over curves. 1Assistant Professor, Department of Civil Engineering, California Baptist University, Riverside, CA 92504. 2Associate Professor, Department of Civil ...
10NCEE
Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 Anchorage, Alaska
COMPARISON BETWEEN SEISMIC DEMAND MODELS AND INCREMENTAL DYNAMIC ANALYSIS FOR LOW-RISE AND MID-RISE REINFORCED CONCRETE BUILDINGS J.-W. Bai1, P. Gardoni2, and M. D. Hueste3 ABSTRACT This paper compares a seismic demand model and incremental dynamic analysis (IDA) for lowrise and mid-rise reinforced concrete building structures. A seismic demand model is one of the essential inputs for seismic fragility analysis. Seismic fragility analysis can be used to assess the vulnerability of structures under earthquake events. It plays an important role in estimating seismic losses and in the decision making process of hazard mitigation planning. Accurate estimates of seismic demands can reduce the uncertainty in seismic vulnerability assessment and loss estimation. IDA is an analysis method used to describe a wide range of structural behaviors under seismic loads. IDA is based on multiple nonlinear dynamic analyses using one or more scaled ground motion records. Through this method, one or more curves of structural responses corresponding to multiple levels of intensity can be developed. The theoretical limitations of IDA are discussed and the results of IDAs on 2-story and 5-story RC flat slab example buildings with one selected ground motion record are compared with the results from seismic demand models already available for these structures. In addition, the IDA curves are compared with traditional push-over curves and the dynamic analysis results using unscaled synthetic ground motions, and the transition points in the demand models are compared with those in the pushover curves.
1
Assistant Professor, Department of Civil Engineering, California Baptist University, Riverside, CA 92504. Associate Professor, Department of Civil and Environmental Engineering, University of Illinois at UrbanaChampaign, Urbana, IL 61801. 3 Professor, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843. 2
COMPARISON BETWEEN SEISMIC DEMAND MODELS AND INCREMENTAL DYNAMIC ANALYSIS FOR LOW-RISE AND MID-RISE REINFORCED CONCRETE BUILDINGS Jong-Wha Bai1, Paolo Gardoni2, and Mary Beth D. Hueste3
ABSTRACT This paper compares a seismic demand model and incremental dynamic analysis (IDA) for lowrise and mid-rise reinforced concrete building structures. A seismic demand model is one of the essential inputs for seismic fragility analysis. Seismic fragility analysis can be used to assess the vulnerability of structures under earthquake events. It plays an important role in estimating seismic losses and in the decision making process of hazard mitigation planning. Accurate estimates of seismic demands can reduce the uncertainty in seismic vulnerability assessment and loss estimation. IDA is an analysis method used to describe a wide range of structural behaviors under seismic loads. IDA is based on multiple nonlinear dynamic analyses using one or more scaled ground motion records. Through this method, one or more curves of structural responses corresponding to multiple levels of intensity can be developed. The theoretical limitations of IDA are discussed and the results of IDAs on 2-story and 5-story RC flat slab example buildings with one selected ground motion record are compared with the results from seismic demand models already available for these structures. In addition, the IDA curves are compared with traditional push-over curves and the dynamic analysis results using unscaled synthetic ground motions, and the transition points in the demand models are compared with those in the push-over curves.
Introduction Incremental dynamic analysis (IDA) is an analysis method to describe a wide range of structural responses based on multiple nonlinear dynamic analyses using one or more scaled ground motion records [1]. Through this method, one or more curves comparing structural response parameters to intensity can be developed. Although there are a few limitations on IDA including a sensitivity of results based on the selection of ground motions [2-4], this method provides useful information to have a better understanding of seismic behavior of structures. Because there is a correlation between transition points from IDA and seismic demand models, IDA is conducted for 2-story and 5-story RC flat slab buildings with one selected ground motion record. In addition, the IDA curves are compared with traditional push-over curves and the dynamic analysis results using 160 unscaled synthetic ground motions. Based on the comparison of the results, it is observed that the IDA curves match well with the results from the nonlinear dynamic analyses used in the demand model. In addition, correlation between transition points from the demand models and push-over curves is also investigated.
1
Assistant Professor, Department of Civil Engineering, California Baptist University, Riverside, CA 92504. Associate Professor, Department of Civil and Environmental Engineering, University of Illinois at UrbanaChampaign, Urbana, IL 61801. 3 Professor, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX, 77843. 2
Case Study Structure and Analytical Models Two representative structures are selected to compare seismic demand models and IDA results: a two-story and a five-story reinforced concrete (RC) flat slab office building typical of those in the Central United States (U.S.). These buildings represent a significant number of low- to midrise structures in this region constructed during the early 1980s. Design load requirements are based on the ninth edition of the Building Officials and Code Administrators (BOCA) Basic/National Code [5], in which St. Louis, Missouri, and Memphis, Tennessee, are considered to be in Seismic Zone 1. The structural member design follows the provisions of the American Concrete Institute (ACI) Building Code Requirements for Reinforced Concrete, ACI 318-83 [6]. The buildings have a moment frame system, not specially detailed for moderate to severe earthquakes. The floor system is composed of an interior flat slab with shear capitals and perimeter moment resisting frames with spandrel beams. Fig. 1 shows a plan view of the selected buildings.
Fig. 1. Plan view of selected buildings. The selected structures are analyzed using ZEUS-NL, a finite element structural analysis program developed for nonlinear dynamic, conventional and adaptive push-over, and eigenvalue analyses [7]. A two-dimensional analytical model is used, which is adequate for the regular floor plan of the selected buildings. Through the eigenvalue analysis, the fundamental periods of the selected structures are computed based on the cracked section properties and the values are 0.914 s and 1.62 s for the two-story and five-story buildings, respectively. More details about the design requirements and analytical modeling are found in Hueste and Bai [8].
Ground Motion Record for IDA One of ground motion records developed by Rix and Fernandez [9] for Memphis is selected for the IDA. This set of ground motions consists of a suite of synthetic records for two earthquake hazard levels, one with 2% and one with 10% probability of exceedance in 50 years. Based on the responses of the structures, the ground motion that provides the closest response to the median response among twenty 10% in 50 years motions is selected (Lowlands, ground motion ID #1). Figs. 2 and 3 show the acceleration time history and the response spectrum of the selected ground record, respectively. The peak ground acceleration (PGA) of this record is 0.157g. The spectral acceleration values corresponding to the fundamental periods for the twostory (T1 = 0.914 s) and five-story (T1 = 1.62 s) buildings are 0.137g and 0.09g, respectively.
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Probabilistic Demand Models Probabilistic demand models have been developed to describe the relationship between earthquake intensity, the spectral acceleration ( S a ), and the overall maximum interstory drift over the height of a building. Ramamoorthy et al. [10-11] found that a probabilistic linear model tends to underestimate the drift demand for small and large values of S a , and to overestimate the
drift demand for intermediate values of S a . In response to this observation, Ramamoorthy et al. [10-11] developed bilinear probabilistic models that provide a better fit over the entire range of S a . The bilinear model can be written as
D1 Sa ; Θ1 10 11 ln Sa 11 D2 Sa ; Θ2 10 11 Sa 21 ln Sa Sa 2 2
S a Sa
S a Sa
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where Di (S a ; Θ i ) ln[ (S a ; Θ i )] natural logarithm of the drift demand for ith branch, Θ i is a vector of unknown parameters ( 10 ,11, 20 , 21, Sa ); i is a random variable representing the error in the model with zero mean and unit standard deviation; and i is the standard deviation of the model error. The logarithmic transformation is used to approximately satisfy the normality assumption (i.e., i has the Normal distribution) and the homoscedasticity assumption (i.e., i is constant). In particular, Fig. 2 illustrates the definition of each model parameter in the bilinear model. ln
21 1
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Fig. 2. Illustration of the parameters in the bilinear model. A probabilistic bilinear demand model is developed to more accurately predict the interstory drift demands. The bilinear models are assessed using the same overall maximum interstory drifts used to assess the linear models. A Bayesian approach is used to estimate the unknown parameters in the demand models. The computation of the posterior statistics for the parameters, Θ1 (10 ,11, 1 ) and Θ2 ( 20 , 21, 2 ) , are carried out using the adaptive Markov Chain Monte Carlo (MCMC) simulation method [12]. Fig. 3 shows the predicted demand model for the two-story structure (on the left) and the five-story structure (on the right). Some maximum interstory drift values determined by the dynamic analysis are relatively large, corresponding to earthquake intensities that are also quite large. Based on experimental data for punching shear failures at the slab-column joints [13], a 5% maximum interstory drift is selected as the threshold for valid data points. Once the maximum interstory drift exceeds this threshold, the responses from the dynamic analysis are deemed no longer reliable because this mode of failure is not included in the nonlinear model. In Fig. 3, the dots (•) represent the equality data, while the triangles () represent the lower bound data. The triangles indicate the data exceeding 5% drift. The top horizontal dashed lines indicate 5% drift and the bottom dashed lines provide
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the location of the transition point between the first and second branches.
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Fig. 3. Overall maximum bilinear models for the case study structures For the two-story building, the overall maximum bilinear demand models provide a good fit to the drift demand over the entire range of S a . The slope for the second branch is steeper than that for the first branch to capture the highly nonlinear behavior of the building at larger values of S a . It is also noted that not only does the bilinear model provide a better fit to the data, but also it provides a more accurate account of the model uncertainties. For the five-story building, the slopes for both branches are similar and the predicted drift values are close to each other. However, the two standard deviation values in the bilinear formulation also better reflect the change in the variability of the data as a function of S a than the traditional linear formulation. A proper account of the uncertainties inherent in the demand models is important because of their effect on the fragility estimates.
IDA Curve Development IDA curves are developed using the selected ground motion with scaling factors. The ground motion is scaled with factors varying from 1 (original) to 10 in increments of 0.2. A total of 46 nonlinear time-history analyses are conducted and the corresponding maximum responses are estimated. Therefore, each IDA curve covers structural responses from the ground motion records having PGA of 0.157g to 1.57g. To be consistent with the demand model development of this study, the IDA curves are developed based on the overall maximum response, compared with the corresponding demand models for both case study structures.
Comparison between IDA and Nonlinear Dynamic Analysis Results The IDA curves are developed based on the overall maximum responses for the two-story and five-story buildings. Using the selected ground motion with scaling factors, the maximum drifts and shear ratios are estimated. Then those data points are connected with linear lines to
complete the IDA curves. To compare with the results from the nonlinear dynamic analyses using a suite of unscaled ground motions, the maximum drifts and shear ratios using 160 ground motion records are also plotted. Fig. 4 shows one of the comparisons between the IDA curve and the results from the nonlinear dynamic analyses. Both the IDA curve and the nonlinear dynamic results are in a logarithmic scale. The vertical axes for IDA curves are based on the spectral acceleration values corresponding to the fundamental period of the structure on the left, and the PGA values on the right. It is noted that the results from the nonlinear dynamic analyses are plotted based on the spectral acceleration (y-axis) and the maximum drift in percentage (x-axis). In addition, the IDA curves in a real space based on the overall maximum responses are provided in Fig. 5. The vertical dashed lines represent the valid limit of estimated data points, which is identified as 5% drift based on the potential for punching shear at the interior slab-column connections. More details for describing the data points are found in Bai et al. [14].
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Fig. 4. IDA and nonlinear dynamic analysis results based on the overall maximum responses in a logarithmic scale. As shown in Fig. 4, the IDA curves and the results from the nonlinear dynamic analyses match well in terms of the responses of the two-story and five-story buildings. Initially, the slope of the IDA curves is similar to the elastic range of data points. Then it is clearly seen that at a certain level of earthquake intensity, the trend changes to a nonlinear response. The corresponding transition point from the IDA also matches well with that of the nonlinear dynamic analyses. The complex behavior due to nonlinearity can be observed more clearly when the IDA curves and the results from the nonlinear dynamic analyses are in a real space (see Fig. 5). As shown, the IDA curves have fluctuating behavior when the earthquake intensity is beyond a certain level. There are several reasons for the fluctuating behavior of the IDA, such as a change in the building period due to stiffness degradation. In addition, the structural behavior at higher intensities is more complicated for the five-story building. This is because the response from the
first story governs the overall behavior for the two-story building, while the story responses are more complex for the five-story building. 1
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Fig. 5. IDA and nonlinear dynamic analysis results based on the overall maximum responses in a real scale.
Comparison between IDA and Traditional Push-Over Curves To compare structural behavior through two different analytical methods, the traditional pushover (nonlinear static) curves are developed for the case study buildings. To be consistent with the probabilistic demand models, push-over curves based on the overall maximum responses are developed. Fig. 6 shows the IDA curves and the push-over curves based on the overall maximum responses for the case study buildings. The results from the nonlinear dynamic analyses are also provided for a comparison purpose. Spectral acceleration on the left vertical axis is used for dynamic results and IDA curves, while shear ratio on the right is used for pushover curves. Because the measures on the vertical axes are different, the initial stiffness of the push-over curve is adjusted to be the same as that of the IDA suggested by Vamvatsikos and Cornell [1]. As shown in Fig. 6, the inflection point of the push-over curves generally correlates with the transition point of the IDA curves. In general, when the slope of the push-over curve becomes negative, there is a change in the slope of the IDA curves. This observation is also provided by Vamvatsikos and Cornell [1]. If the push-over curves from the overall maximum responses are supposed to be the responses from the “equivalent single degree-of-freedom” system, then the initial slope can be comparable to the “elastic” region and the transition can be the “yield” point of the equivalent single degree-of-freedom system. Table 1 shows the comparison of the maximum building drifts at the transition points from the bilinear demand models with the results of nonlinear dynamic analyses, IDA, and pushover curves based on the overall maximum responses. It is noted that the transition points of the
bilinear demand models are determined by the Bayesian updating process. Because the formulation of lower bound data is different from that of equality data, the transition points for the seismic demand models are slightly shifted to smaller ones. This is the reason why there are differences between the values from dynamic analysis and IDA even though they have a good match. However, similar trends are observed from two case studies that the values corresponding to the transition points for seismic demand models are between those for traditional push-over and IDA curves. 0.5
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Fig. 6. IDA and push-over curves based on the overall maximum responses. Table 1. Comparison of overall maximum building drift at transition points. Demand model Buildings Push-over IDA (bilinear) Two-story 2.13% 1.32% 2.36% building Five-story 2.67% 1.77% 3.11% building There is also a difference between IDA and push-over coming from the nature of two different analytical methods. Even though these curves represent demand of the same structure, push-over curves are developed with static lateral load applied to the structure cumulatively, while the results from IDA and nonlinear analyses are more record dependent because these are the results from the nonlinear dynamic analysis using scaled ground motions.
Summary and Conclusions In this study, incremental dynamic analysis (IDA) for low-rise and mid-rise reinforced concrete building structures is conducted and the results of IDAs are compared with seismic demand models. In addition, the IDA curves are compared with traditional push-over curves and the dynamic analysis results using unscaled synthetic ground motions. Based on the comparison, the
followings conclusions are made:
IDA curves and the nonlinear dynamic analysis results for both case study structures match well in terms of the overall maximum responses, in particular for the elastic range of data points. The corresponding transition point from the IDA also matches well with that of the nonlinear dynamic analyses. In the higher intensity range, the IDA curves have fluctuating behavior and the five-story building results show more nonlinearity due to more complex story responses. Based on the comparison between IDA and traditional push-over curves, the inflection point of the push-over curves generally correlates with the transition point of IDA curves. However, there is a difference between IDA and push-over analysis coming from the nature of two different analytical methods. Even though the results are from the same structure, push-over curves are developed with static lateral load applied to the structure cumulatively, while the results from IDA are more record dependent because these are the results from the nonlinear dynamic analysis using scaled ground motions. It is also verified that transition points for the bilinear formulation of seismic demand models are reliable based on the comparison between IDA and the demand models. However, because the formulation of lower bound data is different from that of equality data, the transition points for the seismic demand models are slightly shifted to smaller ones.
Acknowledgment The authors wish to acknowledge the National Science Foundation and the University of Illinois who funded this research through the Mid-America Earthquake Center (NSF Grant Number EEC-9701785). The financial support provided by the Zachry Department of Civil Engineering at Texas A&M University, where this research was conducted, is also appreciated. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the sponsors.
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9.
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