical Hopper

(i) Meridional force: Maximum positive meridional force in conical hopperincreases with the increase of angle of internal friction (Fig. 4-53b). In this case forlower value of p the maximum meridional forces in Type-l and Type-2 is greaterthan that of Type-3. But for higher values of p the reverse is true. The variation ofmaximum meridional forces can be approximated by a straight line.

(ii) Hoop Force: Maximum positive hoop force in Type-l and Type-2 arealways smaller than that of Type-3 but for all the types of silos their pattern ofvariations is similar. Maximum hoop forces in conical hopper increases as theangle of internal friction increases (Fig. 4.54b). Variations of p has no effect onthe location of maximum hoop forces.

(iii)Meridional Moment: Maximum positive meridional moment in Type-3increases more or less linearly with the increase of angle of internal friction (Fig.4.56b). The location of maximum meridional moment is independent of thevariation of p.

(iv) Circumferential moment: Fig. 4-57b shows the variation of maximumcircumferential moment in conical hopper due to variation of angle of internalfriction. In this case only the maximum positive values in Type-3 increases linearlywith the increase of angle of internal friction. The location of maximumcircumferential moment are independent with respect to the variation of p .

4.4.9 Effect of Co-efficient of Wall Friction, p'

(a) Vertical Wall

(i) Meridional force: Fig. 4-59a shows the variation of maxnnummeridional force in vertical wall due to material pressure with respect to co-efficient of wall friction, fJ~ In this case the maximum negative meridional force inType- I and Type-2 are slightly greater than that of Type-3. For all the cases themaximum meridional forces increase with a decreasing rate with respect to the co-efficient of wall friction. Below vertical wall in Type-3 the maximum negative

70

( b ) Conical Hopper

Fig. 4-59 Change in maximum Meridional Force due to change in Ji'

:"':'::"':"':.':'-:"':'::.: :"':,. '. - -,:.t:,,: - :"':"':.,.,

':'-:'":": " :.::.,' :,..:.. •.. :.::.,. ':,.' " :" ..•. " ":'" :.- '"':.' " .•.. :.."'.•..•.•.--" - ", ..,

0.7

0.7

0.6

0.6

( a ) Vertical Wall

0.4 0.5Coefficient of wall friction Ji'

Type-1,Type.2 Type-3Positive PositiveNegative ....••••• Negative

0.4 0.5

Coefficient of wall friction Ji'Type.1,Type-2 Type-3Positive PositiveNegative Negative

0.3

0.3

~

"'- ~

I~ -.... --I--.

o0.2

.600000.2

-40000

.50000

o

.10000

;;:::;; .20000

70000

60000

;;::- 50000,9

'"u 40000~0LL(ij

30000c:0'5.;::'" 20000:2

10000

~o -30000LL(ijc:.Q"0.;::

'":2

meridional force also increases in similar manner to that of vertical wall in pressurezone.

(ii) Hoop Force: Maximum positive and negative hoop force exists in thevertical wall due to material pressure for Type-l and Type-2 (Fig. 4-60a). Nonegative hoop force exists in Type-3 in pressure zone. All the values of maximumpositive and negative hoop forces decreases with a decreasing rate as the co-efficient of wall friction increases. Maximum positive hoop force for all the typesare more or less the same. Below pressure zone in vertical wall in Type-3 themaximum positive hoop force decreases with decreasing rate with increase of co-efficient of wall friction. But in this case as the value of ,u' increases the maximumnegative hoop force also increases (Fig. 4-61b). The locations of the maximum. positive and negative hoop forces are independent with the change of,u~

(iii) Meridional Moment: Maximum negative meridional moments inType-l and Type"2 show a considerable variation with the variation of co-efficientof wall friction (Fig. 4-62a). In this case the maximum negative values decreaseswith decreasing rate as the coefficient of wall friction increases. For Type-3 andbelow pressure zone the maximum negative meridional moment increases with theincrease of ,u'(Fig. 4.64a). Change in,u' has no effect on the location of maximummeridional moment.

(iv) Circumferential moment: Only maxnnum negative circumferentialmoment in Type-l and Type-2 decreases considerably with the increase in co-efficient of wall friction ,u' (Fig. 4.63a). The rate of decrease becomes smaller asthe value of ,u' increases. The variation of maximum circumferential moment forother cases may be neglected. In Type-3 and below pressure zone the maximumpositive and negative circumferential moments are very small and can be neglectedin design (Fig. 4.64b). Variations of,u' has no effect on the locations of maximumcircumferential moments.

(b) Conical Hopper

(i) Meridional force: Maximum meridional forces in conical hopperdecrease exponentially with the increase of ,u ~ For all the types of silos thevariation follows more or less the same path (Fig. 4-59b). The locations ofmaximum meridional forces are independent of the variation of,u~

71

0.7

0.7

0.6

0.6

-- --- --- --

( a ) Vertical Wall

-- -

0.4 0.5Coefficient of wall friction /l'Type-1,Type-2 Type-3Positive PositiveNegative . . . . . . . . . Negative

" •.. •..•..

0.3 0.4 0.5

Coefficient of wall friction /l 'Type-1,Type-2 Type-3Positive PositiveNegative Negative

0.3

""" ...."

"

~""'"-.- -- ---

....... ... .......... ....................................- .... ....................... .......' .. .....

L~

( b ) Conical Hopper

Fig. 4-60 Change in maximum Hoop Force due to change in /l'

80000

60000

20000

o0.2

100000

60000

-400000.2

Q)

~oLL

15- 40000oI

40000

=-- 20000,Q

Q)u~0

0LLC-O0I

-20000

Fig. 4-61 Change in maximum Meridional Force and Hoop Force due tochange in Jl' (Type-3).

0.7

0.7

0.6

0.6

Negative .....•.•..

0.4 0.5Coefficient of wall friction, Jl'

Material PressurePositive

0.3

03 04 05Coefficient of wall friction, Jl'

IMaterial Pressure I

Positive -- Negative - - •••••.

( b ) Hoop Force in Vertical Wall below pressure zone

( a) Meridional Force in Vertical Wall below pressure zone

~

~

t ---~t -t

~~tl-

I-I-~

-- .••~------ - ------ ----- ---- ------- ---------- ------ - - -.

- - --- - - - ---- - - - .-- - -- - - - - - - - - - - - - - - - - -.- - -.- - - -- - -.- .- - -

o

-100000.2

-500000.2

20000

15000

=='-,9 10000

QlU~ 50000lLC-O0 0I

-5000

.10000

=='-,9Ql -20000u~0lLCiJc: -300000'5."Ql

::2;-40000

~ - I- - .-

.... .. .... .. .. ............ .. ... ...... .. ....' .... '.....'

.'.'

'" .

"" .•.. .•.. -- - ---- --------- --- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- -

... .. ..... .. .. .... ....." .- .. .. ..

..... ........... .... .. '

0.7

0.7

0.6

0.6

( a ) Vertical Wall

0.4 0.5Coefficient of wall friction Jl'

Type-1, Type-2 Type-3Posltlve PositiveNegative Negative

( b ) Conical Hopper

0.3 0.4 0.5Coefficient of wall friction Jl'

Type-1,Type-2 Type-3Positive PositlveNegative. Negative

0.3

Fig. 4-62 Change in maximum Meridional Moment due to changein Jl'.

15000

-100000.2

-250000.2

<I:' 10000-<I:'8- 5000c:Q)

E0:2 a(ijc:0'5.;:Q) -5000:2

5000

a<I:'-<I:', -50008~-c:Q) -10000E0:2(ij -15000c:0'5.;:Q)

:2 -20000

-

~ >--. - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .- - - - - - - - - - - - -

... .. ...... .,...... .. .. ....... .. .... ........

..... ...... .

.....

..

"- ...• .... .... .... -- ----- I-.. --- 1-- ---- -- - ---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - -- - - -- - - - - - - - - - ... .. ..... .. ... .. .. ............ .. ............... .. .... ..... ......... ..

\

I I

0.7

0.7

0.6

0.60.4 0.5Coefficient of wall friction Jl'

Type-1,Type-2 Type-3Positive PositiveNegative . . . . . . . . . Negative

( a ) Vertical Wall

0.4 0.5Coefficient of wall friction Jl '

Type-1,Type-2 Type-3Poo~~ poom~Negative Negative

0.3

0.3

( b ) Conical Hopper

Fig. 4-63 Change in maximum Circumferential Moment due tochange in Jl'.

-20000.2

-40000.2

1000

~-~ 0,Q-cCD -1000E0~0;:;::; -2000cCD~~E:J -30000~G

5000

~ 4000-~,Q 3000-cCD 2000E0~0; 1000:;::;cCD~ 0CD.~E:J~ -1000G

-

-

-c

I-~L

-L

C

f-I-~..... ------ ..- -"- .. '. --. --- .. --- .. - ....

'" -------- ---- ------- ..

I-~l-f-~~I-

------ .._~... ", ",". ---- ---- ". ------- ---- . -----------

0.7

0.7

0.6

0.6

Q3 Q4 Q5Coefficient of wall friction, 11'

I P05"'~- Material Pressure I

. '".0 -- Negative .

0.3 0.4 0.5

Coefficient of wall friction, 11'

IMaterial Pressure I

Positive -- Negative" .

( a) Meridional Moment in Vertical Wall below pressure zone

( b ) Circumferential Moment in Vertical Wall below pressure zone

Fig. 4.64 Change in maximum Meridional and Circumferential momentdue to change in 11' (Type-3)

-4000.2

500

-20000.2

100

0:::--0::: 0

,Q-c -100CDE0:2c;; -200:;::;cCD~CD-E -300:J<.>~G

0::: 0--0:::,Q

-500-cCDE0:2 -1000c;;c0'6.", -1500CD:2

(ii) Hoop Force: Maximum hoop forces in Type-l and Type-2 are alwaysless than that of Type-3. But the pattern of variation of these are same for all thetypes of silos (Fig. 4.60b). In every case the maximum positive hoop forcesdecreases exponentially with the increase of fJ~ The locations of maximum hoopforces are independent of the variation of fJ ~

(iii) Meridional Moment: Maximum posItIve and negative meridionalmoment in Type-l and Type-2 decreases with decreasing rate as the co-efficient ofwall friction increases (Fig. 4.62b). Maximum positive meridional moment inType-3 is dominating in this case and also show the same pattern of variation asthat of Type-l and Type-2. Locations of maximum meridional moments are fixedwith respect to the change in fJ~

(iv) Circumferential moment: Only in Type-3 the maxIIDum positivecircumferential moment is considerable in conical hopper due to material pressure. (Fig. 4.63b). It decreases exponentially with the increase of fJ'. In this case themaximum values always occur at the junction of ring beam and conicaI hopper.

4.4.10 Effect of Wind Pressure Intensity, q

Variations of maximum values of all stress resultants for all types of siloswith respect to the variation of wind pressure are always linearly increasing. In thesubsequent paragraph these are discussed briefly.

(a) Vertical Wall

(i) Meridional force: Maximum positive meridional force in Type-l andType-2 increases at a linear rate. Maximum negative meridional force in Type-land Type-2 increases at lower rate. The maximum positive and negative meridionalforces for Type-3 in the pressure zone show the same pattern of increase (Fig.4.65a). Fig. 4.67a shows the linear variations of maximum meridional forces inType-3 below pressure zone due to variation of wind pressure.

(ii) Hoop Forces: Fig. 4-66a shows the linear variations of maximum hoopforces with respect to the variation of wind pressure in all the types. For the verticalwall below pressure zone in Type-3 the linear variations of maximum positive andnegative hoop forces due to increase in the wind pressure are shown in (Fig. 4-67b).

72

120

120

100

100

- .._--

.--_.-_._.-._--

( b ) Conical Hopper

( a ) Vertical Wall

40 60 80

Wind Pressure ( pst)

40 60 80

Wind Pressure ( pst)Type-1 Type-2Positive PositiveNegative Negative

20

Type-1 Type-2 Type-3Positive -- Positive --- Positive --_ ...Negative ......... Negative ----- Negative _ .._ ..-

20

--- .•. 0:.

=:':''l:::;,:,-:::_.:.-; ''':::''_:'::::::''-'';"'':- "::':"::::':-_". __ "_. ._.._ ..

_.-'---_.--_.- -'---'_.-

--------- -- -------= --- -----l.---:::'"- - - -~

---:"'. -- '. ".". '--". '. --.". - .. _- -". '. '- '-.". ". -'- -'. '. -- --.'.". --- -'.'.

". ". '.'.". '. '. '.". '.'. ". '. '.

Fig. 4-65 Change in maximum Meridional Force due to change in windpressure

1500

-1500

-20000

1000

- 500

e a~(ij -500co:g -1000Q)

:2

60000

-= 40000-.cQ) 20000()~0u..(ij 0c0'6'CQ)

:2 -20000

-400000

120

120100".

100

----

.-.-.----

---

.-.------

---

( a ) Vertical Wall

( b ) Conical Hopper

40 60 80Wind Pressure ( pst)

Type-1 Type-2Positive PositiveNegative Negative

40 60 80Wind Pressure ( pst)

-----

Type-1 Type-2 Type-3Positive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

".

---

" .

--

....".

".". ....". ".

20

20

---- - -- ---- -- . ---- ---- ---- ---- -- .---- -----

:::.::.: --------- ----_.--_.

Fig. 4-66 Change in Maximum Hoop Force due to change in Windpressure

-40000

-40000

4000

6000

;:--£~ 0&C-OoI -2000

4000

2000

Q)u~~ 0C-OoI -2000

;:--£ 2000

120

120

100

100

----'- --- -'-'-- --- --.

.. --- - .. _-

40 60 80Wind Pressure ( pst)

I Wind Pressure IPositive -- Negative - - - - -

40 60 80

Wind Pressure ( pst)

IWind Pressure I

~OSitive -- Negative - - - - -

- .. _- ---

- .--- --.--- -'- ---- - .. _- --.

20

20

( b ) Hoop Force in Vertical Wall below pressure zone

( a ) Meridional Force in Vertical Wall below pressure zone

---

" .. - ..

-5000

15000

-100000

Fig. 4-67 Change in maximum Meridional Force and Hoop Force due tochange in wind pressure (Type-3)

5000Q)

2ou.a. 0ooI

~ 10000ol=-,Q

80000

60000ol=-,Q 40000Q)<.>~0u. 20000(ijc:.2"0 0'CQ)

:2-20000

-400000

(iii) Meridional Moment: The increase ofmaximmn positive and maximmnnegative meridional moments in Type-I, Type-2 and Type-3 are shown in Fig.4.68a. For vertical wall below pressure zone in Type-3 the variations are shown inFig.4-70a.

(iv) Circumferential moment: The rates of increase of maximmn positivecircmnferential moments for all the types of silos are virtually the same (Fig. 4-69a). For the negative values the rates of increase of the maximmn circmnferentialmoments for all types of silos are also more or less the same. For Type-3, belowpressure zone the variations of maximmn positive and negative circmnferentialmoments are shown in Fig. 4-70b.

(b) Conical Hopper

In Type-3 the conical hopper is completely separate structure and notsubjected to wind load. So the effect of wind pressure variation on stress conditionof silo is investigated only for Type-l and Type-2 and the linear variations ofresults are shown graphically in Fig. 4.65b, Fig. 4.66b, Fig. 4.68b and Fig. 4.69b.

The variations of wind pressure intensity has no effect on the location ofmaximmn meridional force, hoop force, meridional moment and circmnferentialmoment for all the types and for both vertical wall and conical hopper. In Type-2the hopper is not directly subjected to wind pressure because of continuous wallsupport. But it is monolithically constructed with vertical wall which is subjected towind pressure. Due to the effect of wind pressure in vertical wall some momentsand forces are also developed in the hopper and the maximmn forces and momentsare always smaller than that of Type-I. In Type-I, the due to column support windpressure directly acts on the hopper.

4.4.11 Effect of Height of Hopper Bottom Above Floor Level, h'

Height of hopper bottom has no effect on the values of various stressresultants in Type-l and Type-2. It affects only the behaviour of vertical wall ofsilos ofType-3 below pressure zone.

(i) Meridional force: Fig. 4.71a shows the variations of maxnnmnmeridional forces in the portion mentioned above, but change of h' has little effecton the various stress resultants.

73

120

120100

100

-----.-..........

,-'-'-'--

-- .----......

( a ) Vertical Wall

( b ) Conical Hopper

,-'-'-'--

40 60 80Wind Pressure ( psf)

40 60 80Wind Pressure ( psf )

Type-1 Type-2Positive PositiveNegative Negative

20

Type-1 Type.2 Type-3Positive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..-

20

Fig. 4-68 Change in maximum Meridinal Moment due to changein wind pressure.

2000

""1500-

""£l 1000-c:OJ 500E0::2(ij 0c:0'5.;::OJ -500::2

-10000

2250

~

""- 1500""£l- 750c:OJE0::2 0(ijc:0:g -750Q;::2

-15000

•.....

120

120

100

100

----

...

...

... ... ...

( b ) Conical Hopper

40 60 80

Wind Pressure ( psf )

( a ) Vertical Wall

...

40 60 80Wind Pressure ( psf)

Type-1 Type-2Positive PositiveNegative Negative

--

Type-1 Type-2 Type-3Positive -- Postt!ve --- Positive -----Negative ......... Negative ----- Negative _ .._ ..-

20

20

.........----

--~-

Fig. 4-69 Change in maximum Circumferential Moment due to changein wind pressure.

~•••••••~

~ -------~.

~ l-------," ~--...•.•.•..~ .- •.•..•.....•...f- ..•... ..

..,"l'.~~~':':.:::..~_-

450

01:''- 30001:'

f!- 150cCDE0 0::;;0;:;::;c -150CD~CD-E:l -300u~(3

-4500

-7500 o

3000

6000

01:' 4500'-01:'

_ 1500cCDE 0o~ -1500

'":;::;fii -3000~CDE -4500:l2 -6000(3

120

120

100

100

-. '-

--- -'- --- --. -- .... -'- -----

40 60 80Wind Pressure ( psf )I Wind Pressure I

Positive - Negative _ •• __ :

40 60 80Wind Pressure ( psf )

I Wind Pressure I~ositive -- Negative •. _ •. __ ,

-- ". -'- -."

-- --. --

--- -"- .. ---

20

20

'. "- -.-- '- --

---

--

( a ) Meridional Moment in Vertical Wall below pressure zone

( a) Circumferential Moment in Vertical Wall below pressure zone

1000

-30000

5000

3000

Fig. 4-70 Change in maximum Meridional and Circumferential momentdue to change in wind pressure (Type-3).

~ 4000'*"--'7 3000fa

-20000

~C 2000Ql

Ea~c;;ca 015'CQl

~ -1000

fa~ 1000-CQlEa 0~

'*":;;;2000

c;;:;:;

ffi -1000

~i3 -2000~G

Fig.4-71 Change in maximum Meridional Force and Hoop Force due tochange in height of hopper bottom above floor level (Type-3)

28

28

8 12 16 20 24

Height of hopper bottom above floor level (It )

8 12 16 20 24

.Height of hopper bottom above floor level (It )Se~Weight Material Pressure Wind PressurePositive -- Positive --- Positive _._.-Negative ......... Negative ----- Negative _ .._ ..•

Sel/Weight Material Pressure Wind PressurePositive -- Pos.ve --- Positive -----Negative ......... Negative -_._- Negative _ .._ ..-

( b ) Hoop Force in Vertical Wall below pressure zone

( a ) Meridional Force in Vertical Wall below pressure zone

--'

.._ .._ .._ .._. _ .._ .._ .._ .._. _ .._ .._ .._ ..- '-"-"-"-"- .._..... ..... ." .. ... .. .. .. ... .... ..... .. .... ...... .... .. ..

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

---------- ---------- ---------- 1--------- ---

c. .._ .._ .._ .._ ..c. .._ .._ .._ .._ ..,...._ .._ .._ .._ .. .._ .._ .._ .._. _ ...

..... ...... .. .. .... .. .....

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .

.

15000

10000

""--f1 5000~Q)U~0I!- aC-o0I

.5000

-600004

.10000 4

40000

"" 20000--f1Q) al2ol!-e;;c -20000o'0.;::Q)

::;; -40000

28

28

Material Pressure Wind PressurePosItive PositiveNegative Negative

8 12 16 20 24

Height of hopper bottom above floor level ( It )

8 12 16 20 24Height of hopper bottom above floor level (It )

SeWWeigh! Malerial Pressure Wind PressurePositive Positive PosttlveNegative . . . . . . . . . . Negative Negative

SeWWeigh!PosrnveNegative •.•...•.•

( a ) Meridional Moment in Vertical Wall below pressure zone

Fig. 4-72 Change in maximum Meridional and Circumferential momentdue to change in height of hopper bottom above floor level

. (Type-3)

( b ) Circumferential Moment in Vertical Wall below pressure zone

. •..._._._._. +-._._._._. _._._._._ .f-.--'-'-'-'-'.

l-I-I- -- - -- -- --- f-- --- ---- f--

~'~~'.':":".,....".•.•.•• ......_-- ..•_- _._._ .._ .._.'-'--'-0'_'" '.'~

- ~-- - - - - - - - - - - - - - -- - - - - - - - -- - - - - - - - - - - - -

--_._.-.- -. --'--_._.-.- -,-'-'-'-f-

~ ........ .. ..... ...... ......... ................ ......... .. ...- - -- - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - -f- .._ .._ .._ .._ .-..-..-.._ ..-._ .. .-..- .-..-.._ ..-..- .-.

.

-20004

-15004

2000

;:--;: 1000.0-cQ)E 00::2'0;c0'0 -1000'CQ)

::2'

1500~;:--;: 1000

£- 500cQ)

E0 0::2'0;:;:;c -500Q)~

.l!!E -1000::J()~

<3

(ii) Hoop Force: Maximwn hoop force in vertical wall below pressure zonealso does not vaty with the change of height of hopper bottom above floor level(Fig.4.71b).

(iii)Meridional Moment: Change in h' also has no noticeable effect on themeridional moment in the vertical wall below pressure zone (Fig. 4.72a).

(iv) Circumferential moment: Only maximwn circwnferential moment dueto wind load is sensitive to the variation of height of hopper bottom above floorlevel (Fig. 4.72b). Both the negative and positive maximwn circwnferentialmoment increases approximately linearly with the change in h'.

4.5 REMARKS

From the parametric study a nwnber of important decisions can be maderegarding the design values of stress resultants and their locations required for silodesign. This study also revealed the relative importance of various parameters forthe determination of stress resultants and their locations. From this study it is clearthat the effect of stored material is the most predominant nearly for all forces andmoments. Positive and negative meridional forces due to wind are also dominatingfor silo design. Circwnferential moments due to wind load are considerably high insome cases. Only geometric dimensions influence the locations of maximwn stressresultants for all types of silos. Among geometric dimensions the height of verticalwall and internal diameter of silos have considerably greater effects than any othergeometric parameters. Material properties do not affect the locations of maximwnstress resultants, but they play the most significant role in the determination ofmagnitude of the maximwn forces and moments.

***

74

CHAPTERS

A DESIGN RATIONALE

5.1 GENERAL

Investigations carned out in Chapter 3 and 4 has revealed the.characteristicsand overall behaviour of silo under various loading conditions.

In the conventional method the vertical wall and the conical hopper areconsidered as separate structures which are subjected to membrane action only. Butactually they are not so. The restraint provided either by the ring beam or by theground support have significant effect on the overall behaviour of silo. Due to therestraint, moments develop at different locations. Again negative hoop force ofconsiderable amount develop near the restraint.

Conventional method predicts the maximum hoop force in conical hopper atthe junction of ring beam with hopper. But due to either self weight or storedmaterial pressure the maximum hoop force occurs at a certain distance from thejunction of ring beam and conical hopper. For self weight this distance varies from20% to 50% of the overall length L of conical hopper in Type-1 and Type-2. ForType-3 their range is about 10% to 15% depending on the diameter of the silo.Due to stored material pressure maximum hoop force occurs at a distance of 20%to 30% of "L" in Type-1 and Type-2 and this range is about 10% to 20% for

Type-3.

On the basis of this study an attempt in made to propose a simple and directway of finding the moments and forces required to design the various structural

elements of a silo.

5.2 BASIS OF THE PROPOSAL

On the basis of the comparative study discussed in Chapter 3 it can be saidthat the prediction of various stress resultants at different critical locations byapproximate conventional method may not always be acceptable. Besides,traditional approach of analysis can not predict any type of moments at all. Despiteall such approximations and inaccuracies the conventional method of analysis hasbeen used with success in the past. Conservative design approach combined withhigh factor of safety can be attributed to such success.

With the advancement of the techniques of structural analysis it is nowpossible to deal with complex structures using the Finite Element method.However, the application of Finite Element technique may not always be possible.As a result a straight forward method for analysis which will enable one to carryout the calculations easily but with acceptable accuracy is desirable.

Therefore, on the basis of the extensive study, a set of expressions formoments and forces at the critical sections are suggested in the following articles.

5.3 PROPOSED DESIGN RATIONALE

The design of a reinforced silo structure consists of analysis, selection ofphysical dimensions and calculations and placement of reinforcement. Among theabove three steps analysis is the most important. From analysis one gets variousforces and moments required for design.

Silo is a tall structure. Meridional and hoop forces developed in silo varyvertically. For an economic design, these variations must be taken intoconsideration: There are other stress resultants such as meridional moment orcircumferential moment having very localised effect. In this rationale, expressionsfor maximum forces or moments in terms of different parameters (Art. 4.2, Fig.3-7)of silo are presented in tabular form. These expressions are valid within certainrange of variation of the parameters mentioned above. Attempt has been made tocover the usual ranges.

The expressions of various stress resultants are of empirical nature. Hence,care must be taken to use proper units of measurements. For the proposed designrationale the unit for forces is lb. and the unit of moments is lb-ft.. The forces andmoments given by the equation are for unit linear foot of the respective structural

76

element. The valid range of different geometric parameters and their units areshown in the Table 5-1. This table also shows the symbols used for various

parameters.

Table 5-1 Range and Units of Parameters

Parameters . Notation Range Unit

Height of vertical wall H 40 t0280 . foot

Diameter of silo (Internal) D 10 to 100 foot

Inclination of conical hopper with horizontal a 40 to 75 Degree

Top thickness of vertical wall Ttop 4 to 9 inch

Bottom thickness of vertical wall lboffom 6 to 13 inch

Top thickness of conical hopper Itop 6 to 13 inch

Bottom thickness of conical hopper {bottom 4 to 9 inch

Depth of ring beam d 24 to 96 inch

Unit weight of stored material r 35 to 160 Ib/ft3

Angle of internal mction p IS to 50 Degree

Coefficient of wall mction p' 0.20 to 0.70

Intensity of wind pressure q 10 to 100 Ib/ft2

As stated above, vertical variation of various stress resultants, specially, formeridional force and hoop force are important in silo analysis. Maximum values ofvarious forces and moments and their vertical variations are, therefore, presentedgraphically. In this design guide only those functions which have significant effect

on the silo behaviour are considered.

5.3.1.Maximum Values of Stress Resultants

All the equations presented in this article for the computations of maximum

f9rces and moments are of the same form as given by

Maximum Force or Moment = kjj.ji ..t3.f4 /"

Here n is the number of parameters on which the respective functiondepends. k is a numeric constimt and each of jj, ji, fj /" are factorscorresponding to a particular parameter. Values of jj, .ii, fj f,. are different for

77

78

(5-2)(lb/ft)FmM

where Wg = total weight of conical hopperDc = centre line diameter of conical hopper at top in ft.

Hoop force: Table 5-3 shows the equations for hoop force computation

due to self weight .

ii) Conical Hopper

Meridional{orce: Meridional force in conical hopper due to self weight isalways positive and maximum value exists at the junction of ring beam and hopperwall. Maximum meridional force can be computed as follows:

Wg

Fmax= rcH(Tbollom+ T1op) /24 +Dead loadfrom roof (lb/ft). (5-1)

. where Yc = Unit weight of concrete in lb/ft3

Hoop Force: Maximum negative hoop force in vertical wall is relatively

higher and their values can be computed from Table 5-2.

(a) Stress Resultants due to Self Weight.

i) Vertical wall

Meridional force: In this case the meridional force is always compressiveand at any level meridional force is equal to the weight of concrete above that level.Maximum meridional force occurs at the bottom of vertical wall and it is given by

(A) Type-l and Type-2

different stress resultants and one set offj, /2, fi .... f" are applicable for one function

only.

Table 5-2. Maximum Negative (Compressive) Hoop Force in Vertical Wall

(Type-I, Type-2)

(5-4)

(5-3)

Fmax = 169.5 fi.f2.f3.f4'.f5.f6

II 1.0 _ 3.03xI0.3(H _ 39.6/744

/2 1.0 +0. I477(D _10)'137

f, 1.0 _ 0.026(a _ 40)°914

/4 1.0 + 0.1034(1'01' _ 6)°.986

/s 1.0 + 0.0288(hottom _ 4)"37

/6 0.95 +3.63xlO.4 lid_541}'65

Fmax = - 868 fi.fi ..f3.f4.fj.f6fds

/, 1.0 +6.1 x I 0.3 (H - 39.6)°968

.il 1.0 + 0.0204(D _10)1220

.h 10 _ 0.0140(a _ 40)°.794

/4 1.0 + 0.0393(1;01' _ 4)'°02

I5 1.0 + 0.01 45(1/'ottom- 6)0811

I6 1.0 + 0.0375(1101' _6)°972

/7 1.0 + 0.0160(lbottom- 4/°°4

.f8 0.9638 +2.23xI0.4 lid_361}'658

where q = yD [I _ e-41"kHID ]

m« 4p'k

79

i) Vertical Wall

Meridional force: Meridional force is always compressive and maximummeridional forces can be obtained from the following equations:

(a) Stress Resultants due to Stored Material Pressure

Table 5-3. Maximum Positive (Tensile) Hoop Force in Conical Hopper

(Type-I, Type-2)

Circumferential moment: Table 5-7 shows the equations for computingmaximwn circwnferential moment.

(5-5)

(5-6)

1- sinp=---1+ sinp

k

R = hydraulic radius.

Fnmx = - 1273fj.j2 ..f3.f4.fj.f6.fds

Ji 1.0 + 0.0071(H _ 39.6)°827

h 1.0 + 0.0583(D _ 10)'916

13 1.0 - 0.0226(a _40)'°14

.f~ 1.0 - 0.0392(7/'0"om_ 6)'°53

15 1.0 - 0.0243(d _24)°732

16 1.0 + 00285(y- 35)

1, 10 + 0.008(p _ 15)1.311

j8 1.0 - o 8723(u' _ 020)"511

80

where Pd,""" = CJ.k.gnmCd ,= Over pressm'e factor (Table 2.2)k and gn"" are same as that ofEq. 5-4 and Eq. 5-5.

Hoop force: Maximwn negative hoop force can be computed using theequations of Table 5-4. Maxirnwn tensile hoop force in veltical wall due to storedmaterial pressure is given by

Meridional moment: Table 5-5 and Table 5-6 gives the equations formaximwn meridional moment in vertical wall due to grain load.

Table 5 -4. Maximum Negative (Compressive) Hoop Force in VerticalWall (Type-I, Type-2)

Table 5-5. Maximum Positive Meridional Moment in Vertical Wall(Type-I, Type-2)

M",ox = 91.3 ii.fi.fj.f4.fj:f6f7fil:f9

Ji 1.0 + 0.0331(H _ 39.6)°636

h 1.0 + 0.0574(D _ 10)1739

f3 1.0 -0.0141(a _ 40)°961

./4 1.0 + 0.0138(7;01' _ 4)'°23

h 1.0 + 0.1087(760tro", _ 6)0.988

/6 1.0 - O.Olll(d _ 24)°744

/7 1.0 + 0.0286(y- 35)

i8 1.0 + 00046(p _ 15)'027

/9 1.0 - 1.0073(1/'- 0.20)0614

Table 5-6. Maximum Negative Meridional Moment in Vertical Wall(Type-I, Type-2)

Mnuu = - 362fi:fi:fj:f4:fj:f6f7:filfr;.iio

fi 1.0 + 0.0299(H - 396)""1 ; H s: 140'1.0 + 4.2686(H - 39.6)"°534; H > 140'

h 1.0 + 0.0785439(D _ 10)'850..f, 1.0 - 0.01943(a- 40)'013

.f4 1.0 + 0.0066(7;01' _ 4)'°13

/5 1.0 + 0.15483(760"0"' _ 6)°748

/6 1.0 + 0.0124(1/01' _ 6)0.492;ttop:$ 9"

10187 - 0.0029(1/01' - 9l"6; tlop > 9/1

17 10 - 0.0053(d _ 24)°986

18 1.0 + 0.0286(y - 35)

f9 1.0 + 0.007(p- 15)'181

fio 1.0 - 1.0969(u' _ 0.20)0585

81

Table 5 -7. Maximum Negative Circumferential Moment in Vertical Wall(Type-t, Type-2)

Mmax = - 9Ifi..fi ..f3..f4..fj..f6..fj..f"af9-fio

.Ii 1.0 + 0.0295(H _ 39.6)°544 ; H" 140'

1.0 + 2.8482(H - 39.6)"°.4'9 ; H > 140'

11 1.0 + 0.0574(D _ 10)186

j, 1.0 - OOI92(a- 4ol012

.f< 1.0 + 0.0068(T,ol' _ 4)"989

./5 1.0 + 0.1531 (Tnntlo", _ 6)°.749

16 1.0 + 0.0125(/'01' _ 6)°.481; ftop s: 9"( 9)1413 flOp> 9/11.0189 - 0.0031 /top - ;

1, 1.0 - 0.0047(d-24)

.18 1.0 + 0.0285(y- 35)

19 1.0 + 0.0069(p _ 15)1168

ho 1.0 - 1.0908(u' _ 0.20)°586 .

ii) Conical hopper

Meridionalforce: Table 5-8 shows the equations for computing maximummeridional forces.

Hoop force: Table 5-9 show the equations for computing maximum hoopforces.

Meridional moment: Table 5-10 and Table 5-11 shows the equations forcomputing maximum meridional moments.

Circumferential moment: Table 5-12 to Table 5-13 shows the equations forcomputing maximum circumferential moments.

82

Table 5-8. Maximum Positive (Tensile) Merid,ionalForce in ConicalHopper (Type-I, Type-2)

Fmnx= J 5792fi.fifj.f4fj.f6.f7.f8f9-fio

ii 1.0 + 0.0332(H _ 39.6)°.491; H ~ 230'1.437; H> 230'

/; 10 + 0, I677(D - 10)'517

./3 09569 + 000021 {Ia _ 55.01}B45

f. 1.0 + 00059(T,op - 4)

./3 1.0 - 0.0073(lbottorn _ 6)°83

f6 1.0 - 0.0061 (ItoI'_ 6)°9f, 10 - 2. 7x I0.4(d _ 24)J549

Is 1.0 + 0.0286(r- 35)

f9 1.0 + 0,0082(p _ 15)]219

flO 1.0 - 1.0918(u'- 0.20)°578

Table 5 -9. Maximum Positive (Tensile) Hoop Force in Conical Hopper(Type-I, Type-2)

Fmax= 1999.5 fi.fi.fj.f4jf6..f7jiJg.fio.fi,

ii J.O + 0.D316(H - 39.6t51 ; H~ 180'1.3934 ; H > 180'

/; 1.0 + 0.151984(D _ 10)J581

f3 1.0 - 0.0123(a- 40)°897

j, I.0 + 0 0064(7;0{' _ 41005

h 1.0 - 001385(7bottorn- 6)°731

fr, 1.0 - 0.0186(1101'- 6)"764

f7 1.0 - 0.0046(lbottorn- 4)0.882

j; 1.0 - 0.0024(d _ 24)'°96

f9 1.0 + 0.02861(r- 35)

flO 1.0 + 0.008(p _ 15)]22.1

iii 10 - LlI56(u'- 020)0566 .

83

Table 5-10. Maximum Positive Meridional Moment in Conical Hopper(Type-I, Type-2)

M_ = 82 ft..fj..fj.f4.fj..fr,.j7.fs..f:;'.fJo

.Ii 1.0 + 0.0433(H - 39.6)"385 ; H s: 210'1313 ; H > 210'

f2 10 + 0.1416(D _ 10)'733

j, 10 - 00282(a _ 40)°91

./4 1.0 - 0.04 I2(Tbottom- 6)".833 ;Tbottoms: 8.5"09116; Tbotto11l > 8. 5"

f5 1.0 + O.0848(t,op _ 6) [[6

f6 1.0 + 0.0506(tbuttom _ 4)°.991

I 1.0 ~0.032(d _ 24)"634

js 1.0 + 0.0286(y- 35)

f9 1.0 + 0.0061(p- 15)1305

flO LO - 11307(u' _ 0.20)°539

Table 5-11. Maximum Negative Meridional Moment in Conical hopper(Type-I, Type-2)

Mmax = - 267.6 jj.f]..fj.f4..fj.f6.f7..fS.f:;'.fJo.fj I

Ji LO + 0.0265(H _ 39.6)°684

h 1.0 + 0.0411(D _ 10)'858

13 1.0 - 0.0439(a _ 40)°706

j, LO + 0.0061(1;01' - 4)

is LO + 0.21 67(Tbottom_ 6)°.702

f6 LO + 0.0637(1,o1' _ 6)°515 ; 'top::; 9.5"L lOS - 0.01 56(t,op - 9.5)'478; 1'01'> 9.5"

f7 LO - 0.0139(lbottum _ 4)°998

18 1.0 - 0.1044(d _ 24)°353 ; d s: 72"0.5906 ; d > 72"

f9 LO + 0.0286(y- 35).

flU 10 + 0.0027(p_15)"896; P s: 40°1.0416 - 9.1xlO"\p- 15f127; p> 40°

.Ii I LO - 0.9078(u' _ 0.20)"6)7

84

Table 5-12. Maximum Positive Circumferential Moment in ConicalHopper (Type-I, Type-2)

Mnwx = 14.tififjf4fjf6f7f;'•• .Ii 1.0 + 0.0255(H _ 39.6)°521 ; H s; 180'

1.2911 - 1.9xlO-5(H - 180.0)'474;H> 180'

12 1.0 + 0.2266(D _ 10)1756

f; 1.0 - 0.0583(a _ 40)°755

.t4 1.0 + 0.1328(1'01' _ 6)'°5'

h 1.0 - 0.0344(d _ 24)°651

.16 1.0 + 0.0284(y- 35). 1, 1.0 + 0.0081(p- 15)1238

Is 1.0 - 1.1 034(1" _ 0.20)°544

Table 5-13. Maximum Negative Circumferential Moment in ConicalHopper (Type-I, Type-2)

Mm= = - 41 1,fi.jjf4.jjf6f7 ..t;'.j~.tjo

1, 1.0 + 0.0266(H _ 396)°757

.Ii 1.0 + 0.0243(D _ 10)'913

f; 1.0 - 0.0386(a _ 40)°637

.t~ 1.0 + 0.4128(1;'ottom _ 6)°703

15 1.036 - 0.0259 (It,op _ 7.5»'68

16 1 ( )09'61.0 - 0.0 55 Ibotto",- 4

.f? 1.0 - 0.1236(d _ 24)°31 ; d S; 66"

0.6062 ; d > 66"

j;' 1.0 + 0.0284(y - 35)

19 1.0 - 3.8x10.4(p_ 15)1716

.lio 1.0 - 0.789(1" _ 020)°712

85

(c) Stress Resultants due to Wind Pressure

Wind load analysis of Type-I and Type-2 differs only slightly. So in thisdesign rationale only Type-I and Type-3 only discussed. Effect of wind load onconical hopper is very little and can be ignored for design purpose. Therefore, the

various stress resultants discussed blow are for veltical wall only

Meridional force: Table 5-14 and 5-15 Shows the equations for computing

maximum meridional forces.

Hoop force: Table 5-16 shows the equations for computing maximum hoop

forces.

Meridional moment: Table 5-17 shows the equations for computing

maximum meridional moments.

Circumferential moment: Table 5-18 and Table 5-19 shows the equations

for computing maximum circumferential moments.

Table 5-14. Maximum Positive (Tensile) Meridional Force in Vertical

Wall (Type-I, Type-2)

Fm= = 3200//.f2.13.f4.f5

// 1.0 + 0.0973(H _ 396)°.622; H S 160'

10 + 0.0037(H - 39.6)130\ ; H> 160'

/2 0.616 +23xlO-4{ID _ 20Il238l; D s 60'

1.460 + 0.0187(D _ 60)°686 ; D> 60'

/3 1.0 - 0.075(1;up- 4)".68\

14 1.0 - 0.0365(.hmom _ 6)".948

15 10 + 0 1087(q - 92)

86

Table 5 -15. Maximum Negative (Compressive) Meridional Force inVertical Wall (Type-t, Type-2)

Fmax= - 6525 fi./i.fj.f4.fj

fl 10 + 0.14(H - 39.6)"418 ; H ,; 110'

1.0 + 7.8xl0.6 (H - 396)23"3 ; H > 110'

f2 0.3078 + 00632{\D - 30[}om

J2 I 0 - 0 14(T _ 4)"601. . top

f4 1.0 _ 00507(1hottom _ 6)°855

f5 1.0 + 0.1087(q - 9.2)

Table 5 -t6. Maximum Positive (Tensile) Hoop Force in VerticalWall (Type-I, Type-2)

Mmax = 230 fi.f;.fj.f4.fj

fl 1.0 + 0.1287(H - 39.6)°'°' ; H,; 110'

1.0 + 0.0105(H - 39.6)1300 ; H> 110'

.il 0.9296 - 0.0064{\lJ - 601}1.199.

f, 1.0 - 0.1 I83(1;op _ 4)°.690

f4 1.0 - 0.1962(1hottom _ 6)°.798

f5 1.0 + 0.1082(q - 92)

Table 5-t 7. Maximum Positive Meridional Moment in VerticalWall (Type-t, Type-2)

Mmax = 173.fI.i; ..fj.f4

fl 1.0 + 0.0363(H _ 396)°.489

/2 0.7503 + 0.0163(ID - 201}1.118

f3 1.0 _ 0.0455(1;op _ 4)°745

f, 1.0 + 0.1 084(q - 9.2)

87

Table 5-18. Maximum Positive Circumferential moment in Vertical

Wall (Type-I, Type-2)

Mmax = 155.!J.f2.jj..f4..fj

/, 1.0 + 00464(H _ 396)"947 ; Hs \40'

4.285 + 0.0376(H _ 140)°452; H> 140'

/2 1.0+0.148\ (D_IO)I251; D S 50'

\ 1.5375 -2.4xl0-<CD - 50)2426;J) > 50'

f, 10; T 6'top S

1.0 + 0.0552(T,op- 6) 0.973 ; T > 6'lop

/4 0.8652 + 0.0079{ITbottom - 9.5\}2148

/5 10 + 0 I085(q - 922)

Table 5-19. Maximum Negative Circumferential moment in Vertical

Wall (Type-I, Type-2)

Mmax = - 102fi..fi.jj..f4.fj

/, 1.0 + 0.2336(H _ 39.6)°604 ; Hs 160'

5.18 ; H> 160'

/2 12.397 - 2.2xlO-6{ID - 501)'-31

f, 1.0; T 6'lop S

1.0 + 0.0534(1;op _ 6)°995 ; 1~op> 6'

/4 0.8663 + 0.0071 (llbottom- 9.51}2"8

/5 1.0 +0.1 085(q - 922)

(B) Type-3

(a) Stress Resultants due to self weight

i) Vertical Wall

Meridional jhrce: Maximum meridional force occurs at the bottom of the

vertical wall and this value can be obtained using Eq. 5-1.

88

Hoop Force: Table 5-20 shows the equations for computing maximum

hoop forces.

Table 5-20. Maximum Negative (Compressive) Hoop Force in Vertical

Wall (Type-3)

Fmax = - 890 ji.f2.fj.f4.fj

fl 1.0 + 0.0126(H _ 39.6)°990

f2 1.0 + 0.0099(D - 10)°918

f, 10 + 8.3xI0.4(a _ 40)1452

f4 10 + 0.0737(7~op- 4/°

f5 1.0 + 0.0797(1'bollom_ 6)°.996

ii) Conical Hopper

Meridional Force: Eq. 5-2 can be used for the computation of maximum

meridional force in the conical hopper

Hoop force: Table 5-21 shows the equations for computing maximum hoop

forces.

Table 5-21. Maximum Positive (Tensile) Hoop Force in Conical Hopper

(Type-3)

F = 168 fi.f2.fj.f4.fjmax

fi 1.0 +0.2711(D _ 10)1126

.il 1.0 + 8x 10.7(a _40)338

.13 1.0 + 0.0898(1'01' _ 6)°867

f. 1.0 + 0.0213(loollom _4)1039

f5 1.0 - 0.287(d - 24)°11

89

90

(5-8)

(5-7)

Fmax = 712 Ji.f2.Jj'./4.fj

/1 1.0 + 0.0499(H - 39.6)°61'

/2 1.0 + 0.1519(D - 10)1329

./3 1.0 + 00286(y- 35)

/4 1.0 _1.82xl0.4 (p- 15lo15

/5 1.0 - 1.0907(fl' - 0.20)°707

Table 5-22. Maximum Positive (Tensile) Hoop Force in Vertical WallBelow Pressure Zone (Type-3)

Meridional moment: Table 5-24 and Table 5-25 gives the equations for

maximum meridional moment in vertical wall due to material pressure load.

where Pd.max = Cd-k.qmaxCd = Over pressure factor (Table 2.2)

k and qn"" are same as that ofEq. 5-4 and Eq. 5-5.

Table 5-22 and Table 5-23 gives the equations for computing maximum

positive and negative hoop force in vertical wall below pressure zone.

Hoop force: Maximum tensile hoop force in vertical wall in pressure zone

due to stored material pressure is given by:

Maximum meridional force in vertical wall below pressure zone can be

taken equal to that of pressure zone.

Symbols have the same meanings as those ofEq. 5-3.

i) Vertical Wall:

Meridiana/force: Meridional force is always compressive and maximum

meridional forces can be obtained ii-om the following equations:

(b) Stress Resultants due to Material Pressure

Table 5-23. Maximum Negative (Compressive) Hoop Force in VerticalWall Below Pressure Zone (Type-3)

I Fm= = - 3055 fi.fi.fj.fj.fj

Ii 1.0 + 0.0281(H _ 396)llU5

.h 1.0 + 01783(D - IOt744

j, 1.0 + 00284(y - 35)

14 10 _ 0.0040(p _ 15)1252

.f; 1.0 + 0.8143(et' _ 0.20)0703

Table 5-24. Maximum Positive Meridional Moment in Vertical Wall inPressure Zone (Type-3)

Mm= = 30.5 fi.fi.fj.fj.f5.f6.f7

I, 1.0 + 0.0666(H _ 39.6)U559

12 1.0 + 0.1802(D _ 10)1292

13 . 1.0 + 0.0305(7~up _ 4)'003

14 1.0 + 0.128(Tbullom _ 6)0.952

15 1.0 + 00285(y- 35)

16 1.0 _ 1.06210.4(p- IS) 2.165

17 1.0 - 1.1123(/1'- 0.20)°704

Table 5-25. Maximum Negative Meridional Moment in Vertical WallBelow Pressure Zone (Type-3)

MmQx = - 45.5fi.fi.fj.jj.jj.j6

I, 1.0 + 0.0 134(H _ 39.6)1169

12 1.0 + 0.1773(D _ 10)°822

j, 1.0 + 0.1813(7601l0m_ 6)0994

14 1.0 + 0.0285(y- 35)

Is 1.0 - 0.004(p _ 15)1254

16 1.0 + 6.6257(p' _ 020)208 ; p' " 0.4

1.0 + 0.5591(p' _ 0.20)°649 ; p' > 0.4

91

ii) Conical hopper

Aferidional force: Table 5-26 shows the equations for computingmaximum meridional forces.

Hoop force: Table 5-27 shows the equations for computing maximum hoop

forces.

Meridional moment: Table 5-28 and Table 5-29 show the equations forcomputing maximum meridional moments.

Circumferential moment: Table 5-30 shows the equations for computingmaximum circumferential moments.

Table 5-26. Maximum Positive (Tensile) Meridional Force in ConicalHopper (Type-3)

Fmax = 341.5 fi.j2..f3..f4..fj..fr,./j..fid;

II 1.0 + 0.0379(H _ 39.6)°501 ; H:<:: 230'

1.5257 ; H> 230'

I2 1.0 + 0.6539(D - 10)1556

f, 1.0 _ 0.0038(a _ 40)°929; a:<:: 65°

0.9244 ; a > 65°

J4 1.0 _ 4.9xlO-4(1tup- 6)2507

.is 1 0 ( )1022. -0.0013 Ibollom-4

I6 1.0 - 0.0022(d _ 24)1223

/7 10 + 0.0286(y- 35)

Is 1.0 + 0.01193(p_15)1242

I9 1.0 - 1.1 032(fl , _ 020)°585

92

Table 5-27. Maximum Positive (Tensile) Hoop Force in Conical Hopper(Type-3)

Fm= = 9556 jj.f2.jj.!+.f5.f6.fdS.f9j; 10 +O.0394(H _ 39.6)°501 ; H s; 210'

1.517; H> 210'

h 10 + 03162(D _ 10)1.654

13 1.0 - 0.0054(a _40)1.099

I. 10042 - 49xlO.4{1t,op _ 7.01}2.954

15 10 - 0.005(fbott'Om_ 4)1.02

16 1.0 - 0.0212(d _ 24)°773

17 10 + 0.0286(y- 35)

18 10 + 0.0 12(p _ 15)1.238

19 10 - 1.1233Cu' - 0.20)°572

. Table 5-28. Maximum Positive Meridional Moment in Conical Hopper(Type-3)

Mm= = 55.3 ij.f2.fjj4.j5.f6.fds.f;j; 10 + 0.0428(H _ 39.6)°468 ; H s; 210'

1474 ; H> 210'

12 10 + o 2268(D - loi2117

13 0.7508 - O.OOII{la- 5001}1791

I. 1.0 + 0.2698(1,'01' _ 6)0.77

i, I 0 ( )0986. + 0.0154 fbottom- 4

16 1.0 - 0.1803(d _ 24)°408 ; d s; 72"

0.1251; d> 72".

17 10 + 0.0286(y- 35)

.Ii 1.0 + 0.0 119(p _ 15)1.241

19 10 - 1.l041(,u' - 0.20)°583

93

Table 5-29. Maximum Negative Meridional Moment in Conical Hopper(Type-3)

Mmax = - 602 fifi ..f3I-I.fj.f6.f7.f's.f;

// 10 + 0.0471(H _ 39.6)°467 ; H ,,; 210'

15178 ; H> 210'

/1 10 + O.OIOI(D _ 10)2153

/3 1.0 - 0.0383(a _ 40)°.814 ; a ,,; 70°

0.3897 ; a > 70°

14 10 + 0.1421(ttop _ 6)°.683

15 1.0 + 0.1 153(tbottorn_ 4)1.193

16 10 - 0.1 087(d - 24t519 ; d,,; 84"

0.09 ; d> 84"

17 1.0 + 00287(y- 35)

j,; 10 + 0.0124(p _ 15)1229

19 1.0 - U408(p' _ 0.20)°561

Table 5-30. Maximum Positive Circumferential Moment in ConicalHopper (Type-3)

Mmax = 903.5 /J2.fj.f4.fj.f6f7.f's

1/ 10 + 0.0492(H _ 39.6)°434 ; H,,; 210'

1.4575 ; H > 210'

/2 1.0 + 62x I 0.4(D - 10)2491

f, 10 - 0.0844(a _ 40)°488

j4 1.0 + 0.297(ttop _ 6)°.95

15 10 - 0 I S67(d _ 24)°404 ; d,,; 84"

0.0238 ; d> 84"

/6 10 + 0.0285(y- 35)

/7 1.0 + 0.0118(p _ 15)1242

.f8 10 - U06(p' _ 0.20)°582

94

(c) Stress Resultants due to Wind Pressure

Meridional force: Table 5-31 and Table 5-32 show the equations for

computing maximum meridional forces.

Hoop force: Table 5-33 shows the equations for computing maximum hoop

forces.

Meridional moment: Table 5-34 to Table 5-35 show the equations for

computing maximum meridional moments.

Circumferential moment: Table 5-36 and Table 5-37 show the equations forcomputing maximum circumferential moments.

Table 5-31. Maximum Positive (Tensile) Meridional Force in VerticalWall (Type-3)

IFmax= 5900fifdj.j4.tj I

fl 1.0 + 0.0043(H - 39.6)1085 ; H os; 160'

1.0 + 3.4x I0.4(H _ 39.6)1623; H > 160'

.12 0.635 + 9.0xI0.4 {ID - 201}1849; D OS; 70'

1.0 + 5.9xI0''(D _ IO)1I93 ;.

J) > 70'

f; 10 _ 0.0383(T,op _ 4)°752

f, 1.0 - 0.0405(1botlom _ 6)°891

is 1.0 + 0.1087(q - 9.22)

95

Table 5-32. Maximum Negative(Compressive) Meridional Force inVertical Wall (Type-3)

Fmnx = 9600fJd3.j~.j5

./J 1.0; H<;, 140'

10+3.86xI0.3(H_140.0)1245;H> 140'

f2 0.2688 + 00l33{ID _ 301}1204

f; 1.0 - 0.0755(Ttop- 4)°541

j, 1.0 - 0.0398(7/'ottom - 6)°.748 ; Tbottom <;, II"

0.866 ; Tbottom > 11"

f; 10 + 0 1086(q - 9.2)

Table 5-33. Maximum Positive (Tensile) Hoop Force in VerticalWall (Type-3)

Fmnx = 1260 ./J..fi..fj.j~.j5

fl 1.0 + 6.64x I0.3 (H - 39.6)°9'7; H <;, 160'

10 + 3.5xI0-4 (H - 39.6)1609; H > 160'

j2 0.6649 + 5x 10.5 ID _2012733; D <;, 60'

0.6649 + 0.0425{ID - 201}oS49;D > 60'

f3 1.0 - 0.0355(7;op _ 4)°762

f4 o 1(7' )0895I -0.041 hottom- 6

f5 10 + 0.1085 (q - 9.22)

96

Table 5-34. Maximum Positive Meridional Moment in VerticalWall (Type-3)

Mm= = 23411.fi.jjldj

II 1.0 + 4.2xlO-\H _ 39.6)1362 ; H,s; 180'

1.0 + 1.8x 10-4 (H - 396)1534 ; H > 180'

12 1.0 + o 0283 (ID-301} 1087

13 1.0 - 0.01 82 (Ttop _ 4)°.868

14 1.0 + 0.1658(7bottorn _ 6)°936

15 1.0 + 0.1084(q - 9.22)

Table 5-35. Maximum Negative Meridional Moment in VerticalWall (Type-3)

Mm= = - 33611.ji.jj.f4.fj

j, 1.0 - 0.0 116(H _ 39.6)°637

12 0.459 + 1.24xlO-3{ID _201}1904

13 1.0 - 0.0406(7;op _ 4)°854

14 1.0 + 0.1532(Tbottorn _ 6)°809

Is 1.0 + 0.1 089(q - 92)

Table 5-36. Maximum Positive Circumferential moment in VerticalWall (Type-3)

Mm•x = 29.4jj.h.fj.j4.jj

II 1.0 + 0.0935(H - 39.6)°496 ; H ,s; 160'

1.94 ; H > 160'

12 1.0 + 1.4059(D _ 10)°626

13 0.9734 + 0.0085 (ITIOp-5.51} 1886

.f, 0.9061 + 0.0068(17bottom-91} 1417

15 1.0 + 01087(q - 9.2)

97

Table 5-37. Maximum Negative Circumferential moment in VerticalWall (Type-3)

Mmax = - 29.Mi..fj.fdj

fi 1.0 + 0.0923(H _ 396)°496 ; H::: 140'

1.92; H > 140'

.il 1.0 + 1.4778(D _ 10)°636 ; D ::: 60'

1.0+29.777(D-IO)"0", ; 1) > 60'

/3 0.9732 + 874xl0.3(17;"p _ 5.51}!.'6'

/4 0.9069 + 4.96xlO.3 (17/,ottom - 9.01}3.o78;7/'0"om ::: 11'

0.935 ; Thottom > 11'

/5 1.0 + 01086 (q - 9.2)

5.3.2 Variation of Stress Resultants in Vertical Direction

The variations of various stress resultants in vertical direction are shown inthe Fig. 5-1 to Fig. 5-21. Only those fimctions (stress resultants) are consideredwhich vary considerably in the vertical direction. The value of a function at anyvertical level may now be easily obtained using its maximmn value calculated fromthe previous section.

98

.f}t

12010040 60 80FIF max in percent

20

Fig. 5-1 Positive (tensile) Meridional Force in conical hopper due to selfweight (Type-1, Type-2).

70

60

a

40

90

80

20

30

10

100 a

1::~Ql

a. 50.f:<:!.•••

1201008020 40 60

F/Fmax in percent

o

Fig. 5-2 Hoop Force in conical hopper due to self weight(Type-1 ,Type-2).

-20

o"- t'~'::'::,:",,.'f(...-

"~"-.."" ....~

.•... ~"" :~.~.~.~'. ....~.

' .. ....~,'. ,.~

. '.' .. ",f-

\f-

I"'- / ...//

:X~<,/,' IV - ..... ,/ .,' ,

/..' ,,,/ , ,

/ ,/ 'ao')/.' ,

D = 50' .....40',', ,I ,,

l,' ,,

~,1..- ,f-l.:

,f-

I .... ,,/f- ,' .... ,

,

/•,

I ..•. ,,I: ,

I.:',

/,,'.: ,

/I...."I.... "I,' ,, ,

,.: I...'j. ,: ,. ,

f- I, ,1/

,~ ,

Ii • /,II

.••,-' •,; ,, , , , , ,

90

80

60

40

30

20

70

10

100-40

1::~'"a. 50.5~•••

12010040 60 80

F/Fmax in percent20

Fig. 5-3 Meridional Force in vertical wall due to stored material pressure

(Type-1, Type-2).

20

10

o o

30

40

70

60

80

90

100

1::~Ql

C. 50.s

Fig. 5-4 Hoop Force in vertical wall due to stored material pressure(Type-1, Type-2).

100 12575-25 0 25 50FIF max in percent

-50

~ r%L. ~'.,~ ';". ,

~ \ ....l','.~ \ .... ,L \ .... \~ \ .... \~ ' .'. \, '.~

~

H 80'L " ",

1\ H = 140~ ',;

~H =180'

L , H = 240,

\ \.,1" ", \" ..•. \, ",

, ,,,\' ,\\'~ \'" ,' '.L ' ..,,,

.....\,\ \\'

\\ '. 1\ ',\',:. \

\''1"-: \

"'1'."

. ~. .--k.,

-.~-.~.r.... ~.':'".;::.r.:".:--:-.:-.:-.:-.:".:".:" ,.,.~.:::B.o-100 -75

10

30

20

70

80

40

90

100

1:: 60(J)e(J)a..S 50:c>:

12010040 60 80FIF max in percent

20

Fig. 5-5 Positive (tensile) Meridional Force in conical due to storedmaterial pressure (Type-1, Type-2).

80

70

90

60

o

30

10

20

100 o

'E 40CDeCDa..S 50

<:!.'"

1201008040 60

F{Fmax in percent

20

/ ..../.... ,

/.... ".....,"'30'

/' ,I

/.: ,I." '

I": "1/ ".' ,I.: ,

I.' ,/.... ,... ,. ,

I.: ".... '/:' ,

I...."I "

I ".:,',•... ".. '...I: "~.... '

/...-',/.' ,

/.,' "/ ,,' ,D - 50' / ..... "- / ,,' "

/ .... 0:::',,' ,

-- .....

o

Fig. 5-6 Hoop Force in conical hopper due to stored material pressure(Type-1, Type-2).

o

90

70

80

60

40

30

20

10

100-20

Fig. 5-7 Positive (tensile) Meridional Force in vertical due to windpressure (Type-1, Type-2).

12010080

D = 50'

40 60F/Fmax in percent

D = 20'

..o = 30/~'•.•

...• ..

20

.,. ,,.••••.•••.

••.

20

30

10

o o

70

,,,80 '

60

90

100

1:~ 50

'"a.J:J: 40):'

12010080

,,,,, ,, ,"' ,".•.

,, ,,,"' "',

"'

,,,

60in percent

,,

"" "'D = 30' "

"

40F/Fmax

D= 20'

,,,,

,,,

,,,,,,,,,

20

Fig. 5-8 Negative (compressive) Meridional Force in vertical wall due towind pressure (Type-1. Type-2).

80\ .\ .•\ .•70 •I'.\ '.\ '.\ '.60

30

20

10

90

100

"E 50~~CDC.c.- 40

~

Fig. 5-9 Positive Circumferential Moment in vertical wall due to windpressure (Type-1, Type-2)

12010040 60 80M/Mmax in percent

20

e- oe- o,e- oe- o0

00

••0• I•.' •••.. 00

00f- .. •.. 0

.. .e-o0f-

0.c .

/..' 0

00

/.,D = 50' ....

,,,

~=.. D = 30'.,/

20'..' ,.,,...... ,

../.' ,,.' "

~

.'.' ,,..' ,,..' .,,,.... ,

t .........:.,"~

e-:;.~f<;';",.'oo

90

80

70

60

30

100

10

20

'E~~ 50

.S

~>- 40

12010040 60 80

M/Mmax in percent

20

Fig. 5-10 Negative Circumferential Moment in vertical wall due to windpressure (Type-1, Type-2)

~:1:1:\

.

J

" ,,J ',I ••I :

I •I •I :I ••I ••I,,,

I ,,/ ,/

,/,

/ ,,, /- I ,

- I ,,,- D= 40' I ,.,l- I ...l- I , 'D = 30'

{=20'l- I '

l-I,' ••>" /l- /'

I'I'

" /I- •. .I- •. .'

/I' .'I- ..- / ~,'I- ..- "",•. .

~/ .'

~//10

20

70

o o

30

60

80

90

100

'E~ 50~'"C-.SI40>:-

12010040 60 80

FIF max in percent

20

10

Fig. 5-11 Positive (tensile) Meridional Force in conical hopper due to selfweight (Type-3).

o

30

70

80

90

100o

- 40c:~~QlC-o!: 50...J.•.•.'"

60

12010080

20'

60

FIF max in percent

40

D = 50' / .....;10. •• ••I . ,"""''''30'1 .....,

.:.•:::..:..:. .._..... --- .. -':,:,,:,"-,:,':.:..;. ..:.~...

,.... ,I.' ,: ,,.•.. ". ,

I:' ,I: I

,/ "...,: ,: .I: ,...

20

--

1/,'...• "f.:" "

(.: :..: .

'/ "Ij -''/ ": .

If :':" :I J :..

60

40

30

Fig. 5-12 Hoop Force in conical hopper due to self weight (Type-3)

o

10

20

70

80

90

100 o

E~Qlc.. 50.S<:!.•••

12010040 60 80FIF';'ax in percent

20

Fig. 5-13 Meridional Force in vertical wall due to stored materialpressure (Type-3).

f\\\

I\.

\\

J\.

~

'\"'.

.~

"\

"".~

.

.

10

20

90

30

100

60

70

80

1:~2i 50

.0::I~ 40

o1:~~(])

Co 50c

.t::.">.100 o

125

\\\\

\ .. '" \,,'. ... \'. ...•. \

'. " \.• •.. \'. •... \'. .•... \

.• ",

H = 80'H = 140'H = 180'H = 240'

50 75 100

F/F max in percent

25

~•.......•.

~.~.~~~.~.'. ...... " .

Fig. 5-14 Hoop Force in vertical wall in pressure zone due to storedmaterial pressure (Type-3).

90

o o

80

70

60

40

30

20

10

100

"EQ)u

!50.!:

7550,25 0 25

F/F max in percent

(Fmax = maximum positive hoop force in pressure zone)

,50

Fig. 5-15 Hoop Force in vertical wall below pressure zone due tostored material pressure (Type-3).

0-~

/.

v

.

-.

0-f-l-

.

I- H 80'H= 140'

\ \ .\. /}f- H= 180' /.ycH=240'- --, ,/,' ,

,,' ,

J.. -\::: 1'7.' ,.' ,

-' -...........:,.. '- . ,

o

20

10

30

70

80

90

1°C!y5

40-C'"<.l~'" 50C-.".c>:. 60

12010040 60 80

F/Fmax in percent

20

Fig. 5-16 Positive (tensile) Meridional Force in conical hopper due tostored material pressure (Type-3).

o

10

30

20

70

60

90

80

100o

12010060 80FIF max in percent

4020

,.... ",.... ',.... ,I.. •. ,.. ...,.•.. ".. ,

I: ,,•.•."I. ... "

I... ,.. .I:' "

I:' ,.. ,I.. ,..

I.....". ,/... ,....• ,"...

0=50' / ....~~•••••••/ " , 30't.' ... .

I'" ".. .

.,. ....,.' ,,'

././

/ ...•. ,"I" "I.: ,. ,

Fig. 5-17 Hoop Force in conical hopper due to stored materialpressure (Type-3).

60

90

80

70

30

o

20

10

100 o

40-c::~Qla..5; 50

<:!.•••

10080

D = 40'

'.".......•...•...•...•..•.

D = 3~~~...•~ .....":::" .

40 60F/Fmax in percent

, '..••., ",.•.., ....•.•.' .....

.. ..•.~.•.•..~......•.

20

Fig. 5-18 Positive (tensile) Meridional Force in vertical wall due to windpressure (Type-3).

80

90

70

10

20

100

60

-c:Q) 50()~,QlC-

.5,I 40--,>-

D = 50'30

120100

D = 40'

•.•... ....•.. .:..:,..:.;...

80

..•••. ",...•

..

......0=50'..

......

" ." ., '., '.

..•••. , .

':.....

40 60

F/Fmax in percent

...........

,D = 30'

..

......•..•..

\ •...\ •...\ .•...\ ....\ '.

\\\

20

D = 20'

•.. .80 \",

,": ",\ "\: \

:. \,•.. .

70

90

Fig. 5-19 Negative(compressive) Meridional Force in vertical wall due towind pressure (Type-3).

100

10

20

30

60

-c~Q) 50c..S,~40,>-

1201008060in percent

40M/Mmax

20

Fig. 5-20 Positive Circumferential Moment in vertical wall due to windpressure (Type-3)

, I,,~,

• E•••,,••

.

'/i.

i'. " •/, ,

/, •,

" .,.., ••i/ •

= •...., ,•i, ,

Ie / •D = 50'",-

,Ie ,.. / ,t- ,

.' / ,t- .' / ,,

,,Ie /

,t- D = 40'- .' / ,,t- .; ,,.' ,t- ..' ,

.; ,,

.' ,,

~20'

c- .; ,.; ,Ie

.; ,'D = 30'........•.: '" ,,

,,,, ./

.' .- .,..... /'.'.;

'".' '" ,"1/ ,,

~....

"..../ ,,.''; ::---....;;;.~.•.•~.•

.'" ':.--.$'. .•.•

,

20

10

30

100

60

70

90

80

-cCDe~ 50

.f:,~>- 40

1201008060M/Mmax in percent

4020

Fig. 5-21 Negative Circumferential Moment in vertical wall due to windpressure (Type-3)

v~.\, .~~,,,••,,

j/:/

i'1/...., ,/, ••

•.•J •....1 ,,

...., ,,......1 ,,1 ,

••• J ,,... / ,

.. ,,0=50: ..../ ,

~ ,/ ,.. ,

/ ,.... / ,.

0=40' t-- / ,/ ,... ..

/ ...~ / ,,

~20'

........ ",/,

~'0 = 30'~ .. .•• ,.......<. .•• ,,~ .. .••.••,

... ..- .. V.•• ~---.. .•• ........;~.•• ,,

~.,.,,.

,,'/ e----...?;;//',oo

90

70

80

10

20

100

60

-cQ) 50()~Q)a..£:,

40J:-,>-30

5.4 COMPARISON OF RESULTS OBTAINED FROM PROPOSEDDESIGN RATIONALE WITH THOSE OF FINITE ELEMENTANALYSIS

To show the acceptability of the proposed design rationale, results from thesuggested equations are compared with those of Finite Element analysis. For thispurpose five examples are considered arbitrarily within the scope of the equations.Various parameters for silo analysis used in these examples are shown in the

Table 5-38.

. Table 5 -38. Data (parameters) for Five Example

EX. NO. H D a T"" T ••••• ••• ••••••• 1 P I" d q

FT. FT. DEGREE INCH INCH INCH INCH LEIIT' DEGREE INCH LBIIT'

1 100 30 60 5 10 8 5 60 30 0.40 48 36.86

2 130 30 60 6 11 9 7 50 35 0.50 42 50.18

3 130 25 55 7 9 9 6 80 40 0.45 36 25.60

4 180 35 65 7 . 12 11 4 70 30 0.50 54 57.60

5 160 40 60 6 9 10 5 65 25 0.60 60 43.26

Table 5-39 to Table 5-47 show the comparison of maximum values of stressresultants for various loading conditions and for various Types of silos. Theagreement between the two sets of results are remarkably satisfactory and thevariations of the results obtained from proposed equations are within acceptablelimit in comparison to those of Finite Element analysis.

99

Table 5-39. Maximum Forces and Moments due to Self Weight(Type-I, Type-2)

VERTICAL WALL CONICAL HOPPER

EXAMPLE MERIDIONAL FORCE HOOP FORCE MERIDIONAL FORCE HOOP FORCE

NO, FlNITE PROPOSED F1NITE PROPOSED FlNITE PROPOSED F1NITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

I -10168 10375 -2014 -2018 1320 1224 595 613

2 -14605 14812 -2525 -2489 1582 1485 678 705

3 -13565 13843 -2245 -2240 1110 1074 591.0 620

4 -22440 22531 -3355 -3475 2188 2032 743 728

5 -16377 16313 -3455 -3600 2117 1962 1002 1009

Table 5-40. Maximum Forces and Moments due to Stored MaterialPressure in Vertical Wall (Type-I, Type-2)

EX SIGN MERIDIONAL FORCE HOOP FORCE MERIDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO, F1NITE PROPOSED F1NITE PROPOSED F1NITE PROPOSED F1NITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

I +ve 0,0 0,0 25243 26291 1685 1675 254 -.

-ve 27329 27458 12668 13136 8503 8515 1673 1678

2 +ve 0,0 0,0 19255 19792 1483 1446 212 --ve 32717 32888 10921 11087 6761 6957 1332 1372

3 +ve 0,0 0,0 25492 23588 1704 1616 267 --ve 41142 41688 18747 18256 9018 8703 1765 1710

4 +ve 0,0 0,0 39563 40095 3057 3005 406 --ve 83013 83204 15913 15100 10679 10878 2139 2175

5 +ve 0,0 0,0 41215 41439 2739 2449 456 --ve 81341 81008 24868 21299 12269 10457 2433 2089

100

Table 5-41. Maximum Forces and Moments due to Stored MaterialPressure in Conical Hopper (Type-t, Type-2)

EX. SIGN MERIDIONAL FORCE HOOP FORCE MERIDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED

ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

1 +ve 36116 35750 37675 37466 1457 1413 349 353

1 -ve 0.0 0.0 0.0 0.0 4004 3964 720 716

2 +ve 2m7 27049 28700 27849 1346 1287 323 3ll

2 -ve 0.0 0.0 00 0.0 3540 3468 630 616

3 +ve 36998 35373 40227 38840 2047 1862 546 494

3 -ve 0.0 0.0 0.0 0.0 4262 3922 643 595

4 +ve 53018 51196 50975 48366 2091 1953 476 513

4 -ve 0.0 0.0 0.0 0.0 6610 7103 1199 1266

5 +ve 50910 42431 53008 43565 2392 1901 589 414

5 -ve 0.0 0.0 0.0 0.0 6676 6565 1178 1196

Table 5-42. Maximum Forces and Moments due to Wind Load in VerticalWall (Type-t, Type-2)

EX. SIGN MERIDIONAL FORCE HOOP FORCE MERIDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED

ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

1 +ve 16827 15452 602 824 807 808 1363 1274

-ve 11396 11508 810 - 564 - 1597 1651

2 +ve 22106 22309 627 760 942 1114 2371 2353

-ve 11525 8618 1117 - 610 - 2765 2763

3 +ve 9773 10944 534 560 412 486 936 912

-ve 4084 6949 548 - 230 - 1090 1446

4 +ve 33692 33910 948 909 1226 1482 3887 4108

-ve 15192 18592 1589 - 895 - 4529 4076

5 +ve 25449 31253 1689 2035 1308 1254 1704 3193

-ve 17826 18076 1437 - 960 - 2002 2681

101

Table 5-43. Maximum Forces and Moments Due to Self Weight (Type-3)

VERTICAL WALL CONICAL HOPPER

MERlDIONAL FORCE HOOP FORCE MERIDIONAL FORCE HOOP FORCE

EX. PRESSURE ZONE BELOW PRESSURE ZONE

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

I -9344 '9531 -13619 -13843 -2702 -2678 1236 1186 1083 1076

2 -13726 -13911 -18497 -18744 -3662 -3660 1499 1462 1122 1216

3 -13284 -13535 -16504 -16844 -3260 -3277 1040 1015 903 912

4 -21275 -21346 -28073 -28231 -5542 -5574 2071 2040 1352 1554

5 -15734 -15643 -20660 -20625 -4089 -4031 1967 1954 1320 1761

Table 5-44. Maximum Forces and Moments due to Stored MaterialPressure in Vertical Wall in Pressure Zone (Type-3)

EX. SIGN MERlDIONAL FORCE HOOP FORCE MERlDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

I +ve 0.0 0.0 24755 25315 793 787 88 -

-ve 27891 28162 0.0 0.0 56 - 86 -2 +ve 0.0 0.0 18739 18985 690 631 57 -

-ve 33191 33731 0.0 0.0 40 - 91 -

3 +ve 0.0 0.0 22519 22656 728 680 57 --ve 41835 42757 0.0 0.0 55 - 108 -

4 +ve 0.0 0.0 38678 38710 1555 1473 128 --ve 83699 85338 0.0 0.0 128 - 223 -

5 +ve 0.0 0.0 39945 39813 1235 1032 153 --ve 81983 83086 0.0 0.0 56 - 120 -

102

Table 5-45. Maximum Forces and Moments due to Stored MaterialPressure in Vertical Wall below Pressure Zone (Type-3)

EX SIGN MERIDIONAL FORCE HOOP FORCE. MERIDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

1 +ve 00 0.0 11516 11324 254 - 330 --ve 27954 28162 5527 5577 1170 1174 234 -

2 +ve 0.0 0,0 8886 8297 334 - 42 -

-ve 33263 33731 6560 6576 1494 1423 299 -

3 +ve 0,0 0.0 10607 10027 351 - 45 -

-ve 41922 42757 8251 8363 1506 1444 301 -4 +ve 0.0 0.0 17949 17936 925 - 120 -

-ve 83873 85338 16500 16329 4000 3901 801 -5 +ve 0.0 0.0 18199 16363 703 - 110 -

-ve 82144 83086 16221 16020 2996 2851 600 -

Table 5-46. Maximum Forces and Moments due to Stored MaterialPressure in Conical Hopper (Type-3)

EX SIGN MERIDIONAL FORCE HOOP FORCE MERIDIONAL MOMENT CIRCUMFERENTIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSEDELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

I +ve 33816 33892 46342 46817 4754 5485 1284 1162-ve 0,0 - 0.0 - 260 242 190 -

2 +ve 27151 27063 36718 37750 4926 5905 1394 1230-ve 0.0 - 0.0 - 286 294 7 -

3 +ve 37307 36247 52709 53141 7243 7983 2315 2363-ve 0.0 - 0.0 - 456 430 6.53 -

4 +ve 47914 45m 59372 60641 7987 8820 2191 1695-ve 0,0 - 0.0 - 272 279 8 -

5 +ve 46263 38034 59593 50963 5776 6990 1679 1133-ve 0.0 - 0.0 - 289 260 6.0 -

103

Table 5-47. Maximum Forces and Moments due to Wind Load in VerticalWall (Type-3)

EX. SIGN MERIDIONAL FORCE HOOP FORCE MERIDIONAL MOMENT CIRCUMFEREl'.'TIAL MOMENT

NO. FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED FINITE PROPOSED

ELEMENT EQliATION ELEMENT EQUATION ELEMENT EQUATION ELEMENT EQUATION

1 +ye 18741 18704 3809 3982 1730 1641 1731 1831

-ye 10056 8477 2039 - 886 894 2020 1926

2 +ye 26318 27414 5360 5772 2590 2568 2552 2757

-ye 11234 10854 2273 - 1086 1181 2972 2978

3 +ye 12426 13694 2534 2995 944 1257 924 1193

-ye 5283 7564 1053 - 344 445 1076 1252

4 +ye 42460 42279 8639 8592 4129 4133 4032 3836

-ye 17575 22379 3520 - 1549 1536 4694 4019

5 +ye 32605 35582 6634 7315 2572 2688 2855 3146

-ye 21535 20405 4365 - 1632 1262 3334 3205

***

104

CHAPTER 6

CONCLUSIONS

6.1 GENERAL

Findings regarding the silo behaviour under different loading conditionsare based on the extensive analysis of different types of silos having a widerange of parametric variations, using "Axisymmetric thick shell Finite Elementprogram". A single circular silo has been taken as the prototype. So the fIndingsare basically valid for singular circular silos.

The original program was a general one for the analysis of any type ofaxisymmetric shell with axisymmetric or non-axisymmetric loading. Since theoriginal program required a large number of data for a complex structure like silo,modifications have been made to simplify data input. Modification have also beenmade to obtain the design functions (stress resultants) from the output of theprogram directly.

For the detail investigation of silo behaviour under vanous loadingconditions, thirteen parameters were selected for study. For each parameter anumber of different data sets varying over a wide range were used for the analysisof silo to investigate the influence of the respective parameters on various stressresultants. In this parametric study over one thousand solutions have been madefor silos having wide range of various parameters. On the basis of this extensivestudy a design rationale has been suggested in the previous Chapter.

The computer program developed in this study can analyse various types ofsilos following a number of choice of the user. As for example, either WSD orUSD method of analysis can be followed. One can analyse a silo in FPS or SI unit.Method of stored material pressure computation can also be selected for analysis.

A separate computer program has also been developed for the analysis ofring beam which can deal with ring beam of any type of silos. The forces andmoments required for both ring beam design and column design can be obtaineddirectly using this program.

6.2 FINDINGS FROM THE INVESTIGATION

From this study the actual behaviour of silo under various loadingconditions has become obvious. Conventional method of analysis can notincorporate all of the possible loadings because of its analytical limitations. Asa result it can not predict a number of forces and moments which should beconsidered in the silo design. But the versatile Finite Element method can fmdout all the required stress resultants easily. Conclusions drawn from the resultsof this study are summarised below with regard to the various loadingconditions:

(a) Self weight and stored material pressure.

i) Meridional force and hoop force in the vertical wall:

Conventional method of analysis is in close agreement with the FiniteElement method of analysis of vertical wall for meridional force andhoop force due to self weight and stored material pressure. The twomethods, however, differ widely near the ring beam. Conventionalmethod can not predict any negative hoop force in the vertical wall. Butfrom Finite Element analysis it is found that considerable amount ofnegative hoop force develops at the bottom of vertical wall. This must bedue to the restraint provided by the ring beam at the bottom of verticalwall.

Maximum meridional force always occurs at the bottom of vertical walldue to stored material pressure and self weight. Maximum tensile hoopforce develops at a small distance apart from the bottom of vertical wall.For design purposes maximum tensile hoop force due to stored materialpressure can be assumed to be maximum at the bottom of vertical wall,but the section should be checked for the maximum negative hoop force.

ii) Meridional moment in the vertical wall:

Due to stored material pressure cOlisiderable negative and' positivemeridional moments occur at the bottom of vertical wall for all types of

106

silos. This is also due to the partial fixity provided by the thickened ringbeam and is only obtainable from Finite Element analysis.

iii) Meridional force and Hoop force in the conical hopper:

Conventional method predicts maximum meridional force and hoopforce due to material pressure at the top of hopper. This is true for themeridional force but hoop force is not maximum at the top of hopper innone of the three types of silos. The hoop force due to stored materialpressure is maximum at a distance of 10% to 25% of the total length ofhopper from the junction of ring beam and hopper. This distance variesdepending on the types of silos and diameter of silos. This should betaken into consideration during silo design.

iv) Meridional moment in the conical hopper:

Considerable amount of both positive and negative meridional momentdevelop in conical hopper due to stored material pressure in Type-l andType-2. Maximum negative meridional moment occurs at the top ofhopper and maximum positive values occur at a distance of 12% to 15%of the overall length of the conical hopper wall from top of hopper. Butin Type-3 positive meridional moment of considerable amount occurs atthe junction of ring beam and hopper (hopper top). In this casemaximum negative meridional moment is insignificant.

(b) Wind Load

i) Meridional force in the vertical wall:

Finite Element analysis shows that wind load produces considerableamount of both tensile and compressive meridional forces in verticalwall. Maximum tensile meridional force always occurs at e = 0° for allthe height of the vertical wall on a horizontal section. Negativemeridional force is maximum at e = 180° for most part of the verticalwall except near the ring beam. Near the ring beam negative meridionalforce is maximum between e = 105° to 120° on a horizontal section. Ona vertical section both positive and negative meridional force increaseswith increasing rate as the distance from the top of vertical wallincreases and both become maximum at the bottom of vertical wall.

107

t..-

ii) Circumferential moment in the vertical wall:

Significant amount of Circumferential moment of both posItIve andnegative signs develop in vertical wall due to wind load. Maximumpositive and negative circumferential moments occur near the top ofvertical wall. Conventional method of analysis is completely unable topredict any circumferential moment in the vertical wall.

(c) Temperature effect

Both meridional and circumferential moments develop due totemperatIIre difference between inside and outside of silo. Theconventional method can not consider the effect of ring beam at thebottom. Near the ring beam both the circumferential moment andmeridional moment may be as high as 1.5 times of the value predictedby the conventional method.

6.3 THE DESIGN RATIONALE

Based on the investigation on different types of silos by Finite Elementmethod a simple and straightforward way of analysis and design has beenpresented in Chapter 5. The proposed design guide is applicable for silos having awide range of various parameters. For usual dimensions of silos the proposedequations provide the values of various stress resultants which is in close agreementwith the Finite Element results.

Using the equations in the suggested design rationale one cail easily findout the maximum value of a force or moment. The magnitude of the same functionat any vertical level can then be found using the appropriate design curves. There isno need of an elaborate structIIral analysis. This will relieve a designer from therigorous calculation required even in the conventional method. Besides, theproposed rationale provides a number of important forces and moments which cannot be predicted by conventional method at all.

In this study equations and a set of design curves have been suggested forthe determination of meridional and circumferential moments due to temperaturedifference between inside and outside of silo. Using this equation and curves adesigner can find out the meridional and circumferential moments either in thevertical wall or in the conical hopper wall. Effect of restraint provided by the ringbeam of greater thickness has been taken into consideration in this procedure.

108

..

6.4 SCOPE FOR FUTURE RESEARCH

In this study single circular silo is considered for analysis. Effect of variousloading conditions on this type of silo are investigated in details. On the basis ofthis investigation a design guideline has been fonnulated. However, more remainsto be done in future in this field. Some indications of future study are given below:

i) Only circular silo is considered in this research. But there may be silos ofother shapes including rectangular silos, polygonal silos etc. So further work

may be done for rectangular or polygonal silos.

ii) The behaviour of a single silo and that of a group may be different.Behaviour of silos connected in group of any shape may be investigated in

future.

iii) Nonnally silos are supported by vertical wall. But for other types of silos'the vertical wall and conical hopper may be supported separately (Type-3).In this study For Type-l and Type-2 the support is considered at thejunction of vertical wall and conical hopper. Here,' only verticaldisplacement is considered to be zero and horizontal displacement of ringbeam is not restrained. But the colunm or wall must provide some restraint,in horizontal direction. This may be considered in the future study.

iv) Roofs of silos may be rested on the top of vertical wall in different manner.It may allow free horizontal displacement of the top of vertical wall or itmay' be anchored to the vertical wall by dowel bars so that the horizontal .displacement of vertical wall at top may be fully or partially restrained. Inthis study it is considered that top roof do not affect the horizontaldisplacement of vertical wall at top. So there is a scope of study for othertypes of joirtts between roof and vertical wall. .

v) Conical hopper may be of concrete or steel. In this study silos with concreteconical hopper are investigated in details. In the computer program anoption is included to analysis a silo with steel hopper, but detailinvestigation has not been carried out. Thus there remains a scope of further

study.

vi) The computer program developed in this research can analyse a silo usingeither Janssen's method of pressure computation or Reimbert's method ofpressure computation due to stored material. Detailed study is carried out

109

using Janssen's method of pressure computation. Effect of material pressureon the overall behaviour of silo using Reimbelt's method may be

investigated in future.

vii) Study on full-scale operating silos canied outby Blight G. E. in his researchon "Pressures exerted by materials stored in silos" has shown that thesimple Janssen arching theory provides a good estimate of the hOlizontalpressure with depth in cylindtical silo if used in conjunction with realisticmaterial parameters. For more realistic results, there remains a scope forfurther study of actual data from full-scale operating silos.

viii) In this study wind load is assumed to be unifonnly distributed vertically.Investigations may be earned out using wind pressure varying in vertical

direction.

ix) Types offoundation may affect the behaviour of silo. Also types of silo maydictate the nature of foundation. So effect of foundation may be included in

future study.

***

1.10

REFERENCES

1. ACI Committee 318, "Building Code Requinnents for ReinforcedConcrete (ACI 318-83), "American Concrete Institute, Detroit, 1983.

2. ACI Committee 313, "Recommended Practice for Design and Constructionof Concrete Bins, Silos, and Bunkers for Storing Granular Materials, ACIStandard 313-77 and Commentary," American Concrete Institute, Detroit,revised 1983, 38 pp.

3. Ahmad, S. "Curved Finite Elements in the Analysis of Solid, Shell andPlate Structures," PhD. Thesis, Unviersity of College of Swansea, 1969.

4. Ahmad, S., Irons, B. M., and Zienkiewicz, O. c., "Curved Thick Shelland Membrane Elements with Particular Reference to AxisymmetricProblems," Proc. 2nd Conf on Matrix Methods in Structural Mechanics,Wright-Patterson Air Force Base, Ohio, October, 1968.

5. Ahmad, S., "Axi-Symmetric Thick Shell Element Program (Non-Symmetric Loading) - Listing," Computer Program Report, No. 22,University of Wales, Swansea, 1969.

6. Airy, W., "The Pressure of Grain ,"Minutes of Proceedings, Institution ofCivil Engineers, London, V. 131, 1897, pp. 347-358.

7. Alauddin, Md., "Finite Element Analysis ofAxi-Symmetric Structures withSpecial Reference to Silo," B. Sc. Thesis, Department of Civil Engineering,BUET, September, 1992.

8. Bransby, P.L., Blair-Fish, P.M. and James, R.G., "An Investigation ofthe Flow of Granular Materials," Powder Techno/., Vol.8, 1973, pp.197-206.

9. "Circular Concrete Tanks wuthout Prestressing," Publication ST-57,Structural Bureau, Portland Cement Association, Chicago.

10. Clague, K. and Wright, H., "Pressures in Bunkders," American Society ofMechanical Engineers. Material Handling Division. Paper No. 73-MH-4,1973.

II. Colijn, H., and Peschl, I. A. S. Z., "Non-symmetrical Bin Flow Problems,"Bulk Solids Handling Journal. V. 3, 1981, Trans Tech Publication,Clausthal-Zellerfeld, West Gennany.

12. Cowin, S.c. and Sundaram, V., "The Effect of Material Compressibility onStatic Bin Pressures," Powder Technol., Vol. 25, 1980, pp. 225-227.

13. Finitel, Mark., Hand Book of Concrete Engineering. Vand NostrandReinhold Co., New York, 2nd Edition, 1985

14. Gray, W.S., and Manning, G.P., Concrete Water Towers. Bunkers, Silosand other Elevated Structures, Concrete Publication Ltd. London, FourthEdition, 1964.

15. Huda, N. Md., "Optimum Design of Intze Tanks and Supporting TowersUsing Finite Elements," M Sc. Thesis, BUET, July 1984.

16. Irons, B., and Ahmad, S., Techniques of Finite Elements. Ellis HorwoodLtd., John Wiley & Sons, 1980.

17. Janssen, H. A., "Versuche uber Getreidedruck in Silozellen," Z. Ver. dt.lng., Vol. 39,31 Aug. 1895, pp. 1045-1049.

18. Jenike, A.W. and Johanson,J.R., "Bin Loads," 1. Struct. Div. Am. Soc.Civ. Eng., Vo1.94,No. ST4,ApriI1968, pp. 1011-1041.

19. Jenike, A. W., "Gravity Flow of Bulk Solids," Bulletin lO8. University ofUtah, Engineering Experiment Station Salt Lake City.

20. Jenike, A.W., "Storage and flow of solids," Bul!. Utah Eng. expoStn. No.123, 1964.

21. Jenike, A.W; Johanson, J.R., and Carson, J.W., "Bin Loads part 2,3 and4," Publication No. 72-MH-l,2,3, American Society of MechanicalEngineers. New York 1972.

24. Ketchum, M. S., The Design of Walls, Bins, and Grain Elevators,McGraw-Hili, New York, 1909.

29. Peschl, I. A. S. Z. "Beitrag zur Sicheren Berechnung der Silos," Die Muhle,Oct. 1973 (Germany)

Technicle",Pieper, K, and Stamon, K, "Lesten in Niedrigen Silos'Universitat Braunschweig, Germany, Mar. 1981.

31.

30. Pieper, K, and Wagner, K, "Der Einfluss Verschiedener Auslaufarten aufdie Seitendrucke in Silozellen," Alifbereitungs-Technik, No. 10, Oct. 1968,pp. 542-546.

28. Peschl, I. A. S. Z., "Design of Industrial Bulk Powder Handling Facilities,"Presented at International Conference of Bulk Solids Storage, Handling andFlow, Stratford upon-Avon, England, Nov. 16-18, 1976.

26. Lipnitski, M. E. and Abramovitsch, Sh.P., Zhelezobetonnie Bunkera i

Silosi (Reinforced Concrete Bunkers and Silos), Izdatelstvo Literaturi PoStroitelstvu, Leningrad, 1967.

27. Peschl, I. A. S. Z., "Powder Testing Techniques and their Application on.Solving Industrial Problems, " Presented at International Conference ofBulk Solids Storage, Handling and Flow, Stratford upon-Avon, England,Nov. 16-18, 1976.

25. Khan, M. Amanat, "A Design Rationale for Free Standing Stair SlabBased on Finite Element Analysis," M Sc. Thesis, Dept. of CivilEngineering, BUET, Sep. 1993.

23. Johanson, J. R., "Methods of Calculating Rate of Discharge from Hoppersand Bins," Trans. Am. 1nst. Min Metall. Eng., Vol. 232, March 1965, pp.69-80.

22. Johanson, J.R. "Stress and Velocity Fields in the Gravity Flow of BulkSolids," J. Appl. Mech., Series E, Vol. 86, Sept. 1964, pp. 499-506.

32. Reimbert, M. and Reimbert, A., Si/os-Traite Theorique et Pratique,Editions Eyrollees, Paris, 1956.

33. Reimbert, Marcel, and Reimbert, Andre, "Pressions et Surpressions deVidange des Silos" (unpublished)

34. Reimbert, Marcel, and Reibert, Andre, Silos- Theory and Practice, TransTech Publications, 1st. edition, 1976, Clausthal, Germany.

35. Roberts, I., "Pressure of Stored Grain," Engineering, Lond., Vol. 34, Oct.27, 1882, p. 399.

36. Safarian, Sargis S., and Harris, E. e., Design and Construction of Silosand Bunkers, Van Nostrand Reinhold Company, New york, 1985.

37. Safarian, Sargis, "Design Pressure of Granular Materials in Silos," ACIJournal, proceedings, V. 66, NO.8,Aug. 1969, pp. 647-655.

38. Safarian, Sargis S., and Harris Ernest e., "Determination of Minimumwall Thickness and Temperature Steel in Conventionally ReinforcedCircular Concrete Silos," ACI Journal, Proceedings. Vol. 67, NO.7, July1970, pp. 539-547.

39. "Ukazania Po Proectirovaniu Silosov Dlia Siputshich Materialov"(Instructions for Design of Silos for Granular Materials), Soviet Code CN-302-65, Gosstroy, Moscow, USSR, 1965.

40. Walker, D.M., "An Approximate Theory for Pressures and Arching inHoppers,"Chem. Eng. Sci., Vol. 21, 1966, pp. 975-997.

41. Walters, J. K., "A Theoretical Analysis of Stresses in Axially-SymmetricHoppers and Bunkers," Chem. Eng. Sci., Vol. 28, 1973, pp. 779-789.

42. Williams, J.e., The Rate of Discharge of Coarse Granular Materials fromConical Mass Flow Hoppers, School of Powder Technology, University ofBradford, 1974.

43. Zienkiewicz , O. e., The Finite Element Method, McGraw-Hill BookCompany Ltd., 1977.

APPENDIX

RING BEAM DESIGN

A-l-l INTRODUCTION

Conical hoppers of either reinforced concrete or metal sometimes supportedat their upper edges by closed circular members or ring-beams. Reinforcedconcrete ring-beams are used for both metal and concrete hoppers. The ring-beammay rest on a thickened lower wall or on an independent circular wall. In eithercase, analysis is simple, as the ring-beam is subject only to axial compression andvertical-plane bending moment.

Analysis is more difficult when the ring-beam is supported on columns,(Type-lor Type-3) a common situation more or less dictated by requirements foropenings in the lower silo walls, clearance for equipment below the hopper, and soon.

A-1-2 THEORY AND ANALYTICAL PROCEDURE

Fig. A-l-l shows a typical case, in which a steel conical hopper issupported by a ring-beam of pentagonal cross section, which is supported, in tum,by equally spaced columns. These columns are assumed to be eccentric to the ring-beam centroid, as shown by Fig. A-1-2. Eccentricity as shown, with the columnshifted toward the inside of the ring, is considered positive.

Fig. A-1-3a shows the total Forces and Moments acting on the frame.These are:

(a) An uniformly distributed Torsional MomentM, per unit length of thering beam.

(b) An uniformly distributed Horizontal Force Fxper unit length of ringbeam.

Section A

Silo 'inside dio,

Pion obove hopper

A

l

Fig. A-l-l Concrete ring beam columns supporting a conical steel hopper.

Fig. A-I-2 Ring beam cross section showing total forces acting on beam.

No

.. ' ..... '. ",' '".

. '. '.. .,. .''. ," . " .. ... . .' - . .. . . . .

...:...'..:.-_ .....:...._._'~.::..:... .....:....:: .. , .' '.: .:. ," ...

.: .',:: .... ;::." .. ", '.'". '"'

reol.x R I rr

b I r,r = b + r, I

I: b, ~ISilo inside radius = r5

~ ••

r3Fy

b2 Outline of!y equivalentrectangle

(outside)L

(c) Column free body,showing loads appliedby ring-beam.

( inside)

Ring - beam centroid

MA

(a) Basic frame

Column <l

(b) Free body for segmentof ring-beam, showingforces' applied by column.

Fig. A-1-3 Force system acting on ring beam and columns.

..• ...r' l.. -~~

(A-I-8)

(A-I-6)

(A-I-?)

(A-I-5)

(A-I-4)

(A-I-3)

(A-1-2)

(A-I-I)

a=2y

Fg = self weight per unit length of ring beam.Fv = meridional force in the vertical wall at the bottomMv = meridional moment in the vertical wall at the bottom

(For Type-3 both Fv andMv are zero)

A-2

Fq = q bj

Fp = Pba2

F.,= Fmcosa - Fp + Fa cosa

Fy = Fmsina + Fv + Fq + Fasina + Fg

M, = Fme" + Faes + Mv + AiJ, + Fq (h/2 - x)- Fp (aj - al2 - y) - r~v

where Fa = Va ~a/ +b,'

Fig. A-I-4 shows details of various types ofloadings by which a ring beammay be subjected. Using these loadings Fx• Fy and M, of Fig. A-I-2 can becomputed as follows:

The height a and width b of the equivalent rectangle are

The coordinates of the centroid of the pentagon from the origin 0 (Fig. A-1-2) are:

- a]b]' /2 - (a,b, /2)(b] - b2 /3)x = ---~--~---~A,

(c) An uniformly distributed Vertical Force Fy per unit length of ring-

beam.The cross-sectional area of ring-beam (Fig. A-I-2) is:

qb = Vertical pressure at the top of ring beam

Pb = Lateral pressure at the top of ring beam

q a = Normal pressure on the inclined surface

Va = Friction per unit area

p~= Pb

"8-a."en

b,

y

Fig. A-1-4 Ring beam cross-section with various loadings

X

Centroid~a,

(A-I-9)

(A-I-lO)

(A-I-l1)

(A-I-12)

(A-I-B)

)L'l.\ (due to force Fx) =Fxr 2/ArEr

~2 (due to force HA) =Har 3KI(2Erfry)

82 (due to moments M: applied to the ring-beam by the column)

81 (due to applied uniformly distributed torqueM,)

= 12Mtr/E,a 3 In(rlr[J

Factors C2 and C3 are shown by Table A-I-2 and Table A-I-3. Each factordepends on the number of columns and on the ratio of vertical bending-to-torsional

Rotations of the ring-beam at point A (positive if counterclockwise on Fig.A-I-3b) are:

A-3

With the above limitations, any axis through opposite columns or throughcenters of opposite spaces between columns is a symmetry axis. There are but tworedundants - column shear, HA, and top column moment, MA. The two equationsrequired are merely statements that column and ring-beam have: (1) equal radialdisplacements at point A; and (2) equal rotations atA.

Radial displacements (positive inward) of the ring-beam at point A are:

The following derivation is for even numbers of equally spaced columnswith equal eccentricities and with lower ends fixed.

(Values of K2 for even numbers of support points from 4 to 12 are shown by TableA-I-I. fry is moment of inertia of the ring-beam about its vertical axis).

I':" = meridional force in the conical hopper at the junction of hopper andring beam

Mil = meridional moment in conical hopper at the junction of hopperand ring beam

All forces and moments described above are for unit length of ring beam and thedirections shown are positive.

Fig. A-I-3(b,c) shows free bodies for a portion of ring-beam (extending tothe centers of adjacent spaces between columns) and for the column, respectively.

stiffness, as defined by A = Er Irx /G K. Table A-I-4 gives value of torsion factor Kfor rectangular cross sections.

Table A-I-I. Numerical Coefficients K2

Number of supports K2

4 .012159

6 .003364

8 .001387

10 .000701

12 .000404

Table A-I-2. Values of Factor C2

NUMBER OF EQUALITY SPACED COLUMNS

A 4 6 8 10 12

1.0 0.785 1.047 1.341 1.645 1.954

1.1 0.800 1.056 1.347 1.650 1.959

1.2 0.814 1.065 1.354 1.656 1.963

1.3 0.828 1.074 1.361 1.661 1.967

1.4 0.842 0.091 1.367 1.666 1.972

1.5 0.857 1.092 1.374 1.671 1.976

1.6 0.871 1.102 1.381 1.677 1.98\

1.7 0.885 1.111 1.388 1.682 1.985

A~ E,l~/GK, in which K = torsion constant from Table A-I-4

B;, ~ C2 rM'/E,l~

A-4

, .

(A-I-I8)

(A-I-I6)

(A-I-14)

(A-I-I5)

(A-I-17)

A-5

B,(clockwise on Fig. A -1- 3b) = [ HA~' ] + [MA L ]

2 E col col E coJ col

[ H L3

] [M L' ]/:; (outward - radial) = A + A

, 3EooJ '01 2EooJ,o'

MomentM' = MA - Rec

Eq.A-I-17 and Eq.A-I-I8 may be solved for column shear HA and top-of-columnmomentMA. If the column concrete and ring-beam concrete have the same elasticmodulus, however, the above equations reduce to:

For compatibility, /:;,+ /:;2 + !:i.e = 0, or:

Displacements of the fixed-end column are:

A.=E,I~/GK83= C3Fyr3/E,lrx

Table A-1-3. Values of Factor C3

NUMBER OF EQUALITY SPACED COLUMNS

A. 4 6 8 10 12

1.0 .0191 .00353 .00117 .000496 .0002417

l.l .0200 .00371 .00122 .000510 .0002509

1.2 .0210 .00389 .00128 .000524 .0002604

1.3 .0219 .00407 .00134 .000538 .0002693

1.4 .0229 .00425 .00139 .000553 .0002785

1.5 .0238 .00442 .00145 .000567 .0002877

1.6 .0248 .00460 .00151 .000581 .0002969

1.7 .0257 .00478 .00156 .000595 .0003061

Table A-I-5. Equations for Vertical Bending Moment at Angle e fromSupport (+ = Compression top)

(A-1-20)

(A-1-19)f:R' H [r3K, . L

3] MAL' 0--+ --+-- + -

A, A 21 ry 31 col 2/'01

. No. of supports Equation

4 M ~ M'(V, sinO + 0.500 cosfJ) - Fyr 2(1 - 0.7854 sinO- 0.7854 cosfJ) + M,r

6 M ~ M'(V, sinO + 0.866 cosfJ) - Fyr 2(1 - 0.5236 sinO- 0.9069 cosfJ) + M,r

8 M ~ M'(V, sinO + 1.207 cosO) - Fyf '(1- 0.3927 sinO- 0.9481 cosfJ) + M,r

10 M ~ M'(V, sinO + 1.539 cosfJ) - Fyr 2(1 - 0.3142 sinO- 0.9669 cosO) + M,r

12 M ~ M'(V, sinO + 1.866 cosfJ) - Fyr 2(1 - 0.2618 sinO- 0.9770 cosfJ) + ivl,r

A-6

Section Torsional Con. Torsional Section Points of Maximum Shear Values of Coefficients

stant in' (em)' Modulus in' (em)' Stresses Ib/in' (kg/em') cr,j3,15

(; m~h/a 13 15cr

Middle of long sides 1.0 0.140 0.208 1.00

D Tmux = T / Zt 1.5 0.294 0.346 0.859

K 4 Z, ~ fJa' Middle of shot sides 2.0 0.457 0.493 0.795~0l1

r = aXTmax 3.0 0.790 0.801 0.753

a At comer, T ~ 0 4.0 1.123 1.150 0.745

~

At all points on long side 6.0 1.789 1.789 0.743

h K ~ a'(m - 0.63)/3 Z, ~ a'(m - 0.63)/ except corners, Tm<U" = T / Zt 8.0 2.456 2.456 0.742

a Middle of sort sides, T ~O.74XTmw 10.0 3.123 3.123 0.742

Table A-I-4. Torsional Properties of Rectangular Cross Sections.

Thrust, Shear, Torque, and Bending Moment in Ring-Beam. For design,values of these quantities are required at various locations. Table A-1-5 andTable A-1-6 show equations for vertical bending moment and torque at various

angular positions from the columns.

and

12M r C3Fyr' - C,rRe, HAL' [Cor L]__~,~.-+-~-----+ --+ M --- +- = 0a'ln(rz / r,) I~ 2/'01 A I~ 1'01

Table A-1-6. Equations for Torque at Angle e from Support.

No. of supports Equation

4 M ~M'(!;' cosB- 0.500 sinlJ) + Fyr '(B- 0.7854 cosB + 0.7854 cosB- 0.7854)

. 6 M ~M'(,!: cosB- 0.866 sinlJ) + Fyr '(0_ 0.9069 sinO + 0.5236 cosB- 0.5236)

8 M ~AI'!,!: cosO-1.207 sinlJ) + Fyr '(B- 0.9481 sinO + 0.3927 cos 0- 0.3927)

10 M ~M'!,!: cosO-1.539 sinlJ) + Fyr 2(0 - 0.9669 sinB + 0.3142 cosO - 0.3142)

12 M ~M(!;' 'oosO-1.866 sinlJ) + Fyr 2(B_ 0.9770 sinO + 0.2618 cosO- 0.2618)

Table A-1-7. Summary of Axial Force (Thrust), Shear Force andHorizontal Bending moment in Ring-Beam

NO. OF LOCATION COMPRESSIVE SHEAR HORIZONTAL

SUPPORTS THRUST VERTICAL HORIZONTAL BENDING MOMENT

4 Support Fxr 0.7854 Fy r HAI2 - 0.1366 HA r

Midspan Fxr 0.0 0.0 0.0705 HAr

6 Support F~r 0.5236 r-;'r HAI2 - 0.4550 HA r

Midspan Fxr 0.0 0.0 0.0451 HAr

8 Support }"'xr 0.3927 Fyr HAI2 - 0.773 HA r

Midspan Fxr 0.0 0.0 0.0333 HAr

10 Support Fxr 0.3I42Fyr HAI2 - 1.0920 HA r

'd Fxr 0.0 0.0 0.0265 HArM, span

12 Support Fx"r 0.2618 r-;'r HAI2 - 1.4100 HA r

Midspan Fxr 0.0 0.0 0.0220 HAr

A-1-3 RESULTS OF ANALYSIS OF THE RING BEAM OFPROTOTYPE SILO (ART. 3.1.3)

A FORTRAN program is written following the above method of analysis, asmentioned earlier, and the prototype of art. 3. 1.3 is analysed. In this case theinput data and the output results are the following (Fig. A-I-2):

A-7

(i) Input Data:

Geometric Dimensions:aJ = 3.5 ft.

h = 2.54 ft.b] = 1.54 ft.

al = 2.5 ft.a = 55.0 deg.

r3 = 14.0 ft.L = 20.0 ft.Ical = 3.0 ft4

eco/ = -0.5 ft.

Applied Forces and Moments:F = 31.56. kip/ft.x

F = 69.48 kip/ft.y

M, = 55.20 kip-ft/ft

(ii) Results of analysis (Computer output):

Values of various forces and moments at critical locations required forring beam and column design are presented in Table A-I-8. Variations ofvertical bending moment and torsional moments in the ring beam incircumferential direction are shown in Fig. A-I-5 to Fig. A-I-8. In this figures"8" is the angle subtended by two adjacent column at the centre.

A-8

I.._ .. ._ .. .. .. .._ .. .._ ..

,,-,,-,,-"-"-," "-"-',-, -..

- - - - -- ------- - - -........ ......... ......... ........... ....... ....... ..... .. .. ................. .... ... ......... .. ................ ..... ..

- - .-- - - -- ..- - - -- ---. -- -- '- -, , .'--- -- - ---.....- ....~ -- ~.-...•_- .,-- .--- ", -,---- -------

"/ '-......100

100

(a) Due to M'

(b) Due to Fy

20 40 60 80

Angular Distance from support in % of 0

Fig. A-1-5 Vertical bending moment in ring beam

20 40 60 80Angular Distance from support in % of 0

No. of Column - 4 No. of Column - 6 No. of Column. 8 No. of Column - 10 No. of Column - 12

No. of Column - 4 No. of Column - 6 No, of Column - 8 No. of Column - 10 No, of Column - 12

-500

-3000o

-150o

==, -450c.:i:::- -400:2o-; -350::>"0'E -300

'"Eo:2 -250Clc'6 .200c'"ID

2000

==C. 1000:i:,.,

LL.0 0-'"::>"0 ;.,'

I--'-~-'

100

100

.-"-"-

Fig. A-1-6 Torsion in ring beam

(b) Due to Fy

"';':..'~'~.~~7:':-::::::~"..'::=::::=::.~.""--- ..-.. . ...._- _- .-.-.-

20 40 60 80Angular Distance from support in % of e

(al Due to M'

20 40 60; 80Angular Distance from support in % of e

-~- -~----'.- - .

No, of Column - 4 No, of Column - 6 No. of Column - 8 No. of Column - 10 No. of Column - 12

No. of Column - 4 No. of Column - 6 No. of Column - 8 No. of Column - 10 No. of Column - 12

o

50

-50

-200o

-600o

100

200

600

-150

150

400

<I='

Q. 200:i:,c-

O0-Q)::J"0 -200c0.en~0t- -400

,:::;:o-Q)::J"0Co.~~ -100

2400

100

...•.•...•

80

- ..-.-_.- ..-.--------.-.---

40 60Angular Distance from support in % of e

20

•.", ....

~

.;. ::.:;;.;::~:::~:--:--:-:.-::-::-I:::--:.--:.-..- ..- ..- ..- ..-:.:::.E-:.::::.-=.:::.:::.:::.;"::; ';::.::..t..-'..:!/fJI".e. .,_ .. -" .. _ .._ .. -.. . --.,,~.~".; ". ~ ...•

_.;:.",.~~:~:/ J ------------ --------------< ---------- '\ .•...•.•~~~.::~.~:~..-

-2400 a

1::Q)

Eo:2 -600Clc'6c -1200Q)

III(ijut -1800~ .

~ 1200:;:Q)

(ij"0 600-0-Q):l a"tl

<l:'1800

Q.:s2

CIN~o~.~o~f~CO=,u-m-nc-.74---;N~o-.-of~CO=I~u=m=n~-~6O-;N7o=.=o7f~CO=lu=m~n=.-8:-cN~o~.-o~f~CO=,u=m=nC-.7,O:-cN~o~.=o~f~C~o~lu~m=nC-.='2"

Fig. A-1-7 Vertical bending moment in ring beam due to total effect

I""""

40 60Angular Distance from support in % of a

100

.',.'

80

-----.._---- _----

~~1"~-~--"--"-:;;-:;;-=':=:'::.:.':-::::=:':~.':':':~:~:~:~:'"..'.'...•...•..

......•........-------._--- ...

20

..'.'.. ' :.~"..~:~;;;:.;;;:.;;;:;:J;;;:::~:;.".-".-".-~-=-.

400

-= 300

a.:;;:~ 20013Q):;::Q)

n; 100-0-0-Q) 0:J"0

Em -100.0ClC.C.s -200

c0.iii<5 -300I-

-4000

INo. of Column - 4 No. of Column - 6 No. of Column - 8 No. of Column - 10 No. of Column. 121

Fig. A-l-B Torsion in ring beam due to total effect

.I~ •.

I;~\

Table A-1-8. Forces and Moments in Ring Beam and Column.

Forces and Moments for Column Design Forces and Moments for Ring Beam Design

Number Vertical Horizonta Moment Moment Hoop Hoop Maximum Maximum Maximum

of Axial force at at Top at Bottom Force Stress Bending Bending Torsion

Column Force Top MA MB Moment Moment

R HA (Positive) (Negative)

(kip) (kip) (kip-ft/fl) (kip-ft/fl) (kip) (kip/fl') (kip-ft/fl) (kip-ft/fl) (kip-ft/fl)

4 1541.49 77.53 -1127.63 540.07 -438.94 -60.99 2026.59 -2339.71 346.31

6 1027.66 53.34 -776.93 370.54 -438.94 -60.99 1148.83 -730.80 -131.55

8 770.75 44.00 -642.20 304.20 -438.94 -60.99 789.72 -250.82 -128.49

10 616.60 37.95 -555.39 260.99 -438.94 -60.99 595.41 -64.31 -123.54

12 513.83 33.51 -491.64 229.22 -438.94 -60.99 470.80 0.00 117.36

-0-- -

A-9