Fragility analysis of mid-rise R/C frame buildings

Transkript

1 Engineering Structures 28 (2006) Fragility analysis of mid-rise R/C frame buildings Murat Serdar Kirçil,Zekeriya Polat Department of Civil Engineering, Yıldız Technical University, Istanbul, Turkey Received 26 April 2005; received in revised form 12 December 2005; accepted 9 January 2006 Available online 20 March 2006 Abstract Fragility curves are useful tools for showing the probability of structural damage due to earthquakes as a function of ground motion indices. The aim of this study is to develop the fragility curves for mid-rise R/C frame buildings in Istanbul, which have been designed according to the 1975 version of the Turkish seismic design code, based on numerical simulation with respect to the number of stories of the buildings. Sample 3, 5 and 7 story buildings were designed according to the Turkish seismic design code. Incremental dynamic analyses were performed for those sample buildings using twelve artificial ground motions to determine the yielding and collapse capacity of each sample building. Based on those capacities, fragility curves were developed in terms of elastic pseudo spectral acceleration, peak ground acceleration (PGA) and elastic spectral displacement for yielding and collapse damage levels with lognormal distribution assumption. To investigate the effect due to the number of stories of the building on fragility parameters, regression analysis has been carried out between fragility parameters and the number of stories of the building. It was observed that fragility parameters change significantly due to the number of stories of the building. Finally, using constructed fragility curves and statistical methods, the maximum allowable inter-story drift ratio and spectral displacement values that satisfy the immediate occupancy and collapse prevention performance level requirements were estimated. c 2006 Published by Elsevier Ltd Keywords: Fragility curves; Damage; Yielding; Collapse; Performance levels; R/C frames 1. Introduction The behavior of reinforced concrete structures under the effect of ground motions has always been a subject of investigation, especially in seismic regions. Meanwhile, the damage to buildings from recent earthquakes has emphasized the need for risk assessment of existing building stock to estimate the potential damage from future earthquakes. Seismic risk analysis of a building is important for identifying the seismic vulnerability of a structural system under the effect of potential seismic ground motions. For this purpose, fragility curves are useful tools, since they allow estimation of the probability of structural damage due to earthquakes as a function of ground motion indices or various design parameters, e.g, peak ground acceleration (PGA), elastic pseudo spectral acceleration (S a ),and elastic spectral displacement (S d ).This approach is useful for retrofitting decisions, damage estimation, Corresponding author. Tel.: x2679; fax: address: kircil@yildiz.edu.tr (M.S. Kirçil). loss estimation and disaster response planning. The aim of this study is to develop the fragility curves for mid-rise R/C frame buildings in Istanbul which have been designed according to the 1975 version of the Turkish seismic design code with respect to different numbers of stories of buildings and the estimation of limit values of spectral displacement and inter-story drift ratio that satisfy the immediate occupancy and collapse prevention performance levels. Representative 3, 5 and 7 story buildings were designed according to the former version (1975) of the Turkish seismic design code [1]. Although previous studies have employed different damage levels and corresponding quantities to specify those damage levels, in this study only yielding and collapse are considered, since they can be determined analytically with reasonable accuracy. Under the effect of twelve artificial ground motions, incremental dynamic analyses were performed to determine the yielding and collapse capacity of the sample buildings in terms of S a,pgaands d.those capacities are evaluated by statistical methods to develop the fragility curves. A twoparameter lognormal distribution is assumed for fragility curve construction, as was done traditionally in previous studies, /$ - see front matter c 2006 Published by Elsevier Ltd doi: /j.engstruct

2 1336 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) and a set of fragility curves was developed in terms of S a, PGA and S d.inaddition to the fragility curves, limit values of inter-story drift ratio and spectral displacement that satisfy the immediate occupancy and collapse prevention performance level requirements were estimated by using the constructed fragility curves and statistical methods. 2. Sample buildings Sample 3, 5 and 7 story R/C residential buildings were designed according to the 1975 version of the Turkish seismic design code. Although it was revised in 1998, the former version was used for the design of sample buildings, since most of buildings were constructed before The structural system of the sample buildings consists of R/C frames in two directions. Fig. 1 shows the typical ground floor plan of the sample buildings. A building that is symmetrical in plan may be chosen forsimplicity. However, this type of sample building would not be representative, since most of the buildings in Istanbul are not symmetrical. In Fig. 2, atypical upper floor plan is shown. As is seen from the figures, almost all the perimeter beams of the building are removed at the upper floors. This is a widely practiced structural system in Istanbul and the surrounding area. The typical floor area is 275.4m 2 (16.2m 17 m) and the story height is 2.90 m. Concrete slabs are at story levels with 12 cm thickness. Soil structure interaction was not considered and the base of the columns at the ground floor are assumed to be fixed. Following common practice, the characteristic compressive strength of concrete is assumed to be 16 MPa for the design of the sample buildings. This value is a realistic one, taking structural design practice in Turkey into consideration. Also, two different types of reinforcement steel, Grade 220 and 420, were considered, which have characteristic yield strengths of 220 MPa and 420 MPa, respectively. A combination of 12 sample buildings is considered by varying the number of stories and the type of reinforcement (Table 1). The uncertainty due to the scatter of material properties was not considered. Only mean values of material strength determined by experiment were taken into consideration. Two dimensional nonlinear dynamic analyses were performed separately for each direction of each sample building, since the structural configurations in the X and Y directions are different to each other. 3. Material properties For the nonlinear dynamic analysis of the sample buildings, as-built material strengths determined by experiment were taken into consideration. For concrete, a normal distribution with a mean strength of 13.6 MPa and a standard deviation of 6.6 MPa was used. This distribution is the result of an experimental study conducted by Akcay et al. [2] based on concrete samples of 511 buildings in Istanbul and the surrounding area. The modified Kent & Park [3] model was used to describe the stress strain relationship of the concrete. The confinement effect was neglected, since transverse reinforcement of the columns and beam ends of the existing Table 1 Sample buildings Sample number Fig. 1. Typical ground story plan of the sample buildings. Fig. 2. Typical upper story plan of the sample buildings. Direction Initial period (s) Reinforcement grade 1 X Y X Y X Y X Y X Y X Y Number of stories buildings does not have a significant confinementeffect. Results of another experimental study, carried out by Akyuz and Uyan [4]based on reinforcing bar samples tested in the material laboratory of Istanbul Technical University between 1979 and 1988, show that the mean yield strengths of the Grade 220 and

3 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Fig. 4(a). Ground motion 3. Fig. 3. 5% damped elastic acceleration spectra of generated ground motions. 420 reinforcements are 330 MPa and 440 MPa, respectively. Tri-linear and bilinear stress strain relationships with strain hardening were used for the reinforcement of Grade 220 and 420 respectively. 4. Ground motions The random nature of earthquakes makes the damage estimation problem probabilistic. Shome and Cornell [5] have shownthat, for mid-rise buildings, ten to twenty ground motion records are usually enough to provide sufficient accuracy in the estimation of seismic demand. Twelve artificial ground motions have been used in this study to take the random nature of earthquakes into consideration. Fig. 3 shows the acceleration spectra of the generated ground motions. For the generation of ground motions, a computer program developed in Kandilli Observatory and Earthquake Research Institute (KOERI) at Bogazici University was used that generates artificial ground motions randomly for the specified magnitude, fault distance and duration, considering the given local spectra. Earthquake risk assessment study for the Istanbul metropolitan area [6], carried out in KOERI, considered a magnitude of for earthquake scenarios of Istanbul. Thus, a magnitude of 7.5 was specified for the generation of artificial ground motions used in this study. Thespecifieddurationis 40s forall groundmotions. Fault distance was considered to be 20, 30, 40 and 50 km to obtain ground motions with different characteristics. For each fault distance, three different ground motions were generated. The local spectrum was randomly selected from the available local spectra of the Zeytinburnu region, which is one of the most densely populated regions in the south-western part of Istanbul. Figs. 4(a) and 4(b) show the third and eighth artificial ground motions. Generated ground motion characteristics that have a10% probability of being exceeded in a 50-year period are tabulated in Table Damage levels Yielding and collapse are considered as basic damage levels for this study. Similar recent studies considered different damage levels and corresponding limit values in terms of different damage measures. For instance, for building types of structure, Kircher et al. [7] andsmyth et al. [8] specified four Fig. 4(b). Ground motion 8. Table 2 Properties of generated ground motions Number of ground motion Magnitude Fault distance (km) Duration (s) Peak ground acceleration (g) different damage levels: slight; moderate; major or extensive; and complete or collapse. Maximum inter-story drift ratio was accepted as the damage measure and each damage level has an assumed limit value of inter-story drift ratio. Javanoska [9] used similar damage levels. However, the Park Ang damage index [10] was employed by them as a damage indicator. For bridges, the Park Ang damage index [10] was preferred by Karim and Yamazaki [11 13]. Furthermore, Shinozuka et al. [14,15] and Saxena et al. [16] used the section ductility demand and corresponding limit values as a damage indicator. Almost all damage levels used inprevious studies are related to the assumed limit values of the considered damage measure. Determining those limit values of the considered damage measure using an analytical method is very difficult. They are based on the results of few experiments, engineering judgment, and experience from previous earthquakes. For the present study, only yielding and collapse are considered, since they can be determined analytically with reasonable accuracy.

4 1338 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Fig. 5. IDA curve generated for 7 story sample building with ground motion 2 in the X X direction. 6. Incremental dynamic analysis Incremental dynamic analysis (IDA) is a parametric analysis method that is useful for estimating structural performance under several ground motions. The method is discussed comprehensively by Vamvatsikos and Cornell [17]. It mainly involves producing one or more curves of damage measure versus intensity measure under the effect of scaled ground motions as a result of several non-linear dynamic analyses. For this study, the maximum inter-story drift ratio is assumed as a best damage indicator and a 5% damped elastic spectral acceleration is selected as the ground motion intensity measure. Each ground motion is scaled monotonically with respect to the individual spectral acceleration based on the corresponding elastic fundamental period of each sample building. An increment of 0.05 g in spectral acceleration is selected in order to capture the yield and collapse capacity of the structure with areasonable sensitivity. The IDARC computer package [3] is used for non-linear dynamic analysis, and the maximum interstory drift ratio is recorded at the end of each run. Up to the yield point, the relationship between spectral acceleration and the maximum inter-story drift ratio is linear. The yield capacity of the structure is defined as the spectral acceleration point at which the curve leaves the linear path. When the structure reaches its collapse capacity, practically, an increase in intensity measure produces an infinite increase in damage measure. To determine the collapse capacity of the structure, ground motion is scaled up and several non-linear dynamic analyses are carried out until dynamic instability occurs as a result of a non-converging run. Unless dynamic instability occurs at an inter-story drift ratio lower than 3%, an inter-story drift ratio of 3% and the corresponding intensity measure is considered as the collapse capacity of the structure. Fig. 5 shows the IDA curve generated for the sample 7 story building, with the second generated ground motion in the X X direction. Yield and collapse points are indicated in the figure. Generally, the shape of the IDA curve is different for each ground motion. In Fig. 6, awavyidacurve is shown. As is clear from the figure, sometimes lower damage measure values may be obtained for an increasing value of intensity measure compared to that obtained at the previous step for a lower intensity measure. Using the generated ground motions, for each building, 24 IDA curves were generated: 12 for the X X direction and 12 for the other direction. All of the IDA curves obtained for the sample buildings are shown in Figs Fig. 6. IDA curve generated for 5 story sample building with ground motion 4 in the Y Y direction (a) Reinforcement grade 220. (b) Reinforcement grade 420. Fig. 7. IDA curves generated for 3 story sample building. (a) Reinforcement grade 220. (b) Reinforcement grade 420. Fig. 8. IDA curves generated for 5 story sample building.

5 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) (a) Reinforcement grade 220. Fig. 10. Lognormal probability plot for collapse probability curve of sample 7 story building with reinforcement grade 420. (b) Reinforcement grade 420. Fig. 9. IDA curves generated for 7 story sample building. The definition of the load deformation relationships of the cross-sections has particular importance in defining damage due to the effect of earthquakes. The moment curvature relationships generated by IDARC-2D were used to identify cross-sectional load deformation characteristics. In addition to a skeleton curve, the IDARC-2D needsthreeparameters to identify the hysteretic behavior. The nominal values of the parameters α, β and γ suggested by authors were used to consider the effect of stiffness degradation, strength degradation and pinching. 7. Fragility curves Fragility curves express the probability of structural damage due to earthquakes as a function of ground motion indices. In the present study, fragility curves are constructed in terms of S a, S d and PGA. It is assumed that the fragility curves can be expressed in the form of two-parameter lognormal distribution functions. Based on this assumption, the cumulative probability of the occurrence of damage, equal to or higher than damage level D, isexpressed as ( ) ln X λ P( D) = Φ (1) ζ where Φ is the standard normal distribution, X is thelognormal distributed ground motion index (e.g., S a, S d,pga), and λ and ζ are the mean and standard deviation of ln X. The mean and standard deviation of ground motion indices for each damage level are obtained, as shown in Fig. 10, whichisa lognormal plot of ln X and the corresponding standard normal variable. This method is based on plotting ln X versus the corresponding standard normal variable on a lognormal scale and performing a linear regression analysis to determine the mean and standard deviation of ln X for each damage level [18]. The relationship between the standard normal variable and Fig. 11. Fragility curve of sample 5 story building for reinforcement grade 220 and 420. the mean and standard deviation of ln X can be expressed as follows: s = ln X λ (2) ζ where s is the standard normal variable. Fig. 10 shows the typical lognormal probability plot for the collapse of a sample 7 story building. Table 3 shows the mean and standard deviation of lognormal distributed ground motion indices for each sample building and damage level under consideration. Fragility curves of a sample 5 story building in terms of spectral displacement are shown in Fig. 11. Notethat two different fragility curves were constructed for each building, since two different reinforcement types have been considered. 8. Combined fragility curves Since two different types of reinforcement steel were considered for each sample building, two different reinforcementtype-dependent fragility curves were obtained for each sample building in terms of each ground motion index. The method proposed by Shinozuka et al. [15] wasused for combining reinforcement-type-dependent fragility curves to obtain a unified fragility curve for a mixed set of populations of buildings, in which there are N 1 and N 2 buildings with Grade 220 and Grade 420, respectively. N 1 and N 2 define the ratio of buildings with Grade 220 and buildings with Grade 420 in the population of building stock, respectively. It is highly recommended that one should see Ref. [15]for details of this method.

6 1340 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Table 3 Reinforcement-type-dependent fragility curve parameters Story number Reinforcement grade Indices Parameters Yielding Collapse λ ζ λ ζ S a PGA S d S a PGA S d S a PGA S d S a PGA S d S a PGA S d S a PGA S d Fig. 12. Combined fragility curve of 5 story sample building. The combined fragility curve can be expressed as follows: F c (X) = P 1 F 1 (X) + P 2 F 2 (X) (3) where F c (X) is the combined fragility curve, and F 1 (X) and F 2 (X) are fragility curves of building with Grades 220 and 420, respectively. P 1 and P 2 can be expressed as follows: N 1 P 1 = (4) N 1 + N 2 N 2 P 2 = (5) N 1 + N 2 where N 1 and N 2 are the populations of buildings constructed with reinforcement Grades 220 and 420, respectively. Finally, based on (2), themean and standard deviation of the combined fragility curves can be expressed as follows: λ c = P 1 λ 1 + P 2 λ 2 (6) ζc 2 = P 1ζ1 2 + P 2ζ2 2 + P 1(1 P 1 )λ 2 1 +P 2 (1 P 2 )λ 2 2 2P 1 P 2 λ 1 λ 2 (7) Fig. 13(a). Fragility curves for yielding with respect to spectral acceleration. where λ 1 and λ 2 are the means of buildings with Grades 220 and 420, and ζ 1 and ζ 2 are the standard deviations of buildings with Grades 220 and 420, respectively. Fig. 12 shows the fragility curves presented in Fig. 11 with the combined fragility curve. Since there is not enough information on the populations of buildings constructed with either Grade 220 or Grade 420, in the present study it is assumed that P 1 = P 2 = 0.5. Note that, for the combined fragility curve, the lognormal distribution is no longer valid. However, it is still reasonable to assume that the combined curve is lognormal with the mean and standard deviation estimated by (6) and (7), respectively [15]. All the combined fragility curves for yielding and collapse are shown in Figs. 13 and 14 in terms of elastic pseudo spectral acceleration, peak ground acceleration, and elastic spectral displacement, respectively. 9. Extended fragility curves It is observed from the fragility curves given so far that the fragility curve parameters, mean and standard deviation, change with respect to the number of stories of the buildings. Thus, to extend the fragility curves constructed for 3, 5 and 7 story

7 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Fig. 13(b). Fragility acceleration. curves for yielding with respect to peak ground Fig. 14(c). Fragility curves for collapse with respect to spectral displacement. (a) Regression analysis for mean. Fig. 13(c). Fragility curves for yielding with respect to spectral displacement. (b) Regression analysis for standard deviation. Fig. 15. Regression analysis results. Fig. 14(a). Fragility curves for collapse with respect to spectral acceleration. Fig. 14(b). Fragility curves for collapse with respect to peak ground acceleration. buildings to the fragility curves for 4 and 6 story buildings, linear regression analyses have been performed. The regression models that are used to obtain the relationship between the fragility curve parameters and the number of stories are given below: λ = an + b (8) ζ = cn 2 + dn + e (9) where n is the number of stories of the building, and a, b, c,and d are coefficients obtained from regression analyses. Fig. 15 shows the regression analysis results of λ and ζ for collapse with respect to S a. Table 4 shows the regression coefficients obtained for yield and collapse capacities in terms of S a, S d and PGA. It is worth noting that R 2 values are very high, since the number of data is the minimum number of data required for a regression analysis. Using the regression analyses results, extended fragility curves are shown in Figs. 16 and Performance limits Recently developed performance based design procedures consider the inelastic displacement demand and corresponding deformations as the main performance indicator. Recently

8 1342 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Table 4 List of the regression coefficients Indices Yielding Collapse λ = an + b ζ = cn 2 + dn + e λ = an + b ζ = cn 2 + dn + e a b R 2 c d e R 2 a b R 2 c d e R 2 S a PGA S d Fig. 16(a). Fragility curves for yielding with respect to spectral acceleration. Fig. 17(a). Fragility curves for collapse with respect to spectral acceleration. Fig. 16(b). Fragility curves for yielding with respect to peak ground acceleration. Fig. 17(b). Fragility curves for collapse with respect to peak ground acceleration. Fig. 16(c). Fragility curves for yielding with respect to spectral displacement. published documents ATC 40 [19] andfema 356 [20] recommend several methods to estimate the inelastic displacement of the structures under the effect of earthquakes. Also, some performance levels and corresponding performance criteria are specified by the same documents based on the local inelastic deformation demand and capacity of structural members. Performance based design procedures are more realistic compared to the widely used linear design philosophy. However, these procedures are also time-consuming. Thus, global Fig. 17(c). Fragility curves for collapse with respect to spectral displacement. structural performance criteria in terms of the maximum interstory drift ratio or spectral displacement, which can be used for fast but approximate evaluation of structures, may be useful. In the present study, two main performance levels, immediate occupancy and collapse prevention, are associated with the yielding and collapse probability curves considering the definitions of performance levels given by ATC 40 [19] andfema 356 [20] toestimate the global performance criteria.

9 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Table 5 Spectral displacement limits of performance levels Story number S d (cm) Immediate occupancy Collapse prevention Fig. 18(a). Inter-story drift ratio limit for immediate occupancy performance level. Fig. 18(b). Inter-story drift ratio limit for collapse prevention performance level. The immediate occupancy performance level is the performance level at which the structure has almost pre-earthquake stiffness. Considering this expression, it is assumed that the maximum allowable inter-story drift ratio for this performance level can be defined as the inter-story drift ratio at which there is a high confidence of a low probability of yielding on the yield probability curve. Figs. 18(a) and 18(b) show three different fragility curves constructed for yielding and collapse, respectively. In Fig. 18(a), each curve provides the probability of yielding at varying levels of inter-story drift ratio for a given level of confidence. The fragility curve which has a confidence level of 50% is a typical fragility curve constructed by the methodology used in the present study. It is worth noting that each of those fragility curves was constructed using 144 maximum inter-story drift ratios, regardless of the number of stories of the sample buildings. As mentioned earlier, the immediate occupancy performance level limit can be defined in terms of the maximum inter-story drift ratio at which there is a high confidence of a low probability of yielding on the yield probability curve in Fig. 18(a). For this study, a confidence level of 95% and collapse probability of 5% are selected and a drift ratio of was specified as the maximum allowable inter-story drift ratio for the immediate occupancy performance level. This point is indicated in Fig. 18(a). Thisshows the inter-story drift ratio that causes yielding of 5% of the sample buildings. Based on the same assumptions, a drift ratio of was specified as the maximum allowable inter-story drift ratio for a collapse prevention performance level using the collapse probability curve. This point is indicated in Fig. 18(b) and shows the inter-story drift ratio that causes collapse of 5% of the sample buildings with a confidence level of 95%. Since spectral displacement is a very common response parameter that shows the demand of ground motions on the structures, limit values of spectral displacement were estimated for both immediate occupancy and collapse prevention performance levels. The procedure followed for estimating limit spectral displacement values is summarized below. Following the same procedure used for the estimation of the limit inter-story drift ratio, limit spectral displacement values are estimated for the immediate occupancy performance level at which there is a high confidence of a low probability of yielding on the yield probability curve using each combined yielding probability curve constructed for each number of stories. Fig. 19(a) shows the estimated spectral displacement and combined yielding probability curve for a sample 7 story building. All the limit spectral displacement values estimated for the immediate occupancy performance level are tabulated in Table 5 with respect to the number of stories of the sample buildings. Limit spectral displacement values for the collapse prevention performance level are estimated following the same procedure, but this time using the combined collapse probability curve constructed for each number of stories. Fig. 19(b) shows the estimated spectral displacement and combined collapse probability curve for a sample 7 story building. All the limit spectral displacement values estimated for the collapse prevention performance level are tabulated in Table 5 with respect to the number of stories of the sample buildings. Regression analyses are carried out between the limit spectral displacement values and the story number of the sample buildings to obtain simple equations that produce the story-number-dependent maximum allowable spectral displacement for immediate occupancy and the collapse prevention performance levels, respectively. For the regression analysis, linear equations are assumed to be sufficient for representing the relationship between the number of stories and the maximum allowable spectral displacement. Fig. 20 shows the regression analysis results and confidence intervals for a confidence level of 90%. The following equations were obtained, which produce the limit spectral displacement in cm for immediate occupancy and

10 1344 M.S. Kirçil, Z. Polat / Engineering Structures 28 (2006) Fig. 19(a). Spectral displacement limit of 7 story sample building for immediate occupancy performance level. Fig. 20(a). Relationship between spectral displacement limit of immediate occupancy performance level and story number of building. Fig. 20(b). Relationship between spectral displacement limit of collapse prevention performance level and story number of building. Fig. 19(b). Spectral displacement limit of 7 story sample building for collapse prevention performance level. collapse prevention performance levels, respectively: S d = 0.07n (cm) (10) S d = 1.04n (cm) (11) where S d is the maximum allowable spectral displacement of the corresponding performance level and n is the number of stories of the building. Note that the correlation between n and S d for the immediate occupancy performance level is not as high as the correlation for the collapse prevention performance level. All the regression coefficients are given in Tables 6 and 7 for each confidence levels. 11. Conclusions The uncertain nature of future ground motions is leading to the development of probabilistic structural damage estimation procedures. The fragility curve approach is a useful method for estimating the structural damage for certain types of structure under the effect of potential earthquakes. In this study, a method for obtaining the fragility curves for mid-rise R/C frame buildings is defined and the fragility curves of a mid-rise R/C frame building, typical of the inventory of existing buildings in Istanbul, were constructed in terms of S a,pgaands d under the effect of twelve artificial ground motions with respect to different numbers of stories. The considered damage levels are yielding and collapse, since they can be determined analytically with reasonable accuracy. It is observed from the fragility curves that there is an effect on fragility curve parameters due to the number of stories in the buildings. Regression analysis has been carried out to determine the relationship between the fragility curve parameters and the number of stories, and extended fragility curves were constructed with the help of the results of regression analysis. Furthermore, the maximum allowable inter-story drift ratio and spectral displacement values that satisfy the immediate occupancy and collapse prevention performance level requirements are estimated with respect to the number of stories of the buildings using constructed fragility curves and statistical methods. Simple equations were obtained that show the relationship between the limit values of spectral displacement and the number of stories of the building for the immediate occupancy and collapse prevention performance levels, respectively. Those simple equations may be used for the preliminary evaluation of midrise R/C frame structures that have been designed according to the 1975 version of the Turkish seismic design code. For the collapse prevention performance level, a good correlation between spectral displacement limit and the number of stories is observed. However, the same observation is not valid for the immediate occupancy level. Thus, for this performance level, the maximum allowable spectral displacement may be calculated using the curve of a confidence level of 90% as a lower bound. Acknowledgements The authors are grateful to Professor M.N. Aydinoglu, who provided useful comments that helped to improve this work. The authors would also like to express their gratitude to

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