Models for Estimating the Deformation Capacities of Reinforced Concrete Columns

Transcription

1 ECAS International Symosium on Structural and Earthquake Engineering, October 4,, Middle East Technical University, Ankara, Turkey Models for Estimating the Deformation Caacities of Reinforced Concrete Columns M. Inel Pamukkale University, Deartment of Civil Engineering, Denizli, Turkey M. A. Aschheim Mid-America Earthquake Center, University of Illinois at Urbana-Chamaign, Urbana, IL 68-5, USA ABSTRACT: Five models for estimating deformation caacity of RC columns are reviewed in an attemt to establish reliable inelastic dislacement caacities. Dislacement ductility, eak drift, and lastic hinge rotation arameters are used as measures of inelastic deformation. In many cases the analytical models are observed to overestimate deformation caacity and exaggerate the effect of transverse steel on deformation caacities. A simle, classical model is shown to roduce similar or better estimates of the column deformation caacities than more comlex models. Keywords: deformation caacity, reinforced concrete columns, dislacement ductility ÖZET: Betonarme kolonların deformasyon kaasitelerini elde etmek amacıyla mevcut beş ayrı model incelenmiştir. Elastic ötesi deformasyon ölçütü olarak yer değiştirme sünekliliği, maksimum öteleme ve lastic mafsal dönme arametreleri kullanılmıştır. Birçok durumda analitik modellerin yüksek yer değiştirme kaasiteleri verdiği ve enine donatının deformasyon kaasiteleri üzerindeki etkiyi oluğundan fazla gösterdiği gözlenmiştir. Daha karmaşık modellere oranla, basit ve klasik bir modelle benzer veya daha iyi kolon deformasyon kaasitelerinin hesalanabileceği gösteriliştir. Introduction Inelastic behavior is intended in most structures subjected to infrequent earthquake loading. Reinforced concrete columns are the referred locations of inelastic behavior in many bridges because of their accessibility for insection and reair. Thus, the develoment of dislacement-based design rocedures for bridges requires knowledge of the deformation caacity of the bridge columns. Methods for estimating the deformation caacity of reinforced concrete columns have been the focus of many research studies. Several available models for estimating column deformation caacity include those by Park and Paulay (975), Lehman and Moehle (998), Panagiotakos and Fardis (), and Priestley et al. (996). Excet for the emirical model of Panagiotakos and Fardis ()), these models estimate the

2 deformation caacity at yielding and ultimate based on lumed inelasticity idealization for a cantilever, as shown in Figure. The simlest form of such model is to comute deformation caacities based on flexural contributions, assuming the curvature distributions of Figure, as described by Park and Paulay in 975. This aroach is termed the simle model in this aer. This aer reviews five analytical models in an attemt to establish reliable deformation caacities. Dislacement ductility, drift, and lastic hinge rotation arameters are used as measure of inelastic deformation. A limited set of exerimental data from large-scale tests of reinforced concrete columns having a rectangular cross section are also considered. Estimates of the deformation caacities of the columns are comared to the observed deformation caacities. Nominal deformation caacities for columns with reinforcement as recommended by ATC- (Imroved Seismic Design Criteria for California Bridges) are suggested based on the exerimental data. H a P u y L θ H a L H al+p.5l M u φ y φ u Figure. Lumed inelasticity model for a cantilever column L Evaluation of Inelastic Deformation Caacities The load-deformation behavior of a column is commonly idealized by a bilinear curve that is fit to the resonse comuted analytically, or may be fitted aroximately to the enveloe of exerimental test results. Although a bilinear curve may be defined by two oints, the yield and ultimate dislacements ( y and u ) and the corresonding loads, various definitions of these oints have been used by different researchers. Once the yield and ultimate oints are established, the dislacement ductility, µ δ, lastic dislacement,, lastic hinge rotation, θ, and eak drift, δ d caacities may be derived for the cantilever column of Figure. u µ δ = () u y = () y u y θ = () L. 5L u δ = (4) d L

3 While exerimental data is invaluable without doubt, the design of columns normally relies on calculated estimates of the load-deformation behavior, as illustrated in Figure. The yield and ultimate dislacements may be estimated by including the contributions of flexure, shear, and anchorage sli, as roosed by various researchers. (5) = + + y y, flexure y, sli y, shear (6) = + + u u, flexure u, sli u, shear The five models considered in this aer are: () simle model, () Lehman model (Lehman and Moehle, 998), () Panagiotakos analytical model (Panagiotakos and Fardis, ), (4) Panagiotakos emirical model (Panagiotakos and Fardis, ), and (5) Priestley model (Priestley et al., 996). The calculation of yield and ultimate dislacements according to the five models is summarized in Table. Deformation Index Table. Definition of deformation indices using available models Simle Model Lehman Model Panagiotakos Analytical Model Panagiotakos Emirical Model Priestley Model y,flexure φ yl / φ yl / φ yl / φ yl / φ y(l+.5f yd b) / y,shear NA V yl/(.4e c,sec.8a g) NA.5L () conc, shear+ truss, shear y,sli NA φ ylf yd b/8 f c NA ε yf yd bl/4 f c (d-d ) incl w/ y,flexure θ (φ u-φ y)l (φ u-φ y)l (φ u-φ y)l () θ u- y/l (φ u-φ y)l θ u u/l u/l u/l () u/l θ (L-.5 L ) θ (L-.5 L ) θ (L-.5 L ) u- y θ (L-.5 L ) u y+ y+ y+ θ ul y+ L.5H.5L(M u-m n)/m n +.(f u-f y) d b/4 f c.l+.4f yd b NA.8L+.f yd b NA: Not Alicable () Details for shear contribution to yield dislacement can be found in Priestley et al. (996). () Details for Panagiotakos analytical and emirical models can be found in Panagiotakos and Fardis (). The alication of the models (excet the Panagiotakos emirical model) requires calculation of yield and ultimate curvatures. At the ultimate oint, the contributions to dislacements due to shear and anchorage sli, if considered in the model, are lumed in the lastic hinge length. Since in most cases, the roosed lastic hinge lengths are based on reroducing the exerimental resonse, the definitions of yield and ultimate curvatures used by the investigators become imortant. The Panagiotakos analytical model rovides equations for the calculation of yield and ultimate curvatures. We imlemented the Mander model (Mander et al., 988) in the moment-curvature analyses required for the simle, Lehman, and Priestley models. Figure shows the momentcurvature resonse comuted for a tyical well-confined column. The dashed curve is a bilinear curve fitted to the comuted curve, defined by an effective yield oint (M n, φ y ) and failure of the cross-section (M u, φ u ). The yield oint (M y, φ y ) is defined as the oint when the extreme tension steel yields or the strain in the concrete at the extreme comression fiber reaches., whichever comes first. For any axial load level, the nominal flexural strength, M n, is calculated using a rectangular stress block but with the secified (for analytical study) or reorted (for exerimental data) yield and comressive strengths used without reduction factors. The ultimate curvature is defined as the smallest of the curvatures corresonding to () a reduced moment equal to % of maximum moment, determined from the moment-curvature analysis, () the extreme comression fiber reaching the ultimate concrete comressive strain as determined using the Mander model, and () the longitudinal steel reaching a tensile strain of 5% of ultimate strain caacity.

4 M n M n, φ y (effective yield) M y, φ y (first yield) M u, φ u Moment Comuted Resonse Bilinear Fit Curvature Figure. Tyical moment-curvature resonse of a well-confined column Inelastic Deformation Caacities from Analytical Models Using the five models, the sensitivity of the inelastic deformation caacities was studied for cantilevered columns by varying cross section size, asect ratio, transverse reinforcement amount, and axial load ratio. In one set of analyses, three cross section sizes (5 mm x 5mm, 6 mm x 6 mm, mm x mm) were used, with the asect ratio (cantilever length divided by section deth) held constant at 4. In another set of analyses, asect ratios were changed from to by varying the column length while the cross section was ket constant. Two levels of transverse reinforcement were considered: the amount required er ATC- recommendations and one tenth of the ATC- requirement, termed well-confined and oorly-confined, resectively. Two levels of axial load were considered, equal to. and.5 times A g f c. Material roerties were constant for the cases considered; 4 MPa yield strength for both longitudinal and transverse steel and 7.5 MPa for concrete. The longitudinal reinforcement ratio was.5% for all cases. Neglecting minor differences due to cover requirements, the dislacement ductility, lastic hinge rotation, and ultimate drift caacities were indeendent of the cross section size under the constraint of constant asect ratio of 4. Figure shows how the calculated lastic hinge rotations change with asect ratio for an invariant cross section. The figure also illustrates the effect of the transverse steel amount, axial load ratio, and the influence of the model used to estimate deformation caacity. Similar lots for dislacement ductility and eak drift are available in Inel (). The overall trends exhibited by the collection of models lead to the following observations: (a) excet for the lastic hinge rotation caacity estimated by the simle model, no arameter is invariant with changes in asect ratio, (b) the sensitivity of the inelastic deformation quantities to the models is obvious; different models can result in substantially different estimates of deformation caacity, (c) the effect of axial load ratio on deformation caacity is clear for the oorly-confined case; deformation caacities are smaller for the high axial load case, even though the ATC- comliant transverse steel is greater than for the case of low axial load ratio, and (d) well-confined columns can exhibit substantial calculated deformation caacities, for the cases investigated (axial loads equal to. and.5 times A g f c ). Differences in deformation caacities estimated with the models are greater for the high axial load ratio case. The largest differences are observed in the lastic hinge rotation and drift caacities for high asect ratios, while the largest differences in the dislacement ductility caacity are observed for low asect ratios (Inel, ). Key observations related to the individual models are: (a) the simle model tends to

5 rovide a lower bound estimate of lastic hinge rotation and eak drift for the wellconfined case, (b) the Lehman model is sensitive to the level of confinement. For the oorly-confined case, the deformation caacities estimated by the Lehman model are considerably smaller than those estimated by the Panagiotakos analytical and Priestley models, esecially for the low axial load ratio case, (c) the Lehman model is sensitive to the level of axial load. For the well-confined columns with high axial load ratio, lastic hinge rotation and drift caacities estimated by the Lehman model are considerably higher than those estimated by the other models. The reason for this seems to be that the lastic hinge length suggested by Lehman and Moehle (998) deends exlicitly on the axial load ratio while the other models have lastic hinge lengths that are indeendent of the axial load ratio. For examle, for the well-confined case with asect ratio of 4, when the axial load ratio increases from. to.5, the lastic hinge length estimated by the Lehman model doubles. It should also be noted that although no limitations are identified in the use of the model, Lehman roosed the lastic hinge length equation based on test data for the axial load ratio of., (d) for the Panagiotakos analytical model, the dislacement ductility caacities are nearly indeendent of the asect ratio. This contradicts the generally acceted (and exerimentally verified by Lehman and Moehle (998)) trend that dislacement ductility caacity decreases as asect ratio increases. One reason this occurs is that the lastic hinge lengths, estimated by the Panagiotakos analytical model, are considerably smaller for small asect ratios than those determined by other models such as the Priestley model. Another reason is that the shear dislacement contribution to the yield dislacement for the Panagiotakos analytical model can be substantial (e.g., the shear contribution may exceed the flexural contribution for an asect ratio of, deending on the axial load ratio), resulting in larger yield dislacements. The combination of smaller ultimate dislacement caacity and larger yield dislacements for small asect ratios results in smaller ductility caacities; this leads to results counter to the exected trend in dislacement ductility caacity as a function of asect ratio, and (e) the Panagiotakos emirical model tends to estimate higher dislacement ductility and drift caacities than the other models for the oorly-confined case...6 well-confined P/A gf c =...6 well-confined P/A gf c =.5.. Plastic Hinge Rotation (rad) oorly-confined P/A gf c = oorly-confined P/A gf c =.5. Simle Lehman Priestley. Panagiotakos emirical Panagiotakos analytical Asect Ratio, M/VD Figure. The effect of change in asect ratio on the inelastic measurement quantities comuted using the different models.

6 In summary, the sensitivity study indicates that none of the inelastic deformation caacity arameters (the lastic hinge rotation, dislacement ductility, and eak drift caacities) are a robust, invariant measure of inelastic deformation caacity, for the cases of varying asect ratio considered. Because the analytical study could not identify a single robust measure of inelastic deformation caacity, the following sections investigate results from exerimental tests. Inelastic Deformation Caacities from Exerimental Data Set The exerimental data considered here was obtained from large-scale tests of rectangular reinforced concrete columns subjected to quasi-static reversed cyclic lateral loading, with axial load ratios of varied intensities held constant throughout the tests. Criteria used to establish database were: () a rectangular cross section with minimum dimension of mm, () at least 8 longitudinal bars, each laterally suorted by transverse reinforcement, and () minimum asect ratio (M/VD) of.5. A total of tests with information required were retained among 9 secimens conforming to these criteria. The retained secimens had asect ratios ranging from.86 to 4.8, axial load ratios, P/f c A g, ranging between. and.77, f c between and 47 MPa, longitudinal reinforcement ratios ranged between.5 and.% of the gross area with yield strength of 4 to 5 MPa. Exerimental data was evaluated by identifying an enveloe of the moment at the base of column that includes the alied (actuator) force-deformation lot and the P- contribution arising from the alied axial load. That is, M= H a L + P, where H a = alied horizontal force, P= alied axial load, and L is column height. It should be noted that secondary moment caused by P-δ along the length of member is neglected. The retained secimens had sufficient transverse reinforcement both within and outside otential lastic hinge regions to carry the maximum exerimental shear develoed during testing based on calculation, with the strengths established using the ATC- equations for shear strength. Thus, the inelastic deformation caacity of the secimens was exected to be limited by mechanisms associated with flexural deformation rather than shear strength decay. The retained data is used to observe effects of axial load ratio on exerimentallydetermined deformation caacities and as a basis for examining several roosed relations for estimating deformation caacity. The aarent dislacement ductility, eak drift, and lastic rotation caacities of the secimens were examined using the identified ultimate dislacements in conjunction with the estimated yield dislacements and recommended values of lastic hinge length. The word aarent signifies data that was obtained or derived directly from the exeriments. The ultimate dislacements of the columns were determined by review of the measured resonse data. The ultimate dislacement was defined as the maximum dislacement corresonding to a % reduction of the maximum moment (including P- contributions) develoed during the exeriment. This definition was used by Priestley and Park (984) among others. Since this definition corresonds to a reduction in lateral strength, it may be assumed that vertical load carrying caacity was maintained throughout and beyond the ultimate dislacement caacity as defined here. The use of a % dro is arbitrary and is intended to reresent a substantial remaining flexural caacity for the confined concrete section. From the data set, it is observed that secimens with ATC- comliant transverse reinforcement can achieve a dislacement ductility caacity of 6 or more, a

7 lastic rotation caacity of.4 or more, and a drift caacity of 4.5% or more (Inel, ). Comarison of Aarent and Estimated Deformation Caacities The aarent inelastic deformation caacities relied uon y and L estimated using available models such as the simle, Lehman, Panagiotakos analytical, and Priestley models. These models would have to estimate values of aarent θ in order to accurately estimate the exerimentally determined values of u. This section comares the aarent lastic hinge rotation caacity values with the estimates of θ according to the four models that use the lumed inelasticity model. The Panagiotakos emirical model is also considered for comarison uroses. For this model, the aarent lastic dislacement,aarent = u,aarent -θ y L was comared to the estimated lastic dislacement,estimated = (θ u, -θ y )L, where θ y and θ u were comuted using the roosed equations. The urose of the comarisons of this section is to illustrate the reliability of the aarent inelastic deformation caacities determined from the exerimental data set, rather than showing the accuracy or inaccuracy of the models. The estimated lastic hinge rotation caacities of the models that use the lumed inelasticity model were calculated as θ = (φ u - φ y )L. Comarison between the aarent and estimated deformation caacities shows that differences among the five models are obvious. The ratio of the estimated and the aarent deformation caacities is lotted in Figure 4 against axial load ratio and transverse reinforcement content to identify ossible trends. The figure shows that as axial load ratio increases the differences between models become more noticeable. One obvious reason is the differences in the equations for lastic hinge length calculation. The simle, Panagiotakos analytical, and Priestley models do not consider the axial load ratio in calculating the lastic hinge length while the Lehman model deends exlicitly on the axial load ratio. The effect of transverse reinforcement on the lastic rotation caacity is considered further. As the ercentage of ATC- transverse reinforcement increases, all models excet the Panagiotakos emirical model tend to estimate higher caacities, indicating that the models exaggerate the effect of transverse steel on deformation caacity. The simle model tends to underestimate the lastic Plastic Deformation Caacity: Estimated/Aarent simle Lehman Priestley Panagiotakos analytical Axial Load Ratio, P/A gf c Figure 4. Ratio of the estimated to the aarent lastic deformation caacities of exerimental data set vs. axial load ratio and transverse steel content. Panagiotakos emirical Transverse Steel: Provided/ATC- requirement

8 deformation caacity while the other models, esecially the Lehman and Priestley models, can overestimate the lastic deformation caacity. Conclusions Based on the arametric study of inelastic deformation arameters and the study of the exerimental data, the followings were observed: (a) overall, the simle model tends to give lower bound estimates of deformation caacity, esecially for lastic hinge rotation and drift caacities for columns with asect ratios of or greater, (b) the arametric study on varying cross section size under constant asect ratio showed that the dislacement ductility, lastic hinge rotation, and drift caacities (as ercentage of secimen length) are indeendent of the cross section scaling when asect ratio is ket constant. In the arametric study, minor differences relating to cover requirements and nominal bar diameters were neglected, (c) the scatter in the aarent deformation caacities is similar at low and high axial load ratios, (d) lastic rotation caacity was not clearly deendent on axial load ratio when confinement was rovided satisfying ATC- requirements, (e) analytical models to estimate deformation caacity show large variations, (f) comarison of the aarent and estimated deformation caacities suggests that the analytical models can overestimate deformation caacity, and (g) the analytical models may exaggerate the effect of transverse steel on deformation caacities. References ATC-, Imroved Seismic Design Criteria for California Bridges: Provisional Recommendations, Alied Technology Council, Redwood City, California, 996. Inel, M.,, Dislacement-Based Strategies for The Performance-Based Seismic Design of Short Bridges Considering Embankment Flexibility, Ph.D. Thesis, University of Illinois at Urbana-Chamaign. Lehman, D. E., Moehle, J. P., 998, Seismic Performance of Well-Confined Concrete Bridge Columns, Pacific Earthquake Engineering Research Center, Reort No. PEER-998/, University of California, Berkeley, CA, USA. Mander, J. B., Priestley, M. J. N., and Park, R., 988, Theoretical Stress-Strain Model for Confined Concrete, Journal of Structural Engineering, ASCE, Vol. 4, No. 8, Panagiotakos, T. B., Fardis, M. N.,, Deformation of Reinforced Concrete Members at Yielding and Ultimate, ACI Structural Journal, Vol. 98, No., Park, R., Paulay, T., 975, Reinforced Concrete Structures, John Wiley & Son, Inc, New York, 769. Priestley, M. J. N. and Park, R., 984, Strength and Ductility of Bridge Substructures, Road Research Unit Bulletin #7, National Roads Board, Wellington, New Zealand. Priestley, M. J. N., Seible, F., Calvi, G. M. S., 996, Seismic Design and Retrofit of Bridges, John Wiley & Sons, Inc., New York.