chimney pushover analysis

Finite Elements in Analysis and Design 43 (2007) 879–887www.elsevier.com/locate/finel

3-D pushover analysis of a collapsed reinforced concrete chimney

Wei Huanga,∗, Phillip L. Gouldb

aKPFF Consulting Engineers, Los Angeles, CA 90045, USAbDepartment of Civil Engineering, Washington University, Saint Louis, MO 63130, USA

Received 12 May 2006; received in revised form 6 April 2007; accepted 30 May 2007Available online 17 July 2007

Abstract

During the Izmit (Kocaeli) Earthquake of August 17, 1999, a 115 m high reinforced concrete chimney or heater stack, located at the TüprasRefinery, collapsed. This stack was designed and constructed according to international standards and is representative of similar structuresat refineries throughout the world, including those in earthquake-prone regions. This structure is of particular interest because several similarchimneys at the site survived the shock with only moderate damage. The particular distinction of this chimney appears to be an unusuallylarger rectangular opening, located about 1

3 of the height above the base, which appeared to be the region of collapse initiation.The main focus of the research is the dynamic response of the stack due to an earthquake motion recorded at a nearby site. A new 3-D

pushover analysis procedure is proposed in this paper and the results will be compared with those of a nonlinear dynamic analysis.Results are presented that show the importance of the 3-D interaction effects in the dynamic response of the stack. The results also confirm

that the stack could readily fail under the considered earthquake and are consistent with the debris pattern.� 2007 Elsevier B.V. All rights reserved.

Keywords: Finite element; Chimney; Nonlinear 3-D pushover analysis; Failure

1. Introduction and objectives

This study is focused on a 115 m high reinforced concretechimney that collapsed during the 1999 Kocaeli earthquake.This earthquake caused great damage to inhabited structuresand the regional transportation system that has been well doc-umented. The coincident damage to industrial facilities didnot produce a high death toll, but the economic repercussionswere enormous. Furthermore, many of these facilities weredesigned and constructed to international standards and pro-vide information that is readily transferable to other developedcountries.

The reinforced concrete chimney shown near the center ofFig. 1 collapsed during the earthquake. The debris cut manylines, which fueled fires that shut down the refinery for months.

This structure is of particular interest because severalsimilar chimneys at the site survived the shock with only

∗ Corresponding author. Tel./fax: +1 310 745 1587.E-mail address: [email protected] (W. Huang).

0168-874X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2007.05.005

moderate damage. As shown in Fig. 1, the collapsed heaterstack is shown next to a similar structure that survived. Theparticular distinction of this chimney appears to be an unusuallylarger rectangular opening, located about 1

3 of the height abovethe base, which appeared to be the region of collapse initiation.The remnants of the stack are shown in Fig. 2.

The overall objectives of the study are four-fold:

(1) To evaluate the original design of the collapsed chimney,known as the Tüpras stack, using current analysis tech-niques.

(2) To evaluate the design of a similar size chimney represen-tative of US practice.

(3) To explain why the single stack in question did indeedcollapse while several similar structures in the same vicin-ity survived with minimal damage through the use of ad-vanced seismic evaluation tools.

(4) To extend the pushover analysis procedure for chimneystructures by taking into account the higher modes and the3-D interaction effects.

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880 W. Huang, P.L. Gould / Finite Elements in Analysis and Design 43 (2007) 879–887

Fig. 1. Heater stacks before earthquake.

Fig. 2. Heater stacks after earthquake.

T = 1.0 s

T = 2.0 s

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Mode1 Withhole 0 Deg.

UBC-97

YPTx Mean+1 STDV


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Fig. 3. Mode 1 capacity curves vs. YPTx demand curves.

The input for the study is a single strong motion recordrecorded at a site near the failed stack, named theYPT (A Petro-Chemical Plant in Körfez, Turkey) record. No other nearbyrecord is available, so this record is adopted as the input motionfor the analysis of the Tüpras stack. For the YPT longitudinalspectrum (YPTx) and transverse spectrum (YPTy), along witha modern design code spectrum, UBC 97 (1997 Uniform Build-ing Code) spectrum, several demand curves are plotted in thespectral acceleration vs. spectral displacement domain (ADRS)as shown in Figs. 3 and 4. The ADRS spectrum is convertedfrom the ordinary response spectrum using the relationship be-tween acceleration, displacement and period from simple har-monic motion in order to permit comparisons with the capacitycurve. Note that any radial line from the origin represents aconstant period on all of the intersected demand curves. Alsoincluded on the figures are the first mode pushover capacitycurves for different loading conditions, among which a profileoriented 90◦ to the opening is the critical pushover case. Theconstruction of these curves is explained in later paragraphs.

The damping ratios for intact reinforced concrete chimneystructures are fairly low, typically in the range of 2–5%. Fol-lowing the usual practice for reinforced concrete, 5% dampingwas used.

The first two objectives were addressed earlier [1–3] by a re-sponse spectrum analysis based on the unsmoothed YPT recordas well as the UBC 97 design spectrum. The comparative ca-pacities of the Tüpras stack and a comparably sized US stackwere based on their respective structural designs, which aresomewhat different in approach due to changes in practice overthe years. Elevations of the stacks are shown in Fig. 5.

For the third objective, a contemporary nonlinear techniquewas applied. This method leads to a comparison of demandand capacity as well. In this case, the demand was based onthe YPT record and also a smoothed spectrum derived from

W. Huang, P.L. Gould / Finite Elements in Analysis and Design 43 (2007) 879–887 881

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YPTy Mean+1 STDV Mode1 Nohole Mode1 Withhole 0 Deg.


T = 1.0 s

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20 40 60 80

Fig. 4. Mode 1 capacity curves vs. YPTy demand curves.

Fig. 5. Chimney elevations.

a statistical earthquake simulation, so as not to overemphasizethe local peaks and valleys [4]. The capacities were providedby the pushover curves shown in Figs. 3 and 4. This approachattempted to distinguish the differences in the deformation ca-pacity of a chimney with a large opening hole, as opposed toa chimney without such an opening. Also, the direction of thepush, either perpendicular to the plane of the opening or 90◦to that direction, was studied.

Pushover analysis is a nonlinear static procedure (NSP) inwhich a lateral load pattern is applied to the structure and thenincrementally increased until the structure reaches the targetdisplacement or collapses. Due to its conceptual simplicity andcomputational attractiveness as compared to nonlinear dynamicanalysis, pushover analysis has been gaining popularity as atool for seismic design and performance evaluation of struc-tures. However, it has been shown by many researchers [5–7]


that despite its efficiency and applicability, it also exhibits sig-nificant limitations when higher modes and 3-D interaction ef-fects play important roles in the dynamic response of the struc-ture. For a chimney structure like the Tüpras stack, this may bethe case. How to take those effects into account and developa simple but improved pushover procedure for seismic designand evaluation of a chimney structure is the fourth objective ofthe study and the focus of this paper. Following a summary ofthe 2-D pushover analysis study considering the higher modeeffects, a new 3-D pushover analysis procedure is introducedin this paper and some comparisons with nonlinear dynamicanalysis results are presented.

The stacks were analyzed using the finite element (FE) anal-ysis program ABAQUS. Shell elements having nonlinear con-crete and reinforcement properties have been used for both 2-D and 3-D pushover analyses. A sensitivity study was carriedout in order to obtain the appropriate mesh size for the modelsand a more refined mesh was used around the opening area.The same FE mesh was retained for the nonlinear dynamicanalysis (NDP). In general, ABAQUS provides advanced non-linear concrete and reinforcement modeling features comparedwith some other commercial FE programs. A more detaileddescription of the FE modeling of the stacks can be foundelsewhere [9].

2. 2-D pushover analysis

Nonlinear dynamic analysis has traditionally been regardedas the most precise method for estimating the response of astructure to a particular seismic event. Since the nonlinear prop-erties of the elements of the structure are considered whenmodeling the structure, it should represent the behavior of thestructure at each time during and after the onset of the groundmotion. Therefore this method has been accepted as the bench-mark for seismic structural analysis. However, because of thefollowing reasons, it is generally used only for the evaluationof special, complicated, and important structures: (1) materialmodeling can be complex. For example, the hysteretic behaviormodeling of the concrete components under earthquake load-ing is complicated and involves stiffness degradation, strengthdeterioration, etc.; (2) selection of an appropriate ground mo-tion or suite of ground motions is vital to the analysis. Thenonlinear response of the structure is extremely sensitive to thefrequency contents and phasing properties of the earthquakemotions; (3) the analysis is time consuming and costly sinceit must be performed for small intervals of time to accuratelyfollow the nonlinear response of the structure. Additionally, anumber of runs may be required to get decisive results for asuite of earthquake inputs.

The NSP in ATC-40 [8] is based on the capacity spectrummethod, which has become a standard technique for the eval-uation of existing structures due to its utility in predicting in-elastic dynamic performance. This method requires that boththe capacity of the structure, derived from a pushover analysis,and the demand, given by a selected ground motion spectrum,be compared in the spectral acceleration vs. spectral displace-

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Fig. 6. Tüpras stack pushover load patterns.

ment (ADRS) domain. The NSP results for the Tüpras stackwere directed toward a plausible quantitative explanation of theunique failure of the stack.

In the 2-D pushover analysis studies [3,9], the first step wasto evaluate the collapsed Tüpras stack and identify the possi-ble failure mode during the earthquake. The second step wasto apply the different lateral load patterns commonly consid-ered, including first mode distribution, multimode combination(SRSS) distribution, uniform distribution, triangular distribu-tion, and equivalent lateral force (ELF) distribution as well asthe recently developed modal pushover procedure (MPA) [10].In the MPA procedure, the seismic response of the structuredue to each mode is determined by pushing the structure toits modal target displacement using an invariant modal lateralforce distribution. Then, using an appropriate modal combina-tion rule, e.g. the SRSS rule, the overall peak responses of thestructure is estimated by combining the peak response for eachmode. All of these pushover load patterns are shown in Fig. 6.The analysis then compares the corresponding seismic demandto the “exact” nonlinear dynamic response history analysis (NLRHA) results in order to evaluate the capabilities of the variouspatterns to represent higher mode effects and possible redistri-bution of inertial forces in a structure due to nonlinearity underearthquake load. Since the stack is asymmetric due to the largeopening and since the real seismic loading condition for thestructure is a 3-D earthquake input, the 3-D interaction effectsmay not be negligible. Therefore, the third step is to extend the2-D pushover analysis to a 3-D procedure to account for the 3-Dinteraction effects. This step is addressed in the next sections.

The results for the first two steps of the pushover analysisstated above can be found elsewhere [3,9]. Some of the con-clusions are summarized as follows:

1. A 2-D pushover analysis produced a variety of demand–capacity comparisons, which revealed that the strength ofthe stack, including the effect of the opening, at best barely


met the demand and in some cases fell short. Phase-by-phase cracking patterns for several models under differentloading directions showed clearly that the correspondingdevelopment of cracking patterns changed the capacitiessignificantly. The results provided a plausible explanationfor the failure in the direction 90◦ from the opening, whichis also consistent with the observed debris field.

2. For the 2-D pushover analysis of the model without theopening, the uniform distribution underestimated the targetdisplacement by up to around 40%. The first mode, ELF,and triangle distributions gave fair predictions, with errorsaround 10–20%. Taking into account the higher mode ef-fects, the SRSS distribution showed the ability to predictthe target displacement within about 10%, and the MPAprocedure provided the best estimation of the target dis-placements, for most cases within 10%. As for the peakdeflections, once again, the MPA procedure and SRSS dis-tribution provided better estimations than the other distri-butions.

3. For the 2-D pushover analysis of the model with the open-ing pushed 90◦ to the opening direction shown in Fig. 5,the SRSS distribution gave better predictions for failure dis-placements than other distributions, with around 10% error.

3. 3-D pushover analysis

3.1. A new 3-D pushover analysis procedure

In traditional pushover analysis, only the distribution offorces equivalent to those produced by earthquake action inone direction is applied to the structure to represent the iner-tia forces experienced during the earthquake. This procedurehas provided insightful results for symmetric structures. Butfor asymmetric structures, pushover analysis considering twodirectional earthquake inputs may be more appropriate, sincethe structure has different dynamic properties in each direc-tion. For the Tüpras stack, with the large opening at the 30 mlevel, the stack would have undergone different lateral motionssimultaneously and the 3-D interaction effects may not benegligible. There is very little research focusing on improvingthe pushover analysis by considering 3-D interaction effects[11,12], so the need for developing improved pushover analysisprocedures considering 3-D interaction effects for asymmetricstructures is evident.

In this study, a new 3-D pushover analysis method is pro-posed to extend the traditional 2-D pushover procedure for theanalysis of the asymmetric Tüpras stack. The validity of theproposed method will be assessed by comparing the resultswith those from an “exact” 3-D step-by-step nonlinear dynamicanalysis. The basic procedure is as follows:

1. Carry out a 3-D modal analysis using a FE model with theinitial geometry and material properties. Obtain the naturalfrequencies and fundamental modes for each direction.

2. Now, two types of lateral load patterns may be selected basedon the patterns shown in Fig. 6. One type is a fundamental

Fig. 7. 3-D pushover load pattern.

mode, usually Mode 1, and the other type may be one of thepatterns in Fig. 6 other than Mode 1.

3. For a lateral load pattern other than the fundamental modepatterns, apply the lateral forces to the structure, and performthe pushover analysis for each direction. Plot the pushovercurves in the spectral displacement vs. spectral accelerationdomain (ADRS). The equivalent SDF (single degree of free-dom) period for the lateral load pattern in each direction isthen taken as the initial secant for the pushover curve beforeyielding.

4. For each direction, given the fundamental frequencies for thefundamental modes and equivalent SDF system frequenciesfor the other load patterns, locate the corresponding spec-tral acceleration values from the response spectrum in eachdirection. (In this case, the longitudinal and transverse di-rections of the YPT spectrum.)

5. Apply two directional lateral forces for each load patternto the structure as illustrated in Fig. 7, proportional to thespectral acceleration values obtained from Step 4.

6. For each load pattern, perform the 3-D pushover analysisusing the lateral load forces described in Step 5, and plot thecapacity curve for each direction.

7. Compare the capacity curves with the smoothed mean de-mand curves of the spectra for each direction to obtain thetarget displacement of the structure for the various load pat-terns.

8. Determine the response over the height of the structure usingthe 3-D pushover analysis results for the selected patterns atthe respective target displacements.

The validity of the proposed method will be assessed bycomparing the results with a 3-D nonlinear dynamic analysisof the stack.


Table 1Target displacements for the model without an opening

Method 3-D pushover 2-D pushover

Disp. (m) Error (%) Disp. (m) Error (%)

Mode 1 0.522 −13.1 0.498 −17.1Uniform 0.416 −30.8 0.316 −47.4ELF 0.680 13.1 0.492 −18.1Triangle 0.701 16.6 0.476 −20.8SRSS 0.662 10.1 0.629 4.7MPA 0.565 −6.0 0.528 −12.2NL RHA 0.601 0 – –

3.2. 3-D pushover analysis results

First, the 3-D pushover procedure is applied to the modelwithout an opening and compared to 3-D nonlinear dynamicanalysis with two directional inputs. Then the procedure is ex-tended to predict the failure of the model with the opening. Thefailure analysis for the model with the opening is carried outby 3-D nonlinear dynamic analysis as well. Once again, therecords used in the nonlinear dynamic analysis are from a suiteof simulated earthquakes based on the YPT record [4].

3.2.1. Model without an opening3.2.1.1. Target displacement Similar to the traditional 2-Dpushover analysis, target displacements are calculated by theproposed 3-D pushover analysis based on different lateral loadpatterns as well as the MPA procedure. These target displace-ments using the 3-D and 2-D pushover analysis as given inTable 1 are the magnitudes of the displacement of the stack atthe top. As shown in the table, 3-D pushover analysis resultsare based on the YPT earthquake input in both directions while2-D results are based on YPTy earthquake input in one direc-tion. The 3-D results are calculated by combining the targetdisplacements for each direction. The errors are obtained bycomparison to the NL RHA results.

As seen in Table 1, for the 3-D pushover analysis resultsthe error from the uniform distribution is the largest, whilethe Mode 1 distribution, ELF distribution, and triangle distri-bution errors are less. Taking into account the higher modeeffects, the SRSS distribution gives a good prediction for thetarget displacement and the error from the MPA procedureis less than 10%. In general, the new 3-D pushover anal-ysis provides better estimations for the target displacementwhen compared to 2-D pushover analysis since the pushoverload patterns are simulating the earthquake inputs in bothdirections.

3.2.1.2. Peak deflections The peak deflections over the heightof the stack for each direction are obtained from the structuraldeflection at the target displacement in that direction, as deter-mined by the capacity-demand comparison. The mean valuesof the peak response from the nonlinear dynamic analysis re-sults based on a suite of simulated records are compared withthe pushover results.

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Fig. 8. 3-D peak deflections for the model without an opening.

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Fig. 9. 3-D peak deflection errors for the model without an opening.

The differences between the estimated values using 3-Dpushover analysis and exact values from 3-D nonlinear dy-namic analysis are shown in Figs. 8 and 9.

As seen in Fig. 8, the MPA procedure and SRSS distributiongive better estimates of the peak deflections during the earth-quake than the others, while the uniform distribution underesti-mates the total response up to 30%. In general, the 3-D pushoveranalyses predict the target displacements and the peak deflec-tions of the structure fairly well, as compared to 3-D nonlineardynamic analysis. The MPA procedure and SRSS distributionusing 3-D pushover analysis provide good estimates, especiallyfor the dynamic response of the structure under two directionalearthquake inputs that are not easily predicted using traditional2-D pushover analysis. The ability of the procedure to pre-dict the failure of the stack will be discussed in the followingsection.


Table 23-D failure displacements for the model with the opening

Pattern YPT mean

Failure Disp. (m) Error (%)

Mode 1 0.667 27.0Uniform 0.745 41.8ELF 0.646 23.1Triangle 0.645 22.8SRSS 0.597 13.8NL RHA 0.525 0

3.2.2. Model with the openingSince the stack failed in an earthquake having different lat-

eral loading components acting simultaneously, it is appropri-ate to analyze the structure in multiple directions. The failuredisplacement and the cracking pattern recorded at the failurepoint from 3-D nonlinear dynamic analysis will be used to val-idate the 3-D pushover procedure.

The 3-D pushover procedure was applied to the model withthe opening, where the lateral load patterns in the two directionsare proportional to the response spectrum values based on theequivalent SDF system. Also, a 3-D nonlinear dynamic analysiswas carried out using two directional inputs, a suite of YPTlongitudinal records in the direction 0◦ to the opening, anda suite of YPT transverse records in the direction 90◦ to theopening, based on the orientation of the opening from the sitereference.

The failure displacement at the top and the cracking patternsare compared between the 3-D pushover analysis predictionand the 3-D nonlinear dynamic analysis results. Because thepushover analysis results for the three fundamental modes can-not be combined for failure analysis since the failure displace-ment is determined for each mode by pushing the structure toits capacity in that mode, the MPA procedure is not consideredhere.

3.2.2.1. Failure displacement Incremental lateral loads in twodirections for different loading patterns were applied on thestructure until failure. The magnitudes of the top displacementat the point of failure predicted by the different pushover pat-terns are shown in Table 2. The errors relative to the 3-D NLRHA are listed as well.

As shown in Table 2, where results taking into account highermode effects in both directions are summarized, the SRSS dis-tribution provides the best prediction, with less than 14% error.

3.2.2.2. Cracking pattern In Figs. 10 and 11, the cracking pat-terns for 3-D pushover analysis using different load patternsare plotted at the failure. On the coarse plots of Figs. 10 and11, vertical and diagonal cracks superimposed on the horizon-tal flexural cracks cause the dense crack regions. The failurecracking pattern for the nonlinear dynamic result is shown asNL RHA in Fig. 11.

In the failure cracking pattern from nonlinear dynamic analy-sis, there are more long critical shear cracks around the openingarea than there are flexural cracks along the height. This find-

Fig. 10. 3D cracking patterns for failure of the model with the opening.

ing confirms the initial prediction by 2-D pushover analysis;the critical shear cracks developed at the opening area causedthe stack to fail during that earthquake. The cracking patternsfrom 3-D pushover analysis show the existence of the criticalshear cracks around the top left and bottom right corner of theopening. Considering the limitation of monotonic loading, wewould expect a symmetric cracking pattern for the other direc-tion, so the overall cracking patterns around the opening undercyclic loading match well with the nonlinear dynamic analysisresults. Even though all lateral patterns give good estimationsat the opening level, the SRSS distribution, by taking into ac-count the higher mode contribution, better predicts the cracksdeveloped from the opening level to about the 65 m level.

4. Conclusions

Using a demand–capacity comparison, a nonlinear staticpushover analysis was used to investigate the collapsed Tüprasstack. The demand was represented by an acceleration–displacement response spectrum based on the YPT record mo-tion as well as some smoothed adaptations typical of designspectra. The capacities were calculated from pushover curvesusing a nonlinear reinforced concrete FE analysis. A new3-D pushover analysis procedure was proposed and the results


Fig. 11. 3D cracking patterns for failure of the model with the opening(continued).

were compared with those from a nonlinear dynamic analysis.Based on these pushover analyses, some of the conclusions aresummarized as follows:

1. A new 3-D pushover analysis procedure was proposed andapplied to models of chimneys with and without an open-ing. Various lateral load patterns were considered. For thetarget displacement of the model without the opening, theerror from the uniform distribution was the largest, whilethe Mode 1 distribution, ELF distribution, and triangle dis-tribution provided somewhat better estimates. The SRSSdistribution gave a good prediction, with an error around10% and the error from the MPA procedure was even lessthan 10%. As to the peak deflections, the MPA procedureand SRSS distribution provided the best estimates, whilethe uniform distribution underestimated the total responseby up to 30%. The Mode 1 distribution, ELF distribution,and triangle distribution gave similar estimates. Comparedto a 2-D pushover analysis, the new 3-D pushover analysisprocedure provides a better estimation for target displace-ments.

2. For the 3-D pushover analysis on the model with the open-ing, the failure displacements predicted using different lat-

eral patterns were in an acceptable range. The SRSS distri-bution resulted in the lowest error, between 10% and 20%.All of the lateral load patterns successfully captured theshear cracks developed around the opening, along with flex-ural cracks.

3. From the failure cracking pattern for the 3-D nonlinear dy-namic analysis, there were more long critical shear cracksaround the opening area than there were the flexural cracksalong the height. This confirmed the initial prediction by 2-D pushover analysis that the critical shear cracking aroundthe opening area, along with the concentrated flexural crack-ing, was prominent in the failure. The 3-D nonlinear dy-namic analysis results confirmed that the Tüpras stack couldnot survive the YPT earthquake inputs under both direc-tions.

Acknowledgments

The authors wish to thank the United States National Sci-ence Foundation for the support of this study under GrantCMS-0084737. Additionally the generous support of our of-ficial Turkish collaborator, Prof. Dr. Semih S. Tezcan and hiscolleague Professor Sami A. Kilic is appreciated. The cooper-ation of the designer of the Tüpras stack, Mr. T. Tunca was in-dispensable to our effort and the consent of General ManagerH. Danis and other officials of the refinery to make the photos,calculations and drawings available to us were generous.

References

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[2] P.L. Gould, W. Huang, G.S. Johnson, Nonlinear analysis of a collapsedstack, in: Proceedings of the 13th World Conference on EarthquakeEngineering, Vancouver, British Columbia, Canada, August 2004.

[3] W. Huang, P.L. Gould, R. Martinez, G.S. Johnson, Nonlinear analysis ofa collapsed reinforced concrete chimney, Earthquake Eng. Struct. Dyn.33 (2004) 485–498.

[4] G.I. Schuëller, H.J. Pradlwarter, Estimation of the evolutionary processof strong nonstationary earthquake records, in: Proceedings of the 9thWorld Conference on Earthquake Engineering, Tokyo-Kyoto, Japan,vol. 2, 1988, pp. 777–782.

[5] S. Kim, E. D’ Amore, Push-over analysis procedure in earthquakeengineering, Earthquake Spectra 15 (3) (1999) 427–434.

[6] H. Krawinkler, G.D.P.K. Seneviratna, Pros and cons of a pushoveranalysis of seismic performance evaluation, J. Eng. Struct. 20 (4–6)(1998) 452–464.

[7] R.S. Lawson, V. Vance, H. Krawinkler, Nonlinear static pushoveranalysis—Why, when and how?, in: Proceedings of the 5th USConference on Earthquake Engineering, Chicago, IL, vol. 1, 1994,pp. 283–292.

[8] ATC-40, Seismic Analysis and Retrofit of Concrete Buildings, vol. 1,Applied Technology Council, Redwood City, CA, November 1996.

[9] W. Huang, Nonlinear analysis of a collapsed reinforced concrete chimney,Doctoral Thesis, Washington University, Saint Louis, MO, May 2005.

[10] A.K. Chopra, R.K. Goel, A modal pushover analysis procedure forestimating seismic demands for buildings, Earthquake Eng. Struct. Dyn.31 (2002) 561–582.


[11] A.G. Ayala, E.A. Travera, A new approach for the evaluation of theseismic performance of asymmetric buildings, in: Proceedings of the 7thUS National Conference on Earthquake Engineering, Boston, MA, July2002 (CD-ROM).

[12] A.S. Moghadam, W.K. Tso, Pushover analysis for asymmetricalmultistory buildings, in: Proceedings of the 6th US National Conferenceon Earthquake Engineering, Seattle, EERI, Oakland, 1998.

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