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Engineering Structures 26 (2004) 937­948

Fragility analysis of flat-slab structures

M. Altug Erberik, Amr S. Elnashai Ã

Newmark Civil Engineering Laboratory, Mid-America Earthquake Center, Civil and Environmental Engineering Department, University of Illinois at Urbana­Champaign, 205 Mathews Avenue, Urbana, IL 61801-2352, USA Received 17 June 2003; received in revised form 20 February 2004; accepted 23 February 2004

Abstract Flat-slab RC buildings exhibit several advantages over conventional moment-resisting frames. However, the structural effectiveness of flat-slab construction is hindered by its alleged inferior performance under earthquake loading. Although flat-slab systems are widely used in earthquake prone regions of the world, fragility curves for this type of construction are not available in the literature. This study focuses on the derivation of such fragility curves using medium-rise flat-slab buildings with masonry infill walls. The study employed a set of earthquake records compatible with the design spectrum selected to represent the variability in ground motion. Inelastic response-history analysis was used to analyze the random sample of structures subjected to the suite of records scaled in terms of displacement spectral ordinates, whilst monitoring four performance limit states. The fragility curves developed from this study were compared with the fragility curves derived for moment-resisting RC frames. The study concluded that earthquake losses for flat-slab structures are in the same range as for moment-resisting frames. Differences, however, exist. The study also showed that the differences were justifiable in terms of structural response characteristics of the two structural forms. # 2004 Elsevier Ltd. All rights reserved.

Keywords: Flat slabs; Fragility relationships; Seismic response; Reinforced concrete buildings

1. Introduction The significant social and economic impacts of recent earthquakes affecting urban areas have resulted in an increased awareness of the potential seismic hazard and the corresponding vulnerability of the existing building stock required for estimating seismic risk. Greater effort has been made to estimate and mitigate the risks associated with these potential losses. In order to successfully mitigate potential losses and to aid in post-disaster decision-making processes, the expected damage and the associated loss in urban areas caused by earthquakes should be estimated with an acceptable degree of certainty. Seismic loss assessment depends on the comprehensive nature of estimating vulnerability. The determination of vulnerability measure requires the assessment of the seismic performances of all types of building structures typically

à Corresponding author. Tel.: +1-217-333-8038; fax: +1-217-3339464. E-mail address: [email protected] (A.S. Elnashai).

constructed in an urban region when subjected to a series of earthquakes, taking into account the particular response characteristics of each structural type. The fragility study generally focuses on the generic types of construction because of the enormity of the problem. Hence, simplified structural models with random properties to account for the uncertainties in the structural parameters are used for all representative building types. The flat-slab system, as shown in Fig. 1, is a special structural form of reinforced concrete construction that possesses major advantages over the conventional moment-resisting frames. The former system provides architectural flexibility, unobstructed space, lower building height, easier formwork and shorter construction time. There are however, some serious issues that require examination with the flat-slab construction system. One of the issues which were observed is the potentially large transverse displacements because of the absence of deep beams and/or shear walls, resulting in low transverse stiffness. This causes excessive deformations which in turn cause damage of non-

0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.02.012


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Fig. 1.

Illustration of a typical flat-slab structural form. Fig. 2. The methodology used in the derivation of fragility curves.

structural members even when subjected to earthquakes of moderate intensity. Another issue is the brittle punching failure due to the transfer of shear forces and unbalanced moments between slabs and columns. When subjected to earthquake action, the unbalanced moments can produce high shear stresses in the slab. Flat-slab systems are also susceptible to significant reduction in stiffness resulting from the cracking that occurs from construction loads, service gravity loads, temperature and shrinkage effects and lateral loads. Therefore, it was recommended that in regions with high seismic hazard, flat-slab construction should only be used as the vertical load carrying system in structures braced by frames or shear walls responsible for the lateral capacity of the structure [1]. In such cases, slab­column connections must undergo the lateral deformations of the primary lateral load-resisting structural elements without punching failure in order to sustain the gravity loads. In spite of the above concerns, flat-slab systems are often adopted as the primary lateral load-resisting system and their use has proven popular in seismically active regions, such as in the Mediterranean basin. In these cases, the design of flat-slab buildings is typically carried out in a manner similar to ordinary frames. Where the latter practice is followed, the response under moderate earthquakes indicates extensive damage to non-structural elements even when the code provisions for drift limitation are satisfied [2]. This observation emphasizes the necessity of investigating the vulnerability of flat-slab construction, for which no fragility curves are available in the literature, since the structure exhibits distinct response modes, as compared to conventional moment-resisting frames. 2. Methodology In the construction of the fragility functions, there is no definitive method or strategy [3]. A great degree of

uncertainty is involved in each step of the procedure. This uncertainty is due to variability in ground motion characteristics, analytical modeling, materials used and definition of the limit states. The current study employs accepted procedures whilst attempting to ensure that rational decisions are taken along the route to deriving vulnerability curves for a structural system that has not been dealt with before. The approach used is outlined in Fig. 2. A detailed account is given hereafter of the various steps depicted in Fig. 2.

3. Structural configuration and design A five story flat-slab structure is used as the generic system for this study. Preliminary analysis of three, five and seven story versions had indicated rather insignificant differences in the inelastic dynamic analysis results. This building is considered mid-rise. The reason for choosing a mid-rise building is twofold. Because of the inherent flexibility of flat-slab buildings, it may not be possible to satisfy the drift demands in high-rise construction. On the other hand, low-rise buildings would be sufficiently stiff and may not warrant special consideration. The selected dimensions of the building

Fig. 3. Five-story flat-slab building, (a) elevation, (b) plan.

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are shown in Fig. 3. For simplicity, the building is symmetric in plan with three bays in the horizontal directions. This symmetry enables the use of twodimensional (2-D) models in both design and analysis. The span length represents typical values for this type of construction in earthquake-vulnerable regions, especially in Mediterranean countries [4]. The story height selected is 2.8 m. The building was designed according to the regulations of ACI 318-99 [5] for both gravity and seismic loads. Following common practices, the materials used are 4000 psi (28 MPa) concrete and Grade 60 (414 MPa) reinforcing bars. The gravity load scenario consists of dead load and live load. When calculating the dead load, the weight of the structural members and the masonry infill walls was included. The live load used was 2.5 kN/m2, which is typical for an office building. Other types of loading, such as wind and snow, were not considered. The seismic design is carried out according to FEMA 368, NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures [6]. The flat-slab building is assumed to be located in Urbana, IL. The corresponding spectral response parameters were obtained from the USGS web site (http:// using the ZIP code 61801. The direct design approach was used to determine the slab reinforcement. Since the most significant problem of flat-slab system is the punching shear failure, precautions should be taken in the design stage to prevent this undesirable behavior. The depth of the slab was selected according to the requirements in the code to prevent this type of failure. For the building under consideration, the slab depth was 22 cm which satisfied the punching shear check of ACI 318-99. The slab­ beams reinforcement was detailed to prevent slab failure caused by the combination of forces, including shear, torsion and moment transferred from the column. The bottom reinforcement of the slab was continuous with a reasonable amount passing through the columns. This prevented the progressive vertical collapse of slabs in the event of a local punching failure. The column dimensions used were 40 cm  40 cm throughout the height of the building. Longitudinal and lateral reinforcement were determined according to the ACI regulations. The ACI recommends that the column should have adequate capacity to withstand excessive drift demand to which the building is exposed to under seismic action. Design calculations support the use of 8124 (eight reinforcing bars with a diameter of 24 mm) bars as the longitudinal reinforcement which yields a reinforcement ratio, qt ¼ 0:022. The lateral reinforcement selected is 114 (reinforcing bar with a diameter of 14 mm) bars with a spacing of 10 cm.

4. Analytical model The regularity of the building in terms of mass and stiffness in both plan and elevation enables a 2-D analysis to be used when assessing seismic response. In this study, the building is modeled as a 2-D planar frame with lumped masses. ZEUS-NL [7] is the software program used for the inelastic analysis of the flat-slab structure. The program is a development of previous analytical platforms developed at Imperial College, namely ADAPTIC and INDYAS. The software program was used to perform a static inelastic (pushover) and a dynamic time history analysis. An eigenvalue analysis was performed with the software to yield the periods of vibration of the structure. In order to model the slabs, the portion that will contribute to the frame analysis should be determined as well as the width of the concealed beam within the slab. Two simplified methods, the effective beam width and the equivalent frame method, exist, in which the effect of the slab is accommodated by appropriate modification of the beam width or the column stiffness, respectively. In this study, the effective beam width method is used. For the flat-slab structure being studied, the portion of the slab that will contribute to the frame analysis is determined to be 2.85 m, by using the formulations proposed by Luo and Durrani [8]. To determine the width of the slab-embedded beams, several sources that can also be considered are the Mexico Building Code [9], the Greek Code for RC Structures [10] and ACI 318-99. In this study, the width of the slab beam considered is 100 cm for the first three stories and 90 cm for the top two stories. A typical slab­beam section from the first story of the building is illustrated in Fig. 4. One of the main concerns in flat-slab construction is the control of excessive lateral drift. This concern was addressed by placing masonry infill walls, which have high in-plane stiffness. At low levels of lateral force, frame and infill wall act in a fully composite fashion. However, as the lateral force level increases, the frame attempts to deform in a flexural mode while the infill attempts to deform in a shear mode. As a result, the

Fig. 4. Typical slab­beam section of the flat-slab building (symbol 5112 denotes five reinforcing bars with a diameter of 12 mm and 110/15 denotes tied reinforcement with a diameter of 10 mm and having a spacing of 15 cm).


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frame and the infill separate at the corners on the tension diagonal and a diagonal compression strut on the compression diagonal develops. After separation, the effective width of the diagonal strut is less than that of the full panel [11]. Eigenvalue, pushover and inelastic time history analyses should be based on the structural stiffness after separation. This can be achieved by modeling the infilled frame as an equivalent diagonally braced frame, where masonry infill walls are represented by diagonal compression struts. The stiffness properties of the infill are obtained by using Eqs. (1)­(3), formulated by Stafford-Smith [12] and Mainstone [13]. a p ¼ ð1Þ h 2kh sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 Em tsin2h k¼ ð2Þ 4Es Ic h0 w ¼ 0:175ðkhÞÀ0:4 d ð3Þ

Fig. 5.

Masonry infill frame sub-assemblages.

In the above formulations, w is the equivalent strut width, a is the length of contact, kh is a non-dimensional parameter that represents the relative stiffness of the frame with respect to the infill, Em, t, h0 and d are Young's modulus, thickness, height and length of infill, respectively. Es, Ic and h are Young's modulus, moment of inertia and height of the column, and h is the slope of the infill diagonal to the horizontal. Some of these parameters are illustrated in Fig. 5. The equivalent strut width is calculated to be 73 cm, approximately 11% of the length of the infill diagonal. To obtain the strength of a masonry infill wall, it is necessary to define the most probable failure mode giving the lowest strength from Eqs. (4) and (5) below: Rc ¼ 2 p 0 tf sech 3 2k c fbs dt À lf ðh=l 0 Þ ð4Þ

culation of the confinement factor, the simple relationship proposed by Park et al. [16] is used, and is given below: K ¼1þ qw fyw fc ð6Þ

Rbs ¼



where Rc, and Rbs are the diagonal loads that cause compression and shear failure of the infill, and fc0 and 0 fbs are the compressive and bond shear strength of the infill wall, respectively. The parameter l0 denotes the distance between the centre lines of columns and lf is the coefficient of friction. For the building in the case study, using Eqs. (4) and (5), the strength of the masonry infill is calculated to be 3 MPa. The elastic modulus of masonry is 8250 MPa, in accordance with Paulay and Priestley [11] and FEMA 307 [14]. The concrete is modeled by using the inelastic concrete model with constant (active) confinement in ZEUS-NL. There are four parameters of the model: compressive strength, fc; tensile strength, ft; crushing strain, Eco and confinement factor. This model is based on the formulation by Mander et al. [15]. For the cal-

In the above equation, qw is the volumetric ratio of the transverse reinforcement, fyw is the nominal hoop strength and fc is the unconfined uniaxial concrete strength. For the building used in the case study, the confinement factor values range between 1.18 and 1.37 depending on the amount of lateral reinforcement. Mean values are used for the compressive and tensile strength of concrete. These values are 35.6 and 2.75 MPa, respectively. Unconfined concrete strain at peak stress is equal to 0.002. Steel is modeled with a bilinear elasto-plastic model with kinematic strain hardening. Three parameters are required for the model, namely elastic modulus of steel, Es; yield strength of steel, fy and the strain-hardening parameter ls. In this study, the elastic modulus is 200 kN/mm2. Mean yield strength of steel and the strainhardening parameter is 475 MPa and 0.01, respectively. The finite element mesh, shown in Fig. 6, used is carefully designed to locate integration points at potential critical zones of the structure, with cubic inelastic elements representing reinforced concrete behavior using a fiber analysis approach. Compressions struts are modeled using cubic elasto-plastic elements with a bilinear material model for the infill properties, which were discussed in the previous paragraphs. A 3-D joint element with uncoupled axial, shear and moment actions to simulate the pin joint behavior at the ends of each strut was used. For dynamic analysis, masses in each floor are lumped at the beam column joints. The mesh configuration and mass distribution of the flatslab model is shown in Fig. 6. The small squares indi-

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Fig. 7.

Comparison of elastic spectra with the code spectrum.

Fig. 6. Mesh configuration of the flat-slab model in ZeusNL.

spectrum and the elastic spectra of the selected ground motion records is given in Fig. 7. The characteristics of the selected ground motion records are listed in Table 1. 6. Evaluation of seismic response characteristics Before conducting inelastic dynamic analyses to evaluate the seismic vulnerability of the flat-slab structure, it is necessary to assess the structural response characteristics through eigenvalue and the inelastic static (pushover) analyses. The former provides the dynamic response characteristics whilst the latter provides the necessary `supply or capacity' quantities needed for limit state definition. Performing the eigenvalue analysis, the results of the first three natural periods of the structure are T ¼ 0:38, 0.13 and 0.08 s, respectively. The natural vibration periods seemed reasonable for mid-rise concrete frames with infill panels. The eigenvalue analysis was also conducted on the case in which there were no infill walls, (i.e. bare frame) for comparison. The first three natural periods of vibration of this structure are 0.98, 0.30 and 0.15 s, respectively. The results indicated that the flatslab system was more flexible than moment-resisting

cate the node locations and the big squares symbolize the lumped masses on the floors. 5. Selection of ground motion records Since the current study focuses on the effects of the ground motion variability on the building response, there should be a compromise between the number of ground motions selected and the robustness of the analysis. Bazzuro and Cornell [17] suggested that fiveto-seven input motions are sufficient for representing the hazard in an uncoupled (uncertainty in supply and demand dealt with separately) analysis. Dymiotis et al. [18] state that three ground motions are sufficient if appropriate choices of records and scaling are made. Taking the latter studies into consideration, 10 ground motions with a single criterion; the compatibility of the elastic spectra of these ground motions with the code spectrum used in the seismic design of the building, were selected. The comparison between the design code

Table 1 Characteristics of the selected ground motions Location GM1 GM2 GM3 GM4 GM5 GM6 GM7 GM8 GM9 GM10 Buia Boshroych Cassino Sant'Elia Gukasian Haywaid-MuirSchool Tonekabun L.A.--l5 story government office building El Segundo--l4 story office building Castelnuovo-Assisi Yesilkoy Airport Component Earthquake NS N79E EW NS 90 EW 270 90 EW NS Friuli aftershock Tabas Lazio Abruzzo Spitak Loma Pricta Manjil Northridge Northridge Umbro-Marchigiano Marmara Country Italy Iran Italy Armenia USA Iran USA USA Italy Turkey Date 9/15/1976 9/16/1978 5/7/1984 12/7/1988 10/17/1989 6/20/1990 1/17/1994 1/17/1994 9/26/1997 8/17/11999 Ma 6.1 7.3 5.8 5.8 7.1 7.3 6.7 6.7 5.5 7.8 PGA (m/s2) 1.069 1.004 1.116 1.446 1.36 0.871 1.367 1.281 1.083 0.8771 PGV (m/s) 0.108 0.111 0.079 0.108 0.13 0.091 0.129 0.115 0.109 0.113 Sa,max (g) 0.327 0339 0.395 0.395 0.454 0.302 0.362 0.362 0.405 0.366


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Fig. 8. Pushover analysis results, (a) capacity curve, (b) plastic hinge formation.

frames, and the addition of the infill walls produced a significant increase in the stiffness of the system. It is therefore expected that the response of the building will undergo abrupt change when the deformations exceed the deformational capacity of the infill panels. The static pushover analysis was conducted using the ZEUS-NL. An inverted triangular distribution was used for the lateral loading. Force-controlled analysis was employed up to the point of maximum force resistance to identify the structural deficiencies of the frame, such as soft stories. For post-peak load analysis, the program automatically switches to displacement control, in order that the full ductility capacity of the structure is investigated. The pushover curve of the infilled flat-slab model is shown in Fig. 8(a). The frame was capable of sustaining a lateral load of 1056 kN (0.25% of the weight of the frame). The plastic hinge propagation is shown in Fig. 8(b). The plastic hinges were concentrated at the first three stories, consistent with the large drifts experienced by the same stories. The first plastic hinges were formed at the beam ends of the first two stories, followed by the first story columns.

Fig. 9. Mapping from local limit states to global limit states, (a) story shear versus story drift curve, (b) yield limit state, (c) ultimate limit state.

7. Determination of limit states Definition of limit states plays a significant role in the construction of the fragility curves. Well defined and realistic limit states are of paramount importance since these values have a direct effect on the fragility curve parameters. This is especially true for special systems like flat-slab structures for which the identification of limit states is highly dependent on the characteristics of the structure. It may be misleading to use the performance levels determined for regular concrete frames in the case of the flat-slab buildings without due regard to the inherent flexibility of these structures. The limit states used in this study are defined in terms of interstory drift ratio since the behavior and the failure modes of such structures are governed by deformation. To determine performance levels, the

local limit states of members in an individual story are obtained and then mapped onto the shear force versus drift curve of that story. Local limit states are considered in terms of yield and ultimate curvatures. Then these performance points are used to obtain the limit states of the story in terms of interstory drift. This process is repeated for each story. The performance levels of the most critical story are defined as the global limit states of the structure. The global limit state is illustrated in Fig. 9(a) for the first story of the analysis model used. In Fig. 9, hollow rectangular marks represent the failure of the diagonal struts used to simulate the infill panels, solid circular marks (in gray) denote the local yield criterion and hollow triangular marks represent local ultimate criterion. The yield and ultimate limit state occurrences in the structural members of the first story are illustrated in Fig. 9(b) and (c). When comparing the story shear versus drift curve, it is observed that the infill panels fail sequentially at a low drift level of 3.5 mm. After the failure of the infill panels, the stiffness is significantly reduced. At a drift level of 25­30 mm, the yield limit state is reached at the left end of three beams (Y1) followed by the bottom end of three first story columns (Y2). Two more yield limit states (Y3, Y4) occur at a drift level of approximately 60 mm, in addition to exceeding the ultimate state in one of the beams (U1). At a drift level of 100 mm, the ultimate limit state is exceeded in three columns (U2). Considering this limit state scenario and verifying that the most critical story drifts take place in the first story, the limit states assigned to the frame in terms of interstory drift are shown in Table 2.

M.A. Erberik, A.S. Elnashai / Engineering Structures 26 (2004) 937­948 Table 2 Limit states and corresponding interstory drift ratios Limit state Slight Moderate Extensive Complete Interstory drift (mm) 3.5 28.4 56.1 96.9 Interstory drift ratio (%) 0.1 1.0 2.0 3.5


et al. [18] and Ghobarah et al. [21], COV of the concrete compressive strength is 15%, which represents average quality control. Hence, a normal distribution is used to represent the variability of concrete strength in this study. For a characteristic concrete strength value of 28 MPa, the mean value is calculated as 36 MPa and COV is taken as 15%. 9. Treatment of material uncertainty--sampling Several sampling methods have been proposed in the technical literature to generate values to use as the random variables. Among these sampling methods, the Monte Carlo simulation is the most widely used. Although this method is a powerful tool, there are disadvantages of using very large samples to achieve the required accuracy. The large number of sampling points required in the standard Monte Carlo approach has huge implications in fragility analysis due to the use of inelastic dynamic analysis for each of the sampled strength values under each of the record at each of the scaling levels. Therefore, alternative approaches have been developed to reduce the sample size. One such method developed by McKay et al. [24] is the Latin hypercube sampling (LHS) method. This technique provides a constrained sampling scheme instead of the random sampling used in the Monte Carlo method. In this study, the variability of the yield strength of steel reinforcement in beams and columns are treated separately. Hence, three sets of input values are generated to represent the variability in the compressive strength of concrete, the yield strength of steel reinforcement in beams and the yield strength of steel reinforcement in columns. The variables are indicated as fc, fy,b and fy,c, respectively. Thirty sets of input data are generated to use in the simulation of the dynamic response of the flat-slab structure. To achieve this, the range of each random variable is divided into 30 nonoverlapping intervals on the basis of equal probability. One value from each interval is selected randomly with respect to the probability density in the interval. Thus, the 30 values obtained for fc are paired randomly with the 30 values of fy,b. These 30 pairs are further combined with the 30 values of fy,c to form 30 sets of input data for the response simulation analyses. Using the LHS Method to develop detailed fragility curves of flatslab structures is described in Erberik and Elnashai [25]. 10. Seismic response analysis

8. Material uncertainty One of the main sources that control the response uncertainty of a reinforced concrete structure is the inherent variability of material strength. The mean and standard deviation are used to describe the statistical variation of the material properties. Normal or lognormal distributions are commonly used, for convenience. In this study, the yield strength of steel and the compressive strength of concrete have been chosen as the random variables following a survey of the literature (e.g. [18]) and pilot inelastic analysis using extreme values of material properties. 8.1. Yield strength of steel (fy) Variations in the strength of steel have been studied by researchers in the past [19,20]. In recent studies, Ghobarah et al. [21] and Elnashai et al. [22] used statistical distributions to define the material uncertainty for the yield strength of steel. There is a consensus in terms of using lognormal and normal distributions to represent the material variability in the yield strength of steel with coefficient of variation (COV) ranging between 4% and 12%. Taking the above studies into account, a lognormal distribution is assumed for the yield strength of steel in this study. The mean and COV are 475 MPa and 6 %, respectively. Hence, the lognormal mean and standard deviation parameters take the values of 6.161 and 0.06, respectively by using the following conversion formulae r2 2 n ¼ ln 1 þ 2 ð7Þ l 1 k ¼ lnl À n2 2 ð8Þ

where l and r are the normal distribution mean and the standard deviation parameters, whereas k and n are the lognormal distribution mean and standard deviation parameters. 8.2. Concrete strength (fc) The general agreement in the literature [19,23] is to employ normal distribution to characterize the variability of concrete strength. Based on the aforementioned studies and the studies conducted by Dymiotis

Inelastic response-history analysis is used to evaluate the seismic response and to derive the fragility curves. This approach is the most tedious but it is also the most direct and accurate way to assess the vulnerability of structures. The statistical sample of frames with


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11. Development of the vulnerability functions Response statistics are assessed in terms of interstory drift. The damage versus hazard relationship of the flat-slab structure is illustrated in Fig. 11. The damage axis (y-axis) described as the interstory drift is given in millimeters whereas the hazard axis (x-axis) is described as spectral displacement and is also given in millimeters. Each vertical line of scattered data corresponds to an intensity level. The horizontal lines in the figure represent the limit states used in this study and described in terms of interstory drift. From bottom to top, these are the limits for slight, moderate, extensive and complete damage, respectively. A statistical distribution is fitted to the data for each intensity level on each vertical line. The lognormal parameters, the mean (k) and the standard deviation (n) are calculated for each of these Sd intensity levels. At each intensity level, the probability of exceeding each limit state is calculated. Fig. 12 illustrates the statistical distribution for two different intensity levels (when Sd ¼ 30 and 60 mm, respectively). LS1, LS2, LS3 and LS4 represent the limit states for slight, moderate, extensive and complete damage, respectively, as mentioned above. The mean and standard deviation values of the response data are also given in the figure. The probability of exceedance of a certain limit state is obtained by calculating the area of the lognormal distribution over the horizontal line of that limit state. Hence, the following values are obtained for the two depicted intensity levels: For Sd ¼ 30 mm, PðSd > LS1Þ ¼ 0:999 PðSd > LS2Þ ¼ 0:404 PðSd > LS3Þ ¼ 0:083 PðSd > LS4Þ ¼ 0:011

Fig. 10. Displacement spectra (n ¼ 5%) of the selected ground motions.

random properties described above was subject each in turn to each of the strong-motion records. Each records were scaled at 11 values of ground displacement, thus to total number of inelastic dynamic analyses for a set of fragility curves is 3300. Spectral displacement (Sd) is used as the hazard parameter for constructing the vulnerability curves. For this purpose, the displacement spectra of the selected ground motions are constructed (Fig. 10). The vertical dotted line in the figure denotes the elastic period (T ¼ 0:38 s), which is the fundamental period of the study structure. The scaling procedure employed herein is based on Sd values at this specific period, which is 10 mm. Dynamic analysis gave interstory drift values between the ranges of 0.14% and 0.22% when Sd is equal to 10 mm. The values of the interstory drift range between no damage and slight damage in terms of the limit states determined for the building used in the case study. Scale factors to be applied to the ground motions are selected so that the response of the structure can be monitored over a wide range that includes all damage states. Dynamic analyses are conducted by subjecting random samples of structures to the ground motion records given in Table 1 at each intensity drift level using the corresponding scale factor. Ground shaking intensity was characterized in this study by spectral displacements for two reasons: . For structural response in the period range of the relatively low stiffness flat-slab structures, spectral displacements are more closely correlated to damage than spectral accelerations; the latter would be more suitable for shorter period structures. . The second part of this study, published elsewhere [25], employs the ensuing fragility relationships in HAZUS analysis, which is expressed in terms of spectral displacements for other structural forms. It was therefore necessary to have HAZUS-compatible curves.

Fig. 11. Damage versus motion relationship for the flat-slab structure.

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Fig. 12. Lognormal statistical distributions for two different levels of seismic intensity, (a) intensity level 1 (Sd ¼ 30 mm), (b) intensity level 2 (Sd ¼ 60 mm).

For Sd ¼ 60 mm, PðSd > LS1Þ ¼ 1:000 PðSd > LS2Þ ¼ 0:997 PðSd > LS3Þ ¼ 0:839 PðSd > LS4Þ ¼ 0:338 After calculating the probability of exceedance of the limit states for each intensity level, the vulnerability curve can be constructed by plotting the calculated data versus spectral displacement. As the final step, a statistical distribution can be fitted to these data points, to obtain the vulnerability curves. In this study, a lognormal fit is assumed. The mean and standard deviation parameters of the curves are given as follows: LS1 (none to slight damage): k ¼ 1:85, n ¼ 0:350 LS2 (slight to moderate damage): k ¼ 3:50, n ¼ 0:285 LS3 (moderate to extensive damage): k ¼ 3:85, n ¼ 0:285 LS4 (extensive to complete damage): k ¼ 4:20, n ¼ 0:290

Fig. 13 represents the fragility curves of medium-rise flat-slab structures. The curves become flatter as the limit state shifts from slight to complete because of the nature of the statistical distribution of the response data. Vertical curves would represent deterministic response. The variability of interstory drift at high ground motion intensity levels is much more pronounced relative to the variability at low intensity levels. Hence, small variations in low intensity cause significant differences in the limit state exceedance probabilities. This observation points towards the high sensitivity of the structure to changes in seismic demand. The steep shape of the slight limit state curve is due to the infill panels dominating the response at this low-level limit state. This continues till the panels reach their deformation capacity. Thereafter, the response is dictated by the bare flexible flat-slab system. 12. Comparison of with moment-resisting frames The fragility curves of flat-slab structures derived in the previous section require a form of validation, since no experimental or observational data sets have been hitherto used in the derivation. A possible verification approach is to derive vulnerability curves for familiar moment-resisting frames, establish the realism of the latter by comparison with the literature and hence establish the realism of the new flat-slab curves. This was accomplished by developing the mean fragility curves of a framed structure using the same methodology as for the flat-slab structure. In order to develop the mean fragility curves for the framed structure, modifications were made to the previous analytical model. The slab­beams were replaced by conventional beams of 300 mm  600 mm and a longitudinal reinforcement ratio q of 1.5%. The columns and the infill walls were kept the same as the original flat-slab model. Mean fragility curves for the moment-resisting frame are shown in Fig. 14 alongside the fragility curves for flat-slab structures. It is shown that the flat-slab structure is more vulnerable to seismic damage than the moment-resisting frame across the entire range of seismic hazard. It is also interesting to observe that the difference between the flat-slab structure and the framed structure is more pronounced at the lower limit states. This is because of the inherent flexibility of flat-slab structures, as mentioned in previous sections. Small variations at low levels of seismic intensity can create amplified effects on the fragility curves whereas even large variations at high levels of seismic intensity may not have that much effect on the curves. The next step would be to compare the fragility curves derived for moment-resisting structures to the fragility curves from the literature. This is a challenge because of the dearth of spectral displacement-based

Fig. 13. Vulnerability curves for the flat-slab structure.


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Fig. 14. Comparison of fragility curves for flat-slab and framed structures.

Fig. 16. Comparison of the study curves for framed structure with Hwang and Huo.

vulnerability curves in the literature. Therefore, it was necessary to reconstruct the fragility curves for framed structures using spectral acceleration instead of spectral displacement. This was accomplished by converting the spectral values and then matching the converted values with the corresponding response (interstory drift) values. The spectral acceleration-based fragility curves are shown in Fig. 15. Figs. 16 and 17 show the comparison of the curves obtained for framed structures against the HH curves and the SK curves developed by Hwang and Huo [26] and Singhal and Kiremidjian [27], respectively. The HH curves in Fig. 16 were developed for low-rise (1­3 story) concrete frames. Four damage states were considered in terms of maximum story drift ratio, dmax: (1) no damage, when dmax < 0:2%, (2) insignificant damage, when 0:2% < dmax < 0:5%, (3) moderate damage, when 0:5% < dmax < 1:0% and (4) heavy damage, when dmax < 1:0%. More information about these fragility curves are discussed in Table 3.

The SK curves in Fig. 17 were developed for mid-rise reinforced concrete frames. The Park and Ang damage index was used as the response parameter. Damage states were also identified based on this damage index after calibration with observed damage to several buildings caused by different earthquakes. According to the damage scale, minor damage occurs when the index attains values between 0.1 and 0.2; moderate damage occurs when the index values are between 0.2 and 0.5 and severe damage occurs when the index values are between 0.5 and 1.0. Exceedance of unity for the index value corresponds to the collapse limit state. Additional information about the SK fragility curves is also discussed in Table 3.

Fig. 15. Acceleration-based fragility curves for the framed structure.

Fig. 17. Comparison of the study curves for framed structure with Singhal and Kremidjian.

M.A. Erberik, A.S. Elnashai / Engineering Structures 26 (2004) 937­948 Table 3 Comparison of fragility curve characteristics Derived curves Structure Ground motion Analysis Random variables Damage parameter Hazard parameter Limit states RC frame (MR) Actual, from various earthquakes Time-history fc, fy Interstory drift Sd(T1) 4 SK curves RC frame (MR) Synthetic (for West US region) Time-history fc, fy Park and Ang index Sa(T1) 4 HH curves RC frame (LR) Synthetic (close to NMSZ) Time-history fc, fy Interstory drift Sa(T1) and PGA 3


published studies on the other, makes the case of the reliability of the new curves. The curves are recommended for us in seismic loss assessment in regions where flat-slab structures exist. With regard to the characteristics of the flat-slab fragility curves, they display response features of this special type of construction. The steep light-damage curve reflects the role of the infill panels that dominate the response in the vicinity of the light-damage limit state. When the infill panels are damaged and no longer contribute to the lateral resistance, the buildings reach interstory drift limits more readily than their moment-resisting counterparts. Therefore, using vulnerability curves of moment-resisting frames to assess seismic damage of flat-slab buildings in non-conservative.

The fragility curves developed in this study are a better match with the SK curves than with the HH curves. The study curves seem to result in more damage in the case of the SK curves whereas the opposite is true when compared with the HH curves. As seen in Table 3, ground motion selection is quite different. There are also differences in the characterization of the hazard and the damage parameters. Quantification of the limit states of the SK and the HH curves are discussed in the above paragraphs and the values defined for this study are given in Table 2. In general, methods that different researchers adopt to determine fragility curves can cause significant discrepancies in the vulnerability predictions for the same location, even in cases where the same structure and seismicity are considered [28]. In a statistical context, the agreement between the vulnerability curves derived above for moment-resisting frames and those of Hwang and Huo [26], and Singhal and Kiremidjian [27] is reassuring and lends weight to the curves derived for flat-slab structures.

Acknowledgements The first author was funded by the Mid-America Earthquake Center through the National Science Foundation Grant EEC-9701785. Any findings in this paper do not represent the opinion of NSF but that of the authors. He also received partial funding from TUBITAK in Turkey. The paper is a product of MAE Center Project DS-9 `Risk Assessment Systems'. The authors have benefited from discussions with several colleagues, especially Dan Abrams (University if Illinois and MAE Center Director) and Joe Bracci (Texas A and M, and MAE Center investigator).


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13. Conclusions The purpose of this study is to develop fragility curves for flat-slab structural systems for which no fragility analysis has been undertaken before. A mid-rise flat-slab building is designed and modeled using the structural characteristics typical of the construction type under investigation. The preliminary evaluation of the structure indicates that the model structure is more flexible than conventional frames because of the absence of deep beams and/or shear walls. The reliability of the newly derived vulnerability curves is underpinned by the quality of the models used, methodology adopted and software employed. Moreover, the same approach and tools are used to derive median curves for moment-resisting frames for which there is an abundance of fragility studies in the literature. Comparison between the flat-slab and moment-resisting buildings on the one hand and the latter and two


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