Reinforced Concrete Design to Eurocode 2 (EC2). Authors; (view Pages PDF · Limit State Design. W. H. Mosley, R. Hulse, J. H. Bungey. Pages Reinforced Concrete Design to Eurocode 2. Authors; (view affiliations). Giandomenico Toniolo Engineering book series (SPRTRCIENG). Download book PDF. This textbook describes the basic mechanical features of concrete and explains the main resistant mechanisms activated in the reinforced concrete structures and foundations when subjected to centred and eccentric axial force, bending moment, shear, torsion and prestressing.
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EUROCODE 2: BACKGROUND & APPLICATIONS. DESIGN OF . Exposure classes, structural classes and concrete cover. Conceptual design of slabs. .. Determination of the bending reinforcement for the T- beams. Reinforced Concrete Design to Eurocode 2 - Ebook download as PDF File .pdf), Text File .txt) or read book online. Reinforced Concrete Design to Eurocode [Mosley, Bungey, Hulse].pdf - Ebook download as PDF File .pdf) or read book online.
The dynamit modulus of elasticity, Ed. The standard test is bnscd on determining the resonant frequency or u pri sm spct: It i s al so possible to obtain a good estimate of Ed! The stanclarcl test for fu is on an unstressed specimen. The actual value of E for a concrete depends on many ractor! For limestone aggregates these vul ues should be reduced by a!
Thu magnitude of the modulus of el asticity is required when investigating the de! When considering short-tenn effects. Mi ld steel behaves as an clastic material , with the stmi n proportionul 1o the stress up to the yield, ut whic: After the yield point,! Iligh yield steel. Reinforced concrete design figure 1. All mater. The speciricd strength used in design is based on either the yield stress or a speci ried proof' stress. Removal of the l oad within the pl astic range would result in I he stress- strain di agram following a li ne approximately parallel to the loading pori i on - sec line BC in fi gure 1.
The steel will be left wi th a permanent strain AC. If the steel i s again loaded. Thus, the proportional limit for the second loading i s higher I han for the initial loading. The load deformntion of the steel is also dependent on the length of' time the load is applied.
Creep of the steel is of little significance in normal rei nforced concrete work, hut i1 is an important factor in prestres: Thi s shrinlwge i s liabl e to l: Huse cracki ng of the concrete, but it also has the beneficial c! Tccl of strengthening the bond between the concrete and the steel reinforcement.
Shrinkage begins to take place as soon as the concrete is mixed, and i s cause. Oncrcte and the aggregate. Furrher shrinkage is caused by evaporation of the water whi ch ri ses to the concrete smfnce. Even after the concrete has hardened, shrinkage continues as drying out persist s over many months, and any subsequent wetting and drying can also cause swelling and shrinkage.
Thermal shrinkage may be reduced by restricting the temperature rise during hydration, which may be achieved by the following procedures: Use a mix design with a l ow cement conterll or suitabl e cement replacement e.
Properties of reinforced concrete Avoid rapid hardening and f inely ground cement if possible. Keep aggregates and mixing water cool. Use steel shuttering and cool with a water spray. Strike the shuttering early to allow the heat of hydration to dissipate. A low water-cement ratio will help to reduce drying shrinkage by keeping to a minimum the vol ume of moisture that can be lost. Restraint of the shrinkage. The restraint may be caused externally by fixity with adjoining members or friction against an earth surface, and internnlly by the nction of the steel reinforcement.
For a long wall or fl oor sl ab, the restraint from adjoining concrete may be reduced by constructing successive bays instead of alternate bays. This allows the free end of every bay to contract before the next bay is cast.
Day-to-clay thermal expansion of the concrete can be greater than the movements caused by shrinkage. Thermal stresses and may be controlled by the correct positioni ng of movement or expansion j oints in a strut: When the tensi le stresses caused by shrinknge or thermal movement ext: To control the crack widt hs.
Calculation of stresses induced by shrinkage a Shrinkage restrained by the reinforcement The shrinkage stresses caused by rei nfort: The member shown in figure 1. Is Equating forces in the concrete and steel for equilibrium gives - A,f,c 1. Ac Substituting for in equation I. Es lUS I O: Hence from equ: This feature is accompanied by localised bond breakdown, adjacent to each crnck.
The equilibrium of the concrete and reinforcement is shown in figure 1. Thermal movement As the coefficients of thermal expansion of steel and concrete ar. The di f ferential thermal strain due to a temperature change T may be calculated as T a-r.
A maximum 'restraint factor' of 0. I'aint factor'. Tt is " phenomenon associated with many materia ls, but it is parti cularl y evident with concrete. For such a me1nber, a typical variation of deformati on with rjme is shown by the curve in fi gure 1. The characteristics of creep are 1. The fi nal deformation of the member can he three to four times the shorHerm elastic deformation.
The deformation is roughly proportional to the intensity of loading and to the inverse of the concrete strength. If the load is removed. There is a redistribution or load between the concrete and any steel present.
Thus the compressive stresses i n the steel are increased so that the steel takes a larger propo1tion of the load. The effects of creep are parti cularly important in beams, where the increased deflections may cause tJ1e opening of cracks, damage to finishes. Redistribution of stress between concrete and steel occurs primaril y in the uncracked compressive areas and has li ttle effect on the tension reinforcement other than reducing shrinkage stresses in some instances.
The provision of reinforcement in the compressi ve zone of a nexural member.
The durability of the concrete i s inlluenced by 1. Concrete can be exposed to a wide range of conditi ons such as the soil. The severity or the exposure governs the type of concrete mix required and the minimum cover to the reinforcing steel. Air entrainment is usually specifi ed where i t i s necessary to cater for repeated f'reezing and thawing. Adequate cover is essenti al to prevent corrosi ve agents reachi ng the rei nforcement through cracks and pervious concrete.
The thi ckness of cover required depends on the severity of' the exposure and the quality of the concrete us shown in t. The cover i s al so necessary to protect the reinforcemenL against a rapid ri se in temperature and subsequent l oss of strength during a fi re. Part 1. Durability requirements with rcluted design cul eul alions to check nnd control crack widt hs and depths arc descr ibed in more dctuil in chapter 6.
For exampl e, in the l ower columns of a multi-storey buil di ng a higher- S! These are usually cured, and tested after 28 days according to standard procedures. Table 1. Exposure conditions and durability can also afTect the choice of the mix design and the class of concrete.
Although Class Blast-furnace or sulfate-resisting cement may be used to resist chemical ullack, low-heat cements in massive sections to reduce the heat of hydrati on, or rapid-hardeni ng cement when a high enrly strength is required. These wi ll reduce the heat of hydration and may also lead to a smnlter pore structure and incn.: Generally, naturul aggregates found locnlly preferred: A "designed concrete' is one where the strength class, cement type.
With a 'designated concrete' the producer musr provide a material to satisfy the designated strength class and consistence workability using a particular aggregate site. Tabl e 1. The nominal si7. Grade bars arc hot-rolled m. Thi s type or bar can be more readily bent, so they have in l he past been used where smal l radius bends arc necessary, such as l inks in narrow beams or colunUls, but plai n bars arc not now recogni sed in the Uni on and they nJe no longer avai lable for general use i n the UK.
Square twisted bnrs have inferior bond characteristics and have been used in the past, although they are now obsolete. Deformed bars have a mechani cal bond wi th the conl: The ductility of reinforcing steel i s also classified for design purposes. Ribbed high yield bars may be classified as: This is the lowest ductil ity category and will include on moment redistribution which can be applied sec section 4. Class B - which is most commonl y used for reinforci ng bars.
Cl ass C- high ducti lity whi ch may be used in eart hquake design or simi l ar si tuations. Fl oor slabs. Thi s can give large economi cs in the detai ling of the reinforcement and also in site l abour costs of handling and fi xing. Prefabricated reinforcement bnr assembli es are also becoming increasingly popul ar for si mi lar reasons.
Welded fabric mesh made of ribbed wire greulcr than 6 mm dinmeter may be of any of the cluct. Reinforci ng bars in a member should either be straight or bent to standard shapes.
These shapes must be fully dimensioned and listed in a schedule of the reinforcement which is used on site for the bending and fixing of the bars. Standard bar shapes and a method of scheduling are specified in BS The bar types as previously described are commonly identified by the following codes: Despite the difficulty in assessing the precise loading and variations in the strength of the concrete and steel, these requirements have to be met.
Three basic methods using factors of safety to achieve safe, workable structures have been developed over many years; they are 1. The permissible stress method in which ultimate strengths of the materials are divided by a factor of safety to provide design stresses which are usually within the elastic range.
The load factor method in which the working loads are multiplied by a factor of safety. The limit state method which multipli es the working loads by partial factors of safety and also divides the materials' l. The permissible stress method has proved to be a simpl e and useful method but it does have some serious inconsistencies and is generally no longer In use. Because it is based on an elastic stress distribution, it is not really applicable to a semi-plastic material such as concrete, nor is it suitable when the deformations are not propor- tional to the load, as in slender columns.
It has al so been found to be unsafe when dealing with the stability of structures subject to overturning forces see example 2. As this method does not apply factors of safety to the material stresses, it cannot directly take account of the variability of the materials, and also it cannot be used to calculate the deflections or cracking at working loads. Again, this is a design method that has now been effectively superseded by modern limit state design methods.
The limit state method of design, now widely adopted across Europe and many other parts of the world, overcomes many of the disadvantages of the previous two methods. It does so by applying parlial factors of safety, both to the loads and to the material strengths, and the magnitude of t he factors may be varied so t hat they may be used either with the plastic condi tions in the ultimate state or with the more elastic stress range at working loads.
This fl exibility is particularly important if full benefits are to be obtained from development of improved concrete and steel properties. Thus, ru1y way in which a structure may cease to be fit for use will consti tute a l imit stale and the design aim is to avoid any such condition being reached during the expected life of the structure.
The two principal types of limit stale are the ultimate limit stntc and the serviceabi lity limit state. Deflection - the appearance or effi ciency of any part of the structure must not be adversely affected by nor should the comfort of the building users be adversely affected. Cracking - local damage due to cracki ng and must not affect the efficiency or durability of the struclUrc. Durability- this must be considered in terms of the proposed life of the structure and its conditions of exposure.
Other limit stares that may be reached include: Excessive vibration - which may cau: Fatigue- must be considered if' cyclic loading is li kely. Fire resistance - this must be considered in terms of resistance to collapse. Special ci rcumstances - any special requirements of the structure which are nol covered by any of the more common limit states, such as earthquake resistance, must be taken into account.
The relative importance of each limit stale will vary according to the nature of the structure. The usual procedure is to decide which is the ciUcial limit state for a particular structure and base the design on this, although durabil ity and fire resistance requirements may well innucnce initial member sizi ng and concrete class selection. Checks must also be made to ensure that all other relevant limit states arc satisfied by the results produced. Except in special cases, such as water-retaining structures, the ul timate limit staLe is generally critical for rei nforced concrete although subsequent serviceability checks may affect some of the details of the design.
Prestressed concrete design, however, is generally based on serviceabi li ty conditi ons with checks on the ult. These are called 'characteristic' strengths. The characteristi c strength is taken as that value helow which it is unlikely that more than 5 per cent of lhe results wi ll fa ll. This is given by Jk. The relnrionship between eharHcteristic and mean values aecounts for variations in results of test specimens and wi ll , therefore, reliect the method and cont. Number of test specimens Mean strength f.
These characteristic values represent the limits within which at least 90 per cenl of values will lie in practice. It is to be expected that not more than 5 per cent of cases will exceed the upper limit and not more than 5 per cent wi ll fa ll below the lower limit.
They arc uesign values that take into account the aecmacy with which the structural loading can be predicted. Usuall y. Lnck of adequate data. The strength of the material in an actual member.
This strength wi ll differ from that measured in a carefull y prcparcu test specimen and it is particul arly true l'or concrete where placing, compaction anu curing arc so important to the strength.
Steel, on the other hand, is a relatively consistent material requiring a small partial factor of safety. The severity of the limit state being considered. Thus, hi gher values are taken for the ultimate limit state than for the serviceability limit state. Recommended vallLes for l m are given in table 2. The values in the last two columns be used when the structure i s being designed for exceptional accidental design situations such as tl1e effects of tire or explosion.
It should be noted that desi gn errors and constructi onal inaccuracies have simil ar effects and are thus sensibl y grouped together. These factors will account udequutel y for normal conditions al though gross errors in desi gn or construction obviously cannot he cat ered for.
Acti ons arc cntegori setl us either permanent Od. Qk, ;, where the subscript 'i' indicates the i'th action. The rerms favourable and unfavourable refer to the effect or the action s on the design si tuation under consideration.
For example, if a beam. Limit state design 19 Accidental Reinforcing and Prestressing Steel 1. I shows how the panial safety factors at the ullirnate limit state from tables 2. Variable load - 2. Determine the weight of foundation required at A in order to resist uplift: Investigate the effect on these designs of a 7 per cent increase in the vatiablc action. Thus with a increase in the vnri nbl e action there is u significant increase in the upl i fl and the structure becomes unsafe.
A 7 per ccm increase in the variable action will not endanger the structure, since the acl. In fact in thi s case it would require an increase of 61 per ccm in the variable load before the uplift would exceed the wei ght of a 38 kN foundation.
Parts a and b of example 2. For exampl e, the seU"-wcight of the structure may be considered in combinat ion with the weight of furnishings and people, wi th or wi thout the effect of wind acting on the building which may al so act in more than one directi on In cal'es where actions are to be combined it is recommended that, in determining sui tubl e desi gn values, each characteri stic action i s not only multiplied by the parti al factors of safety, as di scussed above, but al so by a further f act.
Thi s! As can be seen in the table thi s is also dependent on the type of structure being designed. Combination values are used for designing for i the ultimate limit state and i i irreversible servicenbi lity limit states such as irreversible crucking due to temporary hut excessive overloading of the structure.
Tn these cases the vari nhle actions are multi pli ed by a vulue of 1T1 denoted as IJ! Quasi-permanent meaning 'almost. An exampl e of such a loading would be the effect of snow on the roofs of bui ldings at high altitudes where the weight of the snow may have to be sustained over weeks or months. Quasi -permanent combinations of actions are used in the consideration of i ultimate limil Slates involving accidemal actions and ii serviceability limit states attribmable to.
In these cases the variable actions are multiplied by a value of IJ! The values of 'T'2 give an estimation of the proportion of the total variable action that is l ikely to be associated with this particular combi nation of actions.
Limit state design: Figure 2. Design value Ed - factored permanent acti ons combined with factored si ngle leading variable action combined with factored remaining accompanying variable actions The 'factors' wi ll , in all cases, be the appropriate partial factor of safety hr taken together with the appropriate value of iii as given in table 2.
The L: Two other similar equati ons are given in EC2, the least tavourable of which can alternatively be used to give the design value. For accideal design silltations the design value of actions can be expressed in a -,i mi Jar way with the permanent and variable actions being combined with the effect of the accidental design situation such as fire or impact.
As previously indicated. If the single variable act ion is considered to he the leading variable action then wind loading will be the accompanying variable. The reverse may, however. In ' uch cases the factors given in tnble 2. The value of 1. Alternative equations indicated in 2.
Table 2. L but taking account of the different combinations of actions to be used in the three different situations discussed above. The terms in the expressions have the following meanings: Calculate each of the serviceability limit state design values as gi ven by equations 2. From table 2. T he global factor of safety against a particular type of failure may be obtained by multiplying the appropriate partial factors of safety.
Simil arl y, failure by crushing of the concrete in the compression zone hns u f'w.: Thus the basic values of partial factors chosen arc such! Each individual member must be capable of resisting the forces acting on it, so that the determination of these forces is an essential part of the design process.
The full analysis of a rigid concrete frame is rarely simple; but simplified calculations of adequate precision can often be made if the basic action of the structure is understood. The analysis must begin with an evaluation of all the loads carried by the structure, including its own weight. Many of the loads are variable in magnitude and position, and all possible critical arrangements of loads must be considered.
First the structure itself is rationalised into si mplifi ed forms that represent the load-carrying action of the prototype. The forces in each member can then be determined by one of the foll owing methods: Manual calculations are possible for the vast majority of structures, but may be tedious for large or complicated ones.
The computer can be an invaluable help in the analysis of even quite small frames, and for some calculations it is almost indispensable. However, the amount of output from a computer analysis is sometimes almost overwhelming; and then the results are most readily inter- preted when they are presented diagrammatically. Analysis of the structure 29 Si nce t he design of a reinforced concrete member is generally based on the ultimate li mit state, t he analysis is usually performed for loadings corresponding to that state.
Prestressed concrete members, however, are normally designed for serviceability loadi ngs, as discussed in chapter Permanent actions are those whi ch are normally constant during the structure's life. Variable actions, on the other hand. A table of values for some useful permanent loads and variable loads is given in the appendi x.
Once the sizes of all the structural members, and the details or the architectural requi remems and permanent lixtures have been established. For most rcinf'orccd concretes, a typi cal value for the self-weight is 25 kN per cubic merre. In the case of' a building, l.
A minimum parlit. Permanent actions arc generally calculated on a sli ghtl y conservati ve basis, so that a member will not need redesigning because of a small change in its dimensions.
Over- esti mation, however, should be done with care, since the permanent action can often actuall y reduce some of 01e forces i n parts of the structure as will be seen in the case of the hogging moments in 01e conti nuous beam in figure 3. For many of 1hem. Examples of variable actions on buildi ngs arc: A large building is unlikely to be carrying its full variable action simultaneously on all its floors. For this reason EN Similarly from the same code.
Although the wind load is a variable action. The partial factors of safety specified in the code arc discussed in chapter 2. Permanent and wind actions The variable load can usually cover al l or any part or the structure and, therefore, should be:: IITanged to cause the most: A stucly of the deflected shape of the beam would confirm this to be the case. Load combination 2. SOQ, 1.
Thus there is a similar loading pattern for the design hogging moment at each internal Mtpport of a continuous beam. It should be nmcd thai the UK: An el asti c anal ysi s i s generall y used to determine the di stribut ion of these forces within the strucmre; but to some c,xtent - reinforced concrete is a pl astic material, a l imited redi stribution of the cl asti c.: A pl astic yi eld-line theory may be used tn cal culate the moment s in concrete slabs.
The properti es of the materials, such as Young's modulus. The st iiTncsses of the members can be culculatecl on the basis of any one or the following: The concrete cross-section described in 1 is the si mpler to calculme and would normally be chosen. Figure 3. This procedure will be illustrated in the examples for a continuous beam and a building frame.
For these structures it is conventional to draw the bending-moment diagram on the tensi on side of the members. Sign Conventions 1. For the moment-distri hution analysis anti-clockwise support moments arc positive as. For subsequently calculating the moments along the span of a member, moments causing saggi ng nre positi ve, while moments causing hogging are negative, as illustrated in figure 3.
For the ultimate limi t state we need only consider the maximum load of l. A continuous beam is considered to have no fixity with the supports so that the beam is free to rotate. This assumplion is not strictly te for beams framing into columns and for that type of continuous beam it is more accurate to analyse them as part of a frame. A cominuous beam should be anal ysed for the loading arrangements which give the maximum stresses at each section.
The analysis to calculate the bending moments can be curri ed out manually by moment distribution or equi valent methods. For a beam or slab set monolithically into i ts supports, the design mornenl at the support can be taken as the moment at the face of the support. Continuous beams - the general case Having tletermincd the moments at the supports hy, say, moment di stribution, it i s necessary to calcul ate the moments in the spans and also the shear forces on the beam.
For a uniformly distributed load, the equati ons for the shears and the maximum 1-pan moments can be derived from the rollowing analysis. The critical loading pallerns for the ultimate limit state are shown in figure 3.
Table 3. It should be noted that the reduced stiffness of has been used for the end spans. Analysis of the structure: Thus for the first loading a1rangemem and span AB, using the sign convention of figure 3. The individual bending-moment diagrams arc combined in figure 3. Such envelope diagrams arc used in the detailed de! Even so. J67 I "'-J J """""J 9 7. J 8 s "'-J 85 4 [' The values of these coefficients are shown in diagrammatic form in figure 3. They can be analysed as a compl ete space frame or be di vided into a series of plane frames.
Bridge deck-type structures can be anal ysed as an equi valent gri llage. All these methods lend themsel ves to solution by computer. The general procedure for a building i s ro: The slabs can be ei ther one-way spanning or two- way spanning.
The columns and main beums are consi dered as a series of rigid plane f rames whi ch can be divided into two types: I braced frames supporting vertical londs onl y, 2 f rames supporting vertical and lateral loads. Type one frames are in buildings where none of the lateral loads such as wind are lransmitted to the colunUJs and beams but arc resisted by much more sti ffer el ements such as shear walls, lift shafts or stairwells.
Type two frames an; designed to resist the lateral loads, which cause bending, sheari ng and axinl loads in the beams and columns. For both types of frames the axial forces in the columns can generally be calculated as if the beams and slabs were simply supported. The frame shown in figure 3. The substi tute frumc I in figure 3. An analysis of thi s frame wi ll gi ve the bending moments and shearing forces in the beams and columns for Lhe ll oor l evel consiclcrccl.
Substitut e frame 2 i s a single span combined with its connecting columns and two adjacent spans, all li xed at their remote ends. This frame may be Ul'ed to determine the bending moments and shearing forces in the centrul beam.
Provided that the central span is greater than the two adjacent spans, the bendi ng moments in the columns can also be found wi th this frame. Substitute f rame 3 can be used to fi nd the moments in the columns only. It consists of a si ngle j unction, with the remote ends of the members fixed.
This type of subframe woul d be used when bean'ts have been anal ysed as continuous over simple supports. I n frames 2 and 3, the assumption of fixed ends to the outer beams over-esti mates their stiffnesses.
These values are, therefore, halved to allow for the flexibility resulting from continuity. The various critical loading patterns to produce maxjmum stresses have to be considered. In general these loading patterns for the ultimate limit state are as shown in figure 3.
OGk -sec figure 7. The analysis of the sub frame will be carried out by moment distributiou: The moment distribution for the first loading arrangement is shown in table 3. In the table, the distributi on for each upper and lower column have been combined, since thi s simplifi es the layout for the ca.
AB BA Cols. CD LM 0. Reinforced concrete design Figure 3. For the' first loading arrangement and span AB: These diagrams have been combined in figure 3. A comparison of the design envelopes of figure 3. Not only is the analysis of a subframe more precise, but many moments and shears in the beam arc smaller in magnitude. The stiffnesses of these members are ident ical to those calculated in example 3.
A horizontal force should also be appli ed at each level of a structure resulting from u notional incl inati on oft11e vertical members representing imperfections. The value of this depends on building height and number of columns EC2 clause 5. This should be added to any wind l oads at the ultimate limit state An unbraced frame subjected to wind forcel: The vertical-l oading analysis cun be carried out by the methods described previ ously.
The anal ysi s for the l ateral loads ,hould be kept separate. The forces may be cal culated by an elastic computer anal ysis or by a si mplifi ed approximate method. For preliminary design cal culntions.
A suitable approximate anal ysis is the cantilever method.
It assumes that: It is al so usual to assume that ull the in a storey arc of equal cross-sectional area. It shoul d he emphasised that these approximate methods may give quite inaccurate results for irregular or high-rise structures. Applicati on of thi s method is probabl y best Illustrated by an example.
Thi s action i s assumed to be tnmsfcrrecl to the frame as a concentrated loud at each floor level as indicated in the fi gure.
By inspection, there is tension in the two columns w the left and compression in the columns to the ri ght; and by assumption 2 the axi al forces jn columns arc proporti onal to their distances rrom the centre line of the frame.
OkN ' --r C! HkN 5. J --, L OP The analysis of the frame continues by considering a section through the top-storey columns: The forces in this subl'rame arc calculated as follows. Shear wall s are very effective in resisting horizontal loads such as P,. As the walls arc relatively thin they of! The Aoor slabs which arc supported by the walls also act as rigid diaphragms which transfer and distribute the hori zontal forces into the shear wa ll s.
The shear walls act us vertical cantilevers transferring the hori zontal loads to the structural rouncl ations. The relati ve lre: Calculate the proportion of the JOOkN horizontal load carried by each of the walls. The calculation procedure for this case is: Determine the location of the centre of rotation by taking moments of the wal l stiffnesses k about conveni ent axes.
D and E as shown in figure 3. The relative stiffness of each shear wall is shown in the figure in tenus of multiples of k. A 20 0 12 I n the plane frame the second moment or area lc of the columns is equivalent to that of the wall on either side of the Clpenings. The second moment of area lb of the beams i s equi valent to that part of the wall between the openings. The lengths of beam that extend beyond the openings as shown shaded in flgure 3.
The equi valent plane frame would be analysed by computer with a pl ane f rame program. A method of a structure wi th shear walls and structural frames as one equivalent linked-plane frame is illustrated by the example in figure 3. I n the actual structure shown in pl an there arc four frames of type A and two frames of type B whi ch include shear wall s.
Similarly the two type B frames are lumped together into one frame whose member sti ffnesses arc doubled. The two shear wall s are represented by one column having the secti onal properti es of the sum of the two shear walls.
For purposes of analysis tJ1is column is connected to the rest of its frame by beams with a very high bending sti ffness, say I rimes that of the other beams so as to represent the width and rigidi ty of the shear wall.
The link beams transfer the l oads axi ally between the two types of frames A and B sc1 representing the rigid diaphragm action of the concrete floor slabs. These link beams, pi nned at their ends, woul d be given a cross-sectional area of say LOOO times that of the other beams in the frame.
As all the beams in the struclnral frames arc pressing against the ri gid shear wal l in the computer model the effects of axial shortening in beams wi ll be exaggerated. In the computer output the member forces for type A frames would need to be divided by a factor or four and those for type 13 frames by a factor of two.
The assumption of elastic behaviour is reasonably true for low stress level! E Curvature This is recognised in EC2, by allowing redistribution of the clastic moments subject to ccnain limitations. Reinforced concrete behaves in a manner midway between that or steel and concrete. The stress- strain curves for the two materials figures 1. The latter will fail at a relatively small compressive strain. A typical moment- curvature diagram for a reinforced concrete member is shown in figure 3. Provided rotation of a hinge docs not cause crushing of the concrete.
This requirement is considered in more detai l in chapter 4. Usually it is the maximum support moments which arc reduced. The emcnts for applying moment redistribution arc: The cor11inuous beams or sl abs are predominately to fkxure. The rmio of adjacent sptms be in Lhe range of 0. The column design moments must not be reduced. There arc other restrictions on the of moment. This entails limjtations on the grade of rei nforcing steel.. The moment at support B can be calculated.
This exampl e illustrates how, with redistribution the moments al a section of beam can be reduced without exceeding the maximum c. Nevertheless the total des1gn of a structure does depend on the analys1s and design of the individual member sections.
Wherever possible the analysis should be kept simple, yet it should be based on the observed and tested behaviour of reinforced concrete members. The manipula- tion and juggling with equations should never be allowed to obscure the fundamental principles that unite the analysis. The three most important principles are 1. The stresses and strains are related by the material properties, Including the stress- strain curves for concrete and steel.
The distribution of strains must be compa- tible with the d1storted shape of the cross- section. The resultant forces developed by the sect1on must balance the applied loads for static equilibrium.
These principles are true Irrespective of how the stresses and stra1ns are distributed, or how the member is loaded, or whatever the shape of the cross-section This chapter describes and analyses the action of a member section under load. It derives the basic equations used in design and also those equations required for Emphasis has been placed mostly on the analysis a sociated with the ultimate limit state but the behaviour of the section withm the elastic range and the serviceability limit state has also been cons1dered Section 4.
It should be noted that EC2 does not g1ve any explicit equations for the analysis or design of sections. The equations given in this chapter are developed from the principles of EC2 in a form comparable with the Quations formerly given in BS These arc in un idealised form wh1ch can be used in the anCIIysh. The ultimate design:.
Concrete cla:. The behaviour of the steel identical in ten,ion and comprc! Short-term design wess-stmin 1,. However the more commonly used curve shown in figure 4. Cd th1s chapter and throughout the text. It i-; also thar plane of n member remuin atter so thnt the section thl.! The relntlonshtp.
J - d 1' the effective depth of the beam and d' is the depth olthe ment. Inserting these values for and: EC'2 the depth of neutral axis to 0. Thi' When moment redi,tnbuuon '' apphcd these maxunum values of.
Abo the areas of the iaten as When moment redistribution is uppl ied, reference be mudc to 4.
A, lienee 4. Equation 4. R fm the lever arm: The lower limi t of 'l. This is the maximum value allowcc. So that For equilibrium of the compres-. If thi. J2 case if trying. X lHH mrn value of. When this is not the ca"e, reference should he made to 'ection 4. For 1his condition the depth of neutral axis, R7J;dd- d' Muluplying both ;,ide;, or equation 4. II I' cl.. Tbe area of tension steel is calculated from a modified equation 4.
The constant11 r concretes up to clalo. A, d-. Before 1hc equation' can be 1he steel r. Values of t: S lOOAJbd 2. Determine e areas of reinforcement required. For equilibrium of the and compressive forces on the. Stress Block therefore 0.
Also d' j. N m If the depth of neutral axis was such that the or tensile had not yielded. The btcel at halancc would! Flanged section - the depth of the stress block lies within the: The relation between the lever arm. MPLE 4. Lever arm: An alternative procedure is to calculate the moment of resi tance. Hence if the moment. The material strengths arc. Section Stress Block In figure 4.
Applied moment I HO x:: Wmm 0. For the equilibrium of the section F,1 - F,1 I Few or 0. F,1 F" 4 I - R Depth of blocl Applying thi M 11 ul. Analysis of the section 79 4. To allow for thi s. This vnlue of should replace 0.
For a l. S bloc I,. T34 ] l0. In the analy,i'i that follows. For a load the principles ot lt hrium.
The area of concrete m compression nut reduced to allow for the concrete hy the comprc: This could he en tnto uccount by reducing the stress.
I Basic equations and design charts The applied force N mu! Hence, t: C' fckbh II 0. With Iorge effective eccentricity. From the linear strain distribution of figure 4. WiLh cy nnd from equation 4. N Nhut. To calculate N and M at this stage, corrcspondtng to potnt s in ,. Figure 4. These diagram. Thi-; ' nn longer true for unsymmetrical steel area. The ultimate axial lond No acting through the cert1id a unifnrm the section with yielding of all the reinf lrccment.
With uniform:. Stratn The locution of the plasttc centroid determined by of all the stre rewltants about an arhitrary axil. X7 X 0. I Compatibility of used in table 4 3. N and '"' could have been calculated for imcrmediulc val ue! N kN 0, 0 M kNm Figure 4. The equilihrium equauon!
Table 4. The intcntction diagram b constructed figure ! Th1' would ceJtai nly he the. In ligure 4. I"C' J.. L Analysis of the section 5 4.
Fer rcctangular-paraholic: In practice it i' nllt generally used in design calcultHion , cxcept for lrqllld- etaming: With the triangular strc A steel arta f. Analysis of the section 9 a. I rom the ltnear Mrain distribution of figure 4. Ec uations 4. Ll - ' Figure 4. Nimm' and l:. A, bh 0. Punching shear caused by concentrated loads on slabs is covered in section 8 1. The din.! Ncar the support. I he actual hehavillur of com: In EC'2: The use of method allows the designer to seck out economic!.
Even p: The shear capacity of the concrete. VRtt c. The concrete acts the top 1. The bottom chord is the horilCmtal tension steel anu the vertical links are the transverse tension members. It is set by EC2 to h: Con ,iderution of the compressive strength of the diagonal concrete strut and it1- unglc 0; 2. Calculation of the required shear reinfon: Thus the maximum design shear force VK. With reference to figure 5.
I Bb,. R7 cot 0. Otherwise the value for 0 can be calculated from equation 5. EC2 that. Equation 5. Resolving forces hol"iz. In practice. Tim, is discussed further and illuqruted in section 7.
Equntions 5. Figure 5. C2 places a lower and upper limit of 1. This corresponds to limiting B ro 45 and 22 respectively. The fi nal deformation of the member can he three to four times the shorHerm elastic deformation.
The deformation is roughly proportional to the intensity of loading and to the inverse of the concrete strength. If the load is removed. There is a redistribution or load between the concrete and any steel present. Thus the compressive stresses i n the steel are increased so that the steel takes a larger propo1tion of the load.
The effects of creep are parti cularly important in beams, where the increased deflections may cause tJ1e opening of cracks, damage to finishes. Redistribution of stress between concrete and steel occurs primaril y in the uncracked compressive areas and has li ttle effect on the tension reinforcement other than reducing shrinkage stresses in some instances. The provision of reinforcement in the compressi ve zone of a nexural member.
The durability of the concrete i s inlluenced by 1. Concrete can be exposed to a wide range of conditi ons such as the soil. The severity or the exposure governs the type of concrete mix required and the minimum cover to the reinforcing steel. Whatever the exposure, the concrete mix should be made from impervious and chemi cally inert A dense, well -compacted concrete wit h a low water- cement ratio is all important and for some soi l condi tions it is advisabl e to usc a sulfate- resiHti ng cement.
Air entrainment is usually specifi ed where i t i s necessary to cater for repeated f'reezing and thawing. Adequate cover is essenti al to prevent corrosi ve agents reachi ng the rei nforcement through cracks and pervious concrete. The thi ckness of cover required depends on the severity of' the exposure and the quality of the concrete us shown in t. The cover i s al so necessary to protect the reinforcemenL against a rapid ri se in temperature and subsequent l oss of strength during a fi re.
Part 1. Durability requirements with rcluted design cul eul alions to check nnd control crack widt hs and depths arc descr ibed in more dctuil in chapter 6.