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On the modal incremental dynamic analysis of reinforced concrete structures, using a trilinear idealization model
Engineering Structures 33 (2011) 1117–1122
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
On the modal incremental dynamic analysis of reinforced concrete structures, using a trilinear idealization model P. Zarfam, M. Mofid ∗ Civil Engineering Department, Sharif University of Technology, Tehran, Iran
article
info
Article history: Received 2 April 2009 Received in revised form 13 December 2010 Accepted 14 December 2010 Available online 26 January 2011 Keywords: Modal incremental dynamic Nonlinear Trilinear Reinforced concrete Seismic performance
abstract In order to estimate the seismic demands at the performance level, the inelastic behavior of concrete structures should be considered. Incremental dynamic analysis (IDA) based on a nonlinear response time history analysis (NL-RHA) is considered to be the most accurate method in seismic demand calculations. However, modal incremental dynamic analysis (MIDA), based on the equivalent singledegree-of-freedom (SDF) oscillator, is also often used in studying structural engineering performances. As the MIDA method has usually not been applied to reinforced concrete (RC) structures, in this study an attempt is made to investigate the performances of RC frames and to compare the results obtained through the MIDA against those obtained from exact IDA. Furthermore, an innovative suggestion on approximated pushover curves of the corresponding SDF model, by means of a trilinear idealization representation, is also offered. For this purpose, an eight-story concrete frame subjected to 30 different earthquake records is studied with the trilinear idealization model, and the damage measures, important for the seismic vulnerability of buildings, such as the maximum displacement and the interstory drift ratio, are considered. Comparison of the results has shown reasonable and/or acceptable precision and reveals good agreement of the MIDA method with the new idealization behavior model for concrete frames. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Estimating seismic demands at low performance levels, such as life safety and collapse prevention, requires explicit consideration of the inelastic behavior of structures. Although nonlinear response time history analysis (NL-RHA) is one of the most accurate methods for seismic demand calculations, nonlinear static processes are also used in the ordinary performance evaluation of structural engineering. The pushover analysis method gives reliable results in low-rise buildings that respond primarily in the fundamental mode of vibration and have inelastic actions uniformly distributed over their heights. In this method, the contribution of modes, higher than the first mode, is not considered. Therefore, researchers have been increasingly interested in developing new pushover analysis techniques. Bracci, Gupta and Kunnath have presented an adaptive pushover technique, which accounts for the effects of higher modes and time-varying structural stiffness [1]. Elnashai and his co-workers have conducted extensive comparisons between dynamic and pushover analyses in order to identify the domain where a pushover analysis is valid [2]. Chopra and Chintanapakdee determined the seismic demands of irregular frames with high stiffness and strength for 20 earthquakes comparing two methods, modal pushover analysis (MPA) and
∗
Corresponding author. Tel.: +98 21 66014828; fax: +98 21 66014828. E-mail addresses: [email protected] (P. Zarfam), [email protected], [email protected] (M. Mofid). 0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.12.029
nonlinear time history analysis [3]. Hernández-Montes et al. observed that the roof was displaced in the reverse direction of the lateral load in a multimodal pushover analysis. This phenomenon is a potential impediment to MPA procedure application for curves of higher pushover modes. To overcome this difficulty an energy-based pushover analysis (EB-PA) technique was suggested by Hernández-Montes et al. [4], which is recommended in FEMA-440, and the EB-PA formulation has been extended to adaptive pushover analysis by Kunnath [5]. Then in 2006, a new method was presented by Tjhin et al. They considered another feature (displacement upon energy) for studying single-degree-offreedom (SDF) systems instead of roof displacement [6]. Similar to the process of the linear static analysis method moving forward to become the nonlinear static pushover analysis method, the idea of promoting the dynamic analysis method to an incremental dynamic analysis (IDA) method was raised. Apparently, this concept was put forward for the first time by Bertero, in 1977 [7], and he was followed by many other scientists and engineers such as Luco and Cornell [8], Bazzurro and Cornell [9], Yun et al. [10], Mehanny and Deieriein [11], Dubina et al. [12], Nassar and Krawinkler [13], Psycharis et al. [14], Vamvatsikos and Cornell [15]. Aschheim et al. [16] and Mander et al. [17], who worked extensively on this afterwards. Recently, a new technique for the dynamic response of structures has been investigated. This applied procedure can evaluate and predict the approximate seismic demands of structures,
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P. Zarfam, M. Mofid / Engineering Structures 33 (2011) 1117–1122
and it is fast, inexpensive, and the results are reasonably acceptable. In fact, this novel method logically combines two different techniques, IDA and MPA. This method takes advantage of both methods’ ideas, such as the equivalent SDF structure of multidegree-of-freedom (MDF) structures and the implementation of different scaled levels of an earthquake record to the provided equivalent SDF structure. The modal incremental dynamic analysis (MIDA) method was presented by Mofid, Zarfam and Fard in 2005 [18] and by Han and Chopra in 2006 [19]. As the MIDA method has been less applied for concrete structures until now, it is targeted to be investigated in this research. The aims of this research are as follows.
• To observe and compare RC frame performances using IDA and MIDA methods.
Fig. 1. Trilinear idealization of the pushover curve.
• To investigate the damage measures such as maximum displacement and interstory drift ratio.
• Apply a trilinear idealized force–displacement curve for approximated SDF structure behavior. 2. Modal incremental dynamic analysis In this procedure, the IDA curves are not obtained from nonlinear dynamic analysis of an MDF structure. However, the procedure of constructing these curves is based on modeling of the entire structure with several equivalent SDF structures and evaluating them through the modal pushover analysis method. In this method, the capacity curve of the MDF structure can be approximated through an idealized model, and therefore the specifications of the equivalent SDF oscillator will easily be obtained. Then, the MDF structure under a lateral loading pattern will be pushed up to the maximum displacement that was calculated from the SDF oscillator through nonlinear time history analysis. Consequently, the damage indexes will be obtained for each level of imposed scaled earthquake and mode of vibration. The results of different modes have to be combined through the Square Root of the Sum of the Squares, SRSS, method. Hence, the MIDA curve will express the behavior and/or performance of the structure. But, it is fairly understandable that an MIDA curve constructed through one scaled ground motion record cannot solely present the general behavior of structures in a probable earthquake. However, considering different scaled ground motion records and creating multi-MIDA curves through an averaging technique will be more reasonable and practical. This technique possesses all the advantages of the IDA method in studying the performance of structures in different levels of earthquake. In addition, it benefits from easy usage, high solving speed and less computational CPU time and in conforming to the modified idealized curves [20]. 3. Trilinear pushover curve idealization The nonlinear force–displacement relationship between the base shear and the displacement of the control node is replaced with an idealized bilinear relationship. This idealization, considered in the bilinear FEMA-356 idealization [21], has sufficient accuracy to approximate the steel element behavior. However, it has less precision for concrete structures. Therefore, in this study, the trilinear model is presented to characterize a model more similar to the RC frame pushover curve (Fig. 1). Three basic assumptions are taken into account in the trilinear idealization. These are as follows. 1. The slope of the initial part of the bilinear curve is equal to the initial slope of the primary curve. 2. The area underneath the curve is equal to the area of the initial curve. 3. The effective yield shear (vy ) is not greater than the maximum base shear in the initial curve.
Table 1 The characteristics of designed structural elements. Beam
Column
Story
Bottom bars
Top bars
Size (cm)
Bars
Size (cm)
6Φ 20 6Φ 20 6Φ 18 4Φ 18
6Φ 25 6Φ 25 6Φ 25 5Φ 25
45 × 45 45 × 45 40 × 40 35 × 35
12Φ 28 12Φ 22 8Φ 28 8Φ 25
50 × 50 45 × 45 40 × 40 35 × 35
1, 2 3, 4 5, 6 7, 8
The trilinear curve obtained has the following parameters. Initial stiffness (Ke ): The elastic part of the curve is the initial stiffness, located between the original point (0, 0) and the yielding point (uy , Vy ). Hardening stiffness (Ks ): This stiffness is the fraction of initial stiffness located between the yielding point (uy , Vy ) and the peak point of the pushover curve (uc , Vc ). The hardening stiffness is Ks = αs Ke . Post-capping stiffness (Kc ): This stiffness is also defined as the fraction of initial stiffness located between the peak point of the pushover curve (uc , Vc ) and the failure point (uo , Vo ), and it is Kc = αc Ke . The flow-chart shown in Fig. 2 presents the working procedure for the formation of the multi-MIDA curves. 4. Modeling and record selection The test model is a moment-resisting reinforced concrete frame which is designed based on specific ductility principles. The frame has eight stories with 3.2 m height and four bays with 5 m spans. In order to provide the necessary ductility as well as for economical reasons, the reinforcement percentages are limited to 1–3% and 1.7% for columns and beams, respectively. The dimensions and reinforcements of different elements of the frame are given in Table 1. Also, the concrete strength and steel yield strength are considered to be 28 and 300 MPa, respectively. The real modeling of nonlinear behavior of the reinforced concrete is mostly effective in the exactness of the damage assessment processes, structural vulnerability determination, and the accuracy of the results. At the same time, factors such as trilinearity, stiffness degradation, strength deterioration and pinching are considered in the modeling of concrete hysteretic behavior. The pushover curves together with the consequential trilinear curves for different vibration modes are shown in Fig. 3. The ground motion records required for the time history analysis were selected appropriately from reliable sources. In this regard, 30 modified ground motion records were double checked and chosen. The selected records are all from the California region, and some control parameters, such as closest distance to fault, earthquake magnitude, and soil type, were considered in this selection. In order to prevent any resonance phenomenon, the
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In this research, the dynamic instability capacity of the structure occurs when the final point of the local tangent reaches 20% of the elastic slope. It should be mentioned that, in all processes, the form of the IDA curve is considered in order to prevent errors such as negative slope, etc. Single IDA and MIDA curves of the roof displacement and maximum relative displacement of stories (interstory drift ratio) are illustrated for all records in Figs. 5 and 6. 6. Discussion of the resulting errors As mentioned before, MIDA is a new and practical method for studying the seismic behavior and real performance of structures in earthquakes. This method, like others, has some limitations, and the assumptions should be considered prior to any implementation. These limitations are as follows.
• Increasing the number of modes will significantly decrease the
•
•
• Fig. 2. MIDA working procedure flow-chart.
predominant period and vibration mean period are computed accordingly for all records and compared with the first mode period of the structure. All features of ground motion records are presented in Table 2. 5. Results In this section, the results obtained by modal incremental dynamic analysis method are presented and compared with those from the exact analysis. The analyses, charts and results cannot be presented in this paper, regarding their too much detailing. However, a comparison of the bilinear and trilinear idealization methods is shown in Fig. 4 for the maximum displacement and interstory drift ratio.
•
difference between the two techniques, especially in estimation of the interstory drift ratio. However, this is more acceptable for the lower level of the scaled earthquake records, where the structure behaves in the linear region [22]. Approximating the base shear versus maximum displacement with an idealized curve induces some practical errors; however, the new approximated trilinear capacity curve, illustrated in Fig. 4, shows proper improvement in the transformation error. As a structure enters the nonlinear zone, the stiffness matrix changes, and consequently the modal shapes will change as well. However, in the MIDA method, the appropriate modes, calculated in the linear phase, are still used in nonlinear regions [23]. Also, assuming a lateral deformation pattern with no changes in the nonlinear zone will probably induce dramatic errors. To find the maximum damage indexes in the MIDA method, these maximums may occur in different modes, which should reasonably be considered [24]. Conversion of MDF structure specifications into the equivalent SDF oscillator and vice versa is only correct for linear conditions, and the use of these equations for nonlinear conditions induces some errors. Increasing the seismic intensity level does not necessarily increase the errors of the MIDA method. It can simply be observed that if the structure remains in the linear zone, the errors of the MIDA scheme will remain constant in comparison with the IDA method. However, when the structure enters the nonlinear region, on increasing the seismic intensity level, the behavior of the errors changes, i.e. an error may increase or decrease according to the type of earthquake; see Fig. 7.
7. Multi-MIDA curves Studying the single IDA and MIDA curves shows that their extension up to the same levels is not reliable for all records,
Fig. 3. Pushover curve and idealized trilinear curve: (a) first mode, (b) second mode, and (c) third mode.
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Table 2 The features of selected records.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Record
Station
Soil type
Distance (km)
PGA (g)
Imperial Valley 1979 Imperial Valley 1979 Northridge 1994 San Fernando 1971 San Fernando 1971 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Landers 1992 Cape Mendocino 1992 Cape Mendocino Coalinga 1983 Whittier Narrows 1987 Northridge, 1994 Imperial Valley, 1979 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Imperial Valley, 1979 Loma Prieta, 1989 Imperial Valley, 1979 Imperial Valley, 1979 Loma Prieta, 1989 Imperial Valley, 1979 Imperial Valley, 1979 Loma Prieta, 1989
Chihuahua Chihuahua Hollywood Storage Lake Hughes #1 Hollywood Stor Lot Wildlife Liquefaction Arrey Wildlife Liquefaction Arrey Salton Sea Wildlife Refuge Plaster City Calipatria Fire Station Barstow Rio Dell Overpass Rio Dell Overpass Parkfield - Fault Zone 3 Beverly Hills LA, Baldwin Hills El Centro Array #12 Anderson Dam Downstream Anderson Dam Downstream Agnews State Hospital Anderson Dam Downstream Coyote Lake Dam Downstream Cucapah Sunnyvale Colton Ave El Centro Array #13 Westmoreland Fire Station Sunnyvale Colton Ave El Centro Array #13 Westmoreland Fire Station Hollister Diff. Array
C, D C, D C, D –, C C, D –, D –, D D,D C, D C, D B, D C, B C, B –, D B, C B, B C, D B, D B, D C, D B, D B, D C, D C, D C, D C, D C, D C, D C, D –, D
28.7 28.7 25.5 25.8 21.2 24.7 24.7 21.7 21 28.3 36.1 18.5 18.5 36.4 30.3 31.3 18.2 21.4 21.4 28.2 21.4 22.3 23.6 28.8 21.9 15.1 28.8 21.9 15.1 25.8
0.25 0.27 0.23 0.15 0.21 0.13 0.13 0.12 0.19 0.25 0.14 0.39 0.55 0.16 0.13 0.24 0.14 0.24 0.24 0.16 0.24 0.18 0.31 0.21 0.12 0.07 0.21 0.14 0.11 0.27
Fig. 4. Comparing bilinear and trilinear techniques with the NL-RHA method: (a) maximum displacement of record No. 10 and (b) interstory drift ratio of No. 10.
Fig. 5. Maximum displacement of 30 records: (a) IDA method and (b) MIDA method, including three modes.
because of the records’ natural differences. Therefore, after global collapse of the structure (i.e. the IDA and MIDA curves are quite flat), concrete frame responses are omitted in each record. Then, multi-IDA and multi-MIDA curves are prepared for all 30 records after revising the damage indexes data; see Fig. 8. Investigation of the average results reveals that the error in the damage indexes
calculated from the MIDA method is lower than that from the IDA method. In Fig. 7, it is observed that the displacement and interstory drift ratio error of the MIDA method has been significantly increased after Spectra acceleration =1.1g in comparison with the accurate IDA method. Furthermore, the final point of the local
P. Zarfam, M. Mofid / Engineering Structures 33 (2011) 1117–1122
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Fig. 6. Interstory drift ratio of 30 records: (a) IDA method and (b) MIDA method, including three modes.
Fig. 7. Comparing the errors of the MIDA method, including three modes, with those of the IDA method for the average of 30 ground motion records: (a) maximum displacement and (b) interstory drift ratio.
Fig. 8. Comparing the multi-IDA and multi-MIDA methods for the average of 30 ground motion records: (a) maximum displacement and (b) interstory drift ratio.
Fig. 9. Comparing the multi-MIDA, including three modes, and NL-RHA methods for PGA = 0.4g: (a) maximum displacement and (b) interstory drift ratio.
tangent reaches 20% of the elastic slope, seen in the IDA curve, Fig. 8. As mentioned before, this is evidence of unstable phases in the structure. Therefore, the structure is unstable at Spectra acceleration=1.1g which ends to increasing the errors. The average results for the maximum displacement and interstory drift ratio of 30 records for PGA = 0.4g with the multiIDA and multi-MIDA methods are shown in Fig. 9. According to the comparison, in order to calculate the maximum displacement of conventional structures, it would be sufficient to consider only the first mode of vibration. However, calculation of the interstory
drift ratio, especially in concrete buildings, requires at least the first three modes. 8. Conclusion
• The MIDA method with a trilinear idealized model gives reliable results for concrete frames.
• The results obtained by the MIDA method are occasionally underestimates compared with nonlinear time history analysis even with the consideration of higher-mode effects.
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• The difference between the damage measures obtained by •
•
• •
•
MIDA and IDA methods is higher in the upper stories than in the lower ones. The difference between the interstory drift ratio obtained in the first mode and the result of mode combination is higher in the upper stories than in the lower ones. It is observed that the errors of the MIDA procedure are increased with increasing earthquake intensity in comparison with the exact IDA method. Reliable results can be obtained when the contribution of the first dominant mode is high in the frames by using that mode. The effects of higher modes are to decrease the errors which usually occur where the errors are at the highest levels. Several interstory drift ratio irregularities obtained by the IDA procedure cannot be reconstructed by the MIDA method. Modeling multi-story buildings through an equivalent SDF system greatly reduces the costs of performing of time-consuming nonlinear dynamic analysis. This fact is very important in achieving average curves of several seismic records. The error rates of the MIDA method in the near-fault earthquakes are higher than those of the exact IDA method.
References [1] Gupta B, Kunnath SK. Adaptive spectra-based pushover procedure for seismic evaluation of structures. Earthq Spectra 2000;16(2):367–92. [2] Chintanapakdee C, Chopra AK. Evaluation of modal pushover analysis using vertically irregular frames. In: 13th world conference on earthquake engineering. Vancouver (BC, Canada); 2004. Paper 2139. [3] Chopra AK, Goel RK. Role of higher-mode pushover analyses in seismic analysis of buildings. Earthq Spectra 2005;21(4):1027–41. [4] Hernández Montes E, Kwon O-S, Aschheim M. An energy-based formulation for first- and multiple-mode nonlinear static (pushover) analyses. J Earthq Eng 2004;8(1):69–88. [5] Kalkan E, Kunnath SK. Assessment of current nonlinear static procedures for seismic evaluation of buildings. Eng Struct 2006;29(3):305–16. [6] Tjhin T, Aschheim M, Hernández-Montes E. Observation on the reliability of alternative multiple-mode pushover analysis methods. J Struct Eng 2006; 132(3):471–7.
[7] Bertero VV. Strength and deformation capacities of buildings under extreme environments. Struct Eng Struct Mech 1977;211–5. [8] Luco N, Cornell CA. Effects of random connection fractures on demands and reliability for a 3-story pre- Northridge SMRF structural. In: Proceedings of the 6th US national conference on earthquake engineering. Seattle (Washington, EERI, EI Cerrito, California); 1998. [9] Bazzurro P, Cornell CA. Seismic hazard analysis for non-linear structures. I: methodology. ASCE J Struct Eng 1994;120(11). [10] Yun SY, Hamburger RO, Cornell CA, Foutch DA. Seismic performance for steel moment frames. ASCE J Struct Eng 2001. [11] Mehanny SS, Deieriein GG. Modeling and assessment of seismic performance of composite frames with reinforced concrete columns and steel beams. Report no. 136. Stanford: John A. Blume Earthquake Engineering Center Stanford University; 2000. [12] Dubina D, Ciutian A, Stratan A, Dinu F. In: Mazzolani FM, editor. Ductility demand for semi-rigid joint frames. Moment resist connections of steel frames in seismic areas. New York: E & FN Spon; 2000. [13] Nassar AA, Krawinkler H. Seismic demands for SDF and MDF systems. Report no. 95. Stanford: The John A. Blume Earthquake Engineering Center; 1991. p. 62–155. [14] Psycharis IN, Papastramatiou DY, Alexandris AP. Parametric investigation of the stability of classical columns under harmonic and earthquake excitation. Earthq Eng Struct Dyn 2000;29:1093–109. [15] Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [16] Aschheim M, Tjhin T, Hamburger R, Comartin C, Inel M. The scaled nonlinear dynamic procedure. Eng Struct 2007;29(7):1422–41. [17] John ManderB, Dhakal RajeshP, Mashiko Naoto, Solberg KevinM. Incremental dynamic analysis applied to seismic financial risk assessment of bridges. Eng Struct 2007;29(10):2662–72. [18] Mofid M, Zarfam P, Fard BR. On the modal incremental dynamic analysis. The Struct Des Tall Special Build 2005;14:315–29. [19] Han SW, Chopra AK. Approximate incremental dynamic analysis using the modal pushover analysis procedure. Earthq Eng Struct Dyn 2006;35(15): 1853–73. [20] Zarfam P, Mofid M. Evaluation of modal incremental dynamic analysis, using input energy intensity and modified bilinear curve. The Struct Des Tall Special Build 2008. [21] FEMA356. Pre-standard and commentary for the seismic rehabilitation of buildings. Washington, DC: Federal Emergency Management Agency; 2000. [22] Zarfam P, Mofid M. On the assessment of modal nonlinear pushover analysis for steel frames with semi-rigid connections. Struct Eng Mech 2009;32(3). [23] Inel M, Ozmen H. Effects of plastic hinge properties in nonlinear analysis of reinforced concrete buildings. Eng Struct 2006;28(11). [24] Zou X, Chan C. Optimal seismic performance-based design of reinforced concrete buildings using nonlinear pushover analysis. Eng Struct 2005;27(8).
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
On the modal incremental dynamic analysis of reinforced concrete structures, using a trilinear idealization model P. Zarfam, M. Mofid ∗ Civil Engineering Department, Sharif University of Technology, Tehran, Iran
article
info
Article history: Received 2 April 2009 Received in revised form 13 December 2010 Accepted 14 December 2010 Available online 26 January 2011 Keywords: Modal incremental dynamic Nonlinear Trilinear Reinforced concrete Seismic performance
abstract In order to estimate the seismic demands at the performance level, the inelastic behavior of concrete structures should be considered. Incremental dynamic analysis (IDA) based on a nonlinear response time history analysis (NL-RHA) is considered to be the most accurate method in seismic demand calculations. However, modal incremental dynamic analysis (MIDA), based on the equivalent singledegree-of-freedom (SDF) oscillator, is also often used in studying structural engineering performances. As the MIDA method has usually not been applied to reinforced concrete (RC) structures, in this study an attempt is made to investigate the performances of RC frames and to compare the results obtained through the MIDA against those obtained from exact IDA. Furthermore, an innovative suggestion on approximated pushover curves of the corresponding SDF model, by means of a trilinear idealization representation, is also offered. For this purpose, an eight-story concrete frame subjected to 30 different earthquake records is studied with the trilinear idealization model, and the damage measures, important for the seismic vulnerability of buildings, such as the maximum displacement and the interstory drift ratio, are considered. Comparison of the results has shown reasonable and/or acceptable precision and reveals good agreement of the MIDA method with the new idealization behavior model for concrete frames. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Estimating seismic demands at low performance levels, such as life safety and collapse prevention, requires explicit consideration of the inelastic behavior of structures. Although nonlinear response time history analysis (NL-RHA) is one of the most accurate methods for seismic demand calculations, nonlinear static processes are also used in the ordinary performance evaluation of structural engineering. The pushover analysis method gives reliable results in low-rise buildings that respond primarily in the fundamental mode of vibration and have inelastic actions uniformly distributed over their heights. In this method, the contribution of modes, higher than the first mode, is not considered. Therefore, researchers have been increasingly interested in developing new pushover analysis techniques. Bracci, Gupta and Kunnath have presented an adaptive pushover technique, which accounts for the effects of higher modes and time-varying structural stiffness [1]. Elnashai and his co-workers have conducted extensive comparisons between dynamic and pushover analyses in order to identify the domain where a pushover analysis is valid [2]. Chopra and Chintanapakdee determined the seismic demands of irregular frames with high stiffness and strength for 20 earthquakes comparing two methods, modal pushover analysis (MPA) and
∗
Corresponding author. Tel.: +98 21 66014828; fax: +98 21 66014828. E-mail addresses: [email protected] (P. Zarfam), [email protected], [email protected] (M. Mofid). 0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.12.029
nonlinear time history analysis [3]. Hernández-Montes et al. observed that the roof was displaced in the reverse direction of the lateral load in a multimodal pushover analysis. This phenomenon is a potential impediment to MPA procedure application for curves of higher pushover modes. To overcome this difficulty an energy-based pushover analysis (EB-PA) technique was suggested by Hernández-Montes et al. [4], which is recommended in FEMA-440, and the EB-PA formulation has been extended to adaptive pushover analysis by Kunnath [5]. Then in 2006, a new method was presented by Tjhin et al. They considered another feature (displacement upon energy) for studying single-degree-offreedom (SDF) systems instead of roof displacement [6]. Similar to the process of the linear static analysis method moving forward to become the nonlinear static pushover analysis method, the idea of promoting the dynamic analysis method to an incremental dynamic analysis (IDA) method was raised. Apparently, this concept was put forward for the first time by Bertero, in 1977 [7], and he was followed by many other scientists and engineers such as Luco and Cornell [8], Bazzurro and Cornell [9], Yun et al. [10], Mehanny and Deieriein [11], Dubina et al. [12], Nassar and Krawinkler [13], Psycharis et al. [14], Vamvatsikos and Cornell [15]. Aschheim et al. [16] and Mander et al. [17], who worked extensively on this afterwards. Recently, a new technique for the dynamic response of structures has been investigated. This applied procedure can evaluate and predict the approximate seismic demands of structures,
1118
P. Zarfam, M. Mofid / Engineering Structures 33 (2011) 1117–1122
and it is fast, inexpensive, and the results are reasonably acceptable. In fact, this novel method logically combines two different techniques, IDA and MPA. This method takes advantage of both methods’ ideas, such as the equivalent SDF structure of multidegree-of-freedom (MDF) structures and the implementation of different scaled levels of an earthquake record to the provided equivalent SDF structure. The modal incremental dynamic analysis (MIDA) method was presented by Mofid, Zarfam and Fard in 2005 [18] and by Han and Chopra in 2006 [19]. As the MIDA method has been less applied for concrete structures until now, it is targeted to be investigated in this research. The aims of this research are as follows.
• To observe and compare RC frame performances using IDA and MIDA methods.
Fig. 1. Trilinear idealization of the pushover curve.
• To investigate the damage measures such as maximum displacement and interstory drift ratio.
• Apply a trilinear idealized force–displacement curve for approximated SDF structure behavior. 2. Modal incremental dynamic analysis In this procedure, the IDA curves are not obtained from nonlinear dynamic analysis of an MDF structure. However, the procedure of constructing these curves is based on modeling of the entire structure with several equivalent SDF structures and evaluating them through the modal pushover analysis method. In this method, the capacity curve of the MDF structure can be approximated through an idealized model, and therefore the specifications of the equivalent SDF oscillator will easily be obtained. Then, the MDF structure under a lateral loading pattern will be pushed up to the maximum displacement that was calculated from the SDF oscillator through nonlinear time history analysis. Consequently, the damage indexes will be obtained for each level of imposed scaled earthquake and mode of vibration. The results of different modes have to be combined through the Square Root of the Sum of the Squares, SRSS, method. Hence, the MIDA curve will express the behavior and/or performance of the structure. But, it is fairly understandable that an MIDA curve constructed through one scaled ground motion record cannot solely present the general behavior of structures in a probable earthquake. However, considering different scaled ground motion records and creating multi-MIDA curves through an averaging technique will be more reasonable and practical. This technique possesses all the advantages of the IDA method in studying the performance of structures in different levels of earthquake. In addition, it benefits from easy usage, high solving speed and less computational CPU time and in conforming to the modified idealized curves [20]. 3. Trilinear pushover curve idealization The nonlinear force–displacement relationship between the base shear and the displacement of the control node is replaced with an idealized bilinear relationship. This idealization, considered in the bilinear FEMA-356 idealization [21], has sufficient accuracy to approximate the steel element behavior. However, it has less precision for concrete structures. Therefore, in this study, the trilinear model is presented to characterize a model more similar to the RC frame pushover curve (Fig. 1). Three basic assumptions are taken into account in the trilinear idealization. These are as follows. 1. The slope of the initial part of the bilinear curve is equal to the initial slope of the primary curve. 2. The area underneath the curve is equal to the area of the initial curve. 3. The effective yield shear (vy ) is not greater than the maximum base shear in the initial curve.
Table 1 The characteristics of designed structural elements. Beam
Column
Story
Bottom bars
Top bars
Size (cm)
Bars
Size (cm)
6Φ 20 6Φ 20 6Φ 18 4Φ 18
6Φ 25 6Φ 25 6Φ 25 5Φ 25
45 × 45 45 × 45 40 × 40 35 × 35
12Φ 28 12Φ 22 8Φ 28 8Φ 25
50 × 50 45 × 45 40 × 40 35 × 35
1, 2 3, 4 5, 6 7, 8
The trilinear curve obtained has the following parameters. Initial stiffness (Ke ): The elastic part of the curve is the initial stiffness, located between the original point (0, 0) and the yielding point (uy , Vy ). Hardening stiffness (Ks ): This stiffness is the fraction of initial stiffness located between the yielding point (uy , Vy ) and the peak point of the pushover curve (uc , Vc ). The hardening stiffness is Ks = αs Ke . Post-capping stiffness (Kc ): This stiffness is also defined as the fraction of initial stiffness located between the peak point of the pushover curve (uc , Vc ) and the failure point (uo , Vo ), and it is Kc = αc Ke . The flow-chart shown in Fig. 2 presents the working procedure for the formation of the multi-MIDA curves. 4. Modeling and record selection The test model is a moment-resisting reinforced concrete frame which is designed based on specific ductility principles. The frame has eight stories with 3.2 m height and four bays with 5 m spans. In order to provide the necessary ductility as well as for economical reasons, the reinforcement percentages are limited to 1–3% and 1.7% for columns and beams, respectively. The dimensions and reinforcements of different elements of the frame are given in Table 1. Also, the concrete strength and steel yield strength are considered to be 28 and 300 MPa, respectively. The real modeling of nonlinear behavior of the reinforced concrete is mostly effective in the exactness of the damage assessment processes, structural vulnerability determination, and the accuracy of the results. At the same time, factors such as trilinearity, stiffness degradation, strength deterioration and pinching are considered in the modeling of concrete hysteretic behavior. The pushover curves together with the consequential trilinear curves for different vibration modes are shown in Fig. 3. The ground motion records required for the time history analysis were selected appropriately from reliable sources. In this regard, 30 modified ground motion records were double checked and chosen. The selected records are all from the California region, and some control parameters, such as closest distance to fault, earthquake magnitude, and soil type, were considered in this selection. In order to prevent any resonance phenomenon, the
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In this research, the dynamic instability capacity of the structure occurs when the final point of the local tangent reaches 20% of the elastic slope. It should be mentioned that, in all processes, the form of the IDA curve is considered in order to prevent errors such as negative slope, etc. Single IDA and MIDA curves of the roof displacement and maximum relative displacement of stories (interstory drift ratio) are illustrated for all records in Figs. 5 and 6. 6. Discussion of the resulting errors As mentioned before, MIDA is a new and practical method for studying the seismic behavior and real performance of structures in earthquakes. This method, like others, has some limitations, and the assumptions should be considered prior to any implementation. These limitations are as follows.
• Increasing the number of modes will significantly decrease the
•
•
• Fig. 2. MIDA working procedure flow-chart.
predominant period and vibration mean period are computed accordingly for all records and compared with the first mode period of the structure. All features of ground motion records are presented in Table 2. 5. Results In this section, the results obtained by modal incremental dynamic analysis method are presented and compared with those from the exact analysis. The analyses, charts and results cannot be presented in this paper, regarding their too much detailing. However, a comparison of the bilinear and trilinear idealization methods is shown in Fig. 4 for the maximum displacement and interstory drift ratio.
•
difference between the two techniques, especially in estimation of the interstory drift ratio. However, this is more acceptable for the lower level of the scaled earthquake records, where the structure behaves in the linear region [22]. Approximating the base shear versus maximum displacement with an idealized curve induces some practical errors; however, the new approximated trilinear capacity curve, illustrated in Fig. 4, shows proper improvement in the transformation error. As a structure enters the nonlinear zone, the stiffness matrix changes, and consequently the modal shapes will change as well. However, in the MIDA method, the appropriate modes, calculated in the linear phase, are still used in nonlinear regions [23]. Also, assuming a lateral deformation pattern with no changes in the nonlinear zone will probably induce dramatic errors. To find the maximum damage indexes in the MIDA method, these maximums may occur in different modes, which should reasonably be considered [24]. Conversion of MDF structure specifications into the equivalent SDF oscillator and vice versa is only correct for linear conditions, and the use of these equations for nonlinear conditions induces some errors. Increasing the seismic intensity level does not necessarily increase the errors of the MIDA method. It can simply be observed that if the structure remains in the linear zone, the errors of the MIDA scheme will remain constant in comparison with the IDA method. However, when the structure enters the nonlinear region, on increasing the seismic intensity level, the behavior of the errors changes, i.e. an error may increase or decrease according to the type of earthquake; see Fig. 7.
7. Multi-MIDA curves Studying the single IDA and MIDA curves shows that their extension up to the same levels is not reliable for all records,
Fig. 3. Pushover curve and idealized trilinear curve: (a) first mode, (b) second mode, and (c) third mode.
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Table 2 The features of selected records.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Record
Station
Soil type
Distance (km)
PGA (g)
Imperial Valley 1979 Imperial Valley 1979 Northridge 1994 San Fernando 1971 San Fernando 1971 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Super Stition Hills 1987 Landers 1992 Cape Mendocino 1992 Cape Mendocino Coalinga 1983 Whittier Narrows 1987 Northridge, 1994 Imperial Valley, 1979 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Loma Prieta, 1989 Imperial Valley, 1979 Loma Prieta, 1989 Imperial Valley, 1979 Imperial Valley, 1979 Loma Prieta, 1989 Imperial Valley, 1979 Imperial Valley, 1979 Loma Prieta, 1989
Chihuahua Chihuahua Hollywood Storage Lake Hughes #1 Hollywood Stor Lot Wildlife Liquefaction Arrey Wildlife Liquefaction Arrey Salton Sea Wildlife Refuge Plaster City Calipatria Fire Station Barstow Rio Dell Overpass Rio Dell Overpass Parkfield - Fault Zone 3 Beverly Hills LA, Baldwin Hills El Centro Array #12 Anderson Dam Downstream Anderson Dam Downstream Agnews State Hospital Anderson Dam Downstream Coyote Lake Dam Downstream Cucapah Sunnyvale Colton Ave El Centro Array #13 Westmoreland Fire Station Sunnyvale Colton Ave El Centro Array #13 Westmoreland Fire Station Hollister Diff. Array
C, D C, D C, D –, C C, D –, D –, D D,D C, D C, D B, D C, B C, B –, D B, C B, B C, D B, D B, D C, D B, D B, D C, D C, D C, D C, D C, D C, D C, D –, D
28.7 28.7 25.5 25.8 21.2 24.7 24.7 21.7 21 28.3 36.1 18.5 18.5 36.4 30.3 31.3 18.2 21.4 21.4 28.2 21.4 22.3 23.6 28.8 21.9 15.1 28.8 21.9 15.1 25.8
0.25 0.27 0.23 0.15 0.21 0.13 0.13 0.12 0.19 0.25 0.14 0.39 0.55 0.16 0.13 0.24 0.14 0.24 0.24 0.16 0.24 0.18 0.31 0.21 0.12 0.07 0.21 0.14 0.11 0.27
Fig. 4. Comparing bilinear and trilinear techniques with the NL-RHA method: (a) maximum displacement of record No. 10 and (b) interstory drift ratio of No. 10.
Fig. 5. Maximum displacement of 30 records: (a) IDA method and (b) MIDA method, including three modes.
because of the records’ natural differences. Therefore, after global collapse of the structure (i.e. the IDA and MIDA curves are quite flat), concrete frame responses are omitted in each record. Then, multi-IDA and multi-MIDA curves are prepared for all 30 records after revising the damage indexes data; see Fig. 8. Investigation of the average results reveals that the error in the damage indexes
calculated from the MIDA method is lower than that from the IDA method. In Fig. 7, it is observed that the displacement and interstory drift ratio error of the MIDA method has been significantly increased after Spectra acceleration =1.1g in comparison with the accurate IDA method. Furthermore, the final point of the local
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Fig. 6. Interstory drift ratio of 30 records: (a) IDA method and (b) MIDA method, including three modes.
Fig. 7. Comparing the errors of the MIDA method, including three modes, with those of the IDA method for the average of 30 ground motion records: (a) maximum displacement and (b) interstory drift ratio.
Fig. 8. Comparing the multi-IDA and multi-MIDA methods for the average of 30 ground motion records: (a) maximum displacement and (b) interstory drift ratio.
Fig. 9. Comparing the multi-MIDA, including three modes, and NL-RHA methods for PGA = 0.4g: (a) maximum displacement and (b) interstory drift ratio.
tangent reaches 20% of the elastic slope, seen in the IDA curve, Fig. 8. As mentioned before, this is evidence of unstable phases in the structure. Therefore, the structure is unstable at Spectra acceleration=1.1g which ends to increasing the errors. The average results for the maximum displacement and interstory drift ratio of 30 records for PGA = 0.4g with the multiIDA and multi-MIDA methods are shown in Fig. 9. According to the comparison, in order to calculate the maximum displacement of conventional structures, it would be sufficient to consider only the first mode of vibration. However, calculation of the interstory
drift ratio, especially in concrete buildings, requires at least the first three modes. 8. Conclusion
• The MIDA method with a trilinear idealized model gives reliable results for concrete frames.
• The results obtained by the MIDA method are occasionally underestimates compared with nonlinear time history analysis even with the consideration of higher-mode effects.
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• The difference between the damage measures obtained by •
•
• •
•
MIDA and IDA methods is higher in the upper stories than in the lower ones. The difference between the interstory drift ratio obtained in the first mode and the result of mode combination is higher in the upper stories than in the lower ones. It is observed that the errors of the MIDA procedure are increased with increasing earthquake intensity in comparison with the exact IDA method. Reliable results can be obtained when the contribution of the first dominant mode is high in the frames by using that mode. The effects of higher modes are to decrease the errors which usually occur where the errors are at the highest levels. Several interstory drift ratio irregularities obtained by the IDA procedure cannot be reconstructed by the MIDA method. Modeling multi-story buildings through an equivalent SDF system greatly reduces the costs of performing of time-consuming nonlinear dynamic analysis. This fact is very important in achieving average curves of several seismic records. The error rates of the MIDA method in the near-fault earthquakes are higher than those of the exact IDA method.
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