Moment Curvature Characteristics for Structural Elements of RC Building

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1 Moment Curvature Charateristis for Strutural Elements of RC Building Ravi Kumar C M 1,*, Vimal Choudhary 2, K S Babu Narayan 3 and D. Venkat Reddy 3 1 Researh Sholar, 2 PG Student, 3 Professors, Department of Civil Engineering, National Institute of Tehnology Karnataka, Surathkal, Karnataka, India * mravibdt@gmail.om Abstrat The paper highlights the investigation of nonlinear axial-fore and moment-urvature relationships for retangular reinfored onrete (RC) beam and olumn ross setions for the referene reinfored onrete building struture onsidered for Probabilisti Seismi Risk Evaluation. The highly popular model namely Mander s model is used for onrete stressstrain relationship sine it is simple and effetive in onsidering the effets of onfinement. The module determines the expeted behavior of a userdefined ross-setion by first dividing the setion into a number of parallel onrete and steel fibers. Then, the setion fores and deformations are determined from the fiber strains and stresses using fundamental priniples of equilibrium, strain ompatibility, and onstitutive relationships assuming that plane setions remain plane. Index Terms: Mander s Method, Moment-Curvature, Stress-Strain Relationship and Moment Curvature I. Introdution Sine the ountry lie in earthquake prone area and many of the destrutive earthquakes ourred in the history so far resulting in high number of asualties due to ollapse of buildings and dwellings. A major hallenge for the performane based seismi engineering is to develop simple yet effiiently aurate methods for analyzing designed strutures and evaluating existing buildings to meet the seleted performane objetives Elasti analyses are insuffiient beause they annot realistially predit the fore and deformation distributions after the initiation of damage in the building. Inelasti analytial proedures beome neessary to identify the modes of failure and the potential for progressive ollapse. The need to perform some form of inelasti analysis is already inorporated in many building odes. Theoretial moment-urvature analysis for reinfored onrete olumns, indiating the available flexural Journal on Today s Ideas Tomorrow s Tehnologies, Vol. 2, No. 1, June 2014 pp by Chitkara University. All Rights Reserved. 1

Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. strength and dutility, an be onduted providing the stress-strain relation for the onrete and steel are known.

2 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. strength and dutility, an be onduted providing the stress-strain relation for the onrete and steel are known. The moments and urvatures assoiated with inreasing flexural deformations of the olumn may be omputed for various olumn axial loads by inrementing the urvature and satisfying the requirements of strain ompatibility and equilibrium of fores. II. STRESS-STRAIN MODEL FOR CONCRETE Mander s model is highly popular model sine it is simple and effetive in onsidering the effets of onfinement. It onsiders inrease in both the strength and dutility of RC members with onfined onrete. The model is popularly used to evaluate the effetive strength of the olumns onfined by stirrups, steel jaket and even by FRP wrapping as aomplished in Figure 1. Figure 1: Mander s Model for Stress-Strain Relationship for Confined Conrete III. STRESS-STRAIN CURVES FOR REINFORCING STEEL The idealized stress-strain urve for onrete as reommended by IS: is as shown in Figure 2. Stress-strain urve for steel as per British ode CP as shown in Figure 3, aordingly the term 0.7fy is the simplifiation of fy the expression fy. It gives all the simplified general equations whih γ m an be used for any grade of steel. 2

3 Moment Curvature Charateristis for Strutural Elements of RC Building Figure 2: Stress Strain Curve for Conrete Figure 3: Stress-Strain Curve for Steel IV. STRUCTURAL SYSTEM The building is an RC G+3 framed struture. The floor plan is same for all floors. The beam arrangement is different for the roof. It is symmetri in both the diretion. The onrete slab is 120 mm thik at eah floor level. Overall geometry of the struture inluding the beam layout of all the floors is as shown in Figure 4. Figure 5 and 6 shows the size and reinforement details for floor beam and roof beam setions at the olumn fae. Figure 7 shows the size and reinforement details for olumn at the beam fae. V. MOMENT CURVATURE FOR BEAM The most fundamental requirement in prediting the Moment Curvature behaviour of a flexural member is the knowledge of the behaviour of its onstituents. With the inreasing use of higher-grade onretes, the dutility 3

4 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. Figure 4: Overall Geometry of the Struture Figure 5: Details of Roof Beams of whih is signifiantly less than normal onrete, it is essential to onfine the onrete. In a flexure member the shear reinforement also onfines the onrete in the ompression zone. The relationship for the bending member as depited in Figure 8 is as follows, M = f 1 = = ϕ EI E x R 4

5 Moment Curvature Charateristis for Strutural Elements of RC Building Figure 6: Details of Floor Beams Figure 7: Details of olumns at various levels Figure 8: Beam Members in Bending Hene, ϕ = ε = ε = ε + ε s x d x d S 5

6 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. Where, φ = Curvature f = Maximum stress in ompression in onrete ε = Maximum strain in onrete ε s = Maximum strain in steel x = Depth of neutral axis d = Effetive depth of setion Let onsider BF205 beam setion as depited in Figure 9, ρ= A As Figure 9: Detailed BF204 Beam Setion Modular Ratio (m) for elasti analysis = 280 = 14 f k Curvature φ at raking moment just before raking f = 07. f = MPa r k At just before raking moment is resisted by onrete in tension therefore Negleting steel, N.A. = d/2 M r fr1gr = = y KNm Curvature after raking fr M ϕ = = = y EI

7 Conrete does not take any tension so depth of neutral axis x is, 2 2 d k = ( ρ+ ρ ) n + 2n( ρ+ ρ ) ( ρ+ ρ ) n d k = Effetive moment of inertia I eff 3 bx 2 = + Am(D s - x) = mm 3 Φ at M r Mr = = EI ε s strain in steel = f y /E s = The neutral axis at yielding is given as distane kd from extreme ompression fiber, where the ratio k is alulated using expression: eff 4 Moment Curvature Charateristis for Strutural Elements of RC Building 2 2 d k = ( ρ+ ρ ) n + 2n( ρ+ ρ ) ( ρ+ ρ ) n d k = Where ρ A s bd and A s ρ = bd are the tension and ompression steel ratios, n is the modular ratio, and d and d are the distane of ompression and tension steel from extreme ompression fiber. A s Steel in tension, A s Steel in ompression Depth of Neutral axis at yield stage; x = kd =208.65mm =.21m Taking moment about ompressive fore due to onrete, yield moment is given by: M y = kd + kd Af d A f d S y S y 3 3 Sine stress in the tension steel is f y, using similar triangles, stress in ompression steel is alulated as Curvature is then obtained as f = d d y ( ) =. MPa d kd f y ϕ y εy = = d kd

8 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. After yielding of tension steel, its stress remains onstant but strain keeps inreasing until ompressive strain in extreme fiber of onrete reahes the strain value of ε u at maximum stress in onrete f. In order to address the nonlinearity in onrete at high strains, Whitney-blok is used to approximate the paraboli stress distribution in onrete to an equivalent retangular stressblok representation. The alulation of ultimate state requires iteration. For hand alulations, let us assume that strain in ompression steel ε s, exeeds the yield strain ε y. This assumption will be heked later. Equilibrium of tension and ompressive fore gives the depth of neutral axis as; Af s s Af s y A sf s y Af s = = = f b 36 f b β 1 Ultimate moment is then obtained by taking moment about tension steel as: The dutility fator M = C d C d d u S + ( ) 2 = 36f Cbd + A f ( d d ) 2 s s = KNm 0034 ϕ u = = D. F. = ϕ ϕu y = 674. Where is φ u ultimate strain in onrete at maximum stress, whih is as per IS This is first trial value of ε u. Assumption of yielding in ompression is now heked by ensuring: ε d s = ( ) εu εy If the above ondition is satisfied then assumption made is true and obtained value of(m u, φ u ) defines the ultimate state on the moment-urvature urve. If the ondition is not satisfied further iteration is required with new trial strain value as ε ε ε = + y s s 2 VI. CONCRETE PROPERTIES Currently, the entire ross-setion is assumed to be unonfined. The ompression stress-strain relationship of the unonfined onrete is determined using a 8

9 method developed by Mander et al. In this method, the onrete stress, f is given as a funtion of the strain, ε as: f C xr. f = r 1+ x r ε x =, r = ε E o in MPa o E E ε 0 is the strain at peak stress ( f ) and E is the tangent modulus of elastiity of onrete alulated as E = 5000 f MPa Ese, the seant modulus of elastiity, is the slope of the line onneting the origin and peak stress on the ompressive stress-strain urve (i.e., Ese= f ε CO ). Crushing of the unonfined onrete is assumed to our at ε u 2ε o. se Moment Curvature Charateristis for Strutural Elements of RC Building VII. The Moment-Curvature Relationships of the Column Cross-Setions This is an iterative proess, in whih the basi equilibrium requirement P = C-T is used to find the neutral axis loation,, for a partiular maximum onrete ompressive strain, ε m where P = axial fore; C = internal ompression stress resultant; and T = internal tension stress resultant. The total onrete ompressive stress resultant and the loation of its entroid are determined by integrating numerially under the onrete stress distribution. The bending moment is assumed to at suh that the top surfae of the ross-setion is in ompression. The entire proess an be summarized for a ross-setion with two layers of reinforing bars as depited in Figure 10 follows Figure 10: The Cross-Setion with Two Layers of Reinforing Bars For example the olumn setions property in present work is 400mm x 900mm in size with 12-28mm dia bars plaed as shown in the figure 11 below. The transverse reinforement for the olumns is provided 10mm 100mm / as figure shows. 9

10 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. Figure 11: Detailed CL 15 and 19 Column Setion (Ground to 2nd Floor) The alulation of the following four points on a moment-urvature urve will be taken for plotting the moment urvature urve for various olumn setion shown in this example: ε m =0.25 ε u ;ε m = 0.5 ε u ;ε m = 0.75 ε u ; ε m = 1.0ε u (onrete rushing). A. Axial Load, P The axial load onsidered for sample alulations of CL15 is 50% of the balaned failure load. The balaned load, P b, is omputed as follows: εu The neutral axis, = b = d ε + ε, u y Where, ε y = f y /E s With εu = and d = 848m, b = 0.525m. The onrete ompressive resultant, C, is determined by numerially integrating under the onrete stress distribution urve. εcu 0 0 b C = f bdx = C = KN fb dε ε The spaing of eah layer of steel is s = 0.198m The steel fores of eah layer top to bottom wise, F s1 and F s2, F s3, F s4 and F s5, respetively, are alulated using similarity to find the strains in the layers. Balaned failure ondition, by definition, has strain values ε u = ε m = for onrete and ε s5 = ε y = for bottom layer of steel. For the top steel, Similarly for seond and third layer, m ε s1 = ε d =. ss d 10

11 ε ε ε s1 s3 4 5 = ε = ε = ε ε s = ss m m εm d d s = d d s = d 2s = = This implies that the first and fifth layer of steel is at yield stress. Hene, F s1 = f y A s1 = KN (ompression) F s2 = ε s2 E s A s2 = KN (ompression) F s3 = ε s3 E s A s3 =195.47KN (tension) F s4 = ε s4 E s A s4 = KN (tension) and F s5 = f y A s5 = KN (tension) Where, A s1, A s2, A s3 A s4 and A s5 are the total reinforing steel areas in eah layer. As per IS456 onsideration the onrete tensile strength in the tension region reommends the modulus of rupture to be taken as, Moment Curvature Charateristis for Strutural Elements of RC Building f = f MPa for normal weight onrete. r Thus, for f = 20 MPa, f r = 33. MPa. The onrete tension fore 1 C = f A C = KN (tension) t t 2 r r,. Where, A r is the area of onrete in tension alulated based on the linear strain diagram. Then, the balaned axial load is found from equilibrium as, Pb = C + Fs1 + Fs2 Fs3 Fs4 Fs5 Cr Therefore, 50% of the balaned load used is P = kN. B. Instant Centroid The axial load ats at an instant entroid assumes loation that for the alulation of the moment-urvature relationship. The loation of the instant entroid is determined by assuming an initial ondition where only the userseleted axial load ats on the ross-setion without moment. This loading 11

12 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. ondition produes a uniform ompression strain distribution throughout the ross-setion. Let the uniform ompression strain be equal to ε i then, P= fia + fs1as1+ fs2as2 + fs3as3 + fs4as4 + fs5as5 From equilibrium, A trial-and-error solution is needed sine it is not known in advane. Then, the loation of the instant entroid, x, from the top ompression fae is determined as, fah i + f A d + f A ( d s) + f A ( d 2s) + s1 s1 s2 s2 ss ss 2 f A ( d 3s) + f Ad s4 s4 ss ss x fa+ f A + f A + f A + f A + f A i s1 s1 s2 s2 s3 s3 s4 s4 s5 s5 x = m The alulation of the first point on the moment-urvature relationship of the setion an be summarized as follows: 1. ε m = 0.25 ε u = Assume the neutral axis depth, a distane =.4m 3. From the linear strain diagram geometry Similarly for seond and third layer, d εs = ε 1 s 5 d = d s ε = ε = s2 s5 d d s ε = ε s3 m =. d 2s ε = ε s4 m = d ε = ε s5 m = 4. The steel stress resultants are, F s1 = f y A s1 = KN (omp) F s2 = ε s2 E s A s2 = KN (omp) F s3 = ε s3 E s A s3 = KN (tension) F s4 = ε s4 E s A s4 = KN (tension) F s5 = f y As 5 = KN (tension) 12

13 5. Determine C by integrating numerially under the onrete stress distribution urve. C = b C = KN f bdx f b u = ε dε ε 0 0 The onrete that has not raked below the neutral axis ontributes to the tension fore C t. m Moment Curvature Charateristis for Strutural Elements of RC Building 6. Chek to see if f = f r C = KN t MPa P = C + F + F + F F F C b s1 s2 s3 s4 s5 t P = KN b So, the neutral axis must be adjusted downward, for the partiular maximum onrete strain that was seleted in Step 1, until equilibrium is satisfied. This iterative proess determines the orret value of. Trying neutral axis depth =.431m gives, ε s5 = (below yield); εs4 = ; ε s3 = ; ε s2 = ; ε s1 = And C = KN; Ct = 28.51KN; F s1 =285.29KN; F s2 = 95.34KN; F s3 =101.08KN; F s4 = 6.23KN; F s5 = KN Setion urvature an then be found from, εm ϕ = = The internal lever arms for the resultant ompression and tension fores of the onrete measured from the instant entroid. Then, the setion moment an be alulated as, M = C z + F z + F z + F z + F z + F z + C z s1 s1 s2 s2 s3 s3 s4 s4 s5 s5 t t M = knm Similarly urvature and moment value are also alulated at ε m = 0.5ε u ; ε m = 0.75ε u ; ε m = 1.0ε u (onrete rushing) 13

14 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. The moment-urvature plot for olumn has been shown in Figure 12 and for beam as shown in Figure 13. Figiure 12: Moment-Curvature Relationship Curve for Column Figure 13: Moment-Curvature Relationship Curve for Beam Table 1 shows point at moment-urvature urve for olumn. Table 2 shows moment at various points in beams. Table 3 shows urvature at various points in beams. Table 4 shows moment at various points in olumn and Table 5 shows urvature at various points in olumn. 14

15 Table 1. Point At Moment-Curvature Curve For Column Point M(moment) Curvature φ ε u ε u ε u Moment Curvature Charateristis for Strutural Elements of RC Building 1.0ε u (onrete rushing) Beam Table 2: Moment At Various Points In Beams Before Craking After Craking At yield Crushing of Conrete BF BF BF BF BR BR BR BR Beam Table 3: Curvature At Various Points In Beams Before Craking After Craking At yield Crushing of Conrete BF BF BF BF BR BR BR BR

16 Ravi Kumar, C. M. Choudhary, V. Babu Narayan, K. S. Venkat Reddy, D. Table 4: Moment At Various Points In Column Column Origin Yield Ultimate Strain hardening CL 15/19 G/2nd CL 15/19 3rd CL 15/19 4th CL16/20 G/2nd CL 16/20 3/4th Table 5: Curvature At Various Points In Column Column Origin Yield Ultimate Strain hardening CL 15/19 G/2nd CL 15/19 3rd CL 15/19 4th CL16/20 G/2nd CL 16/20 3/4th ViII. Conlusions An analytial model is presented to simulate the moment urvature behavior of reinfored onrete. Based on ontrol of load inrements, the algorithm proposed by Mander enables determination of moment urvature-strain relationship with any geometry and material properties up to the maximum apaity of the setion; however with a onstant axial load or ontrol of deformation inrements, this model an be used to ompute both the asending and desending branhes of the moment-urvature urve. The momenturvature (or moment-rotation) relations play an important part in the study of limit analysis of two or three dimensional reinfored onrete frames. Referenes Bai, Zhizhou, Nonlinear Analysis of Reinfored Conrete Beams and Columns with Speial Referene to Full-range and Cyli, Ph.D Thesis, The University of Hong Kong, China, Cinitha A*, Umesha P. K, Nagesh R. Iyer, Evaluation of Seismi Performane of Existing Steel Buildings, Amerian Journal of Civil and Strutural Engineering, 2014, 1(2): dx.doi.org/ /ajse

17 Filip C. Filippou, Ahmad Issa, Non-Linear Analysis of Reinfored Conrete Frames under Cyli Load Reversals, Report No. UCB/EERC 88/12Earthquake Engineering Researh Center College of Engineering, University of California, Berkeley September Hyo-Gyoung Kwak, Filip C. Filippou, Finite Element Analysis of Reinfored Conrete Strutures under Monotoni Loads, Report No. UCB/SEMM-90/14, Strutural Engineering, Mehanis and Materials Department of Civil Engineering University of California, Berkeley, November1990. Mander, J.B., Priestley, M.J.N., and Park, R. Theoretial Stress-Strain Model for Confined Conrete, Journal of Strutural Engineering, Amerian Soiety of Civil Engineers, Vol.114, No. 8, 1988, pp Sifat Sharmeen Muin, A Parametri Study of RC Moment Resisting Frames at Joint Level by Investigating Moment -Curvature Relations, International Journal of Civil and Strutural Engineering, Vol. 2, No. 1, Varghese P. C, Advaned Reinfored Conrete Design, PHI Learning Private limited, New Delhi, Moment Curvature Charateristis for Strutural Elements of RC Building 17

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