by A.M. Reinhorn, R.E. Valles-Mattox and S.K. Kunnath
The Loss Assessment of Memphis Buildings (LAMB) project is one of the major research projects funded by NCEER. The immediate goal of the project is to assess probable losses and risk levels for specific building types in the Memphis area. However, the long term goal of the project is to develop a unified methodology that can be used for other building types, in other geographical regions. This project studies the response of non-ductile reinforced concrete buildings and unreinforced masonry buildings. The LAMB project integrates research efforts in structures and systems, seismic hazard and ground motions, geotechnical engineering, risk and reliability, and socioeconomic aspects. Questions and comments should be directed to Andrei Reinhorn, University at Buffalo, at (716) 645-2114 ext. 2419; email: email@example.com.
In this article, the performance of a four story building, designed without seismic provisions, is studied when subjected to different intensities of ground motion. The seismic demand in the structure is determined in two ways:
The latter technique was recently developed by the authors (Reinhorn et al., 1996). The deformation demands imposed by the earthquakes are compared to the ultimate deformation capacities, to translate response quantities to damage indices, and the latter to damage states.
The structure described herein was selected as a typical low-rise reinforced concrete building for the Memphis area. The structure, part of the University of Memphis campus, was designed in the mid 1960's without seismic provision, as were most buildings in the area. The four story reinforced concrete structure is used as a classroom building. Plan dimensions, typical at all floors, are 197 feet by 95 feet (60 m x 29 m), with a typical story height of 12 feet (3.66 m) as shown in figure 1. The structural system consists of a ribbed slab supported by columns and shear walls.
Typical column dimensions used in the building are 2 ft. 6 in. (76 mm) by 1 ft. (30.5 mm) columns closely spaced in the perimeter of the building, and 18 in. by 18 in. (46 x 46 mm) in the interior. Similarly, two beam types can be identified: one for the exterior frames, and a second one for the interior frames. All beams, or " slab bands" as referred to in the initial design, are 17 in. (43 mm) deep. The shear walls, delineating the elevator and stairs, have a constant thickness of 1 ft. (30.5 mm) throughout the height of the building. Figure 2 shows elevations of the three frame types identified in the longitudinal (north-south) direction. For a more detailed description of the building, see Valles et al. (1996a).
|Table 1: Story heights and weights for half of the case study building|
To evaluate the seismic response of the building, the computer program IDARC (version 4.0) was used (Valles et al., 1996b). A two dimensional model for the structure was used since no significant torsional amplifications were expected due to the symmetry of the building. Furthermore, only half of the building was modeled in either direction, thus, all results reported consider only half of the actual building. The story weights and story heights for half the building are summarized in table 1.
The lateral loads in the system are mostly carried by the shear walls, due to their higher stiffness, as compared to the column-beam frame action. Nevertheless, the model includes the other frames to ensure that the imposed displacement profile will not threaten the vertical load capacity of the structure. Since the limit state in the building can be reached either by a high force demand in the shear walls, or a significant displacement demand in the moment resisting frames, all structural elements were included. The moment-curvature capacity curves for each structural element were automatically generated by IDARC (Valles et al., 1996b), using the fiber model capabilities.
The Memphis metropolitan area is located in the Mississippi embayment, near the epicenters of the New Madrid series of earthquakes. Attenuation relations for the area have been developed by Nuttli and Hermann (1984). Ground motions for the site, corresponding to different earthquake magnitudes, stress drop, and epicentral distances, were generated by the seismic hazard and geotechnical task group of the LAMB project. The records were generated by combining a deterministic and a probabilistic approach (Horton, 1994). A total of 200 records were generated for the area, out of which five were selected to represent five ground motion intensities, with peak ground accelerations (PGA) of 0.1g, 0.2g, 0.3g, 0.4g, and 0.5g.
Time-History Analysis Response
The structure was subjected to the five earthquake motion accelerations selected. Figure 3 presents the maximum displacements, interstory drifts, and story shears experienced by the structure (NS direction), and for each earthquake intensity. The integration time step for the 0.4g and 0.5g accelerations had to be reduced, since the motion induced significant inelastic excursions in the structural elements.
The nonlinear pushover analysis, or collapse mode analysis, is a simple and efficient technique to study the response of a building. The pushover analysis is carried out by incrementally applying lateral loads, or displacements to the structure. The sequence of component cracking, yielding, and failure, as well as the history of deformations and shears in the structure, can be traced as the lateral loads (or displacements) are monotonically increased (see figure 4). Furthermore, strength and service limit states, such as the failure of an element, the formation of a collapse mechanism, etc., can be identified.
The results from pushover analyses are often presented in graphs that describe the variation of the story shear versus story drift, for an inter-story description of the capacity, and base shear versus top displacement, for a global description. The capacity curve determined from a pushover analysis is influenced by the lateral force (displacement) distribution used to load the structure. The force distribution for pushover analysis was determined as follows:
where VB is the base shear, m is the story mass, is the mass-normalized mode shape, is the participation factor, srss is the root sum of squares superposition, are spectral ratios which can be approximated (see Reinhorn et al., 1996). The same distribution can be simply approximated by the design code distribution:
where h is the height to the story weight, and W and k are a power to simulate the complex behavior described by Equation 1.
Figure 5 shows the overall capacity curves for a constant (k=0), linear (k=1), and parabolic (k=2) distribution. The first shear wall failure, as well as the response obtained from time-history analyses, are shown in figure 4. The capacity curve for k=1 (linear) was found to capture the results from the time history analyses in the deformation range before the first shear wall failure.
Simplified Elastic and Inelastic Response Evaluations
In the simplified design or evaluation process, the capacity of a structure is estimated, and compared to the demand loads. The response is estimated at the intersection of a capacity and a demand curve. The method is an extension of the capacity spectrum method proposed by Freeman (1994). The force deformation capacity curves are determined from pushover analysis. The elastic demand curve is determined from an elastic spectral analysis, modified to account for the hysteretic energy dissipation.
The elastic response evaluation method considers the use of an equivalent linear system to estimate the nonlinear response along with an equivalent damping ratio representing the hysteretic behavior (see figure 6). A summary of different methods to determine the equivalent damping ratio is presented by Iwan and Gates (1979). The average stiffness and energy method seems to give the smallest percentage of error for various ductility ratios (Iwan and Gates, 1979). For this method, the critical damping ratios are defined according to:
The idealized bilinear pushover capacity curve is superimposed with the equivalent linear demand curves. The point where the ductility along the capacity curve coincides with the equivalent ductility of the intersecting demand curves yields an estimate of the inelastic response.
For the inelastic response evaluation method, inelastic spectral curves are generated using a bilinear model for the structure (Reinhorn et al., 1996). The response of the bilinear single-degree-of-freedom system is obtained for a given value of the post-yielding stiffness (), and for different values of the force reduction factor, R, defined as the ratio of the elastic, Ve, to the yield, Vy, force capacity of the system. The point where the demand curve, corresponding to the actual value for R, intersects the capacity curve, is the actual inelastic response of the bilinear structure (figure 7).
The procedure described for an SDOF system can be extended to multi-story buildings by modifying the capacity curve to an equivalent SDOF system. The response can be evaluated considering the overall building response, or the interstory response. In the former case, the top displacement versus base shear is used to characterize the capacity, while in the latter case, the story drift versus story shears are used. Both evaluations are important, since the overall response provides an estimate of the global performance, while the story evaluation will detect undesirable weak stories. Actual overall pushover results (T and Vb) are modified to an equivalent SDOF spectral pushover curve (Sd and Sa/g) according to:
that consider a single mode contribution. The formulas for multiple mode contribution, and for the inter-story response evaluations, can be found in Valles et al. (1996a).
Figure 8 presents a comparison of the overall simplified response evaluations versus the time history analysis. Note that the predictions agree fairly well except for the last two intensities, when the pushover curve for k=1 cannot capture the behavior. The pushover curve for k=2 would yield better estimates for the last two response quantities.
A damage index is a parameter that indicates how close the maximum response is to the maximum ultimate capacity of the structure. Often, damage index models are normalized from a value of zero, indicating negligible response quantities as compared to the ultimate capacity, to a value of one, indicating that the ultimate capacity of the structure has been reached. The response quantities determined for the building are first used to calculate damage indices, which are then correlated to probable damage states. The fatigue based damage model, suggested by Reinhorn and Valles (1996), was used in the study. The damage index is defined as:
where a is the maximum experienced deformation; y is the yield deformation capacity; u is the ultimate deformation capacity; Fy is the yield force capacity; and Eh is the cumulative dissipated hysteretic energy.
|Table 2: Overall damage indices and damage states in the NS direction.|
|0.2g||2.79||35016||> 1.0||Loss of Building|
|0.3g||4.01||64760||> 1.0||Loss of Building|
|0.4g||7.10||161793||> 1.0||Loss of Building|
|0.5g||9.93||316212||> 1.0||Loss of Building|
The fatigue based damage index can be used to quantify the performance of structural elements, stories (or subassemblies), and the overall response of a building. Yield and ultimate capacities were determined using the results from the pushover analyses. Table 2 presents the overall building damage in the NS direction for the five earthquake motion intensities considered. Note that the building is only capable of withstanding an earthquake with a PGA of 0.1g. All other intensities of shaking induce collapse of the structure.
The seismic evaluation of an existing low-rise RC building was summarized. Five ground
motion intensities were considered. The evaluation was carried out using nonlinear
time-history analyses, and simplified elastic and inelastic response evaluation methods.
Results for the three methods show fairly good agreement in the prediction of the
response. However, the simplified methods have the advantage that the evaluation process
involves considerably less
computational effort. Damage quantification of the building response indicates that the structure can withstand an earthquake with a PGA of 0.1g with repairable damage, but an earthquake with a PGA of 0.2g or greater will cause the building to collapse. The example shows various evaluation procedures that can be applied to other building types.
Freeman, S.A. (1994), "The Capacity Spectrum Method for Determining the Demand Displacement," ACI Spring Convention.
Iwan, W.D. and Gates, N.C., (1979), "Estimating Earthquake Response of Simple Hysteretic Structures," Journal of Engineering Mechanics, ASCE, Vol. 105, No. EM3, pp. 391-405.
Horton, S.P. (1994), "Simulation of Strong Ground Motion in Eastern North America," Proc. of the Fifth National Conference on Earthquake Engineering, Chicago, Vol. III, published by EERI, pp. 251-260.
Nuttli, O.W. and Hermann, R.B., (1984), "Scaling and Attenuation Relations for Strong Ground Motion in Eastern North America," Proc. of the Eighth World Conference on Earthquake Engineering, pp. 305-309.
Reinhorn, A.M., Valles, R.E., and Lysiak, M., (1996), "Simplified Inelastic Response Evaluation Using Composite Spectra," Earthquake Spectra, in review.
Valles, R.E., Reinhorn, A.M., and Barrón, R., (1996a), "Seismic Evaluation of a Low-Rise Building in the Vicinity of the New Madrid Seismic Zone," NCEER Technical Report, State University of New York at Buffalo, in preparation.
Valles, R.E., Reinhorn, A.M., Kunnath, S.K., Li, C., and Madan, A., (1996b), "IDARC2D Version 4.0: A Computer Program for the Inelastic Damage Analysis of Buildings," Technical Report NCEER-96-0010, National Center for Earthquake Engineering Research, State University of New York at Buffalo.
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