Estimation of yield curvature for direct displacement-based seismic design of RC columns

Transcription

1 Universit of Wollongong Researh Online Fault of Engineering - Papers (Arhive) Fault of Engineering and Information Sienes 8 Estimation of ield urvature for diret displaement-based seismi design of RC olumns M Neaz Sheikh Universit of Wollongong, msheikh@uow.edu.au Hing-Ho Tsang The Universit of Hong Kong, tsanghh@hku.hku.hk Nelson Lam Universit of Melbourne Publiation Details Sheikh, M. Neaz., Tsang, H. & Lam, N, 8, Estimation of ield urvature for diret displaement-based seismi design of RC olumns. Australian Earthquake Engineering Soiet Conferene (p. [11]). Vitoria: AEES. Researh Online is the open aess institutional repositor for the Universit of Wollongong. For further information ontat the UOW Librar: researh-pubs@uow.edu.au

2 Australian Earthquake Engineering Conferene AEES 8 Estimation of Yield Curvature for Diret Displaement-based Seismi Design of RC Columns M. Neaz Sheikh 1, Hing-Ho Tsang and Nelson T. K. Lam 3 1. Corresponding Author. Leturer, Shool of Civil, Mining & Environmental Engineering, Universit of Wollongong, NSW 5. msheikh@uow.edu.au. Researh Fellow, Department of Civil Engineering, The Universit of Hong Kong, Pokfulam Road, Hong Kong. tsanghh@hku.hk 3. Assoiate Professor and Reader, Department of Civil and Environmental Engineering, The Universit of Melbourne, Parkville, VIC 31, Australia. ntkl@unimelb.edu.au Abstrat Signifiant researh efforts have been devoted in reent ears to the development of displaement-based seismi design methodologies, reognizing the shortomings of traditional, ode-speified fore-based design proedures. Reent advanes in diret displaement-based seismi design of olumns rel on the estimates of ield urvature for the determination of seismi design fores to satisf the speified seismi performane levels. This paper presents simple expressions for estimating the effetive ield urvature for normal- and high-strength irular reinfored onrete (RC) olumns based on moment-urvature anales of a large number of olumn setions. Suh expressions an be programmed into the spreadsheet format and an be used for the displaement-based design of RC olumns. Influene of different parameters on the effetive ield urvature has also been quantified. Effetive ield urvature is presented in terms of the gross diameter of the setion and the ield strain of the longitudinal reinforement together with three modifiation fators that take into aount the effets of the ompressive strength of onrete, the axial load ratio and the quantit of longitudinal reinforement. Kewords: seismi, design, ield, urvature, displaement, performane 1. INTRODUCTION Earthquake engineering researh over the past three deades resulted in fore-based (FB) strutural design and detailing proedures, inorporating apait priniples and ensuring adequate safet of the struture without signifiant damage and asualties when subjet to severe ground shaking [1-3]. Certain fundamental assumptions in the FB design proedures have, however, been found to be not representative of atual strutural behaviour. The stiffness of the struture, whih is used for determining the natural period of the struture and the distribution of fores to different strutural

3 elements, is not known initiall. Different odes take different approahes in estimating their stiffnesses. Moreover, the effetive stiffness of a raked onrete member is not onstant but inreases with inreasing flexural strength [4]. Diffiulties in determining the strutural stiffness of onrete elements result in signifiant inauraies in the estimates of the natural period of the struture. These errors are then translated into errors in the estimation of the strength demand and the distribution of fores alulated from the FB design proedures. Considering the inherent limitations of the FB proedures, displaement-based (DB) proedures have been proposed in reent ears where displaement demand is ompared with the displaement apait of the struture. Suh proedures are transparent as the onsider the true behaviour of the struture and will likel be inorporated in future seismi design odes. In the DB proedures, the design fore levels are based on the estimate of ield displaement of the struture. The ultimate displaement demand is first ompared with the ield displaement for estimating the dutilit demand of a ross-setion. The setion dutilit demand an be used for alulating the effetive damping level whih is then used for alulating the inelasti displaement demand and the effetive natural period based on the elasti displaement response spetrum [5]. The omparison between the ield displaement and ultimate displaement is also important for evaluating the seismi performane of a struture or strutural element under a given level of earthquake shaking. The intended mehanism for the majorit of bridge piers and olumns involves the formation of plasti hinges at ritial loations (ie. plasti hinge regions). The ield displaement of onrete olumns an be estimated using simple expressions that aount for the flexural deformation of the olumn [4]. Suh expressions an be developed based on the ield urvature of the olumn at the ritial loation. Previous studies on ield urvature [4, 6, 7] indiate that ield urvature is not sensitive to the quantit of longitudinal reinforement used in the member. This allows simple expressions to be used for estimating the urvature of the member at ield. One the ield urvature and the ield strength are known, the raked stiffness of the member an be obtained readil. This paper aims at developing simple expressions for estimating the ield urvature of normal- and high-strength irular RC olumns based on moment urvature analis of a large number of olumn setions.. PREVIOUS STUDIES ON YIELD CURVATURE Priestle et al. [6], and later Priestle et al. [4], proposed a formula for alulating the ield displaement for irular bridge olumns taking into aount shear ontribution and strain penetration of the longitudinal reinforement into the foundation. Effetive ield urvature (φ ) has been expressed in terms of the ield strain of the longitudinal reinforement ( ) and the diameter of the gross setion depth (D) of the piers (Equation 1). As mentioned earlier, the effetive ield urvature of a ross-setion does not depend signifiantl on the ratio of the longitudinal reinforement; whereas, effetive stiffness is almost proportional to the ratio of the longitudinal reinforement (given that stiffness is strength divided b displaement at ield). Hene, the effetive

4 urvature should be onsidered as one of the basi properties of a ross-setion. However, in the proposed formula no indiation has been given on the sensitivit of the ield urvature to the ratio of the longitudinal reinforement. Moreover, axial load ratio whih ma affet the ield urvature has not been parameterized and no limitation has been introdued for the appliation of the formula. However, bridge olumns designed aording to most design odes ma have the axial load ratio (P/f A g ; where P=axial load, f =onrete ompressive strength and A g =gross area of the olumn) of around 1%. φ =. 5 (1) D Montes and Ashleim [7] proposed simple expressions for the alulation of the effetive ield urvature based on moment-urvature anales. Yield urvature has been expressed in terms of the ield strain of longitudinal reinforement ( ) and the effetive depth of the setion, d (depth of the extreme tension reinforement) as shown b Equations a,b.less satter is seen with the estimate of the ield urvature when the effetive depth of the ross-setion has been parameterised.. φ =. 4 for steel f = 4 MPa (a) d φ =. 3 for steel f = 5 MPa (b) d One of the most important observations from this stud is the sensitivit of the ield urvature to the level of axial load (n) (Equations 3a, b). P φ =.5 a b. for steel f =4 MPa (3a) d f Ag P φ =.4 a b. for steel f =5 MPa (3b) d f Ag Values of a and b have been found from paraboli urve fitting and have been expressed as funtions of the effetive depth for three ross-setions. For other ross-setions, it is reommended to interpolate the values of a and b. As observed from the stud b Priestle et al. [6] and Priestle et al. [4], there has been no indiation on the sensitivit of the ield urvature to the quantit of longitudinal reinforement. High-strength onrete with ompressive strength of up to 1 MPa is now being inreasingl used in the onstrution of bridge olumns and is also permitted b most design odes. It appears that high strength onrete was not within the sope of the studies ited above (whih have not expliitl parameterised onrete strength in its reommended expressions). The onrete over has also not been parameterised. 3. MODELLING OF COLUMNS FOR YIELD CURVATURE Analtial Modelling of RC Columns Reinfored onrete is a highl non-linear material. Realisti onstitutive law of reinfored onrete is omplex as the non-linearit arising from onrete and the

5 reinforement needs to be appropriatel ombined to auratel simulate the experimentall observed behaviour of reinfored onrete elements. Stress-Strain Relationship of Conrete The uniaxial onfined onrete model of Légeron and Paultre [8], whih is based on strain ompatibilit and transverse fore equilibrium, has been hosen as the onstitutive law of onrete for the analtial modelling of RC olumns. The model has been validated with test results from more than irular and square large-sale olumns tested under slow and fast onentri loading. In the model, the behaviour of onfined onrete is related to the effetive onfinement index, whih takes into aount the amount of transverse onfinement reinforement, the spatial distribution of the transverse and longitudinal reinforement, the onrete strength, and the transverse reinforement ield strength. Stress-Strain Relationship of Longitudinal Bars An aurate model of a stress-strain relationship of steel bars needs to simulate: (i) elasti, ielding and strain hardening behaviour, (ii) ompression behaviour inluding bukling of bars, and (iii) low le fatigue and premature rupture of bars in tension. Gomes and Appleton [9] model has been hosen as the onstitutive law of reinforing bars, sine it is simple and well predits the above harateristis of reinforing bars. The model takes into aount the effet of inelasti bukling of longitudinal reinforing bars in a simplified wa based on the plasti mehanism of a bukled bar. 6 Column 47 1 Column Moment (kn-m) 4 3 Experimental Data [Lehman et al, 4] Analtial Model Moment (kn-m) 6 4 Experimental Data [Lehman et al, 4] Analtial Model Curvature (rad/m) Curvature (rad/m) 15 Column 43 Moment (kn-m) 1 5 Experimental Data [Lehman et al, 4] Analtial Model Curvature (rad/m) Figure 1: Experimental results ompared with analtial preditions

6 Modelling Setional Behaviour In this stud, the omplete moment urvature responses of olumn setions were omputed using omputer program MNPHI [1] whih has inorporated the onstitutive laws of onrete and that of the reinforing bars. The program uses a laered representation of the setion where eah laer is separated into a onfined ore laer and an unonfined over laer with the orresponding material properties. It alulates the moment-urvature response b an inremental analis assuming plane setion remaining plane (before and after bending). The program also takes into aount the spalling of the onrete over. Comparison with Experimental Results To evaluate the apabilit of the developed analtial model, experimental results of a large number of olumns tested under li loading have been ompared. Due to spae restritions, moment urvature preditions of bridge olumns 47, 415 and 43 (reported b Lehman et al., [11]) are presented herein in Figure 1. Exellent agreement has been observed between the experimental results and analtial preditions from this stud. The developed analtial model has been used for the development of moment urvature relationships for this stud. Definition of Yield Curvature Different definitions of ield urvature have been found in the literature based on both experimental and analtial results. For the purpose of design of reinfored onrete olumns, effetive ield urvature rather than true ield urvature is of interest. Priestle et al. [6] defined the effetive ield urvature as the intersetion of the line through the first ield point with the line drawn tangent to the M-φ diagram. This definition of ield urvature an be useful in the presentation of the experimental results (when suffiient data are not available for reliabl estimating the flexural strength of the onrete). However, there is an element of user subjetivit in fitting the tangent line as different tangent lines an be fitted to the softened branh of moment urvature urve. The definition adopted in this paper is based on the first ield point (either onrete or steel) and the maximum flexural strength of the olumn (Equation 4): M max M max φ = Min φ ;φ (4) M M where φ is the urvature when the onrete strain reahes the peak stress of the unonfined onrete,, and φ is the urvature at the onset of ielding of the longitudinal reinforement. The definition of ield urvature adopted herein (Figure ) avoids user subjetivit as it is based on the maximum flexural strength of the olumn. This allows engineers to develop bi-linear moment urvature response funtions whih are of pratial importane to the strutural design of reinfored onrete olumns. 4. ESTIMATION OF YIELD CURVATURE Expressions for the effetive ield urvature presented in this paper are based on the moment-urvature response of olumns aording to methodolog developed in Setion

7 3. It was observed that the effetive ield urvature of the olumn is influened b the overall size of the ross-setion (diameter) D, the axial load ratio, n, the strength of onrete, f and reinforement, f, and to some extent on the ratio of longitudinal reinforement, ρ and the depth of over onrete, (or equivalentl the ratio of area of the gross setion to the area of the ore, A g /A ). 3 5 Moment (kn-m) ielding of onrete ielding of reinforement ield urvature moment urvature (M-φ) response Curvature (rad/m) D=1m f' =3 MPa n=.1 ρ=1% =.5 Figure : Moment urvature (M-φ) response Cirular olumn setions of diameter from.5-.5 m having axial load ratio of -.5 and reinforement ratios of 1-6% have been onsidered. Both normal- and high-strength olumns were within the sope of the stud. Conrete strengths ranging from 3 MPa to 1 MPa, whih overs the wide range of onrete urrentl used for the design of onrete olumns and also the maximum limit permitted b most design odes. Conrete over has been onsidered as.5 m. However, to stud the influene of the onrete over on the effetive ield urvature of the olumns, ross-setions with onrete over varing between.3m and.1m were analed. The ield strength of reinforing bars was assumed to be 4 MPa, 5 MPa and 6 MPa. The elasti modulus of the reinforing bar was taken as GPa. Influene of Setion Dimension Previous researhes on ield urvature proposed expressions for ield urvature in terms of overall ross-setion dimension [4,6] and also in terms of the effetive depth of the ross-setion [7]. In this paper, the effetive ield urvature is presented in terms of the overall ross-setion dimension (D) Ref. [4,6] e=. e=.5 e=.3 φ / 3 1 f' =3 n=. ρ=1% φ / =./D 1.1 Ref. [7] Column Diamter, D (m) Figure 3: Influene of olumn diameter

8 It is evident from Figure 3 that the effetive ield urvature is signifiantl dependent on the diameter of the ross-setion (D) and that the best fit urve an be obtained when it is expressed in term of D The following expression is proposed for the estimation of the effetive ield urvature to design normal strength (f = 3 MPa) onrete olumns whilst ignoring the effets of the axial load (n=). φ =. (5) 1.1 D It is seen from Figure 3 that previous studies have over-estimated the effetive ield urvature, espeiall when the diameter of the setion is small. This over-estimation is also apparent from results presented in Ref. [7]. Intuitivel, the ratio of area of the gross setion to the area of the ore (A g /A or the thikness of onrete over) ma have some influene on the estimates of the effetive ield urvature. Figure 4 presents the influene of onrete over on the estimation of effetive ield urvature. All data points have been normalized with respet to the ield urvature when the onrete over is.5 m. 1.5 Normalized Curvature 1.5 D=.5 m D=1 m D=1.5 m Conrete Cover, (m) Figure 4: Influene of the onrete over It is evident from Figure 4 that onrete over (or A g /A ) does not have signifiant influene on the effetive ield urvature. The influene is even less when the overall diameter of the ross-setion is greater than 1m. However, the onrete over ma have some effets, though not signifiant, when the diameter of the setion is less than.5 m. In Figure 4, onrete over of up to.7 m has been onsidered for a olumn of.5 m in diameter. It is apparent that the gross diameter of the ross-setion is a better parameter to use than the effetive depth of the setion. Influene of the Strength of Conrete Figure 5 presents the influene of the onrete strength (f ) on the effetive ield urvature. The normalized ield urvature (normalized with respet to the ield urvature of 3 MPa onrete) varies from 1. to.9 when the onrete strength varies from 3 MPa to 1 MPa. Thus, ignoring the effets of onrete strength ould onl result in the overestimation of the effetive ield urvature b up to 1% onl (when the onrete strength is 1 MPa). However, the strength of onrete has signifiant influene on the ield urvature and the axial load ratio (refer next sub-setion). A modifiation fator has been proposed to take into aount the effets of the axial load level (Equation 6).

9 .7 ) = 1.5 MF ( f f (6) MF(f' )=1.5x f' -.7 /D n=% ρ=1% D=.5 m D=1.5 m D=.5 m Stength of Conrete, f' (MPa) D=1 m D= m Figure 5: Influene of Strength of Conrete Influene of Axial Load Figure 6 presents the influene of the axial load ratio on the normalized ield urvature (normalized with respet to the ield urvature at zero axial load level). The data points shown in the figure represent the average values of the normalized ield urvature for all the setion dimensions onsidered (.5m-.5m). It is seen that onrete strength influenes the ield urvature when subjet to varing axial load ratio. Yield urvature is shown to inrease with inreasing axial load ratio from -.3 and beond that point effetive ield urvature dereases with inreasing axial load ratio. Similar observations an be found in Ref. [7]. A modifiation fator has been proposed to take into aount these observations (Equation 7). MF ( n) = f.6 n (.43 f +.85) n (7) ( ) where n is the axial load ratio and f is the strength of onrete. It is important to mention that suh modifiation fator ma overestimate the ield urvature for smaller diameter setion (D <.5 m) under high axial load. However, bridge piers are normall subjeted to low level of axial load (around 1%) and usuall have larger diameter ross-setions. Normalized Curvature f'=3 MPa f'=5 MPa f'=4 MPa f'=6 MPa MF(n)=1+ (.41xf'-.6)xn- (.43xf'+.85)xn f'=8 MPa f'=1 MPa Axial Load Ratio (n) Figure 6: Influene of Axial Load Influene of Longitudinal Reinforement Ratio Previous studies on ield urvature have not expliitl studied the effet of axial load ratio as mentioned in Setion. It has been argued that longitudinal reinforement ratio

10 does not have signifiant influene on the effetive urvature. Figure 7 presents the influene of longitudinal reinforement ratio on the normalized effetive ield urvature (normalized with respet to the ield urvature when longitudinal reinforement ratio=1%). The effet of longitudinal reinforement is ver low when the longitudinal reinforement ratio is more than 3%. In most bridge design odes, the permitted amount of longitudinal reinforement varies from 1-6%, although in real pratie longitudinal reinforement ratio less than 3% is usuall provided to avoid ongestion of the reinforement. Results shown in Figure 7 are based on the ondition of no axial load and the differene would be even lower if the axial load level is onsidered (i.e. n > ). A modifiation fator has been proposed to take into aount effets of the longitudinal reinforement ratio (Equation 8)..16 MF ( ρ ) = ρ (8) D=.5 m Normalized Curvature MF(ρ)=ρ.16 D=1 m D=1.5 m Longitudinal Reinforement Ratio ρ (%) Figure 7: Influene of the Longitudinal Reinforement Ratio Proposal for the Estimation of Effetive Yield Curvature Based on the parametri studies onduted on more than olumns, algebrai expressions have been developed (Equation 9) for the estimation of the effetive ield urvature. It was found that ignoring the effets of the longitudinal reinforement ratio would onl marginall underestimate the effetive ield urvature and hene its effets ould be negleted in the preliminar design of the onrete olumn. However, parameterising the effets of longitudinal reinforement ratio an be useful for the aurate evaluation of existing onrete olumns. φ =. MF( f ) MF( n) MF( ρ) 1. 1 (9a) D.7 MF ( f ) = 1.5 f (9b) (.41 f.6) n (.43 f +.85) MF ( n) = 1+ n (9).16 MF ( ρ ) = ρ (9d) The above equations an be easil programmed into an exel spreadsheet for estimating the effetive ield urvature for a wide range of onrete olumns having different axial load ratios, onrete strengths and longitudinal reinforement ratios.

11 Figure 8 indiates that the predited values of the ield urvatures are within 1% of the alulated ield urvatures. Several data points have been observed to be signifiantl different from the alulated values. These data points are for.5 m diameter olumns with axial load ratios of between.3 and.5. It is mentioned earlier that the MF(n) ma overestimate the ield urvature for small diameter olumns under higher axial load ratio. However, the onditions of high axial load ratios in small diameter olumns are not ommon in bridge onstrution.. Predited φ D=.5m; n>.3 n= n=.1 n=. n=.3 n= Caluated φ 5. CONCLUSIONS Figure 8: Calulated Versus Predited Yield Curvature Simple expressions for the estimation of the effetive ield urvature for normal- and high-strength irular reinfored onrete (RC) olumns have been developed based on moment urvature anales of olumn setions. Suh expressions an be easil programmed into an exel spreadsheet and an be ver useful for the preliminar design of onrete olumns and also for the performane evaluation of existing olumns. Previous studies on the effetive ield urvature have been reviewed ritiall and the limitations of the studies in inorporating the influene of axial load, strength of onrete and reinforement have been identified. The present stud is a signifiant improvement over the previous studies as all the parameters that influene the estimate of ield urvature have been quantified. The ield urvature is influened b the size (diameter) of the setion, the axial load ratio, the strength of onrete and to some extent the amount of longitudinal reinforement and the thikness of the onrete over. It has been observed that the best estimate of ield urvature an be obtained when it is expressed in terms of the gross setion depth rather than the effetive depth, as the onrete over has insignifiant influene. The amount of longitudinal reinforement does not have signifiant influene on the effetive ield urvature. Hene, the effetive ield urvature an be estimated reasonabl without expliitl onsidering the influene of the amount of longitudinal reinforement. This is espeiall useful for the preliminar design of onrete olumns. However, the effet of longitudinal reinforement ma be important for the aurate performane evaluation of existing olumns.

12 6. REFERENCES [1] R. Park and T. Paula, Reinfored onrete strutures, Wile, USA, [] T. Paula and M.J.N. Priestle Seismi design of reinfored onrete and masonr buildings, Wile Intersiene, New York, USA, 199. [3] R. Park, A stati fore-based proedure for the seismi assessment of existing reinfored onrete moment resisting frames, Bulletin of New Zealand Soiet for Earthquake Engineering, 3(3), 13-6, [4] M.J.N. Priestle, M.C. Calvi and M.J. Kowalsk, Displaement-based seismi design of strutures, IUSS Press, Pavia, 7. [5] E. Miranda and J.R. Garia, Evaluation of approximate methods to estimate maximum inelasti displaement demand, Earthquake Engineering and Strutural Dnamis, 31, ,. [6] M.J.N. Priestle, J. Ranzo, G. Benzoni and M.J. Kowalsk, Yield displaement of irular bridge olumns, Proeedings of the Fourth Caltrans Researh Workshop, California Department of Transportation, Saramento, [7] E.H. Montes, and M. Ashleim, Estimates of ield urvature for the design of reinfored onrete olumns, Magazine of Conrete Researh, 55(4), , 3. [8] F. Légeron, and P. Paultre, Uniaxial onfinement model for normal- and highstrength onrete olumns, Journal of Strutural Engineering, 19(), 41-5, 3. [9] A. Gomes and J. Appleton, Nonlinear li stress-strain relationship of reinforing bars inluding bukling, Engineering Strutures, 19(1), 8-86, [1] P. Paultre, MNPHI: A program for setional analis of strutural onrete-user manual, CRGP Rep. No Department of Civil Engineering, Universit of Sherbrooke, Sherbrooke, Canada, 1. [11] D. Lehman, J. Moehle, S. Mahin, A. Calderone, and L. Henr. Experimental evaluation of the seismi performane of reinfored onrete bridge olumns, Journal of Strutural Engineering, 13(6), , 4.