Seismic Assessment of RC Frames with Setbacks
1Faculty of Civil Engineering, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia
2UTM-Construction Research Centre, Faculty of Civil Engineering, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia
Behavior assessment of reinforced concrete frame structures with irregularities in elevation compared to regular ones is the objective of this study. Multi-story Reinforced Concrete (RC) frame buildings either regular or irregular in elevation are presented in this paper. The regulations of the International Building Code (IBC 2009) were incorporated in the design of several multi-story Reinforced Concrete Moment Resisting Frames (RCMRFs) with different types of setbacks along with regular frames in elevation. Ten input motions were assigned to all frames to perform nonlinear time history analyses using IDARC 2d v7. Drift ratios and plastic hinge rotations, based on FEMA 356, were the criteria for investigation of seismic performance of all frames. It is concluded from the results that life safety (LS) level requirements in terms of occurrence of setback in elevation will not be satisfied. Also, maximum damage will occur near the setback elements. Hence, strengthening such elements through the use of improved criteria in IBC 2009  along with defining and proposing new indicators and methods that could predict the seismic behavior of vertically irregular buildings is of great importance.
At a glance: Figures
Keywords: RC frames, irregularity in elevation, setbacks, seismic performance
American Journal of Civil Engineering and Architecture, 2014 2 (3),
Received January 13, 2014; Revised January 25, 2014; Accepted May 05, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Faridmehr, Iman, et al. "Seismic Assessment of RC Frames with Setbacks." American Journal of Civil Engineering and Architecture 2.3 (2014): 89-101.
- Faridmehr, I. , Osman, M. H. , Tahir, M. B. M. , Azimi, M. A. , & Hodjati, R. (2014). Seismic Assessment of RC Frames with Setbacks. American Journal of Civil Engineering and Architecture, 2(3), 89-101.
- Faridmehr, Iman, Mohd Hanim Osman, Mahmood Bin Md. Tahir, Mohammad Amin Azimi, and Reza Hodjati. "Seismic Assessment of RC Frames with Setbacks." American Journal of Civil Engineering and Architecture 2, no. 3 (2014): 89-101.
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It is obvious that factors like mass, stiffness and stress distributions either in plan or elevation greatly influence the structural behavior of buildings in case of intense earthquakes. History has proven that damage or collapse-related issues in buildings were revealed once major earthquakes hit structures. It has been concluded that a better performance was expected from regular shaped buildings during earthquakes. Various members of a building with structural irregularities experience non-uniform load distributions in case of an earthquake. The overall seismic assessment of the structures yield a safer design for regular structures or structures with certain mass, stiffness or strength distributions conforming to specific regularity criteria. However, such safe designs could not be guaranteed in case of irregular structures. Evaluation of the previous mentioned parameters on the dynamic behavior of the structures has long been studied [1-6]. Studies have revealed the fact that mass distribution variations in elevation will influence the maximum displacements or ductility demand of structures . Yet, variations in stiffness or strength distribution in elevation will not yield the same results as mentioned for mass distribution variations. Although several provisions have been made in the current design codes regarding the design of structures with irregularities in elevation, they still suffer huge levels of damage and demonstrate inadequate behavior in case of a severe earthquake. Also, several studies on irregularity of structures have been performed including “Seismic Response of Vertically Irregular Frames with Pushover Analysis”, (Chintanapakdee, Chopra, 2004) , “Evaluation of Mass, Strength and Stiffness Limits for Regular Buildings Specified by UBC”, (Valmundsson and Nau, 1997)  and “Seismic Response of RC Frame Buildings with Soft First Story’s”, (Arlekar Jaswant N, Jain Sudhir K. and Murty C.V.R, 1997) . Since the structural conception process involves a multitude of factors, the majority of building structures appear to be irregular either in plan or elevation. This irregularity in plan or elevation might cause significant damage concentrations and deformations in case of an earthquake. Therefore, the behavior of a class of irregular structures has been analyzed and their performance was compared against the behavior of one regular structure.
2. Materials and Methods2.1. Structural Modeling in IDARC-2D
2.1.1. Macro Element Methodology
The computer program IDARC Version 6.1 has been used to perform the inelastic dynamic time-history analysis of all frames that have been incorporated this study . In the modeling process using the IDARC program, almost all structural elements have been modeled through applying the same basic macro formulation. The factors considered in the general structural macro elements include flexural, shear and axial deformations while the beam elements do not possess the last factor. Coupling of flexural and shear components occurs during the deformation in the spread plasticity formulation. Spreading of cracks from the joint interface occurs once the member undergoes inelastic deformations which ends up in a curvature distribution demonstrated in Figure 1 a. Different flexibility characteristics have been exhibited in sections along the elements based on the observed degree of inelasticity. The distribution of flexibility in structural elements is supposed to resemble the one presented in Figure 1 b, where the current flexural stiffness of the sections at ends of the elements are denoted by EIA and EIB; stiffness at the center of the element is denoted by EI0; yield penetration coefficients are denoted by αA and αB; and length of the elements is shown by L. The changes in the force capacity are described by the moment curvature envelope while a nonlinear analysis is in progress. Determination of the moment-curvature is done through the use of IDARC depending on a fiber model analysis of the cross-section.
2.1.2. Hysteretic Modeling Rules
Once the damage index evaluation is desired, hysteretic behavior modeling of structural elements becomes important which is one of the most significant aspects of nonlinear structural analysis,. Strength deterioration, strength degradation, non-symmetric response, slip-lock, and a tri-linear monotonic envelope are the parameters used in the hysteretic model. The hysteretic behavior of an element is traced by the model as it alters from one linear stage to another based on the history of deformations. Park et al.  presents a full description of the hysteretic model. The effect of various degrading parameters on the hysteretic loops shape is shown in Figure 2. A Vertex Oriented hysteretic model has been used to model the elements of the structures in this paper. The hysteretic model parameters are illustrated in Table 1.
2.2.1. Plastic Hinge Rotation Criteria
During the process of modeling the hysteretic properties of actual elements for the purpose of analysis, determination of the properties of cross sections like the initial stiffness, strength, and post-yield force-displacement response must be found depending on the principles of mechanics and/or experimental data taking into account the cyclic loading influences and the interaction between axial, shear, and flexural effects (Ibarra et al.) . An idealized force-deformation relationship in conformance with ASCE 41  to specify the parameters of force and deformation of a nonlinear component model is demonstrated in Figure 3. The four main important strength points and their associated deformations include effective yield point (B), peak strength point (C), residual strength point (D), and ultimate deformation point (E). In order to define the acceptance criteria or performance level for the plastic hinges formed close to joints at the ends of beams and columns, three performance levels namely IO, LS and CP (Immediate Occupancy, Life Safety and Collapse Prevention respectively) are demonstrated in Figure 3. Acceptance Criteria values are derived from rational analyses or experiments considering the flexural, axial and shear interactions (refer to equations 4 and 5 and Table 2 based on FEMA-356 ).
2.2.2. Lateral Drift Limitation
For the purpose of demonstrating the overall structural response to different pre-defined performance levels, FEMA  prescribes drift values regarded as typical values. Although a relationship exists between drift and damage, a complete damage prediction through applying drift alone is not conceivable because of complex interactions between other characteristics of buildings. Moreover, little data controls the creation of established relationships between damage and drift. Therefore, to better refine these guidelines, more and more research and experimental work would be desired. Table 3 demonstrates the indications of FEMA 356  on performance level of structures and damage of vertical elements. Once the first hinge appears in the structure throughout the pushover analysis, the transient drift occurs. Also, permanent drift is connected to target displacement and permanent drift is associated with the target displacement.2.3. Expected Strength Material Properties
Incorporation of expected strength, QCE, would be advantageous when an assessment of the behavior of deformation-controlled actions is expected. The term QCE is also used for the statistical mean value of yield strengths, Qy, which belongs to a population of related components. When the assessment of the force-controlled actions behavior occurs, application of the lower bound component strength, QCL, is permitted. The statistical mean minus one standard deviation of the yield strengths, Qy, for a population of similar components is termed as QCL. When it is preferred to determine the expected or lower-bound strengths of components through calculation, application of expected or lower-bound materials properties respectively is permitted. Table 4 highlights the expected strength material properties factor.
Several moment resisting reinforced concrete frames (MRF) possessing different setbacks have been considered for this study (Figure 4). The occupancy of the structures with assumed moderate level of ductility is less than 500 people and hence, is regarded as Occupancy Category II per apartment use (UNIFIED FACILITIES CRITERIA, 2012) . The design and detailing of the frames have been done in conformance with IBC 2009 (INTERNATIONAL CODE COUNCIL, 2009)  incorporating Response Modification Factor R=8, Site Class = B, Occupancy Importance = 1 and Design Spectral Accelerations (SDS & SD1) = 0.75 and 0.39. Hence, the seismic design category (SDC) D was determined for this design. The material properties are summarized in Table 5 and the schematic position of such stress-strain values are demonstrated in Figure 5.
In order to perform nonlinear dynamic analysis, it is crucial to select some accelerograms proportional to the geotechnical properties and soil conditions of the site compatible with the response spectrum of IBC 2009 code. The intended frames in this study have been designed on rock beds of soil site class B and hence, the selected records were chosen on rock beds (Table 6 and Figure 6). The selection of input motions incorporated in this study was done from Pacific Earthquake Engineering Research Center (PEER) . First, for each of the ten accelerograms of the design spectrum, the response with damping ratio of 5% were required to be determined and for the purpose of having PGAs equal to “g”, all the ten abovementioned accelerograms were required to be divided by their own PGAs (Figure 7). Then, the mean value of these ten accelerograms was calculated. This average response spectra had to be normalized to the intensity of the design spectrum of IBC 2009  for Ss = 3.04 and S1 = 1.17 (site zip code = 94704 “California”). Next, the design spectrum diagram values, derived from mean calculation of 10 response spectrums, needed to be multiplied by a coefficient so that these values became higher than the response spectrum of IBC 2009 . It is noteworthy to mention that the governing period is between 0.2T and 1.5T in all case studies (Figure 8). This study considers four normalized factors due to presence of four frames with different setbacks and heights and consequently, different natural periods. Finally, in order to improve the accuracy of calculations, four normalized factor categories are presented in Table 7.
3. Results and Discussion3.1. Assessment of Drift Ratio
A summary of the inter-story drift ratios of all frames for the “design earthquake” demonstrated in Figure 4 is shown in Figures 9 to 12 where the mean values of the drift ratios of the ten input motions for the purpose of performing the inelastic dynamic time-history analysis are represented. Consideration of mean response values is allowed once at least seven records are available based on IBC 2009 . As shown in these Figures, the LS performance level requirements (limiting drift 2%) are satisfied by the inter-story drift ratios of regular frames (3T0, 6T0, 9T0, 12T0); while different story drifts are expected for irregular frames compared to regular ones.
The inter-story drift ratios are greatly influenced and increased as a result of setback occurring along the height of frames. Even the Collapse Prevention (CP) performance level requirements, with limiting drift of 4%, are not satisfied by the performance levels of the majority of irregular frames (3T3, 6T2, 6T5, 6T6, 9T1, etc). This issue becomes more important as an increase in the number of stories of irregular frames occurs; even bigger inter-story drift ratios compared to CP performance level limiting drift are experienced by all 9 and 12 story frames. In case of the majority of irregular frames, sudden and large changes in the inter-story drift ratios in the vicinity of the irregularity are noticeable. This indicates an urgent need for the members in the setback level to be strengthened which is in agreement with findings of Moehle, J. and G. G. Deierlein (2004) .
Determination of the allowable limit of plastic rotation for LS performance level of each member was done based on their geometry, action, load types and reinforcement as the first step towards the local performance criteria assessment. Based on nonlinear analysis results, a comparison was made between the plastic rotations of the ends of members and their corresponding allowable values. The plastic hinge rotation values and their relevant allowable values in all frame members of Figure 4 are demonstrated in Figure 13- Figure 16 for the design earthquake. It is good to mention that the mean ratio values from the inelastic time-history analysis of ten earthquake records were incorporated in this study. According to column related figures, the maximum plastic rotation of column ends at the same story level is represented by the each story level rotation value. Also, in beam related figures, the maximum plastic rotation of beam ends at the same story level is represented by each story level rotation value. According to Figure 13- Figure 16 for irregular frames, although a violation has been occurred in the LS criteria of the majority of the beams (all the beams in stories 2, 3, 4 and 5 of the 6-story frame, all the beams in stories 6 and 7 of the 9-story frame and all the beams in stories 11 and 12 of the 12-story frame), the LS criteria for columns are satisfied (except for two interior columns in stories 4 and 5 of the 6-story frame and two interior columns of the top floor of the 12-story frame). Also, there has been an increase in number of members where unsatisfactory LS criteria exist as a result of increasing the setback severity. Furthermore, there have been violations in the CP performance level criteria of many irregular frames including 3T3, 6T2, 6T5, 6T6, 9T1, 9T2. According to Figures 13-16, a much more satisfactory seismic behavior is observed for columns compared to beams. A significant finding based on the results indicate the occurrence of a very strong rotation in members adjacent to the setback. Hence, strengthening these members in order to satisfy the local performance criteria is a necessity. Generally, it is concluded that a very poor seismic behavior is expected for the majority of the studied IBC 2009  designed irregular frames. Thus, improved IBC 2009  regulations are required for the purpose of defining and proposing new indicators and methods in order to predict the seismic behavior of vertically irregular buildings.
In this paper, the seismic performance of irregular RC frames in elevation was investigated according to specifications of IBC 2009 . Therefore, several vertically irregular frames were designed based on IBC 2009  regulations and then, ten nonlinear dynamic time-history analyses were performed on them incorporating ten different earthquake records. It was concluded that the LS performance criteria for all regular frames were satisfied. However, it was observed that the seismic performance of the studied IBC 2009  designed multi-story reinforced concrete frames for the high ductility level with setbacks along the height was not satisfactory. In other words, although the IBC 2009  capacity design procedure seemed to be acceptable for regular frames, the LS performance level criteria of irregular frames with setbacks along their height were not satisfactory and most of them collapsed under the design earthquakes. Hence, the IBC 2009  criteria for designing structures need to be improved through defining and proposing new indicators and methods which could actually predict the seismic behavior of vertically irregular buildings.
List of Notations
Cvx: is the base shear of static analysis.
QD: is the dead-load (action).
QL: is the = Effective live load (action), equal to 25% of the unreduced design live load, but not less than the actual live load.
ρ: is the mass density or density of material.
ϒ: is the weight per unit volume.
Ѵ: is the Poisson’s ratio.
Fy: is reinforcement yield stress.
Fu: is reinforcement ultimate stress.
E: is the elastic modulus of the reinforcement.
f’c: is the specified concert compressive strength.
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