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12Reinforced andPrestressedConcrete Design

S C C Bate CBE, BSc(Eng), PhD,C Eng, FIStructE, FICEFormerly at the Building ResearchEstablishment and later, currentlyConsultant to Harry Stanger Ltd

Contents

12.1 Introduction 12/312.1.1 Definitions 12/3

12.2 Behaviour of structural concrete 12/3

12.3 Philosophy of design 12/612.3.1 Criteria for limit state design 12/712.3.2 Characteristics of materials 12/9

12.4 Analytical and design procedures 12/1112.4.1 Objectives 12/1112.4.2 General assumptions 12/1212.4.3 Robustness 12/1312.4.4 Beams and slabs 12/1312.4.5 Continuous and two-way solid slabs 12/1412.4.6 Flat slab construction 12/1412.4.7 Frames 12/1412.4.8 Columns and walls 12/14

12.5 Reinforced concrete 12/1412.5.1 General 12/1412.5.2 Beams 12/1412.5.3 Slabs 12/1712.5.4 Columns 12/1812.5.5 Walls 12/1812.5.6 Bond and anchorage 12/1912.5.7 Cover 12/1912.5.8 Spacing of reinforcing bars 12/1912.5.9 Laps and joints 12/1912.5.10 Curtailment and anchorage of bars 12/2012.5.11 Limits on the amount of reinforcement 12/21

12.6 Prestressed concrete 12/2112.6.1 General 12/2112.6.2 Prestress and serviceability 12/2312.6.3 Losses of prestress 12/2412.6.4 Stress limitations at transfer and for

serviceability conditions 12/2512.6.5 Beams 12/2612.6.6 Other forms of member 12/2712.6.7 Requirements for tendons and

reinforcement 12/28

12.7 Precast and composite construction 12/2912.7.1 General 12/2912.7.2 Structural connections between units 12/2912.7.3 Beams, slabs and frames 12/2912.7.4 Floor slabs 12/3012.7.5 Bearings 12/3012.7.6 Composite concrete construction 12/30

12.8 Structural testing 12/31

12.9 Fire resistance 12/32

References 12/33

Further reading 12/33

12.1 Introduction

The design of reinforced and prestressed concrete has beenincreasingly codified during the past 40 years. Before the SecondWorld War, recommendations for design had been published inthe UK in a Code of Practice prepared by the Department ofScientific and Industrial Research, which was issued in 1934,1

and in the Building By-laws of the London County Council of1938.2 After the war, the DSIR Code was revised and becamethe British Standard Code of Practice, CP 114, in 1948.3

The Institution of Structural Engineers published its FirstReport on Prestressed Concrete in 1951,4 which gave designprocedures for prestressed construction. This report was subse-quently revised and issued as BS Code of Practice, CP 115, in1959.5 The BS Code of Practice for the Design of PrecastConcrete, CP 116,6 appeared in 1965 and supplemented the twoearlier BS codes. By this time, a number of codes dealing withspecialized forms of concrete construction were being prepared.Codes of Practice 114, 115 and 116 have been updated fromtime to time and are currently adopted as deemed-to-satisfydocuments in the Building Regulations.

An important innovation took place in 1972 when a unifiedBS code, CP 110, for the structural use of concrete7 waspublished. This code, which it was intended should supersedeCodes 114, 115 and 116, introduced a new feature in design,namely limit state design in which account was taken directly ofthe possibility of failure or unserviceability occurring during thelife of the structure being designed. The particular factorsconsidered included the risks resulting from variability of thematerials, inaccuracy in design assumptions and construction,variability of loading and the incidence of accidental damage.Whilst the approach to design was modified, many existingmethods of analysis and calculation were retained. Provisionwas made for the incorporation of new data on loading andmaterials, and on structural performance and methods of con-struction as they became available. The basis for this approachhad been developed by the European Committee for Concreteassisted by the International Federation for Prestressing whopublished a jointly prepared code8 in 1978 having previouslyissued separate codes. This code was used in the production of acode for the European Economic Community.9

Code of Practice 110 did not replace the earlier Codes 114,115 and 116, which still remain in force, but it has now beenrevised as BS 8110.10 The approach adopted in CP 110 has beenretained and the content has been brought up to date. Inaddition, a manual" has been prepared by the Institution ofStructural Engineers conforming with its recommendations butpresented in simpler form and dealing with a more limited rangeof construction. The guidance given in this chapter is relateddirectly to the contents of BS 8110.

Whilst these developments were taking place in the UK,somewhat similar changes were occurring elsewhere. Some ideaof the differences between the recommendations adopted in theUK and elsewhere are given in Table 12.1, which makes somecomparisons between BS 8110, the American Building Code,ACI 318-8312 and the EEC Code.

12.1.1 Definitions

This chapter is concerned with the basic approach to design ofreinforced and prestressed concrete. It deals with both cast-in-place and precast concrete whether reinforced or prestressed. Itincludes information on the use of plain or deformed steelreinforcing bars and with tendons which may be either preten-sioned or post-tensioned. In this context some definitions and anindication of limitations may be useful.

(1) Reinforcement which is used to provide the tensile compo-

nent of internal forces in reinforced concrete, generallyconsists of one of three types of material: plain round mild-steel bar produced by hot-rolling; plain square or plainchamfered square twisted mild-steel bar which has had itsyield stress raised by cold-working; ribbed bars, which maybe hot-rolled from steel with high yield stress or cold-worked by twisting from hot-rolled mild-steel.

Since steel reinforcement can only develop an effectivetensile force by extension of the concrete by cracking, thereis a limit on the maximum strength of steel that can be used.In general the yield stress should not exceed 500N/mm2

although higher strength steels may be used if particularcare is taken to avoid excessive cracking or deflection.

(2) Tendons are used to impart a prestress to concrete beforeservice loads are applied which offsets the tensile stresseswhich will later result from the application of these loads.Tendons are usually comprised of plain, indented ordeformed cold-drawn carbon steel wire, of seven-wire ornineteen-wire strand spun from one or two layers respec-tively of cold-drawn carbon steel wire around a core wire, orof high-tensile alloy steel bar. The strength of steel usedmust be high enough for it to be extended sufficiently toavoid excessive loss of tension due to elastic contraction,creep and shrinkage of the concrete. In general it is not oflower tensile strength than about 1000N/mm2.

(3) In prestressed concrete, prestressing may be effected bypretensioning or post-tensioning the tendons. Pretensionedtendons are stressed before the concrete is cast. They arestretched either between temporary anchorages placed suffi-ciently far apart for a number of moulds to be assembled inline around the tendons, i.e. the 'long-line' method, orbetween the ends of specially strong moulds, i.e. the 'indivi-dual' mould method; in each case, concrete is then cast andallowed to harden before the tendons are released from theirtemporary anchorages. The methods are best-suited to massproduction in the factory and usually use wire or the smallersizes of strand as tendons.

With post-tensioning, however, the tendons are stressedafter the concrete has hardened and are usually accommo-dated in ducts within the concrete being held at their ends byanchorages, of which there are various proprietary types.Subsequently the ducts are grouted with cement grout toprotect the tendons from corrosion. This method is mostlyapplied to site construction and tends to use tendons ofrelatively large size.

12.2 Behaviour of structural concrete

The characteristics of concrete that have conditioned its deve-lopment as a structural material are its high compressivestrength and relatively low tensile strength. In consequence itsuse for flexural members did not become practicable until it wasdiscovered that steel reinforcement could be cast in the concreteto carry the bending tensile stresses whilst relying on theconcrete to carry the bending compressive stresses. Experimentshowed that mild steel, when present in the tension zone inrelatively small amounts, provided a material with characteris-tics for deformation and strength which complemented thosefor concrete and provided a practical form of construction.Early research workers concluded that the presence of the steelincreased the extensibility of the concrete. Later experimentsshowed, however, that this was not so. It then became clear thatas the tensile stress in the steel of a beam increased beyond asmall amount, which is appreciably less than that developedunder service loading, cracks developed in the concrete. Thesecracks were controlled in width and numbers by the position ofthe reinforcement relative to the concrete surface and by the size

Status

(A) This national code was prepared bythe British Standards Institution, anorganization with some direct supportfrom Government, and accepted asproviding conformity with BritishBuilding Regulations, but not in itselfmandatory, other authenticated designprocedures may be acceptable.

Design procedure

(A) Limit state procedures (described inthis chapter) are adopted followingclosely the 1964 Recommendations of theEuropean Committee for Concrete,which were used subsequently indeveloping the EEC Code; the basicapproach in the two codes is thereforevery similar.

The ultimate limit states includestrength and stability under dead andimposed loads, wind loads, and earthand water pressure for which partialsafety factors are defined depending onload groupings, and the effects ofaccidental loading and damage.Durability and fire resistance are nottreated as limit states but are included inthe design process, the former beinggiven more emphasis than in previouscodes. As far as possible the analysis ofstructures is based on ultimate behaviourbut, where methods have not beendeveloped, elastic analysis is accepted.The strength of sections is based on thestrength of the materials, as reduced bypartial safety factors, and compatibilitybetween stress and strain using idealizedstress-strain relationships. Simplifyingassumptions relating to thesestress-strain relationships are allowablefor many types of construction.

The serviceability limit states includedeflection and cracking under dead,imposed and wind loads, appropriatepartial safety factors for combinations ofloading being given with limitations ondeflection and crack width. Wherenecessary, allowance is required for theeffects of shrinkage and creep and oftemperature change. For common typesof construction, limits on deflections areimposed by placing limits on span : depth

(B) This national code was prepared bythe American Concrete Institute; it isused extensively in State regulations forbuilding control. It is widely recognizedinternationally and is adopted in part orwholly in the codes of a number of othercountries.

(B) The objectives of the ACI Code aresimilar to those of (A) but the means ofachieving them are different.

Design requires consideration ofultimate strength but a single factor isdefined for relating strength to the loadsto be supported instead of adopting thecombined effects of partial safety factorsfor both loading and strengths ofmaterials, as in (A). The principles forcalculating the strength of sections areotherwise similar in requiringcompatibility between stress and strain.For flexure, the strength of concrete isdefined in terms of 85% of the cylinderstrength reducing as strength increasesinstead of 67% of the cube strengthirrespective of strength, as in (A) andthere are slight differences in the shapeand extent of the stress-strain curveassumed; precautions are introduced toavoid brittle compression failures.

Serviceability with respect to deflectionis ensured either by limiting span : depthratios or by checking that the long-termdeflections do not exceed defined limitingvalues. Cracking is controlled bylimitation of calculated crack width andprovision of reinforcement for bothreinforced and prestressed concrete. In(A), on the other hand, cracking ofprestressed concrete may be controlledby limitation of tensile stress, thenominal tensile stress being related toamount and distribution of secondaryreinforcement.

The ACI Code has an appendix whichgives an alternative method of design forreinforced concrete which is based onpermissible stresses in the materials.

(6) The code has been prepared by theCommission of the EuropeanCommunity for use in member countries,and is one of a number now beingproduced to deal with all commonmaterials and forms of construction. It islikely to be adopted for building controlin those countries and will be recognizedas satisfying the requirements of nationalregulations. The code has drawn on thework of international organizations,which are supported worldwide, andhence it is likely to have an importantinfluence on the formulation and revisionof codes in other countries outside aswell as inside the Community.

(C) Since the developments of the BScode and of the EEC code have drawnon a common source, the two codes, asalready noted, have a common basis.However, the former was drafted by acommittee with a British background indesign and in the development of codes,whereas the latter has incorporatedmulti-national European experience andthere are therefore a number ofdifferences. Also in Britain, codes aregenerally regarded as advisory whereas inEurope they are mandatory. The maindifferences between the two codes are,however', in detail, the EEC Codetending to be more precise.

Thus, the definition of limit states, theloads to be considered, and the strengthsto be adopted with their relevant partialsafety factors are closely similar.

The EEC code is, however, based oncylinder strengths of concrete which givesrise to some differences when comparedwith the BS code. The simplifiedassumptions for calculation of flexuralstrength for each code, for example,show an apparent ratio of cylinderstrength to cube strength of 0.89, whichis appreciably higher than for mostexperimental data.

Table 12.1 Notes on different Codes of Practice (British, American and EEC)(A) BS 8110 - Structural use of concrete10 (B) ACI 318 - Building code

requirements for reinforced concrete12(C) Eurocode No. 2 - 'Common unifiedrules for concrete structures'9

of bars used. Thus with closely spaced bars near the surface, alarge number of small cracks would develop, but with largewidely spaced bars, the cracks would be fewer in number andmuch larger for the same stress in the steel. If the stress in thesteel were increased the size of the cracks increased and their sizewas little influenced by the surface roughness of the steel,although at one time it was thought that roughening of thesurface resulted in appreciably smaller cracks of larger numbers.It was eventually established that the main benefit of using barswith a roughened surface was in developing good end-anchor-age.

Because steel needs to extend to develop stress and hencecauses cracking and deformation of the concrete, there is a limitto the strength of steel that can be used efficiently for reinforce-ment, since unsightly cracking, which could lead to severecorrosion in adverse conditions and unacceptable deflections,must be avoided. The use of steel in prestressed concrete, wherethe stress in the steel is imposed before the concrete member issubjected to external load, avoids this problem, since the initialtensile force is developed without extending the concrete, and sono upper limit is imposed on the strength of steel that can beemployed. This was not, however, appreciated in the earlydevelopment of prestressed concrete. Then, steel of relativelylow strength was used with a small initial tension. The experi-menters found that, although this was effective at the start, theinitial prestress disappeared with time. Eventually, however, itwas established that this nonelastic behaviour was limited inextent and that if a sufficiently large elastic extension wasimparted to the steel, the nonelastic effects of creep and shrink-age of the concrete did no more than reduce the prestress by anacceptable amount. Although for a time there was a tendency tounderestimate the losses of prestress due to contraction of theconcrete and to ignore creep in the steel tendons, research hasnow, however, clearly set the limits on what needs to beconsidered in design.

The performance of reinforced concrete and prestressed con-crete beams under increasing load is characteristically differentsince cracking develops in different ways in each form ofconstruction. This is illustrated by the results of tests on beamsin each form of construction as illustrated in Figures 12.1 and12.2.

Examined in more detail the deformation of the reinforcedconcrete beam under load is linear until cracking occurs;thereafter it approximates to a linear relationship until the steelyields as cracking becomes more extensive for beams of normaldesign. Subsequent deformation leads to the development of ahinge with continued yielding of the steel accompanied bydamage to the concrete. This deformation continues at approxi-mately constant moment until a stage is reached where theresistance reduces. The occurrence of this stage is influenced bythe amount of transverse shear reinforcement in the section.

Central deflection

Figure 12.2 Relationship between applied load and deflection fora prestressed concrete beam showing recovery and reloading

The prestressed concrete beam, however, remains uncrackedusually until the service load is exceeded, and in this range itsdeformation is elastic. Once cracking has occurred deformationincreases disproportionately rapidly with increasing load ascracks widen until the maximum load is reached. Subsequentlythere is a rapid reduction in resistance. Since the prestressedconcrete beam is usually uncracked under service conditions itsstiffness is greater than that of reinforced concrete beams of thesame overall depth.

In continuous construction subjected to applied loads ofshort duration, deformation of both reinforced concrete andprestressed concrete members is elastic or effectively elastic untilservice loads are exceeded. With further loading, as the appliedmoment at any section approaches the resistance moment atthat section, there is a tendency for the moment to be relaxedand redistributed to sections that are less seriously stressed.Thus a loaded beam, built in at each end, may reach its

Stage II Stage IIIStage I

Cracking load

Maximum load

Central deflectionFigure 12.1 Relationship between applied load and deflection fora reinforced concrete beam showing recovery and reloading

Concluding comment: Of necessity, these comparisons are very limited and superficial in character but should serve to show that developments in codes proceeding currentlyin different countries have much in common. This trend is likely to increase through the medium of the extensive international collaboration that now takes place.

Maximum load

Stage IIIStage II

Crackingload

Stage I

Table 12.1 Notes on different Codes of Practice (British, American and EEC)—continued(A) BS 8110 - Structural use of concrete™ (B) ACI 318 - Building code

requirements for reinforced concrete12(C) Eurocode No. 2 - 'Common unifiedrules for concrete structures'9

ratios, and cracking may be controlledfor reinforced concrete by defining theform that the reinforcement should take.

Since many practical engineers havepressed for the retention of permissiblestress methods of design, the previouscode applicable for reinforced concrete3

has been retained in use. It seems likely,however, that it will be withdrawn in thelonger term.

Appl

ied

load

Appl

ied

load

maximum resistance moment at mid-span before the maximumresistance moments at the supports are attained; a hinge thenforms at midspan with the applied moment there remainingsensibly constant whilst the applied moments at the supportsincrease until hinges form at the supports. The beam has thenreached its maximum carrying capacity. The capability ofreinforced and prestressed concrete beams for rotation at hingesis limited, however, and restrictions therefore need to be placedon allowances in design for redistribution of moment. Theseallowances are smaller for prestressed concrete sections than forreinforced concrete sections since their rotational capacities aresmaller.

Under long-term loading, the deflection of reinforced con-crete beams increases usually to about 2 or 3 times the initialdeflection. Although the initial deflection is primarily influencedby the amount of steel in the section and its stress, fhesubsequent deflection is largely the result of creep of theconcrete, breakdown of bond between the steel and the concretein the tension zone between cracks which initially stiffens thebeam, and the effect of the reinforcement in restraining theshrinkage of the concrete.

Since prestressed concrete is usually uncracked under long-term load the initial deflection is mainly due to the deformationof the concrete. The subsequent deflection results mainly fromcreep of the concrete and depends on the combined effects of theprestress and the stresses due to applied load. The former tendto deform the member in the opposite direction to the latter. Inconsequence, a loaded prestressed concrete member mayinitially have an upward deflection which can continue todevelop upwards or downwards depending on how heavily it isloaded.

Under cyclic loading, reinforced concrete members usuallyfail in fatigue by fracture or yield of the reinforcement. Theproperties of most reinforcing steels, provided that they are freefrom welded connections, are, however, such that the ranges ofstress experienced under service loading determined for staticconditions are usually within the fatigue range. Cyclic loadingleads to some increase in deflection of reinforced concretemembers partly due to deformation of the concrete and partlydue to breakdown of bond between cracks. Since prestressedconcrete is uncracked under normal static service load condi-tions, the fluctuations of stress in the steel under cyclic loadingare small. Fatigue failure of the steel only occurs when substan-tial cracks have developed and deflections are generally unac-ceptable. The effect of cyclic loading on prestressed concrete isto increase deflection by a small amount, i.e. 20 to 30% largelyas a result of creep of the concrete. Large numbers of repetitionswithin the normal range of service loading do not reduce theultimate strength of prestressed or reinforced concrete. Becauseof its freedom from cracking, prestressed concrete behavesbetter than reinforced concrete under severe cyclic loading andhas therefore been used extensively for railway sleepers.

Resistance of beams to impact is indicated by the energyabsorbed in deforming which is given by the area of the loaddeflection curves. Referring again to Figures 12.1 and 12.2, thedeformation of prestressed and reinforced concrete beams hasbeen defined in three stages. In stage I, deformation is elasticand largely recoverable; in stage II, deformation is in part elasticbut accompanied by cracking and is partly recoverable; whilst instage IH, deformation is mainly due to permanent damage tothe materials. Since stages I and II represent the largest amountsof absorbed energy for prestressed concrete, this material has aconsiderable capacity for recovery after impact. For reinforcedconcrete, the energy absorbed in stage HI is substantially greaterthan in the other two stages. Thus, reinforced concrete does notshow much recovery after impact but has a high ultimate impactresistance which is appreciably higher than that for prestressedbeams designed for the same static loads. Prestressed concrete

beams are, however, better in resisting repetitions of relativelylight impacts with little residual damage.

So far, performance has been considered mainly in terms ofbending conditions, but conditions of direct stress in compres-sion exist in columns and walls. In such construction, unlesshigh bending moments are also likely to occur, prestressedconcrete would be unsuitable and reinforced concrete should beused with the steel acting in compression. For columns, trans-verse steel in the form of links is essential to contain thelongitudinal steel and ensure ultimate resistance to strains inexcess of those causing failure of plain concrete. Evidence fromlong-term tests also shows that the effect of creep of the concretein a column under load is to raise the stress in the longitudinalsteel to its yield stress and hence there is a need to retain it in itscorrect alignment. Walls when lightly reinforced are slightlyweaker than walls without reinforcement and they can thereforeonly be treated as reinforced when the longitudinal reinforce-ment exceeds a specific minimum.

Other aspects of behaviour which are of importance are shearand torsion. In each case if these cause failure, the mode offailure tends to be brittle and less ductile than bending failures.Hence in design, the procedure is to avoid such failure by theinclusion of sufficient transverse reinforcement to ensure bend-ing or compression failure in the event of severe overloading.

Members subjected solely to tension are relatively rare. If theyare of reinforced concrete, then the role of the concrete is toprotect the reinforcement which is designed to take the wholetensile force. In prestressed members, however, the precom-pressed concrete can sustain the tension until the load exceedsthe cracking loading when the behaviour reverts to that ofreinforced concrete with the steel carrying the whole of thetension, stiffened to some extent between cracks by the concrete.

For most building structures, the Building Regulations definefire resistance requirements, which are expressed in terms of arequired endurance under service load when components aresubjected to a standard heating regime. Both reinforced con-crete and prestressed concrete are primarily influenced in theirbehaviour in fire by the behaviour of the steel at high tempera-ture; as its temperature is raised its strength and yield character-istics are reduced. For reinforcing steels the rate of reduction instrength is lower than for steels used in tendons and hencegreater amounts of protection are needed for prestressed con-crete. This may take the form of concrete cover and the optionaladdition of insulating material. It is often easier, however, toprovide the greater thicknesses of cover needed for tendonswithout loss of efficiency than that needed for reinforcement,since the positioning of tendons is governed by different require-ments.

The need to provide adequate durability also affects theamount of cover required to the reinforcement or tendons. Asconcrete ages, carbon dioxide in the air causes carbonation ofthe concrete which, as it progresses, reduces its capacity forinhibiting rusting of the steel. For dense concrete the rate ofprogress is very low but, since defects exist, experience hasshown that a greater thickness of concrete is required to preventspalling of the concrete caused by expansion of the corrosionproducts on rusting. Cover requirements also affect the width ofcracks that are likely to occur and hence need attention indealing with serviceability.

These characteristics of the behaviour of both reinforced andprestressed concrete are considered in more detail in presentingdesign procedures.

12.3 Philosophy of design

The early developments of the design of reinforced concretewere crystallized in this country by the issue in 1934 of Recom-

mendations for a Code of Practice1 prepared by a committee setup by the Department of Scientific and Industrial Research. Itwas based on the premise that the stresses in the steel andconcrete should not exceed certain permissible values, related tothe strengths of the materials by safety factors, when thestructure was subjected to the maximum loads that it wouldneed to carry in service. The materials were assumed to behaveelastically and compatability of strains between steel and con-crete was ensured by assigning a value for the ratio of theirmoduli of elasticity. Some account was taken of the inelasticeffects of creep of concrete by adopting a low value for themodulus of elasticity of concrete in determining the modularratio for use in the design calculations. No account was taken ofthe effects of shrinkage and no estimate was made of theultimate strength of the structure. When the British StandardsInstitution issued its first Code for Reinforced Concrete, CP114,3 in 1948, it followed the same general approach. In therevision in 1957, however, there was an alternative method fordesign in flexure which limited the stresses to the same permiss-ible values as for elastic design but assumed that they weredistributed as at failure and avoided the use of the modularratio; this was therefore a form of ultimate strength design.

Limitations on the permissible stresses in the steel and onspan :depth ratios were imposed to guard against excessivedeflection or cracking. Thus it could be argued that CP 114provided for safety against failure and for the avoidance ofunserviceability.

The earliest formal presentation of a design procedure forprestressed concrete was contained in the First Report onPrestressed Concrete4 published by the Institution of StructuralEngineers in 1951. Many of the recommendations in that reportfound their way into the British Standard Code of Practice forPrestressed Concrete, CP 115,5 issued in 1959. It conformed withCP 114 in the sense that it was based primarily on the limitationof stresses to permissible values related to the strengths of thematerials with the object of preventing cracking and avoidingexcessive deflection. It also provided for the calculation ofultimate strength and introduced separate requirements forminimum load factors for the dead and imposed loads.

Thus, when the drafting of CP 1107 commenced in 1964 it hadalready been demonstrated that there were a number of limitingconditions or limit states which had to be considered by thedesigner in the overall conception of structural safety andadequacy. These were primarily limits of collapse, deformationand cracking, but other matters such as the effects of vibration,of fatigue, of deterioration with time or as a result of fire, neededattention in the design process.

A further major change in the content of structural codes firstintroduced in CP 110 in 1972 was the move towards consideringthe coordinated design of the structure as a whole for safety andserviceability rather than the separate design of its componentparts with only limited appreciation of their interaction. Thisdevelopment has become necessary partly as a result of theevolution of design philosophy and partly because the utiliza-tion of the materials has become more onerous following thegeneral increase in the levels of stress in both concrete and steelunder service conditions.

12.3.1 Criteria for limit state design

The aim in limit state design is to codify the proceduresnormally adopted by engineers in the design of structures toprovide safe, serviceable and economic construction with areasonable degree of certainty, and to do this with a betterappreciation of the margins of safety and of ignorance involved.As far as possible, it takes into account the variations likely tooccur in the loads on the structure and in the strength of the

materials of which it is comprised; it can allow for inadequaciesof construction and methods of analysis, and should lead todesign being more closely related to the risk of occurrence ofspecific conditions of failure and unserviceability.

For the purposes of design, both loads and strengths areexpressed in terms of characteristic values. For loads, these aredefined loads with a small but acceptable risk that they will beexceeded in service; they are given in the British Standardloadings for buildings,13 in BS 54OO for highway bridges and inother standards for other construction. To meet the needs oflimit state design, there has been a move in recent years awayfrom specifying loads as maximum values and towards express-ing them in terms of their likelihood of occurrence wherepossible determined from observations of their imposition onstructures (see Chapter 19).

The characteristic values of loads allow for normally expectedvariations in loading but not for: (1) unforeseen loading effects;(2) lack of precision in design calculations; (3) inadequacies inthe methods of analysis; and (4) dimensional errors in construc-tion which alter the assumed positions or directions of loads andtheir effects, e.g. incorrect positioning of reinforcement andinaccurate alignment of columns in successive storeys. Thevalues for loads used in design are therefore increased by partialsafety factors to cater for these effects and to provide the marginof safety appropriate to the need for ensuring that a particularlimit state is not reached. Thus, for conditions of failure, highervalues are used than for those of serviceability. Where acombination of loads is assumed to be acting, the partial safetyfactors for each source of loading are smaller since the simulta-neous occurrence of high values for each load is less likely. Theloads for use in the design are therefore the sums of the productsof the appropriate characteristic loads and their partial safetyfactors for the limit states and combination of loads beingconsidered. For simplicity, the structural code for concrete, BS8110,10 reduces the number of situations needing considerationto a minimum, as will be seen later.

Characteristic values for the strengths of materials are usuallygiven in the relevant standard or code. Research on materialsshows that their strengths conform reasonably closely to anormal distribution, and their characteristic strengths can there-fore be stated as follows:

Characteristic strength = mean strength — fc, x standard devi-ation or:

/k= /m-^x^f (12.1)

k} is usually given a value of 1.64, which ensures for a normaldistribution that not more than 5% of strengths are less than thecharacteristic strength. This definition of strength has beenadopted in British Standards for both steel and concrete.

The magnitude of the loads used in design is thereforeincreased by factors, partial safety factors for loads, to cater forthese effects and to provide a margin of safety appropriate to theneed for ensuring that any particular limit state is not reached.Thus, when envisaging conditions of failure, higher values forthe factors are adopted than when considering serviceability.

The strengths of the materials used in the design calculationsare those defined in the specification for the structure, which arechecked by physical tests. The strengths of the materials as theyexist in the structure, however, are likely to differ from thosedetermined from test specimens and some allowance is alsorequired for changes or deterioration with time. Partial safetyfactors for the materials are therefore introduced and thestrengths taken for design are the characteristic strengthsdivided by a partial safety factor, ym, which has a valuedepending on the limit state being considered and the nature ofthe material, being less for steel than for concrete.

An idealized and simplified situation for a homogeneousmaterial is illustrated in Figure 12.3. The provisions for safetyoutlined so far then require:

/vn<A/rm (12.2)

where Fk = the characteristic load

This conforms reasonably closely with what has now becomeaccepted practice in the recent revisions of British Standardscodes, and was first adopted in CP 110 in 1972. Currentthought, however, accepts the view that a further partial safetyfactor should be introduced to take account of the nature of theconstruction and its behaviour under overload conditions, e.g.whether it is capable of sustaining large deformations and sogiving warning of the imminence of collapse, and of the serious-ness of failure in terms of the risks to health, life and property.This factor, yc, might have a value of less than 1 for temporaryconstruction not normally occupied by human beings but ofmore than 1 for buildings with large spans used for publicassemblies. Thus design would then require:

Ft-WUfJr* (12.3)

For an idealized situation, the global factor of safety relatingcharacteristic loads to characteristic strength is then y f -y c ' y m .

If the concept of relating the factors of safety to the nature ofthe construction is not followed, then the global factor is y f -y m .Since reinforced concrete and prestressed concrete are compo-site materials, the value of the global factor for each limit statecannot be expressed as simply as this; it is dependent on the

interaction between steel and concrete, each of which has adifferent value for ym. Also, yf cannot be given a single value foreach limit state since the partial safety factors for dead,imposed, wind and other loads may differ and change withdifferent combinations of loads. Hence, only upper and lowervalues for the global factor can be defined which makes com-parison of the new Code with earlier or other codes imprecise.Nevertheless, in preparing the new Code, the aim has been toavoid substantial changes in the dimensions of the resultingstructures whilst at the same time obtaining more consistentlevels of safety and leaving room for development on morerational lines in the future.

It is convenient to divide the limit states to be considered indesign into two kinds, namely those concerned with collapseand those concerned with serviceability. Limit states of collapsedeal with overturning of the complete structure, failure of thewhole or a large part of the structure as a result of overstressingof a number of sections or buckling of a number of compressionmembers or as a result of a serious accident; the effects of fireand fatigue may also be included. Deflection, cracking, deterio-ration, corrosion and vibration are all aspects of serviceabilityand require limits of acceptability to be set for consideration. Inthe Code for Structural Concrete the limit states specificallydealt with are ultimate conditions in general, and deflection andcracking under the heading of serviceability. The criteria defin-ing the serviceability limits are set out in Table 12.2.

The partial safety factors yf to be used with the characteristicloads for dead, imposed and wind loads obtained from CP 3,Chapter V, or other appropriate specification, are set out inTable 12.3 with notes on interpretation for ultimate and service-ability limit states. The combinations of loading to be taken arethose which create the most severe conditions within the limitsspecified.

The partial safety factors for materials, ym, for the limit statesconsidered are given in Table 12.4 also with notes on theirinterpretation.

The Code for Structural Concrete has special provisions tosatisfy the requirement that, when a building suffers accidentaldamage, the amount of damage caused shall not be inconsistentwith the original cause. It would seem reasonable to apply thissame approach to other structures where safety and avoidanceof excessive damage are necessary considerations in the event ofaccidents. To achieve this in buildings, attention should be givento the choice of an appropriate plan form since this may have alarge influence on the mode of collapse as a result of an accident.When it is necessary to consider the effects of excessive loadsoutside those normally likely to be experienced or the residualstrength of a structure after accidental damage the value of yf

Strength

Load

Figure 12.3 Idealized relationship between load and strength fora structure

Table 12.2 Limits for serviceability conditions

Limit state

Cracking

Deflection

Reinforced concrete

Controlled by detailing rules for sizing and spacingreinforcement (BS 8110, Part 1)For unusual structures or conditions, more specificrecommendations are given in BS 81 10, Part 2

Normally controlled by rules for span : depth ratio(BS 81'10, Part 1). Exceptionally (BS 8110, Part 2),under vertical loads, not more than span/250: or forbrittle finishes and partitions, not more thanspan/500 or 20 mm, or for nonbrittle materials, notmore than span/350 or 20 mm

Prestressed concrete

Class 1 : No flexural tensile stressClass 2: Flexural tensile stresses permitted but no

crackingClass 3: Nominal flexural tensile stresses adopted to

limit cracking to not more than 0. 1 mm forsevere exposures, e.g. sea water and otherwisenot more than 0.2 mm (BS 8001, Part 1)

Normally controlled by limitations of stresses underservice loadings for cracking considerations (BS 8001,Part 1)Exceptionally (BS 8001, Part 2), as for reinforcedconcrete but also applied to upward deflections

Freq

uenc

y of

occ

uren

ce

Table 12.4 Partial safety factors for materials, ym

Limit state values for ym

Material ServiceabilityUltimate

Deflection Cracking

Concrete'in 1.5a 1.0f 1.3d

bending andcompression

Steel 1.15* 1.0 1.0

12ThJs value is related Io the standards of workmanship and supervision advocatedin the code for the production of concrete. If these standards are not applied, ahigher value should be used. It relates primarily to compressive strength ofconcrete.

''This value is for reinforcement in tension or tendons. For reinforcement incompression it is increased to 1.15+/y/2000.

'Calculations of deflection are based on the characteristic strength of the materialand therefore the modulus of elasticity of concrete derived for this strength is lessthan the mean value for the component or structure which strictly speaking wouldbe more relevant. This slightly conservative approach is justified in the interests ofsimplicity.

^This higher value for ym is selected for all calculations of stress for class 2prestressed concrete.

Tor shear without reinforcement, ym should be 1.25 and for bond at the ultimatelimit state should be 1.4.

can be taken as 1.05 for those loads likely to be experienced. Inthese circumstances also, the values for ym for steel and concretemay be taken as 1 and 1.3 respectively. The wind loading shouldbe taken as one-third the characteristic wind load. These lowvalues for the factors are acceptable because the loading con-sidered will not be experienced by most buildings and it wouldtherefore be uneconomic to design for it to be sustained withoutdamage.

12.3.2 Characteristics of materials

The grades of concrete used for reinforced and prestressedconcrete construction in the Structural Concrete Code areexpressed as the characteristic strengths determined from 28-day tests on cubes; they are given in Table 12.5 with theirapplication and properties relevant to design, including theincrease in cube strength with age. No data are given forlightweight aggregate concrete since its properties are dependent

on density in addition to strength as well as on the type ofaggregate. The figures for flexural and indirect tensile strengthrefer to concretes made with smooth gravel aggregates; forcrushed rock aggregates of tough texture, tensile strengths forthe same grades of concrete would be somewhat higher. Gener-ally, the minimum grade of concrete for reinforced concrete willbe grade 25; there are, however, areas in Britain where thenatural aggregates are not of high enough quality for concreteto meet this^grade even though its cement content is sufficient toconform with requirements for durability. Unless there arespecial needs, grades stronger than grade 40 are unlikely to beused for reinforced concrete. When lightweight aggregate isused, a lower grade, grade 15, is acceptable for reinforcedconcrete but it is preferable to use a higher grade for thelightweight aggregates of higher strength. No upper limit needsto be set on the strength for prestressed concrete and highergrades than grade 60 may therefore be used, but only specialcircumstances would justify the much greater cost and need forcontrol and supervision.

Calculations for conformity with ultimate and serviceabilitylimit states require the strength and deformation characteristicsfor concrete to be defined in numerical terms. In particular, dataare required on the relationships between stress and strain incompression under short-term loading and on creep and shrink-age when serviceability in the longer term is being considered.These aspects of behaviour are dealt with in section 12.4 and aresimplified for design later in this section, but it must berecognized that there are substantial variations in the behaviourof concrete, depending on its constituent materials and environ-ment, and that the values given for calculation should only beadopted if more reliable data are not available.

The strength properties of steel reinforcement and steeltendons are defined in British Standards which are summarizedin Tables 12.6 and 12.7. For reinforcing bars of hot-rolled steelthe characteristic strength is derived from the yield stress, butfor cold-worked bars or wire reinforcement, it is derived fromthe 0.2% proof stress. The characteristic strength of tendons forprestressed concrete, however, is derived from their ultimatetensile strengths. In each case these are the relevant strengths forcalculating ultimate strength for structural concrete members.Also in each case, the conformity with the specified characteris-tic strength is determined by ensuring that not more than two inforty consecutive results of tests made during the production ofthe steel falls below the specified value.

The design calculations for serviceability of structural con-crete require information on the modulus of elasticity of steel.

"The figures given in the table are the values for the partial safety factors }>f.Notes:(1) The minimum load for this combination should not be less than 1.0 Gk. When alternate spans are considered loaded in the design of continuous beams, for example, the loaded

spans should be assumed to carry 1.4 Gk + 1.6 0k and the 'unloaded' spans to carry 1.0 Gk.(2) The most serious load condition will usually occur when the design dead load is taken as 1.0 Gk, but for certain cantilevered structures, for example, a more serious situation

may exist when the design dead load for part of the structure is 1.40rk.

Table 12.3 Partial safety factors yf for loads and load effects

Load combinationDead and imposed load

Dead and wind load

Dead imposed and wind load

Limit state design loads0

Ultimate

f 1.4 Gk+ 1.6 Ck I\1.0Gk /

/1.0(7,+ 1.4HO\ 1.4 Gk + 1.4 W^

1.2 Gk+ 1.2 0,+ 1.2H^

Serviceability

See note (1) ( 1.0Gk+1.0gk

{LOG.

See note (2) 1.0Gk+1.0^k

aRecommendations in the Code of Practice for Structural Concrete.

The values adopted in the new code are: for reinforcement forall types of loading 200 kN/mm2, and for short-term loading forwire and strand of small diameter 200 kN/mm2 and for alloybars and strand of large diameter 175 kN/mm2.

In prestressed concrete, considerations of serviceability re-quire allowance not only for the effects of creep and shrinkageof the concrete but also relaxation of the tendons which maymodify the prestress conditions substantially. Appropriate re-quirements are incorporated in the standards which thereforeprovide guidance on values for relaxation to be used in design.

The stress-strain characteristics for concrete and steel may beneeded for calculations of the deformation of structural mem-bers under short-term loading or for assessing ultimate strength.

Table 12.6 British Standards for reinforcing bars for concrete

These are given in Figure 12.4 for concrete, in Figure 12.5 forreinforcement and in Figure 12.6 for tendons.

In interpreting these curves, the value of ym appropriate to thelimit state being considered should be obtained from Table 12.4.The values for the modulus of elasticity given in these figuresshould not be used for estimating the required extension oftendons. These data should be obtained from stress-straincurves for actual material being stressed, which are supplied bythe manufacturers.

The creep and shrinkage characteristics of concrete are con-sidered in section 12.4. Where it is necessary to calculate long-term deformation, the effects of creep can be convenientlyallowed for by adopting an effective modulus:

Type of steel"

BS 4449: 1978Hot-rolled steel barsfor the reinforcementof concrete250 Grade460 Grade

BS 4461: 1978Cold-worked steelbars for thereinforcement ofconcrete460 Grade

BS 4482: 1969Hard-drawn mildsteel wire

Specifiedcharacteristicstrengthb

(N/mm2)

250460

460

485

Elongation atfracture

(%)

2212

12

Diam. for 180°bend test

(no. of bardiam.)

23

3

Rebend test

Upper limit for:carbon content

(%)

0.250.40

0.25

0.25

sulphur content

(%)

0.060.05

0.06

0.06

phosphoruscontent

(%)

0.060.05

0.06

0.06

BS 4461 f 8, 10,12,16, 20, 25, 32 and 40 mm diameter, if a smaller size is required then 6 mm, if larger 50 mm diameter

BS 4482 - 5, 6, 7, 8, 10 and 12 mm diameter.*The characteristic strength is the yield stress below which not more than 5% of results should fall

Table 12.5 Grades and properties of structural concrete

Grade"(characteristicstrength 28days)(N/mm2)

15

2025

30

40

50

Cube strength0 (N/mm2) at the age of:

1 days

13.516.5

20

28

36

28 days

15

2025

30

40

50

2mths

2227.5

33

44

54

3 mths

2329

35

45.5

55.5

6 mths

2430

36

47.5

57.5

1 year

2531

37

50

60

Flexuralstrengthat 28 days

(N/mm2)

2.32.7

3.1

3.7

4.2

Indirecttensilestrengthat 28 days(N/mm2)

1.51.8

2.1

2.5

2.8

Modulus"ofelasticityat 28 days(kN/mm2)

2425

26

30

32

Use"

Reinforced concrete withlightweight aggregate

Reinforced concrete withnatural dense aggregatesPrestressed concrete forpost-tensioning,< 1 5 N/mm2 at transferPrestressed concrete forpretensioning,< 1 5 N/mm2 at transfer

Ec .CfT=^J(I + &£«) (12.4)

where £ci is the short-term modulus of elasticity of concrete and0C is the creep of concrete under a unit stress of 1 N/mm2.

The effects of shrinkage may be treated by assuming that theconcrete contracts without a change in stress except for thatcaused by the effect of the change in strain on the stress in thesteel. Some readjustment of strains then becomes necessary tobalance the forces in the cross-section by assuming that theconcrete is stressed under this strain in proportion to theeffective modulus.

12.4 Analytical and design procedures

12.4.1 Objectives

A recent trend in the approach to the initial design is to placemuch greater emphasis on the requirements for the durability ofconstruction, since experience has shown that deterioration is amore serious cause of failure and of high maintenance costs thanshortcomings in the structural calculations. It has thereforebecome more necessary to treat compliance with requirementsfor the quality of concrete, as placed in the construction and for

Note: These values for relaxation at 100Oh apply in temperate climates and are those obtained at 2O0C. When the prestressing steel is used at higher temperatures toprestress concrete, these percentage values should be increased. The increase may be as much as 2% for each 1O0C increase in temperature.

Table 12.7 British Standards for prestressing tendons for concrete

Type of steel

BS 5896: 1980 High-tensile steel wireand strand for the prestressing ofconcrete

Cold-drawn steel wire in mill coils

Stress relieved and may be crimped orindented and treated to reducerelaxation

Strand seven wire stress-relieved

standardsuperdrawn

BS 4757: 1971 Nineteenwire strand for theprestressing ofconcrete

as spun strand 25.4-31.8normal relaxationstrand 18.0low-relaxation strand 18.0

BS 4486: 1980Hot-rolled andhot-rolled andprocessed high-tensilealloy steel bars forthe prestressing ofconcrete

hot-rolled 20-40hot-rolled and 20-32processed

Range of sizesavailable

(dia. in mm)

3-5

4-7

9.3-15.28.0-15.7

12.7-18.0

659-979

370370

325-1300385-990

Range of specified Othercharacteristic informationbreaking load(kN)

12.2-30.8 t Relaxation: 1000 h\ 60% breaking load* 8%I 70% 10%

21.0-64.3 Relaxation: 1000 h

Class 1 260% b.l. 4.5% 1.0%70% 8.0% 2.5%80% 12.0% 4.5%

Relaxation: 100OhClass 1 2

92-232 \ 60% b.l. 4.5% 1.0%70-265 70% 8.0% 2.5%

209-380 80% 12.0% 4.5%

(Relaxation: 1000 h60% b.l. 9%, 70% b.l. 14%

60% 7.0%, 70% 12%60% 2.5%, 70% 3.5%

Relaxation: 1000 h all bars60% b.l. 1.5%

< 70% 3.5%

80% 6.0%

The breaking load for relaxation testing is the actualbreaking load

the protection of embedded steel by adequate concrete cover orother means, as being at least as important as compliance withrequirements derived from design calculations. Whilst it is notpracticable to define requirements for durability in terms of alimit state, it is nevertheless an aspect of the overall designprocess requiring primary attention.

For somewhat similar reasons, more care is now given to therequirements for fire resistance and information is presented inthe Code (Part 2) which can be used for an analytical approachas an alternative to satisfying somewhat arbitrary requirementsfor concrete cover in order to obtain the necessary fire grading.

Inevitably, structural calculations continue to be a major partof design. Whilst the principles of limit state design require allpossible limit states to be examined in the design of a particularstructure, part of the purpose of the Code is to provide guidanceon containing the effort required in design within reasonablelimits without overlooking significant features, i.e. limit states.In doing this, the Code relies on the experience of the designer toensure that the interpretation is sensible in each instance.

12.4.2 General assumptions

For most forms of concrete construction, with the possibleexception of slabs, it is most convenient at the present time tobase all design on elastic analysis of the structural system. Theanalysis would then apply directly to the serviceability limitstates of deflection and cracking and, with some limited redistri-bution of moments and shear forces to the ultimate state. Forslabs, other than one-way spanning slabs, it will usually be moresatisfactory to use yield line methods or the strip method forultimate design. For most construction it will usually be prefer-able to determine conformity with the serviceability limit statesby using the arbitrary rules given in the Code for span:depthratios and reinforcement detailing instead of calculating deflec-tions and widths of cracks.

The Code recommends procedures in Part 1 for the detaileddesign of beams, solid slabs supported by beams or walls, flatslabs, columns, walls, staircases and bases, which are given atsome length, and generally apply to both reinforced concreteand prestressed concrete. More information dealing with ulti-mate strength, serviceability and deformation due to creep,shrinkage and temperature effects is contained in Part 2. In thisrelatively brief summary, it is only possible to cover the morebasic recommendations, and detailed design therefore requiresreference to the main documents.

Other methods of analysis and design, where experimentalprocedures are used to develop the theoretical approach or todetermine performance, are acceptable but will normally onlybe employed for specially complex structures or where repeti-tion justifies more refinement than is obtained by establishedmethods of calculation. The assessment of stresses in the regionof load concentrations or of holes in continuous constructionmay be determined by photoelastic procedures. Model testingusing special materials or scaled concrete has found applicationsin developing design methods, e.g. in the design of concrete box-girder bridges and pressure vessels for nuclear power stations. Inprecast concrete construction particularly, the behaviour ofjoints can only be established by tests on full-scale assemblies. Itmay also be economic to derive the dimensions of precastcomponents for mass production by testing successively refinedprototypes to obtain the final form; this approach appliesparticularly in dealing with the requirements for fire resistance.The interpretation of test data for design requires the specialcare of experienced engineers since tests cannot embrace all theloads and load effects that may need to be sustained and thecircumstances that exist in actual structures cannot necessarilybe fully reproduced experimentally. When test results are ap-plied, therefore, there is a need to show convincingly the

Strain

E5= 205kN/mm2 for wire to BS 5896section 2 Table 4.

195 kN/mm2 for strand to BS 5896section 3.

206 kN/mm2 for rolled or rolled,stretched andtempered bars toBS 4486.

165 kN/mm2 for rolled andstretched bars toBS 4486

Figure 12.6 Short-term stress-strain curve for normal and lowrelaxation tendons

Figure 12.5 Short-term stress-strain curve for reinforcement

Figure 12.4 Short-term stress-strain curve for concrete

Parabola

Strain

Compression

Tension

Strain

Stre

ss (N

/mm

2)St

ress

(N/m

m2)

Stre

ss

justification for departures from established practice, especiallyso, if these lead to less conservative design. If test data areapplied in contexts for which they were not originally soughteven more caution is necessary.

For the purpose of analysis, the Code offers three alternativemethods for estimating beam and column stiffness: (1) theconcrete section; (2) the gross section; and (3) the transformedsection. The concrete section is the whole concrete sectionexcluding the reinforcement, the gross section is the wholeconcrete section including the reinforcement allowing for themodular ratio, usually taken as 15, and the transformed sectionis the section of concrete in compression together with thereinforcement again allowing for the modular ratio. Generally,the concrete section is most convenient for use in design. Forchecking existing structures or for design in special circum-stances, it would be more appropriate to use the transformedsection for reinforced concrete; in construction where flexuralcracking has occurred, however, the actual stiffnesses obtainedby this assumption will be greater since the concrete exerts sometensile stiffening in the regions between cracks through bondwith the reinforcement. The appropriate section for checkingthe design of existing or special prestressed concrete structures isthe gross section since cracking does not usually occur withelastic deformation under service loads even for class-3 pre-stressed concrete.

12.4.3 Robustness

The Building Regulations require that all buildings of morethan four storeys in height should be designed to resist acciden-tal damage. The Regulations require that these buildings shouldbe capable of sustaining removal of a structural member with-out excessive collapse resulting or should be able to withstandan internal pressure of 34 kN/m2 without collapse.

The layout of the structure and its general form should not besensitive to accidental damage whatever the cause. It is morerealistic to interpret this as meaning that, in the event of anaccident, the resulting damage should not be disproportionateto the magnitude of the cause. Where impact from vehicles is apossibility, buildings should be protected by barriers, such asbollards or earth banks. Greater margins should be allowed indesign when the occupancy of a building may result in a greaterthan normal risk of accident, e.g. in flour mills and bondedstores.

Provisions envisaged in the Code go further in some respectsin dealing with the effects of accidents than the Regulationsrequire. The recommendations for robustness deal with bothexpected and accidental forms of loading, and include thefollowing:

(1) All buildings should be so designed that all dead, imposedand wind loads are safely transmitted to the foundations.

(2) All buildings should be capable of withstanding a horizontaldesign ultimate load applied at roof and each floor levelsimultaneously corresponding to 1.5 % of the dead-weight ofthe structure between the mid height of the storey below andmid height of the storey above for floors, and the surface forthe roof. This, in effect, sets a lower limit for wind loadingfor the first two combinations of loading in Table 12.3.

(3) All buildings should be tied with effectively anchored andcontinuous reinforcement which is capable of withstandingthe notional forces outlined in the following paragraphs.This reinforcement may consist of bars provided to resiststresses due to normal loads, which may be ignored for thispurpose, and it may be assumed to be stressed up to itscharacteristic strength.

Buildings of four storeys or less require tying horizontally

in two directions approximately at right angles with internalties and peripheral ties.

Internal ties, which should be anchored to the peripheralties and should be accommodated in the beams or slabs,should be capable of resisting a notional force of: [(gq + <?k)/7.S](PfS)F1 kN/m or IF1 kN/m width, whichever is thegreater, where (gq + gk) is the sum of the average characteris-tic dead and imposed load in kilonewtons per square metre,and p is the greater of the distances in metres between thecentres of supporting columns, frames or walls of any twoadjacent floor spans parallel to the tie. F1 is the lesser of(20 + 4«0) or 60, «0 being the number of storeys. The spacingof the ties should not be more than \.5p.

Peripheral ties should be provided at each floor and rooflevel and be capable of withstanding a notional force of notless than 1F1 kN. They should be located within 1.2 m of theedge of the building.

Horizontal ties to external columns and walls should beprovided for each external column, in two directions forcorner columns, and for each metre length of external wallat each floor and at roof level. The notional force consideredshould be the greater of the following: 2F1 (or (Ft/2.5) x ceil-ing height m) kN, or 3% of the total design ultimate loadcarried by the column or wall at that level.

Buildings of five storeys or more require additional provi-sion for robustness, which usually will be met by theinclusion of vertical ties in all walls and columns. Theseshould be designed for a notional force corresponding to themaximum design ultimate dead and imposed load receivedby the column or wall from any one storey or roof.

(4) Where there are key elements in a building design, thefailure of which might cause extensive collapse, their designshould take their importance into account if their usecannot be avoided. Where vertical ties cannot be provided(see (3) above), provision should be made for bridging bythe structure above in the event of their removal.

The purpose of these recommendations is to ensure that allstructures are insensitive to damage from localized disturbances.It is therefore important in providing ties, for bridging or anyother action, that the arrangements are sound engineering.

12.4.4 Beams and slabs

The effective span (/) of beams or slabs, which are simplysupported, is taken as either the distance between the centres ofbearings or the clear distance between supports plus the effectivedepth, whichever is the smaller. For continuous members,however, the effective span is the distance between the centres ofthe supports. Whilst for a cantilever which forms part of acontinuous beam or slab, it is to the centre of the support, butfor an isolated cantilever the effective span is to the centre of thesupport plus half the effective depth.

The effective width of a flange to a T-beam may be taken asthe smaller of the width of the web plus one-fifth of the distancebetween points of zero moment or the actual width. Similarly,the effective width of flange for an L-beam is taken as thesmaller of the width of the web plus one-tenth of the distancebetween points of zero moment; for continuous beams thedistance between points of zero moment may be assumed to be0.7 L.

The lateral stability of beams may need attention, usually byproviding for adequate restraints and stiffness. The limitsbetween lateral restraints for simply supported beams or conti-nuous beams should not exceed 606C, or 25Qb2Jd, where d is theeffective depth and bc the breadth of the compression facemidway between supports. For cantilevers restrained only at thesupport, its length should not exceed 25bc or I00b2

cd.

The following loading conditions should usually be con-sidered in the design of continuous beams and slabs: (1) thedesign ultimate load of 1.4Gk + 1.6£k on all spans; and (2) thedesign ultimate load as (1) on alternate spans with I(/k onintermediate spans. When moments at sections are determinedby elastic analysis, the maximum moment may be reduced byredistribution provided that the calculated depth of the neutralaxis is not greater than (/?b - 0.4)d where d is the effective depthand /?b is:

moment at the section after redistributionmoment at the section before redistribution

and that the resistance moment at any section is not less than70% of the moment at that section from elastic analysis.

12.4.5 Continuous and two-way solid slabs

Slabs which are continuous in extent in one or two directionsmay be designed as simply supported, provided that continuousties that may be required for overall stability of the structure areincorporated in the construction. In such cases, cracking willdevelop in the top surface of the floors at their supports andsome provision will be needed for dealing with this in applyingfloor finishes.

Where slabs are required to span in one direction over anumber of supports, they should be designed for moments andshears, calculated in similar manner to those for continuousbeams.

If solid slabs are required to span in two directions, yield lineanalysis or the strip method of design may be used. BritishStandard 8110, however, gives simple methods for the design ofrectangular slabs for simply supported two-way panels and two-way continuous or restrained slabs.

12.4.6 Flat slab construction

Flat slab construction usually consists of a slab which spansbetween columns in two directions without supporting beams.Drops may be provided over the columns by increasing thedepths of the slab and sometimes the column heads may beflared to reduce shear stresses. The slabs may be solid or ribbedin two directions.

British Standard 8110 offers a method of design but does notexclude the use of other methods such as finite element analysisor other procedures. In the BS method, it is assumed that theslab is supported by a rectangular grid of columns in which theratio of the longer spans to the shorter spans is not greater than2. The slabs are divided longitudinally and transversely intocolumn strips and middle strips; the columns and column tripsare designed as frames spanning in each direction. Each frame isthen analysed elastically; a simplified method is given for thesituation where the structure is braced against lateral loadingand the column grid has a regular layout. Procedures are givenfor determining the widths of column strips and for the treat-ment of drops.

12.4.7 Frames

The loads to be adopted in the design of frames with theirfactors have already been given in Table 12.3. When consideringthe ultimate limit state, the forces, shears and moments calcu-lated for design should be the worst combinations of loadingregarded as feasible. British Standard 8110 gives some simpli-fied procedures, which may be used for a number of commonforms of construction. These analyse frameworks by breakingthem down into subframes and make some provision for

redistribution of moments. Two types of frame are dealt with -the no-sway frame, in which bracing, such as shear walls and liftor stair wells, are used to restrain sidesway, and sway frames, inwhich the frame itself provides the lateral restraint. For thelatter, the amount of moment redistribution allowed is restrictedwith further restrictions on frames of four or more storeys inheight to avoid excessive deflection and the possibility of frameinstability.

12.4.8 Columns and walls

The determination of the loads and moments on columns isgiven in BS 8110 to which reference should be made for details.A column is described as slender when the ratio of the effectivelength to the corresponding breadth with respect to either axis isgreater than 12 (10 for lightweight aggregate concrete); if theratio is less than 12, the column is said to be short. The effectivelength is dependent on the length of the column and on thedegree of restraint at the top and bottom connections with thestructure. Generally, the slenderness ratio for a column shouldnot be greater than 60. A distinction is made between bracedand unbraced columns, a column being described as bracedwhen the lateral stability of the whole structure is ensured byproviding walls or bracing to resist all horizontal forces.

The procedures for dealing with walls in BS 8110 have muchin common with those for columns. A concrete component isdefined as a wall when the greater of the lateral dimensions is atleast 4 times the smaller. For plain walls, however, the ratio maybe less (since columns without reinforcement are not recog-nized) and reduction factors are then applied. To be described asa reinforced wall, the area of vertical reinforcement should notbe less than 0.4% of the cross-sectional area of concrete; if theamount of reinforcement is less, the wall should be designed as aplain wall. Some reinforcement may be required in plain walls tocontrol cracking. A stocky wall is one in which the ratio ofeffective length to thickness does not exceed 12 (10 for light-weight concrete), otherwise the wall should be treated as beingslender. As for columns, the effective length is dependent on theheight and conditions of end-restraint. Methods for calculatingthe loads and moments on walls (as for columns) are also givenin some detail in BS 8110 to which reference should be made.

Provided the recommendations in the British Standards arefollowed, the deflections of columns and walls should not beexcessive.

12.5 Reinforced concrete

12.5.1 General

In the design of reinforced concrete to meet the requirements ofthe Code, BS 8110, it will usually be most appropriate toconsider the ultimate limit state first and then check the designagainst the requirements for cracking and deflection. This mightbe inappropriate in exceptional circumstances, e.g. where steelsof characteristic strengths in excess of 500 N/mm2 are being usedor where spans were exceptionally long: in these cases crackingor deflection might govern design. In the sections that follow,design will be treated on the assumptions that normal condi-tions obtain. For these the Code gives simplified treatments fordealing with both cracking and deflection. It also gives methodsmore suited to the exceptional cases for which reference to theCode should be made.

12.5.2 Beams

72.5.2.7 Bending

Ultimate resistance in bending is calculated by assuming that: