MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS USING VARIOUS CONFINEMENT MODELS AND EXPERIMENTAL VALIDATION

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1 ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 8, NO. 3 (2007) PAGES MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS USING VARIOUS CONFINEMENT MODELS AND EXPERIMENTAL VALIDATION M. Srikanth, G. Rajesh Kumar and S. Giri Department o Civil Engineering, National Institute o Tehnology, Waragal , India ABSTRACT This paper presents a proedure or inding the analytial Moment Curvature behaviour o statially determinate reinored onrete beams, taking into onsideration, the oninement oered by shear reinorement to onrete in ompression zone. Six seleted oninement models reported in literature in the last deade are used as a stress blok or onined onrete or generating the omplete analytial Moment Curvature behaviour. The Moment Curvature behaviour obtained using the seleted oninement models are ompared with experimental results. In general it is observed that the results obtained rom the seleted models were lose to the experimental values. However, it is observed that the analytial values obtained using Mendis and Cusson model are loser to the experimental results when ompared to that obtained using the other models. Keywords: stress, strain, moment, urvature, oninement 1. INTRODUCTION The most undamental requirement in prediting the Moment Curvature behaviour o a lexural member is the knowledge o the behaviour o its onstituents. With the inreasing use o higher-grade onretes, the dutility o whih is signiiantly less than normal onrete, it is essential to onine the onrete. In a lexure member the shear reinorement also onines the onrete in the ompression zone. Hene, to predit the Moment Curvature behaviour o a lexural member, the stress strain behaviour o onined onrete in axial ompression is essential. With the development o perormane-based design methods, there is an inreasing need or simpliied but reliable analytial tools apable o prediting the lexural behaviour o reinored onrete members. Design oies will be aed more and more with the need o prediting the deormation apaity o onrete members. A general approah to aount or oninement o onrete and prediting the lexural behaviour o onrete member is needed. In light o the above six oninement models proposed by various authors in the last deade ( ) were seleted and used as stress blok or -address o the orresponding author: garjesri@yahoo.o.in

2 248 M. Srikanth, G. Rajesh Kumar and S. Giri ompression onrete in an RC beam or generating analytial M-φ behaviour. The analytial values obtained using various models were ompared with experimental results. 2. CONFINEMENT MODELS The oninement models used or prediting the Moment Curvature behaviour o RC beams are listed below and the mathematial expressions o the seleted models are given in Table 1. Daniel Cusson and Patrik Paultre (Cusson model) [2] G. Rajesh Kumar and A. Kamasundara Rao (GRK model) [5] Salim Razvi and Murat Saatiglu (Razvi model) [11] P.Mendis, R. Pendyala and S. Setunge (Mendis model) [7] Frederi Legeron and Patrik Paultre (Legeron model) [6] Weena P. Lokunge, J.G. Sanjayan and Sujeeva Setunge (Weena model) [13] Table 1. Mathematial expressions or various models Model Expression or asending branh Expression or desending branh Cusson et al [2] k k 1 + k k [ k ( ) 2 1 ] e GRK et al [5] 2 A + D B + C A b, B b C , D b 2 b b 2 b Same or asending and desending branh b ' C C i ; i ( P b P bb b ' v ) ' Ci b s Razvi et al [11] r ' 1 r r 85 k3ρ1[ k 2(k4 1)] (linear rom peak to the 0.85 o peak stress and 0.2 o peak stress in desending portion ater whih it is residual stress) Mendis et al [7] K ' 2 2 K [1 Z ( )] ' m res Legeron et al [6] k k ' 1 + ' ' k ' e ' k [ k ( ) 2 1 ] Weena et al [13] σ 2τ 1 mp 1 e γ mp + l d + 2 γ mp 1 mp e d + l σ 2τ

3 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS ANALYTICAL MOMENT CURVATURE RELATIONSHIP M-φ In deriving the expressions o the moments and urvatures or onrete setion onined with retilinear ties, the ollowing assumptions were made: a) The stress-strain relationship proposed in a seleted model is taken as a stress blok. b) The tensile strength o onrete is negleted. ) The variation o strain aross the setion is linear upto ailure. d) Idealised stress-strain relation or the tension and ompression steel was used e) The steel is peretly bonded. ) An imaginary leg o stirrup is onsidered at neutral axis to simulate the triaxial state o stress in ompression onrete. In addition to above assumptions, the three basi relationship viz., (i) Equilibrium o ores, (ii) ompatibility o strains and (iii) Stress-strain relationship o the materials have to be satisied. d b nd γnd C α b nd s T Figure 1. Setion strain and stress distribution. where, b width o the beam, d eetive depth o the tension steel, nd neutral axis depth, stress in extreme ompression iber, extreme ompression iber strain, s steel strain and γ & α are redution ators or distane between CG rom neutral axis and area under stress-strain urve respetively. We have, rom igure1, Compressive ore (C ) C b. nd 0 d (1) and, moment o ompressive ore (C ) about neutral axis (M ) M 2 nd b... d (2) 0 Thus, in equation (1) and (2), i the area under onrete stress-strain urve and moment o

4 250 M. Srikanth, G. Rajesh Kumar and S. Giri area under the stress-strain urve is known the ompressive ore (C ) and its moment about neutral axis (M ) an be evaluated. Step-by-Step Calulation Proedure For obtaining the omplete moment urvature relationship or any ross-setion, disrete values o onrete strains ( ) were seleted suh that even distributions o points on the plot, both beore and ater the maximum were obtained. The proedure used in omputation is given below: 1. The extreme iber onrete ompressive strain ( ) was assumed. In present investigation the values o was in the range o to the ailure strain (ie 0.01). 2. The neutral axis depth, nd, was assumed initially as 0.5 times the eetive depth (i.e. 0.5d) 3. For this value o neutral axis depth, the ompressive ore in the onrete, C was alulated rom the respetive stress-strain model. 4. The strain in tension and ompression steel was alulated, based on the strain ompatibility. 5. Based on the strains in tension steels. The orresponding stresses were taken rom stress-stress urve o steel. 6. The total tensile ore (T) in tensile steel was alulated. 7. Same proess was repeated or ompressive steel to alulate the ompressive ore (C s ) in ompression steel. 8. The total ompressive ore C ating in the setion was alulated as C C + C s. 9. I C T, then the assumed value o neutral axis depth (nd) was orret, and the moment (M) and the orresponding urvature (φ) was alulated. Otherwise, the neutral axis depth was modiied until the ondition C T was ahieved. Now, the total moment about the N.A. was given by; M M t (moment o ore in tensile steel about the neutral axis)+ M (moment o ompressive ore in onrete about the neutral axis) +M s (moment o ore in ompression steel about the neutral axis) and the orresponding urvature (φ) was given by; φ. nd 4. EXPERIMENTAL PROGRAMME The analytial moment urvature relation obtained using various models were ompared with experimental results. The experimental programme onsisted o asting six beams o three dierent onrete strengths. For eah onrete strength, one under-reinored (U1, U2, U3) and one over reinored (O1, O2, O3) beams was ast. The details o the beams are given in Table 2. The balaned reinorement required or a partiular strength o onrete was arrived based on the stress strain urve as suggested by IS 456: 2000 [15], without onsidering the partial saety ators.

5 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS Table 2. Details o beams used or experimentation. Beam k (MPa) Balaned steel (%) Tension steel Provided steel (%) U mm O mm mm U mm O mm mm U mm O mm mm Note: mm GI wire were used as hanger bar (Compression steel) in all the beams 2. 8 mm bar was used as stirrups in all the beams with spaing o 125 mm - in under reinored beams(u) and 100 mm - in over reinored beams(o) The size o the beam was 150mm 200mm 2100mm, with eetive span o 1800mm. 53 grade OPC ement onorming to IS 12269:1987 [16], Zone II sand and 20 mm well graded oarse aggregate onirming to IS-383: 1970 [14] was used or asting all the beams. Potable water was used or mixing as well as uring o onrete. The yield strength o 8mm, 12 mm, 16 mm and 20 mm bars were MPa, MPa, MPa, MPa respetively. 8mm dia steel was used or stirrups. The spaing o 125 mm and 100 mm was provided to prevent the shear ailure o beams. The beams thus ast were tested under two-point symmetrial loading, with onstant moment zone o 300 mm, in order to ensure the lexural ailure. The shemati sketh o test setup is given in igure Comparison o analytial behaviour with the experimental behaviour The predited moment urvature obtained using the seleted oninement models were ompared with the experimental moment urvature data both graphially and numerially. The Figures 3 and 4 show graphial omparison o the moment urvature behaviour. For the numerial omparison three signiiant points were hosen namely; ultimate moment and orresponding urvature ( M u and φ u ), moment and orresponding urvature at 85 % o the ultimate moment in asending portion ( M 0.85, a and φ 0.85, a ) and the moment and orresponding urvature at 85% o the ultimate moment in desending portion ( M 0.85, d and φ 0.85,d ). The experimental strain in onrete ( ) and steel ( s ) at the above mentioned points and their orresponding moment and urvature values were taken as the omparison riteria. The analytial moments and urvatures orresponding to the experimental strains in onrete and steel were onsidered or omparison. In general, it was notied that strain in

6 252 M. Srikanth, G. Rajesh Kumar and S. Giri steel was the governing riteria in under-reinored beam while it was the onrete strain in over-reinored beam. The Table 3 shows the experimental moment, orresponding urvature, strain in steel and onrete at ultimate moment, 85 % o the ultimate moment in asending portion and 85 % o the ultimate moment in desending portion. The experimental and analytial values thus obtained were used or the numerial omparison. The ratio o analytial/experimental values was alulated at all the signiiant points. The average o analytial to experimental ratios and mean error in predition was taken or the omparison. The table 4, 5 and 6 shows the omparison o moment and orresponding urvature at the three signiiant points all the models. The average and mean error in predition is listed at the bottom o eah table. 5. DISCUSSION i) Ultimate Moment ( M u ) The table 4 shows the omparison o ultimate moment and orresponding urvature or all the models under onsideration. The seleted models showed a mixed result while prediting the ultimate moment. The Legeron, GRK, Mendis and Weena model slightly overestimated the ultimate moment while the Cusson and Mendis model underestimated the value. However, the average ratios were lose to 1.0. The predition o ultimate moment using Cusson s model had the least mean error o 3.91%. ii) Curvature orresponding to the ultimate moment ( φ u ) All the models under onsideration underestimated the urvature orresponding to the ultimate moment. The variation o mean error in prediting the urvature orresponding to the ultimate moment was slightly high but was within the limit o 15%. The predition made by Mendis model regarding the urvature is better than the other models under onsideration. iii) 85% o Ultimate Moment in asending portion ( M.85, a The table 5 shows the omparison o 85% o ultimate moment and orresponding urvature in asending portion or all the models under onsideration. The preditions made by most o the seleted models were on the higher side. But, the GRK model slightly underestimated the 85% o ultimate moment in asending portion and also had higher value o mean error. The mean error in predition o 85% o ultimate moment in asending portion by Legeron, Cusson and Razvi model were almost same and was around 10%. With a mean error o 9.89% the predition using Mendis model was airly good. 0 ) iv) Curvature orresponding to 85% o Ultimate Moment in asending portion ( φ ) In general, all the seleted models underestimated the value o urvature orresponding to 85% o ultimate moment in asending portion. Exept the GRK model, all the other seleted models were good enough to predit the values with less mean error around 10%. The mean error was more in GRK model or the urvature orresponding to 85% o ultimate moment 0.85, a

7 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS in asending portion. The Legeron, Cusson and Weena model had approximately equal value o mean error. However, the predition made by Cusson model or urvature orresponding to 85% o ultimate moment in asending portion was better. v) 85% o Ultimate Moment in desending portion ( M.85, d The table 6 shows the omparison o 85% o ultimate moment and orresponding urvature in desending portion or all the models under onsideration. In general, all the seleted models overestimated the value, but while the Cusson model slightly underestimated it. The GRK model highly overestimated the value with the highest mean error o 27.63%. The predition made by Weena and Razvi model were slightly on the higher side with a mean error around 14% but was well within the aeptable limit o 15%. The predition made by Legeron and Cusson model was similar with approximately same value o mean error and average value. But, the values o 85% o ultimate moment desending portion estimated by Cusson model were loser to the test results. 0 ) vi) Curvature orresponding to 85% o Ultimate Moment in desending portion ( φ ) The seleted models showed a mixed result while estimating the value o urvature orresponding to the 85% o ultimate moment desending portion. The Legeron, Cusson and Razvi models overestimated the value and had a higher value o mean error around 17%. When ompared to the other models, the preditions made by Mendis and Weena models were better with approximately same value o mean error around 12%. However, the predition o the urvature orresponding to the 85% o ultimate moment desending portion made by Mendis model was ound to be better when ompared to the other models. 0.85, d 6. CONCLUSIONS A proedure or obtaining analytial moment-urvature behaviour taking into onsideration the oninement eet due to shear reinorement was developed. Six dierent oninement models published in literature in the last deade were taken as a stress blok or ompression onrete or generating an analytial moment-urvature urve. The analytial values obtained were validated with experimental results. The analytial to experimental ratio and mean error in predition was used or the omparison. In general, the analytial results obtained using seleted models were loser to the experimental results. However, it was observed that the analytial values obtained using Mendis and Cusson model were loser to the experimental results when ompared to that obtained using the other models.

8 254 M. Srikanth, G. Rajesh Kumar and S. Giri Figure 2. Shemati sketh o the test setup Table 3. Experimental Moment, Corresponding Curvature and Strain in Conrete and Steel at Three Signiiant Points Ultimate 85 % o Ultimate in asending portion 85 % o Ultimate in desending portion Beam M u (knm) φ u ( 10-6 ) ( 10-6 ) s ( 10-6 ) M 0.85,a (kn-m) φ 0.85,a ( 10-6 ) ( 10-6 ) s ( 10-6 ) M 0.85,d (kn-m) φ 0.85,d ( 10-6 ) ( 10-6 ) s ( 10-6 ) U O U O U O

9 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS Table 4. Comparison o ultimate moment and orresponding urvature

10 256 M. Srikanth, G. Rajesh Kumar and S. Giri Table 5. Comparison o 85% o ultimate moment and orresponding urvature in asending portion

11 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS Table 6. Comparison o 85% o ultimate moment and orresponding urvature in desending portion

12 258 M. Srikanth, G. Rajesh Kumar and S. Giri

13 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS

14 260 M. Srikanth, G. Rajesh Kumar and S. Giri Figure 3. will appear in this page Graphial omparison o analytial and experimental moment ~ urvature urve o over-reinored beams

15 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS

16 262 M. Srikanth, G. Rajesh Kumar and S. Giri

17 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS Figure 3. will appear in this page Graphial omparison o analytial and experimental moment ~ urvature urve o over-reinored beams

18 264 M. Srikanth, G. Rajesh Kumar and S. Giri NOTATIONS b Width o beam D Overall depth o beam d Eetive depth o beam nd Depth o Neutral Axis (NA) Extreme ompression iber strain s Strain in steel γ Redution ators or distane between CG rom NA α Redution ators or area under stress-strain urve respetively C Compressive ore k Conrete harateristis ompressive strength s Tie spaing y Yield strength o reinorement steel M Moment φ Curvature T Total tensile ore in tensile steel C s Compressive ore in ompression steel M t Moment o ore in tensile steel about the N.A. M Moment o ompressive ore in onrete about the N.A. M s Moment o ore in ompression steel about the N.A. M u Ultimate moment φ u Curvature orressponding to the ultimate moment M 0.85,a 85 % o the ultimate moment in asending portion φ Curvature orresponding to 85 % o the ultimate moment in asending 0.85,a portion M 0.85,d 85% o the ultimate moment in desending portion φ Curvature orresponding to 85% o the ultimate moment in desending 0.85,d portion M u, ana Analytial ultimate moment M u,exp Experimental ultimate moment φ u,ana Analytial urvature orresponding to ultimate moment φ u,exp Experimental urvature orresponding to ultimate moment M 0.85a, ana Analytial 85% o ultimate moment in asending portion M 0.85a,exp Experimental 85% o ultimate moment in asending portion φ 0.85a,ana Analytial urvature orresponding to 85% o ultimate moment in asending portion φ a,exp 0.85 Experimental urvature orresponding to 85% o ultimate moment in asending portion M.85d, ana 0 Analytial 85% o ultimate moment in desending portion M 0.85d,exp Experimental 85% o ultimate moment in desending portion φ 0.85d,ana Analytial urvature orresponding to 85% o ultimate moment in desending portion

19 MOMENT CURVATURE OF REINFORCED CONCRETE BEAMS φ 0.85d,exp Experimental urvature orresponding to 85% o ultimate moment in desending portion REFERENCES 1. Cheong, H.K., and Hong Zeng, Stress-strain relationship or onrete onined by lateral steel reinorement, ACI Materials Journal, May-June, No. 3, 99(2002) Cusson, D., and Paultre, P., Stress-strain model or onined high-strength onrete, Journal o Strutural Engineering, ASCE, No 3, 121(1995) Han, B.S., Shin, S.W., and Bahn, B.Y., A model o onined onrete in high strength reinored onrete tied olumns, Magazine o Conrete Researh, No. 3, 33(2003) Konstantinidis, D., and Kappos, A.J., Analytial modeling o onined HSC olumns under lexure and axial load, Magazine o Conrete Researh, No. 4, 55(2003) Kumar, G. Rajesh and Rao, A.K., Strength and deormability harateristis o tie onined standard onrete. Journal o Indian Conrete Institute, Otober-Deember, No. 3, 6(2005) Frederi, L., and Paultre, P., Uniaxial oninement model or normal and high strength onrete olumns, ASCE Journal o Strutural Engineering, No. 2, 129(2003) Mendis, P., Pendyala, R., and Setunge, S., Stress Strain Model to Predit the Full Range Moment Curvature Behaviour o High Strength onrete Setions, Magazine o Conrete Researh, No. 4, 52(2000) Serna Ros, P., Yazzar, S.A., and Coa Calvo, A., Inluene o Coninement on High Strength Conrete Behavior, Materials and Strutures, RILEM Publiations, 36(2003) Park, R. and Paulay, T., Reinored Conrete Strutures. Wiley-Inter-Siene, 6(1975) Paultre, P., Legeron, F., and Mongeou, D., Inluene o onrete strength and transverse reinorement yield strength on behaviour o high strength onrete olumns, ACI Strutural Journal, July-August, No. 4, 98(2001) Razvi, S., and Saatioglu, M., Coninement model or high-strength onrete, Journal o Strutural Engineering, ASCE, No 3, 125(1999) Sharma, U., Bhargava, P., and Kaushik, S.K., Comparative Study o Coninement Models or High-Strength Conrete Columns, Magazine o Conrete Researh, No. 4, 57(2005) Weena, P.L., Sanjayan, J.G., and Setunge, S., Stress-strain model or laterally onined onrete, Journal o Materials in Civil Engineering, No. 6, 17(2005) IS:383, Speiiation or oarse and ine aggregate rom Natural Soures or Conrete. Bureau o Indian Standards, IS:456, Code o pratie or plain and reinored onrete. Bureau o Indian Standards, IS: 12269, Speiiation or 53 Grade Ordinary Portland Cement. Bureau o Indian Standards, 1987.