DESIGN TABLES FOR

WATER- RETAINING

STRUCTURES

Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue , Nell' York , NY 10158

@ Longman Group UK Ltd 1991

Design tables for wateryretaining structures

Longman Scientific & Technical. Longman Group UK Limited. Longman House, Burnt Mill, Harlow. Essex CM20 2JE, England and Associated Companies throughout the world.

Copublished in the United States with John Wiley & Sons. Inc .. 605 Third Avenue. New York. NY 10158

© Longman Group UK Ltd 1991

All rights reserved: no part of this publication may be reproduced, stored in a retrieval system. or transmitted in any form or by any means. electronic. mechanical. photocopying. recording. or otherwise without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London WIP 9HE.

First published 1991

Library of Congress Cataloging-in-Publication Data Batty, Ian. 1939-

The design of water-retaining structures / Ian Batty. Roger Westbrook. p. cm.

ISBN 0-470-21846-0 I. Hydraulic structures--Design and construction I. Westbrook.

Roger. II. Title. TC180.B36 1991 627 --dc20 91-43516

Set in Compugraphic Times 10111

Printed and Bound in Great Britain at the Bath Press. A von

CIP

Contents

Preface

Acknowledgements

List of design tables for water-retaining structures (Chapter 9)

1 Standards for the design of water-retaining structures

iv

iv

v

2 Design and constructional aspects 10

3 Design of cantilever walls to retain liquids 29

4 Design of rectangular tanks 54

5 Design of circular tanks 78

6 Design of prestressed concrete circular tanks lOS

7 Design of a flat slab roof and columns for a reservoir 119

8 Design of conical tanks 134

9 Design tables for water-retaining structures 152

Appendix I ,!he analysis of ground-supported open circular concrete tanks 188

Appendix II Metric/Imperial conversion factors 202

Preface

This book provides a comprehensive understanding of the design and construction of water­retaining structures. allowing graduate civil and structural engineering students. as well as the practising engineer. to design with speed and economy. Assuming some familiarity with BS 8110 Structural Use of Concrete the book draws on examples. many of which are based on actual completed structures. and upon extensive tables. related to the analysis of rectangular. circular and conical structures, to develop good working practice. The tables and examples will enable the engineer to check, by hand, the often complex results of computer analysis and output. usually based on the finite element method. for most structures. This is particularly so in those cases where the forces within a structure are affected by the ground upon which they sit. Thus, methods of designing for the soil/structure interaction, which normally require the aid of complex computer programs. are included. The tables and examples will prove to be a good reference for carrying out new work to modern methods and regulations. and will give direction to the student engineer in the use of currrent British Standards for the design of many types of concrete structures.

An essential part of the book are the listed computer programs and output which further assist the designer in obtaining a range of options from which the most effective and economical solution may be determined for a particular structure; whilst a useful appendix covers the analysis of ground-supported open circular concrete tanks.

Acknowledgements

The authors wish to express their appreciation to the BSI and the HSE for permission to use extracts from their publications. In particular they wish to thank the Portland Cement Association of America for permission to use extracts from their tables which assist in the analysis of circular and rectangular tanks. They are also grateful to the editor of Construction Weekly for allowing them to include, as an appendix. the article prepared by Lightfoot and Michael on the design of circular tanks supported by ground having elastic or plastic properties.

The permission by Yorkshire Water to use photographs of construction and the help and encouragement of colleagues in the Central Division of that Authority has been invaluable and is greatly appreciated.

We are grateful to the following for permission to reproduce copyright material:

British Standards Institute for extracts from BS 8007 and BS 8110, also for Fig. 2.2 from BS 8007. Fig. 7.1 from BS 8110, Tables 1.1. 1.2 from CP 2007, Tables 1.3, A.I from BS 8007, Tables 7.1. 7.2. 9.2 from BS 8110, Table 9.3 from BS 4466 (Extracts from British Standards are reproduced with the permission of BSl. Complete copies can be obtained by post from BSI, Linford Wood. Milton Keynes. MKI4 6LE.); the editor, Construction Weekly for Appendix 1 Lightfoot E, Michael D 1965 'The design of ground­supported open circular tanks'; Health & Safety Construction for Fig. 1.2; Portland Cement Association for adapted Tables 9.20,9.21.9.22.9.23.9.24,9.25.9.29,9.30,9.31.9.32. 9.33.9.34,9.35.9.36.

Whilst every effort has been made to trace the owners of copyright material. in a few cases this may have proved impossible and we take this opportunity to offer our apologies to any copyright holders whose rights we have unwittingly infringed.

iv

Table 9.1 Details of (a) bar reinforcement, and (b) fabric reinforcement Table 9.2 Ultimate anchorage bond and lap lengths as multiples of bar size (BS 8110) Table 9.3 Reinforcement scheduling details for (a) preferred shapes, and (b) other shapes Table 9.4 'As' for design crack width 0.2 mm, bar diameter TIO Table 9.5 'As' for design crack width 0.2 mm, bar diameter TI2 Table 9.6 'As' for design crack width 0.2 mm, bar diameter TI6 Table 9,7 'As' for design crack width 0.2 mm, bar diameter T20 Table 9.8 'As' for design crack width 0.2 mm, bar diameter T25 Table 9.9 'As' for design crack width 0.2 mm, bar diameter T32 Table 9.10 'x' and 'z' factors for sections reinforced in tension only -serviceability limit state Table 9.11 'zJd' lever arm factors for ultimate bending moment Table 9.12 Concrete grade C25: permitted values of shear stress 'vc' for a range of As x lOO/(bv x d) and effective depth, d (BS 8110, Table 3.9) Table 9.13 Concrete grade C30: permitted values of shear stress 'vc' for a range of As x lOO/(bv x d) and effective depth, d (BS 8110, Table 3.9) Table 9.14 Concrete grade C35: permitted values of shear stress 'vc' for a range of As x 100/(bv x d) and effective depth, d (BS 8110, Table 3.9) Table 9.15 Shear reinforcement spacing (mm) for beams, where 'v' is greater than (vc + 0.4) Table 9.16 Minimum percentage of reinforcement to resist early thermal cracking (BS 8007 Appendix A) Table 9.17 Deflection - modification factors for tension reinforcement for varying values of Mu/(bdd) and serviceability stresses Table 9.18 Deflection - modification factors for tapered cantilever walls subjected to different types of loads Table 9.19 Values of 'k' factor used for estimating deflections of cantilever walls under hydrostatic pressure Table 9.20 Moment and shear force coefficients for walls subjected to hydrostatic pressure in a three-dimensional rectangular tank, assuming a hinged base, free top and continuous sides Table 9.21 Moment and shear force coefficients for walls subjected to hydrostatic pressure in a three-dimensional rectangular tank, assuming a hinged base, hinged top and continuous sides Table 9.22 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming hinged base, free top and continuous sides Table 9.23 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming fixed base, free top and continuous sides Table 9.24 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming pinned base, pinned top and continuous sides Table 9.25 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming fixed base, pinned top and continuous sides Table 9.26 Deflection of two way spanning slabs with various edge conditions subjected to (a) triangular pressure, (b) rectangular pressure Table 9.27 Ground pressure created beneath a base slab carrying an edge force 'Q'

v

vi

and an edge moment 'M' and supported upon ~ elasdc soil Table 9.28 Bending moments created within a base slab carrying an edge force 'Q' and an edge moment 'M' and supported upon an elastic soil Table 9.29 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a triangular load, assuming a fixed base and a free top Table 9.30 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a triangular load, assuming a pinned base and a free top Table 9.31 (a) tension, and (b) moment coefficients in cylindrical tanks subjected to a moment per m, 'M' applied at base Table 9.32 (a) tension, and (b) moment coefficients in cylindrical tanks subjected to a shear per m, 'V' applied at top Table 9.33 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a rectangular load, assuming a fixed base and a free top Table 9.34 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a rectangular load, assuming a pinned base and a free top Table 9.35 (a) shear at base of cylindrical tanks subjected to: triangular load, rectangular load, moment at edge; (b) stiffness coefficients for cylindrical walls; (c) sti ffness coefficients of circular plates with and without centre support Table 9.36 Supplementary coefficients for values of Lv2/(2 x r x h) greater than 16 Table 9.37 (a,b) coefficients for calculating forces in a conical tank supported at base level, resulting from fixity at the base of the cone Table 9.38 (a,b) coefficients for calculating forces in a conical tank supported at base level, resulting from fixity at the apex of the cone

The necessity to store and supply purified water, and to treat the residual effluents, has been a major source of civil engineering activity for many civilisations. There are many remnants of great structures used for this purpose which demonstrate the skills of those earlier engineers. These indicate that then, as now, if you wish to retain water and prevent it being polluted you had to build well. In more recent times an evolutionary system of Codes of Practice and British Standards were developed, based upon continuing experience and research, in order to help engineers design water-retaining structures more effectively.

The earliest codes, CP 7 (1938) and CP 2007 (1960), considered that if the stresses in the steel and concrete were of a relatively low order then there should be few problems. To minimise cracking those areas of concrete in tension were designed to ensure that the tensile resist­ance of the concrete was greater than the actual tensile force. The permitted design service stresses given in Tables 1. I and 1.2 are extracts from CP 2007 (1960).

Table 1.1 Permissible concrete stresses in calculations relating to the resistance to cracking (CP 2007, Table 2)

Concrete mix

Nominal mixes 1 : 1.6: 3.2 Grade 26 1 : 2: 4 Grade 21

Permissible concrete stresses

Direct (Nlmm2)

1.2 1.3

Tension

Due to bending (Nlmm2)

1.85 1.68

Shear Q

bla (Nlmm2)

1.92 1.71

With the advent of limit state design theory a radical change was introduced into BS 5337 (1976), the water­retaining structures code of practice. The code drafters took into account the experiences of many engineers and essentially permitted three different ways of design:

(a) the limit state method based upon the current level of research;

(b) the alternative method which was similar to the previous code of practice CP 2007;

(c) the limited stress method which incorporates both limit state and elastic theory.

Table 1.2 Permissible steel stresses in strength calculations (CP 2007, Table 4)

Members in direct tension

On liquid-retaining face

Members On face Members less than in remote 225 thick bending from

Members 225 or liquid

more thick

In shear reinforcement

Permissible tensile stress

in steel (mild) (Nlmm2)

82

82

82

125

82

The effect of this standard was to help engineers consider more closely how concrete behaved and how to prevent cracking of the concrete during the construction and work­ing life of the structure. A great deal of attention was focused upon positions an.d types of joints, methods of construction and areas of reinforcement required to prevent early thermal cracking.

Durability of the concrete both in the short and long term was now of as equal importance as the design. The previous design codes tended to result in thick concrete sections with relatively large amounts of mild steel rein­forcement. This, however, did not prevent cracking. The new standard, BS 5337, required engineers to become more involved in the construction process particularly with regard to joint positions and methods of construction. The

limit state design method did lead to thinner sections and deflection under load was more noticeable, particularly with respect to cantilever retaining walls. One other result was that high tensile steel virtually replaced mild steel as the main reinforcement used in construction.

as 8007 (1987) Design of concrete structures for retaining aqueous liquids

As a result of II years of experience with BS 5337, the most recent standard, BS 8007, is now based mainly on the limit state approach to design. Structures are generally designed to restrict crack widths by suitable amounts of reinforcement and appropriate joint spacing. The alterna­tive method given in BS 5337 was removed from the code; a few elements of the limited stress approach, however, did remain. For the first time in a BS design code the designer is required to consider operational safety.

The basic elements of the BS 8007 are now sum­marised, changes and additions to the previous code are highlighted. Where applicable, extracts from the standard are included with kind permission of the BSI.

General: (Section 1 of BS 8007)

Scope: This British Standard provides recommendations for the design and construction of normal reinforced and prestressed concrete structures used for the containment or exclusion of aqueous liquids. The term 'liquid' in this code includes any contained or excluded aqueous liquids but excludes aggressive liquids. The code does not cover dams, pipes, pipelines, lined structures, or the damp-proofing of basements. The term 'structure' is used herein for the vessel that contains or ex.cludes the liquid, and includes tanks, reservoirs, and other vessels. NOTE I The design of structures of special form or in unusual circumstances is a matter for the judgement of the designer NOTE 2 The titles of the publications referred to in this standard are listed on the inside back cover

A design temperature range of 0 °C to 35°C is now specified for containment under normal conditions. Recommendations are also included with regard to structures subject to adverse ground conditions.

Design objectives and general recommendations: (Section 2 of BS 8007)

Design objectives: The purpose of design is the achievement of acceptable probabilities that the structure being designed will not become unfit in any way for the use for which it is intended. This code provides for a method of design based on limit state philosophy that is generally in accordance with the methods employed in BS 8110. Structural elements that are not part of the liquid-retaining structure should be designed in accordance with BS 8110.

Structural design:

(a) It is recommended that the design of sections be based upon crack width limitations initially and then other serviceability and ultimate limit states be checked.

2

(b) The partial safety factor for retained water shall be 1.4 for most situations for ultimate limit state (ULS) and 1.0 for serviceability limit state (SLS).

(c) There shall be a factor of safety of at least 1. I against flotation.

(d) The maximum crack widths shall be: (i) RC - all faces of liquid containing or

excluding structures - 0.2 mm max. RC - where aesthetic appearance is critical - 0.1 mm max.

(ii) PS - limited to requirements of BS 8110; however, refer to Section 4.3 of BS 8007 for particular rules for cylindrical tanks.

(iii) PS - except for the special recommendations for the design of cylindrical prestressed structures (see Section 4.3 of BS 8(07), the tensile stress in the concrete should be limited for prestressed concrete structures in accord­ance with the recommendations of Section 2.2.3.4.2 of BS 8110 : Part 1 : 1985.

(e) Deflection - Walls designed by limit state theory are thinner than those designed by elastic theory and the designer is cautioned to ensure that deflection, due to loading or rotation of the supporting earth, is not excessive. The method of backfilling should be clearly defined. (Where deflection is the sig­nificant factor in the design of a wall the authors of this book recommend that the thickness of the wall be increased rather than the area of steel be increased to satisfy the BS 8110 requirements.)

Loads:

(a) All structures required to retain liquids should be designed for both the full and empty conditions, and the assumptions regarding the arrangement of loading should be such as to cause the most critical effects. Particular attention should be paid to possible sliding and overturning.

(b) ULS condition liquid levels should be taken to the top of the walls for design purposes assuming all outlets blocked. SLS condition liquid levels should be taken to the overflow, or working top level, for design purposes assuming all outlets open.

(c) No relief should be allowed for beneficial soil pres­sures in designing walls subjected to internal water loading.

(d) Thermal movement in roofs should be minimised by appropriate means. It is noted that where a roof is rigidly fixed to a wall, forces will be generated in the wall should the roof expand or contract.

(e) Earth covering roofs should be treated as a dead load, excessive construction loads should, however, be considered in the design.

Analysis of wall and junctions: The code states that bending and direct tension should be taken into account in the design process (refer to examples in Chapters 3 and 4). It is worth noting that significant horizontal bending moments occur at corners of rectangular containers par­ticularly where the walls have a length/height ratio in excess of 2.

SUe condiJiOM:

(a) Ground movements - for subsidence effects. guid­ance is given on methods to limit the damage that may result (see Chapter 2).

(b) Reference is made to the recommendations of BS 8110 regarding the effect of aggressive soils upon concrete.

Causes and control of cracking: Cracking in walls occurs as a result of

(a) external loading and changes in temperature during the working life of the structure;

(b) chemical and physical changes generated particularly by changes in temperature and moisture content as the concrete matures and strengthens;

(c) restraints to movement by adjoining stronger concrete sections;

(d) inadequate detailing of reinforcement and of associated poor construction techniques.

Concrete is particularly weak for the first few days following its construction. Careful thought and super­vision prior to casting, and immediately afterwards, will assist in ensuring a sound structure. The code recommends that the prudent use of reinforcement, movement joints and construction techniques will heip in keeping crack widths within acceptable limits.

The extract below from clause 2.6.2.2 of BS 8007 gives useful advice on particular methods of minimising and controlling cracking resulting from moisture and temperature changes within the structure:

In order to minimise and control cracking thaI may result from temperature and moisture changes in the structure it is desirable to limit the following factors:

(a) the maximum temperature and moisture changes liwing construction by: (I) using aggregates having low or medium coefficients

of thermal expansion and avoiding the use of shrinkable aggregates,

(2) using the minimum cement content consistent with the fe1:juirements for durability and, when necessary. for sulphate resistance,

(3) using cements with lower rates of heat evolution, (4) keeping concrete from drying out until the struc­

ture is filled or enclosed, (5) avoiding thermal shock or over-rapid cooling of

a cone rete surface; (b) restraints to expansion and construction by the provision

of movement joints (see Section 5.3 of BS 8007); (c) restraints from adjacent sections of the work by using

a planned sequence of construction or temporary open sections (see Section 5.5 of BS 8007);

(d) localised cracking within a particular member between movement joints by using reinforcement or prestress;

(el rate of first filling with liquid (see Section 9.2 of BS 8007);

(I) thermal shock caused by filling a cold structure with a warm liquid or vice versa.

Design and detailing recommendations are also given at the end of Section 2.6 of BS 8007 and it is noted that:

(a) where reinforcement is required to control and thermal cracking, it should placed as concrete surface as the cover requirements allow;

(b) unless joints are placed at close centres (see clause 5.3.3 of BS 8007) the amount of reinforcement in each surface zone in both directions shall not be less than the amount shown in Fig. 1.1.

h

---jr-O--;;/?'I' ('--:;/?"'I--:":-.-~ '" .R"" '°.·.0." . . I b~,? 1'':(; ~,,(J.tJ"

h 4500mm

(ilm1 Walls)

Figure 1.1

.. 0 "00°. I I' P

I 1

I 1

h > 500mm

For minimum areas of reinforcement see page 4

3

The reinforcement should be calculated in accordance with Section 5.3.3 and Appendix A of BS 8007. Except as provided for in option 3 in Table 5.1 and Section 5.3.3, the amount of reinforcement in each of two directions at right angles within each surface zone should be not less than 0.35 % of the surface zone cross section. as defined in Figures A.I and A.2 for deformed grade 460 requirement and not less than 0.64 % for plain grade 250 reinforcement. In wall slabs less than 200 mm in thickness the calculated amount of reinforcement may all be placed in one face. For ground slabs less than 300 mm thick (see A.2 of BS 8007). the calculated reinforcement should be placed as near to the upper surface as possible consistent with the nominal cover.

Design life and serviceability: The design life of the structure should be in the range of 40 to 60 years. It is noted that elements of the structure may have a shorter working life than the main structure Uoints. sealants etc}. It is obviously prudent to ensure that replaceable items are accessible without major destruction of other elements.

The designer should explain how often the structure is to be inspected and maintained. In particular the struc­ture should be examined regularly for cracks. rust stains and other signs of deterioration. A schedule of precautions necessary to prevent potential damage to the structure should be written into the commissioning document. For example. if the media in a sunken filter bed is used to prevent flotation then it must not be replaced without first lowering the external water table! Pressure relief valves must be checked before any work is carried out which depends upon their effective operation.

Both faces of a liquid containing or excluding structure. together with internal supports of a containment structure. shall be considered to have a minimum surface exposure rating of 'severe' as defined in clause 3.3.4 of BS 8110. Where exposed concrete is subjected to severe freezing conditions whilst wet. then a 'very severe' rating is to be used.

The concrete design and specification in the code is con­sidered adequate for a structure exposed to 'severe' conditions as defined in BS 8110. However the designer's attention is drawn to the possibility of biological decay resulting from adverse materials contained within the stored liquid or present in the external ground water. Where such conditions arise or where an 'extended design -life' for the structure is required then additional cement content. cover or special reinforcement may be necessary.

Note: All examples in the chapters that follow are designed with 45 mm minimum cover since it is the authors' exper­ience that clients generally expect their structures to have a design life well in excess of 40 to 60 years!

The code stresses the requirement that the concrete should have a low permeability. This is one of the most significant factors in reducing the incidence of chemical attack, erosion, abrasion, frost damage and corrosion of reinforcement.

The nominal cover for reinforcement is given as 40 mm minimum. However, if the cover is increased then surface crack widths resulting from bending and direct tension will also increase (see Appendix B and the design examples in the chapters that follow).

Specification: The designer is asked to ensure that as far

4

as is reasonably practicable the assumptions made at the design stage occur on site and that the quality of both materials and workmanship are satisfactory,

Operational safety considerations: The designer should take into account the requirements given in those sections of the Health and Safety at Works Act (1974). One of the most common 'dangerous occurrences' statistic which happens in the water industry is death or injury resulting from people entering unventilated enclosed structures without first checking that the atmosphere is satisfactory. The code takes this into account by stressing that:

(i) At least two access hatches should be provided at opposite ends of a structure and at least one in each compartment. The hatches should be large enough to enable personnel wearing breathing apparatus to enter.

(ii) Provision should be made to ensure that there is adequate ventilation to limit dangerous accumula­tions of gas or toxic atmospheres to acceptable levels.

Increasing concern over accidents within the construction industry. often resulting from lack of training, has led to the inclusion of the following generalised statement in the contract documents:

'Personnel will only be allowed on site if they have evidence to prove that they have had recent training in the safety requirements necessary for this contract or that they are escorted during their visit by suitably qualified and approved staff. .

The proposed draft HSC Construction Management Regulations includes the following definition of duties for designers under Regulation 7, in Fig. 1.2.

Constnu:tion management Proposals for Regulations and an

Approved Code of Practice

Figure 1.2

Any person who designs a structure shall ensure as far as is reasonably practicable that the structure is so designed that it can be built, maintained (including re-pointed, re-decorated and cleaned), repaired and demolished safely and without risk to health.

Any person who designs a structure shall ensure, so far as is reasonably practicable, that his design shall include adequate information about any aspect of the design or materials which might affect the health and safety of any con­tractor or any other person at work on that structure.

( I ) Designers should consider whether there are any special factors which would affect the health and safety of those doing the work and. if so, should inform prospective contractors in terms at the tender stage and in more detail when specifying design details, construction methods or materials.

(3)

can account of the user's them in the course of the life eventual need to demolish them.

which subsequent work on appropriate information by the designer for future reference.

reinforced concrete:

Design: The basis of design should comply with the requirements of BS 8110, however, those areas of BS 8007 which are not in accordance with BS 8110 are stated.

Methods of limiting crack widths taking into account constructional and design requirements in the immature and mature concrete are listed.

Design and detailing of prestressed concrete: (Section 4 of BS 80(7)

The basis of design is stated. in the same manner as for reinforced concrete above. However, particular rules for cylindrical prestressed concrete structures are included (see Chapter 6). The nominal cover should be such as to satisfy the 'very severe' exposure condition ofBS 8110.

Design, detailing and workmanship of joints: (Section 5 of BS 8007)

General:

Joints in liquid-retaining structures are temporary or permanent discontinuities at sections. and may be formed or induced.

This section describes the types of joint that may be required and gives recommendations for their design and construction. The types of joint are illustrated in Figure 5. I (BS 8007) and are intended to be diagrammatic. Jointing materials are considered in Appendix C of BS 8007.

Joints may be used, in conjunction with a corresponding proportion of reinforcement. to control the concrete crack widths arising from shrinkage and thermal changes to within acceptable limits.

Since the main source of leakage in water-retaining structures occurs at joint positions. considerable attention is given to this subject. The code lists six types of joint:

(a) expansion; (b) complete contraction; (c) partial contraction;

(d) hinged; (e) sliding; (0 construction.

Descriptions and details and method construction.

has to the position and type of joint cOl18id!en~ best for a particular situation.

The spacing of joints is left to Some favour close joints whereas at all and use higher quantities of steel to control crack­ing. Table 1.3. extracted from the code, indicates that both systems are acceptable.

Section 5.4 of the code specifies in some detail how a construction joint may be formed to continuity of strength and resistance to the need of a water bar. Where it is necessary no movement joints to exist such as in tanks where direct tension occurs Section 5.5 of the code refers to the possibility of temporary open sections being left between panels as shown in Fig. 1.3.

Figure 1.3

The benefits are that the amount of reinforcement necessary to control early thermal cracking is minimised. The only thermal effects to be considered are those result­ing from seasonal variations (T2 - see Appendix A­A3. BS 8007).

The section closes with advice on joints in ground slabs, roofs and walls; however, it is noted that for all vertical joints in the walls of circular tanks, including construc­tion joints, it is necessary to provide water bars to prevent leakage.

Concrete: specification and materials: (Section 6 of BS 80(7)

It is recommended that when blended cements are used the maximum proportion of ggbfs should not exceed 50 %. where pfa is used the maximum proportion should not exceed 35 %.

The code specifies a particular concrete mix for general use with water-retaining structures classed as grade C35A with a minimum cement content of 325 kg/m3.

Further comments are made regarding workability. blinding layers and pneumatically applied mortar. It is recommended that since cracking in concrete cannot be

5

Table 1.3 Design options for control of thermal contraction and restrained shrinkage (BS 8007, Table 5.1)

Option Type of construction and method of control

3

Continuous: for full restraint

Semicontinuous: for partial restraint

Close movement joint spacing: for freedom of movement

Movement joint spacing

No joints, but expansion jOints at wide spacings may be desirable in walls and roofs that are not protected from solar heat gain or where the contained liquid is subjected to a substantial temperature range

(a) Complete joints, ~ 15 m (b) Alternate partial and complete joints (by interpolation), ~ 11.25 m (c) Partial joints, ~ 7.5 m

(a) Complete joints, in metres

w ~ 4.8 + -, (b) Alternate partial and complete jOints, in metres

w ~ O.5smax + 2.4-

f

(c) Partial joints

5/eel ratio Comments (see 00/92)

Minimum of Use small size bars at close Peril spacing to avoid high steel

ratios wen in excess of Peril

Minimum of Use small size bars but less Peril steel than in option 1

213 Penl Restrict the joint spaCing for options 3(b) and 3(c)

Note 1 References should be made to Appendix A, BS 8110, for the description of the symbols used in this table and for calculating Peril' smax and, Note 2 In options 1 and 2 the steel ratio will generally exceed Peril to restrict the crack widths to acceptable values. In option 3 the steel ratio of 213 Penl will be adequate

totally avoided, any member that is permanently exposed to view is provided with a profile or type of finish which will minimise the effects of surface marking.

The remaining sections of the code relate to the specifi-

6

cation of reinforcement, prestressing tendons and inspection and testing of the structure for water tightness and liquid retention.

minimum and crack

ten'1pe!ratiure and

This section provides more information than the previous code on the concrete is affected by temperature and moisture. research work has been carried out by such organisations as CIRlA, BCA and many universities, which helps engineers to understand how durable concrete may be produced.

Typical values of the fall in temperature between the hydration peak and the ambient, referred to as T 1 in the code, are given in Table A.I, which is an extract from BS 8007.

Table 14..1 Typical values of T1 for ope concretes, where more particular information is not available (BS 8007, Table A,2)

Section Walls Ground slabs: thickness OPC content, (mm) Steel 18mm (kglm3)

form work: OPC plywood content, form work: (kglm3) OPC content,

325 350 400 325 350 400 325 350 400

300 500 700

1000

11 13 15 23 20 22 27 32 28 32 39 38 38 42 49 42

25 31 35 43 42 49 47 56

15 25

17 21 28 34

Note 1 For suspended slabs cast on flat steel formwork, use the data in column 2 Note 2 For suspended slabs cast on plywood formwork, use the data in column 4. The table assumes the following: (a) that the formwork is left in position until the peak temperature has passed; (b) that the concrete placing temperature is 20°C; (c) that the mean daily temperature is 15°C; (d) that an allowance has not been made for solar heat gain in slabs.

It is noted that the mean daily temperature used in the preparation of this table is 15°C. Once again close co­operation between designer and contractor is necessary to ensure that the estimated TI figure, assumed at the design stage, is valid at the construction stage.

The long term seasonal temperature falls are denoted in Appendix A3 (BS 8(07) by the figure T2. This effect occurs in the mature concrete and is catered for by:

0) where continuous construction is used T2 is added to T I and a greater area of reinforcement is required;

(ii) the use of movement joints to absorb these variations in length.

Further topics in Appendix A give guidance and ~ on:

(i) minimum reinforcement; (ii) the spacing of cracks;

(iii) crack control in thick sections; and (iv) external restraint factors.

Table 9.16 at the rear of this book gives the percentage of steel necessary to comply with Appendix A for varying values ofT! & 1'2, steel diameters and crack widths. For example, for a temperature fall of 40 °C, 16 mm diameter type 2 bar and a crack width of 0.2 mm, 0.64 % steel is required within the zone thickness.

Appendix B Calculations of crack widths In mature concrete

One of the features of BS 5337, the previous 'water­retaining structures' code, was that the cracks resulting from bending stresses should be calculated. Revised equations are given in BS 8007 to comply with BS 8110 requirements. In addition, equations to estimate the crack widths due to direct tension are now included.

Clause 2.2.1 of BS 8007 suggests that the design process commences with the calculation of crack widths based on the Appendix B equations and recommendations.

Tables 9.4 to 9.9 inclusive, are prepared to help the designer to obtain very quickly a range of concrete sections using differing thicknesses, cover and diameters of steel. These will, for a particular service bending moment. generate a crack width equal to or slightly less than 0.2 mm.

In addition Program I.P, given on page 8 allows the designer to input the bending moment, thickness of slab and cover. The output gives a range of diameters of bars, spacing and resulting crack widths.

Example: (Using Tables 9.5 to 9.7, and also using Program I.P,)

Table B.1 Bending moment - 100 kNM; cover to main steel - 45 mm

Table 'Type 2' Thickness Spacing of Crack width bar diameter (mm) (mm) bars (mm) from

Program IP1 (mm)

9-05 T12 500 175 0.20 9-05 T12 400 100 0.19

9-06 T16 500 225 0.18 9-06 T16 400 150 0.20

9-07 T20 500 300 0.18 9-07 T20 400 200 0.20

7

Program 1 P 1 Design of a concrete slab to ensure that the crack width generated does not exc8ed 0.2 mm for a particular bending moment, depth of slab and any cover to steel

4 REM CALCULArES CRACK WIDrHS FOR RC SLABS - BOOKI IS rHE REFERENCE 5 LPRINl''':::::::::::::::::::::::::::::::::::::::::::::::::::::: f: II 6 LPRINr" fHE DESIGN OF R.C.SLABS FOR A CRACK WIDTH OF 0.2mm" 7 LPRINT"::::::::::::::::::::::::::::::::::::::::::::::::::::::::" 9 LPRINl'" " 10 DIM r(12,28) ,S!?(12,28) ,0(12,28) ,M(12,28) ,11.5(12,28) ,HQR(12,28) ,CW(12,28) ,N(12 28) ,OIA(12,2i} 11 LPRINr 12 INPur "1'HE BEN'OINJ MOMENr IS" iB"l 13 INPUT "THE 'rHICKNESS OF THE SIAB IS "iHI 14 INPur "rHE COVER ro THE MAIN REINFORCEMENr I3"iCOV 15 LPRINr II THE DESIGN SERVICE BENDING MOM'll' IS "i 8Mi" kNm" 16 LPRINr" " 11 LPRINr " 18 LPRI~Ii''' " 19 LPRHlr " 20 LPRINl''' " 21 LPRIiH " 22 LPtUtH " 23 LPRIlH" ..

rHE rHICKNESS OF i'RE 3L~a IS ";Ml;" mm"

rHE COVER ro rHE D~SIGN sreEL IS "SCOVi" mm"

ARE~ STEEL DI~

sq.mm mm

30 FOR 3=4 TO 12 STEP 1 40 FOR H=12 "1'') 28 Si'EP .t 42 ~Il\(5,H)=H 45' IF OIA(S,H) = 24 rHE'J !JIA,(S,H) .. 25 46 IF DIA(S,H)= 28 i'HBN OIA(3,H)=32 50 SP(3,H)~ 3*25 60 'r ( s , H) = H 1 70 D(5,H)= HI-COV~DIA(S.H)/2 75 AS(S,H)=!.142* OIA(3,H)-2*.25*lOOO/SP(S,H) 76 HOR(S,H)~M(3,H)*1000/)(S,H)-2 80 GOSua 510 110 NEXT H 120 NE){r S 152 LPRINi'" " 153 LPiUNr" "

mm

Il'\pu~ : BtYld",,'3 ~tKeH t Tk,ekH~$~ of s\ab C.vty to \MaiM stte.\ •

156 LPRINr"::::::::::::::::::::::::::::::::::::::::::::::::::::::::" 180 END ~ 510 ~1(S,H)=9~I*lOOOOJO! .A.J 520 AE '" 15 ~ . .,.:0 ~ D'"" 530 I?=AS(S,H)/(lJOO*')(S,H) CO,,&~):;q 0,,0 .;9 .",-t 510 X=SOR(AE*P* (AE*P+2» -AE*P . ~:;;~ • o~~: 550 Z=1-X!3 ~~--------------------~ 560 FS=:'I(S,H)/(A:3(3,H)*Z*0(S,H) 570 55=F5/(200*1000)

Figure 81

5 !3 0 S E = 5 5 * (r ( 3 , H) - 0 ( '3 , H) * X) I ( D ( :3 , Ii) - 0 ( S , J.j) * x ) 590 SI=1000*(r(S,H)-D(S,H)*X)-1!(600*1000*AS(S,H)*D(S,H)*(I-X» 600 Si1=SE-Sl 510 ~CR=3QR{(.S*3P(S,H»-2+(COVtOIA(S,H)/2)-2)-DIA(S,q)/2 620 CW(S,H).3*ACR*3~/(1+(2*(ACR-COV)/(r(S,H)-D(5,H)*X») G50 IF 2i'l(S,H) <.15 fHEtJ 690 665 IF CW(S,H) ).20l rHEN 690 668 M(S,H)~~(S,H)/(1000*1000) 670 LPRIN£ U5IN':;" ££2£.££" iAS(3,H) ;0I~(S,H) iSP(S,H) ;CI'l(3,H) 674 LPRINf" " 690 RErURN

8

Output

:::::::::: :::::::::::::::;::::: : :::::: ::::: ::::::::: THE DESIGN OF R.C .SLABS FOR A RACK \HDrH OF o. 2mm

:::::: :::::::: : :::::::::::::::: :::::::::: ::::::::::::

THE DESIGN SERVICE BENDING MOt4ENT I3 100 kNm

THE THICKNESS OF rHE SLAB IS 400 mm

THE COVER TO rHE DESIGN STEEL IS 45 mm

AREA srEEL DIA sq.mm mm

1131. 12 12.00

1608.70 16.00

1340.59 16.00

1795.43 20.00

1571.00 20.00

2181. 94 25.00

1963.75 25.00

Appendix C Jointing materials

This section of the code starts by defining the various joint­ing materials. Since the most common source of leakage/entry of water is at joint positions, it then reminds the designer of the need to consider, whilst detailing,. the problems of future maintenance:

The joints described in Section 5 of BS 8007 require the use of combinations of jointing materials, which may be classified as: (a) joint fillers; (b) waterstops; (c) joint sealing compounds (including primers where

required).

These materials are inaccessible once the liquid-retaining structure has been commissioned until the structure is taken out of use. The design uses for these materials in joints should take into account their performance characteristics, both individually and in combination, and the restrictions and difficulties of access to them should the joints not perform as designed.

It is important that acceptable methods of compacting the concrete around the joint are defined prior to the concrete being placed.

As was mentioned at the beginning of this chapter, water-retaining structures must be well built. BS 8007 provides many useful guidelines on how durable concrete may be produced.

SPACING :;1 mm mm

100.00 0.19

125.00 0.16

150.00 0.20

175.00 0.17

'200.00 0.20

225.00 0.16

250.00 0.18

Appendix D Future standards

The advent of the European code for concrete EC 2 is now well under way and the general opinion is that the procedures in the proposed code and those in BS 8110 are similar and the results of using either code will produce little change of any significance. The approach to carrying out the design is different, however, and some of these differences are given below, particularly where they affect the design of water-retaining structures.

The code deals with principles which are mandatory and with rules which contain a method of satisfying these principles but permit alternative methods, ,which must, however, still comply with the necessary requirements. The cover to steel is generally less than that stated in BS 8110 but tolerances for workmanship deficiencies must be added to these values (5-10 mm is the current extra cover recommended for in-situ concrete). The span/ effective depth ratios are of interest in that lightly stressed cantilevers (containing <0.5 % reinforcement) have a permitted slenderness ratio of 10 whereas highly stressed members (containing > 1.5 % reinforcement) have a permitted value of 7. The result is that the designer is encouraged to increase the thickness of the concrete rather than increase the steel areas when deflection is a problem.

The control of cracking resulting from early thermal effects or serviceability tensile stresses is considered in depth by EC 2 and minimum areas of steel will be greater than that specified in BS 8007 in certain situations.

In general terms the individuals and organisations involved with the development ofEC 2 are confident that the effects of the changes will be minimal upon those engineers who are familiar with BS 8110.

9

2 Design and constructional aspects

As with all structures. careful attention to detailing. specification of materials. methods of construction. the supporting element and methods of protectioo from attack by adverse chemicals should result in a structure that will have a satisfactory life. Proposed new safety legislation referred to in Chapter I spells out clearly. however. that the designer should not only ensure that the structure should be built well and safely but also that it can be safely maintained. repaired and demolished! The designer must

5 f

k.

-

not only be skilled in design and construction but also have some understanding of the operational warie that the structure was •• It for and also how it should be main­tained an. repaiFeCil during its working life.

The designer is beililg encouraged to work more closely with those who build the structure and also those who use it. For example. if one is designing a reservoir. a typical design brief prepared by the operations groop would result in requirements similar to those shown in Fig. 2.1.

k J J

.. -

Figure 2.1 (a) full height division wall; (b) minimum slope of floor and roof 1 in 200; (c) all wall/floor, wall/wall, columnlfloor junctions to be haunched; (d) no protrusion of column bases above floor level; (e) smooth internal concrete surfaces; (f) a gap of at least 100 mm between top water level and underside of roof soffit or roof beams; (g) at least two access hatches to each compartment - the sides to extend at least 300 mm above soil level - main access hatch should have a landing 2.5 m below hatch and ideally further descent should be via a flight of steps; (h) corrosion protected ladders but not smooth stainless steel; (j) special 1 m x 1 m sealed access opening for mechanical plant and large equipment placed into compartment by crane sat on hardstand; (k) suitable ventilation inclused to (i) accommodate changes in water level, (ii) prevent local accumulation of stagnant air, (iii) prevent entry of polutants to reservoir; (/) underfloor drainage; (m) roof to be covered with topsoil and grass which is to be cut with the aid of a small tractor and mower; (n) embankment to have a maximum slope of 1 in 2.5.

10

The contractor's preference would probably include:

(a) c1oseconsultation before design details are finalised, based on the understanding that the contractor has specialist knowledge on COfllli:rUC-tion that the designer may not have;

(b) discussions during the construction without the restraint of preconceived solutions;

(c) a combined approach to problem solving; (d) an ~greed performance specification based on design

parameters; (e) simple detailing and sufficient width of section that

enables the concrete to be easily placed and compacted between shutters;

(f) a flat formation level with no downstands for bases or ribs;

(g) a team, rather than adversarial, approach to the contract.

One example where close liaison with the contractor is of value can be shown with the aid of Fig. 2.2. BS 8007, Appendix AS, gives the restraint factors for three differing methods of wall construction. Elevations a, c and d shown in Fig. 2.2 give indications of the valiation ofthe amounts of steel required for each type of construction.

If the designer places sufficient reinforcement for the 'sequential bay wall construction' (type c) but the con­tractor, at estimating stage and often without full detailed drawings, bases his quotation on carrying out the work using a combination of types a and d, the result is that some parts of the wall will be under-reinforced and changes will have to be made either by the designer or by the contractor, or, if not noticed, the wall may crack.

External restraint factors (BS 8007)

Effective external restraint may be taken as 50 % of the total external restraint because of internal creep. Reference was made in A3 (BS 8007) to movement joints that greatlweduce the rigid external restraint assumed for continuous walls. However, there are other situations where the assumed external restraint factor R can be less than 0.5. Some typical situations for thin sections subjected to external restraint are illustrated in Fig. A3 (BS 8007) and allow for any beneficial internal restraints.

Note that no thermal Craclctlig ill likely~)~ ~ 2.4 m of a free edge since experience lw !lOO~~ tl1is is the length of wall or floor slab over w~ ilie~ ~ capacity of the concrete exceeds the increasing ~ contraction, the restraim factor varying betweoo mro 4t the free edge to a maximum of 0.5 a12.4 m from ilie~~. Note that cracking can occur near the eoosif ~li lriIluc.ers such as pipes O"vCur within this 2.4 m length o!~aU of slab. However, if not less than 2/3 Pcr!:, based on the s~ zones, is provided and there are no obvious stress raiseis, it may be assumed that the free ends of the members will move inwards without cracking up to where R "" 0.5. Where this is only a temporary free edge and a subsequent bay is cast against the edge, the larger restraint factor for the sub­sequent bay is shown in parentheses in Fig. A3 (BS 80(7) and should be assumed [4].

The restraint within a wall or floor panel depends not only on the location within the slab but also on the proportions of the slab. The table below shows how the restraint factors vary between opposite edges, one free and one fixed (e.g. for a wall slab the base section is the fixed edge and the top section is the free edge).

Influence of slab properties on the control line restraint factor

LlH Design control line horizontal ratio* restraint factors

>8

Base of panel

0.5t 0.5t 0.5t 0.5t 0.5t

Top of panel

o o 0.05t 0.3t O.5t

* H is the height or width to a free edge L is the distance between full contraction joints

t These values can be less if L<4.8 m

The effective external restraint in ground slabs cast on smooth blinding concrete for the seasonal temperature variation T3 may be taken as being the design restraint factor R = 0.5 at the mid-length. for 30 m lengths and over, and it may be assumed to vary uniformly from 0.5 to zero at the ends.

Where R = 0.25 AS = TI2 at 300 C/C Where R = 0.50 AS = Tl2 at 150 C/C

11

Horizontal r"traint II<lor. Obh;n Ir •• l.bI. A.3 I., lhis unlral ,one

~~it~~ . ...: ------------- ---_. o.s: .OJ.

~'---p~,.nli~;f-- ----WI °1 ~."J °1 ,. UKkl H •

• Wher. H~ L. this factor .0.511- f I

6.0

] (8)

(b)

Figure 2.2 External restraint factors (BS 8007, Appendix 5, Fig. A3). (a) wall on base; (b) horizontal slab between rigid restraints; (c) sequential bay wall construction - with construction joints; (d) alternate bay wall construction - with construction joints

12

llWhtrt LIS 2H, Ih ... rnlrainl faclors

.O.SI1./;' I

NOTE. V.lu" of RUled in 1M dHign Ihould be rtl.ttd to the prKtical distribution of ,einfOfcement.

Figure 2.2 (continued)

(c)

(d)

:,-.-:-

-::..

--._. ~:

-

13

Initial considerations

Prior to the commencement of the design it is first necessary to have information concerning the site con­ditions and then to sketch out essential construction details, i.e., if there is a high water table and flotation is a problem then a decision has to be made whether the design includes for thick slabs and walls, pressure relief valves, ground anchors etc; aggressive soil conditions will affect the specification of the concrete.

Soil investigation

There should be a comprehensive soil report on any major contract, and with the increasing usc of structure-soil interaction, CBR tests should be carried out in order that the modulus of subgrade reaction may be assessed for design purposes.

An example of the influence of the ground upon the structure is shown in Fig. 2.3(a,b) for a circular settlement tank.

<a)

(b)

Figure 2.3 Tank supported by (a) base sat upon rock, (i) a complex conical shell design, (ii) heavily reinforced sections; (b) floor sat upon gravel, (i) simple design, (ii) lightly reinforced sections

14

If flotation is a problem it is beUter, where possible, to have any extra concrete above the external water table since its full weight is used, whereas only approximately 60 % of the weight of the concrete below the water table level is of practical use because of the displacement of the water.

If the base slab extends beyond the wall then not only is a firm support provided for the wall shutters but also the fill above the extension assists in preventing uplift.

Thick base slabs, which are often constructed to prevent flotation, require large quantities of reinforcement to resist thermal cracking and to ccomply with the other recom­mendations of BS 8007. One solution is to have a nominally reinforced layer of 'thick blinding' cast beneath the designed thinner base slab and to tie the two elements together using a detail which permits the upper slab to have an ability to move horizontally but not vertically; it is beneficial that there should be a water seal between the two slabs at the perimeter.

Concrete specification

There are many factors which influence the quality of the concrete used in the construction process, however, the main requirement has always been that the concrete should be durable in the environment it is placed and when sub­jected to the forces it must resist.

Many articles and papers have been published indicat­ing how concrete can be improved or why failures have occurred, but it has been shown that there are certain fundamental factors which must be satisfied in order that a dense impermeable concrete can be produced. The omis­sion of one of these factors may reduce the useful life of the concrete.

The main requirements in obtaining concrete which is easily placed, has a low permeability and adequate durability are:

(a) An adequate cement content. (b) The provision of a consistent, cohesive, well-graded

mix which is easily placeable, does not segregate and does not require a considerable amount of 'working' to achieve a dense outer layer.

(c) The lowest water/cement ratio possible to suit the level of compaction provided.

(d) Sufficient compaction to provide a dense mass of thoroughly compacted concrete particularly around the reinforcement.

(e) The use of a proven satisfactory method of curing the concrete. (Flooding a slab with water for at least seven days is of benefit, however, other considera­tions such as the temperature gradient from the centre of the slab to the upper surface may be the dominant factor.)

(0 The design details, particularly with regard to joint positions, to be well considered in advance of the contract commencing.

(g) The cover to the steel to be at least the minimum recommended to suit the condition of exposure.

(b) Trained and experienced supervision to be provided by the contractor. Experienced engineers inspecting the work on behalf of the client. A proven testing system and an available on-site covermeter.

a particular mix complies with these 'TeQDirements. Information from the materials supplier should include:

(i) The concrete to be designed to resist all forces and known environmental effects and to be inspected and maintained at appropriate intervals of time especially during the early years of its life.

(a) The declared alkali content (DAC) of the cement. (b) The percentage of sodium chloride present in the

coarse and fine aggregate. It is noted that only 76 % of this quantity is active and need be used in calculations .

In BS 8110 clause 6.2.5 precautions are recommended where chemical attack of the concrete is expected. Limits on chloride and alkali content of the concrete are stated and the Graphs 2.1 and 2.2 assist in checking whether

An example of the use of these graphs is given on page 16.

CHLORIDES IN CONCRETE

Design Information required from Concrete Supplier:

i) ii)

iii)

Cement - Declared Alkali Content (DAC) Sodium Chloride Ion % Coarse Aggregate Sodium Chloride Ion % Fine Aggregate

(0.75%) (0.029%) (0.045%)

Note The DAC value for ggbfs and pfa is taken as 0.1% however this figure can increase with finely ground ggbfs - check with supplier.

la) First Design - CONCRETE MIX / CUBIC METRE

Cement Coarse Aggregate Fine Aggregate

ALKALI CONTENT (BS8110 cl.6.2.5.2)

325 kg 1241 kg

687 kg

Coarse Agg. (graph 2.1) (0.029%) 180 * 1241/1000000 Fine Agg.(graph 2.1) (0.045%) 280 * 687/1000000 Cement (graph 2.2) DAC = 0.75%

.22 * .76 • 0.17 kg

.19 * .76 a 0.14 kg .. 2.90 kg

CHLORIDE ION CONTENT (BS 8110 cl.6.2.5.2)

Coarse Agg.(graph 2.1) Fine Agg.(graph 2.1) Cement (Hormal 0.02%)

TOTAL 3.21 kg > 3.00 kg/cubic metre

180 * 1241/1000000 • .22 kg 280 * 687/1000000 a .19 kg

0.2 * 325/100 • .07 kg

TOTAL 0.48 kg % chlorides ion / cement content = 0.48 * 100/325 .. 0.15% < .4% OK

Ib) REDESIGNED MIX TO REDUCE ALKALI CONTENT

Try

ALKALI CONTENT

Cement ggbfs - 1. 5 * (325 -260) coarse aggregate fine aggregate

Coarse Agg.(graph 2.1) ... refer above Fine Agg. (graph 2.1) ••• refer above

260 kg 100 kg

1241 kg 687 kg

Cement (graph2.2) (DAC 0.75%) 260 kg GGBFS (DAC 0.10%) 100 * 0.1/100

TOTAL <3.0 kg/cubic metre

0.17 kg 0.14 kg 2.30 kg 0.10 kg

2.71 kg OK

15

0..,.

.Mf

.... 0.040

... N.CI

0.030

O.OlO

0.010

0

0 100 lOG JOG 400 500

lie of Cbloride loos pcr million k& or Aureea.e

_: AI.-Ie w';'''' ,._ ·IJ. (N.)

ATOMIC WEIGHT CHWItINE • J5.5 (CL.

AJHIk "'Ie'" .0.11 ..... /donM $1.5 (NaCl i

.. CII","", 0/ NaCl 6/).7"

Graph 2.1 Chloride levels in aggregate

16

Cement Cootent pcr rrJor Concrete

380

J60

340

310

.\00

280

260

240

210

0.4 0.5 U 0.7 0.8 0.9 1.0 1.1

Declared Alkali CooteDt 'll> of Cement

'"'": GrapJt Ilttorpol'aU'

AtUilUlMI 10k,I"/ CtllUltl COllk1l1

ii. AddilWlI4I 0.1'1\- 011 DA.C .....

Graph 2.2 Weight of alkali in cement

One of the prime aims of the design of water-retaining structures is that cracks, resulting from any cause, are kept within well defined tolerances.

Cracks can develop as a result of: an unsatisfactory weather and heat environment as the concrete is cast; internal stresses occurring during its early life due to thermal or constructional effects; higher than anticipated forces being generated during its working life and thermal or physical movement occurring after the work is completed.

These causes, and others, have been well investigated and there are many papers which indicate why the problems have occurred and how they may be prevented. The British Cement Association (BCA), in particular, have a special index on worldwide research into these problems which enables the design engineer to have easy access to relevant information.

It is important that the concrete is not affected seriously by extremes of temperature as it is poured and that measures are taken to maintain the concrete temperature within reasonable bounds during its early life.

The designer and contractor must be ready to adjust the specification of the concrete should adverse conditions arise, i.e. certain cement replacement materials slow down the setting time of the concrete in very cold weather and in very hot weather the concrete can set too quickly creating difficulties in compaction.

In general terms concrete should not be placed if the temperature is less than 6 °C unless steps are taken to insulate the concrete during its early life.

At the other extreme measures must be undertaken to keep the concrete cool during hot weather, particularly where the concrete surfaces are exposed to • solar gain' effects from direct sunlight and from drying winds.

To limit the effects of other causes of cracking the engineer has to develop positions and types of joints at the design stage, as well as constructional details and methods.

The final section of this chapter is connected with joint and other details.

Section 5 of BS 8007 considers in some depth the design, detailing and workmanship of joints. The need for movement joints is explained and particular examples are shown in Fig. 5.1 in the code.

Figure 2.4(a-k) is based in principle on the rode requirements and most have been used in pmctice. Various types of water bars are shown, Fig. 2.4(0) shows the centre bulb water bar, Fig. 2.4(b) shows a rearguard type waterstop which must be supported. Fig. 2.4(c) shows a typical PVC waterstop.

If the waterbar's anticipated life is less than the life of the concrete then it would seem prudent to use Ii surface mounted type waterbar as shown in Fig. 2.4(e). The fill material above the joint must be firm but contain a flexible element otherwise a hairline crack could develop which could contain material which might contaminate the stored water. Figure 2.4(j) indicates a type of simple seal which expands when in contact with water and seals the gap in which it is placed and prevents further movement of water.

Some indication of the type and degree of movement each joint can accommodate is also indicated on the diagrams.

The previous water-retaining structures code, BS 5337, introduced the concept of the 'partial contraction joint' (Fig. 2.4(g.h). This idea did not meet with universal approval, however, it was kept in BS 8007 with the recommendation that only half the steel be continuous across the joint. The benefits are that a sealed joint is placed which, though weakened, can still transmit moment and force.

Figure 2.4(k) is a partial contraction joint positioned in a reservoir at the interface between the external retaining wall and the base slab. Plastic coated prestressing cable was used, as shown, with sufficient de-bonded length either side of the joint to permit a limited amount of contraction to occur. The cables were able to tie the slabs together and were strong enough to prevent sliding occurring. The downstand rib, which was usually placed beneath the wall, was no longer required and, as a result, the contractor had a flat site to work on and economies were achieved.

The compressible material wrapped round the cable at the joint also permits some vertical displacement potential.

If a greater degree of rotation or articulation is required then joint Fig. 2.4(j) is capable of this type of movement. Provided the waterbar is capable of accommodating con­siderable extensions, then if joint 2.4(a), the expansion joint, is positioned with a wide gap, this also will accept rotational movement. See also Figs 2.5 and 2.6.

17

Oumb-bc.1I .­water itop

(8)

(c)

(e)

~

+--+ O\O"·Qh.OY~~ ~Ol~t fill .....

~

bpott6l0tt JoIn ~ (Sh,\ dltcol1t,"uous)

Complet~ Cott+racho.., Jo""t (Stu' dlSCOHh,uous)

-(

Complet( COI1+rachol1 JOII1 t ( Shtl OI$COI1"KUOU$)

Figure 2.4 Examples of movement joints

18

(b)

ReargUClrd 'crack. J\lldoc:.eI' '

Wate.r s~op (d)

(f)

eo"'piete Com-octlOO l'O\t1t

(9)

--~-.----C'. L I l. (.'6·0 "D

Partial CO)1tYochot'l jOl)1t

(I)

induced

50fo, CO\\tIMUOIJS s\«.l

Partial CcmtrQchot\ Jontt

(h)

reSl11 Do..d&d cork flU&y

~Yheulo\ed £XPOI1SIOI1 Jm.,+ 0)

s\(.tvtd 1\ \1\11\ .; pla$h~ c.oo~ ,ud

Portlo\ Con\YQcilOt\ 3'o",t (k)

0.,. 61ecvcci ItU. ba ...

-

19

(a) around pipe insert,

(b) at base of wall

Figure 2.5 Water stop details for a waterproofed basement structure

Figure 2.6 Water stop in roof and wall jOint reservoir

20

hPO"SIOI1 JOII1+

(EltYQtcd .lab - t.uppc»'hI19 WQII Sltua+lOt1) (c)

pia she ,lttv.!

pack'!l9

5hdlH9 Roor JoInt (b)

$hd.H9 JO.llt pol\jst!/Ymt

I~-tt-+--- Cot"l"O"Olo1 pyoof&d dow & \

Figure 2.7 Joint details for (a) pinned base joint-tied, (b) sliding roof jOint, (c) expansion joint -slab/supporting wall, (d) sliding roof joint

Typical wall details are given in Fig. 2.7(a-d). Many reservoirs have been damaged by the use of rigid roof slab connections which, when a slab expands or contracts as a result of thermal movement, tends to generate cracks in the walls, particularly at the corners; the roof also cracks.

If the roof is only sat upon a sliding joint at the top of a wall then it has been known for the roof to 'walk off damaging any external rearguard type waterbar in the process. Sliding roof type joint details shown in Fig.

2.7(b,d) have been used which do not restrict reasonable movement of either walls or roof.

There are many benefits which result from making the details as simple as possible for the designer, detailer and contractor. Examples are shown in Figs 2.8 and 2.9 where. although the excavation is greater, the end product is simpler and in many cases results in a quicker and more economical construction.

Typical details of a swimming pool construction are given in Fig. 2.IO(a-b).

21

Simplified Construction Details

(a)

Figure 2.8 External channel above base slab

Dlfflcolt to ot1d detoil

[x+erna\ Cka""e.\ above SQ$f. Slab

(b)

(a)

(b)

Figure 2.9 (a) floor slabs containing ducts, (b) preferred detail

22

(8)

Figure 2.10 Swimming pool details (a) typical section. (b) step details

Differential settlement

It is often necessary to construct buildings in close proximity to d~p tanks and. because the new building is founded upon disturbed ground. there is a possibility of cracks occurring as a result of settlement of the material beneath the foundation of the building. Figure 2.I1(a)

indicates methods of limiting the effects of this occurrence.

Figures 2.12 and 2.13 give details of a screw pumping station where it is essential that both the top and bottom supports of the screw are supported by the same foundation, since any differential movements between these supports will cause the screw to bind upon the

23

~hannel surface in which it sits. Also, in this diagram. the problem of a deep open tank

in waterlogged ground can be observed. This was finally solved by having the walls I m thick and the base slab having a maximum thickness of 2.5 m. The two halves of the base slab are founded upon di fferent strata and care must be taken to ensure that there is minimal rotation of the whole structure particularly as a result of movement of the water table.

Subsidence

Problems of subsidence resulting from mining or other causes are restricted to certain areas and it is preferable not to build sensitive structures in such areas.

Figure 2.14 shows details of the two separate founda­tions built beneath a water tower which would be affected by coal mining activities. To cushion the effect of the mining wave, the lower foundation was sat upon 2 m of stone reinforced with galvanised bars. The adjoining reservoir consisted of a series of 9 m2 raft slabs tied

24

together with 'Tensar' geogrid material which would retain a certain force, but yield if excessive forces occurred. The backfill consisted of rubber tyres to limit any pressure developing against the walls.

The first mining wave has passed beneath the structures and a settlement of approximately I m has occurred. The tower leans an insignificant 50 mm and may be jacked up after the next and final wave occurs. The centrally placed waterbars in the reservoir floor failed in two places and the repairs were carried out using a surface mounted alternative. See Figs 2.15 and 2.16.

For additional information on this type of problem Ref. 2-2 is of benefit.

References

2-1 ijow to make today's concrete more durable. Seminar Institution of Civil Engineers, 1985

2-2 Lackington D W, Robinson B 1973 Articulated service reservoirs in mining subsidence areas. Institute of Water Engineers Journal, Vol. 27, No.4, June

Dlstoybt.d G~oo\1d /

/

/ /

/

/ /

/ /

Figure 2.11 Methods of reducing the possibility of differential settlement

Figure 2.12 Screw pump station

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I ,

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Figure 2.14 Water tower jacking point foundation detail (a) section, (b) plan view, (c) reservoir floor slab joint detail

27

Figure 2.15 Water tower subjected to 1 m subsiC1ence

Figure 2.1 i Upper foundation to water tower showing posi1ions of jacking points

28

Prior to the design of the actual concrete sections of a retaining wall it is necessary to consider the shape of the wall and carry out stability calculations. Using Program 3PJ, page 52, this operation can be carried out simply (see page 30 for computer output using this program for Example 3.1). Obviously the safe bearing pressure of the soil must not be exceeded and there should be adequate resistance against sliding and overturning. Where the wall is built above an embankment it is also necessary to ensure that a 'slip circle' failure does not occur.

Walls built to the limit state theories tend to be more slender in thickness than those built to the elastic theories of earlier codes of practice, and hence the deflection of the walls resulting from earth or water pressure, together with base rotations, should be estimated. Whilst a degree of fixity of the wall at the top by the roof construction will cause a reduction in section sizes and reinforcement quantities, there is a possibility of cracks occurring as a result of restriction to movement particularly where two walls meet. It is also essential to inform the contractor at the time of tendering that the walls will be unstable until the roof is placed.

It is also worth noting that where two long walls meet at a comer a major bending moment (BM) occurs (approximately 2/3 of the maximum BM at the base of the wall) and there will also be direct tension in the walls (see Chapter 4).

Two examples in Fig. 3.1 illustrate the basic procedure for designing cantilever walls and the use of the tables in Chapter 9.

Details of wall used In Example 3.1 Input

filE -fIlICUESS OF filE RE:CTMIGULI>.R WI>.LL IS .3"

THE: IlEI;)Hf OF THE: ill>.LL IS 3.6 II

rHE LE:N;)T!! OF I3I>.SE I N FRONT OF filE WIILL IS 2 III

rHE LENGHr OF 8ME 'fO THE REIIR OF TilE WIILL IS .6 III

THE DE:P ~Il ·OF rHE WIII'ER IS 3 m

THE DEPTH OF THE EIIRTIl IS 3. 2 III

THE: SUPERIMPOSED LOIID ON TilE EIIRTH IS S kN/sq. 'lI

THE Ml:;LE -Of REPOSE OF 'filE EIIRfH IS 35 degrees

rHE OVEl\IILL LENGfH OF rHE 3ASE IS 2. ~ III

29

Output program 3P2 ...... ft ..... """""" kif"" t .. "" k,. •• "" II" *"""" •• ,," k ••• *." " ••••••••

PRESSURES DUE 1'::> SELF WEIGtiT WALL ONLt

rHg ,orAL v'E~TICAL WEIGH'f IS 46.80001 kN

fHE ';:CCENfRICI'U OF {'HE LOAD A80U'1' rHE CENrRE IS .3816922 ID JL 'rliE HORI ZONTAL FORCE IS 0 kN ... " .......... ". rtiE 3~;l.ESS 1'.1' FRQNf IS 3.193345 kN/3l.'1I

fHE S'fRESS 1'.'1' RElI.R [S 29.08252 kN/sq.m

••••• IIl •• kit". If ••• • " •••••• "" ••••••••• " ••••••••••••••••• ".11'.

PRESSURES DUE 1'0 SELF >'IEIGHT WALL AND ElI.RTH

.. " ...................... . l' HE '1'0'1'1'.1.. Vt:RTICAL WEIGH'C IS 84.36 kN

'1'HE ECCENTRICI'r1lroF rHE LOAD 1I.8::>Uf THE CEN'1'RE IS .2169322 m

THe H~tiONfAL FORCE IS 29.49277 kN ." .............. . riiE SfRESS 1'.1' FR;),n IS 16.03349 kN/3'~.'n

'1' HE SfRess 1'.'1' REAR IS 42.14582 kN/sq.m

PRESSURES DUE '1'0 SELF >'IEIGHT WALL AND .HIL'ER

l' HE '1'01'1'.1.. v'ER'fICAL WEIGH'r IS LOS. 56 k;~

Jt THE eCCENrRICI'f'i OF1'HE r~:)AO 0,80U'f1'HE Ce:WfRE IS .464 L822 m

rHE HORI ZONTAL FORCt: IS 44.145 ktl ••••• " ••••• t ••••

raE SfRESS 1'.'1' FROtH IS 1.443551 kN/s1.,n

l' He: S'fRES5 1'.1' RElI.R IS 71.425H kN/sq.m

PRESSURES DUe: TO WALL FULL'i r~;)ADEO 1'.'1' FRONf AND REAR

fH!:: TOTAL v'E:RTICAL .. €I::;.l'l' IS 143.22 k,'/

Jt l' HE "CCE:NfRICI'f'l ::>F rHE LJlI.O A6::>UT l' HE CEN'rRE IS .3435407 m

1'HE: HJRIZO"l1'AL FORCE: IS 14.65223 kN .................. fHE SCRESS 1'.1' FRJtH IS 14. 28319 kN/s'~.n

ftiE SfRESS Af RF.: .. R IS 84.48862 k;</sq.m

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

30

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Figure 3.1 RC details for an open reservoir

41

Deflection of cantilever walls

If it is required to estimate the deflection of a wall due to:

(i) water pressure; and (ii) rotation of the earth beneath the base,

then the following method may be used, (see Fig. 3.2). Deflection of the wall due to external pressure can be

estimated by following the recommendations given in BS 8110: Part 2: Sections 3.6 and 3.7.

Two methods are given, 'a' and 'b', to calculate the estimated curvature at the base of the wall. Method 'a' assumes that the concrete section is cracked whereas method 'b' applies to an uncracked section.

Equation 7 in the BS is derived using method 'a'; however, it assumes that there is some assistance from the concrete in the cracked tensile zone which reduces the curvature. The diagrams below indicate that a maximum concrete tensile stress of I N/mm at the centre line of steel level is peonitted in the short teon. This value reduces, however, to 0.55 N/mm in the long term.

The modulus of elasticity (E) of the concrete is given in Table 7.2 of the BS, this is the short term value which again reduces in the long teon. In this example the E value long teon is taken to be 50 % of the instantaneous E value.

By elastic theory, using the diagrams in Fig. 3.3, the moment (M) about the centre of gravity of the compression zone is:

M = As*fs*(d - x/3) + (fctl2)*(h - x)*(2/3)*h*b

where

(a)

8S above as above

h .. oveY<~\\ depth

p. [lllllllllD pt

W ah~ y prt~iUV'e dlCS"C""

Dtfkchot\ due to WQ~ey p"us"Ye Figure 3.2

Stress

tt ! , I

Od'echol1 doe. ~o rota h.,""

Stress In concrete 1 N/mm1 in shor t term 0.55 Nfinm2 In long term

de.pth +0 V1tutval a~\s mQ)(. COl'tlfY. ~tre$S col1C.

tel1~\\~ ~tve!'5 st. fs I--e-+---+--S tress In (onere te

-"':"-+---1 not more than 1 N/mm 1

(b)

Figure 3.3 (a) section cracked, (b) section not cracked

42

,fct JIIO H N/mm or 0.55 N/mm)*(h-x)/(d-x)

hence

fs = [M - fct*(h - x)*b*hl3]/[As*(d - xl3)]

and since the tensile and compressive forces are equal then:

b*x*fcl2 = fs*As + (h-x)*b*fct/2

hence

fc = 2*[fs*As + (h-x)*b*fct/2]/(b*x)

and, from the strain diagram:

fc/(Ec*x) = fs/[Es*(d - x)]

hence

fc = [fs*Ec*x)/(Es*(d - x)]

To solve the equations a 'trial and error' approach may be used and a value of 'x' is firstly assumed and then refined until the values of 'fc' calculated using the equations above are, within reasonable bounds, approximately equal.

Program 3P2 is given on page 52 which solves the three equations for both short and long term loading. The curvature (l/rb) is also calculated.

Equation 8 in the BS is derived using method 'b' and the value of the estimated curvature, assuming an uncracked section, is:

l/rb = M/(Ec*I)

Whichever method gives the greatest value of curvature then the result of that method is to be used.

The deflection caused by water pressure is finally estimated using equation II and Table 3.1 of the BS. The equation is:

al = k ... e ... e... l/rb

where 'k' is a constant dependent upon the shape of the BM diagram. Examples are given in Table 3.1 of BS 8110: Part 2 (refer also to Table 9.19). 'r is the effective span , I Irb' is the curvature at the base of the cantilever calculated previously.

When the wall thickness varies as shown in Fig. 3.4 then the estimated deflection is obtained again using the factor 'k' obtained from Table 9.19.

Essentially 'k' is calculated by dividing the numerical coefficient for the deflection at the point being considered by the maximum bending moment value.

-t+ lit

.-Hhb +-t---" ..

L P'~P&

Figure 3.4

The deflection caused by base rotation may be estimated as follows:

TAN cp = (PI-P2)/(ks ... B) a2 = (PI-P2)/(ks ... B) ... HI

where PI and P2 are the bearing pressures at the toe and heel of the wall respectively assuming trapezoidal distribution of pressure. 'ks' is the modulus of subgrade reaction (typical values from various sources are listed in the following table).

Type of soil

Stiff clay Medium sand (submerged) Very stiff clay Well graded clayey sand Well graded gravel

23 25 45 54 82

The total deflection is therefore a = al + a2

43

Program 3P1 Check on stability of L-shaped retaining walls subjected to earth and water pressures

44

9 X=O 10 INPUT "THICKNESS OF WALL";A 20 INPUT"HEIGHT OF WALL";B 30 INPUT "TOE LENGTH";F 40 INPUT"HEEL LENGrH"; R 41 INPUT "SUPER LOAD";SL 50 INPUT "DEPrH OF WATER";D 51 INPUT"ANGLE OF REPOSE";TH 57 LPRINT"***************************" 59 LPRINr" " 60 INPur "DEPTH OF EARTH" ;BF 61 LPRINT" rHE THICKNESS OF rHE RECTANGULAR WALL IS "rAJ"H" 62 LPRINI'" " 63 LPRINT" THE HEIGHT OF rHE WALL IS ";B;"M" 64 LPRINi''' II

65 LPRINT" 66 LPRINr" "

rHE LENGrH OF BASE IN FRONr OF rHE WALL IS

67 LPRINr" rHE LENGHT OF 3ASE TO THE REAR OF rHE WALL IS";R;"m" 66 LPRINf" " 69 LPRINT" rHE DEPrH OF i'HE WATER IS ";D;"m" 70 LPRINr" " 71 LPRINT" THE DEP~H OF rHE E~RTH IS ";BF;"m" 72 LPRINl''' " 73 LPRINT" rHE SUPERIMPOSED LOAD ON THE EARrH IS ";SL;"kN/sq.m" 74 LPIUNf" " 75 LPRINT" rHE ANGLE OF REPOSE OF rHE E~RrH IS ";TH;" degrees" 76 LPRINr" " 77 L=A+F+R 79 LPRINT" rHE OVERALL LENGrH OF rHE B~SE IS ";L;"m" SO Cl=A*B*24 Sl LPRINT" n

S3 LPRINT"~**************************" 90 C2=A*L*24 100 WC=Cl+C2 110 MCl=Cl*(F+~/2) III LPRINr " fI

112 LPRINr 113 LPRILH 114 LPRINT 120 MC2-=C2*L/2 130 MC=MCl+MC2 140 EC=MC/WC: 141 E=EC-L/2 142 W=WC 143 LPRINr fl *********************************************************" 144 LPRINr " " 145 LPRINT" PRESSURES DUE ro SELF WEIGHT ~ALL ONLY 14 6 LP RI N r " " 147 LPRINr"**************************" 150 GOSU13 600 ISS IF X=2 GOrO 250 159 IF X=3 Goro 330 160 DE = SF +SL/lS 170 WE~ = DE*lS*R 172 ~EP=rH*3.142/1S0 175 CF=l-SIN(REP) 176 CB=l+SIN(REP) ISO PE=18*CF/CS*DE*DE/2 lSI P=PE: 190 ME=-PE*(DE/3+A) +WEA*(L-R/21 200 W=WC+l1EA 210 M=MC+>lE 220 E=tVW-L/2 225 LPRINf " " 226 LP RI N l' " " 227 LPRINr " "

SL ~H

figure 3.7

~. e F .. SL o !!If

kM/:t

"v\~L. I"«.pflL ....

228 LPRINT"*********************************************************" 229 LPRINT " a

230 LPRINr" PRESSURE:S DUE TO SE:LF ~EIGHT WALL A~D EARrH"

231, LPRtN'l""" • 232 LPRINT"*********************** •• *· 240 ,"OSUS 600 250 WW-9.810001*D*F 260 PW-9.S10001*D*D/2 261 papW 270 MW-PW*{D/3+A)+WW*F/2 280 w-wc+ww 290 M-;.tC+11W 300 E=M/W-L/2 306 LPRINT" II

307 LPRINT" " 308 LPRINT"*********************************************************. 309 LPRINr" " 310 LPRINr" PRgSSURES DUE fa SELF WEISHT WALL AND WArER" 311 LPRINl'" .. 312 LPRINT"***************************" 320 GOSUB 600 325 LPRIN'1'" " 326 LPRINl'" " 327 LPRIW1'" " 328 LPRINT"*********************************************************"' 329 LPRINr" " 330 LPRINT" PRESSURES OUE TO WALL FULLY L0ADEO Ar FRONT AND REAR" 331 LPRINr" " 332 LPRIN£"***************************" 340 w=~qC+WEA+i'M 345 P=P~v-l?E 350 M=MC+ME+Mw 360 E=i1/W-L/2 370 sosua 600 380 LPRHH " " 390 LPRINr " " 39 1 LP RI N'£" : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 392 LPRINr " " 400 LPRINT " " 401 LPRINl' " " 402 LPRINT'" " 403 LPRI~r " .. ~04 LPRIN'1' " " 405 I,PRIl~r " " 406 LPRINT 407 LPRINr " " 408 LPRItH 409 LPRINr .. " 410 END 600 Pl=W/L*(1+6*S/L) 610 P2=W/L*(1-6*E/L) 611 IF E >=0 THEN PF=P2 ELSE PF=Pl 612 IF E < 0 tHEN PR=P2 ELSE PR=Pl 651 LPRINf" rHE rOTAL ~ERrICAL WEIGHT IS ",w," kN" 652 LPRINr" " 653 X=X+l 654 LPRINr" rHE ECCENtRICITY OF THE LO~~ ABOUT rHE CENTRE IS "IE/"m" 655 LPRINr" " 656 LPRIN'T" rHE HORIZONTAL FORCE IS "/p /" kN" 658 LPRINT"****************" 65~ LPRIN'T 660 LPRINT "rHE SrRgSS A'1' FRONr IS",PF;"kN/s::}.·n" 661 LPRINr 662 LPRINT 670 LPRIN'1' "1'HE s'rRESS 1\'1' REAR IS" ;PR;"kN/sq.rn" 571 LPRINl' 672 LPRIN1'''****************'' 6S0 RETURN

45

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Figure 3.5 Reinforcement for cantilevers retaining wall base

Figure 3.6 Cantilever wall

47

48

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19-0

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u.u.. Tlll?'S e '3..&-0 "Ie. ~v7S~. 1'2', e.. 1....-0 c.,t,- "~h~.

51

Program 3P2 Calculation of the curvature of rc cantilever walls to assist in estimation of deflection

52

10 INPur "BREADTH 9 =";B 20 INPUT "THICKNESS H =";H 30 INPUT "EFFECTIVE DEPrH =";0 40 INPUT "AREA OF srEEL =";AS 50 EC=26666 60 ES=200000! 70 1'=1 80 FP=1 90 INPur "BENDING MOMENT = remember *1 000 OOO";M 100 INPUT"INITIAL X VALUE =";X1 110 INPur "FINAL X VALUE =";X2 120 IF T>l rHEN EC=EC/2 130 INPur "SrEP X =";5 140 FOR X = Xl ro X2 STEP S 150 IF 1'>1 THE~ FP=.55 160 IF r>2 THEN 500 170 FC=FP*(H-X)/(D-X) 180 FS=(M-B*H*FC*(H-X)/3)/(AS*(D-X/3)) 190 F1=X/(O-X)*(EC*FS/E5) 200 F2=(FS*AS+3*(H-X)*FC/2)/(B*X/2) 210 R=F1/F2 220 IF R>1 THEN 30SUB 580 240 PRINr "X=";X," Fe1/FC2=";R 491 IF R>l rHEN X = X2 492 NEX"r X 493 'r=T+l 495 LPRINr" " 497 LPRINr" " 498 IF T<3 GOfO 90 500 {mo

b",c,Q,"k tluekM .. ,., .. ".d&~h

I'IOMolC.1\ t ti M" C\t. ,,,,h.\ x AlII4

tlot. ~\tAQ\ ". "' ...

.. te.p vClI", bc.hvu .. lIHh,,1 L h",,,1 X C.~hll\ I)

580 LPRINT"**********kk**k****************k*************** ******k***" 590 IF 1'=1 rHEN LPRINr"rHE RESULTS BELO;J ARE FOR SHORr rERM LOADING" 591 IF 1'=2 rHE,~ LPRItH" rHE RESULTS BELOW ARE F,JR LON:; TERM LOADIN:;" 600 LPRINT"********k**********************k*************************" 610 LPRHlr" " 611 BM=M/1000000! 612 LPRINr" rHE 3READrH OF rHE BEAM OR SLAB IS 613 LPRIN1'" fHE OVERALL DEPTH OF BEAM OR SL'B IS 614 LPRINT" rHE EFFECTIVE DEPrH JF rHE BEAM OR SLAD 615 LPRINT" [HE ARSA OF Sf EEL IS 616 LPRINr" rHE BENDIN3 MOMENr VALUE IS 617 LPRIi~T" rHE DEP'rHfO NEu'rRAL AXIS IS 618 LPRINr" " 613 LPRINf" 620 LPRINT" 621 LPRINr" " 622 LPRINr" " 623 LPRINr" 624 SL=Fl/ (X*EC) 625 LPRINr" 627 LPRIN'f " 628 LPIUN'[" " 629 LPRINr" "

rHE SfRESS IN rHE CONCREfE IS rHE S~RESS IN rHE STEEL IS

rHE RAfIO Fel/Fe2 IS

rHE E {ALUE OF rHE :ONCRETE IS rHE ~URVArURE l/rb IS

It ;3; "mm" lI;Hi"mm"

IS"iD;"mmn "; AS; "mm" ";BM;"kNm" niX;" mm t1

" ;P1;"N/sq.'llm" "; FS; "N/sq.mm"

" ;R

" ;EC;"N/S1· mm" ";SL

630 LPRINf"****k****kk*k**********k*********************************" 640 RETURN

THE BREADTH OF THE BEAM OR SLAB IS THE OVERALL DEPTH OF BEAM OR SLAB IS

THE EFFECTIVE DEPTH OF THE BEAM OR SLAB THE AREA OF STEEL IS THE BENDING MOMENT VALUE IS THE DEPTH TO NEUTRAL AXIS IS

THE STRESS IN THE CONCRETE IS THE STRESS IN THE STEEL IS

THE RATIO FC1/FC2 IS THE E VALUE OF THE CONCRETE IS

THE CURVATURE l/rh IS

1000 mm 300 mm

IS 232 mm 1150 mID 44.2 kNm 91 mm

4.905331 N/sq.mm 57.00568 N/sq.mm

1.012424 26666 N/sq.mm

2.0214 78E-06

**** ••••• ** •••• * •• * ••• * •• * •• *.* ••• *.********* ••• *********

***************.*****.***************.******.**** ••• **.* THE RESULTS BELOW ARE FOR LONG TERM LOADING

******************.******************.**.**.**********.*

THE BREADTH OF THE BEAM OR SLAB IS THE OVERALL DEPTH OF BEAM OR SLAB IS

THE EFFECTIVE DEPTH OF THE BEAM OR SLAB THE AREA OF STEEL IS THE BENDING MOMENT VALUE IS THE DEPTH TO NEUTRAL AXIS IS

THE Sl'RESS IN rHE CONCRE'rE IS THE S'fRESS IN THE STEEL IS

THE RATIO FC1/FC2 IS 'rHE E VALUE OF THE CONCRETE IS

THE CURVATURE l/rb IS

1000 mm 300 mm

IS 232 mm 1150 mm 44.2 kNm 90 mm

4.93261 N/sq.mm 116.7413 N/sq.mm

1.010516 13333 N/sq.mm

4.110611E-06

****** •• ***.******.*.**.*.*.********************.** •••••

53

4 Design of rectangular tanks

The analysis of the elements of the tanks is made considerably easier with the aid of the PCA tables for the walls given in Chapter 9 and the BS 8110 tables for the floor slabs. The effects of tying the floors and walls together and the method of supporting the tank can, how­ever, have a significant effect upon the resulting moments and forces within the structure.

The intensity and distribution of the applied loads, due to the water and self weight, can be easily calculated. The distribution of the reaction forces provided by the support­ing elements is, however, not so readily ascertainable.

In BS 8007 clause 2.4 is written:

The liquid pressure on plane walls may be resisted by a combination of horizontal and vertical bending moments. An assessment should be made of the proportions of the pressure to be resisted by bending moments in the vertical and horizontal planes. Allowance should also be made for the effects of direct tension in walls induced by flexural action in adjacent walls. Reinforcement should be provided to resist horizontal bending moments in all corners where walls are rigidly joined.

Various authors and researchers have considered this problem and to summarise the comments of one author (4-1):

. Application of a rigorous method to the design of continuous slabs often leads to illusory accuracy when one takes into account the more or less indeterminable factors affecting the magnitude of the forces within the slabs.'

54

Such factors include for examples:

(a) the flexibiltty of the floor slab; (b) the nature of the supporting subsoil; (c) the restraining effect of the walls upon the slab.

Following extensive research work one writer recommended that because of the difference between the theoretical and actual results of his experiments a safe design should be adopted by analysing the structure for a range of conditions.

On the basis of the research work of those engaged in this field it would seem prudent to consider a range of possible conditions the results of which, whilst marginally increasing the amount of reinforcement, will ensure that the tank will be structurally satisfactory.

The structural analysis of the wall panels is complex; however, certain authors (4-1 to 4-3) and associations (4-4) have, as mentioned earlier, prepared tables to assist the designer to determine the various forces generated by the water pressure within the container with little dif­ficulty. Extracts are used and are printed in Chapter 9 by kind permission of the PCA.

The design of a base slab supported by the earth has been carried out by modern 'structural/soil interaction' techniques developed as shown in Refs 4-5 and 4-6. Also more simple methods have been used and the results compared.

A flow chart of the design process follows in Fig. 4.1.

~-

1 I,

• 1 ~,

P.u/c ku.bdJ Mu/(.s7.f:tl .

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ComitioM x> Joint ~ xv) eoneretl xl) Cement content T~t

<tl Strllebi-. Analysis . IMtqs etc.

effective Lengths, PeA tables for MsPte,v,

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&or'\d

laps

figure 4.1 Flow chart for the design of rectangular concrete tanks (a) initial consideration, (b) ;tructural analysis

5S

56

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r 01 ")( £ ~ +

..rycyz. '>t G\...E;l"b

"OJ ')< O_'S ~ '>\;/ : '-""'?'1.J-. """'V>'1 ~'Z- ..

_~ <; ~ )( ~. 0 1: .a, 4' ,,--- / r..t o-oz, ::: S ~

. ....., .,.........,., 0 ..... ') '" ~<: L,I' 0 -,') <O/C '0 ) l ~ tJVVI("I~

- 1'0 z; 'H

'-'~ vt') • "H 'c-PI~<O"~

- ........ ~ Z <: 0' - -~ l'"I - O .. "\oAt .. )0 -""" o .. ~ .. '1

~c!)

i ·r->~i 1 I~ . I I

I ""'r-'I'l ~ '-./ 'r--?I':;.s I t I . V'\oo"IIP¥ -1/'0 "1-.~ p ""G., C?<[

V9

} 7I o

'r-~---r---;:------------

f.:f:e~+. tl) ~H";~ L..:IA.A~t- S::r~c.. M , ,,,, '" S"'c )( \.,""" • 10 L.e.- ........

.;

r-t. u .. '10 ><.. 10'"'" :::

f"'" loCI\...... (~~ K IOQ...O)(., e ~"") O·e>'t.2

Fve~ T j(" 9 /11, e :. O·~..>.) K "Z tyS' t; "'2.(:>0 ... _.

IT ~ (M I v ') ::- 'lo K LO ID - {q ,z..~l (O.e:,1)(~\o.<>)(.'1.~\ - -- ~\..f~<:

As. (Fh .. :\ .. ~~I(I.\..JIXIO?> "') os ?~ I~~(p .... ~'" J -- ,.., Odc.

(0·2:>1 X. '+~) ii) C;;~~d.A/

Vv = ~c.: K- I·u )(. lO-G-O .:: 11oo-c N/IAA. v- :> 1100-0 /(Z ~ ~ )(., I~) - O. "2-(., 7 ....... -.. N~'be I MiVl. A-t;. = o. ~ P< 5)'0 ><. l~ ----- l~ --~--

AT. - (P 1'2... ............ "/-

( TI'2. /11 ~ -:. (PULP ............... '- /v.....) Dio A ~ :: (PI_H.",,)(., t9-0 '::> 00. '2 "Z cit.

Ire • (Ol~ ~ O~'- ~"':!:; ( ~~ t" (<;;;t~) /1"2'" Vc & 0 . u (p ,..j / ""'- ...... "" ) o. '2-1...f V- 0.1<.

i Y) be. ft c.c.. t-i 0 "'" ,

SIe"V'lD{t!../~..G--7"-? v~o ... <...-e--e-o '" ~ < '20

to4-'tO """l \.," ..... '3 .

• ") t-•• \) "'I. As (ro f» - 0.0--0 13. ~ ~o x.. le-tr'O • ~_"" ii0 f ...... O'VI-"\ 1 1/'''''-1' ~ ~/l~ J T l ., 3-o·c I IN ... ......,.. .. 0.'2. ........

<YtA.Ao ~r ~ t5""Qv-"'o"'" \,.Ji-n.... ~iL."\.i ~

11'<4 +-....... M......K f<." ~ 0 ,7.~

~"'"'"" 120 : 1oYI..A ....... As .. . 0, loP ,... ~O oK l&'9-0 ,. 5'2.')' J It:r{) 'l. 1..

I>'Yi "'-1 TI1.. """'-'" ",,\, k & O. SA;" I(. 'Y\:"o 1<. ~ ... ~ 19"0 1.. 1-

.0,., t-'\.i~. ~eAV\k'". ,"" t-o~ ~ ~~ rc; (P'2~'"': "1~/1'1" Mi ....... /~;..-t'. j", bt ...... f!Y~ $t~ it Wo-_ .... : TLo/"2'Z.~'"

65

66

t,.,-:. ~ .... o",,", ........

~ If Wo- ~\" .. '20-(.,.

.. "2l~ __ ...... ~ :. S''?> IG-.J,

~1'~ I MS::' $;1 ~, ......... I

H, ::< "3vt - <;'"~ (o''2..1~ - o'll~); 'Zr.,: iG-I, ...... ,

A<;....<. ....... """G Q::.$ Co "2.&-0 ..j/~- ..t c. ~ 0·" J<.. "Z..1 ~

-rl--.~', A s. ~ '.$ l K lo l.co + 1.00 I<. 2..6'l

=. Co \ 1 + '20:><::: -= ~1'Z.. \..

( T ....... 1 11't Ill'>" : As lol-flP ~~ ~)

;. ~-

A-s ~\I c.c,lt L.} ~'l "'-"......, 1,.-

/' C4', !;;; t-'I "'-j = x. ?L-tl&> = <?JI'L.

0/0 A~ Vv,', Sf"; ~ HI - ( .... P .. l x 10.-0 = o· \ '::"<.,., <Q/o

'2.1 0) ><. lo.-o--o

'B,'--1 c.-ov\C/...:>LtA..--bo ..... r $.-\~lo....-/ bo ~ov \;' ~~ \ • I J

d,e.... i 8\/\ U1--1"'4 TI"l. /Il ~- I 'It;- ~ l:>.e.. ~ VI

~AA '-

ltV '" 0.'2.. """,,,,-,.

As VC-4.\' ~oo/' Ulh~-e. M-o~~'\:- ..f ,t)1/<-<",1.:'

.,/ O.l<....

4 I vi '" +-0./ G ~ ~ c:::-a.A ov la,.,.{-e.ol

AnIJIy8I3-01 DUe INaf) aaumlftg conununy wttb waif

Baa seated on an eIastk son r~ This particular analysis is carried out with the aid of Tables 9.27 and 9.28.

As is required for the design of the base slabs of circular tanks, the modulus of subgrade reaction (Ie), alternatively known as the foundation modulus, must have been deter­mined or estimated, usually from the results of CBR tests.

The value of k is not a well-defined constant; however, since the fourth root only is to be used in the calculations. the significance of any reasonable error is considerably reduced.

Figure 4.2

where:

L length of base slab Q concentrated loads at ends of slab due to weight of

walls M concentrated end moments at the base of the wall k modulus of subgrade reaction X proportional distance aIL of load Q from the left-hand

end of the base Y dimensionless distance btL of a position on the base

slab second moment of area of uncracked base slab cross section

E modulus of elasticity of the concrete A foundation - slab stiffness characteristic parameter

~(k/4EI)

The tables give coefficient values for the effect of a unit load at a position X(alL), upon other positions Y(b/L).

These effects are determined for positions up to Y = L for loads placed from X = 0 to X = 0.5 L.

Once the coefficients are determinetl for a series of moveable loads, about a particular position Y, they are adGied together to give the complete value at Y, as illustrated in Figs 4.2 and 4.3.

x • ~ L

Y a b L

1

cocf. cN'eet 0' Q, ortd Qt "po", poStt,on Y a ~ ., lZ, • zal

b

Figure 4.3

67

Since the loads are only placed upon half the span the effect of a load at the right-hand end of the span, i.e. X = I upon the position Y = 0.25, is the same as a load at the left-hand end of the span, i.e. X = 0 upon the position Y = 0.75. This concept is used in Fig. 4.4 and also in Ref. 4-6.

't. 0 o.n 0·$ 0'75 1'0

l( .. t· ~ : d I

I I I y. 0 0.2$ O·SO 0·75 1-0

l( • 0

d

y. po"ho\1 '\1 slab /L

de ~Iecho" I d' Q~ ~. O.ts +0,", load

po.,tlo\1&d Q~ )( • I.

IS Hte. so,"1. a~ H1C. clc"edlo\1 'd'

o~ yo O.7S foW' load po.,tlo \1c.d

at X & 0

X • load fO~lho" I L

Figure 4.4

68

k. : (~) 50~""'~'

C'--'i ~ ~A.V~0 )( (\~h- k.~e-J.)

SO

x. (..-:\"'.~) :::\.0

e- 1: I.;~ ,....,.,)/_ ~

6(... \.of. a-z.,..' ,I(, • W J( ~

... &.-fo.~ 1<."" / ......

G~; ........ ~ ...... ~ Q--d- A.~c-"",; be.<L. i ...... v0. '-I-Is: ;

P.>lv<.ic. c..oeffic.;~ )s..L. C (/ k \.L . '-te!)

0°17 ~L ':. J ~O lo( ~.~ ... -s:. ~·11 t..f ta(. l '{~~ .ool"" --

Not-e l. I.JtL- -r~I~ ~/t.l ~l ~/2~,""""d.. ~~ ~ -..k IN\. 0 ...... .e .......k t-'t ; { V' "'P lll.....l. e-cL ~ AA-"I

~r:~""""-'\e.t-v-ic.~ c..-ovple.. c{ to",c.e,r .~p~d.. o. 0'2.\(' L. ~~~ e.vt'" ~~ ~~ •

....--....M ~ ·o'1~l. -+1: -t-l-.O&C"1. Ii· '-.

N~ 1..

¥~I.. . I Z l

~~~.

I. ~ .. I '1 - ,

I : . ~ e ..• " .# ·ct ... • .. I

-::t. oD1.\' :t. '1\'

I 1"" -h..-e.. ~ c.vIol e.~ I

I h--. G e +-f-c:.-<k'T ~ 62 l. st "11...

1

M- p&-V\hoV\{ '1'2. .. 3>, (;V.r e. d. ",rev """-"'\ ..... .-d... I ~ d....

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69

eef.. --

L

:::!!=

3'5

C \"4ok-e !! et> rC71-1 t"& o.iv<.c.N>" \:..<:> ~ d'o-/oO-.le..

70

i) f./("~v./l- o.i !t""..; b I.d:-\c>V'\ b~~a..-~ ..flIP v>"l.ol~·oCV\ t" q e. V\e,.v-AA e..-oL ~

fOYGe..- t Y'le{,

L ~"O b·":>ll '1- 0

@ ---1

x.. ... \ ~O.2Z <::>

~ ... O

'i +- cpl":>

~ )( .. 0 - (p. ~ 7 \

8) Lj"'o

~~ x. .. 1 +0·2 z.

'1 .. 0

1=£:' ';(.:. .ov.'

+'" '!>e,y-'j:;'O

~ x. .... C!Jl')" -O,?S(,p

'1J.-O

~e.. pvc..{tvvc..

M e.",vc;z(~ '" ~ 1)'

f..-'\. j Q. ( f(.t 4W

+O~e. 'bjJ.7)

~v~vt-e-v ~ M:c! ~ Mv lh'I'''j;''''1 F"~ ..

x. ... o I . e.S"~ ! x.. ,.. 0 .0. Z~t

'1 Co o·zs ~ .. 0· ......

I~I ?<. ". 1 O.Ln; 'l:,. ... l -o . .z~z.,

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~ + I. ::'BCo ~ - o. y.\.pW

-:e. ... 0 -\.~~ ';1::.0 +o.2~2..

~"o·w ~ ",0.';

):.. .. 1 -;c:. ..... I +O·gZ +0 ·Lf13

'j: 0·1..0 y .. 0.'>'

M ~ I J '.>C.;o·o'Z..~

:o:i..-s ZZ 'X. > 0.02.0

+-\.~o \ , -0·\01 1: o .'2.s." ~,. 0- <'"

x..,. ~l} 'X:. : • ~7\'

'j: 0,2-« ~0.(.H7

t.j .... 0.1;:"" -0.\01

gef. -- i') Mo....,e,......r, j"" bA.9<:'- b'j'~ ,~.:,

~ '3~~t:4IL ;"',

( ("c,.te-v- T~le ~/ze, ')

t.ov-c.e ~~ I/'-f ~.-..- MI.d..~ M"lH~:, i~F~

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~ '1 •. 1,.~ '1 .... ~

6(1... ')G" I -0.0'.

><. ... \ / -1,.. '2-\' I~:.· ..,.,. -O·~'

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~ ~~1 +0'0\9-

')(..-,. \ .... o.ol.o'b

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·O2..~"

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'1 =- .2\' -j= .0.:

L. .. to, 0 14 l '" + O. 00 c.,.

M O"",c:,..~ "M-veA'

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tU" ~d.~(M" -0 .0% K~O·\",... ~',~+o.C><)c&>)( t"'loJ'));-~ ~ ..... O'O'2...~

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.:bA..-Se. r4- w M,,{ .

Cj / '20 s f!::, >< £). 6 1 x ';)-..... ..

10...0

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0)0..., .-tAA- I"", O\..c,c..ovd-tJ.-'\ce """if"., o t? ve...-S '~ v\~_;:;'_"""_""--=-"" _______________ _

71

Approximate design of tanks with the base slab continuous with the walls

Many researchers and writers of technical literature concerned with the design of beams or slabs on elastic foundations agree that, because of the nature of the problem, the results are at best a practical approximation and recommend that a 'safe' design be carried out in order to cover a range of possibilities.

This section on the design of rectangular tanks is connected with approximate methods of design which concur with this principle.

One method is to design the wall and part of the floor slab as a complete retaining wall subjected to the bend­ing moment calculated at the base of the wall being resisted by the dead loading of the water and concrete base slab in the traditional manner. as shown in the following example. The supporting earth is assumed to be a rigid mass with no elasticity.

The main object is to ensure that:

(i) the permitted maximum ground bearing pressure is not exceeded;

and

(ii) the base is in contact with the ground throughout and hence there is no uplift.

With the aid of Program 4.P, given at the end of this chapter values of x are assumed and refined until the results conform with (i) and (ii) above. (See Fig. 4.5.)

:. ---- "J :.

'l.".4kK ... 0

u •. 41t1i .... ..;

O.~

II ¥ 0- poYhol jo,"tJ II o. ~

10.0

Figure 4.5

The maximum moment at the base of the wall panel for an effective (length/height) ratio of 2 is, from Table 9.20,

86 * 3.\5 * 3.15 * 9.8\/\000 = 26.4 kNm/m

The joint must have sufficient steel across it to resist a horizontal force of:

(9.8\ * 3 * 312) * factor of safety (2) = 88 kN

72

Figure 4.6 Slab details

less any frictional resistance generated by the self weight of the water and concrete. (See Fig. 4.6.)

For shorter span slabs, such as the one used in the main example of this chapter, the following approach may be used. Essentially the slab spans between the walls, carry­ing a downward load resulting from the weight of the walls only (Ref. 4-6 and Fig. 4.7).

S\ob dt~QI'S

-=

q

-~

Figure 4.7

Design forces on base slab

Forces from wall 4.825 '" 0.35 '" 24 '" 2 5.35

= 15.2 kN/m2

Assume pressure resulting from self weight wall is uniform.

BM Max. (centre of base) - M at edge + Moment from upward pressure -105 + 15.2 • 52/8

= - 58 kNm

,..~ -105 kN",

Al'"r-­- 10SkK",

mnfftt n t t f t ttl n f t t tf t t t

- I05k.N ...

,Ukti/ ... 1 .

pnHurc fro," $.W. walls

(8)

.r+-r---rrH -lOS kti ...

(b)

Figure 4.8 (a) loading diagram, (b) bending moment diagram

The bending moment diagram, Fig. 4Jt, i& smutar lOttie one calculated earlier in this chapter by the more mathe­matical method, the maximum shear force cannot exceed 4O.S kN/m (self weight walls) and, as the span of the slab lengthens. the midspan moment tends to zero (Ref. 4·7) under these loading conditions.

4

.'­~y

.~ ~;;,:;. :,!;'~Q~

Figure 4.9

fl t.4lt5

1 HI ~

orl

,o."o,J

!r~:~ ~il . I

Figure 4.10

73

Output program 4P 1

******************************************************** THE HEEL LENGTH IS 0 METRES THE WIDrH OF THE BASE FROM THE WALL FACE IS 2.4125 METRES

THE THICKNESS OF THE WALL IS .35 METRES THE HEI~HT OF rHE WALL IS 4.825 METRES

********************************************************

++++++++++++++++++++++++++++++++++++++++++++++++++++++++ THE PRESSURE AT THE BASE OF THE WALL IS 169.6878 kN/SQ.M THE PRESSURE AT 'rHE END OF 'rHE BASE IS-40. 87228 kN/SQ.l-t

++++++++++++++++++++++++++++++++~+++++++++++++++++++++++

******************************************************** rHE HEEL LENGEH 13 .5 MErRE3 THE WIorH ~F rHE BASE FR~M rHE WALL FACE IS 2.4125 MErRES

THE rHIC~NESS OF rHE WALL IS .35 METRES rHE HEIGHT OF rHE WALL IS 4.825 METRES

**k*************************************************** **

++++++++++++++++++++++++++++++++++++++++++++++++++++++++ THE PRESSURE Ar THE BASE OF THE WALL IS 109.503 kN/SQ.M THE PRESSURE AT rHE END OF rHE BASE IS 2.145378 kN/SQ.M

+++++++++++++++++++++++++++++++++++++~++++++++++++++++++

If the method of approach given initially (p. 72) is used in the main example for the tank design then, using Program 4.P1, it can be seen that there is uplift at the centre of the base, see Fig. 4.9.

occurring in the base slab are similar (ii) the shear forces are not in close agreement (iii) the sub-soil reactions are also not in agreement.

The forces generated within a concrete container by the contents can be calculated with precision and confidence; however, the reaction of the supporting system. particu­larly when this is provided by the ground, is less predictable and hence many research workers in this field agree that extensive and precise mathematical calculations can be misleading, and recommend a 'safe' design approach which covers a range of possible occurrences.

To overcome this problem and, using the same program. a heel is provided as shown. This is of two­fold benefit since it provides both a firm support for the external wall shutter and, should flotation be a problem. the backfill upon the heel provides additional resistance. see Fig. 4. 10.

Conclusions

In the main example where the walls are continuous with the base, it may be noted that (i) the bending moments

Table 4.1

It is apparent that with only relatively minor increases in steel quantities this concept can be undertaken and Table 4. I gives the approximate percentage of steel reinforce­ment for the various alternatives in the design of the 225 m3 tank used in this chapter. see Fig. 4.11.

Condition at General reinforcement details Reinforcement base of wall %

b

Diameter T20 T12 T16 T12 T12 T20 T12 T12 T16 T12 Hinged Spacing 175 175 175 175 175 225175 175 175175 104

Face both both both both both top btm btm top btm

Diameter T16 T20 T16 T12 T20 T12 T16 T12 Fixed Spacing 175 175 175 175 175 175 175 175 100

Face both both both both both both both both

Fixed but Diameter T20 T12 T16 T12 T20 T12 T16 T12 with wall Spacing 175 175 175 175 175 175 175 175 107 corners Face both both both both both both both both strengthened

74

+1',

_9_ _ .... 8_

a,

·1 (8) (b)

Figure 4.11 Reinforcement details (a) long wall elevation, (b) plan of base slab

A further point is that service failures have occurred in practice as a result of excessive deflections of walls. BS 8007 advises caution with regard to deflection particu­larly where unexpected rotation of the earth occurs beneath a wall.

It is recommended that in the design process, when deflection governs the design, that this is resolved by thickening the wall rather than increasing the steel areas.

Flotation

When the empty tank is almost 'floating' in waterlogged ground then a uniform reaction will occur; however, the external water pressure assists in reducing the slab moments, see Fig. 4.12.

Should this situation arise it is necessary to ensure that

(8)

all elements, including independent floor panels, have a factor of safety of at least 1.1 against flotation.

'L' shaped retaining walls when the tank is empty can become unstable under these conditions and the provision of a thick base slab with an external heel is a simple solution, see Fig. 4.13.

Should the slab span from wall to wall, even though the overall structure may be stable, considerable moments and forces are generated and these can onJy be resisted by:

(i) a heavily reinforced thick concrete slab; (ii) by anchoring the slab down with reliable earth or

rock anchors;

or

(iii) in non-potable water situations by the provision of pressure relief valves.

(b)

Figure 4.12 An empty tank (a) loading diagram. (b) bending moment diagram

75

Program 4P1 Stability calculations for retaining wall panels restrained at edges or top and subjected to a specific bending moment at base of wall

76

10 RE!4 rHIS PROGRA~1 CAL::ULArES rHE BASE LENGTH SUCH THAT MBASE • !\WALL 15 PRIN'r" IS 'fHIS A TWO WAY SLAB? IF SO THEN HtPU'r TWA" 16 PRINr"IF 'fHIS IS NOT A 'rwo WAY SLAB THEN TWA-O' 17 INPUT ·T\~A • VALUE OF BASE MOMENT IF TRUE OR 0 IF PALSE"ITWA 20 INPUT"HEI3HT OF WALL • ; H 40 INPUT"THICKNESS OF WALL "; r 42 INpUT":>\AX PERMISSIBLE 3R'JUNO BEARING PRESSURE • ;GBp 50 INPUT"THE ESTIMA'TE 'JF BASE LENGTH TO BALANCE BASE MOMENT IS'IX 51 IF X >98 GOTO 290 52 IF TNA > 0 'fHEN l1W =rWA 55 IF 'rWA-O THEN MW-9.010001*(HHl"3/6 60 INPUr"the heel len,th "; HL 61 pRINT"MW=" ;:>\W 62 MC=9.810001*H"3/6 70 WA-r*H*l4 71 pRIN,."WA="; WA 80 WB=9. 810001*X*H 81 pRINr"I~B=" ;;~B 90 WCsT* (XI-nUL) *24 91 PRINT"WC=" ;WC 100 Wl=WA+)IBHK 110 Ml=WA*(r/2+HL) +1'13* (X!2.rI-HL) +HC*(X+'frHL)/2 ui PRINT"Ml=" ;111 130 RMxMI-MH 131 Ex (X+r+HL) /2-RM/wl 132 MO'r-Wl"E 135 PRINT"')VERrURNING MOMEN'r x" ;!I'J'r 140 PA-Wl/ (r+X+HL) +10101'*6/ (X+T+HL) "2 141 PRINr"PA~" ;P'" 150 ptl=IIl/(,F+X+IIL) -M'Jr"6/(XHHIL)"2 151 PRINT "PB-" ;PB 160 PT=PB+(PA-Ptl) *X/(X+r+HL) 170 MB-wB*X/2-PB*X" 2/2- (pT-P3) oX· 2/6+r* 24 *)(*X/2 179 IF TNA>O rHEN MC = M\i 160 01 FF=t1C-M9 190 PRINr "X-:";X;".i'~II;T;"H-='·iH: 200 PRINT "Wls";IU;"RM.";RM;"PA=";PA;"PB=";PB 210 PRINT" rHE SLAB BENDlt'l3 MOMENT IS" ;il3 220 PRINr "THE DI FFERENCE :>\C-M3 IS"; 01 FF 221 FX.X 230 Gno 50

,", .. ,t;

/II Q~ bon 'Noll ( M.TWA)

Hf.'9hi Tk,c,k"c.\& Ma •. 'll, .... "d bf.01'11\9 P"Hur'& Elt. b .. u. 1""5"' I\ul lc."stk

Figure 4.13

290 LPRINr ,,******* ••••••••••••••••••••••••••••• **** •••••••••••••••• " 299 LPRINT "THE HEEL LENGrH 13 "; HL; • ME'fRES· 300 LPRINT "rHE WIDTH OF fHE BASE FROM rHE WALL FA(;E IS"; FX; " MErRES" 301 LPRlt'lT " THE 'fHICKNE:SS OF rHE WALL IS ";T; " ME·rRE:S·

~~~ t:~~~~ :: * * * •• *. * ;~~. ~~~~~.;. ~~ .;~; .~~~~*;~ •• * * * * * * * *:! ~! *:. ~~;~~~: 304 LPRI NT" " 305 LPRINr 1\ ++++ t++++++ ... t++++++++ r+++++++++++ +++++++++++++++++++++++" 306 LPRINT "rHE 'fOrAL WEI3H'f OF WALL, SLAB AND WATeR IS";;'Il;" kN" 307 LPRINT " rHE CENrRE OF GRAVUY OF PHe: LOA!) 15";E ;" ~ErRES FROM 2ENPRE LINE 306 LPRHIr" rHE PRESSURE A'r THE BASE OF PHE WALL IS"; PA;" kN/SJ./I" 309 LPRINP" PHE PR8SSURE A'r THE END OF PHE BASE IS"; PB;" kN/SQ.M" 310 LPRINr n +++++++++++ +t+++-+++++++ .. ++ t+ ttt+ t "" rt++ t +++++++ t++ .. ++ ... +++" 317 LPRINr" " 316 LPRINf" " 319 LPRI N"r "++ + t+++++++ t+++++++++++++++ t t++++++++++++++++++++++ "++tt" 320 LPRINT "THE SLAB BENDING :>\'JMENf IS" :M3 330 LPRINT " rHE DIFFERENCE r~C-MB IS";DIFF;" kN'I" 3 31 LP RI NT 11 + t- t+ t t++ t++++++ tt t +++ t++ ... t ... ++ t-+++++" rt+ t t + t-+++ t+ t t+ ...... +++" 332 GOTO 400 335 J=FX/IO 340 FOR I=FX 1'0 0 STEP -J 345 '';0 = H"9.8l0001 +r*24-PB 350 L= F:< - I 355 1'1= (P,'-P31/FX*L 360 :~L=IVD*L* L/2-P l*L* L/6 361 VL=\/f)'L-Pl*LI2 36 2 LP R I N f" . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - "

363 LPRINr" " 365 LPRIN·r ";HE LENGr'H fRC>;.t END r:J "jLi'''1ErRES'' 366 LPRUU "i'HE: BEt'lDING ;\O:1E'lC IS ";ML: "kiM" 367 LPRI:;r "PHE SIICAR FOR::E IS ";VL;"ktl " 360 LPRINP", 370 LPRUll'''------------- --- '---------------------"

330 ~Exr I 4 00 E:ID

4-1 Timoshenko S, Woinowsky-Krieger S 1959 Theory of Plait'S and Shells McGraw-Hill

4-2 Bares R 1971 Tables for A.no.Jysis of Plales, Slabs and Diaphragms &uverlag GmbH, Berlin

4-3 lofreit l C 1975 Design of rectangular concrete tank walls. ACI Journal, July

4-4 Portland Cement Association 1969 (revised 1981) Rectangular Concrete Tanks Illinois

4-5 Iyengar K T S R. Rarnu S A 1979 Design Tables for Beams on Elastic Foundations and Related Structural Problems Applied Science Publishers

4-6 Winterkom H F, Fang H Y 1975 Foundation Engineering Handbook Van Norstrand Reinhold

4-7 Hetenyi M 1958 Beams on Elastic Foundations The University of Michigan Press, Ann Arbor

77

Design of circular tanks

The operational processes within the water and other industries dealing with fluids often require circular structures to ensure their systems of work are carried out efficiently and economically.

The primary stresses set up within the structure are usually a result of the ring tension generated by the con­tained liquid and the main reinforcement, therefore. consists of bands of circular steel hoops. The ability of the cylinder to increase in diameter is resisted, however. at the base where restraint occurs. If outward movement is prevented by a fixed or pinned joint then the ring tension will be zero and vertical bending moments and shear forces will occur.

Tables have been prepared (Refs 5-1, 5-2) to assist the designer and are used in this chapter. More recent research has been involved with 'soil-structure' interaction and the paper (Ref. 5-3) is given as an appendix.

Examples of the design of an open topped 12 m internal diameter concrete tank,S m high. subjected to various base conditions, form the basis of this chapter. The examples. in particular, demonstrate the significant reduc­tion in calculated moments and forces which occur when the design takes into account the fact that the soil pro­vides an elastic rather than plastic reaction.

Three typical base conditions are given, see Fig. 5.1.

(c)

(b)

Figure 5.1 Base conditions of circular tanks (a) free sliding, (b) hinged, (c) fixed

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Using Lightfoot and Michael's tables a circular tank upon an elastic foundation is now designed.

Example 'f takes into account the fact that the bearing pressure is related to the elastic properties of the soil which supports the tank and that the foundation stiffness is a com­posite function of the properties of the slab and the supporting subgrade upon which it bears. Winkler's theory states that:

. At any point the foundation reaction varies linearly with the deflection ..

The surface deflection is related by the formula:

p = k.w

where p pressure, k = modulus of subgrade reaction. w = deflection.

The Modulus 'k' can be determined by such tests as the plate bearing test but the results are affected by:

(i) method of testing; (ii) moisture in the soil;

(iii) compaction of the soil.

It tends not to be a precise figure; however, the results will show a general increase in value as the ground becomes harder. Approximate values obtained are:

Type of soil

Clay of high plasticity Low plasticity clays, silts, poorly­

graded fine sand Well-graded and clayey sands, poorly­

graded and fine gravel Well-graded gravel

13 27

54

82

Figure 5.3 Typical circular tank. Note joint positions and pressure relief valves

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Figure 5.2

De8ign of circular tank with e~ ~

One common problem with the design of circuiar tanks occurs where an external channel is fixed at a high level as shown in Figs 5.3. 5.4 and 5.S. There is a moment generated at the connection of the channel and the wall; and also. the base of the channel will prevent the main wall increasing in diameter and, therefore. the ring tension should be zero at this point. One solution is to design the tank as follows.

(i) Using the tables in Chapter 9 analyse the upper part of the tank above the trough and calculate the moments and shears at the base of the trough level 'a'.

(ii) Design the lower part of the tank for the triangular loads and forces generated at trough level.

(iii) Knowing that the ring tension value at the trough level is zero. calculate the force required to generate an equal and opposite ring compression at this point and then calculate all moments created by this force and add the results to (ii) above for the fmal results.

References

5-1 Timoshenko S, Woinowsky-Krieger S 1959 Theory of Plates and Shells McGraw-Hili

5-2 Portland Cement Association 1965 (revised 1981) Rectangular Concrete Tanks Illinois

5-3 Lightfoot E. and Michael D 1963-1965 (4 Parts) The analysis of ground supported open circular concrete tanks. Construction Weekly

~ ... ~ l ~ c.Q~t(i,,) -+ CO$l\'o') Ruul+

I ... r d 101

(c)

(a)

I (b)

Figure 5.4 Circular tank (8) plan view. (b) cross section, (c) alternative methods of slab reinforcement - note mesh can only be used when slab is flat

102

proggm 5P, Analysis of a fixed base, cylindrical tank

4 LPRINT"***********************************************" 5 LPRINT" .. 6 LPRINT"DESIGN OF A CIRCULAR WATER RETAINING STRUCTURE" 7 LPRINT" II

8 LPRINT" 9 LPRINT"************************************************" 10 LV"5.125 h'put: 11 R"6.125 Lv "" 12 H=.25 t WI

13 Bz(3/(R*R*H*H»~.25 "~ 14 LPRINT"THE RADIUS OF rHE ~ANK IS ";R;" metres" 15 LPRINT" 16 LPRIN'f"THE 'fHICKNESS OFf HE WALL IS" :H; "metres" 17 LPRINf" 18 LPRINT"fHE BETA VALUE IS ";B 19 LPRINT" 38 LPRINT"------------------------------------------------------" 39 LPRINT" 40 LPRINl'"HEIGHT(M) MO~lENl'(kN.M) RING TENSION(kN) 41 LPRINT" " 42 LPR1NT"------------------------------------------------------" 50 LPR1NT" " M t 70 POR 1=11 TO 1 srEP -1 i:bJ i~o~;i~:~:;:::::)*COS(B*X) ~ 1 110 SBX=EXP(-B*X)*S1N(B*X) ~v_ 120 W=9.810001

114300 P::W*R*LV • a. I l I

M=-F*H*.289*(-SBX+(1-1/(B*LV»*rBX) 150 FT=F*(1-X/LV-TBX-(1-1/(B*LV»*SBX) 180 LPR1N'r" 190 LPRINT D;rAB(9) ,:INT(M);" i';:IW1'(P'f) 191 LPR1NT"----------------------------------------------- _______ H

290 NEXf I 295 LPR1Nl'" 296 LPRINr" 297 LPRINr" 298 LPRINT" 300 V=-F*H*.283*(2*B-l/LV) 310 LPRINT" rHE SHEAR FORCE AT rHE BASE OF rHE WALL IS ";V 311 PRHlr" 312 LPRINT" 313 LPRINT"*******************ft**********************************"

103

i

Figure 5.5 RC details for a circular tank

104

Design of prestressed concrete circular tanks

The benefits of the use of prestressed concrete tanks for the storage of water and other liquids include the following.

(i) The concrete is in compression and can be so designed that cracking of the concrete should not occur.

Oi) The sections can be relatively thin which generates savings on costs of materials and reduces foundation loads.

(iii) Prestressed sections often have a greater resistance to ground movements than other forms of construction.

Designers are being encouraged to be more concerned not only about the adequate design of PS structures but also their safe construction, maintenance and repair, and, finally, demolition.

In the design example carried out in this section it is

Table 6.1

Nominal values only

Type Nominal Tensile Steel (kg/m) diameter strength area

(mm) (Rm) (mm2)

(N/mm2)

as 5896 15.2 1670 139 1.09 Standard 12.5 1770 93 0.730

11.0 1770 71 0.557 9.3 1770 52 0.408

BS 5896 15.7 1770 150 1.18 Super 12.9 1860 100 0.785

11.3 1860 75 0.590 9.6 1860 55 0.432 8.0 1860 38 0.298

BS 5896 18.0 1700 223 1.75 drawn 15.2 1820 165 1.295 (Oyform) 12.7 1860 112 0.89

proposed that the post tensioned system of construction used is for the cables to be held horizontally on external vertical hangers and be stressed as shown in Fig. 6.1 from 4 external jacking pilasters. The cables used shall be plastic coated standard strand to BS 5896 (see Table 6.1 below).

In order for the cables to be inspected, maintained or replaced it is preferable that they be visible on the outer surface of the wall and sufficient length of cable be left beyond the anchorage point for future de-tensioning. If this is carried out then further protection in the·form of an outer plastic tube will be necessary.

Should the exposure of the cables be impracticable then the cables and the concrete surface should be sprayed with gunite: but only after the tank is filled. This is to limit the risk of the cracking of the gunite when the tank expands as a result of the internal water pressure.

Specified characteristic values

Mass Breaking 0.1% proof Load at (m/l0G0 kg) load load 1 % elongation

(Fm) (Fp 0.1) (Ft 1.0) (kN) (kN) (kN)

917.4 232 197 204 1369.9 164 139 144 1795.3 125 106 110 2451.0 92 78 81

847.5 265 225 233 1273.9 186 158 163 1694.9 139 118 122 2314.8 102 87 90 3355.7 70 59 61

571.4 380 323 334 772.2 300 255 264

1123.6 209 178 184

105

as 8007 (1987) Design and detailing of prestressed concrete

General: (Section 4.1 of BS 80(7)

This section gives methods of analysis and design that will in general ensure that for prestressed concrete structures the recommendations in section two are met.

Basis of design: (Section 4.2 of BS 80(7)

Design should be in accordance with the recommendations given in section four of BS 8110: Part I: 1985 except where these are at variance with the specific recommendations of this code. In general the design of prestressed concrete members in exposure conditions as defined in Section 2.7.3 of BS 8oo7 is controlled by the concrete tension limitations for service load conditions. but the ultimate limit state should be checked.

Cylindrical prestressed concrete structures: (Section 4.3 of BS 8(07)

The special recommendations for the design of cylindrical concrete structures prestressed vertically and circum­ferentially are as follows.

(a) The jacking force in the circumferential tendons should not exceed 75 % of the characteristic strength.

(b) The principal compressive stress in the concrete should not exceed 0.33/cu.

(c) The temporary vertical moment induced by the circum­ferential prestressing operation in the partially stressed condition should also be considered. The maximum value of the flexural stress in the vertical direction from this cause may be assumed to be numerically equal to 0.3 times the circumferential compressive stress. Where the tensile stress would exceed 1.0 N/mm2• either the vertical prestress should be increased or the circum­ferential prestress should be built up in stages, with each

106

stage involving a progressive application of p~ from one end of the cylinder. '~,?:,

(d) When the structure is full there should be no re6UlUmt tension in the concrete in the circumferential direction, after allowMICC lor all. Josses of prestress and on.fue assumption that the top and bottom edges of the wall are free of all restraint.

(e) The bending moments in the vertical direction should be assessed on the basis of a restraint equal to one-half of that provided by a pinned foot, when the foot of the wall is free to slide. In other cases where sliding at the foot of the wall is prevented, the moments in the vertical direction should be assessed for the actual degree of restraint at the wall foot. The tensile stress arising from vertical moments should not exceed 1.0 N/mm2•

(f) Where the structure is to be emptied and filled at frequent intervals, or perhaps left empty for a prolonged period, the structure should be designed so that there is no residual tension in the concrete at any point when the structure is full or empty.

Prestressing wire may be placed outside the walls, provided that it is protected with pneumatic mortar. However in industrial areas or near the sea, where there is a possibility of corrosive penetration of the covering concrete, the cables should preferably be placed within the walls and grouted. Non-bonded tendons may be used provided that they and their anchorages are adequately protected against corrosion.

Cylindrical concrete structures which are prestressed cir­cumferentially and reinforced vertically should comply generally with the recommendations of this clause, except that 4.3(f) may be relaxed to allow tensile stresses not exceeding 1 N/mm2• The design for the vertical reinforcement should be in accordance with section three.

Other prestressed concrete structures: (Section 4.4 of BS 8(07)

Class 3 prestressed concrete structures as defined in 2.2.3.4.2 of BS 8110 : Part 1; 1985 should be designed in accordance with 4.2 and 4.3. In addition, the nominal cover should satisfy the 'very severe' exposure conditions given in Table 4.8 of BS 8110 : Part I : 1985, and should be not less than 40 mm.

cable He. t-·

t-·

~-.

4-· 5-' ,_. 1-· 6-'

Figure 6.1

Method of stressing

-:-1 I

In general terms, in order to reduce the effects of elastic deformation of the concrete, it is prudent to initially apply only 25 % of the total force to all the cables in the following order:

Figure 6.2

strands no. 1 - 4 - 7 - 10 - ..... . strands no. 2 - 5 - 8 - 11 -strands no. 3 - 6 - 9 - 12 - ..... .

The stressing operation can then be repeated in three further increments of 25 % until the strands are fully stressed. The amount of 'pull-in' of the cable at the anchorage point can be measured and allowed for so that losses due to this effect are zero.

If stressing from one point only, i.e., strands la and Ib tensioned from pilaster A, then, for the next 25 % of force being applied, tensioning should take place from pilaster C, the third 25 % from pilaster A and the final 25 % from pilaster C. It is preferable to use two jacks when stressing from a pilaster since the frictional losses due to curvature are reduced by this approach.

During the stressing operation it is desirable that the design engineer be present and that the anticipated exten­sion of the cable for the maximum force has been cal­culated in order to ensure that the initial design assumptions are justified, ~ Figs 6.2 and 6.3.

107

Figure 6.3 Post tensioning cables around circular filter bed

108

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117

Effect of temperature gradient across the wall

There is a temperature difference between the inner and outer wall surfaces which will create tensile and compres­sive stresses across the wall. This problem has often been ignored in the past usually without serious consequences. In more recent times, however, it has been considered by various researchers.

In a paper by M J N Priestley it is concluded that the design approach for this problem should be based on a serviceability condition whereby, under extreme thermal conditions, a vertical crack less than 0.1 mm wide should be acceptable.

The article concludes by proving a calculated crack width equation which is given below:

(I-x)2 h * fp w = 4 * --x2- --E-

118

fp maximum prestress in concrete h wall thickness a = coefficient of thermal expansion of the

concrete T = temperature differential

In the example used in this section the maximum stress in the concrete is, before long term losses, 2.77 Nlmm, = 0.000 012/oC and T = say 25°C.

x= I 2x2.77 =0.613 ~ 31 000 X 0.000 012x25+2x2.77

4 x (1-0.613)2 X 125 X 2.77 w = 0.6132 X 31 000 = 0.018 mm

(which is acceptable).

Reference

6-1 Priestley M J N 1985 Analysis and design of circular prestressed concrete storage tanks. pel Journal, July-August

Design of a flat slab roof and columns for a reservoir

The columns supporting the roof of a reservoir rarely create a problem with regard to spacing or causing an obstruction. Downstand roof suppport beams, however, can be inconvenient particularly when they restrict the flow of air above the water when the water level is higher than the soffit of the beam. The introduction of ventila­tion holes through the beams permits the cross flow of air but provides a surface area of concrete which cannot easily be cleaned.

A flat slab roof without beams or drops is the .most con­venient form of construction which can be easily main­tained and cleaned and is, therefore, often used in this situation.

The basic definitions, terms and methods of analysis specific to the design of flat slab roofs of reservoirs are extracted from BS 8110 and are given in Tables 7.1 and 7.2. In order that Table 3.19 of the code may be used it is necessary for the roof to have at least three rows of panels of approximately equal span in the direction being considered. If this is not the case a flat slab may stili be used; however. an elastic moment distribution method of analysis or similar must be carried out.

Table 7.2 Distribution of design moments in panels of flat slabs (BS 8110, Table 3.20)

Negative Positive

Apportionment between column and middle strip expressed as percentages of the total negative or positive design moment

Column strip

% 75 55

Middle strip

% 25 45

Note For the case where the width of the column strip Is taken as equal to that of the drop. and the middle strip is thereby increased in width. the design moments to be resisted by the middle strip should be increased in proportion to its increased width. The design moments to be resisted by the column strip may be decreased by an amount such that the total positive and the total negative design moments resisted by the column strip and middle strip together are unchanged.

Table 7.1 Bending moment and shear force coefficients for flat slabs of three or more equal spans (BS 8110, Table 3.19)

Outer support Near First Centre of Interior centre interior interior support

Column Wall of first support span span

Moment -0.04FI' -O.02FI +0.083FI· -0.063FI + O.071FI -O.055FI Shear 0.45F O.4F 0.6F O.5F Total column moments O.04FI O.022FI 0.022F1

'The design moments in the edge panel may have to be adjusted to comply with 3.7.4.3 of BS 8110

Note 1 F is the total design ultimate load on the strip of slab between adjacent columns considered (I.e. 1.4Gk + 1.60J. Note 2 I is the effective span E 11 2hc/3. Note 3 The limitations of 3.7.2.6 (BS 8110) need not be checked. Note 4 These moments should not be redistributed.

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effective dimension of • head

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design ultimate value of the concentr.ted lo.d

design shur tr.nslerred to column

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Design of RC columns

The columns which support reservoir roofs tend to be relatively slender, with regard to BS 81l0. clause 8.8.1.5. given below:

Braced and unbraced columns: A column may be considered braced in a given plane if lateral stability to the structure as a whole is provided by walls or bracing or buttressing designed to resist all lateral forces in that plane. It should otherwise be considered as unbraced.

It is not unreasonable to suggest that since the water loading is generally uniform throughout a reservoir that 'lateral forces' cannot be generated.

Movement of the roof, however. particularly when it is open to the elements. does occur and this movement

50

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V :/ V V V / /. /. /. /.

has been shown to affect walls and columns adversely. The design of the column used in this ex.ample assumes, therefore, that the column is unbraced.

The column is designed firstly for the ultimate load condition using the processes and tables given in BS 8110. The column is then checked for the serviceability state to ensure that the maximum crack width does not exceed 0.2 mm.

In order to carry out this stage of the calculations the elastic theory again has to be used, see Figs 7.1 and 7.2. A graph has been prepared to assist the designer; however, at the end of this chapter the basic equations for the elastic design of columns are given together with a computer program which incorporates these equations and permits the designer to check the column design for serviceability conditions.

10 11 12 13 14 Hi 16

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Figure 7.1 Limit state chart (BS 8110. Chart 39)

127

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le/b ~L~ /<:,.~O<> "" )l"."

ie-II:> ) 10 : C;!.ev-de .... ,-,<:>1.

CaLv ...... "" b~ ~gVr- ::'of:"l,..,

~ e.-1 :-'

I AI,.) "'f-.~.k.",,: " fA co· I ~ j 1<.:. \

()... vi '" o· l~ I<. l x. o.~'

129

130

e ... He;; N

e./L-\ ,. ''2l.#b/5''e--o o.\:~

A. s.~ v """' e- \ ~ 1-e,..r ........ t><' e : ~ £. ~ 15' ~

~c.. {JO"""", P vt-c,..;'

(e../L--\. '" o. ~~ ;:c.

cJ....

""'c:~ o-C 7P, )~ cA--.A. ..... t:- 7/'Z. .J .P~ At:.c- = . $

.", "lC,...:' d .. ~(,'Z. J(. <...f'~ • foi. --.

iV'o",- """"'VI'""'" , T r D • 11_ ~ , 1- Mo. '0 'Z. ~ ~,,. -;.. Co~ h.

b"""" f"", C;; h-"~ i ...... co""c,..

:. tc::..1o ~ b K. lO" / (5&-0 X. 5&-0"- K 0.\0'2) • '·t..b ~ I~: le... .>-t"'<:.4-? ; '"" So. L""l!:.e--1 )

f~:: !>(~. ~ (I - )(, /cA)j():./d.) (?"" F~.. \\'"1< l.u.~ (1- ·4~2. )/C'-t Ie 7-) I;~ ..J/ __ '"

Co) c..Yp-c..k. 1.N'd..Th. -,-,e.,fe", pV'o~",,,--..{ ovr-p",l" if', VI -:- t;' Q-O-....... ; d-. -::. Y'') ~ ___ ; ')C.:: to'?> ........

Z~ = I~/'Z.o-o~ a O.~(p1 oft .... OQ.-o (Q] x. t~] ,,0 .0-9-0& ......

2~

'S. e... ':> &-{) X 't ~ J ,.. • . 99'0 ~'

~ K 'to-o ~ K <Pz.t>)( 2.><0

A c-v' ;t Jc~ '-a f)'" + IS L¥ '\< _ to :. ,~y,-.

W::. S 1<. ,~ Ji( Q.9-Q--O ~ ~ 0"'-_

(I + t (1(1, - '":)"~)/t~1)

, " t.....l <: 0.2.. ...............

OUtput program 1P1

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ THE INITIAL FACTORS TO CHECK THE SERVICE S'l'RESSES IN THE STEEL ANO' CONCRETE,

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ .. +++

01" .8i

NEUTRAL AXIS FACTOR N1 .4701159

M/(B*H*H*FCB) FACTOR = .1023644

Percentage of Reinforcement 2 .5

e/h factor a .5369864

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 'rHE FULL SERVICE CHECK INCLUDING CRACK WIDTH

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ FCB FST W H N M

N/sq.mm N/sq.mm mm '11:n kN kNM +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

7.66 129.49 0.10 500.00 365.00 98.00 +++++++++++++++++++++++++++++++++++++t+++++++++++++++++++++++++++++++

Figure 7.4 Columns in reservoir construction

131

Elastic theory for RC column design and development of computer program

The equations used are based upon the two conditions listed below and upon Figs 7.3 and 7.4.

Condition J: Internal compressive and tensile forces = external forces

[ fcb.x.b & b x-d2 A ] + ----lC.-- sc

2 x

Asc. fsc - AsLfst = N (a)

Condition 2: Internal moment of resistance of section = external moment (Taking moments about centre line of column)

(fCbtb

) (+ _ -}) +

Asc(fcb) (ae-I) (x ~d2) (-i- - d2) +

Ast.fcb.ae(h~X) (-i--d2) =N.e (b)

Noting that

(i) fsc = (x ~d2) ae.fcb,

(ii) fst = (h~X) ae.fcb

and

(iii) p = l()().A&c/b.h

~tI\ :

-~ I

We combine these equauons to p~

fCb~h.h == (x·f (3 - 2.x·f) +

O.O:.p (1-2 .~) [(x - ~) (ae - 1) +

(1 - x)ae )]/12 " (~) Eliminating feb from equations (a) and (c) we obtain:

X.x (~::) (3 - 2.x·f - 6.f) O.~3.p [(ae - l)(x.f -~)

. (2d2 + ~ _ I) + ae. i. (l - x) h h h

e·:2 - 2he - 1)]= 0 (d)

Using fixed values for 'e/h' and 'p' the value of x can be determined from equation (d) and then 'feb' can be determined from equation (c). The value of 'fst' can be obtained by using the equation:

fst = C ~x) ae.fcb

Once all these elements are known then, by using the equations given in BS 8007 the crack width 'w' may be readily determined.

Program 7P1 solves the above equations (page 133).

I M

~, !i rt . ~ 1~~;J]r.~

! 'n/1

~-~-~".'. ~ ~?",---.---t --- -- ~-- '. ,,--.-.---- -.

------ _. -- - -- - ---.--\-t.

Figure 7.3

132

Program 7P1 Design of RC columns for serviceability limit state

I~;

1)1 (,",", t .. 4. tu ...... "MI/II) Fed ... , ('n",dll.)

5 COUIiT • 1 10 IMPUT "D1."II)1 20 02-1-D1 21 INPUT "11.",1 22 INPUT "H- ", B 23 IIIPUT "Ita- ",Ita l4 INPUT "n- ",M 29 8-M*1000/IiS 30 F2-E/H 35 '-1'2 40 INPUT "' OF REINFORCEMENT·" II'

...... -'" -"

SO INPUT "NEUTRAL AXIS. note (If y"" "ish to finish then type 99 ) ",N 51 If' N-99 THEN 400 60 F3- N"N'Ol*ol' (3-2'N'DI-6*F2) -.03"1" (14' (N'01-02) • (2*D2+2 'F2-1) +15*01* (I-N) " 2'02-2*f2-1) ) 70 pRINr "THE V,.Lue OF P'J IS";F3 71 COUNT -COUNT +1 72 IF COUNT> 100 THEN 120 80 IF 1'3>.01 THEN N -N'(1.01 + COUNT/(20'COUNT» 90 IF F3<-.01 THEN N -N/(1.01 + COUNT/(2S'COUNTJ) 100 IF F3> .01 THEN GOTO 60 110 IF f'3<- .01 THEH :;OTO 60 120 Fl" (N'OI* (3-2'N*0IJ +. 03'P/N* (1-2*02 J' ( (N-02J "14+ (I-H) '15) J 112 130 PRINT" " 135 PRIN-r" THE CURRENT NUIIBER OF CIRCUITS IS 140 PRINT "02 0 ";02 150 PRINT "010 " ,01 160 PRINT "Neutral "xi1 ractor n - ",N 170 PRIll1' "Percentage of R.inforc ..... nt ·",P 190 PRINT "M/(b'h'h'fcbJ F,.CTOR - ", f'l 205 COUNT-l 206 LPRlN'Z'" " 210 IF P > 8 -rHEN 400 220 LPRINT" • 221 LPRINT" •

"'COUNT", ~

n~ Ir~ ;: :: D-'-;i-I:" .-=-t

222 LPRIN;r • +++++++.++++++++++++++++++++++++++++ '" t.+++"'++++++++++++++"'++++++.++ +++++++++++ .. 225 LPRIHT" THE II/ITI,.L ,,.:::TORS TO CHECK THE SERVICE STRESSES IN THE STEEL liND

CON;:UTE • 227 LP RI NT • ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++ +++++++++++. 229 LPRlltr" " 250 LPRINT "01-",01 251 LPRINT' " 260 LPRINT "IUUTML IIXIS FIICTOR N1 • ",N 261 LPRINT" • 262 LPRINT "1I/(8'H'H'rCBJ rIlCTOR. ", Fl 263 LPRIN-r" " 270 LPRINr "'PercentAge :>f Reinforcement ·"'IP 271 LPRINT" " 280 LPRIN-r "e/h factor • ": F2 295 INPUT "If' ,.1.1. IS OK rHEN INPUT 1 OTHERWISE INPUT 0 ":OK 296 IF OK-lrHEN 321 300 F2-Z 320 If' OK • 0 THEil 50 321 LPRINT" " 112 LPRI NT • +++ r t r+ tt t ttt .... t t ttttt t +1" t tttt t tttt t t t t tt ... tt ...... tt ... +tf ...... ffftttt+ ttt .. ,. l2J LPRINr" 'THE FULL SER'IICE CHECK INCLUOItI;; CRIICK WIO-fH " )25 INPUT "UAR OI"HETER - ",011\ J27 02R-40+6+0IA/2 329 OSBO ... -1I-2'02R lJO PRIHT" 056011-',056011 J)l INPur "OISr"NC£ BE-r;I£EH Al)JOIHI~G BARS' ",DB 13J FCB - :1'IOOOOOOI/(H*II'H'FI) 135 Fs'r • 15 'FCB'(I-f2}/F2 137 INPUT "'THEW-TAL AREA OF STEEL 15 ";AREAC 3 J9 INPu'r "TilE IIREA OF s'rEEL IN THE TEl/SlOt! ZONE only IS ",lIST 1H X-F2' (H-02RI )-13 ss'r- FS'l'/200000 t HS 51 • SS'T'(H-X)/(H-X-D2R) 141 STEFF-H' I H-X) "2/ (J' 200000 I ''''sr' (H-X-02R) ) 149 SII -51-StEFF 351 "CR-SORI (Oa/2) "2+02R"2) -OIA/2 353 W- l· ... CR·SH/I 1+2' ("'CR-DSR+OIA/2)/(II-X) 154 LPRI NT • t f+t .. tttftt++tt+ tt t ttttf+ t+-t r ttf t ... t ........ t ttt tttttt+-t ttf ttt ttttftt+- ... ," 355 LPRlwr " FeB F3T

)56 LPRINT" " lSI LPRINr" tllsq..... ~/.q.mm mm .... kN • 359 LPRINr .. t t "++tft t t rtttt++-f t-t tttt+ ttt t ttt t t+ tfttttttt tt t t++-ttt++++ t+ t++t++i T"

)60 LPRlItr" • 362 LpRINr" • J65 I.PIUHr USIN:l· Uef.£E ",FCS:Fs-r,W;II,N;H 370 INPUT "CHANGB R~INroRcell~Nr' 1 THEN ZZal O-rHERWISE ZZ-O "lIZ )13 LPRINT" " J 1 5 LP Rl Nt • t t .... t++-+++1' t. t t t t "tt ft+ t++ t+ + t+t"'" t+ tt t+ tt+ t++++++ tt ft++ +tt+t t+++ +i ," lSO lP n • I TUEll 40

133

Design

This type of construction is used for such structures as settlement tanks in water pollution control plants or in water towers. The design of this type of structure is simplified by the use of the tables given in Chapter 9; however, where the tank is built below ground level, the design can be of a relatively simple nature by careful con­sideration of details (see Chapter 2).

Various authors (8-1 to 8-4) have considered the design of conical shells. The tables given to assist in the design of the cone are, as with the circular tanks, mainly con­cerned with the effect of fixity at the base or apex of the cone upon the remainder of the structure. Two examples are given, the first is for a reinforced concrete tank and the second for a steel tank.

134

When building water towers the engineer has to become involved in aesthetics as well as structural design. The tower is usually built at as high a level as possible and may well become the main feature of a locality.

Many tower structures are built on a wide base and become slender and elegant with increasing height. The water tower. however, must store a large quantity of water as high as possible and hence the structure has, at its apex, a large container which can easily look unsightly. It is prudent to take photographs of the site from near and far and, having decided on a range of suitable shapes, to insert an elevation of the tower to scale on the photographs and permit a wider audience than usual to voice their opinions!

j I'1ow CMn for the calculation of moments and forces within the· walls of a conical tank supported at the base

I wJ with additiond restraints at either base or apex of I'he cone, see Fig. 8.1. (An example of the WIe of the flow chart follows on page 140).

A LOADING

(1) Calculate self weight of walls per unit area.

(2) Calculate load from roof on to circumference of upper cone. (See Ref. 8-1 for useful tables)

(3) Calculate any surcharge pressure.

(4) Pressure from contained water increases linearly by 9.81 kN/m.

B CALCULATION OF FORCES AND MOMENTS IN UNRESTRAINED CONE

Using the equations given on page 139 determine the following:

(I) meridional force N:

(2) ring tension force T;

(3) rotational movement of cone 8 at base and apex of cone only (y = f' and y = f);

(4) displacement of cone v at base and apex of cone only (y = t" and y = 0.

(Note: Program 8.P, at the end of .his chapter may be used to calculate all these value" 1I1Oth positions.)

Refer to page 140 for example, also refer t9 page 141 for Output 8P, which confirms the hand calculations. Program 8P I is used to solve the equations given on page 146.

Figure 8.1

Figure 8.2

135

C DETERMINATION OF FORCES TO CREATE FIXITY AT BASE

4 /12*,,2"'cota2

(1) For y = i', calculate x = 2 ~ Ju t

(where t is the thickness of the shell) (2) From Table 9.37(b) select values of CoH', C5M',

COH' and COM'. (3) Calculate Fct = CoH' '" COM' - COH' '" CoM'. (4) Calculate: KMO' = CoH'/Fct.

KHO' = - CoM' IFct. KMo' = - COH' IFct. KH5' = COM' IFct.

(5) Apply equal and opposite 0 and v values as calculated in Section B to create fixity at base of cone. (Note: v = 5')

(6) Calculate the values of the moment M' and the shear force H' which will occur as a result of fixity at base for y = i'. The values required are MO' , Mo'. HO' and Ho' • where

MO' = KMO' '" E '" t '" f' '" 0' I (Tan ci) Mo' '" KMo' :I< E'" t '" 0' I (Sin ex >II Tab ex) HO' = KHO' >II E'" t '" 0' I (Sin ex '" Tan ex) Ho' = KHo' '" E '" t '" 0' I (f' '" Sin ex

2)

(7) Using coefficients given in Table 9.37(a) determine the effects of fixity upon the forces calculated in Sections BI and B2.

136

The coefficients given are for fixity at the base or apex of the cone. The value of 'x' is calculated at the point of fixity and the tables give a series of coefficients in the following way:

For base fixity: For apex fixity: x = n (value at fixed x = n (value at fixed

point) (y=f') point) (y=f) x = (n+1) x = (n-l) x = (n + 2) for y values x = (n - 2) for y values x = (n+3) >f' x = (n-3) <f x = (n+4) x = (n-4)

The coefficients etc indicate how effects of the forces M as one moves further from the point of effects are noted in tables for circular tanks the forces M and V occurring at the ex.trelll1iti'e8 walls.

The 'x' (fil!.ed point) values given in the tables are in increments of 4 and interpolation when 'X'{flJ<edpoint) is not a multiple of 4 is satisfactory.

An example is given on page 140 for a cone fixed at the base but free at the apex. where the X(fixed point) value is 9 and the coefficient values further up the cone are x = 10 - 13 incl.

The bending moments, ring tensions and meridional forces resulting from M' and H' are determined with the aid of the equations given on page 143 and Table 9.37.

It is quite acceptable to use the tables for fixity at both base and apex, however, it is only in short thick cones that the effects of fixity at one end have significant effect on the other end.

For a detailed analysis and solution of the problems of fixity in cones Refs 8-3-8-5 are of benefit.

Once an the forces are known then the design is carried out in a similar manner to the design of a circular tank.

D FOR THE DESIGN OF CONES FIXED AT THE APEX FOLLOW A SIMILAR PROCEDURE TO THAT FOR FIXITY AT THE BASE EXCEPT THAT Y = e AND USE TABLES 9.38

Note: Tables 9.37 and 9.38 have been prepared assuming the Poisson' s ratio' 11-' = O. In the example which follows • 11-' is therefore assumed to be zero throughout the calculations.

M e.1ooA. b .... ~e. .f.p vc..t!- f: i \A c.ov...; CA:V'\. 1-~ "'" ~~» 0 trl:> ~ d..--~ ~e. pA.-S'e. -

') g i""'1 T t!,..1o"\.-(; 0",", -to V"ce. I T (4-) .;t He-v ic:(A 0 fA,........,(

f04"'c.~, ,..,J ("") ,ave ;-01 Ce.l(. t-)c.A~hl:: CioV) ... (

I . TCw)

I~ I 4

1

'\ Cev} ... r'v " 'V+"- b(. 'j" ....... e<:

T (q,.).. 111 t "- ~ ,/£,'"" P< - ~.I

N C tV) '::. L...l<!A' ~..." I:- rl ~·r ...... ...,c.-f"" .......... e. ~ o-ve le...ve.-\ 'j . C ........ (/1.)""'" f<.,vc-v-c:.e .:vi.- \e...v~A l'

.. r -~ ( £ - vj) (~Ii) (LSI' '" 0( 4- 1 ~ )]/r Z ii 'j ~. "" K I t c.ot K '2

- ,\-' (t -'1) ( .t ~ ; V\ ~ + 'j ,; v-. t>() l '2.. ~ '"0 ~ p( ~....... 0( )

< -. "v (1 ~'1) C. R; '1)/ '2,-= "

=-:L.. (.e--'1)(e~1) '" 2':) Cqs, po<

;: _ ~ ( .e"V - 11-) '2~c.or~

N (~) ... - 'V.t (~_~) _ 11>.;z.

~c.v.~ '1 '"

137

138

I-to .... :~o"""r-M p .... e.~ ... "e. t<-k

'~v.e-' 1 ~- w~~ (k:."j rvopov'l-ioiA)J

I' :: ~~ C~) ,{<.~v'J M".t.,e..,..Ve.-l '1 (~ f'vo ~ o."-l-t'Q,,,,) ]

: L (-1) ~ .. ~ K

w~ Ue'1)' ~(-1) ~~« w.e- v (.L'1) , ~, s..;...., K - s.~

~ t

~~. foV',,~ i"", f\..-~e-. Co""e (S-v'j>fD ....... l:"~ a.-t' ~ ~t..:t). --Lo~ ~~I:': .. V'\. rc"-,,e< ~ H OlP\e,&"'I.l:-'t.

l. I'll (- +t (~ - t))/('Z~t~)

1CJJ TI 4 Y <;""'"" t>< +~ ~

~ f4\~ fa- ;«ec:.o<- t.s.-,('("o(~-"Ui .. 'j«~ ~,os .. ",))) ~+' I~~ e- el v zel;

S<:I?, ""e.;~~ ~

11'q,.{"Cj." +A...Aoc:' ('l.~;"", '\.0( -/"")~ ~ +.c.- j( )

1" ..tv

('2!:t:)

N"t - W/(n-1 s.~ 20<) N,*e:. ! ~

~. --

(,.J "* S1~"", ~i"e..,..,

\'~'~7 ",-+rl;e..c 1"0 Dott-o..,

T1.. 0 ~ ~ It. ~ ~CvCJ~dl!.

Yt1"" ";'0"- l::o~ e~c...

~ -W/c21T1 E\:;~(\.K) &t,.

~ .. ~ L .... ~.

V1. r;1 ('2 Tt E t; .c...:n 0<. )

~. N~ - o· 5" f t +~ K ( ~ - 1) V ~1 t~ t>( / "- T3 r ........

Sv".~ ___ ~. e3* (- ~ t t-~ 1- C( (~ +? t )) /(1. et)

v~ (p t v~~~ o(t~ o<'(i ~: +~ (l-f)));{'2eb)

~. Nt.k _wt'v(~- ~1 + ~)~"'C(~ '1 T c- <A

\yJ TIf' w e Y ( 1 ) ( ~ ~ '1) ~j ~ 0(

~ - w e-v~~ k'+~ .c ( ~ ~ l; -~~ "V'((,eb) e .....

IN..ek_ t.l'~ ",..-e . ---- ~~\i~~~~f )(e~~)+r(l- ~f + 1~ \)/(g~e~ Vu

139

140

c~~ 1. V"~ l~ (;-I ... l1..~~.

$v ..... c;.t..._je.. :. 0

~e... ....r;. l,..J~ r .... e£:'"'~ '" 10 kc.J(_

I

M~;.::::\.'\ ~ foV'c.e.. (N) 'l.l..:'3

Nt,. -1'2..b<;r)('I\'42>~(lh.;b!> - 2..~Hy('2.Go~K)" - U':r9.o \ ;Z./p n ll ...... ~

N"l. : -110..0 /(n)(. '2...(,~'!.)(. Si"" 'Z.'() "" -ILk(.g.o

N4 A -La l(. L\·'-i'b~ .... l(.(~ -?)( "t.!.,.:'!> +- '2..l<'2..t.,.>3f iV\t< :: - c...,zt·o Co "2..<&,.:;:; \I.,,",,~~ II·\ .. H;)~'" __ _

Ton...A ,..J bA...('c. :;. - 12tCo fo:.,J •

R..~'fVj T~",·OV\ (T) 2.(p~~ It -= 1"2,· ~ x '2·ro~> )(. Vi", CI< +"'-" C<

o 0

10 ~ ll''-I8~'\.o'l( ('2..(P?~ (ll'\.Jt.;,~-'Z.(O~~))S1""I>(;: 1l-3.·c., II.'-'I~~ II·ub~ j __ _

lordA ,.. b""c:..:: -I--'2oZ.....,.

'Kor~\:--·IQ""' (e) 'l·~~3 '9, .. -I'i·r><;:"x.ll.ut>~ (u.'-${o~)(.I'(o~_'2'l..,,c,(-'2.o1~)): -O.QCX)o~("

'2 )(. e l(, 0.4-

81.'" -1'2.Q<l / ('2.)(.1ix..1...~3~J( exo.4-xCA:lroz.0 .. _o,oooo'l7

9-\ot'" -lO ><: 1\''-1-);)':> 1. x. 0 .e~ )( o.~;. 0 0 ''-4 )(~):;. -o.oooI1~ Tc;>rM e I::;>_te. ,. - o· 0002 ~(,

L:H.f pld...-c...c:..V\,A~"""~ (v-) 1.'!.R !."3

VI :: I~.~"'", tt.-..t~~"l..(o.€X:'2.':."x.."l.~n)A'2.l(f:;(O"""')"'OO~~ o 1& -0000

./4 1: lo.ox: 1I.l.kb~~)(O'O~O(PO~""s,.i"'!...K/(O.y)(~ ... o')" .. p.

Tot-..vI. v- b,,-,>-c. ;: -+o.~<..o (~~)

I

FRSE CONB ANALYSIS - RESULTS

LENGTH OF SHELL WALL ••••••••••••• 8.856001 M LENGTH TO START OF SHELL ••••••••• 11.489 M SHELL WALL THICKNESS .............. 4 M SEMI-CENTftAL ANGLE ............... 48.32 DEGREBS No. OF POINfS FOR ANALYSIS ••••••• 10 STEPS

THE MODULUS OF ELASTICITY IS ••••• 1.5E+07 kN/sq.m POISSONS RATIO ••••••.•••••••••••• 0

SHELL WALL DEAD LOADING •••••••••• 12.85 kN/sq.m THE TOTAL WEI:;H'r OF 'rHE ROOF ••.•• 1200 kN THE SURCHARGE PRESSURE IS •••••••• 0 kN/sq.m 'fHE WATER PRESSURE INCREASES BY •• 10 kN/sq.m

RESUL'l'S FOR COM3INED LOAD CASE

1 OIs'rANeE y ( MERIDIONAL HOOP ! HORIZONrAL 1 1 FROM BASE OF! FORCE FORCE ROTA'rION (DISPLACEMENTI 1 SHELL WALL 1 1 1 1 (M) (kN) (kN) 1 (mm) 1

1----------------------------------------------------------------1 (11.489 -33.5 123.9 +.132E-03 0.1772 1 (10.505 -60.2 190.5 (+.717E-04 0.2491 1

9.521 -97.6 242.6 +.176E-04 0.2875 1 8.537 -148.3 280.3 -.308E-04 0.2978 7.553 -214.1 303.5 (-.741E-04 0.2853 6.569 -302.2 312.2 -.113E-03 0.2553 5.585 -119.3 306.5 -.150E-03 0.2131 4.601 -582.0 286.3 (-.1878-03 0.1640 3.617 -823.4 251.7 -.230E-03 0.1133 2.633 -1226.4 202.5 (-.2968-03 0.0664

Figure '.3 Aeinforcement and upper surface formwork fixing supp;orts for inclined conical slab

141

142

----- - --_. --X. -:: '2. 4'/ 1'2.)( 1.<,~!;,'-c.o!:-~",

'j (o.~ ) ~

fvoV'""\ r ~l e. 0 / ?. 7 ~ cS' ... /:: l,.o.\..kl. ... f '. c:::.Me'=lt>'Ui~/(<.o'U.~J<.'2b'~-I_~.s:"")=+.o"to.f'~ C eM' ~ 1'2 b· ~ : I<. He I ~ -l~. '::' / (\oo'<.-I.~ ~ t1.~.~--;C!>~d=-.ol..g'-f' c..~H': I~.~: I':.H~/"-I~.':)/(<.o.u.~K.I"Ze..f:!>-I~.':(l)",,_.O~3.4-

C9.-tt,. L~·Y: K 11~/ ;:.1'Zt>.t:./(Io.'-I;I....lK.I2.2.> . .!!l-\'!).~"'):a.+ ''22>l

t1 e I .. '01 '-f 1. x. e K . I....\- >< 'Z. ' (p~~ K (+·'0002'?(P)/+JI...-. '-"p( '" +S~.o M\ I:. _'O'-{;l.f K. EK.' Y K (_ 0 .o~)/>1'" "'+""'-'~ ,,+ea.l

+l~·l

I \ (~.-) H e ~ - , oy. ~ }<. e- )(, \...4. >< (+, Q-OO 2 ~ /~V> ~ +~ r>( '" - ~ \. ~ H ~/ ... . '2 ~·n >< ex, u ~ (_o.o~c..) 'Z..,t..:.';~$iVl'loIl(", -1~.1

_ 110.0

(~) O. 'I- := +1~.1 I~ • t I )c 1 ~j ___ '_ ( ......... _ ......... _. o (7

+-;x;t:1 ::. ------------------------

~-e f. Fo"'u!'~ ; ....... (...o",,~ V\e.,.c:l ...... 1d\...U.. ---X ~ Ie II 1"2- I~ Lw"'~~$

r;~W. ~ ( .... ) Z."'~ ;.'l...{l 3, . ':).z I...f·tgl <.,.'",Yb

\l ~~~ .... - 1-1' -+ 1~·1 +13.·1 +1!.·1 +1~·1 +11." MM;O .... ~~I~" H' -\1.0·0 -(1.0.0 -I/o. 0 -tlo.o -110,0 I

Coe.(ft{ CHM ' 1·0 o·S'S 0,20 0·03- -o·oz,.. lS

+"'0"",,\ - ~ t"'-"I~r MM I +1:,'l +I..il,p . .., H'-!·l +- 'Z,.t. - 1... 'Z. i ~

)( -o/~1(-')

---- -- -- -- --- - - - ::t:

CHrt' 0 +0.0'0 +o·o~2. +0 . .01..0 +o.ooq. I - . 'l: r ~

tlc<>,I<.~' -'2.~Z':> -z ~ ~ -t-C) b -'Z~a. U 0

-2~'O II ., -

H ti" r 1:

0 -1;·-:," -( '::,'·2 - (P.o -.0- ~ !: 1"

l.M + l g·l +'2'l.~ - 0·':>- - s· b - ~,I

Cr-J"" ,

0 -1·8>8 -I· ~ -O·l&> _.0'\0

1-'\\+4-."'';'; ~\·'-{"o tS:·I;;"O 21.10 11'10 1':('. \0 ~~ ~:x:

',H-'I' -~o ... '2 :r ~ 0 -<;;'0 . -Il

- - --- --- --- - - - - ~ ~

CNH I ~ iA ).0 o.oL:> -0·l7 -0, (~ -o·o';{"

+- ~

..t Isir"\\>( tY.j t --1'2-1 -IO~ - bs'" -1'2. - G::. \

r "2 "2

Nt-! ' -1'2..1 +l~ +~ -+~ \J U

-w ~ 4 ... ;;

N !....,e.!.I -\'21.1.0 -~<;;o -1'3.{p -S'Co';) -uLkO I 2 2

~N - 12,;,s'!> -1090> -l<:::3 -S;(ol -L{-S~

CT"",' +-l'·~' -+ o· k> --:; -t..-{! -"2 ~~ ~ . )C

)(

TM' -+ '2.-.0. '-t -(I -3.0 t' +- c;. L ....., -\ <=Y'l" ~ Vi - - -- -- -- - -- --- +: -~ CIt; I -+ (p .~ + ~·4 + .Q·3 0 - o.y- _0.4-~ ~-

Tl-\' - b 13 -2.'Z.l -tl -t'2.~ +3.--\ U l-1\

U - u

l~c..LI -t-1..0 'Z. -t '2.'2..~ + 't.-Lo'Z. +'Z.tx+ +~ \...! ~

2,/ ~ +::',0 +11 +lY:O +'Z.4"l. +~ol

143

144

l2.eA.

\0.2>-1'_1-_-+++-_+

10.)1--+- --+++--+

C?J'~'l: .. '

~ ,1(-+ __ +,,+-11--_ f

....1·00

--_._+---

'S11-1<:.-l • . ----

---1--- - ---

'20'2-.,1 .

e: .... , ieAA.'~O V\

Di,..~"~ (~)

1;:-!1c-I .......

~.I-'\.

Di4"''''''''''' eM)

- 1.>0:' ~ IGJ .

'D ; .... c:..c:..1:'

r--\~ ..... tI.A

fo .... c.c:.. bi~ .. ",-......

~y.J

Now I ~t 10 .... ' e.~ kn.Q\.-J"" - d..eA" 6 V\ ~ ~ C-I vc,vL 4...N t" ~ k - ~ c...c.-Ic: M.co Go""", r ~~; vC!-

~rvc...c-f c..-ov-..c..rc:..t-e ....tve +c ,...).

ROOF

CONE

BASE OF CONE AND UPPER SHlI.FT

SHAFr

roUNDATIONS

Figure 8.4 Design of structural elements ef tower

Ref. 8-1

Ref.s 8-1 to 8-5 incl.

rabIes 9/31 , 9/32

J.

Reinforced Concrete J.Faber and F.Mead E.& F.N.Spon Ltd.1965

Reinforced Concrete i) J.Faber and F.Mead

E.&F.N.Spon Ltd 1965

Design of Circular Raft Foundations for Chimneys. S.N.Manohar & S.B.Desai Construction Weekly 1967

and 8-9

145

Program 8P 1 Calculation of forces and movements in forces

unrestrained c'One 'Subjected to external

146

10 E=1.5E+07:U=0:PI=3.1415927£:S=10:W=10 20 CLS:PRINT:PRINT:PRINT 30 PRINT" CALCULAl'ION OF FORCES AND t10ME~l'rS IN FREE 40 PRINT" 50 PRIN£:PRINr" PLE:ASE INPUT 'rHE FOLLOWING Dr,TA :-" 60 PRINf 70 INPU'f" 80 INPUT" 90 INPUT" 100 PRItH

CONE THICKNESS,(M) T·",T TOTAL ROOF WEIGHT ON UPPER CONE PERIMETER W=fl;WR 'THE SURCHARGE PRE:5SURE (KN/M-Z) P=" iF

110 INPUT" THE LENGTH OF CONE FROM IT'S ORIGIN (M) ",L ~ 120 INPUT" rHE Sf ART OF THE: CONE FROM IT'S ORIGIN (M) "'\Ll /' :<. 130 INPUT" fHE: HALF ANGLE OF THE: CONE (deg) ";ALPHA I 140 INPUT" THE NUMBE:R OF POINTS FOR ANALYSIS "IS ~ 150 PRINT ~ 160 INPUT" fHE MOO. ')1" ELASTICITY (DEFAULT=O .15E+08) ., E: ~' ~~g ~~P~~~ fnENP~!~~~:~o;ATIO (DE:FAULT-O) ";U U .0 190 ALPHA = PI*ALPHA/180 200 Q=f*24/{SIN{ALPHA» 210 LC = L-Ll 220 DIM L{S) ,N{S) ,T{S) ,R(S) ,D{S) ,Nl(S) ,N2(S) ,N3(S) ,N4(5) 230 DIM -fIlS) ,T2{S) ,T3{S) ,1'4(5) ,R1(S) ,R2(5) ,R3(5) ,R4{S) ,Dl{S) ,D2(3) ,D3(3) ,04(5)

m ~M~:Q:':~:::::::::::::::::::::: "~ex, , ~~-',~ll 280 N2(I)=-WR/(Pl*Y*SIN(2*ALPHA» 290 N3(I)=-P*L*fAN(ALPHA)*(L/Y-Y/L)/2 ~ 300 N4(I)=-W*L"2*(L/Y-3*Y/L+2*Y"2/L"2)*SIN(ALPHA)/6 ~, 310 N(I)=Nl(l)+N2(I) tN3(I)+N4(1) 320 Tl{I)=Q*Y*SlN(ALPHA)*TAN(ALPHA) P 330 T2(I)=O 340 T3(I)=p*rAN(ALPHA)*Y 350 r4(I)=W*L-2*(Y/L)*(L-Y)/L*SIN(ALPHA) 360 T(l) =1'1 (I) +1'2 (I) tT3 (1) +':'4 (1) 370 Rl(I)=-12.854*L·{L/Y*rAM(ALPHA)*1/COS(ALPHA)-Y/L*~AN(ALPHA)*1/COS(~~HA)*(1 4*(SIN(ALPHA)"2)+2*U*(COS(ALPHA)"2»)/(2*E:*T) 380 R2(I)=-WR/(2*PI*f*E*~*(:::OS(ALPHA)"2» 390 R3(I)=-P*L*(TAN(ALPHA)"2)*(L/Y+3*Y/L)/(2*E*T) 400 R4(I)=-W*L"2*3IN(ALPHA)*TA~(ALPHA)*(L/Y+9*Y/L-16*Y-2/L"21/(6*E:*TI 410 R(I)=Rl(I)+R2(I)+R3(I)+R4(II 420 Dl(I)=Q*L"2*(Y-2/L"2*~A~(ALPHA)*(2*(SIN(ALPHA)"21-U) ~U*~AN(ALPHA»/(2*E*r) 430 D2 (I) =U*ilR/ (2*PI*E*T*COS (ALPHA» 440 D3 (II =p*[," 2*51N (.ALPHA) *TAN (ALPHA) * (? *y" 2/ G" 2 tU* ( l_Y" 2/L" 2) 1/ (2*E* n 450 D4(I)=rl*L"3*(SIN(ALPHA)"2)*«Y"2/L"2)*(L-Y)/L+U*(1-3*Y"2/L"2+2*Y"3/L"3)/6)/ E* r) 460 D(l)-=Dl(I)+D2(IPDJ(I)+D4(I) 470 NEXT I 480 PRINT:PRINT:PRINr" CHE:CK THE: PRINTER IS SWIfCHED ON.":INPUT Z 490 LPRINT" FREE CONE ANALYSIS - RESULfS" 500 LPRINT" ---------------------------- .. 510 LPRINT 520 LPRINT" LENGTH OF SHELL WALL •..•.....•••. ·;L;" M" 530 LPRINT" LENGTH ')1" INITIAL SfE:P ..•...•.... "'Ll, .. M" 540 LPRI N'r" SHELL WALL THICKNE:SS ............... ; r;" M" 550 LPRINT" SE:MI-CENTRAL ANGLE ........•....•. "':LPRINT USING "EEE.E£EB",ALPHA* BO/PI,:LPRINT" deg" 560 LPRINT" NO. OF POINTS F~R ANALYSIS ......... ,5'· STEPS" 570 LPRINl' 590 LPRIN'r" MOD. OF ELASTICI fY ............... "; E," kN 1M" 2" 590 LPRINT" POISSONS RATIO •...•...•.....•.... "; U 600 LPRIlH 610 f"PRINr" SHELL WALL DEAD LOADING .......... ";Q;" kN/tl"2" 620 LPRINT: LPRINT 630 LPRItU" RE:SULf5 FOR COM3INED LOAD :ASE": LPRINl' 640 LPRINT"------------------------------------------------------------------" 650 LPRINl'" I DIsrANCE y ! ME:RIDIONAL! H,)OP! ! HORI Z:)NT,i\L !" 660 LPRINT"!FR')~ BASE: OF! FORCE: FORCE ROTATION !DISPLACEMENT!" 670 LPRINT"! SHE:LL WALL! I" 680 LPRIN'f"! (tl) ! (kNI ! (kNI (mm) I" 690 LPRINf"!----------------------------------------------------------------1" 700 A$="£££.EBE":B$="+£££E£.£":C$="££££.£":D$="+.£E£""-"":E:$="£.££E£" 710 FOR )(=S-l 'ro 0 s'rEP-l 720 LPRIllr"! ";:LPRINT USING A$;L(XI,:LPRINT" ! ',:LPRINT USING B$;N(X);:LP IN'i'" ! n;:LPRINT USING C$,'f(X);:LP!UtH" ! ",:LPRIN'r USING D$;R(X);:LPRINT"

I ""LPRINT USING E$;;)\.\)·lOOO;:LPRINr" 1" 7 30 NEn i{

740 LPRINT .. ------------------------------------------------------------------"

s;f 120

106

106

:;1 94

__ L. 103

89

'OJ

89

89

80

89

t:\..-?.R rlK c;;.' ///1-5' /...15<·,1 ____

OA../.-;.~:O/.,..t;.,.::.,...' A../,t::- .~/

10 - 30

21

o v

Figure 8.5 Enlarged rc details at base of cone

,0

J I

;;;1

147

:".nlr"\ ~I~~I P'Dt {

(}fH.tt~ fO' ~tl.q

\ \

\ \

Figure 8.6 Water tower construction details

148

]'>Or:>m O,~ "1,1(11(, ,'or OUf\,t!'l1a,n

]':to""""d •• ,hHtdt,ron 0v t l,l"'a,l'lfro""o .. t-

Figure 0.7 Plan showing reinforcement in cone walls

Figure 8.8 Detail at base of cone and head of shaft

• 149

<0000

Figure 8.9 Steel conical water tower

150

All ~(.to\ 1'2.~111 thIck.

9yod~ 45E

2m dio. shaft-

Figure 1.10 Completed steel water tower

Design of steel conical water tanks

Figure 8.9 is of a conical steel water tower built at approximately 420 m above sea level in an area subject to high wind speeds and low temperatures. Grade 43E steel was used for the tower and. because of concern over the water freezing in the tank. the wall and roof were insulated externally.

The tank contains 90 m) of water and. because it was small. steel was chosen for both economical and speed of construction reasons. There is no British Standard for the design of this type of structure; however. a very useful guide had been prepared for Australian designers and with the help of this article (Ref. 8-6) and the tables prepared for this handbook the analysis was carried out and the tower built. Although the designed tank wall required relatively little thickness of metal. a minimum thickness of 12 mm plate was used to comply with the recommenda­tions of the Australian article. which took into account experiments on the buckling of cylinders and cones.

One further problem was the possibility af the tower oscillating as a result of wind induced vibrations and also. should the water 'slosh' at the same frequency as the tower, then the results could be serious! Although no specific guidelines had been prepared for this type of structure with regard to this problem. calculations were carried out using the BS for the design of chimneys (Ref. 8-7) and articles from technical journals (Ref. 8-8),

Of considerable value in the design was information provided in an article in the Structural Engineer on wind tunnel tests carried out on a model of a conical water tower built at York (Ref. 8-9). The calculations indicated that

serious oscillations would not occur and that the water, which responded more slowly to wind action, would tend in simple terms. to act as a dampener to any movement. The tower has stood for the past 10 years in often extreme conditions and has shown little signs of structural distress.

References

8-1 Timoshenko S. Woinowsky-Krieger S 1959 Theory of Plates and Shells McGraw-Hill

8-2 Arya D S. 1969-1970 Analysis and design of circular shell structures. Indian Concrete Journal, August 1969-June 1970

8-3 Batty D I 1973 Computerised Design of Conical Shells for Water Towers MSc Bradford University

8-4 Flugge W 1967 Analysis of Shells Springer Verlag 8-5 Taylor A R. Airey E M 1978 The Structural Design,

Construction. Inspection and Repair of Reinforced Concrete Settling Cone Shells used in Coal Preparation Plants Mining Research and Develop­ment Establishment Report No 74

8-6 Ramm D W 1978 Design of Elevated Steel Tanks Australian Institute of Steel Construction (with particular reference to conical tanks) Vol 12 No I.

8-7 BS 4076 Steel Chimneys 8-8 Irish K. Cochrane R G 1971 Wind induced

oscillation of circular chimneys and stacks. Structural Engineer

8-9 Williams G M J, Houghton D S, Moss G M 1967 Design of two unusual structures at York University Structural Engineer, May

151

Design for water .. retaining structures

Table 9.1 Details of (a) bar reinforcement. and (b) fabric reinforcement

Groups of Bars (kg/metre run) Bar S.ze Number 01 bars

mm 1 2 10

, Denoles non·preferred s.zes 6 ' 0.222 0.444 0.666 0.888 1.110 1.332 1.554 1.776 1.998 2.220

0.395 0.790 1.185 1.580 1.975 2.370 2.765 3.160 3.555 3.950 10 0.616 1.232 1.848 2.464 3.080 3.696 4.312 4.928 5.544 6.160 12 0.888 1.776 2.664 3.552 4.440 5.328 6.216 7.104 7.992 8.880

16 1.579 3.158 4.737 6.316 7.895 9.474 11.053 12.632 14.211 15.790 20 2.466 4.932 7.398 9.864 12.330 14.796 17.262 19.728 22.19-4 24.660 25 3.854 7.708 11.562 15.416 19.270 23.124 26.97 30.632 34.686 38.540

32 6.313 12.626 18.939 25.252 31.565 37.878 44.191 SO.504 56.817 63.130 40 9.864 19.728 29.592 39.456 49.320 59.184 69.048 78.912 88.776 98.640

I SO' 15.413 30.826 46.239 61.652 77.065 92.478 107.891 123.304 138.717 154.130 I I I

Weight par m2 in one direction. (Add both Bar Size Spacing of Bars (milhmetres)

directions for total m2 75 100 125 ISO 175 200 225 250 275 300

weight) 6 ' 2959 2.220 1.776 1.480 1.268 1.110 0.986 0888 0.807 0.740

• Denoles non·preferred sIzes 5.261 3.9-46 3.157 2.631 2.255 1.973 1.754 1.578 1.435 1.315 10 8.220 6.165 4.932 4.110 3.523 3083 2.740 2.466 2.242 2.055 12 11.638 8.878 7.103 5.919 5073 4.439 3.9-46 3.551 3.228 2.959

16 21.044 15.783 12.627 10.522 9.019 7.892 7.015 6.313 5.739 5.261

20 32.882 24.661 19.729 16.441 14.092 12.331 10.961 9.865 8.968 8.220 25 51378 38.534 30.827 25689 22.019 19.267 17.126 15.413 14.012 12.845

32 84.178 63133 SO.S07 42.089 36.076 31.567 28.059 25.253 22.958 21.044

40 131.528 98.646 78.917 65.764 56.369 49.323 43.843 39.458 35.871 32.882

SO' 205.512 154.134 123.307 102.756 88.077 77.067 68.504 61.654 56.049 51.378

(a)

152

Table 9.1 (8) (continued)

Sectional A.reap of Groups of liars (mm2)

• Denotes non·prelllrred sizes

Sectional Areas per Metre Width for Various Spacings (mm2/m)

• Denotes non-preferred Siles

6 •

10

12

Number of bats

2 10

28.3 56.5 84.8 113.1 141.4 169.6 197.9 226.2 254.5 282.7

SO.3 78.5

113.1

100.5 lSO.8 201.1

157.1 235.6 314.2 226.2 339.3 452.4

251.3 301.6 351.\1 402.1 452.4 S02.7 392.7 471.2 549.8 628.3 706.9 785.4 565.5 678.6 791.7 904.8 1017.9 1131.0

16 201.1 402.1 603.2 804.2 1005.3 1206.4 1407.4 1608.5 1809.6 2010.6

20 314.2 628.3 942.5 1256.6 1570.8 1885.0 2199.1 2513.3 2827.4 3141.6 25 490.9 981.7 1472.6 1963.5 2454.4 2945.2 34361 39270 4417.9 4908.7

32 604.2 1608.5 2412.7 3217.0 4021.2 4825.5 5629.7 6434.0 7238.2 8042.5

40 1256.6 2513.3 3769.9 S026.5 6283.2 7539.8 8796.5 10053.1 11309.7 12566.4

SO' 1963.5 3927.0 58905 7854.0 9817.5 11781.0 13744.5 15708.017671.519635.0

Bar Size

6 •

10

12

16

20 25

~-::::T

SpaCing of Bars (millimetres)

75 100 125 ISO 175 200 225 2SO 275 300

377.0 282.7 226.2 188.5 161.6 141.4 125.7 113.1 102.8 94.2

670.2 S02.7 402.1 335.1 287.2 251.3 223.4 201.1 112.8 167.6

10472 785.4 6283 523.6 4488 392.7 349.1 314.2 215.6 261.1

lS08.0 1131.0 904.8 754.0 646.3 565.5 502.7 452.4 411.3 377.0

2680.8 2010.6 1608.5 1340.4 1148.9 1005.3 893.6 604.2 731.1 670.2

41888 31416 25133 2094.4 1795.2 1570.8 1396.3 1256.6 1142.4 1047.2 6545.0 49087 3927.0 3272.5 2805.0 2454.4 2181.7 1963.5 1715.0 1636.2

32 107233 8042.5 6434.0 5361.6 4595.7 4021.2 3574.4 3217.0 2924.5 2680.8

40 16755.1 125664 10053.1 8377.6 7180.8 6283.2 5585.0 S026.5 4569.6 4188.8

SO' 26179.9 196349 15708.0 13090.0 11220.0 98175 8726.6 7854.0 7140.0 1545.0

Mesh Sizes Cross· sectional Area Per

Nominal Weight perm'

Sheets Sheets

Type

SOUARE MESH FABRIC

STRUCTURAL MESH FABRIC

LONG MESH FABRIC

WRAPPING FABRIC

BS Nominal Pitch I!eference of Wires

393 252

A 193

142. 98

·s 1131

8 785 8 503 B 385 8 283 8 196

785 S03

C 385 C 283 C 636

98 49

Main

200

200 200

200

200

100 100 100 100 100 100

100 100 100 100 100

200 100

wire Sizes

Cross Main

200 200 200 200

200

200 200 200 200 200

200

400 400 400 400 400

200 100

mm

10 8

12 10

8

10

2.5

Metre Width

Cross Main Cross

mm

10 8

25

mm'

393 252 193 142

98

1131 785 503 385 283 196

785 S03 385 283 636

mm'

393 252 193 142

98

252 252 252 193 193 193

70.8 49.0 49.0 49.0 70.8

98.0 980 49.1 49.1

, A142 also available 3.&m x 1.Om shem. SO sheets per burnal. - merchant slu stocked.

(b)

kg

6.16 3.95 3.02 2.22 154

10.90 814 5.93 4.53 3.73 3.05

6.72 4.34 3.41 2.61 555

154 77

per tonne Sheet per Sq Metres (approx) Wetght bUnale pertonne

15 22 29 40 57

B 11 15 20 24

29

13 20 26

34 16

57

113

leg

70.96 18 45.SO 28 34.79 34 2557 17.74

125.57 93.57 68 31 5219 42.97 35.14

77 41 SO.OO 39.28 30.07

6394

46

60

10 14 18 24 30 36

16

26 30

32 20

17.74 60 8.87 SO

162.34 253.16 331.13 4SO.45 649.35

91.74 122.85 168.63 220.75 268.10 327.87

148.81 230.41 293.26 383.14 180.18

649.35 1298.70

153

Table 9.2 Ultimate anchorage bond and lap lengths as multiples of bar size (BS 8110)

154

25 TenSIon anchor. and lap lengttl

1.4 x tenstOn tap

2.0 x tenSIon tap

39 55 78

32

72 101

143

58

51 71

101 41

40 57 81

32

31

44

62 25 CompresSIOn anchorage IengtI\

Comp<esslon lap IengtI\ 39 72 51 40 31 -~.------"----------

30 Tension anchor. and lap IengtI\ 1.4 x tenSIon lap

36 50

2.0 x tensIOn lap 71

Comp<esslon anchorage IengtI\ 29

. ____ ~p<~lOnlap~ ___ 3_6

35 Tension anchorage and tap length

1.4 x tenSion lap

2.0 x tenSion lap

33 46

66 Comp<esslon anchorage length 27

______ ~.."m~a_ .. slo"_Ia£.I!~th ___ ._ ._~.~.

40 Tension anchorage and lap length

1.4 x lenSlon tap 2.0 x tenSion lap

Comp<esslon anchorage length

Compression lap length

31

43

62 25 31

66 92

131

53 66

61

85 121

49 61

57 80

113

46 57

46 37 64 52 92 74

37 29

46 __ ..:.3_7_

43 34

60 48

85 68 34 27 43 34

40

56 80 32

40

32 45 64 26 32

29 40

57 23

29

27 37 53 21

27

25 35 49

20 25

Table 9.3 Reinforcement scheduling details for (a) preferred shapes, and (b) other shapes

~ of bending dlmeneIona - pnlferN<f fINIpee -lIS 44Gll.

-"""'_"'lIonIIInGoI T_lon;IIIoI ..... (L)

~-_&long_line .....

~ ' __ ... ----l

A

rrnh/~ A+h

~ --A--

~~ A+2h

~ Ire V

A f'/

~ A+n

-.Ilo<Nfl<alldimons.oo allhebobl$O'11lCaI use

n~ "'-codt37

'--- A I

--------

:iij A+2n

INhere the ov.,aa dmen$Ol 01_ bob .. entreat do not ne _~n u.M ''Ir, ShapI

E[~r A+(Bj-',.r-d

ThtsforrnWal$appt"oIlmate 'Nhererl, gteaterrhan It'4

i---- (B) t'ntIWT\Umvaru.!ntable3 "', as U65 1989. UN shape axIo 51

---.J

~ A+S.(Cj-r-2d

r~_ (e)

r~;J a _._.---'

t. , II"ldcates!he mlf'WnUm value In table 3 as 4466 1989

&. The~IO~ar.thefreedarnensiOO$

-... ... bo ....... -Straight

'---A

C-.-::::> A

L-A

L--J A

AL-

A~ a

" To ~ sepal .... equ&tJons Ie< each ..... ~ and bendong 'aOOs. ~mphfled 1o\aI ""'¢>""""-""'-Ie<~CXldO$61. n. 78. 79ard62 n",S<llormulae." ~approxtma.

_of_ofllonlllnGoI T_lonQIIIofbor(L) ~Iobo

~-_&long_line ..... In_

""" rm KangIos_""''''''''DIUI .,...," orIoN.

-- Ie) A.S.(e)

~-I , 0 Seenoce. ~ ~ 8 -' I

.-- A - ,

O?2d

~ " angIM_.".hoN"",",

~ are "S"'OI IctsI (E)

~' A+28.C.(Ej 0

A --1 Seenof ••

~ r " (" 0 r . ___ --1

-~--O?2d

Iijr A+(Sj-",r-d

~ R non standard thIS formula IS ;JClPfOImato

A~ A tt R IS mnmum. UN SI'\ape -'37 nRlS9'8_iNn 200mm. $.Ie nexe 2 to ~use

'(B) 10 as U65 198V

~ 2(A+8j.12d

~ _AnorBarelObeless

'-Bh than 12d Ot lSOmm, whrche_lStno_I8r.Ie<

-0 0: gtade~n$Q"noI

'l~no 20mm not tess Ihanl~I"'''' .. 01~ andover Neltt'lefAnt')lBare lo""lesslhanlOdle<~ 250 WI1h a rTllmmum value of A and 8 of l00mm

See ""'. 3

~~.i If angle WItI'! honzorrtai's.5 or less. r ;. B A.(C) ~8I

i ,-;--/ A_1 -- te) --t '.p

See note "

~--2A.3B.1Bd

1f8ISgfeaferthan400 .. 2tJ. see flOUt 210 dause 10 ... 1

) ? as 4466 1989 c::Ja

(C $4)ef'()le3 T

4. The Iengttl formula IS approlrrnate and when ben:.1Ing ang}es elceed 45 the \enqth should be calculaled ffiOf8 ac:o..trater; artow1OQ to( the difference ~ the specrlJed overall dlfl'l(triSJOnS and \tie true length measured alono lhe central ilXlS ol1t\e bat Of '!We When the bendtno angfes approach 90" >1 15 pcelerab'e to !.4)eC'rl'y 5I"lape c:oda 99 wrth I luftydlmen$)Q(\illskeld"l

Genorol No .. : me unsooaf>ed 0< und>meosoonod portron 01 'he bat sI'<lvId not be 01 • cotrcaI nature as aU tokH"al"CeS of wrtlf"lg am bendlrq etc are taken up on th'" portlOn

(a)

155

Table 9.3 (continued)

o [J

c=::::l- ,-C ___ I

~ .. Ai IE)

~UJtt---, ~ , 0

-C-

156

A.B.(C)-,-2d

a~ C

A

eflJ~O C

A

aD C

(b)

O~ -8-

r- -~-~~ C _ Bl

!

ThoMboIowillbosuppi>O<l O1toq,t_ .... rIldIus,. grNWIhanIhalIl""""'" _~oIBS4466'9!19

2A.3B.1Od

_Arae_bo .... Ihan l2dcr '5Omm. _ ...... ~.tot

grIIIooteoonSlZ""" oxooedlng2Ommra_ Ihan • ...,tot ..... '" 25mm ord fNfII _ A not

SlfOlObo_lhonlOdtot grIIIo 250 """ • monmum vaau.oIAord SoI'OOmm SNr'OI03

A+B+O.57C+(D)-'1tr -2.57d

K C .. grM' .. 1IIan 4OO.2d. _noI02lOdouM'O 854466'9119

_SOSnolgr __ INIn M i' (A-d) (l~l2m) ~, .. ,. A,slhe.ttemal~oI

/leII1,w"nm) ........ """"01_''''"''"1 C ...... <NfII .. hoogItol""". I'" mm) Where B tS ",Nter IhonM ..... lormuIadoosnol II>IJIyTher.shalbo8l1e .... _ilJloumo"' ..... _ NoIo· ......... A .. smaI _lOd."""",-os fl!lbncalold ... dooedtorm andpulle<llO",-an_

.,. ..... Ic __ A_s>.eId>_ be cWawn CMA at. !CNOJle -.mnsAlOE E.-y -.anshalboopoollod ,,,,,-.anlNlltSlO

-~"""""­_1IOnI_bootOcaled

"'~.-""'" ftil:lncalr .. _I0_ """"'-......_-­tot .... '0'."""" "."'­Chaltlgrtenl"llhts\.lilibllOfJn

_'BS'-'9119 .. """""",bula_ -.an .. IO_tot"" ~cIovIabonl."'" ",-_bodr-."'" , II""""INI ",-."..99 & ..... " .. ""'*'""'"_bo orocatod"'~Tho 1OIe<ancoogrven .. _.

BS'-'911901ool1JP1Y

R ~

A

A ,5 the eI'lefNll ..-01_ • oS .... """"01

"... ClSthaovera'

noogIItolheb

Table 9.4 'As' for design crack width 0.2 mm, bar diameter Tio

h ! ~ i ~I ! sp.mgof ...... , .. , iSS H

II II II j

V .10<'

1/ V ~y ~ ..

........ 100

Ii 1/ 1/ 1/ ~

,,11 II I"'~

II .,1"-1--0"

~ I/~ v~ I-' "" •. II~ 7-It(

II 1/ "'"

,I. 2.".2.~ < S2l

Y _. - -

1/ ",I,-- ISO

] .us (I) ~ ITS - l'l J 1-- -0 II' too .., M9 " '1-CD us .a 314 . I- ..

UO 18S I) 1-1-' . -I-

~7f ,"I .. .00

COVER I I I I COVER r Ii 1$ ~r 4r ss 60r 7S as ~s

45 10 \10 )0 .. so .., 70 80 ~o 45 Service Bending Moment (kNm)

"As" For Design Crack Width of O.2mm Bar Diameter Tl0

h !I 0 0 0 :1 ~: Il*III9 of '" ~ moil, .. , ... '" ..

7~

II j.. .. J-1 1/

II I'"

II II II lA I<'" 100

f.<'1'"

(,211 J .-#

..§.

I II 1/

"'~ CI'l r ,- ",,~.II~% ItS < L (If <.1. t.C..'Z..3

5l~

II I I Ci)

III ~ 150 Q)

Ci) 44' , - 17' 0 ('

"3 /. .., zoo Q) .... M' < t~

1>14 I HO i I us

~. us u.1 I

I I I 100

I I II I I II II I II I I I

COVER COVER • IS H ~s ·H· $S "" 7S 85 '!>S

10 7.0 }O 40 SO DO 70 80 ')0

60 60 Service Bending Moment (kNm)

157

Table 9.5 'As' for design crack width 0.2 mm, bar diameter T12

h 0

~I ~ ~ §I ~ ~I ~ J :;~: 0 ,..

1111 \1 L ./ f I I V "

,1/ if II II II V V

II II V

I / II V 100

/i I I / ~ I--t---

// /

I II

1/1/ II i ! ..... ~r "I'm ~ .... v"....

.E I JJ j,r~M'"I\)'/. u.~

'" ~I 1.< •. 1.:-< 164

] I I' II 1 ~/~ I~e

./' "'47

I J II I fJ~ CI)

'AA \-I~ - &101> 0

I I } ~

too G> SO;

I II .< as 46~ II 1"'0 411

11 I 'l.7S ?17

tt!" I I ~oo

10 I ~" I I ! I <,)'0 110 I I'!>t' I I no I I

COVER so "10 ,~¢ 1\)0 COVER 20 ... 0 1>0 to 100 ''1(: .40 ",0 .60

45 Service 8ending Moment (kNm) 45

"As" For Design Crack Width of O.2mm Bar Diameter 112

h ~ ~I ~ it !I ~I ~I ~I 0 spKIl9 of .JI INin 'KIf

1 J I J I I I 75 1I~'1.

II /i 1:/

II v

lit { I V II 100

I I I ~

90~

I III J / I II Ii < r~+.t.r~ .,4

~I..,.,."..

<I: I)<f' Ito", ~_. 'l. It5 I . I (.1. r~-2~

I 754-

I i II~ ,It I I c:;

150 oS CI)

11>41

I I . ..,# I - I~ 0

~ 610b ,V ! I G> -lOY "- SO, c--- --< j I I i H.

¥.)~ I 2;0

411 .I' i I 27')

317

I I I '~oo

I

! I I I I I I i 1M I I I I~ I I~ COVER COVER 10 lI<l 50 70 ';l() 110 ISO

to 40 '-0 sa 100 ,'lo 14CI 1,"0 180

60 Service 8ending Moment (kNm) 60

158

Table 9.6 'As' for design crack width 0.2 mm, bar diameter T16

h 0 ~ ~ ~ ~ ~ ~ §I ~l ~I ! 8 spaciIg If ~ iii IIIIin nil!

J / / / I V V

V / / I / 1S I tOil

II I I I II V V / /

/ I /

II i/ V V / I V I 00

/ / / V / 1"06

I / / J / ~

I ,/ / V ! Vv / V / / I

< il j / I t ~

/ / / \\40 /

:1 L / j / / / / V I V V / L /

ISO ~ j

114' t-- "-J 1/ II / II / -0 / I 7 5 / j J

~ 100<5

I / / / V II / / I tOo .a 6!l4

~ 1/ J j I/A i mill IH "/0 CI.2

1"2.3 t t 5

1104 I F<'

I t so 7)1 - 1/ II( I I t 1 S "'0 U 'i/"

, I :\ 0 0

COVER ED 40 ~ 90 I I I I I I I I

COVER Ito 140 \"0 1&0 no HO t!.O teo no ~o 1.03&0 4to loa too 100 400

45 Service Bencting Moment (kNm) 45

"As" For Design Crack Width of O.2mm Bar Diameter T16

h ~ ~ ~ o 0 ~I § ~ § ~1 ~l 0 ~ I1*fI9 of ::; S ~ mm ,m'

/ / / 1/ v II I J / 75

tOil

l II II J Vl; Vv II V /

IV II V V 100

E / I II / ~ 1'-0$ I I

< II II / II v VI I

\ I I V / II I Z S

Ci) / / I Q.) 1340

!I I V / II V I Ci)

,/ 1 L / I I so

'0 114'3

/f J /1/ II ,/ II Ill! / I fI:I I I 7 5 Q.) IOOS

~ I I ! / / v V / I I ZOO a~4

I

V ./' 1"<'\,'" ~~ 8/0 d. t.~.2..~ Z Z S 304 I I .r ~_l

--i--' I I z 5 ° BI

II/A'I _ I I I I t 7 S (,70

17 r I I I i i ~ 00

COVER I I I I I I I I I I I! I i I I I I

20 40 bO so Ito I4C I"" 190 I 210 240 ZbO lao I 310 >4<> 340 180 410 COVER 100 1.00 300 400

60 Service Bending Moment (kNm) 60

159

Table 9.7 'As' for design crack width 0.2 mm, bar diameter 120

h g ~I ~ ~i ~I ~ I ~I §I ~I I II II 17/ / V '/ / /' / 1i

ll4t

II II

/1/ / I 1;1 IV V V // /

II

Ilil V V V 'IV V

/V V Vv V/,V 100

/ :II I / V V Vi ~513

II V illl II V VV .€ II /1\1 1/1/ lI i/ V V It5

(I) I / II II / / V < 1094

I II II 1/ II IIV II 1/1/ V / I I 1/11 V 1$0

"ai / I 1795 II II V V IV / ~ I // 175 CI') 1$71

Ijl/ /v VI 1/ 1/ - tOO 0 1~9,"

I / II / I 1111 /1 / I us ~ IU,"

I I I II tSO .a 114t / / 7 I ~7S 1047

I~ ~ Vii ~~ IIV VlI I

,00

r/r~ VlI V I I I I

I I COVER so .so ISO lSO 4S0 HO IoSO COVER

10C> 100 ~oo 4(10 500 ,"00

45 Service Sending Moment (kNm) 45

"As" For Design Crack Width of O.2mm Bar Diameter T20

h ! ~ : ~ ~ ~ !I ~ ~ ~ ~I ~ . ~ ~ l S!*fl9 01 main rtf'l'

1/ I 1/ II 1/ 1 / II jI v V V 75

~\4t -

n ~ ~ I / 1/

II 1/ II 1/ / V Vt;' V l/

II II II / II II V V \00

/ V V / I / / / .€ is\!>

II II li/ I I

I CII

II III II / c:x: 1// 1?5

/ / Vii / to'4 II V i/l/ V "'ai I 'IV \50 C2.) I / / I I / en 11&5

I II II II II V VI/ II / /1 175 -0 i511

II Ii / 1111 1/,/ I I / I I I 200 ecJ I~ C2.) If I I II us ... 17.5 .. ct T J 1 / / I 1/ / I I I I ! 150 IIH , r ill I I I I '2.7. 1041

II/~ I;~ II;~ VI' VI I I I

"300 vv V I I I 1 I ! I I

COVER COVER 50 ISO 1$C> lSO 450 SSO '"SO 100 lOO aoo 400 500 (00<:>

60 Service Sending Moment (kNm) 60

160

Table 9.8 'As' for design crack width 0.2 mm, bar diameter T25

II V / Vllr~ VV / too / I/VVV/VI)~/~V+V~~~++++~~~+++-~tt~S~

ill J 1I1I1 / I. tso

SSO

I I I I I so ISO tso

COVER no <.SO .so 1$0

100 100 :\00 ~oo 500 <'00 100 45 Service Bending Moment (kNm)

"As" For Design Crack Width of O.2mm Bar Diameter T25

h

/ V V V 1/ V v

/ II II II II II V V V 15

4')O'!o+-t-+--+-1H;-+I/+H+11 +1 j-ff+-, j-V--I/t---V-tl

/-++I/++-V-¥-+-V l7'11/-+f7"74-I/+v'*l/+v-i:>"t-t-t---;

1 V V ~ V 1/ I V / / I~ I / II II V I / VvV /

~ / II / / / / (/) 3'H1.\--L-l--1--'+-f+-11 hi 1i!-+I1/+17+-+I-+

V-JL-f-ofv +-If+V-lf+V-;f--!V-rf-!-;v.f-t--t-+-+-t-t-t-t----t

~ I Ilil Vv V V/ V / / cu 3m +--+--J-.j.+-+-I-/I-4--¥-I/+f/+I+-jlI/'--l+-1/!-+-I/WI-+V--fA.V-bL!Vi-f-t-+-+-+-++~f-+-t----; ~ Z&OS II IIIII1I1 IV VI/I/VI/~~VI/V : t4~4 I I / / J / IV 17l/l/v 111/ /V ~ 1/"11 VV/I11 !/ ..... ~181 I /

I'~'--'-' of I/I! IVVil I / . I It / /

:::: I I I I / Y 17 ] 7

IN 'jIV r/Vi/VYI/V

IZ5

150

175'

,00

U.s' Z50

• ~iS

300

I COVER '1 so

Service Bending Moment (kNm) 60

161

162

Table 9.9 'As' for design crack width mm, bar diameter T32

COVER

45

~~~~~! I I I I I lOO

I Iii I I I I I I I I I I I i I I 1 I I COVER ~ ~ ~ ~ _ ~ ~ ~ D ~ ~ ~

Service Bending Moment (kNm) 45

"As" For Design Crack Width of O.2mm Bar Diameter T32

COVER

60

I I I I I I I i I I I I I I I I I I I I I 100 ~oo ;;00 400 ~ ~ 100 SOD ~oo 1000 HOC IZOO

Service Bending Moment (kNm)

COVER

60

Table 9.10 'x' and 'z' factors for sections reinforced in tension only - serviceability limit state

******** ******1<** * * * * -.. *

As*100 I x I I z' ---- factor factor

b.d

0.100 0.159 0.947 0.12~ 0.176 0.941 0.150 0.191 0.936 0.175 0.204 0.932 0.200 0.217 0.928 0.225 0.228 0.924 0.250 0.239 0.920 0.275 0.249 0.917

0.300 0.258 0.914 0.325 0.267 0.911 0.350 0.276 0.908 0.375 0.284 0.905 0.400 0.292 0.903 0.425 0.299 0.900 0.450 0.306 0.898 0.475 0.313 0.896 0.500 0.319 0.894 0.525 0.326 0.891

0.550 0.332 0.889 0.575 0.338 0.887 0.600 0.344 0.885 0.625 0.349 0.884 0.650 0.355 0.882 0.675 0.360 0.880 0.700 0.365 0.878 0.725 0.370 0.877 0.750 0.375 0.875 0.775 0.380 0.873

0.800 0.384 0.872 0.925 0.389 0.870 0.350 0.393 0.869 0.875 0.398 0.867 0.900 0.402 0.866 0.925 0.406 0.865 0.950 0.410 0.863 0.975 0.414 0.862 1. 000 0.418 0.861

***1<**** ********* *******

Table 9.11 'zld' lever arm factors for ultimate bending moment

*************** ***** ••• *.***

Mu -------- z/d FACrOR b.d.d.fell

0.000 0.950 0.005 0.950 0.010 0.950 0.015 0.950 0.020 0.950 0.025 0.950 0.030 0.950 0.035 0.950

0.040 0.950 0.045 0.947 0.050 0.941 0.055 0.935 0.060 0.928 0.065 0.922 0.070 0.915 0.075 0.908

0.080 0.901 0.085 0.894 0.090 0.887 0.095 0.880 0.100 0.873 0.105 0.865 0.110 0.857 0.115 0.850

, .120 0.842 0.125 0.833 0.130 0.825 0.135 0.816 0.140 0.807 0.145 0.798 0.150 0.789 0.155 0.779 0.156 0.777

*************** **************

Lever Arm Factors obtained by using CLAUSE 3.4.4.4 BS 8110

l***************x**************

163

Tabie 9.12 Concrete grade C25: permitted values of shear stress 've' for a range of As x 1001 (bv x d) and effective depth, d (Values derive from an equation given in BS 8110. Table 3.9)

164

Effective d~pth 'd' mm 2·~0~0~~2~2~~0--~2~7~5~~37070--~3~5~0----4-0-0-100 125 150 175

0.15 0.47 0.45 0.43 0.41 0.40 0.39 0.38 0.37 0.36 0.35 0.34 "-

0.20 0.52 0.49 0.47 0.45 0.~4 0.43 0.42 0.41 0.40 0.38 0.37

0.25 0.56 0.53 0.51 0.49 0.4~ 0.46 0.45 0.44 0.43 0.41 0.40

0.30 0.60 ~.57 0.54 0.52 0.50 0.49 0.48 0.46 0.45 0.44 0.42

0.35 0.63 0.68 0.57 0.55 0.53 0.51 0.50 0.49 0.48 0.46 0.45

0.40 0.66 0.62 0.60 0.57 0.55 0.54 0.?2 0.51 0.50 0.48 0.47

0.45 0.68 0.65 0.62 0.60 0.58 0.56 0.54 0.53 0.52 0.50 0.48

0.50 0.71 0.67 0.64 0.62 0.60 0.58 0.56 0.55 0.54 0.52 0.50

0.55 0.73 0.69 0.66 0.64 0.62 0.60 0.58 0.57 0.56 0.54 0.52

0.60 0.75 0.71 0.68 0.66 0.63 0.62 0.60 0.59 0.57 0.55 0.53

0.65 0.77 0.73 0.70 0.67 0.65 0.63 0.62 0.60 0.59 0.57 0.55

0.70 0.79 0.75 0.72 0.69 0.67 0.65 0.63 0.62 0.60 0.58 0.56

0.75 0.81 0.77 0.73 G.71 0.68 0.66 0.65 0.63 0.62 0.59 0.57

0.80 0.83 0.78 0.75 0.72 0.70 0.68 0.66 0.64 0.63 0.61 0.59

0.85 0.85 0.80 0.77 0.74 0.71 0.69 0.67 0.66 0.64 0.62 0.60

0.90 0.86 0.82 0.78 0.75 0.73 0.70 0.69 0.67 0.66 0.63 0.61

0.95 0.88 0.83 0.79 0.76 0.~4 0.72 0.70 0.6~ 0.67 0.64 0.62

1.00 0.89 0.85 0.81 0.78 0.7) 0.73 0.71 0.69 0.68 0.65 0.63

1.05 0.91 8.~6 0.82 0.79 0.76 0.74 0.72 0.71 0.69 0.66 0.64

J .10 0.92 0.87 0.83 0.80 0.78 0.75 0.73 0.72 0.70 0.67 0.65

1.15 0.94 0.89 0.85 0.81 0.79 0.76 0.74 0.73 0.71 0.68 0.66

1.20 0.95 0.90 0.86 0.83 0.80 0.78 0.76 0.74 0.72 0.69 0.67

1.25 0.96 0.91 0.87 0.84 0.81 0.79 0.77 0.75 0.73 0.70 0.68

1.30 0.98 0.92 0.88 0.85 0.82 0.80 0.78 0.76 0.74 0.71 0.69

1.35 0.99 0.93 0.89 0.86 0.83 0.81 0.79 0.77 0.75 0.72 0.70

1.40 1.00 0.95 0.90 0.87 0.84 0.82 0.80 0.78 0.76 0.73 0.71

1.45 1.01 0.96 0.91 0.88 0.85 0.83 0.80 0.79 0.77 0.74 0.72

1.50 1.02 0.97 0.92 0.89 0.86 0.84 0.81 0.79 0.78 0.75 0.72

Table 9.13 Concrete grade caO: permitted values of shear stress 'vc' for a range of As x 1001 (bv x d) and effective depth. d (Values derive from an equation given in as 8110, Table 3.9)

Effective depth 'd'mm 100 125 150 175 200 225 250 275 300 350 400

0.15 0.50 0.48 0.46 0.44 0.42 0.41 0.40 0.39 0.38 0.37 0.36

0.20 0.56 0.53 0.50 0.48 0.47 0.45 O. <14 0.43 0.42 0.41 0.39

0.25 0.60 0.57 0.54 0.52 0.50 0.49 0.48 0.46 0.45 0.44 0.42

0.30 0.64 0.60 0.57 0.55 0.53 0.52 0.51 0.49 0.48 0.46 0.45

0.35 0.67 0.63 0.60 0.58 0.56 0.S5 0.S3 0.52 0.51 0.19 0.47

0.40 0.70 0.66 0.63 0.61 0.59 0.57 0.56 0.54 0.53 0.51 0.49

0.45 0.73 0.69 0.66 0.63 0.61 0.59 0.58 0.57 0.55 0.53 0.51

~ 0.50 0.75 0.71 0.68 0.66 0.63 0.62 0.60 0.59 0.57 0.55 0.53 I(

>-:9 0.55 0.78 0.74 0.70 0.68 0.65 0.64 0.62 0.60 0.59 0.57 0.55 -~ < 0.60 0.80 0.76 0.72 0.70 0.67 0.65 0.64 0.62 0.61 0.59 0.57

" o 0.65 0.82 0.78 0.74 0.72 0.69 0.67 0.65 0.64 0.63 0.60 0.58

o ""- 0.70 0.84 0.80 0.76 0.73 0.71 0.69 0.67 0.65 0.64 0.62 0.60

0.75 0.86 0.82 0.78 0.75 0.73 0.70 0.69 0.67 0.66 0.63 0.61

0.80 0.88 0.83 0.80 0.77 0.74 0.72 0.70 0.68 0.67 0.64 0.62

0.85 0.90 0.85 0.81 0.7G 0.76 0.73 0.72 0.70 0.68 0.66 0.64

0.90 0.92 0.87 0.83 0.80 0.77 0.75 0." 3 0.71 0.70 0.67 0.65

0.95 0.93 0.88 0.84 0.81 0.79 0.76 0.74 0.73 0.71 0.68 0.66

1.00 0.95 0.90 0.86 0.83 0.80 0.78 0.76 0.74 0.72 0.69 0.67

1. 05 0.97 0.91 0.87 0.84 0.81 0.79 0.77 0.75 0.73 0.71 0.63

1.10 0.98 0.93 0.89 0.85 0.82 0.80 0.78 0.76 0.74 0.72 0.69

1.15 1.00 0.94 0.90 0.87 0.94 0.81 0.79 0.77 0.76 0.73 0.70

1. 20 1. 01 0.95 0.91 0.88 0.85 0.82 0.80 0.78 0.77 0.74 0.71

1. 25 1. 02 0.97 0.92 0.89 0.86 0.84 0.81 0.79 0.78 0.75 0.72

1. 30 1. 04 0.98 0.94 0.90 0.37 0.85 0.82 0.80 0.79 0.76 0.73

1. 35 1.05 0.99 0.95 0.91 0.88 0.86 0.83 0.82 0.80 0.77 0.74

1.40 1.06 1.00 0.96 0.92 0.89 0.87 0.84 0.83 0.81 0.78 0.75

1.45 1. 08 1. 02 0.97 0.93 0.90 0.88 0.85 0.83 0.82 0.79 0.76

1.50 1. 09 1. 03 0.98 0.95 0.91 0.89 0.86 0.84 0.83 0.79 0.77

165

Table 9.14 Concrete grade C35: per-mil1ed values of shear stress 've' for a of As x ;001 Table 3.9) (bv x d) and effective depth. d derive from an equation given in as 81

166

mm

100 125 150 175 200 225 250 275 300 350 400 -- ------------ ----

0.15 0.53 0.50 0.48 0.46 0.45 0.43 0.42 0.41 0.40 0.39 0.38

0.20 0.58 0.55 0.53 0.51 0.49 0.48 0.47 0.45 0.44 0.43 0.41

0.25 0.63 0.60 0.57 0.55 0 S3 0.51 0.50 0.~9 C 48 0.46 0.45

0.30 O.Li 0.63 0.60 0.58 0.~6 0.55 0.53 0.52 0.51 0.49 0.47

0.35 0.70 0.67 0.64 0.61 0.59 0.58 0,56 0.55 0.54 0.52 0.50

0.40 0.74 0.70 0.67 0.64 0.62 0.60 0.59 0.57 0.56 0.54 0.52

0.45 0.77 0.72 0.69 0.67 0.64 0.63 0.61 0.59 0.58 0.56 0.54

0.50 0.79 0.75 0.72 0.69 0.67 0.65 0.63 0.62 0.60 0.58 0.56

0.55 0.E2 0.77 0.74 0.71 0.69 0.67 0.65 0.64 0.62 0.60 0.58

0.60 0.84 0.80 0.76 0.73 0.71 0.69 0.67 0.65 0.64 0.62 0.60

0.65 0.87 0.82 0.78 0.75 0.73 0.71 0.69 0.67 0.66 0.63 0.61

0.70 0.89 0.84 0.80 0.77 0.75 0.72 0.71 0.69 0.67 0.65 0.63

0.75 0.91 0,86 0.82 0.79 0.76 0.74 0.72 0.71 0.69 0.66 0.64

0.80 0.93 0.88 0.84 0.81 0.78 0.76 0.74 0.72 0.71 0.68 0.66

0.85 0.95 0.90 0.86 0.82 0.80 0.77 0.75 0.74 0.72 0.69 0.67

0.90 0.97 0.91 0.87 0.84 0.81 0.79 0.77 0.75 0.73 0.71 0.68

0.95 0.98 0.93 0.89 0.85 0.83 0.80 0.78 0.76 0.75 0.72 C.70

1.00 1.00 0.95 0.90 0.37 0.84 0.82 0.80 0.78 0.76 0.73 0.71

1 • 05 1 .02 0 .96 0 . 92 0 . 88 0 . 85 0 . 83 0 . 81 0 . 79 ,) .77 0 .74 G . 72

1.10 1.03 0.98 0.93 0.90 0.87 0.84 0.82 0.80 0.78 0.75 0.73

1.15 1.05 0.99 0.95 0.91 0.88 0.86 0.83 0.81 0.80 0.77 0.74

1.20 1.06 1.00 0.96 0.92 0.89 0.87 0.84 0.83 0.81 0.78 0.75

1.25 1.08 1.02 0.97 0.94 0.91 0.88 0.86 0.84 0.82 0.79 0.76

1.30 1.09 1.03 0.99 0.95 0.92 0.89 0.87 0.85 0.83 0.80 0.77

1.35 1.11 1.05 1.00 0.96 0.93 0.90 0.88 0.86 0.84 0.81 0.78

1.40 1.12 1.06 1.01 0.97 0.94 0.91 0.89 0.87 0.85 0.82 0.79

1.45 1.13 1.07 1.02 0.98 0.95 0.92 0.90 0.88 0.86 0.83 0.80

1.50 1.14 1.08 1.03 1.00 0.96 0.93 0.91 0.89 0.87 0.84 0.81

Table 9.15 Shear reinforcement spacing (mm) for beams, where 'v' is greater than (vc + 0.4)

**k***** **************************~********************.*********.****

bv(v-vc DIAMErER OF LINK (mm)

50 100 150 200 250

300 350 400 450 500

550 600 650 700 750

800 850 900 950

1000

1050 1100 1150 1200 1250

6

453 226 151 113

91

75 65 57 50 45

41 38 35 32 30

:'8 27 25 24 23

22 21 20 19 18

a

805 402 268 201 161

134 115 101

89 80

73 67 62 ')7 54

50 47 45 42 40

38 37 35 34 32

10

1257 629 419 314 251

210 180 157 140 1~6

114 105

97 90 84

79 74 70 66 63

60 57 55 52 50

12

1811 905 604 453 362

302 259 226 201 181

165 151 139 129 121

113 107 101

95 91

86 82 79 75 72

16

3219 1610 1073

805 644

537 460 402 358 322

293 268 248 230 215

201 189 179 169 161

153 146 140 134 129

****** ********** **********~*********************~***************** Link diameter and spacing obtained by using equation in TABLE 3.8 BS 8110

FORMULA rABLE 3.8 IS sv = Asv x 0.87 x fyv / (bv(v-vc»

sv Asv fyv bv

= spacing of links along the member = Total cross section of links at neutral axii at a section = characteristic strength of links - 460 N/mm = breadth of section or average width of rib below flange

vc = design concrete shear stress (refer tables 9/12-13 incl. design shear stress at cross section v =

¥¥¥¥*******************************************************************

167

Table 9.16 Minimum percentage of reinforcement to resist early thermal cracking

168

Table 9.17 Deflection - modification factors for tension reinforcement for varying values of Mu/(bdd) and serviceability stresses

******** . *******.********.************* •• **************k******* ***~ Service Stress

100.00 no.oo 120.00 130.00 HO.OO 150.00

160.00 170.00 180.00 190.00 200.00

210.00 220.00 230.00 240.00 250.00

260.00 270.00 280.00 290.00 300.00

0.50

2.00 2.00 2.00 2.00 2.00 2.00

2.00 2.00 2.00 2.00 2.00

2.00 2.00 2.00 1. 96 1. 90

1. 84 1. 78 1.72 1. 66 1. 60

0.75

2.00 2.00 2.00 2.00 2.00 2.00

2.00 2.00 2.00 2.00 1. 95

1. 90 1. 85 1. 80 1. 75 1. 70

1. 65 1. 60 1. 54 1. 49 1. 4 4

2 Hu/(b.d

1. 00 1. 25

2.00 2.00 2.00 1. 97 2.00 1.93 2.00 1. 89 2.00 1. 86 1. 98 1. 82

1. 94 1. 78 1. 90 1. 74 1. 85 1. 70 1. 81 1. 66 1. 76 1. 62

1.72 1. 58 1. 68 1. 55 1. 63 1. 51 1.59 1. 47 1. 55 1.43

1. 50 1. 39 1. 46 1. 35 1. 41 1. 31 1. 37 1. 27 1. 33 1. 24

)

1. 50 1. 75 2.00

1. 86 1.74 1. 63 1. 82 1. 70 1. 60 1. 79 1. 67 1. 58 1. 75 1. 64 1. 55 1. 72 1. 61 1.52 1. 69 1. 58 1. 49

1. 65 1. 55 1. 46 1. 62 1. 52 1.43 1. 58 1. 4 8 1. 40 1. 55 1. 45 1. 37 1. 51 1. 42 1. 35

1. 4 8 1. 39 1. 32 1. 44 1. 36 1. 29 1.41 1. 33 1.26 1. 37 1.30 ' 1. 23 1. 34 1.26 1.20

1. 30 1. 23 1.17 1. 27 1. 20 1.14 1. 2 3 1. 17 1. 12 1. 20 1.14 1. 09 1.16 1.11 1. 06

******** *****************~********~******** ****.***~*****.*.*******

odification factors obtained by using EQUATION 7 TABLE 3.11 BS 8110

*******w********************************************** ***************l

Table 9.18 Deflection types of loads

modification factors for tapered cantilever walls subjected to different

CANrILEVER LOI\DING

THICKNESS 'h' TOP / THICKNESS 'h' BTM

~~~--r-~~~-- -~~~--r-~~~--I-~~~--***** ****** ****** ****** ******

0.75 0.69 0.63 0.58 0.52

-~--~-\D-\-\jL-!O-:"-~-\-\-! --~::~~:I:;:::: :;-:;:fl::-~~:ll:~-::: **Y:;~~;;;~~~~* *** •• l .... ** .** ••••••••••• * •••

----------------0.5 0.4 0.3

******* *********

0.46 0.40 0.33

---------0.68 0.61 0.54

---------0.92 0.85 0.7 S

*.*******

~. + POlnt Load

Modification Factors for oasic span/eff. death ratios TABLE 3.10 958110

**************** •• ********.******* •••••• ******.**************************

169

Table 9.19 Values of 'k' factor used for estimating deflections of cantilever walls under hydrostatic pressure

I 1

O.U

I I '\.0 • ...l--""?"

\.. ... IV'

~ --_ ...... I---'"

~ ... ~A k I-'P"'"

-I-' I---'" --r I I I ~_ L....-l-I- ..-1- ......

~ -l-I--t- l- I-' .......

-I--I-' 1.-1-' I-' I I .10.7 1-'1-' -I-- I--t- \..~ I\,}\ _1--'1--

I O.'l.O

L--t-I--~ I-f-r-n -I-1-1-' l- I I

I I t-t-

\'0 0'8 0·$

170

TOT"L LOAOW

h top I h btm

:alcul~tions based on BS 8110 : P~rt 2 clause 3.7.2, equation 11 and Table 3.1

L .....

EXAMPLE OF

GRAPH CONSTRUCTION

Coefficient k coef. of max. deflection at point being considered

coef. max. Bending Moment

3 Maximum Deflection W x (Lw) x (1 + (5x(Lh - Lw)/(4 x Lw»

15 x E x I

Maximum Bending Moment = W x Lw 1 3

i.e. when Lw = 0.7 x Lh 3

k W x 0.7 x (1 + (5 x 0.3)1(4 x 0.7» h.top/h.btm =1.0

W x 0.7/3

k 0.151 1 ( E x I)

Table 9.20 Moment and shear force coefficients for walls subjected to hydrostatic pressure in a three-dImensional rectangular tank, assuming a hinged base, free top and continuous sides (adapted from peA tables)

LOlGfALL lotUOITAL !!OII!1fS

SlotT v.i.! YllTICAL IIOdm

SlOIf vALL lotlZOITAL !!OIIUH

LOIG vm SlotT VALL

SlwrOICU Slwrmu

_,. cIa LlI'" Lll L,I L,l L,l Ly' SII" SII S,I S,2 S,I S" L.I L.2 L.I "I Sd "I

2.02.0 .... '11. -11. ·4S. ·n. 'l'. -Ii. 'Il. -11; •• S.! -n. '16. 'lI. '10. 'lI •• )1. '10 •• )1.

2.0 I.S ·ai. ·lI. -12 •• SO. -a .• ,!. -60. '11. -12. '11.1 -14. '22. '11. '10. '11. 'lI. .S •• )4.

2.0 1.0 -I'. '11. -SI •• S4. -14 .• 42. -IS. '10. -sa. -II. -64. '0. 'lI. '10. '11. '14. '2. '26.

1.00.S -... '11. -u. ·S4. -14. '44. -IS. '2. -61. -61. -64. -J4. '11. '10. '11. '14 •• 0. '1).

Ult tt.tt .. it .... tu lUlU U .. H UUttt Uttl tUlU tIIIU" ttttt u .... tut .. tit I.S I.S -40. '22. -S2. '21. -6) •• 21. -10. '21. -S2. '21. -II. '21. 'JI. '1. '34. '12 •• S. '14.

1.1 1.0 -.e. ·2S. -JI. -ll. -.,. ·J2. -JS. -10. -ll. -'. -". '10. '12. '1 •• 34. -14 •• 2. ., ••

1.1 0.1 -60. -24. -14. '11. -44. -34. -IS. '2. -J6. -21. -44. -20. ·Il. ·S. '14. 014. -0. 'Il.

1.01.0 -!S. '10. -\'. '10. -li. '11. -JI. '10. -". '10. -J'. ,". '24. '1. '16. '24. '2. '11.

1.00.S -ll. -12. -II. '20. -12. '21. -IS. '2. -II. oS. -J). os. '24. '1. '11. '14. '0. 'Il.

uu ttt •• uu" ttt.. IUU uuu lUU" AUIi

O.i 0.\ -II. '2. -I. '2. -\0. ". -II. '2. -J. '1. -\0. ". '14. '0. ·Il. '14. '0. 'Il.

uttf'IUH UUtl .U*' HUt .... u itHUt UUI

3 MOMENT ~

1000

SHEAR

FORCE

Moment Coefficients

3 ~

1000 ~

1000

Horizontal Span

"U' ...... IUUit ttt.. ttt .. tUttt iUit ttUU UUU itt

~ 1000

• a

2 ~

100

e/2

Fr .. Til'

s,1

I,)

2 ~

100

ell

~----b/--2---------------~

~ rr- /'" \ a/2

ui" Sri

" • a L,) $,1

Moment COi;t ht:ionts Vertical $pan

S,I'

S,I

** (LxI and sxl coefficients only to be used where a fixed base alternative is being considered)

171

Table 9.21 Moment and shear force coefficients for walls subjected to hydrostatic pressure in a three-dimensional rectangular tank, assuming a hinged base, hinged top and continuous sides (adapted from peA tables)

., LOI'VALL YUTIClL lIOIIlITS

~/. cia I.JI Lxl L11

-60. 0. O.

LQIGVllL IOIlZOlUL !!OHms

Ly2 L11 1.,4

O. 59. 20.

nOIf IllL 100IUlUL lIOIIUTS

1,4

20.

21.

11.

-9.

1.01' VlLL UOlt IIlLL

mAl OOU nwoou

hi LvI LVI 'fl h2 h)

ll. O. 16. lO. O. l8.·

ll. O. l6. lO. O. H.·

ll. O. l6. 1I. O. 26 ••

ll. O. J6. 11. O. !l ..

U .. It ... 'Utu ...... uuu HUt tU.u UUt UHH' tu .. HUH t .... UUt tu.U ..... 1.\ I.S -51. 18. O.

I--t--1.\ 1.0 -51. )0. O.

f-- r-1.\ 0.\ -51. l5. O.

Uti IUU H"H HUt tttttt

1.0 1.0 -l2. 11. O. r- '--

1.0 0.5 -J2. 15. O.

tau uu. tt.u .. ..... ... ttt

0.\ 0.1 -11. 2. O.

.... ttlU ,tt.tt. tUH ...... MOMENT COEr .... a

1000

SI1EIIR

rORCE

jr.-- ---. b/2

O. -52.

O. -43.

O. -l1.

.u ... ttttH

O. -l5.

O. -21.

HUH ttttU

O. -10.

ttt ... .tt.tt

3 ~

1000

11.

21.

21.

HUt

16.

8.

..t .. 1.

u ....

----

fl-r IqI4T,,==t

IL,l

t _ Ly4

~Bl;S'

-51. 28.

-ll. 8.

-II. -5.

tttUH tt ...

-32. II.

-u. -J.

.. .... .... t

-II. 1.

...... Httt

b/2

O.

O.

O.

HUH

O.

o. IU.tt

O.

Itttit

1 ,

O. -52.

O. -0.

O. ll.

... ttt u .. tU

O. -J5.

O. -21.

.tt.tt ....... O. -10.

.u.tt ....... ~

1000

11.

11.

-1.

...u 16.

-2.

..... 1.

.....

lO. O. H.

lO. O. H.

lO. O. H.

24. O. 26.

24. O. 26.

Httt

11. ll.

.. tt •

COEI'· ... a --roo-

e/2

~dTop

l2. O. ll ••

24. O. 26.·

U. O. 1l.·

HU .. ttu .. .. u

11. O. 26 ••

U. O. ll .•

...... HUH .. it

U. O. lJ ••

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

2 COEr· ... a

100

e/2 ;I< I

~"'l~ fSYl

-~--i hi

Moment Coefficients W • a

172

Lvi hi Vertical Span

•• (Lxl and Sxl coefficients only to be used where a fixed base alternative is being considered)

,.,t'

si)

Sv)

Table 9.22 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming hinged base, free top and continuous sides (adapted from PCA tables)

b

Vhl)

V¥1

t t. P" MonMnt PoeItIoM Pr._. dlegram P .. st.. Ferce Peeitiane

It

I b/a 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0

I1hl - 3 - 19 - 52 - 91 -13.8 -196 T~_' Fr. Mh2 + 2 + 10 • 27 + 45 + 61 + 70

~. IIhl -10 - 36 - 63 - 89 -115 -137 ~ 11M + 6

• 17 + 27 + 36 + 44 • 49

Vhl 0 + 1 + 4 • 10 + 14 • 17

'1 Vh2

• 13 • 26 32 + 38 • 39 • 41

IE Vh3 + 18

• 31 38 + 41 • 41 + 42

IIvl 0 0 0 0 0 0 j IIv2 · 2 + 10 + 22 + 33 + 42 + 49

vvl • 14 -24 + 32 + 38 + 42 + 45 Vv2 0 0 0 0 0 0 But . Pmed-

Table 9.23 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming fixed base, free top and continuous sides (~dapted from PCA tables)

b/. 0.5 1.0 1.5

Mhl 2 - 18 - 40 Mh2 • 1 + 9 + 21 Mh3 - 9 - 29 - 42 Mh4 + 6 . 13 + 16

Vh1 . 2 + 3 + 11 Vh2 + 13 + 23 + 26 Vh3 + 16 + 19 + 17

Mvl - 15 - 35 - 60 Mv2 + 2 . 9 + 16

Vvl + 20 + 32 + 41 Vv2 0 0 0

3 Moment • Coefficient x 'II x a I 1000

2 Shear Force • coefficient x 'II x a I 100

2.0 2.5 3.0 4.0 5.0

- 62 76 - 89 Tap • Fret

-101 -103 + 27 • 27 + 25 + 14 . 7 - 49 - 53 - 55 - 55 - 55 + 16 · 14 + 10 + 4 0

+ 20 + 27 + 11 + 34 + 35 + 25 + 24 + 23 + 20 + 20 + 15 · 13 + 12 + 10 + 10

- 86 -108 -126 -148 -158 + 15 + 11 + 5 - 6 - 13

• 46 + 48 + 50 + 50 + 50 0 0 0 0 0 Base: fixed

173

Table 9.24 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure, assuming pinned base, pinned top and continuous sides (adapted from peA tables)

b/a 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0

Mh1 0 0 0 0 0 0 0 0 Top . Pinned

Mh2 0 0 0 0 0 0 0 0 Mh3 -10 - 35 - 52 - 59 - 62 - 63 -64 - 64 Mh4 + 5 + 16 + 21 + 20 + 17 + 16 + 16 + 15

Vhl 0 0 0 0 0 0 0 0 Vh2 + 13 + 26 + 34 + 36 + 37 + 38 + 38 + 38 Vh3 + 19 + 32 + 38 + 39 + 39 + 39 + 39 + 39

Mv1 0 0 0 0 0 0 - -Mv2 + 1 + 11 + 28 + 42 + 52 + 57 + 62 + 63

Vv1 + 14 + 24 + 30 + 32 + 33 + 33 + 33 + 33 Vv2 + 4 + 8 + 12 + 14 + 16 + 16 + 16 + 16

~l . . ~ ".."T" ... Base ............ ·"""-"'p ... jm .... "8d,.".,· .... r-.' ..

Table 9.25 Moment and shear force coefficients for wall panels subjected to hydrostatic pressure. assuming fixed base. pinned top and continuous sides (adapted from peA tables)

b/a 0.5 1.0 1.5 2.0 2. S 3.0 4.0 S .0

Mh1 0 0 0 0 0 0 0 Top . Pinned

0 Mh2 0 0 0 0 0 0 0 0 Mh3 -11 - 27 - 34 - 35 - 35 - 35 - 35 - 35 ~ 11M + 6 + 12 + 12 + 9 + 8 + 7 + 6 + 5

Vh1 0 0 O· 0 0 0 0 0

"g

~ If "h2 + 13 t 26 + 34 + 36 + 37 + 38 + 38 + 38 Vh3 + 19 + 32 t 38 + 39 + 39 + 39 + 39 + 39

Mv1 - 12 - 32 - S1 - 60 - 6S - 65 - 65 - 65 Mv2 + 3 + 11 + 21 + 26 + 29 + 30 + 30 + 30

.. .. ~~ ~ en

Vvl + 20 + 32 + 38 + 39 + 39 + 39 + 39 + 39 Vv2 + 3 + 8 + 9 + 10 + 10 + 10 + 10 t 10 Base . Fixed

174

Table 9.26 Deflection of two way spanning slabs with various edge conditions subjected to (a) triangular pressure, (b) rectangular pressure

b/a 0 0 0.50 0.10 0.09 0.60 0.19 0.16 0.70 0.33 0.27 0.80 0.52 0.41

i 0.73 0.90 I 0.57

1.00 0.98 0.76 1.10 1.24 0.96 1. 20 I 1. 57 1.15

1. 30

I 1. 97 1.28

1.40 2.30 1. 40 1. 50 2.63 1.52 1. 60 3. 02 1. 64

1. 70 3.38 1. 78 1.80 3.72 1.90 1.90 3.98 1.98 2. 00 4.22 L04

~ DEFLEC'fION '"

"'.0

b/a 0 0 0.50 0.16 0.16 0.60 0.27 0.33 0.70 0.58 0.53 0.80 0.91 0.83

0.90 1. 33 1.14 1. 00 1.9'2 1. 57 1.10 2.58 1.98 1. 20 3. l4 2.33

1. 30 3.75 2.71 1. 40 4.33 3.18 1.50 5.00 3.42 1. 60 5.74 3.75

1. 70 6.33 3.97 1. 80 7. Of; 4.17 1. 90 7.83 4. 33 2.00 8.44 1.19

I DEFLEcrI'JN '"

P

0 0 [] 0.08 0.02 0.07 0.20 0.05 0.16 0.34 0.08 0.24 0.50 0.20 0.40

0.72 0.36 0.59 1. 04 0.58 0.81 1.44 0.84 1.03 1.92 1. 36 1. 28

2.47 1.92 1.58 3.10 2.70 1. 88 3. 83 3.60 2.20 1.66 4. 65 2.59

5.60

I

5.86 3. 03 6.62 7.14 3.40 7. 80 8.46 3.77 8.96 11. 70 4.14

5 COEFFICIENT * wa ME·rRES -------------------

1000 x E x I

(a)

0 0 0 0.16 0.16 0.15 0.33 0.33 0.10 0.52 0.52 0.48 0.95 0.95 0.91

1. 54 1. 67 1. 33 2.29 2.48 1.138 3.21 3.67 2.50 4.33 5.24 3.22

5.50 6.67 4.18 7.33 9.17 5.18 9.16 12.33 6.33

ll. )0 15.42 7.50

13.66 19.56 8.64 16.16 24.17 9.75 18.91 29.42 10.87 22.56 33.60 11. 84

,.

~ COEFr'IC [ENl' * P .a MErRES -----------------

1000 x E x I

(b)

0 0.02 0.04 0.07 0.19

0.33 0.53 0.80 1.20

1.66 2.16 2.70 3.34

4.12 5.02 6.00 7.05

~ w,e

0 0.16 0.25 0.42 0.83

1. 60 2.58 3.75 5.21

6.74 8.91

11. 25 14.00

17.00 20.19 24 .22 28.47

~ p

POSrTrON WHERE .. DEFLECTION IS TO

BE CALCULATED

PINNED EDGE

(/ (i /1,' f I / (

FUED EDGE

FREE EDGE

175

Table 9.27 Ground pressure created beneath a base slab carrying an edge force '0' and an edge moment 'M' and supported upon an elastic soil

Table for X y values

)..L 0.25 0.5 0.75 1.0

0.1 4.000 2.500 1.000 -0.500 -2.000 ). "'1--~----0.5 4.002 2.S00 0.999 -O.SOO -1.998 4 lC E lC I

1.0 4.040 2.486 0.980 -0.502 -1. 99 6 Where 'k I is the modulus of subgrade reaction

1.5 4.18S 2.469 0.909 -0.506 -1. 861 ~ 0 J 2.0 4.480 2.400 0.722 -0.516 -1.570

" .. 0.""'1 2.5 5.174 2. 285 0.447 -0.522 -1. 168

3.0 6. 019 2.102 0.096 -0.514 -0.663

3; 5 6.971 1.859 -0.232 -0.473 -0.220 b

J 4. a 7.942 1. 562 -0.479 -0.395 0.077 "t ,. "II.

5. a 9.848 0.849 -0.687 -0 .174 0.278 L

6.0 -0.628 0.017 0.288 X • Position of Load '0' 11. 682 0.082 Y • Point under consideration

7.0 13.418 -0.628 -0.458 0.121 0.363 I "''3-'+7 I I

8.0 15.026 -1.222 -0.294 0.172 0.527 I I I I I I

y. O-IIL o.UL O.SoL 0:1'5\. .,OL 9.0 16.484 -1. 682 -0.201 0.217 0.728

, Q ' Coefficient Values Note When X • 0 and Y • 0 read value

Table for X 0.0251. y values

~L 0.25 0.5

0.1 3.850 2.425 1. 000

0.5 3.852 2.425 0.999

1.0 3. 895 2.420 0.980

1.5 4.010 2.399 0.919

2.0 4.328 2.345 0.752

2.5 4.863 2.247 0.508

3.0 5.583 2.098 0.195

3.5 6.384 1.901 -0.101

4.0 7.187 1.663 -0.327

5. a 8.709 1.102 -0.533

6.0 10.110 0.508 -0.508

7.0 11. 386 -0.035 -0.381

8.0 12.529 -0.486 -0.244

9.0 13.534 -0.837 -0.149

Pressure Q coeff. • Q I l +

176

When X • land Y • 0 same va lue as X s 0 and Y s 1 When X • 0.025 and Y • 0 read value When X • 0.975 and Y • 0 same value as X • 0.025 lind Y • 1

Q ~

0.75 1.0 1;M Mt' -1.850 -0.425

-0.425 -1. 848 = J

<l -0.428 -1.797 t -0.506 -1. 726

-0.440 -1.380

-0.450 -1.108 Gl, Q. + Q, 01,

-0.447 -0.656 ft°.0

2.SL O'.~'~

-0.417 -0.256 I

~. C)M.SL X·MlSL

-0.356 0.016 0., " M/c.o.oa51.) -0.178 0.206

-0.020 0.207

0.065 0.240

0.101 0.338

0.124 0.465

2 M Coeff. * ( M I ( 0.025 • l »)

Table 9.28 Bending moments created within a base slab carrying an edge force 'Q' and an edge moment '1',,1' and supported upon an elastic soil

Table fer X 0.01. ~ .. J-~----.

4 x II " I

~ Where 'k' is the 1IIOdululiI ot liIUb<J rade react ion

AI.. 0.25 0.5 0.75 o 0.1 -0.141 -0.125 -0.047 ... 0.}1..

o .S -0.141 -0.12S -0.047

1.0 -0.140 -0.124 -0.046

1.5 -0.137 -0.120 -0.041

2.0 -0.131 -0.109 -0.0]9 L

2.5 -0.120 -0.092 -0.0]0 x • Position of Load '0' 3.0 -0.106 -0.070 -0.020

3.5 -0. 091 -0.048 -0.010

4 :0 -0.077 -0.031 -0. 004

'l • Point under consideration I I

... '3 ..... r ,I f

'1. .to~ Q,~t. •• ~ ~~t. "~L 5.0 -0.054 -0. 010 O. 002

6.0 -0.037 -0. 001 0.002

7.0 -0.024 0.002 0.001

8.0 -0.017 0.002 O. 001

9.0 -0. 008 O. 002 O. 000

, M . Coefficient Values Note When X ~ a and y = 0.25 read value

When X = 1 and y = 0.25 same value as X ~ 0 and Y ~ 0.75

When X 0.025 and Y = 0.25 read value When X = 0.975 and Y = 0.25 same value as X • 0.025 and Y • 0.75

Table for X • O.O25l y values

).1. 0.25 0.5 0.75

0.1 -0.120 -0.11] -0.047

0.5 -0.120 -0.il2 -0.043

La -0.119 -0.111 -0.042

1.5 -0.117 -0.108 -0. 041

2. a -0.112 -0.098 -0.036

2.5 -0.102 -0.083 -0.028

3.0 -0.089 -0.064 -0.019

3.5 -0. 077 -0.045 -0.010

4.0 -0,065 -0.029 -0.004 Q. = M/c o .oa51.)

-0.045 -0.010 0.001

{).O -0.031 -0.002 0.001

7.0 -0. 024 O. 002 0.001

8.0 -0.015 0.001 0.000

9.0 -0.008 0.001 0.000

Moment· Q coot. It Q x L + M coof. )( til I ( 0.025 J( L )

177

::::i Table 9.29 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a triangular 00 load, assuming a fixed base and a free top (adapted from peA tables) Ft = c,)ef. , p.Lv. r k;~ p'?r :n

(Positive sign indicates tensi0n)

Lv' Coefficients a.t point

O.OL 0.1 L 0.2 L O.)Lv

0.4 L 0.5 Lv 0.6 L O.lLv 0.8 Lv 0.9 Lv 2rb v v y y y

0.4 +0.149 .0.1 }!. .0.120 .0.101 .a.082 +0.066 +0.049 .0.029 +0.01" '0.00l. 0.8 +0.26} '0.2}9 .0.215 .0.190 +0.160 .O.I}O .0.096 +0.06} .a.o}!. +0 .. 010 1.2 +0.28} ... 0.271 +0.254 +0.2}!. +0.209 .0.180 +0.142 .0.099 +0.054 +0.016 I 4-1.6 +0.265 +0.268 +0.268 +0.266 .0.250 +0.226 '0.185 .'J.l }!. .0.075 .0.023 2.0 .0.2}!. +0.251 +0.273 .0.285 +0.285 +1J.214 +0.232 +0.172 +0.104 .O.O~l

3.0

I

+0.1}!.

I +0.20}

I .0.267 +0.322 +0.357 +0. )62 +0.3)0 +0.262 +0.157 .0.052

[ 4.0 +0.067 +0.164 +0.256 +0.339 +0.40} +0.429 +0.409 +0.)34 +0.210 .0.073 5.0 +0.025 +0.137 +0.245 +0. }!.6 +0.428 +0.477 +0.469 +0. )98 +0.259 +0.092 6.0 .0.018 +0.119 .0.234 .0. }44 .0.441 +0.504 .0.514 +0.447 .0.)01 +0.112 8.0 -0.011 +0 .. 104 +0.218 +0.}}5 +.1.443 +0.5}!' +0.575 +0.530 .a.)81 +0.151

10.0 -0.011 +0.098 +0.208 +0.323 .0.437 +0.608 +0.589 +0.440 +0.179 12.0 -0.005 +0.097 +0.202 +0. )12 .0.429 +0.628 +0.6)3 .0.494 +0.211 14.0 -0.002 +0.098 +0.200 +0. )06 +0.420 +0.6)9 +0.666 .0.541 .0.241 16.0 0.000 .0.099 .0.199 .0.30l. .0.1.12 '0.641 +0.687 +0.582 +0.265

(a) ~ fIXED BASE - FREE TOP

L 2

...!..... Coefricients at point ;Vei'Jht of Liquid (kN 'lee ",3)

2rh O.IL 0.2L 0.3Ly

o.L.Ly

0.5L y

O.6L y

0.71, v

0.8'[, v

1.0t v Lv Effect ive Height of wall (m) v v

0.4 +.0005 +.0014 +.0021 +.0007 -.0042 - .01,0 -.0302 - .0529 - .1.;'0'! Effective Ra1i'Js of cylinder (111) 0.8 +.0011 +.0037 •. 006) +.0080 +. )010 +.0023 - .0068 -.0224 - .0795 1.2 +.0012 +.0042 + .0077 +.0103 +.0112 +.0090 +.0022 -.0108 -.oGO? rh ickness of cylinder "'311 I'll) 1.6 +.0011 +.0041 +.0075 +.0107 •. 0121 +.0111 +.0058 -.0051 2.0 +.0010 •• 0035 •• 0068 +.0099 +.0120 +.0115 •• 0075 - .0021 Ft Ring ren3i~n IkN)

3.0 +.0006 +.0024 +.0047 +.0071 •• 0090 +.0097 + .0077 +.0012 - .0119 - .0333 Marnen t (kN m)

4.0 +.0003 + .0015 +.0028 +.0047 +.0066 •• 0077 + .0069 .-.0023 -.0080 - .o26e 5.0 +.0002 +.0008 +.0016 + .0029 +.0046 + .0059 +.0059 +.0028 -.0058 - .0222 6.0 +.0001 +.0003 •. 0008 +.0019 +.0032 •• 0046 +.0051 +.0029 -.0041 -.0187 8.0 .0000 •. 0001 + .0002 + .0008 •. 0016 •. 0028 •. 00}8 •. 0029 - .0022 - .0146

10.0 .0000 .0000 +.0001 +.0004 +.0007 +.0019 + .0029 +.0028 - .0012 12.0 .0000 -.0001 •• 0001 +.0002 +.0003 +.0013 +.0023 +.0026 -.0005 14.0 .0000 .0000 .0000 .0000 .... 0001 +.OOOS •. 0019 +.0023 - .0001 16.0 .0000 .0000 -.0001 - .0002 -.0001 +.0004 •• 0013 +.0019 +.0001

I - • '-'~, r ' 3

'1")m;~nt = ':::Jef. x p .. Lv <Nm oer m

(b) (Po.31tive sign indicates t'?ns ion on the f)uto;ide f3ce .. )

::i \0

Table 9.30 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a triangular load, assuming a pinned base and a free top (adapted from peA tables)

L.,.' Coerricient:l at point

2rh 0.0 Lv O. ',ILv 0.2 Lv O.H v 0.4 ,Lv 0.5 Lv 0.( Lv 0.7 Lv o.e Lv O.? Lv

0.4 '0.474 .G.440 .0.J95 .0.J52 .0. }O8 .0.264 .0.215 '0.165 .n.i11 .0.0')7 0.8 .0.42J .0.402 .O.JII, .0.J58 .G.JJO .0.297 .0.249 .0.202 .0. '45 .1.07(, '.2

I .0.350 .0.}55 .0.}6' :~:~; .0.}58 .0.34} +0.J09 +0.256 +0.186 '0.098

1.6 +0.271 +0.J03 ... O4~1 .0.3115 .0.385 +0. }62 .0.}'4 .0.233 .0.1"1 • 2.0 .0.205 • 0.260 .0.}2, .0. }7} .. 1.4" .O.l.}4 .0.419 +0.569 .0.2[10 +').1',1

3.0

I

+0.074

1

.0.'79

1

.0.28' .0.}75 .0.449 .0.506 .0.5'9 .0.479 .O.}75 +-O.?H) 4.0 '0.0' I .G.' }7 .0.25} .0.}67 .1).469 .0.545 .0.579 ").553 0.447 .').7~6 5.0 -0.008 .0. Ill. '0.2}5 '0.}56 .0.469 '0.562 .0.6'7 .0.606 .0.503 .0 .... )4

6.0 -1.0" .\l.10,} .0.22} .a.34} '0.463 .0.566 '0.6}9 .0.64} '0.547 ""I.VI 8.0 -0.0'5 .0.096 .0.208 .a.l24 .(l.44} .0.564 .0.66, '0.697 '0.621 .. n. VI()

0.0

I -0.008

I ").095

I .0.200

I .0.311

I ... ". 1 .0. '"

I ·O.U< 1 '0. nO '0.678 .. 0.4 ~~

2.0 -0.00? .0.097 .0.197 .').}O2 .. 0.1.17 +0.541 .0.664 .0.750 .0. nq .O.J.II 4.0 O.UOO .U.0'l8 .0.197 .~.299 .O.M)!' +O.~51 .0.659 .0.761 .0.71)2 .1). f) 1 ~ 6.0 +1),007 .').100 .0.198 .0,),)9 +0.1.0.5 +O.~21 +tl.6,}{) .0.764 ·').176 .,). t, ~(,

(a)

Lv' Cr)efficients at point

I

I

2rh 0.1 L 0.2 L v 0.3 Lv 0.1. Lv 0.5 Lv 0.6 L v

O.TL v

0.8 L v ).9 Lv 1.0 Lv

v

0.4 +.0020 +.0072 •• 0151 •• 0230 •. 0301 +.0348 '.0}57 •• 0312 •• 0197 0

0.8 +.0019 +.0064 +.0133 •. 0207 •• 0271 +.0}19 •• 0329 + .0292 •• 0187 IJ 1.2 +.0016 •• 0058 +.0111 +.0177 •• 02}7 +.0280 •. 0296 •• 0263 •• 0171 (\

1.6 +.00'2 •• 0044 +.0091 +.0145 •• 0195 •• 0236 +.0255 +.0232 +.0155 G 2.0 +.0009 •• 0033 '.0073 +.0114 +.0158 +.0199 +.0219 +.0205 +.0145 0

3.0

1

•• 0004

1

+.0018 +.0040 +.0063 •. 0092 +.0127 +.0152 •• 0153 +.0111 0

4.0 +.0001 +.0007 +.0016 •• 0033 •• 0057 •• 0083 +.0109 +.0118 +.0092 0

5.0 .0000 +.0001 + .0006 +.0016 •. 0034 +.0057

I +.0080 +.0094 +.0078 0

6.0 .0000 .0000 •• 0002 +.0008 +.0019 •• 0039 +.0062 •• 0078 •• 0068 0

8.0 .0000 .0000 -.0002 .0000 •. 0007 +.0020 +.0038 +.0057 +.0054 0

,"0 1 ~I .oooo

1-=' 1-=' 1 .. ""~ 1 .. 00" I'OO~

•• 0043 + .0045

12.0 .0000 .0000 - .0001 - .0002 .0000 + .0005 +.0017 + .00}2 '.0039 14.0 .0000 .0000 - .0001 - .0001 -.0001 .0000 +.0012 +.0026 +.0033 16.0 .0000 .0000 .0000 - .0001 - .0002 - .0004 •• 0008 +.0022 +.0029

I

(b)

Ft = ::oet'. x ".Lv.r kN :)"c '11

(?')sitlve sig'1 in:ii::ates tension)

\

-0.0'"""

[" ---0.' Lv -·-o.tI ...

:. :::~:~t: .3 -,-()oiLy

---0 .• ""'-,f -·-c."JL'I -'-o.IILv

.. -,-o.~"" .. -- t.c>Lv

PI" IJ\I

PIlE) BASE fREE TOP

3 P = 'Neight)f Li'lui:l (k!; ?er m )

Lv = Effect ive H"i} It ')f '1,,11 (m)

Effect lve Ra 1:'J5 of cylinder (m)

I'hicl;ness 'f ':)'lindec wall (m)

Ft Hin~ J."~ns io~ (!('I)

:1 = t'\,)ment (k:l m)

I ''':lment :: co'!:!'f. x ?Lv :CJP.t O~r m

(Positive 5i911 i'1::jic!ttes ten3 ion l)n the outs id~ f"lce.)

~ Table 9.31 (a) tension, and (b) moment coefficients in cylindrical tanks subjected to a moment per m, 'M' applied at base (adapted from peA tables)

L/ 2rh

0.' 0.8 1.2 1.6 2.0

3.0 '.0 5.0 6.0 8.0

10.0 12.0 1".0 16.0

~ ?Th

0._ n.9

I.' I.h 7.0

l.O _.0 ~.O

h.O •• 0

10.0 11.0 lll.O H •• O

O.OLy

+2.70 +2.02 +1.06 +o.n -0.68

-1.79 -1. 87 -1.5' -1.0' -0.14

+0.21 +0.32 +0.26 +0. '21

n.ll,.

Hl.Ot:l +O.ooq +O.OOf, +0.on3 -0.002

.0.007 -O.OOA _0.001 -0.005 -0.00\

0.000 O.noO

O.ILy

+2.50 +2.06 +1.42 +0.79 +0.22

-0.71 -1. 00 -1.03 -0.86 -0. S3

-0.23 -0.05 +0.014

0.2 Ly

+2.30 +2.10 +1. 79 +1.43 +1.10

+0.43 -0.08 -0.'2 -0.59 -0.73

-0.6' -0 •• 6 -0.28

O.3Ly

+2.12 +2.1' .2.03 +2.0' +2.02

+1.60 +1. 0'" +0.45 -0.05 -0.67

-0.9' -0.96 -0.76 -O,f.4

Coeffici~nt:'C at oointt't

a.li Lv

+1.91 +2.10 +7.46 +2."12 .2.90

+2.95 +2.47 +1.96 +1.71 -0.02

-0.73 -1.15 -1.29 -1. 28

O.SLv

+1.69 +2.02 '2.65 +3.25 +3.69

+-ta,29 ... 4.31 +3.93 +3.3&1 +2.05

+0.82 -0.18 -0.87 .1.30

0.6 Ly

0.7Lv

+1.'1 +1.13 +1.95 +1.75 +2.80 +2.60 +3.56 +3.59 t4.30 +4, Sli

+5.66 1+6.58 +6.34 +9.19 +6.60 .9. 4 1 +6.5. +10.29 +5.87 +11.32

... 79 ,+11.63 +3.52 +11.71 +2.29 +10.55 .1.12 tQ.67

0.8Lv

+0.80 +1.39 +2.22 +3.13 ".08

+6.55 +8.82

+11.03 +13.08 +16.52

+19."6 +21.80 +23.50 +74.53

• ~'lh~n this table is ~sej for moment d?plied at t09, while the t<>p is hinge:J, O.OLv is the bottom of the wall and I.OLv is the top. : .. omcnt a9plied at an edge is "Jositiv+? whcn it causes outward rot:'lt ion at that edge.

O.:;>Lv

+0.051

+0.011 +0.011 -0.002

-o.on -0.016

-0.0"" -0.0\9

_0.007 0.000 0.000

0.3f ....

+0.109 +0. Otto +0.063 +0.03'5 +0. (l} '2

-0.030 -O.Oll~

-O.o"S _O.IlIH)

-o.on

-o.OOq -0.003 0.000

+0.001

(a)

C~ffic j"'''ts .It pointi!

O'''Lv

+0.1')(, .0. If," +0·11 .... +0.019

-0.0'~ -0.0'>1 _0.061 _".OSA _O.Olt"

-0.028 -0.016 _O.OOA -0.003

o. "'!'v

+0.:?'1(,

+0.' .... 3 +0.70r, +Q.lr!7 +o.O'lr,

+0.010 -O.OJ" ·0.0')7 -o.or.s .n,Of!9

-0.053 .0.0"0 -0.01'1 -o,o:?!

(b)

0.6L ..

+0.4116 +0. J7S

+0 .. 316-+0.7S3 +0.1f'l3

+0.087 +0.013 -0.015 -0.037 -0.0,;2

-O,OF.7 -0.0(.16 -o.oc-,q -n.or,l

O.7Lv

.O.Sit7 +0.S03 +O.it'ilt +0.313 +0.340

+0.117 +0.150 H).Oq,)

+0.057

-0.031 _o.O"q .".Ofin -0, or,,;

O.SLv

+0.692 +0.659 .0.616 +0 ..... ,0 +0.51'l

+n,lI1fl

"0.3")" +0.']'1(· +0.152 +0.178

+0.173 "O.O~l +0.04A +0.07""

n."Lv

+0.Att3 +O.R']" +0. A07 +<1. -f7", +0.11IR

.. n,f.'l7

.. o.r,ll')

.n.r·flf. +0. r\77

to. 'I,lI +n.1R?

0.9Lv

+0.44 +0.80 +1.37 .2.01 +2.75

+4,73 +6.81 +9.07

+11.'1 +16.06

+10.07 +25.73 +30.34 +34.fJ5

l.n!.v

+ l.non +J .oon +1. f"lIIO "1.onn +1.nno

+1. nqn +1. non "I.nnn +l.nno "1.nrH1

+1.000 .l.ono +l.nno +t.flnn

Ft = coef. X H.r/Lv kN per m

(Positive sign indicates tension)

\

~ I 1

I

, - __ 0.%.1..01'

---0-,3\.,,­,;.~ ---o·4Lv ;- ---o-SL", -. ---O."Lv ?: -.--O,'lL.V .t Itt -0-81..V' ~

--~:~~';

-O . .,l .... \' . ..1 ~ _, :' •. -- ,-OL"

3 P = Weight of Liquid (kN ;:>er :n )

Lv Effect ive Height of wall (m)

Effective Radius of cylinder (m)

Thickness ~f cylinder wall (m)

Ft R ;:>9 Tens ion (kN)

'I = Moment (kN m)

Moment coef x M kNm per m

00

Table 9.32 (a) tension, and (b) moment coefficients in cylindrical tanks subjected to a shear per m, 'V' applied at top (adapted from peA tables)

L ' Coefficient .. ..'It pointh

2;h f-I -O-.OL--'y

1r--O.I-Ly-'-l-o.-ny-'-,-0-.3L--'v I-n"'-Ly-r-I -o.-~Lv-r-l-o-.Hv--rl-o-.7L-y r-, -0.9-Ly

'T'",-0.---lOLy

I 0.4 0.9 1.7 1.(1

2.0

3.0 4.0 5.0 6.0 8.0

10.0 17.0 1'''.0 16.0

Lv' 2rh

0." 0.8 \.7 \.6 7.0

3.0 -.0 5.0 6.0 8.0

io.o 17.0 1'.0 1( •• 0

-1. 57 -3.0Q

-3.95 -4.57 -5.12

-6.32 -7.34 -8.22 -9.02

-10.42

-11.67 -n.76 -13.77 -J".7l1

0.1 Lv

+n.OQ.1

'0.085 +0.0,.1

+0.079 "0.071

+0.07') +0.0&8 .0.Ofl" .0.0(,2 +0.051

.0.053 to.Oll!}

+0.0'16 +0.0'14

-1.32 -2.55 -3.17 -3.54 -3.83

-4.37 -4.73 -4.99 -5.17 -5.36

-5.43 -5.41 -5.34 -5.n

-1.08 -1.04 -2.q ... -2.hO -2.68

-2.70 -2.60 -2.45 -2.27 -1.85

-1.43 -1.03 -O.M~

·0.33

-0.9h -1.57 -1. 7q -1. 80 -1.7"

-1.43 -1.10 -0.70 -0.50 -0.02

+0.36 +0.63 +0.80 +O.qf,

-0.65 -1.15 -1.25 -1.11 -1.02

-0.58 +o.lQ +0.11 .0.34 .O.~3

+0.78 +0.83 +O.Al +0. 'If.

_o.tJ7 -o.RO

·0.91 _o.r.ll

-0.02 '0.26 .0.47 to.S'} tr).66

+0.h7 to. ~2 ... 0."1 +O.:\?

-0.31 -0.51 -0.48 -0.36 -0.21

+0.15 +0.38 '0.50 +0.53 +0.46

+0.33 '0.21 +0.13 to.OS

-0.18 -O.lA -0.25 -0.16

.0.19 '0.33 +0.37

'0.35 +0.'24

.0.12 +0.04

*.-Jhen this tat:lle is use·":l f-"Jf s~23r ar),lie1 olt !:>1S'?, whil::.o the ton is Eix2'd, 0.0 Lv 13 the b"Jtt,)'ll ')E t:!'e- wa~l 3nd 1.0 Lv i5 the t09. Shear 3c~ing inwarj-:; is )o:;ltiv-e, ~utwarj3 l5 n0gative.

(a)

COf'ff ic j.'nl.-: at (lointn

O.1Lv

.0.117

.0.145 +0.132 +0.122 .0.llS

+0.100 +0.088 +0.078 .0.070 .0.058

+o.~q

+0.042 +0.036 +0.031

n.3Lv

+0.11,10 +0.19S +0.157 'O.lH +0.126

+0.100 +0.081 +O.0~7

+0.0~6

+0.0'11

'0.02~

+0.012 +0.017 '0.012

fl,ll Lv

+0.300 .0.208 +fl.l()t,

'0.IJ9 '0.1I~

+O.OAn

+o.or.J +0.Ott7 +0.036 +0.021

+0.012 +0.007 +0.004 .0.001

n. r, r,v

+0,1,)4 +fl.770 +O.lSI)

.n.12S

+n,nfi6 +O,f)4J .0.029 .0.018 +0.007

+0.001 0.000 -~.OOI

-0.002

(b)

n,r, Lv

+n,40,} .0.724 +0.145 .0.105 +0.080

+0.0144 '0.025 '0.013 .0.006 0.000

_0.002 -0.002 -0.002 -0.002

n.7 Lv

+O.lf4A

'0.223 '0.171 '0.081 '0.056

.0.025 +0.010 +0.003 0.000

-0.002

-0.-02 -0.002 -0.001 -0.001

n.e Lv

+o.t,q::

'0.21Q .0.016 +0.0'\6 .0.031

+0.006 -0.001 -0.003 -0.003 -0.003

-0.002 -0.001 -0.001

0.000

-0.08 -0.02 -0.13 -0.01

-0.10 -0.07 -0.05 -(J,nI '0.01 .0.01

'0.13 +0.19 '0.20 +0.11 +O.Oq

+0.04

.0.06 +O.Of,

+0.01 +0.01

0.00 .0.02 -0.02 _0,,)1

-O.nc, ~o. nt

n. IlLv

+0. ~.3S +0. :?t4 +0.00" +0.0:10 +0.00f)

-0.010 -0.010 _0,001

-~.005

-0.002

-0.001 0.000 0.000

0.000

1.01 .•

tn. \7a tn. ~nA +o.nf.} +o,nol,

-o.nl~

-0.07' -O.OI't -O.lIll -0.01l(,

-n.onl

o.noo n.nno O.noo 0.000

Ft = coef. • V.r/Lv klf/ ..

(Positive sign indicates tension)

[ 4.

" ~o-o"" _._ •• IL" _.- ...... -·-0 .• "" --0 ... '" _._ •. n ... _.- 0·"". -.- o.lLw _.- o.tL.o _.-0._

\oOW

...Jhk­FIXED BASE - FREE TOP

3 :ielght ot Liquid (kN ?er :n )

Lv Ufective Hei'lht 0f .. all (m)

~ffective Radius of ::ylinder (m)

:'hL:kn(!ss Clf cylinjer \1311 <m)

f't = Ri:'l3 :'ension (kN)

fl = :1oment (kif m)

"o_nt • coef. • V. Lv "1IIiLf.

~ Table 9.33 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a rectangular load, assuming a fixed base and a free top (adapted from peA tables)

L ' Coeffir.i('nts a.t point I

y 0.BLv 0.9 Lv I

2rh a.OLy o. I L v 0.2Ly O.}Lv 0. 4Lv 0.5 Lv O·€Lv O.7 Lv

0.4 .0.562 .0.505 '0.431 .0.353 .0.277 .0.206 .0.145 .0.092 .0.046 .0.013

0.8 .1.052 .0.921 .0.796 .().669 .0.542 +1).415 .0.289 .0.179 .0.OH9 +!).O)4

1.2 .1.211\ .1.078 .0.946 +0.808 .0.665 .0.519 .O.}IB '0.246 .0.127 .G.O}4

1.6 .1.257 ".'4' +1.009 +0.1181 .0.742 .0.600 .0.449 .0.294 +Q.1h3 .0.045

2.0 ".25} .1.144 +1.041 .0.929 .o.~o6 .0.661 .0.514 ,o.}45 .').186 .0.055

3.0

I

+1.160 ... 1.112 ".06, .0.998 .G.912 .0.796 .0.646 .• ().459 ").251\ .0.081

4.0 +1.085 .1.073 ".057 .1.029 .1.977 .O.~I .0.746 .0.553 .0.J22 :~: ~~~ I 5.0 ".037 +1.~ ., .047 ".042 .1.015 +').t}J.9 +iJ.f125 .0.629 .0. j79

6.0 ".010 .,.024 .I.OJS .1.045 + I .0}4 +0.9116 .0.819 .(\.694 .. O./dO +11.149

fLO .0.989 .1.005 ".022 ".0}6 +1.0l.4 .1.026 .0.953 .0.1118 ·'J.5'? +0.IU9

to.O

I .a.989 .0.998 .1.0tO .1.023 ".0'9 .'.040 .0.996 .0.859

""U 12.0 .0.994 .0.997 .1.003 .'.014 ... 1 .O~1 .1.('43 +1.072 +0.911 .. ...n .. 7~)? 14.0 +\1.')97 .0.998 .'.0011 .1.007 .'.07? ".()I,O ., .O~') , n.rH~9 +n.;'14

1(,.0 .1.1\()O +)."'1') .. II. 'J~llJ ".003 .'.015 .1 .O.~2 .'.Ott1 .. fl. 'I 1', ~ ). 'I;' I

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

(a)

Lv' Coefficients at point

2rh 0.1 Lv 0.2L v 0.3 Lv O.4Ly o.r) Lv O.6L v 0.7Lv 0.8L v 1),9 L y 1.0 L

v

0.4 -.0023 -.0093 - .0227 ·.0439 -.0710 -.IOIR -.1455 - .2000 - .2593 0.8 .0000 -.0006 -.0025 - .0083 -.0185 - .0362 -.059" -.0917 -.1325 1.2 +.0008 •• 0026 •• 0037 •. 0029 -.0009 -.0089 -.0221 - .0468 - .OB15 1.6 •• 0011 +.00}6 •• 0062 +.0077 '.0068 +.0011 - .0093 -.0267 - .0529 2.0 +.0010 •• 00,6 •• 0066 •• 0088 ·.OQR9 •. 0059 -.0019 -.0148 -.031\9

3.0 •• 0007 •• 0026 •. 0051 + .0074 +.0091 •• 008} + .0042 -.0053 -.0223 -.<\1.0 4.0 +.0004 ·.0015 •• 0033 •• 00~2 '.006R •. 0075 •• 0053 -.0013 -.011.5 - .0365 5.0 •• 0002 +.0008 •. 0019 •• 0035 •• 005 1 •. 0061 •. 0052 +.\)007 -,.'1101 -.02'33 6.0 +.0001 •• 0004 '.0011 •. 0022 '.0036 •. 0049 •• 001.8 ·.0017 -.007} -.0242 8.0 .0000 '.0001 +.no03 +.0008 +.OOH~ •• 0031 .... 1103R +.0024 -.OfJ40 -.OIAi.

10.0 .0000 ,.1021 +.0030 +.0026 12.0 .0000 +.00t11o +.')021. +.0022 14.0 .0000 +.()l)10 •. 0018 +.0021 -.!)OO7

16.0 .0000 •• ()1)()6 .,!){)12 +.0020 -.tj·)05

(b)

Ft = coe f. x !2" r kN ?er m

(Positive sign indicates tens i.)n)

cl

[ -- O-CLv

-._ o·'Lv

-'- G .• Lv --o· .. L. .... _._ ".,Lv

.. _._ O·6.&.v

~. -. - ().1"Lv

-'- O.'L ... -.- Cl~1.N . . . __ \oOLv

j.i/ ..Jh~ RXED BASE . FREE TOP

pi =- uniform ?res!3ure applied vert ically around ?eri .. eter of tank. (kNperm)

Lv Effe.~tive Height )f wall (m)

Effective iladius of cylind"r (m)

h = Thickness .)( cylinder ",~\l tm)

Ft = Rin'~ Tens ion «I,)

:-I = :Iome'lt (kN m)

Iloment = CJeE. ~ o'.Lv 2

kNm per "'

(Positive si'Jn indicates tens ion on the outs ide face.)

~ Vol

Table 9.34 (a) tension, and (b) moment coefficients in cylindrical tanks supporting a rectangular load, assuming a pinned base and a free top (adapted from peA tables)

L/ Coefficients at pOint

2rh O.OLv O. ILv 0.2 L

v 0.3 Lv 0.4 Lv 0.5 L

v 0.6Lv 0.7 Lv O.SLv 0.9 L

v

0.4 .1.474 .1.340 .1.195 .1.052 .0.908 .0.764 +0.615 +0.465 +0.311 ,0.154 0.8 ".423 ".302 +1.181 ".058 .0.930 .0.797 .0.649 .0.502 +0.345 .0.166 1.2 +1.350 ".255 .1.161 ".062 .0.958 +0.84} .0.709 +0.556 .O.}86 +0.198 1.6 ".27' .'.20} ".'4' .' .069 .0.985 .0.885 .0.756 .0.614 .0.433 +0.224 2.0 +1.205 +1.160 +1.121 .1.073 +1.011 +0.934 +J.819 +0.669 .0.480 .0.251

3.0 .1.074 +1.079 .1.081 .1.075 .1.049 .1.006 .0.919 .0.779 +0.575 +0.310 4.0 ".017 .1.037 +1.053 ".067 .1.069 .1.045 .0.979 .0.853 +0.647 .0.356 5.0 .0.992 ".0'4 ".035 ".056 .1.069 .1.062 ".017 +0.906 +0.703 .0.394 6.0 .0.989 ".003 +1.023 .1.043 .1.063 .1.066 +1.039 +0.943 .0.747 .0.427 8.0 • 0.985 .0.996 .1.\)08 .1.024 .1.043 .1.064 .1.061 .0.997 .0.821 • 0.486

10.0 • 0.992 .0.995 +1.000 +1.011 .1.028 .1.(}52 .1.066 .1.030 .0.878 .0.533 12.0 .0.998 .0.997 .0.997 .1.002 ".017 +1.041 .1.064 ".(}50 +0.920 .0.577 14.0 .1.000 ·0.998 .0.997 .0.999 .1.008 .'.0}1 ".059 +1.061 .0.952 .0.61} 16.0 >1.002 _~~~O~~L- .. J'.~L.-tl.OO3 .. 1.021 +1.0S0 +1.064 .. 0.916 +0.6)6

(a)

Lv' Coefficients at point

2rh 0.1 L v

0.2 Ly O.}Lv 0.4 Lv 0.;: Lv 0.6 L v O.TLv 0.8 Ly 0.9 Lv 1.0 Lv

0.4 +.0020 +.0072 +.0151 +.0230 •. 0}01 •• 0346 •• 0357 •• 0312 •• 0197 0.8 •• 0019 +.0064 +.0133 •• 0207 •. 0271 •. 0319 +.0}29 •• 0292 +.0187 1.2 •• 0016 •• 0(}58 +.0111 +.0177 '.02}7 •. 0280 •. 0296 + .0263 •• 0171 1.6 +.0012 +.0044 •• 0091 +.0145 +.0195 •• 0236 +.0255 +.0232 +.0155 2.0 +.0009 I + .0033 •• 0073 +.0114 •• 0158 +.0199 +.0219 +.0205 •. 0145

3.0

1

+.0004

1

+.0018 I··" 1

•• 0063 •. 0092 +.0127 +.0152 •• 0153 •• 0111 4.0 +.0001 +.0007 +.0016 + .0033 •• 0057 +.0083 +.0109 +.0118 +.0092 5.0 .0000 +.0001 +.0006 +.0016 +.0034 +.0057 +.0080 +.0094 +.0078 6.0 .0000 .0000 +.0002 +.0008 +.0019 + .0039 +.0062 +.0078 +.0068 8.0 .0000 .0000 - .0002 .0000 •• 0007 +.0020 +.00}8 +.0057 +.0054

,., I .0000

I .0000 1"- I"'" I··OO~

•• 0011 + .0025 +.oo.} +.0045 2.0 .0000 .0000 -.0001 -.0002 .0000 +.0005 +.0017 +.00}2 +.0039 4.0 .0000 .0000 -.0001 -.0001 -.0001 .0000 •• 0012 +.0026 +.oon 6.0 .0000 .0000 .0000 -.0001 -.0002 -.0004 +.0008 +.0022 +.0029

(b)

Ft • coef. X o·.r kN per II

(Positive sign indicates t"nsion)

\

r r ~ r 1

[ ::::~~ --0.& .... _.-0-.....

'" -·-0-4 ....

• t -'-0-'''' .;I , -.-0 .• 1. .. ", --0." .... .. -'-0..'" _.-0. .....

. • r'lt • -- ,.oL.'1 ..... ~

PNfED BASE • flEE TOP

p' = Uniform pressure applied vert ica lly around per imeter of tank. (kN per a )

Lv • Effective Height of wall (m)

Effective Radius of cylinder (a)

h * Thickness of cylinder wall (a)

Ft • Ring Tens ion (kN)

It • Itoment (kN 111)

2 Moment * coef. x p'. Lv kNa per.

(Positive sign indicates tension on the outside face.)

Table 9.35 (a) shear at base of cylindrical tanks subjected to: triangular load, rectangular load, mo­ment at edge; (b) stiffness coefficients for cylindrical walls; (c) stiffness coefficients of circular plates with and without centre support (adapted from peA tables)

Lv' Tritlngular load, fixed bas~

Rectangular load, Triangular or Moment fixed base rectangular load l at edge

2rh hinged base

0.4 +0.436 ~O. 755 +0.245 -1. 58 0.8 +0.374 +0.552 +0.234 -1. 75 1.2 +0.339 +0.460 +0.220 -2.00 1.6 +0.317 +0.407 +0.204 -2.28 2.0 +0.299 +0.370 +0.189 -2.57

3.0 +0.262 +0.310 to.158 -3.18 4.0 +0.236 +0.271 +0.137 -3.68 5.0 40.213 +0.243 +0.121 -4.10 6.0 +0.197 +0.222 +0.110 -4.49 8.0 +0.174 +0.193 +0.096 -5.18

10.0 +0.158 +0.172 +0.087 -5.81 12.0 +0.145 +0.158 +0.07g -n.38 14.0 +0.135 +0.147 +0.073 -6.88 16.0 +V.l27 +0.137 +0.068 -7.36

V (triangular load) coef. x p.Lv kN

v (rectangular Load) coef. x p' .Lv kN

(Moment at base) coef. x M/Lv kN

(Positive sign indicates shear acting inwards)

(a)

3 k = coef. x E.h ILv

1 ,

~ ,

v Coeff~:':le,,"t

v Coefficient 2rc 21":-,

S ,l.i 0.130 0.713

['.8 C • '27 ~ 0.783 1.2 (',34:- O.9C3

l.E 1[' 1. 010 7 .0 (I .... '" ~ 12 1.108 3. C! 1- 1.198

".0 O. f~ 3:, 1. ?81

(b)

3 k = coC!f, x E.h Ir

hc!2r D.IS 0.20 0.25

Coef. O,2'jC 0.3')9 0.332 0.358 0.387

;1i thout cent re sup!"ort - Coef. = 0.104

Note - he is diameter of c"lumn head.

(c)

184

Table 9.36 Supplementary coefficients for values of Lv2/(2 x r x h) greater than 16 (adapted from peA tables)

"?' TULE 9/29. ~ 2rIl CooN'lol""(8 at point I

• 75"? .SO"? .85 Ly .%L? .951. I 20 .0.716 .0.65l. +0.520 .0.325 +0.115 2l. .0.746 .0.702 ,0.577 .O.}72

::::~l I 32 .0.782 .0.768 .0.66} .0.459 ~o .0.800 .0.805 .0.Hl .0.5JO .0.217 ~ .0.]91 .0.828 .0.765 .0.59} .0.25l. 56 .0.76} .0.8,a .0.824 .0.6}6 .0.285

TENSION IN CIRCULAR RINGS - TRIANGULAR LOAD nXED BASE - ~REE TO\'

:{ TABLE 9/30a -2rh

Coefficient. at pOint

.75 L T

• 80 Lv .85 Lv .90 Lv .95 Lv

20 "'O.~l;" .0.811 .0.756 .o.60} .. O.}W... 24 .0.816 .0.B}9 .0.79} .0.647 .O.}77 }2 .0.814 .0.661 .0.847 .0.721 • 0.436 40 .0.602 .0.666 +0.880 +O.77P. .0.48} 48 .0.791 .0.864 .0.900 +0.820 .0.527 56 .0.781 .0.659 +0.911 +0.8')2 .O.56}

rENSION IN CIRCULAR RINGS - TRIANGULAR LOIIO PINNEO 8ASE - ~REE TO~

~~ rl>.8LE: 9/33a

2rh Coefficients at point

.75 L y .60 Lv .8~y .9().Lv • 95L.

20 .0.949 .0.625 .0.629 .0.}79 .0.128 24 .0.966 .0.879 .0.694 .O.4}O .-0.149 32 .1.026 .0.95} .0.768 .0.519 +0.189 40 ·'.OL..O .0.996 .0.859 .0.591 .0.226 48 .1.04} +1.022 .0.911 .0.652 .0.262 56 +1.040 ".0}5 .0.949 '0.705 +0.294

fENSION IN CIRCULAR RIN::;S - RECTANGULAR LOAD FIXC:O BASE - FREE ro~

r:-:.:. TI>.8LE 9/34a

2rh Coefficients at pOlOt

.75Ly • SOL? .85Lv .9(£v .9,)Ly

20 .1.062 .1.017 .0.906 .0. ?O} .O.}?4 24 .,.066 ".0}9 .O.94} .0.741 .0.427 }2 ·,.064 .1.061 .0.997 .O.A?1 .0.~66 40 ".052 .,.066 ".0}0 .0.878 .0.5}} 48 +1.<:v.1 ".064 .1.050 ... 0.921) .0.577 56 +1.021 ".059 ., .061 +0,')')2 >O.61}

r1::N3[o~1 1:-1 :IRC::ULI>.R RINGS RE::TANGULAR L')I\O

~INNEf) 3A3E - fREE 1'01.'

LT' TMLE 9/29b

2rh Coi!lt'f'lclenta at point

.60 Lv .85 Lv .90 L,. • 'I5L,. 1.()(J. ..

20 +.0015 •• 0014 '.0005 -.0018 -.006} 24 +.0012 •• 0012 •• 0007 -.001} -.~} }2 •• 0007 •• 0009 •• 0007 -.0006 -.oor.o /"0 +.0002 +.0005 +.0006 -.0005 -.00}2 48 .0000 +.0001 •• 0006 -.ooo} -.0026 56 .0000 .0000 "00Cl4 -.000' -.002}

/'laMENTS IN CYLINDRICAL rANKS - TRII>.NGULAR LOAD FIXED 91>.SE - ~REE TOP

:{ rl>.8LE 9/30b

2rh Coerrloient •• t point

• ?SLy .8a.,. .851.,. .9a-y .95L .. 20 •• 0008 +.0014 +.0020 •• 0024 +.0020 24 +.0005 +.0010 •. 0015 +.0020 •• 0017 }2 .0000 ·.0005 •• 0009 +.0011 .. '.001} 40 .0000 "ooo} •. 0006 +.0011 +.0011 48 .0000 ... 0001 '.00Cl4 •• 0008 +.0010 56 .0000 .0000 •• OOO} +.0007 •• 0008

'---

M3MZNT5 IN:YLINORICI>.L rANKS - TRIMIGULI>.R r,ol>.D PINNEO BASE - l"REE rap

~ 'n8LE 9/33b

2rh Coefficients at point

.80 Lv .851.., .9OLy .951.,. 1.00I..y

20 •• 0015 ,.oo'} ,.0002 - .0024 -.007} 24 +.':0'2 •• 0012 •• 0004 -.0018 -.0061 }2 •• 0008 •• 0009 •• 0006 -.0010 -.0046 40 •• 0005 •• 0007 •• 0007 -.0005 -.OO}7 48 '.0004 '.0006 •. 0006 -.ooo} -.00}1 56 ·.0002 +.0004. ·.0005 -.0001 -.0026

~3'IENTS IN CYLINORICAL rANKS - REC'rl>.NGULI>.R LOI>.D fIXED SASE - fREE TOP

:{ fl>.BLE 9/34b

2rh Coefficient! at point

.75Lv .6OL ., .85L? .9OLy • 95Lv

+.0000 .,0014 •. 0020 •• 0024 •• 0020 24 •. 0005 -+.0010 •• 0015 •. 0020 •• 0017 }2 .0000 •• 0005 •. 0009 •. 0014 •• 00l} 40 .0000 •• OOO} •. 0006 +.0011 +.0011 48 .0000 >.0001 '.OOC4 •. 0008 +.0010 56 ,0000 .0000 •• OOO} +.0007 •. 0006

'\J;ontlfS [,1 -:YLINQRICI\[. rANKS - REC'fl>.NGULI>.R U)""O "nltlEO g""SE - fllEE rop

185

Table 9.37 resulting from

CI1M'

6.000 1.000 7. 000 0.509 8.000 0.164 9.000 0.022

10.000 -O.Oll

9.000 1.000 10.000 0.575 11.000 0.11)9 12.000 0.027 11.000 -0.019

\2.000 1.000 \3.000 0.607 1'.000 0.217 15.000 0.029 16.000 -0.02)

16.000 1.000 17.000 0.63) 18.000 0.232 \9.000 0.032 20.000 -0.026

20.000 1.000 21.000 0.645 22.000 0.241 23.000 0.03) 24.000 -0.028

24 .000 1.000 25.000 0.65. 26.000 0.2'7 27.000 0.034 28.000 -0.029

28.000 1.000 29.000 0.660 30.000 0.252 31.000 0.0)5 )2.000 -0.031

32.000 1.000 )3.000 0.665 34 .000 0.255 35.000 0.035 36. 000 -0.0)1

36.000 i

1.000 )7.000 0.668 38.000 0.258 )9.000 0.0)6 40.000 -0.032

'0.000 1.000 41.000 0.671 '2.000 0.260 43.000 O. 036 44 .000 -0.03)

H.OOO 1.000 45.000 0.67) '6.000 0.261 11. 000 0.0)6 '8.000 -0.0))

48.000 1.000 49.000 0.675 50.000 0.263 51.000 0.037 52.000 -0.033

52.000 1.000 ':>).000 0.617 5 •• 000 0.26. OS.OOO 0.037 56. 000 -0.034

-56.000 1.000 57.000 0.678 sa.c')o 0.265 59.000 0.037 60.000 -O.OJ(

60.000 1.000 61.000 0.679 62.000 0.266 63. 000 u.031 6'.000 -0.034

186

C'..'8fficients for calculating fOices in a conical tank supported at base level, at the base of the cone

0.000 0.06' 0.00 0.017 0.003

O.UOO 0.000 0.050 -3.519 0.03S -2.562 0.014 -1.066 O. 002 -0.192

0,000 0.000 0.0<1 -4.442 0.02; -3.252 0.011 -1. 357 (LOO2 -0.243

1.000 0.056

-0.18b -0.149 -0,066

CTM Cl'l/

4.116 1.617 0.212

-0.?59 --[).Z61

8.HO 3.146 0.)45 0.569

-0.529 _._-------

63.867 11.314 3.239 '.321

-13.81& O.HS -11.473 -0.759

-5.312 -0.697 -~-------

1.000 100.261 14.167 O.OS; 5.239 5.)92

-0.191 -21.389 0.548 -0.152 -17.723 -0.914 -0.068 -8. i'll -0.802

----~--- -----"-----D.OOO 0.000 H~.666 17.0ll O.OlS -5.359 7.6110 6.459 0.025 -3.938 -30.577 0.652 0.010 -1.6.8 -25.29' ~ 1 • 127 0.002 -0.294 -11.626 -1.025

----_. -- ---~-----0.000 0.000 1.000 197.076 19.855 O.OlO -6.273 0.053 10.559 1.525 0.022 -4, 623 -0.195 -'1.382 0.751 0.009 -1.938 -0.156 -3'.184 -1.309 0.002 -0.345 -0.069 -15.677 -1.186

0.000 0.000 1.000 257.'90 22.693 0.027 -7.186 0.053 13.877 8.589 0.019 -5.306 -0.196 -53.805 0.863 0.008 -2.228 -0.157 -44.393 -1.409 0.001 -0.396 -0.070 -20.323 -1.347

0.000 0.000 1.000 325.907 25.530 0.024 -6.091 0.053 11.632 9.6S2 0.017 -5.989 -0.197 -!:7.tJ.46 0.969 0.007 -2.518 -0.lS9 -55.921 -1.669 0.001 -0.H7 -0.071 -25.565 -1. 508

0.000 0.000 1.000 402.325 ;;6.365 0.022 -9.007 0.053 21.825 10.715 O.Olb -6.671 -0.198 -83.50' 1.076 0.006 -2.807 -0.159 -68.769 -1.848 0.001 -0.498 -0.071 -31.4D3 -1.669

0.000 0.000 1.000 486.7·H 31.195 0.020 -9.9\6 0.033 26.453 11.777 0.0)4 -7.352 -0.198 -100.780 1.182 0.00. -3.097 -0.160 -82.935 -2.027 0.001 -0.549 -0.071 -37.836 -1.829 _._----0.000 0.000 \.000 579.16, 3'.031 0.018 -10.825 0.053 31.520 12 .8)9 O. OlJ -8.0.13 -0.199 -119.674 1.289 0.005 -3.386 -0.160 -98.421 -2.205 0.001 -0.601 -O.Ott -14.865 -1.989 --_._----

0,000 0.000 1.000 679.584 36.863 0.017 -11.7)4 0.053 37.023 lJ.901 0.012 -8.714 -0.199 -140.186 1.396 0.005 -3,675 -0,161 -115.225 -2.384 0.001 -0.652 -0,072 -52 .490 -2.1.e

-.-~~-~.------.---0.000 0.000 1.000 788.005 39.695 0.016 -12.6'2 O. as) 42.961 14.963 0.011 -9.395 -0.200 -162. 115 1.502 0.005 -3.965 -0.161 -133.349' -2.562 0.001 -0.103 -0.072 -60.710 -2.308

0.000 0.000 1.000 904.'26 42.526 0.015 -11.550 0.053 H.ll8 16.024 0.011 -10.076 -0.200 -lB6.062 1.609 0.004 -4.254 -0.\61 -152.192 -2.740 0.001 -0.755 -0.072 -69.526 -2.468

(a)

MM' n CMM' x M'

HH' =- CMH' x B'x L'x Cose.(

NM'~ CNM'x Mix Tnn~./ y

T!4' CrH ' x M' x Tan~1 y

TH' CTH' x J,' x H' x Sin ... / y

. t~; .. .. ··~;l~~; ...... ~~;~: .... $ .~;~~'," ••• ~~;~ i"" •••• q • •••••••••• ,,.,.,, •••••• , •• * ••••••••• "' ••••••••

6.0 )1.0 8.1 8.1 4.3 9.0 128.8 19.5 19.5 6.4

12.0 308.0 35.5 35.5 B.6 16.0 7)2.3 61.9 6).9 11.5 20.0 1430.5 100,3 100. ) 14.) 24 .0 2410.7 14407 144.7 17.2 28.0 3920.6 197.1 197.1 20.0 32.0 5848.1 257.5 257.5 22.S 36.0 8311.2 325.9 325.9 25.7 40.0

I

11'07. 7 402. ) 402.) "l8.S H.O 15-175.6 486.7 486.7 )l.3 4B.O 19692.) 579.2 579., H.2 52.0 25026.8 679.6 679.6 37.0 56.0 31246.0 788.0 788.0 39.8 60.0 384l8.1 904.4 904.4 42.7

(b)

Table 9.38 (a,b) coefficients for calculating forces in a conical tank supported at base level, resulting from fixity at the apex of the cone

x eMIl CIM CTtt

6.000 1.000 0.000 6.390 5.000 0.1S3 LUI -O.Ut 4.000 0.333 0.931 -1.237 3.000 -0.008 0.544 -0.614 1.000 -0.213 0.332 -0.003

9.000 1.000 0.000 17.277 1.000 0.1S' 1."0 0.377 7.000 0.344 1.50. -3.255 6.000 0.049 0.723 -2.263 5.000 -0.082 0.201 -0.865

12.000 1.000 0.000 32.710 11.000 0.751 2.603 1.219 10.000 0.33) 2.044 -6.205 9.000 0.050 0.936 -4.627 8.000 -0.066 0.209 -1.919

16.000 1.000 0.000 60.297 15.000 0.741 3.543 2.712 14.000 O.)H 2.747 -11.567 1).000 0.0'1 1.225 -1.921 ll.OOO -0.057 0.247 -3.845

20.000 1.000 0.000 ,s.an 19.000 0.734 4.4" 4,63. 18.000 0.313 3.441 -11.554 17.000 0.046 1.515 -14.541 16.000 -0.052 0.292 -6.350

24 .000 1.000 0.000 139.451 23.000 0.729 5.319 6.917 22 .000 0.)08 4.129 -27.160 21.000 0.045 LiaS -21.415 20.000 -0.049 0.3.0 -9.447

28.000 1.000 0.000 191.023 27. 000 0.724 6.304 9.776 26.000 0.304 4.815 -37.383 25.000 O.OH 2.095 -29.740 24.000 -0.048 0.390 -13.138

)2.000 1.000 0.000 250.581 )1.000 0.721 7.217 13.003 )0.000 0.301 5.499 -U.2H 29.000 0.044 2.384 -)9.311 28.000 -0.046 O.HO -17.424

36.000 1.000 0.000 318.145 )5.000 0.718 8.128 16.671 14.000 0.298 6.U2 -62.674 33.000 0.043 2.673 -50.201 )2.000 -0.045 0.490 -22.)05

40.000 1.000 0.000 393.727 )9.000 0.716 9.038 20.779 38.000 0.296 6.164 -77.747 37 .000 0.043 2.963 -62.411 36.000 ·0.0"''' 0.541 -21.182

H.OOO 1.000 -0.000 477.335 43.000 0.114 9.948 25.324 42.000 0.294 7.545 -94.444 41.000 0.043 3.252 -75.945 '0.000 -0.044 0.592 -33.857

48.000 1.000 0.000 568.963 '7.000 0.713 10.857 30.299 46.000 0.293 8.227 -112. 767 45.000 0.042 ). 542 -90.80) 44.000 -0.043 0.643 -40.529

52.000 1.000 0.000 668.593 51.000 0.712 11.767 35.699 50.000 0.292 6.909 -132.717 49.000 0.0'2 J.811 -106.984 48.000 -0.043 0.694 -47.797

56.000 1.000 0.000 776.201 55.000 0.71l 12.676 41.524 54.000 0.291 9.590 -154.2B9 5),000 0.042 4.121 -124 .484 52.000 -0.042 0.745 -55.660

60.000 1.000 0.000 891.763 S9.000 0.710 13.584 47.777 5B.OOO 0.290 10.271 -177.473 57.000 0.042 1.410 -143.297 56.000 -0.042 0.796 -H.1l6

(8)

ell. Cit.

0.000 1.000 -0.164 0.202 -0.159 -0.017 -o.ou -0.114

0.027 -0.101

0.000 1.000 -0.110 0.114 -0.099 -0.163 -0.052 -o.u. -0.013 -0.092

0.000 1.000 -0.082 0.096 -0.070 -0.114 -0.035 -0.170 -0.001 -0.017

0.000 1.000 -0.061 0.010 -0.050 -0.194 -0.024 -0.172 -0.005 -0.084

0.000 1.000 -0.048 0.072 -0.039 -0.199 -0.011 -0.111 -0.004 -0.012

0.000 1.000 -0.040 0.068 -0.032 -0.200 -0.015 -0.171 -0.003 -0.080

0.000 1.000 -0.034 0.065 -0.027 -0.202 -O.OU -0.170 -0.002 -0.010

0.000 1.000 -0.029 0.063 -0.02) -0.202 -0.010 -0 .170 -0.002 -0.019

0.000 1.000 -a.ou 0.062 -0.020 -0.202 -0.009 -0.170 -0.002 -0.078

0.000 1.000 -0.02) 0.061 -0.016 -0.203 -0.008 -0.169 -0.002 -0.078

0.000 1.000 -0.021 0.060 -0.016 -0.203 -0.007 -0.169 -0.001 -0.078

0.000 1.000 -0.019 0.059 -0.015 -0.203 -0.007 -0.169 -0.001 -0.077

0.000 1.000 -0.018 0.059 -0.014 -0.203 -0.006 -0.16B -0.001 -0.077

0.000 1.000 -0.017 0.059 -0.013 -0.203 -0.006 -0.168 -0.001 -0.077

0.000 1.000 -0.015 0.058 -0.012 -0.203 -0.005 -0.168

0.001 077

( 1 -

CTB

-3.561 -1.182 -0.109

0.121 0.054

-5.861 -2.082 -0.239

0.248 0.206

-8.076 -2.924 -0.329

0.391 0.341

-10.973 -4.017 -0.4)9

0.519 0.5ll

-13.141 -5.095 -0.546

0.763 0.678

-16."4 -6.167 -0.653

0.946 0.841

-19.540 -7.236 -0.761

1.127 1.004

-22.382 -8.302 -0.866

1.307 1.165

-25.221 -9.367 -0.975

1.486 1.326

-28.059 -10.431 -1.081

1.665 1.487

-)0.896 -11.495 -1.188

1.845 1.647

-)).732 -12.557 -1.294

2.024 1.807

-36.565 -1),618 -1.401

2.202 1.967

-39.396 -14.680 -1.507

2.381 2.127

-42.226 -15.741 -1.615

2.556 2.286

KK • CKM x '4

MH 2 CKH x H x L x Cos ..

NK • CNK x M x TanllC / 'i

NH • CNH x L x H x Sin .. / 'i

TK • CTM x M x l'an ... / 'i

TH 2 Cl'H x H x L x Sin ... / 'i

.......... fI .................................... .

... ! ...... ~!~~ ......... ;~~ ...... ;;~~ ..... ;~~~ .. , 6.0 32.~ 9.0 117.9

12.0 287.6 16.0 695.1 2t;.O 1311.9 24.0 2)85.6 28.0 3804.2 32.0 5695.4 36.0 8127.0 4\1.0 11161.3 44.0 \4884.3 4LO 19346.5 52.0 44621.8 56.0 30771.6 6~ a 37681.2

6.4 17. ) )2. 7 60.3 95.9

139.5 191.0 250.6 lI8.1 19).7 477.3 569 .0 668.6 776.2 891.8

(b)

6.4 17.) J2.7 60.3 95.9

139.5 191.0 250.6 lI8.1 393.7 477. J 569.0 668.6 776.2 891.6

3.4 5.7 7.9

10.6 11.7 1 •• 5 19.4 22.2 25.1 21'.9 )0.1 n.6 36.4 39.2 42.1

187

I. TANKS ON PLASTIC FOUNDATIONS

The analysis of open circular concrete tanks is explained for cases where the ground support may be assumed uniform. A tank ha ving it, wall monolithic with its circular base is first analysed for bending moments and hoop tensions; the effect of extending the base into a peripheral toe is then considered.

The structural action of a tank having a flexible annular joint in the base is next explained, for tanks with and without a base toe extension. A simple "retaining wall" type of analysis is demonstrated for this mode of construction. The accuracy of this method of analysis has been checked by comparison with an exact method, from which it is found that the simple method is quite adequate for radial moments in the base) though it neglects any account of tangential moments, which may be appreciable.

Several tables are given for circular tank walls and base slabs, 10 enable analY3es to be achieved for all the cases considered.

(Pari 1)

civil engineers today are familiar with the behaviour of circular tank walls under hydrostatic pressure, with

bonom edges either hinged or fixed. The interaction of tank which afe made continuous with circular ba_se slabs is, however, seldom considered in any de-tail in the design office. There is linle information available on this type of analysis in this country, though some engineers are now acquainted with the Portland

• Fellow of Pembroke Col/ege, Oxford. t Assislanl Engineer, ave Arup & Parmers. t FigureJ in parenlheses indicate RtftrenceJ lollowing the Grlicll

Cement Association booklets on the analysis and design of concrete tanks. (These may be obtained very conveniently from the Con­crete Association of India in a single booklet entitled "Reinforced 2nd Prestressed Concrete Tanks" (1):1:.)

The methods described in the P.C.A. booklets are based on the asswnptions that such tanks are supJXlrted either at their edges or on completely plastic foundations. Now the reactive pressure developed by a soil under leading at ground level depends on severa'l factors, but two simple approximations may be used for the purposes of this type of analysis: (a), that of completely plastic behaviour, when the ground pressure is independent of the deflec­tion at the surface, and (b) that of simple elastic behaviour, when the ground pressure, p, is related to the surface deflection, w, by p = kw, where k is the "foundation modulus" of the ground and is taken as a constant. Actuaily k may not be constant even in one particular loading test, and the moisture content, the com­paction, etc., of the soil may affect the value considerab'ly. The foundation modulus does, however, increase with the hardness of the ground and some correlation has already been attempted. For the purposes of this paper asswnption (a) will be accepted, and the consequences of accepting asswnption (b) will be considered in a later conuibution.

It is perhaps worth mentioning that whenever a rationalisation of analysis and design is desired some simple asswnptions must be made. Even with the two assumptions mentioned above the analysis of ground-supported circular tanks is involved enough for any design office, so there seems little likelihood of any future code of practice recommending anything more complicated. It is to be hoped that

TABLE I

VERTICAL BENDING MOMENTS AND RING TENSIONS IN A CIRCULAR TANK WALL WITH THE BonOM END HINGED WHEN THE TANK IS FULL Of LIQUID

M,=cyH' T=~ rHR (lop)

xH 1)=0 0'1 0·2 0') 0-4 I 0'5 0·6 i 0·7 0·8 0'9 1·0

-0'0022 -0·0079 -0,0154 -0'0234 --0'0306 -0-0354 I -0'0361 -0'0315 i -0'0200

-0'0016 -0'0058 -0,0115 -0'0180 -0'0240 -0'0283 --0-0296 -0'0265 -0'0173

-0-0006 -00021 - 0-0047 -0'0079 -0'0116 - 0·0150 -0'0172 -0-0170 -0-0121 0

--0,0002 --0-0008 -0'0020 -0'0039 -0-0063 -0,0088 -0,0100 - 0·0081

+0·0002 +0·0001 -0'0001 -0'0010 -0,0025 -0,0045 -0·0062 -0'0058

i +0-0001 +0·0002 +0·0002 i -0'0001 --0'0008 -0,0023 -0,0039 -0·(}(}44

+0-488) "-0,4427 +0·3971 +0·3511 +0·3045 +0'2569 +02082 +0·J58O +0'1064 +0'0536

+0'3516 +0·3574 +0·3621 +03630 +0·3565 +0'3384 +0·3052 +0·2540 +0·1840 +0'0973

+0·1123 +0'2037 +0-2931 +0·3754 +0-4424 +0'4837 +0-4877 +0·4435 +0'3439 +0·1905

-0'0038 +0-1184 +0·2402 +0'3595 +0'4698 +0'5590 +0·6079 +0-5922 +0·4877 +0·2829

-0,0167 +0·0963 +0·2110 +0·3295 +0·4509 +0'5671 +0·6571 +0·6846 +0'6014 +0·3677

-0,0061 +0'0956 +0·2006 +0·3076 +0-4248 +0'5495 +0·6670 +0·7376 +0-6909 +0·4471

188

detailed IUidance on the ute of these amnnptions will be forth- T coming, in due course, from further reteareh on IJ'OUIId behaviour.

When the ground is supposed to be comp1eteIy plastic, or when water uplift can occur, it is found that the upward bending developed at the centre of a circular bue can be unduly large for tanks with diameten over 30ft or so. This reverse bending can be reduced if the base i. continued outwards beyond the tank walls, and completely removed if a central section of the base is made independent of the rest by means of a suitable joint. When such joints are utilised II simple "retaining wall" method of design can \Ie applied. This method is described here in some detail, for it is extremely useful for design and seems to be little known. It can be just IS easily used when the base is continued outwards to form

= tenaile force in a tank wall, per unit depth. = 'y(k/4EI), also the root of the ditferential equation

a "toe", 15 in the "inverted Tee" type of retaining wall. Tables are required far ease in handling this type of analysis.

iufficient tabulate. Villues are included in the article for the solu­tion of circular tanks continueus with circular lIases, which may be with or without projecting "toes" and internal annular joints. The simple "retaining wall" methe4 of analysis is also campared with _an exact treatment. In the examples the same main tank dimen­~iGm are taken throaghout so that the effect of structural modifica­tions can be appreciated and so that the different methods of analysis CaR be compared.

P k TU

I I

E v D x,l K h,H '1 ",R pv, P~

Notation = loading or pressure. = foundation tacti .. modulus. = narmal 4elleclion of a beam. = thickness of a wall or slab. = second moment of area of a wall or base section,

per unit width, = t'/12. = Young's mMuius. = Poisson's ratio. = plate rigidity, = EI/(l v'). = distance along a beam, length of a beam. = EI/l. = distance down a tank wall, total height of wall. = h/H. = radius of a disc, radius to centre of tank wall. = loading carried by vertical beam and horizontal

ring action in a tank wall. = circumferential and radial strains. = horizontal tensile stress in a tank wall.

for a tank wall. M" M" M. = bending moments; radial, tangential, vertical. MF = fixed-end moment. s, s' = rotationai ~tiffness and stiffness coefficients, s = s'. c, '" = coefficients for bending moments land ring tensions. a, b = "restrained" and "unrestrained" r,adii of an annular

slab.

Circular Tank Wall. of Con.tant ThiclmCIIIII Subjected to Hydroatllltic Preuure

If the water pressure at any depth It of a circtilar tank i. written p = yh, where y is the specific gravity of the contained liquid, then part of this pressure is misted by vertical I bendilll and the rest is cartied by horizontal ring beam action. Fbr a vertical strip of wall,

D(d'w/dx') = ,. = P ~ ..•. (1) where D is the plate rigidity and equals EI'/1 1 - y'), E and v being Young's modulus and Poisson's ratio, res clively, for the material of the wall and I the wall thickness. so ,. and ,~ are the vertically- and horizontally-supported componems of the pressure p.

Now the ring tension in a horizontal strip of unit 4cpth u4 thickness I is, from symmetry, a constant at any particular depth h and equals T = PAR, where R is the tank radius. Thill the horizontal tensile stress is I, = T /1 = pA(l(/I), and the circum­ferential and radial strains are

fc = f, = f,/E = PA(R/Et). The outward radial deflection is therefore

w = £,R = P~(R2/EI) whence p~ = Elw/R'. Thus, from equation 1,

. ... (2)

D(d'w/dx') + (IE/R2)W = P = yh .... (3) This equation is identical in form to the familiar beam on elastic foundations equation

EI(d'w/dx') + kw = P ..•. (4) for which standard solutions are available in terms of the parameter ,\. = '\/(k/4EI). In the case of the circular tank wall,

A' = IE/4R'D = 3(1 - v')/R't> and the dimensionless parameter AH, where H is the total depth of the wall, is

,\.H [Hf\I(Rt)] 'y[3(1 - v'] .... (5)

TABLE II

VEItTICAL BENDING MOMENTS AND RING TENSIONS IN A CIRCULAR TANK WALL WITH THE BonOM END FIXED

WHEN THE TANK IS FULL OF LIQUID

(top)

AH ')=0

+0-0995

+0-2853

+0-1671

+0-0349

-0-0094

-0-0090

0-1

-0-0003

-0-0012

-0-0008

-0-0002

+0-0000

+0-0000

+0-0875

+0-2694

+0-2199

+0-1426

+0·1066

+0-0976

Mv=c_yH'

0-2 :

-0-0005

-0-0042

-0-0027

-0-0008

+0·0000

+0-0001

+0-0754

+0-2527

+0·2699

+0-2480

+0-2230

+0-2061

0-3 i 0-4 0-5 1 +0-0006 ! +0-0040 : +0-0109 i

-0-0076 -0-0104 -0-0113

-0-0054 -0-0081 - 0-0102

-0-0019 -0-0034 - 0-0051

-0-0003 -0-0009 - 0-0020

"0-0001 -0-0001 -0-0007

+0-0634 ! +0-0514 +00395 i -t 0-2332 +0-2090 +0·1782

+0-3110 +0-3346 +0-3322

+0-)448 +0-4219 +0-4640 :

+0-3383 +0-4454 +0-5277

-j 0'3189 +0-4352 +0-5438

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

T=</> yHR

0-6 0-7 I 0-8 ! 0·9 1-0

+0-0224 + 00397 . 1 + 0-0637 I + 0-0957 +0-1366

-0-0089 -0-0019 +0-0111 +0-0315 +0·0608

--0-0105 -0-0078 --0-0004 +0-0136 +0-0362

- 00064 : --00062 -0-0027 +0-0063 +0-0134

- 0-0034 00042 -00030 +0-0027 +0-0160

0-0016 -0-0027 -0-0026 +0-0009 +0-0116

" 0-0281 +0-0176 : +0-0087 +0-0024

.0-1403 ·0-0970 +0-0528 +0-0161

+ 02973 +0-2292 +0-1370 +0-0452

+0-4541 ~0'3799 +0-24t9 +0-0861

-j 0-5586 ~O-506S +0-3537 +0-1351

+0-6160 +0-6035 +0-4570 +0-1888

189

m VERTICAL BENDiNG IN CH~.CULAR WALL DUE TO BoTTOM EDGE

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

Mv=ctM

,H ~=O

-0-003 0012 -0-028 - 0-049

+0-001 0-001 -0-008 --0-025

+ 2-846 ~ 2'600 -+ 2-354 '-2-106 +1-853

+ 1-090 .;. 1-800 +2-134 ..,.2-426

-1-514 +0-641 - 1-779 +2-976

-1-654 -0-331 +0-629 +2-047

-0'455 -0-750 -0-553 +0-347

+ 0-242

The parameter H'/ZRI is used instead of >"H in the P.C.A_ tables, with v taken as 1/5. If coefficients are tabulated against >"H, however, it is easier to allow for any particular ,- value. Often a value of ,. 0 is assumed in the design of concrete structures_

The ven,ical moment per unit width of the tank and the hori­zontal ring tension may be written

M. = eyH" and .. (6) T = rj>yHR . (7)

where c and rj> are coefficients which depend onJy upon the ratio 'I = h/H and on ,\H and can easily be tabulated. Values are given in Tables I and II for circular walls of uniform thickness under hydrostatic pressure to the lOp, with the bottom edge hinged and fixed.

When the wall of a circular tank is continuous with the base it may be subject to an additional edge moment, the effect of which can be obtained as a further solution to the basic differential equa­tion 3 above. The resulting vertical moment and ring tension can be expressed in terms of the edge moment as follows:

M,. = e,M and . (8) T = rj>,MR/H' .. (9)

these coefficients being tabulated in Table III. The final moment acting at the boltom edge of a circular

tank wall can be found as the sum of the fixed-edge moment and a further edge moment obtained by moment distribution. Before

TABLE IV ROTATIONAL Snn"'NESS CoEFFICIENTS fOR THE BoTTOM EI.XJE OF A CiRCULAR TANK WALL. WITH Tllf Top EDGE FREE

->-H 0 I 2

i 3

5t 0 1-017 3-915 J 5'921 12-000

190

TABLE V RADIAL AND TANGENTIAL MOMENTS IN A CIRCULAR SLAB FIXED AT THE EOOES AND SUBJECTED TO A UNIFORM LOADING (POISSON'S

RATIO TAKEN AS 0-2)

r R centre

M, --0-075

-0-075

T=9, MR/H'

0-5 0-6 (}7 0-8 0-9 1-0

+0-304 +0-423 +0-555 +0-697 +0-847 +1

-+ 0-209 +0-319 +0-456 +0-619 +0-S03 +1

+0-038 +0-124 '1-0'261 +0-458 +0-710 +1

-0-052 -0-004 +0-109 +0-313 +0-617 +1

-0-065 -0-056 +0-016 +0-199 +0-532 +1

-0-049 -0067 -0-037 +0-109 +0'453 +1

+1-S92 + 1-318 + 1'028 +(}7IS

+2-636 +2-711 +2·581 + 2-068

+4-184 ~ 5-257 +5-916 +5-713 +4-011

+4-057 +6-566 +9-065 + 10-366 +8·)77

+2-459 +6-159 + 11-1.13 + 15-481 +14·539

+0-491 +4-406 -+ 11-589 +20-213 +22-312

out-of-balance moments can be distributed at a joint, however, the rotational stiffnesses of the connected elements must be known. For a beam AS on elastic foundations, with the end A fixed in position and the other end free, the rotational stiffness at A is found to be s,," 2.\l(sinh >..l cosh >,,1 - sin >..l cos >,,1)/

(cosh' >..l + cos' >,,1)]K.b •• __ (10) where K,," is the El/l value for AB. For the wall of a circular tank D/H replaces El/I and >"H is found from equation 5, to give

- the rotational stiffness per unit width. If the expression above is written s,," s',.K. b , where s',. is noy, defined as a stiffness coefficient, then the value of s',. can be calculated for various >"H values and recorded as in Table IV.

Circular Base Slabs of Constant Thickness Subjected 10 Uniform Loading

The bending of plates is governd by the well-known differential equation \l 'w p / D, where the deflections are small compared to the thickness and the thickness is small compared to the -dimen­sions of the pla,e in its own plane_ For a uniform circular plate, with axial symmetry in loading and restraint, this equation may be written in terms of the polar co-ordinates r, 8 as \l'w (o'/i1r' + (l/rXo/~r)l (f1'w/ilr' (l/r)(ow/or)] -- (l'v.'/~r' + (2/r)«(l'w/ilr') -- (l/r'XiI'w/ilr') + (1/r')(2w/iir)

= p/D . ., (11) When the loading p is uniform and the edge of the circular plate is fixed this equation solves to give

M, (pR'/16)( -(1 + ,) + (3 + vXr'/R')] .. (12) and

M, (pR'/16)[ -(1 + ,) + (1 + 3v)(r'/R')] .. (13) for the radial and tangential bending moments per unit width respec­tively_ Here r is any radius and R the full radius of the circular plate_ As would be expected, M, = M, at the centre of the base and M,/M, = " at the edge. When Poisson's ratio is taken as 0'2 the values of M, and M, can be expressed in terms of pR' for various r/R ')' values, as shown in Table V.

For the special case p 0, equation 11 admits of a -very simple solution, w Cr', which corresponds to unit edge moment only_ Then M, = ZCD(l + ,.) and the slope at the edge is (J ZCr. Thus the rotation stiffness at the edge is

S. = M,/() = D(l + ,-)/R .. (14) Thus s. l'ZD/R, when v = 0'2.

(To be Continued)

REFERENCE (I) Portland Cement Association booklets, reprinted as "Reinforced and

Prestressed Concrete Tanks." Concrete Association of India, 1953.

(ParI 2) Til" first parI of this article appeared IN December 19tH.

Siructural Bf'hllviour of a Circular Tank on a Plaslie Foundalion

Example 1. A circular tank has a mean diameter of 40ft and an internal depth of 16ft. Its wall and base are 10 inches thick and are continuous in construction. The bending moments and tensile forces are required when the tank is full of water and supported on a plastic foundation. Poisson's ratio is to be taken as 0'2, and the densities of water and the concrete in the tank are 62'4 and 150 Ib/cu.ft respectively (Fig. I). (This problem was solved by Amin Ghali (2)i for, = 1/6, during his Ph.D. studies under the senior author's supervision.)

The value of AH for the tank wall is calculated from equation 5 as

(16'42/(20 X 5/6)] '\/[3(1 0'04)] = 5'24. The lixed-end moment at the bottom of the tank wall, at the inter­section with the centre of the base slab, may be found from equation 6, with c obtained from Table II by interpolation. Thus

M/' = 0'0150 x 62'4 X 16'42' = 4,140 Ib.ft/ft.run. Also, the rotational stiffness of the tank wall at the base per

unit width is found from Table 4 as 5,. s,'D/H~(2AH)D/H = 10'48D/16'42 0·638D. If the horizontal continuity of the ground is neglected (as is

usual in simple ground reaction theories), then any uniformly­distributed loading on a plastic foundation induces an equal upward pressure and there is no resulting bending. Now a circular tank full of water would be such a uniformly-distributed loading if the tank walls had the same specilic gravity as water. It is evident that the resultant upward pressure on the base is due to the additional weight of the tank walls over that of water of the same volume. In this case the effective pressure causing bending is p. = (2". X 20 X 5/6 X 16 X (150 62'4)]/[11' X (20'42),]

= 112 Ib/sq.ft. The Iixed-edge moment on the base at the junction with the wall is obtained from Table V as

Mol" = 0'125 X 112 x 20' = 5,600Ib.ft/fLrun. Also, the rotational stiffness of the base slab at its edge per

unit width is found from equation 14 to be ,"~ (D/R)(l + 'J = 1'2D/20 0·06D.

The plate rigidity D equab E('/12(1 - ,.-). In {his example the wall and base thicknesses are equal, so the D values are the same.

• Fcllou' "f Pcml"."kc Collcr:c. OxforJ t AUH/tllIl [!ngllll.:cr. Ot'C A/llp <.:.:.. Pal tlll..'/ \

i. FIJ,:wCJ III pal"clllhc\«( ",d,I..',,,1..' R.:foI..'/lll..'\ /1,IIIIlL'III': lIlt' (Hilde

The Iixed-end moments at the base of the wall and at the edge of the base slab act in the same direction, as shown in Fig. lb for the left bottom comer of a cross-section through the tank. The IOtal out-of-balance moment there equals +4,140 + 5,600 +9,740 lb.ft/ft.run for equilibriwn, therefore, a balancing moment of -9,740 lb.ft/ft.run has to be distributed according to the stiff­nesses of the wall and the base, that is, in the ratio 0'638: 0'060. The distribution factors are therefore 0'914 and 0'086 for the wall and the base respectively and the distribution proceeds as shown bel{lw, without any carryover of moments.

Distribution facwrs Fixed-end moments Balance

Wall 0'914

+4,140 -8,900

Bas, 0'086

+5,600 -840

Final moments -4,760 + 4,760 lb.ft/fuun. The linal bending moments in the tank wall can now be calcu­

lated quite easily, by use of intermediate coefficients from Tables II and III. Alternatively the coefficients in Tables I and III can be used, if the balancing moment is taken as -4,760 instead of - 8,900 lb.ft/ft.run, as from the hinged rather than from the fixed condition. The first method is used in Table VI.

The ring tension in the tank wall can be calculated in a similar way by means of Tables II and III again, or by means of Tables I and III. The first method is used in Table VII.

The radial and tangential moments in the base slab can be calculated for the fixed-edge case, by means of the coefficients in. Table V. Now {he effect of a uniform radial moment on the edge of a uniform circular plate is to produce spherical bending, in which case the radial and tangential moments throughout the plate are constant and equal to the applied edge moment. The calcula­tion of final moments is very easily achieved, as shown in Table VIII.

The linal vertical bending moments and ring tensions in the tank walls, and radial and tangential moments in the base slab, are shown to scale by means of full lines in Fig. Ie. The correspond­ing values for the base of the tank wall being hinged are shown by means of dotted lines. It will be seen that the effect of the con­tinuity of construction is to im;rease the bending moments and ring tensions in the tank wall and over most of the base. The bending of the base slab forces the tank wall to deform outwards beyond the hinged base condition.

EIT .. ct of EXI .. nding Ihe Bue of a Circular Tank The outward rotation of the bottom edge of the circular tank

c:onsidered in Example 1 can be reduced or even reversed if the base is extended outwards beyond the tank wall. Such an "annular

TABLE VI C'\I ('lit" liON elf f-IN'\l VI K' H Al UtNUlN(i MO\tEN rs IN THE TANK VwI"lL IN EXAMPLE I

-------------Distance rrom lOp edge

Coefficients rrom Table II -_ .. _---- --Fi.ed-base tank

Coefficients rrom Table III ·--1-

o 2H

00000

-0009

04H 0·6H

-00007 -00030

-190 -830

-0043 -0·059

0'8H H

-0·002~ -0·0150

-800 4.140

0·177 - -- - - --- - --1-------1---------1-------1-------

Effect or balanCing moment 80 380 530 -1,580 -8.900

Final moments 80 190 -30J -2,380 -4.760

(moments to Ib.rt rt run)

191

(values in ibift run of depth)

toe" is subjected to a comiderably greater uplift pressure effective value calcul2ted before and the fixed-end moment from this acts against the tixed-end moment at the bottom tank wall and at the edge of the inner of the circular Thr effective uplift pressure on the part of the base be reversed if the toe was made large enough, and would further increase of fixing moment to the wall. TIle effect an extension to the base is most easily appreciated in a example. Example 2. The so that the outer required to find moments acting modification.

analysed above has its base extended 43ft shown in Fig. 2a. It is

in the walls, and the as result of this

At the bottom of the tank wall the fixed-end moment and the rotational stiffness are the ll8.Ille as before. The effective upward pressure on the inner part of the base is now due to the differential weight of the tank walls less the weight Qf an annular ring of water over the projecting toe, for the full depth of the tank. ThUll, Pb = [2". X 20 X 5/6 X 16 X (150 - 62-4) - 2". X 20'96

X (13/12) X 16 X 62'4]/[.". X (21'50)'J = (46,700 45,400)/(21'50)' = 1,300/461:::::: 3 Ib/sq.ft.

There is thus a sma\l fixed-end moment at the edge of the inner part of the base equal to 3 X (20)'18 = 150 Ib.ft.

The net upward pressure acnng·on the projecting part of the base is seen to be that due to the depth of water il1llide the tank plus the effective upward pres.,ure, po, calculated for the inner part of the base, In this case, therefore, the upward pressure on the toe is 16 X 62'4 + 3 .= 1,001 ib/sq.ft as ahown in Fig. 2b. Now a vertical strip of the waH of unit width is cOl1llidered in the distri­bution of moments, and a lit width lit 20ft extends to l'048ft at the middle of the projecting toe, at 20ft 1Hin radius. The total upward force on the toe, for unit width of the centre of the lank wall, is therefore

1'048 X (13/12) X 1,001 = l,136lb

.g;~' it1 t---'-- """,to­

(a)

Mt (c)

/WI Uplilt 112 b;"'(I,

(1))

and the moment of this force equals 1,136 X (111/12) = 1,090 Ib.ft (clockwise).

Thus the fixed-cnd moment equals -1,090 Ib.ft (anti-clockwise). It is asswned that the centre of the joint is at the intersection

of the centres of the wall and the base. For the Win width of the wall there is a uniform resultant downward pressure, equal to the excess pressure of the wall above the uniform upward reaction under the base. This provides two equal and opposite moments of

1,399 X (5/12) X (2;;12) = 1201b.ft about the joint centre, thus reducing the fixed-end moments in the base and tbe toe to 30 lb.ft and 970 Ib.ft respectively. This ~mall modification might well be neglected in design.

The stiffness of a free cantilever is known to be zero even if its width is variable, as in this case. It should be noted, however, that a cantilever on an elastic foundation might have an appreciable rotational stiffness, depending on tbe k/EI value. Here, of course, the foundation is plastic and corresponds to a value of k/ EI equal to zero, The restraint of the sides of the cantilever agail1llt rotation, as in this case, can be partly allowed for by taking D in place of EI; but this cannot, in itse'lf, alter the zero stiffness value which applies here, The obliquity of the adjacent sides of the cantilever will, however, cause some rotational stiffness value to develop, but this will be neglected.

The fixed-end moments can now be distributed as before with the same distribution factors operative. Thus:

Distribution factors Fixed-end moments Balance

Toe Wall Base o 0'914 0'086

- 970 + 4,140 + 30 o - 2,920 - 280

Final moments - 970 + 1,220 - 250 lb.ft/ft The vertical moments aod ring tensions in the tank walls, and

the radial and tangential moments in the base, can now be calcu­lated as in Example 1. The final values are sketched in Fig. 2c, which should be compared with Fig. lc for the same tank without

,<1 ~

Ib.k/ft."idln "'./1 •. Mv

:ao i T

j '¥<><> ,,;0 /30

/220 M, it Me (e)

Pig. I. Exampl. 1. Fig. 2. Exampi. 2. (I) kntavre. (hJ loi4lft, aM ftX1ld-end fTIOf'ft4f\Ui. (c) Ven.ral "Mint FnOl'MtltI and rinl tension) in tank wall; r&diaJ aM t::l1nJontial mom.Pltt in slab ban.

192

Effect of lilit A""lular Joint in the -Approximate "Retaining

Construction joints in tanks can be effectively wliterproofed by various methods (3). They can be rnade to transmit moment and/or shear or transmit no structural action at all. It is

practice to construct the cemral part of a circular tank, rests on the ground, at ~ reduced thickness and without any

structural continuity Wilh the outer part of the base and the tank wails. There is then no bending induced in the central part of the floor, which is required only to be waterproof. Special care is required with the annular joint to prevent water seepage and to ensure that no structural interaction occurs across it.

The outer part of the base is usually made continuous with the tank wall, though a construction joint at the bottom of the w~ll would be quite practicable. The outer part of the base may be extended beyond the tank wall as in the "inverted Tee" type of retaining wall. It is evident that the outer parr of the base will both deflect and rotate, in the general case (though it might be possible to design for zero rotation). Now with a plastic foundation the ground pressure is a constant value regardless of the deflection and can easily be calculated in this case from the vertical equilibrium of [he outer part of the base. The fixed-end moment for the projecting toe, per unit width of vertical strip, can therefore be calculated as before. For [he outer part of the base within [he wall the fixed-end moment can be calculated in a similar manner, assuming cantilever action as for the toe. More exactly, in both cases, the theory for annular plates needs to be used-as explained later.

With [he fixed-end moments for the toe (if any) and the outer part of the base inside the tank walls thus calculated the total out­of-balance moment at the junction of the wall and the base can be obtained, since the fixed-end moment at the foot of the tank wall is the same as before. The balancing moment needs to be distri­buted according to the rotational stiffnesses of the connected elements. It can be assumed, as before, that the parts of the base have zero rotational stiffnesses, in which case the balancing moment is allotted directly to the bottom of the lank wall. (This procedure would, of course, be invalid for the straight retaining wall, but is

(0)

because the ring tension in the lank wall prevents

tank shown in Fig. 1& is to be analyserl 30Droxilma"F theory aoove on the supposition that

at a radius of 14ft llin so that the inner independently (Fig. 3a). The

plastic. The vertical bending lanle wall, and the radial moments

are required ioe design. upward bending pressure on the inner

outer p&rr this effective pressure is found

''''.1<1'''1'''''",· - 14'92') 46,700/194 240Ib/sq.ft. Now from a 1ft wide strip at the centre of the tank wall are 0'98ft apart at 19ft 7in radius, aod 0'75ft apart at 14ft 1 lin radius. The effective upward force equals

4'66 X 0'86 X 240 = 9641b, and its moment about the joint is

964 X (20 17'33) = 2,570 Ib.!t. Thus the fixed-end moment for this part of the base is + 2,570 Ib.fr. This means that the fixed-end moment at the bottom of the tank wall is -2,570 Ib.ft. In Example 1 tbe final moment at the bottom of the tank wall equalled - 4,760 Ib.ft, so this new value may be considered an improvement, though it still· means that the tank wall is subieClcd to more vertical bending and ring tension than in the hinged condition. The vertical bending moments and ring tensions in the tank wall are calculated as in the previoUII exampl~s. The resulting values are shown in Fig. 3b, along with the radial moments In the outer part of the base, which are found by simply taking moments. Example 4. The circular tank shown in Fig. 23 is modified by the insertion of an annular joint at 16ft radius, as shown in Fig. 3a. The outer part of the base is thus still 5ft 6in wide, as in the previoUII example, but it now includes a projecting toe as in Example Z. It is to be analysed, as before, when full of water and supported on plastic ground.

In this case the effective upward pressure on the outer part of the base inside the tank walls equals (46,700 45,400)/(21'50' - 16'00') = 1,300/206

= 6'3 Ib/sq.ft. The fixed-end moment on the part of the outer base inside the wall now equals

3'58 X 0'89 X 6'3 X (20 - 17-88) = 40 Ib.ft (clockwise) (neglecting the moment of half the width of the wall). The effective upward pressure on the projecting toe is now 998 + 6 = 1,004 Ib/sq.ft, and the corresponding fixed-end moment is therefore 1'048 X (13/12) X 1,004 X (11l/i2) = 1,090Ib.ft

(anti-clockwise) (again neglecting the moment of half the width of the wall). The fixed-end moment at the bottom of the tank wall therefore equals 1,090 - 40 1,050 Ib.ft (clockwise). The tank wall is therefore restrained between the fixed and the hinged conditions.

t I

Mt''''·'· ~l T '''' I )._

1.eft: FiR. 3 Mv 100 t"- I j) Examples 3 aud 4. (a) Two t~r<:ular tank' anal., .. d by

"r,Ulnln. watr· mfthod

i{ 1S,900

1S10 M~---+-___ Mt ib) .. (c) Momann and nn, (e"llons In

li10 ,,.,

th.unk wall and b'''llab enm,,' .. ~ '(OlIOCfD la"d-4r •• p.cu'tely

IS)O M,. Mt 2"" (b) (al

Right: Fig. 4. ExtJ7Jlphs 3 aPEd 4

,t .,t V.rtlul bend,", moments and (lnl t

'1 i:.""~' tt""oniin unk wall; udiliand tl

"-

i un,.nuII momenn In but slab , 20 (a, Enmpl,,] (b) Exampl, .. M, 10 T

IJ,fOO i 1),001)

I I>''' . l_ it

I

I~ 210 l' 7O<J

9 10 M,. M;'I'IOICID 1100 lifO Mr Me

«! (b)

193

TAlILE VII! C'..ALCULA nON OF !.lAl!l! SLAB IN EXAMPLE I

Distance from joinl 4' B' 12' 16'

M, (fixed edge) + 5,600 +2,370 --DO -].930 i -3,000 -3,360

M (balance) -840 -840 --840 --ll4O -840 -840

FlOal M, + 4,760 1,530 I -970 -2,770 -3,840 -4,200

M, (fixed edge) -+ J,110 --490 --1,750 I -2,640 -3,360

M (balance) -840 -840 -840 i -840 -840 -840

Final M, +280 -1,330 -2,590 ! -3,480 -4,020 -4,200

(values In lb.ft/ft WIdth)

Annular Bue Subjected

The differential equation uniform circular piate, written as equation 11 (Part 1), can solved for an annulus if the appropriate boundary conditions are applied. This has been done for annuli with different ratios of inner to outer radius, the radial and tangential moments being tabulated at various radii. The rotational sriffnesses at the inner and outer radii and the moments due to unit peripheral moments applied separately at these edges have also been obtained for such annuli. These tables are not included here, but with this additional information it is possible to analyse exactly the tanks previously considered in Examples 3 and 4.

The results of these further calculations are shown in Fig. 4. It will be seen that the values are almost identical with those obtained by the simpler method, though the exact treatment now gives appreciable values for the tangential moments, which were ignored before. There does not appear to be any simple way of estimating these tangential moments, but Fig. 4 at least gives the designer some idea of this effect.

194

C..nnclu6ioDe

are this approximate method, as are the the base. The tangential moments 1n base found from an exact analysis to be appreciable, disregarded in the approximate method.

This is a paper explaining methods of analysis forms of circular tanks and though it indicates which types con-struction minimise the bending moments and ring tensions does not purport to teach design. The ground reaction is supposed unifonn, as with a very soft plastic soil. The smlctural behaVIOur of such tanks when supported on elastic foundatIOns will be con­sidered in a further paper. [To be published in future issue.-ED.]

REFERENCES (2) Ghali, Amin. "The Structural Analy.is of Circular and Rectangular

Concrete Tank,." Ph.D. the,is, University of Leeds, 1957. (3) B. S. Code of Puctice, C.P. 2007, 1960. "Design and Construction

of Reinforced and Prestressed Concrete Structures for the Storage 01 Water and other Aourous Liquids."

(4) ~~~' ti:~J:" "~,'bl!J~~tfK~~~:~~~:}re(J;';:~li~op':~e~~~~i): Vol. I, 1952. Kozlekedesi Kiado, Budapest, 1952.

(5) Timoshenko, S. and Woinowsky-Kneger, S. Theory of Plales and She/ls, 2nd Edn., McGraw-Hili, 1959.

The theory of the bending of circular plat.. on el.stic foundations is

~~":OOt!tl~ :.:a:n~' .":'.d~~d,:.men~l~se:u~ circular tanI<s reating on elastic ground with differ.,..t values ascribed to the foundation modulus are compared numerically with the solution for pl",tic ground (which was obtained in Parts I and 2 of this paper (6);). The theory for the bending of an annular slab on an elutie foundation i, referred to and an approximate solution for a very narrow annulus is also dttived; these are used together in the analysis of • circular tanlt with a projecting toe base. Next, the 'retaining wall' method of analysis is ""mined and ita range of applications investigated. Asymptotic solu­tions fot • circular plate and a beam on an elastic foundation are listed lIDd compared; their use is demonstrated for this c[;<ss of structure. Finally, an approximate method of analysi. is' described for u flexible annular :dab b.... A guide to the usage of these various types of analysis is appended.

(Part 3)

The first parts of this article appeared in Ihe December 1963 and 7anuary 1964 issues

II. TANKS ON ELASTIC FOUNDATIONS

THE analysis of the forces and moments in the walls and bases of open circular tanks supported on plastic foundation was

explained in earlier pans of the paper. A plastic foundation has, theoretically, a foundation modulus, k, equal to zero, which means that settlements are infinitely large. This extreme case cannot occur in fact and the general case must be considered; this is governed by the ratio of the foundation modulus k to the flexural rigidity EI of the base. For a strip footing the ratio of k to E1 is defined as 4,\' and calculations proceed in terms of ,\. Similarly, for a plate resting on an elastic foundation the ratio of k to the plate rigidity, D, is defined as f3' and f3 is the corresponding ruling variable. The extreme case of ,\ or f3 equal to zero, which occurs with k zero and EI finite must not, however, be confused with the case when k is finite and EI or D is infinite. Then infinite stiff­ness obtains (except for a rigid cantilever on an elastic foundation) and a different analysis from the plastic case results.

In the earlier parts of the paper it was found that only joint rotation needed to be considered, but once k is finite vertical dis­placements at the joints may have to be introduced as additional variables for some cases (but not for the approximate 'retaining wall' analysis).

Once again the Winkler hypothesis is assumed, viz. that at any point the foundation reaction varies linearly with the deflection. The behaviour of the ground might be represented more realistic­ally by the theory of elasticity, if E and I' values for the soil can be obtained [note the research by Vesic (7) and by Barden (8)], but a complete account of the interaction of the base and soil of a foundation should also take account of whesion, the coefficient of friction, and the effects of voids ratio and moisture content. These further complications will certainly affect the analysis, but the results given here, based on the Winkler theory, are a sub­stantial development beyond the simple plastic theory for soil.

A full explanation of the theory of the bending by circular plates on elastic foundations was published by Schleicher (9) in 1926. Since then Timoshcnko (5) and Hetenyi (10) have modified Schleicher's work in attempts to make the solutions usable in design work. The theory is now presented in summarised form. The notation of srmbols which was used in Part 1 is retained; additional symbols are hsted below:

• Fellow of Pembrol .. College, Oxford. t Smior Engineer, MtHrI. Ove ATilP and Part1Urs.

t Figures in partnthtJtJ indicate References follotL'ing {he art ide Note that references, figures, tables, tec., follow on {rom ParIS I aNd 2 i'~ their numhtrinK.

Additional Notation

f3 = ;IYs~! folun~:ti: bending of a uniform plm on an

Z" etc. = functions for the bending of a unifonn circular plate on an elastic foundation.

Z' = dZ/d(frt). A" etc. = integration constants.

Q normal shear force per unit length of a circumference. w' = dw/dr. S .. ~ rotational stiffness, 0 1, w = O. Seq = Sq, = cross stiffness. Sq. = shear stiffness, w = 1,0 = O.

S' ,. = rotational stiffness without shear restraint, (J 1, W

occurring. R" R" inner and outer radii"of an annulus.

y = weight of water per unit volwne. c" etc. coefficients for radial and tangential bending moments.

Circular Plates on Elastic Foundation. The differential equation for the defonnations of a circular

plate without any foundation support was given in equation 11 (Part 1). The elastic reaction of the foundation, kw, reduces the nett plate loading and the equatiJn becomes

'V'w + (kw/D) = p/D .... (15) The solution of this differential equation, for the R.H.S. zero, may be obtained in terms of the four Bessel functions ber pr, bei f3r, ker f1r and kei f3r, where f1' = kiD = 12(1 - v')k/Et'.

The general solution of equation 15 is, however, most con­veniently written

w A,l,(f1r) + A,Z,(flr) + A,l,(frt) + AiZi(tn') .. (16) with the l functions simply derived from the Bessel functions. These l functions were originally derived and tabulated by Schleicher (9), but they are also to be found in Hetenyi's well­known book (10). They are oscillating functions, somewhat similar to the functions such as sinh x. cos x which occur in the solution of the equation for the bending of a uniform beam on an elastic foundation.

The slope along a radius, dw/dr, is given by w' = (A,l', + A,l', + A,l', + Ail',) ... (17)

where l', = dl/d(flr). These derivatives l'" etc" are not easily derived as with

circular and hyperbolic functions; however, they are also tabulated in Hetenyi's book. The second derivatives can be fairly simply expressed in terms of the original functions and their first deriva­tives, with the first and second, and the third and fourth, functions and their derivatives paired independently.

Four boundary conditions, usually two at the centre and two at the circumference of the plate, give the values of the four inte­gration constants A, to A,. The radial and tangential bending moments in the plate can be written

M, = -D.{3'{A.[l, - (I - l'll',/f3r] + A,[ -Z, (1 1·)l'Jf3r] + A,[l, - (I - I·)l',/pr] + A,[ -l, - (I l'll'./f3r]) . (18)

and M, -D.f3'{A,[,ol, + (1 - 1·)l',ff3r]

+ A,[ -Iol, + (I - 1·)l'Jf3r] + A,[ .. Z, + (l - 1·)l',/f3r] + A.[ -,ol, + (I - 1.)l'./f3rJ) . (19)

The normal shear force at any radius r is Q = D.{3'(A,l', A,l', + A,l'. - A,l',) ... (20)

In each of these three expressions the parameter of the l­functions is f3r, where r is the radius at which the moment or force occurs.

195

When the plate is continuous across its centus the constants A, and A. become zero, since Z, and Z. tend (0 infinity as reaches zero. So for the solid disc there are only two boundary conditions to be found fflf each particular loading case.

Uniform Circular to a Uniform

When the boundary of the plate is given unit radial rotation the boundary conditions are U' = 0 and 10' = 1 at r R. The integration constants are obtained by substituting these values in equations 16 and 17. The values of the constants then become

f3A, = -Z,/(Z,Z', Z,Z',) and f3A, Z,/(Z,Z', - Z,Z',).

Substitution in the expressions for M" M, and Q gives the plate forces for this loading case as M, -(D/R)f3R.

and

{--Z,.[Z" -Z' ,,(l-- .. )/f3r] +Z,. [--Z"

· (21)

+Z,.[ --,Z,,-"-Z'jl-- 'l/f3rl)

--(Z"Z'" + Z,.Z',,) Q = -(D/R').(f3R) ----­

(Z,.Z'" - Z"Z',,) These are the "runou,s" of the bending moments and

· (22)

shears in a uniform circular disc on an elastic foundation a unit edge rotation. Numerical values of the radial and tangential moments are given in Table X. Poisson's ratio has been assumed to be equal to 0'2.

By putting r = R in the expressions 21 and 23 the edge moments and forces required to maintain this edge rotation are obtained. These are the rotational and cross-sliffness values for the plate; they reduce to S" = M. = s •• _(D/R)

and

{-(Z,,'+Z,.')+ [(1--- 1')/f3R](Z',.Z_. -Z'"xZ,x)) -(D/R).f3R--------------

· (24)

10

10

TABLE X

BENDING MOMENTS IN A UNIFORM CIRCULAR PLATE ON AN

ELASTIC FOUNDATION WHEN EDGE ROTATION 6 = 1.

M,=C, x D/R; M, C, >( DjR.

rlR Edge i 0·8 0·6 0-4 ----T-

I 0·2 <i 1'241, ),221 1·201 j·184 1'173 I'Hi9

1·785 1'499 12l] 0·978 0·825 0·772

3208 2'164 J'IB9 I 0·452 0·007 -0,140

Co 4·802 2'689! 0·979 --0'072 --0,583 -0-728

C,

6241 2'820 0·547 - 0419 -0658 -0'678

7·649

9-067

10-49

11·91

13-32

j'208

1·317

j'602

1-920

2208

2-490

2-774

]-057

3-341

3'624

2-661 0·022 -0,594 -0-458 - 0·340

2-319 -0440 -0-608 -0'2IB -0,046

1·86 -0,75 -050 --0,05 0'09

J-34 - 0·90 -0·}2 0·04 0-09

0-79 0·91 --0-16 0-06 0·05

1-197 1-186 1'177 1·17J 1·169

1'150 0997 0-876 0-798 0-772

1-025 0525 0-156 -0-066 -0140

G-~72 0-077 0-412 --0-656 0728

0-713 0-198 --0,576 0-670 0-678

0545 - 0·352

0-374 -0423

0-208 0-430

0-054 - 0-394

-0-085 - 0-332

0501 - 0-401 -0,340

0- 348 - 0-I3J - 0-046

-0-200 0-020 0085

0·087 0-068 0-093

--0-015 0-059 0-052

S", = Q. sq.(D/R') [(Z,RZ',R+Z,RZ'·,R)

= (D/W)(f3R)'-----­(ZIRZ' oR - Z,"Z' 'R»)

These stiffness coefficients, s, have been tabulated for a range _ . _ (25) of f3R values and are given in Table IX, Poisson's ratio again

being taken as 0'2_

Soo

5'00

196

TABLE IX STIffNESS CoEFFICIENTS FOR A CIRCULAR PLATE ON AN ELASTIC FOUNDATION

1·210 1'356

¢

Q~ e= +1 .

M soo.D/R Q = sqo_DjRl

1·836 2-556 3286

1 Q'b~--.--- +-

M = S{jQ_DjRl Q sQq.DjRl

3-987 4-686

w=+l

where D is the flexural rigidity of the plate and EP/l2 (1_1-'); (.- = l.l M and Q are the edge moment and shear I unll cIrcumference;

and 11' k j D where k is the modulus of subgrade reaction.

5-389 6·093

M ~ s·"". D/R Q

10

13·32

Othe!' Boundary ConditioDli Two other set3 of bounduy cooditioru are important in the

analysis of circular tanks; these are firstly II unit edge sway without rotatioo at the circumference, and secondly a unit edge rotation only when no shear restraint is applied on the circumference. The method of ca1cuIation is aimilu to that outlined above for the unit edge rotation.

In the first boundary case U! = 1 and w' = 0 at ,. = R. The bending moments and radial shear in the plate at any radius r become

and Sqq = Q. = s D/R>

=(D/R')(j3R)'{[(Z',JO)'+(Z\.)']iCz,aZ',,,-Z,..z',.)} .. (29) In the case of unit edge rotarion only the boundary conditions

are that w' = 1 and Q = 0 at r = R.

The bending moments and normal shear in the plate then become

(Z',.{Z'r- (l-v)/,Br)Z',r} +Z' .. { -Z,,- [(l-v)/pr)Z',,})

(Z',Il{Z,,-[(l-v)/,BrjZ',,} Mr=-(D/R)(j3R) -Z',ll{ -Z,,- [(l-v)/,Br)Z',,})

M,=-(D/R')(j3R)' -------------(ZtRZ',. - Z,.Z' ,,,)

· ... (26)

[(Z'"e)'+(Z',.)'] · _ .. (30)

(Z',,,{,·Z,,+ [(l-v)/,Br)Z',,} +Z',lI{ -,.Z" + (l-v)/,Br]Z'.,} )

(Z',.{vZ,,+[(l-v)/,BrjZ',r} M (D/R)fQR) -Z""{-'vZ,,+[(l-v)/,8r)Z',,}) ,=- II'

M, = -(D/R')(j3R)'

· _ .. (27) and Q=-(D/R')(j3R)'[(Z',.Z',r+Z',.Z',,)/(Z,.Z',.-Z • ..z'llt)]

· ... (28) Numerical values for Mr and M" with Poisson's ratio at 0-2,

are tabulated in Table XI. In particular the edge moments and forces, which are the cross

and sway stiffness values for a uniform circular plate are

10

10

S.q = M" (= S .. in equation 2S)

TABLE Xl

BENOING MOMENTS IN A UNIFOIlM Cn,euLAIl Pl.An ON AN ELA.STle FOUNDATION WHEN EOOE D£FUCTlON w=1.

, Cj

i C.

-C}074

1·783 i 0·723 -HlI3

6·492 i 2-218 : -o·m; -2-040 . -2,769 - 2·978

13·09 I 2-943 -1·939 i -3'SQ8 - 3·649 -3'56l

21·26 2'019 -40406 -40402 I - 2-833 -2,092

31·49 -0·42\ i -7·167 -40498 : -1,257 i -0,067

43·79 -4,12 i -9,30 -3'69 0·11 1-14

58·09 -8'78 -10,26 ~2'24 0'8S 1·25

74·38 -14-oS -9'98 -0'65 1·00 0·74

92-67 -19·54 I ·~8·66 c}62 I 0·78 0·19

0·025 i -0'011'0-0391-0.058 I -0,070 ·0·074

0'357, -0'158 : -0·544 I -0,808 ! -0·962 .. 1·01l

1·298 : -0'596 -1·818: -2,525 -2,874 2·978

2-618

4-253

6·297

8·757

11·62

-1·268 i -3-062 -3·597 - 3-610 -3,563

-2,155 -3-847: -3'349 -2-468 2-092

-3-2521-4-212 . -2-340 . -0,663 i -0,067

-4-4821-4.173 -1'1771 0'636' 1·139

-5-75 1_3-76 I -0,21 I 1·07 1·25

[(Z',II)'+(Z' .. )'] · ... (31)

Q=-(D/R')(j3R)'{(Z',RZ'"-Z'uZ',.)/(Z',,,)'+(Z',.)'}} · ... (32)

In this case there is only one stiffness value, the edge bending moment, which will produce a unit edge rotation when the edge shear force is zero. This rotational stiffness is S' .. =M.=s' ..D/R

(Z •• Z',.-Z •• Z',lI)

= (D/R)(PR) + [(l-v)/,8R] [(Z'.lI)'+(Z',lI)')}

[(Z/ ,.)' +(Z' ,.)'] _ . __ (33)

TABLE XU BENDING MOMENTS IN A UNIFOIlM CIRCULAIl PLAn ON AN ELASTIC

FOUNDATION FOil EDGE UNIT ROTATION WITH NO EOOE SHEA" RESTIlAINT.

BR r/R Edge 0·8 <i. 1·210 1·208 1·188

1·356 1-325 1·015

1·836 1'695 0·489

2·556 2·184 ' 1·312 0·()4) -0'111

3·286 ' 2·540 1·160 0'193 -0,265 I -C}387

3-987 ' 2·710 ~ 0·856 I ·0·071 i -0'312 -0,332 I I I

4·686 2·730 I 0·491 ! -0·238 ; -0'229 -0,160

5·389 2'631 , 0'150 ! -0,298 i -0,122 -C}024

6·093 -0,040 0-035

10 6·797 0·039

1,202 1'188

1·2.11 1·IK8 f 1-128 1·070 I 1·029 1·015

1·327 1·151 0·909 ! 0'689. 0'541 0·489

C. 1'471 1·089 0·603 ' 0·206 -0-037 -0,117

1·617 1·013 , 0')36 ~O·III -0,327 -0,387

1·757 0'

9231

0·138 -0,229 ":0'324 ·0·332

1·897 0·822 ·0·005 -·0·230 , .. 0'196 -0,160 I

1-0,181 1-0-074 2-038 0·713 -·0·100 -0,024

2·179 ·0'120 -0,002 0-035

10 2·319 -0,066 : I

0·024 0-039

191

The stiffness coefficients which have been developed here are tabulated in Table IX and the runouts of the radial and tan­gential bending moments in the plate, for this last case, are given in Table XII.

Note that the cross stiffness values s.q and Sq. are found to be equal; this might have been expected from Betti's Theorem.

If the foundation modulus k is put equal to zero the moments and shears reduce to those in a circular slab with edge supports only. Considering the circular plate with a unit edge rotation, if fJR = 0 the expressions 21 and 22 reduce to

M, = M, -(D/R)(1 \.), which represents the spherical bending of a circular slab. The case of unit edge sway also reduces to that of a circular slab carrying a uniformly distributed load of p = k.w per unit area. Then the fixing moment reduces to pR' /8 while the tangential moment at the edge becomes vpR' /8; these are identical with the expressions for a uniform pressure, p, on a fixed-edge uniform circular slab.

It is unnecessary to supply tables of fixed-end moments for the only common case of internal loading on a disc, that of a uniform pressure on the plate', since these are simply the ronouts of moments for a unit edge sway multiplied by the free deflection of the slab, p/k, where p is the unifOimly-distributed loading. (It is supposed that the deflection p/k first freely occurs, then has to be restored.)

EXAMPLE 5. It is required to analyse the structural action of the tank which was used in the first example in Part 1 when the tank is supported on an elastic foundation. A very soft subgrade will be considered with the modulus of reaction k equal to 3 tons/sq.ft/in. Young's modulus for concrete is taken as 3 X 10" Ib/sq.in.

4760

/740 I j Me Ie)

/ 4760

Fig. S. Circular tank 011 ela$lic foundation in Example 5. (0) jlrl.lcrurt, (b) elaslic and planic Joils compared) and (c) values for a range 0/ para­

meters.

198

TABLE XII! RADIAL

Position acrosssiab Centre

Final radial moments -140

Final tangential moments -140

Then fJ' kiD = 3 X .2,240 X 12 X 12 X 0·96/3 X 10' X 144 X 0·833'.

ThusfJ= 0·25, and fJR = 5·0. From Table IX the plate rotational stiffness is

5' •• = 3·286 X D/20 0·164D. Since the uniform vertical load due to the head of water

causes no bending of the base slab, only the edge vertical shear due to the differential weight of the wall needs to be considered in calculating the fixing moments in the base. This force has a value of 16·0 X 0·833 x (150 - 62"4) = 1,169Ib/ft run. Now Sq. = 153·0 D/20' 0·0191D and 58. = 21·26D/.20' =

0·0532D. (Table IX) The fixing moment at the edge of the slab when it deflects with no edge rotation is therefore

M'" = 1,169 X (0"0532/0·0191) = + 3,250 Ib.ft/ft. The rotational stiffness of the tank wall is Sw = 0·638D and the fixed-end moment in the wall is Mr ll = +4,140 Ib.ft/ft. Since the plate provides the vertical support at the joint and S' 88 is now used for it, there no need to consider the vertical force equilibrium further and the joint is analysed by simple moment-distribution, thus:

Distribution factors Fixed-end moments Balance

Wall 0·796

Base 0·204

+3,250 -1,510

Final moments -1,740 + 1,740 Ib.ft/ft run. With the joint now in equilibrium the final bending moments in the slab can be found as the sum of the fixed-joint moments (Table XI) and the joint-release moments (Table XII). These slab moments are summarised in Table XIII.

The curves of the bending moments in the base are shown in Fig. 5b, along with the moments and hoop tensions in the tank wall. It is seen that the elastic foundation reduces the rotation of the base joint and gives a large reduction in the bending moments near the centre of the base slab, as compared with a plastic foundation. The tangential bending moments remain of significant magnitude and must be provided for in the base slab reinforcement. This large reduction of the bending moments has been produced by a very soft foundation; stiffer soils produce correspondingly greater effects. The forces in the tank wall for a wide range of foundation conditions are shown in Fig. 5c.

The exact vlhlue attributed to the foundation modulus is not critical in the calculation of the forces in the tank structure as the bending moments are proportional to its fourth root. Thus 100 per cent change in the modulus value alters fJR by only 19 per cent, and the final moments to a similarly small degree. The numerical values of the body deflections of tbe structure would be inversely proportional to the modulus value, but the deflections would not usually be estimated in the course of this type of computation.

(T a be Continued)

REFERENCES (6) Lightfoot, E. and Michael, D. "The Analysis of Ground-Supported

Open Circular Concrete Tanks; I, On Plastic Foundations." Cit'il Engineering oud Public Works Review, Dec. 1963 and Jan. 1964

(7) Vesic, A. B. "Beams on Elastic Sub-grade and the Winkler's Hypothesis." Proc. 5th 1>11. COllf. Soil Mech. F. Eltg. Paris, 1961

(8) Barden) L. "Distribution of Contact Pressures under Foundations." Geotechnique Vol. 12, 1962.

(9) Schleicher, F. "Kreisplatten auf Elasticher Unterlage.'· B'lin, 1926. (10) Hetenyi, M. "Beams on Elastic Foundation." Michigan Univ

Press, 1946.

(PIIrl 4) Parts 1, 2 and 3 appeared in 1M Dec. 1963, 7an. 1964 and Sept.

1965 issues resputiveiy.

Annular ~ on Ewtie Foundationa The theory of the bending of annular slabs on elastic founda­

tions is similar to the theory for the solid disc which has been outlined above, but four boundary constants mUlt now be evaluated. Tabulation of the stiffness coefficients for the annulus involves a further variable, the ratio of the innet' and outer radii R;/R., in addition to the parameter f3(R. - R;). Tables of edge stiffness coefficients have been published elsewhere (11):1: for the far end fixed and the far end free conditions, with a range of the two parameters fi(R. - R;) and R;/R •.

A shon toe is often provided round the perimeter of a circular base slab to reduce the joint rotation. While in an exact analysis the toe should be considered as an annulus on an elastic foundation it can often be treated as a rigid beam cantilevered out from the tank wall. It is found from a comparison with the analytical solutions that the rigid beam stiffnesses are accurate to within 10 per cent at f3(RQ - R;) = 1, with smaller errors at lower para­meter values. At greater parameter values the rigid beam asswnp­tion gives excessiv~ stiffness values. The expressions for a rigid cantilever on an elastic foundation are now derived.

Consider a rigid cantilever of length i,Alnd unit width. When the restrained end is given unit normal deflection, end rotation being prevented, the edge balancing shear and moment are

Sq. k X I X 1 = kl . (34) and

S", = kl x ;1 = lkP . . . (35) Similarly when the beam end sustains unit rotation and nonnll deflection at the suppon is prevented the rotational and cross stiffnesses are obtained. These are

Sq. = ; X I X I X k = 1kl' = S", as in equation 35 and

S .. = !kl' X (2/3)1 = 1kf3 .... (36) The rotational stiffness of the rigid cantilever when no end shelr restraint occurs is found to be

S'H = 1 X 11 X 11 X k x (2/3)1 = kl'/12 .. (37) For this two-member tank base account must be taken of the

venical deflection and shear forces at the wall joint, as the applied loads are divided between the inner and outer pans of the slab to a statically indeterminate degree. The slope deflection method of analysis then becomes the most suitable technique.

EXAMPLE 6 In Example 2 a tank with its projecting base slab supported on • plastic foundation was analysed. The analysis when the founda­tion is elastic, with a modulus k = 33 tons/sq.ft/in, is now described. The foundation constant f3' =

33 X 2,240 x 12 X 12 X 0'96

3 X 10' X 144 X 0'833' giving p == 0'45, and pR = 9'00. The stiffness values of the inner plate are from Table IX:­

S .. = ll'9ID/20 = +0'596D, S .. = 74'38D/20' = +0'1859D

and S .. = 951-9D/2O' == +O'l190D. The stiffness values for the cantilever toe are __ S~~ kl'/3 = +~6ID, _________ _

• Fellow of Pembrolu Colleg" Oxford. t Senior Engin,er, MetsrJ. OV4 Arup and Partners. * Firures in parenth4J.s indicate Refer,nCts following Ihe arliclt.

S .. = ;kl' = -01)461D and S .. = kl = +0'0615D. The rotational stiffness of the wall is Sw = O'638D.

Assuming zero joint rotation initially, the fixed-end moment at the joint at the base of the wall due to the horizontal water pressure is M'w = 4,140 Ib.ft/ft run, as before.

Now the entire base of the tank alone would produce a uni­form settlement without any bending being induced; this can be ignored. Similarly, if the pan of the base-up to the centre-line of the walls was subject only to the pressure of water above it would senle a distance yH /k and no bending would be induced, This displacement, however, would cause the cantilevered toe to be subject to an upward pressure of yH and on unit width (at the joint) the resulting fixed-end moment and vertical force would be M', = -yHI'/2 and V', = +yHI (uniform width being asswned). The values in this case are -62-4 X 16 X 11'/2 = -1,125 Ib.ft and 62'4 X 16 X Ii = 1,500lb per foot run of structure at the joint.

The downward force V w due to the weight of the wall should have half the weight of the wall as water deducted, since this has already been allowed for. In this example, V", = 150 X 16 X 5/6 - 62'4 X 16 X 5/12 = 2,000 - 417 = l,5831b.

If the subsequent venical displacement of the joint is w and the joint rotation is IJ, then the following equations can be estab­lished for moment and venical force equilibriwn at the joint: (0'638 +0'596 +0'046)DIJ +(0'186-0'046)Dw }

+4,140-1,125 =0 (38) (0'186 -0'046)DIJ +(0'119+0'061)Dw+ 1,500= 1,583 These equations solve to give IJ = -2,628/D and w = 2,500/D. The balanced joint members are then calculated by back­substitution and are:

M w ," = 4,140 - 1,678 = 2,462, M,o. = -1,125 - 121 - 115 = -1,361, M b ... = -1,568 + 465 = -1,103 Ib.ft/ft run.

The variations of the radial and tangential moments in the base slab are shown in Fig. 6. From a comparison with the values for the plastic foundation in Example 2 it is seen that the effect of

.~

'(a)

,uo

'(b)

-- -- Plastic -- EI~'ti', {1lR. 9

12600 15000

Fig. 6. Circular ta"k with exltndtd ball 011 an f1aure foundation, Example 6. (a) rtructUrl, (h) moments and forctJ.

199

the elastic medium is not so great as when there is no toe on the base slab. The joint rotation remains relatively small because of the balanced proportions of the base.

For other values of the foundation modulus these comments remain valid, the chief effect of modulus variation for this type of tank being on the rate of damping of the joint disturbances.

Retaining Wall Method of Analysis When the base slab of a circular tank is divided by a circular

flexible joint (see Figs. 3 and 4) a section through the tank resembles a retaining wall. One method used to treat the structure is to design the base slab to carry the tank wall fixed-end moment, bu\ this method can be correct only fortuitously when the joint rotation is zero. Now the vertical cantilever of a straight retaining wall has no stiffness against outward lotation and it is unnecessary to consider the joint deformations in the analysis. However, a circular tank wall has considerable rotational stiffness and th ~ circular retaining wall type of structure is statically indeterminate. The moment carried by the base depends on the amount of joint rotation.

In this retaining wall method the base slab will first be assumed to be rigid. Then the complete deflection pattern of the foundation is defined by the joint displacements and the inner slab and toe can be treated as one member. A sector of the tank witb unit width at the wall radius is considered. The error of treating this area as rectangular in plan when in fact it is almost trapezoidal is usually small. The method of analysis based on these assump­tions is now described.

The structure is allowed to deflect vertically while the bas~ joint is held fixed against rotation. Vertical equilibrium is achieved and needs no further consideration. The foundation reactive pres­sure is uniform and is equal to the vertical load divided by the total annular area. The three unbalanced moments about the centre of the base are caused by the weights of the water and the wall and by the equilibrant of the fixed-end moment. This moment is balanced between the wall and the base in the ratio of the stiffnesses. The rotational stiffness of the base slab, without vertical force restraint, is found to be S' •• = (l/12)kL' (for any point on the base). The final moments in the wall and base are calculated as before.

This simple method can be applied with accuracy only where the base is very stiff, where {J(R. - R.) < I. Comparison of the exact and "retaining wall" rotational stiffnesses of the inner base of the tank used in Example 4, when {J(R, - R.) = 1'0 gives

Exact "Retaining Wall" S.. O'0777D O'0828D S'.. O'0295D 0·0207D.

The differences reptesent in part the rotational stiffness of the annulus itself and also the error in assuming a rectangular base plan.

EXAMPLE 7 It is required to analyse the structural action of the tank given in Example 4 when the foundation is an elastic medium with a value of foundation modulus of k = 3 tons/sq.ft/in and the base is assumed to be rigid. The values for the tank wall are as before: -

S ... = O'638D and

M"" = +4,140D lb.ft/ft run. The stiffness of the base beam is

S. = (1/12) x 3 x 2,240 x 12 x 5'5" = O'054D. The moments of the weight of the water and the wall about the centre of the base are M. = 62'4 x 3'58 x 16 x 0'96 - 150 x 0'833 x 16 x 1'25

= +9301b.ft/ft. The moment distribution proceeds as shown: -

Distribution factors Fixed-end moments Balance

Wall 0'922

+4,140 +2,960

Base 0'078

- 930 - 250

Fin~l moments + 1,180 -1,180 Ib.ft/ft run. Th·: runout of moments is shown in Fig. 7 along with the

200

,~

t 0/0'

Ib!/ft'Jj .. (b)

Mt 240

-- Retainin9 1Ii,,1I vAI"¢5 - - - - -. Andlyr.c~1 v41..... ( )

b,,.cketed

liJooo; 13200

Fig. 7 Circular {auk alla/yuJ by Retaining WrJII U/OJII'J II' Exump!e 7. (0) JlrHcllIft, (b) momellts and I(lfas

results from the exact analysis. The difference in the values of the wall base moment, 280, is about 8 per cent of the wall fixed-end moment. If the base is assumed to be trapezoidal, the difference is reduced to 2 per cent.

Provision of a toe on the base gives a structure which is approximately statically balanced. Thus similar joint moments occur in the above example even when the foundation is assumed to be plastic. When the tank section is L-shaped this effect is lost ~d the accuracy of thc retaining wall analysis is reduced. For a stiff foundation, when 131 becomes large, the retaining wall analysis should not be used, particularly when there is no projecting toe However, the base can then be assumed in many cases to be a finite, uniform beam on an elastic foundation and the retaining wall analysis can then be followed in a modified form.

Asymptotic Approximations It is possible to obtain fairly simple approximate solutions for

the behaviour of various structural elements on elastic foundations when the parameters AI, {JR, {J(R, - R.), etc., become sufficiently large. In the case of a beam with constant EI, for example, the behaviour approximates to that of a semi-infinite beam once Al exceeds 3 or 4 (the lower limit for AI depends on the accuracy required). There is then no effective "carry-over" from one end to the other.

The behaviour of a finite beam on an elastic foundation is thus seen to be asymptotic to that of a semi-infinite beam when AI is large enough. For a circular plate, however, the solution for infinite radius is not so easily derived, but an asymptotic solution is possible from the asymptotic values for the Schleicher functions and their derivatives (10). A close parallel exists between the extreme solutions for these two cases, as can be seen from Table XIV, where the expressions are all wrinen in terms of A. (It is apparent that {J can be wrinen as \/2,>. since 13' = 4A' = k/D.)

The stiffnesses are seen to be identical in the two cases, except for the terms {I - (I v)/2>.R) and {I - (1 - I)/AR) which have to be included in the expressions for S .. and S' .. for a uniform circular plate. The run-outs of moments are writtcn in terms of the B

A" CAr and DAr functions which apply for the semi­

infinite uniform beam, (where for the circular plate x = R - r) The annular plate can be considered in a similar manner,

depending on the value of {J(R. - R.). If, however, this is less than unity, tabulated functions should be normally used. If, in

addition, R./CR. R.) is large, such II IWTOW annulus may be comidered as II rigid toe, When PCR. - R.) is not very high, each &eCtor may be treated as II finite beam (see Reference 12) for stiff­neHeS and run-outs). When P(R. - R;) exceeds 6 asymptotic lIpptoximations may be used (12). Two examples of these asymp­totic approximadons are given in Table XV, with exact valu!s coming first.

TABLE XIV

ASYMPTOTIC SoLUTIONS FOR THE UNIFOIlM BEAM AND THE UNIFORM CIRCULAR

PLATE ON ELASTIC FOUNDATIONS

Sliffnesse-; and Uniform beam Uniform circular plate run-outs

See 2DA

S.6 = S6q 2DA'

S •• 4DA'

S'ee DA

0=1. w=O. Mr 2DA,O •• 2DAy'(R/r) [o".-(I-v)Ch/2Arj at end/edge M. t 2DA.vD),x 2DAy' (R/rl [vO .. + (1- v)CA,/2Ar]

w=l. 0=0. Mr 2DA'.C .. 2DP v'{R/r) [C •• +(I-V)Bblkj at end/edge M. t 20A'.v.C)" 20Aly'(R/r) [>C •• -{I-v)B,,!Ar.j

6=1. Q=O. M. D),.Ah DAy'(R/r) [Ah-(l-.·)o"./),r.] at end/edge M. t D), .• ·.Ah DAy'(R/r) [vA),x +(I-v)Dh/Ar.j

t Only if edges restrained to remain paralic I.

TABLE XV COMPARISON Of EXACT AND ASYMPTOTIC VALUES

Stiffness coefficients Uniform circular plate (xD for stiffnesses) BR = 10

Sq6 = Seq

Sq.

,'ee

13·32

92·7

1J16

6'80

13·34

Condu.ions

Uniform annulus B(Ro-R,) = 4 (in Example 3)

0·975 1·022

0·517

0·536

0·524

This paper explains the theory required for the analysis of ground-supported open circular tanks when the reactive pressure is proportional to the deflection of the foundation. The Tables published here enable these tanks to be analysed quite easily,

provided some value of the foundation modulus is known. The structural behaviour of such tanks resting on very soft ground with the foundadon modulus taken as zero--the extreme plastic case-was explained in Part 1. It is seen that the complications resulting in taking an estimated value for the foundation mo:lulus are nor at all difficult and that considerable economies may be obtained if elastic rather than plastic foundations are allowed for.

The circular retaining wall method of analysis, which was explained in Part 1 for plastic ground, is still very quid and attractive even for an elastic foundation, provided the base is assumed rigid. Even allowing for flexure in the base the analysis is not very difficult, so it is hoped that designen will utilise this, as well as the simpler methods. A guide to the various types of tank and the various methods of analysis considered here is in­cluded in the Appendix.

REFERENCES (11) Michael, O. "The Structural Actions of Circular Tonk, and Pris­

matic Bunkers." Ph.D. Thesis, Leeds Univ., 1962. (12) Li~htfoot. E. "Moment Distribution." Spon, London, 1961.

TABLE XVI SUMMARY Of ANALYTICAL METHOOS FOR VAIUOUS TyPES OF GROUND­

SUPPORTED CIRCULAR TANKS

Structural element

Circular wall

Circular base

Circular toe

Annular base inside wall

i.H = 0 to 6 ),H>4

Aid for analysis

, BR = 0 I Plastic solution.

~(R.-R,) < I I ~:.: ~iid base ~(R.-R.» I . Auume tinite beam. Ref. (12). Use

annular slab tables. Ref. (I I) 'l(R.-R,»6 Asymptotic solulion-Table XIV

=--~-'=-=-~- ~-~-.~===========

Appendix GUIDE TO THE ANALYSIS OF THE VARIOUS TYPES OF GROUND­SUPPORTED CIRCULAR TANKS CoNSIDERED IN THIS ARnCLE For ease of reference the different kinds of tank and the different methods of analysis ~re classified in Table XVI which also identifies the examples given.

Simple moment distribution solutions were possible in all examples except Example 6.

201