Structural analysis methods/design approaches, assumptions and Approximations for practical life line structure

Transcription

1 Chapter 6 Structural analysis methods/design approaches, assumptions and Approximations for practical life line structure 6.1 Introduction This chapter deals with different types of structural analysis and design different types of connections of members, different design assumptions and approximations different design philosophies etc. In the analysis of structure i.e. determination of the internal forces like axial compression, bending moment, shear force, twisting moment etc. in the component members for which these members are to be designed, under the action of given external loads. This process requires the knowledge of Structural Mechanics which includes mechanics of rigid bodies (i.e mechanic of forces, mechanics of deformable bodies (i.e. mechanics of deformations) and theory of structures (i.e. the science dealing with response of structural system to external loads). A brief review is taken of structural analysis to refresh the basic principles. 6.2 Methods of Analysis The various approaches to structural analysis are mentioned below. 1) Elastic Analysis based on Elastic theory. 2) Limit Analysis based on Plastic Theory or Ultimate load theory. Generally, elastic analysis is utilised in working stress or permissible stress Method of design ( WSM) moreover the Limit analysis is utilised in Ultimate strength or Ultimate Load method of design (ULM). Modified version related ultimate load method is known as Limit state method. Therefore Limit state method of design comprise design for ultimate limit state for ultimate load theory is applicable, and also for service state at this point elastic theory applies, consequently requiring study of both the theories.

2 Simultaneously, one should not be confused amid limit state philosophy of design with limit analysis. Latter is a process or method of structural analysis collapse whereas the former is a process or method of design on behalf of different limit states. In this section, both these approaches of structural analysis will be briefly reviewed Elastic Analysis It the study of behaviour plus strength of the structures and members at working loads. This is based on following assumptions which is related to Hook's law: 1. Relation amid force and displacement is linear (Hooke's Law is applicable). 2. Displacement are extremely small compared to geometry of structure in the sense that they does affect analysis Methods of elastic analysis broadly can be classified as below: a) Classical Methods: i. Consistent deformation Method ii. Slope-deflection Method iii. Methods of strain energy. b) Relaxation/Iterative Methods: i. Moment Distribution method. ii. Kani's Method c) Computer Methods i. Matrix Method, ii. Finite Difference Method, iii. Finite Element Method. d) Approximate Methods:

3 i. Substitute Frame Method, ii. Portal Method. iii. Cantilever Method, Coefficient method iv. Coefficients mentioned in design hand books or IS codes are utilised to get force bending moment, etc With the and unproblematic access computers, we can divide the above methods into two main groups, First group consist of methods which are more suitable for manual. Method of consistent deformations, Kani's Method, moment distribution method coefficient method and approximate methods come in this group or category. Substitute Frame method be appropriate for analyzing a building frame for loads, while the cantilever method and the portal appropriate for analyzing the horizontal loads effects on frames. Second group comprises of methods need the utilisation of computers. Matrix as well as computer methods described in arts (iii) and (iv) at the top come within this group. The scope of this thesis work is restricted to design of Practical life line structure. As per investigation it is (G+3) storied building, hence discussion can be and hence is limited to utilization substitute frame method for analysis of building frames intended for vertical loading only. Coefficient method or approach of determination design forces e.g. axial loads, bending moment, shear force, etc. by utilization of coefficient obtainable for standard cases of loading, is extremely common in building design generally for analyzing simple or easy frames and standard beams such as simply supported, cantilever, and continuous beams as well as slabs, and single storied single bay rectangular portal frames. Since in many cases redistribution of moments is carried out therefore it is necessary to know the limit analysis and redistribution of moments.

4 6.2.2 Limit Analysis: It is a type of analysis dealing within the study and behaviour of structure and members at collapse. This is totally plastic theory based, for structure made up of perfectly plastic it is based mainly on ultimate load theory meant for structure of reinforced concrete, the behaviour of it is characterized by yielding of steel and crushing of concrete at collapse. Therefore It have to be borne in brain that this type of ultimate state is never accepted to be attained with utilisation of suitable safety factors. However knowledge related to strength and behaviour on collapse is extremely essential to know the safety margin. Consider the behaviour of astatically indeterminate fixed beam of span L, fixed at both ends A,B and carrying a uniformly distributed load of intensity w. The maximum bending moment occurs at ends A and B rather than at the mid-span. Now, the load is gradually increased till the collapse occurs. Initially the beam behaves elastically till the stress at any section reaches its yield value at ultimate moment capacity. The plastic hinges develop at ends due to plasticisation of concrete in compression, and cracking of concrete accompanied by yielding of steel in tension The UDL has increased from w to this stage -1. The plastic hinges destroy full fixity at A and B. At this stage the collapse does not occur the beam behaves as a with constant moment Mu acting at A and B see Fig M A =M B =w 1 L 2 /12=M u i.e M ua = M ub = w 1 L 2 /12=M ur The corresponding moment at mid-span is just w 1 L 2 /24 On loading further, the moments at the fixed ends remain unchanged as the beam section at their locations cannot offer any additional resistance, but the moment in the span region increases. (Thus, additional load is resisted by span region only)

5 Fig Redistribution of Moments Fig Redistribution of Moments In fixed Beam Stage- I In fixed Beam Stage- II A stage-ii is reached when the bending moment at mid-span reaches its ultimate moment capacity with the formation of hinges at C as shown in Fig This causes division of beam into two segments AC and occurring at A,B and C. At this stage mechanism is supposed to be formed or created which lead the collapse of beam. The bending moment at mid-span = M uc =M ur = w 1 L 2 /8 M ua = w 1 L 2 /8 - w 1 L 2 /12 But M ua = M ur r, M ur = w 1 L 2 /8 M ur or M ur = w 1 L 2 /16 But M ua is also equal to w 1 L 2 /12, w 1 L 2 /16 = w 1 L 2 /12 w u = 1.33w 1 Thus, there is increase in load carrying capacity of the beam by 33% over the load w1 obtained by elastic analysis at stage - i. The merit of analysis lies fact that it gives higher load carrying capacity for an indeterminate structure due to redistribution of moments

6 Advantages of Redistribution of Moments. (1) In case of indeterminate structures, it helps to reduce bending moments in the peak regions such as beam column junction or supports of continuous beams, thereby the congestion of reinforcement is reduced making detailing and concreting easier. (2)Reduction in support moment not only helps in reducing steel at supports but utilizes higher moment resisting capacity of a flanged section in the span region. (3) It ensure reinforced failure since the depth of the neutral axis decreases with the increase in the percentage redistribution of moments. (4) It gives distribution of moments along the length of the member and makes detailing easier and gives economic design. (5) It is not only reduces the moment at support but, many times, it also does not increase the design moment at mid-span. This can be seen in the case of continuous beam designed of maximum moments decided by bending moment envelop i.e. diagram for maximum moment got by all likely loading arrangements consideration. The procedure limit analysis involves the following 1. Elastic Analysis for ultimate (factored) loads 2. Redistribution of moments subject to following conditions a. Equilibrium shall always be maintained i.e the sum of the support moment and the span moment shall be equal to the maximum span moment for a simply supported beam. Thus, for beam subjected to uniformly distributed load. M sup + Mspan = w u L 2 /8

7 In brief,in support moment by dm must be accompanied corresponding increase in the mid-span moment by dm/2. b. Amount of redistribution (dm=elastic moment M EU assumed design moment MDU) shall not exceed prescribed percentage given below. -30% in Limit state design The limitations of 30% redistribution of moments has been imposed to avoid large rotation, and hence excessive deflection and cracking which affect the serviceability of the structure. -15% to 20% redistribution may be normally taken as reasonable limit. -10% in case of structures over 4 storey height to prevent lateral instability. (c) The design moment MDU shall not be less than 0.7 times the elastic moment MEU at ultimate state. Since M EU =1.5 x working moment M EW, it means that M DU 0.7 X 1.5 M EW i,e M DU 1.05 M EW. In other words the design moment after redistribution shall not be less than the working moment (i,e MEW) at service load. The depth of neutral axis shall be limited to k u.limit (=x ulimit /d) < = (0.6-dM/100) or k u.max whichever is less. This is required satisfying the requirement of ration capacity at a point where redistribution of moment is done. 6.3 Elastic analysis of Building frame General The structural frame of a building consists of floor and roof slabs and supporting beams and columns. All the components of the frame are usually

8 cast together forming a monolithic construction. The resulting frame acts as one integral unit. The monolithic casting of members and proper detailing enables to have rigid connections between the members so that every member acts integrally with the connected members. The continuity between the members help to distribute the forces to number of connected members. This enhances the reserve strength of the structure and eliminates the possibility of collapse of the structure due to failure of any component member on account of effect of localized loads and actions safety of the building as a whole is increased. The rigidly of the connection is also desirable or rather essential for resisting horizontal loads like wind load or earthquake load. A typical frame of a multi-storeyed building is shown in Fig say the frame consists of continuous one-way slab S1 cast monolithically with secondary beams B1, B2 and main beams B3, B4, B5.The main beam is continuous over columns and is rigidly connected to them above and below that floor. The frame as a whole consists of number of rigid joints. The structure is highly statically indeterminate. The exact analysis of the entire frame by use of classical methods is beyond the capacity of manual/hand calculations. It can only be done by computer. Besides formulation will involve large number of unknown displacements and large computer memory. The solution is also, therefore costly and beyond there ach of a common designer. In fact such rigorous computer analysis will be required only for tall and unconventional irregular structures. Besides in R.C. buildings of ground plus three storeys does not demand use of exact classical analysis. The approximate methods are more than adequate.

9 Plan Elevation Fig. 6.3 Plan showing the type of Slab. Approximate methods as are based on principle of dividing the structure into parts and analyzing only part of interest, disregarding the effect of member resistances and their loads which are distant away from the interested member. The above simplification is based on the fact that a load on any member and its stiffness hardly affect two storeys or two spans beyond existing member. For example, suppose unit moment is applied at one joint end of continuous beam produces a moment of only 27% at next i.e. second joint, 7% at 3rd joint and only 2% at 4th joint. The assumed approximate reduces the computational efforts to a great extent without much affecting the accuracy, and results obtained are on the safer side. The approximate method, therefore adopts some standardized small portions of the whole frame, known as substitute Frames or sub-frames principally consisting of members of interest and another adjacent members connected to it. The method of analysis is known is Substitute Frame Method. Since the scope of this thesis is restricted to Practical life line structure i.e. the buildings of up to storey height for which the effect of horizontal loads is not worth considering the substitute frame method, suitable for vertical loads only will be discussed here.

10 Fig Figure showing analysis outcomes beyond supports Substitute frames: Analysis for vertical loads A building frame, in fact is a three dimensional frame i.e. A space frame show in Fig. practical life line structure of G+3 storied commercial building analysis of a space frame is complex, laborious and also time consuming. Besidess it is also not necessary or not even justified for degree of accuracy required R.C. construction. Therefore as a first level or degree of Fig approximation, threee dimensional space frame is divided into a number of two dimensional plane frames. Each plane frame Fig 6.6. analysed the loads horizontal or vertical in the plane of frame assumed to behave independently i.e. disregarding its interconnection with the adjacent frames. This assumption holds, so that there is no relative deformation between in the loading conditions and the structural properties say stiffness of adjacent frames, the relative deformations which are caused due to them are ignored in the analysis by this first degree of approximation. As an illustration, the torsion and/or lateral bending stiffness of members (cross beams) at right angles are ignored. However these members are assumed to give lateral support to the plane frame (i.e. the cross members are assumed to be very rigid), with the result, the vertical frame which is plane beforee loading remains plane after loading.

11 Fig. 6.5 A three dimensional figure of space frame Thus, the basic frame considered for analysis of a building is a vertical plane frame Fig.6.6. shows the front elevation of vertical frame marked B-B in Fig.ÿ6.5. This existing plane frame which is subdivided in substitute frames into different manners discussed below making further approximations. The method assumes that forces (i.e. B.M. and SF.) in the beam of any floor are influenced by the loading on that floor ignoring the effect of loading on the lower and upper floor. As earlier, the explanation of substitute frames provided which is restricted for those used in analysis for vertical loads only.

12 a) Substitute Frame - I:Frames Fig 6.6 Elevation of Frame In the second degree of approximate, the complete vertical frame which is subdivided in several two storey frames within each floor. The floor frame or substitute frame - I at any floor consists of beams at the floor under consideration together with all connected columns in adjacent storeys, assumed to be fixed at their far ends. Fig. 6.7 Substitute Frame I Floor Frame

13 The substitute frames for the top floor and intermediate floors are shown in Fig. 6.7.For different cases of loading frame can be analysed with any method to produce the maximum forces for design of members. The analysis is carried out for each floor frame, and moments and shears in all beams and columns are determined b) Substitute Frame ii Bay Frames Substitute Bay approximation at the level of third degree, instead of taking entire columns and entire beam segments, in adjacent two storeys, this frame is further subdivided into separate bay frames each one consisting of interested beam along the connected columns and adjacent spans beams only, fixed on their distant ends as shown in Fig.ÿ6.8. This type of frame is known as substitute Bay Frame. While beams are not considered beyond their adjacent spans however assumed as fixed, stiffness of these get extra estimated. Thus, their stiffness is lessened by half to allow for flexibility on account of continuity.

14 Fig Substitute Frame- II: Bay Frame This third degree approximate holds well, symmetric frames for symmetric loadings The results are likely to differ from exact values in case of unsymmetrical frames and /or unsymmetrical loading. However, the difference is hardly beyond 10%. Such frame is also analysed for different loading cases to get maximum forces in columns beams as usual. c) Substitute Frame iii Beam and column systems: A very conservative alternative to the preceding substitute consists of only continuous beam at each floor level with ends simply supported providing no restraint to rotation as shown in Fig 6.9. Since rigidity offered by column is totally ignored, very large beam moments come, even at times to the extent of 30% to 50%the actual moments. Thus, though the analysis is simplified, the design proves to be very I.S. Code does not permit to use this method However BS: Code 3.8 still permits this method of approach.

15 Fig 6.9. Substitute frame beam system Even though, interconnection amid columns and beams are to be ignored calculation, moments in columns gets induced by actual rigidity which is not to be ignored in design. Therefore the moments in columns are obtained by considering only column systems made up of upper and lower column at a joint together with connecting beams fixed at their far ends as shown in Fig The far ends of beams and columns are assumed to be fixed stiffness of beams are reduced to half to compensate for the effects of bay beyond. The columns sub-frames are analysed for such loading on beams so as to cause maximum column moments. The results of this totally approximate method are required to be brought nearer to those of substitute I the effect of moments on beam shear and moments must further be taken into consideration and beams shear with moments be modified. Fig Substitute Frame- III : Beam Column System

16 In the frame analysis discussed above whatever approximation is adopted, fundamental fact on which the analysis is based is that the joints between members are rigid. It is, therefore necessary to know how rigid connection is obtained in a R.C. construction and what are the types of connections. Besides the analysis which is based on the significant structural property, namely, the stiffness (k) which depends upon the ration I/L and the nature of support conditions of the member at the far ends. In all substitute frames discussed above, it is assumed that the far end of member is fixed. If far end hinged i.e. rotation free, the stiffness of the member is taken equal to 0.75 I/L. From above, it is evident that for computing the stiffness, it is necessary to decide whether the support is rotation free or not. The different types of support in R.C. construction and the different alternative methods of computation of stiffness (I/L) are discussed in subsequent sections Types of connections between two members When two members (viz slab-beam or beam-column are to be connected, no relative translator movement can be allowed between them. Therefore, connection between the two members are only of two types.. i. A simple of hinged connection: It allows relative rotation between the connected members. It does not transfer moment from one component to other but transverse shear will not get transferred with axial load. ii. Rigid connection : It does not allow the relative rotation connected members. It transfer the moment besides shear and axial force from one member to other. For transfer of moment and hence for joint to be rigid, the following conditions are required to be satisfied.

17 a) Between the two members there should be tension steel which is to interconnecting on the tension face through area adequately enough to effect transfer of forces (i.e B.M. and S.F.) b) The interconnecting steel should be adequately anchored in both the connected members either by requisite development or by mechanical anchorage. If any one of the above conditions is not satisfied, the joint will not act as a rigid joint. For a joint which is to be rotation free. It should be seen that above conditions are not satisfied. If they are satisfied partially, the joint will act as a partially rigid (i.e. semi-rigid) joint. For illustration, consider a beam column connection shown in Fig For rigid connection enough area of interconnecting steel `Ast' to resist moment `M' at support must be provided on the tension face (i.eat top, for vertical downward loading) for a length equal to development length and at the same time it must be extended further in the column through a distance BA equivalent to anchorage length in turn development length Ld will be same as anchorage length. If it is found that this length is large, then instead of extending beam reinforcement into the columns, the columns bars should be bent and extended in the beam through a distance BC equal to the development length. On the contrary, for simple connection between column and beam, no steel other than anchor bars be provided at top in the beam, and furthermore, these bars could be simply continued straight in the column through column width only, just for getting sufficient lateral support at top.

18 Fig Figure showing joint connections When the beam deflects it rotates at support. This type of connection allows to rotation with the development of crack which is vertical at the face of column or at junction of upper part of column cast with beam and lower part of the column top cast earlier. It should be kept in mind that merely casting two components monolithic will not ensure continuity of structure. The structural continuity is obtained only by rigid connection. As an illustration, consider a monolithically cast slap-beam connection shown in Fig Connection shall not be a rigid one till adequate tension steel is existing at top portion of slab and also it to be sufficiently anchored.

19 Fig Figure showing beam column connection Enlarging it with a distance of bonder mechanically hooked round the beam bar through 180o.ÿIf at the top separate tension steel is not provided the slab and if it is simply leftover the beam, the connection will be a simple one. The slab only will rotate (and not the beam) by development of crack at the top of slab just at the beam face. If the connection is rigid and beam itself is simply supported and it will also rotate (due to torsion along with the slab) Types of supports or End conditions There are about three different kind of ideal support exists. a. Simple support: In this support it allows member to move in the direction of the plane of support. It also allows rotation though, it does it does not allow the movement in direction perpendicular to supporting plane. b. Hinged support: It is a support which permit the supported member for rotation and does not allow any translatory movement. This support offers reaction in any direction but does not resist moment. Support is called by the name rotation free support. c. Fixed support : It is the support which not only resists translation but also rotation, but also. It resists moment and offers reaction in any direction. Thus, fixed support does not rotate.

20 A slab or abeam embedded in the rigid wall then it gives a rigid support. Fixed support means it is a type of support in which a resisting moment offered for prevention of rotation. A rigid connection should not be taken as to give fixed end condition. Rigid connection implies zero relative rotation between the connected members. It does not imply zero rotation of the joint or of the supported member. Thus, it must be noted that a simple connection between two members if it gives simply rotation free end condition to the supported member. But rigid connections members not essential to provide a fixed end condition for supported member. Consider a two span continuous beam carrying equal U.D load on both the spans. The beam is simply supported over three supports i.e. it is not even interconnected with the support. Still, the symmetry of the loading span and end condition of zero rotation at the intermediate support which can be taken into consideration like rotation fixed support condition for the purpose of analysis. However, still the support will be simple as rotation is likely due to change in the loads on two spans. Question of end condition for column near the end of footing is a typical one. Consider a column subjected to axial load P and a moment M at the top. It is observed whether the footing end might be termed fixed or hinged. A stated above, for the column base to be fixed, footing should be capable of offering a resisting moment equal to M/2 besides, axial compression. This resisting moment can be made available either by nonuniform pressure division at the footing base in case of a concentric footing or alternatively, resisting moment it can be available for eccentric footing having pressure distribution uniformly at base of it which is shown in Fig If the bearing capacity of the soil is low, it is many times not necessary provide a fixed base i.e. moment resistance footing. In such a case, only for axial load footing could be designed. Footing cannot offer resisting moment, the column has a tendency to rotate at the base. This type of rotation is possible if supporting soil yields more on one side and less at the other edge, therefore rotation free condition is created as shownÿfig This rotation free condition is possible only with soils having low or medium bearing capacities.

21 For soil with large bearing capacity, rotation of footing is not possible as such footing cannot be designed for axial loads only. They have to be designed as moment resistant or fixed. The moment at the footing can also be avoided or reduced by providing a heavy plinth beam in the plane bending near the footing. This practice is common in R.C. building construction when the depth of footing below plinth is very large i.e when the cost of wall below plinth works to be greater than the cost of plinth beam. The plinth beam also helps in reducing the effective length of column at ground floor. Fig Concrete Footing Fig.6.14 End Column Conditions for Footing

22 6.3.5 Members Stiffness Calculating the stiffness of a member, the moment of inertia (I) of the members meeting at a joint are required. Code allows taking any one of the following definitions of moment of inertia for determining the stiffness. (1) Moment of inertia of Gross- section : Moment section (igr) ignoring reinforcement is given by : Rectangular section of size b x D, I gr = bd 3 /12 (2) Transformed gross section Moment of inertia Moment of inertia (MI ) of transformed gross concrete section including the reinforcement transformed on the basis of modular ratio, is obtained as under: In the case of column of size b x D with the neutral axis lying outside the section. The while section is under compression. In such a case all the steel will be in compression and the moment of inertia is given by : n I = bd3/12 + (m 10) A si x x i 2 i =1 Where, m = modular ratio reinforcement Asi compression xi = distance of the steel at the level from C.G of station 3) Transformer cracker section Moment of Inertia: it is the moment of inertia concrete compression (Ir) including area of reinforcement transformed on the basis ratio. Whatever may be the basis adopted for calculation of I, it is required to be applied consistently to all members. In preliminary design, since neither the moments are known not the reinforcement, the question of using the second or third method of finding I (described above) does not arise at all. The common practice is, therefore, to take I of concrete gross section (Igr) ignoring reinforcement.

23 The moment of Inertia of gross concrete section excluding reinforcement may be obtained using the following equations: Rectangular section : I gr = bd 3 /12 Flanged section : = Depth of N.A x _ 2 2 b w D / 2 + (b f b w ) D f / 2 b xd + (b f b f w )xd w (I) _ 3 I gr = b f x / 3 (b f b w )(x D f ) 3 / 3 + b (D x) 3 / 3 w (II) or I gr = k f b w D 3 /12 Where k f = [k _ 3 x / 3 (b f b w )(x D f ) 3 / 3 + b (D x) 3 / 3] w (III) k 1 = b f b w k 2 = D f / D and _ k = x/ D However, main difficulty in calculating arises when the beam is continuous at both ends as in the case often frames, because in that case the beam acts as rectangular beam in the negative moment region and a flanged beam in positive moment region. Thus such a beam is of varying moment of inertia along the length. A single equivalent or effective value of I to be taken for stiffness would depend up on the ratio of region of positive moment (Lo) to length L of the beam. However, since value of Lo depends upon the end conditions, the loading and the moment developed at supports, it is not constant. It varies from 0.58 L for a fixed at both ends at L for a beam simply supporter at both ends. Since 100% fixity at supports is hardly ever possible because of rotational at supports due to flexibility of supports, the ratio Lo/L is normally large, hence some designers take Igr of a flanged section ignoring the difference between Lo and L. The other method is to use moment of inertia of T-section equal to

24 twice that of rectangular section. The multiplying factor 2 corresponds of flanged section having bf/bw = 6 and Df/D=0.2 Since the section acts as a flange section in the major portion of beam span normally the ratio is bf/bw is nearly 6, authors consider I gr of beam = 2xb w D 3 /12 two times that of a rectangular section to be more appropriate. Institute of structural Engineers Manual recommends considering actual flange width of T-beam or (0.14L+bw) whichever is less. In book say by `wang at all' it is assumed as an equivalent system approximating T section having flange width equal to twice web width over the entire section. This will over estimate the beam stiffness giving higher moments in beam. Alternatively, the assumption of rectangular section (instead of flanged section) is easy for calculation, given higher moments in the column The length of the member, in calculation of stiffness (k = I/L), which is considered equivalent to centre to centre distance in between supports for beams and floor to floor height for columns except in case of ground floor column for which the length is taken from the top of footing to the top of floor level when there is no plinth beam, and the top of plinth beam when the same exists Effect of stiffness on Distribution of Moment in Beams columns When a beam is connected to columns, fixed end moments Me is calculated on the assumption of zero rotation or clamping of the joint. This moment is known as unbalanced moment acting at the joint. The joint is then released unclamped by applying an opposite moment. This moment is now distributed to various members meeting at the joint in proportion of their stiffness. Thus, moment in columns are obtained as follows M = cal.a ca k ca k + k + k cb b xm e M cal.b = k cb k ca + k cb + k b xm e M = beam ca k b k + k + k cb b xm e

25 Where, k ca = stiffness of columns Above the joint = I ca /L ca k cb = stiffness of columns below the joint = I ca /L ca k b = stiffness of Beam = I b /L b If the beam is continuous beyond, the stiffness of beam kb is reduced to half to account for the effect of loads on spans beyond. It will be observed from the above relations, that if the column has large cross-section and is short compared to beam, its stiffness kc = Ic/Lc will be large While, if a beam is of smaller cross-section and has a large span. Its stiffness will be small Consequently, negligible small rotation will occur at the joint and the column is said to offer practically full fixity to beam. The bending moments in the beam and column will both be nearly wl2/12. The fixity offered is more are two in number while the beam is only one. The situation is common in lower storeys of multi-storeyed frames having large span bays. On other hand of the cross-section of beam is large and span short, its stiffness will be large. Simultaneously, if the column cross-section is small and length large its stiffness will be small and consequently joint will rotate and practically no fixed end moment will develop either in the column or in the beam Total quantity of fixed end moment get released at the joint with no moment remaining in the beam This type of condition offers simple support of beam which is rotation free, thus, the bending moment in the beam and column at the joint lies between o and wl in general; the actual magnitude is dependent on beams relative stiffness and the column for satisfying the equilibrium condition, summation of moments in the beams meeting at a joint must be equal and opposite to the sum of the moments in the column meeting at that joint. 6.4 Assumptions and approximations of Design In practical design, a designer is many times required to make certain assumptions and approximations the analysis simple to save computational

26 effort and time. The design assumptions, of course, should be such as to make the designer on the safer side. If at all they are found to be on the unsafe side at certain places, an allowance should be made in the analysis based on earlier observations and judgment. Some of the assumptions are given below Assumptions regarding support condition The first and the assumptions that required to be made is about the support condition or the type is support for slabs and beams. Normally, though slab is cast monolithically with the beam, it is not necessary that it should be connected rigidly to supporting beams. Such a rigid connection does not necessarily ensure fixed end condition. It may cause rotation of the beam if the beam itself is simply supported at its ends. If the beam is fixed at the ends, the rigid connection between the slab itself is simply supported at its ends. If the beam is fixed at the ends, the rigid connection between the slab and the beam induces torsion in the beam giving condition of partial fixity and not full fixity. Therefore, it is commonly assumed that a slab is simply supported at discontinuous end and continuous over intermediate support Same assumption holds good for beams hence applicable, it is because whether a to a supporting column beam is connected simply or rigidly, it generally at the ends rotation free, therefore assumption is that it is at the ends simply supported if one is unsure of rotational condition of restraint at the ends. The above assumption is for ends continuous beams or Similarly, assumption is many times, required to be made for an intermediate support too. For example, the exact analysis of a continue slab or beam contain spans large in number (say more than 4) is extremely laborious, and besides continuity of beam/slab for more than spans has advantage. Simplicity in analysis can be brought via introduction of discontinuity at suitable intermediate support (like the discontinuity at support in an multi span bridge). As an illustration, of a public building considering a beam revealed in Fig.6.15.

27 A structural discontinuity can be introduced at supports D and also at E, total beam be divided into 3 separate ABCD-freely supported A and D, DEsimply supported at D and E, and EFGH freely supported at E and H. As the structural discontinuity is assumed at D and E, the same condition can be on obtained by not allowing the top bars to extend from CD to DE or from FE to ED. Fig Introduction of Discontinuity in a long unequal span continuous beam Assumption of giving for beams a simple support which is backed by age-old practice of well-known post-lintel construction adopted before since many centuries. However it should not be extended to each span of a continuous beam. Alternatively, beam in Fig can be divided three segments ABCD,DEEFGH for analysis purpose only and each section analyzed separately assuming fixity fully at D and E. This assumption of treating structural continuity as fixity, though may not be rigorously correct, especially if beams have unequal spans and also loads, can still be accepted as the difference because of this approximate is found to be well within the degree of accuracy expected in reinforced concrete structures It simplifies the analysis to a great extent. If a physically continuous beam cast monolithically is designed as it is made up of many simply supported beams, it is quite likely that may not be any steel or there may be very little insufficient steel over the top of intermediate support extending through anchorage length in both the spans.

28 The negative moment that may structurally develop at intermediate support due to physical continuity of beam and/ or rigidity of intermediate walls/columns/supports would cause at the face of support concrete cracking at top which is quite objectionable though it may not be a sign of structural unsoundness or lack of safety. As cracking occurs, the moment at support is transferred to the mid-span. The load is fully sustained of the mid-span section is designed corresponding to moment for a simply support condition. However, this is a crude design practice which makes the beam heavy and misuses the principle of compensatory resistance. Especially, such design though may not be unsafe for vertical loads, is definitely unsound for resisting horizontal loads and unacceptable from viewpoint of serviceability or performance of the structure and it will not give any reserve strength at collapse, and consequences of collapse, whenever it may occur, are likely to be serious Approximations regarding bending moments in Beams and slabs The analysis of a multi span can be further simplified by analyzing and designing each beam span) separately considering approximate moments at continuous end based on redistribution of moments. In this approach, simplicity is achieved together with the desired economy, safety and a satisfactory structural performance. Exact bending moments computation for single span beams or slabs will not be a problem. Directly they can be obtained using the coefficient for standard loading cases available in various design aids. Normally difficulty arises in finding out of bending moments into continuous slabs/beams having equal spans approximately (deviation amid short and short span does not exceed 15% of long span) along with carrying uniformly distributed loads. in various design aids coefficients for equal span continuous beams slabs for other standard loadings like equal point loads at 1/4 th, central point load or1/3rd span points are too available.

29 As the continuous beams/slabs can be approximately designed via considering a continuous beam/slab as made up of number of independent single span beams or a group of typical multi span beams. This approximation is an extension of the principle used in substitute frame method related to analysis in which division of big structure in parts for the reason of analysis, and then analyzing each part independently. Here, it is applied to continuous beams of approximately equal spans. Each one is analyzed independently by utilising standard bending moment co-efficient which are dependent on the ordinates of bending moment's envelope and allowing redistribution of moments. They provide values within 30% of exact theoretical values. It may be noted, that redistribution of moments is allowed the extent of 30% hence variation of 30% is tolerable amid the design moment and elastic moment within mid-span moment. The results got from the approximations in discussion prevail between elastic moments and those got by limit analysis allowing redistribution of moments. The design moment coefficients used for typical beams are as follows: Moment at supports as well as at mid-span Uniformly distributed load (w) End spans ± wl 2 /10 Intermediate spans ± wl 2 / Assumptions regarding Beam section Another important assumption made in design of R.C. beams is about the type of the section (whether flanged or rectangular). A beam having a slab acting as flanged on compression side and having minimum transverse reinforcement at top and which has stirrups running through total depth (including thickness of slab portion will structurally act as a flanged beam. Since, design of a flanged beam is complicated and laborious as compared to the design of rectangular beam the design could be done assuming beam to

30 be of rectangular section only when the design moment does not exceed Mur.max of the balanced section The beam designed on the basis of this assumption is always in the safer side when design moment is less than M ul which is the value of Mur of flanged section for xu = Df (i.e when x u <D f ), of assumption of rectangular section is always greater than that required for flanged section because the lever arm of the rectangular section is always less than that of a flanged section since x u in a rectangular section is always greater than x u for flanged section (for required total compression). Fig Fig Assumption Regarding Beam Section The area of steel increases with in the value of and is maximum when x u =Df in case of flanged beam, the quantity of steel required to balance the compression in outstanding flanged portions depends upon the ratio D f /d and b f /b w and is a variable quantity. However to get rough ides, it may be mentioned that the maximum additional steel required due to assumption of rectangular section in place of flanged section is about 10% to 20% for x u D f However, when the design moment M ur.max of assumed rectangular section, this assumption of rectangular section should not be made i.e. the area of steel should not be worked out on this assumption of rectangular section because that would necessitate the rectangular section will be designed as a doubly reinforced, area of steel required for doubly reinforced rectangular section will be much more than that required for flanged section. In such a case, the beam, shall be designed as flanged beam only. The following example will clearly will clearly bring out the difference between the

31 area of steel required when design is based on assumption of rectangular section in place of a flanged section. Illustrative example: Data : A beam continuous at both ends, Span = 4m, Slab 110mm thick, Section 230mm x 380mm, M20-Fe415, UD. Load (a)w u = 48kN/m 2 (b) w u = 72 kn/m 2 Required : A st Solution : Let d = 40mm, M ur.max = R ur.max d= = 340mm x bd 2. For M20-Fe415, R ur.max = 2.76 N/mm2 M ur.max = 2.76 x 230 x x 10-6 = 73.3 kn.m a) M u = w u L 2 /12 = 48 x 4 2 /12 = 64kN.m For T-section : L o = 0.7 x L = 0.7 x = 2500mm b f = L o /6 + 6D f + b w = 2800/6+6 x = 1357mm (M url )x u = Df = 0.36f ck b f D f (d-0.42d f ) = 0.36 x 20 x 1357 x 110 x ( x 110) x 10-6 = kn.m > kn.m x u < D f Case I : M u < M ur.max For Flanged section :

32 6 Required A st = = 534.5mm 2 For rectangular section : Since M u < M ur.max, the section is singly reinforced 6 Required A st = % increase over flanged section = = 17% Case II : M u = M ur.max = 73.3 kn.m For flanged section : By Eq Required A st = 614.4mm 2 For rectangular section : By Eq b Required A st = 744.5mm % increase over flanged section = 100 = 21% Case III : M u = M ur.max /2 = 37 kn.m (This possibility is very less) For flanged section : Required A st = 305.7mm 2 For rectangular section :

33 Required A st = 330.5mm 2 % increase over flanged section = = 8% Case IV : w u = 72 kn.m M u = 72 x 4 2 /12 = 96 kn.m>m ur.max For flanged section : Required A st = 812 mm 2 For rectangular section : Since M u > M ur.max, the rectangular section will be designed as doubly reinforced section only. Design of rectangular section as doubly reinforced: X ur.max = 0.48 x 340 = 163.2mm, M ur.max = 73.3kN.m as obtained earlier A st1 = ( ) = 747.9mm2 A st2 = ( ) (340 40) = 209.6mm2 Total area of tension steel = = 957.5mm 2 A sc = d c /d= 40/340 = 0.12, f sc = 348mm x209.6 ( ) = 223mm2 Total area of steel = = mm 2 as against 812 mm 2 % increase over flanged section = = 45% 812

34 Comments: It will be observed that percentage of increase of steel over flanged section varies from between 12% to 20%. But when Mu > Mur.max the doubly reinforced section becomes very costly is 45% in this case. 6.5 Various Design Philosophies Reinforced concrete structures can be designed by using one of philosophies. 1. Working stress method (WSM) 2. Ultimate load method (ULM) 3. Limit state method (LSM). 1. Working stress method (WSM) It is also known as Modular Ratio Method: This is the traditional method of design, utilised in both for reinforced concrete as well for steel structures Close to about hundred years old, the method is based on linear elastic theory or classical elastic theory. This method of design was around 1900 and was the first theoretical method accepted by the National Codes of Practice for the of reinforced concrete sections. Method ensures adequate safety by suitably restricting the stresses in the materials (i:e concrete and steel) induced by the expected working on the structure. Herein assumption of linear elastic behavior is taken into consideration justifiable since the specified permissible or allowable stresses kept well below the ultimate strength of the material. The ratio of yield stress of the steel reinforcement or the cube strength of the concrete the corresponding permissible or working stress is usually called the factor of safety. The WSM uses a factor of safely of about 3 the cube strength of concrete and a factor of safety of about 1.8 with respect to the yield strength of steel. Reinforced concrete is a composite material. The WSM assumes strain compatibility; in which the strain in Reinforcing steel is assumed to be equal to that in the adjoining to which it is bounded Consequently the stress in steel linearly related to the stress in adjacent concrete by an invariable factor known as modular ratio which is defined as the ratio of modulus of elasticity of steel to that of concrete. The WSM is therefore also known ratio method.

35 Demerits of WSM Many of the structures designed from WSM have been normally performing suitably for many years the method has the following demerits: 1. The WSM does not show real strength nor gives the true factor of safety failure. 2. The modular ratio design results in larger percentage of compressive steel than that given by the limit state design thus leading uneconomic design. 3. Because of creep and non linear stress strain relationship concrete does not have definite modulus of elasticity. 4.Fails to discriminate between different types of loads so as to act concurrently but contain different uncertainties. Merits of WSM In above defects the WSM has the advantage of its simplicity both in concept as usual as in design generally results in relatively sections of structural members in comparison to the ULM. Due to this, structures designed by WSM give better serviceability performance example (i.e. less deflection, less track width etc,) under working loads, WSM is the only method available when one has to investigate the R.C. selection for service stresses and for the serviceability states of deflection and cracking. It is essential to have knowledge of WSM since it forms a part of limit design (LSD) for a serviceability condition. 2. Ultimate load method (ULM) The ultimate load method (ULM) evolved in 1950 as the WSM. The method is resting on the reinforced concreters ultimate strength at ultimate load. The ultimate load by service load by some factor referred to as load factor for giving a desired

36 margin of safety. Hence the method is also referred to as the factor method or the ultimate strength method. The ULM was introduced like alternative to the WSM within ACI Code in 1956, the British Codes the in In the ULM method, stress condition at the state of impending collapse [se of the structure is analysed, thus using the nonlinear stress strain curves meant for concrete, steel also, the safety measure in the design is obtained by the use of proper load factor. This makes it possible to use different load factors under combined loading conditions. It is to be carefully noted that satisfactory strength presentation ultimate loads will not promise satisfactory serviceability in plastic region ( inelastic region) and of ultimate strength of resulting section is very slender or This gives rise to excessive deformation and cracking. Also, the method does not consideration the affects of creep and shrinkage. Merits of ULM 1. While the WSM uses only the nearly linear part of stress-strain curve, uses fully the actual stress-strain curve. In other words, the stress block parameters are defined by the actual stress-strain curve. 2. The load factor give the exact margin of safely against collapse. 3. The method allows to use different load factors for different types of combination thereof. 4. The failure load computed by ULM matches with the experimental results..5. The method is based on the ultimate strain failure criteria. The method utilises the reserve of strength in the plastic region. Demerits of ULM 1. The method does not take the serviceability criteria of deflection and cracking. 2. The utilisation of high strength reinforcing steel and concrete results in increase and crack width.