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Engineering Structures 33 (2011) 3341–3350

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Toward an economic design of reinforced concrete structures againstprogressive collapse

H.M. Salem a, A.K. El-Fouly b,∗, H.S. Tagel-Din b

a Department of Structural Engineering, Cairo University, Giza, Egyptb Applied Science Int., 2012 TW Alexander Drive, Durham, NC 27709, USA

a r t i c l e i n f o

Article history:Received 22 December 2009Received in revised form21 June 2011Accepted 22 June 2011Available online 28 July 2011

Keywords:Applied Element MethodNumerical analysisProgressive collapseGSAUFCASCE

a b s t r a c t

A three-dimensional discrete crack model based on the Applied Element Method is used to performeconomic design for reinforced concrete structures against progressive collapse. The model adopts fullynonlinear path-dependent constitutive models for concrete and reinforcing bars. The model applies adynamic solver in which post-failure behavior, element separation, falling and collision are predicted.First, the model is used to study the behavior of multi-story reinforced concrete buildings designed in atraditional manner according to the ACI 318-08 and subjected to accidental removal of one or two centralcolumns at the ground level. In an iterative way, the model is then used to investigate a safe designagainst progressive collapse for such extreme loading case. Based on the analytical results of the AEM,it can be concluded that the collapse of only one column would not lead to any progressive collapse ofthe studied reinforced concrete structure. However, the collapse of more than one column may lead to aprogressive collapse of a considerable part of it. It is concluded also that theAEMcould be successfully usedas an analytical tool to suggest economical designs that are safe against progressive collapse of reinforcedconcrete structures.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The spread of an initial local failure from element to element,eventually resulting in the collapse of an entire structureor a disproportionately large part of it has been known as‘‘progressive collapse’’ [1]. Progressive collapse of a structure takesplace when the structure has its loading pattern or boundaryconditions changed such that structural elements are loadedbeyond their ultimate capacities and fail. When any element fails,the remaining elements of the structure seek alternative load pathsto redistribute the load applied to it. As a result, other elementsmay fail, causing failure mechanism. It first drew the attention ofstructural engineers after the accidental collapse of the 22-storyRonan Point tower in Canning Town, UK on May 1968. The causeof the collapse was a human-error gas explosion that knocked outthe precast concrete panels near the 18th floor causing the floorsabove to collapse [2].

The Oklahoma City Murrah Fedral Building was not designedto resist progressive collapse. Half the building was collapsed in1995 due to the destruction of only one column by the blast.

∗ Corresponding author. Tel.: +1 9196454090; fax: +1 9196454085.E-mail address: [email protected] (A.K. El-Fouly).

Even though the structure met all code provisions, researchconducted after the disastrous event showed that alternativesto the building design, such as different reinforcement detailingand addition of some reinforcement, would have prevented thecollapse without a significant increase in construction costs [3].

Structural progressive collapse has been the focus of extensiveresearch during the past few years because of the increasing rateof victims resulting from natural disasters (e.g., earthquakes andhurricanes) or human-made disasters (e.g., bomb blasts, fires andvehicular impacts) [4–8]. Structural designers have traditionallyfocused on optimizing the cost of constructed facilities whilemeeting code requirements. Unfortunately, most of the structureshave been designed to resist gravity loads and lateral loadsresulting from wind or moderate earthquakes. The structuralbehavior of a constructed facility when subjected to loads beyondconventional design is not typically addressed.

1.1. Progressive collapse design in current codes and standards

The cause of the initiating damage to the primary load-bearingelement is unimportant; the resulting sudden changes to the struc-ture’s geometry and load-path are what matter. This means thatthe analysis is threat independent. Design codes, therefore, in-corporates a threat independent approach to progressive collapseanalysis.

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.06.020


3342 H.M. Salem et al. / Engineering Structures 33 (2011) 3341–3350

Fig. 1. Progressive collapse analysis by Kaewkulchai and Williamson [14].

The ACI Code 318-08 [9] rely on structural integrity require-ments to prevent progressive collapse of structures. This is basedon the assumption that improving redundancy and ductility bygood detailing of reinforcement can help to localize the damageand prevent it from propagating to other members and thus, theoverall stability of the structure can be satisfied.

The ASCE code for minimum design loads for buildings andother structures [1] specifies two alternative design approachesfor increasing resistance against progressive collapse: direct designand indirect design. The direct design approach basically considersresistance to progressive collapse, explicitly during the designprocess, by either the alternative load-path method or the specificlocal resistance method. The alternative load-path method allowslocal failure to occur but the progressive collapse mechanism isaverted or bridged over with alternate load paths to distributethe load from the missing member to other redundant membersso that the effect of the damage can be absorbed. The specificlocal resistance method does not allow local failure to occur byproviding sufficient strength on the ‘‘key’’ element to resist thefailure of a structural member. While the direct design approachoffers a more explicit design solution, the indirect design methodtakes a different methodology approach. It considers resistance toprogressive collapse, implicitly during the design process, throughthe provisions of minimum levels of strength, continuity, andductility. It is also stated that structures can be designed to sustainor minimize the occurrence of progressive collapse by limiting theeffects of a local collapse from spreading out to other membersexcept for special protective structures where extra protection isneeded.

General Services Administration (GSA) Guidelines [10] statesthat redundancy, detailing to provide structural integrity andductility, and capacity for resisting load reversal need to beconsidered in the design process tomake the structuremore robustand thus enhance its resistance against progressive collapse. Itstipulates an analysis procedure of removing vertical load-bearingelements to assess the potential of progressive collapse to occurin a structure. The guideline also gives requirement on maximumallowable collapse area that can occur if one vertical membercollapses.

Unified Facilities Criteria (UFC) [11] provides the detail forstructural design against progressive collapse. The indirect designapproach is applied with the tie forces method. Minimum tie forcecapacity requirements must be satisfied in the horizontal, vertical,or horizontal and vertical directions of the building. Similar toGSA requirements, the direct design approach is applied with thealternate-path method for buildings assigned to medium and highlevels of protection.

1.2. Progressive collapse analysis

Although the FEM is a robust and well established structuralanalysis method, it is not the optimum solution for the scopeof progressive collapse analysis. Many drawbacks are associatedwith the FEM progressive collapse analysis. The element damage,

separation, falling and collision with other elements are verydifficult. The GSA and UFC specifications tried to approximate andsimplify the progressive collapse analysis procedure so as to beable to be carried out by FEM; however, the process became socomplicated and need many trials of analysis. Hartmann et al. [12]showed that the computations associated with the simulation ofcollapses of real world structures based on conventional FEM arevery costly, and therefore followed another approach based onmultibody models. Researchers have used the FEM for progressivecollapse analysis of frame structures [13–15]. As an example,Fig. 1 shows a model for progressive collapse analysis of framesby Kaewkulchai and Williamson [14]. To represent the initialdeformations before column removal, forces equal and opposite tothe member forces of the failed column are applied to the nodeconnecting to the failed column. In this case, both the uniformload w and the applied forces (P, V , and M) are slowly applied tothe frame so that static deformations are obtained. At a time stepwhere all loads reach their peak value, the applied forces (P, V , andM) are removed to simulate an initiating collapse event. However,the analysis does not follow the behavior to the complete collapse.On the other hand, the Applied Element Method (AEM), adoptedin the current study, proved to be capable of incorporating thesefeatures and following the structure to its total collapse [16–24].

In the current study, the AEM is used to investigate the behaviorof RC structures under the removal of one or two central columnsof the ground floor. Suggestions to prevent the entire progressivecollapse in an economic way are numerically investigated.

2. Research significance

The significance of the current study arises from the need fora reliable analytical tool and a methodology that can be used fordesign of RC structures against progressive collapse. Such a toolshould be able to analyze a dynamic problem, to allow differentparts of a structure to separate and fall, to predict collision ofdifferent parts of a structure, and to account for the collision forcesbetween different parts. According to the available literature, allthese capabilities do not exist in the current numerical methodsbased on the FEM. On the other hand, the AEM was proved tohave such capabilities. In fact, progressive collapse analysis is abig challenge because, for trustable results and for simulating suchhigh nonlinearity, fully detailed nonlinear structural modeling iscrucial. The current study introduces three-dimensional numericalsimulation, based on the AEM, for progressive collapse analysisof reinforced concrete structures subjected to accidental removalof ground columns. Based on an iterative-analysis, an optimumdesign against progressive collapse is suggested.

3. The Applied Element Method (AEM)

The Applied Element Method [16,17,21–23] is an innovativemodeling method adopting the concept of discrete cracking. Inthe Applied Element Method (AEM), the structures are modeledas an assembly of relatively small elements, made by dividing the


H.M. Salem et al. / Engineering Structures 33 (2011) 3341–3350 3343

Fig. 2. Modeling of a structure with the AEM.

Shear spring in XShear spring in y Normal Spring

Contact normal spring

ContactShearsprings

(a) Corner-to-face or corner-to-ground contact. (b) Edge-to-edge contact.

Fig. 3. Different types of element contact.

structure virtually, as shown in Fig. 2. The elements are connectedtogether along their surfaces through a set of normal and shearsprings. Each adjacent element is assumed to be connected bynormal and shear springs located at contact points, which aredistributed on the element faces. Normal and shear springs areresponsible for transfer of normal and shear stresses. Springsrepresent stresses and deformations of a certain volume as shownin Fig. 2. Two adjacent elements can be separated once the springsconnecting them are ruptured.

Fully nonlinear path-dependent constitutive models areadopted in the AEM as shown in Fig. 2. For concrete in compres-sion, an elasto-plastic and fracture model is adopted [25]. Whenconcrete is subjected to tension, linear stress–strain relationshipis adopted till the cracking of concrete springs occurs, where thestresses drop to zero. Since the method adopts discrete crack ap-proach, the reinforcing bars are modeled as bare bars for the en-velope while the model of Ristic et al. [26] is used for the interiorloops.

The AEM is a stiffness-based method, in which an overall stiff-ness matrix is formulated and the equilibrium equations includ-ing each of stiffness, mass and damping matrices are nonlinearlysolved for the structural deformations (displacements and rota-tions). The solution for equilibrium equations is an implicit onethat adopts a dynamic step-by-step integration (Newmark-betatime integration procedure [27,28]. One of the main valuable fea-tures in the AEM is the automatic detection of element separa-tion and contact. Two neighboring elements can separate fromeach other if the matrix springs connecting them are ruptured. El-ements may automatically separate, re-contact again or contactother elements. Fig. 3 illustrates the different types of element con-tact, where contact springs are generated at contact points. In this

study, the Extreme Loading for Structures (ELS) software [29], thatis based on the AEM, is used.

TheApplied ElementMethod (AEM)was proven to be capable ofanalyzing the structure to its total collapse [16–24]. For validatingthe AEM ability to analyze the progressive collapse-resistanceof structures, the authors performed a structural analysis tothe experimental study carried out by Wei et al. [30]. In thisexperiment, Wei et al. [30] investigated progressive failure of areinforced concrete frame due to the loss of a lower story column.A four-bay and three-story one-third scale model representing asegment of a larger planar frame structure was tested. A constantvertical load of 109 kNwas applied to the top of themiddle columnby a servo-hydraulic actuator to simulate the gravity load of theupper floors and the failure of the middle column of the firststory was simulated by unloading a mechanical jacking system.Fig. 4 shows the layout of the experiment, while Fig. 5 showsconcrete dimensions and reinforcement details for both beamsand columns. Fig. 6 shows the ELS model for the structure, whereall the reinforcement details have been taken into consideration.Fig. 7 shows the force versus downward displacement of the lowermiddle column obtained from the experimental results comparedto the AEM analytical results, where, the AEM results are validated.Fig. 8 shows the collapse limit state of the frame predicted by AEMcompared to the experimental results, which also verifies the AEMresults.

4. Progressive collapse analysis of a multistory RC building

A five-story reinforced concrete building, with a total heightof 15 m and footprint dimensions of 27 × 17.6 m, is considered



Table 1Material properties used in analysis.

Material Young’s modulus Yield stress Tensile strength (MPa) Compressive strength (MPa)

Concrete 22,135 – 2 25Reinforcing bars 210,000 360 MPa 540 540

Fig. 4. One-third scale progressive collapse experiment for a reinforced concrete frame [30].

Fig. 5. Concrete dimensions and reinforcement details for beams and columns [30].

Fig. 6. ELS model for the experiment of Wei et al. [30].

in this study as shown in Fig. 9(a). Two different cases of loadingare considered. The first one incorporates a sudden removal ofone central column (C1), while the other incorporates a suddenremoval of two central columns (C1 and C2) as shown in Fig. 9(b).At first, a Linear Elastic Finite ElementAnalysis (LEFEM)was carriedout under gravity loads (dead loads and live loads) using theSAP 2000 package [31], and the internal forces of all members

Fig. 7. Middle column load versus unloading displacement of failed columnpredicted by AEM compared to the experimental results of Wei et al. [30].

were obtained. Thereafter, a complete design of the buildingusing ACI [9] was performed. Typical reinforcement details areshown in Fig. 10. With this design, the building was analyzedusing the AEM. The AEM model is shown in Fig. 11, where allthe reinforcement details of the slabs, beams and columns areexplicitly introduced into the analysis. The reinforcing bars arenot modeled as independent elements, however, they represent aportion of the connecting springs. The total number of elements inthe analysis was 10,885. The material properties used in analysisare shown in Table 1.

4.1. Case #1: removal of one central column

In this case, the buildingwas subjected to an accidental removalof one central column (C1) at the ground floor level. The analysisconsisted of two stages. The first stage was a static stage beforethe removal of the column, where the structure experiences theoriginal deformations due to gravity loads before removing anyelements. The second stage was a dynamic one in which thecolumn was suddenly removed. The time step of analysis was



(a) Experiment [30]. (b) AEM.

Fig. 8. Collapse limit state of the frame predicted by AEM compared to the experimental results of Wei et al. [30].

(a) Perspective view. (b) Removed columns.

Fig. 9. Studied multi-story RC building.

Fig. 10. Typical reinforcement details of the multi-story building.

0.001 s. This case did not show any progressive collapse. It wasobserved that the structure was capable of redistributing theinternal forces after the removal of the column and was thenable to resist progressive collapse. Fig. 12 shows the history ofthe deflection of the floor just above the removed column. Asseen, the floor exhibited a sudden deflection of 43 mm but thenvibrates up and down around a residual deflection of 38 mm,which represents a final stable state for the floor. Fig. 13 shows

the evolution of normal forces in the columns in a longitudinalframe passing through the removed column. The normal forcesare shown for three stages; just before column removal, at 0.04 s,and at 0.2 s after column removal. As shown, the normal forces inthe columns above the removed one suddenly reduced almost tozero and their loads are conveyed to the surrounding columns byflexure,where the twobays above the removed columnbehave likea Vierendeel girder.



Fig. 11. AEM mesh for the concrete and reinforcing bars.

Fig. 12. Floor vibration after a sudden removal of column C1.

4.2. Case #2: removal of two central columns

This case was similar to case #1 except that two neighboringcolumns (C1 and C2) were simultaneously removed. In this case,a progressive collapse of about one-third of the building wasobserved as shown in Fig. 14. The removal of these columns causedexcessive deformations of the supported slabs and beams, andfinally led to a partial collapse. The collapsed area represents ananswer to the requirements of GSA and UFC codes. In this case,the maximum allowable area is 142 m2 (the smaller of 334 m2

and the area directly associated with the instantaneously removedcolumns). The collapsed area is 142 m2 which lies within theallowable limits.

Fig. 15 shows a close-up view (inside view) for the two removedcolumns and the supported floors, where the major principalstrain contours are drawn. These strains are a good indicator ofconcrete cracking and plastic hinge visualization. The damage,which initiated the progressive collapse, was localized in themiddle sections of the floor beams above the removed columnsas shown in Fig. 15. The locations of the plastic hinges are shownat time = 0.4 s. With further deformations, strains in reinforcingbars at the location of plastic hinges increase and eventually the

bars are ruptured as shown at time = 0.8 s. Thus, propagatingin a disproportionate way, the excessive deformations and theiraccompanied damage lead, finally, to the collapse of a huge part ofthe structure (one-third).

Fig. 16 shows the stress evolution in both the upper and lowerbars of one of the beams above one of the removed columns asshown in Fig. 16(a). Just above the removed column, and beforeits removal, the stresses were tensile for the upper bars andcompressive for the lower ones. Once the column was removed,and due to the downward deformations at this location, thetensile stresses in the upper bars started to decrease and becamecompressive, while the compressive stresses in the lower barsdecreased and changed into tension at t = 0.04 s. With furtherdeformations, stresses in the lower bars kept increasing, whilethose in the upper bars changed again into tension. Meanwhile,the compressed concrete was tensioned again and does not fail incompression. This is attributed to the catenary action of the beam.The tensile stresses in both the upper and lower bars then keptincreasing with further deformations and eventually they wereruptured at t = 0.8 s at four locations for the upper bars and twolocations for the lower bars, where the stresses in the bars droppedto zero.

Fig. 17 shows the stress history of the upper and lower bars inthe beam section adjacent to column just above a removed column.As seen, before column removal, the upper bars were subjected totension,while the lower oneswere subjected to compression. Afterremoval of the column, the stresses in the upper bars changed intocompression, while that in the lower ones changed into tension.At t = 0.05 s, the stresses in the upper one were tensionedagain, revealing the activation of the catenary action, where bothbars were tensioned with the lower one having higher tensilestresses due to bending action. The catenary action was provedby integrating the normal stresses in the beam and the center-to-center part of the slab as shown in Fig. 18. The normal force historyshowed clearly a transformation of a compressive force of 50 kNinto a tensile one of 600 kN at t = 0.4 s. At t = 0.7 s, the catenaryaction started to drop due to collapse of beams as shown inFig. 18.



(a) Time = 0.0 s (Before column removal).

(b) Time = 0.04 s.

(c) Time = 0.2 s.

Fig. 13. Normal force evolution in columns due to column removal in case # 1 (in kN).

5. Can we prevent progressive collapse?

In this section, a proposed design to prevent progressivecollapse of the multi-story RC structures due to removal of groundcolumns in an optimum and economic way is introduced. Theproposed method has an iterative design, in which, additionalbottom reinforcement is used above removed columns as shownin Fig. 19. The steps of the design procedures are shown ina flow chart in Fig. 20, where AEM analysis is carried outseveral times till getting the minimum amount of additionalreinforcement that help preventing the progressive collapse. Theminimum amount of additional reinforcement was decided fromthose trials as shown in Fig. 21. In Fig. 21, the history of floordeflection above removed columns is drawn for different amountsof additional reinforcement. The non-collapsing structures arethose, whose floor deflection is eventually stabilizing withoutexcessive increase. The amount of additional reinforcement inFig. 21 is expressed as a percentage of the required additionalreinforcement calculated based on a static linear FEM analysisin which the two ground columns are missing. As can be seen,the static LFEM overestimates the required reinforcement. Usingthe AEM, a proper economic amount of reinforcement is obtained(about 50% less than the amount of reinforcement calculated using

LFEM). With this additional bottom reinforcement, progressivecollapse was prevented with a floor deflection of about 240 mm(span/20). This deflection may be considered reasonable for suchan extreme loading case knowing that the beam span is 5.4 m andhence retrofitting process would not be so difficult.

6. Conclusion

Based on the analytical results of the AEM, the following con-clusions are obtained.

1. TheAEM is an efficient, accurate, and simple tool for progressivecollapse analysis compared to FEM. In one analysis, thecollapse area can be obtained by taking into considerationmaterial nonlinearity, large deformations, element failureand separation, and collision between different parts of thestructure.

2. For the studied RC structure, the collapse of only one centralground column would not lead to any progressive collapse.However, the collapse of two central ground columns wouldlead to a progressive collapse of one-third of the studiedstructure.

3. TheAEMcanbe successfully used as an analytical tool to suggesteconomical designs that are safe against progressive collapse



(a) Time = 0.0 s. (b) Time = 0.5 s. (c) Time = 1.5 s.

(d) Time = 2.5 s.

Fig. 14. Progressive collapse of the five-story reinforced concrete building in case #2.

(a) Time = 0.0 s. (b)Time = 0.4 s.

(c) Time = 0.8 s. (d) Time = 1.2 s.

(e) Principal contours scale.

Fig. 15. Inside view for the deformations and principal strain contours in case #2.



(a) Position of bars rupture. (b) Stresses along upper bars.

(c) Stresses along lower bars.

Fig. 16. Stress distribution along bars in a beam above removed column.

Fig. 17. Stresses in the upper and lower bars in the section adjacent to column in the beam above a removed column.

Fig. 18. Normal force in a slab strip and a beam above a removed column.

of reinforced concrete structures. For the studied case, it wasproved that using the AEM can bring about 50% reductionof the amount of additional reinforcement compared toLEFEM.

Fig. 19. Additional bottom reinforcement above removed columns.

Acknowledgment

The authors would like to express their gratitude to AppliedScience International, LLC, for the financial support to the currentresearch.



Fig. 20. Flow chart for the proposed design procedure.

Fig. 21. Floor deflection history for different additional RFT ratios.

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