Progressive collapse of framed structures:...

Scientia Iranica A (2014) 21(2), 329{338

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Research Note

Progressive collapse of framed structures: Suggestionsfor robustness assessment

H.R. Tavakolia;� and F. Kiakojourib

a. Department of Earthquake Engineering, Babol University of Technology, Babol, P.O. Box 484, Iran.b. Department of Structural Engineering, Islamic Azad University, Takestan Branch, Iran.

Received 14 January 2013; received in revised form 6 July 2013; accepted 7 September 2013

KEYWORDSProgressive collapse;Robustness;Column loss;Alternative load path;Dynamic nonlinearanalysis.

Abstract. The term \progressive collapse" has been used to describe the spread of localfailure in a manner analogous to a chain reaction that leads to partial or total collapse of astructure. Robustness is de�ned as a fundamental property of structural systems to preventdamage propagation and to mitigate the potential of progressive collapse. In this paper, theprogressive collapse capacity of steel moment-resisting frames was �rst investigated usingthe alternative load path method, then suggestions are made for assessment of structuralrobustness, and the robustness of frames is quanti�ed. According to the results, therobustness and progressive collapse potential of the frames varied signi�cantly, dependingon the location of the initial local failure and number of building stories.c 2014 Sharif University of Technology. All rights reserved.

1. Introduction

The term \progressive collapse" has been used todescribe the spread of an initial local failure in amanner analogous to a chain reaction that leads topartial or total collapse of a building. The underlyingcharacteristic of progressive collapse is that the �nalstate of failure is disproportionately greater than theinitial failure [1]. Progressive collapse �rst attractedthe attention of researchers from the partial failureof Ronan Point, a 22-story apartment in London,UK, in 1968. After the event of 11th September,2001, more researchers around the world have refocusedon the causes of progressive collapse. After such adisaster, concepts of progressive collapse and structuralrobustness have been re ected in new guidelines andcodes [2,3]. Robustness is the ability of a structureto resist damage without premature or brittle failure,due to events like impact, blast, �re or consequences

*. Corresponding author. Tel.: 0111-3231707, 0911-1214285E-mail addresses: [email protected] (H.R. Tavakoli);[email protected] (F. Kiakojouri)

of human error, because of its vigorous strength andtoughness [2]. According to this de�nition, robustnessis a structural property, de�ned as the insensitivity of astructure to local failure. Parameters such as ductility,redundancy, continuity and energy absorption havean in uence on progressive collapse resistance andare listed as factors that in uence the robustness ofstructures [4].

Among di�erent approaches to analyzing anddesigning buildings against progressive collapse, theguidelines recommend the alternative load pathmethod. In this method, the building is analyzedand designed, such that, if one structural elementfails, alternative paths are available for the loads and,therefore, collapse does not occur. The alternative loadpath method is a threat-independent methodology.This method does not consider the type of triggeringevent, but, rather, considers the structural responseafter the initial local failure.

Most of the published progressive collapse analy-ses are based on the alternative load path method withsudden column removal, as recommended in previouslymentioned guidelines [2,3]. In most of the published

330 H.R. Tavakoli and F. Kiakojouri/Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 329{338

numerical studies of progressive collapse, commercialand open source nonlinear FEA packages are used, suchas Abaqus [5-7], SAP2000 [8-10] and Opensees [11-13]. Most considerations are con�ned to 2D framesusing a beam element. Detailed 3D numerical studiesusing a shell element are rare, due to the requiredcomputational time and the poor pre-processing abilityof most general purpose FEA packages. An exampleof complete 3D �nite element modeling by Ls-Dyna isprovided in [14]. As mentioned above, all these papersare based on numerical study, but, in recent years, someparametric [15,16] or experimental [17-19] studies alsohave been presented in the literature.

The potential abnormal loads that can triggerprogressive collapse are categorized as: aircraft impact,design error, construction error, �re, gas explosions,accidental overload, hazardous materials, vehicularcollision, and bomb explosions, etc. [1]. As mentionedbefore, most of the published papers use a threat-independent methodology, but, in recent years, moreresearch has been focused on progressive collapse due tocertain triggering events, such as �re-induced progres-sive collapse [20-22], seismic progressive collapse [23-25], blast-induced progressive collapse [7,26,27] andimpact-induced progressive collapse [28,29].

To date, there is no uniform theory of robustnessassessment. A large variety of di�erent approachesfor quanti�cation of structural robustness has beensuggested. These approaches include both determin-istically de�ned and probabilistically de�ned. Mostof them are based on assuming damage and eithercomparing the property of damaged and undamagedstructures [30,31] or examining the response of a struc-ture after initial local failure [32]. An exception to thementioned approaches is presented in [33]. Starossekclassi�ed these approaches into three basic categories:Sti�ness-based, damage-based and energy-based mea-sures of robustness, and compared the applicability ofthese approaches to di�erent structures and di�erenttypes of progressive collapse [4].

In this paper, the progressive collapse capacityof steel moment resisting frames is �rst investigatedusing the alternate load path method. The structuralresponses of models under sudden loss of columns underdi�erent scenarios of column removal were studied.Since progressive collapse is inherently a nonlinearand dynamic event, nonlinear dynamic analysis ismore desirable when investigating progressive collapsepotential and the collapse mechanism of frames. Ac-cordingly, in this study, the nonlinear dynamic methodwas used for progressive collapse analysis. The lineardynamic analysis method was used for comparison.Then, suggestions are made for assessment of therobustness of steel frames. Using these approaches,structural robustness is quanti�ed, and the results arecompared and contrasted.

2. Finite element model

In this study �nite element analysis is performedusing the general purpose �nite element package,Abaqus/Explicit, version 6.10. An explicit methodsolves dynamic response problems using an explicitdirect-integration procedure. In an implicit dynamicanalysis, the integration operator matrix must beinverted and a set of nonlinear equilibrium equationsmust be solved at each time increment. On the otherhand, in an explicit dynamic analysis, problems aresolved incrementally, and displacements are calculatedin terms of quantities that are known at the beginningof an increment. There is no need to form or invertsti�ness matrices, which means that each incrementis relatively inexpensive compared to the incrementsin an implicit integration. Therefore, the explicitmethod is very robust and great for highly nonlinearproblems and short-term events, such as blast, impactand collapse [34].

2.1. Analytical modelThe model structures are the 3, 5, and 10-story steelmoment resisting frames, the oor height is 3.2 m andthe span length is 5m, as shown in Figure 1. Box and Isections are used for columns and beams, respectively.More input data can be found in [7]. Connectionsbetween beams and columns are perfectly rigid, andthe bottoms of the �rst story columns are �xed. Incolumn removal analysis, beam to beam continuity isassumed to be maintained across a removed column,according to [2]. The structures were assumed to belocated in a high seismic zone, and the steel momentframes were designed to carry gravity and seismicloads. The seismic design was performed using anequivalent static method, according to the IranianBuilding Code [35].

In this study, the beam element in the Abaquselement library was used to model the beams andcolumns. The selection of the type of element to beused is based on the fact that the study considersthe global response of the structures; therefore, beamtheory is su�cient. All beam elements in Abaqus

Figure 1. Elevation of model structure and columnremoval cases.

H.R. Tavakoli and F. Kiakojouri/Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 329{338 331

are beam-column elements, and means that they allowaxial, bending, and torsional deformation [34]. How-ever, torsion is not applicable to the in-plane behaviorof the 2D frames. The beam properties are inputby de�ning the cross-section from the Abaqus cross-section library. At each increment of the analysis,the stress over the cross-section of the elements isnumerically integrated to de�ne the beams response asthe analysis proceeds [34]. The in uence of mesh sizehas been studied, and it is su�ciently �ne to ensurethe accuracy of the model structure. The analyseswere conducted with 5% mass proportional damping,which is common for analysis of structures subjectedto extreme loads [36].

3D and slab e�ects were not involved in this study.As found by Qian and Li [37], 3D and slab e�ectsare important in progressive collapse analysis, but, inthis paper, these e�ects are ignored. Also, the speedof column removal will a�ect the dynamic response.Sudden column removal provides a larger structuralresponse than gradual column removal [38,39]. Thesee�ects are not considered in this study and columns areremoved suddenly.

2.2. Material propertyThe adopted material properties were: Young's mod-ulus, E = 210 GPa, Poisson coe�cient, v = 0:3,and density � = 7850 kg/m3. The static yield stresswas fy = 240 MPa. The plastic property is shownin Figure 2. Abaqus provides the classical metalplasticity; the elastic part being de�ned by Young'smodulus and Poisson's ratio. The plastic part is de�nedas the true stress and logarithmic plastic strain. Duringthe analysis, Abaqus calculates the values of yieldstress from the current values of plastic strain. Itapproximates the stress-strain behavior of the materialwith a series of straight lines joining the given datapoints to simulate the actual material behavior. The�rst piece of data given de�nes the initial yield stress ofthe material and, therefore, should have a plastic strainvalue of zero. In this study, bilinear curves were used.The material will behave as a linear elastic materialup to the yield stress of the material. After this stage,

Figure 2. Plastic property.

Figure 3. Applied loads for column removal analysis.

it goes into the strain hardening stage until reachingultimate stress [34].

2.3. Applied loads for dynamic columnremoval analysis

For nonlinear dynamic analysis, load DL+0.25LL wasuniformly applied to the entire span of frame as avertical load [2]. To carry out dynamic analysis, theaxial force acting on a column is determined before itsremoval. Then, the column is removed and replacedby the concentrated load equivalent of its forces. Tosimulate the phenomenon of progressive collapse, themember forces are removed after a certain time elapses,as shown in Figure 3, where the variables, R, denotethe reaction forces and G is the vertical gravity load.In this paper, the forces were increased linearly for �veseconds until they reached their maximum amounts.Then, they were kept unchanged for two seconds untilthe structure reached a stable condition, and theconcentrated forces were suddenly removed at sevenseconds to simulate the dynamic e�ect caused by thesudden removal of the column [36]. More informationabout dynamic column removal is presented in [39].Di�erent cases for column removal are presented inFigure 1.

3. Results and discussion

Nonlinear dynamic analysis is performed using the gen-eral purpose �nite element package, Abaqus/Explicit,version 6.10. In this paper, the words \displacement"and \response" are used to refer to the \verticaldisplacement of the column removal point".

The nonlinear analysis method is more sophis-ticated than the linear method in characterizing theresponse of a structure under extreme loading con-ditions. When this method is used, the codes allowless restrictive acceptance criteria. In this paper, thenonlinear dynamic method was performed for progres-sive collapse analysis, and the linear dynamic analysismethod was used for comparison.

3.1. Column removal analysisNonlinear dynamic time-history analyses were carriedout by removing the selected column, as shown in


Figure 4. Collapse mode of 3-story frame: a) Cornercolumn is removed; and b) second column is removed.

Figure 1. When the corner column in the �rst storyof a 3-story structure was suddenly removed, theentire corner of the building collapsed, as shown inFigure 4(a). When the second column in the �rst storywas suddenly removed, again, collapse occurred (seeFigure 4(b)). Collapse modes are drastically dependenton the location of removed columns. This is relatedto the a�ected members of the structure after columnremoval. Figure 5 shows the vertical displacementsof the model structures obtained from time-historyanalyses, when the �rst and second columns in the �rststory of a 3-story structure were removed.

In a 5-story structure, for case 1, when the cornercolumn in the �rst story (as shown in Figure 1),was suddenly removed, the node on the top of theremoved column vibrated and reached a peak verticaldisplacement of 98 mm in the nonlinear procedure and70 mm in the linear procedure. For case 2, whenthe second column in the �rst story was suddenly

removed, the node on the top of the removed columnvibrated and reached a peak vertical displacement of59 mm in the nonlinear procedure and 51 mm inthe linear procedure. From the comparison of case1 and case 2, it can be seen that the building ismore vulnerable to the removal of corner columns.The time history of the column removal point verticaldisplacement for the two mentioned cases is shown inFigure 6(a) and (b), respectively. It is obvious thatmaximum vertical displacements obtained by linearanalysis are meaningfully smaller than those obtainedby nonlinear analysis. It also can be observed that, incomparison with the linear analysis results, the resultsof nonlinear analysis vary signi�cantly, depending onmodel parameters such as the location of the removedcolumn and the number of building stories.

When a column at a higher story was removed,the vertical displacement of the column removal pointsigni�cantly increased, because less structural memberscontributed to energy absorption after column removal.In this analysis, when the corner column in the thirdstory of a 5-story structure was suddenly removed(case 3, 5-story structure), the node on the top of theremoved column vibrated and reached a peak verticaldisplacement of 186 mm in the nonlinear procedureand 96mm in the linear procedure. For case 4, whenthe second column in the third story was suddenlyremoved, the node on the top of the removed columnvibrated and reached a peak vertical displacement of81mm in the nonlinear procedure and 64 mm in thelinear procedure. This conclusion can be obtained forother higher stories; column removal at a higher levelwill induce larger vertical displacement than columnremoval in the �rst story. This conclusion is consistentwith the �ndings presented in [6]. Displacements of thecolumn removal point for cases 3 and 4 are shown inFigure 6(c) and (d), respectively.

It was also observed that as the number of storiesincreases, the displacement of the column removalpoint decreases, because more structural membersparticipate in resisting collapse and, therefore, moreload path is available. In a 10-story structure, for

Figure 5. Displacement time history in a 3-story structure: a) Case 1; and b) case 2.


Figure 6. Displacement time history in a 5-story structure: a) Case 1; b) case 2; c) case 3; and d) case 4.

case 1, when the corner column in the �rst story (asshown in Figure 1), were suddenly removed, the nodeon the top of the removed column vibrated and reacheda peak vertical displacement of 47 mm in the nonlinearprocedure and 45 mm in the linear procedure. Forcase 2, when the second column in the �rst storywas suddenly removed, the node on the top of theremoved column vibrated and reached a peak verticaldisplacement of 32 mm in the nonlinear procedure and31 mm in the linear procedure. It can be concludedthat the progressive collapse potential decreased as thenumber of stories increased. The �ndings obtained areconsistent with the �ndings presented in [36]. The timehistories of the column removal point vertical displace-ment for two mentioned cases are shown in Figure 7(a)and (b), respectively. It is obvious that maximumvertical displacements obtained by linear analysis aresmaller than those obtained by nonlinear analysis inall considered cases. But, it was also observed that thedi�erence in results obtained by the two approachesdecreased as the number of stories increased.

The overall results obtained in the previous struc-tures are also true in the case of a 10-story structure.That means that column removal at a higher levelwill induce larger vertical displacement than columnremoval in the �rst story, and that the building ismore vulnerable to the removal of corner columns. Itcan be concluded that, as long as an alternative loadpath is available in damaged structures, the aboveresults will be true. The time histories of the column

removal vertical displacement of cases 3-6 are presentedin Figure 7(c) to (f).

3.2. Robustness analysisRobustness indicates the overall performance of thedamaged structure after initial local failure. Theprogressive collapse of the building is more likely tooccur for lack of structural robustness. Although therobustness of structures in abnormal events such asexplosion and impact has become a worldwide researchtopic, there has been neither a uniform theory ofstructural robustness assessment nor a methodology forquanti�cation of robustness in the progressive collapsescenario [4].

The usefulness of the measure of robustness islinked to certain requirements. The measure shouldquantify the structure's robustness with one singlevalue. It should be possible to derive the measure fromthe property or response of the structure, and the inputdata must be quanti�able. The measure should bede�ned in as simple a manner as possible and applicableto any kind of structure, as far as possible [4].

In this section, three simple approaches are pro-posed for robustness assessment, and, using these ap-proaches, structural robustness is quanti�ed and resultsare compared. A simple measure of robustness fromexamining the sti�ness of the structure is presented inEq. (1):

Rs =Kd

Ki; (1)


Figure 7. Displacement time history in a 10-story structure: a) Case 1; b) case 2; c) case 3; d) case 4; e) case 5; and f)case 6.

where Rs is the sti�ness-based measure of robustnessand Kd and Ki are the sti�ness of the damaged andintact structure, respectively. In this paper, a simplefrequency approach was used for sti�ness assessment.Since, in the column removal scenario, the mass of theframes do not change considerably, changes in the fre-quency are due to changes in the sti�ness of the frames.Using Eq. (1), the robustness value is automatically inthe range of 0 and 1. In this formulation, the value of1 represents a complete robust structure, while value0, represents total lack of robustness.

Results show that the robustness value does notchange meaningfully when di�erent scenarios of columnremoval are considered. As discussed before, a 3-storystructure collapsed in both column removal scenarios.But, as shown in Figure 8(a), the robustness value is

examined very highly using Eq. (1). Furthermore,the most vulnerable case in a 5-story structure is case3, according to nonlinear dynamic column removalanalysis. But, as shown in Figure 8(b), less structuralrobustness is obtained in case 1. It can be concludedthat robustness indexes obtained by Eq. (1) do notcorrelate well with corresponding column removal sce-narios, and, therefore, this method is not suitable forrobustness assessment, at least in the current simpleform.

As mentioned before, structural responses ob-tained by linear analysis are smaller than those ob-tained by nonlinear analysis in all considered cases, but,it was also observed that the di�erence in results ob-tained by the two approaches decreased as the numberof stories increased. Using this idea, another simple


Figure 8. Sti�ness-based measure of robustness: a) A3-story structure; and b) a 5-story structure.

Figure 9. Robustness based on dynamic column removalresults.

formulation is proposed for robustness assessment. Asshown in Eq. (2), robustness is assumed to be theratio of maximum structural response in linear dynamicanalysis to maximum structural response in nonlineardynamic analysis:

Rd =max(dld)max(ddn)

; (2)

where Rd is robustness based on dynamic analysisand dld and dnd are structural responses in linear andnonlinear dynamic analyses, respectively. As shownin Figure 9, the robustness index obtained by thismethod does correlate well with the corresponding

column removal scenario. As expected, more struc-tural robustness is obtained in case 2 of the 10-storystructure, while less structural robustness is obtainedin the 3-story structure.

As a third method for estimating structural ro-bustness, the energy of the structure is consideredand robustness is assumed to be the ratio of di�erentenergies in the column loss scenario, as presented inEq. (3):

RE = 1� max(Ep)max(EI)

; (3)

where RE is the energy-based assessment of robust-ness and Ep and EI are plastic dissipation energyand internal energy in the column removal scenario,respectively. Internal energy was calculated using thefollowing de�nition:

EI =Z Z

�d"

!dv: (4)

Comparison of the total internal energy historyand the total plastic dissipation history is shown inFigure 10 for two di�erent cases.

This method focused on the amount of energythat is dissipated in a progressive collapse scenario.The ratio of this energy to total energy is a goodindex for robustness assessment. As shown in Fig-ure 11, the obtained results for structural robustness

Figure 10. Comparison of the total internal energyhistory and the total plastic dissipation energy: a) Case 1of a 5-story structure; and b) case 1 of a 10-storystructure.


Figure 11. Robustness based on energy of structure.

are compatible with existing expectations for the pro-gressive collapse potential. More value for robustnessis obtained in �rst story column removal cases in a 10-story structure. This method is also consistent withthe previous method.

It should be noticed that the energy-based ap-proach in its current form is only applicable to theredistribution-class of progressive collapse. When thestructure is subjected to pancake-type or domino-typecollapse, the rigid motion and impact of the structure'sparts or members have a signi�cant in uence on energyconversion in the collapse scenario. Therefore, kineticenergy should be included in the formulation.

4. Conclusions

In this paper, the progressive collapse capacity of steelmoment frames was �rst investigated using the alterna-tive load path method. A nonlinear dynamic methodwas performed for progressive collapse analysis, andthe linear dynamic method was used for comparison.Using these methods, the structural response of 3-,5- and 10-story steel moment resisting frames underthe sudden loss of columns for di�erent scenarios ofcolumn removal was assessed. Then, suggestions weremade for assessment of the robustness of steel frames,and structural robustness in di�erent column removalscenarios was quanti�ed. It could be concluded thatthe proposed approaches o�er the advantages of com-putational simplicity and practicality for robustnessassessment in framed structures.

The results of this study can be summarized asfollows:

� It was observed that as the number of story in-creases, the displacement of the column removalpoint decreases, because more structural membersparticipate in resisting progressive collapse. There-

fore, it can be concluded that the progressive col-lapse potential decreased as the number of storiesincreased.

� Potential for progressive collapse is highest when acorner column is suddenly removed, either in the�rst, or higher, story.

� Column removal at a higher level will induce largervertical displacement than a column removal inthe �rst story, because less structural memberscontributed to energy absorption when a column ata higher level was removed.

� It was observed that su�ciently tall buildings, de-signed according to seismic design speci�cations,have enough strength to resist progressive collapsedue to column removal.

� It is obvious that maximum vertical displacementsobtained by linear analysis are smaller than thoseobtained by nonlinear analysis in all consideredcases. But, it was also observed that the di�erencein results obtained by the two approaches decreasedas the number of stories increased or the initial localfailure location changed.

� Three approaches for structural robustness assess-ment were proposed in this paper. It can beconcluded the methods based on dynamic columnremoval analysis have a good capability for robust-ness assessment, especially when the energy of themodel is considered, while the methods based onstatic sti�ness are not suitable.

The common structures are usually modeled byeither the brace or shear wall or moment resistingframe. However, in this study, only the momentframe has been used for studying sudden column lossand, therefore, the results apply only to the steelmoment-resisting systems with almost the same height.However, some general conclusions may be applicableto other framed structures. It should be noted thatthe current methodology, presented in this paper forassessment of structural robustness due to column lossin steel moment resisting frames, can be easily extendedto include other steel framed structures.

Any measure of structural robustness should pro-vide a clear distinction between robust and non-robuststructures. Therefore, suggested approaches must benormalized or calibrated for each structural systemand each progressive collapse type. For this purpose,the structural analysis of more structures and on morecollapse scenarios is necessary.

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Biographies

Hamid Reza Tavakoli was born in 1978, in Nour,Iran. He obtained his MS and PhD degrees in Earth-quake Engineering, in 1996 and 2008, respectively, fromIIEES. He is currently Assistant Professor and Deanof the School of Civil Engineering at Babol Universityof Technology, Iran. His main research interestsinclude explosion/earthquake progressive collapse, per-formance based earthquake engineering, dynamic soil-structure interaction modeling and energy dissipationin composite materials.

Foad Kiakojouri received his MS degree in StructuralEngineering from the Islamic Azad University, Iran, in2012. His main research interests include progressivecollapse, blast e�ect on structures and sandwich struc-tures. Currently, he is working as a civil engineer.

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Progressive collapse of framed structures:...

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