EVALUATION OF EXISTING REINFORCED CONCRETE COLUMNS

13 th World Conferene on Earthquake Engineering Vanouver, B.C., Canada August 1-6, 2004 Paper No. 579 EVALUATION OF EXISTING REINFORCED CONCRETE COLUMNS Kenneth J. ELWOOD 1 and Jak P. MOEHLE 2 SUMMARY An idealized bakbone model defining the key damage states of an existing reinfored onrete olumn (flexural yielding, shear failure, and axial load failure) is developed using drift apaity models. Assumptions and limitations of the drift apaity models are disussed. To demonstrate the appliation of the idealized bakbone model to the evaluation of existing reinfored onrete olumns, the model is ompared to data from shake table tests and measured drifts from an instrumented building damaged during the Northridge Earthquake. The need for probabilisti drift apaity models is also disussed. INTRODUCTION Experimental researh and post-earthquake reonnaissane have demonstrated that reinfored onrete olumns with light or widely spaed transverse reinforement are vulnerable to shear failure during earthquakes. Suh damage an also lead to a redution in axial load apaity, although this proess is not well understood. As the axial apaity diminishes, the gravity loads arried by the olumn must be transferred to neighboring elements, possibly leading to a progression of damage, and in turn, ollapse of the building. To assess the ollapse potential of a building struture, engineers require models apturing the key damage states of a olumn during seismi response, namely: flexural yielding; shear failure; and axial load failure. Models have been proposed by Elwood and Moehle [1, 2] to estimate the drifts at shear and axial load failure for existing reinfored onrete olumns. This paper will summarize these models and disuss how they an be inorporated into an idealized bakbone response for existing reinfored onrete olumns. This idealized bakbone response has been implemented in an analytial model [3], enabling the nonlinear analysis of existing building frames inluding the influene of shear and axial load failures. The models for shear and axial load failure have been developed for olumns whih are expeted to experiene flexural yielding prior to shear failure. As suh, the idealized bakbone model presented here should only be applied to olumns with similar harateristis. To demonstrate its appliation, the bakbone model has been applied to two olumns: one from dynami laboratory tests, and a seond from an instrumented building damaged during the Northridge Earthquake. 1 Assistant Professor, Department of Civil Engineering, University of British Columbia, Vanouver, Canada 2 Professor and Diretor, Paifi Earthquake Engineering Researh Center, University of California, Berkeley, USA

L Figure 1. Fixed-fixed olumn subjeted to lateral loads V Drift Ratio, δ = /L IDEALIZED BACKBONE RESPONSE The idealized bakbone aptures signifiant hanges in the shear-drift response of a olumn subjeted to yli lateral loads; namely, flexural yielding, shear failure (defined by a 20% loss of shear strength), and axial load failure (defined by the initial redistribution of axial load supported by the olumn). The bakbone presented herein is developed for a olumn whih is assumed to be fixed against rotation at both ends and deforming in double urvature, as illustrated in Figure 1. The shear, V, is onstant over the height of the olumn, and the drift ratio, δ, is defined as the displaement of the top of the olumn divided by the lear height of the olumn. 400 300 200 V p Figure 2 shows an example of the idealized bakbone response ompared with pseudo-stati test data by Sezen [4]. The bakbone is defined by the following oordinates approximating the ritial damage states during yli response: Shear (kn) 100 0 100 200 300 δ a δ s δ y δ y δ s δ a -V p 400 0.06 0.04 0.02 0 0.02 0.04 0.06 Drift Ratio Figure 2. Calulated idealized bakbone plotted with olumn test data (Column data from referene [4]) Flexural yielding at (δ y, V p ) Shear failure at (δ s, V p ) Axial load failure at (δ a, 0.0) where V p is the alulated plasti shear apaity, and δ y, δ s, and δ a, are the alulated drift ratios at yielding, shear failure, and axial load failure, respetively. Figure 2 illustrates the appliation of the idealized bakbone for a olumn with approximately onstant axial load, and hene, the bakbone is assumed to be symmetri about the origin. For olumns with varying axial load, the yli response will not be symmetri about the origin and the oordinates of eah damage state must be determined using the appropriate axial load. The following setions desribe how the oordinates for eah of the damages states an be alulated. Flexural Yielding The idealized bakbone approximates the gradual yielding of longitudinal reinforement with an elastiperfetly-plasti response. Yielding is assumed to our one the shear demand reahes the plasti shear apaity, V p. For the assumed boundary onditions shown in Figure 1, the olumn plasti shear apaity an be determined as follows: V p 2M p = (1) L

where M p is the plasti moment apaity based on standard setion analysis (assuming plane setions remain plane). The drift ratio at yielding of the longitudinal reinforement an be onsidered as the sum of the drifts due to flexure, bar slip, and shear: δy = δ flex + δslip + δshear (2) Assuming the olumn is fixed against rotation at both ends and assuming a linear variation in urvature over the height of the olumn, the drift ratio at yield due to flexure an be estimated as follows: δ flex L = φy (3) 6 where φ y is the urvature at yielding of the longitudinal reinforement. The drift ratio at yield due to bar slip an be estimated as follows [5]: db f yφy δ slip = (4) 8u where d b is the diameter of the longitudinal reinforement, f y is the yield stress of the longitudinal reinforement, and u is the bond stress between the longitudinal reinforement and the beam-olumn joint onrete. A bond stress of 0.5 f (MPa units) [ 6 f in psi units] an be assumed for the alulation of δ slip [5]. Finally, assuming the olumn is fixed against rotation at both ends, the drift ratio at yield due to shear deformations an be estimated by idealizing the olumn as onsisting of a homogeneous material with a shear modulus G: where A g the gross area of the olumn setion. 2M p δ shear = (5) (5 / 6) A GL The three drift omponents desribed above an be estimated using the olumn ross setional dimensions and material properties. Given the drift omponents, Equations 1 and 2 an be used to estimate the point of flexural yielding for the idealized bakbone, (δ y, V p ). Shear Failure Sine the idealized bakbone has been developed for olumns whih are expeted to yield prior to shear failure, some limited dutility an be expeted in the yli response prior to a signifiant degradation in shear strength. To apture this limited dutile response prior to shear failure, the idealized bakbone is given a horizontal slope between the drift at flexural yielding and the drift at shear failure, and hene, V p is g

used to approximate the shear demand at shear failure. The following paragraphs desribe the evaluation of the drift at shear failure, δ s. Several models have been developed to represent the degradation of shear strength with inreasing inelasti deformations [4, 6, 7, 8]. While these shear strength models are useful for estimating the olumn strength as funtion of deformation demand, they are less useful for estimating displaement at shear failure. For example, the model by Sezen [4] represents shear strength as a funtion of displaement dutility, µ δ, using the relation shown in Figure 3, where V 0 = initial shear strength. A small variation in shear strength (shown by mean plus or minus one standard deviation bounds) or similar variation in flexural strength (not shown) an result in large variation in estimated displaement apaity, µ δ. These models also suggest misleading trends in the relation between some ritial parameters (suh as axial load) and displaement apaity at shear failure. V V o Sezen shear strength model 1.0 0.7 slope = 3/40 µ+σ µ µ σ 1.0 µ δ µ δ Figure 3. Displaement at shear failure as funtion of model variability Pujol et al. [9, 10, 11] have proposed drift apaity models for olumns failing in shear. These models make an important ontribution by fousing attention diretly on displaement apaity and by analyzing data for model development. The database of Pujol et al. inludes olumns with transverse reinforement ratios exeeding 0.01, whih is larger than that whih is of interest in the present study. To evaluate the response of olumns with low transverse reinforement ratios, a database of 50 olumn tests with observed shear distress at failure and tested in single or double urvature was ompiled [4]. Columns in the database had the following range of properties: shear span to depth ratio: 2.0 a / d 4. 0 onrete ompressive strength: 13 45 MPa [ 1900 6500 psi] f longitudinal reinforement yield stress: 324 f yl 524 MPa [ 47 f yl 76 ksi] longitudinal reinforement ratio: 0.01 ρ l 0.04 transverse reinforement yield stress: 317 f yt 648MPa [ 46 f yt 94 ksi] transverse reinforement ratio: 0.0010 ρ " 0.0065 f

v v maximum shear stress: 0.23 0.72 [ 2.8 8.6 ] f, MPa f, psi P axial load ratio: 0.0 0.6 A f g The model of Sezen [4] an be used to estimate mean shear strength as a funtion of displaement dutility. As suggested by Figure 3, the intersetion of the shear orresponding to flexural strength with mean shear strength an be interpreted to indiate the expeted displaement at shear failure. Figure 4 ompares results obtained by this proedure with those atually observed during the tests inluded in the database. The results show relatively high bias and dispersion (mean of the measured drift ratio at shear failure divided by the alulated drift ratio is 1.78 and the oeffiient of variation is 0.63). 0.07 0.06 0.06 0.05 drift ratio alulated 0.05 0.04 0.03 0.02 0.01 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 drift ratio measured Figure 4. Measured versus alulated drift at shear failure, interpreted from Sezen shear strength model. drift ratio alulated 0.04 0.03 0.02 0 ρ < 0.0011 0.0011 ρ < 0.002 0.01 0.002 ρ < 0.004 0.004 ρ < 0.0055 0.0055 ρ < 0.008 0 0 0.01 0.02 0.03 0.04 0.05 0.06 drift ratio measured Figure 5. Measured versus alulated drift at shear failure, Equation 6 Elwood and Moehle [1] reevaluated the database test results from a displaement-apaity perspetive, and proposed the following relationship to estimate drift ratio at shear failure: 3 1 v 1 P 1 δ = + 4 ρ" (MPa units) (6) s 100 40 f 40 Ag f 100 3 1 v 1 P 1 = + (psi units)] 100 500 40 100 [ δs 4 ρ" f A g f where ρ = transverse steel ratio, v = nominal shear stress, f = onrete ompressive strength, P is the axial load on the olumn, and A g is the gross ross-setional area. Figure 5 ompares Equation 6 with the results from the database. The mean of the measured drift ratio divided by the alulated drift ratio is 0.97, the oeffiient of variation is 0.34. The orrelation is improved ompared with that shown in Figure 4. Equations 1 and 6 an be used to determine the point of shear failure for the idealized bakbone response. When omputing the drift ratio at shear failure using Equation 6, the shear stress, v, an be estimated based on the plasti shear apaity, i.e. v = V p /(bd).

Figure 6. Free-body diagram of olumn after shear failure Axial Load Failure Experimental studies have shown that axial load failure tends to our when the shear strength degrades to approximately zero [12]. Hene, the final point on the idealized bakbone assumes a shear strength of zero, and therefore, only requires the alulation of the drift ratio at axial load failure, δ a. Elwood and Moehle [2] have proposed a shear-frition model to represent the general observation from experimental tests that the drift ratio at axial failure of a shear-damaged olumn is inversely proportional to the magnitude of the axial load. Figure 6 shows a free-body diagram for the upper portion of a olumn under shear and axial load. The inlined free surfae at the bottom of the free-body diagram is assumed to follow a ritial inlined rak assoiated with shear damage. The ritial rak is one that, aording to the idealized model, results in axial load failure as shear-frition demand exeeds the shear-frition resistane along the rak. Several assumptions are made to simplify the problem. Dowel fores from the transverse reinforement rossing the inlined rak are not shown; instead, the dowel fores are assumed to be inluded impliitly in the shear-frition fore along the inlined plane. Shear resistane due to dowel ation of the longitudinal bars depends on the spaing of the transverse reinforement, and reasonably an be ignored for the olumns onsidered in this study. Relative movement aross the shear failure plane tends to ompress the longitudinal reinforement. Given the tendeny for bukling, espeially in the limit, the axial fore apaity of the longitudinal reinforement will be assumed equal to zero. Finally, the horizontal shear fore is assumed to have dropped to zero in the limit following shear failure as noted above from experimental evidene [12]. Equilibrium of the fores shown in Figure 6 results in the following expression for the axial apaity of the olumn: Ast fytd 1+ µ tanθ P = tanθ s tanθ µ (7) in whih all variables are shown in Figure 6 exept µ = effetive shear-frition oeffiient. An empirial approah was used to define the rak angle θ and effetive shear-frition oeffiient µ, with data from twelve olumns tested at the University of California, Berkeley [4, 13]. These full-sale shear-ritial reinfored onrete olumns were subjeted to onstant or varying axial load with yli lateral load

reversals in one plane only. Tests were ontinued past shear failure to the point where axial load ould no longer be sustained. From these relations, Elwood and Moehle [2] developed relations among axial load, transverse reinforement, and drift ratio at axial load failure as: 2 4 1+ tan θ δ a = 100 s tan θ + P A st f yt d tanθ (8) in whih θ = ritial rak angle assumed = 65 deg, P = axial load, A st = area of transverse reinforement parallel to the applied shear and having longitudinal spaing s, f yt = transverse reinforement yield stress, and d = depth of the olumn ore measured parallel to the applied shear. Results of tests reported in [4] and [13] are ompared with Equation 8 in Figure 7. While useful as a design hart for determining drift apaities, Figure 7 must only be used with a full appreiation for the inherent satter in the results and the limitation that the results are based on unidiretional, pseudo-stati tests. 40 30 model +/ σ test data P/(A st f yt d /s) 20 10 0 0.00 0.02 0.04 0.06 0.08 Drift Ratio at Axial Failure Figure 7. Drift apaity urve based on shear-frition model Disussion of Idealized Bakbone Model Given olumn details and the axial load demand, Equations 1, 2, 6, and 8 an be used to alulate the idealized bakbone response shown in Figure 2 for a reinfored onrete olumn experiening shear and axial load failure after flexural yielding. However, before the idealized bakbone model an be used to assess the apaity of an existing reinfored onrete olumn, it must be determined if the olumn is expeted to experiene shear failure after yielding of the longitudinal reinforement, a ommon harateristi of all the olumn tests used to develop the drift apaity models desribed above.

V V o V p 0.7V o Equation 6 Idealized flexural response Sezen shear strength model δ y δ s δ Figure 8. Comparison of Sezen shear strength model and model for drift ratio at shear failure (Equation 6) To determine if a olumn is likely to experiene shear failure after flexural yielding, the plasti shear apaity, V p, should be ompared with an appropriate shear strength model, suh as that proposed by Sezen [4]. As shown in Figure 8, the plasti shear apaity should fall in the range between the initial shear strength, V o, and the final shear strength, 0.7V o. Beause there is some dispersion between atual and alulated shear strength, some olumns with shear demand less than the alulated shear, or plasti shear apaity greater than V o, may still experiene shear failure after flexural yielding. Currently, engineering judgment is required to selet whih olumns with V p outside the V o to 0.7V o range are still expeted to experiene shear failure after flexural yielding, and hene, an be evaluated using the proposed idealized bakbone model. To redue the reliane on judgment alone, there is a need to develop probabilisti apaity models aounting for the variability in the test results relative to the mean response aptured by the model. As desribed later in this paper, given suh probabilisti apaity models, it is possible to define the probability of a given olumn experiening shear failure after flexural yielding. It must be reognized that the axial failure model desribed above is based on data from only twelve olumns. All of the olumns were onstruted of normal strength onrete, had the same height to width ratio, and were designed to yield the longitudinal reinforement prior to shear failure. Only limited variation in the spaing and type of transverse reinforement was possible. The axial failure model presented here may not be appropriate for olumns for whom the test speimens are not representative. Furthermore, all twelve olumns were tested under unidiretional lateral load parallel to the one fae of the olumn. With the exeption of two tests, the loading routine was standardized, with eah olumn subjeted to nominally onstant axial ompression and a series of lateral displaements at inreasing amplitude (three yles at eah amplitude). During earthquake exitation olumns an experiene bidiretional loading and a wide variety of loading histories, whih may onsist of a single large pulse or many smaller yles prior to shear and axial load failure. It has been demonstrated that an inrease in the number of yles past the yield displaement an result in a derease in the drift apaity at shear failure [11]. Equation 6 for the drift ratio at shear failure does not aount for the influene of the number of signifiant loading yles. It is also antiipated that an inrease number of yles will redue the drift apaity at axial failure, although not enough test data is available to support or refute this hypothesis. Further testing of reinfored onrete olumns to the stage of axial failure is needed to supplement the urrent database.

The proposed models for the drift ratios at shear and axial load failure (Equations 6 and 8) provide estimates of the mean drift ratio at failure for the olumns inluded in the respetive databases. As suh, approximately half of the olumns from the databases used to develop the models experiened failure at drifts less than those predited by the models. Furthermore, atual olumns have onfigurations and loadings that differ from those used in the tests, so that some additional satter in results seems likely. A model used for design or assessment might be adjusted aordingly to provide additional onservatism as may be appropriate. Probabilisti apaity models, desribed later in this paper, may assist in seleting an appropriate degree of onservatism. APPLICATIONS OF THE IDEALIZED BACKBONE MODEL Two appliations of the idealized bakbone model developed above will be disussed. First, the idealized bakbone model will be ompared with shake table test results [14]; and seond, the bakbone model will be used to assess the apaity of a olumn from an instrumented building damaged during the Northridge Earthquake. Comparison with shake table test results Shake table tests were designed to observe the proess of dynami shear and axial load failures in reinfored onrete olumns when an alternative load path is provided for load redistribution. The test speimens were omposed of three olumns fixed at their base and interonneted by a beam at the upper level (Figure 9). The enter olumn had wide spaing of transverse reinforement making it vulnerable to shear failure, and subsequent axial load failure, during testing. As the enter olumn failed, shear and axial load would be redistributed to the adjaent dutile olumns. Two test speimens were onstruted and tested. The first speimen supported a mass of 300 kn, produing olumn axial load stresses roughly equivalent to those expeted for a seven-story building. The seond speimen also supported a mass of 300 kn, but pneumati jaks were added to inrease the axial load arried by the enter olumn from 128 kn (0.10 f A g ) to 303 kn (0.24 f A g ), thereby amplifying the demands for redistribution of axial load when the enter olumn began to fail. Transverse torsional beam 1830 1.5 m wide beam designed to model bending stiffness of typial spandrel beam (masses not shown for larity) #4 orner bars 230 #5 enter bars 1475 Designed to support gravity loads after failure of enter olumn Shear-Critial Column 230 Setion A W2.9 wire @ 152 o.. A B 255 #4 #3 spiral @ 50 Fore transduers to measure reations at base of olumns Setion B Figure 9. Shake table test speimen (units in mm).

Center Column Shear (kn) Figure 10a. Top of enter olumn, Speimen 1 at 24.9 se 100 P = 0.10f A g 50 0 50 Center Column Shear (kn) Figure 10b. Top of enter olumn, Speimen 2 at 24.9 se 100 P = 0.24f A g 50 0 50 100 0.08 0.04 0 0.04 0.08 Drift Ratio 100 0.08 0.04 0 0.04 0.08 Drift Ratio Figure 11. Center olumn hystereti response ompared with idealized bakbone The shear-ritial enter olumn was designed as a one-half sale reprodution of the 2.95 m tall, 457 mm x 457 mm square-ross-setion olumns reported in [4] and [13]. From those previous tests, it was expeted that the enter olumn would sustain flexural yielding prior to shear failure. Axial load failure was expeted to be more gradual for the olumn with low axial load and more sudden for the olumn with higher axial load. The remaining frame elements (i.e., the beams, outside olumns, and footings) were not saled from prototype designs, but instead were designed to ahieve the desired response. Both speimens were subjeted to one horizontal omponent from a saled ground motion reorded at Viña del Mar during the 1985 Chile earthquake. The first speimen with lower axial load sustained severe diagonal raking (Figure 10a) and a loss of lateral load apaity, however, was able to maintain most of its axial load demand. The enter olumn from the seond speimen with moderate axial load, also failed in shear, however, was unable to support most of its axial load demand, resulting in shortening of the enter olumn (Figure 10b). Further results from the shake table tests are reported elsewhere [14]. Figure 11 indiates that the idealized bakbones, alulated using Equations 1, 2, 6, and 8, provide a reasonable estimate of the measured hystereti response of the enter olumns. The model underestimates the drift ratio at shear failure for the speimen with low axial load, but provides a good estimate of the drift ratio at shear failure for the speimen with moderate axial load. These results suggest that the model for the drift ratio at shear failure (Equation 6) may underestimate the influene of the axial load on the drift ratio at shear failure, partiularly for low axial loads. Figure 11 also indiates that the observed

response of the test speimens was onsistent with the alulated drifts at axial failure based on Equation 8. The measured drifts for the speimen with low axial load (and no axial failure) remain below the alulated drift ratio at axial load failure, while the measured drifts for the speimen with moderate axial load (and experiening axial failure) exeed the alulated drift ratio at axial load failure. Figure 12. Building frame with olumns damaged during Northridge Earthquake #2 @ 305 6 - #7 Diretion of shear failure 508 Column C4 Properties: f y long = 496 MPa f y trans = 345 MPa f = 28 MPa P = 0.13 A g f L = 2.08 m A A 356 Setion A-A 356 Figure 13. Details and assumed properties for fourth story olumn Column damaged during the Northridge Earthquake In this setion the idealized bakbone model will be used to assess the apaity of a olumn from a building damaged during the Northridge Earthquake. The seven-story moment frame building, shown in Figure 12 and loated in Van Nuys, California, experiened signifiant ground shaking during both the San Fernando Earthquake and the Northridge Earthquake. Instrumentation distributed throughout the struture prior to both earthquakes has provided very valuable data on the inelasti response of moment frame buildings. Extensive literature on the struture is available elsewhere [15, 16, 17, et.]. Only details on the damaged fourth-story exterior olumns will be disussed here. An exterior olumn from the fourth

story of the building (C4, irled in Figure 12) will be evaluated using the idealized bakbone model. Typial details for this olumn are shown in Figure 13. The axial load was estimated based on the dead load from a three-dimensional stati analysis. Signifiant shear distress at the top of the olumn was observed after the Northridge Earthquake (Figure 14). Based on this damaged state, it is reasonable to expet that the olumn has lost the majority of its lateral load apaity due to shear failure. However, it is unlear whether the olumn experiened loss of axial load apaity resulting in gravity load redistribution to neighboring elements. Based on interpolation of the reorded motions on the third and sixth floors, the maximum drift ratio at the fourth story during the Northridge Earthquake was estimated to be 0.018. Note that 0.018 is expeted to be a lower-bound value sine the drifts were likely higher at the fourth story due to the onentration of damage at this level. Figure 14. Damage to Column C4 after Northridge Earthquake [17] As emphasized earlier, the idealized bakbone model has been developed for olumns whih are expeted to experiene shear failure after flexural yielding. To determine if olumn C4 experiened shear failure after flexural yielding, the plasti shear apaity (assuming the olumn is fixed against rotation at both ends), V p, is ompared with the shear strength aording to the Sezen model, V o, as illustrated in Figure 8. The ratio V p /V o is found to be 1.04, indiating that the Sezen shear strength model suggests that the olumn may have failed in shear prior to flexural yielding. Currently no model exists to determine the drift at axial load failure for olumns experiening this mode of failure. As will be disussed in the subsequent setion, due to the variability in the olumn shear strength (and, indeed, in the flexural strength), finding V p /V o greater than unity does not prelude the possibility of shear failure ourring after flexural yielding. To illustrate the appliation of the idealized bakbone model to olumn C4, it will be assumed that there exists suffiient probability that shear failure ourred after flexural yielding to warrant the investigation of this failure mode. This ase highlights the importane of developing drift apaity models for olumns expeted to fail in shear prior to flexural yielding. Applying Equations 2, 6, and 8, the following drift apaities were determined for olumn C4: δ y = 0.010 δ s = 0.023 δ a = 0.022 Note that the alulated drift at shear failure is slightly larger than the drift at axial load failure. This result is inonsistent with the expeted behavior, sine Equation 8 assumes shear failure has already ourred. Due to the lak of oupling between the models for the drift at shear failure (Equation 6) and the drift at axial failure (Equation 8), there is nothing requiring δ s < δ a when the drift ratios are alulated.

Due to the variability assoiated with both drift apaity models, the results an be interpreted to imply that axial load failure is expeted to our almost immediately after shear failure at a drift of approximately 0.022. The idealized bakbone response and the estimated maximum drift ratio during the Northridge Earthquake are shown in Figure 15. Given that the maximum drift ratio of 0.018 is a lower-bound value, and the variability in the drift apaities, these results suggest that olumn C4 may have experiened axial load failure and redistributed some load to neighboring elements. The results also reinfore the need to diretly aount for the variability in the apaity models during the evaluation of existing reinfored onrete olumns. 250 δ max =0.018 200 Shear (kn) 150 100 50 0 0 0.01 0.02 0.03 0.04 Drift Ratio Figure 15. Idealized bakbone model for olumn C4. PROBABILISTIC CAPACITY MODELS The apaity models desribed above only represent the mean response from the tests used to develop the models. Probabilisti apaity models, defining not just the mean response but also the variation in apaity about the mean value, are required to assess the probability of exeeding the olumn apaity. Given suh a probability, an engineer will be better equipped to make an informed deision regarding the need for seismi retrofit. The appliation of a probabilisti apaity model for the drift at shear failure is shematially represented in Figure 16. As disussed above, Equation 6 an be used to estimate the mean drift apaity at shear failure, δ s. If the drift demand on the olumn, δ dem, is less than δ s, as shown in Figure 16, then, without a probabilisti drift apaity model it is be left to the engineer s judgment to determine if the olumn has suffiient safety against shear failure. With the development of a probabilisti drift apaity model (shown shematially in Figure 16 as a probability density funtion, f(δ), with a mean δ s and standard deviation σ), it is possible to determine the probability of shear failure, P f, by finding the shaded area under the probability density funtion: δ dem Pf = f( δ) dδ 0 (Note that for illustration purposes, Figure 16 ignores variability in the drift demand or plasti shear apaity).

Gardoni et al. [18] have introdued a methodology to develop probabilisti apaity models. Ongoing work by Zhu et al. [19] will apply this methodology to develop probabilisti drift apaity models for shear and axial load failure. δ s σ δ s +σ V f(δ) V p Equation 6 δ dem δ s δ Figure 16. Illustration of the appliation of a probabilisti apaity model for the drift at shear failure CONCLUSIONS An idealized bakbone model to assess the apaity of existing reinfored onrete olumns with light transverse reinforement has been presented. The model approximates the true bakbone response of a olumn by apturing only the key damage states during yli response, namely, flexural yielding, shear failure, and axial load failure. Comparison of the model with shake table test data indiates that the idealized bakbone adequately represents the yli behavior and aounts for the hange in response with a hange in axial load. Appliation of the model to a olumn damaged during the Northridge Earthquake highlights the need to aount for the variability in the apaity models when evaluating the response of reinfored onrete olumns. The development of probabilisti apaity models will enable the alulation of the probability of failure, and provide the engineer with better tools to assess the seismi response of a struture. REFERENCES 1. Elwood, K.J., and Moehle, J.P. (2004) Drift Capaity of Reinfored Conrete Columns with Light Transverse Reinforement, Earthquake Spetra, (aepted for publiation). 2. Elwood, K.J., and Moehle, J.P. (2004) An Axial Capaity Model for Shear-Damaged Columns, ACI Strutural Journal, (submitted for review). 3. Elwood, K.J. (2004) Modelling failures in existing reinfored onrete olumns, Canadian Journal of Civil Engineering, (submitted for review). 4. Sezen, H. (2002) Seismi response and modeling of lightly reinfored onrete building olumns, Ph.D. Dissertation, Department of Civil and Environmental Engineering, University of California, Berkeley.

5. Sozen, M.A., Monteiro, P., Moehle, J.P., and Tang, H.T. (1992) Effets of Craking and Age on Stiffness of Reinfored Conrete Walls Resisting In-Plane Shear, Proeedings, 4th Symposium on Current Issues Related to Nulear Power Plant Strutures, Equipment and Piping, Orlando, Florida. 6. Watanabe, F., and Ihinose, T. (1992) Strength and dutility of RC members subjeted to ombined bending and shear, Conrete Shear in Earthquake, Elsevier Applied Siene, New York, pp. 429-438. 7. Ashheim, M., and Moehle, J. P. (1992) Shear strength and deformability of RC bridge olumns subjeted to inelasti displaements, UCB/EERC 92/04, University of California, Berkeley. 8. Priestley, M.J.N., Verma, R., and Xiao, Y. (1994) Seismi Shear Strength of Reinfored Conrete Columns, Journal of Strutural Engineering, 120 (8), pp. 2310-2329. 9. Pujol, S., and Ramirez, J.A., Sozen, M.A. (1999) Drift apaity of reinfored onrete olumns subjeted to yli shear reversals, Seismi Response of Conrete Bridges, SP-187, Amerian Conrete Institute, Farmington Hills, Mihigan. 255-274. 10. Pujol, S., Sozen, M.A., and Ramirez, J.A. (2000) Transverse reinforement for olumns of RC frames to resist earthquakes, Journal of Strutural Engineering, 126 (4). 461-466. 11. Pujol, S. (2002) Drift Capaity of Reinfored Conrete Columns Subjeted to Displaement Reversals, Ph.D. Thesis, Shool of Civil Engineering, Purdue University. 12. Yoshimura, M., and Yamanaka, N. (2000) Ultimate Limit State of RC Columns, Seond US- Japan Workshop on Performane-Based Earthquake Engineering Methodology for Reinfored Conrete Building Strutures, Sapporo, Japan, PEER report 2000/10. Berkeley, Calif.: Paifi Earthquake Engineering Researh Center, University of California, pp. 313-326. 13. Lynn, A.C., (2001) Seismi Evaluation of Existing Reinfored Conrete Building Columns, Ph.D. Dissertation, Department of Civil and Environmental Engineering, University of California, Berkeley. 14. Elwood, K.J. and Moehle, J.P., (2003) Shake Table Tests and Analytial Studies on the Gravity Load Collapse of Reinfored Conrete Frames, PEER report 2003/01. Berkeley, Calif.: Paifi Earthquake Engineering Researh Center, University of California. 15. Browning, J., Li, Y. R., Lynn, A. C., and Moehle, J. P. (2000) Performane Assessment of a Reinfored Conrete Frame Building, Earthquake Spetra, Vol. 16, No. 3, pp. 541-555. 16. Trifuna, M.D. and Hao, T.Y. (2001) 7-Storey Reinfored Conrete Building in Van Nuys, California: Photographs of the Damage from the 1994 Northridge Earthquake, Report No. CE 01-05, Department of Civil Engineering, University of Southern California. 17. Gardoni, P., Der Kiureghian, A., and Mosalam, K.M. (2002) Probabilisti Capaity Models and Fragility Estimates for Reinfored Conrete Columns based on Experimental Observations. Journal of Engineering Mehanis, 128(10): 1024-1038. 18. Zhu, L., Elwood, K.J., Haukaas, T., and Gardoni, P. (200x) Appliation of a probabilisti drift apaity model for shear-ritial olumns, Amerian Conrete Institute (expeted to be submitted for Speial Publiation in 2004)